Advances in Optical Physics: Volume 3 Advances in Nonlinear Optics 9783110304497, 9783110304305

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Xianfeng Chen, Qi Guo, Weilong She, Heping Zeng, Guoquan Zhang Advances in Nonlinear Optics

Advances in Optical Physics

| Editor-in-Chief Jie Zhang

Volume 3

Xianfeng Chen, Qi Guo, Weilong She, Heping Zeng, Guoquan Zhang

Advances in Nonlinear Optics | Edited by Xianfeng Chen

Physics and Astronomy Classification Scheme 2010 42.65.-k, 42.65.Re, 42.65.Tg, 42.25.Ja, 78.20.Jq, 42.25.Lc Editor Prof. Xianfeng Chen Department of Physics and Astronomy Shanghai Jiao Tong University 800 Dongchuan RD 200240, Minhang District Shanghai China

ISBN 978-3-11-030430-5 e-ISBN (PDF) 978-3-11-030449-7 e-ISBN (EPUB) 978-3-11-038282-2 Set-ISBN 978-3-11-030450-3 Library of Congress Cataloging-in-Publication Data A CIP catalog record for this book has been applied for at the Library of Congress. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2015 Shanghai Jiao Tong University Press and Walter de Gruyter GmbH, Berlin/Munich/Boston Cover image: Ellende/iStock/Thinkstock Typesetting: le-tex publishing services GmbH, Leipzig Printing and binding: CPI books GmbH, Leck ♾ Printed on acid-free paper Printed in Germany www.degruyter.com

The series: Advances in Optical Physics Professor Jie Zhang, Editor-in-chief, works on laserplasma physics and has made significant contributions to development of soft X-ray lasers, generation and propagation of hot electrons in laser-plasmas in connection with inertial confinement fusion (ICF), and reproduction of some extreme astrophysical processes with laser-plasmas. By clever design to enhance pumping efficiency, he and his collaborators first demonstrated saturation of soft-X-ray laser output at wavelengths close to the water window. He discovered through theory and experiments that highly directional, controllable, fast electron beams can be generated from intense laser plasmas. Understanding of how fast electrons are generated and propagated in laser plasmas and how the resulting electron beams emit from a target surface and carry away laser excitation energy is critical for understanding of the fast-ignition process in ICF. Zhang is one of the pioneers on simulating astrophysical processes by laser-plasmas in labs. He and his collaborators used high-energy laser pulses to successfully create conditions resembling the vicinity of the black hole and model the loop-top X-ray source and reconnection overflow in solar flares. Because of his academic achievements and professional services, Professor Zhang received Honorary Doctors of Science from City University of Hong Kong (2009), Queen’s University of Belfast (2010), University of Montreal (2011) and University of Rochester (2013). He was elected member of CAS in 2003, member of German Academy of Sciences Leopoldina in 2007, fellow of the Third World Academy of Sciences (TWAS) in 2008, foreign member of Royal Academy of Engineering (FREng) of the UK in 2011 and foreign Associate of US National Academy of Sciences (NAS) in 2012. He is the President of Shanghai Jiao Tong University, and also a strong advocate and practitioner of higher education in China.

Preface After a three years’ effort by many top-tier scientists, the book series Advances in Optical Physics (English version) is completed. Optical physics is one of the most active fields in modern physics. Ever since lasers were invented, optics has permeated into many research fields. Profound changes have taken place in optical physics, which have expanded tremendously from the traditional optics and spectroscopy to many new branches and interdisciplinary fields overlapping with various classical disciplines. They have further given rise to many new cutting-edge technologies: – For example, nonlinear optics itself is an interdisciplinary field, which has been developing since the advent of lasers and it is significantly influenced by various technological advances, including laser technology, spectroscopic technology, material fabrication and structural analysis. – With the rapid development of ultra-short intense lasers in the past 20 years, high field laser physics has rapidly developed into a new frontier in optical physics. It contains not only rich nonlinear physics under extreme conditions, but also has the potential of many advanced applications. – Nanophotonics, which combines photonics and contemporary nanotechnology, studies the mechanisms of light interactions with matter at the nanoscale. It enjoys important applications such as in information transmission and processing, solar energy, and biomedical sciences. – Condensed matter optics is another new interdisciplinary field, which is formed due to the intersection of condensed matter physics and optics. Here, on the one hand, lasers are used as probes to study the structures and dynamics of condensed matter. On the other hand, discoveries from condensed matter optics research can be applied to produce new light sources, detectors, and a variety of other useful devices. In the last 20 years, with the increasing investment in research and development in China, the scientific achievements by Chinese scientists also become increasingly important. These are reflected by the greatly increased number of research papers published by Chinese scientists in prestigious scientific journals. However, there are relatively few books for a broad audience – such as graduate students and scholars – devoted to this progress at the frontiers of optical physics. In order to change this situation, three years ago, Shanghai Jiao Tong University Press discussed with me and initiated the idea to invite top-tier scientists to write the series of “Advances in Optical Physics”. Our initial plan was to write a series of introductory books on recent progresses in optical physics for graduate students and scholars. It was later expanded into its current form. The first batch of the series includes eight volumes:

viii | Preface – – – – – – – –

Advances in High Field Laser Physics Advances in Precision Laser Spectroscopy Advances in Nonlinear Optics Advances in Nanophotonics Advances in Quantum Optics Advances in Ultrafast Optics Advances in Condensed Matter Optics Advances in Molecular Biophotonics

Each volume covers a number of topics in the respective field. As the editor-in-chief of the series, I sincerely hope that this series is a forum for Chinese scientists to introduce their research advances and achievements. Meanwhile, I wish these books are useful for students and scholars who are interested in optical physics in general, one of these particular fields, or a research area related to them. To ensure these books could reflect the rapid advances of optical physics research in China, we have invited many leading researchers from different fields of optical physics to join the editorial board. It is my great pleasure that many top tier researchers at forefronts of optical physics accepted my invitation and made their contributions in the last three years. Almost at the same time, De Gruyter learned about our initiative and expressed their interest in introducing these books written by Chinese scientists to the rest of world. After discussion, De Gruyter and Shanghai Jiao Tong University Press reached the agreement in co-publishing the English version of the series. At this moment, on behalf of all authors of these books, I would like to express our appreciation to these two publishing houses for their professional services and supports to sciences and scientists. Especially, I would like to thank Mr. Jianmin Han and his team for their great contribution to the publication of this book series. At the end of this preface, I must admit that optical physics itself is a rapidly expanding forefront of science. Due to the nature of the subject area, this series can never cover all aspects of optical physics. However, what we can do – together with all authors of these books – is to try to pick up the most beautiful “waves” from the vast science ocean to form this series. By publishing this series, it is my cherished hope to attract minds of younger generation into the great hall of optical physics research. Professor Jie Zhang Editor-in-chief

Contents Preface | vii Guoquan Zhang, Daohong Song, Zhibo Liu, Mengxin Ren, Zhigang Chen, Jianguo Tian, Jingjun Xu 1 Recent progresses on weak-light nonlinear optics | 1 1.1 Ultraviolet photorefraction in LiNbO3 | 2 1.1.1 Ultraviolet photorefractive materials, their effects and applications | 3 1.1.2 Ultraviolet photorefractive effects in LiNbO3 | 4 1.1.3 Ultraviolet band edge photorefractive effects | 12 1.1.4 Enhancement on ultraviolet light-induced absorption | 14 1.1.5 Absorption edge and related defect structures in LiNbO3 | 16 1.1.6 Other effects and applications in the ultraviolet spectral range | 20 1.2 Incoherent nonlinear optics and discrete spatial solitions | 20 1.2.1 From coherent to incoherent light | 21 1.2.2 From bulk to discrete media | 22 1.2.3 Introduction to spatial optical solitons | 22 1.2.4 Nonlinear optics with incoherent light | 23 1.2.5 Spatial solitons in discrete systems | 28 1.3 Nonlinear optical properties of novel carbon-based materials | 39 1.3.1 Carbon-based materials | 40 1.3.2 Progress of optical nonlinearities of carbon-based materials | 42 1.3.3 Optical nonlinearities of carbon nanotube and its hybrid materials | 44 1.3.4 Optical nonlinearities of graphene and its hybrid materials | 53 1.4 Nonlinear optics from metallic plasmonics | 65 1.4.1 Introduction to surface plasmonic polariton | 67 1.4.2 Nonlinear processes in metals | 72 1.4.3 Metallic plasmon-enhanced second-harmonic generation | 74 1.4.4 Metallic plasmon-enhanced third-order nonlinear optics | 75 1.4.5 Metallic plasmon-enhanced nonliner optical activity | 81 1.5 Summary | 87

x | Contents Xianfeng Chen 2 Polarization coupling and its applications in periodically poled lithium niobate crystal | 105 2.1 Introduction | 105 2.2 Polarization-coupling theory based on transverse electro-optical effect | 108 2.2.1 Transverse electro-optical effect | 108 2.2.2 Jones matrix method | 114 2.2.3 Polarization-coupling mode theory | 119 2.3 Tunable wavelength filter and optical switch based on polarization-coupling effect | 124 2.3.1 Solc-type filter based on polarization-coupling effect in PPLN crystal | 124 2.3.2 Tunable multiwavelength filter by local-temperature-control technique | 127 2.3.3 Flat-top bandpass Solc-type filter and optical switch in PPLN crystal | 133 2.4 Control of linear polarization and its applications | 138 2.4.1 The evolution of the polarization state in the PPLN and the control of linear polarization state via electro-optical effect | 138 2.4.2 Linear polarization state modulator | 140 2.4.3 Electro-optic chirality control in PPLN | 144 2.4.4 Optical isolator based on the electro-optic effect in HPPLN | 147 2.4.5 Polarization based all-optical logic gates | 149 2.5 Control of angular momentum and its applications | 155 2.5.1 The angular momentum of light | 155 2.5.2 The evolution of polarization state of light by changing electric field | 155 2.5.3 Control of spin angular momentum of light | 159 2.5.4 Control of orbital angular momentum and its applications | 162 2.6 Polarization-coupling cascading and its applications in PPLN crystal | 169 2.6.1 Second-harmonic generation (SHG) cascading | 169 2.6.2 Polarization-coupling cascading and nonlinear phase shift | 171 2.6.3 Cross-modulation | 175 2.6.4 Fast and slow light | 179

Contents | xi

Heping Zeng 3 Ultrafast nonlinear optics | 191 3.1 Introduction | 191 3.2 Cascaded quadratic nonlinearity and spatiotemporal modulational instability | 192 3.2.1 Two-dimensional (2D) multicolored transverse arrays | 192 3.2.2 2D multicolored up-converted parametric amplification | 195 3.2.3 Colored conical emission (CCE) | 198 3.2.4 Seeded amplification of colored conical emission (SAC) | 199 3.2.5 Carrier envelope phase stabilization via difference frequency generation | 200 3.3 Interaction of intense ultrashort filaments | 201 3.3.1 Filament-interaction-induced nonlinear spatiotemporal coupling | 202 3.3.2 1D plasma channels | 205 3.3.3 Plasma grating | 207 3.3.4 2D plasma grating | 209 3.3.5 Third harmonics generation enhancement | 211 3.4 Molecular alignment assisted filament interaction | 214 3.5 Ultrafast optical gating by molecular alignment | 217 3.6 Conclusions | 222 Qi Guo, Daquan Lu, Dongmei Deng 4 Nonlocal spatial optical solitons | 227 4.1 Introduction to optical soliton research | 227 4.1.1 Historical background of optical solitons | 227 4.1.2 Optical Kerr effect and its spatial and temporal nonlocality | 238 4.1.3 Nonlinear propagation model of optical envelope: The nonlocal nonlinear Schrödinger equation | 241 4.1.4 Soliton solutions and their physical essence | 250 4.2 The phenomenological theory of the nonlocal spatial solitons | 260 4.2.1 The classification of the nonlocality | 261 4.2.2 The Snyder–Mitchell model | 262 4.2.3 The weak-nonlocality | 269 4.2.4 The general nonlocality | 270 4.2.5 Interaction of double solitons | 272 4.3 Nematicons | 278 4.3.1 Nonlinear propagation models of beams in the nematic liquid crystal | 279 4.3.2 The voltage-controllable nonlinear characteristic length and the nonlinear refractive index coefficient | 283

xii | Contents 4.3.3 4.3.4 4.4 4.4.1 4.4.2

The propagation model of the liquid crystal in strong nonlocal condition | 285 The approximate analytical solution of a single nematicon | 287 The thermal nonlinear nonlocality | 291 Spatial optical solitons in the lead glass | 292 Other thermal nonlinear materials | 295

Weilong She, Guoliang Zheng 5 Wave coupling theory and its applications of linear electro-optic (EO) effect | 307 5.1 Introduction | 307 5.2 Wave coupling theory of linear EO effect in transparent bulk crystal | 309 5.3 Wave coupling theory of linear EO effect in absorbent medium | 319 5.4 Wave coupling theory of mutual action of linear EO, OA, and Faraday effects | 325 5.5 Wave coupling theory of QPM linear EO effect | 330 5.5.1 Wave coupling theory of Linear EO effect in periodically poled crystals | 333 5.5.2 Linear EO effect in linear chirped-periodically poled crystals | 338 5.5.3 Wave coupling theory of united effect of QPM linear EO effect, second-harmonic generation, and sum/difference frequency generation | 344 5.5.4 Wave coupling theory of QPM linear EO effect for a focused Gaussian beam | 355 Index | 367

Guoquan Zhang, Daohong Song, Zhibo Liu, Mengxin Ren, Zhigang Chen, Jianguo Tian, Jingjun Xu

1 Recent progresses on weak-light nonlinear optics

Soon after the invention of laser in the 1960s, Franken et al. [1] observed the secondharmonic generation (SHG) in crystalline quartz, which declared the birth of modern nonlinear optics. Traditionally, optical nonlinear effect is significant only when the strength of the incident light field is comparable to the internal binding electric field strength of atoms [2, 3]. In this case, the light fields induce distortion of the atomic electron cloud in the media, therefore, modify the optical properties such as the refractive index, absorption, polarization and susceptibility of the materials. On the other hand, these changes in the optical properties of materials will inversely influence or change the polarization, strength, and frequency of the light propagating in the materials. Laser, characterized by its good coherence and high intensity, provides a kind of excellent light sources for the study of nonlinear optics, and the development of nonlinear optics is always related to the advances of laser technology. These historical facts give one the impression that nonlinear optical (NLO) effects take place only at high light intensities. With the development of laser technology, nonlinear optics at high light intensities has achieved great success in the literature. In fact, almost at the same time of the observation of the SHG, photorefractive effect, a typical NLO effect which can take place at weak light, was observed in Bell laboratory [4]. In contrast to the NLO effects at high light intensities, weak-light NLO effects can be observed at very low light intensities. Typical material systems include photorefractive materials, electromagnetically induced transparency media, micro-/ nanostructures and microcavity, where NLO effects can be observed at a milli-watt power level or even at a single photon level through enhancement mechanisms such as quantum transport, quantum coherence, slow light, and light localization [5–13]. More interestingly, weak-light NLO effects can also be observed even with incoherent light sources other than lasers at appropriate conditions [14, 15]. In this chapter, we will briefly introduce several typical weak-light NLO effects and their recent advances, including ultraviolet (UV) photorefraction in lithium niobate (LiNbO3 ) crystals, nonlinear optics with incoherent light and discrete spatial solitons, NLO properties of novel carbon-based materials, and nonlinear optics from metallic plasmonics. While weak-light NLO effects in other systems such as electromagnetically induced transparency media, micro-/nanostructures, and microcavities, and

Guoquan Zhang, Daohong Song, Zhibo Liu, Mengxin Ren, Zhigang Chen, Jianguo Tian, Jingjun Xu: The MOE Key Laboratory of Weak Light Nonlinear Photonics, School of Physics and TEDA Applied Physics Institute, Nankai University, Tianjin 300457, China

2 | Guoquan Zhang et al. biosystems are not included due to the space limit and the knowledge scope of the authors. The content of this chapter is organized as follows: Section 1.1, written by Guoquan Zhang and Jingjun Xu, discusses the UV photorefractive effect, the defect structures, and the light-induced charge carrier transport in LiNbO3 crystals. Section 1.2 introduces the incoherent optical spatial solitons and discrete spatial solitons in the continuous and discrete noninstantaneous nonlinear materials by Daohong Song and Zhigang Chen. In Section 1.3, Zhibo Liu and Jianguo Tian introduce the state-of-the-art development of the fabrication, optimization, NLO mechanism, and related applications of carbon-based materials. While Section 1.4 introduces the plasmonic enhancement on the light fields and NLO effects in metamaterials based on metallic structures in subwavelength scales and surface plasmonic polaritons by Mengxin Ren and Jingjun Xu. Finally, the contents are compiled by Guoquan Zhang and Jingjun Xu.

1.1 Ultraviolet photorefraction in LiNbO3 Photorefractive effect is the abbreviation of the light-induced refractive index change of materials [5, 16–19]. In general, the photorefractive effect occurs when appropriate materials are illuminated by a spatially inhomogeneous beam, and electrons or holes are photo-excited from the donor impurity centers or defect centers to the conduction band or valence band, where these photo-excited charge carriers transport to dark areas by means of diffusion due to concentration gradient, drift due to external electric field and/or photovoltaic effect, and are then trapped by the acceptor impurities or defect centers there. Such photo-excitation, transportation, and trapping of charge carriers are repeated again and again until a dynamic equilibrium arrives, and finally a spatially inhomogeneous distribution of charge carriers forms, which results in a space charge field according to the spatially inhomogeneous distribution of the light intensity. The space charge field modulates the refractive index through the electrooptic effect of the material, which is, in general, of noncentrosymmetry. Normally, the photorefractive effect can be observed at a milli-Watt level or even at a μW-level incident power as long as the light–matter interaction time is long enough. It is worth noting that, for noninstantaneously responded photorefractive materials, remarkable photorefractive effect can be observed even with an incoherent calorescence light. It is evident that the photorefractive effect is a typical weak-light NLO effect. The photorefractive effect was discovered in LiNbO3 and lithium tantalate (LiTaO3 ) crystals by Ashkin et al. [4] in 1966, and was named as optical damage at the very beginning because its induced refractive index change degrades the phase-matching condition of SHG. This optical damage is reversible and it will disappear after a certain time in dark or under appropriate treatments (e.g., thermal treatment). In 1968, Chen et al. [20] demonstrated that this reversible light-induced refractive index change can be used for optical storage, and proposed in 1969 the formation mechanism of the

1 Recent progresses on weak-light nonlinear optics

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photorefractive effect [21]. In 1979, Kukhtarev et al. [22] proposed the band transport model for the photorefractive effect, which was widely accepted thereafter. Great progresses on photorefractive materials, effects, and their applications have been made up to now, and various photorefractive materials such as inorganic crystals including LiNbO3 , LiTaO3 , BaTiO3 , KNbO3 , SBN, Sn2 P2 S6 , semiconductor quantum well, and organic photorefractive materials like liquid crystals and photopolymer were developed and systematically characterized [16, 23], a series of novel effects such as coherent light amplification, phase conjugate wave, spatial solitons, space charge wave, and slow light were demonstrated, and fruitful applications such as holographic data storage, phase conjugate mirror, dynamic holographic interferometry, holographic filter, correlator, and photorefractive waveguide devices were demonstrated [6, 18, 24–26]. However, although the photorefractive effects in the visible and near-infrared were extensively studied, the photorefractive effect in the UV spectrum was sparely reported in the literature due to the availability of appropriate materials and light sources. With the recent development of UV photorefractive materials and UV light source, and the attractive high spatial resolution and fast response rate of UV photorefractive effect, in combination with the advantages of high-energy UV photons in the characterization of material properties, the UV photorefractive effect gradually attracted intensive attentions and made progresses rapidly.

1.1.1 Ultraviolet photorefractive materials, their effects and applications Early in 1978, Krätzig et al. [27] studied the photorefractive effect and the light-induced charge carrier transport in the UV spectrum in LiNbO3 :Fe and LiTaO3 :Fe crystals, they found that there is electron–hole competition in LiNbO3 :Fe and the hole will be the dominant charge carrier under the illumination of UV light in the oxidized crystals. The absorption in the UV is very strong (> 50 cm−1 ) in LiNbO3 :Fe, which seriously limits the practical applications of UV photorefractive effect in LiNbO3 . In the same year, Fridkin et al. [28] also reported the UV photorefractive effect in KDP-type ferroelectric materials. Later, Montemezzani et al. carried out a series of studies on UV photorefractive effect in crystals such as Bi4 Ge3 O12 [29], KNbO3 [30, 31] and LiTaO3 [32–34], and proposed the interband photorefractive effect. Different from the traditional photorefractive effect, which relies on the defect or impurity centers in the band gap of the materials, the interband photorefractive effect is based on the photo-excitation of electron directly from the valence band to the conduction band with high-energy photons; therefore, it is of relatively fast response rate and high recording sensitivity. For example, the response time τ, the magnitude of the refractive index change Δn, and the photorefractive recording sensitivity S n ≡ Δn/(2Iτ) were reported to be 5 μs, 2 × 10−5 and 2 cm2 /J in KNbO3 crystal illuminated by a 1 − W/cm2 , 351-nm UV light, where I is the incident light intensity. In LiTaO3 crystal, the operating wavelength is shifted downward to 257 nm, and the response time is shortened to be tens of μs for interband

4 | Guoquan Zhang et al. photorefractive effect, which is improved by three orders of magnitude as compared to the traditional photorefractive effect based on defect or impurity centers in band gap of materials [35]. It is seen that, due to the high absorption at the recording wavelength because of the interband photo-excitation, the thickness of the interband photorefractive gratings is usually of the order of hundreds of μm, and therefore the grating is read out transversely to increase the diffraction efficiency. In addition, Xu et al. studied the UV photorefractive effect in α-LiIO3 crystals [36, 37]. In 1992, Jungen et al. [38] reported efficient photorefractive recording at 351 nm with a two-wave coupling gain coefficient Γ ∼ 13.94 cm−1 in nominally pure congruent LiNbO3 . It was found that the photorefractive effect in the UV is very different from those in the visible and near-infrared for LiNbO3 crystals. It is well known that, electron is the dominant light-induced charge carrier in LiNbO3 in the visible 4+ and near-infrared, intrinsic defect centers such as bipolarons Nb4+ Li -NbNb and small 4+ 2+ 3+ electron polarons NbLi , and extrinsic impurities such as Fe /Fe , Cu+ /Cu2+ and Mn2+ /Mn3+ are identified as the photorefractive centers, and the photovoltaic effect is regarded to be the main transport mechanism of the light-induced charge carriers. However, Jungen et al. found that hole is the dominant photo-excited charge carrier and the photovoltaic field is only of the order of 550 V/cm in nominally pure congruent LiNbO3 , therefore, diffusion becomes the dominant light-induced charge carrier transport mechanism, which leads to an efficient unidirectional energy transfer in the two-wave coupling process [38]. Laeri et al. [39] demonstrated the self-pumped phase conjugate wave and lensless image transmission with a spacial resolution of 2800 lines/mm based on the UV photorefraction in LiNbO3 . More interestingly, Barkan et al. [40] achieved irreversible light-induced refraction index change up to −0.2 in LiNbO3 with a 5 ns, 20-MW/cm2 pulsed UV laser at 249 nm. This large light-induced refractive index change was observed only in a very thin layer near the crystal surface because of the extremely high absorption at 249 nm in LiNbO3 , which might have potential applications in the fabrication of surface structures in the micro-/nanoscale and integrated optics.

1.1.2 Ultraviolet photorefractive effects in LiNbO3 Lithium niobate is regarded as one of the candidates for “silicon of photonics,” which offers excellent electro-optic, piezoelectric, pyroelectric, acousto-optic, and nonlinear optic properties [41–43]. The nominally pure congruent LiNbO3 is in a Li-deficiency state with a Li/Nb ratio of 48.6/51.4. According to the Li-vacancy model [44], there are a large amount of antisite NbLi and Li vacancy VLi in congruent LiNbO3 ; therefore, congruent LiNbO3 is capable of accommodating a large amount of extrinsic impurities, which provides a convenient way to modify the crystal properties through doping or composition modification. It is known that dopants such as Fe, Cu, Mn, and Ce tend to enhance the photorefractive effect, while those such as Mg, Zn, In, and Sc

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Table 1.1. Typical parameters of UV photorefraction in LiNbO3 :Mg crystals with different MgO doping concentrations. The grating was recorded by two extraordinary polarized, 351-nm recording beams with a crossing angle 2θ = 70° and a total recording intensity I = 0.2 W/cm2 . The grating wavevector was set to be along the crystalline c-axis. All crystals were y-cut sheet with a dimension of 10 mm × 3 mm × 10 mm. The data are taken from Ref. [52]. MgO doping concentration (mol%) (cm−1 )

Absorption coefficient α Specific photoconductivity σ ph /I(×10−12 cm/V2 ) Two-wave mixing gain Γ (cm−1 ) Grating diffraction efficiency η (%) Refractive index change Δn (×10−5 ) photorefractive response time τ e (s) Recording sensitivity S I (×10−5 cm2 /J)

0.0

3.0

5.0

9.0

10.2 0.13 6.8 7.2 1.0 18.4 2.6

5.1 0.25 10.0 13.0 1.3 8.4 4.6

2.2 1.8 13.8 24.6 1.9 1.15 15.3

1.2 2.9 15.1 30.3 2.1 0.6 21.6

will suppress the photorefractive effect of LiNbO3 [42, 45]. In 1980, Zhong et al. [46] found that the so-called light-induced optical damage could be suppressed by two orders of magnitude in LiNbO3 by doping the crystal with 4.6 mol% MgO. This was confirmed by Bryan et al. [47], and they pointed out further that the main reason for the observed optical damage resistance is the increase in the photoconductivity in LiNbO3 :Mg which results in a reduction of the photovoltaic field. Due to the potential applications of optical-damage-resistance LiNbO3 on nonlinear frequency inversion devices such as SHG, extensive studies have been carried out to improve the damage-resistance intensity threshold of LiNbO3 , and subsequently more dopants such as Zn2+ [48], In3+ [49, 50], Sc3+ [51] were found to be capable of resisting the optical damage with doping concentrations higher than the threshold values of 7 mol%, 3–5 mol%, and 1 mol%, respectively. In 2000, Xu et al. [52] reported that although the optical damage of highly Mgdoped LiNbO3 is significantly suppressed in the visible and near-infrared, the photorefractive effect in the UV is enhanced greatly in LiNbO3 :Mg with the increase of Mgdoping concentration. Table 1.1 shows the typical UV photorefraction performance of LiNbO3 :Mg with different Mg-doping concentrations. One sees that, with the increase of Mg-doping concentration, the absorption edge of LiNbO3 :Mg is shifted gradually toward short wavelength and the absorption coefficient at 351 nm decreases gradually. For example, the absorption coefficient at 351 nm of a nominally pure congruent LiNbO3 is ∼ 10.2 cm−1 , while that of LiNbO3 :Mg (9.0 mol%) is ∼ 1.2 cm−1 . The grating diffraction efficiency η = I d /(I d + I t ) is improved from 7.2% for a 3-mm congruent LiNbO3 to 30.3% for a 3-mm LiNbO3 :Mg (9.0 mol%); here I d and I t are the diffracted and transmitted light intensities, respectively. The light-induced refractive index change Δn and the two-wave coupling gain coefficient Γ are increased to 2.1 × 10−5 and 15.1 cm−1 , respectively, for a LiNbO3 doped with 9.0 mol% MgO, which is increased by a factor of 2 as compared to those of congruent LiNbO3 . In addition, due

6 | Guoquan Zhang et al. Table 1.2. Parameters of LiNbO3 crystals. The data are taken from Ref. [54]. Sample

Doping element

Doping concentration (mol%)

Dimension

CLN CZn5 CZn7 CZn9 CIn1 CIn3 CIn5

– Zn Zn Zn In In In

0 5.4 7.2 9.0 1.0 3.0 5.0

10 × 3 × 10 4×2×5 4×2×5 7×2×4 7 × 3.5 × 7 5 × 3.5 × 5 7 × 3.5 × 7

Table 1.3. Photorefractive properties of Zn- and In-doped LiNbO3 crystals at 351 nm. The gratings were recorded by two extraordinary polarized coherent beams with a crossing angle 2θ = 40° in air and recording intensities of I S = 121.7 mW/cm2 and I P = 176.9 mW/cm2 , respectively. The grating period Λ was 0.5 μm and its wavevector was set to be along the crystalline c-axis. The response time τ e was measured with a 351-nm erasing beam of 70.8 mW/cm2 . The signal to pump beam intensity ratio I S /I P was set to be 1:100 in the measurement of two-wave mixing gain coefficient Γ. The data are taken from Ref. [54]. Sample

CLN

CZn5

CZn7

CZn9

CIn1

CIn3

CIn5

Photoconductivity σ ph (×10−12 cm/ΩW) Diffraction efficiency η (%) Response time τ e (s) Two-wave mixing gain coefficient Γ (cm−1 ) Recording sensitivity S (cm/J) Dynamic range M/#

3.32

10.6

25.2

57.3

1.59

7.46

12.9

9.05 12.4 1.32

16.9 1.97 11.0

22.3 1.01 15.2

25.3 0.88 21.7

10.1 13.9 1.16

15.9 3.06 11.8

17.7 1.68 17.0

0.99 0.14

4.0 0.11

8.85 0.12

11.1 0.14

0.86 0.26

2.85 0.19

3.88 0.15

to the increased photoconductivity in highly Mg-doped LiNbO3 , the photorefractive response time constant τ e is shorten to be 0.6 s and the recording sensitivity S I ≡ dΔn/d(It) is improved up to 2.16 × 10−4 cm2 /J for a LiNbO3 :Mg (9.0 mol%). Unidirectional energy transfer toward +c-axis of crystal was reported in the two-wave mixing process at 351 nm, indicating that hole is the dominant light-induced charge carrier and diffusion is the main charge transfer mechanism in the UV photorefractive process. The results clearly show that highly Mg-doped LiNbO3 is a kind of excellent UV photorefractive material, which is very different from the photorefractive behavior in the visible and near-infrared. The UV photorefraction, especially the high two-wave mixing gain coefficient, was successfully applied to a micro-lithographic authentification system [53]. Later, Qiao et al. studied extensively the UV photorefractive effects of Zn- and Indoped LiNbO3 crystals at 351 nm. Table 1.2 lists the parameters, including the doping concentrations and the dimensions, of the samples used in Ref. [54], and their UV photorefractive properties at 351 nm are shown in Table 1.3, where σ ph = εε0 /τ e is the

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photoconductivity, ε0 and ε are the vacuum dielectric constant and the relative dielectric constant of the crystal, respectively, τ e is the erasing time constant of the grating, and η = I d /(I d + I t ) is the diffraction efficiency of the grating with I d and I t being the diffracted and the transmitted intensity of the readout beam through the gratings, respectively. The grating recording sensitivity S and the dynamic range M/# are defined as S ≡ (1/Id)∂√ η/∂t|t=0 and M/# ≡ τ e ∂√η/∂t|t=0 , where d is the crystal thickness, I is the total recording intensity, ∂√η/∂t|t=0 is the temporal slope of the square root of the diffraction efficiency at the very beginning of the grating buildup. Γ ≡ 󸀠 󸀠 󸀠 󸀠 (1/d)ln((I S I P )/(I S I P )) is the two-wave mixing gain coefficient with I S (I S ) and I P (I P ) being the transmitted intensities of the signal and pump beams, respectively, without (with) the coupling effect. It is seen that, similar to the case in LiNbO3 :Mg, the UV photorefractive effect of LiNbO3 :Zn is enhanced with the increase of Zn-doping concentration. For example, the diffraction efficiency η, the photoconductivity σ ph , the recording sensitivity S, the response time τ e , and the gain coefficient Γ reach 25.3%, 57.3 × 10−12 cm/ΩW, 11.1 cm/J, 0.88 s and 21.7 cm−1 , respectively, in a congruent LiNbO3 :Zn crystal with a Zn-doping concentration of 9.0 mol%. Note that the gain coefficient Γ is even larger than those in LiNbO3 :Mg. Although diffusion was also confirmed to be the dominant charge transport mechanism in highly Zn-doped LiNbO3 , Qiao et al. reported that electron is the dominant light-induced charge carrier in these crystals, which is different from that in LiNbO3 :Mg reported in Ref. [52]. Similarly, LiNbO3:In also offers good UV photorefractive effect when the In-doping concentration is higher than the so-called threshold value (3–5 mol%). In a LiNbO3 :In (5.0 mol%) crystal, the UV photorefractive parameters such as the diffraction efficiency η, the photoconductivity σ ph , the recording sensitivity S, the response time τ e , and the gain coefficient Γ were measured to be 17.7%, 12.9 × 10−12 cm/ΩW, 3.88 cm/J, 1.68 s, and 17.0 cm−1 , respectively, which are much better than those in nominally pure congruent LiNbO3 [54]. Again, electron is the dominant UV-lightinduced charge carrier and diffusion is the dominant charge transport mechanism in LiNbO3 :In. In addition, the dark conductivity σ d increases with the increase of doping concentration in LiNbO3 :M (M = Mg, Zn, and In), therefore, the gratings recorded with

Fig. 1.1. The temporal dark decay dynamics of UV photorefractive gratings in (a) nominally pure LiNbO3 , (b) LiNbO3 :Zn, and (c) LiNbO3 :In. The data are taken from Ref. [54].

8 | Guoquan Zhang et al. UV light beams decay quite quickly even in dark in highly doped LiNbO3 , as shown in Figure 1.1. Recently, Kokanyan et al. [55] reported that the photorefraction of LiNbO3 in the visible is effectively reduced by doping Hf4+ with an optical-damage-resistance doping threshold concentration of ∼ 4.0 mol%. Soon after, Kong et al. [56, 57] reported that highly Zr4+ - and Sn4+ -doped LiNbO3 with doping concentrations higher than the threshold value of 2.0–2.5 mol% are capable of resisting the optical damage in the visible. Note that the doping concentration thresholds of tetravalent ions Hf4+ , Zr4+ , and Sn4+ are lower than that of Mg2+ , and the distribution coefficients of these tetravalent ions in LiNbO3 are close to unit (the distribution coefficient of Mg2+ is around 1.2 in LiNbO3 ), which is in favor of growing crystals with high optical quality and large size. Therefore, Hf4+ -, Zr4+ - and Sn4+ -doped LiNbO3 are considered to be very promising for NLO applications [58]. It is seen that when the doping concentration is higher than the respective threshold, the photorefraction of Zr4+ -, Hf4+ - or Sn4+ -doped LiNbO3 in the visible is significantly suppressed [55–58], however, this is not true in the UV and the UV photorefractive behavior is quite complicated. Yan et al. [59] studied the UV photorefractive effect of LiNbO3 :Hf at 351 nm, and revised the concentration threshold of Hf4+ to be 2.0– 2.5 mol%. Their results show that, similar to the case of LiNbO3 :Mg, the UV photorefractive effects of LiNbO3 :Hf, for instance, the photoconductivity σ ph , the recording sensitivity S, the response time τ, and the UV-light-induced refractive index change Δn, are enhanced when the Hf4+ -doping concentration is higher than the threshold. It was also reported that electron is the dominant charge carrier and diffusion is the dominant charge carrier transport mechanism. On the other hand, LiNbO3 :Hf also exhibits properties that are different from those of LiNbO3 :Mg. For example, it was found that the UV photorefractive grating in LiNbO3 :Hf is stable in dark, but can be erased by 633-nm red light or UV light. Yan et al. [59] attributed the difference to the different occupation sites of Fe2+/3+ in LiNbO3 :Hf and LiNbO3 :Mg. It was reported [60] that Fe2+/3+ ions still occupy Li-sites even when the Hf4+ -doping concentration is higher 2+/3+ than the threshold, therefore, the UV photorefractive gratings are recorded on FeLi 2+/3+ centers. However, Fe ions are repelled to the Nb-sites in LiNbO3 :Mg, resulting in a reduced electron trapping capability of Fe3+ Nb . Kong et al. [56] reported that, when the Zr4+ -doping concentration is higher than 2.0 mol%, the intensity threshold against optical damage at 514.5 nm of the congruent LiNbO3 :Zr reaches 2.0 × 107 W/cm2 , and the light-induced refractive index change Δn is only of the order of ∼ 10−7 , which is of the same order as that of highly doped near-stoichiometric LiNbO3 :Mg prepared by vapor transport equilibration (VTE) treatment [61]. More interestingly, highly Zr4+ -doped LiNbO3 is of resistance against optical damage not only in the visible and near-infrared, but also in the UV [62]. It was confirmed experimentally that the intensity threshold against optical damage at 351 nm is larger than 105 W/cm2 in congruent LiNbO3 :Zr (2.0 mol%), and the light-induced refractive index change Δn is only of the order of 10−6 , which is essentially different

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Fig. 1.2. Transmitted beam spots through near-stoichiometric LiNbO3 :Zr crystals after a 5-min laser irradiation. The first row is the results with a Zr4+ -doping concentration of 0.5 mol%, and the second row shows the cases with a Zr4+ -doping concentration of 1.0 mol%, the crystalline c-axis is oriented horizontally. (a) and (d): 2.0 × 107 W/cm2 at 514.5 nm, CW laser beam; (b) and (e): 80 GW/cm2 10-ns, 532-nm pulsed laser beam; (c) and (f): 120 kW/cm2 at 351 nm, CW laser beam. The data are taken from Ref. [63].

from other so-called damage-resistance ions such as Mg2+ , Zn2+ , In3+ , and Hf4+ . Liu et al. [63] confirmed that the intensity threshold of optical damage resistance could be further improved in near-stoichiometric LiNbO3 :Zr prepared by the VTE technique. For example, in a near-stoichiometric LiNbO3 :Zr crystal with a Li/Nb ratio of 0.998 in crystal and a Zr4+ -doping concentration of 0.5 mol%, the intensity threshold against optical damage is larger than 2 × 107 W/cm2 for a continuous-wave (CW) laser beam at 514.5 nm, 80 GW/cm2 for a 10-ns, 532-nm pulsed laser beam, and 120 kW/cm2 for a CW laser beam at 351 nm (see Figure 1.2), respectively. Moreover, the ferroelectric domain inversion field of this near-stoichiometric LiNbO3 :Zr is only 1 kV/mm. Nava et al. [58] reported that the optoelectric properties such as the refractive index and the optoelectric coefficient of LiNbO3 do not change very much as a function of Zr4+ doping concentration. Note that, up to now, LiNbO3 :Zr is the only kind of LiNbO3 that is of resistance against optical damage in both the visible and UV spectral ranges. Considering its low doping concentration threshold, low ferroelectric inversion field, high resistance against optical damage in a broadband spectral range, and a distribution coefficient close to unit, LiNbO3 :Zr is regarded as a promising candidate for NLO applications. Dong et al. [64] found that LiNbO3 :V is also a promising UV photorefractive material. In a LiNbO3 :V (0.1 mol%) crystal, the grating recording time constant τ b , the recording sensitivity S, and the two-wave coupling gain coefficient Γ were measured to be 160 ms, 10.2 cm/J, and 12.6 cm−1 , respectively, with a total recording intensity 583 mW/cm2 at 351 nm, while the absorption coefficient α of the crystal at 351 nm was

10 | Guoquan Zhang et al.

Fig. 1.3. Photochromic effect in a LiNbO3 :V (0.1 mol%) crystal under the illumination of an incoherent beam peaked at 365 nm. The data are taken from Ref. [64].

only about 1.7 cm−1 . Also, electron is the dominant light-induced charge carrier, and diffusion is the dominant charge transport mechanism in the UV photorefractive process in LiNbO3 :V. Different from dopants Mg2+ , Zn2+ , In3+ , Hf4+ , Sn4+ , and Zr4+ , which are in the single valence state in LiNbO3 , the element V can be in multiple valance states V3+ , V4+ , and V5+ in LiNbO3 , which is confirmed by the x-ray photoelectron spectroscopy and the electron paramagnetic resonance measurement. The absorption bands of V3+ and V4+ in LiNbO3 are peaked at 420 nm and 475 nm, respectively. Efficient photochromic effect was observed in LiNbO3 :V under the illumination of an incoherent beam centered at 365 nm, and the color of the illuminated region changed from green to reddish brown. All these results indicate that V3+/4+ and V4+/5+ are the effective photorefractive centers in LiNbO3 :V. One notes that the valences of all impurities mentioned above are lower than +5, which is the valence of Nb in LiNbO3 , and all these impurities tend to occupy Li sites when the doping concentrations are low. Recently, Tian et al. [65] reported that the impurity may tend to occupy Nb site in LiNbO3 when its valency is higher than +5, which leads to very different photorefractive effects as compared to those doped with impurities with the valence state lower than +5. For example, the element Mo can be in valence state Mo4+ , Mo5+ , and Mo6+ in LiNbO3 with their absorption bands peaked at 326 nm, 337 nm and 461 nm, respectively, which together lead to a broad absorption band covering from UV to visible spectral ranges. Therefore, LiNbO3 :Mo shows a very good photorefractive effect in a broad spectral range, as shown in Figure 1.4. The holographic recording time constant τ b and the grating diffraction efficiency η were measured to be 0.35 s and ∼ 60%, respectively, in a 3-mm LiNbO3 :Mo (0.5 mol%) crystal with a total recording intensity of 640 mW/cm2 at 351 nm. The photorefractive 4+ centers are ascribed to O2−/− -V−Li , O2−/− -Mo+Nb and bipolarons Nb4+ Li -NbNb in the visible, + 2−/− − 2−/− and O -VLi and O -MoNb in the UV, respectively. It is seen that the photorefractive properties of LiNbO3 in the UV are quite different from those in the visible; this is also true for the light-induced charge transport mechanism. In the visible, the dominant charge transport mechanism is the photovoltaic effect, while in the UV diffusion is the dominant charge transport mechanism, which

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Fig. 1.4. The saturated diffraction efficiency (a) and the recording time constant (b) of the photorefractive gratings recorded in LiNbO3 :Mo crystals versus the Mo-doping concentration. The recording wavelength was set at 351 nm, 488 nm, 532 nm and 671 nm with a total recording intensity of 640 mW/cm2 , 800 mW/cm2 , 800 mW/cm2 and 6000 mW/cm2 , respectively. The two recording beams were of equal intensity. The empty circle, empty downward triangle, empty upward triangle represent the results for a LiNb3 :Fe (0.03 mol%). The data are taken from Ref. [65].

leads to very different light-induced scattering patterns in the visible and UV, as shown in Figure 1.5. The light-induced scattering (fanning noise) is symmetrically distributed in the visible due to the multiple three-wave interaction [66–68], while it is asymmetrically distributed and the energy is transferred toward crystalline -c-axis in the UV due to the two-wave mixing [38]. Ellabban et al. [69] studied the asymmetric distribution of the light-induced scattering of LiNbO3 at different recording wavelengths, and confirmed that the contribution of photovoltaic effect decreases while that of diffusion increases with the increase of incident photon energy. Based on the unidirectional energy transfer in LiNbO3 :Mg in the UV, Qiao et al. [70] observed the stimulated photorefractive scattering and achieved self-pumped phase conjugate wave at 351 nm in a c-cut LiNbO3 :Mg (5.0 mol%). It is known that the resistance capability against the optical damage of LiNbO3 can be tested by observing the light spot distortion of transmitted beam after the crystal. In the visible, in general, when the symmetric light-induced scattering is strong, the transmitted light spot distortion will also be strong. This is because the underlying mechanism for the fanning noise and the light spot distortion is the same, that is the photovoltaic effect in LiNbO3 . On the other hand, as already mentioned above, in the UV spectral range, the asymmetric, unidirectional light-induced scattering is strong in highly doped LiNbO3 :M (M=Mg2+ , Zn2+ , In3+ , Hf4+ , Sn4+ , and Zr4+ ), but the light spot distortion of transmitted beam can be observed only when the incident light intensity is higher than the intensity threshold against optical damage. This is because the charge transport mechanisms responsible for the fanning noise and the light spot

12 | Guoquan Zhang et al.

Fig. 1.5. (a) The asymmetric intensity distribution of light-induced scatterings in LiNbO3 :Mg (5.0 mol%) at 351 nm and (b) the symmetric intensity distribution of light-induced scatterings in LiNbO3 :Fe (0.01 wt%) at 633 nm.

distortion effect are different. The fanning noise is mainly based on diffusion of electrons in a crystal, while the light spot distortion effect is mainly due to the photovoltaic effect. This relationship between the fanning noise and the light spot distortion is also true for the UV band edge photorefraction in LiNbO3 [71]. Another basic issue in the UV photorefractive effect of LiNbO3 is the polarity of the light-induced charge carrier. In the visible, electron is considered as the main light-induced charge carrier in LiNbO3 [5, 16, 42]. In the UV spectral range, hole is considered as the dominant charge carrier at the early stage [27, 38, 52]. In 2004, however, Qiao et al. [54] reported that electron is the dominant light-induced charge carrier in highly doped LiNbO3 :Zn and LiNbO3 :In at 351 nm. Subsequently, a series of studies on the UV photorefractive effect in highly doped LiNbO3 :Hf [59], LiNbO3 :Zr [62], and LiNbO3 :Sn [71] support the conclusion that electron is the dominant charge carrier in the UV photorefractive process. This is also true for the UV photorefraction in LiNbO3 :V [64] and LiNbO3 :Mo [65]. It was reported that the polarity of the lightinduced charge carrier in LiNbO3 is not only dependent on the dopants, but also on the oxidation/reduction state of the crystal and the recording wavelength [5, 27].

1.1.3 Ultraviolet band edge photorefractive effects Xin et al. [72] studied the UV band edge photorefractive effects at 325 nm in LiNbO3 with various dopants (see Table 1.4), and the results are listed in Table 1.5. It is seen that the UV band edge photorefraction at 325 nm is improved as compared to the UV photorefraction at 351 nm, especially for those crystals with doping concentrations higher than the so-called damage-resistance threshold concentration of Mg2+ , Zn2+ , In3+ , and Hf4+ . For instance, in the UV band edge photorefraction at 325 nm, the twowave mixing gain coefficient Γ and the grating recording sensitivity S reach 38 cm−1 and 37.7 cm/J in a LiNbO3 :Zn (9.0 mol%) crystal, the grating buildup time constant τ b is only 73 ms in a LiNbO3 :Mg (9.0 mol%) crystal with a total recording intensity of 614 mW/cm2 at 325 nm. In addition, the recording sensitivity S reaches 7.75 cm/J and the recording dynamic range per unit thickness (M/#)/d is 5.63 cm−1 in a dou-

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Table 1.4. Parameters of LiNbO3 with various dopants. Sample

Dopant

Concentration (mol%)

Dimension (x × y × z, mm3 )

CLN CMg1 CMg2 CMg4 CMg5 CMg7 CMg9 CZn5 CZn7 CZn9 CIn1 CIn3 CIn5 CHf4 CHf6 M22

– Mg Mg Mg Mg Mg Mg Zn Zn Zn In In In Hf Hf Mg, Fe

– 1.0 2.0 4.0 5.0 7.8 9.0 5.4 7.2 9.0 1.0 3.0 5.0 4.0 6.0 Mg: 6.3 mol% Fe: 0.01 wt%

8.0 × 0.5 × 16 8.0 × 0.7 × 16 8.0 × 0.7 × 16 8.0 × 4.3 × 10 19 × 2.7 × 10 16 × 5.2 × 15 10 × 3.3 × 14 6.0 × 2.0 × 9.0 6.0 × 2.0 × 8.0 7.0 × 2.0 × 4.0 18 × 3.4 × 16 11 × 3.4 × 10 18 × 3.4 × 16 10 × 0.5 × 7.0 10 × 0.5 × 7.0 11 × 1.9 × 9.0

bly doped LiNbO3 :Fe,Mg crystal. In all these crystals, electron is the dominant lightinduced charge carrier, and diffusion is the main charge transport mechanism in the UV band edge photorefractive process. It is seen from Table 1.5 that the UV band edge photorefraction of LiNbO3 doped with Mg2+ , Zn2+ , In3+ , and Hf4+ shows the doping concentration threshold effect. Xin et al. [71] studied the UV band edge photorefractive effect, including the saturated diffraction efficiency of grating ηst , the grating buildup time constant τ b , the recording sensitivity S, the specific photoconductivity σ ph /I e and the two-wave mixing gain coefficient Γ, in LiNbO3 :Sn, which also shows the doping concentration threshold effect. It is worthy to mention that, although Hf4+ and Sn4+ are of the same valence state, the UV photorefractive grating recorded at 351 nm in LiNbO3 :Hf is erasable by a 633-nm red beam; this is not the case for the UV band edge photorefractive gratings recorded at 325 nm in LiNbO3 :Sn. Xin et al. [71] found that the UV band edge photorefractive grating in LiNbO3 :Sn tends to be nonvolatile under the illumination of a 633-nm red beam after an initial quick decrease by an amount of ∼16%, as shown in Figure 1.6. This indicates that there are at least two kinds of different photorefractive centers involved in the UV band edge photorefractive process in LiNbO3 :Sn. The absorption spectra of LiNbO3 :Sn with Sn4+ -doping concentrations higher than 2.0 mol% at a cryogenic temperature of 3.8 K show a broadband absorption shoulder near to the absorption band edge of crystal. The related defect centers may also be responsible for the UV photorefraction in LiNbO3 , which will be discussed in more detail in Section 1.1.5.

14 | Guoquan Zhang et al. Table 1.5. The UV band edge photorefraction at 325 nm of LiNbO3 with various dopants. Here α is the absorption coefficient at 325 nm, η st is the saturated diffraction efficiency of gratings, σ ph /I e is the specific photoconductivity, τ b is the buildup time constant of gratings, Γ is the two-wave mixing gain coefficient, Δn is the UV-light-induced refractive index change, S is the recording sensitivity, and (M/#)/d is the recording dynamic range per unit thickness, respectively. The data are taken from Ref. [72]. Sample

CLN CMg1 CMg2 CMg4 CMg5 CMg7 CMg9 CZn5 CZn7 CZn9 CIn1 CIn3 CIn5 CHf4 CHf6 M22

M/# d −1

α (cm−1 )

η st (%)

σ ph /I e ×10−12 (cm/ΩW)

τb (s)

Γ (cm−1 )

Δn ×10−5

S (cm/J)

(cm )

7.79 5.81 5.32 2.47 5.75 1.72 2.07 3.49 3.03 2.90 10.80 6.38 5.02 2.47 2.51 17.75

1.52 1.18 1.73 2.10 46.8 75.9 81.7 13.6 61.4 67.7 9.76 15.4 49.5 1.23 2.77 50.7

2.31 9.22 9.57 10.8 82.6 110 278 52.7 237 255 1.42 33.7 98.8 105 120 3.64

4.38 2.85 2.33 1.90 0.35 0.43 0.073 0.33 0.20 0.19 8.60 0.24 0.27 0.48 0.28 3.08

4.75 4.53 5.41 1.69 23.5 20.6 25.3 11.9 27.6 38.0 2.46 11.6 22.9 23.1 29.3 29.2

2.39 1.50 1.82 0.33 2.75 1.97 3.35 1.75 4.26 4.67 0.91 1.16 2.24 2.15 3.23 4.03

4.29 2.78 2.86 0.45 8.60 9.13 33.1 10.9 24.5 37.7 1.44 7.35 25.5 5.30 28.5 7.75

4.76 0.78 0.77 0.11 0.28 0.22 0.31 0.54 0.28 0.39 2.62 0.57 0.69 0.13 0.38 5.63

The UV band edge photorefraction of LiNbO3 is also dependent on the crystal composition. It is found that the UV photorefraction of LiNbO3 is enhanced significantly with the increase of the Li/Nb ratio in crystal [73]. For example, the two-wave mixing gain coefficient Γ and the diffraction efficiency of grating η were measured to be ∼ 9.98 cm−1 and ∼ 5.3% at 325 nm in a 0.5-mm pure near-stoichiometric LiNbO3 with a Li/Nb ratio of 49.9%, while those in a 0.5-mm pure congruent LiNbO3 were measured to be ∼ 3.95 cm−1 and ∼ 1.42%, respectively.

1.1.4 Enhancement on ultraviolet light-induced absorption LiNbO3 grown by the Czochralski method is usually in the Li deficient state, and there are plenty of intrinsic anti-site NbLi defect center and its related ones such as small po4+ 4+ larons Nb4+ Li and bipolarons NbLi -NbNb in pure congruent LiNbO3 [44, 74–76]. These intrinsic defect centers participate in the light-induced charge transport process and play a key role in the photorefractive behaviors of LiNbO3 , which directly leads to a series of effects and applications such as the light-induced absorption change and

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Fig. 1.6. The normalized decay dynamics of the UV band edge photorefractive gratings in LiNbO3 :Sn (4.0 mol%) under the sequential erasure of a red beam of 82.7 mW/cm2 at 633 nm and a UV beam of 32.6 mW/cm2 at 325 nm. The data are taken from Ref. [71].

two-color nonvolatile holographic recording [77–81]. Therefore, the photorefractive properties of LiNbO3 can be modified by adjusting the concentration of intrinsic defect centers such as NbLi . According to the Li-vacancy model [44], the concentration of anti-site NbLi in crystal decreases with the increase of the doping concentration of impurities such as Mg2+ , Zn2+ , In3+ , Hf4+ , Zr4+ , and Sn4+ . When the doping concentration of these so-called damage-resistent ions is higher than the respective concentration threshold, the anti-site NbLi disappears, and the photorefractive effect and the light-induced absorption change are dramatically suppressed in the visible and nearinfrared spectral range. However, similar to the case of photorefractive effect, the UV-light-induced absorption change is also enhanced significantly in LiNbO3 highly doped with the socalled damage-resistent ions [82–84]. Zhang et al. [82] showed that, under the illumination of incoherent UV light centered at 365 nm, a broad light-induced absorption band peaked at 760 nm appears in the congruent LiNbO3 :Mg with the Mg-doping concentrations lower than its concentration threshold. This absorption band is unstable in dark at room temperature, and it decays quickly with a time constant in the order of μs–ms in congruent LiNbO3 and ms in near-stoichiometric ones, which can be attributed to the formation of small polarons Nb4+ Li . On the other hand, when the Mg-doping concentration is higher than the concentration threshold, a very broad absorption band covering the visible and the near-UV spectral ranges appears under the illumination of an incoherent UV beam centered at 365 nm. This broad light-induced absorption band also decays in dark, but with a time constant of tens of seconds in highly doped congruent LiNbO3 :Mg, and the temporal dark decay dynamics can be well described by the empirical function proposed by Schirmer et al. [85, 86] which was used to describe the dark decay dynamics of small hole polarons O− . Therefore, one may ascribe the appearance of this broad UV-light-induced absorption band to the formation of small hole polarons in highly doped LiNbO3 :Mg. Due to the enhanced UV-light-induced absorption change in the visible and near-infrared, the recording sensitivity in the visible and near-infrared is significantly improved in highly doped LiNbO3 :Mg after an appropriate pre-exposure to

16 | Guoquan Zhang et al.

Fig. 1.7. The temporal evolutions of the two-color holographic recording, optical fixing, and grating erasing processes in LiNbO3 :Mg with doping concentrations of 4.0 mol% (CMg4), 5.0 mol% (CMg5), and 9.0 mol% (CMg9), respectively. The holographic gratings with a grating period of 1.1 μm were recorded by two 633-nm red beams, each with a recording intensity of 122.9 mW/cm2 , under the simultaneous illumination of a 120-mW/cm2 incoherent sensitizing UV beam centered at 365 nm. Then the gratings were read out by one of the recording red beams in the optical fixing stage, and finally the gratings were erased by the incoherent UV beam. The data are taken from Ref. [90].

the UV light [82]. For instance, the photorefractive recording sensitivity at 780 nm is only about 10−5 cm/J in LiNbO3 :Mg (5.0 mol%), while it is enhanced significantly up to 0.5 cm/J after a pre-exposure to the incoherent UV light centered at 365 nm with an intensity of 0.81 W/cm2 . This may be applicable to two-color nonvolatile holographic storage with high recording sensitivity [87–90]. As shown in Figure 1.7, the two-color holographic recording sensitivity can reach 1.0 cm/J at 633 nm in a LiNbO3 :Mg (5.0 mol%) crystal sensitized by a 120-mW/cm2 incoherent UV light centered at 365 nm. However, the dynamic range of the holographic grating M/# decreases (down to 0.3 in Figure 1.7) because of the erasure by the sensitizing UV light. This also indicates that there are several photorefractive defect centers involved in the UV-lightinduced charge transport process in doped LiNbO3 . More detailed discussions on the defect structures near the band edge of LiNbO3 will be presented in Section 1.1.5.

1.1.5 Absorption edge and related defect structures in LiNbO3 In the past four decades, the defect centers in LiNbO3 involved in the photorefractive effect in the visible and near-infrared spectral ranges have been intensively studied, and the visible and near-infrared photorefractive properties of LiNbO3 can be effectively adjusted through the control of photorefractive defect centers [42, 43, 74]. However, the progress on UV photorefractive effect is slow, and the defect centers and the related light-induced charge carrier transport processes in LiNbO3 involved in the photorefractive effect in the UV and band edge spectral ranges are still not clearly clarified.

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Fig. 1.8. (a) The temperature dependence of the effective band energy gap E g of a pure congruent LiNbO3 . The dependence of (b) the fundamental band gap E gBE (0) at T = 0 K, (c) the average energy E pBE of the Einstein oscillator, and (d) the strength of the electron–phonon interaction on the Mg2+ -doping concentration, respectively. The data are taken from Ref. [91].

Recently, Xin et al. [73, 91] studied the spectral property of the absorption band edge and the related defect centers in a series of LiNbO3 crystals with different dopants and doping concentrations. The results show that the absorption band edge is red shifted significantly with the increase of crystal temperature, and the effective band energy gap E g , defined as the photon energy at which α = 70 cm−1 , obeys the Bose– Einstein expression [92] E g (T) = E gBE (0) −

2a B , exp(E pBE /k B T) − 1

(1.1)

where T is the absolute temperature, k B is the Boltzmann constant, E gBE (0) is the fundamental band gap at T = 0, E pBE is the average energy of the Einstein oscillator corresponding to the most active phonons, and a B represents the strength of electron– phonon interaction. Figure 1.8 (a)–(d) shows the temperature dependence of E g (T) of a pure congruent LiNbO3 , the Mg2+ -doping concentration dependence of the fundamental band gap E gBE (0), the energy of the Einstein oscillator E pBE , and the strength of the electron–phonon interaction a B , respectively. It is seen that E gBE (0) increases with increasing Mg2+ -doping concentration, which is in good accordance with the blue shift of the absorption band edge of LiNbO3 :Mg with increasing Mg 2+ -doping con-

18 | Guoquan Zhang et al. centration. More interestingly, the electron–phonon interaction in LiNbO3 also shows the Mg2+ -doping concentration threshold effect, and the Mg2+ -doping concentration threshold is the same as that in optical damage resistance of LiNbO3 :Mg. For example, with the increase of Mg2+ -doping concentration over the concentration threshold, the energy of the Einstein oscillator E pBE decreases from ∼ 400 cm−1 to ∼ 250 cm−1 , and the strength of the electron–phonon interaction a B changes from ∼ 0.45 to ∼ 0.28 eV, respectively. In addition, at temperature higher than 400 K, the absorption edge tail of LiNbO3 obeys the Urbach rule [93] α = α 0 exp [

ς (ℏω − ℏω0 )] , kB T∗

(1.2)

where α 0 , ς, and ω0 are constants, T ∗ = (ℏω p /2k B ) coth(ℏω p /2k B T) is the effective temperature, and ω p is the phonon frequency at which the most active interaction occurs with electrons, which also changes with the increase of Mg2+ -doping concentration in a similar tendency to that of E pBE . It is known that the photoconductivity of crystal will increase with the decrease of the electron–phonon interaction [94], which would give rise to the optical damage resistance in highly doped LiNbO3 :Mg. Interestingly, as the temperature goes down below ∼400 K, a new broad absorption band shows up at the bottom of the Urbach tail in LiNbO3 , as shown in Figure 1.9 [91]. This broad absorption band enhances with the increase of Mg2+ -doping concentration, and it can be decomposed into two Gaussian components peaked at 3.83 eV and 4.03 eV, respectively, in highly doped LiNbO3 :Mg (higher than 5.0 mol%). In addition, under the illumination of a 325-nm UV beam, a light-induced broad absorption band Δα li appears even at room temperature (298 K) in highly doped LiNbO3 :Mg (> 5.0 mol%), as shown in Figure 1.10. This light-induced absorption band can be decomposed into two Gaussian-type spectral profiles with typical characteristics of small hole polarons O− [95] peaked at 2.64 and 3.45 eV and with the finger-

Fig. 1.9. The absorption spectra of LiNbO3 :Mg crystals with various doping concentrations at 3.8 K. Here CMg1, CMg2, CMg3, CMg4, CMg5, CMg6.5, CMg7.8, and CMg9 represent LiNbO3 :Mg crystals with Mg2+ -doping concentrations of 1.0, 2.0, 3.0, 4.0, 5.0, 6.5, 7.8, and 9.0 mol%, respectively. The data are taken from Ref. [91].

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Fig. 1.10. The light-induced absorption spectrum of LiNbO3 :Mg (5.0 mol%) at room temperature under the illumination of a 0.26-W/cm2 , 325-nm UV light beam. The solid black curve is the measured one, while the dashed and dotted curves are the theoretical fits of two small hole polaronic components peaked at 2.64 and 3.45 eV, and the dash-dotted curve is the sum of two polaronic components, respectively. The data are taken from Ref. [91].

print ratios W 2 /M being 0.15 and 0.04 eV, respectively, where W is the half-width at half-maximum of the polaronic spectrum and M is the peak spectral energy of the respective small hole polarons. The spectral component peaked at 2.64 eV and with a fingerprint ratio of W 2 /M = 0.15 eV should be ascribed to the small hole polarons (O− -VLi ) near the Li vacancy in LiNbO3 , while that peaked at 3.45 eV and with a ratio of W 2 /M = 0.04 eV may be attributed to the small hole polarons (O− -MgNb ) near the negatively charged defect center MgNb in highly Mg2+ -doped LiNbO3 . Correspondingly, the spectral components peaked at 3.83 and 4.03 eV in Figure 1.9 may be ascribed to the defect centers (O2− -VLi ) and (O2− -MgNb ), respectively. Li et al. [96] reported that the absorption band edge of LiNbO3 may be related to the Li-vacancy VLi . Herth et al. [97] observed the evidence for small hole polarons induced by 532-nm laser pulses in LiNbO3 :Fe. More recently, Luedtke et al. [98] suggested a hidden reservoir of photoactive electrons in LiNbO3 . These may provide further experimental evidences for the UV-light-induced small hole polarons. It is worthy to mention that, absorption bands at cryogenic temperature similar to those in Figure 1.9 were also observed in Zn2+ -, In3+ -, Hf4+ -, Sn4+ -, or Zr4+ -doped LiNbO3 , but the absorption is much weak in LiNbO3 :Zr [71, 73]. This may give a reasonable explanation on why LiNbO3 :Zr is of resistance against optical damage even in the UV spectral range. Therefore, the defect centers such as O2−/− near the negatively charged centers VLi and MgNb may play an important role in the UV photorefraction and the UV-light-excited charge carrier transport process in LiNbO3 .

20 | Guoquan Zhang et al. 1.1.6 Other effects and applications in the ultraviolet spectral range LiNbO3 , as a promising candidate for “silicon of photonics,” has important practical applications such as nonlinear frequency conversion, guided and integrated optics, and optical information processing. With the development of high-quality crystal growth and the quasi-phase-matching techniques and the extension of the spectral range of optical damage resistance of LiNbO3 , the fabrication of high quality periodically poled LiNbO3 (PPLN) becomes the key issue. In 2003, Fujimura et al. [99] and Buse et al. [100] reported the ferroelectric domain inversion field could be reduced significantly under the illumination of UV light near to the absorption band edge of LiNbO3 . However, Eason et al. reported that the domain inversion will be suppressed under the illumination of UV beam at 244 nm, at which the photon energy is much higher than the absorption band edge of LiNbO3 . The light-induced assistance or suppression of the ferroelectric domain inversion may be ascribed to the formation of additional new light-induced space-charge fields in LiNbO3 [99–102]; nevertheless, the mechanism and the involved defect centers in the light-induced charge carrier transport processes are still in debate. In addition, Eason et al. [103] reported the fabrication of optical channel waveguides in congruent LiNbO3 by direct writing with a 244-nm CW laser beam. These effects and techniques may provide effective ways for light-assisted ferroelectric domain engineering and laser direct writing waveguide structures, which would have important potential applications in ferroelectric domain engineering, nonlinear optics, and integrated photonics.

1.2 Incoherent nonlinear optics and discrete spatial solitions Nonlinear optics is a field of history as venerable as that of lasers, yet in the past decades there has been a surge of research activities on incoherent nonlinear phenomena and nonlinear phenomena in periodic or discrete systems. Both incoherent and periodic systems are abundant in nature. In optics, a typical example of incoherent phenomena involves a partially coherent optical beam such as from an incandescent light bulb or sunlight, while a typical example of periodic systems arises from a closely spaced waveguide array in which the collective behavior of wave propagation exhibits many intriguing phenomena. Solitons, as one of the examples, have been actively considered as a typical nonlinear wave phenomenon in many and diverse branches of physics such as optics, plasmas, condensed matter physics, fluid mechanics, particle physics, and astrophysics, but in recent years the field of solitons and related nonlinear phenomena have been substantially advanced and enriched by research and discoveries in nonlinear optics. In particular, advances in the study of incoherent solitons and discrete spatial solitons in optical periodic media not only led to advances in our understanding of fundamental processes in nonlinear optics and photonics, but also impacted a variety of other disciplines in nonlinear science. In this part, we

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provide an overview of optical incoherent solitons and discrete spatial solitons. This chapter will cover a variety of issues pertaining to incoherent self-trapped waves as well as different families of spatial lattice solitons such as coherent discrete solitons, gap solitons, surface solitons, and incoherent discrete solitons.

1.2.1 From coherent to incoherent light In optics, coherence is a property of light that determines whether a stationary (i.e. temporally and spatially coherent) interference can be formed. More generally, it describes all properties of the correlation between physical quantities describing a single light wave or between several light waves or wave packets, such as amplitude and phase. The most typical example of a coherent light source is “single-frequency” lasers which produce the laser beam when a group of atoms are pumped into an excited state and, upon their de-excitation, light is emitted collectively, thus making the output coherent. (All emissions are in phase and are of the same wavelength.) The invention of the lasers has greatly promoted the development of optics and especially nonlinear optics. However, nature is full of incoherent light. A typical example is the light emitted from a white light source (i.e. a light bulb) which is highly incoherent. Intuitively, one can consider that a white light source emits a light beam with random phase fluctuations across the whole beam in space (spatial incoherence) and with many varying wavelengths all at random intervals (temporal incoherence). Moreover, the incoherent light has a wide range of applications in many areas of optics such as microscopy, displays, and solar cell technology. In nonlinear optics, the study of the nonlinear dynamics of incoherent light and related soliton phenomena have, however, attracted little attention. In particular, it was believed that solitons and related nonlinear wave phenomena can only be generated and studied with coherent light waves such as light beams from the lasers. This belief was not changed for a long time until the discovery of the incoherent spatial solitons in 1996, which opened a new era for nonlinear optics especially for incoherent nonlinear optics. Incoherent optical solitons only exist in some kind of nonlinear media with special nonlinear response, such as noninstantaneous saturated nonlinearity and instantaneous nonlocal nonlinearity. The nonlinear phenomena resulting from the incoherent light are very different from those resulting from the coherent light. This is because the coherence plays an important role in the nonlinear process, and it can be considered as a new degree of freedom to manipulate and control the propagation dynamics of light waves. Furthermore, the incoherent light beams can be viewed as a representative of all weakly correlated particle systems, and the phenomena and conclusions drawn from the incoherent light can be extended to other weakly correlated particle systems.

22 | Guoquan Zhang et al. 1.2.2 From bulk to discrete media Light propagation in bulk media can be described by wave functions that follow the continuum model. There are situations, however, in which the evolution of an optical field must be described by a discrete model. This happens when the field can be described as a sum of discrete modes. Such systems are called discrete systems which exist in different branches of physics such as optics, solid state physics, Bose– Einstein condensation, and so on. All these discrete systems can be described by the similar mathematical equations and thus share similar physical dynamics. In optics, the most typical example is photonic crystals [104], which have a spatial periodicity in their dielectric constant. There are photonic band gap structures in such photonic materials, i.e. a frequency window in which propagation through the crystal is inhibited. Recently, another different type of periodic structures called waveguide arrays or photonic lattices has attracted great attentions [105–107]. Waveguides are the basic elements of the waveguide arrays, and light propagation in waveguide arrays or waveguide lattices can be described by a sum of the individual modes in each waveguide. Due to the coupling between nearby waveguides, light propagation in waveguide arrays exhibits many novel and interesting physical phenomena which are unique to discrete systems, such as Bloch waves, discrete diffraction, anomalous refraction and diffraction, and photonic band gap structures [108, 109]. Furthermore, the nonlinearity in photonic lattices can be easily controlled, and the balance between the nonlinearity and discrete diffraction results in various novel spatial solitons which do not exist in bulk media [107, 110]. Moreover, since the evolution of the optical wave packets in photonic lattices can be directly observed, photonic lattices have turned into a convenient platform to study a variety of intriguing linear and nonlinear discrete phenomena.

1.2.3 Introduction to spatial optical solitons Optical solitons are self-trapped wave-packets in nonlinear media which result from the delicate balance between diffraction or dispersion and the nonlinearity. Typically there are two kinds of optical solitons, spatial and temporal solitons, where the nonlinear effect balances the diffraction and dispersion of the optical wave packets, respectively. Optical spatial and temporal solitons share the same physical mechanism, although they may emerge in different domains or different physical systems. There is a simple and intuitive physical picture to understand spatial solitons. When a narrow optical beam linearly propagates in a medium without affecting its properties, the beam undergoes natural diffraction and broadens with distance. The narrower the initial beam, the faster its diffraction. In nonlinear materials, the presence of light modifies the refractive index. For instance, in a Kerr-type nonlinear material, the refractive index change resembles the intensity profile of an incident Gaussian beam, forming

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Fig. 1.11. Experimental demonstration of an optical spatial soliton propagating through a nonlinear photorefractive crystal. Top: side-view of the soliton beam taken from scattered light; bottom: normal diffraction of the same beam when the nonlinearity is “turned off” [113].

an optical lens due to the increase of the index in the beam’s center which will focus light. When the self-focusing effect exactly balances beam divergence (diffraction), the beam becomes self-trapped at a very narrow width and is called an optical spatial soliton as shown in Figure 1.11. Solitons can also be understood from the waveguide picture: when the localized wave packet induces a potential (via the nonlinearity) and “captures” itself, it becomes a bound state of its own induced potential. A spatial optical soliton forms when a very narrow optical beam induces a waveguide structure and guides itself in its own induced waveguide. Both bright and dark spatial solitons can exist in bulk media with different types of nonlinearity. More specifically, self-focusing media can support bright spatial solitons [111], while dark spatial solitons exist in homogenous media with a self-defocusing nonlinearity [112]. The study of optical spatial solitons started already in 1964 with the discovery of nonlinear self-trapping of CW optical beams in a bulk nonlinear medium [114]. Selftrapping was not linked to the concept of spatial solitons because of its unstable nature. During the 1980s, stable spatial solitons were observed in Kerr nonlinear media in which diffraction spreading was limited to only one transverse dimension [115]. Later on, spatial solitons in two transverse dimensions were observed by use of photorefractive screening nonlinearity [113]. Since then photorefractive spatial solitons have become the most ideal candidate for studying various interesting physical properties of solitons. In fact, the study of photorefractive spatial solitons has greatly advanced the development of soliton science, which led to many discoveries of solitonrelated nonlinear phenomena, including the incoherent spatial solitons and discrete spatial solitons.

1.2.4 Nonlinear optics with incoherent light Before 1996, all experimental studies of solitons in optics as well as in fields beyond optics were based on coherent waves. In fact, for decades, solitons have been exclusively considered to be coherent entities. Yet, nature is full of incoherent (or partially coherent) radiation sources. One can simply focus a light beam from a natural radia-

24 | Guoquan Zhang et al. tion source such as the sun or an incandescent light bulb into a narrow spot, thereby creating a partially coherent light beam. Can incoherent light also self-trap and form optical solitons? More generally, can an ensemble of weakly correlated particles such as those representing partially coherent wavefronts form a self-trapped entity? This intriguing and challenging question has motivated several earlier experiments on selftrapping of incoherent light. By now, a series of experimental demonstrations along with theoretical studies have provided clear evidence that incoherent bright and dark spatial solitons are indeed possible in nonlinear materials provided that these materials have noninstantaneous self-focusing/defocusing nonlinearity [116–123]. This brings about the interesting possibility of using low-power incoherent light beams to form solitons, which in turn can guide and control other high-power coherent laser beams. The key ingredient for the generation of incoherent solitons is the noninstantaneous nature of the nonlinearity. That is, for incoherent solitons to exist the nonlinearity must have a temporal response much slower than the characteristic time of the random fluctuation that makes the beam incoherent. Photorefractive nonlinearity is a typical kind of noninstantaneous nonlinearity, and the material response time can be controlled by the intensity of the beam. This fact, combined with the fairly large nonlinearities offered by photorefractive crystals, makes them a convenient choice for experiments with incoherent solitons. In fact, most incoherent solitons and related incoherent NLO phenomena are experimentally demonstrated in photorefractive crystals, although recently incoherent solitons are also realized in instantaneous medium with nonlocal nonlinearity. The former is averaged out by time while the latter takes advantage of the spatial averaging effect. In the following, we will briefly summarize the experimental progress and advancement of various incoherent spatial solitons in noninstantaneous photorefractive crystals including incoherent bright solitons, incoherent dark solitons, and incoherent soliton arrays.

Incoherent bright solitons In fact, incoherent bright solitons are the firstly experimentally realized incoherent localized wave packets in 1996 by Segev’s group [116]. The first experiment on selftrapping of a bright partially incoherent optical beam was done in a photorefractive SBN:75 crystal, which provided a noninstantaneous self-focusing photorefractive nonlinearity. The incoherent beam was created by sending a coherent laser beam passing through a rotating diffuser which creates the spatially incoherent beam. The diffuser rotates on a time scale much faster than the response time of the SBN crystal. Then such a spatially incoherent beam focused by a circler lens was sent into the SBN crystal, at linear propagation, such a incoherent beam diffracts more compared with the coherent beam of the same size, due to the diffraction is dominated by the speckle size not the beam size in such a spatially incoherent beam. However, when a positive biased voltage was applied, the diffraction was balanced by the self-focusing nonlin-

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earity, resulted in self-trapping of the beam, which maintained a constant width when a background beam was illuminated. This first demonstration of optical spatial solitons from partially coherent light soon followed by experimental demonstrations of self-trapping of a “fully” incoherent white light beam from a simple incandescent light bulb as well as self-trapping of a “dark” incoherent light beam [117, 118]. These experimental observations surely motivated the theoretical studies toward understanding of incoherent solitons. Soon thereafter, several theoretical approaches were developed, including the coherent density theory, the modal theory, the theory describing the propagation of mutual coherence, and the simplified ray-optics approach [119–123, 123]. The real success of the theories explaining incoherent solitons was in their ability to come up with exciting new predictions, some of which suggested truly fascinating phenomena. It became very clear right from the beginning that incoherent solitons are not some esoteric creatures specifically related to photorefractives, but rather form a general and rich new class of solitons, whose existence is relevant to diverse fields even beyond nonlinear optics. Indeed, in a number of subsequent experiments, novel phenomena arising from incoherent self-trapping and incoherent modulation instability [124] were observed, including incoherent anti-dark solitons [125], incoherent pattern formation [126], soliton clustering in weakly correlated wavefronts [127, 128], and photonic lattices induced by partially coherent light [127]. The underlying physics of the observed phenomena may relate to other weakly correlated wave systems with a noninstantaneous nonlinearity. We note that a recent study showed that the bright incoherent solitons can also exist in instantaneous nonlinear media but with strong nonlocal nonlinearity [129].

Incoherent dark solitons “Dark beams” are nonuniform optical beams that contain either a one-dimensional dark stripe or a two-dimensional dark hole resulting from a phase singularity or an amplitude depression in their optical field. For decades, self-trapped dark beams (dark solitons) have been observed using coherent light only. The first experimental demonstration of self-trapped dark incoherent light beams (self-trapped dark incoherent wave packets) came in 1998. Both dark stripes and dark holes nested in a broad partially spatially incoherent wavefront were self-trapped to form dark solitons in a host photorefractive medium, which in turn created refractive-index changes akin to planar and circular dielectric waveguides [118]. Based on the knowledge on coherent dark solitons, one may speculate that the transverse phase plays a crucial role also for incoherent solitons. Fundamental 1D coherent dark solitons require a transverse phase shift at the center of the dark stripe, whereas an initially uniform transverse phase leads to a Y-junction soliton [112]. Furthermore, 2D coherent dark solitons (vortex solitons) require a helical 2 m π transverse phase structure. Extending the idea of dark coherent solitons to dark incoherent solitons raises several questions. If dark incoherent solitons were to exist, is their phase

26 | Guoquan Zhang et al. structure important (as for coherent dark solitons) or irrelevant (as for bright incoherent solitons, upon which the phase is fully random)? And, if the phase does play a role, how can it be “remembered” by these incoherent entities throughout propagation? Altogether, even though the bright incoherent solitons have been experimentally demonstrated, the possibility for dark incoherent solitons was not at all clear. Motivated by these questions, the coherent density approach was employed first for the study of propagation of an incoherent dark beam in biased photorefractives [130, 131]. Surprisingly, it was found that, although an arbitrary dark stripe-bearing incoherent beam experienced considerable evolution during propagation, it eventually stabilized around a self-trapped solution. Such a self-trapped dark incoherent soliton required an initial transverse phase jump, and after self-trapping it was always “gray” (instead of “black,” as expected for a coherent dark soliton with phase jump). This theoretical work, although not providing clear answers to the questions raised above, did suggest that dark incoherent solitons should exist. Indeed, almost in parallel with this work came the first experimental demonstration of self-trapping of a dark incoherent light beam [118]. Shortly thereafter, the modal theory of incoherent dark solitons was developed, which revealed the underlying mechanism of such solitons. It was found that incoherent dark solitons were actually associated with self-induced waveguides involving both bound states (guided modes) and the continuous belt of radiation modes. This 1D theory of incoherent dark solitons provided a qualitative explanation of the experimental observations such as why a phase jump was needed to initiate the dark incoherent solitons and why the observed dark incoherent solitons were always gray [132]. The first experiment results of dark incoherent solitons are shown in Figure 1.12, where the top row is for a 1D dark incoherent soliton and the middle and bottom rows for a 2D vortex incoherent soliton [118]. The dark notch at the output facet of the crystal is broader than at the input facet due to the linear diffraction when nonlinearity was not present. By applying self-defocusing nonlinearity, self-trapping of the dark notch to its initial size was achieved. Compared with coherent dark solitons, it was observed that incoherent dark solitons were always gray (with their grayness depending on the beam coherence). This observed behavior was confirmed by numerical simulations, and was also in agreement with theoretical predictions. Figure 1.12 (b) and (c) displays experimental results obtained using a 2D incoherent dark soliton (a 2D dark “hole” on a uniform intensity background). The self-trapped “hole” was again gray, and it became less visible when the beam was made more incoherent. However, even when the grayness was large and the self-trapped “hole” was almost invisible, one could still monitor its presence by its induced waveguide. Following this first experiment, a few other experiments were performed for the study of dark incoherent solitons. In particular, dark incoherent soliton Y-splitting and associated “phase-memory” effects were investigated, and it was found that dark incoherent solitons are characterized by strong “phase-memory” effects that are otherwise absent in the linear region [133]. Using an amplitude mask (which provides the

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Fig. 1.12. Self-trapping of a dark partially coherent beam. Shown are photographs of the transverse intensity patterns from (a) a dark stripe beam and (b, c) from a dark vortex beam taken at crystal input face (left), output face with linear diffraction (middle), and output face with nonlinearity (right) [118].

“even” input conditions for Y-junction dark solitons), a dark incoherent Y-splitting soliton was experimentally observed. Interestingly enough, as the coherence of the dark beam decreases, the grayness of the soliton increases, but the spacing between the branches of the soliton Y-splitting at the crystal output remains the same as discussed in Ref. [133]. In separate experiments, such a dark incoherent Y-junction soliton was shown to be able to induce a Y-splitting waveguide that could be used for guiding other beams [134]. Although a quasi-monochromatic spatially incoherent light source was employed, the experiments suggest that spatial solitons formed from “fully” (temporally and spatially) incoherent light sources (e.g., incoherent white light) might also be able to induce waveguides capable of guiding other coherent and incoherent beams.

Incoherent soliton arrays Pixel-like spatial solitons and soliton-based waveguide arrays are of particular interest because of their potential applications in signal processing and information technology [135]. Yet, it has always been a challenge to create (or fabricate) a closely spaced two-dimensional soliton array or a 2D waveguide array [136]. Pixel-like spatial solitons from partially coherent light were successfully demonstrated [136]. Just as shown in Figure 1.13, these soliton pixels were created under critical experimental conditions (e.g., the degree of beam coherence, the strength of non-

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Fig. 1.13. Experimental results of 2D spatial soliton pixels of partially incoherent light (3D intensity plot) [136].

linearity, and the spacing between nearby pixels) which relate to the growth of the induced incoherent modulation instability [137]. However, when a spatially coherent extraordinarily polarized beam was used instead, even at a much lower strength of nonlinearity, the beam tends to break up into many disordered filaments rather than ordered soliton arrays due to strong coherent modulation instability. In general, a broad, uniform, and partially coherent extraordinarily polarized beam tends to break up into many disordered filaments due to incoherent modulation instability. However, under certain conditions, ordered patterns as well as clusters of quasi-solitons in incoherent (weakly correlated) wavefronts were observed. When the incoherent beam was periodically modulated initially, robust 2D pixel-like spatial solitons were observed. This experiment brings about the possibility for optically inducing reconfigurable photonic lattices with low-power incoherent light. Pixel-like solitons and nonlinear waveguide arrays are of particular interest apart from their potential applications. This is because the collective behavior of light propagation in the waveguide arrays induced by the pixel-like solitons exhibits many intriguing phenomena, such as discrete solitons which also exist in other nonlinear discrete systems beyond optics. The interesting results of discrete phenomena in photonic lattices induced by partially incoherent light will be discussed in the next section.

1.2.5 Spatial solitons in discrete systems Linear and nonlinear discrete or periodic systems are abundant in nature. In optics, a typical example of such an arrangement is that of a closely spaced waveguide array, in which the collective behavior of wave propagation exhibits many intriguing and unexpected phenomena that have no counterpart in homogeneous media. In the past decades, waveguide arrays have been considered as a convenient platform to study discrete solitons and various related discrete phenomena. In particular, the study of spatial solitons in photonic lattices has extended the soliton research from continuous media to discrete media which has become the hottest topic in soliton science in the past decade. Discrete solitons arise from the interplay between the discrete diffraction and nonlinearity. Since the first prediction [106] and experimental demonstra-

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tion [138] of discrete solitons in 1D AlGaAs waveguide arrays, this field has been greatly developed. In the following, we will provide a brief overview and summary of some experiment results of the formation of different types of discrete solitons in 2D square photonic lattices induced by partially spatially incoherent light in nonlinear photorefractive crystals.

Optically induced photonic lattices With today’s nanofabrication technology, to create a closely spaced 1D waveguide array on a substrate material is not a problem. As an example, such waveguide structures have been fabricated with AlGaAs semiconductor materials or LiNbO3 crystals. In fact, the first experimental demonstration of discrete solitons was carried out in fabricated 1D semiconductor waveguide arrays [138]. Yet, it has always been a challenge to create or fabricate 2D or 3D waveguide arrays in bulk media. However, discrete solitons in higher dimensions will lead to many interesting phenomena which do not exist in 1D photonic lattices. Later, it has been suggested that 2D waveguide lattices could be optically induced in a photorefractive crystal [139]. The physical idea of this method is to transfer the nondiffracting beams with different periodic intensity patterns to the corresponding refractive index change in photo-sensitive nonlinear bulk media. Indeed, experimental observations of discrete solitons in such waveguide lattices were established with optical induction by sending multiple interfering beams into the nonlinear photorefractive crystal [140, 141]. The coherent multiple-beam interference method has many disadvantages for creating photonic lattice structures. For instance, the induced lattice tends to be more sensitive to ambient perturbation. Furthermore, when the lattice beam itself experiences an appreciable nonlinearity, it becomes considerably more susceptible to modulation instability and the lattice structure cannot be stable either when the lattice spacing is too small or the nonlinearity is too high. With a partially coherent lattice beam, such a problem can be somewhat avoided due to reduced nonlinear interference/interaction between the lattice sites. Moreover, more complicated lattice structures such as binary lattices or lattices with structured defects or surfaces cannot be created by the interference method. In view of that, we proposed a new method which is based on the amplitude modulation of a partially incoherent optical beam [136]. The method of optical induction based on amplitude modulation of partially coherent light provides an effective way for creating stable nonlinear photonic lattices and later on for inducing various reconfigurable lattices with structured defects and surface [142–144], even 3D photonic lattices [145]. We emphasize that the method of optical induction of waveguide lattices and nonlinear self-trapping of discrete solitons in SBN photorefractive crystals is directly related to the anisotropic property of the photorefractive nonlinearity. In general, in an anisotropic photorefractive crystal, the nonlinear index change experienced by an optical beam depends on its polarization as well as on its intensity. Under appreciable bias conditions, i.e. when the photorefractive screening nonlinearity is domi-

30 | Guoquan Zhang et al. nant, this index change is approximately given by Δn e = [n3e r33 E0 /2](1 + I)−1 and Δn o = [n3o r13 E0 /2](1 + I)−1 for extraordinarily polarized (e-polarized) and ordinarily polarized (o-polarized) beams, respectively [146, 147]. Here E0 is the applied electric field along the crystalline c-axis, and I is the intensity of the beam normalized to the background illumination. Due to the difference between the nonlinear electro-optic coefficients r33 and r13 , r33 is more than 10 times larger than r13 under the same experimental conditions in a SBN:60 crystal. Thus, if the lattice beam is o-polarized while the probe beam is e-polarized, the lattice beam would experience only weak nonlinear index change. We also mention that, if the external applied electric field E0 is orthogonal rather than parallel to the crystalline c-axis, the nonconventionally biased crystal will exhibit anisotropic hybrid nonlinearity, where the self-focusing and self-defocusing nonlinearity coexist [148].

Discrete solitons in 2D self-focusing photonic lattices In this section, we will show some experimental results of different types of discrete solitons and discrete phenomena in 2D photonic lattices optically induced by amplitude modulation of partially incoherent light. Discrete solitons are formed when discrete diffraction of the probe (soliton-forming) beam which results from the coupling between the nearby waveguides is balanced by the self-focusing nonlinearity, whose propagation constants reside in the total internal reflection gap of the photonic band gap structures [106]. First, we present our experimental results on 2D fundamental discrete solitons as shown in Figure 1.14. A stable waveguide lattice is induced by an o-polarized partially coherent beam. Then, a probe beam is launched into one of the waveguide channels, propagating collinearly with the lattice. Due to weak coupling between closely spaced

Fig. 1.14. Experimental demonstration of a discrete soliton in a partially coherent lattice. (a) Input, (b) diffraction output without the lattice, (c) discrete diffraction at 900 V/cm, and (d) discrete soliton at 3000 V/cm. Top: 3D intensity plots; Bottom: 2D transverse patterns. [146].

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waveguides, at a low bias field the Gaussian-like probe beam exhibits discrete diffraction with most of the energy flows from the center toward the diagonal directions of the lattice. While without the optically induced photonic lattices, the linear diffraction pattern of the probe beam is still Gaussian like. Even more importantly, a discrete soliton is observed at a high bias field, with most of energy concentrated in the center and the four neighboring sites along the principal axes of the lattice, and they are in phase with each other [146]. Soon after, high-order discrete solitons with unique phase structures and multicomponent solitons are also realized in optically induced photonic lattices. The first example is the 2D discrete vector solitons, when two mutually incoherent beams propagating along the same lattice site can lock into a fundamental vector soliton, although each beam alone would experience discrete diffraction under the same conditions. Such mutually trapped two-component vector solitons are attributed to the intensity-dependent nonlinearity [149]. By launching two mutually coherent beams (with controlled phase relationship) into two neighboring lattice sites of the square lattice rather than overlapping them in the same lattice site, we have also demonstrated the formation of discrete dipole solitons in a 2D optically induced photonic lattice [150]. Moreover, discrete soliton trains are also experimentally realized by sending a stripe beam into the square photonic lattices, such self-trapped states are only localized in one direction while extended in the perpendicular direction, which can be considered as trains of in phase fundamental discrete solitons [147]. In addition to on-site excitation of the probe beams, we have also studied offsite excitations in weakly coupled lattices created by optical induction. When a weak Gaussian-like probe beam is launched between two lattice sites, its energy switches mainly to the two closest waveguide channels evenly, leading to a symmetric beam profile. However, as the intensity of the probe beam exceeds a threshold value, the probe beam evolves into an asymmetric beam profile, akin to that resulting from the symmetry breaking in a double-well potential. Should the probe beam itself experience no or only weak nonlinearity, such symmetry breaking in the beam profile does not occur regardless the increase of its intensity. When two probe beams are launched in parallel into two off-site locations, they form symmetric or antisymmetric (dipolelike twisted) soliton states depending on their relative phase [151]. Our experimental and theoretical studies show that both symmetric states (corresponding to a single beam on-site) and antisymmetric states (corresponding to two out-of-phase beams on two different sites) can be linearly stable. Photonic lattices also support other interesting self-trapped nonlinear localized sates while they do not exist in nonlinear continuous media such as vortex solitons which only exist in self-defocusing media as dark solitons. The vortex beam typically breaks up into filaments in self-focusing continuous media due to the azimuthal modulation instability [152]. However, such instability can be suppressed by the periodic refractive index potential in photonic lattices, which leads to the observation of stable bright discrete vortex solitons under self-focusing nonlinearities [153, 154].

32 | Guoquan Zhang et al. Discrete gap solitons in 2D self-defocusing photonic lattices The formation of gap solitons is a fundamental phenomenon of wave propagation in nonlinear periodic media. In optics, gap solitons are traditionally considered as a temporal phenomenon in 1D periodic media such as an optical fiber with periodic refractive-index variations [155]. Recently, gap solitons have been extended to the spatial domain [156]. Different from discrete solitons whose propagation constants are located in the total internal reflection gap, the gap solitons reside in the Bragg reflection gap. Spatial gap solitons in 2D photonic lattices can arise from Bloch modes in the first band (close to high-symmetry M points (Figure 1.15 (i))), i.e. the edges of the first Brillouin zone (BZ)), where anomalous diffraction is counteracted by self-defocusing nonlinearity, or the second band (close to X points),where normal diffraction is balanced by self-focusing nonlinearity. Since gap solitons typically bifurcate from the Bloch modes of the band top or band edge, their phase structure and spectrum always characterizes the features of the corresponding Bloch modes, which provides an effective way to experimentally identify the relationship between the gap solitons and the high symmetry points of the photonic band gap structures. In what follows, we will summarize the experimental demonstration of families of spatial gap solitons bifurcated from the edge of the first Bloch band (the M point) in 2D optically induced “backbone” lattices with saturable self-defocusing nonlinearity. These include fundamental gap solitons, dipole-like gap solitons, gap soliton trains, and vortex gap solitons. First, we summarize our results of on-axis excitation of a single 2D gap soliton as shown in Figure 1.15. Different from previous experimental observations in which

Fig. 1.15. Formation of a 2D fundamental gap soliton by single-beam on-axis excitation. Experimental results (a)–(h) show lattice pattern with the waveguide excited by the probe beam marked by a circle (a) and its spectrum with the first BZ and high-symmetry points marked (b), probe beam at input (c) and its linear output spectrum (d) through the lattice, output pattern of the gap soliton (e), its interferograms with a plane wave titled from two different directions (f), (g) and its nonlinear output spectrum (h), (i) the first Bloch band marked with high symmetry points (top) and the chessboardlike phase structure of the Bloch modes of the M point. The gray and black colors are corresponding to 0 and π phase, respectively [157].

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either the probe beam was launched off-axis to match the edge of the first BZ or its input phase or spectrum was engineered, we demonstrate the on-axis excitation of a 2D gap soliton without a priori phase or spectral engineering. In fact, we show that nonlinear trapping of the probe beam leads to spectrum reshaping even though its initial spectrum is nearly uniform in the entire first BZ. The probe beam is focused into a 2D circular beam and launched into one of intensity minima (index maxima) of the “backbone” lattice. Under linear propagation, the probe beam experiences discrete diffraction, and its spectrum covers the first BZ with most of the power concentrating in the center. However, when the 2D gap soliton formed under nonlinear propagation, its power spectrum reshapes to have most of its power located in the four corners of the first BZ where diffraction is anomalous. The interferograms clearly show that the gap soliton has a staggered phase structure (i.e. the central peak is out of-phase with neighboring peaks). The nonlinear spectrum and the staggered phase structures clearly show that such fundamental gap solitons bifurcate from the edge of the first band where diffraction is anomalous [157]. High-order gap solitons such as dipole-like gap solitons in 2D self-defocusing lattices which are composed of two fundamental gap solitons with in-phase or out-ofphase relationship are also experimentally realized. And their spectrum and stability strongly depend on the initial excitation conditions. In particular, the out of phase dipole gap solitons are also bifurcated from the M points of the first band (the first row in Figure 1.16) [158]. Note that we have demonstrated gap solitons localized mainly in a single spot or two spots corresponding to the waveguide lattice sites. Is that possible to create a train of gap solitons that would populate many lattice sites? The answer can be found in our recent experimental demonstration of a gap soliton train excited by a uniform stripe beam [157]. Typical experimental results are shown in the middle row of Figure 1.16. In the linear case, the spectrum of the vertically oriented stripe beam forms nearly a horizontal line extended to only two diagonal M points of the square lattice. Under nonlinear conditions, the stripe beam evolves into a gap soliton train whose spectrum covers four M points of the first BZ, similar to the above-mentioned gap solitons. The staggered phase structure of the soliton beam is confirmed by its interferograms with a tilted plane wave, where the breaking and interleaving of interference fringes suggests the out-of-phase relation between the central stripe and two lateral stripes. Careful experimental and theoretical studies indicate that the power spectrum of the probe beam has been drastically reshaped by nonlinear trapping. This work is important not only because the gap soliton train we generated could be considered as nonlinearity-induced line defects in photonic band gap structure, but also because we have predicted and demonstrated the formation of a new type of “unexpected” gap soliton trains due to nonlinear transport and spectrum reshaping of a stripe beam in 2D induced lattices. The soliton trains arise from Bloch modes from the high-symmetry M points of the first photonic band, although some of these modes are initially not or only weakly excited from lattice scattering or diffraction.

34 | Guoquan Zhang et al. All the above-mentioned gap solitons are bifurcating from the M point of the first Bloch band, this can be clearly seen both from their phase structure and spectrum in k-space. A natural question arises: Are there any other kinds of gap solitons which do not bifurcate from the M points? When a singly charged (m = 1) donut vortex beam self-traps into a stable gap vortex soliton in 2D photonic lattices with self-defocusing nonlinearity, we found most of the spectrum power is located alongside the first BZ, but it would not concentrate to the four corner points (corresponding to four highsymmetry M points) which mark the edge of the first Bloch band [159] as shown in the bottom row of Figure 1.16. These results suggest that although the m = 1 vortex can evolve into a gap vortex soliton, while it does not bifurcate from the edge of the first Bloch band, quite differently from all previously observed fundamental, dipole gap solitons, and gap soliton trains in self-defocusing lattices. Intuitively, this might be attributed to the nontrivial helical phase structure of the vortex, which cannot be expressed as a simple superposition of linear Bloch modes of the M point. Our experimental results agree well with the theoretical results [159]. Furthermore, the nonlinear propagation dynamics of the vortex beam in self-defocusing lattices depend on the topological charge. If a doubly charged (m = 2) vortex beam was sent into the self-defocusing lattices under four-site excitation, it will evolve into a quadrupole gap soliton with no angular momentum which also bifurcates from the M point of the first band. Is it possible to realize stable doubly charged vortex gap solitons in self-defo-

Fig. 1.16. Experimental demonstration of dipole-like gap solitons (top row), gap soliton trains (middle row) and vortex gap solitons (bottom row). (a) Lattice beam and the superposed dots, stripe and circle indicate the corresponding positions of the probe beam. (b) Output intensity pattern of the gap soliton, (c) interferogram of the gap soliton with an inclined plane wave, (d) spectrum of the gap soliton. The dashed square indicates the first BZ [157–159].

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cusing lattices? We found if the input vortex beam covered eight lattices sites, under appropriate nonlinear conditions it will evolve into a stable doubly charged gap vortex soliton [160]. For more detailed results about the gap lattice solitons and vortex solitons, please refer to the review papers [161, 162].

Discrete surface solitons Discrete surface solitons form an important family of discrete solitons that exist at interface between the semi-infinite photonic lattices and homogenous media. They have attracted great interests recently due to their link with the Tamm and Shockley electric surface waves in solid state physics. However, such electric surface waves are difficult to observe due to sample fabrication. While the photonic lattice systems provide a controlled and clean platform to experimentally explore the surface wave dynamics which lead to the observation of a variety of discrete surface solitons. Onedimensional discrete surface solitons were first predicted to exist at the edge of nonlinear self-focusing waveguide lattices by Stegeman’s and Christodoulides’s groups [163],

Fig. 1.17. Experimental demonstration of two-dimensional surface solitons. (a) Microscope image of laser-written array with excited waveguide marked by a circle. (b–d) Output intensity distributions for progressively increasing input power levels. Observation of surface soliton (middle row) and surface gap soliton (bottom row) in optically induced photorefractive lattice. (e), (i) Lattice patterns with the waveguide excited by the probe beam marked by a cross. (f), (j) Surface soliton intensity patterns. (g), (k) Interference pattern between the soliton beam and a tilted plane wave. (h) 3D intensity plots of an in-phase surface soliton and (l) the corresponding pattern when its intensity is reduced significantly under the same bias condition. In all plots, dashed lines mark the interface [143, 166].

36 | Guoquan Zhang et al. and demonstrated experimentally shortly after by the same group in AlGaAs arrays with the dominant Kerr nonlinear effect [164]. In the 2D domain [165], a direct experimental observation of 2D surface solitons remained a challenge due to experimental difficulties in fabricating 2D nonlinear lattices with sharp surfaces or interfaces. In 2007, two independent experimental demonstrations of 2D surface lattice solitons were reported utilizing different materials and settings: One was accomplished by Wang et al. in optically induced lattices in a photorefractive crystal [143], while the other was done by Szameit et al. in femtosecondlaser written waveguide arrays in bulk fused silica [166]. Figure 1.17 shows typical experimental results of 2D discrete surface solitons obtained from these two independent studies. In the fs-laser written waveguide experiment, focused laser pulses were sent into bulk fused silica which created a localized permanent increase in the refractive index of the material. Consequently, when moving the sample transversely with respect to the beam, a longitudinal extended index modification (a waveguide) was written. A microscope image of the facet of such a laser-written 5 × 5 waveguide array is shown in Figure 1.17 (a). While for low input peak powers a clear spreading of the light into the array was observed, for a high input peak power almost all of the light was localized in the excited waveguide. In the photorefractive induction experiment, the lattice pattern was generated by a periodic modulation of a partially incoherent optical beam with an amplitude mask, which was then sent to an SBN crystal to induce a square lattice featuring sharp edges or corners. With an appropriate high bias field, the spreading of a probe beam was suppressed at the lattice interface to form a discrete surface soliton and a surface gap soliton, while the beam at reduced intensity displayed significant diffraction under the same lattice conditions. Other types of surface solitons have also been theoretically investigated such as incoherent, polychromatic, and spatiotemporal surface solitons, etc. [107]. Given that the geometries of the lattice surfaces and interfaces can greatly vary, discrete surface solitons were observed in a number of different settings including the interface between two dissimilar periodic media and superlattice surfaces [167]. In addition to nonlinear Tamm-like surface states, linear optical Shockley-like surface states were first introduced and observed in optically induced photonic superlattices [144], and in subsequent experiments, transitions between Shockley-like and nonlinear Tamm-like surface states were also demonstrated. Furthermore, surface states that do not belong to the same family of Tamm or Shockley states have also been demonstrated as a new type of defect-free surface states in fs-laser written curved waveguide arrays [168, 169]. More detailed results related to the interesting properties of discrete surface solitons can be found in the review paper [170].

Discrete solitons with hybrid nonlinearity In previous sections, the discrete solitons or gap solitons are supported by either only self-focusing or defocusing nonlinearities, which can be established by changing the

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polarity of the bias field in the same photorefractive crystal. Recently, a novel nonlinear system with the coexistence of both self-focusing and defocusing nonlinearity under the same experimental setting was identified by Zhang et al. in nonconventional biased photorefractive crystals, such hybrid nonlinearity leads to a controlled 1D soliton transition from different band edges or subband edges [171]. This new setting enables the reconfiguration of photonic structures and Brillouin zones (BZs) for band-gap engineering and light manipulation, including Bragg reflection controlling the interplay between normal and anomalous diffraction/refraction under the same excitation conditions [172, 173]. Novel discrete solitons can exist in such periodic systems comparing with systems with only self-focusing or defocusing nonlinearity. For example, in an optically induced 2D square lattice, the high-symmetry X-point in the first Bloch band is akin to a “saddle” point in the diffraction spectrum, where normal and anomalous diffractions co-exist along the orthogonal directions. At this X-point, a quasi-one-dimensional soliton train can be excited provided that an appropriate type of nonlinearity is used to balance beam diffraction in one particular direction, whereas in the orthogonal direction it is an extended plane wave. The propagation constant of such a 1D soliton train could reside within the first Bloch band, thus termed “in-band” or “embedded” solitons. However, to simultaneously balance normal and anomalous diffractions in different directions, one needs such orientation-dependent hybrid nonlinearity. Indeed, Hu et al. observed such “saddle” solitons by balancing the saddle-shaped bi-diffraction with hybrid focusing/defocusing in an optically induced 2D ionic-type lattice [174], typical experimental results are shown in Figure 1.18, both the phase structure and the spectrum of the soliton are in accord with the Bloch modes of the X-point. These “saddle” solitons have a propagation constant residing in the Bragg reflection gap, but they differ from all previously observed solitons supported by a

Fig. 1.18. Experimental (a–d) and numerical (e–h) results of 2D saddle solitons. The top row shows the intensity pattern (a), interferograms (b, c) with a tilted plane wave at two orthogonal directions, and Fourier spectrum (d) of the soliton. The second row shows corresponding numerical results [174].

38 | Guoquan Zhang et al. single focusing or defocusing nonlinearity. You can refer to our review paper for more detailed results of discrete solitons with hybrid nonlinearity [175].

Incoherent discrete solitons All the above-mentioned discrete solitons are coherent entities, a nature question arises whether incoherent solitons exist in discrete systems, since the incoherent solitons have been demonstrated in continuous media. If it does, and what are the new features unique to the incoherent discrete solitons? Actually, the propagation dynamics of light in photonic lattices are dominated by periodic scattering and the mutual interference effects; there should be new physical phenomena if the incoherence is introduced into the discrete systems. In 2004, Segev’s group first theoretically predicated the existence of incoherent discrete solitons in photonic lattices [176]. They found that such incoherent discrete solitons can only exist in periodic structures with noninstantaneous nonlinearity, just as the incoherent solitons in continuous media. The difference is that the intensity, spectrum, and coherence length of the incoherent discrete solitons should coincide with the lattice period. Soon after, they experimentally observed such incoherent discrete solitons (random phase solitons) in optically induced photonic lattices in photorefractive crystals under a self-focusing nonlinearity [177], such random phase solitons have multipeaks in spectrum, which locate at different parts of Brillouin zone where the diffraction is normal. Later, random phase gap solitons with their propagation constants residing in the Bragg reflection gap in optically induced photonic lattices under self-defocusing nonlinearities have also been theoretically proposed and experimentally observed [178], they also showed the multipeak spectrum structures of the random phase gap solitons in k-space, but most of the spectrum energy located in the regions with anomalous diffraction. In 2006, incoherent discrete solitons based on the white light were also theoretically proposed and studied [179]. Besides the discrete random phase solitons, other discrete nonlinear phenomena have also been extended from the coherent to incoherent regime such as discrete incoherent modulation instability [180]. In particular, one advanced technique based on the incoherent light-Brillion zone spectroscopy [181], has became a powerful tool to study the linear and nonlinear light propagation dynamics in periodic structures in momentum space. We have provided a brief overview of spatial beam dynamics in both bulk media and optically induced photonic lattices, with emphasis on the novel incoherent solitons, coherent discrete solitons, and incoherent discrete solitons. The study of incoherent solitons and discrete solitons expands our research of spatial beam dynamics from coherent systems to incoherent systems, and from continuous to discrete systems. All these results will not only broaden our understanding of the fundamental phenomena in nonlinear optics and photonics, but they will also bring about many possibilities for potential applications in signal routing and processing. Furthermore, the conclusions drawn from the studies of spatial optical solitons might have signifi-

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cant impacts on similar phenomena in other nonlinear and periodic systems beyond optics.

1.3 Nonlinear optical properties of novel carbon-based materials In the past few decades, people were paying unremitting efforts to look for and prepare NLO materials with fast response time and large NLO coefficient. In many materials, it was discovered that carbon structural materials have a rich structure, in addition to the three-dimensional diamond and graphite, as well as zero-dimensional fullerenes [182], and one-dimensional carbon nanotubes (CNTs) [183–185]. The discovery of two-dimensional graphene in 2004 [186], further improved the dimensional structure of the carbon structure material system from zero-dimensional to threedimensional complete structure. These materials, zero-dimensional fullerene [187] and one-dimensional CNT [188] have a large optical nonlinearity, while the twodimensional graphene material also proved to be a novel NLO materials [189, 190]. For the study of the mechanism of optical nonlinearity of graphene materials, as well as through appropriate physical and chemical modification to improve carbon structural material of the different dimensions of optical nonlinearity is particularly important. The studies of the NLO material and its modification are usually focused on a single structure, such as porphyrin, phthalocyanine, etc. [191, 192]. It can be by changing their molecular structure to achieve the improved performance of the optical nonlinearity. However, for a practical application, in addition to having a large optical nonlinearity, the NLO materials should also have good optical quality, stability, easy preparation and low-cost characteristics. It is often difficult for the unitary structure to further improve the nonlinear performance and meet these requirements. Then people began to blend two nonlinear material. Nonlinear performance improvement can be obtained by a nonlinear mechanism complementary [193–195]. Then physical blending is upgraded to the chemical covalent link. The hybrid materials constituted by the covalent bond can effectively integrate the optical nonlinear performance of the different materials. The strong interaction makes hybrid materials not only contain some properties of each component, but also exhibit some novel properties that respective components do not have. For example, electron transfer and energy transfer in hybrid materials improve the optical nonlinearity [196, 197]. Therefore, hybrid materials for NLO materials modification and improvement of the optical nonlinearity provides a better idea.

40 | Guoquan Zhang et al. 1.3.1 Carbon-based materials Carbon is one of the most common elements in the natural world, the carbon atom has six extranuclear, filled with two electrons in the 1s orbital, and the remaining four electrons can be filled in sp3 , sp2 , or sp hybrid orbitals [198], it forms the bonding structures such as diamond, graphite, carbon nanotube, or fullerene. In diamond, each carbon atom of the four valence electrons occupy the sp3 hybrid orbital, forming a four-equivalent σ covalent bond, no delocalized π bond, so the diamond is electrical insulator; in graphite, each carbon atoms has three outer electrons occupy the planar sp2 hybrid orbital to form a three-plane σ bonds, and the remaining one-ofplane π orbital (π bond). Such a bond causes the formation of a plane hexagonal grid structure. van der Waals forces of these hexagonal mesh sheet layer in parallel to each other together, the surface from 0.34 nm; fullerene (C60 ) from 20 six-membered rings and 12 five-membered rings form [182]. The bond of carbon atoms also belong to the sp2 , although it is characterized with the same sp3 highly curved; nanotubes can be viewed as a hollow cylinder formed by the graphite sheet [198], and its bond sp2 ; however, such cylindrical bending will result in quantum confinement and σ-π then heteroaryl, wherein three σ slightly offset from the plane, and the delocalized π orbital more biased toward the outside of the tube. Further, the hexagonal mesh structure also allows the presence of a defect of the five-membered ring and seven-membered ring topology, and to form a closed, curved, annular, and spiral nanotubes. Due to the redistribution of π electrons, the electrons will be localized in the five-membered ring and seven-membered rings. As a traditional elemental carbon, diamond and graphite are three dimensional. In 1991, Iijima et al. found 1D carbon nanotubes [183]. According to the composition of the graphite sheet layers of carbon nanotubes, carbon nanotubes can be classified into single-walled carbon nanotubes (SWNTs) and multiwalled carbon nanotubes (MWNTs). With different chirality, SWNTs can behave as a semiconductor or a conductor. Since carbon nanotubes are mainly of sp2 -hybridized carbon atoms, sp2 hybrid orbital forms a number of highly nonlocalized π electrons. This structure determines the carbon nanotubes having excellent electrical properties, and electron transports along nanotube sidewalls by π bond conjugated with high speed. On the other hand, there was a strong intermolecular interactions of carbon nanotubes, so that it is easy to aggregate to form the carbon nanotube bundles. Carbon nanotube sidewalls are smooth and highly polarized, so the carbon nanotube aggregates due to the strong van der Waals force that is about 0.5 eV per nanometer carbon nanotube [199]. Carbon nanotubes are considered to have a huge molecular weight of the rigid structure and the resulting carbon nanotube bundles are insoluble in water and common organic solvents, and difficult to disperse. However, research and applications require it could be dissolved in a solvent or dispersed, so as to have stronger workability. The appropriate modification of the carbon nanotubes can improve the dispersibility in an

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organic solvent, ability to make it easy for characterization and further applications. At present, the modification of carbon nanotubes includes covalent and noncovalent modification, specifically in the functional defects, covalent sidewall functionalization, noncovalent modification of the tube [200]. In 2004, two scientists have discovered a single atomic layer of the new twodimensional atomic crystals-graphene. The studies of graphene no longer remain in the theoretical stage [186]. The basic structural unit of graphene is the most stable organic material benzene six-membered ring. It is the ideal two-dimensional nanomaterials, and is the world’s thinnest two-dimensional material. Its thickness is only 0.35 nm. Graphene’s strength is the highest among known materials, up to 130 GPa, 100 times that of steel; their carrier mobility rate of 15 000 cm2 V−1 s−1 is twice as much as than that of InSb material, and 10 times than commercial silicon. Under certain conditions, such as low temperature quenching, etc., the mobility can reach 25 0000 cm2 V−1 s−1 , three times that of the diamond; graphene also has a room temperature quantum Hall effect and ferromagnetism and other special properties [201]. Graphene manifested unique electronic and physical properties, and has important applications in molecular electronics, micro-nano-devices, composites, field emission materials, sensors, batteries, and hydrogen storage materials, and other fields. However, the structural integrity of the graphene is a benzene contain any unstable bonds combination of six-membered ring, has high chemical stability, the surface with inert state, and weak interactions with other media (such as solvents, etc.). Between the graphene sheets, there is a strong van der Waals forces, so it is difficult to dissolve in water and common organic solvents, which will hinder graphene further research and application. Graphene oxide contains a lot of functional groups, such as hydroxyl, carboxyl, carbonyl group, epoxy group, etc., wherein the hydroxyl and epoxy groups are mainly located in the base surface, and carbonyl and carboxyl groups in the edge of graphene [202, 203]. Since the introduction of these groups, one part of graphene carbon atom is sp2 -hybridized, and the other part is sp3 -hybridized, thus losing the electronic conductivity. However, these groups give the graphene oxide number of new features, such as dispersibility, hydrophilicity, compatibility with the polymer, which is functionalized graphene provided for convenience. In fact, one can think that graphene is the basic material for other carbon allotropes as shown in Figure 1.19 [204]. The graphene can be wraped to becomes zerodimensional sphere C60 ; graphene surface can be a straight axis, to the crimp 360° and a seamless hollow tube becomes into a one-dimensional carbon nanotube; addition, if the graphene placed parallel, stacked together to form a three-dimensional graphite. So the discovery of graphene allows us to understand from zero to threedimensional structure of carbon materials.

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Fig. 1.19. Graphene, fullerenes, carbon nanotubes, and graphite structural relationship [204].

1.3.2 Progress of optical nonlinearities of carbon-based materials Since 1985, Prof. Kroto from the University of Sussex and Prof. Smalley from Slice University found C60 , there have been a lot of researches on carbon-based materials and have opened a new curtain – from the macro to the micro world into the nanoworld. This time, it is the stage of the rapid development of nonlinear optics. Looking for good NLO materials is imperative. The emergence of carbon nanostructured materials brings new choices for NLO material research. Since the early 1990s, Tutt et al. [187] first reported the optical limiting effect of fullerenes, scientists have begun to study the NLO effects of carbon nanomaterials. Subsequently, Sun [188] and Vivien [205] published their first NLO research of multiwalled carbon nanotubes and single-walled carbon nanotube suspensions in 1998 and in 1999, respectively. Unlike C60 , nonlinear mechanism of carbon nanotubes exhibits strongly nonlinear scattering, which comes from the laser scattering center formation, including two processes: when the energy is low, the heat transfer from the solute to the solvent forms the solvent microbubbles; when the energy is high, the vaporization of carbon nanotubes, and microplasma ionized is formed. Thus there is the formation of an effective scattering centers. Nonlinear scattering properties of carbon nanotubes can also be used to prepare the optical limiting devices, which limit effect is more obvious than the C60 . There are some advantages such as a good linear transmittance and wide-band availability. But there have also been some problems. The earliest formation of scattering centers is a few nanoseconds, it is not shorter ef-

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fective limits than the pulse width of the laser pulses. Meanwhile, due to the strong van der Waals forces between the carbon nanotubes and gathered into bundles, poor dispersion, insoluble in common solvents, and because inert surface, the interface of the base material is poor and difficult to make device. After studying the various mechanisms based on various types of optical limiting materials, it was found difficult to meet practical application using a single-structure material that exhibits a variety of problems. The emergence of these problems did not hinder the pace of explorers, people put research extended to nonlinear mechanism based on a variety of composite materials in the optical limiter, and expected to get stronger nonlinear, other better properties. Currently, most studies of optical limiting material focus on reverse saturable absorption (RSA) and two photon absorption (TPA) material. For the nonlinear enhancement process, scientists think that the nonlinear absorption material mixed with the nanocarbon material can enhance the its nonlinearity. Dispersion of carbon nanotubes in the solution is poor; however, it can be modified using a variety of methods, such as hydroxy-modified carbon nanotubes (MWNTs-OH). Without affecting the basis of the nature of the carbon nanotubes, modification will improve the surface properties of the carbon nanotube, and greatly improve its solubility. In the study on optical nonlinearity of the hybrid material, the researchers also made a lot of achievements. Izard et al. [206] studied the combination of SWNT and organic two-photon absorber with broadband optical limiting. Webster et al. [194] reported that the MWNT with one RSA phthalocyanine dyes HITCI hybrid, NLO properties of the material obtained in a wide band have been optimized: with the lower energy, the RSA dye plays a major role, inducing a lower threshold; when energy is greater than the limiting threshold of carbon nanotubes, its strong scattering makes hybrid materials withstand high incident energy without the risk of saturation. In these studies, the soluble carbon nanotubes and the optical limiting effect were increased simultaneously. Since the emergence of graphene, its optical nonlinearity has also been extensively studied. Wang et al. [190] observed the NLO response of graphene in organic solvents at 532 and 1064 nm, indicating that graphene is a relatively good broadband optical limiting materials. Feng et al. [207] studied the optical limiting effect of the different forms of graphene family (including graphene/graphene oxide nanosheets, graphene oxide/graphene nanoribbons). In 2009, we first discovered that graphene oxide in dimethyl formamide (DMF) suspension presented optical nonlinearity [189] in both nanosecond and picosecond time domains. For lower energy nanosecond and picosecond pulses graphene oxide suspension in DMF showed a saturable absorption. For higher energy pulses, graphene oxide has good TPA and excited state absorption in the nanosecond time domain, while having TPA only under picosecond. Thus graphene oxide should have a good optical limiting effect because of its TPA and RSA performances. However, we found that the graphene oxide has not stronger optical limiting effect than C60 both in the nanosecond and picosecond pulses. As a new NLO materials, optical nonlinearity and optical limiting of graphene oxide needed to be improved

44 | Guoquan Zhang et al. urgently. Thus we begin preparation and study graphene hybrid materials, including fullerenes–graphene hybrid materials [208, 209], porphyrin–graphene hybrid materials [210], polythiophene–graphene hybrid materials [211, 212], iron oxide–graphene hybrid materials [213, 214], and so on. In nonlinear optics research of graphene hybrid material, the enhancement of optical nonlinearities obtained by light-induced electron transfer between acceptor and donor becomes a research hotspot [215–218].

1.3.3 Optical nonlinearities of carbon nanotube and its hybrid materials Because carbon nanotubes have a high surface energy, it is prone to aggregate in a suspension. This limits the practical application of carbon nanotubes. In order to improve the dispersibility of carbon nanotubes in a solvent, people developed a variety of methods for the modification of carbon nanotubes, including covalent and noncovalent modifications. Among the many covalent modifications of the carbon nanotubes, the preparation method of MWNTs modified hydroxyl groups is simple, and it has a good dispersion in some solvents. Hydroxyl group on carbon nanotubes and other small modification can improve the surface of carbon nanotubes to increase its dispersion, while some properties of carbon nanotube themselves have little effect. On the other hand, porphyrin covalently modified carbon nanotube hybrid materials (CNT-porphyrin) not only it can improve the dispersibility of carbon nanotubes, but also it is expected to enhance optical nonlinearities because of electronic transfer or energy transfer between carbon nanotubes and porphyrin, and the complementarity of nonlinear absorption and nonlinear scattering mechanism [219].

Optical nonlinearities of hydroxyl groups modified MWCNTs The hydroxyl groups modified MWNTs (MWNTs-OH) with diameters of < 8 nm (MWNTs8 -OH), 10–20 nm (MWNTs10–20 -OH), 20–30 nm (MWNTs20–30 -OH), 30–50 nm (MWNTs20–30 -OH), > 50 nm (MWNTs50 -OH), and MWNTs with the diameter of 20–30 nm (MWNTs20–30 ) were dispersed in water, N,N-dimethylformamide (DMF) and chloroform, with the same initial concentration of 0.2 mg/mL. Generally, the NLO properties of CNT suspensions are mainly attributed to the solvent and/or carbon vapors bubble-induced nonlinear scattering, while a nonlinear absorption mechanism was proposed for soluble CNTs. The strong dependence of solvent and pulse width for the three kinds of dispersions indicates nonlinear scattering is the dominant mechanism, which is induced by the solvent and/or carbon vapors bubble as mentioned above. For a longer pulse, the solvent and/or carbon vapors bubble have longer time to grow to larger size and interact with the pulse, and then scatter the light more strongly. The superior NLO properties of chloroform dispersion may be due to the lowest boiling point in combination with the lowest surface tension and

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thermal conductivity of chloroform. Furthermore, the largest ratio of bundle due to the poor dispersion may also have contributions to nonlinear scattering. Nonlinear scattering due to the nonlinear thermal process is a complex, many of thermal process parameters, such as the boiling point of the solvent, the surface tension, the concentration of the suspension, the pulse laser energy, the structure of carbon nanotubes may affect the nonlinear scattering. Therefore, we next examined a number of experimental parameters on the suspension of nonlinear effects, in order to find MWNTs-OH suspensions exhibits strong NLO properties under optimal conditions. Figure 1.20 (a) shows the Open-aperture Z-scan curves of the three dispersions with 5-ns-pulse. All of dispersions exhibit decreased transmittances as they are brought closer to focus. Among the three dispersions, the normalized transmittance of chloroform dispersion shows the largest reduction at focus, indicating that it has the best NLO performance. DMF dispersion shows larger optical NLO properties than water dispersion. The different NLO properties of these dispersions imply the significant solvent effect. As shown in Figure 1.20 (b) chloroform dispersion shows the enhanced NLO properties when the pulse is prolonged to 11.7 ns with the same pulse energy as 5-ns-pulse. The water and DMF dispersions also show the similar enhancement behaviors as the chloroform dispersion, indicating the pulse-width dependence of NLO properties for the dispersions. The effective nonlinear absorption model and effective nonlinear absorption coefficients β eff are successful to evaluate or compare the NLO properties of the dispersions quantitatively [221]. The theoretical fitting curves (solid lines) deviate from the experiments data in some figures, which can be attributed to the difference between the effective nonlinear absorption model and the nonlinear scattering process of the dispersions.

Fig. 1.20. The Open-aperture Z-scan curves of MWNTs20−30 -OH in water, DMF and chloroform with the pulse width of 5 ns (a) and in chloroform with the pulse width of 5 ns and 11.7 ns (b) [220].

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Fig. 1.21. (a) The Open-aperture Z-scan curves of MWNTs-OH in DMF with the pulse width of 5 ns. (b) Normalized transmittance and the effective nonlinear absorption coefficient as functions of onaxis fluence at focus [220].

Although the chloroform dispersion shows the largest NLO properties among the three dispersions, but the poor quality of the dispersion may limit the practical application as an optical limiter. Comparatively, MWNTs20−30 -OH in DMF shows both better NLO performances and more stable form of dispersion. Hence, it is very necessary to study the NLO performances of DMF dispersion in detail. Figure 1.21 (a) and 1.21 (b) shows the Open-aperture Z-scan curves, the normalized transmittance at focus (NTmin ) and β eff of MWNTs20−30 -OH in DMF for different F0 , where F0 is on-axis fluence at focus. As shown in Figure 1.21 (a) and (b), all the Open-aperture Z-scan curves show that the normalized transmittance decreases as the sample is moved near to the focus. NTmin decreases from 0.89 to 0.47 and β eff increases from 5 to 14 cm/GW when F0 increases from 1.179 to 4.323 J/cm2 . The increase of β eff with F0 indicates that the stronger nonlinear scattering occurs at high input fluence. So, MWNTs20−30 OH in DMF can serve as an effective optical limiter for high input fluence. Results show that the NLO properties of all the dispersions exhibit pulse-width and solvent dependence. MWNTs20−30 -OH in DMF shows good dispersion and large NLO properties. MWNTs20−30 -OH in DMF exhibits enhanced NLO properties for the higher concentration or higher on-axis fluence at focus, but the concentration which exceeds to a certain level can lead to the saturation of optical limiting performance. The diameter has no significant influences on the NLO properties of MWNTs-OH. MWNTs-OH with different diameters in DMF dispersion shows stronger NLO properties than C60 at the higher fluence due to nonlinear scattering. Therefore, MWNTs-OH may be good candidates for broadband optical limiters.

Optical nonlinearities of MWCNT–porphyrin hybrid materials Progress in NLO materials in the past decade was stimulated by the developments of organic materials that offer the advantage of flexible chemical synthesis. An important

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Fig. 1.22. The syntheses of three MWCNT–TPPs.

aspect in the developments of organic NLO materials is the attempt to simultaneously combine the primary NLO properties of two kinds of materials by noncovalent or covalent interaction. In these NLO materials, functionalization of carbon nanotubes (CNT) through surface modification has attracted significant interest recently. Covalent and noncovalent functionalization strategies involving reactions of organic or polymeric molecules onto CNTs have primarily focused on optical limiting and NLO properties. Unlike CNT, porphyrins generally show strong excited state absorption (ESA), high triplet yields and long lived excited states together with the wide transmission window between the main ground state absorption bands (Q and B), these make them considered as good NLO materials [223]. On the other hand, excited-state lifetimes of porphyrins also affect the ability of electron transfer in porphyrin donor–acceptor system when pulse laser is used. Improved and pulse-width dependent NLO behavior is therefore to be expected from porphyrin–carbon nanotube composite solutions. Here we present three MWNTs covalently functionalized with tetraphenylporphyrin (MWCNT– TPP). Three kinds of MWCNTs with diameter of < 10 (MWCNTs10 ), 10–30 (MWCNTs30 ) and 40–60 nm (MWCNTs60 ) were used to produce three MWCNT–TPP nanohybrids: I, II, and III, respectively. The syntheses of MWCNT–TPPs (Figure 1.22) were carried out using amine functionalized porphyrin (TPP-NH2 ) and MWCNTs in DMF following standard chemistry. The NLO properties of these materials were measured by the Z-scan technique with a linearly polarized laser at 532 nm generated from a frequency doubled Q-switched Nd:YAG laser, with a pulse repetition rate of 10 Hz. The laser pulse width was tuned at 5 ns and 11.7 ns, respectively, for our experiments. The spatial profile of the pulsed beam was of nearly Gaussian distribution after spatial filtering. The pulsed beam was split into two parts: the reflected part was used as reference, and the transmitted part was focused onto samples by using a 25-cm focal length lens and the beam waist radius was about 25 μm at focus. All the dispersions were poured into 5-mm quartz cuvettes and adjusted to the linear transmittance of 75% at 532 nm. Figures 1.23 and 1.24 give Open-aperture Z-scan results of MWCNT–TPPs I (a), II (b), III (c), pristine MWCNTs30 , TPP, and a controlled blend sample (MWCNT+TPP) of MWCNTs30 with TPP (1 : 1 weight ratio) at 532 nm with 5.6 and 11.7 ns pulses. All Z-scans performed in this study exhibited a reduction in transmittance near the focus. It can be clearly noted that the NLO properties of MWCNT–TPPs are increased when

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Fig. 1.23. The Open-aperture Z-scan curves of MWCNT–TPPs I (a), II (b), and III (c) at 532 nm with 5.6 and 11.7 ns pulses [222].

Fig. 1.24. The Open-aperture Z-scan curves of MWCNTs30 (a), TPP (b), and a controlled blend sample (MWCNT+TPP) of MWCNTs30 with TPP (c) at 532 nm with 5.6 and 11.7 ns pulses [222].

the laser pulses width increased from 5.6 to 11.7 ns. With the increase of laser pulse width, a little increase of normalized transmittance valley at focus can be obtained in the sample of MWCNTs30 , while there is a decrease of normalized transmittance valley in the sample of TPP. Comparing the Z-scan curves in Figures 1.23 and 1.24, we can see that as the pulse width increases from 5.6 to 11.7 ns, the changes of normalized transmittance of porphyrin covalently functionalized MWCNTs are obviously larger than those of the controlled blend sample as well as the individual TPP and MWCNT. Different mechanisms exist for optical nonlinearities, such as nonlinear absorption (multiphoton absorption, ESA), nonlinear refraction (electronic or thermal effects), and nonlinear light scattering [211]. Three mechanisms have been suggested to contribute to the NLO properties of CNT nanohybrids functionalized with ESA chromophore:

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RSA from chromophore, nonlinear scattering from CNT, PET from chromophore to CNT. Hence, the greatly enhanced NLO properties of MWCNT–TPPs may arise from these three mechanisms above. To compare quantitatively, we theoretically fitted the experimental results to obtain the effective nonlinear absorption coefficients β eff [219]. Firstly, we can see that three MWCNT–TPP hybrids have larger effective nonlinear absorption coefficients β eff than the individual MWCNT and TPP, and their blended sample for both 5.6 and 11.7 ns pulses. Considering the covalent donor–acceptor structure and the efficient fluorescence quenching of these nanohybrids, we conclude that besides the combination of RSA from TPP and nonlinear scattering from MWCNT, the PET from electron donor TPP to acceptor MWCNT should also play an important role for greatly enhanced NLO performance. Secondly, with the increase of diameter of MWCNT moiety, the values of β eff gradually decrease for MWCNT–TPPs I, II, and III. Since the same linear transmittance was set for all of samples, this result implies that the MWCNT–TPP with the shorter diameter has a better optical limiting performance, which is opposite to the result of pristine CNT. The reason may be that the NLO properties of MWCNT–TPPs originate from a combination of RSA, nonlinear scattering and PET effect. As we have discussed above, the larger the diameter of MWCNT is, the lower the content of porphyrin in these nanohybrids will be. The decrease of porphyrin content causes the weakening of covalently functionalized reaction, which will reduce the increases of NLO properties. Finally, as the laser pulse width increased from 5.6 to 11.7 ns, the values of β eff of three MWCNT–TPP hybrids increased more than that of MWCNT and TPP. This indicates that the NLO properties of MWCNT–TPP hybrids strongly depend on pulse width than their parents. As the pulse width increases, the efficiency of PET can be improved, and so lead to significant increase of the NLO properties. Based on this viewpoint, it can be expected that large increase of the NLO properties can be found in the blended system of MWCNT and TPP, in which a weak PET effect exists [221]. To get a better insight into the change of the NLO properties of MWCNT–TPPs with pulse width, Z-scan experiments of MWCNT–TPP (II) carried out with the pulse width of 5.6, 5.9, 7.0, 9.2, and 11.7 ns. Figure 1.25 gives the transmittance valleys (Tmin ) of Zscan curves with different pulse width. There is a fast decline of Tmin from 5.6 to 7.0 ns, while a slowly changing can be observed from 9.2 to 11.7 ns. Since the PET takes place between singlet excited porphyrin and MWCNT, the singlet-excited state lifetime τ s1 of TPP, which is 6.6 ns, will play an important role in the PET of MWCNT–TPPs [199]. Hence, when the pulse width is less than or comparable to τ s1 , the efficiency of PET during laser pulse irradiation will be quickly changed with the pulse width, causing a fast decline of Tmin . On the contrary, when pulse width is much greater than τ s1 , the dependence of PET efficiency on pulse width is weak because the PET has reached saturation during laser pulse irradiation, and so the change of NLO properties is also small.

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Fig. 1.25. Transmittance valleys (T min ) of Z-scan curves of MWCNT–TPP (II) with different pulse width of 5.6, 5.9, 7.0, 9.2, and 11.7 ns [222].

Optical nonlinearities of SWCNT–porphyrin hybrid materials The results can be seen from the above, using a special role to a functional group of nanotube structure of a certain chemical modification is necessary. Meanwhile, the functional group modification of SWNTs can also increase the stability of the carbon nanotubes in the solution, and even to make it soluble, which is very helpful for the quantitative studies on their electronic and optical properties. We synthesized three SWNTs covalently functionalized with porphyrin. SWNTs were obtained by the arc discharge method. The diameter of carbon nanotubes is approximately 1.4–1.7 nm. Figure 1.26 shows molecular structures of three SWNT-porphyrins. Since the experimental structure of optical limiting is similar to the Z-scan experiments, we the measured and analyzed optical nonlinearities of three compounds I, II, III using Z-scan methods. Porphyrins and SWNTs have good NLO properties, the two covalently linked together, nonlinear performance can be better than a single material. RSA of porphyrin and nonlinear scattering of SWNTs in the Z-scan showed the

Fig. 1.26. Structures of porphyrin-covalently functionalized SWNTs I, II, and III [219].

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Fig. 1.27. Z-scan curves of SWNTs, SWNT+TPP, I, II, and III.

absorption valley, i.e. the focal point at z = 0 with the smallest normalized transmittance. SWNT-porphyrin compounds I, II, and III should have RSA and nonlinear scattering, and the combination of these two mechanisms can further improve the nonlinear performance. We carried out Open-aperture Z-scan experiments of SWNTs, SWNT+TPP, I, II, and III with the same concentration of 20 mg L−1 , and results are shown in Figure 1.27. The linear transmittances of SWNTs, I, II, and III are 64%, 70%, 68%, and 75%, respectively. The composite I has a largest dip among the transmittance curves of these materials, indicating that it should have the best optical limiting effect. Although the combination of nonlinear scattering with RSA can improve the optical limiting effect effectively, we believe that the enhanced limiting performance arises from not only the combination of nonlinear mechanism, but also from the photoinduced electron or energy transfer from electron donor TPP moiety to acceptor SWNTs. Figure 1.28 gives the Open-aperture Z-scan curves of I and II for different input fluence. In general, the value of nonlinear absorption coefficient will decrease as input fluence increases for the RSA process because of the saturation of RSA, and however it will keep unchanged for the two-photon absorption (TPA) process. The increase of nonlinear absorption coefficient with input fluence implies that besides nonlinear absorption the observed optical limiting performance is also influenced by nonlinear scattering in the high-fluence regime. The similar phenomena were also observed in the composites III. In SWNTs and porphyrin noncovalent blended system, the nonlinear absorption coefficient will decrease as intensity increases. This may be due to the saturation of the accessible energy levels under high-intensity pumping.

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Fig. 1.28. Z-scan curves of I and II for different input fluence.

For RSA when the incident energy reaches a certain level, there will be RSA saturation phenomenon, namely saturated absorption changes. It is difficult to obtain a large nonlinear effects only depending on nonlinear scattering. If the effective combination of both nonlinear mechanisms can be overcome the restrictions of a single nonlinear process can improve the nonlinear performance. We have previously observed that if the two components have different nonlinear mechanisms (porphyrins and SWNTs), the supermolecular system with covalent coupling can effectively improve the nonlinear properties. Also, another linear combination of different mechanisms is to simply mix the two components together. We performed the nonlinear transmittance measurements to compare optical limiting effects of porphyrin-functionalized SWNTs. The characteristics of output fluence vs input fluence for these samples are shown in Figure 1.29. C60 solution in toluene was employed as a reference. For comparison, all of the samples were set to have the same linear transmittance of 75% at 532 nm by adjusting their concentration. From Figure 1.29, it can be seen that the optical limiting effects of three porphyrins functionalized SWNTs (I, II, and III) are much better than not only C60 and SWNTs, but also than individual porphyrins (TPP and Sn(OH)2 DPP). The optical limiting thresholds of I, II, and III are approximately 70, 100, and 150 mJ/cm2 , respectively, all of which are much smaller than those of C60 (300 mJ/cm2 ) and SWNTs (250 mJ/cm2 ). At the highest fluence (95 mJ/cm2 ) used in our experiments, the transmittance has decreased to 4.3%, 5.1%, and 6.6% for I, II, and III, respectively, while the transmittance is 11.3%, 9.9%, 21%, and 33.5% for C60 , SWNTs, TPP, and Sn(OH)2 DPP, respectively. This illustrates that enhanced optical limiting effects can be obtained by functionalizing SWNTs with RSA chromophore porphyrins. Functionalizing SWNTs with RSA chromophores porphyrins can enhance optical limiting performance. Through combination of nonlinear mechanism and the photoinduced electron or energy transfer between porphyrin moiety and SWNT, the por-

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Fig. 1.29. The optical limiting of C60 , SWNTs, SWNT+TPP, TPP, Sn(OH)2 DPP, I, II, and III for 5-ns pulsed laser at 532 nm [219].

phyrin covalently functionalized SWNTs offer superior performance to C60 , the individual SWNTs and porphyrins.

1.3.4 Optical nonlinearities of graphene and its hybrid materials Graphene as a new material discovered only a few years back, has a single atom thick, and contain sp2 -hybridized carbon with two-dimensional structure. Pure graphene has a full composition of C=C double bond in the crystal structure, a good thermal and electrical properties. Since the first discovery in 2004, the nature of research on graphene has been everywhere, including reports on their NLO properties. However, between the graphene layers there is a strong attraction, which makes it difficult to uniformly disperse in water and common organic solvent solution. This greatly restrict the application of graphene. To overcome the shortcomings of graphene to find a better performance optical limiting materials, we have conducted in-depth research on optical nonlinearity of a number of graphene-based nanostructured hybrid materials. In order to improve the optical nonlinearity of graphene, we use an organic nonlinear material porphyrin, fullerenes, and inorganic oligothiophene Fe3 O4 nanoparticles modified by covalent bonds of graphene oxide (GO) forming the hybrid material. Through the combination of materials, nonlinear mechanism complementary, and even electronic or energy transfer, nonlinear scattering produced to achieve the nonlinearity and optical limiting performance improvement of GO materials [208, 209].

54 | Guoquan Zhang et al. Optical nonlinearities of GO–porphyrin and GO–C60 hybrid materials The synthesis of the porphyrin–graphene nanohybrid, GO–TPP (Figure 1.30), was carried out using an amine functionalized prophyrin (TPP–NH2 ) and graphene oxide in DMF following standard chemistry. For fullerene-graphene nanohybrid, GO–C60 (Figure 1.30), pyrrolidine fullerene (C60 (OH)x ) and graphene oxide in DMF were used to the synthesis via a mild coupling reaction between the −OH group of pyrrolidine fullerene and the −COOH group of GO. The NLO properties of these materials were measured by Z-scan technique in the regime of nanosecond and picosecond. Figure 1.31 (a) shows Open-aperture Z-scan results of GO–TPP, TPP–NH2 , graphene oxide, and a controlled blend sample of TPP– NH2 with graphene oxide (1 : 1 weight ratio) at 532 nm with 5-ns pulses. The Open aperture Z-scan measures the transmittance of sample as it translates through the focal plane of a tightly focused beam. As the sample is brought closer to focus, the beam intensity and nonlinear effect increases, which leads to a decreasing transmittance for RSA, TPA, and nonlinear scattering. As shown in Figure 1.31 (a), the GO–TPP had the largest dip among the transmittance curves of the studied materials: GO–TPP, TPP– NH2 , graphene oxide, the controlled sample. At focal point where the input fluence is maximum, the transmittances of GO–TPP, TPP–NH2 , graphene oxide, and the controlled sample drop down to 44.8%, 75.6%, 93.8%, and 83.5%, respectively. Therefore, GO–TPP demonstrated much better NLO properties compared with the controlled sample and the individual components (TPP–NH2 and graphene oxide) of the hybrid. Under the same experimental conditions, we also carried out the nanosecond Open-aperture Z-scan experiments to study the NLO performance of GO, GO–C60 , GO/pyrrolidine fullerene blend and pyrrolidine fullerene, as shown in Figure 1.31 (b).

Fig. 1.30. Synthesis schema of GO–TPP and GO–C60 [208].

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Fig. 1.31. (a) Open-aperture Z-scan curves of GO–TPP, TPP–NH2 , graphene oxide and a controlled blend sample of TPP–NH2 with graphene oxide (1 : 1 weight ratio). (b) Open-aperture Z-scan curves of GO–C60 , pyrrolidine fullerene, GO and a controlled blend sample of pyrrolidine fullerene with graphene oxide (1 : 1 weight ratio) at 532 nm with 5-ns pulses [208].

At focal point, the transmittances of GO–C60 , pyrrolidine fullerene, graphene oxide, and GO/pyrrolidine fullerene blend drop down to 35.4%, 71.3%, 90.1%, and 79.8%, respectively. These results demonstrate that while excellent NLO properties of all the samples were observed, the largest dip among the transmittance curves indicates that the GO–C60 hybrid is the best one. In the meantime, compared with Z-scan results of GO–TPP shown in Figure 1.31 (a), the larger dip of Z-scan curves of GO–C60 indicates that it exhibits better NLO response than GO–TPP. From Figure 1.31, we can obviously see that both GO–TPP and GO–C60 have larger NLO properties than the blended samples and their two parents. Therefore, the graphene hybrid materials covalently functionalized with porphyrin and fullerene are better candidates for applications in optical limiting than the individual graphene, porphyrin, or fullerene. In the experiment, when the samples of GO–TPP and GO–C60 close to the focus, we observed a strong nonlinear scattering signal. Since the scattering spot/ transmission light intensity of the central region of the spot is high, and the weak spot of the edge of the light intensity, in order to make better scattering spot imaged, in close-lens place a circular central shielding plate. The shooting incident and scattered light spot shown in Figure 1.32. When the input fluence is low (< 0.1 J/cm2 ), no obvious scattering signal was observed as shown in Figure 1.32 (b). When a large input fluence (7.96 J/cm2 ) was used, strong nonlinear scattering signals were observed for GO–C60 (Figure 1.32 (c)) and GO–TPP (Figure 1.32 (d)), which may be assigned to the incandescence and submission of graphitic particles leading to the fast growth of hot carbon vapor bubble, similar to the mechanism of CNTs. Different mechanisms exist for NLO, such as nonlinear absorption (multiphoton absorption, RSA), nonlinear refraction (electronic or thermal effects), and nonlinear light scattering. In addition, CNTs have also been reported to have strong optical lim-

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Fig. 1.32. Images of input laser beam (a) and scattering light (b, c, d) when the samples GO–C60 and GO–TPP were fixed on the focus [208].

iting effects, which arise from strong nonlinear light scatterings due to the creation of new scattering centers consisting of ionized carbon microplasmas and solvent microbubbles. Due to graphene and carbon nanotubes are the main component of carbon, and the micrometer scale are satisfied, so the GO–TPP and GO–C60 suspensions may be similar mechanisms as suspended carbon nanotube liquid. Feng et al. and Balapanuru et al. have observed strong nonlinear scattering in graphene hybrid materials [215, 224]. Figure 1.33 gives the Open-aperture Z-scan curves of GO–C60 (a) and GO–TPP (b) with different input fluence. Using the Crank–Nicolson finite-difference scheme, we fitted the Z-scan curves numerically (the solid lines) and obtained the values of the effective nonlinear absorption coefficient, β eff , as shown in Figure 1.33 (b). In general, the value of β eff will decrease as input fluence increases for RSA process because of the saturation of RSA, and but it will keep unchanged for TPA process. However, the increase of β eff with input fluence implies that besides nonlinear absorption from GO and porphyrin (or fullerene), the observed NLO performance is also influenced by nonlinear scattering in the high-fluence regime for two graphene hybrid materials. The similar phenomena have been observed in the porphyrin covalently functionalized SWNTs.

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Fig. 1.33. Open-aperture Z-scan curves of GO–C60 (a) and GO–TPP (b) with different input fluence at 532 nm in the nanosecond regime [208].

From the above discussion, we can see that besides nonlinear absorption the graphene hybrids covalently functionalized with porphyrin and fullerene have strong nonlinear scattering performance in the nanosecond regime, which is different from the individual components of GO, porphyrin and fullerene. Furthermore, another possible reason for this enhanced NLO performance may be attributed to the possible photoinduced electron and/or energy transfer mechanism between graphene and fullerene or porphyrin.

58 | Guoquan Zhang et al. Optical nonlinearities of GO–oligothienylene hybrid materials Oligothienylene is a polymer containing π electrons. It is widely used in areas such as photovoltaic cells, and is a good photoelectric material [225, 226]. In this section, we will study the oligothienylene covalent modification of graphene oxide hybrid materials; the results show that this hybrid material has a strong optical nonlinearity. Figure 1.34 gives the structure of 6THIOP and GO-6THIOP. The synthesis of the graphene-oligothiophene nanohybrid was carried out using an amine functionalized oligothiophene (6THIOP-NH2 ) and graphene oxide in o-dichlorobenzene (ODCB). The oligothiophene and graphene in the nanohybrid act as a donor and an acceptor, respectively. The covalent functionalization of graphene oxide with oligothiophenes has changed graphene oxide from hydrophilic to hydrophobic, and the hybrid can be dissolved in organic solvents such as ODCB. This makes it possible that this graphene hybrid can be homogeneously dispersed (together with other organic materials) in organic solvents needed for various organic electronic applications. Based on the elemental analysis, 6THIOP unit was estimated to be one for every 108 graphene carbons in the hybrid.

Fig. 1.34. The structure of 6THIOP and GO-6THIOP.

Fig. 1.35. Open aperture Z-scan of GO-6THIOP, 6THIOP, GO, the blend sample and C60 [211].

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The NLO properties of these materials were measured by Z-scan technique at 532 nm with 5-ns pulses. Figure 1.35 shows Open-aperture Z-scan results of GO-6THIOP (in ODCB), 6THIOP (in ODCB), GO (in DMF), the blend sample of 6THIOP and GO with a mass ratio of 1 : 1 (in component solvent of ODCB and DMF, because of the poor solubility of GO in ODCB), and C60 (in toluene). The linear transmittance of GO-6THIOP, 6THIOP, GO, the blend sample, and C60 are 45%, 98%, 84%, 91%, 95%, respectively. As shown in Figure 1.35, the transmittance of GO and 6THIOP exhibits negligible decrease at the focus, indicating that no obvious NLO properties of them were shown in this experiment. The blend sample shows stronger NLO properties than the individual components (GO and 6THIOP) but weaker than GO-6THIOP and C60 . The GO-6THIOP has a largest dip among the transmittance curves of these materials, indicating that it should have the best NLO properties. So the covalent hybrid material of GO-6THIOP shows the enhanced NLO properties compared to the individual components and the blend sample. In the experiments, when the GO-6THIOP is at closer focus, we observ a strong nonlinear scattering, and similar for the GO–TPP and GO–C60 . We use the Z-scan methods to measure the nonlinear scattering material. The experimental apparatus used is similar to the literature [227], as shown in Figure 1.36. The enhanced nonlinear scattering behavior was observed for GO-6THIOP. Figure 1.37 (a) shows the scattered light signal versus the sample position with C60 toluene solution for reference. As shown in Figure 1.37 (a), the scattered light appears a peak as the sample of GO-6THIOP moved to the focus, which means that the scattered light increases with the increasing input fluence. There is also a similar peak for the blend sample, but it is much weaker and emerges at higher fluence, indicating the weaker scattering. For 6THIOP, GO and C60 , there are no such obvious peaks in the scattering curves, so there are no nonlinear scattering for them in our experiments. Compare Figures 1.36 and 1.37 (a), GO-6THIOP have the largest dip of transmittance and the most intensity scattered signal at the focus, so nonlinear scattering plays an important role for the enhanced optical nonlinear properties of GO-6THIOP.

Fig. 1.36. Experimental setup of Z-scan measurement for nonlinear scattering.

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Fig. 1.37. (a) Scattering of GO-6THIOP, 6THIOP, GO, the blend sample and C60 with fluence (Z position) at an angle of 7°. (b) Scattering of GO-6THIOP at three different forward angles with fluence (Z-position) [211].

Figure 1.38 gives the Open-aperture Z-scan curves of GO-6THIOP for different input fluence. Using the Crank–Nicolson finite-difference method, we fitted the Z-scan curves numerically (the solid lines) and obtained the values of the effective nonlinear absorption coefficient β eff , as shown in Figure 1.38. In general, the value of β eff will decrease as input fluence increases for RSA process because of the saturation of

Fig. 1.38. Open-aperture Z-scan curves of Graphene-6THIOP for different input fluence [211].

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Fig. 1.39. Open-aperture Z-scan and the optical limiting of graphene-6THIOP, GO, and C60 [211].

RSA, and however it will keep unchanged for TPA process. The increase of β eff with input fluence implies the existence of nonlinear scattering in the high-fluence regime and the good optical nonlinear properties of GO-6THIOP. So GO-6THIOP can serve as a good scatterer at high fluence and may lead to the good optical limiting properties. For the optical limiting properties, Figure 1.39 gives the characteristics of output fluence vs input fluence of GO-6THIOP, GO, and C60 . For comparison, all of the samples were set to have same linear transmittance of 65% at 532 nm by adjusting their concentration. From Figure 1.39, it can be seen that the optical limiting effects of GO6THIOP are much better than GO and C60 . For example, at the highest input fluence (7.22 J/cm2 ) used in our experiments, the output fluence are 3.58, 1.68, 1.01 J/cm2 for GO, C60 , and GO-6THIOP, respectively. So GO-6THIOP offers the superior optical limiting effect. GO-6THIOP exhibits enhanced NLO and optical limiting properties compared to the individual components (GO and 6THIOP) and C60 . The enhanced nonlinear scattering of GO-6THIOP is observed and photoinduced electron or energy transfer mechanism is proposed.

Optical nonlinearities of GO–Fe3 O4 hybrid materials Recently, Fe3 O4 nanoparticles modified GO hybrid materials has attracted great interest. This material is expected to be applied to wastewater treatment, sensing, drug delivery and other areas [228–230]. From the perspective of nonlinear optics, Fe3 O4 nanoparticles typically have a strong excited state absorption and nonlinear scattering properties [231, 232], while the GO has nonlinear absorption, so we expect the optical nonlinearity of GO–Fe3 O4 hybrid material may be greatly enhanced due to the introduction of Fe3 O4 nanoparticles. This section examines the GO–Fe3 O4 hybrid materials and optical limiting properties of nonlinear optics, and with the GO and fullerene C60 in toluene were compared. The statistical analysis using atomic force microscopy

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Fig. 1.40. Scheme of the synthesis of GO–Fe3 O4 hybrid material.

shows that the size of GO sheets is mainly distributed between 200 and 500 nm. The synthesis of GO–Fe3 O4 hybrid was prepared by chemical deposition of iron ions using water soluble GO as carriers and Fe3 O4 is bound onto GO surface by the coordination interaction between the −COOH and Fe3 O4 [214, 228], as shown in Figure 1.40. The size of Fe3 O4 nanoparticles is 2–4 nm with a narrow size distribution, and some Fe3 O4 aggregation is also observed. Figure 1.41 shows the Open-aperture and Closed-aperture Z-scan results of GO and GO-Fe3 O4 with the same intensity and concentration. For GO, the obvious peak-valley feature of the closed-aperture Z-scan curves indicates the strong negative nonlinear refraction, while the peak of the curve is seriously suppressed for GO-Fe3 O4 , suggesting that the stronger nonlinear absorption/nonlinear scattering exists. By theoretical fitting, the effective TPA coefficients β eff and the effective nonlinear refraction coefficients n2eff were obtained as 7.8 cm/GW, 9.74 × 10−14 cm2 /W for GO and 26 cm/GW, 2.83 × 10−13 cm2 /W for GO-Fe3 O4 , respectively. So both the effective TPA and nonlinear refraction were enhanced in GO-Fe3 O4 compared with the pristine GO. Since

Fig. 1.41. Open-aperture and closed-aperture Z-scan curves of GO and GO-Fe3 O4 [213].

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the beam waist radius at focus is about 23 μm, the buildup time of the thermally induced optical nonlinearities is about 16 ns. Compared with the pulse width of 5 ns, the thermally induced optical nonlinearities is highly transient. So the observed negative nonlinear refraction should be attributed to the transient thermally induced optical nonlinearities and the intrinsic nonlinear refraction of the samples. To evaluate the NLO properties and optical limiting effect of GO-Fe3 O4 , we measured the optical limiting effect of GO-Fe3 O4 , compared with C60 and the pristine GO with the same linear transmittance of 49% and 87%, respectively. The high and low linear transmittance was obtained by adjusting the mass concentration of the samples. As shown in Figure 1.42 (a) and (c), with the linear transmittance of 49%, GO-Fe3 O4 exhibits enhanced optical limiting effect, compared with GO, but weaker than C60 . For example, at the input fluence of 20 J/cm2 , the output fluence are 1.32, 3.30, and 0.56, and the optical limiting threshold (defined as the input fluence at which the transmittance falls to 50% of the normalized linear transmittance) are 2.82, 10.19, and 0.41 J/cm2 , for GO-Fe3 O4 , GO and C60 , respectively. The lowest output fluence and optical limiting threshold of C60 indicate that C60 exhibits the best optical limiting effect at high concentration. As shown in Figure 1.42 (b) and 1.42 (d),

Fig. 1.42. The optical limiting of GO-Fe3 O4 , GO, and C60 with the same linear transmittance of 49% and 87% with nanosecond pulses. (a) and (b) show output fluence vs input fluence. (c) and (d) show nonlinear transmittance and scattered signals spectra vs input fluence [213].

64 | Guoquan Zhang et al. C60 shows the lowest output fluence and normalized transmittance for input fluence lower than 2.27 J/cm2 , but it shows the higher output fluence and normalized transmittance than that of GO-Fe3 O4 for input fluence higher than 2.27 J/cm2 . At the input fluence of 20 J/cm2 , the output fluence are 2.81, 5.06, and 3.33 J/cm2 , the optical limiting threshold are 3.70, 10.38, and 8.58 J/cm2 , for GO-Fe3 O4 , GO and C60 , respectively. This indicates that GO-Fe3 O4 shows the best optical limiting effect at low concentration and high input fluence. In our experiments, nonlinear scattering signals were also measured for the samples. From Figure 1.42 (c) and (d), we can see that the scattered intensity increase along with the decrease of normalized transmittance for the three samples at high input fluence, indicating that nonlinear scattering exist and is responsible for the optical limiting at high input fluence. However, we noticed that the onset of the growth of nonlinear scattering is higher than that of the decrease of normalized transmittance, which is much pronounced for C60, indicating the existence of other nonlinear mechanisms, such as nonlinear absorption and/or nonlinear refraction. For the linear transmittance of 49% as shown in Figure 1.42 (c), we can see that GO shows the weakest scattered intensity and the weakest optical limiting effect, GO-Fe3 O4 shows the stronger scattered intensity but weaker optical limiting effect than C60 . For the linear transmittance of 87% as shown in Figure 1.42 (d), C60 exhibits significant scattered signals and lead to the near constant output fluence for input fluence higher than 10 J/cm2 , but GO-Fe3 O4 shows the stronger scattered intensity and lower output fluence than C60 for input fluence higher than 2.33 J/cm2 . So GO-Fe3 O4 exhibits better optical limiting performance than C60 at low concentration and high input fluence due to the strong nonlinear scattering properties combined with negative nonlinear refraction and TPA. Since a practical optical limiter requires high linear transmittance, large broadband NLO properties, fast response time, considering the strong scattering properties even at low input fluence and low concentration, the strong negative nonlinear refraction and the obvious nonlinear absorption under picosecond pulses for GO–Fe3 O4 . It is expected that the hybrid material GO–Fe3 O4 may be a good candidate for optical limiter. In this section, we studied optical nonlinearity of carbon nanotubes, graphenebased hybrid structures, and new carbon structural materials. In order to better study their photophysical processes and nonlinear mechanisms, we used nanosecond, picosecond pulsed laser to test and analyze. During the experiments on the hybrid materials, the individual component was compared to the hybrid materials and the reference materials. First, We comprehensively studied the NLO properties of hydroxyl groups modified MWNTs suspension. To obtain the optimal experimental configuration for application, NLO properties of MWNTs20−30 -OH with different solvents, pulse widths, on-axis fluences at focus and concentrations were studied. Secondly, we studied a series of porphyrin-modified SWNTs and MWNTs with large optical nonlinearity. The NLO properties of the hybrid materials come from nonlinear scattering of CNTs moiety, the RSA of porphyrin moiety, the photon induced electron transfer between

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CNTs and porphyrin also influence the NLO properties. Finally, the studies of optical nonlinearity of GO hybrid material with fullerenes, porphyrins, and oligothienylene and Fe3 O4 magnetic nanoparticles showed enhanced NLO properties and optical limiting compared to individual component, blend sample and reference materials. Looking for new NLO materials, nonlinear mechanism analysis and tuning optical nonlinearity is the main direction of people trying. The new carbon structure nonlinear material is dependent on materials science and technology development. Both carbon nanotube and graphene material have unique structure and excellent mechanical, heat, light, electrical properties, while carbon structural material also has good optical nonlinearity, which laid the foundation for its practical applications in optical limiting, optical switching, saturable absorber, and other aspects.

1.4 Nonlinear optics from metallic plasmonics¹ Nonlinear optics is one of the most charming topics that have fascinated researchers for many years. Nevertheless, when talking about conventional nonlinearity, one comes at a price of intense pulsed laser sources and bulky materials. Novel solutions are highly demanded to realize less light consuming miniature devices that are capable of controlling light in subwavelength nanoscales. Clearly, the stronger the nonlinear effect, the much lower power operation meanwhile smaller the material required. However, nonlinear interactions between photons and matters are naturally weak, so some additional tricks are usually resorted to overcome this limitation and reinforce the nonlinear responses. For example, a resonator could be used to spatially squeeze the optical energy and enhance the nonlinear process of materials inside; photonic crystal structures were used to enhance supercontinuum generation in optical fibers [233]; slowing light in waveguide promoted stronger light–matter interaction and improved nonlinear process efficiency [234]. But the above techniques seem to be inadequate in realizing optical switchable nanodevices since they need dimensions lager than or similar to wavelength scales. Plasmonics, by confining the optical fields to the surface of metal, on the other hand, provides a possibility of inducing and manipulating nonlinear responses in nanoscales and extends the nonlinear optics from bulk to subwavelength. Plasmonic mode results from the special dispersion characteristics of metals which have negative permittivity under plasma frequency. Distinguishable from “ordinary” photons, plasmonic wave has a much smaller wavelength, which makes nanoscale optical integrated circuits or devices that are smaller than optical wavelength achievable. Waveguides [235], splitters [236], filters [237], and even amplifiers [238] for plasmon

1 The results shown in this section were accomplished under collaboration with group of Prof. Nikolay Zheludev from the University of Southampton.

66 | Guoquan Zhang et al. waves have been manufactured, in addition, plasmon was used to enhance the extraction efficiency of light from LED [239, 240], improve efficiencies of batteries and solar cells [241], all of which are essentially indispensable components in realizing future nanointegrated plasmonic chips. Based on the subwavelength advantages of plasmon, a kind of new artificial materials were proposed at the beginning of 21st century, now known as “Metamaterials.” They are a new class of manmade ordered composites that exhibit extinct properties not readily observed in natural materials. The most common metamaterials are formed by nanoscopic patterning of plasmonic metal films or arrangement of conducting nanoparticles to compose artificial periodic structures with lattice constants that are much smaller than the wavelength of the incident radiation, thus acts as an effective continuous material under the light field. Rather than directly from composition, metamaterials gain their properties from their structures. By properly designing the “artificial atoms/molecules” inside a metamaterial, one can achieve optical properties which are not analogous among natural substances, e.g. a negative index of refraction [242, 243], huge refractive index [244], large magnetic response [245], etc. These artificial composites can achieve performances beyond the limitations of conventional composites, such as perfect imaging beating diffraction [246], cloaking [247–249], etc. Metamaterials have opened a whole new era of photonics connected with novel concepts and potential applications. Plasmonics offers an exciting route to subwavelength photonics; however, in some applications such as integrated modulators, switches, and sensors, it is desirable to have the ability to dynamically tune the plasmonic wave or metamaterials preferably by applying an external light, i.e. the nonlinear plasmonics. Due to the compression of light mode at the interface of metal/dielectric, the propagation of plasmon wave is ultra sensitive to the minor change of metal surface or properties of neighboring dielectric. Imagining the change is introduced by another control light, one will arrive at a sensitive optical plasmonic switch to turn the propagation of plasmon wave on or off [250]. In addition, strong electric fields will be generated within the subwavelength dimensions at the interface of metal, it is not surprising that the nonlinear responses of materials nearby can be dramatically reinforced, thus permitting the nonlinear phenomena to occur under lower laser power and in nanoscales. Combining these enhanced nonlinear responses with novel optical properties of metamaterials, a variety of optical controllable devices with fascinating functionality are achievable. This section will focus on the introduction to the plasmonic enhancement in different nonlinear processes in different metallic systems. Introductory Section 1.4.1 outlines the basic properties of surface plasmon polariton (SPP) and its role in enhancing field distributed nearby. Section 1.4.2 will move to the major nonlinear mechanisms in metals. The following three subsections will be devoted to describe the plasmonic enhancement in harmonic generation, third-order nonlinear absorption, and nonlinear optical activity (NOA); some typical experimental works will also be introduced.

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1.4.1 Introduction to surface plasmonic polariton The existence of surface electromagnetic wave was first predicted by Sommerfeld in 1909 [251]. Surface electromagnetic waves are propagating modes that are confining at the interface with its amplitude decaying exponentially in the direction perpendicular to the interface [2]. Here we will only discuss the SPP at the interface of metallic conductor and dielectrics, which arises via the coupling of the electromagnetic fields to oscillations of the conductor’s electron plasma. Rather than directly reflecting the incident photons away and leave electric fields almost nil at the surface due to the deconstructive canceling of incident and reflected light, SPPs make the photons concentrate near the surface and generate strong electric fields which are useful in reinforcing light–matter interactions, optical signal amplification, sensing, surface-enhanced absorption, and nonlinearity enhancement. A planar boundary that separates free space from a semi-infinite conducting space is the most simple geometry to support the propagation of SPP, as shown in Figure 1.43 [252]. The optical constants of conducting and free space are ϵ m and ϵ d , respectively. The interface is assign as plane z = 0, the surface wave that propagates along the x direction can be described by E(x, y, z) = E(z)eiβx , in which β is called the propagation constant corresponding to the wave vector along the propagation direction and the imaginary part shows the exponential decay of the electrical field while propagating. For a propagating SPP, the Helmholtz function ∇2 E + k 20 ϵE = 0 can be written as ∂2 E(z) + (k 20 ϵ − β 2 )E(z) = 0 , (1.3) ∂z2 where k 0 is the wave vector in vacuum, and ϵ is the optical constant of optical medium. In the single interface system as shown in Figure 1.43 (a), the SPP can only exist in the transverse magnetic (TM) modes, where the field components E x , E z , and H y are nonzero, and no surface modes exist for TE polarization [252]. Under the TM mode, the field components in two media can be described by the following equations: d H y (z) = A d e { { { E x (z) = iA d ωϵ10 ϵ d k d e iβx−k d z { { { β iβx−k d z {E z (z) = −A d ωϵ0 ϵ d e

iβx−k z

(z > 0)

H y (z) = A m e iβx+k m z { { { E (z) = −iA m ωϵ10 ϵ m k m e iβx+k m z { { x { β iβx+k m z , {E z (z) = −A m ωϵ0 ϵ m e

(z < 0) ,

(1.4)

where k i = k zi (i = m, d) is the wave vector component along the normal of the interface, whose reciprocal δ = 1/|k z | gives the penetration depth giving the distance that the field attenuates to 1/e of its initial amplitude, which quantitatively describe the compression of electromagnetic energy at the interface.

68 | Guoquan Zhang et al.

Fig. 1.43. (a) The SPP propagating mode at a gold/air interface, and electric field attenuates exponentially along z direction. The confinement of electromagnetic energy in air and gold are described by δ m and δ d , respectively. (b) The dispersive curve of SPP on the gold–air interface and light cone in air, the wave vector mismatch is shown by Δk. The optical constant of gold comes from Ref. [253].

The continuous boundary conditions of E x (z = 0+ )E x (z = 0− ) and H y (z = kd ϵd − y (z = 0 ) lead to A m = A d and k m = − ϵ m . The confinement of field to the interface demands k i > 0, determining that the dielectric constants of the adjacent media must differ in sign, which can be satisfied at the air/metal interface as the real part of the optical constant of metal remains negative (Re{ϵ m } < 0) below its plasmonic frequency ω p . Solutions (1.4) must fulfil the wave equation (1.3) resulting in

0+ )H

k 2m = β 2 − ϵ m k 20 k 2d = β 2 − ϵ d k 20 . Further considering interface is

kd km

(1.5)

= − ϵϵmd , the dispersion relation of SPPs propagating at the β = k0 √

ϵm ϵd . ϵm + ϵd

(1.6)

The confinement property of the SPP mode results in a larger propagation constant β than the wave vector in the free space leading to the SPP dispersion curve lying to the right of the light cone of the dielectrics (see Figure 1.43). Thus, the SPP excitation cannot be accomplished by direct shinning on flat metal surface and some special phase-matching techniques must be adopted to compensate the mismatch Δk between β and k. The most common optical techniques for SPP generation include prism coupling, grating coupling, and excitation using highly focused optical beams, which have been introduced in many literatures [252, 254–257] and will not be discussed here. The existence of SPP tightly localizes the electric field in the subwavelength scale near the interface. For instance, for 1000 nm infrared light, the optical constants of air and gold are ϵ d = 1 and ϵ m = −46.51 + 3.51i, respectively, following Equations (1.5) and (1.6), the penetration depths on the gold and air sides are δ m = 1/ Re |k m | = 23 nm

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and δ d = 1/ Re |k d | = 1076 nm, i.e. the electromagnetic energy is concentrated in the 1/100λ ∼ λ scale. The resulting mode compression generates strong electric fields which are useful in the reinforcement of light–matter nonlinear interactions. The electrical field density could be amplified through confinement in one dimension forming propagating SPP waves at the 2D air/conductor interface as mentioned above. In metallic nanoparticles, plasmonic fields could be squeezed further forming even a tighter, hence, stronger localized field distribution. When light is incident on the particle the oscillating electric field of the light produces a force on the mobile conduction electrons in the metal and the redistribution of charges acts to provide a restoring force on the displaced electrons associating with a resonant frequency. For metallic particles with size a much smaller than wavelength λ, where the quasi-static approximation is valid, the interaction between light and particle could be regarded as the driven oscillation of free electrons under a uniform time-varying field and only plasmon modes containing electric dipole moments can be excited by and coupled with the incident electromagnetic fields. We shall start with this situation to describe the electric field enhancement and the resonant frequency properties of this localized plasmon excitation. Assuming a homogeneous, isotropic conducting sphere of radius a with an optical constant ϵ m is immersed in an isotropic and lossless medium with dielectric constant ϵ d . It is worth to notice that for a small enough particle (a < 10 nm), its optical constant differs from bulk metal [258, 259] due to the quantum size effect. The sphere center is assigned to be the origin of system and the external static electric field is parallel to the z-direction E = E0 ẑ (as shown in Figure 1.44). The inner electrons move freely about in particle and will be attracted toward the −z side of sphere under the force exserted by the external field, leaving positive charges due to the unmovable nuclei on the other side, forming an equivalent dipole p which will create its own electric field opposing the external one. The electrons continue to move until an equilibrium is reached in which the induced charges are exactly the right position to screen out the external electric field throughout the interior of the metal object, and the spatial field is determined afterward. Under the static field approximation, the potential distribution is determined by the Laplace equation ∇2 Φ = 0, and electric field can be deduced by E = −∇Φ.

Fig. 1.44. The homogeneous, isotropic conducting sphere of radius a (optical constant ϵ m ) locates in an isotropic and nonabsorbing medium (dielectric constant ϵ d ). The external static electric field E0 is parallel to the z direction.

70 | Guoquan Zhang et al. Following solving process in the literature [260], and furthermore taking the convergence of solutions and boundary conditions into consideration, the potential distributions inside and outside the sphere are 3ϵ d E0 r cos θ ϵ m + 2ϵ d cos θ ϵm − ϵd = −E0 r cos θ + E0 a3 2 , ϵ m + 2ϵ d r

Φin = − Φout

(1.7)

in which −E0 r cos θ is the potential distribution of the external static electric field. ϵ m −ϵ d 3 d − ϵ m3ϵ +2ϵ d and ϵ m +2ϵ d a describe the influence of metallic sphere to the Φin and Φout . Equation (1.7) tells that Φout could be regarded as the superposition of the external field E0 and field induced by dipole p, Φout = −E0 r cos θ + in which p = 4πϵ0 ϵ d a3

p⋅r , 4πϵ0 ϵ d r3

ϵm − ϵd E0 . ϵ m + 2ϵ d

(1.8)

(1.9)

Via p = ϵ0 ϵ d α s E0 , we derive the polarizability of the sphere α s α s = 4πa3

ϵm − ϵd . ϵ m + 2ϵ d

(1.10)

Fig. 1.45. The normalized scattering cross-section C sca and absorption cross-section C abs of gold nanosphere.

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Using E = −∇Φ, we arrive at the electric field distributions inside and outside the sphere 3ϵ d E0 ϵ m + 2ϵ d 1 3ϵ d 3n(n ⋅ p) − p = E0 + E0 a3 3 (−2 cos θ e⃗ r − sin θ e⃗ θ ) = E0 + , ϵ m + 2ϵ d r 4πϵ0 ϵ d r3

Ein = Eout

(1.11)

in which e⃗ r and e⃗ θ are the unit vectors along the radial r and the azimuth θ direction. From the scattering theory, the scattering cross-section Csca and absorption crosssection Cabs of the gold nanosphere are [252] k4 8π 4 6 󵄨󵄨󵄨󵄨 ϵ m − ϵ d 󵄨󵄨󵄨󵄨2 |α s |2 = k a 󵄨󵄨 Csca = 󵄨󵄨 6π 3 󵄨󵄨 ϵ m + 2ϵ d 󵄨󵄨 (1.12) ϵm − ϵd Cabs = k Im{α s } = 4πka3 Im { }. ϵ m + 2ϵ d The scattering cross section is proportional to the square of the particle volume (V ∝ a3 ); however, absorption cross section is proportional to the particle volume, so for a small metallic particle, absorption dominates, while the size increases, the scattering cannot be negligible. Most interesting, when ϵ m = −2ϵ d , i.e. Fröhlich condition is satisfied, Ein and Eout experience resonant enhancements since |ϵ m + 2ϵ d | is minimum. For a perfect conductor, the enhancement approaches infinity; however, for a real conductor, the enhancement is finite limited by nonzero Im{ϵ m } due to loss. Meanwhile, scattering and absorption cross sections are resonantly amplified and could be figured out by the appearance of peaks in scattering and absorption spectra (Figure 1.45). Intense energy is localized inside and the region nearby. It is not surprising that nonlinear processes in metals and neighboring dielectrics can be strongly reinforced. The above discussion predicts that for a small conducting particle where quasistatic approximation is valid, the wavelength where the surface plasmonic resonance happens is determined by the Fröhlich condition, which is only related to the metal itself and the environment and is independent of the sphere size a. The width of the resonance is determined by the imaginary part of ϵ m . Spurred by state-of-the-art nanofabrication techniques, nano-optical characterization methods, and sophisticated numerical modeling capabilities, research studies on the optical properties of particles with different shapes or geometries and various featured 2D or 3D nanostructures on the nanometer scale have flourished over the past decade. Plasmon resonances were found to be tunable sensitively on the geometrical shape of the structures. An important topic is the appearance of “Fano” resonances which was proved to result in much larger electromagnetic field enhancements and theoretically investigated for a variety of structures ranging from particle lattices and split ring-type structures to nanowire arrays and particle dimers [261–263]. The “Fano” resonance is caused by the interaction of broad bright dipole modes with nondipole active multipolar or magnetic modes by introducing symmetry breaking to metallic nanostructures or utilizing the retardation effects under glancing shinning.

72 | Guoquan Zhang et al. For nanoparticles of a size comparable to a quarter of the wavelength of the incident light, the electric field can no longer be assumed uniform across the nanoparticle in which the higher multipolar components of the incident wave can directly excite the corresponding “dark” multipolar plasmon modes. Symmetry breaking can furthermore enable hybridization of plasmons of different multipolar symmetry and hence can strongly enhance its intensity. As the “Fano” resonance was found to exhibit narrower resonances compared to dipolar modes, it was widely used in the area of sensitive detection, sensing [264–266] and nonlinear enhancements which will be further demonstrated in Section 1.4.4. Regarding to the surface plasmon enhancement to the nonlinear processes, let us go back to the 1970s, when M. Fleischman and his co-workers observed surfaceenhanced Raman scattering (SERS) from pyridine adsorbed on electrochemically roughened silver. Despite they did not recognize it as an enhancement effect originally, the studies related to the surface plasmonic enhancement to nonlinearity were initialized [267], and during 1980s the enhancement effects were systematically studied in gold colloids through four-wave-mixing technique (FWM) and proved the role of plasmonic resonance in reinforcing the occurrence of nonlinear processes [258, 268]. The enhancement effect could be divided into two kinds: first, the nonlinear response comes from dielectrics neighboring to metal nanostructure and is reinforced, for example, SERS. Second, the intrinsic nonlinear susceptibility of the metal itself is enhanced. However, both of them happen at frequencies within the lineshape of the plasmon resonance. We will focus in this section on a description of the later case in the form of exploiting intense electric fields of resonant plasmon modes to enhance SHG, or thirdorder nonlinear absorption, or even novel giant NOA of metal nanostructures. In the following section, we shall start from a brief introduction to the nonlinear processes in metals.

1.4.2 Nonlinear processes in metals Until now, people has deeply investigated various nonlinear processes in dielectrics, for example the SHG from crystals such as KDP, KTP, and LiNbO3 , the multiphoton absorption and saturable absorption from semiconductors and chemical solutions. However, rare systematic introductions were made to the nonlinearity of metals. In metallic conductors, such as gold, silver, copper and aluminum, et al., there are no band gap as dielectrics. Hybrid chemical bonds between metal atoms result in a valance band formed by localized d-band states and a conduction band partially filled by quasi-freeelectron up to the Fermi level (E F ). Both the electrons from valence and conduction bands will contribute to the nonlinear properties of metal. Metals typically have nonlinear susceptibilities that are orders of magnitude (typical ∼ 106 ) larger than those of dielectrics; however, due to their large attenuation

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constants, render samples that are just tens of nanometers thick essentially opaque for light and their nonlinear properties are effectively inaccessible. [269]. Multilayer metal-dielectric structures were used to maximize the transmission while keeping the absorption at an acceptable level; hence enhance the nonlinear polarization [269]. In other works, the reflected light rather than transmitted signal were used to explore the nonlinear properties of metals [270, 271]. Now coupling and compressing the incident electromagnetic energy into the near field region of metal structures through excitations of SPP could act as another solution to reinforce the nonlinear interactions between electromagnetic field and metals. In the following, second-order and third-order nonlinearities of metals will be introduced. It is well know that SHG is a forbidden process for a material that possesses a center of inversion symmetry, such as metal. However, the surface of a material clearly lacks inversion symmetry, and thus SHG can occur at the surface of a material of any symmetry group. The SHG on reflection of a pulse laser beam from the surface of a silver mirror was first observed in 1965 [270]. The origin of the metal nonlinearity has been widely discussed using free-electron-gas model and hydrodynamic model, which attributed the double-frequency signal to surface and volume contributions. The bulk nonlinearity arises from magnetic dipole and electric quadrupole interactions, and the Lorentz and convective forces exerted on the free electrons [272–276]. The surface contributions are proved to dominate over the bulk component [276, 277]. The induced nonlinear polarization at the surface is P si (2ω) = χ ijk E j (ω)E k (ω) ,

(1.13)

in which χ ijk is the second-order surface nonlinear susceptibility tensor arises from variations of electron density and field at the surface along the surface normal and (2) (2) (2) (2) symmetry arguments predicts the nonvanishing elements are χ zzz , χ zxx = χ zyy , χ xxz = (2) (2) (2) χ xzx = χ yyz = χ yzy (the surface normal is assumed to be in the z-direction) [278]. Despite there is no band gap in metals, direct interband electronic transition still can be triggered by incident photons with energy larger than the energy gap ΔE between spatially localized d-band and free-electron conduction band above Fermi level E F (for gold ΔE= 2.4 eV)[279], resulting in the Fermi distribution smearing, which induces a very strong cubic optical nonlinearity and saturation of the two-level transition (β ∼ 10−5 m/W for gold [280]) peaking at plasma wavelength (516 nm for gold) [258, 259]. However, this nonlinearity is relatively slow as it depends on the thermalization of the hot electron ensemble, which occurs over a period of several picoseconds [281], as illustrated in the left part of Figure 1.46. For bulk metal, the intraband transition is forbidden due to the large slope of the conduction band where momentum conservation rule is broken; however, in metal particles of nanometer dimensions, the free-electron states are quantized in discrete levels due to the confinement and the intraband transitions between filled and empty states in the conduction band are available, which arise nonlinear response to the applied light [258, 259]. Apart from the absorbed energy that promotes inter- and intraband transitions, the

74 | Guoquan Zhang et al.

Fig. 1.46. The band structure of gold and comparison between Fermi smearing and twophoton nonlinear responses in gold (redrawn from Ref. [285]).

rest of it is absorbed by the conduction electrons. The specific heat of these conduction electrons are weak, they can easily be raised to high temperatures [282] breaking the thermal equilibrium with the lattice. Their Fermi–Dirac distribution is therefore modified, part of the electron levels below the Fermi level being emptied whereas part of the levels above the Fermi level become occupied. This leads to a modification of the dielectric constant ϵ and is the origin of the so called the hot-electron contribution (3) to χ Au , it takes a few picoseconds for these hot electrons to come into thermal equilibrium with the lattice [283]. For infrared incident light, a single photon energy is too low to promote interband absorption from d-band to conduction band, however, under irradiation of intense light direct TPA takes place without a real intermediate level as there are no empty states in the Fermi sea. It occurs through a virtual state when the energy of two incident photons is combined to bridge the gap, ℏω1 + ℏω1 > ΔE (see right part of Figure 1.46). The process is less efficient (β ∼ 10−8 m/W), but due to the short life time of virtual state ( 90° at the filter output, the transmission spectrum will evolve into flat-

Fig. 2.16. The filter curves when A = π/4.

134 | Xianfeng Chen

Fig. 2.17. The transmission spectrums are simulated theoretically in PPLN with the product A fixed at 2.24.

top type at some critical angles. In order to demonstrate it, we designed the following experiment. The sample with a dimension of 30 mm (L)× 10 mm (W)× 0.5 mm (T) consists of 2857 domains with the period of 21 μm. With the increment of the electric field, the transmission of the fundamental wavelength attends to maximum at 3 kV/cm (A) and subsequently declines when the electric field keeps on rising. Our study reveals that with a higher electric field, the transmission spectrum correspondingly evolves into a flat-top waveform at the critical point 4.2 kV/cm (B), which is very attractive. The corresponding transmission spectra at A and B are, respectively, shown in Figure 2.18. A flat-top waveform with a 1 nm flat-top width is obtained in the experiment. Based on the above, we demonstrate that a 1 × 2 precise electro-optic switch was demonstrated in a PPLN crystal. Optical switches play a significant role in optical communication and optical information applications. Different technologies have been proposed and demonstrated to realize optical switches. However, thermal optical switches and the acousto-optic switches usually have a switching time which is longer than a few microseconds [15, 16], all-optical switches are usually complicated

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Fig. 2.18. Transmission spectra at electric fields of 3 and 4.2 kV/cm.

and costly at the current stage of development [17]. Comparing to the above switches, electro-optical switches with high speed and related technology have been widely used in optical communication. We knew that the transmission of central wavelength could be modulated from 0 to 100% by changing the extra electric field. Based on this, we demonstrate precise electro-optic switches including the 1 × 2 narrow-band switch and 1 × 2 flat-top switch [18]. The schematic of the experimental setup is shown in Figure 2.19. The arrows inside the PPLN indicate spontaneous polarization directions. The PBS is employed to

Fig. 2.19. Experimental setup for a PPLN electro-optic switch.

136 | Xianfeng Chen separate the light which has passed through PPLN into two channels. In channel A, the polarization direction of the light wave at the output of the PBS is along the Z-axis of the PPLN sample, and in the channel B, the polarization direction is along the Yaxis. An electric field is applied along the Y-axis. If each domain serves as a half-wave plate, after passing through the stack of half-wave plates, the optical plane of polarization of the input light rotates continually and emerges finally at an angle of 2Nθ, where N is the number of plates. Therefore, when 2Nθ = 0 at the output, for channel A, the light does not experience loss and the switch is “ON”; for channel B, the light is forbidden and the switch is “OFF.” When 2Nθ = π/2 at the output, for channel A, the switch is “OFF” and for channel B, the switch is “ON.” As θ can be extremely small (10−6 –10−5 radians), precise control of the final rotation angle at the output is accessible. Thereby, the switching state of “ON” and “OFF” can be very precise which enables it to achieve a high extinction ratio. Figure 2.20 (a) and (b) is the experimental observation of the transmission spectra for A and B channels at electric fields of 2.1 and 0.5 kV/cm. For A channel, with the electric field of 0.5 kV/cm, the light intensity of the 1541.17 nm wavelength attends a

Fig. 2.20. The experimental transmission spectrums at electric fields of 0.5, 2.1, and 4.3 kV/cm for A and B channels.

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maximum value of 2.716 μW and the switch is “ON.” When the electric field is shifted to 2.1 kV/cm, the light intensity falls sharply down to nearly zero (15.25 nW) which means “OFF.” For B channel with electric field of 2.1 kV/cm, the light intensity of the 1541.17 nm wavelength attends a maximum value of 2.489 μW and the switch is “ON.” When it shifts to 0.5 kV/cm, the light intensity falls sharply down to nearly zero (3 nW) which means “OFF.” Thus, we can control this switch by shifting the electric field between 2.1 and 0.5 kV/cm. When the electric field is 0.5 kV/cm, only the OSA from A channel can receive the light with the polarization direction which is parallel to the polarizer, which means that the switch is “ON” for A channel, and “OFF” for B channel. The same as above, the switch is “ON” for B channel, and “OFF” for A channel, when the electric field is shifted to 2.1 kV/cm, and the polarization direction of the emergent light is perpendicular to the polarizer. It could not be ignored that the drift of the central wavelength would reduce the precision of such a narrowband switch. When we increase the voltage of the electric field to 4.3 kV/cm, the spectra evolve into broadband with flat top and the width is nearly 2 nm. Figure 2.20 (c) and (d) is the experimental observation of the transmission spectra for A and B channels at electric fields of 4.3 and 0.5 kV/cm. This means that the drift of the working wavelength up to certain extent has almost no effect on the precision of this 1 × 2 flat-top electro-optic switch. As we know, the precise electro-optic switch desires a low crosstalk. In our experiment the crosstalk level is lower than −20.98 dB between A and B channels at the three critical electric fields, which is similar to the theoretical results. Compared with other kind of electro-optic switch, the crosstalk of this switch is at the same level. Another important performance of the precise electro-optic switch is extinction ratio. In our experiment, the extinction ratio (on/off) is more than 22.32 dB, which is a little higher than the switch realized in other material. Although the critical electric fields are little higher than theoretical ones because of the voltage loss in our setup, the proposed 1 × 2 precise electro-optic switch is still very attractive. It should be noted that the PPLN waveguide has been successfully proposed recently, where the gap between the electrodes can be as short as 10 μm, so that only several Volts is enough to switch the light for this kind electro-optic switch. In this section, we analyze the Jones matrix method and coupling-mode theory, and present the Solc-type filter and electro-optical switch based on PPLN crystal. From the experiments, we know that the incident of light could be modulated by the swing domain structure, and the central wavelength could be modulated by the temperature. What is more, the flat-top spectrum can be realized by higher electric filed, which enhances the stability of the filter and the switch.

138 | Xianfeng Chen

2.4 Control of linear polarization and its applications 2.4.1 The evolution of the polarization state in the PPLN and the control of linear polarization state via electro-optical effect In this section, we first discuss the evolution of the optical polarization state in PPLN crystal. In a birefringent crystal, a polarized light decomposes into the ordinary wave (OW) and the extraordinary wave (EW), and the two waves, in general, do not exchange energy with each other. However, in PPLN the rocking angle of the optical axis behaves as a periodical small perturbation, in which case the coupling of energy between OW and EW will be yielded. Considering E1,2 = A1,2 (z) exp[i(k 1,2 z − ωt)], the Jones vectors, which represent the polarization state of a light, can be given as a function of the distance inside the PPLN by ⃗ E(z) = {[cos(sz) − iΔβ/(2s) sin(sz)]A1 (0) − i(κ/s) sin(sz)A2 (0)} e iΔβz/2 ], [ ∗ {(−iκ /s) sin(sz)A1 (0) + [cos(sz) + iΔβ(/2s) sin(sz)]A2 (0)} e−iΔβz/2 e i(k1−k2 )z (2.106) for a simplified case where the domain angle vanishes (κ = 0), Equation (2.106) is hence derived as A1 (0) ⃗ E(z) =[ ] . (2.107) A2 (0)e i(k1 −k2 )z which describes the evolution of the polarization state in a birefringent crystal. The orthogonal circularly polarized modes A+ and A− can be obtained by means of the following relation: {A+ = (A1 + iA2 )/√2 (2.108) { A− = (A1 − iA2 )/√2 . { The polarization state is then determined by the complex ratio ξ = A+ /A− , with the azimuth of the polarization ellipse being θ = 1/2 arg(ξ); ellipticity being e = (|ξ| − 1)/ (|ξ| + 1). The evolution of the polarization state of the light beam during propagation can be represented by a variety of graphic methods. Two particularly useful representations are the Poincare sphere and the phase plane. The latter was selected here to describe the evolution of the polarization state of a light. From Equation (2.107), it is easy to see that the beat length L0 in this case is the minimum common multiple of L0 = 2π/(k 1 − k 2 ). Assume that θ = 30°, e = 0, then the evolution of the polarization state is shown in Figure 2.21 (a), the evolution of polarization undergoes a single closed path.

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Fig. 2.21. Phase-plane trajectories of the polarization state.

When κ ≠ 0, the evolution of polarization is more complicated. Considering Δβ = 0, we get A1 (0) cos(|κ| z) − A2 (0) sin(|κ| z) ⃗ E(z) =[ ] . (2.109) {A2 (0) cos(|κ| z) + A1 (0) sin(|κ| z)} e i(k1−k2 )z In this condition, L0 is the lowest common multiple of L1 = 2π/(k1 − k 2 ) and L2 = 2π/|κ|. Starting from a linearly polarized light, for instance, θ = 0, e = 0, the trajectory of the evolution of the polarization under specific conditions L2 = nL1 , n = 1, 2, 3, are shown in Figure 2.21 (b)–(d). When κ ≠ 0, Δβ ≠ 0, by adjusting the operating wavelength, the beat length L0 is the minimum common multiple of L1 = 2π/(k1 − k 2 ), L2 = 2π/s and L3 = 2π/Δβ. In general, the magnitudes of L1 , L2 , and L3 differ dramatically with each other, and the evolution of polarization splits into considerable discrete paths, which are shown in Figures 2.21 (e) and (f). These discrete paths even develop into areas, covering more states of polarization. In the last section, we have discussed the transverse electro-optical effect in PPLN, the result shows that, the positive and negative domains would rotate with an angle

140 | Xianfeng Chen of +θ and −θ, respectively. Here θ=

γ51 E [(1/n e )2 − (1/n o )2 ]

.

(2.110)

Meanwhile, the optical axis of each domain will rotate continually by increasing the extra electric field. The coupling-mode equation is { dA1 /dx = −iκA2 e iΔβx { dA /dx = −iκ ∗ A1 e−iΔβx { 2

(2.111)

n2 n2 γ E

ω o e 51 y i(1−cos mπ) with Δβ = k 1 − k 2 − m( 2π (m = 1, 3, 5. . .), where Λ ), and κ = − 2c √n o n e mπ A1 is the normalized complex amplitude of OW, and A2 is the normalized complex amplitude of EW. Λ is the period of the PPLN, γ51 is the electro-optical coefficient, E y is the electric field intensity, and n o and n e are the refractive indices of the ordinary and extraordinary waves, respectively. When the incident light wavelength satisfies the QPM condition, the solution of the coupled-mode equation is given as

{A1 (L) = cos (|κ| L) A1 (0) − sin (|κ| L) A2 (0) { A (L) = cos (|κ| L) A2 (0) + sin (|κ| L) A1 (0) . { 2

(2.112)

From the solution, we can find that the normalized complex amplitudes of OW and EW are totally determined by transverse electric field, meanwhile the polarization state |κ|L changes periodically. After passing through N plates the final azimuth angle of the light is 2Nθ. Therefore, the polarization direction of incident light can be rotated based on electro-optical effect. Here we will introduce several linear polarization-controlled devices based on polarization-coupling theory.

2.4.2 Linear polarization state modulator Here we propose a simple configuration capable of rotating a linear polarization state of light by a certain angle with high precision for a series of wavelengths [36]. In contrast with previous approaches for linear polarization state generators by using birefringence plates, liquid-crystal material [37] and magneto-optic crystals [38] the present scheme takes advantage of high precision with 0.04° and a compact one-chip integration in lithium niobate. Besides, the rotation angle is controlled by the external electric field, which is faster and more convenient. The rocking angle θ here is proportional with the electric field and is given by θ ≈ γ 51 E/[(1/n e )2 − (1/n o )2 ] .

(2.113)

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Fig. 2.22. Experimental setup for linear polarization-state generator.

After passing through the stack of half-wave plates, the optical plane of polarization of the input light will rotate continually and emerge finally at an angle of 2Nθ, where N is the number of domains. Because the operating wavelength, which satisfies the condition that each domain serves as a half-wave plate is given by λ0 = Λ(n o − n e ) .

(2.114)

The operating wavelength can be extended to a series of different wavelengths by changing the temperature of PPLN The schematic of the experimental setup is shown in Figure 2.22, during the experiment we first attempted to find out the working wavelength (central wavelength). After that, we observed the rotation angle of the input light at different electric fields. In the above discussion, we knew that the operating wavelengths will remain in the linear state of polarization after passing through the PPLN crystal, because each domain serves as a half-wave plate with respect to them. Therefore, the operating wavelengths can be determined by checking whether the state of polarization of the output light is linear by rotating the analyzer. With this method, the operating wavelength at the temperature of 15 °C is found out to be 1543.47 nm. The corresponding rotation angle at each electric field is shown in Figure 2.23. The electric field intensity is tuned from 0 to 3 kV/cm, with the step of 0.1 kV/cm each time and the rotation angle varies between 0° and 100°. From the figure, we can see that the rotation angle has a linear relation with the external electric field, which shows agreement with the theory. The experimental result also indicates that there is already a rocking angle between the optical axes of the positive and negative domains when electric field is not applied. From the figure, we can also learn that for obtaining a given rotation angle, the theoretical results need less electric field than the experimental results, which is due to several reasons. First, the real refractive indices of this PPLN sample employed in the experiment inevitably have a deviation with the theoretical values calculated by the Sellmeier equation and the real EO coefficient also varies with different PPLN crystals in practice, which consequently contribute to the discrepancy. Second, the external electric

142 | Xianfeng Chen

Fig. 2.23. Experimental measurement of the rotation angle of the output light, when varying the applied electric field from 0 to 3 kV/cm.

field is generated by use of a pair of parallel copperplates, which requires extreme closeness to the PPLN crystal. During the experiment, we found that by slightly pressing the copperplates toward the PPLN crystal, less external electric field was required to obtain the same rotation angle. However, for consideration of possible damage to the PPLN, we did not apply strong pressure on them, which resulted in incomplete closeness and consequently led to the discrepancy. Actually, by using more accurate EO coefficient and indices in the theoretical calculation and achieving complete closeness between the copperplates and the PPLN in the experiment, the discrepancy will vanish. The greatest advantage of such a linear polarization-state generator is capable of rotating a linear polarization state of light by a certain angle with high precision, which is shown in Figure 2.24. In the experiment, different precision has been achieved by reducing the electric field. In order to obtain higher precision, between (A) 0.100 and (B) 0.200 kV/cm, we reduce the electric field step to 0.010 kV/cm each time and the precision is improved to 0.4°, which is shown in Figure 2.24 (a). Similarly, by further reducing the electric field step to 0.001 kV/cm between (A) and (C), a higher precision with 0.04° is achieved in Figure 2.24 (b) limited by the accuracy of the measurement system. Actually, higher precision is still possible if we continue to reduce the electric field step. During the experiment, we also found the operating wavelengths at different temperatures by using a tunable laser as shown in Figure 2.25. By tuning the temperature from 10 to 50 °C, the operating wavelength varies from 1546.02 to 1525.62 nm, with a

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Fig. 2.24. Experimental measurement of the rotation angle of the output light.

band of 20 nm, limited by the output range of the tunable laser. If we change the temperature to a wider range of 300 °C, a broader band of 150 nm is still available. It should be noted that the Ti-indiffusion PPLN waveguide has been successfully proposed recently [39]. In the waveguide configuration, the gap between the electrodes can be as short as 10 μm, so that only several voltages are enough for such linear polarization-state generator.

144 | Xianfeng Chen

Fig. 2.25. Experimental measurement of the operating wavelengths at different temperatures.

2.4.3 Electro-optic chirality control in PPLN [40] Optical activity is the turning of the polarization plane of linearly polarized light about the direction of motion as light travels through certain materials. It occurs in solutions of chiral molecules such as sucrose, spin-polarized gases of atoms or molecules, and solids with rotated crystal planes such as quartz. It is widely used in the sugar industry to measure syrup concentration [41]; in optics to manipulate polarization [42]; in chemistry to characterize substances in solution, and in optical mineralogy to help identify certain minerals in thin sections. The rotation angle of the polarization plane in an optically active material such as quartz is β = αL, where α is the specific rotation, and L is the path length of light in the material. The specific rotation of a pure material is an intrinsic property of that material at a given wavelength and temperature. A positive value corresponds to dextrorotatory rotation while a negative value is related to levorotatory rotation. When an external voltage V is applied along the +Y-axis of the PPLN, looking along the +X-axis, the Y- and Z-axes of the index ellipsoid rotate an angle of θ left-handedly and right-handedly in the positive and negative domains, respectively. When QPM condition is satisfied, the polarization plane of an incident linearly polarized light traveling along the −X-axis will rotate an angle of 2Nθ right-handedly at the output side. However, looking along the −X-axis, we see that the Y- and Z-axes of the index ellipsoid rotate an angle of h right-handedly and left-handedly in the positive and negative domains, respectively. The overall effect is that a linearly polarized light traveling along the +X-axis also rotates an angle of 2Nθ right-handedly at the output side. Thus, the polarization plane twists in the same sense during the forward and

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Fig. 2.26. Experimental measurement of the specific rotation versus external electric field.

backward pass. Optical propagation is therefore reciprocal in PPLN, which is similar to optically active material such as quartz. When the external voltage V is applied along the −Y-axis, the rotation directions of the Y- and Z-axes of the index ellipsoid also reverse, the polarization plane then rotate left-handedly during the forward and backward pass. The chirality of PPLN is thus controlled by the external electrical field. The rotation angle of the polarized light after passing through the PPLN is β = γ E 2 ΛL (1/ne )251 = αL, where Λ and L are, respectively, domain thickness and length −(1/no )2 γ E

of the PPLN. The specific rotation, defined as α = Λ2 (1/n )251 2 , is relevant to the e −(1/n o ) wavelength, temperature, and material. In addition, it is also electric field adjustable, which shows great advantage over optically active material in that the specific rotation can be adjusted according to practical demand. Large optical rotation in materials with small size is then at hand. The PPLN sample used in our experiment is MgO doped with 3582 domains. The domain period is 20.1 μm with the duty cycle of 1 : 1. We measured the specific rotation under different electric field with the working wavelength of 1568.5 nm at 22 °C, as depicted in Figure 2.26. The specific rotation increases linearly with the electric field, and reaches 0.87°/mm under an external electric field of 3 kV/cm. In the waveguide configuration, the width of the MgO:PPLN can be as small as 10 μm. The specific rotation can be as large as 2.43°/mm under an electric voltage of 1 V, which is very attractive. We designed an experiment to demonstrate the chirality of MgO:PPLN, as shown in Figure 2.27. A tunable laser worked as the light source. Two polarization-beam splitters (PBSs) were set perpendicularly to work as polarizer and analyzer. A MgO:PPLN crystal and a 45° dextrorotatory quartz were placed between the two PBSs. The

146 | Xianfeng Chen

Fig. 2.27. Experimental setup for studying the optical activity of MgO:PPLN.

MgO:PPLN and working environment were the same as that in the specific rotation measurement. We measured the output power of light traveling along the Xaxis with an external electric field applied along the Y-axis, which is expressed as T = sin2 (π/4 ± 2Nθ). The results are presented in Figure 2.28. The square and round dots represent transmissions of light traveling along the +X (forward wave) and −X direction (backward wave), respectively. Applying the electric field along the +Y-axis, we get the transmissions as shown in Figure 2.28 (a). With the help of the dextrorotatory quartz, transmission curves of light present a form of sinusoidal function. It gets the maxima first with the increment of the electric field, which means a 45° right-hand rotation of the polarization plane from the MgO:PPLN. While when an electric field along the −Y-axis is employed, the transmission curves appear a cosine-function shape. They reach the minima first when the electric field increases from 0, which indicates that light undergoes a left-hand rotation from the MgO:PPLN, as shown in Figure 2.28 (b). From Figure 2.28 (a) and (b), we get the conclusion that the change of chirality of MgO:PPLN can be achieved by altering the direction of the applied electric field.

Fig. 2.28. Normalized transmissions of light traveling along the +X (forward wave) and the −X-axis (backward wave) in MgO:PPLN.

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Comparing the square and round dots in Figure 2.28 (a) and (b), we find that they are almost of the same shape, which reveals that light experiences the same rotation process, verifying that optical propagation is reciprocal in MgO:PPLN. The small shift between these two curves is probably caused by the temperature fluctuation during the experiment, which induces the variation of the QPM wavelength. The electric field where the maximum and minimum transmissions occur is actually larger than the theoretical expectation due to several reasons. The overriding reason is that the external electric field is generated by use of a pair of parallel copperplates, which requires extreme closeness to the MgO:PPLN crystal. During the experiment, we found that by slightly pressing the copperplates toward the MgO:PPLN crystal, less external electric field was required to obtain the same rotation angle. However, we did not perform strong pressure on them and the pressure was different for each time, which resulted in incomplete closeness between the PPLN sample and the copperplates and consequently led to the discrepancy. Second, the 45° dextrorotatory quartz we used is fabricated at the wavelength of 1550 nm. When the wavelength is 1568.5 nm, the rotation angle is 43.88°, which can be obtained from the formula φ = α ∗ d, where φ is the rotation angle, α represents the specific rotation, and d stands for the thickness of quartz. Thereupon, the shift of the rotation angle from the quartz gives rise to a shift in the applied electric field from the theoretical anticipations. The temperature fluctuation mentioned above is also responsible for the shift. In summary, we demonstrate the chirality control of MgO:PPLN by the external electrical field. The MgO:PPLN can be dextrorotatory or levorotatory under the transverse electric field applied along different directions, which makes it an optically active material with multipurpose.

2.4.4 Optical isolator based on the electro-optic effect in HPPLN In nature, there are many materials having optical rotation. But, most of them cannot be used to realize optical isolation, because rotation cannot be accumulated. Here we propose an optical isolator based on the EO effect in PPLN crystal. A domain with half the domain thickness is equivalent to a quarter-wave plate in the same manner as a complete domain which is equivalent to a half-wave plate. Linearly polarized light becomes circularly polarized after passing through a quarterwave plate, and vice versa. Meanwhile, light passing through a quarter-wave plate twice has the same effect as that of light passing through a half-wave plate once. Thus, an additional domain with half the domain thickness is added to the normal PPLN (the positive and negative domains appear as a pair); this newly formed PPLN with an additional half-domain is henceforth referred to as HPPLN. As shown in Figure 2.29, the polarization direction of the reflected light, incident on the PPLN in the backward

148 | Xianfeng Chen

Fig. 2.29. Schematic diagram of the optical isolator based on the EO effect of HPPLN. A positive halfdomain is added to the normal PPLN to form HPPLN. Under the QPM condition, each domain serves as a half-wave plate, and the half-domain serves as a quarter-wave plate.

Fig. 2.30. Polarization evolution process of light in HPPLN. Y and Z represent the principal axes of the index ellipsoid, and P represents the polarization state of light. The incident light satisfies the QPM condition.

pass, has changed by the time it is transmitted out of the PPLN in the forward pass. Optical rotation is therefore accumulated. Figure 2.30 shows a detailed description of the polarization evolution process of light in HPPLN. Specifically, when the electric field is applied along the −Y-axis, for incident light with an azimuth angle of y o = xθ, the light undergoes right rotation in the forward pass; the azimuth angle, after passing through N periods, is yin = (4N + x)θ. The light becomes circularly polarized after the half-domain and transmits out. On the other hand, the reflected light becomes linearly polarized with an azimuth angle of −(4N + x + 2)θ after the half-domain. It then undergoes right rotation and transmits out with an azimuth angle of yref = −(8N + x + 2)θ. The angle between the reflected and incident light can be easily calculated as |yref || − yin | = 8Nθ + 2θ. According to Equation (2.21), under a suitable electric field, it is easy to make the reflected light perpendicular to the incident light, i.e. |yref | − |yin | = 8Nθ + 2θ = 2π , 3π 2 . . .. The reflected light is thus blocked. Interestingly, the angle between the reflected and incident light |yref | − |yin | is independent of the incident azimuth angle y o = xθ, which implies that the isolation effect is independent of the incident azimuth angle. Thus, the isolator can be employed for all linearly polarized light.

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Fig. 2.31. (a) Theoretical transmittance of the incident light (solid line) and reflected light (dashed line) and (b) isolation contrast, as a function of the external electric field. The QPM wavelength is 1550 nm.

For +Z polarized light, the transmittance of the incident and reflected light along the +Z-axis is Tin = cos2 (4Nθ) and Tref = cos2 (8Nθ + 2θ), respectively. For HPPLN that consists of 3650 domains, transmittance of the incident and reflected light at room temperature can be calculated as a function of the external electric field, as shown in Figure 2.31 (a). It is obvious that under a suitable electric field, such as E = 0.5 kV/cm or E = 1.5 V/cm, the transmittance of the reflected light can be zero, i.e. a total block of the reflected light. To provide a straightforward view of the isolation effect, we define the isolation contrast as C = (Tin − Tref )/(Tin + Tref ). Figure 2.31 (b) shows the contrast ratio versus the external electric field. The contrast ratio is smoothly tuned from 1 to −1, with the increment of the electric field, and is equal to 1 under a suitable electric field where complete optical isolation occurs. In contrast to the early attempts to realize nonmagnetic isolators based on photonic crystals, where isolation is only achievable for strong optical intensity or restricted polarization states, the optical rotation studied here is linear with respect to the incident light and depends on the electric field. By properly controlling the external electric field, it can be used in a weak-light system for all linearly polarized light. In the waveguide configuration, the width of HPPLN can be as small as 10 μm, that 1 V is enough to make the polarization rotate by 45°, which is very attractive.

2.4.5 Polarization based all-optical logic gates The all-optical network means the transmission and switching of information flow always exist in the form of light, without electrical-to-optical and optical-to-electrical conversions [43]. Therefore, it has good transparency, wavelength routing features, compatibility and scalability. The present all-optical network is not the whole of the optical network, but rather refers to the light transmission and exchange of informa-

150 | Xianfeng Chen tion in the form of the presence of light, and it is realized by circuit control section. In the past decade, all-optical switching and all-optical signal processing have attracted many scholars participated in the study. With the unprecedented rapid development of photonic technology, all-optical signal processing is more feasible than ever a time. All-optical signal processing, including light switches, judgments, regeneration and calculation, and all-optical logic gate is one of the key components. Except all-optical networks, all-optical logic device is also the basis of optical computing. Based on amplification and suppression, three basic logic operations of “AND,” “OR” and “NOT” could be realized. Adder, bidirectional oscillator, monostable and bistable flip–flops and other logic devices could be made to achieve optical computing. Photon propagation speed is 3 × 108 m/s, which is 500 times the speed of propagation of the electron, therefore, photonic computer has ultrahigh speed of operation. The computing speed of a photonic computer can reach 1023 times/s theoretically. Computing speed can be achieved technically 1012–1015 times/s and 100 Gb/s transmission capacity [44]. Although the photonic computer has not come into existence, but with the development of photonics and photonic technology, people have recognized the enormous potential of photonic computers. Optical logic development is the basis to achieve all-optical data exchanging. Therefore, the development of alloptical logic gates will have far-reaching implications for the future in terms of alloptical network optical packet switching, optical transport, and optical computing. Based on electro-optic Pockels effect of PPLN, different polarization-based binary all-optical logic functions, controlled-NOT, XOR, and XNOR gates are demonstrated. The experimental setup is shown in Figure 2.32. We selected the QPM wavelength 1540.8 nm as operating wavelength, at a given experimental temperature about 26 °C. In the experiment, we set polarizer parallel to the Y-axis, so that the input optical signal is of a horizontal polarization state and the analyzer parallel to the Z-axis, as

Fig. 2.32. Experimental setup for all-optical polarization-based logic gates.

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Fig. 2.33. Two schematic configurations with different polarization in Figure 2.33 (a) and (c), which have the similar performances in Figure 2.33 (b) and (d).

shown in Figure 2.33 (a). In Figure 2.33 (c), similarly, we set polarizer parallel to the Z-axis, so the input optical signal is of a vertical polarization state and the analyzer parallel to the Y-axis. At last, we measured the change of T ‖ and T⊥ with the increment of transverse external electric field E from 0 to 4.5 kV/cm, as shown in Figure 2.33 (b) and (d). The experimental results have shown that the polarization state of output optical signal could be switched between horizontal and vertical polarization state when the transmission T ‖ and T⊥ got their maximum at the applied electric fields, of 3.6 and 3.9 kV/cm, respectively. In our experiment, there was a difference of the intensity between the input optical signals with different polarization states in Figure 2.33 (a) and (c). Then the variation of the transmission T in Figure 2.33 (d) was not evident when the external electric field is small, because of the limitation by the sensitivity of measurement instrument. Table 2.2. Experimental results. Electric field E (kV/cm)

Input polarization

Input intensity (μW)

Output polarization

Output intensity (μW)

T (%)

3.14 2.71

→ ↑

120 500

↑ →

106 480

88.4 96.0

152 | Xianfeng Chen By optimizing the experimental conditions, we got the maximum transmission T ‖ and T⊥ about 88.4% and 96.0%, respectively, as shown in Table 2.2. The maximum transmission T was relative large, which means that the depletion of polarization encoded signal induced by coupling and propagating process was very small and could be neglected. Because of the insertion and propagation loss of optical signal in bulk device of PPLN, and also due to the reflection on the incidence planes of used optical devices, the measured transmission T in our experiment was smaller than theoretical results. Here we discuss how to realize all-optical logic gates.

1. Controlled-NOT According to the above experimental results, we can realize the controlled-NOT gate by representing two orthogonal linear polarization states of optical signal as logic 0 and logic 1, as is shown in Figure 2.34 (a). When the applied electric field is on, the input optical signal can be switched between the two orthogonal linear polarization states; otherwise, it can keep its polarization state. Because the logic NOT function can be obtained by switching ON or OFF the applied electric field, it is called the controlledNOT gate.

2. XOR and XNOR In order to obtain XOR and XNOR gates as shown in Figure 2.34 (b), we give the definition of the input and output signals for XOR and XNOR. Above all, we choose the input optical signal and the applied voltage as two binary logic signals, with the presentation shown in Table 2.2. For XOR gate, the input optical signal with horizontal polarization state as signal 1 is logic 0; and that with vertical polarization state is logic 1. At the same time, we suppose the applied voltage as signal 2, with logic 0 and logic 1, which corresponds to the electric voltage 0 and V (i.e., E is not equal to 0), respectively, while for XNOR logic gate, the definition of logic 0 and logic 1 is contrary to that for XOR gate. Accordingly, based on the experimental results in Table 2.3 and the definition in Table 2.2, we obtain the truth tables of our XOR and XNOR gates as shown in Table 2.3. The presentation of output optical logic signals is the same as that of Table 2.3. Presentation of signals for XOR and XNOR gates. XOR

Signal 1 Signal 2 Output

XNOR

Logic 0

Logic 1

Logic 0

Logic 1

→ 0 →

↑ V ↑

↑ V ↑

→ 0 →

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Fig. 2.34. Schematic diagram of logic gates: (a) controlled-NOT gate and (b) XOR/XNOR gate. Table 2.4. Experimental results and truth table for XOR and XNOR. Experiment Signal 1 Signal 2 Output

→ 0 →

→ V ↑

↑ 0 ↑

XOR ↑ V →

0 0 0

0 1 1

1 0 1

XNOR 1 1 0

1 1 1

1 0 0

0 1 0

0 0 1

the input optical logic signals, and the output logic signals are totally identical to the truth table of XOR and XNOR gates. By cascading the controlled-NOT gate above, it is convenient to switch the function between XOR and XNOR gates by switching the presentations of input optical signal in Table 2.2.

3. AND and OR In order to obtain AND and OR gates as shown in Figure 2.34 (c), we give the definition of the input and output signals for AND and OR, with the presentation shown in Table 2.4. We obtain the truth tables of our AND and OR gates as shown in Table 2.5. The presentation of the output optical logic signals is the same as that of the input optical logic signals, and the output logic signals are totally identical with the truth table of AND and OR gates. By cascading the controlled-NOT gate above, it is convenient to switch the function between AND gate and OR gate by switching the presentations of input optical signal in Table 2.6.

154 | Xianfeng Chen

Fig. 2.34c. Schematic diagram of AND/OR gates. Table 2.5. Presentation of signals for AND and OR gates. AND

Signal 1 Signal 2 Output

OR

Logic 0

Logic 1

Logic 0

Logic 1

→ V →

↑ 0 ↑

↑ 0 ↑

→ V →

Table 2.6. Experimental results and truth tables for AND and OR. Experiment Signal 1 Signal 2 Output

→ 0 →

→ V →

↑ 0 ↑

AND ↑ V →

0 1 0

0 0 0

1 1 1

OR 1 0 0

1 0 1

1 1 1

0 0 0

0 1 1

In summary, by utilizing electro-optic modulation of polarization in PPLN, we propose an approach to demonstrate binary all-optical polarization-based logic gates, including controlled-NOT, XOR, XNOR, AND, and OR. Through processing optical signal encoded in the polarization state, where the intensity of the optical signal itself carries no information, this system is more suitable for cascading gates infinitely to implement some complex Boolean functions. Additionally, the realization of all-optical logic gates in PPLN enriches its unique advantage, which can be as a compact onechip integrating multiple optical processing functions such as optical routing [45] and optical convertor [46, 47], etc. At last, PPLN crystal also has a wide range of applications in quantum communication; the report of single-photon polarization-control based on electro-optic effect in communication band (around 1550 nm) has attracted wide attention [48]. On the other hand, the related report of Jin Yun’s group of Shanghai university [49] shows that the all-optical logic gates based on PPLN can play a greater role in the development of future optical computers.

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2.5 Control of angular momentum and its applications 2.5.1 The angular momentum of light In 1909, Poynting proposed the concept that light has angular momentum (AM) after investigating the rotation character of circularly polarized light, and pointed out that it is concerned with the polarization [50]. In 1936, Beth from Princeton university used mechanical experiment to prove that the AM of left-handed photon is ℏ, while the AM of right-handed photon is −ℏ [51]. In 1992, Allen of Leidon University theoretically predicted the existence of orbital AM of photons, and then AM of light has become a hot spot of optics research [52]. Current research of orbital AM is mainly focused on two aspects. One is the field of optical tweezers [53–55]. Light has AM, when particle interacts, light could control the particle by transferring the AM, which could be used to realize optical spanner. Spin AM could be used as the driving force of optical wrench to make the particles around their own axis, while orbital AM could be used as the driving force wrench to make the particles rotate around the axis of the beam [56]. This kind of optical wrench has been used to drive the metal particles [57], anisotropic dielectric sphere [54, 58], particles having an absorption character and some biological macromolecules [59]. The other important application is the modern quantum information technology based on quantum entanglement [60]. Spin AM of light has two eigenstates which are similar to the two polarization states of light. It could be used in two-dimensional quantum communication systems. In 2001, Zeilinger from Vienna University observed that photons with spontaneous parametric down conversion can achieve high dimensional orbital AM entanglement, which could be used to encode quantum information, and improve information capacity [61, 62].

2.5.2 The evolution of polarization state of light by changing electric field In the last section, we have demonstrated a linear polarization-state generator for the QPM wavelengths; here we filled the vacancy of the investigation of the wavelengths disagreeing with the QPM condition (we call it the NQPM wavelengths) [63]. We experimentally discovered that for the NQPM wavelengths the polarization follows successive continuous but discrete paths on the Poincare sphere with increasing electric field. However, for the QPM wavelengths, the discrete paths suddenly degenerate and are parallel to the equator plane. To get an insight into the behavior of the NQPM wavelengths, a polarization coupled-mode theory is established to track the polarization state of light propagation along PPLN. At the output of PPLN, the solutions

156 | Xianfeng Chen

Fig. 2.35. Theoretical results of the degenerated paths of the QPM wavelength.

of the coupled-wave equations of the ordinary and extraordinary waves are given by {A1 (L) = exp[i(Δβ/2)L] {[cos(sL)−iΔβ/(2s) sin(sL)]A 1 (0)−i(κ/s) sin(sL)A2 (0)} { A (L) = exp[−i(Δβ/2)L] {(−iκ ∗ /s) sin(sL)A1 (0)+[cos(sL)+iΔβ(/2s) sin(sL)]A2 (0)} , { 2 (2.115) 2 2 ω n o n e γ 51 E y i(1−cos mπ) where, Δβ = (k 2 − k 1 ) − G m , G m = 2πm/Λ, κ = − 2c (m = 1, 3, 5. . .), mπ √n n o e

s2 = κκ ∗ + (Δβ/2)2 ; where A1 and A2 are the normalized amplitudes of the ordinary and extraordinary waves, respectively, k1 and k 2 are the corresponding wave vectors; G m is the mth reciprocal vector corresponding to the periodicity of poling, L is the length of PPLN, Λ is the period of PPLN; no and ne are the refractive indices of ordinary and extraordinary waves, respectively, γ51 is the electro-optical coefficient, and E y is the electric field intensity. Specifically, for the QPM wavelength, the output polarization state will periodically change with |s|L, resulting in the degenerated paths of evolution along the Poincare sphere (shown in Figure 2.35). But for the NQPM wavelength, the evolution of the polarization state starts and ends at the same polarization state (the initial polarization state) but experiences different paths along the Poincare sphere. Thereby, when the evolution is getting across the initial point each time and about to enter a new round of period, the new coupling coefficient κ will provoke the evolution to bypass the pretrack into a different path and eventually give birth to the interesting phenomena of split paths along the Poincare sphere (shown in Figure 2.36). Figure 2.35 presents the numerical results using the Jones method for the QPM wavelengths, which reveals that the evolution of the polarization states periodically circles along a degenerated path with the external electric field, showing agreement with the analysis based on the coupled-mode theory. For the input light linearly polarized (A–C), the path is always right in the equator, which means that each linear polarization state on the path comprises a group, in which the element can be transformed to any one included in the same group by modulating the external electric

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Fig. 2.36. Theoretical results of the discrete paths of the NQPM wavelength.

field. Besides the group of linear states of polarization, distinctive other groups can also be discovered on the Poincare sphere. There, D and E, the left- and right-hand circular states of polarization are two groups with only one element, respectively, which means that the path on the Poincare sphere is just a point, and F (or G) is an arbitrary elliptical polarization state whose path is parallel to the equator plane. For the linear polarization-state group, we are capable of designing devices behaving as the linear polarization-state generator, and for other groups, it is also interesting to achieve orthogonal elliptical states of polarization such as states F and F 󸀠 , which can be used in the satellite communication for doubling the capacity of the services. It is interesting to note that in an input light with a specific polarization state of one group, if the incident state undergoes mutation but still belongs to the same group, the polarization state of the output light can still be pulled back with the external electric field. However, if the mutation makes the incident state jump into another group, it will never be restored but circle along another path forever. These different groups with different

158 | Xianfeng Chen paths can be compared to the so-called energy levels along the Poincare sphere and the behavior of the path shift can be considered as the energy transition by analogy. Then we move to investigating the NQPM wavelengths. Figure 2.36 reveals that the polarization evolution of such wavelengths does take on the discrete paths along the Poincare sphere and each path corresponds to a specific quantum number. Consider an incident light with state A, when no electric field is applied, the output state is situated at point A (Figure 2.36 (a)). Then with the increment of the electric field, the output state begins to circle along the inner path (n = 1) and when passing by point A second time, the evolution alters the way and jumps into another path (n = 2) and the rest may be deduced by analogy. The phenomena suggest that by modulating the electric field and altering the incident states the Poincare sphere can be basically covered, which may play a vital role in the test of the polarization-dependent loss. Uniquely, we discover that, when the incident state is the C state, the output state can be shifted between dual linear states of polarization, which is attractive when designing devices that function as the broadband electro-optical switch or laser-Q switch. Besides, for incident light circularly polarized, the state of the output light periodically shifts between circularly and linearly polarized, which can be used for precise quantum coding. To make conclusions more convincing, we also verified them with experiments. Figure 2.37 is the experimental setup for investigating the evolution of the polarization. The applied electric field is modulating from −0.3 to 0.3 V/μm. By the use of the λ/4 plate and the analyzer the polarization state of the output light can be determined. Experimental measurements for a given wavelength are shown in the Poincare sphere in Figure 2.38. The dissimilarity between Figure 2.38 (a) and (b) is the different incident states of polarization; Figure 2.38 (a) is for light linearly polarized along the Z-axis (state A) and Figure 2.38 (b) is for light linearly polarized along the Z-axis (state B). By changing the sign of the electric field from Y to −Y-axis, the path correspondingly shifts from the upper sphere (left-handed spin) to the lower sphere (right-handed spin) by virtue of the change of sign of κ, which is coherent with the state of the output light. The mere deviation between the experiment and numerical results is the differ-

Fig. 2.37. Experimental setup for investigating the polarization behavior of light.

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Fig. 2.38. Experimental results of the polarization state evolution with the external electric field for both the QPM and NQPM wavelengths.

ent starting points (point O) along the Poincare sphere, which results from the inner electric field existed in such a periodically poled ferroelectric domain structure that may be caused by the strain-optic effect produced in the process of polarization or the photovoltaic effect engendered by the input light.

2.5.3 Control of spin angular momentum of light As with other physical realities, the light also carries energy, linear momentum (LM), and angular momentum (AM). Generally, the AM of the light field could be separated into the orbital angular momentum (OAM) and spin angular momentum (SAM). OA is related to the helical phase fronts of light beams. While the SAM is associated with circular polarization and arises from the spin of an individual photon, the light at the output facet of PPLN can be expressed as the superposition of the LHCP and RHCP ones or the superposition of the ordinary and extraordinary ones: E1 (z) 1/√2 1/√2 ⃗ E(z) =[ ] = {Elef (z) [ ] + Erig (z) [ ]} , E2 (z) −i/√2 i/√2

(2.116)

we obtain Elef = { [cos (sL) + (−iΔβ + 2κ ∗ exp (−iΔβL) √n1 /n2 ) sin (sL) / (2s)] E1 (0) + [(−2iκ √ n2 /n1 − Δβ exp (−iΔβL)) sin (sL) / (2s) + i cos (sL) exp (−iΔβL)] E2 (0)}√2/2 exp [i (Δβ/2) L]

160 | Xianfeng Chen Erig = { [cos (sL) + (−iΔβ − 2κ ∗ exp (−iΔβL) √n1 /n2 ) sin (sL) / (2s)] E1 (0) + [(−2iκ √n2 /n1 + Δβ exp (−iΔβL)) sin (sL) / (2s) − i cos (sL) exp (−iΔβL)] E2 (0)}√2/2 exp [i (Δβ/2) L]

(2.117)

According to the quantum theory, the energy of each photon is ℏω, so the numbers of LHCP and RHCP photons transmitted at the output surface per unit area per second are, respectively, average Poynting energy flow divided by Nlef (z) = cε0 |Elef (z)|2 /2ℏω, and Nrig (z) = cε0 |Erig (z)|2 /2ℏω. Each LHCP photon contains the AM of ℏ and the RHCP one −ℏ. The total SAM is hence given by [64] M(z) =

󵄨 󵄨2 cε0 (|Elef (z)|2 − 󵄨󵄨󵄨Erig (z)󵄨󵄨󵄨 ) 2ω

.

(2.118)

When A1 (0) = 1, A2 (0) = 0, if κ = 0, we can get M(z) = 0. If κ ≠ 0, the SAM oscillates in the PRS. When Δβ = 0, incident light contains linear polarization, and M(z) = 0: when Δβ > 0, beam consists of more LHCP, and M(z) ≥ 0: when Δβ < 0, beam consists of more RHCP, and M(Z) ≤ 0. The SAM of lights with different wavelengths at a given distance (z = 2.1 cm) was also investigated and has been shown in Figure 2.39 (a). The period of the PPLN is

Fig. 2.39. SAM as a function of some parameters.

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21 μm. At a temperature of 20 °C, the wavelength that satisfies Δβ = 0 is calculated as 1540 nm by aid of smaller equation. Figure 2.39 (a) indicates that the normalized SAM can be transferred from −1 to 1 by adjusting the wavelength in the vicinity of the QPM condition (Δβ = 0), which can behave as a controllable “spanner.” Light with wavelength beyond 1540 nm can rotate a nanoscale particle clockwise, while light with wavelength below 1540 nm can exert an anticlockwise rotation on the particle. Thus, by modulating the wavelength from above 1540 nm to below 1540 nm, the normalized SAM oscillates from 1 to −1, and hence, a particle spins from clockwise to anticlockwise with different AM. Figure 2.39 (a) also suggests the QPM condition can be detuned by temperature, and so does the SAM. About 2 °C is enough to realize the transfer of normalized SAM from 1 to −1. Figure 2.39 (b) shows that the SAM is also modulated with the applied electric field when Δβ ≠ 0, and about 2 kV/cm is able to continuously completely transfer the SAM from −ℏ to ℏ. Because the SAM can be controlled precisely by adjusting operating wavelength or temperature as discussed, it takes lots of convenience over the traditional method such as by rotating a one-fourth wave plate mechanically. The normalized LHCP and RHCP photon numbers and the total SAM of the output light with the incident light linearly polarized along the Y-axis are shown in Figure 2.40. From Figure 2.40 (a) and (b), we find that the distributions of the LHCP and RHCP photon numbers are both centrosymmetric, and their superposition will lead to a unitary value 1. Figure 2.40 (c) shows that the total SAM varies with the operating wavelength and external electric field. It is very interesting to notice that some regions exist, which possess the same sign of SAM (A and B), and we call it the spin cell. It is worth emphasizing that the transition of SAM is consecutive, which means that the continuous manipulation of SAM and dynamic control of particles are available by changing the external electric field and operating wavelength. The phase modulators on the market, mainly based on the longitudinal electro-optic effect of lithium niobate crystal, can also be used as SAM controllers. However, the external electric field applied on the lithium niobate monodomain crystal for the modulation of SAM of light is about 7.23 kV/cm, which is much higher than the 0.44 kV/cm in the PPLN crystal with the same length. The electric field needed in our method is also much lower than that in the Chen et al. article, about 66 kV/cm with crystal as long as 1 cm. Because the PPLN consists of thousands of cascaded waveplates with micrometer thickness, it has advantages of being compact and requiring ultralow voltage, compared with the conventional multiple bulk crystalline waveplates, i.e. lithium niobate crystal waveplates. And the modulation of the wavelength is a convenient means for SAM management [65]. The investigations on manipulation in the microscale and spin-orbital AM entanglement based on this approach are necessary in the future.

162 | Xianfeng Chen

Fig. 2.40. Normalized photon numbers of (a) LHCP light, (b) RHCP light, and (c) the total SAM of the output light.

2.5.4 Control of orbital angular momentum and its applications Light beam may carry both SAM and OAM [66]. The SAM is associated with circular polarization and arises from the spin of individual photon with a value of ℏ or −ℏ, for the left- or right-handed circular polarized light, respectively [51]. In 1992, OAM arises from the spiral phase distribution at the wavefront of a beam [52]. The helical phase structure of light, commonly called as an optical vortex, is described by a phase cross section of exp(ilθ). A Laguerre–Gaussian beam is a very typical optical vortex field in the paraxial approximation, the solution of the Helmholtz equation is 1 ∂ ∂E ∂E 1 ∂2 + 2ik (ρ ) + 2 =0. ρ ∂r ∂r ∂z ρ ∂φ2

(2.119)

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Fig. 2.41. Laguerre–Gaussian beam intensity and phase distribution with different values of l and p.

Laguerre–Gaussian modes of a mathematical expression for the complex amplitude is E lp (r, θ, z) =

l

d lp 1/2

(1 + z2 /z2R ) exp [

−r2 ω2

(

2 r√2 ikr2 |l| 2r ) L p ( 2 ) exp [−ilθ] exp [ ] 2 ω ω 2 (z2 + z2 ) R

] exp [−i (2p + |l| + 1) ϕ] tan−1 (

z ) , zR (2.120)

kω20

where ϕ = tan−1 ( zzR ) is the Guoy phase, z R = 2 is the Rayleigh length, ω0 is Waist ω0 radius, ω = cos(ϕ) is the beam radius at z, R = z2 is the radius of curvature at z, |l|

sin ϕ

d lp is the normalization factor, L p is the Laguerre polynomial, and l and p are the characteristic quantum numbers. Radial quantum number p represents the number of concentric rings of the beam cross-section (p + 1), angle (phase) quantum number l represents the phase singularity of the order, which is the physical meaning of revolution around the singularity, the phase change 2lπ. The phase factor means that the light is a kind of beam with a spiral structure. (a) shows the Laguerre–Gaussian beam intensity distribution with different values of p and l, while (b) shows the phase distribution of the light beam with different values of l. The beam AM density vector is M = ε0 r × E × B ,

(2.121)

where E and B are the electric field and magnetic induction field strength, respectively, and ε0 is the permittivity of vacuum. The total AM of the light field is J = ε0 ∫ r × (E × B) dr

(2.122)

164 | Xianfeng Chen and J =L+S

(2.123)

in Lorentz norms, linearly polarized vector potential A can be expressed as A = u (x, y, z) exp (ikz) x⃗ .

(2.124)

Based on the Lorentz specification approximation it can be obtained i ∂u ⃗ exp (ikz) z] k ∂y i ∂u ⃗ exp (ikz) . z] B = μ 0 = ik [u y⃗ + k ∂y E = ik [u x⃗ +

(2.125) (2.126)

Finally, we get the Poynting vector ε0 E × B: p=

ε0 ∗ ε0 (E × B + E × B∗ ) = iω (u∇u ∗ − u ∗ ∇u) + ωkε0 |u|2 z⃗ . 2 2

(2.127)

Then we can obtained the beam AM component along the z-axis direction j z = (r × ε0 ⟨E × B⟩)z = rε0 ⟨E × B⟩θ .

(2.128)

For optical vortex, its complex amplitude of the light field can be expressed as u (r, θ, z) = u 0 (r, z) exp (ilθ) .

(2.129)

Put Equation (2.129) into Equation (2.127), we get p θ = ε0 ⟨E × B⟩θ = [iω

ε0 (u∇u ∗ − u ∗ ∇u) + ωkε0 |u|2 z]⃗ = ωε0 l |u|2 /r 2 θ

(2.130)

and put it into Equation (2.128), we get j z = (r × ε0 ⟨E × B⟩) z = cωε0 k |u|2 .

(2.131)

We know that the light field energy density along the propagation direction z is w = cp z = cε0 ⟨E × B⟩z = cωε0 k |u|2

(2.132)

finally we get l jz = . w ω The ratio of AM and energy per unit length is ∬ r × ⟨E × B⟩z rdrdθ J l lℏ = = = . W ω ωℏ c ∬ ⟨E × B⟩z rdrdθ

(2.133)

(2.134)

Research on vortex physics has a great significance in physics and scientific applications such as light control, single cell cure, high-density optical storage, quantum information processing, quantum coding, therefore, this aspect of research has attracted

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more and more attention [53, 60–62, 67, 68]. In addition, the vortex particle beam can imprison “low refractive index” particles [69]. In 2001, Zeilinger from Vienna University observed that photons with spontaneous parametric down conversion can achieve high-dimensional orbital AM entanglement, which could be used to encode quantum information, and improve information capacity [70]. Spiral phase plate (SPP) is a transparent plate with refractive index of n0 . Its thickness is proportional to the center of the phase plate azimuth angle θ, the surface structure are flat and spiral, the spiral surface is similar to a rotating stage, with step height of h. Transmitted beam will have a spiral phase characteristics by passing thought the SPP. Assume that the complex amplitude of the incident light is u(r, θ, z), and the complex amplitude of emergent light is u 󸀠 = u exp(−iΔlθ), here Δl = Δn0 h s /λ [72]. When a light beam passes through the transparent modulator, the difference of phase is (n 0 −1)h s φ(θ) = 2π θ], and the vortex beam generates. 2π λ [ Compuer generated hologram (CGH) is an effective method for the generation of optical vortices. Computer is used to generate the object and reference lights’ interference pattern, then the pattern is written to form holographic grating or printed directly. In 1992, laser beams that contain phase singularities can be generated with computer-generated holograms, which is the simplest case of the form of spiral Fresnel zone plates [73]. Under paraxial conditions, the optical vortex in polar coordinates with the topological charge of 1 is expressed as E (r, θ, z) = E0 r exp (iθ) exp (2 ln

kA/2 r2 + iπ − ) . z + ikA/2 A + 2z/ik

(2.135)

When φ = 0, the intensity distribution is I (r, θ, z) = E20 + [E0 r exp (p)]2 + 2E20 r exp (p) cos (Φ) ,

(2.136)

where p = ln

(kA/2)2 2

− r2

A

+ (kA/2) + (2z/k)2 2z/k Ak Φ = θ − r2 − arctan ( ) + π . 2 2 2π A + (2z/k) z2

A2

The great intensity fringes condition is cos(Φ) = 1 or Φ = 2nπ, n = 1, 2, 3 . . . . Finally, we get the interferometer pattern of holographic grating, and the spiral direction is shown by the symbols of exp(iθ). When the spiral wave is coherent with plate wave at an angle of φ, the maximum condition for the interference fringe is arctan (y/x)−(x2 + y2 )

2z/k A2 + (2z/k)2

−arctan (Ak/2z)−kx sin(2φ)−2kz sin φ+π = 2nπ. (2.137)

166 | Xianfeng Chen Some nonlinear processes such as SHG [76, 77], parametric down conversion [78, 79], can also be used to control the optical vortex of topological charge. Bahabad et al. proposed that optical vortices can be generated from the base mold without vortex [80]. Here a kind of voltage-controlled optical vortex converter is proposed in a helical-periodically poled ferroelectric crystal, for example, helical-periodically poled LiNbO3 (HPPLN), marked with a TC l󸀠 . When the incident optical vortex with TC l is ordinary light, thanks to the transverse electro-optics effect, the external electric field can add the TC of crystal to extraordinary light, identified by l + l󸀠 ; hence, the HPPLN works as an optical vortex adder. On the other hand, the HPPLN can also work as an optical vortex substractor for the extraordinary incidence, and the TC of the output ordinary light is identified by l − l󸀠 . If there is no external electric field, the TC of output optical vortex remains l. Meanwhile, the optical vortex coming from HPPLN can be served as a voltage-controlled optical spanner. Assuming that the input light is an ordinary light (this could be achieved by putting a horizontal polarizer in front of the HPPLN), the initial condition at x = 0 is given by A1 = exp(ilθ), A2 = 0. When the QPM condition is satisfied (Δβ = 0), the solution can thus be simplified into { A1 (L) = cos (󵄨󵄨󵄨󵄨κ q 󵄨󵄨󵄨󵄨 L) exp (ilθ) { A (L) = −i exp (i (l󸀠 + l) θ) sin (|κ ∗ | L) . { 2

(2.138)

From Equation (2.138), we can see that the output extraordinary light (A2 ) possesses both the information of ordinary light and structure of material. The condition that l󸀠 = l = 0, where the incident light is plane wave, and the nonlinear material is normal PPLN, has extensively investigated; in this case, Equation (2.138) is reduced to that in PPLN [81–83]. In the following, without the loss of generality, the length of HPPLN is fixed as 2.1 cm and the period is fixed as 21 μm. We also set m = 1, corresponding to a QPM wavelength of 1.540 μm, and = 32.6 pm/V for LiNbO3 crystal. Without the external electric field, there is no coupling between ordinary light and extraordinary light, and the output light is still the ordinary one with its original TC. However, when the external electric field is applied on HPPLN whose TC is l󸀠 , the situation changes. Figure 2.42 (a) shows the phase distributions of incident ordinary light and the output extraordinary light. The first row shows the phase distributions of incident ordinary lights with different TC identified by l. The second, third, and fourth rows show the phase distribution of output extraordinary light from HPPLN with l󸀠 = 1, 2, 3, respectively. We can see that the TC of output extraordinary light is given by l + l󸀠 . Hence, we achieve a kind of optical vortex adder controlled by the external electric field, through which the helical poled property of material could be added to the light. We note that this method can be used to generate optical vortex from nonvortex beam and change the TC of already present optical vortex.

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Fig. 2.42. Phase distributions of incident ordinary light and output extraordinary light with external electric field applied. The HPPLN works as an optical vortex adder.

If the input light is an extraordinary light which can be realized, for example, by putting a vertical polarizer in front of the HPPLN, the initial condition at x = 0 is given by A1 = 0, A2 = exp(ilθ) . When the QPM condition is satisfied, the solution is be simplified to {A1 (L) = −i exp (i (l − l󸀠 ) θ) sin (󵄨󵄨󵄨󵄨κ q 󵄨󵄨󵄨󵄨 L) (2.139) { 󵄨 󵄨 A2 (L) = cos (󵄨󵄨󵄨κ q 󵄨󵄨󵄨 L) exp (ilθ) . { From Equation (2.139), one finds that the output ordinary light possesses the information of incident extraordinary light and the structure of material. Figure 2.43 plots the phase distributions of incident extraordinary light and output ordinary light. The first row shows the phase distributions of incident extraordinary lights with different TC identified by l. The second, third, and fourth rows show the phase distribution of output ordinary light from HPPLN with l󸀠 = 1, 2, 3, respectively. One finds that the TC of output ordinary light is determined by l − l󸀠 . Hence we achieve a kind of optical vortex substractor controlled by external electric field, through which the helical poled property of material could be used to substract the TC of the incoming light. The coupling between ordinary and extraordinary light is controlled by external electric field. When the ordinary light is launched into the HPPLN, the amplitude of extraordinary light coupled from ordinary light is controlled by external electric

Fig. 2.43. Phase distributions of incident extraordinary light and output ordinary light with external electric field applied. The HPPLN works as an optical vortex substractor.

168 | Xianfeng Chen

Fig. 2.44. (a) The normalized light intensity of output ordinary and extraordinary lights controlled by external electric field. (b) The averaged OAM of output light beam passing through the HPPLN with different TC controlled by external electric field.

field, as shown in Figure 2.44 (a). We can see that, when the external electric field rises to 0.831 kV/cm, the ordinary vortex light with TC l can be fully transferred to extraordinary vortex light with TC l + l󸀠 . Although the OAM of each photon in extraordinary light is given by (l + l󸀠 )ℏ, for the total light beam, the averaged OAM is given by (N0 l0 ℏ + N g l g ℏ)/(N0 + N g ), where N0 and N g are the numbers of extraordinary photons and ordinary photons, respectively. Note that the averaged OAM can be controlled by the external voltage, as shown in Figure 2.44 (b), where the dependence of the averaged OAM on the external electric field is plotted for the incoming vortexless beam. One observes that the averaged OAM of the resulting light continuously increases with the increase of external electric field, and a maximum value of averaged OAM is achieved under a suitable electric field when the whole incoming beam is fully transferred into extraordinary beam. Thus, our system can be used in a highly tunable optical spanners where the torques due to the transfer of OAM of photons into particles can be controlled by the external field. Experimentally, a possible solution for constructing such HPPLN is by lapping and polishing electric-field poled ferroelectric materials into thin χ(2)-modulated planar plates, which has been reduced to the thickness as thin as 6.2 μm, and stacking them together. Compared to other means of nonlinear modulation to optical vortex [80], the intensity of optical vortex in our scheme could be much lower because the coupling between ordinary light and extraordinary light is just determined by external electric field. Hence this method can be used to modulate low intensity light or single photon in a high-dimensional quantum communication system [84] as an optical vortex modulator. Since our method is based on the electro-optics effect, it can operate accurately and stably at a high speed up to a multigigahertz region [85]. In addition, due to their different polarized directions, the output optical vortices with different TC are readily to be separated with the help of polarizing beam splitter or analyzer, which seems more convenient than the method using complex spatial mod-

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ulation technique [86]. Meanwhile, in our scheme, the total OAM of light beam is controlled by the external voltage, which implies that the torque acted on particle is highly tunable; thus, this method takes some advantages over traditional optical tweezers. In summary, we have proposed a kind of external voltage-controlled optical vortex converter in HPPLN based on electro-optic effect. According to different incident condition, the HPPLN can be used as a topological charge adder or a substractor. The converter features highly voltage-afforded tunability and thus may serve as a promising candidate for the vortex generation and transformations in diverse applications.

2.6 Polarization-coupling cascading and its applications in PPLN crystal 2.6.1 Second-harmonic generation (SHG) cascading Nonlinear optics generally originated in the 1960s and 1970s. Nonlinear optics has traditionally been discussed in terms of second- and third-order nonlinearities and the effects to which they lead. For example, second-order nonlinearities are well known for phenomena such as frequency conversion, parametric amplification, etc. Thirdorder nonlinearities, on the other hand, are usually associated with an irradiancedependent refractive index, four-wave mixing, solitons, etc., phenomena in that the beam frequencies are degenerate. However, it was recognized in the early stages of NLO that second-order phenomena could effectively contribute to third-order nonlinearities, therefore quasi-phase theory has been proposed. It indicates that when the QPM is satisfied (Δk = k 2w − 2k w − G m = 0), the second-harmonic light has the maximum conversion efficiency. The existence of nonlinear phase shifts in the fundamental beam during SHG was first discussed by Ostrovskii in 1967, and the existence of soliton-like (henceforth called simply solitons) waves was predicted in 1974 [86, 87]. The importance of cascading, however, was not fully appreciated until 1996, when Stegeman studied cascading systematically and considered it as a promising direction to explore optical phenomena [88]. The physical picture of the classic SHG cascading is shown in Figure 2.45. As is known, SHG cascading occurs via up-conversion (ω + ω → 2ω) followed by down-conversion (2ω − ω → ω). Due to the phase velocity of the new fundamental frequency photon is inconsistent with the input one, the nonlinear phase shift is yielded via cascaded processes, as shown in Figure 2.45. The nonlinear phase shift occurs when energy flew back to fundamental frequency light. From Figure 2.45, we can see that the NPS could be modulated by Δk.

170 | Xianfeng Chen

Fig. 2.45. The physical pictures of SHG cascading processes.

In the slowly varying amplitude approximation, the coupling-mode equation of SHG process is given by dE2 ω = −i χ (2) (2ω; ω, ω)E1 E1 exp(iΔkz󸀠 ) , dz󸀠 2cn2ω dE1 ω (2) = −i χ (ω; 2ω, −ω)E2 E∗1 exp(−iΔkz󸀠 ) . dz󸀠 4cn ω

(2.140) (2.141)

To simplify these equations, we define Γ Γ=

ωdeff |E0 | , c√n2ω n ω

(2.142)

where deff = |χ (2) (2ω; ω, ω)|/2, E0 is the intensity of fundamental frequency light. Assuming that there is no double frequency light in incident light, we get dE1 d2 E1 + iΔk 󸀠 − Γ 2 (1 − 2 |E1 /E0 |2 )E1 = 0 , dz dz󸀠2

(2.143)

when Δk = 0, it change into E1 = E0 sec h(ΓL), there is no NPS in fundamental frequency light under this condition. When Δk ≠ 0, with approximation it can be obtained |E1 | ≅ |E0 |, and E1 (z󸀠 ) = |E0 |×exp[−iΔΦ NL (z󸀠 )], put them into Equation (2.143); we get the NPS of the fundamental frequency light at z󸀠 = L: ΔΦ NL ≅

ΔkL 1/2 {1 − [1 + (2Γ/Δk)2 ] } , 2

(2.144)

for large phase mismatch or low external electric field, we have |Δk| ≫ |Γ|, we get ΔΦ NL ≅ −

Γ 2 L2 . ΔkL

(2.145)

NPS changes linearly with the light intensity I, which is similar to the optical Kerr effect which can be expressed as n = n0 + n2 I. Following the example of changing

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in refractive index of the Kerr effect, an “effective” nonlinear refractive index by PC cascading can be introduced by 2

neff 2 =−

1 4π L deff . 2 cε0 λ n2ω n ω ΔkL

(2.146)

In the realm of NLO, the polarizability of a medium modified by optical fields can be eff given by P = P0 + χ1 E + χ2 EE + χ eff 3 EE + χ 3 EEE . . . , where χ 3 EE corresponds to the effective third-order nonlinearity which is induced by second-order nonlinearity χ2 in SHG cascaded processes.

2.6.2 Polarization-coupling cascading and nonlinear phase shift By following the example of SHG cascading, we propose a new cascaded phenomenon – PC cascading under non-QPM (NQPM) condition [89]. These cascaded processes are divided into two steps: Taking OW incidence for example, the energy of OW flows to EW, but does not cause the complete depletion of OW; then, the energy flows back from EW into OW, after approximately one coherence length. Because the regenerated OW is no longer in phase with the nonconverted, a net OW phase is yielded (shown in Figure 2.46 (b)). The vectors in Figure 2.46 (b) show the energy variation of OW and EW during the coupling processes. From Figure 2.46 (c), the increment of the nonlinear phase primarily occurs during the cycle wherein energy strongly exchanges between the two beams. The relative azimuth angle between the dielectric axes of two adjacent domains is very small so that the periodic alternation of the azimuth can be considered as a periodic small perturbation. In this case, the coupled-mode equations of the ordinary

Fig. 2.46. The physical pictures of SHG cascading processes and PC cascading processes.

172 | Xianfeng Chen and extraordinary waves are dA1 /dz = −iκA2 exp(iΔβz) ,

(2.147)

∗

(2.148)

dA2 /dz = −iκ A1 exp(−iΔβz) , with Δβ = k 1 − k 2 − G m ,

G m = 2πm/Λ

and κ=−

ω n2o n2e γ51 E y i(1 − cos mπ) , 2c √n o n e mπ

(m = 1, 3, 5, 7. . .) ,

where A1 and A2 are the normalized complex amplitudes of OW and EW, respectively. Δβ is the vector mismatch, k 1 and k 2 are the corresponding wave vectors, G m is the mth reciprocal vector corresponding to the periodicity of poling, Λ is the period of the PPMgLN, γ51 is the electro-optical coefficient. E y is the transverse electric field intensity, n o and n e are the refractive indices for the ordinary and extraordinary waves, respectively, and with the initial condition A1 (0) = 1, A2 (0) = 0 (Assuming that the incident beam is OW). The solution is given by A1 (z) = exp[i(Δβ/2)z][cos(sz) − iΔβ/(2s) sin(sz)] , ∗

A 2 (z) = exp[−i(Δβ/2)z](−iκ /s) sin(sz) ,

(2.149) (2.150)

with s2 = κκ ∗ +(Δβ/2)2 . For perfect QPM (Δβ = 0), the solution is simplified to A1 (z) = cos(|κ|z), A2 (z) = sin(|κ|z). However, the cascading phenomenon of phase mismatch has not yet been understood. Thus, this study concentrates on the non-QPM solution (Δβ ≠ 0), by studying the phase of OW and EW, in which a rich variety of cascaded phenomena can occur. From Equation (2.150), we have A2 = C exp[−i(Δβ/2)z], with C = (−iκ ∗ /s) sin(sz). A2 is a real number. The phase of EW is easily obtained: {− Δβ 2 z, C > 0 ΔΦ NL e = { Δβ − z ± π, C < 0 ; { 2

(2.151)

NL with ΔΦ NL e confined to interval [−π, π]. Equation (2.151) suggests that ΔΦ e of EW varies linearly with phase mismatch, and the “half-wave loss” happens. The phase of OW is more complicated. Thus, obtaining the approximate solution for a large phase mismatch or low external electric field was attempted. Assume |A1 | ≅ 1, and hence A1 (z) = exp[−iΔΦ NL o (z)]. From Equations (2.147) and (2.148), 2

dΔΦ NL dΔΦ NL d2 ΔΦ NL o o o −i( ) − iΔβ + iκ ∗ κ = 0 , 2 dz dz dz

(2.152)

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Fig. 2.47. Typical variation of the nonlinear phase shifts with external electric field for EW and OW.

when z = L,

ΔβL (2.153) (1 − √1 + (2κ ∗ κ/Δβ)2 ) . 2 For a large phase mismatch or low external electric field, we have |Δβ| ≫ |κ|. The variation is similar to the classic electro-optical Kerr effect and can be shown as ΔΦ NL o =−

ΔΦ NL o ≅

|κL|2 . ΔβL

(2.154)

Following the example of changing in refractive index of the EO Kerr effect, an “effective” EO nonlinear refractive index by PC cascading can be introduced by Δneff o =

ωn3e n2o γ251 2 Ey . π2 cΔβ

(2.155)

Similarly, during PC cascading the index of refraction modified by cascaded electro-optical effect can be given by 1 1 1 n = n0 + γn30 E + seff n30 E2 + sn30 E2 + ⋅ ⋅ ⋅ . 2 2 2

(2.156)

eff Just like χ2 inducing an χ eff 3 in SHG cascading, γ leads to an s in PC cascading. Here seff is referred to as the effective EO Kerr constant which could be enhanced during the effective electro-optical Kerr effect. Supposing Δβ = 1π/m, λ = 600 nm, n e = 2.1930, n o = 2.2829 and γ51 = 32.6 pm/V, the effective Kerr constant seff will be 1.04 × 10−14 m2 /V2 , [90] which is three orders of magnitude larger than the classical EO Kerr constant of lithium niobate, 3.39×10−17 m2 /V2 [91–98]. It should be noted that the enhanced EO Kerr constant is an effective effect which has nothing to do with the particular material but mainly governed by the periodical index modulation. The precise phase variations with external electric field of EW and OW governed by Equations (2.151) and (2.152) are shown in Figure 2.47, which indicates the potential of nonlinear EO phase modulators. The four solid lines present the variation of

174 | Xianfeng Chen

Fig. 2.48. Experimental setup for demonstrating the nonlinear phase shifts yielded in PC cascading.

the nonlinear phase shifts with external electric field at different vector mismatches, Δβ = 121π/m, 16π/m, −16π/m and −121π/m, while the dash lines present the transmission with electric fields, showing the energy coupling between OW and EW in PPMgLN. By keeping an appropriate nonzero value of the phase mismatch, the transmission of EW and OW varies from 0 to 0.99 and 0.01 to 1, respectively. Just by adjusting the vector mismatch, the magnitude and sign of the effective nonlinearity, i.e. nonlinear phase can be varied. As shown in Figure 2.47 (a), the large phase shifts only occur when the EW experiences a “half-wave loss,” which means that the phase maintains constant except at some critical electric fields. While in Figure 2.47 (b), in the vicinity of some critical electric fields, tiny changes of the electric field can cause large changes of phase for OW. It should be noted that the conventional phase modulators based on the Pockels effect only realize the linear phase shift and are unable to obtain sharp phase shift or half-wave loss. The schematic of the experimental setup is shown in Figure 2.48. A scheme of March–Zehnder interference was utilized to investigate the phase shifts. The wavelength of the He–Ne laser is 632.8 nm, which almost satisfies the third-order phasematching condition with the poling period of PPMgLN to be 21.1 μm. The laser power is 8 mW. First, the horizontally polarized incident beam was separated by a beam splitter (BS) with one beam passing through PPMgLN and the other in free space. Thus, the incident beam in PPMgLN is OW. The experimental results are shown in Figure 2.49 (a) and (b). By varying the transverse electric field from 0 to 0.56 V/μm, we found that the interference fringes’ “light-dark” changed at different temperatures (different vector mismatches). When the vector mismatch was around −1000π/m (23 °C, Figure 2.49 (a)), Newton’s rings had hardly any change in varying electric field. It concludes that the nonlinear phase shift cannot be accumulated to π at very large vector mismatch. When the vector mismatch was changed to approximate −150π/m by lowering temperature to 21.3 °C (Figure 2.49 (b)), the interference fringes experienced three times “light-dark” alternation at some certain electric fields, 0.28, 0.44, and 0.56 V/μm. Each change means a π phase shift. By this token, transverse electric field is not the determinant of the large phase shifts. In a PPMgLN, the classical transverse electro-optical effect is only

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Fig. 2.49. The comparison of experimental results and theoretical simulation for demonstrating the enhanced phase shift yielded in PC cascading.

able to rotate the optical axis of the lithium niobate, which indicates the large phase shift cannot be the classic EO Pockels effect. Figure 6.5 (c) shows the comparison of the simulated curve and the experimental results at the temperature of 21.3 °C. The critical electric fields leading to π phase shifts with the vector mismatch of approximate −150π/m (in Figure 2.49 (b)) are shown as the color dots in Figure 2.49 (c). Thus, we can see that the experimental data well fit the theoretical curve. Because of some unavoidable errors, for instance, the refractive indices of the PPMgLN sample we used are not consistent with the simulated values; the experimental data cannot be in full agreement with the theoretical ones. In summary, PC cascading was demonstrated in PPMgLN through experiment and theory. Nonlinear phase shifts generated from the cascaded processes between a pair of orthogonal beams, i.e. OW and EW have been investigated. The results provide a method which can be used to achieve the enhanced EO Kerr effect. It should be noted that the PC cascading proposed here is different from the SHG cascading phenomenon, because the former belongs to linear optics, and the latter is classified to NLO. With a different physical understanding, PC cascading may trigger interest in a wide range of fields.

2.6.3 Cross-modulation In this section, we propose cross-modulation like based on polarization-coupling cascading. Compared with the traditional nonlinear optical cross-phase modulation, under this cross-modulation, the transmission, phase and the dispersion of one beam depend on the power ratios of the two beams instead of relying on the power values, which means that the active control of light we discussed here may open a door for broader scopes of applications in weak-light optical operation.

176 | Xianfeng Chen

Fig. 2.50. Schematic model of polarization-cascaded process.

Consider two orthogonally polarized lights with one OW and the other EW into this structure. The cascaded process of energy coupling between OW and EW happen in these folded domains (see Figure 2.50 (a)). Consider an incident EW with the almost phase-matched case. The cascaded process is divided into two steps. The OW grows with distances from the input, but does not completely deplete the EW. After approximately one coherence length, the energy flows back from the OW to the EW. Because the regenerated EW is no longer in phase with the nonconverted, a net EW phase is yielded, either advanced or retarded relative to that of the unconverted EW, depending on which velocity is larger. The phase shift variation along the Z-direction in PPLN is shown in Figure 2.50 (b), indicating the prominent characteristic of it is a stepwise change. From Figure 2.50 (b), the increment in nonlinear phase occurs primarily during the cycle wherein the energy strongly exchanges between the two beams. Using the coupled-wave Equations (2.147) and (2.148), the phase of the two orthogonally polarized beams can be derived as Φ1 (z) =

Δβz 2 + arctan [

Φ2 (z) = −

−√I1 /I2 (Δβ/2s) sin(sz) + Re(κ/s) sin(sz) sin(δ 0 ) ] , √I1 /I2 cos(sz) + Re(κ/s) sin(sz) cos(δ 0 )

(2.157)

Δβz 2

+arctan [

(Δβ/2s) sin(sz) cos(δ0 ) + cos(sz) sin(δ0 ) ], cos(sz) cos(δ0 )−(Δβ/2s) sin(sz) sin(δ0 )− √ I1 /I2 Re(κ/s) sin(sz) (2.158)

where I1 and I2 are input light intensities of OW and EW, respectively, and δ0 is the initial relative phase between the two beams. From Equations (2.157) and (2.158), the phase of one beam is determined by both itself and the other one. The former functions as an SPM, while the latter function as an XPM. The key feature is that they are determined by the relative power ratio rather than the absolute power value. Without losing generality, a 2.1-cm-long PPLN with the period of 21 μm is employed to inves-

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Fig. 2.51. Nonlinear phase shift as a function of the initial relative phase and power ratio.

tigate such polarization-coupling cascading. The phase and transmission of the two orthogonally polarized lights shown in Figures 6.7 and 6.8 were calculated using MATLAB software and with a–c for OW and d–f for EW. Besides, a and d shares the same phase mismatch, Δβ = 190/m; b, e, Δβ = 120/m; c, f, Δβ = 60/m. The applied electric field is 0.15 V/μm. It is interesting to see at some critical points (P1 , P2 , P3 , P4 , P5 , P6 ) in Figure 2.51, tiny change of the relative power ratio causes large nonlinear phase change. The fea-

Fig. 2.52. Transmission as a function of the initial relative phase and power ratio.

178 | Xianfeng Chen

Fig. 2.53. Variation of the nonlinear phase shifts against wavelengths at different power ratios.

ture may be used to design a new kind of all-optical phase modulators. However, comparison with the curves of Figure 2.52 indicates that the “price” for obtaining sharp phase shifts of the two beams is effectively the “loss” of transmission. Figure 2.52 suggests that the transmission can be shifted between exactly 0 and almost 100% by simply modulating the relative power ratio, serving as an all-optical intensity modulator. The nonlinear phase shift as a function of the wavelengths is shown in Figure 2.53. During the simulation, δ0 = 0, E = 0.17 V/μm, and A2 = √I2 /(I1 + I2 ). Besides, Figure 2.53 (a) and (c) show the results of OW and b, and d presents the results of EW. Just by adjusting the wave-vector mismatch condition, both the magnitude and the sign of the nonlinear phase shift can be changed. That the nonlinear phase shift (Figure 6.9 (a) and (b)) or its derivative (Figure 2.53 (c) and (d)) can be detuned by the power ratio is considered to be very feasible and attractive, as the derivative of nonlinear phase shift usually represents the group velocity, indicating that slow light with its time delay controllable by optical powers is possible. We can note that the bandpass in Figure 2.53 is merely of a few nanometers. This may not be useful for ultrashort pulses or ultrafast pulses because the bandpass of them are in general tens of nanometers. Our previous research studies on Solc-type filters revealed that larger bandpass can be created if properly reducing the length

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Fig. 2.54. Variation of the nonlinear phase shifts and its derivative against wavelengths for PPLN samples with different lengths.

of the PPLN sample [99]. Figure 2.54 presents the nonlinear phase shift, i.e. induced dispersion at different sample lengths. During the simulation, δ0 = 0, E = 0.17 V/μm, and A2 = 0.99. The bandpass of nonlinear phase shift is increasing with the decrease of the sample length. Figure 2.54 (c) and (d) shows the bandpass can be as large as tens of nanometers. Thereby, a femtosecond pulse or picoseconds pulse is able to be delayed or advanced via such a wideband nonlinear phase shift. However, the quest for wider bandpass is with the reduction on transmittance as well as on the phase shift. A balance on them is necessary when considering the ultrashort pulses.

2.6.4 Fast and slow light The group velocity modulation of light has attracted significant interest recently [100– 102] as a potential solution for optical delay lines and time-domain optical signal processing, and the enhancement of nonlinear optical effects [103, 104]. However, so far most of these methods bear inherent limitations that may hinder their practical deployment. For example, electromagnetically induced transparency using ultracold atomic gas rather than solid material [105]; Quantum dot semiconductor optical amplifiers can only achieve limited signal delay [106]; the band of coherent beam oscillations [101] and stimulated Raman scattering [107] are very narrow; Surface plasmon waves on the surface smoothness of the material is very sensitive, and is difficult to excite [108]; photorefractive effect has slow response [109]; and SHG cascading needs extremely high power [110]. In this section, a method was demonstrated to rapidly control the group velocity at room temperature in electro-optical PPLN, where the group velocity of input optical beam can be modulated from subluminal to superluminal by simply adjusting the applied external electric fields.

180 | Xianfeng Chen The relative azimuth angle between the dielectric axes of two adjacent domains is very small so that the periodic alternation of the azimuth can be considered as a periodic small perturbation. In this case, the coupled-mode equations of the ordinary and extraordinary waves are dA1 /dz = −iκA2 exp(iΔβz) ,

(2.159)

∗

(2.160)

dA2 /dz = −iκ A1 exp(−iΔβz) , with Δβ = k 1 − k 2 − G m ,

G m = 2πm/Λ

and κ=−

ω n2o n2e γ51 E y i(1 − cos mπ) , 2c √n o n e mπ

(m = 1, 3, 5, 7. . .) ,

where A1 and A2 are the normalized complex amplitudes of OW and EW, respectively. Δβ is the vector mismatch; k 1 and k 2 are the corresponding wave vectors; G m is the mth reciprocal vector corresponding to the periodicity of poling; Λ is the period of the PPLN, γ51 is the electro-optical coefficient. E y is the transverse electric field intensity, n o and n e are the refractive indices for the ordinary and extraordinary waves, respectively, and with the initial condition A1 (0) = 1, A2 (0) = 0 (Assuming that the incident beam is OW). The solution is given by A1 (z) = exp[i(Δβ/2)z](−iκ/s) sin(sz) ,

(2.161)

A2 (z) = exp[−i(Δβ/2)z][cos(sz) + iΔβ/(2s) sin(sz)] ,

(2.162)

with s2 = κκ ∗ + (Δβ/2)2 . Considering E1,2 = A1,2 (z) exp[i(β 1,2 z − ωt)], the phase of EW is derived as Φ2 (z) =

π (3β 2 − β 1 ) Δβ z+ z + arctan [ tan(sz)] . Λ 2 2s

(2.163)

The inverse of the effective group velocity of EW is given by 1 1 dΦ 3v1 − v2 dk = = = + Vg dω z dω 2v1 v2 1+

1 Δβ 2 4s 2

tan2 (sz)

[

v2 − v1 tan(sz) Δβ tan sz 󸀠 + ( )] , 2v1 v2 sz 2 sz

(2.164) where v1 and v2 are the group velocity of OW and EW in the medium of lithium niobate, respectively. And the transmission of EW is I2 = [cos2 (sz) +

Δβ 2 sin2 (sz) ] . 4s2

(2.165)

For QPM condition (Δβ = 0), the effective group velocity of EW is Vg =

2v1 v2 3v1 − v2 + (v2 − v1 ) tan(sz) sz

,

(2.166)

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Fig. 2.55. Theoretical results on the group velocity as a function of sz.

if sz = π/2 + mπ, m = 0, 1, 2 . . . , V g approaches 0, seeming that the light is brought to a complete standstill. However, as the transmission declines to zero, it cannot be considered as an approach for storing lights. The group velocity V g exceeds c and can even become negative by choosing an appropriate “sz.” A negative group velocity of light has been demonstrated, though it seems counterintuitive. The detailed variation in the group velocity as a function of sz is shown in Figure 2.55, which also reveals a critical point that is very interesting, where the group velocity is extremely sensitive to sz. This critical point satisfies the equation of 3v1 − v2 + (v2 − v1 ) tan(sz) = 0. sz In the experiment, we fabricated a periodically poled grating in a z-cut LiNbO3 crystal with a dimension of 30 mm (L) × 10 mm (W) × 0.5 mm (T). The corresponding grating period of the PPLN is 21 μm. The scheme designed to measure the group velocity of light near the band edges of the transmission spectrum was shown in Figure 2.56. A high-voltage source with the maximum of 1.5 kV is used to generate the electric field along the transverse direction of PPLN. An optical test system is employed in our experiment, which includes a broadband-amplified spontaneous emission (ASE) source with an output wavelength ranging from approximately 1530 to 1560 nm, as well as an OSA for observing the transmission spectrum. The light signal was launched from a tunable laser modulated by an intensity modulator with a function generator, and the output light signal was observed by a photoelectric detector connected with an oscilloscope. By utilizing the above experimental setup, we observed the slow light at the room temperature. Many studies have demonstrated that the group velocity of light exhibits strong dispersion near the band edges in photonic band crystal resulted from Bragg reflection [109]. Our scheme here engendered a forbidden band using folded dielectric axes structures. The formation of this forbidden band can be described as follows: when a uniform electric field is applied along the Y-axis of PPLN, based on the electrooptic effect, the optical axis of each domain is alternately aligned at the angles +θ and −θ with respect to the plane of polarization of the input light. The angle θ is called the rocking angle and is proportional to electric field intensity. For wavelength which satisfies the case that each domain serves as a half-wave plate with respect to it, after passing through the stack of half-wave plates, the optical plane of polarization of such

182 | Xianfeng Chen

Fig. 2.56. Experimental setup for slowing light signals.

wavelength rotates continually and emerges finally at an angle of 2Nθ, where N is the number of plates. Therefore, when 2Nθ = 90° at the filter output, light of the wavelength experiences giant loss in passing through the analyzer which is parallel to the polarizer, and a forbidden band gap can be formed. In the vicinity of the band gap, the optical signal can be modulated by electric field and wavelength. Figure 2.57 (a) presents the transmission spectrums at a given electric field of 0.15 V/μm observed by the OSA. Near the band gap a light at 1546.8, 1547.5, and

Fig. 2.57. Modulation of slow optical signal with wavelength.

2 Polarization coupling and its applications in periodically poled lithium niobate crystal

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1547.1 nm was launched from a tunable laser. This continuous wave was modulated by an intensity modulator as well as a function generator with a frequency of 10 MHz. Figure 2.57 (b) presents the signal waveforms at different wavelengths observed by an infrared photodetector. The delay can be as large as 20 ns, which is one-fifth of the period of the signal with a frequency of 10 MHz. Wavelength of the band gap was also investigated. Figure 2.57 (c) shows the measured delay with respect to wavelength, indicating the delay only experience dramatic variation near the bottom of the forbidden gap. Figure 2.58 (a) presents the theoretical transmission spectrum as a function of the external electric fields, which suggests that a light can be trapped in a forbidden band gap or released out of it by simply modulating the electric fields. Figure 2.58 (b) presents the experimental transmission spectrums at electric fields of 0 and 0.15 V/μm, showing a dip for zero applied field while the theory in Figure 2.58 (a)

Fig. 2.58. Modulation of slow optical signal with wavelength.

184 | Xianfeng Chen shows that such a gap should not exist. The deviation between the experimental result and the theoretical result is due to an initial domain angle existing in such a periodically poled structure which may be caused by the strain-optic effect produced in the process of polarization or the photovoltaic effect engendered by the input light. Figure 2.58 (c) presents the signal waveform at electric fields of 0, 0.05, and 0.15 V/μm, where the delay also approaches 20 ns. As the length of the sample is 3 cm, a light without slowing the process will spend about 0.2 ns in passing through this PPLN sample. The effective group velocity here can be about 1.5 × 106 m/s and the effective group index can be about 200. The large effective group index facilitated the exclusion of the impact resulted from the classical Pockels EO effects which are too weak to produce such an amount. In summary, a method was demonstrated for slowing light signals. The group velocity of a light near the filter band gap can be delayed via changes in electric field strength or wavelength, which are attractive for EO signal processing and all-optical signal processing. A negative group velocity or that exceeds c helps to investigate physics on fast light. In this section, we investigate the modulation of phase based on PPLN crystal. We propose a new phenomenon named “polarization-coupling (PC) cascading,” which is modeled after the SHG cascading. PC cascading effect could be defined as the effective EO Kerr effect. Interestingly, the Kerr constant of this effective EO effect is several orders of magnitude larger than that in the classic counterparts, and could lead to large nonlinear phase shifts. Based on PC cascading, we propose a concept of crossmodulation-like. The parameters of one beam as transmittance, phase, and dispersion are conditioned on a relative ratio of these two beam powers. We also proposed a method of group velocity modulation, with a maximum delay of 20 ns in the experiment, which is attractive for electro-optical signal processing and all-optical signal processing. This section presents that the PC cascading can also be extended to the field of ultrafast optics, to achieve the group velocity modulation of femtosecond and picosecond lights, pulse compression and shaping; in this section, the electro-optic Pockels effect induced effective the electro-optical Kerr effect, in this inspired, by changing the method of domain angle production, for example, magneto-optical effect or elastic effect could also be used to realized PC cascading, and during this process, effective second-order magneto-optical effect or elastic-optical effect could be induced.

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190 | Xianfeng Chen [110] M. Marangoni, C. Manzoni, R. Ramponi, G. Cerullo, F. Baronio, C. De Angelis and K. Kitamura, “Group-velocity control by quadratic nonlinear interactions,” Opt. Lett. 31, 534–536 (2006).

Heping Zeng

3 Ultrafast nonlinear optics

3.1 Introduction The propagation of intense ultrafast laser pulse induced quite a lot novel nonlinear effects, such as temporal, spatial and spatiotemporal modulational instability, cascaded second-order nonlinearity, intense phase-modulation-induced collapse of optical wave-packet, and filamentation in third-order nonlinear materials, which accompanied with series of novel nonlinear phenomena, such as spectrum broadening, supercontinuum generation, conical emission, pulse self-compression, spatial self-cleaning, THz, and harmonics generation. This brings about a new era in the field of ultrashort laser pulse generation, compression, broadband spectral remote sensing, LIDAR, time-resolved spectroscopy, ultrashort laser pulse diagnostics, and optical coherence tomography. The study of all the listed novel nonlinear effects and phenomena has become a new branch and extension of conventional nonlinear optics that named ultrafast nonlinear optics. In this review chapter, we present the experimental observation of some unique nonlinear effects in second and third-order nonlinear materials. The nonlinear multifilament interaction and control, impulsive molecular alignment, and applications in broadband light pulse characterization are discussed. The first part of this chapter reviews our recent experiments on optical wave-packet collapse induced by cascaded second-order nonlinearity and spatiotemporal modulation instabilities, generation of two-dimensional multicolored transverse arrays and colored conical emission (CCE), formation and control of transient gratings, and multicolored up-converted parametric amplifications. Assisted by the spatiotemporal modulation instabilities, seeded amplification of colored conical emission exhibits a widely tunable range in wavelength and high-energy conversion efficiency that can be further applied as an alloptical method to generate carrier-envelope-phase (CEP) stabilized pulse by nonlinear frequency mixing with the fundamental pump pulse. Intense femtosecond laser-pulses-induced filamentation has been so far demonstrated quite useful for remote sensing, lightening guiding, and pulse self-compression down to a few cycles in duration. We show that coalescence of interfering noncollinear intense femtosecond pulses assisted periodic wavelength scale selfchanneling, which functioned as a novel and effective plasma waveguide with the advantages of ultrahigh damage threshold, relatively long lifetime, period tunablity and convenient multidimension extension. Moreover, attraction, repulsion, and fusion of Heping Zeng: State Key of Precision Spectroscopy, East China Normal University, Shanghai 20062, China

192 | Heping Zeng two parallel propagated femtosecond filaments, induced by Kerr effect, plasma, and molecular alignment are also presented in the second part. In the third part, we demonstrate that field-free alignment of gaseous molecules could function as an ultrafast polarization optical gating for laser pulse diagnostics. Originated from the transient birefringence induced by molecular alignment, this cross-correlation frequency-resolved optical gating technique (M-XFROG) exhibits advantages of no phase-matching constraint, high sensitivity, and applicability to pulses at any wavelength ranging from ultraviolet to far-infrared. Ultrashort pulse measurements of ultraviolet, supercontinuum, and sub-10-fs pulses were experimentally performed by using the M-XFROG technique.

3.2 Cascaded quadratic nonlinearity and spatiotemporal modulational instability Cascaded quadratic nonlinear process describes the energy transfer between the fundamental wave (FW) and second harmonic (SH) due to frequency doubling or mixing in propagation of light [14, 15]. Nonlinear phase shift, which can be obtained from cascaded quadratic processes in the presence of group-velocity mismatch, can be two orders higher than that of the third-order nonlinear processes and its sign and magnitude can be easily controlled by changing the mismatch conditions. On the other hand, spatiotemporal modulational instability exists in the nonlinear interaction between ultraintense laser pulses and nonlinear medium [1, 4, 6, 8]. Temporal modulational instability occurs due to the coupling between self-phase modulation and abnormal group-velocity dispersion, while spatial modulational instability happens in the presence of self-focusing and beam diffraction. From this aspect, spatiotemporal modulational instability exists with the help of dispersion, diffraction, and nonlinear effects. Cascaded quadratic nonlinearity and the related processes, originated from the strong spatiotemporal coupling of FW and SH when the ultrashort laser pulse is propagated through a cascaded material, give rise to novel nonlinear wave-packet collapse, two-dimensional multicolored transverse arrays, CCE, transient grating, and multicolored up-converted parametric amplifications. CCE supports efficient seeded amplification in a widely tunable wavelength range. Such a seeded amplification can be further applied to generate CEP-stabilized pulses.

3.2.1 Two-dimensional (2D) multicolored transverse arrays Spatiotemporal modulation instabilities during propagation have been experimentally explored and breakup of intense wave-packets, 2D stable multicolored transverse arrays, CCE, and multicolored up-converted parametric amplifications (MUPAs) in a

3 Ultrafast nonlinear optics

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quadratic nonlinear medium under the pump of two crossly overlapped femtosecond beams are reported. Various nonlinear phenomena can happen when incident a 50-fs laser pulse from Ti:sapphire system (800 nm, 1 kHz, 700 μJ) to a type-I (4 mm, 29.18° cut) BBO crystal. As shown in Figure 3.1 (a), the femstosecond pulse was split into two noncollinear beam k 1 and k 2 with the crossing angle θ p . The k 1 beam was focused by an f = 100 cm high-reflection spherical concave mirror, leading to an elliptically spatial profile with beam width ratio of 1.58. The k 2 beam was round and collimated down to the BBO crystal. Wave-package breakup was clearly visualized as additional spatial patterns beside the original beam k 1 due to the mutual interactions between the FW and SH pulses [34–36]. Weak SH beam along the k 1 beam was experimentally confirmed to control the formation of this solution like wave-package breakup [Figure 3.1 (b)].

Fig. 3.1. (a) Schematic for generation of 2D multicolored transverse arrays in BBO. (b) Scheme for generation and weak beam control of one-dimensional multiple quadratic spatial solitary waves. Insets: experimentally observed beam breakup and corresponding transverse intensity profiles. See Ref. [36] for details.

2D multicolored transverse arrays were generated when both k 1 and k 2 were temporally synchronized inside the BBO crystal. As shown in Figure 3.2 (a), two groups of well-visualized red and blue dotted arrays were generated based on the cascaded quadratic processes with the implementation of the off-axis k 2 beam. This 2D array was a direct evidence of enhanced spatial breakup and a direct result of energy transfer involved in the cascaded quadratic processes among different arrays. This novel approach could be served as a new mechanism for 2D array generation and provided us the possibility to explore the fundamental dynamics of wave-packet collapse during wave propagation in quadratic nonlinear media, which on the other hand might bring about various new applications [32]. Since the generation of 2D multicolored transverse arrays was originated from noncollinear cascaded quadratic processes, the effective refractive index of the BBO

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Fig. 3.2. (a) Experimentally observed 2D multicolored transverse arrays, (b) (c) the corresponding intensity profiles labeled as transverse and column dashed line in (a). See Ref. [36] for details.

Fig. 3.3. (a) Schematic of multicolored up-converted amplification with a white light probe k3 . (b) Principle of diffraction and amplification of the white light probe beam by the cascaded noncollinearly quadratic processes induced two-dimensional transient grating. (c) Observed multicolored up-converted amplification patterns as the white light was incident at R(1, 2) in Figure 3.2 (a). (d) Spectra of multicolored up-converted amplification spots along the G(0,3)column. Refer to [36] for details.

crystal was also modulated and gave rise to 2D transient gratings. We experimentally confirmed the existence of the transient gratings by using a synchronized white probe light. The corresponding spatiotemporal characteristic was studied as well.

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3.2.2 2D multicolored up-converted parametric amplification As illustrated in Figure 3.3 (a), a white light, generated by shot another off-axis beam k 3 to a 2 mm thick sapphire, was collimated and focus to the BBO crystal and spatially overlapped with k 1 and k 2 . When all these three beams were temporal synchronized, the white light probe was diffracted by the transient grating formed by k 1 and k 2 . However, unanticipated amplification along the diffraction directions was also observed, leading to the production of 2D multicolored up-converted parametric amplified arrays, as shown in Figure 3.3 (c). Spectra of multicolored up-converted amplification spots along the G(0,3) column in Figure 3.3 (a) are plotted in Figure 3.3 (d). Different multicolored up-converted amplification patterns are anticipated to generate under either different incident angle or various geometries. Figure 3.4 (a) and (b) gives examples of 2D multicolored up-converted parametric amplification patterns at different white light incident angles, where (a) and (b) correspond to the case of incident at spot R(2, 1), and in the middle of the square formed by R(1, 1), R(1, 2), R(2, 1), and R(2, 2), respectively. Figure 3.4 (c) presents the spectra of selected spots from Figure 3.4 (b), indicating that the generated multicolored up-converted amplification patterns covered visible region from 490 to 700 nm. The dependence of 2D multicolored up-converted amplifications on the cascaded quadratic-processes-induced transient gratings could be tested by changing the relative delays among the grating-forming and probe beam. Strongest amplification patterns were observed when the beam k 1 and k 2 were synchronized. While blocking k 2 , the observed 2D multicolored pattern degraded into a 1D patterns similar to Figure 3.1 (b). We could conclude that the multicolored up-converted amplifications was

Fig. 3.4. Pictures of 2D multicolored upconverted parametric amplifications when seeded at (a) R(2, 1) and (b) in the middle of R(1, 1), R(1, 2), R(2, 1), and R(2, 2), respectively. (c) The spectra of selected spots of multicolored up-converted parametric amplifications as shown in (b). Refer to [36] for details.

196 | Heping Zeng produced by the quadratic-nonlinearities-induced transient gratings’ diffraction and amplification of the incident white light probe pulse. Moreover, spectral coherence would be expected if the phase was preserved in the nonlinear amplifications along different directions, since they were from the same supercontinuum pulses. We measured phase difference between two selected spots by using an f –2f spectral interferometer. Experimentally, a stable interference pattern was observed, indicating that phase coherence Δφ = φ2 − φ1 was preserved during the diffraction and amplification of the probe pulse at different directions. Since up-converted 2D transverse patterns were generated due to cascaded quadratic and multiple parametric processes, it exhibited quite different features from the standard optical parametric amplifications. As the probing white light pulses had different spatiotemporal chirps in different spectral regions, the dynamics of the cascaded quadratic nonlinearity and spatial distribution of the transient grating could be well described by the 2D multicolored up-converted patterns. Apart from a white light probe, the existence of the transient grating could also be proved by using a weak 800-nm FW pulse, as illustrated in Figure 3.5 (a). Similar to that in the white light probe case, the FW probe pulse was diffracted and amplified by the k 1 - and k 2 -induced transient grating to form a 2D multicolored transverse arrays that contained almost 180 spots with different colors, corresponding to different wavelengths. The difference with respect to the white light probe was that the entire three noncollinearly propagated beams were temporally synchronized. Two separate cascaded quadratic processes between k 1 and k 2 , k 3 were generated, and the further coupling of the two cascaded quadratic processes gave rise to the multicolored radioactive-like arrays. The observation of such improved multicolored arrays further specified that spatial wave-packet breakup and the followed cascaded processes played key roles in 2D arrays generation. Phase coherences between different spots were also demonstrated by using spectral interferometer technique in the frequency domain, indicating such 2D patterns could be considered as multicolored coherent optical sources with broadband spectra.

Fig. 3.5. (a) Experimental sketch for observation of enhanced 2D multicolored transverse arrays with a weak FW pulse seed. (b) Photo of the observed 2D multicolored transverse arrays. Refer to Ref. [36] for details.

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This stimulates an all-optical control of spectrum synthesis by using different interacting pulses with different colors, energies, or pulse durations. We combined some generated spots in the coherent 2D patterns into a hollow fiber (50 mm long, 1.0 mm diameter, f = 60 mm) with a scheme as shown in Figure 3.6 (a). Figure 3.6 (b) gives the spectrum spread from 590 to 820 nm at the output of the hollow fiber. By using customized chirped mirror pairs or spatial light modulators to compensate the dispersion, one can compress this broadband spectrum to produce ultrashort pulse, as shown in Figure 3.6 (c) with a transform-limited FWHM of 1.3 fs for the main peak and pulse energy of 44 μJ. High-peak power ultrashort pulse generation is primarily significant for a variety of fundamental applications, such as single attosecond pulse generation by high harmonic generation [39], ultrafast nonlinear spectroscopic studies, and exploring nonlinear optical phenomena.

Fig. 3.6. (a) Experimental diagram for spectra synthesis by using a hollow fiber. (b) Spectrum of the combined beam after the hollow fiber. (c) Corresponding transform-limited pulse from (b). Refer to [36] for details

Moreover, the coherent 2D multicolored arrays could also be used as pumps for multicolored impulsive stimulated Raman scatterings [40]. A beam with pulse duration shorter than the period of the Raman-active vibrational mode was used as pump, which excited the molecules in a vibrational state. The refractive index modulation could be explored by delayed coherent pulses at different frequencies from the 2D arrays. Each pulse itself would lead to a coherent broadened spectrum and an overall ultrabroadband spectrum could be obtained for sub-fs pulses generation [41].

198 | Heping Zeng 3.2.3 Colored conical emission (CCE) CCE is originated from spatiotemporal modulation instabilities, which is a result of exponential growth of noise during propagation along nonlinear media in the presence of dispersion and diffraction under high-intensity pump [10, 13, 33]. Conical emission describes a noise-like amplification of various spectral elements at different angles, as shown in Figure 3.7. Apart from second-harmonic generation (SHG) in quadratic crystals, extra photon pairs with frequencies of ω ± δ can be produced under spatiotemporal modulation instabilities, and bright colored cones around the second-harmonic wave could be observed. The group velocity mismatch (GVM) is compensated by the strong coupling and phase-locking between FW and SHG.

Fig. 3.7. Generation schematic of CCE in a thick quadratic medium.

A 800 nm, 1 kHz repetition rate, 700 μJ, 50 fs laser system was used for the experimental test. The femtosecond pulse was focused to a 6-mm-thick β-BBO crystal by a concave mirror with focal length of 1000 mm. An iris was used to control the diameter and divergence angle of the pump beam. As shown in Figure 3.8 (a), a bright blue green conical emission with conical angle of 5° and central wavelength of 500 nm was observed when SHG was tuned to reach its maximum. Figure 3.8 (b) and (c) illustrated different conical emission under different pumping conditions with different pump pulse durations and beam divergences. Red semiconical with central wavelength of 650 nm appeared when the pumping diameter was reduced [Figure 3.8 (c)] or small negative chirps were introduced [Figure 3.8 (b)] to the pump pulse.

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Fig. 3.8. (a) Bright blue-green CE and the corresponding spectra observed at the optimum chirp and angular spectra of the fundamental pulses. (b) CE rings excited by a negatively chirped fundamental pulse, and typical spectra of the red semiconical emission. (c) CE rings observed with the fundamental pulses passing through a small iris, and typical spectra of the red semiconical emission.

3.2.4 Seeded amplification of colored conical emission (SAC) Since the observation of CCE was originated from exponential noise growth due to strong coupling between the FW and SHG pulses, and synchronized seeded amplification was thus expected. In experiments, 800-nm, 700-μJ, and 50-fs laser pulses were split to two paths, a weaker FW pulse was focused into a sapphire plate for white light generation, while the stronger one was used for CCE generation. The white light seed was filtered, focused, and collimated to have a spatial overlap with the pump in BBO crystal. Experimentally, we demonstrated the CCE enhancement by using a white light seed pulse similar to the white light probe used in the multicolored up-converted amplification in Section 3.2.2. The injected seed attained exponential gain along a certain conical angle of CCEs and the nonlinear gain was dependent on the FW pump intensity [13, 33]. Strong couplings between FW and SH pulses produced further balance the GVM and dispersion, enabling long interaction length and a quite high pulse energy up-converted amplification output of ∼ 150 μJ, centered at 500 nm by using a 80-fs, 600-μJ FW pulse (∼ 14 mJ/cm2). Comparing with standard optical parametric amplification (OPA), SAC enabled one order higher energy with comparable pump energy. Moreover, we found that the central wavelength of the output spectrum exhibited red shift from 496 to 504 nm when reducing the pump intensity from 10.6 to 1.1 mJ/cm2 .

200 | Heping Zeng This can be understood as modification of FW and SHG coupling and change of the phase-matching condition under different pump intensities. SAC was demonstrated to support broadband up-conversion, and a spectrum with the FWHM of ∼ 60 nm was obtained with 10.6 mJ/cm2 pump intensity. Therefore, both the central wavelength and SAC bandwidth was dependent on the pump intensity, SAC with shorter central wavelength and broadband tenability were anticipated by using higher intensity pump. As the CCE had different spectra at different conical angles, SAC with different wavelengths could be obtained by changing the seeding angle. The smaller the incident angle, longer the central wavelength of the SAC. When the seed angle was varied from 0.68° to 5.0°, the central wavelength of the SAC could be tuned from 798 to 500 nm. As self-phase-modulation-induced spectral chirp in the white light pulse, the wavelength tuning of SAC could also be established by adjusting the delay between pump and seed pulses.

3.2.5 Carrier envelope phase stabilization via difference frequency generation As discussed above, CCE due to the spatiotemporal modulational instability in SHG of ultrashort laser pulses. As a consequence, the strong nonlinearity between FW- and SHG-induced phase-locking balanced their GVD, GVM, and diffraction [42, 43], leading to the great enhancement of energy conversion. SAC was also demonstrated as a promising method for high-energy tunable and ultrabroadband optical parametric amplification. CEP of SAC preserved during the amplification process. Besides, the white light seed itself, generated from FW-pulse-induced self-phase modulation, was a replica of the pump FW. So the SAC differed from FW pump pulse in CEP by a constant value of π/2. We proposed a CEP self-stabilization method via difference frequency generation (DFG) between the FW and SAC [47]. In experiments, 1.4 mJ, 1 kHz Ti:sapphire laser with randomly distributed CEP was applied. Part of the energy (690 μJ) was used to generate SAC, and the remained FW energy of 710 μJ was used for DFG in a 2-mmthick β-BBO. The SAC was tuned to 533 nm (∼ 25 μJ), so as to generate an idler pulse of 1600 nm via DFG. CEP-stabilized 800-nm pulse could be generated by using a BBO crystal to frequency-double the idler at 1600 nm. Once again, an f − 2f spectral interferometer was used to verify whether the CEP of the generated 800-nm pulse was stabilized or not [48]. A stable interference pattern was observed, as shown in Figure 3.9 (a), clearly indicating a stabilized CEP of the DFG signal pulse. Shot-to-shot drifts of the measured CEP, originated from the instability of laser system and precision of the f − 2f measurement, were calculated and illustrated in Figure 3.9 (b). Moreover, the CEP drifts of the laser pulses directly from the laser system are plotted in Figure 3.9 (c), showing a random CEP from shot to shot without stabilization.

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Fig. 3.9. (a) Spectral interference result of the generated CEP stabilized pulses. (b) and (c) are the shot-to-shot carrier envelope phase drifts of the CEP-stabilized pulses and the pulses emitted directly from Ti:sapphire laser, respectively. (d) The measured FWHMs of the spectra of CEP-stabilized idler pulses as the central wavelength was tuned. Refer to [36] for details.

By changing the incident angle of the seed pulse, various CEP-stabilized pulses at different wavelengths could be generated via DFG. The CEP-stabilized DFG pulses could be tuned from 660 to 800 nm when the SAC central wavelength was tuned from 500 to 798 nm, as shown in Figure 3.9 (d), corresponding to a tunable idler range from 1320 to 1600 nm.

3.3 Interaction of intense ultrashort filaments Intense laser pulses experience dramatic self-focusing, multiphoton ionization, spectral broadening, and phase modulation when propagating in atmosphere. Counterbalance between Kerr self-focusing and plasma defocusing in neutral media, brings about robust self-guided channels that facilitate abundant self-action nonlinear processes [16]. Filamentary propagation of intense femtosecond laser pulses has been so far demonstrated quite useful for light detection and atmosphere remote sensing, atmosphere pollutant detection, lightening guiding, laser-induced water condensa-

202 | Heping Zeng tion and pulse self-compression down to a few cycles in duration [18–24]. Interaction of multiple filaments was demonstrated to induce new nonlinear effects, such as light fusion, fission, and spiraling [49–53]. By controlling noncollinear interaction of filaments in air, wavelength-scale, 1D periodic plasma structures with nanosecond lifetime could be generated. Such periodic plasma structures not only supported efficient coupling and guiding of incident intense femtosecond pulses, but also influenced nonlinear frequency conversions. Two orders of magnitude enhancement of third-harmonic generation (THG) was observed in the presence of such plasma structures. As a result of the periodic refractive index modulation, long-lifetime plasma gratings were expected and experimentally confirmed by the Bragg diffraction and in-line holographic imaging. Moreover, periodic plasma structure could be used as plasma photonic components with ultrahigh damage thresholds to sustain intense laser fields in ultrafast laser generation, propagation, and subsequent applications like harmonics generation, laser-induced chemical reaction, and so on.

3.3.1 Filament-interaction-induced nonlinear spatiotemporal coupling Interference occurs if two intense femtosecond filaments are spatially noncollinearly overlapped and temporally synchronized in air. The interference fringes reallocate the intensity and intensity peaks will be located around the constructive interference peaks. Self-focusing around the interference intensity peaks further increase the local intensity and is then counterbalanced by multiphoton-ionization-induced plasma defocusing and higher order optical nonlinear effects, forming periodically localized plasma structures along the bisector of the two interacting filaments [54, 55]. Different from common fibers or photonic bandgap fibers, where the reflection or diffraction is applied for guiding laser pulses, the plasma microstructures presented here induce tight wave guiding for intense pulses, i.e. input pulses are guided into bundles of plasma channels of local decrease in the refractive index encircled by air. The experiments were done with femtosecond laser pulses from a Ti:sapphire laser system (1 kHz, 800 nm). 50-fs, 2-mJ pulses were equally split into two beams that were then focused with two f = 800 mm focus lenses, generating two 40-mm-long noncollinear filaments with the crossing angle of ∼ 6°. Figure 3.10 (a) and (b) presents fluorescence pictures of the two filaments without and with interaction. The corresponding far-field beam profiles are shown in Figure 3.10 (c) and (d). Strong nonlinear interaction induced fluorescence enhancement when two filaments were synchronized. As shown in Figure 3.10 (b), a bright white fluorescence enhancement emerged due to filament fusion, which sustained a finite propagation distance along the bisector direction. As shown in Figure 3.11 (a), the fusion length increased from 0.2 to 4.2 mm when the crossing angle varied from 16.0° to 2.0°. Besides, far-field distributions were modulated with a wing expansion of each filament and thread fringes between the two filaments [54]. In order to investigate the

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Fig. 3.10. Photo of the intersected noncollinear filaments (a) without and (b) with filament interaction, and the observed far-field spatial distributions, indicating no filament interaction (c) and strong spatiotemporal couplings (d), respectively. Refer to [55] for details.

details about the observed far-field patterns, we chose and measured the corresponding spectrum of five selected points in Figure 3.11 (b). Gradually broadened spectrum and spectral breakups from A to D, ascribed to strong spatiotemporal coupling, as well as the complicated self- and cross-phase modulations in the interaction region, were observed. Note that the spectrum of the wing patterns spanned from 650 to 830 nm, while the inner threads ranged from 775 to 825 nm. Such spectral differences directly implied different originalities of the wing and thread patterns. Strong nonlinear interaction was observed in filament interaction and periodic intensity distributions were formed in the overlapping region. Such an intensity modulation could thus in turn modulate the refractive index of the interaction region. Throughout this review, techniques with direct cross-section imaging, fluorescence detection and in-line holographic recording [56] were applied to observe the periodic distributed plasma structures.

Fig. 3.11. (a) The length of the filament fusion as a function of the crossing angle between two intersected filaments. (b) The spectrum distribution at five selected positions in the wing pattern. Refer to [55] for details.

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Fig. 3.12. (a) The schematic of the direct imaging of the interaction region with a grazing angle inserted thin plate. (b) The transverse fringe periods and the corresponding intensity distributions (insets) of the noncollinearly intersected filaments at different crossing angles. Refer to [55] for details.

As shown in Figure 3.12, the laser intensity distribution could be directly imaged by inserting a grazing-angle-placed thin plate, a 4f imaging system and a CCD camera. Typical intensity distribution of a single filament is shown in the left inset of Figure 3.12 (a). Di Trapani et al. [43] studied the measured intensity interference modulation periods at different three crossing angles of 0.7°, 2°, and 4° (red dotted), which fits well with the calculated interference fringe (blue-squared curve) [Figure 3.12 (b)]. Multiphoton ionization of air molecules occurs during intense femtosecond filamentation and part of the ionized molecules are excited to highly lying electronic states, which then undergoes electronic transitions to emit characteristic fluorescence ranging from 300 to 450 nm, corresponding to transition of N2 and N2 + energy levels. As shown in Figure 3.13 (a), fluorescence microscope with an optical imaging system and a UV-intensified CCD camera were used to monitor the plasma structure near the interaction region. Figure 3.13 (b) and (c) showed the fluorescence images of the interaction region without and with interactions. Note that the fluorescence was greatly enhanced in the presence of filament interaction, which was consistent with the observation in Figure 3.10 (b). Fluorescence imaging provided a direct way to study the structure variation with different crossing angles. As the crossing angle increased, the number of the plasma channels increased, while the channel width decreased. Furthermore, this method could be used to study the attraction, repulsion, and fusion of parallel distributed filaments interaction under Kerr effect, plasma defocusing, and molecular alignment [57, 58]. Another application of this method could be found in Ref. [59], where 5-μm single filament was demonstrated by interacting two noncollinear ultraviolet filaments. Different from fluorescence detection, the in-line holographic imaging technique records the refractive index modulation by using a perpendicularly propagated weak probe beam and a 4f imaging system. As compared with the above two methods, holographic recording provides an approach to decode the temporal characteristic of

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Fig. 3.13. (a) The schematic of the fluorescence detection of the femtosecond filament structures. Measured N2 fluorescence profile of the interaction region (b) without and (c) with filament interactions. Refer to [55] for details.

the plasma structures by changing the time delay of the probe pulse with respect to that of the plasma-forming pulse. The phase shift, refractive index change, and the plasma electron density induced by the plasma channels could be retrieved afterwards by applying certain reconstruction algorithm on the recorded CCD image. Figure 3.14 presents two typical holographic images of the 1D plasma structures observed at different time delays of 0.25 and 2.0 ps, respectively. [54].

Fig. 3.14. CCD captured typical holographic images of the 1D plasma structures at time delays of (a) 0.25 ps and (b) 2.0 ps.

3.3.2 1D plasma channels As mentioned above, strong spatiotemporal couplings induced 1D plasma channels, together with observable fluorescence enhancement and far-field beam pattern modification in nonlinear interaction of two noncollinear filaments. According to the experimental observations, the observed thread fringes should not be originated from diffraction of the incident pulses, as diffraction made no changes of the beam pro-

206 | Heping Zeng file. Besides, the thread fringes exhibited similar structures with the observed periodic plasma structures, suggesting their intrinsic origin as self-guiding within the plasma channels. In order to confirm our clarification, the fluorescence image of the whole interaction region was captured by moving the optical imaging system and CCD camera, as shown in Figure 3.15 (a). Most of the filaments energies propagated along their original direction, while part of the energies were guided along the bisector of the interfering filaments. Figure 3.15 (b) and (d) shows the fluorescence images of the beginning and ending parts of the filaments interaction, which show clearly guiding of the input pulses by the plasma channels [Figure 3.15 (b)] along the bisector. The energy ratio of the thread fringes and original filament far-field spots was measured to be 1 : 50, indicating ∼ 2% of the incident pulses went off the original propagation direction and were guided by the plasma channels.

Fig. 3.15. (a) Measured fluorescence profiles of the interaction region of the intersected filaments. (c) The observed fringe spatial distribution inside the filament interaction region. (b) and (d) The fluorescence images of the leading and trailing part of the interaction, illustrating guiding of the incident pulse. Refer to [54, 55] for details.

The time evolution of the 1D plasma channels was studied by holographic recording of the interaction region with different time-delayed weak probe. Two imaging lenses with f1 = 40 and f2 = 200 mm were used to magnify the image with a factor of 5 to have a high spatial resolution. The images of plasma channels at six time-delays are shown in Figure 3.16 (a)–(f). Periodic plasma channels were clearly visualized at a probe time-delay of 0.25 ps [Figure 3.16 (a)]. Moreover, the plasma channels moved forward at larger time delays of 2.0 and 5.0 ps [Figure 3.16 (b and c)]. Figure 3.16 (f) shows the plasma channels image at 100 ps time-delays after its formation. Periodic structure was visualized even though the images became blurred at larger time delays. This could be ascribed to plasma expansion that weakened the plasma modulation.

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Fig. 3.16. The time evolution of the 1D plasma channels at different time-delays (a) 0.25 ps, (b) 2 ps, (c) 5 ps, (d) 10 ps, (e) 50 ps, and (f) 100 ps. Refer to [55] for details.

We anticipate that such long-lifetime plasma channels (of ∼ 100 ps or ns) would stimulate appealing applications in high-intensity optics and lattice solitons.

3.3.3 Plasma grating Spatially modulated plasma density would periodically modulate the local refractive index, equivalent to a diffraction grating with modulated refractive index that could diffract and split the incident light beams. As demonstrated above, the period of the plasma grating is determined by Λ = λ c /2 sin(θ/2), indicating a tunability of grating period by simply changing the intersection angle θ. The typical plasma grating thickness was experimentally measured to be ∼ 100 μm, and a thin grating that follows the Bragg diffraction condition could be formed when the crossing angle θ less than 4.6° with 800 nm grating forming pulses. The diffraction formula in the case of a thin grating can be written as Λ[sin(α + ϕ m ) − sin(α)] = mλ, where pulses with central wavelength λ and incident angle α were diffracted to angle ϕ m , determined by the diffraction order m. Experiments have shown that filaments supported efficient THG with nonlinear phase matching. Thus, the third-harmonic pulse generated from filaments could be applied to study the characteristics of plasma gratings. We experimentally generated three crossed air filaments by using three beams A, B, and C split from the same femtosecond laser. Plasma density modulation and plasma grating were formed by synchronizing pulses A and B with a crossing angle of θ, while beam C was located along the bisector of beam A and B, enabling a 0°-incident angle onto the grating. The third-harmonic pulse from pulse C will be diffracted according to φ mTH = 2m sin(θ/2)λ TH /λ FM , where m = 0, ±1, ±2, . . ., denote the diffraction order, φ mTH is the corresponding diffraction angles, λ TH and λ FW are the wavelengths of the TH and FW pulses, respectively.

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Fig. 3.17. The 3D distribution of the diffracted third-harmonic pulse by 1D plasma grating with different cross angles of (a) θ = 4.3°, (b) θ = 5.5°, and (c) θ = 6.8°, respectively. (d). The comparison of the experimental result (red-circular line) and calculated (blue-squared line) dependences of the +1 order diffraction angle on the cross angle θ. Refer to [55] for details.

The blue-squared line in Figure 3.17 (d) gives the prediction of the dependence of the first-order diffraction angle ϕ±1 on the cross angle θ. In the experiment, we chose θ = 4.3°, 5.5°, and 6.8° to investigate the diffraction property of the plasma grating. The 3D distributions of the filtered diffraction patterns of the THG from the pulse C, taken by a camera on a paper located 150 cm away from the interaction position, are shown in Figure 3.17 (a)–(c). Apparently, the diffracted ±1 orders moved apart from the mother THG pulse and the corresponding measured diffraction angles are shown as red-circular spots in Figure 3.17 (d). The agreement between the calculation and experiment thus gives clear explanation of the generation of the side spots, while on the other hand verifies the existence of plasma grating. The time characteristic of the plasma grating could be investigated by using a time-delayed THG pulse and recording the THG diffraction patterns at different timedelays with respect to the 4.3°-crossed grating-forming pulses. Figure 3.18 plots the experimentally measured integrated spatial distribution of the diffracted THG pulse with time-delays from 1.0 to 65.0 ps. The first-order intensity of the diffracted THG pulse decreased gradually as the time delay increased. However, weak diffraction still existed and could be observed at long time-delays (65.0 ps), which further confirmed the relatively long lifetime of the plasma grating. In contrast with the instantaneous Kerr

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Fig. 3.18. The measured time evolution of the plasma grating diffracted THG intensity with time delay varied from 1.0 to 65 ps. The THG pulse was generated from the time-delayed pulse C. Refer to [60] for details.

gratings that only exist within the pulse duration, the plasma grating presented here exhibited advantages of long lifetime and temporal flexibility.

3.3.4 2D plasma grating By adding more interacting filaments, multidimensional plasma gratings could be generated, and here we demonstrated the formation of 2D plasma grating by interacting three air filaments. Figure 3.19 schematically shows the geometry of the three crossed filaments at different planes as well as the interference-induced intensity localization and subsequent 2D periodic plasma density modulation. When the three interacting filaments were temporally synchronized, hexagonal far-field THG diffraction array could be observed, as shown in Figure 3.21 (a). Once again, in-line holographic recording was used to directly observe and prove the existence of the plasma density modulation inside the interaction region by

Fig. 3.19. The sketch geometry for 2D plasma grating generation by three noncollinearly intersected pulses A, B, and C. The insets schematically demonstrated the intensity localizations within the 1D and 2D plasma gratings.

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Fig. 3.20. The measured holographic images of the (a, c) 1D and (b, d) 2D plasma gratings from different views. Fourier domain filtering was applied to optimize the signal to noise ratio. Refer to [60] for details.

recording the modified spatial distribution of probe beams, propagating from below to top and left to right through the interaction region, respectively. The images were magnified by a factor of 5. Figure 3.20 (a, c) and (b, d) depicts the holographic images of the 1D and 2D plasma gratings when the probe pulse was delayed ∼ 5.0 ps after the plasma grating formation, in which (a) and (b) represent the top view (bottom to top) while (c) and (d) are the side view (left to right). For the case of two interacting filaments (A and B), the weak probe beam pattern clearly verified the 1D plasma lattice structure in the top view, while no periodic structures were observed in the side view. The image of 2D plasma lattice structures in the top view became somehow vague [Figure 3.20 (b)], while some periodic structures emerged in the side view [Figure 3.20 (d)], clearly indicating the formation of a 2D plasma grating within the interaction region. Unlike the 1D grating generated via nonlinear interaction of two noncollinearly crossed filaments, the plasma grating induced by three interfering filaments was originated from collective superposition of the three pulses, i.e. 1D gratings generated by two of the three filaments and 2D grating performed by all the three filaments. Similar but more complicated hexagonal THG diffraction arrays were observed. Note that the THG pulses were amplified and meanwhile diffracted [Figure 3.21 (a)] due to the strong interaction among the three intersecting filaments. The observed six diffraction spots near the A, B, and C were the ±1 diffraction orders of the THG pulses diffracted by the three 1D plasma gratings. Experimentally, the measured ±1 orders diffraction angle was ±2.58°, which agreed with the calculated ±2.67°. Figure 3.21 (b) schematically shows the observed 2D far-field THG diffraction pattern, where the dashed ellipses indicate the diffraction of the 1D plasma gratings as discussed above. Besides, we concluded that A2 , B2 , and C2 were initiated from additional diffraction of 2D plasma grating. Specifically, we explained such diffraction by assuming equivalent plasma grating, marked with white lines, which diffracted the

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Fig. 3.21. (a) The measured diffraction pattern of the generated THG pulses as a 2D plasma grating created by the interaction of three femstosecond pulses. (b) The sketch of the observed diffraction pattern from the 2D plasma grating. Refer to [60] for details.

THG and the +1 diffraction order led to the observed A2 , B2 , and C2 . The period of the equivalent plasma grating was estimated to be ∼ 4.6 μm based on the measured THG spot at C2 . Accompanied with A2 and B2 that generated with similar equivalent plasma gratings, we explained the originality of the far-field THG diffraction array and the concluded that such 2D plasma grating could be treated as either the superposition of independent 1D plasma gratings or a complicated volume grating.

3.3.5 Third harmonics generation enhancement As mentioned above, THG could be enhanced by filament-interaction-induced plasma gratings [61]. Here we focus on air plasma grating-assisted THG enhancement, and compare with THG from single filaments. 1D plasma grating was formed with two 3°-crossed filaments. As shown in Figure 3.22 (a), the energy and spatial distribution was, respectively, recorded by a photomultiplier tube (PMT) and a CCD camera. Significant THG enhancement induced by filament interaction was observed in comparison with that from single filaments. In the presence of filament interaction,

Fig. 3.22. (a) The schematic setup for the third-harmonic energy and spatial distribution measurement. (b) the measured spectra of the THG pulses with and without filament interaction [61].

212 | Heping Zeng

Fig. 3.23. The measured THG pulse energy as a function of propagation distance in the cases with and without filament interaction. The THG spatial distributions at positions of 100, 102, and 120 cm are presented in the insets. Refer to [55] for details.

the peak spectral intensity of the THG pulse depicted in Figure 3.22 (b) was observed to have an enhancement of ∼ 770 times. The corresponding pulse energy and spatial profiles of the THG are illustrated in Figure 3.23. Stable THG could be generated along with the formation of single filaments due to the nonlinear phase locking between the FW and THG in the presence of clamping intensity of 1013 –1014 W/cm2 . However, a rapid THG decrease occurred when the balance between self-focusing and plasma defocusing was collapsed, indicating the termination of filament (red-diamond curve, Figure 3.22). Such a termination could be also visualized by recording the spatial profiles of the THG along the propagation direction. Clear multiring structures with the divergence angles of 2.25, 4.09, and 5.85 mrad were observed in the propagate position of 100 mm. The outer ring (5.85 mrad) was originated from the off-axis THG, while the inner two rings were identified to be the firstand second-order diffraction of the on-axis THG. With the propagation of the filament (120 cm), the multiring structure degenerated into single outer ring with a blurred THG core. However, in the case of filament interaction, the THG preserved and maintained its energy and spatial profiles along a relatively long distance compared with single filaments as shown by the solid-circled curve in Figure 3.23. Meanwhile, the multiring structure was effectively sustained, indicating interaction-assisted elongation of filaments. We thus concluded that filament interaction would effectively increase the nonlinear phase locking between the FW and TH during filamentary propagation of femstosecond laser pulse, resulting in a prolonged filament due to the modification of dynamic balance between self-focusing and plasma defocusing. Intensity redistribution and localization, occurred from the interference of the interacting filaments, greatly tailored the peak intensity within the interaction region that gave rise to strong nonlinear effects and spatiotemporal modulation. We systematically studied the dependence of THG enhancement on incident pulse polarization,

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crossing angle, peak intensity, and time delay. Figure 3.24 (a) plots the THG enhancement as a function of pump intensity in the case of 9°-crossed filament interaction. The total THG intensity increased slowly at small pump intensities, while dramatic THG enhancement was observed at the pump intensity of 64 GW/cm2. Almost a linear increase of the THG energy was observed as the pump intensity varied in the range from 100 to 220 GW/cm2 , and the THG conversion efficiency decreased as the phase matching failed. As shown in Figure 3.24, the crossing angle and relatively time-delay between the two interfering filaments played vital roles in the THG enhancement. A THG enhancement factor up to 174 was obtained with a crossing angle of 13° under the pump intensity of 220 GW/cm2 .

Fig. 3.24. (a) Spectrally integrated THG intensity under different pump intensities at a noncollinear crossing angle of 9°. (b) Integrated THG enhancement factor as a function of control intensity at different noncollinear crossing angles. (c) Integrated TH enhancement factor measured at different noncollinear crossing angles. (d) Integrated TH intensity as a function of time delay. Inset: the delay dependence of the THG near zero time-delay. Refer to [55] for details.

The relative polarization of the interfering filaments also affected the formation of the plasma structure and subsequent THG enhancement, and the largest THG enhancement would be obtained when the two filaments had the same polarization di-

214 | Heping Zeng rection. As shown in the inset of Figure 3.24 (d), the THG enhancement reached its optimum as the interfering filaments were temporally synchronized, while THG enhancement still existed for tens of picoseconds delays.

3.4 Molecular alignment assisted filament interaction The intense ultrafast filament not only leaves ionization-induced plasma that contributes a negative variation of the air refractive index, but also impulsively aligns the diatomic molecules with periodical field-free revivals [62]. The aligned molecules show orientation-dependent refractive index changes as δnmol⊥ (r, t) = 2π(ρ 0 Δα/n0 )[≪ cos2 θ⊥ (r, t) ≫ −1/3] ,

(3.1)

where θ⊥ is the angle between the field polarization of the pump pulse and molecular axis, ρ 0 is the initial molecule density, Δα = α ‖ − α ⊥ is the polarizability difference between the components parallel and perpendicular to the molecular axis, and n0 is the linear refractive index of the randomly orientated molecules. The molecular alignment degree is characterized by the metric ≪ cos2 θ ≫, which is greater, less, and equal to one-third for parallel, perpendicular, and random molecular alignments, respectively. Different from the above discussion on nonlinear interaction of crossed filaments, we concern here alignment-induced interaction of parallel propagated filaments. Nonlinear interaction of the parallel filaments [57, 58] in the presence of Kerr, plasma and molecular alignment is schematically illustrated in Figure 3.25 (a–c), corresponding to parallel, perpendicular, and random molecular alignment with respect to the polarization of the excitation pulse. Local increase or decrease of the refractive index induced by parallel or perpendicular molecular alignment along with the plasma-induced refractive index decrease would lead to overall net increase or decrease of the local refractive index that gives rise to the attraction or repulsion of

Fig. 3.25. Schematic of light kick by prealigned molecules. The probe filament is, respectively, tuned to match the (a) parallel revival, (b) perpendicular revival, and (c) random orientation with prealigned molecules excited by an advancing pump filament. The relative refractive index variation induced by Kerr effect, field-free molecular alignment, plasma, and the sum of them (total) are plotted as curves. Refer to [58] for details.

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the two parallel filaments, as shown in Figure 3.25 (a) and (b). On the other hand, the plasma wake in randomly aligned molecules could also induce repulsion when the two filaments were close enough [Figure 3.25 (c)]. Here we experimentally demonstrated the filaments attraction and repulsion by manipulating the alignment status and relative delays after the aligning pulse. Experiments was performed with a 35-fs, 800-nm Ti:sapphire laser, and the pulse energy of the pump and prove were set to be 1.6 and 0.8 mJ, focused with two independent lenses with focal length of 100 cm. Two parallel propagated filaments with diameter of 108 and 50 μm and initial separation of 106 μm were generated in air [Figure 3.26 (a)]. Fluorescence imaging technique was applied to directly monitor the relative spatial position of the two interacting filaments under different interaction conditions. Figure 3.26 (b) shows the measured revival structure of the field-free air molecular alignment by using the well-known weak field polarization spectroscopy technique. Delayed Raman response with respect to the pump is shown in the inset of Figure 3.26 (b), indicating a field-free alignment process. The revival periods of N2 and O2 could be identified as 8.3 and 11.6 ps from Figure 3.26 (b). Figure 3.26 (c) shows the simulated alignment signal of air molecules, giving direct sense of the alignment status related to the polarization direction of the excitation pulse. Air molecules were first aligned perpendicularly and then parallel to the probe polarization when the time-delay of the probe filament was tuned to be the half revivals of N2 or O2 , and the separation modulation representing filament attraction or repulsion is shown in Figure 3.27 (a). The probe filament was first kicked away of 15 μm

Fig. 3.26. (a) The fluorescence measurement of the pump and probe filaments in air. The (b) measured and (c) simulated molecular alignment revivals of air molecules. The inset of (b) shows the delayed rotational Raman response of air molecules with respect to the excitation pulse. Refer to [58] for details.

216 | Heping Zeng and then got 12 μm close to the pump filament for N2 half revival, while a kick away of 14 μm and attraction of 9 μm were observed with half revival of O2 . Figure 3.27 (b) plots the fluorescence intensity of molecular ions inside the filament around the molecular alignment half-revivals of N2 or O2 . Evident fluorescence increase or decrease was observed along with the attraction or repulsion of the two filaments around the parallel and perpendicular revivals of the prealigned air

Fig. 3.27. Measured separation and fluorescence intensity of the probe filament, manipulated by prealigned air molecules. The pump probe filaments separation and fluorescence intensity of the probe filament as it was tuned to match the (a) and (b) half-revivals and (c) and (d) full- and threequarter revivals of the aligned molecular N2 and O2 molecules in air. (e) The corresponding fluorescence intensity profiles of the pump and probe filaments. Refer to [58] for details.

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molecules. Similar attraction, repulsion, and fluorescence intensity modulation were also experimentally demonstrated by tuning the delay of the probe filament to the full revival period of N2 (8.3 ps) and 3/4 revival of O2 around 8.7 ps. Detailed repulsion, attraction and fluorescence intensity modulation of the intense probe filaments can be clearly visualized in Figure 3.27 (e), corresponding to various time delays labeled in the points A–H. The filament control by rotational Raman-excitation-induced field-free molecular alignment presented here may stimulate varies of potential applications of remote optical light control, light bullet interaction, atmosphere pollutant detection, and remote sensing.

3.5 Ultrafast optical gating by molecular alignment Field-free alignment of gaseous molecules could function as an ultrafast polarization optical gating with periodic revivals originated from quantum wakes of the impulsively excited molecular wave-packets. Here, we review our experimental explorations on molecular alignment induced optical gating techniques of molecular alignment based cross-correlation frequency-resolved optical gating (M-XFROG). Such molecular alignment induced optical gating has already been demonstrated to support unique applications like ultrafast optical switching for weak pulses, ultrafast optical memories and revivable optical buffers, time-encoding for ultrafast holographic processing of optical information, and so on [63–68]. The M-XFROG technique employs the impulsive alignment of gaseous molecules as a gate function to characterize the ultrashort pulse and exhibits advantages of no phase-matching constraint and applicability to pulses at any wavelength ranging from ultraviolet to far infrared. Ultrashort pulse measurements of ultraviolet, supercontinuum and sub-10 fs pulses were experimentally performed by using the M-XFROG technique. As discussed above, field-free molecular alignment with periodic revivals induce a variation of the refractive index, characterized by the metric ⟨⟨cos2 θ⟩⟩ and is dependent on the relative alignment status with respect to the polarization direction of the excitation pulse. As illustrated in Figure 3.28, a probe pulse that propagates through the prealigned molecules encounters transient birefringence, i.e. the parallel and perpendicular components to the alignment direction undergo different refractive indices. Therefore, the polarization direction of the probe pulse will be rotated. Thus, the prealigned molecules could be considered as a transient gaseous wave-plate that functioned as a polarization gating. Similar to the conventional polarization gating FROG (PG-FROG), the polarization direction of the excitation pulse was tuned to be 45° with respect to that of the target pulse to obtain maximum polarization rotation signal under the same excitation intensity. Sketch of the M-XFROG is presented in Figure 3.29, where a target pulse passed

218 | Heping Zeng

Fig. 3.28. Schematic of birefringence induced by molecular alignment. Refer to [68] for details.

through two crossed polarizers located before and after the molecular gas enabling no transmitted target pulse in the case of random alignment. In the presence of molecular alignment with revival period, a polarization of the target will be rotated and a portion of the target could transmit to the spectrometer. In principle, any molecular alignment revival period could be applied as a gate for pulse measurement. Hereby, the molecular alignment around zero time delay was used since the maximum signal of molecular alignment enabled higher signal-to-noise ratio that helped the precise characteristic reconstruction of the unknown target pulse. Note that the width of such molecular gate is in principle decided by the excitation pulse bandwidth, molecular gas temperature, and species. FROG collects a spectrogram by spectrally resolving the target pulse versus the time-delay that can be used for uniquely retrieve the amplitude and phase for an unknown pulse [69–71]. FROG spectrogram could be expressed as 󵄨󵄨 ∞ 󵄨󵄨2 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 IFROG (ω, τ) = 󵄨󵄨 ∫ ETarget (t)G(t − τ) exp(−iωt)dt 󵄨󵄨󵄨 , 󵄨󵄨 󵄨󵄨 󵄨󵄨−∞ 󵄨󵄨

(3.2)

Fig. 3.29. Schematic of molecular alignment-based frequency-resolved optical gating (M-XFROG). Refer to [68].

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where ETarget (t) is the electric field of the unknown pulse and G(t − τ) is the timedelayed gate function and ESig (t, τ) = ETarget (t)G(t − τ). In order to reconstruct the target pulse ETraget (t) from the measured spectrogram, iterative Fourier-transform algorithm with a generalized projection method is commonly applied [70]. Briefly, the signal ESig (t, τ) is first generated by a guessed field ETarget_Guess (t) and the gate function G(t). Then ESig (t, τ) is Fourier transformed to its frequency domain ESig (ω, τ), whose amplitude is replaced by the square root of the measured FROG trace while keeping the original phase. The modified field E󸀠Sig (ω, τ) is then transformed back to time domain by inversely Fourier transformation to obtain a new signal E󸀠Sig (t, τ) for the next projection. The process repeated unless the error between the measured and reconstructed FROG reaches a satisfactory minimum. Since no nonlinear crystals are required, the M-XFROG technique exhibits unique features for any wavelength and broadband ultrashort pulse characterization. A series of measurements were carried out to characterize the UV, tunable OPA, and supercontinuum (SC) pulses, confirming that M-XFROG is a powerful tool for various ultrashort laser pulse measurement at wavelengths ranging from UV to the far-infrared. Figure 3.30 (a) and (b) presents experimentally measured spectrograms for UV pulses around 267 nm (third harmonic, TH) and 200 nm (fourth harmonic, FH), with the reconstructed intensity envelope and phase and spectrum shown in Figure 3.30 (b and c) and (e and f), respectively. Interestingly, ultrashort extreme UV pulses could be effectively characterized by the M-XFROG technique. Broadband SC pulse generation normally involves various nonlinear processes including self-phase modulation, four-wave mixing, stimulated Raman scattering, self-

Fig. 3.30. Measurements of UV pulses by M-XFROG. (a) and (d) are measured XFROG traces. (b) and (e) are the retrieved temporal intensity and phase.(c) and (f) are the measured and retrieved spectra of the target pulse. TH: third harmonic; FH: fourth harmonic. Refer to [68] for details.

220 | Heping Zeng steepening, and material intrinsic dispersions, which inevitably bring about complicated phase and temporal structures, and makes it a challenge to fully characterize ultrabroadband SC pulses. The difficulty becomes more serious if the generated SC has quite weak field intensity. Traditional techniques require broadband phase-matching for SC characterization and the large time-bandwidth products additional constraints. Some of the recent experiments proposed to overcome the broadband phase-matching problem by angle-dithering a specifically cut thin nonlinear crystal [30, 31]. Nevertheless, high-speed rotation may induce detrimental vibration. As a direct comparison, M-XFROG exhibits obvious convenience in measuring SC pulses since no phasematching is required. A typical measured result is shown in Figure 3.33 (a). Figure 3.31 presents our experimental and retrieved results for a positively chirped SC pulse and a compressed SC pulse [Figure 3.31 (a–c)] by using two pairs of chirped mirrors [Figure 3.31 (d–f)]. It was clearly demonstrated that the M-XFROG can be not only used to characterize the complicated long pulse, but also versatile for diagnosis of ultrashort laser pulses of ∼ 10 fs in various spectral regions. Moreover, the polarization gating essential of M-XFROG enabled the possibility of simultaneous measurement of pulses at one spectrogram with the same polarization direction but consisting of different wavelengths. As a demonstration, a target pulse composed of the FW pulse at 800 nm and third-harmonic pulse at 267 nm was experimentally measured [65] and the results are shown in Figure 3.32.

Fig. 3.31. M-XFROG for the measurement of SC pulses. (a), (b), and (c) are experimental and retrieved results for a positively chirped SC pulse, respectively, corresponding to the measured M-XFROG trace, the retrieved temporal intensity and phase, and the measured and retrieved spectra. (d), (e) and (f) are experimental and retrieved results for a compressed SC pulse by using two pairs of chirped mirrors, respectively, corresponding to the measured M-XFROG trace, the retrieved temporal intensity and phase and the measured and retrieved spectra. Refer to [68] for details.

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Fig. 3.32. M-XFROG for the simultaneous measurement of pulses at 267 nm (TH) and 800 nm (FW). (a) The measured M-XFROG trace containing SH and FW pulses. (b) and (d) are retrieved M-XFROG trace and temporal intensity of the SH pulse. (c) and (e) are retrieved M-XFROG trace and temporal intensity of the FW pulse. Refer to [65, 68] for details.

Fig. 3.33. (a) Schematic setup of the SC pulse measurement. (b) The directly measured spectrum of the SC pulse and the time marginal of the M-XFROG trace. (c) Wavelength response of the M-XFROG. (d) The molecular-alignment-induced polarization rotation efficiency as a function of the excitation intensity. Inset: Example power variation of the polarization modulated pulse as a function of incident pulse power. Refer to [68] for details.

222 | Heping Zeng During the M-XFROG measurement on the UV and SC pulses, important issues such as reliability, noise process and reduction, wavelength response, modulation depth, and sensitivity must be carefully addressed. For example, since the SC measurements were performed noncollinearly, one should pay attention to the time smearing effect. For a small crossing angle of 4°, the time smearing δ t was estimated to be ∼ 0.6 fs, which made negligible influence on the measurement. Moreover, owing to the broadband frequencies of the SC pulse, a reliable measurement requires extremely careful concerns on the wavelength response of the involved optical components. The wavelength response, acquired by comparing the directly measured spectrum of the SC pulse (blue-squared curve) and the time marginal of the M-XFROG trace (orangecircular curve) in Fig 3.33 (b), is shown in Figure 3.33 (c). Wavelength calibration was implemented in each retrieval process. Figure 3.33 (d) plots the nonlinear increase of the modulation efficiency as a function of the intensity of the molecular aligning pulse. The inset of Figure 3.33 (d) presents a nearly linear polarization modulation response to the incident target power under the excitation intensity of 9.1 × 1013 W/cm2 .

3.6 Conclusions As a conclusion, this chapter reviews our recent experimental explorations on the propagation of intense ultrashort laser pulse in terms of ultrafast nonlinear optics. The main results are summarized as follows. (1) Experimental studies on spatiotemporal collapse of ultrashort pulse propagation in quadratic nonlinear media due to modulational instability are reviewed. 2D multicolored transverse arrays are experimentally observed in a quadratic nonlinear medium based on cascaded noncollinearly quadratic couplings and spatial breakup of input pump beams. Furthermore, twodimensional multicolored up-converted parametric amplifications are realized with preserved coherence by using an incident weak white light seed pulse. As a result of spatiotemporal collapse of femstosecond pulses, CCE are observed under strong phase-matched fundamental pump, which are further amplified to support tunable broadband femtosecond pulse generation. Difference in frequency generation between pump and seeded amplification of CCE pulses was verified to exhibit a constant CEP, which can be used as an all-optical control of CEP. (2) Intense laser pulses experience dramatic self-focusing, multiphoton ionization, spectral broadening, and phase modulation when propagating in atmosphere. Filamentary propagation of intense femtosecond laser pulses, due to counterbalance between Kerr self-focusing and plasma defocusing, has been so far demonstrated quite useful for various interesting applications. We show that coalescence of interfering noncollinear intense femtosecond pulses assisted periodic wavelength-scale self-channeling with encircling air molecules. Such a periodic modulation can be functioned as plasma grating that support diffraction of intense incident pulses, experimentally confirmed by the co-axis propagated third-harmonic enhancement and diffraction. The generated

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plasma grating exhibits unique features of ultrahigh damage threshold, long lifetime, easy period tunablity, and convenient multidimension extension. Besides, an efficient approach to explore and control nonlinear interactions between two orthogonally polarized femtosecond filaments launched parallel in air by field-free molecular alignment was proposed, resulting in attraction and repulsion of parallel filaments with different spatiotemporal proximities. (3) We further demonstrate such field-free alignment could function as an ultrafast polarization optical gating with periodic revivals. Our experimental explorations of M-XFROG for ultrashort pulse characterization are reviewed. The M-XFROG technique exhibits promising advantages of no phase-matching constraint and applicability to pulses at various wavelengths from UV to far-infrared. Ultrashort UV pulse, long supercontinuum, and ∼ 10 fs pulses are successfully characterized by the M-XFROG technique. Acknowledgment: The experiments reviewed in this chapter are financially supported by National Key Scientific Instrument Project (2012YQ150092), National Basic Research Program of China (2011CB808105), and National Natural Science Fund (61127014, 10990101, 11004061, and 91021014). The author thanks Dr. Jia Liu for offering valuable helps in editing this chapter. Experiments reviewed here include collaborating contribution from Profs. Jian Wu, Wenxue Li, Liangen Ding, and graduate students working or have worked in the group.

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Qi Guo*, Daquan Lu, Dongmei Deng

4 Nonlocal spatial optical solitons

4.1 Introduction to optical soliton research As an introduction to the subject of nonlocal spatial optical solitons, in this section we briefly introduce the history of optical soliton research and the related basic concepts, which include the optical Kerr effect and its spatial and temporal nonlocality, the model of nonlinear slowly varying optical envelopes, the solution of the model, and its physical connotation, etc. This section is the basis to read and understand the following sections.

4.1.1 Historical background of optical solitons Almost all of the literatures related to solitons mention the story that the famous British engineer Russell observed the soliton (water) wave for the first time in a canal in 1834 [1]. However, the history of optical soliton is not as long as that of the mechanical soliton. In fact, since the Japanese scholar Hasegawa theoretically predicted the possibility of the existence of optical solitons in 1973 [2, 3], the research on optical solitons has just stepped into its forties. So what is “soliton” exactly? We usually call the local traveling-wave solutions¹ of the nonlinear wave equation as “solitary waves,” and the stable solitary waves, which do not disappear after mutual collision and are without any change or with only slight change in the shape and propagation speed (like the case of the two-particle collisions), are called solitons. The so-called optical solitons are local optical waves (optical envelopes) that propagate in optical nonlinear media, and include temporal optical solitons, spatial optical solitons, and spatiotemporal optical solitons. On one hand, the optical pulse broadens when propagating in an optical waveguide due to dispersion effect; correspondingly, the optical beam diverges during its propagation because of the diffraction effect. On the other hand, due to the self-induced nonlinear refractive index, the optical pulse is compressed (or the optical beam is focused). The

1 Local solutions refer to the solutions of nonlinear differential equations in the confined space (or time) area, and such solutions tend to zero or constant at infinity. Qi Guo (*Corresponding Author), Daquan Lu, Dongmei Deng: Laboratory of Nanophotonic Functional Materials and Devices, School of Information and Photoelectronic Science and Engineering, South China Normal University, Guangzhou 510631, People’s Republic of China Email: [email protected]

228 | Qi Guo, Daquan Lu, Dongmei Deng temporal (or spatial) optical soliton is a stable propagation state for the optical pulse (or optical beam) when the linear dispersion (or diffraction) effect balances the nonlinear effects precisely and thus the pulse duration (or beam width) keeps invariant. As the name suggests, the spatiotemporal optical soliton is a stable propagation state of optical pulsed beam whose shape keeps invariant in three-dimensional spatiotemporal coordinates (three dimensions are left when one dimension of propagation-direction is removed from the four-dimensional space-time).

A. Temporal optical solitons The optical solitons firstly predicted theoretically [2] and observed experimentally [4] are temporal optical solitons.² In 1973, Hasegawa and Tappert published their pioneering work on the optical soliton research. They studied the propagation of optical pulse in an optical fiber constructed by the self-focusing Kerr medium,³ and derived the propagation equation that describes the evolution of the optical pulse. Although this equation has the structure of the (1 + 1)-dimensional⁴ nonlinear Schrödinger equation (NLSE) (in fact it is the NLSE with the term of dissipation), Hasegawa and Tappert did not associate this equation with the (1 + 1)-dimensional NLSE that had been solved by the former Soviet Union mathematician Zakharov and Shabat via the inverse scattering method. Hasegawa later recalled that [6]⁵ they had not read the paper before completing the theoretical prediction of temporal optical solitons, although Zakharov and Shabat’s paper [7] had been translated into English in 1972 and was listed in the references. Both the analytical result and the numerical result show that there exists a bright pulse solution⁶ with the temporal waveform of hyperbolic secant [sech(t)]

2 Based on the following two reasons, the authors think that the discovery of temporal optical solitons should be regarded as the beginning of the optical soliton research. First, although the theory and experiment of optical beam self-trapping are earlier than the temporal optical solitons, the early theory and experiment do not support each other [5]. In fact, associating the beam self-trapping phenomenon with soliton and reaching the experiment of spatial optical soliton which is consistent with the theory, is after the successful experiment of temporal optical solitons and is inspirited by it. Second, the first research boom of optical solitons is on the temporal optical solitons, rather than on the spatial optical solitons. The spatial optical solitons had not been a research hotspot until the experimental verification of photorefractive solitons in 1993. 3 The refractive index of Kerr nonlinear media is given by n = n 0 + n 2 |E|2 , where E is the electric field, n 0 is the linear refractive index of the media, n 2 is the nonlinear refractive index coefficient (Kerr coefficient). n 2 > 0 corresponds to self-focusing Kerr media, n 2 < 0 corresponds to self-defocusing Kerr media, detailed in the discussion in Section 4.1.2. 4 The number in front represents the propagation (longitudinal) direction, the number in the back represents the dimensionality of cross section which is perpendicular to the longitudinal direction, and the same below. 5 See preface in Ref. [6] 6 Bright pulse is the convex waveform along the time coordinate axis, and is a bright spot in dark background. Dark pulse is the concave waveform along the time coordinate axis, and is a dark spot

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in anomalous group velocity dispersion regime, which is stable during its propagation [2], while there is a stable dark pulse solution with the temporal profile of hyperbolic tangent [tanh(t)] in normal group velocity dispersion regime [3]. Hasegawa and Tappert named the bright pulse solution as envelope soliton [2], which is bright temporal optical soliton in fact, and called the dark pulse solution as envelope shock⁷ [3], which is actually dark temporal soliton. Hasegawa and Tappert associated the stable nonlinear optical envelope solution with the soliton for the first time, although they made a minor and trivial mistake in naming the dark pulse solution. In order to propagate the bright temporal optical soliton stably, the order of input power is about 1 W for the pulse duration of 1 ps. The temporal optical solitons in fiber waveguides are formed by the interaction between the nonlinear self-phase modulation effect and the linear group velocity dispersion effect [8]. When an optical pulse propagates inside the optical fiber, due to the nonlinear self-phase modulation effect, the carrier frequency changes (frequency modulation), and the degree and sign of the change are dependent on the pulse waveform. For a bright pulse, the pulse leading edge has lower carrier frequency than the trailing edge due to the self-phase modulation; on the other hand, in the anomalous group velocity dispersion regime ∂v g /∂ω > 0, the pulse leading edge moves more slowly than the trailing edge, then the pulse is compressed. This compression effect is exactly opposite of the pulse broadening effect induced by linear dispersion effect alone. If the pulse has the appropriate amplitude and waveform, the compression effect can exactly balance the broadening effect, and the pulse will propagate steadily without deformation, then the bright temporal optical soliton forms. Dark pulse is just the opposite, the pulse leading edge has higher carrier frequency than the trailing edge due to the self-phase modulation, so the pulse compression only occurs in the normal group velocity dispersion regime ∂v g /∂ω < 0, which would balance the pulse broadening effect induced by linear dispersion effect alone; thus the dark temporal soliton forms. In 1980, Mollenauer et al. [4] first successfully observed the stable propagation of the bright temporal optical soliton of which the wavelength is 1.55 μm (in the anomalous group velocity dispersion regime) and the pulse duration is 7 ps (full width at half maximum) at the end of a 700 m long standard single mode fiber. The experimental results are highly consistent with Hasegawa’s theoretical prediction. In the late 1970s, there were two major progresses for the research in the field of photoelectrics which were the premise for the success of bright temporal optical soliton experiment. First, the low loss (less than 0.2 dB/km) optical fiber around 1.55 μm wavelength was carried out successfully. Second, the mode-locked color center laser whose wavelength is tunable (adjustable range is 1.4–1.6 μm) had been developed in the same wavelength

in bright background. Similarly, the bright beam is the convex spot along the spatial coordinates, and dark beam is the concave scotoma along the spatial coordinates. 7 Shock waves are the wave packets whose envelope edge becomes very steep and then makes its slope infinite during their propagation, see Ref. [9] and Section 4.3.1 in Ref. [8].

230 | Qi Guo, Daquan Lu, Dongmei Deng range. The first observation of dark temporal soliton was independently completed by Emplit et al. [10] and Krökel et al. [11], respectively. Because the input pulse has no phase jump which is needed by single dark soliton solution, what they observed was actually a pair of coexistent dark solitons. In order to form the propagation of single dark soliton, the input dark soliton with an appropriate phase jump must be constructed. Such work was completed by Weiner et al. later [12]. The parameters used in the experiment of Emplit et al. are listed as follows: the background pulse duration is 26 ps, the carrier wavelength is 0.595 μm, the dark soliton pulse duration is 5 ps, and the fiber length is 52 m [10]. And the parameters for the experiments of Krökel et al. and of Weiner et al. are 100 ps, 0.532 μm, 0.3 ps, 10 m [11] and 1.76 ps, 0.620 μm, 185 fs, 1.4 m [12], respectively. The experimental results above are the dark solitons under a limited bright background, rather than the tanh(t) waveform dark solitons theoretically predicted by Hasegawa [3], which are under the infinite bright background. The true experiment of dark soliton under the infinite bright continuous wave background was not successfully accomplished until 1990 [13]. Under the stimulus of the successful experiment of observing bright temporal optical solitons in 1980, Hasegawa et al. studied the possibility of using bright temporal optical soliton as the communication carrier in optical fiber communication system, and put forward the concept of optical fiber soliton communication in 1981 [14]: to replace the linear pulse information carrier in the conventional optical fiber communication system, bright temporal optical solitons whose pulse duration remains constant during their propagation in optical fiber is used as the communication carrier in optical fiber communication system. The linear pulse broadening during propagation in fiber caused by dispersion effect is the limit of increasing the communication speed in conventional optical fiber communication system. But in the optical fiber soliton communication system, the dispersion becomes the necessary condition for the stable propagation of bright temporal optical solitons. The calculation given by Hasegawa et al. shows that the communication speed of optical fiber soliton communication system is higher than conventional optical fiber communication system by one or two orders of magnitude [14]. From the late 1980s to the mid-1990s, the optical soliton communication system [6] had aroused people’s widespread attention, and had been considered as one of the effective solutions of achieving high speed, large capacity, long distance all-optical communication. At that time, many countries invested a lot of manpower and money to research the optical soliton communication system and its related key technology, which formed the first round of research climax of the optical soliton. However, because of the unexpected breakthrough and mature of dense wavelength division multiplexing (DWDM) technology, the conventional linear communication systems can also meet the needs of large capacity, high-speed communications. With the mature of DWDM technology, the study enthusiasm of optical fiber soliton communication system has been decreased since the end of the last century. In recent decades, there are only scattered researches on the temporal soliton and the optical fiber soliton communication system. The international mainstream of optical

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soliton research turns to the spatial optical solitons and its application in all-optical information processing.

B. Spatial optical solitons Although both temporal optical solitons and spatial optical solitons are mathematically described by the NLSE and its deformation equation (adding a number of additional terms on the basis of the NLSE), the physical mechanisms of their formation are completely different. The temporal soliton is a result of the interaction between the linear dispersion effect and nonlinear effect; however, the spatial soliton is a result of the interaction between the linear diffraction effect and nonlinear effect. Due to the multidimension of spatial diffraction and the diversity of materials, there are many types of spatial optical solitons, and the research contents about spatial optical solitons are much more rich compared with that of temporal optical solitons.⁸ According to the physical mechanism about generating the nonlinear effect, spatial optical solitons can be classified into Kerr and Kerr-like (or quasi-Kerr) solitons [15, 16], photorefractive solitons [15, 16], quadratic solitons (also called cascading solitons or parametric solitons) [15, 16], nematicons [17, 18], etc. Kerr solitons and Kerr-like solitons are the spatial optical solitons that exist in materials whose refractive index can be expressed as the function of intensity.⁹ In Kerr materials, the nonlinear refractive index originates mainly from the contribution of electrons. Photorefractive solitons are the spatial optical solitons produced in photorefractive materials. The photorefractive effects are as follows: under the optical radiation, the material first forms a charge field corresponding to the light intensity because of the photoconductive effect, thereby the refractive index changes with the light intensity because of the linear electro-optic effect. Quadratic solitons are the spatial optical solitons produced in the process of second-order nonlinear effects rather than in the process of third-order self-action nonlinear effects, so in the process of producing the quadratic solitons, it is inevitably accompanied with the energy exchange among optical fields with different frequencies. The nematicon, which is the theme of this chapter, is the spatial optical soliton produced in nematic liquid crystals because of the reorientation process of liquid crystal molecules. Regardless of the mechanism of the nonlinearity for forming solitons [19], according to different intrinsic properties, solitons can be classified into several different

8 At present, temporal optical solitons are only obtained in the quartz glass optical fiber, and optical fiber waveguide’s dispersion effect is one dimensional. 9 Strictly speaking, these material should be called as the local Kerr (or Kerr-type) nonlinear material. Due to historical reasons, people did not begin to gradually realize that the Kerr effect should be classified into two types: the local type and the nonlocal type, until 1997. The difference between local Kerr nonlinearity and nonlocal Kerr nonlinearity is shown in Section 4.1.2. However, people are now used to the past name, ignoring the rigor of concept.

232 | Qi Guo, Daquan Lu, Dongmei Deng couple of categories, such as bright solitons and dark solitons, incoherent solitons and coherent solitons [15, 16], single component solitons and vector solitons (multicomponent solitons) [5, 20, 21], continuous solitons and discrete solitons [15, 16], bulk material solitons and surface-wave solitons [22, 23], traveling-wave solitons and cavity solitons (standing-wave solitons) [15, 16], local spatial optical solitons and nonlocal spatial optical solitons, etc. Compared with the coherent solitons produced by coherent beams, incoherent solitons are produced by partially or completely incoherent beams. For the single component solitons with a single frequency and a single polarization direction, the vector solitons (also called multicomponent vector solitons) can be classified into two types: narrowly defined vector solitons and generalized vector solitons. The former is a symbiotic state (both components propagate with the state of solitons) that formed because of the nonlinear coupling between the two independent components of the electric field (such as the two components polarized in the direction of the fast axis and slow axis of the uniaxial crystal, respectively). But the latter is a soliton symbiotic state that formed because of the nonlinear coupling between the optical fields with different frequencies. Relative to the continuous solitons in continuous and homogeneous materials, the discrete solitons are the spatial optical solitons that exist in discrete structures which are transversely periodical and discontinuous (such as periodic waveguide array structures), and the spatial optical solitons generated in the periodic structures with continuous change can also be generally called discrete solitons. Different from the physical mechanism of continuous spatial optical solitons, the formation of discrete spatial optical solitons is the result of the balance between the nonlinear modulation effects and the discrete diffraction effects which are generated by the energy coupling of the periodic discrete structures. Compared with the bulk material solitons in continuous bulk (relative to the solitons’ scale) materials, the surface-wave solitons are the spatial optical solitons that exist near the interfaces of two materials with different optical properties, and they are new solitons just found in recent years. Relative to the traveling-wave solitons formed when propagating in media, cavity solitons are the standing-wave solitons that formed in the nonlinear medium which is put into the optical cavity structure. In fact, the Kerr solitons are the local spatial optical solitons, and all of the quadratic solitons, photorefractive solitons and nematicons are nonlocal spatial optical solitons. The nonlocal spatial optical solitons are the theme of this topic and will be introduced in detail in the following sections. Although the leading role in the first round of climax for the optical soliton research is the temporal optical soliton instead of the spatial optical soliton, the theoretical and experimental work of the spatial optical soliton is earlier than the temporal optical soliton. However, theoretical model of the spatial optical soliton was not consistent with experimental phenomena until the year 1985, which was later than the time when the consistence for temporal optical solitons was reached. As early as 1962, the former Soviet Union scholar Askar’yan have already predicted that strong electromagnetic fields can form self-induced waveguides and thereby result in nondiffraction

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propagation in plasma [24]; this was the earliest discussion of the self-trapping phenomenon of beam. Later, Talanov [25] and Chiao [26] , respectively, independently made quantitative discussion on the nonlinear propagation of electromagnetic fields in plasma and Kerr self-focusing media, and found out the self-trapping phenomenon of optical beam. Both of them got the conclusion that the self-trapped solution of the (1 + 1)-dimensional beam has a hyperbolic secant spatial distribution, and in addition Chiao et al. got the numerical solution of the self-trapped (1 + 2)-dimensional beam. But soon Kelley [27] proved that the self-trapping mode of (1 + 2)-dimensional beam in Kerr self-focusing nonlinear medium is unstable and it will be sure to appear catastrophic collapse: the beam diameter will tend to zero and the beam intensity will be infinity. The equation obtained by Kelley at that time is the NLSE. Although Kelley did not link this equation with the NLSE, this was the first time to get the correct equation of slowly varying optical beam propagating in nonlinear Kerr medium. Of course, this is a digression. Actually, the quasi-(1 + 1)-dimensional beam in (1 + 2)-dimensional structure is also unstable [5, 6].¹⁰ As a result, the self-trapped beam in Kerr selffocusing medium can only propagate stably in (1 + 1)-dimensional structure. In a (1 + 1)-dimensional structure, a dimension of the two-dimensional diffraction space is restrained in some way (such as through the structure of planar dielectric waveguide), therefore the (1 + 1)-dimensional beam’s propagation model in Kerr medium has the same mathematical structure with the temporal optical pulse’s model in optical fiber, and thereby makes self-trapping optical beams propagate as stable as the temporal optical solitons. A few years later, both the approximate analytical results [28] and the numerical results [29, 30] proved that the saturable nonlinearity¹¹ is the key factor for restraining the catastrophic collapse and forming self-trapped (1 + 2)-dimensional beam. Almost at the same time, the former Soviet Union scholar also independently obtained the similar results [31].¹² Until the end of last century, saturable nonlinearity was the key to all sorts of newfound (1 + 2)-dimensional stable spatial optical solitons [5]. In recent years, people have realized that the nonlinear nonlocality is also one of the factors which suppress the catastrophic collapse of (1 + 2)-dimensional beams [32]. The root cause for the stability of (1 + 2)-dimensional photorefractive solitons is the saturable nonlinearity [5]; that of quadratic solitons and nematicons is nonlinear nonlocality [32, 33]. Is it possible to prove in principle that the saturable nonlinearity is equivalent to the nonlocal nonlinearity on restraining the catastrophic

10 More discussion on this problem see Section 14.1 in Ref. [6]. 11 The saturable nonlinear medium is described by the model n = n 0 + n 2 |E|2 /(1 + |E|2 /E 2s ) [E s is the saturation parameter (a real constant)], which means that its nonlinear refractive index is of the upper limit. 12 Although the English translation of Ref. [31] is not found, the following facts can support the author’s view: the introduction of Ref. [30] indicates that their results and the results of Ref. [31] are “very similar numerical results,” the authors of Ref. [30] also compared their results with the results of Ref. [31].

234 | Qi Guo, Daquan Lu, Dongmei Deng collapse of (1 + 2)-dimensional beams? In our point of view, this would be an open theoretical problem. The spatial optical soliton is a result of the precise balance between the nonlinear effect and the linear diffraction effect under their combined action. For the bright beam propagating in Kerr self-defocusing medium, the nonlinear refractive index (proportional to the beam intensity) sensed by the beam center is greater than sensed by the beam edge; therefore, a self-induced convex lens is formed. With the same principle of converging beams with convex lens, the phase velocity (c/n) at the beam center is lower than that at the beam edge, which makes the cophasal surface concave and results in the convergence of the beam. The beam’s convergence effect generated by the nonlinear effect is the exact opposite of the divergence effect which appears when linear effect exists alone. When the two effects reach balance, the beam gets self-trapped and becomes bright spatial optical soliton. It is the same principle for the dark optical beam to form the dark spatial optical soliton in self-defocusing media. The first experimental works include the stable (1 + 2)-dimensional self-trapped beam [34] in lead glass in 1968 and the stable (1 + 2)-dimensional self-trapped beam [35] in sodium vapor in 1974. In the lead glass the stable self-trapped beam, of which the power is 3 W, the beam width is approximately 50 μm, the length of the lead glass is 15 cm (working wavelength is 514 nm, n0 = 1.75, the Rayleigh range is about 5.3 cm), is observed. Also in the sodium vapor the stable self-trapped beam, of which the power is 20 mW, the beam width is approximately 70 μm (full width of half-power points), the length is 12 cm (working wavelength is 589 nm, n0 ≈ 1,¹³ the Rayleigh range is about 1.9 cm) is observed. The nonlinear mechanism of lead glass is the thermal nonlinearity, while the nonlinear mechanism of sodium vapor comes from the saturable nonlinearity nearby the electronic absorption peak in the two-level energy system. Both the two types of nonlinearity mentioned above are not typical Kerr nonlinearity, therefore the catastrophic collapse of self-focusing does not appear (but at that time there was no reasonable and quantitative theoretical explanation on the reason of its stability). Although the purpose of the experiments in lead glass is to investigate the self-focusing of optical beam, the stably propagating self-trapped pattern obtained in experiment actually is the (1 + 2)-dimensional spatial optical soliton that formed because of the nonlocal nonlinearity, and almost 40 years later did people realize this truth [36]. Perhaps because of lacking of theoretical support, the two early experimental works do not cause much response. However, in the history of the spatial optical soliton research, it is still an event that worth being mentioned. Over 10 years later, people did not begin to realize that the (1 + 1)-dimensional self-trapped beam has soliton’s characteristics until the temporal optical soliton experimental work was highly consistent with the theory. In 1985, Barthelemy et al. first clearly associated the

13 The data about the linear refractive index of sodium vapor is not found, but the linear refractive index of gas is close to 1.

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(1 + 1)-dimensional self-trapping phenomenon of beam with soliton [37], and obtained the experimental result of spatial optical soliton which is consistent with the theory. Their first experiment was finished in the (1 + 2)-dimensional CS2 liquid, where they used a special method to realize a quasi-(1 + 1)-dimensional propagation.¹⁴ Until their second experiment, they obtained the true (1 + 1)-dimensional spatial optical soliton in a “sandwich” structure of the planar waveguide which consists of two glasses and CS2 liquid. Soon afterward, the stable propagation of the (1 + 1)-dimensional spatial optical soliton is successively realized in glass [38], semiconductor [39], and polymer [40]. Similar to the principle for dark temporal optical solitons existing inside the optical fiber in normal group velocity dispersion regime, in the self-defocusing medium (n2 < 0), there are also dark spatial optical solitons existing [5, 19]. Dark spatial optical solitons differ from bright spatial optical solitons in two aspects. First, the dark spatial optical solitons are stable in (1 + 2)-dimensional self-defocusing Kerr medium [41], but the bright spatial optical solitons do not exist in (1 + 2)-dimensional self-focusing Kerr medium.¹⁵ Second, the phase of dark spatial optical solitons has discontinuity (singularity) [5, 19]. However, that of bright spatial optical solitons is continuous. The event with epoch-making meanings in the history of spatial optical soliton research is the theoretical prediction [42] and experimental validation [43] of photorefractive solitons. The importance of photorefractive solitons mainly manifests in the following aspects [5, 19]. First, the power to generate photorefractive solitons is very low (even small to microwatt magnitude), so only using the continuous-wave laser source can realize the spatial optical soliton propagation, and there is no need to use pulsed laser source which is used in early experiments to get higher power [37–40]. Second, in photorefractive materials, the low-power spatial optical solitons can induce waveguide structure which can be erased by the changes of electric field and temperature. The induced waveguide can make high power beams with other frequencies propagate along this waveguide and realize the function of light controlling light. Finally, photorefractive nonlinear is not only nonlocal, but also saturable. Because of these unique features, the photorefractive materials become the ideal platform to realize the propagation and the (1 + 2)-dimensional interactions of all kinds of solitons (such as incoherent solitons and discrete solitons). Just like the photorefractive nonlinearity, nonlocal nonlinearity is also one of the two important factors of stable propagation of spatial optical solitons. Although the previous studies had more or less involved the problem for propagation in nonlocal nonlinear medium, the systematic study of nonlocal spatial optical soliton began with an article [44] published in “Science” in 1997 by Snyder, an international famous ex-

14 Because the paper is published in French, we cannot read the original text and thereby cannot give a description. But Ref. [5] has a detailed explanation on the principle of the experiment. 15 The bright self-tapped beams are unstable in the (1 + 2)-dimensional self-focusing Kerr medium, they, therefore, cannot be considered as the spatial optical solitons.

236 | Qi Guo, Daquan Lu, Dongmei Deng pert in guide-wave optics. Under the strongly nonlocal condition, Snyder and Mitchell simplified the NNLSE into a linear model, i.e. the Snyder–Mitchell model, and found that there is spatial optical soliton solution. They called this type of spatial optical soliton as “accessible soliton.”¹⁶ It is really a great creation to transform the nonlinear problem to a linear problem. The famous nonlinear optics expert, Professor YR Shen gave a high evaluation to this. In the same issue of “Science,” he published a comment [45] in which he thought the Snyder–Mitchell model is “invaluable.” But he also expressed concern that: until then, the correlated length (nonlinear characteristic length) of the known materials was only in the micron range. For conventional beams whose width is also in the micron range, the known materials are all weakly nonlocal; it seems that the materials with strong nonlocality had not be found. However, he optimistically predicted that Snyder and Mitchell’s prediction may stimulate experimentalists to try to find the method to extend the material’s characteristic length. Sure enough, after 6 years, Assanto and his team first made the theoretical prediction (in 2003) [46] and confirm (in 2004) [47] that the reorientation mechanism of nematic liquid crystal can realize the strong nonlocality under certain conditions. In the following year, another material with strongly nonlocal nonlinearity was confirmed [36]. This material is lead glass, in which the stable propagation of elliptical solitons and optical vortex-ring solitons is realized. In recent years, the research on nonlocal spatial optical solitons has aroused wide concern [17, 18, 48–50].

C. Spatiotemporal optical solitons: light bullets The spatiotemporal optical solitons, known as light bullets, result from the balance among the group velocity dispersion effect (linear temporal effect), the diffraction effect (linear spatial effect), and the nonlinear effect during propagation, which makes the pulsed beam invariant in temporal dimension and spatial dimensions (cross section) perpendicular to the direction of propagation. Since the possibility of its existence is predicted by Silberberg [51], the light bullet is a frontier problem not merely in nonlinear optics, but in nonlinear science [52]. Like the (1 + 2)-dimensional selftrapped beam, the self-tapped spatiotemporal pulse beam can produce spatiotemporal self-focusing in (1 + 3)-dimensional Kerr self-focusing nonlinear medium until the catastrophic collapse happens [51]. It has been theoretically proved that [53] the saturable nonlinearity, the cascading quadratic nonlinear process, the self-induced transparency effect, the nonlocal nonlinearity, the waveguide array structure the refractive index of which periodically changes on the cross section, etc., can prevent the catastrophic collapse, resulting in the stable propagation of the light bullet.

16 Because the governing equation is a linear equation (although its essence are still nonlinear), accessible solitons are easier to deal with, compared to the soliton described by the NLSE.

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The stable propagation of light bullets has been experimentally observed in cascading quadratic nonlinear process [54], waveguide array structure [52], and bulk material in the normal group velocity dispersion regime [55]. Liu et al. first achieved stable propagation of the light bullet [54], but the light bullet they obtained is (1 + 2)dimensional quasi-light bullet¹⁷ in the anomalous group velocity dispersion regime instead of (1 + 3)-dimensional real light bullet. The (1 + 3)-dimensional real light bullet was observed in waveguide array structure by Minardi et al. [52] in the anomalous group velocity dispersion regime. Koprinkov et al. [55] observed the stable propagation of light bullet in three-dimensional bulk medium of different materials (argon gas, krypton atoms gas, CH4 molecular gas and fused silica solid) in normal group velocity dispersion regime. The existing theory suggests that [51, 53], light bullets only exist in the anomalous group velocity dispersion regime of the self-focusing Kerr medium, and are impossible in the normal group velocity dispersion regime. Therefore, it needs new theory to explain the experimental results obtained by Koprinkov in the normal group velocity dispersion regime of self-focusing Kerr medium. The mechanism may be derived from the dominated higher order (fifth-order) nonlinear effects [55] induced by super strong laser field. These continuously completed experimental works show that the door of light bullet research has already been opened, waiting for the arrival of younger generation of scholars with curiosity.

D. Breathers and solitons At the end of this section, we would like to discuss the difference of two terms – breatherss and solitons. Breathers and solitons are twins in most of the nonlinear system, and are also two different states but closely related in the evolution process of nonlinear system. A breathers [56, 57] is the local traveling-wave solution with periodic oscillation along the direction of propagation (or temporal evolution coordinate) in nonlinear system, and a soliton is the local traveling-wave solution whose shape (waveform and its temporal width or spatial width) keeps unchanged along the direction of propagation (or temporal evolution coordinate) in nonlinear system. Although these two kinds of solutions of light envelopes are called optical solitons in many literatures of nonlinear transmission optics generally, but actually they are physically different: the optical soliton forms because of the precise balance between the dispersion effect (or diffraction effect) and the nonlinear effect, while the breathers [58] is the result of partial balance between the dispersion effect (or diffraction effect) and the nonlinear effect. Under this concept, the author thinks that the higher order

17 They obtained the stable propagation of light bullet in the case of one-dimensional dispersion and one-dimensional diffraction (1 + 2D), rather than in the case of one-dimensional dispersion and twodimensional diffraction (1 + 3D).

238 | Qi Guo, Daquan Lu, Dongmei Deng soliton solutions [8, 59] of the NLSE should be strictly called optical breatherss rather than optical solitons.

4.1.2 Optical Kerr effect and its spatial and temporal nonlocality Optical soliton is a physical phenomenon that is closely related to the optical Kerr effect. Before discussing the characteristics of the optical soliton, we first introduce the concepts of optical Kerr effect and its spatial and temporal nonlocality. Optical Kerr effect belongs to the third-order nonlinear optical effects, and is one of the main effects in the field of nonlinear optics. When a strong laser beam propagates in the medium, it changes the refractive index of the medium; meanwhile, the change of the refractive index affects the propagation behavior of the laser. This is the so-called optical Kerr effect (optical Kerr nonlinear process) [60, 61].¹⁸ The main difference between optical Kerr effect and other optical effects (such as optical frequency doubling effect, optical frequency mixing effect (sum frequency mixing and difference frequency mixing), and optical parametric amplification, optical parametric oscillation) is that the latter effects produce new frequency during the nonlinear effect process, while the former does not. In this sense, the optical Kerr effect is a self-action nonlinear effect, i.e. the high-intensity laser beam influences its own propagation behavior through stimulating the nonlinear response of the medium. The physical mechanism that can produce the optical Kerr effect include [60, 61], molecular reorientation, thermal nonlinearity, photorefractive effect, electronic contribution, electrostriction, etc. No matter what the physical mechanism is, the refractive index caused by the optical Kerr effect is changed; therefore, the refractive index can always be phenomenologically represented as n(r, z, t) = n0 (r, z) + Δn(r, z, t) ,

(4.1)

where n0 is the linear refractive index of the medium (supposing that the linear dispersion of the medium is neglected, n0 is independent upon the time, but may be a function of space coordinates), Δn is the variation of the nonlinear refractive index that caused by the optical field (shortly called nonlinear refractive index). r is the Ddimensional transverse coordinate vector that perpendicular to the coordinate z (the propagation direction of beams) (when D = 1, r = xex ; but when D = 2, r = xex + yey ). Generally speaking, the nonlinear refractive index Δn is a function of spatial and temporal coordinates, and shows spatial and temporal nonlocality. Spatial nonlocality means that Δn at a certain point relates not only to the optical field at the point, but also to the optical field at adjacent points. Similarly, the temporal nonlocality means that Δn at a certain time is related not only to the optical field of this time, but also re-

18 About the optical Kerr effect, see Ref. [60] and Chapters 4 and 7 of Ref [61].

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239

lated to that before this time. If do not consider the spatial and temporal nonlocality, one gets (4.2) Δn(r, z, t) = n2 |E(r, z, t)|2 , where n2 is the nonlinear refractive index coefficient (also called the Kerr coefficient), E is the amplitude of the electric field. The above equation shows that the nonlinear refractive index at the space point (r, z) only depends on the electric field at this point and has nothing to do with electric field of the other points in the space (spatial locality); the nonlinear refractive index of the moment t is only related to the electric field at this time and has nothing to do with the electric field of other moment (temporal locality). When the planar electromagnetic wave¹⁹ propagates in a Kerr medium, the optical Kerr effect can be regarded to be localized temporally and spatially. However, a real electromagnetic field must be one of the three listed below: the optical beam which is the optical wave distributed in the limited space (the spacially localized one), the optical pulse propagating in the optical waveguide and having limited temporal distribution (the temporally localized one), and the optical pulsed beam localized both temporally and spatially. For a real electromagnetic field mode, when the time scale of the optical field (represented as pulse duration) is much longer than the response time (relaxation time) of material’s optical Kerr effect, and the space scale of the optical field (represented as beam width) is much larger than the space characteristic length of the material’s optical Kerr effect, the optical Kerr effect can be regarded to be spatiotemporally localized and the nonlinear refractive index can be represented as Equation (4.2). Otherwise, the spatial nonlocality (when the beam width is closed to or smaller than the space characteristic length of the optical Kerr effect), the temporal nonlocality (when the pulse duration is closed to or shorter than the response time of the optical Kerr effect), or the spatiotemporal nonlinearity should be taken into account. When the spatial nonlocality of the optical Kerr effect needs to be considered, Δn in general is described as the differential equation w2m ∇2D Δn(r, z) − Δn(r, z) = −n2 |E(r, z)|2 ,

(4.3)

where ∇D is the D-dimensional differential operator vector of the transverse coordinates, ∇2 = ∂/∂xex + ∂/∂yey , but ∇1 = ∂/∂xex or ∂/∂yey , and w m is the nonlinear characteristic length of the material, which has the dimension of length, and represents the characteristic parameter for the scale of space occupied by the nonlinear response function. In an infinite space, the equivalent integral expression of Equa-

19 Planar electromagnetic wave is a mathematical idealized electromagnetic wave propagation mode, which fills infinite time and space.

240 | Qi Guo, Daquan Lu, Dongmei Deng tion (4.3) is

∞

Δn(r, z) = n2 ∫ R(r − r󸀠 )|E(r󸀠 , z)|2 dD r󸀠 ,

(4.4a)

−∞

where R(r) is the spatial nonlinear response function of the material²⁰, dD r is the D-dimensional infinitesimal of the point (r, z). R(r) satisfies the normalizing condi∞ tion ∫−∞ R(r)dD r = 1. It is the physical requirement to normalize R(r), and the purpose is to make the nonlinear refractive coefficient n2 in Equation (4.4a) have the same dimension with the nonlinear refractive coefficient of the local optical Kerr effect [8, 60, 61],²¹ given by Equation (4.2). The response function of the system described by Equation (4.4a) in the infinite space is only a function of the distance between the source point r󸀠 and the field point r, which is said to be of the translation invariance, and is also symmetric. For the physical systems without the translation invariance, however, their response functions have the form of R(r, r󸀠 ), and depend not only on the field point but also on the source point itself. For example, the system (4.4a) within a finite space will lose translation invariance because of the existence of the boundaries [62]. In this case, Δn(r, z) = n2 ∫ R(r, r󸀠 )|E(r󸀠 , z)|2 dD r󸀠 ,

(4.4b)

V

where V represents the whole range of the finite space. When it needs to consider the temporal nonlinearity of the optical Kerr effect, Δn satisfies the following Debye relaxation equation [60] τ

∂ Δn(r, z, t) + Δn(r, z, t) = n2 |E(r, z, t)|2 , ∂t

(4.5)

where τ has the time dimension, and is the response time (relaxation time) of the Kerr material, i.e. the characteristic time of the nonlinear temporal response function. The equivalent integral equation of Equation (4.5) is t

Δn(r, z, t) =

t − t󸀠 n2 ∫ exp (− ) |E(r, z, t󸀠 )|2 dt󸀠 . τ τ

(4.6)

−∞

Comparing Equation (4.4a) with Equation (4.6), one can see that the response function of the temporal nonlinear system has the following asymmetric form: { 1 exp (− τt ) , R t (t) = { τ 0, {

t>0 t≤0

.

(4.7)

20 The concrete expression of R(r) can be found in Sections 4.3 and 4.4, and the process how to obtain R(r) is detailed in Appendix A of this chapter. 21 About the local optical Kerr effect, see [8, 60] and Chapters 4 and 7 of Ref. [61].

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The asymmetry of the temporal response function originates from the causality [61].²² Considering both the temporal and the spatial nonlocality, one gets the following mode of Δn: w2m ∇2D Δn(r, z, t) − τ

∂ Δn(r, z, t) − Δn(r, z, t) = −n2 |E(r, z, t)|2 . ∂t

(4.8)

Related research on the physical effects described by this model still remains unexplored so far. The condition that the space scale occupied by optical field is much larger than spatial characteristic length of material’s optical Kerr effect (or the time scale occupied by optical field is much longer than temporal characteristic length of material’s optical Kerr effect), is mathematically equivalent to w m → 0 (or τ → 0). It can be seen from Equation (4.3) [or Equations (4.5) and (4.8)] that there is Δn(r, z, t) → n2 |E(r, z, t)|2 under this condition. Thus, w m → 0, τ → 0 or the both are satisfied corresponds to the local optical Kerr effect. On the other hand, when w m → 0 (τ → 0), there is R(r) → δ(r) [R t (t) → δ(t)], where δ(⋅) represents the δ-function. Using the properties of the δ-function,²³ we can see from Equations (4.4a) and (4.6) that in local case the response function becomes the δ-function. According to the sign of the nonlinear refractive index coefficient n2 , we can classify the optical Kerr media into two categories, i.e. self-focusing media (corresponding to n2 > 0) and self-defocusing media (corresponding to n2 < 0). Self-focusing media focus the beam just as the optics convex lenses do; the self-defocusing media play an opposite role, i.e. defocus the beam, just as the concave lenses do.

4.1.3 Nonlinear propagation model of optical envelope: The nonlocal nonlinear Schrödinger equation Optical envelopes are localized optical waves, including optical pulses, optical beams, and optical pulsed beams. In this section, we only discuss the propagation model for slowly varying envelope of which the spatial effect and the temporal effect are separated, such as narrow-band optical pulses and paraxial optical beams; the pulsed beams which involve spatiotemporal coupling effect are not taken into account.

22 See Sections 1.6 and 1.7 in Ref. [61]. ∞ 23 Properties of δ function: ∫−∞ δ(r − r󸀠 )f(r)d D r = f(r󸀠 ), f(r) is an arbitrary continuous function.

242 | Qi Guo, Daquan Lu, Dongmei Deng A. Optical beams Suppose that there is a transversely linearly polarized time-harmonic electric field E(r, z, t) in a homogeneous linear medium,²⁴ it can be written as 1 e0 E0 (r, z) exp(−iωt) + C.C. , 2 E0 (r, z) = Ψ(z, r) exp(ikz) ,

E(r, z, t) =

(4.9a) (4.9b)

where e0 is the unit vector in the direction of polarization, E0 (r, z) is a complex number which represents the spatial part of the electric field, Ψ(z, r) is a paraxial beam (i.e., spatial slowly varying envelope function of the electric field), k = ωn0 /c is the wave number, ω is the frequency, c is the speed of light in vacuum, and C.C. represents the complex conjugate of the previous term. Substituting E(r, z, t) into the Maxwell’s equations, one gets the Helmholtz equation [64] ∂2 E0 ω2 n2 + ∇2D E0 + E0 = 0 , 2 ∂z c2

(4.10)

where n is the refractive index of medium given by Equation (4.1). Substituting E0 (r, z) in Equation (4.9b) into the equation above, using the paraxial approximation |∂Ψ/∂z| ≪ k|Ψ|, and ignoring the square of the nonlinear perturbation of the refractive index △n yields i

k△n ∂Ψ 1 2 Ψ =0. + ∇ Ψ+ ∂z 2k D n0

(4.11)

Substituting △n in Equation (4.4a) (which is spatially nonlocal) into the equation above, we obtain the nonlocal nonlinear Schrödinger equation (NNLSE) which models the dynamics of the paraxial optical beam Ψ in nonlocal Kerr media [44, 66, 67], i.e. ∞

∂ 1 2 i Ψ(z, r) + ∇ Ψ(z, r) + γ b Ψ(z, r) ∫ R(r − r󸀠 )|Ψ(z, r󸀠 )|2 dD r󸀠 = 0 , ∂z 2k D

(4.12)

−∞

where γ b = kn2 /n0 . In the local case [R(r) = δ(r)], Equation (4.12) becomes the NLSE²⁵ [60, 61, 65]:²⁶ i

∂Ψ 1 2 + ∇ Ψ + γ b |Ψ|2 Ψ = 0 . ∂z 2k D

(4.13)

24 Strictly speaking, the assumption that linearly polarized harmonic electric field has a limited space distribution in linear homogeneous medium is inconsistent with the law that the divergence of the electric field equals zero (∇ ⋅ E = 0). But as a lowest order approximation, the electric field of limited spatial distribution can be regarded to be linearly polarized. A detailed discussion about the problem see Ref. [63, 64] and Chapter 4 in Ref. [65]. 25 Directly substituting the expression of △n in local case, i.e. Equation (4.2) into Equation (4.11) can also get the same result. 26 See Chapter 17 in Ref. [60], Chapter 7 in Ref. [61] and Chapter 10 in Ref. [65].

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As early as 1965, Kelley has obtained this equation [27], although he did not connect this equation with the NLSE at that time. In the case that D = 2, Equations (4.12) and (4.13) are (1 + 2)-dimensional equations which describe propagation of beams in three-dimensional bulk media. When D = 1, (1 + 1)-dimensional equations (4.12) and (4.13) become models which describe propagation of beams in slab waveguide [38, 65]. In this case, the three-dimensional space is reduced to the two-dimensional space, because one-dimensional degree of freedom is constrained by the slab waveguide. Equations (4.12) and (4.13) have several integral invariants (conserved quantities) [68], two of these are the power integration²⁷ ∞

P0 = ∫ |Ψ(z, r)|2 dD r ,

(4.14)

−∞

and the momentum integration ∞

i M= ∫ (Ψ∇D Ψ ∗ − Ψ ∗ ∇D Ψ)d D r , 2k

(4.15)

−∞

where ∗ means complex conjugate. Using the Ehrenfest theorem in quantum mechanics, one can get the trajectory equation for the mass center of the beam rc from Equation (4.12) [69] dr c (z) M , (4.16) = dz P0 where the mass center of the beam is defined as ∞

1 ∫ r|Ψ(z, r)|2 d D r . r c (z) = P0

(4.17)

−∞

Because M and P0 are conserved quantities, one has r c (z) =

M z + r c0 , P0

(4.18)

where r c0 = rc (0) is the mass center at the entrance plane z = 0. Equation (4.18) indicates that the trajectory for the mass center of the beam is a straight line, of which the slope is M/P0 .²⁸ Equation (4.18) will be used in Section 4.2 where the NNLSE is simplified into the Snyder–Mitchell model. ∞

27 In the (1 + 1)-dimensional case, P 0 = ∫−∞ |Ψ(z, x)|2 dx is the linear power density per unit length along the y direction (the thickness direction of the slab waveguide). 28 This conclusion is only true for infinite bulk material medium (infinite medium is of conservation of momentum integral). For limited medium, the boundary effect will result in the oscillation of the beam centroid [17].

244 | Qi Guo, Daquan Lu, Dongmei Deng It should be noted that the power which is defined by Equation (4.14) is not the real power carried by electromagnetic field actually, but is proportional to it. For an optical beam, the real power carried by electromagnetic field is ∞

ε0 n0 c Pp = ∫ |Ψ(z, r)|2 dD r , 2

(4.19)

−∞

where ε0 is the permittivity in vacuum. Therefore, relationship between P p and P0 is: P p = ε0 n0 cP0 /2.

B. Optical pulses Compared with the derivation of the propagation equation for optical beams, the derivation of the propagation equation for optical pulses from Maxwell’s equations is much more complex, because it involves basic knowledge of the optical waveguide theory. In addition, the subject of this chapter is not about temporal optical solitons but about spatial optical solitons. Given these two reasons, this section gives only the preconditions and the final results of the derivation of the governing equation for the evolvement of optical pulses. Readers are referred to Refs. [8] and [65] for detailed derivation of the equation.²⁹ When the laser propagates in the fiber (two-dimensional waveguide structure), the transverse distribution of the field is constrained by the waveguide structure and the guided wave mode (waveguide mode field) comes into being. Under this condition, the electromagnetic field can propagate as pulse form. Assume that the optical waveguide is in the single mode condition (i.e., only the dominant mode propagates in the waveguide), the electric field of the dominant mode can be regarded as a linearly polarized wave under the condition of weak guiding, thus the electric field propagating in pulse form can be expressed as E(r, z, t) =

1 e0 F(r)A(z, t) exp[i(β 0 z − ω0 t)] + C.C. , 2

(4.20)

where A(z, t) is the pulse function, F(r) is the modal field distribution function of the waveguide’s dominant mode [F(r) is a dimensionless function, which approximates well to the Gaussian function for weak-guiding fiber], ω0 is the carrier frequency, β 0 = β(ω)|ω=ω0 , β(ω) is the propagation constant of the waveguide’s dominant mode. In the co-moving reference frame T = t − β 1 z = t − z/v g , the propagation of a narrow-band signal pulse A is governed by the NNLSE [70]: ∞

i

∂ β2 ∂2 A(z, T) − A(z, T) + γ p A(z, T) ∫ R t (T − T 󸀠 )|A(z, T 󸀠 )|2 dT 󸀠 = 0 , ∂z 2 ∂T 2 −∞

29 Chapter 2 in Ref. [8] and Chapter 10 in Ref. [65].

(4.21)

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where R t is the temporal nonlinear response function described by Equation (4.7), v g = 1/β 1 is the group velocity of the pulse, β 1 = dβ(ω)/dω|ω=ω0 , β 2 = d2 β(ω)/dω2 |ω=ω0 , γp =

ω20 n0 n2 kn2 ≈ , n0 Seff c2 β 0 Seff

Seff =

∫ |F(r)|2 d D r ∫ |F(r)|4 d D r

.

(4.22)

In order to get the final result of γ p in Equation (4.22), the approximation β ≈ ωn0 /c (which is satisfied in the weak-guiding waveguide) is utilized. It should be noted that the nonlinear coefficient of pulse equation γ p (Equation (4.22)) is the same as the corresponding coefficient γ b of beam equation (4.12) except for the integral item Seff ; this fact is helpful for remembering the coefficient expressions. In the local case, Equation (4.21) becomes the NLSE [8, 65]: i

∂A 1 ∂2 A − β2 + γ p |A|2 A = 0 . ∂z 2 ∂T 2

(4.23)

The above equation is the standard equation which governs the propagation of picosecond pulse in the single-mode lossless fiber [8]. For the optical pulse expressed in Equation (4.20), the power of the electromagnetic field is ∞

ε0 n0 c |A(z, t)|2 ∫ |F(r)|2 dD r . Pp = 2

(4.24)

−∞

C. Dimensionless transform of propagation model In the nonlocal nonlinear Schrödinger equations (4.12) and (4.21) (reduced to Equations (4.13) and (4.23) in local case) which describe the propagation evolution process of beams and pulses, respectively, some physical parameters are explicit, such as k, n2 , n0 , etc. Except for the explicit parameters, there are some physical parameters are included implicitly in the equations. For example, the maximum amplitude of beams (or pulses), the beam width (or pulse duration), etc. In fact, these parameters are not fully independent, there is some kind of intrinsic connection existing among them. The connection is the dimensionless transformation which will be discussed below.³⁰ Dimensionless transformation changes concrete physical variables into dimensionless and abstract variables with the same order of magnitude. Through the dimensionless transformation, one can not only make the equation more concise, but also accurately grasp the physical essence of the problem.

30 Most of the literatures (including mine) published so far called the transformation process which is discussed in this section as “normalized transform.” But in recent years, I think it is better to call it the dimensionless transform. Marked by Ref. [71], thereafter all my published literatures use the name “dimensionless transform.”

246 | Qi Guo, Daquan Lu, Dongmei Deng Let us first discuss this problem based on the beam propagation equation. Introducing the dimensionless variable transformations ξ=

z , L

ηx =

x , w0

ηy =

y , w0

U(ξ, η) =

Ψ(z, r) , Ψ0

R̄ = w0D R ,

(4.25)

where ξ , η, U, and R̄ are, respectively, dimensionless longitudinal (propagation direction) coordinate, dimensionless D-dimensional transverse coordinate vector [η = η x eη x + η y eη y (for D = 2) or η = η x eη x (for D = 1)], dimensionless optical beam function and dimensionless response function, Ψ0 is the maximum amplitude of the optical beam, L is the propagation distance, and w0 is the characteristic parameter that represents the width of beams (the scale of space that is occupied by optical fields), Equation (4.12) is reduced to ∞

i

∂ L 1 L 2 ̄ − η󸀠 )|U(ξ, η 󸀠 )|2 d D η󸀠 = 0 , ∇ U(ξ, η) + sgn(n2 ) U(ξ, η) ∫ R(η U(ξ, η) + ∂ξ 2 Ldif D Lnl −∞

(4.26) where ∇2 = ∂/∂η x eη x + ∂/∂η y eη y , ∇1 = ∂/∂η x eη x , Ldif = kw20 ,

Lnl =

n0 . k|n2 |Ψ02

(4.27)

Both Ldif and Lnl have the dimension of length, so Ldif is usually called as diffraction length (also called as Rayleigh length or confocal parameter), Lnl is called as nonlinear length. They represent the strength of diffraction effect and nonlinear effect, respectively. The shorter Ldif and Lnl are, the stronger the corresponding effects will be. For given wavelength and material, Ldif and Lnl are dependent on the beam width and amplitude (the power carried by beam), respectively. The narrower the light beam width, the shorter the diffraction length, and the stronger the diffraction effect, the stronger the power (amplitude), the shorter the nonlinear length, and the stronger the nonlinear effect. In addition, the larger the nonlinear refractive coefficient n2 is, the smaller the power is needed for the same nonlinear effect. In Equation (4.26), except for L, Ldif , and Lnl , all the parameters are dimensionless and have the same order of magnitude (It is assumed that the beam has a smooth enough spatial profile so that all derivative has the order of O(1)). According to relative sizes of the propagation length, the diffraction length, and the nonlinear length, the beam propagation can be classified into four different types: near field propagation, linear propagation, nonlinear propagation, and combined action propagation. 1. Near field propagation. When the propagation distance L satisfies conditions L ≪ Ldif and L ≪ Lnl at the same time and ignoring the last two terms therein, Equation (4.26) is simplified into ∂ ξ U = 0. This demonstrates that in this region neither the diffraction effect nor the nonlinear effect affects the propagation; therefore, the beam shape keeps invariant during propagation. The region where conditions

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2.

| 247

L ≪ Ldif and L ≪ Lnl are simultaneously satisfied is the so-called near-field region³¹ [65]. Linear propagation. When the propagation distance is far less than the nonlinear distance and is of the order of the diffraction distance, i.e. when the conditions L ≪ Lnl and L ∼ Ldif are satisfied, the last term in Equation (4.26) is negligible, thereby Equation (4.26) is reduced into i

∂U 1 L 2 ∇ U =0. + ∂ξ 2 Ldif D

(4.28)

Equation (4.28) is the paraxial propagation equation in uniform linear media [65]. According to the relation Lnl ≫ Ldif , one gets Ψ02 w20 ≪

3.

n0 n0 , ⇒ w2−D 0 Pp ≪ 2 I , k 2 |n2 | k |n2 |

where the power carried by the beam is P p ∼ ε0 n0 cΨ02 w0D ³². As can be seen, when the power is sufficiently weak, the material’s nonlinear effect on the beam can be ignored; the propagation characteristics of the beam are decided solely by the linear diffraction effect, and the beam broadens during propagation. Nonlinear propagation. Under the condition L ≪ Ldif , L ∼ Lnl , Equation (4.26) is simplified into (here only gives the expression under the condition that the response function is the δ-function) i

∂U L |U|2 U = 0 . + sgn(n2 ) ∂ξ Lnl

(4.29)

Equation (4.29) describes the spatial self-phase modulation of the beam in bulk media [72], and its precondition can be obtained from the relation Lnl ≪ Ldif : w2−D 0 Pp ≫ 4.

n0 2 k |n2I |

.

Combined action propagation. When L, Ldif , and Lnl have the same order of magnitude, both the diffraction effect and the nonlinear effect affect the propagation of the beam, and the three terms in Equation (4.26) have the same order of magnitude. The phenomena of self-focusing, self-defocusing, and spatial optical solitons happen in this case. On this occasion, P p (or Ψ0 ) is determined by the relation

31 See Chapter 4 in Ref. [65]. 32 Two ways can be used to define the nonlinear refractive index coefficient n 2 . One is the way used in this chapter, i.e. n = n 0 + n 2 |E|2 , the dimension for n 2 is m2 /V2 . The other is n = n 0 + n 2I I, where I is the light intensity, the relationship between I and E is I = ε 0 n 0 c|E|2 /2. The dimension for n 2I is m2 /W, and the relationship between n 2I and n 2 is n 2I = 2n 2 /ε 0 n 0 c (A detailed discussion about this problem, see Appendix B in Ref. [8]). On the other hand, based on the expression for the power P p , i.e. Equation (4.19), and with the dimensionless transformation (4.25), one gets P p = κ P ε 0 n 0 cΨ 02 w 0D /2, ∞ where κ P = ∫−∞ |U|2 d D η is a constant determined by the waveform of beam function Ψ, the magnitude is O(1). Take the Gaussian wave beam {Ψ|z=0 = Ψ 0 exp[−(x 2 + y2 )/2w 20 ]} as an example, κ P = π.

248 | Qi Guo, Daquan Lu, Dongmei Deng Ldif ∼ Lnl : Ψ02 ∼

n0 2 k w20 |n2 |

(w2−D 0 Pp ∼

n0 ) 2 k |n2I |

⇒

|Δn| ∼ σ2 , n0

where σ = 1/kw0 = λ/2πw0 n0 , λ is the wavelength in vacuum. The parameter σ is very important for describing the evolution of beams during propagation [64], even in the case that the beam width is focused to w0 = λ/n0 , one still has σ ≈ 0.16. Generally speaking, the relation σ 2 ≪ 1 is always satisfied. The above inequality |Δn|/n0 ≪ 1 suggests that Δn is only the perturbation compared with n0 . Later in this chapter, we will focus on discussing the case that both the diffraction effect and the nonlinear effect affect the beam propagation. For this purpose, reintroduction of a new dimensionless transformation ξ=

z , Ldif

η=

r , w0

u b (ξ, η) = kw0 √

|n2 | Ψ(z, r) , n0

R̄ = w0D R

(4.30)

makes Equation (4.26) evolve into the dimensionless NNLSE ∞

∂ 1 2 ̄ − η󸀠 )|u b (ξ, η 󸀠 )|2 d D η󸀠 = 0. (4.31) i u b (ξ, η)+ ∇D u b (ξ, η)+sgn(n2 )u b (ξ, η) ∫ R(η ∂ξ 2 −∞

̄ For spatial local nonlinear system [R(η) = δ(η)], Equation (4.31) becomes i

∂u b 1 ∂2 ∂2 ) u b + sgn(n2 )|u b |2 u b = 0 , + ( 2 + ∂ξ 2 ∂η x ∂η2y

or i

∂u b 1 ∂2 u b + sgn(n2 )|u b |2 u b = 0 , + ∂ξ 2 ∂η2x

(D = 2) ,

(D = 1) .

(4.32)

(4.33)

Because sgn(n2 ) = ±1, Equation (4.32) in fact corresponds to two different equations, and so does Equation (4.33). Similarly, for the optical pulse, we can analogically introduce two characteristic parameters, the dispersion length Ldis and the nonlinear length Lnp : Ldis =

T02 , |β 2 |

Lnp =

n0 Seff , k|n2 |A20

(4.34)

where A0 is the maximum amplitude of the optical pulse, T0 is the characteristic parameter that represents the optical pulse width. The propagation of the pulse can be classified into two types according to the relative size of Ldis and Lnp .³³ When the relation Ldis ∼ Lnp is satisfied, through dimensionless transform ξ=

z Ldis

,

ηt =

T , T0

u p (ξ, η t ) = (

1/2 k|n2 | ) T0 A(z, T) , |β 2 |n0 Seff

R̄ t = T0 R t , (4.35)

33 This classification and the classification of spatial beams are analogous, see Section 3.1 in Ref. [8].

4 Nonlocal spatial optical solitons

|

249

one can transform Equation (4.21) to the dimensionless equation i

1 ∂2 ∂ u p (ξ, η t ) − sgn(β 2 ) 2 u p (ξ, η t ) + sgn(n2 )u p (ξ, η t ) ∂ξ 2 ∂η t ∞

× ∫ R̄ t (η t − η󸀠t )|u p (ξ, η 󸀠t )|2 dη󸀠t = 0 .

(4.36)

−∞

Corresponding to the temporally local nonlinear case, the dimensionless equation above is simplified to i

∂2 u p ∂u p 1 + sgn(n2 )u p |u p |2 = 0 . − sgn(β 2 ) ∂ξ 2 ∂η2t

(4.37)

According to the combination of plus or minus symbols of β 2 and n2 , it seems that Equations (4.36) and (4.37) have four kinds of forms, but in fact there are only two in nature. Moreover, when comparing Equations (4.33) with (4.37), we can see that the former equation and the latter equation which meet the condition for β 2 < 0 have the same mathematical structure, although their physical meanings are totally different. Therefore, we can remove the subscript of the variables, and Equations (4.33) and (4.37) that under different conditions can be unified as the following two equations: i

∂u 1 ∂2 u + + |u|2 u = 0 , ∂ξ 2 ∂η2

(4.38)

i

∂u 1 ∂2 u + |u|2 u = 0 . − ∂ξ 2 ∂η2

(4.39)

and

Equation (4.38) is (1 + 1)-dimensional beam’s propagation model in the self-focusing nonlinear media (n2 > 0), and is also the propagation model of the optical pulse in the self-focusing nonlinear media if the carrier frequency is in the anomalous group velocity dispersion regime (β 2 < 0)³⁴; Equation (4.39) is the propagation model of optical pulse in the self-defocusing media and its carrier frequency is in the normal group velocity dispersion regime. Through the spatial and temporal coordinates inversion transform r = −r󸀠 , z = −z󸀠 , t = −t󸀠 , we can prove that Equation (4.38) is also the equivalent propagation model of optical pulse in the self-defocusing nonlinear media (n2 < 0) when its carrier frequency is in the normal group velocity dispersion regime.³⁵ Moreover, Equation (4.39) is the propagation model of the (1 + 1)-dimensional beam in

34 The dispersion properties of materials, as well as the division of normal and anomalous dispersion regime can be found in Section 1.2.3 of Ref. [8]. 35 The propagation model of optical pulse whose carrier frequency is in normal dispersion regime in self-defocusing nonlinear media is i∂u/∂ξ − (1/2)∂2 u/∂η 2 − |u|2 u = 0. After inverse transformation of spatial coordinate z, ∂u/∂ξ = −∂u/∂ξ 󸀠 . While other inverse transformation has no effect on the equation, so the equation evolves into Equation (4.38); similarly, after inverse transformation,

250 | Qi Guo, Daquan Lu, Dongmei Deng the self-defocusing nonlinear media, and is equivalent to the propagation model of the optical pulse in the self-defocusing nonlinear media when its carrier frequency is in the anomalous group velocity dispersion regime. The nonlinear Schrödinger equations (4.32), (4.38), and (4.39) are of the following transform invariance [73]: ξ 󸀠 = B2 ξ ,

η󸀠 = Bη ,

u󸀠 =

u , B

(4.40)

where B is an arbitrary constant, that is, the NLSEs keep unchanged after the transformation above. By extension, the invariant transformation for the NNLSE is [74] ξ 󸀠 = B2 ξ ,

η󸀠 = Bη ,

u󸀠 =

u , B

R󸀠 =

R . BD

(4.41)

Due to the transform invariance, we can observe that the power integration ∫ |u(η x , η y )|2 dη x dη y is an invariant after transform for the 1 + 2D problem; while for 1 + 1D problem the invariant is the integration ∫ |u(η)|dη rather than the power integration ∫ |u(η)|2 dη.

4.1.4 Soliton solutions and their physical essence A. Mathematical tools of optical soliton research To discuss the propagation characteristics of slowly varying envelope (paraxial optical beam and the narrow band signal optical pulse) in nonlinear Kerr medium, we need to solve the nonlinear partial differential equations first. Compared with linear partial differential equations, nonlinear partial differential equations are very difficult to solve. Solving Equation (4.31), which will be discussed in detail in Sections 4.2–4.4, is the topic of this subject. However, solving Equation (4.36) is just starting and needs further investigation [70]. Therefore, this section mainly discusses the problem associated with Equations (4.32),(4.38), and (4.39). (1) Exactly analytical method: Inverse scattering method Fortunately, mathematicians provide us with a powerful tool, i.e. the inverse scattering method. This method is found in solving the KdV equation by Gardner et al. in 1967. In the effort of some applied mathematicians like Lax, Zakharov, and Shabat et al., it has been generalized to a lot kinds of nonlinear partial differential equations, and becomes a common method of accurately solving these equations. It is said that

i∂u/∂ξ + (1/2)∂2 u/∂η 2 − |u|2 u = 0 becomes Equation (4.39). Furthermore, after the simultaneous spatial and temporal coordinate inverse transformation, the electric field expressions (4.9) and (4.20) still keep propagating in positive direction. Therefore the equations before and after the inverse transformation are equivalent to each other.

4 Nonlocal spatial optical solitons

| 251

inverse scattering method is one of the greatest discoveries in the field of mathematical physics in the 20th century. The inverse scattering method is very convenient for solving Cauchy problem (initial value problem) of nonlinear partial differential equations. The main characteristic of this method is that a complex nonlinear partial differential equation can be solved exactly through the combination of a plurality of linear equations (the combination is the so-called Lax pair³⁶). This solving process is similar to the Fourier transformation method of the partial differential equations with constant coefficients. The latter is to change a partial differential equation into an infinite set of linear ordinary differential equations, and the inverse scattering method is to map the coefficients of several linear differential operators into scattering data set. The mapping plays the same role as the Fourier transformation. The former Soviet mathematician Zakharov and Shabat successfully constructed the Lax pairs of the (1 + 1)-dimensional NLSEs (4.38) (1971 [7]) and (4.39) (1973 [75]), respectively. And thereby the accurate analytical solutions of the two equations were found by inverse scattering method. Subsequently, Satsuma accurately integrated the initial-value problem u(ξ = 0, η) = Nsech(η) (where N is a positive integer) of Equation (4.38) by utilizing the inverse scattering method, and gave out the accurate analytical expressions of N = 1 and N = 2 [59]. An accurate analytical expression of N = 3 had also been obtained [76]. It is difficult to obtain the accurate analytical expressions of N > 3 because a system of 2N linear algebra equations is needed to be solved and the process becomes very complicated and tediously long. Due to historical reasons, people have got accustomed to call the solution of N = 1 as the first-order soliton (or fundamental soliton), and the solutions when N ≥ 2 the higher order solitons [8, 76]. But, as a matter of fact, the solutions are the periodic oscillation functions of ξ when N ≥ 2; therefore they are actually breatherss rather than solitons, as discussed in part D of Section 4.1.1. (2) Approximately analytic approach: The variational method For a problem that a closed-form solution cannot be found, an approximately analytical solution can be obtained by various of perturbation methods. There are many methods to obtain the approximate solution of the NLSE [6],³⁷ but the most widely used method is the variational method [77]. Here is a brief introduction to the basic idea of variational method.

36 The Lax pair is a pair of differential operators which are introduced in the process of solving nonlinear partial differential equation by inverse scattering method. If the Lax pair is found, the nonlinear partial differential equation can be solved by the inverse scattering method. 37 See Chapter 5 in Ref. [6].

252 | Qi Guo, Daquan Lu, Dongmei Deng The problem associated with the variation of the functional ∞ ∞

I[u, u ∗ ] = ∫ ∫ L(u, u ∗ , u z , u ∗z , u η x , u ∗η x , u η y , u ∗η y )d D ηdz ,

(4.42)

0 −∞

where I[u, u ∗ ] is the functional of the complex function u and its complex-conjugate function u ∗ is described as δI = 0 , (4.43) which is equivalent to the following two Euler–Lagrange equations [78]: ∂ ∂ ∂L ∂ ∂L ∂L ∂L )+ ( )+ ( )− =0, ( ∂z ∂u ∗z ∂η x ∂u ∗η x ∂η y ∂u ∗η y ∂u ∗

(4.44)

∂ ∂L ∂ ∂L ∂L ∂ ∂L =0, )+ ( )+ ( )− ( ∂z ∂u z ∂η x ∂u η x ∂η y ∂u η y ∂u

(4.45)

and

where L is said to be the Lagrangian density, and the integral ∞

L = ∫ Ld D η

(4.46)

−∞

is said to be the Lagrangian. If the Lagrangian density L of the functional expression (4.42) takes the form of ∞

L=

∂u ∂u ∗ 1󵄨 i 󵄨2 1 ̄ − η 󸀠 )|u(ξ, η 󸀠 )|2 d D η󸀠 , −u (u ∗ ) − 󵄨󵄨󵄨󵄨∇D u 󵄨󵄨󵄨󵄨 + |u|2 ∫ R(η 2 ∂ξ ∂ξ 2 2

(4.47)

−∞

̄ as well as the function R(η) is symmetric about argument η, Equation (4.31) can be deduced from Equation (4.44), while the complex-conjugate equation of Equation (4.31) can be deduced from Equation (4.45) [79]. How to find out a trial function, of which the dependency on the cross-section coordinates is known, is the key step to solve the NNLSE by variational method. Looking for the trial function is completely dependent on experience. In general, the trial function can be given by the exact analytical solution or the approximate solution obtained in similar conditions. For Equation (4.31), the trial function can be selected as u(ξ, η) = q A (ξ) exp [−

η2x + η2y 2q2w (ξ)

] exp [iq c (ξ)(η 2x + η2y ) + iq θ (ξ)] .

(4.48)

Substituting the above expression into Equation (4.46) and integrating over the crosssectional area, Equation (4.43) becomes ∞

δ ∫ L(q A , q w , q c , q θ , q̇ A , q̇ w , q̇ c , q̇ θ )dξ = 0 , 0

(4.49)

4 Nonlocal spatial optical solitons

| 253

where q i̇ = dq i /dξ (i = A, w, c, θ). The variational equation (4.49) corresponds to the four Euler–Lagrange equations (four ordinary differential equations) with generalized coordinates q A , q w , q c , and q θ , d ∂L ∂L )− = 0, ( dξ ∂ q i̇ ∂q i

(i = A, w, c, θ) .

(4.50)

After solving these four ordinary differential equations, we can obtain the variational approximate solution of the nonlocal nonlinear Schödinger equation (4.31). ̄ It should be noted that the condition in which the response function R(η) should be of symmetry about the argument η [79] means that only the problem associated with the spatial nonlocality for the bulk media could be approximatively solved by the variational method. Whether the variational method, however, can still work on the problem associated with the spatial nonlocality for the media with boundary and the problem associated with the temporal nonlocality is an open question. The reason is that the spatial response function of the bulk media without the boundary is of the symmetry, which is only the function of the distance between the source point and the field point, but the spatial response function of the finite media with the boundary loses the symmetry due to the boundary [62], so does the temporal response function of the temporal nonlocality due to the causality [61]. ̄ If the response function R(η) of the Lagrangian density function (Equation (4.47)) is replaced by the δ-function δ(η), the above process is a standard approach to solve the NLSE and its modified equations by the variational method [77]. (3) Numerical method: Because the equations are nonlinear, only in some special cases one can obtain the accurate analytical solution by the inverse scattering method, or find the approximate analytical solution by variational method or else. Generally we obtain the result by numerical simulation directly. There are two types of problems in solving numerically the NNLSE (including nonlinear Schrödinger equation). The first is the numerical simulation of the evolution of the envelope in case that the transverse distribution of the envelope at the cross section ξ = 0 is given. This in mathematics is the initial value problem of differential equations. The second is finding out the soliton solution by solving equations numerically. For the first type of problem, many algorithms have been suggested. These algorithms can be divided into two categories: the finite-difference method and the pseudo spectral method. Generally speaking, to achieve the same precision, the computing speed of the latter is faster than the former by one or more magnitude. The split-step Fourier method [8]³⁸ is the most widely used pseudo spectral method. For solving

38 See Section 2.4 in Ref. [8].

254 | Qi Guo, Daquan Lu, Dongmei Deng the NNLSE (including nonlinear Schrödinger equation), both finite-difference method and split-step Fourier method have their merits. For the problem of infinite space, splitstep Fourier method should be used absolutely; but for the problem of limited space with boundary, split-step Fourier method has its limitations. For the second type of problem, we need to find an solution which has the form of u(ξ, η) = u A (η) exp(iβ s ξ), where u A (η) is the real function to be determined, β s is the soliton propagation (real) constant. Taking Equation (4.32) for example, the function u A (η) satisfies the equation (assuming n2 > 0) 1 2 ∇ u A (η) − β s u A (η) + u 3A (η) = 0 . 2 D

(4.51)

The numerical methods of solving Equation (4.51) include spectral reforming method [80], Newton-conjugate-gradient method [81], squared-operator method, modified squared-operator method, and power-conserving squared-operator method [82]. The first four methods are to compute the iterative solution u A (η) for a given propagation constant β s , the other method is to iteratively solve the propagation constant β s and u A (η) for a given power integration of the solution. As for the computation speed, the first two methods are fast, and the other three methods are much slower. As for the range of application, the first two methods are suitable for solving both onedimensional and two-dimensional soliton problem, but the other three methods do not apply to solving the two-dimensional bright soliton problem. The spectral reforming method is more suitable for solving the unipolar soliton problem but is difficult to obtain the multipole soliton, the other four methods are suitable for solving the multipole soliton and the vortex soliton with complex structure. After solving u A (η) numerically, according to the power defined in Equation (4.19) and the dimensionless transformation (4.30), the power of the optical beam (the power of optical pulse can be obtained analogically) can be given by Pp =

ε0 c

∞

∫ u 2A (η)dD η , 2k 20 w2−D n 2 0 −∞

(4.52)

where k 0 = ω/c is the wave number in vacuum. According to the invariant transformation of nonlinear Schrödinger equation (4.40), the power of (1 + 2)-dimensional structure is irrelevant to the selection of the propagation constant β s , but in (1 + 1)dimensional case the power is dependent on β s . This is because the power in (1 + 2)dimensional case is independent on the beam width parameter w0 , whereas in (1 + 2)dimensional case it is proportional to w−1 0 .

B. (1 + 1)-Dimensional bright soliton solutions of the nonlinear Schrödinger equation For the optical beams in the slab waveguide consisting of self-focusing nonlinear medium, and for optical pulses in the optical fiber of which the core is composed

4 Nonlocal spatial optical solitons

| 255

of self-focusing nonlinear medium and the carrier wavelength is in the anomalous dispersion regime, the propagation process can be described by Equation (4.38). The accurate analytical solution of this equation can be obtained by using the inverse scattering method [7, 59]. A particular solution is [8]³⁹ i u(ξ, η) = q0 sech[q0 (η − ξ0 ξ − η 0 )] exp [ (q20 − ξ02 )ξ + iξ0 η + iϕ0 ] , 2

(4.53)

where q0 , ξ0 , η0 , and ϕ0 are integration constants determined by the initial condition at ξ = 0. If we assume that ξ0 = 0, η0 = 0 and ϕ0 = 0, the solution can be simplified to u(ξ, η) = q0 sech(q0 η) exp(iq20 ξ/2). It can be seen that the amplitude |u| of the solution is not the function of ξ . Equation (4.53) is called as the fundamental soliton solution (or first-order soliton solution) of the NLSE (4.38), the intensity distribution (|u|2 ) is shown in Figure 4.1 (a). Each variable of the solution are abstract mathematical variables without physical meaning. To know the physical connotation of the variables for different physical problems, one needs to transform the abstract mathematical variables into specific physical parameters through the dimensionless transformation. In the following, we discuss the case of beams and the case of pulses, respectively.

Fig. 4.1. The intensity distribution of the bright soliton (a) and the dark solitons [black soliton (b), gray soliton (c)].

(1) Bright spatial optical soliton solution: Substituting the dimensionless transformation into the solution (4.53), then putting this into Equation (4.9), yields E(r, z, t) = e0 √

n0 q0 q0 ξ0 sech [ (x − z − w0 η0 )] n2 kw0 w0 kw0

× cos [(

q20 − ξ02 2kw20

+ k) z +

ξ0 x − ωt + ϕ0 ] , w0

(4.54)

where the constants of q0 , ξ0 , ϕ0 , and η0 are, respectively, determined by the amplitude, the incident angle, the phase, and the position of the maximum amplitude of

39 See details in Section 5.2.2 of Ref. [8].

256 | Qi Guo, Daquan Lu, Dongmei Deng the input electric field at z = 0. ϕ0 is the initial phase at z = 0, and has no physical significance for a single beam during propagation (but we need to consider the constant initial phase when discussing the interaction of two or more beams, see Section 4.2.5), thus we can assume ϕ0 = 0. η0 is the position in the x coordinate for the maximum amplitude of the input electric field at z = 0. If we choose a proper x coordinate to make the maximum amplitude appear at x = 0, we have η 0 = 0. For the electric field shown in Equation (4.54) [the beam width is w󸀠0 = w0 /q0 ], the wave vector is⁴⁰ k0 = k x e x + k z e z =

q2 − ξ 2 ξ2 ξ0 ξ0 ex + (k + 0 20 ) ez ≈ ex + (k − 0 2 ) ez . w0 w0 2kw0 2kw0

(4.55)

This represents an oblique incident beam, of which the incident angle (the angle between the wave vector and the z-axis) is ϑ ≈ sin ϑ = k x /k = ξ0 /kw0 , and the z component of the wave vector is k z = k cos ϑ ≈ k(1 − ϑ2 /2) = k − ξ02 /2kw20 . In addition, from Equation (4.54), one can obtain the movement of the beam center in the x-axis at the propagation distance of z, which is given by z/kw0 − x0 = 0 (x0 = w0 η0 ), where the angle between the beam trajectory and the z-axis is ϑ ≈ tan ϑ = (x−x0 )/z = ξ0 /kw0 . Thus the tilt angle of the wave vector is identical to that of the beam center displacement. Now we can draw a conclusion: Equation (4.54) describes the propagation process of a beam with incident angle ϑ in nonlinear Kerr medium. Because the diffraction effect (linear effect) balances the self-phase modulation (nonlinear effect), the beam width keeps invariant during the propagation, this is the spatial optical soliton. And we can also find out that the beam width of the spatial optical soliton is inversely proportional to the integral constant q0 and that the soliton width is inversely proportional to the amplitude is the essential characteristic of soliton solution for the (1 + 1)dimensional NLSE. It originates from the transformation invariance of the equation (see Equation (4.40)). (2) Bright temporal optical soliton solutions: Similarly, using the dimensionless transformation (4.35), the electric field expression of the optical pulse can be given by (without loss of generality, the initial parameters are selected as η 0 = 0, ϕ0 = 0) E(r, z, t) = e0 F(r)√

|β 2 | q0 q0 q0 ξ0 |β 2 | sech [ t − (β 1 + ) z] γ p T0 T0 T0 T0

× cos [(

q20 − ξ02 2T02

|β 2 | + β 0 −

ξ0 ξ0 β 1 ) z − (ω0 − ) t] . T0 T0

(4.56)

According to the above equation, the frequency change of the electric field is δω = ω−ω0 = −ξ0 /T0 (i.e., for the pulse, the constant ξ0 is determined by the frequency shift 40 In order to get the final result of the following formula, we need to use the paraxial approximation [64]: σ 2 = 1/(kw 󸀠0 )2 ≪ 1.

4 Nonlocal spatial optical solitons

| 257

of the input electric field and the pulse duration together: ξ0 = −δωT0 ), the change of the propagation constant is δβ = β − β 0 = −

q2 − ξ 2 ξ2 ξ0 ξ0 β 1 + 0 2 0 |β 2 | ≈ − β 1 + 02 β 2 . T0 T0 2T0 2T0

(4.57)

To get the final result of the above equation, the narrow-band pulse condition q20 |β 2 |/β 0 T02 ≪ 1 is used, and the carrier frequency is in the negative group velocity dispersion regime, |β 2 | = −β 2 . The velocity of the wave envelope v g is obtained by taking the derivative of the variable for the sech function in the equation above with respect to t, dz 1 1 vg = = . (4.58) = dt β 1 + |β 2 |ξ0 /T0 β 1 − β 2 ξ0 /T0 The physical interpretation of the results in Equations (4.57) and (4.58) is: because the pulse propagation constant β is a function of frequency ω (material dispersion), the frequency change is bound to cause the change of propagation constant, which will result in the change of the group velocity, i.e. ξ2 1 ξ0 β 1 + 02 β 2 , β 2 δω2 = β 0 − 2 T0 2T0 v g (ω0 ) 1 1 1 ≈ = = . v g (ω0 + δω) = dβ(ω)/dω|ω=ω0 +δω β 1 + δωβ 2 β 1 − β 2 ξ0 /T0 1 − β 2 ξ0 /T0 β 1 β(ω0 + δω) ≈ β 0 + β 1 δω +

Equation (4.56) is the bright temporal optical soliton solution of optical pulse propagating in the fiber. The bright temporal optical soliton in the fiber results from the precise balance between the group-velocity dispersion (linear effect) and self-phase modulation (nonlinear effect). Its group velocity is v g = 1/(β 1 − β 2 ξ0 /T0 ), and the pulse duration keeps invariant.

C. (1 + 1)-Dimensional dark soliton solutions of the nonlinear Schrödinger equation The propagation of optical pulses, of which the carrier frequency is in the normal dispersion regime, is governed by Equation (4.39) in self-focusing nonlinear media. At the same time, this equation is also the equivalent propagation model of the (1 + 1)-dimensional beam and of the optical pulse of which the carrier frequency in the anomalous dispersion regime, in the self-defocusing nonlinear media. The inverse scattering method can be used to find the exact analytical solutions of Equation (4.39) [75]. The dark soliton solution is [6]⁴¹ u(ξ, η) = q0 {1 − a20 sech2 [a0 q0 (η − η0 )]}1/2 exp[iϕ(ξ, η)] ,

41 See Chapter 16 in Ref. [6].

(4.59)

258 | Qi Guo, Daquan Lu, Dongmei Deng where the phase is { { a0 tanh[a0 q0 (η − η0 )] } } 1 2 + ϕ0 , q0 (3 − a20 )ξ + q0 √1 − a20 (η − η 0 ) + arctan { } { } 2 √1 − a20 { } (4.60) where a0 , q0 , η0 , and ϕ0 are the integration constants determined by the input condition. Different to the bright soliton, the dark soliton has a new parameter a0 called the modulation depth. When a0 = 1 (without loss of generality, assume η0 = 0), we obtain ϕ(ξ, η) =

π u(ξ, η) = q0 | tanh(q0 η)| exp [iq20 ξ + isgn(η) ] = q0 tanh(q0 η) exp(iq20 ξ) . 2

(4.61)

To distinguish the two different types of dark solitons, we call it the black soliton when a0 = 1, and gray soliton when a0 ≠ 1. Their intensity distributions are shown in Figure 4.1 (b) and (c), respectively. The most obvious difference between dark solitons and bright solitons is that the phase of the former is discontinuous, while that of the latter is continuous. According to the dimensionless transformation in Equations (4.30) and (4.35), we can also understand the physical connotation of the parameters in dark soliton solutions for optical beam and optical pulse. Due to space constraints, the discussion is omitted here.

D. (1 + 2)-Dimensional nonlinear Schrödinger equation solutions: Beam self-focusing and stable dark solitons Different from the governing equation for the propagation of nonlinear optical pulse in optical fiber which is (1 + 1)-dimensional, the governing equation for the propagation of optical beams can be (1 + 1)-dimensional or (1 + 2)-dimensional. The (1 + 1)dimensional NLSE (Equation (4.33)) describes the propagation of optical beams in the dielectric slab waveguide the guiding layer of which consists of local nonlinear Kerr medium. The (1 + 2)-dimensional NLSE is the model for the propagation of optical beams in the infinite local nonlinear Kerr medium. Although all of them are described by the NLSE, the propagation characteristics of beams for different number of dimensions essentially differ to each other not only in mathematics but also more importantly in physics. In mathematics, we can find some explicit analytical solutions of the (1 + 1)-dimensional equation through the inverse scattering method, but this method cannot extend to the (1 + 2)-dimensional, at least has not succeeded now. In physics, the stable spatial optical soliton can be formed in (1 + 1)-dimensional local Kerr medium beam (as discussed previously); however, the self-trapped optical beam is unstable in the (1 + 2)-dimensional local self-focusing Kerr medium.

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In general, the (1 + 2)-dimensional NLSE should be solved numerically. In order not to fall into the miscellaneous mathematical process, here we only discuss the related problem about the physical concept qualitatively. We assume that the Gaussian beam Ψ(z = 0, x, y) = Ψ0 exp[−(x2 + y2 )/2w20 ] (Ψ0 is a real number) enters the local self-focusing bulk Kerr medium with normal incidence. According to the beam’s power expression P p = (ε0 n0 c/2)Ψ02 πw20 (see Equation (4.19)), the Gaussian beam can be equivalently regarded as a uniformly distributed circular facula whose radius is w0 and amplitude is Ψ0 . Due to the diffraction effect, this circular facula would become an Airy disk whose radius is 0.61λz/w0 n0 and divergence angle is θ d ≈ 0.61λ/w0 n0 during propagation. On the other hand, because the refractive index inside the facula [n0 + n2 Ψ02 (n2 > 0)] is greater than that at the surrounding (n0 ), the nonlinear effect can balance the diffraction effect and the beam would be trapped by the self-induced cylindrical waveguide, as long as it meets the total reflection condition π/2 − θ d ≥ arcsin[n0 /(n0 + n2 Ψ02 )] (critical angle of total reflection), i.e. θ2d ≤ 2n2 Ψ02 /n0 . Therefore, the critical power of the self-trapped beam is P pc = 1.222 πλ2 ε0 c/16n2 . When the power of the beam is equal to the critical power, the beam can be self-trapped. After such a clear physical picture, Chiao estimates the critical power for the self-trapped beam in local self-focusing bulk Kerr material [26]. Through numerical computation of Equation (4.51), we obtain ∞ ∫−∞ u 2A (η)dη x dη y = 5.85. By using Equation (4.52), the precise critical power is given by 5.85λ2 ε0 c 1.222 πλ2 ε0 c P pc = ≈ . (4.62) 64n2 8π2 n2 The precise value is consistent with Kelley’s result [27], and four times smaller than Chiao’s estimation [26].⁴² Kelley pointed out this fact [27], but did not analyze the reason of the difference. Chiao quantitatively calculated the critical power numerically, but two mistakes in the derivation of the quantitative calculation model made the computation four times bigger. First, when calculating the expression of the refractive index, he used the time average of electric field which makes the nonlinear refractive index be decreased by two times, so the power is increased by two times. Second, the time average of the power is not used in computing the power, so the power is increased by two times again. Therefor, it is the first equation of Equation (4.62) be the correct result of the critical power of the self-trapped beam. The self-trapped beam solution is also called the Townes soliton [53, 83, 84], Equation (4.62) consistent with the critical power of the Townes soliton given in Ref. [84].⁴³ The self-trapped optical beam is unstable in the (1 + 2)-dimensional local selffocusing bulk Kerr medium. Any disturbance, which deviates power P p from the catas-

42 In addition, on page 312 of Ref. [60], the critical power is two times bigger than in Equation (4.62). 43 In Ref. [84], the dimension for nonlinear refraction index coefficient n 2 is m2 /W, namely n 2I in this book.

260 | Qi Guo, Daquan Lu, Dongmei Deng trophic critical power Ppc , either the beam focuses continuously until the width becomes zero and the catastrophic collapse happens, or the beam expand continuously until the width becomes infinite. We can easily understand this instability in (1 + 2)dimensional case in the view of energy conservation (Equation (4.14)) [6]⁴⁴. According to the energy conservation, Ψ02 w0 is conserved for the (1 + 1)-dimensional beam, and Ψ02 w20 is conserved for the (1 + 2)-dimensional beam. Since the nonlinear refractive index of local self-focusing Kerr medium is proportional to the intensity of beam Ψ02 , the nonlinear refractive index is proportional to w−1 0 for the (1 + 1)-dimensional structure, and to w−2 0 for the (1 + 2)-dimensional structure; however, the diffraction effect is always proportional to w−2 0 , and has nothing to do with the dimension of the structure. Therefore, when the nonlinear effect plays the leading role in the (1 + 1)dimensional structure, the beam focuses, and the diffraction effect (∼ w−2 0 ) will eventually exceed the nonlinear focusing effect; on the contrary, when the beam expands due to the diffraction effect, the diffraction effect (∼ w−2 0 ) decreases much faster than the nonlinear effect (∼ w−1 ), and the beam refocuses finally. This is the reason why the 0 (1 + 1)-dimensional beam is stable. However, in the (1 + 2)-dimensional case both the nonlinear effect and the diffraction effect increase or decrease with the same speed. When the diffraction effect plays the dominant role and makes the beam diverge, the nonlinear effect would not replace the diffraction effect to play the dominant role, therefore the divergence would not be prevented. When the beam focuses at the beginning, the diffraction effect is impossible to become strong enough to balance the nonlinear effect. This is the root for the instability of the (1 + 2)-dimensional beam in Kerr self-focusing material. Contrary to the instability of the self-trapped bright optical beam in the (1 + 2)dimensional self-focusing Kerr medium, the dark spatial optical soliton is stable in the (1 + 2)-dimensional self-defocusing Kerr medium [41].

4.2 The phenomenological theory of the nonlocal spatial solitons In the first section (Section 4.1.4), we have discussed the soliton-like solutions of the nonlinear Schrödinger equation and their physical intrinsic properties. The evolution of the nonlinear slowly varying envelop (paraxial beams or narrow-band signal pulses) in the local Kerr media whose nonlinear index of refraction Δn equals to n2 |E|2 can be described by the nonlocal nonlinear Schrödinger equation. As described in Section 4.1.2, the nonlinear refraction index in the Kerr medium is generally space-time nonlocal. In this section, we will discuss the theme of this special topic, namely, the space nonlocality and its influence on the propagation of the space beams in the Kerr medium. Although people refers to more or less the beam propagation problems in

44 See Section 14.1 in Ref. [6].

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the nonlocal nonlinear medium, for example, the photorefractive material actually is a kind of the nonlocal nonlinear medium which has a weaker nonlocality, the systematic research on optical solitons in the nonlocal media should start from an article published in SCIENCE in 1997 [44]. Now, people have realized that the molecular reorientation mechanism of a nematic liquid crystal [46] and the nonlinear response in thermotropics [36] may have the large nonlinear characteristic length; thus, the strongly nonlocal propagation condition can be easily satisfied.

4.2.1 The classification of the nonlocality In the nonlocal Kerr-type nonlinear medium, the nonlinear refraction index Δn given by Equation (4.4a) is ∞

Δn(r, z) = n2 ∫ R(r − r󸀠 )|Ψ(z, r󸀠 )|2 dD r󸀠 .

(4.63)

−∞

The evolution of the paraxial beams in the nonlocal Kerr media can be phenomenologically described by the nonlocal nonlinear Schrödinger equation (4.12). The integration in Equations (4.12) and (4.63) includes the product of the beam function and the material nonlinear response function R(r − r󸀠 )|Ψ(z, r󸀠 )|2 , which is determined by the relative sizes of spatial features of functions |Ψ|2 and R, plays a crucial role in finding the solutions of Equation (4.12) and their properties. According to the relative size of the spatial scale of the beam function Ψ (the beam width, w) and that of the nonlinear material response function R (the material characteristic length, w m ), the degree of the nonlocality can be divided into four categories [32, 67], such as the local, the weakly nonlocal, the general, and the strongly nonlocal (strongly nonlocal⁴⁵) response. In the limit of a singular response, R(r) = δ(r), the degree of the nonlocality is local (see Figure 4.2 (a)). For a weak nonlocality, the width of the nonlinear material response function is much less than the beam width (see Figure 4.2 (b)). On the contrary, in the limit of a strongly nonlocal response, the width of the nonlinear material response function is much broader than the beam width (see Figure 4.2 (d)). The another case is the general nonlocal response except (a), (b), and (d). The degree of the nonlocality is a relative conception, which represents the relative strength of the spatial scale of the nonlinear material response function and that of the beam propagation in it. From Equation (4.4a) (Equation (4.63)), the nonlinear nonlocality means that the nonlinear polarization (nonlinear response) of the media at a certain point with a small volume of radius r0 (r0 is far less than any wavelength involved) depends not only on the value of the electric field inside this volume, but also on the

45 Some references called highly nonlocal, for example, [44, 47] and other references published by Prof. G. Assanto et al.

262 | Qi Guo, Daquan Lu, Dongmei Deng

Fig. 4.2. The degree of the nonlocality. Shown is the local (a), the weakly nonlocal (b), the general (c), and the strongly nonlocal (d) cases (after O. Bang et al., Physical Review E, Volume 66, 046619 (2002)).

electric field outside the volume under consideration. The stronger the nonlocality, the more fields are involved to contribute to the polarization. Contrariwise, the nonlinear polarization at a certain point is only determined by the electric field at that point in the local nonlinear media. In other words, the nonlinear response induced at a certain point is carried away to the surrounding regions in the nonlocal nonlinear media. In this way, the electric field at a certain point can impact on the behavior of the other electric fields near the surrounding by its inducing spatially broad response. The stronger the nonlocality, the more areas the field can impact on. The weaker the nonlocality, the less areas the field can impact on. The nonlinear locality means the field can do nothing on the areas, i.e. the light field at a given point can only control itself, on the contrary, the infinite nonlocality means the light field in one point can “remote control” other light field in an infinite area.

4.2.2 The Snyder–Mitchell model The Snyder–Mitchell model [44, 45] is the simplified model of the NNLSE (4.12) for the case of the strong nonlocality and the condition that the response function R(r) is symmetrical and regular (or at least twice differentiable) at r = 0. The Snyder–Mitchell model is a linear model but can describe the evolution of the beam propagation in the nonlinear medium. In this subsection, the procedure from the NNLSE to the Snyder–Mitchell model will be developed, and the method to solve the Snyder–Mitchell model will be discussed.

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A. From the nonlocal nonlinear Schrödinger equation to the Snyder–Mitchell model For the strongly nonlocal case w/w m ≪ 1, if R(r) is symmetrical and twice differentiable at r = 0, then one can expand R(r) into Taylor’s series. As a result, Equation (4.12) can be reduced to the following strongly nonlocal model [50, 85]: ∞

i

R󸀠󸀠 ∂Ψ 1 2 + ∇D Ψ + γ b Ψ ∫ [R0 + 0 (r − r󸀠 )2 ] |Ψ(z, r󸀠 )|2 d D r󸀠 = 0 , ∂z 2k 2

(4.64)

−∞

2 2 where R0 = R(0), R󸀠󸀠 0 = ∂ R(r)/∂ x|r=0 . Because R0 is the biggest point of R(r), so R󸀠󸀠 0 < 0. By adding −r c and r c defined in Equation (4.17), Equation (4.64) can be changed into

i

1 ∂Ψ 1 2 2 + ∇ Ψ + γ b R0 P0 Ψ + γ b R󸀠󸀠 0 P 0 (r − r c ) Ψ ∂z 2k D 2 ∞

+

1 󸀠 2 󸀠 2 D 󸀠 γ b R󸀠󸀠 0 Ψ ∫ (r − r c ) |Ψ(z, r )| d r = 0 . 2

(4.65)

−∞

By introducing the coordinate and the function transforms [69, 71, 86], respectively, {s = r − rc (z) , { ζ = z, { Ψ(z, r) = ψ(ζ, s) exp[iϕ(ζ, s)] ,

(4.66) (4.67)

where the phase ϕ(ζ, s) is expressed as ϕ=k

M M2 ⋅ [s + rc (ζ)] − k 2 ζ + γ b {R0 P0 ζ P0 2P0 ζ

∞

0

−∞

R󸀠󸀠 2 + 0 ∫ dζ 󸀠 ∫ [r󸀠 − r c (ζ 󸀠 )] |ψ(ζ 󸀠 , r󸀠 )|2 d D r󸀠 }, 2

(4.68)

we can show that ψ(ζ, s) satisfies the following equation: 2ik

∂ψ + ∇2D ψ − κ m s2 ψ = 0 , ∂ζ

(4.69)

where κ m = k 2 n2 (−R󸀠󸀠 0 )P 0 /n0 . When M = 0 and r c0 = 0, Equation (4.69) is simplified into the Snyder–Mitchell model [44] 2ik

∂ψ + ∇2D ψ − κ m r2 ψ = 0 . ∂z

(4.70)

Equation (4.69) is the same form as Equation (4.70), but they are in the different coordinate systems. The reference frame for the Snyder–Mitchell model (4.70) is at rest (a laboratory coordinate system), while the reference frame for Equation (4.69) moves with

264 | Qi Guo, Daquan Lu, Dongmei Deng the mass center rc . The trajectory of the mass center of the solution to Equation (4.70) is a straight line along the z-axis, while that to Equation (4.69) is a straight line with a slope described by Equation (4.18). In this sense, Equation (4.69) can be called a generalized Snyder–Mitchell model. When M = rc0 = 0, the generalized Snyder–Mitchell model is reduced to the Snyder–Mitchell model. The function transform (4.67) gives the relationship between Ψ(z, r) and ψ(ζ, s), we can conclude that there is a phase difference ϕ (4.68) defined by Equation (4.67) between the solution of Equation (4.12) and that of the generalized Snyder–Mitchell model (4.69). Even though for the case when M = r c0 = 0, the phase difference (ϕ(z) ≠ 0) between Equation (4.12) and the Snyder–Mitchell model (4.69) can be expressed as z

∞

R󸀠󸀠 ϕ(z) = γ b [R0 P0 z + 0 ∫ dζ 󸀠 ∫ r󸀠2 |ψ(ζ 󸀠 , r󸀠 )|2 d D r󸀠 ] . 2 −∞ 0 [ ]

(4.71)

For the case of the strongly nonlocal condition, the essence of the physics of simplifying the nonlocal nonlinear equation to the linear Snyder–Mitchell model is equaling the nonlinear refractive index of the convolution form with the linear square refractive index (also called the parabolic refractive index), that is, Δn can be expressed in the form of [44] (4.72) Δn(r, z) = R0 P0 − α p r2 , where α p = |R󸀠󸀠 0 |P 0 /2 represents the attenuation index of the parabolic refractive index. In the strongly nonlocal condition, the distribution area of the nonlinear refractive index Δn is larger than that of the beam Ψ, Ψ(z, r) can only “sample” a very small part of near the center (r = 0), i.e. in Δn(r, z)⁴⁶, then, we can expand Δn(r, z) to the variable vector r in r = 0, and keep the first two terms which are not zero and obtain Equation (4.72). The Snyder–Mitchell model (including the generalized model) is the linear partial equation, but the nonlocal nonlinear equation is the nonlinear partialintegral equation. In relative terms, the mathematic solution of the Snyder–Mitchell model is more simpler and easier. In physics, the Snyder–Mitchell model transforms the complex nonlinear propagation problem to the propagation problem of the square refractive index medium [87], or the quantum wave function in the potential energy of the linear harmonic oscillator [88]. If we got the accurate analytical solution ψ of the Snyder–Mitchell model ((4.69) or (4.70)), through the function transformation (4.67), we can get the similar analytical solution Ψ of the nonlocal nonlinear equation in the strongly nonlocal condition (4.12).

46 The last term in Equation (4.11) is directly proportional to Δn(r, z)Ψ(z, r), the nonzero area of the product term is determined by the nonzero area of the “narrow” function Ψ(z, r).

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B. An accessible soliton of the Snyder–Mitchell model Suppose a solution to Equation (4.70) of the Gaussian function form ψ(z, r) =

√P0 exp[iφ(z)] [√πw(z)]D/2

exp [−

r2 + ic u (z)r2 ] , 2w(z)2

(4.73)

where φ(z) is the phase of the complex amplitude of the solution, w is the beam width, c u is the phase-front curvature of the beam, and they are all allowed to vary with the propagation distance z. The real amplitude of the solution has the form √P0 /[√πw]D/2 , owing to the conservation of the power. Inserting the above trial function into Equation (4.70), and from the zeroth-order coefficient of r, we obtain the first-order ordinary differential equation for φ(z) dφ(z) D + = 0. dz 2kw2

(4.74)

In the same way, from the real and the imaginary parts of the quadratic coefficient of r, one can yield the two equations for the parameters w and c, respectively, dw 2c u w − =0, dz k

(4.75)

2c2 κ m dc u 1 + u + − =0. 4 dz k 2k 2kw

(4.76)

The combination of Equation (4.76) with the derivative form of Equation (4.75) yields 1 κm d2 y − + y=0, dz2 k 2 w40 y3 k 2

(4.77)

where the normalization w(z)/w0 = y(z) is introduced, and w0 = w(0) is the initial beam width of the Gaussian beam. It has been found that [85] Equation (4.77) is equivalent to Newton’s second law in the classical mechanics for the motion of a one-dimension particle with the equivalent mass 2k acted by the equivalent force F = 1/k 2 w40 y3 − κ m y/k 2 , while y and z are equivalent to the spatial and temporal coordinates of the particle, respectively. The first term of F makes the particle accelerate, and the particle’s velocity dy/dz becomes bigger and bigger, which means that the beam is being expanded or has a trend to be expanded, depending upon the initial velocity dy/dz|z=0 ≥ 0 or < 0. It is obvious that this term is the effect of diffraction. By contraries, the second term of F depends upon the sign of n2 . For self-focusing medium (n2 >0), F acts as an elastic force following Hooke’s law that always drives the particle back to its initial state, decelerates the particle, and presents the compression effect of the nonlinear-induced refraction. When the diffractive force and the refractive force have the same amplitude, the total force will be zero, and the particle will keep its velocity unchangeable. Then the particle with the initial zero-velocity keeps rest, and its spatial coordinate y is always 1:

266 | Qi Guo, Daquan Lu, Dongmei Deng this is a spatial soliton state. Letting the two forces equal and y = 1, we obtain the critical (input) power for the soliton propagation Pc =

n0 4 2 k (−R󸀠󸀠 0 )n2 w0

.

(4.78)

For the self-focusing medium, then the integration of Equation (4.77) reads w = w0 [cos2 (Ωz) +

1/2 Pc sin2 (Ωz)] , P0

(4.79)

where Ω=√

P0 1 . P c kw20

(4.80)

Substituting w into Equations (4.74) and (4.76), one can obtain φ and c u . Pc D tan(Ωz)] , arctan [√ 2 P0 ] [ Ωk(P c /P0 − 1) sin(2Ωz) . cu = 4[cos2 (Ωz) + (P c /P0 ) sin2 (Ωz)] φ = φ(D) = −

(4.81) (4.82)

Equation (4.73) with Equations (4.79)–(4.82) of w, φ, and c u is the exact analytical solution of the Snyder–Mitchell model (4.70). Applying Equation (4.67), we can obtain the approximate analytical solution of the nonlocal nonlinear equation (4.12) which is symmetric to the z-axis: Ψ(z, r) =

√P0 (√πw)D/2

exp (−

r2 ̄ , ) exp[i(c u r2 + φ)] 2w2

(4.83)

where Pc P0 z D tan(Ωz)] + σ p arctan [√ 2 P0 P c kw20 ] [ P0 D 1 P0 (1 − + [ ) sin(2Ωz) − ( + 1) z] , Pc Pc 8kw20 2Ω

φ̄ = −

(4.84)

2 σ p = R0 /(−R󸀠󸀠 0 w0 ). In the z = 0 plane, this initial input solution is

Ψ(z, r)|z=0 =

√P0 (√πw0 )D/2

exp (−

r2 ) , 2w20

(4.85)

where w is given by Equation (4.79). Equation (4.79) shows that when P0 < P c , the beam diffraction initially overcomes the beam-induced refraction, and the beam initially expands, when the beam width broadening to a certain extent, the diffraction effect which is inversely proportional to the beam width becomes weaker than the

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nonlinear effect, thereby, the beam width begins to compress, the process can be repeated periodically (for a period π/Ω), the beam width with w2 /w20 vibrating between the maximum value P c /P0 and the minimum value 1. Whereas when P0 > P c , the reverse happens and the beam initially contracts, with w2 /w20 breathing between a maximum 1 and a minimum P c /P0 . When P0 = P c , diffraction is exactly balanced by nonlinearity, and the Gaussian-shaped beam preserves its width as it travels in the straight path along the z-axis. When P0 = P c , this is a soliton. When P0 ≠ P c , this is a breathers. The evolution of the single Gaussian beam for the different power P0 is shown in Figure 4.3.

Fig. 4.3. Comparison of the analytical solution (solid curves) with the numerical simulation (dashed curves) for the (1 + 1)-dimensional beam propagation in the Gaussian-shaped response (Equation (4.87)) material when w 0 /w m = 0.3. The initial conditions are (a) P 0 /P c = 0.70, (b) P 0 /P c = 1.00, and (c) P 0 /P c = 1.55.

When P0 = P c , it can be deduced that w = w0 , c u = 0, and Ω = 1/z R (z R = kw20 is the Rayleigh distance, namely the diffraction length defined by Equation (4.27)), then Equation (4.83) is simplified to an accessible (spatial) soliton Ψ s (z, x, y) =

n0 √ 2+D/2 (−R󸀠󸀠 kπ D/4 w0 0 )n2 1

exp (−

r2 ) exp(iφ s z) , 2w20

(4.86)

where φ s = (σ p − 3D/4)/kw20 . φ s is the phase shift after the beam propagating a distance z. Ψ s (Equation (4.86)) is the exact expression⁴⁷ of the strongly nonlocal spatial optical soliton from the Snyder–Mitchell model [44]. An accessible (spatial) soliton with any width can propagate in the media as long as its power P0 equals exactly the 2 2 2 critical power P c defined in Equation (4.78). Because −R󸀠󸀠 0 ∼ R0 /w m , so σ p = νw m /w0 , where the dimensionless coefficient ν here is only determined by the nature of the material, has an order of O(1). For example, if the response of the material is assumed to

47 The phase expression of the soliton was not obtained because Snyder and Mitchell discussed mainly the evolution of the beam width in the Ref. [44].

268 | Qi Guo, Daquan Lu, Dongmei Deng be the Gaussian function R(r) =

1 (√2πw m )D

exp (−

r2 ) . 2w2m

(4.87)

It can be shown that ν = 1, then we can conclude that the parameter σ p is determined by the initial beam width of the spatial optical soliton and the material property, has nothing to do with other parameters [85]. When the strong nonlocal condition is satisfied, we have w m /w ≫ 1, then it is observed that φ s z ≈ νw2m z/w40 k. On the other hand, for the local Kerr soliton, it has been shown that the phase shift is φ ns z = z/2kw20 (Equation (4.54)).⁴⁸ Comparing the results between the strongly nonlocal and the local cases, one observes that the phase shift for the former is (w m /w0 )2 times (about two order) larger than that for the latter. Therefore, one concludes that [85] the phase shift of the accessible (spatial) soliton can be very large during its propagation comparable to its local counterpart. One can estimate the specimen length L π for a π phase shift in such media: L π = 2π2 w40 n0 /νλw2m . Taking ν ≈ 1, w m ≈ 10w0 , n0 ≈ 1, w0 ≈ 20λ, and λ ≈ 0.5 μm, we get that L π ≈ 40 μm. In the visible spectrum region, therefore, the specimen length for the π phase shift in the strongly nonlocal media has an order less than 0.1 mm. The effective generation of the π phase shift is a key issue for the modification, manipulation, and control of optical fields based on the principle of interference, especially in the optical switching based on the Mach–Zehnder interference principle; hence, this phenomenon might be of the potential value in applications of integrated all-optical devices, such as optical switches and couplers. The big phase shift phenomenon of the strong nonlocal spatial optical soliton has been experimentally observed in the lead glass [89]. The comparison of the analytical solutions with the numerical simulations of Equations (4.12) and (4.85) for different w0 /w m and P0 shows that [85] the analytical predictions are still close approximation (the absolute values of the relative errors are within 10%) to the simulations till w0 /w m is about 0.5. For the same w0 /w m , higher the input power, better is the approximation. Some of the comparison results are shown in Figure 4.3.

C. Other accurate analytical solutions of the Snyder–Mitchell model The Snyder–Mitchell model has other accurate analytical solutions, including the Hermite–Gaussian breathers and soliton solution [90, 91] in the rectangular coordinate system, the Laguerre–Gaussian breathers and soliton solution [92, 93] in the cylindrical coordinate system, the Ince–Gaussian breathers and soliton solution [94, 95] in the elliptic coordinate system, the Hermite–Laguerre–Gaussian solution cluster [96] and the complex-variable-function Gaussian breathers and soliton solution cluster [97, 98]. The spatial propagation process described by the Snyder–Mitchell 48 Let ξ 0 = 0, ϕ 0 = 0 in Equation (4.54), and delete the factor exp[i(kz − ωt)], the result is the same as that in Ref. [38].

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model can also be looked as the self-induced fractional Fourier transform [99]. In view of the focus of this section is the Snyder–Mitchell model and its single soliton solutions, you can refer to the relevant literatures and monographs by the author and other colleagues to learn the details of other contents [71].

4.2.3 The weak-nonlocality For the weak nonlocal case, the characteristic length of the nonlinear response function of a material is much narrower than the width of the beam propagating in the material (Figure 4.2 (b)), so, we can expand |Ψ(z, r)|2 which is in the integral form of the NNLSE (4.12), and not the response function R(r) like the strong nonlocal case. In the (1 + 1)-dimensional case, one can expand |Ψ(z, x󸀠 )|2 by means of Taylor’s expansion around the point x󸀠 = x, to the fourth order. Equation (4.12) can be reduced to the weak nonlocal model [100] i

∂2 |Ψ|2 1 ∂2 Ψ ∂Ψ 2 + γ Ψ (|Ψ| + κ )=0, + b w ∂z 2k ∂x2 ∂x2

(4.88)

where the weak nonlocality parameter κ w (> 0) is given by the following equation: ∞

κw =

1 ∫ R(x)x2 dx . 2

(4.89)

−∞

Suppose a bright soliton solution to Equation (4.88) is of the function form in the self-focusing medium (γ b > 0) Ψ(z, x) = G(x) exp(iΓz) ,

(4.90)

where G is the symmetric real function, Γ(> 0) is the propagation constant. Substitution of Equation (4.90) into Equation (4.88), and after a complex set of calculations, we can get the implicit function expression G(x) of [100] x=

σ w (x) 1 tanh−1 [ ] + 2√κ w tan−1 [2√κ w kγ b σ w (x)] , G0 (kγ b )1/2 G0

and Γ=

1 γ b G20 , 2

(4.91)

(4.92)

where σ w (x) = {[G20 − G2 (x)]/[1 + 4κ w kγ b G2 (x)]}1/2 , G0 is the maximum value of G. The implicit function (4.91) is the bright soliton solution of the nonlocal medium in the case of the weak nonlocality. In the limit case of κ w → 0, the weak nonlocality approaches to the locality, one can obtain Ψ = G0 sech[(kγ b )1/2 G0 x] exp(iγ b G20 z/2) from Equations (4.91) and (4.92), which is just the expression (4.54) of local spatial

270 | Qi Guo, Daquan Lu, Dongmei Deng

Fig. 4.4. The normalized intensity (|Ψ(x)|2 |/|Ψ|2max ) distributions of the bright spatial optical solitons with different parameters κ w for the case of the weak nonlocality (after W. Krolikowski et al., Physical Review E, Volume 63, 016610 (2001)).

optical solitons [38].⁴⁹ Figure 4.4 shows the intensity distribution of the soliton given by the implicit function (4.91) with the different nonlocal parameter κ w . Obviously, the beam width becomes broader with the increase of the degree of the nonlocality. The dark soliton solution of Equation (4.88) has been found in a self-defocusing medium (γ b < 0), and it has been proved that both bright soliton solutions and dark soliton solutions are stable [100].

4.2.4 The general nonlocality In the case of the general nonlocality, except for some special nonlinear response functions, such as the logarithmic nonlocal response function [66] with the Gaussian function kernel, it is generally difficult to obtain the accurate analytical solution to the NNLSE with an arbitrary nonlocal response function. But by the numerical simulation, we can get numerical solutions of spatial optical solitons to the NNLSE with different degrees of the nonlocality [101, 102]. In order to simplify the numerical procedure, we first need to remove the independent “redundant” parameters in Equation (4.12). Therefore, by using the dimensionless transformation equation (4.30), the NNLSE can be transformed into the following dimensionless NNLSE (n2 > 0): ∞

i

1 2 ∂ ̄ − η󸀠 )|u(ξ, η 󸀠 )|2 d D η󸀠 = 0 . u(ξ, η) + ∇D u(ξ, η) + u(ξ, η) ∫ R(η ∂ξ 2

(4.93)

−∞

49 If the fast variable scale factor exp[i(kz − ωt)] is removed and let ξ 0 = ϕ 0 = 0, Equation (4.54) is exactly the same as the result here.

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The Gaussian response function (4.87) in dimensionless system can be expressed as α 2 (η2x + η2y ) α D ̄ R(η) =( ) exp [− ] , 2 √2π

(4.94)

where the parameter α = w0 /w m stands for the degree of the nonlocality of the nonlinear response, the smaller α is, the stronger the degree of the nonlocality is; conversely, the larger α is, the weaker the degree of the nonlocality is. Figure 4.5 shows the numerical properties of the single soliton solution [101, 102] of the (1 + 1)-dimensional Equation (4.93) with a phenomenological regular Gaussian response function (4.94). The characteristics of the (1 + 1)-dimensional nonlocal spatial optical soliton can be summarized as, regardless of the degree of the nonlocality, the (1 + 1)-dimensional spatial beams can propagate stably in the single-peak (unimodal) optical soliton form in the nonlocal nonlinear medium. The waveform of the single optical soliton is changed from a Gaussian wave packet in the case of the strong nonlocality to a wave packet of a hyperbolic secant form in the local case (Figure 4.5 (a)), the critical power for the single soliton decreases with the degree of the nonlocality decreasing (Figure 4.5 (b)), the phase of the optical soliton increases linearly with the distance, the weaker the nonlocality is, the small the slope of the soliton phase (the derivative of the soliton phase with respect to the propagation distance) (Figure 4.5 (c)). The numerical results for different nonlinear response functions show [102] that the properties of the nonlocal spatial optical soliton in Figure 4.5 are universal, are not dependent on the specific form of the nonlinear response function, but for the different nonlinear response function, the quantitative relationship of the results is different. Section 4.1.4 has pointed out that the stable transmission state of the (1 + 2)dimensional bright spatial optical soliton cannot form in the local self-focusing Kerr nonlinear material. When the carrying power of the beam is equal to the critical power defined by Equation (4.62), the (1 + 2)-dimensional light beam in the local Kerr self-

Fig. 4.5. The properties of the nonlocal spatial optical solitons vs the degree of nonlocality. Left: the normalized intensity profiles [|u|2 /u 20 (u 0 maximum amplitude)] of the optical solitons with the same beam width for the different degrees of nonlocality α, middle: the critical power P c as a function of the degree of nonlocality α (the α-coordinate is the logarithm coordinate.), Right: the slope of the soliton phase as a function of the degree of nonlocality α (the α-coordinate is the logarithm coordinate.).

272 | Qi Guo, Daquan Lu, Dongmei Deng focusing material will form the self-trapping state. But as long as any disturbance makes the beam power greater than the critical power of the beam self-trapping, the beam will continue to focus until the beam width is zero, the great crash appears. However, the nonlocal nonlinearity is the same as the saturation nonlinearity, which is one of the important factors to stop the great crash of the (1 + 2)-dimensional beam to appear. Bang et al. [32] have shown that the (1 + 2)-dimensional NNLSE with an arbitrary nonlinear response function are stable, its bright soliton solution would not appear in the great crash. The only condition of the stability is that the nonlinear response function is to have the band-limited positive Fourier spectrum, this property will be satisfied if the nonlinear response function has the physical significance.

4.2.5 Interaction of double solitons When two solitons co-propagate together, their interaction will occur. According to the distance between the two solitons, the interaction of solitons can be divided into the short-range interaction and the long-range interaction. When the distance between the two solitons is so close that their light fields overlap, the interaction in such a case is the short-range interaction; conversely, when the spacing between the two solitons becomes large enough so that there is no overlap between optical fields of the solitons, the interaction in such a case is the long-range interaction. There is significantly different characteristics between the short-range interaction and the long-range interaction: the former is closely related to the phase difference of two solitons, and the latter has nothing to do with the phase difference; the long-range interaction only happens when the degree of the nonlocality is strong enough, but the short-range interaction occurs with an arbitrary degree of the nonlocality.

A. The centroid trajectory of the co-propagating double beams To better understand the interaction of the double beams, we first discuss the changing rule of the centroid trajectory for the co-propagating double beams. Suppose the two simultaneously obliquely incident Gaussian solitons are coplanar on the y–z plane into the nonlocal nonlinear medium, with a beam width w0 , a phase difference ϕ I , and a separation 2h, and the incident angles (the angle with respect to the z-axis), respectively ϑ I and −ϑ I , that is Ψ(z, x, y)|z=0 = Ψ0 exp [−

x2 + (y + h)2 + ik(y + h) tan ϑ I ] 2w20

+ Ψ0 e iϕ I exp [−

x2 + (y − h)2 − ik(y − h) tan ϑ I ] , 2w20

(4.95)

where Ψ0 is the amplitude of the beam, we assume that the beam amplitude is large enough so that the critical power of the soliton propagation can be gener-

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ated [44, 46, 47]. For incident conditions (4.95), the initial power and momentum are, respectively, P0 = 2πΨ02 w20 [1 + cos ϕ I exp (− M = ey

h2 − k 2 w20 tan2 ϑ I )] , w20

2πhΨ02 h2 exp (− 2 − k 2 w20 tan2 ϑ I ) sin ϕ I , k w0

(4.96) (4.97)

and the initial beam centroid r c0 = 0. Because the NNLSE (4.12) has the characteristics of the power conservation and the momentum conservation, the beam centroid trajectory can be determined as a straight line in the y–z plane by applying the initial power, the initial momentum and the beam centroid trajectory (Equation (4.18)), the angle Θ y [86] with respect to the z-axis satisfies exp [− (h/w0 )2 − (tan ϑ I /Θ d )2 ] sin ϕ I tan Θ y h =( ) , Θd w0 1 + exp [− (h/w0 )2 − (tan ϑ I /Θ d )2 ] cos ϕ I

(4.98)

where Θ d = 1/kw0 is the far field divergence angle of the Gaussian beam. As shown in Equation (4.98), there is no relationship between the changing rule of the centroid trajectory and the amplitude of the beam (i.e., power). Figure 4.6 shows the centroid straight slope of two parallel incident beams as a function of the beam spacing 2h and the phase difference ϕ I . As the figure shows, the slope of the line for the trajectory of the mass center is highly dependent on the separation 2h and the phase difference ϕ I . Figure 4.6 shows that Θ y is closely related to the phase difference ϕ I , only when ϕ I = 0 or π there is Θ y = 0 for h/w0 ≤ 2, Θ y goes to zero when h/w0 > 2, Θ y has a significant nonzero value when h/w0 ≤ 1. It is important to emphasize that the above analytical result for the movement of the mass center is universal and independent of the form of the nonlinear response function R. As Equation (4.98) shows, when the soliton spacing 2h is greater than 6w0 , Θ y ≡ 0, the change of the centroid trajectory for two soliton beams is no longer associated with their phase difference, the centroid will remain the same. The interaction of the beams is the long-range interaction when h ≥ 3w0 . In this condition, the intensity waveform of two beams have no effective overlap part, when the degree of the nonlocality is weak enough, two beams would not be affected by the nonlinear refractive index produced by each other, thus the interaction does not occur, and two beams propagate independently. Therefore, only under the condition of the strong enough nonlocality, there exists the long-range interaction between two beams. Conversely, the interaction of the beams is the short-range interaction when h < 3w0 . The short-range interaction of the double beams occurs under any degree of the nonlocality, and is closely related to the phase difference of two beams.

B. The short-range interaction To discuss the short-range interaction features of the solitons, it is necessary to solve the NNLSE (4.12) with the initial inputting double beams (4.95) in the case of h/w0 < 3.

274 | Qi Guo, Daquan Lu, Dongmei Deng

Fig. 4.6. Dependence of the slope on the distance h (a) and the phase difference ϕ I (b) for two parallel-injected solitons (ϑ I = 0) (after W. Hu et al., Physical Review A 77, 033842 (2008)).

For an arbitrary degree of the nonlocality, this problem generally cannot be solved analytically, can only be solved numerically. Figure 4.7 shows the contour map [86] of the numerical results obtained from the (1 + 1)-dimensional NNLSE in the different degrees of the nonlocality. The (1 + 1)-dimensional model makes it possible to compare propagations in the nonlocal and the local nonlinearity, and provides a sufficiently accurate description of the (1 + 2)-dimensional coplanar propagation. Some interesting consequences can be seen in Figure 4.7. First, for the local system described by the NLSE, the two solitons (in the first column of Figure 4.7) attract only for the inphase case (ϕ I = 0) and otherwise repel, as mentioned in Ref. [20]. There is also a power transfer between the solitons when the phase difference ϕ I does not equal 0 or π. The force between the in-phase solitons is attractive and independent of the degree of the nonlocality, as shown in the first row, whereas the repulsion between the solitons (from the second row to the fourth row in Figure 4.7) for the other phase difference cases weakens as the degree of the nonlocality increases. As a result, two solitons with an arbitrary phase difference can attract when the nonlocality becomes sufficiently strong. For all of the propagations, however, the movement of the mass center obeys the same regulation, a straight-line trajectory with a slope given by Equation (4.18), the slope is a straight line determined by the initial momentum and the initial power M y /P0 . It is clear that for a strong enough nonlocality, the two spatial solitons trap each other and propagate together as a whole (The distance between them is too close to distinguish the two closely spaced beams.) along the trajectory of the mass center. Therefore, in the case of the strong nonlocality, by applying the short-range interaction characteristic of the solitons, we can realize controlling the propagation together as a whole of two spatial solitons through the phase difference of two spatial optical solitons. It is a very complicated mathematical problem to solve the nonlinear wave interaction problem. It is impossible to obtain the analytical results for the problem of the interaction with an arbitrary degree of the nonlocality, but under the

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Fig. 4.7. Contour graph of the numerical propagation of Equation (4.93) for the two parallel-injected solitons. The local case (α = ∞) is shown in the first (from left to right) column, and the two different nonlocal cases are shown in the second (α = 2) and third (α = 0.1) columns, respectively. The dashed lines show the movement of the mass center of the two solitons. The results from the modified Snyder–Mitchell model, i.e. Equation (4.69), with the same initial condition are presented in the fourth column for comparison. The phase differences between the two solitons are 0, π/2, π, and 3π/2, respectively, from top to bottom (after W. Hu et al., Physical Review A 77, 033842 (2008)).

condition of the strong nonlocality, the analytical interaction solution of the double solitons can be obtained from the generalized Snyder–Mitchell model. Due to the linearity of the Snyder–Mitchell model, the nonlinear interaction problem of the beams can be analyzed using the linear superposition principle in the strongly local cases. As Ref. [44] has pointed out: if ψ(z, r) is a solution of Equation (4.70), when the necessary condition that r0 satisfies the harmonic oscillator equation r0̈ (z)+Ω2 r0 (z) = 0, μ(z) and φ I (z), respectively, decided by the μ(z) = k r0̇ (z) and φ İ (z) = k[Ω 2 r20 (z) − r20̇ (z)]/2 is satisfied, Ξ± (z, r) = ψ(z, r ± r0 ) exp(∓iμ ⋅ r + iφ I ) is also the solution of Equation (4.70). Because Equation (4.70) is linear, we can construct the analytical in-

276 | Qi Guo, Daquan Lu, Dongmei Deng teraction solution [103] of the double beams by using the superposition principle, ψ± (z, r) = C± [Ξ+ (z, r) ± Ξ− (z, r)] ,

(4.99)

and Ξ± =

√P 0 exp[iφ(z)] [√πw(z)]D/2

exp (−

x2 + [y ± y0 (z)]2 + ic u (z){x2 + [y ± y0 (z)]2 }) 2w(z)2

× exp[∓iμ(z)y + iφ I (z)] ,

(4.100)

where w, Ω, φ, and c u are, respectively, determined by Equations (4.79)–(4.82). The functions y0 (z), μ(z) and φ I (z) which are determined by the initial condition at z = 0, i.e. y0 (0) = h, ẏ 0 (0) = 0 are given by y0 (z) = h cos(Ωz), μ(z) = −kΩh sin(Ωz), φ I (z) =

1 kΩh2 sin(2Ωz) . 4

(4.101)

Equations (4.99)–(4.101) describe the interactions of two initial in-phase (ϕ I = 0) or initial out-of-phase (ϕ I = π) parallel incidence (ϑ I = 0) Gaussian beams. C ± √P 0 x2 + (y + h)2 x2 + (y − h)2 {exp [− ] ± exp [− ]} (√πw0 )D/2 2w20 2w20 (4.102) The total power of two beams is P0 , where the constant C± of Equation (4.102) is determined by ∫ |ψ± (0, r)|2 dD r = P0 . When D = 2 (1 + 2-dimensional case), we can get C± = {2[1 ± exp(−h2 /w20 )]}−1/2 from Equation (4.96). For the state of the soliton ψ± (z, x, y)|z=0 =

y0 = h cos (

z ), zR

μ=−

h z sin ( ) , 2 z w0 R

φI =

h2 2z sin ( ) . 2 zR 4w0

(4.103)

The above analytic solutions are suitable for describing both the short-range interaction and the long-range interaction of the strongly nonlocal spacial light solitons. Though the above solutions are the interaction results of two initial in-phase or initial out-of-phase beams, it is not difficult to generally extend this method to discuss the interaction of strongly nonlocal spacial optical solitons with an arbitrary phase difference, as is shown in Ref. [69]. The difference between the short-range interaction and the long-range interaction is, the trajectory of the effective mass center of interacting beams for the former is a straight line whose slope depends on the incident conditions like the phase difference, we can control the linear slope of the centroid trajectory by the phase difference (see the fourth row of Figure 4.7), but the latter is because the spacing is so big that the initial momentum is zero, then the slope of the trajectory of the effective mass center always equals zero (more details are as follows).

C. The long-range interaction When the spacing meets h ≥ 3w0 , the interaction mode of the strongly nonlocal solitons is independent of the phase difference.

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Let D = 1 and x = 0 in Equations (4.99)–(4.102), we can get the interaction solution of the (1 + 1)-dimensional two beams ψ± (z, y) in the strongly nonlocal case, that’s just the long-range interaction solution [44, 104]⁵⁰ of the strongly nonlocal solitons obtained by Snyder and Mitchell in 1997. Based on the above results, we can see that: two beams with the same amplitude and parallel to the incidence propagation under the strongly nonlocal condition will occur periodic collision; the collision period is π/Ω (in terms of the light intensity |ψ|2 ), the period is inversely proportional to the input power of the beam; the two beams will recover to their initial states after every collision, as Figure 4.8 shows. We can see from Figure 4.8 that the long-range interaction of the strongly nonlocal spatial optical solitons is independent of the relative phase of the two solitons, two solitons always attract each other no matter how the phase difference of two solitons. Snyder and Mitchell also discussed the long-range interaction character [44] of one strong soliton and one weak soliton. Suppose two Gaussian beams have the same beam width, but one has the power of P c − ΔP, and the other has the power of ΔP, and P c ≫ ΔP. So we can find that the strong bright solution with the power of P c − ΔP will propagate along a straight line, just like exist alone; but the beam width of the weak bright beam with the power of ΔP will also stays the same (also the soliton state), but its propagation trajectory will be a sine trajectory along the trajectory of the strong bright soliton. But when the weak bright beam propagates alone, it will move along a straight line not a curve, and its beam width will be expanded during the propagation. This shows that, in the strongly nonlocal nonlinear medium, the weak bright beams

Fig. 4.8. The long-range interaction of the two strongly nonlocal spatial solitons. y is the transverse coordinate, z is the longitudinal coordinate (only two interaction cycles are given in the figure.), |ψ|2 is the total intensity, and the initial spacing between the two beams is 6w 0 . (a) the initial in-phase, (b) the initial out-of-phase. The intensity distributions are independent on the phase difference except the overlapping regions, where the intensity distribution has a little bit difference due to the in-phase constructive and the out-of-phase destructive interference.

50 The difference between the long-range interaction and the short-range interaction has not realized at that time, which was advanced in Ref. [86].

278 | Qi Guo, Daquan Lu, Dongmei Deng can be controlled to veer by the strong bright solitons in the far distance. This kind of the long-range interaction is the characteristic of the optical beams in the nonlocal nonlinear medium only for the case of the strong nonlocality. The long-range interaction of the strongly nonlocal spatial optical solitons is independent on their phase difference, and the short-range interaction of the strongly nonlocal spatial optical solitons depends strongly on their phase difference, which had already proved qualitatively [105] and quantitatively [86], respectively, by experiments in the nematic liquid crystal. It is interesting to note that people has still not realized the nematic liquid crystal is the nonlinear material with which it is possible to realize strongly nonlocal propagation conditions even after the phase-independenceinteraction of the solitons was experimentally observed in the nematic liquid crystal by Assanto et al. It might be just because the experiment of the phase-independenceinteraction that makes Assanto to realize this problem, which leads to the subsequent prominent work in Physical Review Letters [46, 47] and Nature [106]. It were experimentally demonstrated as the long-range interactions between the strongly nonlocal solitons with an initial separation 10 times larger than the soliton width in the lead glass [107], and with an initial separation 70 times larger than the soliton width in the nematic liquid crystal [108]. The interaction characteristics of the strongly nonlocal soliton are completely different from the weakly nonlocal soliton. Whatever the phase difference is, two strong nonlocal solitons are always attracted to each other, but the centroid trajectory of the short-range interaction is closely related to their phase difference [86], the centroid trajectory of the long-range interaction has nothing to do with the phase difference [44, 103], and is always parallel to the straight line of the propagation direction. The local soliton has no long-range interaction and the short-range interaction of two local solitons is very sensitive to the relative phase of two solitons, and the in-phase solitons attract each other, out-of phase solitons repel each other [20].

4.3 Nematicons Nematicons are self-trapping optical beams which propagate in the nematic liquid crystal (NLC) when the diffraction is exactly balanced by the nonlinearity, including spatial optical breatherss and spatial optical solitons. Nematicons belong to the nonlocal spatial optical soliton (spatial optical breathers) family, which can be described by the NNLSE under certain conditions. The NLC is the first Kerr nonlinear material whose characteristic length is much larger than the beam width [46, 47]. The spatial optical soliton in the NLC is the first strong nonlocal spatial optical soliton observed in experiments [47], and its discoverer Assanto et al. specially invented a new English word “nematicon” to name this spatial optical soliton [110]. The liquid crystal is the special phase state of the material, and the intermediate state between the solid and the liquid state [109]. The liquid crystal is the anisotropy

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of the liquid and can flow, its molecular orientation is of the long-range order, which is similar to the molecules ordering in crystals and is different from the conventional isotropic liquid. The NLC is a kind of the liquid crystal, its optical properties are uniaxial anisotropy. Under an applied light field or an applied low-frequency electric field, the elongated rod-shaped liquid crystal molecules of the NLC will be induced to generate the electric dipole moment, and the direction of the dipole moment is parallel to the rod axis of the long rod-shaped molecules. This induced electric dipole moment will tend to be in parallel with the polarization direction of the applied electric field, which makes the long rod-shaped molecules of the liquid crystal rotate. Such a lightinduced reorientation of its molecules is the physical mechanism for the optical Kerr nonlinearity in the NLC. The nonlocality of the nonlinearity comes from the elastic deformation and its transfer in the liquid crystal that is considered as the continuum medium. The light field makes the liquid crystal molecules within the area occupied by the light to generate the splay, twist, and bend elastic deformation, and such an elastic deformation energy will be transferred to the area that is not occupied by the light in the liquid crystal as the continuum medium, resulting in the molecules in the no-light-area to orientate again, which leads to the nonlinear nonlocality. In this section, we will mainly discuss the propagation characteristics of the optical beams in the sample cell of the nematic liquid which is taken as the equivalent infinite medium (ignore the influence of the boundary effect), including the establishment of the beam propagation model, an approximate analytic solution of a single nematicon, etc. More contents about the research of the nematicon, such as the impact of the boundary effects on propagation characteristics of the nematicon, you can refer to the monographs [17] of the nematicon problems.

4.3.1 Nonlinear propagation models of beams in the nematic liquid crystal The structure of the sample cell of the liquid crystal in the research plan is shown in Figure 4.9, where the y direction is infinity, the polarization direction of the electric field is the x direction, and the propagation of the light wave is along the z direction, the thickness H of the liquid crystal cell is much wider than the beam width of the beam propagation in the liquid crystal cell. The static (low frequency) voltage (preset voltage) of the upper and lower electrodes is to make the liquid crystal molecules have a certain preset bias angle (the preset voltage is greater than the Freederichsz threshold voltage when the liquid crystal molecules start twirling), to reduce the laser input power, and to avoid the thermotropic nonlinear effect. Due to technical difficulties, sample cells used in early experiments [47, 105, 111–113] did not have the output panel (see Figure 4.9 (a)). Because there is no glabrous output surface, we can only measure the light beam width by collecting the scattered light of the beam in the upper observation plane, and cannot measure the cross section waveform of the beam. The latest experiment [114] has adopted the structure with the output panel (Figure 4.9 (b)). As-

280 | Qi Guo, Daquan Lu, Dongmei Deng

Fig. 4.9. The schematic diagram of the sample cell of the NLC: (a) without smooth output panel structure, (b) with smooth output panel structure.

suming that the NLC (the long-rod-shaped liquid crystal molecules denoted by arrows) full of the sample cell is the uniaxial crystals, which satisfies n⊥ < n‖ , where n⊥ is the refractive index of the ordinary light, n‖ is the refractive index of the extraordinary light, which is anchoring along the z direction on the border. After the preset voltage is put in the liquid crystal cell, the paraxial optical beam Ψ of the extraordinary light which polarizes in the x direction and propagates along the z direction is described by the following coupling equations [46, 113]: ∂Ψ op + ∇2D Ψ + k 20 ϵ a (sin2 θ − sin2 θ0 )Ψ = 0 , ∂z 2 ∂2 θ rf op |Ψ| ) sin(2θ) = 0 , 2K ( 2 + ∇2D θ) + ε0 (ϵ a E2rf + ϵ a 2 ∂z 2ik

(4.104) (4.105)

where θ is the angle between the director vector of the liquid crystal molecules and the z-axis, θ0 (0 ≤ θ0 ≤ π/2) is the maximum preset angle in the center of the liquid crystal cell with only the low-frequency electric field, K is the average elastic constant op rf of the NLC, ϵ a (= n2‖ − n2⊥ ) and ϵ a (= ϵ‖ − ϵ⊥ ) respectively represent the anisotropy of the dielectric constant of the medium in the optical frequency and the low-frequency domain, E rf is the electric field intensity consistent with the low-frequency voltage. It has been proved [113, 115] that the derivative argument ∂2z θ of the z coordinate in Equation (4.105) is negligible compared with ∇2D θ. Equation (4.104) describes the paraxial beam evolution in the liquid crystal, while Equation (4.105) describes the interaction between the light field and the liquid crystal molecules. The boundary condition θ|x=−H/2 = θ|x=H/2 = 0 is determined by anchoring on the border along the z directions (the source point of the x coordinate is the center of the liquid crystal cell). When no laser field in the liquid crystal, the preset bias angle θ̂ of the liquid crystal molecules produced by the low-frequency electric field is symmetric to the center of the liquid crystal cell (x = 0), and is the only function of x [113], 2K

∂ 2 θ̂ rf + ε0 ϵ a E2rf sin(2θ)̂ = 0 . ∂x2

(4.106)

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In order to simplify the above coupling equations (4.104) and (4.105), assume θ = θ̂ + ̂ 0 )Φ, where Φ characterizes the orientation angle disturbance of the liquid crystal (θ/θ molecular induced by the light (Φ ≪ 1). When the beam width is far less than the thickness of the sample cell of the liquid crystal, θ̂ ≈ θ0 and ∂ x θ̂ ≈ 0 near the center of the sample cell, such coupling equations (4.104) and (4.105) can be simplified to the coupling equations about Ψ and Φ [113, 116] ∂Ψ op + ∇2D Ψ + k 20 ϵ a sin(2θ0 )ΦΨ = 0 , ∂z op ε0 ϵ a sin(2θ0 )w2m |Ψ|2 = 0 , w2m ∇2D Φ − Φ + 4K 2ik

where 1

wm =

(0)

E rf

{

2θ0 K rf

ε0 ϵ a [sin(2θ0 ) − 2θ0 cos(2θ0 )]

(4.107) (4.108)

1/2

}

,

(4.109)

which has the dimension of length, is the nonlinear characteristic length of the (0) NLC [112] (E rf is the electric field at the low frequency at the center of the liquid crystal cell). w m > 0 when 0 < θ0 ≤ π/2. If the boundary influence of the sample cell of the liquid crystal is ignored, the liquid crystal cell is equivalent to the infinite liquid medium. For the infinity space without boundaries, Equation (4.108) has the specific solution with the convolution (the detailed derivation process can be found in Appendix A of this chapter) ∞

op

Φ(r, z) =

ε0 ϵ a sin(2θ0 )w2m ∫ R(r − r󸀠 )|Ψ(z, r󸀠 )|2 d D r󸀠 , 4K

(4.110)

−∞

where R is the nonlinear response function of the NLC. For the (1 + 1)-dimensional case, R is the exponential decay function [115] R(x) =

|x| 1 exp (− ) . 2w m wm

(4.111)

And for the (1 + 2)-dimensional case, one has [112] R(r) =

√x 2 + y 2 1 K ( ) , 2 0 wm 2πw m

(4.112)

where K0 is the zeroth-order correction Bessel function of the second kind. So Equations (4.107) and (4.108) can be combined into the NNLSE (4.12), the nonlinear refractive index coefficient n2 is given by [108, 116]⁵¹ op

n2 =

(ϵ a )2 θ0 sin(2θ0 ) rf

(0)

4n0 ϵ a (E rf )2 [1 − 2θ0 cot(2θ0 )]

.

(4.113)

51 Both results reported (Equation (4) of Ref. [116] and Equation (2) of Ref. [108]) missed a factor 1/2n 0 .

282 | Qi Guo, Daquan Lu, Dongmei Deng And the nonlinear response function R is determined by Equation (4.111) or (4.112). It can be seen from Equation (4.113) that for the NLC with positive dielectric anisotropy, rf ϵ a > 0 (ϵ‖ > ϵ⊥ ), so n2 > 0 (0 ≤ θ0 ≤ π/2), in the same way, for the NLC with negative dielectric anisotropy (ϵ‖ < ϵ⊥ ), the nonlinear refractive index coefficient n2 < 0. From what has been discussed above, under the condition of the beam width is far less than the thickness of the liquid crystal cell, the central region of the liquid crystal cell shown in Figure 4.9 can be equivalent to the infinite liquid medium, the director of the equivalent liquid medium is the constant vector, but its direction can be controlled by the applying the bias voltage in some way. The nonlinear refractive index coefficient n2 of the equivalent liquid medium is given by Equation (4.113), the nonlinear response function is the exponential decay function in the (1 + 1) dimensional case (Equation (4.111)) or the zeroth-order modified Bessel function of the second kind in the (1 + 2)-dimensional case (Equation (4.112)), and the evolution of the paraxial optical beams propagating in the region is described by the NNLSE (4.12). The corresponding dimensionless NNLSE is Equation (4.93), and the dimensionless nonlinear response functions R̄ are, respectively, α ̄ R(η) = exp (−α|η x |) (D = 1) , 2

(4.114)

and

α2 ̄ R(η) = (4.115) K0 [α(η 2x + η2y )1/2 ] (D = 2) . 2π The above rigorous theory can be easily understood via its physics behind, which is discussed here in this paragraph. The extraordinary light in the middle region of the NLC-cell samples the refractive index n0 which can be expressed as ̂ ̂ n0 = n(θ)| θ=θ 0 =

󵄨󵄨 󵄨󵄨 op 󵄨󵄨 ≈ (n2⊥ + ϵ a sin2 θ0 )1/2 . 󵄨 2 (n2‖ cos2 θ̂ + n2⊥ sin θ)̂ 1/2 󵄨󵄨󵄨󵄨θ=θ ̂ 0 n⊥ n‖

(4.116)

Suppose that it is strong enough, the extraordinary light will induce the NLC molecular to re-orientate, and the director generates a small change of the angle Δθ. As a result of this reorientation, an incident laser (the extraordinary light) experiences a refractive ̂ ̂ ̂ , which reads ̂ ̂ index change given by Δn = [n(θ̂ + Δθ) − n(θ)]| θ=θ 0 ≈ Δθdn(θ)/d θ| θ=θ 0 op

Δn =

ϵ a sin(2θ0 ) op 2√n2⊥ + ϵ a sin2 θ0

Δθ .

Δn expressed above results from the light-induced angle-change of the NLC molecular, and is the nonlinear refractive index sampled by the extraordinary light. Moreover, only the extraordinary light (with polarization in the direction of the electric field in the main plane, i.e. the plane constructed by the director and the light propagation direction) can “see” the nonlinear refractive index, but the ordinary light cannot (with polarization in the direction of the electric field perpendicular to the main

4 Nonlocal spatial optical solitons

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plane) since its refractive index is nothing to do with the angle of the NLC director. Noting the fact that Δθ ≈ Φ in the middle region of the NLC-cell, one can directly obtain the expression (4.113) of n2 by substitution of Equation (4.110) into the expression of Δn above and by use of the nonlocal relationship between the nonlinear refractive op index Δn and the optical intensity |E|2 (Equation (4.4a)). Because ϵ a > 0 [109] and op sin(2θ0 )/√n2⊥ + ϵ a sin2 θ0 > 0 (0 ≤ θ0 ≤ π/2), we can conclude that Δn > 0 if Δθ > 0, ̂ θ̂ otherwise Δn < 0 when Δθ < 0. We can also observe that the derivative dn(θ)/d ̂ reaches its maximum when θ = π/4, therefore the reason can be understood why the nonlinearity of the NLC gets strongest and the critical power of spatial solitons reaches its minimum at θ̂ = π/4 (see Figure 4.10 (b) and the discussion in Section 4.3.4).

4.3.2 The voltage-controllable nonlinear characteristic length and the nonlinear refractive index coefficient When E rf ≤ E FR (E FR is the Freederichsz threshold electric field)⁵², E rf is a constant, θ̂ ≡ 0; Conversely, E rf is a function of x, the function relationship of θ̂ and E rf (x) is (0) given by the differential equation (4.106). At this moment, θ0 and E rf as a function of the voltage V can, respectively, be expressed as (the detail for the derivation of the function, see Appendix B) √1 + κ a sin θ0 1 √κ a ] [ F [arcsin ( ), ] √1 + κ a + κ a ) 0.32 sin θ0 √1 + κ a sin2 θ0 [ ] (4.117) ϕ rf (0) and E rf = V/Heff , where κ a = ϵ a /ϵ⊥ , F(ϕ e , k e ) = ∫0 e dx/√1 − k 2e sin2 x is the elliptic integrals of the first kind, Heff is the equivalent thickness of the liquid crystal cell, given by Equation (4.153) (see Appendix B). Equation (4.117) gives the approximate analytic implicit function expression of θ0 (V). Therefore, it is easy to find that w m and n2 are only decided by the preset voltage V or the equivalent maximum preset angle θ0 , its function relationship is shown in Figure 4.10. When the preset voltage increases from the threshold voltage, θ0 monotonously increases from 0 to π/2, and w m monotonously drops from the positive infinite to 0, n2 also decreases monotonically. It is thus clear that, for this kind of the liquid crystal material in the liquid crystal cell structure, the characteristic length of the nonlinear response function can change when the bias voltage changes, that is to say, which can be controlled by the bias voltage. So, for a laser beam with the beam width w0 , it is convenient to control the change of the degree of the nonlocality by the preset voltage. The properties of the V =√

K(1 + κ a sin2 θ0 ) rf ε0 ϵ a (1

rf

52 E FR = V FR /H, V FR is the Freederichsz threshold electric voltage, V FR = K 1/2 π/(ε 0 ϵ a )1/2 [118]. When the bias voltage (electric field) is greater than the threshold voltage, the liquid crystal molecules will start to rotate.

284 | Qi Guo, Daquan Lu, Dongmei Deng

Fig. 4.10. (a) The nonlinear characteristic length w m and the nonlinear refractive index coefficient n 2 of the NLC vs the bias voltage V, (b) the nonlinear characteristic length w m and the critical power of a single soliton P nc vs the pretilt angle θ0 . The NLC is TEB30A, whose parameters are n ‖ = 1.692, n ⊥ = 1.522 (λ = 0.589 μm), ϵ ‖ = 14.9, ϵ ⊥ = 5.5, K = 10−11 N, V FR = 1.09 V, and the thickness of the liquid crystal cell is H = 80 μm [112].

characteristic length of the nonlinear response function can be controlled by the applied voltage, which is unique, and the extremely important feature of the NLC [116]. In addition, it can be seen from Figure 4.10 (a) that the value of the nonlinear refractive index coefficient n2 is in the range of 10−5 –10−3 cm2 /W, almost the same order of the magnitude with the value of the nonlinear refractive index coefficient in the NLC given in Ref. [117].⁵³ (0) Substitution of Equation (4.152) in Appendix B for E rf into Equation (4.109), the nonlinear characteristic length w m can be re-expressed as θ

−1

0 1 + κ a sin2 θ̂ H √ θ0 (1 + κ a sin2 θ0 ) ̂ d θ) (∫ √ w m (θ0 ) = √2 sin(2θ0 ) − 2θ0 cos(2θ0 ) sin2 θ0 − sin2 θ̂

.

(4.118)

0

When θ0 = π/4 (the nonlinear refractive index of the soliton is the largest and the critical power is the minimum), w m is about 30 μm.⁵⁴ As long as the beam width of the light is controlled in about 2–4 μm, the propagation condition of the strong nonlocality can be achieved, it is relatively easy to obtain the beam width of this kind experimentally.

53 The magnitude of the nonlinear refractive coefficient of the NLC which is originated from the purely optically induced is 10−4 –10−3 cm2 /W in Table 1 of Ref. [117]. 54 For commercial undoped E7 crystal (The corresponding parameters are [48, 111, 116] ϵ ‖ = 19.6, ϵ ⊥ = 5.1, K = 1.2 × 10−11 N, H = 75 μm), when θ0 = π/4 (The corresponding preset voltage is V = 2.1V), w m ≈ 30.0 μm, but for TEB30A liquid crystal (The corresponding parameters [112] are: ϵ ‖ = 14.9, ϵ ⊥ = 5.5, K = 10−11 N, H = 80 μm), θ0 = π/4, (The corresponding preset voltage is V = 2.0 V), w m ≈ 30.6 μm.

4 Nonlocal spatial optical solitons

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The feature of the nonlinear characteristic length of the liquid crystal can be controlled through the bias voltage of the liquid crystal cell; this has been verified indirectly in the experiments [112] on the interaction of the nonlocal solitons and the dependencies of the degree of the nonlocality.

4.3.3 The propagation model of the liquid crystal in strong nonlocal condition As proved in Section 4.2.2, if the response function R(r) about the origin (r = 0) is symmetric and analytical, the NNLSE for the case of the strong nonlocality can be simplified to the Snyder–Mitchell model. On the other hand, it has been pointed out in the last section (Section 4.3.2) that the propagation of the nematicon is described by the NNLSE (4.12), and the nonlinear characteristic length of the NLC can reach the order of the magnitude of tens of microns. For the laser beam with the beam width of a few microns order of the magnitude, it is easy to realize the conditions of the strong nonlocality. Then, the analytic response function of the phenomenological assumptions (such as the Gaussian function) has the essential difference with the real response function of the NLC: the response function of the NLC in its symmetric origin r = 0 is always singular. In fact, the analytic response function of the phenomenological assumptions (such as the Gaussian function) cannot describe the real physical materials, though many guiding significant results can be obtained by using the analytic response function. From the fact of the nonlinear response function of the NLC (Equations (4.111) and (4.112)) and the nonlinear response function (Section 4.4.1) of the lead glass which will be discussed later, it is easy to find that the response function of the real physical material always seems to have singularity, to say the least, at least now the response function of the real physical material which is not singularity still have not been found. In this section, we will explain (but that is not the proof in the strict mathematical sense), if the nonlinear response function R of the material has a singularity at a symmetric point, then no matter how strong nonlocal degree, the nonlinear refractive index △n given by Equation (4.63) generally cannot be equivalent to the square refractive index, except in special cases; But if R is analytical, △n is available equivalent to the square refractive index under the condition of the strong nonlocality, △n is convergence to the square refractive index when the degree of the nonlocality tends to infinity. In other words, under the condition of R has singularity, the NNLSE (Equation (4.12)) generally cannot be simplified to the Snyder–Mitchell model (Equation (4.70) or (4.69)). In order to be simple but without loss of generality, the (1 + 1) dimensional case will be discussed. It is more complex for the (1 + 2)-dimensional case, but it is easy to extend the (1 + 1) dimensional case to the (1 + 2)-dimensional case. When the response function is symmetry and strong nonlocality, one can expand △n (Equation (4.63)) in

286 | Qi Guo, Daquan Lu, Dongmei Deng the origin x = 0 as a series, and get 1 △n(x, z) 2 = △n0 + △n󸀠0 x + △n󸀠󸀠 0x + ⋅⋅⋅ , n2 2

(4.119)

where △n0 = △n(x, z)/n2 |x=0 = ∫ |Ψ(z, ξ)|2 R(ξ)dξ ≈ R0 P0 , △n󸀠0 = ∂△n(x, z)/∂x/n2 |x=0 = − ∫ |Ψ(z, ξ)|2 R󸀠 (ξ)dξ = 0 [Ψ(z, ξ) is even or odd symmetry about ξ ], and △n󸀠󸀠 0 = ∂2 △n(x, z)/∂x2 /n2 |x=0 = ∫ |Ψ(z, ξ)|2 R󸀠󸀠 (ξ)dξ . If R(x) is analytical, one can get △n󸀠󸀠 0 ≈ R󸀠󸀠 (0)P0 , then △n can be expressed as the square function of x. But if R(x) in x = 0 has singularities, such as the NLC, R󸀠󸀠 (x) satisfies the equation (see Equation (4.142) in Appendix A) R(x) δ(x) R󸀠󸀠 (x) − 2 = − 2 , wm wm then one can obtain △n󸀠󸀠 0 ≈

R0 P0 − |Ψ(z, 0)|2 , w2m

where the second term actually is far greater than the first, because (R0 P0 )/|Ψ(z, 0)|2 ∼ (R0 |Ψ(0, 0)|2 w0 )/|Ψ(z, 0)|2 ∼ w0 /w m ≪ 1 (strong nonlocality),⁵⁵ then Equation (4.119) becomes △n(x, z) 1 ≈ R0 P0 − |Ψ(z, 0)|2 x2 . n2 2w2m

(4.120)

It is thus clear that only even symmetry soliton solution (|Ψ(z, 0)|2 is not equal to zero, and is not as a function of the z coordinate), Equation (4.120) is possible to be square of the function x, otherwise, △n(x, z) will be the function of z (for the breathers solution), or in expansion (4.119), the first order of the nonzero high-order term must be considered (for the singular symmetric soliton solution, |Ψ(z, 0)|2 = 0). In order to better understand this problem, we will research each coefficient of the expansion concretely, so the expansion will be re-expressed as [74, 119–121] △n(r, z) = χ 0 + χ 2 r2 + χ 4 r4 + χ 6 r6 + ⋅ ⋅ ⋅ , n2

(4.121)

where χ n = Q(n) (r)/n!|r=0 , Q(r) = ∫ R(r − r󸀠 )|Ψ(z, r󸀠 )|2 d D r󸀠 . If the response function is the Gaussian analytic function given by Equation (4.87), then the coefficient of the expansion is independent on the cross-sectional dimension, respectively [74, 119]: χ0 = χ4 =

1 + 2w2m /w20 2w20

χ2 = −

,

1 4w60 (1 + 2w2m /w20 )

,

χ6 = −

1 2w40

, 1

12w80 (1 + 2w2m /w20 )2

(4.122) .

55 For the (1 + 1)-dimensional case, P 0 ∼ |Ψ(x, 0)|2 w 0 (Section 4.1.4), at the same time R 0 ∼ 1/w m [Because R(x) needs to satisfy the normalized relation ∫ R(x)dx = 1].

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It is thus clear that χ 4 and χ 6 will tend to zero when w m → ∞ (for a given beam width w0 , the degree of the nonlocality will tend to be infinity), △n will tend to be the square function of r. So at this moment, the NNLSE (4.12) will tend to the Snyder– Mitchell model (4.70) (or (4.69)) when the degree of the nonlocality becomes infinity. But for the response function of the NLC (Equation (4.111) or Equation (4.112)), the coefficient of the expansion is associated with the cross-sectional dimension. In the case of the (1 + 1) dimension, the coefficients are, respectively [74] χ0 =

√πw m 2w30

,

χ2 = −

1 2w40

χ4 =

,

1 12w60

,

χ6 = −

1

.

(4.123)

1 . 36w80

(4.124)

60w80

But in the case of the (1 + 2) dimension, the coefficients are [120] χ0 =

w2 1 Γ ( 02 ) , 2 2w0 4w m

χ2 = −

1 2w40

,

χ4 =

1 8w60

,

χ6 = −

∞

where Γ(x) = ∫0 e−x /xdx is the Γ function. Because the response function has singularity, which makes the coefficients χ4 and χ6 be independent on the nonlinear characteristic length w m . Even if the nonlinear characteristic length w m tends to infinity, χ 4 and χ6 are still limited values, and will not tend to zero like the case of the analytical response function. From Equations (4.123) and (4.124), the ratio of the third and the second in expansion (4.121) can be obtained as, respectively χ 4 r2 r2 = − 2 (D = 1) , χ2 6w0

χ 4 r2 r2 = − 2 (D = 2) . χ2 4w0

This means that in r ≥ w0 region, the third term in the expansion (4.121) will not be ignored. In other words, in the edge of a laser beam area, even if the degree of the nonlocality is infinity, the effects of χ4 will appear, so the wave form of a single nematic will be different with the wave form of a Gaussian function (The more detailed discussion will be given in Section 4.3.4).

4.3.4 The approximate analytical solution of a single nematicon A. Breather solution Professor Assanto and his students technically obtained the approximate analytical breathers solution [47] in coupling equations (4.107) and (4.108) with a profound understanding of the physical background of the light transmission problems in the liquid crystal. Here their solving process from Equation (4.125) to Equation (4.127) will briefly be introduced.⁵⁶

56 Although the corresponding process is inconsistent with Ref. ([47]), but they are precisely equivalent.

288 | Qi Guo, Daquan Lu, Dongmei Deng For the infinite boundary, the perturbation distribution function Φ(r) of the director angle of the liquid crystal molecules induced by a laser is symmetric about the origin. In strongly nonlocal conditions, the spatial scale of Φ(r) should be much larger than the beam width, so we can expand Φ(r) into series, and just take the first two nonzero terms.⁵⁷ 1 Φ(r) ≈ Φ0 + ∇2D Φ0 r2 , (4.125) 4 where the subscript 0 indicates the function Φ and its derivatives in r = 0. Under the condition of the strong nonlocality (w m → ∞), the second term in Equation (4.108) can be neglected, substituting the result to Equation (4.125) and make θ0 = π/4, one can obtain op ε0 ϵ a Φ(r) ≈ Φ0 − (4.126) |Ψ(z, 0)|2 r2 . 16K Substituting Equation (4.126) into Equation (4.107), one can obtain op

2ik

k 2 ε0 (ϵ a )2 ∂Ψ + ∇2D Ψ − 0 |Ψ(z, 0)|2 r2 Ψ = 0 . ∂z 16K

(4.127)

op

Equation (4.127) would also include the linear term k 20 ϵ a Φ0 Ψ about Ψ, but as Section 4.2.2 described, the linear term of Ψ only modulates the phase (corresponding to the large phase shift phenomenon [89] of the strongly nonlocal soliton), does not affect the waveform of the amplitude function. Now that we only solve the waveform of the amplitude, the last term need not write out. In fact, by the transformation of the exponential function similar to Equation (4.67), we can absorb the linear term. Because the function |Ψ(z, 0)|2 is dependent on the z coordinate, the analytical solution actually cannot be obtained in Equation (4.127). But as already discussed in the last section, for a soliton solution, |Ψ(z, 0)|2 ≡ Ψ02 (Ψ0 is the maximum amplitude of the soliton) is independent with the z coordinate. If the input power of a laser beam is in the vicinity of the soliton critical power, then |Ψ(z, 0)|2 will be approximately equal to the constant number Ψ02 . So, when the special condition which is the input power is in the vicinity of the soliton critical power is satisfied, Equation (4.127) can be expressed as op k 2 ε0 (ϵ a )2 Ψ02 2 ∂Ψ 2ik + ∇2D Ψ − 0 r Ψ =0. (4.128) ∂z 16K The above equation is exactly the Snyder–Mitchell model. From the solving process of the Snyder–Mitchell model in Section 4.2.2, the approximate solution of Equation (4.127) [namely the exact solution of Equation (4.128)] is the Gaussian function given by Equation (4.73) (only considering the amplitude without considering the

57 Expand Φ(r) with respect to r about r = 0 to the second order, one can obtain Φ(r) = Φ 0 + r ⋅ ∇D Φ 0 + (1/2)(r ⋅ ∇D )2 Φ 0 + ⋅ ⋅ ⋅ . Since Φ(r) is symmetric with r = 0, one has ∂ x Φ 0 = ∂ y Φ 0 = 0, ∂2xy Φ 0 = 0, and ∂2x Φ 0 = ∂2y Φ 0 , then the final results can be obtained Φ(r) = Φ 0 +(1/2)∂2x Φ 0 r⋅r+⋅ ⋅ ⋅ = Φ 0 + (1/4)∇2D Φ 0 r 2 + ⋅ ⋅ ⋅ .

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phase), the oscillation beam width is given by Equation (4.79), for convenience, the consistent result with Ref. [47] is re-expressed as w2 = w20 [1 + (

P Gnc − 1) sin2 (Ωz)] , Pp

(4.129)

where P p is the power carried by the light field [given by Equation (4.19)], P Gnc is the critical power of the Gaussian soliton in the NLC, the expression of Ω is the same as that in Equation (4.80), but should replace P0 and P c with P p and P Gnc , namely, Ω=√ and P Gnc =

Pp P Gnc

1 , kw20

8πn0 cK 2 2 op 2 k 0 w0 (ϵ a )

(4.130)

.

(4.131)

The oscillation evolution features of the beam width described in Equation (4.129) has already been discussed in detail in Section 4.2.2, the oscillation period Λ is Λ=

2√2(πn0 )3/2 w0 √cK 1 π . = op Ω (ϵ a )2 √P p

(4.132)

In addition, we can get the maximum beam width (P p < P Gnc ) or the minimum value (P p > P Gnc ) W m which meets (the related discussion in Section 4.2.2) 2 = Wm

w20 P Gnc 2cn0 Kλ2 1 = . op Pp π(ϵ a )2 P p

(4.133)

It is thus clear to see from the above two equations that Λ and W m are inversely pro−2 have a linear portional to the square root of the beam power √P p , and Λ−2 and W m relationship with P p . The dependency relationship between the oscillation period Λ of the beam width and the extremum (maxima or minima) W m and the carrying power P p of the beam has been proved in the quantitative experiment (the comparison between the experimental results and the theoretical results [47] is seen in Figure 4.11), which is the important evidence to prove the nematic liquid can become the nonlocal nonlinear media of the strongly nonlocal propagation condition.

B. Soliton solution In the above solving process of the breathers solution, the nonlinear refractive index of the NLC is equivalent with the square refractive index; thus, the amplitude of the waveform is still a Gaussian function. It is pointed out in Section 4.3.3, on the other hand, due to the nonlinear response function of the NLC has a singularity, it will bring error to take the nonlinear refractive index of the NLC to be equivalent with the square

290 | Qi Guo, Daquan Lu, Dongmei Deng

Fig. 4.11. The experimental results of the strongly nonlocal soliton in the NLC. (a) The maximum beam width vs input power, (b) the inverse square of the maximum beam width vs the input power, (c) the period of the beam width vs the input power (error bars are negligibly small), (d) the inverse square of the period of the beam width vs the input power. The solid lines are the best fitting curves of the theoretical results (Equations (4.131) and (4.132)) (after C Conti et al., Physical Review Letters, Volume 92, 113902 (2004)).

refractive index, especially in the two ends of the amplitude waveform. The approximation analytical solutions of the more accurate soliton will be discussed in this section. Equation (4.93) is a differential-integral equation, and it is difficult to find its accurate analytical solution. But in the strong nonlocality condition, the approximate analytical solution can be obtained by using the perturbation method [88] widely used in the study of the quantum theory, and this method has nothing to do with whether the nonlinear response function has a singularity. Because the solving process is complicated [74, 119, 121], here only the final results are given. In the (1 + 2)-dimensional case, the perturbation solution of Equation (4.115) whose response function is given by Equation (4.93) (see Equation (36) in Ref. [121], here only the amplitude of the solution is discussed and its phase is not important) is |u(ξ, η x , η y )| =

r2d r4d r6d r8d r2d σn ) (1 + a + b + c + d ) , exp (− μ2 α 2μ2 μ2 μ4 μ8 μ6

(4.134)

where σ n ≈ 1.44, a ≈ 0.076, b ≈ 0.022, c ≈ 0.00022, d ≈ 0.00037 are the perturbation parameters in the calculation (all are independent of the degree of the nonlocality parameter α), r2d = η2x + η2y , μ is the beam width parameter in the dimensionless system. By using the dimensionless transformation equation (4.30), the single soli-

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Fig. 4.12. Comparison of the intensity profiles of the quasi-Guassian soliton (Equation (4.135)) (solid line) and the Gaussian soliton (Equation (4.86)) (dashed line). The maximum normalized value is 1, and the beam widths are the same.

ton expression of the NLC can be obtained in the laboratory coordinate system [takes μ = 1, σ n ≈ √2 and ignores the last two terms in Equation (4.134)] |Ψ(z, x, y)| =

4n0 K 1/2 1/2 op

kw20 ε0 ϵ a sin(2θ0 )

exp (−

r2 r2 r4 ) (1 + a + b ) . 2w20 w20 w40

(4.135)

The waveform in Equation (4.135) is not a Gaussian function unless a = b = 0, the full width at half maximum (FWHM) of its intensity is wFWHM ≈ 1.841w0 , and the G corresponding beam width of the Gaussian beam is wFWHM = 2(ln 2)1/2 w0 ≈ 1.665w0 . The compared result of the waveform of Equation (4.135) and the Gaussian waveform is shown in Figure 4.12. As shown in the figure, the middle part of two waveforms is consistent, but the edge has some differences. The waveform given by Equation (4.135) is even more accurate than the Gaussian waveform [71]. According to Equation (4.19), the critical power of the soliton in Equation (4.135) can be obtained by Pmin c P nc = , (4.136) sin2 (2θ0 ) where Pmin = (1 + 2a + 4b)P Gnc = 1.24P Gnc , P Gnc is given by the Gaussian waveform corc responding the soliton critical power in Equation (4.131), only the linear terms a and b are kept and the higher order terms are ignored in the expression Pmin c . P nc is the function of the maximal angle θ0 , its function relationship is shown in Figure 4.10 (b), when θ0 = π/4, P nc reaches the minimum value Pmin c . When the beam width is the min G is more presame, the relative error of P c and P nc reaches up to 50%, and Pmin c cise [71]. Although the difference between the waveform of Equation (4.135) and the Gaussian waveform seems to be not quite obvious (Figure 4.12), but the corresponding power error of two waveforms is so big.

4.4 The thermal nonlinear nonlocality When a laser propagates through an optical material, some of the energy which is absorbed by the medium will convert into the heat, and the temperature will rise. Due to the temperature changes which changes the refractive index of the medium,

292 | Qi Guo, Daquan Lu, Dongmei Deng in turn, affects the transmission behavior of the laser. The variation of the temperature is the light intensity |E|2 , rather than a function of the electric field intensity E itself. Therefore, this effect is nonlinear, and this is thermally induced nonlinear optical effects [61].⁵⁸ The nonlocality of the thermal nonlinearity is derived from the processes of the heat transfer: the temperature rises in the region of the laser irradiation due to the absorption of the light energy, and the heat will transfer gradually from the high temperature to the low temperature of the nonirradiated region and makes the temperature rise. The conversion of the temperature in the nonirradiated region will make the refractive index change, thus the nonlocality of a nonlinear refractive index is generated. For the continuous-wave laser beams, the relationships between the amount of the temperature Q change △Q and the light intensity |Ψ|2 , and between the nonlinear refractive index △n and △Q are [61] κ(

∂2 + ∇2D ) △Q = −ρ|Ψ|2 , ∂z2

(4.137)

△n = β Q △Q ,

(4.138)

where κ is the thermal conductivity of the material, ρ is the absorption coefficient (linear) of the material, and β Q = dn/dQ is the temperature coefficient of a refractive index for a given material (the temperature dependence of the refractive index). Equation (4.137) is equivalent to the Poisson’s equation in the electrostatics in the form, the equivalent charge density is ε0 ρ|Ψ|2 /κ, and △Q is equivalent to the scalar electric potential.

4.4.1 Spatial optical solitons in the lead glass The lead glass is the second nonlocal nonlinear media [36] which is proved to achieve the strong nonlocal transmission condition. The nonlinear mechanism of the lead glass is the thermal nonlinearity. Although more than 40 years ago, the stable transmission of the (1 + 2)-dimensional self-trapping beam have been achieved in the lead glass [34] (This was actually the first experiment on the spatial optical soliton transmission.), but they did not understand the physical mechanism of this phenomenon and relate to the spatial optical soliton. When the laser transmits through the lead glass, the light beam is slightly absorbed and becomes into a heat source. The generated heat diffuses to the low temperature of the nonirradiated region, thus a temperature gradient distribution field is formed. If the absorption of the light by the lead glass is weak enough, we can consider that the small changes of the light intensity occur in the transmission process

58 About a thermal-induced nonlinearity, see Section 4.5 of Ref. [61].

4 Nonlocal spatial optical solitons

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293

of the beam. Therefore, the light intensity is not a function of the distance z, and the temperature changes which are determined by the light intensity are not a function of the transmission distance [122]. Thus, the (1 + 2)-dimensional Poisson equation (4.137) can be converted to the (0 + 2)-dimensional Poisson equation [36, 62, 107], and the nonlinear refractive index which satisfies the differential equation can be obtained ∇2D △n = −

ρβ Q 2 |Ψ| . κ

(4.139)

Comparing Equations (4.3) and (4.108), it can be seen that Equation (4.139) is the special case when the nonlinear characteristic length of the material w m tends to infinity. That is to say, in the case of the absorption of the material is small, and the thermal nonlinear has the nonlinear characteristics length which is close to infinity. It is easy to understand the physical image of the result: Formula (4.139) describes the nonlinear refractive index distribution of the steady state (the time goes to infinity). When the time goes to infinity, the stable heat source has transferred the heat to the infinite point of the space. So the nonlinearity of the refractive index may also have changed in the infinite point. For the infinite material, in the case of ignoring the material absorption, the response function of the thermal nonlinear material is the symmetric logarithmic function [123], but the response function will lose its symmetry at a limited space. For the rectangular lead glass material (assuming its edge length in the x-direction is a, the y-direction is b), by using the Green’s function method, the solution of Equation (4.139) can be obtained: a/2

b

−a/2

0

ρβ Q △n(x, y, z) = ∫ dx0 ∫ R(x, y, x0 , y0 ) |Ψ(z, x0 , y0 )|2 dy0 . κ

(4.140)

Then we can use the conformal transformation to obtain the response function R(x, y, x0 , y0 ) of the rectangular lead glass [62] R(x, y, x0 , y0 ) =

1 (g − g0 )2 + (v + v0 )2 , ln 4π (g − g0 )2 + (v − v0 )2

(4.141)

here the parameters g and v are given by g= v=

sn(2Kx/a, k e )dn(2Ky/a, k 󸀠e )

, 1 − dn2 (2Kx/a, k e )sn2 (2Ky/a, k 󸀠e ) cn(2Kx/a, k e )dn(2Kx/a, k e )sn(2Ky/a, k 󸀠e )cn(2Ky/a, k 󸀠e ) 1 − dn2 (2Kx/a, k e )sn2 (2Ky/a, k 󸀠e )

,

where sn(x, k e ), dn(x, k e ), and cn(x, k e ) are the Jacobian elliptic functions. k e and k 󸀠e are the modulus and the complementary modulus of the Jacobi elliptic functions, they satisfy k 2e +k 󸀠2 e = 1, K is the complete elliptic integral of the first kind of the modulus k e , 1 namely K(k e ) = ∫ dx/√1 − x2 √1 − k 2e x2 . And the value of the modulus k e is deter0

mined by the aspect ratio of the rectangular 2K(ke )/K(k󸀠e ) = a/b. Thus, the nonlinear

294 | Qi Guo, Daquan Lu, Dongmei Deng

Fig. 4.13. Experimental results demonstrating the coherent elliptic spatial optical solitons in the lead glass at different propagation distances: 50 mm (a)–(c), 33 mm (d)–(f), and 17 mm (g)–(i), showing the input beam (left column), the diffracted output beam at low (center column), and the output soliton beam at high power (right column) (After C. Rotschild et al., Physical Review Letters, Volume 95, 213904 (2005)).

response function of the limited space is the functions of the source point (x0 , y0 ) and the field point (x, y). It has no translational invariance or the symmetry of the response function in the infinite space. Since the thermal nonlinear of the weak absorption material has a very large nonlinear characteristic length, the nonlinear refractive index generated by formula (4.140) will be influenced greatly by the geometry structure of the materials, and the asymmetric anisotropic geometric structure will produce the anisotropic nonlinear refractive index.When the anisotropic nonlinear effects balance with the isotropic diffraction effects which derived from the isotropic linear refractive index, the noncentro symmetric elliptic spatial optical soliton will be generated. The experimental results of the elliptic spatial optical soliton in the lead glass are showed in Figure 4.13. The figure shows that the beam will be widened due to diffraction effects in the case of the low power. For example, the beamwidth will be widened from 80 to 110 μm after the beam propagates 50 mm (Figure 4.13 (b)); In the high power of about 1 W, the beamwidth will keep a substantially constant 80 μm at different propagation distances, this is a soliton. This is the coherent elliptical spatial optical solitons which are obtained for the first time in experiment [36]. The above experiments show that: if the nonlinear characteristic length of the nonlinear material is very large, the beam localized in a limited area will produce a nonlinear refractive index distribution which is much larger than its occupied scale.

4 Nonlocal spatial optical solitons

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295

So the beam can affect the “very far away” place relative to its occupied scale through the nonlinear refractive index. Conversely, the “very far away” geometric structure and boundary conditions can also make an enormous impact on the transmission behavior of the beam itself by the nonlinear refractive index generated by the beam, or the two “very far away” beams will produce long-range interactions. This means that the nonlinear strongly nonlocality makes it possible to “remote control” the propagation behavior of the beam. A variety of “remote control” mode has been observed experimentally in the lead glass [107], including the coplanar (two-dimensional) remote interaction of the distance-beam width ratio 10 times, the three-dimensional rotation remote interaction of the distance-beam width ratio five times, and the “no optical connection” interaction of the soliton which propagates in the two medium connected by a thin metal sheet.

4.4.2 Other thermal nonlinear materials It can be seen from Equation (4.138) that the thermal nonlinearity can be both the selffocusing nonlinearity and the self-defocusing nonlinearity, which depends on the refractive index temperature coefficient β Q of β Q > 0 or β Q < 0. For the gas, in the constant pressure, the change of the refractive index always decreases with the temperature increasing [61], namely dn/dQ < 0. Therefore, under the constant pressure conditions, the thermal nonlinear of the gas is always self-defocusing. However, liquids and solids are not; their self-focusing or self-defocusing depends on the internal structure of the material [60, 61].⁵⁹ This section will discuss two examples of the selfdefocusing nonlinear nonlocal liquid material impacts on the beam propagation. For the effect of the dark soliton interaction, as pointed out in Section 4.1, there exists the bright spatial soliton in the self-focusing nonlinear medium, while the dark spatial soliton exists in the self-defocusing nonlinear medium. The local bright soliton interaction is closely related to the phase difference between two solitons, in-phase solitons attract, out of phase solitons repel. While the interactions of the local dark solitons always repel, there is no mutually attractive interaction. However, these are the same modes to change the bright soliton-like interaction, the nonlinear nonlocality is also completely changed the dark soliton interaction modes: the dark solitons will attract each other in the nonlocal self-defocusing nonlinear medium. In experiment, it was the first time to observe dark solitons attracting each other in the thermal nonlinear self-defocusing paraffin oil dyed with iodine [124], the experimental results had a good agreement with the numerical calculation. For the influence of the collisionless shock, as has been discussed earlier, the bright spatial soliton (the (1 + 1)-dimensional case) or the self-focusing (the (1 + 2)-dimensional case) is formed

59 See Section 17.6 of Ref. [60] and Section 4.5 of Ref. [61].

296 | Qi Guo, Daquan Lu, Dongmei Deng when the light beam propagates in the local self-focusing medium. However, when the bright light beam propagates in the local self-defocusing medium, the situation is completely different: the diffraction of the central part of the high intensity will be more serious than that of the edge, which makes the beam profile become steeper, eventually it forms the steep oscillation waveform. This is a collisionless shock, [it is also known as a dispersive shock] [9]. The nonlinear nonlocality cannot stop the formation of the shock wave, but it can delay the distance of the shock forms and produce the nonoscillatory motion steep waveform. In the thermal nonlinear self-defocusing rhodamine aqueous solution, the experimental results agree well with the theoretical ones [122]. Since the rhodamine aqueous solution has the strong absorption of the light intensity, the nonlinear refractive index model cannot use the Poisson equation model (Equation (4.139)) whose nonlinear characteristic length is infinite, and must use the model of the limited nonlinear characteristic length (Equation (4.3)) [122]. Acknowledgment: The Chinese version of this chapter is written by Qi Guo. Section 1 of the chapter is translated into English by Daquan Lu, and Sections 2–4 as well as the appendixes are translated by Dongmei Deng. Professor Zhigang Chen (San Francisco State University, USA) has read the manuscript of the Chinese version of Section 1, and put forward some constructive suggestions, to who Qi Guo would like to express his thanks. The authors’ research works were successively supported by the following research Funds, including the National Natural Science Foundation of China (Grant Nos. 10474023, 10674050, 10904041, 11074080, 11174091, 11274125, and 11374108), the Specialized Research Fund for the Doctoral Program of Higher Education (Grant Nos. 20060574006 and 20094407110008), and the Natural Science Foundation of Guangdong Province of China (Grant Nos. 04105804, 05005918, and 10151063101000017).

Appendix A: The derivation of the nonlinear response function in the nematic liquid crystal In order to obtain the solution of Equation (4.108), we can first find the solution of the following equation: − w2m ∇2D R(r, r󸀠 ) + R(r, r󸀠 ) = δ(r − r󸀠 ) .

(4.142)

The function R(r, r󸀠 ) satisfied Equation (4.142) is known as the fundamental solution of Equation (4.108) [125]. We will observed via the following process that the nonlinear response function of the equivalent infinite NLC does be the fundamental solution of Equation (4.108) for the infinite space, that is, the solution of Equation (4.142) for the infinite space. If the solution R(r, r󸀠 ) of Equation (4.142) for the infinite space is obtained, the solution of any function Ψ in Equation (4.108) for the infinite space can be expressed by Equation (4.110).

4 Nonlocal spatial optical solitons

| 297

In the following, we integrate Equation (4.142) by using the Fourier transform method. The Fourier transform of Equation (4.142) yields 󸀠 ̃ r󸀠 ) = exp(ir ⋅ f) , R(f, w2m f 2 + 1

(4.143)

̃ r󸀠 ) is the Fourier transform of R(r, r󸀠 ) about the variables r. The Fourier where R(f, transform and its inverse transform are defined as follows: ∞

̃ r󸀠 ) = ∫ R(r, r󸀠 ) exp(ir ⋅ f)d D r , R(f,

(4.144)

−∞ ∞

R(r, r󸀠 ) =

1 ̃ r󸀠 ) exp(−ir ⋅ f)d D f . ∫ R(f, (2π)D

(4.145)

−∞

Substituting Equation (4.143) into Equation (4.145), we can obtain for the case of D = 2 ∞

R(x, y, x󸀠 , y󸀠 ) =

exp[−i(x − x󸀠 )f x − i(y − y󸀠 )f y )] 1 df x df y ∫ (2π)2 w2m (f x2 + f y2 ) + 1 −∞

=

√(x − x󸀠 )2 + (y − y󸀠 )2 1 K ( ) . 2 0 wm 2πw m

(4.146)

When D = 1, one has ∞

exp[−i(x − x󸀠 )f x ] 1 1 |x − x󸀠 | ∫ = exp (− ) . R(x, x ) = df x 2π 2w m wm w2m f x2 + 1 󸀠

(4.147)

−∞

One can conclude from Equations (4.146) and (4.147) that the nonlinear response function (the solution of Equation 4.142), for the infinite space is of translation invariance, which depends only upon the distance between the source point r󸀠 and the field point r, and can be expressed by the symbol R(r − r󸀠 ), while the nonlinear response function for the finite space is without translation invariance due to boundaries, which depends upon not only the source point r󸀠 but also the field point r, and can only be expressed by the symbol R(r, r󸀠 ). From Equation (4.142), one can also observe that R(r, r󸀠 ) 󳨀󳨀󳨀󳨀󳨀→ δ(r−r󸀠 ). This conclusion can be proved simply by integrating w m →0

Equations (4.146) and (4.147) and by having their limits in w m → 0. When D = 2, for ∞ example, firstly one has ∫−∞ R(r)d D r = 1, then one can also obtain R(r) 󳨀󳨀󳨀󳨀󳨀→ ∞ w m →0

when r = 0 and R(r) = (1/2πw2m )K0 (r/w m ) ∼ exp(−r/w m )/(2w m )3/2 π1/2 󳨀󳨀󳨀󳨀󳨀→ 0 when r ≠ 0.⁶⁰

60 The asymptotic expression for K0 (x) when x → ∞ is K0 (x) ∼ √π/2x exp(−x).

w m →0

298 | Qi Guo, Daquan Lu, Dongmei Deng

Appendix B: The derivation of the function relationship between the maximum pre-bias angle and the bias voltage The literature [116] gives the empirical formula θ0 ≈ (π/2)[1 − (E FR /E rf )3 ] of the functional relationship between the maximum low-frequency electric field E rf and the preformed angle θ0 in a liquid crystal cell. However, the electric field strength E rf within the liquid crystal is a function of the coordinate x, which makes this empirical equation actually no significance. On the other hand, the empirical equation and the actual low-frequency electric field also exist a large deviation, after comparison with the exact integral equation (see Equation (4.152)) of a low-frequency electric field E rf , which shows that the relative error between the exact value of the low-frequency electric field and the approximate value of the empirical equation is over 20% in a large angle range at the center of the liquid crystal cell. The practical significance function should be the function relationship between the maximum pre-bias angle θ0 and the voltage V of the liquid crystal cell. The derivation of the function θ0 (V) will be given in the following. When there is no light field only the low-frequency electric field, the liquid crystal system is described by Equation (4.106), all variables are the only function of the x coordinate. From the relationship equation E = −∇Π(x) on the electric field intensity vector potential E and the scalar equation Π(x), it shows the low-frequency electric field Erf only has the x component, Erf = E rf ex . Further, by applying the relationship rf between the D = ε0 ϵ⊥ E + ε0 ϵ a (n ⋅ E)n of the electric displacement vector D and E in ̂ x +cos θe ̂ z , one can get Drf = D x (x)ex +D z (x)ez , relation to the position vector n = sin θe rf 2 ̂ and D x (x) = ε0 (ϵ⊥ + ϵ a sin θ)E rf . Because ∇ ⋅D = 0, dD x (x)/dx = 0, then get D x = C1 , (C1 is the integration constant), and the low-frequency electric field can be expressed rf ̂ Substituting the expression of E rf into Equation (4.106) as E rf = C1 /ε0 (ϵ⊥ + ϵ a sin2 θ). and integrating, one can obtain the following first-order differential equation: 2

K(

C21 d θ̂ = C2 , ) − rf dx ε0 (ϵ⊥ + ϵ a sin2 θ)̂

(4.148)

̂ where C2 is the integration constant. There is a maximum pre-bias angle θ(x)| x=0 = θ0 ̂ which satisfies d θ(x)/dx|x=0 = 0 in the liquid crystal center (x = 0), thus obtains rf C2 = −C21 /ε0 (ϵ⊥ + ϵ a sin2 θ0 ). Substituting the relationship of E rf and θ̂ into C1 , and considering E rf = −dΠ/dx, Equation (4.148) becomes rf 2 ̂ ̂ ⊥ + ϵ rf d θ̂ dΠ ε0 ϵ a (sin2 θ0 − sin2 θ)(ϵ a sin θ) √ . =± rf dx dx K(ϵ⊥ + ϵ sin2 θ0 )

(4.149)

a

̂ By applying the boundary conditions Π(x)|x=−H/2 = 0, Π(x)|x=H/2 = V and θ(x)| x=±H/2 ̂ = 0, and taking the symmetry of θ(x) into account, after integrating Equation (4.149), one can obtain the function relationship between the bias voltage V and the pre-bias

4 Nonlocal spatial optical solitons

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299

maximum angle θ0 [118]⁶¹ V(θ0 ) = 2√

K(1 + κ a sin2 θ0 ) rf ε0 ϵ a

θ0

∫ 0

d θ̂ ̂ √(sin2 θ0 − sin2 θ)(1 + κ a sin2 θ)̂

,

(4.150)

rf

where κ a = ϵ a /ϵ⊥ . In order to obtain the analytical expression of the integral (4.150), approximating the integral function to 1/sin θ0 √1 + κ a sin2 θ̂ and then dividing the coefficient 0.64, the following approximate empirical formula can be expressed as: V(θ0 ) = 2√

K(1 + κ a sin2 θ0 ) rf ε0 ϵ a (1

+ κa)

√1 + κ a sin θ0 1 √κ a ] [ F [arcsin ( ), ] , 0.64 sin θ0 √ 1 + κa √1 + κ a sin2 θ0 [ ] (4.151)

x where F(x, k e ) = ∫0 dt/√1 − k 2e sin2 t is the elliptic integral of the first kind.⁶² After calculation, in the range of 1 < κ a < 15 and 0 < θ0 < 2π/5, the relative error of the accurate integral equation (4.150) and the empirical formula (4.151) is about ±10%. Therefore, Equation (4.150) and the empirical formula (4.151) give the exact implicit analytical function expressions of the maximum pre-bias voltage as the bias voltage function θ0 (V) and the approximate implicit analytical function expressions, respectively. On the other hand, the exact integral expression of the low-frequency electric field at the center of the liquid crystal cell is [118]

(0) E rf

=

2 H √1 + κ a sin θ0 2

√

θ0

K rf

ϵ0 ϵ a

∫√ 0

1 + κ a sin2 θ̂ sin2 θ0 − sin2 θ̂

d θ̂ .

(4.152)

Then one can get the relationship between the pre-bias voltage of the liquid crystal cell and the low-frequency electric fields E rf (0) = V/Heff at the center of the liquid crystal cell, where Heff is the equivalent thickness of the liquid crystal cell θ0

(1 + κ a sin2 θ0 ) ∫0 Heff = H

θ ∫0 0

√

d θ̂ 2 ̂ ̂ √(sin2 θ 0 −sin2 θ)(1+κ a sin θ)

1+κ a sin2 θ̂ sin2 θ 0 −sin2 θ̂

d θ̂

.

(4.153)

61 The corresponding formula in the literature [118] is set the saddle-splay elastic constants k11 and flexural elastic constants k33 to be equal, namely, k11 = k33 = K, then, one can obtain the equivalent Equations (4.150) and (4.152). √κ a dθ 1 62 From the indefinite integral formula ∫ = √1+κ F[arcsin ( √1+κa sin θ ), √1+κ ] (See √1+κ a sin2 θ

a

√1+κ a sin2 θ

a

Gradshteyn I S and Ryzhik I M, Table of Integrals, Series, and Products [M], Sixth Edition p. 199) and the properties of the elliptic integral F(0, k e ) = 0, and from the approximate expression of the Equation (4.150), one can obtain (4.151).

300 | Qi Guo, Daquan Lu, Dongmei Deng

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Weilong She, Guoliang Zheng

5 Wave coupling theory and its applications of linear electro-optic (EO) effect

5.1 Introduction There are two kinds of electro-optic (EO) effects in nonlinear optics: the Pockels effect and the Kerr effect, which are usually taken as the phenomena having a perturbation of refractive index in a medium induced by an applied electric field. In other words, the Pockels effect (or linear EO effect) is considered to be that having a perturbation of refractive index linearly proportional to the applied electric field, and the perturbation of refractive index with Kerr effect (or quadratic EO effect) is proportional to the square of the electric field. However, this opinion would sometimes give one such impression that EO effect is a response of first-order susceptibility to an applied electric field. From the viewpoint of nonlinear optics, the linear EO effect is really a secondorder nonlinear effect, which is a mutual action of light wave and external electric field, and can be depicted by the second-order polarization [1] (2)

P i (ω) = 2ε0 ∑ χ ijk (−ω, ω, 0)E j (ω)E k (0) ,

(5.1)

jk (2)

where χ ijk (−ω, ω, 0)[i, j, k = x, y, z] is the second-order susceptibility tensor of the linear EO effect and ε0 is the permittivity of vacuum. E(ω) = (E x (ω), E y (ω), E z (ω)) and E(0) = (E x (0), E y (0), E z (0)) are the light field and applied dc electric field, respectively. In contrast, the Kerr effect is the third-order nonlinear effect, for which the related polarization is (3)

P i (ω) = 3ε0 ∑ χ ijkl (−ω, ω, 0, 0)E j (ω)E k (0)E l (0) ,

(5.2)

jkl (3)

where 𝛸 ijkl (−ω, ω, 0, 0) is the third-order susceptibility tensor of the Kerr effect. The EO effect discussed in this chapter is linear EO effect unless it is especially stated. The linear EO effect only occurs in noncentrosymmetric materials, and it is the base of many optical devices, such as electro-optical modulator [5], optical filter [6, 7], polarization controller of light wave [8, 9], tunable optical spanner [10], and so on. The linear EO effect was first studied in 1893, but there is no a satisfied theory of it for a long time. The traditional theory for describing the effect is the index elWeilong She: State Key Laboratory of Optoelectronic Materials and Technologies, Sun Yat-sen University, Guangzhou 510275, China Guoliang Zheng: College of Electronic Science and Technology, Shenzhen University, Shenzhen 518060, China

308 | Weilong She, Guoliang Zheng lipsoid theory. The refractive index ellipsoid theory has been extensively utilized in many applications of linear EO effect. However, the standardization of refractive index ellipsoid equation, which is the key step of the theory is in general not an easy task. Furthermore, even if the equation is standardized, it is not so easy to calculate the phase of polarized light propagating along an arbitrary direction when an external electric field is applied along an arbitrary direction, especially in biaxial crystals. In 1973, Yariv developed a coupled-mode theory, which considers the linear EO effect as an energy coupling between TE mode and TM mode [11]. This theory is widely used in solving the problem about energy exchange between different modes of electromagnetic wave caused by EO effect. However, the coupled-mode theory only focuses on the energy coupling, while the phase changes of independent components of light field caused by EO effect are not taken into account. In addition, the theory is useless in some EO modulation like phase modulation and frequency modulation. In 1975, Nelson presented a perturbation theory for the plane-wave eigenvector equation of crystal optics [12]. He gave the general expressions for the refractive-index changes that result from a homogeneous dielectric perturbation for isotropic solids, uniaxial crystals, and biaxial crystals. The expressions can be, in principle, applied to the linear EO effect for any direction of light propagation, for either state of light polarization, and for any direction of electric field application. Therefore, this theory was called “general solution theory.” However, this theory is difficult to use. For example, the expressions for the refractive-index changes in biaxial crystals are very complex and hard to be simplified. Also the expressions for the refractive-index changes in uniaxial crystals are not accurate, and it will bring serious error in some cases (for example, when the wave vector of light is close to the optical axis). For isotropic solids, it is necessary to solve a set of quadratic equations with three unknown in order to obtain the expressions for the refractive-index changes. In 1998, M. J. Gunning and R. E. Raab presented an algebra theory, which is basically the same as the refractive index theory in nature [13]. In the theory, a systematic algebraic approach was presented for deriving some expressions for the principal refractive indices and dielectric axes of a nonmagnetic crystal in a uniform electric field. This approach is applicable for an arbitrary field and for any symmetry point group. But the calculation is rather complicated except for the situation that the applied electric field is along a special direction. As early as 1962, Armstrong and Bloembergen presented a wave coupling theory for nonlinear optical processes including linear EO effect [1]. Like the coupled mode theory by Yariv, the phase changes of independent components of light field caused by EO effect are not taken into account in this theory. Till 2001, She and Lee presented a wave coupling theory for linear EO effect in bulk crystals [14]. Starting from Maxwell’s equations, considering the second-order nonlinearity as a perturbation, and thinking over the relation between light waves involved and applied electric field, they derived the wave-coupling equations for linear EO effect, and gave the general solution for the resultant equations. The theory is applicable for any direction of light propagation

5 Wave coupling theory and its applications of linear electro-optic (EO) effect |

309

with any polarization state, for an arbitrary applied electric field and for any symmetry point group of crystals. This approach is more accurate in explaining physics picture than traditional theories, and the complex mathematical manipulation is not needed. The theory has its obvious advantage in optimal design and thermal stability study of EO modulator, and shows broad application future. So far, the wave coupling theory of linear EO effect has been generalized to focused Gaussian beam [15], ultrashort laser pulses [16], optically active crystals/magneto-optic crystals [17], and absorbent crystals [18]. In 2006, She and Zheng developed a wave coupling theory for quasiphase-matched (QPM) linear EO effect [19], and studied the noncollinear quasi-phasematched linear EO effect and its application in periodically poled LiNbO3 [20, 21]. In 2009, She and Huang presented a united wave coupling theory for parametric down conversion (PDC), EO effect, and second-harmonic generation, and presented a possible high-flux photon-pair source constructed by single lithium niobate optical superlattice with a combined quasi-periodically and periodically poled structure [22]. In 2012, She and Li presented an electrically control different frequency and received high-efficient conversion utilized cascade effect [23]. In this chapter, we will systematically introduce the wave coupling theory of linear EO effect and its generalization. First, we will introduce the wave coupling theory of linear EO effect in bulk crystal under plane wave approximation, including EO effect in transparent medium and absorbent medium, mutual action of EO effect, optically active effect and magneto-optic effect. Second, we will introduce the linear EO effect and mutual action of EO effect, second-harmonic generation and sum/different frequency effect in periodically poled/linearly chirp poled optical superlattice. Finally, we will introduce the wave coupling theory and its applications of linear EO effect for focused Gaussian beam in optical superlattice.

5.2 Wave coupling theory of linear EO effect in transparent bulk crystal In 2001, She and Lee presented a wave coupling theory of linear EO effect [14], which provided an efficient method for analyzing this nonlinear optical phenomenon. Importantly, the theory seems more coordinated with the theories for other nonlinear optical effects. Besides, the theory is the fundamental of other wave coupling theories related to EO effect. For example, the wave coupling theories of EO effect in absorbent crystal [18] and QPM materials [19–21], the wave coupling theory of mutual action of EO effect, optically active effect and magneto-optic effect, and the wave coupling theory of mutual action of EO effect, second-harmonic generation, and sum/different frequency effect [17, 22, 23].

310 | Weilong She, Guoliang Zheng In a lossless nonmagnetic medium without net free charge, we can derive from Maxwll’s equations that ↔

∇ [∇ ⋅ E(t)] − ∇2 E(t) +

∂2 PNLS (t) 1 ∂2 [ ε ⋅ E(t)] = −μ0 , 2 2 c ∂t ∂t2

(5.3)

↔

where ε is the relative dielectric tensor, μ 0 and c are the magnetic susceptibility and the speed of light in vacuum, respectively, E is the electric field and PNLS the nonlinear polarization. Suppose all other second-order nonlinearity effects are so weak because of phase mismatch that only linear EO effect has to be considered and the electromagnetic wave considered is a plane one propagating along the r direction. The total electric field participating in EO effect can be expressed as E(t) = E(0) + [ 12 E(ω) exp (−iωt) + c.c.] ,

(5.4)

where [ 12 E(ω) exp (−iωt) + c.c.] is the monochromatic light wave of frequency ω, c.c. denotes the complex conjugate, E(0) the static electric field or slow varying electric field of frequency ≪ ω. In general, for a monochromatic light wave of frequency ω, there exist two independent plane electromagnetic wave components in a birefringent crystal. So we have E(ω) = E1 (ω) + E2 (ω) = E1 (r) exp (ik1 ⋅ r) + E2 (r) exp (ik2 ⋅ r) ,

(5.5)

where k1 and k2 are the wave vectors of E1 (ω) and E2 (ω), respectively. For k1 = k2 , E1 (ω) and E2 (ω) denote two perpendicular components of the light field. And for k1 ≠ k2 , E1 (ω) and E2 (ω) denote two independent light field components travelling through the crystal with different refractive indices. For example, E1 (ω) and E2 (ω) are electric fields of o- and e-rays in a uniaxial crystal, respectively. Since the light waves are propagating along the r direction, Equation (5.5) can be simplified to E(ω) = E1 (ω) + E2 (ω) = E1 (r) exp (ik 1 r) + E2 (r) exp (ik 2 r) .

(5.6)

The second-order polarization is then P(2) (t) = 12 P(2) (ω)e−iωt + c.c. = ε0 χ (2) (ω, 0) : E(ω)E(0)e−iωt + c.c. . = ε0 χ (2) (ω, 0) : E1 (r)E(0)e−ik1 r e−iωt + ε0 χ (2) (ω, 0) : E2 (r)E(0)e−ik2 r e−iωt + c.c. ,

(5.7)

where χ(2) (ω, 0) is the second-order susceptibility tensor of the linear EO effect and ε0 is the permittivity of free space. We denote P(2) (ω) = 2ε0 χ (2) (ω, 0) : E1 (r)E(0)e−ik1 r + 2ε0 χ (2) (ω, 0) : E2 (r)E(0)e−ik2 r .

(5.8)

We are interested in the perpendicular component of E(t) [denoted by E⊥ (t), which is the main part of E(t)]. From Equations (5.3), we obtain ∇2 E⊥ (ω) +

ω2 ↔ (2) [ ε ⋅ E(ω)]⊥ = −μ0 ω2 P⊥ (ω) . c2

(5.9)

5 Wave coupling theory and its applications of linear electro-optic (EO) effect |

311

Fig. 5.1. The relationship of vectors considered.

Putting Eqs. (5.6) and (5.8) into (5.9), treating the second-order nonlinearity as a perturbation, under slow varying amplitude approximation, we get ∂E1⊥ (r) ∂E2⊥ (r) + ik 2 e ik2 r ∂r ∂r ω2 ω2 (2) = − 2 [χ (ω, 0) : E1 (r)E(0)]⊥ e ik1 r − 2 [χ (2) (ω, 0) : E2 (r)E(0)]⊥ e ik2 r . c c

ik 1 e ik1 r

(5.10)

Let E1⊥ (r) = E1⊥ (r)a ,

E2⊥ (r) = E2⊥ (r)b ,

E(0) = E0 c ,

(5.11)

where a, b, and c are the unit vectors and a ⋅ b = 0. Taking inner product on both sides of Equation (5.10) with a or b, and rearranging, we get ∂E1⊥ (r) ω2 = − 2 a ⋅ [χ (2) (ω, 0) : E1 (r)E(0)]⊥ e ik1 r ∂r c ω2 − 2 a ⋅ [χ (2) (ω, 0) : E2 (r)E(0)]⊥ e ik2 r , c 2 ∂E (r) ω 2⊥ ik 2 e ik2 r = − 2 b ⋅ [χ (2) (ω, 0) : E1 (r)E(0)]⊥ e ik1 r ∂r c ω2 − 2 b ⋅ [χ (2) (ω, 0) : E2 (r)E(0)]⊥ e ik2 r . c ik1 e ik1 r

(5.12a)

(5.12b)

It can be proved that the subscript “⊥” in Equations (5.12a) and (5.12b) can be dropped. For example, a ⋅ [χ(2) (ω, 0) : E1 (r)E(0)]⊥ = a ⋅ [χ(2) (ω, 0) : E1 (r)E(0)], and so on. The simple course of this proof is as follows: let P⊥ to be the perpendicular component of P which is perpendicular to wave vector k; and P⊥ , a and b are evidently in the same plane. Suppose that P⊥ and P make an angle δ, P⊥ and a make an angle ξ , P and a make an angle ψ, the detailed relationship as shown in Figure 5.1. It is easy to prove that P⊥ ⋅ a = |P⊥ ||a| cos ξ = |P| cos δ|a| cos ξ = |P||a| cos ξ cos δ. Since cos ψ = cos δ ⋅ cos ξ , we have P⊥ ⋅ a = |P||a| cos ξ cos δ = |P| ⋅ |a| cos ψ = P ⋅ a, and then P⊥ ⋅ a = P ⋅ a. In the same way, one can get P⊥ ⋅ b = P ⋅ b. Besides, the angle between E1⊥ and E1 (E2⊥ and E2 ) is very small, so that we can ignore the walk-off effect and substitute E1⊥ (E2⊥ ) with E1 (E2 ) in Equations (5.12a) and (5.12b) without causing

312 | Weilong She, Guoliang Zheng obvious error. Therefore, (5.12a) and (5.12b) can be simplified to ∂E1 (r) ω2 a ⋅ χ(2) (ω, 0) : bcE2 (r)E0 e iΔkr =i ∂r k1 c2 ω2 +i a ⋅ χ(2) (ω, 0) : acE1 (r)E0 k1 c2 ∂E2 (r) ω2 b ⋅ χ (2) (ω, 0) : acE2 (r)E0 e−iΔkr =i ∂r k2 c2 ω2 +i b ⋅ χ (2) (ω, 0) : bcE1 (r)E0 , k2 c2

(5.13a)

(5.13b)

where Δk = k 2 − k 1 . Since the medium is lossless, χ(2) (ω, 0) is real and satisfies the full permutation symmetry [3], so that a ⋅ χ(2) (ω, 0) : bc = b ⋅ χ(2) (ω, 0) : ac .

(5.14)

Let n1 and n2 be the refractive indices corresponding to two independent wave components, respectively, then k i = n i k0 = n i

ω , c

i = 1, 2 .

(5.15)

The well-known relationship between the EO tensor elements r jkl and the secondorder nonlinear susceptibilities are 1 (2) χ jkl (ω, 0) = − (ε jj ε kk )r jkl , 2

ε jj = n2jj ,

ε kk = n2kk ,

j, k, l = 1, 2, 3 ,

(5.16)

where ε jj and ε kk are the diagonalized electric permittivity tensor elements, n jj and n kk are the principal refractive indices. Note that for transparent medium, due to the full permutation symmetry, the EO tensor elements r jkl can be written as r xxl = r1l , r yyl = r2l , r zzl = r3l , r zyl = r yzl = r4l , r zxl = r xzl = r5l and r xyl = r yxl = r6l , respectively. For simplifying Equations (5.13), here we define three effective EO coefficients reff1 = ∑ (ε jj ε kk )(a j r jkl b k c l ) j,k,l

reff2 = ∑ (ε jj ε kk )(a j r jkl a k c l ) j,k,l

(5.17)

reff3 = ∑ (ε jj ε kk )(b j r jkl b k c l ) . j,k,l

And we obtain the resultant equations as follows: dE1 (r) = −id1 E2 (r)e iΔkr − id2 E1 (r) dr dE2 (r) = −id3 E1 (r)e−iΔkr − id4 E2 (r) , dr

(5.18)

5 Wave coupling theory and its applications of linear electro-optic (EO) effect |

where

k0 reff1 E0 2n1 k0 d3 = reff1 E0 2n2 d1 =

k0 reff2 E0 , 2n1 k0 d4 = reff3 E0 . 2n2

313

d2 =

(5.19)

Note that these wave-coupling equations are different from those developed by Armstrong et al. [11]. There is only one term on the right-hand side of either of the equations in Ref. [11], while there are two terms in each of our equations. Let E1 (0) and E2 (0) be the incident fields corresponding to two independent polarizations, we can derive the analytic solutions of Equation (5.18). The solutions can be distinguished, according to the value of reff1 , into two (and only two) different cases: Case 1: reff1 = 0, so that d1 = d3 = 0, the two complex amplitudes of the light field can be easily obtained from Equation (5.18): E1 (ω) = E1 (r)e ik1 r = E1 (0)e i(k1 −d2 )r E2 (ω) = E2 (r)e ik2 r = E2 (0)e i(k2 −d4 )r .

(5.20)

Case 2: reff1 ≠ 0, the solution of Equation (5.18) is then E1 (ω) = E1 (r)e ik1 r = ρ 1 (r)e i(k1 +β)r e iϕ1(r) E2 (ω) = E2 (r)e ik2 r = ρ 2 (r)e i(k1 +β)r e iϕ2(r) ,

(5.21)

where γE1 (0) − d1 E2 (0) 2 ] sin2 (μr) μ γE1 (0) − d1 E2 (0) sin(μr)] ϕ1 (r) = arg [E1 (0) cos(μr) + i μ

(5.22)

γE2 (0) + d3 E1 (0) 2 ] sin2 (μr) μ −γE2 (0) − d3 E1 (0) sin(μr)] ϕ2 (r) = arg [E2 (0) sin(μr) + i μ

(5.23)

d4 − d2 − Δk 2 Δk − d2 − d4 β= 2 √(Δk + d2 − d4 )2 + 4d1 d3 μ= . 2

(5.24)

ρ 1 (r) = √E21 (0) cos2 (μr) + [

ρ 2 (r) = √E22 (0) cos2 (μr) + [

and

γ=

Equations (5.21)–(5.24) are the general solution we are looking for. They can be used to describe linear EO effect of a light wave propagating along an arbitrary direction with an applied dc electric field along any direction in any crystal. In many (and the most important) cases, Equations (5.21)–(5.24) can further be simplified as follows.

314 | Weilong She, Guoliang Zheng ̄ (A) Δk = 0, corresponding to linear EO effect in crystals of 43m and 23 symmetry, or a light wave propagating along the optic axis of a crystal. Without loss of generality, we set E1 (0) = 0. Then d4 − d2 γ= 2 d2 + d4 β=− (5.25) 2 √(d2 − d4 )2 + 4d1 d3 , μ= 2 and Equations (5.22) and (5.23) become ρ 1 (r) = |E2 (0)| √ φ1 (r) = ± 2π

4d21 4d21 + (d2 − d4 )2

sin2 (μr)

,

(5.26)

(sign determined by − d1 )

(d4 − d2 )2 sin2 (μr) (d4 − d2 )2 + 4d21 −γE2 (0) ϕ2 (r) = arg [E2 (0) sin(μr) + i sin(μr)] . μ ρ 2 (r) = |E2 (0)| √cos2 (μr) +

(5.27)

Fig. 5.2. Amplitude modulation using orthogonal polarizers system: output light intensity output I out /I 2 (0) vs (d 2 − d 4 )/d 1 for Δk = 0.

5 Wave coupling theory and its applications of linear electro-optic (EO) effect |

315

Applying this result to amplitude modulation with two orthogonal polarizers, the output light intensity can be easily obtained by using Equations (5.21) and (5.27): Iout =

√(d2 − d4 )2 + 4d21 nε0 c I2 (0) 2 sin ( r) , |E1 (r)|2 = −d 4 )2 2 2 1 + (d24d 2

(5.28)

1

nε 0 c 2 2 |E 2 (0)| .

We see, in general, that the maximum of I1out does not where I2 (0) = equal I2 (0) unless d2 − d4 = 0. Figure 5.2 shows the variation of output light intensity Iout /I2 (0) as a function of (d2 − d4 )/d1 for Δk = 0. (B) Δk ≠ 0, which is the case when two independent components of E(ω) experience different refractive indices. Since the change of the refractive index, Δn, is about 10−5 , and for most birefringent crystals |n o − n e | is 10−3 to 10−1 , we have Δk ≫ d i (i = 1, 2, 3, 4) in most cases except the case where k1 and k2 are very close to the optic axis. For Δk ≫ d i , it is easy to obtain, from Equations (5.21) to (5.24), that Δk − d2 − d4 2 |Δk + d2 − d4 | , μ≈ 2

(5.29)

ΔkE1 (0) + d1 E2 (0) 2 ] sin2 (μr) Δk −ΔkE1 (0) − d1 E2 (0) ϕ1 (r) = arg [E1 (0) cos(μr) + i sin(μr)] |Δk|

(5.30)

β=

ρ 1 (r) = √E21 (0) cos2 (μr) + [

ΔkE2 (0) − d3 E1 (0) 2 ] sin2 (μr) Δk (5.31) ΔkE2 (0) − d3 E1 (0) ϕ2 (r) = arg [E2 (0) cos(μr) + i sin(μr)] . |Δk| For amplitude modulation, one can choose E1 (0) = E2 (0), which leads to (d1 /Δk)E1,2 (0) ≪ E1,2 (0), and thus ρ 2 (r) = √E22 (0) cos2 (μr) + [

E1 (ω) ≈ E1 (0)e i(k1 r−d2 )r , E2 (ω) ≈ E2 (0)e

i(k2 r−d 4 )r

.

(5.32a) (5.32b)

These results are the same as those for reff1 = 0, meaning that, in most cases, we can omit terms containing Δk in Equations (5.18) for Δk ≠ 0. Let L be the effective path length of the light ray that has experienced the EO effect. From Equations (5.32), the modulated output intensity is Iout = I0 sin2 (Γ/2) ,

(5.33)

where Γ = [k 0 (n1 − n2 ) − (d2 − d4 )]L. If the condition Δk ≫ d i does not hold, we must use the exact expression ρ 21 (r) + ρ 22 (r) − 2ρ 1 (r)ρ 2 (r) cos[ϕ1 (r) − ϕ2 (r)] , 2 where ρ 1 (r), ρ 2 (r), ϕ1 (r), and ϕ2 (r) are given by Equations (5.22) and (5.23). Iout =

(5.34)

316 | Weilong She, Guoliang Zheng For phase modulation, one usually set up such a system so that E1 (0) = 0 or E2 (0) = 0. In this case only one equation of Equations (5.18) should be used. For example, for E2 (0) = 0, one should use the second equation of Equations (5.18). Here an athermal design for the KTP EO modulator is presented to illustrate the application of wave coupling theory. As is well known, one of primary limitations in the EO application is zero-field leakage resulted from natural birefringence. Although the natural birefringence can be compensated by tuning temperature or using a common Soleil–Babinet compensator, a constant temperature system is indispensable because of its sensitive temperature dependence. Therefore, it is necessary and meaningful to make the KTP EO devices insensitive to temperature. In fact, Ebbers has verified the existence of athermal static phase retardation (ASPR) directions in KTP theoretically and experimentally and discussed the EO application of the ASPR orientation [24]. Taking account of thermal expansion, the more accurate ASPR directions in KTP are found to be at 32.5° from the z-axis in the x–z plane for a 1064 nm light wave [24, 25]. In our numerical calculations, some relevant parameters are required. The three principal refractive indices of KTP are, respectively, n x = 1.7377, n y = 1.7453, and n z = 1.8297 for 1064 nm at room temperature [24]; the refractive index temperature derivatives of KTP (°C−1 ) are given by Δn x = 6.1 × 10−6 , Δn y = 8.3×10−6 and Δn z = 14.5×10−6 ; the temperature expansion coefficients (°C−1 ) from −10 to 100 °C, along the (x, y, z) crystallographic direction are a1 = 7.81 × 10−6 , a2 = 9.80 × 10−6 and a3 = −0.65 × 10−6 , respectively [25]; and the nonvanishing EO coefficients of KTP (in pm/V) are r13 = 9.5, r23 = 15.7, r33 = 36.3, r42 = 9.3, and r51 = 7.3 [26]. Figure 5.3 (a) shows the schematic of the athermal KTP amplitude modulator, in which a working crystal KTP and a Soleil–Babinet compensator also made of KTP are sandwiched between two cross polarizers. The 1064 nm light wave propagates along the ASPR direction (θ = 32.5°, φ = 0) of both working crystal and compensator, whose 󸀠 effective lengths are l and l󸀠 (l0 and l0 at room temperature), respectively. For convenience, here we define a laboratory coordinate system (X, Y, Z) with Z-axis parallel to the wave vector k1 (or k2 ), X and Y-axes perpendicular and parallel, respectively, to the optical table. The relation between the laboratory coordinate system and crystal coordinate system (x, y, z) is shown in Figure 5.3 (b), where the Y-axis is parallel to the y-axis and the Z-axis makes an angle of θ with respect to the z-axis. Also the transverse electric field applying on the KTP lies in the X–Y plane and makes an angle of ξ with the X-axis. For the case that the wave vector of light is directed along the “athermal natural birefringence direction,” the relation, (i =1,2,3,4), is always satisfied. And the wavecoupling equations become dE 1 (r) ≈ −id2 E1 (r) , dr dE2 (r) ≈ −id4 E2 (r) , dr

(5.35) (5.36)

5 Wave coupling theory and its applications of linear electro-optic (EO) effect |

317

Fig. 5.3. (a) Schematic of transverse amplitude modulator, and (b) relation between the laboratory coordinate system (X, Y, Z) and crystal coordinate system (x, y, z).

where d2 and d4 are found to be k 0 sin θ cos2 θ cos ξ E0 (n4x r13 + 2n2x n2z r51 + n4x r33 tan2 θ) , 2n1 k 0 n4y r23 d4 = − E0 sin θ cos ξ . 2n2

d2 = −

(5.37) (5.38)

For amplitude modulation, one can choose E1 (0) = E2 (0). And the solutions of Equations (5.35)–(5.36) can be easily obtained as follows: E1 (ω) ≈ E1 (0)e i(k1 −d2 )r E2 (ω) ≈ E2 (0)e

i(k2 −d 4 )r

(5.39) .

(5.40)

Then we can get the output intensity from the modulator as follows: Iout = I in sin2 ( 2Γ ) ,

(5.41)

where I in is the intensity of input light, and Γ = (d2 − d4 )l + Γ0 , Γ0 = k 0 (n1 − n2 )(l + l󸀠 ) .

(5.42) (5.43)

To lower the external applied voltage, we should make |d2 − d4 | as large as possible. From Equations (5.37)–(5.38), we find that |d2 − d4 | is maximal when δ = mπ (m = 0, ±1, ±2, ±3. . .), which indicates that the optimized direction of the external electric field should be along or opposite to the X-axis. From Equations (5.41)–(5.42), one can see that the natural birefringence disappears when Γ0 = 2mπ (m = 0, ±1, ±2, ±3. . .).

318 | Weilong She, Guoliang Zheng

Fig. 5.4. Temperature dependence of zero-field leakage.

Assuming the length l0 of KTP crystal is 2.5 cm, we can achieve very excellent contrast ratio (∼ 105 ) by tuning the total length (l0 + l󸀠0 ) to 2.7219 cm in room temperature. For (l0 +l󸀠0 ) = 2.7219 cm, we consider the temperature dependence of zero-field leakage of the modulator. The numerical result is shown in Figure 5.4. From Figure 5.4, one can see that the zero-field leakage is almost zero (about 10−4 ) when temperature varies from 25 °C to 75 °C. We further study the thermal stability of the modulator. Within a large range of field strength, we always have Γ0 ≫ (d2 − d4 )l, that is to say, the thermal stability mainly depends on the static phase retardation, Γ0 . So we can predict that the modulator can work in a very high thermal stability if the light propagates along the ASPR direction. Figure 5.5 (a) shows the dependence of output intensity on the external electric field when the temperature is at 25 °C and 75 °C, respectively. It clearly shows that the two curves have little difference and are overlapped to each other, namely, the thermal stability of the modulator is very excellent. To show the difference between the two curves clearly, we let ΔI = Iout (25 °C) − Iout (75 °C), and its dependence on the external field is shown in Figure 5.5 (b). One can also see that the required electric field corresponding to the half-wave voltage (2 kV/mm) is much higher than that in the commonly used configuration, because the ASPR orientation nearly coincides with the “zero effective EO coefficient” orientation. Besides of this application, the wave coupling theory can have many other applications, such as the analysis of THz EO sampling [27, 28], design of polarization inde-

5 Wave coupling theory and its applications of linear electro-optic (EO) effect |

319

Fig. 5.5. (a) Output intensity at 25 °C (solid curve) and 75 °C (dashed curve) and (b) difference between them (ΔI) as a function of external electric field when the light propagates along one of the ASPR directions (θ = 32.5°, φ = 0).

pendent EO modulator [29, 30], Mini-rotating angle measurement based on the linear EO effect [31], and so on.

5.3 Wave coupling theory of linear EO effect in absorbent medium As is well known, all the previous theories of EO effect are built on the postulate of no absorption with the media [2, 12–14]. While in many EO materials like polymers, the influence of absorption could not be neglected since they are more or less absorbent [32–35]. So it is very meaningful to generalize the above theory to the case of absorbent medium. She and Zheng found that there exist four absorption coefficients, which affect the EO effect in low symmetry media [36]. In the following, we will develop a wave coupling theory of linear EO effect in absorbent medium. Considering the contributions of electric conductivity and nonlinear polarization, we can derive from Maxwell’s equations that ↔

∇[∇ ⋅ E] − ∇2 E + μ0 σ ⋅ ↔

↔

∂2 PNLS ∂E 1 ∂2 ( ε ⋅ E) = −μ , + 2 0 ∂t c ∂t2 ∂t2

(5.44)

where ε and ↔ [σ] are the relative permittivity tensor and electric conductivity tensor of the medium, c and μ 0 are the speed of light and magnetic susceptibility in vacuum, respectively, E is the total electric field participating in the process of linear EO effect

320 | Weilong She, Guoliang Zheng and PNLS is the nonlinear polarization. Suppose all other high-order nonlinearities are so weak because of phase mismatch that only linear EO effect has to be considered and the electromagnetic waves considered are plane waves propagating along the r direction. The total electric field E can be expressed as E(t) = E(0) + [ 12 E(ω) exp (−iωt) + c.c.] ,

(5.45)

where [ 12 E(ω) exp (−iωt) + c.c.] is the light field with frequency ω, c.c. denotes the complex conjugate, and E(0) is the static electric filed or slow varying electric field of frequency ≪ ω. And the second-order polarization relative to linear EO effect can be expressed as P(2) (t) =

1 (2) P (ω)e−iωt + c.c. = ε0 χ (2) (ω, 0) : E(ω)E(0)e−iωt + c.c. , 2

(5.46)

where χ (2) (ω, 0) is the second-order susceptibility tensor of the linear EO effect and ε0 is the permittivity of vacuum. As discussed in Section 5.2, we now consider the component of E(ω) perpendicular to the propagation direction of light and denote it as E⊥ (ω). From Equation (5.44), we get ↔

↔

− ∇2 E⊥ (ω) + μ 0 [ σ ⋅

∂2 P⊥NLS (ω) ∂E(ω) 1 ∂2 [ ε ⋅ E(ω)]⊥ = −μ . ] + 2 0 ∂t c ∂t2 ∂t2 ⊥

(5.47)

Putting Equations (5.45) and (5.46) into Equation (5.47), we have ↔

∇2 E⊥ (ω) + iωμ0 [ σ ⋅ E(ω)]⊥ +

ω2 ↔ (2) [ ε ⋅ E(ω)]⊥ = −μ0 ω2 P⊥ (ω) . c2

(5.48)

Supposing that the condition ↔

ωμ0 [ σ ⋅ E(ω)]⊥ ≪

ω2 ↔ [ ε ⋅ E(ω)]⊥ c2

(5.49)

is satisfied, one can treat the item iωμ0 [↔ [σ] ⋅ E(ω)]⊥ as a perturbation. According to Fresnel’s equation, there always exist two independent plane wave components corresponding to a monochromatic light wave with frequency ω propagating in a birefringent crystal. So, in general, the light wave in crystal can be expressed as E(ω) = E1 (ω) + E2 (ω) = E1 (r) exp (ik 1 r) + E2 (r) exp (ik 2 r) ,

(5.50)

where k 1 and k 2 are all real here, and the imaginary parts are included in E1 (r) and E2 (r). Accordingly, P(2) (ω) can be written as P(2) (ω) = 2ε0 χ (2) (ω, 0) : E1 (r)E(0)e ik1 r + 2ε0 χ (2) (ω, 0) : E2 (r)E(0)e ik2 r .

(5.51)

Putting Equations (5.50) and (5.51) into Equation (5.48), and treating the second-order nonlinearity and absorption as perturbations as done in Ref. [12], under slow varying

5 Wave coupling theory and its applications of linear electro-optic (EO) effect |

321

amplitude approximation, we obtain ik 1 e ik1 r

∂E1⊥ (r) ∂E2⊥ (r) + ik 2 e ik2 r ∂r ∂r

ω2 ω2 (2) [χ (ω, 0) : E1 (r)E(0)]⊥ e ik1 r − 2 [χ (2) (ω, 0) : E2 (r)E(0)]⊥ e ik2 r 2 c c ωμ 0 ↔ ωμ0 ↔ ik1 r −i −i [ σ ⋅ E1 (r)]⊥ e [ σ ⋅ E2 (r)]⊥ e ik2 r . 2 2

=−

(5.52)

Let E1⊥ (r) = E1⊥ (r)a,

E2⊥ (r) = E2⊥ (r)b,

E(0) = E0 c ,

(5.53)

where a, b, and c are three unit vectors and a ⋅ b = 0. Taking inner product on both sides of Equation (5.52) with a or b, we get ∂E1⊥ (r) ω2 = − 2 a ⋅ [χ (2) (ω, 0) : E1 (r)E(0)]⊥ e ik1 r ∂r c ω2 ωμ0 ↔ − 2 a ⋅ [χ(2) (ω, 0) : E2 (r)E(0)]⊥ e ik2 r − i a ⋅ [ σ ⋅ E1 (r)]⊥ e ik1 r 2 c ωμ0 ↔ −i (5.54a) a ⋅ [ σ ⋅ E2 (r)]⊥ e ik2 r 2 ∂E2⊥ (r) ω2 ik 2 e ik2 r = − 2 b ⋅ [χ (2) (ω, 0) : E1 (r)E(0)]⊥ e ik1 r ∂r c ω2 ωμ 0 ↔ − 2 b ⋅ [χ (2) (ω, 0) : E2 (r)E(0)]⊥ e ik2 r − i b ⋅ [ σ ⋅ E1 (r)]⊥ e ik1 r 2 c ωμ0 ↔ b ⋅ [ σ ⋅ E2 (r)]⊥ e ik2 r . −i (5.54b) 2 ik1 e ik1 r

As done in Section 5.2, it is easy to prove that the subscript “⊥” can be dropped on the right-hand sides of Equations (5.54a) and (5.54b). Since the angle between E1⊥ and E1 (E2⊥ and E2 ) is very small, so that we can ignore the walk-off effect and substitute E1⊥ (E2⊥ ) with E1 (E2 ) in Equations (5.12a) and (5.12b) without causing obvious error. Let Δk = k 2 − k 1 and ωμ 0 ↔ ( σ ⋅ a) ⋅ a , k1 ωμ 0 ↔ = ( σ ⋅ a) ⋅ b , k2

ωμ 0 ↔ ( σ ⋅ b) ⋅ a , k1 ωμ 0 ↔ = ( σ ⋅ b) ⋅ b , k2

α 11 =

α 21 =

α 12

α 22

(5.55)

and then Equations (5.54) can be simplified as dE1 (r) = − (id1 + dr dE2 (r) = − (id3 + dr

α 21 α 11 ) E2 (r)e iΔkr − (id2 + ) E1 (r) , 2 2 α 12 α 22 ) E1 (r)e−iΔkr − (id4 + ) E2 (r) , 2 2

(5.56a) (5.56b)

where d i is listed in Equation (5.19). Note that the coupling equation (5.56) includes four absorption coefficients α ij (i, j = 1, 2) listed in Equation (5.55). In the case of low symmetric crystal such as biaxial crystal, one can also prove that the cross-terms (α21

322 | Weilong She, Guoliang Zheng and α 12 ) do not vanish. So there exist four coefficients α 11 , α 22 , α 21 , and α 12 in the resultant equations in general. With the initial values of E1 (r) and E2 (r), i.e. E1 (0) and E2 (0), the solutions of the Equation (5.56) can be obtained as follows: E1 (ω) = E1 (r) exp (ik 1 r) = ρ 1 (r) exp [i(k 1 + β)r + iϕ1 (r) − (

α 11 + α 22 ) r] , 4

(5.57)

E2 (ω) = E2 (r) exp (ik 2 r) = ρ 2 (r) exp [i(k 1 + β)r + iϕ2 (r)r − (

α 11 + α 22 ) r] , 4

(5.58)

where ρ 1 (r) = √l21 + m21

(5.59)

ϕ1 (r) = arg(l1 + im1 )

(5.60)

ρ 2 (r) = √l22 + m22

(5.61)

ϕ2 (r) = arg(l2 + im2 )

(5.62)

l1 =

gE1 (0) cos(pr) s[uE1 (0) − (d1 p + 12 α 21 q)E2 (0)] cos(pr) + 2 2(p2 + q2 ) −

m1 =

s[vE1 (0) + (d1 q + 12 α 21 p)E2 (0)] cos(pr) 2(p2 + q2 )

(5.64)

gE2 (0) cos(pr) s[−uE2 (0) − (d3 p + 12 α 12 q)E1 (0)] cos(pr) + 2 2(p2 + q2 ) +

m2 =

(5.63)

sE1 (0) sin(pr) g[uE1 (0) − (d1 p + 12 α 21 q)E2 (0)] sin(pr) + 2 2(p2 + q2 ) +

l2 =

g[vE1 (0) + (d1 q + 12 α 21 p)E2 (0)] sin(pr) 2(p2 + q2 )

g[vE2 (0) − (d3 q + 12 α 12 p)E1 (0)] sin(pr) 2(p2 + q2 )

(5.65)

sE2 (0) sin(pr) g[−uE2 (0) − (d3 p + 12 α 12 q)E1 (0)] sin(pr) + 2 2(p2 + q2 ) +

s[−vE2 (0) + (d3 q + 12 α 12 p)E1 (0)] cos(pr) , 2(p2 + q2 )

(5.66)

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323

and u = pγ + qα, p=

v = pα − qγ,

s = exp (−qr) − exp (qr), 2γα +

1 2 d 1 α 12

+

g = exp (−qr) + exp (qr) 1 2 d 3 α 21

√−2(γ 2 − α 2 + d 1 d 3 − 14 α 12 α 21 ) + 2√(γ 2 − α 2 + d 1 d 3 − 14 α 12 α 21 )2 + (2γα + 12 d 1 α 12 + 12 d 3 α 21 )2 √2 √ −(γ 2 − α 2 + d 1 d 3 − 1 α 12 α 21 ) + √(γ 2 − α 2 + d 1 d 3 − 1 α 12 α 21 )2 + (2γα + 1 d 1 α 12 + 1 d 3 α 21 )2 4 4 2 2 2 α 11 − α 22 d 4 − d 2 − Δk Δk − d 2 − d 4 α= , γ= , β= . (5.67) 4 2 2 q=

From Equation (5.56), one can see that the cross-terms α ij ≠ 0 (i ≠ j) do not affect the linear EO effect obviously when Δk ≫ α ij (i ≠ j). And when the light propagates along or close along the optical axis, α ij ≠ 0 (i ≠ j), which will play their roles evidently. Here we take the doped KTP crystal as an example to illustrate the influence of absorption on EO modulation. The nonvanishing EO coefficients of KTP (in 10−12 m/V) are r13 = 9.5, r23 = 15.7, r33 = 36.3, r42 = 9.3, and r51 = 7.3 [26], respectively. The three principle refractive indices are: n x = 1.7416, n y = 1.7496, and n z = 1.8323 at 1 μm [36]. Suppose the wave propagates along such a direction that θ = Ω + 0.00012π and φ = 0 (close to the optical axis), and the external electric field is along the y-axis, i.e. c = (0, 1, 0). Fix the effective path length of the EO crystal L = 2.5 cm, and λ0 = 1 μm. For convenience, we let E1 (0) = E2 (0) = 1 V/m. Fix E0 = 2500 V/cm and α 11 = 25, α 22 = 20 (in m−1 ). Then we can get the |E1,2 (L)| as the functions of α 21 and α 12 as shown in Figures 5.6 and 5.7. One can see that α 21 and α 12 influences both |E1 (L)| and |E2 (L)| evidently, and they do not weaken the intensity of light as do α ii (i = 1, 2) [35]. When α21 (α 12 ) becomes large, |E1 (L)| increases, while |E2 (L)| decreases. There is energy exchanged between the two independent components when α ij ≠ 0(i ≠ j).

Fig. 5.6. |E 1 (L)| as the function of α 21 and α 12 .

324 | Weilong She, Guoliang Zheng

Fig. 5.7. |E 2 (L)| as the function of α 21 and α 12 .

Fig. 5.8. ϕ 1 (L) as the function of α 21 and α 12 .

Similarly, we get ϕ1,2 (L) as the functions of α 21 and α 12 , shown in Figures 5.8 and 5.9. They show that ϕ1 (L) depends obviously on α 21 but slightly on α 12 , while ϕ2 (L) depends obviously on α12 but slightly on α 21 . It is easy to prove that α21 = α 12 = 0 in high symmetric crystal like isotropic and pure uniaxial crystals. The coefficients α 11 and α 22 weaken the intensity of light as do traditional absorption coefficients [18].

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Fig. 5.9. ϕ 2 (L) as the function of α 21 and α 12 .

5.4 Wave coupling theory of mutual action of linear EO, OA, and Faraday effects The authors of “Handbook of Optics” stated [37], “crystals of quartz are optically active (OA), and this complicates use of the material as an EO modulator.” For the case of an optically active medium with applied electric field and magnetic field simultaneously, the theory of the interaction between light and medium will become more complicated affirmatively. She and Chen adopted the wave coupling idea involved in She and Lee’s theory [14] of EO effect, which avoids the difficulty from traditional method, and solved this problem successfully [10]. In general, a monochromatic plane light wave with frequency ω in an optically active medium possessing birefringence can be decomposed into two independent polarized components, i.e. E(ω) = E1 (ω) + E2 (ω) = E1 (r) exp (ik 1 r) + E2 (r) exp (ik 2 r) ,

(5.68)

where E1 (ω) and E2 (ω) denote two cross components of the light field when k 1 = k 2 , while they denote two independent components experiencing different refractive indices when k 1 ≠ k 2 . The polarization relative to spatial dispersion, responsible for the optical activity, can be written as [38] ω ω Pω μ = ε0 χ μαβ ∇ β E α = iε0 χ μαβ E α k β .

(5.69)

Putting Equation (5.68) into (5.69), we can easily get (2)

(2)

(2)

P1 (ω) = i2ε0 κ 1 : E1 (r)k1 exp (ik 1 r) + i2ε0 κ 2 : E2 (r)k2 exp (ik 2 r) ,

(5.70)

326 | Weilong She, Guoliang Zheng (2)

(2)

where κ 1 and κ 2 denote the second-order susceptibility tensor of natural optical (2) (2) activity and generally κ 1 ≠ κ 2 . Besides, there exist another two second-order polarizations responsible for the Pockels effect and Faraday effect, when the medium is simultaneously subject to slow varying electric field E(0) and magnetic field B(0). They are, respectively, (2)

P2 (ω) = 2ε0 χ (2) (ω, 0) : E1 (r)E(0) exp (ik 1 r) + 2ε0 χ (2) (ω, 0) : E2 (r)E(0) exp (ik 2 r) (5.71) (2)

P3 (ω) = i2ε0 η(2) : E1 (r)B(0) exp (ik 1 r) + i2ε0 η(2) : E2 (r)B(0) exp (ik 2 r) ,

(5.72)

where χ(2) and η(2) denote the second-order susceptibility tensors of Pockels effect and Faraday effect, respectively. Then, the total second-order polarization can be written as (2)

(2)

(2)

P(2) (ω) = P1 (ω) + P2 (ω) + P3 (ω) (2)

(2)

= i2ε0 κ 1 : E1 (r)k1 exp (ik 1 r) + i2ε0 κ 2 : E2 (r)k2 exp (ik 2 r) + 2ε0 χ (2) (ω, 0) : E1 (r)E(0) exp (ik 1 r) + 2ε0 χ (2) (ω, 0) : E2 (r)E(0) exp (ik 2 r) + i2ε0 η(2) : E1 (r)B(0) exp (ik 1 r) + i2ε0 η(2) : E2 (r)B(0) exp (ik 2 r) . (5.73) Starting from Maxwell’s equations, only considering the second-order nonlinearity and neglecting the high-order nonlinearity as well as the linear absorption, we can derive, under slow varying amplitude approximation and no walk-off approximation, the wave-coupling equations of the mutual action of EO effect and Faraday effect in an optically active medium possessing birefringence as follows: dE1 (r) f0N f0B =( + − id1 ) E2 (r) exp (iΔkr) − id2 E1 (r) dr n1 n1 f0N f0B dE2 (r) − − id3 ) E1 (r) exp (−iΔkr) − id4 E2 (r) , = (− dr n2 n2

(5.74) (5.75)

where Δk = k 2 − k 1 and d i (i = 1, 2, 3, 4) is listed in Equation (5.19); and f0N = (2) (2) − ∑jkl (k 20 ) ⋅ (a j n2 κ 2jkl b k k̂ l ) = ∑jkl (k 20 ) ⋅ (b j n1 κ 1jkl a k k̂ l ) (the second equation is the re(2)

quirement of the law of conservation of energy), f0B = − ∑jkl (k 0 B0 ) ⋅ (a j η jkl b k m l ), ˆ paralwhere a j , b k , m l , and k̂ l are the components of four unit vectors a, b, m, and k lel to E1 , E2 , B(0) and k, respectively. It should be noticed that in derivation of Equa(2) (2) (2) (2) tions (3a) and (3b), the relation κ jkl = −κ kjl and η jkl = −η kjl [24] has been used, (2) (2) (2) which leads to ∑ a j κ a k k̂ l = 0, ∑ b j κ b k k̂ l = 0, ∑ a j η a k m l = 0, and jkl

(2)

1jkl

jkl

2jkl

jkl

jkl

∑jkl b j η jkl b k m l = 0. Equations (5.74) and (5.75) can describe the propagation of a light with an arbitrary polarization state traveling along an arbitrary direction in any birefringent optically actively media with slow varying electric field and magnetic field applied along arbitrary directions, respectively.

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As an example, an electrically and magnetically controlled optical spanner is presented here to illustrate the application of this theory. Optical spanner is a light beam that can exert a torque on an object. We now use Equations (5.74) and (5.75) to discuss the manipulation of the transfer of spin angular momentum of light in an optically active crystal Ce:BTO (Ce:Bi12 TiO20 ) and the electrically and magnetically controlled optical spanner on this crystal. Ce:BTO crystal exhibits not only optical activity, but also Pockels effect and Faraday effect as crystalline quartz, besides, its EO coefficients and refractive index are larger than those of crystalline quartz. In the following, an Ce:BTO whisker crystals with 5 μm radius and 100 μm (or 200 μm) length is taken considered. For simplicity, suppose that the propagation of light and the magnetic ̄ field are all along the [110] direction of the whisker crystal, while the electric field along [1 1 0] direction, i.e. a = (0, 0, 1), b = (1/√2, 1/√2, 0), c = (1/√2, 1/√2, 0), m = (−1/√2, 1/√2, 0) and Δk = 0. According to the nonvanishing elements and symmetry of the second-order tensor of Ce:BTO crystal (belonged to 23 class) [40],we (2) get d1 = d3 = k 0 n30 E0 γ63 /2 = d, d2 = d4 = 0 and fN = f0N /n0 = −k 20 κ xyz , fB = (2) −k 0 η xyz B0 /n0 . Subsequently, Equations (5.74) and (5.75) read dE1 (r) = (fN + fB − id)E2 (r) , dr dE2 (r) = (−fN − fB − id)E1 (r) . dr

(5.76) (5.77)

Assuming fN > 0 and the incident beam is linearly polarized with initial values E1 (0) = E in and E2 (0) = 0, then we have the solutions of (5.76) and (5.77): E1 (r) = E in cos(√(fN + fB )2 + d2 r) ,

E2 (r) = E in sin(√(fN + fB )2 + d2 r) exp [i(θ + π)] , (5.78) where θ = arg(fN + fB + id). Generally, this light field can also be expressed as the superposition of left- and right-handed circularly polarized lights: T

T

E(r) = [E1 (r), E2 (r)]T = E in (α L [1/√2, −i/√2] + α R [1/√2, i/√2] ) ,

(5.79)

where α L = [cos(√(fN + fB )2 + d2 r) − i sin(√(fN + fB )2 + d2 r) exp (iθ)]/√2 ,

(5.80)

α R = [cos(√(fN + fB )2 + d2 r) + i sin(√(fN + fB )2 + d2 r) exp (iθ)]/√2 .

(5.81)

Let e−1 = [1/√2, −i/√2]T , e+1 = [1/√2, i/√2]T be the eigenstates of left- and right-handed circularly polarized lights, one can get e+1 ⋅ e∗−1 = 0 and eμ × e∗μ󸀠 = ˆ μμ󸀠 (μ, μ 󸀠 = ±1). According to Ref. [42], in the basis of circular polarization the −iμ kδ ˆ n̂ k,+1 − n̂ k,−1 ), where n̂ k,+1 and spin angular momentum operator is [40]: ˆJs = ∑k ℏk( n̂ k,−1 are the right- and left-handed circularly polarized photon number operators, respectively. On the other hand, it is known that the ratio of spin angular momentum flux to the energy flux of light is σℏ/ℏω[42], where σ = ±1 for right- and left-handed

328 | Weilong She, Guoliang Zheng circularly polarized light, respectively. Suppose the input and output surfaces of the Ce:BTO whisker crystal have perfect antireflection coatings; then at the output surface we have cε0 |α L E in |2 cε0 |α R E in |2 , NR = , (5.82) NL = 2ℏω 2ℏω where N L and N R are the numbers of left- and right-handed circularly polarized photons transmitted per unit area per second. Therefore, the total spin angular momentum of light transmitted per unit area per second is MS = ℏ(NR − NL ) = =−

cε0 E2in (|α R |2 − |α L |2 ) 2ω

cε0 E2in sin θ sin(2√(fN + fB )2 + d2 r) . 2ω

(5.83)

Assume an He–Ne laser beam at 632.8 nm is used, then we know the corresponding refractive index n0 = 2.54, optical rotatory power fN = 0.1103 rad/mm and Verder’s constant V = 0.0611 rad/(T mm); the nonvanishing EO coefficients γ41 = γ52 = γ63 = 7.37 (in 10–12 m/V) [43]. From these parameters, we can get fB = 0.0611B0 (in mm−1 )

Fig. 5.10. Normalized photon numbers of left- or right-handed circularly polarized light controlled by applied electric and magnetic fields: (a) and (b) for whiskers with lengths of 100 μm; (c) and (d) for whiskers with lengths of 200 μm.

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Fig. 5.11. Total spin angular momentum normalized of light controlled by applied electric and magnetic fields: (a) for the whisker with length of 100 μm; (b) 200 μm.

and d = 5.9 × 10−2 E0 (in mm−1 ), where E0 and B0 are with the units of kV/cm and T, respectively. The numerical results are showed in Figure 5.10 and Figure 5.11, respectively. From Figures 5.10 and 5.11, we find that both the left- and right-handed circularly polarized photon numbers and the total spin angular momentum are functions of the applied electric and magnetic fields, and they vary with the applied electric field in a sinusoidal way if the magnetic field is fixed at −1.8 T. For a 200 μm whisker crystal (Figure 5.10 (c), (d), and Figure 5.11 (b)), when the electric field changes from −66 to 66 kV/cm, the transfer of spin angular momentum per photon can change from −ℏ to ℏ. However, the field may damage the crystalline structure of the medium when E0 > 50 kV/cm. For a 100-μm whisker crystal, a 40 kV/cm (− 40 kV/cm) electric field will make 46% right-handed (left-handed) circularly polarized photons translate to left-handed (right-handed) ones, which corresponds to transfer of 0.46ℏ angular momentum “per photon.” While without the aid of the electric field, the magnetic field cannot change the polarization state of the light any more. Obviously, the present way has much superiority over that only using an electric field to manipulate the transfer of spin angular momentum of light. Figure 5.10 clearly demonstrates that the total photon number is conservative although the left- or right-handed circularly polarized one varies with the applied electric field and magnetic field. Different from total photon number, the total spin angular momentum of light is not a constant and varies with the applied electric field and magnetic field (see Figure 5.11). This indicates that

330 | Weilong She, Guoliang Zheng there should be a transfer of angular momentum from medium to light induced by the applied fields since the total spin angular momentum of the incident light is zero. According to the law of conservation of angular momentum, we know that light should transfer an angular momentum to the medium in return and exert a torque on it when light passes through the whisker crystal. According to Equation (5.83), a larger torque can be achieved by using a high intensity light beam since the torque is proportional to intensity. According to the Einstein box treatment [45], the rotation velocity of the microdisk is Ω=

sε0 E2in 1 ) sin θ sin [2√(fN + fB )2 + d2 s] . (n0 − 2Iω n0

(5.84)

Equation (5.84) indicates that the rotation of the disc, resulted from the torque exerted by light, can be driven and manipulated by tuning the electric field and magnetic field, which is similar to the manipulation of the transfer of spin angular momentum. The magnetic field considered here is easy to obtain from an Nd–Fe–B permanent magnet, whose magnetic field can be over 2 T [46]. On the other hand, since the electric field is applied transversely, we can use a pair of electrodes well polished with a separation of 50 μm and then put the whisker crystal in it. The maximal voltage needed is less than 250 V. The further calculation shows that if a crystal with much larger EO coefficients is used, the length of the sample can be shortened remarkably and the maximal electric field needed can be further lowered. As a special case, for example, if a SBN crystal (whose γ33 is 1340 pm/v) is used, we only need a 24 μm long crystal and 8.4 kV/cm electric field to result in the change of 2ℏ per photon when B = 0 and, for a pair of electrodes with 50 μm separation, the voltage needed is only 42 V. In addition, the wave coupling theory can also be applied to the optimal design for EO modulator and polarization controller made of optically active crystals [44].

5.5 Wave coupling theory of QPM linear EO effect Before introducing the “quasi-phase matching (QPM)” linear EO effect, let us review the wave coupling theory of EO effect proposed by She and Lee (see Section 5.2), dE1 (r) = −id1 E2 (r)e iΔkr − id2 E1 (r) , dr (5.85) dE2 (r) = −id3 E1 (r)e−iΔkr − id4 E2 (r) . dr When conditions Δk ≠ 0 (for the case that the two independent components of light field have experienced different refractive indices) and Δk ≫ d i (i = 1, 2, 3, 4) are satisfied, Equation (5.85) can be simplified as dE1 (r) ≈ −id2 E1 (r) , dr dE2 (r) ≈ −id4 E2 (r) . dr

(5.86)

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Equation (5.86) clearly indicates that E1 (r) and E2 (r) do not influence each other. That is to say, when Δk ≠ 0 and Δk ≫ d i there is no energy exchange between the two independent components of light field. This case often occurs in birefringent crystal. For example, when a light beam does not propagate along the optical axis of uniaxial crystal, the EO effect will not result in the energy exchange between o-ray and e-ray due to phase mismatch. Therefore, for the case of Δk ≫ d i , one should compensate the mismatch Δk in order to achieve the effective energy exchange between the two independent components of light field. And it can be approached by the QPM technique proposed by Bloembergen et al. in 1962 [1]. The QPM technique provides an efficient method to make compensation to the phase mismatch of a nonlinear process. In QPM materials (optical superlattices), the sign of the nonlinear coefficient is modulated periodically or aperiodically along the propagation path of the light beam. So the QPM materials can offer proper reciprocal vector for compensating the phase mismatch of a nonlinear optical process (including linear EO effect), and then a high conversion efficiency can be achieved. Considering a nonlinear optical process with mismatch Δk and an optical superlattice with structure function f(r) = ∑+∞ −∞ F m exp (iGm r) (where Gm is the mth-harmonic grating or mth-order reciprocal vector, and Fm is the corresponding Fourier coefficient), the “quasi-phase-matching” is achieved when Δk + Gm = 0. Figure 5.12 shows the diagrammatic sketch for collinear and noncollinear QPM sum-frequency generation and linear EO effect, and some common poling structures and applications of 1D and 2D optical superlattices are listed in Table 5.1. In fact, the concept and technique of QPM has been widely used for the linear EO effect in the materials whose EO coefficients are periodically modulated, and some interesting phenomena have been observed [7, 8, 19, 21, 48–51]. Lu et al. used the

Fig. 5.12. The diagrammatic sketch for collinear and noncollinear QPM sum-frequency generation and linear EO effect.

332 | Weilong She, Guoliang Zheng Table 5.1. Common poling structures and applications of 1D and 2D optical superlattices. Poling structure and pattern

Independent reciprocal vectors

Application

Periodic

Discrete:1

Single QPM (collinear)

Quasi-periodic

Discrete:2

Multiple QPM (collinear)

Chirp-periodic

Continuous: multiple

Broadband QPM (collinear)

Aperiodic

Discrete: ≥ 2

Multiple QPM (collinear)

Complex structure with different poling units and lattices

Discrete or Continuous: ≥ 2

Noncollinear multiple QPM

1D

2D

coupled-mode theory developed by Yariv [11] to investigate the EO coupling in a periodically poled lithium niobate (PPLN) crystal, and predicted that PPLN may be used as a precise spectral filter or an EO switch [7]. After that, Chen et al. demonstrated an EO Šolc-type wavelength filter in PPLN, experimentally [8]. They also found that the central wavelength of the filter made of PPLN can be tuned by temperature, electric field, and illumination [8, 49–51]. Although the coupled-mode theory is widely used in QPM linear EO effect, it is not valid in some cases, since it does not consider the influence of EO effect on the phases of the independent wave components as mentioned in the introduction. In 2006, She and Zheng derived a more general wave coupling theory of QPM linear EO effect [19]. In 2010, She and Zeng utilized the wave coupling theory to study the linear EO effect in linear chirped-periodically poled crystal, and achieved a broadband optical filter [52]. In addition, QPM materials can provide multiple reciprocal vectors to match multiple nonlinear optical processes in phase, which make the united QPM nonlinear optical processes possible. In 2009, She and Huang used the QPM concept and presented a possible high-flux photon-pair source constructed by single lithium niobate optical superlattice (OSL) with a combined quasi-periodically and periodically poled structure, in which an electrically induced PDC takes place after second-harmonic gener-

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ation (SHG). The principle of such a photon-pair source is from a united theory, in which SHG, PDC, and EO (EO) effect are treated as the coequal two-order nonlinear effects [22].

5.5.1 Wave coupling theory of Linear EO effect in periodically poled crystals For the QPM materials, the total electric field E participating in the process of linear EO effect can be expressed as 1 E(t) = E(0) + [ E(ω) exp (−iωt) + c.c.] , 2

(5.87)

where [ 12 E(ω) exp (−iωt) + c.c.] is the light field with frequency ω, E(0) is the static electric filed or slow varying electric field. In general, for a monochromatic light wave of frequency ω propagating in a birefringent crystal, there are two independent plane wave components, i.e. E(ω) = E1 (ω) + E2 (ω) = E1 (r) exp (ik 1 r) + E2 (r) exp (ik 2 r) ,

(5.88)

where E1 (ω) and E2 (ω) denote two perpendicular components when k 1 = k 2 , or two independent components experiencing different refractive indices when k 1 ≠ k 2 . Let E1 (r) = √ω/n1 A1 (r)a,

E2 (r) = √ω/n2 A2 (r)b,

E(0) = E0 c ,

(5.89)

where a, b, and c are three unit vectors and a⋅b = 0, A1 (r) and A2 (r) are the normalized amplitudes of the two independent wave components, n1 and n2 are the unperturbed refractive indices of the two wave components. Similarly to the derivation in Section 5.2, starting from Maxwell’s equations and taking the linear EO effect as a perturbation, we can obtain the following wave-coupling equations under the slow varying amplitude approximation: dA1 (r) = −iκ ⋅ f(r)A2 (r) exp (iΔk 󸀠 r) − iv1 ⋅ f(r)A1 (r) dr dA2 (r) = −iκ ⋅ f(r)A1 (r) exp (−iΔk 󸀠 r) − iv2 ⋅ f(r)A2 (r) , dr

(5.90a) (5.90b)

where f(r) is the structure function of the materials, Δk 󸀠 = k 2 − k 1 , and κ=

k0 reff1 E0 , 2√n1 n2

v1 =

k0 reff2 E0 , 2n1

v2 =

k0 reff3 E0 , 2n2

where reffi (i = 1, 2, 3) is the effective EO coefficient [14] (see also Section 5.2).

(5.91)

334 | Weilong She, Guoliang Zheng If f(r) is a periodic function of r with period Λ due to periodical modulation of EO coefficient, it can be written as a Fourier series +∞

f(r) = ∑ F m exp (iG m r) ,

(5.92)

m=−∞

where G m = 2πm/Λ is the mth-harmonic grating wave vector. Assume that the optical superlattice is so designed that there is a G m󸀠 (still denoted by G m ) very close to −Δk 󸀠 , and Fm is the corresponding Fourier coefficient. Substituting Equation (5.92) into Equation (5.90), and neglecting those components which make little contributions to EO effect because of phase mismatch, we have dA1 (r) ≈ −iκ q A2 (r) exp (iΔkr) − iv1q A1 (r) , dr dA2 (r) ≈ −iκ ∗q A1 (r) exp (−iΔkr) − iv2q A2 (r) , dr

(5.93a) (5.93b)

where Δk = Δk 󸀠 + G m , and κ q = κF m ,

κ ∗q = κF−m ,

v1q = v1 F0 ,

v2q = v2 F0 .

(5.94)

Equations (5.93) are the resultant coupling equations of QPM linear EO effect. If the initial normalized amplitudes of the two waves are A1 (0) and A2 (0), respectively, the solutions of Equations (5.93) can be obtained as follows: A1 (r) = ρ 1 (r) exp [iβr + iϕ1 (r)]

(5.95a)

A2 (r) = ρ 2 (r) exp [i(β − Δk)r + iϕ2 (r)] ,

(5.95b)

where γA1 (0) − κ q A2 (0) 2 +[ ρ 1 (r) = ] sin2 (μr)} μ γA1 (0) − κ q A2 (0) ϕ1 (r) = arg [A1 (0) cos(μr) + i sin(μr)] μ

1/2

{A21 (0) cos2 (μr)

ρ 2 (r) =

{A22 (0) cos2 (μr)

+[

γA2 (0) + κ ∗q A1 (0)

ϕ2 (r) = arg [A2 (0) cos(μr) + i

μ μ

(5.96b)

1/2

2 2

] sin (μr)}

−γA2 (0) − κ ∗q A1 (0)

(5.96a)

sin(μr)]

(5.97a) (5.97b)

1√ 2 (Δk + v1q − v2q ) + 4κ q ⋅ κ ∗q 2 (5.98) 1 1 γ = (v2q − v1q − Δk) , β = (Δk − v1q − v2q ) . 2 2 These analytical solutions can be used to describe the EO effect in QPM materials with an external electric field applied along arbitrary direction. μ=

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Here we take tunable Šolc-type EO filter as a example to illustrate the application of wave coupling theory of QPM linear EO effect [21]. Compared to traditional Šolc-type filter, the PPLN EO Šolc-type filter can be realized in one chip of lithium niobate, and the output intensity of the filter can be controlled electrically, which lead to simultaneous achievements of modulator and filter in one piece of PPLN. Instead of heating or UV-light illumination, we find that under certain conditions the tuning of transmitted central wavelength can be achieved by applying a dc electric field along the z-axis of PPLN with a biased one along the y-axis. Since the tuning is based on the linear EO effect, it would have a much faster response than those of thermal and UVillumination ones. Because the QPM condition is very sensitive to the wavelength and the largest EO coefficient utilized, the filter has narrowband spectrum and broad tuning range (above 16 nm). Besides, the dependence of transmitted central wavelength shift on the control electric field shows a nearly linear relation with a tuning rate of 0.95 kV/mm per nm [21]. Figure 5.13 shows the schematic diagram of an electrically tunable PPLN filter. The filter consists of a 2.5 cm z-cut PPLN placed between two cross polarizers, in which the polarization direction of the front one is set parallel to the z-axis and the other parallel to the y-axis of the PPLN. The duty cycle of the PPLN used here is 0.75, i.e. the ratio of the neighboring positive- and negative-domain widths is 3 : 1, which is the optimum value of second-order quasi-phase matching. The poling period is Λ = 2λ0 /[n o (λ0 ) − n e (λ0 )](λ0 = 1550 nm), which means that the second-order (m=2) QPM condition is satisfied for a light with λ0 = 1550 nm, propagating along the xaxis. For this case, we have F2 = 1/iπ and F0 = 0.5. The nonzero EO coefficient of LiNbO3 are: r22 = 3.4, r23 = 8.6, r33 = 30.8, and r51 = 28 (in 10−12 m/V) [47], and the refractive indices can be calculated by Sellmeier equations [53]. According to the conditions mentioned above and Figure 5.13, we have r = x, A1 (0) = 0, and A2 (0) = 1. The solution to Equation (5.93) becomes A1 (x) = −i exp (iβx)

κq sin(μx) μ

(5.99a) γ

A2 (x) = exp [i(β − Δk)x] ⋅ [cos(μr) − i μ sin(μx)] .

Fig. 5.13. The schematic diagram of EO PPLN Šolc-type filter.

(5.99b)

336 | Weilong She, Guoliang Zheng And the output intensity of the o-ray I o is thus given by 󵄨󵄨 󵄨󵄨2 󵄨κ q 󵄨 I o = 󵄨 2󵄨 sin2 (μL) , μ

(5.100)

where L is the effective length of the crystal. In our design, two electric fields are used, one of which is along the y-axis and another is along the z-axis. The total external electric field is Ey Ez E(0) = E0 c = E0 ( j + k) , (5.101) E0 E0 where c is the unit vector of E(0); E y and E z are the amplitudes of external electric fields along the y-axis and the z-axis; E0 = √E2y + E2z ; and j and k are the two corresponding unit vectors pointing to positive directions of the y-axis and the z-axis. When E y and (or) E z change, both the direction and amplitude of the total external filed will change. Therefore, in the presence of the external electric field E(0), we have κ q = −i

n20 n2e k 0 r51 E y , 2π√n o n e

κ ∗q = i

n20 n2e k 0 r51 E y 2π√n o n e

(5.102)

and v2q − v1q = − 41 n3o r22 k 0 E y + 14 (n3e r33 − n3o r23 )k 0 E z .

(5.103)

Equation (5.102) clearly shows that κ q (κ ∗q ) is independent of E z , and it is controlled by E y . On the other hand, the value of v2q −v1q depends strongly on E z through the largest EO coefficient r33 . It should be emphasized here that the nonzero v2q − v1q appears only when the duty cycle is not equal to 0.5. From Equations (5.98) and (5.100), one can see that the maximal conversion efficiency occurs at Δk + v1q − v2q = 0 . (5.104) Since the refractive indices are the functions of wavelength, relation (5.104) reads [n e (λ) − n o (λ)]

1 2 n3o (λ) 1 3 + + r22 E y − [n (λ)r33 − n3o (λ)r23 ]E z = 0 . λ Λ 4λ 4λ e

(5.105)

The wavelength that satisfies this condition is called “transmitted central wavelength.” From Equation (5.105), one can see that the transmitted central wavelength will shift when E z is applied. For a fixed electric field E y , the output intensity of the filter almost keeps unchanged although the wavelength shifts by tens of nanometers, because sin(|κ|L) is insensitive to the wavelength. But, the output intensity of the filter can be controlled by varying the electric field E y , which is reflected by sin(|κ|L). This characteristic is applicable in EO modulation, which leads to simultaneous achievements of modulator and filter in one piece of PPLN. We find that when E y = 0.33 kV/m, the filter has the largest output intensity. So, we fix the external electric field E y at 0.33 kV/mm and study the dependence of output

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Fig. 5.14. The output intensity of EO PPLN Šolc-type filter as a function of wavelength λ. The solid line, dashed line, and dashed-dot line correspond to E z = 0 kV/mm, E z = 1.9 kV/mm and E z = −1.9 kV/mm, respectively.

Fig. 5.15. The output intensity of EO PPLN Šolc-type filter as function of control field E z and incident light wavelength λ.

338 | Weilong She, Guoliang Zheng intensity of o-ray on the wavelength of light. Figure 5.14 shows the results corresponding to E z = 0 and E z = ±1.9 kV/mm, respectively. One can see that the transmitted central wavelength shifts as much as 2 nm when E z = ±1.9 kV/mm. The full width at half maximum (FWHM) of the filter is about 1 nm, which can even be narrowed to 0.5 nm if the length of PPLN is doubled. In fact, FWHM is inversely proportional to the crystal length, so expected much narrower spectrum filter can be achieved by the employment of a long enough PPLN crystal. The relation between transmitted central wavelength and control electric field E z would be useful for practical applications. The numerical result is shown in Figure 5.15, in which the dependence of transmitted central wavelength on the control electric field E z shows a nearly linear relation with a tuning rate of about 0.95 kV/mm per nm. The stronger the control electric field E z , the larger the shift of transmitted central wavelength is. However, for an undoped PPLN the electrical breakdown limits the electric field to a maximum value of 16.8 kV/mm [21]. In our calculations, the electric field E z is set in a safe region from −7.6 to 7.6 kV/mm, which gives the transmitted central wavelength from 1558 to 1542 nm.

5.5.2 Linear EO effect in linear chirped-periodically poled crystals In periodically, qusasi-periodically and aperiodically poled crystals, the perfect EO coupling happens only in a narrow bandwidth (∼1 nm) since the QPM condition is very sensitive to the wavelength. However, in some special cases, such as the EO effect of an ultrashort laser pulse, it requires broadband for coupling. It has been found that in a chirped-periodic optical superlattice, the nonlinear optical processes based on QPM can be realized in a broad wavelength range [54–60]. She and Zeng have also found an EO coupling of wide wavelength range with a high coupling efficiency in a linear chirped-periodically poled LiNbO3 (LCPLN) [52]. Here we describe this kind of EO coupling in detail. Figure 5.16 shows a linear chirped periodically poled lithium niobate (LCPLN) poled along the z-axis with a monochromatic light propagating along the x-axis and an electric field applied along the y-axis.

Fig. 5.16. EO coupling in LCPLN.

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According to Section 5.5.1, the EO wave coupling of ordinary-ray (o-ray) and extraordinary-ray (e-ray) in a LCPLN can be described by the following equations: dA1 (x) = −iκ(x)A2 (x) exp [iϕ(x)] − iv(x)A1 (x) , dx dA2 (x) = −iκ ∗ (x)A1 (x) exp [−iϕ(x)] , dx

(5.106a) (5.106b)

where κ(x) = −

π (n o n e )3/2 r51 E y F1 (x) , λ0

v(x) =

π 3 n r22 E y F0 (x) , λ0 o

x

ϕ(x) = ∫ Δk 󸀠 (u)du ,

Δk 󸀠 (x) = Δk (λ0 ) + G1 (x) ,

(5.107)

0

Δk(λ0 ) =

2π (n e − n o ) , λ0

G1 (x) = G0 − αx ,

where A j (x) (j = 1, 2) are the complex amplitudes corresponding to o- and e-ray, respectively; λ0 is the vacuum wavelength of incident light; n o and n e are the refractive indices of o-ray and e-ray, respectively; r51 and r22 are the linear EO coefficients of LiNbO3 ; E y is the applied electric field along the y-axis; F0 (x) and F1 (x) are the zeroorder and first-order Fourier coefficients of LCPLN; Δk(λ0 ) denotes the wave-vector mismatch between o-ray and e-ray. In a LCPLN, G1 (x) = 2π/Λ(x) is the first-order reciprocal vector, where G0 is the reciprocal vector at x = 0 and α is a constant chirp coefficient. Λ(x) is the poling period of LCPLN, which is different from that of PPLN or QPLN. By Λ(x) = 2π/G1 (x), one can see that Λ(x) ≈ Λ0 (1 + βx) ,

(5.108)

where Λ0 = 2π/G0 is the poling period at x = 0 and β = α/G0 . When Δk 󸀠 (xpm ) = 0, perfect QPM is reached. Here, xpm is called the perfect phase matching point. We first consider a LCPLN of length L, with a duty cycle of 0.5, F0 (x) = 0, F1 (x) = 2/(iπ), and 2 κ (x) = κ = (n o n e )3/2 r51 E y , v (x) = 0 . (5.109) iλ0 The reciprocal vector G1 (x) = G0 − αx varies continuously from G0 to G0 − αL. Also the wave-vector mismatch between o-and e-ray is Δk(λ0 ) = 2π(n e − n o )/λ0 . So the wavelength range, in which QPM can be satisfied, is approximatively from λ1 to λ2 (λ2 > λ1 ), with its center at λ0.5 , where λ0.5 is the wavelength whose perfect phase matching point is at xpm = 0.5L. Here, 2π (n o − n e ) . G0 − 0.5αL (5.110) Since αL is usually much smaller than G0 , λ1 ≈ λ0.5 (1 − 0.5αLλ0.5 /G0 ), λ2 ≈ λ0.5 × (1 + 0.5αLλ0.5 /G0 ), and the FWHM of EO coupling is about λ1 =

2π (n o − n e ) , G0

λ2 =

2π (n o − n e ) , G0 − αL

λ0.5 =

Δλ = λ2 − λ1 ≈ ξλ0.5 ,

(5.111)

340 | Weilong She, Guoliang Zheng where ξ = |αL| G 0 . Obviously, one can increase the bandwidth by increasing ξ . For instance, for λ0.5 = 1550 nm, when ξ increase from 0.01 to 0.1, Δλ changes from 15.5 to 155 nm. If one wants to obtain a broadband quasi-phase matched EO coupling from λ1 to λ2 , then one can get G0 = −Δk (λ1 ) , G0 − αL = −Δk (λ2 ) , αL = Δk (λ2 ) − Δk (λ1 ) .

(5.112)

Then, from Equation (5.112), one can get the proper parameters K0 and αL for the LCPLN. One sees from Equations (5.106) that, when the chirp coefficient is small enough so that the LCPLN can be taken as a PPLN, |κ|L = π/2, which is just the condition for the strongest EO coupling. Generally, it is not an easy task to qualitatively solve Equations (5.106). We notice that Equations (5.106) are linear coupling equations, which are much similar to the equations of two-level quantum system discussed by Landau and Zener [62, 63]. Without losing generality, we consider that the incident light is an o-ray, i.e. A1 (0) = 1, A2 (0) = 0. Then the conversion efficiency from o- to e-ray is η = |A2 (L)|2 . By the way similar to Ref. [63], one can further find η = 1 − exp (−

2π |κ|2 ), |α|

(√|α|L → ∞) .

(5.113)

It should be pointed out that Equation (5.113) is an asymptotic result when the crystal length √|α|L of LCPLN is infinite. For the case of finite length of LCPLN, the conversion efficiency can also be described approximatively by Equation (5.113) as long as √|α|L ≫ 1. From Equation (5.113), one can get |κ|/√|α| = [− ln(1 − η)/(2π)]1/2 , which is independent of √|α|L. For high conversion, here we set η = 0.99, then |κ|/√|α| = 0.8561, which can be taken as the condition for perfect conversion. We choose η = 0.99 since in this case the EO coupling requires only a rather smaller |κ|/√|α|. To show this, here we give some numerical results. Also the relation of |κ|/√|α| versus √|α|L for different conversion from o- to e-ray is shown in Figure 5.17. One sees from Figure 5.17 (a), when √|α|L ≤ 1, |κ|/√|α| is inversely proportional to √|α|L for 100% conversion. For example, for a 100% conversion, the parameter |κ|/√|α| = 157, 15.7, 1.57, corresponding to √|α|L = 0.01,0.1,1. We find that |κ|/√|α| × √|α|L = |κ|L = π/2, which behaves very like that of perfect OE coupling in a PPLN. In contrast, when √|α|L is large enough (≥ 10), the conversion efficiency is almost independent of √|α|L. For this case, we find that the conversion can also be as high as 99% when |κ|/√|α| = 0.86 [Figure 5.17 (c)], which are consistent with the above analyses. In the following, we will focus mainly on the case of larger √|α|L (∼10), and calculate the electric field for 99% conversion from equation |κ|/√|α| = 0.8561, i.e. E y = 0.4281

λ0 √|α| (n o n e )3/2 r51

.

(5.114)

In simulation, the incident light is assumed to be an o-ray, i.e. A1 (0) = 1, A2 (0) = 0. The Sellmeier equations of LiNbO3 used are from Ref. [53]. Also the EO coefficients of

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Fig. 5.17. |κ|/√|α| vs √|α|L for different conversion efficiency from o- to e-ray. The incident light is oray, and the wavelength has a perfect phase matching point at 0.5√|α|L. The insert in (a) is a curve for 100%. conversion.

LiNbO3 are r51 = 28 pm/V, r22 = 3.4 pm/V [47], respectively. The temperature is set at T = 298 K. Now we consider an example of EO coupling covering 40 nm of C band (1528– 1568 nm) in ITU DWDM specification [64]. As mentioned above, the LCPLN can be designed as G0 = |Δk(1528)|, G1 = G0 − αL = |Δk(1568)|, then we have G0 = 0.3110 μm−1 , αL = 0.0088 μm−1 ; while the length of LCPLN is chosen to be 2 cm, then α = 4.4 × 10−7 μm−2 . From (5.114), we can get E y = 1.52 kV/mm for the center wavelength 1548 nm. Under these conditions, we get the conversion efficiency of o- to e-ray as a function of wavelength, shown in Figure 5.18 (a), where the FWHM is 40 nm, from 1528 to 1568 nm, which agrees well with the expectation. Similarly, we make the calculations for the LCPLNs with lengths of 3, 4, and 5 cm, respectively. The results are shown in Figure 5.18 (b)–(d), which indicate that, for the same bandwidth, as the length of LCPLN increases, both α and E y will decrease. In fact, the bandwidth of LCPLN is not limited to be 40 nm. Figure 5.19 shows the conversion efficiency of several LCPLNs that cover 20, 80, 120 nm. The parameters are in Table 5.2. By similar way, one can also obtain the bandwidth larger than 120 nm by choosing a larger ξ . However, a larger coefficient means a larger E y , which should be considered in practical use. For practical use, the ripples reduction of the conversion efficiency is important. One can reduce the ripples by changing the duty cycle of LCPLN, called apodization

342 | Weilong She, Guoliang Zheng

Fig. 5.18. Numerical simulations of EO coupling covering 40 nm in LCPLNs with different lengths. (a) L = 2 cm, α = 4.4 × 10−7 μm−2 , E y = 1.52 kV/mm; (b) L = 3 cm, α = 2.9 × 10−7 μm−2 , E y = 1.24 kV/mm; (c) L = 4 cm, α = 2.2 × 10−7 μm−2 , E y = 1.08 kV/mm; (d) L = 5 cm, α = 1.8 × 10−7 μm−2 , E y = 0.96 kV/mm.

[65]. Several apodization profiles can be used. Here we use the tanh profile. The Fourier coefficients and the duty cycles corresponding to tanh profile are 1 2ax L { { { 2 tanh ( L ) , 0 ≤ x < 2 , D(x) = { { { 1 tanh [ 2a (L − x)] , L ≤ x ≤ L , L 2 {2 F0 (x) = 2D (x) − 1 1 F1 (x) = {1 − cos [2πD (x)] + i sin [2πD (x)]} , iπ

(5.115a)

(5.115b) (5.115c)

where a is an apodization parameter and is set at 3. Figure 5.20 shows the |F1 (x)| corresponding, respectively, to unapodized LCPLN and apodized LCPLN with tanh profile. One can see that the curve of tanh profile has a FWHM of 0.88. As is well known, the coupling coefficient κ(x) is proportional to |F1 (x)|. Therefore, the FWHM of apodized

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Table 5.2. The parameters of LCPLN covering 20, 80, and 120 nm. Wavelength range (nm) 1540–1560

G0 ( μm−1 )

L (cm)

0.3083

1510–1590

0.3151

1490–1610

2

0.3198

α( μm−2 )

E y (kV/mm)

2.2

× 10−7

1.08 (1550 nm)

8.8

× 10−7

2.16 (1550 nm)

1.3

× 10−6

2.64 (1550 nm)

Fig. 5.19. Numerical simulations of conversion efficiency of broadband EO coupling in LCPLN with FWHM 20 nm, 80 nm and 120 nm, respectively. The length of LCPLN is 2 cm. Solid line: 1540– 1560 nm; dash line: 1510–1590 nm; dotted line: 1490–1610 nm.

LCPLN will decrease to about 0.88 for a unapodized LCPLN. That is, Equations (5.111) and (5.112) should be modified as Δλ ≈ τξλ0.5 , G0 −

1 2

(1 − τ) αL = −Δk (λ1 ) ,

(5.116a) G0 −

1 2

(1 + τ) αL = −Δk (λ2 ) ,

(5.116b)

where τ = 0.88 for the apodized LCPLN with tanh profile (a = 3). Figure 5.21 shows the ripples reduction with apodized LCPLN, where the bandwidths designed are 20 (1540–1560), 40 (1528–1568), 80 (1510–1590), and 120 nm (1490–1610 nm), respectively and the lengths of LCPLN are set as 2 cm. It should be noticed that the parameters of the crystals are now obtained from Equation (5.116b). One sees that the ripples are effectively reduced by using apodized LCPLN the tops of the conversion efficiency curves become much flat. And the FWHMs, however, become 16, 36, 76, and 116 nm, respectively, slightly smaller than those of design. The apodized χ(2) grating has been experimentally reported [56], meaning the fabrication of such grating is feasible. But there

344 | Weilong She, Guoliang Zheng

Fig. 5.20. |F1 (x)| corresponding respectively to unapodized LCPLN (D(x) = 0.5) and an apodized LCPLN with tanh profile (a = 3).

exists a limitation to the smallest duty cycle. For example, the domain width of 0.5 m or less is hard to fabricate. The EO coupling in LCPLN with wide wavelength range would be useful for some devices, such as broadband EO polarization converters and bandpass filters. For example, when the LCPLN is inserted into the interspace of two polarizers (y- and z-polarized), then the device can work as a broadband Šolc-type bandpass filter in optical communication. The Šolc-type bandpass filters in LCPLN is similar to that with temperature-gradient-control technique but can work much faster [66].

5.5.3 Wave coupling theory of united effect of QPM linear EO effect, second-harmonic generation, and sum/difference frequency generation The QPM technology has been widely used in nonlinear optical frequency conversion, and the QPM linear EO effect has also made its progress in both theory and application. However, nonlinear optical frequency conversion and linear EO effect have never been involved in a united theory for a long time. When there are multiple second-order nonlinear optical effects including linear EO effect simultaneously in a material, people always take the linear EO effect as an adjunctive one of other second-order nonlinear effects. In 2007, a wave coupling theory describing the united effects of QPM-based electrically controllable second-harmonic generation (SHG) and PDC was developed by She and Huang [22] on the basis of the wave coupling theory of QPM linear EO effect. They used the QPM concept simultaneously for multiple second-order nonlinear

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Fig. 5.21. Ripples reduction by using the apodized LCPLNs with tanh profile (a = 3).

optical effects including linear EO effect, and considered all these effects as the equivalent ones. Their theory gives a new way for solving the problem of multiple nonlinear optical effects. To establish a wave coupling theory describing the QPM-based united effect of EO–SHG-sum frequency (SF)/difference frequency (DF), let us consider the following case first: there are two monochromatic plane light waves being incident onto a nonoptically actively nonlinear crystal. According to the principle of nonlinear optics, all the SHG, SF, and DF processes would occur simultaneously in the crystal. To make the problem more clear and for the convenience of discussion below, we assume that the two incident waves propagate collinearly along the x-direction; the new waves generated through the second-order nonlinear effects also propagate along the same direction. Generally, each monochromatic light wave (either the incident one or the generated one) will have two independent linear polarization components: yand z-ones. Further suppose that two linearly polarized light waves with frequency ω1 and ω2 makes a coupling and generates a linearly polarized wave with frequency ω3 through the second-order nonlinear effect (it is just SHG if ω1 = ω2 and ω1 + ω2 = ω3 ). Here, we only discuss the second-order nonlinear optical effects. This means that the high-order nonlinear effects are ignorable. If there is no external electric and magnetic fields, there may exist six independent linearly polarized wave components simultaneously in the crystal. For the quasi steady state, similar to the derivation of threewave-coupling equations describing second-order nonlinear optical effects, we can

346 | Weilong She, Guoliang Zheng obtain the following general six-wave-coupling equations: dE1y D iω1 d23 E∗2z E3z e i(k3z −k2z −k1y )x + d24 E∗2z E3y e i(k3y −k2z −k1y )x ] = [ dx 2 n1y c +d24 E∗2y E3z e i(k3z −k2y −k1y )x + d22 E∗2y E3y e i(k3y −k2y −k1y )x

(5.117a)

dE1z D iω1 d33 E∗2z E3z e i(k3z −k2z −k1z )x + d34 E∗2z E3y e i(k3y −k2z −k1z )x ] = [ dx 2 n1z c +d34 E∗2y E3z e i(k3z −k2y −k1z )x + d32 E∗2y E3y e i(k3y −k2y −k1z )x

(5.117b)

dE2y D iω2 d23 E∗1z E3z e i(k3z −k2y −k1z )x + d24 E∗1z E3y e i(k3y −k2y −k1z )x ] = [ dx 2 n2y c +d24 E∗1y E3z e i(k3z −k2y −k1y )x + d22 E∗1y E3y e i(k3y −k2y −k1y )x

(5.117c)

dE2z D iω2 d33 E∗1z E3z e i(k3z −k2z −k1z )x + d34 E∗1z E3y e i(k3y −k2z −k1z )x ] = [ dx 2 n2z c +d34 E∗1y E3z e i(k3z −k2z −k1y )x + d32 E∗1y E3y e i(k3y −k2z −k1y )x

(5.117d)

dE3y D iω3 d23 E1z E2z e−i(k3y −k2z −k1z )x + d24 E1z E2y e−i(k3y −k2y −k1z )x = ] [ dx 2 n3y c +d24 E1y E2z e−i(k3y −k2z −k1y )x + d22 E1y E2y e−i(k3y −k2y −k1y )x

(5.117e)

dE3z D iω3 d33 E1z E2z e−i(k3z −k2z −k1z )x + d34 E1z E2y e−i(k3z −k2y −k1z )x = [ ] , (5.117f) dx 2 n3z c +d34 E1y E2z e−i(k3z −k2z −k1y )x + d32 E1y E2y e−i(k3z −k2y −k1y )x where {1 ω 1 = ω 2 D={ 2 ω 1 ≠ ω 2 { and E1y , E1z correspond to ω1 , E2y , E2z to ω2 and E3y , E3z to ω3 , respectively. In the above six wave-coupling equations, there exist eight different mismatches of wave vector, namely Δk 1 = k 3z − k 2z − k 1z , Δk 2 = k 3z − k 2z − k 1y , Δk 3 = k 3z − k 2y − k 1z ,

Δk 4 = k 3z − k 2y − k 1y ,

Δk 5 = k 3y − k 2z − k 1z ,

Δk 6 = k 3y − k 2z − k 1y ,

Δk 7 = k 3y − k 2y − k 1z ,

Δk 8 = k 3y − k 2y − k 1y .

(5.118)

The phase-matching conditions for these wave vector mismatches cannot be met simultaneously with the conventional birefringent phase matching technology. And for a particular nonlinear crystal, its available nonzero nonlinear optical coefficients are restricted, which also make some phase matching impossible. Therefore, only three linearly polarized waves can meet the phase-matching conditions, for example, the phase-matching condition for E1y + E2y ↔ E3z is satisfied when Δk 4 = 0. Thus, only these three linearly polarized waves can take part in the second-order nonlinear effects. Also Equation (5.117) will be simplified into the well-known three-wave-coupling equations dE1y D iω1 = d24 E∗2y E3z e i(k3z −k2y −k1y )x dx 2 n1y c dE2y D iω2 = d24 E∗1y E3z e i(k3z −k2y −k1y )x dx 2 n2y c dE3z D iω3 = d32 E1y E2y e−i(k3z −k2y −k1y )x . dx 2 n3z c

(5.119a) (5.119b) (5.119c)

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However, in an optical superlattice, two or more of phase mismatches can be compensated simultaneously by using QPM technology, so that the multiwave coupling processes can occur simultaneously in the optical superlattice. Furthermore, if we add an external electric field onto the optical superlattice, multiwave coupling process will become more colorful. As mentioned above, She and Huang have studied the cascading effect of electrically controllable SHG and PDC in a quasi-periodic/periodic poled lithium niobate optical superlattice, and have also presented the corresponding wave-coupling equations [22]. However, the equations proposed are yet not so universal. Here we further generalize the wave-coupling equations so that they can describe the QPM-based united effect of EO–SHG–SF/DF for wider applications. Based on the above discussion, we assume that there are three frequencies of light waves with six independent linear polarized components taking part in the second-order nonlinear effects, and they are collinear and propagate along the x-axis of the optical superlattice. That is, each wave with given frequency has two independent linearly polarized components vibrating along the y- and z-axis of the crystal, respectively. And in the crystal a couple of linearly polarized light waves with frequencies ω1 and ω2 will result in another linearly polarized wave with frequency ω3 due to the second-order nonlinear effect. If an external electric field E(0) = E(0)c [where c = (c1 , c2 , c3 ) is a unit vector] is applied onto the optical superlattice, the second-order polarization with nonzero frequency should include not only the items describing the well-known SHG–SF/DF, but also the items describing the linear EO effect. For the EO effect, the second-order polarization is (2)

PEO (ω i ) = 2ε0 χ (2) (ω i , 0) : Ei y (x)E0 exp (ik i y x) + 2ε0 χ (2) (ω i , 0) : Ei z (x)E0 exp (ik i z x)(i = 1, 2, 3) .

(5.120)

Here, Ei y (x) and Ei z (x) represent the amplitudes of monochromatic light waves, k i y and k i z (i = 1, 2) means the corresponding wave vectors. Because of the presence of external electric field, the linear EO effect (ω1z ↔ ω1y , ω2z ↔ ω2y , ω3z ↔ ω3y ) and SHG–SF/DF effect (ω1y + ω2y ↔ ω2y , ω1y + ω2z ↔ ω2y , ω1z + ω2z ↔ ω2y , ω1y + ω2y ↔ ω2z , ω1y + ω2z ↔ ω2z , ω1z + ω2z ↔ ω2z ) will occur simultaneously in the optical superlattice. Then we can obtain the wave-coupling equations describing the united effect of EO–SHG–SF/DF by synthesizing the methods for deducing sixwave-coupling equations and the quasi-phase-matched EO wave-coupling equations. These equations are as follows: dE1y = −id1 (x)E1z (x)e−iΔk1 x − id2 (x)E1y (x) dx d23 (x)E∗2z E3z e i(k3z −k2z −k1y )x + d24 (x)E∗2z E3y e i(k3y −k2z −k1y )x D iω1 + ] [ 2 n1y c +d24 (x)E∗2y E3z e i(k3z −k2y −k1y )x + d22 (x)E∗2y E3y e i(k3y −k2y −k1y )x (5.121a)

348 | Weilong She, Guoliang Zheng dE1z = −id3 (x)E1y (x)e iΔk1 x − id4 (x)E1z (x) dx D iω1 d33 (x)E∗2z E3z e i(k3z −k2z −k1z )x + d34 (x)E∗2z E3y e i(k3y −k2z −k1z )x + ] [ 2 n1z c +d34 (x)E∗2y E3z e i(k3z −k2y −k1z )x + d32 (x)E∗2y E3y e i(k3y −k2y −k1z )x (5.121b) dE2y = −id5 (x)E2z (x)e−iΔk2 x − id6 (x)E2y (x) dx D iω2 d23 (x)E∗1z E3z e i(k3z −k2y −k1z )x + d24 (x)E∗1z E3y e i(k3y −k2y −k1z )x ] [ + 2 n2y c +d24 (x)E∗1y E3z e i(k3z −k2y −k1y )x + d22 (x)E∗1y E3y e i(k3y −k2y −k1y )x (5.121c) dE2z = −id7 (x)E2y (x)e iΔk2 x − id8 (x)E2z (x) dx D iω2 d33 (x)E∗1z E3z e i(k3z −k2z −k1z )x + d34 (x)E∗1z E3y e i(k3y −k2z −k1z )x ] [ + 2 n2z c +d34 (x)E∗1y E3z e i(k3z −k2z −k1y )x + d32 (x)E∗1y E3y e i(k3y −k2z −k1y )x (5.121d) dE3y = −id9 (x)E3z (x)e−iΔk3 x − id10 (x)E3y (x) dx D iω3 d23 (x)E1z E2z e−i(k3y −k2z −k1z )x + d24 (x)E1z E2y e−i(k3y −k2y −k1z )x + ] [ 2 n3y c +d24 (x)E1y E2z e−i(k3y −k2z −k1y )x + d22 (x)E1y E2y e−i(k3y −k2y −k1y )x (5.121e) dE3z = −id11 (x)E3y (x)e iΔk3 x − id12 (x)E3z (x) dx D iω3 d33 (x)E1z E2z e−i(k3z −k2z −k1z )x + d34 (x)E1z E2y e−i(k3z −k2y −k1z )x + ] , [ 2 n3z c +d34 (x)E1y E2z e−i(k3z −k2z −k1y )x + d32 (x)E1y E2y e−i(k3z −k2y −k1y )x (5.121f) where D is the same as that defined above, but (5.122a)

Δk 2 = k 2y − k 2z

(5.122b)

Δk 3 = k 3y − k 2z k 10 n1y n21z r4l c l E0 f(x) , 2 k 10 n21y n1z r4l c l d3 (x) = − E0 f(x) , 2 k 20 n2y n22z r4l c l d5 (x) = − E0 f(x) , 2 k 20 n22y n2z r4l c l d7 (x) = − E0 f(x) , 2 d1 (x) = −

2π (n1y − n1z ) λ1 2π = (n2y − n2z ) λ2 2π = (n3y − n3y ) λ3

Δk 1 = k 1y − k 1z =

d2 (x) =

k 10 n31y r2l c l 2

(5.122c)

E0 f(x)

k 10 n31z r3l c l E0 f(x) 2 k 20 n32y r2l c l d6 (x) = E0 f(x) 2 k 20 n32z r3l c l d8 (x) = E0 f(x) 2 d4 (x) =

(5.123a) (5.123b) (5.123c) (5.123d)

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Fig. 5.22. Experimental setup of electronically controllable DF generation in a QPPLN. The electric field is applied along y-axis. x, y, and z represent of the three primitive axes of crystal, the arrows indicate the polarized direction in each domain. p, s, i denote the pumping, signal, and DF beams, respectively.

k 30 n3y n23z r4l c l E0 f(x) , 2 k 30 n23y n3z r4l c l d11 (x) = − E0 f(x) , 2 d9 (x) = −

d10 (x) = d12 (x) =

k 30 n33y r3l c l

E0 f(x)

(5.123e)

k 30 n33z r3l c l E0 f(x) 2

(5.123f)

2

d22 (x) = d22 f(x) ,

d23 (x) = d23 f(x)

(5.124a)

d24 (x) = d24 f(x) ,

d32 (x) = d32 f(x)

(5.124b)

d33 (x) = d33 f(x) ,

d34 (x) = d34 f(x) .

(5.124c)

It should be noted that in the above equations, l must be summed up for 1, 2, 3 for the product terms containing the same subscript l; k l0 = 2π/λ l0 (l = 1, 2, 3), λ l0 is the vacuum wavelength of each monochromatic light waves. In addition, d11 (x) and d12 (x) are the EO coupling coefficient instead of effective nonlinear coefficient. Just pay attention for it, no confusion will be caused. In the following, we take electrical generation of DF in a quasi-periodically poled lithium niobate (QPPLN) as an example to illustrate the application of the wavecoupling equations of united effect of EO–SHG–SF/DF [23]. Figure 5.22 is the experimental setup. Consider that a 1064 nm pumping beam and a 1550 nm signal beam propagate through the QPPLN crystal; and a beam of 3393.4 nm is created through DF effect. In order to make use of the largest nonlinear coefficient d33 of lithium niobate, we let the light waves propagate along the x-axis of crystal. And the pumping beam, signal beam, and the new beam generated by DF process are all polarized along the z-axis of crystal. In addition, we add an external electric field to the crystal, so that the DF e- and o-rays can make their coupling, which modules the efficiency and polarization of output DF beam. To depress the voltage, the applied electric field is applied along the y-axis. The QPPLN is designed in such a way that it can provide two reciprocal vectors, which compensate the mismatches of wave vector from DF process (pumping e-, signal e-, and DF e-waves) and EO effect between DF e- and o-rays. Here p, s and i are used to label the pumping, signal, and DF beams, respectively. For clarity, we further set 1 → s, 2 → i, 3 → p. Using the above wave-coupling equations describing united effect of EO–SHG–SF/DF, and keep in mind that the mismatches of wave vector for the

350 | Weilong She, Guoliang Zheng EO effects between pumping e- and o-rays, signal e- and o-rays and r32 = 0 (reff3 = 0) have never been compensated, one can obtain the wave-coupling equations of the cascading DF (ω pz − ωsz ↔ ω iz ) and EO effects (ω iy ↔ ω iz ) dE pz (x) ω p d33 f(x)E iz (x)E sz (x)e iΔk D x =i dx cn pz dE sz (x) ω s d33 f(x)E pz (x)E∗iz (x)e−iΔk D x =i dx cn sz

(5.125) (5.126)

k 0 n3iy r22 k 0 n iy n2iz r42 dE iy (x) =i E0 f(x)E iz (x)e−iΔk2 x − i E0 f(x)E iy (x) (5.127) dx 2 2 k 0 n2iy n iz r42 ω i d33 dE iz (x) f(x)E pz (x)E∗sz (x)e−iΔk1 x , (5.128) =i E0 f(x)E iy (x)e iΔk2 x + i dx 2 cn iz where n iz n sz n pz + − ), λi λs λp

2π (n iy − n iz ) . λi (5.129) Here E jμ , E0 , λ j , ω j , k jμ , and n jμ [j = p, i, s; μ = y, z] are the light fields, electric field, wavelengths, frequencies, wave numbers, and the refractive indices, respectively; k 0 is the wave number of DF beam in vacuum; d33 is nonlinear coefficient; r22 and r42 are the EO coefficients; f(x) is the structure function. Obviously, when the external electric field is zero, Equations (5.125)–(5.129) reduce to the familiar wave-coupling equations describing the generation of DF. We assume that the QPPLN crystal is composed of fundamental blocks A and B, which is aligned quasi-periodic. And each block contains a pair of positive and negative domains. The width of A and B are l A = l+A + l−A and l B = l+B + l−B , respectively. The width of positive domains of A and B are designed to be the same as each other, i.e. l+A = l+B = l; the length of the superlattice L = 40 mm, the temperature T = 100 °C. By the Sellmeier equation of lithium niobate [53], the calculated mismatches of DF and EO effects are |Δk D | = 0.21 μm−1 and Δk 2 = 0.1110 μm−1 , respectively. For the general form of quasi-periodic structure [67], the structure function f(x) in Equations (5.125)–(5.129) can be expanded into a Fourier series Δk D = k iz + k sz − k pz = 2π (

Δk 2 = k iy − k iz =

f(x) = ∑ Fm,n exp (iG m,n x) ,

(5.130)

m,n

where G m,n = 2π m+nτ D L , D L is the average lattice parameter given by D L = τl A + l B , τ is the golden ratio, and the Fourier coefficients are [68] F m,n = 2(1 + τ)lD−1 L sinc(G m,n l/2) sin c(X m,n ) ,

(5.131)

πD−1 L (1

with X m,n = + τ)(ml A − nl B ). According to the phase-matching condition G m,n = 2π(m + nτ)/D L = |ΔkD | G m󸀠 ,n󸀠 = 2π(m󸀠 + n󸀠 τ)/D L = Δk 2 .

(5.132)

5 Wave coupling theory and its applications of linear electro-optic (EO) effect |

We obtain

m󸀠 ⋅ Δk D − m ⋅ Δk 2 n ⋅ Δk 2 − n󸀠 ⋅ Δk D 2 ⋅ π ⋅ (n ⋅ m󸀠 − m ⋅ n󸀠 ) DL = . n ⋅ Δk 2 − n󸀠 ⋅ Δk D

351

τ=

(5.133)

The domain boundary condition of the quasi-periodic superlattice is xn = n ⋅ a + (

n⋅a n⋅a −⌊ ⌋) , b b

(5.134)

where a and b are the structural parameters of the quasi-periodic structure, ba = τ + 1, τ ≥ 1. According to (5.134), we can determine the position of blocks A and B and the cycle period of superlattice. Here, we give the structure consisting of 17 blokes: ABABABABABABABABA. Thus, under the conditions of QPM, (5.125)–(5.129) can be reduced to the following form: dE pz (x) ω p d33 =i F1,1 E iz (x)E sz (x) , dx cn pz ω s d33 dEsz (x) =i F1,1 E pz (x)E∗iz (x) , dx cn sz k 0 n3iy r22 k 0 n iy n2iz r42 dE iy (x) =i E0 F0,1 E iz (x) − i E0 f(x)E iy (x) , dx 2 2 k 0 n2iy n iz r42 ω i d33 dE iz (x) F1,1 E pz (x)E∗sz (x) . =i E0 F0,1 E iy (x) + i dx 2 cn iz

(5.135) (5.136) (5.137) (5.138)

The intensity ratio between the input signal and pumping beams is defined as r = Is0 /I p0 , where I p0 and I s0 are the initial conditions used in simulation, representing the intensities of pumping and signal beams at the input surface of QPPLN. The conversion efficiency of DF is then η = I i (L)/I p0 × 100%, where I i (L) represents the intensity of DF beam at the output surface of QPPLN. At first, we study the relationship between the DF conversion efficiency η and the intensity of pumping beam I p0 for different r (r = 0.01, 0.1, 1) in the absence of applied electric field. There is only DF effect occurs in this case. So only the pumping, signal, and e-polarized DF beams exist. The numerical results are shown in Figure 5.23, from which one sees that, when r = 0.01, η gradually increases with I p0 firstly, and reaches its maximum (31.35%) at 9.5 MW/cm2 . And then, η decreases with the increase of I p0 , since the energy of signal and DF beams reflow back to the pumping beam. When r = 0.1, η reaches a maximum value (31.35%) at I p0 = 4.15 MW/cm2 , and decreases to the minimum (0) at 16.5 MW/cm2 , and then η gradually increases with I p0 . When r = 1, η reaches its maximum value (31.35%) first time at I p0 = 1.05 MW/cm2 . It decreases to the minimum value (0) first time at 4.05 MW/cm2 , and then repeats the previous process. Therefore, in the absence of applied electric field E0 , for a given r, the DF efficiency cannot achieve the ideal value for an arbitrary intensity of pumping beam. We note that, for different r, the largest

352 | Weilong She, Guoliang Zheng

Fig. 5.23. The dependence of DF conversion efficiency η on the intensity of pumping beam I p0 in the absence of external electric field. r is the ratio between input signal light and pumping beams, for which three different values are used here: r = 0.01 (dashed line), r = 0.1 (dashed line), and r = 1 (solid line). In the calculation, L = 40 mm.

DF efficiencies are the same, i.e. 31.35%; the largest efficiencies are obtained when pumping beam is exhausted; and η oscillate with the increase of I p0 . One thing is evident, when r is larger, the variation of η with I p0 is faster; the energy backflow is also faster; and the range of I p0 for high-energy output is narrower. Therefore, in order to obtain maximum DF efficiency (31.35%) at the output plane of QPPLN, we have to select the appropriate amplitude and ratio of the pumping intensity and signal one, when having no an applied electric field. Figure 5.24 shows the dependences of pumping, signal, and e-polarized DF beams on the propagation distance without applied electric field, for r = 0.1 and I p0 = 16.5 MW/cm2 . As can be seen from Figure 5.24, the intensities of DF and signal beams increase with the propagation distance gradually at first, and reaches its maximum at 20 mm, where the pumping beam is exhausted, DF beam occupies 28.5% of total energy, and the DF conversion efficiency is maximum (31.35%). And then, due to the energy reflows back to the pumping beam, the intensities of DF and the signal beams decrease gradually with the increase of propagation distance, and reach their minimum at 40 mm, where the intensity of DF beam is zero, and the intensity of pumping beam is maximum. In this process, the energy flows back and forth among the pumping, signal, and DF beams. As shown in Figure 5.23, the output intensities of pumping and signal beams can be, respectively, the same as the input ones when DF efficiency become zero. Thus, for a given r and I p0 , the DF efficiency varies with crystal length.

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Fig. 5.24. The dependences of pumping, signal, and e-polarized DF beams on the propagation distance in the absence of external electric field, for r = 0.1 and I p0 = 16.5 MW/cm2 . Dotted, dashed, and solid lines represent the pumping, signal, and e-polarized DF beams, respectively.

When an external electric field is applied, the couple between DF and EO effects will result in the energy exchange among the pumping, signal, and DF o- and e-rays. Therefore, the output DF beam can be controlled by the external electric field. The maximum achievable DF efficiency and the corresponding external electric field are shown in Figure 5.25 (a) and (b), respectively, for different I p0 and different r. The lines A and B depict the positions of first and second times of η reaching its maximum value (31.35%) and the corresponding external electric fields. They divide both Figure 5.25 (a) and (b) into the three regions I, II, and III. In the region I, the maximum efficiency always occurs at E0 = 0 for a given I p0 . This is because the energy backflow has not yet occurred, the energy flow unidirectionally from pumping beam to signal and DF beams in the superlattice. Then η increases gradually with I p0 , until it reaches its maximum value (31.35%). At this time, the EO modulation will reduce DF efficiency. On the A line, there is no external electric field. For 1.01 MW/cm2 < I p0 < 20 MW/cm2 , one can find an r, for which the energy of pumping beam completely converted into signal and DF beams. However, in both regions II and III, the DF efficiency always varies in the range of 0 and the maximum (31.35%) for arbitrary I p0 and r, if the applied electric field is absent. This is because the energy backflow from DF e-ray and signal beam to the pumping beam occurs, thus, the output DF efficiency cannot reach its maximum value (31.35%), which is evident shown in Figure 5.23. However, in the region II, a suitable applied electric field can make the DF efficiency maximum. For example, when r = 0.1, for three input intensities of the pumping beam: 6 MW/cm2 ,

354 | Weilong She, Guoliang Zheng

Fig. 5.25. (a) The maximum DF efficiencies η for different I p0 and r. (b) The external electric field required for maximum DF efficiency. The lines A and B represent the positions of first and second times of η reaching its maximum value (31.35%) and the corresponding applied electric fields. They separate (a) and (b) into three regions I, II, and III. (c) The dependence of DF efficiency on I p0 without the external electric field (solid lines) and with the external electric field (red dotted line) for r = 1. (d) The dependence of the phase different between the output DF o- and e-rays on I p0 for r = 1. In the calculation, L = 40 mm.

0.69 kV/mm, and 18 MW/cm2 , the DF efficiencies can increase from the original values of 24.44%, 0, 0.19% to 31.35%, 31.35%, 31.35%, respectively, if the applied electric fields are 0.615, 0.69, and 0.635 kV/mm, respectively. When r < 0.324 and the input intensity of pumping beam is larger than a certain value, only in regions I and II, the energy of pumping beam can completely converts into signal and DF beams by the use of an appropriate applied electric field. However, for r > 0.324, not only in region I and II, but also in region III one can always obtain the maximum conversion efficiency (31.35%) by using an appropriate electric field. In regions I and II, the situation are the same as those of r < 0.324, but in the region III, the situation is obviously different. EO modulation cannot increase the DF conversion efficiency in a certain zone of the region III (in red one of the region III). Except for this zone, the external electric field can improve the conversion efficiency even up to its maximum value. The dependence of DF efficiency on I p0 without the applied electric field (solid lines) and with the applied electric field (red dotted line) for r = 1 are shown in Figure 5.25 (c). Figure 5.25 (d) shows the phase difference between the output DF o- and e-rays in the present of E0 , where points C and D correspond to the phase difference of the first, the second times when DF efficiency reaches its maximum value without applied electric field, points C and E are the critical points. Before point C, it belongs to region I, where the maximum output of η is obtained at E0 = 0, and the output DF beam is linearly polarized along

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the z-axis of crystal. From points C to D, it belongs to region II, where the maximum output DF efficiency can be obtained through EO modulation, and the output DF in this region is elliptically polarized. After point D, it is region III. From points D to E, EO modulation cannot improve the DF efficiency, and the greater efficiency appears when the external electric field is zero, with the output DF beam being linearly polarized along the z-axis of crystal. Point E is the critical point, where the DF efficiency for zero applied electric field is equal to the maximum efficiency with an applied electric field. The difference is only that in the former case the output DF beam is linearly polarized along the z-axis of crystal, while in the latter case is right-handed-elliptically polarized. After point E, an appropriate electric field can increase the DF efficiency, which can be up to 31.35% for optimization. The output DF beam is right-handed elliptically polarized. Besides the above example, the wave-coupling equations of united effect of EO– SHG-SF/DF can also be used to design and analyze the electrically tunable production of circularly polarized beam [69] and the creation of high brightness photon pair [22].

5.5.4 Wave coupling theory of QPM linear EO effect for a focused Gaussian beam The principles of all the above wave coupling theories of linear EO effect are based on the plane-wave model. As is well known, the plane-wave model is valid only when the length of crystal is much shorter than the confocal parameter of light beam so that the beam width remains approximately a constant within the crystal. In fact, in the nonlinear interaction processes, especially the cascading of linear EO effect and other second-order nonlinearity effect, a focused laser beam is usually used to improve the efficiency or the signal intensity. In this case, the plane-wave model is invalid, and the transverse distribution of optical field should be taken into account [70–73]. On the other hand, the formation of spatially inhomogeneous polarization beam has also attracted considerable interest [74–78]. Various methods have been proposed to form the spatially inhomogeneous polarization beam, which can be classified into two kinds: direct and indirect ones. The direct method is by novel lasers with specially designed laser resonators [79–83]. And the indirect one is based on the wavefront reconstruction of the output field from the traditional lasers, with the help of specially designed optical elements [76, 84–88]. However, the formation of spatially inhomogeneous polarization is still a great challenge and paramount issue, due to the expectation of high flexibility in manipulating the spatially inhomogeneous polarization beam and in developing novel photonic devices and optical systems [76]. It is found that due to the EO effect of an appropriate applied electric field, the output beam will form a spatially inhomogeneous polarization, changing continuously in transverse section of beam. In order to explain the formation of spatially inhomogeneous polarization, a wave coupling theory of EO effect for QPM of focused Gaussian beam in an OSL was presented by She and Tang [15].

356 | Weilong She, Guoliang Zheng

Fig. 5.26. The experimental schematic diagram of EO effect for QPM of focused Gaussian beam in an OSL. The arrows indicate the directions of the polarizations of crystal domains. x, y and z stand for the three principal axes of the crystal. The applied electric field E 0 is along the y-axis of the OSL. a, b, c are three unit vectors.

Figure 5.26 shows the experimental schematic diagram of EO effect for QPM of focused Gaussian beam in an OSL. The applied electric field is along the y-axis of the OSL and the monochromatic light wave propagates along the x-axis of the OSL. In a cylindrical coordinate system, the total electric field participating in the process of linear EO effect can be expressed as [14] E(r, x, t) = E(0) + [ 12 E(ω) exp (iωt) + c.c.] ,

(5.139)

where [E(ω) exp (iωt)/2 + c.c.] is the optical field with frequency ω; c.c. denotes the complex conjugate; E(0) is the dc electric field or slow varying electric field; r is the radial distance from the propagation axis x. According to Refs. [89–90], the paraxial approximation is determined by the parameter g = 1/(k 0 /W0 ), where k 0 is the wave number of the optical field in vacuum and W0 is the waist radius at the input surface. For a wavelength λ = 632.8 nm, when g = 1/(k 0 /W0 ) ≤ 0.01, namely W0 ≥ 10.07μm, the paraxial approximation condition holds. And the transverse field of optical field is too small so that it can be neglected. But, there still exist two independent electromagnetic wave components of a monochromatic light wave propagating in the OSL, i.e. E(ω) = E1 (ω) + E2 (ω) = E1 (r, x) exp (−ik 1 x) + E2 (r, x) exp (−ik 2 x) ,

(5.140)

where E1 (r, x) and E2 (r, x) denote two perpendicular components of the electric field when k 1 = k 2 , or two independent electric field components experiencing different refractive indices when k 1 ≠ k 2 . For a Gaussian beam, the light field can be expressed as Ej (r, x) = Dj (x)u j (r, x) (j = 1, 2) [70–72], where Dj (x) are the expansion coefficients of the Laguerre–Gaussian modes of zero order, and u j (r, x) are the Gaussian modes. Here, the waist of the incident Gaussian beam is set at the input surface of OSL, then the two independent polarization components of optical fields have the same waist radius, i.e. W01 = W02 = W0 . Therefore, u j (r, x) (j = 1, 2) read [70–72] u j (r, x) = √

2 r2 1 exp {− 2 } , π W0 [1 − i(2x/b j )] W0 [1 − i(2x/b j )]

(5.141)

where b j = k j W02 are the confocal parameters and b 2 = n2 /n1 b 1 , with n1 and n2 being the unperturbed refractive indices of two wave components of different polarizations.

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Let D1 (x) = √ω/n1 A1 (x)a ,

D2 (x) = √ω/n2 A2 (x)b ,

E(0) = E0 c ,

(5.142)

where a, b, and c are three unit vectors and a ⋅ b = 0; A1 (x) and A2 (x) are the normalized amplitudes of the two wave components. Starting from Maxwell’s equations and taking the EO second-order nonlinearity as a perturbation, we derive the wavecoupling equations under the slow varying amplitude approximation and the paraxial approximation, as follows: dA1 (x) 1 = −id1 A2 (x)f(x) exp (iΔkx) − id2 f(x)A1 (x) dx 1 + i(x/b 1 )(1 − n1 /n2 ) (5.143) dA2 (x) 1 = −id3 A1 (x)f(x) exp (−iΔkx) − id4 f(x)A2 (x) , dx 1 − i(x/b 1 )(1 − n1 /n2 ) where f(x) = 1 and −1 correspond to the positive and negative domains of OSL, respeck0 reff2 E0 , tively; Δk = k 1 −k 2 is the two vector mismatch; and d1 = 2√nk0 n reff1 E0 , d2 = 2n 1 1 2

k0 reff3 E0 , where reffi (i = 1, 2, 3) are the same as those in d3 = 2√nk0 n reff1 E0 , d4 = 2n 2 1 2 Equation (5.17). To compensate for the wave vector mismatch perfectly, we consider such a structure of OSL, f(x) = sgn(Re{[1 + i(x/b 1 )(1 − n1 /n2 )]−1 exp (iΔkx)}), where Re represents the real part; sgn is the sign function, sgn(x) = 1 when x ≥ 0, sgn(x) = −1 when x < 0. Under the condition of QPM, Equations (5.143) can be simplified as

dA1 (x) 1 − id2 f(x)A1 (x) = −id1 A2 (x)f1 2 dx 1 + (x/b 1 ) (1 − n1 /n2 )2 dA2 (x) 1 − id4 f(x)A2 (x) , = −id3 A1 (x)f1 dx 1 + (x/b 1 )2 (1 − n1 /n2 )2

(5.144)

where f1 (G) = ∫0L f(x) exp [iGx + φ(x)]dx/L is the Fourier coefficient for chosen structure, and L is the length of OSL; φ(x) = arg{[1 + i(x/b 1 )(1 − n1 /n2 )]−1 }. For plane-wave interactions, φ(x) becomes a constant, and the OSL will degenerate to a periodic one. Equations (5.144) are those describing the linear EO effect for QPM of focused Gaussian beam in an OSL, which are different from the coupled equations of the linear EO effect for QPM of plane wave [19] (also see Section 5.5.1). And the main difference is that there is now a coefficient of [1 ± i(x/b 1 )(1 − n1 /n2 )]−1 for each item on the right-hand side of Equation (5.144). The factor [1 ± i(x/b 1 )(1 − n1 /n2 )]−1 depends on x, which therefore causes a continuously phase variation, so-called Gouy phase shift. When x ≪ b1 , Equations (5.144) reduce to the familiar wave-coupling equations under the plane-wave approximation (see Section 5.5.1). Compared with the EO effect of QPM for the plane wave, a significant character of present case is that, due to the EO effect, the polarization of output beam will form a transversely inhomogeneous distribution in space. Generally, the description of the polarization state (ellipse) requires two parameters: azimuth angle ψ ∈ [−90°, 90°]

358 | Weilong She, Guoliang Zheng and ellipticity e ∈ [−1, 1] (the positive and negative correspond to right- and lefthanded polarizations, respectively). ψ and e can be obtained by the relations [91] tan(2ψ) =

2Re(X) 1 − |X|2

,

sin(2arctan e) =

2Im(X) 1 + |X|2

,

(5.145)

where X = E2 (r, x)/E1 (r, x) = √ω/n2 A2 (x)u 2 (r, x)/[√ω/n1 A1 (x)u 1 (r, x)]. For a Gaussian beam, the polarization of output beam does not depend on the coordinate azimuthal angle in a cylindrical coordinate system [72–74]. However, the polarization of output beam varies with propagation distances. And more interesting is that at a fixed x, the output beam will form a spatially inhomogeneous polarization, changing continuously in transverse section of beam. It is obviously different from the EO effect of plane wave, for which the output beam has a polarization with homogeneous distribution transversely in space. The reason is that two independent polarization components of Gaussian beam have different confocal parameters, i.e. b 1 ≠ b 2 , which result in a phase difference between two independent wave components in OSL. The following numerical results will illustrate this further. As an example of calculation, the wavelength λ, the temperature T, the length of the OSL L and the beam waist W0 are set to be 632.8 nm, 298 K, 2.5 cm, and 15 μm, respectively; the nonvanishing EO coefficients of lithium niobate used are r22 = 3.4 and r51 = 3.4 (in 10−12 m/V) [14]; the Sellmeier equations for lithium niobate are from Ref. [53]. Please note that all the parameters used here satisfy the paraxial approximation. For an extraordinary incident beam with initial condition A1 (0) = 0, A2 (0) = 1, we obtain the numerical results shown in Figure 5.27, which demonstrate the spatial distribution of polarization of output beam for different applied electric field E0 . One sees from Figure 5.27 (a) and (e) that, when E0 = 0 or 64 V/mm, the output beam is linearly polarized. This is because that when E0 = 0, it has no EO effect and the output beam is an extraordinary one; and when E0 = 64 V/mm, |A2 (L)|2 = 1 the output beam has become an ordinary one fully. More interesting is that, when E0 takes other values, for example, E0 = 15, 30, or 45 V/mm, the polarization of output beam becomes spatially inhomogeneous. To further identify the relative change of polarization for output beam, we plot the dependence of ψ and e on r at different applied electric filed, as shown in Figure 5.28. One sees from Figure 5.28 that, when E0 = 15 V/mm [corresponding to Figure 5.27 (b)], a ψ varying from −0.10° to −8.63° (Δψ = 8.53°) and an e from −0.40 to −0.34 to (Δe = 0.06) with r increasing from 0 to 150 μm. When E0 = 45 V/mm [corresponding to Figure 5.27 (d)], ψ varies from 0.14° to 13.82° (Δψ = 13.68°) and e from −0.48 to −0.46 (Δe = 0.02) with r. Though Δe in Figure 5.27 (b) and (d) is both small, Δψ is both great; thus the spatial inhomogeneity of polarization of output beam is evident. Especially E0 = 30 V/mm [corresponding to Figure 5.27 (c)], ψ varies from −1.60° to 34.23° (Δψ = 32.63°) and e from −0.93 to −0.66 (Δe = 0.27) with r. Δe and Δψ are both great, which means that the spatial inhomogeneity of polarization of output beam is very evident.

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Fig. 5.27. The spatial distribution of polarization of output beam for different applied electric field E 0 and with λ = 632.8 nm, T = 298 K, L = 2.5 cm, W0 = 15 μm fixed. (a) E 0 = 0; (b) E 0 = 15 V/mm; (c) E 0 = 30 V/mm; (d) E 0 = 45 V/mm; (e) E 0 = 64 V/mm.

Fig. 5.28. Dependence of ψ and e on r for different applied electric field E 0 and with λ = 632.8 nm, T = 298 K, L = 2.5 cm, W0 = 15 μm fixed. Solid, dashed, short dashed lines, respectively, represent at E 0 = 15, 30, 45 V/mm. (a) ψ on r; (b) e on r.

We find that the transverse spatial inhomogeneity of polarization of output beam is not only controlled by the applied electric field E0 , but also affected by the confocal parameters b 1 and b 2 . To demonstrate this, we fix E0 at 30 V/mm, and change b 1 (b 2 = n2 /n1 b 1 ). The numerical results are shown in Figure 5.29. One sees from Figure 5.29 (a) that, when b 1 = 5.11 mm (W0 = 15μm), the spatial inhomogeneity of polarization of output beam is very evident. With the increase of b 1 , however, Δψ or Δe or both of them will become smaller. This means that the transverse polarization of output beam will vary gradually from spatial inhomogeneity to spatial homogeneity.

360 | Weilong She, Guoliang Zheng

Fig. 5.29. The spatial distribution of polarization of output beam for different b 1 and with λ = 632.8 nm, T = 298 K, L = 2.5 cm, E 0 = 30 V/mm fixed. (a) b 1 = 5.11 mm; (b) b 1 = 4 × 5.11 mm; (c) b 1 = 16 × 5.11 mm; (d) b 1 = 64 × 5.11 mm.

To further identify the relative change of polarization for output beam, we also plot the dependence of ψ and e on r at different b 1 , as shown in Figure 5.30. One sees from Figure 5.30 that, when b 1 = 4× 5.11 mm [corresponding to Figure 5.29 (b)], ψ varies from −6.62° to −20.40° (Δψ = 13.78°) and e from 0.94 to 0.68 (Δe = 0.26) with r, where Δψ is much smaller than that at b 1 = 5.11 mm. And when b 1 = 16 × 5.11 mm [corresponding to Figure 5.29 (c)], ψ varies from −6.25° to 4.83° (Δψ = 11.08° ) and e from −0.92 to −0.84 (Δe = 0.08) with r. Both of Δψ and Δe are much smaller than those at b1 = 5.11 mm. Further b 1 = 64×5.11 mm [corresponding to Figure 5.29 (d)], ψ varies from −1.91° to 0.87° (Δψ = 2.78°), and e from −0.916 to −0.913 (Δe = 0.003) with r. Compared with b1 = 5.11 mm, the Δψ here is very small and Δe is almost unchanged. The polarization is almost spatially homogeneous. We also investigate the effect of the confocal parameter b 1 on the half-wave voltage V π = E󸀠0 d, where E󸀠0 is the applied electric field for turning an extraordinary light into an ordinary fully; d is the thickness of OSL along the direction of applied electric field. The dependence of the applied electric field E󸀠0 on the confocal parameter b 1 for the output intensity of o-ray reaching at its maximum value is shown in Figure 5.31, from which one sees that E󸀠0 (or V π ) continually decreases as b 1 increases from 2.3 to 21.14 mm. And when b 1 ≥ 21.14 mm, E󸀠0 (or V π ) almost keeps a constant because

5 Wave coupling theory and its applications of linear electro-optic (EO) effect |

361

Fig. 5.30. Dependence of ψ and e on r for different b 1 and with λ = 632.8 nm, T = 298 K, L = 2.5 cm, E 0 = 30 V/mm fixed. Solid, thinner solid, dashed, short dashed lines, respectively, represent at b 1 = 5.11 mm, 4 × 5.11 mm, 16 × 5.11 mm, and 64 × 5.11 mm. (a) ψ on r; (b) e on r.

Fig. 5.31. Dependence of the applied electric field E 󸀠0 on the confocal parameter b 1 when the output intensity of o-ray obtains its maximum value for λ = 632.8 nm, T = 298 K, L = 2.5 cm.

(x/b 1 )2 (1− n1 /n2 )2 in Equation (5.144) is close to zero. For this case, it is easy to obtain V π = (π√n1 n2 d)/(k 0 reff1 f1 L) according to Ref. [14,19]. The output intensity of o-ray, |A1 (L)|2 as a function of b 1 and E0 is shown in Figure 5.32, which exhibits a recurrence of o-ray to its maximum intensity with E0 for a fixed b 1 . It can be understood as follows: according to Equation (5.144), the ratio A2 (L)/A1 (L) determines the polarization of light field. Figure 5.32 shows that A1 (L) and A2 (L) have some periodicity vs E0

362 | Weilong She, Guoliang Zheng

Fig. 5.32. The output intensity of o-ray |A 1 (L)|2 as a function of the confocal parameter b 1 and the applied electric field E 0 , for λ = 632.8 nm, T = 298 K, L = 2.5 cm.

for a fixed b 1 , which means that the dependence of the polarization on E0 has some periodicity. Besides the wave coupling theories of linear EO effect mentioned in 5.2–5.5, She and Zhong have generalized the wave coupling theory of linear EO effect for the ultrashort laser pulse. Due to the limitation of space, the detailed treatment of this topic is not shown here. One can turn to Ref. [16] for further understanding.

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Index A absorbent 309, 319 absorption edge 5, 16, 18 B band edge photorefraction 12, 13 bipolarons 4, 10, 14 breather 237, 238, 251, 267, 268, 278, 286, 287, 289 bright soliton 232, 258, 278 C carbon nanotube 40, 41, 43, 44, 56, 65, 78, 81 carbon nanotubes 39–45, 47, 50, 56, 64, 87 cascading 107, 153, 154, 169–175, 177, 179, 184 concentration threshold 8, 9, 13, 15, 18 D dark soliton 230, 232, 255, 257, 258, 270, 295 defect structures 2, 16 diffraction efficiency 4, 5, 7, 10, 13, 14 dimensionless transformation 245, 247, 248, 254–256, 258, 270, 290 discrete soliton 31 discrete solitons 21, 28–32, 35–38 E Einstein oscillator 17, 18 electric dipole 69, 75, 83, 84 electro-optical effect 107, 108, 123, 138–140, 173, 174 extrinsic chirality 83, 84 F fanning noise 11, 12 filter 107, 113–119, 121, 122, 124–127, 129–133, 137, 182, 184 filters 178 G gap soliton 32, 33, 36 gap solitons 21, 32–34, 36, 38 Gaussian 355–358 general nonlocality 270 graphene 39, 41, 43, 44, 53, 55–58, 65, 87 graphene oxide 41, 43, 53–55, 58

H holographic recording 10, 15, 16 hybrid material 43, 44, 53, 59, 61, 64, 65 hybrid materials 39, 44, 46, 50, 53–56, 58, 61, 64 I incoherent soliton 24, 26, 27 incoherent solitons 20, 21, 24–26, 38 interband photorefractive effect 3 L lattice solitons 21, 35, 36 LCP 81 lead glass 234, 278, 292, 293, 295 left-circular polarized (LCP) 81 light bullet 236, 237 light localization 1, 87 light-induced absorption 14, 15, 18 light-induced charge carrier 2–4, 6, 7, 10, 12, 16, 20 light-induced refraction index change 4 light-induced refractive index change 2, 5 linear chirped-periodically 338 linear EO effect 307–310, 313, 314, 319, 320, 323, 330–335, 338, 344, 345, 347, 355–357, 362 lithium niobate 1, 4 local field enhancement 80, 81 long-range interaction 272, 273, 276–278, 295 M magnetic dipole 73, 83, 84 magnetic dipoles 84 metamaterials 2, 66, 74, 78, 79, 81, 84, 87 multiwalled carbon nanotubes 40, 42 N nematic liquid crystal 231, 236, 261, 278, 279, 296 nematicon 231–233, 278, 279, 285, 287 NLC 278–287, 289–291, 296 NLO 105, 107, 169, 171, 175 NLSE 228, 231, 233, 236, 238, 242, 243, 245, 250, 251, 253, 255, 256, 258, 259, 274

368 | Index NNLSE 236, 242–244, 248, 250, 252–254, 262, 269, 270, 272–274, 278, 281, 282, 285, 287 NOA 86 nonlinear absorption 43–45, 48, 49, 51, 55–57, 60–62, 64, 66, 72, 77, 79–81 nonlinear bleaching 81 nonlinear characteristic length 236, 239, 261, 281, 283–285, 287, 293, 294, 296 nonlinear optics 105, 169 nonlinear phase shift 169, 171, 174, 177–179 nonlinear phase shifts 169, 173–175, 178, 179, 184 nonlinear refraction 48, 55, 62–64 nonlinear refractive coefficient 284 nonlinear refractive index 227, 231, 233, 234, 238, 239, 241, 247, 259, 260, 264, 282–285, 289, 292–296 nonlinear refractive index coefficient 228, 281, 282, 284 nonlinear scattering 42, 44–46, 49–57, 59, 61, 62, 64 nonlinear Schrödinger equation 228, 254, 257 nonliner optical activity (NOA) 82 nonlocal nonlinear Schrödinger equation 242 nonlocality 227, 233, 238, 239, 241, 253, 260–262, 270–274, 279, 283, 285–288, 290–292, 295, 296 O optical damage resistance 5, 8, 9, 18, 20 optical limiting 42, 43, 46, 47, 49–53, 55, 61, 63–65, 87 optical nonlinearity 39, 43, 53, 58, 61, 64, 65, 73 P PC 107, 171, 173–175, 184 periodically poled lithium niobate crystal 105 photonic band gap structure 33 photonic band gap structures 22, 30, 32 photonic lattice 29, 31, 35 photonic lattices 22, 25, 28–32, 34, 35, 38 photorefractive effect 1–5, 10, 15, 16 photorefractive grating 8, 13 photorefractive gratings 4 photorefractive nonlinearity 24, 29 photovoltaic effect 2, 4, 10–12 polarization-coupling 107, 108, 119, 124, 140, 169, 171, 175, 177, 184 porphyrin 39, 44, 46–57, 64, 65

porphyrins 47, 50, 52, 53, 65 propagation equation 244 Q QPM 309, 330–335, 338, 339, 344, 345, 347, 351, 355–357 quasi-phase-matched 347 R RCP 81 response function 239–241, 245–247, 253, 261, 262, 269–273, 281–287, 289, 290, 293, 294, 296, 297 reverse saturable absorption 43 right-circular polarized (RCP) 81 S second-harmonic generation 105, 169 self-defocusing 228, 234, 235, 241, 247, 249, 250, 257, 260, 270, 295, 296 self-focusing 228, 233–237, 241, 247, 249, 254, 255, 257–260, 265, 266, 269, 271, 272, 295, 296 self-trapping 228, 233, 235, 272, 278, 292 SHG 105, 166, 169–171, 173, 175, 179, 184 short-range interaction 272–274, 276–278 single-walled carbon nanotube 42 single-walled carbon nanotubes 40 small polarons 14, 15 Snyder–Mitchell model 236, 243, 262–269, 275, 285, 287, 288 soliton 21, 23, 25, 27, 28 soliton array 27 soliton arrays 28 solitons 1, 2, 20–25, 28, 31 spatial optical soliton 227, 228, 231–236, 244, 247, 255, 256, 258, 260, 267, 268, 270, 271, 274, 277, 278, 292, 294 spatial soliton 23 spatial solitons 3, 20–25, 27, 28, 87 spatiotemporal optical soliton 227, 228 strong nonlocality 236, 274, 275, 278, 284, 285, 290 surface plasmon polariton 66 T temporal optical soliton 227–236, 244, 256, 257 thermal nonlinearity 234, 238, 292, 295 two-photon absorption 51

Index | 369

two-wave coupling gain 4, 5, 9 two-wave mixing 11 two-wave mixing gain 5–7, 13, 14

U ultraviolet band edge photorefractive effect 12 ultraviolet photorefraction 2 united effect 344, 347, 349

V variational method 251–253 W wave coupling theory 308, 309, 316, 318, 319, 330, 332, 335, 344, 345, 355, 362 weak nonlocality 269 weak-light NLO 1, 2, 88 weak-light nonlinear optics 1

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