A First Course in Laboratory Optics 1108488854, 9781108488853

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A First Course in Laboratory Optics

An optics experiment is the product of intricate planning and imagination, best learned through practice. Bringing forth the creative side of experimental physics through optics, this book introduces its readers to the fundamentals of optical design through eight key experiments. The book includes several topics to support readers preparing to enter industrial or academic research laboratories. Optical sources, model testing and fitting, noise, geometric optics, optical processes such as diffraction, interference, polarization, and optical cavities are just some of the key topics included. Coding tutorials are provided in the book and online to further develop readers’ experience with design and experimental analysis. This guide is an invaluable introduction to the creative and explorative world of laboratory optics. Andri M. Gretarsson is Associate Professor at Embry-Riddle Aeronautical University. A member of the LIGO Collaboration for the detection of gravitational waves, he helped commission the initial detectors and researched the effect of optical coatings on a range of current and future detectors. He has 15 years’ experience teaching laboratory optics.

A First Course in Laboratory Optics ANDRI M. GRETARSSON Embry-Riddle Aeronautical University

University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 314–321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi – 110025, India 79 Anson Road, #06–04/06, Singapore 079906 Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781108488853 DOI: 10.1017/9781108772334 c Andri M. Gretarsson 2021  This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2021 Printed in the United Kingdom by TJ Books Limited, Padstow Cornwall A catalogue record for this publication is available from the British Library. ISBN 978-1-108-48885-3 Hardback Additional resources for this publication at www.cambridge.org/gretarsson Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet web sites referred to in this publication and does not guarantee that any content on such web sites is, or will remain, accurate or appropriate.

Contents

Preface

page ix

1 Light Waves

1.1 1.2 1.3 1.4

Maxwell’s Equations Huygens’ Principle The Paraxial Approximation Coherence

2 Components and Methods

2.1

2.2

2.3

2.4

Safety 2.1.1 Laser Safety 2.1.2 Good Safety Habits 2.1.3 Non-Beam Hazards Optical Components 2.2.1 Light Sources 2.2.2 Lenses 2.2.3 Mirrors, Reflective Prisms, and Non-Polarizing Beamsplitters 2.2.4 Polarizers, Wave Plates, and Optical Isolators 2.2.5 Electro-Optic and Acousto-Optic Modulators 2.2.6 Cameras and Photodetectors 2.2.7 Optomechanics Measurement Error 2.3.1 Propagating Uncertainties 2.3.2 Testing Models against Data Experiment: Components of an Optics Chain

3 Geometric Optics

3.1 3.2 3.3

3.4 v

3.5

The Geometric Optics Approximation Refraction Basic Imaging 3.3.1 Principal Rays 3.3.2 A Simple Model of the Eye 3.3.3 Magnifying Devices Paraxial Geometric Optics 3.4.1 Effect of an Optic on Rays Emanating from a Point Experiment: Imaging Optics

1 1 2 6 9 13 13 13 15 16 16 17 21 22 24 25 26 30 32 33 35 44 49 49 49 52 54 55 55 58 62 66

Contents

vi

4 Physical Optics

4.1

4.2 4.3

Paraxial Physical Optics 4.1.1 Self-Similar Beams Propagating in Free Space: TEMmn Modes 4.1.2 Propagating a Beam through an ABCD Optic in the Paraxial Approximation: A Fourier Transform Gratings and Spectrometers Experiment: Diffraction and Spectroscopy

5 Interferometry

5.1

5.2

5.3

Interferometer Types 5.1.1 Two-Beam Interferometers 5.1.2 Multiple-Beam Interferometers Selected Applications of Interferometers 5.2.1 Fourier-Transform Spectroscopy 5.2.2 Gravitational Wave Astronomy Experiment: The Michelson Interferometer

6 Lasers

6.1

6.2 6.3

How Lasers Work 6.1.1 Stimulated Emission 6.1.2 The Two-level Model 6.1.3 A Four-level Laser: The HeNe 6.1.4 Rate Equations 6.1.5 Lasing Threshold Etendue and Radiance Experiment: Helium–Neon Laser

7 Optical Cavities

7.1 7.2 7.3

7.4

Use of Optical Cavities Plane-Wave Cavity Resonant Modes of a Cavity 7.3.1 Cavity Stability 7.3.2 Spatial Modes Experiment: Resonant Optical Cavity

8 Polarization

8.1 8.2 8.3 8.4 8.5

Polarized Light Fresnel Equations Jones Vectors 8.3.1 Linear, Circular, and Elliptical Polarization Jones Matrixes 8.4.1 Optical Isolators Experiment: Investigation of Polarized Light

73 73 74 78 81 85 91 91 91 96 99 99 103 110 115 115 115 116 121 122 124 124 129 133 133 133 138 139 142 146 152 152 153 156 158 158 160 164

Contents

vii

9 Optical Noise

9.1 9.2 9.3

9.4

Characterizing Noise 9.1.1 Power Spectral Density Quantum Fluctuations in Optical Power Measurements Technical Sources of Optical Noise 9.3.1 Parasitic Interferometers and Scattering 9.3.2 Intrinsic, Noise-generating, Laser Dynamics Experiment: Shot Noise

Appendix A

A.1 A.2 A.3 A.4 A.5

Analyzing and Displaying Data with Matlab and Python Matlab versus Python Basic Coding Loops and Branching Functions Putting It Together

Appendix B

B.1 B.2 B.3 B.4

Computer Code

Image Processing Functions Miscellaneous Functions Curve Fitting Fourier Propagation

References Index

168 168 168 170 175 176 178 183 187 187 188 191 193 194 200 200 202 204 207 211 213

Preface

The aim of this book is to prepare you for work in a field requiring the use of laboratory optics. Through the experiments, you will build and investigate several of the most important fundamental optical systems. While each experiment is preceded by several pages of theory, this book is not intended to be a comprehensive textbook about optics. There are already many excellent such texts. The idea here is to learn optics by doing optics with theory as a backdrop. The theory is presented “purposefully” in order to support the experiments and covers the core elements of each topic, enough for you to really understand what’s going on. The idea is that after reading this book and doing the experiments, you’ll be able to walk into an industrial or academic research lab and at least know how to get started! The book has one experiment per chapter (except for the introductory first chapter). Each experiment should take about eight hours to complete. You’re given the goals of each experiment and just enough information to get started on each part. After that it’s up to you to figure out how to proceed based on the theory and other information you may gather. I’ve tried hard to avoid giving you “cookbook style” experiments so that you have the opportunity to do your own creative experimental design work. As a result, you’ll need to give yourself time to investigate different approaches to a problem. Be careful not to get too deep into an experiment before checking whether your approach is yielding sensible data. Before starting an experiment, make sure you study the relevant chapter(s), do some of the problems and make a preliminary design/plan for each section of the experiment. If you can, visit the lab during the design phase and look at the equipment you’ll be using. The experiments are all fairly open-ended. Extending them to include follow-on investigations is always possible and may be expected. A few ideas for follow-on work are included at the end of each experiment but you will need to tailor your investigations to the equipment available/obtainable. In addition to the equipment listed in the experiments, you will need access to an optics bench (either a sizable optical breadboard or an optical table), a set of mirrors and lenses, and a selection of mounting hardware for these and other optical components. Most of the experiments rely on using a computer with Matlab, Python, and so on, for analysis tasks. A small selection of Matlab and Python 3 code for planning and analysis tasks is provided in the appendixes. The Matlab code has also been tested in Octave. While the code can be used verbatim, it’s primarily intended as a framework upon which to expand. The Matlab and Python versions of the code are available to download at https://github.com/CambridgeUniversityPress/FirstCourseLaboratoryOptics. Additional support materials are also placed there and errata as they arise. It’s worth mentioning the word “intensity.” In much of physics, intensity is synonymous with energy flux density. However, in the subfield of optics known as radiometry, ix

x

Preface

“intensity” is sometimes used as a shorthand for “radiant intensity,” which has a different meaning. This causes confusion and I try to avoid using the unqualified term “intensity” altogether. I use the term irradiance for energy flux density associated with light, that is, “power per unit area perpendicular to the direction of the light’s travel.” I generally use the letter I for irradiance.

Instructors I recommend using the book as part of a laboratory-based course in optics with co-requisite lectures. Advanced students will be able to do the labs without formal lectures and the relevant theory sections can be assigned for self-study. In order to reduce the amount of equipment needed for a class, it is helpful to run three or four experiments simultaneously. That way, only two or three setups are required for each experiment. I recommend assigning the relevant theory sections and several exercises as preparation for each experiment. The exercises vary widely in difficulty and some rely on moderate coding ability. Although the number of exercises provided is modest, I hope it is possible to find a mixture that suits the student level and available time. The text is aimed at upper-level undergraduates and beginning graduate students. The book assumes preparation in vector calculus and at least one physics laboratory course. Some prior programming experience is helpful but an introduction to Matlab and Python is also included in Appendix A. Scientists in other fields embarking on experiments requiring a significant optical component should also find the text useful. By and large, the theory is covered in a survey mode. However, the propagation of laser beams through optics chains is so central to experiments in optics that it is covered more thoroughly. Specialty topics like nonlinear optics, laser cooling, advanced imaging, holography, scattering theory, and so on, are not addressed. The notation and theory development largely follows that of popular optics and laser books, such as Optics (Hecht 2017); Principles of Physical Optics (Bennett 2008); Introduction to Modern Optics (Fowles 1989); Principles of Lasers (Svelto 2010); and Introduction to Optics (Pedrotti 2007). I’ve found all of these books to be good supporting texts for the laboratory-based optics class that I teach. As for more advanced texts, I recommend Lasers (Siegman 1986); Photonics: Optical Electronics in Modern Communications (Yariv 2007); and of course Born and Wolf’s Principles of Optics (Born and Wolf 2019). There are two other books on laboratory optics I recommend as complementary reading to this text. They are An Introduction to Practical Laboratory Optics (James 2014), and Laboratory Optics: A Practical Guide to Working in an Optics Lab (Beyersdorf 2014), which is a multimedia book.

Acknowledgments This book was motivated by a decade of teaching laboratory optics to third- and fourthyear undergraduate students. I’m grateful to the students who came through my course and tried hard to get their experiments to work even when the instructions were impossibly

xi

Preface

vague. Thanks to them, I have settled on what I hope is “just the right amount” of information to give students who are embarking on an experiment. I owe thanks to the students who pointed out significant manuscript errors and/or gave constructive feedback about the course and/or manuscript content. They include Harsh Menon, Billy Nollet, Aaron and Amy Rose, Marina Koepke, Calley Tinsman, Ashley Elliot, and Brennan Moore. I have surely forgotten some contributions, for which I apologize. Thanks to Professor Darrel Smith, who as the department chair encouraged me to create a serious laboratory optics course for upper-level undergraduates and allowed me a free hand in its design. Thanks to my mother, Anna Garner, for carefully proofreading the manuscript. Special thanks to my wife, Ellie Gretarsson, for testing experiments, reading the manuscript, and giving invaluable feedback at all stages.

1

Light Waves 1.1 Maxwell’s Equations The field of optics describes the behavior of light as it propagates through space and materials. To understand the behavior of light, we start with the fundamental classical physics model describing it: Maxwell’s equations of electrodynamics. Maxwell’s equations show that the electric and magnetic fields can travel as waves. In a source-free region, Maxwell’s equations in linear media are1  · E = 0, ∇  ·B  = 0, ∇   × E = − ∂ B , ∇ ∂t   ×B  = µ ∂E , ∇ ∂t

(1.1) (1.2) (1.3) (1.4)

 is the magnetic field, µ is the permeability of the medium, where E is the electric field, B and  is the permittivity of the medium. If we take the curl of both sides of Eq. (1.3), apply  ×∇  × E = ∇(  ∇  · E)  − ∇2 E to the left-hand side and exchange the order the vector identity ∇ of the time derivative and the curl on the right-hand side, we get    ∇  · E)  − ∇2 E = − ∂(∇ × B) . ∇( ∂t

(1.5)

Then substitute from Eqs. (1.1) and (1.4) to get ∇2 E = µ

∂2 E . ∂t2

(1.6)

√ This is the wave equation in three dimensions where the wave speed is v = 1/ µ. Taking the curl of Eq. (1.4) and performing similar algebra shows that the magnetic field also satisfies the wave equation with the same wave speed. Thus Maxwell’s equations allow for √ electromagnetic waves. In vacuum, the speed is v = 1/ µ0 0 ≡ c, the speed of light in vacuum. Light is indeed an electromagnetic wave. We now look for solutions to Eq. (1.6) and its magnetic field counterpart. We actually only need to solve for the electric field because the magnetic field can always be found  = 1 kˆ × E,  where kˆ is the direction of travel. (See Exercise 8.1.) We’ll assume from B c 1

1

A source-free region has no net free charge and no net free current. Linear media include vacuum and dielectrics: air, glass, and so on. For an excellent introduction to electrodynamics, see Griffiths (2017).

Light Waves

2

a single linear polarization and single-frequency (monochromatic) electromagnetic wave. The electric field should then be in the form    r, t) = nˆ E(r) cosωt + φ(r) = Re nˆ E( ˜ r) ei(ωt) , E( (1.7) ˜ r) = E(r)e iφ . Generalizing to the complex plane, we look for solutions to Eq. (1.6) where E( of the form  r, t) = nˆ E( ˜ r)eiωt , (1.8) E( with the anticipation that at the end we will take the real part to get the actual physical field. Substituting Eq. (1.8) into Eq. (1.6) allows us to eliminate nˆ , reducing it to a scalar equation. Also, the time derivatives can be performed explicitly bringing down an ω2 from the eiωt . This results in a second-order partial differential equation known as the Helmholtz equation ˜ r) = 0, (1.9) (∇2 + k2 ) E( where k = ω/v is the wave number. If we can solve Eq. (1.9) for the complex scalar ˜ r), then we can get the actual physical electric field by multiplying our solution by field E( iωt nˆ e and taking the real part. Since Eq. (1.9) is a second-order partial differential equation in three spatial coordinates (e.g. x, y, z), we will need to specify appropriate boundary conditions for the field on some surface in order to obtain explicit solutions.

1.2 Huygens’ Principle In many cases of interest in optics, Eq. (1.9) is solved to a good approximation by Huygens’ integral. The field is assumed to be known on a “source plane” S 1 perpendicular to the z-axis and is only nonzero in some finite region of that plane. The values of the field on S 1 serve as a boundary condition for solving Eq. (1.9). The solution is given by Huygens’ integral for the complex scalar field at any desired point x, y, z. ˜ y, z) = i E(x, λ

 S1

−ik r

˜  , y , z ) cos φ e E(x

r

dS  .

(1.10)

The integration over the source plane S 1 is performed using the integration variables x , y , z . The vector r joins points in the source plane S 1 with the point (x, y, z) at which we are calculating the field. The angle between r and the z-axis is φ (see Figure 1.1). The solution represented by Huygens’ integral is satisfying because it encapsulates an intuitive understanding of how light waves behave that was understood long before the formal mathematics was fully worked out. The intuitive description of Eq. (1.10) is known as Huygens’ principle, due to Christiaan Huygens (1629–1695), a Dutch mathematician and scientist. Under Huygens’ principle, every point in the source is considered to be emitting light with spherical wavefronts propagating outward – the so-called Huygens’ wavelets. These wavefronts are represented by

3

1.2 Huygens’ Principle

Figure 1.1

The electric field in the source plane S1 is propagated to the field point in plane S. The source plane is located at z = z1 and the field plane is at z. The circle on the source plane indicates a typical source point involved in the integral. The complex scalar field ˜ y, z1 ). Similarly, the circle on the field plane indicates a typical field at this point is E(x, ˜ y, z). point with complex scalar field E(x,

t

−ik r

the factor cos φ e r . They are emitted preferentially in the direction perpendicular to the source plane due to the presence of cos φ. The constant λi out-front contributes 90◦ of phase and the λ in the denominator serves to keep the units the same on both sides of the equa˜ y, z1 ) sets the relative amplitudes tion. The complex scalar field in the source plane E(x, ˜ y, z) is then simply the linear and phases of these tiny spherical emitters. The field E(x, superposition of all the spherical wavefronts emitted from the source.

Example 1.1 Single-Slit Diffraction A typical use of the Huygens’ integral solution 1.10 is to find the diffraction pattern from a small aperture of some specific shape. Consider the diffraction of a plane wave from a rectangular aperture of width 2b and height 2d viewed on a screen at a distance L downstream from the slit. The screen distance is much larger than either dimension of the aperture b, d