Living waveguides

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Living waveguides. T. Tulaikova, A. Michtchenko *, E.Yurkov. Institute for Problems in Mechanics Russian Academy of Sciences, 101 prosp. Vernadskogo, 119526 Moscow, Russia. Fax: 7-095-938-20-48, e-mail: [email protected] *Institute Politechnico National, ESIME-SEPI, Zacatenco, C.P.07738, Mexico, D.F. ABSTRACT. New waveguide effects are considered on the basis on various variants for self-focusing of the living cells of aquatic plant by laser beam irradiation. It is studying the process of forming of the ‘living multy-mode fiber-optic’ into the road of laser's beam in water with motion of green living cells. Analysis gives that the wide set of different profiles of the refractive index in this ‘living optical fiber’ could be easy produced depending the frequency of applied lasers. Optical waveguide approach gives high sensitivity in living layer readout and control, this waveguides could be apply to environmental processing or for a cell’s studying, also different medical application could be realized according described mechanism. Keywords: fiber optics, waveguide, environment, sensors, aquatic plant.

1. INTRODUCTION. It was found that an additional living waveguide layer is forming itself according to the mechanism of the diffusion motion of the living particles of single-cell aquatic plant (Chlorella) near the quartz sample due to its inside irradiation by laser or surface heating [1]. Waveguide effect based on the fact that the refractive index of living cells is larger then water one. The same effect was discovered earlier due to the semiconductor’s chemical etching [2]. The complete analysis was presented in this work for optical fiber with additional living-cell’s cladding as the girdle one of the quartz core. The areas of operation of this liquid waveguide could include two large fields, it is appeared near solid surface by increasing of the concentration of admixture of liquid. At first, it is medical applications for the dissolution of the medicaments in a blood by changed doses with the instantaneous control of this process in the living cells with high efficiency. Also second is a pollution control in environmental water. Simple remote sensors are investigated on the basis on various variants of amalgamation of optical fibers with living sensitive element as they applied to environmental processes. Due to slow diffusion motion but fast processes into the cells of aquatic plant this living waveguides are stable during jointly propagating of nano-second laser's pulses or more faster ones, fast readout of changes in living cells in their natural environment could be received by modern lasers. The living aquatic plant serves as the natural sensitive element of sensor of water pollution. Optical waveguide approach provides of high sensitivity in living layer readout and control by involving of the set of common methods from the standard fiber optic. Different variants and compositions of artificial and additional living elements were considered and developed earlier as most convenient approaches to study of the environment. Additional ‘living’ fiber cladding could be formed by laser-beam irradiation according to the mechanism of the diffusion movement of the particles or cells to the fiber quartz side-surface due to its heating, Fig.1. This additional waveguide layer is stable to shake, because waveguide connection is very strong. Analogous diffusion phenomena is waiting for the drag solution. Also living aquatic plants are very promising as the sensitive elements of this composite fiber-optic chemical sensor with additional cell’s cladding, Fig.2. The operation of this waveguide could be controlled with reverse connection by fiber long-period grating, for example [1]. Positive role of seaweed optical losses was analyzed due to the amalgamation of standard quartz optical fiber with aquatic plant as additional living fiber cladding. It was received that the imaginary field components was increased with the cladding mode’s presents into the core region, this effect provides the increase of the number of dips in the transmission spectrum in experiments with long-period grating at the quartz fibe and aquatic plant as this fiber’s cladding. The theory of optical fiber with ‘living’ formed cladding was presented earlier.

The process of forming of the living multy-mode fiber-optic by self focusing of the living cells into the axis of road of laser's beam in water will be analyzed in this paper, see Fig.3.

2. BACKGROUND. There is a large number of the wide-spread sensor’s methods which are not acceptable for water study, such as high-sensitive devices based on the laser spectroscopy, for example. The most well-known non-laboratory method for the measurement of admixture concentrations in water is the Clark electrodes. So, there are a few methods in oceanography which don’t demand the preliminary preparing of any probe. Fiber optical sensors seem to have future possibilities for the remote control of a water pollution in real time. Laser-beam could be used to create new kinds of devices for operation in a liquid medium as more simple sensor’s method. The oil spill water pollution are wide-spread in cities, rivers and seas. Mechanisms of photo synthesis of the chlorophyll were considered in many papers and monographs, for examp le in [3] the spectral region as 800-870 nm of absorption of chlorophyll molecule was discussed. For example, our multy-mode laser-beam sensor could serve as an instrument for very sensitive excitation of one of the chlorophyll centers in its natural environment in water. Laser beam in a liquid seems as living fiber optic, it has the discrete number of propagated optical mode. So, various future schemes of such unification of fiber-optic with single cells of aquatic plants must bring new kind of nature-artificial chemical sensors for water pollution monitoring. The most feature of this approach is high sensitivity of the readout such fiber mode’s selection due to small changes in behavior of cells or their feeling with small refractive index changes as a res ult. Its amalgamation with fiber optic is very promising for the creation of the sensor systems for utilization in water, due to its simplicity, convenience, insensitivity to electromagnetic and radiation fields, explosion safety and corrosion resistance. On the other hand, there is some large classes of aquatic plant, they are water inhabitants in the rivers. Some of them can live only in clean water as it proved in a work [4]. Let’s consider this aquatic plant as a sensitive element of our chemical sensor which must operate in its natural environment. The dead of this aquatic plant due to water pollution means the breaking of our sensitive element. The optical fiber has to be used [1] to get current information by transmission of the optical power through this living sensitive element to the equipment for measurement and data processing. There are different structures of the aquatic plants which have single cells or colonies. There is seaweed with threads and branches, and the multy cell as spiral as lengthened. The effect of the damage of the living cells by pollution was studied in the science literature. So, the wide spread mechanism is mechanical damage of the cells and their membranes, as well as different chemical and biological processes. Many authors believe that the aquatic plants are more stable than water in controlling pollution. On the other hand, some aquatic plants help to the natural processes of splitting and process of the oil spill which gives cleaning functions to the seaweed. Such processes could be controlled also by laser beam readout. Some fundamental processes and appropriate optical properties are studied in many papers and monographs. Another method are based on the recording of the fluorescence, they are convenient and not expensive [5]. This fluorescence process is the result of the reverse reactions of the primary delivered charges in the photosynthetic centers. A number of aquatic plants such as chlorella, as platymonas viridis porich, etc. were studied by optic methods in works mentioned above. The irradiation influence was studied for adhesion of the green cells (chirococcum minutum) to the pentone [6]. According to these results, the irradiation of the water with the spores of chirococcum minutum increases considerably the adhesion to pentone substrate from 90 units per single square (in dark) up to 4100 (with light). The spectral influence was also investigated. This fact make the background for the technology development for the seating or planting of the thick seaweed garden at the side surface of our fiber. It is as thick as optical power is large. The physical basis for the high efficiency of such devices was developed in the work [7] where the optical properties of the chlorella were studied in details. The refractive index of the seaweed is the complex value n∃ = n Re + i ⋅ nIm , there nRe =1.36 (more exactly, it’s from 1.37 up to 1.42), and nIm ≈0.1. The amplitude value of the seaweed refractive index exceeds that of water (n water=1.33), so the optical power will be focused into the seaweed waveguide layer during propagation. Such living optical way could be formed by diffusion motion of green cells to the axis of laser beam during propagation in a water. Due to slow diffusion motion but fast processes into the cells of aquatic plant this living waveguides will be stable during jointly propagating of nano-second laser's pulses or more faster ones, fast readout of changes in living cell in their natures will received by modern lasers.

3. ANALYZES OF DIFFERENT REGIMES OF TEMPERATURE DISTRIBUTIONS INTO LASER LIVING ROAD. Different kind of solid state or gas lasers could be used for irradiation of aquatic plants in their natural environment. As laser power is higher as the way of beam is longer. The fact of absorption of different laser’s beams in a water and aquatic plant’s body are determine of the length of resulting laser road in the liquids. There are known the absorption coefficients α of a water for different types of laser’s irradiation, so the better propagation is known for Nd:YAG - laser as it characterized by wavelength length λ =1.06 µm and α=10-5 sm-1 . The same high propagation appropriates to the semiconductor laser’s beam due to their wavelengths λ =0.8 - 1 µm. For Holmium and Erbium solid-state lasers the beam propagation is worse because absorption coefficient is about α∼10-4 sm-1 , also it tends to α∼10-3 sm-1 for ultraviolet- and CO2 –laser’s wavelengths. Three main time-dependence regimes of laser optical power are interesting for their analyses here. First is the timeconstant irradiation by cw lasers, for example it could be He-Ne laser or argon laser, but there are not optimal one due to visible wavelengths and its great absorption into the water. The second is the wide class of pulse lasers with different possibility of changes of parameters of modulations of output optical power, for example they could be used the most common YAG:Nd-lasers with Q-switches as LiF:F2 to produce the sets of laser pulses [8], the pulse duration τ∼1 ns and average powers about 103 -104 W. The last considered regime of irradiation is single laser’s pulses, they could be produced by YAG:Nd-lasers in free generation, f~50 Hz. So, the modulated optical power seems the best regimes of laser irradiation, as frequency more higher (f ~ 108 – 109 Hz) as better. The semiconductor lasers or miniature Nd-laser with Q-switches generation are the best optical sources for such experiments. Let’s analyze the process of forming of the living multy-mode fiber-optic into the road of laser's beam in water with green single-cells aquatic plant (Chlorella). Such living optical way will be formed by diffusion motion of green cells to the axis of laser beam during its propagation in a water, because it’s warm and irradiated central area in more preferable for aquatic plant locations. So, the green living cells are moving to laser beam’s center by diffusion motion, but their resulting concentrations profile could be different for different laser’s utilization according appropriate cw or pulse lasers. We will demonstrate below some concentration profiles of the family of ‘living fibers’ which could be produced by irradiation of aquatic plant as Chlorella with different lasers. There are only small heating by laser irradiation is possible in this our consideration. Strong heating predicts the dead of living cells or aquatic plant particles. Limit temperature for albumen is ~ 400-500 C as a biological restriction, the limit possible heating of the Chlorella particles is the same level. This low interval of heating is expedient in such experimental study, the temperatures around T∼20-400C are correspond to vitality of living particles.

Periodical heating regime is considered below, it is correspond to the modulated optical power which is the irradiation of the most modern lasers. The circular frequency of modulated optical power is -ω-. We will consider the temperature distribution in the infinite fiber with cylindrical cross section. The heat equation [9] will have the form of Bessel equation with zero order ν=0:

d 2 T ( r ) 1 dT ( r ) iω + − ⋅ T( r ) = 0 r dr χ dr 2

(1)

here T=T(r,t) is the temperature in the fiber cross section as a function of the fiber radius -r-, and time -t-, χ is the temperature coefficient. There are considered the central symmetry heating without any dependence of the azimuth angle ϕ. Also there are not any dependence on the z-direction coincident with laser beam’s direction. We consider the boundary temperature at the cylindrical surface of irradiated area in liquid remains equal to the initial temperature of liquid T0. It is appropriate to good heat conduction of surrounding water, also we suggest small interval of laser heating up to a few Calcium degrees physically. The common solution of considered heating process has the form : T(r,t)=T(r)⋅exp(iωt) .

(2)

The common solution of equation (1) will be as the combination of modified Bessel function I0 and Macdonald function K0 :

T ( r ) = PI0 ( kri 1/ 2 ) + QK0 ( kri 1 / 2 )

(3)

here k=(ω/χ)1/2 , but P and Q are the constants. Let’s consider the boundary conditions for the heat equation (1). The physical demands of the decreasing of the temperature at infinity, so we put the constant P=0 in this model. So, the only K0(kri1/2) should be using as the fundamental solution of the heat equation (1). The constant Q we will find from second boundary condition, it could be forming as the equality of the environmental temperature to T0 at the beam radius r=a:

T( r ) r =∞ = 0

(4)

T ( r ) r = a = T0

The first boundary condition demands of the equality to zero of the constant P=0 to avoid of the heating at the infinite radius r→∝. It gives the next form of solution from (3):

T ( r ) = Q ⋅ K0 ( kr i )

(5)

After joint solution of the last equation with the second boundary equation we have the formula for radial dependence of the temperature:

T ( r ) = T0 ⋅

( ) K (k ⋅ a i ) K0 k ⋅ r i

(6)

0

Result solution for the temperature describes the heated cylinder into the area of laser beam 0≤ r ≤a . These cylindrical functions with the imaginary arguments are presented through Kelvin functions:

i − n K n ( kr i ) = {kern ( kr ) + i ⋅ kein ( kr )}

(7)

Let’s consider this function in the most interesting region. The pulse’s frequency is f=109 Hz, typical radius of laser beam is r≈a=1mm=10-3 m, for a water ÷=0.14 10-2cm2/sec. The argument of cylindrical functions (k⋅a) will have typical value ka=(ω/χ)1/2=21.092>>1 for this cylindrical boundary. So, this argument is great enough for all mentioned regimes of modulated optical power of typical laser: ka=(ω/χ)1/2=21.092>>1

(8)

This temperature according (4) could be decreased through the small arguments as at the beam axis (ka1), this regime could be realized by focusing of the laser beam into irradiated liquid medium up to beam radius r=a or the less one. For the temperature outside the laser beam r>a the temperature could be found through the heat flow -q- by the next equation [ 9]. This area of heating locates outside the laser beam, really heating is small here, also it can not take part at the forming of the optical modes. So we will not take into account this outside heating at this simplest consideration. Lets consider at first this high frequency regime f∼108 with ka>>1, because its more perspective for the heating of Chlorella and forming appropriate living optical waveguide. Common temperature formula with account (6,7) is analyzed below in the next form:

T( r ) =

ker( kr ) ⋅ ker( ka ) + kei ( kr ) ⋅ kei ( ka ) + i ⋅ [ kei ( kr ) ⋅ ker( ka ) − ker( kr ) ⋅ kei ( ka )] ker 2 ( ka ) + kei 2 ( ka )

(9)

Let’s divide our further consideration into 5 parts according to 5 radial areas for the optical mode description and presentation for the Kelvin functions depending of radius r. First one is the axial beam area, the next area locates near cylinder side boundary, the 2th, 3th and 4th are located at the intermediate interval for the beam radius. The most

important areas are the cylinder axis with concentration of the most optical energy in it and the side boundary with no heated water. V. Firstly, we should introduce the presentation of the Kelvin functions by the following formulas for the case of large arguments ka>>1 near the cylindrical boundary r=a. These presentation are made through the modules Nν(x) and arguments φν(x) [10] :

kerν ( x ) = N ν ( x ) ⋅ cos φν ( x ) Nν ( x ) =

[ ker

( x )] + [ kei ν ( x )] 2

ν

kei ν ( x ) = N ν ( x ) ⋅ sinφν ( x )

, 2

,

tgφν ( x ) = kei ν ( x ) / kerν ( x )

(10)

For large values of the arguments x=kr≈ka we were used the next formulas with the accuracy as x--2 :

exp(ka / 2)  1 1  ⋅ 1 − ⋅ + O(( ka) −2 )  2πka  8 2 ka  ka 1 1 1 φ0 ( ka) ≈ − − ⋅π + ⋅ + O(( ka) −2) 2 8 8 2 ka π 1 1   ker 2 ( ka) + kei 2 ( ka) ≈ ⋅ exp − ka ⋅ 2  ⋅ 1 − ⋅ + O(( ka) −2 )  2ka  4 2 ka  N 0 ( ka) ≈

(11)

here the order of Kelvin functions was accepted as ν=0 according to initial equation. These equations were used to find presentation of the Kelvin functions ker(kr), kei(kr) near the beam cylinder at its side surface. Near the beam boundary r≈a we used the Kelvin function’s presentation for K0(ka) and for K0(kr) by equation (11) the only. We received the temperature distribution formula near the beam cylindrical boundary , where k⋅r≈k⋅a >> 1:

T (r)

a   k ( a − r)    k ( a − r )  1 − ( ka ⋅ 8 2) −1 ⋅ (1 + a / r ) exp  ⋅ ⋅ exp i ⋅    −1 r 2  (1 − ( ka ⋅ 4 2) ) 2    

= T0 ⋅ kr ≈ka

(12)

By neglecting of the small terms we will get more simple approximate formula for amplitude of the temperature:

T ( r ) ≈ T0 ⋅

a  k (a − r )  exp   , ka ≈ kr >> 1 r 2  

(13)

I. For the beam axis we have kr→0, ka>>1 . In the cylinder axis due to the neglecting of small addendums as xn from the standard Kelvin functions we get this another form for small arguments:

 kr  π ( kr ) 2  kr  r → 0 : ker(kr ) ≈ − ln   + ⋅ − γ + O { ( kr ) 2 } ≈ − ln   , 2  4 4 2 π  kr   ( kr ) 2  π kei ( kr ) ≈ − ln   ⋅  − ≈−  4  2  4  4 ber ( kr ) ≈ 1

bei( kr ) ≈

,

,

lim

[ − ln( kr /2) ]′ → 0  4/(kr )  ′

(14)

2

( kr ) 2 ≈0 4

The temperature formula by the reorganization of the common one (9) will have the next form:

T (r )

r→ o

 π  ka       ⋅ exp   ⋅  ker(kr ) ⋅ exp −i ⋅ ϕ ( ka )  + kei (kr ) i ⋅  − ϕ (ka )   ⋅ π    2   2

2 ka

= T0 ⋅

( ) ⋅ 1 − ( ka4 2 ) 1 − ka ⋅ 8 2

−1

−1

(15)

Let’ consider the logarithm in ker(kr) by the radius drawing near zero. So, let’s represent the logarithm function as a first terms of the series:

ker ( kr )

r→ 0

2 3 4 5  kr  kr 1 kr 1 kr 1 kr 1 kr = − ln   ≈ −  −1 −  −1 +  −1 −  −1 +  −1 ...   2  2 2  3 2  4 2  5 2    2

(16) By substitution of last approximation into the temperature equation (15) we will get :

T (r)

= T0 ⋅ r→ o

2k ⋅ a  π 1 − (8 2 ⋅ k ⋅ a )−1 ⋅ 7 ⋅ exp [− i ⋅ ϕ( ka ) ] − exp [ i ⋅ (π / 2 − ϕ( ka ) ] ⋅ 4 π   1 − (4 2 ⋅ k ⋅ a ) −1

(17)

But with neglecting of small terms as ∼1/ka we will get the simple approximate formula:

T

r =0

= T0 ⋅

2ka   ka π   π   ka 5π     ka   ⋅ exp  ⋅ 7 ⋅ exp i ⋅  +   − ⋅ exp  −i ⋅  +    π  2     2 2  4   2 8   

(17a)

But there is logarithm determination in the bases functions in (14) by arrival r to zero, als o N0(ka)=N(ka). The modules of the amplitude for temperature should be expressed by the next equation:

T

r→ 0

 ka   kr  ≈ T0 ⋅ 3,6 ⋅ exp   ⋅ ln    2  2

(18)

This approximation could gives at the cylinder axis r=0 the next value ker(kr)~7 for example, according to appropriate number of addendums into the series (16) for logarithm. This function numeral ker(kr)=7 correspond to the argument’s value for x=kr 0.001 r 0.047 ì m. The physics sad that not axial but intermediate radial areas are more interesting for our further consideration, because there is very small part of optical energy could be located at fiber radial area determined as 0 r 0.047µm. So, this axial area will be described by constant=7 for logarithm in equation (18). II. The same formulas (14-18) could be used for the temperature description in the next radial interval as 0.047