Light propagation through a monolayer of discrete scatterers: analysis of coherent transmission and reflection coefficients

Light propagation through a monolayer of discretescatterers: analysis of coherent transmission andreflection coefficients

Valery A. Loiko and Alexander A. Miskevich

An investigation of the coherent transmission and reflection coefficients of a monolayer of sphericalscatterers as a function of their size, optical constants, and concentration is carried out. An analysis isperformed of the quasi-crystalline approximation of the multiple-wave scattering theory and on thesingle-scattering approximation (SSA). The results permit determining the limits of applicability of theSSA to the layers with the partial ordering of spherical scatterers in analyzing the phases of thetransmitted and the reflected waves. The phase of the transmitted and the reflected waves is investigatedin the conditions of the quenching effect. It is shown that in such a case small changes in the refractiveindex of particles can cause dramatic phase changes. This effect can be used to modulate the light-wavephase, e.g., by electrically controlled composite liquid-crystals films. © 2005 Optical Society of America

OCIS codes: 030.1670, 290.2200.

1. Introduction

There are a great number of natural and artificialdisperse media with a high concentration ofscatterers.1–4 Among them are monolayers of discretescatterers. In particular these are cell membranes,island films, photonic crystals, films for optical imagerecording, and polymer-dispersed liquid-crystalfilms.

Layers with rarely spaced and closely spaced scat-terers have different optical characteristics. First wedeal with the independent scattering regime. Secondwe deal with the dependent scattering regime.5 Theoptical properties of these layers receive muchconsideration.6–13

In a small concentration the spatial distribution ofscattering particles in the monolayer is random. Theintensities of scattered light that are due to individ-ual particles are additive. In this case, to determinethe characteristics of transmitted and reflected light,

a single-scattering approximation (SSA) can beused.13

As the concentration of scatterers increases, a par-tial ordering of scatterers arises. As a result the in-fluence from the interference effects increases. At thedependent scattering regime, when the distances be-tween the particles are comparable with their sizes,to determine coherent transmission and reflection co-efficients, the quasi-crystalline approximation (QCA)is used.1,14–18 With this approximation the spatialdistribution of particles is described by the pair dis-tribution function.1,13 The QCA takes into accountsequential scattering events, excluding forward–backward scattering events. With this approximationthe hierarchy of the Foldy–Lax equations is restrictedto a system of two equations. It is assumed that awave field with two fixed particles does not differfrom a wave field with one fixed particle.1,13,19–21 Theconcentration dependences of the optical characteris-tics (extinction coefficient, phase function, the meanfree path, the transport mean free path, etc.) of thedense scattering media differ from those for the rar-efied media.1,13

In this paper the problem of coherent transmissionand reflection by a monolayer of isotropic sphericalscatterers is considered. The amplitudes and phasesof the transmitted and the reflected waves are inves-tigated. The calculations are carried out with the SSAand the QCA. The results can be used to develop newcomposite materials, films for optical image record-

The authors are with the B. I. Stepanov Institute of Physics,National Academy of Sciences of Belarus, F. Scaryna avenue 68,Minsk, 22072, Belarus (e-mail for A. Miskevich, [email protected]).

Received 2 September 2004; revised manuscript received 13 De-cember 2004; accepted 13 January 2005.

0003-6935/05/183759-10$15.00/0© 2005 Optical Society of America

20 June 2005 � Vol. 44, No. 18 � APPLIED OPTICS 3759

ing, and electrically controlled polymer-dispersedliquid-crystal light modulators.

2. Basic Relations

Consider a system of N homogeneous spherical scat-terers with a refractive index m � n � i� and diam-eter D distributed on the surface area S. Thescatterers form a monolayer with a surface density ofparticles ns � N�S. Let the monolayer be illuminatedby an external wave with electric vector Einc�r�. Eachscatterer creates a scattered wave T̂ · Einc�r�, where T̂is the particle scattering operator. Each ith scatterer�i � 1, 2, . . . , N� is in the field E�r|ri, r1, . . . , rN�formed by the external wave Einc�r� and wavesscattered by the �N � 1� particles centered atr1, r2, r3, . . . , rN on the z–y plane. The average fieldfar beyond the layer is formed by the wave Einc�r� andthe waves scattered by all particles,

E(r) � Einc(r) ��S

drins(ri)T̂ · E(r|ri). (1)

Here T̂i · E�r|ri� is the function describing the fieldscattered by the ith particle; E�r|ri� is the field inci-dent on the ith scatterer averaged over the positionsof the �N � 1� scatterers.

At normal incidence the function T̂i · E�r|ri� can bewritten in the far-field region as a divergent trans-verse spherical wave,1,17

T̂i · E(r|ri) � f(n̂)h0(k|r � ri), (2)

where n̂ is the unit vector in the direction r� ri, hn�x� is the spherical Hankel functions22,23 of theorder of n, and f�n� � f1��, ��e� � f2��, ��e� is theamplitude scattering function. Here and below vec-tors e with subscripts are the corresponding unit vec-tors.

To take into account the pair correlations of thescatterers and the multiplicity of the wave scatteringin the layer, we used the QCA.1,15,17 In this approxi-mation the field E�r|ri� is written in the form

E(r|ri) � Einc(r) ��S

drjn(rj|ri)T̂j · E(r|rj), (3)

where n�rj|ri� � �N � 1�p�rj|ri�, p�rj|ri� is the prob-ability density of the finding particle j at point rj ifparticle i is at point ri.

Let a wave with circular polarization Einc�r� � �ez

� iey�exp�ik · r� be incident on the layer (ez, ey, ex isthe right-hand system). To solve Eq. (3), we expandthe functions entering Eq. (3) into vector sphericalfunctions Mn1

�i� and Nn1�i�.17,22 We write the incident

wave in the form

Einc(r) � (ez � iey)exp[ik · (r � ri)]exp(ik · r)

� exp(ik · ri) �n�1

�

Cn1[Mn1(i)(r � ri)

� Nn1(i)(r � ri)], (4)

Cn1 � in�12n � 1

n(n � 1), (5)

where ri determines the position of the ith scatterer.Assume that the coordinate origin is in the mono-

layer plane; so for a normal incidence of the incomingwave k · ri � 0 and

Einc(r) � �n�1

�

Cn1[Mn1(i)(r � ri) � Nn1

(i)(r � ri)]. (6)

The wave scattered by a spherically symmetricscatterer can be written as18

T̂i · E(r|ri) � ��n

[bnzn1Mn1(3)(r � ri) � anyn1Nn1

(3)(r

� ri)]. (7)

Here coefficients an and bn are coefficients of the Mie-scattering series for a sphere.22 For convenience infurther presentation we replace subscript n with sub-script i:

ai �m�i(mx)�i�(x) � �i(x)�i�(mx)m�i(mx)i�(x) � i(x)�i�(mx) , (8)

bi ��i(mx)�i�(x) � m�i(x)�i�(mx)�i(mx)i�(x) � mi(x)�i�(mx) , (9)

where x � D�� is the dimensionless size parameter;� is the wavelength; �i�x� � xji�x�, i�x� � x�ji�x�� iyi�x�� are the Riccati–Bessel functions; and ji�x�,yi�x� are the spherical Bessel functions22,23 of the or-der of i.

We seek the solution of Eq. (3) in the form

E(r|ri) � �n

[zn1Mn1(1)(r � ri) � yn1Nn1

(1)(r � ri)].

(10)

The amplitudes of the coherent transmission andreflection coefficients are defined by

T � 1 ��

x2 �i�1

N

(2i � 1)(zi � yi), (11)

R � ��

x2 �i�1

N

(�1)i(2i � 1)(zi � yi), (12)

Coefficients zi, yi are found from the solution of thefollowing system of equations:

3760 APPLIED OPTICS � Vol. 44, No. 18 � 20 June 2005

zi � bi � bi

�k2

x2 �j�1

N

(Aijzj � Bijyj)

yi � ai � ai

�k2

x2�j�1

N

(Aijyj � Bijzj) , (13)

where

Amn �2n � 1

21

[m(m � 1)n(n � 1)]1�2

�p�0, 2

N

i�p(2p � 1)[m(m � 1) � n(n � 1)

� p(p � 1)]Pp(0)m n p0 0 0� m n p

1 �1 0�Hp,

(14)

Bmn �2n � 1

21

[m(m � 1)n(n � 1)]1�2

�p�0, 2

N

i�p(2p � 1)[(p � m � n)(p � m � n)

(m � n � 1 � p)(m � n�1 � p)]1�2

Pp(0)m n p � 10 0 0 � m n p

1 �1 0�Hp; (15)

j1 j2 j3

m1 m2 m3�

is the Wigner 3j symbol,24 N � x � 4x1�3 � 2 is thenumber of terms of the scattering series,22

� � nsD2�4 is the filling coefficient, and

Hp � 2 �D

�

dRRg2(R�D)hp(1)(kR), (16)

where hp�1��x� � jp�x� � iyp�x� is the spherical Hankel

function. The notation p � 0, 2 means that p takes onthe values of 0, 2, 4, . . . .

The radial distribution function1,13 g2 of particles inthe monolayer is calculated by

g2(R) � 1 �1

8� �0

�C2(z�2)

1 � C(z�2) J0(zR)zdz, (17)

where the function C�t� is found by19

C(t) �4�

1 � �

2J1(2t)2t �

4�2

(1 � �)2 J0(t)2J1(t)

t

� � �2

(1 � �)2 �2�3

(1 � �3) �2J1(t)t 2

, (18)

where Jn�t� is the cylindrical Bessel function of ordern. Note that the use of Eq. (18) for C�t� essentiallysimplifies the radial distribution function [Eq. (17)]calculation compared with the commonly used meth-od.20

The summation in Eqs. (14) and (15) is carried outover even p, since at odd p the polynomials Pp�0� areequal to zero. For even p one has25

Pp(0) � ip(p � 1)!!

p!! , (19)

where p!! is the double factorial of p;p!! is a product ofthe even numbers for p even and odd numbers for podd: p!! � p�p � 2��p � 4�. . . .

We represent Eq. (16) as the sum of two integrals.To this end let g2 be in the form g2 � 1 � �g2 � 1�. Inthis case Hp � 2�H1p � H2p�, where

H1p ��D

�

dRRhp(1)(kR), (20)

H2p ��D

�

dRR[g(R) � 1]hp(1)(kR). (21)

Using the recurrent relations for the sphericalBessel function,25 we can write the integral H1p in theform

H1p � � k�2��kDhp�1(1)(kD) � �

q�0, 2

p

[2(p � q) � 1]

p !! (p � q � 1)!!

(p � 1) !! (p � q)!! hp�q(1)(kD)�, (22)

where k � 2��.The integral H2p is easy to calculate numerically

since the quantity g2 � 1 tends to zero and is otherthan zero at R � 7D.13

To find the 3j symbols of the form

m n p1 �1 0�,

we make use of24

m n p1 �1 0��

12 �(p � m � n)(p � m � n)(m � n � 1 � p)(m � n � 1 � p)

m(m � 1)n(n � 1) 12m n p � 1

0 0 0 � (23)


at odd m � n � p values and

m n p1 �1 0��

12

m(m � 1) � n(n � 1) � p(p � 1)[m(m � 1)n(n � 1)]

m n p0 0 0� (24)

at even m � n � p values.A 3j symbol of the form

m n p0 0 0�

is found from24

m n p0 0 0�� (�1)p�2m

1

(2p � 1)1�2 �m n p0 0 0 . (26)

Here d is an integer;

� j1 j2 j3

m1 m2 m3

is the Wigner coefficient.In this section we have considered the main equa-

tions of the QCA that we use for numerical calcula-tions. The results are described below.

3. Results

For convenience in representing the calculation re-sults we denote the amplitude coherent transmissionand reflection coefficients calculated in the QCA withsuperscript ms:

Tms � 1 ��

x2 �i�1

N

(2i � 1)(zi � yi), (27)

Rms � ��

x2 �i�1

N

(�1)i(2i � 1)(zi � yi), (28)

and in the SSA with superscript ss:

Tss � 1 ��

x2 �i�1

N

(2i � 1)(ai � bi), (29)

Rss� ��

x2 �i�1

N

(�1)i(2i � 1)(ai � bi). (30)

The phases of the transmitted and the reflectedwave calculated by the QCA ��t

ms, �rms� and by the

SSA ��tss, �r

ss� are defined by

�tms � arctan

Im(T ms)

Re(T ms), (31)

�rms � arctan

Im(Rms)

Re(Rms), (32)

�tss � arctan

Im(Tss)

Re(T ss), (33)

�rss � arctan

Im(Rss)

Re(Rss). (34)

An analysis of the coherent transmission and re-flection coefficients and transmitted and reflectedwave phases has been carried out. We have investi-gated their dependences on the refractive indices.

�m n p0 0 0 ��

0, m � n � p � 2d � 1,(�1)d�p(2p � 1)1�2d!

(d � m) ! (d � n) ! (d � p)!

�(2d � 2m) ! (2d � 2n) ! (2d � 2p)!(2d � 1)! 1�2

, m � n � p � 2d,

(25)

Fig. 1. Transmitted wave phase versus the refractive index n atvarious size parameters: solid curves, results of the calculationsfrom the QCA ��t

ms�; dashed curves, from the SSA ��tss�. � � 0.5,

� � 5 10�5.


Figure 1 shows the dependence of the phase of thetransmitted wave on the refractive index at variousvalues of the size parameter x calculated by the QCAand by the SSA. Small-size parameters in the consid-ered range of the refractive index are characterizedby a substantially linear dependence of the phase ofthe transmitted wave on n. The dependence of thephase transmitted through the monolayer wave onthe concentration and the size parameter correlateswith the results for the effective phase velocity26 inthe dense three-dimensional medium in the low-frequency limit.

As the size of the particles increases the depen-dence becomes nonlinear, and large phase changestake place for smaller changes in the refractive index.The phase fluctuation frequency also changes. In par-ticles with large-size parameters the differences be-

tween calculations by the QCA and the SSA decreaseand become insignificant when x � 10.

The dependence of the transmission coefficient onthe refractive index at various size parameters isshown in Fig. 2. The SSA can underestimate or over-estimate the transmittance calculated with the QCA.This result agrees with the data of Hong.17 An in-crease in the size parameter of the soft particles leadsto a sharp change in the transmission coefficient.With a particle size parameter of x � 10 the results ofthe calculations of the QCA and the SSA are close toeach other. The larger the value of x the more justi-fied is the use of the SSA for the coherent transmit-tance calculations. This is also evident from acomparison with the experimental results.13,27

Figure 3 shows the dependence of the reflectedwave phase on the refractive index at various valuesof the size parameter x calculated with the SSA and

Fig. 2. Dependence of the coherent transmission coefficient on therefractive index n at various size parameters: solid curves, resultsof the calculations with the QCA �|T ms|2�; dashed curves, with theSSA �|T ss|2�; � � 0.5, � � 5 10�5.

Fig. 3. Dependence of the phase of reflected wave on the refractive index n at different size parameters: solid curves, results of thecalculations from the QCA ��r

ms�; dashed curves, from the SSA ��rss�; � � 0.5, � � 5 10�5.

Fig. 4. Dependence of the reflection coefficient on the refractiveindex n at various size parameters: solid curves, results of calcu-lations from the QCA �|Rms|2�; dashed curves, from the SSA�|Rss|2�; � � 0.5, � � 5 10�5.


the QCA. When the small-size parameters are in therange of the refractive index that is being considered,a small phase shift takes place [see the curve for x� 0.1 in Fig. 3(a)]. As the particle size increases, thephase shift increases. The difference between the cal-culation results with the QCA and the SSA decreases.

The dependence of the reflection coefficient on therefractive index at various size parameters is given inFig. 4. At the small-size parameters the dependenceof the reflection coefficient on the refractive index isweakly marked. (At x � 0.1 there is a change in |R|2

in the third decimal digit after the point.) With anincreasing size parameter this dependence becomesmore pronounced and acquires a nonmonotonic char-acter. The greatest difference between the calculationresults with QCA and SSA is observed at large re-fractive indices.

At some values of the size parameter and the refrac-tive index the transmission coefficient of the high con-centrated monolayers has clearly defined minima. Thedependences of the transmission coefficient and thephase of the size parameter are illustrated in Fig. 5.Figures 6 and 7 present the dependences of the phaseof the transmitted wave and the coherent transmissioncoefficient on the refractive index at parameters ofdifferent size. The values of the size parameter and thevalues of the refractive index correspond to the param-eters at which the transmission minimum takes place(Fig. 5). This minimum is due to interference from theincident and the transmitted waves when their ampli-tudes are close in value and the phases differ by �. Inthese conditions the quenching effect takes place.7,13,28

The calculation was performed in the QCA. The closerthe values of the coherent transmission coefficient areto zero, the steeper is the phase change caused by achange in the refractive index of the particles. Thephase of the transmitted wave is changed by � witha very small change in the refractive index. When

there is a small (at the fourth decimal digit) devia-tion in the size parameter of the particles from thevalue corresponding to zero transmission, thetransmitted wave phase can also change by � or�. This is due to the specific behavior of the am-plitude scattering function in the parameter rangebeing considered.28 For minimal transmission theimaginary part of the amplitude transmission coef-ficient is zero. Figure 8 shows the dependences ofthe imaginary and the real parts of the transmis-sion coefficient and its amplitude in this range ofthe values of the refractive index, the filling coeffi-cient, and the size parameter of the layer particles.

Fig. 5. Solid curve, dependence of the coherent transmission co-efficient |T ms|2; dashed curve, phase of transmitted wave �t

ms onsize parameter x in the region of size parameters corresponding tothe transmission minimum, which is due to interference from theincident and the transmitted waves. Calculation is from the QCA;� � 0.5, n � 1.6, � � 5 10�5.

Fig. 6. Dependence of the phase of transmitted wave �tms on the

refractive index n in the region of size parameters corresponding tothe transmission minimum, which is due to interference from theincident and the transmitted waves. Calculation is with the QCA;� � 0.5, � � 5 10�5.

Fig. 7. Dependence of the coherent transmission coefficient|T ms|2 of the monolayer on the refractive index n in the size-parameter region corresponding to the transmission minimum,which is due to interference from the incident and the transmittedwaves. Calculation is with the QCA. � � 0.5, � � 5 10�5.


Fig. 8. Dependence of the amplitude transmission coefficient |Tms|, Re Tms, and Im Tms on the refractive index n in the size-parameterregion corresponding to the transmission minimum that is due to interference from the incident and the transmitted waves. Calculationis from the QCA; � � 0.5, � � 5 10�5. (a), (b) Change in the phase by �. (c), (d) Change in the phase by �.

Fig. 9. Solid curve, dependence of the coherent reflection coeffi-cient |Rms|2; dashed curve, phase of transmitted wave �r

ms on thesize parameter x in the size-parameter region corresponding to thereflection minimum. Calculation is with the QCA; � � 0.5, n� 1.2, � � 5 10�5.

Fig. 10. Dependence of the phase of reflected wave �rms on the

refractive index n in the size-parameter region corresponding tothe reflection minimum. Calculation is with the QCA; � � 0.5, �

� 5 10�5.


The results in Figs. 8(a) and 8(b) correspond to thecase of the phase change by �. The minimum ofthe transmission coefficient is attained at the pointat which the imaginary part of the amplitude trans-mission coefficient is equal to zero. The real part ofthe amplitude transmission coefficient at this pointis greater than zero. The results in Figs. 8(c) and8(d) correspond to the case of the phase change by�. As in the previous case the minimum of thetransmission coefficient occurs for the zero imagi-nary part of the amplitude transmission coefficient,but the real part at this point is less than zero.

The dependence of the reflection coefficient on thesize parameter also has clearly defined minima. Theycan be due to the interference of the singly backre-flected wave and the wave arising from multiple scat-tering from the monolayer. Figure 9 illustrates thedependences of the reflection coefficient and the back-reflected wave phase on the size parameter. Thebackreflected wave phase (Fig. 10) and the amplitudereflection coefficient (Fig. 11) as a function of therefractive index have been investigated in the vicinity

Fig. 11. Dependence of the coherent reflectance coefficient |Rms|2

of the monolayer on the refractive index n in the size-parameterregion corresponding to the reflection minimum. Calculation isfrom the QCA; � � 0.5, � � 5 10�5.

Fig. 12. Dependences of the amplitude reflection coefficient |Rms|, Re Rms, and Im Rms on the refractive index n in the size-parameterregion corresponding to the reflection minimum. Calculation is from the QCA; � � 0.5, � � 5 10�5: (a), (b) Change in the phase by �;(c), (d) change in the phase by �.


of the size parameters where a deep reflection mini-mum takes place. The results of the calculations forthe phases are in Fig. 10. The behavior of the phasesof the reflection waves has the same characteristicfeatures as those for transmission. With very smallchanges in the size parameters in the range of valuesclosest to the size parameter corresponding to zeroreflection, the phase can be changed by either � or�. The closer the size parameter is to the size pa-rameter corresponding to the deepest reflection min-imum, the steeper is this change. Figure 12 shows thedependences of the amplitude, the imaginary, andthe real parts of the reflection coefficients on the re-fractive index in the region of the reflection � and�. The phase change by �, as for transmission,takes place in the case in which the real part of theamplitude reflection coefficient is greater than zero atthe point at which its imaginary part vanishes. Thephase change by � takes place in the case in whichthe real part of the amplitude reflection coefficient isless than zero and its imaginary part is equal to zero.

Figure 13 shows the dependences of the coherenttransmission coefficient and the transmitted wavephase on the refractive index. The most prominentchanges in the transmitted wave phase take place ina monolayer with optically soft particles. The calcu-lation was performed with the QCA. From Fig. 13 onecan see that monolayers composed of large particleswith small relative refractive indices efficiently mod-ulate the phase of transmitted light. The reason isthat for such particles a small change in the refrac-tive index (e.g., in films with droplets of liquid crys-tals upon a control field) can lead to a considerablechange in the transmitted wave phase.

4. Conclusions

In the quasi-crystalline approximation of the theoryof the multiple scattering of waves, the phase changein the wave transmitted through a monolayer of scat-

terers and in the wave reflected from it has beenanalyzed. We have determined the limits of applica-bility of the single-scattering approximation for cal-culating the phases of the transmitted and thereflected waves. The results permit estimating pa-rameters at which an effective phase modulation ofthe transmitted wave can be realized by changing therefractive index of the particles. They can be gener-alized to the finite-thickness layers in which thetransmitted and the reflected wave phases can becontrolled by varying not only the characteristics ofthe particles and binding matrix but the layer thick-ness as well.

This research was supported in part by the BelarusScientific Research and Development program,coherence.

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Light propagation through a monolayer of discrete scatterers: analysis of coherent transmission and reflection coefficients