Vol. 4, No. 1/January 1987/J. Opt. Soc. Am. A 27

Analysis of diffraction in periodic liquid crystals: the opticsof the chiral smectic C phase

Katsu Rokushima and Jiro Yamakita

Department of Electrical Engineering, College of Engineering, University of Osaka Prefecture,Sakai, Osaka, fapan

Received December 26, 1985; accepted August 7, 1986

A method for analyzing diffraction properties of periodic liquid crystals is presented that uses the previous theoryfor anisotropic dielectric gratings [J. Opt. Soc. Am. 73, 901 (1983)]. The method applies to any type of periodicliquid-crystal structure with arbitrary orientation of the helical axes. As an example of the present analysis, bothforward and backward diffraction properties of a chiral smectic C phase are given for the incidence of linearlypolarized and circularly polarized waves. An approximate two-wave analysis is also derived and is compared withthe rigorous results.

INTRODUCTION

Optical diffraction by spatially periodic dielectric media hasbeen studied extensively for many years. These periodicstructures play an important role in the fields of diffractiongratings, acousto-optics, integrated optics, and liquid crys-tals.

A unified theory was formulated for anisotropic dielectricgratings.1 This method applies to any isotropic or aniso-tropic gratings, any polarization of the incident plane wave,and any orientation of the grating vector. We show how thismethod can be applied to treat periodic liquid-crystal struc-tures.

Several liquid-crystal structures have been reported, suchas the helical cholesteric liquid crystal (CLC), the chiralsmectic C liquid crystal (CSC), and the periodically bentnematic liquid crystal (PBNLC). Wave propagation in suchliquid crystals has been studied extensively, and the solu-tions are given by the rigorous numerical 4 X 4 matrix meth-od,2' 3 approximate dynamic theory,4-6 and exact and approx-imate analytical methods.7 -10 However, those analyses arelimited to the case when the helical axis is perpendicular tothe surface.

The rigorous approach presented here analyzes the for-ward and backward diffraction properties of periodic liquid-crystal structures with arbitrary orientation of the helicalaxes. As a numerical example, diffraction properties of theCSC are given for the incidence of a linearly or circularlypolarized wave. An approximate two-wave analysis is alsoderived from the rigorous one, and its accuracy is discussed.

ANALYSIS

As shown in Fig. 1, we consider the optical diffraction in aperiodic liquid crystal bounded by two isotropic media withrelative permittivity es (I = 1, 3). The helical axis of thecrystal is inclined to the surface normal (x axis) with anangle 00 in the xz plane. For the incidence of a generallypolarized wave in the xz plane with an incident angle Oj, allthe field components of the incident wave have the formexpU[wt - Jj(x cos Oi + z sin 0i)]}, where space variables

normalized by ko = - = 27r/X are used and kor - r, kox -x, and koz - z for simplicity.

Electromagnetic Field in a Periodic Liquid CrystalThe relative permittivity tensor of a local uniaxial liquidcrystal rotating along the u axis with pitch P is given by3

where

FEuu 'U

f(u, V, W) = [c 1j(u)] = IEVU er

LEWU Ewv

Ewl(1)

eu = E(1 + ( cos 20),

EVV = E[(1 - 6 cos2 0) - sin2 0 cos 20],

Ew. = E(1 - 6 cos20) + 6 sin2 6 cos 201,

Euv = Evu = E( sin 20 sin 0,

Euw = ewu =-eb sin 20 cos 0,

evw = ew, = -e5 sin2 0 sin 20, (2)

with

E = (Eu + Ev)/2, 6 = (eu - ev)/(Ec + Ev),

eu = need Ev = E = n02. (3)

Here the tilt angle 0 and the azimuth angle 4 are given by

0 = 27ru/A, 0 < 6 < 7r/2, P=A (CSC),

20 = 27ru/A, 0 7r/2, P = 2A (CLC),

20 = 27ru/A, 0 = 0, P = 2A (PBNLC). (4)

The permittivity tensor for the xyz coordinate is then givenby 2(x, y, z) = R-•~(u, v, w)R, where R is the rotation tensorof the coordinate, and each tensor element Eij(r) (i, j = x, y, z)can be expressed by (Appendix A)

Ei1(r) = E j, exp(jlnk * r),

0740-3232/87/010027-07$02.00 © 1987 Optical Society of America

K. Rokushima and J. Yamakita

28 J. Opt. Soc. Am. A/Vol. 4, No. 1/January 1987

X

region 1 E1 0i +1

region 2 - EE2y o 2 -

region 3 E3 X +1

0

l 3

Fig. 1. Configuration of optical diffraction by a periodic liquidcrystal with inclined helical axis. Either tilt angle 0 or azimuthangle 0 rotates along the u axis.

20= n = lnk(X, cos 0 + z sin 0) = px + sz,20J

Inkl = nk = A/A, (5)

where eij,l is the 1th-order Fourier coefficient. Usually thehelical axis may be either perpendicular (00 = 0) or parallel(O0 = 7r/2) to the surface, and then ;(x, y, z) = ;(u, v, w) or;(X, y, Z) = e(-w, v, u). From the Floquet theorem, theelectric and magnetic fields can be expressed by the summa-tion of the space harmonics as

* Y0E = E em(x)exp(-jnm.- r),m

FAoH = E hn(x)exp(-jnf * r), (6)m

where Yo = liZO = 1-7i and

nm r = pmx + smz, Pm = Po + mP, Sm so + ms.

(7)

Here ei = ei(x) and hi = hi(x) (i = y, z) are column matriceswith elements etm(x) and hi,,,(x) (m = 0, ±1, ±2, .. .), eij =(Eijim) = (Eij,m-1) are (2m + 1) X (2m + 1) submatrices, Eij-1 isthe inverse matrix of Eij, and p = [b(impm], s = [bimsm], and 1 =[6im] are (2m + 1) X (2m + 1) diagonal submatrices, where a1m

is the Kronecker delta.By transforming

f, = Tg (11)

Eq. (8) is reduced to

d = jKg.dx

(12)

Here K = (Omkn,) is a diagonal matrix, Kn = Kno are eigen-values of the n X n matrix C, where n = 2n' = 4(2m + 1), T =(T+, T-) = (ul+ . . . un,,+ U1 ... u,'nI is the diagonalizer of Cwith eigenvectors unit corresponding to Knos and g is a col-umn matrix with elements go+. Equation (8) has eigensolu-tions un+ exp(QKnx) gn," that correspond to the opticaleigenmodes in the liquid crystal traveling along the ax di-rections.

Electromagnetic Fields in a Uniform Isotropic MediumIn the uniform isotropic region, eij = bijEl and p = 0 in Eq.(10), and then C becomes Cu with all the diagonal subma-trices. The diagonalizer Tu for Cu in this two-dimensionalcase is explicitly given by

s ° s 0s 0 -~ 0ITu - ° wets ° arts= (Tu+, Tu), (13)

°- as re t g

with diagonal submatrices s = ((31,,J) and t = ((Imem), whereKmu+ = H= I:- are the eigenvalues ofCu, Sm = sm/ISmI, and the corresponding g is expressed as

E[+

g[ 9 ] mg+

Ug-j

Here so = eg sin 0i, and P0 is arbitrary and is set equal toSubstitution of Eqs. (6) into Maxwell's equations yie

the following coupled-wave equation in matrix form':

d ft =jCft,dx

where

-he]

ft = hz,

Lhyj

0.

,ldsEgI = (Eg+ +... Eg0+ ... Eg+)t,

Mg9 = (Mg- ... Mg0 + ... Mg z)t. (15)

(8) Here the field amplitudes of each eigenmode in the uniformregion are normalized to lEeml = IMemi = 1.

The superscripts E and M refer to TE and TM waves,respectively, and i refer to the propagation along the Atxdirections.

(9) Boundary ConditionsWithin region I (I = 1, 2, 3), the solution of Eq. (12) is givenby

-1 0

,E~ + s 2 P I, EX1X

XY0 sE 'exx 1x + P

r1:iEXy 0 62Z - 6XZEXX' 16X2

01-1 -exyfxx S

-exXXJ's + 1 J (10)

,Exzfxx IS + pI

p

seXX_1e

yZ- EXZ6X

K. Rokushima and J. Yamakita

(14)

Vol. 4, No. 1/January 1987/J. Opt. Soc. Am. A 29

f 9g(0) l g1 (0)

T 92 (-d)

g93 (-d)

x = 0

x = -d

Fig. 2. Boundary surfaces and unknowns g 1(xo).

(I = 1,3) are given, for both linearly and circularly polarizedwaves, by

EMp im: = ReLjR)jEmMg:': =LRp I ' = Re(t~mjlLRg~m-'2.

(20)

Then, for the incidence of a TE or TM wave,

EMg 1-(0 ) = (0. 1 ... *... ) MEg 1 (°) = 0; (21)

and for the incidence of a LC or a RC wave

Eg-(0) = (O ... 1/X2 . . * 0), Mg 9 (0) = (O ... Ij/F2 .. . 0)'.

(22)(x)1 = [expUK1+(x - xo)] 0 1 [g 1+(x) 1

LgF(X)J L 0 exp)jJ i(x-x LgI-(xo)J

(16)

where expUKI-(x - xo)] = {(mn expUKxt'(x - xo)]j are diagonalmatrices and g 1l(xo) are constant column matrices at x = xo.

At the boundary surfaces (x = 0 and x = -d), tangentialcomponents of the fields jYOE and WZH are continuous.Since K,,' have complex values [Im(K,,4 ) z 0], the boundaryconditions have been expressed in somewhat modified formin order to avoid overflow problems in the numerical compu-tation. Thus, from Eqs. (6), (9), and (16), and taking g2-()and g 2 +(-d) as the unknowns in region 2 of Fig. 2, we get

B12 ° 01B2 3 T3UJ [ 0

(17)

with

= 2 expUK2'd] 0]B12+ = T2[ IL 1

0 1 0

B23- = -EXP(jP2 d)T2 [ exp0-jKc-d]' (18)

where EXP[Up2 d] is a diagonal matrix with four diagonalsubmatrices exp(jp 2d) = [6Im exp(jp 2md)].

Therefore mth-order reflected and transmitted diffractionefficiencies flmr and flmt are given by Pim-/PIo- for an arbi-trary orientation of the waves. For a lossless medium, thepower-conservation relation requires that ,m ZE,M (f7mr +7lmt) = Zm, FL,R (f1mr + 77rt) = 1, which can be used as a checkfor the numerical calculations.

The Special Case O0 = 0For 00 = 0, only zeroth-order diffraction can take place in theuniform external regions, as shown in Fig. 3. In this case,f(X' y, Z) = f(U. V, W), s = 0, p = nk sm = so, and then it becomesem(x) = em exp(jKx) and hm(x) = hm exp(jKx), that is,

ft = exp(jKX)ft, ft = (23)ez

Here ft is a constant column matrix with constant elementsei and hi, in contrast to ft(x) in Eq. (9). Therefore Eq. (8)becomes

Kft = CfU, (24)

where C is the 4(2m + 1) X 4(2m + 1) matrix given by settings = sol in Eq. (10). Equation (24) means that K is an eigen-value of C. However, the number of independent eigen-values is reduced to four, and the others are given by

Kkm = Kko -MP (k = 1, 2), (25)

where Kko' are the four independent eigenvalues correspond-ing to the propagation constants of the four eigenmodesalong the ±x directions.

Diffraction EfficiencyFrom Eq. (17), diffracted waves g1+(O) and g 3 (-d) areobtained for the incident wave g1j(0) = [Egji(0)Mgji(0)]t.The linearly polarized wave (TE or TM) and the left- orright-circularly polarized (LC and RC, respectively) waveare related by

Lg' = 1/5f2 (Eg' - jMg'),

Eg+ = 1/a-2(Lg± + Rg+),

Rg± = 1i-2 (Eg+ + jMg±)

MgI = j/4-(Lg± - Rg±), (19)

where the superscripts L and R refer to LC and RC waves,respectively, about the +x direction. (We define LC and RCabout the +z direction.) Therefore the reflected LC or RCwave is the LC or RC wave, while incident and transmittedLC or RC waves are reversed RC or LC waves in the usualdefinition.

The mth-order powers along the ±x directions in region I

region 1

S ~~~-zregion 2- e2 -9g2 Y k d

region 3 6 3

30Fig. 3. Configuration of optical diffraction by a periodic liquidcrystal with perpendicular helical axis.

region 1

region 2

region 3

K. Rokushima and J. Yamakita

-Tlu+

0

30 J. Opt. Soc. Am. A/Vol. 4, No. 1/January 1987

the external regions were then chosen to be cu = 2.89, ew = Ew

= 2.1904, e = 2.5402, 6 = 0.1377056, and Eext = El = E3 = 2.4025so they could be compared with the other results.6

(i = 1 - 4) (26)

be the eigenvector corresponding to KkW', where iUk' is a (2m+ 1) X 1 column matrix with elements iUkm ; then the char-acteristic field for Kk0o is given from Eqs. (23) by

ftk(X) = Uk exp(jKkoX)gk', (27)

where gas is a constant. Fields in the liquid crystal (region2) are given from Eqs. (23) and (6) by

+i = I

+ 92k iUkm exp0jKk. X)] i. =

i = 4

where g2hX (k = 1, 2) are four unknowns in region 2.In regions 1 and 3, only the zeroth-order waves exist; they

are given by

} = g10 exp(-jtjox) + Eglo exp(j/ 1 ox),

YOEZl/tlo]

H }Y~ I = I =i [Mg10+ exp(-j< 1 0 x) i Mg10 exp(j~1 0 x)],

(29)and

= +Eg3 0 exp(it 30x),

nt

FZO3/430} = f3 Mg93o exp(/j 30 x). (30)

At the boundary surfaces (x = 0, x = -d), tangential fieldcomponents are continuous, giving eight equations for eightunknowns for the incidence of known Eg1 o- and Mglo. Thezeroth-order diffraction efficiencies nor and got are thenreadily obtained. This is a simpler calculation than that forthe general case of 0o Fd 0. Moreover, variations of Kk0l withnh directly give the dispersion characteristics for the opticaleigenmode of the liquid crystal. The present analysis is analternative method to Berreman's 4 X 4 matrix method.

Helical Axis Perpendicular to the Surface (00 = 0)When the helical axis is perpendicular to the surface, f(x, y,z) = f(u, v, w), p = nk and sm = so, and then ex = e(1 + ( cos20)and ey =E = f(1-6 cos2 0), where the tilt angles 0 are chosento be 0 = 0.005, 0.16, and Ir/8, respectively, and the angle ofincidence 0i = r - 1.136519.

For the incidence of a TE wave, both forward and back-ward diffractions can take place, as shown in Fig. 4. Thevariations of the transmitted diffraction efficiency t7t withthe normalized thickness ko'yod at the first-order Bragg con-dition are shown in Fig. 5. The result of an approximate

Backward x, Forward

p1

Z

RPOP. < 0 POP-1 >0Fig. 4. Forward and backward TE-TM diffractions in the chiralsmectic C structure for the incidence of a TE wave.

1.0

0.8

0.6

0.4

0.2

0.0ri/2 UI

NUMERICAL EXAMPLES

The method described above was used to calculate severaldiffraction properties of the CSC that have been consideredby a few authors. 3'5 6 The parameter values of the CSC and

Fig. 5. Variations of transmitted diffraction efficiencies with nor-malized thickness of the CSC for the incidence of a TE wave at thefirst-order Bragg condition. Here 0 = 0.005, 0.16, 7r/8, respectively.The corresponding nk = 0.37, 0.356582, 0.293009, respectively.

Let

uklL [uk H iA

V Iz

YoEz:; To E,2

= E [ 2k Uk/ expQKkm+X)k m

ZHZ./3301

0

K. Rokushima and J. Yamakita

Vol. 4, No. 1/January 1987/J. Opt. Soc. Am. A 31

1.0

0.8t

Ti

0.6

0.4

0.2

0.0

0.40

0.45

0.50

XkO.

0.55 , X , +

2.0 2.2 2.4 2.6 2.8 3.0 3.2A/X

Fig. 6. Transmission spectra for linearly polarized wave and corre-sponding dispersion relation for optical eigenmode near the first-order Bragg condition. Here d/A = 8 and 0 = 0.16.

two-wave analysis (Appendix B) is shown by the dashed line.The rigorous results deviate from two-wave analysis withincreasing 0. However, even for small values of 0 = 0.005,fine pulsations exist (as shown in the inset) because of themultiple reflection at the boundary surfaces. This finestructure was missing in the work reported in Ref. 6 becauseof the rough spacing between the sampling points.

Figure 6 shows the transmission spectra together with thecorresponding dispersion curves near the first-order Braggcondition, where d/A = 8 and 0 = 0.16. The effect of multi-ple reflection is apparent, and eigenvalues KkOA split awayfrom the unperturbed values in the vicinity of their intersec-tion point. The angular dependence of diffraction efficien-cies was also calculated, showing fair agreement with that ofRef. 6.

Figures 7 and 8 shows the backward diffractions in con-trast to the forward diffractions of Figs. 5 and 6, where 0 = 7r/8. The effect of multiple reflection is also shown in thesefigures. In particular, an appreciable number of reflectedTE waves exist, although the direct diffraction between TEwaves cannot occur at the first-order Bragg condition.3 Thedispersion curves show complex eigenvalues that correspondto the first-order TE-TM and the second-order TM-TMstop bands.

Helical Axis Parallel to the Surface (00 = r/2)5When the helical axis is parallel to the surface,5 i(x, y, z) =

v(-W , u), nk = S, Pm = po, Ez = dE( + t cos20), and Ex = cy =

(l-6 cos2 0). Only forward diffraction can take place, andeach higher-order diffraction appear in the external regions,as in Fig. 1.

Figure 9 shows the variations of transmitted diffractionefficiencies with the normalized thickness koyod for the inci-dence of both TE and RC waves at the second-order Braggcondition, where O0 = 3nr/4, 0 = ir/4, and yo = s2',I*2Fb sin 2 0/4PoEx corresponds to Eqs. (B5) below. Since TE-TE andTE-TM diffractions take place simultaneously in this case,

1.0

0.8r

0.6

0.4

0.2

0.0

0 I/2 UII

k Y d

Fig. 7. Variations of reflected diffraction efficiencies with normal-ized thickness of the CSC for the incidence of a TE wave at the first-order Bragg condition. Here 0 = 7r/8 and nk = 1.321509.

1.0

0.8r

0.6

0.4

0.2

0.0

0.2

0.4

0.6XkO

0.8

1.0Xk0k0

0.4 0.6 0.8

0.0

0.1

1.0 1.2 1.4

A/X

Fig. 8. Reflection spectra for linearly polarized wave and corre-sponding dispersion relation for optical eigenmode near the first-order Bragg condition. Here d/A = 8 and 0 = 7r/8.

K. Rokushima and J. Yamakita

32 J. Opt. Soc. Am. A/Vol. 4, No. 1/January 1987

1.0

0.8t

Tim

0.6

0.4

0.2

0.0

0

1.0

0.8t

Tim

0.6

1/2 Ui

(a) Incidence of TE wave

042

02 / \LC...2 '~ ~ LC

0 Is/2

koyod

(b) Incidence of RC waveFig. 9. Variations of transmitted diffraction efficiencies with nor-malized thickness of CSC at the second-order Bragg condition forthe incidence of (a) a linearly polarized TE wave and (b) a circularlypolarized RC wave. Here Oi = 37r/4, 0 = 7r/4, and nk = 1.114002 forthe TE-TM Bragg condition.

simple two-wave analysis cannot predict even approximateresults, as seen in Fig. 9(a), and the rigorous analysis isindispensable. For the incidence of a RC wave, the complexdiffraction properties are as shown in Fig. 9(b) because ofboth spiraling and anisotropic effects of the liquid-crystalstructure.

Accuracy of the Numerical CalculationsTable 1 shows the accuracy of the solution by the truncationof the matrix C in the case of Fig. 7 with a fixed value ofko'yod and with 0 as a parameter, where m is the order ofspace harmonics taken into the calculations; that is, m = 0,+1, ... am. From this table, it can be seen that the solu-tions have almost converged at m = 3. Since the otherexamples showed similar results, most of the calculationswere performed by retaining up to m = 3-4 with the error inthe power-conservation relation of the order of 10-4-10-7 inall our results.

CONCLUSIONS

A rigorous analysis of diffraction in a periodic liquid crystalhas been presented. The solution is reduced to an eigenval-ue problem of the coupling matrix C whose elements aregiven in a unified form so that other liquid-crystal struc-tures, such as the CLC and the PBNLC, can be treated in asimilar way by systematic matrix calculations. Boundaryconditions have been given in somewhat modified form to beapplicable to an arbitrarily thick or absorbing medium with-out causing overflow problems in the computations. As anumerical example, optical diffraction in the CSC has beenconsidered, and the result is compared with the approximatetwo-wave analysis. Any arbitrary level of accuracy can beobtained by increasing the number of space harmonics re-tained in the analysis. However, convergence is so rapidthat the retention of only a few harmonics is sufficient forpractical cases.

APPENDIX A: ELEMENTS OF PERMITTIVITYTENSOR

The rotation tensor about the v(y) axis is

C [ SR = 0 1 0 ,

-_S 0 C_

C = cos 00, S = sin 00. (Al)

Table 1. Accuracy of the First-Order ReflectedDiffraction Efficiencies with 0 as a Parameter for the

Incidence of TE Wavesa

nr' (TE-TM)0/r 0.005 0.01 0.05 0.125 0.25nk 1.2936 1.2938 1.2989 1.3215 1.3471

m1 0.744268 0.750622 0.724238 0.682435 0.6657422 0.747725 0.756682 0.756669 0.745050 0.7616323 0.747730 0.756701 0.755995 0.746491 0.7638924 0.747730 0.756701 0.755992 0.746458 0.7636565 0.747730 0.756701 0.755992 0.746457 0.7636496 0.747730 0.756701 0.755992 0.746457 0.763649

0i = 7r - 1.136519, koyod = o.5r.

Thus the elements of (X, y, z) are given by

exx = 4[1 + b(C 2 cos 20 - S2 cos 2 0)]

+ 6{2SC sin 20 cos 0 + S2 sin 2 0 cos 2 ]},

Eyy = e[(1 + a cos 2O) - &sin2 0 cos 20],

e 22 = el[1 + 5(S 2 cos 20 - C2 cos 2 0)]

- 6[2SC sin 20 cos 0 - C2 sin 2 0 cos 203},

exy = E6(C sin 20 sin 0 + S sin2 0 sin 20),

ex, = E5[SC(cos 20 + cos2 0)-(C 2-S 2 )sin 20 cos 0

-SC sin20 cos 20],

y= E6(S sin 20 sin 0 - C sin2 0 sin 20). (A2)

K. Rokushima and J. Yamakita

Vol. 4, No. 1/January 1987/J. Opt. Soc. Am. A 33

Therefore a theoretical analysis can be performed for anarbitrary orientation of the helical axis.

APPENDIX B: APPROXIMATE TWO-WAVEANALYSIS

For the incidence of a TE wave, if we retain only the m = 0(eo, h,,) and m = +1 (e+i,, hy+,) terms in the rigorousequation (8) and eliminate him (i = y, z), we get the second-order differential equations. Neglecting further the termsd2eim/dx2, bdeim/dx, and 52eim, we get the following first-order differential equations:

dex +0 P l1 c 0,dx 2p0 1 - S0

2/,Ex 2 1 =0

-y = (..O2 :F A2 )1 /2, c ( P±1

H=2 po

1 1/2

1- sO2/EX)* (B5)

Since the powers along the x direction are EPo = Po leyo02 andMp~l = pijjez2/(1 - so 2/ex), and x represents the normal-ized space variable (kox - x), the transmitted and reflecteddiffraction efficiencies are given, respectively, by,

cult = (yo sin koyd/y) 2 , (B6)

1al= 1 + (','/^y sinh koyd) 2 (B7)

These correspond to dynamic or kinematic (ko yd << 1)theory. 6

d _ 2jAe,.l - 2 eyo = 0O (Bi)

c = - 6 sin 20,2 Ex

(B2)

Here A and c represent the deviation from the Bragg condi-tion and the effective coupling coefficients, respectively.The solutions of Eqs. (Bi) under the boundary conditions ofeo(0) = 1 and e,_i(0) = 0 for forward diffraction as well asei(-d) = 0 for backward diffraction become, respectively,

e = (cos x - J- sin yx)exp(jAx),YO~~~

e,_1 = C sin -yx exp(jAx),2,y

-y coshy(x + d) - jA sinh y (x + d)-y cosh yd - jA sinh yd

c sinhy(x + d)2(-y cosh yd - jA sinh yd)

(B3)

exp(jAx),

), (B4)

ACKNOWLEDGMENTS

The authors wish to express their thanks to K. Tominaga forhis assistance in the numerical calculations. Thanks arealso due S. Mori and M. Kominami for their valuable sugges-tions.

REFERENCES

1. K. Rokushima and J. Yamakita, "Analysis of anisotropic dielec-tric gratings," J. Opt. Soc. Am. 73, 901-908 (1983).

2. D. W. Berreman, "Optics in stratified and anisotropic media: 4X 4-matrix formulation," J. Opt. Soc. Am. 62, 502-510 (1972).

3. D. W. Berreman, "Twisted smectic C phase: unique opticalproperties," Mol. Cryst. Liq. Cryst. 22, 175-184 (1973).

4. V. A. Belyakov and V. E. Dmitrienko, "Theory of the opticalproperties of cholesteric liquid crystals," Sov. Phys. Solid State15, 1811-1815 (1974).

5. V. A. Belyakov, V. E. Dmitrienko, and V. P. Orlov, "Optics ofcholesteric liquid crystals," Sov. Phys. Usp. 22, 63-88 (1979).

6. S. Garoff, R. B. Meyer, and R. Barakat "Kinematic and dynamiclight scattering from the periodic structure of a chiral smectic Cliquid crystal," J. Opt. Soc. Am. 68, 1217-1225 (1978).

7. M. A. Peterson, "Light propagation and light scattering in cho-lesteric liquid crystals," Phys. Rev. A 27, 520-527 (1983).

8. C. Oldano, E. Miraldi, and P. Taverna Valabrega, "Dispersionrelation for propagation of light in cholesteric liquid crystals,"Phys. Rev. A 27, 3291-3299 (1983).

9. H. L. Ong and R. B. Meyer, "Electromagnetic propagation in aperiodically bent nematic liquid crystal," J. Opt. Soc. Am. 73,167-176 (1983).

10. H. L. Ong and R. B. Meyer, "Geometrical-optics approximationfor the electromagnetic fields in layered-inhomogeneous liquid-crystalline structures," J. Opt. Soc. Am. A 2, 198-201 (1985).

where

A = (Eso 2 /E, + p 12 - z)14p±,,

Po = 2so, Pt=po J: nk.

and

where

K. Rokushima and J. Yamakita