Elena Georgieva Vol. 12, No. 10 /October 1995 /J. Opt. Soc. Am. A 2203

Reflection and refraction at the surface of anisotropic chiral medium: eigenvalue–eigenvector

solution using a 4 333 4 matrix method

Elena Georgieva

Department of Physics of Condensed Matter, Faculty of Physics, University of Sofia,5 J. Bourchier Boulevard, Sofia 1126, Bulgaria

Received August 9, 1994; revised manuscript received February 27, 1995; accepted March 30, 1995

Fresnel reflection amplitude coefficients (fractional amplitudes) at the surface of an isotropic, intrinsicallynonmagnetic chiral medium are derived on the basis of the Drude–Condon model of optical activity. Theeigenvalue–eigenvector solution is obtained with use of Berreman’s 4 3 4 matrix method. Self-consistentresults are obtained when the calculations are based on a new 4 3 4 matrix for reflection from an isotropicchiral medium. 1995 Optical Society of America

1. INTRODUCTIONOptical activity1,2 (rotation of the polarization plane) andcircular dichroism3 (differential absorption of left andright circularly polarized light) have been used for a longtime to study the gyrotropy of matter by transmission.In this way information on internal molecular structureand gross molecular arrangements has been obtained.The field eigenstates in chiral media are generally ei-ther left circularly polarized or right circularly polarized,with each polarization having a distinct wave number.The medium, either natural or induced, distinguishes leftand right circularly polarized light. Gyrotropy is a prop-erty of a chiral medium, and it can be induced in aninitially achiral medium by an external magnetic field,as in the Faraday effect4; by an applied electric field, asin the electrogyratory effect5,6; or by mechanical stress.7

A three-dimensional chiral object is an object that can-not be brought into congruence with its mirror image bytranslation or rotation, so that a collection of chiral ob-jects will form a medium that is characterized by right orleft handedness8; i.e., gyrotropy is a property of a specialnoncentrosymmetric medium.

Gyrotropy also manifests itself in the reflection ofelectromagnetic waves from a chiral medium, and inrecent years attention concerning this reflection hasbeen revived. Reflection of electromagnetic waves atachiral–chiral planar interfaces has received particularattention.9 – 31

Silverman,10 – 12 Lalov,14 Lalov and Miteva,15 Lakhtakiaet al.,16 and Bassiry et al.17 derived the Fresnel reflectionamplitude coefficients for specular reflection at the sur-face of a chiral medium. Expressions for the reflectanceand transmittance of a wave normally incident on anisotropic, optically active slab situated in a dielectricmedium were obtained by Bokut and colleagues.18,19

Specular reflection and transmission from an opticallyactive medium have also been studied theoretically.20

Lakhtakia et al.21 – 26 did a parametric study of the mi-crowave reflection characteristics of the interface and thereflection and transmission of normally incident waves

0740-3232/95/102203-09$06.00

on a chiral slab with linear property variations. Lalovand Miteva27,28 and Cory and Rosenhonse29 studied thereflection and transmission of waves at chiral–achiralplanar interfaces. Also, experimental configurations tomeasure chiral effects in specularly reflected light, withuse of a photoelastic modulator have been proposed.30,31

Such experiments will test some fundamentals of chi-ral electrodynamics that cannot be obtained with trans-mission studies. Various applications also have beenproposed.32 – 36

The objectives of this paper are to derive the Fresnel re-flection amplitude coefficients (the fractional amplitudes)by use of a different approach that is based on a cor-rected Berreman matrix for reflection from an isotropicchiral medium. The new 4 3 4 matrix is based on theDrude–Condon model of optical activity and leads to self-consistent results. Jaggard and Sun37 derived a 4 3 4matrix technique for chiral multilayers that is also ap-plicable to a single chiral interface. The technique pre-sented in this paper is different from that of Jaggard andSun in that the reflection amplitudes are calculated forcircular polarization states rather than for linear polar-ization states.

2. DRUDE–CONDON MODEL OF OPTICALACTIVITY AND THE CONSTITUTIVEEQUATIONS OF A CHIRAL MEDIUMWe use the time-dependence factor exps1ivtd and the twoMaxwell equations in the SI system of units:

rot E ­ 2≠By≠t , (1a)

rot H ­ ≠Dy≠t , (1b)

where E, D, and H, as usual, are the vectors of the elec-tromagnetic field. To include the effect of the field onmatter, the constitutive (material) equations are supple-mented:

D ­ e0E 1 P , B ­ m0sH 1 Md , (2)

where P and M are the electric and the magnetic polar-

1995 Optical Society of America

2204 J. Opt. Soc. Am. A/Vol. 12, No. 10 /October 1995 Elena Georgieva

izations, respectively, and e0 and m0 are the permittivityand the permeability of vacuum, respectively.

Furthermore, Maxwell’s equations are generally held tobe inviolable, as also are the boundary conditions on thecontinuity of the tangential E and H fields at the bimate-rial interface. Hence the properties of matter must entersolely through the constitutive equations.

Condon38 noted that the essential feature of the ma-terial relations is that part of the polarization P be de-pendent on ≠Hy≠t and part of the magnetization M bedependent on ≠Ey≠t:

D ­ eE 2 g≠H≠t

, (3a)

B ­ mH 1 g≠E≠t

, (3b)

where g is the gyrotropic parameter. Hence the con-stitutive equations of optical activity must contain bothpolarization and magnetization terms.

According to Paul Drude,39

rot H ­ e≠E≠t

1gm

≠

≠trot E ­ e

≠E≠t

1gm

≠

≠t

√2

≠B≠t

!,

or

2rot H ­ 2iveE 1 s1ivgdsivHd . (4a)

To take into account the coupling between the electricand magnetic field quantities, which results in the polar-ization and magnetization of the medium as a result of theapplied fields, and also to fulfill the breakdown, we write

rot E ­ 2≠B≠t

1ge

≠

≠trot H ­ 2m

≠H≠t

1ge

≠

≠t

√≠D≠t

!,

or

2rot E ­ ivmH 1 s2ivgdsivEd . (4b)

The Drude–Condon model of optical activity is widelyaccepted by physicists. Microscopic models, both classi-cal and quantum mechanical,40 have been presented tojustify it, and in this work our calculations are based onthis model.

Other sets of constitutive relations are also used.When Eqs. (3a) and (1a) are combined, one obtains

D ­ esE 1 b=== 3 Ed , (5a)

and similarly,

B ­ msH 1 b=== 3 Hd , (5b)

where b ­ gyem is a pseudoscalar. Equations (5a) and(5b) are also used by some authors, and it is evident thatthe value of D (respectively, B) at any given point dependsnot only on the value of E (respectively, H) at that par-ticular point but also on the behavior of E (respectively,H) in the vicinity of this point; i.e., D (respectively, B)depends on the derivatives of E (respectively, H). Thisrelation between D and E (respectively, between B andH) is called spatial dispersion.41

Since D and E are polar vectors and B and H are axialvectors, it follows that e and m in the above equations are

true scalars and that g is a pseudoscalar; i.e., the handed-ness of the medium is manifested by the quantity g.

3. SOLVING THE PROBLEM OF SPECULARLIGHT REFLECTION AND TRANSMISSIONAT AN ACHIRAL–CHIRAL INTERFACEWITH USE OF THE 4 333 4 MATRIX METHODWe apply the general and powerful 4 3 4 matrixmethod,42 – 48 using the Drude–Condon model, to solve theproblem of reflection and transmission of polarized lightat the surface of an isotropic, spatially dispersive, non-ferromagnetic s m ­ 1d, lossless, optically active medium.

Usually the polarized light incident on the reflectivesurface is resolved into components parallel ( p light) andperpendicular (s light) to the plane of incidence, and thesetwo problems are solved independently. In the case ofreflection from gyrotropic media, the incident light withits electric vector entirely in the plane of incidence ( plight) gives rise to reflected p light and also to s lightwith a component of its electric vector perpendicular tothis plane. In much the same way, the incident s light onreflection gives rise to s and p light. Thus the reflectionproblem should be solved as a single problem and not astwo independent problems, and the 4 3 4 matrix methodis suitable for the solution of such problems because itwas developed for this purpose. On the other hand, thereflection properties of a substance can be easily repre-sented with this method by a 2 3 2 complex matrix, usedin the Jones calculus.49,50

In Cartesian coordinates Eqs. (1) can be combined intoa single matrix equation (or six differential equations, as-suming no macroscopic charge density and no macroscopiccurrent density):26666666664

0 0 0 0 2≠y≠z ≠y≠y0 0 0 ≠y≠z 0 2≠y≠x0 0 0 2≠y≠y ≠y≠x 00 ≠y≠z 2≠y≠y 0 0 0

2≠y≠z 0 ≠y≠x 0 0 0≠y≠y 2≠y≠x 0 0 0 0

37777777775

3

26666666664

Ex

Ey

Ez

Hx

Hy

Hz

37777777775­ iv

26666666664

Dx

Dy

Dz

Bx

By

Bz

37777777775, (6)

or, in brief,

RG ­ ivC . (60 )

Here R, the rotor matrix, is a 6 3 6 symmetric matrixoperator. It can be divided into four 3 3 3 matrices:

R ­

"0 rot

2rot 0

#. (7)

If we ignore nonlinear effects, a linear relation existsbetween G and C as

C ­ MG , (8)

Elena Georgieva Vol. 12, No. 10 /October 1995 /J. Opt. Soc. Am. A 2205

Fig. 1. Schematic diagram of polarized light reflected from anisotropic, optically active medium, showing the coordinate systemfor the surface and the directions of the electric fields for incidentand reflected light. Generally, two elliptical polarizations aretransmitted in an optically active medium with different anglesof refraction and wave vectors k1 and k2.

where the 6 3 6 matrix M possesses all the optical prop-erties of the medium. The matrix M , called the opticalmatrix, may be written in the form

M ­

"e r

r 0 m

#, (9)

where e ­ sMij d and m ­ sMi13,j13d are the dielectric andthe permeability tensors, respectively, and r ­ sMi,j13dand r0 ­ sMi13,j d are the optical rotation tensors si, j ­1, 2, 3d.

For our case of isotropic media the constitutive parame-ters e, m, r, and r0 are scalar quantities.

According to Eqs. (4a) and (4b),

r ­ 1ivg ­ 1ig, r0 ­ 2ivg ­ 2ig , (10)

which place the constitutive equations in the form putforward by Jaggard et al.,37 where g is the chirality

admittance.When C from Eq. (8) is replaced in Eq. (60 ),

RG ­ ivMG . (11)

On the other hand, G can be divided into the time-dependent part expsivtd and the spatial part G. ThenEq. (11) becomes

RG ­ ivMG , (12)

which is the spatial wave equation for frequency v.We assume that the plane of incidence is the x–z plane,

so the propagation vector in the y direction ky ­ 0, and

any changes of the field components in the y directionare absent s≠y≠y ­ 0d. Our problem is the reflection andtransmission of a monochromatic plane electromagneticwave, incident obliquely at an angle u1 from an opticallyinactive medium sZ , 0d with coefficient of refraction n1

on an isotropic optically active medium sZ . 0d with coef-ficient of refraction n2, for which M is a function only ofZ (Fig. 1). We are dealing with monochromatic solutionswith a time dependence exps1ivtd and a dependence onx of exps2ivjxycd; i.e., exps2ikxxd ­ exps2ivjxycd andj ­ ckxyv ­ ck sin u1yv ­ n1 sin u1. So

≠y≠x ­ 2ijvyc . (13)

Therefore Eq. (12) becomes26666666664

0 0 0 0 2≠y≠z 00 0 0 ≠y≠z 0 ivjyc0 0 0 0 2ivjyc 00 ≠y≠z 0 0 0 0

2≠y≠z 0 2ivjyc 0 0 00 2ivjyc 0 0 0 0

37777777775

3

26666666664

G1

G2

G3

G4

G5

G6

37777777775­ iv

26666666664

´ 0 0 ig 0 00 ´ 0 0 ig 00 0 ´ 0 0 ig

2ig 0 0 1 0 00 2ig 0 0 1 00 0 2ig 0 0 1

37777777775

26666666664

G1

G2

G3

G4

G5

G6

37777777775.

(14)

Equation (14) may be written as four first linear differen-tial equations and two linear algebraic equations in which

G1 ­ Ex, G2 ­ Ey , G3 ­ Ez, G4 ­ Hx, G5 ­ Hy , G6 ­ Hz .

Then we follow Berreman’s work46 to construct the 4 3

4 matrix D that contains the optical parameters of thereflecting material. For D we find

D ­

266666666664

0 1 2n2

2 sin2 u2

n22 2 g2

2ig

0@1 1n2

2 sin2 u2

n22 2 g2

1A 0

n22 0 0 2ig

ig 0 0 1

0 ig

0@1 1n2

2 sin2 u2

n22 2 g2

1A n22

0@1 2n2

2 sin2 u2

n22 2 g2

1A 0

377777777775. s15d

Matrix (15) is different from Berreman’s matrix,46,49

which is based only on Drude’s equation (4a) and does nottake into account Eq. (4b); i.e., it describes only one sideof the problem. Berreman’s matrix leads to inequality ofthe off-diagonal Fresnel reflection amplitude coefficients,which is not correct (see Appendix A).

If we use the two algebraic equations to eliminate Ez

and Hz, the remaining four differential equations can bewritten as a single matrix equation:

≠

≠z

266664Ex

Hy

Ey

2Hx

377775 ­ 2ivD

266664Ex

Hy

Ey

2Hx

377775 , (16)

2206 J. Opt. Soc. Am. A/Vol. 12, No. 10 /October 1995 Elena Georgieva

which will be abbreviated

≠cy≠z ­ 2ivDc , (160 )

with the column vector

cszd ­

266664Ex

Hy

Ey

2Hx

377775 .

D is the 4 3 4 matrix that depends on the elements ofe, the elements of the optical rotation tensors r and r0,and j.

Equation (16) contains only the tangential componentsof E and H; and, as we know, they are usually used inmatching boundary conditions when one is calculatingreflection coefficients.

We denote the dependence of c on the z coordinate asexps2ikzzd ­ exps2ivzzycd, and the solution of Eq. (16)is zc ­ Dc. So our problem is an eigenvalue problem.The eigenvalues z will allow us to calculate the z compo-nent of the propagation vector skzd, and the eigenvectorsc will give us the tangential (x and y) components of thefields for each polarization state. Then the z componentsof the fields may be calculated from the two linear alge-braic equations.

In the case of complex n2 (i.e., absorption), angle u2

will also be complex, and it cannot be interpreted asthe angle of refraction for complex n2. No mathematicaldifficulties arise in that case.51

From matrix D we find for our problem the followingeigenvalues:

z ­ 6sn22 cos2 u2 1 g2 6 2gn2d1/2. (17)

From Eq. (17) we need to take only the two positivevalues of z , because these represent light with a positivez component of velocity. To simplify our future calcula-tions we do the substitutions

d ­ n22 cos2 u2 1 g2 2 2gn2 , (18)

s ­ n22 cos2 u2 1 g2 1 2gn2 . (19)

Therefore the two eigenvalues that we need may bewritten in the abbreviated form

zd ­ d1/2, zs ­ s1/2. (20)

The eigenvectors associated with these eigenvalues arefound from the nonlinear system of equations based onEq. (15). The system has a nonzero solution because itsdeterminant is zero:

For d polarization

r1 ­ cd1 ­ Ex ­ 1,

r2 ­ cd2 ­ Hy ­ n2sn2 2 gdyp

d ,

r3 ­ cd3 ­ Ey ­ 2isn2 2 gdyp

d ,

r4 ­ cd4 ­ 2Hx ­ 2in2 ;

For d polarization

r1 ­ cs1 ­ Ex ­ 1 ,

r2 ­ cs2 ­ Hy ­ n2sn2 1 gdyp

s ,

r3 ­ cs3 ­ Ey ­ isn2 1 gdyp

s ,

r4 ­ cs4 ­ 2Hx ­ in2 . (21)

The first column of these eigenvectors we call the d

polarization, and it represents left elliptically polarizedlight; i.e., it has an electric vector that rotates counter-clockwise for an observer looking into the beam. Thesecond column of eigenvectors (the s polarization) rep-resents right elliptically polarized light. However, thedifference in the amplitudes of the r1 and r3 compo-nents arises because of oblique incidence. The modes arecircularly polarized at normal incidence. In this waythe problem of reflection and refraction is solved as aneigenvalue–eigenvector problem.

The linear combination of these two eigenvectors givesthe total transmitted light into the sample:

ct ­ k1cd 1 k2cs , (22)

where ct denotes the vector containing the components ofthe total transmitted fields and constants k1 and k2 areproportional to the amplitudes of the waves for the twocomponent polarizations.

The next step of our problem is to include the boundaryconditions in it; i.e., at the boundary between the achiraland the chiral materials the tangential components of theE and the H fields must be continuous:

ci 1 cr ­ ct . (220 )

Usually the reflection problems at oblique incidence aresolved by resolving the incident and reflected fields intocomponents parallel to the plane of incidence ( p light)and components perpendicular to this plane (s light). Soif ci stands for the incident light, Ep,i and Es,i are theamplitudes of the electric fields of incident light for thesetwo polarizations (p and s, see Fig. 1). Therefore wewrite

ci ­ Ep,i

266664cos u1

n1

00

377775 1 Es,i

266664001

n1 cos u1

377775 . (23)

Similarly for the reflected light the two polarizations( p and s) may be represented as

cr ­ Ep,r

2666642 cos u1

n1

00

377775 1 Es,r

266664001

2n1 cos u1

377775 . (24)

If we apply the four boundary conditions in order torelate the six quantities Ep,i, Es,i, Ep,r, Es,r, and the co-efficients k1 and k2, we obtain four linear homogeneousequations. These may be represented in matrix form:

Elena Georgieva Vol. 12, No. 10 /October 1995 /J. Opt. Soc. Am. A 2207

266664cos u1 0 2 cos u1 0 cd1 cs1

0 1 0 1 cd3 cs3

n1 0 n1 0 cd2 cs2

0 n1 cos u1 0 2n1 cos u1 cd4 cs4

377775

3

26666666664

Ep,i

Es,i

Ep,r

Es,r

2k1

2k2

37777777775­ 0 . (25)

In the large matrix the second and third rows are de-liberately interchanged. Also, for the next calculationsthe large matrix is broken into six 2 3 2 matrices. SoEq. (25) may be written in abbreviated form:

"M1 M2 M3

M4 M5 M6

#264Ei

Er

k

375 ­ 0 , (250 )

where

Ei ­

"Ep,i

Es,i

#, Er ­

"Ep,r

Es,r

#, k ­

"2k1

2k2

#.

The matrix that we need is the reflection matrix; it isfound by elimination of k:

M1Ei 1 M2Er 1 M3k ­ 0 ,

M4Ei 1 M5Er 1 M6k ­ 0 ,

or, finally,

fM3M621M5 2 M2g21fM1 2 M3M6

21M4gEi ­ Er .

So the reflection matrix r in Er ­ rEi is

r ­ fM3M621M5 2 M2g21fM1 2 M3M6

21M4g . (26)

Carrying out the detailed calculations, we obtain thefollowing results for the Fresnel reflection amplitudecoefficients:

rpp ­2

pds n1n2 1 fsn2

pd 1 n2

ps 1 g

pd 2 g

ps dsn1

2 2 n22d 2 2n1n2 cos u1sn2

2 2 g2dgcos u1

2p

ds n1n2 1 fsn2p

d 1 n2p

s 1 gp

d 2 gp

s dsn12 1 n2

2d 1 2n1n2 cos u1sn22 2 g2dgcos u1

, s27d

rps ­ i2n1n2fsd 2 sdsn2

2 1 g2d 1 2gn2sd 1 sdgcos u1

2p

ds n1n2 1 fsn2p

d 1 n2p

s 1 gp

d 2 gp

s dsn12 1 n2

2d 1 2n1n2 cos u1sn22 2 g2dgcos u1

, s28d

rsp ­ 2i2n1n2fsd 2 sdsn2

2 1 g2d 1 2gn2sd 1 sdgcos u1

2p

ds n1n2fsn2p

d 1 n2p

s 1 gp

d 2 gp

s dsn12 1 n2

2d 1 2n1n2 cos u1sn22 2 g2dgcos u1

, s29d

rss ­22

pds n1n2 1 fsn2

pd 1 n2

ps 1 g

pd 2 g

ps dsn1

2 2 n22d 1 2n1n2 cos u1sn2

2 2 g2dgcos u1

2p

ds n1n2 1 fsn2p

d 1 n2p

s 1 gp

d 2 gp

s dsn12 1 n2

2d 1 2n1n2 cos u1sn22 2 g2dgcos u1

. s30d

We note that when g ­ 0, then d ­ s, rss and rpp

reduce to the usual p and s Fresnel reflection amplitudecoefficients, and the off-diagonal elements of the reflectionmatrix (rsp and rps) become zero. Figures 2–7 are plotsof rss, rss

2, rpp, rpp2, rsp ­ rps, and rsp

2 ­ rps2 versus angle

of incidence u1, with Eqs. (27)–(30) used for Bi12SiO20

(g ­ 1025, n ­ 2.583, l ­ 550 nm).

4. DISCUSSIONThe 4 3 4 matrix analysis shows that two elliptical wavesare refracted into the chiral medium. These two waveshave different indices of refraction and therefore propa-gate with different velocities. The two indices of refrac-tion for the gyrotropic medium must satisfy

n62 ­ j2 1 z 2 ­ n1

2 sin2 u1 1 n22 cos2 u2 1 g2 6 2gn2

or

n6 ­ sn22 1 g2 6 2gn2d1/2 ­ 6sn 6 gd , (31)

where the plus in front of the parentheses has physicalsignificance only.

The difference in refractive indices is

Dn ­ n1 2 n2 ­ 2g . (32)

Using the two algebraic equations, we can resurrect thez components of the fields. For d polarization,

Ez ­ 2in2g sin u2

n22 2 g2

Ey 2n2 sin u2

n22 2 g2

Hy ­ 2n2 sin u2p

d,

Hz ­n2

3 sin u2

n22 2 g2 Ey 2 i

gn2 sin u2

n22 2 g2 Hy ­ 2i

n22 sin u2p

d.

(33)

Finally, the components of the electric and magnetic fieldsfor d polarization are

Ex ­ 1, Hx ­ in2 ,

Ey ­ 2in2 2 g

pd

, Hy ­n2sn2 2 gd

pd

,

Ez ­ 2n2 sin u2p

d, Hz ­ 2i

n22 sin u2p

d. (34)

For s polarization,

Ez ­ 2in2g sin u2

n22 2 g2

Ey 2n2

2 sin u2

n22 2 g2

Hy ­ 2n2 sin u2

ps

,

Hz ­n2

3 sin u2

n22 2 g2

Ey 2 in2g sin u2

n22 2 g2

Hy ­ 1in2

2 sin u2p

s.

(35)

2208 J. Opt. Soc. Am. A/Vol. 12, No. 10 /October 1995 Elena Georgieva

Fig. 2. Coefficient of reflection rpp versus angle of incidence u1for Bi12SiO20 (g ­ 1025, n ­ 2.583, l ­ 550 nm).

Fig. 3. Coefficient of reflection srppd2 versus angle of incidenceu1 for Bi12SiO20 (g ­ 1025, n ­ 2.583, l ­ 550 nm).

Fig. 4. Coefficient of reflection rss versus angle of incidence u1for Bi12SiO20 (g ­ 1025, n ­ 2.583, l ­ 550 nm).

Fig. 5. Coefficient of reflection srssd2 versus angle of incidenceu1 for Bi12SiO20 (g ­ 1025, n ­ 2.583, l ­ 550 nm).

Fig. 6. Coefficient of reflection srpsd2 ­ srspd2 versus angle ofincidence u1 for Bi12SiO20 (g ­ 1025, n ­ 2.583, l ­ 550 nm).

Fig. 7. Coefficients of reflection rps and rsp ­ 2rps versus angleof incidence u1 for Bi12SiO20 (g ­ 1025, n ­ 2.583, l ­ 550 nm).

Elena Georgieva Vol. 12, No. 10 /October 1995 /J. Opt. Soc. Am. A 2209

So the components of s polarization are

Ex ­ 1, Hx ­ 2in2 ,

Ey ­ in2 1 g

ps

, Hy ­n2sn2 1 gd

ps

,

Ez ­ 2n2 sin u2

ps

, Hz ­ 1in2

2 sin u2p

s. (36)

For both polarizations E ? H ­ 0.Now we find the angle between the propagation vector

and the surface normal for the two refracted waves:

tan u6 ­j

z­

n2 sin u2

sn22 cos2 u2 1 g2 6 2gn2d1/2

­tan u2p

1 1 sg2 6 2gn2dysn22 cos2 u2d

. (37)

Finally, we calculate the Poynting vector for each po-larization, taking the cross product of the real parts ofthe complex fields and remembering that each field com-ponent is multiplied by exps1ivtd.

For d polarization we find that

Sx ,n2

2sn2 2 gdsin u2

d,

Sy , 0 ,

Sz ,n2sn2 2 gd

d1/2,

but for s polarization they are

Sx ,n2

2sn2 1 gdsin u2

s,

Sy , 0 ,

Sz ,n2sn2 1 gd

s1/2.

5. CONCLUSIONUsing the 4 3 4 matrix method we derived the Fresnelreflection amplitude coefficients for an achiral–chiral in-terface at an arbitrary angle of incidence. The electricand magnetic amplitudes for the two circular polariza-

tions propagating in the chiral medium, their angles ofrefraction, indices of refraction, and Poynting vectors werealso found. The 4 3 4 matrix for reflection from an op-tically active medium is a corrected Berreman matrix,with the same coupling in the constitutive relation for themagnetic field as appears for the electric field taken intoaccount.

APPENDIX AIn this appendix are given the corresponding formulasderived from Berreman’s matrix for reflection from anisotropic optically active medium:

D ­

26666640 cos2 u2 2ig sin2 u2 0

n22 0 0 2ig

0 0 0 1

0 ig n22 0

3777775 . (A1)

The corresponding eigenvalues are

z ­ 6

0@2n02 cos2 u2 1 g2 6

p4n0

2g2 1 g4

2

1A1/2

, (A2)

d ­12

s2n02 cos2 u2 1 g2 2

p4g2n0

2 1 g4 d , (A3)

s ­12

s2n02 cos2 u2 1 g2 1

p4g2n0

2 1 g4 d , (A4)

zd ­ d1/2, zs ­ s1/2, (A5)

r1 ­ cd1 ­ 2d1/2sd 2 g2 2 n02 cos2 u2d ,

r2 ­ cd2 ­ n02sd 2 n0

2 cos2 u2d ,

r3 ­ cd3 ­ 2in02g ,

r4 ­ cd4 ­ 2id1/2n02g ,

r1 ­ cs1 ­ Ex ­ 2s1/2ss 2 g2 2 n02 cos2 u2d ,

r2 ­ cs2 ­ Hy ­ 2n02ss 2 n0

2 cos2 u2d ,

r3 ­ cs3 ­ Ey ­ in02g ,

r4 ­ 2Hx ­ is1/2n02g .

So with the use of Eq. (A1) we find for the Fresnelreflection amplitude coefficients

rss ­

n1ahfn22scos2 u1 2 cos2 u2d 2 an1 cos u1 2 bgb 1 n2

4 cos2 u1 cos2 u2j 1 n22 cos u1bc 1 f 2 n1

2 cos u1sdn22 cos2 u2 2 ed

n1ahfn22scos2 u1 1 cos2 u2d 1 an1 cos u1 1 bgb 1 n2

4 cos2 u1 cos2 u2j 1 n22 cos u1bc 1 f 1 n1

2 cos u1sdn22 cos2 u2 2 ed

,

rpp ­

n1ahfn22scos2 u1 2 cos2 u2d 1 an1 cos u1 2 bgb 1 n2

4 cos2 u2j 2 n22 cos u1bc 2 f 1 n1

2 cos u1sdn22 cos2 u2 2 ed

n1ahfn22scos2 u1 1 cos2 u2d 1 an1 cos u1 1 bgb 1 n2

4 cos2 u1 cos2 u2j 1 n22 cos u1bc 1 f 1 n1

2 cos u1sdn22 cos2 u2 2 ed

,

rsp ­

i2gn1n2

2 cos u1sb 1 n22 cos2 u2d

n1ahfn22scos2 u1 1 cos2 u2d 1 an1 cos u1 1 bgb 1 n2

4 cos2 u1 cos u2j 1 n22 cos u1bc 1 f 1 n1

2 cos u1sdn22 cos2 u2 2 ed

,

rps ­ i2n1 cos u1sb 1 n2

2 cos2 u2dsdn22 cos2 u2 2 ed

ghfn1ab 1 n22 cos u1sb 1 n2

2 cos2 u2d gfn1 cos u1a 1 b 1 n22 cos2 u2g 1 n1

2 cos u1sdn22 cos2 u2 2 b2 2 bg2dj

,

2210 J. Opt. Soc. Am. A/Vol. 12, No. 10 /October 1995 Elena Georgieva

where

a ­ s1/2 1 d1/2,

b ­ s1/2d1/2,

c ­ s1/2d1/2 1 2n22 cos2 u2 ,

d ­ s 1 d 2 g2 2 n22 cos2 u2 ,

e ­ sd 1 g2s1/2d1/2,

f ­ n26 cos u1 cos4 u2 .

At g ­ 0, rsp and rps become zero, while rss and rpp re-duce to the usual Fresnel amplitude reflection coefficients.

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