Vol. 6, No. 1/January 1989/J. Opt. Soc. Am. A 23

What happens to plane waves at the planar interfaces ofmirror-conjugated chiral media

Akhlesh Lakhtakia, Vasundara V. Varadan, and Vijay K. Varadan

Department of Engineering Science and Mechanics, The Pennsylvania State University, University Park,Pennsylvania 16802

Received May 23, 1988; accepted September 21, 1988

The plane-wave reflection and transmission characteristics of bimaterial interfaces between chiral and chiral-achiral interfaces have been extensively explored. We report on the curious characteristics of the interface formedby two chiral half-spaces, one of which is the mirror image of the other; this is referred to as problem 1. It is shownthat these characteristics are related to the reflection of plane waves on the interface of a chiral half-space and aperfectly conducting one, which constitutes problem 2.

INTRODUCTION

The lack of geometric symmetry between an object and itsmirror image is referred to as chirality,"12 and the mirrorimage of a chiral object cannot be made to coincide with theobject itself by any operation involving only rotations and/ortranslations. The most commonly investigated chiral ob-jects are the L- and the D-type stereoisomers, which arefamiliar to organic chemists. The basis for the difference inthe physical properties of the mirror conjugates lies in thehandedness, or the chirality, possessed by their molecularconfigurations. When an electromagnetic disturbance trav-els through a medium consisting of chiral molecules, it isforced to adapt to the handedness of the molecules. In otherwords, linearly polarized plane waves cannot be made topropagate through such a medium, whereas left-circularlypolarized (LCP) and right-circularly polarized (RCP) planewaves, traveling with different phase velocities, are perfectlyacceptable solutions of the vector wave equation for thisclass of medium.

The usual constitutive equations D = EE and B = ,uH donot hold for chiral media; instead, the equationsl"2

D=eE+I3eVXE, B=,uH+fl VXH (1)

are deemed applicable, with A being the chirality parameter.The regular time-harmonic Maxwell equations [exp(-jwt)]are now utilized along with Eqs. (1), and, following Bohren,3the electric and the magnetic fields are transformed into

E = QL + aRQR, H = QR + aLQL, (2)

where the LCP and the RCP fields, QL and QR, respectively,must satisfy the conditions

(V2 + kl2 )QL = 0,

V X QL = klQL,

V X QR = -k2QR,

(V2 + k22 )QR = 0,

V -QL=0,V -QR =-

In these equations, k, = k/(1 - ko) and k2 = k/(1 + ko), whilek = w(Et)"1/2 is a convenient abbreviation; aR = -j](/e)1/

2 is animpedance, and aL = -j(E/$)l/

2 is an admittance.

The plane-wave reflection and transmission characteris-tics of bimaterial interfaces between chiral and chiral-achiral interfaces have been extensively explored recently.4 -6

Here we report on the curious characteristics of the interfaceformed by two chiral half-spaces, one of which is the mirrorimage of the other; this is referred to as problem 1. We showthat these characteristics are related to the reflection ofplane waves on the interface of a chiral half-space and aperfectly conducting one; this constitutes problem 2.

PROBLEM 1

Let the space z S 0 be occupied by the chiral medium (E, ,), while the half-space z > 0 is occupied by the mirror-

conjugate medium (E, aL, -O). Either a LCP or a RCP planewave is incident upon the interface z = 0 from the zone z ' 0.It is appropriate to express the fields in the zone z < 0 by thefields

QL = AL[ey + j(-e.,al + ezK)/kl]expjj(Kx + a1Z)]

.+ BL[ey + j(eal + ezK)/kl]exp[(KX -FZ)],

z < O, (4a)

QR = AR[eY + j(ea 2 - ezK)/k2]expUj(Kx + a 2Z)]

+ BR[ey - j(exa 2 + ezK)/k 2]exp(Kx -2Z)]

z < O. (4b)

In these equations, AL and AR represent the incident planewaves, while BL and BR are the amplitudes of the reflectedones; K is the horizontal wave number required by Snell'slaws to satisfy the phase-matching condition on the inter-face z = 0; a1 = +(k, 2 - k2 )1/2 and a 2 = +(k2

2 - k 2)1/2 ; and ex,

etc. are the unit Cartesian vectors.The half-space z > 0 is occupied by the mirror-conjugate

medium; this means that phase velocities of the LCP and theRCP plane waves here are, respectively, those of the RCPand the LCP plane waves in the medium of incidence. Con-sequently, an acceptable representation of the fields in themedium of transmission is given by

0740-3232/89/010023-04$02.00 © 1989 Optical Society of America

Lakhtakia et al.

24 J. Opt. Soc. Am. A/Vol. 6, No. 1/January 1989

QL = CL[ey + j(-e.,a 2 + eK)/k 2]expU(Kx -

QR = CR[eY + j(excel - ezK)/kl]expU(Kx +

with CL and CR being the transmission coefficieThe solution of the boundary-value problem

the convenient matrix forms

FBL1 RLL RLR1 [ALl

BR RRL RRR AR

[CL1[ TLL TLR1 AL

CR[ ] TRL TRR] AR

with the R's constituting the reflection matrixconstituting the transmission matrix. The uEqs. (2), (4), and (5) in ensuring that there are ruities in the tangential E fields and the tangeracross the interface z = 0 yields

RLL =-RRR = (clk2 - a2k,)/(alk2 + a!

TLL = 2alk2 /(alk 2 + a2k,),

TRR = 2a2k,/(alk 2 + a2k,),

RLR = RRL = TLR = TRL = 0.

I-a2Z)], This curious result should be noted:, If the incident planez > 0, (5a) wave is LCP (RCP), then the reflected and the transmitted

z _ 0, (5a) waves are also LCP (RCP). There are no waves of the

axlz)], opposite handedness generated at the planar boundariesbetween mirror-conjugated chiral media. Thus, the ar-

z 2 0, (5b) rangement of problem 1 acts somewhat like a beam splitter;!nts. an incident LCP (RCP) plane wave is broken into two LCPis sought in (RCP) plane waves, which leave the interface in opposite z

is soght in directions with different amplitudes and phase velocities.

(6a) PROBLEM 2

Let the medium in the zone z 2 0 be perfectly conducting.Then a solution of the form

(6b)[LL i= rL rLR AL(8

and the T's r ][L RR] ARtilization of is to be sought. For this purpose, Eqs. (2) and (4) areio discontin- utilized to ensure the nulling of the tangential E field at theLtial H fields impenetrable surface z = 0, and the result obtained is

rLL= -rRR = (alk 2 - a2k,)/(alk2 + a2k,),

rRL= 2aLalk 2 /(aAk2 + a 2kl),

rLR= -2aRa 2k1/(aok2 + a2k,). (9)

We observe, therefore, that when a LCP or a RCP plane(7) wave hits a perfectly conducting surface, the reflected field

- 0-- QL

LCP INCIDENCE

LL

Ijj~3RL

r

RCP INCIDENCE

,1

/S.1

r LR

r RR

P , A i

rRR

Fig. 1. Illustration of the correspondence between problems 1 and 2 in accordance with Eqs. (10). Problem 1 considers the interface of a chiralmedium and a perfect conductor, while in problem 2 the two chiral media on either side of the interface are mirror conjugates.

Lakhtakia et al.

.

Vol. 6, No. 1/January 1989/J. Opt. Soc. Am. A 25-** QL - - 0 QR

LCP INCIDENCE

1

, g, fP

Perfect Conductor

rLL

+ 8E, JI

rRL

/1

/ \akyLL

aL aLrI±aRrL

RCP INCIDENCE

r; rRR

Perfect Conductor -4-

aLrLR

Fig. 2. Illustration of the imaging concept for problem 2 by using two specializations of problem 1. See Fig. 1 for the descriptions of problems1 and 2.

consists of components of both circular polarization states.However, when K = ki, then rRL = 0, rLL = -1, rLR = 2ap,and rRR = 1, which gives rise to the trivial case of no reflec-tion for the grazing LCP incidence. Likewise, when K = k2,

then rLR = 0, rRR = -1, rRL = 2 aL, and rLL = 1.

DISCUSSION

The correspondence between the solutions [Eqs. (7) and (9)]of problems 1 and 2 should be noted. That is,

RLL = rLL, RRR = rRR,

TLL = rRL/aL, TRR = -rLR/aR. (10)

For illustration, let us consider the case of LCP incidence.In problems 1 and 2, rLL is the amplitude of the reflectedLCP wave that travels with a phase velocity '/kj. In prob-lem 2, rRL is the amplitude of the reflected RCP wave with aphase velocity k/h2, but in problem 1, rRL/aL is the amplitudeof the transmitted LCP wave, which also travels with aphase velocity W/k2 because the medium of transmission isthe mirror conjugate of the medium of incidence and reflec-tion. Analogous comments also apply to the case of RCPplane-wave incidence, and both cases are schematically il-lustrated in Fig. 1. It should be noted that Fermat's princi-ple is equally well satisfied in problems 1 and 2, and inidentical fashion.

The correspondences given in Eqs. (10), coupled with thecomplementary relations

rLL(-O) = rRR(O), rRR(-() = rLL(W),

rLR(-1) = aR rRL(), rRL(-1) = aL2rLR(O) (11)

point to the imaging concept. Consider problem 2 againwith an incident LCP plane wave of unit amplitude. Rele-vant to the zone z S 0, this problem is equivalent to thesuperposition of two problems, each of which is a specializa-tion of problem 1. These two problems are (i) a problem 1 inwhich a LCP plane wave is incident upon the interface fromthe zone z ' 0 with unit amplitude and (ii) a problem 1 inwhich a RCP plane wave is incident upon the interface fromthe zone z ' 0 with an amplitude equal to aL. The case of anincident RCP plane wave in problem 2 can also be handled inthis way, and both cases are schematically illustrated in Fig.2. However, as becomes clear from Fig. 2, the use of animaging theory for chiral media is complicated for scatteringproblems in general: not only do the sources get imaged butthe medium does also.

The authors are also with the Center for the Engineeringof Electronic and Acoustic Materials, The PennsylvaniaState University, University Park, Pennsylvania.

REFERENCES

1. C. F. Bohren and D. R. Iluffman, Absorption and Scattering ofLight by Small Particles (Wiley, New York, 1983).

2. A. Lakhtakia, V. V. Varadan, and V. K. Varadan, "Field equa-tions, Huygens's principle, integral equations, and theorems for

rRR1

C, ,

rLR

, a -r

El , f

- aR

Lakhtakia et al.

1 rLL rRL

El g, 0 , - - 3

26 J. Opt. Soc. Am. A/Vol. 6, No. 1/January 1989

radiation and scattering of electromagnetic waves in isotropicchiral media," J. Opt. Soc. Am. A 5, 175-184 (1988).

3. C. F. Bohren, "Light scattering by an optically active sphere,"Chem. Phys. Lett. 29, 458-462 (1974).

4. A. Lakhtakia, V. V. Varadan, and V. K. Varadan, "A parametricstudy of microwave reflection characteristics of a planar achiral-chiral interface," IEEE Trans. Electromag. Compat. EC-28, 90-95 (1986).

Lakhtakia et al.

5. M. P. Silverman, "Reflection and refraction at the surface of achiral medium: comparison of gyrotropic constitutive relationsinvariant or noninvariant under a duality transformation," J.Opt. Soc. Am. A 3, 830-837 (1986).

6. V. K. Varadan, V. V. Varadan, and A. Lakhtakia, "On the possi-bility of designing anti-reflection coatings using chiral composi-tes," J. Wave Mater. Interact. 2, 71-81 (1987).