JOSA COMMUNICATIONSCommunications are short papers. Appropriate material for this section includes reports of incidental researchresults, comments on papers previously published, and short descriptions of theoretical and experimental tech-niques. Communications are handled much the same as regular papers. Galley proofs are provided.

Reflection of plane waves at planar achiral-chiral interfaces:independence of the reflected polarization state from the

incident polarization state

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

Center for the Engineering of Electronic and Acoustic Materials, Pennsylvania State University, 227 HammondBuilding, University Park, Pennsylvania 16802

Received September 21, 1989; accepted March 16, 1990

Reflection of plane waves at planar achiral-chiral interfaces has been examined in order to broaden, and toreinterpret, the Brewster law. New relations between the Fresnel reflection coefficients for this interface have alsobeen obtained.

INTRODUCTION

In a celebrated paper published in 1815,1 Sir David Brewsterdescribed his experiments on the reflection of unpolarizedlight from planar dielectric-dielectric interfaces. Data col-lected by him gave rise to what is now called the Brewsterangle and was condensed by him into the Brewster law: Ifunpolarized light is incident under this angle, the reflectedlight is plane polarized, and the angles of reflection andtransmission add up to 90 deg.

It is the purpose of this Communication to broaden theconcept of the Brewster law by examining the reflection ofplane waves at planar achiral-chiral interfaces.2 3 Planewaves propagating toward (or away from) a planar interfacecan be expressed in terms of two distinct and orthogonaleigenmodes.4 For the case studied here, it is shown that acondition exists when the ratio of the amplitudes of the twoeigenmodes of the reflected wave is independent of the ratioof the amplitudes of the two eigenmodes of the incidentwave. This condition, obtained in terms of a horizontalwave number K, corresponds to the usual Brewster law forachiral-achiral interfaces.5 Additionally, new relations be-tween the Fresnel reflection coefficients for this interfacewill be obtained.

ANALYSIS

Consider the interface z = 0: an isotropic, homogeneous,lossless, achiral medium occupies the half-space z < 0; whilethe half-space z ' 0 is filled with an isotropic, homogeneous,lossless, chiral medium.

The constitutive relations in the achiral half-space are D= £aE and B = uzH, where Ea and dua respectively, are thepermittivity and the permeability. Plane waves in theachiral half-space are linearly polarized.4 5 Thus the appro-priate representation of the electromagnetic fields in thisregion is given as

E = [A, ey + All(-aaex + ce,)/k]exp[i(KX + aaZ)I

+ [Bey + Bll(aaex + Kez)/kaJexp[i(Kx -aaZ)I,z < 0, (la)

7aH = [-Alley + A, (-aaex + Kez)/ka~exp[i(KX + craZ)]

+ [-Biley + B, (aaex + Keez)/ka]exp[i(Kx - aaZ)],

z < 0. (lb)

It is noted that the major premise of this Communicationwill not be affected if the fields in the achiral half-space areexpressed in terms of circularly polarized eigenstates.

In Eqs. (la) and (lb) and hereafter, the impedance a -

[(ia/ea)/2, the wave number ka = [(,uaea) 1 2, while anexp(-icot) time dependence has been assumed. The coeffi-cients A and All represent plane-wave eigenmodes incidenton the interface, while B 1 and B represent plane-waveeigenmodes reflected off the interface. Finally, K is thehorizontal wave number required by Snell's law to satisfythe phase-matching condition of the interface z = 0; aa =+[(ka2

- K2)]1/

2; and e, etc., are the unit Cartesian vectors.

On the other hand, the chiral medium has the constitutiverelations D = (E + BV X E) and B = s(H + BV X H); here e,Au, and j3 are the permittivity, the permeability, and thechirality parameter in the Drude-Born-Fedorov represen-tation.2 6 It is noted that other constitutive relations arealso possible but are equivalent to the ones used here fortime-harmonic fields.2 6 The chiral medium is circularlybirefringent; hence the plane waves in the region z > 0 arecircularly polarized and may be represented by using thevectors2 ' 7

Q = B[ey + i(-ale + Ke)/yl]exp[i(Kx + az)]

+ Al[ey + i(ale + Kez)/l]exp[i(K - az)],

z 0, (2a)

0740-3232/90/091654-03$02.00 © 1990 Optical Society of America

Vol. 7, No. 9/September 1990/J. Opt. Soc. Am. A 1655

Q2 = B21ey + i(a 2eX - KeC)/-y2]exp[i(KX + a2Z)]

+ A2[ey - i(C2 e, + Ke)/72]exp[i(KX-X2Z)]

z 0. (2a)

Here, Q, and Q2, respectively, represent left-circularly andright-circularly polarized plane waves. The wave numbersare given by 71 = k/(1 - K/) and Y2 = k/(l + k3); k = [(p)]1/2is merely a shorthand notation, while a1 = +[(7y2- 2)11 2and Ce2 = +[(Y2

2- K2

)]1/2

. The coefficients A, and A2 repre-sent plane-wave eigenmodes incident on the interface, whileB1 and B2 denote the plane-wave eigenmodes reflected offinto the chiralhalf-space. The electromagnetic fields in thisregion are given by2 6

E = Q1 -iQ 2 , H Q 2 -(i/ 7 7 )Q1, Z 0 (2c)

with -o = [E]1/2

The boundary-value problem is solved by ensuring thecontinuity of the tangential components of the E and the Hfields across the interface z = 0. For a given K, the resultingsolution is best stated in matrix notation as follows:

r_ 1F, T1T1 2 Al1[BLl 1 r21 r22]LAII + [T21 T22][A2]' (3a)

LB2 I L 2 1 R 2 LA1 [t 2 l t 22 Al (3b)The various Fresnel reflection and transmission coefficientsinvolved in the foregoing matrices are given as follows:

Arl, = -(77a - 72)(4 + 42) - 277a7l(4142 - 1),

Ar22 = (77a2 - 72)(4 + 42) - 27a(412 - 1),

Ar12 = -Ar 2 l = 2Vna77(4i - 42)-

At,, = 277(7742 + 7a),

At12 = 2 i-q(1a42 + ),

At 2 l = 2i(q7t1 + 77a),

At2 2 = 2 (77ati + ),

AR,1 = (7a2 + 772)( - 42) + 27a7(4142 - 1),

AR2 2 (77a2 + ?2)(4- 42) + 27a7(4142 - 1),

AR1 2 -2i74 2 (.qa - 772),

AR21 = 2ij( 7 a2 - 772)/f,?

AT,, = 4-7a1q(7742 + 'la),

AT12 = -4iq7 %7 a 2(7 174 + fla),

AT21 = 4 ina41 (Ma42 + 7),

AT2 2 = -4ja2(7a41 + 7),

A = (1a+2 + 2) ( +42) + 2ia77(M142 + 1),

4m = (ambym)I(ajaka), m = 1,2.

Suppose now that Al = A 2 = 0, so that incidence is fromthe achiral side only. The condition on the horizontal wavenumber K such that the ratio (BI/B1M) is independent of the

ratio (AI/Al) can be obtained easily by following Ref. 4 (p.246) and is given by

(4)rl2r2l = rr22,

which can be succintly expressed as

(71a2 + 72)(41 + 42) - 2?a77(4142 + 1) = .

Therefore, if R satisfies Eq. (5), the field reflected into theachiral half-space will be linearly polarized regardless of thestate of the polarization of the plane wave that is incidentfrom the achiral side. In other words, the reflection ratio(B/B[l) is completely independent of the incidence ratio(AI/All), if Eq. (5) is satisfied. If Eq. (5) holds, then thereflection ratio is fixed at

B11Bl = (7a2 - 2)(1 + 42) + 27a77(4142 - 1)]

X [2ina7 (4i - 42)1 (6)

In accordance with the usual practice, the angle arctan(x/aa), if real, can be called the Brewster angle for the achiral-chiral interface.5 It is a simple matter to verify that Eq. (5)does indeed give rise to the usual Brewster angles5 9 forachiral-achiral interfaces by setting ,B = 0.

Pertinent to the present purposes, however, Eq. (5) comesas the answer of another question too! Let A1 = All = 0, sothat incidence is from the chiral side only. In order that thereflection ratio (Bl/B2 ) be independent of the incidence ratio(A1 /A2), the condition

(7)

must be satisfied. But Eq. (7) also reduces to Eq. (5), and, ifEq. (5) holds, the reflection ratio turns out to be fixed at

B1B2 = [(71a2 + n2)(4 - 42) + 2?nlf(4142 1)]

X [2itj(72

- n2)/n]-1. (8)

Thus Eq. (5) broadens the concept of the Brewster law forachiral-chiral interfaces, regardless of which half-space theincidence is from.

In passing, it is noted here that Eq. (5) reduces to themuch simpler form

(41-1)(42-1) = (9)for impedance-matched achiral-chiral interfaces,'( i.e., if fa= 77.

FRESNEL REFLECTION COEFFICIENTS

For dielectric-dielectric interfaces, Azzam1 has given a rela-tionship between the Fresnel reflection coefficients that isindependent of the angle of incidence. This relationshiphas been broadened for the more general achiral-achiralinterfaces as well.9

After some algebra, by using the Fresnel coefficients de-rived in the previous section, we can show that

(rj, - r22)(1 - rr 22 + r 2 r2dY 1 = (,02 - a2) (72 + n 2)l

(10)

for any horizontal wave number K. Of course, if / = 0, thenr12 = r2l = 0, and Eq. (10) reduces to the expression given inRef. 9.

JOSA Communications

(6)

R12R2 = R1R22

1656 J. Opt. Soc. Am. A/Vol. 7, No. 9/September 1990 JOSA Communications

For incidence from the chiral side, the analog of this rela-tion [Eq. (10)] is only slightly more complicated and takesthe form

[(R12/i) + (iqR21)][ 1 - R1R22 + (R12/in)(i 7R 2 1)]

= (72 - 77a 2)(72 + 77a2Y1. (11)

ACKNOWLEDGMENT

The authors are also with the Department of EngineeringScience and Mechanics, Pennsylvania State University, 227Hammond Building, University Park, Pennsylvania 16802.

REFERENCES

1. D. B. Brewster, "On the laws which regulate the polarization oflight by reflection from transparent bodies," Philos. Trans. R.Soc. London 105, 105-129 (1815).

2. A. Lakhtakia, V. K. Varadan, and V. V. Varadan, Time-Har-monic Electromagnetic Fields in Chiral Media (Springer-Ver-lag, Berlin, 1989).

3. A. Lakhtakia, V. V. Varadan, and V. K. Varadan, "A parametricstudy of microwave reflection characteristics of a planar

achiral-chiral interface," IEEE Trans. Electromagn. Compat.EC-28, 90-95 (1986).

4. H. C. Chen, Theory of Electromagnetic Waves (Wiley, NewYork, 1983).

5. G. P. Sastry and S. Chakrabarti, "The generalised Brewstercondition from the extinction theorem," Eur. J. Phys. 8, 125-127 (1987).

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

7. A. Lakhtakia, V. V. Varadan, and V. K. Varadan, "Field equa-tions, Huygens's principle, integral equations, and theorems forradiation and scattering of electromagnetic waves in isotropicchiral media," J. Opt. Soc. Am. A 5, 175-184 (1988).

8. S. Bassiri, C. H. Papas, and N. Engheta, "Electromagnetic wavepropagation through a dielectric-chiral interface and through achiral slab," J. Opt. Soc. Am. A 5, 1450-1459 (1988).

9. A. Lakhtakia, V. V. Varadan, and V. K. Varadan, "Relations forthe Fresnel reflection coefficients of a bimaterial interface inde-pendent of the angle of planewave incidence," Int. J. InfraredMillimeter Waves 9, 631-634 (1988).

10. A. Lakhtakia, V. K. Varadan, and V. V. Varadan, "Influence ofimpedance mismatch between a chiral scatterer and the sur-rounding chiral medium," J. Mod. Opt. 36, 1385-1392 (1989).

11. R. M. A. Azzam, "Relationship between the p and s Fresnelreflection coefficients of an interface independent of angle ofincidence," J. Opt. Soc. Am. A 3, 928-929 (1986).

Grating Diffraction

FEATURE EDITORS

Thomas K. Gaylord

Elias N. Glytsis

School of Electrical EngineeringGeorgia Institute of TechnologyAtlanta, Georgia 30332-0250

GRATING DIFFRACTIONAnalysis of dielectric lamellar gratings coated with anisotropic layers Shizuo Mori, Kazuhiro Mukai, 1661

Jiro Yamakita,Katsu Rokushima

Electromagnetic theory of gratings: some advances and some comments on R. Petit, M. Cadilhac 1666the use of the operator formalism

Improvement of the Kirchhoff approximation for metallic and dielectric H. Faure-Geors, D. Maystre 1675surfaces

Theoretical and numerical study of gratings consisting of periodic arrays of thin R. Petit, G. Tayeb 1686and lossy strips

Computer-aided algorithm based on the Yasuura method for analysis of T. Matsuda, Y. Okuno 1693diffraction by a grating

Solving slab lamellar grating problems by the singular-value-decomposition Neal C. Gallagher, 1701method Allen J. Whang

Analysis of diffraction from doubly periodic arrays of perfectly conducting Amir Boag, Yehuda Leviatan, 1712bodies by using a patch-current model Alona Boag

Diffraction from metal strip gratings with high spatial frequency in the infrared Thomas Schimert, 1719spectral region Robert Magnusson

Analysis of a strip-grating twist reflector S. Sohail H. Naqvi, 1723Neal C. Gallagher

Spectral behavior of anomalies in deep metallic gratings E. Popov, L. Tsonev, E. Loewen, 1730E. Alipieva

Perfect blazing for transmission gratings M. Neviere, D. Maystre, 1736J. P. Laude

Diffraction of Hermite-Gaussian beams from a sinusoidal conducting grating Toshitaka Kojima 1740Optical modulation by a traveling surface acoustic wave and a holographic D. A. Larson, T. D. Black, 1745

reference grating M. Green, R. G. Torti,Y. J. Wang, R. Magnusson

System characterization of apodized acousto-optic Bragg cells Ronald . Pieper, 1751Ting-Chung Poon

Linearly focusing grating coupler for integrated-optic parallel pickup S. Ura, Y. Furukawa, 1759T. Suhara, H. Nishihara

Experimental study of local and integral efficiency behavior of a concave E. G. Loewen, E. K. Popov, 1764holographic diffraction grating L. V. Tsonev, J. Hoose

Deviation of second-order focal curves in common plane-symmetric Christopher Palmer 1770spectrometer mounts

Constant-dispersion grism spectrometer for channeled spectra Wesley A. Traub 1779Photonic band structure: the face-centered-cubic case E. Yablonovitch, T. J. Gmitter 1792

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