Field equations, Huygens’s principle, integral equations, and theorems for radiation and scattering of electromagnetic waves in isotropic chiral media

Vol. 5, No. 2/February 1988/J. Opt. Soc. Am. A 175

Field equations, Huygens's principle, integral equations, andtheorems for radiation and scattering

of electromagnetic waves in isotropic chiral media

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

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

Received December 22, 1986; accepted September 29, 1987

Several aspects of electromagnetic wave propagation and scattering in isotropic chiral media (D = eE + OEV X E, B= uH + 03tV X H) are explored here. All four field vectors, E, H, D, and B, satisfy the same governing differential

equation, which reduces to the vector Helmholtz equation when /3 = 0. Vector and scalar potentials have been

postulated. Conservation of energy and momentum are examined. Some properties, consequences, and computa-

tionally attractive forms of the applicable infinite-medium Green's function have been explored. Finally, themathematical expression of Huygens's principle, as applicable to chiral media, has also been derived and employedto set up a scattering formalism and to establish the forward plane-wave-scattering amplitude theorems. Several of

the results given that pertain to the field equations and Green's dyadic are available for constitutive equations other

than those mentioned above; these results, along with some others, have been given now for the above-mentioned

constitutive equations. The derivations of Huygens's principle and other developments described here have notbeen given earlier, to our knowledge, for any pertinent set of constitutive equations. With advances in polymer

science, the formalisms developed here may be useful in the utilization of artificial chiral dielectrics at subopticaland microwave frequencies; application to vision research is also anticipated.

1. INTRODUCTION

A chiral medium is characterized by either a left-handednessor a right-handedness in its microstructure. As a result,left- and right-hand circularly polarized fields propagatethrough it with differing phase velocities,1-3 the field withthe latter polarization propagating through a right-handedmedium faster than the left-hand circularly polarized fieldand vice versa. In the optical frequency range, many organ-ic molecules exhibit what is termed optical activity, which isa manifestation of the native chirality of these molecules.4 5

Since the measures of optical activity are material specific,in the past century and a half they have been extensivelyused by physical chemists to characterize molecular struc-ture.

In order to describe the electromagnetic properties of iso-tropic chiral media, the usual constitutive equations, D EEand B = /,u are simply inadequate because they admit of asingle phase velocity that is generally frequency dependent.Several sets of constitutive equations have, however, beenproposed in order to surmount this difficulty; it will beshown below that transformations exist between the varioussets, generally for the time-harmonic case.

Considerations of optical activity by Born 6 have lead tothe proposal that

D = (E + i1V X E),

B =,uH,

(la)

(lb)

being the chirality parameter. A medium following Eqs.(la) and (lb) is nonreciprocal.7 8 Also, after studying thereflection and transmission characteristics of planarachiral-chiral interfaces, Silverman 9 has shown that the fail-ure of Eqs. (la) and (lb) to satisfy the standard boundary

conditions results in physically unacceptable amplitudesaround the critical angles of incidence. Born's proposal forisotropic chirality has been modified to

D = e[E + 03V X E],

B = A[H + f3V X H],

(2a)

(2b)

a set that not only is symmetric under time-reversality8 andduality transformations9 but also satisfies the tests set up bySilverman9 10 to distinguish between Eqs. (la) and (lb) andEqs. (2a) and (2b); 3 is the measure of chirality here. Thevalidity of Eqs. (2a) and (2b) has been affirmed by studiescarried on optically active molecules 1 ' as well as from theexamination of light propagation in optically active crys-tals.12-1 4 In the past decade, Bohren has used these consti-tutive relations to compute the scattering responses of chiralspheres,1 5 spherical shells,'6 and infinitely long, right circu-lar cylinders.1 7 Recently, Lakhtakia et al. followed up onthe work of Bohren and investigated the interaction of elec-tromagnetic fields with planar achiral-chiral interfaces2 aswell as those with nonspherical chiral objects embedded inachiral host media.18

It is to be noted here that Condon's equations 4"19 for opti-cal activity,

D = EcE - H/at,

B = ,pcH + X9E/Ot,

(3a)

(3b)

are equivalent to Eqs. (2a) and (2b), provided that harmonictime dependence, exp[-iwt], is assumed7 ; in that case, thecorrespondence between Eqs. (2a) and (2b) and Eqs. (3a)and (3b) is given by x = egu,/(1 - W2 /2), EC = e/(1 - W2 EgI32),

and /'c = ,/(1 - w2CA

2). Furthermore, a takeoff from Telle-gen's formulation for the gyrator 20 gives rise to yet another

0740-3232/88/020175-10$02.00 © 1988 Optical Society of America

Lakhtakia et al.

176 J. Opt. Soc. Am. A/Vol. 5, No. 2/February 1988

set of constitutive equations 2 '-2 3 for isotropic reciprocalchiral media,

D = TE + H, (4a)

B = MH - E. (4b)

This set is also equivalent to Eqs. (2a) and (2b) for the time-harmonic case, provided that the chirality parameter =iWepo/( - w2qi132), eT = E/(1 - 2e,0 2), and UT = /(1 -o02(eA,2).

A remarkable set of equations was deduced by Post,' whodid not consider the medium microstructure at all but sim-ply required all equations to be generally covariant. Hisproposal,

D = epE + iB, (5a)

B = y[H - iE], (5b)

was also obtained 2 4 phenomenologically by considering achiral medium composed of a dilute suspension of perfectlyconducting, short helices embedded in an otherwise achiralhost medium. The validity of Eqs. (5a) and (5b) has alsobeen tested by studies conducted on optically active mole-cules." 25 These constitutive equations were recently usedby Bassiri et al.

2 6 to derive an infinite-medium Green's func-tion and to calculate the radiation field of a short electricdipole embedded in a chiral medium. Equations (5a) and(5b) are similar to those set up by Cheng and Kong2 7 forbianisotropic media. The correspondence between Eqs.(2a) and (2b) and Eqs. (5a) and (5b) holds with ' = WEo, ep =e, and up = ,/(1 - W

2eC1 2) for the time-harmonic fields.

Perhaps the most complete exposition of the electromag-netic field theory has been done by Kong and his co-workers(e.g., Refs. 27 and 28) for bianisotropic media, whose consti-tutive equations are in tensor form and were given by Post'as

cD = 3 E + c B (6a)

H=V-E+c B, (6b)

in which c is the speed of light in vacuum and the Germanletters denote 3 X 3 matrices in a Cartesian coordinate sys-tem. Most of this work is summarized in a paper by Kong,28

wherein he has shown that Eqs. (6a) and 6(b) can be recast ina bianisotropic Tellegen form also.29 For time-harmonicfields, reciprocity and symmetry conditions have been ex-amined, and conditions for lossless propagation have beendetermined. The conservation of energy and electromag-netic momentum has also been established. Last, an infi-nite-medium Green's dyadic has also been formulated, al-though it is given as an integral in the (three-dimensional)spatial-frequency space. The reduction of the various ex-pressions derived in this paper2 8 for isotropic chiral mediawith the isotropic Post constitutive equations is certainlypossible. Most of this work, however, has been applied tomoving30 or stationary, isotropic or biaxial or uniaxial media[e.g., Refs. 31-34].

The merits of the various constitutive equations givenabove are debatable, and they have been shown to be equiva-lent to each other for time-harmonic fields: whereas theequations of Post, Condon, and Tellegen preserve the per-meability IBI/IHI, Eqs. (2a), (2b), (5a), and (5b) have the

same permittivity, IDI/IEI. As was mentioned above, thesubstantial work of Kong and his colleagues with bianiso-tropic media can certainly be applied to the study of waves inisotropic chiral media, although the resulting isotropic equa-tions will be of the Post type [Eqs. (5a) and (5b)]. However,the asymmetry associated with optical activity is immedi-ately apparent in Eqs. (2a) and (2b) as opposed to the otherconstitutive equations: the curl is not a vector under areflection of coordinate systems. 7' 35 Hence in the presentstudy and for our future work we decided to use Eqs. (2a)and (2b).

Although the quantum-mechanical aspects of chiralityhave been given much thought, 4 5"19 a systematic study of theclassical electromagnetic field theory in isotropic chiral me-dia has been lacking, except in the work of Kong and hiscolleagues, which, however, is for bianisotropic media [Eqs.(6a) and (6b)]. With modern advances in polymer science,however, there is reason to believe that artificial chiral di-electrics, active at the millimetric wave frequencies, maybecome feasible.1836-38 This possibility motivated Enghetaand Mickelson39 to explore the transition radiation causedby a chiral plate, since they believe that antennas coatedwith chiral materials may have significantly interesting radi-ation characteristics. If the utilization of artificial chiraldielectrics at suboptical frequencies is realized, then it willbe necessary to examine all aspects of the pertinent field-theoretic relationships. Thus the progress in constructingchiral media that are active at suboptical frequencies is theprime mover for the present work.

The results described in this paper may have some impacton vision research as well. Several different types of mediain the eye have been shown to be anisotropic 40 ; in particular,the retina fibers are thought to be uniaxial, 4 ' whereas arecent model of the cornea has posited the cornea to besimilar to biaxial crystals.42 The structural anisotropy hasbeen used by de Vries et al.

4 0 to explain the difference insensitivity of eyes to left and right circularly polarized light.However, it has been shown that when a rod is illuminatedend-on with linearly polarized light, dichroism is not ob-served because the chromophore dipole of rhodopsin rotates;both vertically and horizontally polarized light componentsare absorbed equally (Ref. 43, p. 36). In addition, biologicalmedia are made up of macromolecules, which are generallyhanded. Thus the retinylidene chromophore of rhodopsinhas two distinct circular dichroism bands at 335 and 487nm.44'45 In fact, it turns out that the circular dichroismmeasurements of rhodopsin suggest that the visual pigmentscontain a high a-helical content (Ref. 43, Chap. 8). Further-more, the primary stroma of the cornea is composed of colla-gen, and it is known that the individual collagen moleculehas chirality.46 The triple helix formed from three collagena chains is right handed; hence fibrils and most tissues,layers, and bundles must be handed. However, the issue isprobably even more complex, possibly with the handednessalternating from left to right in consecutive layers.47 Thusnot only should any treatment of the physics of the eyeinclude the anisotropy that is due to the microstructure-rods, cones, ganglions, etc.-but it should also consider thenanostructure, e.g., the helicities of the component mole-cules whose dimensions may be significant fractions of theoptical wavelengths.

In Section 2 it will be shown that all the four field vectors

I I

Lakhtakia et al.


satisfy the same governing differential equation, which re-duces to the vector Helmholtz equation when 13 = 0; vectorand scalar potentials will be derived along with the specifica-tion of a Coulomb-like gauge condition48 ; the reaction theo-rem will be set up; and the conservation of energy and mo-mentum will also be examined. In Section 3 an expressionof the infinite-medium dyadic Green's function will be ob-tained for the constitutive equations (2a) and (2b) from thatof Bassiri et al.

2 6 for the isotropic Post media [Eqs. (5a) and(5b)]. Some properties and consequences of the Green'sfunction will be then explored, and computationally tracta-ble forms will be set up. Finally, in Section 4 the mathemat-ical expression of Huygens's principle as applicable to chiralmedia will be calculated and used to set up a scatteringformalism and to derive the forward-scattering-amplitudetheorems.4 9 It should be mentioned here that several of theresults derived in Sections 2 and 3 are available for constitu-tive equations other than Eqs. (2a) and (2b), but not all ofthem have been derived for any one set of constitutive equa-tions. Thus Sections 2 and 3 form a compendium of results,some old and the others new, but all for the same set of theconstitutive equations [Eqs. (2a) and (2b)]; in this sense,these sections should be regarded as the counterpart ofKong's comprehensive paper on bianisotropic media.28 Onthe other hand, the derivations of Section 4, to our knowl-edge, have not been given previously for any set of constitu-tive equations.

2. FIELD EQUATIONS

A. PreliminariesConsider an unbounded, source-free region V occupied by anisotropic chiral medium in which the constitutive relations[Eqs. (2a) and (2b)] hold in accordance with the discussion inSection 1. It is necessary to realize that a chiral volume willexhibit its handedness only while interacting with time-varying fields, although chirality can be introduced into anotherwise achiral medium simply by subjecting it to a staticmagnetic field (Ref. 5, Chap. 1). In addition, it should benoted that E, ,, and 1 may be frequency dependent; unlessotherwise stated, a harmonic time dependence exp(-iwt)has been assumed.

Use of the first two monochromatic Maxwell equations, inconjunction with Eqs. (2a) and (2b), easily yields the factthat all fields, E, H, D, and B, are divergenceless.5 0 Fur-thermore, the latter two Maxwell equations, when usedalong with the constitutive equations, yield the equivalentmonochromatic constitutive relations,

(1 - k212)D = eE + i(1/cw)k2H, (7a)

(1 - k212 )B = AH - i(/c)k 2E, (7b)

where k = co[ey]l/2 is simply a shorthand notation and doesnot represent any wave number. It must be mentioned thatEqs. (7a) and (7b) are similar to the isotropic Tellegen equa-tions (4a) and (4b).

Four other relations dealing with the circulation of thefield quantities may also be derived for later use from Eqs.(2a), (2b), (7a), and (7b):

V X E = y2OE + iWt(,/k) 2 H, (8a)

V X H = 'y2OH - ice(/k) 2E, (8b)

(9a)

(9b)

V X D = y2 D + iE('Yk) 2B,

V X B ='Y 20B - it(-y/k) 2 D.

The parameter -y is given by

y2 = k2[1 - k 2 12]-' = 02(E[1 - . 202E,]-1. (10)

It is easy to show that when the curl of both sides of Eq.(8a) is taken, the relation

V2 E + y213V X E + i/(NY/k) 2 V X H = 0 I (11)

is obtained. Some more manipulations of Eq. (11) involvingEqs. (8a) and (8b) finally yield the differential equation thatgoverns the electric field:

V2E + 2y 20,V X E+ -2 E =0. (12)

In fact, by utilizing similar procedures,5 it can be shownthat Eq. (12) is satisfied by all the fields, viz.,

V2 U + 2,Y20V X U + y 2U = 0, U = E, H, D, B. (13)

It should be mentioned that hereafter U will denote either ofthese four fields. Also to be noted is the fact that Eq. (13)reduces to the vector Helmholtz equation, V2U + k2U = 0,when 13 = 0.

B. Vector and Scalar PotentialsSeveral radiation problems for achiral media are considera-bly simplified by the use of scalar and vector potentials. 48"52Although the potentials pertaining to chiral media will notbe needed in the subsequent parts of this paper, they arebriefly discussed here for the sake of completeness.

By proceeding in the usual manner,4 8 it is observed thatthe Maxwell equations, V X E = iwB and V X H = -iwD, canbe satisfied jointly by a vector magnetic potential A and ascalar electric potential V; consequently,

B = V X [A + OV X A],

E = iw[A + 3V X A] - VV,

(14a)

(14b)

D = ie[A + 2V X A + 2V XV X A]-eVV, (14c)

H = y-'V X A. (14d)

The use of Eqs. (14c) and (14d) in the equation V X H =-icoD then gives rise to the differential relation

(k/y)2 V2 A + 2k2 V X A + k2A

+ V[iow V - (k/y) 2 V - A] = 0. (15)

Provided that the gauge condition,

iott/eV -(k/y) 2V A = 0, (16a)

is satisfied, it can be shown that A satisfies the same vectordifferential governing Eq. (13) that U conforms to and thatV satisfies the modified wave equation

[V2 + Y2] V = 0. (16b)

To be noted here is the fact that A is not divergenceless,unlike the other fields U. It should also be mentioned herethat the Coulomb gauge for isotropic achiral media is simplyV *A = 0.

In the same fashion, it is also easy to prescribe a vectorelectric potential F and a scalar magnetic potential W as

Lakhtakia et al.


E = e'V X F, (17a)

H=-i[F+OVXF] +VW. (17b)

Provided that F and W also satisfy the gauge condition(16a), it can be shown that F satisfies Eq. (13) while Wsatisfies Eq. (16b).

C. ReciprocityConsider a chiral medium (, /1, d) in which the electric andthe magnetic sources -Ja, -Kaj, distributed or localized,give rise to the fields Ea, Ha, Ba, Dal; consider another set ofsources {-Jb, -Kb} that independently create the fields Eb,

Hb, Bb, Dbl. As dictated by Eqs. (8a) and (8b), provided thatboth sets of sources operate at the same frequency,

V X Ea b = y2OEa b + iu(-y/k) 2Hab + Kab, (18a)

-V X Ha b = 'Y 21Ha,b + ie(y/k) Ea,b + Jab- (18b)

Next, consider the same sets of sources but in an alteredmedium (e', ,u', 1'), whence

V X Ea,b' = zy203'Ea~b' + i,'(y'/k') 2Ha b' + Ka b, (19a)

-V X Hab' =- 'Y'2o Ha,b' + iWfE('Y'/1k)2Eb' + Jab- (19b)

From (19) and (20) it can be shown that

-V (Ha' X Eb) = [iw'('/k') 2Ea' Eb + i ('Y/k)2 Ha' Hb

+ (2 - y,'2 1')Ha' Eb]+ J. Eb + Kb * Ha', (20a)

-V (Hb X Ea') = [iwf(y/k)2E.' * Eb + i'W'(,'y/k')2Ha' Hb

- (20 - 'y'20)Ea 0 * Hb]

+ Jb Ea' + Ka * Hb- (20b)

If E = e', ,u = u', and 1 = 1', then the terms grouped insidethe square brackets in Eqs. (20a) and (20b) are equal to eachother, implying that an isotropic chiral medium with theconstitutive properties [Eqs. (2a) and (2b)] is reciprocal orself-complementary in the Krowne sense.21 Consequently,after dropping the primes and integrating over all space, weget 4 9

J I llspace d'x(Jb . Ea - Kb . Ha)

= f f f d3x(Ja Eb-Ka Hb)- (21)JJall space

The derivation of Eq. (21) involves the radiation conditionthat at the boundary of all space the fields are zero. Thislast equation is nothing but Rumsey's reaction theorem.53It is therefore clear that in the application of the reactiontheorem, an isotropic chiral medium does not differ from anisotropic achiral medium.54

D. Conservation of Energy and MomentumFormally, on an instantaneous basis, the electromagneticenergy conservation law should have the form

V * $ + dW/at - Pmech 0, (22)

in which the first term, V , represents the energy flowingout of a given (differential) volume per unit time; aW/at is

the time rate of change of the stored electromagnetic energywithin the same volume; and Pmech represents the total workdone by the fields on the sources included in that volume. is the Poynting vector. When the medium is nonchiral, i.e.,when = 0 and and ,u are assumed to be real, then

(23a)V3 = E x H,

W = (1/2)[E D + H B]

= (1/2)[D D/e + B B/,u],

Pmech = J * E = J - D/e,

in which J = pv is an impressed source current densityconsisting of a charge density p, moving with a velocity v.

In order to derive similar expressions for chiral media, weproceed in the usual fashion48 with the later two Maxwellequations to obtain

V (E X H) + H (B/Ot) + E (D/t) + J - E = 0. (24)

Now, after substitution of Eqs. (2a) and (2b) and some ma-nipulation, it can be shown that

H (B/dt) = (1/2){a/,9tI[H * B] + (1/2),40V [H/at} X H],(25a)

E (D/at) = (1/2)d/at[E D] + (1/2)E3V [E/at} X E],

(25b)

and Eq. (24) can be recast as

V [E X H - (/2)(aH/at} X H + e{dE/dt} X E)]

+ (d/dt)[(1/2){E * D + H B}] + [J E] = 0. (26a)

This equation is identical in form to Eq. (22), but the inter-pretation of $ is different from that in Eq. (23a) because ofthe inclusion of the time derivatives of E and H; further-more, W and Pmech now have definitions that are same as thefirst set of definitions in Eqs. (23b) and (23c), respectively.Further manipulation of Eq. (26a), however, leads to theelimination of the time derivatives on which the divergencealso operates, yielding

V [E X H] + (/at)[(1/2){D * D/e + B B/,u}]

+ [J D/E] = 0. (26b)

In this second expression, it should be noted that the form of$ has been restored to that in Eq. (23a); however, W andPmech now have definitions that are the same as the secondset of definitions in Eqs. (23b) and (23c), respectively.Whereas the two sets of definitions of I3, W, and Pmech fromEqs. (26a) and (26b) are identical if 1 = 0, they are notobviously so for isotropic chiral media.

For time-harmonic fields, the conservation-of-energyprinciple can be similarly obtained. Corresponding to Eqs.(26a), this principle can be stated as

V P + (1/2) Reliw(E * D* - H - B*)} + (1/2) Re{E * J*} = 0,(27a)

whereas, corresponding to Eq. (26b), it is given as

V P + (1/2) Reliw(D - D*/e - B B*/,u)}

+ (1/2) Re{D * J*/E} = 0, (27b)

(23b)

(23c)

Lakhtakia et al.


where P = (1/2) ReIE X H*i is the usual time-averaged-harmonic Poynting vector.55 56

Finally, in this section the conservation of electromagneticmomentum is considered.4 8 It can be shown from Maxwell'sequations and the constitutive relations (2) that

JXB+ (V-D)E+ (a/8t)[D X B] =e[E(V-E) -EX (VXE)]

+ ,u[H(V H)-H X (V X H)]. (28)

However, V - D = P,, the charge density, whereas J = pvv.Hence

JXB+(V-D)E=pJE+VXB], (29)

which is simply an expression of the Lorentz force. Conse-quently, any volume integral of the left-hand side of Eq. (29)

must be interpreted as the time derivative of the sum of themechanical momenta of all charges included in that volume,i.e.,

(a/at)Gmech = f d'xp[E + v X B]. (30)

This immediately suggests that the volume integral of (a/at) [D X B] must be the electromagnetic momentum Gem,

0 2(r, r') = (k/87r)[y23 + 72-VV - V X a]g(y2; R), (34c)

(34d)g(r; R) = exp[icR]/R,

with the wave numbers

/1 = k[1 - k1]-',

72 = k[1 + ko]-l.

This is the Green's dyadic in three dimensions, and the one-and the two-dimensional forms are given in Appendix A. Itmust be mentioned that only values of 1 that cause both yjand 2 to have finite, positive real parts are to be consideredhere. In the limiting cases in which ko = 1, the Green'sdyadic is given by

@(r, r')k,=,, = -(1/82rk)[-(k/2)3 - (k/2)-1VV ± V X 3]

X g(k/2; R). (36)

Two reciprocity properties of the dyadic Green's functionshould be noted, viz.,

.(r, r') = [(r', r)]T, (37a)

V X (r, r') = [V' X @(r', r)]T,

(35a)

(35b)

Gem = J d3x[D X B]. (31)

Finally, the right-hand side of Eq. (28) can be used to

define the Maxwell stress tensor Z as applicable to chiralmedia. The Cartesian components of £ are defined as

(£)nm = e[EnEm - (1/2)E * Eb,,]

+ A[HnHm - (1/2)H - Hnm], n, m = 1, 2, 3,

(32)

6nm being the Kronecker delta function. In terms of thedefinitions (30)-(32), the principle of conservation of mo-mentum can then be compactly set down as

(a/at) [Gmech + Gem]n = 1 d 3 x( /aXm)()nm,m=1,2,3

n = 1, 2,3, (33)

which will reduce to the usual statement for nonchiral mediaif = 0.

3. PROPAGATION IN UNBOUNDED CHIRALMEDIA

A. Dyadic Green's FunctionA study of electromagnetic wave propagation requiresknowledge of the infinite-medium Green's function, whichis, of course, nothing but a field propagator. In order to finda dyadic Green's function for the constitutive equations (5a)and (5b), Bassiri et al.

2 6 utilized dyadic algebra.57 By usingthe mapping existing between Eqs. (5a) and (5b) and Eqs.(2a) and (2b) mentioned in Section 1, it is easy to show thatthe Green's dyadic, pertinent to Eqs. (2a) and (2b) is given as

®(r, r') = ®i(r, r') + 0 2(r, r'), (34a)

@,(r, r') = (k/87r,) ['y,3 + ,yl-'VV + V X $]g(y,; R), (34b)

(37b)

in which the superscript T denotes transpose. Further-more,58

(38a)V X @1(r, r') = yl®,(r, r'),

V X 232(r, r') = -72@ 2(r, r'),

which shows that the isotropic chiral media are circularlybirefringent.

The rotational properties [Eqs. (38)] can be utilized toyield yet another decomposition of the dyadic Green's func-tion. By using Eqs. (34a) and (38), it can be shown that

3(r, r') = rI(r, r') + r2 (r, r'),

F,(r, r') = (1/8ir)[ + y-2VV]g(,yl; R),

r 2(r, r') = (1/8r) [ + 7y2-2VV]g('y2; R);

(39a)

(39b)

(39c)

along with this decomposition, however, the auxiliary condi-tions

V X F,(r, r') = yPr(r, r'),

V X I2 (r, r') = -Y2r 2 (r, r')

(40a)

(40b)

must be explicitly mentioned, since they are not built intothe definitions (39).

Obviously, if 1 = 0, then, either from Eq. (34) or from Eq.(39), the Green's dyadic will have the familiar form,

5(r, r') = (1/47r)[3 + k- 2VV]g(k; R), (41)

for isotropic, achiral media. It must be pointed out that thecircular birefringence of the chiral media [see Eqs. (38) and(40)] has been obtained here not from examining the plane-wave propagation, as is commonly done, but from theGreen's function; this implies that all kinds of fields-nearas well as far fields and fields with spherical, cylindrical,planar, or any other wave fronts-are circularly birefringent.

The admissible plane-wave solutions of Eq. (13) are, re-

(38b)

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spectively, left circularly polarized (LCP) and right circular-ly polarized (RCP):

U1 = ( + i)exp(i-yjz), (42a)

U2 = (x - iy)exp(iy 2z)- (42b)

If dl is assumed to have a positive real part (right-handedmedium), then U1 propagates with a slower phase velocity;otherwise U2 is the slower of the two waves. In addition, inunbounded, isotropic chiral media, these plane waves aretransverse electromagnetic (TEM). The properties of U,and U2 were examined in great detail by Silverman andSohn,3 to which source we refer the interested reader. Itmust, however, be emphasized here that combining U1 andU2 in any fashion cannot lead to linearly polarized planewaves unless a = 0, because the LCP and RCP plane wavestravel with different phase velocities. In an unboundedchiral medium, the LCP and RCP plane waves can propa-gate without interfering with each other. But, when a waveof either polarization encounters a boundary, mode conver-sion takes place; the scattered field then consists, in general,of waves of both (circular) polarizations. 15-18 Unattenuatedpropagation of both the LCP wave and the RCP wave occurs,provided that both k and are real. If Im(k) + Iki2 Im(0) =0, then 'yj is real, and U, traverses the chiral medium withoutlosing any energy. On the other hand, if Im(k) - kJ2 Im(0)= 0, then 7Y2 is real, and U2 propagates through the mediumwithout suffering any attenuation.

B. Useful Forms of Green's DyadicFor numerical computational procedures such as the spec-tral-domain iteration method59 and the method of mo-ments,60 the dyadic Green's function can be exploited easilyby using the relations

VVg(cr; R) = [iuR- -R-21(- 3RRR- 2 )

- (/R) 2RR] g(a; R),

V X [g(a; R)] = iR' -R-21g(a; R)R X a(43)

(44)

in the definitions [Eqs. (34)]. The resulting expressions for8(r, r') are too lengthy for inclusion here; nevertheless, they

are relatively straightforward to work with.For semianalytical procedures, such as the T-matrix

method,18 6' the Green's function must be expanded in termsof global basis functions. In that case, it turns out to bemore convenient to use the decomposition [Eqs. (39)] ofGreen's dyadic, which yields

rP(r, r') = (iyl/27r) E DnmL,()(r,)L,(1)(rj,

r2 (r, r') = ( 2/27r) V DnmR,(3)(r>)Rv(')(r<),

where MW(i) and N( 1) are the vector spherical wave func-tions.63 It should be noted that V X L,(O) = yLi(j) and V XRVP) = - 2RW); hence Eqs. (45) automatically satisfy Eqs.(40).

4. SCATTERING IN CHIRAL MEDIA

A. Huygens's PrincipleTo investigate the scattering response of an obstacle embed-ded in a chiral medium, it is necessary to formulate Huy-gens's principle for such media. Therefore consider thesourceless volume V bounded by the surfaces S and S.Using the second Green's identity for vectors U and W =@(r, r') - a in Ve, we obtain

J J J|| d 3xjU (V X V X W) - (V X V X U) W

= | |ss d'xj~en X (V X U) W + ( X U) (V X W)J, (47)

where en is the unit normal shown in Fig. 1. By letting a be aconstant vector, this equation can be stated in a more rele-vant form as

J J J d3x{U. (V X V X 3)-(V X V X U) .

=| fs+ d2xjnX(VXU) @3+(enXU)-(VXX)1. (48)

It is assumed that in Ve there are no sources; consequently,(V X V X U) = y2 [U + 2V X U] [see Eq. (13)], and (VXVX

) = y2[@ + 20V X @] + S6(r-r'). By using these relationsin Eq. (48), it is easy to show that

J J | d3x{U(r) ~S(r -r')

=-2 72# J | d2x. [U(r) X @(r, r')]

+ J J d2xIpe X [V X U(r)] @(r, r')

s+sX

+ [ X U(r)] [V X @(r, r')]}.

(45a)

(45b)

(49)

So.

where r> and r<, respectively, are the greater and the lesserof r and r' in magnitude. The functions LWJ) and R,(j) aredefined in terms of the vector spherical harmonics MW andNV(j) and are given by15,16,62

L^(')(r) = M,(i)(yr) + N,(i)(-yjr),

R,(')(r) = Mf'j)(72r) - N(j)(72r),

(46a)

Fig. 1. Relevant to Huygens's principle.

Lakhtakia et al.

(46b)


However, the integral on the infinitely large and far-awaysurface S. goes to zero, since the Green's dyadic decays withincreasing distance; furthermore, en [U(r) X @(r, r')] = [nX U(r)] @(r, r'). In addition, the Dirac delta function canbe used to eliminate the integral on the left-hand side of Eq.(49). Finally, by also utilizing the properties [Eqs. (37)] ofthe Green's dyadic [Eqs. (39)], Huygens's principle can bestated simply by the twin relations

The properties of the scatterer must also be taken intoaccount. If Vi were to be impenetrable, then the reductionsof Eqs. (51) and (52) are self-evident. For a permeable Vi,on the other hand, let the discussion specialize to U = E orH. In this case, the boundary conditions should be suchthat S remains charge neutral as well as current neutral.Consequently,

e X [V+ X U(r)] = en X [V X Uint(r)], (53a)

U(r') = -2Y2/B J | d2x@(r', r) - [n X U(r)]

+ f f d 2 xI@(r', r) (en X [V X U(r)])

+ [V' X @(r', r)] [ X U(r)]}, r' E V, (50a)

0 = -2y 2o fJ | d2x@3(r', r) [ X U(r)]

+ | | d2x 1@3(r', r) - (en X [V X U(r)])

+ [V' X @(r', r)] [ X U(r)]}, r' e Vi. (50b)

Thus it i's possible to compute the field U in any source-freeregion, provided that the tangential components of U and VX U(r) are known on the surfaces enclosing the volume ofinterest.

B. Scattering FormalismThe development of the Huygens's principle contains theseeds of a scattering formalism pertinent to isotropic chiralmedia. In Section 3, we let Vi denote a scatterer upon whicha field Uinc is incident. In accordance with the equivalenceprinciple,64 the incident field will cause the creation of sur-face currents, en X [V+ X U(r)] and e, X U+(r), on S+, theexterior surface of the obstacles. If Vi were to be perfectlyconducting and U were to represent either E or H, then &, XE+ would be identically zero. On the other hand, if Vi wereto be electromagnetically permeable, then these currentswould be merely equivalent surface currents. 18 64 In eithercase, the effect of these surface currents would be to null theincident field inside the obstacle. Hence

-Uinc(r') = -2y 20 J J| d2 x@(r', r) [en X U+(r)]

+ J f d 2x(@(r', r) * {en X [V+ X U(r)]}

+ [V' X @3(r', r)] * [en X U+(r)]), r' e Vi (51)

constitutes an integral equation for the surface fields; assuch, it is the extinction theorem 61'65 for scattering in iso-tropic chiral media. Once it is solved, the scattered field canbe simply computed through Eq. (50a) as

USc(r') = -2y2J J | d2x@(r', r) [en X U+(r)]

+ J J d2 x(@(r', r) - {en X [V+ X U(r)]}

+ [V' X @(r', r)] * [en X U+(r)]), r' e Vi. (52)

e X U,(r) = en X Uint(r), r S (53b)

should be substituted into Eqs. (51) and (52). The resultingsets of equations can then be solved most easily, for example,by the T-matrix procedure.18,65

C. Far-Zone Scattered Field and the Plane-Wave-Scattering MatrixSince Huygens's principle [Eqs. (50a) and (50b)] applies toany field that does not have sources in Ve in Fig. 2, it can beused to obtain a scattering matrix that characterizes thefrequency response of the obstacle volume Vi. Thus, byusing Eqs. (8), (37), (39), and (40) in Eq. (50a), it is easy toshow that

Es(r/) = (y 2/k) J J d2x{F1 (r', r) - F2(r', r) [ X ESC(r)]

+ (iCOYy2/k2) | | d2 xIl (r', r) + P2 (r', r))

[n X Hsc(r)], r' e Ve. (54)

Now, in the far zone, the scattered field must be complete-ly TEM; it will consist, in general, of both LCP and RCPwaves, as is evident from Eq. (54). To evaluate the radiationfield in a direction , the asymptotic forms4 9 of F, and F2 areto be used in Eq. (54):

]P,(r', r) S(1/87r){exp[iylr']/r'}exp(-i-l * r), (55a)

F2(r', r) (1/8r)exp(iY2r')/r'exp(-i-y2 * r), (55b)

n being a vector of unit magnitude. Substitution of relations(55) into Eq. (54) gives rise to the simple relationship,

ESC(r') = exp(iyr')/r'1Ft 1(i) + exp(iY2 r')/r'}Ft 2 (i0), (56)

where

Fig. 2. For the far-zone scattering amplitude.

Lakhtakia et al.


Fti(n) = (iy 2/8wE'7r) | | d2 xen X JHS(r) - (iwe/k)ESC(r))

(57a)

Ft21) = (2

/8We7r) J J d2 xen X HSC(r) + (iwe/k)ESC(r)I

X exp[-i-y 2 r] (57b)

are the far-zone amplitudes of the LCP and RCP scatteredfields, respectively, expressed as functions of the scatteredfield on the surface S of the obstacle.

The scattered fields on the surface are simply parts of thesurface current densities, e X H+ and & X E+, excitedbecause of the irradiation of Vi by an incoming field, as inSubsection 4.B. Let the incident field be a LCP plane wave,where

Elinc(r) = e, exp[i-yll r], (58a)

H1i1n(r) = -(i1e/k) 1 exp(iyll r), (58b)

el * 11 = , (58c)

11 X el = -ie 1 , (58d)

a RCP plane wave, where

E2 inc(r) = 2 exp[iY2172 - r], (59a)

H2inc(r) = (ie/k)e 2 exp[i' 2YA2 * r], (59b)H2 ~~~~~~~~~~(59c)

e2 - e12 = , (59c)

fl2 X 2 = ie2 , (59d)

or a combination of both. From the far-zone point of view,then, plane-wave-scattering matrices nm, (n, m = 1, 2) canbe construed 4 9 in the following fashion:

F,,(1) = Sl(fIMI) * e + S12(0le1k2) (60a)

FtAn) = 21( 1 hi) * e1 + S 22 I172) *2- (60b)

An interesting property of these scattering matrices comesfrom the reciprocal nature of the chiral medium. Let {E, HIbe the total fields existing in V owing to the sum of theincident fields Eminc, HminCIlm=1,2 as in Eqs. (58) and (59); let$E", H"} be the total fields existing in V independentlyowing to the sum of the incident fields Em//inC, Hm'inCIIm=1,2,which resemble Eqs. (58) and (59). By using the reciprocityrelations, it can be shown that4 9

E |, | d2X[{en X H(r)l -Em"inc(r)

+ Ign X E`(r)j * Hm.-inc(r)]

J J ' d2 x[{&n X H"'sc(r)I * Eminc(r)m=1,2

+ 1~n X E-E1 c(r)j. Hminc(r)]. (61)

Substitution of the explicit expressions [Eqs. (58) and (59)]for the incident fields and the use of Eqs. (60) transforms Eq.(61) into

[e," * (21(-1''1l) * P - P, * 1251(-111') * e,']

+ [Pl 12(fll 1) -e2 (521(fl21'7l) * P11

+ [P2' * e21(-12 i1 1) -el-e12(-1O12 ) * Pll]+ [e2l (*22(fl2 I'7 2 - 2(-172I2 ) b2] = 0,

(62)

which implies that

S22( 12 112) = [22(-2 )1 72) ],

S12( 11 '172) = [(21( 12 I l )] T

S21(V172IllM) = [512( 1l2 )]T.

(63a)

(63b)

(63c)

(63d)

The essence of this argument is that the supermatrix S,given by [ S11 12

% = 21 (R22J(64)

is symmetric, and the scattering formalism of Subsection 4.Bsatisfies reciprocity constraints imposed on it by the self-complementary nature of the medium filling the externalvolume V, regardless of the the kind of medium that consti-tutes the scattering object Vi. It should be noted that i12 =(21 = 0 and (25 = 22 are symmetric matrices if = 0, andthey conform to the result of de Hoop66 for isotropic achiralmedia.

D. Forward-Plane-Wave-Scattering AmplitudeTheoremsAs in the case of the isotropic achiral media, here, too, theforward-plane-wave-scattering amplitude contains informa-tion about the loss suffered by an incident plane wave in theforward direction. Now, the time-averaged power scatteredby the obstacle volume Vi is given by

Pr = (1/2)Re {J | d 2 x * ESC X Hsc*}, (65a)

whereas the time-averaged power absorbed in the scattereris

Pabs =-(1/2) Re {J J d2 xen. [EsC + Einc] X [HsC* + Hinc*]};

(65b)

consequently, it can be shown that the total power extractedfrom the incident wave is

Pext = -(1/2)Re {J J d 2 xe. * [Einc X HsC* + ESC XHinc*]

(66)

with the asterisk denoting the complex conjugate.First, let the scatterer be irradiated by the LCP plane

wave [Eqs. (58)]. From Eq. (66), it then turns out that thetotal power extinguished by the obstacle is given by

Plext = (1/2)Re 1* | J d2 xen

X Hs(r) - (iwe/k)Ec(r)Jexp[-izy 1 1 r]) (67a)

Lakhtakia et al.

X exp(-i-y,1 r),


whereas the extinction cross section is

Ciext = (2kwe)Ie1- 2 P lext

REFERENCES AND NOTES

(67b)

Comparison of (76a) and (86a) leads to the forward ampli-tude theorem for LCP plane-wave incidence:

Clext = (87rh/y2 )ImIel* Ftl ( 1)e11-J. (68)

Likewise, for RCP plane-wave incidence of the type inEqs. (59), the total power extracted can be set down as

P2ext = (1/2)Re e2* J J d2xen

X H"'(r) + (icoE/k)Es'(r)Jexp[-iyA2 r]}) (69)

whence the forward amplitude theorem for RCP plane-waveincidence,

C2ext = (87rk/_y2 )ImIP 2 * Ft2(1 2)1 21j. (70)

The similarity in form of Eqs. (68) and (70) on one hand andthat of either one of them with the forward-amplitude theo-rem6 7 for linearly polarized waves in isotropic achiral mediashould be noted.

APPENDIX A

Equations (34) define a three-dimensional Green's dyadic.If the electromagnetic problem were to be independent ofthe z coordinate, then d/z 0. In that case, the two-dimensional Green's dyadic is given as

@(p, p') = (ik/8y 2 ) [_1la + ^yl-VV + V X aJ]Ho(y1Ip - p'l)

+ (ik/8- 2 )[72a + y2-VV - V X a] Ho(7 2 IP -Pl)

(71)

where Ho(.) is the cylindrical Hankel function of the firstkind and zero order. Next, if O/Oz 0 and a/ly 0, then theone-dimensional Green's dyadic is

@(x, x') = (ih/4y 1 y'2)h'yCS + l-7VV + V X a]

x exp(iyllx - x'l) + (ik/472 yy 2)[ 7 23

+ y2 '1VV - V X a]exp(iy 21x - x'I). (72)

Finally, it is a simple matter to verify that V X @(r, r') isthe appropriate Green's dyadic for the equation

[VV- V 2 3 - y2 - 2y21V X a] - U(r) = V X U0 (r) (73)

and that it will be needed frequently for the solution ofradiation problems, U0 being a source term.

ACKNOWLEDGMENTS

We thank S. Bassiri, N. Engheta, and C. H. Papas of Califor-nia Institute of Technology for supplying us with a copy oftheir report and paper (Ref. 26). The comments providedby the reviewers were greatly appreciated. Thanks are alsodue to K. E. Oughstun for his patience and understandingduring the review process.

1. E. J. Post, Formal Structure of Electromagnetics (North-Hol-land, Amsterdam, 1962), Chap. 8.

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

3. M. P. Silverman and R. B. Sohn, "Effects of circular birefrin-gence on light propagation and reflection," Am. J. Phys. 54, 69-76 (1986).

4. E. Charney, The Molecular Basis of Optical Activity (Krieger,Tampa, Fla., 1979).

5. L. D. Barron, Molecular Light Scattering and Optical Activity(Cambridge U. Press, Cambridge, 1982).

6. M. Born, Optik (Springer-Verlag, Heidelberg, 1972), p. 412.7. C. F. Bohren, "Light scattering by optically active particles,"

Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1975).8. R. A. Satten, "Time-reversal symmetry and electromagnetic

polarization fields," J. Chem. Phys. 28, 742-743 (1958).9. M. P. Silverman, "Reflection and refraction at the surface of a

chiral medium: comparison of gyrotropic constitutive relationsinvariant or noninvariant under a duality transformation," J.Opt. Soc. Am. A 3, 830-837 (1986).

10. M. P. Silverman, "Specular light scattering from a chiral medi-um: unambiguous test of gyrotropic constitutive equations,"Lett. Nuovo Cimento 43, 378-382 (1985).

11. C. F. Bohren, "Angular dependence of the scattering contribu-tion to circular dichroism," Chem. Phys. Lett. 40, 391-396(1976).

12. F. I. Fedorov, "On the theory of optical activity in crystals. I.The law of conservation of energy and the optical activity ten-sors," Opt. Spectrosc. (USSR) 6, 49-53 (1959).

13. F. I. Fedorov, "On the theory of optical activity in crystals. II.Crystals of cubic symmetry and plane classes of central symme-try," Opt. Spectrosc. (USSR) 6, 237-240 (1959).

14. B. V. Bokut' and F. I. Fedorov, "On the theory of optical activityin crystals. III. General equations of normals," Opt. Spec-trosc. (USSR) 6, 342-344 (1959).

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

16. C. F. Bohren, "Scattering of electromagnetic waves by an opti-cally active spherical shell," J. Chem. Phys. 62, 1566-1571(1975).

17. C. F. Bohren, "Scattering of electromagnetic waves by an opti-cally active cylinder," J. Colloid Interface Sci. 66, 105-109(1978).

18. A. Lakhtakia, V. K. Varadan, and V. V. Varadan, "Scatteringand absorption characteristics of lossy dielectric, chiral, non-spherical objects," Appl. Opt. 24, 4146-4154 (1985).

19. E. U. Condon, "Theories of optical rotatory power," Rev. Mod.Phys. 9, 432-457 (1937).

20. B. D. H. Tellegen, "The gyrator: a new electric network ele-ment," Phillips Res. Rep. 3, 81-101 (1948).

21. C. M. Krowne, "Electromagnetic theorems for complex aniso-tropic medium," IEEE Trans. Antennas Propag. AP-32, 1224-1230 (1984).

22. Ll. G. Chambers, "Propagation in a gyrational medium," Q. J.Mech. Appl. Math. 9, 360-370 (1956). This paper containsseveral errors, chiefly as follows: Eqs. (4.6) are not a necessaryconsequence of Eqs. (4.5), and Sec. 8 appears to imply a Green'sfunction of the kind a exp(ikR)/R, which is incorrect.

23. H. Unz, "Electromagnetic radiation in drifting Tellegen aniso-tropic medium," IEEE Trans. Antennas Propag. 12, 573-578(1964).

24. D. L. Jaggard, A. R. Mickelson, and C. H. Papas, "On electro-magnetic waves in chiral media," Appl. Phys. 18, 211-216(1978). The isotropic Post equations have been derived here byconsidering the medium to be a suspension of small helices in anachiral host medium.

25. H. Eyring, J. Walter, and G. E. Kimball, Quantum Chemistry(Wiley, New York, 1944), Chap. 17.

26. S. Bassiri, N. Engheta, and C. H. Papas, "Dyadic Green's func-tion and dipole radiation in chiral media," Antenna Lab. Rep.no. 118, (California Institute of Technology, Pasadena, Calif.,

V

Lakhtakia et al.


1985); "Dyadic Green's function and dipole radiation in chiralmedia," Alta Freq. 55, 83-88 (1986).

27. D. K. Cheng and J. A. Kong, "Covariant descriptions of bianiso-tropic media," Proc. IEEE 56, 248-251 (1968). The covariantconstitutive equations given here are precisely the same as thoseof Post.,

28. J. A. Kong, "Theorems of bianisotropic media," Proc. IEEE 60,1036-1046 (1972). An infinite-medium Green's dyadic, ex-pressed as an integral in the three-dimensional spatial-frequen-cy space, for the bianisotropic Tellegen media has been givenhere.

29. The transformation of the bianisotropic Tellegen equations intothe bianisotropic Post equations involves a redefinition of thepermittivity and the permability tensors; this is quite clear fromEq. (5) of Ref. 28.

30. J. van Bladel, Relativity and Engineering (Springer-Verlag,Berlin, 1984). To a stationary observer, even a moving isotropicmedium appears to be bianisotropic. Thus moving media canbe described by the bianisotropic Post Eqs. (6a) and (6b), whichare Lorentz covariant. This monograph contains an excellentaccount of the work done in this area since the turn of thiscentury, although it should be noted that the work of Kong andhis colleagues has not been cited here.

31. J. A. Kong and D. K. Cheng, "Wave reflections from a conduct-ing surface with a moving uniaxial sheath," IEEE Trans. Anten-nas Propag. 16, 577-583 (1968).

32. D. K. Cheng and J. A. Kong, "Time-harmonic fields in source-free bianisotropic media," J. Appl. Phys. 39, 5792-5796 (1968).Normal incidence of plane waves on stationary isotropic ormoving uniaxial media has been considered here. The movinguniaxial medium has been modeled by the bianisotropic PostEqs. (6a) and (6b).

33. J. A. Kong, "Optics of bianisotropic media," J. Opt. Soc. Am. 64,1304-1308 (1974). Plane-wave propagation in and dispersionequations for bianisotropic Tellegen media have been consid-ered here.

34. J. A. Kong and D. K. Cheng, "Wave behavior at an interface of asemi-infinite moving anisotropic medium," J. Appl. Phys. 39,2282-2286 (1968). Plane-wave reflection and refraction char-acteristics of a plane interface between free space and a bianiso-tropic Post medium have been derived here.

35. C. Altman, A. Schatzberg, and K. Suchy, "Symmetry transfor-mations and reversal of currents and fields in bounded (bi)ani-sotropic media," IEEE Trans. Antennas Propag. AP-32, 1204-1210 (1984).

36. L. S. Corley and 0. Vogl, "Optically active polymers," PolymerBull. 3, 211-217 (1980).

37. W. J. Harris and 0. Vogl, "Synthesis of optically active poly-mers," Polymer Preprints 22, 309-310 (1981).

38. G. Heppke, D. Ldtzsch, and F. Oestreicher, "Chirale dotier-stoffe mit ausserwbhnlich hohem verdrillungsvermbgen," Z.Naturforsch. Teil A 41, 1214-1218 (1986).

39. N. Engheta and A. R. Mickelson, "Transition radiation causedby a chiral plate," IEEE Trans. Antennas Propag. AP-30,1213-1216 (1982).

40. H. L. deVries, A. Spoor, and R. Jielof, "Properties of the eyewith respect to polarized light," Physica 19, 419-432 (1953).

41. L. J. Bour and N. J. Lopes Cardozo, "On the birefringence of theliving human eye," Vision Res. 21, 1413-1421 (1981).

42. G. J. Van Blokland and S. C. Verhelst, "Corneal polarization inthe living human eye explained with a biaxial model," J. Opt.Soc. Am. A 4, 82-90 (1987).

43. H. Shichi, Biochemistry of Vision (Academic, New York, 1983).44. T. I. Shaw, "The circular dichroism and optical rotatory disper-

sion of visual pigments," in Handbook of Sensory Physiology,H. J. A. Dartnall, ed. (Springer-Verlag, Berlin, 1972), Vol. 7,Part 1.

45. H. Shichi, "Circular dichroism of bovine rhodopsin," Photo-chem. Photobiol. 13, 499-506 (1971).

46. K. A. Piez, "Molecular and aggregate structures of the colla-gens," in Extracellular Matrix Biochemistry, K. A. Piez and A.H. Reddi, eds. (Elsevier, New York, 1984), pp. 1-39.

47. R. L. Trelstad, "Multistep assembly of type I collagen fibrils,"Cell 28, 197-198 (1982).

48. J. D. Jackson, Classical Electromagnetics (Wiley, New York,1975), Sec. 6.4.

49. J. van Bladel, Electromagnetic Fields (Hemisphere, New York,1985), Chap. 8.

50. It is easy to derive this result for any of the constitutive equa-tions, provided that the media are homogeneous; for example, itbe obtained for the Condon media [Eqs. (3a) and (3b)] from Eqs.(2a) and (2b) of Ref. 9.

51. Governing differential equations of the type of Eq. (13) werederived for the isotropic Post media [Eqs. (5a) and (5b)] byBassiri et al.,

2 6 for the bianisotropic Post media by Cheng andKong, 3 1 for the bianisotropic Tellegen media by Kong, 28 and forthe isotropic Tellegen media [Eqs. (4a) and (4b)] by Cham-bers.2 2

52. Vector and scalar potentials were also prescribed by Cham-bers22 for the isotropic Tellegen media [Eqs. (4a) and (4b)].

53. V. H. Rumsey, "Reaction concept in electromagnetic theory,"Phys. Rev. 94, 1483-1491 (1954).

54. The satisfaction of reciprocity constraints has been extensivelyinvestigated by Krowne 2 l for the bianisotropic Tellegen mediaand by Post' and Kong2 8 for the bianisotropic Post media [Eqs.(6a) and (6b)].

55. Losslessness conditions for the isotropic Post media [Eqs. (5a)and (5b)] have been considered by Jaggard et al.,

2 4 by Kong28

for the bianisotropic Tellegen and Post media, and by Cham-bers2 2 for the isotropic Tellegen media [Eqs. (4a) and (4b)].Fedorov12 has considered the conservation of energy for theconstitutive equations (2a) and (2b) used here.

56. The conservation of energy and electromagnetic momentum hasalso been considered by Kong 28 for bianisotropic media, usingNoether's theorem.

57. H. C. Chen, Theory of Electromagnetic Waves (McGraw-Hill,New York, 1983).

58. The rotational properties of @ were not identified by Bassiri etal. 26

59. R. Kastener and R. Mittra, "A spectral-iteration technique foranalyzing scattering from arbitrary bodies. I. Cylindricalscatterers with E-wave incidence," IEEE Trans. Antennas Pro-pag. AP-31, 499-506 (1983).

60. R. F. Harrington, Field Computation by Moment Methods(McGraw-Hill, New York, 1968).

61. P. C. Waterman, "Scattering by dielectric obstacles," Alta Freq.38, 348 (1969).

62. The functions L and R are defined in the spirit of the vectorspherical-harmonic representations of the fields QL and QR

introduced by Bohren 7 for bounded media.63. P. M. Morse and H. Feshbach, Methods of Theoretical Physics

(McGraw-Hill, New York, 1953).64. R. F. Harrington, Time-Harmonic Electromagnetic Fields

(McGraw-Hill, New York, 1964), Chap. 2.65. V. K. Varadan and V. V. Varadan, eds., Acoustic, Electromag-

netic and Elastic Scattering-Focus on the T-Matrix Ap-proach (Pergamon, New York, 1980).

66. A. T. de Hoop, "A reciprocity theorem for the electromagneticfield scattered by an obstacle," Appl. Sci. Res. B 8, 135-140(1960).

67. A. T. de Hoop, "On the plane-wave extinction cross-section ofan obstacle," Appl. Sci. Res. B 7, 463-469 (1959).

Lakhtakia et al.

Documents

Field equations, Huygens’s principle, integral equations, and theorems for radiation and scattering of electromagnetic waves in isotropic chiral media