CHIRALITY

INTRODUCTION

Chiral media have the ability to discriminate betweenleft-handed and right-handed electromagnetic (EM) elds.These media can be classied into two types: (1) isotropicchiral media and (2) structurally chiral media. Themolecules of a naturally occurring isotropic chiral mediumare handed, while an articial isotropic chiral medium canbe made by randomly dispersing electrically small, handedinclusions (such as springs) in an isotropic achiral hostmedium. The molecules of a structurally chiral medium,such as a chiral nematic liquid crystal, are randomly posi-tioned but have helicoidal orientational order. Structurallychiral media can also be articially fabricated either asstacks of uniaxial laminae or using thin-lm technology.Whereas considerable theoretical and experimental workon isotropic chiral media has been reported at microwavefrequencies during the 1980s and the 1990s, microwaveresearch on structurally chiral media remains in an em-bryonic stage at the time of this writing (1). Therefore, themajor part of this article is devoted to isotropic chiral me-dia.

NATURAL OPTICAL ACTIVITY

Ordinary sunlight is split into its spectral components bya prism. A spectral component is monochromatic (i.e., ithas one and only one wavelength 0 in vacuum). The wave-length 0 of one of the visible spectral components lies any-where between 400 nm (violet) and 700 nm (red).A spectralcomponent can be almost isolated from other spectral com-ponents by carefully passing sunlight through a series oflters. Although ltering yields quasimonochromatic light,many experiments have been and continue to be performedand their results analyzed, assuming that the ltered lightis monochromatic.

Light is an EM wave with spectral components to whichour retinal pigments happen to be sensitive, and the con-sequent images, in turn, happen to be decipherable in ourbrains. All optical phenomena can be generalized to otherelectromagnetic spectral regimes.

Suppose that a monochromatic EM wave is propagat-ing in a straight line in air, which is synonymous with vac-uum (or free space) for our present purpose. Its electriceld vector vibrates in some direction to which the prop-agation direction is perpendicular; the frequency of vibra-tion is f=c/0, where c=3 108 m/s is the speed of light invacuum. Its magnetic eld vector also vibrates with thesame frequency, but is always aligned perpendicular to theelectric eld vector as well as to the propagation direction.Suppose that we x our attention on a certain plane that istransverse to the propagation direction. On this plane, thelocus of the tip of the electric eld vector is the so-calledvibration ellipse, which is of the same shape as the locusof the tip of the magnetic eld vector. A vibration ellipse isshown in Fig. 1. Its shape is characterized by a tilt angleas well as an axial ratio; in addition, it can be left-handed

Figure 1. The tip of the electric eld vector of a plane-polarizedmonochromatic electromagnetic wave traces the socalled vibra-tion ellipse in a plane transverse to the propagation direction.

if the tip of the electric eld vector rotates counterclock-wise, or right-handed if otherwise. Similarly, an EM waveis said to be elliptically polarized, in general; however, thevibration ellipse can occasionally degenerate into a circle(circular polarization) or even a straight line (linear polar-ization).

The shape of the vibration ellipse of monochromaticlight is altered after traversal through a certain thicknessof a so-called optically active medium. This phenomenon,known as optical activity, was discovered around 1811by F. Arago while experimenting with quartz. Crystalsare generally anisotropic, but J.-B. Biot observed around1817 the optical activity of turpentine vapor, denitely anisotropic medium. Isotropic organic substances were be-lieved to have exclusively biological provenances, and in1860 L. Pasteur argued that turpentine vapor exhibitednatural optical activity, but the optical activity of crystalscould not be similarly qualied. Pasteur was unduly re-strictive. Isotropic optically active media, of biological orother origin, are nowadays called isotropic chiral media,because EM elds excited in them necessarily possess aproperty called handedness (Greek cheir=hand). Facsimilereproductions of several early papers are available (2).

CHIRAL MEDIA: NATURAL AND ARTIFICIAL

The molecules of an isotropic chiral medium are mirrorasymmetric (i.e., they are noncongruent with their mir-ror images). A chiral molecule and its mirror image arecalled enantiomers (3). As examples, the two enantiomersof 2-butanol are shown in Fig. 2. Enantiomers can havedifferent properties, although they contain identical atomsin identical numbers. One enantiomer of the chiral com-pound thalidomide may be used to cure morning sickness,during pregnancy, but its mirror image induces fetal mal-formation. Aspartame, a common articial sweetener, isone of the four enantiomers of a dipeptide derivative. Ofthese four, one (i.e., aspartame) is sweet, another is bit-ter, while the remaining two are tasteless. Of the approx-imately 1850 natural, semisynthetic, and synthetic drugsmarketed these days, no less than 1045 can exist as two ormore enantiomers;but only 570were beingmarketed in the

J. Webster (ed.),Wiley Encyclopedia of Electrical and Electronics Engineering. Copyright 2007 John Wiley & Sons, Inc.

2 Chirality

Figure 2. The two enantiomers of 2-butanol are mirror imagesof each other, as shown by the directed circular arrangements ofthe OH, CH2CH3, and CH3 groups.

late 1980s as single enantiomers, of which 61 were totallysynthetic. But since 1992, the U.S. Food and Drug Admin-istration (FDA) has insisted that only one enantiomer ofa chiral drug be brought into the market. Biological chi-rospecicity, once the subject of speculations by Pasteur onthe nature of the life force (vis viva), is now the topic ofconferences on the origin of life (4).

An isotropic chiral medium is circularly birefringent(i.e., both left-handed and right-handed circularly polar-ized light can propagate in a region lled with a homoge-neous isotropic chiral medium, with different phase veloc-ities and attenuation rates). Therefore, when monochro-matic, elliptically polarized light irradiates an isotropicchiral slab, the tilt angle and the axial ratio of the transmit-ted light are different from those of the incident light. Thechange in the tilt angle is quantied as optical rotation(OR) and alteration of the axial ratio as circular dichro-ism (CD). Both OR and CD depend on the wavelength 0,and the dependences are reasonably material-specic thatspectroscopies based on theirmeasurements have long hadindustrial importance. Biot himself had pioneered these at-tempts by cataloging the OR spectra of a large number ofsyrups and oils, and went on to found the science of sac-charimetry for which he was awarded the Rumford Medalin 1840 by the Royal Society of London. The rst edition ofLandolts tables on optical activity appeared in theGermanlanguage in 1879; the English translation of the second edi-tion of 1898 appeared in 1902.

AlthoughMaxwells unication of lightwith electromag-netism during the third quarter of the nineteenth centurycame to mean that natural optical activity is an EM phe-nomenon, the term optical rotation persisted. By the end ofthe nineteenth century, several empirical rules had evolvedon OR spectrums of isotropic chiral mediums. Then, in thelate 1890s, two accomplishments of note were reported:

1. J. C. Bose constructed several articial chiral materi-als by twisting jute bers and laying them end to end,and experimentally veried OR at millimeter wave-lengths. These materials were anisotropic, but Bosewent on to infer from his experiments that isotropicchiralmaterials could also be constructed in the sameway (5). Thus, he conclusively demonstrated the geo-metric microstructural basis for optical activity, andhe also constructed possibly the worlds rst arti-cial anisotropic chiral medium to alter the vibrationellipses of microwaves.

2. P. Drude showed that chiral molecules can be mod-eled as spiral oscillators and theoretically veried arule Biot had given regarding OR spectra (6).

Experimental verication of Drudes spiral oscillatorhypothesis had to wait for another two decades. As elec-tromagnetic propositions can be tested at lower frequen-cies if the lengths are correspondingly increased and otherproperties proportionally adjusted, K. F. Lindman made2.5-turn, 10-mm-diameter springs from 9-cm-long copperwire pieces of 1.2 mm cross-sectional diameter. Springs arehanded, as illustrated in Fig. 3. Each spring was wrappedin a cotton ball, and about 700 springs of the same hand-edness were randomly positioned in a 262626-cm card-board box with an eye to achieving tolerable isotropy. Thenthe box was irradiated with 13-GHz (30 cm010 cm)microwave radiation and the OR was measured. Lindmanveried Drudes hypothesis remarkably well. He also de-termined that (1) the OR was proportional to the numberof (identically handed) springs in the box, given that thedistribution of springs was rather sparse; and (2) equalamounts of left-handed or right-handed springs broughtabout the same OR, but in opposite senses (7). Lindmansexperiments were extensively repeated during the 1990sby many research groups in several countries (8, 9), andseveral patents have even been awarded on making arti-cial isotropic chiral mediums with miniature springs.

CONSTITUTIVE RELATIONS OF AN ISOTROPIC CHIRALMEDIUM

Electromagnetic elds are governed by the Maxwell postu-lates, in vacuum as well as in any material medium. Thesefour postulates have a microscopic basis and are given invacuum as follows:

Thus, E(r, t) and B(r, t) are the primitive or the fundamentalEM elds, both functions of the three-dimensional positionvector r and time t; 0=8.8541012 F/m and 0=4107H/m are, respectively, the permittivity and the permeabil-ity of vacuum; tot(r, t) is the electric charge density andJtot(r, t) is the electric current density.

Equations (1a), (1b), (1c) and (1d) apply at any lengthscale, whereas the charge and the current densities mustbe specied not continuously but over a set of isolatedpoints. Electromagnetically speaking, matter is nothingbut a collection of discrete charged particles in vacuum.As per the HeavisideLorentz procedure to get a macro-scopic description of continuous matter, spatial averagesof all elds and sources are taken, while both tot(r, t) andJtot(r, t) are partitioned into matter-derived and externallyimpressed components.Then theMaxwell postulates at the

Chirality 3

Figure 3. An enantiomeric pair of springs. An articial isotropicchiral medium can be made by randomly dispersing springs in anisotropic achiral host medium, with more springs of one handed-ness than the springs of the other handedness.

macroscopic level can be stated as

Here, (r, t) and J(r, t) are the externally impressed sourcedensities, while the new elds

contain two matter-derived quantities: the polarizationP(r, t) and the magnetization M(r, t)

Constitutive relations must be prescribed to relate thematter-derived elds D(r, t) and H(r, t) to the basic eldsE(r, t) and B(r, t) in anymaterial medium.The constructionof these relations is primarily phenomenological, althoughcertain epistemologically mandated proprieties must beadhered to. The constitutive relations appropriate for ageneral, linear, homogeneous, material medium with time-

invariant response characteristics may be stated as

Four constitutive property kernels appear in these equa-tions; the dyadic-

e(t) is the dielectric susceptibility kernel,

-m(t) is themagnetic susceptibility kernel,while the dyadics

-em(t) and -

em(t) are called the magnetoelectric kernels. Al-though a dyadic may be understood as a 33 matrix forthe purpose of this article, Chens textbook (10) is recom-mended for a simple introduction to the use of dyadics inEM theory.

All four dyadic kernels in Eqs. (4a) and (4b) are causal[i.e., -

e(t) -0-0 for t0, etc.], because all materials mustexhibit delayed response. In addition, when we substituteEqs. (4a) and (4b) in Eqs. (2c) and (2d), respectively, a re-dundancy emerges with respect to Eqs. (2a) and (2b). Elim-ination of this redundancy leads to the constraint (11)

which has never been known to be violated by a physi-calmaterial. Finally, crystallographic symmetriesmay alsoimpose additional constraints on the constitutive kernels.Amedium described by Eqs. (4a) and (4b) is said to be bian-isotropic, since the constitutive kernels indicate anisotropy,and both D(r, t) and H(r, t) depend on both E(r, t) and B(r,t)

Suppose next that the linear mediums constitutiveproperties are direction-independent. Equations (4a) and(4b) then simplify to

in consequence of Eq. (5), where the scalar chi(t) is the chi-rality kernel. Equations (6a), (6b) describe the isotropicchiral mediumthe most general, isotropic, linear electro-magnetic material known to exist (12, 13).

Most commonly, EM analysis is carried out in thefrequency domain, not the time domain. Let all time-dependent quantities be Fourier-transformed; thus

and so on, where w = 2 p f is the angular frequency. Inthe remainder of this article, phasors such as D(r, ) are

4 Chirality

called elds, following normal practice. The four Maxwellpostulates Eqs. (2a), (2b), (2c) and (2d) assume the form

while the constitutive equations [Eqs. 6] for an isotropicchiral medium simultaneously transform into

Using Eqs. (8b) and (8d) with J(r, ) = 0 in Eqs. (9a)and (9b), respectively, we obtain the DrudeBornFedorov(DBF) constitutive relations of an isotropic chiral medium:

Their great merit is that the necessary mirror asymmetryis transparently reected in them, because E(r, ) and H(r, ) are not true vectors but only pseudovectors. Achiral medium is thus described by three constitutive prop-erties; the permittivity and permeability in Eqs. (10a) and(10b), respectively, may be formally dened as the ratios

but the chirality parameter () can be regarded as either

or

where the asterisk denotes the complex conjugate. Equa-tions (11a), (11b), (11c) and (11d) make it clear that while() and() are true scalars,() has to be a pseudoscalarsince the numerator in either of its two denitions containsa pseudovector. Other constitutive relationsequivalentto Eqs. (9a) and (9b) and Eqs. (10a) and (10b)are alsoused in the frequency-domain EM literature, but this arti-cle is restricted to theDBF constitutive relations Eqs. (10a)and (10b), as they bring out the essence of chirality at thevery rst glance. An isotropic chiral medium and its mir-ror image share the same () and (), and their chiralityparameters differ only in sign.

The time-averaged Poynting vector

denotes the direction of power ow. In any linear medium,the monochromatic Poynting theorem reads as

For specialization to an isotropic chiralmedium,we have tosubstitute Eqs. (10a) and (10b) in Eq. (12b). The resultingexpression is not particularly illuminating.

An isotropic chiral medium is Lorentz-reciprocal. Sup-pose that all space is occupied by a homogeneous isotropicchiral medium and all sources are conned to regions ofbounded extent. Let sources labeleda radiate eldsEa(r, )andHa(r, ), while sources labeled b radiate elds Eb(r, )andHb(r, ), all at the same frequency. Then the relations(12)

arise in a source-free region, in consequence of the Lorentzreciprocity of the medium.

ARTIFICIAL ISOTROPIC CHIRAL MEDIA

That matter is discrete has long been established. Further-more, when we probe matter at length scales at which itappears continuous,whether themicrostructure ismolecu-lar ormerely comprises electrically small inclusions is of noconsequence.The linear dimensions of an electrically smallinclusion are less than about a tenth of themaximumwave-length, in the media outside as well as inside the inclusion,at a particular frequency.Articial isotropic chiralmediaactive at microwave frequenciescan be constructed withthis thought in mind. Consider a random suspension ofidentical, electrically small, inclusions in a host medium,which we take here to be vacuum for simplicity. The num-ber of inclusions per unit volume is denoted by N, and thevolumetric proportion of the inclusions in the compositemedium is assumed to be very small. Our objective is tohomogenize this dilute particulate composite medium andestimate its effective constitutive properties (13). Homog-enization is much like blending apples into apple sauce ortomatoes into ketchup.

Any inclusion scatters the EM wave incident on it. Faraway from the inclusion, the scatteredEMeld phasors canbe conceptualized, equivalently, as being radiated by an en-semble ofmultipoles.Multipoles are necessarily frequency-domain entities; and adequate descriptions of electricallylarger inclusions require higher-order multipoles, but ho-mogenizing composite media with electrically large inclu-sions is fraught with conceptual perils.

The lowest-ordermultipoles are the electric dipolep andthe magnetic dipole m. In formalisms for isotropic chiralmedia, both are accorded the same status. As all inclu-

Chirality 5

sions in our composite medium are electrically small, wecan think that an inclusion located at position r is equiva-lent to the colocated dipoles characterized by the followingrelations:

Here, Eexc(r, ) and Hexc(r, ) are the elds exciting theparticular inclusion; while and -hh() are the four linearpolarizability dyadics that depend on the frequency, theconstitution, and the dimensions of the inclusion. As theinclusions are randomly oriented and any homogenizablechunk of a composite medium contains a large number ofinclusions, -ee() and other terms in Eqs. (14) can be re-placed by their orientationally averaged values. If the ho-mogenized composite medium is isotropic chiral, this ori-entational averaging process must yield

The polarizability dyadics of electrically small, handed in-clusions (e.g., springs) may be computed either with stan-dard scattering methods such as the method of moments(14) or using lumped-parameter circuit models (15). Pro-vided that dissipation in the composite medium can be ig-nored, at a certain angular frequency, ee(), hh(), andchi() are purely real-valued.

On applying the Maxwell Garnett homogenization ap-proach, the constitutive relations of the homogenized com-posite medium (HCM) are estimated as follows (12):

where

Equivalently

are the DBF constitutive relations of the HCM, with

as the constitutive parameters. Clearly, if chi() =0 thecomposite medium has been homogenized into an isotropicchiral medium. In passing, other homogenization ap-proaches are also possible for chiral composites (1, 13).

BELTRAMI FIELDS IN AN ISOTROPIC CHIRAL MEDIUM

In a source-free region occupied by a homogeneousisotropic chiral medium, (r, ) = 0 and J(r, ) = 0. Equa-tions (8a), (8c), and (10a) and (10b) then show that E(r, ) = 0 and H(r, ) = 0. Thus all four eldsE(r, ),H(r, ),D(r, ) andB(r, )are purely solenoidal.Next, Eqs. (8a), (8b), (8c) and (8d) and (10a) and (10b) to-gether yield the following vector Helmholtz-like equations:

In the limit ()0 the medium becomes achiral andthese equations reduce to the familiar vector Helmholtzequation, and so on.

In lieu of the second-order differential equations [Eqs.(23)], rst-order differential equations can be formulated.Thus, after dening the auxiliary elds

and using the wavenumbers

we get the two rst-order differential equations

6 Chirality

Figure 4. Optical rotation (OR) and circular dichroism (CD) spec-tra of a simple isotropic chiral medium. When the OR changessign, the CD records either a maximum or a minimum,which phe-nomenon is called the Cotton effect.

which are easier to analyze than Eqs. (18). The denomina-tors on the left sides of Eqs. (20a) and (20b) suggest that2()()2() = 1 is not permissible for an isotropic chi-ral medium, as both wavenumbers must have nite mag-nitudes.

According to Eqs. (21a) and (21b),Q1(r, ) and Q2(r, )are Beltrami elds (12). A Beltrami eld is parallel to itsown circulation. The concept arose early in the nineteenthcentury, and has often been rediscovered. The easiest wayto think of a Beltrami eld is as a spiral staircase or atornado.

WhileQ1(r, ) is a left-handed Beltrami eld, the nega-tive sign on the right side of Eq. (15b) means thatQ2(r, )is a right-handed Beltrami eld, because the two complex-valued wavenumbers 1() and 2() must have positivereal parts. Both wavenumbers also must have positiveimaginary parts in a causal material medium, since causalmaterials must exhibit delayed response in the time do-main and therefore must demonstrate EM loss (or attenu-ation) in the frequency domain.

As an isotropic chiral medium displays two distinctwavenumbers at a specic frequency, it is birefringent.More specically, becauseQ1(r, ) andQ2(r, ) have plane-wave representations possible only in terms of circularlypolarized plane waves, an isotropic chiral medium is of-ten said to be circularly birefringent. The difference be-tween 1() and 2() gives rise to natural optical activity.While OR is proportional to the real part of [1() 2()],CD is proportional to the imaginary part of [1() 2()].The OR and CD spectra must be consistent with theKramersKronig relations (16). The CD spectrum has a lo-cal maximum or minimum at the frequency where the signof the OR changes; this feature is labeled as the Cotton ef-fect after H. Cotton, who reported it in 1895 (2). The ORand CD spectra of a simple chiral medium are illustratedin Fig. 4.

REPRESENTATION OF BELTRAMI FIELDS

A Beltrami eld is represented in terms of toroidal andpoloidal elds because the curl of a toroidal eld is poloidal

and vice versa (17). Thus, the decomposition

is possible, as the rst parts on the right sides of Eqs. (22)are toroidal and the second parts are poloidal. The scalarfunctions v(r, ) satisfy the scalar Helmholtz equation asfollows:

Solutions of Eqs. (23) in theCartesian, the circular cylindri-cal, and the spherical coordinate systems are commonplace(18).

Beltrami plane waves propagating in the +z directionmay be represented as

with A as the amplitudes, while x, y, and z are the Carte-sian unit vectors.

In the circular cylindrical coordinate system (, , z) Bel-trami elds with an exp(iz) dependence may be expressedas the sums

for regular behavior as while the expansions

are well behaved at = 0 with An and Bn as the coef-cients of expansion. The vector cylindrical wavefunctionsare given as

where = +(2 2)1/2; , and z are the unit vectors inthe cylindrical coordinate system; Jn () are the cylindri-cal Bessel functions of order n, and Jn () are the respec-tive rst derivatives with respect to the argument; whileH (1)n () are the cylindrical Hankel functions of the rstkind and order n, and H (1)n () are the rst derivatives withrespect to the argument. For quasi-two-dimensional prob-lems, = 0 because /z = 0 Parenthetically, in this para-graph denotes the radial distance in the xy plane andshould not be confused with the use of for charge densityelsewhere in this article.

Finally,withAsmn andBsmn as the coefcients of expan-sion, in the spherical coordinate system (r, , ), we have

Chirality 7

for elds regular as r , and

for elds regular at r = 0. The well-known vector sphericalwavefunctions, M( j)smn (r) and N

( j)smn (r) are stated for j = 1,

3 as

where the angular functions

have been used. In these expressions, r, , and are theunit vectors in the spherical coordinate system; Pmn (cos )are the associatedLegendre functions of ordern anddegreem; jn(r) are the spherical Bessel functions of order n; andh(1)n (r) are the spherical Hankel functions of the rst kindand order n.

Boundary-value problems involving scattering byisotropic chiral half-spaces, cylinders, and spheres can beanalytically solved using Eqs. (24,25a) and (25b), (26a),(26b) and (27b), (27a) and (27b), (28a) and (28b), (29a) and(29b). Boundary-value problems involving more compli-cated geometries generally require numerical treatment,which necessitates the use of Green functions.

Isotropic chiral waveguides for use at microwave fre-quencies have been theoretically studied extensively, al-though no practical realization thereof has yet come tolight. Theoretical investigations on propagation in the so-called chirowaveguides generally consist of decomposingthe Beltrami elds into axial and transverse componentsas

where the z coordinate is measured on the waveguide axiswhile two other mutually orthogonal coordinates are spec-ied in the transverse plane. Assuming that all elds havean exp(iz) dependence on z, and making use of Eqs. (21a)

and (21b), we get

where-I is the identity dyadic.The axial components satisfythe reduced scalar Helmholtz equations

appropriate solutions of which are commonly worked outin many different ways for waveguides of different cross-sectional geometries (18).

SOURCES IN AN ISOTROPIC CHIRAL MEDIUM

Let us now assume the existence of a magnetic chargedensity m(r, ) and a magnetic current density Jm(r,), because they assist in the solution of dual problems(19). In addition, let us dene the intrinsic impedance() =

()/() as well as the auxiliary wavenumber

k() =

()/() and drop the explicit indication of de-pendences on for notational simplicity. NowEqs. (8), (8b),(8c) and (8d) may be written as

which yield the relations

for a chiral medium, where

are the Beltrami source current densities (12).Since Eqs. (34) are linear, they can be solved using stan-

dard techniques. Their complete solution can be compactlystated for all r as

where

are the particular solutions due to the source densitiesW(r) which are wholly conned to the regionVs, andQcf (r)are the complementary functions satisfying the relations

8 Chirality

identically. Substituting Eqs. 3638 in Eqs. (34), we obtainthe dyadic differential equations

where () is the Dirac delta function.The solutions of Eqs. (39) are the BeltramiGreen

dyadic functions

wherein

is the familiar dyadic Green function for free space. Asthe properties of -Gfs(, r, r0) can be found in almost anygraduate-level EM textbook (20, 21), those of-G(r, r0) canbe easily determined, as illustrated in Ref. 12.

As an example of the use of Eqs. (37), let us consideran electric dipole moment Plocated at the origin: J(r) =ip(r) and Jm(r) = 0 The radiated Beltrami elds turnout be

which show clearly that the radiation eld of a point elec-tric dipole in an isotropic chiral medium consists of left-handed as well as right-handed components. If we haveinstead a point magnetic dipole m located at the origin,the source current densities are specied as J(r) = 0 andJm(r) = im(r) so that

are the corresponding radiated Beltrami elds. A majordifference between isotropic chiral and achiral media isshown by the two sets of radiated elds, Eqs. (42a), (42b)and (43a), (43b). Without loss of generality, let the sourcedipole moments be aligned parallel to the z axis. Then, ifthe dipole moments are radiating in an achiral medium(i.e., =0), there is no magnetic eld due to p and there isno electric eld due to m at any point on the z axis. On theother hand, the wavenumber difference between the left-handed and the right-handed Beltrami elds guaranteesthat, in an isotropic chiralmedium,bothErad(r) andHrad(r)are not generally null-valued on the z axis, regardless ofwhich one of the two dipole moments is radiating.

Canonical sources of Beltrami elds are possible. Ifthere is a source distribution such that J(r) (1/i)Jm(r)for all r, then Qrad2(r) 0 from Eqs. (35a) and (35b) and(37). Likewise, a source distribution containing electric and

magnetic current densities in the simple proportion J(r) =(1/i)Jm(r) for all r radiates only a right-handed eld, be-cause Qrad1(r) 0 emerges from the same equations.

Radiation by complex sources has to be generallytreated using integral equations. Both the Maue and thePocklington integral equations for radiation in a homoge-neous isotropic chiral medium are available (12). Cerenkovradiation in an isotropic chiral medium has also been de-scribed using Beltrami elds (12).

The foregoing developments make it clear that a de-scription involving differentials of only the rst order suf-ces for monochromatic radiation and propagation in anisotropic chiral medium. True, there are terms in-G1(r, r0) and -G2(r, r0) but dyadic Green functions are notelds, being instead solutions of dyadic differential equa-tions.

Finally, although the left-handed and the right-handedBeltrami elds are capable of being independently radiatedand propagated as per Eqs. (34), they do indeed couple inan isotropic chiral medium. This coupling takes place onlyat bimedium boundaries where conditions on the tangen-tial components of E(r) and H(r) must be satised; that is,the boundary conditions are specied not on Q1(r) or Q2(r)singly, but on the tangential components of the combina-tions E(r) = Q1(r) iQ2(r) and H(r) = Q2(r) + (1/i)Q1(r).

THEOREMS FOR SCATTERING IN AN ISOTROPICCHIRAL MEDIUM

Equations 36-39 sufce to set up certain often-used princi-ples for monochromatic scattering and radiation problems,when all space is lled with a homogeneous isotropic chiralmedium.

The sourceregion Beltrami elds can be obtained fromEqs. (37) using the Fikioris approach (22). Let S be thesurface of the convex-shaped source region Vs, where n0 isthe unit outward normal at r0 S (see Fig. 5). Then, Eqs.(37) and (40) yield the following relations:

The depolarization dyadic

in Eqs. (44) is dependent on the shape of the region Vs,while

If the maximum linear extent of the region Vs timesthe magnitude of the greater of the two wavenumbers, 1

Chirality 9

Figure 5. For the evaluation of elds in the region Vs, when thesources are also conned to the same region and all space is occu-pied by a homogeneous chiral medium.

and 2, is much smaller than unity, we may make the qua-sistatic approximation: W1(r0)=W1(r) and W2(r0)=W2(r)for all r0 VS . Then, Eqs. (44) simplify to

where the dyadics

depend on the shape as well as on the size of Vs. Finally,the Rayleigh approximation requires that we ignore thedyadics -M(|r) and -N(|r) completely to obtain the esti-mates

when Vs is an extremely small region. The right sides ofEqs. (47) and (49) are useful in homogenizing isotropicchiral composites as well as for devising the method ofmoments and the coupled dipole method for scattering bybianisotropic objects in isotropic chiral environments (12,23).

Turning now to the mathematical realizations of theHuygens principle and its progeny, we suppose that allspace is divided into two regions, as shown in Fig. 6. Theexternal regionVext extends to innity in all directions butis separated from an internal regionVint by the convex andonce-differentiable surface S. Then the Huygens principlein a homogeneous isotropic chiral medium reads as follows(12):

Thus, the Cauchy data for the elds in a chiral mediumcomprise the components of the Beltrami elds that aretangential to a boundary. When these data are prescribedon the surface S, we can nd the Beltrami elds every-where in the region Vext.

Figure 6. Relevant to the Huygens principle, the exterior surfaceequivalence principle, and the EwaldOseen extinction theorem,when all space is occupied by a homogeneous chiral medium.

The Huygens principle allows the enunciation of the ex-terior surface equivalence principle. Consider a problemin which surface Beltrami current densities WS1(r) andWS2(r) exist on the exterior side of the surface S (see Fig.6). As per Eqs. (37), these surface current densities act assources of the radiated elds

On comparingEqs. (50a) and (51) to ensure the equivalenceQrad(r) Q(r) for all r Vext, we obtain the relationships(12)

as the exterior surface equivalence principle for Beltramields and sources, r0 in Eqs. (52) lying on the exterior sideof S.

The EwaldOseen extinction theorem is a cornerstoneof the extended-boundary-condition method (12, 24). Forscattering in an isotropic chiral medium, this theoremmaybe stated as

where Qcf (r) play the role of the incident Beltrami elds.Once Q(r0), r0 S have been determined from Eqs. (53),the total elds in the exterior region may be determined as

From Eqs. (53) and (54), the plane-wave scattering dyadicsfor an object in an isotropic chiral environment can be de-rived, as can the forward plane-wave scattering amplitudetheorems (12).

STRUCTURALLY CHIRAL MEDIA

The molecules of a naturally occurring isotropic chi-ral medium are mirror-asymmetric, and so are the in-clusions in an articial isotropic chiral medium. As a

10 Chirality

Figure 7. Schematic depiction of the arrangement of needle-likemolecules in a chiral nematic liquid crystal. The gaps betweenthe consecutive sheets as well as the sheets are ctitious, as theyare merely aids to visualization. Only half of the electromagneticperiod is shown.

randomly dispersed and randomly oriented collectionof mirror-asymmetric molecules or inclusions is alsomirror-asymmetric, isotropic chiral media emerge withdirection-independent constitutive properties. In contrast,the molecules or inclusions of a structurally chiral mediumare not mirror-asymmetric, but their orientation is.

In chiral nematic liquid crystals (CNLCs)also calledcholesteric liquid crystalsneedle-like molecules are ran-domly positioned on parallel sheets, with all molecules onany one sheet oriented parallel to one another and with theorientation rotating helicoidally as one moves across con-secutive sheets. The situation is schematically depicted inFig. 7. From 1850 to 1888, several scientists came acrossCNLCs but were unable to capitalize on their observa-tions (25). Then in 1888 the biochemist F. Reinitzer ob-served that a CNLC named cholesteryl benzoate has twodistinct melting pointsit is a solid at temperatures be-low 145.5C, a clear liquid at temperatures above 178.5C,and a cloudy liquid in between. Reinitzers observation ofthe mesophasewhen positional order is absent as in aliquid, but orientational order is still strong as in a solidopened up the area of liquid crystal research in continuummechanics as well as in optics (2628).

Earlier, however, (in 1869), E. Reusch had anticipatedthe CNLC structure as a laminate of uniaxial dielectricsheets, with the crystallographic axes of any two adjacentsheets offset in the transverse plane by a xed small an-

gle. At a low enough frequency, this laminate appears as acontinuously nonhomogeneousmediumwhose constitutiveproperties vary helicoidally. Thus

are the frequency-domain constitutive relations of a CNLC,where

is the relative permittivity dyadic in a reference plane des-ignated as z=0. The rotation dyadic

denotes that the CNLC structure varies helicoidally in theaxial (i.e., z) direction with a period 2 ; however, the elec-tromagnetic period is . The upper sign in Eq. (57) appliesfor structural right-handedness; the lower, for structuralleft-handedness.

Reuschs model of a CNLC has been often implementedwith either uniaxial crystals or brous laminae, and ap-pears promising formicrowave and RF applications as well(29). More recently, thin-lm technology has been pressedinto service to realize the CNLC structure by releasing adirected evaporant ux toward a rotating substrate (30,31). The reference permittivity dyadic of these chiral sculp-tured thin lms (STFs) differs from Eq. (56), being

instead, and the electromagnetic period is 2 .The reference permittivity dyadics in Eqs. (56) and (58)

are uniaxial and biaxial, respectively; that is, they have ei-ther one or two crystallographic axes. Biaxial-ref () is dis-played by chiral smectic liquid crystals also (26, 27). Thusin general -ref () displays orthorhombic symmetry (32).Moreover, particularly with advances in thin-lm technol-ogy, there is no reason for a chiral STF to be necessarily di-electric only. These considerations led to the proposal of thehelicoidal bianisotropic medium (HBM), whose frequency-domain constitutive relations may be stated as (33)

subject to the constraint

The launching and propagation of EM waves in HBMs isbest studied using a 44 matrix differential equation for-malism (31, 34).

Chirality 11

Figure 8. Scanning electron micrograph of a 10-period chiralsculptured thin lmmade of silicon oxide. (FromProfessor RussellMessier, Pennsylvania State University, with permission.)

Although chiral STFs made of uorites, and single-frequency OR measurements on them, were reported in1959 (35), systematic experimental studiesalong withscanning electron microscopic verication of the mi-crostructural geometryappear to have begun only in1995 (30). Figure 8 shows the scanning electron micro-graph of a chiral STF made of silicon oxide. As typicalvalues of realized today range from 30 nm to 10 m,microwave applications of these lms are yet not feasible,but are likely to become an active area of research oncelms with 100 m become available. Many possible ap-plications have been anticipated as the concept of STFsfor biological, optical, electronic, chemical, and other appli-cations is beginning to take root, while many optical andrelated applications have already been implemented (30,31). Large-scale production appears feasible as well, withadaptation of ion-thruster technology (36).

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AKHLESH LAKHTAKIAPennsylvania State University,

University Park, PA