Anomalous axial propagation in helicoidal bianisotropic media

1 December 1998

Optics Communications 157 Ž1998. 193–201

Full length article

Anomalous axial propagation in helicoidal bianisotropic media Akhlesh Lakhtakia Computational and Theoretical Materials Sciences (CATMAS) Group, Department of Engineering Science and Mechanics, PennsylÕania State UniÕersity, UniÕersity Park, PA 16802-1401, USA Received 28 May 1998; revised 23 July 1998; accepted 27 July 1998

Abstract Anomalous axial propagation, characterized by axially propagating Voigt waves, in a general linear helicoidal bianisotropic medium ŽHBM. is examined. A fourth-order ordinary differential equation is derived, from which the conditions for anomalous axial propagation emerge. The analysis is exemplified by application to supercholesteric media, which include cholesteric liquid crystals. In addition, purely dielectric HBMs are considered in detail, with later specialization to chiral smectic dielectric media, and light is shed on the effects of reciprocity and dissipation. q 1998 Elsevier Science B.V. All rights reserved. Keywords: Axial propagation; Voigt wave; Helicoidal bianisotropic medium

1. Introduction Many colored minerals, such as iolite and biotite, are pleochroic. Depending on its orientation, a pleochroic mineral absorbs photons in different parts of the visible spectrum. Most pleochroic minerals are biaxial dielectric media, but some are uniaxial ones w1x. Woldemar Voigt w2x showed in 1902 that electromagnetic waves traveling along any one of the two singular axes of certain pleochroic crystals do not have the familiar form D Ž r . sD1 exp Ž ik 1 ua Pr . qD2 exp Ž ik 2ua P r . ,

Ž1.

but D Ž r . sD1exp Ž ik 1ua Pr . q ik 2 Ž ua Pr . D2 exp Ž ik 2ua Pr . ,

k1 s k 2

Ž2.

instead. Here, DŽr. denotes the electric displacement phasor, D1 and D2 are complex amplitude vectors, ua is a unit vector parallel to a particular singular axis, k 1 ' k 1Žua . and k 2 ' k 2 Žua . denote wavenumbers, and an expŽyi v t . time-dependence is assumed. The extension of Voigt’s anomalous propagation result to various other crystals was expertly reviewed recently by Borzdov w3x. Perusal of his earlier work on Voigt waves – delineated by Eq. Ž2. – in uniaxial and biaxial crystals w4,5x is also highly recommended. Literature on Voigt waves is mostly confined to homogeneous media. The sole exception is a 1998 paper w6x, wherein anomalous propagation along the axis of rotational nonhomogeneity of a locally uniaxial dielectric medium was treated. This medium exemplifies the so-called thin-film helicoidal bianisotropic medium ŽHBM., which is now the subject of both theoretical and experimental investigations w7,8x. The objective of the present communication is to delineate anomalous axial propagation in general linear HBMs. Both thin-film HBMs and chiral liquid crystals w9,10x typify the HBMs conceptualized only a few years ago w11,12x. HBMs are periodically nonhomogeneous substances with a preferred axis of rotational nonhomogeneity – say, the z axis. In addition, HBMs must be locally anisotropic, and can display dielectric, magnetic as well as magnetoelectric response properties w7x. 0030-4018r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 9 8 . 0 0 4 2 0 - 9

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The plan of this paper is as follows: The constitutive relations of HBMs are presented in Section 2, followed by the derivation of a 4 = 4 matrix differential equation for axial propagation Ži.e., ErE x ' 0 and ErE y ' 0. in Section 3. The characteristics of a fourth-order ordinary differential equation for axial propagation are examined in Section 2, and the conditions for anomalous axial propagation Žof Voigt waves. are discussed. In addition, the developed theory is applied to supercholesteric media w13,14x in Section 4. Finally, Section 5 is devoted to purely dielectric HBMs with later specialization to chiral smectic dielectric media; light is shed on the effects of reciprocity and dissipation; and a derived condition for Voigt waves is shown to reduce to the one available for a simpler thin-film HBM w6x. A note about notation: Vectors are underlined once and dyadics w15x are underlined twice; column vectors are underlined once and enclosed by square brackets, while matrices are underlined once and also enclosed by square brackets.

2. Helicoidal bianisotropic media Let us adopt the cartesian coordinate system, with Žux , uy , uz . as the corresponding triad of unit vectors. Then, the frequency-domain electromagnetic constitutive relations applicable to a general linear HBM can be set down as w11,12x D Ž r . s e 0 a11Ž z , v . PE Ž r . q a12 Ž z , v . PH Ž r .

Ž3.

B Ž r . s m 0 a21Ž z , v . PE Ž r . q a22 Ž z , v . PH Ž r .

'

Here and hereafter, e 0 and m 0 are the permittivity and the permeability of free space Ži.e., vacuum., while k 0 s v e 0 m 0 is the free-space wavenumber. The four dyadics in these equations are the following: a11Ž z, v . is the relative permittivity dyadic, a22 Ž z, v . is the relative permeability dyadic, while a12Ž z, v . and a21Ž z, v . are the two magnetoelectric dyadics. All four nonhomogeneous constitutive dyadics are factorable as

aln Ž z , v . sSz Ž z . P bln Ž v . PSzT Ž z . ,

l s 1,2,

n s 1,2.

Ž4.

The unitary dyadic Sz Ž z . s Ž ux ux quy uy . cos Ž p zrV . q Ž uy ux yux uy . sin Ž p zrV . quz uz

Ž5.

STz Ž z .

is responsible for the rotational nonhomogeneity of the HBM, is the transpose of Sz Ž z ., and 2 V is the structural period as it is the pitch of a helix described by Sz Ž z ., The four dyadics bln Ž v . s aln Ž0, v . must be consistent with the Kramers–Kronig relations to be causal w16x. Furthermore, the constraint y1 Trace by1 22 Ž v . P b21Ž v . q Ž e 0rm 0 . b12 Ž v . P b22 Ž v . s 0

Ž 6.

is mandated by the structure of modern electromagnetic theory w17,18x. Reciprocity, if applicable, imposes additional requirements on bln Ž v . w19x. At least one of the four dyadics bln Ž v . must not be isotropic in order to ensure that the chosen HBM is indeed rotationally nonhomogeneous w20x. Let us reiterate that the constitutive relations outlined in this section are sufficient to cover chiral liquid crystals w9,10x; ferrocholesteric w21x, ferrosmectic w22,23x, and supercholesteric w13,14x media; as well as a canonical class of sculptured thin films w7x.

3. Matrix differential equations for axial propagation Our focus lies here on axial propagation so that we set ErE x ' 0 and ErE y ' 0; accordingly, EŽr. s EŽ z . and HŽr. s HŽ z.. In this and the following sections, mention of dependences on the angular frequency v is suppressed. Substituting Eq. Ž3. in the Maxwell curl postulates, = = EŽr. s i v BŽr. and = = HŽr. s yi v DŽr., we obtain four differential equations of the first order and two algebraic equations. The latter two can be solved for Ez Ž z . and Hz Ž z ., provided the condition uz P b11 Puz

uz P b22 Puz / uz P b12 Puz

uz P b21 Puz

Ž7.

holds. In that event, the four differential equations may be simplified and compactly stated together as d z fŽ z . s i LŽ z . P fŽ z . s i

½

p

ž

L0 q exp 2 i

V

p z

/

ž

L1 q exp y2 i

V

z

/ 5 L2

fŽ z . ,

Ž8.

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with d z ' drd z. In this 4 = 4 matrix ordinary differential equation ŽODE., f Ž z . s Ex Ž z . , E y Ž z . , Hx Ž z . , H y Ž z .

T

Ž9.

is a column 4-vector, while the 4 = 4 matrices w Lm x Ž m s 0,1,2., are independent of z. All three matrices are generally too unwieldy to be reproduced here, but an interested reader can obtain explicit expressions using a symbolic manipulation program such as MATHEMATICA. The right side of Eq. Ž8. is complicated, since expŽ"2 iŽprV . z . appear in the kernel matrix w LŽ z .x. A simpler 4 = 4 matrix ODE emerges on using the auxiliary column 4-vector w11,12x X

f Ž z.

EXx Ž z . , EXy Ž z . , H xX Ž z . , H yX Ž z .

s

p cos

p

ž / ž / V

z

sin

p

ysin

V

s

T

z

ž / ž /

cos

V p

V

z

0

0

z

0

0

p

ž / ž /

0

0

cos

0

0

ysin

P fŽ z . .

p z

V

p

V

z

sin

Ž 10.

ž / ž /

cos

V p

V

z

z

X The column 4-vector w f Ž z .x satisfies the matrix ODE X

X

dz f Ž z . si A P f Ž z . ,

Ž 11.

where the kernel matrix A11 A 21 A s A 31 A 41

A12 A 22 A 32 A 42

A13 A 23 A 33 A 43

A14 A 24 A 34 A 44

Ž 12.

does not involve z at all. An explicit expression for w Ax may also be obtained using MATHEMATICA, etc., but is not reproduced here because of its enormous bulk in general. Once Eq. Ž11. has been solved, the actual Ži.e., the unprimed. field phasors may be recovered using Eq. Ž10..

4. Normal and anomalous axial propagation Matrix procedures for solving Eq. Ž11. under normal – not anomalous – conditions are available elsewhere w12x, but a different approach is more illuminating for anomalous axial propagation w7x. Eq. Ž11. yields H xX Ž z .

A13 sy A 23 H yX Ž z .

A14 A 24

y1

½

P id z

EXx Ž z . EXy

Ž z.

A11 A 21

q

EXx Ž z . A12 P X A 22 Ey Ž z .

5

.

Ž 13.

Eliminating H xX Ž z . and H yX Ž z . from Eq. Ž11., we obtain the 2 = 2 matrix ODE d 2z

EXx Ž z . EXy Ž z .

s B Pdz

EXx Ž z . EXy Ž z .

q C P

EXx Ž z . EXy Ž z .

,

Ž 14.

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with B

B11 B21

s

i

s

½

B12 B22

A11 A 21

A12 A13 q A 22 A 23

A14 A 33 P A 24 A 43

A 34 A13 P A 44 A 23

Ž 15.

y1

A14 A 24

5

and C s

C11 C21

C12 A13 s C22 A 23

A14 P A 24

½

A 33 A 43

A 34 A13 P A 44 A 23

A14 A 24

y1

P

A11 A 21

A12 A 31 y A 22 A 41

A 32 A 42

5

Ž 16.

as two auxiliary matrices. Next we eliminate EXy Ž z . by noting that the relation 3

EXy Ž z . s

Ý

gm d zm EXx Ž z .

Ž 17 .

ms 0

emerges from Eq. Ž14.. The following four coefficients are involved in Eq. Ž17.:

g0sy g2sy

2 B12 C21 y C11Ž B12 B22 q C12 . 2 B12 C22 y C12

Ž B12 B22 q C12 .

B11 B12 q B12 B22 q C12 2 B12 C22 y C12 B12 B22 q C12

Ž

.

Ž

,

g1 s y

,

g3s

.

B12 Ž B12 B21 q C11 . y B11Ž B12 B22 q C12 . 2 B12 C22 y C12 Ž B12 B22 q C12 .

B12 2 B12 C22 y C12

Ž B12 B22 q C12 .

.

.

Ž 18.

Thus, EXx Ž z . satisfies the 4th-order ODE 4

m Ý Žyi . a m d zm EXx Ž z . s 0

Ž 19.

ms 0

with constant coefficients a0 s

g 0 C12 q C11 g 3 B12

, a1 s i

g 0 B12 q g 1C12 q B11 g 3 B12

, a2 s y

g 1 B12 q g 2 C12 y 1 g 3 B12

, a 3 s yi

g 2 B12 q g 3 C12 g 3 B12

, a 4 s 1.

Ž 20. Eq. Ž19. can be solved using standard methods w24x. For instance, the characteristic equation corresponding to it is given by 4

m Ý Žyi . a m s m s 0,

Ž 21.

ms 0

which may be factored as 4

Ł Ž s y iqn . s 0,

Ž 22.

ns 1

with qn , Ž1 F n F 4., being the four eigenvalues of w Ax w12x. Because axial propagation occurs in the yz as well as the qz directions, and because HBMs are passive, it is reasonable to expect that Re  q1 4 ) 0,

Re  q2 4 ) 0,

Re  q3 4 - 0,

Re  q4 4 - 0,

Im  q1 4 G 0,

Im  q2 4 G 0,

Im  q3 4 F 0,

Im  q4 4 F 0.

Ž 23 .

4.1. Normal propagation Provided the eigenvalues qn , Ž1 F n F 4., are all different, the solution of Eq. Ž19. is straightforward. Thus, 4

EXx Ž z . s

Ý Enx exp Ž iqn z . ns 1

with four unknowns Enx to be determined eventually using boundary conditions prescribed on z s constant planes.

Ž 24.

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197

X Eqs. Ž17. and Ž13. then assist in determining the remaining components of w f Ž z .x. We get

4

EXy Ž z . s

4

Ý Eny exp Ž iqn z . ,

H xX Ž z . s

ns 1

4

Ý Hnx exp Ž iqn z . ,

H yX Ž z . s

ns1

Ý Hny exp Ž iqn z . ,

Ž 25.

ns1

where the coefficients 3

Eny s Enx

m Ý Ž iqn . gm

,

1 F n F 4,

P

qn y A11 yA 21

Ž 26.

ms 0

Hnx Hny

A13 s A 23

A14 A 24

y1

Enx yA12 P , qn y A 22 Eny

1FnF4

Ž 27.

The field expressions thus obtained are in consonance with the matrix method exposed earlier for HBMs w12x, and finally lead to expressions for w f Ž z .x, Ez Ž z . as well as Hz Ž z .. 4.2. Anomalous propagation The issue of anomalous axial propagation arises on the fulfilment of either one or both of the following conditions: Ži. q1 s q2 and Žii. q3 s q4 . For the sake of illustration, let us assume in this subsection that q1 s q2 but q3 / q4 . Then, the solution of Eq. Ž19. is w24x EXx Ž z . s Ž E1x q iq1 z E2x . exp Ž iq1 z . q E3x exp Ž iq3 z . q E4x exp Ž iq4 z . ,

Ž 28.

with unknown coefficients Enx , Ž1 F n F 4.. In consequence, Eqs. Ž17. and Ž13. yield EXy Ž z . s Ž E1y q iq1 z E2y . exp Ž iq1 z . q E3y exp Ž iq3 z . q E4y exp Ž iq4 z . , H xX Ž z . s Ž H1x q iq1 z H2x . exp Ž iq1 z . q H3x exp Ž iq3 z . q H4x exp Ž iq4 z . , H yX Ž z . s Ž H1y q iq1 z H2y . exp Ž iq1 z . q H3y exp Ž iq3 z . q H4y exp Ž iq4 z . .

Ž 29.

Expressions Ž26. and Ž27. can still be used in Eq. Ž29., but only for 2 F n F 4. The remaining coefficients in Eq. Ž29. are as follows: 3

E1y s E1x

m Ý Ž iq1 . gm

3

q E2x

ms 0

H1x H1y

s

A13 A 23

m Ý m Ž iq1 . gm

,

Ž 30 .

ms0

A14 A 24

y1

P

q1 y A11 yA 21

E1x yA12 A13 P q q1 q1 y A 22 A 23 E1y

A14 A 24

y1

P

E2x E2y

.

Ž 31.

Comparison with Eq. Ž2. shows that the first parts of the right sides of Eqs. Ž28. and Ž29. constitute a Voigt wave. It is delineated by the two independent amplitudes E1x and E2x. Although separate in Eq. Ž28., both amplitudes blend together in Ž29., as evidenced by Eqs. Ž30. and Ž31.. The structure of the axially propagating Voigt wave in a HBM is far more complicated than in a homogeneous medium. This becomes clear on inverting the transformation Ž10. to get the actual electric and magnetic field phasors. For example, the transverse components of the electric field phasor may be stated as Ex Ž z . s

Ž E1x q iq1 z E2x . exp Ž iq1 z . q E3x exp Ž iq3 z . q E4x exp Ž iq4 z . cos y

Ey Ž z . s

V

Ž E1y q iq1 z E2y . exp Ž iq1 z . q E3x exp Ž iq3 z . q E4x exp Ž iq4 z sin

Ž E1x q iq1 z E2x . exp Ž iq1 z . q E3x exp Ž iq3 z . q E4x exp Ž iq4 z . sin q

p

ž / . ž / ž / . ž / z

p

V

z ,

Ž 32.

z .

Ž 33.

p

V

Ž E1y q iq1 z E2y . exp Ž iq1 z . q E3x exp Ž iq3 z . q E4x exp Ž iq4 z cos

z

p

V

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The vibration ellipse of the Voigt wave ŽEq. Ž2.. changes with propagation distance in a homogeneous medium w2,3x, which is true also of the axially propagating Voigt wave in a HBM. Additionally, as the vibration ellipse of any axial propagation mode in a HBM precesses along the z axis w25x, the vibration ellipse of the axially propagating Voigt wave in a HBM also precesses similarly. This precession – indicated by coswŽprV . z x and sinwŽprV . z x on the right sides of Eqs. Ž32. and Ž33. – is a direct consequence of the rotational nonhomogeneity of any HBM, and would not be evident in a homogeneous medium. 4.3. Supercholesteric media As a simple but non-trivial example of media for which the preceding theory may be used for, let us consider the supercholesteric media conceived as particulate composites by Weiglhofer and Lakhtakia w13,14x. These media can also be constructed from planar arrays of aligned electrically small helices w26,27x. Experiments conducted by Gerritsen and Yamaguchi w28x easily lead to a suggestion for fabricating supercholesteric media as stacks of appropriately metalized dielectric plates. A supercholesteric medium is bianisotropic. All four of its constitutive dyadics have the cholesteric liquid–crystalline structure, for which reason the alternative name cholesteric bianisotropic media w12x is justified for supercholesteric media. Thus, with I denoting the identity dyadic,

bln s bln a Ž Iyuxux . q bln bux ux ,

l s 1,2,

n s 1,2,

Ž 34 .

and the condition Ž6. applies. For axial propagation in a supercholesteric medium, the matrix w Ax turns out to be as follows:

vm 0 b 21a 0 yve 0 b 11a 0

0 yvm 0 b 21b A s 0 ve 0 b 11b

vm 0 b 22 a 0 p 1 0 qi yve 0 b 12 a V 0 0 0

0 yvm 0 b 22 b 0 ve 0 b 12 b

0 0 0 1

y1 0 0 0

0 0 . y1 0

Ž 35.

Accordingly, we get B11 s B22 s 0, B12 s i yvm 0 b 21b q B21 s i yvm 0 b 21b q

b 22 b

C22 s yk 02 b 11a b 22 b q

g 0 s g 2 s 0, g 3 s

b 22 a b 22 b b 22 b b 22 a 1

B12 C22

b 22 b

p

ž

ve 0 b 12 b q i

p

ž

b 22 a

C11 s yk 02 b 11b b 22 a q

b 22 a

ve 0 b 12 a q i

qi

V

,

p

/ /ž /ž

qi

V p

ž ž

V

p

/

ve 0 b 12 b q i

V

ve 0 b 12 a q i

V

p

V

,

Ž 36. p

ve 0 b 21 b y i

V

ve 0 b 21 b y i

V

p

/ /

, C12 s C21 s 0, ,

Ž 37.

, g 1 s yg 3 Ž B12 B21 q C11 .

Ž 38.

and a0 s C11C22 ,

a1 s a 3 s 0, a 2 s B12 B21 q C11 q C22

Ž 39.

The foregoing expressions indicate that satisfaction of the condition 2

2 2 B12 B21 q 2 B12 B21Ž C11 q C22 . q Ž C11 y C22 . s 0

Ž 40.

leads to the reduction

'

q4 s q3 s yq2 s yq1 s yi a 2r2 .

Ž 41 .

Voigt waves then propagate in both directions along the z axis as per EXx Ž z . s E1x y etc.

a2

ž / 2

1r2

z E2x exp y

a2

ž / 2

1r2

z q E3x q

a2

ž / 2

1r2

z E4x exp

a2

ž / 2

1r2

z ,

Ž 42.

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5. Dielectric HBMs Let us now specialize to dielectric HBMs, a class that includes chiral liquid crystals w9,10x as well as dielectric thin-film HBMs w6x. This class may be the most populated one of all HBMs – because of the preponderance of dielectric substances over magnetic as well as magnetoelectric substances – and a detailed exposition is in order. A dielectric HBM is characterized by the frequency-dependent constitutive dyadics

b11 s e r e f ,

b12 s0,

b21 s0,

b22 sI,

Ž 43.

where 0 is the null dyadic, while the relative permittivity dyadic

eref s e x xux ux q e y yuy uy q e z zuz uz q e x yux uy q e y xuy ux q e x zux uz q e z xuz ux q e y zuy uz q e z yuz uy

Ž 44.

is arbitrary. The matrix w Ax for the dielectric HBM may be stated as

p 0

0

vm 0

yvm 0

0

yi

V

p i A s

0

V

ž

yve 0 e y x y

ž

ve 0 e x x y

e yz ez x ez z

ex z ez x ez z

/

ž

yve 0 e y y y

ž

/

ve 0 e x y y

e yz ez y ez z

ex z ez y ez z

p

/

0

,

yi

Ž 45.

V

p

/

i

0

V

while

p

B s2

0 V y1

Ž 46.

2

p

ž / V

1 , 0

ž

y k 02 e x x y

C yk 02

ž

eyx y

ex z ez x ez z

e yz ez x ez z

/

ž

yk 02 e x y y

ž / V

ez z

2

p

/

ex z ez y

y k 02

ž

ey y y

/

e yz ez y ez z

Ž 47.

/

are the two auxiliary matrices. Finally, we need the following two sets of expressions, wherein the elements of wCx are not substituted for the sake of transparency: 2

g0sy

g2sy

4 Ž prV . C21 y C11C12 2

2 4 Ž prV . C22 y C12

C12 2

2 4 Ž prV . C22 y C12

,

p ,

g3sy p

a0 s C11C22 y C12 C21 , a1 s 2 i

g 1 s y2

V

C11 y 4 Ž prV .

2

2 V 4 Ž prV . 2 C22 y C12

2prV 2

2 4 Ž prV . C22 y C12

,

,

Ž C12 y C21 . , a2 s C11 q C22 y 4 Ž prV . 2 , a3 s 0, a4 s 1.

Ž 48 . Ž 49 .

The characteristic equation Ž21. simplifies for the dielectric HBM when Ž49. are substituted therein. As the result a0 y ia1 s y a 2 s 2 q s 4 s 0

Ž 50.

is a quartic equation with its cubic term missing Žbecause a 3 s 0., the relation q1 q q 2 q q 3 q q4 s 0

Ž 51.

follows therefrom. Very possibly, two roots of Eq. Ž50. are identical – so that a Voigt wave propagates in either the yz or the qz direction – but the left side of that equation is much too complicated to be explored analytically.

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5.1. Conditions for bidirectional anomalous axial propagation Let us, however, assume that a1 s 0, i.e., ex z ez y y e yz ez x ex yseyxq . ez z

Ž 52.

Then, Eq. Ž50. simplifies to a biquadratic equation and we conclude that

 a1 s 0 4 ´  q3 s yq1 ,q4 s yq2 4 .

Ž 53.

In addition, if a 22 s 4 a 0 , Eq. Ž50. simplifies further, leading to the conclusion that

 a1 s 0,a22 s 4 a0 4 ´  q4 s yq3 s yq2 s yq1 s yi'a2r2 4 .

Ž 54 .

Thus, the simultaneous satisfaction of the conditions a1 s 0 and a22 s 4 a 0 guarantees that Voigt waves propagate in both directions along the z axis as per Eq. Ž42., etc. 5.2. Chiral smectic dielectric media The condition Ž52. is satisfied automatically by all Lorentz-reciprocal dielectric media, because their permittivity dyadics are symmetric w19x. This is exemplified by any medium characterized by

eref s e a Iq Ž e b y e a . ut ut q Ž e c y e a . uy uy ,

ut sux cos x quz sin x .

Ž 55 .

These equations represent a chiral smectic liquid crystal in general w29x, a cholesteric liquid crystal when e b s e a w9x, and a dielectric thin-film HBM when e c s e A w6,30x. The angle x is called the tilt angle in liquid crystal literature and the angle of rise in the literature on thin-film HBMs. Eq. Ž50. simplifies to a0 y a 2 s 2 q s 4 s 0,

Ž 56.

where 2

p a0 s

ž / V

y k 02 e c

p

ž / V

2

y k 02 e˜d ,

p a 2 s y2

ž / V

2

y k 02 Ž e c q e˜ .

Ž 57.

and

e˜d s

eae b

Ž 58.

e a cos 2x q e b sin2x

is a convenient shorthand notation. When 1 2

ž

k0 2prV

2

/

Ž e c y e˜ . e c q e˜d

2

s y1,

Ž 59.

the condition a 22 s 4 a0 is satisfied, Voigt waves propagate in both directions along the z axis as per Eq. Ž42.. The condition Ž59. simplifies to the parallel result given elsewhere w6x for anomalous axial propagation in dielectric thin-film HBMs Ž e c s e a .. Clearly, Eq. Ž59. cannot be satisfied if the medium is non-dissipative Ži.e., e a , e b and e c are real-valued., but physically possible solutions exist when the medium is dissipative. The truth of the latter conclusion was numerically demonstrated elsewhere w6x for the reduced case e c s e a , and the more general case is thereby vindicated.

6. Concluding remarks Anomalous axial propagation has been shown to be possible in a general linear HBM, which is a rotationally nonhomogeneous medium. The anomalous nature is similar to that of the so-called Voigt waves in homogeneous, anisotropic, dielectric media w2,3x. However, the precession of the vibration ellipses of axially propagating Voigt waves in HBMs is a direct consequence of the rotational nonhomogeneity, and would not be evident in a homogeneous medium. Whether HBMs can be realized to satisfy the twin conditions a1 s 0 and a 22 s 4 a0 is an area of future experimental research, now that techniques for fabricating thin-film HBMs are being refined w31–33x. Let us also note that realizations of

A. Lakhtakiar Optics Communications 157 (1998) 193–201

201

HBMs not based on thin-film technology are possible w13,14,26,27x. Hopefully, these and other developments in material sciences will allow the examination and the technological exploitation of the peculiarities of wave propagation in the uncommon substances that HBMs are.

Acknowledgements This paper is dedicated to a future treaty for global elimination of nuclear weapons.

References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x w11x w12x w13x w14x w15x w16x w17x w18x w19x w20x w21x w22x w23x w24x w25x w26x w27x w28x w29x w30x w31x w32x w33x

C.D. Gribble, A.J. Hall, Optical Mineralogy: Principles and Practice, UCL Press, London, 1992. W. Voigt, Nachricht. Gesell. Wissen. Gott. ¨ 5 Ž1902. 269. G.N. Borzdov, Pramana-J. Phys. 46 Ž1996. 245. G.N. Borzdov, J. Modern Optics 37 Ž1990. 281. G.N. Borzdov, Optics Comm. 75 Ž1990. 205. A. Lakhtakia, J. Phys. D 31 Ž1998. 235. A. Lakhtakia, Optik 107 Ž1997. 57. A. Lakhtakia, R. Messier, Mat. Res. Innovat. 1 Ž1997. 145. S. Chandrasekhar, Liquid Crystals, Cambridge Univ. Press, Cambridge, 1992. P.G. de Gennes, J. Prost, The Physics of Liquid Crystals, Clarendon Press, Oxford, 1993. A. Lakhtakia, W.S. Weiglhofer, Microw. Opt. Tech. Lett. 6 Ž1993. 804. A. Lakhtakia, W.S. Weiglhofer, Proc. R. Soc. Lond. A 448 Ž1995. 419. W.S. Weiglhofer, A. Lakhtakia, J. Phys. D 26 Ž1993. 2117. ¨ W.S. Weiglhofer, A. Lakhtakia, Arch. Elektr. Uber. 48 Ž1994. 339. H.C. Chen, Theory of Electromagnetic Waves, TechBooks, Fairfax, VA, USA, 1993. ¨ W.S. Weiglhofer, A. Lakhtakia, Arch. Elektr. Uber. 50 Ž1996. 389. E.J. Post, Formal Structure of Electromagnetics, Dover Press, New York, 1997. A. Lakhtakia, W.S. Weiglhofer, IMA J. Appl. Math. 54 Ž1995. 301. C.M. Krowne, IEEE Trans. Antennas Propagat. 32 Ž1984. 1224. A. Lakhtakia, Int. J. Appl. Electromagn. Mater. 3 Ž1992. 101. F. Brochard, P.G. de Gennes, J. Phys. ŽParis. 31 Ž1970. 691. A. Lakhtakia, W.S. Weiglhofer, Optics Comm. 111 Ž1994. 199. A. Lakhtakia, W.S. Weiglhofer, Optics Comm. 113 Ž1995. 570. H. Hochstadt, Differential Equations: A Modern Approach, Dover Press, New York, 1975. S.F. Nagle, A. Lakhtakia, W. Thompson Jr., J. Acoust. Soc. Am. 97 Ž1995. 42. G. Busse, J. Reinert, A.F. Jacob, in: Proc. ICARSM ’97: III Int. Conf. Antennas, Radiocommun. Sys. Means, Voronezh, Russia, May 1997, pp. 134–140. J.H. Cloete, S.A. Kuehl, M. Bingle, Int. J. Appl. Electromag. Mech. 9 Ž1998. 103. H.J. Gerritsen, R.T. Yamaguchi, Am. J. Phys. 39 Ž1971. 920. I. Abdulhalim, Opt. Comm. 64 Ž1987. 443. V.C. Venugopal, A. Lakhtakia, Optics Comm. 145 Ž1998. 171. R. Messier, T. Gehrke, C. Frankel, V.C. Venugopal, W. Otano, ˜ A. Lakhtakia, J. Vac. Sci. Technol. A 15 Ž1997. 2148. A. Lakhtakia, K. Robbie, M.J. Brett, J. Acoust. Soc. Am. 101 Ž1997. 2052. K. Robbie, J.C. Sit, M.J. Brett, J. Vac. Sci. Technol. B 16 Ž1998. 1115.