Green function for radiation and propagation in helicoidal bianisotropic mediums

A.Lakhtakia W.S. Weig I hofer

Indexing terms: Chiral liquid crystals, Sculptured thin jilms, Thinlfilm HBMs, Complex media, Electromagnetic theory, Green function

Abstract: Radiation by prescribed sources in a helicoidal bianisotropic medium (HBM) is investigated. The Oseen transformation and spatial Fourier transforms are used to obtain an inhomogeneous, first-order, matrix differential equation with nonconstant coefficients. As these coefficients are analytic functions of z and can be expanded as polynomials in that variable, the solution of an auxiliary homogeneous differential equation is explicitly and simply obtained. On using this auxiliary equation's solution, the radiated fields are determined and an appropriate Green function is formulated as a 4 x 4 matrix. Novel results apply to thin-film HBMs as well as to chiral liquid crystals.

1 Introduction

The first thin-film realisation of a helicoidal bianiso- tropic medium (HBM) took place recently [l] and its optical activity experimentally confirmed [2], while the well-known chiral liquid crystals [3] are examples of purely dielectric HBMs. The electromagnetic constitu- tive relations of a HBM are stated in the frequency domain as [ 3 ]

D(r) = .O[B11(2) . E(r) + B l 2 ( Z ) . H(r) ( l a ) ( I b )

where = 8.854 x 10-I2F/m and ,uo = 4x x 10-7H/m are the permittivity and the permeability of free space, respectively. The electromagnetic response of a HBM is quantified in terms of four dyadics: _all(z) is the relative permittivity dyadic, _az2(.) is the relative permeability dyadic, and aI2(z) and ~ ~ ~ ( 2 ) are the magnetoelectric dyadics. While -al ( z ) and ~ ~ ~ ( 2 ) are dimensionless, - alz(z) has the dimension of an impedance and Q~(Z)

that of an admittance. All four inhomogeneous consti- tutive dyadics &z) of a HBM are factorisable as

B(r) = P O [ ~ ~ ~ ( X ) . E(r) + B,,(z) . H(r)

axv(z) = S(z) . pxv . ST(4, (A = 1 , 2 ; = 1 , 2 ) (2)

0 IEE, 1997 IEE Proceedings online no. 19970979 Paper first received 8th January 1996 and in revised form 12th November 1996 A. Lakhtakia is with the Department of Engineering Science and Mechanics, The Pennsylvania State University, University Park, PA

W.S. Weiglhofer is with the Department of Mathematics, University of Glasgow, Glasgow G12 SQW, Scotland, UK

16802-1401, USA

The unitary dyadic

S ( z ) = (u,u, + u,u,) c o s ( T x / n ) + ( U ~ U , - u,uy) sin(m/R) + u,u, ( 3 )

is responsible for the periodic inhomogeneity of the HBM; S'(z) is the transpose of S(z), the z-axis is the spirality axis, 2Q is the helical pitch, while (ux, uy, U,) is the triad of unit Cartesian vectors. The frequency- dependent dyadics &, are independent of z and must be consistent with the Kramers-Kronig relations to be causal. In addition, they must satisfy the constraint [4]

mandated by the Lorentz-Hcaviside visualisation of the structure of electromagnetic theory. Reciprocity, if applicable, imposes additional requirements on DLhv [5]. Finally, are interpreted in terms of 3 x 3 matrices as

(5) I [B*,,, BXu,y P h Z t

Lauz, P x v z y P x v z z

pxv = P X V Y Z Pxvyy P x v y z > (A = 1 , 2 ; v = 1 , 2 )

Propagation of electromagnetic fields parallel to the spiral axis of HBM has been comprehensively handled recently [3]. Following that work, the dyadic Green functions for axially varying fields were also set up [6]. In this short paper we go on to investigate radiation by arbitrarily prescribed sources in a general HBM. For that purpose we make use of the Oseen transformation introduced in [3] and a Fourier representation of all fields and sources with respect to x and y , and then obtain an inhomogeneous, first-order, matrix differen- tial equation with nonconstant coefficients. As the coefficients are analytic functions of z [7] and can be expanded as polynomials in that variable, the solution of an auxiliary homogeneous differential equation is explicitly available to an arbitrary degree of precision. Using this auxiliary equation's solution we determine the fields radiated by prescibed sources and formulate an appropriate Green function as a 4 x 4 matrix. Our analysis is applicable in particular to chiral liquid crys- tals, for which materials the presented results are also novel.

2 Matrix differential equation

With an exp(-iwt) time-dependence, the Maxwell curl postulates may be stated as

V x E(r) = z w B ( r ) - Jm(r) 0 x H(r) = -iwD(r) + Je(r)

( 6 a ) (6b)

57 IEE Proc -Microw Antennus Propug., Vol. 144, No. 1, Februury 1997

where J,(r) and Jm(r), respectively, are the electric and the magnetic source current densities. We substitute the constitutive relations eqn. 1 in eqn. 6 and then imple- ment the Oseen transformation [3]

E’(r) = $ T ( z ) E(r) (7a) H’(r) = s T ( z ) . H(r) (7b)

therein. As a result eqn. 6 may be written as

- S’(x) (V x 1) S(x) . E’(r) = i w o [ & , . E’(.) + gz2 . H’(4l - JX.) ( 8 4

- S T ( z ) . (V x L) . S(z) . H’(r) = --ZWEO[&, . E’(r) + gl2 . H‘(r)] + JL(r) (8b)

where 1 is the identity dyadic, while

JL(r) = sT(z). Je(r) (9a) JA(r) = ST(.). Jm(r) (9b)

The differential operator on the left sides of eqn. 8 is given by

- S*(X) . (V x 1) .S(X) = (V x 1) - (7r/O)[1 - uzu,]

- [l - c0s(7rx/O)](Ot x I ) - sin(?rz/R)(u, x V,) x I

(10) where V, = V - u,(d/dz). On using the spatial Fourier transform

0 0 0 0

A’(r) =/ / ’ ~ ~ f i ~ z e z f i u y {a’, ( K z , K y , Z ) U z + , Ky, .)U,

(11)

-00--00

+U’,(K,, Ky,Z)Uz} dKzdKy

for all fields and sources eqn. 8 decompose into two algebraic equations and four ordinary differential equa- tions. The two algebraic equations can be used to elim- inate e\(K,,Ky,Kz) and ~ \ ( K , , K ~ , K , ) from the four differential equations 181, provided

Assuming eqn. 12 holds true, we rearrange the four differential equations as

P l l z z P 2 2 z z # P 1 2 t z P 2 l z z (12)

(d/dz)[f’(fiz, K y , 4 1 = i [ P ’ ( K z , K y , 4 l [ f ’ ( K . z , K y , 4 1

+ [ j ’ b z , KY, 41 (13)

which is a 4 x 4 matrix ordinary differential equation. The column 4-vector

[ f ’ ( K z , K y , .)I = [e; (Kz, K y , z ) , e$(,z, KGy, 21,

h: (&> K,, 2 ) ) h&, Ky, .)IT (14) contains the four field components to be determined, while the source 4-vector [~’(K,,K~,Z)] and the 4 x 4 matrix function [p’(~, ,K~,z)] are too cumbersome to reproduce here. Exact expressions for the elements of [~’(K,,K~,Z)] and [P’(K,,K~,Z)] have been determined using Mathematica’”. For an arbitrary HBM, it suffices to state here that

[P’(Kz, Ky, 211 = [A] + (6: + .;)[Dol

+ { e i 2 n z / Q ( K z - i ~ , ) 2 [ ~ 2 ~ + e--i2nz/Q ( K z + i K y ) 2 [ O - 2 ] }

+ {eZTz”(Kz - iKV)[Ol] + + iKy)[O-l]}

(15) where [A], [Do], [Dkl] and [ D 4 are 4 x 4 matrices that are independent of z, K~ and K~ but not of Q.

58

3 Auxiliary differential equation

We begin the solution of eqn. 13 by determining a nonsingular matrix function [&~‘(K,,K~,Z)] that satisfies the 4 x 4 matrix differential equation

( ~ l d 4 [ M ’ ( K z , KY,Z) l = i[P’(Kz, K Y , 4 1 [ W K % K Y , 4 1

IM’(Kz, KY, 011 = 111

(16)

(17)

subject to the boundary condition

where [d is the 4 x 4 identity matrix. The matrix func- tion [P‘(K,,K~,Z)] defined in eqn. 15 is an analytic func- tion of z [7]. This becomes clear on rewriting eqn. 15 as the polynomial expansion

00

[ P ’ ( K Z 3 K y , .)I = [ P X K . . , K y ) l Z m (18) m=O

the right side of eqn. 18 being convergent for all z . The constant matrices used in this expansion are as follows:

[ G ( K z , K y ) l = [AI + (2 + 4 P O l + {(Kz - i K J D 1 1 + (KZ + -ZKy)[D-lI}

+ { ( K z - i K , ) 2 [ 0 2 ] + (& + iK,) [ D - a ] } (19a) (Kz, KY )I

+ 2m ( ( K z - z K y ) 2 [ 0 2 ] + (6% + i K y ) 2 [ D - 2 ] ) }

(m = 2,4,6, . .) (19c)

[ ~ ’ ( & z , ~ y , z ) ~ = [Rm(nz, 6 y ) l z m (20)

The solution of eqn. 16 is the polynomial expansion [7] 00

m=O

which is also convergent for all z. The matrices [R,(K,,K~)] have to be calculated from the recurrence relation

m

(m+l)[RTn+l(~z, = 7“C[eLhz, K y ) l [ R n ( K z , Ky)l TL=O

(m = 0,1,2,3, . . .)

[Ro(Kz, KY)] = [Wb Icy, 011 = [ I]

(210)

(21b)

beginning with

In practice, the infinite summation on the right side of eqn. 20 would have to be suitably truncated. However, [M’(K,,K~,Z)] can be computed easily to an arbitrary degree of precision because the right side of eqn. 20 is uniformly convergent for all z. Several examples of [M‘(O, 0, z)] are available elsewhere [3], and numerical examination of [M’(O, 0, z)] for cholesteric-dielectric HBMs has been explicitly carried out as well [9]. Appli- cation of convergence acceleration techniques to com- pute [M’(K,,K~,Z)] is currently underway.

4 Matrix Green function

We are now ready to solve eqn. 13 to determine ~ ( K , , K ~ , Z ) ] radiated by a given L~’(K,,K~,Z)]. Differenti-

IEE Proc.-Microw. Antennas Propag., Vol. 144, No. 1, Febuuary 1997

and therefore on substituting eqn. 16,

( d l W [ ~ ’ ( n z , Ky, 4-l = -i[M’(Kz, Ky, 4 ] - l [ f q K z , Ky, 41 (24)

Eqn. 24 constitutes the adjoint of eqn. 16, as may be inferred from Hochstadt [7]. Using eqns. 13 and 24, we next reduce eqn. 22 to

( d / W { W ’ ( k Kyr ~ ) l - l [ . f ’ ( k > K Y , 41) = [M’(K~,Ky,Z)l-l[j’(K~,KY,Z)l (25)

which is a simple first-order differential equation with a given right side. The solution of eqn. 25 may be stated as

[M’bz, Ky, 4 - l [ f ’ ( K z , K y , 4 1

= [M’(&, K y , O ) l - l [ f ’ ( K z , Ky, 011

+ [M’(J% Icy, <)I-’[.?(&, K y , 014 (26) r 0

where [~(K,,K~,O)] is the boundary value; hence

[ f ’ ( K , , K y , Z ) ] = [ M ’ ( & , K y , 4 l [ f ’ ( K z , K,>0)1

as a result of eqn. 17. Thus eqn. 27 is the solution of eqn. 13. The first term on the right side of eqn. 27 is the standard complementary function, while the second term is the particular integral. Once the right side of eqn. 27 becomes available we can recover the remain- ing two primed field components e\(Kx,Ky,z) and h’, (K,,K~,Z). Inversion of the spatial Fourier transform (eqn. 11) and the Oseen transformation (eqn. 7) finally yields the actual fields E(r) and H(r).

The value of z in eqn. 27 can be negative, zero or positive, while the plane z = 0 can be positioned as desired. Therefore we are able to define a Green func- tion in the form of a matrix as

[ G ’ ( b Icy, X,O] = [M’(Kz, K y , 41[M’(b Ky, O1rl (28) and compactly state the complete solution of eqn. 13 as

+ [G‘(Kz, Ky, ~,Ol[j’(&c, K y , 014 (29) .i 0

The first term on the right side of eqn. 29 is appropri- ate for propagation problems, the second for radiation problems.

5 Concluding remarks

The radiation problem in any HBM has been solved for the first time. The obtained results apply to chiral liquid crystals as well as to thin-film HBMs. Eqn. 29 the chief result, employs a matrix function [M’(K,,K~,z)] with a polynomial representation that can be ascer- tained to an arbitrary degree of precision simply by adding and multiplying 4 x 4 matrices. Many potential applications of thin-film HBMs have been described in a recent concept paper [ 101 and experimental investiga- tions by several colleagues are currently going on. Finally, a more detailed study of general wave propa- gation, which extends the results obtained in [3], has meanwhile appeared in [l 11.

6 Acknowledgments

The first author thanks the Board of Trustees, Pennsyl- vania State University for the grant of a sabbatical leave of absence. This work was also supported in part by a Visiting Fellowship Research Grant from the Engineering and Physical Sciences Research Council of Great Britain (EPSRC GR/K76979).

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