green dyadic and dipole fields for a medium with anisotropic chirality

Green dyadic and dipole fields for a medium with anisotropic chirality

I.V. Lindell W.S. Weiglhofer

Indexing terms: Anisotropic chirality, Green dyadic, Reciprocal bianisotropic medium

Abstract: A closed-form expression is derived for the Green dyadic corresponding to a reciprocal bianisotropic medium with isotropic permittivity and permeability and arbitrary anisotropic chirality. Such a medium can be approximated, for example, by inserting in an isotropic host medium an equal number of similar right- and left-handed metal helices with suitable orientations. The solution is applied to determine the electric field of an electric dipole embedded in such a medium. Evaluation of its far-field shows that the wave is split into two plane waves with different wave- normal directions.

1 Introduction

Isotropic chiral media have been under intensive study during the last decade or so because of their promising applications in antenna, microwave and radar engineering [l, 23. Also, nonisotropic chiral media have shown interesting properties in their power of transforming the polarisation of electromagnetic waves [3, 41. To be able to solve electromagnetic problems in varius media, the Green dyadic of the medium in question should be known, because boundary value problems can then be turned into integral equations which can be efficiently solved through numerical procedures.

Closed-form Green dyadic expressions are known to a limited class of media including isotropic chiral [SI. busotropic [SI, uniaxially nonchiral anisotropic [7-91 as well as axially chiral uniaxial bianisotropic media [lo], all of which fall in the class of uniaxial bianisotropic media. Also, a closed-form Green dyadic can be obtained for media which are related to the above through affine transformations (‘affine uniaxial media’) [l 13. However, there does not seem to exist a single example of a pure nonuniaxial bianisotropic medium with a closed-form Green dyadic. Such a solution is presented here.

In the present paper, we consider certain reciprocal bianisotropic media belonging to a class which is more general than uniaxial. The constitutive equations are assumed to be of the form

D = EE - j J(po zO)RH = E(E - jqR, H)

B = pH + jJ(pozo)RE = p H -k

(1)

(2) K E ( i - r

where the permittivity and permeability are isotropic

0 IEE, 1994 Paper llOOH (Ell) , first received 14th June and in revised form 22nd December 1993 The authors are with the Electromagnetics Laboratory, Helsinki Uni- versity of Technoloa, Otakaari 5A, Espoo, Finland 02150

IEE Proc.-Microw. Antennas Propng., Vol. 141, No. 3, June 1994

because of the scalar parameters E and p and the chirality parameter R is a dyadic, which is assumed symmetric because of reciprocity [I 11. The relative chirality dyadic R, and the wave impedance q are defined by

(3)

A possible realisation of this kind of medium can be visualised by inserting small metal helices (modelled as uniaxial inclusions with handedness) in an isotropic host medium. To obtain anisotropic chirality, the helices should have some preferred orientations. However, the permittivity and permeability will then be also anisotropic, in general. In principle, this could be balanced by a proper inclusion of electric and magnetic dipoles. However, there is a more practical method to fabricate a medium like this. Noting that the permittivity and the permeability generated by the helical inclusions do not depend on the handedness, we can distribute an equal amount of right- and left-handed helices so that, without handedness taken into account, they prefer no direction in space. However, if the right-handed helices prefer some direction and the left-handed helices another direction in space, the chirality becomes inevitably anisotropic.

As a concrete example we may consider two sets, 3N right-handed and 3N left-handed helices, randomly dis- tributed in a host medium. Let 2N of the right-handed helices have their axes parallel to the x axis, 2N of the left-handed helices parallel to the y axis and the remain- ing 2N helices (N left-handed and N right-handed) parallel to the z axis. Then, the permittivity and permeability are isotropic and the dyadic R has no z component. In fact, R, must have the form

E, = K,(U, U, - U, U,) (4) This simple model presumes that the helices are uniaxial objects, which is an idealisation [12]. The practical realisation remains yet to be tested.

Anyway, it is clearly seen that the medium with a con- struction as outlined above is not uniaxial but of a more general bianisotropic nature, and it appears to be the first truly nonuniaxial bianisotropic medium to have a closed- form Green dyadic expression. The nonuniaxiality is seen very clearly from the wave-number surfaces of the medium with the chirality dyadic (eqn. 4), because, instead of one or two optical axes, it turns out to have two planes of directions in space where the wave numbers of the eigenwaves coincide.

2 Field analysis

2.1 Basic equations Let us start from the Maxwell equations which can be written in the form

211

(V x i - kri-,)E = -jkqH - J,,,

(V x r - kZ,)H=j- E + J

(5)

(6) k q

where k = w J ( p ~ ) , J and J,,, are the electric and magnetic current sources and 7 is the unit dyadic. Eliminating the magnetic field leaves us with the Helmholtz equation

(7) R(V)E = +jkqJ + (V x i - key,,,

with the Helmholtz operator

R(V) = -(V x 1- kEJ2 + k 2 i

= -L+(V)E-(V)

The two first-order operators L+(V), z-(V) are defined as

E,(V) = V x fT kE, E , = T + 17,

E+(V) - L ( V ) = -2ki

(9) They commute with each other: E+ E- = 1- E + , and obey the simple relation

(10) 2.2 The Green dyadic problem Let us consider the electric Green function G(r) defined by the Helmholtz operator

R(v)G(r) = - 6(r)T (11) It can be shown that the solution to eqn. 11 can be found in terms of two Green dyadics G+ , 5- , each satisfying a first-order differential equation of the form

E,(V)C,(r) = -6(r)J (12)

R(V)(C+ - G-) = - E - @ + G+) + E+(E- e-) By using eqns. 8 and 12 it can be seen that

= (E- - L+)S(r) = 2k6(r)T (13) Comparison of eqn 13 with eqn. 11 (uniqueness assumed through suitable radiation conditions) then yields the sought-after Green dyadic as

1 2k (14) q r ) = - - [C+(r) - G-(r)]

It remains to solve eqn. 12 for the Green dyadics c, ,

Written more explicitly, we have the equations

t,(V)G,(r) = (V x fT kE,)c,(r)

= -6(r)i (15) This type of Green dyadic problem has been solved previously [l 11 and the same method is applied here. Recal- ling that, because of reciprocity, the dyadics 8, are symmetric, whence the two Green dyadics can be expressed as

(16) where the scalar Green functions G,(r) satisfy the equations

G,(r) = [VV +_ k(V.8,) x r + k2B(:’]G,(r)

(17)

The dyadic E(’ ) (the ‘adjoint’, or ‘second‘, or ‘double-cross square’ of 8) is defined as

(18) and it can be readily evaluated applying dyadic oper- ations like and : products. For details, see Reference 11.

212

1 k

[ E , : VV + k2 det E,]G,(r) = k- 6(r)

$2) = ‘8” E = 8-1 det E 2 x

Applying an affne transformation to eqn. 17 ( E , is assumed positive definite, in which case the square-root operation produces a unique dyadic.)

(19) V‘ = E;/2v rt = E , 1/2r

which entails v r ’ = v r = T

k; = k,/(det a,) (20) Thus, eqn. 17 is transformed into the differential equation for the Green dyadic of an isotropic medium

(21) 1

( V 2 + kj)G’*(r’) = +y 6(r’) k*

with the well-known solutions

where

D + = J(det B ,)J(r’ . r’) = J(rE(:)r) (23) Substituting eqn. 22 into eqn. 16 yields

1 1 G, ( r )=Tk [i2 - V V + - ( b + . V ) x f + E ~ ) k g r (24)

Finally, putting eqn. 24 into eqn. 14 provides us with the expression for the Green dyadic in the form

1 1 G ( r ) = g V V ( g + +g-)+-(E+Vg+ 2k - ~ - v g - )

2.3 Bi-isotropic special case To check the result of eqn. 25 let us consider the biisotropic medium for which the Green dyadic expression is known. Setting E, = K, 7, we have

8 , = (1 K,) f

$2) + - - (1 K,)’T

det 8 , = (1 +_ K , ) ~

D, = (1 & K,)T

1 e - i k + r

g * = = - Z

k, = k(l & K,)

Inserting these relations into eqn. 25 gives

which coincides with that given elsewhere [6, 111. In the isotropic special case with K, = 0 we have E , =

E($) = I, g + = g- = g and the well-known Green dyadic for isotropic media is seen to result

I E E Proc.-Microw. Antennas Propag., Vol. 141, No. 3, June 1994

2.4 Uniaxial special case Another, more interesting case, is the medium with axial chirality

where TI = u,u, + uyuy is the transverse unit dyadic and p = J(x2 + y'), the cylindrical radial coordinate. The Green dyadic (eqn. 25) has now the form

(33)

This expression coincides with one derived earlier for uniaxial chiral media [lo], provided the uniaxial permittivity and permeability are reduced to isotropic, with some minor differences in notation.

3

Let us apply the novel Green dyadic expression to find the electric field in a medium with the chirality dyadic

Dipole fields in a special medium

E, = .,(Ux U, - uy uy) (34) This kind of a dyadic can be realised, at least in principle, by a mixture of similar right- and left-handed helices as explained in the Introduction. The dyadics E , are of the form

E , = 7k E, = (1 * K,)UxU, + (1 f K,)UyUy + u,u, (35) and the derived quantities are

det E , = 1 - ~f (36)

(37)

$2' - + - (1 T K,)U,U, + (1 f Kr)UyUy 4- (1 - Kf)U,U, (38) The two distance functions can be written in Cartesian and spherical coordinates as

D&) = ,/[(I f K,)X2 + (1 k Kr)y2 + (1 - Kf)Z2]

=f& 4)r (39) with

f,(O, 4) = J(l - ~f cos2 0 f K, sin2 0 cos 24) (40) The Green dyadic (eqn. 25) for the medium has the form

1 1 :[lz k G(r) = - - vv + - (E,. V ) x i+ i- K:U,U,

X ( g + + g - ) + - - V x l - E , ( g + - g - ) (41) :[: - 1 3.1 A dipolar electric current source J = U, IL6(r) gives rise to an electric field which can be written in terms of the

Field from an electric dipole

I E E Proc.-Microw. Antennas Propag., Vol. 141, No . 3, June I994

Green dyadic (eqn. 25) as

E(r) = -jk&r)uz1L = E , + E - (42)

The two terms corresond to two radiating waves which produce the total field. The two scalar Green functions g , have different functions D , = f*(0 , 4)r . Denoting

(44) - $2' * - * U , .f* =J(w,.u,) r = u , r their derivatives outside the origin can be written as

(45)

3.2 The far field In the far field, at any fixed point r = ro = U, r,,, the radiation field can be approximated by a combination of two plane waves. For a nonisotropic medium, the directions of propagation of the plane waves do not coincide with the radial direction U,, in general, [lS].

To find the far-field expression in the limit r + CO, we can approximate

V g , U - jkg , !!k f* w* w i VVg, U -k2g+ ~

f: which inserted in the field expressions (eqn. 43) gives

jkqlL 2

E , Z - (1 - Kf)g,

(49)

(50)

On the xy plane with 0 = 4 2 we have U, = up and the two field components reduce to

The polarisations are transversal to the radial direction up and elliptic with the axial ratios f, . They are circular in the directions 4 with

(53) f, = J(1 T K, cos 24) = 1 which requires cos 24 = 0, or 4 = * 4 4 + nx.

213

3.3 Local plane waves Let us consider the local plane-wave behaviour of the two field components E,, not limited to the xy plane. Close to a point ro with r = ro + Ar in the far field, the exponential functions in the Green functions g, can be approximated by

(54) This is a plane-wave function of the small distance Ar and of the form exp ( - jk , Ar) with the wave vectors defined as

e - j k D f(i) - - e - j k D i ( r o I e - jkVD+ (ro) AP

k, = k V D , = k w . (55) f, Thus, the two local plane waves have wave-propagation directions parallel to the vectors w, = @)U, , which, in general, differ from the radial direction U,. Expressing the unit vectors of the directions as U , = w,/ l w, 1 , we can find a relation for the wave numbers of plane waves propagating in the directions U ,

This result can also be directly obtained from the plane- wave analysis of the present medium, a subject treated in another paper [13].

The wave normal directions w* of the two plane-wave components of the far field of the radiating dipole in the xy plane are exemplified in Figs. 1 and 2. The normalised

1 5 , ..

10

0 5

0

-0 5

-1 0

-1 5 Fig. 1 Polar diagrams of normalised distance functions f, = D J r corresponding to two components o f f e ld radiatedfrom dipole in anisotropic chiral medium, for some values of normalised chirality factor K , given in Figure Only diagrams off, are depicted in xy plane; diagrams off.(r,) equal those of f+( -x,); x axis is horizontal and y axis vertical in Figure

distance function f+(x /2 , +), is given in polar form for some values of the K, parameter in Fig. 1. Note that, because of the symmetry relation f+(~,) = f - ( - ~ , ) , f+ andf- produce the same figures but rotated by the angle n/2.

The wave-normal vectors of the two plane-wave components are normal to the corresponding f, diagrams, which fact is depicted in Fig. 2. Both plane waves travel in the same radial direction when it coincides with one of the artesian axial directions 4 = 0, 4 2 , n, 31112. In the intermediate directions 4 = n/4, 3x14, 5n/4, 7n/4, the wave numbers are the same but the wave directions of the two plane waves are not except when the chirality vanishes. In the cases K, = 1, the patterns in space have

214

a double spherical form which means that the waves do not propagate in the xz or yz plane because of the nulls in the corresponding f, patterns, since f, = J(2) sin e sin 4 or = J(2) sin e cos 4.

-151- ~- -I-I Fig. 2 Wave-normal direction vectors w * , corresponding to two plane-wave components of far field radiated from dipole, are normal to respective polar diagrams of normalised distance functions f, Figurer correspond to relative chirality K, = -0.7; a~es as in Fig. 1

It is to be noted that, even if the two plane-wave components of the radiated field do not propagate in the radial direction, in general, their polarisations are transversal to the radial direction. This is also the case outside the xy plane.

4 Summary

A specific example of a reciprocal bianisotropic medium with isotropic permittivity and permeability but anisotropic chirality was considered in the present paper. It was seen that the Green dyadic can be solved in closed form, which enables one to solve the electromagnetic fields of any sources in terms of spatial integrals. This appears to be the first truly nonuniaxial medium for which a closed-form Green dyadic expression exists. The expression obtained was checked by two special cases for which solutions have been obtained previously. As an application, radiation fields from an electric dipole were derived for a special medium which can be approximated in practice by inserting similar right-handed and left- handed helices in a host medium. In the far field, the radiation field was seen to consist of two plane waves propagating in different directions, but polarised transverse to the radial direction. In the ongoing process of developing novel microwave applications of chiral media, the solution given in the present paper may prove useful.

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green dyadic and dipole fields for a medium with anisotropic chirality

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