Russian Physics Journal, Vol. 41, No. 8, 1998

FUNDAMENTAL PROBLEMS OF RADIOWAVE PROPAGATION IN INHOMOGENEOUS MEDIA AND INTERACTION OF ELECTROMAGNETIC RADIATION WITH MATTER

D I F F R A C T I O N AND PROPAGATION OF ELECTROMAGNETIC

WAVES IN T H E P R E S E N C E O F C H I R A L MEDIA

UDC 537.874

A. G. Dmitrenko and V. V. Fisanov

The propagation of electromagnetic waves in infinite, semiinfinite, and laminar chiral structures is investigated, along with the diffraction of electromagnetic waves on semiinfinite objects in a chiral medium and three-dimensional objects consisting entirely or partially of chiral materials. Particular attention is paid to diffraction on three-dimensional uniform chiral bodies and ideally conducting bodies covered by chiral shells.

INTRODUCTION In the last ten years, there has been increasing interest in UHF electromagnetic wave processes in chiral media [1]. In the simplest case, a complex artificial medium of this type is assumed to be macroscopicaUy homogeneous and isotropic, consisting of noncentrosymmetric objects of the same mirror form (for example, segments of spiral wires), which are uniformly distributed and randomly oriented within an isotropic medium. A chiral medium is anisotropic if the microobjects introduced are in a particular order or are used in combination with objects of different shape, like omega particles, or if the surrounding medium is initially anisotropic. At the phenomenological level, the difference between chiral and traditional media is determined by the presence of one or two parameters (in the general case, tensor parameters) ensuring an additional (magnetoelectric) relation between the induction and field-strength vectors in the material equations. Electromagnetic processes in chiral media have a series of properties different from those in ordinary isotropic media. For example, chiral properties are clearly indicated by the presence of a cross-polarized component in the field scattered from a chiral object when the field scattered from an ordinary isotropic object does not include this component. Chiral properties lead to intensification of the mechanisms of electromagnetic-energy absorption within the body, to complication of the spectrum, and to hybridization of the intrinsic waves. The use of such effects may permit the creation of UHF waveguide components with fundamentally new properties, as well as nonreflecting coatings and protective screens. In this context, the electrodynamics of chiral structures is under active investigation around the world, and international conferences on the electromagnetism of chiral and bianisotropic media are held annually. In the present work, some results on the electrodynamics of chiral structures obtained in recent years are described, relating to the propagation of electromagnetic waves in infinite, semiinfinite, and laminar chiral media and structures, the diffraction of electromagnetic waves on semiinfmite objects in a chiral medium, and three-dimensional scattering on objects consisting entirely or partially of chiral materials. Particular attention is paid to diffraction on three-dimensional uniform isotropic chiral bodies and ideally conducting bodies covered by chiral shells.

Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 50-61, August, 1998.

780

1064-8887/98/4108-0780520.00

9

Kluwer Academic/Plenum Publishers

~e,

~te//

~

- ~ ~ \ \

/ (o, ,--

,

,

Mn, i ~

~nji

Fig. 1

e r ~

"101"

""

3

_..

"%

-20

.s.~ ~

~

-30 o

"

""

1 ,o

I

0, do :

Fig. 2

1. WAVES IN REGULAR CHIRAL MEDIA AND STRUCTURES The study of intrinsic waves propagating in a uniform infinite chiral medium is the essential first stage in investigating wave phenomena in finite regular and irregular electrodynamic structures. At the phenomenological level, a generalized bianisotropic chiral medium is characterized by a system of material equations

D=~.E +i~.B,

(1)

H=i~. E+O q- B,

(2)

if a harmonic time dependence of the form exp (-ioa) is considered. In Eqs. (1) and (2), the electric E and magnetic H fieldstrength vectors are linearly related to the electric D and magnetic B induction vectors by the dielectric permittivity ~ and permeability/2 tensors and the magnetoelectric interaction parameter ~ (a pseudotensor). The following special types of chiral media are considered here: an isotropic chiral medium (e, # are scalars, ~ is a pseudoscalar); a chiroferrite (e is a scalar, fi is a uniaxial gyrotropic tensor, ~ is a pseudoscalar); a bigyrotropic medium (~ and /2 are gyrotropic tensors, ~ is a pseudoscalar); a gyrotropic omega medium (~ or/2 is a uniaxial gyrotropic tensor, ~ is a pseudotensor with zero spur). Another example of a chiral medium is a generalized biisotropic medium additionally characterized by a nonmutuality parameter.

781

dB

2 ..

-10

%''",, 9~

9 ~.

~3

..3....~."\ ",,

-20

"~.

-30

~ 50

0

~ 100

~ O, dee

Fig. 3

c00/;L2~ dB -4 -6 -8 -10 0

50

100

O, deg

1O0

0, deg

Fig. 4

dB

3

-10 -15 -20 -25 50 Fig. 5

In gyrotropic chiral media, the propagation of intrinsic waves in a plane transverse to the gyrotropy axis of the medium (the z axis of a rectangular coordinate system) is considered. After substituting an expression in the form of a plane wave E(r) = E0ex p 0k-r), where k is a two-dimensional wave vector ( ~ = 0), into the Helmholtz vector wave equation

(3) and taking account of the rotational symmetry relative to the z axis, the problem reduces to a system of algebraic equations for the scalar components of the phasor E 0, and the dispersional equation for the modulus of k is derived from the condition

782

Sd,,./~\ o!2) / /

/~.~.'X

ti

De \\,e

7

t-

Fig. 6 of existence of a nontrivial solution. The dispersional equation determines the constants of propagation of the two hybrid volume waves, with left and right elliptical polarization, respectively, as a rule. For example, in chiroplasma (a gyroelectric version of a Faraday ehiral medium [2]) (4)

where ez and e • are the dielectric permittivities for ordinary and extraordinary waves propagating in a nonchiral hydroelectric medium transverse to a constant magnetic field [3]. The conditions of free propagation (real k+) and damping (imaginary or complex k+) of intrinsic waves are studied in the frequency plane in the coordinates (I], R), where fl and R are the ratios of the field frequency and the cyclotron frequency to the electron plasma frequency within a gyrotropic medium and the ratios of the field frequency and gyromagnetic-resonance frequency to the saturation-magnetization frequency in the ease of a ferrodieleetrie, respectively. In this formulation, plane waves in a ehiroferrite [3], in a bigyrotropic medium, and in uniaxial gyrotropic omega media of gyroelectric and gyromagnetic types may also be studied. Surface electromagnetic waves represent a significant addition to the intrinsic-wave spectrum of semiinfinite chiral media. Such waves are possible in some conditions at the plane surface of an isotropic chiral halfspace [4, 5]. A halfspace consisting of gyrotropie chiral material has improved waveguide properties: it permits the propagation of surface waves along an ideally conducting screen (electrical, magnetic wall) [3] and also along the interface with a nonchiral isotropic magnetodielectdc and along the boundary between two enantiomorphic chiral Faraday media [6]. Surface waves at the boundary of a halfspace consisting of a gyroelectric gamma medium have much in common with the analogous waves of chiral Faraday media: they are unidirectional (which is typical for surface waves with a Voigt configuration: propagation occurs transverse to the external magnetic field, which is applied along the boundary of the medium) and hybrid. The difference is determined by the conditions of existence of these waves. Frequency regions where the surface waves have a real constant of propagation are bounded by resonance lines and segments of surface waves. In gyroelectric media, there may be a generalized surface wave characterized by unit damping decrement of the wave transverse to the boundary under the condition that bulk waves do not propagate in this frequency range. A thread-like rectilinear source with a traveling electric-current wave parallel to a plane isotropic chiral layer with an ideally conducting substrate excites a complex quasi--three-dimensional electromagnetic field. In comparison with a nonchiral structure, a cross-polarized component appears in the radiation field above the layer; the discrete-spectrum wave propagating along the layer becomes hybrid, retardation of the waves increases, and the spectrum is broadened [7, 8]. When there is also a system of close rectilinear conductors at the layer surface, so that an anisotropically conducting surface is formed, the discrete wave spectrum maintained by the layer is enriched on account of two nonsymmetric surface waves. The propagation constants of these waves are functions of the geometric and material parameters of the layer and also the phasing parameter of the source. In addition, waves predominantly due to the waveguide of the system consisting of linear conductors and an 783

dB

f

~~176176

\~

0

30

..:..-.

....

9.......

-15

9,4,-

,

. . ............. :;,

"~:.,....-'"3..,,." ~ '

-10

"-'"

60

,

,

90

120

0, deg

0

30

60

Fig. 7

90

120

O, deg

Fig. 8

dB

5

"~176176 ~ .,~ ~176176

9,~-..

-

.

.

~

~176176

.

-.-~

0

...:~":'~"

0

J.,

-5 -10

6"

-20

-1~ 0

30

60

90

120

0,

deg

"~'''~'f 0

I

I

30

60

Fig. 9

"%,/4 I

90

I

120

I

0 deg

Fig. 10

ideally conducting screen are formed [9, 10]. If the screening surface is moved to infinity, the surface-wave spectrum is signifi~ntly depleted, but the structure now permits the excitation of lateral waves [9, 11]. When a two-dimensional electromagnetic field is excited by a system of synphase delta generators located at breaks in wires running along the surface of the plane screened layer, an znalogy is seen between the structure conducting along logarithmic spirals and the framework of rectilinear wires. By changing the twist parameter of the spiral and the chiral admittance, the retardation coefficient of the waves directed by the layer may be regulated.

2. ELECTROMAGNETIC-WAVE DIFFRACTION IN CHIRAL MEDIA ON OBJECTS W I T H A SHARP EDGE

Scattering obstacles of simple form with a sharp edge (a halfplane, a wedge, etc.) are canonical Objects of wavediffraction theory. For complex chiral media, the classical results on wave diffraction must be reexamined; generalization both in methodological terms and in the physical interpretation of the wave components of the diffraction field is required. The electromagnetic-field components transverse to the rib may be characterized by singular behavior in its vicinity. The presence of media of different physical types has a significant influence on the field singularity at the rib, precise knowledge of which is required for physically correct formulation of the boundary problems and the construction of mathematical models at the electrodynamic level. Various rib structures with material media and conducting boundaries may be obtained as a result of considering a generalized multisectorial structure with a common rib. According to the method of [12], the short-range field within each sector must be represented by a generalized power series of the form

A(r,q))=r'-' ~ aj(q))r j , j=0

784

(5)

where A(r, ~o) is one of the scalar components of the electromagnetic field close to the rib, which is located at the origin of a cylindrical coordinate system, in the direction of the z axis; r is the field-singularity index. Substitution of Eq. (5) into the Maxwell equation, taking account of Eqs. (1) and (2), leads to a system of differential and algebraic equations in aj(~o), which are valid within each partial sector with specified material parameters. Matching of the boundary values of the electromagnetic-field components tangential to the boundaries is ensured by introducing the characteristic matrix of the partial sector, which is unimodular. This approach provides a universal method of deriving the characteristic equations for ~- in a set of angular structures, with and without external impermeable boundaries (electrical or magnetic walls). This approach, first developed for the sectors of isotropic chiral media, has been extended to structures combining chiral and gyrotropic sectors [13] and to structures with a gyrotropic chiral medium [14] (under the condition that the axis of the medium is oriented parallel to the rib). For several simple configurations commonly encountered in electrodynamic models, we obtain relatively simple relations, which are the accurate solutions of transcendental equations in r. For example, if two biisotropic sectors of the same angular dimension $ (0 < ~I, _ w) have one boundary in common and the other boundaries are ideally conducting electrical walls, we may write [15] (6)

r,l +(., where Yj = Ikj + i~j (j = 1 or 2) is a generalized isotropic magnetoelectric coupling parameter; it includes the chirality parameter ~j and the nonmutuality parameter ~kj. The solution in Eq. (6) must not disrupt the condition at the rib, according to which Re ~" > 0. In chiral media, if the losses are disregarded, Re ~-may be zero with certain combinations of the material parameters; this must he taken into account in the mathematical modeling of such media in the presence of a rib [19, 20]. Suppose that an ideally conducting halfplane is in a uniform isotropic chiral medium. Since the medium sustains two intrinsic waves of right and left circular polarization with different wave numbers, the key problem is the diffraction of these waves at the halfplane. The result of the diffraction of a primary electromagnetic wave may be obtained as the superposition of the solutions of diffraction problems for both intrinsic waves. Solution by the Wiener--Hopf method (a variant of the Jones method) [17] includes the stages of obtaining the system of functional equations (they have an accurate explicit solution) and factorizing functions containing two pairs of branching points. An alternative method of solving diffraction problems for a halfplane, which has been specially developed for a wedge of arbitrary vertex angle, is the Malyuzhints method using a Sommerfeld two-loop path integral [18]. Solutions have been obtained for the diffraction of intrinsic waves in an isotropic chiral medium [19], a biisotropie medium [20], and a gyroelectric chiral medium [21]. The total field is expressed as the superposition of two path integrals with kernels corresponding to plane waves of the medium. The Malyuzhints system of functional equations is not expressed in closed form but is reduced to integral equations; it permits the localization of singular points of the integrands, which permits the separation of characteristic wave components of the scattered field in the far zone. The solution includes plane waves of both intrinsic polarizations of the medium that have been reflected and transformed by the boundaries; the number of such waves is determined by the wedge vertex angle and the angle of incidence of the primary wave, the two cylindrical waves from the rl'b, and two lateral waves, the phase fronts of which are inclined to the faces of the wedge at the angle of total internal reflection. In the case of a gyroelectric chiral medium (chiroplasma), a unidirectional surface wave is also excited at one of the surfaces.

3. ELECTROMAGNETIC-WAVE DIFFRACTION AT A CHIRAL BODY BOUNDED BY AN ARBITRARY SURFACE After studying the diffraction of electromagnetic waves on semiinfmite (two-dimensioual) structures, the logical next step is to analyze diffraction on three-dimensional objects of chiral material. This is a considerably more complex problem, requiring the use of numerical methods. In this section, we consider the diffraction of an external electromagnetic field on a three-dimensional isotropic chiral object bounded by an arbitrary smooth surface. The solution is obtained by a n original numerical method, which is a generalization of the method developed in [22-25] for diffraction on three-dimensional magnetodielectric bodies. The physical features of scattering due to the chiral properties of the object.

785

As illustrated in Fig. 1, suppose that a uniform chiral body Dil characterized by dielectric permittivity el, permeability /~i, and chiral parameter ~ and bounded by an arbitrary smooth surface S is within an in/mite uniform medium De with electrodynamic parameters ee,/%. The body is excited by an external harmonic field {E0, Ito} with a time dependence of the form exp (-iwt). We want to f'md the scattered field {Ee, lie} in De. Since the scatterer is transparent to electromagnetic radiation, the field {Ee, He} in De will be accompanied by another unknown field {Ei, Hi}. These fields must satisfy Maxwell's equation

V x E e =imPeH , , V x H e ~--io)s,,E~. in D~.,

(7)

V•

i =i(o~ti(U i +~V•

, V• H i =

-io~e,(E i +[3Vx El) in Di,

(8)

the boundary conditions

nx(ri-E~)=n•

n•215

o on S

(9)

and the radiation conditions (10) where n is the unit vector normal to S; R is the distance from body ~ to the point of observation; a x b is a vector product. The surface S is assumed to be a Lyapunov surface, Im ei,/z i _> 0. Note also that Eq. (8) takes into account that the material equations of the ehiral medium are chosen in the form [26] Di = ~i(Ei + 13V x E i ) ,

B i = lai(H i + ~V x

H i).

(11)

The solution is obtained as follows. We introduce two auxiliary surfaces Se = YeS and Si = KiS , similar to surface S. Surface Se with similarity coefficient Ye < 1 is within Di; surface Si with similarity coefficient Ki > 1 is outside D i. The unknown field {E i, Ill} in D i is represented as the sum of fields of auxiliary elementary electric dipoles at points {in,i}n=lNi on surface Si; the dipoles are oriented tangentially to S i. Using the dyadic Green's function [26] for an infinite uniform chiral medium with parameters e i, #i, B, we obtain

Z~, ,'(

Ei(M)=(m2~ilkiXT/ki) 2 N, On i M ,Mn,i)+ Y20;'i(M,M.,i)]" R~ i, n=l

(12)

Iti(M)=(-io~X~tlk,)2Y'.~tGt'i(M,M..i)-~12G;'i(M, Mn,i)]. R~ i ,

M ~ Di ,

n=[

where k i = o J ~ ) , 3,2 = ki2/(1 _ ki2/32); 3"1 = ki/(1 - ki/3), 3'2 = ki/(1 + ki/3) are the wave numbers of the right- and left-polarized waves; pr n,i (n = 1, 2 . . . . . N i) are unknown dipole moments consisting of two independent components prl n,l and Pr2na in the directions %1n,~ and er2na chosen in the tangential plane to S i at the point Mn,i; 1,2n 'i are elements of the dyadic Green's function

M.,,) = ^hi

+O/Y,) v v

+v •

M..,),

(13)

G 2" (M, Mn.i)=(kil2y2~ll+(ll'[ )VV+VxI]W~(M, Mni ) . In Eq. (13), I is a unit dyad; XItl,2i(M, Mn,i) = exp (iT1,2RMn'i)/47rRMn,i are scalar Green's functions; RMn,i are the distances from the points Mn, i to point M in D iBy analogy with the internal field {Ei, Hi} , the scattered field {Ee, He} in D e is expressed as the sum of the fields of tangentially oriented electric dipoles with unknown moments pr n,e (n = 1, 2 . . . . . Ne) at points {Mn,e}n=lNe on surface Se (Ne is the number of dipoles) 786

/ Ve

s

^

E(k2
,'i=l

iv< H<.(J,t)= (-i<,>)Y. (v x ,~)',,'<(M, M,.< ). t,~" ,

(14) M~ Z~<,

n=l

where ke = to( ~ e ~ e ) ; le(M, Mn,e) = exp (ikeRMn'e)/41rRMn'e, RMn'e is the distance from point Mn, e to point M in De; pr n,e consists of two independent components prl n,e and Pr2n,e along the directions erl n,e and er2n,e, which are chosen in the tangential plane to Se at point Mn,e. The fields in Eqs. (12) and (14) satisfy Eqs. (8) and (7), respectively. In addition, the field in Eq. (14) satisfies Eq. (10). To satisfy the boundary conditions in Eq. (9), we need to determine the unknown vector constants pr n'i (n = 1, 2 . . . . . Ni) and pr n'e (n = 1, 2 . . . . . Ne). We use collocation for this purpose. Suppose that Mj (j = 1, 2 . . . . . L) are the collocation points on S. Then we obtain the following system of linear algebraic equations with a complex matrix of dimensions 4L x 2(Ne + Ni) for the unknown complex constants Prl n,l, Pr2n,~ (n = 1, 2 . . . . . Ni) and Prl n,e, pr2n,e (n = 1, 2 . . . . . N e)

The solution of Eq. (15) is determined by minimizing the functional

<,>=

I,'•

gl:+

)1,,, • (Hi- -:' )-,,'• ,,al }

(16)

on the basis of the conjugate-gradient method. The accuracy of solution is monitored by calculating

I,,- •

+

}.

(17)

where ~ ' is the value of the functional in Eq. (16) at the grid of intermediate points; L' is the number of intermediate points. The method here outlined is implemented in a Fortran program for calculating the scattering-fieM components and monitoring the accuracy of the solution. The input variables are the scatterer geometry, its wave dimensions, the exciting fieM, the material parameters e i, gi, ~ and also the parameters of the method: the similarity coefficients Ke, Ki, the number of dipoles Ne, N i, and the number of collocation points L. The functional in Eq. (16) is minimized by the conjugate-gradient method. This program may be used for computational experiments to investigate the influence of the chirality parameter on the scattering properties of bodies of various shapes. The characteristics investigated are the bistatic scattering cross sections for the basic and cross-polarized components of the scattered field (18)

where Ee,0 and Ee,~, are the corresponding components of the field in Eq. (14), in a spherical coordinate system. Some calculation results are given in Figs. 2-5. The variation in bistatic scattering cross sections with intensification of the chiral properties of the scatterer is shown in Figs. 2 and 3, for a sphere of radius kro = 1.5, k = 2x/),, where X is the wavelength in free space; the parameters of the sphere material are ei/e e = 4.0, /zi//ze = 1.0. The sphere is excited by a linearized plane wave propagating in the positive direction of the z axis in a Cartesian coordinate system (Fig. 1); the planewave vector E 0 is directed along the x axis. The copolarized component of the scattered field is shown in Fig. 2, and the cross-polarized component in Fig. 3. The results are given for the x0z plane; 0 is the angle between the observation vector k and the direction of observation in the x0z plane. Curves 1-4 correspond to ki/3 = 0.1, 0.3, 0.5, and 0.0, respectively. The parameters of the method are as follows: Ke = 0.3, K i = 3.0, M e = N i = 60. The dipoles and collocation points are located on the corresponding surfaces of identical form; six dipoles (collocation points) are uniformly distributed in each of the half cross sections r = const at intervals of Ar = 36 ~ In these conditions, the error in Eq. (17) is no more than 4%. Comparison of curves 1-4 shows that the chiral properties of the scatterer significantly influence the shape of the scattered-

787

field diagram. Note that increase in the chiral properties does not always lead to increase in the cross-polarized component of the scattered field (compare curves 1 and 3). The bistatic scattering cross sections in the x0z plane of triaxial ellipsoids with Ei/ee = 4.0,/xi/Pe = 1.0, ki/3 = 0.3 but different semiaxis ratios are shown in Figs. 4 and 5. The semiaxes ka, kb, and kc are directed along the x, y, and z axes of the Cartesian coordinate system with its origin at the center of the ellipsoids: ka = (1 - ~)kb, kb = 1.5, kc = (I +/~)kb. This method of determining the semiaxes allows the shape of the scatterer to be changed simply by changing the parameter ~; if ~ = 0, the scatterer is spherical. The ellipsoids are excited by a plane wave propagating along the z axis so that the electric field E 0 is oriented along the x axis. The parameters of the method are chosen as in Figs. 2. and 3. Curves I-3 correspond to ~ = 0, 0.2, and 0.3, respectively. As is evident from Fig. 5, the cross-polarized c o t ~ e n t of the scattered field in directions close to direct scattering (0 = 0 ~ is more sensitive to the shape of the chiral scatterer than is the copolarized component. This effect may be used for diagnostics of the shape of chiral objects. However, the cross-polarized component of the scattered field decreases significantly in directions close to inverse scattering (0 = 1800C).

4. DIFFRACTION OF ELECTROMAGNETIC WAVES AT AN IDEALLY CONDUCTING BODY IN A CHIRAL SHELL In this section, the diffraction of electromagnetic waves at an ideally conducting body of arbitrary shape covered with a chiral shell is considered. Analysis of the scattering processes at such structures is of great interest in order to reduce the radar visibility of objects and to develop protective screens on the basis of chiral materials. The method of solution adopted is a generalization of the method adopted in [27] for scatterers with isotropic magnetoelectric shells. As illustrated in Fig. 6, we consider the steady diffraction of an electromagnetic field {Eo, Ho} at an ideally conducting body D coated by a uniform chiral layer Di; the time dependence is chosen in the form exp (-kot). The layer is characterized by dielectric permittivity ei, permeability/zi, and chiral parameter/~. Layer D i is bounded on one side by the surface Se of the ideally conducting body D and on the other side by the surface Sa, beyond which is region De, consisting of an infinite uniform medium of dielectric permittivity ee and permeability/ze. Suppose that surfaces Sd add Sc are mutually similar, in the sense that they are homothetic relative to the point 0 within region D, which is also the origin of the Cartesian coordinate system. It is required to find the scattered field {Ee, He} in region De. In addition to the field {Ee, He} in De, the field {Ei, Hi} exists within layer D i. These fields must satisfy Eqs. (7) and (8) in the external region De and within the layer Di, respectively, as well as Eq. (10) and the boundary conditions n c x E i = 0 on S,,

.,•215

(19)

(20)

where n c and n d are unit vectors normal to surfaces Sc and Sd, respectively. Suppose that Sc and Sd are Lyapunov surfaces, Im e i,/.ti -> 0. It is expedient to introduce three auxiliary surfaces Se = KeSc, Si(1) = K~(I)Sc, and Si(2) = Ki(2)Sc, which are homothetic with respect to surface Sc of the ideally conducting core (and hence also the layer surface Sd). Surface Se with the similarity coefficient Ke < 1 lies within the region DUDi; surface Si(1) with the similarity coefficient Ki(D < 1 lies within region D; and surface Si(2) with the similarity coefficient Ki(2) > 1 covers the magnetodielectric layer and lies within the external region De. As in the preceding section, we write the unknown scattered field {Ee, He} in De as the sum of the fields of auxiliary, tangentially oriented electric dipoles with unknown moments pr n,e (n = 1, 2 . . . . . Ne) at points {Mn,e}n=lNe on surface Se (N e is the number of dipoles), i.e., in the form in Eq. (14). The field {~, Hi} in D i is expressed as the sum of the fields of analogous dipoles at points {Mn,i,1}n=l N1 and {Mn,i,2}n=l N2 on auxiliary surfaces Si(1) and Si(2), radiating in a uniform chiral medium with parameters ei, gi, ~ (N1 and N 2 are the numbers of dipoles at Si(1) and Si(2), respectively)

788

El(M)= (,:)~, / k3 r / k i ) 2 [n=l 4""'1111 n'=l

~,

Ui(M)-(-'~9

6,1 M , Mn,i 3 + Y 2

I

n,i,2]

)

j~l M , M n i l ' '

T2G2 (M, Mn,i,2 "P,

,,'"

" Ar~

+

,

"n,i,I ^ n,i,l {nZ=~IGI (M,Mn,iJ)-T2G2 (M, Mn,i,t)]'P,Fm,i,I +

2 NI

_

+ ~ L &.,.21u M

(21)

~l . . ~ l

~ ~ &'2~M U

where k i = o J ~ ) ; ,y2 _ ki2/( 1 _ ki2~2); "fl = ki/(l - ki3); ")12= ki/(1 4- ki~); p~n,i,1 (n = 1, 2 . . . . . N1) and pr n,i,2 (n = 1, 2 . . . . . N 2) are unknown dipole moments consisting of the independent components Prl n'i'l, pr2n,i,l and prl n,i,2, Pr2n,i,2 along the directions erl n'i'l, er2n'i'l arid erl n,i,2, er2n,i'2 chosen in the planes tangential to Si(l) and Si(2) at points Mn,i, 1 and Mn,i,2, respectively; CJ1,2n,i,l and (~1,2n,i,2 are the elements of dyadic Green's functions

G~'i"(M, M~.O) = (ki 12y2)[y, I + (I/y, )VV + V xi]~(M,M~.i.,), O~'i"(M, Mn.ij)=(ki/2u

2)[Y2I + (1/ y2)VV + V xI]Wi~(M,M#.ij),

O~'i'2(M, Mn.i.2)=(k, / 2T2)[~/1i + (I/u

+ Vx[]W~(M,

(22)

Mn.i.2),

M,.,,)=(k, / 2:)[yg + (l/y,)vv + v • ?],I,~(M, M,.,.~).

0p'(M,

In Eq. (22), ~l,2i(M, Mn,i,l) = exp (i'Y1,2RMn,i,I/4~'RMn,i,l, ~l,2i(M, Mn,i,2) - exp (i'Yi,2RMn,i,2/41rRMn,i,2 are scalar Green's functions; RMn,i'l and RMn,i,2 are the distances from the points Mn,i, 1 on Si(1) and Mn,i, 2 on Si(2) to point M in D i. The fields in Eqs. (14) and (21) satisfy Eqs. (7) and (8), respectively. To satisfy the boundary conditions in F_,qs. (19) and (20), the vector constants pr re,e, pr n'i,1, and pr n,i,2 must be chosen appropriately. As in the preceding section, we use collocation for this purpose. Suppose that Mdj (j = 1, 2 . . . . . LI) and McJ (j = 1, 2 . . . . . L 2) are collocation points on S d and Sc, respectively. Then, to determine the unknown complex constants prl n,e, Pr2n'e (n = 1, 2 . . . . . Ne), Prl n'i'l, pr2n'i'l (n = 1, 2 . . . . . NI), prl n,i,2, Pr2n'i'2 (n = 1, 2 . . . . . N2), we obtain the following system of linear algebraic equations with a complex matrix of dimensions (4L 1 + 2L2) x (2N e + 2N 1 + 2N2)

- ~,A I- "a "-'o.a,

J

n~xE/,#=O,

"x HJo.,t,

j = 1,2,..., L I ,

(23)

j = l , 2 ..... L2.

Subscripts d and c denote the values of the corresponding field components at surface S d of the chiral layer and Sc of the conductor, respectively. The solution of Eq. (23) is determined by minimizing the functional

9

=

{I-,

-

'+

•

-z.,l'}+

(,.r ,'..,

(24)

which is the square of the norm of the discrepancy vector for Eq. (23). The conjugate-gradient method is used to minimize the functional in Eq. (24). The accuracy of solution is monitored by calculating the relative value of the functional in Eq. (24) at the grid of collocation points and at intermediate points chosen both at the surface S d and at the surface Sc e--(*'/*o)"'

9|

= ~,

(

.2

E~,~ 2 + (~e / ~e) . 2

.:)

>, H0,~

,

(25)

789

where ,I,' is the value of the functional in Eq. (24) at the given set of points and ~0 is the value of the corresponding norm of the incident field at the set of collocation points and intermediate points on S d with m = 1, 2 . . . . . 1_,3. On the basis of the method outlined, a program is written to calculate the scattered-field components and monitor the accuracy of the solution obtained. This program permits the solution of a broad class of diffractional problems and, on that basis, the investigation of various aspects of diffraction at the given structures. We will now examine some results of numerical experiments regarding the influence of chiral properties of the energy characteristics of the scattered fields with shells characterized by predominantly electric or magnetic absorption. Consider triaxial ellipsoids excited by a linearly polarized plane wave propagating along the z axis; the plane-wave vector E o is directed along the x axis; the semiaxes ka, kb, and Ice are directed along the x, y, and z axes, respectively. The characteristics investigated are the bistatic scattering cross sections in Eq. (18) in the E plane (xz plane) for the basic and cross-polarized components of the scattered field. The calculations show that, for the given structures, the bistatic scattering cross sections for the cross-polarized components are no greater than - 1 2 dB; therefore, the corresponding diagrams are not shown here. Some calculation results are shown in Figs. 7-10. Spherical structures differing in the values of the dielectric permittivity and permeability of the shell are considered in Figs. 7 and 8. For both structures, the radius kea of the ideally conducting core is 1.5, and the shell radius keao is 1.8. The structure with a shell characterized by relative dielectric permittivity ei/ee = 3.0 + il.5 and relative permeability gi/ge = 1.0, i.e., by electrical absorption, is shown in Fig. 7, and the structure characterized by parameters ei/ee = 2.5 and gi/#e = 1.5 + i0.75, i.e., by magnetic absorption, is shown in Fig. 8. Triaxial ellipsoids that are flattened in the direction of exciting-wave propagation and differ in the dielectric permittivity and permeability of the shell are considered in Figs. 9 and 10. The semiaxes of the ideally conducting core of these structures are: kea = 2.0, keb = 1.2, kec = 0.8. The shell is ellipsoidal and is characteristics by semiaxes kead = 1.2kea, kebd = 1.2keb, kecd = 1.2kec. A structure with ei/ee = 3.0 + il.5 and/zi/#e --- 1.0, i.e., as in Fig. 7, is considered in Fig. 9, and a structure with r,i/ee = 2.5 and/~i//% = 1.5 + i0.75, i.e., as in Fig. 8, is considered in Fig. 10. The same parameters of the method are chosen in every case. The number of dipoles at the auxiliary surfaces S e, Si(1), and Si(2) is the same: Ne = N 1 = N 2 = 40. They are distributed as follows: the total number of dipoles in each half cross section ~, = const is 8; five dipole locations are chosen uniformly with respect to O (0 and ~, are the conventional spherical angular coordinates). The number of collocation points at the surfaces Sc of the ideally conducting body and Sd of the shell is also the same: L l = L2 = 80; the number of half cross sections ,t, = const in which the collocation points lie is twice is great as for the dipole locations; the distribution of the collocation points with respect to 0 is the same as the dipole distribution. A preliminary numerical experiment shows that the optimal (corresponding to minimzl norm of the boundarycondition discrepancy) positions of the auxiliary surfaces with the chosen numbers of dipoles and collocation points are determined by the values of Ke, Ki(1), and Ki(2) within the intervals 0.3 < Ke <_ 0.6, 0.3 <_ Ki(1) _< 0.5, and 3.0 -< Ki(2) < 5.0. In this ease, we select: Ke = 0.5, Ki({) = 0.4, and Ki(2) = 5.0. The chiral properties of the structure are characterized by the real parameter B. Curve 1 in Figs. 7-10 corresponds to/3 = 0 (an ordinary magnetodielectric layer), curve 2 to /3 = 0.01)`, curve 3 to /~ = 0.02)`, and curve 4 to /3 = 0.03)`. With the chosen method parameters, the discrepancy in Eq. (25) is no more thzr~ 8% for spherical structures (Figs. 7 and 8) and 16% for elliptical structures (Figs. 9 and 10). Analysis of the results in Figs. 7-10 yields the following conclusions. Overall, if all possible scattering directions are considered, there is a tendency to reduction in the scattered field with increase in the chiral properties of the layer. However, the effect of the chiral properties is not simply to reduce the level of scattering, but also to redistribute the energy of the scattered field in space; consequently, the level of the scattered field may be significantly increased in some directions. This is the case, in particular, for the spherical structures considered in Figs. 7 and 8. As shown by these figures, increase in/~ for the shell is accompanied by significant (by 7 dB for/3 = 0.03),) increase in the inverse-scattering (/9 = 180 ~ cross section in comparison with an ordinary magnetodielectric shell (fl = 0). Thus, in terms of the inverse-scattering cross section, the presence of chiral properties results in undesirable increase in radar visibility for the given example. In all the cases considered, the reduction in the scattered field is greatest in directions close to lateral scattering (/9 ~ 90~ the effect is less in the direction of exciting-wave propagation (/9 -~ 0~ Thus, chiral properties in the shell lead to redistribution of the scattered-field energy in space, in the general case; preliminary analysis is required to determine whether the use of chiral materials is expedient in any particular case.

790

CONCLUSIONS Electromagnetic-wave propagation in infinite, semiinfmite, and laminar chiral structures is considered in the present work, along with the diffraction of electromagnetic-wave diffraction at semiinfinite objects in a chiral medium and at threedimensional objects consisting entirely or partially of chiral materials. Particular attention is paid to diffraction at threedimensional uniform chiral bodies and ideally conducting bodies covered by chiral shells. The influence of chiral properties on the energy characteristics of the field scattered by a uniform ckiral object is investigated. It is shown that the cross-polarized component of the scattered field in directions close to direct scattering is more sensitive to the shape of the chiral scatterer than is the copolarized component. This effect may be used for diagnostics of the shape of chiral objects. Data on the scattering of conducting objects covered with chiral shells show that the effect of the ehiral properties is not only to reduce the overall level of scattering but also to redistribute the scattered-field energy in space; in some directions, the scattered field may be significantly increased. In all the cases considered, the effect of the ehiral properties (the reduction in the scattered field) is greatest in lateral directions and less notable in the direction of exciting-wave propagation. In the general case, chiral properties in the shell are accompanied by a redistribution of the scattered-field energy in space; preliminary analysis is required to determine whether it is expedient to use chiral materials in any particular case.

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