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Electromagnetic Field Representations in UniaxialBianisotropic-Semiconductor Material by Cylindrical

Vector Wave FunctionsDajun Cheng

To cite this version:Dajun Cheng. Electromagnetic Field Representations in Uniaxial Bianisotropic-Semiconductor Ma-terial by Cylindrical Vector Wave Functions. Journal de Physique III, EDP Sciences, 1997, 7 (1),pp.1-12. <10.1051/jp3:1997106>. <jpa-00249562>

https://hal.archives-ouvertes.fr/jpa-00249562

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J. Phys. III France 7 (1997) 1-12 JANUARY 1997, PAGE 1

Electromagnetic Field Representations in UniaxialBianisotropic-Semiconductor Material by Cylindrical Vector

Wave Functions

Dajun Cheng

Wave Scattering and Remote Sensing Center,, Department of Electronic Engineering,

Fudan University, Shanghai 200433, People's Republic of China

(Received 25 June1996, revised 3 October1996, accepted 11 October1996)

PACS.03 50 -zClassical field theory

PACSAI.10 Electromagnetism; electron and ion optics

Abstract. Uniaxial bianisotropic-semiconductor material isa

generalization of the well-

studied semiconductor material and uniaxial medium. It could be realized by arranging chiral

objectsin a

host semiconductor-medium, with a preferred direction. This class of material prob-ably opens up the opportunity to realize artificial intelligence

mliving organisms In the present

study, basedon

the concept of characteristic waves and the method of angular spectral expansion,electromagnetic field representations m

this class of materialsare

developed. The analysis in-

dicates that solutions of source-free Maxwell equations for uniaxJal bianisotropic-semiconductormaterial can be represented in sum-integral forms of cylindrical vector wave

functions. Addi-

tion theorem of vector wave functions for uniaxial bianisotropic-semiconductor materialcan

be

straightforwardly derived from that of vector wavefunctions for isotropic medium. An applica-

tion of the proposed theory in scattering is presented to show how to usethese formulations in

apractical way.

1. Introduction

The direct solution method for Maxwell equations in isotropic medium with vector wave func-

tions was first developed by Hansen ill in the middle of1930s This approach, which has

been followed by Stratton [2], Morse and Feshbach [3], and Tai [4] in solving electromagneticbonndaij-vaIne problems, seems to be of increasing interest and importance. The vector wave

functioni have found versatile applications and presented great advantages compared with

other methods'(e.g., three-dimensional moment method [5], coupled-dipole method [6], and

integral-equation technique [7]). For instance, the circular cylindrical vector wave functions

have been successfully used in studying the scattering properties of a chiral-coated perfectconducting cylinder [8], and the radiation characteristics of a dipole antenna in the proximityof a gyroelectric cylinder [9]. Unfortunately, to the best knowledge of the author, only in a bi-

isotropic medium [10] (the most general linear isotropic medium) and a gyroelectric medium [9],have electromagnetic waves been represented in terms of the vector wave functions. Therefore,electromagnetic field representations in complex media need to be developed in the forms of

eigenfunction expansion of the vector wave functions, so as to provide methodological conve-

nience in investigating the electromagnetic properties of these materials.

@ Les iditions de Physique 1997

JOURNAL DE PHYSIQUE III N°1

With recent advances in polymer synthesis techniques, increasing attention has been at-

tracted to the analysis of interaction between electromagnetic waves and complex materials.

From a macroscopic point of view, a uniaxial bianisotropic medium is a generalization of the

well-studied chiral medium ill,12]. With increases probing into the origins of life, more and

more evidence has indi~ated that the natural propensity of life is closed associated with opti-cally active materials [13,14], where chirality occurs. On the other hand, the chips utilized in

analogue and digital integrated circuits are basically attributed to the physical properties of

semiconductors, where diffusion process offree carrier (electron and hole) would take effect [15].Here, a uniaxial bianisotropic-semiconductor material is proposed to'generalize the nniaxial

bianisotropic medium and semiconductor material. The uniaxial bianisotropic-semiconductormaterial, due to its semiconductor properties and analogy with the microstructures of biolog-ical substances, probably opens up the opportunity to realize artificial intelligence in livingorganisms.

In practice, auniaxial bianisotiopic-semiconductor material with uiiaxial linear magneto-

electric interaction and diffusion process of free carrier could be fabricated by immersing chiral

objects in a host semiconductor medium, wi,th a preferred direction. Recent advances in poly-

mer synthesis techniques and hetArostructures of solid-state semiconductor make it possible to

realize such a speculhtive composite material. This material is a-generalization of the wider

class referred to as bianisotropic medium. Excellent wqks in general bianisotropic medium

have been reported by Post [16], Kong [17], and Chen [18], among others. In contradistinction

to these general considerations, the present contribution is intended to develop electromagneticfield representations in this class of materials in term§ of the circular cylindrical vector wave

functions. The formulations are considerably simplified by using the concept of characteristic

waves and the method of angular spectrum expansion. In addition, to make the efficient recur-

sive algorithm developed by Chew [19] availible to multiscatterers and multilayered structures

consisting of uniaxial bianisotropic-semiconductor materials, an outline to derive the addition

theorem of the vector wave functionsjis described. As an application example to illustrate how

to use the present theory in apractilal way,

electromagnetic,scatterini of a normally incident

plane wave by aninfinitely~ial bianisotropic-sem,ic~~iod is studied.

In the following analysis, the harn~onic e~"~ time dependence is assumed and suppressed in

'what follows. In the notations we adopted, double undirline is used tb represent dyadics and

bold face is used for vecto- ~~£~

2. Theory

From a phenomenological point of view, a uniaxial bianisotropic-semiconductor material can

be characterized by the set,of,'constiiutive relations ';.' i,+GQQ m-g

~

D= E E + ( H + TVV E la)

B=

( .E+ ~t.H (16)

where e = Et It

+ Ezezez and ~t = ~lt It

+ ~tzezez are the permittivity and permeability

dyadics, respectively. (=

i(eo~to)~/~ l-a It

+flez x It

-'fezez) and (=

i(eollo)~/~ lo It

+

flez x It

+'fe~ez) are the magnetoelectric pseudo-dyadics. Here, It = e~e~ + e~e~ denotes

the transverse unit dyadic, and ej represents the unit vector in the j direction. eo and ~lo are

the permittivity and permeability of free space, respectively. It is seen that these constitutive

relations (la) and (16) recover those ofa uniaxial bianisotropic medium, and the introduction

of the gradient term in equation (1a) has a more profound significance: it accounts for diffusion

N°1 ELECTROMAGNETIC FIELDS IN SEMICONDUCTOR MATERIAL 3

effects of free carrier (electric and hole). However, it should be pointed out that to simplifythe analysis of the application example in the next section, the material we will try to tackle

is idealized so that T # 0 for field point inside the region occupied by the material, andT =

0

for field point positioned at the boundary interface. From a physical point of view, within the

framework of our idealized model, the diffusion process of free carrier is taken into account

inside the region occupied by the material and ignored at the boundary interface.

To have an idea of a medium with constitutive parameters of the above forms, we first

note the special case with T =fl

= +~ =0, corresponds to the transversely chiral uniaxial

bianisotropic medium studied earlier [20]. This medium can be created by suspending metal

helices in a host dielectric in such a way that the axes of all helices are perpendicular to

the z-axis, but possess arbitrary orientations and locations. In aiother special case with

T = o = +~ =0, the present medium becomes the uniaxial omega medium [21], which may be

fabricated by embedding two ensembles of orthogonally positioned Q-shaped particles in a host

isotropic medium. WhenT = o =

fl=

0, the medium is called a uniaxial chiral medium [22],which can be realized by mixing randomly oriented conductive helices with an isotropic base

medium in such a manner that the axes of all helices are parallel to the z-axis. As T = +~ =0,

the medium becomes the uniaxial chiro-omega medium [23].Substituting the constitutive relations (1a) and (1b) into the source-free Maxwell equations,

a E-field vector wave equation in the uniaxial bianisotropic-semiconductor material is obtained,

Vx ~1~~.VXE+iuJ(Vx ~1~~. ( .E- ( ~1~~.VXE)

-w~(e (

~1~~ ( E ~~TVV E

=0. (2)

The characteristic waves corresponding to equation (2) can be examined in the Fourier domain

by introducing the three-dimensional transformation

« « «

Elr)=

/ / / Elk)e-~k rdk~dkydkz 13)

-« -« -«

where k=

k~e~ + k~e~ + kzez. Substituting equation (3) into equation (2) and after proper

vector algebraic manipullion,we have

'/~ /~ /~

W E(k)dk~dk~dk~=

0 (4)1« -« -«

where (he spectral matrix W is determined as

lk) + p'kj uk(a -(~1 + ~1')kx kg bkz -(u 1)k~kz + ck~ dk~

W=

-(u + ~1')k~k~ + bkz k) + ~t'k] ~lkj ~ a"(u + 1)k~kz dk~ =-dk~

-(~1+1)k~kz=

cky + dk~ -(~ + l)kykz ck~ + dk~ k] + kj + uk] a'

(5 ).

with the notations '

~ ~ ~a = w

[etllt Eollo la + fl Ii' b

=21koo

C =iko IO + /1'~)

j~)d

=ikofl

U #Ld~T/lt

a'=

w~(/1tEz Eo/1oi~/1'I

and ko=

w(Eollo)~/~, /l'=

/lt/Pz. For nontrivial solutions of equation (4), the determinant of

matrix W operating onE(k) must be equal to zero Straightforward algebraic manipulation

4 JOURNAL DE PHYSIQUE III N°1

leads to the characteristic equation

(~~l'k( + [(~1' ~1)(k( al + ~t'(d~ ~la' 2~lkz k( c~~l]k)+(k( al ([c~ + d~ a + (~ ~t'la' ~(~l'k( + 2)k(] + bk(16 2c(~ + 1)] )k) (7)

-(ttk( + a')[(k( al + b~k(]=

0

where k)=

kj + kj.

In the following analysis, the roots of equation ii) are designated as kp = kpq (q=

1, 2,

3, 4, 5, 6), which are functions of the kz variable. The eigenvector corresponding to kpq,expressed in a circular cylindrical coordinate system, can be easily obtained from equation (4)in conjunction with the rectangular-circular cylindrical coordinate transformation,

Eqjkz, dzi=

iAqjkz) cosji ik) + Bjkz) sinji itjjep

+i-Aqjkzj sinjd ikj + Bqjkzj cosji ikjjej + ez j8j

where the spectral parameters are

Aq ikz)-

~~~ '~~ ~ ~~~~ ~ ~llllli ~ ~~ ~~ ~ ~~~~~19a)

jy ~~PQ~~~~~ ~~(q ~) ~~Z ((~ ~ ~j~Z ~ 4 )

jg~jz ~ ~

z

Dq(kz)=

(k( ttk)~ a)(~l'k)~ + k( a) + b~k( (9c)

and #k=

tan~~(ky/k~), #=

tan~~(y/x).Returning to equation (3) and noting kp4 "

-kpi, kps "-kp2 and kp6 "

-kp3, the electric

fieldm

the circular cylindrical coordinate system can be represented m terms of the above-

mentioned eigenvectors,

E(r)=

~j dkz d#ke~~l~z~+~PQP~°~l'~~~~lEq(kz, #k )Eqz(kz, #k) (10)~~ ~/~~o

",

where p =(x~ +y~ )~/~, and Eqz (kz, #k is the amplitude of the spectril longitudinal componept

of electric field. The symmetric roots kp4, kps and kp6 are not included in the summation of

equation (10), since they are automatically taken into account as thy spectral azimuthal angle,#k spans from 0 to 27r.

Substituting the explicit expression of the eigenvector (8) into equation (10), the solu-

-tion E(r) of the siirce-free vector wave equation (2) for aninfinite'uniaxial bianisotropic-

semiconductor material is expressed m terms of the scalar cylindrical wave functions. How-

ever, in-order to have tractable solutions for the boundary-Value prbblems involving uniaxial

bianisotropic-semiconductor material,-.it_is required to-transform the expansion of equation'(10) into a form resembling the vector wave solution for an isotropic medium. To this end,

applying the angular spectral expansion for the Eqz(kz, #k) compon/nt under the hypothesis-that E,z(kz, #k is continuniis fin( §eparable with respect to t[e ~j;;,ijd kz .,~ariables,

«

Eqz(kz,4k)=

~j e~~jkz)e-m'L iii)

n=-«

we have3

co co

El~l"

~j dkz ~j ~qnlkz)EQn(kz) l~~lq=1~"n=-co


where~~

Eqn(kz)=

d#ke~4~~~+~P~ ~°~l'~'~+"'4Eq(kz, #k). (13)~"0

Substituting into equation (13) the well-known identity

«

e-~kp~PC°s<~-~L)=

~j i-i)mJ~ik~~pje-~mi«-#L) i14j

m=-«

and its derivatives with respect to p and #, after cumbersome mathematical manipulation bygrouping properly the terms involving in the integration for the #k variable and introducingthe circular cylindrical vector wave functions, we end up with (see the Appendix for details)

Eiri=

~rf /°~~~f

~

l-?I"eqnikz)iAeikzim(i)jk~ ~~

~"~ ~°~n=-oc

~ n , a

+B((kz)N(~)(kz, kpq) + t7((kz)L(~~(kz~ kpq)> (15)

where the vector wavefunctions are defined as

~(j)j~ ~_~i~~(j)j~ j~ _~~~~(~POPj~ ~-~(n~+kzz) j~~~)

~ ~' ~~P ~

~~~~ 0P ~

N<J)jk kI ~~ 0Zi~(kpqp) nkz

j~~~ ~' ~~ k, ~ op ~P )~n (kPqP)e~ (16b)

~Q~~~(kPqP)e~le~~(n~+k~z~

L(~~ (kz, kpq)=

~~~j~~~~~ep

flZ(J~ (kpqp)e~

P P

ikzZ(J~(kpqp)ezle~4~~+~~~~ (16c)

with kq = (k)~ + k()~/~ land

,---

_,_,-_~-~_

Jn (kPa RI-,,-. j.Ei~),~~

.=.. m~~~~is~/, ~t;, i'i?'iiti" ~'~~

:i» Yn(kp,p) jwiZ(J~(kp,p)

=

(16d)

~

H~~~ (kPqP) J "3

H(~~(kpqp) j =4

In equation (15), the weighted coefficients of the vector wave functions are found to be

A((kz)=

~~~~~~~~ (17a)kPq

~

-2k~A~(kz) 2[1 + kp~~%>

_~~~~~~lkPqkz)

~kq

~~~ ~~

2ikz[1 +~§~]

t7((kz)= ~~

~ (17c)q


The representation for the magnetic field can be easily obtained from the relation

H=T+. ~1~~.T (I(VXE)- T+. ( .TEj (18)~ " ~

where

CDs ( sin # 0

~g)T=

sin j cos j 0~

0 0 1

is the coordinate transformation matrix [19>, and the superscript (+) denotes the transpose of

the matrix. Straightforward mathematical manipulation leads to

3cc CO

Hlrl= ~

~ /dkz ~ l-il~eqnlkzllA[(kz)M[~~(kz, kpql

q=i -C°n=-«

+B(lkzlNf~lkz> kPql + C)lkzlLf~(kz, kPql> (2°1

where the weighted coefficients of the vector wave functions are determined as

~~~~~~ ~~~~~~~~ /~t ~°~~~~~~ ~~~~~~~~~

~~~~~~~~ ~~~~~

B) (kz" ~) ~j (flkz k0 + ikl + ~/~'k)q

/ltq

~~~~(~~~~~~ ~'~~~'~~ ~

~~~~~~~~° ~'~~ ~~~~~

~~~~~~~~~ ~j~~~~~~~~

~~~~~~~~

-~~°°[t7((kz

~~~B((kz11.

,,

(21c)~pt k~ "-~'~

,,,c'~Since Bessel, Neumann, aqd_Hqnkel functions of the same ojjer satisfy the identical differential

equation, the first kind 4vector wave functions in equatiiiis (15) arid (10)

canbe generalized

,

Jo three other ones, corretzqnjjnz to Neumann and Hankei fiinctia+q.,_~'The resulting equatioiis'(15i irid"(20) indithte'that solutions of the

squrii-~e&laxwell

equations for uniaxial bianisotropic-semiconductor material, which can be represented in terms

of the circular cylindrical vector wave functions, are superpositions of two transversal waves

and a longitudinal wave of isotropic media.

Straightforwardly, the addition theorem of circular cylindrical vector wave functions for uni-

axial biamsotropic-semiconductor material can be directly obtained by using the counterpartof isotropic medium [19> in equations (15) and (20).

3. An Application Example

To illustrate how to use the present theory m a practical way, we study the electromagneticscattering of a normally incident plane wave by an

infinitely long rod, which consists of a

homogeneous uniaxial bianisotropic-semiconductor material.

Let usfix the coordinate system so that the scatterer is bounded by the surface p = po, and

its axis is coincident with the z-axis. The surrounding medium is taken to be free space.

N°1 ELECTROMAGNETIC FIELDS IN SEMICONDUCTOR MATERIAL

In the case of a TMz polarized incident plane wave illuminating along the +x-axis, the

incident electromagnetic fields can be expanded in terms of the circular cylindrical vector wave

functions [2,8>

Einc(~)"

~~~°~ ~~~~ ~~~~co ~~~~~~~~~~~~ ~~~~ ~~~

~~~~~

Hmc(~)" ~~i~

~~~~ ~~~n~co~~~ ~~~~~~~~~~~~~~~~

~~~~~

where kp =(k] k()~/~, and b is the Dirac delta function. ko and ~o =

(llo/Eo)~/~ representthe wave number and wave impedance of free space, respectively. The electromagnetic fields

induced in the scatterer can be expressed in terms of the circular cylindrical vector wave

functions in the way we have presented in the previous section, and the scattered fields may

have TMz and TEz components and should be expanded as [2,8>

co CC

Escalr)#

/dkz ~ l~~)~l~nmf~lkz~kp)+bnNf~lkz>kp)> 1~3~)

-co~=_~

~~~~~~~ ~~~ n~«~o [a~N[~)

kz,~k~)~) «M[~~(kz, i-p)<

~~~~~

In the expressions (22) and (23), the functions M)~(kz,kp) and N)~(kz,kp)(j=

1,4) are the

circular cylindrical vector wave functions defined in the previous section.

Remembering the idealized model of the present case (T =0 at the boundary interface), the

conventional boundary conditions still hold (i.e., continuity of the tangential components of

electric and magnetic fields). Applying the mode-matching method [8,15> to have the boundaryconditions satisfied at the outer surface of the scatterer p = po, the expansion coefficients of

the scattered fields an and bn can straightforwardly be obtained from a set of equations:

3

koanH@~~ (koPo# ir

~j eqnvj~ j24~~

q=1

3

Eob(k~)Jn(koPo) + kobnJn(koPo)" 7r

~ eqm~t(~ (24b)

q=1

~°~l~~~~~l~°~°l ~ ~°~n~i~'lkoPo)" -12r1Jo

feqnv~ j~4~j

qn

q=1

3

koanH(~)(koPo)"

-i7r~o~eqnX)~ (24d)

q=i

~~~Hf)jkopojon" 7r~eqn(~~w(n~~Q0lCYwj~+pvjlj (~4~)

p0~

~

E0

for q =1, 2, 3 and p = e, h; and the primes over the Bessel functions denote derivatives with

respect to the arguments. In equations (24a)-(24e),

~j(=

~4((kz)kp,J[(kpqpo) + ~C((k~)Jn(kpqpo) (25a)Po


10

~s

CQ2~

~c o$iIli

-10~~§

-20

~

i-30

0 45 90 135 180

scattering angle (degree)

Fig. I Scattering pattern of aninfinitely long uniaxial bianisotropic-semiconductor rod due to a

normally incident TMz polarized planewave.

The scatterer hasa

radius of po =0.710, and constitutive

parameters of et =(3.I +131)eo, ez =

(2 5 +1.b)eo,T =

7.I x10~~~ Fm, pt "(24 + 0.31)po,

pz =(2.2 + 0.21)po,

a =I-1+101, fl

=1.9 +10.3, and j =

1.4 +10.2

x(n"

B(lkz)JnlkPqp0) l~sb)

iV(~ =~~A((kz)Jn(kpqpo) kpqt7((kz)J[(kpqpo). (25c)Po

The bistatic echo width, which represents the density of the power scattered by the cylindricalobject, is defined

as

-4a(§i)"

lim 27rp~~~~~~~~~~ ~(ca(rl' eP

~~~~'E~~~l~l ~ Hlncl~l>'~P

l~~l

where the superscript asterisk indicates complex conjugate, and Ill represents the real partof complex value.

Due to the emergence of the Dirac delta function b(kz), the infinite integration of the kzvariable for the scattered fields actually disappears. Invoking the asymptotic expressions of

Bessel functions for truncation of series appearing in this process, numerical computation is

straightforward. It should be mentioned that previous to the actual computation, a convergencetest was made to check the validity of the numerical results.

Figure I shows the scattering pattern of the bistatic echo width, which is normalized with

respect to the wavelength ~o of the incident wave. In this case, the scatterer has a radius of po "

0.710, and constitutive parameters of et "(3.1+1.31)eo, Ez =

(2.5+1.11)Eo,T =

7.1x10~~5 Fm

(for point inside the scatterer, andT =

0 at the boundary interface), ~lt "(2.4 + 0.31)~lo.

jiz =(2.2 + 0.21)~lo, a =

I.1+10.I, fl=

I-g +10.3, and+~ =

1.4 +10.2. Since the incident wave

isilluminated along the #

=180° line, the pattern have a ma~~imum in the forward direction,

i-e-, #=

0°.


So far there is a gap in associating the microscopic structure parameters with the macroscopicmedium parameters and the present study does not address this point but simply assume the

medium parameters known. However, a uniaxial bianisotropic-semiconductor material could

be realized in the ways as described in Section I, and the constitutive parameters for the

calculated example are typically chosen without sacrificing both generality and the purpose of

this section.

For the present example,it should be pointed out that the mode-matching method employedhere is easily formulated and the calculation time for one point of the figure is short, comparedwith other means, by integral equation and point matching, for example.

4. Conclusions

In the present study, a uniaxial bianisotropic-semiconductor material is proposed and electro-

magnetic field representations in this class of materials are developed in terms of the circular

cylindrical vector wave functions. Diffusion process of free carrier is taken into account in

the modellini of constitutive relations. The formulation is greatly facilitated by using the

method of angular spectral expression for the spectral expansion coefficient of the electric

field. The theory developed here generalizes the canonical solutions of vector wave functions

for an isotropic medium [1-4>, and recovers the cases of semiconductor material [15>, trans-

versely chiral uniaxial bianisotropic medium [20>, uniaxial omega medium [21], uniaxial chiral

medium [22], and uniaxial chiro-omega medium [23]. An application of the proposed theory in

investigating the electromagnetic scattering ofa plane wave by an infinitely long circular cylin-der composed of a homogeneous uniaxial bianisotropic-semiconductor material is given, which

illustrates the applicability of the proposed formalism. Moreover, it is of interest to note that

the cylindrical vector wave functions can be expanded as discrete sums of the spherical vector

wave functions [24], therefore the formulation presented here could be extended to solve the

problems of spherical structures. Although only the mode-matching method is applied m the

present example~ it is estimated that the electromagnetic field representations proposed here

lay foundations to the point-matching method and T-matrix method in tackling the problemsinvolving uniaxial bianisotropic-semiconductor material~ and simplify the process of solving

nontrivial problems with other means, such as the integral equation method. It is believed the

present contribution would be helpful in simplifying the analytical and numerical solutions to

boundary-value problems of layered structures as well as multiscatterers consisting of uniax-

ial bianisotropic-semiconductor materials. Further applications of the present theory will be

reported in the near future.

Acknowledgments

The author appreciates the help and encouragement of Professors Weigan Lin and Ya-Qiu Jin,

and is greatly indebted to one of the referees for providing constructive comments which lead

to the improvement of the manuscript This work was partially supported by the National

Natural Science Foundation of China, and Shanghai Research and Development Foundation

for Applied Materials.


Appendix

Derivation of equations (15) and (17)

Substituting expressions (8) and (13) in equation (12), we have the electric field expanded in

terms of eigenvectors in the circular cylindrical coordinate system,

Ej~)=

( ~dk~ f~~jk~)

~~

dj~~-~lkzz+kp~pcosjj-j~j+nj~j

,=i

~oc~=_~

~ ~k"0

x llAqlkz CDs(< 4kl + Bqlkz Sin(< dk )lep

+[-Aq(kz sin(( #k + B,(kz) cos(# #k))ej + ez). (A.1)

Then, taking the derivatives of equation (14) with respect to p and #, respectively, we have

~

~ ~j~j~~~p)~ ~~jj-jL) (A.2)_~kpqPcosjd-4k)=

~ (~?~~kpa0P

j j~ )e

~-_oc

~~~

~~~14 4 )e-~kpqpcos j-~~~«

~~

m=_~~ ~~~~))j~~~e~~mld-j~j

Inserting equations (A.2), (A.3), and (14) into equation (A.I), we obtain

2cc CO

E(r)=

~j dkz ~ eqn(kz)[Pn(kz)ep + Qn(kz)ej + Rn(kz)ez) (AA)=1~°~

n=-oc

where

~~j~~j- ii~

~

d<k

~

"

~iiii~-~ °llk~~ ~~~~~~

-(-i)~~~~~~~~~~B,(kz]e~4~~~+~')+4m-n)4~kPaP

~~~ ~~~~~~~~~$/~~~~~ ~~~~i~~~~~~~~

~~~~~~~"~~ (A.5)

Qn(kz)= /~~ d#~

f j(-ijm-i°~m(kPqP)B(k~)~~=o

~~_~

kpq0p ~

~~ ~~~~i~)~~~~~q(kz))e~4kzz+m~)+4m-n)~~

~~~ ~~~~~~"~)~j~°~~~~

+~n(kPqP)Aq(k~)j

~~ ~ kpqp~

~~~~~~"~~ (A,6)

and

~~~~~~~ /2~ ~<~

fi-i)mJ~it~~p)ie-~ikzz+m~~+~i~~~'~

~~-~~~~

=27r(-1)"Jn(kp,p)e~~~~z~+~'~. IA.?)

N°I ELECTROMAGNETIC FIELDS IN SEMICONDUCTOR MATERIAL 11

By introducing the circular cylindrical vector wave functions (16a)-(16c) and recalling the com-

plete property of this set of functions, it is legitimate to let the electric field in the umaxial

bianisotropic-semiconductor material be represented in the form of equation (15). Then, com-

paring the coordinate components of equation (15) with those of equation (AA) where Pn(kz),Qn(kz), and Rn(kz)

are determined by equations (A.5)-(A.7), we derive a set of equations

-4[(kzl= -)Bq(kzl (A.81

~~B((k-)+1t7((kz)=

~A,(kz) (A.9)

kq kpq

and~2fB(lkzl ~kzc((kz)

"2. (A.lo)

q

The solutions to this system of linear equations list as(A.8)-(A.10) give rise to the expressions

of equations (17a)-(17c).

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