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Separation of Variables for Electromagnetic Sea ttering by Spheroidal Particles

loan R. Ciric Department of Electrical and Computer Engineering University of Manitoba Winnipeg, Manitoba Canada R3T SV 6

I. Introduction II. Spheroidal Coordinate Systems

III. Spheroidal Wave Functions A. Spheroidal Angle Functions B. Spheroidal Radial Functions

IV. Spheroidal Vector Wave Functions

V. Electromagnetic Scattering by a Coated Lossy Spheroid A Field Expressions B. Boundary Value

Problem Solution C. Scattered Far Field D. Quantitative Results

Francis R. Cooray CSIRO Telecommunications and Industrial Physics Epping, New South Wales 1710 Australia

VI. Scattering of Electromagnetic Waves by a Chiral Spheroid A. Field Expansions B. Boundary Conditions C. Scattering Cross Sections and

Numerical Results VII. Scattering by Systems of

Arbitrarily Oriented Spheroids A. Problem Formulation B. Imposing the

Boundary Conditions C. Scattering Cross Sections D. Numerical Results for

Two Arbitrarily Oriented Spheroids

Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications Copyright© 2000 by Academic Press. All rights of reproduction in any form reserved. 89

90 loan R. Ciric and Francis R. Cooray

I. INTRODUCTION

The method of separation of variables can be used to obtain exact analytic solu-tions to problems involving scattering of electromagnetic waves by objects whose surfaces can be made to coincide with coordinate surfaces in certain curvilinear orthogonal systems. Spheroidal coordinate systems are such coordinate systems and, because spheroidal surfaces are appropriate for approximating the surfaces of a large variety of real-world, nonspherical objects, in this chapter we derive exact expressions for the electromagnetic fields scattered by single spheroids and also by systems of spheroids in arbitrary orientation. These expressions represent benchmark solutions that are useful for evaluating the accuracy and efficiency of various simpler, approximate methods of solution.

Exact solutions for the scalar Helmholtz equation in spheroidal coordinates, obtained by using the method of separation of variables, are expressed in terms of scalar spheroidal wave functions. These functions and their vector counterparts, both of which are defined later, are the key to obtaining exact analytic solutions to scattering problems involving single or multiple spheroids. For a historical survey on the application of spheroidal wave functions, the reader is referred to Flam-mer' s monograph (Flammer, 1957).

Even though various applications of spheroidal wave functions have been pre-sented since 1880, it is in the work of Schultz (1950) that a formulation was given for the first time for obtaining an exact solution to the problem of scatter-ing of electromagnetic waves by a perfectly conducting prolate spheroid at ax-ial incidence. Based on Schultz's formulation, Siegel et al. (1956) carried out quantitative calculations of the backscattering from a prolate spheroid and plot-ted the variation of the backscattering cross section with the size of the spheroid, for a prolate spheroid of axial ratio 10. An exact solution for the more general case of the scattering of electromagnetic waves by a perfectly conducting prolate spheroid for arbitrary polarization and angle of incidence was given by Reitilinger ( 1957), but no numerical results were published. Numerical results for this prob-lem, in the form of scattering cross sections, were presented by Sinha and Mac-Phie (1977), whose formulation also eliminated some drawbacks in the analysis given by Reitilinger ( 1957). An exact solution to a similar problem involving a di-electric spheroid was obtained by Asano and Yamamoto (1975) using a different type of vector wave functions than that used in the work of Sinha and MacPhie (1977). Analytic solutions for scattering by a dielectric spheroid with a confocal lossy dielectric coating (Cooray and Ciric, 1992) and by a chiral spheroid (Cooray and Ciric, 1993) were first given by the authors. A similar solution for a prolate spheroid with a lossy dielectric coating was also presented by Sebak and Sinha ( 1992), but only for axial incidence. Recent calculations using the method of sep-aration of variables for the scattering by single homogeneous spheroids and by coated spheroids were also reported by Voshchinnikov ( 1996). On the other hand,

Chapter 4 Separation of Variables for Electromagnetic Scattering 91

exact analytical results for a single spheroidal particle have been used to derive its T matrix in spheroidal coordinates (Schulz et al., 1998a). The T matrix expressed in spherical coordinates is applied to evaluate optical properties of ensembles of randomly oriented spheroidal particles by performing the averaging analytically (Mishchenko, 199la).

Research on the application of the method of separation of variables to the elec-tromagnetic scattering by systems of two spheroids is not as extensive as in the case of a single spheroid because of the fact that the analysis is now more compli-cated by the necessity of implementing translational and rotational-translational addition theorems for spheroidal wave functions. An exact solution for electro-magnetic scattering by two parallel conducting spheroids was first obtained by Sinha and MacPhie (1983). A similar solution, but using again a different type of vector wave function was given by Dalmas and Deleuil (1985). The scatter-ing from two parallel dielectric spheroids was analyzed by Cooray et al. (1990). Analytic solutions to the problems of electromagnetic scattering by two conduct-ing and two dielectric spheroids of arbitrary orientation were first obtained by the authors (Cooray and Ciric, l989b; Cooray and Ciric, 1991a) using the rotational-translational addition theorems for vector spheroidal wave functions previously derived by them (Cooray and Ciric, 1989a) and independently by Dalmas et al. (l 989). The authors also presented a solution for the case of an arbitrary number of dielectric spheroids in arbitrary orientation (Cooray and Ciric, 1991 b ), with numerical results given for the case of two spheroids. The case of scattering by two homogeneous lossy dielectric spheroids of arbitrary orientation was recently considered by Nag and Sinha (1995).

In the following sections of this chapter, we present the spheroidal coordinate systems, define different spheroidal wave functions used in the analysis of scat-tering problems, and present exact solutions to the problems of electromagnetic scattering by a single spheroid (coated dielectric and chiral) and by systems with an arbitrary number of dielectric spheroids in arbitrary orientation, with illustra-tive numerical results computed for spheroids of different sizes. A time-harmonic dependenceexp(jwt), where w is the angular frequency and j 2 = -1, is assumed throughout in the formulas and derivations given in this chapter (cf. Section IV of Chapter 1).

II. SPHEROIDAL COORDINATE SYSTEMS

The prolate and oblate spheroidal coordinate systems are constructed by rotat-ing a two-dimensional elliptic coordinate system, consisting of confocal ellipses and hyperbolas, about the major and minor axes of the ellipses, respectively. In the following definition of the spheroidal coordinates, the z axis of the Carte-sian coordinate system attached to the center of the confocal spheroids has been


considered as the axis of revolution. Let F be the corresponding semi-interfocal distance. The prolate spheroidal coordinates ( T/, ~,


Srnn (h, ry) and Rrnn (h, g) satisfy the differential equations

d [( 2) d J [ 2 2 m2 J - 1 - ry -Srnn(h, ry) + Arnn - h ry - --2 Srnn(h, ry) = 0 dry dry 1 - ry (6)

and

d [( 2 ) d J [ 2 2 m2 J dg g - 1 dg Rrnn(h, g) - Arnn - h g + g2 _ l Rrnn(h, g) = 0, (7)

where Arnn and m are separation constants, with Arnn being a function of h. The discrete values of Arnn (n = m, m + 1, m + 2, ... ) for which the differential equation (6) gives solutions that are finite at ry = ±1 are the desired eigenval-ues, with m being an integer including 0 (Flammer, 1957). For a discussion on how to evaluate the eigenvalues the reader is referred to Flammer ( 1957) and Sinha et al. (1973). The equations satisfied by the corresponding oblate functions Srnn (- jh, ry) and Rrnn(- jh, jg) are obtained from Eqs. (6) and (7), respectively, by replacing h by - j hand g by jg.

A. SPHEROIDAL ANGLE FUNCTIONS

The prolate spheroidal angle functions are the eigenfunctions Srnn(h, ry) cor-responding to the eigenvalues Arnn of Eq. (6). The solution to Eq. (6) yields two kinds of angle functions s;,f2 (h, ry) and s:J2 (h, ry). It is only s;,f2 (h, ry) that is used in physical problems, because of its regularity throughout the interval -1 :=: ry :=: 1. Thus we simplify the notation by writing Srnn (h, ry) to mean the an-gle functions of the first kind. This remark is also valid for the oblate spheroidal angle functions Srnn ( - j h, ry).

The spheroidal angle functions can be expressed in the form of an infinite series of associated Legendre functions of the first kind as (Flammer, 1957)

00

Srnn(C, ry) = L' d';'n(c)P:::+r(ry), (8) r=0,1

in which c is either h or - j h, depending on whether we consider the prolate or the oblate cases, and the prime indicates that the summation is over even values of r when n - m is even and over odd values of r when n - m is odd. The evaluation of the spheroidal expansion coefficients d;"n ( c) is described in detail in the previous reference.

94 Joan R. Ciric and Francis R. Cooray

A power series expansion can also be used to evaluate the angle functions, which for prolate functions is given by (Flammer, 1957)

00

Smn(h, 71) = (1 - 71 2t 12 I:c2tCh)(l - 71 2)k, (n - m) even, (9) k=O

00

smn(h, 71) = 71(1 - 71 2t 12 I>2tCh)(l - 712)k, (n - m) odd, (10) k=O

where

mn 1 ~(2m+2r)! ( 1) mn c2k (h) = 2mk!(m + k)! ~ (2r)! (-r)k m + r + 2 k d2r (h),

r=k (n-m)even, (11)

mn h 1 ~ (2m + 2r + 1) ! ( 3) mn c2k ( ) = 2mk!(m + k)! ~ (2r + 1)! (-r)k m + r + 2 k d2r+1 (h),

(n - m) odd, (12)

in which

(a)k = a(a + l)(a + 2)- .. (a+ k - 1), (a)o = 1. (13)

A recursion formula for calculating czt(h) is also given in this reference. The power series expansion for the oblate case is obtained from Eqs. (9) and (10) by replacing h by - jh.

An important property of spheroidal angle functions is the orthogonality in the interval -1 :S 71 :S 1 that results from the Sturm-Liouville theory for the differ-ential equations, which can be expressed as

where 8nn' is the Kronecker delta and

Nmn(c) = 2 f,1 (2m + r)!{d;'n(c)}2 (2m + 2r + l)r!

r=0,1

(14)

(15)

is the normalization constant, with c = h or c = - j h depending on whether we consider prolate or oblate functions.


B. SPHEROIDAL RADIAL FUNCTIONS

The prolate spheroidal radial functions are solutions of the differential equa-tion (7), and the range of the coordinate; is 1 ::::; ; < oo.

In physical problems one usually requires the spheroidal radial functions of

both the first kind R~~ (h, ;) and the second kind R~h (h, ;), which are indepen-dent solutions of Eq. (7). The functions of the third and fourth kind, R;;_h (h, ;) and R~h(h, ;), are linear combinations of R~h (h, ;) and R~h (h, ;).

Like the spheroidal angle functions, the spheroidal radial functions R~h (h, ;) and R~h(h, ;) can also be expanded as an infinite series in the form (Morse and Feshbach, 1953; Flammer, 1957; Sinha, 1974)

(;2 _ l )m/2 oo ~ L' a,(hJmn)}m+r(h;),

r=O,I

R~h(h, ;) (16)

(;2 _ l )m/2 oo

R~2(h, ;) = --2 - L' a,(hJmn)Ym+r(h;), ; r=O,I

(17)

where }m+r (h;) and Ym+r (h;) are spherical Bessel and Neumann functions, re-spectively, of order m + r and argument h;, and a, (h Jmn) are the expansion coefficients, whose calculation is given in the three references mentioned previ-

ously. The series representation of R~1h (h, ;) has good convergence, whereas the one of R~h (h, ;) is an asymptotic series that is very slowly convergent for small values of h;. In such cases, R~h (h, ;) can be calculated using an integral method that is presented in Sinha (1974) and Sinha and MacPhie (1975).

Prolate spheroidal radial functions can also be expressed as a series of powers

of ; 2 - 1 (Flammer, 1957). Series of this type are particularly advantageous and provide an alternate method for the calculation of R~h (h, ;) for small values of h;. The series expansions of R~h(h, ;) are of the form

00

R~2(h, ;) = [K~12(h)r 1 (;2 - 1r12 2.:C-l)kc2tCh)(;2 - l)k, k=O

(n - m) even, (18) 00

R~2ch, ;) = [K~2Ch)r'H;2 - 1r12 I:c-1lc2tch)(;2 - i)k, k=O

(n - m) odd, (19)


where

00 (2m + r)'

(2m + l)(n + m)! ~'d;:'in(h) · ~ r!

K~lJ (h) = -------'-,--=_0_--,---..,.--..,.-2n+md0m (h )hmm! ( n ~ m ){n ~ m )! ' (n - m) even, (20)

00 (2m + r)I

(2m + 3)(n + m + l)! ~' d;."n(h) · ~ r!

K~lJ(h) = r=I 2n+mdjn(h)hm+lm!(n - ~ - 1}(n+~+1}'

(n - m) odd, (21)

and c1,1Jt (h) are given in Eqs. (11) and (12). The series expansions for Rf;~ (h, ~) are of the form

(2) - 1 (1) ~ + 1 Rmn (h, ~) - - Qmn Rmn (h, ~)log -- + 8mn (h, ~), (22)

2 ~ -1

where

[K~IJ(h)]2 m mn h (-l)m-r+1(2m - 2r)! Qmn = h l..:.>~r ( ) r![2m-r(m - r)!]2 '

r=O (n - m) even, (23)

- [K~1J(h)]2 m amnh (-l)m-r+l(2m-2r+l)! Qmn - h ~ r ( ) r![2m-r(m - r)!]2 '

(n - m) odd, (24)

in which

(25)

and

00

8mn(h,~) = H~2 - l)-m/2Lb;"n(~2- l)', (n - m) even, (26) r=O

00

8mn(h, ~) = (~2 - irm/2 Lb;"n(~2 - 1)', (n - m) odd. (27) r=O


Recurrence formulas for the evaluation of the coefficients b';1n and the explicit forms of a';1n(h) for r = 0, 1, 2, 3, 4 are also given in Flammer (1957). The accuracy of the calculated prolate spheroidal radial functions can be checked by using the Wronskian relation

The oblate spheroidal wave functions can be accurately calculated using a power series expansion in terms of ~2 + 1 (Flammer, 1957). For radial functions of the first kind, this expansion is given by

00

R1;;2(-jh, j~) = u-mK,;,1~(- jh) rl (~2 + lr/2 I>2tc-jh)(~2 + l)k, k=O

(n - m) even, (29) 00

R1;;2C-jh, j~) = [rm-1K,;,1~(-jh)r1~(~2 + lr12 I>2tc-jh)(~2 + 1)\ k=O

(n -m) odd, (30)

where Kf;J~(- jh) and c2/2(- jh) are obtained from Eqs. (20), (21) and Eqs. (11), (12), respectively, by changing h in these equations to - jh. The power series expansion for radial functions of the second kind is given by

where

u-mK,;,1~(-jh)]2 ~ amn(- 'h) (2m - 2r)! Qmn = h L., r 1 r![2m-r(m _ r)!]2'

r=O (n - m) even, (32)

- [ ·-m-1 (!)(- 'h)]2 m (2 - 2 + 1)1 Q _ _ } Kmn } ""' ;:xmn (- . h) m r . mn - h ~ r } r![2m-r(m - r)!]2,

(n - m) odd, (33)

in which

(34)


and

00

gmn (- jh, j~) = (~2 + 1)-m/2 L Bi,.n~2r+I, (n - m) even, (35) r=O

00

gmn(-jh,j~) = (~2+1fm/2LB2rn~2r, (n -m) odd. (36) r=O

Recursion relations for calculating the coefficients Bi,.n are given in Flammer (1957). The Wronskian relation in this case is

R;;2(-jh, j~) :~ R~2(-jh, j~) - R~2(-jh, j~) :~ R;;2(-jh, j0 1

= h(~2 + 1) . (37)

IV. SPHEROIDAL VECTOR WAVE FUNCTIONS

By the application of vector differential operators to the scalar spheroidal wave function given in Eq. (5), the vector spheroidal wave functions Mand N are de-fined as (Flammer, 1957)

Mmn = Vif!mn x a,

Nmn = k-I (\1 X Mmn),

(38)

(39)

where a is either an arbitrary constant unit vector or the position vector r. None of the coordinate unit vectors 1/, ~, or


M( - j h; T/, j ~,


V. ELECTROMAGNETIC SCATTERING BY A COATED LOSSY SPHEROID

To illustrate the application of the method of separation of variables, we derive in this section an exact solution to the problem of scattering of electromagnetic waves by a homogeneous dielectric spheroid coated with a confocal lossy dielec-tric material of arbitrary thickness. The electric and magnetic fields inside the spheroid, within the coating, and outside the coating are expressed in terms of a set of vector spheroidal wave functions, and the solution is obtained by imposing the appropriate boundary conditions at each spheroidal surface.

A. FIELD EXPRESSIONS

Consider a linearly polarized monochromatic plane electromagnetic wave with an electric field of unit amplitude, incident on a dielectric spheroid with a confocal lossy dielectric coating of arbitrary thickness, as shown in Fig. 1. The materials of the spheroid and the coating layer are assumed to be linear, homogeneous, isotropic, and lossy, with permittivities £2 and £1 and permeabilities µ2 and µ1, respectively. The medium outside the coating layer is assumed to be linear, ho-mogeneous, and isotropic, with permittivity £ and permeability µ. The semiaxial lengths of the spheroidal core are denoted by a1 and b2, and those of the outer sur-face of the confocal layer by a1 and bi. The inner and outer spheroidal surfaces

µ,e

x

' ' ' ' ' ' ' ' ' ' ' ' ' ' ' --+:b2 ;.._:

b1

Figure 1 Geometry of the scattering system.

y


are defined by~ = ~2 and~ = ~1, respectively (see Fig. 1), with~ being the radial coordinate of a spheroidal coordinate system whose origin is at the center 0 of the spheroid. The major axes of the spheroidal surfaces are along the z axis of the Cartesian system Oxyz.

Without any loss of generality, the incident plane can be considered to be the xz plane (cpi = 0). The incident propagation vector k makes an angle !'.'i with the z axis. A linearly polarized incident wave can in general be resolved into trans-verse electric (TE) and transverse magnetic (TM) components. These components are usually defined in terms of the polarization angles, which is the angle between the direction of the incident electric field vector and the direction of the normal to the plane of incidence. For TE polarization s = 0 and for TM polarization s = n/2.

The incident electric field Ei and magnetic field Hi can be expressed in terms of the vector spheroidal wave functions M and N defined in Section IV as (Sinha and MacPhie, 1983; Cooray and Ciric, 1989b)

00 00

~ ~ ( + M+Ol + - M-Ol) ~ ~ Pmn mn Pmn mn ' (50)

m=-oon=lml 00 00

Hi = j(c/µ,) 112 L L (P;tnN;t~I) + P;;;nN;;;~'l), (51) m=-OOn=/ml

where

P!n = Smn(h, cos Oi) -- =F j sins , 2r-1

( coss ) kNmn(h) COS!'.'i

(52)

in which Nmn(h) is the normalization constant of the spheroidal angle function Smn (h, cos !Ji) defined in Section III.A and h = k F, with k being the wavenumber in the medium outside the coating layer and F the semi-interfocal distance of the spheroid. Equation (50) can now be rewritten in matrix form

(53)

with the overbar denoting a column matrix and T denoting the transpose of a matrix, where

... ] '

and

-T [-+(l)T MiO = M_ 1

M-(l)T] I ,

jT = [.PJ

M-(l)T -(a-1)] for a ?': 1,

(54)

(55)


with

M±(l)T r

- [ ±(1) . ±(1) ±(1) - Mr,lrl (h, rJ, ~, cp) Mr, Ir I+ I (h; r], ~, cp) Mr,lrl+2(h; r], ~, cp)

and

-T [- +T Po = P-1 P1-T],

-T [- +T Pa = Pa-1

- -T Pa+I

-+T P-(a+ll

- -T ] P-(a-1) for CT ;? 1,

with

-±T [ ± Pr = Pr,lrl ± Pr,lrl+l

± Pr,lrl+2 ... ].

... ], (56)

(57)

(58)

Equation (51) can similarly be written in matrix form (Cooray et al., 1990)

(

E )1/2 ff = . - N~l)T i I ) I ' µ

(59)

with the elements of Nil)T obtained from the corresponding elements of :M}1)T by replacing the vector wave functions M by N.

The scattered field Es for~ > ~I can be expanded in terms of vector spheroidal wave functions of the fourth kind as (Cooray and Ciric, 1989b)

00 00

Es= L L(a!nM!~4) +&~+1,n+IM~~l,n+l) m=On=m

00

""'(-+ M+(4J -7 Mz(4)) + L...., a-1,n+I -1,n+I + aon On n=O

00 00

""'""'(-- M-(4) -7 Mz(4) ) + L...., L...., a_mn -mn + a::_(m+l),n+I -(m+l),n+I ' m=On=m

(60)

where auv are the unknown expansion coefficients. This equation can now be written in matrix form

E _ M-(4)L:

s - s a, (61)

where

... ], (62)

... ],


in which

M;o = [M~i4JT M~(4JTJ,

- T _ [ - +(4)T M-z(4)T M--(4)T Mser - Mer-I er -(er-I)

(63) for a ;? 1,

with

M~


wave functions of the fourth kind by those of the first kind and the second kind, respectively, and h by h1 = k1F, with k1 = (Jµ1/µ,J£1f8)k. The elements

of the matrix


those in Cooray and Ciric (1991 b). The details regarding the calculation of the in-tegrals that appear as a result of applying the orthogonality of the spheroidal wave functions are given in Sinha and MacPhie (1977) and Cooray and Ciric (l99lb). The solution ofEq. (75) can now be written in the form

- -S = [G]/, (76)

where [G] = [Gdr 1[Rd] is the system matrix, whose elements are independent of the direction and polarization of the incident wave. Once the expansion coef-ficients a, fi' ji' and 8 are known, it is possible to calculate the fields inside the spheroid, inside the coating layer, and outside the coating layer, by substituting back in the appropriate series expansions of the fields.

C. SCATTERED FAR FIELD

The scattered electric field in the far zone can be written by using the asymp-totic forms of the vector spheroidal wave functions M as

where

e-Jkr E5 (r, ff, cp) = --[ Fo (ff, cp )iJ + F


with r, {},

Chapter 4 Separation of Variables for Electromagnetic Scattering

-10

1il -20 :£. c 0 -30 ti CD (/) Cl) e -40 (.) -50 Q

~ -60 1i5 -70

-80 0 30

0

1il -5 :s, c 0 ti -10 CD (/)

gi e -15 (.) Cl c ":i5 -20 ~ al Q

~ 1il -25 co

-30 0 15

E-plane (alb=2) --H-plane (alb=2) ---------

E-plane (alb=10) ··········· H-plane (a/b=10) ·

(a)

60 90 120 150 180

Scattering Angle (degrees)

TE (alb=2) TM (alb=2)

TE (alb=10) TM (a/b=10)

-----------

(b)

30 45 60 75

Angle of Incidence (degrees)

90

107

Figure 2 Normalized scattering cross sections of a dielectric prolate spheroid of semi-major axis length A/4 and relative permittivity 3, coated with a material of relative permittivity 2.13 - }0.055 and thickness 0.02A, for two axial ratios.

spheroids, the magnitude of the backscattering cross section in the TM case is larger than that of the corresponding TE case.

In Fig. 3 we show the variation of the normalized backscattering cross section with the angle of incidence for a lossless dielectric spheroid and for a perfectly

108

10

iii' 5 :s c: 0 0 tS Cl)

-5 (/) (fJ (fJ

e -10 (.) Cl c:

-15 ·~

~ 0 -20 gi 0 ca co -25

-30

-20

iii' :s c: -25 0 ·u Cl) (/) (fJ (fJ

e -30 (.) Cl c:

·53 ~ 0 -35 gi 0 ca

CD

-40

0 15

0 15

Joan R. Ciric and Francis R. Cooray

30 45 60


30 45 60


(a)

75

TE-diel --TM-diel ---------

(b)

75

90

90

Figure 3 Backscattering cross sections for prolate spheroids of axial ratio 5 and semi-major axis length J../4. (a) Conducting spheroid and (b) dielectric spheroid of relative permittivity 3.

conducting spheroid, for both TE and TM polarizations of the incident uniform plane wave. The relative permittivity of the lossless material of the dielectric spheroid considered is 3, and the plots are given for both spheroids having a semi-major axis length .A./4. The axial ratio of the spheroids in both cases is 5. The results for the case of the dielectric spheroid have been obtained from those corresponding to a coated dielectric spheroid, as a special case when the confocal


layer has the material constants reduced to those of a free space. On the other hand, the conducting spheroid has been simulated from the dielectric spheroid by using a very high value of relative permittivity. The variation of the backscatter-ing cross section in the TE case is almost the same for both types of spheroids. However, in the TM case the magnitude of the scattering cross section for the conducting spheroid increases quite rapidly as compared to that for the dielectric spheroid.

VI. SCATTERING OF ELECTROMAGNETIC WAVES BY A CHIRAL SPHEROID

In this section, an exact solution to the problem of plane wave scattering by an optically active (or chiral) spheroid is derived using the same method of separation of variables. Fields outside as well as inside the spheroid are expanded in terms of vector spheroidal wave functions, and a set of simultaneous linear equations is obtained by imposing the boundary conditions on the surface of the spheroid. The solution to this system of equations yields the unknown coefficients in the series expansions of the corresponding fields.

A. FIELD EXPANSIONS

Consider a homogeneous, isotropic, and chiral spheroid, with chirality admit-tance ~c, permittivity es, and permeability µs, located in a medium assumed to be linear, homogeneous, and isotropic, with permittivity e and permeability µ. An arbitrarily polarized monochromatic uniform plane electromagnetic wave with an electric field intensity of unit amplitude is incident at an arbitrary angle on the spheroid whose center is at the origin 0 of a Cartesian coordinate system and whose axis of symmetry is along the z axis. The same time dependence exp(} wt) is used in the derivations and formulas given later. The plane of incidence is as-sumed to be the xz plane. Then, the expansion of the incident electric field in terms of the radial vector spheroidal wave functions defined in Section IV is given by (Cooray and Ciric, 1993)

00 00

Ei = L L [Pmn(h, Oi)M;;,~)(h; r) +qmn(h, Oi)N;;,~)(h; r)], (81) m=-OOn=[m[

where r denotes the coordinate triad T/, ~,


in which h = k F, with k being the wavenumber in the region outside the spheroid and F the semi-interfocal distance of the spheroid, and ~ is the polarization angle defined in Section V. The coefficients f mn (h, '!Ji) and gmn (h, '!Ji) in Eqs. (82) and (83) are given by (Cooray and Ciric, 1993)

2r 00 dmn(h) d f, (h JJ·) - - ""'' r pm (cos JJ·) (84)

mn ' 1 - Nmn(h) ~ (r+m)(r+m+l)diJi m+r 1 ' r=O,I

2mr ~' d;1n(h) P:;;+r(cos '!Ji) gmn(h, '!Ji) = - Nmn(h) ~ (r + m)(r + m + 1) sin '!Ji

r=O.I

(85)

When '!Ji= 0, only fln(h, 0) and g1n(h, 0) remain nonzero, and are given by

·n oo

f1n(h, 0) = g1n(h, 0) = --1- L' d]n(h). N1n(h) r=O,I

(86)

Equation (81) can now be written in a matrix form similar to that ofEq. (53) as

(87)

with the overbar denoting a column matrix and T denoting the transpose of a matrix, where

M(l)T - [M.(l)T r i - 0

M(l)T I

M(l)T -I

M(l)T 2

M(l)T -2 ... ] , (88)

with

M(l)T = [Mr(I) (h· ) w w,lwl 'r M~.11~1+1 (h; r) M~c.11~1+2(h; r) ... ] , (89)

and

-T [ -T rip = lo

-T 11

-T [_I

-T Iz

-T I_z ... ], (90)

with

-T lw = [Pw,lwl Pw,lwl+I Pw.lwl+2 ... ] . (91)

- (l)T - - (l)T -The elements of rNi and rlq are obtained from those of rMi and rip, re-spectively, by replacing M by N and p by q.

The electric field scattered by the spheroid is expanded in terms of radial vector spheroidal wave functions of the fourth kind in the form

00 00

Es= L L [amnM~~)(h; r) + /3mnN~~)(h; r)], (92) m=-OOn=lml


where


The elements of y and 8 are obtained from those of a and f> by replacing a and f3 by y and 8, respectively.

The expressions for the incident and scattered magnetic fields can be obtained from those of the corresponding electric fields as in the previous section:

. c - (l)T - - (l)T -( )

1/2

Hi = } µ, [rNi rip+ rMi rlq], (102)

H = . _ [ N-(4)T _ M-(4)T ii] ( c) 1/2 s } r s a+r s I-'· µ (103) Similarly, the magnetic field inside the chiral spheroid is obtained from Ee, in terms of ER and EL, as

He = j (::) 112

(ER - EL). (104)

Using Eqs. (95) and (96), He can now be expressed in matrix form as

( c ) 1/2 H = . ___..


equation yields the unknown expansion coefficients, and thus the electromagnetic field inside and outside the spheroid.

C. SCATTERING CROSS SECTIONS AND NUMERICAL RESULTS

The scattered electric field in the far zone can be written as in Eq. (77):

e-Jkr E5 (r, fJ,


The normalized bistatic and backscattering cross sections are given, respec-tively, by

and

Numerical results are again presented in the form of normalized bistatic and backscattering cross sections in the far zone, for a few chiral spheroids. The bistatic cross section is calculated for the case of an axial incidence. To obtain numerical results with a required accuracy, the series and the corresponding ma-trices involved in the field expressions need to be truncated appropriately. For the examples considered, a two-digit accuracy in the computed bistatic cross section is obtained by only retaining the terms corresponding tom = 1 and n = lml, Im I + 1, ... , Im I + 12. However, to obtain the same accuracy for the backscatter-ing cross section, it is necessary to use terms with m = 0, 1, 2, 3, and the same n as before. All the results presented are for the case µs = µ, but the formulation and the software used for the calculations are valid for any µs =I= µ, as long as the material is linear. Numerical experiments performed to validate the software and the accuracy of the results obtained are described in Cooray and Ciric (1993).

Figure 4 shows the variation of the normalized bistatic cross section with the scattering angle and the variation of the normalized backscattering cross section with the angle of incidence for both prolate and oblate chiral spheroids, under a uniform plane wave illumination. Each spheroid has a relative permittivity given by Fr = 1.33, a semi-major axis length of A./4, and a chirality admittance ~c = 0.00lS. The axial ratio of the spheroids is 2. It can be seen that the bistatic cross section patterns are similar for the prolate and the oblate spheroids. However, the magnitude of the cross section at a given scattering angle is lower for the prolate spheroid than for the oblate spheroid. The behavior of the backscattering cross section is completely different for the two spheroids. In the case of the prolate spheroid the magnitude of the cross section increases with the angle of incidence, whereas in the case of the oblate spheroid it decreases.

VII. SCATTERING BY SYSTEMS OF ARBITRARILY ORIENTED SPHEROIDS

An exact analytic solution to the problem of scattering of a monochromatic uniform plane electromagnetic wave of arbitrary polarization and angle of in-cidence by a system of dielectric prolate spheroids of arbitrary orientation can be obtained by expanding the incident, the scattered, and the transmitted elec-


iD -5

~ c: 0 t5 (])

-10 (/) (/J (/J e

-15 (.) 0

ii Ui iXi -20

-25 0

(a)

30 60 90

E-plane (prolate) --H-plane (prolate) E-plane (oblate) H-plane (oblate) - - ---

120 150 180


0 ------------~---~---~--~ TE (prolate) --

c: 0

-5

~ -10 (/)

~ (.) -15

(b)

---------------- ..•. ,., '·'·'·'·'·"·-.......... :.-.. :;:.:

·---... ······-~ ..

Cl j -~ ~-_.,=--~------------------------------------· ~ -25

TM (prolate) ---------TE (oblate) ---····· ·· TM (oblate)

-30 '-------'-----'-----'----_,_ ___ _,___ __ __, 0 15 30 45 60 75 90


115

Figure 4 Scattering cross sections of a chiral prolate spheroid and of a chiral oblate spheroid, with a refractive index of 1.33 and a chirality admittance of O.OOlS, illuminated by a uniform plane wave. The axial ratio is 2 and the semi-major axis length is A./4 for both spheroids.

tromagnetic fields in terms of appropriate vector spheroidal wave functions. The boundary conditions at the surface of a given spheroid are imposed by express-ing the electromagnetic field scattered by all the other spheroids in terms of the spheroidal coordinates attached to the spheroid under consideration, employing


the rotational-translational addition theorems for vector spheroidal wave func-tions. The solution of the resultant set of algebraic equations yields the unknown scattered and transmitted field expansion coefficients, in terms of which the elec-tromagnetic field can be evaluated at any given point.

A. PROBLEM FORMULATION

Consider N arbitrarily oriented prolate spheroids, with their centers located at the origins Or of the local Cartesian coordinate systems Orxr Yr Zr, r = 1, 2, ... , N, attached to the N spheroids as shown in Fig. 5. The major axes of these spheroids are along the z axes of the respective local Cartesian systems. The position of each of the origins Or with respect to the global Cartesian coordinate system Oxyz is given by the spherical coordinates dr, iJo,,


angle ~, with respect to the x z plane, as described in Sections V and VI. The medium in which the spheroids are embedded is assumed to be homogeneous and nonconducting, of permittivity e and permeability /L

The direction of the incident wave vector k with respect to the global system Oxyz is specified by the angular spherical coordinates iJi and 'Pi = 0 (see Fig. 5). Thus,

k = -k(sin iJix +cos iJiz), (114)

where x and z are the unit vectors along the x and z axes of the system 0 xy z and k is the wavenumber of the medium outside the spheroids. If the direction of k with respect to the local system Orxr Yr Zr is specified by the angular spherical coordinates iJi, , 'Pi, , then

k = -k(sin iJi, cos 'Pi,Xr +sin iJi, sin Zr are the unit vectors along the Xr, Yr, and Zr axes of the system OrXr YrZr· The unit vectors x, y, z are related to Xr, Yr> Zr by

a= r CaxXr + r CayYr + r CazZr, a= x, y,z, where

r c xx = cos


where s is the polarization angle. rEi can now be expanded in terms of vector spheroidal wave functions rM associated with the system OrXr Yr Zr as

00 00

rEi =exp(- jk. dr) L L r p;t/M;t~l) + r p;;;/M;;;~I) + r P~nrM~~l),

where

rp± mn

m=-OOn=lml

1 Smn(hr, cos fJi,) exp(- jmcpj,) rcyx =i= j rCyy) sins

2 ·n-1 [

kNmn(hr)

{

r . r COSS J ( Cxx =f) Cxy )--.

COS u· + I r COS { J

-( Czx =f FczY)~ Sllluj

l r cos{ J Cxz--

r . COS fJi x [cyz sm{ + J r cos{

- Czz-.--sm fJi

for fJi =F rr /2,

for fJi =F 0, rr,

for fJi =F rr /2,

(120)

(121)

(122)

Smn(hr, cosfJi,) and Nmn(hr) are the spheroidal angle function and the nor-malization constant associated with it, respectively, as defined in Section III.A, hr = k Fr with Fr being the semi-interfocal distance of the rth spheroid, and dr is the position vector of Or relative to 0. In Eq. (120), the argument (hr; 1Jr, ~r, cpr) of each rM, where 1Jr, ~r,


00 00

qEs - " "(q R+ qM+(4) + q RZ qMz(4) ) - ~ ~ Pmn mn Pm+l,n+I m+l,n+I

m=On=m 00

+ "(q R+ qM+(4) + q RZ qMz(4)) ~ "'-1,n+l -1,n+l f'On On

00 00

+ L L(qfJ=mnqM=~~ +qfJ~(m+I),n+lqM~~+l),n+l), (124) m=On=m

where q f3uv are the unknown expansion coefficients. Arranging the terms in this expansion in the sequence ei0 , e±Jcpq, e±2J'Pq, ... yields

(125)

- (4) -where qMs and q f3 are column matrices whose elements are prolate spheroidal vector wave functions of the fourth kind expressed in terms of the spheroidal co-ordinates associated with the system OqXqYqZq and the corresponding unknown expansion coefficients, respectively. These matrices are similar to the correspond-ing ones defined in Section V.A.

To impose the boundary conditions at the surface of the rth spheroid, the elec-tromagnetic fields scattered by all the other N - 1 spheroids have to be expressed as incoming fields with respect to the rth spheroid. This can be achieved by employing rotational-translational addition theorems. Consider the qth spheroid whose scattered field is denoted by qE5 • To express this field as an incoming field with respect to the rth spheroid, the vector spheroidal wave functions of the fourth kind associated with the system OqXqYqZq have to be expanded in terms of the vector spheroidal wave functions of the first kind associated with the sys-tem Orxr Yr Zr, using the appropriate rotational-translational addition theorems for vector spheroidal wave functions (Cooray and Ciric, 1989a; Dalmas et al., 1989),

00 v

qM;:;~4)(hq; rq) = L L WQ~i(aqr• f3qr, Yqr; dqr) v=O µ=-v

x [qr C1 rM+(l)(h · r ) +qr C2 rM-(l)(h · r ) µv r' r µv r' r

+qrc3rMt~)(hr;rr)], (126) 00 v

qM;;;~4)(hq; rq) = L L ~;!Q~~(


00 v

qM~~)(hq; rq) = L L ~;?Q:~(aqr. ,Bqr, Yqr; dqr) v=Oµ=-v

x[qrc rM+(l)(h · r) +qrc*rM-(l)(h · r) 4 µv r, r 4 µv r• r

+qrcsrM~~)(hr;rr)], (128)

which are valid for rr _::: dqr, where the asterisk denotes the complex conjugate, rq and rr denote the coordinate triads (T/q, ~q,


both sides of Eq. (130) and substituting for q]\f~4lT in Eq. (125) yield

qrEs = qrM(l)T [f qr ]T q iS forq=l,2, .. .,r-1,r+l, .. .,N. (134)

These N - 1 secondary incident fields and the primary incident plane wave field determine the electric field rEs scattered by the rth spheroid, which is expanded as in Eq. (125), in the form

(135)

- (4) - - (4) The column matrices rMs and r fJ have the same form as those of qMs and q ~' respectively, with the vector wave functions evaluated with respect to the spheroidal coordinate system attached to the rth spheroid.

The electric field rEt of the electromagnetic field transmitted inside the rth spheroid can be expanded in terms of a set of vector spheroidal wave functions as (Cooray et al., 1990)

rE - rM-(l)T r -1- 1 a, (136)

where the unknown coefficients in the series expansion are contained in the col-

umn matrix r a and the elements of the column matrix rJ\1;1l are prolate spheroidal wave functions of the first kind expressed in terms of the spheroidal coordinates associated with OrXr Yr Zr, taking into account the permittivity of the material in-side the rth spheroid, respectively.

The expansions of the different magnetic fields in terms of vector spheroidal wave functions can be obtained from those of the corresponding electric fields as in the previous sections, by replacing M by N, where N = k-1(\7 x M), and by multiplying each expansion by the appropriate value of j ( E / µ) 112 . This yields

(137)

(138)

(139)

( )

1/2 ru - . !.!_ rN- (l)T r -t - J 1 a, µ,

(140)

with E, Er being the permittivities and µ, µr the permeabilities of the media outside and inside the rth spheroid, respectively. The elements of the matrices

- (l)T - (l)T - (4)T - (l)T rNi , qrN , rNs , and rN1 are obtained from the elements of the matri-

122 Ioan R. Ciric and Francis R. Cooray

- (l)T - (l)T - (4)T - (l)T ces rMi , qrM , rMs , and rM1 , respectively, by replacing the vector spheroidal wave functions M by the corresponding N functions.

B. IMPOSING THE BOUNDARY CONDITIONS

On the surface of each dielectric spheroid ~r = ~ro, r = 1, 2, ... , N, the tan-gential components of both E and H fields must be continuous across the bound-ary. With Eqs. (123) and (134)-(140), these conditions can be expressed in the form

(rM(l)T r i + ~[(1 - 8 ) qrM(l)T[r ]T + 8 qM(4)T] q/3-) x l! I -

1 L., qr qr qr s Sr ~r-~ro q=l

( )

1/2 = Er ft (r:N(l)T r -) X l! J _

,. t Ci Sr ~r-~r0 c.ftr (142)

for r = 1, 2, ... , N, where 8qr is the Kronecker delta and ~r is the unit vector normal to the surface of the rth spheroid. Taking the scalar prod-uct of both sides of Eqs. (141) and (142) by q/l11 Sm,lml+K(hr, lJr)e±J(m±l)


y'

d x'

x Figure 6 Geometry of two spheroids in arbitrary orientation and the associated coordinate systems.

of each of the spheroids become very high (theoretically infinite). Because there is no field inside the spheroids, the boundary condition in Eq. (141) now becomes

(rM:~l)T r j + ~[(1 - 8 ) qrM(l)T[r ]T + 8 qM(4)T] qf3-) x I: I - = 0 1 L qr qr qr s Sr l;r-l;r0 q=l

(144) for r = 1, 2, ... , N. Following a similar procedure as in the case of dielectric spheroids, a system of algebraic equations is obtained in a matrix form as

(145)

where [ G'tv] is the system matrix, which has the same properties as those of [ G N] in Eq. (143), and Sfl is the column matrix of the unknown coefficients. The struc-ture and the elements of [ G'tv] are given in Cooray ( 1990).

C. SCATTERING CROSS SECTIONS

To illustrate the derivation of the normalized scattering cross section expres-sions, we consider a system of two spheroids A and B in an arbitrary config-uration. The Cartesian system Oxyz attached to the spheroid A and the system O' x' y' z' attached to the spheroid B are shown in Fig. 6. The observation point has


spherical coordinates r, iJ,


phase factor exp(jks · d) with ks given by

ks= k(sin 7'J cos


0

(a) -5

m -10 ~ c: 0 -15 ~ Q)

en CJ) CJ)

-20 e (.) -25 0

E CJ) -30 iD

-35

-40 0 30 60 90 120 150 180


0

(b)

m -5 ~ c: -10 0 ts Q)

-15 en CJ) CJ)

e -20 (.) Cl c:

-25 TE (1l ·55

~ TM (1 -30 TE (2l ~ TM(2 0 al cc -35

-40 0 30 60 90 120 150 180


Figure 7 Normalized scattering cross sections for two identical dielectric prolate spheroids of semi-major axis length >../4, axial ratio 2, and relative permittivity 2, with one spheroid rotated by the Euler angles 30°, 45°, 90° and displaced by a distance (1) >../2 and (2) >.. along the major axis of the other spheroid.


CD :E.. c 0 u C]) en (/) (/) e

(.) 0

~ iii

CD :E..

-10

-15

-20

-25

-30

-35

-40

-45

-50

§ -15 ts ~

~ (.) -20 O> c '5i =a 0 -25 j

(a)

0 30 60 90 120


E-plane (1) H-plane (1) E-plane (2) H-plane (2)

150

;~m === TE (2) ··········· TM(2) --··

180

-30 L--~~~'--~~--L~~~-'-~~~__..._~~~-'-~~----' 0 30 60 90 120 150 180


127

Figure 8 Scattering cross sections of two different dielectric prolate spheroids A and B, each of semi-major axis length J.c/4, but with axial ratios 2 and JO, and relative permittivities 2 and 3, respec-tively. The rotation and the displacements of B relative to A are the same as those in Fig. 7.

128

4

ID :!:?.. 2 c 0

~ gi 0 e

(.) 0

~ -2 iii

.4

4

2

0

-2

-4

(a)

0 30

(b)

~ .... ;~-~ ......

\

\ ' ... /

0 30

loan R. Ciric and Francis R. Cooray

60 90 120


60 90 120


E·plane H-plane

150

TE TM

180

//-/

/ \_,'

150 180

Figure 9 Scattering cross sections for two identical conducting prolate spheroids A and B, each of semi-major axis length )./4 and axial ratio 5, with B rotated by the Euler angles 45°, 90°, 45° relative to A and displaced by a distance )./2 along the x axis of the coordinate system attached to A.


ro ~ c 0

u Q) en (Jj (Jj e 0 0 :a en iii

ro ~ c 0 u Q)

en (Jj (Jj 0

0 Ol c ·~

1ii 0 ~ 0

"' al

-10

-20

-30

-40

-50

-60

-70

(a) E-plane --H-plane ---------

-80 ._ __ ___, ___ ___._ ___ _,_ ___ _._ ___ __J_ __ ___J

0 30 60 90 120 150 180


-10 (b)

-20

-30

-40

-50

-60 0 15 30 45 60 75 90


129

Figure 10 Scattering cross sections for two identical dielectric prolate spheroids of semi-major axis length 1./4, axial ratio 5, and relative permittivity 3, in a parallel configuration, displaced by a distance 1./2 along the x axis.


and TM polarizations have the same behavior for both cases, but the magnitude for the TM case is higher than the magnitude for the TE case.

In Fig. 9 the normalized bistatic and backscattering cross sections are plotted for two identical conducting spheroids A and B of axial ratio 5 and semi-major axis length )../4, separated by a center-to-center distance of )../2 along the x axis of A, with the rotation of B relative to A specified by the Euler angles 45°, 90°, 45°. The behavior of both the £-plane and the H-plane patterns is quite sim-ilar. However, the backscattering cross section pattern for TM polarization has more oscillations as compared to the pattern for TE polarization, with two min-ima around !Ji = 35°, 145° and a maximum around !Ji = 90°.

Numerical results for two identical dielectric spheroids A and B of axial ra-tio 5, semi-major axis length )../4, and relative permitivity 3, in a parallel configu-ration, are given in Fig. 10. The two spheroids are separated by a center-to-center distance of )../2 along the x axis. In this case the £-plane pattern has a sharp min-imum around iJ = 90°, whereas the H -plane pattern presents some oscillation about the same minimum point. The backscattering cross section has a minimum around !Ji = 30° for both polarizations and exhibits a similar behavior, with a higher magnitude for the TM case than for the TE case at higher angles of in-cidence, as expected. Because of symmetry about !Ji = 90°, the variation of the backscattering cross section is shown only for the range from !Ji = 0° to !Ji = 90°.

References

Abdelazeez, M. K. (1983). Wave scattering from a laxge sphere with rough surface. IEEE Trans. Antennas Propag. 31, 375-377.

Ackerman, T. P., and Toon, 0. B. (1981). Absorption of visible radiation in atmospheres containing mixtures of absorbing and nonabsorbing paxticles. Appl. Opt. 20, 3661-3668.

Acquista, C. (1976). Light scattering by tenuous paxticles: A generalization of the Rayleigh-Gans-Rocaxd approach. Appl. Opt. 15, 2932-2936.

Adams, C. N., and Kattawar, G. W. (1970). Solutions of the equations of radiative transfer by an invariant imbedding approach. J. Quant. Spectrosc. Radiat. Transfer 10, 341-366.

Adams, C. N., and Kattawax, G. W. (1993). Effect of volume-scattering function on the er-rors induced when polarization is neglected in radiance calculations in an atmosphere-ocean system. Appl. Opt. 32, 4610-4617. [Errata: 33, 453 (1994).]

Aden, A. L., and Kerker, M. (1951). Scattering of electromagnetic waves from two con-centric spheres.! Appl. Phys. 22, 1242-1246.

Agarwal, G. S. (1976). Relation between Waterman's extended boundary condition and the generalized extinction theorem. Phys. Rev. D 14, 1168-1171.

Aitchison, J., and Brown, J. A. C. (1963). "The Lognormal Distribution." Cambridge Univ. Press, Cambridge.

Allan, L. E., and McCormick, G. C. (1978). Measurements of the backscatter matrix of dielectric spheroids. IEEE Trans. Antennas Propag. 26, 579-587.

Allan, L. E., and McCormick, G. C. (1980). Measurements of the backscatter matrix of dielectric bodies. IEEE Trans. Antennas Propag. 28, 166--169.

Al-Rizzo, H. M., and Tranquilla, J.M. (1995a). Electromagnetic scattering from dielec-trically coated axisymmetric objects using the generalized point-matching technique. J. Comput. Phys. 119, 342-373.

Al-Rizzo, H. M., and Tranquilla, J.M. (1995b). Electromagnetic wave scattering by highly elongated and geometrically composite objects of large size parameters: The general-ized multipole technique. Appl. Opt. 34, 3502-3521.

Anderson, R. (1992). Measurement of Mueller matrices. Appl. Opt. 31, 11-13.

603

604 References

Anderson, D. G. M., and Barakat, R. (1994). Necessary and sufficient conditions for a Mueller matrix to be derivable from a Jones matrix. J. Opt. Soc. Am. A 11, 2305-2319.

Anderson, T. L., Covert, D. S., Marshall, S. F., et al. (1996). Performance characteristics of a high-sensitivity, three-wavelength, total scatter/backscatter nephelometer. J. Atmos. Oceanic Technol. 13, 967-986.

Andreae, M. 0. ( 1986). The ocean as a source of atmospheric sulfur compounds. Jn ''The Role of Air-Sea Exchange in Geochemical Cycling" (P. Buat-Menard, Ed.), pp. 331-362. Reidel, Dordrecht.

Andrew, W. V., Balanis, C. A., Tirkas, P.A., et al. (1997). Finite-difference time-domain analysis of HF antennas on helicopter airframes. IEEE Trans. Electromagn. Compat. 39, 100-113.

Andsager, K., Beard, K. V., and Laird, N. S. ( 1999). A laboratory study of oscillations and axis ratios for large raindrops. J. Atmos. Sci. 56, 2673-2683.

Ansmann, A., Wandinger, U., Riebesell, M., et al. (1992). Independent measurement of extinction and backscatter profiles in cirrus clouds by using a combined Raman elastic-backscatter lidar. Appl. Opt. 31, 7113-7131.

Ansmann, A., Bosenberg, J., Brogniez, G., et al. ( 1993). Lidar network observation of cir-rus morphological and scattering properties during the International Cirrus Experiment 1989: The 18 October 1989 case study and statistical analysis. J. Appl. Meteorol. 32, 1608-1622.

Arakawa, E. T., Williams, M. W., and Inagaki, T. (1977). Optical properties of arc-evaporated carbon films between 0.6 and 3.8 eV. J. Appl. Phys. 48, 3176-3177.

Arfken, G. B., and Weber, H. J. ( 1995). "Mathematical Methods for Physicists." Academic Press, San Diego.

Arnold, S., Liu, C. T., Whitten, W. B., and Ramsey, J.M. (1991). Room-temperature mi-croparticle based persistent spectral hole burning memory. Opt. Lett. 16, 420-422.

Arnold, S., Communale, J., Whitten, W. B., eta/. (1992). Room-temperature microparticle-based persistent hole-burning memory spectroscopy. J. Opt. Soc. Am. B 9, 4081-4093.

Arnold, S., Ghaemi, A., and Fuller, K. A. (1994). Morphological resonances detected from a cluster of two microspheres. Opt. Lett. 19, 156-158.

Amott, W. P., and Marston, P. L. (1991). Unfolded optical glory of spheroids: Backscat-tering of laser light from freely rising spheroidal air bubbles in water. Appl. Opt. 30, 3429-3442.

Amott, W. P., Dong, Y. Y., and Hallett, J. (1995). Extinction efficiency in the infrared (2-18 µm) of laboratory ice clouds: Observations of scattering minima in the Chris-tiansen bands of ice. Appl. Opt. 34, 541-551.

Asano, S. (1979). Light scattering properties of spheroidal particles. Appl. Opt. 18, 712-723.

Asano, S. ( 1983). Light scattering by horizontally oriented spheroidal particles. Appl. Opt. 22, 1390-1396.

Asano, S., and Sato, M. ( 1980). Light scattering by randomly oriented spheroidal particles. Appl. Opt. 19, 962-974.

Asano, S., and Yamamoto, G. ( 1975). Light scattering by a spheroidal particle. Appl. Opt. 14, 29-49.

Ashkin, A., and Dziedzic, J. (1980). Observation of light scattering from nonspherical particles using optical levitation. Appl. Opt. 19, 660-668.

References 605

Aspens, D. E. ( 1982). Local-field effects and effective-medium theory: A microscopic per-spective. Am. J. Phys. 50, 704-709.

Atlas, D. (1964). Advances in radar meteorology. In "Advances in Geophysics" (H. E. Landsberg and J. Van Miehem, Eds.), pp. 317-478. Academic Press, New York.

Atlas, D. ( 1990). "Radar in Meteorology." Am. Meteorol. Soc., Boston. Atlas, D., and Ulbrich, C. W. ( 1977). Path- and area-integrated rainfall measurements by

microwave attenuation in the 1-3 cm band. J. Appl. Meteorol. 16, 1322-1331. Atlas, D., Harper, W. G., Ludlam, F. H., and Macklin, W. C. ( 1960). Radar scatter by large

hail. Quart l R. Meteorol. Soc. 86, 468-482. Atlas, D., Matrosov, S. Y., Heymsfield, A. J., et al. (1995). Radar and radiation properties

of ice clouds. l Appl. Meteorol. 34, 2329-2345. Awaka, J., and Oguchi, T. ( 1982). Bistatic radar reftectivities of Pruppacher-and-Pitter form

raindrops at 34.8 GHz. Radio Sci. 17, 269-278. Aydin, K., and Daisley, S. ( 1998). Effects of raindrop canting and oscillation on rainfall

rate estimates from 35 GHz differential attenuation. Proc. Int. Geosci. Remote Sens. Symp., Seattle, WA, 1998, 153-155.

Aydin, K., and Giridhar, V. (1991). Polarimetric C-band radar observables in melting hail: A computational study. Prepr. Vol., Conf Radar Meteorol., 25th, Paris, 1991, 733-736.

Aydin, K., and Giridhar, V. ( 1992). C-band dual-polarization radar observables in rain. J. Atmos. Oceanic Technol. 9, 383-390.

Aydin, K., and Seliga, T. A. ( 1984 ). Radar polarimetric backscattering properties of conical graupel. J. Atmos. Sci. 41, 1887-1892.

Aydin, K., and Tang, C. (1997a). Millimeter wave radar scattering from model ice crystal distributions. IEEE Trans. Geosci. Remote Sens. 35, 140-146.

Aydin, K., and Tang, C. ( l 997b ). Relationships between IWC and polarimetric radar mea-surands at 94 and 220 GHz for hexagonal columns and plates. J. Atmos. Oceanic Tech-nol. 14, 1055-1063.

Aydin, K., and Walsh, T. M. (1996). Computational study of millimeter wave scattering from bullet rosette type ice crystals. Proc. Int. Geosci. Remote Sens. Symp., Lincoln, NE, 1996, 563-565.

Aydin, K., and Walsh, T. M. (1998). Separation of millimeter wave radar reftectivities of aggregates and pristine ice crystals in a cloud. Proc. Int. Geosci. Remote Sens. Symp., Seattle, WA, 1998, 440-442.

Aydin, K., and Zhao, Y. ( 1990). A computational study of polarimetric radar observables in hail. IEEE Trans. Geosci. Remote Sens. 28, 412-422.

Aydm, K., Seliga, T. A., and Bringi, V. N. (1984). Differential radar scattering properties of model hail and mixed phase hydrometeors. Radio Sci. 19, 58-66.

Aydin, K., Seliga, T. A., and Balaji, V. (1986). Remote sensing of hail with a dual linear polarization radar. J. Clim. Appl. Meteorol. 25, 1475-1484.

Aydin, K., Direskeneli, H., and Seliga, T. A. ( 1987). Dual polarization radar estimation of rainfall parameters compared with ground-based disdrometer measurements: 29 Octo-ber 1982, Central Illinois Experiment. IEEE Trans. Geosci. Remote Sens. 28, 834-844.

Aydin, K., Lure, Y. M., and Seliga, T. A. ( 1990a). Polarimetric radar measurements of rainfall compared with ground-based rain gauges during MAYPOLE'84. IEEE Trans. Geosci. Remote Sens. 28, 443-449.

606 References

Aydin, K., Zhao, Y., and Seliga, T. A. ( 1990b ). A differential reflectivity radar hail measure-ment technique: Observations during the Denver hailstorm of 13 June 1984. J. Atmos. Oceanic Technol. 7, 104-113.

Aydin, K., Tang, C., Pazmany, A., et al. (1994). 95 GHz polarimetric radar measurements in a cloud compared with model computations. Atmos. Res. 34, 135-144.

Aydin, K., Bringi, V. N., and Liu, L. (1995). Rain-rate estimation in the presence of hail using S-band specific differential phase and other radar parameters. J. Appl. Meteorol. 34, 404-410.

Aydin, K., Park, S. H., and Walsh, T. M. (1998). Bistatic dual-polarization scattering from rain and hail at S- and C-band frequencies. J. Atmos. Oceanic Technol. 15, 1110-1121.

Azzam, R. M.A., and Bashara, N. M. (1987). "Ellipsometry and Polarized Light." North-Holland, New York.

Bacon, N. J., Swanson, B. D., Baker, M. B., and Davis, E. J. (1998). Laboratory studies of light scattering by single levitated ice crystals. Proc. Conj. Cloud Phys., Everett, WA, 1998, 427-428.

Bahar. E., and Chakrabarti, S. (1985). Scattering and depolarization by large conducting spheres with rough surfaces. Appl. Opt. 24, 1820-1825.

Bahar, E., and Fitzwater, M. A. (1986). Scattering and depolarization by conducting cylin-ders with rough surfaces. Appl. Opt. 25, 1826-1832.

Balakrishnan, N., and Zrnic, D. S. ( 1990a). Estimation of rain and hail rates in mixed-phase precipitation. J. Atmos. Sci. 47, 565-583.

Balakrishnan, N., and Zrnic, D. S. (1990b ). Use of polarization to characterize precipitation and discriminate large hail. J. Atmos. Sci. 47, 1525-1540.

Balch, W. M., Holligan, P. M., Ackleson, S. G., and Voss, K. J. (1991). Biological and optical properties of mesoscale coccolithophore blooms in the Gulf of Maine. Limnol. Oceanogr. 36, 629--643.

Baltzer, F., Jett, S. D., and Rubahn, H.-G. (1998). Alkali cluster films on insulating sub-strates: comparison between scanning force microscopy and extinction data. Chem. Phys. Lett. 2'.17, 273-280.

Bantges, R. J., Russell, J.E., and Haigh, J. D. (1998). Cirrus cloud radiance spectra in the thermal infrared. Prepr. Vol., Conf Light Scat. Nonspher. Part.: Theory Meas. Appl., New York, 1998, 305-308.

Barabanenkov, Yu. N., Kravtsov, Yu. A., Ozrin, V. D., and Saichev, A. I. (1991). Enhanced backscattering in optics. Jn "Progress in Optics" (E. Wolf, Ed.), Vol. XXIX, pp. 65-197. Elsevier, Amsterdam.

Barakat, R. (1981). Bilinear constraints between elements of the 4 x 4 Mueller-Jones transfer matrix of polarization theory. Opt. Commun. 38, 159-161.

Baran, A. J., Foot, J. S., and Mitchell, D. L. (1998). Ice-crystal absorption: A comparison between theory and implications for remote sensing. Appl. Opt. 37, 2207-2215.

Barber, P. W. (1977). Resonance electromagnetic absorption by nonspherical dielectric objects. IEEE Trans. Microwave Theory Tech. 25, 373-381.

Barber, P. W., and Hill, S. C. (1990). "Light Scattering by Particles: Computational Meth-ods." World Scientific, Singapore.

Barber, P. W., and Wang, D.-S. (1978). Rayleigh-Gans-Debye applicability to scattering by nonspherical particles. Appl. Opt. 17, 797-803.

http:Commun.38

References 607

Barber, P., and Yeh, C. (1975). Scattering of electromagnetic waves by arbitrarily shaped dielectric bodies. Appl. Opt. 14, 2864-2872.

Barber, P. W., Miller, E. K., and Sarkar, T. K., Eds. (1994). Feature issue on Scattering by Three-Dimensional Objects. J. Opt. Soc. Am. A 11, 1379-1545.

Barkey, B., Liou, K. N., Takano, Y., et al. ( 1999). An analog light scatte1ing experiment of hexagonal icelike particles. II. Experimental and theoretical results. J. Atmos. Sci. 56, 613-625.

Barrick. D. E. (1970). Rough surfaces. In "Radar Cross Section Handbook" (G. T. Ruck, D. E. Barrick, W. D. Stuart, and C. K. Krichbaum, Eds.), Chap. 9. Plenum, New York.

Barton, J.P. (1998). Electromagnetic field calculations for a sphere illuminated by a higher-order Gaussian beam. II. Far-field scattering. Appl. Opt. 37, 3339-3344.

Barton, J. P., Alexander, D. R., and Shaub, S. A. (1989). Internal fields of a spherical particle illuminated by a tightly focused laser beam: Focal point positioning effects at resonance. J. Appl. Phys. 65, 2900-2906.

Bates, R.H. T. (1975). Analytic constraints on electromagnetic field computations. IEEE Trans. Microwave Theory Tech. 23, 605-623.

Bates, R.H. T., and Wall, D. J. N. (1977). Null field approach to scalar diffraction. Philos. Trans. R. Soc. London, Ser. A 287, 45-114.

Bates, T. S., Calhoun, J. A., and Quinn, P. K. (1992). Variations in the methanesulfonate to sulfate molar ratio in submicrometer marine aerosol particles over the South Pacific Ocean. J. Geophys. Res. 91, 9859-9865.

Battaglia, A., Muinonen, K., Nousiainen, T., and Peltoniemi, J. I. (1999). Light scattering by Gaussian particles: Rayleigh-ellipsoid approximation. J. Quant. Spectrosc. Radiat. Transfer 63, in press.

Battan, L. J. (1973). "Radar Observation of the Atmosphere." Univ. of Chicago Press, Chicago.

Batten, C. E. ( 1985). Spectral optical constants of soots from polarized angular reflectance measurements. Appl. Opt. 24, 1193-1199.

Bauer, P., and Schluessel, P. (1993). Rainfall, total water, ice water, and water vapor over sea from polarized microwave simulations and special sensor microwave/imager data. J. Geophys. Res. 98, 20,737-20,759.

Bauer, P., Schanz, L., Bennartz, R., and Schliissel, P. (1998). Outlook for combined TMI-VIRS algorithm for TRMM: Lessons from the PIP and AIP projects. J. Atmos. Sci. 55, 1644-1673.

Baum, W. A., Kreidl. T. J., and Schleicher, D. G. (1992). Cometary grains. Astrophys. J. 104, 1216-1225.

Bayiiss, A., and Turkel, E. (1980). Radiation boundary conditions for wave-like equations. Commun. Pure Appl. Math. 33, 707-725.

Beard, K. V., and Chuang, C. ( 1987). A new model for the equilibrium shape of raindrops. J. Atmos. Sci. 44, 1509-1524.

Beard, K. V., and Kubesh, R. J. (1991). Laboratory measurements of small raindrop distor-tion. 2. Oscillation frequencies and modes. J. Atmos. Sci. 48, 2245-2264.

Beard, K. V., Johnson, D. B., and Jameson, A. R. (1983). Collisional forcing of raindrop oscillations. J. Atmos. Sci. 40, 455-462.

Beard, K. V., Kubesh, R. J., and Ochs, H. T., III ( 1991 ). Laboratory measurements of small raindrop distortion. l. Axis ratios and fall behavior. J. Atmos. Sci. 48, 698-710.

608 References

Beardsley, G. F. ( 1968). Mueller scattering matrix of sea water. J. Opt. Soc. Am. 58, 52-57. Bebbington, D. H. 0., McGuinness, R .• and Holt, A. R. (1987). Correction of propaga-

tion effects in S-band circular polarisation diversity radars. IEEE Proc. H: Microwaves, Antennas Propag. 134, 431-437.

Beckering, G., Zilker, S. J., and Haarer, D. (1997). Spectral measurements of the emission from highly scattering gain media. Opt. Lett. 22, 1427-1429.

Beckmann, P., and Spizzichino, A. ( 1963). 'The Scattering of Electromagnetic Waves from Rough Surfaces." Macmillan, New York.

Ben-David, A. (1998). Mueller matrix for atmospheric aerosols at C02 laser wavelengths from polarized backscattering lidar measurements. J. Geophys. Res. 103, 26,041-26,050.

Benson, M. C., McDougal, D. C., and Coffey, D.S. (1984). The application of perpendic-ular and forward light scatter to access nuclear and cellular morphology. Cytometry 5, 515-522.

Berenger, J.-P. (1994). A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys. 114, 185-200.

Berenger, J.-P. (1996). Three-dimensional perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys. 127, 363-379.

Berkman, R. M .• and Wyatt, P. J. (1970). Differential light scattering measurements of heat-treated bacteria. Appl. Microbiol. 20, 51~512.

Berkman, R. M., Wyatt, P. J., and Phillips, D. T. (1970). Rapid detection of penicillin sensitivity in Staphylococcus aureus. Nature (London) 228, 458-460.

Berntsen, S., and Homsleth, S. N. (1994). Retarded time absorbing boundary conditions. IEEE Trans. Antennas Propag. 42, 1059-1064.

Beuthan, J., Minet, 0., Helfmann, J., et al. (1996). The spatial variation of the refractive index in biological cells. Phys. Med. Biol. 41. 369-382.

Bhandari, R. ( 1985). Scattering coefficients for a multilayered sphere: Analytic expressions and algorithms. Appl. Opt. 24, 19~1967.

Bickel, W. S., and Stafford, M. E. (1980). Biological particles as irregularly shaped scat-terers. In "Light Scattering by Irregularly Shaped Particles" (D. W. Schuerman, Ed.). pp. 299-305. Plenum, New York.

Bickel, W. S., Davidson, J. F., Huffman, D. R., and Kilkson, R. (1976). Application of polarization effects in light scattering: A new biophysical tool. Proc. Natl. Acad. Sci. U.S.A. 73, 486-490.

Blaschak, J. G., and Kriegsmann, G. A. (1988). A comparative study of absorbing bound-ary conditions. J. Comput. Phys. 77, 109-139.

Bohren, C. F. (1974). Light scattering by an optically active sphere. Chem. Phys. Lett. 29, 458-462.

Bohren, C. F. (1978). Scattering of electromagnetic waves by an optically active cylinder. J. Colloid Interface Sci. 66, 105-109.

Bohren, C. F. ( 1986). Applicability of effective-medium theories to problems of scattering and absorption by nonhomogeneous atmospheric particles. J. Atmos. Sci. 43, 468-475.

Bohren, C. F., and Battan, L. J. (1980). Radar backscattering by inhomogeneous precipita-tion particles. J. Atmos. Sci. 37, 1821-1827.

Bohren, C. F., and Battan, L. J. ( 1982). Radar backscattering of microwaves by spongy ice spheres. J. Atmos. Sci. 39, 2623-2628.

References 609

Bohren, C. F., and Huffman, D.R. (1983). "Absorption and Scattering of Light by Small Particles.'' Wiley, New York.

Bohren, C. F.. and Singham, S. B. (1991). Backscattering by nonspherical particles: A re-view of methods and suggested new approaches. J. Geophys. Res. 96, 5269-5277.

Bolin, F. P., Preuss, L. E., Taylor, R. C., and Ference, R. J. (1989). Refractive index of some mammalian tissues using a fiber optic cladding method. Appl. Opt. 28, 2297-2303.

Borghese, F., Denti, P., Toscano, G., and Sindoni, 0. I. (1979). Electromagnetic scattering by a cluster of spheres. Appl. Opt. 18, 116-120.

Borghese, F., Denti, P., Saija, R., et al. (1984). Multiple electromagnetic scattering from a cluster of spheres. I. Theory. Aerosol Sci. Technol. 3, 227-235.

Borghese, F., Denti, P., Saija, R., et al. (1987). Extinction coefficients for a random dis-persion of small stratified spheres and a random dispersion of their binary aggregates. J. Opt. Soc. Am. A 4, 1984-1991.

Borghese, F., Denti, P., Saija, R., and Sindoni, 0. I. (1992). Optical properties of spheres containing a spherical eccentric inclusion. J. Opt. Soc. Am. A 9, 1327-1335.

Borghese, F., Denti, P., and Saija, R. (1994). Optical properties of spheres containing sev-eral spherical inclusions. Appl. Opt. 33, 484-493. [Errata: 34, 5556 (1995).]

Borghese, F., Denti, P., Saija, R., et al. (1998). Optical resonances of spheres containing an eccentric spherical inclusion. J. Opt. (Paris) 29, 28-34.

Born, M., and Wolf, E. (1970). "Principles of Optics." Pergamon, Oxford. Bottcher, C. J. F., and Bordewijk, P. (1978). "Theory of Electromagnetic Polarization."

Elsevier, Amsterdam. Bottiger, J. R., Fry, E. S., and Thompson, R. C. (1980). Phase matrix measurements for

electromagnetic scattering by sphere aggregates. In "Light Scattering by Irregularly Shaped Particles" (D. W. Schuerman, Ed.), pp. 283-290. Plenum, New York.

Bouligand, Y., Soyer, M.-0., and Puiseaux-Dao, S. (1968). La structure fibrillaire et !'orientation des chromosomes chez !es dinoftagelles. Chromosoma (Berlin) 24, 251-287.

Bourrely, C., Chiappetta, P., and Torresani, B. (1986). Light scattering by particles of arbi-trary shape: A fractal approach. J. Opt. Soc. Am. A 3, 250-255.

Bourrely, C., Chiappetta. P., and Lemaire, T. (1989). Electromagnetic scattering by large rotating particles in the eikonal formalism. Opt. Commun. 70, 173-176.

Bowell, E., Hapke, B., Domingue, D., et al. (1989). Application of photometric models to asteroids. In "Asteroids II" (R. Binzel, T. Gehrels, and M. Matthews, Eds.), pp. 524-556. Univ. of Arizona Press, Tucson, AZ.

Breon, F.-M., Deuze, J.-L., Tanre, D., and Herman, M. (1997). Validation of spaceoome estimates of aerosol loading from Sun photometer measurements with emphasis on polarization. J. Geophys. Res. 102, 17,187-17,195.

Briggs, R. 0., and Hatchett, G. L. (1965). Techniques for improving underwater visibility with video equipment. Ocean Sci. Ocean Eng. 1&2, 1284-1308.

Bringi, V. N., and Hendry, A. (1990). Technology of polarization diversity radars for me-teorology. In "Radar in Meteorology: Battan Memorial and 40th Anniversary Radar Meteorology Conference" (D. Atlas, Ed.), pp. 153-198. Am. Meteorol. Soc., Boston.

Bringi, V. N., and Seliga, T. A. ( l 977a). Scattering from axisymmetric dielectrics or perfect conductors imbedded in an axisymmetric dielectric. JEE£ Trans. Antennas Propag. 25, 575-580.

610 References

Bringi, V. N ., and Seliga, T. A. ( 1977b ). Scattering from non-spherical hydrometeors. Ann. Telecommun. 32, 392-397.

Bringi, V. N., Seliga, T. A., and Mueller, E. A. (1982). First comparisons of rainfall rates derived from radar differential reflectivity and disdrometer measurements. IEEE Trans. Geosci. Remote Sens. 20, 201-204.

Bringi, V. N., Seliga, T. A., and Cherry, S. M. (1983). Statistical properties of the dual polarization differential reflectivity (ZDR) radar signal. IEEE Trans. Geosci. Remote Sens. 21, 215-220.

Bringi, V. N., Seliga, T. A., and Aydin, K. (1984). Hail detection with a differential reflec-tivity radar. Science 225, 1145-1147.

Bringi, V. N., Rasmussen, R. M., and Vivekanandan, J. (1986a). Multiparameter radar mea-surements in Colorado convective storms. I. Graupel melting studies. J. Atmos. Sci. 43, 2545-2563.

Bringi, V., Vivekanandan, J ., and Tuttle, J. D. ( l 986b ). Multi parameter radar measurements in Colorado convective storms. II. Hail detection studies. J. Atmos. Sci. 43, 2564-2577.

Bringi, V. N., Burrows, D. A., and Menon, S. M. (1991). Multiparameter radar and aircraft study of raindrop spectral evolution in warm-based clouds. J. Appl. Meteorol. 30, 853-880.

Bringi, V. N., Chandrasekar, V., and Xiao, R. (1998). Raindrop axis ratios and size distri-butions in Florida rainshafts: An assessment of multiparameter radar algorithms. IEEE Trans. Geosci. Remote Sens. 36, 703-715.

Britt, C. L. ( 1989). Solution of electromagnetic scattering problems using time domain techniques. IEEE Trans. Antennas Propag. 37, 1181-1191.

Bronk, B. V., Smith, M. J., and Arnold, S. (1993). Photon-correlation spectroscopy for small spherical inclusions in a micrometer-sized electrodynamically levitated droplet. Opt. Lett. 18, 93-95.

Bronk, B. V., Druger, S. D., Czege, J., and van de Merwe, W. P. (1995). Measuring di-ameters of rod-shaped bacteria in vivo with polarized light scattering. Biophys. J. 69, 1170-1177.

Brownlee, D. E. (1985). Collection of cosmic dust: Past and future. In "Properties and Interactions of Interplanetary Dust" (R. H. Giese and P. Lamy, Eds.), pp. 143-147. Reidel, Dordrecht.

Bruggeman, D. A. G. ( 1935). Berechnung vershiedener physikalischer Konstanten von het-erogenen Substanzen. 1. Dielektrizitatskonstanten und Leitfahigkeiten der Mischkorper aus isotropen Substanzen. Ann. Phys. (Leipzig) 24, 636-664.

Bruning, J. H., and Lo, Y. T. (1971). Multiple scattering of EM waves by spheres. IEEE Trans. Antennas Propag. 19, 378-400.

Brunsting, A. (1974). Can light scattering techniques be applied to flow-through analysis? J. Histochem. Cytochem. 22, 607--615.

Brunsting, A., and Mullaney, P. F. (1972). Light scattering from coated spheres: Model for biological cells. Appl. Opt. 11, 675-680.

Brunsting, A., and Mullaney, P. F. ( 197 4 ). Differential light scattering from spherical mam-malian cells. Biophys. J. 10, 439-453.

Bryant, H. C., and Cox, A. J. (1966). Mie theory and the glory. J. Opt. Soc. Am. 56, 1529-1532.

References 611

Buettner, K. (1963). Regenortnung vom Wettersatelliten mit Hilfe von Zentimeterwellen. Die Naturwissenschaftler 50, 591-592.

Buratti. B. J., Hillier. J. K.. and Wang. M. (1996). The lunar opposition surge: Observations by Clementine. Icarus 124, 490-499.

Bums, M. M., Fournier, J.-M., and Golovchenko, J. A. (1990). Optical matter: Crystalliza-tion and binding in intense optical fields. Science 249, 749-754.

Burrows, M. L. ( 1969). Equivalence of the Rayleigh solution and the extended-boundary-condition solution for scattering problems. Electron. Lett. 5, 277-278.

Bustamante, C., Maestre, M. F., and Tinoco, I., Jr. (I 980a). Circular intensity differential scattering of light by helical structures. I. Theory. J. Chem. Phys. 73, 4273-4281.

Bustamante, C., Maestre, M. F., and Tinoco, I., Jr. (1980b). Circular intensity differential scattering oflight by helical structures. II. Applications. J. Chem. Phys. 73, 6046-6055.

Bustamante, C., Tinoco, I., Jr., and Maestre, M. F. (1981). Circular intensity differential scattering of light by helical structures. III. A general polarizability tensor and anoma-lous scattering. J. Chem. Phys. 74, 4839-4850.

Bustamante, C., Tinoco, I., Jr., and Maestre, M. F. ( 1982). Circular intensity differential scattering of light by helical structures. IV. Randomly oriented species. J. Chem. Phys. 76,3~3446.

Bustamante, C., Maestre, M. F., Keller, D., and Tinoco, I., Jr. ( 1984 ). Differential scattering of circularly polarized light by dense particles. J. Chem. Phys. 80, 4817-4823.

Cai, Q., and Liou, K.-N. (1982). Polarized light scattering by hexagonal ice crystals: The-ory. Appl. Opt. 21, 3569-3580.

Cai, J ., Lu, N., and Sorensen, C. M. ( 1993). Comparison of size and morphology of soot ag-gregates as determined by light scattering and electron microscope analysis. Langmuir 9, 2861-2867.

Cameron, B. D., Rakovic, M. J., Mehrtibeoglu, M., et al. (1998). Measurement and calcu-lation of the two-dimensional backscattering Mueller matrix of a turbid medium. Opt. Lett. 23, 485-487.

Carder, K. L., Beardsley, G. F., and Pak, H. (1971). Particle size distributions in the Eastern Equatorial Pacific. J. Geophys. Res. 76, 5070-5077.

Cariou, J., Le Jeune, B., Lotrian, J., and Guem, Y. ( 1990). Polarization effects of seawater and underwater targets. Appl. Opt. 29, 1689-1695.

Carslaw, K. S., Wirth, M., Tsias, A., et al. ( 1998). Particle microphysics and chemistry in remotely observed mountain polar stratospheric clouds. J. Geophys. Res. 103, 5785-5796.

Carswell, A. I. (1983). Lidar measurements of the atmosphere. Can. J. Phys. 61, 378-395. Carswell, A. I., and Pal, S. R. (1980). Polarization anisotropy in lidar multiple scattering

from clouds. Appl. Opt. 19, 4123-4126. Caylor, I. J., and Chandrasekar, V. (1996). Time-varying ice crystal orientation in thun-

derstorms observed with multiparameter radar. IEEE Trans. Geosci. Remote Sens. 34, 847-858.

Caylor, I. J., and Illingworth, A. J. (1987). Radar observations and modeling of warm rain initiation. Quart. J. R. Meteorol. Soc. 113, 1171-1191.

Cesar, S. L., Vasconcelos, E. F., Freire, V. N., and Farias, G. A. (1994). Scattering coef-ficients for a spherical particle with stationary stochastic roughness. Prepr. Vol., Conj. Atmos. Rad., 8th, Nashville, TN, I994, 350-352.

612 References

Chandrasekar, V., and Bringi, V. N. (1988a). Error structure of multiparameter radar and surface measurements of rainfall. I. Differential reflectivity. J. Atmos. Oceanic Technol. 5, 783-795.

Chandrasekar, V., and Bringi, V. N. (1988b). Error structure of multiparameter radar and surface measurements of rainfall. Il. X-band attenuation. J. Atmos. Oceanic Technol. 5, 796-802.

Chandrasekar, V., Cooper, W. A., and Bringi, V. N. (1988). Axis ratios and oscillations of raindrops. J. Atmos. Sci. 45, 1323-1333.

Chandrasekar, V., Bringi, V. N., Balakrishnan, N., and Zrnic, D.S. (1990). Error structure of multi parameter radar and surface measurements of rainfall. Ill. Specific differential phase. J. Atmos. Oceanic Technol. 7, 621-629.

Chandrasekar, V., Gorgucci, E., and Scarchilli, G. (1993). Optimization of multiparameter radar estimates of rainfall. J. Appl. Meteorol. 32, 1288-1293.

Chandrasekar, V., Hubbert, J., Bringi, V. N., and Meischner, P. F. (1994). Transformation of dual-polarized radar measurements to arbitrary polarization bases. J. Atmos. Oceanic Technol. ll, 937-949.

Chandrasekhar, S. (1950). "Radiative Transfer." Oxford Univ. Press, London (also Dover, New York, 1960).

Charalampopoulos, T. T., and Chang, H. (1991). Effects of soot agglomeration on radiative transfer. J. Quant. Spectrosc. Radiat. Transfer46, 125-134.

Charlson, R. J., Lovelock, J.E., Andreae, M. 0., and Warren, S. G. (1987). Oceanic phy-toplankton, atmospheric sulphur, cloud albedo and climate. Nature (London) 326, 655-661.

Chen, H. Y., Iskander, M. G., and Penner, J.E. (1991). Empirical formula for optical ab-sorption by fractal aerosol agglomerates. Appl. Opt. 30, 1547-1551.

Cheng, L., and English, M. (1983 ). A relationship between hailstone concentration and size. J. Atmos. Sci. 40, 204-213.

Chepfer, H., Brogniez, G., and Fouquart, Y. (1998). Cirrus clouds' microphysical prop-erties deduced from POLDER observations. J. Quant. Spectrosc. Radiat. Transfer 60, 375-390.

Chemyshev, A. V., Prots, V. I., Doroshkin, A. A., and Maltsev, V. P. (1995). Measurement of scattering properties of individual particles with a scanning ftowcytometer. Appl. Opt. 34, 6301-6305.

Chew, W. C. (1990). "Waves and Fields in Inhomogeneous Media." Van Nostrand-Reinhold, New York.

Chew, W. C., and Wagner, R. L. (1992). A modified form of Liao's absorbing boundary condition. IEEE AP-S Int. Symp. Digest, 1992, 536-539.

Chew, W. C., and Weedon, W. H. (1994). A 3D perfectly matched medium from modified Maxwell's equations with stretched coordinates. Microwave Opt. Technol. Lett. 1, 599-604.

Chew, W. C., Lu, C. C., and Wang, Y. M. (1994). Efficient computation of three-dimensional scattering of vector electromagnetic waves. J. Opt. Soc. Am. A 11, 1528-1537.

Chiappetta, P. (1980a). Multiple scattering approach to light scattering by arbitrarily shaped particles. J. Phys. A: Math. Gen. 13, 2101-2108.

References 613

Chiappetta, P. ( l 980b ). A new model for scattering by irregular absorbing particles. Astron. Astrophys. 83, 348-353.

Chuang, C., and Beard, K. V. ( 1990). A numerical model for the equilibrium shape of electrified raindrops. J. Atmos. Sci. 47, 1374-1389.

Chylek, P., and Hallett, J. (1992). Enhanced absorption of solar radiation by cloud droplets containing soot particles in their surface. Quan. J. R. Meteorol. Soc. 118, 167-172.

Chylek, P., and Klett, J. D. (199Ia). Extinction cross sections of nonspherical particles in the anomalous diffraction approximation. J. Opt. Soc. Am. A 8, 274-281.

Chylek, P., and Klett, J. D. (199lb). Absorption and scattering of electromagnetic radiation by prismatic columns: Anomalous diffraction approximation. J. Opt. Soc. Am. A 8, 1713-1720.

Chylek, P., and Srivastava, V. (1983). Dielectric constant of a composite inhomogeneous medium. Phys. Rev. B 21, 5098-5106.

Chylek, P., and Videen, G. (1994). Longwave radiative properties ofpolydispersed hexag-onal ice crystals. J. Atmos. Sci. 51, 175-190.

Chylek, P., and Videen, G. (1998). Scattering by a composite sphere and effective medium approximations. Opt. Commun. 146, 15-20.

Chylek, P., Grams, G. W., and Pinnick, R. G. (1976). Light scattering by irregular randomly oriented particles. Science 193, 480-482.

Chylek, P., Ramaswamy, V., Cheng, R., and Pinnick, R. G. (1981). Optical properties and mass concentration of carbonaceous smokes. Appl. Opt. 20, 2980-2985.

Chylek, P., Ramaswamy, V., and Cheng, R. J. (1984a). Effect of graphitic carbon on the albedo of clouds. J. Atmos. Sci. 41, 3076-3084.

Chylek, P., Gupta, B. R. D., Knight, N. C., and Knight, C. A. (1984b). Distribution of water in hailstones. J. Clim. Appl. Meteorol. 23, 1469-1472.

Chylek, P., Srivastava, V., Pinnick, R. G., and Wang, R. T. (1988). Scattering of electro-magnetic waves by composite spherical particles: Experiment and effective medium approximations. Appl. Opt. 21, 2396-2404.

Chylek, P., Pinnick, R. G., and Srivastava, V. (1991 ). Effect of topology of water-ice mix-ture on radar backscattering by hailstones. J. Appl. Meteorol. 30, 954-959.

Chylek, P., Videen, G., Ngo, D., et al. (1995). Effect of black carbon on the optical prop-.erties and climate forcing of sulfate aerosols. J. Geophys. Res. 100, 16,325-16,332.

Chylek, P., Lesins, G. B., Videen, G., et al. (1996). Black carbon and absorption of solar radiation by clouds. J. Geophys. Res. 101, 23,365-23,371.

Cialdi, Cmdr. A. (1866). Sul moto ondosodel mare e su le correnti de osso specialmente auquelle littorali, pp. 258-288. Cited in ON/ Transl. A-655, l. Hydrographic Office, i955.

Clark, R. E. H. ( 1978). Intermediate coupling collision strengths from LS coupled R-matrix elements. Comput. Phys. Commun. 16, 119-127.

Clebsch, R. F. A. (1863). Ueber die Reflexion an einer Kugelfliiche. Crelle's J. 61, 195-251.

Clothiaux, E. E., Miller, M.A., Albrecht, B. A., et al. (1995). An evaluation of a 94-GHz radar for remote sensing of cloud properties. J. Atmos. Oceanic Technol. 12, 201-229.

Cloude, S. R. (1986). Group theory and polarisation algebra. Optik 75, 26-36. Cloude, S. R. (1989). Conditions for the physical realisability of matrix operators in po-

larimetry. Proc. SPIE 1166, 177-185.

614 References

Cloude, S. R. ( 1992a). Uniqueness of target decomposition theorems in radar polarimetry. In "Direct and Inverse Methods in Radar Polarimetry" (W.-M. Boerner et al., Eds.), pp. 267-296. Kluwer Academic, Dordrecht.

Cloude, S. R. (1992b). Measurement and analysis in radar polarimetry. In "Direct and Inverse Methods in Radar Polarimetry" (W.-M. Boerner et al., Eds.), pp. 773-791. Kluwer Academic, Dordrecht.

Coffeen, D. L. (1979). Polarization and scattering characteristics in the atmospheres of Earth, Venus, and Jupiter. J. Opt. Soc. Am. 69, 1051-1064.

Cole, J.B. (1995). A high accuracy FDTD algorithm to solve microwave propagation and scattering problems on a coarse grid. IEEE Trans. Microwave Theory Tech. 43, 2053-2058.

Cole, J.B. (1998). Generalized nonstandard finite differences and applications. Comput. Phys. 12, 82-87.

Coletti, A. (1984). Light scattering by nonspherical particles: A laboratory study. Aerosol Sci. Technol. 3, 39-52.

Cooper, J., Rombach, V., and Schiavoni, A. (1996). Comparison of computational electro-magnetic codes applied to a sphere canonical problem. IEE Proc. Microwave Antennas Propag. 143,309-316.

Cooray, M. F. R. (1990). Electromagnetic scattering by systems of arbitrarily oriented spheroids. Ph.D. Thesis, Univ. of Manitoba, Winnipeg.

Cooray, M. F. R., and Ciric, I. R. (1989a). Rotational-translational addition theorems for vector spheroidal wave functions. COMPEL 8, 151-166.

Cooray, M. F. R., and Ciric, I. R. (1989b). Electromagnetic wave scattering by a system of two spheroids of arbitrary orientation. IEEE Trans. Antennas Propag. 37, 608-618.

Cooray, M. F. R., and Ciric, I. R. (1990). Scattering of electromagnetic waves by a sys-tem of two conducting spheroids of arbitrary orientation. In "Radar Cross Sections of Complex Objects" (W.R. Stone, Ed.), pp. 343-355. IEEE Press, New York.

Cooray, M. F. R., and Ciric, I. R. (199la). Scattering of electromagnetic waves by a system of two dielectric spheroids of arbitrary orientation. IEEE Trans. Antennas Propag. 39, 680--684.

Cooray, M. F. R., and Ciric, I. R. (199lb). Scattering by systems of spheroids in arbitrary configurations. Comput. Phys. Commun. 68, 279-305.

Cooray, M. F. R., and Ciric, I. R. ( 1992). Scattering of electromagnetic waves by a coated dielectric spheroid. J. Electromagn. Waves Appl. 6, 1491-1507.

Cooray, M. F. R., and Ciric, I. R. (1993). Wave scattering by a chiral spheroid. J. Opt. Soc. Am. A 10, 1197-1203.

Cooray, M. F. R., Ciric, I. R., and Sinha, B. P. (1990). Electromagnetic scattering by a system of two parallel dielectric prolate spheroids. Can. J. Phys. 68, 376-384.

Cox, C., and Munk, W. (1954). Measurement of the roughness of the sea surface from photographs of the sun's glitter. J. Opt. Soc. Am. 44, 838-850.

Crepe!, 0., Gayet, J. F., Fournol, J. F., and Oschepkov, S. (1997). A new airborne po-lar nephelometer for the measurements of optical and microphysical cloud properties. II. Preliminary tests. Ann. Geophys. 15, 460-470.

Cross, J. D. (1968). Study of the surface of ice with a scanning electron microscope. In "Physics of Ice," pp. 81-94. Plenum, New York.

References 615

Cruzan, 0. R. (1962). Translational addition theorems for spherical vector wave functions. Quart. Appl. Math. 20, 33-40.

Czekala, H. ( 1998). Effects of ice particle shape and orientation on polarized microwave radiation foroff-nadir problems. Geophys. Res. Lett. 25, 1669-1672.

Czekala, H., and Simmer, C. ( 1998). Microwave radiative transfer with nonspherical pre-cipitating hydrometeors. J. Quant. Spectrosc. Radiat. Transfer 60, 365-374.

Dalmas, J., and Deleuil, R. (1985). Multiple scattering of electromagnetic waves from two infinitely conducting prolate spheroids which are centered in a plane perpendicular to their axes of revolution. Radio Sci. 20, 575-581.

Dalmas, J., Deleuil, R., and MacPhie, R.H. (1989). Rotational-translational addition the-orems for spheroidal vector wave functions. Quart. Appl. Math. 47, 351-364.

d'Almeida, G. A., Koepke, P., and Shettle, E. P. (1991). "Atmo

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