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LIGHT SCATTERING BY OPTICALLY ACTIVE PARTICLES

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Authors Bohren, Craig F., 1940-

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BOHREN, Craig Frederick, 1940-LI6HT SCATTERING BY OPTICALLY ACTIVE PARTICLES.

The University of Arizona, Ph.D., 1975 Physics, optics

Xerox University Microfilms , Ann Arbor, Michigan 48106

THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED.

LIGHT SCATTERING BY OPTICALLY

ACTIVE PARTICLES

by

Craig Frederick Bohren

A Dissertation Submitted to the Faculty of the

DEPARTMENT OF PHYSICS

In Partial Fulfillment of the Requirements For the Degree of

DOCTOR OF PHILOSOPHY

In the Graduate College

THE UNIVERSITY OF ARIZONA

19 7 5

THE UNIVERSITY OF ARIZONA

GRADUATE COLLEGE

I hereby recommend that this dissertation prepared under my

Craig Frederick Bohren direction by

entitled Light Scattering by Optically Active Particles

be accepted as fulfilling the dissertation requirement of the

Doctor of Philosophy degree of

Dis!

0 • IIU- |if i v

Date

/<f 7 5~~

;rtation Director

Dissert at ion Co-Director Date

After inspection of the final copy of the dissertation, the

following members of the Final Examination Committee concur in

its approval and recommend its acceptance:""

J ^ / / / "T* 7̂

/K !IU,U f-Vsf

This approval and acceptance is contingent on the candidate's

adequate performance and defense of this dissertation at the

final oral examination. The inclusion of this sheet bound into

the library copy of the dissertation is evidence of satisfactory

performance at the final examination.

STATEMENT BY AUTHOR

This dissertation has been submitted in partial fulfillment of requirements for an advanced degree at The University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library.

Brief quotations from this dissertation are allowable without special permission, provided that accurate acknowledgment of source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in his judgment the proposed use of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author.

SIGNED:

ACKNOWLEDGEMENTS

Any dissertation is to some extent a cooperative

venture. Therefore, I would be remiss if I did not thank

the following persons who helped me in various ways.

Dr. Robert Parmenter encouraged me to complete my

doctoral work at a time when I was only too ready to abandon

a seemingly hopeless undertaking. He also gave me helpful

suggestions when I was frantically trying to purge the sign

errors from my analysis.

Drs. John 0. Kessler and Donald R. Huffman jointly

undertook the task of supervising the research for this

dissertation. Unlike the professor who supposedly sees his

graduate student for the first time at the final oral, my

advisers were almost always available for consultation. I

should like to thank them for remaining cool every time I

burst into their offices (unannounced) with a new crisis.

But most of all I thank them for behaving in such a manner

that I will always have the best of memories about the time

I spent as their student.

Dr. Arlon Hunt took time he could barely afford

from the completion of his own dissertation to teach me

the intricacies of his apparatus and to share his knowledge

iii

about light scattering. I certainly would have spent a

much longer time on my research if I had not been able to

freely exploit Dr. Hunt's expertise.

Dr. George Holzwarth gave me several valuable

suggestions and directed me to obscure references in the

literature. More importantly, he received my work in a

spirit of scientific cooperation.

Dr. Donald Bourque supplied biological samples for

the initial experimental stages of my research and Drs.

Karen Warren and E. Philip Krider helped with the computer

programming.

Mr. Herold Miller took time from his well-earned

retirement to exhaustively review the first draft of the

dissertation and the final version is the better for his

comments.

Dr. David B. Thorud gave me much encouragement

and support.

Finally, I should like to thank my wife, my family

and my friends for being more patient with me than I would

have been with them if our situations had been reversed.

TABLE OF CONTENTS

Page

LIST OF ILLUSTRATIONS . vii

ABSTRACT ix

I. INTRODUCTION 1

II. LIGHT SCATTERING BY OPTICALLY ACTIVE PARTICLES, GENERAL CONSIDERATIONS 3

Constitutive Relations for an Optically Active Medium 4

Electromagnetic Field Equations for an Optically Active Medium 7

III. SCATTERING BY AN OPTICALLY ACTIVE SPHERE ... 10

Solution of the Problem 10 Expansion in Vector Spherical Harmonics 10

Amplitude Transformation Matrix ... 14 Cross Sections 18 CD and ORD for Suspensions of Particles 21

Scattering Matrix 26 Small Particle Limit 28

Derivation of Approximate Amplitude Transformation Matrix 30

Comparison With Theory of Gordon and Holzwarth 33

Angular Dependence of the Scattering Contribution to Circular Dichroism ... 44

IV. SCATTERING BY AN OPTICALLY ACTIVE SPHERICAL SHELL 51

Solution of the Problem 52 Expansion in Vector Spherical Harmonics 52

Scattering Coefficients 54

v

vi

TABLE OF CONTENTS--Continued

Page

V. SCATTERING BY A PLASMA SPHERE 58

Electromagnetic Field Equations 59 Boundary Conditions 62 Solution of the Problem 64 ,

Expansion in Vector Spherical Harmonics 64

Scattering Coefficients 66

APPENDIX A: NORMAL BOUNDARY CONDITIONS 68

APPENDIX B: REFRACTIVE INDICES 71

APPENDIX C: COMPUTER PROGRAMS 77

CDXORD and CMIE 81 ANGLE and AMIE 83

LIST OF REFERENCES 97

LIST OF ILLUSTRATIONS

Figure Page

1. Calculated CD for a 0.03 ym radius sphere embedded in a medium with refractive index 1.4 36

2. Calculated ORD for a 0.03 ym radius sphere embedded in a medium with refractive index 1.4 37







9. Angular dependence of the scattering contribution to circular dichroism for an optically active sphere with radius 0.03 ym . . 48


vii

viii

LIST OF ILLUSTRATIONS--Continued

Figure Page


12. Calculated refractive index difference 74

13. Calculated real part of the average refractive index 75

14. Calculated imaginary part of the average refractive index 76

ABSTRACT

The scattering of electromagnetic waves by a sphere

with intrinsic optical activity is calculated exactly from

electromagnetic theory through the use of the constitutive

relations for an optically active, isotropic medium. In

addition, the optically active spherical shell is similarly

treated. The solutions so obtained reduce to the standard

solutions in the limit of equal refractive indices for left-

and right-circularly polarized waves.

Expressions are given for single-particle cross

sections and amplitude transformation matrix elements as

well as circular dichroism and optical rotation for a

suspension of particles. These expressions are potentially

applicable to the calculation of optical rotatory dispersion

and circular dichroism spectra for particles of biological

origin. Sample calculations are made for a suspension of

spheres of poly-L-glutamic acid (PGA).

The angular dependence of the scattering contri

bution to circular dichroism is calculated for PGA spheres.

This scattering contribution tends to be peaked in the

forward direction, even for small particles. Therefore, it

appears from these calculations that it is possible to

ix

X

correct for scattering contributions to circular dichroism

through the use of instruments with a relatively small

acceptance angle.

The solution to the problem of scattering by a

sphere in which longitudinal waves may propagate is also

obtained. It is shown that the scattering coefficients

obtained in the Mie theory must be modified when longi

tudinal waves are present.

CHAPTER I

INTRODUCTION

Measurement of optical rotatory dispersion (ORD)

and circular dichroism (CD) spectra is a commonly used

method in the structural analysis of organic molecules

(Djerassi, 1960; Ciardelli and Salvadori, 1973). Extensive

measurements of ORD and CD for solutions have been followed,

in recent years, by measurements for suspensions of

particles, including particles of biological origin (Lenard

and Singer, 1966; Maestre and Tinoco, 1967; IVrigglesworth

and Packer, 1968). However, the interpretation of spectra

for suspensions is complicated, in part, by unequal scat

tering of left- and right-circularly polarized waves.

Consequently, there has been theoretical interest in the

effect of scattering on CD and ORD spectra (Urry and Ji,

1968; Urry and Krivacic, 1970; Ottaway and Wetlaufer, 1970;

Gordon and Holzwarth, 1971a; Schneider, 1973). Some of the

general features of the theory of such scattering were dis

cussed by Schneider (1971) and detailed calculations based

on approximations involving the standard Mie theory (van de

Hulst, 1957, chap. 9; Kerker, 1969, chap. 3) for spheres

and spherical shells were made by Gordon and Holzwarth

1

(1971b), Gordon (1972), and Holzwarth et_ aJL (1974). How

ever, there has been no theoretical analysis for scattering

by optically active particles that is consistent with the

electromagnetic field equations for an optically active

medium. These equations are discussed in Chapter II. In

Chapter III the exact solution of the problem of scattering

by an optically active sphere is obtained. The optically

active spherical shell is similarly treated in Chapter IV.

Finally, in Chapter V, the Mie theory modifications, which

are necessary when longitudinal waves are present, are

briefly discussed.

CHAPTER II

LIGHT SCATTERING BY OPTICALLY ACTIVE PARTICLES, GENERAL CONSIDERATIONS

Circular dichroism and optical rotation in homo

geneous media, which shall herein be referred to collec

tively as optical activity, are manifestations of unequal

refractive indices for left- and right-circularly polarized

electromagnetic waves. If a horizontally polarized beam is

incident on an optically active, isotropic, homogeneous slab

of thickness h, the azimuth $,p and ellipticity 0^, of the

outgoing beam are defined as the optical rotation and

circular dichroism, respectively, of the slab. These

quantities are related to the refractive indices by

= hTr(n^ - nj^)/X

(1)

©T = hir (n£ - nj£)/X ,

where X is the wavelength of the beam and the (complex)


waves, respectively, are n^ and n^ , where

nL = nL + inL

nR = nR + inR •

4

The common features of all problems of scattering by

optically active particles are discussed in this chapter.

Constitutive Relations for an Optically Active Medium

The usual constitutive relations D - eE and B =

yff where the permittivity e and the permeability y may

be real or complex scalars or tensors, are not compatible

with the observed phenomenon of optical activity. However,

from a phenomenological point of view, the following consti

tutive relations are sufficient for a description of optical

activity:

S = eE + yeVx E

and (2)

B = yH + 3yVx ft ,

where e , y , y , and 3 are phenomenological coefficients.

The unbounded propagation of waves, as well as reflection

and refraction at plane boundaries, for media with the con

stitutive relations (2), have been extensively investigated

(Fedorov, 1959a, 1959b; Bokut' and Fedorov, 1959; Rama-

chandran and Ramaseshan, 1961; Mathieu, 1957). The rela

tions (2), or others with similar form, have been obtained

from the analysis of both classical (Caldwell and Eyring,

1971, chap. 1) and quantum-mechanical (Condon, 1937) models.

Moffitt and Moscowitz (1959) obtained the constitutive

relations

5

5 = e£ + fH - gaifyst

and (3)

B = yH + fE + g9E/3t

If harmonic time dependence, exp(-iwt), is assumed, and if

the Maxwell equations that relate the curl of the fields

to time derivatives of the fields are used, Eqs. (3) are

equivalent to Eqs. (2), where

2f = iwey(y - 3)(1 - YBeyw2)"1

and

2g = ey(y + 3)(1 - Y3eyw2)-1 •

Satten (1958) showed that, in the absence of an externally

applied magnetic field, time-reversal symmetry requires

that f vanish, a condition that implies the equality of

Y and 3 • Therefore, it will be assumed throughout this

dissertation that y and 3 are equal.

Although Eqs. (2) and (3) are equivalent, the

asymmetry associated with optical activity is immediately

apparent from inspection of the former constitutive rela

tions. The curl is not a vector under a reflection of

coordinate systems and it is this reflectional asymmetry at

the molecular level that is the underlying cause of optical

activity.

Homogeneous, plane electromagnetic waves,

£ exp(ikz - iwt), with wave number k and frequency w , can

6

be propagated in media described by Eqs. (2) only for circu

larly polarized waves of either handedness. In an isotropic

medium the phenomenological coefficients are scalars and the

wave numbers k^ and kR for left- and right-circularly

polarized waves, respectively, are given by

k^ = to(ey)2 [1 + 3w(ey)^] (1 - 62w2ey)-1

and (4)

kR = wCey)"2" [1 - $w(ey)2] (1 - B2w2ey)_1 .

The difference An between left- and right-handed refrac

tive indices relative to the average refractive index

n = |(nL + nR) is given by

An/'n = 2(nL - nR)/(nL + nR)

= 2(kL - kR)/(kL • kR)

= 23o)(ey)E .

Optical activity is usually associated with small differ

ences between refractive indices for left- and right-

circularly polarized waves. For example, a difference of

10"3 in the real part of the refractive indices produces an

optical rotation of about 101* degrees per centimeter of path

length for an ultraviolet (~200 nm) beam. Therefore, for

any real system the quantity Bw(ey)E may be assumed to be

much less than unity without loss of generality. If powers

of $w(ey)s higher than the first are neglected in Eqs. (4),

one obtains

and

kT =

'R

oj(ey)? [1 + 3w(ey)®]

w(ey)E [1 - $w(ey)p]

(5)

Electromagnetic Field Equations for an Optically Active Medium

If harmonic time dependence, exp(-iwt), is assumed,

the field equations for an optically active, isotropic

medium with constitutive relations (2) are compactly written

in matrix form

V'

Vx

i -> E

+ K2

a. + K2 ->•

H

t r >

£ = K

H 4 . -

= 0

= 0 ,

where

K = iw(l - 32w2ey)_1 -iBeyw

- e

V

- iBcyo)

If the opposite sign convention, exp(iwt), is chosen, all

expressions in this dissertation are replaced by their

complex conjugates. This fact must be kept in mind if

comparison is made with expressions of other authors ivho

use the opposite sign convention.

A linear transformation of the electromagnetic

field

( 6 )

rg" % } = A L

V H qr

diagonalizes K :

A = A" 1KA ,

where

A =

A =

R

R

and

aR = -i(o)e)"1 [kR(l - g2w2ey) + $eyw2]

iL = -i((oy)-1 [kL(l - $2w2 ey) - geyw2]

(7)

If the same assumptions which led to Eqs. (5) are used,

one obtains

aR = -i(u/e)2

and (8)

aL = -i(e/y)r •

The transformed fields, an^ > independently

satisfy equations of the following form:

V2§ + k2§ = 0 (9)

Vx($ = k$ (10)

V-$ = 0 , (11)

where k = k^ when ^ H <§ and k = -k^ when $ E •

Equations (9-11) are the fundamental field equations for an

isotropic, optically active medium. The separation of the

electromagnetic field (h,H) into tt^o independent fields,

and , is useful in the analysis of scattering by

optically active particles.

CHAPTER III

SCATTERING BY AN OPTICALLY ACTIVE SPHERE

The problem under consideration in this chapter is

the determination of the electromagnetic field scattered by

a homogeneous, isotropic, optically active sphere that is

surrounded by a nonactive medium with (real) wave number

A plane harmonic wave, E exp(ik?z - iwt) , with electric o ~

field linearly polarized along the x axis and propagating

in the z direction is incident on the sphere. In this

chapter the subscripts 1 and 2 refer to quantities inside

and outside the sphere, respectively. A condensed version

of this chapter was recently published (Bohren, 1974).

Solution of the Problem

Expansion in Vector Spherical Harmonics

A divergence-free vector field that satisfies the

wave equation (9) and that is azimuthally symmetric can be

expanded in an infinite series of vector spherical harmonics

Mn and Nn (Morse and Feshbach, 1953, chap. 13; Stratton,

1941, chap. 7), where

K = Vx(?V '

10

11

kN = Vx $L , (12) n n

and

kM = Vx N n n

The generating function satisfies the scalar wave

equation in spherical polar coordinates and has the form

sin <i> -i *n(r) = Pn(cos e) z (kr) , (k > 0) (13)

COS (|)

I where is an associated Legendre polynomial of the first

kind, r is the position vector, and (r, 0, cf>) are spherical

polar coordinates. The choice of spherical Bessel function

zn(kr) is determined by subsidiary conditions that the

solution must satisfy. In the following expansions the

subscripts o or e (odd or even) are appended to vector

harmonics generated by the sine or cosine form, respectively,

of Eq. (13). Vector harmonics of types 1, 2, and 3 are

generated by the spherical Bessel functions zj;"^ , zj;^ , and

z<3>» where

n ' n

(kr) = (ir/2kr)^Jn+|(kr) = jn(kr)

z^2^(kr) = (ir/2kr)2Y i(kr) n n+ 2

z^3)(kr) = (7r/2kr)^H^i(kr) = h^ (kr)

= (kr) + iz^2^(kr) ,

and J i , Y .1 , are half-integral order Bessel n+a n+2 n+2

functions of the first, second, and third kinds, respec

tively (Watson, 1966, chap. 3).

The fields an^ > which satisfy Eqs. (9-11)

in the sphere 0 < r < a , are expanded in vector harmonics

of type 1 (harmonics of type 2 lead to singularities at

r = 0) :

$L " En VW*™ <kL> + C'Vl

* + Sei5(kL>]}

(14)

5r " En (kR) " ̂ CkR) ]

where En = inEQ(2n+l)/(n2+n). The expansion coefficients

fon>.",gen are determined by application of the boundary

conditions. The incident and scattered fields (^.,fi.) i I

and CES,I?S), respectively, are similarly expanded:

fii " E„ ^ " iSe"<k2>]

Bi = -k2(w2«.)-' En[i5^(k2) * iSW(k2)]

and

13

£ = E E [a i^3)(k?) - ib Nf^Cko) s n n n on v 2 J n en lJ

* - id„Son}(k2>]

i?s = -k2(u2»)-' Xn Bn[bn8»>Ck2) -

+ d r^3) (k7) + ic (k9) ] . n on *• 2 J n en v 2 J J

The scattered field is expanded in harmonics of

type 3 so that the field is an outgoing spherical wave at

large distances (k2r >> 1) from the sphere. At the boundary

of the sphere (r=a), the fields must satisfy the conditions

C&2 - 2X) x r = 0

and |r| = a (15)

($2 " x r = 0 ,

where = + an<* ^2 = ^i + ^s ' T^e inside

the sphere, E^ , , are obtained from Eqs. (6) and (14).

The preceding boundary conditions lead to eight

equations in the eight expansion coefficients which can be

solved to obtain the coefficients a , b , c , and d n ' n n n

of the scattered field. The results of this solution, the

details of which are omitted, are given in the following

section. The fields obtained by the application of condi

tions (15) also satisfy the usual boundary conditions for the

normal components of 5 and 5 (see Appendix A).

14

Amplitude Transformation Matrix

It is convenient to write the relation between

incident field and scattered field in the far zone >> 1)

in matrix form ( v a n de Hulst, 1957, p. 34):

E,

Jj.s

fs0 SJ E„ ."l expil<2 (r - z)

2 3 II i

- ik0r z S. S i E • 4 1 ^ .

JLl

(16)

where subscripts II and ± indicate field components parallel

and perpendicular, respectively, to the plane defined by

the directions of incident and scattered waves. The

elements of the amplitude transformation matrix are given by

S, = £ (2n+l)(n2+n)~1(-a T - b IT ) 1 n v ' v ' v n n n n y

S0 = £ (2n+l) (n2+n) ~1 (-a TT - b T ) 2 n v J v n n n n 7

S- = Z (2n+l)(n2+n)~1(c IT - d T ) 3 n v v n n n n J

(17)

S, = Z (2n+l) (n2+n) ~1 (d IT - c T ) , 4 n v n n n n J '

1 1 where TT = P /sin0 and X = dP /d0 , and the scattering n n n n

coefficients are

an = -4;' fi(l-aRaL)CXLnXRnY2nY4n+ULnURnYlnY3n)

• (XLnURn+XR„UL„) (aL"2uYlnY4„/k2+aRk2Y2nY3n'''I2l»' '

bn = -V ̂ ^-V^WznVWlnV

+ ^XLnURn+XRnULn) (^aLu2wY2nY3n//k2 + aRk2YlnY4n/y2

cn = An1 Aln(Y2nY3n"YlnY4n^

dn An* A2n(Y2nY3n"YlnY4n) '

where

an " iC1-aRaLHxLnxRnY4n+ULnURnY3„)

+ ^XLnURn+XRnULn-) ̂ aLy2w^k2 + aRk2/'y2w^Y3nY4n

A- XT Un *** Ut X-pv In Ln Rn R L Ln Rn

A2n ULnXRn + aRaLXLnURn

XLn = jn(kLa)

XRn ° Jn^kRa'

ULn ° (V)"1[Vj|.tV']'r-«

URn "

Yln = Jn(k2a^

V 2 n = C k 2a)-'[k2 r j n ( k 2 r ) ] ' r = a

16

Y3„ * hn15 t*2a>

Y4n = Ck2a)"1[k2rhi1)ck2r)1'r=a " (18)

The prime indicates differentiation with respect to the

argument kr .

The preceding solution to the scattering problem

[Eqs. (16-18)] is exact, subject only to the restriction

that Eqs. (2) are the appropriate constitutive relations

for an optically active medium. If 6 is zero, an and

bn reduce to the standard Mie coefficients (Stratton, 1941,

p. 565) and the coefficients cn and d^ vanish. The

amplitude transformation matrix in Eq. (16), evaluated in

the forward direction (0 = 0°), satisfies the symmetry

condition [7] of van de Hulst (1957, p. 57).

In the preceding analysis of scattering by an

optically active sphere no assumption was made about the

order of magnitude of Bw(ey)2 . However, in Chapter II, A

it was shown that $w(ey)^ will generally be much less than

unity (say 10"3, or smaller). The expressions (18) for the

coefficients of the scattered field are greatly simplified, A

with little loss of generality, if powers of 3aj(ey)e

greater than the first are neglected. If this assumption is

made, Eqs. (5) and (8) may be used for kT , k^ , a^ , and

instead of Eqs. (4) and (7). It will also be assumed

that all media are nonmagnetic (y^ \^2 ** yQ) • In addition

17

to simplification, these assumptions allow the solution to

be expressed in terms of kL and lcR rather than the

three phenomenological coefficients e , y , and 3 • It is

also convenient to introduce the Riccati-Bessel functions

and £n , the size parameter x, relative refractive

indices mT and mn , and the average relative refractive L K

index m :

^n(z) = ^n^

5n(z) = zh|^(z)

x = k2a

mL = nL/n2

mR = nR/n2

m = + mR) ,

where n^ is the refractive index of the medium surrounding

the sphere and n^ and nR are the refractive indices of

the sphere.

If the assumptions and notation discussed in the

preceding paragraph are introduced into Eqs. (18), one

obtains

/Wn(L)AnCR) + Wn(R)An(L) an

Wn(L)VR) + VL)WnCR)

n

'n

VR)BnCL) + VnCL;)BnCR)

WNCL)VN(R) + VN(L)WN(R)

• FWNCL)BN(R) - WN(R)BN(L)|

WN(L)VN(R) + VN(L)WN(R)J -d , n '

where

Wn(L) = miI>n(mLx)^(x) B, (X)IIJ1 (MTx 'RNV L

Wn(R) = mi|>n(mRx)^(x) - 5n(x)^CmRx

Vn^L^ = " m^n^x^A^mLx

vn(R) = ^n(mRx)?n(x) " m?nCx)^(mRx

An(L) = i|>n(mLx)i|/^(x) - m.t|>n(x)^CmLx

An(-R') =

Bn(L) = m^n(mLx)^(x) - ^n(x)^CmLx

Bn(R) = mi|>n(mRx)^(x) - (x)^(mRx) , (19)

and the prime indicates differentiation with respect to the

argument in parentheses.

Cross Sections

Extinction and scattering cross sections for left-

and right-circularly polarized waves will generally be

different because of optical activity. The electric field

may be expressed in terms of circularly polarized compon

ents El and Er , where

(20)

In the circular basis the relation between incident and

scattered field is written

1 = 2 s

1 i Ell

% »

1 - i «

E j.

•<

ELS EXPIK2(r-Z)

S2c S3c En

ERS

-ik2r

S4c cn

i-^ o

*

ERi •

where

- |-(S9 + S-| - iSA + iS^) '2 "1

S2c ~ ̂ S2 + S1 + iS4 " iS3-'

C — JL. '3c

= i(So Sq + iS* + iS,)

S4c " ̂ S2 " S1 " iS4 " iS3^ '

and the S^ are given by Eqs. (17). The scattering cross

sections for left- and right-circularly polarized waves

a i and o D , respectively, are given by S J L S J K

IT F^(0)sin0 d0 s,L

20

and

"s.R = 2rt2 - 2

TT F^(0)sin0 d0 ,

where

FL • S2c '4c

and

FR " lSlc|2 + lS3c

The scattering cross sections can be written in terms of

the scattering coefficients

"s.L = 2rt-2Zn(2n+l) [|an|2 • |bj2 • | cn | 2

ft . ft * + Id I2 + i(a c -a c +bd -bd)] 1 n 1 ^ n n n n n n n n ^ J

and (21)

°s,R = 2"k22Zn(2nn) [|an|2 * b |2 + |c |2 1 n1 1 n1

+ |d |2 - i(a c - a c + bd - b d)] , 1 n 1 nn nn nn n nJ

where we have used the relations

| (TnTm + *nVsine de " «n>n,2n2 (n+l)2 (2n-l) - 1

r* (IT T + IT X )sin0 d0 = 0 , v m n n nr *

and an asterisk denotes the complex conjugate. The average

scattering cross section is

21

0s = ^(as,L + as,R)

= 27rk~2Zn(2n+l) (|an|2 + |bn|2

+ 'cn12 + ldn'2) *

The extinction cross sections and for

left- and right-circularly polarized waves, respectively,

can be obtained from the optical theorem or by integration

of the Poynting vector over a large sphere surrounding the

particle. The result is

= 4 7rk 2 2 Re{S^}

= 2trk;2 Re{Z (2n+l) (-a - b_ - ic + id )} 1 n n n n n

and (22)

aR = 4Trl<22 Re{SR}

= 2irko2 Re{E (2n+l)(-a - b + ic - id )} , 2 n^ J v n n n n' '

where = S2c CO) and SR = S^c CO) are the amplitude

transformation matrix elements in the circular basis for the

forward direction.

CD and ORD for Suspensions of Particles

It is inappropriate to define circular dichroism

and optical rotation for a suspension of particles in terms

of refractive indices, as in Eq. (1), because there is no

well-defined refractive index for such a medium. However,

22

CD and ORD for a particulate medium can be operationally

defined in terms of the Stokes parameters. The Stokes

parameters (Walker, 1954) for a polarized beam can be

written

1 = eleL - eReR

Q = eLeR + eReL

( 2 3 )

U = i(E*ER - E^)

V = ERER " eL< '

If the beam is partially polarized, the Stokes parameters

are expressed as time averages of the quantities on the

right-hand side of Eq. (23). The azimuth X and the

ellipticity tan x °f "the vibration ellipse are given by

tan 2\ = UQ - I

and (24)

:R|/|ELHE,I/|ER| tan 2x = V(Q2+U2)"* - i(|E„|/|E, |-|E, |/|Ej) .

I is the intensity, and the handedness of the ellipse is

specified by the sign of V .

Consider a slab of thickness h composed of N

particles per unit volume embedded in a nonactive, nonabsorb'

ing medium with wave number k2 • If optical rotation

for such a system is defined as the change in azimuth of a

23

horizontally polarized incident beam (X^ = 0) after it

passes through the slab, one obtains

$T E At - A. = At « -|tan 2At . (25)

The subscripts i and t designate incident and trans

mitted beams, respectively. Similarly, if circular

dichroism 0,p is defined as the change in ellipticity of a

horizontally polarized incident beam (x^ = 0) after it

passes through the slab, one obtains

0T = tan xt " tan Xj = tan xt * ftan 2xt • (26)

If the particles are spherically symmetric and they all are

illuminated by approximately the same horizontally polarized

incident beam with amplitude E , the amplitude of the beam

transmitted (incident + forward scattered) is given by

(van de Hulst, 1957, p. 32)

El = E(1 - 2iTk22NhSL)

and (27)

Er = E(1 - 2TT]<22NhSR) .

If Eqs. (23) through (27) are combined, one obtains

= irk^Nh Im{SR - S^}

and (28)

0pp = nk^Nh Re{S^ - SR} ,

provided that the optical density ^k^Nh Re{-|(S^+SR)}

is much less than unity. Equations (28) are applicable to

any suspension of particles which have an amplitude trans

formation matrix in the circular basis that is diagonal for

the forward direction. It is assumed in the preceding

derivation that the particles are identical, that they are

separated by sufficiently large distances such that each

particle scatters independently of the others, and that

the optical thickness of the slab is sufficiently small

such that multiple scattering is negligible.

Circular dichroism and optical rotation for a de

polarizing medium (nonspherical particles or a distribution

of particle sizes) or for a multiple scattering medium can

also be operationally defined through the use of Eqs. (23-

26). However, for these instances, CD and ORD are not

expressed as simple functions [Eqs. (28)] of the single-

particle matrix elements.

The forward matrix elements for the optically active

sphere are

SL = S1(0) - iS3(0)

and (29)

sr = S1(0) + is3(0) ,

25

where

SjCO) = S2(0) = |ZnC2n+l)(-an - bn)

S3(0) = -S4(0) - |Zn(2n+l)(cn - dn) ,

because

*n(0) = Tn(°) = |n(n+l) .

The number of particles per unit volume N is related to

the total mass of particles per unit volume (concentration)

c by

N = 3c/ 4 T r a 3 p , (30)

where p is the density of the particle. If Eqs. (28-30)

are combined, one obtains

$ 5 p$,j,/hc = 3 Im{iSg(0) }/2x2a

and (31)

0 = p0T/hc = -3 Re{iS3(0)}/2x2a .

If Eqs. (19) are used for the scattering coefficients

S3C0) = Zn(2n+l)cn ,

and Eqs. (31) can be written

$ = 3 Im{Zn(2n+l)(ic ) }/2x2a

and (32)

0 = -3 Re{En(2n+l)(icn)}/2x2a .

Scattering Matrix

The elements of the (4x4) single-particle scatter

ing matrix are coefficients in the linear transformation

between incident and scattered Stokes parameters (van de

Hulst, 1957, p. 44), as follows:

1 :S

Qs 11

Us

Vs

» d

11

'21

12 S13 S14

S22 S23 '24

S31 S32 S33 S34

S41 S4 2 S43 44

\

<5i

u. 1

vi

These scattering matrix elements are functions of the

amplitude transformation matrix elements:

S11 = i(lsj12 + |S2I2 - |S3|2 • |s4! | 2 >

S12 = if|S2|2 - Is^2 * |S4|2 - 1 s3 1 !

2 >

S13 = Re{S,S* + S1SJ)

S14 = Im{S2S2 - siV

S21 =• 4<|S2|2 - Is-J2 - |S4|2 + 1 s3 1

2 >

S22 - |{|s2|2 + Is-J2 - |S4|2 - 1 S3 1

2 }

S23 = Items' - siV

S24 + S1S4>

27

Sji = Re{S2S^ + S^Sg)

S32 = Re{SoS^ - S1S*}

Sjj = Re{S^S2 + SjS^}

S34 = Im{S2S1 + S^Sj}

541 = Ii»^S2S4 + S3Si^

542 = Im^S2S4 ~ S3S1^

543 = Im^sis2 " S3S4^

544 = Re^S* - S3S4> . (33)

If the spacing between particles in a suspension is random,

the Stokes parameters of the light scattered by a suspension

are the sum of the Stokes parameters of the light scattered

by each particle. Therefore, the scattering matrix of a

suspension is merely the sum of scattering matrices of the

individual particles. In the general case there are 16

non- zero independent elements S^. .

If the suspension consists of identical optically

active spheres, there are only ten independent matrix

elements. From Eqs. (17) and (19) it follows that

S3 = -S4 . (34)

28

If Eq. (34) is substituted in Eqs. (33), one obtains the

six following relations:

513 = "S31

514 = S41

523 = "S32 (35)

524 = S42

S12 = S21

S34 = ~ S43 *

The functional form of the scattering matrix defined by

Eqs. (35) is predicted from general symmetry considerations

(van de Hulst, 1957, p. 49). Therefore, the derivation of

Eqs. (35) from the solution to the problem of scattering

by an optically active sphere [Eqs. (19)] provides an

additional check on the correctness of this solution.

Small Particle Limit

When |z| is << 1, the Riccati-Bessel functions and

their derivatives may be approximated by the expressions

i|>n(z) « 2nn!zn+1/(2n+l) !

ip^(z) ^ (n+l)2nn! zn/(2n+l) !

£n(z) ~ -i(2n)!/2nn!zn

^(z) » in(2n) !/2nn!zn+1 . (36)

l£ Eqs. (36) are substituted in Eqs. (19), and if only the

first term in the series expansion in Eqs. (32) is retained,

one obtains

* + i0 = §(kL - kR)[3/(2+m2)] . (37)

Equation (37) gives the CD and ORD for an optically active

sphere with a size parameter much less than unity. The

quantity ~ i-s the intrinsic CD and ORD of the

particle, and 3/(2+m2) may be interpreted as the effect of

the surroundings, i.e., a "solvent correction." It should

be noted that Eq. (37) is not identical with the result that

is obtained from molecular optics except in the special

instance of a refractive index near unity for particles and

their surroundings. Some of the differences between

molecular optics and small particle optics are discussed by

van de Hulst (1957, pp. 32-39).

The amplitude transformation matrix elements in the

small particle limit are given by

= -3(a^cos6 + b^)/2

52 = -3(a^ + b-^cos0)/2

53 = 3c^(l + cos0)/2 ,

30

where

= -ix3(mL - mR)2/18(2+m2)

b^ = -2ix3(l - mz)/3(2+m2)

= x3(^l - mR)/3(2+m2) (38)

Derivation of Approximate Amplitude Transformation Matrix

The amplitude transformation matrix elements Sg

•and [Eqs. (17)] both vanish for a nonactive sphere.

Consider an optically active sphere that is illuminated by

a left-circularly polarized wave with amplitude E^ and

assume that the amplitude of the scattered field in the far

zone (^2r >> 1) is given by

V(1)

pen

where

and

4" = f(r)

* \

S2L 0

0 S1L k .

4i}

» ^

1 1 " JL

1 11

p (1) EJ.I

j - i i

1

f(r) = expik2(r-z)/(-ik2r) .

j.i

E Li

(39)

31

Equation (39) is obtained through inversion of Eq. (20), and

the matrix elements S2L and are obtained through

substitution of the refractive index for the refractive

index in the expressions for the scattering coefficients in

the ordinary Mie theory. The circularly polarized compo

nents of the scattered field are

and

(1) _ Ls

= ICS 2L + SlL>ELi£C*)

(40)

C D _ JRs

= i(S 2L S1L^ ELi£

Similarly, consider the incident beam to be right-circularly

polarized with amplitude E^^ and assume that the amplitude

of the scattered field is given by

where

EF2) II s

= f (r)

S2R 0

E(2) J.S

.

0 S1R .

E 0.1

Ei( ̂ II i 1 1

» *

0 _ i

= 2 2

P ( 2 ) E±i - i

.

i ERi .

32

The circularly polarized components of the scattered field

are

ELs') = ^S2R " SlR)ERi£(;r)

and (41)

ERS} ^S2R + SlR^ERi£(^r^

An arbitrary polarized beam may be considered to be a

superposition of left- and right-circularly polarized com

ponents. Therefore, if Eqs. (40) and (41) are added, one

obtains the relation

JLs

JRs

= f ( r ) 2

S2L+S1L

S2L"S1L

S2R~S1R

S2R+S1R

*

EU

ERi *

(42)

where

and

ELs " ELS' +

= (1) + (2) nRs Rs Rs

The corresponding relation between linearly polarized

components of the scattered and incident fields is given by

E,

E xs

f(r) 2

S2L+S2R

1(S1R"S1L)

1^S2L~ S2R^

S1R+S1L

E it . II l

E • j.1 *

(43)

The amplitude transformation matrix in Eq. (43) satisfies

the following necessary conditions: (1) it reduces to the

result of the ordinary Mie theory in the limit of equal


waves (m^=m^) and (2) in the forward direction (0 = 0°) it

satisfies the symmetry condition [7] of van de Ilulst (1957 ,

p. 57). However, this matrix is not consistent with the

field equations (9-11) and the boundary conditions (15);

nevertheless it may be a good approximation for many compu

tational purposes. In addition, the amplitude transforma

tion matrix obtained by the preceding heuristic arguments is

not unique: it is possible to use the results of the

ordinary Mie theory to obtain different matrices that

satisfy the two preceding conditions. One of these matrices

is discussed in the following section.

Comparison With Theory of Gordon"~~and Holzwarth

Gordon and Holzwarth (1971b) and Gordon (1972)

calculated CD and ORD for suspensions of optically active

poly-L-glutamic acid (PGA) spheres. The starting point for

their calculations was the amplitude transformation matrix

(circular basis)

( ,

£

S2L+S1L ^S2L~SlL+S2R"SlR-)

^^S2L"S1L+S2R"S1R) S2R+S1R

(44)

The matrix in Eq. (42) and the matrix (44) have identical

diagonal elements but different off-diagonal elements.

However, these two matrices are identical in the forward

direction because the off-diagonal elements vanish. Thus,

in the forward direction, (44) reduces to

0

R

where

SL = F^n(2n+1) [-a°(mL) - b°(mL)]

and

SR = f^(2n+l) [-a°(mD) - b°(mB)] n nv R; nu"R;

(45)

The coefficients a0 and b° are those obtained in the n n

ordinary Mie theory. Therefore, CD and ORD calculated

according to the method of Gordon and Holzwarth are

and

$T = irlc^Nh Im{SR - S^}

®T = irk^Nh Re{S^ - S^} .

(46)

In the small particle limit, Eqs. (46) reduce to

<i> + i0 = £-(k^ - k^) 9m/ (m2 + 2) (47)

Equation (47) is similar to Eq. (37), with the exception

that the "solvent correction" is different.

The figures on pages 36-43 show CD and ORD cal

culated according to the exact theory [Eqs. (19) and (32)]

and calculated [Eqs. (45) and (46)] according to the method

of Gordon and Holzwarth. The computer programs CDXORD and

CMIE that were used to calculate the various scattering

coefficients, are described in Appendix C. In Figs. 1 and

2, calculated CD and ORD are shown for a 0.03 ym radius

sphere with optical properties similar to PGA (see Appendix

B for the refractive indices). The sphere is surrounded

by a medium with refractive index 1.4. There is very good

agreement between the exact theory and the theory of Gordon

and Holzwarth (GH). In Figs. 3 and 4, the index of refrac

tion of the surrounding medium is reduced to 1.2. However,

this reduction does not change the generally good agreement

between the two theories. The CD and ORD for a 0.10 ym

sphere surrounded by a medium with a refractive index of

1.4 (Figs. 5 and 6) are considerably distorted in comparison

with the intrinsic values (by intrinsic values is meant

the CD and ORD in the molecularly dispersed state). Again,

there is very good agreement with the GH theory (not

shown). If the sphere radius is increased to 0.50 ym

(Figs. 7 and 8), the CD and ORD bear little resemblance to

the intrinsic values.

36

Radius ® 0-03 ,m

1.60

Gordon & Holzwarth 1.20

Present Work

0.80

S

WAVELENGTH (nm

250 240 230 190 200 210 220 180

-0.40

0.80

Fig. 1. Calculated CD for a 0.03 ym radius sphere embedded in a medium with refractive index 1.4.

The refractive indices used in this calculation are given in Appendix B.

Radius = o .03/4m

160. rr2 « 1. 4

1.20"

— 0 . 8 0 -S s 5 0.40« oc N o

0.00-

& -0.40-

- 0 8 0 -

~ ~ ~ G o r d o n & H o l z w a r t h

Present Work

WAVELENGTH (nm)

250

Fig. 2. Calculated 0RD for a 0.03 ym radius sphere embedded in a medium with refractive index 1.4.


Radius • 0.03^um

1.60

Gordon & Holzwarth 1.20

Present Work

_ 0.80 S $ < 0.40 or

WAVELENGTH (nm) o — 0.00

2S0 200 220 240 190 230 X

<D 180

- 0.40

-0.S0



Radius - 0.03yUm

1.60

Gordon & Holiwarth 1.20

Present Work

0.80

2

0.40 O < on

A WAVELENGTH (nm 0.00

^30 250 190 240 200 210 220 180 X

-0.40

0.80

Fig. 4. Calculated ORD for a 0.03 ym radius sphere embedded in a medium with refractive index 1.2.


Radius = o. I0^<m 1-60

1.20

0.80

< 0.40 cc

WAVELENGTH ' nm

0.00- VA 250 240 230 210 220 190 200 100

-0.40

-O.BO



Radius ? oioyttr

1.20-

-0.40-

-O.BO

240 250

WAVELENGTH (nm )

Fig. 6. Calculated ORD for a 0.10 ym radius sphere embedded in a medium with refractive index 1.4.


42

Radius = 0.50 jJLm

20-

"""" Gordon & Holzwarth IS-

Present Work

s ,0-\ a < 5 -

O WAVELENGTH (nm)

180 240 200 210 220 230 250

- 5 -

-10-

Fig. 7. Calculated CD for a 0,50 ym radius sphere embedded in a medium with refractive index 1.4.


43

Radius * o .50 um

20 -

Gordon & Holzwarth

Present Work

5*

210 230 200 250 190 220 240

WAVELENGTH (nm)

180

-10-

Fig. 8. Calculated ORD for a 0.50 pm radius sphere embedded in a medium with refractive index 1.4.


It is possible to choose for the refractive indices

arbitrary values that will result in significant differences

(factors of 2 or more) between the exact theory and the GH

theory. However, for the realistic data that have been used,

there is very good agreement, even for spheres with radii

up to 1.0 ym . Calculated differences are about 5-10 per

cent.

Angular Dependence of the Scattering Contribution to Circular Dichroism

The circular dichroism 0^ of a suspension of N

optically active spheres per unit volume of a slab with

thickness h may be written

©T = ^hN(aL - aR) . (48)

The extinction cross section is the sum of an absorption

cross section aa and a scattering cross section <?s :

aT = a T + a ,

Li 3) L s j L (49)

°R = aa,R + as,R '

Thus, if Eqs. (30), (48), and (49) are combined, the

circular dichroism may be expressed as a sum of contribu

tions due to Scattering and absorption:

0 = p0T/ch = 0a + 0g ,

where

®a = 3("a,L " "a.R^16™'

and

0. 3(OS,L " "s.R^16™' (SO)

It was implicitly assumed in the preceding paragraph

that no light scattered by the spheres (except in the for

ward direction) is collected by the instrument used to

observe the circular dichroism. However, any real instru

ment has a finite acceptance angle that can be varied by the

use of apertures and stops. Therefore, the observed circu

lar dichroism is

®obs ®a + ®s,obs '

where

0 s,obs = 3(as,L " "scobs'16"'

and (51)

(os,L - Os.R'obs = F°S,L " °S,r' "

(da T/dfi)dfi - (da D/df2)dfi S , JJ S , K acc acc

The quantity das/dfi is the differential scattering cross

section, and the integration is taken over the acceptance

solid angle subtended by the instrument. 0S Q^s reduces to

0g in the limit of zero acceptance angle, and vanishes for

an instrument that collects all the scattered light.

It is convenient to define the quantity :

Zs E (das,L/dfi " das,R/cm;)^as,L " as,R^

which is normalized in the sense that

Z dfl = 1 . s

4 TT

Therefore, Eqs. (51) may be written

0s,obs V1 Esdft) .

acc

may be interpreted as the (normalized) angular depen

dence of the scattering contribution to circular dichroism,

The differential scattering cross sections for left- and

right-circularly polarized waves are

das,l/dfi = ^22 ̂ Sll " S14^

and

das,R/'dfi = k2Z(Sll + S14^ '

their difference is

das,l/dfi " das,R^d^ "2k22 S14

Therefore, may be written

Es = 2S14/„2(QS;R - QS;L) , (52)

47

where the efficiency factors for scattering are

and

Qs,R = °s,R^Tra2

There has been recent interest in the construction

of instruments that will reduce the scattering contribution

to CD spectra (Dorman, Hearst, and Maestre, 1973). The

analysis presented in the preceding paragraphs provides some

useful guidelines in the development of such instruments.

The figures on pages 48-50 (Figs. 9-11) show

calculated as a function of scattering angle for several

particle radii. The wavelength of the incident light is

0.190 v>m , and the refractive index of the medium surround

ing the particle is 1.4. The computer programs ANGLE and

AMIE that were used to calculate the amplitude transforma

tion matrix elements, are described in Appendix C. Appendix

B contains the refractive indices. It is interesting to

note that even for small particles, tends to peak in the

forward direction. For particles above 0.10 ym radius, this

forward peaking is highly pronounced. For sm^ll particles,

however, 0 is small compared with 0 . Therefore, it 3 3.

appears from these calculations that it is possible to

correct for scattering contributions to circular dichroism

through the use of instruments with a relatively small

acceptance angle (about 20°).

48

Radius = 0.03jj,m

/I = 0.190 ytem

0.2s

0.00 0 60 140 20 40 80 100 120 160

SCATTERING ANGLE

Fig. 9. Angular dependence of the scattering contribution to circular dichroism for an optically active sphere with radius 0.03 ym.

49

0.80

0.70

Radius * 0 10 /im

0.60

0.50

0.40

0.30

0.20

0.10

0.00 60 80 100 120 20 40

SCATTERING ANGLE

Fig. 1 0 . Angular dependence of the scattering contribution to circular dichroism for an optically active sphere with radius 0.10 ym.

50

Radius = O.SO yum

A ' 0.190y£4m

so «A w

40

20

120 80 100 60 40

SCATTERING ANGLE

Fig. 11. Angular dependence of the scattering contribution to circular dichroism for an optically active sphere with radius 0.50 ym.

CHAPTER IV

SCATTERING BY AN OPTICALLY ACTIVE SPHERICAL SHELL

The problem under consideration in this chapter is

the determination of the electromagnetic field scattered by

an optically active spherical shell. This problem is of

interest because many particles of biological origin consist

of a membrane surrounding a nucleus. The shell surrounds a

nonactive medium (inner sphere) with wave number kj and is

surrounded by a nonactive, nonabsorbing medium with wave

number k . The inner and outer radii of the shell are a

and B , respectively; the shell has wave numbers kL and

k^ for left- and right-circularly polarized waves. In this

chapter the subscripts I, II, and III refer to the inner

sphere, shell, and surrounding medium, respectively. A

plane harmonic wave, EQ exp(ikz - iwt), with electric field

linearly polarized along the x axis and propagating in the

z direction is incident on the shell. Because of symmetry,

the expressions for the amplitude transformation matrix

elements [Eqs. (17)], cross sections [Eqs. (21-22)], and

circular dichroism and optical rotation [Eqs. (28)] have the

same form as those for the optically active solid sphere.

51

52

The only differences between the shell problem and the solid

sphere problem are the expressions for the scattering

coefficients a , b , c , and d . The results in this n n n n

chapter were recently published (Bohren, 1975).



The fields anc^ which satisfy Eqs. (9-11)

in the spherical shell, a < r < b , are expanded in vector

harmonics of types 1 and 2:

Sl = Z „ E n { f o""^[^(kL) • i^CV]

+ - S(«(kL)]

* f o n } (kL) • N^CkL)]

+ fen' (" ^0 + ̂ (kL> 1 >

and (53)

5r = En{foP ™ tkR> "

+ ̂ (R5

+ f V " CkR) 1

+ f e n ' < k R > " ̂ ( k R ) ] }

where En = inEQ(2n+l)/(n2+n). The expansion coefficients

fon^L^ ' £en^L^' ' * " ' £en^R) are determine(i by application

of the boundary conditions. The fields in the inner sphere

0 < r < a , that satisfy the usual field equations, are

expanded in vector harmonics of type 1:

®I = Sn VSoiAlm'cV " iPen^en1

* " iPon^fV1

and

^ + nrr W^l) Hj = -kjCyjU.)- £n En[penM^n'(k1) • igonN^(kj)

* Pernio" + '

The incident and scattered fields (it.,?!.) and ) , 11 So

respectively, are similarly expanded:

6i " En

i?i = -i(»in»)-' En[i5W(k) • ifi^tk)]

and

®s • En VanCm -

+ 'n0™ W " "n^on' ® 1

K " \ w C ® • i a „ s ™ t o

+ ̂ on'to - lcn^^(k)] .

At r = b the fields must satisfy the conditions

ÎII " EII-) x r = 0

and |r| = b

C^ni - ? " 0

where = ̂ + anc* ÎII = î + ^s " "îe

inside the spherical shell, and , are obtained

from the transformation equation (6) and Eqs. (52). At

r = a the fields must satisfy the conditions

(îl " Sj) x r = 0

and |r| = a

(IÎI - Sj) x r = 0

The preceding boundary conditions lead to 16 equations in

the 16 expansion coefficients that can be solved to obtain

the coefficients a , b , c , and d of the scattered field. n n n n

The results of this solution, the details of which are

omitted, are given in the following section.

Scattering Coefficients

The approximate equations (5) and (8) are used for

^L' ̂ R' aL* an<^ aR i-nstead of the more general equations

(4) and (7) and all media are assumed to be nonmagnetic

55

(Pi »» vijj & Vijjj ^ yQ) • It is also convenient to define

the quantities:

kII = ^kL + kR^

v = kb

a = ka

NL = kLk~1

NR = kRk 1

NII = kIIk 1

Nj = kjk"1

and the functions

nn ( p ) = P " 1 d [ p z n ( p ) ] / d p .

The scattering coefficients, obtained by solving the sixteen

linear algebraic equations in the expansion coefficients,

are given by

a = -A_1(AD WT + AT WD ) n n v Rn Ln Ln Rn-^

bn " -V<BLnVRn + BRnVLn>

cn = lAn'(ALnVRn " ARnVLn'

d„ " 1An'tBRnWL„ " BLnWRn>

An WI.nVRn + WRnVLn '

where

56

Rn

Ln

W Ln

xRnC-5"n1}M " NIIURn

X^On^Cv) - N:IULn

XLbC-'NII^3:I(^ " ULn

w Rn (3)

Ur

V Ln

'Rn

B Ln

B Rn

XRnMNIl"n tv> " "Rn

X^Mn^Cv) - NnULn

XRn(-)"n3)Cv5 - NIIURn

" " L „

XRn(+'NIInn " URn

The functions in the expressions (54) are given by

- ) , ( 3 ' ( v ) J n v

+ )z£3)(v)

+ )Z^(V)

-)z^ (v) 7 n ^ J

- ) z £ 1 3 ( v )

+ )zf15(v) (54)

+ D4n42)(NRv' 1

XLn^ z»>(Nlv) + DlnZ(^(NLv) ± D5nz(^(NRv)

"Rn^l = + D4n^2)(NRv' 1 D2n^2''V

u T c ± ) LNV J N« (NLV) • DLNR,£2) (NLV) ± D3NN F21 (NRv) ,

57

where

Dln = -"^WW + WW^

D2n = Dn'lFnCVW " WW1

D3„ " VIWW '

D4n " "^WW + WW'

Dn " WW + WW

Fn(N) = NjjZ^fNojn^'fNjO) - Njn^tNcOz^CNjCO

Gn(N) = NjZ^2' (NcOn^1' (NjCi) - Njjii^2'(No) (Nja)

Hn(N) - NjjZ^1'(Na)n'1'(Njtt) - Nji^1'(Na)z^1'(Njcc)

Kn(N) » NjZ^1-1 (Najn^1-* (Nja) - Njjn^1-1 (Na) z^1'(Nj-a) .

When the shell is nonactive (k^ = k^) , the

coefficients c and d vanish, and the coefficients a n n ' n

and b reduce to the solution of Aden and Kerker (1951) . n

In the limit of vanishing inner sphere radius (a -> 0) , the

preceding solution reduces to that for an optically active

solid sphere.

CHAPTER V

SCATTERING BY A PLASMA SPHERE

In the previous chapters it was implicitly assumed

that longitudinal waves could not be propagated in the

media under consideration. The mathematical statement of

this condition is that the divergence of the electric field

vanishes. Melnylc and Harrison (1970) showed that the usual

Fresnel theory of reflection and refraction at plane inter

faces (Born and Wolf, 1965, pp. 36-51) is .incorrect when

applied to materials in which longitudinal waves may propa

gate. Except for geometry, the Fresnel problem is similar

to the problem of scattering by a sphere. Therefore, it is

expected that the Mie theory should be similarly modified

for media that can support longitudinal waves.

The solution to the problem of scattering by a

plasma sphere, briefly discussed in this chapter, was ob

tained by the writer in collaboration with A. Hunt (private

communication, 1974). Unfortunately, soon after this

problem was solved, they discovered that an essentially

identical solution had been published a few months previously

by Ruppin (1973). Nevertheless, the solution will be

included in this dissertation, even though it does not

59

strictly satisfy the criterion of originality. (The preced

ing chapters are independent of this chapter.)

Electromagnetic Field Equations

The field equations satisfied by the electric field

E and the magnetic induction § may be written

v x 2 = -a§/at (55)

V • $ = 0 (56)

eoV * 2 = ptotal

v * S = y03t0tal , (S8)

where

J _ j + J total ~ true polarization

+ ̂ + t magnetization displacement '

and

3 , . = a?/at polarization '

J V magnetization x ̂

Jj. , . = e aS/at . displacement o

? is the electric dipole moment per unit volume, S is the

magnetic dipole moment per unit volume, Ptotai is the total

charge per unit volume, and eQ and are the permit

tivity and permeability, respectively, of free space. The

magnetic field H is defined by

60

H = B/uq - M .

Therefore, Eq. (58) can be written

V x S = 3 t r u e + 8 P / 3 t + e 0 3 E / 9 t . ( 5 9 )

If the following constitutive relations are introduced

3. = oS true

? - xe S A o

s = nfi ,

where a is the conductivity and x i-s electric

susceptibility, and if harmonic time dependence, exp(-iwt),

is assumed, one obtains

V x S = -iweS (60)

V x S = iwy$ (61)

V • H = 0 , (62)

where

e = e + i cf/w

e = (1 + X) e0 •

If the curl of Eq. (60) is taken, it follows from Eqs. (61)

and (62) that H satisfies the wave equation

61

V2H + k£ £ = 0 , (63)

where the transverse wave number k^, is

kT = w(ey)

The Helmholtz theorem (Morse and Feshbach, 1953, p. 52)

states that any vector field that is finite, uniform, and

continuous, and that vanishes at infinity may be expressed

as the sum of a field with zero divergence and a field with

zero curl. Therefore, the electric field may be written

$ = ST + S , (64)

where

and

V • Et = 0 (65)

7x1, = 0 . (66) L

It is assumed that the transverse field and the longi

tudinal field E^ both satisfy wave equations

V2Et + k2 £j. = 0 (67)

and

V2El + k[ = 0 (68)

where k^ is the longitudinal wave number. It follows

from Eqs. (61), (64), and (66) that

62

H = (V x E)/iwy = (V x ET)/iooy . (69)

It is also assumed that Eq. (60) can be replaced by

tT = (V x ff)/(-iwe) . (70)

The equations (62) , (63) , (65), (66) , (67) , (68) , (69) , and

(70) are the field equations for a medium that can support

longitudinal waves.

Boundary Conditions

The current density J is defined by

J = 3true + e09?/3t + 3P/3t (71)

and Eq. (59) can be written

V x S = 5 . (72)

The divergence of the curl of any vector field vanishes;

therefore,

V • J = 0 . (73)

If the usual "pillbox" argument (Panofsky and Phillips,

1962, p. 31) is applied to Eqs. (56) and (73) across the

boundary between dissimilar media, one obtains the

boundary conditions

(J2 - Jx) • ft = 0

63

and (74)

($2 - • ft = 0 ,

where fi is the vector that is normal to the surface

defined by the intersection of media 1 and 2. The charge

current density "^charge 1S "efined as

J , = Q + 3P/3t . (75) charge true v J

Equations (57), (71), (73), and (75) can be combined to

yield a continuity equation

V ' ̂charge + 3ptotal/3t = 0 ' t76'

If it is assumed that the surface total charge density

vanishes, then the pillbox argument applied to Eq. (76)

yields

I (^charge) 2 " ^charge^l •" = <>• (")

It follows from Eqs. (71), (74), (75), and (77) that the

electric field satisfies the normal boundary condition

[ (9E/3t)2 - (3S/3t)1] • n = 0 .

If the usual "loop" argument (Panofsky and Phillips, 1962,

p. 32) is applied to Eqs. (55) and (72) and if it is assumed

that there is no surface total current density, one obtains

the tangential boundary conditions

64

($2 " x n = 0

and

(•^2 " ) x n = 0


The problem under consideration is the determination

of the electromagnetic field scattered by a plasma sphere

surrounded by a nonabsorbing medium. A plane harmonic wave,

Eq expfil^z - iwt) , with electric field linearly polarized

along the x axis and propagating in the z direction, is

incident on the sphere. The subscripts 1 and 2 refer to

quantities inside and outside the sphere, respectively.


-> ->•

The set of vector spherical harmonics {M ,N } 1 n' n

[Eqs. (12)] is not sufficient for an expansion of the

electric field in a medium that can support longitudinal

waves. The curl of Mn and Nn does not, in general,

vanish. However, the curl of the vector harmonics L , n '

where

^ = ™n •

and is given by Eq. (13), does vanish. Therefore,

these vector harmonics are suitable for an expansion of the

longitudinal electric field E^ .

The field E^ that satisfies Eqs. (66) and (68) in

the sphere 0 < r < a , is expanded in vector harmonics of

type 1 (harmonics of type 2 lead to singularities at r = 0):

1 = E E [f + g L^(kT)] , L n nL n on 1/ 6n en v 1/J

where E = E in(2n+l)/(n2+n). The transverse electric n o '

field Erp that satisfies Eqs. (65) and (67) in the sphere

is expanded:

L = E E [p ift^dr) - iw i^ f l )(k T)] . T n n L*n on v TJ n en ^ T^

The magnetic field in the sphere is given by

S1 - " C VW V W VS'(V + '

The incident and scattered fields (E.,H.) and (E ,H ) , XX 3 s

respectively, are expanded:

£. = EE [M^ (k9) - iN(1)(k9)] l n n on ^ 2' en v 2-'J

fii - + iRo" (k2^ •

and

K ' WV1™ Ck2> " ib„fien}J

K - "CV^WV^C1^ * IAN€ (K2>' '

At the boundary of the sphere (r = a), the fields must

satisfy the conditions

66

(E2 - £]_) x r = 0 ,

(fi2 - Sj) x r = 0 ,

and

( S 2 - Sp • ? = o

± r* where E2 = iL + Eg , H2 = + Hg , and E^ = E^ + E^ .

The preceding boundary conditions lead to six equations in

the six expansion coefficients that can be solved to obtain

the coefficients an and bn of the scattered field. The

results of this solution, the details of which are omitted,

are given in the following section.

Scattering Coefficients

The scattering coefficients for the plasma sphere

are

where

a = <V^UnYln - CV^XnY2n C78)

(k2/ii2)xnY4n " (kT/yl)UnY3n

b = (kTXnY2n " k2UnYlnul/y2^ " dnYln

n " CViVs^ " kTXnY4n) + dnY3n

dn = V2nkTn(n+l)(l/k2 - p^/y2kf)/V^a2 ,

and

67

Yln " V k2a)

Y2n = (k2a5"'[k2r3n(k2r']'r=a

Y3n " hnV (k2a>

Y4n = (Vl'^rf'tVU'r-a

Xn =

Un - fkTa)-'[Vjn(kTr']'r=a

Vln = »nCkLr)]'r=a

V2n = W.5 '

The prime indicates differentiation with respect to the

argument in parentheses.

The coefficient a is the same as that obtained in n

the ordinary Mie theory, and the coefficient bn is modi

fied by the quantity dn in the numerator and denominator

of Eq. (79). In the limit of infinite , the quantity

dn vanishes, and bn reduces to the expression given by

the ordinary Mie theory. Ruppin (1973) used the Lindhard

(1954) theory of the transverse and longitudinal dielectric

constants to obtain the extinction spectrum of small metal

lic spheres. Ruppin predicts the existence of additional

structure in the extinction spectrum for frequencies above

the plasma frequency.

APPENDIX A

NORMAL BOUNDARY CONDITIONS

There are two types of conditions that the electro

magnetic field must satisfy at every point on the boundary

between dissimilar media: the tangential boundary conditions

(E2 - E^) x fi = 0

(Al)

(fi2 - Sj) x ft = 0

and the normal boundary conditions

($2 - • n = 0

(A2)

(S2 - 61) • n = 0 ,

where n is the vector that is normal to the surface

determined by the intersection of media 1 and 2. In most

problems of light scattering by small particles, as well as

in problems of reflection and refraction at plane inter

faces, the tangential boundary conditions (Al) are applied;

and it is assumed that the solution so obtained is consis

tent with the normal boundary conditions (A2). However, the

problem of scattering by an optically active sphere is suf

ficiently different from the ordinary Mie theory to require

68

69

a determination whether the solution does satisfy the nor

mal boundary conditions.

In the form given above in (A2), the normal boundary

conditions are somewhat unwieldy to apply to the optically

active sphere. If harmonic time dependence, exp(-iwt), is

assumed, Maxwell's equations give

V x £ = iw§

(A3)

V x H = -iwS

The vector ft is fixed at a point on the boundary between

media 1 and 2. Therefore, the following identities are

obtained:

(V x E) • n = V • (E x fi)

( A 4 )

(V x ft) • ft = V • (H x n) .

If Eqs. (A3) and (A4) are substituted in Eqs. (A2), one

obtains

V • cs2t - Slt) - 0 (A5)

v • (i?2t - filt) = 0 ,

where the tangential fields are defined as

= S x fi

S t = f i x f i

70

"V -V Thus, continuity of the normal components of B and D is

equivalent to continuity of the divergence of the tangential

components of E and H . The solution of the problem of

scattering by an optically active sphere given in Chapter

III satisfies conditions (A5).

APPENDIX B

REFRACTIVE INDICES

In addition to the radius and refractive index of

the surrounding medium, four quantities are required in a

scattering calculation for an optically active sphere: the

real and imaginary parts of the refractive indices for left-

and right-circularly polarized waves. An equivalent set of

data is the refractive index difference An and the average

refractive index n , where

An = An' + iAn"

n = n' + in"

and

An' = - n£

An" = n£ - n£

n' = |(n£ + n£)

n" = i(nj; + nj-) . (Bl)

For a given optically active compound, it is difficult to

find measured values for all four quantities (Bl) in the

71

spectral region of interest (usually around 200 nm). The

approach of using theoretical expressions to obtain realis

tic and dispersively self-consistent refractive indices for

scattering calculations has therefore been adopted.

The Natanson-Drude equations (Lowry, 1964, p. 425;

Condon, 1937) may be written

K - "r = w(x2-xi>/[c**-^ + ria2]

(B2)

n" - n" = E.D.r.A2/[(X2 - A?) + r?A2] , L R x l l L ^ iJ l J '

where is the rotatory strength and is the width of

the ith optically active absorption band centered about the

wavelength A^ . The average refractive index may be

calculated from (Ziman, 1964, p. 228)

(n')2 - (iT")2 = k'

(B3)

2n'n" = k" ,

where the dielectric function is

k' = kc + EjFjA2CA2 - A?)/ [ (A2 - A?) + G?A2]

(B4)

k" = E.F.G.A3/[(A2 - A2) + G? A2] . J J J 3 J

F. is the strength, G. is the width, and addition of the J j

constant kc is an approximate means of accounting for

transitions that lie outside the spectral region under

consideration.

7 3

The refractive indices used for scattering calcu

lations in this dissertation, shown in Figs. (12-14), are

obtained from Eqs. (B2-B4); widths and strengths are chosen

so that the calculated real parts of n and An are

similar to those tabulated by Urry and Krivacic (1970) for

poly-L-glutamic acid (PGA).

7 4

A j ( n m )

0.016

- 0.007 - 0.009

190 208 224

Urry & Krivacic ( 1970 )

WAVELENGTH (nm )

o • -ft-220 \\ 230 190 250 200 240 210 180

Fig. 12. Calculated refractive index difference.

7 5

SINGLE ABSORPTION BAND

/ \ Al 190 nm

* G: 20nm F = 0.051 2.95

S Urry & Krivacic 0970)

Vh —T~ t— 230 240

i 250

i 1 180 190 200

i 210

—r— 220

WAVELENGTH Cnm)

Fig. 13. Calculated refractive

real part index.

of the average

7 6

0.12 •

0.10 • SINGLE ABSORPTION BAND

AT 190 nm

0 08 - G= 20 nm F = O.OSI

0.06 •

0.04 •

0.02 •

0.00 t t T T T T ISO 190 200 210 220 230 240

WAVELENGTH CnnO

Fig. 14. Calculated imaginary part of the average refractive index.

APPENDIX C

COMPUTER PROGRAMS

In Chapter III the analytical solution to the prob

lem of scattering by an optically active sphere is obtained.

The scattering parameters, such as cross sections, CD and

ORD, and amplitude transformation matrix elements, are

expressed as series expansions. In this appendix the com

puter programs that are used to calculate the coefficients

in these expansions and to sum the various series are

briefly described. These programs are based on the .program

of Dave (1968) for the standard Mie theory.

After considerable manipulation, the coefficients

an' ̂ n' an<* cn [Eqs. (19) 1 maT be written as follows:

b. n

Bn(L)Vn(R) + Vn(L)Bn(R)

Wn(L)Vn(R) + Vn(L)Wn(R)

7 7

7 8

where

A n (R ) = m 2 [D n (m R x )Re{^ n ( x ) } /m] + Re (T n ( x ) }

A (L ) = m 2 [D (m T x )Re{£ ( x ) } /m] + Re{T ( x ) } t1 11 li ii 11

B^(L ) = D n (m L x )Re{C r i ( x ) } /m + Re{T r i ( x ) }

Bn(R) = Dn(mRx)Re{?n(x) }/m + Re{Tn(x)}

W n (L ) = D n (m L x ) C n ( x ) /m + T n ( x )

wn(r) = dn^mrx) ?n(x)/m + tnw

?n(R) = m2[Dn(mRx) £n(x)/m] + Tn(x)

Vn(L) = m2[Dn(mLx) £n(x)/m] + Tn(x)

Tn(x) = n Cn(x)/x - •

The logarithmic derivative (Aden, 1951) is given by

Dn(z) = d In i|>n(z)/dz .

The coefficients a°(mL), a°(mR) , b°(m^), and b°(mR) i-n

the theory of Gordon and Holzwarth (see pages 33-34) are

given by

m l d n ^ m l x ^ + r e ^ t t , ( x ) ^ a°(mT) = nv W

n n

mT D (mT x) £ (x) + T (x) l n v l ^ n v ' n v j

a >R> "

mRDn(mRx)ReUn(x)} + Re{Tn(x)}

m R D n ( m R x ) S n C x ) + T n ( x '

7 9

b>L> "

b°(mr)

Dn(mLx)Re{^Ti (x) } + mT RetT^ (x) } n n

D (m,x)£ (x") + m. T (x) n L ;snv • L nv J

Dn(mRx)Re{^n(x)} + mRRe{Tn(x)}

. D n ( r n R x U n ( x ) + "W 0

The Riccati-Bessel functions £ satisfy the recursion n J

relation

5n(z) = (2n-l)5n-1 (z)/z - ?n_7(z) 'n-1 n- 2

where

(CI)

£Q(z) = sin(z) _ icos(z)

£_^(z) = cos(z) + isin(z) .

The logarithmic derivatives satisfy the recursion relation

Dn(z) = (n+1)/z - 1/ [ (n+1)/z + DR+1(z)] (C2)

The amplitude transformation matrix elements for

the optically active sphere [Eqs. (17)] are given by

S1(y) = -Sn(2n+1) (n2+n)"1 [anTn(y) + bnirn(y)]

S 2 ( y ) = -E n ( 2n + l ) (n 2+ n) - 1 [ a n TT n ( y ) + b^y ) ]

S3(^) = -Zn(2n+1) (n2+n)-1 c^ i r^y ) + T n ( y ) ] , (C3)

where y = cos 0 , and the functions -rrn and xR satisfy

the following recursion relations:

8 0

un = (2n-1) (n-1) ~1 - n(n-l)_17rn_2

(C4)

Tn = yC\i " V2> ' (2n-l)(l-y2)7rn_1 + Vz ,

where

7 T = 0 o

T =0 o

IT = 1

T j = y .

The functions irn and xn are alternately even and odd

functions of y :

%OvO = C-i)n+1^n(y)

Tn(-y) = (-l)nTnCy) .

Therefore, it is only necessary to compute irn and in for

positive values of y (angles 0 less than 90°) because the

matrix elements for the complementary angles 180° - 0 are

given by

Sl(-U) = EnC2n+l)(n2+n)-'(-l)n[anTnCv) - Vn(w)i

S2(-u) = £n(2n+l)Cn2+n)"lC-l)n[bnTn(u) - anirn(p)]

Sj(-u) = En(2ntl)(n2-m)-1(-l)ncn[Tn(u) - *n(iO] .

(CS)

CDXORD and CMIE

The listings of subroutine CMIE and its calling

program CDXORD are given on pages 85-89. CMIE computes

the coefficients an, bn, cn, a°(mL) , a°(mR), b°(mL),

b°(mR) and the following quantities:

QSCATL = QSJL = CRS)L/^A:

QSCATR - Qs>r - °SjR/ia:

QEXTL = QL = AL/IRA:

QEXTR = Qr = or/tra:

Eqs. (21)

Eqs. (22)

CD

ORD

0

CDSCA = 0,

QSCALG = QS L

QSCARG = QS>R

CDGH = 0

ORDGH = $

}

Eqs. (32)

Eq. (50)

All calculated according to

the method of Gordon and

Holzwarth (see pages 33-34),

CDSCAG = 0,

8 2

It is necessary to compute the logarithmic deriva

tive D by downward recursion because of round-off error n

accumulation for sufficiently large spheres (Kattawar and

Plass, 1967). At a value of n = NMX , where

NMX = 1.1|m|x + 1 ,

the logarithmic derivative is taken to be (Dave, 1968)

DNMX = + i0'° '

Starting with comPutec* an& stored using the

downward recursion relation (C2). The Riccatti-Bessel

function £ is computed from the upward recursion relation

(CI).

The various series expansions are considered to be

converged when the following criteria are satisfied:

|a |2 + |b |2 + 21c |2 < 10~14 i n 1 1 n1 1 n1

and

|§[a°0nL) + a°(mR)]|2 + l|[b°CmL) * b°(mR)]|2 < lO"" .

In the special case when m^ = mR , CMIE should

produce the results of the ordinary Mie theory. Therefore,

sample calculations for several different radii were made

in which the refractive indices were set equal to each

other. The resultant computed scattering and extinction

8 3

cross sections agreed with those tabulated by Wickramasinghe

(1973). As an additional test of the consistency of CMIE,

circular dichroism and optical rotation were calculated by

hand using Eq. (37) and the results were compared with

those computed in CMIE for small size parameters (10~2 -

10"4). Again, there was very good agreement between the

two results.

ANGLE and AMIE

The listings of subroutine AMIE and its calling

program ANGLE are given on pages 90-96. AMIE computes

the coefficients a , b , and c in a manner identical to n' n' n

that of CMIE. In addition, CMIE computes the amplitude

transformation matrix elements , S2 , [see Eqs. (C3)

and (C5)] and the following quantities:

S l l = s n

S14 - S14

S14REL = sn/s14

DIFSCA = Z s

The functions tt and t are calculated from the n n

recursion relations (C4).

Eqs. (33)

} Eq. (52)

8 4

As a test of the consistency of AMIE, the quantities

S-li , S14 , and were calculated by hand using Eqs. (38)

and the results were compared with those computed by AMIE

for small size parameters. There was very good agreement

between the two results.

PROGRAM CDXORD (INPUT,OUTPUT) DIMENSION WAVING(75),DRFR(2,75),DRFI(2»75)» SPF(75)»RFR(2),RFI(2)

10 FORMA T(3F10.M 20 FORMAT (6FIC06) 33 FORMAT (1H1^SCATTERING 3Y OPTICALLY ACTIVE SPHERE*/

11X , ̂OUTPUT REFRACTIVE INDICES AND SIZE PARAMETER** 2* ARE RELATIVE TO SURROUNDING MEDIUM*/ 31X,'WAVELENGTHS AND RADII IN MICRONS*)

U G FORMAT (1H0/1H0,*RADIUS =*,^1-3.6) 50 FORMAT <1X, 80 (1H* }) 63 FORMAT {lHu,*WAVtLENGTH =*, F10 «, ft , * SIZE PAPAMETER = *, F10.6, 1* SRF =*, F10.6 ?/ 21X,*LEFT INDEX (PEAL) =*,F13.6,* LEFT INDEX (IMAG) =*»F1G.6» 3* RIGHT INDEX (RFAL» =*,FI0.6,* &IGHT INDEX (IMAG)=*,F1C. U S )

70 FORMAT (1HC,*QSCATL =*,E13.6*6X,*QSCATR =*,E13.6, LIIX,*0SCALG=*,E13.6,6XT*QSCARG=*,E13.6,/ 21X,*QF.<TL = *,E1 3.6,6X,*QEXTR =*,E13.6,/ *1X1*C O =*,E13.O,9X,*CQGH = *,EL 3.6,1 OX,*PFRCFNT DIFF =*,£13.6*/ <+lX,*CRD =*,E13.6,BX,*DRDGH =*,E13.&, 1CX,*PERCENT DIFF =*,E13.6,/ 51X ,*CDSCA =*,E13,6,6X,*CDSCAC- = *, E13. 6 ,1 GX , *?ERCEN T DIFF =*, E13. 6)

C C RADII AND WAVELENGTHS IN MICRONS C

PEAO 1J, RAOL, RADINC, PR J = 1

100 READ 20 , WAVLNG(J),DRFR(1,J),DRFI(1,J)TQRFR(2,J),DRFI(2,J>,SRF(J) C C DRF(l) = LEFT HANDED, 0RF<2) = RIGHT HANDED C SRF=REFRACTIVE INDEX OF SURROUNDING MEDIUM C

IF (WAVLNG(J).EQ.O.) GO TO 110 J = J + 1 GO TO 100

110 N = J - 1 00 PRINT 30 W

RAO = RAOl PI = ACOS(-1.0) NR = PR

115 PKINT I+Q, RAD PRINT 53

120 00 13 0 L = i,N XO = 2.*PI*RAD/WAVLNG(L) X = SRF(L)*XO

C C x = SIZE PARAMETER OF SPHERE IN SURROUNOING MEDIUM C

00 125 K=1.2 RFR < K) = DRFR(KtL)/SRF(L)

123 RFHK) = DRFI(K,L)/SRF(L) C c RFR(K) = REAL PART 0^ REFRACTIVE INDEX REL. TO SURROUNDING MEDIUM C RFI(K) = IMAG. PART OF REFRACTIVE INDEX REL. TO SURRGUNDING MEDIUM C

CALL CMIE(X,RFR,RFI,QSCATL, QSCATR,QEXTL,QEXTR,RAD.CD.ORD,CDGH, 10RDGH,CnSCA,CDSCAG,QSCALG,QSCARG) PDIFF1 = ((CO-CDGHI/Gn)*10 0. PDIF^2 = { ( O^D-ORDGH) /ORD1*1CO . PDIFF3 = <(COSCA-COSCAG)/CDSCA)*100. PRINT 60, WAVLNG(L) ,X,SRF(L) ,RFR(1) ,RFIll) ,RFR(2>,RFI<2) PPINT 7:, QSCATL,QSCATR.QSCALG,QSCARG,QEXTL-»QEXTR,C0,CDGH,PDIFF1,

10RD*0RQGH,PDIFF2,CDSCA,CDSCAG,PDIFF3 130 CONTINUE

NR = NR - 1 IF (NR.LT.1J GO TO 1<»0 RAD = RAD + RAOINC GO TO 115

ItfO STOP END

oo a\

SUBROUTINE CHIE FX,RFR,9FT,QSCATL,OSC«TR,GEXTL.QEXTR,RAQ,CD,ORH, 1CDGH,ORDGH,CDSCA,CDSCAG,QSCALG,QSCARG) COMPLEX R(2),9F,AN,PN,CNI,NCAF>{2,3a00l , XI, X11, X 12 , X 110 , X12 0 , SUM <5,

1 RKF ( 2 ) , ACP<2> ,F3CP(2) TVCP(2) ,WCP(2) , TCD* D ENOM , SU M 2, S A N 0 ( 2 > , S BNG ( 2 > DIMENSION R'R'(2), ?FI(2)

15 FORMAT (1HC,*THE UPPER LIMIT FO^ OCAP IS TOO SHALL*) 0 0 1 6 1 = 1 , 2

16 R ( I) = CMPLX(F?FR(I) , RF I (I)) °F = (R(1)+R(2>)/2. . N!1X = 1. 1*CABS (RF > *X + 1. IF (NMX,LE.30G3I GO TO 130 print 1 '3 STOP

1J C IF (N M X . G T.151) GO TO 110 MHX = 151

11C N N = N M X - 1 ou 120 1=1,2 DCAP (I,NMX) = CMPLX<0.3,0.Q) RNF(I) = 1•/(R (I) * X) do 12g n=1,nn r;i = nhx-,m4-1 OCAP(I,MHX-M)=RN*RPF(I)-1./(DCAP<I,NMX-N+1)+RN*R°F(I))

120 CONTINUE XI10=CMPLX(SIN(X),-COS(X)) XI20 = CMT-»LX (COS ( X) , SIN(X ) ) XI1=XI10 xi2=xi 2 3 pn=1. N = 1 S U M1 = C . G SUM 2 = ONPLX(G.0,0.0) S U M 3 = 0 . 0 SUM«+=0. 0 SUM5=CMPLX(0.0,0.0) SUM6 = G • 0 SUM7 = 0.0

50 XI=<<2.*FN-i.)/X)*XIl-XI2 TCP=( f?M*XT)/X-XI1 no a:c 1=1,2 WCP(I)=(OCAP(I,N)*XI)/RF VCPtI )=(RF**2) *WCP(I) WQP(I)=WCP(I)«-TCP VC3(I I = V/r.P (I) + TCP RCP(T)=(DCA^(I,N)*REAL(XI))/PF ACP< I) = (BF**2) *BCPCI) BCPd ) = BCP ( I)*REAL (TCP) ACp(I)=ACP(I)+ RE AL(T C°) SANO(I)=P(I);r-r)CAP(I,M)*pEAL(XI)''-pEAL(TC?5) SA.'JO {I)=-SA.'Jo (I) / (R( I) *OCAP(I, N) *XI+TCP) S-3NG(I)=nCAP{I,N)*RFAL(XI)+R(I)*REAL(TCD) S3N j ( I) =-SBMG {I »/< OCAP (I, N ) *XI + t? (I) *TCP)

200 CONTINUE ntNOM=WGP(1) *VCP (2 ) <-VCP <1) *WOP (2) AM=- < WC? (i) *AC» (2) + WCP C 2) *ACP (i) ) PN = -( BCP (1) *VC° < 2) + VC° (1) *BCP< 2 ) ) CNl=RF*(PCP(2I*WCP<1)- 3CP(i)*MCP(2)) A N = A N / Q E N 0 M nn=bn/r)eh0m CNI=CNI/PFN D M SUM1 = SU;-I1 + REAL ((2.*RN+1.>*(-AN-3N)> SUM2=SUM2+(2.*RN*1.)*CNI TEST=(CAbS (AN )>**?+( CASS(3N))**2 + 2 . * ( C A 3 S ( C N i n * * 2

SUH3=SUM3*TEST* ( 2.*RN«-1. ) S'Ji1^= SUM ̂ + <2.*f?N + l . I *Rt"AL ( (CON JG< AN H-CON JG { 8N } ) »CNIJ SUM5=SUM5+( (2«*RN«-l»)/2» ) * (SANG (2) - SANG (1) + SBN *• (2) - SBN3 (1) ) SUMb=SU:t6M2.*RN*i .) * (CABS (3 AND (1) ) **2 4-CA3S (SBNO (1) )**2) SUM7 = SUM7«-(2.»RN«-i.)*(CA:3S(SANij (2) ) **2+CABS ( SBNG (2) )»»2) TESTGH = ( CABS ((SANG (1) + S A N 0 (2) ) / 2. > ) **2 TESTG H=TESTGH * ( CABS((SBNC(i)+SBMQ(2)>/2.))* *2 IF (TEST.LT.l.SE-iU .AND. TESTGH. LT. 1. G E-i<* > GO TO 205 XI2 = X II ' yn=xi c»

RN=RN+1 • N = RN GO TO 5 0

205 QFCF X TL =(2./X**2)*SUM1 GT XT'=QEXTL OUIFF = (<+./X**2)*REFLL(SUM2L NEXTL=QEXTL-GOI'R':" QFXTR^LXTR+QDIFF QSCATL=(Z./X**2!*SUM3 QSCAT^OSCATL NSNI(R = (T+./X**2) *SUMU RSCATL=QSCATI>QSDIF QSCATR=QSNATP-LSDIF F=3»/(2»*RAD*X**2) CD=-F*REAL(SUM?) ORD = F * AIMA G{SUM 2) CL)GH= (3 . *RE AL( SUM5) )7(«T .*RAD*X**2> ORDGH = -(3.*AIMAG(SUM5))/(*F.*RAD*X**2> QSCALG = (2./X»»£)»SUM6 OSCARG = <2./X**2) *SUM7 COSCA = (3./(L<S.*C?AD) ) * IQSCATL-QSCATR) COSCAG = (3./(16.*RAQ>1 *(QSCALG-OSCARG) pcturn E N 3

oo to

PROGRAM ANGLE <INPUT,OUTPUT) nIMENSI OH WAVLNGC7 5),DRFR(2,75),DRFI<2,75) , SRF<75>,RFR(2) ,RFI<2), 1 THETA(irO) COMPLEX S1<1QQ,2),S2?10u,2),S3<1QQ,2)

10 FORMAT (tFlJ.6) 15 FORMAT (15F5.1I 2G FORMAT (6FlD.fi) 30 FORMAT (1H1,*SCATTERING BY OPTICALLY ACTIVE SPHERE*/

11X,*0UTPUT REFRACTIVE INDICES AND SIZE PARAMETER*, 2* APE RELATIVE TO SURROUNDING MEDIUM-/ 71X ,*WAV'7 LENGTHS AND RADII IN MICRONS*)

kU FORMAT (1HQ/1H0 »*RADIUS = *,F10.fi) 50 FORMAT (IX, HG(1H*I) 60 FORMAT (1HC,*WAVELENGTH = #, FIG.6,* SIZE FARAMETER=*, F10.6,/

11X,*LEFT INDEX {REAL) =*,F1Q.6,* LEFT INDEX (IMAG) =*,F10*fi, 2* RIGHT INDEX {REAL} =*,F10.6»* RIGHT INDEX {IMAG)=*,F1C. 36)

70 FORMAT (1H0,*QSCATL=*,E13.6,fiX,*OSCATR=*,E13.&, / 11X » *QEX TL =*,E13.6» 5X,*QEXTR =*,E13.6,/ 21X,*CD =*,E13.o,/ 31X » *0 RO =*,E13.fi)

80 FORMAT (1X,*ANGLE = *,F«5.1,* SI1=*,E13.6,* S1U=*,E13.6, 1* S1WEL=*, E13.6,* DIFSCA=*,El3.fi)

C C RADII AND WAVELENGTHS IN MICRONS C

READ 10, RAOl, RADINC, PR, AMAX C C RADi=INITIAL RADIUS, RADINC=RADIU3 INCREMENT C PR=NUM3ER OF RADII AMAX=NUMBER OF ANGLES BETWEEN 0 AND 90 DEGREES C

JX = AM AX JJX=2 *JX

95 READ 15,(THETA<J),J=l,JX) C THETA(J)- ANGLES BETWEEN 0 AND 90 DEGREES

J = 1 133 PEAO 23, WAVLNG(J) ,0RFO<l,J) ,DBFI(1,J) ,DRFR(2,JJ ,0RFI(2,J} ,SR^(J>

C C OkF(l) = LEFT HANDED. DRF{2) = RIGHT HANDED C SRF=R£FRACTIVE INDEX OF SURROUNDING MEDIUM C

IF (WAVLNS(JI.EQ.Q.) GO TO 110 J = J + 1

GO TO ICC lie N = J - 1

PRINT 3 0 pad = rad1 PI = Acos(- i .a) NR = PR

115 PRINT uG, RAD DRI NT 52

120 DO 130 L = 1,N yq = 2. *PI*RAO/WAVLNG(L) X = SRF(L)*XO

C C X = SIZE PARAMETER OF SPHERE IN SURROUNDING MEDIUM C

GO 125 K = 1, 2 cPRtKJ = DRFR(<,L)/SPF(L)

125 RFU'O = DRFICK,L)/SRFfL)

C C RFR(K) = REAL PART OF REFRACTIVE INDEX REL. TO SURROUNDING MEDIUM C RFI(K) = I MAG. PART OF REFRACTIVE INDEX REL. TO SURROUNDING MEDIUM r

CALL AMIE(X,RFR,RFI,OSCATL,QSCATR,QEXTL*GEXTR,RAD,CD,0RD, 1 S1,S2,S3,JX.THETA5 PRINT 60, WAVLNG(L) ,X,RFR( 1) ,RFItl) ,RFR(2) ,RFI (2) PRINT 73, GSCATLtOSCATR,QEXTL,QEXTR,CD,ORD DO 175 J=1,JJX IP (J.GT.JX) THETA(J)=180. -THETA(JJX-J+l) IF (J.GT.JX) SKJ, 1)=S1 (JJX-J + 1,2)

IP (J.GT.JX) S2 < J, 1>=S2 (JJX-J+1,21 IF (J.GT.JY) S? (J, II=S3(JJX-J + 1,21 Sll = CARS(SHJtl) ) **2 + CABS<S2(J,l))**2 Sll=Sll/2. +CA3S(S3(J,1>>**2 Siu =AIMAG((Sl(J,l>+S2(J,i))*C0NJG(S3(JT 1) ) ) Si 4RE L= Sl^/Sll DIFSCA=J . 0 IF (OSCATL.tQ.QSCATR) GO TO 175 0IFSCA = 2.*S1<4/ (3.1^15q2&5*(X**2)*tQSCATl?-QSCATL> )

175 PRINT 8c, THETAIJJ,S11,Sl^,SlUREL,DIFSCA 133 CONTINUE

NR = M - I IF CNP.LT.l) GO TO 1^0 PAD = RAD + RAD INC GO TO 115

1*+G STOP r NO

SUBROUTINE AMIE(X,RFR,RFI,QSCATL,QSCATR,QEXTL,QCXTR,RAO,CO,ORO, 1 S1,S2,33,JX,THETA> COMPLcX R(2),PF,AN,3N,CNI,OCAP(2,30G0),XI,XIl,XI2,XI10,XI2a,

1 P R F {2 ) , A CP (2), TCP (2) ,VCP(2J ,WCP(2) » TCP ,0 ENOM, SU M 2 COMPLEX CNfAI,Sl(lQG.2>,S2(lCC»2),S3(lCa,2) 01 MENSION PFR(2)tRFI(2>,THETA<10C)»A^lj<10Q), PI(lu0),TAU(lOG),

1 PIK133 ) ,PI?< 1GC) ,TAU1 (ICC),TAU?(1C0) 15 FURHAT (lhG,*THE UPPER LIMIT FOR OCAP IS TOO SMALL*)

AI = CMnL X ( C . 3 , 1. C- ) no 16 1=1,2

16 P(I) = C-/PLXlRFRd) .RFI (I) ) RF = ir, ti> + R(2) ) /2. NMX-1ol*CABS(RF)*X + 1. IF (NMX.LE.3CGC) GO TO 1GC PRINT 15 STOP

IOC IF (NMX.GT.151H GO TO 110 NHX = 151

110 NN=NMX-1 do 12 c 1 = 1,2 OCAP (I,NMX) = CMPLX(0.0,0.0) RPP(I) = l./(R(I)*X) OO 12C N=1,NN PM=NMX-N+1 nCAP(I,NMX-N)=RN*RRF(I)-l«/(DCAP(I,NMX-N+l)+RN+RPF(I))

12G CONTINUE no 125 J=1,JX ANGRA0=(3.1iil59265*THFTA (J))/18 0.Q

125 AMU(J)=CCS(ANGRAO) XI13=CMPLX(SIN/X),-COS(XI) VI20=CMPLX(COS(X),SIN(X))

XIl=XIlu XI 2 = XI2Q no 135 J=i,JX pii(j)=0 . 0 PI2(J)=0.0

TAU1< J)=C.0 T AU2(J)=C • G PI (J > = 1. C

135 TAU(JI=4MU(J» RN=1. n = 1 SUM 1=3.0 SUM2 = CMPLX(G.0,0.0) SUM 3=G. 0 SUMU=C . C no 13 G L = l,2 DO 13G J = 1»JX SKJ,L> = CM-°LX(G.0,0.3) S2(J,L)=S1(J,L)

133 s3(J, L>=S2(J,L> 50 XI={(2.*PN-1.I/XJ*XI1-XIZ

TCP=(PN*XI)/X-XI1 DO 2Li 3 1 = 1,2 W C P ( I ) = (DGAc,{I,N)*XI)/RF VCP(I) = (RF ̂^2)»WCP(I) WCP(I) =WCP (I) «-TCP VC-P (I )=VCP(I)+TCP 3CP(I> = (nCA°<I,N)* SEAL (XI) ) / RF A C P (I ) = ( RF * * 2 J *<3CP (I ) 3CP (T) =5CP ( + AL (TCP) AC0 (I ) = A CP ( I ) ̂EAL (TCP)

2 00 CONTINUE nENOr1='.-iCP (1) *VCP (2 ) +VCP(1)*WCP<2> AM = - < WCP (1) • fiCP I ? i + WC°< 2 ) * ACP < in P.N=-( BCP (1) *VCP ( 2? f-'/CP (1) *8CP (2 n CNI = RF*(BCp(2)* WC°(1)- BCP C1)*WCP{2)) AH= AN/OFNO M BN=QN/DENOM CNI=CNI/DENOM CN=-AT*CNI Rl=(2.* RN+1•>/(RN*(RN+1.))

p2=f?l* c-l. ) n o 3 1 5 J = 1 , J X

S 1 ( J , 1 ) = S I ( J , 1 ) - R i * ( A N * T A I J ( J ) + B N J * P I ( J > )

SI ( J, 2) =S1 (J,2)-R2*< A N!»Tflu { J) -BN*PI ( J) ) S (£: < J * 1) = S 2 (J ? 1) -Rl*(ANJ,lPI (J) + 8<N*TAU fj) ) S2(J,2)=S2(J,2> -R2* <3M*TAU(J)-AM*PI<J)) S 3 ( J » l ) = S 3 ( J , i » + « l*CN * ( P I ( J ) + T f t U ( J ) l

315 S3(J,2)=S3(J,£)+R2*CN*(TAU<J>-^1(J)) S U M 1 = S U ;-11 * 0 E A L ( (?.. )* (-AN-4M) )

SUM?=SU'124- ( 2.*RN+1.) *CNI T F S T = < C A 3 5 ( A N > > » * ? + < C A ^ S C i M > > * * 2 * 2 . * ( C A 1 S < C N I > » * * 2

S U M o = S' . J H 3 + T E S T ' ( 2 . * R N + 1 . >

S U M h = S U ^ ' . H + ( ? . * R N + 1 . ) * R E A L ( ( O O N J G < A N ) + C O N J G < B N ) > * C N I )

I F (TEST. L T . l . O F - i a ) G O T O 2 0 5 X I 2 = X I I

X 1 1 = X I

°m-?ii + l. N = RN t1 = 2.*pn-1. T 2 = ; < f J - i .

HO 1S5 J=i,JX p 12 ( J > = p 11 ( J) PI1(J)=PI(J» T A U 2 ( J ) = T A U 1 ( J ) T A U 1 ( J ) = T A U ( J ) P I < J > = { T 1 / T 2 ) * A M U ( J ) * P T 1 ( J ) - ( R N / T 2 ) * P I 2 ( J )

18 5 TAU{J) = AMU(J)*(PI( J)-PI2( J ) } - T l*(l.-A,-IU( J)**2>*DIl{ J)+TAU2< J > G O T Q 5 3

2 3 5 Q r X T L = { ? . / X * * 2 > * S U M 1

O t X T ^ s Q F X T L

O D I F F = {< + , / X * * 2 > * R E A L ( S U M 2 )

C t X T L = ^ E X T L - 3 n i F F

Q c X T R = Q E X T R + Q D I F F

Q S C A T L = ( 2 . / X * * 2 ) • S U M S

C S C A r * = Q S C A T L

O S Q I F = { ^ . / X * * 2 > * S U M * +

QSCATL=OSCATL*QSDIF QSCATS=OSCATR-QSDIF F = 3 . /(2.*^AQ*X**2) C D = - F * ? E A l _ (SUM2I nRD=F*AIMA&(SU'"12l pttu^m end

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