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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
3. Calculated CD for a 0.03 ym radius sphere embedded in a medium with refractive index 1.2 38
4. Calculated ORD for a 0.03 ym radius sphere embedded in a medium with refractive index 1.2 39
5. Calculated CD for a 0.10 ym radius sphere embedded in a medium with refractive index 1.4 40
6. Calculated ORD for a 0.10 ym radius sphere embedded in a medium with refractive index 1.4 41
7. Calculated CD for a 0.50 ym radius sphere embedded in a medium with refractive index 1.4 42
8. Calculated ORD for a 0.50 ym radius sphere embedded in a medium with refractive index 1.4 43
9. Angular dependence of the scattering contribution to circular dichroism for an optically active sphere with radius 0.03 ym . . 48
10. Angular dependence of the scattering contribution to circular dichroism for an optically active sphere with radius 0.10 ym . . 49
vii
viii
LIST OF ILLUSTRATIONS--Continued
Figure Page
11. Angular dependence of the scattering contribution to circular dichroism for an optically active sphere with radius 0.50 ym . . 50
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)
refractive indices for left- and right-circularly polarized
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
refractive indices for left- and right-circularly polarized
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.
The refractive indices used in this calculation are given in Appendix B.
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
Fig. 3. Calculated CD for a 0.03 ym radius sphere embedded in a medium with refractive index 1.2.
The refractive indices used in this calculation are given in Appendix B.
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.
The refractive indices used in this calculation are given in Appendix B.
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
Fig. 5. Calculated CD for a 0.10 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 ? 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.
The refractive indices used in this calculation are given in Appendix B.
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.
The refractive indices used in this calculation are given in Appendix B.
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.
The refractive indices used in this calculation are given in Appendix B.
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).
Solution of the Problem
Expansion in Vector Spherical Harmonics
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
^III " EII-) x r = 0
and |r| = b
C^ni - ? " 0
where = ̂ + anc* ^III = ^i + ^s " "^ie
inside the spherical shell, and , are obtained
from the transformation equation (6) and Eqs. (52). At
r = a the fields must satisfy the conditions
(^il " Sj) x r = 0
and |r| = a
(I^II - 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
Solution of the Problem
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.
Expansion in Vector Spherical Harmonics
-> ->•
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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