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Basic Scattering Parameters Scattering Amplitudes and Cross Sections Scattering Amplitude Matrix Rayleigh Scattering Rayleigh Scattering by a Small Particle Rayleigh Scattering by a Sphe

Basic Scattering Parameters

Scattering Amplitudes and Cross Sections

Scattering Amplitude Matrix

Rayleigh Scattering

Rayleigh Scattering by a Small Particle

Rayleigh Scattering by a Sphere

Rayleigh Scattering by an Ellipsoid

Scattering by a Disk Based on the Infinite Disk Approximation 46

41

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Scattering of Electromagnetic Waves: Theories and Applications

Leung Tsang, Jin Au Kong, Kung-Hau Ding Copyright  2000 John Wiley & Sons, Inc ISBNs: 0-471-38799-1 (Hardback); 0-471-22428-6 (Electronic)

2 1 ELECTROMAGNETIC SCATTEZUNG BY SINGLE PARTICLE

A major topic in this book is the study of propagation and scattering

of waves by randomly distributed particles We first consider scattering by a single particle This chapter and the next discuss and derive the scattering characteristics of a single particle Both exact and solutions are studied Scattering by a single particle is an important subject in electromagnetics and optics There exist several excellent textbooks on this subject [van de Hulst, 1957; Kerker, 1969; Bohren and Huffman, 19831 We will treat those topics that are pertinent to later chapters of multiple scattering by random discrete scatterers

1.1 Scattering Amplitudes and Cross Sections

Consider an electromagnetic plane wave impinging upon a particle which has permittivity eP(?;) that is different from the background permittivity c (Fig 1.1.1) The finite support of Q(?;) - E is denoted as V

The incident wave is in direction & and has electric field in direction & that is perpendicular to & The electric field of the incident wave is

(1.1.1) where

In the far field, the scattered field is that of a spherical wave with de- pendence eik’ /T, where r is the distance from the particle In general, the particle scatters waves in all directions Let E, be the far field scattered field

in direction of kS Since Maxwell’s equations are linear, we write

E s = 2?,f(&, ki)Eoe”‘” (1.1.4)

r where & is perpendicular to iS The proportionality f &, &) is called the scattering amplitude from direction & into direction &

5 1.1 Scattering Amplitudes and Cross Sections 3

The magnetic field associated with the incident wave is

whereq=d p E is t e wave impedance /-’ h The Poynting vector denoting power flow per unit area is

1 ELECTROMAGNETIC SCATTERING BY SINGLE PARTICLE

and &

Then the differential scattered power dP, through dA is

dP S = I$1 dA = lSsl r2dS1, (1.1.12) Putting (1.1.9) in (1.1.12) gives

dPs = If(i&, &),2y dR, (1.1.13) Using the Poynting vector of the incident wave, from (1.1.6), we have

dP

i The dimension of equation (1.1.14) is area It is convenient to define a dif- ferential scattering cross section a&&, &) by

ad(L, b) = If (b, &)I2 (1.1.16) Integrating (1.1.14) over scattered angle gives

P S

- =

II - S i J d&If (s, &)I2 (1.1.17)

$1.1 Scattering Amplitudes and Cross Sections 5

Thus the scattered power

-

P s= o PI s i where us is the scattering cross section which is

(1.1.18)

(1.1.19)

The geometric cross section CT~ of a particle is its area projected onto a plane that is perpendicular to the direction of incident wave & Thus the power

“intercepted” by the particle, pT, from a geometric optics standpoint, is the product of the geometric cross section and the magnitude of the incident Poynting vector:

OS -E

which is known as the geometric optics limit It is important to remember that the scattering cross section us also depends on the contrast between Q and e In the case of weak scatterers when Q, N e, we have

6 I ELEC’I’ROMAGNETIC SCATTERING BY SINGLE PARTICLE

The particle can also absorb energy from the incoming electromagnetic wave Let

fz (4 P- = c;(T) + i(p) (1.1.25) From Ohm’s law, the power absorbed

(1.1.26) where Eint(r) denotes internal field which is the electric field inside the particle, di;; = dx dy dz, and the integration in (1.1.26) is over the three- dimensional volume of the particle The absorption cross section oa is defined

and the albedo of the particle is

(1.1.29) Thus 0 < G < 1 The albedo is a measure of the fraction - - of scattering cross section in the total cross section

We next generalize the concept of scattering amplitudes to include polariza- tion effects For the incident wave, the electric field Ei is pe.rpendicular to the direction of propagation & There are two linearly independent vectors that are perpendicular to & Let us call them & and & Then

- E i= ( &Eai + &Ebi > - eiki-5: (1.1.30) where Ei = kki The directions of &, iti, and & are such that they are orthonormal unit vectors following the right-hand rule

Similarly for the scattered wave, let k,, G,, and 6, form an orthonormal system Then

e ikr

E s = itsEas + b,gEbs -

$1.2 Scattering Amplitude Matrix 7

The scattered field components, EaS and I&, are linearly related to I& and I& The relation can be conveniently represented by a 2 x 2 scattering amplitude matrix

(1.1.32)

There are two common choices of the orthonormal unit systems (;ii, &, R,> and (&, &,, is) that describe scattering by a particle

Let the angle between & and ks be 0 (Fig 1.1.3) h h

incident direction ki and the scattered direction Ic,

b i= 2 = ki x ii i

The plane containing the

is known as the scattering

plane and let

(1.1.33)

(1.1.34)

(1.1.35)

I ELECTROMAGNETIC SCATTEH-NG BY SINGLE PARTK’LE

2 i = 5!? and the incident wave is 2i polarized The inconvenience is that Ei

is 2 polarized and propagation in 2 direction, and yet it is & or 2i polarized h depending on whether the scattered direction ks is 5 or 5%

Some useful relations for this orthonormal system are

In many problems there is a preferred direction, for example the vertical direction that is labeled 2 In geophysical probing and earth remote sensing problems, that will be the vertical axis which is perpendicular to the sur- face of the earth Then we can have vertical polarization Ci and horizontal polarization iLi that form an orthonormal system with ii We choose

52 Rayleigkz Scattering

X

terized by the angles & and q& in spherical coordinates (Fig 1.1.4), then

2.1 Rayleigh Scattering by a Small Particle

In Rayleigh scattering, the particle size D is much less than wavelength A

In such a case an oscillatory dipole with moment p is induced inside the particle The field radiated by the dipole is the scattered field

The far field radiated by a dipole 1, in the direction kS is

10 1 ELECTROMAGNETIC SCATTERING BY SINGLE PARTICLE

Let the electric field inside the particle be denoted by ant, where the sub- script int

2.2 Rayleigh Scattering by a Sphere

Consider a sphere of radius a < X centered at the origin Then the internal field inside the particle is

52.2 Rayleigh Scattering by a Sphere 11

We can calculate the scattering amplitude matrix using the scattering plane orthonormal system

First let Eli = 1 and J!& = 0, thus Ei = ii Then El, = fll and

E2s = f21 From (1.2.9a)-( 1.2.9b), we obtain El, = fO( i, l ii) = f0 and Ezs = f&& &) = 0 Hence fll = f0 and f21 = 0 Next, let Eli = 0 and

E2i = 1 so that Ei = 2i Then El, = fr2 and Egs = f22 From (1.2.9a)- (1.2.9b), El, = fo(is l g> = 0 and Ezs = fO( 2S 2i) = f cos 0 Therefore

f12 = 0 and f22 = f co& Thus the scattering amplitude matrix assumes following simple form in the scattering plane system of coordinates:

f vv = f( 0 vs h l q = fo [cos 0, cos & cos(& - q&) +sin&sin&] (1.2.12~)

To get fvh and fhh, we let Evi = 0 and Ehi = 1 so that Ei = & Then from (1.2.9u)-(1.2.9b), Evs = fo(Cs hi) and Ehs = fo(ks ii) Thus,

To calculate the scattering cross section, suppose that the incident wave i.s horizontally polarized The scattered power is determined by integration over all scattered angles of the sum of the scattered vertical and horizontal polarizations:

12 1 ELECTROMAGNETIC SCATTERING BY SINGLE PARTICLE

If we compare the scattering cross section to the geometric - cross section

= mu2 of a sphere, we obtain

The power absorbed by a small particle is

The absorption cross section is

P (Ta = -

Thus the relative magnitude of CT~ to oa depends on the relative values of

2.3 Rayleigh Scattering by an Ellipsoid

Let a, b, and c be the half axes length of the ellipsoid respectively in $b, ‘&,, and zb directions where a, b, and c are all much less than A Then the internal electric field Eint induced by the incident electric field Ei is

where

(1.2.21)

5 2.3 Rayleigh Scattering by an EIlipsoid

14 1 ELECTROMAGNETIC SCATTERING BY SINGLE PARTICLE

In view of equation (1.2.20), it is convenient to define the scattering dyad

F - k2v,(E, - E) - ii?&) %Yb h +,&,

47E 1+ v&a + 1+ V&b + 1+ vd& > (1.2.35) From (1.2.4) and (1.2.20), we have

(1.2.40a)

15

a,h(L h) = 4~lfvh12

~hv(L k) = 4~lfhv12 ahh(L, &> = 4Qhh12

In the backscattering direction (monostatic radar)

(1.2.40b) (1.2.40~) (1.2.40d)

fhh = ii, l fo? + ii = -f

(1.2.46~) (1.2.46d) Thus

(1.2.47)

(1.2.48)

16 1 ELECTROMAGNETIC SCATTERING BY SINGLE PARTICLE

Thus a characteristic of backscattering from a sphere is that gVV = ahh and there is no cross-polarization

If the incident wave is right-hand circularly polarized, then Ei = Gi + iiLi

so that & = 1 and Ehi = i Substituting into (1.2.47) gives & = f0 and Ehs = -if0 so that E, = f& - iiL,) The scattered wave is left-hand circularly polarized

imat ion

3.1 Integral Expression for Scattering Amplitude

In this section, we derive the integral expression for the exact scattering amplitude

Recall from equation (1.2.4) that the far-field scattered field from a small particle is

$3.1 Integral E x p ression for Scattering Amplitude 17

the far-field scattered field cE, from dv’ is

where R = 1’; - ~‘1 is the distance between F’ and observation point E If the observation point is far away, then

We approximate the phase term in (1.3.2) by

and the amplitude by

is dependent on the coherent wave interaction among different parts of the particle

18 1 ELECTROMAGNETIC SCATTERING BY SINGLE PARTICLE

Instead of using the internal electric field, one can use the surface tan- gential electric and magnetic field Given tangential electric and magnetic fields, the far-field radiation field is

In (1.3.11), we have extended the integration limits over all space because outside the particle eP(F’) = e and the integrand is zero Let

00 S(&, ii) = -L

VO Ill dz’ dy’ dz’ (ET(+) - l)eircAar’

(1.3.13)

(1.3.14)

However, in order to reconstruct E,(?) - 1 by taking the inverse Fourier transform of S( k,, ii>, we need, as from (1.3.16), S(R,&) for all values of kdx, kdy, kda, that is from -oo to 00 However, since Ed = k&i - kks, we have max I& 1 = 2k Thus we can know S only within the spherical volume

of k-space of radius 2k The reconstructed E@) - 1 will be limited to a low-pass version, i.e., to spatial frequencies < 2k

If 0 is the angle between ki and k,, then

kd = l&l = 2ksin ; (1.3.17) The maximum kd is at 0 = 180”, backscattering, when kd = 2k = y and the minimum is at 0 = O”, forward scattering, when kd = 0

The polarization of the scattered field is ks x (k, x &) and depends only

on scattered direction and incident polarization & Thus the polarization

20 1 ELECTROMAGNETIC SCATTERING BY SINGLE PAKf-TCLE

dependence contained in Born approximations is “uninteresting” because it does not depend on the properties of the particle

Example:

For a sphere of radius a and homogeneous permittivity Q, we have

To perform the integration, we choose Zd to be along the 2’ axis without loss of generality, so that Ed = k&Z’ Then exp(& l F’) = exp(ik& cos 0’) Performing the integration in spherical coordinates gives

3( 6 -

d

(1.3.18)

S(L,, &) has a maximum at 0 = 0 that is kd = 0 The maximum is equal

to (Er - 1) It then oscillates with a decreasing envelope with increasing kda The angular spread of scattering A@ is such that

A(k&=A(%zsin%) =r The angular spread indicates that the Born approximation predicts a strong forward scattering with most of the scattered power within angular spread A@ For large ka, we have

x

If we use the scattering plane coordinate system to represent the scattering amplitude matrix and note that ks x (k, x l?i) = -is and ks x (& x Q = -2s co&, then using (1.3.13) we obtain

$4 Plane Waves, Cylindrical Waves and Spherical Waves

where

3

3

and S(& &) is given by (1.3.18)

21

(1.3.22~)

(1.3.223)

The scalar wave equation is

4.1 Cartesian

The scalar plane

Coordinates: Plane Waves wave solution is

$,(T) = A$*T where

is tne propagation vector Substitution of (1.4.3) in (1.4.1) gives

k; + ky” + k; = k2 For vector electromagnetic waves, let

E = &iE*T Substitution of equation (1.4.6) in (1.4.2) gives

-E x (x x To) - k2i!& = 0 Dot multiplying (1.4.7) with Ic gives

L&=0 From (1.4.7) and (1.4.8), we have

(x*x)&-k2&=0 which yields Eq (1.4.5)

22 I ELECTROMAGNETIC SCATl-‘ER.lNG BY SINGLE PARTICLE

The solution in cylindrical coordinates (p, 4, z) that is regular at the origin

is

Rg$,(kP, kz, r) = Jn(k,p)eikzZSin4 (1.4.12) with n as an integer, that is, n = 0, fl, f2, f3, In (1.4.12), Jn is the Bessel function of order n and Rg stands for regular The outgoing cylindrical wave is

$n(kp, k,,F) = Hi” (kpp)eikzz+in4 where Hn (‘) is the Hankel function of the first kind

The behavior of the functions as k,p -+ 0 is

H$l)( kPp) N 2 ln(k,p) + 1

7r and for n > 0

eik,x+ilc,y = e’lC,pCOS(~-~k) = x Jn(kpP)e in((b-&&-inr/2 (1.4.27)

24 1 ELECTROMAGNETIC SCATTERING BY SINGLE PAKf’ICLE

-

= ilc X 2e iii s; _ - ilcpii(t9k, q5k)eikV (1.4.29) where iC is the horizontal polarization vector Hence, we have expressed the vector electromagnetic plane wave in terms of vector cylindrical waves:

(1.4.30) Taking a curl of both sides next gives

ik O”

ix X ii@&, Cj+#‘* = r 7: RgXn(kp, kZ,!?)i”e-in’k (1.4.31)

P n= Then

1

$(ek,$k)$’ = -k 7: RgIVn(k,, k,, S;)ineSin4” (1.4.32)

P n=-m where 6 is the vertical polarization vector Thus for a general vector plane wave, the transformation to vector cylindrical waves is

Ed = (6i Evi + KQ Ehi) eixi*’

In spherical coordinates, the scalar wave equation is

[$g (r2j$)+ &g (sineg)+ r2s;n2e$j + k2]+(F)zo

(1.4.34)

It has the outgoing wave solution

+mn (kr, 8,+) = h~‘)(kr)P~(cosB)eim4 (1.4.35) with 72 = 0, 1,2, ., m = 0, &l, , &n In (1.4.35) hi’) is a spherical Hankel function of the first kind and Pr is the associated Legendre function We also define regular wave functions by

Rg$mn(kr, 0, qb) = j,(kr)Pp(cos B)eim4 (1.4.36) where Rg stands for regular and jn is the spherical Bessel function The prefix regular is used to denote the fact that h$&‘) is replaced by jn which is

n

negative values of m The relation between positive and negative values of

(1.4.41a)

(1.4.41b)

(1.4.41c) (1.4.41d) (1.4.41e)

(1.4.42a)

(1.4.423)

(1.4.42~)

26 1

For spherical Hankel

$kr

The spherical harmonic is defined as

with orthogonality relation

The completeness relation is

g f: (-l)m2Z1 -q-“(O’, qs)y,m(o, qi) = S($ - qs)6(cos 8 - cos 0’)

(1.4.47) The expansion of plane waves in spherical waves is given by

eiK*’ = F h (-1)“i1(21 + l)$(kr)q”(@, +)q-“(&, &) (1.4.48)

where (ok, +k) denote the angular variables describing the direction of z