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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, 1983]. We will treat those topics that are pertinent to later chapters of multiple scattering by random discrete scatterers.
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Basic Scattering Parameters
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1.1 Scattering Amplitudes and Cross Sections Consider an electromagnetic plane wave impinging upon a particle which has permittivity Ep(r) that is different from the background permittivity (Fig. 1.1.1). The finite support of Ep(r) - E is denoted as V. The incident wave is in direction k: i and has electric field in direction ei that is perpendicular to ki . The electric field of the incident wave is (1.1.1) where
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is the position vector, and
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wJIiE =21r oX
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is the wavenumber. In (1.1.3), w is the angular frequency, /1 is the permeability, and oX is the wavelength. In (1.1.1), Eo is the amplitude of the electric field. The time harmonic dependence exp( -iwt) has been suppressed. In the far field, the scattered field is that of a spherical wave with dependence eikr Ir, where r is the distance from the particle. In general, the particle scatters waves in all directions. Let E s be the far field scattered field in direction of k: s . Since Maxwell's equations are linear, we write
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E s = esf(k s, ki)Eo-
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where es is perpendicular to k s . The proportionality f(k s , ki ) is called the scattering amplitude from direction ki into direction ks
1.1 Scattering Amplitudes and Cross Sections
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Figure 1.1.1 Scattering of a plane electromagnetic wave Ei(f) by a particle occupying volume V and having permittivity Ep(r). The scattered field is Es(r).
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The magnetic field associated with the incident wave is 1, Hi = -ki X E i
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where 'fJ = is the wave impedance. The Poynting vector denoting power flow per unit area is
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1 IEo l , S =-Re (E xH =--k 2 21]' Similarly, for the scattered wave, the magnetic field is 1, Hs=-ksxE s 1]
(1.1.9) Consider a differential solid angle dO s in the scattered direction (Fig. 1.1.2). In the spherical coordinate system
dO s = sin Os dOs d<ps
At a distance r, the surface area subtended by the differential solid angle dO s is (1.1.11)
r sin O.d</>.
0 ...... ... cPs
... ...........
Figure 1.1.2 Differential solid angle dO s = sin Os dOs dcPs in spherical coordinates r, Os, and </>s.
Then the differential scattered power dPs through dA is
dPs =
ISsl dA = ISsl r 2 dO s
Putting (1.1.9) in (1.1.12) gives
dPs = II(ks, ki)1
21 E ol 2
2;] dns
Using the Poynting vector of the incident wave, from (1.1.6), we have (1.1.14) The dimension of equation (1.1.14) is area. It is convenient to define a differential scattering cross section (Td(ks);i) by
ISil = (Td(k s, ki ) dOs
Comparing (1.1.14) and (1.1.15) gives
(Td(k s, ki ) = II(ks, k )1 2 i
Integrating (1.1.14) over scattered angle gives
dnsII(k s, ki )1
1.1 Scattering Amplitudes and Cross Sections
Thus the scattered power is
Ps = O"slSil where o"s is the scattering cross section which is
dOslf(ks, ki )1
dOsO"d(ks, ki)
Scattering Cross Section and Geometric Cross Section
The geometric cross section 0"9 of a particle is its area projected onto a plane that is perpendicular to the direction of incident wave lei' Thus the power "intercepted" by the particle, Pr , from a geometric optics standpoint, is the product of the geometric cross section and the magnitude of the incident Poynting vector:
We can compare O"g to O"s and Pr to Ps . Let D be the size of the object (the maximum distance between two points inside the object). When the size of the object D is much less than the wavelength, the results of Rayleigh scattering theory indicate that
O"s = O"g
0(D4) >.4
where 0 denotes the order of magnitude. Thus
P 14 P = 0 (D
in Rayleigh scattering. This shows that when the particle is small compared with the wavelength, the power scattered by the particle is much less than the product of geometric cross section and incident Poynting vector. In the short wavelength limit, D >.. Then
as = 0(1) (1.1.23) O"g which is known as the geometric optics limit. It is important to remember that the scattering cross section 0"s also depends on the contrast between Ep and E. In the case of weak scatterers when Ep ~ E, we have
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