Final Exciting Field and Multiple Scattering Equation in .NET

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2.1 Final Exciting Field and Multiple Scattering Equation
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'l/Jinc
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Figure 7.2.1 An incident field
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'l/Jinc("F)
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impinges on a collection of N particles in volume
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and includes near-field and far-field effects. Thus given a single particle j and a field 'l/Jf impinging upon a particle j, the scattered field from particle j is
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(7.2.1)
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where the superscript E denotes exciting field and Go is the Green's function (propagator) . Next consider two particles j and l with the incident field 'l/Jinc upon them. The total field includes
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(7.2.2)
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where GOTPPinc and GOTj'l/Jinc are the scattered fields from particle land particle j, respectively, that "directly" scatters 'l/Jinc' However, there can be second-order scattering which are GoTzGoTj'lj;inc and GoTjGoTl'l/Jinc. The second-order field GoTtGOTj'l/Jinc scatters the incident field from j to l which further scatters the field. Similarly, GoTjGoTl'l/Jinc consists of scattering first from particle I and then from particle j. Thus we have
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'l/J = 'l/Jinc+GoTt'l/Jinc+GoTj'l/Jinc+GoTtGoTj'l/Jinc+GoTjGoTl'l/Jinc+' .. (7.2.3)
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We can keep repeating for the third order, the fourth order and so on, up to an infinite number of terms
'Ij;
= 'lj;inc + GOTz'lj;inc + GOTj'lj;inc + GOT1GOTj'lj;inc + GoTjGoT1'lj;inc
+ GoTtGoTjGoTt'l/Jinc + GoTjGoT1GoTj'l/Jinc + Go'TzGoTjGoT1GoTj'lj;inc + GoTjGo'TzGoTjGoTz'lj;inc + ...
(7.2.4)
7 VOLUME SCATTERING SIMULATIONS
The infinite series in (7.2.4) can be rearranged as follows
= 'l/Jinc + GoTz (1/Jinc + GOTj'l/Jinc + GoTjGoTz'l/Jinc
+ GoTj GoTzG oTj1/Jinc
+ ...) + GoTj (1/Jinc + GOTz'l/Jinc
(7.2.5)
+ GoTzGoTj'l/Jinc + GoTzGoTjGoTZ~)inc + ...)
The first sum in (7.2.5) went through particle l as the last particle, and the second sum in (7.2.5) went through particle j as the last particle. We define the exciting field 1/Jf to be the sum in the first parentheses and define the exciting field 1/Jf to be the sum in the second parenthesis. Thus
'l/Jf
1/Jinc + GOTz'l/Jinc + GoTzGoTj'l/Jinc + GoTzGoTjGoTz'l/Jinc + GoTzGoTjGoTzGoTj1/Jinc + ... (7.2.6) 1/Jf = 'l/Jinc + GOTj'l/Jinc + GoTjGoTZ'l/Jinc + GoTjGoTzGoTj'l/Jinc + GoTjGoTzGoTjGoTz1/Jinc + ... (7.2.7)
We also let
1/Jl = GoTz~)f 'l/Jj = GoTj'l/Jf
be the scattered fields from particles I and j, respectively. Then,
(7.2.8a) (7.2.8b)
'l/J
We also have
1/Jinc + 'l/Jl
+ 'l/Jj
(7.2.9)
(7.2.10)
'l/J = 'l/Jinc + 'l/Js 1/Js = 1/Jt + 1/JJ
(7.2.11)
Thus'l/Jf of (7.2.6) represents the "final" exciting field that excites particle j. It expresses the idea that after going through multiple scattering between the two particles, this is the field that is finally exciting the particle j. Similarly 1/Jt is the "final" scattered field from particle I. Next, the infinite series in (7.2.6) can be manipulated as follows:
1/Jf
'l/Jinc +GoTz (1/Jinc+GoTj'l/Jinc+GoTjGoTz1/Jinc+GoTjGoTzGoTj1/Jinc+' ..)
(7.2.12) Comparing with (7.2.7), it is clear that the term in the parentheses in (7.2.12) is 1/Jf. Thus
'l/Jf = 'l/Jinc + GoTz'l/Jf
Similarly from (7.2.7) we obtain
(7.2.13) (7.2.14)
1/Jf = 1/Jinc + GoTj 1/Jf
2.2 Foldy-Lax Equations for Point Scatterers
As derived above, the concepts of final exciting field and final scattered field are rigorous. Note that GoTj'l/Jf does not appear on the right-hand side of (7.2.13) because the final exciting field of particle j does not excite itself. Equations (7.2.13) and (7.2.14) are the self-consistent multiple scattering equations for the exciting fields of two particles. After the exciting fields are solved, (7.2.8) and (7.2.11) are used to calculate the total scattered field. Note that the Foldy-Lax multiple scattering equations for the two particles as given by (7.2.13) and (7.2.14) are exact. It is also important to recognize that Tz and T j are only the single-particle scattering transition operator. The equations of two-particle scattering can be readily generalized to N particles. For N particles
'l/Jf = 'l/Jinc
where j
GoTz'ljJr
(7.2.15)
l,,'j
= 1, ... ,N.
The total scattered field is
1j!s
L 1j!J
(7.2.16) (7.2.17)
1j!J = GoTj 1j!f
The total field is
'l/J = 1j!inc + 1j!s
(7.2.18)
Equations (7.2.15)-(7.2.18) are the Foldy-Lax self-consistent multiple scattering equations. They are derivable from Maxwell's equations and are exact relations without approximations [Peterson and Strom, 1973; Tsang et al. 1985]. Generally, (7.2.15) consists of N equations with N unknowns 1j!f, j = 1, ... ,N. In principle they can be solved numerically to yield the exact result of the multiple scattering equations.
2.2 Foldy-Lax Equations for Point Scatterers
For the cases of point scatterers, the transition operator Tj is simple so that the Foldy-Lax multiple scattering equations assume a simple form. The point scatterer has a simple single-particle scattering relation of
eikr
(7.2.19)
where
f is an isotropic quantity. To obey the optical theorem, f must be
7 VOLUME SCATTERING SIMULATIONS
complex. The scattering cross section is
dD s ll1
= 47r111 2
(7.2.20)