IEEE - Aerospace and Electronic Systems - April 2020 - 64

Equivalence of Classical and Quantum Electromagnetic Scattering in the Far-Field Regime

c¼À

iev3 ðdab Á eki Þ
8p2 c3 0

ðsÞ

dððk À ks Þ Á rÞek eikÁr sin ududf:

(19)
We now transform our coordinate system such that the
elevation angle aligns with the vector r. This allows us to
write k Á r as jkjjrj cos u. Thus, we have
c¼À

iev3 ðdab Á eki Þ
8p2 c3 0

ZZ
dðkjrjð cos u À cos us ÞÞ
4p

ðsÞ

Â ek eijkjjrj cos u sin ududf:

(20)

Next, we use the following identity:
dðgðxÞÞ ¼

X dðx À xi Þ

dgðxi Þ
i
dx

(21)

where xi denote the roots of gðxÞ. In our case, we obtain
dðkjrjð cos u À cos us ÞÞ ¼

dðu À us Þ
kjrjj À sin us j

(22)

where we note that we will have one zero occurring at
u ¼ us . Our wave function now becomes
c¼À

iev3 ðdab Á eki Þ
8p2 c3 k0 jrj

ðsÞ

dðu À us Þek eijkjjrj cos u

sin u
dudf:
sin us

(23)
We have dropped the absolute value bars on the sine function, as in the domain of ½0; p, sine is always positive.
Evaluating the integral now gives us
iev ðdab Á eki Þ ðsÞ ikÁr
ðsÞ
ek e ¼ iE 0 ek eikÁr
c¼À
4pc2 0 jrj

current density J, and the resulting fields from that current
density are E. This is related by Ohm's Law
J ¼ sE

(26)

where s is the conductivity of the material. We can think
of an electron moving between energy levels in an analogous way. The motion of the electron can be viewed as a
small current, and the resultant fields are the fields we
have obtained in this analysis thus far, since the photon
wave function c is the transition amplitude of the electric
field. We see here that the E field is in the same direction
as the current density J.
Therefore, by not assuming that the scattered polarization vector is the same as the incident polarization vector
(as is normally done in Wigner Weisskof Theory [18]),
we are inherently taking into account target geometrical
effects, in other words, the lattice of atoms making up the
ðsÞ
target may not allow eki to be equal to ek , such as on the
edge of a flat surface. For our purposes, we will not delve
into the details on determining the directions of all possible currents over the entire surface of a target. We only
ðsÞ
need to realize that ek describes the direction of the electron motion ^je in an analogous way to the wire antenna
ðsÞ
example. We therefore replace ek notationally by
ðsÞ

ek ! ^je :

(27)

Therefore, we can describe the scatted electric field from
an atom by
E ¼ iE 0^je eikÁr :

(28)

(24)

where we have defined
E0 ¼ À

ev2 ðdab Á eki Þ
:
4pc2 0 jrj

(25)

Note that in this construction, we have made no assumptions on the direction of the scattered polarization vector
ðsÞ
ek . The evaluation of the integral with such a general
assumption is only possible in the far field where the density function g approaches a Dirac delta function. This is
important because the scattered polarization vector is
related to the direction of the induced surface current
density. Since we know from classical scattering theory
that the induced current direction changes as a function
of position on the surface, our construction allows
for that.
As an example to elucidate this point, suppose we had
a group of electrons oscillating in one dimension. Such a
situation is analogous to a wire antenna. From antenna
theory, the direction of electron oscillation is directly proportional to the direction of polarization of the incident
field. The bulk electron oscillation can be viewed as some
64

Note that these arguments straddle the fuzzy line
between atomic electron movements, and macroscopic
current densities. In the next section, we will make this
transition between regimes much more concrete.

BUILDING THE SCATTERING INTEGRAL
Our goal in this section is to construct a scattering integral
in identical mathematical form as that shown in (3). In
other words, we want to obtain classical expressions from
a quantum construction. This section hinges on the conjecture that mathematically c behaves as if it is an electric
field, and we will treat it as such. It will be shown that
when we make this conjecture, we are successful in deriving the classical scattered electric field equation. However,
we make no assumptions about the physical nature of c,
and will henceforth refer it as a pseudoelectric field. Also,
as we will see, this pseudoelectric field only appears as
jEj2 , which corresponds to the localization probability of
the photon.
We begin by rewriting the expression for the scattered
pseudoelectric field (or equivalently, the photon wave

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APRIL 2020

IEEE - Aerospace and Electronic Systems - April 2020

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