IEEE - Aerospace and Electronic Systems - April 2020 - 66

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

Figure 7.
Electric field impinging on a target. In the classical interpretation, the field induces surface currents on the target. In the quantum interpretation,
these currents are caused by the impinging field pushing electrons to different energy levels throughout the atomic lattice. We zoom in to the scale
of the infinitesimal to analyze the equivalence. a) Line defining the surface current (dashed) and the electron transitions (purple arrows) caused by
the impinging field. The charges that pass through the line contribute to the current amplitude. b) Projection of the electron transitions (blue arrows)
along the normal of the line defining the current. c) Accounting of charge entering and leaving the surface element.

electron paths will follow very similar trajectories based on the incident field polarization and
wavelength.
Keeping all of these points in mind, we need to recall
how current density is defined. We first assume that the
incident photons do not penetrate very far into the target.
In other words, we restrict all atomic transitions to atoms
on the surface.
In the most general sense, we can define a current density as the rate of charges in a volume passing through a
surface [see Figure 6(a)]. Since we are interested in a surface current density based on our restriction of only allowing atomic transitions along the surface, we must recall
the analogous definition of the surface current density,
namely, the rate of charges along a surface passing
through a line [see Figure 6(b)].
Therefore, for a unit vector ^l describing the normal
direction of this two-dimensional line on the surface (and
therefore the direction of the current), the projection of the
electron motion along this direction is
^j Á ^l
e

(33)

66

Â

i
Ãh
deff Á eki ^je Á ^l ^l:

(35)

For each atomic transition along the length of the line, there
will be a term of the form of (35). Each transition will cross
the line in differing directions and in differing magnitudes.
The total combined contribution from each atomic transition along the length of the line defines the total distance Dl
the electrons have moved in a bulk manner
Dl ¼ Dl^l ¼

X
n

ðnÞ

deff Á eki

h
i
^jðnÞ Á ^l ^l:
e

(36)

This process is shown pictorially in Figure 7(a) and (b).
We can write this Dl as being equal to some drift
velocity ud multiplied by some length of time Dt
Dl ¼ uDt:

(37)

Thus, based on the preceding arguments, we have obtained
the microscopic-to-macroscopic transformation as
À

and the vector describing this motion is
h
i
^j Á ^l ^l:
e

Thus, the transformed projection term from the atom-photon
coupling is given by

i
h
X ðnÞ
Á
^^
dab Á eki ^je !
deff Á eki ^jðnÞ
e Á l l ¼ Dl ¼ uDt:
n

(34)

IEEE A&E SYSTEMS MAGAZINE

(38)
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IEEE - Aerospace and Electronic Systems - April 2020

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