Page 48 - Electromagnetics Handbook

and the behavior of p-n junctions in solid-state rectifiers. The invention of the laser
extended interest in nonlinear effects to the realm of optics, where phenomena such as
parametric amplification and oscillation, harmonic generation, and magneto-optic inter-
actions have found applications in modern devices [174].
Provided that the external field applied to a nonlinear electric material is small com-
pared to the internal molecular fields, the relationship between E and D can be expanded
in a Taylor series of the electric field. For an anisotropic material exhibiting no hysteresis
effects, the constitutive relation is [131]

3 3
(1) (2)
D i (r, t) =
0 E i (r, t) + χ E j (r, t) + χ E j (r, t)E k (r, t) +
ij ijk
j=1 j,k=1
3
(3)
+ χ E j (r, t)E k (r, t)E l (r, t) +· · · (2.43)
ijkl
j,k,l=1
where the index i = 1, 2, 3 refers to the three components of the fields D and E. The
first sum in (2.43) is identical to the constitutive relation for linear anisotropic materi-
als. Thus, χ (1) is identical to the susceptibility dyadic of a linear anisotropic medium
ij
considered earlier. The quantity χ (2) is called the second-order susceptibility, and is a
ijk
three-dimensional matrix (or third rank tensor) describing the nonlinear electric effects
quadratic in E. Similarly χ (3) is called the third-order susceptibility, and is a four-
ijkl
dimensional matrix (or fourth rank tensor) describing the nonlinear electric effects cubic
(2) (3)
in E. Numerical values of χ and χ are given in Shen [174] for a variety of crystals.
ijk ijkl
When the material shows hysteresis effects, D at any point r and time t is due not only
to the value of E at that point and at that time, but to the values of E at all points and
times. That is, the material displays both temporal and spatial dispersion.

2.3 Maxwell’s equations in moving frames

The essence of special relativity is that the mathematical forms of Maxwell’s equa-
tions are identical in all inertial reference frames: frames moving with uniform velocities
relative to the laboratory frame of reference in which we perform our measurements.
This form invariance of Maxwell’s equations is a specific example of the general physical
principle of covariance. In the laboratory frame we write the differential equations of
Maxwell’s theory as
∂B(r, t)
∇× E(r, t) =− ,
∂t
∂D(r, t)
∇× H(r, t) = J(r, t) + ,
∂t
∇· D(r, t) = ρ(r, t),
∇· B(r, t) = 0,
∂ρ(r, t)
∇· J(r, t) =− .
∂t

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