Page 47 - Electromagnetics Handbook

The geometrical structure of a crystal determines the relationship between
x ,
y , and

z .If
x =
y <
z , then the crystal is positive uniaxial (e.g., quartz). If
x =
y >
z ,
the crystal is negative uniaxial (e.g., calcite). If
x
=
y
=
z , the crystal is biaxial (e.g.,
mica). In uniaxial crystals the z-axis is called the optical axis.
If the anisotropic material is dispersive, we can generalize the convolutional form of
the isotropic dispersive media to obtain the constitutive relations

t

D(r, t) =
0 E(r, t) + ¯ χ e (r, t − t ) · E(r, t ) dt , (2.34)
−∞

t

B(r, t) = µ 0 H(r, t) + ¯ χ m (r, t − t ) · H(r, t ) dt , (2.35)
−∞
t

J(r, t) = ¯ σ(r, t − t ) · E(r, t ) dt . (2.36)
−∞
Constitutive relations for biisotropic materials. A biisotropic material is an
isotropic magnetoelectric material. Here we have D related to E and B, and H related to
E and B, but with no realignment of the fields as in anisotropic (or bianisotropic) mate-
rials. Perhaps the simplest example is the Tellegen medium devised by B.D.H. Tellegen
in 1948 [196], having
D =
E + ξH, (2.37)
B = ξE + µH. (2.38)
Tellegen proposed that his hypothetical material be composed of small (but macroscopic)
ferromagnetic particles suspended in a liquid. This is an example of a synthetic mate-
rial, constructed from ordinary materials to have an exotic electromagnetic behavior.
Other examples include artificial dielectrics made from metallic particles imbedded in
lightweight foams [66], and chiral materials made from small metallic helices suspended
in resins [112].
Chiral materials are also biisotropic, and have the constitutive relations
∂H
D =
E − χ , (2.39)
∂t
∂E
B = µH + χ , (2.40)
∂t
where the constitutive parameter χ is called the chirality parameter. Note the presence
of temporal derivative operators. Alternatively,
D =
(E + β∇× E), (2.41)
B = µ(H + β∇× H), (2.42)

by Faraday’s and Ampere’s laws. Chirality is a natural state of symmetry; many natural
substances are chiral materials, including DNA and many sugars. The time derivatives
in (2.39)–(2.40) produce rotation of the polarization of time harmonic electromagnetic
waves propagating in chiral media.

Constitutive relations in nonlinear media. Nonlinear electromagnetic effects have
been studied by scientists and engineers since the beginning of the era of electrical tech-
nology. Familiar examples include saturation and hysteresis in ferromagnetic materials

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