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72 Chapter 2

′
(check (2.68) – (2.69)) that the displacement current = ()(, ) much exceeds
0
the sum of conduction current = (, ) and the current induced by the polarization
′′
()
effect = ′′ ()(, ). In good conductors tan = ≫ 1 meaning that the
0
′ ()
conductivity current much exceeds all other currents such as displacement and polarization.
7
Everything between can be treated as semiconductor. Typically, (ω) > 10 S/m for the
−7
4 −6 (ω) < 10
conductors, 10 S/m > (ω) > 10 S/m for the semiconductors, and
S/m for the insulators. This classification combines in Table 2.3. Evidently, similar
classification can be provided for magnetic materials.
2.4.3 Linearity and Nonlinearity
Our next topic is the nonlinearity that we will define as any violation of the linearity principle.
It is worth to mention that all known natural and human-made material including ferro-materials
are more or less nonlinear, generally less. As such, we need to describe first a linear material.
The definition is quite straightforward

()
= (1 + )()

0 � (2.75)
()
= (1 + )()

0
It means that the displacement and inductance field vectors in linear materials are always
strictly proportional to the electric and magnetic vector at any moment of time. What about
linearity in frequency domain? Following to [20], we need to assume additionally that the
relation (2.75) is held not only at each moment of time but each frequency. For example, the
displacement field is strictly proportional to the applied electric field or (, ) =
()(, ) at all frequencies. In other words, the frequency spectrum of vector-signals
0
of (, ) and (, ) always contains the same set of frequencies. The violation of this
condition leads to generation of multiple harmonics and beat frequencies distorting the
propagating through media signals. It turns out that the vast majority of natural and artificial
dielectric materials used in engineering practice is linear to a high degree of accuracy at all field
strengths commonly attained. Explicitly, we can define in similar manner the linearity of
magnetic material through the constitute relation between vectors B and H as (, ) =
()(, ).
0

2.5 BROADBAND COMPLEX-VALUED MATERIAL PARAMETERS

2.5.1 Introduction
Now, we are going to attend more interesting and common phenomena, namely, the physics of
material parameter deviations in the frequency domain. Let us start from the single practical
result. The copper, which we know as excellent conductor, has dielectric constant = -16.7 -

j1.74 at frequency of 430 THz (wavelength 700 nm). Everything here is against our common
′
′′
sense. First, the real part is negative. The second that tan = ⁄ = 0.1 only. The copper

(see Table 2.3) comes to be some sort of semiconductor but certainly not a good conductor.
Furthermore, at such frequencies the thin copper layers become optically transparent. It means
that we definitely missed something in our understanding and description of materials
properties. The following analysis demonstrates the remarkable possibility to get quite accurate
prediction of conductivity effect in wide variety of materials, metals and dielectrics. Our task