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104 Chapter 2
Step 1. Looking back at (2.81) and (2.89) we can come to conclusion that according to Drude-
Lorentz’s model the relative dielectric or magnetic constant with single resonance frequency
can be represented as
2
′
′′
() = () − () = 1 − (2.106)
2
2
− 0 −(/)
Here () = () or () = () depending on which parameter is described.
Definitely, the set of constants { , , } is unique for each material but the frequency
0
dependence is exactly the same.
Step 2. We can apply to (2.106) the frequency transform called Hilbert transform
1
̂
() = − ∫ ∞ ()−1 (2.107)
−∞ −
The improper integral in (2.107) is understood in the sense of its major value. Note that Hilbert
transform plays a significant role in the theory and practice of signal processing and is deeply
related to Fourier transform defined in Chapter 1. But both topics are out of our book theme. A
lot of information can be found in the specialized literature [12, 13].
The integral in (2.107) must be handled cautiously because the integrand has a singularity at
the point = and the limits of integration are infinite. Omitting non-critical for our analysis
mathematical details we obtain
1 ∞ ()−1
− ∫ = ( () − 1) (2.108)
−∞ −
Inserting () = () − () in the left-hand side of (2.108) and () = () − ()
′′
′′
′
′
in the right-hand side and equating the real and imaginary parts of both sides we get the K-K
relations
′
1 ∞ ()−1
′′
() = − ∫
−∞ − (2.109)
′′ �
′ 1 ∞ ()
() − 1 = ∫
−∞ −
It is important to note that these equations hold not only for the function (2.106) but for any
complex function that diminishes to zero as | | → ∞ at last as fast as 1/ and analytic
(indefinitely complex differentiable) in the lower half-plane. Practically, any function
describing the passive structures where the loss of absorbed EM energy is not compensated by
some sources satisfies such requirements . As illustration we can mention:
32
a) () → n()=� () () = () − () is the complex index of refraction,
′
′′
b) () → () = () + () is the complex impedance of passive lumped circuit,
() and () is the real and imaginary part of this impedance,
c) () → () = |()| () is the complex transmission coefficient, |()| and ()
is the magnitude and phase shift,
d) () → () = |()| () is the complex reflection coefficient, |()| and () is
the magnitude and phase shift, and much more.
32 The K-K relations are the direct consequence of the physical causality principle or the existence of
relationship between cause and effect.