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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.
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