Page 217 - Electromagnetics
P. 217

We have obtained the second form of each of these expressions using the property (4.3)

                        for the transform of a real function, and by using the change of variables ω =−ω.
                        Multiplying the two forms of the expressions and adding half of each, we find that
                                  1     ∞  dω     ∞
                            ∂w e               dω      ˜ ∗     ˜        ˜    ˜ ∗        − j(ω −ω)t

                                =                   jωE (ω ) · D(ω) − jω E(ω) · D (ω ) e   .   (4.57)
                             ∂t   2  −∞ 2π  −∞ 2π
                          Now let us consider a dispersive isotropic medium described by the constitutive rela-
                                  ˜ ˜
                                          ˜
                             ˜
                        tions D = ˜ E, B = ˜µH. Since the imaginary parts of ˜  and ˜µ are associated with power
                        dissipation in the medium, we shall approximate ˜  and ˜µ as purely real. Then (4.57)
                        becomes
                                ∂w e   1     ∞  dω     ∞  dω    ˜ ∗     ˜               − j(ω −ω)t

                                    =                  E (ω ) · E(ω) jω˜ (ω) − jω ˜ (ω ) e  .
                                 ∂t    2  −∞ 2π  −∞ 2π
                        Substitution from (4.55) now gives
                                    1     ∞  dω     ∞
                              ∂w e               dω


                                  =                   jω˜ (ω) − jω ˜ (ω ) ·
                              ∂t    8  −∞ 2π  −∞ 2π
                                     ˇ  ˇ ∗ ˜      ˜           ˇ  ˇ ∗ ˜      ˜

                                  · E · E F(ω − ω 0 )F(ω − ω 0 ) + E · E F(ω + ω 0 )F(ω + ω 0 )+
                                                             ˇ ∗ ˇ ∗ ˜
                                                                            ˜

                                     ˇ
                                       ˇ ˜
                                                  ˜

                                  + E · EF(ω − ω 0 )F(ω + ω 0 ) + E · E F(ω + ω 0 )F(ω − ω 0 ) e − j(ω −ω)t .


                                                     ˜


                        Let ω →−ω wherever the term F(ω + ω 0 ) appears, and ω →−ω wherever the term
                                                        ˜
                        ˜
                                                ˜

                        F(ω + ω 0 ) appears. Since F(−ω) = F(ω) and ˜ (−ω) = ˜ (ω), we find that
                                 1     ∞  dω     ∞  dω
                           ∂w e                   ˜        ˜
                               =                  F(ω − ω 0 )F(ω − ω 0 ) ·
                           ∂t    8  −∞ 2π  −∞ 2π
                                                         j(ω−ω )t                       j(ω −ω)t

                                                                 ˇ
                                  ˇ

                                     ˇ ∗




                                                                    ˇ ∗
                               ·  E · E [ jω˜ (ω) − jω ˜ (ω )]e  + E · E [ jω ˜ (ω ) − jω˜ (ω)]e  +



                                    ˇ
                                  ˇ
                                                                ˇ ∗ ˇ ∗


                               + E · E[ jω˜ (ω) + jω ˜ (ω )]e j(ω+ω )t  + E · E [− jω˜ (ω) − jω ˜ (ω )]e − j(ω+ω )t  .


                                                                                               (4.58)

                        For small α the spectra are concentrated near ω = ω 0 or ω = ω 0 . For terms involving
                        the difference in the permittivities we can expand g(ω) = ω˜ (ω) in a Taylor series about
                        ω 0 to obtain the approximation

                                               ω˜ (ω) ≈ ω 0 ˜ (ω 0 ) + (ω − ω 0 )g (ω 0 )
                        where

                                                            ∂[ω˜ (ω)]
                                                    g (ω 0 ) =          .
                                                              ∂ω
                                                                    ω=ω 0
                        This is not required for terms involving a sum of permittivities since these will not tend
                        to cancel. For such terms we merely substitute ω = ω 0 or ω = ω 0 . With these (4.58)

                        becomes
                              ∂w e  1     ∞  dω     ∞  dω    ˜  ˜
                                  =                  F(ω − ω 0 )F(ω − ω 0 ) ·
                              ∂t    8  −∞ 2π  −∞ 2π
                                                          j(ω−ω )t                     j(ω −ω)t
                                     ˇ
                                                                  ˇ


                                        ˇ ∗


                                                                     ˇ ∗
                                  ·  E · E g (ω 0 )[ j(ω − ω )]e  + E · E g (ω 0 )[ j(ω − ω)]e  +


                                       ˇ

                                     ˇ
                                                                ˇ ∗ ˇ ∗


                                  + E · E˜ (ω 0 )[ j(ω + ω )]e j(ω+ω )t  + E · E ˜ (ω 0 )[− j(ω + ω )]e − j(ω+ω )t  .
                        © 2001 by CRC Press LLC
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