By H. M. Nussenzveig
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Additional info for Causality and Dispersion Relations
40) where 9'"-(0) denotes a polynomial of degree 6 n - I in w . 40) is the general form of a dispersion relation with n subtractions. 14), -2 (n-2)! 18). 9. The Kramers-Kronig Relation Let us now return to the discussion of light propagation in a dielectric medium and let us find out the implications of the above results. 4 that, as a consequence of causality applied to the dielectric polarization at one point of the medium, the complex dielectric susceptibility ~ ' ( w= ) [ ~ ' ( w-) 1]/4n = [n" - 1]/47c (0) is holomorphic in I , .
8)]. , they are dense in 9+' one would write f ( t ) = I f ( ~ ) d ( t - - ) d ~ ]Then, . 2) for all ji E 9+'. 2) always exists, because gr E 9+', J; E 9+' (cf. Section A7). Let us now make the additional assumption thatf,, x,, andg, are temperate distributions, so that they have Fourier transforms (cf. Section A10) F, = Ff,, X, = SX,, G, = Sg,. 2) yields X, = G, F, . 4) Some sufficient conditions for the validity of the convolution theorem have been given in Section A10. In particular, if we restrict the input f, to be a rapidly decreasing distribution, f, E OC', Green's function gr can be any temperate distribution: g , E 9'.
14)], and it has also been linked [cf. 5)] to the phase velocity and the extinction coefficient of a wave traveling through the medium. We shall now consider the relation between these two definitions of n’(w),by discussing the mechanism whereby the change in phase velocity and the extinction of the incident wave arise, in terms of the microscopic processes associated with the polarization of the medium. When the incident wave falls upon the medium, it gives rise to oscillating dipole moments in each atom (or molecule).