There are several other ways to prove the Kramers-Kronig relations. For example, a more direct way is to state the causality condition in terms of the signum function sign (t) . Indeed, because u(t) = ( 1 + sign (t) ) / 2, Eq. (1.17.4) may be written in the equivalent form χ(t) = χ(t) sign (t) . Then, Eq. (1.17.7) follows by applying the same frequency-domain convolution argument using the Fourier transform pair: sign (t) P 2 jω (1.17.10) Alternatively, the causality condition can be expressed as u( − t)χ(t) = 0. This ap- proach is explored in Problem 1.12. Another proof is based on the analyticity properties of χ(ω) . Because of the causality condition, the Fourier integral in (1.17.3) can be re- stricted to the time range 0 < t < ∞ : χ(ω) = ∞ −∞ e − jωt χ(t)dt = ∞ 0 e − jωt χ(t)dt (1.17.11) This implies that χ(ω) can be analytically continued into the lower half ω -plane, so that replacing ω by w = ω − jα with α ≥ 0 still gives a convergent Fourier integral † The right-hand side (without the j ) in (1.17.7) is known as a Hilbert transform. Exchanging the roles of t and ω , such transforms, known also as 90 o phase shifters, are used widely in signal processing for generating single-sideband communications signals.
1.18. Group Velocity, Energy Velocity 29 in Eq. (1.17.11). Any singularities in χ(ω) lie in the upper-half plane. For example, the simple model of Eq. (1.11.7) has poles at ω = ± ¯ ω 0 + jγ/ 2, where ¯ ω 0 = ω 2 0 − γ 2 / 4. Next, we consider a clockwise closed contour C = C + C ∞ consisting of the real axis C and an infinite semicircle C ∞ in the lower half-plane. Because χ(ω) is analytic in the region enclosed by C , Cauchy’s integral theorem implies that for any point w enclosed by C , that is, lying in the lower half-plane, we must have: χ(w) = − 1 2 πj C χ(w ) w − w dw (1.17.12) where the overall minus sign arises because C was taken to be clockwise. Assuming that χ(ω) falls off sufficiently fast for large ω , the contribution of the infinite semicircle can be ignored, thus leaving only the integral over the real axis. Setting w = ω − j and taking the limit → 0 + , we obtain the identical relationship to Eq. (1.17.5): χ(ω) = − lim → 0 + 1 2 πj ∞ −∞ χ(ω ) ω − ω + j dω = 1 2 π ∞ −∞ χ(ω ) lim → 0 + 1 j(ω − ω ) + dω An interesting consequence of the Kramers-Kronig relations is that there cannot exist a dielectric medium that is purely lossless, that is, such that χ i (ω) = 0 for all ω , because this would also require that χ r (ω) = 0 for all ω . However, in all materials, χ i (ω) is significantly non-zero only in the neighborhoods of the medium’s resonant frequencies, as for example in Fig. 1.11.1. In the frequency bands that are sufficiently far from the resonant bands, χ i (ω) may be assumed to be essentially zero. Such frequency bands are called transparency bands . 1.18 Group Velocity, Energy Velocity Assuming a nonmagnetic material ( μ = μ 0 ), a complex-valued refractive index may be defined by: n(ω) = n r (ω) − jn i (ω) = 1 + χ(ω) = (ω) 0 (1.18.1) where n r , n i are its real and imaginary parts. Setting χ = χ r − jχ i we have the condition n r − jn i = 1 + χ r − jχ i . Upon squaring, this splits into the two real-valued equations n 2 r
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