Question

Let f(ω) be the scattering amplitude for forward scattering of light at an individual scattering center in an optical medium. If the amplitude for incoming and outgoing light waves are denoted by A_in(ω) and A_out(ω) respectively, one has A_out(ω) = f(w) A_in(ω) Suppose the Fourier transform

vanishes for x - t > 0.

(a) Use the causality condition (no propagation faster than the speed of light c = 1) to show that f(ω) is an analytic function in the half-plane Imω > 0.

(b) Use the analyticity of f(ω) and the reality of \(\bar A\)_in(ω) and \(\bar A\)_out (ω), and assume that f(w) is bounded at infinity to derive the dispersion relation

with \(\varepsilon\) arbitrarily small and positive.

Answer 1

(a) \(\bar A\)_in(x - t) = 0 for t < x means \(\bar A\)_out(x - t) = 0 for t < x. Then

is a regular function when Im ω > 0, since when \(\tau\) < 0 the factor exp (Im ωT) of the integrand converges. As A_out(ω) = f(ω) A_in(ω), f(ω) is also analytic when Imω > 0.

(b) For w → \(\infty\), 0 ≤ argw ≤ π, we have If (ω)l < M, some positive number.

Assume that f(0) is finite (if not we can choose another point at which f is finite). Then x(ω) = \(\frac{f(\omega) - f(0)}{\omega}\) is sufficiently small at infinity, and so

where P denotes the principal value of the integral.