Classical Electrodynamics · Problem 13.13

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · transition-radiation-dielectric-foil

Problem

Consider the transition radiation emitted by a relativistic particle traversing a dielectric foil of thickness

a

perpendicular to its path. Assuming that reflections can be ignored because

|[n(\omega) - 1]/[n(\omega) + 1]|

is very small, show that the differential angular and frequency spectrum is given by the single-interface result (13.84) times the factor,

\mathcal{F} = 4\sin^2\Theta, \quad \text{with} \quad \Theta = \nu\left(1 + \frac{1}{\nu^2} + \eta\right)\frac{a}{4D}

Here

D = \gamma c/\omega_p

is the formation length,

\nu = \omega/\gamma\omega_p

, and

\eta = (\gamma\theta)^2

. Provided

a \gg D

, the factor

\mathcal{F}

oscillates extremely rapidly in angle or frequency, averaging to

\langle\mathcal{F}\rangle = 2

. For such foils the smoothed intensity distribution is just twice that for a single interface.