Classical Electrodynamics · Problem 13.14

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · transition-radiation-stack-foils

Problem

Transition radiation is emitted by a relativistic particle traversing normally a uniform array of

N

dielectric foils, each of thickness

a

, separated by air gaps (effectively vacuum), each of length

b

. Assume that multiple reflections can be neglected for the whole stack. This requires

\left|\frac{n(\omega) - 1}{n(\omega) + 1}\right| \approx \frac{\omega_p^2}{4\omega^2} \ll \frac{1}{N}

(a) Show that if the dielectric constant of the medium varies in the

z

direction as

\epsilon(\omega, z) = 1 - (\omega_p^2/\omega^2)\rho(z)

, the differential spectrum of transition radiation is given approximately by the single-interface result (13.84) times

\mathcal{F} = \left|\mu \int dz\, \rho(z) e^{i\omega z/v} \exp\left(-i\cos\theta \int^z k(z')\, dz'\right)\right|^2

where

\rho(0) = 1

by convention,

\mu = \omega/v - k(0)\cos\theta

, and

k(z) = (\omega/c)\sqrt{\epsilon(\omega, z)}

. (b) Show that for the stack of

N

foils

\mathcal{F} = 4\sin^2\Theta \frac{\sin^2[N(\Theta + \Psi)]}{\sin^2[\Theta + \Psi]}

where

\Theta

is defined in Problem 13.13 and

\Psi = \nu(1 + \eta)(b/4D)