On the Čerenkov threshold associated with synchrotron radiation in a dielectric medium

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Abstract
A formal exact expression for the radiation field of a charged particle executing a circular orbit in a dispersive dielectric medium is obtained by means of a dyadic Green's function approach. The field is expressed as an infinite sum of vector spherical harmonics. The radiated power P is computed numerically and a set of universal curves of nP/q2Ω2 vs n β is obtained where q is the charge on the particle, n is the index of refraction of the medium, Ω is the orbital angular velocity of the particle, and β=v/c, where v is the particle velocity and c is the speed of light in vacuum. The Čerenkov threshold phenomenon is shown to be manifest primarily in the high‐frequency portion of the radiated spectrum. For an index of refraction of unity, the result is shown (at low velocities) to be identical with both Larmor's formula for the radiated power and Schwinger's result for the angular distribution of the radiation.