Impact Parameter Profile of Synchrotron Radiation

The horizontal impact parameter profile of synchrotron radiation, for fixed vertical angle of the photon, is calculated. This profile is observed through an astigmatic optical system, horizontally focused on the electron trajectory and vertically focused at infinity. It is the product of the usual angular distribution of synchrotron radiation, which depends on the vertical angle ψ, and the profile function of a caustic staying at distance bcl = (γ −2 +ψ2)R/2 from the orbit circle, R being the bending radius and γ the Lorentz factor. The classical impact parameter bcl is connected to the Schott term of radiation damping theory. The caustic profile function is an Airy function squared. Its fast oscillations allow a precise determination of the horizontal beam width. keywords: synchrotron radiation, beam diagnostics, radiation damping. 1 Theoretical starting point Classically [1, 2], a photon from synchrotron radiation is emitted not exactly from the electron orbit but at some distance or impact parameter bcl ≡ Rphot − R = R v cosψ −R ≃ γ + ψ 2 R , (1) from the cylinder which contains the orbit. R is the orbit radius, ψ is the angle of the photon with the orbit plane and γ = ǫ/me = (1− v ) ≪ 1 is the electron Lorentz factor. In this paper we will assume that the orbit plane is horizontal. Eq.(1) is obtained considering the photon as a classical pointlike