Synchrotron radiation representation in phase space

Abstract

The notion of brightness is efficiently conveyed in geometric optics as density of rays in phase space. Wigner has introduced his famous distribution in quantum mechanics as a quasiprobability density of a quantum system in phase space. Naturally, the same formalism can be used to represent light including all the wave phenomena as originally done by Walther and for synchrotron radiation by Kim. It provides a natural framework for radiation propagation and optics matching by transferring the familiar ''baggage'' of accelerator physics ( function, emittance, phase-space transforms, etc.) to synchrotron radiation. More specifically, the use of Wigner distribution formalism allows a rigorous description of partially coherent non-Gaussian sources, which is generally the case for synchrotron radiation from an undulator with a high degree of transverse coherence. This paper reviews many of the properties of the Wigner distribution starting from quantum mechanics and provides examples of how its use enables physically insightful description of partially coherent synchrotron radiation in phase space. The concepts of diffraction limit and coherence are given an exact correspondence to their quantum mechanical counterparts. In particular, it is shown that the undulator radiation on resonance by a single electron is not diffraction limited though fully coherent. An extension of how to account for practically important cases of electron beams with nearly diffraction limited emittances is presented along with a discussion of appropriate figures of merit suitable for comparing future light sources.

Demonstration of cathode emittance dominated high bunch charge beams in a DC gun-based photoinjector

“…Here σ px,0 is the momentum variance intrinsic to the cathode material, which can be expressed in terms of the mean (kinetic) energry of the photoemitted electrons (MTE), and σ x,0 is the spatial variance of the laser distribution. The rms emittance motivates a simple definition for the average transverse (normalized) brightness, defined generally as the particle flux per unit 4D transverse phase space volume [15,16]:…”

“…To characterize the contribution of the central core of the phase space to the emittance, as well as provide a pratical means to compare non-Gaussian beams, we define the emittance vs. fraction curve (see [15,17] for details): for every area in phase space a, we find a bounding contour D(a) which maximizes the enclosed fraction f of beam particles. The rms emittance computed for the particles inside D(a) defines the corresponding fractional emittance n,x (f ).…”

Compact representations of partially coherent undulator radiation suitable for wave propagation

“…(24) transforms in the same manner as does the phase space distribution of geometrical optics [18]. Hence, the brightness is a physically appealing representation of second-order coherence, and it has recently received renewed interest [24][25][26]. The brightness can be used to define the source strength and purity in a number of ways.…”

Numerical analysis of brilliance and coherent photon flux of segmented undulator radiation based on statistical optics

“…The SR wave character becomes noticeable for small electron beam emittances, especially for a diffraction-limited light source. The SR properties in a low-emittance ring have therefore attracted many investigations (Geloni et al, 2008;Bazarov, 2012;Tanaka, 2014). A typical method is to compute the WDF directly from the electron beam providing a general way to analyse the wave properties of SR without a Gaussian approximation (Kim, 1986).…”