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 ($\ensuremath{\beta}$ 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.
Highlights
The concept of phase space plays an important role in accelerator physics
The approach allows light characterization of arbitrary degree of coherence [6] and polarization [7] in phase space, though its application by the accelerator community has so far been mostly limited to the simplest cases of Gaussian or Gauss-Schell beams [8,9], despite the fact that the non-Gaussian nature of undulator radiation in phase space has been long recognized [4,5,8,10]
This foundation allows one to demonstrate quite how the Wigner distribution function (WDF) can be used with much physical insight to describe synchrotron radiation sources with an arbitrary degree of coherence and polarization
Summary
The concept of phase space plays an important role in accelerator physics. Useful tools such as Twiss parameters, emittance, and phase space propagation have been in long use in the accelerator community. Dealing with non-Gaussian sources requires one to differentiate between the concepts of the diffraction limit and the transverse coherence: e.g., a source can be fully coherent but not diffraction limited [12] The distinction between these two important concepts has been generally vague or lacking in the accelerator literature. One of the main goals of this paper is to provide a more rigorous foundation for the Wigner distribution formalism by reviewing many of its useful properties beginning with their connection to quantum mechanical description of states This foundation allows one to demonstrate quite how the WDF can be used with much physical insight to describe synchrotron radiation sources with an arbitrary degree of coherence and polarization. Since neither synchrotron radiation nor electron beam in this case have Gaussian phase-space density, some consideration is given to generalizing the concepts of emittance and brightness to describe a non-Gaussian source
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