Brightness, coherence, and propagation characteristics of synchrotron radiation

Abstract A formalism is presented by means of which the propagation and imaging characteristics of synchrotron radiation can be studied, taking into account the effects of diffraction, electron beam emittance, and the transverse and longitudinal extent of the source. An important quantity in this approach is the Wigner distribution of the electric fields, which can be interpreted as a phase-space distribution of photon flux, and thus can be identified with the brightness. When integrated over the angular variables, the brightness becomes the intensity distribution in the spatial variables and when integrated over the spatial variables, it becomes the intensity distribution in angular variables. The brightness so defined transforms through a general optical medium in exactly the same way as in the case of a collection of geometric rays. Finally, the brightness of different electrons adds in a simple way. Optical characteristics of various synchrotron radiation sources — bending magnets, wigglers and undulators — are analyzed using this formalism.


I. Introduction
Calculations in Gaussian optics, i.e., geometrical optics with a paraxial approximation, are greatly facilitated by working with the phase-space variables associated with each ray. The density distribu tion of the rays in phase space, called the brightness, is a useful quantity in describing the propagation properties of a collection of rays through an arbitrary optical medium. The brightness is also important as an invariant characterization of source strength, since the phase-space area is conserved under optical transformation.
The importance of the brightness has been widely recognized in connection with the planning and design of next-generation synchrotron radiation sources. However, the use of brightness in this context has been only qualitative. Thus, the source brightness 38 of a synchrotron radiation device is sometimes calculated by means of the following approximate formula: (1) <r* / 3 In this paper, we introduce the brightness as a certain Fourier transform of a mutual coherence function of electric fields. The brightness so defined satisfies the same transformation properties as in Gaussian optics and is thus useful in studying the propagation properties of radiation through optical media, taking a full account of the diffraction effects. The formalism permits a quantitative cal culation of the source brightness of a synchrotron radiation device. Section II reviews the properties of Gaussian optics, which serves as a model for later discussions. In Section III, the brightness in a general case is defined in terms of the electric fields, and its transformation properties are described. Section IV discusses the transverse coherence properties in terms of our generalized bright ness. A simple application of the formalism to the fundamental opti cal resonator mode appears in Section V. Section VI establishes an important theorem concerning synchrotron radiation due to a random collection of electrons. Section VII derives an approximate expres sion for the brightness of undulator radiation by approximating the brightness due to a single electron by the brightness due to a laser mode. Section VIII contains a more rigorous discussion of the source brightness of bending magnets, wigglers, and undulators. Finally, Section IX concludes the paper.

Explicit derivations of formula are often neglected here. They
Mill appear in a future publication.

II. Gaussian Optics
In this section, we review some well-known properties of Gaussian optics as an introduction to Uhe brightness concept.

4
Consider the problem of finding the intensity distribution at the image plane of the optical system shown in Fig. (1). In Gaussian optics, only those rays that stay close to a reference trajectory, called the optical axis, are considered. The position along the opti cal axis will be specified by the z-coordinate. A ray passing through a plane transverse to the optical a.;is at z can be characterized by (x,t), where x is the position of the ray in the plane and t> is the angle between the ray and the optical axis. Although x and t are twodimensional vectors, we shall suppress the vector notation whenever possible throughout this paper. In passing through an optical medium, a ray changes according to Here M. is a 2 x 2 matrix which characterizes a component of the medium and which is given by M 1 * ^o 0 f°r free s P ace of length i , Equation (7) gives the transformation properties of the brightness through an arbitrary optical medium. When slits are present, rays that hit the opaque parts should be removed.
A luminous object, or source, can be specified by a function S(x,|6;z), defined by S(x,ib;z)dxd0dz • the number of rays generated by an (8) infinitesimal section dz into the phase space element dxdp.
A more convenient way to characterize the source is to provide the brightness Jf(x,(6:0) referred to a transverse plane within the source at z -0. It is easy to show that ^(x,ft;0) . /dz S(x -zM;z) .
This will be referred to as the source brightness.
The solution of the general optical problem illustrated in Physically measurable quantities, such as the flux density <re7d x and the angular distribution dVjd 2 *, are obtained by integration as follows:

III. General Definition of Brightness
The general discussion of brightness, taking into account diffraction effects, starts with the electric field E. Throughout this paper, we shall always consider a narrow bandwidth am about a given frequency u>. (In this sense, the brightness we discuss here is in fact the "spectral" brightness). Also, we shall limit our discus sion to a single polarization component because, in synchrotron radia tion, usually only the horizontal component is important. Effectively, therefore, the electric field is considered to be a scalar quantity.
Electric fields and brightness are always referred to a certain transverse plane. For notational simplicity, the z depedence will be suppressed whenever possible. Thus, the electric field will be repre sented by either E(x) or E(x;z). It is convenient to introduce the following Fourier transform pair: Here It « «t/c * 2*1 k n \ being the radiation wavelength. As In Gaussian optics, m make the paraxial approximation * c « I, so that The wave propagation in free space is then described by the Fresnel diffraction formula given by The brightness is defined as a bilinear function of the electric field as follows: mechanics. Since then, the representation has been rediscovered and studied by several authors [3] in the context of optics.

Ej^Jfu
We shall now establish that the brightness defined here shares many properties of Gaussian optics. First, one obtains by integration The right-hand sides of above equations, which are positive definite, can clearly be identified as the fluxes, and thus Eq. In other words, we found the intuitively reasonable result that the total flux is coherent for the fundamental mode.

VI. Synchrotron Radiation Due to Many Electrons and an Addition Theorem
The electric field at z«0 associated with a fast-moving electron . ji-_|i/cdt " x (n " fi , t))e 1.<t -D-r/c) m m Here e is the vacuum dielectric constant, n is the direction vector whose transverse component is i> and whose axial component is The produce a c, is known as the emittance. Equation (25) gives the probability distribution of electrons in phase space (x ,6 ).
The theorem, which will be called the addition theorem, will be where N is the total number of electrons. The conditions necessary e for Eq. (26) are that different electrons are statistically independent and that the variation of the magnetic guide field across the electron beam dimensions is negligible. Both of these are well-satisfied by the usual synchrotron radiation sources.

VII. Approximation of Undulator Radiation by Laser Mode
The expression for the source brightness of undulator radiation is somewhat involved, and it is useful to approximate it by t.ie laser mode discussed in Section V. To identify a and o , with the undulator parameters, we first determine «, from the undulator angular distri bution and write [7] v-Jt-J 1 The source strength of an undulator may be characterized by the peak brightness^(0,0). Reference [8] contains the brightness of various synchrotron radiation sources computed in this way.

VIII. Source Brightness of Synchrotron Radiation
Let us now turn to a more rigorous derivation cf the source brightness of synchrotron radiation due to a single electron based on Eq. (24) and Eq. (14). The electron trajectory is assumed to lie in the horizontal plane, and the ccordinate system is explained in Fig. (3).

Bending Magnets and Wigglers
Using the same approximation that Schwinger [9] uses for his derivation of the radiation from bending magnets, one finds that the where o is the instantaneous radius of the curvature. Equation (31) is analogous to, but simpler than, Green's classical analysis [10], in which he incorporated the diffraction effect on the basis of an intui tive argument.

Undulators
Although the general expression for undulator brightness is rather complicated, one obtains the following approximate expression when N » -: ,