Synchrotron radiation representation in phase space

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 ($\beta$ 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.

Figure 1

Potential for an electron $V(x)$. The initial state $\psi (x,t=0)$ and the first 40 energy eigenstates ${\psi }_n(x)$ offset by their eigenvalues are shown.

Figure 2

Motion of a classical (left) and quantum (right) electron in phase space. In both cases the initial position is 3 nm with no velocity. Points show time steps from 0 to $T$ with steps of $T/16$, where $T=3.8\mathrm{fs}$ is the natural period of the motion.

Figure 3

Time evolution of the Wigner distribution. Solid curves show projections of the WDF, probability densities in position and momentum (velocity), respectively. The electron’s classical counterpart is depicted by $\times$.

Figure 4

First three states (left) of a simple harmonic oscillator along with the radial Wigner distribution (right).

Figure 5

The Wigner distribution for coherent (left) and incoherent (right) superposition of two quantum states with equal weights. Solid curves show projections of the WDF, probability densities in position and velocity, respectively. The states are of the form $\psi (x)\propto e^{-\Delta x^2/2{\sigma }_x^2}{e}^{i(m/\hslash )[v_0\Delta x+(1/2)(dv/dx)\Delta x^2]}$, where $\Delta x=x-x_0$. The parameters $\{x_0,{\sigma }_x,v_0,dv/dx\}$ are equal to $\{3\mathrm{nm},0.3\mathrm{nm},0\mathrm{nm}/\mathrm{fs},4{\mathrm{fs}}^{-1}\}$ and $\{2\mathrm{nm},0.5\mathrm{nm},3\mathrm{nm}/\mathrm{fs},1.5{\mathrm{fs}}^{-1}\}$ for the two Gaussian states depicted.

Figure 6

The Wigner distribution for a mixed state of two Gaussians of equal weight (left column) and corresponding orthogonal mode decomposition (middle and right columns). The preparation states are of the form $\psi (x)\propto e^{-\Delta x^2/2{\sigma }_x^2}{e}^{i(m/2\hslash )(dv/dx)\Delta x^2}$, where $\Delta x=x-x_0$. The parameters $\{x_0,{\sigma }_x,dv/dx\}$ are equal to $\{3\mathrm{nm},0.3\mathrm{nm},4{\mathrm{fs}}^{-1}\}$ for fixed and $\{1\mathrm{\text{−}}2\mathrm{\text{−}}3\mathrm{nm},0.4\mathrm{nm},1{\mathrm{fs}}^{-1}\}$ for displaced Gaussians. The reconstructed WDF is recovered through $W_{(\widehat{\rho })}={\lambda }_1{W}_{({\varphi }_1)}+{\lambda }_2{W}_{({\varphi }_2)}$.

Figure 7

An example of a vertical trajectory in a planar undulator and the magnetic field as seen by the particle. The angular offset is taken to be rather large to illustrate the effect of cosh-like dependence of the field on vertical position.

Figure 8

Radial phase-space distributions (left) and corresponding emittance vs fraction curves (right). All distributions are scaled to have $ϵ=1$. Core fraction and emittance for different distribution types are shown as well.

Figure 9

Radial phase-space distributions (left) and corresponding emittance vs fraction curves (right). Both distributions are scaled to have $ϵ=1$. Core fraction and emittance for the two distributions are shown as well.

Figure 10

Comparison of calculated (dots) vs theoretical values (solid curves) for undulator radiation: (a) angular flux density, (b) central cone flux, (c) on-axis 4D brightness, (d) on-axis 2D brightness. The first three odd harmonics are shown. The undulator is planar with $N_u=250$. Refer to the text for other parameters.

Figure 11

Wigner-Stokes density distribution functions computed for a helical undulator. The x-ray phase space is backpropagated to the undulator center. Refer to the text for details.

Figure 12

(a) Wigner 2D distribution for a planar undulator with $N_u=250$ on resonance of the 1st harmonic at 8 keV photon energy along with (b) the emittance of the light vs fraction curve. The WDF is backpropagated to the undulator center.

Figure 13

Trajectories inside a segmented undulator with a quadrupole focusing. The undulator has two segments each with $N_u=100$ periods. The separation between the two is 0.486 m. A single horizontally focusing quadrupole of length 0.3 m is located at the center with $3.5\mathrm{T}/\mathrm{m}$ gradient.

Figure 14

Radiation of the 1st harmonic on 8 keV energy resonance from the segmented undulator with quadrupole focusing of Fig. 13. Different rows correspond to different trajectory offsets, $x_{\mathrm{ini}}$, as shown. The left column is 2D WDF backpropagated to the center of the undulator, which corresponds to the right column showing the radiation spectral flux density at the detector 30 m away from the undulator center.

Figure 15

Scanning around the 1st harmonic resonance of 8 keV for $N_u=250$ period planar undulator: (a) angular flux on axis, (b) integrated (central cone) flux. Red circles denote analytical values.

Figure 16

The WDF (top row) corresponding to Fig. 15 for different detuning off the resonance. The WDF is backpropagated to the undulator center. Solid lines show position and angular intensity projections. The radiation pattern used to calculate WDF (bottom row) is obtained at 50 m from the undulator center.

Figure 17

Light 100% and core emittances (left) and beta function (right) at the undulator center corresponding to Fig. 15 for different detuning off the resonance.

Figure 18

The average 2D brightness ${\mathcal{B}}_{avx}$, and ${\mu }_{gx}^2$ corresponding to Fig. 15.

Figure 19

Electron phase space at the center of 25 m long undulator corresponding to two different currents: 25 mA (left) and 100 mA (right).

Figure 20

X-ray phase space corresponding to zero emittance and zero energy spread (left) and nonzero energy spread ${\sigma }_{{\delta }_e}=2\times {10}^{-4}$ (right). The beam current is 100 mA.

Figure 21

X-ray phase space including the effects of emittance and energy spread for the two different currents.