Coherence properties of the high-energy fourth-generation X-ray synchrotron sources

Coherence properties of the fourth-generation high-energy storage rings with low-emittance values are discussed.

distribution at the far field position (Eq. (34)). Transverse positional distribution at the source position was obtained by taking the Fourier transform of the angular field distribution in the far field. The size of the source was determined as the variance value of the intensity distribution at the source position in the middle of the undulator (Eq. (34)). Integration in Eqs. (34) was performed over the finite area defined as a size of the detector in the far-field or size of the source at the source position. In both cases the area corresponded to 95% of the total intensity. Total photon emittance was calculated as the product of the source size and source divergence (Eq. 28) (see Fig. 1 (triangles)). The error of the performed simulation with the XRT software for each natural electron emittance value was calculated as the standard deviation of the corresponding value.

b) Coherent-mode representation
To determine independent coherent modes of radiation or perform coherent mode decomposition one needs to solve Fredholm integral equation according to Eq. (10). We performed mode decomposition of the CSD according to Eqs. (9-11) using the XRT software.
In the XRT software, CSD is calculated according to Eqs. (7,8) for the radiation in the far field region for a single electron. The CSD of radiation is, in general, a four-dimensional (4D) matrix (the CSD function depends on coordinates x 1 , y 1 ; x 2 , y 2 ). Effectively in the XRT software, one more dimension is added by including a number of electrons so that the correlation function is saved for each electron, which gives additional dimension. The next step is to compose the 4D CSD matrix from the 5D field distribution and to solve the CSD for eigenvalues and eigenfunctions.
The eigenvalue problem for the CSD matrix in XRT software is reduced to twodimensional (2D) by applying principal component analysis (PCA) (see Fig. S1). After the diagonalization of this matrix, the eigenvalues (mode weights) and eigenfunctions (modes) of CSD were determined (see Fig. 3). If the mode weights are known than the global degree of coherence and the coherent fraction of radiation may be calculated according to Eqs. (12) and (13) (see Fig. 5 (triangles)). It should be also noted, that the global degree of coherence according to the PCA method can be obtained without solving this huge eigenvalue problem.
The global degree of coherence, in this case, can be calculated as the ratio between traces of the CSD matrix ζ DC = Tr(W PCA 2 )/ �Tr(W PCA )� 2 , which is equivalent to Eq. (12), where W PCA is the rearranged CSD matrix according to the PCA method.
The number of electrons used in the coherent-mode simulations is given in Table S1.
The first four coherent modes and their normalized weights for high photon energies of 24 keV and 50 keV for different energy spread values are shown in Fig. S2. The behavior of these modes is similar to the ones observed at 500 eV and 12 keV case (see Fig. 3). The values of the global degree of coherence are given in Table 4.

c) Degree of transverse coherence
Simulations of amplitudes and correlation functions in one transverse direction were performed by XRT software as well. One dimensional CSD function at given photon energy was simulated in the far field region as average over an ensemble of electrons according to Eq. (8) in the horizontal direction. Parameters of the simulation are given in Table 5 Table 5.

S2. Details of analytical simulations a) Field amplitude simulation
In the frame of the analytical approach, source properties were calculated as well (Geloni et al., 2015;Geloni, 2018). In this case, the amplitude � , � in the far field region from a single electron was calculated according to Eq. (33). Total amplitude in the far field region for the electron bunch was defined as with (2)

b) Source parameters simulation
Calculations of source parameters were done for zero, 1 • 10 −3 , and 2 • 10 −3 relative energy spread values. Amplitude and intensity distribution in the far field region at a distance of 30 m from the source for different photon energies were simulated on the same angular mesh, as in the case of XRT simulations (see Table S1). Total photon emittance was calculated in the same way as it was done previously according to Eq. (34) and Eq. (28) of the main text (see Fig. 1 (dots)). Due to the narrowing of divergence of the radiation higher is the photon energy more electrons were needed to sample the radiation on the virtual detector in order to obtain uniform intensity distribution. In order to accumulate good statistics and get a uniform intensity distribution number of electrons used in this calculation was increased (see Table 5). This adjustment is also valid for larger energy spread values.
c) Coherent-mode representation, coherent fraction and degree of transverse coherence.
The same parameters of the analytical simulations of the CSD function were used for XRT simulations (see Table S1). Correlation functions were calculated and saved for each electron according to Eq. (8), with the amplitude distribution calculated according to Eq.
(32). The eigenvalue problem for the 5D CSD matrix was also reduced to two-dimensional by applying PCA as it was done previously by XRT software. In this case, a coherent fraction of the radiation as a function of electron emittance is shown in Fig. 5 (dots). In order to accumulate good statistics the number of electrons used in this calculation was also increased (see Table S1).
Simulations of the amplitude and CSD were also performed in one transverse direction in the frame of the analytical approach (see Figs Table 4. These results are very close to the results obtained from simulations by XRT software. Due to the fact that analytical analysis matches very well to the simulation of correlation functions performed by XRT software (see Fig. 6 and Figs. S5-S8), it has been extended to other natural electron emittances in a range from 1 to 100 pm rad. The degree of transverse coherence (Eq. (19) of the main text) calculated in the frame of the analytical approach as a function of natural electron emittance for various relative energy spread values is shown in Fig. S9. We observe the same tendencies for the degree of transverse coherence as for coherent fraction of the radiation shown above in Fig. 2 and Fig. 5. At 500 eV and 12 keV we reach diffraction-limited case already at 10 pm rad and 1 pm rad respectively. However, at 24keV and 50 keV even 1 pm rad electron emittance is not sufficient to reach diffraction limit.  c) Cross-spectral density matrix W PCA obtained by multiplication of matrix 1 − by its complex conjugated and transposed matrix.

Figure S2
First four modes and their normalized weights obtained from the coherent mode