## Abstract

In this work we point out that slope errors play only a minor role in the performance of a certain class of x-ray optics for X-ray Free Electron Laser (XFEL) applications. Using physical optics propagation simulations and the formalism of Church and Takacs [Opt. Eng. 34, 353 (1995)], we show that diffraction limited optics commonly found at XFEL facilities posses a critical spatial wavelength that makes them less sensitive to slope errors, and more sensitive to height error. Given the number of XFELs currently operating or under construction across the world, we hope that this simple observation will help to correctly define specifications for x-ray optics to be deployed at XFELs, possibly reducing the budget and the timeframe needed to complete the optical manufacturing and metrology.

© 2015 Optical Society of America

## 1. Introduction

The recent availability of x-ray sources with laser-like quality [1, 2] and the growing demand for sub-micrometer size focal spots [3, 4] have stimulated much debate in the x-ray optics community. From a practical point of view the question is whether the state-of-the-art in optic manufacturing and metrology is compatible with the requirements imposed by the experimentalists working at XFELs. While x-ray optics with astonishingly low height errors and slope errors have been made available on the market [5], it is apparent that this has triggered a drastic change in the timeframe typical for the deployment of these optics. In fact, besides the extra time needed to polish such extreme optics, the need often arises on the customer side for the development of dedicated characterization facilities, where full suites of metrology tools can confirm such tight specifications, often employing time consuming cross-calibration procedures among instruments. In particular, the issue of slope errors seems to be at the heart of many discussions among optics designers and manufacturers. A geometric argument is usually used as a rough metric to compute the largest slope error an optic can tolerate while still guaranteeing diffraction limited spot sizes at the focal plane. As we will show in the next section, such a geometric argument often leads to slope errors as small as 60 nrad for micrometer size focal spots. This represents a significant challenge from both the manufacturing and metrology point of view. In this paper we challenge the need for such extremely low slope errors when dealing with optics that are by design diffraction limited, which happen to be ubiquitous in XFEL settings. We perform physical optics propagation simulations with the XFELsim package under development at LLNL, and support our findings with the theory of Church and Takacs [6]. We show that for a class of optics routinely deployed at XFEL facilities, the extremely small slope errors satisfying the geometric argument for diffraction limited focal spot sizes do not offer a real advantage over slightly larger slopes that are cheaper and easier to achieve.

## 2. Diffraction limit for an XFEL mirror

As a specific example we consider a mirror focusing a *λ* = 1.5 Å XFEL gaussian source, the latter having an intensity distribution profile diameter (FWHM) *d*_{source} = 28 *μ*m at the undulator exit, and a corresponding divergence *ρ*_{FWHM} = 4.75 *μ*rad. The mirror is *x* = 374.7 m downstream of the undulator exit, measures *L* = 40 cm in length, operates at a grazing angle *θ _{i}* = 3.35 mrad, and focuses x-rays at a focal plane located at a distance

*x*′ = 8.3 m from its center. The system's magnification is $M=\frac{{x}^{\prime}}{x}$. These parameters match focusing optics currently in use at the Linac Coherent Light Source [7] (LCLS), at the SLAC National Accelerator Laboratory. The angular width with respect to the optical axis of the intensity distribution profile at the focal plane subtended by the mirror is the sum of three contributions [6]

In this equation the first term is the diffraction-limited angular width of the image, caused by the finite length of the mirror *L* or the finite beam footprint *L*′, whichever is less,

The second term is the angular width of the x-ray source subtended by the mirror,

Finally, the last term is the angular width of the detector subtended by the mirror, which is negligible in most experimental setups. In fact, at the focal plane the beam is “detected” at the intrinsic length scale of the interaction between the sample and the beam itself. Therefore, it is clear that, disregarding any height error, the optic in question is by design diffraction limited

Given that the chosen parameters are similar for several beamlines under operation at the LCLS and other XFELs, the conclusions that follow from this work are quite general. By erroneously using a simple geometrical argument, one could come to the conclusion that the rms slope error *μ*_{rms} the optic must meet in order to fulfill the diffraction limited requirement is

## 3. Physical optics propagation simulations

Our physical optics simulations are one-dimensional, in that the complex electric field is defined along the x-axis, and it propagates along the z-axis. This allows us to use a large numerical grid to correctly handle high frequencies in the power spectral density of the mirror's height error. We assume that the height error is isotropic making our one-dimensional height profile a reliable representation of a true two-dimensional surface. The LCLSsim code uses the PROPER library of functions [9], translated to the Python programming language. The *λ* = 1.5 Å source is gaussian, with a waist located 13 m upstream of the undulator exit, and divergence *ρ*_{FWHM} = 4.75 *μ* rad. This value matches the measured value of the LCLS beam divergence at *λ* = 1.5 Å [10]. The beam is propagated 374.7 m downstream of the undulator exit slit, where it is reflected by a single concave optic with grazing angle *θ _{i}* = 3.35 mrad, length

*L*= 40 cm, and focal length

*F*= 8.12 m. During propagation, the only phase error imparted to the wavefront is caused by the height error on the surface of the focusing optic. The numerical pixel size is 0.5 mm, and is limited by the numerical grid size used in our simulations. Consequently, the largest frequency correctly handled without aliasing is 1/1.0 mm

^{−}^{1}. The goal of our simulations is to determine whether, for constant height error, lower slope errors yield lower FWHM for the beam intensity profile at the focal plane, as predicted by geometric optic.

A random height error profile is generated over the 40 cm length of the focusing mirror, and is then filtered through a measured height profile of a real x-ray optic to replicate the fractal form of the power spectral density (PSD) typical of highly polished surfaces. Such profile is then Fourier transformed, and selected frequencies are removed in order to obtain different values of slope error *μ*_{rms}, while keeping the height error *σ*_{rms} constant. For this simulation we use two profiles, both with *σ*_{rms} = 1.0 nm, and *μ*_{rms} = 320 nrad and 30 nrad respectively. The profiles are shown in Fig. 1. The beam intensity profiles calculated at the focal plane are shown in Fig. 2, where we also show for reference the profile intensity obtained using an ideally flat mirror with no height errors. As a measure of the intensity profile quality at focus we consider its FWHM, and the on-axis Strehl ratio. Table 1 reports the simulation results.

It can be readily seen that the lower slope error case does not outperform the higher one. In disagreement with geometric optics, we observed no appreciable difference in the numerically computed FWHM, and no appreciable difference in on-axis Strehl ratio. Additionally, the mirror with lower slope error causes an undesired increase in the intensity of the wings at both sides of the central spot. This effect is not surprising if we consider that lowering the slope error while keeping the height error constant results in a power spectral density that is heavily weighed towards low frequencies. These are responsible for low angle scattering and contribute to the increased intensity near the focal spot center. In other words, for a constant height error, lowering the slope error enhances scattering near the focal region of interest, which is generally undesirable. To make sense of this seemingly unintuitive results it is necessary to look back at the theory by Church and Takacs [6]. In their work they define the critical length

Spatial wavelengths greater than *W* diffract into the image core, while spatial wavelengths less than *W* diffract out of it. The former represents the slope error contribution, while the latter the finish contribution. According to the formalism of Church and Takacs, the on-axis Strehl ratio can be written as

*I*(0) and

*I*

_{0}(0) are the on-axis intensities of the beam at focus for an imperfect and an ideally flat mirror respectively,

The upper and lower limits of the integrals define the regions over which slope error and finish matter. Here the PSD (*f _{x}*) is non-zero only for 1/

*L*<

*f*< 1/λ. Given the parameters used in our simulation, the critical length

_{x}*W*~ 0.48 m. We point out that for any optic satisfying Eq.(4) the critical length will always be larger than the mirror size, measuring at most $W~\sqrt{2}L$. Then, since the PSD is zero for frequencies lower than 1

*/L*, the integral for

*μ*reduces to zero, and only the rms height error contributes to the image degradation. Since this is kept constant for the two height error profiles used in this simulation, no appreciable difference is observed in the on-axis Strehl ratio. According to Eq.(7), the Strehl ratio for both the 30 nrad case and the 320 nrad case is 0.926, in good agreement with our numerical results shown in Table 1.

It is important to point out that slope error specifications can play a role for certain XFEL applications. Consider for instance the Tender X-ray beamline planned for LCLS II. This beamline is supposed to cover photon energies from 0.4 keV to 6 keV. This broad range requires large angles of incidence to collect the lower photon energies. Additionally, in order to collect the light from both the soft and hard x-ray undulators planned for LCLS II [11], while being able to focus at different experimental stations, the focal distance of the optic should be adjustable and in the 6 m range. In the currently studied configuration such beamline could have an optic ~ 118 m downstream of a *d*_{source} ~ 30 *μ*m source, focusing x-rays 6 m downstream of its center with a grazing angle of 7 mrad. Assuming the length of the mirror is *L* = 40 cm, Eq.(2) and Eq.(3) show that this configuration is not diffraction limited as the source term dominates. In this case *W* = 0.15 m, and according to Eq.(7) the on-axis Strehl ratio for the *μ*_{rms} = 320 nrad and the *μ*_{rms} = 30 nrad case measures 0.86 and 0.91 respectively. We confirmed these values by performing physical optics propagation simulations for these two configurations. In Fig. 3(a) we plot the intensity profiles at focus in linear scale and log scale (inset). Our numerical simulations yield a Strehl ratio of 0.86 and 0.92 for the *μ*_{rms} = 320 nrad and the *μ*_{rms} = 30 nrad case respectively. We point out that the reduced Strehl ratio for the higher slope error case does not correspond to an increase in FWHM of the intensity profile. In fact, as we can see from the log plot, most of the intensity is lost at large scattering angles far away from the central spot. This suggests that the on-axis Strehl ratio is a reasonable metric of performance even in this source-limited case. In order to further investigate the performance of this proposed optical design, we perform additional simulations for the *μ*_{rms} = 320 nrad case varying the grazing angle of the focusing mirror. Increasing the latter reduces the diffraction contribution in Eq.(1), and should therefore enhance the contribution of slope errors to the performance of the optic. In Fig. 3(b), we compare the Strehl ratio we obtain via physical optics propagation simulations to that of Eq.(7). As predicted by the model, angles larger than 7 mrad cause further intensity loss on-axis, and should be ruled out if possible from the final design of the beamline. The good agreement between the simulations and the model suggests that the model still provides a reasonable metric to define optics for XFEL applications.

## 4. Conclusions

When specifying the slope error requirement for diffraction limited x-ray optics, a simple geometrical argument fails to take into account the frequency-dependent contribution of an optic height error to the intensity profile at focus. In this work we suggest that the model of Church and Takacs can still be used to define height error specifications for XFEL optics. We have shown that for optics satisfying Eq.(4) a looser tolerance can be adopted for slope errors, while on the other hand special attention must be paid to the surface finish, Eq.(9), which now spans the entire frequency spectrum of the surface from 1*/λ* to 1*/L*. Eliminating the need for slope errors well below 300 nrad rms could lead to a considerable saving in both the optical manufacturing and metrology, expediting deployment. Additionally, we suggest that for optics working in the geometric regime, the model gives a reasonable estimate of the on-axis Strehl ratio which can be a useful metric for a variety of experiments.

## Acknowledgments

This work was performed under the auspices of the U.S. Department of Energy (DOE) by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344, and under the auspices of the U.S. Department of Energy at SLAC under Contract No. DE-AC02-76SF00515. Document Release Number LLNL-JRNL-677641.

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