Diffraction gratings are key elements of soft X-ray synchrotron beamlines. Besides wavelength dispersion, specific parameters can be tailored to adjust the energy dependent efficiency and focusing, and to correct wavefront aberrations. As key elements of a beamline, any departure from the design values can severely reduce the overall performance. On the other hand, known non-conformities can often be corrected by slight adjustment of the alignment parameters. A careful and accurate metrology is therefore required before installation on the beamline. After presenting what variable line spacing gratings, variable groove depth gratings, and alternate multilayer gratings are, the use of the SOLEIL long trace profiler for the measurement of groove density variation along the surface and of the atomic force microscope for the groove geometry and roughness characterizations will be discussed. A few examples of grating metrology will be presented and analyzed with the help of optical simulations.

## I. INTRODUCTION

V-UV and soft X-ray beamlines usually rely on grating monochromators for photon energy selection. At SOLEIL, most of them are Variable Line Spacing (VLS) gratings. VLS gratings allow compensating the tuning defocus when the gratings are not used in perfectly collimated beams, and also coma aberration correction. In some cases, they may even provide the monochromator focusing. We talk about the “VLS law” which is expressed as a polynomial variation form of the grating line density along the propagation axis of the synchrotron beam and therefore, in most cases, along the longitudinal axis of the grating. The line density K(x) is thus expressed as follows:

where K is in l/mm and x is the position on the substrate in mm (x = 0 at the center). Each term of the polynomial is related to a specific wavefront aberration: K_{0} is the central groove density, K_{1} is the focus term, K_{2} is the coma correction, and K_{3} is the correction of spherical aberration. In most of the synchrotron applications, the aperture angle is kept very small and therefore the influence of K_{2} and K_{3} is usually quite small. At SOLEIL, most of VLS are holographic ion-etched lamellar gratings because the positions of the holographic recording sources offer little control on these coefficients, one only tries to keep them low. That is why we usually optimize these coefficients directly on holographic source points.

Many of the gratings used at SOLEIL also have a Variable Groove Depth (VGD) along their width, which provides good tunability of the efficiency. Therefore they have a groove height that varies laterally to better adjust the diffraction efficiency and to allow higher harmonic rejection (Fig. 1). The properties of VGD and VLS gratings can be combined, and this is the case for most of the gratings installed in SOLEIL beamline monochromators.

Alternate Multilayer (AML) gratings, which are special kinds of multilayer coated gratings where the groove depth is close to one half of the multilayer period, are used in the tender X-ray range, up to 5 keV; they have been the subject of a special development effort at SOLEIL.^{1,2} The AML grating presents a double periodicity at the nanometer scale. In the plane of the surface, the period is the pitch p of the laminar grating, while it is the 2d spacing of the multilayer in the vertical direction (Fig. 2). Therefore, an AML grating has properties similar to a crystal with the advantage of the freedom of choosing the periodicities. AML gratings make a connection between grating and crystal spectral domains, and offer a very high diffraction efficiency at intermediate energies over 1 keV where blazed gratings would require quite shallow and difficult to produce blaze angles. Each step of the AML manufacturing process (grating etching and multilayer deposition) requires several metrology controls: blank measurement, AFM characterization, and diffraction efficiency measurement.

More than 35 VLS/VGD, blazed, and AML gratings are mounted into SOLEIL monochromators, and all of them have been thoroughly characterized for geometrical parameters before installation in the Optical Metrology Lab of the Optics Group^{3} and on the METROLOGIE beamline for diffraction efficiency.^{4}

VLS laws are characterized with our Long Trace Profiler (LTP). Independently, atomic force microscopy enables us to measure the groove depth and the profile shape (duty cycle and slopes), and their variations over the grating surface. The geometric parameters being known, a model of the grating can be entered in computation codes based on electromagnetic field propagation^{5,6} to predict the diffractive properties. Finally, the predicted properties are compared with “at wavelength” diffraction efficiency measurements performed on a synchrotron beamline. If this is not possible, the analysis of the resolving power of the monochromator where the grating will be installed can be used. But the influence of the grating itself on the result is difficult to separate from the other optical components.

## II. LONG TRACE PROFILER FOR VLS LAW MEASUREMENTS

In the case of diffraction gratings used in synchrotron monochromators, the local line density variation can be extracted from the measured slope variation. At the 0 diffraction order (in normal incidence), the specular reflection of the grating and therefore its topographic profile is measured. On the other hand, by measuring the diffracted beam at the first order (or other higher order) in Littrow mode, the line density variation can be easily extracted. It is obviously necessary that the diffraction efficiency in visible light (in our case at 532 nm) is sufficient so that the signal on the detector has sufficient intensity and/or that the incident light intensity is sufficient (in our case, the laser light used is 50 mW). Obviously, the diffracted beam measurement must be made at the Littrow angle α_{0}, which corresponds to the case where the diffracted angle is equal to the incident angle.^{10} The beam thus goes back on itself in the opposite way (Fig. 3).

The beam is adjusted to the Littrow angle at a grating position where the line density is K_{0}; the incoming (α) and diffracted (β) beams have the same direction α = β = α_{0}, hence

where N is the order of diffraction. When the beam is scanned over the grating, the line density K changes and the diffraction angle becomes β = α_{0} + θ, such as

The LTP is able to measure slopes with a repeatability uncertainty smaller than 0.5 *µ*rad, hence, deflection θ with an uncertainty smaller than 1 *µ*rad. However the absolute accuracy is given by the LTP calibration and evaluated as ΔK/K = Δθ/θ = 10^{−5} (see the Appendix for uncertainty calculation). For practical reasons, the Littrow angle cannot usually exceed 75° (K limit = 2/N λ), corresponding to a maximum measurable line density on the order of 3600 l/mm. A drawback of the method: the footprint of the measuring beam varies according to the inverse of the cosine of the incidence angle. The measuring beam size varies then from 2 mm (normal incidence) up to 8 mm (75° incidence). This increase of the integrated measuring surface may change the local apparent line density^{11} and even adds the focusing or defocusing effect. But taking into account the low values of the second order and higher terms of most polynomial VLS laws, the effect of the variation of the beam size, in most cases, can be neglected.

The aim is not to present the different measurement principles used by synchrotron metrology laboratories for determining the topography of an optical surface by measuring its local slope, but to propose a method to adapt it to the measurement of variation of density of lines. These concepts and properties have been presented many times: in short, there are two classes of instruments that measure the local slope by deflectometry. The first one is based on a modified high resolution auto-collimating telescope, the Nanometer Optical component measuring Machine^{7} (NOM). These instruments are simple to implement with a few optical components. They achieve reproducibilities in the order of 0.03 *µ*rad rms and 0.05 *µ*rad rms accuracy.^{8,9} However the calibration of the instrument is rather sensitive and may depend on the diameter of the measuring beam which may vary from 0.8 to 5 mm with an optimized aperture of 2.5 mm.^{12}

The second class Long Trace Profilers (LTPs) measure the direction of a parallel beam (usually about 2 mm in diameter) reflected back by the surface, typically by means of CCD or CMOS detector placed in the focus of the Fourier transform lens, as described later. The detection sensitivity of the measurement position is increased by the use of an interferometric system which multiplies the signal detected on the detector by a fast changing sine function. This gives a dark fringe at the center. The displacement is then better evaluated by measuring the position of the minimum. The repeatability of such instruments is of the order of 0.1 *µ*rad rms with an overall accuracy of 0.2 *µ*rad.^{13} Calibration can be done *in situ* by simple comparison with inclinometer measurements. Once aligned, the LTPs can operate for weeks without any additional adjustment or calibration.

The SOLEIL LTP has been developed since 1998, and it has been constantly improved to keep pace with the demands of optical surface measurement quality. It was constructed from a custom 1 m translation stage on air bearings powered by a linear motor that allows us to make “on the fly” measurements along the X axis. The beam incidence on the Surface Under Test (SUT) is classically stabilized by a mirror penta prism. It is limited by a small iris. The sinusoidal fringe pattern is produced by a polarization interferometer based on the Nomarski principle.^{14} The detector is a CMOS camera (2048 × 2048 pixels).

The VLS grating measurement configuration of the SOLEIL LTP is the one shown in Fig. 3(c) with the inclined beam. It seemed easier to rotate the measuring beam rather than the grating. Indeed, for high line densities, the inclination of the grating increases the distance between the measuring head and the SUT. In addition, the measurement position of the grating no longer corresponds to that of the stage, but this configuration also allows setting the optical pupil as close as possible from the SUT, which limits the effects of vignetting and contributes to the stability of the measurement position. The device does not allow a precise measurement of the Littrow angle. However its value can be determined with sufficient precision from the knowledge of the central line density, which is usually given by manufacturers with an accuracy of less than 0.5 l/mm (supplier’s metrology reports). This gives, in the case of a grating of 2400 l/mm, an uncertainty of 0.21° on the Littrow angle evaluation (α_{0} = 39.67°). The interpretation of the variation of slope in variation of line density passes by the knowledge of K_{0}. This uncertainty generates indirectly uncertainty on the conversion of the slopes (*μ*rad) into number in line density (l/mm) which will be added to the uncertainty budget. Thus, the evaluation of the polynomial VLS law carried out using a polynomial fit will give an uncertainty on each coefficient of the polynomial calculated from a variance analysis (ANOVA, available on most fitting plot softwares) (see Sec. IV A).

One of the key parts of the LTP optical head is the focusing lens also called Fourier Transform Lens (FTL)^{9} since it should be aberration free. Basically, the FTL converts the deflection of the measurement beam coming from the SUT into a displacement read by the detector. For curved mirrors, the beam scans the entire diameter of the lens (50 mm) for an observable angular field of 8 mrad. Our original optical system used an ordinary plano-convex lens of 0.5 m focal length. The optical simulations using the OSLO^{®} ray tracing software carried out that, at its center (normal incidence), the lens is used under optimal conditions and thus was only limited by diffraction. On the other hand, used at the edge of the diameter (at ±4 mrad), the field distortion could reach 1% or more. From an optical design initially meant to read barcodes, we have developed a doublet of lenses to avoid distortion to less than 0.002% over a field of 10 mrad (Fig. 4). The LEEP (Linearity Error Elimination) method^{15} was used to validate this new optical system and characterize the instrument error function. Figure 4(d) shows the instrument error functions for the two types of FTLs. The distance between the 2 lenses of the doublet was finely adjusted by LEEP analysis as well [Fig. 4(e)]. It can be seen that the distance between the 2 lenses has not much influence on the instrumental function. Due to the doublet, we could increase the size of the detector, and therefore the observable field, which has been extended 8 mrad (1024 pixels CCD camera) to 16 mrad (2048 pixels CMOS camera).

This new optical system was originally developed for curved mirror measurements but was also very useful for measuring VLS laws with high variation without using stitching methods. For example, for a grating length of 200 mm, we can measure K_{1} coefficients up to 0.170 l mm^{−2} and 0.340 l mm^{−2} with the LEEP method beyond which the calibration of the instrument is no longer valid. Stitching VLS law measurements is indeed, quite possible, but requires extra care. The stitching procedure must be applied on measurements already converted into line density, with a Littrow angle correction since each measurement trace is obtained with a different tilt of the SUT. One must also take into account the shift of the beam position on the SUT generated by this tilt.

## III. ATOMIC FORCE MICROSCOPY FOR GRATING CHARACTERIZATIONS

Until the 1990s, atomic force microscopes were dedicated to specialists. Then, step by step, these instruments were democratized and thanks to the improvement of their control systems, servo loops, and user-machine interfaces, these instruments have become usable by a large community of scientists.^{16} Since the 2000s, many instruments have been available on the markets which were easy to use after few weeks of practice. The SOLEIL optics group has early acquired this type of instrument for the characterization of gratings. The selection criteria were as follows: the capacity of the instrument to accommodate optical pieces around 200 mm long, 40 mm wide, and 60 mm thick with a weight of several kg, the linearity quality of piezo-movements, and their servo control speed, low measurement noise, easiness of calibration, and also user-friendliness of the software interface. From 2002 to 2015, SOLEIL used an instrument from Pacific Nanotechnology Company (Nano I). Then, in 2015, a NX 20 from Park Company was acquired. This AFM has only one direct piezo-movement with a 15 *µ*m stroke (Z) and 2 flexure stages for the X and Y the SUT displacements and thus offers an excellent linearity quality. The thermal and vibration shielding enclosure and the X, Y, and Z movements of the instrument have been customized to accommodate large gratings. The measurement noise is less than 12 pm rms over 90 × 90 *µ*m^{2} scans. This instrument allows non-contact topographic measurement which limits the tip wear and therefore limits errors due to change in the tip geometry during a measurement session. Before any grating characterization, a calibration procedure of the instrument is performed.

In order to validate our procedures, we participated in an inter-comparison campaign between AFM users organized by the Laboratoire National d’Essais.^{17} A careful calibration procedure was applied to ensure a high dimensional accuracy. Thanks to this inter-comparison, we have been able to optimize and to validate our measurements and data analysis in accordance with the international standard ISO 11952. The analysis of the data is performed on scanned images sized to display 4–6 grating grooves. These images are used to extract the measurement of the line heights, duty cycles, blaze angles, and the apparent roughness (Fig. 5). Due to the small observed field (2 *µ*m × 2 *µ*m for a grating of 2400 l/mm), it is not possible to accurately extract a local line density measurement and to be able to go back to the absolute line density all along the grating and therefore to the VLS law.

## IV. MEASUREMENT RESULTS

### A. AML/VLS characterization

A Multilayer Grating Monochromator (MGM) has been recently developed for the SIRIUS beamline of SOLEIL, which is dedicated to the study of thin films, nanostructures, and advanced materials using X-ray diffraction and spectroscopy between 1.4 and 12 keV.^{18} This monochromator features a variable line spaced holographic grating with a line density of 2400 lines/mm with 35 periods of a Cr (2.5 nm)/B4C (4.1 nm) multilayer and includes, as a second optical element, two matched multilayer mirrors. This monochromator can provide high throughput through the whole 1-4 keV range. The metrology steps applied to control the grating fabrication were as follows: the first step was the characterization of the topography of the silicon substrate before etching in order to evaluate the slope error and, after etching to check whether this slope error had evolved during the process. This step was performed using the LTP by measuring the local slope at normal incidence on the substrate (Fig. 6). The VLS law was then measured under Littrow incidence (39.67° for 2400 l/mm), with a measuring beam diameter of 2.6 mm. Then, the etching depths of the groove were evaluated on a large number of sampling points. From this depth map, a grating simulation code was used to determine the optimal ML period. In this case, an average height of 3.3 nm was measured with the AFM. The thickness of the bilayer (B4C + Cr) was therefore 2 × 3.3 nm, as presented below. Then, the final profile of the multilayer grating was checked with the AFM (Fig. 7). A rounding of the grating profile was observed, which can be entered in the simulation code to estimate the AML efficiency. An ultimate characterization step was to determine the diffraction efficiency of the grating at the working wavelength. This step was carried out on the SOLEIL METROLOGIE beamline; these results and the nice match with metrology deduced simulations will be published later.

### B. VGD gratings

The characterization of a VGD grating is performed using the AFM. Indeed, the heights of the lines measured by the manufacturer are usually performed before the coating deposition. If the thickness of the coating is not homogeneous, the resulting height may be different (Fig. 8). As with all gratings, special attention is paid to the measurement of roughness in the line lands but also in the valleys because the deposition of Pt or Au coating can, sometimes, alter the native roughness of the substrate.

### C. VLS law analysis

The SOLEIL metrology lab is open to make grating measurement for other laboratories. It was for instance asked to measure the VLS law, blazed angle, and roughness of one of the gratings of the SIX beamline at NSLS II. This beamline is dedicated to soft X-ray inelastic scattering and was especially designed to achieve a resolving power of 70 000 at 1 keV.^{19} The VLS law measurement of this grating, which is a ruled blazed grating of 500 l/mm, was performed at a Littrow angle of 7.6°. In Fig. 9, the line density of this grating is plotted after a best third order polynomial subtraction. At 73 mm from the center, one observes a peak. This artifact results from an interruption of the ruling machine at this position on the grating and its later resuming at this point. In 15 years of grating metrology and more than fifty gratings measured, this was the first time we observed this kind of defect. It was therefore very interesting to analyze thoroughly the consequences of using this grating in a monochromator. Unfortunately, it was not possible to measure the “at wavelength” diffraction efficiency which would have been the best solution to analyze the consequences of the observed defect. In order to determine if the grating was acceptable in that state and to determine if the resolution of the monochromator would be decreased, 2 types of numerical simulations were tested and presented as follows.

#### 1. Simulation of grating performance

If the grating is illuminated along the longitudinal direction by incoherent illumination, then the light is only sensitive to the position and profile of neighboring grooves over a short distance. In this limit, the usual ray tracing assumption that the grating diffracts according to the local line density is a good approximation. However, if the grating is illuminated by fully coherent light, then the light becomes sensitive to the position and profile of every groove covered by the envelope of the light incident on the grating. Modern synchrotron sources are diffraction limited in the soft X-ray region in the vertical dimension, corresponding to the longitudinal direction along the grating. It is therefore anticipated that a coherent description of the diffraction will be required to properly simulate the grating performance.

The LTP measures the local line density as a function of the longitudinal position along the grating. In the ray tracing approximation, the LTP data can therefore be used directly. Due to the fact that it can accept an arbitrary line density profile, the simulation package XRayTracer^{®}^{20} has been employed to do the ray tracing simulations. For coherent simulations, knowledge of the position of the grooves is required. The position of the individual grooves is determined by integrating the line density obtained by the LTP data. The spatial resolution of the LTP is ∼2 mm which is clearly not sufficient to resolve individual grooves; we simply linearly interpolate the data between data points. Any higher spatial frequency line placement errors are not accounted for, although we do not anticipate significant higher frequency errors from the interferometric control of the ruling process. To do the coherent simulations, the python code was developed. To keep the code simple and yet capture the essential physics, we used kinematic diffraction theory with several reasonable assumptions to model the resolution of the gratings. The grating is placed into the identical arrangement as it will be in operation; r_{1} and r_{2} represent the input and output arm, as measured from the center of the grating, and α and β are set to the correct values for the desired energy (Fig. 10). The beam will then focus at the focal plane, which is at the exit slit for a perfect grating. The intensity at any point along the focal plane x is then given by

In this equation, A_{i} is the electric field amplitude at the *i*th groove on the grating and k = 2π/λ. This is known from the source divergence (21 *µ*rad σ). The summation is taken over all grooves.

#### 2. Discussion of the results

Figure 11 shows the results of the coherent and ray tracing simulations. Figure 11(a) shows the results of coherent simulations with an idealized point source. Shown are the monochromatic images at the exit slit. While this is an unphysical situation, we present this to emphasize the small effects of the ruling errors, which otherwise would be washed out by the finite source size. The solid black line shows the results for an ideal grating for reference. The black dotted line shows the same except for the real grating, using the LTP data. The two have nearly the same peak width, which points to the quality of the ruling. There is a small amount of increased scattering in the tails due to the ruling imperfections.

The coherent simulations also allow us to address the issue of the phase slip. The LTP data show a clear singularity at ∼73 mm from the grating center. This is due to a phase slip in the grating at this point, most likely due to a mechanical interruption during the ruling process. Under normal operations, LTP measurements indicate the presence of a phase slip on a grating by a singularity in the LTP trace but do not measure the magnitude. In this case, for the simulations we simply assume the worst case phase slip of π radians. The red trace shows the results for the real grating when the phase slip is included. Under these assumptions, the phase slip contributes more to the peak width and scattering in the tails than the line placement error. However, once these results are convolved with the real source and exit slit size, the degradation in the resolution is still small. This is because the phase slip occurs far from the grating center, and the grating is illuminated by a rather Gaussian intensity distribution so that the effect of the phase slip on the resolution is minor. The effect of the phase slip would be greater if it were located near the center of the grating, depending on the magnitude of the phase error.

Figure 11(b) shows the results of the coherent simulations once the effect of the real source and exit slit size is included. A source size of 24.0 *µ*m σ and an exit slit size of 30 *μ*m are used to represent the actual conditions at the beamline at 400 eV. The source effects are included as a convolution over the source intensity distribution. An ideal grating would yield a resolution of 11.8 meV, while the inclusion of the line placement errors and phase slip lead to a slight degradation of the resolution to 12.7 meV.

Figure 11(c) shows the results of ray tracing simulations. In the case of an ideal grating, the resolution is predicted to be 11.5 meV or essentially the same as the coherent simulations predict. However, once the effect of the line placement errors is included, the ray tracing predicts a more severe resolution degradation of 28.5 meV. Experimentally, a resolution of 22.4 meV for the N_{2} spectrum at 400 eV was measured.^{16} We consider this as an upper limit of the resolution, due to the limitations of measuring such a small resolution with gas phase absorption measurements, and expect to get a better measurement of the resolution when the inelastic spectrometer is commissioned. In addition, this measurement includes all error contributions to the resolution along the beamline, not just those due to imperfect ruling. Thus, our results show that in order to correctly predict the performance of high resolution soft x-ray beamlines, physical optics simulations of the grating, as well as accurate metrology data of the real grating are required.

## V. CONCLUSION AND PERSPECTIVES

35 gratings are installed on SOLEIL beamlines and have been carefully characterized for slope errors before and after groove etching and the VLS law by LTP measurements. AFM was used for groove geometry and roughness measurements and for most of them, the efficiency was measured at wavelength of utilization of the grating. Grating metrology is a specificity of our optics group, and we endeavor to keep the highest level of accuracy. In order to improve the mechanical stability of the measuring head and to be able to keep the pupil of the optical system (aperture stop) as close as possible from the SUT, thus avoiding vignetting and making it possible to have a constant measuring beam, a new optical head is under development (see Fig. 12). The optical principle will remain unchanged. The Wollaston prism support will be made more flexible to allow easier adjustment of the shearing interference pattern which can be unbalanced when the grating modifies the diffracted beam polarization. For the development of future monochromator generations, we are working on the development of a new calculation code for the treatment of grating diffraction. This code is able to calculate the diffraction grating properties from a height mapping measured with the AFM. For high groove densities, the AFM image is small (less than 5 × 5 *µ*m). We are working on the possibility of stitching on AFM scans. Due to the excellent linearity of our instrument and its low noise, it is possible to connect recorded images in the measuring field accessible by the flexure stage of 90 × 90 *µ*m.

Grating metrology at SOLEIL is a very important research topic for maintaining monochromators at the highest level of performance. That is why we are always looking for ways to improve our methods.

## ACKNOWLEDGMENTS

A portion of this work was done at the SIX beamline of the National Synchrotron Light Source II, a U.S. Department of Energy (DOE) Office of Science User Facility operated for the DOE Office of Science by the Brookhaven National Laboratory under Contract No. DE-SC0012704.

The authors would like to thank the SOLEIL Design Group and especially Marc Ribbens and Alexandre Carcy for the design of this new optical head.

### APPENDIX: UNCERTAINTY BUDGET

We propose to calculate the uncertainty of type A^{22} (repeatability) on the determination of the value of the line groove density measurement K. The influence quantities are the measurement of the angle variation (θ) viewed by the LTP and the uncertainty on the knowledge of the laser diode wavelength.

We assume that the Littrow condition is satisfied at the center point of the grating where the line density is K_{0}, Eq. (2) in Sec. II. At a position where the line density is K, the return beam is deflected by an angle θ given by Eq. (3). Combining these two equations yields

Derivating this equation with respect to θ and λ gives the following relation on uncertainties:

From Eq. (1), we have

Then

For a 2400 l/mm grating, α_{0} is 39.67°, Δθ = 1 *µ*rad (see Sec. II), θ is typically 10 mrad, Δλ for laser diode depends on the device used.^{21}

where n is the index of refraction and L is the laser diode chip length. Taking typical values for YAG n = 1.8 and cavity length L = 1 mm, Δλ = 0.08 nm at 532 nm. For N = 1,

In conclusion, the contribution of these 2 terms is of the same order of magnitude and smaller compared to calibration uncertainty.

## REFERENCES

_{4}C/Mo

_{2}C alternate multilayer grating for monochromators in the photon energy range from 0.7 to 3.4 keV

_{4}C multilayer mirrors: Study of interfaces and x-ray reflectance