To match the evolution of the light source facilities (synchrotron radiation and free-electron lasers), X-ray optical elements must be at the diffraction limit to deliver the high-quality X-ray beam to the end station for scientific research. As a type of widely used X-ray optics, X-ray mirrors are required to be fabricated at the sub-100 nrad Root Mean Square (RMS) level for residual slope errors or the sub-nm RMS for residual height errors to preserve the wavefront of the incoming X-ray beam and produce a diffraction limited focal spot. Special dedicated optical metrology systems were developed to characterize such high-precision long rectangular X-ray mirrors. The Long Trace Profiler (LTP) [1] and the Nanometer Optical component measuring Machine (NOM) [2] are two classical optical slope profilers widely used in the light source facilities all over the world [3–8]. With slightly different configuration, the Nano-accuracy Surface Profiler (NSP) [9] was developed with two separate beam arms. The sample beam arm (with the sample beam autocollimator) scans the test mirror surface (x-scan) with a fixed working distance, while the reference beam arm (with reference beam autocollimator) monitors the carriage wobble.

To meet the increasing need of the scientific research, more strongly curved focusing mirrors have been proposed for soft X-ray beamline. A soft X-ray nanoprobe will offer nano-imaging and spectroscopy tools non-destructive capabilities to study advanced materials using Nano ARPES and Nano RIXS experimental techniques [10]. To produce a diffraction-limited spot size for low energy (high λ), X-ray mirrors with high numerical aperture are required. These mirrors can have a total slope range larger than the current measuring range of the LTP/NOM/NSP (10 mrad). To overcome this problem, it is possible to stitch partial slope data recorded by measuring the mirror at different pitch angles. Based on this idea, Polack et al. proposed the Linearity Error Elimination Procedure (LEEP) algorithm [11]. This algorithm can reconstruct not only the mirror slope profile, but also the instrument error of the optical head at the same time. A modified LEEP algorithm was proposed for the NSP setup to reconstruct the instrument error of the sample-beam autocollimator [12]. The ambiguity in the algorithm was addressed by proposing several regularizations in the data acquisition and the algorithm constraints.

On the experimental side, when the LEEP algorithm or any other stitching algorithm is used where the pitch angle must be adjusted between different x-scans, the rotation center of the pitch needs to be known to a certain extent. The error on the position of the rotation center leads to pitch-dependent discrepancies of the sampling positions on the test mirror. For circular cylinders and shallow elliptical mirrors (few mrad total slope), the pitch rotation center does not need to be known with high accuracy, because the sampling position errors introduce little systematic slope errors in the final results (<10 nrad RMS). However, when highly curved and asymmetric elliptical mirrors must be measured, it becomes critical to know the pitch rotation center location with enough accuracy as this error leads to nonnegligible total systematic slope error contribution. In addition, the commonly applied “flat assumption” in the LEEP algorithm or the multi-pitch NSP technique as shown in Figure 1, assuming that the mirror sag is negligible, may no longer be valid when measuring such “extreme” ellipses.

In this work, the theory of the multi-pitch NSP technique is briefly reviewed, followed by the analysis of the influence of the pitch rotation center location with respect to the surface of the mirror under test. We propose a method to overcome the invalid “flat assumption” issue in the reconstruction of “extreme” ellipses. Then we focus on the measurement tolerance of the pitch rotation center location in the horizontal and the vertical directions. Series of simulations with different ellipse geometries are carried out to study the trend between the measurement tolerance of the pitch rotation center location and the curvature variation of the aspherical mirrors. We implement a Monte Carlo simulation with a real mirror design. Taking the uncertainty of the pitch rotation center measurement and the uncertainty of the angle measurement as two inputs, the multi-pitch NSP technique with the Monte Carlo simulation can give the slope residuals and an estimation of the grazing angle θ and the chief ray location x o with their respective uncertainties.

Based on the redundant dataset acquired from all the x-scans for different pitch angles, the multi-pitch NSP technique can simultaneously calculate the mirror surface slope, the instrument error of the sample-beam autocollimator, and the introduced pitch angles [12]. For simplicity, we define the origin of the world coordinate system ( x w ,   z w ) = ( 0 ,   0 ) at the rotation center of the pitch angle α as shown in Figure 2. The direction of the x-translation in the NSP is defined as the direction of X w axis in the world coordinate system, while the Z w axis in the world coordinate system points upwards.

The center point of the x-scan range on the mirror surface is defined as the origin of the mirror coordinate system ( x m ,   z m ) = ( 0 ,   0 ) . The x-axis of the mirror coordinate system X m is along its tangential direction and shares the same direction of X w when the pitch angle α = 0 as shown in Figure 2A, but they are different for other pitch angles as illustrated in Figure 2B. The Z m axis of the mirror coordinate system is defined along the surface normal and pointing outwards the mirror surface. When the pitch angle α = 0 , the Z m axis shares the same direction of Z w . The coordinates of the pitch rotation center, expressed in the local frame of the mirror coordinate system, are given by ( x m ,   z m ) = ( x c ,   z c ) . It is illustrated in Figure 2 that, for the same world abscissa x w , the mirror abscissa x m can be different at different pitch angles α . Moreover, as shown in Figure 2B, there is a clear difference in the x m calculation with and without “flat assumption” which will be addressed in Iterative Reconstruction.

After a rotation in pitch with an angle α , one point ( x m ,   z m ) in mirror coordinate system can be transferred to the world coordinate system as [ x w z w ] = [ cos ⁡ α − sin ⁡ α sin ⁡ α cos ⁡ α ] [ x m − x c z m − z c ] . (1) The x position in world coordinate system can be calculated as x w = ( x m − x c ) cos ⁡ α − ( z m − z c ) sin ⁡ α . (2) For a particular measurement ( x c , z c , and z m are determinate), the x m can be expressed as a function of x w and α . x m ( x w ,   α ) = x w + ( z m − z c ) sin ⁡ α cos ⁡ α + x c . (3) Under the n th actively-introduced pitch angle α n , the angular signals captured in the sample-beam arm s n ( x w ) and the reference-beam arm r n ( x w ) can be explained by the mirror slope m ( x m ) at the actual sampling position x m and the instrument error of the sample-beam autocollimator e ( s n ( x w ) ) at its current reading s n ( x w ) as well as the uncorrelated additive random noise n a . s n ( x w ) + r n ( x w ) ≈ m ( x m ( x w , α ) ) + e ( s n ( x w ) ) + α n + n a , n ∈ [ 0 , N − 1 ] . (4) Here the instrument error of the reference-beam autocollimator is ignored in our model, as the reference-beam angle r n ( x w ) varies in a small range, usually <10 μrad, if an air-bearing translation stage is used.

The mirror slope m ( x m ) and the instrument error of sample-beam autocollimator e ( s n ( x w ) ) are then represented by two independent uniform cubic B-splines M ( x m ) and E ( s ( x w ) ) as m ( x m ) = M ( x m ) c m and e ( s n ( x w ) ) = E ( s n ( x w ) )   c e , respectively, where c m and c e are column vectors containing the B-spline coefficients. Therefore, the mathematical model of multi-pitch NSP becomes s n ( x w ) + r n ( x w ) ≈ M ( x m ( x w , α n ) ) c m + E ( s n ( x w ) )   c e + α n + n a ,     n ∈ [ 0 ,   N − 1 ] , (5) Once the coefficients c m and c e are determined, we can reconstruct the mirror slope m ( x m ) and the sample-beam instrument error e ( s n ( x w ) ) . The multi-pitch NSP algorithm is essentially to optimize the B-spline coefficients c m and c e , and the pitch angles α = [ α 0 , α 1 , … ,   α N − 1 ] ⊤ . In addition, to avoid the ambiguities in the optimized results, several regularizations are applied in the designs of the data acquisition and the constraints in algorithm [12]. Putting the constraints on the first pitch angle, the instrument error intercept, and the linear term, the optimization can be expressed as c ^ m , c ^ e ,   α ^ = arg min   c m ,   c e ,   α  ∑ n = 0 N − 1 ∑ x w ( s n ( x w ) + r n ( x w ) − M ( x m ( x w , α n ) ) c m − E ( s n ( x w ) ) c e − α n ) 2 s . t . α 0 = 0 , E ( 0 ) c e = 0 , s ⊤ E ( s ) c e = 0. (6) where the first pitch angle α 0 is constrained to 0. When the slope signal s = 0 , the instrument error intercept is constrained as e ( 0 ) = E ( 0 )   c e = 0 . If we fit the instrument error e ( s ) up to the linear terms, we have e ( s ) = E ( s )   c e ≜ e ( 0 ) + ε s , where s is the column vector of all slope sample values, the symbol ≜ stands for equal in a least squares sense. and ε is the linear term coefficient. Since we have constrained the instrument error intercept e ( 0 ) = 0 , we have E ( s )   c e ≜ ε s . As the third constraint, the linear term coefficient ε is also constrained to 0, so we have ε = ( s ⊤ s ) − 1 s ⊤ E ( s ) c e = 0 . Because s ⊤ s = | s | 2 ≠ 0 is a nonzero scalar value, this constraint is simplified as s ⊤  E ( s )   c e = 0 .

One thing to highlight is that nonuniform pitch steps Δ α n are necessary in data acquisition to avoid the periodic errors in the optimization results. Technical details on the ambiguities and the regularization can be found in Ref. [12] with more simulations and discussions.

To calculate the sampling position x m ( x w ,   α n ) in the mirror coordinate system from the NSP x-scanning position x w and pitch-scanning angle α by Eq. 3, we need to know the profile of the mirror height z m and where the pitch rotation center ( x c ,   z c ) is in the mirror coordinate system. In the following sections, we are discussing the necessary accuracy of these different values.

The pitch rotation center ( x c , z c ) can be determined using additional metrology instruments. The measurement error of the pitch rotation center can be expressed as e x c = x ^ c − x c , and e z c = z ^ c − z c , where ( x ^ c ,   z ^ c ) is the measured pitch rotation center. If the estimated mirror height profile is symbolized as z ^ m , the estimation error is e z m = z ^ m − z m . In a real measurement, the dwell position of the x-stage x ^ w in the scans are defined as x ^ w = ( x m − x ^ c ) cos ⁡ α − ( z ^ m − z c ) sin ⁡ α . (7) If we have a measurement error of e x c when determining x c , then the true dwell position x w is x w = x ^ w + e x c = x ^ w + x ^ c − x c . (8) Based on Eq. 3, the estimated sampling position on the mirror x ^ m is x ^ m = x ^ w + ( z ^ m − z ^ c ) sin ⁡ α cos ⁡ α + x ^ c , (9) And the true sampling position x m is x m = x w + ( z m − z c ) sin ⁡ α cos ⁡ α + x c . (10) The sampling position error e x m = x ^ m − x m in the mirror coordinate system is e x m = x ^ m − x m = ( x ^ w + ( z ^ m − z ^ c ) sin ⁡ α ) cos ⁡ α − ( x ^ w + x ^ c − x c + ( z m − z c ) sin ⁡ α ) cos ⁡ α + x ^ c − x c . (11) With some simplifications, it becomes e x m = [ ( z ^ m − z m ) − ( z ^ c − z c ) ] tan ⁡ α − ( x ^ c − x c ) 1 − cos ⁡ α cos ⁡ α . (12) Finally, we have the sampling position error as e x m ≈ e z m α − e z c α − e x c α 2 2 . (13) Considering the pitch angle α ≪ 1 rad, the error e z c will contribute a sampling position error of − e z c α in Eq. 13, while e x c only gives − e x c α 2 / 2 error on x m . It means that the error e z c has more impact on the multi-pitch NSP results than the error e x c .

By using a simple measurement tool, such as a metric ruler, and the mechanical tolerances of the pitch rotation system, it is not difficult to determine the pitch rotation center location with a few mm accuracy. For a circular cylinder, this level of uncertainty is enough. The slope profile of a circular cylinder is linear with x m . In this case, the reconstruction is not sensitive to the measurement error e z c , because the slope error with a constant sampling error − e z c α n at the n th pitch α n due to e z c in Eq. 13 will almost be equivalent to a slope offset on this linear slope profile. This slope offset will be treated as a pitch offset and will not affect the slope result by using the reconstruction algorithm. However, if the test mirror become strongly aspherical, the tolerance of the measurement on the pitch rotation center location becomes tighter, especially for the vertical distance z c . Since the slope profile of an aspherical mirror has nonlinear terms, the sampling error e x m will lead to slope errors from these nonlinear terms. This becomes an important error source of the reconstruction in the multi-pitch NSP technique. A tolerance value is needed to control this type of error in the reconstruction result.

For simplicity, the mirror height profile z m is usually assumed as a flat surface, i.e., z ^ m = 0 . This “flat assumption” can be used when the mirror sag is much smaller than the tolerance of e z c . However, if the tolerance for e z c starts to be comparable to the mirror sag, its contribution e z m α to the sampling position error e x m in Eq. 13 cannot be ignored and this will severely affect the reconstruction accuracy. A better estimate of the mirror height profile z m is necessary. Two options can be used: the known mirror shape parameters or an iterative reconstruction with an initial guess.

To conduct a tolerance study, we need to set a threshold. Here we set the threshold on the systematic slope error contributed from e x c and e z c . If 100 nrad slope accuracy is expected, the systematic slope error contribution from e x c and e z c should be less than 10 nrad RMS.

The error contribution due to e x c and e z c is isolated by setting the additive random noise on slope in the model to zero. Therefore, by meeting the reconstruction error <10 nrad RMS on the slope residuals, we can give a tolerance to the pitch rotation center error e x c and e z c .

From what precedes we can infer that a correlation exists between the error committed on z c and the resulting errors in estimating the fit parameters. We use Monte Carlo simulations to study this correlation considering that the uncertainty of the two inputs, coming from the two separate metrology tools, are uncorrelated. The standard deviation of the additive angular noise σ n a is still set to 70 nrad, as previously.

M1 is selected as the test mirror in this Monte Carlo simulation. The average pitch steps are about 0.2 mrad, and 76 pitch angles are performed in the multi-pitch NSP simulation. As shown in Figures 8, 9, the tolerance of the e z c for M1 is about ± 2.2 mm if the RMS threshold of slope residuals is set at 10 nrad, so we will need a metrology technique to measure z c with an accuracy of σ z c = 2.2 / 3 ≈ 0.7  mm. We simulated a sequence of 1,000 independent measurements of M1 mirror with the multi-pitch NSP technique, taking σ z c = 0.7  mm and σ n a = 70  nrad for standard deviation of the two random inputs, as shown in Figure 11.

After the best fit of ellipse (optimization of θ and x o ), as shown in the right panel of Figure 11, the Monte Carlo simulation can give the uncertainty of the measurement in θ and x o , and the slope residuals from the best fit as the result of the measurement and data analysis. Taking this simulation as an example, if we measure the vertical distance z c with σ z c = 0.7  mm and perform the angle acquisition for multi-pitch NSP with σ n a = 70  nrad, the slope residuals will end up at 10 nrad RMS level (the RMS of slope residuals due to e z c is less than 10 nrad RMS threshold). The grazing angle is estimated with a ± 3 σ θ uncertainty as θ ^ = 30.7178 ± 0.006  mrad and the chief ray location, with a ± 3 σ x o uncertainty, as x ^ o = 16 ± 0.036  mm. The Monte Carlo simulation offers a better picture of what we are expecting to get from the multi-pitch NSP measurement of a specific test mirror with a particular measurement of the pitch rotation center.

To take the challenge of measuring extreme off-axis elliptical mirrors, we revisit the multi-pitch NSP technique from its basic theory to the reconstruction algorithm. The true abscissa x m of the measured position in the mirror coordinate system is needed in the reconstruction algorithm. This abscissa x m in the mirror coordinate system is derived as a function of recorded position in the instrumental word coordinate system, the mirror height profile z m and the measurement error on the pitch rotation center ( x c , z c ). We discuss the failure of the “flat assumption” ( z ^ m = 0 ) when measuring “extreme” ellipses and propose a solution with iterative reconstruction in multi-pitch NSP system.

The tolerance of the measurement of the pitch rotation center location ( x c , z c ) is studied with simulations on a series of ellipse parameters. We find out from both theoretical analysis and simulations that the tolerance of the x c measurement is large, and the requirement can easily be satisfied. When measuring “extreme” ellipses, the tolerance of the z c measurement is much tighter. We proposed a practical framework to determine the needed measurement accuracy of the z c measurement. For some extreme cases, the tolerance of the e z c is so small that the measurement of z c can be a challenge.

A Monte Carlo simulation on a real design of an “extreme” elliptical mirror shows that, with a known accuracy of the pitch rotation center measurement, we can estimate θ and x o with their uncertainties and get the slope residuals from the best fit. The RMS value of the slope residuals due to measurement error e z c is controlled by meeting the tolerance of the e z c .

This simulation study can guide the real multi-pitch NSP measurement for “extreme” elliptical mirrors. The tolerance of e z c can be calculated before the design of the measurement. A proper way to measure z c becomes the key to the “extreme” elliptical mirror characterization with the multi-pitch NSP technique.