# Long trace profiler downward- and sideways-facing measurement simulations

T. Sieg-Letessier, A. Vivo, F. Perrin, J. Marque, M. Ferrari, R. Barrett

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T. Sieg-Letessier, A. Vivo, F. Perrin, J. Marque, M. Ferrari, et al., Long trace profiler downward- and sideways-facing measurement simulations. Review of Scientific Instruments, 2026, 97 (1), pp.011301. 10.1063/5.0291270. http://hal-05477317

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# Long trace profiler downward- and sideways-facing measurement simulations

Special Collection: The 8th International Workshop on X-Ray Optics and Metrology

T. Sieg-Letessier ; A. Vivo ; F. Perrin ; J. Marque ; M. Ferrari ; R. Barrett

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Rev. Sci. Instrum. 97, 011301 (2026)

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# Long trace profiler downward- and sideways-facing measurement simulations

Cite as: Rev. Sci. Instrum. 97, 011301 (2026); doi: 10.1063/5.0291270

Submitted: 16 July 2025 • Accepted: 7 December 2025 •

Published Online: 5 January 2026

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T. Sieg-Letessier,<fnref n="1" />,<fnref n="2" />,ᵃ) A. Vivo,<fnref n="2" /> ID F. Perrin,<fnref n="2" /> ID J. Marque,<fnref n="3" /> ID M. Ferrari,<fnref n="1" /> ID and R. Barrett<fnref n="2" /> ID

## AFFILIATIONS

Note: This paper is part of the Special Topic on The 8th International Workshop on X-Ray Optics and Metrology.

ᵃ) Author to whom correspondence should be addressed: theo.sieg-letessier@esrf.fr

## ABSTRACT

Maintaining optimal x-ray beam quality in accelerator-based light-sources requires mirrors with exceptional shape and surface precision. In practice, mirrors are often mounted in diverse orientations—such as in Kirkpatrick-Baez systems—where gravitational and mechanical effects may differ significantly from upward-facing metrology configurations. In this study, we use optical ray tracing to investigate alternative metrology geometries suitable for measuring side-facing and downward-facing mirrors. We propose a modified profilometry approach for use with the European Synchrotron Radiation Facility long trace profiler, in which beam folding optics are integrated beneath the scanning head to redirect the measurement beam. The approach combines a detailed optical model of the instrument implemented using ray-tracing, enabling accurate evaluation of the impact of motion-induced errors under realistic acquisition conditions. For downward-facing configurations, a promising solution was found experimentally, then characterized, and investigated with an analysis of the impact of alignment tolerance. While preliminary insights are provided for the side-facing setup, all proposed configurations exhibited sensitivity to motion errors of the scanning head, highlighting challenges for robust implementation.

© 2026 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0291270

# I. SIMULATION FRAMEWORK AND METHODOLOGY

## A. Context and motivation

X-ray mirror metrology at synchrotron facilities relies on a variety of instruments to assess mirror shape with a minimal systematic bias. Among these, the Long Trace Profiler (LTP) is widely recognized for its speed and accuracy in measuring slope profiles.<fnref n="1" /> Originally developed by Takacs and with continual enhancements,<fnref n="2" />⁻⁴ the LTP remains a key instrument at the European Synchrotron Radiation Facility (ESRF), where this study is based.⁵

While most LTP instruments are optimized for upward-facing mirrors, many mirrors are operated in non-horizontal configurations, as, for example, in Kirkpatrick-Baez (KB) geometries. In this context, metrologists aim to extend measurement capabilities to match the operational orientation of the optics, thereby improving the evaluation of the in-service slope errors. Downward- and side-facing measurements have been demonstrated using

instruments such as penta-prism based Nano-Optical Measuring (NOM) machines,⁶ by reorienting the measurement beam by a rotation of the penta-prism. Similar configurations have also been implemented in stitching interferometry approaches.⁷ For the LTP, reorientation of the measurement probe beam is generally relatively complicated (requiring reorientation and alignment of several optical systems). Therefore, it is useful to investigate solutions for the measurement of alternative mirror orientations using simple beam redirecting optics. Here, we assess different optical systems, which can be added temporarily to the LTP to allow measurement in alternative SUT attitudes (face-down and face-sideways). Although this particular study presented relates to the ESRF LTP⁸, the principles should be adaptable for other LTP designs.

## B. Optical simulation model and simulation protocol

A natural strategy for enabling measurements in orientations other than face-up with the Long Trace Profiler (LTP) is to

<footnote n="1">¹Laboratoire d'Astrophysique de Marseille (LAM), 13388 Marseille, France</footnote>

<footnote n="2">²European Synchrotron Radiation Facility (ESRF), 38043 Grenoble, France</footnote>

<footnote n="3">³Bertin Technologies, 13290 Aix-en-Provence, France</footnote>
integrate a beam-folding optical extension beneath the scanning head. However, it is important to assess how such modifications might impact the sensitivity of measurements to pitch, yaw, and roll errors of the carriage motion. A full 3D optical model of the ESRF LTP was implemented in ray tracing software (Ansys Zemax Optic-Studio) to assess the impact of beam-folding geometries. The model (Fig. 1) includes the scanning head, beam path extensions, and rotational perturbations corresponding to pitch, yaw, and roll errors of the movable head (and its rigidly attached extension). Throughout the study, these three rotational errors refer to the movable head motion and not to the SUT orientation.

While the use of an open penta-prism on the movable head of the ESRF LTP limits the impact of pitch, yaw, and roll errors of the head translation on the tangential slope measurements, the addition of optical extensions can reintroduce sensitivity. Translational motion errors of the head do not significantly affect the measured slope errors. To evaluate the robustness of different beam-folding configurations, angular motion errors were introduced into the ray tracing process using a Python code to simulate the measurement of a 500 mm long mirror. This work builds upon previous optical modeling in this area,²,³ but here, it is expanded to a full simulation of the measurement process. Based on the measurements of the motion errors of the ESRF LTP movable head, the amplitude of the typical worst case angular motion errors is ±5 µrad. To assess the sensitivity of each measurement configuration to these motion errors, we introduce a controlled perturbation function for pitch, yaw, and roll applied during simulated scans. For each acquisition point n ∈ [0, 50], corresponding to regularly spaced positions u_n ∈ [-250, 250] mm along the movable head translation axis, we define the pitch error as

$$pitch(n) = 5 * (-1)^n \mu rad. \quad (1)$$

Using this approach, rather than simulating the specific motion error profile of the movable head, allows a straightforward evaluation of the peak-to-valley amplitude of errors introduced into the SUT measurement by the angular errors at different positions of the head. Tests of the simulations showed that beyond a certain noise floor, a few nrad PV, set by model assumptions and simulation accuracy, the relationship is approximately linear, so doubling the motion error doubles the induced bias. The cyclic errors described by Eq. (1) allow the simulations to capture the bias amplitude, sign, and spatial dependence along the scan path. Initial simulations assume a perfectly flat 500 mm SUT and perfect

optical elements, ensuring that systematic errors arise solely from motion artifacts. Similarly, the optical elements of the source, movable head, and detector module are perfectly aligned. Although the method does not capture medium- or high-frequency sensitivities, it provides a high-contrast diagnostic of peak-to-valley error contributions and a practical tool for assessing design robustness. This allows the errors introduced by the extension, either through its design or through eventual misalignment, to be evaluated independently of other contributions of from the basic instrument.

## II. DOWNWARD-FACING MEASUREMENTS

### A. Proposed solution and alignment tolerancing

A previously implemented ESRF solution for downward-facing mirrors⁸ involves combining a roof prism with a flat folding mirror, forming a “separated” corner cube reflector. We note that the roof prism adopts an open design using two reflective surfaces, rather than a solid glass element with coated interfaces. The addition of this optical configuration essentially creates an offset inverted measurement beam, which is parallel to the measurement beam used for upward-facing optics. The offset distance can be adjusted by varying the distance between the roof prism and folding mirror to allow access to the reflective surface of wide inversed mirrors and should not modify the LTP sensitivity to pitch, yaw, and roll motion errors. The present work extends this concept by conducting a quantitative tolerancing analysis to evaluate the impact of relative misalignments between the prism and folding mirror.

Due to the self-compensating optical properties of the corner cube geometry, the absolute alignment of the extension assembly relative to the movable head is not critical. When the roof prism and folding mirror are perfectly aligned, the motion errors of the movable head do not impact any of the slope error measurements of the SUT. However, relative misalignments between the two—characterized by rotations θ_x, θ_y, and θ_z defined in Fig. 2—can introduce measurable biases, depending on the direction and magnitude of motion-induced angular errors. We note that similar studies for alignment tolerances were done on penta-prisms.⁹,¹⁰ To assess this, we simulate various misalignment scenarios and compute the resulting deviations in measured slope errors. The results, summarized in Table I, show the individual contributions to the evaluated tangential slope error of the SUT due to the pitch, yaw, and roll errors. The misalignments were chosen to represent realistically achievable values.

<figure><img src="image_13.png" /><figcaption>FIG. 1. Optical simulation of the ESRF LTP operating in a downward-facing configuration (Zemax Optic Studio). The instrument is divided into three optical heads: a fixed source head sending a ray to scan the SUT, a movable head to redirect the beam, and a sensor head for acquisition purposes. In this study, we focus on the design and tolerancing of the extension mechanically coupled to the movable head, whose designs are presented below.</figcaption></figure>
<figure><img src="image_14.png" /><figcaption>FIG. 2. Definition of roof prism misalignment parameters relative to the motion-error axes. The extension introduced in Fig. 1 includes the roof prism and folding mirror, both rigidly mounted to the movable head. The whole group translates along the translation rail to cover the surface under test full length.</figcaption></figure>

In each case, the surface under test (SUT) is reoriented to ensure that the beam reaches the detector, mimicking experimental alignment conditions. As shown, a combined misalignment of 0.9 mrad around all three axes results in a peak-to-valley (PV) slope error of ~30 nrad—within acceptable tolerances for all but the highest-precision applications. The separation of the roof prism and the folding mirror was 200 mm. Figure 3 illustrates the resulting

error patterns for each perturbation axis under these worst-case conditions.

## B. Tolerancing for curved SUTs and unwanted curvature of the extension's folding mirror (spherical case)

All previous ray-tracing analyses were conducted under the simplifying assumption of ideal reflection and transmission through the ESRF LTP optics with a flat SUT. In practice, the SUT or optical elements in the extension assembly may have curvature that can accumulate and generate significant systematic bias, perhaps exceeding the sensitivity to alignment tolerances. These effects may be accentuated when the return beam from the SUT follows a different optical path.

To evaluate such contributions, we simulated measurements of a cylindrical SUT with a 100 or 2500 m radius of curvature. To remain within the representative operating conditions of the ESRF LTP—specifically slope variations below ±100 µrad—the mirror length for the 100 m radius of curvature was set to 20 mm. In practice, for larger angular variations, stitching measurement strategies are applied.⁵ Over such small scan lengths, the worst-case angular motion errors are ±0.5 µrad. In addition, a spherical deformation was introduced on the folding mirror of the downward extension, with a curvature radius of 250 km. The results are summarized in Table II.

## C. Conclusion on the tolerancing requirements for the extension

In summary, the impact of curvature of the folding mirror of the extension is non-negligible: a radius of 250 km can introduce

<table><caption>TABLE I. Tangential slope errors introduced by relative misalignment of the roof prism with respect to the folding mirror. The errors are given in peak-to-valley nanoradians with ±5 µrad pitch, yaw, and roll motion perturbations. The overall error denotes the slope error introduced if the three simultaneous perturbations combine.</caption><thead><tr><th>Roof prism misalignment parameter</th><th>Amplitude (mrad)</th><th>Overall error introduced (nrad PV)</th><th>Pitch contribution (nrad PV)</th><th>Yaw contribution (nrad PV)</th><th>Roll contribution (nrad PV)</th></tr></thead><tbody><tr><td>θ<sub>z</sub></td><td>0.9</td><td>3.1</td><td>0.1</td><td>0</td><td>3</td></tr><tr><td>θ<sub>y</sub></td><td>0.9</td><td>23.9</td><td>2.8</td><td>2.7</td><td>18.4</td></tr><tr><td>θ<sub>x</sub></td><td>0.9</td><td>32.6</td><td>4.3</td><td>21.7</td><td>6.6</td></tr><tr><td>Combined misalignment (θ<sub>z</sub>, θ<sub>y</sub>, θ<sub>x</sub>)</td><td>0.9</td><td>34.7</td><td>0.2</td><td>18.4</td><td>16.1</td></tr><tr><td>Combined misalignment (θ<sub>z</sub>, θ<sub>y</sub>, θ<sub>x</sub>)</td><td>0.2</td><td>9.4</td><td>0.1</td><td>3.7</td><td>5.6</td></tr></tbody></table>

<figure><img src="image_15.png" /><figcaption>FIG. 3. Simulated slope error profiles induced by a 0.9 mrad misalignment (θz, θy, θx) of the roof prism relative to the folding mirror, with ±5 µrad pitch, yaw, and roll perturbations of the movable head. The resulting measurement bias remains below 30 nrad peak-to-valley, confirming the robustness of the “separated” corner cube geometry under realistic alignment deviations.</figcaption></figure>
<table><caption>**TABLE II.** Simulated slope error contributions (PV, nrad) from motion errors (pitch, yaw, and roll), SUT curvature, relative misalignments, and folding mirror curvature in the ESRF LTP configuration.</caption><thead><tr><th colspan="10">Tolerancing parameter</th></tr><tr><th rowspan="2">Rc (m)</th><th colspan="2">SUT</th><th colspan="2">Motion error</th><th colspan="2">Extension</th><th rowspan="2">Pitch<br>contribution<br>(nrad PV)</th><th rowspan="2">Yaw<br>contribution<br>(nrad PV)</th><th rowspan="2">Roll<br>contribution<br>(nrad PV)</th></tr><tr><th>Length<br>(mm)</th><th>Amplitude<br>(μrad PV)</th><th>Combined<br>misalignment<br>introduced<br>(θz, θy, θx)</th><th>Folding<br>mirror<br>curvature (km)</th></tr></thead><tbody><tr><td>Flat</td><td>500</td><td>±5</td><td>0.9 mrad</td><td>Flat</td><td>0.2</td><td>18.4</td><td>16.1</td><td></td><td></td></tr><tr><td>100</td><td>20</td><td>±0.5</td><td>...</td><td>Flat</td><td>8.9</td><td>3.3</td><td>1.6</td><td></td><td></td></tr><tr><td>100</td><td>20</td><td>±0.5</td><td>0.9 mrad</td><td>Flat</td><td>8.3</td><td>0.9</td><td>2.2</td><td></td><td></td></tr><tr><td>Flat</td><td>500</td><td>±5</td><td>...</td><td>250</td><td>30.5</td><td>0.0</td><td>3.0</td><td></td><td></td></tr><tr><td>Flat</td><td>500</td><td>±5</td><td>0.9 mrad</td><td>250</td><td>30.4</td><td>18.5</td><td>16.1</td><td></td><td></td></tr><tr><td>100</td><td>20</td><td>±0.5</td><td>...</td><td>250</td><td>5.9</td><td>3.3</td><td>1.6</td><td></td><td></td></tr><tr><td>100</td><td>20</td><td>±0.5</td><td>0.9 mrad</td><td>250</td><td>5.3</td><td>0.9</td><td>2.2</td><td></td><td></td></tr><tr><td>2500</td><td>500</td><td>±5</td><td>...</td><td>Flat</td><td>3.7</td><td>1.4</td><td>3.4</td><td></td><td></td></tr><tr><td>2500</td><td>500</td><td>±5</td><td>0.9 mrad</td><td>Flat</td><td>6.0</td><td>19.9</td><td>18.4</td><td></td><td></td></tr><tr><td>2500</td><td>500</td><td>±5</td><td>...</td><td>250</td><td>27.3</td><td>1.5</td><td>3.5</td><td></td><td></td></tr><tr><td>2500</td><td>500</td><td>±5</td><td>0.9 mrad</td><td>250</td><td>29.2</td><td>20.0</td><td>18.4</td><td></td><td></td></tr></tbody></table>

up to 30 nrad PV bias in slope error measurements, whether the SUT is flat or curved. Using a simple trigonometric analysis under the small-angle approximation, if the curvature present on the folding mirror of the extension is increased, the measurement error would be multiplied by the same factor. Consequently, if the folding mirror exhibits a radius of curvature of 10 km instead of 250 km, the induced measurement error is theoretically amplified by a factor of 25. The origin of these errors is primarily due to the angular dependent movement of the reflected beam from the SUT across the folding mirror; if the latter is curved, the measurement beam acquires an additional tilt. The curvature of any of the reflecting surfaces encountered by the measurement beam, including the roof prism and, indeed, other optics in the mobile head, can also contribute to these re-trace errors. Some of such curvatures can be compensated by angular calibration of the LTP after installation of the extension. Without such recalibration, in the specific configuration studied here (with the center of the folding mirror 3 cm below the SUT, for instance), a measurement accuracy of the level of 20 nrad RMS (corresponding to ~120 nrad peak-to-valley) requires extension optics with radii of curvature greater than 100 km.

As long as the measurement errors introduced by the extension—whether originating from the surface quality of the optics or from residual angular misalignments—remain within the dynamic range of the instrument (±100 μrad measurement included for the ESRF long trace profiler), these contributions will be directly measured by the instrument without amplification. Any biases introduced by the extension are measured in the same manner as the surface under test. Consequently, ensuring that the fabrication tolerances and alignment errors of the extension stay well within this range is sufficient to prevent any distortion beyond the nominal measurement accuracy.

### III. SIDEWAYS-FACING MEASUREMENTS

Side-facing mirror geometries were also investigated using similar ray-tracing simulations, using the same Python-based acquisition routine described in Sec. I B. Despite extensive testing, all evaluated optical configurations exhibited sensitivity to rotational motion errors, particularly pitch and yaw, which could not be compensated optically. These limitations are analyzed to inform future design strategies. The optical extension designs considered are shown in Fig. 4. Table III summarizes the optical extensions studied. The criterion “Longitudinal measurement” indicates with a checkmark (✓) whether the design successfully measures slope along the mirror’s tangential axis (as required for x-ray mirror optics). Configurations failing this criterion measure sagittal slope, which is of lesser relevance for most beamline applications. The remaining columns—“Pitch,” “Yaw,” and “Roll” sensitivities—quantify the bias introduced by each configuration (a–g) paired with the respective angular carriage motions to slope error measurement, expressed as peak-to-valley (PV) value.

None of the tested side-facing extensions provided sufficient robustness against motion-induced artifacts. Moreover, increasing optical complexity introduces alignment tolerances that could significantly degrade measurement accuracy. As such, further exploration of these configurations was discontinued. Alternative strategies are considered. One possibility is to implement a single folding mirror in combination with post-processing corrections based on externally measured carriage motion—potentially via autocollimation. For classic LTP architectures where both source and detector are integrated within the scanning head (as in the original design by Takacs), a more viable solution could involve rotating the entire head to achieve the desired orientation, thus preserving measurement integrity.
<figure><img src="image_16.png" /><figcaption>FIG. 4. Simulated side-facing extension configurations for the LTP. (a) 45° folding mirror, (b) 45° roof prism, (c) penta-prism, (d) penta-roof prism, (e) penta-prism with roof prisms, (f) hybrid configuration, and (g) double penta-prism.</figcaption></figure>

<table><caption>TABLE III. Performance summary of simulated optical extension configurations for sideway slope metrology under angular motion errors (10 µrad PV).</caption><thead><tr><th>Configuration</th><th>Longitudinal measurement</th><th>Pitch sensitivity (PV)</th><th>Yaw sensitivity (PV)</th><th>Roll sensitivity (PV)</th></tr></thead><tbody><tr><td>(a) Folding mirror</td><td>✓</td><td>10 µrad</td><td>10 µrad</td><td>3.0 nrad</td></tr><tr><td>(b) Roof prism</td><td>✓</td><td>10 µrad</td><td>10 µrad</td><td>3.0 nrad</td></tr><tr><td>(c) Penta-prism</td><td>✓</td><td>10 µrad</td><td>10 µrad</td><td>3.0 nrad</td></tr><tr><td>(d) Penta-roof prism</td><td>✓</td><td>10 µrad</td><td>10 µrad</td><td>3.0 nrad</td></tr><tr><td>(e) Penta-prism with roof prisms</td><td>✓</td><td>10 µrad</td><td>10 µrad</td><td>3.0 nrad</td></tr><tr><td>(f) Hybrid configuration</td><td>X</td><td>10 µrad</td><td>3.0 nrad</td><td>10 µrad</td></tr><tr><td>(g) Double penta-prism</td><td>X</td><td>10 µrad</td><td>3.0 nrad</td><td>10 µrad</td></tr></tbody></table>

## IV. CONCLUSION AND SUMMARY

This study demonstrates that for downward-facing measurements, splitting a corner cube into a roof prism and a flat folding mirror provides a robust, compact, and experimentally viable solution. This configuration preserves the intrinsic insensitivity to angular carriage errors, as observed in standard upward-facing measurements. Even with a relative misalignment of 0.9 mrad between the optical components, the induced slope error remains ~30 nrad (peak-to-valley) across a pitch error of the movable head of 10 µrad PV—compatible with current state-of-the-art tolerances for synchrotron optics. The impact of curvature of the folding mirror is not negligible, as a curvature radius of 250 km can introduce up

to 30 nrad PV of slope error bias with ±5 µrad motion errors. By contrast, for curved mirrors measured with the ESRF LTP, stitching algorithms restrict slope variations to within ±100 µrad, reducing the relative influence of equivalent motion errors. Extending this study to other LTP configurations could be fruitful.

In contrast, all considered side-facing metrology configurations, based on beam-folding optics mounted beneath the scanning head, introduce measurement errors due to unmitigated sensitivity to pitch, yaw, and roll. These errors, inherent to the geometry, limit the feasibility of such designs for high-precision applications. However, for profilometers where both source and detector are integrated into the movable head—such as in classical LTP or NOM instruments—a 90° rotation of the entire head (or, in some cases,
only the penta-prism) about the scanning axis offers a promising alternative that preserves error immunity.

Beyond specific design outcomes, this work highlights the strength of combining full 3D optical modeling using ray-tracing with *Python*-controlled simulation of scanning processes. This approach offers a generalizable and powerful framework for diagnosing, tolerancing, and mitigating motion-induced artifacts, enabling the design, optimization, and future development of next-generation optical metrology instruments.

## ACKNOWLEDGMENTS

<figure><img src="image_17.png" /></figure>

This work was part of the Nano-PtoV Lab-Com program, supported by the French ANR under Grant No. 21-LCV1-0013-01.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**T. Sieg-Letessier:** Conceptualization (lead); Investigation (equal); Methodology (lead); Software (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (equal). **A. Vivo:** Investigation (equal); Validation (lead). **F. Perrin:** Investigation (supporting); Validation (supporting). **J. Marque:** Investigation (supporting); Methodology (supporting); Supervision (equal); Validation (supporting). **M. Ferrari:** Funding acquisition (lead); Investigation (supporting); Methodology (supporting); Supervision (equal); Writing – review & editing (supporting). **R. Barrett:** Conceptualization (supporting); Funding acquisition (lead);

Investigation (supporting); Methodology (supporting); Supervision (lead); Visualization (supporting); Writing – original draft (supporting); Writing – review & editing (lead).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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