# A First-Principle Approach to X-ray Active Optics: Design and Verification

Dezhi Diao  Institute of High Energy Physics, Chinese Academy of Sciences,
Beijing 100049, China  University of Chinese Academy of Sciences, Beijing
100049, China  Han Dong  Institute of High Energy Physics, Chinese Academy of
Sciences, Beijing 100049, China  University of Chinese Academy of Sciences,
Beijing 100049, China  Fugui Yang<EMAIL_ADDRESS>Institute of High Energy
Physics, Chinese Academy of Sciences, Beijing 100049, China  Ming Li
Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049,
China  Weifan Sheng  Institute of High Energy Physics, Chinese Academy of
Sciences, Beijing 100049, China  Xiaowei Zhang<EMAIL_ADDRESS>Institute
of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China

###### Abstract

This paper presents the first-principle design approach for X-ray active
optics, using the simulation-modulation cycle in place of the measurement-
modulation feedback loops used in traditional active optics. Hence, the new
active optics have the potential to outperform the accuracy of surface-shape
metrology instruments. We apply an X-ray mirror with localized thermal elastic
deformation to validate the idea. Both the finite element simulations and
surface shape measurements have demonstrated that the active optics modulation
accuracy limit can be achieved at the atomic layer level. It is believed that
the implementation of the first-principle design strategy has the capacity to
revolutionize both the manufacturing processes of X-ray mirrors and the
beamline engineering of synchrotron radiation.

††preprint: APS/123-QED

At the cutting edge of engineering and technology, obtaining large-size,
ultra-high-accuracy mirrors has proven to be a recurring difficulty [1, 2].
This involves ultra-precision machining and measurement techniques.
Nevertheless, these optics often need to operate under extreme conditions,
where factors such as thermal deformations induced by the beam and
deformations due to mirror clamping surpass the precision of mirror
manufacturing. Simply striving for accuracy during production is no longer
adequate to fulfill realistic usage demands. Therefore, the introduction of
new technological elements like active optics becomes imperative to address
these challenges. Successful practices on large-scale ultra-precision optical
instruments such as LIGO and Telescope [3], along with advances in traditional
optics and optical engineering, demonstrate that active optics is a
fundamental strategy in system construction.

Synchrotron radiation facilities and their x-ray mirrors are categorized as
large-scale precision equipment. The low grazing incident angle of synchrotron
X-ray mirrors poses challenges in achieving precise surface shape during both
fabrication and operation in the beamline. However, the elongated dimension of
X-ray mirror in tangential direction provide ample space for mirror surface
modulations, and the one-dimensional adjustment requirement makes the approach
to X-ray mirror surface shape correction relatively straightforward. To make
X-ray active optics work better, different methods have been developed since
the 1990s. These include piezoelectric [4, 5], mechanically bent [6, 7], and
heating technologies [8]. The primary constituents of active optics systems
consist of deformable mirror mechanics and metrology. Due to the challenges in
high-precision surface shape metrology and the non-linear response of
deformable mirrors to the drive, the traditional operation mode requires
iterative procedures which includes measuring surface shape, adjusting
deformable mirrors based on feedback, and ultimately achieving convergence.
Hence, the level of accuracy achieved in active optics is dependent upon the
precision of surface shape measurements and the intrinsic precision of the
actuator. Compared to metrology in the manufacture of X-ray mirrors, this in-
situ high-precision surface-shape metrology technique presents more
significant challenges. The need for high-precision devices, including
actuators and metrology equipment, unavoidably contributes to the total cost.
Hence, it is pragmatically crucial to develop robust, efficient, and cost-
effective methodologies and technologies for X-ray active optics.

Figure 1: The first-principles-designed X-ray active optics system. (a) the
model of an X-ray mirror with thermal deformation modulation; (b) and (c)
depict the temperature field distribution and thermo-elastic deformation
calculated through FEA, respectively. A demon’s active mirror (d) is divided
into different areas for heating and measurement with details in (e). (f) is
measurement setup of the surface shape modulation test experiment with long
trace profiler.

In this paper, we introduce and explore the application of a first-principles
approach for designing X-ray active optics. The strategy involves simplifying
technical elements within the system and ensuring deterministic control as a
fundamental assurance for achieving the target precision with low risk. To
achieve deterministic control, a design rooted in fundamental principles is
crucial, encompassing the careful selection of controllable physical processes
and the utilization of appropriate engineering implementation methods. In
addition, significant progress in computer science has enhanced human
capabilities and provided solutions for complex engineering challenges,
enabling the implementation of first-principles design. In contrast to the
normative ”copying principle,” which prioritizes precision in all technical
elements and aspects, we believe that the computational-assisted first-
principles design approach not only propels technological advancement but also
significantly reduces costs and development cycles.

To realize the concepts of first-principles design, we have developed a system
for X-ray active optics that utilizes the thermo-elastic deformation of the
x-ray mirror, namely the thermal-deformable mirror. The modulation system in
concern employs an array of space-distributed thermal sources, such as laser
heating, electrical heating, or cooling-heating, to stimulate localized
thermal deformation on the mirror surface. This process effectively modulates
the one-dimensional shape of the X-ray mirror surface. The system
implementation is illustrated in Fig. 1. Fig. 1 (a) shows the configuration of
heating devices on the mirror surface. These components are strategically
placed next to or partially covering the area of the X-ray reflecting mirror
to enhance the modulation performance. ”First-principles design” is also the
accurate prediction of the necessary configuration of heating elements and
amounts of deformation using thermal elasticity theory. The general heat
conduction equation for a temperature distribution $T$ within a body has the
form [9]:

$\nabla^{2}T(\mathbf{r},t)=-\frac{1}{k}g(\mathbf{r},t)$ (1)

where $\alpha=k/\rho c_{p}$ is the thermal diffusivity and $g(\mathbf{r},t)$
is the heat generation or source term. If we call $\vec{u}(r,z)$ the
displacement vector, i.e., the difference between the coordinates of a given
atom before and after heating, the classically relevant quantities are the
strain tensor $\varepsilon_{ij}$ and the stress tensor $\theta_{ij}$. In the
presence of a temperature field $T$, the two tensors are related by the
generalized Hooke law for isotropic media via the Lamé coefficients
$\lambda,\mu$:

$\theta_{ij}=\delta_{ij}(\lambda\varepsilon-\nu T)+2\mu\varepsilon_{ij}$ (2)

where $\nu$ is the stress temperature modulus and $\varepsilon$ is the trace
of the strain tensor. $\delta_{ij}$ is the Kronecker tensor. The strain tensor
$\varepsilon_{ij}$ is related to the displacement field $\vec{u}(r,z)$ through
the displacement gradient $(\nabla\vec{u})$ as follows:

$\varepsilon_{ij}=\frac{1}{2}\left(\frac{\partial u_{i}}{\partial
x_{j}}+\frac{\partial u_{j}}{\partial x_{i}}\right)$ (3)

These equations, when appropriately coupled and solved along with suitable
initial and boundary conditions, can describe the evolution of temperature and
displacement fields in an object undergoing elastic deformation due to
heating. We solve these equation using either analytical methods, which
involve deriving formulas [Hello 1990, Vinet 2009], or numerical methods,
which utilize computer computation. The choice between these methods depends
on the complexity of the model. The temperature response function of the
mirror is determined by the power drive of the heating unit, and the thermal
deformation response function $f_{i,j}=f_{i}(x_{j})$ along the sampling
position $x_{j}$ is computed accordingly. The target modulation shape
$h_{j}=h(x_{j})$ can be generated by

$h_{j}=\sum f_{j,i}p_{i}$ (4)

Where $p_{i}$ is the driving power of $i-th$ heat unit. To derive the inverse
solution for this equation, the driving power $p_{i}$ can be determined using
the non-linear least squares method implemented in SciPy, an open-source
Python library. Fig. 1(b) and 1(c) illustrate thermal fields and thermal
deformations modified through a simulation experiment using Finite Element
Analysis (FEA). Considering disparities between the boundary conditions and
system characteristics in the real system and the simulated model, the
response coefficient of simulation data needs to be calibrated at the initial
stage of the application. It should be noted that this system does not exactly
require precise measurements of every system parameter, as the experiment that
follows will demonstrate.

The active optics verification test system is illustrated in Fig. 1(d)-(f).
Following the simulation model, heating elements are arranged on the top of an
optical fused quartz glass mirror with a surface flatness of $\lambda$/4, as
shown in Fig. 1(d)(e). The size of the fused quartz mirror is 205 mm in
length, 50 mm in width, and 20 mm in thickness. The X-ray reflection area (3
mm * 200 mm) is located in the center portion along the tangential direction.
There are 18 pairs of electrical heating elements (2.5 mm * 7.8 mm) with a
10mm period on both sides of the X-ray reflection zone. In order to prevent
electrical conduction among the heating elements, a precise insulation gap of
0.2 mm is maintained between each unit. To simplify the fabrication process, a
single mask is utilized to deposit both types of layers onto the substrate
simultaneously, resulting in the formation of a 200 nm Cr coating. The Copper
pad layer, depicted as the purple area in Fig 1(e), has a thickness of 2
$\mu$m. This thickness is chosen to ensure that the major thermal load is on
the Cr resistor, resulting in uniform heating. Simulation calculations and
physical intuition suggest that achieving greater surface heating deformation
modulation performance can be achieved by symmetrically spreading the heating
elements on both sides of the reflective coating and positioning them as close
as feasible to the reflective coating. The MEMS (micro-electromechanical
systems) process utilizes modern semiconductor microfabrication techniques to
achieve precise micrometer-level accuracy. This results in cost-effective and
consistent heating elements, each having a resistance of 13.3 ±0.1$\Omega$.

The heating elements are powered by six units of programmable DC power supply
(Keithley 2230G1). The minimum division of a single heating element can
approach 0.02 mW (current 1 mA) based on the measured resistance of the
heating elements and the power supply precision. This thermal-deformable
mirror ensures the reliable implementation of the first-principles design for
two specific reasons: (1) precise model of heating source. the heating
elements function by utilizing joule heating directly on the mirror surface.
The structure of the mirror is simple, consisting only of the mirror body and
coating. The manufacturing precision of the heating elements is at the
micrometer level; (2) simple boundary conditions. The heat generated by the
heating elements primarily transfers through the substrate via heat conduction
and through convective heat exchange with the surrounding air, which implies
that the boundary conditions for FEA simulations are explicitly defined.
Hence, the thermal deformation characteristics (shape) of the entire system
are deterministic due to its precise structure and distinct boundary types.
Adjusting the computational parameters enables a precise estimation of actual
operational conditions.

Finally, a pp-LTP (penta-prism long-trace profiler) shown in Fig.1 (f) was
constructed and employed to measure the extremely small thermal deformation
resulting from the heating elements on the mirror. The thermal-deformable
mirror is housed in a small stainless steel chamber because high-precision
measurements require a very steady environment. With the use of a scanning
pentaprism inside the chamber and a lab-built optical head outside, the pp-LTP
allows for accurate scanning measurements of the surface shape of large
mirrors. The coating in the reflection area simultaneously serves as the
measurement condition for the pp-LTP. When considering the practical uses of
X-ray reflective mirrors, several coating materials can be selected according
to specific criteria.

Figure 2: Characterization the thermo-elastic deformation response, the 9th
unit was selected. (a) compares the results of the first-principles
calculation with the measured curve of the response function. The simulation
conditions considered different boundary conditions: air convection and
vacuum. Inset: 3D shape of the deformed mirror. The deformation amplitude
determined by FEA is normalized to the measured shape by a factor of 3.94% in
vacuum and 0.01% in air. The residual images (Y-axis 2) exhibit shape residual
proportions of 0.75 nm rms and 0.13 nm rms, respectively. (b) shows the linear
relationship between the thermal deformation response amplitude and the
heating power, with a residual fitting at the $10^{-3}$ nm PV level.

The simulation and surface shape measurement results of the heating element
deformation response are shown in Fig. 2. The calculation model involves
applying a power of 0.2 W to the 9th pair of elements (see Supplementary
Material 1.1). The thermal power contribution not contributing to the outward
thermal expansion of the substrate surface is excluded, and the amplitude of
the performance is scaled down (with a correction of 30%). The test results
indicate that, when suitably modifying a parameter, the deformation response
of a single heating unit matches well with the FEA calculation results for the
measured surface shape, exhibiting a residual of 0.75 nm rms(root mean
square). Furthermore, we can see that the surface shape measurement data
exhibits high-frequency oscillation characteristics, which is a common problem
in surface shape metrology (due to high spatial frequency features or ambient
measurement noise). In order to evaluate the influence of boundary conditions,
the thermal deformation of the mirror is computed under vacuum conditions,
employing side cooling and removing air convection. After normalization, it
becomes apparent that the two-dimensional distributions exhibit
dissimilarities. However, the deformations along the center line, which are
particularly significant in X-ray applications, display an almost identical
pattern. The difference between these deformations is 0.13 nm rms, observed at
a deformation amplitude of 48 nm. To find out if the amplitude of the thermal
deformation response is linear while taking into account the limits of
measuring the surface shape accurately, a set of finite element simulation
experiments was carried out to find the largest deformation response for
various power drivers. The results, as depicted in Fig. 2(b), provide a linear
property of 0.28 nm/mW for this heating component, accompanied by a fitting
residual of about $10^{-3}$ nm PV. Based on FEA calculations, it can be proved
that for a given mirror structure, deviations in material properties and
fluctuations in heat transfer or thermal boundary conditions have minimal
impact on the shape of thermal deformation at low levels of deformation.
However, they have an effect on the magnitude of the thermal deformation
response. The amplitude of the thermal deformation response can be precisely
adjusted by calibration tests, similar to with other linear systems.

Figure 3:  Results of the thermal deformation modulation experiment. (a) the
thermal deformation response curve calculated using FEA under air conditions.
(b) resulting modulation curve for a sinusoidal target with an amplitude-
period of 80 mm, FEA simulation and measured results are presented with the
boundary condition in the calculations being the mirror’s bottom cooling.
Theoretical calculations for vacuum condition are also given for comparison.

To characterize the shape modulation capability of the thermal-deformable
mirror, we experimentally measured the modulated surface shape. The FEA and
experimental results are shown in Fig. 3. Firstly, the response functions for
each unit are calculated in FEA and depicted in Fig. 3(a), with a loading
power of 0.2 W. Due to the wider heat dissipation path at the ends of the
reflective mirror compared to the middle, the thermal deformation at these
heating elements becomes smaller and asymmetric. Considering that the thermal
elastic deformation decreases with increasing distance, modulation units at
the edge region contribute less and pose a challenge to achieving modulation
accuracy compared to the central region, introducing the edge effects of
thermal deformation. The surface modulation target is configured as a
sinusoidal deformation with a period of 80 mm. The pairs of heating elements
are powered up in accordance with the computation to generate sinusoidal
deformation. In order to reduce the influence of edge effects on the precision
of modulation, we have shortened the merit-function optimization region (the
range of $x_{j}$) in our experiment by 40 mm compared to the complete length
of the deformable mirror. Note that all 18 pairs of heating units are involved
in the optimization calculation. The results of FEA simulation experiments and
PP-LTP surface shape measurement experiments are shown in Fig. 3(b), with a
total electrical power of 1W for the 18 pairs of heating elements. Similar to
the single-point response test, considering the differences between the
physical system parameters and the simulation calculation system, the target
deformation in the calculation is scaled to fit the measured surface shape.
The final adjustment coefficient consistent with the measured sinusoidal shape
is 70%, with an amplitude of ±14 nm and modulation capability of 0.028 nm/mW.
The surface results optimized based on finite element data and the surface
detection results based on pp-LTP show differences from the target surface
shape of 0.39 nm rms and 0.65 nm rms, respectively. The main contribution to
the residual of the former comes from the discrete arrangement of a finite
number of heating units and the corresponding sampling phase adaptation for
the given sinusoidal wave deformation. The optimization of the phase
discrepancy between the heating element arrangement and the modulation target
function is not addressed in this work. The high-frequency features of the
residuals in the measured data (circled in Fig. 3(b)) show that the disparity
between the theoretical modulation residuals and the observed data is due to
defects in the surface profiling system. With respect to the influence of
boundary conditions, Fig. 3(b) concurrently displays the results under vacuum
conditions in finite element simulations, akin to the computations shown in
Fig. 2. In this case, the adjustment coefficient is 0.6, and the residual
value of 0.37 nm rms is basically consistent with the test results in the
atmosphere. This suggests that the changes in these boundary conditions solely
impact the magnitude of deformation.

Figure 4:  Thermal deformable mirror optimization taking into account the
number of heating elements $N$ and the period of target sinusoidal shape $S$.
(a) residual shape errors $R$; (b) total driving power $P$. The power function
fit results are also plotted, with R-squared values of 0.987 and 0.995,
respectively, for $log(P)=7.78-1.76log(T)+0.09logN$ and
$log(R)=10.9-1.85log(T)-1.71logN$.

In order to ensure accuracy in a system, it is essential to allocate an
acceptable error budget for each stage of the processes involved, such as
fabrication, assembly, and correction. Optimizing the thermal deformation
modulation to achieve optimum efficiency, particularly focusing on precision
and power consumption, is significantly important for system design. The
studies conducted indicate the significance of limiting the magnitude of
deformations in a certain heating unit design when we consider the linear
response characteristics of thermal elastic deformation. Another approach
entails adjusting the dimension of heating units by considering the
distribution of the desired deformation in the spatial frequency domain. In
the following study, we investigated the impact of modifying the size and
power of heating elements on the ability to form sinusoidal target shapes with
different periods but the same amplitude. The residual shape error required as
well as the total heating power were assessed. This analysis is depicted in
Fig. 4. The amplitude of the sinusoidal target functions with varied periods
is set to 5 nm, taking into account the typical polishing precision of an
X-ray mirror. Fig. 4(a) shows that the accuracy of modulation improves
progressively as either the number of heating elements N or modulation period
S increases. It is found that the negative power function of
$R=54176/(S^{1.85}N^{1.71})$ can be used to fit the curves with R-squared
value of 0.995. The fit error becomes large for high-frequency shape
modulation (i.e. small $S$). The power required and minimum residual value for
various sinusoidal modulations $S$ are shown in Fig. 4(b). The result shows
that higher-frequency surface modulation needs a lot of heating power and
leaves a big shape error, which is in line with how general linear systems
respond. The low power requirement in the case of N = 12 can be explained by
the low modulation accuracy capability. The power function of
$P=2392N^{0.09}/(S^{1.76})$ with R-squared value of 0.987 is also a good fit
for each curve when the heating element number $N$ is larger than 14. Note
that the overall heating power required for surface modulation in the entire
system should be controlled considering the challenge of the cooling
technique. To achieve a match between the number of heating elements (power
investment) and modulation accuracy, we can fit the minimum number of elements
required to achieve the corrected accuracy for the target. By the fitting
function in Fig.4 (a), the residual error is approximately proportional to
$(SN)^{-1.8}=(S*L/S_{heating})^{-1.8}$, which means that the requirement of
the heating elements within a single modulation period $S/S_{heating}$ is more
fundamental. For instance, to achieve a low-frequency surface correction
residual of 0.1 nm rms, $S/S_{heating}$ should be larger than 7.7 in this
system where $L$ = 200 mm. Overall, the final mirror design should consider
the characteristics of the desired shapes. For a particular shape and the
design of the heating components, raising the heating power can result in
large deformation but reduced modulation accuracy.

In summary, we present an attempt to build an X-ray active optical system
through the application of first-principles computations. We have successfully
demonstrated the effectiveness of our design approach in thermal elastic
deformation mirrors, where heating elements are directly and accurately
manufactured on the mirror’s surface. The agreement between the simulations
and the real modulation performance of this technology is confirmed through
optical testing on the long trace profiler. In comparison with other active
optical systems that have limitations because of complex, indirect correction
mechanisms and the requirement of accurate surface shape metrology, the
deterministic modulation capability of X-ray active optics system shows
promise in achieving precise sub-nanometer-level form modulation at a low
cost. It is crucial to emphasize the universal value of this precision
instrument design methodology based on first principles. In an age of computer
science, this approach enables the efficient development of precision
instruments and efficiently manages costs. In particular, the first-
principles-based active optics system design is a paradigm shift, representing
a technique freed from the limitations of conventional machining and
polishing. New-generation synchrotron radiation facilities can benefit from
this X-ray modulation technology that enables precise wavefront control over
diffraction-limited focusing and beam shaping.

###### Acknowledgements.

This work is supported by the National Science Foundation of China (No.
11505212, 11875059). We thank Prof. Zhongliang Li, Prof. Le Kang, Prof.
Hongxin Luo for their experimental help and useful discussions.

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