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Abstract
The nested Wolter-I type focusing mirror is widely used in the field of X-ray astronomy. The thin-shell mirrors produced by the electroforming replication method will introduce various shape errors during the fabricating and assembling process. This study introduces a non-analytical 3D geometrical ray tracing algorithm capable of predicting optical performance for large mirror deformations. The algorithm's implementation involves error reconstruction, light source and ray simulation, and optical performance calculation. Experimental and simulation validation underscores the algorithm's precision and effectiveness. The results also indicate that edge deformation can seriously affect imaging contrast which is generally considered to be determined only by surface scattering. Applying the 3D ray tracing algorithm, a range of low-frequency fabrication and assembly errors are simulated, such as absolute radius, taper, roundness, edge effects, mirror posture, and hoisting deformation errors, and their effects on imaging quality are analyzed and discussed.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
X-ray telescopes are the core equipment for observing temporary sources and explosive celestial bodies in the universe, and the Wolter-I structure is widely used as their focusing optical components. Wolter-I type focusing mirror is composed of coaxial confocal paraboloid and hyperboloid sections. In order to balance the field of view requirements and the satellite load limitations, Wolter-I type telescopes are often nested with multiple thin-shell mirrors. Electroforming replication is an effective method of reproducing high-precision mirrors through mold fabrication, mold coating, electroforming and mirror separating processes [1]. The metal focusing mirror with thin shell structure has low stiffness and is very prone to deformations. Therefore, it is very challenging to maintain the surface accuracy of the mold for the mirror during the replication process. The mold fabrication needs to consider reducing full-band frequency errors, while the mirror fabrication needs to focus more on low-frequency errors, including mold surface error and deformation errors caused by electroforming stress, demolding, assembling, etc. Therefore, in order to achieve better optical performance, it is necessary to simulate the influence of various errors to allocate process chain errors and better guide fabrication and assembly processes.
There are two main methods for simulating optical performances of X-ray mirrors. The analytical estimation is a method that directly establishes the relationship between mirror errors and optical performance by establishing analytical expressions. D. Spiga et al. [2] established an analytical algorithm for the influence of alignment errors on the effective area and simulate the ATHENA mirror assembly. The calculation results are consistent with the McXtrace ray tracing result. Saha et al. [3] and Shen Zhengxiang et al. [4] studied the mirror tolerance estimation of Wolter-I telescope and microscope respectively by constructing the transverse ray aberration equations. Though the analytical algorithm has high calculation efficiency, the form of the analytical formula will become extremely complex with the increase of the input error numbers. At the same time, all the analytical algorithms are derived based on the small deviation assumption, so they are not suitable for simulating the impact of large surface errors.
Ray tracing is widely used in error evaluation and optical performance prediction. The inherent reasons of commercial ray tracing software such as Zemax and CodeV lead to low efficiency and difficulty in modeling and simulation Wolter-I optical systems. Therefore, the vast majority of related research is achieved through self-developed algorithms [5–11]. G. Sironi et al. [6] used long trace profilometer to measure the three-dimensional surface shape of ATHENA mirrors and use TraceIT developed by Brera Astronomical Observatory for performance prediction. The obtained HEW (Half Energy Width) accuracy was less than 1”. D. Spiga et al. [8,9] used MT_ RAYOR to simulate the stray light of the ATHENA module and McXtrace to calculate the effective area and the influence of assembly error. Niels Westergaard et al. [10] used MT_ RAYOR to predict the PSF (Point Spread Function) of NuSTAR mirrors and compared the results with ground testing for verification. The above works only simulate a single or entire fabrication error without subdividing the individual influence of errors in each process, thus are unable to provide effective feedback on each process of mirrors fabrication and assembly. Jun Yu et al. [11] established a semi analytical three-dimensional ray tracing algorithm for simulating the optical performance of five nested thermally collapsed Wolter-I mirrors. Although this algorithm takes different kinds of fabrication errors into account, it assumes the mirror shape as a cone and partially uses analytical formulas for calculation, such as the calculation of the axial ray propagation position and angular resolution. Therefore, it sacrificed simulation accuracy for computational efficiency.
In this paper, we established a multi-parameter 3D geometric ray tracing algorithm that is focused on predicting imaging quality with various fabrication and assembly errors. The software package, named RTrace_3D, version 2.1.2, was programmed by MATLAB. The implementation process of the code based on geometrical optics is described in the following steps: surface error reconstruction and simulation, light source simulation, ray tracing, and optical performance calculation. Then, low frequency error measurement, ray tracing calculation and X-ray optical performance testing have been implemented to verify the accuracy and effectiveness of the algorithm. Finally, the common fabrication and assembly low-frequency errors, such as absolute radius, taper, roundness, edge effect, mirror posture and hoisting deformation errors are simulated by the algorithm. The influences of these errors on optical performance are analyzed and discussed in detail.
2. 3D ray tracing algorithm
The flow chart of the 3D ray tracing algorithm established in this paper is shown in Fig. 1. The algorithm is divided as follows: 1) Surface error input, read or generate surface error data, calculate the error point cloud and surface normal vectors; 2) Light source generation, generate corresponding point-light source coordinates or parallel light vectors; 3) Ray tracing, generate incident points in pupil entry area, cycle through calculation of incident direction vectors, reflection points and reflected lights until meeting the exit or lost conditions; 4) Optical performance calculation, based on the coordinates and directions of the exit points, calculate scatter points on the image planes at different positions and calculate the corresponding optical performance.
2.1 Reconstruction method for mirror surface error
There are three main measurement methods for reconstructing the surface shape of general rotational symmetric parts, as shown in Fig. 2: (a) Generatrix measurement is carried out along the generatrix profile, and multiple contours at different azimuth reconstruct the surface. (b) Perform roundness measurement along the azimuth direction and reconstruct the surface by roundness of different cross-sections. These two methods can obtain accurate generatrix profile or roundness data, but the efficiencies are relatively low. (c) The spiral path measurement has the highest efficiency, but requiring good positioning accuracy and following errors of the linear axis, as well as small radial runout of the rotating axis.
Fig. 2. Three surface measurement methods. (a) Generatrix path measurement; (b) Roundness measurement; (c) Spiral path generatrix measurement.
Section 3 and 4.2.1 indicate the edge effects must be fully considered. However, roundness measurement cannot effectively characterize the effects. Therefore, it is not appropriate to use roundness measurement to reconstruct the surface error. For spiral path measurement, it is necessary to restore the edge errors through data extension and interpolation. The generatrix path measurement can effectively characterize the edges and generatrix profiles, and complement to spiral measurement.
The corresponding data processing methods are shown in Table 1. The edge data of spiral path measurement is first determined by linear operation to calculate the control points outside the edge, and then fitted with the NURBS (Non Uniform Rational B-Splines) toolbox to interpolate the edge data. Due to the fact that the density of generatrix points of spiral path measurement is much lower than that of generatrix path, in order to prevent data distortion, the wavelet packet denoising order of the spiral path is smaller than that of the generatrix path.
Table 1. Data Preprocessing Methods of Mirrors Surface Reconstruction
The measurement data needs to be interpolated and filtered to form a surface error matrix ${E_{Gene}}$. Finally, the actual surface R used for ray tracing calculation needs to be overlaid with the ideal surface matrix R0:
where ${R_{mn}}$ is the radius at the surface grid node [m, n] (the m-th section, n-th generatrix). Similarly ${E_{mn}}$ is the surface error at the corresponding grids which is the superposition of generatrix error ${E_{Gene}}$, absolute radius error ${E_{AbsR}}$, taper error ${E_{Taper}}$, roundness error ${E_{Roundness}}$, etc. $R{0_{mn}}$ is the ideal radius matrix.Absolute radius error ${E_{AbsR}}$ is a constant matrix shown in Eq. (2), where $Ones$ is the matrix with all elements equaling 1, and ${\varepsilon _{AbsR}}$ is the absolute radius error value.
The taper error ${E_{Taper}}$ is the slope error ${\varepsilon _{Taper}}$ of the total length superimposed on each generatrix, as shown in Eq. (3). Where ${Z_M}$ is the axial discrete point sequence, L is the mirror length. The default taper error at the paraboloid edge is 0.
Roundness error ${E_{Roundness}}$ is determined by the amplitude and phase of the head and tail roundness, and is quadratic in azimuth direction by default. As shown in Eq. (4), $({\varepsilon _{RndH}},{\theta _{RndH}})$ and $({\varepsilon _{RndT}},{\theta _{RndT}})$ is the roundness amplitude and phase of the mirror edge at paraboloid and hyperboloid, respectively. $\varphi$ is the azimuth interval sequence. Specifically, when ${\theta _{RndH}}$ equals ${\theta _{RndT}}$, ${E_{Roundness}}$ is the in phase roundness error, otherwise the out of phase roundness error which manifests as mirror torsion.
Considering posture error, coordinate transformations need to be performed on the surface shape matrix ${R_{mn}}$ as shown in Eq. (5), where ${T_i}$ is the coordinate transformation matrix and $R_{mn}^{\prime}$ is the transformed surface shape matrix.
In Fig. 3, the actual surface with the pitch angle error ${\theta _{pitch}}$ becomes:
The characteristics of common fabrication errors in this paper are shown in Table 2.
Table 2. Characteristics of common fabrication errors
Calculate the unit internal normal vector matrix ${N_{surf}}$ at each node according to $R_{mn}^{\prime}$ as shown in Eq. (7), where $Nx$, $Ny$ and $Nz$ are the directional components of ${N_{surf}}$ and are all m ×n numerical matrix.
2.2 Light source simulation and ray tracing
2.2.1 Light source simulation algorithm
Firstly, simulate the light source and calculate the initial incident light. Each group of incident rays is at the same azimuth angle θ. The initial incident point set P is generated based on the aperture setting, $Rin$ is the uniform random radial length set of P.
The initial direction vector of the point light source is shown in Eq. (10). Note that ${P_{ki}}$, ${P_S}$ and ${V_{ki}}$ are all 1 × 3 matrix, r is the number of reflections, and r equals 0 for the initial incident light. As for parallel light, the initial direction vector can be directly defined as ${}^{(0)}{V_{ki}} = [0,0, - 1]$.
Finally, the initial incident light $Vp$ is generated based on the incident position and direction as shown in Eq. (11),
2.2.2 Ray tracing algorithm
The ray tracing algorithm is divided into three parts, namely the intersection calculation, reflection judgment and reflection calculation. The intersection calculation is used to calculate the intersection point between the incident ray and the actual mirror surface, the reflection judgment is used to determine whether the ray intersects and the behavior of the intersecting ray (reflection, exit, loss), and the reflection calculation is used to calculate the reflection direction and generate the next cycle of incident ray propagation vector.
The essence of intersection calculation is to calculate the intersection point between a spatial line segment and any continuous surface as shown in Fig. 4. The algorithm process is as follows: Firstly, employ axial coordinate ${}^{(r)}{z_{ki}}$ along the ray vector ${}^rV{p_{ki}}$ to interpolate the continuous curved surface and calculate the curve contour $Rs{|_{1 \times n}}$ corresponding to cross-sectional profile. Then calculate the radius $R{p_{ki}}{|_{p \times 1}}$ at each azimuth ${}^{(r)}{\theta _{ki}}{|_{p \times 1}}$ of the cross-section. When the radial distances corresponding to the points on the ray and the surface under the same cross-sectional profile and azimuth are equal, that point serves as an intersection point between the ray and the curved surface. Construct the difference function Eq. (12), where $j = 1\ldots p$ is the number of discrete points. The root of ${F_{ki}}(j) = 0$ is the intersection ${}^{(r)}P{c_{ki}}$.
The interval located the intersection points is found by bisection method and then the intersection coordinates are calculated by interpolation. The denser the surface grid data and the more discrete points of a ray are obtained, the higher accuracy of the algorithm will be achieved. However, increasing the ray points density cannot improve the accuracy any more when the density of ray points is greater than the axial density of the surface mesh. Similarly, more interpolation processes and higher density will increase the computational burden and reduce the simulation efficiency.
The ray behavior judgment is divided into three situations: the existence of the intersection, the absence of the intersection and the loss of light. The conditions met in each case are as follows: the presence of the intersection is shown in Fig. 4 ① and Eq. (13).
The situation of ray emission is shown in Fig. 4 ② and Eq. (14). If a ray is emitted, record the coordinates of the incident point and the direction of the incident ray as the exit point $P{o_{ki}}$ and direction $V{o_{ki}}$. Jump out of the current cycle, initialize and start tracing the next ray.
The situation where ray strikes the outer surface of the mirrors and cannot continue to propagate is shown in Fig. 4 ③ and Eq. (15).
After calculating the intersection points, the grazing incidence angle is calculated to determine whether the ray can undergo reflection. Two-dimensional interpolation is performed to calculate the normal vector $^{(r)}V{n_{ki}}{|_{1 \times 3}}$ at the intersection based on the normal surface vector ${N_{surf}}$ and intersection coordinates ${}^{(r)}P{c_{ki}}$. Then, the grazing incidence angle ${}^{(r)}{\gamma _{ki}}$ at this point is calculated by Eq. (16). The reflected rays must meet the condition ${}^{(r)}{\gamma _{ki}} \in (0,{\theta _{grazing}})$, where ${\theta _{grazing}}$ is the critical angle of grazing incidence.
Only rays that intersect with the surface and meet the reflection conditions can be used for the reflection calculation. Equation (17) is the corresponding Householder transformation [12] to calculate the reflected direction vector $^{(r + 1)}V{c_{ki}}{|_{1 \times 3}}$, where, ${}^{(r)}{H_{ki}}$ is the Householder transformation matrix, ${\textrm{I}_3}$ is the 3rd identity matrix.
The current ray tracing loop ends. The outgoing ray parameters, ${}^{(r)}P{c_{ki}}$, $^{(r)}V{c_{ki}}$ and ${}^{(r)}P{c_{ki}}$, are taken as the incoming ray parameters, ${}^{(r + 1)}{P_{ki}}$, $^{(r + 1)}{V_{ki}}$ and ${}^{(r + 1)}Z{p_{ki}}$ respectively, to recalculate the next incident ray ${}^{(r + 1)}V{p_{ki}}$ and commence the subsequent iteration.
2.3 Optical performance calculation
Given the exit point $Po$ and the direction $Vo$, the image points can be calculated to obtain the optical performance. The scattered point coordinates ${P_I}$ on the image plane with axial coordinates ${Z_f}$ in Fig. 3 are shown in Eq. (18), where q is the total number of emitted rays, ‘$./$’ represents the division of the corresponding elements of the two matrixes.
The focal point coordinate is the first order origin moment of the scattered point intensity, or the barycenter. Due to the usage of Monte Carlo method, each initial incident point is random, so it is assumed that the probability density of each scattered point on the image plane is the same and equal to 1/q. The ray intensity is considered the same according to geometrics, and the focal coordinates ${P_{Focus}}({x_{Focus}},{y_{Focus}})$ are calculated as:
When X-rays and surface roughness are both at low levels, the influence of surface scattering can be neglected [13,14]. Assuming that X-rays are monochromatic, the encircled energy E can be calculated based on the distance between ${P_{Focus}}$ and ${P_I}$, thus angular resolution index such as HPD (Half Power Diameter) and W90 (90% Energy Width) can be calculated. Note that Eq. (18), (19) have high computational efficiency, it is possible to search for the focal plane position by scanning the axis near the design focal plane (Z = 0 in Fig. 3), that is, to calculate at multiple positions with small axial intervals and select the position with the best optical performance as the focal plane. However, this method is susceptible to the influence of overly discrete points because their origin moments are longer and easier to affect the results of the barycenter calculation.
Simulating the ideal surface can to some extent verify the algorithm. As shown in Fig. 5, the optical performance of EP (Einstein Probe [15]) # 1, # 27, and # 54 mirrors is calculated with ideal surface under parallel light and point light conditions, the design parameters of EP are shown in Table 3. The axial interval of the surface grid is 0.5 mm and the circumferential interval is 1.8°. All point-light mentioned in this paper are used to simulate the IHEP (The Institute of High Energy Physics of the Chinese Academy of Sciences) X-ray calibration devices located in Beijing. The object distance of the device is set to 100 m [16] and the off axis distance is 0.
Fig. 6. Measurement and surface reconstruction of #52 thick mirror. (a) Measurement method; (b) Surface reconstruction; (c) Six generatrix errors calculated by interpolation.
Table 3. Some design parameters of EP mirrors
It can be seen that the HPD and W90 are on the order of 0.01” under ideal surface and parallel light conditions. The degradation of angular resolution caused by point light varies with the grazing incidence angles and the maximum can reach 1.13 “HPD and 2.42” W90 under ideal surface condition. For well fabricated electroformed mirrors, the typical HPD is around 10-15” [14]. Meanwhile, the simulation results for the ideal surface exhibit a value of 0.01”, considerably lower than 30”. It can be preliminarily asserted that the algorithm meets the precision requirements for optical performance simulation, demonstrating a certain level of accuracy. However, relying solely on zero-error surface simulations is insufficient. Consequently, we proceeded with the experimental validation in Section 3 to provide further verification.
3. Experimental verification of the tracing algorithm
For demonstration, an EP #52 mirror with a thickness of 1 mm and radius of 76 mm was fabricated. The mirror is measured on the DRL2000 ultra precision drum roll lathe machine and its angular resolution is simulated using the 3D ray tracing algorithm. Due to the narrow entrance pupil of the #52 mirror, visible light test is affected by diffraction, so the actual optical performance is tested by the IHEP X-ray facility.
The paraboloid end is fixed on a ring with an inner diameter slightly larger than the outer diameter of the mirror. The two are spliced using optical adhesive with low curing shrinkage. Three-points clamping was used to fix the ring onto the fixture in order to minimize deformations and the coaxiality of the mirrors is adjusted. The drum roll lathe has been proven to have sufficient motion and machining accuracy [17]. The method of in situ measurement using ultra-precision machine tools has been proven to be accurate and effective [18,19]. The surface error was measured with a white confocal displacement sensor (CCS PRIMA). The probe has a resolution of 20 nm, a sampling frequency of 200 Hz, and an allowed slope error of < 7 °. As shown in Fig. 6 (a), the measurement error is reduced by adjusting the probe to be at the same height as the spindle axis. The #52 mirror is measured through a spiral path, and the axial interval of the measured points on the same generatrix is 1 mm.
The ray tracing simulated spot pattern and angular resolution on the focal plane are shown in Fig. 7. The HPD of the #52 mirror is within the expected range of around 30”, but the W90 degradation is very severe. The reason for this phenomenon is the existence of a 4 mm diameter halo. The intensity of the halo is about one thousandth of the focal point, so it has no significant impact on HPD. But the halo is in-field scattering and seriously degrades W90. It's important to distinguish this scattering from that induced by surface roughness. The halo's internal energy is uniformly distributed, with minimal energy scattering beyond its boundaries.
This phenomenon is caused by the edge error of the hyperboloid. It can be seen from Fig. 6 (b) (c) that the electroforming stress causes the hyperboloid edge to resemble a ‘bottleneck’. The hyperboloid edge corresponds to a large localized slope error. The influence area of the edge error is limited, and the scattering range is also limited. Therefore, the scattering caused by the edge is at low intensity, limited in range, and relatively uniform. The incident light of a point light source can be seen as the superposition of an initial slope error by a parallel light, which is equivalent to amplifying the external edge error. So the W90 simulated under the point light is greater than that simulated by a parallel light. The influence of edge errors will be discussed in detail in section 4.2.1.
The roughness of electroformed replica mirror is typically slightly larger than that of the mold [14], and is always obtained by measuring the mold roughness. The Chotest SuperView WX100 white light interferometer is used as the measurement instrument, offering a height resolution of 0.1 nm and a bandwidth ranging from 1µm to 240µm. Based on the roughness, the total integrated scatter (TIS) equation for the Wolter-I system, as presented by Harvey [20], along with Henke's algorithm [21,22], was employed to compute the specular reflectance under grazing incidence conditions, as illustrated in Fig. 8. It should be noted that TIS represents the scattering component, while (1-TIS) represents the specular reflection component. For the EP #52 mirror with a grazing incidence angle of 0.35° and a roughness of 0.5nmRMS(Root Mean Square) (corresponding to a mold roughness of 0.294nmRMS), the relative scattering intensity at Al-Kα X-Rays is less than 7%, indicating that the surface scattering effect can be neglected. Consequently, Al-Kα X-Ray radiation is selected to assess the optical performance of EP #52 to validate the geometric 3D ray tracing algorithm.
Fig. 8. #52 surface roughness and the corresponding specular reflectivity. (a) Mold roughness testing result. (b) Relative intensity of specular reflection based on TIS under 0.5nmRMS surface roughness. (c) Relative intensity of specular reflection based on CXOB algorithm under 0.5nmRMS surface roughness. Parameters are as follows: top surface roughness 0.5nmRMS, layer material Au, layer thickness 200 nm, substrate material Ni.
X-ray testing experiment of #52 thick mirrors is shown in Fig. 9 (a), where four support screws are used to adjust the mirror posture and a coated aluminum sheet is used to block direct light. The ray directly incident on hyperboloid will not be reflected to the camera. The X-ray testing results are shown in Fig. 9 (b), and there is also a uniform scattered halo around the focal spot. The diameter of the halo is about 4 mm and the intensity is about one thousandth of the center, which is consistent with the simulation results. However, there are also some differences between the test and the simulation results. X-ray testing exhibits higher W90 compared to ray tracing. The X-ray halo demonstrates distinct boundaries, while the simulation boundaries are relatively blurry. There may be several reasons for these differences: 1) The ray tracing can only simulate the low-frequency errors without considering the surface scattering effect. 2) The stray light in X-ray testing will increase W90. 3) Spiral measurement cannot effectively restore edge errors, and data filtering can also reduce edge error values.
Fig. 9. #52 thick mirror X-ray testing method and results. (a) X-ray testing method; (b) X-ray testing results.
Consequently, generatrix path measurement was used for comparison. The axial interval of generatrix path measurement data is 15µm. Considering the limited number of measured generatrix, the small deformation of thick mirror and a relatively consistent nature of the generatrices, in order to obtain a complete error surface, all the generatrix errors are set to the mean value of all generatrix measurements. The generatrix error before and after denoising is shown in Fig. 10, and the ray tracing results are shown in Fig. 11. The simulated angular resolution and X-ray testing result under the point light are close enough to indicate the effectiveness of the ray tracing algorithm. The premise is that the surface error data can accurately reflect the mirror error and the simulation parameters are consistent with experimental conditions.
Fig. 11. #52Thick mirror ray tracing results of generatrix path measurement data. (a) Parallel light; (b) Point light.
Comparing the outcomes of the two measurement paths, it becomes evident that the spiral path effectively captures roundness characteristics but inadequately represents edge attributes. On the other hand, the generatrix path excellently portrays edge attributes, thus yielding ray tracing results for generatrix measurements that exhibit a more distinct and well-defined circular halo. From Fig. 10 and Fig. 11, it is now apparent that the degradation of W90 in X-ray testing is an outcome of the combined effects of hyperboloid edge error, taper error, and point light. Under parallel light, the W90 degradation caused by edge error is lesser than that under point light, signifying that there is a difference between the operational performance in space and the ground-based testing results. This underscores the necessity to account for mirror errors and point light influence during ground testing. The absence of cross-validation could potentially yield misleading results.
In fact, accurately predicting imaging quality through fabrication errors is a difficult task [23]. The above experiments demonstrate that the measurement method, instrument bandwidth, data processing methods and the limitations of the algorithm itself can all affect the final calculation results. Through the simulation and experiments of mirror #52, it is sufficient to indicate that the 3D ray tracing algorithm established in this paper has a certain level of accuracy, sensitivity, and reliability, and is suitable for low-frequency errors simulation. Specifically, the algorithm does not rely on analytical formulas, and there are no limitations on the magnitude of errors and the geometry size of mirrors. The input of algorithm fully considers various factors in processing, assembling and testing. It is suitable for systematic research and analysis of the influence of fabrication errors on optical performance.
4. Study of the influence of mirror errors on imaging quality
RTrace_3D is employed in this section to simulate and analyze several common mirror errors, including absolute radius error, taper error, roundness error, edge effects, posture misalignment, and hoisting deformation errors. The simulation parameters in this section are: ray spacing of 2µm; 36 groups of incident ray uniformly distributed in the azimuthal direction; CCD (Charge Coupled Device) size of 10 × 10 mm (the size of the plane used to receive light); repeated calculations more than 10 times for each situation.
It should be noted from Fig. 8 that, when X-ray energy, grazing incidence angle, or surface roughness is relatively high, scattering effects cannot be ignored. However, simultaneously accounting for these independent factors would considerably complicate the analysis of common mirror errors. Therefore, this section solely discusses the influence of low-frequency errors on optical performance while neglecting scattering effects. Considering that the actual PSF is a convolution of the specular reflection PSF and the scattering PSF, and that large-angle surface scattering has a limited impact on HPD but a more substantial effect on W90, the calculation outcomes of this section can only provide qualitative insights when scattering effects cannot be ignored.
4.1 Influence of absolute radius, taper, and roundness error
4.1.1 Absolute radius error
The absolute radius error is corresponds to each generatrix being close to or far from the optical axis in the radial direction. As shown in Fig. 12, a linear trend illustrates the influence of absolute radius error on focal length under varying grazing incidence angles and sources. The overall shifting of the point light curve from the parallel light is -26.7 mm, which is exactly the focal coordinate value under ideal surface. Smaller grazing incidence angles amplify focal length error for a given absolute radius error. Positive radius errors corresponds to negative focal length errors, implying longer focal lengths, aligning with empirical knowledge. Focal length is notably sensitive to absolute radius error due to the small tilt angle of outgoing rays, allowing minimal radial disturbances to lead to significant axial errors.
Accounting for distinct focal lengths, the influence of absolute radius error on angular resolution is shown in Fig. 13. It can be seen that the absolute radius error has little effect on the size of the focal spot, but can significantly alter the focal length. As a result, the focused rays can still form a point at a certain axial location but the image pattern on the designed focal plane is a ring. If multiple mirrors with different absolute radius errors are assembled into a module, due to the different focal lengths, the image on the focal plane will be the accumulation of multiple concentric rings with different widths. The defocus ring's radius expands proportionately to focal length increase, concurrently exhibiting concentrated energy, seriously degrading the module's W90.
Fig. 13. Influence of absolute radius error on angular resolution. The solid line represents the angular resolution at the focal plane, the dashed line represents that at the designed focal plane. (a) Parallel light; (b) Point light.
4.1.2 Taper error
According to Eq. (3), the taper error is equivalent to adding a slope over the entire generatrix. The typical precision lathe accuracy of the mold is about 10µm (6.88”) of taper error. Without loss of generality, a larger range of taper error is added on the ideal surface to study its influence on image performance as shown in Fig. 14. It illustrates that when the taper error <10µm, the degradation of angular resolution is small, with 1” HPD and 3” W90 respectively. Comparison between solid and dashed lines shows that the taper error indirectly affects angular resolution by affecting focal length. When the taper error reaches ± 50µm (34.38”), the degradation caused by focal length is 3∼5”. Afterwards, it quickly rises and when the taper error reaches ±100µm (68.75”), the degradation can reach the order of 10” which is already unacceptable.
Fig. 14. The influence of taper error on optical performance. As for (a) and (b), the solid line represents the angular resolution at the focal plane, the dashed line represents that at the designed focal plane. (a) Angular resolution under parallel light; (b) Angular resolution under point light; (c) Focal length error.
Under the same taper error, differences in angular resolution between the two sources exist, spanning from fractions of a micrometer to several micrometers. The taper error shows minimal sensitivity to shifts in grazing incidence angles. With taper error constrained within 100µm, the maximum divergence in angular resolution between the two sources approximates 1”. This observation suggests a relatively minor influence of taper error on ground testing when compared to the designated angular resolution of 30” HPD.
4.1.3 Roundness
The deformation of thin shell structure mirrors during demolding and hoisting tend to be a large ellipse in the circumferential direction. Here we only discusses the influence of this overall deformation behavior. As shown in Eq. (4), for the convenience of the research, only the case where ${\varepsilon _{RndH}}$ equaling ${\varepsilon _{RndT}}$ is calculated as a reference for other situations.
When the roundness error is in the same phase, the axial cross-sections of the mirror are all ellipses, which is equivalent to adding different taper and absolute radius errors on each generatrix. Especially, taper error is zero when ${\varepsilon _{RndH}}$ equaling ${\varepsilon _{RndT}}$. Figure 15 illustrates that the influences of in-phase roundness error on angular resolution all rapidly increase with the increase of roundness amplitude under different conditions. The mean focal Z-coordinate for both #1 and #54 mirrors under parallel light conditions is 0, and under point light, it is -26.7 mm and -26.4 mm, respectively. This implies the limited influence of in-phase roundness error on focal length. The reason can be roughly explained as follows: the generatrices corresponding to the major axis of each ellipse cross-section increase focal length, while the ones corresponding to the short axis decreases focal length. The final comprehensive effect is that the focal length remains almost unchanged, but the angular resolution degrades.
Out of phase roundness is equivalent to a torsional deformation of the mirror shown in Fig. 16 (a). The focal spot shape of #27 and the simulated results under parallel are shown Fig. 16 (b). The error causes the focal spot to become a diamond shape to greatly expand the radius and result in sharp decrease in angular resolution.
Fig. 16. Out of phase roundness error surface and simulated results. (a) Input #27 error surface of 100 µm amplitude and 30 ° phase difference roundness error; (b) #27 Simulated result under parallel light.
After extensive calculations, it was found that the both roundness have little influence on the focal length when the magnitudes are less than100µm. So ignoring the influence of focal length, the angular resolution at the design focal plane is calculated shown in Fig. 17. Because the default roundness error is quadratic, the roundness is the same when $\Delta {\theta _{Rnd}}$ equals 180°. $\Delta {\theta _{Rnd}} \in ({90^ \circ },{180^ \circ })$ is equivalent to $\Delta {\theta _{Rnd}} - {90^ \circ }$, so $\Delta {\theta _{Rnd}} \in ({0^ \circ },{90^ \circ })$ is able to consider all situations.
Fig. 17. Influence of Out of Phase Roundness Error. The horizontal axis represents the phase difference $\Delta {\theta _{Rnd}} = {\theta _{RndT}} - {\theta _{RndH}}$. The solid line represents the error of 10µm amplitude, the dashed line represents that of 100µm amplitude.
According to Fig. 15 and Fig. 17, the influence of 10µm roundness is on the order of 1”, that of 100µm roundness is on the order of 10”. Under the same amplitude, the influence of in-phase roundness is greater than that of out of phase roundness. As $\Delta {\theta _{Rnd}}$ increases, the angular resolution gradually decreases. When $\Delta {\theta _{Rnd}}$ reaches 90 °, the angular resolution comes to the minimum value. The cumulative influence of different phase cross-sections mitigates the effect of roundness error, causing minimal change in focal length and diminishing its influence on angular resolution. Simultaneously, it also indicates that for roundness errors, the ground test can reflect the situation under parallel light.
4.2 Influence of edges and intersections
4.2.1 Edge error
Edge effect is a problem that must be faced in almost all engineering fields. Many fabricating processes of focusing mirrors introduce edge errors. The ultra-smooth polishing of the mold can cause collapse at the ends and the middle intersection of the mold [24], which can be replicated to the mirrors. In order to achieve better surface quality, small polishing removal amounts and longer polishing times are often required. The discontinuities of the surface profile results in an increase in polishing pressure and removal amount, leading the occurrence of surface collapse. The magnitude of this phenomenon is usually within 1µm, up to a maximum of 10µm. The internal stress of electroforming can also cause deformation of the edges. The residual tensile stress on the outer surface causes the edge to be flipped outward, conversely flipped inward. The maximum size of the flipping can reach tens of micrometers [25]. In addition, mirror demolding may also cause mechanical deformation at the pressure points.
Based on references and our measurement results (Fig. 6 (c)), edge error can be approximated by spline fitting. A regular cubic spline curve is used to simulate the corresponding error. The discrete control points and their positions on the spline curve are based on the influence area and amplitude of the edges. The axial interval of the control points is 15 mm and the generated edge error are shown in Fig. 18. The 3D error surface utilized for ray tracing simulations is constructed by combining simulated edge errors with the ideal surface and assuming all the generatrix errors are the same.
The angular resolutions of ± 1, 10, and 100µm edge error respectively are illustrated in Fig. 19. The proportion of once-reflected light and effective area is shown in Fig. 20. The effective area refers to the ratio of rays received by the CCD to the total incident rays. The influence of edge error on HPD remains restricted; even at an edge error of 100µm, HPD degradation is within the arcsecond range, approximately 3”. However, with a 10µm edge error, W90 degradation escalates to the scale of 100”, indicating that the impact of edge error on imaging contrast is catastrophic.
Fig. 20. Influence of edge error on once-reflected light percentage and effective area. (a) Parallel light; (b) Point light.
Negative edge error on paraboloid will block some incident light, resulting in the decrease in effective area. Meanwhile, this part of light cannot be received by the CCD and contribute to the optical performance calculation. Consequently, for significant errors (<-10µm), W90 may even exhibit a declining tendency. The positive edge error on paraboloid will cause a large angle scattering and induce the rays to become once-reflected or lost. A comparison between effective area and the single-reflection curve reveals that the lost rays mainly originate from single reflections.
The influence of hyperboloid edge error is similar to that of paraboloid. The W90 under point light is slightly smaller than that of parallel light. When the absolute value of the hyperboloid edge error is large, the influences of different sources are quite different. Under point light, the reflection ray slope and edge error combine, resulting in a simulation outcome akin to the negative edge error on the paraboloid. When the edge flanging is large, part of the rays are blocked, leading to the decrease of the effective area. However the change trend of W90 isn't obvious, explaining the flat left end of the hyperboloid W90 curve in Fig. 19(b). Similarly, large edge concavity prevents some rays from achieving a second reflection, causing a rise in once-reflected rays. These rays doesn't contribute to W90 since they can't reach the CCD, rendering the right end of the hyperboloid W90 curve in Fig. 19(b) flat. Furthermore, smaller grazing incidence angles reduce sensitivity to edge errors.
In conclusion, the influence of edge errors of paraboloid or hyperboloid is different. The once-reflected ray caused by the edge slope will seriously degrade W90. The once-reflected ray with large angle and the blocking effect of the edge will reduce the effective area. It is generally believed that the large angle scattering introduced by surface roughness is the main accuse for affecting W90 [14,26]. However, this section demonstrates that aside from high-frequency errors, edge effect errors can also significantly degrade W90, which requires special attention during the production of focusing mirrors.
4.2.2 Intersection error
Long time polishing can result the collapse of the paraboloid and hyperboloid intersection. Finite element simulation revealed this intersection to be the first place where the mirror separate from the mold. In specific scenarios, after the separation of the intersection, the remaining mirror may still adhere to the mold and deforms as the mold contracts at low temperature. Both these factors can introduce a maximum error of 10µm at the intersection point. The error is modeled by a Gaussian function with a half width 2 mm and a height corresponding to the error value. The simulated outcomes are shown in Fig. 21.
The intersection exhibits a discernible influence on angular resolution, but it is not the main contributing factor. When the absolute value of the error is greater than 50µm, W90 will be deteriorated by 30”. Maintaining the error within 10µm results in a deterioration of HPD and W90 by around 3” and 5” respectively.
4.3 Influence of mirror posture and hoisting deformation
4.3.1 Mirrors posture
The deformation of the thin shell structure mirror is reduced by splicing multiple hoisting pieces at the end of the hyperboloid. Varied lengths of hoisting wires will introduce off-axis deviation. Due to the mirror’s rotational symmetry, the mirror coordinate system (${X_M}{Y_M}Z$ in Fig. 3) can be rotated along the Z-axis to identify a position with an off-axis angle equaling to a pitch angle or a swing angle. To streamline the analysis, the off-axis angle is substituted with the pitch angle in this section, without differentiation between these two angles.
Figure 22 provides an intuitive explanation of the influence of pitch angles on the focal spot shape. For minimal pitch angles, the focal spot becomes fan-shaped. As the error increases, the focal spot gradually becomes butterfly shaped. Substantial error results in once-reflection the hyperboloid as depicted by the green scatterings in Fig. 22④. A large off-axis angle error can cause serious once-reflected phenomenon and a sharp decrease in effective area. Even though the spot shape changes notably, it can be found that the energy on each focal plane remains relatively concentrated.
Fig. 22. #27 Focal spot corresponding to different pitch angle errors under parallel light. Pitch angle errors: ① 30”, ②1’, ③ 0.5°, ④ 1°. (a) Overall view of the focal spot; (b) Partial enlarged view of the focal spot.
Further computations reveal significant variation in focal length due to different off-axis errors and between different mirrors. To investigate the influence of mirror posture on angular resolution during the assembly process, it is necessary to maintain a consistent focal length and size. By setting the image plane to the ideal focal position with the size of 10 × 10 mm, the influence of diverse pitch angles ${\theta _{pitch}}$ is illustrated in Fig. 23.
Fig. 23. Influence of pitch error. (a) Influence on angular resolution; (b) Influence on once-reflected and effective Area.
For ${\theta _{pitch}}$<1’, the trend of angular resolution deterioration shows a gradual change with minimal magnitudes, approximately 1’. As ${\theta _{pitch}}$ increases beyond 1’, angular resolution rapidly deteriorates, particularly at smaller grazing incidence angles. When ${\theta _{pitch}}$ reaches 10’, the HPD and W90 is around 10” and 20” respectively. Upon reaching 0.5°, the maximum angular resolution degradation is observed. This phenomenon is analogous to the influence of edge error, where excessive scattered rays evade CCD detection and cannot participate in the angle resolution calculation. Thus, as off-axis angles reaches a specific threshold, angular resolution peaks while the effective area drastically declines. When ${\theta _{pitch}}$ equals 0.5°, EP #54 loses approximately 20% of its effective area, and when ${\theta _{pitch}}$ reaches 1°, the effective area loss exceeds 80%. In order to minimize the loss of optical performance, it is required to keep the off-axis angle below 1’.
4.3.2 Hoisting deformation error
Uneven tension of the hoisting wires can introduce deformation during the assembly process. To explore this scenario, an assumption is made that tension is concentrated on a few cyclically symmetric hoisting segments that bonded at the hyperboloid's end. Utilizing ANSYS finite element simulation, the mirror's deformation is assessed. As depicted in Fig. 24, concentrated force on few hoisting points results in substantial deformation. Using only two hoisting segments yields deformations ranging from 20µm to 0.18 mm, while three segments exhibit deformations ranging from 1 to 15µm. Consequently, it is imperative to rigorously manage hoisting forces during assembly, mitigating potential deformations through adjustment of mirror position and posture. Due to the challenge of directly measuring the hoisting force, indirect calculation based on measuring the force on the hoisting hook should be employed. Hence, the connection approach between hoisting segments and wires, along with force computation, requires careful consideration. Figure 24 (a, b, c) illustrates that with limited segments under force, paraboloid deformation surpasses that of the hyperboloid. When force distribution is relatively uniform, a coherent wave pattern forms along the circumferential direction, as seen in Fig. 24 (d, e). Notably, a strong correspondence between deformation characteristics and focal spot shape is evident.
Fig. 24. Deformation and focal spot shape of EP # 27 mirror of 0.3 mm thick under different numbers of hoisting points. (a) 2 points; (b) 3 points; (c) 4 points; (d) 6 points; (e) 12 points.
The deformation calculated by ANSYS is then imported into the RTrace_3D for optical performance calculation depicted in Fig. 25. Diverse mirror diameters exhibit significant angular resolution differences under the same hoisting points, which diminish as hoisting point numbers increase. Larger grazing incidence angles offer larger effective area contributed by the mirror and much easier to deformation, demanding more hoisting points for force uniformity. To ensure HPD and W90 remain below 3” and 10” due to hoisting-induced deformation, EP mirrors with diameter greater than #27 require a minimum of 12 hoisting points, while those below #27 require at least 6 points. The considerable W90 of #1 under 12 hoisting points is due to the inner fold of the hyperboloid end (Fig. 24 (e)) which will increase W90 as elucidated in section 4.2.1.
Fig. 25. Influence of different number of hoisting points under force on optical performance. (a) Angular resolution. (b) Once-reflected ray and effective area.
5. Conclusions
We developed a geometric 3D ray tracing algorithm for predicting the optical performance of Wolter-I focusing mirrors. The algorithm's validity was confirmed through simulations and X-ray testing. Based on this, we conducted analyses on common fabrication errors in the electroforming replication process, leading to the following conclusions:
- (1) Surface reconstruction data precisely representing various errors is crucial for accurate ray tracing simulations. Generatrix path and spiral path measurements can complement each other. Enhancing the algorithm's accuracy requires comprehensive consideration of the effects of operational parameters during fabrication and testing. Cross-validation with diverse measurement and simulation methods is necessary for definitive conclusions.
- (2) Absolute radius and taper errors indirectly influence HPD by affecting focal length. The influence of absolute radius error exceeds that of taper error. Roundness error minimally affects focal length, mainly altering focal spot shape and degrading angular resolution. Out-of-phase roundness has a milder effect compared to in-phase roundness, and angular resolution degradation diminishes with phase difference increase.
- (3) Mirror edge errors may have a significant impacts on the optical performance, with varying effects on paraboloid and hyperboloid. There are differences between the on-ground test results and the actual operational performance for edge errors. The once-reflected ray caused by the error will seriously degrade W90. Edge blockage and once-reflected rays with large scattering angles will diminish the effective area.
- (4) Non-uniform hoisting force during assembly can lead to substantial deformation. Deformation characteristics correlate with focal spot shape. Minimizing angular resolution degradation caused by deformation necessitates maximizing hoisting points and ensuring even force distribution.
The current limitations of the algorithm lie in its reliance on geometric ray tracing, making it applicable solely to simulations of low-frequency errors. It can only perform quantitative calculations under conditions of relatively small roughness, grazing incidence angles, and X-ray energy. The further improvements will focus on accounting for surface scattering effects and developing an optical performance prediction algorithm for full-band frequency errors.
Funding
National Key Research and Development Program of China (2022YFB3206700, 2022YFB3403300); The Strategic Priority Research Program of the Chinese Academy of Sciences (XDA15310000).
Acknowledgments
The authors would like to thank Chunsheng Wu for his help in the measurement of mirror profile.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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