1. INTRODUCTION

Optical imaging systems are widely used in biomedicine [1–3], materials detection [4,5], astronomical observation [6–8], integrated circuit fabrication [9,10], and other fields [11,12]. As a result of the demand for scientific research and major engineering applications, a series of high-performance optical systems have gradually been proposed, including laser fusion devices [13], lithography systems [10,11], the James Webb Space Telescope [8,14], the Thirty Meter Telescope (TMT) [15,16], and the Giant Magellan Telescope (GMT) [17,18]. The optical surfaces in these systems, including both spherical and aspherical surfaces that require large apertures and high precisions, are extremely difficult to manufacture [19]. In addition, freeform surfaces are now gradually being applied in high-precision and high-performance optical imaging systems [12,20,21] due to their advantages, such as higher degrees of freedom for aberration correction, and ensuring more compact structures than conventional surfaces [20,22–25]. However, the nonrotational symmetry of freeform surfaces also increases their manufacturing difficulty [26–30]. The fabrication of these high-precision, large-aperture, and complex surface-shaped optical components is often a bottleneck for the development of optical imaging systems, which is an enduring research topic [26,31,32].

Currently, the root-mean-square (rms) or peak-to-valley (PV) value of the entire surface is used to evaluate optical component manufacturing accuracy and to describe the tolerance requirement provided by the conventional tolerance analysis theory [33–37]. This tolerance theory is a probabilistic and statistical theory based on the Monte Carlo method [38–40] and is suitable for tolerance analysis of optical components manufactured in large quantities. It has been considered to be the only effective theory for over 50 yr.

However, optical systems are complex and have local characteristics in many aspects. First, for any given field, the nominal wave aberrations have a spatial distribution at different aperture positions. Thus, the nominal wave aberrations of rays that are incident at different positions on the surface also vary. Second, different fields use different working areas on the surface, which may also demonstrate localized characteristics in wave aberrations. For the above two points, to meet a specified imaging quality, the permissible wave aberration changes of different rays caused by surface figure errors are different. Third, rays incident at different positions on the surface generally have different angles of incidence. Therefore, the change in the wave aberration of each ray caused by the same manufacturing error is different, i.e., the surface-figure-error sensitivity has local characteristics. Moreover, the above phenomena are more pronounced in off-axis optical systems. Thus, to meet a permissible imaging performance range, the tolerances for different areas on the surface in an optical system also vary, i.e., the tolerance for a surface should be described by a spatial map. However, the conventional tolerance analysis theory only provides a generic parameter for the entire surface and cannot indicate the tolerance differences in different areas on the surface.

The component quality is generally evaluated by the rms or PV value of the surface-figure error, which is directly obtained from the surface measurement data. However, the current figure of merit, e.g., rms or PV, cannot accurately indicate the system imaging quality when the manufactured component is implemented in the system. Components with the same rms or PV value of figure errors are likely to produce different imaging performances. The imaging performance of the system with the manufactured component is of interest, rather than the surface geometric deviation of the component itself. Therefore, a merit function of component quality that can be linked to imaging performance is highly desired. The merit function should have different values for the same component when implemented in different systems.

In this paper, a local tolerance model for optical surfaces in imaging systems is proposed, which can provide different tolerances for different areas on the surface accordingly, and allows different areas to be manufactured with specific precision requirements. The local tolerance model is demonstrated through examples of different optical imaging systems.

A merit function of optical component quality linked to imaging performance is further proposed. The nominal wave aberration, the surface-figure-error sensitivity, the irradiance, and the component geometric accuracy are comprehensively considered in this merit function. With partial information about the designed system, the quality of an individual component can be accurately evaluated by the manufacturer. Taking the Cassegrain system as an example, the quality of primary mirrors with different figure errors is evaluated.

2. METHOD

In this section, an isolated-point-jump-off model is proposed at first to simulate figure errors of optical surfaces. Then the surface-figure-error sensitivity analysis is introduced with the aid of the isolated-point-jump-off model, which is a forward tolerancing process. Next, the local tolerance analyses for a single surface and multiple surfaces are illustrated in Sections 2.C and 2.D, both of which are inverse tolerancing processes. Finally, a merit function of component quality is further proposed.

A. Isolated-Point-Jump-Off Model and Perturbed Ray

Manufacturing errors of the optical surface can cause data points to deviate from those on the nominal surface. An isolated-point-jump-off model is proposed to simulate the figure errors of optical surfaces. Here, a reflective surface is used to illustrate the model in Fig. 1; whereas, this model also applies to refractive surfaces. The manufacturing error at a point $P$ on the surface causes point $P$ to jump to point $P^\prime$. The manufacturing error at point $P$ is simulated by the jumping distance $d$ along the surface normal at this point. If sufficient points are sampled on the surface, the values of the jump distance $d$ for all sampled points along their normal directions can then be used to model the surface-figure errors. The positive and negative manufacturing errors of a point on the surface can be simulated using its positive and negative jumping distances, respectively. For reflective surfaces, the sign of a jump is positive when the point jumps toward the side on which the ray is incident; otherwise, its sign is negative.

 

Fig. 1. Schematic of the isolated-point-jump-off model and perturbed ray.

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The surface errors cause the ray propagation paths passing through points on the surface to change. For any field $F$, the ray that passes through point $P^\prime$ with perturbation is called the perturbed ray (solid red line) as shown in Fig. 1.

B. Surface-Figure-Error Sensitivity Analysis

The sensitivity analysis is a forward tolerancing process in which the changes in imaging performance are calculated after the nominal surface is perturbed with figure errors. Here the wave aberration is chosen as the performance metric. The wave aberration of a ray is the optical path difference of the actual wavefront along that ray from the reference spherical wavefront. In this section, the relationship between the wave aberration change in the perturbed ray and the corresponding manufacturing error is established to analyze the surface-figure-error sensitivity.

For one field, the manufacturing error causes point $P$ to jump away to point $P^\prime$, and point $A^\prime$ on the actual wavefront also deviates from point $A$ on the nominal wavefront as illustrated in Fig. 2, which causes the wave aberration of the perturbed ray that passes through $P^\prime$ to change. Point $O^\prime$ represents the exit pupil center of this field. The wave aberrations of the ray that passes through points $P$ and $P^\prime$ are the nominal wave aberration and the perturbed wave aberration, denoted by ${W_0}$ and ${W_E}$, respectively. The difference between them is $\Delta W$, which is the change in wave aberration of the nominal value and the as-built value. Since the surface-figure errors are small, the isolated-point-jump-off model is utilized to calculate the wave aberration change, denoted by $\Delta W$ of each perturbed ray [41].

 

Fig. 2. Schematic of wave aberration changes in the perturbed ray.

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When the manufacturing errors at all points are equal, a larger $\Delta W$ indicates that the imaging quality is more sensitive to the manufacturing error at that point. The $\Delta W$ ’s at different points caused by the equal errors are different, i.e., the surface-figure-error sensitivity has local characteristics.

 

Fig. 3. (a) Layout of a three-mirror off-axis freeform imaging system. (b) Footprints of different fields on the tertiary mirror. (c) Nominal wave aberration map of the central field. (d) Changes in the wave aberration for the center field caused by equal errors at each point on the tertiary mirror.

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C. Local Tolerance Analysis of a Single Surface

In this section, the local tolerance analysis of a single surface is introduced, which can be considered as an inverse tolerancing process. To meet a specified imaging performance, the tolerance for the surface in an optical system is local. The three-mirror off-axis freeform imaging system shown in Fig. 3(a) is used as an example; the system specifications are introduced in Section 3.A. First, the working areas on the surface for different fields are not completely overlapped as illustrated in Fig. 3(b). The black circle represents the surface aperture, and the colored circles represent different working areas for different fields. For one field, the nominal wave aberration of each ray is different. Figure 3(c) shows the nominal wave aberration map for the different rays in the central field on the tertiary mirror. Thus, to ensure a specified imaging quality, the wave aberration change allowed for each ray caused by the surface-figure error is different. Second, the surface-figure-error sensitivity also has local characteristics. When the manufacturing error at each point on the tertiary mirror is 0.3 µm, the wave aberration change in each ray in the central field is different as shown in Fig. 3(d).

The analysis process of the local tolerance for a single surface includes the following three steps.

First, the wave aberration is chosen to be the performance metric, and the permissible range of wave aberrations, denoted by ${W_{{\rm Exp}}}$, is provided according to the expected imaging performance of the as-built system.

Second, the tolerance for the surface is calculated when one field is considered. Here, ($x$, $y$, $z$) is the coordinate of any point $P$ located in the working area of field $F$ on the surface. The jumping distance will cause the wave aberration change in the perturbed ray of field $F$, denoted by $W_E^{\;(F)}(x,y,z)$. To achieve the expected as-built performance, $W_E^{\;(F)}(x,y,z)$ should not exceed the permissible upper limit and lower limit as shown in Eq. (1),

Third, the local tolerance for the surface is obtained. The tolerances for each field obtained in step 2 are then intersected to obtain their overlapped zone (indicated by pink), which is the final local tolerance as illustrated in Fig. 4.

 

Fig. 4. Schematic of local tolerance.

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As the permissible performance range gets smaller, the imaging quality of the as-built system gets closer to that of the nominal system. If the upper and lower limits of wave aberrations for each field are the peak-and-valley values of wave aberration of field $F$ in the nominal system, then a system with imaging quality reaching the nominal value can be obtained.

D. Local Tolerance Analysis of Multiple Surfaces

The tolerances of multiple surfaces in an optical system affect each other. If we make the tolerance for one surface looser, then the tolerances for other surfaces will become tighter. Next, the local tolerance analysis of multiple surfaces is introduced.

First, the effect of figure errors of multiple surfaces on the imaging quality is analyzed. The ray $R$ of field $F$ intersects with each surface $S$ in turn at point ${P_s}(x,y,z)$, and the nominal wave aberration for this ray is $W_0^{(F)}(x,y,z)$. When one surface has a figure error, the intersection points of this ray with each surface are considered approximately unchanged. The wave aberration change in the perturbed ray $R^\prime$ caused by the figure errors for all surfaces is equal to the summation of that caused by the figure error for each surface individually as shown in Eq. (2),

Here, $n$ is the number of surfaces, $W_E^{(F)}(x,y,z)$ is the wave aberration of the perturbed ray, $\Delta {W^{(F)}}(x,y,z)$ is the wave aberration change caused by the figure errors for all surfaces, and $\Delta {W^{(S,F)}}(x,y,z)$ is the wave aberration change caused by the figure error for each surface individually. With the figure errors for all surfaces, the final wave aberration should meet the requirement shown in Eq. (3),

For a two-surfaces system, a specified wave aberration limit is first assigned to one surface according to the actual requirement, and the local tolerance for this surface is analyzed. Next, the local tolerance for the other surface is calculated for each field using Eqs. (2) and (3), and the final local tolerance is obtained by taking the intersection of that of all fields.

The local tolerance analysis method proposed here solves the tolerance at each point on the surface by ensuring the final wave aberration of each ray meets the requirement. This local tolerance provides a spatial map that fundamentally differs from the rms or PV.

E. Merit Function of Optical Component Quality

Here, a merit function is proposed to evaluate the individual component quality. This merit function comprehensively considers the nominal wave aberrations of all rays along with the changes in the wave aberration caused by surface-figure errors. In the optical system, If the manufacturing error is $d(x,y,z)$, the changes in wave aberration of the perturbed ray of a specific field $F$, denoted by $\Delta {W^{(F)}}(x,y,z)$, can be calculated by

The quality of an optical component can be evaluated using the RWE, which is the average value of ${{\rm RWE}^{(F)}}$ over the field of views (FOVs) of the optical system as shown in Eq. (7),

Equations (4)–(7) show that the RWE, which is a function of $d(x,y,z)$, ${\theta ^{(F)}}(x,y,z)$, and $W_0^{(F)}(x,y,z)$, reflects the average rms wave aberration over the full FOV of the optical system when using the specific component. A larger RWE for an optical component corresponds to poorer imaging quality for the final optical system when using that component. The correctness of this RWE merit function is verified in Section 3.C. The nominal wave aberration, the angle of incidence, and the normalized irradiance are independent data that can be separated from the designed system. After the figure error of a component is measured, the component quality can be evaluated by the manufacturer with the above-mentioned information.

Here, the RWE merit function is presented as the average ${{\rm RWE}^{(F)}}$ value over full FOV. Additionally, the RWE merit function can be modified according to the design requirements. For example, the weight ${\mu ^{(F)}}$ of each field can be introduced as shown in Eq. (8),

3. EXAMPLES

According to the Rayleigh wavefront criterion, the imaging is considered to be perfect when the PV value of the wave aberration does not exceed $\lambda /{4}$. Here, the expected performance ${W_{{\rm Exp}}}$ is specified as $\lambda /{4}$. In Sections 3.A and 3.B, the proposed local tolerance model is used to analyze the local tolerances of two example systems. Next, the quality of the primary mirror with figure error in the second example is then evaluated in Section 3.C using the proposed merit function of the component quality.

A. Local Tolerance Analysis of a Three-Mirror Off-Axis Freeform Imaging System

The first example is a three-mirror off-axis freeform reflective system, which operates in the long-wave-infrared band (8–12 µm). It has a focal length of 100 mm, a $F$ number of 1.5, and a FOV of ${3}^\circ \times {4}^\circ$. This system is symmetrical about the YOZ plane in which the aperture stop is located on the secondary mirror. All mirrors are freeform surfaces that can be described using $XY$ polynomials with terms up to the sixth order. The system layout is shown in Fig. 3(a). The field map of the rms wave aberrations is shown in Fig. 5(a) where the average value is ${0.0102}\lambda$ ($\lambda = {10}\;{\unicode{x00B5}{\rm m}}$), the maximum value is ${0.014}\lambda$, and the minimum value is ${0.007}\lambda$.

 

Fig. 5. (a) Field map of the rms wave aberrations for the three-mirror off-axis system. (b) Sampled points on the tertiary mirror.

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Fig. 6. Local tolerance for tertiary mirror in the three-mirror off-axis system. (a) Upper variation. (b) Lower variation. (c) Local tolerance. (d) Enlarged edge area from part (c).

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The local tolerance for the tertiary mirror with a circular diameter of 130 mm was analyzed. The data points were sampled uniformly in the form of rectangular grid points where thesampling interval was 0.5 mm in the $X$ and $Y$ directions as shown in Fig. 5(b). The full FOV was sampled at intervals of 0.3° in the horizontal direction and 0.4° in the vertical direction. The local tolerance for the tertiary mirror was calculated as shown in Fig. 6. The upper and lower variations are shown in Figs. 6(a) and 6(b), respectively. Figure 6(c) shows the local tolerance map, which is the difference between the upper and the lower variations.

It can be seen from Fig. 6 that the local tolerance varied from 0.9 µm to 1.4 µm, which were marked in different colors. Comparing Figs. 6(a) and 6(b), it can be seen that the upper and lower variations were different in the same area. The edge area of the surface in Fig. 6(c) is enlarged and is shown in Fig. 6(d). The continuous change in the tolerance in the edge area was more dramatic than that in the center area.

In this paper, computations were implemented in MATLAB 2020a on a workstation with an Intel i9-10900 K CPU and 32 GB (RAM), running Windows 11 (64 bit). The calculation of this example took around 6 h.

To verify this local tolerance, the figure errors that met the local tolerance for the tertiary mirror were generated by 36 terms of standard Zernike polynomials. Details are described in Supplement 1. A total of 3000 figure errors were generated and were then separately added onto the tertiary mirror utilizing CODEV’s interferogram file (INT file) to obtain optical systems with figure errors, i.e., as-built systems. For each as-built system, the PV values of the wave aberrations of 25 sampled fields were calculated, and the maximum PV wave aberrations were obtained. The maximum PV wave aberrations for 3000 as-built systems are shown in the histogram of Fig. 7, all of which do not exceed $\lambda /{4}$. The result demonstrates the effectiveness of the local tolerance model.

To compare the local tolerance with the conventional tolerance, the rms values of the above 3000 figure errors were calculated, and the maximum value was ${0.032}\lambda$. Then, the conventional tolerance analysis theory based on the Monte Carlo method was used to analyze the tolerance for the tertiary mirror. Some 21 groups with 3000 random figure errors each were generated, and each group had the same rms value ranging from ${0.012}\lambda$ to ${0.032}\lambda$ with an interval of ${0.001}\lambda$. Also, these figure errors were generated by the 36 terms of standard Zernike polynomials. Then, in each group, each figure error was separately added to the tertiary mirror to obtain the corresponding as-built system group. In each as-built system group, the yield is the ratio of the number of as-built systems with maximum PV wave aberration not exceeding $\lambda /{4}$ to the total number of as-built systems. The blue line in Fig. 8 illustrates the relationship between the yield and the rms values of these figure errors. The red line shows the relationship when the figure errors meet the local tolerance.

It can be known from the blue line that when the yield was 58%, the rms value of the figure errors was required to be no more than ${0.022}\lambda$. If the yield was expected to increase to 97.8% and even to 100%, the requirements for the rms values would rise sharply to ${0.016}\lambda$ and ${0.012}\lambda$ respectively. However, in the case of local tolerance, the figure errors with ${0.032}\lambda$ rms value can ensure a 100% yield as long as the figure error met the local tolerance.

Next, the conventional tolerance was calculated by taking the rms wave aberrations as an imaging quality criterion. In this system, the ratio of the PV and rms values of wave aberrations for each field was between 5.2 and 8. Assuming a ratio of 7, if the PV wave aberrations do not exceed $\lambda /{4}$, the rms wave aberrations would not exceed ${0.0357}\lambda .$ A total of 36 terms of standard Zernike polynomials were selected as tolerance terms, and the tolerance for the tertiary mirror was calculated by the conventional method. The result showed that when the yield was 97.7% and even to 100%, the rms values of figure errors were ${0.034}\lambda$ and ${0.0288}\lambda$, respectively.

B. Local Tolerance Analysis of a Cassegrain System

The second example is a Cassegrain system that operates in the middle-wave-infrared band (3–5 µm) with a $f$-number of 5, a FOV of ${0.3}^\circ \times {0.3}^\circ$, and a focal length of 1.5 m. The aperture stop is on the primary mirror, which is a quadric surface. The secondary mirror is an aspheric surface described by up to eighth-order aspheric terms. The system layout is shown in Fig. 9(a). The field map of the rms wave aberrations is shown in Fig. 9(b) where the average value is ${0.0171}\lambda$ ($\lambda = {3}\;{\unicode{x00B5}{\rm m}}$).

 

Fig. 7. Histogram of the maximum PV wave aberrations for 3000 as-built systems with figure errors.

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Fig. 8. Yield versus rms values of the surface figure error.

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Fig. 9. Cassegrain system. (a) System layout. (b) Field map of the rms wave aberration.

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In total, 81 fields were sampled uniformly for the tolerance analysis. The diameters of the primary and secondary mirrors are 300 mm and 100 mm. Data points on both mirrors were sampled uniformly in the form of rectangular grid points with a sampling interval of 2 mm for the primary mirror and 0.5 mm for the secondary mirror. The local tolerances of the primary and secondary mirrors were separately calculated when the other mirror was nominal as shown in Figs. 10 and 11 where the local characteristics were shown apparently. The tightest tolerance for the primary mirror was 0.25 µm; whereas, the loosest one was 0.35 µm. For the secondary mirror, the tightest and the loosest tolerances were 0.24 µm and 0.38 µm, respectively. Additionally, the maps of the local tolerance for both mirrors were symmetrical about the YOZ plane and the XOZ plane, respectively, but not rotationally symmetric. The shape of the FOVs of the optical system affects the local tolerance maps of the surfaces. If a rotationally symmetric system has a circular FOV, then the final local tolerance map is rotationally symmetric. The calculation of local tolerance for the primary and secondary mirrors took around 1.5 h and 2.5 h, respectively.

 

Fig. 10. Local tolerance for primary mirror in the Cassegrain system when the secondary mirror was nominal. (a) Upper variation. (b) Lower variation. (c) Local tolerance.

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Then, after assigning the permissible performance range to the primary and secondary mirrors, the local tolerances for them were calculated. The final expectation ${W_{{\rm Exp}}}$ of the system was still $\lambda /{4}$. The expectation of PV wave aberrations assigned to the primary mirror was specified as 3 $\lambda /{16}$ here, considering the diameter and manufacturing difficulty of the primary mirror. The local tolerances for both mirrors were shown in Fig. 12. This calculation time was around 20 h.

In Figs. 12(a)–12(c), the local tolerance for the primary mirror was similar to that shown in Fig. 10. The local tolerance map of the secondary mirror shown in Figs. 12(d)–12(f) was different from and tighter than that shown in Fig. 11. This indicates that the tolerances of multiple surfaces are not independent. The local tolerance map of one surface changes with that of the other surfaces.

Totally 3000 random figure errors that met the local tolerance for the primary and secondary mirrors in Fig. 12 were generated randomly for each mirror, and they were then combined to obtain 3000 optical systems with figure errors on both surfaces. The maximum PV wave aberrations for all as-built systems did not exceed $\lambda /{4}$ (reaching a maximum of ${0.248}\lambda$), which verifies the effectiveness of local tolerance in Fig. 12.

 

Fig. 11. Local tolerance for secondary mirror in the Cassegrain system when the primary mirror was nominal. (a) Upper variation. (b) Lower variation. (c) Local tolerance.

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Fig. 12. Local tolerances for the primary and secondary mirrors. (a)–(c) are upper variation, lower variation, and local tolerance for the primary mirror, respectively. (d)–(f) are upper variation, lower variation, and local tolerance for the secondary mirror, respectively.

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The radius $R$ of an optical surface reflects the optical power of that surface. The deviation of radius $R$ caused by manufacturing errors can be compensated by adjusting the surface position. The local tolerance was then analyzed with the position compensation. Here, the position of the reflective surface with a radius error was shifted to achieve a constant conjugate distance between the object and the image planes of the surface. Taking the Cassegrain system above as an example, the focal length of a reflective surface was expressed approximately as $f^\prime = R/{2}$. The surface position shift, i.e., the position compensation $\Delta l$, can be obtained from Eq. (9),

In this example, the primary mirror radius was ${-}{1123.55}\;{\rm mm}$. If the primary mirror’s radius error $\Delta R$ was equal to 1 mm, after a 0.5-mm position compensation, the system imaging quality remained almost unchanged. Then, the local tolerance for the primary mirror was recalculated with the secondary mirror nominal, and the results are shown in Fig. 13.

 

Fig. 13. Local tolerance for position compensated primary mirror in the Cassegrain system with the secondary mirror nominal. (a) Upper variation. (b) Lower variation. (c) Local tolerance.

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A comparison of Figs. 13 and 10 shows that the maps of the two local tolerances were highly similar where the differences in the tolerance at each point did not exceed 0.5 nm. This difference was negligible when compared with the tolerance for the primary mirror. If the radius error $\Delta R$ increased to 5 mm, the primary mirror was then compensated by a shift of 2.5 mm. The local tolerance map for the primary mirror was still highly similar to that in Fig. 10 with the maximum tolerance difference equal to 3.2 nm at each point. As the radius error $\Delta R$ increased, the local tolerance for the primary mirror would change more.

$\Delta {R_{\rm{max}}}$ is the maximum radius error of the optical surface during the manufacturing process. If a surface with any $\Delta R \le \Delta {R_{\rm{max}}}$ can meet a specific tolerance, the system imaging quality can be ensured using position compensation. This specific tolerance should be the intersection of the local tolerance $s$ of the surfaces of which the radii are $R + \Delta {R_{\rm{max}}}$ and $R - \Delta {R_{\rm{max}}}$ after position compensation. When $\Delta {R_{\rm{max}}}$ increases, this local tolerance becomes stricter.

 

Fig. 14. Histograms of the RWE values for six groups of primary mirrors with figure errors. For (a)–(c), the PV values of figure errors are ${0.1}\lambda$, ${0.08}\lambda$, and ${0.05}\lambda$, respectively. For (d)–(f), the rms values of figure errors are ${0.02}\lambda$, ${0.016}\lambda$, and ${0.01}\lambda$, respectively.

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C. Quality Evaluation of Cassegrain System’s Primary Mirror

The quality of the primary mirror in the Cassegrain system was evaluated using the RWE merit function with 25 uniformly sampled fields. First, for each field, the nominal wave aberrations and the angles of incidence of the rays as well as the normalized irradiance were obtained via ray tracing. Then, six groups of random figure errors were generated by the linear combination of 36 terms of Zernike polynomials, each of which consisted of 3000 figure errors with the same PV or rms value. The PV values of the three groups were ${0.1}\lambda$, ${0.08}\lambda$, and ${0.05}\lambda$, and the rms values of the other groups were ${0.02}\lambda$, ${0.016}\lambda$, and ${0.01}\lambda$. Next, these figure errors were separately added onto the primary mirror, and the RWE values of the primary mirrors with each error were calculated using Eqs. (4)–(7).

The group of 3000 primary mirrors with a PV value of ${0.1}\lambda$ for figure errors was ranked by the RWE values. Additionally, these mirrors were put into the Cassegrain system separately to calculate the imaging quality of as-built systems via ray tracing and were then ranked according to the average rms wave aberrations for full FOV. The two ranking results are consistent, which indicates that the RWE merit function can reflect the imaging quality of systems with these components accurately without the lens data.

The RWE values of each group of primary mirrors with figure errors were analyzed statistically as shown in Fig. 14. The maximum value, minimum value, and distribution range of RWE for each group are given in Fig. 14, and they are denoted by ${{\rm RWE}_{\rm{max}}}$, ${{\rm RWE}_{\rm{min}}}$, and $\Delta {\rm RWE}$, respectively. Comparison of Figs. 14(a)–14(c) shows that as the PV value of the figure errors group decreased, the overall RWE values of the primary mirrors in the group became smaller, and the ${{\rm RWE}_{\rm{max}}}$ and ${{\rm RWE}_{\rm{min}}}$ values also decreased. This indicates that primary mirrors with higher surface geometric accuracy had a higher probability of producing better imaging quality. Simultaneously, the distribution range of the RWE, i.e., $\Delta {\rm RWE}$, became narrower, indicating that the difference in imaging quality of the actual system was also gradually narrowing. Figures 14(d)–14(f) also show similar conclusions in which the three groups of primary mirrors had figure-error rms values of ${0.02}\lambda$, ${0.016}\lambda$, and ${0.01}\lambda$.

As can be seen from any of the subplots in Fig. 14, even if a group of primary mirrors had the same rms or PV value of figure errors, their RWE values were different. This indicates that surfaces with the same rms or PV value may produce different performances. In addition, a primary mirror with a larger PV or rms figure error may have a smaller RWE value than that with a smaller PV or rms.

To provide a further illustration of the conclusions above, six primary mirrors with quite different figure errors were selected from the four groups above, and their figure errors are shown in Fig. 15. The spatial maps of their figure errors were quite different. A comparison of Figs. 15(a) and 15(b) shows that, although their figure-error PV values were both ${0.1}\lambda$, the RWE of the latter was less than half of the former, i.e., the latter would produce much better imaging quality. From Figs. 15(b) and 15(c), the primary mirror with a larger figure-error PV value had a smaller RWE value and can produce better imaging quality. In this case, the result given by the PV value would be misleading. Figures 15(d)–15(f) indicate that the rms value had the same problems. The figure-error map of each manufactured component is different. Additionally, for the designed system, the wave aberration of each ray, the surface-figure-error sensitivity, and the normalized irradiance values at different surface positions are different. However, the conventional merit function does not consider these local characteristics, which causes its result to be inconsistent with the actual quality of components evaluated from a performance perspective sometimes.

 

Fig. 15. Figure errors of the primary mirrors. For (a)–(c), the PV values are ${0.1}\lambda$, ${0.1}\lambda$, and ${0.08}\lambda$, respectively. For (d)–(f), the rms values are ${0.02}\lambda$, ${0.02}\lambda$, and ${0.016}\lambda$, respectively.

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4. DISCUSSION AND CONCLUSION

The conventional tolerance analysis theory provided a single or few parameters through rms or PV requirement for the entire surface, which did not reflect the local characteristics of the tolerance. To overcome the above shortcoming, a local tolerance model for optical surfaces was derived, which can obtain the local tolerance by solving the accurate tolerance at each point on the surface.

The tolerances can vary considerably across different areas on the surface. Therefore, for optical components, large figure errors may not necessarily exceed the local tolerances in the corresponding areas; whereas, small ones may not necessarily meet the local tolerances. The local tolerance points out that different areas on the surface have different requirements for manufacture, which means that not the entire surface must be manufactured with high precision. Also, the local tolerances can pinpoint the locations and values of figure errors that exceed the tolerance and guide further surface shape refinishing during the manufacturing process, which is impossible when using the conventional method. In addition, interferometers can measure the local surface shape, and some machines, such as magnetorheological finishing, ion-beam polishing, and Bonnet polishing can locally change the surface shape. Local tolerances are particularly valuable for the manufacture of components in high-precision large-aperture optical systems.

In the conventional tolerance theory, the tolerance was given for a specified yield. As the yield increased further, the tolerance tightened dramatically; whereas, the local tolerance model calculated the tolerance for each surface based on the permissible wave aberration change assigned to each surface. As long as the figure error for each surface met its local tolerances, the system imaging quality 100% met the requirement, and the rms or PV values of the surface figure errors were not necessarily increased significantly.

Moreover, the local tolerances of individual surfaces in an optical system affected each other. The map of the local tolerance for one surface changed with that of other surfaces. Also, the shape of the FOV of the optical system affected the map of the local tolerance for the surface. If a rotationally symmetric system had a nonrotational FOV, then the local tolerance maps for its surfaces were nonrotationally symmetric. In addition, for two designs with the same given specification, even if they had the same rms or PV wave aberration, the local tolerance maps for their surfaces were different due to the differences in the map of nominal wave aberrations.

In this paper, wave aberration was used as the imaging performance criterion for tolerance analysis. If other performance criteria were chosen, such as modulation transfer function and spot size, they should be first converted to wave aberration requirements [42–44] before the local tolerance analysis.

Furthermore, component quality had always been conventionally evaluated through the rms or PV value of the entire surface figure error. In this paper, a merit function of component quality linked to imaging performance was proposed. Additionally, the merit function was application specific, and a component may have different RWE values in different systems.

The proposed merit function had a few advantages. First, the quality of a component was directly linked to the imaging performance, rather than a geometric deviation. A smaller RWE value corresponded to a higher imaging quality in the system. However, components with the same rms or PV value of figure errors might have different imaging qualities in the system. Even a component with a larger rms or PV value of figure errors might produce better imaging quality. Second, the imaging qualities of different fields can be reflected by the ${{\rm RWE}^{(F)}}$ values when a component with a figure error was implemented in the system. Therefore, the proposed merit function is an effective method to determine the quality of components. Third, using the proposed merit function, manufacturing, and metrology teams can independently evaluate the quality of individual components when partial information about the designed system was provided.

With the advancement of more complex optical systems, this framework provided new perspectives to the manufacture of these optical systems, especially high-precision large-aperture systems and had transformative impacts on theories, technologies, and criteria in the fields of optical design, manufacture, and metrology.