Comprehensive performance domain tolerance analysis methodology for freeform imaging spectrometers

Yujie Xing; Yujie Xing; Yujie Xing; Yujie Xing; Jun Yu; Jun Yu; Jun Yu; Jun Yu; Xuquan Wang; Xuquan Wang; Xuquan Wang; Xuquan Wang; Hongmei Li; Hongmei Li; Hongmei Li; Hongmei Li; Chunling He; Chunling He; Chunling He; Chunling He; Zhiyuan Ma; Zhiyuan Ma; Zhiyuan Ma; Zhiyuan Ma; Dongfang Wang; Zhanshan Wang; Zhanshan Wang; Zhanshan Wang; Zhanshan Wang; Xinbin Cheng; Xinbin Cheng; Xinbin Cheng; Xinbin Cheng; Xiong Dun; Xiong Dun; Xiong Dun; Xiong Dun

doi:10.1364/OE.519818

1. Introduction

Imaging spectrometers, integral for simultaneous acquisition of spatial and spectral data, have become indispensable in a multitude of domains, such as earth observation [1–3], deep space exploration [4,5], intelligent agriculture [6,7], food safety inspection [8,9], criminal detection [10], and biomedical measurement [11,12]. The increasing range of applications is driving the demand for miniaturization of imaging spectrometers. One technological approach to achieve this is the on-chip imaging spectrometer, which utilizes computational on-chip spectral coding techniques [13,14] and micro 3D printing technology [15]. However, there exists a trade-off between system volume and either spectral or imaging performance. Another approach involves utilizing freeform technology to compact the optical system, such as employing freeform mirrors [16] or freeform gratings [17]. The advent of freeform technology has significantly enhanced these instruments, particularly through the development of the three-mirror anastigmat (TMA) configuration [18,19] and the Schwarzschild configuration [20]. These configurations, characterized by their compact size and high performance, employ plane gratings for specialized detection. These gratings serve as cost-effective alternatives to the convex gratings used in Offner [21–23] and concave gratings in Dyson configurations [24–26]. The Schwarzschild configuration is especially promising due to its reduced grating size and minimal stray light.

Recent advancements in the optical design of Schwarzschild imaging spectrometer have been driven by research exploring the use of spherical [27], aspherical [28], and freeform [29–31] mirrors. These studies aim to decrease system volume while enhancing performance. For instance, Liu [29,30] demonstrated that the Schwarzschild configuration with freeform mirrors excels in minimizing spot size and distortion than other types. Additionally, Liu [31] successfully employed freeform surfaces to counterbalance astigmatism aberrations, resulting in a more compact optical system.

Despite these advancements in freeform optical design, freeform Schwarzschild imaging spectrometer prototypes have rarely been reported due to the absence of an effective tolerance analysis method. The existing tolerance analysis techniques, such as the approach proposed by Deppo [32] for off-axis surface tolerance analysis, enable rotational adjustment relative to the optical element center through coordinate conversion. Additionally, a novel opto-mechanical tolerance analysis method presented by Lin [33] and Lamontagne [34] employed conjoint analysis with optical and mechanical software. Furthermore, Chen [35] and Hu [36] proposed a freeform machining tolerance analysis method that utilizes Zernike freeform surfaces to conduct surface irregularity analysis. However, those tolerance analysis methods primarily rely on a single criterion, such as the MTF, making them unsuitable for freeform imaging spectrometers. In this type of optical system, the keystone/smile distortion and spectral resolution are equally critical to the system MTF. If the tolerance analysis solely relies on the system MTF as the criterion, the keystone/smile distortion and spectral resolution requirements may not be met, thereby impeding the efficacy of such approaches in guiding machining and assembly processes.

This paper introduces a comprehensive performance domain tolerance analysis method for freeform imaging spectrometers, encompassing exhaustive tolerance criteria such as the system MTF, keystone/smile distortion, and spectral resolution. These tolerance criteria were evaluated by performing conjoint calculations in Zemax and Matlab based on performance rendering. Our method was applied to the tolerance analysis of a freeform Schwarzschild imaging spectrometer with an F-number of 3.2 and a spectral range of 400–1700 nm. The tolerance analysis results identified the first (M1) and third (M3) freeform mirrors as critical components. To improve the assembly efficiency, we developed and implemented a preassembly process for these mirrors by utilizing a CGH for alignment purposes. Subsequent assembly and rigorous performance evaluation of the spectrometer confirmed the effectiveness of our comprehensive performance domain tolerance analysis method.

2. Comprehensive performance domain tolerance analysis method

Tolerance analysis is indispensable for assessing the feasibility of manufacturing and assembling optical systems. Nevertheless, a significant hurdle in conducting tolerance analysis for freeform imaging spectrometers stems from the inability of the conventional single tolerance criterion to fulfill the demands of comprehensive performance assessment for these systems. To address this limitation, we developed comprehensive tolerance criteria by performing conjoint analysis with Zemax and Matlab, as illustrated in Fig. 1. This approach consists of four key components: tolerance parametric model unit, comprehensive performance evaluation unit, sensitivity analysis unit, and tolerance values adjustment unit. Firstly, the machining and assembly tolerance values, derived from general machining and assembly abilities, are transformed into the boundary of simulation parameters through the tolerance parametric model unit. Secondly, we employ comprehensive performance evaluation and sensitivity analysis based on the boundary of simulation parameters, resulting in the worst each normalized tolerance criterion value and identification of the top three sensitivities for each criterion. Finally, if the normalized criterion values fail to meet the specified requirements, the top three sensitive tolerance item values undergo meticulous adjustments based on the rules in the tolerance values adjustment unit. Then, the analysis process is iterated until the normalized criterion values satisfy the specified requirements. The specific implementation methods and details of each module are elaborated below.

Fig. 1. Framework of the comprehensive performance domain tolerance analysis method for freeform imaging spectrometer. (a) Comprehensive performance domain tolerance analysis units. (b)-(d) The units for tolerance parametric model, comprehensive performance evaluation, and sensitivity analysis, respectively.

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2.1 Comprehensive performance evaluation unit

The comprehensive performance evaluation unit encompasses four principal steps, as illustrated in Fig. 1(c): Monte Carlo sampling of parameters from the parametric model, ray tracing, performance rendering and recording, and output the worst normalized value of each tolerance criterion. For an imaging spectrometer, the performance evaluation includes the system MTF, keystone/smile distortion, and spectral resolution. The essence of these steps lies in establishing a numerical calculation method for evaluating the performance of imaging spectrometers.

System MTF:

For the system MTF, a numerical calculation method was established based on modulation contrast, as shown in Fig. 2(a). The sensor image $g(x, y)$ is obtained by convolution of the target pattern $f(x, y)$ with the point spread function (PSF) of the optical system $h(x, y)$, which can be expressed as the formula $g(x, y)=h(x, y) \otimes f(x, y)$. The target pattern includes a series of the equidistant rectangular holes. The numerical calculation method for system MTF can be represented as shown in formula (1):

(1)$$M T F=\frac{\pi}{4} \frac{\left(D N_1-D N_{\text{dark }}\right)-\left(\left(D N_2-D N_{\text{dark }}\right)+\left(D N_3-D N_{\text{dark }}\right)\right) / 2}{\left(D N_1-D N_{\text{dark }}\right)+\left(\left(D N_2-D N_{\text{dark }}\right)+\left(D N_3-D N_{\text{dark }}\right)\right) / 2},$$

where $DN_{1}$ is the digital number of the center equidistant rectangular hole; $DN_{2}$ and $DN_{3}$ are the digital numbers of the right and left holes around the center equidistant rectangular hole, respectively; and $DN_{dark}$ is the dark digital number of the detector.

Fig. 2. Numerical calculation methods for comprehensive evaluation criteria of an imaging spectrometer. (a)-(c) Numerical calculation method for the system MTF, keystone/smile distortion, and spectral resolution, respectively.

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Keystone/smile distortion:

For keystone/smile distortion, a specific numerical calculation method has been established. This method is based on analyzing the spot centroid coordinates across different wavelengths and fields. Keystone distortion is defined as the maximum deviation of the image of a slit point across the spectral range from linearity [37]. Similarly, smile distortion refers to the maximum deviation of the image of a slit for a single wavelength from linearity [37], as demonstrated in Fig. 2(b). Consequently, the numerical calculation methods for keystone/smile distortion in imaging spectrometers are represented by formula (2) and (3), respectively.

(2)$${Keystone}_\lambda=\frac{\max \left(x_{\lambda, w}\right)-\min \left(x_{\lambda, w}\right)}{d} \frac{\lambda_{\text{end }}-\lambda_{\text{start }}}{N},$$

(3)$$Smile_w=\max \left(y_{\lambda, w}\right)-\min \left(y_{\lambda, w}\right),$$

where $Keystone_{\lambda }$ is the keystone distortion with different wavelengths, $x_{\lambda,w}$ is the x-coordinate with wavelength $\lambda$ and field $w$, d is the detector size, $\lambda _{end}$ is the ending detection wavelength, $\lambda _{start}$ is the starting detection wavelength, $N$ is the number of spectral channels, $Smile_{w}$ is the smile distortion with different fields, and $y_{\lambda,w}$ is the y-coordinate with wavelength $\lambda$ and field $w$.

Spectral resolution:

In the context of spectral resolution, a numerical calculation method has been developed based on the energy transfer response function. Spectral resolution is typically quantified by the full width at half maximum, representing the bandwidth where the spectral intensity falls to 50% of its maximum. As illustrated in Fig. 2(c), in an ideal imaging spectrometer without aberrations, the system spectral response is a convolution of the spectral responses of the slit width and the detector size, both modeled as rectangular functions, as indicated in formula (4).

(4)$$\operatorname{rect}\left(\lambda-\lambda_i\right)=\left\{\begin{array}{lr} 1 & \left|\lambda-\lambda_i\right|<\Delta \lambda / 2 \\ 0 & \text{ other } \end{array}\right.,$$

When the slit width is equivalent to the detector size and in the absence of system aberrations, the system spectral response transforms into a trigonometric function, as expressed in formula (5).

(5)$$\Lambda\left(\frac{\lambda-\lambda_i}{\Delta \lambda}\right)=\left\{\begin{array}{lr} 1-\left|\frac{\lambda-\lambda_i}{\Delta \lambda}\right| & \left|\frac{\lambda-\lambda_i}{\Delta \lambda}\right| \leq 1 \\ 0 & \text{ other } \end{array}\right.,$$

where $\lambda _{i}$ is the center wavelength, and $\Delta \lambda$ is the spectral sampling of the imaging spectrometer.

To incorporate system aberration effects, the spectral response at the detector is modeled as the convolution of this trigonometric function with the PSF. This modified spectral response is then fitted with a Gaussian function, as shown in formula (6). Finally, the spectral resolution can be determined using formula (7), which calculates it based on the Gaussian fit.

(6)$$\mathrm{y}=\mathrm{A} e^{-\left(\frac{x-b}{c}\right)^2}.$$

(7)$$R =2 \sqrt{\ln 2} c.$$

Comprehensive tolerance criteria:

After conducting N rounds of Monte Carlo sampling analysis, the most adverse values of each tolerance criterion are identified. By utilizing these values, the comprehensive tolerance criteria for the imaging spectrometer are formulated, as expressed in formula (8). The comprehensive tolerance criteria include the four normalized criterion values $\varepsilon _{M T F}$, $\varepsilon _{Keystone_{\lambda }}$, $\varepsilon _{Smile_{w}}$, and $\varepsilon _{\delta \lambda }$. The values of these four normalized criteria should be smaller than 1, indicating that the system performance exceeds the specified requirements.

(8)$$\begin{array}{c} \varepsilon_{M T F}=\frac{\min \left(M T F_d\right)-\min \left(M T F_t\right)}{\min \left(M T F_d\right)-M T F_r}<1, \\ \varepsilon_{\text{Keystone }}=\frac{\max \left(Keystone_{\lambda t}\right)-\max \left(Keystone_{\lambda d}\right)}{Keystone_{\lambda r}-\max \left(Keystone_{\lambda d}\right)}<1, \\ \varepsilon_{\text{smile }}=\frac{\max \left(Smile_{w t}\right)-\max \left(Smile_{w d}\right)}{Smile_{w r}-\max \left(Smile_{w d}\right)}<1, \\ \varepsilon_{\delta \lambda}=\frac{\max \left(\delta \lambda_t\right)-\max \left(\delta \lambda_d\right)}{\delta \lambda_r-\max \left(\delta \lambda_d\right)}<1, \end{array}$$

where $*_{t}$ is the tested results of for the tolerance criterion of interest, $*_{r}$ is the requirement for the tolerance criteria according to the specifications, and $*_{d}$ is the performance in the design state.

By utilizing the previously described numerical calculation methods for the system MTF, keystone/smile distortion, and spectral resolution, the comprehensive performance evaluation was conducted through a joint analysis utilizing Zemax and Matlab. Initially, the PSF and spot ray data across various fields and wavelengths in the imaging spectrometer were obtained from Zemax. Subsequently, Matlab was employed to render the system performance, including the MTF, keystone/smile distortion, and spectral resolution.

2.2 Tolerance parametric model unit

In the tolerance analysis of imaging spectrometers, the tolerance items typically encompass surface irregularity, tilt, decenter, thickness, and slit width. While establishing a connection between tilt, decenter, and thickness tolerances with simulation parameters is relatively straightforward during the development of a parametric model for these items, correlating freeform surface irregularity and slit width with such simulation parameters poses a significant challenge.

Parametric model for freeform surface irregularity:

In the realm of freeform systems, the xy extended polynomial surface type is commonly utilized, as described in formula (9). Our approach uses this xy extended polynomial surface as a representative example, and it is applicable to other freeform surfaces as well. To prevent the introduction of additional tilt and power components, it is imperative not to alter the polynomial coefficients associated with x, y, $x^2$, and $y^2$ or the standard surface parameters of curvature (c) and the conic constant (k). Consequently, the surface sag error $\Delta z_i(x, y)$, resulting from each polynomial coefficient change $\Delta A_i$, is expressed in formula (10). This formula indicates that the change in each polynomial coefficient is a constant value, independent of the standard deviation calculation, as described by in formula (11).

(9)$$\mathrm{z}(x, y)=\frac{c\left(x^2+y^2\right)}{1+\sqrt{1-(1+k)\left(x^2+y^2\right) c^2}}+\sum_{i=1}^N A_i E_i(x, y),$$

where $z$ is the sag of the surface, $c = 1/R$ is the curvature (the reciprocal of the radius), $k$ is the conic coefficient, $N$ is the number of polynomial coefficients in the series, $A_i$ is the coefficient on the $i^{\text {th }}$ extended polynomial term, and $E_i (x,y)$ is the $i^{\text {th }}$ extended polynomial term.

(10)$$\Delta z_i(x, y)=\Delta A_i E_i(x, y),$$

(11)$$\operatorname{std}\left(\Delta z_i(x, y)\right)=\Delta A_i \operatorname{std}\left(E_i(x, y)\right),$$

where $std(*)$ is the calculated standard deviation.

Considering that the final surface sag error is a linear superposition of each polynomial contribution, the model enables the calculation of the boundaries for each polynomial coefficient by inputting a target surface irregularity value into the RMS and considering the number of polynomial coefficients involved in the analysis. The boundaries are then derived based on formula (12):

(12)$$\Delta A_i=\frac{R M S_{\text{target }}}{K \times s t d\left(E_i(x, y)\right)},$$

where $RMS_{target}$ is the target surface irregularity in RMS, and $K$ is the number of polynomial coefficients in the analysis.

Parametric model for slit width:

In most imaging spectrometers, the optical system exhibits a magnification factor of 1 $\times$, resulting in an equivalent slit width equal to the size of the detector. In a specific imaging spectrometer, the slit width $d$ corresponds to the spectral response range $\Delta \lambda$. When considering a tolerance for slit width of $\Delta d$, the spectral response range $\Delta \lambda ^{\prime }$ can be calculated in formular (13):

(13)$$\Delta \lambda^{\prime}=\frac{d+\Delta d}{d} \Delta \lambda.$$

Parametric model for tilt, decenter, and thickness:

In the tolerance analysis, the simulation of tolerance items such as tilt, decenter, and thickness is conducted using coordinate break surface in Zemax software. The parameters of tilt in the x, y, and z directions, decenter in the x and y directions, and thickness of each optical element are generated according to formular (14):

(14)$$\begin{array}{c} \text{ tilt }_{x, i}, \text{ tilt }_{y, i}, \text{ tilt }_{z, i}=\text{ tilt }_{\text{target }, i}, \\ \text{ decenter }_{x, i}, \text{ ddecenter }_{y, i}=\text{ decenter }_{\text{target }, i}, \\ \text{ thickness }_i=\text{ thickness }_{\text{target }, i}, \end{array}$$

where $\text { tilt }_{x, i}$, $\text { tilt }_{y, i}$, and $\text { tilt }_{y, i}$ are the tilt tolerance parameters of the $i^{\text {th }}$ optical element in x, y, and z directions respectively, $\text { tilt }_{\text {target }, i}$ is the target tolerance of tilt in the $i^{\text {th }}$ optical element, $\text { decenter }_{x, i}$ and $\text { decenter }_{y, i}$ are decenter tolerance parameters of the $i^{\text {th }}$ optical element in x and y directions respectively, $\text { decenter }_{\text {target }, i}$ is the target tolerance of decenter in the ith optical element, $\text { thickness }_i$ is the thickness tolerance parameters of the $i^{\text {th }}$ optical element, $\text { thickness }_{\text {target }, i}$ is the target tolerance of thickness in the $i^{\text {th }}$ optical element.

As depicted in Fig. 1(b), the generation of boundary parameters generated by Matlab software involves three main steps. Firstly, utilizing the formular (12), the parameter boundaries for polynomial coefficients can be accurately determined using Matlab when provided with the RMS irregularity of a freeform surface. Then, those parameter boundaries are fed to xy extended polynomial surface in Zemax software with an application programming interface (ZOS-API). Secondly, based on formular (13), the tolerance of slit width is converted into a parameter representing the spectral response range, which is then used in comprehensive performance evaluation unit to evaluate system spectral resolution. Thirdly, employing formula (14), boundary parameters for tilt, decenter, and thickness of each optical element are easily generated and fed to coordinate break surface in Zemax software with ZOS-API.

2.3 Sensitivity analysis unit

Sensitivity analysis plays a pivotal role in tolerance analysis, guiding the adjustment of values for each tolerance item. This unit encompasses four critical steps: setting the boundary parameters for each tolerance item, ray tracing, performance rendering and recording, and calculating the sensitivities for each tolerance criterion. Finally, according to the numerical calculation methods mentioned in section 2.1, the sensitivity relationship between each performance criterion and the individual tolerance items was established.

2.4 Tolerance values adjustment unit

According to the results from, the sensitivity analysis unit and comprehensive performance evaluation unit, the process culminates in the meticulous adjustment of the tolerance values with dividing them by 2, guided by the top three sensitivities of each tolerance criteria. This modification process continues until all four tolerance criteria meet the requirements according to the system specifications.

3. Application example

In the following, we show an example that applied the comprehensive performance domain tolerance analysis method to facilitate the development of a freeform Schwarzschild imaging spectrometer, featuring an F-number of 3.2 and a spectral range of 400–1700 nm.

3.1 Optical design

The object performance specifications of the proposed freeform Schwarzschild imaging spectrometer are an MTF at the Nyquist frequency (25 lp/mm) exceeding 0.3, keystone/smile distortions below 4 $\mathrm{\mu}$m / 0.52 nm, and a spectral resolution under 4 nm, respectively. To meet these specifications, we utilized the xy extended polynomial surface described in formula (9) to correct system aberrations. After several optimizations, the final layout of the freeform Schwarzschild imaging spectrometer was as shown in Fig. 3(a), consisting of three freeform mirrors, a plane grating, and a fold mirror. Figures 3(b)–(e) present the performance evaluation during the design phase. The system MTF exceeds 0.74 at 25 lp/mm; the maximum keystone/smile distortion is 2.3 $\mathrm{\mu}$m / 0.26 nm; and the maximum spectral resolution is 2.81 nm. The aspheric departure of the three freeform mirrors is illustrated in Fig. 3(f). It should be noted that the maximum aspheric departure of these mirrors measures approximately 0.5 mm, a value well within the achievable range using conventional machining techniques. Thus, the theoretical performance of the freeform imaging spectrometer surpasses the specification requirements and allows a certain margin for tolerance analysis.

Fig. 3. Optical design results of the freeform Schwarzschild imaging spectrometer. (a) Layout of the freeform Schwarzschild imaging spectrometer. (b) System MTF. (c) Keystone distortion. (d) Smile distortion. (e) Spectral resolution. (f) Aspheric departure of three freeform mirrors.

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3.2 Tolerance analysis

Based on the comprehensive performance domain tolerance analysis method, we conducted the tolerance analysis of a freeform imaging spectrometer. Firstly, the tolerance values for the surface irregularity, tilt, decenter, thickness and slit width were initially set at 0.2 $\lambda$ in RMS, 0.1 ${^\circ }$, 0.1 mm, 0.1 mm, and 4 $\mathrm{\mu}$m respectively based on the general machining and assembly capabilities. Secondly, these parameter boundaries were generated by the tolerance parametric model unit using initial input tolerance values. Additionally, based on the parameter boundaries of polynomial coefficients, we randomly generated 100 sets of freeform surface errors and calculated the PV value for each set. Ultimately, we selected the maximum PV value as the tolerance for surface irregularity in terms of PV. As an example, the boundaries of polynomial coefficients for M1 in $x^3$, $x^2y$, $xy^2$, and $y^3$ were determined to be $0 \pm 1.14 \times 10^{-3}$, $0.223 \pm 8.76 \times 10^{-4}$, $0 \pm 2.65 \times 10^{-4}$, and $-2.947 \pm 1.51 \times 10^{-4}$ respectively. Thirdly, the parameter boundaries were incorporated into both the comprehensive performance evaluation unit and sensitivity analysis unit. In the comprehensive performance evaluation unit, the worst normalized tolerance criterion values of system MTF, keystone/smile distortion, and spectral resolution were found to be 1.44, 5.34, 4.23, and 3.48 respectively, as shown in Fig. 4(a), which indicating that the system performance failed to meet the specified requirements within these tolerance values. In the sensitivity analysis unit, we conducted comprehensive sensitivity analysis for each tolerance criterion by considering each tolerance item and identified the top three most sensitive tolerance items for each tolerance criterion that failed to meet the specified requirements. These tolerance items were the tilt of M3, tilt of M1 and decenter of M3 for the system MTF; tilt of M3, decenter of M3 and thickness of M1 for the keystone distortion; tilt of M3, tilt of M1 and decenter of M1 for the smile distortion; and tilt of M3, tilt of M1 and thickness of M3 for the spectral resolution. Finally, based on the rules outlined in tolerance values adjustment unit, we halved the values of the corresponding tolerance items mentioned above and then started a new iteration.

Fig. 4. Iterative process of comprehensive performance domain tolerance analysis for a freeform imaging spectrometer. (a)-(d) The normalized criterion values in each iteration.

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After two additional iterations of tolerance analysis, as depicted in Fig. 4(c), the system MTF successfully fulfills the specified requirements. In traditional tolerance analysis, one would typically conclude the analysis and the tolerance items for manufacturing and assembly of the optical elements are listed in Table 1, resulting in imprecise outcomes and ultimately failing to meet the desired performance targets as the keystone/smile distortion still fails to satisfy the given specifications. However, in our proposed method for comprehensive performance domain tolerance analysis, the tolerance analysis would be extended until all tolerance criteria meet the specified requirements. As illustrated in Fig. 4(d), all normalized tolerance criterion values are below 1 after three iterations of analysis, indicating that the system performance surpasses the specified requirements, as summarized in Table 2. The final values of the tolerance items for manufacturing and assembly of the optical elements are listed in Table 3. Based on the tolerance analysis results, all optical components except for M1 and M3 possess loose tolerances that can be easily achieved using current technology. Therefore, particular emphasis should be placed on the assembly process of M1 and M3.

Table 1. Fabrication and assembly tolerances in traditional tolerance analysis

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Table 2. Performance comparison of the design and tolerance values

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4. Experimental verification

To validate the efficacy of our comprehensive performance domain tolerance analysis method, we completed the manufacturing, assembly, and performance evaluation of the freeform Schwarzschild spectrometer.

4.1 Optical elements machining and system assembly

According to the tolerance results, the machining requirements for optical components exhibit a moderate level of stringency and can be effectively met through conventional machining techniques. In this study, we employed computerized numerical control machining technology in conjunction with magnetorheological polishing technology to achieve the precision finishing of freeform mirrors. The metrology results of M1, M2, and M3 were 0.053 $\lambda$, 0.061 $\lambda$, and 0.055 $\lambda$ in RMS respectively, as illustrated in Figs. 5(a)-(c). Additionally, the plane grating was fabricated using advanced electron-beam lithography technology, ensuring a wavefront error of 0.015 $\lambda$ at the −1 order, as depicted in Fig. 5(d). More tested results about the plane grating can be found in Supplement 1. Furthermore, the slit was manufactured using laser cutting technology and the slit width achieves an accuracy of approximately $\pm$ 1 $\mathrm{\mu}$m, as shown in Fig. 5(e). These measurement results confirm that the performance of each optical element aligns precisely with the requirements established in our tolerance analysis.

Fig. 5. Optical element performances. (a)-(c) Surface errors of the M1, M2, M3. (d) Wavefront error of the plane grating. (e) Slit width.

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According to our tolerance analysis, the alignment requirements for M2 and the grating are not excessively stringent, enabling easy alignment using an interferometer. However, M1 and M3 are the most critical components, requiring utmost precision with a maximum tilt of 0.012 ${^\circ }$ and a decenter of 0.025 mm. To address this challenge, we developed an effective assembly method for the freeform spectrometer, utilizing an alignment CGH for the preassembly of M1 and M3. The assembly process of the freeform Schwarzschild imaging spectrometer comprises two main steps.

Step 1: Preassembly of M1 and M3 with an alignment CGH, depicted in Fig. 6(a). The alignment CGH fabricated through laser writing technology, including patterns for the M1 and M3 interferometers and an alignment pattern, as demonstrated in Fig. 6(b). Initially, the CGH is aligned using a standard sphere (0.75 F/#) of the interferometer, facilitated by the alignment pattern on the CGH. Then, M3 is installed onto the spectrometer frame, which is positioned on a five-dimensional adjustment platform. This platform enables easy positioning of M3, which is then locked in place. M1 is subsequently adjusted using spacers between its frame and the spectrometer frame. The final interferograms for M1 and M3, nearly showing a single fringe pattern as illustrated in Fig. 6(d), indicate precise alignment to their designed positions.

Fig. 6. Assembly of the imaging spectrometer. (a) Layout of M1 and M3 alignments. (b) Patterns of the CGH. (c) Preassembly setup in the laboratory. (d) Alignment results of M1 and M3. (e) Aassembled of imaging spectrometer. (f) System wavefront maps of three points (normalized 0 and $\pm$1 fields), normalized 0 field is located at the center of the silt, normalized $\pm$1 fields are positioned at the edges of the silt, $\pm$6.4 mm away from the center.

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Step 2: Assembly of the imaging spectrometer according to the system wavefront error. The assembly of the imaging spectrometer is conducted with the system wavefront error measured by an interferometer. The resulting interferogram guides the adjustments of the second freeform mirror and the plane grating, ensuring that the wavefront errors at the center and margin slit positions aligns with the design specifications. The final system wavefront maps for three points, as shown in Fig. 6(f), confirm that the RMS wavefront errors across all fields are better than 0.223 $\lambda$, validating the successful alignment of the spectrometer, as shown in Fig. 6(e).

4.2 Performance evaluation

The freeform Schwarzschild imaging spectrometer underwent extensive performance evaluation in a laboratory setting. A schematic diagram of the optical path for performance evaluation is presented in Fig. 7(a), comprising five key components: halogen tungsten lamp, monochromator, focusing lens, testing slit, and freeform imaging spectrometer. The measurement results are presented in Table 3 and Figs. 7(b)-(e). The system MTF across the entire imaging plane exceeds 0.43 and the maximum keystone/smile distortion is 3.6 $\mathrm{\mu}$m / 0.48 nm. Moreover, the spectral resolution throughout the imaging plane is less than 3.7 nm. The performance evaluation results exceed the expectations set by the tolerance analysis, fulfilling the requirements according to the specifications, as shown in Table 4. This outcome validates the effectiveness of the comprehensive performance domain tolerance analysis method.

Fig. 7. Performance evaluation of the freeform Schwarzschild imaging spectrometer. (a) Schematic of performance evaluation in the laboratory. (b)-(e) Measured results of system MTF, keystone/smile distortion, and spectral resolutions, respectively.

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Table 3. Fabrication and assembly tolerances in our proposed tolerance analysis

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Table 4. Comparison of the measurement and tolerance analysis results

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Finally, we present an application of camouflage recognition with our imaging spectrometer. As shown in Fig. 8(a), the true and false leaves are indistinguishable in the RGB image. However, the spectral curve of the true leaf exhibits an absorption peak around the 1400 nm band, aligning with the characteristic water absorption peak, as depicted in Fig. 8(b). This distinction is absent in the spectral curve of the false leaf. Leveraging this spectral difference, we selected three specific bands—1450 nm, 1200 nm, and 735 nm—to create a fused color map, as illustrated in Fig. 8(c). This color map distinctly differentiates between the true and false leaves, rendering them easily distinguishable.

Fig. 8. Application for the identification of true and false leaves. (a) RGB picture of true and false leaves. (b) 3D data cube of true and false leaves and their spectral curves. (c) Color map fused by three bands (1450 nm, 1200 nm, and 735 nm).

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5. Conclusion

In conclusion, we proposed a comprehensive performance domain tolerance analysis method specifically tailored for freeform imaging spectrometers, considering vital aspects such as the MTF, keystone/smile distortion, and spectral resolution. Based on this method, we conducted a detailed tolerance analysis of a freeform Schwarzschild imaging spectrometer with an F-number of 3.2 and a spectral range of 400–1700 nm. The analysis revealed stringent tolerance items and employed an alignment CGH for the preassembly of M1 and M3, resulting in successful system assembly. The measurement performance results closely aligned with our tolerance analysis predictions, affirming the efficacy of the comprehensive performance domain tolerance analysis method. Moreover, our proposed tolerance analysis method holds significant potential for application in other optical systems, necessitating the consideration of additional performance aspects beyond the system MTF.

Funding

Fundamental Research Funds for the Central Universities; Youth Innovation Promotion Association of the Chinese Academy of Sciences (Y1K4H0FKG1); National Natural Science Foundation of China (62105243, 62192774, 62305250).

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.

Supplemental document

See Supplement 1 for supporting content.

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Optical element	Surface irregularity	Tilt	Decenter	Thickness	Width
Slit	$/$	0.1 $^{\circ}$	$/$	0.1 mm	2 $μ$ m
Fold mirror	RMS < 0.2 $λ$ $/$ PV < 1.48 $λ$	0.1 $^{\circ}$	$/$	0.1 mm	$/$
M1	RMS < 0.2 $λ$ $/$ PV < 1.52 $λ$	0.025 $^{\circ}$	0.05 mm	0.05 mm	$/$
M2	RMS < 0.2 $λ$ $/$ PV < 1.60 $λ$	0.1 $^{\circ}$	0.05mm	0.05 mm	$/$
Plane grating	RMS < 0.2 $λ$ $/$ PV < 1.52 $λ$	0.1 $^{\circ}$	$/$	0.1 mm	$/$
M3	RMS < 0.1 $λ$ $/$ PV < 0.68 $λ$	0.025 $^{\circ}$	0.025mm	0.025 mm	$/$

Performance	Specifications	Design	Tolerance
MTF	$\geq$ 0.300	> 0.741	> 0.324
Keystone	$\leq$ 4.00 $μ$ m	< 2.30 $μ$ m	< 3.90 $μ$ m
Smile	$\leq$ 0.52 nm	< 0.26 nm	< 0.51 nm
Spectral resolution	$\leq$ 4.00 nm	< 2.81 nm	< 3.95 nm

Optical element	Surface irregularity	Tilt	Decenter	Thickness	Width
Slit	$/$	0.1 $^{\circ}$	$/$	0.1 mm	2 $μ$ m
Fold mirror	RMS < 0.2 $λ$ $/$ PV < 1.48 $λ$	0.1 $^{\circ}$	$/$	0.1 mm	$/$
M1	RMS < 0.1 $λ$ $/$ PV < 0.71 $λ$	0.012 $^{\circ}$	0.025 mm	0.025 mm	$/$
M2	RMS < 0.1 $λ$ $/$ PV < 0.86 $λ$	0.05 $^{\circ}$	0.05mm	0.05 mm	$/$
Plane grating	RMS < 0.2 $λ$ $/$ PV < 1.52 $λ$	0.05 $^{\circ}$	$/$	0.1 mm	$/$
M3	RMS < 0.1 $λ$ $/$ PV < 0.68 $λ$	0.012 $^{\circ}$	0.025mm	0.025 mm	$/$

Performance characteristic	Specifications	Tolerance	Measured
MTF	$\geq$ 0.300	> 0.332	> 0.430
Keystone	$\leq$ 4.00 $μ$ m	< 3.90 $μ$ m	< 3.60 $μ$ m
Smile	$\leq$ 0.52 nm	< 0.51 nm	< 0.48 nm
Spectral resolution	$\leq$ 4.00 nm	< 3.90 nm	< 3.70 nm

Optical element	Surface irregularity	Tilt	Decenter	Thickness	Width
Slit	$/$	0.1 $^{\circ}$	$/$	0.1 mm	2 $μ$ m
Fold mirror	RMS < 0.2 $λ$ $/$ PV < 1.48 $λ$	0.1 $^{\circ}$	$/$	0.1 mm	$/$
M1	RMS < 0.2 $λ$ $/$ PV < 1.52 $λ$	0.025 $^{\circ}$	0.05 mm	0.05 mm	$/$
M2	RMS < 0.2 $λ$ $/$ PV < 1.60 $λ$	0.1 $^{\circ}$	0.05mm	0.05 mm	$/$
Plane grating	RMS < 0.2 $λ$ $/$ PV < 1.52 $λ$	0.1 $^{\circ}$	$/$	0.1 mm	$/$
M3	RMS < 0.1 $λ$ $/$ PV < 0.68 $λ$	0.025 $^{\circ}$	0.025mm	0.025 mm	$/$

Comprehensive performance domain tolerance analysis methodology for freeform imaging spectrometers

Abstract

1. Introduction

2. Comprehensive performance domain tolerance analysis method

2.1 Comprehensive performance evaluation unit

2.2 Tolerance parametric model unit

2.3 Sensitivity analysis unit

2.4 Tolerance values adjustment unit

3. Application example

3.1 Optical design

3.2 Tolerance analysis

4. Experimental verification

4.1 Optical elements machining and system assembly

4.2 Performance evaluation

5. Conclusion

Funding

Disclosures

Data availability

Supplemental document

References

Supplementary Material (1)

Data availability

Cited By

Figures (8)

Tables (4)

Equations (14)

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