1. Introduction

Optical freeform surface refers to a type of optical surface that does not have an axis of rotational symmetry, either within or outside the range of the optical part [1]. Freeform surfaces can improve the overall performance of imaging systems. When the system specifications are similar, use of a freeform surface can reduce the system’s volume, enhance its compactness, and improve the imaging quality when compared with a system based on conventional optics [2]. Freeform surfaces are also used to realize systems with novel functions, e.g., imaging systems with as-designed field-dependent characteristics [3][4], imaging systems with multiple integrated performance settings [5], and freeform lenses that enable image shifting and rotation [6]. Freeform surfaces have thus been receiving increasing attention in optical design [7][8][9][10].

Studies have shown that the application of freeform surfaces to imaging spectrometer design can improve the overall system performance comprehensively. Reimers et al. studied the F/3.8 imaging spectrometer [11]. As the number of degrees-of-freedom (DOFs) of the surface shape within the system increased, from spherical surface to off-axis aspheric surface, to anamorphic aspheric, and finally to freeform surface, the system’s imaging quality gradually improved. The average root-mean-squared value of the wavefront error (AVG WFE RMS) over the full field within their system was 58% lower than that of the spherical system. Further study showed that use of freeform surfaces can also improve the compactness and the performance of imaging spectrometers [12]. Freeform surfaces can reduce the AVG WFE RMS by up to 65% under the condition that the slit length and the spectral bandwidth are the same; or enable a 5× compactness improvement when the performance specifications are similar; or enable a 3× wider spectral bandwidth or 2× slit length enhancement when a similar system structure is used, while also simultaneously improving the imaging quality and correcting distortion. In an imaging spectrometer that contains a freeform concave grating as the only system component, it has been shown that a variable-line-spacing (VLS) freeform grating can improve the system’s spectral dispersion and spectral resolving power significantly; freeform surfaces with figures that have higher DOFs can further improve the effect of the VLS grating in enhancing the system’s performance [13]. Therefore, freeform surfaces are promising for application to systems containing diffraction gratings.

All-freeform surface design can be realized by starting with an all-spherical imaging spectrometer system and optimizing this system using commercial software, but this approach will require significant human interaction, and it is also inefficient when exploring new imaging spectrometer system structures. Point-by-point direct design methods have been proposed for freeform imaging spectrometer design in which the freeform surface figure is calculated based on the system’s object-image relationships in both the spectral and spatial dimensions [14][15]. In [15], the proposed design method for freeform systems that contain diffraction gratings can obtain initial solutions with high imaging quality and low distortion when starting from a planar system with a geometry that is given by folding lines, or from a spherical system in which the optical power for each mirror has been assigned in advance.

These direct design methods reduce the participation and effort of human designers in the design process significantly, and the output results can be regarded as good starting points for further optimization. However, this type of design method has disadvantages from two perspectives. First, an initial unobscured system must be provided in advance. Second, because the imaging quality and the object-image relationship in both the spectral and spatial dimensions serve as the primary metrics in the calculations, the optical power distribution among all the mirrors in the system can only be determined when the system’s geometry and the order in which to calculate the mirror shapes are given. Consequently, the quality and number of the systems that can be obtained using these methods are limited.

For the hundreds of years since the birth of optical design, human participation in the design process has been essential, and it is the common aim of optical designers to realize fully automated optical design that provides high imaging quality without the need for human involvement. Design methods have thus been developed with the intention of removing the uncertainty from the optical design process or realizing automatic design of freeform optics [16][17][18]. One design method has been proposed that requires the desired system’s specifications as the only input and has allowed diffraction-limited freeform imaging systems with various optical power distributions and various structures to be achieved directly [18].

In this work, a design method is introduced that is able to generate good initial design solutions for three-mirror freeform imaging spectrometers automatically, with the system’s specifications being the only inputs required. The systems obtained using this method have various optical power distributions and various structures, and can serve as good starting points for further design optimization. The method is realized by combining the point-by-point method for direct design of freeform systems containing diffraction gratings from [15] with the automatic design method for freeform imaging systems that has been used to generate various high-quality systems [18]. Systems containing three mirrors and a single diffraction grating placed at the secondary mirror are used as design examples.

Additionally, based on the various solutions obtained by the approach above, which have different structures and different optical power distributions, the t-distributed symmetric neighbor embedding (t-SNE) dimensionality reduction algorithm is used to reduce the dimensionality of the solution space for the optical system and then visualize it. This visualization step shows that systems with different imaging qualities gather around different vicinities within this solution space. In recent years, it has been reported that the technology of artificial intelligence, such as neural network based deep learning and reinforcement learning, has been implemented in non-diffractive imaging system's design, in the aspect of generating initial designs according to the system's specifications [19][9], and discovering optimal policies for optimizing optical systems [20]. In this work, a large number of imaging spectrometer systems that contains diffraction grating are obtained first, and then, in order to utilize the design results and predict the quality of systems with unknown structure and optical power distribution, a neural network is used to fit the relationships among the systems’ structures, optical power distributions, and imaging qualities. Cross-verification based on a training set and a test set shows that the two-layer feedforward neural network used here is sufficient to predict the imaging quality of a system that has an unknown structure and unknown optical power distribution.

2. Methods

When an imaging spectrometer is to be designed, the required system specifications must be provided, including the slit length, the numerical aperture, the magnification, and specifications with regard to the system’s dispersion capabilities, including the working spectral band, the spectral resolution, and the pixel pitch of the detector that is to be used. In this case, the magnification refers to the ratio of the hyperspectral image width to the length of the slit in the object space. The design method consists of the following five steps, which should be considered as an integrity and the calculation parameters generally does not need user determination.

Step 1.

In accordance with the given magnification for the system, construct a series of coaxial all-spherical mirror systems, denoted by P₁, P₂, …, P_m, …, P_M, while disregarding the obscurations. These systems have various optical power distributions. The diversity of optical power distributions is created in this step and remained in the output systems of this method. For convenience in the discussion here, these systems are collectively denoted by {P}.

Following propagation of light rays through this system, the radii of curvature of the spherical surfaces are indicated by r₁, r₂, r₃, …; the corresponding object distances and image distances for each spherical mirror are indicated by l₁, l₂, l₃,… and l'₁, l'₂, l'₃,…, respectively. According to the imaging equation for a spherical mirror, the magnification of the entire coaxial system is:

For systems that contain three spherical mirrors, k = 3, n₁ = 1, and n₃’ = −1.

The distances from the slit to the primary mirror, from the primary mirror to the secondary mirror, from the secondary mirror to the tertiary mirror, and from the tertiary mirror to the image plane, are indicated sequentially by d₀, d₁, d₂, and d₃, respectively. Because d₀ and d₁ are known and l₁ = −d₀, based on the paraxial imaging equation for a spherical mirror, then l'₁ = l₁r₁/(2l₁−r₁). Similarly, because l₂ = l'₁a−d₁, then l'₂ = l₂r₂/(2l₂−r₂). Then,

For convenience when describing systems with different optical power distributions, the system is indicated in the form of the vector P = [d₀, r₁, d₁, r₂, d₂, r₃, d₃], in which the signs and the values of the vector elements are constrained as follows. Following the sign conventions in [18], we set d₀ > 0, d₁ < 0, d₂ > 0, and d₃ < 0. Because the imaging spectrometer works at a finite object distance, the overall optical power of the system is relatively large when compared with that of a system working at an infinite object distance. To avoid a requirement for a large secondary mirror and an oversized system volume, the primary mirror must be concave. To balance the optical power among the mirrors and acquire a flat image, the secondary and tertiary mirrors should be convex and concave, respectively. Therefore, r₁ < 0, r₂< 0, and r₃ < 0. Finally, the absolute values of r₁, r₂, r₃, d₀, and d₁ must be constrained to ensure that the mirror distances and the system volume do not exceed the constraints.

In summary, when solving for coaxial spherical systems with various optical power distributions in Step 1, the ranges and intervals of the values of r₁, r₂, r₃, d₀, and d₁ are set up in advance. Depending on the system magnification, the values of d₂ and d₃ can be solved with respect to every combination of r₁, r₂, r₃, d₀, and d₁. A series of coaxial spherical systems {P} that satisfy the system magnification requirement can then be obtained.

Step 2.

In order to automatically generate a large number of diversified unobscured geometries, the following parameterized presentation of system geometry is introduced. For every coaxial system in {P}, adjust the position and the tilt angle of each mirror, and then a series of off-axis systems can be obtained, where the mirrors may or may not be unobscured. For example, for system P_m, the series of off-axis systems are denoted by C_m,1, C_m,2,…, C_m,s,…, C_m,Sm, and can be recorded in a simplified form as {C}_m. For an arbitrary off-axis system that belongs to {C}_m, the system’s structure can be represented by the vector C = [θ₁, θ₂, θ₃], where θ₁, θ₂, and θ₃ are the chief ray’s bending angles at each of the mirrors sequentially. If the chief ray’s light is reflected in the direction of incidence, then θ = −180°; if the light ray is not deflected and continues to travel along the direction of incidence, then θ = 0°.

Because the systems in set {C}_m do not currently contain diffraction gratings, a diffraction grating should then be placed at the secondary mirror before inspecting a system in {C}_m to determine whether it is unobscured or not, because the direction in which the light bundles travel will be shifted after diffraction. In this work, a diffraction grating that has straight grooves with constant line-spacing is used. Using the system’s spectral band, its spectral resolution, and the detector’s pixel pitch, the line-spacing of the diffraction grating can be determined using the method given in [15], and real ray tracing is then performed to establish whether or not obscuration occurs in the system. In this work, first-order diffracted light is used for spectral imaging, including the +1 order and the −1 order. Because the diffracted light of these two orders will travel along different sides of the reflected light, four possible conditions may occur when inspecting a system in {C}_m to determine whether it has obscuration: (1) light rays of both orders are unobscured; (2) the +1 order light is obscured, and the −1 light is unobscured; (3) the +1 light is unobscured, and the −1 light is obscured; or (4) light rays of both orders are obscured. Therefore, one system in {C}_m may generate a maximum of two unobscured systems, working in the +1 and −1 diffraction orders.

When the procedures above are followed with respect to a coaxial spherical system P_m that has a specific optical power distribution, a series of off-axis spherical mirror systems can be obtained that have various structures and operate in different diffraction orders (i.e., +1 and −1). The systems corresponding to P_m are denoted by $\overline C$_m,1, $\overline C$_m,2,…, $\overline C$_m,r,…, $\overline C$_m,Rm, and the set of these systems is denoted by {$\overline C$}_m, with a total number of elements R_m. Based on {$\overline C$}_m, a series of compact systems can then be found, in which the distances between the mirrors and the light beam are the smallest when the unobscured condition is maintained. The selected compact systems are denoted by $\widetilde C$_m,1, $\widetilde C$_m,2,…, $\widetilde C$_m,t,…, $\widetilde C$_m,Tm, and the set of these systems is denoted by {$\widetilde C$}_m, with a total number of elements T_m.

Step 3.

Based on the systems included in {$\widetilde C$}_m, construct freeform systems and correct the optical power of the entire system. The optical power of the system was broken after Step 2, in which the mirrors in the system were replaced or tilted and a diffraction grating was added to the system. In this step, according to the imaging requirements in both the spatial and spectral dimensions, the optical power of the entire system is corrected, and meanwhile a new unique optical power distribution is generated corresponding to the original one. From the system’s magnification, its spectral band and the spectral resolution, the object-image relationship can then be found. Light rays that are orientated from different slit positions, located at different pupil positions, and have different wavelengths must satisfy this relationship. Using this object-image relationship, and starting from every system in {$\widetilde C$}_m, the point-by-point direct design method for freeform systems containing a diffraction grating [15] can be implemented to calculate the freeform surface figure for every mirror in the system. In this process, the surface shape is calculated in a point-by-point manner and the points are fitted into XY polynomials added to a conic base, which can realize high imaging quality [18]. Through the calculations performed in this step, a series of freeform imaging spectrometers can be obtained with a variety of structures and these spectrometers operate in various diffraction orders (+1 or −1). These systems are denoted by $\tilde{\boldsymbol{F}}_{m, 1}$, $\tilde{\boldsymbol{F}}_{m, 2}$,…, $\tilde{\boldsymbol{F}}_{m, t}$,…, $\tilde{\boldsymbol{F}}_{m, Tm}$, and the set of these systems is denoted by $\{\tilde{\boldsymbol{F}}\}_{m}$.

Step 4.

For every system in {F˜}_m, iteratively calculate the shapes of the freeform surfaces and thus improve the system’s imaging quality as far as possible. During this iterative process, the RMS value of the distances between the target image points, which are determined using the object-image relationship, and the actual interaction points, which are acquired via real ray tracing in the system, is calculated and used as a metric to evaluate the system quality. The RMS value above is denoted by σ, which can be considered to be a combination evaluation metric for both the transverse aberration and the distortion. The evaluation metric σ decreases as the iteration continues and will ultimately converge. In this work, an improvement rate τ is defined as τ = |σ'−σ|/σ, where σ’ and σ represent the metrics for the results of the current and previous rounds of iterations, respectively. When τ is lower than a given threshold value τ_itr, the iteration is then terminated and the system obtained is output. Using the procedures described above, a series of good solutions for freeform imaging spectrometers can be obtained.

Step 5.

Through the automatic steps described above, a large number of freeform imaging spectrometer systems with various optical power distributions and various structures can be acquired automatically. Next, the imaging quality metric σ is evaluated for every output system. In this work, σ is regarded as the primary metric for evaluation of the quality of an initial solution, and systems with smaller σ values are considered to be better initial solutions for further optimization.

3. Results

In this work, two design examples of freeform imaging spectrometers are presented that have two different magnifications in the spatial dimension. Note that one is able to assign other items of system specifications in another design case, such as numerical aperture, spectral band, and spectral dispersion.

Design example 1.

The first freeform imaging spectrometer design has a magnification of 1, which means that the image on the detector has a length that is equal to that of the slit. The system has a slit size of 10 mm and a numerical aperture of 0.167. The operating spectral band of this system is 400–1000 nm and the spectral dispersion is 1 nm. Both the smile and keystone distortions are less than 20% of the pixel pitch of the detector, which is assumed to be 8 μm.

The values of the elements in every vector P, i.e., d₀, r₁, d₁, r₂, d₂, r₃, and d₃, along with the values of the elements in vector C, i.e., θ₁, θ₂, and θ₃, are set up in advance and listed in Table 1, which does not need user's determination. The intervals of curvature radii, mirror distances, and bending angles can be rougher or finer, depending on the amount of computation and level of discretization of the solution space. Note that the subscripts of both P and C are suppressed for generalization of the discussion here. The image quality improvement rate threshold is set at τ_itr = 0.5%.

Table 1. Ranges and intervals of the values of the elements in vectors P and C in design example 1

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The calculation of the designed example was completed on a computer workstation that has 96 cores and 192 threads operating at 2.29 GHz. After a calculation time of 76.4 h, a total of 11704 freeform imaging spectrometer systems were calculated with various structures, corresponding to 1226 different optical power distributions. These systems were then sorted in order of σ from small to large. The top ten systems are shown in Fig. 1, and the WFE RMS and distortion values of these systems are listed in Table 2.

 

Fig. 1. Top ten systems with the smallest σ values in design example 1.

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Table 2. WFE RMS and distortion values of top ten systems with smallest σ values in design example 1

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The systems presented in Fig. 1 have AVG WFE RMS values that vary between 0.47λ and 0.85λ (where λ is the wavelength), which can be regarded as good initial solutions for further design optimization. Among these output results, System P322-C9 showed the best imaging quality and very low distortion, with a smile distortion that already satisfied the design requirements. All of the systems obtained by the automatic design method can be used as the starting point for further optimization, as long as one considered it appropriate in the aspect of structure, volume, imaging quality, distortion, etc. We then select system P322-C9 and optimize it to meet the final requirements of diffraction limited operation and low distortion. The system obtained after optimization is shown in Fig. 2(a) and its WFE RMS over the full field and the full spectral band is shown in Fig. 2(b). The optical power distribution and the system structure of this optimum system were not greatly changed when compared with the initial solution of System P322-C9. To provide the optimum system, the maximum WFE RMS is 0.068λ and the maximum smile and keystone distortions are 1.6 μm and 1.5 μm, respectively.

 

Fig. 2. (a) System after optimization starting from System P322-C9 and (b) this system’s WFE RMS map over the full field and the full working spectral band.

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Design example 2.

The second freeform imaging spectrometer design has a magnification of 2, which means that the image on the detector has a length that is twice that of the slit. The rest of the system specifications are the same as those used in design example 1.

The calculation of design example 2 was also conducted on the computer workstation. After a calculation time of 25.6 h (96 cores, 192 threads, 2.29 GHz), a total of 3947 freeform imaging spectrometer systems with various structures were calculated, corresponding to 438 different optical power distributions. These systems were then sorted in order of σ from small to large. The top ten systems are shown in Fig. 3, and the WFE RMS and distortion values of these systems are listed in Table 3.

 

Fig. 3. Top ten systems with the smallest σ values in design example 2. Note that in System 0005-P75-C5 the slit is close to mirrors and gratings, but the system is unobscured.

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Table 3. WFE RMS and distortion values of top ten systems with the smallest σ values in design example 2

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The systems presented in Fig. 3 have AVG WFE RMS values that vary between 0.33λ and 0.86λ and can be considered to be good initial solutions for further design optimization. Among the output results, System P100-C8 showed the best imaging quality and very low distortion, with a smile distortion that already satisfied the design requirements. System P100-C8 was then optimized to meet the final requirements of diffraction limited operation and low distortion. The system obtained after optimization is shown in Fig. 4(a) and its WFE RMS over the full field and the full spectral band is shown in Fig. 4(b). The optical power distribution and the system structure of the optimum system are again not greatly changed when compared with the initial solution in System P100-C8. To provide the optimum system, the maximum WFE RMS is 0.072λ, and the maximum smile and keystone distortions are 1.6 μm and 1.6 μm, respectively.

 

Fig. 4. (a) System after optimization starting from System P100-C8 and (b) this system’s WFE RMS map over the full field and the operating spectral band.

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4. Discussion

Using the design method introduced in Section 2, large numbers of design results can be obtained as long as system specifications for which the function has never been realized are provided, and these results can be used to explore novel system structures for imaging spectrometer systems. Based on this result-diversified design method, systems with various structures and various optical power distributions can be obtained, and an overview of the distribution of the systems in the solution space that have different imaging qualities can thus be obtained. To visualize and observe the distribution of the systems in the solution space, a dimensionality reduction visualization method for the optical system solution space is proposed, as follows.

Systems obtained via the design method introduced above can be regarded as samples within the solution space for the imaging spectrometer system under the given design specifications. The set of these samples is denoted by {S_m,t}, and an arbitrary sample is represented by S_m,t = [P_m, $\widetilde C$_m,t] = [d₀, r₁, d₁, r₂, d₂, r₃, d₃, θ₁, θ₂, θ₃], which contains the information comprising the mirror curvature radii, the mirror distances, and the chief ray’s bending angle at the mirrors. Because the values of d₂ and d₃ can be obtained from d₀, r₁, d₁, r₂, and r₃ via Eq. (2) and Eq. (3), they are redundant values and should thus be removed. Therefore, each sample is indicated by an eight-dimensional row vector:

Using the design method that was introduced in Section 2, the desired freeform imaging spectrometer has a slit length of 10 mm, a numerical aperture of 0.05, spectral resolution of 1 nm, a spectral band of 400–1000 nm, and a pixel pitch of 8 μm. Through 61.3 h of calculation time (96 cores, 192 threads, 2.29GHz), a total of 17316 freeform imaging spectrometer systems were calculated with various structures, corresponding to 2207 different optical power distributions. These systems were then sorted in order of σ from small to large and the set of the top 1000 systems is denoted by {S}_1k. The t-SNE algorithm was then implemented and the dimensionality of the samples was reduced to three, using the sample distances given by the Euclidean distance. The distribution of these systems in the three-dimensional space after dimensionality reduction is shown in Fig. 5. Each point in Fig. 5 represents a system in {S}_1k and its color indicates the system’s imaging quality ranking among the 1000 systems. Red indicates a high ranking and good imaging quality, while blue means a low ranking and relatively poor imaging quality.

 

Fig. 5. Distribution of samples in the solution space after dimensionality reduction using the t-SNE algorithm. The colors indicate the rankings of the systems given by their σ values, as shown by the color bar. Red indicates that the σ value is small and the imaging quality is higher; blue indicates that the σ value is large and the imaging quality is lower.

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Figure 5 shows that systems with different imaging qualities are clustered and distributed over several different positions in the solution space. At some positions, there are more solutions with better imaging quality (indicated by the red spots), which means that it is easier to find good solutions at these positions; alternatively, there are more solutions with relatively poor imaging quality (indicated by the blue spots) at several specific positions, which means that it is more difficult to find good solutions in the vicinity of these positions. In some locations, solutions with good imaging quality and poor imaging quality are mixed together, which means that if a system in this area is optimized, the optimization process may be unstable and solutions of different qualities may appear successively.

The distribution of samples in the solution space can be observed after dimensionality reduction using the t-SNE method, and systems with different imaging qualities are clustered, but we are unable to tell the quality of an unknown system using Fig. 5. Finally, by taking the eight-dimensional row vector of each sample as the input and the σ value of the corresponding system as the output evaluation parameter, a neural network was trained. This neural network uses vector that describe system's optical power distribution and structure as the input, and outputs the σ value. The training set of this neural network should include systems that have various imaging qualities, such that the trained model can predict the quality of unknown systems. Using this neural network, the large number output design results of the result-diversified design method can find the application on predicting the imaging quality of systems with different optical power distribution forms and different structural forms directly, without the need to actually solve fully for the imaging spectrometer system. Along with the introduced result-diversified automatic design method, unobscured high-quality imaging spectrometers that have novel structures could be obtained.

For the system designed in this section, all the systems were sorted in order of σ from small to large. The set comprising the top 10,000 systems is denoted by {S}_10k and the set of corresponding σ values is denoted by {σ}_10k (units: mm). The relationship between {S}_10k and {σ}_10k is fitted using the two-layer feedforward neural network model, as illustrated in Fig. 6, using the regression training tool in MATLAB. Although systems in {S}_10k are not optimized, it is sufficient to use them to predict the quality of the corresponding optimized system, as is shown in the above design examples that the optical power distribution and the system structure of this optimum system were not greatly changed when compared with the initial solution. {S}_10k is divided randomly into training set, validation set and testing set, the number of which are 60%, 20% and 20% of the total sample number. Since the imaging qualities of samples in the testing set is already accessible, the validation of the trained neural network is instantaneous. The test cases using this trained network should be three-mirror imaging spectrometer of the same optical specifications, with the diffraction grating placed at the secondary mirror, and the values of curvature radii, mirror distances, and bending angles within the ranges of corresponding items listed in Table 1. In the fitting process, the training algorithm is the Levenberg-Marquardt algorithm, and a one-dimensional search process is used to search for the best setting of hidden layer nodes.

 

Fig. 6. Graphical diagram of the two-layer feedforward neural network.

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In the final training model, the optimal number of hidden layer nodes is set as 50, as determined based on the premise of avoiding overfitting. Over the whole data set, the root mean square error (RMSE) is 2.5525, the R-squared value is 0.9244, the mean absolute percentage error (MAPE) is 0.1194, and the weighted mean absolute percentage error (wMAPE) is 0.0866. The accuracy of the model on the training set, the validation set, and the test set are shown in Table 4. The scale of data set will influence the performance of the model, which is related to system's type, specifications, number of elements, volumes, etc., but result shows 10K samples is feasible to realize the function of using neural network to estimating system quality. The results show that the accuracy is consistently high on the testing set, thus indicating that the generalization performance is good, which enables the model to be applied to predict the imaging quality of a system with unknown optical power distributions and structures. With the new optical power distribution and structure as the input, the quality of these systems that have not been designed can be predicted without calculation in detail.

Table 4. Accuracy of the trained neural network on the training set, the validation set, and the testing set

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5. Conclusion

In this work, a result-diversified automatic design method for freeform imaging spectrometers is introduced. When using this method, it is only necessary to provide the performance specifications of the desired system, including the slit length, numerical aperture, magnification, spectral range, spectral resolution, and detector pixel pitch, and a series of freeform imaging spectrometer systems with various optical power distributions and various structures can then be obtained. Imaging spectrometer systems with appropriate optical power distributions and novel structures can be obtained automatically. Large number of output results would be beneficial for one to observe and understand the distribution of high-quality solutions in the solution space. Moreover, one can compare the various initial designs and select the most preferred one as the starting point for optimization.

To demonstrate use of this method, two design examples that have magnifications of 1 and 2 are presented, while all other specifications remain identical. The program is deployed on a high-performance workstation and a large number of initial solutions are obtained automatically without human interaction. In each design example, a system is selected and is then optimized to verify the quality of the initial solution; finally, the system is diffraction-limited over the full field and the full operating band, with distortion that is lower than the 20% of the pixel pitch. The final optimization result has smaller differences in its power distribution, volume, and structure, thus indicating that the system calculated using the proposed method can be used as a good starting point for design optimization. The method proposed in this paper provides an effective, automatic, and multi-solution approach to solve the problem where the optical power distribution and the appropriate structure are difficult to determine when designing an optical system.

Large number of systems are obtained via the design method and the potential utilization of the output results are discussed. In order to allow the distribution of the solutions of different qualities within the solution space to be observed, the t-SNE algorithm can be used to reduce the solution space dimensionality and then map the systems in the high-dimensional space down to the three-dimensional space and visualize them. By utilizing the large number of design results, a two-layer feedforward neural network model is used to for regression, with 50 hidden layer nodes. In the final training model, on the whole data set, the wMAPE is 0.0866, thus showing that the model can predict the quality of the initial solution with an unknown system structure and an unknown optical power distribution. In future work, better fitting models and advanced fitting method will be studied to obtain more accurate predictions of the imaging quality of the designed optical system, or generate novel systems with higher imaging qualities beyond the results obtained by the result-diversified automatic design method.