1. Introduction

Optical systems containing diffraction gratings have broad applications in the field of biomedical study, earth remote sensing, and space exploration. Freeform optics is regarded as a revolution in optical design [1], which involves optical designs that contain at least one freeform optical surface with no translational or rotational symmetry [2]. Benefiting from the high degrees-of-freedom of freeform surfaces, novel and high-performance optical systems have now been achieved, which include systems having an ultra-wide field-of-view (FOV) [3] or a low F-number [4] or a combination of the two [5], as well as freeform optical designs for spectrometers [6]. Freeform surfaces can increase the overall performance of imaging spectrometers in the aspects of compactness, spectral and spatial bandwidth [7]. Identifying good starting points is essential to design freeform systems. However, the conventional design approach, such as first-order optics and primary aberration theory, requires considerable human efforts to produce novel high-performance freeform systems, because of the complexity of the surface shapes and the lack of off-axis patents. Sometimes the initial solution will deviate largely from the result and it will be time consuming to recalculate a new geometry.

Point-by-point design methods for optical systems calculate discrete points on optical surfaces based on a given object-image relationship [8–12]. These methods are developed to solve the “field-pupil-wavelength” (FPW) problem in optical design. To explain this problem, a point P on the optical surface of an ideal optical system is considered, as illustrated in Fig. 1. Light rays incident on this point have multiple pupil coordinates and are from multiple fields with multiple wavelengths. The aim is to solve for every point in the system that deflects these light rays towards their corresponding target points in the image space that meet the mapping relationship between the object and the image. In this work, such points are called the ideal image points. The FPW problem for systems containing dispersive elements becomes more difficult compared to that for non-dispersive systems, because there is an independent object-image relationship in the spectral dimension in addition to that in the spatial dimension.

 

Fig. 1 Illustration of the FPW problem.

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In this paper, we propose a design method for freeform optical systems containing diffraction gratings, which gives an approximated solution for the FPW problem. This method is based on Fermat’s principle and Ludwig’s grating ray-tracing equation [13] and can be extended to design freeform systems with other dispersive elements. Here, reflective three-mirror systems are used to demonstrate the design approach, which is also feasible for systems containing fewer or more surfaces. Two near-diffraction-limited freeform imaging spectrometers are achieved directly without software optimization within about 110 minutes (Intel Core i7-4790K 4.00GHz, 8GB RAM, NO GPU, 128G SSD, 1TB HDD).

2. Method

There are three steps in the proposed method. Firstly, a non-dispersive spherical imaging system with the same slit object as the final design is constructed. Then, an initial spherical imaging spectrometer is established, which meets the spectral specifications at the chief ray of the central field. Finally, all surfaces are calculated in freeform shapes and an iterative process is implemented to improve the image quality.

2.1 Construction of a non-dispersive spherical imaging system

The first step involves the construction of a non-dispersive spherical imaging system that has the same spatial object-image relationship as the desired design. In this step, we implement the construction-iteration (CI) method for non-dispersive systems raised by Yang et al. [8] to calculate the shape of the spherical surface. The CI method enables our method to start from geometries given by folded lines, which will be explained later in this section. It will be seen that this step is essential for the design of systems with novel geometries. Since the planar system is one of the simplest form of input, it will make this method require the least optical design knowledge and experience from the user.

For a better understanding, it is essential to review the CI method [8]. As shown in Fig. 2, the feature light rays are defined at every polar grid point on the aperture stop over the field. The intersection points for each feature light ray with every surface are defined as feature data points that have coordinates and normal direction information. Each feature light ray is expected to intersect the image plane at the ideal image point determined by the object-image relationship. Based on the feature light rays, the tilt/decenter and shape of each surface is calculated following the procedures below. Starting from an initial data point on the original surface, the coordinates of the next data point, e.g., P₁⁽¹⁾ in Fig. 2, are calculated using the Nearest-Ray algorithm [8]. Then, the path of the feature light ray, which is P₁⁽¹⁾P₂⁽¹⁾P₃⁽¹⁾T⁽¹⁾ in Fig. 2, is determined based on Fermat’s principle. Because the incident and emerging directions of the light rays are known, the normal of the surface at P₁⁽¹⁾ can then be obtained by Snell’s principle. This procedure is repeated until all data points have been calculated. The discrete points are then fitted into a freeform [8], spherical or aspherical shape [12] using a method that considers both the coordinates and the surface normals [14].

 

Fig. 2 Definitions of the feature light rays and feature data points. P_i^(j) represents a single data point and T^(j) is the corresponding ideal image point, where i indicates the surface number and j indicates the light ray number.

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However, experimental results show that the CI method, which starts from an initial planar system, will fail in the case that the object is at a finite distance, especially when the system’s numerical aperture (NA) is large. For example, starting from the initial planar system shown in Fig. 3(a), we implement the CI method following the calculating order of tertiary-primary-secondary mirror (T-P-S). In the output system shown in Fig. 3(b), the tertiary mirror, which is first calculated, has larger optical power than other surfaces, and has an extreme large mirror size. Note that the system shown in Fig. 3(b) has an obviously different scale than other systems shown in Fig. 3. This is not conductive to eliminating aberrations and controlling system size. In order to fix this problem and improve the feasibility of CI, a calculating strategy is proposed and the procedure is described as follows:

 

Fig. 3 Illustration of the strategy in Step 1. (a) The initial geometry. (b) Output of CI method following the T-P-S order. (c) Initial planar system whose NA = 0.05. (d) System after calculating the tertiary mirror. (e) System after the NA is expanded. The black lines indicate the areas on the mirror that have been used before the aperture is extended, while the blue lines indicate the areas that are used by the newly added feature light rays. (f) System after calculating the primary mirror. (g)(h) Outputs of the proposed calculating strategy in T-P-S and P-T-S order, respectively.

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However, there are still two major problems to be considered: the calculating order and the initial location of surfaces. We studied the effect of calculating order on the optical power of surfaces by computational experiments. Results indicate that the surface that is first calculated is positive. For example, when the calculating order is S-T-P, the output system has a positive secondary mirror as shown in Fig. 3(h), while in Fig. 3(f) the secondary mirror is negative. One can imagine that the results are close when the calculation starts from the same mirror, such as S-T-P and S-P-T, P-T-S and P-S-T, T-S-P and T-P-S. Based on experiments, we find that the order of T-P-S and P-T-S are most feasible and effective when implementing our method, indicating that the surface shape of the dispersive element located at the aperture should be calculated at the end. There is no obvious difference between the results of these two orders.

Inappropriate initial location of surfaces will cause obscuration in the system, which should be removed before constructing the freeform surfaces in order to fully utilize its degree-of-freedom and make the design process efficient. In general, the users can easily judge where the obscuration would happen in the system. To avoid obscuration, the simplest way is to leave enough space between surfaces when deciding the initial geometry. The optical power of each mirror can be controlled by multiplying the sphere radius by a factor ε before expanding the NA in Step 1 [12], such that the mirror size can be changed and obscurations can be eliminated. When ε>1, the absolute value of power is reduced; when 0<ε<1, the absolute value of power is increased. However, there are cases that surfaces could overlap with each other. We provide a method to solve this problem. The initial geometry can be parameterized into seven parameters, which is L₀₁, L₁₂, L₂₃, L₃₄, α₀₁, α₁₂, and α₂₃, as shown in Fig. 3(a). Sequences of these parameters can be input into the program, which presents the results later. Then the unobscured system can be selected.

2.2 Establishment of the initial spectrometer system

In this step, a spherical spectrometer system is established, which meets the dispersion specifications while using exactly the same surface shape as the non-dispersive spherical system obtained in the last step. A diffraction grating is placed on the secondary mirror. As shown in Fig. 4, it is defined by the intersection of an optical surface with a series of parallel planes. Starting from a single point on the grating surfaces, G is the normal of the grating generation surface and N is the normal of the optical surface. The nominal ruling space d is the distance between pairs of adjacent generation surfaces.

 

Fig. 4 Grating defined by the intersection of an optical surface with a series of parallel planes.

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Normal direction of the generation surfaces G and the ruling space d need to be determined in this step. G is parallel to a plane (the meridian plane) that is perpendicular to the object slit of the imaging spectrometer such that the system is a planar symmetrical system about this plane. The ruling space d is determined by the spectral specifications and the non-dispersive spherical system.

By real ray tracing, we obtain the focal length of the optics between the secondary mirror and the image plane (denoted by f ') and the angle of incidence on the secondary mirror of the chief ray at the central field (denoted by θ_i). The spectral image height h_spec is given by:

Consideration of the chief rays of the central field at wavelengths λ₁ and λ₂ gives the equations:

Substituting θ₁ and θ₂ from Eq. (4) into Eq. (5), Eq. (5) into Eq. (3), we then obtain the grating space d. The calculation of d is based on the chief ray of the central field and it is an approximation over the field and the aperture.

2.3 Calculation of the freeform shape of surfaces

For a similar reason to that of the case that CI fails when starting from planar system with a large NA, our method cannot skip Step 1 and 2 to perform Step 3, unless there is already a fine geometry like that of Offner, the design example of which will be presented later. Because the system after the dispersive element is non-dispersive, the CI method can be implemented to calculate the shapes of surfaces that are situated after the dispersive element, i.e. the diffraction grating. However, as for the dispersive element and those surfaces located before that, there are two major problems that are solved by the proposed method: ray-tracing of the multi-wavelength feature light rays through the diffraction grating and construction of the freeform surface shapes of the diffraction gratings.

2.3.1 Ray-tracing method for multi-wavelength feature light rays

This method is used when calculating the shapes of the surfaces situated before the dispersive element. As an example, the data point P₁ on the primary mirror is considered, as shown in Fig. 5. The corresponding feature light rays have N wavelengths denoted by λ₁, λ₂, …, λ_N, where N = 3. The paths of the light rays are coincident with each other before the diffraction grating, but disperse and travel along different paths after that. Using the calculation method in Step 1, the coordinates of P₁ are first obtained, while the normal vector N₁ is yet to be determined using the path P₁P₂P_3wT_w, where T_w is the ideal image point of the feature light ray with wavelength λ_w. Assume that the refractive index of the medium is 1. Then, the sum of the optical path lengths of the feature light rays with different wavelengths from P₁ to T_w is given as

 

Fig. 5 Path of a feature light ray with three wavelengths λ₁, λ₂, and λ₃. The light rays with different wavelengths disperse at P₂ on the secondary mirror and are expected to finally reach the corresponding target points in the image space.

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Based on the generalized grating ray-tracing equations given by Ludwig [13], the ray-tracing equation for multi-wavelength feature light rays of the system shown in Fig. 5 is given as:

2.3.2 Construction method for the freeform surface shapes of diffraction gratings

This method is used when calculating the surface shape of the diffraction grating. Let’s consider P₂ on the grating surface in Fig. 5. Because the normal N₂ determines the paths of feature light rays with different wavelengths, it needs to be calculated to make the light rays finally reach their corresponding ideal image points. At the same time, with respect to different wavelengths, the paths and the normal must be as close as possible to the solution of Ludwig’s equation as presented by Eq. (8). A method is proposed to calculate N₂ by using the quasi-newton algorithm.

Consider feature light rays with N wavelengths, denoted by λ₁, λ₂, …, λ_w, …, λ_N, which are expected to arrive at T_w. The emerging light rays’ directional vectors R_w', where w = 1, 2, …, N, are obtained independently following the object-image relationship. From Ludwig’s equation [13], with respect to N₂, one has

The normal N₂ can be given in the form of direction cosines as N₂ = (cos α, cos β, (1 - cos²α - cos²β)^1/2), where α and β represent the direction angles in the global Cartesian coordinates ℊ. We then substitute N₂ into Eq. (8) and take the sum of the squares as the cost function Γ for the quasi-newton algorithm, which is given as

2.3.3 Iteration process

We are now able to begin calculation of the freeform shapes of every optical surface in the system. To further improve the imaging quality, we implement an iteration process in which the coordinates of each data point are retained but the normal is recalculated, and the results are then used for fitting into a new shape. We use the following root-mean-square (RMS) value σ_RMS to evaluate the process:

3. Design examples

3.1 Design starting from a planar system

Using the proposed method, we designed a freeform imaging spectrometer operating over the 400-1000 nm range that has a slit length of 10 mm and the NA of 0.16. The smile and keystone distortions are both smaller than 20% of the pixel pitch of 8 μm. The spectral resolution is 2 nm. Following the procedures given in Step 1, a planar system with an NA of 0.05 is first set, as shown in Fig. 6(a). As the aperture size increases, one of the in-progress systems with a NA of 0.10 is shown in Fig. 6(b). To reduce the size of the grating surface and eliminate the obscuration, the power of the secondary mirror is controlled by a factor of ε = 0.8 as described in Step 1. The non-dispersive spherical imaging system is obtained as shown in Fig. 6(c).

 

Fig. 6 In-progress and final results of the example design starting from a planar system.

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Next, the ruling space d of the diffraction grating is calculated to be 0.007 mm in Step 2. The initial spherical spectrometer system is established as shown in Fig. 6(d) with σ_RMS = 0.79. Then, following the order of T-P-S, all the optical surfaces are calculated into freeform shapes. In the example design, the XY polynomial surface up to fourth order with a conic base is used to describe the freeform surface, which is written as:

The iterative process is then implemented and the σ_RMS finally converges to 0.0074, as shown in Fig. 6(g). The output system, as shown in Fig. 6(e), is diffraction-limited over part of the field and the spectral band without optimization. The RMS wavefront error (RMS WFE) is shown in Fig. 6(h). The modulation transfer function is more than 0.6 at 62.5 cyc/mm over 400–820 nm. The keystone and smile distortions are smaller than 18 μm and 10 μm, respectively. We finally use CODE V to optimize the system to the diffraction-limited. The optimization process becomes surprisingly easy because of the good results obtained by our method. The final system and the RMS WFE are shown in Fig. 6(f) and 6(i), respectively.

3.2 Design with the Offner geometry

Another design of an Offner spectrometer with a NA of 0.19 is also given as an example. The other specifications are the same as those in the previous design. Because of the good aberration correction properties of the Offner spectrometer, we can begin to calculate the freeform shapes of its surfaces without performing Step 1. As shown in Fig. 7(a), two concentric mirrors with radii of −100 mm and −50 mm are used. The slit is placed at an arbitrary distance of 22 mm from the center of curvature of the mirrors.

 

Fig. 7 Example design of a freeform Offner imaging spectrometer starting from a concentric geometry.

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The ruling space of the diffraction grating is calculated to be 0.0067 mm and σ_RMS for the initial spherical spectrometer is 0.14. The surfaces are then calculated in the freeform design and σ_RMS converges to 0.018 when the iteration is complete, as shown in Fig. 7(b). The final output from the system and its RMS WFE are shown in Fig. 7(c) and 7(d), respectively, and are diffraction-limited over all fields near the central wavelength. The keystone and smile distortions are 1.4 μm and 11.1 μm, respectively. The system reaches its limit through a rapid optimization using CODE V. The final design and the RMS WFE are shown in Fig. 7(e) and 7(f), respectively.

4. Discussion

The proposed method can design freeform systems that are near-diffraction-limited over partial field and spectral band in a short time. Such results are almost impossible to achieve directly by the conventional approaches such as first-order optics and primary aberration theory. Though commercial optical design software can optimize spherical or freeform systems, it requires experience and knowledge from the designer. The proposed method simplified the input and only planar system is required. The calculation process is completed by computer, during which requires no human interactions; thus the method is efficient and time consuming. As for the fine initial geometries that already exists, e.g. Offner, this method can yield equally good results with high performance in a rapid and high-efficiency manner.

The two examples show that direct result of our method is a fine starting-point for further optimization, no matter it starts from a planar geometry or fine geometries like that of the Offner. However, perfect (diffraction-limited) imaging quality and the specifications of distortions are not achieved directly. There are aspects that may lead to imperfect results, such as inappropriate number, location, and shape of surfaces. One major consideration is the initial geometry. Possible approaches to solve this problem are such as the automatic design framework [9] and the starting geometry creation and design method [15]. Another consideration is that there are fitting errors for the freeform surfaces, thus making the surface shape deviating from the raw data points. This problem needs further investigation in the future.

The proposed method has broad potential applications. It can be used to design reflective, refractive, or hybrid systems with an arbitrary number of optical surfaces that have spherical, aspherical, or freeform shapes. Because Fermat’s principle is valid no matter how many surfaces are presented in the system, this method can design systems containing an arbitrary number of surfaces. The only modification would be adding more terms to the ray-tracing equations. Furthermore, it is of importance to note that the diffraction grating can be placed on the primary (or the tertiary mirror). The modification needed would be removing equation terms that are related to the surfaces located before (or after) the diffraction grating. This method can be extended to become a general method for systems containing dispersive elements, such as prisms and DOEs, as long as there is a generalized ray-tracing equation as that of the Ludwig’s equation.

5. Conclusion

We propose a design method for freeform optical systems that contain diffraction gratings based on Fermat’s principle and Ludwig’s equation, giving an approximated solution to the FPW problem in the design of freeform imaging spectrometers. The effectiveness and efficiency were demonstrated by two design examples, starting from a planar system and Offner geometry, respectively. A calculating strategy was proposed to enable the calculation of optical systems to start from planar systems. Two methods were proposed: the ray-tracing method for multi-wavelength feature light rays and the construction method for the freeform surface shapes of diffraction gratings. These methods make the feature light rays be calculated to fit the Fermat’s principle, the diffraction principle, as well as the given object-image relationship at the same time. They are the key to achieve fine results that have near-diffraction-limited imaging quality directly. The proposed method can be modified to become a design method for dispersive elements in addition to diffraction gratings, such as prisms and DOEs, as long as there is a generalized real ray-tracing equation. With the benefits of faster computation speed and the development of artificial intelligence, the design method will serve users well by offering massive quantities of fine initial designs that cannot only be used for theoretical studies in optics, but also for high-performance optical system design.