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The coordinates and sign conventions are defined first. We initially define the global coordinates O-XYZ, as shown in Fig. 4a for a coaxial system and Fig. 4b for an off-axis system. The number of optical surfaces in the system is N. These optical surfaces are denoted by S1, S2, …, Si, …, SN. The system contains another two special surfaces, S0 and SI, where S0 is a virtual surface located in the front of the optical system and SI is the image plane surface. Let the light rays in the central field (0°) travel along the direction of the unit axial vector OZ. These light rays emerge from S0 and reach SI. As in the coaxial system shown in Fig. 4a, the system is rotationally symmetrical about the optical axis OZ. The center of the sphere with surface Si is located on the optical axis and denoted by Oi (not marked in the figure). Si intersects the optical axis at the vertex point Vi, as indicated in the figure. The radius of curvature of Si is denoted by ri. When the vector ViOi and the unit axial vector OZ are oriented in the same direction, the sign of ri is positive; otherwise, its sign is negative. The distance between surfaces Si and Si+1 is denoted by di, which is equal to the vector length |ViVi+1|. When the vector and the unit axial vector OZ are oriented in the same direction, the sign of di is positive; otherwise, its sign is negative. In the off-axis system shown in Fig. 4b, in which there is no axis of rotational symmetry for the system, local coordinates must be set up at every optical surface and the surface shapes are described using these local coordinates. The chief ray (denoted by CR) of the central field is set as the reference in the off-axis systems. The point of incidence of the CR of the central field on surface Si is denoted by Vi. Local coordinates Vi-XYZ are defined for surface Si, with the origin point being Vi. The direction of the unit axial vector ViZ lies parallel to the direction of the normal vector at point Vi on surface Si. The unit axial vector ViY lies parallel to the surface O-YZ and lies perpendicular to the vector ViZ. The unit axial vector ViX is oriented in the same direction as the unit axial vector OX. Specifically, the CR of the central field intersects surface S0 at point V0 and intersects surface Si at VI. Unless otherwise stated, in the coordinates VI-XYZ, the unit axial vector VIZ lies perpendicular to the image plane; the unit axial vector VIY is oriented parallel to the surface O-YZ and perpendicular to VIZ; and the unit axial vector ViX is oriented in the same direction as the unit axial vector OX.
The flow chart of the key procedures of the proposed design method is shown in Fig. 5 and the detailed phases of this method are described as follows.
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Solve for a series of coaxial spherical systems with various OP distributions. Using the matrix approach for first-order optics, the reflection matrix Ri for reflection of the light ray at surface Si is:
$$ R_i = \left[ {\begin{array}{*{20}{c}} 1 & {\left( {n_i - n_{i - 1}} \right)/r_i} \\ 0 & 1 \end{array}} \right] $$ (1) where ni represents the refractive index of the medium between Si and Si+1. The transfer matrix Di of a light ray propagating from Si to Si+1 is:
$$ D_i = \left[ {\begin{array}{*{20}{c}} 1 & 0 \\ { - d_i/n_i} & 1 \end{array}} \right] $$ (2) The system matrix for the complete system T is:
$$ T = R_ND_{N - 1}R_{N - 1} \ldots R_2D_1R_1 = \left[ {\begin{array}{*{20}{c}} B & A \\ D & C \end{array}} \right] $$ (3) where A, B, C, and D are functions of ri, di, and ni. Therefore, the image focal length of the optical system can be obtained using A(ri, di, ni) and is
$$ f^{\prime} = \frac{{n_N}}{{A\left( {r_i,d_i,n_i} \right)}} $$ (4) Because f′ is given, Eq. (4) represents the equation that the spherical curvature radii ri (i = 1, 2, …, N) and surface distances di (i = 1, 2, …, N−1) must satisfy. In a reflection system, the condition ni = −ni−1 applies; therefore, all the refractive indexes are canceled by each other. Eq. (4) has an infinite number of solutions and thus it is impossible to discuss all solutions to Eq. (4). Given that there are manufacturability limits in practice, some solutions to Eq. (4) should be disregarded and constraints should be set to narrow the solution space; this will be discussed in the following part.
In Eq. (4), there is a total of 2N−1 parameters for the radii of curvature and mirror distances, which are r1, r2, …, rN, and d1, d2, …, dN−1. As long as 2N−2 parameters out of the 2N−1 parameters are given, it is possible to solve for the last remaining parameter. After the 2N−1 parameters are obtained, an additional parameter dN can be determined using first order optics (not shown in Eq. (4)), where dN represents the distance between the last optical surface and the image plane. Therefore, there are a total of 2N parameters that describe the coaxial system. The 2N parameters are placed together in a row vector P = [r1, r2, …, rN−1, rN, d1, d2, …, dN−1, dN], which is used to represent a coaxial spherical system with a specific OP distribution. In this work, we assume that the given 2N−2 parameters are r1, r2, …, rN−1, rN, d1, d2, …, dN−2. Sequences of the radii of curvature ri (i = 1, 2, …, N) are given as rmin, rmin+Δr, rmin+2Δr, …, rmax, with the range [rmin, rmax] and with interval Δr. Sequences of mirror distances di (i = 1, 2, …, N−2) are given as dmin, dmin+Δd, dmin+2Δd, …, dmax, with the range [dmin, dmax] and the interval Δd. For every combination of ri (i = 1, 2, …, N) and di (i = 1, 2, …, N−2), the corresponding dN−1 can be solved using Eq. (4) and dN can then be obtained using first-order optics. Following the procedure described above, with a series of 2N parameters obtained, a series of coaxial spherical systems with focal length f′ and various OP distributions are obtained and denoted by P1, P2, …, Pm, …, PM; the set of these distributions is denoted by the symbol {P}.
As stated above, constraints are required to limit the range of values of some coefficients in the vector Pm. By changing the range of the radius of curvature ri (i = 1, 2, …, N), the values of the radii of curvature and the positive/negative state of the OP of that optical surface can be controlled. In this work, the range for ri is [−1000, 1000] (the units are millimeters hereinafter, unless otherwise stated). For the range of di, three aspects must be considered. First, the values should not be too large to avoid large system volumes. Second, the values should not be too small or it may be impossible to ensure that the system is unobscured in the subsequent phases. Third, the differences between two arbitrary mirror distances should not be too large to guarantee system compactness. Because the overall optical system size is usually comparable to the entrance pupil size, we use the entrance pupil diameter (EPD) as the unit length to describe the range of mirror distances, e.g., EPD≤|d1|≤4 × EPD. When determining the interval values Δr and Δd, we must consider the balance between the computation time and the number of output results. If the values of Δr and Δd are too high, there will be fewer output results; however, values that are too small will increase the number of output results but will also consume more computation time.
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For each coaxial spherical system Pm obtained in the previous step, tilt and reposition every surface in the system to obtain a series of noncoaxial systems while maintaining the direction of incidence of the CR of the central field in the object space. The systems obtained have various structures and can be considered to be field-biased, or off-axis, or a combination of the two. As shown in Fig. 6, where the three-mirror system is used as an example to explain the principle and the notation when solving for noncoaxial systems, the system corresponds to a coaxial system with a specific OP distribution P = [r1, r2, r3, d1, d2, d3]. Every mirror is tilted and repositioned by following the principle described below: the distances between the original points of the local coordinates are equal to the surface distances, which means that |V1V2| = d1, |V2V3| = d2, and |V3VI| = d3; at the same time, every mirror is tilted by a specific angle around the unit axial vector ViX in each local coordinate system.
For convenience of description, we use a vector C to represent the system structure. Noting that the path of the CR of the central field is a fold line that can be used to describe the system structure, we define C as a vector that contains the following two types of information: (1) the lengths of each segment of the fold line, which represents the absolute value of the surface distances |di|; and (2) the angles between every adjacent pair of segments of the fold line. As shown in Fig. 6, the vector Vi−1Vi rotates by an angle θi (i = 1, 2, …, N) to coincide with the vector ViVi+1. Therefore, θi represents the deflection angle of the CR of the central field at each mirror, which should have the range −360° < θi < 0° or 0° < θi < 360°. For clarity of description, the sign convention for θi is given as follows: if the vector Vi−1Vi rotates clockwise to coincide with the vector ViVi+1, θi is negative; otherwise, θi is positive. Thus, as shown in Fig. 6, θ1 < 0, θ2 > 0, and θ3 < 0. The vector C can now be written as C = [θ1, |d1|, θ2, |d2|, … θi, |di|, …, θN, |dN|]. For systems with the same OP distribution, the mirror distances are the same for the different structures; therefore, only the angles in the vector C are maintained, which make it look like C = [θ1, θ2, … θi, …, θN]. Specifically, the structure of the coaxial system is denoted by a symbol with a subscript of 0, i.e., C0 = [θ10, θ20, θ30] = [−180, 180, −180].
In vector C, when the values of the angles (θ1, θ2, … θi, …, θN) change continuously, the corresponding system structure also varies and may be obscured or unobscured. However, it is only when the angles vary within a specific range that the system remains unobscured. Because the system may contain different unobscured structure types, there may be multiple ranges within which θi can change in vector C. In other words, the range of θi in C is not continuous to guarantee that the system is unobscured. To find as many unobscured structures as possible, as many systems with different structures as possible can be listed, regardless of whether they are obscured or not, and the obscured systems can then be filtered out.
As stated above, there are angle ranges of −360° < θi < 0° or 0° < θi < 360°. If two structures C and C′ satisfy C =− C′, then the two structures are identical. Therefore, in the case of repeated consideration of the same structure, the range for θ1 should be within (−180°, 0°), while the ranges of θi (i = 2, 3, …, N) should be within (−360°, 0°). In this work, the range of θ1 is given by (−180°, −120°]; the range for θ2 is [120°, 240°]; and the range for θ3 is [−240°, − 120°]. For the angle interval Δθ, the diversity of the structures of the output results must be considered and balanced with the computation time. By following the above steps, with respect to the system with the OP distribution Pm, a series of spherical systems ${\boldsymbol{C}}_{{\boldsymbol{m, }}{\bf{1}}}{\mathrm{, }}{\boldsymbol{C}}_{{\boldsymbol{m, }}{\bf{2}}}{\mathrm{, \ldots, }}{\boldsymbol{C}}_{{\boldsymbol{m, s}}}{\mathrm{, \ldots, }}{\boldsymbol{C}}_{{\boldsymbol{m, S}}_{\boldsymbol{m}}}$ are obtained, where the set of these systems is denoted by {C}m with a total number of elements Sm. Next, all the unobscured systems in each set {C}m are found and denoted by ${\bar{\boldsymbol C}}_{{\boldsymbol{m, }}{\mathrm{1}}}{\mathrm{, }}{\bar{\boldsymbol C}}_{{\boldsymbol{m, }}{\mathrm{2}}}{\mathrm{, \ldots, }}{\bar{\boldsymbol C}}_{{\boldsymbol{m, r}}}{\mathrm{, \ldots, }}{\bar{\boldsymbol C}}_{{\boldsymbol{m, R}}_{\boldsymbol{m}}}$; the set of these systems is denoted by $\left\{ {{\bar {\boldsymbol{C}}}} \right\}_m$ with a total number of elements Rm. When a variety of unobscured systems with various OP distributions and various structures has been obtained, we can proceed directly to the next step and construct the freeform systems.
In this phase, filters could be implemented on the system structure or the volume limit because the system structure and the OP distribution will only change slightly in the coming phases. The systems in set $\left\{ {{\bar {\boldsymbol{C}}}} \right\}_m$ can be classified into several categories based on the geometry of the system structure; in each category, the value of θi in vector C is varied. By defining the new vector $\Delta {\bar{\boldsymbol C}}_{{\boldsymbol{m, r}}} = {\bar{\boldsymbol C}}_{{\boldsymbol{m, r}}} - {\boldsymbol{C}}_{\bf{0}} = \left[ {\Delta \theta _1, \Delta \theta _2, \ldots \Delta \theta _i, \ldots, \Delta \theta _N} \right]$, where C0 represents the coaxial system structure, the geometry of the system structure is classified using the positive/negative sign of Δθi in the vector $\Delta {\bar{\boldsymbol C}}_{{\boldsymbol{m, r}}}$. For each category of the system structure geometry, the absolute value of Δθi can be regarded as a metric to evaluate the system compactness. A smaller absolute value of Δθi represents a system with high compactness. In this work, for each category of the system structure geometry in the set $\left\{ {{\bar{\boldsymbol C}}} \right\}_m$, we obtain systems with compact structures by minimizing the absolute value of Δθi in the vector $\Delta {\bar{\boldsymbol C}}_{{\boldsymbol{m, r}}}$. The systems obtained are denoted by ${\tilde{\boldsymbol C}}_{{\boldsymbol{m, }}{\bf{1}}}{\mathrm{, }}{\tilde{\boldsymbol C}}_{{\boldsymbol{m, }}{\bf{2}}}{\mathrm{, \ldots, }}{\tilde{\boldsymbol C}}_{{\boldsymbol{m, t}}}{\mathrm{, \ldots, }}{\tilde{\boldsymbol C}}_{{\boldsymbol{m, T}}_{\boldsymbol{m}}}$ and the set of these systems is denoted by $\left\{ {{\tilde{\boldsymbol C}}} \right\}_m$ with a total number of elements Tm. In this phase, the system volume can be evaluated by calculating the volume of the space occupied by the light bundle and systems with volumes that exceed the limit can be removed from the set $\left\{ {{\tilde{\boldsymbol C}}} \right\}_m$.
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Construct freeform systems based on the systems in the set $\left\{ {{\tilde{\boldsymbol C}}} \right\}_m$. After the process to eliminate the obscured structures in phase 2, the OP of the entire system has been changed. In this phase, the freeform shapes for every optical surface are calculated by following the object-image relationship of the system, so that the OP of the entire system is corrected while the OP of each mirror is changed only slightly. Correction of the OP of the entire system should follow these principles: first, it must be realized automatically; second, the system structure must remain unchanged after the correction; third, the change in the OP of each mirror is small. Any method that satisfies these three rules can be implemented in this phase.
In this work, we use the point-by-point construction method for freeform systems26 to correct the OP of the entire system. The point-by-point construction method calculates freeform surface shapes based on feature light rays and feature data points. Feature light rays are defined at every field position and are located at different positions over the entrance pupil. Feature data points are defined as the intersection points of the feature light rays with the optical surfaces and contain the information of the point coordinates and the normal direction on the optical surface. Because the field-of-view angle for each feature light ray is known, and based on the object-image relationship that the system provides perfect imaging, corresponding image point coordinates (target image point coordinates) can be obtained on the image plane. When calculating the shape of the surface Si, intersection point coordinates on surface Si−1 and the propagation direction towards Si are obtained for all feature light rays by real ray tracing. Starting from a given initial feature data point on surface Si, the next feature light ray is then determined. Based on the coordinates and the normals of the feature data points that have already been calculated, the corresponding feature data point's coordinates are obtained via the nearest-ray algorithm26. Next, when the ideal image point coordinates and the corresponding feature data point coordinates on Si are known, the direction in which the feature light ray leaves Si can be resolved using Fermat's principle; then, by knowing both the direction of incidence and direction of departure of the feature light ray on Si, the normal directions of the corresponding feature data points can be obtained based on the law of reflection. The procedures above are repeated until the coordinates and normals of all feature data points are solved. Finally, the mathematical expressions for the freeform surface Si are obtained via a fitting method that considers both the coordinates and the normals of all feature data points31. In this work, XY polynomials with up to sixth order terms are used to describe the shape of the freeform surfaces. The results in the design examples show that the precision of this fitting method is high enough to achieve high imaging quality of diffraction-limited or near-diffraction-limited. Following the procedures above, the shapes of the freeform surfaces in the system are all calculated in a given order (e.g., tertiary-secondary-primary mirrors) and the construction of the freeform system is completed.
By implementing the method described above, the following series of freeform systems is obtained: ${\tilde{\boldsymbol F}}_{{\boldsymbol{m, }}{\bf{1}}}, {\tilde{\boldsymbol F}}_{{\boldsymbol{m, }}{\bf{2}}}, \ldots, {\tilde{\boldsymbol F}}_{{\boldsymbol{m, t}}}, \ldots, {\tilde{\boldsymbol F}}_{{\boldsymbol{m, T}}_{\boldsymbol{m}}}$, corresponding to the systems ${\tilde{\boldsymbol C}}_{{\boldsymbol{m, }}{\bf{1}}}, {\tilde{\boldsymbol C}}_{{\boldsymbol{m, }}{\bf{2}}}, \ldots, {\tilde{\boldsymbol C}}_{{\boldsymbol{m, t}}}, \ldots, {\tilde{\boldsymbol C}}_{{\boldsymbol{m, T}}_{\boldsymbol{m}}}$. The set of freeform systems obtained is denoted by $\left\{ {{\tilde{\boldsymbol F}}} \right\}_{\it{m}}$.
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Improve the imaging quality of the freeform system. By following the phases above, a series of unobscured freeform systems $\left\{ {{\tilde{\boldsymbol F}}} \right\}_{\it{m}}$ with various OP distributions and structures has been obtained, but the system imaging qualities still require improvement. The method used in this phase must satisfy the three principles stated in Phase 3. In this work, we use a point-by-point iteration method to improve the imaging quality of the freeform systems26.
The point-by-point iteration method is the same as the construction method from the perspective that the shape of each optical surface is resolved by following the object-image relationship, which is also based on the feature light rays and feature data points. The difference is that, during the iteration process, the feature data point coordinates on Si are obtained and retained by tracing the feature light rays incident on Si, while the surface normals are newly solved. When the traced data point coordinates and newly solved normal directions are known, a new freeform surface can be obtained by fitting.
In the point-by-point iteration method, an iteration round consists of calculation of the shapes of all optical surfaces in the system in a given order. Multiple iteration rounds can be performed until the imaging quality reaches the required value or it stops improving. In this work, the root-mean-square (RMS) values of the distances between the actual imaging points and the target imaging points are calculated at different field points and the average value of these distances (denoted by σ) is used as the metric to evaluate the imaging quality of the result of each round of iteration. As the iteration proceeds, the value of σ decreases and then gradually converges. When σ is smaller than a specified threshold σitr, the iteration process is terminated. The rate at which this value decreases after each round of iteration, τ, is defined as τ = |σ′−σ|/σ, where σ′ and σ are used to evaluate the imaging qualities of the results of the previous and current rounds of iterations, respectively. When τ decreases below a specific threshold τitr, the iteration process is terminated.
After an inspection of all available degrees-of-freedom for design of the freeform systems, we found that the tilt angle of the image plane has not been considered yet; therefore, the optimal tilt angle of the image plane must be determined to achieve the best possible imaging quality for the optical system. In this work, a one-dimensional search process is implemented. The image plane tilt angle is defined by the angle between the Y axis of the local coordinates and the Y axis of the global coordinates and denoted by β. In particular, when the CR of the central field is perpendicular to the image plane, the image plane tilt angle is denoted by β0. In a round of one-dimensional searching, a series of freeform systems with different image plane tilt angles is iterated to improve the system imaging quality until the iteration stops. In the first round of one-dimensional searching, the image plane tilt angles are given by a sequence varying within the range [β0−βr, β0+βr] with the interval Δβ. In the subsequent round of one-dimensional searching, the image plane tilt angles are given by a sequence varying within the range [βopt−βr, βopt+βr] with the interval Δβ, where βopt is the image plane tilt angle of the system with the best imaging quality from the previous round of searching. As multiple rounds of one-dimensional searching are performed, the system's imaging quality improves and the imaging quality metric σ converges. When σ is smaller than the specified threshold σsrh, the search process is terminated. When the improvement rate τ = (σ′−σ)/σ is lower than the specified threshold τsrh, the search process is terminated. σ′ and σ above are the imaging quality metrics for the results of the previous and current rounds of searching, respectively.
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By following the steps above, a series of freeform systems with various structures and various OP distributions is obtained. For each freeform system obtained, the imaging quality metrics are calculated, including the spot diameter of the imaging points, the modulation transfer function, and the RMS WFE over the field. In this work, systems where the AVG RMS WFE is lower than 0.075λ are eventually presented to the designers as the output results. The designers can then analyse the systems obtained and select their preferred designs. The specific parameters used for the design example 1 and 2 are shown in Table S7. The framework for the automatic design method is shown in Fig. S6.