Design method for assembly-insensitive freeform reflective optical systems

Xinyu Liu; Tongtong Gong; Guofan Jin; Jun Zhu

doi:10.1364/OE.26.027798

1. Introduction

Generally, an as-built optical system has a lower performance than its design, because factors such as assembly errors will affect the image quality. The assembly sensitivity indicates the influences of assembly errors on the imaging quality of a system. For an optical system with a given specification, multiple results can be designed that have different assembly sensitivities, among which the one that is most insensitive to assembly errors is favored since it is less difficult to assemble.

The conventional design process by commercial software requires initial systems as the starting point for further optimization. There are three approaches to find the initial systems. The first is to search in a lens database. But it is almost impossible to find an initial system in the lens database for the freeform system with desired specifications and loose tolerance for assembly errors. The second is to use the paraxial theory to calculate a coaxial system. But freeform surfaces are often applied to off-axis systems. An off-axis system can be created by tilting the surfaces of a coaxial system, but the off-axis system will generally deviate from the paraxial solution, thus making it difficult to optimize successfully. The third is to calculate the system using direct design methods, such as the Wassermann-Wolf differential-equations method [1,2], the simultaneous multiple surfaces method [3,4], and the Construction-Iteration method [5–7]. Freeform systems that meet the specifications can be designed directly by these methods. However, to the best of our knowledge, no design method for freeform systems has been proposed to find an initial system that considers both the system specifications and the low-assembly-sensitivity requirement at the same time. If the assembly sensitivity of the system is so high that its alignment requirements cannot be met, the system must be redesigned, which is human effort consuming. The closer an initial system is to the system desired, the easier the optimization process will be, and the possibility of achieving success in optimization is increased. Therefore, a design method for initial system of an assembly-insensitive freeform system that requires less human effort is expected.

Freeform surfaces are surfaces without rotational symmetry. In comparison with the spherical and aspheric surfaces, freeform surfaces provide more degrees of freedom for design and have stronger ability to correct aberrations. As these features yield great advantages in reducing the number of components required by the system as well as reducing the volume and weight of the system, freeform surfaces have been successfully applied to LED illumination [8,9], beam shaping [10,11], head-mounted displays [12,13], spectrometers [14], and off-axis reflection systems [15–18].

Freeform surfaces have been applied to correct asymmetric aberrations and improve the image quality of off-axis systems [5]. Off-axis reflection systems provide the advantages of no chromatic aberration, no central obscuration, and little absorption, which have been applied to cameras and telescopes [19,20]. However, the assembly of an off-axis system is difficult [20,21]. Since a freeform surface does not have an axis of rotational symmetry, the assembly of an off-axis system that contains freeform surfaces is generally more difficult than the assembly of an off-axis system comprised of surfaces with symmetry of revolution. Hence, a design method of off-axis freeform systems with low assembly-sensitivity will greatly drive the application of freeform surfaces in optical systems.

This paper proposes an automated design method for initial systems of assembly-insensitive freeform reflective systems, which requires no human interaction. Rather than trying to reduce the alignment sensitivity during the optimization process of commercial software, this method aims to find an assembly-insensitive solution before the optimization. Starting from plane systems, this method firstly constructs spherical systems with different optical power distributions. Then, the spherical systems are evolved into mixed-surface-type systems, which are defined as systems that have at least two types of surfaces out of sphere, aspheric and freeform surfaces. Next, the assembly-insensitive one is selected as the initial system for optimization. During the evolution, a principle is obeyed that the assembly-insensitive spherical surfaces are preferentially evolved to surfaces with higher degrees of freedom. This paper illustrates how to obtain an assembly-insensitive system by designing freeform off-axis reflective imaging systems. The system operates at F/2.5 with a 60 mm entrance-pupil diameter and a 0.6° × 8° field of view. Among the systems obtained by the automated design process, three systems were optimized, and their assembly sensitivities were analyzed. The result shows that, among the three systems, the total assembly sensitivity of the system that is most assembly-insensitive is approximately one third of that of the system that is most assembly-sensitive, and an assembly-insensitive system can be effectively obtained by the method proposed. The proposed method can also be applied to the design of other systems, such as assembly-insensitive off-axis aspheric imaging systems.

2. Design method of assembly-insensitive optical systems

This section shows how to obtain an assembly-insensitive initial system through the automated design process. The key procedures of the design method are as follows: (1) Establish an initial plane system, and construct a series of spherical systems with different optical power distributions through a point-by-point design method [22]. Then, analyze the assembly sensitivities of these spherical systems. (2) Evolve resulting spherical systems into mixed-surface-type systems, and adopt a method of evolution that preferentially transforms the assembly-insensitive spherical surfaces to surfaces whose type owns higher degrees of freedom (DOF), such as aspheric surfaces or freeform surfaces. (3) Take different plane systems as initial plane systems, and repeat steps (1) – (2). (4) Select the system with lower assembly sensitivity among the resulting mixed-surface-type systems.

2.1 Construction of spherical systems with different optical power distributions

The optical power distribution can have a great influence on the assembly sensitivity of an optical system. By distributing the optical power reasonably, the assembly sensitivity of a system can be effectively reduced. In this paper, we adopt the point-by-point construction design method to obtain spherical systems [22,23]. The optical-power-distribution method proposed in [23] can effectively distribute the optical power, but it may cause a drop of image quality. When constructing a surface before the aperture stop by the point-by-point construction method, if the constructed surface changes too much, the entrance pupil may change significantly. It will result in the changes of the rays incident on the surfaces for which constructions have been completed, leading to a drop in design accuracy. Based on the method proposed in [23], we propose an optical-power-distribution method with better design accuracy that changes the optical power distribution step-by-step. The specific steps are as follows:

1. Establish an initial plane system P₁ with no obscuration.
2. Select F fields and chose a certain number of feature rays in each field. The total number of the feature rays is T. Determine the ideal image point for each field.
3. Calculate the feature data points on the surface Ω_s₁. Determine the initial feature ray R₁, and take its intersection with the surface Ω_s₁ as the first data point D₁. According to Fermat's principle and the object-image relationships, the intersection points of the initial feature ray R₁ with the surfaces adjacent to surface Ω_s₁ can be determined [22], as shown in Fig. 1. The normal vector N₁ at D₁ can be determined using Snell’s law: r₁′ × N₁ = r₁ × N₁, where r₁ and r₁′ are unit vectors in the direction of incidence and reflection, respectively. After determining the initial data point D₁ and its normal vector N₁, the next data point D₂ can be determined using the nearest-ray algorithm [22]. All feature data points can be obtained by repeating this process.
4. Fit the data points to a sphere of radius r_s₁ using the least-squares algorithm. Then, the construction of surface Ω_s₁ is completed.
5. Construct all other surfaces Ω_s₂, Ω_s₃,…, Ω_S of the system with the same method in steps 3 and 4, thus obtaining a spherical system sph⁽¹⁾ (S is the total number of surfaces in the system).

Fig. 1 The calculation of N₁.

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For spherical system sph⁽¹⁾, the surface that is constructed first will receive most of the optical power. To obtain spherical systems with different optical power distributions, the optical power needs to be redistributed. Based on the spherical system sph⁽¹⁾, spherical systems sph⁽²⁾, sph⁽³⁾,…, sph⁽ⁿ⁾, sph⁽ⁿ⁺¹⁾,…, sph⁽^N⁾ are obtained by the redistribution of the optical power and reconstruction, where N is the total number of spherical systems with different optical distributions. The specific steps are as follows:

1. Start from spherical system sph⁽¹⁾, and let the radius of surface Ω_s₁ that is reconstructed first be r_s₁ × ε_s₁ to change the optical power of the surface Ω_s₁ (ε_s₁ is the power distribution coefficient for the s1-th surface). Reconstruct the next surface Ω_s₂ with the same method used in the construction of the system sph⁽¹⁾.
2. Redistribute the optical power of surface Ω_s₂, and reconstruct the next surface Ω_s₃ with the same method in step 1. Then, redistribute the optical power of surface Ω_s₃ and reconstruct surface Ω_s₄. Repeat this process until the redistribution of optical power of surface Ω_S_-1 and reconstruction of surface Ω_S is completed. Then, spherical system sph⁽²⁾ is obtained.

Starting from the system sph⁽²⁾, system sph⁽³⁾ can be obtained with the method described above. Similarly, systems sph⁽⁴⁾ – sph⁽^N⁾ can be obtained.

We adopt the root-mean-square (RMS) deviation σ_RMS of the distance between the intersection of a feature ray with the image plane and its corresponding ideal image point to evaluate the design accuracy [5]:

σ_{RMS} = \sqrt{\frac{\sum_{m = 1}^{K} σ_{m}^{2}}{K}},

where K is the total number of feature rays, and σ_m is the distance between the ideal and actual intersections of the m-th feature ray with the image plane.

A series of subsystems with different σ_RMS can be obtained by letting a structural parameter of a system, such as the radius of the s-th surface, be a series of values near the original value of the structural parameter. Let the parameter of the system be the value that corresponds to the lowest σ_RMS. This process is called the search of the structural parameter of the system. Spherical system SPH⁽ⁿ⁾ with better image quality is obtained after searching the radius and tip of each surface of system sph⁽ⁿ⁾ (n = 1, 2,…, N).

2.2 Analysis method of the assembly sensitivity

The effect of assembly errors on optical performance can be evaluated by the aberrations [24,25], modulation transfer function [26,27], RMS wavefront error [28–30], and the RMS spot diameter [28]. In this paper, the assembly sensitivity of a surface is evaluated by the change in the RMS wavefront error of feature fields caused by a slight perturbation of the position parameters of the surface. It is described in detail as follows. To simulate assembly errors, a given perturbation is applied to the i-th position parameter of the s-th surface. After the focus compensation, the maximum change in the RMS wavefront error of the f-th feature field caused by this perturbation is Δw_i waves, where i = 1, 2, …, 6 correspond to perturbations in the surface’s X-decenter, Y-decenter, Z-decenter, tip, tilt and clocking, respectively. For the s-th surface, the influence of the assembly errors on the f-th feature field is evaluated by RSS_f, which is the Root-Sum-Square (RSS) of the Δw_i corresponding to all the position parameters of the s-th surface:

R S S_{f} = \sqrt{\sum_{i = 1}^{6} {(Δ w_{i})}^{2}}

Let the total number of feature fields be F. Then the assembly sensitivity of the s-th surface is defined as the average of the quantities RSS_f over all of the feature fields:

R S S_{s} = \frac{\sum_{f = 1}^{F} R S S_{f}}{F}

Let the total number of surfaces in the system be S. The following formula is adopted to evaluate the total assembly sensitivity of the system:

R S S_{a l l} = \sum_{s = 1}^{S} R S S_{s}

The difficulty of aligning a system is not only related to the assembly sensitivities of its surfaces, but also connected with the surface type of each surface. For instance, the assembly of a freeform surface is generally harder than the assembly of a spherical surface. Therefore, the following equation can be used to evaluate the difficulty of assembly for a system:

D i f f i c u l t y_{a l l} = \sum_{s = 1}^{S} W_{s} \cdot R S S_{s},

where W_s is the weight of assembly difficulty for the s-th surface, which can be related to the s-th surface’s type, area, and off-axis magnitude, according to the fabrication.

Using the analysis method described above, the assembly sensitivities of the spherical systems SPH⁽¹⁾–SPH⁽^N⁾ obtained in section 2.1 can be determined.

2.3 Evolution of the spherical systems into mixed-surface-type systems

The aberration-correction ability of a spherical system is limited, so it may need to be evolved into mixed-surface-type optical system [23]. Generally, the assembly of a freeform surface is more difficult than that of an aspheric surface, and the assembly of an aspheric surface is harder than that of a spherical surface. Therefore, to reduce the assembly sensitivity of surface whose type owns higher DOF, it is better to evolve the spherical surface for which the assembly sensitivity is lower into a surface whose type owns more DOF. If the image quality of the system reaches the design requirements after optimization, the design process is completed. Otherwise, the DOF of the surface can be further increased or the remaining spherical surface with lower assembly sensitivity can be evolved into a surface whose type owns more DOF. Then, the system is optimized. Repeat this process till the image quality meets the given requirements. If the number of surfaces whose type owns more DOF required can be determined, we can adopt iteration [5,23] to obtain mixed-surface-type systems for which the assembly sensitivities of surfaces whose type owns higher DOF are low:

1. Determine the surface type (sphere, aspheric, freeform) for each surface of the system, and the principle of preferentially evolving the assembly-insensitive spherical surfaces into surfaces whose type owns higher DOF is followed.
2. Obtain the intersection point D_i of the feature ray R_i with the s-th surface of the spherical system SPH⁽ⁿ⁾ obtained in section 2.1. Calculate the normal vector N_i at each data point D_i with the same method in section 2.1 (i = 1, 2,…, T; T is the total number of feature rays). Fit the acquired data points and normal vectors into a surface with the type desired using the fitting method that considers both the coordinates and normal vectors [31]. Then, the s-th surface is regenerated.
3. Repeat step 2 until all surfaces of the system have been fitted. Calculate σ_RMS of the resulting system.
4. Repeat step 2 and step 3 until the σ_RMS of the resulting system no longer decreases significantly.

2.4 Construction and evolution with different initial plane systems

Taking different plane systems as initial plane systems, a designer can obtain several series of mixed-surface-type systems with different assembly sensitivities by repeating the steps in sections 2.1–2.3. The total assembly sensitivity of each of these systems can be calculated from Eq. (4). From these mixed-surface-type systems, an assembly-insensitive initial system can be selected. After optimization, an assembly-insensitive system can be designed. The initial system can also be selected according to other evaluation indicators, such as the assembly difficulty, as given by Eq. (5).

The design process proposed is shown in Fig. 2, where D is the total number of initial plane systems, ε_s is the power-distribution coefficient for the s-th surface, and N is the total number of the optical power distributions. As shown in Fig. 2, after inputting initial plane systems and optical power distribution coefficients, the design process can be automatically completed by a computer. Finally, the mixed-surface-type system with lower assembly sensitivity is output.

Fig. 2 The flow diagram for the design process.

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3. Design of an assembly-insensitive freeform system

This section took the design of an assembly-insensitive freeform off-axis three-mirror optical system as example to show the effectiveness of the method proposed in this paper. The specifications of the system are listed in Table 1. The system had an 8° FOV in tangential direction. During the design process, the wavelength 10 μm was taken to evaluate the performance of the system.

Table 1. Specifications of the Optical System

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For convenience of discussion, in this paper, the optical systems are represented by symbols that consist of three characters in order, e.g. SAF_d⁽ⁿ⁾, which is explained as follows: “S” represents spherical surface; “A” represents aspheric surface; “F” represents freeform surface; “SAF” indicates that the system has a spherical primary mirror, an aspheric secondary mirror and a freeform tertiary mirror; the superscript (n) denotes the identifier of the systems that have different optical power distributions; the subscript d = 1 (or d = 2), denoting that this system takes the first (or the second) plane system as initial plane system. (Two plane systems were input to show the effectiveness of the method in this paper.)

3.1 Construction of spherical systems

This section shows the design of spherical systems with different optical power distributions. The assembly sensitivity of each surface of these spherical systems was analyzed, which would be used to determine which surfaces to be evolved into freeform surfaces according to the evolution principle of the automated process.

An initial plane system P₁ was established firstly, as shown in Fig. 3. The secondary mirror was the aperture stop of the system. Since the optical system we designed was symmetric about the YOZ plane, the central field in sagittal direction was equal to 0°. The system was asymmetric about the XOZ plane, the central field in tangential direction was selected flexible. During the point-by-point construction process, six fields were sampled, and 112 feature rays were chosen in each field. Then, the ideal image points of the sampled fields were determined according to the specifications expected.

Fig. 3 The initial plane system P₁.

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Construct the tertiary mirror, the secondary mirror and the primary mirror successively, and let the optical power distribution coefficients ε_s of the primary mirror, secondary mirror and tertiary mirror be 1, 1 and 1.1 respectively. Twenty five spherical systems SSS₁⁽¹⁾ – SSS₁⁽²⁵⁾ with different optical power distributions were obtained using the method described in section 2.1.

With the assembly sensitivity analysis method of a surface [as given by Eq. (3)] described in section 2.2, the assembly sensitivity of each surface of the system SSS₁⁽ⁿ⁾ was analyzed, as shown in Fig. 4 (applying ± 100 μm decenter perturbations and ± 1′ perturbations to the tip, tilt and clocking). The abscissa is the system number n. Systems with different system number n have different optical power distributions. The ordinate gives the assembly sensitivity of the primary, secondary or tertiary mirror of the system SSS₁⁽ⁿ⁾ (n = 1, 2,…, 25).

Fig. 4 Assembly sensitivities of the primary, secondary and tertiary mirrors of the spherical system SSS₁⁽ⁿ⁾ (n = 1, 2,…, 25).

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3.2 Discussion on the evolution principle

This section studies the influence of selection of surface to be evolved into a freeform surface on the assembly sensitivity of the freeform surface, showing the reasonableness of the evolution principle. The type of freeform surfaces was selected as XY polynomials in this paper. Since the system we designed was symmetric about the YOZ plane, the coefficients of odd items of x in XY polynomials were zero. In the local coordinate system of a freeform surface, the freeform surface can be written as:

\begin{array}{l} z (x, y) = \frac{c (x^{2} + y^{2})}{1 + \sqrt{1 - (1 + k) c^{2} (x^{2} + y^{2})}} + A_{2} y + A_{3} x^{2} + A_{5} y^{2} + A_{7} x^{2} y \\ + A_{9} y^{3} + A_{10} x^{4} + A_{12} x^{2} y^{2} + A_{14} y^{4}, \end{array}

where c is the curvature, k is the conic constant and A_i is the coefficient of the x-y terms.

It is shown in Fig. 4 that the assembly sensitivity of the tertiary mirror is generally lower than that of the secondary mirror for the spherical system SSS₁⁽ⁿ⁾. To discuss how to reduce the assembly sensitivity of freeform surface, two series of systems were designed by evolving the tertiary or secondary mirror of the spherical system SSS₁⁽ⁿ⁾ into a freeform surface. Evolving the tertiary mirror of the system SSS₁⁽ⁿ⁾ into a freeform surface, we obtained system SSF₁⁽ⁿ⁾, for which the tertiary mirror was a freeform surface and the primary and secondary mirrors were spherical surfaces (n = 1, 2,…, 25). Then, evolving the secondary mirror of the system SSS₁⁽ⁿ⁾ into a freeform surface, we obtained system SFS₁⁽ⁿ⁾, for which the secondary mirror was a freeform surface and the primary and tertiary mirrors were spherical surfaces (n = 1, 2,…, 25). The process of evolving the system SSS₁⁽ⁿ⁾ into the systems SSF₁⁽ⁿ⁾ and SFS₁⁽ⁿ⁾ is show in Fig. 5. The assembly sensitivities of freeform surfaces of the systems SSF₁⁽ⁿ⁾ and SFS₁⁽ⁿ⁾ are shown in Fig. 6. The abscissa is the system number n, and the ordinate gives the assembly sensitivity of the freeform tertiary mirror of the system SSF₁⁽ⁿ⁾ or freeform secondary mirror of the system SFS₁⁽ⁿ⁾ (n = 1, 2,…, 25). From Fig. 6, it was found that the freeform surface of system SSF₁⁽ⁿ⁾, for which the insensitive tertiary mirror was evolved into a freeform surface, is insensitive to assembly errors than the freeform surface of system SFS₁⁽ⁿ⁾, for which the sensitive secondary mirror was evolved into a freeform surface. The assembly of a freeform surface is generally harder than a spherical surface. Therefore, the evolution principle of the proposed automated design method is reasonable and can lower the assembly sensitivities of surfaces whose type owns higher DOF.

Fig. 5 Process of evolving system SSS₁⁽ⁿ⁾ into systems SFS₁⁽ⁿ⁾ and SSF₁⁽ⁿ⁾ (n = 1, 2,…, 25).

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Fig. 6 Assembly sensitivities of freeform surfaces of mixed-surface-type systems SSF₁⁽ⁿ⁾ and SFS₁⁽ⁿ⁾ (n = 1, 2,…, 25).

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3.3 Obtainment of assembly-insensitive initial systems

This section shows the design of an assembly-insensitive initial system with the automated design method proposed.

In this paper, two plane systems were input to show the effectiveness of the method. If more plane systems are selected, it is more likely to obtain a system that is more insensitive to assembly errors. The first initial plane system is P₁, as shown in Fig. 3. With the system P₁ as an initial plane system, the design of spherical systems SSS₁⁽¹⁾–SSS₁⁽²⁵⁾ obtained by the point-by-point construction method has been shown in section 3.1. After trials, we found that the system requires two freeform surfaces and one spherical surface for this performance. As shown in Fig. 4, for spherical system SSS₁⁽ⁿ⁾, the assembly sensitivity of the secondary mirror was generally the highest among the primary, secondary and tertiary mirrors. Therefore, the secondary mirror was remained unchanged, the primary and tertiary mirrors were evolved into freeform surfaces for system SSS₁⁽ⁿ⁾. Adopting this way of evolution and using the method described in section 2.3, mixed-surface-type system FSF₁⁽ⁿ⁾ was obtained by the evolution of the spherical system SSS₁⁽ⁿ⁾ (n = 1, 2,…, 25).

For the secondary initial plane system P₂ (as shown in Fig. 7), adopting the method described in section 3.1, spherical systems SSS₂⁽¹⁾-SSS₂⁽²⁵⁾ with different optical power distributions were obtained. After analyzing the assembly sensitivity of each surface of spherical system SSS₂⁽ⁿ⁾, it was found that the assembly sensitivity of the secondary mirror was generally the highest for the system SSS₂⁽ⁿ⁾. Therefore, for system SSS₂⁽ⁿ⁾, the secondary mirror was remained unchanged, and the primary and tertiary mirrors were evolved into freeform surfaces. Adopting this way of evolution and using the method described in section 2.3, mixed-surface-type system FSF₂⁽ⁿ⁾ was obtained by the evolution of the spherical system SSS₂⁽ⁿ⁾ (n = 1, 2,…, 25). The process of construction and evolution to obtain systems FSF₁⁽ⁿ⁾ and FSF₂⁽ⁿ⁾ is shown in Fig. 7.

Fig. 7 Construction and evolution to obtain systems FSF₁⁽ⁿ⁾ and FSF₂⁽ⁿ⁾ (n = 1, 2,…, 25).

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The total assembly sensitivities [as given by Eq. (4)] of mixed-surface-type system FSF₁⁽ⁿ⁾ which took system P₁ as initial plane system and mixed-surface-type system FSF₂⁽ⁿ⁾ which took system P₂ as initial plane system are shown in Fig. 8. The abscissa is the system number n, and the ordinate gives the total assembly sensitivity of system FSF₁⁽ⁿ⁾ or FSF₂⁽ⁿ⁾ (n = 1, 2,…, 25). As shown in Fig. 8, mixed-surface-type systems evolved from spherical systems with different optical power distributions or different surface positions have different total assembly sensitivities. Therefore, searching for reasonable optical power distribution and surface positions can effectively reduce the total assembly sensitivity of a mixed-surface-type system. Besides, for this system, when the tertiary mirror receives most of the optical power, the total assembly sensitivity of the system tends to be lower.

Fig. 8 Total assembly sensitivities of mixed-surface-type systems FSF₁⁽ⁿ⁾ and FSF₂⁽ⁿ⁾ (n = 1,2,…,25).

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After obtaining mixed-surface-type systems FSF₁⁽¹⁾ - FSF₁⁽²⁵⁾ and FSF₂⁽¹⁾ - FSF₂⁽²⁵⁾, the initial system for assembly-insensitive system was selected from them. The process in this section was automatically completed by a computer.

3.4 Optimization

The image qualities of the initial mixed-surface-type systems obtained in section 3.3 are not sufficiently good. In this section, some of them were optimized to obtain systems for which the image qualities are similar and close to the diffraction limit. The optimization was conducted in optical software, Code V [32]. During the optimization process, the changes in the surface positions and optical power distributions of the systems were limited to a certain range.

Among the mixed-surface-type systems FSF₁⁽¹⁾ - FSF₁⁽²⁵⁾ and FSF₂⁽¹⁾ - FSF₂⁽²⁵⁾, the total assembly sensitivity of the system FSF₁⁽⁵⁾ was relatively low. And it could be optimized to a system whose image quality is close to the diffraction limit when the changes of the optical power distribution and surface positions were small. Its geometry is shown in Fig. 9(a). The system FSF₁⁽⁵⁾_opt was obtained by optimizing the system FSF₁⁽⁵⁾, and the geometry and RMS spot diameter of the system FSF₁⁽⁵⁾_opt are shown in Fig. 9(b).

Fig. 9 Initial systems and systems after optimization. (a) System FSF₁⁽⁵⁾. (b) The geometry and RMS spot diameter for system FSF₁⁽⁵⁾_opt. (c) System FSF₁⁽²⁵⁾. (d) The geometry and RMS spot diameter for system FSF₁⁽²⁵⁾_opt. (e) System FSF₂⁽⁵⁾. (f) The geometry and RMS spot diameter for system FSF₂⁽⁵⁾_opt.

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The systems FSF₁⁽²⁵⁾ and FSF₁⁽⁵⁾ have different optical power distributions, and the positions of their surfaces are similar. To study the influence of optical power distribution on the total assembly sensitivity of the system, we optimized the system FSF₁⁽²⁵⁾, for which the total assembly sensitivity was the highest among the systems FSF₁⁽¹⁾ - FSF₁⁽²⁵⁾. The geometry of the system FSF₁⁽²⁵⁾ is shown in Fig. 9(c). The system after optimization was FSF₁⁽²⁵⁾_opt, and its geometry and RMS spot diameter are shown in Fig. 9(d).

The systems FSF₂⁽⁵⁾ and FSF₁⁽⁵⁾ have similar optical power distributions, and the positions of their surfaces are different. To compare the influence of the positions of surfaces on the total assembly sensitivity of the system, we optimized the mixed-surface-type system FSF₂⁽⁵⁾, which is shown in Fig. 9(e). The system after optimization was FSF₂⁽⁵⁾_opt, as shown in Fig. 9(f).

The assembly sensitivities of the surfaces of the initial systems and systems after optimization are shown in Table 2 and Table 3, respectively.

Table 2. Assembly Sensitivities of Initial Systems

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Table 3. Assembly Sensitivities of Systems after Optimization

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The systems FSF₁⁽²⁵⁾_opt and FSF₁⁽⁵⁾_opt have different optical power distributions, and the positions of their surfaces are similar. The total assembly sensitivity of the system FSF₁⁽²⁵⁾_opt is approximately 3.5 times that of the system FSF₁⁽⁵⁾_opt. The systems FSF₂⁽⁵⁾_opt and FSF₁⁽⁵⁾_opt have similar optical power distributions, and the positions of their surfaces are different. The total assembly sensitivity of the system FSF₂⁽⁵⁾_opt is approximately 1.7 times that of the system FSF₁⁽⁵⁾_opt. Thus, by reasonably distributing the optical power and selecting the surface positions, the total assembly sensitivity of the system can be effectively reduced. By taking the system output by the automated design process as the initial system, a final design that is assembly-insensitive can be effectively obtained.

The parameters of the system FSF₁⁽⁵⁾_opt, which is the most insensitive system in the systems optimized, are listed in Table 4. Since the system we designed was symmetric about the YOZ plane, the coefficients of odd items of x in XY polynomials, tilt and clocking are zero.

Table 4. Surface Parameters of System FSF₁⁽⁵⁾_opt

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4. Conclusion

This study aims to propose an automatic design method for initial systems of assembly-insensitive freeform reflective systems, which could greatly increase the efficiency of design activity. The initial system that is closer to the system desired is beneficial for the final design. Therefore, for the design of an assembly-insensitive system, this method can provide a better initial system than the traditional direct design methods that only consider expected specifications. The automated design process takes plane systems as input, and searches for both reasonable optical power distribution and surface positions to reduce the assembly sensitivity of a system. To reduce the assembly sensitivity of surfaces whose type owns higher DOF, the principle is followed of preferentially evolving the surfaces with lower assembly sensitivities into surfaces whose type owns higher DOF. An assembly-insensitive system with satisfactory image quality can then be obtained through optimizing the assembly-insensitive initial system obtained by the automated design process. We take the design of a freeform off-axis three-mirror system, which contains two freeform surfaces and one spherical surface, as an example to illustrate the design process of an assembly-insensitive system. The results show that the principle of evolution can effectively reduce the assembly sensitivities of surfaces whose type owns higher DOF, and an assembly-insensitive final design can be effectively obtained through the design method proposed.

The work was completed by inputting two plane systems into the automatic design process. If much more plane systems are input, the method is able to output a global insensitive solution after searching for reasonable optical power distribution and surface positions globally. A reflective system was taken as example, but the method proposed has the potential to be applied to refractive systems. Since the method is based on CI method, which has been applied to the design of a refractive system [33]. This automated design method can also be applied to the design of assembly-insensitive off-axis aspheric imaging systems, and it provides a new approach for studying how to reduce the assembly sensitivity of a system, supporting further studies on the reduction of assembly sensitivities.

Funding

National Natural Science Foundation of China (61775116).

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System	Primary mirror (freeform surface)	Secondary mirror (spherical surface)	Tertiary mirror (freeform surface)	Total assembly sensitivity
FSF₁⁽⁵⁾	0.026	0.041	0.014	0.081
FSF₁⁽²⁵⁾	0.118	0.153	0.016	0.287
FSF₂⁽⁵⁾	0.046	0.062	0.019	0.127

System	Primary mirror (freeform surface)	Secondary mirror (spherical surface)	Tertiary mirror (freeform surface)	Total assembly sensitivity
FSF₁⁽⁵⁾_opt	0.026	0.041	0.015	0.082
FSF₁⁽²⁵⁾_opt	0.116	0.148	0.020	0.284
FSF₂⁽⁵⁾_opt	0.048	0.072	0.020	0.140

Parameters	Primary mirror	Secondary mirror	Tertiary mirror
c	9.524172e-04	1.198329e-03	−2.285945e-03
k	−3.574448	0	−1.990818e-01
A₂	−4.117447e-04	0	−1.707697e-04
A₃	1.047440e-05	0	−5.030712e-06
A₅	−1.028819e-05	0	2.111107e-06
A₇	4.424928e-07	0	4.340029e-07
A₉	6.270639e-07	0	2.661306e-07
A₁₀	1.604491e-09	0	2.469729e-09
A₁₂	9.653518e-09	0	−9.877302e-10
A₁₄	4.242421e-09	0	−5.190270e-10
x-decenter	0	0	0
y-decenter	8.605239491	−89.97639046	−82.08317251
z-decenter	181.0253908	17.62307611	196.7883686
Tip	3.153785061	16.81270045	11.42552399

System	Primary mirror (freeform surface)	Secondary mirror (spherical surface)	Tertiary mirror (freeform surface)	Total assembly sensitivity
FSF₁⁽⁵⁾	0.026	0.041	0.014	0.081
FSF₁⁽²⁵⁾	0.118	0.153	0.016	0.287
FSF₂⁽⁵⁾	0.046	0.062	0.019	0.127

System	Primary mirror (freeform surface)	Secondary mirror (spherical surface)	Tertiary mirror (freeform surface)	Total assembly sensitivity
FSF₁⁽⁵⁾_opt	0.026	0.041	0.015	0.082
FSF₁⁽²⁵⁾_opt	0.116	0.148	0.020	0.284
FSF₂⁽⁵⁾_opt	0.048	0.072	0.020	0.140

Design method for assembly-insensitive freeform reflective optical systems

Abstract

1. Introduction

2. Design method of assembly-insensitive optical systems

2.1 Construction of spherical systems with different optical power distributions

2.2 Analysis method of the assembly sensitivity

2.3 Evolution of the spherical systems into mixed-surface-type systems

2.4 Construction and evolution with different initial plane systems

3. Design of an assembly-insensitive freeform system

3.1 Construction of spherical systems

3.2 Discussion on the evolution principle

3.3 Obtainment of assembly-insensitive initial systems

3.4 Optimization

4. Conclusion

Funding

References

Cited By

Figures (9)

Tables (4)

Equations (6)

Optics Express

Parameter	Specification
Field of view	0.6° × 8°
F-number	2.5
Entrance pupil diameter	60 mm
Wavelength	8-12 μm