Design of Off-Axis Four-Mirror Optical Systems Enabled by Freeform Optics

Abstract

The off-axis reflective optical system offers several advantages, including the elimination of central obstruction, zero chromatic aberration, and a compact structure. These features make it highly valuable in the domain of space remote sensing. Freeform surfaces transcend the limitations imposed by rotational symmetry, providing significant design flexibility that is particularly effective for correcting non-rotationally symmetric aberrations present in off-axis systems. In this paper, we propose the averaged seed curve extension (A-SCE) method, which facilitates the direct design of an initial structure for freeform off-axis reflective systems. Both focal and afocal off-axis four-mirror freeform optical systems are designed utilizing the A-SCE method, demonstrating an enhanced capability for initial structure design. The results indicate excellent optical performance while maintaining relatively loose processing and assembly tolerances for both systems, thereby enhancing the facilitation of practical implementation.

1. Introduction

With the advancement of space technology, there is an increasing demand for enhanced resolution, a wider field of view, superior imaging quality, and cost-effective solutions in space optical systems. This trend has positioned lightweight, low-cost, and high-resolution space optical systems as a prominent research focus within the domain of space remote sensing, driven by innovative design concepts [1]. Off-axis reflective optical systems present several advantages, including a compact structure, the absence of chromatic aberration, and the elimination of central obstruction [2,3,4,5], which render them widely utilized in various space optical applications. Given the long operational distances—typically exceeding 160 km—these systems generally require longer focal lengths to achieve adequate observational resolution. When the ratio of the system length to focal length ranges between 0.5 at relative apertures ranging from F/2.5 to F/3, off-axis three-mirror configurations can effectively meet performance requirements. However, when this ratio decreases to between 0.25 and 0.3 at relative apertures around F/5 to F/6, off-axis three-mirror systems encounter difficulties in fulfilling specifications, thus necessitating the adoption of off-axis four-mirror optical systems with smaller ratios [6]. Nevertheless, traditional spherical or aspherical off-axis reflective optical designs often face challenges in accommodating large fields of view while maintaining lightweight characteristics due to inherent limitations in design flexibility. Consequently, addressing how to expand the system’s field of view while ensuring a compact structure without compromising high resolution and imaging quality has emerged as an urgent challenge that must be resolved for contemporary space optical systems.

Freeform surface optical components represent an innovative category of optical surface geometries that have emerged in conjunction with continuous advancements in computational power and optical processing techniques. In contrast to traditional designs, freeform surfaces lack rotational symmetry, thus providing enhanced degrees of freedom for optical design. The implementation of freeform surfaces can effectively correct asymmetric aberrations within optical systems, resulting in improved imaging performance, elevated system specifications, more compact configurations, and a reduction in the number of required optical components. The integration of freeform surfaces into optical systems is considered to be a significant revolution within the field of optical design [7,8]. To date, freeform surfaces have been successfully applied across both imaging and non-imaging domains. In terms of imaging applications, they include head-up displays (HUDs) [9], hyperspectral imagers [10], ultra-short focus projectors [11], and extreme ultraviolet (EUV) lithography objectives [12]. In non-imaging contexts, their applications extend to illumination systems [13], laser shaping [14], and solar light collection systems [15].

For optical design, the identification of an appropriate initial structure is essential, as it forms the basis for subsequent optimization efforts. In addition to examining existing patents, commonly utilized methods for constructing the initial structure of off-axis reflective freeform optical systems include simultaneous multiple surfaces (SMS) [16], partial differential equations (PDEs) [17], construction iteration (CI) [18], seed curve extension (SCE) [19], nodal aberration theory (NAT) [20], power series [21], field-of-view expansion [22], and deep learning techniques [23]. Each of these methodologies has specific applicability and limitations [24,25,26,27]; at present, there is no unequivocally superior option available for designers.

One can obtain all necessary sample points for constructing the freeform surface using the SCE method. Subsequently, the initial shape of the freeform surface can be derived by fitting these data points. However, it has been observed that the selection of initial construction point positions significantly influences the resulting surface shape and consequently affects the performance of optical systems [28]. To enhance the reliability of the SCE method, multiple initial points can be selected simultaneously, enabling the separate generation of data points based on this approach. All obtained data points can then be utilized to fit the freeform surface through a least squares method. This averaged seed curve extension (A-SCE) method mitigates, to some extent, the impact of randomness in initial point selection on optical system performance. In this paper, we construct initial structures for focal and afocal off-axis four-mirror freeform optical systems utilizing the A-SCE method, demonstrating a relatively good capability for designing initial structures for freeform surface off-axis reflective systems.

2. Methods

Here is a concise explanation of the SCE method, as illustrated in Figure 1. The light emitted from the aperture surface (S) is reflected by the freeform mirror (M) and converges at the image point (T). We initiate our process by selecting m × n uniform feature points on the aperture surface (S). From these feature points, we generate m × n parallel feature rays. Given both the aperture surface (S) and image point (T), it is crucial to identify an initial point M₁₁ (x₁₁, y₁₁, z₁₁), which represents the intersection of the first characteristic ray r₁₁ originating from the aperture surface (S) with the freeform mirror (M). At this intersection, $\overrightarrow{r_{11}}$ serves as the incident vector while $\overrightarrow{M_{11}T}$ acts as the exit vector at point M₁₁. By applying Snell’s law, we can directly compute $\overrightarrow{N_{11}}$, which denotes the normal vector of the tangent plane at point M₁₁.

$$\overrightarrow{N_{11}}={\displaystyle \frac{n_2·\overrightarrow{M_{11}T}-n_1·\overrightarrow{r_{11}}}{\left|n_2·\overrightarrow{M_{11}T}-n_1·\overrightarrow{r_{11}}\right|}}$$

(1)

where n₁ and n₂ denote the refractive indices of the incident object space and the outgoing image space, respectively. In scenarios involving a single mirror positioned in the air, both n₁ and n₂ are equal to 1. The equation that governs the tangent plane is expressed as follows:

$$\overrightarrow{N_{11}}·\left(\left(x-x_{11}\right),\left(y-y_{11}\right),\left(z-z_{11}\right)\right)=0$$

(2)

The intersection point M₁₂ between the second characteristic ray $\overrightarrow{r_{12}}$ and the tangent plane at point M₁₁ is designated as the second characteristic data point on the freeform mirror (M). The incident vector r₁₂ and the exit vector $\overrightarrow{M_{12}T}$ at point M₁₂ can be readily calculated. Similarly, by applying Snell’s law, one can derive both the normal vector $\overrightarrow{N_{12}}$ of the second data point M₁₂, along with the equation of the tangent plane at this location. The third characteristic ray $\overrightarrow{r_{13}}$ intersects with the tangent plane at point M₁₂ yielding a new intersection point denoted as M₁₃. By analogy, one can obtain intersection points denoted as M_1j (j = 1, 2, …, n) between the first row of characteristic rays and the surface; these points serve as a foundation for constructing the seed curve of the freeform mirror (M).

Afterwards, the intersection points between the feature ray $\overrightarrow{r_{21}}$ and the tangent plane at point M₁₁ are utilized to derive point M₂₁. Similarly, the intersection point between feature ray $\overrightarrow{r_{22}}$ and the tangent plane at point M₁₂ is employed to obtain point M₂₂. In this manner, all data points M_2j (j = 1, 2, …, n) in the second row can be generated; this constitutes the process of expanding the seed curve. Using an analogous calculation method, data points in the third row can be derived from those in the second row, while data points in the mth row can similarly be obtained from their corresponding (m−1)th row counterparts. Ultimately, all data points of the freeform surface (M) can be systematically acquired by expanding upon the initial seed curve.

It is evident that when applying the SCE method to obtain data points on freeform surfaces, both the position of the aperture surface (S) and the image points (T) are predetermined. However, there exists a range of options for selecting the initial point M₁₁ (x₁₁, y₁₁, z₁₁), which introduces uncertainty into the design outcomes of freeform surfaces. For identical application requirements—specifically, when utilizing the same aperture surface (S) and corresponding image points (T)—the freeform mirrors generated from two distinct initial points M₁₁ and M′₁₁ yield different results. As illustrated in Figure 2a, the data points for these two configurations are represented by red circles and blue circles, respectively. It is clear that these data points do not completely overlap. Moreover, Figure 2b distinctly highlights the differences in z-coordinates between these two sets of data points.

The SCE method utilizes an approximate solution technique to identify sampling points on a freeform surface. The data points on the freeform surface are obtained by intersecting the incident ray with the tangent plane of adjacent points. However, given that the actual optical surface is continuous and smooth, it cannot be accurately represented as a collection of numerous small tangent planes assembled together. Consequently, this results in a certain degree of error between the calculated points and the ideal points on the freeform surface, with variations in error occurring at different locations. As the seed curve expands, these errors tend to accumulate and increase in magnitude. For example, when selecting M_1n as the initial point illustrated in Figure 1, it becomes evident that the resulting data points for the freeform surface do not perfectly align with those from previous cases; similarly, there is also an absence of complete congruence between the fitted freeform surfaces derived from both scenarios. Therefore, modifications to the SCE method are necessary to mitigate errors arising from initial point selection to some extent.

In this paper, we employed the A-SCE method to address the inherent limitations of the SCE method. As illustrated in Figure 3, we initiated our approach by selecting an initial point, denoted as M₁₁, and applying the SCE method to generate a corresponding set of data points referred to as the first set (M₁₁-M_m1-M_mn-M_1n). The expansion direction and sequence are indicated by red arrows and red circled numbers, respectively. Subsequently, we utilized M_m1 as the new initial point M′₁₁ for generating a second set of data points. Again, employing the SCE method allowed us to obtain this second dataset (M′₁₁-M′_m1-M′_mn-M′_1n), with its expansion direction and sequence represented by green arrows and green circled numbers. Next, using M′_m1 as the initial point M″₁₁ for generating a third set of data points, we continued to apply the SCE method to derive this third dataset (M″₁₁-M″m₁-M″m_n-M″1_n). The corresponding expansion direction and sequence are depicted through blue arrows and blue circled numbers. Following that, we took M″m₁ as the starting point M‴1₁ for generating a fourth set of data points. By once again utilizing the SCE method, we obtained this fourth dataset (M‴1₁-M‴m₁-M‴m_n-M‴1_n), with its respective expansion direction and sequence illustrated through purple arrows and purple circled numbers. Finally, all four sets of data points were aggregated, from which their average was calculated to form the dataset (D), utilized for generating the freeform surface.

$$D: {\left\{(x_i,y_i,{\displaystyle \frac{1}{4}}{\sum }_{j=1}^4{z}_{ij})\right\}}_i$$

(3)

Subsequently, we employed the least squares method to approximate the freeform surface. This study utilized widely adopted XY polynomials for fitting purposes. The application of XY polynomials has proven effective in correcting asymmetric aberrations and aligns well with expressions used in numerical control (CNC) machining. The expression for an 8th order XY polynomial that represents the sag of the freeform surface is presented as follows:

$$z\left(x,y\right)={\displaystyle \frac{c{r}^2}{1+\sqrt{1-\left(1+k\right)c^2{r}^2}}}+{\sum }_{i=1}^{24}{C}_i{\left({\displaystyle \frac{x}{R_0}}\right)}^m{\left({\displaystyle \frac{y}{R_0}}\right)}^n$$

(4)

where c represents the vertex curvature, k denotes the conic constant, r indicates the radial coordinates, and R₀ is the normalization radius. Additionally, C_i refers to the coefficient of the polynomial ${\left({\displaystyle \frac{x}{R_0}}\right)}^m{\left({\displaystyle \frac{y}{R_0}}\right)}^n$ (m ≥ 0, n ≥ 0, and 1 ≤ m + n ≤ 8). Given that the off-axis four-mirror optical systems discussed in this study exhibit symmetry about the YOZ plane, it follows that the coefficients of odd terms in ${\left({\displaystyle \frac{x}{R_0}}\right)}^m$ are zero. Consequently, this leads to a total of 24 higher-order terms.

The surface fitting process for the data points was performed using MATLAB software. The configuration yielding the smallest root mean square error was selected as the initial profile. Subsequently, the vertex curvature and coefficients of these polynomials were further optimized utilizing commercial ZEMAX software. Throughout the optimization process, the conic coefficient remained zero, indicating that the base surface is a spherical surface. Generally, an optical system typically requires a certain FOV to be effective. It is worth mentioning that the A-SCE method presented in this work was employed solely for constructing the initial structure of an off-axis four-mirror freeform optical system tailored to the center FOV. This initial structure was subsequently imported into ZEMAX software for further optimization. During the optimization process, the FOV was gradually expanded through iterative adjustments [29], ultimately leading to the acquisition of the desired optical system.

3. Results and Discussions

In this section, we present the design of both focal and afocal off-axis four-mirror freeform optical systems utilizing the A-SCE method. The initial structure of the off-axis four-mirror freeform imaging system was established using a methodology analogous to that employed for constructing the initial configuration of a single mirror, as illustrated in Figure 1. By defining the (virtual) image points, the four-mirror system was decomposed into four individual single-mirror systems. Subsequently, data point construction for each of the four freeform mirrors ws executed according to the A-SCE method, as depicted in Figure 4. The detailed design process is elaborated upon below.

I. By utilizing the aperture surface (S), the initial point (M₁₁) on the first mirror (M), and the virtual image point (T₁), all relevant data points pertaining to the first mirror (M) were determined using the A-SCE method.

II. The emergent beam from the first mirror (M) acts as the incident beam onto the second mirror (N). By knowing both the initial point (N₁₁) on the second mirror and its corresponding virtual image point (T₂), all data points for this second mirror were derived through the application of the A-SCE method.

III. In a similar manner, considering that the emergent beam from the second mirror (N) serves as an incident beam onto a third mirror (P), we employed known parameters, including an initial point on this third mirror designated as (P₁₁) and its corresponding image point (T₃). As a result, all relevant data points for this third mirror can also be derived using the A-SCE methodology.

IV. Finally, considering that the emergent beam from the third mirror (P) is directed towards a fourth mirror designated as Q, and given our established knowledge of an initial point on this fourth mirror referred to as Q₁₁ along with its corresponding image point denoted by T₄, it follows that all relevant data points for this fourth mirror could also be acquired through the application of A-SCE method.

Afterwards, the XY polynomial was utilized to fit the data points of each freeform mirror. These fitted data points were subsequently input into ZEMAX software as the initial structure of the optical system for further optimization. The geometric spot diameter on the image plane served as the merit function during this process. During optimization, unobstructed constraints were established to ensure that the solid stop, reflector, and image plane did not obstruct light or interfere with one another. The specification of these unobstructed constraint conditions is represented by L₁~L₈ in Figure 4. It is crucial to control these parameters to eliminate light obscuration and prevent surface interference; it is essential that these constraints are neither excessively large—leading to an oversized optical system—nor too small, which could complicate engineering implementation.

The system parameters of the off-axis four-mirror freeform imaging system are detailed in

Table 1. The system parameters of the off-axis four-mirror freeform imaging system.

Parameters	Mid-Infrared Band
FOV	1.6° × 0.6°
Entrance pupil diameter	180 mm
F number	9.33
Wavelength	0.45 μm~0.9 μm
System volume	230 mm × 340 mm × 220 mm

. The field of view (FOV), entrance pupil diameter, and wavelength range are specified as 1.6° × 0.6°, 180 mm, and 0.45 μm~0.9 μm, respectively. Due to the small FOV, the system stop is located prior to the first mirror. The F-number is established at 9.33, enabling a focal length of up to 1680 mm.

The wavefront error was utilized as the criterion for performance evaluation. Figure 5a,b illustrate the distribution of wavefront error for the initial structure obtained using the SCE method and A-SCE method, respectively. In Figure 5a, the peak-to-valley (PV) value is measured at 5.33 λ, while the root mean square (RMS) value is recorded at 1.37 λ; in contrast, Figure 5b presents a PV value of 1.34 λ and an RMS value of 0.37 λ for the A-SCE method. Thus, it can be concluded that the performance of A-SCE surpasses that of SCE. Following optimization based on structures designed by the A-SCE method as initial configurations, we achieve a wavefront PV value of 0.098 λ and an RMS value of 0.022 λ for the optimized system, as depicted in Figure 5c. The light path diagram after optimization is shown in Figure 6. The overall dimensions of the system measure 230 mm × 340 mm × 220 mm.

The aperture sizes of the four freeform mirrors in the off-axis four-mirror freeform imaging system are Φ226 mm, Φ63 mm, 42 mm × 41 mm, and 87 mm × 58 mm, respectively. The surface sag and the residual surface sag after removing the best-fit sphere are illustrated in Figure 7a–h, respectively. The maximum values of surface departure from the best-fit sphere are recorded as 338 µm, 167 µm, 188 µm, and 114 µm for each mirror.

Figure 8 presents the modulation transfer function (MTF) curves for all FOVs, which approach the diffraction limit. Given that the off-axis four-mirror freeform imaging system lacks rotational symmetry, the distribution of aberrations across the entire FOV will also demonstrate a lack of rotational symmetry. Consequently, it is imperative to analyze the imaging quality throughout the full FOV in order to accurately assess performance. The distribution of spot radii for all FOVs is illustrated in Figure 9, with a maximum value reaching 1.84 μm, thereby achieving the diffraction limit. Additionally, Figure 10 provides a distortion grid; here, the maximum relative distortion is recorded at 1%, which can be further mitigated through image calibration.

The initial structure of the off-axis four-mirror freeform afocal system was constructed using a method analogous to that employed in developing the initial structure of a single mirror, as illustrated in Figure 1. By defining (virtual) image points, the four-mirror system was effectively decomposed into four individual single-mirror systems. Subsequently, data point construction for the four freeform mirrors was conducted according to the A-SCE method, as depicted in Figure 11. The detailed design process is outlined below.

I. By utilizing the aperture surface (S), the initial point (M₁₁) on the first mirror (M), and the virtual image point (T₁), all data points associated with the first mirror (M) were determined through the A-SCE method.

II. The emergent beam from the first mirror (M) acts as the incident beam for the second mirror (N). By utilizing the known initial point (N₁₁) on the second mirror and its corresponding virtual image point (T₂), all necessary data points for this second mirror were obtained through the application of the A-SCE method.

III. In a similar fashion, considering that the emergent beam from the second mirror (N) serves as an incident beam onto a third mirror (P), we identified both an initial point on this third mirror, referred to as P₁₁, and its corresponding image point (T₃). Therefore, all data points for this third mirror were obtained through the application of the A-SCE method.

IV. Finally, considering that the emergent beam from the third mirror (P) is directed towards a fourth mirror (Q), and given the known parameters, including an initial point on this fourth mirror designated as Q₁₁ and an exit unit vector represented by $\overrightarrow{V}$, it can be concluded that all data points related to this fourth mirror could also be derived utilizing the A-SCE methodology.

It is essential to recognize that, owing to the afocal system, there exists no definitive image point. Consequently, it was imperative to provide the unit vector of the exit light from the fourth mirror in order to compute all relevant data points. In a similar manner, the configuration of unobstructed constraint conditions is denoted as L₁~L₈ in Figure 11.

The system parameters of the off-axis four-mirror freeform afocal system are presented in

Table 2. The system parameters of the off-axis four-mirror freeform afocal system.

Parameters	Mid-Infrared Band
FOV	3.6° × 2°
Entrance pupil diameter	120 mm
Magnification	0.25
Wavelength	0.4 μm~1 μm
System volume	230 mm × 340 mm × 220 mm

. The FOV, entrance pupil diameter, magnification, and wavelength are specified as 3.6° × 2°, 120 mm, 0.25, and 0.4 μm~1 μm, respectively. To ensure optimal pupil matching with subsequent optical paths, the system stop was strategically positioned at the end of the optical assembly.

The wavefront error was utilized as the performance evaluation criterion. Figure 12a,b illustrates the distribution of the wavefront error for the initial structure obtained using the SCE method and the A-SCE method, respectively. In Figure 12a, the PV value is measured at 2.32 λ, while the RMS value is recorded at 0.62 λ; conversely, Figure 12b reveals a PV value of 0.68 λ and an RMS value of 0.21 λ for the A-SCE method. This observation indicates that the optical performance of systems constructed using the A-SCE method significantly surpasses that achieved with the SCE method, establishing it as a favorable starting point for subsequent optimization efforts while reducing optimization complexity. Following optimization based on structures designed by the A-SCE method as initial configurations, we achieve a wavefront PV value of 0.139 λ and an RMS value of 0.03 λ for the optimized system, as depicted in Figure 12c. The light path diagram after optimization is presented in Figure 13a. The overall dimensions of the system are measured 170 mm × 340 mm × 250 mm.

The aperture sizes of the four freeform mirrors in the off-axis four-mirror freeform afocal system are as follows: Φ161 mm, 62 mm × 52 mm, 56 mm × 41 mm, and 107 mm × 78 mm. The surface sag and the residual surface sag after subtracting the best-fit sphere are depicted in Figure 14a–h. The maximum values of surface deviation from the best-fit sphere are recorded as 83 µm, 36 µm, 51 µm, and 107 µm, respectively.

Since the off-axis four-mirror freeform afocal system lacks rotational symmetry, the aberration distribution across the entire FOV will similarly reflect this asymmetry. Therefore, it was crucial to analyze imaging quality throughout the full FOV to accurately evaluate performance. The wavefront error distribution for all FOVs is depicted in Figure 15a, with a maximum value approaching approximately 1/15 λ, which approaches the diffraction limit. The distortion grid is presented in Figure 16, where the maximum relative distortion measures 0.85%. This distortion can be further minimized through image calibration.

A paraxial lens with a focal length of 75 mm was added at the location of the system stop, specifically at the exit pupil, as illustrated in Figure 13b. The MTF curves for all FOVs are presented in Figure 17, demonstrating an approach to the diffraction limit. The distribution of spot radii across all FOVs is depicted in Figure 15b, where the maximum value reaches 0.59 μm, coinciding with the diffraction limit. These findings indicate that the off-axis four-mirror freeform afocal system exhibits optimal optical performance.

For freeform surface fitting, a different number of datasets was selected each time, allowing us to observe the relationship between the wavefront error of the initial structure of the constructed off-axis four-mirror freeform optical system and the number of datasets used in fitting the freeform surface. There are twelve different methods for selecting datasets, including four cases of choosing one dataset at a time, six cases of selecting two datasets simultaneously, four cases of opting for three sets concurrently, and one instance involving the selection of all four sets together. Here, the selection of a single dataset corresponds to the SCE method, while selecting four datasets pertains to the A-SCE method employed above. Figure 18 presents the fitting results of the off-axis four-mirror freeform imaging and afocal systems. It is evident that the wavefront error of the initial system, derived from employing an equal number of diverse datasets for freeform surface fitting, does not exhibit significant variation. Furthermore, as the quantity of datasets utilized in fitting freeform surfaces gradually increases, a corresponding decrease in wavefront error for the initial system is observed; however, this rate of reduction progressively diminishes. This phenomenon may be attributed to limitations in the accuracy inherent in the fitting process between discrete data points and XY polynomials within the dataset. Consequently, one can infer that while increasing numbers of datasets involved in freeform surface fitting using the A-SCE method improve the accuracy of the optical system’s initial structure, this enhancement will eventually reach a point of diminishing returns with further increments in dataset size. Therefore, an indiscriminate pursuit of expanding dataset quantities within A-SCE methodology not only has limited efficacy in enhancing fit quality but also escalates design workload. Thus, the selection of the number of datasets necessitates a balance between precision and efficiency.

Single-point diamond turning (SPDT) technology has been employed for the machining of freeform mirror surfaces [30]. However, various factors—including cutting residual errors in the feed direction, linear interpolation errors in the cutting direction, cutter vibration, and tool wear—inevitably lead to certain inaccuracies in surface machining [31]. Therefore, the analysis of machining tolerances becomes crucial. Surface tolerance is defined as the deviation between the actual surface shape achieved after processing and the ideal surface shape specified during design; this discrepancy can negatively impact the performance of optical systems. Consequently, a comprehensive analysis of machining tolerances for freeform mirrors is essential. While ZEMAX software is capable of performing surface tolerance analyses on traditional spherical and aspherical surfaces, it lacks functionality for assessing surface tolerances specific to freeform geometries. To address this limitation, a random statistical method was utilized to analyze the surface tolerance of freeform mirrors [32]. The sag height of a freeform surface influenced by machining errors can be expressed as follows:

$$z\left(x,y\right)=f\left(x,y\right)+∆\left(x,y\right)$$

(5)

where $f\left(x,y\right)$ represents the ideal sag height, while $∆\left(x,y\right)$ indicates the machining error:

$$∆\left(x,y\right)=K·h_m$$

(6)

where K represents a random number within the range of [−1, 1], and h_m denotes the maximum surface shape error.

To evaluate the impact of machining errors on the optical performance of freeform surfaces, the previously mentioned equations were utilized to generate data points for freeform surfaces that incorporate random errors. Subsequently, a new freeform surface with these machining errors was refitted and imported into the ZEMAX software to assess the system’s optical performance. In this study, all freeform mirrors within the off-axis four-mirror configuration were uniformly assigned an identical value of h_m. By varying hm, we observed changes in the RMS wavefront error for both focal and afocal optical systems. The results are depicted in Figure 19, where each data point represents an average derived from 20 sets of experimental outcomes. It is evident that when the PV value 2·h_m of the machining error on the freeform mirror does not exceed 100 nm (~1/7 λ, λ = 632.8 nm), the optical system exhibits satisfactory performance. This level of machining accuracy can typically be achieved using the SPDT method.

4. Conclusions

In conclusion, the A-SCE method has been introduced as a means to directly construct freeform surfaces in off-axis reflective imaging systems. By averaging multiple sets of data points, the A-SCE method can partially alleviate errors that arise from the random selection of initial points commonly observed in traditional SCE methods. This approach was employed to develop both focal and afocal off-axis four-mirror freeform optical systems. The initial structures generated using the A-SCE method were subsequently optimized using ZEMAX software. The results indicate that, while achieving machining accuracy, both the focal and afocal off-axis four-mirror freeform optical systems are capable of meeting diffraction limitations concerning performance. By comparing the performance of the SCE method and the A-SCE method in constructing off-axis four-mirror freeform imaging and afocal systems, it is evident that the initial structure of the optical system developed using the A-SCE method exhibits lower wavefront aberration and greater construction precision. This phenomenon can be attributed to the rectification of random errors following the application of multiple averaging processes. Therefore, the A-SCE method is deemed a superior construction approach compared with the traditional SCE method, which can provide a significant reference for the design of the initial structure of freeform off-axis reflection optical systems.