Correction of Wavefront Distortion in Common Aperture Optical Systems Based on Freeform Lens

Abstract

The common aperture optical system enhances light utilization efficiency during the imaging process by utilizing a single shared aperture. This approach not only facilitates independent synchronous multi-band imaging across various applications but also reduces the complexity, size, and cost of optical systems. However, conventional common aperture optical systems typically employ inclined plates or prisms for spectral splitting, which can introduce wavefront distortion in the transmission light path, an issue that is particularly problematic in imaging systems with a large field of view. In this work, we propose employing a freeform lens to correct wavefront distortion arising from imperfections within an optical system. We present a design methodology for the freeform lens based on ray tracing techniques. The application of this freeform lens effectively mitigates wavefront distortion in an infrared dual-band composite detection system, resulting in commendable optical performance across both mid-infrared and far-infrared bands.

1. Introduction

With the advancement of sophisticated information acquisition technologies, many optoelectronic detection systems are increasingly employing composite detection methods [1,2], enabling simultaneous imaging or detection of multi-band information from targets. Furthermore, extracting spectral information from these targets allows for a more accurate and comprehensive understanding of their characteristics. The optimal strategy for achieving compact size, light weight, low power consumption, and cost-effectiveness in optoelectronic imaging or detection systems is through the utilization of multi-band common aperture optical systems. Such systems facilitate multi-band imaging or detection by sharing all or part of the optical aperture [3].

Currently, common aperture optical systems have found widespread application across various fields including confocal microscopic imaging [4,5,6], multi-band composite detection [7,8,9], and Raman spectroscopy [10,11,12]. A fundamental characteristic of these systems is that light from different wavelength bands passes through a shared front objective lens before being separated and ultimately imaged on distinct detectors via separate correction optical paths. To achieve this separation between different light bands, inclined plates or prisms are typically employed as beam splitters within common aperture optical configurations. However, it is crucial to note that wavefront distortion can occur after light traverses such beam splitters [13]. The degree of asymmetrical aberration increases with greater deviation of the incident light from the normal surface orientation of the beam splitter. To address this issue, several approaches are currently available. Generally, a complex rear optical system is often used to correct for wavefront distortions arising from the use of beam splitters. For instance, one or more aspheric lenses are usually incorporated for aberration correction in optical systems with large FOV [14,15,16]. However, this will lead to the components in the optical path becoming complex, which is not conducive to ensuring processing and assembly accuracy in engineering. In addition, reducing the thickness or tilt angle of the beam splitter in the optical path is also conducive to minimizing its impact on the wavefront [17]. Nevertheless, reducing the thickness of the beam splitter will diminish its mechanical strength, thereby making it arduous to guarantee the surface shape accuracy and mechanical performance. And the tilt angle of the beam splitter is also restricted by the optical–mechanical structure, and usually the adjustable angle is not large enough. Typically, the adjustable magnitude is not significant, and the application effect is restricted. Therefore, introducing a single optical element capable of effectively regulating the wavefront into the optical path is beneficial for enhancing the performance of the common aperture optical system.

Compared to traditional rotationally symmetric surfaces, freeform surfaces present numerous advantages regarding optical performance, size, and weight. Freeform surfaces are typically defined as optical surfaces that do not possess axial rotational symmetry or translational symmetry properties [18]. The parametric representations of freeform surfaces are varied [19,20,21], including XY polynomials, Zernike polynomials, Forbes polynomials, Legendre polynomials, Chebyshev polynomials, Bernstein polynomials, Gaussian functions, NURBS surfaces, among others. Due to their enhanced flexibility in surface shape design, freeform surfaces can be characterized by a greater number of parameters. This increased parameterization offers more freedom in the optical design processes. They demonstrate strong capabilities for describing surface shape and effectively correcting aberrations. These attributes render freeform surfaces particularly well-suited for correcting aberrations within optical systems—especially those that exhibit asymmetry (such as astigmatism induced by inclined plates)—while simultaneously minimizing the quantity and mass of optical components required within the system. The introduction and implementation of freeform surfaces signify a revolutionary advancement in the field of optical design [22].

In this work, a freeform lens was employed to correct wavefront distortion through the regulation of light. By tracing incident rays that are uniformly distributed across the pupil, data points on the freeform surface can be calculated point by point. Subsequently, the initial shape of the freeform surface was derived by fitting these data points according to a specific polynomial function. This preliminary structure can then be imported into optical design software (Ansys Zemax OpticStudio^® 2024 R1.00) for further optimization. The proposed method was validated using an infrared dual-band composite detection system, which included an inclined plate positioned after the front objective optics. The mid-infrared (3 μm~5 μm) reflected light path can be designed based on conventional rotationally symmetric optical elements. In contrast, in the far-infrared (8 μm~12 μm) transmission light path, asymmetric aberrations introduced by the inclined plate were corrected utilizing a freeform lens. Imaging results for both mid-infrared and far-infrared bands approached the diffraction limit with a field of view (FOV) of 5° × 5° and an F-number (F/#) of 1, indicating that effective detection performance was achieved with the assistance of a freeform lens. The incorporation of freeform optical elements can significantly reduce the wavefront aberration in common aperture optical systems with large FOV, and optical performance can approach the diffraction limit. Furthermore, asymmetric aberrations can also be substantially mitigated. The tolerance analysis results showed that the optical system has good tolerance characteristics after adding a freeform lens.

2. Methods

In the following paragraphs, we propose a method for the direct construction of a freeform surface designed to correct wavefront distortion. As illustrated in Figure 1, incident light traverses the front optical path, resulting in a distorted wavefront characterized by asymmetric aberrations. To rectify this distorted wavefront, a freeform lens was integrated into the optical path, facilitating the achievement of an optimal image on the final image plane.

The method primarily comprises six steps: (i) Selecting points on the entrance pupil for ray tracing. (ii) Establishing the initial configuration of the light path. (iii) Constructing a seed curve to generate a freeform surface. (iv) Expanding the seed curves to encompass all data points on the freeform surface. (v) Fitting the data points using an XY polynomial. (vi) Utilizing the FOV expansion method to complete optimization.

(i) Acquisition of points on the entrance pupil for ray tracing. To obtain data points on the freeform surface, a ray tracing method can be utilized for direct computation. A sufficient number of points were sampled from the entrance pupil, with each point serving as the origin of a tracing ray directed along the central FOV. Sampling was conducted along m concentric rings and n radial arms within the pupil. As illustrated in Figure 2, these rings and arms were evenly distributed both radially and angularly, respectively. The entrance pupil was segmented into n planes S_i (i = 1, 2,..., n) in the angular dimension. Consequently, data points on the freeform surface were also organized into n planes corresponding to this angular division.

(ii) Establishing the initial configuration of the light path. To elucidate the methodology for constructing a freeform surface, we analyzed the ray tracing model of all rays on the first slice (S₁) of the entrance pupil. As illustrated in Figure 3, the left side depicts incident light exhibiting wavefront distortion after passing through the front optical system. The rays denoted as $\overline{r_{11}}$, $\overline{r_{12}}$, …, $\overline{r_{1m}}$ originate from this front optical system and can be traced from the entrance pupil to the freeform surface. The region between M₁₁ and M₁₂ represents a freeform lens with a refractive index n, where M₁₁ is characterized as a freeform surface and M₁₂ as a planar surface. The intersections of $\overline{r_{11}}$, $\overline{r_{12}}$, …, $\overline{r_{1m}}$ with M₁₁ yield points P₁₁, P₁₂, …, P_1m, respectively. Similarly, for refracted rays intersecting with M₁₂, we obtain intersection points Q₁₁, Q₁₂,…, Q_1m. Point T’ is defined as the intersection of extended lines connecting pairs $\overline{P_{11}{Q}_{11}}$, $\overline{P_{12}{Q}_{12}}$, …, $\overline{P_{1m}{Q}_{1m}}$. This point serves both as a virtual image point following its passage through M₁ and a virtual object point prior to reaching surface M₂. In an ideal scenario devoid of aberrations, point T’ should ideally coincide at one singular location. Point T represents the final image position derived from T’ after being imaged by surface M₂.

(iii) Constructing a seed curve for generating freeform surfaces. Utilizing the known characteristic rays ($\overline{r_{11}}$, $\overline{r_{12}}$, …, $\overline{r_{1m}}$) on surface S₁ and the image point T, all data points (P₁₁, P₁₂, …, P_1m) on the freeform surface M₁₁ can be computed using Snell’s law. Initially, we assume that the back focal length of the optical system is denoted as L, which corresponds to the image distance of the final plane M₂. Consequently, according to Gaussian optics principles, the object distance L’ for this final plane can be derived, as illustrated in Figure 4.

$$L^{\prime }=nL$$

(1)

where n is the refractive index of the freeform lens. Secondly, it is crucial to define the initial point P₁₁ (x₁₁, y₁₁, z₁₁) on M₁, which signifies the intersection of the first characteristic ray $\overline{r_{11}}$ and the line $\overline{P_{11}{Q}_{11}{T}^{\prime }}$. In accordance with Snell’s law, the normal vector $\overline{N_{11}}$ of the tangent plane at point P₁₁ can be computed directly.

$$\overline{N_{11}}={\displaystyle \frac{n_2·\overline{P_{11}{T}^{\prime }}-n_1·\overline{r_{11}}}{\left|n_2·\overline{P_{11}{T}^{\prime }}-n_1·\overline{r_{11}}\right|}}$$

(2)

Here, n₁ = 1 and n₂ = n denote the refractive indices of the incident object space and the outgoing image space, respectively. The equation for the tangent plane is expressed as follows:

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

(3)

Then, the intersection point P₁₂ between the second characteristic ray $\overline{r_{12}}$ and the tangent plane at point P₁₁ is identified as the second characteristic data point on the freeform surface M₁. Consequently, the exit vector $\overline{P_{12}{T}^{\prime }}$ at point P₁₂ was determined. Similarly, the normal vector $\overline{N_{12}}$ can be computed from both $\overline{r_{12}}$ and $\overline{P_{12}{T}^{\prime }}$, utilizing Equation (2). The third characteristic ray $\overline{r_{13}}$ intersects with the tangent plane at point P₁₂ enabling the calculation of point P₁₃, again employing Equation (2). By analogy, all intersection points denoted as P_1j (j = 1, 2,..., m) on the freeform surface M₁ can be obtained to form a seed curve.

(iv) Expanding the seed curves to encompass all data points on the freeform surface. After acquiring the seed curve, we intersect the tangent plane at each point (P₁₁, P₁₂, …, P_1m) along the seed curve with the corresponding incident vectors ($\overline{r_{21}}$, $\overline{r_{22}}$, …, $\overline{r_{2m}}$), which are directed towards the second slice (S₂). This process generates data points (P₂₁, P₂₂, …, P_2m) on M₁. Similarly, we determine the tangent planes at each of these points (P₂₁, P₂₂, …, P_2m) on M₁ using Equations (2) and (3). Consequently, all data points on the freeform surface (M₁) can be computed in accordance with the direction indicated by the blue arrow in Figure 1; these datasets will subsequently be employed for fitting the freeform surface.

(v) Fitting the data points to an XY polynomial. In this study, we employ the least-squares method to fit the freeform surface and utilize XY polynomials for this purpose. The application of XY polynomials proves particularly effective in correcting asymmetric aberrations and aligns seamlessly with expressions commonly used in CNC machining. The expression for the sag of the freeform surface using 8th-order XY polynomials 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 signifies the radial coordinates, and R₀ is the normalization radius. Additionally, C_i refers to the coefficients of the polynomial term ${\left({\displaystyle \frac{x}{R_0}}\right)}^m{\left({\displaystyle \frac{y}{R_0}}\right)}^n$, where m ≥ 0, n ≥ 0, and 1 ≤ m + n ≤ 8. Given that the common aperture optical systems examined in this study exhibit symmetry about the YOZ plane, it follows that all coefficients corresponding to odd powers of ${\left({\displaystyle \frac{x}{R_0}}\right)}^m$ are zero. Consequently, there exists a total of 24 higher-order terms. The surface fitting process for data points is conducted using MATLAB R2024a software. The result yielding the smallest root mean square error was selected as the initial profile; this serves as an effective starting point for subsequent optimization of the system. During this optimization phase, it is important to note that the conic coefficient remains fixed at zero, indicating that the base surface is spherical. Furthermore, both vertex curvature and polynomial coefficients can be further optimized utilizing commercial ZEMAX software (Ansys Zemax OpticStudio^® 2024 R1.00).

(vi) Utilizing the FOV expansion method for optimization. The fitting result obtained from the central FOV serves as the initial configuration of the optical system, which is subsequently optimized through an iterative process that involves gradually expanding the FOV. This FOV expansion strategy partitions the entire field into X and Y directions, allowing for selecting appropriate step sizes for each direction during expansion [23]. The choice of expansion step size plays a crucial role in determining the number of iterations required for effective optimization. A step size that is too small may lead to an excessive number of FOV expansions, thereby diminishing the design efficiency. Conversely, if the step size is excessively large, it can complicate the optimization process and impede achieving optimal image quality.

3. Results and Discussions

Since the beam splitter functions solely as a plane mirror within the reflection optical path, it does not introduce any aberrations along this pathway. The entire reflective optical path is devoid of non-rotationally symmetric optical elements aligned with the direction of the optical axis. Consequently, it will not generate asymmetric wavefront distortions and can achieve a relatively ideal spherical wavefront through optimization. Conversely, in the transmission optical path, the beam splitter lacks rotational symmetry about the optical axis. As light traverses this tilted plate, it induces asymmetric wavefront distortions that are challenging to correct effectively using rotationally symmetric components such as spherical or aspheric lenses. Therefore, freeform lenses possessing non-rotationally symmetric characteristics can leverage their multiple degrees of freedom to rectify these asymmetric wavefront distortions present in the transmission optical path.

In order to verify the effectiveness of utilizing freeform lenses for correcting asymmetric aberrations, an infrared dual-band composite detection system has been designed, as schematically illustrated in Figure 5. The incident light first passes through the front objective lens and reaches the inclined plate for splitting. The mid-infrared band is reflected and directed into the reflection optical path, resulting in a perfect wavefront that exhibits no asymmetric distortion; this wavefront is ultimately focused onto the corresponding image plane. Meanwhile, the far-infrared band traverses through the inclined plate into the transmission optical path. The resulting distorted wavefront is subsequently corrected using a freeform lens before being focused onto its respective image plane.

The system parameters of the infrared dual-band composite detection system are summarized in

Table 1. The parameters of the infrared dual-band composite detection system.

Parameters	Mid-Infrared Band	Far-Infrared Band
Wavelength	3 μm~5 μm	8 μm~12 μm
FOV	5° × 5°
Entrance pupil diameter	60 mm
F-number	1.8	1
Pixel size	12 μm × 12 μm	24 μm × 24 μm
System volume	70 mm × 110 mm × 295 mm

. The wavelength ranges for the mid-infrared band and the far-infrared band are 3 μm~5 μm and 8 μm~12 μm, respectively. Both bands feature an FOV of 5° × 5° and an entrance pupil diameter of 60 mm. The optimization process for the infrared dual-band composite detection system is delineated as follows: (i) Initially, the far-infrared band with full FOVs is optimized, reflecting off the front surface of an inclined plate without subsequent wavefront correction. At this stage, the optical parameters for both the front objective optics and the reflection optical path are established. (ii) Subsequently, the transmission optical path for the mid-infrared band with central FOV is directly optimized while neglecting any influence from tilted flat plates, thereby maintaining fixed optical parameters for the front objective optics. (iii) Following this step, after incorporating the inclined plate, asymmetric wavefront distortion is corrected using a freeform lens according to methodologies outlined in previous sections. (iv) Building upon these initial conditions, rapid acquisition of the transmission optical path can be achieved through ZEMAX software optimization. Thereafter, the FOV of this transmission optical path is gradually expanded to finalize design aspects for the entire system. The light path diagram post-optimization is illustrated in Figure 6. Overall, the layout remains relatively compact with dimensions measuring approximately 70 mm × 110 mm × 295 mm.

The front objective optics of the common aperture optical system consists of four lenses (L₁~L₄). The front surfaces of these lenses are designed as an eighth-order even asphere, while their rear surfaces are spherical. The mathematical expression for the sag of the eighth-order even asphere is presented below:

$$z\left(x,y\right)={\displaystyle \frac{c{r}^2}{1+\sqrt{1-\left(1+k\right)c^2{r}^2}}}+{\sum }_{i=1}^4{\alpha }_i{r}^{2i}$$

(5)

where c represents the vertex curvature, k denotes the conic constant, r indicates the radial coordinates, and ${\alpha }_i$ refers to the coefficients of $r^{2i}$. In Equation (5), the coefficients ${\alpha }_1$ and k exert a similar influence on the surface shape. Consequently, only the conic constant and high-order coefficients ${\alpha }_2$~${\alpha }_4$ were utilized to characterize the surface shape of the 8th-order even asphere. The materials employed for lenses L₁~L₄ were silicon (Si), germanium (Ge), Si, and Si, respectively. The system stop was positioned within the front objective lens, located very close in front of L₃; this configuration aids in minimizing the size of the detection system. An inclined plate (P₁), made of Si, was placed behind the front objective optics at an angle of 45°. The transmission optical path comprises five lenses (L₅ ~ L₉, Ge, Si, Si, Ge, Ge). Lens L₅~L₈ share an identical shape with that of lenses L₁ ~L₄. Lens L₉ is designed as a freeform lens featuring a freeform front surface and a flat back surface, as illustrated in Figure 3. The reflection optical path consists of one plane reflector (M₁) and five lens (L₁₀~L₁₄, Si, Ge, Si, Si, Ge), which also possess shapes consistent with those of lenses L₁~L₄. Both bands utilize a cooled infrared detector matched with a cold stop to enhance the suppression of stray radiation. The optimal parameters for the freeform lens (L₉) are detailed in

Table 2. The optimal parameters for the freeform lens L₉.

Parameters	Values	Parameters	Values
c	−5.086 × 10−3	x0y5	−3.518 × 10−8
d (center thickness)	3	x6y0	−3.039 × 10−8
D (diameter)	46	x4y2	−6.213 × 10−8
k	0	x2y4	−4.823 × 10−8
x0y1	−9.204 × 10−3	x0y6	−6.863 × 10−9
x2y0	1.417 × 10−3	x6y1	−2.005 × 10−10
x0y2	1.812 × 10−3	x4y3	1.969 × 10−10
x2y1	3.436 × 10−6	x2y5	8.417 × 10−11
x0y3	5.406 × 10−6	x0y7	1.208 × 10−10
x4y0	6.718 × 10−6	x8y0	1.162 × 10−10
x2y2	1.157 × 10−5	x6y2	2.64 × 10−10
x0y4	4.847 × 10−6	x4y4	2.76 × 10−10
x4y1	1.836 × 10−8	x2y6	1.712 × 10−10
x2y3	−4.468 × 10−8	x0y8	−9.71 × 10−12

.

The aperture size of the freeform lens L₉ is Φ46 mm. The surface sag and the residual surface sag after removing the best-fit sphere are depicted in Figure 7. The maximum values of surface deviation from the best-fit sphere reach 119 µm. Single-point diamond turning (SPDT) technology can be employed to fabricate the freeform surface [24]. Typically, SPDT can achieve a depth cut of 2 μm when utilizing fast tool servo (FTS). Consequently, the spherical Ge lens is initially processed using traditional polishing methods, followed by machining of the freeform surface via SPDT, thereby enabling rapid manufacturing of the freeform lens. The conventional machining accuracy of freeform surfaces processed by SPDT can reach 1/35λ@632.8 nm.

To ensure the machining accuracy of freeform lenses, a computer-generated hologram (CGH)-based interferometric null test method is commonly employed for high-precision metrology of freeform surfaces [25]. The CGH generates a complex wavefront through diffraction to compensate for the wavefront introduced by a freeform mirror. As illustrated in Figure 8a, the CGH consists of three regions: the main region, the alignment region, and the fiducial region. The metrological principle underlying the CGH is depicted in Figure 8b, where alignment is achieved using an autocollimation method. A 632.8 nm laser interferometer is commonly utilized for testing optical paths. The spherical wavefront produced by the interferometer is retroreflected by the alignment CGH. Subsequently, this retroreflected wavefront interferes with the reference wavefront generated by the interferometer. By analyzing the resulting interferogram, we can determine whether there are any issues such as tilt, defocus, or decentering. If the CGH is positioned correctly within its nominal parameters, a null fringe will be observed, indicating that proper alignment has been achieved between the CGH and the interferometer. The fiducial CGH serves to project specific spots onto predetermined locations on the freeform surface to facilitate adjustments in its position. The methodology employed for calculating the phase of the CGH through ray tracing is illustrated in Figure 8b. Based on the direction of light rays, phase distributions across these three regions can be determined using ray tracing techniques [26]. The phase distribution corresponding to the main CGH is illustrated in Figure 8c as an integer multiple of 2π. Figure 8d depicts the binary fringe pattern for an amplitude CGH element, with one line representing 30 fringes in reality. The minimum linewidth of these fringes measures 5.7 µm and can be fabricated utilizing a direct-laser-writing machine. Figure 8e presents the optical path for testing with CGH within ZEMAX software, emphasizing the relative positional relationships among the interferometer, CGH, and freeform lens. Furthermore, as demonstrated in Figure 8f, the residual wave aberration measured by this CGH for freeform lens L₉ does not exceed PV ~1/204 λ, where λ denotes a wavelength of 632.8 nm. This suggests that when the processing of the CGH and the alignment of the testing optical path achieve an optimal state, the wavefront error solely resulting from the CGH design will not exceed this value. Nevertheless, in actuality, errors (e.g., CGH substrate’s profile, thickness, and refractive index; stripe shape and distortion) are inevitable during the manufacturing process of the CGH, and there are also errors in the relative positions of the components in the testing optical path. Consequently, in practice, this scheme can typically achieve a detection accuracy of PV ~1/12λ value for the freeform surface shape.

The modulation transfer function (MTF) plots for the final infrared dual-band composite detection system at various FOVs are presented in Figure 9. The MTF for the mid-infrared band exceeds 0.48 at a spatial frequency of 41.67 lp/mm, which corresponds to its Nyquist frequency. In contrast, the MTF for the far-infrared band surpasses 0.48 at a spatial frequency of 20.83 lp/mm, also aligning with its Nyquist frequency. It is evident that the performance of the mid-infrared band approaches the diffraction limit, while that of the far-infrared band remains sufficiently robust. As illustrated in Figure 10, the maximum root mean square (RMS) radius for all spot diagrams across different FOVs is measured at 2.952 μm for the mid-infrared band and 10.381 μm for the far-infrared band; neither value exceeds their corresponding Airy radii of 10.62 μm and 14.78 μm, respectively. Furthermore, distortion grids are provided in Figure 11, where it can be observed that the maximum relative distortions for both mid-infrared and far-infrared bands do not exceed −0.4% and −0.1%, respectively. These results demonstrate that freeform lenses can effectively correct asymmetric aberrations in optical systems.

Consequently, following the wavefront compensation achieved through a freeform lens, the far-infrared band has demonstrated commendable imaging quality, effectively meeting general detection criteria (with spot sizes on the image plane not exceeding 2 × 2 pixels). In order to facilitate an intuitive comparison of the contribution made by the freeform lens, we extracted it from the optical path and re-optimized the far-infrared band while keeping the size of the optical path basically unchanged. The wavefront distributions for the full FOVs, both with and without the freeform lens, are presented in Figure 12a,b, respectively. It is evident that the introduction of the freeform corrector significantly reduced the wavefront error of the far-infrared band, both in terms of the central FOV and the edge FOV. Furthermore, the spot diagram at various FOVs for the optical path without the freeform lens is illustrated in Figure 12c. By comparing with Figure 10b, it can be observed that the spot diagram of the optical system lacking freeform surface correction exhibits pronounced astigmatism characteristics. In contrast, the astigmatism features present in the spot diagram of the corrected optical system are significantly diminished. As a result, the implementation of freeform lens effectively mitigates wavefront distortions induced by inclined beam splitters in common aperture optical systems. This advancement facilitates the achievement of superior detection and imaging performance. In addition to common aperture optical systems, the incorporation of freeform elements—such as lenses or mirrors—for the correction of wavefront distortion is also applicable to a wider range of fields, especially asymmetric aberrations arising from an extensive FOV. However, the addition of freeform lens will also cause some adverse effects on the entire common aperture optical system, mainly a reduction in the far-infrared band’s transmittance. Hence, it is crucial for the freeform lens to be coated with an anti-reflection film designed for wavelengths ranging from 8 μm to 12 μm, akin to that of other lenses (L₅~L₈) in the far-infrared band. Typically, such an anti-reflection coating can ensure an average transmittance of approximately 98% for a single surface of a lens. When incorporating the freeform corrector, however, there is a reduction in transmittance for the far-infrared band by a factor of (98%)² ≈ 0.96. This decrease has minimal impact on overall performance.

A tolerance analysis is conducted to assess the feasibility of the practical machining and assembly processes for the optical system. The impact of minor fabrication and installation errors on the imaging quality of the system was thoroughly examined. The evaluation criterion is the RMS spot radius both for the mid-infrared and the far-infrared band. Compensation is achieved through the utilization of rear intercept of [−1 mm, 1 mm], image plane tilt of [−1′, 1′], and image plane descent of [−0.02 mm, 0.02 mm]. Based on empirical evidence and the current level of process technology, it is advisable to initially establish relatively lenient tolerance preset values for each parameter. Subsequently, a thorough tolerance analysis should be conducted on the design outcomes to identify particularly sensitive tolerances, followed by a redistribution of these tolerances. The tolerance allocations for the mid-infrared and the far-infrared band are presented in

Table 3. The tolerance allocations for the mid-infrared band.

Tolerance Type		L1	L2	L3	L4	P1	L10	M1	L11	L12	L13	L14
Machining tolerance	Radius (fringe)	4	4	4	4	3	4	3	4	4	4	4
	Thickness (mm)	0.02	0.02	0.02	0.02	-	0.025	-	0.025	0.025	0.025	0.025
	Decenter X/Y (mm)	0.015	0.015	0.015	0.015	-	0.02	-	0.02	0.02	0.02	0.02
	Tilt X/Y (′)	1	1	1	1	-	1	-	1	1	1	1
	Surface irregularity (λ)	1/30	1/30	1/30	1/30	1/35	1/30	1/35	1/30	1/30	1/30	1/30
Assembly tolerance	Decenter X/Y (mm)	0.015	0.015	0.015	0.015	0.015	0.02	0.015	0.02	0.02	0.02	0.02
	Tilt X/Y (′)	1	1	1	1	1	1	1	1	1	1	1
Materials tolerance	Index	0.0005	0.0005	0.0005	0.0005	-	0.0005	-	0.0005	0.0005	0.0005	0.0005
	Abbe (%)	0.3	0.3	0.3	0.3	-	0.3	-	0.3	0.3	0.3	0.3

and

Table 4. The tolerance allocations for the far-infrared band.

Tolerance Type		L1	L2	L3	L4	P1	L5	L6	L7	L8	L9
Machining tolerance	Radius (fringe)	4	4	4	4	3	4	4	4	4	3
	Thickness (mm)	0.02	0.02	0.02	0.02	0.025	0.025	0.025	0.025	0.025	0.02
	Decenter X/Y (mm)	0.015	0.015	0.015	0.015	0.02	0.02	0.02	0.02	0.02	0.015
	Tilt X/Y (′)	1	1	1	1	0.5	1	1	1	1	0.5
	Surface irregularity (λ)	1/30	1/30	1/30	1/30	1/35	1/30	1/35	1/30	1/30	1/30
Assembly tolerance	Decenter X/Y (mm)	0.015	0.015	0.015	0.015	0.015	0.02	0.02	0.02	0.02	0.015
	Tilt X/Y (′)	1	1	1	1	1	1	1	1	1	0.5
Materials tolerance	Index	0.0005	0.0005	0.0005	0.0005	0.0005	0.0005	0.0005	0.0005	0.0005	0.0005
	Abbe (%)	0.3	0.3	0.3	0.3	0.3	0.3	0.3	0.3	0.3	0.3

, respectively. It can be observed that the results of the tolerance allocations are not particularly stringent. Although the tolerance requirements for freeform lens are more rigorous than those for other optical components, it is still possible to achieve satisfactory outcomes by relying on conventional accuracy.

Five hundred Monte Carlo sensitivity analyses are conducted for the mid-infrared and the far-infrared band, respectively. The results of the tolerance analysis concerning the RMS spot radius (average value across the full FOVs) are presented in

Table 5. RMS spot radius tolerance analysis.

Probability	RMS Spot Radius (μm)
	Mid-Infrared Band	Far-Infrared Band
90%	6.912	20.045
80%	5.836	18.434
50%	4.195	15.193
20%	3.373	12.679
10%	3.071	11.963

. The probability that the average diameter of the energy dispersion spot in the mid-infrared band is less than 13.824 μm across the full FOVs exceeds 90%. Similarly, for the long-wave infrared band, there is also a greater than 90% probability that the average diameter of the energy dispersion spot remains below 40.09 μm within the full FOVs. Notably, both cases do not exceed 2 × 2 pixels, thereby fulfilling conventional detection requirements. Consequently, after adding a freeform lens, the optical system still has good tolerance characteristics and can be realized in engineering.

4. Conclusions

In conclusion, a freeform lens can effectively correct asymmetric wavefront distortion in common aperture optical systems. The data points for the freeform surface are computed point by point through ray tracing, enabling the derivation of initial surface shape of the freeform lens via data fitting for subsequent optimization. This work presents a wavefront correction method that employs a freeform lens within an infrared dual-band composite detection system, which incorporates an inclined plate positioned after the front objective optics. The initial light path is established, and the initial parameters of the freeform lens—expressed as 8th-order XY polynomials—are obtained through ray tracing and least-squares fitting. Subsequently, this initial structure is imported into ZEMAX software for further optimization. Following the integration of a freeform lens into the transmission light path, both reflection in the mid-infrared band and transmission in the far-infrared band exhibited favorable detection performance. The integration of freeform optical elements has the potential to significantly diminish wavefront aberrations in common aperture optical systems featuring a large FOV. Consequently, the optical performance can approach the diffraction limit. Furthermore, asymmetric aberrations can also be effectively mitigated. The results of the tolerance analysis indicate that the optical system exhibits favorable tolerance characteristics following the incorporation of the freeform lens, and this can be effectively implemented in engineering practices. These findings highlight that freeform lenses are exceptionally effective tools for correcting asymmetric aberrations in a variety of optical systems.