Efficient Testing Light Path for Aspherical Surfaces Based on Secondary Imaging

Abstract

At present, off-axis three-mirror optical systems mostly adopt aspherical mirrors with small apertures and small F/# to meet the development requirements of remote sensing payloads towards high precision, small volume, and lightweight design. However, current references rarely provide the derivation, design, and detection of the testing light path for aspherical mirrors with small apertures and small F/#. Aiming at the existing gap, this paper proposes a method of decomposing the compensation optical path into two imaging light paths and derives the initial structure of the compensation optical path. Furthermore, specific solutions are proposed from two aspects: the design of the null compensator and the establishment of the testing light path. Finally, the compensation optical path design and detection are carried out for the primary mirror and the tertiary mirror of the self-calibrating real entrance pupil imaging spectrometer, guiding the completion of the system processing, assembly, and adjustment. The detection results show that the RMS of the surface shapes of the primary mirror and the tertiary mirror is 1/40λ (λ = 633 nm). This derivation method and the design method of the initial optical path have the characteristics of simple calculation, rapid optimization, and universal applicability, and are applicable to the detection of all quadratic concave surfaces.

1. Introduction

The off-axis three-mirror anastigmatism (TMA) optical system has the characteristics of being achromatic, small in size, and compact in structure. Moreover, through the aspheric design of the mirrors, it can achieve features such as a large field of view, high resolution, and excellent image quality. Therefore, it is widely applied in the design of remote sensing payloads and lithography machine systems [1,2,3,4,5,6,7]. Examples include China’s Gaofen-7 [2], Ziyuan-3 [3], the United States’ EO-1 [4], WorldView-3 [5], the European Space Agency’s High-Resolution Imaging Spectrometer (HRIS) [6], and France’s Pleiades-HR [7], all of which employ off-axis TMA optical systems. Currently, off-axis TMA systems predominantly utilize small-aperture, small-F/# aspheric mirrors to meet the development demands of remote sensing payloads for high precision, compact size, and lightweight design. However, the testing accuracy of high-precision aspheric mirrors directly determines manufacturing precision, which in turn affects the imaging quality of the entire system [8]. Therefore, research on testing light paths for small-F/# and small-aperture aspheric surfaces is of significant importance for future aerial remote sensing payloads.

Current aspheric surface measurement techniques can be primarily categorized into scanning measurement methods and optical interferometry methods [9,10,11,12,13,14,15,16,17]. Scanning measurement methods are further divided into contact and non-contact measurements [9,10]. Contact measurement requires a probe to physically touch the mirror surface, which can cause scratches and is thus limited to the initial mirror grinding stage [9]. Non-contact probe profilometry can measure both concave and convex aspheric surfaces, with measurement accuracy at the sub-micron level. However, its detection accuracy is limited, and its efficiency is relatively low [10]. Optical interferometry methods include null and non-null testing. For aspheric surface testing, commonly used null interferometry methods include the aplanatic point method, null compensation, and computer-generated holography (CGH) [11,12,13,14,15,16,17]. The aplanatic point method uses spherical or planar mirrors combined with an interferometer for autocollimation testing. However, this method cannot detect the central region of the mirror under test [11,12]. The null compensation method requires the design of a null compensator to construct a testing light path. The null compensator generally employs small-aperture spherical lenses. Due to the mature processing technology and high precision of spherical lenses, this method is widely used in aspheric surface shape detection [13,14]. The CGH method can achieve the measurement of deeply aspheric surfaces. The manufacturing cost of the CGH element is relatively high, and both the processing precision and the positioning precision have a certain impact on the measurement accuracy. The development of this method in aspheric surface shape detection is restricted [15]. Non-null testing methods use a single compensator to measure aspheric surfaces with varying parameters, allowing for retrace errors between the modulated test wavefront and the aspheric surface. However, non-null methods inherently suffer from unavoidable errors, and different retrace error handling approaches can yield inconsistent results [16,17]. Considering current optical fabrication and assembly capabilities, null compensation stands out for its low manufacturing difficulty, cost-effectiveness, and ability to achieve high-precision aspheric surface measurements.

The fundamental design principle of null testing involves creating a null compensator to counteract the aspheric deviation, ensuring the test wavefront perfectly matches the theoretical surface profile of the aspheric mirror under test. The detection wavefront modulated by the compensator reaches the surface to be measured and is reflected. The detection light beam, after reflection from the surface to be measured, carries the surface shape information of the surface to be measured and interferes with the reference light beam when they meet, so that the surface error data of the surface to be measured can be obtained. The accuracy of the null compensator directly determines the processing accuracy of the mirror surface to be measured. Therefore, designing the null compensator and constructing the testing light path are the key to testing the surface shape of the aspheric surface [13,14,18].

Most of the current references are comprehensive introductions to aspheric surface detection technologies [13,14]. There are a few derivations and designs of the initial optical paths of null compensators. According to the investigation, the current method for deriving the initial light path of a null compensator is based on the traditional third-order aberration theory. To solve the initial structure of the compensation system, it is necessary to calculate the initial parameters according to the asphericity, normal angle, and normal aberration of the aspheric surface to be measured. Moreover, most of the initial parameters are obtained based on empirical values, and the entire calculation process is rather cumbersome and complex, with mutual nesting. Finally, when optimizing with the optical design software, it is still necessary to try values by continuously changing the radii of curvature and the central distances of the compensating lens and the field lens on the basis of the initial structure, and the process is relatively cumbersome [19]. The secondary imaging method only requires knowing the parameters of the measured mirror and does not require empirical values, and the derivation process is relatively simple. The parameters that need to be known for deriving the initial optical paths of the two methods are shown in

Table 1. Comparison of the initial parameters of the two methods.

	D0 The Aperture of the Test Mirror	R0 The Radius of Test Mirror	n Compensator Lens Refractive Index	e02 Conic Coefficients of Test Mirror	Obscuration Ratio		Magnification		Aberration Sharing Factor
					α1 Compensating Lens	α2 Field Lens	β1 Compensating Lens	β2 Field Lens	t1 Compensating Lens	t2 Field Lens
Third-order aberration	√	√	√	√	√	√	√	√	√	√
Secondary imaging	√	√	√	×	×	×	×	×	×	×

. As can be seen from Table 1, the three-order aberration method requires more known parameters than the secondary imaging method, and the derivation process is more cumbersome.

To address the current limitations in the optical design of compensators for null testing optical paths, this study proposes specific solutions for achieving high-precision testing and deterministic manufacturing of small-F/#, small-aperture secondary aspheric surfaces. The approach focuses on two key aspects: the design of the null compensator and the testing optical path. The compensation optical path is decomposed into two imaging optical paths, and the initial structure of the testing system is derived using Newtonian and Gaussian optical formulas. This method is applied to design and test the compensation optical path for the off-axis aspheric surfaces of the primary and tertiary mirrors in a self-calibrating real entrance pupil imaging spectrometer (SCREPIS), guiding the system’s manufacturing and alignment [20]. The new method offers simplicity in calculation, rapid optimization, and universal applicability, making it suitable for testing all secondary concave surfaces.

2. Design Principle of the Testing Light Path

Within the range of primary aberrations, compared with a spherical surface whose curvature is the vertex curvature of the aspheric surface, the angles of intersection and the positions of the intersection points between the normals of each zone on the aspheric surface and the optical axis are different. That is, normal aberrations are generated, which is equivalent to a beam of light with spherical aberration existing on the optical axis. The null compensator makes this beam of light with spherical aberration become a perfectly convergent beam, which eliminates the spherical aberration [13,14]. The key to the null compensation method lies in making the interferometer, the null compensator, and the mirrors to be tested form a compensating self-return optical path, so as to complete the inspection of the mirror to be tested. Therefore, an important principle should be followed in the design process of the null compensator: the normal at any point on the curved surface of the mirror to be measured is perpendicular to the tangent plane at that point [13]. If a ray of light is incident on the mirror to be measured along the normal, the ray will return along the original path. The schematic diagram of the design principle of the testing light path is shown in Figure 1.

When designing the null compensation optical path, the entire testing light path is decomposed into two imaging optical paths. In the first imaging optical path, the mirror to be measured is imaged on the compensating lens through the field lens, as shown in Figure 2a. In the second imaging optical path, the center of curvature B of the mirror to be measured is imaged at the focal point A of the interferometer through the combined system consisting of the field lens and the compensating lens, as shown in Figure 2b. By combining the object–image relationship with the Gaussian formula and the Newton formula [21,22], the initial structure of the detection system can be solved.

First, the D₂ is determined based on the D₀, as shown in Formula (1). Since the field lens is typically positioned near point B, the L₁ can be determined, which is given by (2). Finally, based on the first imaging optical path, the L₁′ can be obtained, as shown in Formula (3). According to Formulas (1) to (3), the f₁ can be obtained, as shown in Formula (4).

$$D_2=m_1\cdot D_1$$

(1)

$$L_1=\left(1+m_2\right)\cdot R_0$$

(2)

$$L_1^{\prime }={\displaystyle \frac{D_2\times L_1}{D_0}}=m_1{R}_0\left(1+m_3\right)$$

(3)

$${\displaystyle \frac{1}{m_1{R}_0\left(1+m_3\right)}}+{\displaystyle \frac{1}{R_0\left(1+m_3\right)}}={\displaystyle \frac{1}{f_1}}\Rightarrow f_1={\displaystyle \frac{m_1{R}_0\left(1+m_3\right)}{1+m_1}}$$

(4)

where m₁ and m₂ are proportionality coefficients, which are selected according to the reflector under test. D₀ is the aperture of the mirror under test. D₂ is the aperture of the compensating lens. R₀ is the radius of curvature of the mirror under test. L₁ is the distance between the field lens and the test mirror. L₁′ is the distance between the compensation lens and the field lens. β₁ is the transverse magnification of the field lens. f₁ is the object-side focal length of the field lens.

The field lens is a single lens composed of two spherical surfaces [17]. According to the imaging properties of a single lens, as shown in Formula (5), one side of the field lens is designed to be a plane, so R₂ is taken as infinity. The R₁ can be obtained, given by (6).

$$-{\displaystyle \frac{1}{f_1}}=\left(n_1-1\right)\cdot \left({\displaystyle \frac{1}{R_1}}-{\displaystyle \frac{1}{R_2}}\right)+{\displaystyle \frac{{\left(n_1-1\right)}^2{d}_1}{n_1{R}_1{R}_2}}$$

(5)

$$R_1=\left(1-n_1\right)f_1$$

(6)

where n₁ represents the refractive index of the field lens, R₁ is the radius of the left refractive spherical surface of the field lens, and R₂ is the radius of the right refractive spherical surface of the field lens. d₁ is the center thickness of the field lens.

In the second imaging optical path, the beam divergence angle θ typically ranges from 5° to about 16°, and the L₂′ is given by Formula (7). The f₂ can be derived using the Gaussian formula for imaging position relationships, as shown in Formula (8). An ideal optical system can be formed by combining the compensating mirror and the field lens. Using the ideal optical combination system formula, the f₃ can be calculated, as shown in Formula (9). The compensating lens is a single lens. According to Formula (5), the expression (10) of the spherical radius R₃ can be obtained.

$$L_2^{\prime }={\displaystyle \frac{D_2}{m_3}}$$

(7)

$${\displaystyle \frac{m_2}{m_1{D}_0}}+{\displaystyle \frac{1}{\left(m_1+m_1{m}_3+m_3\right)R_0}}={\displaystyle \frac{1}{f_2}}\Rightarrow f_2={\displaystyle \frac{m_1{D}_0\left(m_1+m_1{m}_3+m_3\right)R_0}{m_2\left(m_1+m_1{m}_3+m_3\right)R_0+m_1{D}_0}}$$

(8)

$$f_2={\displaystyle \frac{f_1{f}_3}{\Delta }}\Rightarrow f_3={\displaystyle \frac{m_1{D}_0\left(m_1+m_1{m}_3+m_3\right)R_0\left(1+m_1\right)}{m_2\left(m_1+m_1{m}_3+m_3\right)R_0+m_1{D}_0}}$$

(9)

$${\displaystyle \frac{1}{f_3}}=\left(n_2-1\right)\left({\displaystyle \frac{1}{R_3}}-{\displaystyle \frac{1}{R_4}}\right)+{\displaystyle \frac{{\left(n_2-1\right)}^2{d}_2}{n_2{R}_3{R}_4}}\Rightarrow R_3=\left(n_2-1\right)f_3$$

(10)

where m₃ are proportionality coefficients, which are selected according to the θ. L₂′ is the distance from the compensating lens to A. f₂ is the object-space focal length of the null compensator composed of the compensating lens and the field lens. f₃ is the object-space focal length of the compensating lens. In the second imaging optical path, ∆ is approximately equal to the distance L₁′. n₂ represents the refractive index of the compensating lens. R₃ is the radius of the left refractive spherical surface of the compensating lens, and R₄ is the radius of the right refractive spherical surface of the compensating lens. d₂ is the center thickness of the compensating lens.

3. Design of the Testing Light Path

3.1. Compensator Design

Design and detection of the compensation optical paths for the off-axis aspheric surfaces of the primary mirror and the tertiary mirror of the self-calibrating real entrance pupil imaging spectrometer (SCREPIS) was conducted. The parameters of the off-axis aspheric surfaces of the primary mirror and the tertiary mirror are shown in

Table 2. Structure parameters of primary and tertiary mirrors.

Title	Radius of Curvature/mm	Diameter/mm	Off-Axis Amount/mm	The Aperture of the Off-Axis Mirror/mm	Conic
Primary mirror	−106.736	58	13	32	−2.394
Tertiary mirror	−69.432	42	6	30	−1.294

[20].

Generally, the same lens material is selected for the field lens and the compensating lens, so the refractive index n₁ of the lens is equal to n₂. The above-mentioned principles and processes of initial light path derivation are programmed into software, and the interface is shown in Figure 3. By inputting the diameters and radii of curvature of the primary mirror and the tertiary mirror, as well as the refractive indices of the materials of the field lens and the compensating lens used, the initial compensating light path for detecting the surface shape of the primary mirror can be obtained.

Optimization is carried out based on the initial light path obtained above. Taking the primary mirror as an example, the programming operands for the entire light path are set as shown in

Table 3. Programming operands for optimizing the testing light path.

	Op#	Type	Surf1	Surf2	Target		Weight	Value
Control the length of the optical path	1	TTHI	A	Compensating lens	OPGT	35	1	129.445
					OPLT	200	1
	2	TTHI	Compensating lens	Field lens	OPGT	20	1	56.349
					OPLT	100	1
	3	TTHI	Field lens	Primary mirror	OPGT	50	1	129.296
					OPLT	135	1
Control the system aberration	4		Field		Freq	Target	Weight	Value
	5	GMTS	1		40	1	1	0.851
	6	GMTT	1		40	1	1	0.851
	7		Term		Field	Target	Weight	Value
	8	ZERN	Astigmatism X		1	0	1	7.27 × 10−18
	9	ZERN	Astigmatism Y		1	0	1	−1.59 × 10−19
	10	ZERN	Coma X		1	0	1	6.23 × 10−19
	11	ZERN	Coma Y		1	0	1	5.85 × 10−18
	12	ZERN	Primary Spherical		1	0	1	−2.23 × 10−7
	13	ZERN	Trefoil X		1	0	1	−1.80 × 10−19
	14	ZERN	Trefoil Y		1	0	1	−9.25 × 10−18

. In the optimization process, the wavefront is selected as the optimization function. Smaller ring and arm parameters are set for hammer-shaped optimization. When the wavefront parameters of the system remain unchanged, larger ring and arm parameters are set to continue the optimization. The aim is to eliminate the residual aberrations at different positions on the mirror surface to be measured.

3.2. Design Results and Tolerance Analysis

According to the light path design and optimization method in Section 3.1, the null compensating testing light paths for the primary mirror and the tertiary mirror are shown in Figure 4a,b, and the parameters of the testing light paths are shown in

Table 4. Parameters of primary and tertiary testing light paths.

	D1/mm	D2/mm	L1/mm	L1′/mm	L2/mm	L2′/mm	R1/mm	R2/mm	R3/mm	R4/mm	d1/mm	d2/mm	Mirror Material
Primary mirror	16	40	129.296	56.349	78.909	129.445	28.152	∞	31.603	∞	5.623	14.705	H-K9L
Tertiary mirror	20	30	78.193	37.858	46.619	177.192	19.482	∞	24.857	∞	7.794	9.813	H-K9L

. After the aberration compensation and correction by the compensating lens and the field lens in the testing light path of the primary mirror, the wavefront RMS of the entire optical path system reaches 0.0007λ (λ = 633 nm), as shown in the wavefront diagram of the testing light path system in Figure 4c. After the aberration compensation and correction by the compensating lens and the field lens in the testing light path of the tertiary mirror, the wavefront RMS of the entire optical path system reaches 0.0006λ (λ = 633 nm), as shown in the wavefront diagram of the testing light path system in Figure 4d. The design results show that the null compensator designed by this method can correct the aberration of the mirror to be measured effectively.

Considering the machining and alignment errors of the mirrors in the detection system, a tolerance analysis is carried out on the detection optical path. According to the current machining and alignment levels, the tolerance parameters are set for each mirror [23,24,25].

A total of 200 Monte Carlo analyses are carried out for the primary mirror and the tertiary mirror to simulate the situation of the system’s testing light path caused by the machining and alignment errors of the compensating lens and the field lens. The worst deviations introduced by the machining and alignment of the compensating lens and the field lens in this testing light path are shown in

Table 5. The worst deviation of primary and tertiary mirrors testing light paths.

	Type	Suf	Value	Criterion	Change
Primary mirror	TEDX	R3 of compensating lens	±5 × 10−3	0.8709	−0.0169
	TEDY
	TEDX	R1 of field lens	±5 × 10−3	0.8748	−0.0129
	TEDY
	TIND	Compensating lens	5 × 10−3	0.8736	−0.0141
			−5 × 10−3	0.8739	−0.0138
Tertiary mirror	TEDX	R3 of compensating lens	±5 × 10−3	0.7810	−0.0247
	TEDY
	TEDX	R1 of field lens	±5 × 10−3	0.7887	−0.0171
	TEDY
	TIND	Compensating lens	5 × 10−3	0.7912	−0.0146
	TRAD	R3 of compensating lens	−5 × 10−3	0.7929	−0.0129

. The analysis is conducted with the Modulation Transfer Function (MTF) as the standard, and the results are shown in

Table 6. Tolerance analysis results of primary and tertiary mirrors testing light path.

	Primary Mirror	Tertiary Mirror
Nominal MTF	0.8878 (40 lp/mm)	0.8058 (40 lp/mm)
Simulation result	10% > 0.8833	10% > 0.7991
	50% > 0.8714	50% > 0.7870
	90% > 0.8451	90% > 0.7564

.

Based on the tolerance analysis results, for the testing light paths of the primary mirror and the tertiary mirror, the sensitive tolerances are mainly reflected in the eccentricity tolerance of the side with a radius of curvature of the compensating lens and the field lens, as well as the refractive index tolerance of the compensating lens. In addition, the detection optical path of the tertiary mirror is also quite sensitive to the tolerance introduced by the radius of curvature of the compensating lens. Since the relative position accuracy between the compensating lens and the field lens in the alignment instrument’s alignment is relatively high, the tolerance analysis reveals that the system is not sensitive to the eccentricity and tilt of the compensating lens and the field lens. After 200 Monte Carlo calculations and analyses, the maximum deviations of the MTF of the primary mirror and the tertiary mirror caused by machining and alignment errors are 4.81% and 6.13%, respectively. The impact on the entire testing light path can be neglected, and it does not affect the detection results of the primary mirror and the tertiary mirror. The machined compensating lens and field lens are placed into a mechanical device, aligned, and assembled to form a null compensator, as shown in Figure 5.

4. Light Path Construction and Detection

As can be seen from Table 1, the structural parameters of the primary and tertiary mirrors indicate that the object to be detected is an off-axis aspherical mirror, which means only a part of the entire surface to be measured is taken. The design principles of the null compensator for off-axis aspherical mirrors parallel those of a non-off-axis mirror. The key difference lies in the testing light path setup: during the construction of the testing light path, it is crucial to align the center of the off-axis mirror blank, rather than the geometric center of the off-axis mirror itself, with the central axis of the optical path, as illustrated in Figure 6.

The machined compensating lens and field lens are combined into a null compensator through a mechanical device. The testing light path is constructed using the null compensator, the off-axis mirror to be measured, and the interferometer to conduct surface profile detection on the primary and tertiary mirrors of the self-calibrated real entrance pupil imaging spectrometer. The testing light paths for the surface profiles of the primary and the tertiary mirrors are shown in Figure 7a and Figure 8a, respectively. The wavefront diagrams of the mirrors to be measured, Figure 7b and Figure 8b, show that the RMS of the surface profiles of the primary and the tertiary mirrors are 1/40λ (λ = 633 nm). The system MTF (Nyquist frequency) diagrams, Figure 7c and Figure 8c, show that the testing light path systems of primary and tertiary mirrors reach the diffraction limit. The system wavefront diagram and the system MTF indicate that the surface profiles of the mirrors to be measured are of good quality and meet the usage requirements of the entire system [20].

5. Conclusions

In view of the current deficiencies in the design of the testing light path for small-F/# and small-aperture aspherical mirrors, this paper proposes a new initial light path design method. The null compensation optical path is decomposed into two imaging optical paths, based on which the initial structure of the detection system is derived. This method is used to design the testing light path for the off-axis aspheres in the SCREPIS, guiding the completion of system processing and alignment. The results show that the RMS of the surface profiles of the primary and the tertiary mirrors are better than 1/40λ (λ = 633 nm). This derivation method and the design method of the initial light path have the characteristics of simple calculation, rapid optimization, and universal applicability, and are applicable to the detection of all quadratic concave surfaces.