## Abstract

A null lens moving back and forth relative to a point source can generate variable spherical aberration for flexible test of aspheres. Different from the previous methods, variable spherical aberration null theory was developed by us to optimize the null lens. The optimized null was a plano-convex singlet containing a high order even asphere. Its attractive advantages are the simple structure and the broad range of testable surfaces. Most concave prolate conic and near conic surfaces with *k∙R* value varying between 0 and about 70000mm and with smaller relative aperture than that determined by each *k∙R* value can be tested. The testable asphericity range is between 0 and about 230λ. Relations among these testable surfaces were revealed as different groups of equidistant surfaces. To explicitly show the ability of the null, the measurable surfaces range map that contains all parameters defining a conic surface was offered. A practical near-null test system using this null was established. Alignment, near-null data processing, and error sources are analyzed in detail. To verify the broad testable surfaces range, three surfaces with widely varying amounts of asphericity were tested. Cross tests were provided to verify the test system accuracy.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Aspherical optical elements have been widely used in modern optical systems. To fabricate these elements with high accuracy, surface figure metrology is crucial. Among various test techniques, interferometry is preferred for its higher accuracy, and less measurement time compared with coordinate measurement [1], deflectometry [2,3], etc. However, interferometry has very limited dynamic range. A null corrector is usually required. Because each asphere requires a unique null, the flexibility is limited. Developing a flexible interferometric test method for aspheres is meaningful to enhance the test efficiency and reduce the cost. Many publications are focused on this issue.

Longer wavelength [4], two-wavelength [5], shear interferometry [6] and Sub-Nyquist interferometers [7] can improve flexibility to some extent but with loss of accuracy and sensitivity. Tilt-Wave-Interferometry can also improve the dynamic range [8]. However, it is currently a little complicated and difficult to calibrate. Stitching interferometry [9–12] can also realize flexible test for aspheric surfaces by dividing the full aperture into subapertures which are within the dynamic range of interferometers. However, it is time consuming and it requires delicate algorithms to estimate and compensate the uncertainties of the motion. Utilizing a variable null (VN) capable of generating variable aberrations is a prevailing way to realize flexible test. The VN can be used to compensate the total aberration of the test surface to realize flexible null test. Alternatively, the aberrations of the test surfaces are partially corrected by the VN, it is also referred to as near-null test [13], known from the null test and non-null test. Chen *et al.* [13,14] and QED Technologies [15] proposed to generate variable coma and astigmatism by utilizing a pair of computer generated holograms and optical wedges, respectively. However, these VNs are used to compensate variable aberrations of off-axis subapertures for near-null stitching test of aspheres. Another kind of booming VNs is adaptive optics (AO) elements or a combination of AO elements with traditional null. Pruss and Tiziani [16], He *et al.* [17], Huang *et al.* [18], and Zhang *et al.* [19] investigated using deformation mirrors (DM) alone as VN for aspheres or free-forms with moderate asphericity. Fuerschbach *et al.* [20], and Zhang *et al.* [21] proposed to utilize DM together with a partial null to test relatively steep free-forms. Cao *et al.* [22], Kacperski *et al*. [23], Ares *et al.* [24] Cashmore *et al*. [25], and Xue *et al.* [26] utilized the spatial light modulator as VN. These techniques are promising for flexible test of free-forms. However, the commercially available AO elements have limited accuracy and modulation range/strokes. Extending the testable range and improving the test accuracy of these methods are still under researched. The partial compensation null is a more sophisticated technique for flexible test of aspheres. Outstanding works were done [27–30]. However, it utilizes the null in a collimated beam. The flexibility is achieved by only moving the test surface back and forth relative to the null. The testable aspherical surfaces are a group of equidistant surfaces in essence. Hence, the testable aspherical surfaces range is a little narrow.

By moving the null back and forth relative to the point source and varying the distance from the null to the test surface, a much wider range of aspherical surfaces are testable from the center of curvature. This idea is not new. Hilbert and Rimmer [31] reported this thought nearly half century ago. The null they used was a combination of two/three aspheric phase plates. Shafer [32] done the similar work by utilizing a zoom lens composed of three lenses. Illuminating work was done by them. However, they did not explain in theory why the test system can generate variable spherical aberration. The relations among the testable surfaces were also not expounded. Hence, a concise measurable surfaces range map was not offered. Moreover, their null was not optimized for obtaining a broader testable aspherical surfaces range. The tolerance control of their null was also difficult for the complex null structure. Due to the poorly developed non-null test technique, a practical test system was not established by them yet.

Based on the same idea, this paper developed the variable spherical aberration null theory for aspheres to troubleshoot the above problems. Utilizing this theory, a null was optimized to be a plano-convex singlet containing a high order even asphere (HOEA). Different from the above phase plates combinations or zoom lens, the null has a simple structure as a singlet. Moreover, it can generate a much broader range of variable spherical aberration. Concave prolate conics such as paraboloids, hyperboloids, ellipsoids and near conics with widely varied asphericity, aperture and shape parameters can be tested. By investigating relations of these testable surfaces according to the theory, the testable surfaces were regarded as different groups of equidistant surfaces. Furthermore, an explicit and concise testable surfaces range map was offered to show the ability of the null. The range map is useful to evaluate whether a given conic surface is testable. Moreover, preliminary parameters for designing the test optical layout for each testable surface can be easily obtained. A practical near-null test system based on this method was established. To enhance the test accuracy, alignment and near-null data processing were analyzed. Experiments were conducted to verify the broad measurable surfaces range and test accuracy. In the last section, the development of the work was discussed.

## 2. Principle

#### 2.1 Variable spherical aberration null theory and the optimization design result

It is well known that the majority aberration of concave conic and near conic surfaces is primary spherical aberration (PSA). A null can be called as broad range variable spherical aberration null (VSAN) if it can generate a variable amount of nearly pure PSA over a wide range. Utilizing this null, a wide range of aspheric surfaces can be tested. This subsection is to reveal the theory of VSAN, and to optimize the VSAN based on the theory.

To demonstrate the principle of the broad range VSAN, analysis of the algebraic solution for Dall null [33] is a good start. Dall null can obtain sufficient aberration correction for concave conic and near conic surfaces with moderate relative aperture tested at the center of curvature. The prolate conic surfaces, i.e., ellipsoid, paraboloid, and hyperboloid are more usually used than the oblate ones. Testing prolate conic surfaces requires a positive lens to generate under-corrected PSA. Hence, a plano-convex spherical singlet type Dall null is analyzed by us. The Dall test principle is shown as Fig. 1. Based on Seidel aberration theory [1], the first Seidel sum *S*_{I} of the Dall null can be obtained under thin lens approximation as shown in Eq. (1).

*h*denotes the semi-aperture of the Dall null;

*φ*as shown in Eq. (2) is power of the lens. In Eq. (2),

*r*

_{1}and

*r*

_{2}are radii of curvature (RoC) of the two surfaces, respectively.

*P*as shown in Eq. (3) is a function of the refractive index

*n*and conjugate ratio

*m.*Conjugate ratio means the ratio of image distance to object distance.

In the paraxial region, *u* = tan*u*, where *u* is the aperture angle. Hence, $h\text{=}su$, where *s* denotes the object distance as shown in Fig. 1. Substituting this into Eq. (1), Eq. (4) can be obtained as

The first Seidel sum of the conic surfaces is

Null test condition means the following equation should be satisfied,

Substituting Eqs. (4) and (5) into Eq. (6), the principle equation for the spherical singlet type VSAN can be obtained as

Note that *R* value of concave surface is set as negative, thus a minus sign is removed from Eq. (7). According to Eq. (7), it is inferred that, for a fixed null structure and a fixed *s* value, a conic surfaces group can be tested with third order aberration accuracy theoretically. Moreover, the *k*∙R value remains constant for this group. In fact, it has been verified that the transmitted wavefronts at different propagation distances *l* after the singlet null change shape [34] according to

*t*means advancing a distance

*t*of the transmitted wavefront towards the test surface.

*k*and

*R*are shape parameters of the transmitted wavefront fitted by conic equation.

After comprehensive analysis of Eqs. (7) and (8), a spherical singlet type VSAN can be obtained theoretically. Two rules for the VSAN are shown as follows. Note that these rules are verified to be right by the following simulations.

Rule 1: For a fixed object distance *s*, at different distances *l* of the null to the test surface, a group of equidistant conic and near conic surfaces can be tested. The surfaces of the group have a nearly constant *k*∙R value.

Rule 2: Moving the null back and forth relative to the point source, i.e., varying the *s* value, different equidistant surfaces groups with different *k*∙R value can be tested. The *k*∙R value is a quartic polynomial function of object distance *s*.

To obtain a practically useful spherical singlet type VSAN, following analysis is further conducted. Conic surface can be defined by three parameters, i.e., *k*, *R*, and relative aperture *A* (or aperture *D*). According to Rule 2 and Eq. (7), for surfaces with larger *k*∙R value, *s* must be relatively larger. It is obvious that for a null with fixed aperture, the larger *s* is, the smaller numerical aperture (NA) of the test beam is. Hence the testable relative aperture is small for surfaces with a large *s* value. If the *k*∙R value increases fast as *s* increases within a relatively short range, measurable surfaces range will be broadened. To realize this goal, increasing the coefficient of *s*^{4} in Eq. (7) is a straightforward way. This coefficient is *φ*^{3}*P* in Eq. (7). For plano-convex singlet with common BK7 material, *n ≈* 1.516, *r*_{2} = ∞. Conjugate ratio *m* usually varies between 1.2 and 2. A fixed mean value 1.6 is chosen for *m* in the following analysis. Substituting these values into Eqs. (2), (3), and (7) yields

From Eq. (9), it seems that choosing *r*_{1} as small as possible is helpful to increase *g*_{4}. However, it is impractical because too small *r*_{1} will introduce much higher-order spherical aberration. To choose an appropriate *r*_{1}, simulations are conducted. The particular test surface in the simulations is a concave parabolic surface with RoC of 3416mm and aperture of 1010mm. This surface is chosen because its majority aberration is PSA, and relatively small higher-order spherical aberration is included. A plano-convex null lens is simulated to test this surface. The thickness and clear aperture of the null are chosen as moderate values. They are 29mm, and 88mm, respectively. In the simulations, the *r*_{1} value ranges from 80mm to 360mm with 20mm increment. In each simulation with certain *r*_{1} value, optimization is conducted. The optimization variables are *s* and *l*. Optimization goal is to minimize the root mean square (RMS) value of the residual wavefront error. Simulation results show that the residual higher-order spherical aberrations all increase with the decrease of *r*_{1}. An appropriate value for *r*_{1} is chosen when the interferogram caused by the residual aberration can be resolved by the CCD with a common pixel density. Also, *r*_{1} should be as small as possible to increase *g*_{4}. *r*_{1} = 220mm is finally chosen with residual fifth-order spherical aberration Zernike Fringes (ZF) [35] coefficient of 9.94λ (double pass, λ = 632.8nm) and PSA ZF coefficient 1.19λ. Substituting this *r*_{1} value into Eq. (9) yields *g*_{4} = 4.981e-7 mm^{−3}. The design of the spherical singlet type VSAN is completed.

To evaluate the measurable surfaces range of this spherical singlet type VSAN, simulations are conducted. In the simulations, *s* ranges from 50mm to 200mm with increment of 5mm, and *l* from 100mm to 3500mm with increment of 20mm. For every *s* and *l* value, optimization is conducted to find the measurable conic surface. Conic surface parameters *k* and *R* are optimization variables. Optimization goal is to minimize the RMS value of the residual wavefront error. Testability criterion is that the residual wavefront error is within the dynamic range of a typical CCD with 256 × 256 pixels. Dynamic range means the maximum allowable fringe frequency equaling the Nyquist frequency of the CCD. For every fixed *s* value, the *k*∙*R* value variation with *l* is plotted in Fig. 2. It is obvious that, for a fixed *s*, the testable surfaces group has a nearly constant *k*∙R value. This simulation result verifies the above Rule 1. To explicitly represent the measurable conic surfaces range, the mean *k*∙*R* value and *A* value varying with *s* are plotted in Fig. 3. The blue dotted line represents the result of fitting the mean *k*∙*R* value with *s* curve by a quartic polynomial function. The fitting curve matches the raw *k*∙*R* value to *s* curve very well. This verifies the Rule 2.

Based on the above principle and the designed spherical singlet type VSAN, a plano-convex singlet containing a high order even asphere (HOEA) is further studied. It is called HOEA singlet type VSAN hereafter. The HOEA surface has much more shape parameters than that of the spherical VSAN. The initial purpose of replacing the spherical surface of the null with an HOEA surface is to further increase *g*_{4} value. Obtaining a broader testable surfaces range by optimizing these parameters is expected.

To obtain the PSA contribution of a general HOEA surface, Taylor series expansion is conducted on the HOEA surface equation and the spherical surface equation. Corresponding results are shown in Eqs. (10), and (11), respectively.

Comparing Eq. (10) with Eq. (11), the difference can be represented as

*α*,

*β*,

*χ*are deformation coefficients of HOEA surface relative to spherical surface.

Hence, HOEA singlet can be regarded as a combination of a weak aspherical plate represented by Eq. (12) with a spherical singlet as shown in Fig. 4.

Hence, the HOEA singlet will introduce an additional wavefront aberration to spherical singlet. If only PSA is considered, the additional wavefront aberration is

This is, of course, a Seidel aberration and can be related to first Seidel sum by*A*

_{4}is the coefficient of

*h*

^{4}as shown in Eq. (10). It is obvious that

*g*

_{4}value of the HOEA singlet type VSAN has an increment of (

*n*–1)

*αc*

_{1}

^{3}compared with that of the spherical singlet type VSAN. This means, by choosing propriate

*k*and

*A*

_{4}values, the testable surfaces range can be broadened.

To obtain a practically useful HOEA singlet type VSAN with broadened measurable surfaces range, following analysis is further conducted. The center thickness, refractive indices, *r*_{1}, *r*_{2}, and clear aperture are chosen the same with those of the above spherical singlet type VSAN. To make *g*_{4} about seven times that of the spherical singlet type VSAN, *k* = −1, and *A*_{4} = 6.5e-7mm^{−3} are chosen as preliminary values. The same test surface (RoC = 3416mm, D = 1010mm, *k* = −1) is utilized to minimize the high-order spherical aberration. Optimization variables are *A*_{i} (i = 4, 6, 8, 10, 12). Optimization goal is to minimize RMS value of the residual wavefront error. After the optimization, the residual wavefront error RMS value is 0.034λ. The parameters of the HOEA surface after optimization are shown in Table 1. Note that *r*_{1} and *A*_{i} have units to make the sag value *z*^{(hoean)} have the unit of millimeter. Substituting these values into Eq. (16), *g*_{4} = 3.138e-06 mm^{−3} is obtained. It is about 6.3 times that of the spherical singlet VSAN. This means the maximum *k*∙*R* value of the testable surfaces is about 6.3 times that of the spherical singlet VSAN theoretically. The design of the HOEA singlet type VSAN is completed.

To verify the theoretical analysis result, similar simulations are conducted to find all testable conic surfaces. The measurable surfaces range map is shown in Fig. 5. The blue solid line represents the result of fitting the HOEA singlet type VSAN mean *k*∙*R* value with *s* curve by a quartic polynomial function. The red solid line is the curve of mean *A* value with *s*. To compare its measurable range with that of the spherical singlet VSAN, the curve of mean *k∙R* value with *s* of the spherical singlet type VSAN is also included in Fig. 5. It is represented by the blue dotted line. Figure 5 shows that the measurable surfaces maximum *k∙R* value of the HOEA singlet type VSAN is about seven times that of the spherical singlet VSAN. This result is similar with the theoretical analysis result, i.e., 6.3 times. As the range map shows, when *s* varies between 50mm and 200mm, most conic surfaces with *k∙R* value varying between 0 and about 70000mm can be tested. For each surface, the maximum mean testable *A* is determined at the corresponding *s* value. Hence, the HOEA singlet type VSAN is called broad range VSAN by us. Some typical testable surfaces with asphericity varying between 0 and about 230λ are listed in Table 2.

The measurable surfaces range map contains all the three parameters defining a conic surface. It visually and explicitly shows the test ability of the HOEA singlet type VSAN. Whether a given test conic surface is testable can be easily judged by this map. Moreover, it can offer preliminary parameter values for designing the test optical layout. For a given test surface, its *k∙R* value is firstly calculated. Then it is judged whether the value is within the measurable *k∙R* value range. If it is, the estimated objective distance *s*_{p} value is determined at the corresponding *k∙R* value of Fig. 5. Then, the estimated maximum testable relative aperture *A*_{m} at *s*_{p} is obtained. If the relative aperture of the test surface is smaller than *A*_{m}, the surface is probably testable. Finally, simulation is conducted to make the final judgement whether this surface is testable. In the simulation, preliminary *s* value is chosen as *s*_{p}, preliminary *l* value is chosen as *l*_{p} = *R* - *T* - *s*_{p}, where *T* is the thickness of the null. Optimization variables are *s* and *l*. Optimization goal and testability criterion are the same with the above simulation. Near-null test optical layout design is completed. Surface figure error reconstruction and alignment are two keys of near-null test. These are described in the following subsection.

#### 2.2 Test system, alignment, data processing, and error considerations

The test system based the HOEA singlet VSAN is shown in Fig. 6. It can be based on a commercially available Fizeau interferometer such as Zygo GPI. The *f*/# of the Transmission Sphere (TS), distance values of *s* and *l* are varied to test different surfaces according to the above principle.

To obtain high-accuracy test result, alignment of the HOEA singlet VSAN and the test surface is important. As for the HOEA singlet VSAN, misalignment degrees of freedoms (dofs) are tilt, transverse displacement (TD), and axial displacement (AD). Many simulations show, to control the misalignment-induced aberrations within 0.02λ RMS (with piston and tilt aberration removed), mean tolerance requirement on tilt and TD is about ± 50”, and ± 10*μ*m, respectively.

To align tilt and TD of the HOEA singlet VSAN, an alternative thought is to make the convergence point locate on the optical axis of the VSAN. Firstly, coarse alignment at the test position is conducted with the collimated beam of the interferometer. That is, the TS is removed. An interferogram is formed by the reflected beams from the center region of the HOEA surface (can be regarded as a small flat) and the flat surface of the VSAN. If the optical axis of the VSAN is parallel with the collimated beam of the interferometer and it locates at the center of the CCD, the fringes are Newton's rings. Moreover, the center of the Newton's rings coincides with the center of CCD, as shown in Fig. 7(a). In Fig. 7(a), a white crosshair locates at the center of the CCD image. When slight misalignments exist, the interferogram is different from Fig. 7(a). Figures 7(b) and 7(c) show the interferograms when the VSAN tilts around *y* and *x* axis, respectively. When the VSAN translates along *x* and *y* axis, the interferograms are shown in Figs. 7(d) and 7(e), respectively. Coarse alignment of the VSAN is completed by tilting (conducted by adjusting the tip-tilt stage 1) and translating the VSAN to obtain the interferogram of Fig. 7(a). Whether the Newton's rings are concentric and aligned with the center of the crosshair is evaluated by image processing technique. This can be completed at pixel-level accuracy [36]. Secondly, adjust the movement axis of AST1 to be parallel with the optical axis of VSAN. If the two axes are parallel, the center of the Newton's rings always coincides with the center of crosshair when the VSAN is axially translated. If not, the center of the Newton's rings will deviate from the center of crosshair. Iteratively translate the VSAN by the axial translation stage 1 (AST1), and adjust tip-tilt stage 1 and 2 until the two axes are parallel. Finally, install the TS back to the interferometer and translate the VSAN to the cat’s eye position. To make the convergence point locate at the vertex of HOEA surface, adjust tip-tilt of the TS and axial translation of the VSAN until cat’s eye interferogram appears. Then axially translate the VSAN back to the test position. The vertex of HOEA surface is on the optical axis of VSAN in nature. Hence, the vertex of HOEA surface at the cat’s eye position and the test position define the optical axis of VSAN considering that the movement axis of the AST1 parallels with the optical axis of VSAN. Thus, the convergence point is on the optical axis of VSAN. That means tilt and TD are aligned.

The AD of the HOEA singlet VSAN is monitored by a distance measuring set LenScan LS600 [37]. LenScan LS600 can measure center thickness of optical elements and air gaps along the optical axis based on low coherence interferometry. Its measurement range is 600mm with absolute accuracy of ± 1*μ*m. Simulation shows the misalignment aberration is negligible if the AD error is ± 1*μ*m. Moreover, when the aspherical parameters accuracy (i.e., *k* and R) are considered during the test, simulations show the test error of aspherical parameters caused by ± 1*μ*m AD error is negligible. These are the entire alignment procedures for VSAN. Frankly, it is a little complex and experiences dependent. However, it practically works well if it is carefully conducted. Experiment in the following section on a test surface with about 0.07λ RMS surface error can obtain good cross test result after this alignment process.

Alignment of the test surface is similar with the common null test of aspheres when the theoretical fringes are sparse. However, when the theoretical fringes are dense, slight misalignment-induced aberrations become imperceptible. High-order misalignment-induced aberrations will remain if we still follow the null test adjustment procedures. For these surfaces, a better way is to utilize the specific misalignments calculated from the selected low-order aberrations to predict all the misalignment-induced aberrations by ray tracing [38,39]. They can be removed along with retrace error from the test result.

Near-null data processing is essential to reconstruct the surface figure from the test result. It mainly includes mapping error correction and retrace error correction. Mapping error can be corrected by reverse transform between the test surface coordinates and the CCD coordinates according to the ray tracing result. Retrace error correction is a little complex. Due to the deviation from null configuration, rays from the test and the reference arms will not follow the same path and retrace error exists. Many publications have been focused on this issue [40–45]. Shi *et al.* have compared these methods [46]. Among these methods, we prefer to use theoretical reference wavefront (TRW) method [43] for surfaces with small figure error and moderate theoretical residual wavefront error. While for surfaces with relatively large figure error or large theoretical residual wavefront error, the reverse optimization reconstruction (ROR) method [42,44] is better.

Accuracy of both methods is mainly limited by the modeling accuracy. The modeling error of our test system is analyzed. It includes inaccurate modeling of the HOEA singlet VSAN and the test surface, which include the inaccurate element structure parameters and element position inconsistences with the actual system. The VSAN structure parameter errors mainly include surface figure error of both surfaces of the VSAN and center thickness of the singlet. The HOEA surface figure error is tested by a Zygo verifire asphere^{TM} interferometer. The test result is about 0.024λ RMS. Because the test beam refracts at this surface, its contribution to test result is about (*n*-1)∙0.024λ ≈0.012λ RMS, where *n* is the refractive index. The surface figure error of flat surface is about 0.016λ RMS. Similarly, its contributions to test result is about 0.008λ RMS. The thickness of the VSAN is measured by LenScan LS600. The measurement result is 28.578mm. This test value is utilized in the practical modeling of the test system. Considering the absolute measurement accuracy of LenScan is ± 1*μ*m, many simulations are conducted to show its contribution is negligible. As for the VSAN and test surface position modeling error, their contribution to measurement result is about 0.02λ RMS as analyzed above. Note that the LenScan measurement results of *s* and *l* values are utilized in the ray tracing model simulation, hence their contributions to modeling error are also negligible. To sum up, the total modeling error influence is about λ/40 RMS. Modeling error is the dominant error source of the method. Although there exist other error sources such as reference wavefront error from the interferometers and noise error from environment, their contribution is weak. Therefore, measuring accuracy of the method is also about λ/40 RMS.

## 3. Experiments

The test surfaces include two paraboloids and one ellipsoid with different amounts of asphericity. All these surfaces are primary mirrors of practical systems fabricated at our laboratory. Surface shape parameters of these three surfaces are shown in the second, third and fourth columns of Table 3, where *Obs*. means the obscuration diameter size, *D* means the diameter of the aperture. With the aid of measurable surfaces range shown in Fig. 5, near-null test optical layout design for the three surfaces is completed. *s* and *l* values for the three surfaces are shown in the fifth column of Table 3. Residual wavefront errors when fringe density of interferograms is moderate for the three surfaces are shown in the last column of Table 3. For 1^{#} surface, a large amount of power aberration is deliberately induced to reduce the maximum slope of the residual wavefront error. It is helpful to resolve the interferogram. The test system is set up according to these design results. For simplicity, the test process for 1^{#} surface is stated in detail. For the other two surfaces, only test results are provided.

The near-null test system for 1^{#} surface is presented in Fig. 8. The interferometer is a Zygo GPI 4” with a *f*/0.75 TS. Firstly, the HOEA singlet VSAN is aligned according to the procedures stated in the above section. Then, axially translate the VSAN until the LenScan LS600 measurement result of the distance between the vertex of HOEA surface and the vertex of the TS is *s*_{1} + *R*_{ts,} where *R*_{ts} = 48.24mm is the nominal radius of aplanatic surface of the *f*/0.75 TS, and *s*_{1} is shown in Table 3. Secondly, the test surface is adjusted by the five-dof stage. After careful adjustments, the near-null test result can be acquired. Figures 9(a) and 9(b) are one of the acquired phase-shifting interferograms and the extracted phase map (single pass, i.e., halving the phase map that calculated from the phase-shifted fringes maps), respectively. Finally, TRW retrace error correction method is utilized to extract the surface figure. Figures 9(c) and 9(d) are the predicted interferogram and residual wavefront error of the simulation system. The simulated and practical interferograms are very similar. The surface figure is obtained by subtracting Fig. 9(d) from Fig. 9(b) with the low order misalignment-induced aberration (piston, tilt, and coma) removal and mapping error correction. Figure 10(a) shows the surface figure error with PV 1.447λ, and RMS 0.288λ. For cross test, the surface is tested by autocollimation method with a 38mm diameter high accuracy retro-sphere. The test system is shown in Fig. 11. Due to the large aperture of the test surface, Zygo MST 24” interferometer is utilized. The test result is shown in Fig. 10(b) with PV value of 1.565λ, and RMS value of 0.249λ. Surface figure distribution and PV/RMS value of the two methods are very similar. The two methods utilize two different interferometers with transmission sphere and transmission flat, respectively. Unknown different mapping error exists between the two test results due to the unknown different imaging systems inside the commercial interferometers. Moreover, this error cannot be neglected because the residual wavefront error for 1^{#} surface is relatively large. The map of point to point difference between the two results may be unreliable due to the very different coordinates mapping. Hence, it is not offered for 1^{#} surface. It is only offered for 2^{#} surface as stated following.

Following the similar procedures, near-null test utilizing HOEA singlet VSAN for 2^{#} surface is conducted. The result is shown in Fig. 12(a). Cross test is performed by autocollimation method with a high accuracy retro-flat. Figure 12(b) shows the test result. The same interferometer with the same TS is utilized by both methods, and the residual wavefront error of the near-null test is small. Hence the coordinate mapping difference between the two results caused by the unknown imaging system and TS configurations is small. A point to point difference map can be relatively reliable to evaluate the difference of the two results. Figure 12(c) shows the difference map with RMS value of 0.027λ. The experiment layouts of the two methods for 2^{#} surface are shown in Fig. 13(a) and 13(b), respectively.

For 3^{#} surface, result of the near-null test is shown in Fig. 14(a). Cross test is conducted utilizing an Offner null [47], which is composed of two spherical lenses. The design and manufacture of the Offner null follows the regular method [48]. The test optical layout is shown in Fig. 15(a). The collimated beam emitted from the interferometer is transformed by the Offner null to aspherical wavefront which matches the ideal test surface. Figure 15(b) shows the residual wavefront error. The RMS value of the residual wavefront error is about 0.004λ. Figure 14(b) shows the Offner null test result. By comparison, surface figure distribution and PV/RMS value of the two methods results are also very similar. The experiment layouts of the two methods for 3^{#} surface are shown in Figs. 16(a) and 16(b), respectively.

## 4. Discussion

The conventional partial compensation null [27–30] has its great merits as a sophisticated technique. For example, the alignment of the null is easier than our method since it utilizes the null in a collimated beam. Compared with the sophisticated partial compensation null method, the prominent merits of the proposed method are the broader measurable surfaces range and the theoretical design process. Our utilized aspherical singlet null is designed according to the proposed variable spherical aberration null theory. By moving the aspheric null lens relative to the convergence point and moving the test surface relative to the null, our method can test different equidistant surfaces groups with widely varied *k∙R* values. However, the conventional partial compensation null can only test one equidistant surfaces group theoretically. This is because that the flexibility of the partial compensation null is achieved by only moving the test surface back and forth relative to the null. Axial translation of the null relative to the interferometer is useless to enhance the flexibility because a collimated beam is utilized. Although the surfaces of one equidistant surfaces group can also be different kinds of conic surfaces, such as paraboloid, hyperboloid, and ellipsoid, they share a nearly constant *k*∙*R* value. Moreover, the conventional partial compensation null utilizes a spherical singlet as the null. It has been verified as above that the spherical singlet null has a very narrow testable surfaces range compared with the aspherical singlet null proposed in our paper.

Prospects are provided as follows for further development of the proposed method.

- 1) The null we designed was a positive lens for testing concave prolate conic and near conic surfaces. For oblate surfaces, a negative lens is required to generate over-corrected PSA. Such a null can be designed based on the similar design procedures stated in Section 2. In fact, a series of nulls can be designed to cover a wider range of testable surfaces than that of the single null reported in this paper. Another improvement may be to optimize the design's higher-order aberration so that it gives very good nulls for a variety of particular surfaces of interest, instead of aiming for the all-purpose design shown here. This deserves further investigation.
- 2) Combining the proposed VSAN with AO element will be fascinating. Firstly, the residual wavefront aberration is usually within the modulation range/strokes (about tens of microns) of the AO element. Hence it can be compensated by the AO element. Near-null test turns into null test. The drawbacks of near-null test such as retrace error can be overcome. Secondly, AO element can further extend the testable surfaces range. A variety of free-forms whose basic shape is a concave conic or near conic surface can be tested. The VSAN compensates the dominant aberration, i.e., basic shape of the free-form deviating from spherical surface. The residual aberration of the free-from deviating from its basic shape is compensated by the AO element. Thirdly, these testable free-form or aspheric surfaces can be in-process surfaces with unknown severe local surface figure error that beyond dynamic range of conventional interferometers. This is realized by combining the adaptive wavefront interferometry (AWI) technique we reported recently [26]. AWI utilizes an AO element to iteratively generate adaptive wavefronts for compensating the unknown severe surface figure error. The AO element is close-loop controlled by wavefront sensor-less optimization algorithms to monitor the compensation effects for guaranteeing convergence. Invisible fringes turn into resolvable ones within the local region of severe surface figure error. The VSAN combined with AO element technique is under research by us. We hope to report it soon.
- 3) As stated above, the mapping error correction and the retrace error correction both rely on the accurate system modeling. However, the TS and imaging system of the commercial interferometer we utilized in this paper were unavailable. The incomplete system modeling introduced error to the test results. The error cannot be neglected when the residual wavefront error of the near-null test is large. To improve the test accuracy, an interferometer established in the laboratory with complete system modeling should be used. Further investigation is required.

## 5. Conclusion

Utilizing the developed variable spherical aberration null theory, the variable spherical aberration null (VSAN) is optimized as a plano-convex singlet containing a high order even asphere (HOEA). The HOEA singlet type VSAN is theoretically proved to have measurable surfaces range about 6.3 times that of the spherical counterpart. A measurable surfaces range map containing all parameters defining a conic surface is obtained to visually and explicitly show the broad testable surfaces range. Most concave prolate conic surfaces with *k*∙*R* value varying between 0 and about 70000mm and with smaller relative aperture than the corresponding value at each *k*∙*R* value can be tested. The adaptive asphericity range is between 0 and about 230λ. Theoretical error analysis shows the established practical near-null test system can obtain accuracy of λ/40 RMS. Three surfaces with different amounts of asphericity are tested. Difference map between the proposed method and the traceable cross test results for one of the experimental surfaces shows a difference of 0.027λ RMS. The test flexibility and efficiency are enhanced with reduced cost for concave conic and near conic surfaces.

## Funding

Hunan Provincial Natural Science Foundation of China (2016JJ1003); Science Challenge Program of China (TZ2018006).

## Acknowledgments

We would like to thank Mr. Jinfeng Lu for his earlier contributions testing the aspherical surface of the null. Dr. Chaoliang Guan and Prof. Xiaoqiang Peng at NUDT are appreciated for providing the test surfaces for our experiment. Ms. Wanxia Deng at NUDT has provided background knowledge and codes on image processing technique.

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