Non-null testing for aspheric surfaces using elliptical sub-aperture stitching technique

Zixin Zhao; Hong Zhao; Feifei Gu; Hubing Du; Kaixing Li

doi:10.1364/OE.22.005512

1.Introduction

The aspheric surfaces are extremely important in optical systems and have been applied in various kinds of fields for their great ability in correcting aberrations, improving image quality and reducing the size and weight of the system [1]. As the prevalent use of the aspheres in the optical systems, the need for developing precise and efficient measurement techniques are growing. Among them, one of a promising measurement method is interferometry. Due to high resolution, high sensitivity and reproducibility, the method may become the standard tool for testing optical surfaces and wavefronts. In interferometry, the null testing [1] has been always a dominating test configuration for the goal of the aspheric surfaces measurement. But the configuration is inconvenient and time consuming because an auxiliary element (for example the CGH or null lenses) is required during measurement. Therefore, the non-null testing [1] is used to overcome these disadvantages.

It should be noted that, when testing the aspheres using the non-null configuration, affected by the failure to common path argument, some errors(retrace errors) [2] are introduced into the existing commercial interferometers, which are often designed for null testing. Special methods [3] have been proposed to compensate the errors. But the prescription of how to use them for a commercial interferometer is not clear to us. For this reason, the compensation method may not be practical in some cases. So a further study for the non-null testing is necessary.

Another important issue for the measurement of the aspheres by a commercial interferometer is that the resulting fringe spacing is too fine to analyze for the large deviations between the test and reference wavefronts. But the problem can be solved by the sub-aperture stitching technique [4]. According to the different shapes of the sub-aperture, there are two major stitching methods: one is the annular stitching method [5–8], the other is the circular stitching method [9–11]. The Annular sub-aperture stitching (ASAS) method is an effective way to extend the vertical dynamic range of a conventional interferometer. Moreover its motion mechanism, theoretically one degree of freedom, is much simpler. The circular sub-aperture stitching (CSAS) method needs six degrees of freedom. However the ASAS method can only measure the rotationally symmetric aspheric surfaces while the CSAS method could test flats, spheres and a variety of aspheres.

In this paper, we propose an elliptical sub-aperture stitching (ESAS) technique to test aspheres in non-null configuration. The proposed method is simple and fast. But it is more efficient than the technique above when small and middle aspheric optics are tested because it only needs several simple adjustment (i.e. different tilts)at one testing position. In the proposed technique, the aspheres are tested with the vertex reference sphere. Then, by fine-tuning the optical adjusting stage, the different tilts are added to the testing wavefronts, and the different nearly null areas (nearly elliptical areas) are obtained with the elliptical masks. Finally, all the nearly null testing results are stitched together to get the full aperture testing result. The paper is organized as follows. In Section 2, the principle of our technique is given. In Section 3, the effectiveness of our method is shown by simulation. In Section 4, the testing ability of our proposed method is analyzed. In Section 5,we demonstrate the performance of our technique by testing a concave ellipsoid and convex hyperboloid surface. Finally, in Section 6,our conclusions are presented.

2.Theory

2.1 Principle of local nearly null testing

The null zone with broad interference fringes will always occur when the rays strike the aspheric surface in normal direction [12]. This is the case, when the center of the spherical reference surface coincides with the point, where the normal of the aspheric surface strikes the symmetry axis. The geometric diagram of the nearly null zone and its corresponding simulated interference pattern are shown in Fig. 1. It shows that the local nearly null zone moves from the center to the edge of the aperture when dynamic tilt is added. The retrace errors can be neglected in those nearly null areas, because the rays reflected from those areas follow nearly the same path through the optical system as the reference rays.

Fig. 1 Sketch of the local nearly null zone and fringe patterns. (a) no tilt. (b)with tilt.

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2.2 Stitching algorithm

In order to obtain the full aperture result, a stitching algorithm is developed to stitch all the phase data together. Currently, there are two main stitching algorithms. One is based on phase compensation using the least square method [13], and the full-aperture data is reconstructed by compensating the piston, tilt and power of each sub-aperture. The other is based on three-dimensional coordinate transformation using the iteration method [10].The least square method is simple and faster while the iteration method is immune to fairly big parameter uncertainties. As the adjustment in our method is simple, the phase compensation stitching algorithm has been used with minor modification. Different to the tip-tilt-power compensation stitching algorithm, the algorithm we proposed is the same as the plane stitching algorithm when only the piston and tilt are corrected. Consider the adjacent null zone $A$ and $B$ , their testing results can be described as follows,

{\begin{cases} ϕ_{A} (x, y) = ϕ_{O} (x, y) + P_{A} + T_{A x} \cdot x + T_{A y} \cdot y \\ ϕ_{B} (x, y) = ϕ_{O} (x, y) + P_{B} + T_{B x} \cdot x + T_{B y} \cdot y \end{cases},

where

ϕ_{A} (x, y)

and

ϕ_{B} (x, y)

are the testing results,

ϕ_{O} (x, y)

is the ideal results.

P_{A}

and

P_{B}

,

T_{A x}

and

T_{B x}

,

T_{A y}

and

T_{B y}

are the piston,

x

direction tilt,

y

direction tilt respectively for each zone. Based on Eq. (1), a merit function

F

is defined as follow,

\begin{array}{l} F = u_{1} \cdot \sum_{j = 1}^{M} \sum_{m = 1}^{N - 1} \sum_{n = m + 1}^{N} \iint_{σ_{j}} [ϕ_{m j o} (x, y) - ϕ_{n j o} (x, y) - ((P_{m x} - P_{n x}) + (T_{m x} - T_{n x}) x + (T_{m y} - T_{n y}) y]^{2} d x d y / N_{j} \\ + u_{2} \cdot \sum_{i = 1}^{N} \iint_{s_{i}} {(ϕ_{i} (x, y) - (P_{i} + T_{i x} x + T_{i y} y))}^{2} d x d y / N_{i}, \end{array}

where

u_{1}

and

u_{2}

are positive weights and sum of them is 1.

M, N

are the number of overlap areas and sub-apertures respectively.

ϕ_{m j o} (x, y)

and

ϕ_{n j o} (x, y)

are the

j -th

overlap phase data coming from the

m - t h

and

n - t h

sub-aperture respectively.

$N_{j}$ is the number of effective points in the $j - t h$ overlap area and $N_{i}$ is the number of effective points in the $i - t h$ sub-aperture. $σ_{j}, s_{i}$ represent the region of integration for the $j - t h$ overlap area and the $i - t h$ sub-aperture respectively. The first term in Eq. (2), scaled by $u_{1}$ , means the root mean squares(RMS) error of the overlapping points and the second term, scaled by $u_{2}$ , means the RMS error of all the sub-aperture data relative to vertex reference sphere. Let $E_{i} = {(P_{i}, T_{i x}, T_{i y})}^{T}$ ,thus $F$ is a function of $E_{i}$ .Then the normal Eq. (3) is obtained by calculating partial derivative that $\partial F / \partial E_{i} = 0$ .

A {[E_{1}^{T} ... E_{N}^{T}]}^{T} = b,

Let

U_{i} = {[O_{1} ... O_{i - 1}, I, O_{i + 1} ... O_{N}]}^{T}

. According to Eq. (2), the data matrix

A

can be split into

A R_{j}

and

A F_{j}

. Accordingly, the observation vector

b

can be split into

b r_{j}

and

b f_{j}

. For the overlap area, the matrix forms are as follows,

A R_{j} = \sqrt{u_{1}} \cdot [\begin{matrix} (1, 1) & (1, x) & (1, y) \\ (x, 1) & (x, x) & (x, y) \\ (y, 1) & (y, x) & (y, y) \end{matrix}],

b r_{j} = \sqrt{u_{1}} \cdot [\begin{matrix} (1, ϕ_{m j o} (x, y) - ϕ_{n j o} (x, y)) \\ (x, ϕ_{m j o} (x, y) - ϕ_{n j o} (x, y)) \\ (y, ϕ_{m j o} (x, y) - ϕ_{n j o} (x, y)) \end{matrix}],

where

(x, y)

denotes the inner product of the variable

x

and

y

.

For the individual nearly null zone, the matrix forms are as follows,

A F_{i} = \sqrt{u_{2}} \cdot [\begin{array}{l} (1, 1) & (1, x) & (1, y) \\ (x, 1) & (x, x) & (x, y) \\ (y, 1) & (y, x) & (y, y) \end{array}],

b f_{i} = \sqrt{u_{2}} \cdot [\begin{array}{l} (1, ϕ_{i} (x, y)) \\ (x, ϕ_{i} (x, y)) \\ (y, ϕ_{i} (x, y)) \end{array}],

Thus the data matrix

A

and observation vector

b

in Eq. (3) can be described as follows,

A = \sum_{j = 1}^{M} [U_{m j} A R_{j} U_{m j}^{T} + U_{n j} A R_{j} U_{n j}^{T} - U_{m j} A R_{j} U_{n j}^{T} - U_{n j} A R_{j} U_{m j}^{T}] + \sum_{i = 1}^{N} U_{i} A F_{i} U_{i}^{T},

b = \sum_{j = 1}^{M} [(U_{m i} - U_{n i}) \cdot b r_{i}] + \sum_{i = 1}^{N} U_{i} b f_{i},

After the data matrix and observation vector are obtained, using all the nearly null zone data and overlapping data, the relative piston, tilt of each sub-aperture can be estimated.

3. Simulation

The purpose of the simulation is to demonstrate the effectiveness of the method that we proposed. Generally, taking the z axis as the axis of revolution, the rotationally symmetrical aspheric surface can be described as follow [1],

Z (s) = \frac{c \cdot s^{2}}{1 + \sqrt{1 - (1 + K) \cdot c^{2} \cdot s^{2}}} + A \cdot s^{4} + B \cdot s^{6} + C \cdot s^{8} + D \cdot s^{10} + \dots,

where

s = \sqrt{x^{2} + y^{2}}

,

c

is the vertex curvature,

K

is the conic constant.

A, B, C

and

D

are the coefficients of the high order aspheres. If the coefficients are all zero, the surface is a conic surface of revolution.

In the first simulation, the tested optics is a concave ellipsoid surface with an aperture of 100 mm and a radius of vertex curvature of −348.6 mm. Besides, the conic constant is −0.266.The radius of the reference sphere equals the radius of vertex curvature of the tested optics. By adding the different tilts, nine interference patterns are obtained as shown in Fig. 2(a) and the corresponding local nearly null interference patterns are extracted through the masks as shown in Fig. 2(b). After the phase data, shown in Fig. 2(c), are extracted, we can stitch them together to get the full aperture result as shown in Fig. 2(d). Figure 2(e) is the ideal result, and the differences are shown in Fig. 2(f).The PV and RMS of the differences are 0.000002 wave(wave = 0.6328 um) and 0.0000005 wave respectively, and are mainly the result of the numerical error.

Fig. 2 The first simulation result. (a)global fringe patterns .(b) local fringe patterns with different masks. (c) extracted phase data. (d) stitching result. (e) ideal result. (f) differences between stitching and ideal result.

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In order to examine the impact of the departure on the stitching algorithm. Another simulation is performed, the tested optics is a concave surface with an aperture of 150 mm and a radius of vertex curvature of −600 mm. The maximum departure from the base sphere is about 15 waves. Twenty five sub-apertures are used and the simulation results are shown in Fig. 3(a)-(d). The differences, as shown in Fig. 3(e), are quite small and can be neglected as the first simulation result. The two simulation results show that our stitching algorithm is effective.

Fig. 3 The second simulation result.(a) fringe patterns with different tilts .(b) extracted phase data. (c) stitching result. (d) ideal result. (e)differences between stitching and ideal result.

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4.Measurability analysis

In this section, the testing ability of our proposed method is analyzed. In order to get the off-axis nearly null zone, different tilts are added. When the aspheric departure is very large, no matter how many tilts are added, only the part of the surface, as shown in Fig. 4(b), can be tested for the Nyquist limit of one fringe per two pixels. As a result, the proposed ESAS method is strictly limited by the aspheric departure of the surface.

Fig. 4 Testing ability of our proposed method with respect to conic constant $(K)$ and numerical aperture $(N A)$ . (a) the full aperture could be tested with dynamic tilts for the fringe aliasing are not occurred. (b)only the area in the red circle could be tested due to the fringe aliasing in the area outside of the red circle. (c)the area filled by oblique line can be measured and the red circle locates the turning point with $K = - 1$ .

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From Eq. (10), it can be seen that the key parameters of an asphere are $K$ , $c$ and its aperture size $d$ . In our simulation, the radius of reference sphere is equal to $1 / c$ , thus we analyzed the testing ability of our method with respect to the parameter $K$ and numerical aperture $(N A = 0.5 \cdot d \cdot c)$ .Three situations representing three different kinds of conic surfaces are considered for the $K = - 0.1, - 1, - 10$ and $N A = 0.1, 0.18, 0.3, 0.5$ respectively, the corresponding fringe patterns are shown in Fig. 4(a) and (b).

In order to avoid the fringe aliasing, the Nyquist condition, expressed mathematically by Eq. (11), should not be violated [14].

\frac{\partial W (x, y)}{\partial x} \leq \frac{λ}{2 \cdot (Δ x)},

where

W (x, y) = 2 \cdot [Z (s) - (Z (s) |_{K = 0})]

is the wavefront deformation,

Z (s)

is defined by Eq. (10),

λ

is the wavelength,

Δ x

is the distance between the two consecutive pixels and can be set equal to one for the simulation. Figure 4(a) illustrates the maximum

N A

of the optics, that is allowed when using our proposed method. It can be seen that for the different conic constant

(K)

, the measurable

N A

is different. Based on Eq. (11), the testing ability of our method was analyzed with respect to arbitrary

K

(here between −10 and 0) as shown in Fig. 4(c). It can be seen that the measureable

N A

is increasing rapidly when

K

is larger than −1, when the surface becomes close to a sphere

(K = 0)

.

It should be noted that the analysis above is restricted to conic surfaces, i.e. vanishing aspheric coefficients. As the ESAS method is strictly limited to the middle apheres, only the aspheric coefficient $A$ is considered. For the different $A$ ,the measurable $N A$ , varied with the conic constant $(K)$ , are shown in Fig. 5. It can be seen that the effect of the aspheric coefficient is not significant when it is smaller than 1e-10. Along with the increasing coefficient, the effect is more significant and it becomes a dominating factor than $K$ after larger than 1e-8.

Fig. 5 The testing ability of our method with respect to the different aspheric coefficient $A$

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5. Experiment

Two experiments were carried out to test the performance of our method further. The experiment setup is shown in Fig. 6. Besides, two null testing for the tested optics, using the Hindle Sphere method [15], were carried out for the cross testing.

Fig. 6 Experimental setup: an adjustment stage with 5 degrees of freedom is used to add dynamic tilt and an attenuation filter is used to obtain the interferograms with high contrast.

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5.1 Testing procedures

The testing procedures are as follows,

1. choosing the appropriate transmission sphere(TS) according to the numerical aperture( $N A$ ) of the specimen,
2. aligning the TS and the specimen,
3. moving the vertex of the specimen to the location of the cat eye of the interferometer,
4. moving the adjusting stage to get the center local nearly null interference pattern based on the nominal radius of vertex curvature of the tested optics,
5. fine-tuning the adjustment stage to get the other local nearly null interference patterns,
6. obtaining the phase data and stitching.

5.2 Application to the ESAS test of two conic surfaces

In the first experiment, a concave ellipsoid surface with an aperture of 100 mm is tested. The nominal parameter of radius of vertex curvature is −348.6 mm. The testing and stitching results are shown in Fig. 7(a) and Fig. 7(b) respectively. In another experiment, a convex hyperboloid surface with the aperture of 60 mm is tested. The nominal parameter of radius of vertex curvature is 428.5 mm. The testing and stitching results are shown in Fig. 8(a) and Fig. 8(b) respectively. Besides, the null testing results are shown in Fig. 7(c) and Fig. 8(c). Note that the surface sag with respect to the vertex reference sphere in normal direction was included in the stitching result. In order to obtain the surface figure and make a comparison with the null testing result, the surface sag was removed using the data subtracting technique in the MetroPro software provided by ZYGO Corporation. Moreover, the differences between the corresponding pixels of the stitching result and the full-aperture null testing result are shown in Fig. 7(d) and Fig. 8(d) respectively. For the purpose of comparison, here the central obscuration area in the stitching results is excluded by the circular masks.

Fig. 7 Testing result of concave ellipsoid. (a) nine fringe patterns. (b)stitching result with surface sag removed(PV = 0.2213wave, RMS = 0.0176wave). (c)null testing result(PV = 0.198 wave, RMS = 0.015wave). (d)residual error(PV = 0.0929 wave, RM S = 0.0117wave). (wave = 0.6328 um)

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Fig. 8 Testing result of convex hyperboloid. (a) nine fringe patterns. (b)stitching result with surface sag removed(PV = 0.1872wave, RMS = 0.0211wave). (c)null testing result(PV = 0.190 wave, RM S = 0.022wave). (d)residual error(PV = 0.062 wave, RM S = 0.065wave). (wave = 0.6328 um)

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The PV of the residual error between the stitching result and the null testing result are 0.0929 wave and 0.062 wave respectively and RMS of the residual error are 0.0117 wave and 0.0065 wave respectively. It can be seen that the residual error and the manufacturing error of the transmission sphere in our experiment (PV is 0.049 wave and RMS is 0.007 wave) are at the same level. Therefore, the residual error is acceptable and the stitching result is valid and of satisfactory accuracy.

6. Conclusion

An elliptical sub-aperture stitching (ESAS) method is proposed to measure the aspheric surfaces in non-null configuration. Testing the aspheric optics against the reference sphere with its radius equals to that of the vertex curvature of the tested optics by adding dynamic tilt, then combining all the local nearly null testing results to obtain the full-aperture result. Simulation results show that the ESAS method is very effective when measuring small and middle aspheric optics. It should be noted that the number of sub-apertures in our simulation is nine and twenty five respectively. However, in our experiment nine sub-apertures is used for the aspheric departure is small and a satisfied result is obtained. The number of sub-apertures required, depends on the optics to be tested. Generally speaking, a larger departure will result in a higher number, For a certain $N A$ , a minimum number of sub-apertures is required, in order to obtain a valid result. Thus, a further research is needed for the optimal number.

Acknowledgments

We are grateful to Ms. Zheng Xue at Xi'an Research Institute of Applied Optics, China, for her kind help about the null testing and for providing us with partial experimental data used in this paper. This work is supported by the Program of Chang Jiang Scholars and Innovative Research Team in University (grant no. IRT1033).

References and links

1. D. Malacara, Optical Shop Testing, (Wiley 2007).

2. A. E. Lowman and J. E. Greivenkamp, “Interferometer errors due to the presence of fringes,” Appl. Opt. 35(34), 6826–6828 (1996). [CrossRef] [PubMed]

3. D. Liu, Y. Yang, C. Tian, Y. Luo, and L. Wang, “Practical methods for retrace error correction in nonnull aspheric testing,” Opt. Express 17(9), 7025–7035 (2009). [CrossRef] [PubMed]

4. C. J. Kim and J. C. Wyant, “Subaperture test of a large flat or a fast aspheric surface,” J. Opt. Soc. Am. 71, 1587 (1981).

5. Y. M. Liu, G. N. Lawrence, and C. L. Koliopoulos, “Subaperture testing of aspheres with annular zones,” Appl. Opt. 27(21), 4504–4513 (1988). [CrossRef] [PubMed]

6. M. Melozzi, L. Pezzati, and A. Mazzoni, “Testing aspheric surfaces using multiple annular interferograms,” Opt. Eng. 32(5), 1073–1079 (1993). [CrossRef]

7. F. S. Granados-Agustin, F. Escobar-Romero, and A. Cornejo-Rodriguez, “Testing a paraboloid mirror using annular subapertures without auxiliary optics,” Proc. SPIE 4829, 44–45 (2003). [CrossRef]

8. X. Hou, F. Wu, L. Yang, and Q. Chen, “Experimental study on measurement of aspheric surface shape with complementary annular subaperture interferometric method,” Opt. Express 15(20), 12890–12899 (2007). [CrossRef] [PubMed]

9. P. Murphy, G. Forbes, J. Fleig, P. Dumas, and M. Tricard, “Stitching interferometry: a flexible solution for surface metrology,” Optics & Photonics News 14(5), 38–43 (2003). [CrossRef]

10. S. Chen, S. Li, Y. Dai, and Z. Zheng, “Lattice design for subaperture stitching test of a concave paraboloid surface,” Appl. Opt. 45(10), 2280–2286 (2006). [CrossRef] [PubMed]

11. P. F. Zhang, H. Zhao, X. A. Zhou, and J. J. Li, “Sub-aperture stitching interferometry using stereovision positioning technique,” Opt. Express 18(14), 15216–15222 (2010). [CrossRef] [PubMed]

12. M. F. Küchel, “Interferometric measurement of rotationally symmetric aspheric surfaces,” Proc. SPIE 7389, 738916 (2009). [CrossRef]

13. J. Fleig, P. Dumas, P. E. Murphy, and G. W. Forbes, “An automated subaperture stitching interferometer workstation for spherical and aspherical surfaces,” Proc. SPIE 5188, 296–307 (2003). [CrossRef]

14. Z. Malacara, Interferogram Analysis for Optical Testing, (The Chemical Rubber Company 2010).

15. M. A. Abdulkadyrov, A. N. Ignatov, V. E. Patrikeev, V. V. Pridnya, A. V. Polyanchikov, A. P. Semenov, Y. A. Sharov, E. Atad-Ettedgui, I. Egan, R. J. Bennett, and S. C. Craig, “M1 and M2 mirrors manufacturing for VISTA telescope. in Astronomical Telescopes and Instrumentation,” Proc. SPIE 5494, 374–381 (2004). [CrossRef]

Non-null testing for aspheric surfaces using elliptical sub-aperture stitching technique

Abstract

1.Introduction

2.Theory

2.1 Principle of local nearly null testing

2.2 Stitching algorithm

3. Simulation

4.Measurability analysis

5. Experiment

5.1 Testing procedures

5.2 Application to the ESAS test of two conic surfaces

6. Conclusion

Acknowledgments

References and links

Cited By

Figures (8)

Equations (11)

Optics Express