## Abstract

In off-axis subapertures of most aspheres, astigmatism and coma dominate the aberrations with approximately quadratic and linear increase as the off-axis distance increases. A pair of counter-rotating Zernike plates is proposed to generate variable amount of Zernike terms Z4 and Z6, correcting most of the astigmatism and coma for subapertures located at different positions on surfaces of various aspheric shapes. The residual subaperture aberrations are then reduced within the vertical dynamic range of measurement of the interferometer. The plates are fabricated with computer generated holograms and the experimental results show the variable aberration correction effect without ghost fringes. The same plates are reconfigurable by counter-rotating to enable near-null test of various aspheres flexibly.

© 2014 Optical Society of America

## 1. Introduction

Null optics such as null lens and computer generated holograms (CGHs) are nowadays widely used in surface figure metrology for aspheres. Since they are designed to balance the aspheric wavefront aberration perfectly, null optics are used in “one-to-one” mode. That means different surface shape requires different null optics. Another problem specially exists in null test of large convex aspheres is the limited range of measurement. In contrast, subaperture stitching interferometry shows advantages of extended lateral and vertical range of measurement [1]. But it is impractical when applied to steep aspheres because too many subapertures are required to reduce the suabperture aspheric departure effectively [2]. Null optics is therefore incorporated to balance the subaperture aberration. However, subapertures at different off-axis distances require different null optics because the aberration changes dramatically. QED technologies proposed the variable optical null [3] technology combined with subaperture test. A counter-rotating Risley prism pair with adjustable overall tilt is utilized to generate variable astigmatism, coma and trefoil (not completely independent) for aberration correction. It enables subaperture test of various aspheric shapes. Since the aberrations are partially corrected, it is also referred to as near-null test, known from the null test and non-null test.

We show here a pair of counter-rotating Zernike plates (Fig. 1) can be used to generate variable astigmatism and coma, and hence enables subaperture test of different aspheres. The counter-rotating angle is the only degree of freedom. Without overall tilt, the compact design makes it easy to fit the short space between the interferometer and the test mirror. Furthermore, the plates can be fabricated with CGHs including alignment patterns to facilitate calibration and alignment.

## 2. Aberrations of off-axis aspheric subapertures

Consider an off-axis subaperture of a convex conic surface:

where*R*is the radius of curvature at vertex (negative for convex), and

*e*is the eccentricity. The coordinates of the geometric center of the subaperture are (

*x*

_{0}, 0,

*z*

_{0}) in the parent frame. The optical axis of the testing system is approximately the surface normal at the geometric center when testing the subaperture. A local frame is hence built as shown in Fig. 2.

By simple coordinate transformation, the local coordinates of measuring points are related as follows:

*β*is the off-axis angle between the optical axis of the testing system and that of the parent surface. Therefore coordinate

*z*can be solved as a function of (

*x*,

*y*). With spherical component

*z*subtracted from

_{s}*z*, we then get the analytical description of the wavefront aberration. By using the Maclaurin series expansion up to third-order terms, the wavefront aberration is described as follows:

The other terms disappear because the off-axis is purely in x direction without loss of generality. The coefficients are explicitly related to Seidel aberrations [4], and Zernike polynomial terms Z4 (astigmatism at 0 degree and focus), Z6 (coma and x-tilt) and Z9 (trefoil) in Cartesian coordinates:

The corresponding coefficients are obtained as below:

*e*

^{2}sin

^{2}

*β*is far less than 1. Moreover

*P*

_{4},

*P*

_{6}, and

*P*

_{9}are quadratically, linearly and cubically proportional to

*β*, respectively, if

*β*is small. Therefore in testing of off-axis subapertures, we may correct mostly the astigmatism and coma, while leaving the trefoil uncorrected.

## 3. Phase function design of the Zernike plates

As shown by Acosta and Bará [5], variable amounts of pure Zernike modes can be generated by rotating a pair of Zernike plates, which can be used to calibrate ocular aberrometers. Mills et. al. [6] also proposed the idea of using a pair of counter-rotating phase plates for conformal dome aberration correction. The plates are thickness variant, basically structured as free-form surfaces described by Zernike polynomials. Implementation of the idea for surface figure metrology has not yet been reported to the best of our knowledge. We take advantage of this property to propose a new design of reconfigurable optical null.

Suppose the phase function of one plate comprises two terms Z5 (astigmatism at 45 degrees and focus) and Z7 (coma and y-tilt) of Zernike polynomials:

*ρ*is normalized radial pupil coordinate,

*θ*is angular coordinate,

*a*and

*b*are coefficients of Z

_{5}and Z

_{7}, respectively. The other plate has complementary phase function, i.e., –

*aZ*

_{5}–

*bZ*

_{7}. Then variable aberrations comprising Z4 and Z6 terms are generated by counter-rotating these two plates by an angle

*α*:

We demonstrate the capability of aberration correction for some convex mirrors. The first one is convex hyperbolic (mirror 1). The aperture is 360 mm, the conic constant is *K* = –2.1172 and the radius of curvature at vertex is Roc = –772.48 mm. Its aspheric departure is about 151μm and requires up to 142 subapertures in non-null test [2]. While in near-null test using 4-inch beam, three rings of subapertures are arranged with *β* = 3.8°, 7.6° and 11.4°, respectively. The total number of subapertures is 44 as shown in Fig. 3(a).

Mirror 2 is a 6-order even asphere (Roc = –1023.76 mm, *K* = 0, clear aperture is 320 mm) with about 34μm in aspheric departure. It also requires three rings of subapertures with *β* = 2.5°, 5° and 7.5°, respectively. Figure 3(b) shows the optical layout of near-null subaperture test of the two surfaces with the phase plates.

Due to the rotational symmetry, aberrations of those subapertures lying on the same ring (with equal off-axis distance) are identical. Therefore only aberrations of the three off-axis subapertures along x direction are calculated. Then according to Eq. (8), we obtain the coefficients *a* and *b* of Z5 and Z7 for the phase function and the counter-rotating angles *α _{i}* by solving the system of nonlinear equations:

*P*

_{4}

*and*

_{i}*P*

_{6}

*are calculated coefficients of Z4 and Z6 for single-pass wavefront aberrations of different subapertures.*

_{i}The aberrations of the central subaperture are generally small enough to be resolved directly by the interferometer. Hence the counter-rotating angle for the two plates is zero. Once all subapertures are tested in such a near-null configuration, specific stitching algorithm can be applied to bring them together [7,8].

Allowing for the double pass of the two plates, care must be taken for the efficiency of diffraction and disturbance orders. Phase-type CGHs are suggested with generally about 40% diffraction efficiency achieved at + 1st order. The fringe contrast will be better for silicon carbide test mirrors or coated mirrors. For uncoated glass materials, 4-level CGH is used to approximately achieve efficiency of 80% [9]. The major disturbance orders are combinations of −3rd and + 5th orders with theoretical diffraction efficiency of 9.01% and 3.24%, respectively. For example, (−3, + 5,-3, + 5) orders for double pass through the two CGHs consequently produce negligible ghost fringe because the efficiency is less than 0.001%. In order to separate disturbance orders of diffraction, a power carrier is introduced. The two plates have different power carrier so that a transmission flat can be used to simplify the alignment of CGHs regarding the interferometer. We will address the thorough design of the CGH phase function with power carrier and alignment patterns in another article.

The final design of the phase plates is represented by Zernike standard polynomials (terms 4, 5, and 7) as listed in Table 1. The two plates have different power and a transmission flat is used to facilitate the alignment. Figure 4 shows the simulated interferograms of different subapertures in near-null test which are definitely resolvable by a standard interferometer. In contrast, the non-null interferograms of the outmost subapertures at the edge of the two mirrors contain more than 130 fringes and 50 fringes, respectively. The CGHs are nominally tipped to avoid the disturbance reflected by the substrates. As a result, all off-axis subapertures along x-axis of the test surface are decentered by about 0.2 mm in y-axis. That’s why the simulated fringe patterns in Fig. 4 are not symmetric about the horizontal axis.

The near-null optics is also applicable to some other aspheres of different shape. Most of subaperture aberrations can still be corrected by properly arranging the subaperture layout and adjusting the counter-rotating angles of the plates, while keeping the phase function unchanged. For example, the secondary mirror (Roc = –954.5 mm, *K* = –1.280, aperture is 352 mm) of Stratospheric Observatory For Infrared Astronomy (SOFIA) [10] can be tested with the same pair of phase plates. Limited by the 4-inch aperture, it requires four rings of subapertures. All residual subaperture aberrations are confirmed to be less than 5 λ. In contrast, the aberration of the outmost subaperture is about 40 λ before correction.

When the same plates are applied to concave aspheres, the off-axis direction is reversed (–x direction). Consider a concave parabolic surface with 540 mm in clear aperture and the radius of curvature is 1400 mm. Similarly three rings of subaperture are arranged. All residual aberrations are again confirmed to be less than 5 λ with the same phase plates. By virtue of the variable aberration correction capability, the Zernike plate-based reconfigurable optical null may also be extended to test of cylindrical or even free-form optics, though different phase pattern will be written on the CGHs in different applications.

## 4. Experimental verification

In the experimental setup, the two phase plates are mounted on two center-through rotary tables and then inserted between the interferometer and the test surface, as shown in Fig. 5. The surface of mirror 2 is used in this experiment. The single-pass wavefronts of the subapertures at the center, rings 1, 2 and 3 are shown in Fig. 6 with the CGHs counter-rotating by proper angles. It is easy to see the aberration correction effect when we compare Figs. 6(c) and 6(d) with Fig. 7 which is obtained without CGHs counter-rotating (*α* = 0). The interferogram of the outmost subaperture shown in Fig. 7(b) is irresolvable while the counter-rotating CGHs successfully reduce it to about 8 fringes (double pass). In addition, the interferogram has good fringe contrast for uncoated glass surface and there is no visible ghost fringe.

By employing the subaperture stitching algorithm [8] to optimally correct the misalignment induced aberrations, all subapertures are stitched together and the final surface error map is shown in Fig. 8. Although the stitching optimization is effective, the results are to be verified by other testing methods, e.g. through-the-back test with null lens, which will be one of our future research focuses. In addition, a more precise aligning mechanism is expected for subaperture measurement to reduce the misalignment aberrations.

## 5. Concluding remarks

Based on the counter-rotating Zernike plates, we present a reconfigurable optical null capable of testing various aspheres flexibly. It is fundamentally different from the traditional “one-to-one” null test mode. By incorporating subaperture stitching interferometry it is particularly advantageous for test of convex aspheres since the full aperture is divided into a series of smaller subapertures which are measurable with reconfigurable optical null. The phase plates are realized with CGHs and the experimental results show the effect of variable aberration correction with good fringe contrast.

## Acknowledgments

This project is supported by National Natural Science Foundation of China (Grant No. 51375488) and National Basic Research Program of China (Grant No. 2011CB013200).

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