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# Near-null optics

In off-axis subapertures of most aspheres, astigmatism and coma dominate the aberrations with approximately quadratic and linear increases as the off-axis distance increases. Therefore, the function of near-null optics is to mostly correct the astigmatism and coma. Khchel30 summarized some typical solutions capable of compensating astigmatism and coma. However, solutions such as the tilted two-mirror design or three-mirror design do not meet the requirements of compact space and easy alignment. QED Technologies proposed the VON26 technology combined with a subaperture test. A counter-rotating Risley prism pair with an adjustable overall tilt is utilized as the near-null optics to generate variable astigmatism, coma, and trefoil (not completely independent) for aberration correction.

As shown by Acosta and Bara,58 variable amounts of pure Zernike modes can be generated by rotating a pair of Zernike plates that can be used to calibrate ocular aberrometers. Mills et al.59 also proposed the idea of using a pair of counterrotating phase plates for conformal dome aberration correction. The plates are thickness variant, basically structured as freeform surfaces described by Zernike polynomials. We take advantage of this property and show here that a pair of counter-rotating Zernike plates (Fig. 12) can be used to generate variable astigmatism and coma, which enables the subaperture testing of different aspheres. The counter-rotating angle is the only dof. Without an overall tilt, the compact design makes it easy to fit the short space between the interferometer and the test mirror.

Figure 12 Schematic diagram of the counter-rotating Zernike plates.

Furthermore, the plates can be fabricated with CGHs, including alignment patterns to facilitate calibration and alignment.

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

where p is the normalized radial pupil coordinate, 0 is the angular coordinate, and a and b are the coefficients of Z5 and Z7, respectively. The other plate has a complementary phase function, i.e., -aZ5 - bZ7. Variable aberrations comprising Z4 and Z6 terms are then generated by counter-rotating these two plates by an angle a:

We demonstrate the capability of aberration correction for some convex mirrors. The first one is a convex hyperbolic (mirror 1). The aperture is 360 mm, the conic constant is K = -2.1172, and the radius of curvature at the vertex is Roc = -772.48 mm. Its aspheric departure is about 151 pm and requires up to 142 subapertures in a non-null test, as shown in Fig. 10. While in a near-null test using a 4-in. beam, three rings of subapertures are arranged with p = 3.8, 7.6, and 11.4 deg, respectively. The total number of subapertures is 44, as shown in Fig. 13.

Mirror 2 is a sixth-order even asphere (Roc = -1023.76 mm, K = 0, and the clear aperture is 320 mm) with about 34 pm in aspheric departure. It also requires three rings of subapertures with p = 2.5, 5, and 7.5 deg, respectively. Figure 14 shows the optical layout of a 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 distances) are identical. Therefore, only aberrations of the three off-axis subapertures along the x direction are calculated. Then, according to Eq. (18), we obtain the coefficients a and b of Z5 and Z7 for the phase

Figure 13 Near-null subaperture layout.

Figure 14 Optical layout of near-null subaperture test.

function and the counter-rotating angles a by solving the system of nonlinear equations

where P4; and P;* are the calculated coefficients of Z4 and Z6, respectively, for single-pass wavefront aberrations of different subapertures.

The aberrations of the central subaperture are generally small enough to be resolved directly by the interferometer. Therefore, the counter-rotating angle for the two plates is zero.

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 +first order. The fringe contrast will be better for silicon carbide test mirrors or coated mirrors. For uncoated glass materials, a four-level CGH is used to achieve an approximate efficiency of 80%.60 The major disturbance orders are combinations of -third and +fifth orders with theoretical diffraction efficiencies of 9.01% and 3.24%, respectively. For example, (-3, +5,-3, +5) orders for a double pass through the two CGHs consequently produce a negligible ghost fringe because the efficiency is <0.001%. In order to separate the disturbance orders of diffraction, a power carrier is introduced. The two plates have different power carriers so that a TF can be used to simplify the alignment of CGHs regarding the interferometer.

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 powers, and a TF is used to facilitate the alignment. Figure 15 shows the simulated interferograms of different subapertures in a near-null test, which are

Table 1 Phase description of the two plates.

 Normal radius (mm) Term 4 Term 5 Term 7 50 -201.7 28 9.1 50 -302.6 -28 -9.1

Figure 15 Near-null subaperture interferograms at different off-axis distances (X = 632.8 nm): (a) Mirror 1 (peak-to-valley residual aberrations are 3.2X, 5.7X, 3.8X, and 9.0X, respectively) and (b) mirror 2 (peak-to-valley residual aberrations are 3.1X, 4.6X, 5.6X, and 4.0X, respectively).

definitely resolvable by a standard interferometer. In contrast, the non-null inter- ferograms of the outmost subapertures at the edge of the two mirrors contain more than 130 fringes and 50 fringes, respectively.

The near-null optics is also applicable to some other aspheres of different shapes. Most of the 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, and the clear aperture is 352 mm) of the Stratospheric Observatory for Infrared Astronomy61 can be tested with the same pair of phase plates. Limited by the 4-in. aperture, it requires four rings of subapertures. All residual subaperture aberrations are confirmed to be <5Z. In contrast, the aberration of the outmost subaperture is about 40Z before correction.

When the same plates are applied to concave aspheres, the off-axis direction is reversed (-x direction). By virtue of the variable aberration correction capability, the Zernike plate-based reconfigurable optical null may also be extended to test cylindrical or even freeform optics, though different phase patterns will be written on the CGHs in different applications.

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