A technique for testing large, highly aspheric convex secondary mirrors is being pursued at the Steward Observatory Mirror Lab. This new test will use a full aperture test plate with a computer-generated hologram (CGH) fabricated onto a concave spherical reference surface. Fringes of interference will be viewed through the test plate, which will be supported several millimeters from the secondary. The hologram will consist of annular rings of metal spaced at intervals as small as 80 µm and as large as 500 µm. The accuracy of the surface measurement using this technique is expected to be better than 60 nm P-V and 10 nm rms.

The holographic test plate is very attractive for testing secondary mirrors for several reasons listed below:

1. Inherent accuracy. The accuracy of the test is limited only by the quality of a concave spherical surface and the accuracy of the ring locations. Both may be verified to high accuracy. Also, since this is a full aperture test, azimuthal errors in the holographic test plate may be separated from those in the secondary mirror by using a rotation test. For further verification, an interferometric measurement of the holographic test plate using a null corrector or Silvertooth test may be performed.

2. High testing efficiency. Once the test facility is in place, this test will allow rapid testing to very high accuracy. Vibration and atmospheric effects that often make interferometric testing difficult will be negligible for the holographic test plate. Highly accurate alignment of the secondary with its test plate may be easily performed. Using phase-shifting interferometry on a null fringe, high resolution maps of the surface shape will be obtained in a matter of minutes.

3. Moderate cost. There is no inexpensive way to accurately test convex aspheres. Testing with a holographic test plate is likely to cost less than conventional optical tests or mechanical profilometry, although this has not yet been determined. The cost advantage of using a test plate is more extreme for situations where more than one identical secondary will be made.

4. Reasonable time scale. It is expected that this method can be tested on a small scale in three months and implemented for a 1- to 1.8-meter secondary mirror in less than a year.

5. Potential spin-offs. The technology developed for fabricating large holograms may have uses in fields other than optical testing. The ability to produce 2-m class diffractive optical elements could open up exciting new research possibilities.

The interferometric testing of secondary mirrors using a holographic test plate is a hybrid of conventional test plate methods and common CGH testing. Convex spherical surfaces are commonly tested with matching concave test plates. When the test plate is placed on the convex optic and illuminated with sufficiently coherent light, fringes of interference show the shape difference between the two parts. Since the concave sphere may be tested independently from its center of curvature, this shape difference is used to measure the convex surface. To test an aspheric convex surface this way, an aspheric concave test plate must be fabricated and tested.

Optical testing using computer-generated holography has proven to be accurate and cost effective for measuring aspheric surfaces. This test works by imaging the aspheric surface onto a CGH and using diffraction to alter a collimated reference wavefront to give it the shape expected by the test wavefront. The CGH is a diffraction grating that uses a spatial variation in ruling frequency to create a desired change in wavefront. The CGH may be equivalently interpreted as changing ray directions according to the grating equation or as directly changing the wavefront phase. When used in m ^th order, the CGH effectively adds m waves of optical path to the wavefront for each ruling cycle. The CGH, which usually consists of an opaque pattern on a glass substrate, may be fabricated in the same way integrated circuit masks are made. A large-scale master may be plotted and photographically reduced. Since the CGH will generally produce many orders of diffraction, it is critical that all unwanted diffracted orders are blocked with an aperture. To fully separate the orders, a large amount of tilt is usually incorporated into the CGH. This causes a lateral fanning out of the orders enabling the isolation of a single order.

Light scattered from the grating will propagate in directions where constructive interference causes maximal intensity. This constructive superposition occurs when the phase difference between rulings is an integral number of wavelengths, or mathematically

d_o + d _m = s (sin ₀ + sin _m ) = m

where

d₀, d _m = incident and diffracted path length shown in Fig 1.
₀, _m = incident and diffracted angle
m = order of diffraction
s = local ruling spacing.

Testing with a CGH may be thought of in terms of a moiré effect. The CGH is a binary representation of the expected fringe pattern formed by an aspheric test beam and a reference beam. When the live interference pattern is superimposed on the CGH, a moiré, or spatial frequency "beating" effect is observed. When properly filtered in the frequency domain, the moiré fringes directly give the shape difference between the two wavefronts, thus the shape error in the asphere.

The CGH test plate uses a fringe pattern drawn directly onto the spherical reference surface of a test plate. This makes the test beam and reference beam fully common-path, eliminating any possibility of error due to illumination and imaging optics. The common-path implementation also makes the test insensitive to vibration and atmospheric effects.

The principle of this test, shown schematically in Fig. 2, uses the concave spherical surface as a beamsplitter and reference surface. The test plate is illuminated with light that refracts from the reference sphere to strike the secondary mirror at normal incidence. This light reflects back onto itself to form the test beam. Any figure errors in the secondary mirror will be imparted to this test wavefront. The reference wavefront is formed by diffraction from the ring pattern on the reference sphere. The CGH is designed to make this reference beam match the test wavefront so it also retraces its path. (This is known as a Littrow configuration for a grating.) To perform a test, laser light from a point source will be used so the desired wavefronts will converge to a conjugate image point.

Fig. 2. Diagram of CGH test plate method showing the relationship between the test beam, the reference beam, and spurious diffracted orders.

Rather than using tilt in the CGH to separate the diffracted orders, the circular symmetry of the test is preserved by using power or focus to separate the orders. The orders are separated by coming to focus at different axial positions. By designing the test to make the desired order focus to a point and placing a small aperture at the focal point, a single order is largely isolated. The out of focus orders will also pass through this stop and cause spurious fringes, but only in a small central region of the secondary. By optimally choosing the amount of power in the CGH and the size of the stop, this leakage is limited to the central unused region of the mirror that is ignored during testing.

There are several feasible methods currently being studied to fabricate a rotational CGH onto a curved surface. These methods generally use a precise rotary stage to spin the test plate under a linear stage that controls the radial position of an optical device that draws the pattern. The stages need only several-micron accuracy, which is readily available in coordinate measuring machines. The pattern may be drawn by exposing photoresist or actually ablating a metallic coating with a high-energy laser. It may be useful to fabricate the CGH as a single spiral rather than concentric rings.

The accuracy of the wavefront created by CGH is directly related to the positional accuracy of the bands. The error in the diffracted wavefront is simply

( W / ) = (x / s)

Where x is the error in band position, s is the local band spacing. The CGH test plate may have spacings as small as 80 µm so an accuracy of 5 µm in the band position will guarantee /16 wavefront quality.

The optics used to illuminate the test plate are not required to be very accurate. Since the optical path difference between the reference and test surfaces is only several millimeters, wavefront slope errors as large as 1 mrad will cause measurement errors of several nanometers. However, slope errors larger than 1 mrad will affect the ability to block unwanted diffracted orders.

Test implementation

A preliminary design for a facility to perform this test on all of the proposed secondary mirrors (See Table 1) has been performed. The design of the imaging system was driven by the most difficult mirror to be tested, the MMT F/5 secondary. The illumination system needs only to send the light in the correct direction to about one mrad and to create an undistorted image of the test optic. The orientation of the test has not yet been decided. A two-mirror illuminator was chosen to keep the system small and to preserve the circular symmetry. The illuminator, shown in Fig. 3, consists of a 2.5-m diameter primary mirror with a 3.8-m radius of curvature and a small secondary. The 2.5-m primary is an ellipsoid with conic constant K near -0.35 and will work for the illumination of all proposed secondary mirrors. The secondary mirror is a convex sphere that will be mounted onto the test plate. The test plate will have a concave reference spherical surface with the CGH and a low-accuracy spherical back surface. Each secondary tested will need its own test plate and convex secondary illuminator mirror. For all tests, spherical surfaces are used for the test plate and small secondary.

Table 1. Proposed secondary mirrors to be tested with CGH test plate.

mirror	Sloan	MMT F/9	MMT F/15	MMT F/5	Col F/15	Col f/4	ARC F/8
outer diameter (mm)	1200	996	620	1653	747	1235	792
radius of curvature (mm)	7194	2806	1663	5022	1776	3690	3167
conic constant	-12.161	-1.749	-1.397	-2.640	-1.357	-3.737	-2.185

The wavefront aberrations due to the illuminating system are required to be less than 1 mrad. This loose tolerance allows errors in the surface figures and a large refractive index inhomogeneity in the test plate.

There are several ideas for inexpensively fabricating the 2.5-m illuminator. The required accuracy of this optic is very low by optical standards. The mirror could be made of metal, plate glass, or epoxy and mounted to a steel support. It could be machined or lapped to the correct shape and polished smooth. Since it is a concave ellipse, the mirror can be optically tested, although it should not be necessary to do so.

Optimization of test for rejection of stray orders

The optimization of the test involves several tradeoffs with the order rejection. An increased amount of power in the CGH spreads the orders out, making them easier to separate. It also causes an increase in the number of fringes, increasing cost and decreasing accuracy. The use of a smaller aperture blocks the unwanted orders better, but it also may cause a decrease in measurement resolution and requires higher quality illumination optics. These two trade-offs are treated separately below using the Sloan secondary as an example.

A free parameter in the design of this test is the amount of power in the CGH. The wavefront slope for different amounts of power is shown in Fig. 4. The amount of power for each curve was chosen by matching the slope at the edge to the slope at other radial zones. The diffracted orders next to the desired orders will vary in slope by the amount shown in Fig. 4. A two mrad radius aperture and the power set at to match the 0.3 zone will allow the test of the Sloan secondary outside the 0.2 zone.

The wavefront function generated by the CGH is shown in Fig. 5 for the same amounts of power. Since it is used in first order, the CGH would be made to have one ring per wave OPD. Choosing power to match the edge slope at the 0.3 zone, 3000 rings will be required. The ring spacing corresponding to the OPD is shown in Fig. 6. The measurement error produced by a band position error in the CGH is a function of spacing. Fig. 7 shows the variation in measurement error across the pupil for a 5 µm position error in the CGH.

The aperture used to block unwanted light is specified in terms of ray slope at the CGH. The absolute size of the stop depends on the focal length of the imaging system. This aperture will be placed at the focal point or Fourier transform plane of the imaging system. The size of this stop determines the magnitude of allowable errors in the illuminator. For the above example of the Sloan secondary, a 4 mrad diameter pinhole could be used when the power is set for the edge slope to match the 0.3 zone.

Optimization of diffractive efficiency

The fringe contrast is maximized by matching the intensity of the test and reference wavefronts. This requires either very thin bands of highly reflective aluminum, or the use of a highly absorptive metal.

The efficiency of the diffraction into each order for this test was modeled using Fourier optics. The use of metal bands on uncoated glass was assumed for the test plate and uncoated glass assumed for the secondary. The metal layer will be thick enough onto the glass to prohibit any transmission through the metal. The test beam uses the 0-order transmitted through an opaque binary hologram. The reference beam uses the -1 diffracted order from the CGH. This wavefront is modulated in amplitude by the reflectivity of the band material and in phase by a p shift between the light reflected from the metal and that from the bare glass.

A free parameter in the design of the CGHs is the duty cycle: the ratio of the width of the bands to the center-to-center spacing. Fig. 8 shows how the fringe contrast depends on the duty cycle for metal bands with reflectivity R of 2% to 90%. A duty cycle of 0.5 is desired because it makes the fabrication and verification of the rulings easier and also eliminates the even diffractive orders. A highly absorptive metal is required to match the beam intensities for a duty cycle of 0.5. The reticle industry uses black chrome with reflectivity between 2% and 5%. This material has excellent adhesion, is readily etched, and easily applied.

Conclusion

A CGH test plate technique may be a highly accurate and efficient method for testing large convex aspheres. The fabrication of the holograms, consisting of annular rings of metal, can be performed to machine-tool accuracy. By optimizing the test geometry, the CGH metal, and the ruling duty cycle, the test can give surface data with a high signal-to-noise ratio over the entire tested region.