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Talbot imaging is a well-known effect that causes sinusoidal patterns to be reimaged by diffraction with characteristic period that varies inversely with both wavelength and the square of the spatial frequency. This effect is treated using the Fresnel diffraction integral for fields with sinusoidal ripples in amplitude or phase. The periodic nature is demonstrated and explained, and a sinusoidal approximation is made for the case where the phase or amplitude ripples are small, which allows direct determination of the field for arbitrary propagation distance. Coupled with a straightforward method for calculating the effect in a diverging or converging beam, the Talbot method provides a useful approximation for a class of diffraction problems. OCIS codes: 050.1940, 070.6760.

A software configurable optical test system (SCOTS) based on the geometry of the fringe reflection or phase measuring deflectometry method was developed for rapidly, robustly, and accurately measuring large, highly aspherical shapes such as solar collectors and primary mirrors for astronomical telescopes. In addition to using phase shifting methods for data collection and reduction, we explore the test from the point view of performing traditional optical testing methods, such as Hartmann or Hartmann-Shack tests, in a reverse way. Using this concept, the slope data calculation and unwrapping in the test can also be done with centroiding and line-scanning methods. These concepts expand the test to work in more general situations where fringe illumination is not practical. Experimental results show that the test can be implemented without complex calibration for many applications by taking the geometric advantage of working near the center curvature of the test part. The results also show that the test has a large dynamic range, can achieve measurement accuracy comparable with interferometric methods, and can provide a good complement to interferometric tests in certain circumstances. Avariation of this method is also useful for measuring refractive optics and optical systems. As such, SCOTS provides optical manufacturers with a new tool for performing quantitative full field system evaluation. OCIS codes: 080.4228, 120.6650, 150.3045.

The LOTIS Collimator provides scene projection within a 6.5 m diameter collimated beam used for optical testing research in air and vacuum. Diffraction-limited performance (0.4 to 5 [micro]m wavelength) requires active wavefront control of the alignment and primary mirror shape. A hexapod corrects secondary mirror alignment using measurements from collimated sources directed into the system with nine scanning pentaprisms. The primary mirror shape is controlled with 104 adjustable force actuators based on figure measurements from a center-of-curvature test. A variation of the Hartmann test measures slopes by monitoring the reflections from 36 small mirrors bonded to the optical surface of the primary mirror. The Hartmann source and detector are located at the f/15 Cassegrain focus. Initial operation has demonstrated a closed-loop 110 nm rms wavefront error in ambient air over the 6.5 m collimated beam. OCIS codes: 110.0110, 120.0120, 220.0220, 110.6770, 120.1680, 220.1080.

Interferometers accurately measure the difference between two wavefronts, one from a reference surface and the other from an unknown surface. If the reference surface is near perfect or is accurately known from some other test, then the shape of the unknown surface can be determined. We investigate the case where neither the reference surface nor the surface under test is well known. By making multiple shear measurements where both surfaces are translated and/or rotated, we obtain sufficient information to reconstruct the figure of both surfaces with a maximum likelihood reconstruction method. The method is demonstrated for the measurement of a 1.6 m flat mirror to 2 nm rms, using a smaller reference mirror that had significant figure error. OCIS codes: 120.3180, 120.3940, 120.4800.

The optical surface of a large optical flat can be measured using an autocollimator and scanning pentaprism system. The autocollimator measures the slope difference between a point on the mirror and a reference point. Such a system was built and previously operated at the University of Arizona. We discuss refinements that were made to the hardware, the alignment procedure, and the error analysis. The improved system was demonstrated with a 1.6 m flat mirror, which was measured to be flat to 12 nm rms. The uncertainty in the measurement is only 9 nm rms. OCIS codes: 120.3940, 220.4840.

Aspheric optical surfaces are often tested using computer-generated holograms (CGHs). For precise measurement, the wavefront errors caused by the CGH must be known and characterized. A parametric model relating the wavefront errors to the CGH fabrication errors is introduced. Methods are discussed for measuring the fabrication errors in the CGH substrate, duty cycle, etching depth, and effect of surface roughness. An example analysis of the wavefront errors from fabrication nonuniformities for a phase CGH is given. The calibration of these effects for a CGH null corrector is demonstrated to cause measurement error less than 1 nm. OCIS codes: 050.1380, 090.2880, 120.2880.

The performance of wide-field multiple-aperture imaging systems is dominated by easily understood, low-order errors. Each aperture produces an individual image, each pair of apertures produces a set of fringes under a diffraction envelope, and the system bandwidth produces a coherence envelope. For wide-field imaging, each of these elements must be coincident in the image plane as the field angle changes. We explore the causes of image degradation, derive first-order rules for preserving image quality across field, and give an example design that enforces some of the rules to achieve a relatively wide-field interferometric imaging telescope.

A binary diffraction model is introduced to study the sensitivity of the wavefront phase of binary computer-generated holograms on groove depth and duty-cycle variations. Analytical solutions to diffraction efficiency, diffracted wavefront phase functions, and wavefront sensitivity functions are derived. The derivation of these relationships is obtained by using the Fourier method. Results from experimental data confirm the analysis. Several phase anomalies were discovered, and a simple graphical model of the complex fields is applied to explain these phenomena. OCIS codes: 050.1380, 090.2880, 120.2880.

Interferometers that use different states of polarization for the reference and the test beams can modulate the relative phase shift by using polarization optics in the imaging system. Thus the interferometer can capture simultaneous images that have a fixed phase shift, which can be used for phase-shifting interferometry. As all measurements are made simultaneously, the interferometer is not sensitive to vibration. Fizeau interferometers of this type have an advantage compared with Twyman--Green-type systems because they are common-path interferometers. However, a polarization Fizeau is not strictly common path when both wavefronts are transmitted by an optic that suffers from birefringence. The two polarized beams see different phases owing to birefringence; as a result, an error can be introduced in the measurement. We study the effect of birefringence on measurement accuracy when different polarization techniques are used in Fizeau interferometers. We demonstrate that measurement error is reduced dramatically and can be eliminated if the reference and test beams are circularly polarized rather than linearly polarized. OCIS codes: 120.3180, 260.5430, 260.1440.

A circular null computer-generated hologram (CGH) was used to test a highly paraboloidal mirror (diameter, 90 mm; f number, 0.76). To verify the null CGH test a classic autocollimation test with a flat mirror was performed. Comparing the results, we show that the results of the null CGH test show good agreement with results of the autocollimation test. OCIS codes: 120.3180, 220.4840, 120.6650.

There is a great demand for new telescopes that use larger primary mirrors to collect more light. Because of the difficulty in the fabrication of mirrors larger than 8 m as a single piece, they must be made with numerous smaller segments. The segments must fit together to create the effect of a single mirror, which presents unique challenges for fabrication and testing that are absent for monolithic optics. This is especially true for the case of a highly aspheric mirror required to make a short two-mirror telescope. We develop the relationship between optical performance of the telescope and errors in the manufacture and operation of the individual segments. OCIS codes: 220.4840, 120.3940.

We developed the pupil astigmatism criteria for correcting the quadratic field-dependent aberrations. These criteria provide an elegant way to determine and correct aberrations that have quadratic field dependence and arbitrary pupil dependence in the same way that the Abbe sine condition is used for aberrations with linear field dependence. Like the sine condition, the pupil astigmatism criteria involve only the properties of the rays originating from the on-axis object point, so it is convenient to implement them in optical design. We introduce an algorithm to apply the criteria in designing new well-corrected optical systems. Some example designs are presented. OCIS codes: 220.1010, 220.3620, 220.4830.

The aberrations of axisymmetric imaging systems can be calculated to third order by use of the Seidel formulas. The Coddington equations give aberrations that have quadratic dependence on the pupil, for all field points. The pupil astigmatism conditions were recently developed to predict and control aberrations that have quadratic field dependence and arbitrary pupil dependence. We investigate the relationship between the exact pupil astigmatism conditions and the classical Seidel treatment of pupil aberrations. OCIS codes: 220.1010, 220.3620.

An advanced interferometer was built for surface metrology in environments with severe vibration. This instrument uses active control to compensate for effects of vibration to allow surface measurement with high-resolution phase-shifting interferometry. A digital signal processor and high-speed phase control from an electro-optic modulator allows phase measurements at 4000 Hz. These measurements are fed back into a real-time servo in the digital signal processor that provides a vibration-corrected phase ramp for the surface measurements taken at video rates. Unlike fringe locking, which compensates vibration to keep the phase constant, we show a true phase servo that allows the phase to be stabilized while it is ramped, enabling surface measurements using phase-shifting interferometry that requires multiple images with controlled phase shifts. OCIS codes: 120.3180, 120.3930, 120.5050, 220.4830.