A self-scanned 1024 element photodiode array and minicomputer are used to measure the phase (wavefront) in the interference pattern of an interferometer to lambda/100. The photodiode array samples intensities over a 32 x 32 matrix in the interference pattern as the length of the reference arm is varied piezoelectrically. Using these data the minicomputer synchronously detects the phase at each of the 1024 points by a Fourier series method and displays the wavefront in contour and perspective plot on a storage oscilloscope in less than 1 min (Bruning et al. Paper WE16, OSA Annual Meeting, Oct. 1972). The array of intensities is sampled and averaged many times in a random fashion so that the effects of air turbulence, vibrations, and thermal drifts are minimized. Very significant is the fact that wavefront errors in the interferometer are easily determined and may be automatically subtracted from current or subsequent wavefrots. Various programs supporting the measurement system include software for determining the aperture boundary, sum and difference of wavefronts, removal or insertion of tilt and focus errors, and routines for spatial manipulation of wavefronts. FFT programs transform wavefront data into point spread function and modulus and phase of the optical transfer function of lenses. Display programs plot these functions in contour and perspective. The system has been designed to optimize the collection of data to give higher than usual accuracy in measuring the individual elements and final performance of assembled diffraction limited optical systems, and furthermore, the short loop time of a few minutes makes the system an attractive alternative to constraints imposed by test glasses in the optical shop.

Digital wavefront measuring interferometer for testing optical surfaces and lenses

Diffraction characteristics of general dielectric planar (slab) gratings and surface-relief (corrugated) gratings are reviewed. Applications to laser-beam deflection, guidance, modulation, coupling, filtering, wavefront reconstruction, and distributed feedback in the fields of acoustooptics, integrated optics, holography, and spectral analysis are discussed. An exact formulation of the grating diffraction problem without approximations (rigorous coupled-wave theory developed by the authors) is presented. The method of solution is in terms of state variables and this is presented in detail. Then, using a series of fundamental assumptions, this rigorous theory is shown to reduce to the various existing approximate theories in the appropriate limits. The effects of these fundamental assumptions in the approximate theories are quantified and discussed.

Analysis and applications of optical diffraction by gratings

A system of two aspheric lenses is described, which efficiently converts a collimated Gaussian beam to a flattop beam. Departing from earlier designs, both aspheric surfaces were convex, simplifying their fabrication; the output beam was designed with a continuous roll-off, allowing control of the far-field diffraction pattern; and diffraction from the entrance and exit apertures was held to a negligible level. The design principles are discussed in detail, and the performance of the as-built optics is compared quantitatively with the theoretical design. Approximately 78% of the incident power is enclosed in a region with 5% rms power variation. The 8-mm-diameter beam propagates approximately 0.5 m without significant change in the intensity profile; when the beam is expanded to 32 mm in diameter, this range increases to several meters.

Design and performance of a refractive optical system that converts a Gaussian to a flattop beam.

Since Gabor's first hologram, the meanings of that term have grown with increased use of the invention. Computers expanded applications further, and this paper surveys the methods and techniques of computer generated holograms.

Computer generated holograms: an historical review

We describe a generalized phase-shifting interferometry for which the reference phases are directly evaluated at each time that the interference fringe data are read. The reference phases are obtained from the additional straight fringes on the interfering plane by the fast-Fourier-transform method. According to error estimation, the repeatabilities in the measurements of optical surfaces are λ/500 rms, when the generalized algorithm with eight data acquisitions is used.

Generalized phase-shifting interferometry

A set of holographic filters was developed to convert the Gaussian intensity distribution of a collimated laser beam into a uniform one. The design and the fabricating method of the holographic filters are presented and experimental results are shown.

Reshaping collimated laser beams with Gaussian profile to uniform profiles.

A digital phase-measuring interferometer with a laser-diode source has been developed that is based on a fringe-scanning technique with a stepwise wavelength change by variation of the laser injection current. The phase is changed to produce a relative phase difference between the beams in the two arms of the interferometer. Calibrated phase shifts used for a phase-extraction algorithm are derived from one-dimensional least-squares fits to cosine fringe functions to achieve accurate results. Experimental results are presented.

Digital phase-measuring interferometry with a tunable laser diode.

A digital phase measuring interferometer with a frequency-modulated laser diode using the integrated-bucket technique is described. The injection current is continuously changed to introduce a time-varying phase difference between the two beams of an unbalanced Twyman-Green interferometer. The intensity of the interference patterns is integrated with a CCD array sensor for intervals of one-quarter period of the fringe. Using the intensity data a microcomputer calculates the phase to be detected. Some experimental results with the interferometer are presented; the rms repeatability obtained was λ/80.

Heterodyne interferometry with a frequency-modulated laser diode

A polygonal holographic scanner that combines a reflection volume hologram on dichromated gelatin with a computer-generated hologram (CGH) is described. Such a scanning system allows for a compact folded version and highly efficient Bragg diffraction into a single order. A design of the scanner with a nonspherical wave front based on ray tracing and the damped least-squares optimization technique allows a flat-field linearized scan to be used. A suitable trade-off between field flatness and position linearity of scan is adopted. The optimized wave front designated by polynomials is encoded into a CGH. Experiments demonstrating the feasibility of this scannner are presented. Additional data on the diffraction efficiency and the spectrally diffracted intensity of the reflection volume gratings are also shown.

Flat-field linearized scans with reflection dichromated gelatin holographic gratings.

Color-coded, simultaneous recognition of several characters is experimentally performed using wavelength-triplexed, reflection-type holographic filters made with the primary wavelengths, 632.8, 514.5, and 488.0 nm. Helpful data on the diffraction efficiency and the spectrally diffracted intensity of reflection-type holograms are presented.

Yukihiro Ishii

Papers

Reshaping collimated laser beams with Gaussian profile to uniform profiles.

Digital phase-measuring interferometry with a tunable laser diode.

Heterodyne interferometry with a frequency-modulated laser diode

Flat-field linearized scans with reflection dichromated gelatin holographic gratings.

Color-coded character-recognition experiment with wavelength-triplexed, reflection-type holographic filters