A fast-Fourier-transform method of topography and interferometry is proposed. By computer processing of a noncontour type of fringe pattern, automatic discrimination is achieved between elevation and depression of the object or wave-front form, which has not been possible by the fringe-contour-generation techniques. The method has advantages over moire topography and conventional fringe-contour interferometry in both accuracy and sensitivity. Unlike fringe-scanning techniques, the method is easy to apply because it uses no moving components.

Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry

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

Digitally sampled in-line holograms may be linearly filtered to reconstruct a representation of the original object distribution, thereby decoding the information contained in the hologram. The decoding process is performed by digital computation rather than optically. Substitution of digital for optical decoding has several advantages, including selective suppression of the twin-image artifact, elimination of the far-field requirement, and automation of the data reduction and analysis process. The proposed filter is a truncated series expansion of the inverse of that operator that maps object opacity function to hologram intensity. The first term of the expansion is shown to be equivalent to conventional (optical) reconstruction, with successive terms increasingly sup-pressing the twin image. The algorithm is computationally efficient, requiring only a single fast Fourier transform pair.

Digital decoding of in-line holograms

We introduce and experimentally demonstrate a fast and accurate method for quantitative imaging of the dynamics of live biological cells. Using a dual-channel interferometric setup, two phase-shifted interferograms of nearly transparent biological samples are acquired in a single digital camera exposure and digitally processed into the phase profile of the sample. Since two interferograms of the same sample are acquired simultaneously, most of the common phase noise is eliminated, enabling the visualization of millisecond-scale dynamic biological phenomena with subnanometer optical path length temporal stability.

https://www.eng.tau.ac.il/~omni/PhaseMicroscopy-OL2009.pdf

Dual-interference-channel quantitative-phase microscopy of live cell dynamics

A general method for reducing the influence of vibrations in phase-shifting interferometry corrects the surface phase map through a spectral analysis of a “phase-error pattern,” a plot of the interference intensity versus the measured phase, for each phase-shifted image. The method is computationally fast, applicable to any phase-shifting algorithm and interferometer geometry, has few restrictions on surface shape, and unlike spatial Fourier methods, high density spatial carrier fringes are not required, although at least a fringe of phase departure is recommended. Over a 100× reduction in vibrationally induced surface distortion is achieved for small amplitude vibrations on real data.

Suppressing phase errors from vibration in phase-shifting interferometry

We present a new type of quadrature phase-shifting interferometer, which utilizes wave plates, a diffraction grating, and two lasers with different wavelengths, in order to acquire two sets of two quadrature fringe patterns in each wavelength formed on a single image sensor. This method for calculating with four phase-shifted fringe patterns gives us the phase sum and difference distributions between the phases in two wavelengths. This is also substantiated by results of our experiments.

Simultaneous formation of four fringes by using a polarization quadrature phase-shifting interferometer with wave plates and a diffraction grating

We present a new type of phase-shifting interferometer, which utilizes a polarizing prism to form quadrature phase-shifted fringe patterns onto a single imaging sensor. By changing the direction of linear polarization of the incident light orthogonally, four phase-shifted fringe patterns in quadrature are obtained by taking images twice; thus it is possible to reduce phase errors caused by mechanical vibrations and air turbulence that occur in temporal phase-shifting interferometers. We present the principle of this interferometer with its theoretical analysis, using the Jones matrix, along with experimental results.

Phase-shifting interferometer based on changing the direction of linear polarization orthogonally.

An alternative method for digital holography using a quadrature phase-shifting interferometer for high-speed measurement is presented. We show that it has image quality equal to the four-bucket method. In addition, it requires fewer imaging devices. Two quadrature phase-shifting fringe patterns are acquired in each state of an object changed temporally. The phase calculation method with these four fringe patterns gives the phase distribution of the hologram. This digital phase hologram is reconstructed to yield an object image by the Fresnel transform using digital convolutions with the fast Fourier transform algorithm. Verification results of simulations and experiments are given.

Digital holography with a quadrature phase-shifting interferometer.

Although wavefront aberration in stepper projection optics has been classified with respect to its spatial frequency on the pupil, the physical meaning of this classification has not been clarified. In this paper, we show that wavefront aberration can be classified into figure aberration, random aberration, and nonconserved aberration, by taking into consideration the theoretical effect of undulated wavefront aberration with respect to its spatial frequency. We also show that the predictions of this theoretical classification coincide with both the results of numerical simulations for random aberration and the experimentally measured values of local flare size. Since our classification has a clear physical meaning, it will be valuable and applicable in developing not only stepper projection optics but also other more general optics.

Classification of undulated wavefront aberration in projection optics by considering its physical effects

Digital holography using a quadrature phase-shifting interferometer is presented. Two quadrature phase-shifed holograms are acquired to give the digital phase hologram. The object image is reconstructed by digital convolutions with the Fast Fourier Transform algorithm.

Tomohiro Kiire

Papers

Simultaneous formation of four fringes by using a polarization quadrature phase-shifting interferometer with wave plates and a diffraction grating

Phase-shifting interferometer based on changing the direction of linear polarization orthogonally.

Digital holography with a quadrature phase-shifting interferometer.

Classification of undulated wavefront aberration in projection optics by considering its physical effects

Quadrature Phase-shifting Digital Holography