Photon-noise limits to the detection of the closure phase in optical interferometry

Michelson stellar interferometers with long baselines have been proposed as a means for obtaining high-resolution images of space objects. The fringes measured in such interferometers move randomly owing to atmospheric turbulence effects. For overcoming turbulence effects the fringe phase at any instant is summed around groups of three or more aperture pairs to create the so-called closure phase. The closure phase is insensitive to atmospheric turbulence effects; however, it is corrupted by photon-noise effects. The probability-density function of the error in the closure-phase estimate that is due to photon noise is derived as a function of the fringe visibility and is evaluated. It is shown that, for dim objects and low fringe visibility, several hundred to several thousand independent realizations of the closure phase must be averaged to obtain acceptable closure-phase variance.     

Fringe visibility, irradiance, and accuracy in common path interferometers for visualization of phase disturbances

Common-path interferometers have been used to perform phase visualization for over 40 years. A number of techniques have been proposed, including dark central ground, phase contrast (π/2 and π), and field-absorption interferometers. The merits of the interferometers have been judged ad hoc by either tests with a small number of phase objects or by computer simulation. Three standardized criteria, which consolidate the work of others, are proposed to evaluate common-path interferometers: fringe visibility, fringe irradiance, and fringe accuracy. The interferometers can be described as one generic class of Fourier-plane filters and can be analyzed for all input conditions. Closed-form expressions are obtained for visibility and irradiance under the forced condition that little inaccuracy is tolerated. This analysis finds that the π-phase-contrast interferometer is a good choice if the optical phase disturbance is at least 2π; for smaller disturbances, the Π/2 filter selected by Zernike is near optimum. It is shown mathematically that the resulting fringe visibility is highly object dependent, and good results are not ensured. By allowing the optical beam to be 50% larger than the phase object, the interferometer performs well under all conditions. With this approach and a combination π-phase/field-absorption filter, interference fringe visibility is greater than 0.8 for all phase objects.     

Application of geometric phase techniques to stellar interferometry

In stellar interferometry the fringe visibility can be measured by modulating the optical path difference between the two arms of an interferometer. This approach yields accurate estimates of the fringe visibility only if the bandwidth is small, and this limits the sensitivity of the technique. We propose using a geometric phase modulator that is achromatic and does not suffer from bandwidth restrictions. Fringe detectors using geometric phase modulation have the potential of greatly increasing the sensitivity of optical stellar interferometers.     

Optimal phase contrast in common-path interferometry

We have developed an analytical model for the design and optimization of common-path interferometers (CPI's) based on spatial filtering. We describe the mathematical analysis in detail and show how its application to the optimization of a range of different CPI's results in the development of a graphical framework to characterize quantitatively CPI performance. A detailed analytical treatment of the effect of curvature in the synthetic reference wave is undertaken. We show that it is possible to improve the linearity and fringe accuracy of certain standard interferometers by a modification of the Fourier filter, and we propose and analyze a dual CPI system for the unambiguous mapping of phase to intensity over the complete input phase range.     

Effect of wavefront corrugations on fringe motion in an astronomical interferometer with spatial filters

Numerical simulations of atmospheric turbulence and adaptive optics (AO) wavefront correction are performed to investigate the time scale for fringe motion in optical interferometers with spatial filters. These simulations focus especially on partial AO correction, where only a finite number of Zernike modes are compensated. The fringe motion is found to depend strongly on both the aperture diameter and the level of AO correction used. In all the simulations the coherence time scale for interference fringes is found to decrease dramatically when the Strehl ratio provided by the AO correction is < or = 30%. For AO systems that give perfect compensation of a limited number of Zernike modes, the aperture size that gives the optimum signal for fringe phase tracking is calculated. For AO systems that provide noisy compensation of Zernike modes (but are perfectly piston neutral), the noise properties of the AO system determine the coherence time scale of the fringes when the Strehl ratio is < or = 30%.     

Calibration standard for laser beam profilers: method for absolute accuracy measurement with a Fresnel diffraction test pattern

The absolute accuracy of a clip-level laser beam profiler is measured to the 0.3% level, by comparison of the profiler's reading to the known width of a Fresnel diffraction test pattern. A pair of opposed knife edges, illuminated by a quasi-uniform and quasi-plane wave, generates the pattern whose width is determined by the 50% cut points in translating the edge pair across a tightly focused beam. The convolution of the scanning aperture with the diffraction fringe pattern is modeled to remove the effect of the aperture size from the accuracy test and to give a means of measuring the aperture width. Discussions of the experimental aspects of this test method show it to be an acceptable calibration standard for optical profilers, of use to those working on the International Standards Organization draft standard for laser beam parameter measurements.     

Analysis of coherent symmetrical illumination for electronic speckle pattern shearing interferometry

In an effort to find a non-contact technique capable of providing measurements of in-plane strain, a speckle shearing interferometer was designed using symmetrical coherent illumination. This paper presents an analysis of the sensitivity to displacement and strain of this interferometer, together with an analysis of the phase-stepping of the resultant fringe patterns. New notation is introduced alongside this analysis to define the interference components in speckle shearing interferometers using multiple illumination beams. Experimental results show fringe patterns and phase stepping in support of the theoretical analysis.     

Resolving quadrature fringes in real time

In many interferometers, two fringe signals can be generated in quadrature. The relative phase of the two fringe signals depends on whether the optical path length is increasing or decreasing. A system is developed in which two quadrature fringe signals are digitized and analyzed in real time with a digital signal processor to yield a linear, high-resolution, wide-dynamic-range displacement transducer. The resolution in a simple Michelson interferometer with inexpensive components is 5 x 10(-13) m Hz(-1/2) at 2 Hz.     

Closed-loop phase stepping in a calibrated fiber-optic fringe projector for shape measurement

Active homodyne feedback control can be used to stabilize an interferometer against unwanted phase drifts introduced by, for example, temperature gradients. The technique is commonly used in fiber-optic sensors to maintain the fiber at its most sensitive (quadrature) position. We describe an extension of the technique to introduce stabilized, pi/2-rad phase steps in a full-field interferometer. The technique was implemented in a single-mode, fiber-optic interference fringe projector used for shape measurement and can be easily applied to other fiber- or bulk-optic interferometers, for example, speckle pattern and holographic interferometers. Fresnel reflections from the distal fiber ends undergo a double pass in the fibers and interfere at the fourth port of a directional coupler. The interference intensity (and hence phase) is maintained at quadrature by feedback control to a phase modulator in one of the fiber arms. Stepping between quadrature positions (separated by pi rad for light undergoing a double pass) introduces stabilized phase steps in the projected fringes (separated by pi/2 rad for a single pass). A root-mean-square phase stability of 0.61 mrad in a 50-Hz bandwidth and phase step accuracy of 1.17 mrad were measured.     

Automated Precision Polarimeter for the HF-VHF Range

An automated system for Stokes parameter-polarization analysis over the HF-VHF range is described. Axial ratio, orientation angle, polarization fraction, and polarization sense are determined by amplitude measurements using a conventional fieldintensity receiver. Six amplitude measurements from four crossed nonresonant dipoles, including quadrature sum and difference, eliminate the requirement for phase measurement. The antenna does not use active components and is adaptable for mobile or stationary operation. VSWR measurements on the antenna output cables show less than 1.2:1 (50 ohms) over the 2-70 MHz range. The antenna aperture increases from 1 × 10-5 square meters at 2.0 MHz to 0.019 square meters at 70 MHz. A solid-state sequencer processes each amplitude measurement separately through the receiver and digital conversion circuits (providing BCD output) to an incremental tape recorder. The Stokes parameter analysis is performed by an off-line digital computer using the magnetic tape data. This analysis permits computation of total received power from either set of orthogonal element measurements. When combined with the measured antenna aperture, power density (or field strength) also can be derived. Polarization fraction measurements for locally controlled signals show a mean of 1.02 as compared to a theoretical value of 1.00 (standard deviation of 0.1) over the 2-70 MHz range and polarization results consistent with propagation predictions.     

Calibration of a computer-controlled precision wavemeter for use with pulsed lasers

The design of a pulsed wavemeter to monitor the high-precision tuning of pulsed (as well as cw) laser sources is presented. This device is developed from a combination of silver-coated Fabry-Perot etalons with various plate spacings. These etalons provide stepwise refinement of the wavelength to be measured. The wavemeter is controlled by a computer through a CAMAC interface, which measures the absolute wavelength in the visible with an accuracy of 2 parts in 10(8). The time required for data acquisition and computation to measure the refined wavelength with a single 2-MHz CPU is less than 100 ms. We describe the calibration of the instrument over the wavelength range 400-850 nm. We obtain the required calibration lines by locking lasers on hyperfine transitions of iodine, uranium, rubidium, and cesium. Methods to reduce the number of calibration lines required for calibration of the system are described. The expected wavelength-dependent phase shift of the silver coatings is compared with that measured for the etalon following calibration. The differences are larger than expected because of either optical aberations or the use of centroids to measure the fringe position.     

Fringe detection in noisy complex interferograms

A new algorithm to estimate the two-dimensional local frequencies of phase interferometric data is described. With a complex sine-wave model, demonstration is given that a conventional multiple-signal classification (MUSIC) algorithm can be used in spite of multiplicative noise perturbations. A faster algorithm dedicated to the processing of interferograms is developed and a measure of confidence in the estimate is proposed. We studied numerical performances using synthetic fringes. As a result of the frequency estimation, knowledge of the fringe local width and orientation can be applied to restore noisy phase data. Results of a complex phase filter are presented for real interferograms obtained from synthetic aperture radar images.     

Fast,common-path,switchable, achromatic phase-shifter for polarization interferometry

Abstract

A fast switchable phase-shifter using a pair of liquid-crystal devices with a switching angle of 60° is described. This phase-shifter can be placed in the common path traversed by the two orthogonally polarized beams emerging from a polarization interference microscope and used for digital phase measurements over a wide range of wavelengths. It can also be used in a stellar interferometer with orthogonally polarized beams for measurements of fringe visibility with white light.     

Grating-assisted demodulation of interferometric optical sensors

Accurate and dynamic control of the operating point of an interferometric optical sensor to produce the highest sensitivity is crucial in the demodulation of interferometric optical sensors to compensate for manufacturing errors and environmental perturbations. A grating-assisted operating-point tuning system has been designed that uses a diffraction grating and feedback control, functions as a tunable-bandpass optical filter, and can be used as an effective demodulation subsystem in sensor systems based on optical interferometers that use broadband light sources. This demodulation method has no signal-detection bandwidth limit, a high tuning speed, a large tunable range, increased interference fringe contrast, and the potential for absolute optical-path-difference measurement. The achieved 40-nm tuning range, which is limited by the available source spectrum width, 400-nm/s tuning speed, and a step resolution of 0.4 nm, is sufficient for most practical measurements. A significant improvement in signal-to-noise ratio in a fiber Fabry-Perot acoustic-wave sensor system proved that the expected fringe contrast and sensitivity increase.     

Some Aspects of Fringe Counting in Laser Interferometers

A number of arrangements are possible for deriving signals in phase quadrature from interferometers, so that bi-directional counting of the interference fringe movements can be carried out. Arrangements which avoid splitting the image field are preferred for laser light sources, and can give a better signal-to-noise ratio. In environments where the interferometer is subjected to vibration, the fringe counting system may be required to handle rapid reversals in the direction of movement. In these circumstances the rate of production of rapid counting signals of alternate direction may be considerably reduced by using an appropriate logic system between the interferometer and the counter.     

Increasing the range of unambiguity in step-height measurement with multiple-wavelength interferometry--application to absolute long gauge block measurement

An instrument for step-height measurement by multiple-wavelength interferometry is described. The addition of a 1152-nm wavelength to a multiple-wavelength scheme applying wavelengths of 633, 612, and 543 nm relaxes the tolerance range of the required preliminary measurement to +/- 140 microm, if the total uncertainty in the fringe fraction measurement can be kept below 2%. For larger fringe fraction measurement uncertainty, numerical simulations show that the integer number of interference orders can still be determined unambiguously if the range in the preliminary knowledge of the length has been correspondingly reduced. The interferometer instrument is described, and experimental data are presented in the context of long gauge block calibration at the National Research Council of Canada.     

Local frequency estimation for the fringe pattern with a spatial carrier: principle and applications

A local frequency estimation approach for the fringe pattern with a spatial carrier by which the 2D spatial frequencies at a certain pixel are estimated from its neighborhood is presented. The applications of this approach in the fringe pattern analyses are also introduced. First, a 2D spatial carrier phase-shifting algorithm is derived. With it the detuning errors induced by frequency mismatching are avoided, and the stronger phase deformations can be successfully coped with. Second, a novel aperture extrapolation method is developed by which the phase accuracies of the Fourier-transform method at the aperture boundaries are effectively improved.     

Intensity range extension method for three-dimensional shape measurement in phase-measuring profilometry using a digital micromirror device camera

Phase-measuring profilometry is an accurate and effective technique for performing three-dimensional (3D) shape and deformation measurements of diffuse objects by fringe projection. However, phase analysis cannot be performed in underexposed or overexposed areas of the detector when an object with wide reflectance is measured. A novel intensity range extension method using a digital micromirror device (DMD) camera is proposed. In the optics of the DMD camera, each pixel of the CCD corresponds exactly to each mirror of the DMD. The phase-shifted fringe patterns with high contrast can be easily captured by programming an inverse intensity pattern that depends on the reflectance of the object. Our method can provide a wider intensity range and higher accuracy for 3D shape measurement than other conventional methods in both underexposed and overexposed areas. The measurements of a replica of a metallic art object and a flat plane are analyzed experimentally to verify the effectiveness of our method. In the experiment, the percentage of invalid points due to underexposure and overexposure can be reduced from 20% to 1%.     

Aberration effects in two-beam laser interferometers

In displacement measurements by two-beam interferometers, the wavefront curvature of a laser beam causes a systematic increase of the fringe period. This increase depends on beam collimation: It is null for a plane wave and proportional to the squared divergence of the beam. With interfering beams not perfectly recombined, an additional fringe-period error is caused, with the effect of counteracting and also of compensating for and prevailing over the usual error. We describe this hitherto unsuspected effect and give a correction equation.     

Measurement of higher harmonics in periodic vibrations using phase-modulated TV holography with digital image processing

We separately measure the higher harmonics vibration patterns of a periodic vibrating object by using time-average TV holography and phase modulation. During measurements the frequency of the phase modulation is adjusted to each harmonic component while the excitation of the object is set low enough to record all components on the linear part of the fringe function. Using acoustical phase stepping and calibration of the fringe function, we compute the amplitude and phase distributions of the frequency component. We measure components up to the 65th harmonic by using square-wave excitation.