Regularized quadrature and phase tracking from a single closed-fringe interferogram

A new sequential phase demodulator based on a regularized quadrature and phase tracker system (RQPT) is applied to demodulate two-dimensional fringe patterns. This RQPT system tracks the fringe pattern's quadrature and phase in a sequential way by following the path of the fringes. To make the RQPT system more robust to noise, the modulating phase around a small neighborhood is modeled as a plane and the quadrature of the signal is estimated simultaneously with the fringe's modulating phase. By sequentially calculating the quadrature of the fringe pattern, one obtains a more robust sequential demodulator than was previously possible. This system may be applied to the demodulation of a single interferogram having closed fringes.     

Estimation of the orientation term of the general quadrature transform from a single n-dimensional fringe pattern

The spatial orientation of the fringe has been demonstrated to be a key point in the reliable phase demodulation from a single n-dimensional fringe pattern regardless of the frequency spectrum of the signal. The recently introduced general n-dimensional quadrature transform (GQT) makes explicit the importance of the fringe orientation in the demodulation process. The GQT is a quadrature operator that transforms cos phi into -sin phi--where phi is the modulating phase--and it is composed of two terms: an orientation factor directly related to the fringe's spatial orientation and an isotropic n-dimensional generalization of the one-dimensional Hilbert transform. We present a method for the determination of the orientation factor in a general n-dimensional case and its application to the demodulation of a single fringe pattern by the GQT. We have tested the algorithm with simulated as well as real photoelastic fringe patterns with good results.     

Recovery of absolute phases for the fringe patterns of three selected wavelengths with improved anti-error capability

Abstract

In a recent published work, we proposed a technique to recover the absolute phase maps of fringe patterns with two selected fringe wavelengths. To achieve higher anti-error capability, the proposed method requires employing the fringe patterns with longer wavelengths; however, longer wavelength may lead to the degradation of the signal-to-noise ratio (SNR) in the surface measurement. In this paper, we propose a new approach to unwrap the phase maps from their wrapped versions based on the use of fringes with three different wavelengths which is characterized by improved anti-error capability and SNR. Therefore, while the previous method works on the two-phase maps obtained from six-step phase-shifting profilometry (PSP) (thus 12 fringe patterns are needed), the proposed technique performs very well on three-phase maps from three steps PSP, requiring only nine fringe patterns and hence more efficient. Moreover, the advantages of the two-wavelength method in simple implementation and flexibility in the use of fringe patterns are also reserved. Theoretical analysis and experiment results are presented to confirm the effectiveness of the proposed method.     

Dynamic pressure sensing with a fiber-optic polarimetric pressure transducer with two-wavelength passive quadrature readout

We describe the combination of a polarimetric pressure sensor with a two-wavelength passive quadrature demodulation system allowing for dynamic pressure sensing in the 10-MPa range with unambiguous fringe counting. Furthermore, continuous phase measurement with the arctan method applied to the quadrature interference signals after automatic offset subtraction is demonstrated for the first time, to our knowledge. A single low-coherent superluminescent diode is used as a light source, and a polarizing beam splitter in combination with two adjustable interference filters of slightly different central wavelengths serves for the creation of the quadrature signals. Results of initial experiments with 60-ms pressure relaxation-time constants with the fringe-counting technique demonstrate the performance that was predicted theoretically. The measured pressure sensitivity exhibits excellent agreement with the previous research of Bock and Urbanczyk [IEEE Trans. Instrum. Meas. 44, 694-697 (1995)] using a polarimetric readout. The fringe-contrast variation and the measurement range obtained experimentally show the fiber dispersion to influence dephasing (deviation from quadrature) and visibility decrease significantly with increasing pressure.     

Passive directional discrimination in laser-Doppler anemometry by the two-wavelength quadrature homodyne technique

We report a method for passive optical directional discrimination in laser-Doppler anemometers. For this purpose frequency-shift elements such as acousto-optic modulators, which are bulky and difficult to align during assembly, have traditionally been employed. We propose to use a quadrature homodyne technique to achieve directional discrimination of the fluid flow without any frequency-shift elements. It is based on the employment of two laser wavelengths, which generate two interference fringe systems with a phase shift of a quarter of the common fringe spacing. Measurement signal pairs with a direction-dependent phase shift of +/- pi/2 are generated. As a robust signal-processing technique, the cross-correlation technique is used. The principles of quadrature homodyne laser-Doppler anemometry are investigated. A setup that provides a constant phase shift of pi/2 throughout the entire measurement volume was achieved with both single-mode and multimode radiation. The directional discrimination was successfully verified with wind tunnel measurements. The complete passive technique offers the potential of building miniaturized measurement heads that can be integrated, e.g., into wind tunnel models.     

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

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.     

Natural demodulation of two-dimensional fringe patterns. II. Stationary phase analysis of the spiral phase quadrature transform

Utilizing the asymptotic method of stationary phase, I derive expressions for the Fourier transform of a two-dimensional fringe pattern. The method assumes that both the amplitude and the phase of the fringe pattern are well-behaved differentiable functions. Applying the limits in two distinct ways, I show, first, that the spiral phase (or vortex) transform approaches the ideal quadrature transform asymptotically and, second, that the approximation errors increase with the relative curvature of the fringes. The results confirm the validity of the recently proposed spiral phase transform method for the direct demodulation of closed fringe patterns.     

Phase recovery from a single fringe pattern using an orientational vector-field-regularized estimator

Recent studies have demonstrated that the phase recovery from a single fringe pattern with closed fringes can be properly performed if the modulo 2pi fringe orientation is estimated. For example, the fringe pattern in quadrature can be efficiently obtained in terms of the orientational phase spatial operator using fast Fourier transformations and a spiral phase spectral operator in the Fourier space. The computation of the modulo 2pi fringe orientation, however, is by far the most difficult task in the global process of phase recovery. For this reason we propose the demodulation of fringe patterns with closed fringes through the computation of the modulo 2pi fringe orientation using an orientational vector-field-regularized estimator. As we will show, the phase recovery from a single pattern can be performed in an efficient manner using this estimator, provided that it requires one to solve locally in the fringe pattern a simple linear system to optimize a regularized cost function. We present simulated and real experiments applying the proposed methodology.     

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.     

Fringe pattern demodulation with a two-dimensional digital phase-locked loop algorithm

A novel technique called a two-dimensional digital phase-locked loop (DPLL) for fringe pattern demodulation is presented. This algorithm is more suitable for demodulation of fringe patterns with varying phase in two directions than the existing DPLL techniques that assume that the phase of the fringe patterns varies only in one direction. The two-dimensional DPLL technique assumes that the phase of a fringe pattern is continuous in both directions and takes advantage of the phase continuity; consequently, the algorithm has better noise performance than the existing DPLL schemes. The two-dimensional DPLL algorithm is also suitable for demodulation of fringe patterns with low sampling rates, and it outperforms the Fourier fringe analysis technique in this aspect.     

Fringe pattern demodulation with a two-frame digital phase-locked loop algorithm

A novel technique called a two-frame digital phase-locked loop for fringe pattern demodulation is presented. In this scheme, two fringe patterns with different spatial carrier frequencies are grabbed for an object. A digital phase-locked loop algorithm tracks and demodulates the phase difference between both fringe patterns by employing the wrapped phase components of one of the fringe patterns as a reference to demodulate the second fringe pattern. The desired phase information can be extracted from the demodulated phase difference. We tested the algorithm experimentally using real fringe patterns. The technique is shown to be suitable for noncontact measurement of objects with rapid surface variations, and it outperforms the Fourier fringe analysis technique in this aspect. Phase maps produced withthis algorithm are noisy in comparison with phase maps generated with the Fourier fringe analysis technique.     

Subwavelength resist patterning using interference exposure with a deep ultraviolet grating mask: Bragg angle incidence versus normal incidence

Interference lithography using a deep-ultraviolet (DUV) laser is instrumental in the manufacture of subwavelength patterns used at visible wavelengths. We investigated a grating mask strategy for exposure in terms of how to set and illuminate masks. To obtain high aspect ratio patterns, high fringe visibility, and high exposure uniformity are essential, and for that purpose the use of only two beams with liquid immersion is necessary but not sufficient. It needs to be addressed whether the grating should face air or liquid to achieve index matching without affecting its beam-splitting properties. Currently, the most feasible solution to produce sub-200 nm periods requires the use of a fused-silica grating under Bragg geometry (not normal incidence geometry) and filling the gap between the grating and resist with a high-index liquid.     

Direct determination of isochromatic fringe order by use of two wavelengths

A method of direct determination of photoelastic fringe order is proposed. The relations between the integral fringe order for one wavelength and the fractional fringe orders of two wavelengths are derived. The exact fringe order of the whole field is automatically calculated. The usefulness of this method is demonstrated through experimental result.     

Spatial carrier fringe pattern demodulation by use of a two-dimensional continuous wavelet transform

A novel technique that uses a fan two-dimensional (2D) continuous wavelet transform (CWT) to phase demodulate fringe patterns is proposed. The fan 2D CWT algorithm is tested by using computer generated and real fringe patterns. The result of this investigation reveals that the 2D CWT technique is capable of successfully demodulating fringe patterns. The proposed algorithm demodulates fringe patterns without the requirement of removing their background illumination prior to the demodulation process. Also, the algorithm is exceptionally robust against speckle noise. The performance of the 2D CWT technique in fringe pattern demodulation is compared with that of the 1D CWT algorithms. This comparison indicates that the 2D CWT outperforms its 1D counterpart for this application.     

Control of phase of fringes in speckle interferometry for application of fringe scanning method

The speckle interferometry is an effective technique in the displacement measurement of a structure with a rough surface. However, when the fringe scanning technique is introduced to speckle interferometry for improving the measurement resolution, generally two speckle patterns before and after the deformation of the measurement object and another speckle pattern obtained under different conditions from these two speckle patterns are required at least. So, three speckle patterns are generally required for precise fringe analysis as a minimum condition. In this paper, a method for introducing the fringe scanning method is proposed by controlling the phase of the specklegram as a fringe image using filtering techniques. Then, the temporal fringe analysis method that uses only two speckle patterns are proposed for speckle interferometry. As the result of experiments, it is shown that high precise fringe analysis can be realized by the fringe scanning methods using only two speckle patterns for the displacement measurement with a large deformation.     

Fringe visibility improvement using an asynchronous image-subtracting optically addressed spatial light modulator

We demonstrate the application of an asynchronous image-subtraction optically addressed spatial light modulator to particle image velocimetry fringe processing. The device comprises an amorphous silicon p-i-n-i-p photosensor and a ferroelelectric liquid-crystal light-modulating layer. The images to be subtracted are encoded on two separate wavelengths. The operation of the device is described, and characterization shows a frame rate of 100 Hz, a resolution of 3 line pairs/mm, and a write-light sensitivity of ≈1 mW/cm(2) at a wavelength of 514 nm. The device is read by the use of light with a 633-nm wavelength whereas the subtraction light is at a wavelength of 670 nm. Using this device to subtract a nonuniform pedestal from the optically computed power spectral density function (the Young's fringe pattern), we find we can improve the signal-to-clutter ratio of peaks in the image-transmittance autocorrelation function of particle image velocimetry transparencies. The device also permits processing of very low-visibility fringe patterns, generated from doubly exposed images, in which one image has half the transmittance of the other. These could not be processed with a nonsubtracting, binary, liquid-crystal optically addressed spatial light modulator.     

Multiwavelength shearography for quantitative measurements of two-dimensional strain distributions

We report on the development of a multiwavelength speckle pattern shearing interferometer for the determination of two-dimensional strain distributions. This system is based on simultaneous illumination of the object with three diode lasers that emit at different wavelengths between 800 and 850 nm. Wavelength separation and image acquisition were performed with a special optical arrangement, including narrow-bandpass filters and three black-and-white cameras. The shearographic camera with a variable shearing element, in combination with the appropriate illumination geometry, permitted us to isolate all six displacement derivatives from phase-stepped fringe patterns. The optical system and the measurement procedure were validated with two different experiments. First, the shearographic sensor head was used for the determination of in-plane displacements, and, second, in-plane strain distributions of an aluminum block caused by temperature expansion were measured.     

Noise-free normalized fringe patterns and local pixel transforms for strain extraction

Noise reduction is one of the most exciting problems in automatic fringe processing. We propose a two-dimensional (2-D) envelope transform for normalization of fringe patterns, coupled with spin filtering, to construct so-called noise-free normalized fringe patterns. The 2-D envelope transform uses correct fringe intensity envelopes for normalization of fringe patterns, i.e., for making the fringe background and amplitude constant over the whole field. Spin filtering is applied to fringe patterns for removal of random noise taking into account fringe flow. With spin filtering and the 2-D envelope transform, a noise-free normalized fringe pattern is constructed for postprocessing. Based on this improved fringe pattern, two local pixel transforms for strain extraction from a single moiré pattern are developed, in which the digital pure secondary moiré method is improved and the strain-field image method with division is developed.     

Two-wavelength laser-diode interferometer with fractional fringe techniques

A two-wavelength interferometer with a fractional fringe technique (the method of coincidence) has been constructed by using dual frequency-ramped laser diodes. The respective wavelengths of two optical phases were measured by the heterodyne technique. The detected two phases are employed with real-time electronic processing to produce two signals that correspond to the integer and the fractional fringe numbers at a single wavelength. These summed signals can yield a synthetic phase having a single-wavelength resolution. The upper limits for the measurement accuracy are theoretically analyzed.     

Robust spatiotemporal quadrature filter for multiphase stepping

A robust algorithm for phase recovery from multi-phase-stepping images is presented. This algorithm is based on the minimization of an energy (cost) functional and is equivalent to the simultaneous application of a fixed temporal quadrature filter and a spatial adaptive quadrature filter to the phase-stepping pattern ensemble. The algorithm, believed to be new, is specially suited for those applications in which a large number of phase-stepping images may be obtained, e.g., profilometry with a computer-controlled fringe projector. We discuss the selection of parameter values and present examples of its performance in both synthetic and real image sequences.