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.     

Phase-shifting moireí method with an atomic force microscope

Using a phase-shifting technique with an atomic force microscope (AFM), we propose a phase-shifting AFM scanning moiré method. The phase shifting is realized in four steps from 0 to 2pi by a piezoscanner in the AFM. The measurement method and experimental techniques are described in detail. For demonstration this method is applied to determine the phase distribution in the AFM moiré of a 1200-line/mm holographic grating used to measure the thermal deformation in a Quad FlatPack electronic package.     

Thin-film achromatic phase shifters for nulling interferometry: design approach

Nulling interferometry in the thermal IR is the most promising technique for direct detection of Earth-like exoplanets. This technique requires a pi phase shifter for the parent star of the planet to be completely extinguished by destructive interference. We investigate how thin films can be used to design pi achromatic phase shifters. The design approach that we propose works on reflection and can be carried out by two steps, namely, the design of a mirror and an antireflection structure with no constraint on the phase properties of the thin film stacks. Phase-shift accuracy is derived analytically, and a numerical example illustrates this concept.     

Spiking voltages for faster switching of nematic liquid-crystal light modulators

Electrical address circuits developed for driving fast-switching ferroelectric liquid-crystal spatial light modulators (SLM) can be programmed to increase the speed of much slower responding nematic liquid-crystal SLMs. Using an addressing circuit that can switch as fast as 0.164 ms, voltages are programmed for values of phase that exceed the desired phase, and when the phase reaches the desired value, the voltage is switched to the required steady-state voltage. For a SLM that has a phase range of 3.5pi and that is programmed over a 2pi range, switching speed is reduced from 400 ms to between 71 and 77 ms. The speedup algorithm is applied to each pixel of the SLM together with a digital correction for a spatially nonuniform phase.     

Modulo 2pi fringe orientation angle estimation by phase unwrapping with a regularized phase tracking algorithm

The fringe orientation angle provides useful information for many fringe-pattern-processing techniques. From a single normalized fringe pattern (background suppressed and modulation normalized), the fringe orientation angle can be obtained by computing the irradiance gradient and performing a further arctangent computation. Because of the 180 degrees ambiguity of the fringe direction, the orientation angle computed from the gradient of a single fringe pattern can be determined only modulo pi. Recently, several studies have shown that a reliable determination of the fringe orientation angle modulo 2pi is a key point for a robust demodulation of the phase from a single fringe pattern. We present an algorithm for the computation of the modulo 2pi fringe orientation angle by unwrapping the orientation angle obtained from the gradient computation with a regularized phase tracking method. Simulated as well as experimental results are presented.     

Analysis of a dephased complex-coupled distributed-feedback laser by the power-series method

A novel power-series method to solve the coupled-wave equations is introduced. The method is used to calculate the threshold gain margins of a complex-coupled distributed-feedback laser as functions of the ratio of gain coupling to index coupling (|kappa(g)|/|kappa(n)|) and of the phase difference between the index and the gain gratings. For coupling coefficient |kappa|l < ., the laser shows a mode degeneracy at specific values of the ratio |kappa(g)|/|kappa(n)| for cleaved facets. At phase differences pi/2 and 3pi/2 between the gain and the index gratings, an antireflection-coated complex-coupled laser becomes multimode, and a different mode starts to lase. The effect of facet reflectivity (both magnitude and phase) on the gain margin of a complex-coupled DFB laser is also investigated. Although the gain margin varies slowly with the magnitude of the facet's reflectivity, it shows large variations as a function of the phase. Spatial hole burning was found to be minimum at phase difference npi, n =, ..., and maximum at phase differences pi/2 and 3pi/2.     

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.     

Symmetrical periods in antireflective coatings for plastic optics

Plastic optical parts require antireflective as well as hard coatings. A novel design concept for coating plastics combines both functions. Symmetrical three-layer periods with a phase thickness of 3/2pi are arranged in a multilayer to achieve a step-down refractive-index profile. It is shown mathematically that the equivalent index of symmetrical periods can be lower than the lowest refractive index of a material used in the design, if the phase thickness of the symmetrical period is set equal to 3/2pi instead of the usual pi/2. The straightforward application of the concept to the design of antireflection coatings in general is demonstrated by example.     

Digital recording and numerical reconstruction of holograms: an optical diagnostic for combustion

Holographic interferometry (HI) has proved to be a useful tool for nonintrusive temperature measurements in flames (and thereafter for inference of the local composition based on the state relationship approach) with high spatial and temporal resolution. Digital holographic interferometry (DHI) is a relatively new imaging and measurement technique that electronically records a hologram (e.g., on a CCD) and reconstructs it by a numerical method. Cumbersome chemical processing of the hologram is avoided in DHI, which thereby provides greater flexibility, speed, and the potential for real-time processing. In conventional holography, fringes that are neither bright nor dark on a hologram cannot be accurately resolved. The DHI technique has not yet to our knowledge been used for combustion applications. Herein we evaluate its efficacy for making temperature measurements in flames and assess its applicability through a simulation. Each part of a double exposure associated with the holographic technique is considered to be recorded by a hypothetical CCD sensor at a separate time from the other part. We applied the principles of Fourier optics to develop two numerical methods for hologram reconstruction, and we show that both methods provide an accurate reconstruction of the phase image associated with a flame. Because of the periodic nature of the wave function, the reconstructed phase values are limited to the interval [-pi/2, pi/2]. Thus an unwrapping algorithm is provided that produces a continuous phase distribution based on the condition that the reconstructed phase is jumped by a value of -pi or pi. We have also developed an iterative calculation method to adjust the value of the digital reference wave parameters that determines the phase imaging reconstruction in DHI.     

Extending the dynamic range of phase contrast magnetic resonance velocity imaging using advanced higher-dimensional phase unwrapping algorithms.

Phase contrast magnetic resonance velocity imaging is a powerful technique for quantitative in vivo blood flow measurement. Current practice normally involves restricting the sensitivity of the technique so as to avoid the problem of the measured phase being 'wrapped' onto the range -pi to +pi. However, as a result, dynamic range and signal-to-noise ratio are sacrificed. Alternatively, the true phase values can be estimated by a phase unwrapping process which consists of adding integral multiples of 2pi to the measured wrapped phase values. In the presence of noise and data undersampling, the phase unwrapping problem becomes non-trivial. In this paper, we investigate the performance of three different phase unwrapping algorithms when applied to three-dimensional (two spatial axes and one time axis) phase contrast datasets. A simple one-dimensional temporal unwrapping algorithm, a more complex and robust three-dimensional unwrapping algorithm and a novel velocity encoding unwrapping algorithm which involves unwrapping along a fourth dimension (the 'velocity encoding' direction) are discussed, and results from the three are presented and compared. It is shown that compared to the traditional approach, both dynamic range and signal-to-noise ratio can be increased by a factor of up to five times, which demonstrates considerable promise for a possible eventual clinical implementation. The results are also of direct relevance to users of any other technique delivering time-varying two-dimensional phase images, such as dynamic speckle interferometry and synthetic aperture radar.     

An online triple-frequency color-encoded fringe projection profilometry for discontinuous object

Abstract

An online triple-frequency color-encoded fringe projection profilometry is proposed to measure the complex and discontinuous object at straight-line movement. N frames of color fringe patterns are specially designed. Three grayscale sinusoidal grating patterns with geometric progression frequency growth are encoded into red (R), green (G), and blue (B) channels separately to compose a color-encoded fringe pattern. If these three grayscale sinusoidal grating patterns are phase-shifted N steps with an equivalent shift phase of 2π/N, they can compose the corresponding N frames of color-encoded fringe patterns as above respectively. In order to avoid the movement’s interference to the phase shifting, position adjustment should be done to guarantee the phase-shifting direction to be perpendicular to the moving direction. While these N frames of specially designed color-encoded fringe patterns are projected onto the moving object one by one, the corresponding deformed color patterns are captured by a CCD camera in real time. By color separating, color crosstalk compensation, pixel matching, and phase calculation, three wrapped phase at different frequencies can be extracted. The unwrapped phase can be solved by a simplified algorithm based on temporal phase unwrapping method from the relationship of the three wrapped phase at the same pixel. Thus, it is very suitable to measure the online complex and discontinuous objects at straight-line movement. The experimental results show the feasibility and the validity of the proposed method.     

Surface profilometry with laser-diode optical feedback interferometer outside optical benches

A simple and robust interferometer with a laser diode subject to optical feedback from the interferometer is presented for surface testing of a spherical mirror. The fringe phase can be locked by the optical feedback within less than 0.2pi (peak-to-valley value) even when the interferometer is placed on a wooden table. The fringe locking is caused by the change of lasing wavelength that suppresses the net phase change to be much less than 2pi. The locked fringe pattern with spatial carriers can be analyzed by a fringe analyzer at a video rate, and the measurement results of the spherical mirror showed the same result as on an optical bench.     

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.     

Modulation light efficiency of diffractive lenses displayed in a restricted phase-mostly modulation display

We present an analysis of the diffraction efficiency of diffractive lenses displayed on spatial light modulators that depends on the modulation response of the display. An ideal display would produce continuous phase-only modulation, reaching a maximum phase-modulation depth of 2pi. We introduce the concept of modulation diffraction efficiency that accounts for the effect of nonlinearities only in the phase modulation of the display. We review a diffractive model with which to evaluate this modulation efficiency, including modulation defects such as nonlinear phase modulation, coupled amplitude modulation, phase quantization, and a limited modulation depth. We apply this diffractive model to Fresnel lenses and show that these modulation defects produce a lens multiplex effect. Finally we demonstrate that the application of a minimum Euclidean projection principle leads to high modulation diffraction efficiency even if the phase-modulation depth is much less than 2pi. We demonstrate that the modulation efficiency can exceed 90% for a modulation depth of 1.4pi and can exceed 40% (the equivalent for a binary phase element) for a modulation depth of only 0.7pi. Experimental results from use of a twisted nematic liquid-crystal display are presented to confirm these conclusions.     

Two-dimensional grating light modulator for projection display

A novel two-dimensional (2D) phase grating light modulator for projection display is proposed. It consists of an upper moveable grating, a bottom mirror, and four supporting posts between them. After the driving voltage is applied to the modulator, the upper grating will move down, which induces a phase difference and, therefore, leads to a controlled variation of its diffraction pattern. Optical characteristics of the modulator and the modulator array are analyzed with Fourier optics theory. The analysis shows the incident light will be switched from its zero order diffraction fringe to the first order diffraction fringe when the phase difference between the moveable grating and the bottom mirror changes from 2 pi to pi. The diffraction pattern of the light modulator array is the coherent superposition of all single modulators. A 16 x 16 modulator array is fabricated by surface micromachining technology. The test result shows that the device works well when it is actuated by a voltage with a 1 kHz frequency and 10V amplitude. Both theoretical analysis and experiment results indicate that the 2D phase grating light modulator has potential application in a projection display system.     

Sinusoidal-wavelength-scanning interferometer with double feedback control for real-time distance measurement

In addition to a conventional phase a the interference signal of a sinusoidal-wavelength-scanning interferometer has a phase-modulation amplitude Zb that is proportional to the optical path difference L and amplitude b of the wavelength scan. L and b are controlled by a double feedback system so that the phase alpha and the amplitude Zb are kept at 3pi/2 and pi, respectively. The voltage applied to a device that displaces a reference mirror to change the optical path difference becomes a ruler with scales smaller than a wavelength. Voltage applied to a device that determines the amplitude of the wavelength scan becomes a ruler marking every wavelength. These two rulers enable one to measure an absolute distance longer than a wavelength in real time.     

New regularization scheme for phase unwrapping

A new, to our knowledge, algorithm for the phase unwrapping (PU) problem that is based on stochastic relaxation is proposed and analyzed. Unlike regularization schemes previously proposed to handle this problem, our approach dispells the following two assumptions about the solution: a Gaussian model for noise and the magnitude of the true phase-field gradient's being less than pi everywhere. We formulate PU as a constrained optimization problem for the field of integer multiples of 2pi, which must be added to the wrapped phase gradient to recover the true phase gradient. By solving the optimization problem using simulated annealing with constraints, one can obtain a consistent solution under difficult conditions resulting from noise and undersampling. Results from synthetic test images are reported.     

Generation of Phase-Modulated Periodic Optical Signals by use of a Polarization Interferometer

Polarization phase-shifting interferometry is an established technique in optical metrology. In the present study it is shown that, by use of this technique, not only is it possible to realize any discrete magnitude of a predetermined phase difference (from 0 to 2pi) between two light beams but also phase-modulated periodic optical signals can be generated simply by rotation of a polarizer or a retarder or both placed at the input of a conventional two-beam interferometer. Some representative linear and nonlinear periodic polarization-induced phase-modulated optical signals are shown. A linear phase modulation of 0-2pi with constant output intensity is obtained in some cases. The Poincaré sphere representation is introduced as a convenient tool for visualizing the dynamics involved in the generation of polarization-phase-modulated waveforms and as a possible aid to intelligent modification of the generated waveform as required. This all-optical technique of continuous and periodic phase variation should prove useful for introducing phase modulation without the need for electro-optic devices.     

Adjustable spiral phase plate

A spiral phase retarder phi(r, theta) = mtheta has been constructed with use of a deformed cracked plexiglass plate. By changing the degree of deformation, the retarder can be adjusted for use at any wavelength, and the value of the phase step 2pim at theta = 2pi can be chosen.     

Phase-slope and group-dispersion calculations in the frequency domain by simple optical low-coherence reflectometry

We report a new method by which phase slope and group dispersion can be calculated with a simple optical low-coherence reflectometer to quantify physiological conditions. A discrete-time signal processing algorithm based on the first and second derivatives of the phase with respect to wave number was developed from discrete-time Fourier properties. The algorithm avoids the 2pi ambiguity associated with most phase unwrapping. Experimental data collected by use of well-characterized optical materials validated the algorithm, which was minimally sensitive to phase noise. The group dispersion of bovine cornea was measured at various hydrations and was significantly dependent on hydration. The results suggest that group dispersion is an indicator of corneal alterations.