Three-dimensional phase step profilometry with a multicore optical fiber

This paper demonstrates the feasibility of using phase stepping and a multicore optical fiber to calculate an object's depth profile. An interference pattern is projected by an optical fiber onto the object. The distorted interference pattern containing the object information is captured by a CCD camera and processed using a phase step interferometry method. The phase step method is less computationally intensive compared to two-dimensional Fourier transform profilometry and provides more accuracy when measuring objects of high frequency spatial variations.     

Speckle decorrelation in surface profilometry by wavelength scanning interferometry

The accuracy and the measurement range of surface profilometry by wavelength scanning interferometry applied to diffusely reflecting surfaces are investigated. The influences of surface roughness and the imaging system in the interferometer are theoretically analyzed by derivation of the autocorrelation function of interferograms arising from wavelength scanning. By using a dye laser with a tuning range of 4.2 nm to a yield resolution of 39.1 mum, we have observed interferograms and their Fourier transforms and autocorrelations to study effects of defocusing and the size ratio of speckle to the CCD pixel for a plane diffuse object positioned normal to the incident beam.     

Measurement of the wavelength dependence of birefringence for optical disk substrates

The in-plane and vertical birefringence of polycarbonate plastic substrates of optical disks are measured for wavelengths between 360 and 860 nm, which covers the full range of interest for blue as well as for the current red and infrared recording. It is found that the birefringence generally decreases as the measurement wavelength is increased. In a typical case, the in-plane birefringence, Δn‖ goes from 1.7 × 10(-5) to 1.2 × 10(-5), and the vertical birefringence, Δn⊥, drops from 7.5 δ 10(4-) to 5.7 × 10(4-) in the wavelength range from 360 to 860 nm.     

Reliability-guided phase unwrapping in wavelet-transform profilometry

The phase unwrapping algorithm plays a very important role in many noncontact optical profilometries based on triangular measurement theory. Here we focus on discussing how to diminish the phase error caused by incorrect unwrapping path in wavelet transform profilometry. We employ the amplitude value map of wavelet transform coefficients at the wavelet-ridge position to identify the reliability of the phase data and the path of phase unwrapping. This means that the wrapped phase located at the pixel with the highest amplitude value will be selected as the starting point of the phase unwrapping, and that pixels with higher amplitude value will be unwrapped earlier. So the path of phase unwrapping is always in the direction of the pixel with highest amplitude value to the one with lowest amplitude value. Making full use of the amplitude information of wavelet coefficients at the wavelet-ridge position keeps the phase unwrapping error limited to local minimum areas even in the worst case. Computer simulations and experiments verify our theory.     

Effects of phase change on reflection in phase-measuring interference microscopy

We show by analytical and numerical calculations that the phase change on reflection that occurs in interference microscopy is almost independent of the numerical aperture of the objective. The shift of the microscope interferogram response due to the phase change on reflection, however, increases with the numerical aperture. Measurements of the interferogram shift are made with a Linnik interference microscope equipped with various numerical-aperture objectives and are reported and compared with theory.     

Effect of a change in phase of the reflected wave on measurements of the shape of a surface in optical profilometry

We examine the effect of a change in phase of the reflected electromagnetic wave when measuring a surface profile using a scanning white-light interference microscope and a phase-shifting coherent light interference microscope. We present the results of experiments measuring the thickness of an aluminum coating deposited on a silicon wafer, without any phase shift correction and with phase shift correction, calculated from handbook data and from the results of ellipsometric measurements. We compare with the results of measurements on the same object on an atomic force microscope.     

Unified calibration technique and its applications in optical triangular profilometry

A unified calibration technique based on ray tracing for optical triangular profilometry is presented. The proposed technique based on the inherent geometric relation between depth and a distorted signal is capable of speedy and accurate measurement without the determination of geometric parameters. The technique can also reduce calibration error caused by the lens distortion of the projector and the camera owing to the reasonable assumption that mapping in a small local area is a linear transformation and the coefficients of the linear transformation may be varied in different local areas. Three classical systems of triangular profilometry, spot inspection, a single-line system, and a projection-grating system, are discussed and demonstrated by experiment.     

Precise and fast phase wraps reduction in fringe projection profilometry

Phase wraps in a 2D wrapped phase map can be completely eliminated or greatly reduced by frequency shifting. But it usually cannot be optimally reduced using conventional fast Fourier transform (FFT) because the spectrum can be shifted only by a integer number in the frequency domain. In order to achieve a significant phase wrap reduction, we propose a fast and precise two-step method for phase wraps reduction in this paper, which is based on the iterative local discrete Fourier transform (DFT). Firstly, initial estimate of the frequency peak is obtained by FFT. Then sub-pixel spectral peak with high resolution is determined by iteratively upsampling the local DFT around the initial peak location. Finally, frequency shifting algorithm that operates in the spatial domain is used to eliminate phase wraps. Simulations and experiments are conducted to demonstrate the superb computing efficiency and overall performance of the proposed method.     

Real-time tricolor phase measuring profilometry based on CCD sensitivity calibration

A real-time tricolor phase measuring profilometry (RTPMP) based on charge coupled device (CCD) sensitivity calibration is proposed. Only one colour fringe pattern whose red (R), green (G) and blue (B) components are, respectively, coded as three sinusoidal phase-shifting gratings with an equivalent shifting phase of 2π/3 is needed and sent to an appointed flash memory on a specialized digital light projector (SDLP). A specialized time-division multiplexing timing sequence actively controls the SDLP to project the fringe patterns in R, G and B channels sequentially onto the measured object in one over seventy-two of a second and meanwhile actively controls a high frame rate monochrome CCD camera to capture the corresponding deformed patterns synchronously with the SDLP. So the sufficient information for reconstructing the three-dimensional (3D) shape in one over twenty-four of a second is obtained. Due to the different spectral sensitivity of the CCD camera to RGB lights, the captured deformed patterns from R, G and B channels cannot share the same peak and valley, which will lead to lower accuracy or even failing to reconstruct the 3D shape. So a deformed pattern amending method based on CCD sensitivity calibration is developed to guarantee the accurate 3D reconstruction. The experimental results verify the feasibility of the proposed RTPMP method. The proposed RTPMP method can obtain the 3D shape at over the video frame rate of 24 frames per second, avoid the colour crosstalk completely and be effective for measuring real-time changing object.     

Analysis of the effects of bias phase and wavelength choice on the design of dual-wavelength diffractive optical elements

Diffractive optical elements (DOEs) are often used in pattern formation for display purposes. Constructing these images from two or more colors greatly enhances their visual effect. To achieve this with DOEs is not simple, as they are inherently wavelength specific. We discuss an algorithm for designing quantized elements that produce distinct intensity patterns in the far field for two wavelengths. The benefits of applying bias phase to the dual-wavelength problem are investigated. The difference between the best and the worst choice of bias phase is shown to produce a variation of up to 2% in the efficiency. The mean square error can vary by up to a factor of 2 between the best and the worst case. It is also critically important to understand how the values of the two wavelengths affect the result. We present an analysis of how choosing different pairs of wavelengths in the design process affects the quality of our results.     

Out-of-plane reflection and refraction of light by anisotropic optical antenna metasurfaces with phase discontinuities

Experiments on ultrathin anisotropic arrays of subwavelength optical antennas display out-of-plane refraction. A powerful three-dimensional (3D) extension of the recently demonstrated generalized laws of refraction and reflection shows that the interface imparts a tangential wavevector to the incident light leading to anomalous beams, which in general are noncoplanar with the incident beam. The refracted beam direction can be controlled by varying the angle between the plane of incidence and the antenna array.     

Improved algorithm for phase-to-height mapping in phase measuring profilometry

An improved algorithm for phase-to-height mapping in phase-measuring profilometry (PMP) is proposed, in which the phase-to-height mapping relationship is no longer restricted to the condition that the optical axes of the imaging system must be orthogonal to the reference plane in the basic PMP. Only seven coefficients independent of the coordinate system need to be calibrated, and the system calibration can be accomplished using only two different gauge blocks, instead of more than three different standard planes. With the proposed algorithm, both the phase measurement and system calibration can be completed simultaneously, which makes the three-dimensional (3-D) measurement faster and more flexible. Experiments have verified its feasibility and validity.     

Generic nonsinusoidal fringe model and gamma calibration in phase measuring profilometry

Gamma distortion is a dominant error source of phase measuring profilometry. It makes a single frequency for the ideal sinusoidal waveform an infinite width of spectrum. Besides, the defocus of the projector-camera system, like a spatial low-pass filter, attenuates the amplitudes of the high-frequency harmonics. In this paper, a generic distorted fringe model is proposed, which is expressed as a Fourier series. The mathematical model of the harmonic coefficients is derived. Based on the proposed model, a robust gamma calibration method is introduced. It employs the multifrequency phase-shifting method to eliminate the effect of defocus and preserve the influence of gamma distortion. Then, a gamma correction method is proposed to correct the gamma distortion with the calibrated gamma value. The proposed correction method has the advantage of high signal-to-noise ratio. The proposed model is verified through experiments. The results confirm that the phase error is dependent on the defocus and the pitch. The proposed gamma calibration method is compared with the state of the art and proves to be more robust to pitch and defocus variations. After adopting the proposed gamma correction method, the phase precision is much enhanced with higher quality in the measured surfaces.     

Coherence and polarization of electromagnetic beams modulated by random phase screens and their changes through complex ABCD optical systems

The change of coherence and polarization of an electromagnetic beam modulated by a random anisotropic phase screen passing through any optical system is found within the framework of complex ABCD-matrix theory This means that the formalism can treat imaging and Fourier transform and free-space optical systems, as well as fractional Fourier transform systems, with finite-size limiting apertures of Gaussian transmission shape. Thus, the current paper shall be considered as a continuation, extension, and generalization of a previous work by Shirai and Wolf [J. Opt. Soc. Am. A21, 1907 (2004)]. It will be shown that the inclusion of apertures in the optical system strongly influences not only the propagation of spatial coherence but also the degree of polarization of a propagating field. Analytical expressions of coherence and polarization propagation will be given in terms of the matrix elements for any complex optical system.     

Age-related changes in wavelength discrimination

Wavelength discrimination functions (420 to 620-650 nm) were measured for four younger (mean 30.9 years) and four older (mean 72.5 years) observers. Stimuli consisted of individually determined isoluminant monochromatic lights (10 Td) presented in each half of a 2 degrees circular bipartite field with use of a Maxwellian-view optical system. A spatial two-alternative forced-choice method was used in combination with a staircase procedure to determine discrimination thresholds across the spectrum. Small but consistent elevations in discrimination thresholds were found for older compared with younger observers. Because the retinal illuminance of the stimuli was equated across all observers, these age-related losses in discrimination are attributable to neural changes. Analyses of these data reveal a significant change in Weber fraction across adulthood for a chromatically opponent pathway receiving primarily antagonistic signals from middle-wavelength-sensitive and long-wavelength-sensitive cones but not for a short-wavelength-sensitive cone pathway.     

Polarization phase shifting cyclic interferometer for surface profilometry of non-birefringent phase samples

A rectangular path cyclic interferometer has the unique property that the counter-propagating wavefronts travelling in the interferometer arms are folded with respect to each other in the plane of the interferometer although the two wavefronts finally emerge from the interferometer unfolded. A phase disturbance introduced in one lateral half of the interferometer arm is therefore manifested in complementary lateral halves of the observed interference pattern. This phenomenon is utilized to evaluate the surface profile of a reflecting sample placed on one of the interferometer mirrors. The sample phase is retrieved using polarization phase shifting. Experimental results showing three-dimensional surface morphology of a small scale integrated circuitry directly etched on silicon are presented.     

Multi-spatial-frequency and phase-shifting profilometry using a liquid crystal phase modulator

Optical profilometry is widely applied for measuring the morphology of objects by projecting predetermined patterns on them. In this technique, the compact size is one of the interesting issues for practical applications. The generation of pattern by the interference of coherent light sources has a potential to reduce the dimension of the illumination part. Moreover, this method can make fine patterns without projection optics, and the illumination part is free of restriction from the numerical aperture of the projection optics. In this paper, a phase-shifting profilometry is implemented by using a single liquid crystal (LC) cell. The LC phase modulator is designed to generate the interference patterns with several different spatial frequencies by changing selection of the spacing between the micro-pinholes. We manufactured the LC phase modulator and calibrated it by measuring the phase modulation amount depending on an applied voltage. Our optical profilometry using the single LC cell can generate multi-spatial frequency patterns as well as four steps of the phase-shifted patterns. This method can be implemented compactly, and the reconstructed depth profile is obtained without a phase-unwrapping algorithm.     

Center wavelength shift dependence on substrate coefficient of thermal expansion for optical thin-film interference filters deposited by ion-beam sputtering

Single-layer films of Ta2O5 and multilayer thin-film filters of Ta2O5 and SiO2 were deposited by ion-beam-sputter deposition. Postdeposition annealing of the structures resulted in increased optical thickness of the films, resulting in an upward shift in the wavelength of the transmission-reflection spectra. Modeling of the single-layer films by means of the effective media approximation indicates an increase in the void fraction of the film after annealing. This increase is consistent with an observed decrease in refractive index and an increase in physical film thickness. The multilayer structures, deposited on substrates of varying coefficient of thermal expansion (CTE), were annealed at various temperatures, and the change in the center wavelength was measured. The measured change is dependent on the annealing temperature and the substrate CTE, indicating that the increase in the void fraction is caused in part by thermally induced stress during the annealing process. A simple model is proposed that relates the void fraction present in the films after annealing with the substrate CTE and the annealing temperature.     

Evidence for wavelength dependence of the scattering phase function and its implication for modeling radiance transfer in shelf seas

More than 90% of stations from the Irish and Celtic Seas are found to have significantly higher back-scattering ratios in the blue (470 nm) than in the red (676 nm) wave band. Attempts to obtain optical closure by use of radiance transfer modeling were least successful for stations at which backscattering ratios are most strongly wavelength dependent. Significantly improved radiance transfer simulation results were obtained with a modified scattering correction algorithm for AC-9 absorption measurements that took wavelength dependency in the scattering phase function into account.     

Wigner-related phase spaces for signal processing and their optical implementation

Phase spaces are different ways to represent signals. Owing to their properties, they are often used for signal compression and recognition with high discrimination abilities. We present several recently introduced Wigner-related sets of representations that have improved signal processing performance, and we introduce an optical implementation. This study deals with the generalized Wigner spaces, the fractional Fourier transform, and the x-p and the r-p representations. The optical implementations are demonstrated and discussed.