Wavenumber Axis Research Articles

Hyperspectral interferometry for single-shot absolute measurement of two-dimensional optical path distributions

We propose a method that we call hyperspectral interferometry (HSI) to resolve the 2π phase unwrapping problem in the analysis of interferograms recorded with a narrow-band light source. By using a broadband light source and a hyperspectral imaging system, a set of interferograms at different wave numbers are recorded simultaneously on a high resolution image sensor. These are then assembled to form a three-dimensional intensity distribution. By Fourier transformation along the wave number axis, an absolute optical path difference is obtained for each pixel independently of the other pixels in the field of view. As a result, interferograms with spatially distinct regions are analysed as easily as continuous ones. The approach is illustrated with a HSI system to measure 3D profiles of optically smooth or rough surfaces. Compared to existing profilometers able to measure absolute path differences, the single-shot nature of the approach provides greater immunity from environmental disturbance.

Hyperspectral interferometry for single-shot absolute measurement of 3-D shape and displacement fields

We propose a method that we call Hyperspectral Interferometry (HSI) to resolve the 2π phase unwrapping problem in the analysis of interferograms recorded with a narrow-band light source. By using a broad-band light source and hyperspectral imaging system, a set of interferograms at different wavenumbers are recorded simultaneously on a high resolution image sensor. These are then assembled to form a three-dimensional intensity distribution. By Fourier transformation along the wavenumber axis, an absolute optical path difference is obtained for each pixel independently of the other pixels in the field of view. As a result, interferograms with spatially distinct regions are analysed as easily as continuous ones. The approach is illustrated with a HSI system to measure 3-D profiles of optically smooth or rough surfaces. Compared to existing profilometers able to measure absolute path differences, the single shot nature of the approach provides greater immunity from environmental disturbance.

Instability of coupled geostrophic density fronts and its nonlinear evolution

Instability of coupled density fronts, and its fully nonlinear evolution are studied within the idealized reduced-gravity rotating shallow-water model. By using the collocation method, we benchmark the classical stability results on zero potential vorticity (PV) fronts and generalize them to non-zero PV fronts. In both cases, we find a series of instability zones intertwined with the stability regions along the along-front wavenumber axis, the most unstable modes being long wave. We then study the nonlinear evolution of the unstable modes with the help of a high-resolution well-balanced finite-volume numerical scheme by initializing it with the unstable modes found from the linear stability analysis. The most unstable long-wave mode evolves as follows: after a couple of inertial periods, the coupled fronts are pinched at some location and a series of weakly connected co-rotating elliptic anticyclonic vortices is formed, thus totally changing the character of the flow. The characteristics of these vortices are close to known rodon lens solutions. The shorter-wave unstable modes from the next instability zones are strongly concentrated in the frontal regions, have sharp gradients, and are saturated owing to dissipation without qualitatively changing the flow pattern.

TWO-WAY COUPLED-MODE SOLUTIONS IN THE COMPLEX HORIZONTAL WAVENUMBER PLANE

A coupled-mode formalism based on complex Airy layer mode solutions is presented. It is an extension into the complex horizontal wavenumber plane of the companion article [Stotts, J. Acoust. Soc. Am.111 (2002) 1623–1643], referred to here as the real horizontal wavenumber version, which accounted for general ocean environments but was restricted to normal modes on the real horizontal wavenumber axis. A formulation of the expressions for both trapped and continuum complex coupling coefficients is developed to dramatically reduce computer storage requirements and to make the calculation more practical. The motivation of this work is to demonstrate the numerical implementation of the derivations and to apply the method to an example benchmark. Differences from the real horizontal wavenumber formalism are highlighted. The coupled equations are solved using the Lanczos method [Knobles, J. Acoust. Soc. Am.96 (1994) 1741–1747]. Comparisons of the coupled-mode solution to a parabolic equation solution for the ONR shelf break benchmark validate the results.

Perturbation-Correlation Moving-Window Two-Dimensional Correlation Spectroscopy

A new method of analysis, perturbation-correlation moving-window two-dimensional (PCMW2D) correlation spectroscopy, is proposed. For a spectral data set collected under an external perturbation, this method provides a pair of synchronous and asynchronous two-dimensional correlation spectra plotted on a plane between a spectral variable (e.g., wavenumber) axis and a perturbation variable (e.g., temperature) axis. One of the advantages of this new correlation analysis method is that complicated spectral variation along the perturbation direction can be monitored. It has been found that the synchronous and asynchronous PCMW2D correlation spectra are similar to the first perturbation derivative and negative second perturbation derivative spectra of the original data, respectively. To demonstrate the potential of PCMW2D correlation spectroscopy, it has been applied to temperature-dependent infrared (IR) spectra of a poly(vinyl alcohol) (PVA) film. The thermal behavior of the PVA film has been revealed by the PCMW2D correlation analysis. Two characteristic cross-peaks are observed in the synchronous PCMW2D correlation spectra generated from the temperature-dependent IR spectra between the crystalline phase C-O stretching band at 1141 cm-1 and the melting temperature of 209 degrees C and between the amorphous phase C-O stretching band at 1095 cm-1 and another specific temperature of 233 degrees C. This specific temperature of 233 degrees C corresponds to the thermal degradation temperature due to the elimination of the hydroxyl group attached to the main chain.

Effects in the Solarp‐Mode Power Spectrum from Scattering on a Turbulent Background Flow with Stochastic Wave Sources

This work demonstrates how the scattered wave field, in combination with stochastic wave sources, influences the p-mode power spectrum. I adopt a zone with random fluctuations in sound speed and velocity (characterized by the respective correlation functions). I present a general formalism (for plane-parallel conditions) to calculate the expectation value of the p-mode power spectrum. The power due to the direct field from the sources dominates, and depends only on the source correlation function. Smaller, but significant, are due to scattered wave field components. These corrections depend in general on the correlation functions of the fluctuations in the medium. I adopt a simple waveguide for the purpose of clear demonstration, and show that the corrections generate three important effects: (1) the line profiles are shifted along the wavenumber (frequency) axis, with varying effects depending on which turbulent physical quantity we consider; (2) the shapes and widths of the line profiles are altered; and (3) the troughs between the line profiles are to some extent filled in, contributing to the solar background power. It is demonstrated that boundary zones can generate important contributions to the power spectrum.

Phase-Shifting Fringe Analysis for Laser Diode Wavelength-Scanning Interferometer

A method for analyzing the fringe patterns obtained from a wavelength-scanning interferometer is proposed. It is based on phase retrieval with the phase-shifting method and derivation of its slope along the wave number axis. The accuracy and the error sources are estimated by experiments, in which a laser diode under small wavelength-modulation (-95pm) is used as the light source. A conventional piezoelectric phase shifter was used for a phase shifter. This technique can be applied to profile measurement of stepped objects located across the zero-path position.

Tracing a metameric match to individual variations of color vision

A metameric match made by an observer under established experimental conditions might not be the same as made by another, indicating that the color vision differs between the two observers. We have analyzed the variations of normal color vision using metameric surfaces. First, using a set of cone fundamentals, we modeled the matches of a theoretical observer with normal color vision on the Davidson & Hemmendinger (D&H) Color Rule. We also derived deviate cone fundamentals by changing the macular pigment density and the lens density, and by shifting the long-wave-sensitive photopigment along the wave-number axis. The results showed shifts in the matches for the deviate observer of no more than one sample on the D&H Color Rule, the largest shifts being due to lens density. Second, we modeled the matches made on the D&H Color Rule of 8 observers by computing their personalized cone fundamentals using their independently recorded measures of macular pigment density and lens density, and their Rayleigh matches. The results show that the use of personalized cone fundamentals provides a better prediction than the use of data from a theoretical observer. © 1998 John Wiley & Sons, Inc. Col Res Appl, 23: 379–389, 1998

Tracing a metameric match to individual variations of color vision

A metameric match made by an observer under established experimental conditions might not be the same as made by another, indicating that the color vision differs between the two observers. We have analyzed the variations of normal color vision using metameric surfaces. First, using a set of cone fundamentals, we modeled the matches of a theoretical observer with normal color vision on the Davidson & Hemmendinger (D&H) Color Rule. We also derived deviate cone fundamentals by changing the macular pigment density and the lens density, and by shifting the long-wave-sensitive photopigment along the wave-number axis. The results showed shifts in the matches for the deviate observer of no more than one sample on the D&H Color Rule, the largest shifts being due to lens density. Second, we modeled the matches made on the D&H Color Rule of 8 observers by computing their personalized cone fundamen- tals using their independently recorded measures of macu- lar pigment density and lens density, and their Rayleigh matches. The results show that the use of personalized cone fundamentals provides a better prediction than the use of data from a theoretical observer. © 1998 John Wiley & Sons, Inc. Col Res Appl, 23, 379 -389, 1998

Wavelength-tuning interferometry of intraocular distances

We describe basic principles of wavelength-tuning interferometry and demonstrate its application in ophthalmology. The advantage of this technique compared with conventional low-coherence interferometry ranging is the simultaneous measurement of the object structure without the need for a moving reference mirror. Shifting the wavelength of an external-cavity tunable laser diode causes intensity oscillations in the interference pattern of light beams remitted from the intraocular structure. A Fourier transform of the corresponding wave-number-dependent photodetector signal yields the distribution of the scattering potential along the light beam illuminating the eye. We use an external interferometer to linearize the wave-number axis. We obtain high resolution in a model eye by slow tuning over a wide wavelength range. With lower resolution we demonstrate the simultaneous measurement of anterior segment length, vitreous chamber depth, and axial eye length in human eyes in vivo with data-acquisition times in the millisecond range.

Energy balance for radiated waves and propagating vibration in a fluid-loaded beam under point excitation

Using a compensating solution method [S. Kovinskaya and I. Krasnov, Acta Acust. 82, 237 (1996)] a field of flexural waves propagating along a submerged beam under a point excitation and a radiated acoustical field are represented analytically. The method is based on the Fourier integral transform of the vibration equation and is similar to the hybrid method [J. M. Cuschieri and D. Feit, J. Acoust. Soc. Am. 95, 1998–2005 (1994)]. The main and compensating solutions are determined as a result of the inverse Fourier transform along a real wave number axis of nonuniform and uniform equations, correspondingly. A sum of the main and compensating solutions must supply the vibration radiation condition. Let the last equation determine an indefinite coefficient in the compensating solution. The energy balance equation based on the solution of the problem leads to an energetic equivalence between a decreasing vibration and radiated spherical wave and the same between a cylindrical acoustical wave and an energy difference in the excitation point and infinity for propagating flexural wave.

A technique for analyzing elastodynamic responses of anisotropic laminated plates to line loads

A new technique is presented to modify the hybrid numerical method (HNM) proposed earlier by the authors for analyzing the responses of anisotropic laminated plates subjected to time-step and time-pulse line loads. In the modified HNM, eigenfrequencies and modal factor functions for wave modes in the plate are computed at equally spaced points on the wavenumber axis. In each interval of the points, the eigenfrequencies and modal factor functions are replaced by straight lines, and the inverse Fourier integrations are then carried out analytically. The proposed modification can significantly reduce the sampling points in the inverse integrations. The modified HNM is much more efficient than the original HNM, and can be used to compute not only near-field and short-time but also far-field and long-time responses for anisotropic laminated plates, without increasing the sampling points. Numerical examples are presented to demonstrate the modified HNM.

Two-dimensional time-harmonic elastodynamic Green's functions for anisotropic media

An exact matrix formulation for 2D time-harmonic elastodynamic Green's functions for anisotropic media is presented. In this formulation, a Fourier transform with respect to spatial coordinates is employed. The displacement and stress in the Fourier transform domain is obtained by using the method of modal expansion. A technique is suggested to evaluate the 2D inverse integral in the polar coordinate system to obtain the solution for the displacement and stress. In this technique, the inverse integration is carried out in a very efficient way along the wave number axis over a semi-infinite region by changing integral paths. Therefore a numerical integration is needed only for the polar angle over a finite region. Numerical examples are presented to demonstrate the present method, and wave fields for the displacement are investigated for isotropic and anisotropic media.

Compact Raman Spectrometers: Data Handling Methods

Measurements have been made on a compact dispersive Raman spectrometer of the type often proposed for on-line measurements. Quantitative use of the details of spectra that are produced by this system require different data treatments than those which have been described for longer-focal-length instruments. The data sets are sparsely sampled, and they cover a wide spectral window. Owing to sparse sampling, spectra as recorded have the appearance of piecewise linear approximations. It is necessary to interpolate additional data points in order to observe details of peaks, such as exact positions of maxima. With wider spectral windows, procedures for calibration of the wavenumber axis which have been previously described do not meet our performance criterion: an accuracy of 0.1 cm−1 or better. A model that involves a fifth-order polynomial is needed. On the basis of the performance of this model, atomic lines which give satisfactory results when used as wavenumber references have been identified. Details of a self-modeling procedure for removal of broad-band spectral interferences are described, and its performance is demonstrated.

A hybrid numerical and analytical solution for the Green’s function of a fluid-loaded elastic plate

A fundamental problem in structural acoustics is the scattering or radiation of sound from a fluid-loaded plate with discontinuities. In calculating the response of the plate, as well as the radiated pressure field, the traditional approach has been to evaluate the transform representation of the response or radiated pressure solution using a contour integral. The contour integral evaluation of these representations is numerically inefficient when compared to an approach that utilizes an FFT algorithm to compute the inverse transform by a direct integration along the real wave-number axis. The latter approach however can run into problems because of the presence of singularities along the real wave-number axis. In previous work by other investigators, this problem has been overcome by artificially including structural damping in their formulations, which in some instances may obscure the true effects of the radiation damping. In the present work, a direct integration approach is used after first analytically extracting from the solution the pole-like contributions on or near the real axis, and thus removing the problem of the singularities. The results only represent the influence of the radiation damping. The Green’s functions obtained in this way for the fluid-loaded plate are compared to results obtained by the other techniques. The resulting Green’s functions can be used to solve for the radiation and scattering problems for inhomogeneous fluid-loaded plates.

Compact Table for the Publication of Infrared Spectra That are Quantitative on Both Intensity and Wavenumber Axes

A Compact Table format is presented for the publication of infrared spectra that are quantitative on both intensity and wavenumber axes. The format is illustrated with a molar absorption coefficient spectrum, Em(ν˜) vs. ν˜, and with infrared real and imaginary refractive index spectra, n(ν˜) vs. ν˜ and k(ν˜) vs. ν˜, respectively. The algorithm consists of two steps: first, the number of spectral points is reduced by using larger wavenumber spacings than appear in the original spectrum; second, the resulting spectral points are presented in a compressed table format. The Compact Table is about one tenth the size required for the original spectrum to be presented in a conventional XY table. The essential criterion for increasing the wavenumber spacing is that it must be possible to recover the original spectrum by interpolation to an accuracy better than that of the original spectrum. Nearly all the recovered imaginary refractive index and molar absorption coefficient values are within 1% of the original values, and for each quantity the average of the magnitudes of the accuracies of recovery is 0.2%. The real refractive index spectrum is most accurately recovered by Kramers-Kronig transformation of the recovered imaginary refractive index spectrum. Nearly all the recovered real refractive index values are within 0.02% of the original values, and the average of the magnitudes of the accuracies of recovery is 0.005%. The real and imaginary infrared dielectric constant spectra, ɛ′(ν˜) vs. ν˜ and ɛ″(ν˜) vs. ν˜, can be calculated from the recovered data with an accuracy in ɛ′ that is about one half of that of the real refractive index and an accuracy in ɛ″ that is approximately that of the imaginary refractive index. The detailed method is outlined and is applied to infrared intensities of chlorobenzene. Computer programs are presented for the construction of the Compact Table and for the recovery of the full spectrum from the tabulated information.

Submolecular dynamics of amorphous polymers probed by two‐dimensional infrared (2D IR) spectroscopy

AbstractTwo‐dimensional infrared (2D IR) spectroscopy was used to probe the submolecular dynamics of atactic polystyrene. 2D IR is a powerful analytical technique especially suited for the elucidation of localized motions of polymer segments. In 2D IR, a polymer sample is excited by a small‐amplitude oscillatory strain at a frequency in the acoustic range. The fluctuation of IR dichroism signals resulting from the strain‐induced reorientation of electric dipole‐transition moments is monitored with a time‐resolved spectrometer. Spectra defined by two independent wavenumber axes are constructed by applying a correlation analysis to such signals. The 2D spectra provide detailed information about the local dynamics of submolecular constituents of the system. From the sign of cross peaks in the synchronous 2D IR spectrum of glassy polystyrene, it is shown that the main‐chain backbone of polystyrene reorients in the direction of applied strain. Cross peaks in an asynchronous 2D IR spectrum reveal highly localized reorientational motions of phenyl side groups occurring more or less independently of the main‐chain realignment. In the glassy state, the phenyl ring tends to fold back along the main‐chain, indicating that there exists a highly constrained local distortion of side groups during deformation.

Mirror Velocity Sampling Errors in a Michelson Interferometer

ABSTRACT Mirror velocity variations in a continuous mirror drive Michelson interferometer result in erroneous interferogram sampling. The effects of both systematic and random mirror fluctuations are evaluated. Four methods of evaluation provide a means to quantify the importance of mirror velocity errors in the Fourier spectrometer's performance as a spectroradiometer. The four methods are time interval signal analysis, sine wave test, optical chopper modulation, and examination of consecutive interferograms. Examination of consecutive interferograms includes the monitoring of a constant blackbody infrared radiation source as well as an interposed ammonia sample cell for wavenumber axis registration. Each method furnishes a known input to the interferometer modulation equation (i.e. 2uv=f) and permits an internal validation of the interferometer mirror velocity evaluation results.

Enhancing the information content of vibrational spectra through sample perturbation

Infrared and Raman band frequencies, intensities and line shapes are often sensitive to the local molecular environment determined by molecular conformation, surrounding matrix, temperature, pressure, etc. The variety of local environments experienced by a condensed-phase molecule can lead to vibrational spectra with broad bands containing many overlapped spectral features. The spectral resolution of these overlapped features can be enhanced by making perturbations to the sample environment. Examples of perturbations which can be applied to the sample to enhance the information content of infrared spectra are changes in temperature, concentration and mechanical strain. In each instance, the spectra obtained as a function of the perturbation can be cross-correlated to produce a two-dimensional correlation map defined by two independent wavenumber axes. in this representation, infrared bands which respond to the perturbation in a similar or different manner can be clearly identified. This information can be used to help resolve overlapped bands and make unambiguous band assignments.

Doppler Sonar Observations of Internal Waves: The Wavenumber-Frequency Spectrum

Abstract In May 1980 an 18-day sequence of velocity profiles of the top 600 m of the sea was collected off the coast of Southern California. The measurements were obtained using a pair of Doppler sonars mounted on the Research Platform FLIP. From these data, estimates of the wavenumber-frequency spectrum of the oceanic internal wavefield are obtained. The spectra are characterized by a series of ridges, which occur at near-internal and tidal frequencies as well as higher harmonics and sums of these fundamentals. The ridges run parallel to the wavenumber axis. There is a pronounced near-inertial spectral peak. The near-inertial motions are dominated by a few identifiable wave groups. There is a net downward energy propagation in the near-inertial frequency band. The vertical-wavenumber dependence of a the spectrum is decidedly asymmetric in this region. The asymmetry extends to five times the inertial frequency, making much of the so-called continuum asymmetric. A high-wavenumber cutoff at approximately 60...