Antonio L. Guerrero

Spectrally-resolved white-light interferometry as a profilometry tool

Phase-shifting interferometry and white-light interferometry are reliable techniques for surface analysis in which the optical path difference has to be changed by some transducer to evaluate the phase. We present here a different procedure in which optical path modulation is completely avoided. This technique is based on the spectral analysis of white-light interferograms. By means of a spectroscopic device, a non-visible interferogram is split into its monochromatic components and absolute, unambiguous values of the phase are obtained along the spectral axis. Only one interferogram is required to obtain the profile of one-dimensional surfaces with nanometric resolution.

Optical implementation of frequency domain analysis for white-light interferometry

The purpose of recent developments of profilometry by using white light interferometry is to provide new tools for the analysis of rough samples which when studied by monochromatic phase-shifting interferometry, may cause phase calculation ambiguities. The usual way to perform depth measurements by white light interferometry is to analyze the coherence-limited interference fringes while the optical path difference is scanned. The method proposed here does not use optical path difference scanning. A spectroscopic device is used instead to separate the interference intensities associated to each spectral component of the light source. Phase variations due to wavelength change are proportional to the optical path difference and allows depth measurement to be performed without axial scanning. The profile of one line of the inspected sample is obtained from only one 2D interferogram. In this 2D interferogram one direction corresponds to the inspected direction of the surface while the other one is the chromatic axis which allows phase to change with wavelength. Experimental results show the ability of the proposed method to obtain the profile of 1D surface with nanometric resolution.

Refractive index distribution measurements by means of spectrally-resolved white-light interferometry

Abstract A new procedure to measure the spatial distribution of the refractive index in transparent media is presented. It is based on the spectral analysis of optical interferograms obtained from a wide, continuous-spectrum light source. The method yields fairly high precision (up to 10-8) in the measurements of local values of differential refractive index, Δn (Δn=n−nref), along a line in the sample. By means of a CCD TV-camera linked to a microcomputer, fast recording and automatic data processing are achieved. As an application, we present an experimental study of a thermal gradient in a liquid sample.

Fractional approximations to the Bessel function J0(x)

A method to obtain fractional approximations to the Bessel function J0(x) is reported here. This method improves a recently published one principally in that all the parameters are uniquely determined by linear equations. Our approximations give fairly good accuracy for all real, positive values. The maximum absolute error for the first‐degree approximation is about 0.0035, and for the fourth‐degree one, about 2.8×10−6.

Higher order two-point quasi-fractional approximations to the Bessel functions J o ( X ) and J 1 ( X )

On cherche des solutions approchees pour les fonctions de Bessel Jν(x) (ν=0, 1) valables pour des valeurs de x petites et grandes

Multi-channelled white-light interferometry for real-time dispersion measurements

Abstract A white-light source, a two-wave interferometer and a spectrometer are associated in order to analyze the dispersion (refractive index as a function of wavelength) of a prismatic sample. This set-up behaves like a multi-channelled device: simultaneous monochromatic interferograms are stored in a single bi-dimensional interferogram in the hybrid spatial–spectral domain. Real-time, direct determination of refractive index at every resolved wavelength is provided with high precision (up to 10 −6 ).

A solution to the problem of stationary phase in double spectral modulation profilometry

Abstract A solution to a problem frequently found in interferometric surface profilers working with spectral modulation is presented. The phase of the signal which stores the surface profile depends both on the surface depth and on the wavenumber. In those regions where the phase function reaches an extreme value (stationary phase points) phase determination from intensity measurements is not possible. A procedure which overcomes this problem is presented: two recordings of the interference pattern are required; the second one is recorded after the introduction of an (arbitrary) additional path difference. The method is self-contained in the sense that no piezoelectric transducers, nor calibrated mirror shifts are required.

Real-time dispersion curve measurement from spectrally resolved white-light interferometry

Spectrally-resolved white-light interferometry (SRWLI) is used for real-time measurement of dispersion functions. SRWLI consists in the spectroscopic analysis of the interferograms which are produced when a wide, continuous- spectrum light-source is used to illuminate a 2-wave interferometric device. This produces incoherent superposition of many monochromatic interferograms, one for each resolved wavelength in the source spectrum. Each monochromatic interferogram at wavelength (lambda) stores the optical delay in the interferometer at that particular wavelength. When a transparent, dispersive specimen is introduced in the interferometer, the optical delay becomes a function of (lambda) , and this function is stored in the incoherent superposition of the whole set of monochromatic interferograms. We propose in this paper the use of a prismatic specimen with a linear variable thickness. In this way the interferometry gives rise, for each wavelength, to a classical Young fringe pattern whose spatial frequency stores the dispersion function. The spectroscope, in series with the interferometer, splits the incoherent superposition of the different monochromatic patterns. The recorded image is processed by measuring the frequency of the fringes at each resolved wavelength. Precision in refraction index is about 10-6. The method is well adapted for measuring dispersion curves of evolving specimens because only one image is sufficient to determine the dispersion curve in the useful spectral range. Experimental results are presented for an optical glass in the visible spectrum.

Crossed Data Processing in Spectrally-Resolved White-Light Interferometry

Spectrally resolved white-light interferometry has proved to be an absolute and precise tool for measuring optical delays. Hitherto, the bidimensional interferogram in the hybrid spatial-spectral domain, has been processed row by row as function of (sigma) , in order to obtain the map of the optical delay (Delta) (y). In this way, nanometric precision is obtained in profilometric studies. In this paper we show that more information an be obtained from the same interferogram when it is additionally processed column by column. This corresponds to the analysis of a sequence of monochromatic interferograms which are free from the classical ambiguities in the phase, since the preliminary row-by-row data processing along the spectral axis removed them. The absolute phase function thus obtained for each column at constant (sigma) yields the profile z(y). Statistical treatment of the overall information provided by this row by row plus column by column data processing increases precision in one order of magnitude. This paper presents the method and experimental results as well.

Surface profilometry in the spectral domain

Spectrally resolved white light interferometry (SRWLI) is applied to 1-D profilometry. The technique allows us to deal with discontinuous profiles without any ambiguity. Experimental results show good agreement with phase shifting profilometry; nanometric resolution is attained. In order to extend the method to 2-D samples, double spectral modulation (DSM) is applied using a new experimental set-up which enhances luminosity.