Polychromatic Speckle Patterns Research Articles

Roughness Measurements with Polychromatic Speckles on Tilted Surfaces

Surface light scattering enables contactless and fast measurements of surface roughness. A surface inclination alters the direction of the scattering beam and thus the measured surface roughness is calculated from the detected intensity distribution. Hence, an accurate sensor–surface alignment is necessary. In order to achieve tilt-independent roughness measurements, a model-based evaluation approach for polychromatic speckle patterns is presented. By evaluating the shape of the superposed speckles, which occur for polychromatic illumination, with regard to the distance to the scattering centre, surfaces with an Sa roughness value in the range of 0.8–3.2 μm are measurable. Experimental investigations demonstrate that the influence of a surface tilt up to ± 1.25° on the roughness measurement is reduced by 90%. As a result, the robustness of the polychromatic speckle roughness measurement is improved, which allows to speed up the adjustment of the measurement system or the surface sample, respectively.

Analysis on Combination of Different Wavelengths in Surface Roughness Measurement by Polychromatic Laser Speckle Patterns

Based on the method of polychromatic speckle autocorrelation measuring surface roughness, this paper discussed the influence of the combination of different wavelengths on speckle patterns and speckle elongation ratio when rough surface and polychromatic speckle patterns were simulated. The results show that the smaller the difference of wavelengths combination is, the further peak of the speckle elongation ratio is from the speckle field centre, for the same surface roughness. And the wider area of speckle elongation effects extends. Therefore, influence of combination of wavelengths on measurement precision should be taken into consideration during measuring process.

Using a zone model to incorporate the influence of geometry on polychromatic speckle contrast

A simple physical zone model was developed to explain the formation of polychromatic speckle patterns within the Fresnel region. This model represents a reasonable compromise between complex theoretical formulation and simple estimations for practical needs, and allows the speckle contrast to be calculated as a function of geometric parameters for the optics and coherence length of the light source. The model was experimentally verified, and the results are consistent with our previous rigorous theoretical formulation.

Contrast of polychromatic speckle patterns and its dependence to surface heights distribution

A polychromatic speckle pattern can be understood as an addition of N independent speckle patterns. We use the simple model for polychromatic speckle contrast, with some modifications, to evaluate the number of independent speckle patterns that contribute in the image plane. These speckle patterns may have different mean intensities depending on the surface heights distribution function. We derive a formula for the addition of multiple independent speckle patterns with different mean intensities and calculate the contrast for a surface with Gaussian statistics versus a surface with uniform distribution of the surface heights.

Aspect ratio of elongated polychromatic far-field speckles of continuous and discrete spectral distribution with respect to surface roughness characterization.

Here polychromatic speckle patterns generated either by a polychromatic light source that emits at discrete frequencies or by a light source showing a continuous narrow-band spectral distribution are studied. The purpose here is the application of polychromatic speckle-pattern analysis to an in-process surface roughness characterization. To compare the coherence properties of the different polychromatic light sources, first a modified definition of the coherence length is introduced. Furthermore, the relevant optical phenomena, namely, the speckle elongation caused by the angular dispersion and the roughness-dependent speckle decorrelation, are summarized. It is shown that light sources with a continuous spectral distribution have essential advantages in comparison with discrete wavelength sources. The theoretical results are confirmed by experimental investigations based on a digital algorithm for the evaluation of CCD images of polychromatic speckle patterns, which are recorded in the Fourier plane of a Fourier-transforming lens.

Simple model for image-plane polychromatic speckle contrast

A simple geometrical model was developed for calculation of the contrast of a polychromatic image-plane speckle pattern from a source of light with high spatial coherence. It is based on counting the number of independent speckle patterns that contribute to a given point in the image plane. This results in a simple equation for the contrast as a function of imaging geometry; relative orientation of the projection direction, observation direction, and specimen normal; bandwidth of the light source; and surface roughness. Its validity was established by comparison with an exact solution: rms errors in the calculated contrast were only 0.033 over a wide range of parameter values likely to be encountered in practice.

Dependence of polychromatic-speckle-pattern contrast on imaging and illumination directions

An analytical relationship between the contrast in polychromatic image speckle patterns and the positions of the diffuse object's surface normal, the illumination, and the axis of the imaging system has been obtained. When the axis coincides with the direction in which the mirror reflects the illumination about the normal, the contrast in the image speckle pattern is the greatest. Some qualitative experimental results are also given.

The autocorrelation function of polychromatic laser speckle patterns near the image plane

The autocorrelation function of polychromatic speckle patterns produced near the image plane of a double diffraction imaging system is experimentally studied. The condition under which the polychromatic speckle field obeys Gaussian statistics is further investigated as a function of the numberN of scattering cells, using the average contrast of the speckle intensity fluctuations. The profile of the autocorrelation function is next investigated as a function ofN and the defocused distance from the image plane. It is found that the average contrast and autocorrelation function of the polychromatic speckle intensity fluctuations converge less rapidly with an increase inN than those of the monochromatic ones.

Surface Roughness Dependence of the First-order Statistics of Polychromatic Speckle Patterns

A theoretical formula for the first-order probability density function of polychromatic speckle is presented, based on recent results for integrated laser speckle. We consider both completely and partially coherent incident light. An experimental set-up for accurate measurements of speckle statistics is described. Experimental results for the constrast and the probability density of polychromatic speckle in the far field show a significant dependence on surface roughness. Except from some high contrast values they agree fairly well with the theory.

The contrast variation of polychromatic speckle patterns at the far-field diffraction plane

The contrast variation of polychromatic speckle patterns at an arbitrary point of the far-field diffraction plane is studied theoretically and experimentally. Gaussian statistics are assumed for the formation of speckles. The average contrast of speckle patterns at the far-field diffraction plane is actually theoretically evaluated and experimentally measured for diffuse objects having a wide range of surface roughnesses under illumination of the polychromatic Gaussian laser beam. This contrast is found to depend strongly on the surface roughness of objects, the number of scattering cells within an illuminated area over the object, and a radial distance from the center of the far-field diffraction plane. Good agreement between the theoretical and experimental results is confirmed.

Use of dominant eigenvalues in evaluating the probability density function of the intensity for a polychromatic speckle pattern

A method has been described to determine a priori the optimum number of samples required to estimate the eigenvalues in the Karhunen–Loeve expansion of a polychromatic speckle pattern without solving the relevant integral equation. It is found that choosing samples less than this number leads to an incorrect estimate of the eigenvalues, while choosing a larger number of samples does not change the eigenvalues significantly. Thus the method allows the use of the Karhunen–Loeve expansion with minimum effort and leads to an accurate estimate of the probability density function of the intensity. Owing to the general nature of this method, it is also applicable to a large class of problems where the Karhunen–Loeve expansion is used.

Surface roughness measurements by means of polychromatic speckle patterns

A new method for surface roughness measurements is described based on the fibrous, roughness-dependent structure of polychromatic speckle patterns in the far field. The patterns are recorded on photographic film and analyzed by an optical Fourier transform system with a double aperture in the film plane. We have calculated the theoretical roughness dependence of a parameter in the Fourier spectrum. Our experimental results agree fairly well with the theory.

Measurement of an arbitrary parallel displacement of a rigid body by using polychromatic speckle patterns

A novel speckle method for measuring the parallel displacement of a rigid body in an arbitrary direction is presented. The amount of displacement as well as its angle made with the optical axis is to be measured from the Fourier transform patterns of a doubly exposed photographic plate.

Comparing roughness dependent contrasts of polychromatic speckle patterns in light of different spectral densities

An exact solution is obtained for the roughness dependent speckle contrast in the case of a rectangular spectrum. This is compared to the simple formula previously obtained for a gaussian spectrum. The agreement is shown to be good if the formulas are fitted to the same asymptotic dependence in the incoherent limit. From this we conclude that the simple gaussian formula may be a useful approximation even for other non-gaussian spectra, and a fitting procedure is indicated.

The scattering of polychromatic light from rough surfaces: first order statistics

We consider the theoretical form of the probability density function of the intensity in a polychromatic speckle pattern. Measurements using a multimode argon laser are compared with theory.

Second-order Statistics of Light Diffracted from Gaussian, Rough Surfaces with Applications to the Roughness Dependence of Speckles

The second-order statistics of light diffracted from gaussian rough surfaces is shown to be given by a locally stationary degree of coherence of scattered amplitudes in the reciprocal object space. The corresponding degree of speckle correlation is derived by analogy with intensity interferometry. This unified approach is applied to describe the roughness dependence of spectral and angular speckle correlations, as well as the first- and second-order statistics of polychromatic speckle patterns. For general scattering geometries a simple formula for the polychromatic speckle contrast is derived. For normal incidence and paraxial scattering a formula is derived that describes the roughness dependent fibrous structure of a polychromatic pattern.

On the Contrast of Polychromatic Speckle Patterns and Its Dependence on Surface Roughness

It is shown how the contrast of a polychromatic speckle pattern can be calculated from the spectral coherence of the scattered amplitude. For gaussian surfaces and relatively narrow spectral profiles an analytical relationship between speckle contrast and r.m.s. surface roughness is derived. The theoretical results are compared with available experimental data and discrepancies are explained.