Determination of height distribution on a rough interface by measuring the coherently transmitted or reflected light intensity

Laser light source limited uncertainty of speckle-based roughness measurements

Surface microtopography measurement (e.g., form, waviness, roughness) is a precondition to assess the surface quality of technical components with regard to their applications. Laser speckle-based roughness measurement is an optical scattered light measuring technique that provides field of view dimensions of some square millimeters and measuring frequencies in the kilohertz domain enabling in-process roughness characterization of even moving part surfaces. However, camera exposure times of microseconds or less and a high detector pixel density mean less light energy per pixel. This affects the minimal achievable measurement uncertainty, which has not been clarified yet for almost plain sample shapes. For this reason, the measurement uncertainty limit of the surface roughness parameter Sa due to fundamental, inevitable noise sources such as laser shot noise and detector noise is analytically estimated and compared to experimental data. The results show a mainly shot-noise-limited measurement uncertainty contribution of less than 0.033 nm. In addition, a significant influence of laser beam profile variations on the achievable roughness measurement uncertainty is identified for the current experimental setup, which is generally below 0.3 nm and can be improved in future setups. The already achieved low measurement uncertainty offers ideal preconditions for in-process roughness measurements on samples with a similar surface structure in industrial environments.

Measurement of roughness based on the Talbot effect in reflection from rough surfaces

In the present work, the Talbot effect of a square grating is analyzed when light is reflected from a rough surface. It is shown theoretically that the scattered light intensity in the Fresnel diffraction limit depends on statistical properties of the rough surface, the angle of incidence of the light, the grating period, and a geometric coefficient, related to the ratio of distance of the rough surface and the observation plane from the grating. At Talbot distances of the grating, the surface height difference function, in terms of multiplication of the Talbot number, the grating period, and the geometric coefficient is the modulation transfer function (MTF) of the scattering in reflection from the rough surface. If the argument of the height difference function is larger than twice the surface correlation length, the height difference function is constant for different spatial frequencies. Therefore, the square wave is reproduced with smaller contrast. The surface roughness can be obtained by measuring the contrast at different incident angles. It is also shown that the contrast measurements in both reflection and transmission, provide the refractive index of transparent samples with a rough surface. In experimental studies, the roughness of three metal standard rough surfaces are determined at different angles of incidence. Also, the refractive index of a sheet glass with a rough surface is obtained. The results are quite consistent.

Mirage is an image in a flat ground surface

Mirage is a fascinating phenomenon that has attracted many scientists to report their observations and descriptions about it. There are two different approaches to mirage formation. The more popular one is attributed to total internal reflection that occurs in the near ground air layers on hot sunny days. According to the other approach, mirage is an image in a rough surface that is observed at grazing angles of incidence. Most of the existing descriptions are qualitative and some include calculations based on guessing temperature change with no concrete experiments. In this report, first we show that Fermat's principle also concerns the wave nature of light and covers the constructive and destructive interference that is essential for image formation. Then, we provide a brief review of the image formation theory in a rough plane and demonstrate by experiments in the lab and deserts that the temperature gradient in the near ground air layers does not lead to mirage formation.

Determination of the rough interface parameters using the self-imaging effect

In this work, a linear grating is used to project a periodic light intensity distribution on a rough interface, and the near field transmitted light scattering is studied. It is shown theoretically that the intensity in the Fresnel regime depends on statistical properties of the rough interface and the light intensity period. The self-image contrast exponentially depends on the interface height-height correlation function. The correlation is obtained in terms of multiplication of the self-image number and the period of the light intensity distribution. Therefore, the roughness and the correlation length of the interface can be obtained by determining the contrast of the self-images when the light intensity period is smaller than the interface correlation length. For periods longer than twice the correlation length, the contrast measurements only provide the interface roughness. In experimental studies, the roughness of interfaces is determined by square gratings with periods much longer than the correlation lengths. The rough interfaces are prepared by roughening sheet glass by powders of different grit numbers. The results for different gratings and light wavelengths are quite consistent.

Spectral Modification by Diffraction and Scattering

Different effects induce spectral changes, for example, correlation of source fluctuations, propagation of light, random changes in optical properties of a medium, diffraction and scattering from objects, and rough interfaces. We review the spectral changes that occur as a result of light diffraction from phase steps, and particularly we discuss the spectral changes in the neighborhood of phase singularities. We also review the redshift and blueshift in the spectra of the lights coherently and diffusely scattered from rough interfaces. In addition, we study the effects of roughness and incident angle on the spectral profiles of scattered lights in reflection and transmission modes.

Redshift and blueshift in the spectra of lights coherently and diffusely scattered from random rough interfaces

We show theoretically and experimentally that the spectrum of coherently scattered light from a randomly rough interface in reflection and transmission is redshifted with a shrinkage in spectral width. In reflection mode the amounts of the redshift and the shrinkage depend on interface roughness, incident angle, and the spectral width of the illuminating light. In transmission mode they also depend on the refractive indices of the surrounding media. The redshift and width shrinkage increase with decrease of the coherently scattered light intensity. This study shows that the spectrum of the diffusely scattered light is blueshifted in the specular direction and in directions with small scattering angles only in situations with appreciable intensity of the coherently scattered light. With decrease of the latter intensity the blueshift reduces and turns into redshift. Also, the redshift and blueshift decay with increase of the scattering angle. An experimental investigation has been carried out, on sheet glasses with different roughness on one side, in reflection and transmission modes. The experimental results and theoretical predictions are quite consistent.

Optical diffractometry

Interference of light has numerous metrological applications because the optical path difference (OPD) can be varied at will between the interfering waves in the interferometers. We show how one can desirably change the optical path difference in diffraction. This leads to many novel and interesting metrological applications including high-precision measurements of displacement, phase change, refractive index profile, temperature gradient, diffusion coefficient, and coherence parameters, to name only a few. The subject fundamentally differs from interferometry in the sense that in the latter the measurement criterion is the change in intensity or fringe location, while in the former the criterion is the change in the visibility of fringes with an already known intensity profile. The visibility can vary from zero to one as the OPD changes by a half-wave. Therefore, measurements with the accuracy of a few nanometers are quite feasible. Also, the possibility of changing the OPD in diffraction allows us to use Fresnel diffraction in Fourier spectrometry, to enhance or suppress diffracted fields, and to build phase singularities that have many novel and useful applications.