Surface finish measurements are usually fitted to models of the finish correlation function which are parametrized in terms of root-mean-square roughnesses, σ, and correlation lengths, l. Highly finished optical surfaces, however, are frequently better described by fractal models, which involve inverse power-law spectra and are parametrized by spectral strengths, Kn, and spectral indices, n. Analyzing measurements of fractal surfaces in terms of σ and l gives results which are not intrinsic surface parameters but which depend on the bandwidth parameters of the measurement process used. This paper derives expressions for these pseudoparameters and discusses the errors involved in using them for the characterization and specification of surface finish.

Fractal surface finish.

Previously published vector equations describing angle-resolved scattering from single-layer- and multilayer-coated optics have been integrated numerically and analytically over all angles in the reflecting hemisphere to obtain numerical results and analytical expressions for total integrated scattering (TIS). The effects of correlation length, polarization, angle of incidence, roughness height distribution, scattered light missed by the collecting hemisphere, and roughness cross-correlation properties of the multilayer stack on the TIS expression are considered. Background material on TIS from optics coated with single opaque reflecting layers is given for completeness and comparison to corresponding multilayer TIS results. It is shown that errors can occur in calculating the true rms surface roughness from actual TIS measurements; ways to correct these errors are discussed.

Relationship of the total integrated scattering from multilayer-coated optics to angle of incidence, polarization, correlation length, and roughness cross-correlation properties

Interfacial roughness in multilayer films may be random or correlated, ie, replicated from layer to layer It is shown that these can be separated and quantified using x‐ray diffraction rocking curves and a straightforward analysis The lateral correlation length along the interfaces can additionally be determined A quantitative evaluation for W/C multilayers shows that correlated roughness contributes significantly to the total roughness, even at length scales that are surprisingly short, of the order 2–6 nm

Determination of roughness correlations in multilayer films for x‐ray mirrors

A theoretical method is described for calculating the angular scattering curve for any given thin-film multilayer in which the various surfaces and interfaces are assumed to be rough. The energy that is scattered in any particular direction depends on two factors. The first is solely a function of the characteristics of the ideal multilayer and the observation conditions. The second depends on the roughnesses of the various surfaces, that is, on their autocorrelation and cross-correlation functions. The expressions that are obtained are completely general because no restrictive hypothesis has been developed on the relationship that may exist between the roughnesses of the various interfaces. The principal features of the apparatus used for the scattering measurements are briefly described. Next, several results of calculation and measurement are given as illustrations. The measurements show that polished surfaces of high optical quality often show marked anisotropy.

Scattering from multilayer thin films: theory and experiment

Optical techniques have great potential for non-destructive and on-line measurements of surface roughness during manufacturing. This paper reviews the state-of-the-art in a number of optical techniques including specular reflectance, total integrated scatter, diffuseness, angular scattering distributions, speckle, ellipsometry, and interferometry. The distinction is drawn between the more quantitative but slower profiling techniques and less quantitative, parametric techniques, which are faster and hence more useful for high-speed monitoring of surfaces. Overall, no currently available technique combines accuracy and speed and is therefore suitable as an on-line metrological tool for roughness measurement of engineering surfaces. Speckle techniques hold perhaps the greatest potential as accurate, high-speed metrological tools

Optical techniques for on-line measurement of surface topography

Measurements of angular scattering due to surface roughness were taken from a 24-layer dielectric mirror and compared to theory. In addition, the top surface roughness of the multilayer stack is analyzed from Talystep profilometer measurements. These roughness data are used to obtain a roughness spectral density function to be used in a vector multilayer scattering theory. The theory uses three multilayer stack models to incorporate possible effects of different degrees of correlation between interfaces of the stack. It was found that the angular scattering calculated using the experimentally obtained roughness spectral density function agreed remarkably well with the measured angular scattering data. This is especially true if care is taken to differentiate between particulate and roughness scattering. For the sake of comparison, the angular scattering from an aluminum film is also given, and differences from scattering from the multilayer mirror are noted.

Light scattering from multilayer optics: comparison of theory and experiment.

Although it is possible to accurately predict angle-resolved scattering from surfaces covered with multilayer dielectric films and some of the scattering from metal-coated surfaces, there are still areas of disagreement between theory and experiment. In particular, the ratio of p- to s-polarized scattered light at a given scattering angle should be a function only of the optical thicknesses and optical constants of the surface layers (for dielectric films) or optical constants (for opaque metal films) , and the angles of incidence and scattering; it should be independent of surface topography. It was found experimentally that the p/s polarization ratio for 60-degree incidence, 60-degree retroscattering at λ6328 A varied widely from the theoretical values for silver- and aluminum-coated glass samples and for polished and diamond-turned copper samples. In addition, theory predicts no cross-polarized scattered light (p-incident/s-scattered or s-incident/p-scattered), but in some cases large amounts were measured experimentally. It was found that the polarization ratio could be increased to approach the theoretical value by heating the films in vacuum, with consequent recrystallization and roughening of the films. It was hypothesized that some of the measured effects could be caused by fluctuations in the optical constants in the plane of the film; a first-order vector perturbation theory was derived to cover this situation. Calculations made using the theoretical equations look encouraging, but the theory does not predict any cross-polarized scattered light. More theoretical work needs to be done and, in addition, some structural characterization of the films.

J. M. Elson

Papers

Relationship of the total integrated scattering from multilayer-coated optics to angle of incidence, polarization, correlation length, and roughness cross-correlation properties

Light scattering from multilayer optics: comparison of theory and experiment.

Angle-Resolved Scattering: Comparison Of Theory And Experiment