This paper addresses problems associated with the development of widely applicable non-Gaussian noise models, particularly with reference to the statistical properties of scattered waves. A combination of phenomenological arguments and exact solutions of specific scattering problems are used to elucidate the significance of one model—K-distributed noise—which has several attractive features and has already found many applications. A full statistical-mechanical formulation is developed for non-Gaussian compound Markov processes, with this model as a special case. The implications for numerical simulation of correlated non-Gaussian noise are explored and comparisons made with experimental data. A brief review of current applications of the K-distribution model is given.

Non-Gaussian models for the statistics of scattered waves

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

Some important types of instrumentation for measuring surfaces both past and present are reviewed. Exhaustive lists of instruments and performance are not presented; rather more emphasis is placed on the philosophy of measurement. An attempt is made to classify the surface features and also the function of surfaces as a pre-requisite to measurement. It is revealed that, as the push towards miniaturization is being taken beyond the nanometrology scale, some theoretical restrictions are likely to be encountered. 1. Why measure surfaces? In recent years surface texture has been recognized as being significant in many fields. In particular the surface roughness is an important factor in determining the satisfactory performance of the workpiece, in tribology for example or in coatings. Also in engineering applications the surface roughness has been found useful in machinetool monitoring. These aspects will be discussed presently. It is, however, pertinent to consider how the importance of surface roughness is changing with the passage of time and how the importance of roughness depends on the actual scale of size of the workpiece and the process used to make it. In very general terms the requirements for energy transfer and information transfer and storage have dominated the development of technology. This will no doubt also be true in the future but the factors governing such transfer depend themselves on size. As objects get smaller changes emphasizing the importance of the surface take place. The energy and force equations experience a change of balance, figure 1( a), as the scale of size of the moving object decreases. Information storage in 3D is possible but still very difficult to achieve; also, data retrieval is a major problem. On the other hand, storage of data on surfaces is still actively being extended. This capability is obviously a function of the surface area. There is no problem with accessibility of the stored data like there is with volume storage. Notice that information storage trends tend to have the opposite trend to the energy equations with respect to the effect of the scale of size, figure 1( b). In both situations the critical regime is that of area. Consider figure 1( a). In the force diagram momentum effects quickly decrease as the size is reduced, in fact by a factor of L3. Damping is proportional to the area and decreases as L2. Elastic forces only decrease as a linear factor of L so that they become progressively more important than the others as the scale of Figure 1. The importance of surface properties with scales of size. size decreases. In rotational situations the dependence on scale is even more pronounced. For example the moment of inertia decreases by a factor of L5. Of the above factors, the elastic properties can be controlled by varying the properties of materials, which are well understood, and inertial forces can be controlled by design. This leaves areal or damping forces which cannot be controlled easily. Energy losses and transfer and information storage and transfer are all sensitive to uncontrolled areal properties and the areal property which is most likely to be influential is the surface roughness. 0957-0233/97/090955+18$19.50 c © 1997 IOP Publishing Ltd 955

REVIEW ARTICLE: Surface metrology

Publisher Summary This chapter discusses the statistics of speckle patterns. The statistical properties of a speckle pattern depend, in general, on both the coherence of the incident light and the statistics of the scattering surface or medium. In a large number of practical situations normal speckle patterns are produced; the statistics of these speckle patterns are well-established both theoretically and experimentally, and in particular, they do not depend on the detailed scattering properties of the scattering medium. When examining the effects of polychromatic illumination on speckle statistics, it was found that the value of the r.m.s. height variation of a scattering surface strongly influenced the statistical properties of the scattered intensity. In addition, the chapter shows that the general relationship between the scattered intensity and the scattering surface may be very complicated, even for surfaces with a Gaussian distribution of surface heights.

I The Statistics of Speckle Patterns

The relations between surface statistics and the near- and far-field scattered light statistics are surveyed, mainly using scalar diffraction theory. The methods which have been proposed and used are grouped and described according to whether they operate in the near-field or the far-field and whether they use monochromatic or polychromatic light. Several methods emerge as promising for practical application to non-contacting and non-destructive testing.

Optical estimation of statistics of surface roughness from light scattering measurements

Beckmann's theory is restated in a form suitable for scattering in the optical region and some applications to the estimation of surface roughness are indicated.

A re-formulation of some results of P. Beckmann for scattering from rough surfaces

A weak scattering medium produces random phases in a transmitted or reflected wavefront distributed over less than 2π. The far-field speckle pattern has a central maximum. The statistics of intensity in this maximum are calculated and the applications of the results to surface roughness measurement are indicated.

W. T. Welford

Papers

Optical estimation of statistics of surface roughness from light scattering measurements

A re-formulation of some results of P. Beckmann for scattering from rough surfaces

First order statistics of speckle produced by weak scattering media