The in situ measurement of surface roughening is important to the characterization or control of a variety of dynamic processes, including polishing, etching, film growth, strain relaxation, phase transitions and interdiffusion. Solid and liquid surface roughness statistics are being studied using optical, x-ray, electron and atom diffraction. While average roughness can be measured by specular scattering of x-rays, for example, only non-specular, diffuse diffraction can detect lateral correlations in atomic scale roughness and quantify the surface height-height correlation function or power spectral density function in real time. In this paper, the capabilities and application of diffuse diffraction of photons for in situ measurement of roughening statistics are reviewed.

It is pointed out that the von-Mises-distribution can replace the Gaussian distribution for circular or spherical vector fields, i.e. BRDF data obtained from a variety of technical surfaces by stray light measuring or sensing. For the purpose of in line quality control formulae for the parameters corresponding to mean and variance in Gaussian distributions as well as parameter tests and confidence intervals for circular unimodal vector fields are given. Finally, measurement and simulation results are compared to circular statistical inference.

We study theoretically the scattering of a beam of s-polarized light from a one-dimensional random metal surface with a localized deterministic defect. We carry out numerical calculations of the far field intensity, using a formally exact technique based on Green's second integral identity and statistical ensemble averaging. Our results obtained for very rough surfaces show that the backscattering peak in the differential reflection coefficient for the scattered light almost does not change for small angles of incidence. However, for larger angles it undergoes significant enhancement due to the presence of the defect, which phenomenon we attribute to a form of the corner cube effect. We also consider how the presence of a deterministic defect changes the behavior of the angular intensity correlation function for the scattered light. We are primarily interested in the changes of the memory effect and the time-reversed memory effect peaks. We show that the defect can enhance or suppress these peaks depending on the relative positions of the light sources and the points of observation. The explanation of these results is again associated with the corner cube effect.

We report the experimental results which show that the far- field correlation function is sensitive to a small local change of the rough surface geometry, where the speckle spatial correlation is adopted rather than the sample ensemble average. The angular cross-correlation function of the far field speckles scattered by one-dimensional random rough surfaces is measured, when a polarized beam of light is incident on the rough surface from vacuum, where one part of the surface used is a thin dielectric film deposited on a glass substrate and the other part is identical to the first one except for a localized defect on it.

This method uses a coherent laser beam to scan a point on the sample from all azimuth angles, at either a fixed or a variable angle of incidence, completing one revolution in approximately 8 milliseconds. The data, collected by reflection from the sample, is reduced by special algorithms to obtain quantitative data related to light detected on axis. For crystals which have been suitably processed, it is feasible to determine the Miller Index and the lattice plane orientation of the atoms and to quantitatively measure other crystal parameters such as microtwinning. For thin films, this method can detect anomalies in reflectance with azimuth direction to help identify characteristics and for quality control. For single point diamond machined surfaces, the signatures obtained can help determine tool wear non- destructively. A program has also been developed to perform area scans which yield information on extended surface roughness characteristics using back reflection and interference phenomena. This information is recorded and displayed in graphics and data printouts in a number of formats appropriate for use in both research and manufacturing applications.

A light scattering model, based on scalar perturbation diffraction theory, has been used to derive surface roughness information from measurements on transparent thin film samples. The method utilizes the spectral behavior of the diffusely scattered reflectance (transmittance) as compared to the total reflectance (transmittance). By studying interference effects within the film, i.e. correlated and uncorrelated interface roughness contributions, it is possible to separate the origin of the scattering and extract statistical data of the boundaries. In this study, sputtered tin oxide films deposited onto glass substrate has been investigated. Optical characterization was made with a spectroscopic total integrating sphere (TIS) instrument in the wavelength range 0.4 less than lambda less than 1.0 micrometer. Surface roughness data from the light scattering model was compared with atomic force microscope (AFM) measurements through the use of power spectral density (PSD) functions. The AFM measurements made it possible to determine surface roughness scaling properties of sputtered tin oxide thin films with respect to film thickness and scan length.

An apparatus for total integrated backscattering measurement is described that operates in the UV to IR spectral region. Background levels smaller than 0.1 ppm at 633 nm have been achieved. During the measurement, the sample surface is scanned automatically, yielding one- or two-dimensional scattering diagrams. From the latter, small defects on supersmooth surfaces can be localized. Results are reported of measurements on samples with different surface qualities such as supersmooth Si-wafers with sub-angstrom roughness, CaF₂ substrates, thin film optical coatings and rough engineering surfaces. The equipment is involved in standardization project ISO/CD 13696'.

Laser light scattering from holographic sinusoidal gratings has been investigated with a view to its use in the calibration of the linearity of BRDF instruments, a task that requires a wide dynamic range in the scattered intensity. An aluminum-coated grating of an amplitude of approximately 90 nm and a spatial wavelength of 6.67 micrometer was used. Measurements and calculations were performed for an angle of incidence of 6 degrees and for light incident from a HeNe laser (lambda equals 0.6328 micrometer). Experimental results are compared with the predictions of two theories: Beckmann's scalar theory and Rayleigh's vector theory applied to sinusoidal gratings. Both theories, which apply to perfectly conducting scatterers, produce nearly identical results. However, these predictions differ significantly from some of the experimental results. The measured scattering pattern has a large background of scattered light and the higher-order peak intensities are larger by several orders of magnitude than the computed ones. The measured peak intensities are polarization dependent. The large background scattering is shown to be due to the residual surface roughness. The profile of the grating was measured using a stylus instrument with a 1-micrometer-radius tip and a 0.1-micrometer-radius tip, and it appears that the profile does not contain significant harmonics that might be responsible for the large higher-order peak intensities. Scattering from a gold-coated specimen with the same specifications was also measured and compared with that from the aluminum-coated one to determine the effect of non-topographic scatter. Possible causes of the discrepancy between the measured and the computed magnitude and polarization dependence of the higher-order peak intensities are discussed.

Currently, there are several techniques available for measuring microroughness. However, the results tend to be qualitative. Until recently, there was no metrology standard available to correlate the accuracy of various instruments at extremely low levels of surface texture. This paper describes a metrology standard that is useful for calibrating instruments for the levels of microroughness encountered in semiconductor, disk drive, and related industries today. In advanced applications, this level is about 5 angstroms rms in a 0.01 - 1.0 micrometer^-1 spatial bandwidth range. This standard uses a one-dimensional square wave to reduce the effects of instrument spatial bandwidth. The standard has a 20 micrometer pitch with feature depths as small as 10 angstroms. The overall theoretical design guidance for this standard has been describe previously.

The dressed Rayleigh expansion is used to calculate the scattering of electromagnetic radiation from a randomly rough surface. This expansion was developed for grating calculation by Agassi and George [Phys. Rev. B 33, 2393-2400, 1986] to eliminate the numerical overflow, due to the evanescent waves, when the grating height is large. This dressed expansion involves redefining the expansion coefficient so that the matrix element is multiplied by an exponential damping factor. We find that this damping is too strong causing numerical underflow. As the root mean square (rms) roughness increases, the size of the expansion must decrease to avoid this numerical problem.

We present a numerical simulation of photon scanning tunneling microscopy based on the Rayleigh approach. Our non- perturbative formalism is able to solve the electromagnetic field near to a dielectric surface with deterministic one- dimensional sub-wavelength structures. The constant-height near-field image in transmission mode through two triangular ridges placed on surface has been calculated for p-polarized incident light at normal incidence. The influence on the signal by the dielectric function of the probe, the polarization of the incident light, and the separation of the objects are investigated.

The classical diffraction limit of conventional optical systems of image formation can be beaten in near field optical microscopy, and several schemes have been proposed for this purpose. Most of the arrangements employ a tapered optical waveguide to illuminate or/and collect the light reflected or transmitted by the sample. The resolution is, in this case, primarily determined by the size of the tip, and efforts are being made to produce sharper ones. An alternative technique that has also been considered consists of using scatterers in the near field of the sample under study. Although the interpretation of the images becomes more difficult, this technique appears promising for extending the current resolution of near-field optical microscopes. In this paper, we consider the problem of image formation in a near field scatter-probe optical microscope, applied to the visualization of metallic samples. In our study, the problem is approached with a combination of numerical and experimental techniques.

Over twenty years ago a linear systems approach to modeling surface scatter phenomena was developed by considering it to be a scalar diffraction process resulting from random phase variations in the exit pupil of an optical system. This led to the derivation of a surface transfer function (STF) that relates the scattering behavior to the surface topography. The resulting model was analogous to, and an extension of, the highly successful application of linear systems theory to the understanding of image forming systems. Experimental angular scattering data was shown to be shift-invariant in direction cosine space with respect to incident angle (this led to a modest following among the radiometric community of BRDF curves plotted in the Harvey-Shack (beta) -(beta) _o format). During the 1980s this STF was generalized to include: (1) the effects of small-angle scatter caused by 'mid' spatial frequency surface irregularities which span the gap between the traditional 'figure' and 'finish' errors, and (2) the extremely large incident angles inherent to grazing incidence Wolter Type I x-ray telescopes. Since no explicit smooth surface approximation is imposed, this STF can be utilized to predict the scattering behavior of rough surfaces not accurately modeled by vector perturbations techniques considered to be more rigorous by many investigators. Also, the scattering function is normalized by the total reflectance of the surface. Hence, the dominant polarization effects are included (in spite of the fact that this is basically a scalar treatment) by using the Fresnel reflectance coefficients for the desired polarization. In this paper it is emphasized that scattered radiance (not irradiance or intensity) is shift- invariant in direction cosine space paper to explain some non- intuitive scattering behavior reported in the literature.

Rayleigh-Rice or Beckmann-Kirchoff theories are commonly used to predict scatter results. However, in order to apply these theories in practice, inherent assumptions must be made that either limit the roughness of the surface under test or limit the predictions to small, paraxial incident and scatter angles. Various published reports show experimental scatter results and diffraction efficiencies that do not agree with these theories. One possible explanation for these discrepancies is that there is some confusion between whether the data being plotted is intensity or radiance. The quantity intensity is usually measured in the laboratory, not radiance. Using the Harvey-Shack theory, a Fourier linear systems theory based on using a surface transfer function, we show excellent agreement between experimental results and theoretical predictions. This holds true for scatter from rough surfaces as well as large scatter angles and angles of incidence.

The enhanced backscattering of light from randomly rough metal surfaces, which manifests itself as a well-defined peak in the retroreflection direction in the angular distribution of the intensity of the light scattered incoherently has attracted a great deal of attention recently. The backscattering phenomenon is attributed to the coherent interference of multiply-scattered surface plasmon polaritons excited on a metal surface with their time-reversed partners. The coherent interference of multiply-scattered lateral waves excited in the scattering of light from strongly rough dielectric surfaces is known to lead to an enhanced backscattering peak in the angular distribution of the intensity of s-polarized light scattered from them. In this paper we present an analytical theory of the scattering of light from a one- dimensional randomly rough interface between two media. One of the media is a dipole-active medium that is characterized by a frequency-dependent dielectric function, that is negative in a restricted frequency range, while the other is characterized by a frequency-independent, real, positive dielectric constant. We assume that the interface profile function is a single-valued function of the coordinate in the mean plate of the interface that is normal to its grooves and ridges, and constitutes a zero-mean, stationary, Gaussian random process. We assume that either p- or s-polarized electromagnetic waves are incident on the interface from the medium whose dielectric constant is frequency-independent. We study the angular distribution of the light that has been scattered incoherently as a function of the frequency of the incident light. The evolution of the enhanced backscattering peak in the case of p-polarized incident light as the frequency of the incident light is tuned through the frequencies of the dipole-active excitations in the medium whose dielectric function is frequency-dependent, is studied. Different mechanisms for the formation of the enhanced backscattering peak in different frequency regions are discussed.

A numerical algorithm is used to generate two-dimensional surfaces defined by x₃ equals (zetz) (x), with x equals (x₁, X₂), where (zetz) (x) is a single-valued function of x that constitutes a zero- mean, stationary, isotropic, Gaussian random process defined by the properties <(zetz) (x)> equals 0, <(zetz) (x)(zetz) (x¹)> equals (sigma) ²W(x - x¹), and (sigma) ² equals <(zetz) ²(x)>. The angle brackets here denote an average over the ensemble of realizations of the surface profile function (zetz) (x). The results are used to compute the probability density P₁(x)[P₂(x)] that the nearest maximum (minimum) to a given maximum (minimum) is at a distance x from the latter; and the probability density P₃(x) that the nearest minimum to a given maximum is at a distance x. Results are presented for random surfaces defined by surface height autocorrelation functions W(x) equals exp(-x²/a²), a²/(x² + a²), and 2[(k₂² - k₁²)x²]^-1[k₂xJ₁(k₂x) - k₁xJ₁(k₁x)], where J₁(z) is a Bessel function. Results are also presented for a novel type of one-dimensional random surface used in recent experimental studies of enhanced backscattering.

An experimental investigation of the angular distribution of the light scattered by randomly rough, two-dimensional, isotropic dielectric surfaces is presented. The surfaces, whose profiles constitute good approximations to Gaussian random processes with Gaussian correlation functions are fabricated in photoresist and characterized by means of a mechanical profilometer. The substrates employed in the fabrication of the samples consist of thick parallel plates of filter glass that absorb the incident light and whose refractive index is close to that of photoresist. This allows us to approximate experimentally a situation in which the light is scattered by a randomly rough interface separating two semi-infinite dielectric media, illuminated from the air side. With the rougher surfaces, we have observed enhanced backscattering effects in both, the s and p cases of incident polarization. Small but important cross-polarized components of the scattered light have also been observed.

We calculate the elements of the Stokes matrix for the scattering of light from a one-dimensional randomly rough metal surface in the general case when the plane of incidence is not perpendicular to the generators of the surface. By using Green's second integral identity one can obtain a system of four coupled integral equations for the components of the electric and magnetic fields parallel to the generators of the surface, and their normal derivatives, evaluated on the surface. The components of the scattered electric and magnetic fields are given in terms of integrals containing these four source functions. The system of four coupled integral equations is solved numerically for each of 2000 realizations of the surface profile function, and the results are used in calculating the elements of the Stokes matrix for the scattering geometry assumed. It is found that all elements of the Stokes matrix are nonzero, in contrast to the case when the plane of incidence is perpendicular to the generators of the one-dimensional surface. The results of this study provide complete information about the diffuse scattering properties of one-dimensional randomly rough metal surfaces.

By means of numerical simulation, we study the scattering of p-polarized light from, and its transmission through, a thin, free-standing, metal film. The illuminated face of the film is a one-dimensional, randomly rough surface, whose generators are perpendicular to the plane of incidence; the back surface is planar. The random roughness of the illuminated surface is characterized by a West-O'Donnell power spectrum that is nonzero in only a narrow range of wave numbers k_min less than k less than k_max that includes the wave numbers q₁((omega) ) and q₂((omega) ) of the surface plasmon polaritons supported by the film at the frequency (omega) of the incident light. The existence of two surface electromagnetic waves leads to the appearance of two satellite peaks in the angular dependence of the intensity of the incoherent component of the light scattered from the film at scattering angles (theta) _s given by sin (theta) _s equals - sin (theta) _i plus or minus (c/(omega) )[q₁((omega) ) - q₂((Omega) )], where (theta) _i is the angle of incidence of the light, in addition to the enhanced backscattering peak in the retroreflection direction (theta) _s equals -(theta) _i. At the same time satellite peaks occur in the angular dependence of the intensity of the light transmitted incoherently through the film at angles of transmission (theta) _t given by sin(theta) _t equals - sin (theta) _i plus or minus (c/(omega) )[q₁((omega) ) - q₂((omega) )], in addition to the enhanced transmission peak in the antispecular direction (theta) _t equals -(theta) _i. These results are compared with those for a metal film whose rough surface is characterized by a Gaussian power spectrum yielding the same rms height and rms slope as the West-O'Donnell power spectrum.

Recently, Surface Optics Corporation has designed and manufactured a field portable bidirectional reflectometer that measures the bidirectional reflectance of samples in place without the need to take samples into the laboratory. The instrument consists of a measurement head, power supply box, and a PC. The measurement head weighs approximately sixty pounds and it contains the source, detector, stepper motors for varying the incident and reflected angles, and a filter wheel. All of these components are software controlled for measuring the BRDF of samples from 400 nm to 1100 nm (VIS-NIR configuration) or 3.0 micrometer to 12.0 micrometer (IR configuration) at incident polar angles of 0 to 60 degrees. The detector can map the BRDF of a sample from 0 to 85 degrees polar angle and 0 to 180 degrees in azimuth. The instrument configuration is reviewed and measured data presented on a blue krylon paint sample.

The final design, fabrication, and testing of a space-based scatterometer has been completed and is described in this paper. The instrument, part of the optical properties monitor (OPM) experiment developed for NASA by AZ Technology in Huntsville, was designed to fly for an extended period in low earth orbit to monitor the effects of the orbital environment on various materials and coatings. The scatterometer measures the total integrated scatter (TIS) of various samples at wavelengths of 532 and 1064 nm. The instrument is able to distinguish between surface roughening and surface contamination and operates with an accuracy of plus or minus 10% and a repeatability of plus or minus 2%. The instrument is now attached to the outer hull of the MIR space station and is scheduled to operate for nine months or more before being returned to Earth.

A goniometric optical scatter instrument has been developed at the National Institute of Standards and Technology which can readily perform out-of-plane measurements of optical scatter as well as polarimetric measurements. This paper uses the description of this instrument as a platform to discuss key issues that must be addressed when developing either out-of- plane measurement capabilities or polarimetric capabilities, or both at the same time. The transformation from the sample coordinates to the instrument coordinates has been carried out, including the rotation of the polarization coordinates for out-of-plane measurements. The out-of-plane instrument signature that results from Rayleigh scatter in air is calculated and compared with measurement. Finally, the results of some out-of-plane Mueller matrix BRDF measurements of the backside of a silicon wafer are presented.

Experimental results are presented for the angular correlation function of far field speckle patterns scattered in the double passage of waves through a one-dimensional random phase screen. The experiment for the correlation measurement was set up to use a CCD camera to obtain the image of the speckle patterns in the scattering directions for each given angle of incidence, the cross correlation function is then calculated from the digitized images. The theoretical analysis of the motion of the speckle, as the source is moved, made by Escamilla is verified experimentally. It is found that in contrast with the memory effect line of speckle motion, the speckle pattern produced in the region of observation tracks the backscattering direction.

Diagrammatic perturbation theory is used to compute the angular intensity correlation function C(q,kq',k') equals <[I(qk) - <I(qk)>][I(q'k') - <I (q'k')]> for light scattered from a dielectric film on a perfectly conducting substrate and light scattered from and transmitted through a thin metallic film. The illuminated surface in each of these systems is taken to be a weakly rough, one-dimensional random surface, I(qk) is the squared modulus of the scattering matrix for the system, and q, q' and k,k' are the projections on the mean scattering surface of the wave vectors of the scattered and incident light, respectively. Contributions to C include: (1) a short range memory effect and time-reversed memory effect terms associated with the resonant excitation of guided or surface waves in the films, C⁽¹); (2) an additional short range term of comparable magnitude C⁽¹⁰); (3) a long range term C⁽²); (4) an infinite range term C⁽³); (5) and a terms C^(1.5) that along with C⁽²) displays peaks associated with the excitation of guided or surface waves.

We present the experimental results of the angular correlation function of far field speckle patterns scattered by a one- dimensionally random rough surface of a thin dielectric film on a glass substrate when a polarized beam of light is incident on the rough surface from vacuum. This surface, which separates the vacuum and the dielectric, is rough enough that only diffused speckles are observed. The experiment for the correlation measurement was set up to use a CCD camera to obtain the image of the speckle pattern in the specular direction for each given angle of incidence; the cross- correlation function is then calculated from the digitized images. It is found that the intensity correlation functions exhibit two distinct maxima, one arises from the auto- correlation and the other from the reciprocity condition. It is also found that different scattering processes give rise to quite different correlation functions, where multiple- scattering processes produce narrow peaks with secondary maxima, while single-scattering processes produce relatively broad peaks. The error analysis is also presented.

The application of analytical light scattering techniques for virtual prototyping the optical performance of paint coatings provides an effective tool for optimizing paint design for specific optical requirements. This paper describes the phenomenological basis for the scattering coatings computer aided design (ScatCad) code. The ScatCad code predicts the bidirectional reflectance distribution function (BRDF) and the hemispherical directional reflectance (HDR) of pigmented paint coatings for the purpose of coating design optimization. The code uses techniques for computing the pigment single scattering phase function, multiple scattering radiative transfer, and rough surface scattering to calculate the BRDF and HDR based on the fundamental optical properties of the pigment(s) and binder, pigment number density and size distribution, and surface roughness of the binder-interface and substrate. This is a significant enhancement to the two- flux, Kubelka-Munk analysis that has traditionally been used in the coatings industry. Example calculations and comparison with measurements are also presented.

Fluorescence or Raman emission can be used in characterizing particles inside or near a surface, e.g., a biological cell or spore on a filter, or a contaminant particle on a silicon wafer. Here we model the emission from a sphere on a surface. The internal fields in a sphere on a surface are known for plane-wave excitation. These fields induce dipole moments in molecules in the sphere. These oscillating dipoles are the sources of the incident radiation at the shifted frequency. The Green function for emission is found by using the reciprocity theorem for Green functions along with the internal fields generated by a plane wave at the shifted frequency. Reciprocity provides a simple method for obtaining the far fields for systems for which the near/internal fields are known for plane-wave excitation.

A scanning-mode microfocused total integrated scatter (TIS) instrument, the scanning scattering microscope (SSM), can produce two-dimensional images of very small surface and subsurface features and of variations in surface and buried interface roughness. In its present configuration, the lateral resolution is approximately 10 micrometer and the minimal detectable rms-roughness is 0.05 nm (with a bandwidth of 0.096 micrometer^-1 to 1.56 micrometer^-1). The performance of the SSM has been demonstrated using calibration gratings, Ge samples and ultra-smooth Si(100) and SiO₂/Si(100) samples. Intercomparison has also been made with atomic force microscope (AFM) measurements. The results indicate that this scanned optical technique is a very sensitive, non-contact optical method for evaluating surface microroughness. Our measurements also indicate that in some cases, e.g. for ultrathin (less than 10 nm) SiO₂ on Si, this optical method can be used to directly image microroughness of buried interfaces. Due to the small beam spot size, compared to a standard TIS, the SSM is applicable to the TIS measurements of rms roughness of small areas (of submillimeter diameter), e.g. craters made by secondary ion mass spectrometry (SIMS) techniques.

The formalism to study the scattering by a planar interface with spatial variations in the refraction index is developed. The results obtained are applied to two cases of theoretical interests, which are not currently implemented with this method: thermal gratings and spatial variations in the magnetic permeability. Surface impedance is plotted as a function of the spatial coordinate and the angular component of incident field. Also, scattered intensity is calculated for each case using a Gaussian beam as incident field.

A new model to treat the interactions of scintillation photons with dielectric surfaces is being implemented in DETECT, a public domain optical photon transport simulator for the design of scintillation counters. Inspired from the initial work of Nayar et al., the new UNIFIED model has the particular advantage of merging, into a single parameterization, models that usually apply over a very limited range of surface roughness values. Its flexibility is ensured by using the standard deviation of the surface slope as an input parameter that can be extracted from simple surface profile data. Here, we start by presenting the implementation of the UNIFIED model into DETECT. A prescription to constrain its free parameters with a simple set of characterization data is also discussed. The model is then tested for its capacity to predict the photoelectron yields measured for BGO crystals with different surface finish and reflective coats. Current results indicate that the transport of scintillation photons internally trapped within the volume of a highly polished crystal is well accounted for. However, significant discrepancies are noted between predictions and measurements when considering a corrugated finish or when the surface is coated by a diffuse reflector. Possible explanations are discussed and subject to further investigations.

The deep space one (DS-1) probe is a mini-tour photo reconnaissance mission of Earth-Mars space. Target bodies include an asteroid, a comet, and the planet Mars. Central to this mission is an accurate measurement of the reflectance of the primary mirror of the imaging system. Knowledge of this reflectance will allow calculation of the absolute albedos of the target bodies encountered. Scattering measurements were also made on the so-called 'diffuser plate' of the optical train. This component was intended to be used only for the solar occultation experiment which measures the solar intensity as a function of atmospheric depth as Mars comes between the Sun and the probes optical detectors. The diffuser plate was designed to reduce the solar intensity by scattering light in an isotropic fashion. However, it has been found that the diffuser plate is not an isotropic scatterer. This report describes the equipment used to make the reflectance and scattering measurements, as well as the results obtained from the measurement program.