Surface Height Fluctuations Research Articles

Holographic Method for Reconstruction of Random Rough Surface

A holographic principle for computer reconstruction of random rough surface (RRS) and determination of the surface parameters is used. The random function describing the surface is expanded in a Fourier series. The scattering from the RRS light field is approximated as a field, obtained as result of diffraction from sinusoidal gratings. The mixed field (obtained from the mixing of the diffracted field and the field of the laser beam) carries information about the phase of each sinusoidal grating. Sections of the diffracted and mixing fields are correspondingly registered by photodiode array. The first, the second and the third diffraction orders from each grating are taken into account for the surface reconstruction. The statistical distribution and the autocorrelation function of the surface height fluctuation are calculated. The surface parameters of mean roughness, root mean square and correlation length are determined. The studied surface, having a Gaussian statistic, is measured [from the work of Daintyet al.Talystep profilometer. The values of the above-mentioned surface parameters calculated in Daintyet al.are in good agreement with those obtained in the present work.

Surface morphology of GaN films determined from quantitative x-ray reflectivity

The surface of two GaN films grown under Ga-rich conditions by molecular beam epitaxy was characterized using x-ray reflectivity, assuming a self-affine/fractal surface morphology. The surface height fluctuations were similar for both samples at the largest lateral length scales at which the fractal description is valid, although this lateral length was significantly greater and the “jaggedness” significantly smaller for the sample grown under higher Ga flux. Previous atomic force microscopy images revealed a higher density of large features on the surface for the sample grown under lower Ga flux. The lateral size of the features are dominated by a convolution of the atomic force microscopy tip shape and the actual features on the surface, which precludes an accurate determination of the surface structure at length scales smaller than the tip radius. This study illustrates the importance of using different techniques to evaluate the film surface morphology at different length scales.

Simulations of roughening of amorphous carbon surfaces bombarded by low-energy Ar-ions

Roughening of amorphous carbon surfaces bombarded by 5 keV Ar-ions is studied by using Monte Carlo simulations, where sputtering is treated in detail by simulating the entire collision cascades and the surface relaxation is described by a Wolf-Villain-type model. The simulations show that bombarded surfaces have self-affine topography on a submicrometer scale. The roughness exponent is found to decrease from α ≈ 0.47 to α ≈ 0.25, when the angle of incidence is varied from 0° to 60°, respectively. In the studied cases the roughness exponent α is nearly independent of the relaxation of the surface. Local surface height fluctuations are monitored to gain insight of the spatio-temporal correlations in the process of erosion.

Invalidity of the spatiotemporal white noise assumption for a stochastic diffusion-type equation

I point out that the Gaussian white noise assumption, where the random variables have the spatiotemporal \ensuremath{\delta}-function-type correlation, for the Edwards-Wilkinson equation with periodic boundary condition is not valid in dimensions higher than two, because it leads to divergent fluctuation of surface height. It is stressed that the physical solution should be irrelevant to the choice of the cutoff. One finds through exploring a continuum limit that the correlation should be less singular than the \ensuremath{\delta} function and we have a finite solution with the vanishing roughness exponent.

Simulations of submicron-scale erosion and ripple formation on ion bombarded solid surfaces

We have performed Monte Carlo simulations of roughening and erosion of amorphous carbon surfaces bombarded by 5 keV Ar-ions. The maximum fluence reached in the simulations is 2.8×10 17 ions/cm 2. The statistics of surface height fluctuations suggests that roughened surfaces are self-affine, with roughness exponent α ≈ 0.6 ± 0.2. In the simulations “laguna” and ripple-like patterns are observed. These patterns, however, are not stable. Only in one studied case regular and stable ripples are observed. The scale of the ripples is comparable to cascade dimensions, as expected when there is no surface diffusion.

Open‐ocean validation of TOPEX/POSEIDON sea level in the western equatorial Pacific

During the verification phase of the TOPEX/POSEIDON radar altimeter mission a rigorous open‐ocean validation experiment was conducted in the western equatorial Pacific Ocean. From August–September 1992 to February–March 1993, two Tropical Ocean and Global Atmosphere (TOGA) Tropical Atmosphere Ocean moorings at 2°S–156°E (1739 m depth) and 2°S–164.4°E (4400 m depth) were outfitted with additional temperature, salinity, and pressure sensors to measure precisely the dynamic height from the surface to the bottom at 5‐min intervals directly beneath two TOPEX/POSEIDON crossovers. Bottom pressure gauges and inverted echo sounders were deployed as well. A predeployment design study using full depth conductivity‐temperature‐depth casts, subsequently confirmed by postdeployment analyses, indicated this suite of instruments was capable of measuring sea surface height fluctuations to within 1–2 cm. The validation experiment also benefited from the comprehensive set of ocean‐atmosphere measurements that were made in the region during the TOGA Coupled Ocean‐Atmosphere Response Experiment intensive observation period of November 1992 to February 1993. The surface relative to bottom dynamic height fluctuations observed in situ during the 6–7 month experiment had a standard deviation of 5 cm with excursions of order ±15 cm. Energetic steric sea level variability was found to exist on short timescales of order hours to a few days, most notably, the quasi‐permanence of strong semidiurnal internal tides. Such internal tides were noted to induce changes in surface dynamic height with a standard deviation of 2 dynamic centimeters. At the shallower of the two sites, 2°S–156°E, a possible nonlinear rectification of the internal tide was observed occasionally to change the dynamic height by as much as 30 cm over less than an hour. On timescales longer than the 10–day repeat of the TOPEX/POSEIDON satellite, the low‐frequency fluctuations of dynamic height were related to interannual variations corresponding to the 1991–1993 El Niño‐Southern Oscillation, to the seasonal cycle, and to intraseasonal variations associated with the 40‐ to 60‐day oscillations of the equatorial zonal wind field. Instantaneous comparisons between the 1‐s TOPEX/POSEIDON altimeter retrievals and the 5‐min dynamic height were performed with regard to several tide models, the barotropic tide measured in situ, European Centre for Medium‐Range Weather Forecasts surface air pressure, and the surface air pressure measured in situ. Depending on the choice of altimeter and of the environmental corrections applied to the altimeter data, the rms differences between the satellite and the in situ measurements of sea level were as low as 3.3 cm at 2°S–156°E and 3.7 cm at 2°S–164.4°E. When additional satellite data in the general vicinity of the mooring are included and after the use of a 30‐day low‐pass filter, the satellite and in situ data were found to be highly correlated, with correlation coefficients of about 0.95 and rms differences around 1.8 cm.

Kuroshio sea surface height fluctuations observed simultaneously with inverted echo sounders and TOPEX/POSEIDON

Sea surface height anomalies measured by inverted echo sounders with pressure gauges (IES/PGs) and TOPEX/POSEIDON are compared at six locations in the northwest Pacific near the Kuroshio where it separates from the Japanese coast. The instruments are positioned along a TOPEX/POSEIDON descending ground track near 35°N, 143°E. Good correlation is found between measurements, with best agreement near the Kuroshio, where the range of sea surface height change is largest. Correlations between IES/PG and TOPEX/POSEIDON sea surface height variations range from 0.70 to 0.97. Tides modeled and removed from the TOPEX/POSEIDON data at the inverted echo sounder sites compare well with tides measured by the IES/PGs. TOPEX/POSEIDON and IES/PG data are complementary in describing mesoscale variability in the Kuroshio region.

Variability and Energetics of the Kuroshio Extension and Its Recirculation Gyre from the First Two-Year TOPEX Data

Abstract Altimetry data from the first 21-month TOPEX mission (September 1992–June 1994) are analysed to investigate the sea surface height (SSH) fluctuations in the Kuroshio Extension and its southern recirculation gyre regions (25°–40°N, 136°E–180°). To separate the time-dependent (SSH) signals associated with the Kuroshio Extension from those associated with the westward recirculating flows, and to study the energetics of these currents, the author first estimated the mean SSH profiles along individual ground tracks by assuming the velocity profile of the Kuroshio Extension to be Gaussian shaped and by successively fitting the synthetic current's height profile to the time-dependent SSH data. The mean SSH field, including the influence from the recirculating flows, is then derived through the constraint from climatological hydrographic data. During the 21-month period of the TOPEX mission, the eddy kinetic energy (EKE) of the Kuroshio Extension had relatively uniform values during three separate stages...

Intrinsic roughness of glass surfaces

It is shown that the spectrum of the thermal surface height fluctuations on supercooled liquid close to the glass transition contains a narrow quasielastic component, which desribes slow fluctuation freezing in at the glass transition. The spectrum is obtained from a dynamic susceptibility which is calculated using the theory of viscoelasticity.

Thermal and kinetic surface roughening studied by scanning tunneling microscopy and atomic force microscopy

Abstract Recent studies of thermal roughening on Si surfaces and kinetic roughening of some growing films, copper and tungsten, by using scanning tunneling microscopy and atomic force microscopy are reviewed. A logarithmic divergence of the surface height fluctuations of Si(111) vicinal surfaces is confirmed, in agreement with the theoretical prediction of rough surface in thermal equilibrium. For the kinetically formed rough surfaces, power law dependences of the interface width on the system size are clearly observed. Furthermore, the tungsten films show a short-range scaling regime and a long-range “smooth” regime. The roughness exponents α are compared with theoretical predictions: for the typical Cu electrode position condition (α=1/2), the exponent appears to be close to that found for local growth models, and for tungsten films (0.7∼0.8), it is consistent with recent predictions for growth where surface diffusion is predominant.

Optical remote sensing of surface roughness through the turbulent atmosphere

Using a CO(2), cw, coherent lidar, we have measured the surface roughness of diffuse targets through 1000 m of turbulent atmosphere. The technique measures the phase fluctuations of the speckle field at the receiver and relates the measured phase variance to the surface roughness. Measurements were made with aluminum targets that had been sandblasted with 8-, 16-, and 30-grit material and also with a flame-sprayed aluminum target. It was found that a linear relationship exists between the standard deviation (SD) of the unwrapped phase fluctuations and the SD of the target surface-height fluctuations. Good results were obtained with modest transmitter power and small receiver optics in just a few seconds of averaging time.

Effective reflection coefficients for the mean acoustic field between two rough interfaces

A formalism is presented that demonstrates that the mean Green’s function for the acoustic field between two rough interfaces can be expressed as a Green’s function associated with two flat interfaces with effective reflection coefficients. This result incorporates all orders of the fluctuations in the half-space scattering amplitudes associated with each interface considered separately. From the mean Green’s function modal attenuations can be found. To lowest order in the surface height fluctuations it is shown that it is not sufficient to use mean half-space scattering amplitudes as effective reflection coefficients. The formalism is designed to provide approximations for the Green’s function in layered media which are based on previously developed approximations for half-space scattering amplitudes.

Ge-Au eutectic bonding of Ge {100} single crystals

We present preliminary results on the eutectic bonding between two {100} Ge single crystal surfaces using thin films of Au ranging from 900A/surface to 300A/surface and Pd (10% the thickness of Au). Following bonding, plan view optical microscopy (OM) of the cleaved interface of samples with Au thicknesses ≤ 500A/surface show a eutectic morphology more conducive to phonon transmission through the bond interface. High resolution transmission electron microscopy (HRTEM) cross sectional interface studies of a 300A/surface Au sample show epitaxial growth of Ge. In sections of the bond, lattice continuity of the Ge is apparent through the interface. TEM studies also reveal heteroepitaxial growth of Au with a Au-Ge lattice mismatch of less than 2%. Eutectic bonds with 200A/surface Au have been attained with characterization pending. An optical polishing technique for Ge has been optimized to insure intimate contact between the Ge surfaces prior to bonding. Interferometry analysis of the optically polished Ge surface shows that surface height fluctuations lie within ±150A across an interval of 1mm. Characterization of phonon transmission through the interface is discussed with respect to low temperature detection of ballistic phonons.

Sea surface height fluctuations observed simultaneously with inverted echo sounders and Geosat

Time series of sea surface height anomaly as measured by inverted echo sounders with pressure gauges (IES/PGs) and Geosat altimetry are compared at six locations in the northwest Atlantic near 40°N, 55°W. Three IES/PG sites lay along each of two adjacent Geosat‐Exact Repeat Mission ground tracks. Better agreement was found along the westernmost ground track; pressure sensor noise is the probable source of the larger errors along the easternmost ground track. The overall correlation is 0.78; along the westernmost ground track it is 0.91. Decorrelation scales are about 100 km along track and about 5 days. Uncorrelated errors dominate, but along the westernmost ground track, IES/PG‐derived sea surface height variability is systematically smaller than that derived from Geosat data; this factor, determined by regression analysis, is about 0.81. However, a consistent adjustment of IES calibration removes the systematic discrepancy.

Coherence of acoustic scattering from a dynamic rough surface

Numerical studies of the angular and temporal coherence of rough-surface acoustic scattering are presented at kh=12, 20, and 32, where k is the acoustic wave number and h is the root-mean-square surface height fluctuation. The computations are based on point-source illumination of a one-dimensional pressure-release Pierson–Moskowitz dynamic rough surface, and the Kirchhoff approximation. In the region near specular, for a source-location grazing angle of 20°, the angular variation of the scattered field coherence is found to predominantly depend on the product of k and a fixed length scale. The computed results show only a mild dependence on kh. For the same geometry, the temporal variation of the scattered field coherence is found to depend predominantly on (g Δt/U)kh, where g is the acceleration of gravity, Δt is the time shift, and U is the wind speed. A general method for scaling the scattered field coherence in terms of k, Δt, U, and the angular separation of the field points is suggested based on the findings. The effects of wind direction and surface-wave frequency cutoff on acoustic scattering are also noted.

Recirculation and seasonal change of the Kuroshio from altimetry observations

Surface height fluctuations across an eastward flowing boundary current, such as the Gulf Stream and the Kuroshio, can be caused by changes both in the upstream inflow and in the neighboring recirculation gyres. Clarifying these causes is important in understanding the time‐dependent nature of the boundary currents and their relationship to the surface wind and buoyancy forcing. Using satellite altimetry data to identify these causes, however, requires mean surface height field information, which is not readily available from the altimetry observation owing to lack of accurate geoid data. The present study presents a method to estimate the mean surface height profile across a boundary current system (including recirculation gyres) by combining altimetrically measured residual height data and historical hydrographic data. Applying this method to the Kuroshio revealed that the absolute surface height profiles thus estimated agree well with the results from quarterly, in‐situ hydrographic observations. By separating the signals of the surface height fluctuations into those of the eastward flowing jet and the recirculation gyres, we found that the seasonal cycle in the surface transport of the Kuroshio, which has a maximum in July and August, is primarily due to the seasonal change in the intensity of the recirculation gyre south of the Kuroshio.

Scintillation index of the acoustic field forward scattered by a rough surface for two- and three-dimensional scattering geometries

Scintillation indices of acoustic waves forward scattered by a rough surface are obtained from numerical techniques leading to both accurate and approximate results in two-dimensional (2-D) and three-dimensional (3-D) geometries. The Kirchhoff approximation with a Gaussian spectrum and an empirical model for ocean surface height fluctuations is used. It is found that in the case of a Gaussian spectrum, the 3-D accurate values obtained are lower than the 2-D ones when the focusing–defocusing phenomenon does not occur. However when this phenomenon occurs, the 3-D results become higher than the 2-D ones. In the case of the empirical spectrum, both 2-D and 3-D accurate scintillation indices decrease with the wind speed on the ocean surface. The 2-D model predicts larger values than the 3-D model. When an ensonification factor or finite incident beam size is introduced, the scattering moves from the saturation limit into the focusing–defocusing range.

Comparison of the Two-dimensional Fraunhofer and the Two-dimensional Fresnel Approximations in the Analysis of Surface Roughness by Angle Speckle Correlation

Abstract In this paper we discuss some new theoretical results which we reached in the analysis of a random surface roughness in the framework of the two-dimensional (2D) Fresnel approximation. A comparison with 2D Fraunhofer approximation results is made. It is shown for experimentally reasonable values of the geometrical arrangement parameters of the method that the surface roughness measurement can be interpreted in the 2D Fraunhofer approximation only for surfaces with high slopes of surface irregularities. Surfaces with small slopes of irregularities require interpretation of the surface roughness measurement in the 2D Fresnel approximation. In this case, not only the standard deviation of surface height fluctuations but also the correlation length of the surface can be determined.

Fourth moments of acoustic waves forward scattered by a rough ocean surface

Abstract Three types of statistical fourth moments of acoustic waves forward scattered by a randomly rough ocean surface are derived and numerically evaluated. The first one is related to the scintillation index which characterizes intensity fluctuations. The second one is the two-position intensity correlation function which describes the spatial correlation of wave intensity. The third is the fourth-moment two-position coherence function which carries information on the phase fluctuations of the scattered wave. In the range of weak scattering, the ratio of the absolute value of the fourth-moment two-position coherence function over the two-position intensity correlation exactly describes the mean-square fluctuation of the relative phase between the two positions. The acoustic frequency is high so that the Kirchhoff approximation can be used. Two types of spectral functions for surface-height fluctuations are considered: a Gaussian spectrum and the Donelan-Pierson spectrum. The latter is obtained from a model for the fluctuations of the ocean surface height which are controlled by the wind speed at the ocean surface.

Growth interfaces of Si1−xGex/Si heterostructures studied by i n s i t u laser light scattering

Elastic scattering of laser light has been used to monitor surface topography in real time during growth of Si1−xGex heterostructures. Large increases in scattered intensity were caused by surface height fluctuations in the alloy layers which, after growth, could be imaged in cross section by transmission electron microscopy and in plan view by scanning optical microscopy. For pseudomorphic alloy layers (x=0.19±0.01) grown at 750 °C the surface height fluctuations were periodic, the wavelength of approximately 270 nm giving strong laser scattering. Plan view transmission electron micrographs on such a layer showed periodic strain fluctuations with the same wavelength, whereas an equivalent layer grown at 610 °C showed strain fluctuations which were irregular in shape and spatial distribution. The intensity of light scattered from the surface of this layer was much lower. Thick alloy layers grown beyond the critical thickness for appearance of misfit dislocations showed strong scatter which increased more quickly in intensity at higher growth temperature.