Surface Slope Profile Research Articles

Tidal resurfacing model for (99942) Apophis during the 2029 close approach with Earth

ABSTRACT We numerically investigate tidally induced surface refreshing on Apophis during its close approach with Earth within a perigee distance of 5.96 Earth radii on 2029 April 13. We implement a tidal resurfacing model with two stages: dynamics modelling of the entire body to determine time-varying accelerations and surface slope profiles felt by each surface patch during the 6-h-long closest encounter, and discrete-element method modelling to track motions of surface grains in localized patches. The surface slope profiles and measured grain motions are combined to statistically extrapolate the ‘expected’ percentage of resurfaced area. Using the tidal resurfacing model, we present surface maps showing the total expected resurfacing on Apophis given three representative encounter orientations. Our simulation results indicate that tidal resurfacing, limited to certain localized regions, will likely occur half an hour before perigee and on the scale of 1 per cent of Apophis’s entire surface area. Our models indicate that the most likely locations to detect tidal resurfacing are: initially high-sloped regions (>30°) regardless of the encounter orientation of Apophis, and mid-sloped regions (15°–30°) that experience a significant positive slope variation (>0.5°), which is mainly controlled by the encounter orientation. Expected data from ground-based observations of the 2029 flyby will help us better constrain the targeted locations likely to experience tidal resurfacing. We thus expect to find evidence supporting tidal resurfacing via further analysis of post-encounter surface images or albedo changes at the expected resurfaced areas.

Super-resolution surface slope metrology of x-ray mirrors.

We present experimental, analytical, and numerical methods developed for reconstruction (deconvolution) of one-dimensional (1D) surface slope profiles over the spatial frequency range where the raw data are significantly perturbed due to the limited resolution of the measurement instrument. We characterize the spatial resolution properties of a profiler with the instrument's transfer function (ITF). To precisely measure the ITF, we apply a recently developed method utilizing test surfaces with 1D linear chirped height profiles of constant slope amplitude. Based on the results of the ITF calibration, we determine parameters of an analytical model for the ITF that is used in the original reconstruction software. Here, we treat surface slope metrology data obtained with the Optical Surface Measuring System (OSMS), using as a sensor an electronic autocollimator (AC) ELCOMAT-3000. The spatial resolution of the OSMS is limited by the size of the AC light-beam-collimating aperture. For the purposes of this investigation, the OSMS is equipped with a circular aperture with a diameter of 2.5 mm. This is a typical arrangement of most AC-based slope profilers developed for surface slope metrology of state-of-the-art x-ray mirrors. Using the example of surface slope metrology of two state-of-the-art elliptically shaped x-ray focusing mirrors, we demonstrate that the developed data reconstruction procedure allows us to significantly improve the accuracy of surface slope metrology with the OSMS over the spatial wavelength range from ∼1.6 mm to 7 mm. Thus, the amplitude of the quasi-periodic error characteristic of the deterministic polishing process used appears to be higher by a factor of ∼2 than is apparent from the rough metrology data. Underestimation of the surface slope errors in this spatial wavelength range can lead to serious errors in the expected performance of x-ray mirrors in synchrotron beamlines, especially at modern light sources utilizing coherent x rays, where the perturbations can lead to increased speckle-like intensity variation.

Optimization of the size and shape of the scanning aperture in autocollimator-based deflectometric profilometers.

Deflectometric profilometers based on industrial electronic autocollimators (ACs), as the ELCOMAT-3000, have become indispensable tools for precision form measurements of optical surfaces. A growing number of labs at synchrotron and free electron laser x-ray facilities are going for BESSY-II NOM-like versions of the AC-based profilometers. These tools have proven capable of characterizing state-of-the-art aspherical x-ray optics with an accuracy on the level of 100 nrad (root-mean-square) over the spatial frequency range limited by the size of the aperture used in the profilometer. Typically, a round aperture with a diameter of about 2.5 mm is used. Previous investigations have shown that with the optimally aligned 2.5-mm aperture, the spatial resolution of a NOM-like profilometer corresponding to the first zero-crossing of the optical transform function (OTF) is ∼1.2 mm. In this paper, we investigate the performance of an AC ELCOMAT-3000 for a slope profilometer with different aperture sizes and shapes. The results of angular calibration of the AC equipped with circular and rectangular apertures placed at different distances from the AC are discussed. The calibration was performed at the Physikalisch-Technische Bundesanstalt using the original experimental arrangements, also discussed in the paper. The OTF measurements with the specially developed test sample with chirped surface slope profiles were performed at the Advanced Light Source X-Ray Optics Laboratory (XROL) in application to a new optical surface measuring system under development at the XROL. In the OTF measurements, we have shown that application of a rectangular aperture with dimensions of 1.5 mm × 3 mm improves the spatial resolution in the tangential direction by a factor of ∼1.4 compared to that of the standard circular aperture of 2.5-mm diameter. We believe that the results of our investigations are crucial for reaching fundamental metrological limits in deflectometric profilometry utilizing state-of-the-art electronic autocollimators.

New figuring model based on surface slope profilefor grazing-incidence reflective optics.

Surface slope profile is widely used in the metrology of grazing-incidence reflective optics instead of surface height profile. Nevertheless, the theoretical and experimental model currently used in deterministic optical figuring processes is based on surface height, not on surface slope. This means that the raw slope profile data from metrology need to be converted to height profile to perform the current height-based figuring processes. The inevitable measurement noise in the raw slope data will introduce significant cumulative error in the resultant height profiles. As a consequence, this conversion will degrade the determinism of the figuring processes, and will have an impact on the ultimate surface figuring results. To overcome this problem, an innovative figuring model is proposed, which directly uses the raw slope profile data instead of the usual height data as input for the deterministic process. In this paper, first the influence of the measurement noise on the resultant height profile is analyzed, and then a new model is presented; finally a demonstration experiment is carried out using a one-dimensional ion beam figuring process to demonstrate the validity of our approach.

Simulation and control of aggregate surface morphology in a two-stage thin film deposition process for improved light trapping

This work focuses on the development of a model predictive control algorithm to simultaneously regulate the aggregate surface slope and roughness of a thin film growth process to optimize thin film light reflectance and transmittance. Specifically, a two-stage thin film deposition process, which involves two microscopic processes: an adsorption process and a migration process, is modeled based on a one-dimensional solid-on-solid square lattice. The first stage of this process utilizes a uniform deposition rate profile to control the thickness of the thin film and the second stage of the process utilizes a spatially distributed deposition rate profile to control the surface morphology of the thin film. Kinetic Monte Carlo (kMC) methods are used to simulate this two-stage thin film deposition process. To characterize the surface morphology and to evaluate the light trapping efficiency of the thin film, aggregate surface roughness and slope corresponding to length scale of visible light are introduced as the root-mean squares of the aggregate surface height profile and aggregate surface slope profile. An Edwards–Wilkinson (EW)-type equation with appropriately computed parameters is used to describe the dynamics of the surface height profile and predict the evolution of the aggregate root-mean-square (RMS) roughness and aggregate RMS slope. A model predictive control algorithm is then developed on the basis of the EW equation model to regulate the aggregate RMS slope and the aggregate RMS roughness at desired levels. Closed-loop simulation results demonstrate the effectiveness of the proposed model predictive control algorithm in successfully regulating the aggregate RMS slope and the aggregate RMS roughness at desired levels that optimize thin film light reflectance and transmittance.

Dependence of film surface roughness and slope on surface migration and lattice size in thin film deposition processes

This work focuses on the study of the dependence of film surface roughness and slope on lattice size in thin film deposition processes. Two different models of thin film deposition processes, in both 1D and 2D, are considered: random deposition with surface relaxation model and deposition/migration model. Surface roughness and surface slope are defined as the root-mean-squares of the surface height profile and of the surface slope profile, respectively. Both surface roughness and slope evolve to steady-state values at large times but are subject to different dynamics and scaling properties. A linear and a logarithmic dependence of surface roughness square on lattice size are observed in the 1D and 2D lattice models, respectively, in both the random deposition with surface relaxation model and the deposition/migration model with zero activation energy contribution from each neighboring particle. Furthermore, a stronger lattice-size dependence is found in the deposition/migration model when the migration activation energy contribution from each neighboring particle becomes significant. On the other hand, a weak lattice-size dependence is found for the surface mean slope in all growth models considered, especially at large lattice sizes. Finally, the dynamics of surface roughness and surface slope is studied with respect to different characteristic length scales.

Elliptically Bent X-Ray Mirrors with Active Temperature Stabilization

We present details of design of elliptically bent Kirkpatrick-Baez mirrors developed and successfully used at the advanced light source for submicron focusing. A distinctive feature of the mirror design is an active temperature stabilization based on a Peltier element attached directly to the mirror body. The design and materials have been carefully optimized to provide high heat conductance between the mirror body and substrate. We describe the experimental procedures used when assembling and precisely shaping the mirrors, with special attention paid to laboratory testing of the mirror-temperature stabilization. For this purpose, the temperature dependence of the surface slope profile of a specially fabricated test mirror placed inside a temperature-controlled container was measured. We demonstrate that with active mirror-temperature stabilization, a change of the surrounding temperature by more than 3 K does not noticeably affect the mirror figure. Without temperature stabilization, the rms slope error is changed by approximately 1.5 μrad (primarily defocus) under the same conditions.

Predictive control of surface mean slope and roughness in a thin film deposition process

This work focuses on the development of a model predictive control algorithm to simultaneously regulate the surface slope and roughness of a thin film growth process to optimize thin film light reflectance and transmittance. Specifically, a thin film deposition process modeled on a one-dimensional triangular lattice that involves two microscopic processes: an adsorption process and a migration process, is considered. Kinetic Monte Carlo (kMC) methods are used to simulate the thin film deposition process. To characterize the surface morphology and to evaluate the light trapping efficiency of the thin film, surface roughness and surface slope are introduced as the root mean squares of the surface height profile and surface slope profile. An Edwards–Wilkinson (EW)-type equation is used to describe the dynamics of the surface height profile and predict the evolution of the root-mean-square (RMS) roughness and RMS slope. A model predictive control algorithm is then developed on the basis of the EW equation model to regulate the RMS slope and the RMS roughness at desired levels by optimizing the substrate temperature at each sampling time. The model parameters of the EW equation are estimated from simulation data through least-square methods. Closed-loop simulation results demonstrate the effectiveness of the proposed model predictive control algorithm in successfully regulating the RMS slope and the RMS roughness at desired levels that optimize thin film light reflectance and transmittance.

Multivariable Model Predictive Control of Thin Film Surface Roughness and Slope for Light Trapping Optimization

This work focuses on the development of a multivariable model predictive controller that simultaneously regulates thin film surface roughness and mean slope to optimize light reflectance and transmittance during thin film manufacturing by manipulating substrate temperature and deposition rate. Surface roughness and surface slope are defined as the root-mean-squares of the surface height profile and the surface slope profile, respectively. The dynamics of the evolution of the thin film surface height profile are assumed to be described by an Edwards-Wilkinson-type equation (a second-order stochastic partial differential equation) in two spatial dimensions. Analytical solutions of the expected surface roughness and surface slope are obtained on the basis of the Edwards-Wilkinson equation and are used in the controller design. The model parameters of the Edwards-Wilkinson equation depend on the substrate temperature and deposition rate. This dependence is used in the formulation of the predictive controller to predict the influence of the control action on the surface roughness and slope at the end of the growth process. The model predictive controller involves constraints on the magnitude and rate of change of the control action and optimizes a cost that involves penalty on both surface roughness and mean slope from the set-point values. The controller is applied to the two-dimensional Edwards-Wilkinson equation and is shown to successfully regulate surface roughness and mean slope to set-point values at the end of the deposition that yield desired film reflectance and transmittance.

Kinetic Hierarchies and Macroscopic Limits for Crystalline Steps in $1+1$ Dimensions

We apply methods of kinetic theory to study the passage from particle systems to nonlinear partial differential equations (PDEs) in the context of deterministic crystal surface relaxation. Starting with the near-equilibrium motion of N line defects (“steps”) with atomic size a, we derive coupled evolution equations (“kinetic hierarchies”) for correlation functions $F_n^a$, which express correlations of n consecutive steps. We investigate separately the evaporation-condensation and the surface diffusion dynamics in $1+1$ dimensions when each step interacts repulsively with its nearest neighbors. In the limit $a\to0$ with $Na=\mathcal{O}(1)$, where a is appropriately nondimensional, the first equations of the hierarchies reduce to known evolution laws for the surface slope profile. The remaining PDEs take the form of simple continuity equations, which we solve exactly and, thereby, connect continuous limits of $F_n^a$ with the slope profile. In addition, we construct a particular example of $F_n^a$ asymptotically for small but finite a by regularization of measures. In the limit $a\to0$, this construction yields singular correlation functions.

Ultra-wideband radar studies of steep crested waves with scanning laser measurements of wave slope profiles

We report results of ultra wide-band radar sea spike experiments using steep and weakly breaking non-linear water surface features in a wave tank. To generate these features we used a 1 s paddle wave and wind waves for a sequence of wind speeds. A scanning laser was used to measure synchronously the surface slope profile across 12 cm along the wave propagation direction once per radar pulse. A time domain reflectometer (TDR) radar transmitted short horizontally polarized pulses at X-band, several hundred picoseconds long, to give a range resolution of 10 cm. A radar range of 36 cm was digitally sampled so that surface feature echoes could be tracked through the area continuously with 5 ms temporal resolution with each instrument. We report results considering the wave slope component in the propagation direction and the corresponding curvature component. For the conditions studied, two types of features which produce sea spike radar echoes were generated–a non-linear feature near the crest front of the wind wave, caused by extreme steepening as a result of the passage of the paddle wave, and a steepened blocked wind wave in the trough of the paddle wave, caused by the local orbital current of the 1 s wave being nearly equal to and opposite the phase velocity of the wind wave.

Self-affine scaling and subsurface response to snowmelt in steep terrain

The use of local scaling transformations is proposed as a framework for modeling the geometry, and dynamic response of subsurface flow from steep hillslopes. The study site is the Upper Sheep Creek Watershed, a first-order sub-basin in the Reynolds Creek Experimental Watershed, southwest Idaho. The goal of the research is to gain insight into the evolution of snowmelt, subsurface flow, and runoff generation in this rugged volcanic terrain. In this phase of the study rescaling geometrically proportional (self-affine) surface and bedrock slope profiles provide an analytical framework for defining the shape, volume and dynamic changes of saturated flow for this hillslope system. Rescaling self-affine hillslope trajectories along a stream reach, collapses the flow-domain to a unit cube. A dynamic model for hillslope runoff is proposed, and implications for modeling topographic controls on the snowmelt process are suggested.

Variations of global mesoscale eddy energy observed from Geosat

The global distribution of eddy kinetic energy has been synoptically observed from analysis of the Geosat Exact Repeat Mission (ERM) altimeter data collected for a 2‐year period from November 1986 through November 1988. Using a technique developed by Sandwell and Zhang (1989), altimeter data from forty‐four 17‐day repeat cycles (2 years) were processed into sea surface slopes along the satellite ground track, averaged, and filtered to produce a mean sea surface slope profile having an estimated accuracy of 0.2 μrad (2 cm sea level change over 100 km distance). A series of global eddy kinetic energy maps, each averaged over 3 months, and their mean were then generated. The maximum mean eddy kinetic energy per unit mass exceeds 2000 cm2/s2 for most of the western boundary currents; however, it only reaches approximately 500 cm2/s2 for the Antarctic Circumpolar Current (ACC). More than 65% of the world ocean has relatively low variability with an eddy kinetic energy of less than 300 cm2/s2. Results obtained from this study are in general agreement with other Geosat ocean variability studies (e.g., Zlotnicki et al., 1989). However, significantly higher variability is found when compared with either Seasat or ship drift data. Significant seasonal variations were found in the Gulf Stream and Kuroshio currents. The ACC system exhibits no apparent seasonal variation.

Global mesoscale variability from the Geosat Exact Repeat Mission: Correlation with ocean depth

We have developed a new technique for extracting global mesoscale variability from satellite altimeter profiles having large radial orbit error (∼3 m). Long‐wavelength radial orbit error, as well as other long‐wavelength errors (e.g., tides, ionospheric‐atmospheric delay, and electromagnetic bias), are suppressed by taking the derivative (slope) of each altimeter profile. A low‐pass filter is used to suppress the short‐wavelength altimeter noise (λ<100 km). Twenty‐two repeat slope profiles are then averaged to produce a mean sea surface slope profile having a precision of about 0.1 μrad. Variations in sea surface slope, which are proportional to changes in current velocity, are obtained by differencing individual profiles from the average profile. Slopes due to mesoscale dynamic topography are typically 1 μrad (i.e., a 0.1‐m change in topography over a 100‐km distance). Root‐mean‐square (rms) slope variability as low as 0.2 μrad are found in the southeast Pacific, and maximum slope variations up to 6–8 μrad are found in major western boundary currents (e.g., Gulf Stream, Kuroshio, Falkland, and Agulhas) and Antarctic Circum‐polar Current (ACC) systems. The global rms variability map shows previously unknown spatial details that are highly correlated with seafloor topography. Over most areas, the rms slope variability is less than 1 μrad. However at mid‐latitudes, areas of higher variability occur in deep water (>3 km) adjacent to continental shelves, spreading ridges, and oceanic plateaus. Variability is low in shallower areas (<3 km). Along the ACC, the meso‐scale variability appears to be organized by the many shallow areas in its path. We do not see convincing evidence that variability is higher downstream from topographic protrusions. Instead, the areas of highest variability occur in the deep basins (>4km).

The Response of Closed Channels to Wind Stresses

The surface velocity, surface slope, and velocity profile produced by the application of a wind stress to the smooth surface of a closed channel have been determined by adapting the empirical laws of flow in smooth tubes. The estimated responses agree well with the available experimental data.