Guido Sandri

Ion-beam machining of millimeter scale optics

An ion-beam microcontouring process is developed and implemented for figuring millimeter scale optics. Ion figuring is a noncontact machining technique in which a beam of high-energy ions is directed toward a target substrate to remove material in a predetermined and controlled fashion. Owing to this noncontact mode of material removal, problems associated with tool wear and edge effects, which are common in conventional machining processes, are avoided. Ion-beam figuring is presented as an alternative for the final figuring of small (<1-mm) optical components. The depth of the material removed by an ion beam is a convolution between the ion-beam shape and an ion-beam dwell function, defined over a two-dimensional area of interest. Therefore determination of the beam dwell function from a desired material removal map and a known steady beam shape is a deconvolution process. A wavelet-based algorithm has been developed to model the deconvolution process in which the desired removal contours and ion-beam shapes are synthesized numerically as wavelet expansions. We then mathematically combined these expansions to compute the dwell function or the tool path for controlling the figuring process. Various models have been developed to test the stability of the algorithm and to understand the critical parameters of the figuring process. The figuring system primarily consists of a duo-plasmatron ion source that ionizes argon to generate a focused (approximately 200-microm FWHM) ion beam. This beam is rastered over the removal surface with a perpendicular set of electrostatic plates controlled by a computer guidance system. Experimental confirmation of ion figuring is demonstrated by machining a one-dimensional sinusoidal depth profile in a prepolished silicon substrate. This profile was figured to within a rms error of 25 nm in one iteration.

Generalized Helmholtz-Kirchhoff Model for Two-Dimensional Distributed Vortex Motion ∗

The two-dimensional Navier–Stokes equations are rewritten as a system of coupled nonlinear ordinary differential equations. These equations describe the evolution of the moments of an expansion of the vorticity with respect to Hermite functions and of the centers of vorticity concentrations. We prove the convergence of this expansion and show that in the zero viscosity and zero core size limit we formally recover the Helmholtz–Kirchhoff model for the evolution of point vortices. The present expansion systematically incorporates the effects of both viscosity and finite vortex core size. We also show that a low-order truncation of our expansion leads to the representation of the flow as a system of interacting Gaussian (i.e., Oseen) vortices, which previous experimental work has shown to be an accurate approximation to many important physical flows [P. Meunier, S. Le Dizes, and T. Leweke, C. R. Phys., 6 (2005), pp. 431–450].

Vorticity dynamics and sound generation in two-dimensional fluid flow

An approximate solution to the two-dimensional incompressible fluid equations is constructed by expanding the vorticity field in a series of derivatives of a Gaussian vortex. The expansion is used to analyze the motion of a corotating Gaussian vortex pair, and the spatial rotation frequency of the vortex pair is derived directly from the fluid vorticity equation. The resulting rotation frequency includes the effects of finite vortex core size and viscosity and reduces, in the appropriate limit, to the rotation frequency of the Kirchhoff point vortex theory. The expansion is then used in the low Mach number Lighthill equation to derive the far-field acoustic pressure generated by the Gaussian vortex pair. This pressure amplitude is compared with that of a previous fully numerical simulation in which the Reynolds number is large and the vortex core size is significant compared to the vortex separation. The present analytic result for the far-field acoustic pressure is shown to be substantially more accurate than previous theoretical predictions. The given example suggests that the vorticity expansion is a useful tool for the prediction of sound generated by a general distributed vorticity field.

Statistical performance evaluation of electrostatic microactuators for a deformable mirror

This paper describes a study to characterize the performance of surface micromachined electrostatic actuators, with a brief introduction to some MEMS (microelectromechanical systems) mirrors developed at BU incorporating these actuators. Fixed-fixed actuators were extensively tested to determine suitability for optical applications, and specifically for an adaptive optics imagining system. The critical issues relating to device performance, namely yield (indicating robustness and process reliability), position repeatability, precision and frequency response, were quantified in the research effort described here. The study demonstrated 95% device yield, 10 nm position repeatability (99% confidence levels), and greater than 66 kHz frequency bandwidth.

Orthonormal wavelet representation using Butterworth filters

Butterworth wavelets are introduced and it is shown that they constitute a large class of orthonormal wavelets. Advantages of this approach are the simplicity of the analyzing wavelet design, connections with the digital filter design techniques, FIR and IIR type of implementations and computational savings in the IIR case which gives rise to fast wavelet transform algorithms. A mirror representation and nonorthogonal wavelet expansion are discussed in this context.

The general solution of the three-dimensional acoustic equation and of Maxwell's equations in the infinite domain in terms of the asymptotic solution in the wave zone

The general solution of the three‐dimensional scalar wave equation (or acoustic equation) and of Maxwell’s equations in the infinite spatial domain is given in terms of the asymptotic forms for large times in the future and in the past, or, equivalently, in terms of the fields in the wave zone. One is therby able to obtain the exact solutions from arbitrary solutions in the wave zone. It is shown that the exact fields computed from an arbitrary wave zone solution always satisfy an initial value problem, and that, therefore, they are always physical. In contrast to earlier derivations of related results which required the use of Radon transforms and the introduction of somewhat sophisticated geometrical concepts, the derivations are simple and use only elementary properties of the Fourier transform.

Data compression using the deconvolution algorithm CLEAN

We describe an application of the nonlinear deconvolution algorithm CLEAN in which a priori knowledge of the point‐spread function allows transmission of nonredundant information. We refer to this as CLEAN compression. The point‐spread function is viewed as a redundancy function. The data set may be regarded as a convolution of the nonredundant information with the redundancy function. Since the nonredundant data is a small subset of the overall data set, images or telecommunication messages may be transmitted over narrowband channels using CLEAN. Effective analog data compression is maximized; the analog signal may be significantly compressed with CLEAN even before any additional compression or digital encoding algorithms are applied.©1994 John Wiley & Sons Inc

Convolution and deconvolution with Gaussian kernel

SummaryIn this paper we prove the following theorem. Suppose an imageI and an object Ω are related by the convolution equationI=Ω*δλF, whereδλF is a Gaussian kernel with widthλF. Suppose further that the imageI is expanded in a series of Gaussian derivatives asI=Σan▽nδΛ, whereδΛ is a Gaussian with widthΛ>λF, and where ∇represents then-th derivative ofδΛ. Then the object Ω is given by Ω = Σan∇nδλ, whereλ2-Λ2-λF2, and where the coefficientsan are exactly the coefficients obtained in the expansion of the imageI. The expansion in Gaussian derivatives can therefore be used to develop a simple and efficient deconvolution method for images which have been convolved with a Gaussian filter. We consider both one- and two-dimensional problems, and give a discussion of the error caused by truncation of the expansion of the image. We also give a two-dimensional numerical example which shows how our deconvolution method can be used in the restoration of digitized gray-scale images.

Stress and Strain Analysis in Molecular Dynamics Simulation of Solids

Molecular dynamics simulation of solid materials is applied to a two-dimensional indentation problem. Methods are presented for calculating the stress and strain tensors at interior points within the model. The stress calculations corresponding to the elastic deformation portion of the indentation process are compared with an analytical continuum solution. Stress calculations are also presented for the plastic deformation portion of the indentation process. The methods are derived independently so that future work can be directed at determining the constitutive relationship between stress and strain throughout the development of plastic flow and fracture in a solid material.

Laboratory study of pollutant detention times in wake cavities downwind of low-rise buildings

Abstract A physical modeling study was performed of the transient dispersion of air pollutants “trapped” in recirculating wake-cavity regions downwind of low-rise buildings. Measurements were made of the characteristic detention times of a methane tracer as a function of both the aspect ratio of rectangular building models and also the ratio of the model height to boundary layer thickness. The measurements were made with a standard total-hydrocarbon analyzer. A novel deconvolution technique was developed to determine the exponential decay constant of the methane concentration since the hydrocarbon analyzer had a response rate slower than the concentration decay. The technique involved curve matching on a personal computer using a standard spread-sheet program. Results of the study, after the appropriate standard scaling, compared favorably with those of similar studies in the literature of both field tests and laboratory tests using fast-response instrumentation.