A Langevin analysis of fundamental noise limits in Coherent Anti-Stokes Raman Spectroscopy

We use a Langevin approach to analyze the quantum noise in Coherent Anti-Stokes Raman Spectroscopy (CARS) in several experimental scenarios: with continuous wave input fields acting simultaneously and with fast sequential pulsed lasers where one field scatters off the coherence generated by other fields; and for interactions within a cavity and in free space. In all the cases, the signal as well as the quantum noise due to spontaneous decay and decoherence in the medium are shown to be described by the same general expression. Our theory in particular shows that for short interaction times, the medium noise is not important and the efficiency is limited only by the intrinsic quantum nature of the photon. We obtain fully analytic results \emph{without} making an adiabatic approximation, the fluctuations of the medium and the fields are self solved consistently.Comment: 12 pages, 1 figur

Backpropagation Imaging in Nonlinear Harmonic Holography in the Presence of Measurement and Medium Noises

In this paper, the detection of a small reflector in a randomly heterogenous medium using second-harmonic generation is investigated. The medium is illuminated by a time-harmonic plane wave at frequency omega. It is assumed that the reflector has a non-zero second-order nonlinear susceptibility, and thus emits a wave at frequency two omega in addition to the fundamental frequency linear scattering. It is shown how the fundamental frequency signal and the second-harmonic signal propagate in the medium. A statistical study of the images obtained by migrating the boundary data is performed. It is proved that the second-harmonic image is more stable with respect to medium noise than the one obtained with the fundamental signal. Moreover, the signal-to-noise ratio for the second-harmonic image does not depend neither on the second-order susceptibility tensor nor on the volume of the particle.Comment: 36 pages, 18 figure

Detection of Electromagnetic Inclusions using Topological Sensitivity

In this article a topological sensitivity framework for far field detection of a diametrically small electromagnetic inclusion is established. The cases of single and multiple measurements of the electric far field scattering amplitude at a fixed frequency are taken into account. The performance of the algorithm is analyzed theoretically in terms of its resolution and sensitivity for locating an inclusion. The stability of the framework with respect to measurement and medium noises is discussed. Moreover, the quantitative results for signal-to-noise ratio are presented. A few numerical results are presented to illustrate the detection capabilities of the proposed framework with single and multiple measurements.Comment: 31 pages, 5 figure

Material parameter estimation and hypothesis testing on a 1D viscoelastic stenosis model: Methodology

This is the post-print version of the final published paper that is available from the link below. Copyright @ 2013 Walter de Gruyter GmbH.Non-invasive detection, localization and characterization of an arterial stenosis (a blockage or partial blockage in the artery) continues to be an important problem in medicine. Partial blockage stenoses are known to generate disturbances in blood flow which generate shear waves in the chest cavity. We examine a one-dimensional viscoelastic model that incorporates Kelvin–Voigt damping and internal variables, and develop a proof-of-concept methodology using simulated data. We first develop an estimation procedure for the material parameters. We use this procedure to determine confidence intervals for the estimated parameters, which indicates the efficacy of finding parameter estimates in practice. Confidence intervals are computed using asymptotic error theory as well as bootstrapping. We then develop a model comparison test to be used in determining if a particular data set came from a low input amplitude or a high input amplitude; this we anticipate will aid in determining when stenosis is present. These two thrusts together will serve as the methodological basis for our continuing analysis using experimental data currently being collected.National Institute of Allergy and Infectious Diseases, Air Force Office of Scientific Research, Department of Education, and Engineering and Physical Sciences Research Council

Localization, Stability, and Resolution of Topological Derivative Based Imaging Functionals in Elasticity

The focus of this work is on rigorous mathematical analysis of the topological derivative based detection algorithms for the localization of an elastic inclusion of vanishing characteristic size. A filtered quadratic misfit is considered and the performance of the topological derivative imaging functional resulting therefrom is analyzed. Our analysis reveals that the imaging functional may not attain its maximum at the location of the inclusion. Moreover, the resolution of the image is below the diffraction limit. Both phenomena are due to the coupling of pressure and shear waves propagating with different wave speeds and polarization directions. A novel imaging functional based on the weighted Helmholtz decomposition of the topological derivative is, therefore, introduced. It is thereby substantiated that the maximum of the imaging functional is attained at the location of the inclusion and the resolution is enhanced and it proves to be the diffraction limit. Finally, we investigate the stability of the proposed imaging functionals with respect to measurement and medium noises.Comment: 38 pages. A new subsection 6.4 is added where we consider the case of random Lam\'e coefficients. We thought this would corrupt the statistical stability of the imaging functional but our calculus shows that this is not the case as long as the random fluctuation is weak so that Born approximation is vali