Inverse scattering with diffusing waves.

We consider the problem of imaging the optical properties of a highly scattering medium probed by diffuse light. An analytic solution to this problem is derived from the singular value decomposition of the forward-scattering operator, which leads to explicit inversion formulas for the inverse scattering problem with diffusing waves. Computer simulations are used to illustrate these results in model systems.

Inverse scattering for the diffusion equation with general boundary conditions.

We consider the inverse scattering problem for the diffusion equation. A solution to this problem in the form of an explicit inversion formula is derived. Computer simulations are used illustrate our approach in model systems.

Inverse problem in optical diffusion tomography. I. Fourier-Laplace inversion formulas.

We consider the inverse problem of reconstructing the absorption and diffusion coefficients of an inhomogeneous highly scattering medium probed by diffuse light. Inversion formulas based on the Fourier-Laplace transform are used to establish the existence and uniqueness of solutions to this problem in planar, cylindrical, and spherical geometries.

Near-field tomography without phase retrieval.

We investigate the near-field inverse scattering problem with evanescent waves. An analytic solution to this problem within the weak-scattering approximation is used to show that the usual Rayleigh limit may be overcome even when measurements are made without phase information. Applications to a novel form of three-dimensional microscopy with subwavelength resolution are described.

Three-dimensional total internal reflection microscopy.

We investigate the inverse-scattering problem that arises in total internal reflection microscopy. An analytic solution to this problem within the weak-scattering approximation is used to develop a novel form of three-dimensional microscopy with subwavelength resolution.

Spectral quantitation by principal component analysis using complex singular value decomposition.

Principal component analysis (PCA) is a powerful method for quantitative analysis of nuclear magnetic resonance spectral data sets. It has the advantage of being model independent, making it well suited for the analysis of spectra with complicated or unknown line shapes. Previous applications of PCA have required that all spectra in a data set be in phase or have implemented iterative methods to analyze spectra that are not perfectly phased. However, improper phasing or imperfect convergence of the iterative methods has resulted in systematic errors in the estimation of peak areas with PCA. Presented here is a modified method of PCA, which utilizes complex singular value decomposition (SVD) to analyze spectral data sets with any amount of variation in spectral phase. The new method is shown to be completely insensitive to spectral phase. In the presence of noise, PCA with complex SVD yields a lower variation in the estimation of peak area than conventional PCA by a factor of approximately 2. The performance of the method is demonstrated with simulated data and in vivo 31P spectra from human skeletal muscle.

Generalized reciprocity.

The remarkable theorem of reciprocity as described by D. I. Hoult and R. E. Richards (J. Magn. Reson. 24, 71 (1976)) may be generalized to account for the near, intermediate, and radiation zone fields of a magnetic dipole. This form of reciprocity may be important when the wavelength of the NMR signal is not large compared to the distance scale of the system. In these situations the effects of interference may be significant. In addition, both the frequency dependence and distance dependence of the NMR signal amplitude are altered. In general, the distance dependence of the signal follows a weighted sum of 1/r3, 1/r2, and 1/r dependence. The frequency dependence of the signal amplitude is a function of omega, omega2, and omega3. Finally, the signal reflects the full vector field nature of the magnetic dipole. The mathematical expression of generalized reciprocity is completely equivalent to that of Hoult and Richards if the appropriate retarded potential form of the magnetic field is utilized.

Photon hitting density.

Optical and near-IR spectroscopy and imaging of highly scattering tissues require information about the distribution of photon-migration paths. We introduce the concept of the photon hitting density, which describes the expected local time spent by photons traveling between a source and a detector. For systems in which photon transport is diffusive we show that the hitting density can be calculated in terms of diffusion Green's functions. We report calculations of the hitting density in model systems.

Long-time behavior of photon diffusion in an absorbing medium: application to time-resolved spectroscopy.

We have investigated the motion of photons after the injection of a light pulse into a highly scattering, inhomogeneous absorbing medium. We show that the terminal slope of the logarithm of the transmittance curve does not depend on the positions of the source or of the detector. We use numerical calculations to follow the motion of photons in a model system to further understand the physical process that gives rise to this result.