Inverse scattering of 2d photonic structures by layer-stripping

Design and reconstruction of 2d and 3d photonic structures are usually carried out by forward simulations combined with optimization or intuition. Reconstruction by means of layer-stripping has been applied in seismic processing as well as in design and characterization of 1d photonic structures such as fiber Bragg gratings. Layer-stripping is based on causality, where the earliest scattered light is used to recover the structure layer-by-layer. Our set-up is a 2d layered nonmagnetic structure probed by plane polarized harmonic waves entering normal to the layers. It is assumed that the dielectric permittivity in each layer only varies orthogonal to the polarization. Based on obtained reflectance data covering a suitable frequency interval, time-localized pulse data are synthesized and applied to reconstruct the refractive index profile in the leftmost layer by identifying the local, time-domain Fresnel reflection at each point. Once the first layer is known, its impact on the reflectance data is stripped off, and the procedure repeated for the next layer. Through numerical simulations it will be demonstrated that it is possible to reconstruct structures consisting of several layers. The impact of evanescent modes and limited bandwidth is discussed

Discrete Gel'fand-Levitan and Marchenko matrix equations and layer stripping algorithms for the discrete two-dimensional Schrödinger equation inverse scattering problem with a nonlocal potential

We develop discrete counterparts to the Gel'fand-Levitan and Marchenko integral equations for the two-dimensional (2D) discrete inverse scattering problem in polar coordinates with a nonlocal potential. We also develop fast layer stripping algorithms that solve these systems of equations exactly. The significance of these results is: (1) they are the first numerical implementation of Newton's multidimensional inverse scattering theory; (2) they show that the result will almost always be a nonlocal potential, unless the data are `miraculous'; (3) they show that layer stripping algorithms implement fast `split' signal processing fast algorithms; (4) they link 2D discrete inverse scattering with 2D discrete random field linear least-squares estimation; and (5) they formulate and solve 2D discrete Schrödinger equation inverse scattering problems in polar coordinates.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/49103/2/ip8322.pd

Inversion of the Bloch transform in magnetic resonance imaging using asymmetric two-component inverse scattering

In magnetic resonance imaging, the relation between the radio-frequency modulation of the magnetic field and the desired final magnetisation state is called the Bloch transform. Selective excitation then amounts to inverting this transform, which is highly nonlinear. Previous attempts to formulate this problem as an inverse scattering problem have restricted attention to solutions using reflectionless potentials. The author uses fast numerical algorithms for inverse scattering problems to obtain a much larger set of solutions. Numerical examples are included.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/49094/2/ipv6i1p133.pd

Discrete matrix Schrodinger equation equivalents of discrete two-component matrix wave systems

Multichannel discrete lossless two-component wave systems describe physical scattering for multiple coupled transmission lines, for elastic media, and for electromagnetic wave propagation in layered media. Such a wave system is transformed into an equivalent discrete matrix Schrodinger equation that includes the effects of transmission losses and transmission scattering. These effects are not included in the continuous Schrodinger equation, so that discretisation of the latter omits significant effects. Applications include fast algorithms for synthesis and inversion of the reflection responses of multichannel scattering media. These algorithms require roughly half as many matrix multiplications as previous algorithms that employ the multichannel two-component wave system.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/49091/2/ipv5i3p425.pd

Inverse scattering for time-varying one-dimensional layered media: algorithms and applications

We formulate and present algorithms and applications for the inverse scattering problem for a one-dimensional layered medium whose reflection coefficients vary with time. We show that given the reflection responses to impulsive plane waves incident on the medium at different times, this problem can be solved either by solving a series of nested non-Toeplitz systems of equations, or by propagating a set of coupled layer-stripping algorithms with reflection coefficients varying in time. All multiple scattering effects are included here. We show that our results reduce to previous results in the special case of time-invariant media. Applications include the two-dimensional (2D) inverse resistivity problem of reconstructing the 2D resistivity of a 2D medium from surface measurements, and the reconstruction of rapidly curing layered media. We also present a time-varying discrete Miura transform linking the time-varying discrete wavesystem inverse scattering problem to a time-varying discrete Schrödinger equation, and a new important feasibility condition on the reflection responses, which seems to be new in the context of the 2D inverse resistivity problem. A numerical example is provided.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/49101/2/ip7318.pd

COMMENT: Reply to a comment by R Newton

Following Newton's comment (see ibid., vol.5, p.437 (1989)), the author points out that: (1) the major contribution of his previous paper (see ibid., vol.4, p.549 (1988)) is the generalised fast Cholesky algorithm, not the integral equations; (2) the Marchenko integral equation is correct as written, but does not determine the scattered field; and (3) it is implicitly assumed that the regular solution exists. The last two points suggest that the generalised fast Cholesky algorithm may be the only viable solution to the non-local problem.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/49092/2/ipv5i3p439.pd

One-dimensional inverse scattering problems: an asymmetric two-component wave system framework

Many one-dimensional inverse scattering problems can be formulated as a two-component wave system inverse problem, including inverse problems for lossless and absorbing acoustic and dielectric media. The advantage of doing so is that well known signal processing algorithms with good numerical stability properties can be used to reconstruct such media from either reflection or transmission responses to impulsive or harmonic sources. If the system is asymmetric, i.e. has different reflectivity functions in different directions, transmission data as well as reflection data are required. The author summarises algorithms for a wide variety of one-dimensional inverse problems, derives some new ones, and presents a simple framework that reveals much about these problems.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/49093/2/ipv5i4p641.pd

Blind deconvolution of sources from the transmission responses of one-dimensional inhomogeneous continuous and discrete layered absorbing media

Non-destructive testing and reconstruction of lossy non-uniform transmission lines require solving inverse scattering problems for inhomogeneous one-dimensional (1D) lossy media. Often the source waveform used to probe the medium is unknown, since it cannot be measured separately. This paper shows that the blind deconvolution problem of reconstructing both an unknown source and an unknown lossy 1D medium can be solved, provided the medium absorption is sufficiently large. We generalize the minimum-phase property of the transmission response to a more severe property for lossy media. We also show how partial knowledge of the source can be used to reconstruct low-loss and lossless 1D media. Both discrete and continuous 1D layered media are considered. Numerical examples are presented.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/49102/2/ip7417.pd

Connections between three-dimensional inverse scattering and linear least-squares estimation of random fields

The three-dimensional Schrödinger equation inverse scattering problem with a nonspherically-symmetric potential is related to the filtering problem of computing the linear leastsquares estimate of the three-dimensional random field on the surface of a sphere from noisy observations inside the sphere. The relation consists of associating an estimation problem with the inverse scattering problem, and vice-versa. This association allows equations and quantities for one problem to be given interpretations in terms of the other problem. A new fast algorithm is obtained for the estimation of random fields using this association. The present work is an extension of the connections between estimation and inverse scattering already known to exist for stationary random processes and one-dimensional scattering potentials, and isotropic random fields and radial scattering protentials.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/41642/1/10440_2004_Article_BF00046966.pd

Discrete layer-stripping algorithms and feasibility conditions for the 2D inverse conductivity problem

We develop a discrete layer-stripping algorithm for the 2D inverse conductivity problem. Unlike previous algorithms, this algorithm transforms the problem into a time-varying 1D Schrödinger equation inverse scattering problem, discretizes this problem and then solves the discrete problem exactly. This approach has three advantages: (i) the poor conditioning inherent in the problem is concentrated in the solution of a linear integral transform at the beginning of the problem, to which standard regularization techniques may be applied and (ii) feasibility conditions on the transformed data are obtained, satisfaction of which ensures that (iii) the solution of the discrete nonlinear inverse scattering problem is exact and stable. Other contributions include solution of discrete Schrödinger equation inverse potential problems with time-varying potentials by both layer-stripping algorithms and solution of nested systems of equations which amount to a time-varying discrete version of the Gel'fand-Levitan equation. An analytic and numerical example is supplied to demonstrate the operation of the algorithm.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/49105/2/ip0504.pd