# Inverse medium scattering for the Helmholtz equation at fixed frequency

@article{Bao2005InverseMS, title={Inverse medium scattering for the Helmholtz equation at fixed frequency}, author={Gang Bao and Peijun Li}, journal={Inverse Problems}, year={2005}, volume={21}, pages={1621 - 1641} }

Consider a time-harmonic electromagnetic plane wave incident on a medium enclosed by a bounded domain in . In this paper, existence and uniqueness of the variational problem for the direct scattering are established. An energy estimate for the scattered field is obtained on which the Born approximation is based. The Fréchet differentiability of the scattering map is examined. A new continuation method for the inverse medium scattering, which reconstructs the scatterer of an inhomogeneous medium…

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## References

SHOWING 1-10 OF 18 REFERENCES

### Inverse Medium Scattering Problems for Electromagnetic Waves

- MathematicsSIAM J. Appl. Math.
- 2005

In this paper, existence and uniqueness of the variational problem for forward scattering are established and an energy estimate for the scattered field with a uniform bound with respect to the wavenumber is obtained in the case of low frequency on which the Born approximation is based.

### Inverse scattering via skin effect

- Mathematics
- 1997

We present a stable method for the inverse scattering problem of the Helmholtz equation in two dimensions. The algorithm requires single-frequency scattering data, and is an iterative procedure…

### A propagation-backpropagation method for ultrasound tomography

- Mathematics
- 1995

Ultrasound tomography is modelled by the inverse problem of a 2D Helmholtz equation at fixed frequency with plane-wave irradiation. It is assumed that the field is measured outside the support of the…

### Regularity and Stability for the Scattering Map of a Linearized Inverse Medium Problem

- Mathematics
- 2000

The main goal of this paper is to study the linearization of an inverse medium problem. Regularity and stability results are established for the near-field scattering map (or scattering matrix) which…

### Inverse Acoustic and Electromagnetic Scattering Theory

- Mathematics, Physics
- 1992

Introduction.- The Helmholtz Equation.- Direct Acoustic Obstacle Scattering.- III-Posed Problems.- Inverse Acoustic Obstacle Scattering.- The Maxwell Equations.- Inverse Electromagnetic Obstacle…

### Generalized linear inversion and the first Born theory for acoustic media

- Mathematics
- 1983

A procedure is derived which incorporates a generalized linear inverse viewpoint within a multidimensional Born inversion method. The method we present is a more general Born theory which can…

### Near Field Microscopy and Near Field Optics

- Physics
- 2003

The early eighties have experienced a revolution in the perception of physical phenomena. This revolution is the birth of a new generation of imaging systems based on the detection of non-radiating…

### Elliptic Partial Differential Equa-tions of Second Order

- Mathematics
- 1977

Chapter 1. Introduction Part I: Linear Equations Chapter 2. Laplace's Equation 2.1 The Mean Value Inequalities 2.2 Maximum and Minimum Principle 2.3 The Harnack Inequality 2.4 Green's Representation…

### Near-Field Tomography

- Mathematics

We consider the inverse scattering problem for wave fields containing evanescent components. Applications to near-field optics and tomographic imaging with subwavelength resolution are described.

### Three-dimensional total internal reflection microscopy.

- MathematicsOptics letters
- 2001

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.