Optical Tomography for Media with Variable Index of Refraction

Optical tomography is the use of near-infrared light to determine the optical absorption and scattering properties of a medium M ⊂ Rn. If the refractive index is constant throughout the medium, the steady-state case is modeled by the stationary linear transport equation in terms of the Euclidean metric and photons which do not get absorbed or scatter travel along straight lines. In this expository article we consider the case of variable refractive index where the dynamics are modeled by writing the transport equation in terms of a Riemannian metric; in the absence of interaction, photons follow the geodesics of this metric. The data one has is the measurement of the out-going flux of photons leaving the body at the boundary. This may be knowledge of both the locations and directions of the exiting photons (fully angularly resolved measurements) or some kind of average over direction (angularly averaged measurements). We discuss the results known for both types of measurements in all spatial dimensions

Stability of the Gauge Equivalent Classes in Inverse Stationary Transport in Refractive Media

In the inverse stationary transport problem through anisotropic attenuating, scattering, and refractive media, the albedo operator stably determines the gauge equivalent class of the attenuation and scattering coefficients

An Electromagnetic Inverse Problem in Chiral Media

We consider the inverse boundary value problem for Maxwell\u27s equations that takes into account the chirality of a body in R3 . More precisely, we show that knowledge of a boundary map for the electromagnetic fields determines the electromagnetic parameters, namely the conductivity, electric permittivity, magnetic permeability and chirality, in the interior. We rewrite Maxwell\u27s equations as a first order perturbation of the Laplacian and construct exponentially growing solutions, and obtain the result in the spirit of complex geometrical optics