Diffraction Tomography in Dispersive Backgrounds Tony Devaney Dept. Elec. And Computer Engineering Northeastern University Boston, MA A.J. Devaney, “Linearized inverse scattering in attenuating media,” Inverse Problems 3 (1987) Other approaches discussed in: A. Schatzberg and A.J.D., ``Super-resolution in diffraction tomography, Inverse Problems 8 (1992) K. Ladas and A.J.D., ``Iterative methods in geophysical diffraction tomography, Inverse Problems 8 (1992) R. Deming and A.J.D., ``Diffraction tomography for multi-monostatic gpr, Inverse Problems 13 (1997) 29-45

Experimental Configuration n(  ) s0s0 s O(r,  ) Generalized Projection-Slice Theorem E. Wolf, Principles and development of diffraction tomography, Trends in Optics, Anna Consortini, ed. [Academic Press, San Diego, 1996]

Born Inverse Scattering Ewald Spheres Forward scatter data Back scatter data z Limiting Ewald Sphere Ewald Sphere k 2k k=real valued

Born Inversion for Fixed Frequency Inversion Algorithms: Fourier interpolation (classical X-ray crystallography) Filtered backpropagation (diffraction tomography) Problem: How to generate inversion from Fourier data on spherical surfaces A.J.D. Opts Letts, 7, p.111 (1982) Filtering of data followed by backpropagation: Filtered Backpropagation Algorithm Fourier based methods fail if k is complex: Need new theory

Pulse Propagation in a Dispersive Background n(  ) s0s0 s O(r,  )

Fourier Transformed Scattered Field Choose a complex frequency  0 such that k (  0 ) is real valued There is no reason a priori to dismiss this possibility, but will it work? Close in u.h.p. Roots of dispersion relationship with real k are in l.h.p.

Simple Conducting Medium Real valued Complex in l.h.p. Complex  plane Desired frequency  0 Im  Re  X  <0 Will not be able to close in u.h.p.: can only drop contour to branch points X Branch point

Lorentz Model b 2= 20x10 32  0= 16x10 16  =.28x10 16 Real n Imag n K.E. Oughstun and G.C. Sherman, Electromagnetic Pulse Propagation in Causal Dielectrics [Springer-Verlag, 1994, New York]

Lorentz Medium X Complex  plane Branch Cuts Im  Re  Desired frequency  0  <0 Roots of dispersion relationship must lie above branch points -- Im  0 >-  x x Poles of n(  ) -- ++

Contour Plot of Re ik(  ) Real k Branch point Re  Im 

Mesh Plot of Re ik(  )

Exciting the Plane Wave s0s0 O(r,  ) n(  ) Non-attenuating mode of medium Close in l.h.p.

The Complete Pulse X X Complex  plane Branch Cuts Precursors Im  Re  Can the non-attenuating plane wave be excited; i.e., is it dominated by the precursors? 00 -0-0

Asymptotic Analysis K.E. Oughstun and G.C. Sherman, Electromagnetic Pulse Propagation in Causal Dielectrics [Springer-Verlag, 1994, New York] XX Complex  plane Im  Re  00 -0-0 X X X XX Plane wave excited Plane wave not excited Steepest Descent Contour Saddle point

Summary and Questions Have reviewed one possible approach to inversion in dispersive backgrounds Method is based on computing the temporal Fourier transform of pulsed data at complex frequencies for which the wavenumber of the background is real Method will not work for simple conducting media but appears feasible for Lorentz media The idea behind the approach suggests that it may be possible to excite non-decaying, plane wave pulses using complex frequencies Asymptotic analysis is required to determine the feasibility of the theory