Diffraction Tomography in Dispersive Backgrounds Tony Devaney Dept

Diffraction Tomography in Dispersive Backgrounds Tony Devaney Dept. Elec. And Computer Engineering Northeastern University Boston, MA 02115 Email: tonydev 2@aol. com A. J. Devaney, “Linearized inverse scattering in attenuating media, ” Inverse Problems 3 (1987) 389 -397 Other approaches discussed in: • A. Schatzberg and A. J. D. , ``Super-resolution in diffraction tomography, Inverse Problems 8 (1992) 149 -164 • K. Ladas and A. J. D. , ``Iterative methods in geophysical diffraction tomography, Inverse Problems 8 (1992) 119 -132 • R. Deming and A. J. D. , ``Diffraction tomography for multi-monostatic gpr, Inverse Problems 13 (1997) 29 -45

Experimental Configuration n( ) O(r, ) s 0 s Generalized Projection-Slice Theorem E. Wolf, Principles and development of diffraction tomography, Trends in Optics, Anna Consortini, ed. [Academic Press, San Diego, 1996] 83 -110

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

Born Inversion for Fixed Frequency Problem: How to generate inversion from Fourier data on spherical surfaces Inversion Algorithms: Fourier interpolation (classical X-ray crystallography) Filtered backpropagation (diffraction tomography) 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( ) O(r, ) s 0 s

Fourier Transformed Scattered Field Close in u. h. p. 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? Roots of dispersion relationship with real k are in l. h. p.

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

Lorentz Model b 2=20 x 1032 0=16 x 1016 =. 28 x 1016 Real n Imag n K. E. Oughstun and G. C. Sherman, Electromagnetic Pulse Propagation in Causal Dielectrics [Springer-Verlag, 1994, New York]

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

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

Mesh Plot of Re ik( )

Exciting the Plane Wave n( ) s 0 Close in l. h. p. Non-attenuating mode of medium O(r, )

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

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

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 theory