Diffraction Tomography in Dispersive Backgrounds

Tony DevaneyDept. Elec. And Computer Engineering

Northeastern UniversityBoston, MA 02115

Email: tonydev2@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()

s0 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] 83-110

Born Inverse Scattering

Ewald SpheresForward scatter dataBack scatter data

z

Limiting Ewald SphereEwald Sphere

k2k

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()

s0 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 valuedComplex 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

XBranch point

Lorentz Model

0 2 4 6 8 10 12 140

0.5

1

1.5

2

2.5

3real (red) and imaginary (blue) parts of index of refraction

b2=20x1032

0=16x1016

=.28x1016Real 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()

- +

0 2 4 6 8 10 12 14-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

Contour Plot of Re ik()

Real k

Branch point

Re

Im

Mesh Plot of Re ik()

Exciting the Plane Wave

s0 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

Asymptotic AnalysisK.E. Oughstun and G.C. Sherman, Electromagnetic Pulse Propagation in

Causal Dielectrics [Springer-Verlag, 1994, New York]

X X

Complex plane

Im

Re 0-0

X

X XXX

Plane wave excited Plane wave not excited

Steepest Descent Contour

Saddle point Saddle point

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