Diffraction Tomography in Dispersive Backgrounds
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Transcript of Diffraction Tomography in Dispersive Backgrounds
Diffraction Tomography in Dispersive Backgrounds
Tony DevaneyDept. Elec. And Computer Engineering
Northeastern UniversityBoston, MA 02115
Email: [email protected]
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