Diffraction Tomography in Dispersive Backgrounds

1
Diffraction Tomography in Dispersive Backgrounds
Tony Devaney Dept. Elec. And Computer
Engineering Northeastern University Boston, MA
02115 Email tonydev2_at_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

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Experimental Configuration
n(?)
O(r,?)
s0
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
3
Born Inverse Scattering
kreal valued
Ewald Spheres
k
2k
Ewald Sphere
Limiting Ewald Sphere
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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
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Pulse Propagation in a Dispersive Background
n(?)
O(r,?)
s0
s
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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.
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Simple Conducting Medium
Complex in l.h.p.
Real valued
Im ?
Complex ? plane
Branch point
X
?lt0
X
Desired frequency ?0
Re ?
Will not be able to close in u.h.p. can only
drop contour to branch points
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Lorentz Model
b220x1032 ?016x1016 ? .28x1016
Real n
Imag n
K.E. Oughstun and G.C. Sherman, Electromagnetic
Pulse Propagation in Causal Dielectrics
Springer-Verlag, 1994, New York
9
Lorentz Medium
Im ?
Complex ? plane
Re ?
?lt0
-?
Desired frequency ?0
X
?-
?
x
x
Poles of n(?)
Branch Cuts
Roots of dispersion relationship must lie above
branch points
Im ?0gt-?
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Contour Plot of Re ik(?)
Im ?
Re ?
Real k
Branch point
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Mesh Plot of Re ik(?)
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Exciting the Plane Wave
n(?)
s0
O(r,?)
Close in l.h.p.
Non-attenuating mode of medium
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The Complete Pulse
Im ?
Complex ? plane
Re ?
?0
-?0
X
X
Branch Cuts
Precursors
Can the non-attenuating plane wave be excited
i.e., is it dominated by the precursors?
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Asymptotic Analysis
K.E. Oughstun and G.C. Sherman, Electromagnetic
Pulse Propagation in Causal Dielectrics
Springer-Verlag, 1994, New York
Im ?
Steepest Descent Contour
Complex ? plane
Saddle point
X
Re ?
?0
-?0
X
X
X
X
X
Saddle point
X
Saddle point
Plane wave excited
Plane wave not excited
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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