1 identifiability of scatterers in inverse obstacle scattering jun zou department of mathematics the...
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Identifiability of ScatterersIn
Inverse Obstacle Scattering
Jun Zou Department of Mathematics
The Chinese University of Hong Kong
http://www.math.cuhk.edu.hk/~zou
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Inverse Acoustic Obstacle Scattering
D : impenetrable scatterer
Acoustic EM
Underlying Equations
• Propagation of acoustic wave in homogeneous isotropic medium / fluid : pressure p(x, t) of the medium satisfies
• Consider the time-harmonic waves of the form
then u(x) satisfies the Helmholtz equation
with
Direct Acoustic Obstacle Scattering
• Take the planar incident field
then the total field solves
the Helmholtz equation :
• satisfies the Sommerfeld radiation condition:
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Physical Properties of Scatterers
Recall
Sound-soft : (pressure vanishes) Sound-hard : (normal velocity of wave vanishes)
Impedance : (normal velocity proport. to pressure) or mixed type
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Our Concern : Identifiability
Q : How much far field data from how many incident planar fields can uniquely determine a scatterer ?
This is a long-standing problem !
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Existing Uniqueness Results
A general sound-soft obstacle is uniquely determined by the far field data from :
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For polyhedral type scatterers :
Breakthroughs on identifiability for both
inverse acoustic & EM scattering
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Existing Results on Identifiability
• Cheng-Yamamoto 03 : A single sound-hard polygonal scatterer is uniquely determined by at most 2 incident fields
• Elschner-Yamamoto 06 :
A single sound-hard polygon is uniquely determined by one incident field
• Alessandrini - Rondi 05 : very general sound-soft polyhedral scatterers in R^n by one incident field
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Uniqueness still remains unknown in the following cases for polyhedral type scatterers :
sound-hard (N=2: single D; N>2: none), impedance scatterers;
when the scatterers admits the simultaneous presence of both solid & crack-type obstacle components;
when the scatterers involve mixed types of obstacle components, e.g., some are sound-soft, and some are sound-hard or impedance type;
When number of total obstacle components are unknown a priori, and physical properties of obstacle components are unknown a priori .
A unified proof to principally answer all these questions.
Summary of New Results (Liu-Zou 06 & 07)
One incident field: for any N when no sound-hard obstacle ;
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Inverse EM Obstacle Scattering
D : impenetrable scatterer
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Reflection principle :
hyperplane
Reflection Principle For Maxwell Equations
(Liu-Yamamoto-Zou 07)
Then the following BCs can be reflected w.r.t. any hyperplane Π in G:
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• Results:
(Liu-Yamamoto-Zou 07)
Far field data from two incident EM fields :
sufficient to determine
general polyhedral type scatterers
Inverse EM Obstacle Scattering
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Identifiability of Periodic Grating Structures
(Bao-Zhang-Zou 08)
• Diffractive Optics:
Often need to determine the optical grating structure, including geometric shape, location, and physical nature
periodic structure
Time-harmonic EM Scattering
s
q
S
q: entering angle
downward
S:
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Identification of Grating Profiles
S
q: entering angle
Q: near field data from how many incident fields can uniquely determine the location and shape of S ?
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Existing Uniqueness for Periodic Grating
Hettlich-Kirsch 97:
C2 smooth 3D periodic structure, finite number of incident fields
Bao-Zhou 98:
C2 smooth 3D periodic structure of special class; one incident field
Elschner-Schmidt-Yamamoto 03, 03: Elschner-Yamamoto 07: TE or TM mode, 2D scalar Helmholtz eqn All bi-periodic 2D grating structure: recovered by 1 to 4 incident fields
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New Identification on Periodic Gratings(Bao-Zhang-Zou 08)
For 3D periodic polyhedral gratings : no results yet
We can provide a systematic and complete answer ;
by a constructive method.
For each incident field :
We will find the periodic polyhedral structures unidentifiable ;
Then easy to know
How many incident fields needed to uniquely identify any given grating structure
Forward Scattering Problem
Forward scattering problem in
Radiation condition : for x3 large,
With
Important Concepts
S
A perfect plane of E , PP :
PP: always understood to be maximum extended, NOT a real plane
Technical Tools
(1) Extended reflection principle :
hyperplane
(2) Split decaying & propagating modes :
CRUCIAL :
lying in lying in
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Technical Tools (cont.)
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Technical Tools (cont.)
Crucial Relations
Equiv. to
Find all perfect planes of
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Find all perfect planes of E
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Need only to consider
Find all perfect planes of E
Part I.
Part II.
Then
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Find all perfect planes of E
Part II.
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Find all perfect planes of E
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Find all perfect planes of E
The above conditions are also sufficient.
Have found all PPs of E, so do the faces of S .
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Class I of Gratings Unidentifiable
Have found all PPs of E, so do the faces of S .
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Class 2 of Gratings Unidentifiable
By reflection principle & group theory, can show
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Class 2 of Gratings Unidentifiable
Have found all PPs of E, so do the faces of S .
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Uniquely Identifiable Periodic Gratings
IF
IF
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