Transcript Find all perfect planes of E

Identifiability of Scatterers
In
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
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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
the Helmholtz equation :
•
solves
satisfies the Sommerfeld radiation condition:
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.
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Summary of New Results
(Liu-Zou 06 & 07)
One incident field:
for any N when no sound-hard obstacle ;
Inverse EM Obstacle Scattering
D:
impenetrable
scatterer
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Reflection Principle For Maxwell Equations
(Liu-Yamamoto-Zou 07)
Reflection principle :
hyperplane
Then the following BCs can be reflected w.r.t. any
hyperplane Π in G:
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Inverse EM Obstacle Scattering
(Liu-Yamamoto-Zou 07)
• Results:
Far field data from two incident EM fields :
sufficient to determine
general polyhedral type scatterers
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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
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Time-harmonic EM Scattering
q:
downward
q
s
S
S:
entering angle
Identification of Grating Profiles
q:
entering angle
S
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
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Forward Scattering Problem
Forward scattering problem in
Radiation condition : for x3 large,
With
Important Concepts
A perfect plane of E , PP :
S
PP:
always understood to be maximum extended, NOT a real plane
Technical Tools
(1) Extended reflection principle :
(2) Split decaying & propagating modes :
CRUCIAL :
lying in
lying in
Technical Tools (cont.)
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Technical Tools (cont.)
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Crucial Relations
Equiv. to
Find all perfect planes of
Find all perfect planes of E
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Find all perfect planes of E
Need only to consider
Then
Part I.
Part II.
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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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