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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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