1 Identifiability of Scatterers In Inverse Obstacle Scattering Jun Zou Department of Mathematics The Chinese University of Hong Kong

2 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:

5 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

6 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 !

7 Existing Uniqueness Results A general sound-soft obstacle is uniquely determined by the far field data from :

8 For polyhedral type scatterers : Breakthroughs on identifiability for both inverse acoustic & EM scattering

9 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

10 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 ;

12 Inverse EM Obstacle Scattering D : impenetrable scatterer

13 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:

14 Results: (Liu-Yamamoto-Zou 07) Far field data from two incident EM fields : sufficient to determine general polyhedral type scatterers Inverse EM Obstacle Scattering

15 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:

17 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 ?

18 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

19 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

23 Technical Tools (cont.)

24 Technical Tools (cont.)

Crucial Relations Equiv. to

Find all perfect planes of

27 Find all perfect planes of E

28 Need only to consider Find all perfect planes of E Part I. Part II. Then

29 Find all perfect planes of E Part II.

30 Find all perfect planes of E

31 Find all perfect planes of E The above conditions are also sufficient. Have found all PPs of E, so do the faces of S.

32 Class I of Gratings Unidentifiable Have found all PPs of E, so do the faces of S.

33 Class 2 of Gratings Unidentifiable By reflection principle & group theory, can show

34 Class 2 of Gratings Unidentifiable Have found all PPs of E, so do the faces of S.

35 Uniquely Identifiable Periodic Gratings IF

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