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Identifiability of Scatterers In Inverse Obstacle Scattering

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. Inverse Acoustic Obstacle Scattering. D : impenetrable scatterer. Acoustic EM.

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Identifiability of Scatterers In Inverse Obstacle Scattering

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  1. Identifiability of ScatterersIn Inverse Obstacle Scattering Jun Zou Department of Mathematics The Chinese University of Hong Kong http://www.math.cuhk.edu.hk/~zou

  2. Inverse Acoustic Obstacle Scattering D : impenetrable scatterer Acoustic EM

  3. Underlying Equations • Propagation of acoustic wave in homogeneous isotropic medium / fluid : pressurep(x, t)of the medium satisfies • Consider the time-harmonic waves of the form then u(x) satisfies the Helmholtz equation with

  4. 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 byone 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.

  11. 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 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:

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

  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

  16. Time-harmonic EM Scattering q: entering angle downward q s S S:

  17. 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 ?

  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 ; bya 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

  20. Forward Scattering Problem Forward scattering problem in Radiation condition : for x3 large, With

  21. Important Concepts A perfect plane of E , PP : S PP: always understood to be maximum extended, NOT a real plane

  22. Technical Tools (1) Extended reflection principle : hyperplane (2) Split decaying & propagatingmodes : CRUCIAL : lying in lying in

  23. Technical Tools (cont.)

  24. Technical Tools (cont.)

  25. Crucial Relations Equiv. to

  26. Find all perfect planes of

  27. Find all perfect planes of E

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

  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 IF

  36. Thank You

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