Equivalence problem

1
Equivalence problem

• Numerical methods in field theory and propagation
• MOB9-1, October 22nd, 2002
• Persa Kyritsi (persa_at_cpk.auc.dk)

2
Outline

• Huygens?
• Equivalence principle
• Sources and scattered field formulation
• Example Radar backscatter from a flat conducting
  plate
• Near-to-far transformation
• Applications

3
Huygens principle
Source

• Every point on a wavefront gives rise to a new
  wavelet
• Limitations
• Wavelets do NOT propagate backwards
• Polarization is not described
• Obstructed view? Fresnel?

4
NON uniqueness of sources

• Obvious example Method of images

5
Equivalence principle

• Two different source configurations, on or
  within a dividing surface S, that produce the
  same E, H outside (inside) of S are said to be
  equivalent in the region outside (inside) S

Original problem
Example of equivalent problem
Ms E x n
n
E, H
0, 0
S
Js n x H
6
General equivalence problem formulation
Problem II
Problem I
n
EI,HI
n
EII,HII
EI,HI
EII,HII
Js
Js
S
Ms
S
Ms
Ms (EI EII) x n
EI,HI
n
EII,HII
S
Js n x (HI-HII)
7
Scattered field formulation

• Total field Incident field Scattered field
• Simplified for
• perfect electric conductor only magnetic current
• perfect magnetic conductor only electric current

8
Radar cross-section from a flat conducting plate
x
Discontinuities! Edge effects! One vs both sides!
z
y

• Radar cross section Gain x Effective aperture
• Solve on the board. Hints
• Incident field gives rise to magnetic current
• Look at E-plane (yz plane, ?00) and at H-plane
  (xz plane, ?00)

9
Intuition from Maxwells equations

• If you know the field on inner surface, you can
  find the field on the outer surface

10
Idea!!!

• Remember
• Source ? Vector potential ? Field
• Original problem source
• Near to far field transformation
• Instead of considering the original source as
  the source of the field, we consider the
  near-field distribution as the source

11
Near to far field transformation
Original geometry Intermediate eq.
Problem Final eq. problem
(E, H),(?0, ?0)
(0, 0),(?0, ?0)
(E1, H1),(?0, ?0)
S
(E1, H1),(?0, ?0)
(E1, H1),(?0, ?0)
S
S
(E1, H1),(?0, ?0)
12
Applications of equivalent problems

• Slits/ Gaps
• Space subdivision
• Apply different methods to different subspaces
• Find field at boundaries
• Predict the field in the rest of the space

13
Applications of scattering

• Radar astronomy
• Radar
• Sonar
• Remore sensing
• Environmental monitoring
• Imaging
• Wireless