1. Kirchhoff Approximation for multi-layer rough surface Noppasin Niamsuwan By ElectroScience Laboratory, Ohio State University

2. Motivation Transmitted Wave Receiving Wave ?

3. Huygen’s principle Observer known Total field on the surface Green’s function

4. Huygen’s principle (cont.) Huygen’s principle Func. of distance between surface and observer

5. Tangent plane approximation We know the “reflected field” from the “flat” surface At each point on the surface, we evaluate the reflected field (E_reflected) as if it is on the flat surface. Tangent plane This is our FIRST “APPROXIMATION” - Surface needs to be relatively smooth.

6. Single interface Single Interface Observer (reflected) Observer (transmitted) interface

7. Single interface Single Interface

8. Multiple interfaces ??? Observer

9. Multiple interfaces (cont.) Observer SECOND “APPROXIMATION” - How many orders of reflection we need to keep ??? MULTIPLE REFLECTION

10. Multiple interfaces (cont.) - Not a problem for deterministic case SHADOWED REGION Not directly illuminated - However, for statistical case, we need some function to “approximate” this effect

11. Multiple interfaces (cont.) - Straight forward Computation: (Ray Tracing) - Too expensive

12. Multiple interfaces (cont.) 1. Compute the scattered field from the uppermost interface Computation: (Layer by Layer) 2. Those fielded produced by the upper interface become “incident” field of lower interface 3. Group the incident field that has the “same incident angle” 4. Solve for the scattered field 5. Repeat (4) with the other inc. angle

13. Multiple interfaces (cont.) Computation: (Layer by Layer) STEP 1: (result) ~ 20 deg