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

Motivation Transmitted Wave Receiving Wave ?

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

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

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.

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

Single interface Single Interface

Multiple interfaces ??? Observer

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

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

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

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

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