Electromagnetic Sensing for Space-borne Imaging

1. Electromagnetic Sensing for Space-borne Imaging Lecture 4Fourier Optics: Kirchoff Integral, Huygens-Fresnel Principle

2. Geometric Optics Approximation to Wave Optics • If the scale, L, of the ripples in the wavefronts are much larger that the wave length and if we are not concerned with what happens within several wave lengths of sharp edges, then we can depict light propagation by tracing the lines normal to the wave fronts (the rays). The behaviour of rays is analogous to the propagation of a stream of particles: • In a homogeneous medium, rays travel in straight lines. • When encountering a mirror surface, a ray is deterred according to the law of reflection. • When meeting the boundary of another, transparent, homogeneous medium, a ray is bent according to Snell’s law. • In general, if s denotes the unit vector tangent to the rays, then: L s 

3. The Law of Reflection

4. Medium 2 n2 > n1 Medium 1 n1 The law of Refraction

5. Here’s what the actual wave field does:

6. Law Reflection Applied to Surfaces of Revolution O z F0 F1 C (z =r) O z C (z = -r) F1 F0

7. Now Back to Wave Optics:Helmholtz Equation – Plane waves revisited

8. Radiation Due to a Point Source

9. y x x’ z x - x’ s Radiation Due to a Point Source on a Plane

10. (x, y, z)- s v P S Determination of the field when it is known on an enclosing surface S  a closed surface enclosing a volume, v, containing no sources. P  a point inside S (we use “P” also to refer to the spatial coordinates of the point) s  distance from P to a point on S. unit vector normal to S and pointing inward. Then it is a fundamental result for the Helmholtz equation that: This is the integral theorem of Helmholtz and Kirchhoff. It expresses U(P) in terms of both U and on the surface

11. The field determined from its values on the image surface zP Radius  d x P I s s z O y Q Extended source, s The basic imaging situation. "image surface" Hemisphere with Radius  "observation surface" We have, as illustrated above, an extended source, s, centered at the coordinate origin of extent ~ d from the origin. Applying the integral theorem, we get:

12. Hemisphere of arbitrarily large radius x P I s z O y Q no source Imaging propagation processes in reverse time. Image surface field in terms of values on an observation surface When we reverse time, we need to change the factor eiks/s to e-kis/s. This represents waves coming in from infinity and converging near the original source.

13. Huygens-Fresnel Principle

14. The very same time-reversal principle has been applied to accoustics. See Scientific American, November 1999.

15. The Acoustic Time-Reversal Mirror

16. I O Q s s d P D Huygens-Fresnel Principle Extended incoherent source, s Huygens-Fresnel Principle: Meaning: At any point, the field is just a combination of point-sources-on-a-plane distributions, each having a strength = the local field value Inverse H-F Principle:

17. Surface S Extended source Qk P Huygens’ Principle

18. Don't panic! There's light at the end of all this!