1. EM spherical waves
…in any coord. system
Differentials depend on coord system: z q r y f x

2. 1st derivatives
2nd derivatives
“Laplacian”
Field:

3. The vector spherical Laplacian:
(This is a good time for a scalar wave equation!)
: Our scalar wave which represents the field strength.
The (non-vector, scalar) spherical Laplacian:

4. Unless you are a masochist ..... put the spherical source at the origin!!!!

5. Put back into wave equation:
Wave equation for ry
Guess a solution:
Note: y (the field strength) and yo (the field amplitude) don’t have the same unit.
As r increases, the field strength decreases

6. a r
For: r >> a, spherical waves can be approximated as a plane waves.

7. Geometrical Optics: light is a “ray” that travels in straight lines
(the “ray” is essentially the wave vector of a plane wave)

8. Huygens’ and Fermat’s Principles
All of Geometrical optics boils down to…
Law of Reflection:
normal
Snell’s Law: qr qi
vph = c/n1
vph = c/n2 qt

9. Huygens’ Principle
Every point on a wavefront may be regarded as a secondary source of wavelets.
Planar wavefront:
c Dt

10. Huygens’: Point source through an aperture:
Ignore the peripheral and back propagating parts!
In geometrical optics, this region should be dark.

11. Huygens’: Reflection L

12. Huygens’: Refraction x
vi Dt Qi Qt
vt Dt

13. Fermat’s Principle
The path a beam of light takes between two points is the one which is traversed in the least time.
Isotropic medium: constant velocity. A B
Minimum time = minimum path length.

14. Fermat’s: Refraction A
normal a qi n1 O n2 x qt b B c

15. Huygens’ Principle and Fermat’s principle provide a qualitative (and quantitative) proof of the law of reflection and refraction within the limit of geometrical optics.