1. PHYS 408 Applied Optics (Lecture 5)
Jan-April 2017 Edition
Jeff Young
AMPEL Rm 113

2. Quick review of key points from last lecture
The time-averaged (over many, but not too many optical cycles) Poynting vector, S(r,t)=E(r,t) x H(r,t), specifies the local energy flux, or intensity of the electromagnetic field, and the direction in which that flux is flowing.
For harmonic fields it’s time averaged value can be related to the complex field amplitudes E(r) and H(r) as Re[S=1/2 E x H*]
Spherical wave solutions of the scalar wave equation have spherical shells as wavefronts.

3. Actual spherical wave solution of Maxwell Equations
Point-like dipole oscillating at frequency w, in x direction: E(r) and B(r)? z x
Difficult to derive/solve wave equations for E & B, can you suggest why?
Make analogy with electrostatics, where typically easier to solve for potential phi, rather than E directly
Can’t average and set r =0, and the curl of j is difficult with microscopically small sources.

4. The vector potential A In the Lorentz Guage and if then
Once you have A, then H and E follow from:

5. The vector potential A
So each component of the A vector satisfies the wave equation with a driving term proportional to the same component of the current density. If you solve the inhomogeneous Helmholtz equation for an harmonic dipole motion:
Do you recognize U(r)?
you find that the following is a solution:

6. The Fields due to point-like dipole
Find E and H for large r for a) y=z=0 and b) x=y=0: z x
Give them a fair amount of time on this

7. Very characteristic and useful pattern to understand/recognize
How does the amplitude of the Poynting vector behave at large z?
How is the field polarized at large z? z x
Again, give them a fair amount of time on this. Can neglect change of 1/r with respect to r at large r (goes like 1/r^2), so to a good approximation is a good plane wave, with an amplitude 1/r, so PV goes as 1/r^2

8. Approximate expressions for spherical wavefronts far from the source
Using Cartesian coordinates, look at how exp(-jkr)/r behaves at large z, for small x,y q

9. l Zi(x) Label which is which? x z
Not too long on this, but some, focus on the algorithm for how to solve the problem
To be a decent approximation, Zi(x)-Zj(x)<<l/2

10. Regions of Validity?

14. Spherical, Plane, Paraboloidal type waves… More Generally?
Light with orbital angular momentum
Cavity modes
Refer to Fourier Optics lab for orbital ones, and to cylindrical symmetry version of standing waves for modes

15. Spherical, Plane, Paraboloidal type waves… More Generally? (Con’t)
Many fascinating and distinctly different types of solutions. One important set, although not the most exotic, are plane waves with Slowly Varying Envelopes (paraboloidal wave was an example).

16. Conditions on A(z)?
Want the percentage change of A from wavefront to wavefront to be very small: translate into an equation.
Which implies
Which also implies

17. Conditions on A(z)?
To be safe, insist that the percentage change of the z derivative of A from wavefront to wavefront also be very small…show that this implies:
Not too long on this

18. Substitute into Helmholtz Eqn
Refer to homework and Piazza

19. Exercise
A fair mount of time on this, permitting…leave about minutes for last slide

20. Preview of next day’s topic

21. New Module: Optical Elements
What happens when one of our plane waves strikes a flat interface between two media with different dielectric constants?
What is the boundary condition for field components parallel to surface?
qin qr
kin kr kt z x
Write related equation qt