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

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.

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.

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

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:

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

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

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

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

Regions of Validity?

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

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).

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

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

Substitute into Helmholtz Eqn Refer to homework and Piazza

Exercise A fair mount of time on this, permitting…leave about 10-15 minutes for last slide

Preview of next day’s topic

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