# E field phase matching at interface, deriving Snell's Law

1. Sep 14, 2010

### Cruikshank

I'm studying optics on my own and have gotten very stuck on page 75 of Klein and Furtak, the unnumbered equation. Specifically, I don't understand why there aren't arbitrary phase factors inserted; the equation does not look general enough. I checked in Jackson, 2nd edition Classical Electrodynamics, and on page 279 Jackson simply states in equation 7.34 that the phases have to match at the origin. There isn't much there, just a declaration, and I don't understand why it should be true. Any pointers?

I've gotten nearly everything else in that derivation, but really I must have written 50 pages of equations trying to solve these boundary conditions.

Incoming wave vector k = n - iK (propagation and attenuation) Reflected wave has double prime, transmitted wave has single prime. Klein and Furtak simply write (for E_t, t is tangential, consider it a variable for which x or y can be substituted:)

E_t*e^-i(k_x x) + E"_t*e^-i(k"_x * x + k"_y*y) = E'_t*e^-i(k'_x * x + k'_y * y)

The interface is the z=0 plane, so I see why there are no z factors. But I would have inserted arbitrary phases on E' and E". Why are they not needed?

2. Sep 14, 2010

### gabbagabbahey

IIRC, Griffiths gives a decent argument (middle of chapter 7 I think).