# Optics, waveguide coupling

1. Oct 5, 2008

### buttersrocks

1. The problem statement, all variables and given/known data
We have a 1-D optical waveguide of width 2a (with axis z=0), with a propagating wave (propogating in $$\hat{z}$$) in the lowest order mode, approximated by:

$$E(x)=ACos(\frac{\pi x}{2a})$$

We insert a dielectric slab half-way into the guide such that the slab ends at z=0, it has optical path length $$nd=\frac{\lambda}{2}$$

Looks something like this:
*************z=-d
**----------------|-----|------------------------------
x=0--------------- ''''''''''''' ------------------------------
**----------------|-----|------------------------------
******************z=0

(sorry about the *'s, but it was the only way to line it up.)
Where the ''''''s represent the dielectric.

What is the fraction of power coupled to the lowest order m=0 mode for z>0?

What is the fraction of power coupled to the m=1 mode for z>0?

2. Relevant equations
None other than the usual ones

3. The attempt at a solution
We haven't really done any coupling. Is this just basically 1/2 of the power couples to the m=0 mode since the dielectric is only in half the guide and then I use reflection/transmission in the dielectric to calculate the power that makes it out of that? (where the nd=1/2 wavelength implies that it completely out of phase with the original field... this doesn't seem physically right to me.)

My other ideas is to derive all sorts of boundary conditions based on Maxwell's equations. Or, lastly, something with polarization.

I don't want to reinvent the wheel here, I just want to solve the problem. I don't think it's intended to be difficult. Maybe I'm missing something simple?