[Slide 1] Hi everyone.
This lecture 2.15.
In the last lecture, we just started talking about how to use CAMFR.
And of course we said you can download it for
free from CAMFR.sourceforge.net and you can also download Python for
free and install it in Python if you don't already have it.
[Slide 2] So continuing on the basic discussion we had earlier,
we already showed how you can use it to calculate a 1D waveguide performance.
Now we're going to actually look at a what we'll call a 2D waveguide,
where basically it's a waveguide with a finite length and so this formulation here
for our waveguide in 1D is actually almost identical as what we had earlier.
But then we're actually going to also insist that we have a spacial region
on either side that has air, and has a total thickness of 4.5.
And this essentially is going to be semi-infinite,
on this side and then on the far right side.
And then we're actually going to allow the length of this gallium arsenide
slab to vary, in this direction.
So you can see this length here,
from one end of the gallium arsenide to the other is just defined as L.
And then we allow L to go over a whole range.
From a very small value to a slightly larger value of 0.1.
So it's still very short of course but this is just a brief of concept.
And then we can actually stock these three layers, the air region and
then the gallium arsenide slab and then the air again.
And the we can perform the calculation.
And then we can extract this so-called R12 parameter which
corresponds to the reflection starting from one and
then reflecting back from the two region, if you will.
And then of course there is a reciprocal term, RT1.
[Slide 3] And so for a lot of problems of course these should match.
Then here's the results for the 2D waveguide that we calculated,
not surprisingly if L is zero then there's no reflection
because there's nothing physically there really to reflect it.
But as L gets larger of course the reflection starts to go up and
you can see that it doesn't go up just in a straight line forever.
It does start to roll over because we're
physically limited to a certain amount of reflection.
In this particular case it's drawing around over 35 percent.
Which very roughly corresponds to the Fernel reflection of an error gallium
arsenide interface.
So it kind of seems logical.
[Slide 4] Now we're not necessarily actually limited just to Cartesian geometries.
Another possibility is to look at cylindrical stacks of materials.
So in this case it's actually very analogous to what we just did.
Except that instead of having essentially a 1D layer,
or 1D stack, that was here, we actually create a circularly symmetric stack.
So while it looks almost the same from the side, you can see from the top
that actually it looks like a round piece of gallium arsenide.
So you can kind of think of it as a very high Index fiber optic
waveguide that's cut off into very short stub, right, so
it's a very short and squat cylinder that we're calculating here.
And so you can see, basically, the way that we set this up is,
first of all we have to have these circular PML boundary conditions,
which stands for perfectly matched layers.
And so then this ensures that if the fields get too far away
In the radial direction, then they got absorbed.
And then we have to set up an air circle, of course, to go above and
below the gallium arsenide, and we have to set what's called
the circular order, which is basically the symmetry of the input mode.
And so, in this kind of problem we can probably assume
that we have m = 0, or m = 1.
M = 1 would correspond usually to a plane wave, so that's also a common choice.
But here, we chose m = 0.
So it's like a circularly symmetric source.
And then we can actually calculate, based on this structure,
which is a stack of an air circle and
then gallium arsenide surrounded by air circle and then another air circle.
So adding these three things together, calculate it and
then print out the reflection.
[Slide 5] And so then we actually get something like this.
And so, probably not surprisingly, again,
if the radius of this cylinder is very small, It doesn't really do much, but
as the radius increases, then you can see that it has a lot of peaks and valleys.
And so this probably corresponds to certain resonant modes
that create a stronger quality factor mismatch, and thus stronger refraction.
In the next lecture, we'll go through another example of how to use CAMFR.