Pentration of light in cladding from a waveguide

[Lee]

As part of my project I want to calculate the theoretical value of the best thickness of cladding to use on my waveguide to prevent light from leaking through the cladding to the next layer, I'm aware I need to play with Maxwells equations and it's pretty much particle in the box where I'll have a negative exp function in the cladding, but I'm not sure how to approach the problem, does anyone have a useful link or advice?

 

[ZapperZ]

Why is this a "quantum physics" topic?

Zz.

 

[Lee]

Feel free to throw it wherever.

 

[Lee]

anyone?

 

[Claude Bile]

You need to solve the EM wave equation separately in the core and cladding regions and match them at the boundary by calculating appropriate values for the arbitrary constants that pop up in your solution to the diff. eqn.

The process of solving the equations will depend on what the shape of your waveguide is, specifically what symmetries it possesses. This will then determine what coordinate system you should use to expand the Laplacian.

For a full derivation you can expect to write five or six pages minimum. It might be a sledgehammer approach to a problem that has a more elegant solution. Essentially your problem reduces down to finding the Mode field Diameter of your waveguide mode(s). See if you can't find an equation for the MFD that is suitable for you waveguide shape first before tackling the not-insignificant task of deriving a full field solution.

Claude.

 

[Lee]

So the waveguide is rectangular, and my only concern is the penetration in one dimension, so would I be able to discount the other dimensions and solve in 1-D making the problem much simpler? Making this very similar to a 1st year Schroedinger equation particle in a box?

 

[Claude Bile]

Yes, you can discount other dimensions if you are just after the shape of the fields, the amplitude of the fields though will depend on the entire solution.

The thing you have to be mindful of is that the electric and magnetic fields are vector fields, not scalar fields. If the refractive index contrast between your core and cladding is reasonably small (say, less than 0.5) you can apply the weak guiding approximation and reduce the problem to solving for a scalar field - otherwise you're stuck with working out all 3 vector field components.

It just struck me that if you're only interested in the penetration depth of the cladding, you don't need to bother with solving for the fields inside the core or matching the solutions at the boundary (since this would only change the amplitude of the field in the cladding).

Claude.

 

[Lee]

So I can simply apply the equation to the cladding and be able to come out with the negative exponential function for the light in the cladding, and from that work out the penetration at different distances?

 

[Claude Bile]

Yes, you need to evaluate the exponent, which, for a rectangular slab waveguide is;

$$\beta^2 + n_{clad}k_0^2$$

Where;

- $\beta$ is the propagation constant of the guided mode
- $n_{clad}$ is the refractive index of the cladding
- $k_0$ is the magnitude of the wavevector in free space (i.e. $2\pi/\lambda_0$)

The hard bit is finding the propagation constant, to do this you need to solve for the modes of the guide. Fortunately you probably don't need to find the full field solutions, you will probably be fine deriving the modes from ray theory.

When I have time, I'll say a bit on how to solve for the modes using ray theory.

Claude.

 

[Lee]

I had a bash using Maxwell's equations and BCs, and I'm down to making the wave function continues at the boundaries. So I've got down to a set of equations that need to be equal for which I now have to solve. Though I'm currently normalizing my wave function to get terms for my Constants.

 

[Claude Bile]

You don't need to worry about making them continuous for your purposes, since you are only interested in the decay length. Making things continuous only changes the relative amplitudes of the core and cladding solutions.

Claude.

 

[Claude Bile]

Further to post #9 - to solve for the modes of a slab waveguide, you need to find the angles that satisfy the following transverse resonance condition;

$$2dn_1k_0cos(\theta_m) = 2m\pi$$

Where

- $d$ is the diameter of the guide.
- $n_1$ is the refractive index of the core.
- $k_0$ is the magnitude of the wavevector in free space = $2\pi/\lambda$.
- $\theta$ is the angle of incidence of the totally-internally-reflected beam for mode $m$.
- $m$ is the mode number.

Once you know the values of $\theta_m$, you can calculate $\beta_m$ as follows;

$$\beta_m = n_1k_0sin(\theta_m)$$

Claude.

 

[Lee]

Excellent Claude, thanks very much for the input.

So now I have worked out the propagation constant where does that then go in my exponent function? Between talking to my adviser and reading literature on line I'm now rather confused.

http://ocw.mit.edu/NR/rdonlyres/Electrical-Engineering-and-Computer-Science/6-772Spring2003/2D4523FC-BDB0-4A02-B01A-8D4B81C9AB14/0/Lecture17v2.pdf

 

[Claude Bile]

Sorry, I made an error, in post #9, the top equation should be;

$$w = -\sqrt{\beta^2 - n_{clad}k_0^2}$$

Where $w$ is your exponent.

Claude.

 

[Lee]

Thanks buddy, I finally got round to creating the graph I wanted and I'm really happy with it, and it looks like it agrees with my results (though I would of liked to create more samples or test the 540nm layer of silica).

http://img440.imageshack.us/img440/9472/awesomenesscs8.jpg

Do you have the reference I could use? As this is going to be in my report and I'll need one if I can include the results in my paper.

 

[Claude Bile]

The numbers look sensible, nice work (though I would change the units on the vertical axis to microns rather than meters).

"Theory of Optical Waveguides" by A. Snyder and J. Love is what I use but it is pretty full on.

"Lasers and Electro-Optics - Fundamentals and Engineering" by C. Davis is more digestible for the non-expert.

Claude.