I How to calculate the ring resonator radii via Laplace?

AI Thread Summary
Two approaches for calculating ring resonator dimensions yield slightly different results. The first method uses the relationship between the waveguide wavelength and the ring's circumference, which provides consistent results. The second method involves solving the Laplace equation in cylindrical coordinates, leading to a more complex condition for the radii. The discrepancy in results may be due to neglecting surface charges at the waveguide's end. Clarification is sought on how the waveguide's length affects field presence and charge at the intersection with the ring.
Leopold89
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Dear forum,

I was trying out two different approaches to calculate the dimensions of a ring resonator (sketch below) and got two slightly different solutions.
The first approach is to assume that the waveguide wavelength has to fit n times onto the circumference of the ring (taking the average of outer and inner radius): $$2\pi r = n \lambda_{waveguide}$$, with the longer waveguide wavelength. This works well.
The second approach is to solve the Laplace equation in cylindrical coordinates, yielding the condition $$J_m(\alpha r_{in})Y_m(\alpha r_{out})=J_m(\alpha r_{out})Y_m(\alpha r_{in})$$ for the radii, with $$\alpha=\frac{2\pi}{\lambda}$$ as wave number.
I noticed that the solutions to the second condition are more of less slightly off, so I wanted to ask why the second approach does not work.
My guess is that I need to take the surface charges at the end of the waveguide into account. But what I don't understand is that I would want to have the waveguide as long as half a waveguide wavelength, so wouldn't I get no field and therefore no charge at the intersection of waveguide and ring?
ring_resonator.jpg
 
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