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I am looking for assistance with calculating the fabry-perot standing modes in a resonator which has a non-static refractive index.

For a resonator with perfectly reflective mirrors only the standing modes experience sustained population. For a resonator with no internal medium the wavelengths at which a standing mode occurs can be written easily:

$$\lambda_m = \frac{mL}{2}$$

In a material with a refractive index the length of the cavity can be modified to have an effective length given by the product of the refractive index with the real length. The inclusion of a frequency-dependent refractive index causes an issue though. The frequencies that the standing modes appear can no longer be simply found because I believe the result is a non-analytic transcendental equation;

$$E_m(w) = \frac{2 hc}{m\ n(w)\ L}$$

Here the left is the energy of the $m$-th standing mode, and on the right h is Planck's constant, $c$ is the speed of light, m is the mode index, $L$ is the length of the cavity, and $n(w)$ is the frequency dependent refractive index. Written entirely in terms of angular frequency (instead of energy) the equation would be:

$$w_m = \frac{4 \pi c}{m\ L\ n(w)}$$

Would anyone be able to suggest a computational technique to try to solve this problem? Generally $n(w)$ is some complicated function of frequency, $w$, but it could always be fit to some polynomial if necessary. Any and all insight would be appreciated.

nicoguaro
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speeze
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  • Welcome to SciComp.SE. I have edited your equations using MathJax, so they can be read easily, please do so next time. Regarding your questions, is "issue" a synonym for "problem"? Are you asking how to solve the nonlinear equation for $w$? – nicoguaro Feb 27 '18 at 21:40
  • Thank you kindly for your edits. I attempted to use LaTex but that was not supported. Yes I meant issue as being synonymous with problem; I am not so familiar with computational techniques, and am looking for some guidance of techniques that would be appropriate for solving this equation, given some frequency-dependent refractive index profile. – speeze Feb 27 '18 at 22:27

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I agree with your general assessment .. Fabry-Perot modes of a constant material are easy to compute/emumerate, but introducing dispersion (frequency dependent index) leads to a nonlinear solve. If you are able to differentiate your dispersion relation (write down the equation for dn/dw analytically) I would suggest you try Newton's method. If the dispersion relation is reasonably well behaved (some average/background index, plus a small perturbation that varies slowly/smoothly with w), you can probably use that to warm-start the iteration with a good initial guess (based on the constant index case, just use the average).

You might also want to draw inspiration from the procedures used to find cutoff frequencies for e.g. dielectric slab waveguide/optical fiber. Their dispersion relationships are non-linear, so you end up doing similar things .. graphical solution, Newton methods, and the like.

rchilton1980
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