Hi,
I need to estimate the overlap between optical modes and acoustic modes of a optically guiding and acoustically antiguiding waveguide (silica core suspended in air).
I setup a Comsol model solving for 2 coefficient PDEs, one for the optical modes and the other for the acoustic modes. First the optical modes were found. Then the optical propagation constants were plugged into the acoustic PDE (because the acoustic field is generated via the interation of the optical fields via SBS) to solve for the acoustic modes.
Initially I tried the simpler case of both optically and acoustically guiding waveguide (GeO2 core and SiO2 cladding), 4 um core and 62.5 um cladding, boundary condition is “zero field at the claddingbecause the field is confined in the core, and search for the eigenvalues of the acoustic PDE within the REAL value from 1/V_cladding to 1/V_core. The solutions looked ok, matched with some published papers.
When I tried for the case of V_core much smaller than V_cladding (air and silica), PML layer outside of the cladding, and search in the same range (1/V_core ~ 1/V_cladding) but now in IMAGINARY value. The solution does look anything like a leaky mode with most of field in the core due to strong sound impedence difference, such as in this paper
https://arxiv.org/pdf/1308.0382.pdf
What did I get wrong in the mode? Would be very grateful for your suggestions.
Presumably where you say "IMAGINARY value" you mean COMPLEX value. You have to search for leaky modes in the complex plane, not along the imaginary axis.
A weakly leaky mode will be close to the real axis in the complex plane. A very leaky mode will be further from the real axis, in the complex plane, but not along the imaginary axis.
Modes that are on the imaginary axis are evanescent, not leaky. Evanescent modes decay quickly with distance (do not propagate), and their eigenfunctions are well behaved in depth. Leaky modes propagate with weaker exponential decay, and their eigenfunctions grow exponentially in depth making their normalization problematic.
Also, when looking for leaky modes, you have to take good care of your branch cuts and Reimann sheets arising from the vertical wavenumbers (or slowness). The search for the mode eigenvalues in wavenumber space for instance, can head toward and across a branch cut between Riemann sheets, taking you from one Riemann sheet to another, from a proper Riemann sheet to an improper Riemann sheet, where the leaky modes are found. Your branch cuts must be chosen so that they do not intercept and block the mode (eigenvalue) search algorithm in its effort to cross a branch cut to find leaky modes. Perhaps COMSOL manages branch cuts for you. They might speak of Sommerfeld versus Ewing-Jardetsky-Press (EJP) branch cuts. This takes you far into complex variable theory, which may be unavoidable when working with leaky modes.
Note that weakly leaky modes can be transferred from an improper Riemann sheet to a proper Riemann sheet by adding absorption losses to a half-space or thick layer participating the eigenfunction of the leaky mode. In effect, the exponential attenuation of the absorption loss of the medium overcomes the exponential growth of the leaky mode's eigenfunction, converting a leaky mode into a proper mode with notable imaginary part to its eigenvalue. So you might try adding artificially high absorption loss to your thicker media to trick leaky modes into being proper modes for the sake of finding them, normalizing them, and using them.
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