Coupling of resonant modes in a 1D Photonic Crystal

[jgk5141]

In 1D Photonic crystals, a defect can be introduced to create a defect/resonance mode and enable transmission. At first considerations, the thickness of the single defect layer determines the transmission frequency. Moreover, if it is a half-wavelength layer it will enable a resonance condition (this is where the analogy of fabry-perot comes in) at that wavelength and allow transmission in the forbidden band (photonics band gap) of the original photonic crystal. However, when performing simulations (Transfer Matrix Method), and the simulation starts to vary farther from the ideal, then it becomes clear that the performance of the structure is dictated by a much more complex scattering problem. For example, if multiple defects are introduced at different points within the photonic crystal the multiple transmission peaks appear within the stop-band. Even if all of these defects are at the same thickness, the resonances are at different frequencies. I have attached a screen shot of double-cavity structure that is a good example of something that would display this.

An analogy that first comes to mind is the tight-binding model and this seems to be a good starting point, but I cannot find a good starting point for this idea within the field of photonics. Trying to understand this phenomenon, I came across several topics such as Couple-mode theory (CMT), quasi-modes, quasi-normal modes, Wannier functions, and more.

What I am trying to understand is first, the fundamental question of why do these optical modes couple? And second, how can I predict the frequencies of these coupled modes prior to any simulation? I want to develop this theoretical intuition without just brute-forcing my simulations until I get the desired results. Any help our guidance in this area would be greatly appreciated. Please let me know if more information is needed.

 

[Valerie Park]

A:The coupled mode theory (CMT) is the most common approach to model the interaction of optical modes in a photonic structure. In essence, CMT assumes that the optical modes in the structure are coupled through an interaction matrix $M$. This matrix is usually determined by solving Maxwell's equations for the structure and calculating the overlap integrals between the optical modes. By solving the eigenvalue problem $\mathbf{M}\mathbf{q} = \lambda \mathbf{q}$, where $\mathbf{q}$ are the eigenvectors of the matrix and $\lambda$ are the eigenvalues, one can obtain the frequencies of the coupled modes and their corresponding eigenvectors. In the case of two defects, the CMT reduces to two coupled harmonic oscillators, and the coupled modes can be calculated analytically. The exact form of the coupled mode equations depends on the type of interactions between the defects, for example, whether they are interacting through a direct dipole-dipole interaction or an evanescent field coupling.In general, CMT is a powerful tool for predicting the frequencies of coupled modes in photonic structures. It is also possible to use CMT to predict the transmission spectra of photonic structures, as well as their scattering properties. It should be noted, however, that CMT only provides an approximate description of the interactions between optical modes, and more accurate results can be obtained by performing full-wave simulations.