Coupling of resonant modes in a 1D Photonic Crystal 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.

 A: Your observations are mostly correct.

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*"a defect can be introduced to create a defect/resonance mode and enable transmission" - that is, when the photonic crystal is of finite thickness. In an infinite PhC, the defect only introduces localised states centered at that defect.


*"if it is a half-wavelength layer it will enable a resonance condition" - any defect will do, though λ/2 layer is more efficient


*"allow transmission in the forbidden band" - this is basically what you wrote before. Transmission = coupling into defect mode + coupling out of the defect mode.


*"the performance of the structure is dictated by a much more complex scattering problem" - absolutely. Every layer in the structure contributes to the finite PhC's transmission.


*"if all of these defects are at the same thickness, the resonances are at different frequencies" - this is analogous to well-known coupled oscillators, where anticrossing (or avoided crossing) is observed when their frequency is the same, or close. [Also note your example does not even put both red defect layers into exactly same conditions. Your structure is asymmetrical. Removing one layer, you get different spectrum than when removing the other one.]


*"Trying to understand this phenomenon" - I always perceived PhC theory as counter-intuitive, or more exactly, somewhat orthogonal to any intuition I have gotten from other fields of physics. Textbooks which build parallels with solid-state physics are correct, but a student should not get overwhelmed by the complex and overlapping terminology coming from there.


*"why do these optical modes couple" - because their energy is never fully localized inside the red boxes; it also penetrates the surrounding PhC. Solving for eigenstates, you always find part of energy in each of the oscillators. The lower frequency eigenstate corresponds usually to symmetric wave residing in both defects, the higher frequency to antisymmetric one. Coupled oscillator theory is useful here.


*Having spent part of my productive life with simulations of PhCs and metamaterials, I admit I have a very little intuition to infer structure behaviour without numerical simulation.
At any rate, I can recommend the "Photonic Crystals: Molding the Flow of Light" book that can be legally downloaded as PDF online. But you probably know this.
