Abstract
We present a formulation to analyze photonic periodic structures from viewpoints of sources and gain. The approach is based on a generalized eigenvalue problem and mode expansions of sources which sustain optical fields with phase boundary conditions. Using this scheme, we calculate power spectra, dispersion relations, and quality factors of Bloch modes in one-dimensional periodic structures consisting of dielectrics or metals. We also compare the results calculated from this scheme with those from the complex-frequency method. The outcomes of these two approaches generally agree well and only deviate slightly in the regime of low quality factors.
Highlights
Photonic periodic structures [1,2,3,4] have been widely utilized in optical fibers [5], laser cavities [6,7,8], and gratings [9] due to the effect of photonic bands
Ωb c εr(r)E(r), E(r) = eik·rF(r), (1a) εr(r + Rm) = εr(r); F(r + Rm) = F(r), m = 1 − 3, (1b) where εr(r) is the relative permittivity tensor of the structure, which is approximately dispersionless in the frequency range of interest; c is the speed of light in vacuum; Rm (m = 1 − 3) are primitive vectors of the unit cell; k is the wave vector corresponding to the crystal momentum; and F(r) is the Bloch periodic part of E(r)
We apply the formulation to a few one-dimensional (1D) periodic structures including the nondispersive and lossless dielectric/dielectric, nondispersive but lossy metal/dielectric, as well as dispersive and lossy metal/dielectric bilayers
Summary
Photonic periodic structures [1,2,3,4] have been widely utilized in optical fibers [5], laser cavities [6,7,8], and gratings [9] due to the effect of photonic bands. The presented formalism is more like a resonance calculation based on the variation of complex permittivity rather than the direct perturbation of the resonance frequency With these points, we apply the formulation to a few one-dimensional (1D) periodic structures including the nondispersive and lossless dielectric/dielectric, nondispersive but lossy metal/dielectric, as well as dispersive and lossy metal/dielectric bilayers. Whitesource power spectra, and Q factors of Bloch modes are calculated based on this approach and compared to those from the dispersive complex-ω method similar to Eq (1a). The results from these two approaches in general agree with each other and only deviate slightly as Q factors are low.
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