16 February 2009 A new fundamental equation for the band spectra of dielectric layer films
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Abstract
This paper derives a new fundamental equation for the frequency spectra ω(q) of one-dimensional photonic crystals as a function of Brillouin wave vector q in the form of a novel factored expression, tan2 qd / 2 = tan(kNaNN) × tan(kNaN - βN), where N the number of layers per period is, d is the unit cell width, and ki = niω/c is the local wave vector in the ith layer of width 2ai and refractive index ni. Angles(αNN) depend on the parameters of all N layers but are independent of aN . For two layers, (α2, β2) correspond to the even/odd parity solutions at the center and the edge of the Brillouin zone. The derived spectral expression provide separate eigenvalue conditions for consecutive band edges at the center and the edge of the Brillouin zone for any N and is useful in finding the Bloch phase that is necessary in finite crystal calculations. The formalism is convenient for tailoring band gaps and for calculating impurity modes in dielectric stacks.
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Frank Szmulowicz, Frank Szmulowicz, } "A new fundamental equation for the band spectra of dielectric layer films", Proc. SPIE 7223, Photonic and Phononic Crystal Materials and Devices IX, 72230S (16 February 2009); doi: 10.1117/12.809262; https://doi.org/10.1117/12.809262
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