A new fundamental equation for the band spectra of dielectric layer films

Frank Szmulowicz

doi:10.1117/12.809262

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16 February 2009 A new fundamental equation for the band spectra of dielectric layer films

Frank Szmulowicz

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Proceedings Volume 7223, Photonic and Phononic Crystal Materials and Devices IX; 72230S (2009) https://doi.org/10.1117/12.809262
Event: SPIE OPTO: Integrated Optoelectronic Devices, 2009, San Jose, California, United States

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(k_Na_N -α_N) × tan(k_Na_N - β_N), where N the number of layers per period is, d is the unit cell width, and k_i = n_iω/c is the local wave vector in the ith layer of width 2a_i and refractive index n_i. Angles(α_N,β_N) depend on the parameters of all N layers but are independent of a_N . 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 "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); https://doi.org/10.1117/12.809262

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