Recursion relations for the impedance of periodic layers

Previous numerical work is extended by deriving simple analytic expressions for the impedance of periodic layers over a wide frequency range within the reflection stop band (not just the center Bragg frequency) for an arbitrary number of periods in the structure, for arbitrary layer thicknesses (not just quarter-wave layers), for sizable absorption, and for arbitrary sizes of the refractive index differences. When the number of periods in the structure is infinite, exact expressions for impedance, which are valid for all frequencies in the reflection stop band, are derived. For a finite number of periods in the structure, it is shown that the asymptotic approach of the impedance toward its value for an infinite structure has a decaying exponential dependence. It is shown that the characteristic number of periods in this decaying exponential dependence is determined by the condition number of the transverse field matrix. Simple analytic expressions for the phase shift throughout the reflection stop band are derived, as well as simple analytic expressions to show that a small fractional error in the VCSEL cavity mode frequency can still result from a large fractional error in the cavity thickness if the layers in the Bragg mirror have a small refractive index difference. These simple analytic expressions are useful for design.