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Published on: December 27, 2012
Recursion relations for the impedance of periodic layers
1Yale University, New Haven, Connecticut 06520-8284, USA. janet.pan@yale.edu
Applied Optics
|March 27, 2007
Summary
Simple analytic expressions for periodic layer impedance are derived, extending previous work. These formulas are valid across a wide frequency range and useful for designing optical structures like VCSELs.
Area of Science:
- Optics and Photonics
- Materials Science
Background:
- Periodic dielectric structures are crucial in optical devices.
- Previous analyses of their impedance were limited in scope.
Purpose of the Study:
- To derive simple analytic expressions for the impedance of periodic layers.
- To cover a wide frequency range within the reflection stop band.
- To account for arbitrary parameters like layer thickness and absorption.
Main Methods:
- Extending previous numerical work with analytic derivations.
- Analyzing impedance behavior for infinite and finite periodic structures.
- Investigating the condition number of the transverse field matrix.
Main Results:
- Analytic expressions for impedance across a wide frequency range within the reflection stop band.
- Exact impedance expressions for infinite periodic structures.
- Demonstration of exponential dependence for finite structures, linked to the condition number.
- Analytic expressions for phase shift and error propagation in VCSEL cavity modes.
Conclusions:
- The derived analytic expressions simplify the design of periodic optical structures.
- Understanding impedance and phase shift is critical for optimizing devices like VCSELs.
- Error analysis highlights the impact of refractive index differences on cavity mode frequency.
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