Light propagation in layered media in a total reflection geometry: a transfer matrix method using virtually linear basis functions to handle critical conditions

Reinhard Sigel

doi:10.1364/JOSAA.472361

Journal of the Optical Society of America A
Vol. 39,
Issue 12,
pp. 2142-2152
(2022)
•https://doi.org/10.1364/JOSAA.472361

Light propagation in layered media in a total reflection geometry: a transfer matrix method using virtually linear basis functions to handle critical conditions

Reinhard Sigel

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Reinhard Sigel, "Light propagation in layered media in a total reflection geometry: a transfer matrix method using virtually linear basis functions to handle critical conditions," J. Opt. Soc. Am. A 39, 2142-2152 (2022)

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Table of Contents Category
- Scattering and Propagation
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About this Article
History
- Original Manuscript: August 3, 2022
- Revised Manuscript: September 28, 2022
- Manuscript Accepted: October 11, 2022
- Published: November 3, 2022

Abstract

For light propagation in a layered refractive index profile, critical conditions occur when the wave vector perpendicular to the layering becomes zero. Such conditions can occur in a total reflection geometry. Conventional transfer matrix methods become singular, and geometrical optics concepts break down at critical conditions. We introduce two new, to the best of our knowledge, basis systems using virtual linear functions that obey the Helmholtz equation and turn to linear functions required for critical conditions.

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Basis Set	$‖ b_{j}^{\pm} ‖^{2}$			$⟨ b_{j}^{+}, b_{j}^{-} ⟩_{j}$			$lim_{k_{j} \to 0} \| \cos (γ_{j}) \|$
ET	$\frac{e^{2 k_{j}^{''} d_{j}} - e^{- 2 k_{j}^{''} d_{j}}}{8 k_{j}^{''} d_{j}} \pm \frac{\sin (2 k_{j}^{'} d_{j})}{4 k_{j}^{'} d_{j}}$	→	$\frac{1}{2} [1 \pm 1]$	$\frac{1 - \cos (2 k_{j}^{'} d_{j})}{4 k_{j}^{'} d_{j}}$
				$+ i \frac{\cos (2 k_{j}^{''} d_{j}) - 1}{4 k_{j}^{''} d_{j}}$	→	0	$\frac{\sqrt{3}}{2}$
CT	$\frac{e^{k_{j}^{''} d_{j}} - e^{- k_{j}^{''} d_{j}}}{4 k_{j}^{''} d_{j}} \pm \frac{\sin (k_{j}^{'} d_{j})}{2 k_{j}^{'} d_{j}}$	→	$\frac{1}{2} [1 \pm 1]$	0			0
EE	$\pm \frac{1 - e^{\mp 2 k_{j}^{''} d_{j}}}{2 k_{j}^{''} d_{j}}$	→	1	$\pm \frac{1 - e^{- 2 i k_{j}^{'} d_{j}}}{2 i k_{j}^{'} d_{j}}$	→	1	1
CE	$\frac{e^{k_{j}^{''} d_{j}} - e^{- k_{j}^{''} d_{j}}}{2 k_{j}^{''} d_{j}}$	→	1	$\frac{\sin (k_{j}^{'} d_{j})}{k_{j}^{'} d_{j}}$	→	1	1
SV	$‖ b_{j}^{+} ‖^{2} = \frac{e^{k_{j}^{''} d_{j}} - e^{- k_{j}^{''} d_{j}}}{4 k_{j}^{''} d_{j}} + \frac{\sin (k_{j}^{'} d_{j})}{2 k_{j}^{'} d_{j}}$	→	1	0			0
	$‖ b_{j}^{-} ‖^{2} = \frac{1}{{\| k_{j} \|}^{2} d_{j}^{2}} [\frac{e^{k_{j}^{''} d_{j}} - e^{- k_{j}^{''} d_{j}}}{4 k_{j}^{''} d_{j}} - \frac{\sin (k_{j}^{'} d_{j})}{2 k_{j}^{'} d_{j}}]$	→	$\frac{1}{12}$
TV	$\frac{e^{k_{j}^{''} d_{j}} - e^{- k_{j}^{''} d_{j}}}{4 k_{j}^{''} d_{j}} [1 + \frac{1}{{\| k_{j} \|}^{2} d_{j}^{2}}]$			$\frac{e^{k_{j}^{''} d_{j}} - e^{- k_{j}^{''} d_{j}}}{4 k_{j}^{''} d_{j}} [1 - \frac{1}{{\| k_{j} \|}^{2} d_{j}^{2}}]$
	$+ \frac{\sin (k_{j}^{'} d_{j})}{2 k_{j}^{'} d_{j}} [1 - \frac{1}{{\| k_{j} \|}^{2} d_{j}^{2}}]$	→	$\frac{13}{12}$	$+ \frac{\sin (k_{j}^{'} d_{j})}{2 k_{j}^{'} d_{j}} [1 + \frac{1}{{\| k_{j} \|}^{2} d_{j}^{2}}]$	→	$\frac{11}{12}$	$\frac{11}{13}$

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Journal of the Optical Society of America A