Transversely bounded DFB lasers*

Charles Elachi; Gary Evans; Cavour Yeh

doi:10.1364/JOSA.65.000404

Journal of the Optical Society of America
Vol. 65,
Issue 4,
pp. 404-412
(1975)
•https://doi.org/10.1364/JOSA.65.000404

Transversely bounded DFB lasers

Charles Elachi, Gary Evans, and Cavour Yeh

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Charles Elachi, Gary Evans, and Cavour Yeh, "Transversely bounded DFB lasers*," J. Opt. Soc. Am. 65, 404-412 (1975)

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Abstract

Bounded distributed-feedback (DFB) lasers are studied in detail. Threshold gain and field distribution for a number of configurations are derived and analyzed. More specifically, the thin-film guide, fiber, diffusion guide, and hollow channel with inhomogeneous-cladding DFB lasers are considered. Optimum points exist and must be used in DFB laser design. Different-modes feedback and the effects of the transverse boundaries are included. A number of applications are also discussed.

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PERIODICITY	F	ψ	χ_pq
GUIDE INDEX	^*EVEN MODES: ${tan}^{2} (sw) [1 + \frac{2 sw}{sin (2 sw)}]$	$\frac{k}{β c} (c_{1} A + ∊_{2} B)$	$\begin{matrix} η \frac{∊_{1}}{4} k^{2} {(P_{pq} P_{qp})}^{1 / 2} \\ \begin{array}{l} P_{ij} = 2 \frac{δ_{i}}{β_{i}} \frac{B_{i}}{+ δ_{j}} FOR i \neq j & P_{ij} = A_{i} / β_{i} FOR i \neq j \end{array} \end{matrix}$
SUBSTRATE INDEX		^*EVEN MODES: $\begin{matrix} A = \frac{δ w + si n^{2} (sw)}{1 + δ w} \\ B = \frac{{cos}^{2} (sw)}{1 + δ w} \end{matrix}$	$\begin{array}{l} η \frac{∊_{2}}{4} k^{2} {(P_{pq} P_{qp})}^{1 / 2} \\ P_{ij} = 2 \frac{δ_{i}}{β_{i}} \frac{B_{i}}{δ_{i} + δ_{j}} \end{array}$
SUBSTRATE INDEX	^*ODD MODES: $- cota n^{2} (sw) [1 - \frac{2 sw}{sin (2 sw)}]$
BOUNDARY		^*ODD MODES:	$\begin{matrix} η \frac{∊_{1} - ∊_{2}}{2} k^{2} w {(P_{p} P_{q})}^{1 / 2} \\ P_{i} = β_{i} / δ_{i} B_{i} \end{matrix}$
		sin² → cos²
		cos² → sin²
BOUNDARY	$\frac{δ}{s} [\frac{J_{0} / J_{1} + sw {(J_{0} / J_{1})}^{'}}{K_{0} / K_{1} + δ w {(K_{0} / K_{1})}^{'}}]$	$\frac{k}{β c} (\frac{∊_{1} A + ∊_{1} B}{A + B})$	AS JUST ABOVE WITH $P = \frac{β J_{0}}{δ J_{1}} [sw \frac{K_{1}^{'}}{K_{1}} + δ w \frac{J_{1}^{'}}{J_{1}} - \frac{s^{2} + δ^{2}}{s δ}]$
BOUNDARY		WHERE $\begin{matrix} A = - δ [δ w \frac{J_{1}^{'} K_{0}}{J_{1} K_{1}} + \frac{J_{0}}{J_{1}} - sw] \\ B = s [sw \frac{K_{1}^{'} J_{0}}{J_{1} K_{1}} + \frac{K_{0}}{K_{1}} - δ w] \end{matrix}$
BOUNDARY	γδdR/ν	$[\frac{\sqrt{a}}{c} (1 + 2 δ d) \frac{J_{ν}^{'}}{J_{ν}} + γ \frac{\sqrt{a}}{c} \frac{J_{ν}^{″}}{J_{ν}} - ∊_{2} \frac{k}{c δ} \frac{\partial J_{ν}}{\partial ν}] / P$ →	AS ABOVE WITH $P = \frac{2 β d}{ν} (γ + 2 δ d) \frac{\partial J_{ν} / \partial ν}{J ν} + β / δ$
	WHERE:
	$\begin{matrix} γ = 2 kd \sqrt{a} \\ R = \frac{\partial}{\partial ν} (\frac{J_{ν - 1} - J_{ν + 1}}{J_{ν}}) \end{matrix}$
BOUNDARY	^*EVEN MODES:	$\begin{array}{l} [\frac{\sqrt{a}}{c} (1 + 2 δ d tanh δ w) \frac{J_{ν}^{'}}{J_{ν}} + γ \frac{\sqrt{a}}{c} \frac{J_{ν}^{″}}{J_{ν}} \\ - ∊_{2} \frac{k}{c δ} \frac{\partial J_{ν}}{\partial ν} {tanh δ w ∊_{2} \frac{kw}{c} sec h^{2} δ w}] / P \end{array}$ →	AS ABOVE WITH $\begin{array}{r} P = \frac{2 β d}{ν} [γ + 2 δ d tanh δ w] \frac{\partial J_{ν} / \partial ν}{J ν} \\ + \frac{β}{δ} (tanh δ w + δ w sec h^{2} δ w) \end{array}$
	γδdR/ν (tanh δw + δw sech²δw)
	^*ODD MODES:
	tanh → cotanh
	sech² → cosech²

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Journal of the Optical Society of America