First principles derivation of microcavity semiconductor laser threshold condition and its application to FDTD active cavity modeling

Adam Mock

doi:10.1364/JOSAB.27.002262

Journal of the Optical Society of America B
Vol. 27,
Issue 11,
pp. 2262-2272
(2010)
•https://doi.org/10.1364/JOSAB.27.002262

First principles derivation of microcavity semiconductor laser threshold condition and its application to FDTD active cavity modeling

Adam Mock

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Adam Mock, "First principles derivation of microcavity semiconductor laser threshold condition and its application to FDTD active cavity modeling," J. Opt. Soc. Am. B 27, 2262-2272 (2010)

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Abstract

A laser threshold condition derived from the instantaneous form of Maxwell’s equations is presented. This derivation incorporates the passive quality factor of the resonator and is particularly amenable to semiconductor microcavity lasers. The optical confinement factor is derived and compared to previous reports. The threshold condition derived here is compared to the results of active cavity finite-difference time-domain calculations, and excellent agreement is found.

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Reference	$Γ_{g}$	Cavity Type
[2]	$\frac{\int_{active} ϵ (x) {\| E_{y} (x) \|}^{2} d x}{\int_{- \infty}^{\infty} ϵ (x) {\| E_{y} (x) \|}^{2} d x}$	1D slab waveguide

[3]	$\frac{\int_{active} \sqrt{ϵ (x)} {\| E_{y} (x) \|}^{2} d x}{n_{eff} \int_{- \infty}^{\infty} {\| E_{y} (x) \|}^{2} d x}$	1D slab waveguide

[4, 7, 9, 10]	$\frac{\int_{active} {\| E_{y} (x) \|}^{2} d x}{\int_{- \infty}^{\infty} {\| E_{y} (x) \|}^{2} d x}$	1D slab waveguide

[6]	$\frac{n_{g} \int \int_{active} {\| \vec{E} \cdot \vec{E} \|}^{2} d x d y}{n \int \int_{- \infty}^{\infty} {\| \vec{E} \cdot \vec{E} \|}^{2} d x d y}$	2D rib waveguide

[8]	$\frac{n \int \int \int_{active} {\| E^{+} + E^{-} \|}^{2} d V}{n_{eff} \int \int \int_{- \infty}^{\infty} ({\| E_{T}^{+} \|}^{2} + {\| E_{T}^{-} \|}^{2} +) d V}$	2D rib waveguide

[5]	$\frac{\int \int \int_{active} {\| \vec{E} \cdot \vec{E} \|}^{2} d V}{\int \int \int_{- \infty}^{\infty} {\| \vec{E} \cdot \vec{E} \|}^{2} d V}$	VCSOA

This work	$Γ_{g} = \frac{\int (\frac{1}{n}) \frac{g (x, y, z)}{g_{max}} ϵ ⟨ \vec{E} \cdot \vec{E} ⟩ d V}{\int [\frac{1}{2} ϵ ⟨ \vec{E} \cdot \vec{E} ⟩ + \frac{1}{2} μ ⟨ \vec{H} \cdot \vec{H} ⟩] d V}$	Applies in general

	$Γ_{g} = \frac{n_{active} \int_{active} ⟨ \vec{E} \cdot \vec{E} ⟩ d V}{\int [\frac{1}{2} ϵ ⟨ \vec{E} \cdot \vec{E} ⟩ + \frac{1}{2} μ ⟨ \vec{H} \cdot \vec{H} ⟩] d V}$	Piecewise constant gain distribution

Gain Distribution	$g_{max}^{thr}$ from Fig. 2	$g_{max}^{thr}$ from Eq. (34)	Error (%)
Full slab	0.1129	0.1132	0.3
1/2 slab	0.1855	0.1855	$< 0.1$
1/4 slab	0.3526	0.3544	0.5

Gain Distribution	$g_{max}^{thr}$ from Fig. 4	$g_{max}^{thr}$ from Eq. (34)	Error (%)
$2 a$ diameter spot	0.2610	0.2627	0.6
$1 a$ diameter spot	0.6134	0.6165	0.5

Reference	$Γ_{g}$	Cavity Type
[2]	$\frac{\int_{active} ϵ (x) {\| E_{y} (x) \|}^{2} d x}{\int_{- \infty}^{\infty} ϵ (x) {\| E_{y} (x) \|}^{2} d x}$	1D slab waveguide

[3]	$\frac{\int_{active} \sqrt{ϵ (x)} {\| E_{y} (x) \|}^{2} d x}{n_{eff} \int_{- \infty}^{\infty} {\| E_{y} (x) \|}^{2} d x}$	1D slab waveguide

[4, 7, 9, 10]	$\frac{\int_{active} {\| E_{y} (x) \|}^{2} d x}{\int_{- \infty}^{\infty} {\| E_{y} (x) \|}^{2} d x}$	1D slab waveguide

[6]	$\frac{n_{g} \int \int_{active} {\| \vec{E} \cdot \vec{E} \|}^{2} d x d y}{n \int \int_{- \infty}^{\infty} {\| \vec{E} \cdot \vec{E} \|}^{2} d x d y}$	2D rib waveguide

[8]	$\frac{n \int \int \int_{active} {\| E^{+} + E^{-} \|}^{2} d V}{n_{eff} \int \int \int_{- \infty}^{\infty} ({\| E_{T}^{+} \|}^{2} + {\| E_{T}^{-} \|}^{2} +) d V}$	2D rib waveguide

[5]	$\frac{\int \int \int_{active} {\| \vec{E} \cdot \vec{E} \|}^{2} d V}{\int \int \int_{- \infty}^{\infty} {\| \vec{E} \cdot \vec{E} \|}^{2} d V}$	VCSOA

This work	$Γ_{g} = \frac{\int (\frac{1}{n}) \frac{g (x, y, z)}{g_{max}} ϵ ⟨ \vec{E} \cdot \vec{E} ⟩ d V}{\int [\frac{1}{2} ϵ ⟨ \vec{E} \cdot \vec{E} ⟩ + \frac{1}{2} μ ⟨ \vec{H} \cdot \vec{H} ⟩] d V}$	Applies in general

	$Γ_{g} = \frac{n_{active} \int_{active} ⟨ \vec{E} \cdot \vec{E} ⟩ d V}{\int [\frac{1}{2} ϵ ⟨ \vec{E} \cdot \vec{E} ⟩ + \frac{1}{2} μ ⟨ \vec{H} \cdot \vec{H} ⟩] d V}$	Piecewise constant gain distribution

Gain Distribution	$g_{max}^{thr}$ from Fig. 2	$g_{max}^{thr}$ from Eq. (34)	Error (%)
Full slab	0.1129	0.1132	0.3
1/2 slab	0.1855	0.1855	$< 0.1$
1/4 slab	0.3526	0.3544	0.5

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Journal of the Optical Society of America B