Ab initio theory of quantum fluctuations and relaxation oscillations in multimode lasers

Adi Pick; Alexander Cerjan; Steven G. Johnson

doi:10.1364/JOSAB.36.000C22

Journal of the Optical Society of America B
Vol. 36,
Issue 4,
pp. C22-C40
(2019)
•https://doi.org/10.1364/JOSAB.36.000C22

Ab initio theory of quantum fluctuations and relaxation oscillations in multimode lasers

Adi Pick, Alexander Cerjan, and Steven G. Johnson

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Adi Pick, Alexander Cerjan, and Steven G. Johnson, "Ab initio theory of quantum fluctuations and relaxation oscillations in multimode lasers," J. Opt. Soc. Am. B 36, C22-C40 (2019)

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History
- Original Manuscript: November 1, 2018
- Revised Manuscript: January 11, 2019
- Manuscript Accepted: January 11, 2019
- Published: February 19, 2019

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Abstract

We present an ab initio semi-analytical solution for the noise spectrum of complex-cavity microstructured lasers, including central Lorentzian peaks at the multimode lasing frequencies and additional sidepeaks due to relaxation-oscillation (RO) dynamics. In Phys. Rev. A 91, 063806 (2015) [CrossRef] , we computed the central-peak linewidths by solving generalized laser rate equations, which we derived from the Maxwell–Bloch equations by invoking the fluctuation–dissipation theorem to relate the noise correlations to the steady-state lasing properties. Here, we generalize this approach and obtain the entire laser spectrum, focusing on the RO sidepeaks. Our formulation treats inhomogeneity, cavity openness, nonlinearity, and multimode effects accurately. We find a number of new effects, including new multimode RO sidepeaks and three generalized $α$ factors. Last, we apply our formulas to compute the noise spectrum of single-mode and multimode photonic-crystal lasers.

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Quantity	Symbol	Definition
SALT permittivity	$ε (x, ω)$	$ε_{c} (x, ω) + \frac{γ_{⊥} D_{0} F (x)}{ω - ω_{a} + i γ_{⊥}} {[1 + \sum_{μ} \frac{γ_{⊥}^{2}}{{(ω_{μ} - ω_{a})}^{2} + γ_{⊥}^{2}} {\| a_{μ 0} \|}^{2} {\| E_{μ} \|}^{2}]}^{- 1}$
Nonlinear restoring force	$c_{μ ν} (x)$	$\frac{- i ω_{μ}^{2} \frac{\partial ε (x, ω_{μ})}{\partial {\| a_{ν 0} \|}^{2}} E_{μ}^{2} (x)}{\int d x \frac{d}{d ω} [ω^{2} ε (x, ω)] \|_{ω_{μ}} E_{μ}^{2} (x)}$
Dressed decay rate	$γ (x)$	$γ_{∥} (1 + \sum_{μ} \frac{γ_{⊥}^{2}}{{(ω_{μ} - ω_{a})}^{2} + γ_{⊥}^{2}} {\| a_{μ 0} \|}^{2} {\| E_{μ} (x) \|}^{2})$ .
Noise amplitude	$R_{μ ν} (ω)$	$2 ℏ ω_{μ}^{4} \frac{\int d x {\| E_{μ} (x) \|}^{2} Im ε (x, ω) \coth (\frac{ℏ ω_{μ} β (x, ω)}{2})}{{\| \int d x \frac{d}{d ω} [ω^{2} ε (x, ω)] \|}_{ω_{μ}} E_{μ}^{2} (x) \|^{2}} \cdot δ_{μ ν}$

Quantity	Symbol	Definition
Phase diffusion coefficient	$Γ_{0} (ω)$	$R_{μ μ} (ω) / 2 a_{μ 0}^{2}$
RO frequency	$Ω$	${[2 a_{0}^{2} \int d x Re c_{μ μ} (x) γ (x)]}^{1 / 2}$
RO decay rate	$Γ$	$\int d x γ (x) / 2$
Sideband linewidth	$Γ_{SB}$	$Γ_{0} (α_{1}^{2} + 1) + Γ$
Linewidth enhancement	$α_{1}$	$\frac{\int d x Im [c_{μ μ} (x)]}{\int d x Re [c_{μ μ} (x)]}$
Sideband power fraction	$α_{2}$	$\frac{\int d x γ (x) Im [c_{μ μ} (x)]}{\int d x γ (x) Re [c_{μ μ} (x)]}$
Asymmetry factor	$α_{3}$	$\frac{\int d x γ {(x)}^{2} Im [c_{μ μ} (x)]}{[\int d x γ (x) Re [c_{μ μ} (x)]] [\int d x γ (x)]}$

$A_{μ ν} (x) = 2 a_{μ 0} a_{ν 0} Re [c_{μ ν} (x)]$ , $A = \int d x A (x)$	$Q_{+ σ} = \sum_{ℓ m n} \frac{P_{+ σ} P_{- n} R_{+} P_{- ℓ}^{†} P_{+ m}^{†}}{(ω_{+ σ} - ω_{- n}) (ω_{+ σ} - ω_{+ ℓ}^{}) (ω_{+ σ} - ω_{- m}^{})}$
$B_{μ ν} (x) = 2 a_{μ 0} a_{ν 0} Im [c_{μ ν} (x)]$ , $B = \int d x B (x)$	$Q_{- σ} = \sum_{ℓ m n} \frac{P_{+ n} P_{- σ} R_{-} P_{- ℓ}^{†} P_{1 m}^{†}}{(ω_{- σ} - ω_{+ n}) (ω_{- σ} - ω_{+ ℓ}^{}) (ω_{- σ} - ω_{+ m}^{})}$
$M_{\pm} \equiv \pm \sqrt{\int d x A (x) γ (x)} + \frac{i}{2} \int d x γ (x) 1$	$S^{σ} = Re \frac{2 i}{a_{0}^{2}} [\int d x γ (x) B (x)] (\frac{Q_{+ σ}}{ω_{+ σ}^{2}} + \frac{Q_{- σ}}{ω_{- σ}^{2}}) {[\int d x γ (x) B (x)]}^{T}$
${(ω 1 - M_{\pm})}^{- 1} = \sum_{σ} \frac{P_{\pm σ}}{ω - ω_{\pm σ}}$	$T^{σ} = - Im \frac{2 i}{a_{0}^{2}} [\int d x γ (x) B (x)] (\frac{Q_{+ σ}}{ω_{+ σ}^{2}} - \frac{Q_{- σ}}{ω_{- σ}^{2}}) {[\int d x γ (x) B (x)]}^{T}$
$ω_{\pm σ} = \pm Ω_{σ} - i Γ_{σ}$	$U^{σ} = Re (\frac{2 i}{a_{0}^{2}} ω_{+ σ}^{2} Q_{+ σ} + ω_{- σ}^{2} Q_{- σ})$
$Γ_{μ ν} \equiv \frac{2 {[R_{0}]}_{μ μ} δ_{μ ν}}{a_{μ 0} a_{ν 0}} + \frac{2 {(B A^{- 1} R_{0} A^{† - 1} B^{†})}_{μ ν}}{a_{μ 0} a_{ν 0}}$	$V^{σ} = - Im \frac{2 i}{a_{0}^{2}} (ω_{+ σ}^{2} Q_{+ σ} - ω_{- σ}^{2} Q_{- σ})$
$Γ_{SB}^{μ ν σ} \equiv \frac{Γ_{μ ν}}{2} + Γ_{σ}$	$X^{σ} = [\int d x γ {(x)}^{2} B (x)] \frac{Q_{+ σ}}{ω_{+ σ}} + \frac{Q_{- σ}}{ω_{- σ}}$
	$Y^{σ} = 2 i [\int d x γ {(x)}^{2} B (x)] \frac{Q_{+ σ}}{ω_{+ σ}} - \frac{Q_{- σ}}{ω_{- σ}}$

Quantity	Symbol	Definition
SALT permittivity	$ε (x, ω)$	$ε_{c} (x, ω) + \frac{γ_{⊥} D_{0} F (x)}{ω - ω_{a} + i γ_{⊥}} {[1 + \sum_{μ} \frac{γ_{⊥}^{2}}{{(ω_{μ} - ω_{a})}^{2} + γ_{⊥}^{2}} {\| a_{μ 0} \|}^{2} {\| E_{μ} \|}^{2}]}^{- 1}$
Nonlinear restoring force	$c_{μ ν} (x)$	$\frac{- i ω_{μ}^{2} \frac{\partial ε (x, ω_{μ})}{\partial {\| a_{ν 0} \|}^{2}} E_{μ}^{2} (x)}{\int d x \frac{d}{d ω} [ω^{2} ε (x, ω)] \|_{ω_{μ}} E_{μ}^{2} (x)}$
Dressed decay rate	$γ (x)$	$γ_{∥} (1 + \sum_{μ} \frac{γ_{⊥}^{2}}{{(ω_{μ} - ω_{a})}^{2} + γ_{⊥}^{2}} {\| a_{μ 0} \|}^{2} {\| E_{μ} (x) \|}^{2})$ .
Noise amplitude	$R_{μ ν} (ω)$	$2 ℏ ω_{μ}^{4} \frac{\int d x {\| E_{μ} (x) \|}^{2} Im ε (x, ω) \coth (\frac{ℏ ω_{μ} β (x, ω)}{2})}{{\| \int d x \frac{d}{d ω} [ω^{2} ε (x, ω)] \|}_{ω_{μ}} E_{μ}^{2} (x) \|^{2}} \cdot δ_{μ ν}$

Quantity	Symbol	Definition
Phase diffusion coefficient	$Γ_{0} (ω)$	$R_{μ μ} (ω) / 2 a_{μ 0}^{2}$
RO frequency	$Ω$	${[2 a_{0}^{2} \int d x Re c_{μ μ} (x) γ (x)]}^{1 / 2}$
RO decay rate	$Γ$	$\int d x γ (x) / 2$
Sideband linewidth	$Γ_{SB}$	$Γ_{0} (α_{1}^{2} + 1) + Γ$
Linewidth enhancement	$α_{1}$	$\frac{\int d x Im [c_{μ μ} (x)]}{\int d x Re [c_{μ μ} (x)]}$
Sideband power fraction	$α_{2}$	$\frac{\int d x γ (x) Im [c_{μ μ} (x)]}{\int d x γ (x) Re [c_{μ μ} (x)]}$
Asymmetry factor	$α_{3}$	$\frac{\int d x γ {(x)}^{2} Im [c_{μ μ} (x)]}{[\int d x γ (x) Re [c_{μ μ} (x)]] [\int d x γ (x)]}$

Abstract

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Equations (133)

Journal of the Optical Society of America B