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Photonics Research
Vol. 5, Issue 6, B1 (2017)

Unidirectional lasing in semiconductor microring lasers at an exceptional point [Invited]

Stefano Longhi^1、* and Liang Feng²

Author Affiliations

¹Dipartimento di Fisica, Politecnico di Milano and Istituto di Fotonica e Nanotecnologie del Consiglio Nazionale delle Ricerche, Piazza L. da Vinci 32, I-20133 Milano, Italy

²Department of Electrical Engineering, The State University of New York at Buffalo, Buffalo, New York 14260, USA

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DOI: 10.1364/PRJ.5.0000B1 Cite this Article Set citation alerts

Stefano Longhi, Liang Feng. Unidirectional lasing in semiconductor microring lasers at an exceptional point [Invited][J]. Photonics Research, 2017, 5(6): B1 Copy Citation Text

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Abstract

Recent experiments demonstrated that chiral symmetry breaking at an exceptional point (EP) is a viable route to achieve unidirectional laser emission in microring lasers. By a detailed semiconductor laser rate equation model, we show here that unidirectional laser emission at an EP is a robust regime. Slight deviations from the EP condition can break preferential unidirectional lasing near threshold via a Hopf instability. However, above a “second” laser threshold, unidirectional emission is restored.

Keywords

(140.3560) Lasers (140.3945) Microcavities.ring (270.3430) Laser theory

\frac{d E_{1}}{d t} = κ (1 + i α) [N (1 - s {| E_{1} |}^{2} - c {| E_{2} |}^{2}) - 1] E_{1} + i κ_{1} E_{2},

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\frac{d E_{2}}{d t} = κ (1 + i α) [N (1 - s {| E_{2} |}^{2} - c {| E_{1} |}^{2}) - 1] E_{2} + i κ_{2} \exp (i Θ) E_{1},

(2)

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\frac{d N}{d t} = γ [μ - N - N (1 - s {| E_{1} |}^{2} - c {| E_{2} |}^{2}) {| E_{1} |}^{2} - N (1 - s {| E_{2} |}^{2} - c {| E_{1} |}^{2}) {| E_{2} |}^{2}],

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E_{1} = \sqrt{μ - 1} \cos (\frac{θ}{2} + \frac{π}{4}) \exp (i ϕ_{1}),

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E_{2} = \sqrt{μ - 1} \sin (\frac{θ}{2} + \frac{π}{4}) \exp (i ϕ_{2}),

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\frac{d θ}{d t} = J κ_{1} \sin θ \cos θ + (1 - \sin θ) [κ_{2} \sin (ψ - Θ) - κ_{1} \sin ψ] + 2 κ_{1} \sin ψ,

(6)

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\frac{d ψ}{d t} = α J κ_{1} \sin θ + κ_{2} cotg (\frac{θ + π / 2}{2}) \cos (ψ - Θ) - κ_{1} tg (\frac{θ + π / 2}{2}) \cos ψ,

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J \equiv \frac{κ}{κ_{1}} c (μ - 1) (1 - \frac{s}{c})

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\frac{d θ}{d t} = J κ_{1} \sin θ \cos θ + κ_{1} (1 + \sin θ) \sin ψ,

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\frac{d ψ}{d t} = α J κ_{1} \sin θ - κ_{1} tg (\frac{θ + π / 2}{2}) \cos ψ .

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θ (t) = - π / 2, ψ (t) = - α κ_{1} J t .

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\frac{d δ ψ (t)}{d t} = - \frac{κ_{1}}{2} δ θ (t) \cos (α κ_{1} t),

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\frac{d δ θ (t)}{d t} = - κ_{1} J δ θ,

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J_{1} ≃ \frac{3.33}{\sqrt{1 + α^{2}}} .

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D = \frac{{| E_{1} |}^{2} - {| E_{2} |}^{2}}{{| E_{1} |}^{2} + {| E_{2} |}^{2}},

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D = - \sin θ .

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tg ψ_{0} = \frac{\cos θ_{0} [κ_{2} cotg (\frac{θ_{0}}{2} + \frac{π}{4}) - κ_{1} tg (\frac{θ_{0}}{2} + \frac{π}{4})]}{2 α κ_{1} - α (κ_{1} - κ_{2}) (1 - \sin θ_{0})},

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J = \frac{(κ_{1} - κ_{2}) (1 - \sin θ_{0}) \sin ψ_{0} - 2 κ_{1} \sin ψ_{0}}{\sin θ_{0} \cos θ_{0}} .

(18)

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\frac{d}{d t} (\begin{matrix} δ θ \\ δ ψ \end{matrix}) = (\begin{matrix} M_{11} & M_{12} \\ M_{21} & M_{22} \end{matrix}) (\begin{matrix} δ θ \\ δ ψ \end{matrix}),

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M_{11} = J \cos (2 θ_{0}) - (κ_{2} - κ_{1}) \cos θ_{0} \sin ψ_{0},

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M_{12} = (κ_{2} - κ_{1}) (1 - \sin θ_{0}) \cos ψ_{0} + 2 κ_{1} \cos ψ_{0},

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M_{21} = α J \cos θ_{0} - \frac{κ_{2} \cos ψ_{0}}{2 \sin^{2} (\frac{θ_{0}}{2} + \frac{π}{4})} - \frac{κ_{1} \cos ψ_{0}}{2 \cos^{2} (\frac{θ_{0}}{2} + \frac{π}{4})},

(22)

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M_{22} = κ_{1} \sin ψ_{0} tg (\frac{θ_{0}}{2} + \frac{π}{4}) - κ_{2} \sin ψ_{0} cotg (\frac{θ_{0}}{2} + \frac{π}{4}) .

(23)

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ω_{Hopf} = 2 \sqrt{κ_{1} κ_{2}} .

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References (27)

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Stefano Longhi, Liang Feng. Unidirectional lasing in semiconductor microring lasers at an exceptional point [Invited][J]. Photonics Research, 2017, 5(6): B1

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