Researching | Universal single-mode lasing in fully chaotic two-dimensional microcavity lasers under continuous-wave operation with large pumping power [Invited]

Journals >Photonics Research >Volume 5 >Issue 6 >Page B39 > Article

Photonics Research
Vol. 5, Issue 6, B39 (2017)

Universal single-mode lasing in fully chaotic two-dimensional microcavity lasers under continuous-wave operation with large pumping power [Invited]

Takahisa Harayama^1、*, Satoshi Sunada², and Susumu Shinohara¹

Author Affiliations

¹Department of Applied Physics, School of Advanced Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan

²Faculty of Mechanical Engineering, Institute of Science and Engineering, Kanazawa University, Kakuma-machi, Kanazawa, Ishikawa 920-1192, Japan

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DOI: 10.1364/PRJ.5.000B39 Cite this Article Set citation alerts

Takahisa Harayama, Satoshi Sunada, Susumu Shinohara. Universal single-mode lasing in fully chaotic two-dimensional microcavity lasers under continuous-wave operation with large pumping power [Invited][J]. Photonics Research, 2017, 5(6): B39 Copy Citation Text

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Abstract

For a fully chaotic two-dimensional (2D) microcavity laser, we present a theory that guarantees both the existence of a stable single-mode lasing state and the nonexistence of a stable multimode lasing state, under the assumptions that the cavity size is much larger than the wavelength and the external pumping power is sufficiently large. It is theoretically shown that these universal spectral characteristics arise from the synergistic effect of two different kinds of nonlinearities: deformation of the cavity shape and mode interaction due to a lasing medium. Our theory is based on the linear stability analysis of stationary states for the Maxwell–Bloch equations and accounts for single-mode lasing phenomena observed in real and numerical experiments of fully chaotic 2D microcavity lasers.

Keywords

(000.1600) Classical and quantum physics.(140.3410) Laser resonators (270.3430) Laser theory (140.3945) Microcavities

\frac{\partial}{\partial t} \tilde{E} = \frac{i}{2} (\nabla_{x y}^{2} + 1) \tilde{E} - α_{L} \tilde{E} + \frac{2 π N κ ℏ}{ϵ} \tilde{ρ},

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\frac{n_{out}^{2}}{n_{in}^{2}} \frac{\partial}{\partial t} \tilde{E} = \frac{i}{2} (\nabla_{x y}^{2} + \frac{n_{out}^{2}}{n_{in}^{2}}) \tilde{E},

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\frac{\partial}{\partial t} \tilde{ρ} = - {\tilde{γ}}_{⊥} \tilde{ρ} - i Δ_{0} \tilde{ρ} + \tilde{κ} W \tilde{E},

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\frac{\partial}{\partial t} W = - {\tilde{γ}}_{∥} (W - W_{\infty}) - 2 \tilde{κ} (\tilde{E} {\tilde{ρ}}^{*} + {\tilde{E}}^{*} \tilde{ρ}),

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\tilde{E} = \sum_{i} E_{i} (t) e^{- i Δ_{i} t} U_{i} (x, y),

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\tilde{ρ} = \sum_{i} ρ_{i} (t) e^{- i Δ_{i} t} V_{i} (x, y),

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\frac{d E_{i} (t)}{d t} + \sum_{j \neq i} \frac{d E_{j} (t)}{d t} e^{- i Δ_{i j} t} U_{i j} = [i (Δ_{i} + \frac{1}{2}) - (α_{L} + {\tilde{γ}}_{i i})] E_{i} (t) + \sum_{j \neq i} {[i (Δ_{j} + \frac{1}{2}) - α_{L}] U_{i j} - {\tilde{γ}}_{i j}} E_{j} (t) e^{- i Δ_{i j} t} + \frac{2 π N κ ℏ}{ε} \sum_{j} e^{- i Δ_{i j} t} ρ_{j} (t) \int_{D} U_{i}^{*} (x, y) V_{j} (x, y) d x d y,

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{\tilde{γ}}_{i j} \equiv - \frac{i}{2} \int_{D} d x d y U_{i}^{*} (x, y) \nabla^{2} U_{j} (x, y) .

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ρ_{j} (t) V_{j} (x, y) = \frac{\tilde{κ} W}{{\tilde{γ}}_{⊥} - i Δ_{0 j}} E_{j} (t) U_{j} (x, y) .

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W = W_{\infty} / {1 + [\sum_{i} \sum_{j} \frac{2 {\tilde{κ}}^{2} E_{i} E_{j}^{*} U_{i} U_{j}^{*} e^{- i Δ_{j i} t}}{({\tilde{γ}}_{⊥} + i Δ_{0 j}) ({\tilde{γ}}_{∥} - i Δ_{j i})} + c.c.]} .

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L (x, y) \equiv 1 + \sum_{m} a_{m} {| U_{m} |}^{2},

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C (x, y) \equiv \sum_{\binom{l, j}{l \neq j}} \frac{2 {\tilde{κ}}^{2} E_{l} E_{j}^{*} U_{l} U_{j}^{*} e^{- i Δ_{j l} t}}{({\tilde{γ}}_{⊥} + i Δ_{0 j}) ({\tilde{γ}}_{∥} - i Δ_{j l})} + c.c .

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\frac{d E_{i} (t)}{d t} + \sum_{j \neq i} \frac{d E_{j} (t)}{d t} e^{- i Δ_{i j} t} U_{i j} = [i (Δ_{i} + \frac{1}{2}) - (α_{L} + {\tilde{γ}}_{i i})] E_{i} (t) \sum_{j \neq i} {[i (Δ_{j} + \frac{1}{2}) - α_{L}] U_{i j} - {\tilde{γ}}_{i j}} E_{j} (t) e^{- i Δ_{i j} t} + ξ W_{\infty} \sum_{k} \frac{e^{- i Δ_{i k} t} E_{k}}{{\tilde{γ}}_{⊥} - i Δ_{0 k}} \times \int_{D} d x d y \frac{U_{i}^{*} U_{k}}{L (x, y)} {1 - \frac{C (x, y)}{L (x, y)} + {[\frac{C (x, y)}{L (x, y)}]}^{2} - \dots},

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\frac{d E_{i}}{d t} ≃ [i (Δ_{i} + \frac{1}{2}) - (α_{L} + {\tilde{γ}}_{i i})] E_{i} + \frac{ξ W_{\infty} E_{i}}{{\tilde{γ}}_{⊥} - i Δ_{0 i}} \int_{D} d x d y \frac{{| U_{i} |}^{2}}{L (x, y)} - ξ W_{\infty} E_{i} \int_{D} d x d y \frac{{| U_{i} |}^{2}}{{[L (x, y)]}^{2}} \sum_{\binom{k}{k \neq i}} \frac{2 {\tilde{κ}}^{2} {| E_{k} |}^{2} {| U_{k} |}^{2}}{({\tilde{γ}}_{⊥} - i Δ_{0 k}) ({\tilde{γ}}_{∥} - i Δ_{k i})} \times (\frac{1}{{\tilde{γ}}_{⊥} + i Δ_{0 k}} + \frac{1}{{\tilde{γ}}_{⊥} - i Δ_{0 i}}) .

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\frac{d {| E_{i} |}^{2}}{d t} = \frac{d}{d t} (E_{i} E_{i}^{*}) = E_{i}^{*} \frac{d E_{i}}{d t} + E_{i} \frac{d E_{i}^{*}}{d t} = (- 2 α_{L} + {\tilde{γ}}_{i i} + {\tilde{γ}}_{i i}^{*}) {| E_{i} |}^{2} + 2 ξ W_{\infty} g (Δ_{i}) \int_{D} d x d y {[L (x, y)]}^{- 2} \times {| E_{i} |}^{2} {| U_{i} |}^{2} {L (x, y) - \sum_{k, k \neq i} 2 {\tilde{κ}}^{2} g (Δ_{k}) g_{∥} (Δ_{i} - Δ_{k}) \times [2 {\tilde{γ}}_{⊥} + (Δ_{i} - Δ_{0}) (Δ_{i} - Δ_{k}) / {\tilde{γ}}_{⊥} + (Δ_{i} - Δ_{k}) (Δ_{i} + Δ_{k} - 2 Δ_{0}) / {\tilde{γ}}_{∥}] {| E_{k} |}^{2} {| U_{k} |}^{2}},

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L (x, y) ≃ 1 + \sum_{k, | Δ_{k} - Δ_{i} | ≪ {\tilde{γ}}_{∥}} \frac{4 {\tilde{κ}}^{2}}{{\tilde{γ}}_{∥}} g (Δ_{k}) {| E_{k} |}^{2} {| U_{k} |}^{2} .

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2 {\tilde{γ}}_{⊥} ≫ (Δ_{i} - Δ_{0}) (Δ_{i} - Δ_{k}) / {\tilde{γ}}_{⊥} + (Δ_{i} - Δ_{k}) (Δ_{i} + Δ_{k} - 2 Δ_{0}) / {\tilde{γ}}_{∥} .

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\frac{d {| E_{i} |}^{2}}{d t} = (- 2 α_{L} + {\tilde{γ}}_{i i} + {\tilde{γ}}_{i i}^{*}) {| E_{i} |}^{2} + 2 ξ W_{\infty} g (Δ_{i}) \int_{D} d x d y {[L (x, y)]}^{- 2} {| E_{i} |}^{2} {| U_{i} |}^{2} \times [1 + \sum_{k, | Δ_{k} - Δ_{i} | ≪ {\tilde{γ}}_{∥}} \frac{4 {\tilde{κ}}^{2}}{{\tilde{γ}}_{∥}} g (Δ_{k}) {| E_{k} |}^{2} {| U_{k} |}^{2} - \sum_{k, k \neq i, | Δ_{k} - Δ_{i} | ≪ {\tilde{γ}}_{∥}} \frac{4 {\tilde{κ}}^{2}}{{\tilde{γ}}_{∥}} g (Δ_{k}) {| E_{k} |}^{2} {| U_{k} |}^{2}] = (- 2 α_{L} + {\tilde{γ}}_{i i} + {\tilde{γ}}_{i i}^{*}) {| E_{i} |}^{2} + 2 ξ W_{\infty} g (Δ_{i}) \int_{D} d x d y {[L (x, y)]}^{- 2} {| E_{i} |}^{2} {| U_{i} |}^{2} \times [1 + \frac{4 {\tilde{κ}}^{2}}{{\tilde{γ}}_{∥}} g (Δ_{i}) {| E_{i} |}^{2} {| U_{i} |}^{2}] .

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\frac{d I_{i}}{d t} ≃ S_{i} I_{i},

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S_{i} \equiv - 2 (α_{L} + γ_{i}) + 2 ξ W_{\infty} g (Δ_{i}) \int_{D} d x d y \frac{{| U_{i} |}^{2} L_{i} (x, y)}{{[L (x, y)]}^{2}},

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γ_{i} \equiv - \frac{i}{4} \oint_{\partial D} d s (U_{i}^{*} \frac{\partial U_{i}}{\partial n} - U_{i} \frac{\partial U_{i}^{*}}{\partial n}),

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M_{i j} \equiv {S_{j} + 2 ξ W_{\infty} b_{j}^{2} \int_{D} d x d y \frac{{| U_{j} |}^{4}}{{[L (x, y)]}^{2}}} δ_{i j} - 4 ξ W_{\infty} b_{i} b_{j} \int_{D} d x d y \frac{{| U_{i} |}^{2} {| U_{j} |}^{2} L_{i} (x, y)}{{[L (x, y)]}^{3}},

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{\tilde{M}}_{j j} = - 2 ξ W_{\infty} b_{j}^{2} \int_{D} d x d y \frac{{| U_{j} |}^{4}}{L_{s, j}^{2}} < 0,

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{\tilde{M}}_{i i} = 2 ξ W_{\infty} g (Δ_{i}) \int_{D} d x d y {| U_{i} |}^{2} \frac{L_{s, i} - L_{s, j}^{2}}{L_{s, j}^{2} L_{s, i}},

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M_{i i} = 2 ξ W_{\infty} b_{i}^{2} \int_{D} d x d y \frac{{| U_{i} |}^{4} L_{i} (x, y)}{{[L (x, y)]}^{3}} [\frac{L (x, y)}{L_{i} (x, y)} - 2],

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M_{i j} = - 4 ξ W_{\infty} b_{i} b_{j} \int_{D} d x d y \frac{{| U_{i} |}^{2} {| U_{j} |}^{2} L_{i} (x, y)}{{[L (x, y)]}^{3}} .

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\int_{D} d x d y ε_{i} (x, y) {| U_{0} (x, y) |}^{2} = 0,

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{\tilde{M}}_{i i} = 2 ξ W_{\infty} g (Δ_{i}) \int_{S} d x d y (1 + ε_{i}) {| U_{0} |}^{2} \times \frac{a_{s, i} (1 + ε_{i}) - a_{s, j}^{2} {(1 + ε_{j})}^{2} {| U_{0} |}^{2}}{a_{s, j}^{2} {(1 + ε_{j})}^{2} (1 + ε_{i}) a_{s, i} {| U_{0} |}^{6}} {| U_{0} |}^{2} = \frac{2 ξ W_{\infty} g (Δ_{i})}{a_{s, j}^{2} a_{s, i}} \int_{S} d x d y {[1 + O (ε_{i} (x, y) {| U_{0} (x, y) |}^{2})] \frac{a_{s, i}}{{| U_{0} |}^{2}} - [1 + O (ε_{j} (x, y) {| U_{0} (x, y) |}^{2})] a_{s, j}^{2}} ≃ \frac{2 ξ W_{\infty} g (Δ_{i})}{a_{s, j}^{2} a_{s, i}} (a_{s, i} \int_{S} d x d y \frac{1}{{| U_{0} |}^{2}} - A a_{s, j}^{2}),

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{(\frac{a_{s, j}}{A})}^{2} > \frac{a_{s, i}}{A} \int_{S} d x d y \frac{1}{A^{2} {| U_{0} |}^{2}} .

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L_{i} (x, y) ≃ (1 + ε_{i}) a_{i} {| U_{0} |}^{2},

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L (x, y) ≃ \sum_{m = 1}^{N} (1 + ε_{m}) a_{m} {| U_{0} |}^{2}

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M_{i i} ≃ - 4 ξ A W_{\infty} b_{i}^{2} \frac{a_{i}}{a_{tot}^{3}} (1 - \frac{a_{tot}}{2 a_{i}}),

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M_{i j} ≃ - 4 ξ A W_{\infty} b_{i} b_{j} \frac{a_{i}}{a_{tot}^{3}},

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M^{'} (N) \equiv - \frac{4 ξ A W_{\infty}}{a_{tot}^{3}} diag (\sqrt{a_{1}}, \dots, \sqrt{a_{N}}) \times B R B diag (\sqrt{a_{1}}, \dots, \sqrt{a_{N}}) .

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| M^{'} (k) | = {(\frac{2 ξ A W_{\infty}}{a_{t o t}^{2}})}^{k} (1 - \sum_{i = 1}^{k} \frac{2 a_{i}}{a_{tot}}) \prod_{i = 1}^{k} b_{i}^{2} .

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Takahisa Harayama, Satoshi Sunada, Susumu Shinohara. Universal single-mode lasing in fully chaotic two-dimensional microcavity lasers under continuous-wave operation with large pumping power [Invited][J]. Photonics Research, 2017, 5(6): B39

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