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Journals >Matter and Radiation at Extremes >Volume 6 >Issue 5 >Page 054402 > Article

Matter and Radiation at Extremes
Vol. 6, Issue 5, 054402 (2021)

Generating optical supercontinuum and frequency comb in tenuous plasmas

Kenan Qu^a) and Nathaniel J. Fisch

Author Affiliations

Department of Astrophysical Sciences, Princeton University, Princeton, New Jersey 08544, USA

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DOI: 10.1063/5.0052829 Cite this Article

Kenan Qu, Nathaniel J. Fisch. Generating optical supercontinuum and frequency comb in tenuous plasmas[J]. Matter and Radiation at Extremes, 2021, 6(5): 054402 Copy Citation Text

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Abstract

There are several mechanisms by which the frequency spectrum of a laser broadens when it propagates at near-relativistic intensity in a tenuous plasma. Focusing on one-dimensional effects, we identify two strong optical nonlinearities, namely, four-wave mixing (FWM) and forward Raman scattering (FRS), for creating octave-wide spectra. FWM dominates the interaction when the laser pulse is short and intense, and its combination with phase modulation produces a symmetrically broadened supercontinuum. FRS dominates when the laser pulse is long and relatively weak, and it broadens the laser spectrum mainly toward lower frequencies and produces a frequency comb. The frequency chirping combined with group velocity dispersion compresses the laser pulse, causing higher peak intensity.

(\partial_{t t} - c^{2} \partial_{z z}) a = \frac{ω_{p}^{2}}{γ} \frac{n}{\bar{n}} a ≅ ω_{p}^{2} (1 + \tilde{n} - \frac{a^{2}}{2}) a,

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(\partial_{t t} + 2 ν ω_{p} + ω_{p}^{2}) \tilde{n} = \frac{c^{2}}{2} \partial_{z z} a^{2},

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a = \sum_{j} a_{j} e^{- i ω_{j} t + i k_{j} z} + c . c .,

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\tilde{n} = δ n e^{- i ω_{p} t + i k_{p} z} + c . c .,

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\begin{align} (\partial_{t} + v_{g, j} \partial_{z}) a_{j} = - \frac{ω_{p}^{2}}{2 i ω_{j}} (δ n^{*} a_{j + 1} + δ n a_{j - 1}) + \frac{3 ω_{p}^{2}}{4 i ω_{j}} \\ \times [\sum_{k} | a_{k} |^{2} a_{j} + \sum_{m, n}^{m \neq n} a_{m}^{*} a_{n} a_{m - n + j} + T S F G], \end{align}

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(\partial_{t} + i ν) δ n = - \frac{ω_{p}}{2 i} \sum_{j} a_{j + 1} a_{j}^{*},

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\begin{align} [\partial_{τ} + (\frac{c}{v_{g, j}} - 1) \partial_{ζ}] a_{j} = - \frac{ω_{p}^{2}}{2 i c k_{j}} (δ n^{*} a_{j + 1} + δ n a_{j - 1}) \\ + \frac{3 ω_{p}^{2}}{4 i c k_{j}} \sum_{l} χ_{l} a_{j - l}, \end{align}

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\begin{align} (\partial_{ζ} + i ν) δ n = & - \frac{ω_{p}}{2 i} \sum_{j} a_{j + 1} a_{j}^{*}, \end{align}

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\begin{align} χ_{l} = & \sum_{j} a_{j + l} a_{j}^{*} . \end{align}

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ω_{p} τ ≳ \frac{ω_{j T}}{χ_{0 M}} .

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\partial_{τ} \sum_{j} ω_{j} | a_{j} |^{2} = \partial_{ζ} \sum_{j} (1 - \frac{c}{v_{g, j}}) ω_{j} | a_{j} |^{2},

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\partial_{τ} \sum_{j} ω_{j}^{2} | a_{j} |^{2} = \partial_{ζ} \sum_{j} (1 - \frac{c}{v_{g, j}}) ω_{j}^{2} | a_{j} |^{2} - ω_{p}^{2} \partial_{ζ} | δ n |^{2} .

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\begin{align} (\partial_{ζ} - i ν) \partial_{τ} δ n = \partial_{ζ} \sum_{j} (2 - \frac{c}{v_{g, j}} - \frac{c}{v_{g, j + 1}}) a_{j}^{*} a_{j + 1} \\ + \frac{ω_{p}^{3}}{4} \sum_{j} (\frac{1}{ω_{j - 1}} - \frac{1}{ω_{j + 1}}) | a_{j} |^{2} δ n . \end{align}

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\begin{align} \partial_{ζ τ} δ n & = \frac{ω_{p}^{3}}{4} \sum_{j} (ω_{j - 1}^{- 1} - ω_{j + 1}^{- 1}) | a_{j} |^{2} δ n \\ \approx \sum_{j} \frac{ω_{p}^{4}}{ω_{j}^{2}} \frac{| a_{j} |^{2} δ n}{2} . \end{align}

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\partial_{t t} δ n \approx \partial_{ζ τ} δ n = \frac{ω_{p}^{3}}{ω_{0} - ω_{p}} \frac{| a_{0} |^{2} δ n}{4} .

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Γ_{FRS, 3} = {(\frac{ω_{p}^{3}}{ω_{0} - ω_{p}})}^{1 / 2} \frac{a_{0}}{2} .

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\begin{align} [\partial_{τ} + (\frac{c}{v_{g, j}} - 1) \partial_{ζ}] (ω_{j} | a_{j} |^{2}) = Re [ω_{p}^{3} (a_{j + 1}^{*} δ n - a_{j - 1}^{*} δ n^{*}) a_{j}] \\ - Im [3 ω_{p}^{2} \sum_{k} a_{j} a_{j + k}^{*} χ_{k}], \end{align}

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\partial_{τ} \sum_{k} | χ_{k} |^{2} = \sum_{j, k} χ_{k} (1 - \frac{c}{v_{g, j}}) \partial_{ζ} (a_{j} a_{j + k}^{*} + a_{j - k} a_{j}^{*}),

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\begin{align} \partial_{τ} (χ_{k} \pm χ_{- k}) = \sum_{j} (1 - \frac{c}{v_{g, j}}) \partial_{ζ} [a_{j} (a_{j + k}^{*} \pm a_{j - k}^{*})] \\ + \frac{3 ω_{p}^{2}}{4 i} \sum_{j, k^{'}} (\frac{1}{ω_{j}} - \frac{1}{ω_{j - k - k^{'}}}) χ_{k^{'}} a_{j - k^{'}} a_{j - k}^{*} \\ \pm \frac{3 ω_{p}^{2}}{4 i} \sum_{j, k^{'}} (\frac{1}{ω_{j}} - \frac{1}{ω_{j + k - k^{'}}}) χ_{k^{'}} a_{j - k^{'}} a_{j + k}^{*} . \end{align}

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\begin{align} a_{j} (τ, ζ) = \sum_{o} a_{o} (0, ζ) e^{i (j - o) (\frac{π}{2} - ψ)} \\ \times \exp (\frac{3 i ω_{p}^{2} χ_{0} τ}{4 ω_{j}}) J_{j - o} (\frac{3 ω_{p}^{2} | χ_{1} |}{2 ω_{o}} τ), \end{align}

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ω_{j} | a_{j} (τ, ζ) |^{2} ≅ ω_{j} \sum_{o} | a_{o} (0, ζ) |^{2} J_{j - o}^{2} (\frac{3 ω_{p}^{2} | χ_{1} |}{2 ω_{o}} τ) .

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Δ ω_{FWM} \sim \frac{3 ω_{p}^{2} | χ_{1} |}{ω_{o}} τ = \frac{3 ω_{p}^{2} | a_{o} a_{o - 1}^{*} |}{ω_{o}} τ .

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a_{j} (τ, ζ) ≃ \sum_{o} a_{o} (0, ζ) {(- 1)}^{j - o} J_{j - o} (\frac{ω_{p}^{2} | δ n |}{ω_{o}} τ),

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ω_{j} | a_{j} (τ, ζ) |^{2} ≃ ω_{j} {[\sum_{o} {(\mp 1)}^{o} | a_{o} (0, ζ) | J_{j - o} (\frac{ω_{p}^{2} | δ n |}{ω_{o}} τ)]}^{2},

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Δ ω_{FRS} \sim \frac{ω_{p}^{2} | δ n |}{ω_{o}} τ = \frac{ω_{p}^{3} τ}{2 ω_{o}} |\int_{0}^{ζ} a_{o} a_{o - 1}^{*} e^{- ν ζ^{'}} d ζ^{'}| .

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\begin{align} a = 2 Re \sum_{o, j} a_{o} (0, ζ) i^{j - o} \exp (\frac{3 i ω_{p}^{2} χ_{0} τ}{4 ω_{j}}) J_{j - o} (\frac{3 ω_{p}^{2} | χ_{1} |}{2 ω_{o}} τ) e^{- i ω_{j} ζ} \\ \approx 2 Re \sum_{o} a_{o} (0, ζ) \exp [\frac{3 i ω_{p}^{2} χ_{0} τ}{4 ω_{o}} + i \frac{3 ω_{p}^{2} | χ_{1} |}{2 ω_{o}} τ \cos (ω_{p} ζ) + i ω_{o} ζ] \\ = \sum_{o} a_{o} (0, ζ) \cos [\frac{3 ω_{p}^{2} χ_{0}}{4 ω_{j}} τ + \frac{3 ω_{p}^{2} | χ_{1} |}{2 ω_{o}} τ \cos (ω_{p} ζ) + ω_{o} ζ] . \end{align}

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\begin{align} a ≃ 2 Re \sum_{o, j} a_{o} (0, ζ) {(- 1)}^{j - o} J_{j - o} (\frac{ω_{p}^{2} | δ n |}{ω_{o}} τ) e^{- i ω_{j} ζ} \\ \approx 2 Re \sum_{o} a_{o} (0, ζ) \exp [i (\frac{ω_{p}^{2} | δ n |}{ω_{o}} τ) \sin (ω_{p} ζ) + ω_{o} ζ] \\ = \sum_{o} a_{o} (0, ζ) \cos [(\frac{ω_{p}^{2} | δ n |}{ω_{o}} τ) \sin (ω_{p} ζ) + ω_{o} ζ] . \end{align}

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Kenan Qu, Nathaniel J. Fisch. Generating optical supercontinuum and frequency comb in tenuous plasmas[J]. Matter and Radiation at Extremes, 2021, 6(5): 054402

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