Second and third order susceptibilities mixing for supercontinuum generation and shaping

Baronio, Fabio; Conforti, Matteo; Angelis, C. De; Modotto, D.; Wabnitz, Stefan; Andreana, Marco; Tonello, Alessandro; Leproux, Philippe; Couderc, Vincent

doi:10.1016/j.yofte.2012.07.001

Cited by 19 publications

(16 citation statements)

References 38 publications

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“…We model the dynamics of optical frequency comb generation by means of an Ikeda-like map [29] involving the SEE for describing the evolution of the broadband envelope [23][24][25][26][27] A ffiffiffiffiffi W p ∕m of the real electric field E within a waveguide with both quadratic and cubic nonlinearities [i.e., the total nonlinear polarization is P NL P 2 NL P 3 NL ε 0 χ 2 E 2 χ 3 E 3 , where χ 2 and χ 3 are the quadratic and cubic nonlinear susceptibilities, respectively, and ε 0 is the vacuum permittivity]. The map is constructed by combining the SEE with boundary conditions that relate the fields between successive round trips and the input pump field [30,31], viz.,…”

Section: Modelmentioning

confidence: 99%

“…In Eq. (2), the nonlinear polarization p NL is given by the sum of the quadratic and cubic contributions [24][25][26][27] p 2 NL ε 0 χ 2 2 2jAj 2 expiψt; z A 2 exp−iψt; z; (4)…”

Section: Modelmentioning

confidence: 99%

“…In this work, we present a more general model of ultrabroadband frequency comb generation in cavities where several nonlinearities may act simultaneously, based on the single envelope equation (SEE) [23][24][25][26][27][28]. Compared to previous models where a single dominant nonlinear process (e.g., singly or doubly resonant cavity SHG) is involved [21,22], the SEE model has its relative strength in its greater generality, as all nonlinear processes are simultaneously included.…”

Section: Introductionmentioning

confidence: 99%

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Single envelope equation modeling of multi-octave comb arrays in microresonators with quadratic and cubic nonlinearities

et al. 2016

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We numerically study, by means of the single envelope equation, the generation of optical frequency combs\ud ranging from the visible to the mid-infrared spectral regions in resonators with quadratic and cubic nonlinearities.\ud Phase-matched quadratic wave-mixing processes among the comb lines can be activated by low-power continuous\ud wave pumping in the near infrared of a radially poled lithium niobate whispering gallery resonator. We examine\ud both separate and coexisting intracavity doubly resonant second-harmonic generation and parametric oscillation\ud processes and find that modulation instabilities may lead to the formation of coupled comb arrays extending over\ud multiple octaves. In the temporal domain, the frequency combs may correspond to pulse trains or even to a single\ud pulse circulating in the cavity

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Section: Modelmentioning

confidence: 99%

“…In Eq. (2), the nonlinear polarization p NL is given by the sum of the quadratic and cubic contributions [24][25][26][27] p 2 NL ε 0 χ 2 2 2jAj 2 expiψt; z A 2 exp−iψt; z; (4)…”

Section: Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Single envelope equation modeling of multi-octave comb arrays in microresonators with quadratic and cubic nonlinearities

et al. 2016

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“…In this work, we present an ultra-broadband model of OFC generation in nonlinear cavities with both quadratic and cubic nonlinear response, based on the single envelope equation (SEE) [8]. The SEE model permits to describe situations where the different parametric combs start to overlap, or when mixing between the primary combs and higher-order harmonic and phase matching processes occurs, since all processes can be modeled at once with a single equation.…”

Section: Introductionmentioning

confidence: 99%

Single envelope equation modelling of frequency comb generation in quadratic and cubic nonlinear resonators

Hansson

Léo

Erkintalo

et al. 2016

Conference on Lasers and Electro-Optics

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Abstract:We introduce the single envelope equation for the numerical modelling of ultra-broadband optical frequency comb generation via phase-matched parametric mixing in a resonator including both quadratic and cubic nonlinearities. There is currently much interest in developing microscale, low pump power light sources based on nonlinear parametric frequency conversion. Low threshold parametric mixing is enabled by the high field confinement in quasiphase-matched (QPM) whispering gallery modes (WGMs) of optical microresonators [1]. By using radially poled resonators, low power and highly tunable optical parametric oscillation (OPO) has been demonstrated [2,3]. Very interestingly, recent experiments have demonstrated that macro-cavity second-harmonic generation (SHG) may also lead to optical frequency comb (OFC) generation [4,5].Nonlinear microresonator based sources of OFCs are a promising alternative to mode-locked lasers for several applications, ranging from high-precision metrology, coherent communications to biomedical and environmental spectroscopy [6]. We recently developed a narrowband theoretical description of OFC generation by cavity SHG, based on separate time domain evolution equations for coupled OFCs, centered around the fundamental frequency and the second harmonic, respectively [7].In this work, we present an ultra-broadband model of OFC generation in nonlinear cavities with both quadratic and cubic nonlinear response, based on the single envelope equation (SEE) [8]. The SEE model permits to describe situations where the different parametric combs start to overlap, or when mixing between the primary combs and higher-order harmonic and phase matching processes occurs, since all processes can be modeled at once with a single equation. We apply here the SEE approach to OFC generation in a doubly resonant, radially poled lithium niobate microring resonator. OFC dynamics is modeled by combining the SEE with proper boundary conditions [9], viz.where Eq. (1) is written in the Fourier domain, Ω = ω − ω 0 , ω 0 is a reference frequency, andHere the independent variables are the evolution variable z, which is the longitudinal coordinate measured along the waveguide, and t which is the (ordinary) time. Eq. (1) is the boundary condition that determines the intracavity field A m+1 (t, z = 0) at the input of roundtrip m + 1 in terms of the field from the end of the previous roundtrip A m (t, z = L) and the pump field A in . The path length of the resonator is assumed to be equal to L.Additionally,θ (Ω) is the frequency dependent transmission coefficient between the resonator and the bus waveguide, and δ 0 = 2πl − φ 0 ≈ (ω R − ω 0 )t R is the linear phase-shift, with δ 0 the detuning from the cavity resonance ω R (assumed to correspond to the longitudinal mode number l) which is the closest to the pump frequency ω 0 , and t R is the cavity round-trip time. Moreover, ρ 0 = ω 0 /2n 0 cε 0 , ε 0 is the vacuum permittivity, n 0 = n(ω 0 ), ω 0 is the pump frequency, n is the linear refractive index, and the shock ...

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“…In these situation, it is possible to numerically study the dynamics of optical frequency comb generation by means of a map 22 involving the so-called single-envelope equation (SEE) [23][24][25][26][27] for the envelope A m of the real electric field E. We consider here a waveguide with both quadratic and cubic nonlinearity, i.e., the total nonlinear polarization is…”

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confidence: 99%

Nonlinear dynamics of optical frequency combs

et al. 2017

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Second and third order susceptibilities mixing for supercontinuum generation and shaping

Cited by 19 publications

References 38 publications

Single envelope equation modeling of multi-octave comb arrays in microresonators with quadratic and cubic nonlinearities

Single envelope equation modeling of multi-octave comb arrays in microresonators with quadratic and cubic nonlinearities

Single envelope equation modelling of frequency comb generation in quadratic and cubic nonlinear resonators

Nonlinear dynamics of optical frequency combs

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