Raman-Kerr frequency combs in microresonators with normal dispersion

A. V. Cherenkov; N. M. Kondratiev; V. E. Lobanov; A. E. Shitikov; D. V. Skryabin; M. L. Gorodetsky

doi:10.1364/OE.25.031148

1. Introduction

High-Q optical microresonators opened a new pathway to the development of Kerr frequency comb generators with characteristics unattainable in the comb systems based on mode-locked lasers [1]. Microresonator combs are generated by coupling a CW laser light into a resonator that converts the pump frequency into a broadband comb through the cascaded four-wave mixing (FWM) due to Kerr nonlinearity of the medium boosted by the strong light confinement [2]. The spacing between the comb lines corresponding to the free spectral range (FSR) of the resonator is determined by the inverse of the resonator round-trip time and can range from few GHz to THz [3]. The threshold pump power which can be below 1 mW level [4] is determined by the resonator quality-factor Q and mode volume. The microresonator group velocity dispersion (GVD) plays a major role in the comb generation processes since it determines the bandwidth of the four-wave mixing gain [5]. In particular, microresonators with the anomalous GVD have been used to demonstrate octave spanning combs [6,7]. Finally, highly-coherent mode-locked combs based on bright dissipative Kerr solitons [8] promise a plethora of applications. However, most of the dielectric materials used for microresonators have normal GVD in the visible and mid-IR spectral ranges, prohibiting four-wave mixing gain and hence coherent and broadband soliton combs. Comb generation in the normal GVD range, therefore, is governed by other physical mechanisms and it has also been explored actively in recent years [9,10]. In particular, local dispersion perturbation due to linear coupling between various families of the resonators’ modes [11–13] allows excitation of a new type of dissipative soliton structures, the so-called platicons [14–16], associated with normal GVD combs. Platicons represent the bound states of the opposing switching waves [17] connecting the upper and lower states of the bistability loop and satisfying periodic boundary conditions in a microresonator. Interestingly, quite similar structures were discovered for complex Ginzburg-Landau equation, where bound kink–antikink complexes can also form top-flat pulse [18]. Properties of platicons still have not been explored to the same degree of details as the physics of the bright comb solitons. In particular, the influence of the Raman effect on the normal GVD combs and platicons remains largely unknown.

The interplay between the Kerr and Raman effects in microresonators has attracted significant interest in recent years, especially in the context of bright comb solitons in the anomalous GVD regime [19–25]. The above works reported the soliton self-frequency shift [19, 22–24], interplay of the Cherenkov radiation and Raman effects [19] and the generation of the two-color Stokes solitons [25]. Most of these studies are based on the generalized Lugiato-Lefever equation (LLE) [26] incorporating an approximation of the Raman gain by the first derivative of the pulse intensity in the physical space [27–29], corresponding to an unrealistic linear increase of the gain with frequency detuning. In this paper, we derive a model from the first principles, describing optomechanical coupling with Raman mechanical oscillator [19,30] and proceed to solve it with the coupled-mode approach [31], that incorporates both Kerr and Raman nonlinearities. The resulting equations are used to investigate comb generation within the gain line of the Raman scattering and the associated dynamics of platicons. Our present approach appears to be useful for combs with a small number of modes and complex dispersions and also for the interpretation of the interplay between Raman and Kerr effects. We demonstrate that a significant excess of the Raman gain over losses leads to instabilities of platicons appearing as a branching of the parent platicon into a bifurcation tree with newly generated stable and unstable platicons. The associated comb generation appears to be qualitatively different from the one known for bright solitons where the Raman effect mostly results in self-frequency shift [22,23].

2. Theory

A part of the total susceptibility χ of the microresonator’s material, which is induced by the Raman effect, is described by the equation for a classical mechanical oscillator with generalized amplitude q driven by the light intensity, see, e.g., [19, 30]: χ = χ⁽¹⁾ + χ⁽³⁾|E|², where χ⁽¹⁾ includes the linear susceptibility $χ_{0}^{(1)}$ and the amplitude q of the Raman oscillator:

χ^{(1)} = χ_{0}^{(1)} + \frac{\partial χ^{(1)}}{\partial q} q,

\frac{d^{2} q}{d t^{2}} + 2 γ \frac{d q}{d t} + Ω_{0}^{2} q = w F_{q},

where w is the inverse of the oscillator mass, Ω₀ corresponds to a Stokes resonance with a width of 2γ at half-maximum [see Fig. 1], and generalized force F_q corresponding to optical field:

F_{q} = \frac{∊_{0}}{2} \frac{\partial χ^{(1)}}{\partial q} {| \vec{E} |}^{2} V_{eff},

where ∊₀ is the vacuum permittivity and V_eff is the mode volume.

Fig. 1 A schematic representation of Raman scattering in microresonators. The Raman peak covers several free spectral ranges (FSR) in a large diameter microresonator. Ω_R is the detuning of the Raman peak from the pump frequency ( $\frac{Ω_{R}}{2 π} = 9.66 THz$ for CaF₂ and 12.3 THz for MgF₂).

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For the optical part we start from the nonlinear wave equation [31]:

\nabla \times \nabla \times \vec{E} + \frac{n^{2}}{c^{2}} \frac{\partial^{2} \vec{E}}{\partial t^{2}} = - \frac{1}{∊_{0} c^{2}} \frac{\partial^{2} {\vec{P}}_{p}}{\partial t^{2}} - \frac{1}{c^{2}} \frac{\partial χ^{(1)}}{\partial q} \frac{\partial^{2}}{\partial t^{2}} q \vec{E} - \frac{χ^{(3)}}{c^{2}} \frac{\partial^{2} (\vec{E} {| E |}^{2})}{\partial t^{2}},

where

{\vec{P}}_{p} = ∊_{0} χ ℜ ({\vec{f}}_{p} (r) e^{- i ω_{0} t})

is the polarization induced by the pump field. We expand the electric field E⃗ in terms of the resonator modes e⃗_μ(r):

\vec{E} (r, t) = ℜ (e^{- i ω t} \sum_{μ} A_{μ} (t) {\vec{e}}_{μ} (r)),

where we accounted detunings of the mode eigenfrequencies ω_μ from the pump frequency ω in time dependent amplitudes A_μ(t) expecting that all modes are oscillating with the pump frequency. |μ| = 0, 1, 2, . . . is the mode numbers, where μ = 0 corresponds to the mode closest to the pump. We assume, that the sum goes from μ = −N to N, with unspecified summation limits. The mode profiles are defined as

\nabla \times \nabla \times {\vec{e}}_{μ} - \frac{n^{2} ω_{μ}^{2}}{c^{2}} {\vec{e}}_{μ} = 0

with the orthogonality relation

\int {\vec{e}}_{μ}^{*} {\vec{e}}_{ν} = δ_{μ ν}

. The Raman force F_q is expressed in terms of the modal amplitudes as:

F_{q} = \frac{∊_{0}}{4} \frac{\partial χ^{(1)}}{\partial q} ℜ \sum_{μ ν} ({\vec{e}}_{μ} {\vec{e}}_{ν} A_{μ} A_{ν} e^{- 2 i ω t} + {\vec{e}}_{μ} {\vec{e}}_{ν}^{*} A_{μ} A_{ν}^{*}) V_{eff} .

Neglecting fast-oscillating terms in Eq. (7) and using

q = \sqrt{w} ℜ \sum {\vec{e}}_{ν} (r) {\vec{e}}_{ν^{'}}^{*} (r) q_{ν ν^{'}} (t)

we transform the equation for the mechanical oscillator (2) to the form

\frac{d^{2} q_{ν ν^{'}}}{d t^{2}} + 2 γ \frac{d q_{ν ν^{'}}}{d t} + Ω_{0}^{2} q_{ν ν^{'}} = \sqrt{w} \frac{∊_{0}}{4} V_{eff} \frac{\partial χ^{(1)}}{\partial q} A_{ν} A_{ν^{'}}^{*} .

We can think of q_νν′ as an amplitude of oscillations of the Raman oscillator excited by a force with a form of the dot product of ν and ν′ modes. After tedious, but straightforward algebra, which includes the standard omission of the second derivatives of the slow varying amplitudes and of the fast oscillating exponential terms, we derive the system of equations for the modal amplitudes coupled to the Raman oscillator amplitudes:

\begin{array}{l} {\dot{A}}_{μ} = & - i \frac{ω_{μ}^{2} - ω^{2}}{2 ω} A_{μ} + i \frac{ω f_{μ} χ^{(1)}}{2 n^{2}} δ_{μ 0} + i \frac{3 χ^{(3)} ω}{8 n^{2}} \sum_{μ^{'} μ^{″} {μ^{'}}^{″}} Λ_{μ^{'} μ}^{μ^{″} {μ^{'}}^{″}} A_{μ^{'}} A_{μ^{″}} A_{{μ^{'}}^{″}}^{*} \\ + i \frac{\sqrt{w}}{2 n^{2}} \frac{\partial χ^{(1)}}{\partial q} ω \sum_{ν^{'} ν μ^{'}} Λ_{μ^{'} μ}^{ν ν^{'}} q_{ν ν^{'}} (t) A_{μ^{'}} . \end{array}

Here

Λ_{μ^{'} μ}^{ν ν^{'}} = \int ({\vec{e}}_{ν} {\vec{e}}_{ν^{'}}^{*}) ({\vec{e}}_{μ^{'}} {\vec{e}}_{μ}^{*}) d V

. We then can multiply Eq. (8) by

Λ_{μ^{'} μ}^{ν ν^{'}}

and sum over ν and ν′ to simplify the Raman term sum in (9). To further simplify

Λ_{μ^{'} μ}^{ν ν^{'}}

we should have some insight into mode forms and structure. We presume that we get nonzero Λ ≈ 1 only for ν′ = ν + μ′ − μ. This relation is supposed to be correct for all cavities, which modes can be represented by complex exponent in one direction (standing or traveling waves) and fixed field distribution in the others. For illustration we consider whispering gallery modes (WGMs). First, using the cylindrical symmetry of the WGM problem we extract the azimuthal dependence as e⃗_ν(r⃗) → e⃗_p,q,ν(r, z)e^iνφ. Here p and q are the transverse mode numbers that were previously implicit in ν and μ, while ν and μ are now pure azimuthal indexes. If ν is very large (which is usually the case), then the transverse profiles of all the modes can be assumed to be similar and independent of ν. Performing azimuthal integration we get nonzero Λ ≈ 1 only for ν′ = ν + μ′ − μ and the same p and q of the four modes. Note that due to the latter the following applies only to each single mode family separately. Renormalizing (

A \to \sqrt{\frac{2 ℏ ω}{∊_{0} ∊ V_{eff}}} A

) and introducing the loss terms κ_μ into Eq. (9) we obtain a system of equations for modal amplitudes

\frac{\partial A_{μ}}{\partial t} = - i (ω_{μ} - ω) A_{μ} - \frac{κ_{μ}}{2} A_{μ} + i G ω \sum_{μ^{'}} q_{μ - μ^{'}} (t) A_{μ^{'}} + i g \sum_{{μ^{'}}^{″} = μ^{'} + μ^{″} - μ} A_{μ^{'}} A_{μ^{″}} A_{{μ^{'}}^{″}}^{*} + F δ_{μ 0}

and Raman oscillations amplitudes

\frac{d^{2} q_{μ - μ^{'}}}{d t^{2}} + 2 γ \frac{d q_{μ - μ^{'}}}{d t} + Ω_{0}^{2} q_{μ - μ^{'}} = G ℏ ω \sum_{ν} A_{ν} A_{ν + μ^{'} - μ}^{*} .

Here,

g = \frac{ℏ ω_{0}^{2} c n_{2}}{n^{2} V_{eff}}

is the nonlinear coupling coefficient, n₂ is the nonlinear refractive index, V_eff is the effective mode volume,

κ = \frac{ω_{0}}{Q} = κ_{0} + κ_{c}

denotes the cavity decay rate as the sum of intrinsic decay rate κ₀ and coupling rate κ_c, Q is the total quality factor,

F = \sqrt{\frac{κ_{c} P_{0}}{ℏ ω}}

is the pump amplitude,

G = \frac{\sqrt{w}}{2 n^{2}} \frac{\partial χ^{(1)}}{\partial q}

is the Raman gain coefficient. Now, Eq. (10) coincides with the model used in [8] with the additional Raman term.

Expanding the eigenfrequencies $ω_{μ} = ω_{0} + μ D_{1} + \frac{μ^{2} D_{2}}{2}$ and introducing the field envelope $A = \sum_{μ} A_{μ} e^{i μ ϕ}$ , where ϕ is the polar angle, we find that Eq. (10) transforms to the Lugiato-Lefever equation:

\frac{\partial A}{\partial t} = - i [(ω_{0} - ω) - i \frac{κ}{2}] A - D_{1} \frac{\partial A}{\partial ϕ} + i \frac{D_{2}}{2} \frac{\partial^{2} A}{\partial ϕ^{2}} + i g A {| A |}^{2} + i G q ω A + F,

where

\sum q_{μ - μ^{'}} A_{μ^{'}}

is replaced by q(ϕ)A(ϕ), where

q (ϕ) = \sum q_{η} e^{i η ϕ}

. The set of Raman equations (11) is replaced under these conditions with

\frac{d^{2} q (ϕ, t)}{d t^{2}} + 2 γ \frac{d q (ϕ, t)}{d t} + Ω_{0}^{2} q (ϕ, t) = G ℏ ω {| A (ϕ, t) |}^{2} .

Therefore, the modal approach presented above is equivalent to the system of the Lugiato-Lefever and the Raman oscillator equations. A formal solution of Eq. (13) is given by the convolution of its right hand side with the oscillator’s response function,

h (t) : q (ϕ, t) = \int_{- \infty}^{+ \infty} h (t - t^{'}) {| A (ϕ, t^{'}) |}^{2} H (t - t^{'}) d t^{'}

, where

h (t - t^{'}) = G ℏ e^{- γ (t - t^{'})} sin \sqrt{Ω_{0}^{2} - γ^{2}} (t - t^{'})

and H (t − t′) is Heaviside step function. A linear in frequency approximation for

h (Ω) = \int_{- \infty}^{+ \infty} h (t) e^{i Ω t} d t ≃ G ℏ (1 - Ω τ_{R})

leads to the Raman shock term −iG²ħωτ_RA∂_t|A|² in the time domain. This approach may be suitable to study the wave forms with the spectra centered away from the maximum of the Raman gain, see, e.g., [23], but can not be applied for the type of dynamics reported below, which is initiated by the sidebands growing at the frequencies determined by the maximum of the gain and by the width of the Raman line.

For the numerical simulation Eq. (10) is converted to a dimensionless form through substitutions $A_{μ} = \sqrt{D_{1} / g} a_{μ} e^{- i μ D_{1} t}$ and τ = tD₁, so that modes are now expected to be separated by FSR, and time is dimensionless:

\frac{\partial a_{μ}}{\partial τ} = - (\frac{κ}{2 D_{1}} + i ζ_{μ}) a_{μ} + R_{μ} + i \sum_{μ^{'} \leq μ^{″}} (2 - δ_{μ^{'} μ^{″}}) a_{μ^{'}} a_{μ^{″}} a_{μ^{'} + μ^{″} - μ}^{*} + f δ_{μ 0},

All mode numbers μ are defined relative to the pump mode μ = m − m₀ with the initial azimuthal number

m_{0} \approx \frac{2 π r n_{0}}{λ}

, where

λ = \frac{2 π c}{ω_{0}}

. The cold cavity dispersion law is

ω_{μ} = ω_{0} + \sum_{k} \frac{D_{k}}{k!} μ^{k} : D_{1} ≃ \frac{c}{n_{0} R}

is the FSR (R is the radius of the resonator) and D₂ < 0 (normal GVD),

ζ_{μ} = \frac{ω_{μ} - ω - μ D_{1}}{D_{1}}

is the normalized detuning,

f = \sqrt{\frac{g η κ P_{0}}{D_{1}^{3} ℏ ω}}

is the dimensionless pump amplitude and

R_{μ} = i ω \frac{G}{D_{1}} \sum_{μ^{'}} q_{μ - μ^{'}} a_{μ^{'}} e^{- i (μ^{'} - μ) D_{1} t}

is the Raman term.

To this end, we rewrite Eq. (11) for the Raman oscillator. First, we note that due to the resonant character of the equation we can reduce the summation only to those terms that lie inside the resonance band |μ − μ′|D₁ ∈ Ω₀ ± 2γ. Second, as Eq. (14) has constant parameters, only slowly-varying parts of Raman term will influence the solution. Hence one does not need to solve separate equations in (11) for each η = μ − μ′, and we can use the single equation obtained after the summation over η:

\frac{d^{2} \bar{q}}{d τ^{2}} + 2 \tilde{γ} \frac{d \bar{q}}{d τ} + {\tilde{Ω}}_{0}^{2} \bar{q} = \sum_{η D_{1} \in Ω_{0} + 2 γ, ν} a_{ν} a_{ν - η}^{*} e^{- i η τ},

where

\bar{q} = \frac{g D_{1}}{G ℏ ω} \sum q_{η}

. Here we also performed the same renormalization of time as in (10):

\tilde{γ} = \frac{γ}{D_{1}}

and

{\tilde{Ω}}_{0} = \frac{Ω}{D_{1}}

and made q̄ dimensionless. So the Raman term in Eq. (14) becomes

R_{μ} = i G_{R} \bar{q} \sum_{μ^{'}} a_{μ^{'}} e^{i η τ},

where

G_{R} = \frac{G^{2} ℏ ω^{2}}{D_{1}^{2} g} = \frac{w V_{eff}}{4 n^{2} c n_{2} D_{1}^{2}} {(\frac{\partial χ^{(1)}}{\partial q})}^{2}

is the normalized Raman gain coefficient.

3. Numerical simulation

Our numerical model is based on a system of dimensionless coupled mode equations (14) and the oscillator equation (15) with the Raman term (16). In our simulations, we used at least 1024 modes and solved the system using the adaptive Runge-Kutta integrator. The integration time was large enough for all transient effects to disappear. Nonlinear terms were calculated using a fast Fourier method proposed in [32]. Parameters used below correspond to a crystalline microresonator [14–16]: f = 0.0014, $\frac{κ}{2 D_{1}} = 0.005$ , $\frac{D_{2}}{κ} = 0.01$ . For the frequency and damping rate of the Raman oscillator we assumed Ω̃₀ = 200 and γ̃ = 1.5. These parameters do not correspond to a specific material, but are realistic for crystalline materials. However qualitatively the same results were observed for other parameters of the Raman oscillator.

We studied the comb generation assuming normal GVD and applying the pump detuning corresponding to the upper branch of the steady-state intracavity field perturbed by noise. The Raman gain was used as a control parameter. For a given pump frequency providing that the Raman gain G_R is large enough we observed combs with a spacing defined by the Raman frequency. At the initial stage of evolution, we observed the generation of primary Stokes and anti-Stokes lines at μ_s = −198 and μ_s = 202, which then evolved due to a cascaded generation of secondary lines [see Fig. 2]. We note that the position of the Raman peaks does not exactly match Ω̃ = 200 due to non-zero GVD responsible for the non-equidistance of the spectrum. A mode with μ = 198 for the $\frac{D_{2}}{κ} = 0.01$ approximately corresponds to 200D₁. This distinguishes this state from a soliton comb, where the distance between the lines is accurately determined by FSR due to mode locking. One can also see that the Stokes part of the generated comb is stronger than the anti-Stokes one (Fig. 2). We have checked numerically that the Kerr part of the four-wave mixing has weak effect on the generation of the frequency comb. However, the fact that anti-Stokes lines are non-negligible strongly suggests that the comb generation was also influenced by the Raman part of FWM process. The Kerr nonlinearity manifests itself mainly in self-action effects.

Fig. 2 Optical spectrum for the primary Stokes line formation for G_R = 120 (a) and G_R = 160 (b). The Raman peak overlaps the mode with μ_s = −198, the width of the Raman gain γ̃ = 1.5, ζ = 0.02. The anti-Stokes line is strongly suppressed. The Raman FWM between the pump mode and the Stokes mode creates an anti-Stokes peak at the mode number μ_as = 202 and cascades a peak with frequency such as ω_cascade is divisible by the Raman frequency. Spectral lines around the Raman peak are shown in the insets.

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Lines around the pumped mode line result from nondegenerate FWM interactions between the primary Stokes and anti-Stokes components [see Fig. 3]. The number of the generated comb lines depends on the Raman line-width γ̃ and gain G_R. The increase in the Raman gain value leads to the growth of the lines number. The detuning range providing comb generation also becomes wider with the increase of G_R. It is also worth noting that the Raman line-width in a very small microresonator (diamond or silicon) can be much narrower than the FSR. If the peak of the Raman gain is far detuned from any cavity resonance, then the Raman effect will be either strongly suppressed or practically non-existent [33]. Contrary, in large resonators and for materials with wider Raman lines, the Raman gain can cover many FSRs. For example, Raman line overlaps tens of FSRs for crystalline microresonators (full-width at half maximum line-width $\frac{γ}{2 π} = 450 GHz$ for calcium fluoride, $\frac{γ}{2 π} = 210 GHz$ for magnesium fluoride).

Fig. 3 The comb spectrum is generated by a combination of the Raman scattering, which downshifts its frequency and nondegenerate four-wave mixing processes in which all four photons have different frequencies. Insets show energy level diagrams for the Raman scattering and four-wave mixing processes. The Raman part of the FWM plays a dominant role in the spectrum formation.

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The generation of a Raman comb is also possible if a pump frequency scan is applied. This approach can be more efficient since in this case, the system can stay in the upper state for larger detunings, which corresponds to larger intensities of the pumped mode providing more efficient Raman process. In contrast, starting from the noise-like inputs at large values of detuning one may generate only low-intensity steady-state solutions not always supporting efficient generation of cascaded Raman peaks.

4. Platicon dynamics

In this section, we study the generation of platicons in the presence of Raman effect. We do not expect that the Raman effect on its own can lead to the platicon generation since platicons are known to require an application of either the pumped mode shift method [14] or the pump modulation [15]. Before looking at non-trivial effects in normal dispersion regime, we first simulated the process of bright soliton propagation in the anomalous regime for a range of the Raman gain values, while fixing Ω̃₀ = 100 and γ̃ = 1.0. As it was demonstrated (e.g. in [19]) the presence of the Raman gain leads to the modification of the soliton profile and to the appearance of nonzero “transverse” (drift) velocity. It was found that the drift velocity increases with the growth of Raman gain coefficient G_R. Such a drift may be explained by the fact that low-frequency components of the soliton spectrum may experience significant Raman gain. Our simulations in anomalous regime were in agreement with known results, demonstrating stable propagation of bright solitons with different velocities and frequency shifts corresponding to selected values of the Raman gain.

To study the platicon dynamics for the same set of parameters, we modified the conventional dispersion law as $ω_{μ} = ω_{0} + δ_{0 μ} Δ + D_{1} μ + \frac{D_{2}}{2} μ^{2}$ by including the pumped mode shift Δ = 2κ and taking D₂ < 0. Our numerical simulations revealed that the Raman effect strongly influences the platicon dynamics in a way very different from that of bright solitons [see Fig. 4]. In particular, the velocity and frequency shifts typical for bright solitons do not appear for platicons. Platicons seem to be insensitive to the Raman process up to some threshold value of the Raman gain [see Fig. 4(d)]. Note, that the third order dispersion, unlike the Raman effects, acts on platicons [16] in a way qualitatively similar to its action on the bright solitons [19] and leads to a non-zero drift velocity.

Fig. 4 Wide platicon (ζ₀ = 0.02) evolution for different values of the Raman gain: (a) G_R = 150; (b) G_R = 175; (c) G_R = 200; (d) G_R = 220; (e) G_R = 300; (f) G_R = 575.

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If G_R exceeds some critical value, then platicons lose their stability and give birth to various spatiotemporal patterns [Figs. 4(b)–4(d)]. As it was shown in [14] the width of the platicon spectrum varies with pump frequency detuning. Using different values of ζ₀ we studied the dynamics of different platicons in Raman scattering band. Numerical simulations demonstrate that the Raman induced instability of the relatively wide platicons typically leads to splitting (branching) [Figs. 4(b)–4(d), Fig. 5], while the narrow ones decay during propagation [Figs. 5(d)–5(f)]. In some cases a kind of quasi-stable two-hump solutions was observed as seen in Fig. 4(d).

Fig. 5 Platicon dynamics for intermediate width (ζ₀ = 0.03, a–c) and narrow (ζ₀ = 0.04, d–f) platicon with different values of the Raman gain: (a) G_R = 200; (b) G_R = 250; (c) G_R = 300, (d) G_R = 400; (e) G_R = 575; (f) G_R = 700.

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Further growth of the Raman gain leads to a blurring of high-intensity patterns, [Fig. 4(e)], and to a transformation of a platicon into a complex spatiotemporal pattern [Fig. 4(f)]. Obviously, the greater the Raman gain is, the faster the instability develops. The instability also leads to a significant distortion of the platicon spectrum. For parameters used at Fig. 4(b) one can compare initial platicon spectrum [Fig. 6(a)] and the spectrum in the point of instability at τ = 5000 [Fig. 6(b)] which demonstrates the appearance of cascaded Raman peaks. The threshold of the platicon instability increases with the growth of the detuning and decrease of platicon width [compare Fig. 4(c) and Fig. 5(a), and Fig. 5(d)]. Also, for the same Raman gain, narrower platicons produce more structured intensity distribution profiles than the wider ones [compare Fig. 4(e) and Fig. 5(c)]. Since platicons are formed by binding of two counter-propagating fronts, [17], [34], it is natural to expect that the instabilities discussed here result from the Raman effect influence on the relative velocities of these fronts frustrating robustness of platicons.

Fig. 6 Degradation of the platicon optical spectra for ζ₀ = 0.02, G_R = 175: (a) τ = 0; (b) τ = 5000. The Stokes mode corresponds to μ ≈ 100. The waveforms for the corresponding spectra are shown in the (c) and (d).

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The situation dramatically changes at some critical value of the platicon width (determined by pump frequency detuning). Under the influence of the high Raman gain exceeding some critical value, narrow platicons do not split into several parts and do not form complex structures during the propagation. For narrow platicons, the instability manifests itself as a decay of the initial profile to the low-intensity background, with the stronger Raman gain leading to the shorter platicon lifetimes [see Figs. 5(e)–5(f)].

5. Conclusion

We have developed the coupled mode approach to the Kerr frequency comb generation accounting for the Raman scattering in optical microresonators. Using this model, we demonstrated the generation of Kerr-Raman comb in the normal GVD regime and studied the platicon dynamics featuring branching instabilities leading to the formation of complex spatiotemporal structures.

Funding

Russian Science Foundation (17-12-01413).

Acknowledgments

D.V.S. acknowledges support from the ITMO University Visiting Professorship Scheme via the Government of Russian Federation Grant 074-U01. D.V.S. and M.L.G. acknowledge support from EU H2020 (691011, Soliring) for exchange visits. A.V.C. acknowledges scholarship from the Foundation for the Advancement of Theoretical Physics “BASIS”.

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Raman-Kerr frequency combs in microresonators with normal dispersion

Abstract

1. Introduction

2. Theory

3. Numerical simulation

4. Platicon dynamics

5. Conclusion

Funding

Acknowledgments

References and links

Cited By

Figures (6)

Equations (16)

Optics Express