## Abstract

We report theoretical and numerical study of the dynamical and spectral properties of the conservative and dissipative solitons in micro-ring resonators pumped in a proximity of the zero of the group velocity dispersion. We discuss frequency and velocity locking of the conservative solitons, when dissipation is accounted for. We present theory of the dispersive radiation emitted by such solitons, report their Hopf instability and radiation enhancement by multiple solitons.

© 2014 Optical Society of America

## Corrections

C. Milián and D.V. Skryabin, "Soliton families and resonant radiation in a micro-ring resonator near zero group-velocity dispersion: erratum," Opt. Express**22**, 8068-8068 (2014)

https://www.osapublishing.org/oe/abstract.cfm?uri=oe-22-7-8068

Spectral broadening effects in optical micro-ring cavities are attracting a considerable interest, in particular, in the context of the frequency comb generation [1, 2]. In the standard scenario, an external cw pump triggers the cascaded four wave mixing (FWM), or modulational instability (MI), process resulting in the ultrabroad comb [3]. It has been recently proposed that the comb generation in micro-rings can be linked to the formation of cavity solitons [5–7], though no direct experimental proof of this link has been so far demonstrated. Suppressed cavity dispersion is expected to enhance phase matching of the FWM cascade across the wider spectral range. Therefore using cavities with the zero’s of the group velocity dispersion (GVD) in the proximity of the pump frequency or with the flattened dispersion has been considered, see, e.g. [8, 9]. Under these conditions the role of the higher order dispersions in FWM and soliton formation becomes pronounced [7, 8], which is also directly linked with the extensive recent research on spectral broadening in optical fibers [10]. Both dark [14, 15] and bright [7, 15] cavity solitons have been reported in presence of the higher order dispersion in coherently pumped resonators. Regarding bright solitons, which are a subject of this paper, the results have been limited by numerical studies of the associated dispersive wave emission [7] leading to the characteristic asymmetry of the comb [8].

Below we present new insights and results into the role played by third order dispersion in the dynamics of the bright cavity solitons. In particular, we demonstrate how velocity and frequency of solitons existing in the Hamiltonian limit lock to specific values when losses are introduced. We also describe properties of the resonant radiation and elaborate on the role of the cavity resonances and of the soliton background, which makes the difference with the free propagation settings [10]. We trace branches of dissipative solitons and report their spectral and stability properties. Note, that solitons in micro-ring resonators are very closely linked with the ones in optical fiber loops, where the experimental data on the soliton existence have been gathered [16].

There exist two approaches to describe nonlinear dynamics of the modal quasi-continuum in optical resonators. One is the modal expansion, when a set of ordinary differential equations is used to describe time evolution of the modal amplitudes [3]. An alternative to the above is to solve the propagation problem over the resonator period and apply an appropriate pumping, loss and periodicity conditions after each round trip [16]. For the coherently pumped resonators both of these formalisms can be reduced to an initial value problem for a partial differential equation representing various generalizations of the so-called Lugiato-Lefever equation (gLLE) [4, 7, 8, 14, 15, 17]. In the resonators made of waveguides arranged into closed loop geometries, spectral representation of the quasi-continuum limit of the modal approach assumes expansion of the frequency *ω*(*β*) in the waveguide propagation constant series [4], while the ’round trip’ approach assumes the equivalent expansion of the propagation constant *β*(*ω*) into the frequency series [8]. For the case of a semiconductor waveguide ring resonator, considered below, the former method is more straightforwardly connected to a set of formally well defined modal evolution equations [3]. Their continuum approximation leads to the following equation for the envelope *E* of the intracavity field *Ee*^{iβ0z} + *c.c.*, where *β*_{0} = *β*(*ω*_{0}),

*ω*

_{0}is the frequency of the reference cavity resonance. Γ is the rate of photon loss from the cavity and

*A*is the dimensionless amplitude of the pump field,

*r*is the pump coupling rate and

*ω*is the pump frequency, which is detuned from

_{p}*ω*

_{0}roughly within the cavity linewidth 1/Γ. |

*E*|

^{2}is the dimensionless intracavity intensity, see Refs. [3, 4] for scaling. $g=\frac{{n}_{2}c}{2{n}^{2}}\frac{\overline{h}{\omega}_{0}}{V}$ is the dimensionless nonlinear parameter, where

*n*is the refractive index,

*n*

_{2}is the Kerr coefficient,

*c*is the vacuum speed of light and

*V*is the modal volume [3].

*z*is the coordinate along the cavity varying from 0 to 2

*πR*, where

*R*is the radius:

*E*(

*z*= 0) =

*E*(

*z*= 2

*πR*). Dispersion of nonlinearity is disregarded.

*ω*

^{(}

^{n}^{)}coefficients are related to the usual dispersion parameters of the waveguide ${\beta}_{m}={\partial}_{\omega}^{m}\beta :{\omega}^{\prime}=1/{\beta}_{1}=c/{n}_{g}$, where

*n*is the group index at

_{g}*ω*

_{0},

*ω″*= −(

*c/n*)

_{g}^{3}

*β*

_{2},

*ω′″*≃ −(

*c/n*)

_{g}^{4}

*β*

_{3}. For the relatively narrow band problem considered here it suffices to terminate the expansion at the third order dispersion. In the linear and no-pump limits, any mode

*e*generated in the cavity acquires the frequency ${\omega}_{Q}={\omega}_{0}+{\omega}^{\prime}Q+\frac{1}{2!}{\omega}^{\u2033}{Q}^{2}+\frac{1}{3!}{{\omega}^{\prime}}^{\u2033}{Q}^{2}$, where

^{iQz}*Q*=

*β*(

*ω*) −

_{q}*β*(

*ω*

_{0}) is the wavenumber offset, which is linked to the modal number offset

*q*=

*QL*/(2

*π*),

*q*= 0, ±1, ±2, ±3,....

Seeking solution of Eq. (1) in the form
$E=\frac{1}{\sqrt{g{\omega}_{0}\tau}}\mathrm{\Psi}{e}^{-i{\omega}_{p}t}$ and normalizing time, *t* = *Tτ*, *τ* = *c*/[2*πRn _{g}*] (so that the group velocity coefficient is unity), and distance

*z*=

*ZL*,

*L*= 2

*πR*, we get a more handy form of the dimensionless gLLE:

*γ*= Γ

*τ*,

*δ*= (

*ω*

_{0}−

*ω*)

_{p}*τ*, $h=r\tau \sqrt{g{\omega}_{0}\tau}A$, ${B}_{2}=-{v}_{g}^{2}{\beta}_{2}/(2\pi R)/2!$, ${B}_{3}=-{v}_{g}^{3}{\beta}_{3}/(2\pi {R}^{2})/3!$. Soliton solutions of Eq. (2) can be sought in the form Ψ(

*T*,

*Z*) =

*ψ*(

*Z*− (

*v*+ 1)

*T*), where

*ψ*obeys

*Q*= 0, found from (

*iγ*−

*δ*+ 2|Ψ

_{0}|

^{2})Ψ

_{0}+

*h*= 0, is multivalued (bistable) in the soliton existence range and the solitons are nested on the background given by the root with the smallest value of |Ψ

_{0}|

^{2}, see Fig. 1(a).

The limit *γ* = 0 corresponds to the Hamiltonian (conservative) case and is a convenient starting point to understand physical properties of solitons and their radiation. The field momentum
$M=-\frac{i}{2}{\int}_{0}^{1}dZ({\mathrm{\Psi}}^{*}{\partial}_{Z}\mathrm{\Psi}-c.c.)$ is conserved in this limit, while solitons constitute a family continuously parameterized by the shift *v* of the group velocity, while all other parameters are kept fixed [18]. The *v*-family is known in the analytical form for *h* = *B*_{3} = 0:

Note, that for the anomalous group velocity dispersion (GVD) *β*_{2} < 0 and *B*_{2} > 0. Numerically calculated plots of *M* vs *v* for several pump values are shown in Fig. 1(b). All branches are plotted only up to their turning points, *∂ _{v}M* → ∞, where the multi-hump solitons emerge, since our interest is focused here on the single hump solitons only (see Ref. [18] for further details). Typical spatial and spectral profiles of the solitons for

*h*= 5 × 10

^{−4}are shown in Fig. 2(a–d). Spectra of these solutions provide a useful physical insight. As

*v*increases, the soliton core spectrally separates from the pump, so that the pump ability to supply energy into the soliton diminishes. Thus, when losses are accounted for, the soliton adiabatically decays in time without any change in its momentum, see Fig. 2(g,h). If, however, the spectral offset is moderate to small, then the soliton is fed by the pump efficiently, so its velocity shift

*v*evolves towards zero and its momentum converges to the pump momentum, see Fig. 2(e,f). For

*γ*≠ 0, the momentum is not conserved and there exists only one soliton state with

*v*= 0 [19]. Note, that so far we have kept

*B*

_{3}= 0. Before we proceed further, we describe a physical system, which is used to present numerical data in this work including the ones in the already discussed Fig. 2. We consider a silicon nitride ring resonator of radius

*R*= 0.9 mm with the cross section 500 × 730 nm

^{2}, see Fig. 1(c). The zero GVD wavelength (

*β*

_{2}= 0) is at ≈ 1586 nm.

Accounting for the third order dispersion, *β*_{3} ≠ 0, in optical fibers leads to the emission of the resonant radiation by solitons [10]. It is natural to expect a similar effect in our case. To find the resonance wavenumber and the corresponding frequency, we apply the perturbation technique originally developed in the fiber context [9–12]. We assume Ψ = *ψ*(*x*) + *g*(*t*, *x*), where
$g={G}_{1}{e}^{-i[Qx-\mathrm{\Omega}t]}+{G}_{2}^{*}{e}^{i[Qx-\mathrm{\Omega}t]}$ is the radiation field. Linearizing Eq. (2) for small *g* we find

The wavenumbers of the radiation resonant with the soliton satisfy Ω(*Q _{r}*) = 0. The corresponding physical frequencies are found as

*ω*

_{0}+

*Q*[

_{r}*v*+ 1]

*c*/[2

*πRn*]. Graphical solutions of Ω(

_{g}*Q*) = 0 are shown in Fig. 3(a). One can see that there exist two resonances symmetrically located with respect to the pump. The existence of two real nonzero roots is common to all radiating nonlinear waves which are embedded in a non-vanishing background and in particular this has been reported in the case of fiber dark solitons [11–13]. The fact that there is periodicity in

_{r}*x*(equivalently, in

*z*) makes the exact resonance possible only for

*Q*=

_{r}*q*/(2

*π*) (since dimensionless cavity length is 1),

*q*= 0, ±1,..., implying that in practice we are dealing with the quasi-resonant radiation in this system.

The relative strength of the two resonances,
$\left|{G}_{2}/{G}_{1}\right|=\left|W/2{\mathrm{\Psi}}_{0}^{2}\right|$ is shown in Fig. 3(c), here *W* = *Ŵ* (*Q* = *Q _{r}*,

*∂*= 0). Note, that for the wavelength interval considered here

_{x}*β*

_{3}< 0 and the short wavelength resonance is driven through the nonlinear mixing of the soliton background with the primary resonance, and therefore its amplitude is relatively small and tends to zero together with

*h*. The long wavelength resonance is much stronger and it retains the non-zero amplitude in the limit

*h*= 0, corresponding to the case of free (no background) propagation [10]. For

*β*

_{3}> 0 the short wavelength resonance becomes domineering.

Changes in the location of the primary resonance with varying *δ* are shown in Fig. 3(b). While the pump frequency varies roughly within the cavity free spectral range (FSR), the resonance frequency swipes across many FSRs. The question to consider now is will the periodic boundary conditions imposed by the cavity have an appreciable impact on the radiation amplitude, when its frequency resonates with the cavity mode. In quantitative terms, this problem is best addressed numerically, since the soliton core is expected to be altered when the radiation feeds back on the soliton through the periodic boundaries and the radiation frequency is going to be influenced by the soliton being on its way. Therefore we solve Eq. (3), for *γ* = 0, numerically and find the soliton solutions nesting on top of the radiation wave, see Figs. 4(a,b). Conservative solitons shown here are numerical continuation along *B*_{3} and *δ* of those at *h* = 5 × 10^{−4} and *v* = −1.25 in Fig. 1(b) (*v* is also a free parameter for *B*_{3} ≠ 0). The plot of the radiation amplitude, *α*, vs *δ* is shown in Fig. 3(d), one can see the series of peaks that appear for the values of *δ* producing the resonant radiation nearly matching the cavity resonances, see Fig. 3(b). Matching is not exact, primarily due to the effect of the soliton core on the effective length of the cavity. The maxima of the radiation amplitude corresponds to the amplitude of the soliton background, which drops with *δ*.

Including nonzero losses *γ* ≠ 0 for *β*_{3} ≠ 0 has the two fold impact on solitons. First, it selects the value of the group velocity shift *v* to be non-zero, unlike *v* = 0 selection for *β*_{3} = 0, see Fig. 5 (note *γ* = 0 for *t* ≤ 500 and *γ* > 0 for *t* > 500). Second, it damps the radiation in space, so that solitons propagate together with the extended, but localized, radiation tail attached to them, see Fig. 6. *v* ≠ 0 implies that the soliton carrier frequency is detuned from the pump. This can be interpreted as the spectral recoil effect from the radiation on the soliton [10]. From the soliton and radiation spectra in Fig. 6, one can see that the radiation amplitude increases as one approaches *β*_{2} = 0 wavelength. It is also important to notice that the soliton state persists, when the pump frequency shifts into the normal GVD range, while the soliton spectral maximum remains in the range of anomalous GVD, see Fig. 6(c), 7(a). The corresponding changes of the soliton velocity with the wavelength of the reference cavity mode approaching *β*_{3} = 0 are shown in Fig. 7(b), where the pump frequency is assumed to change simultaneously, so that *δ* is kept constant.

Figure 8 demonstrates evolution of the spectral maximum of the soliton and radiation, when the lossy and cw-pumped resonator is triggered by a short pulse with the frequency coinciding with the pump frequency (no radiation at *t* = 0). One can see the recoil effect on the soliton core and frequency/momentum locking effect. A notable difference between Figs. 8(a,b) and 8(c,d) is that (a,b) show pronounced oscillations of the soliton and radiation amplitudes with time, while in (c,d) these oscillations are initially small and progressively decaying. To understand this dynamics, we have numerically computed spectrum of linear perturbations around the soliton and found that the soliton can be unstable with respect to the Hopf instability (instability with complex eigenvalues) [18, 20]. The growth rate of the Hopf instability is shown in Fig. 7(c), as one can see the third order dispersion has a strong stabilizing influence on this instability, since the instability is suppressed sufficiently close to *β*_{2} = 0. Finally, we provide numerical evidences of the radiation enhancement through the excitation of soliton trains [6], see Fig. 9. The issues related to multiple re-scattering of the radiation on solitons trains deserve further investigation, which goes beyond our present aims.

In summary, we have families of the soliton solutions in a microring cavity in the presence of the third order dispersion. We have discussed spectra of the resonant radiation forming soliton tails. We have studied and compared solitons in the Hamiltonian and dissipative cases and demonstrated frequency and velocity selection effect induced by the arbitrary small losses.

## Acknowledgments

We acknowledge support from EPSRC UK: EP/G044163/1.

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