Multi-frequency radiation of dissipative solitons in optical fiber cavities

New resonant emission of dispersive waves by oscillating solitary structures in optical fiber cavities is considered analytically and numerically. The pulse propagation is described in the framework of the Lugiato-Lefever equation when a Hopf-bifurcation can result in the formation of oscillating dissipative solitons. The resonance condition for the radiation of the dissipative oscillating solitons is derived and it is demonstrated that the predicted resonances match the spectral lines observed in numerical simulations perfectly. The complex recoil of the radiation on the soliton dynamics is discussed. The reported effect can have importance for the generation of frequency combs in nonlinear microring resonators.

For the propagation of an initial field A 0 (x) in terms of Eq. (1) in the main text we use numerical methods that rely on (i) a pseudospectral scheme implementing t-propagation using a fourth-order Runge-Kutta method, and, (ii) a pseudospectral scheme implementing t-propagation using a split-step approach. In our simulations, the x-domain was bounded by −160 ≤ x < 160 and discretized using N x = 2 13 equidistant mesh-points. The temporal evolution the fields A(x, t) was followed up to a maximal propagation time t = 16 using at least N t = 4 × 10 4 equidistant steps.
To verify reliability of our findings we repeated a subset of the numerical simulations for cavities of different size and for different numbers of discretization points in both, x and t.

S-2. CONSTRUCTION OF INITIAL CONDITIONS FOR EQ. (1)
The initial conditions used for t-propagation in terms of Eq. (1) in the main text are obtained by an analysis of the corresponding standard Lugiato-Lefever equation (LLE). The standard LLE, obtained by setting d 3 = 0 in Eq. (1), exhibits field solutions that describe localized dissipative states satisfying ∂ t A(x, t) = 0 for a wide range of control parameter values, including the particular choice θ = 15 and P = 8 considered in our study.
Here we compute such initial conditions by finding a complex-valued solution A 0 (x) such that F (A 0 (x)) = 0 with As ansatz for an approximate trial solution for this rootfinding problem we take wherein A HSS denotes a homogeneous steady-state solution (HSS), and where A LDS is a localized dissipative structure (LDS). The homogeneous steady-state solution of the standard LLE for given control parameters θ and P reads [1] Re[ with I HSS = |A HSS | 2 being a solution of obtained by setting all derivatives of the LLE to zero [2]. As an approximate trial function for a possible localized dispersive structure, the similarity of the LLE to a perturbed nonlinear Schrödinger equation suggests [3,4] resembling a standard nonlinear-Schrödinger soliton. In the considered case, i.e. for a constant control parameter P , the constant phase ζ = arccos[ √ 8θ/πP ] is found as a consequence of the conservation law d dt |A LDS (x)| 2 dx = 0 under propagation with the standard LLE [4].
For the control parameter ranges considered in our numerical simulation we found that a trial solution given by Eq. (S2), with A HSS and A LDS (x) as detailed above, is already close to a proper stationary solution. Consequently, a root-finding procedure for Eq. (S1), implemented via standard numerical tools, converges quite fast. In Fig. S1 we demonstrate that the stationary solution used as initial condition for the LLE with nonzero  To unravel the x and k relationships within the complex valued field A(x, t) at a given propagation time t we use a spectrogram [5,6], as illustrated in Figs. 1(c,f) in the main text. To calculate a spectrogram we here consider a Gaussian window function h(x) = exp(−x 2 /2σ 2 ) to localize the field in the coordinate x. The Fourier transform of the localized field then provides a spectrogram P S (x, k) = |S x (k)| 2 by sweeping over x. The choice of the root-mean-square (rms) width σ naturally limits the resolution in x and k, achieved by the spectrogram [5]. An exemplary spectrogram for the field A(x, t) at propagation distance t = 10 for control parameter values d 3 = 0.02, θ = 15, and P = 8 is shown in Fig. S2(b). The spectrogram is restricted to the range (x min , x max ) = (−20, 130) and (k min , k max ) = (−30, 60). The frequency resolution of the pump at k = 0 is due to the finite rmswith σ = 2 of the Gaussian window function used for field-localization. The figure further shows the dissipative soliton at x ≈ 3 and the resonantly generated radiation at large values of k, i.e. the synchrotron part of the spectrum, enclosed by a red dashed box. The cyclotron part of the radiation at small values of |k| is also well visible in the spectrogram.
In Fig. S2(a) we show the normalized intensity P 1 (x) = I(x)/max [I(x)] with I(x) = | kmax kmin A k e −ikx dk| 2 . Evolution of the field in terms of intensity and spectral intensity up to t = 10 is shown in Figs. 1(a,b) of the main text. The restriction to frequency components in range (k min , k max ) allows to emphasize features of the intensity that might otherwise be obscured by features that are present in the full field A(x, t) at the selected value of t. This becomes evident by inspection of P 1 (x) in Fig. 1(c) in the main text, which allows to clearly distinguish the resonantly generated radiation with modes in the range (k min , k max ) = (47, 54). In Fig. S2(a) these are immersed on a constant background contributed by the pump, and further dominated by the intensity of the dissipative soliton at x ≈ 3.
In Fig. S2(c) we show the normalized intensity P 2 (k) = I k /max [I k ] with I k = |A k | 2 restricted to the range (k min , k max ). While this emphasizes the constant pump in Fig. S2(c), it reveals the small-scale polychotomous structure of the resonantly generated radiation for (k min , k max ) = (47, 54) in Fig. 1(c) of the main text.

S-4. DESCRIPTION OF THE SUPPLEMENTARY VIDEO
As supplementeray material we also provide a supplementary video (see supplementary video SV1), detailing the propagation dynamics of an oscillating soliton for controll parameters θ = 15, P = 8, and d 3 = 0.02, in the propagation range t = 5 − 8. Figure S3  a caption detailing the different panels of the video.

S-5. SUPPORTING INFORMATION FOR FIG. 3(A)
As discussed in the main text, Fig. 3(a) of the main text shows the radiation field of a non-oscillating soliton for the parameter values P = 6 and d 3 = 0.02. For completeness, in Fig. S4 we show the temporal evolution of the intensity [ Fig. S4(a)], spectral intensity [ Fig. S4(b)], and a close up view of the resonantly generated radiation in terms of a spectrogram [ Fig. S4(c)]. As evident from the spectrogram, the Cherenkov radiation of a nonoscillating soliton is a slowly decaying dispersive wave. In Fig. 5 in the main text we detailed the dynamics of oscillating solitons for selected values of third-order dispersion in the range d 3 = 0.02 through 0.18. Therein, the qualitative change in their dynamics was illustrated using the temporal evolution of the peak intensity I max and phase ϕ. Here, for completeness, we show the corresponding temporal evolution of the intensity and spectral intensities in Fig. S5.

S-7. DEPENDENCE OF THE RESONANT RADIATION ON d3
We studied the peak amplitude I RR max = max(I RR k ) of the resonant radiation and its spectral width, for which we here use the definition of the root-mean squared width in more detail. In our analysis we focused on two regions of third order dispersion. These are the range of small values d 3 = 0.02 . . . 0.04 (see Fig. S6a-c), and, the range of large values d 3 = 0.18 . . . 0.195 (see Fig. S6df). As evident from Fig. 5(a) of the main document, in both these ranges the peak intensity of the oscillating soliton alternates between a unique minimum and maximum. Above, I RR max refers to the spectral intensity of the dominant pulse of resonant radiation, taken at a point in the steady-state regime where the amplitude of the temporal oscillation of the soliton assumes its minimum value. At that point, the soliton is spectrally narrow and the spectral width of the dominant pulse of resonant radiation can be well distinguished. For an example, see the spectrogram in Fig. 1c