Bound states of solitons which are stable solutions of the nonlinear Schrödinger equation (NLSE) are often referred to as soliton molecules. Such molecules occur in various systems, such as dispersion-managed fibers [1], twin-core fibers [2], optical microresonators [3], the complex Ginzburg–Landau equation [4], and for envelope solitons in the perturbed NLSE [5]. Moreover, recent advances in dispersive-Fourier-transformation-based imaging techniques have revealed the formation of soliton molecules in a variety of mode-locked lasers [6–8]. Based on this technique, the current research aims to observe the dynamics under the impact of a collision with further solitons [9,10]. In these schemes, different collision scenarios can be distinguished [11,12]. In contrast to usual soliton molecules, which exhibit a double-hump structure in the time domain, we theoretically proposed a fundamentally different kind of photonic molecule governed by a generalized nonlinear Schrödinger-type equation [13]. These two-frequency photonic molecules are characterized by a double-hump structure in the frequency domain and a single localized state in the time domain. They evolve stably and their spectrum can reach over two separate regions of anomalous dispersion and a region of normal dispersion. Such molecule states have also been investigated as a part of a larger class of “generalized dispersion Kerr soliton” [14] and were recently demonstrated experimentally in a mode-locked laser cavity [15,16]. Two-color soliton microcomb states with a similar pulse structure were also demonstrated in microresonators modeled by the Lugiato–Lefever equation [17,18].

Collision events between dispersive waves (DWs) and solitons give deeper insight into interaction processes and reveal effective methods to manipulate light [19], soliton–soliton collisions across a normal dispersion region generate molecules [13], and collisions of DWs with photonic molecules give rise to new frequency components [20]. Here, we present the interaction of solitons and DWs with a two-frequency molecule to gain knowledge about the internal molecule state characteristics. We study the stability of the two-frequency molecule, observe molecule-like processes, and synthesize modified photonic molecules via the collision of the two-frequency molecule with solitons and DWs, which can be described as a molecule state transition.

For our investigations we use the propagation equation

 

Fig. 1. Collision of two-frequency soliton molecule with a projectile soliton. (a) Group delay $\beta _1$, (b) group-velocity dispersion $\beta _2$. Region of normal dispersion ($\mathsf {ND}$) is gray shaded. (c) Temporal profile of initial ($M_{in}$) and resulting ($M_{out}$) molecule state. (d) Evolution in the time domain with projectile (blue arrow) and ejected (red arrow) soliton and trajectories of initial and resulting molecule state in time domain. (e) Evolution in frequency domain. Dotted lines mark zero dispersion frequencies. (f) Phase-matching analysis (see text for details). (g) Selected output spectra of total spectrum ($out$), resulting molecule state ($M_{out}$), and ejected soliton ($S_{out}$). Vertical dashed lines mark the location of selected phase-matched components. (h) Input spectra with initial molecule ($M_{in}$) and projectile soliton ($S_P$).

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As a first controlled interaction scenario, we investigate the collision of a photonic molecule with a fundamental soliton. Figure 1 shows a collision process with a projectile soliton with $\omega _P=2.945\,\mathrm {rad/fs}$, and $t_P=20\,\mathrm {fs}$. The photonic molecule was seeded using Eq. (2) with $t_{1,2}=30\,\mathrm {fs}$ resulting in a molecule state with a duration $t_M=8.6\,\mathrm {fs}$. As evident from the propagation shown in Fig. 1(d), the collision results in the ejection of a different soliton, accompanied by residual, non-solitonic radiation as well as a modified molecule state. Let us emphasize that while the projectile soliton is located in $\mathsf {A2}$ (blue arrow), the ejected soliton emerges in $\mathsf {A1}$ (red arrow) [see discussion of Figs. 1(f) and 1(g) below]. Figure 1(c) compares the initial molecule state (gray) with the resulting one (magenta) and reveals a collision-induced state transition manifesting through a temporal compression. For the compressed molecule we find $t_M=6.4\,\mathrm {fs}$, corresponding to a compression factor $F_C=1.3$. The dynamics in the frequency domain reflect the transition of the molecule state: both subpulses experience a slight frequency shift, visible as a velocity change of the molecule in Fig. 1(d); the decrease of the duration is further reflected by a spectral broadening, which also leads to a higher spectral density within the region of normal dispersion ($\mathsf {ND}$), see Fig. 1(g). Also a collision with the leading edge of the molecule state is possible, leading to similar behavior, but with an opposed frequency shift. We investigate in Figs. 1(f), 1(g), and 1(h) the soliton spectral tunneling [28], where the projectile soliton tunnels from $\mathsf {A2}$ to $\mathsf {A1}$. The spectra in Fig. 1(g) display the profiles of the entire output (gray; $out$), the resulting soliton molecule (magenta; $M$), and the ejected field components, including a soliton in $\mathsf {A1}$ (red; $S_{out}$). For investigation of different wavenumber matching conditions [Fig. 1(f)], we introduce a wavenumber for a molecule subpulse, which is approximated by the wavenumber of a fundamental soliton and a correction from a secondary soliton: $k_n(\omega ) \approx \tfrac {1}{2} \gamma (\omega _n) (a_n^2 + 2 a_m^2)+\beta (\omega _n)+\beta _1(\omega _n)(\omega -\omega _n)$ [20]. Indices $n,m \in (1,2), m \neq n$, label the different subpulses. Potential resonant frequencies $\omega _R$ are found where the wavenumber curve $\beta (\omega )$ intersects with wavenumber combinations of the subpulses and further incident fields. The phase-matching conditions including the interaction with an additional field at $\omega _P$ [20]: $\beta (\omega _R)=2k_1(\omega _R) - \beta (\omega _P)$ and $\beta (\omega _R)=k_1(\omega _R) - k_2(\omega _R) + \beta (\omega _P)$ are in our case responsible for the generation of new frequencies. As indicated by the vertical lines in Figs. 1(f) and 1(g) four intersection points give rise to the generation of new frequencies. The most prominent component is located at $\omega =1.32$ rad/fs corresponding to the frequency of the ejected soliton. Further components are visible around $\omega =1.07$ and $\omega =1.39$ rad/fs with a less efficient excitation.

To highlight the changes induced by the collision event, spectrograms of the states before ($z_0$) and after collision ($z_{end}$) are shown in Figs. 2(a) and 2(b). Blue and red arrows mark the projectile and the ejected soliton. Figure 2(e) summarizes a parameter study in which the effect of the projectile soliton duration and frequency on the molecule duration after collision was probed. As might be expected, the outcome of the collision process is influenced by the values of $t_P$ and $\omega _P$. For instance, Fig. 2(c) shows a collision resulting in a single molecule and non-solitonic radiation, whereas the scenario in Fig. 2(d) results in two molecules.

 

Fig. 2. Parameter study for different projectile solitons. (a) Spectrogram of the field in Fig. 1(d) at $z_0$. (b) Corresponding spectrogram at $z_{end}$. Arrows in (a),(b) indicate projectile and ejected soliton, respectively. (c),(d) Collision scenarios for two different projectile solitons [parameters detailed in (e)]. Trajectories of the molecule states before (gray) and after (magenta) the collision. (e) Molecule compression factor $F_C$ for different combinations of projectile soliton frequency and duration.

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The results in Figs. 1 and 2 demonstrate that the collision with a single projectile soliton results in a temporal compression, broader spectra, and higher brightness regarding the increase of energy in $\mathsf {ND}$. Aiming for even broader spectra and higher brightness, in Fig. 3 a cascaded collision process with multiple projectile solitons is shown. The temporal profiles and evolution in Figs. 3(a) and 3(b) reveal that a stepwise compression of the photonic molecule down to $t_M=1.1$ fs can be achieved, (succession of resulting widths: $8.6\,\rightarrow 6.4\,\rightarrow 4.4\,\rightarrow 2.5\,\rightarrow 1.1$ fs) corresponding to $F_C=7.8$. Each collision is accompanied by the ejection of radiation and a change of the molecules velocity. The spectral properties in Figs. 3(c) and 3(d) also highlight the stepwise characteristic of spectral broadening, where the energy located in region $\mathsf {ND}$ is strongly increased. The red and blue framed boxes show the filtered view of the molecule state after first and third collision, respectively, meaning that the background radiation as well as projectile and ejected solitons are filtered out. The filtered view after the third collision in Figs. 3(c) and 3(d) reveals the increased spectral energy in $\mathsf {ND}$. The temporal evolution shows that with the two last collisions, the velocity of the photonic molecules slightly increases again. This can be explained by the increased transfer of energy to the second phase-matched component at lower frequencies [see also Figs. 1(f) and 1(g)] after the third collision. This, in turn, leads to a rearranged energy distribution within the molecule state, which leads to a slight frequency shift, also visible in Fig. 3(c). It is noticeable that even if the molecule is subject to several collision impacts, the stable state does not decay. When generating a spectrally broad state with high brightness by multiple impacts, we induce an inhomogeneity of the state where energy within $\mathsf {A2}$ is continuously increased but decreased in $\mathsf {A1}$. When considering a further scenario (Scenario B) with access to shorter soliton durations, a collision between a soliton molecule with $t_M=4.7\,\mathrm {fs}$ and a single projectile soliton with $t_P=15$ fs results already in a significantly increased brightness [teal line in Fig. 3(c)]. This offers an alternative route to highly coherent supercontinuum generation, especially for ultrashort pulses.

 

Fig. 3. Cascaded collision process. (a) Profiles of initial (gray) and resulting (magenta) molecule state after a collision with four projectile solitons. (b) Evolution in time domain. (c) Spectra with initial (gray), resulting molecule states after the first (red), third (blue), and fourth (magenta) collisions, and an additional scenario (teal; details in text). (d) Evolution in frequency domain. The boxes show filtered evolution of resulting molecule state after first (red) and third (blue) collision.

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The outcome of the collision of a DW with a two-frequency molecule state was studied in Ref. [20]. Therein, the focus was on deriving and verifying resonance conditions that predict the excitation of scattered waves. However, the dynamics are much more complicated such that we complement this earlier study by investigating how the structure and behavior of a stable molecule state is affected by a DW impinging on its leading and trailing edges. In Fig. 4 the collision process of a compound state with initial duration $t_M=14.7$ fs with different DWs is shown. The amplitudes $a_{DW}$ of a colliding DW can be scaled via a parameter $N_{DW}$. For our case we chose $N_{DW}=2$ and $N_{DW}=4$. For each choice of a DW’s amplitude, we induce a collision with the leading and trailing edge of the molecule state. An exemplary collision event with the leading edge ($-$) with $N_{DW}=4$ is shown in Figs. 4(a) and 4(b) in time and frequency domain, respectively. Additionally, three further scenarios are investigated. In the time domain the molecules trajectory of each scenario is included; in the frequency domain the trajectories of the subpulse centroids in $\mathsf {A1}$ are highlighted as well as the respective gv-matched component in $\mathsf {A2}$. The exemplary case displayed in Fig. 4(b) shows that the trajectories of the gv-matched component (solid magenta lines) match very well with the subpulses centroids, revealing that the molecule state stays stable throughout the collision process. One can notice that due to the collision with the leading/trailing edge of the molecule state, the frequency centroids [Fig. 4(b)] of the subpulses are upshifted/downshifted resulting in an increased/decreased velocity of the molecule state [Fig. 4(a)]. This behavior is similar to the collision process of a single soliton with a DW [19]. To get a deeper insight into the inter-connectivity of the constituents, we approximate the single subpulses by fundamental soltions to mimic the interaction of each constituent in an unbound case. The parameters for the solitons are chosen to fit the intensities of the respective subpulse. Under this condition we get the soliton parameters $\omega _S=1.2/2.94\,\mathrm {rad/fs}$ with $t_S=33/25\,\mathrm {fs}$ approximating the subpulse in $\mathsf {A1}$/$\mathsf {A2}$. In Fig. 4(c) the comparison between the trajectories of the molecule (scenarios $N_{DW}=4, +/-$) and the respective trajectories of the approximated solitons is presented. It can be seen that different soliton trajectories match the molecules trajectories, e.g., for collision at the leading edge ($-$) the approximated $\mathsf {A2}$-soliton fits well, while for collision at the trailing edge ($+$) the $\mathsf {A1}$-soliton fits. This is in line with previous observations of inter-connectivity [24], where properties are transferred among binding partners, such that an omitted frequency downshift for the subpulse in $\mathsf {A2}$ due to spectral recoil, stops also the frequency shift for the subpulse in $\mathsf {A1}$.

 

Fig. 4. Interaction of a two-frequency molecule state with DW. (a) Evolution of collision scenario ($N_{DW}=4$, $\omega _{DW}=1.95\,\mathrm {rad/fs}$) in time domain. Trajectories with index $-/+$ (magenta/cyan line) indicate the collision with the leading/trailing edge of the molecule state, respectively. (b) Evolution in frequency domain. Lines in $\mathsf {A1}$ indicate the trajectory of the first subpulse, lines in $\mathsf {A2}$ denote the respective gv-matched component. (c) Trajectories in time domain of scenarios “$N_{DW}=4, +/ -$” and of unbound, approximated solitons (see text for details) colliding with the DWs.

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In conclusion, we demonstrated that two-frequency photonic molecules exhibit high robustness when subjected to collision with a projectile soliton as well as a DW. The collision with a projectile soliton leads to a molecule state transition, i.e., temporal compression, shift of frequencies, and spectral broadening. Our study of the influence of the projectile parameters reveals that for some scenarios energy can be exchanged among separated regimes of anomalous dispersion. We can cascade the collision process leading to a stepwise expansion of the spectrum, eventually leading to an octave spanning supercontinuum. In addition, our findings demonstrate that the interaction with a DW can lead to steering of the molecule by imposed frequency shifts on both subpulses. Even under the impact of a strong DW, the molecule is seen to remain stable, highlighting the strong inter-connectivity between the subpulses, where one subpulse can determine—through the binding partner—the behavior of the entire photonic molecule [24]. The presented interaction schemes offer new methods of controlling light and a broad range of possible applications including spectroscopy and effective supercontinuum generation. The possibility of energy transfer among the constituents and the influence of the frequency locations might be useful for logical gate operation.