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Abstract
We report experimental measurements of structural soliton pairs in a mode-locked Yb-doped fiber laser. The quantization of the temporal separation and relative phase between the pair of dissipative solitons is clearly observed and, most interestingly, the transitions between different states as the pump power is varied show abrupt jumps. This is a clear signature of the discrete nature of the stable pair states. The typical separations are of the order of 1 ps and the separation changes are of the order of 100 fs, while the relative phase of the pulses jumps between π/2 → π/4 → π/2 as the pump power is increased.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The mode-locked operation of fiber lasers is strongly determined by the intrinsic intracavity optical nonlinearities. As a result, these systems are particularly rich in a multitude of different nonlinear dynamic phenomena they may present. Mentioning few examples, deterministic chaos [1], rogue waves [2], soliton explosion [3] and multi-pulsing regime [4,5] have all been observed in this kind of system. The interest in mode-locked fiber lasers (MLFL) is also justified by the many applications they find in different areas, enhancing the importance of a better understanding of the generation and manipulation of optical solitons. It has been demonstrated that solitons can be manipulated by controlling the MLFL cavity parameters, like intensity and dispersion. MLFL are dissipative systems and therefore can generate dissipative solitons. This kind of soliton can exist in several regimes, ranging from single to multi-pulse ones [5]. This last case is also refereed to as multi-soliton bound state, or more oftenly soliton molecules, and has attracted much interest since Malomed [6] first predicted the existence of bound solitons in dissipative systems.
Even being fundamentally different, the analogy between solitons and atoms is interesting because of the similarities in behavior of waves and particles. Single solitons can be bounded and form two [7], three [8], and even $2 + 2$ soliton molecular complexes [9]. For optical soliton molecules, the relevant internal degrees of freedom are temporal separation, $\tau$, and relative phase, $\Delta \phi$ [4,9]. Observing how $\tau$ and $\Delta \phi$ evolve is important because it gives further information about the bound state of the soliton molecules, such as quantization of the temporal separation [7] and the oscillations observed in the relative pulse phase [10]. To measure these parameters, the Dispersive Fourier Transform (DFT) [11] is a real time technique that provides $\tau$ and $\Delta \phi$ as the solitons evolve per round trip in the laser cavity [9,10]. Despite being a powerful technique, it requires relatively fast oscilloscope and photodetectors, which may not be available. On the other hand, performing average measurements with pulse trains is simpler to implement and still gives reliable information about $\tau$ and $\Delta \phi$. These parameters can be measured as function of the laser pump power, which will be the focus of this work.
The quantized temporal separation in optical soliton pair molecule was first reported in an Er MLFL by Soto-Crespo et al. [7]. Wang et al. [12] reported this effect in a Tm MLFL. Both references explain this phenomenon using Kelly side bands, where the energy of the pulses is redistributed in the laser cavity as the amplitude of the pulse gets higher. Gui et al. [13] reports discrete behavior of $\tau$ in an Er MLFL with a saturable absorber of carbon nanotubes. They consider the effect as a result from the balance of repulsive and attractive forces caused by nonlinear effects (cross phase modulation, saturated gain and saturated absorption) and dispersion. They also claim the change of $\tau$ as the pump power grows but they do not show its evolution nor a clear signature of the quantized jumps from one separation to the next. None of these references claim that the phase was also quantized.
In this work we present the evolution of temporal separation and relative phase for a bound soliton pair in an Yb MLFL, showing discrete jumps for both parameters. The optical spectra and autocorrelations are measured by averaging over pulse trains, as the laser pump power varies. Three different values for the distance between the solitons were observed, with the relative phase following the same behavior.
2. Experiment
The experimental setup is schematically represented in Fig. 1(a). The laser cavity has a ring configuration with unidirectional operation, ensured by an optical isolator (FI). The gain medium is a highly Yb-doped fiber CorActive Yb214, with absorption coefficient of 1348 dB/m at 976 nm and length of 22 cm. The pump source is a fiber-coupled diode laser (DL), driven by a temperature and current controller, emitting at 976 nm. The pump is coupled into the gain medium through a wavelength division multiplexer (WDM), with an efficiency of 89%. A pair of GRIN collimators is used to couple the beam from the fiber medium to the free-space section and then back to the fiber medium, providing optical grating with an efficiency of 35%. To manage the total intracavity dispersion, a grating pair with 600 lines/mm and reflectivity of 65% per pass is used. The proper adjustment of the distance between the grating pair sets the operation of the laser at either normal or anomalous net dispersion regime. For the results presented here, the gratings are separated by 3.1 cm, leading to a negative overall cavity dispersion of $\approx -0.004$ ps$^{2}$, which is slightly anomalous. Passive mode-locking is achieved (Fig. 1(b) and (c)) via nonlinear polarization rotation (NPR), although other mode-locking techniques are also available using 2D material-based such as graphene as ultrawide-band saturable absorbers [14–16]. A set of waveplates and a polarizing beamsplitter (PBS) are included to control the polarization of the light, where the PBS also acts as an output coupler. When mode-locked, the laser presents a fundamental repetition rate of 120 MHz and optical spectrum centered in 1017 nm.
Fig. 1. (a) Experimental setup: diode laser (DL), wavelength division multiplexing (WDM), Yb-doped fiber (YDF), quarter-waveplate ($\lambda$/4), half-waveplate ($\lambda$/2), polarizing beamsplitter (PBS), grating pair (GP), Faraday isolator (FI), GRIN collimators (C). (b) RF spectrum and (c) time series when the laser is in mode-locked regime.
In order to obtain the double pulse regime the procedure is the following: first, the Yb laser must be in the mode-locked regime, with a slightly negative net group dispersion. The pump power must be high compared to the minimum power necessary to achieve mode-locking since it is widely known that multiple pulses per round trip are favored by this condition. There is a relatively small range of values for the waveplates where this happens for our laser. In the sequence, the half-waveplate is tuned to increase the output power (therefore decreasing the feedback to the laser cavity). The spectrum of the output pulses is continuously monitored. Finally, at a certain point, the output power suffers an abrupt decrease and the spectrum will show a transition from one to two pulses with the appearance of the fringes, which are characteristic of this regime.
To initiate data acquisition, $I_{\textrm {pump}}$ is decreased to the verge of the single pulse mode-locked regime. From there, $I_{\textrm {pump}}$ was systematically incremented at regular steps of 4 mA, keeping all the others parameters fixed, such as the orientation of the waveplates and the distance between the grating pair. Meanwhile, the average optical spectrum and the interferometric autocorrelation of the signal were continuously monitored. Data acquisition was automated by a custom LabVIEW program, involving an optical spectrum analyzer (Ocean Optics HR4000), an 1 GHz oscilloscope (Keysight DSO7104B) and a home-made interferometric autocorrelator connected to the oscilloscope. At each step, the optical spectrum and the oscilloscope trace of the autocorrelation were acquired and processed. The experimentally controlled parameter is $I_{\textrm {pump}}$ with current to power conversion factor of 0.605 mW/mA, already taking into account the coupling efficiency through the WDM. In the discussion that follows, all the results will be presented as a function of pump power, $P_{\textrm {pump}}$.
3. Results and discussion
Proper adjustment of the waveplates realizes double-pulse bound state for a pump power of 315 mW. This regime is repeatable within $\pm 2^{0}$ of the waveplates position and for the dispersion of $-0.004$ ps$^{2}$. This regime leads to a strong modulation in the optical spectrum, which is a consequence of the coherence between the pulses. In the time domain, two pulses delayed by $\tau$ and with relative phase $\Delta \phi$ can be described as:
where $E_{1}(t)$ and $E_{2}(t-\tau )$ are the amplitude of the slowly varying envelope of leading and trailing pulses, respectively. $\omega _{0}$ is the central angular frequency of the laser. In the frequency domain, this corresponds to:Figures 2(a)-(c) show the optical spectra of two bound solitons. The dots correspond to experimental data, while the red lines are numerical fits using Eq. (3). The optical spectrum is recorded as the pump power is increased. An interesting phenomenon is observed when $P_{pump}$ is raised from 315 mW to 460 mW (Fig. 2(d)). At first, the modulation period is $\Delta \lambda \approx$ 3.15 nm, corresponding to a distance between the solitons of 1.09 ps. This modulation remains practically constant until $P_{\textrm {pump}}$ reaches 345 mW and an abrupt transition indicates that the temporal separation has changed. The solitons are now closer ($\tau =$ 0.97 ps) and the spectrum presents a modulation period of $\Delta \lambda \approx$ 3.57 nm. Increasing $P_{\textrm {pump}}$ further leads to another transition at 411 mW, when the period jumps to $\Delta \lambda \approx$ 3.88 nm, indicating that $\tau$ stabilizes at 0.89 ps. The sudden changes in the spectrum reveal the discrete nature of the separation between the bound solitons with respect to the total intracavity energy. It is in agreement with theoretical predictions of the quantization of the temporal separation between soliton molecules, as well as of the binding energy of two bound solitons [7,19]. The temporal overlap of the pulse pairs exiting the cavity is very small, but one must remind that our laser operates in the stretched-pulse regime [20], and the pulse duration varies dramatically during one round trip. In the region of maximum stretch the pulse overlap increases and the nonlinear cross-phase modulation (XPM) leads to an effective attractive force between the pulses [13,21]. Hence, increments in the pump power foments a more pronounced XPM, bringing the pulses closer as $P_{\textrm {pump}}$ increases.
Fig. 2. Average spectra recorded by the OSA: (a) - (c) Experimental data (dots) and fitting results (red lines) for different pump powers; (d) Evolution of optical spectrum with pump power.
The interferometric autocorrelation of the output signal is also monitored during the experiment, presenting an excellent agreement with the temporal separations indicated by the modulated spectra. The oscilloscope autocorrelation traces, normalized to 8 (arbitraty units), are presented in Fig. 3(a)-(c) for some representative cases. The side peaks are the interferometric cross-correlation of the bound pulses, which are expected to have a peak value of 4.5 in the case of pulses presenting the same amplitude and fixed phase relationship [22]. That is not the case for the presented data, which may be an indication that the pulses do not have the same amplitude. In this situation, they present different phase velocities due to the Kerr effect, followed by different phase shifts per round trip, which could be responsible for the reduced optical visibility observed in the optical spectra [17]. Figure 3(d) shows how the autocorrelation evolves with $P_{\textrm {pump}}$. The discrete nature of the distance between the pulses is evident. The values of $P_{\textrm {pump}}$ for which the jumps occur are in full agreement with those reported in the analysis of the optical spectrum. This experimental observation is in conformity with the theoretical predictions reported in [19], in which the relative positions allowed for bound states are separated by approximately the same temporal distance, in this case, approximately 100 fs.
Fig. 3. Interferometric autocorrelation for different pump powers, showing temporal separation of approximately (a) 1.09 ps, (b) 0.97 ps and (c) 0.89 ps. (d) Evolution of interferometric autocorrelation with pump power.
The phase of optical soliton molecule is also studied for nonconservative systems in order to infer how stable these molecules can be. Using standard perturbative analysis for soliton interaction, Malomed [6] has shown that stationary stable solutions for two solitons in the form of bound states can be in-phase (0) or out-of-phase ($\pi$). Using a variational method for the cubic-quintic Ginzburg Landau equation, Akhmediev et al. [4] has reported that there are stable solutions for two solitons with a $\pi /2$ phase difference between them. Komarov et al. [19] reported that the temporal distances and the relative phases between solitons in a steady state have quantized values to minimize nonlinear losses of the field structure. They also verified that phase values of 0, $\pi$ and $\pi /2$ between a soliton pair can coexist.
In Fig. 2(a)-(c), the average optical spectra are asymmetrical, meaning that relative phase is neither zero nor $\pi$. A phase difference of $\pi$/2, which leads to spectral asymmetry, is often related to bound solitons in anomalous dispersion regime, by both experimental observation and numerical analysis [4,7,22,23]. But any phase other than an integer multiple of $\pi$ will also result in asymmetries in the spectrum. The evolution of $\tau$ and $\Delta \phi$ with $P_{\textrm {pump}}$ we obtain from our experimental data shown in Fig. 4. As indicated by the evolution of the optical spectrum and the autocorrelation, $\tau$ remains essentially constant at each window, before the sudden transitions that occurs at 345 mW and 411 mW. The relative phase follows similar behavior. At first, when the pulses are 1.09 ps apart, the average relative phase is close to but less than $\pi /2$. Increasing $P_{\textrm {pump}}$, as the pulses jump to a separation of 0.97 ps, the phase difference also changes to $\pi$/4. Only in the last scenario, when the pulses stabilize at 0.89 ps, the relative phase has values around $\pi$/2. The polar representation in Fig. 5 provides a better visualization, making clear the discrete nature of the transitions.
Fig. 4. (a) Dependence of temporal separation and (b) relative phase between solitons as function of pump power. The black circles are the experimental data and the red lines are the average values for $\tau$ (1.09 ps, 0.97 ps and 0.89 ps) and $\Delta \phi$ (0.43$\pi$, 0.22$\pi$ and 0.51$\pi$) in each pump power window.
Fig. 5. Polar representation for temporal separation and relative phase between solitons, as the pump power is varied. $\tau _1 = 1.09$ ps and $\left \langle \Delta \phi \right \rangle = 0.43\pi$ (blue), $\tau _2 = 0.97$ ps and $\left \langle \Delta \phi \right \rangle = 0.22\pi$ (red), and $\tau _3 = 0.89$ ps and $\left \langle \Delta \phi \right \rangle = 0.51\pi$ (green). The big symbols, blue diamond, red circle and green triangle are the average values of $\Delta \phi$.
Despite the averaging process over approximately $10^{5}$ bound solitons, which is inherent to our data acquisition, we can say that the values of the relative phase fluctuate around an approximately constant value $\left \langle \Delta \phi \right \rangle$ for each window of $P_{\textrm {pump}}$. As shown in Fig. 4(a)-(b), the deviation in $\Delta \phi$ is greater than in $\tau$. It is a consequence of the limitation in determining the center frequency, $\omega _{0}$. The evolution of $\omega _{0}$ with $P_{pump}$ obtained from our fitting procedure presents fluctuations around an average value of $\omega _{0} = 1852.8(2) \times 10^{12}$ rad/s. These fluctuations are insufficient to account for the magnitude of the discrete jumps observed in $\Delta \phi$. Moreover, the jumps happen at two specific points that coincide for both $\Delta \phi$ and $\tau$. Inside each window, there is a clear tendency around the previously claimed values for $\Delta \phi$ and $\tau$, stressing the discrete nature that both parameters present with increments in total intracavity energy. None of the previous works concerning quantized temporal separation between soliton molecules have shown clearly this discrete evolution.
4. Conclusion
In conclusion, we report the experimental observation of quantized temporal separation between two bound solitons in an Yb doped mode-locked fiber laser. As the total intracavity energy is increased, we observe transitions between the bound soliton states accompanied by transitions in the relative phase of the pulses. We observe that the temporal separation $\tau$ undergoes jumps of $\approx 100$ fs and $\Delta \phi$ takes average values close to $\pi /2$, $\pi /4$ and $\pi /2$ for each $P_{pump}$ window. Both $\tau$ and $\Delta \phi$ take discrete values to minimize nonlinear losses in the solitons field structure [19]. As the laser operates in the stretched-pulse regime, our experimental results indicate that the overlap of the pulses promotes a direct nonlinear interaction responsible to bring the pulses closer at larger intracavity energy.
Funding
Conselho Nacional de Desenvolvimento Científico e Tecnológico (441668/2014-3, INCT-IQ 465469/2014-0); Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (23038.003382/2018-39, PRONEX 534/2018); Fundação de Amparo à Ciência e Tecnologia do Estado de Pernambuco (APQ - 1178 - 1.05/14).
Acknowledgments
We thank M. D. Croitoru and B. A. Malomed for fruitful discussions.
Disclosures
The authors declare no conflicts of interest.
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