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Abstract
The optical spectrum of mode-locked lasers can exhibit multiple peaks resulting from different mechanisms such as modulation instability, dispersive waves (DWs), and coupling between continuous waves (CWs) and DWs. The latter was recently reported in a mode-locked fiber laser. Here we show that besides the coupling between single-wavelength CW and DWs, dual-wavelength CWs can also couple with DWs giving rise to quite different spectral peaks in a mode-locked fiber laser. In particular, we find that the sidebands of one CW can couple with the other CW, leading to an enhancement of the CWs.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Mode-locked fiber lasers are building blocks of many photonics systems, ranging from material processing, ranging, and optical communications, to name a few [1,2]. Besides, they are also of great fundamental interest due to the rich physics involved [3–15]. Mode-locked lasers can emit solitons when the net cavity dispersion is anomalous. In this case, the soliton spectrum exhibits Kelly sidebands [16], as a result of interference between solitons and dispersive waves (DWs) which are emitted by perturbed solitons (the solitons are periodically perturbed by discrete intra-cavity components such as an output coupler and gain fiber). There is also another kind of spectral sidebands caused by modulation instability (MI) in the anomalous dispersion regime [17]. These sidebands exhibit as sub-sidebands on both sides of each Kelly sidebands, showing a threshold behavior. Although the MI is generally absent in a single-pass system with normal dispersion, cavity boundary conditions add another degree of freedom such that MI can also be excited in mode-locked lasers with normal dispersion [18]. Nonlinear effects play a key role in the formation of these spectral sidebands.
Due to the soliton energy saturation effect, the excess energy could transfer to continuous waves (CWs) when the soliton laser is pumped above a threshold. Recently, it is found that in a regime where the laser is mode-locked emitting solitons together with CW emission, there are spectral sidebands on both sides of CWs in a mode-locked fiber laser. This type of sidebands is generated by the interference between CWs and DWs, and nonlinear effects are not involved in the phase-matching condition between them. The positions of the sidebands depend on dispersion and the central wavelength of the CW components [19].
In this work, we report on the observation of complex spectral sidebands in erbium-doped mode-locked fiber lasers. The spectral sidebands are a result of the coupling between three waves. One is DW, and the others are two CWs with different central wavelengths. In particular, we find that the sidebands of one CW can couple with the other CW, leading to an enhancement of the CWs. Demonstrating the universality of coupling between different spectral components in a soliton laser, the results have the potential for wavelength-tuning of the fiber laser.
2. Experimental setup
Figure 1 shows the configuration of our mode-locked fiber laser. The gain medium consists of a 1.1 m Er-doped fiber (EDF) (OFS-EDF-80) with second-order dispersion of 61 ps2/km. The EDF is pumped by a 980 nm laser diode (LD) with a maximum output power of 450 mW through a 980/1550 wavelength division-multiplexer (WDM). The nonlinear polarization rotation (NPR) technique, which consists of two polarization controllers (PC) and a polarization-dependent isolator (PD-ISO), is used to start and sustain mode-locking. The pulses are extracted via an optical coupler (OC) with a 10% output ratio. The pigtail of all the components is single-mode fiber (SMF-28) with second-order dispersion of -21.7 ps2/km. The total length of the cavity is about 14.3 m. The net dispersion around 1550 nm of the cavity is -0.2193 ps2. The output pulses are measured by a 1-GHz oscilloscope (Tektronix, MDO3104), an optical spectrum analyzer (Deviser AE8600), and a commercial autocorrelator (APE, pulsecheck150).
3. Results and discussion
The mode-locking threshold of the cavity is about 30 mW. The pulse trains are shown in Fig. 2(a) with a repetition period of 69.75 ns, corresponding to a repetition rate of 14.336 MHz, which is determined by the cavity length. The optical spectrum of the pulse is shown in Fig. 2(b), with a 3-dB bandwidth of 6.476 nm and a center wavelength of 1570 nm. Strong Kelly sidebands presented on the spectrum shown in Fig. 2(b) are typical in the net anomalous dispersion regime due to the periodical perturbations, indicating the stable mode-locking operation [16]. We notice that the central wavelength of soliton shown in Fig. 2(b) is 1569 nm, which deviates from 1550 nm, a wavelength that has the largest gain in EDF. That is to say, the gain on the longer wavelength side is lower than that on the short wavelength side [20,21]. This explains the asymmetric Kelly sidebands shown in Fig. 2(b). The auto-correlation trace measured over a larger time window of 50 ps with a Sech fitting, as shown in Fig. 2(c). The AC trace is perfectly symmetric. The spurs on the AC trace are due to the single-shot measurement. The pulse duration is 1 ps, which is 2.5 times the Fourier transform-limited pulse (400 fs for a hyperbolic secant pulse with a 3 dB spectral width of 6.476 nm), indicating that the pulse is slightly chirped, which is attributed to the dispersion of the fiber pigtails. The mode-locking pulse train has a signal-to-noise ratio(SNR) up to 68 dB, as shown in Fig. 2(d), which further confirms the stable mode-locking.
Fig. 2. Experimental results of fundamental mode-locking operation: (a) optical spectrum; (b) pulse train; (c) autocorrelation trace; (d) RF spectrum.
The occurrence of the Kelly sidebands in the spectrum of mode-locked pulse in anomalous dispersion regime is due to the resonant coupling between the soliton and dispersion waves at some frequencies. Therefore, the average dispersion of the laser cavity can be retrieved from the position of Kelly sidebands in the optical spectrum. According to the phase-matching condition, one can calculate the net dispersion of the cavity following the formula $D = 4\pi ({\omega_2^2 - \omega_1^2} )$, where ${\omega _2}$ and ${\omega _1}$ are the angular frequencies offset of the two adjacent Kelly sidebands to the central wavelength of the soliton. For instance, the central wavelength of soliton is 1569.84 nm. Thus, the net cavity dispersion around 1570 nm is calculated to be -0.2201 ps2, which is close to the value of -0.2193 ps2 that is calculated based on the fiber dispersion parameters.
It is natural that in a mode-locked laser using NPR technique, the tunable transmission provided by NPR allows one to mode lock the laser at different wavelengths via adjusting the polarization states and the pump power. The wavelength tunability of the laser enables us to calculate the net cavity dispersion at different wavelengths according to the soliton center wavelength and its associated Kelly sidebands. As illustrations, we present three different states of optical spectrums in Fig. 3 (a)–(c), under a pump power of 230 mW, with different central wavelengths of solitons and hence different locations of Kelly sidebands. Note that we observed the coexistence of the soliton spectrum and CW components in Fig. 3(a) and 3(b) and dual-wavelength mode-locking as shown in Fig. 3(c). All these phenomena are quite normal in mode-locked lasers in high pump power conditions [22,23]. The inverted triangle indicates the frequencies of the Kelly sidebands that we use to calculate in the cavity dispersion retrieve process. Figure 3(d) summarizes the calculated net cavity dispersion versus wavelengths. The value of the calculated dispersion is crucial for us to figure out the mechanism of the appearance of the CW and its spectral sidebands later. The RF spectra of Fig. 3(b) and Fig. 3(c) are shown in Fig. 3(e) and Fig. 3(f), respectively. It worthy noting that the the operation of soliton with CW yields a RF spectrum with a low signal noise ratio, indicating large noise background due to the CW component. As a comparison, the RF spectrum in the dual-wavelength mode-locking regime features higher SNR due to the coherence of solitons. Thus, the RF spectra can help us distinguish different operation regimes.
Fig. 3. Different states of mode-locking operation: (a) CW with sidebands on the right of the soliton spectrum; (b) CW with sidebands on the left of the soliton spectrum; (c) dual-wavelength soliton spectrum; (d) the net cavity dispersion at different wavelengths calculated by the Kelly sidebands; (e) RF spectrum of (b); (f) RF spectrum of (c). The insets in (a)-(c) are the pulse train. The insets in (e) and (f) are the broad span RF output spectrum.
The coexistence of the CW with multiple spectral sidebands in the optical spectrum in a mode-locked laser is attributed to the phase matching between the CW and DW at certain frequencies [14]. Depending on the polarization states, the CW and its associated spectral sidebands can either appear on the left side (Fig. 3(a)) or the right side (Fig. 3(b)) of the spectrum. Different than the MI-induced sidebands, the location of the CW-induced spectral sidebands is insensitive to the intensity of the CW components, suggesting that these spikes are CW-induced resonant sidebands [17,18]. Figure 4(a) shows the spectrum of the CW and spectral sidebands under different pump power. Figure 4(b) shows the interval between the sidebands that is insensitive to the pump power, indicating that they are not MI-induced sidebands. To further confirm that these spectral sidebands are caused by the interference between DWs and CWs [19], we calculate the orders of each sideband using a phase-matching condition below [19]:
where m is the order of the CW sidebands, L is the length of the cavity, ${\beta _2}$ is the average dispersion of the cavity, ${\omega _m}$ and ${\omega _c}$ are angular frequency offset of the CW sidebands and CW from the soliton central wavelength, respectively. The sideband orders calculated from Eq. (1) are summarized in Table 1. The integer orders indicate that these sidebands are generated by constructive interference between DWs and CWs.Fig. 4. Enlargement of the CW sidebands in Fig. 3(b): (a) The spectra of CW sidebands under different pump power; (b) the interval between sidebands and CW under different pump power.
Table 1. The sideband orders calculated from the frequency (wavelength) offset of the CW and its sidebands from the central wavelength of solitons; C refers to the wavelength (frequency) offset between the CW and the soliton, Ln (Rn) refers to the offset between the n-order sideband on the left (right) of the CW.
Besides single CW-induced sidebands, we firstly observed double CWs induced sidebands, as shown in Fig. 5. For clarity, Fig. 5(b) is a magnified version of Fig. 5(a). The two CWs are marked as CW1 and CW2 with a wavelength interval of 4.336 nm. There are two small peaks denoted as CW2L1 and CW1R1 between CW1 and CW2, which refer to the left first-order sideband of CW2 and right first-order sideband of CW1, respectively. This relationship is identified by using Eq. (1). According to the net dispersion shown in Fig. 3(d), it is reasonable to assume the dispersion around 1593.53 nm (CW1) and 1597.85 nm (CW2) is -0.2203 ps2 and -0.2242 ps2, respectively. Table 2 shows the wavelength (frequency) offset (${\omega _m}$ and ${\omega _c}$) and the calculated sideband order. The sideband orders are close to integers, which confirms that these spectral sidebands are caused by phase-matching resonance between CW and DWs.
Table 2. The sideband orders calculated from the frequency (wavelength) offset of the CW and its sidebands from the central wavelength of solitons; C refers to the wavelength (frequency) offset between the CW and the soliton, Ln (Rn) refers to the offset between the n-order sideband on the left (right) of the CW.
Furthermore, by adjusting the PC, several wavelength intervals between CW1 and CW2 can be achieved. As shown in Fig. 6, the interval is 1.452 nm. One can see that there are two strong peaks marked as CW1 (1599.45 nm) and CW2 (1600.90 nm). Based on the net dispersion of -0.2262 ps2 at 1599.45 nm and -0.2272 ps2 at 1600.9 nm and using Eq. (1), we calculate the sideband orders, shown in Table 3. Again, the integer of the order for different sidebands confirms the phase matching between CW and its associated sidebands.
Table 3. Frequency offset from the soliton and phase-matching orders to the double CWs of the CWs sidebands in Fig. 6
Table 4. Measured and calculated wavelengths of CW1 and CW2
The overlap of the two sets of the spectral sidebands induced by CW1 and CW2 shown in Fig. 6 stimulates us to consider the interaction between CW1 and CW2. To confirm, we calculate the position of different orders of spectral sidebands of CW1(CW2) based on the measured spectrum using the phase-matching condition in Eq. (1), as shown in Table 4. For instance, the first-order spectral sideband of CW2 on the left-band side is calculated to be located at 1599.349 nm, which is perfectly superimposing on the CW1. The frequency offset between the CW1 and the CW2L1 is only 0.09 nm. Likewise, all the position of the calculated N-th order spectral sidabands of CW1 (CW2) are overlapped with the position of the N+1 order spectral sidebands of CW2 (CW1), showing an error less than 0.11 nm. These results confirm the interaction between the two independent sets of spectral sidabands.
4. Conclusion
In conclusion, we report for the first time the observation of complex spectral sidebands in an Er-doped mode-locked fiber laser, which is due to the resonant coupling of double CWs and DWs. The phase matching between the CWs and spectral sidebands is confirmed by calculating the spectral order based on the phase-matching conditions. In particular, by adjusting the polarization states and pump power, we observe an overlap of the two sets of spectral sidebands induced by two different CW components and reveal the enhancement of the spectral sidebands caused by the sidebands of the other CW.
We believe that our work is significant in the sense that it confirms that the spectral sidebands generated by interference between continuous waves and dispersive waves are universal, as our work demonstrates that double continuous waves also exhibit such dynamics and such a phenomena can be observed in a quite different laser parameters in terms of cavity lengths (the length of our laser is only 1/5 of that in Ref. [19]). Our work also opens the possibility of studying such a phenomena with more continuous waves, and since breathing solitons can also exhibit strong dispersive waves [24], our work also stimulates parallel research in the breathing soliton regime. Furthermore, it can stimulate the numerical simulations of such phenomena using a discrete model based on nonlinear Schrödinger equation. Finally, our work again shows that mode-locked fiber lasers contain a wide range of complex regimes with different spectral dynamics. An interesting study could be using machine learning to actively control theses rather different laser dynamics [25, 26}.
Funding
National Natural Science Foundation of China (61665002).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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