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Abstract
Phase evolution of soliton and that of first-order sidebands in a fiber laser are investigated by using nonlinear Fourier transform (NFT). Development from dip-type sidebands to peak-type (Kelly) sidebands is presented. The phase relationship between the soliton and the sidebands calculated by the NFT are in good agreement with the average soliton theory. Our results suggest that NFT can be an effective tool for the analysis of laser pulses.
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1. Introduction
Fiber lasers have been widely applied and are considered among the first-choice platforms for ultrashort pulse generation [1,2]. Ultrashort pulses generated in fiber lasers are most often described as optical solitons. Strictly speaking, fiber lasers are dissipative systems. Pulses generated from a fiber laser is a mixture of solitons and sidebands [3]. The formation of optical solitons in a fiber laser is a balance of fiber dispersion and nonlinearity, cavity loss, laser gain saturation, and spectral filtering [4]. The output soliton spectrum of net-anomalous dispersion mode-locked fiber lasers contains a series of discrete sidebands [5]. Solitons periodically experience amplification and loss as they cycle in the cavity and adapt to the periodic disturbances by releasing dispersive waves (DW) [6]. DW is regarded as a buffer by which solitons modify their properties as well as interact with each other [7]. If the phase difference between a soliton and a DW accumulated over a round trip is a multiple of 2π.the DW will interfere constructively with the soliton and appear symmetrically as peak-type coherent sidebands (Kelly sidebands) in the laser pulse spectrum [8]. The formation of these coherent sidebands requires specific phase matching condition. At the same time, there are also incoherent sidebands when the mode lock operation is imperfect. Incoherent sidebands, appearing as background in the time domain, do not have a locked phase relationship with solitons [9]. Theoretically, a DW and a soliton can also interfere destructively, then dip-type sidebands emerge on the spectrum when the phase difference meets odd multiples of π [10–12]. Dip-type sidebands have been observed in different laser cavity structures [12–14] and in various mode-locked schemes [12,13,15], indicating that dip-type sidebands are a universal phenomenon in fiber lasers. An interesting feature of sidebands is their dependence on intra-cavity position. Because the phase difference between soliton and DW varies along pulse propagation inside the cavity, soliton and DW can interfere constructively or destructively at different cavity locations [7,12]. Understanding the dynamic characteristics of laser pulse inside the cavity is crucial for possible applications. For example, sidebands should be removed when the fiber laser is used a seed source for amplification [16].
The Kelly and dip-type sidebands are both examples of resonant DWs. The solitons and sidebands are coherent, making them difficult to be separated. So far, studies on the separation of resonant DW are few. Du et al. proposed a model to separate sidebands in which a DW is considered as the product of soliton-shaping rather than a dissipative process, and the model only applies to high pump strength situations [17]. To the best of our knowledge, no other method has been proposed to separate solitons from the resonant DW except for the nonlinear Fourier transform (NFT).
Developed from the inverse scattering method, the NFT has been widely applied as a powerful tool in optical communication systems to compensate for nonlinear damage [18,19]. In addition, NFT is also an effective approach to analyze nonlinear dynamics processes in nonlinear systems, including rogue waves [20], optical spectral combs [21], cavity solitons [22], optical micro-resonators [23] and fiber laser radiation [24–28]. As an analysis tool, NFT decomposes signals into nonlinear spectrum, including a continuous spectrum and a discrete spectrum, which represent DW and soliton components, respectively [24]. The continuous spectrum is defined on the real axis of the complex plane and, at the low power limit, converges to the Fourier transform. However, some low-order coherent sidebands, which also belong to the DW, appear in the discrete rather than continuous spectrum due to the locked phase relationship and high spectral intensity [26]. The discrete spectrum consists of two parts, which are discrete eigenvalues and the corresponding norming constants. In the nonlinear domain, the dynamics of soliton components can be characterized by the discrete spectrum [23]. NFT can separate soliton components into discrete spectrum. Although the discrete eigenvalue distribution always contains both soliton and low-order sidebands eigenvalues [26–28], their eigenvalues distributions are different. Sidebands are a kind of DW, which are coherently resonant with solitons. All the other non-coherent DWs can be denoted by the continuous spectrum. Consequently, by removing the continuous spectrum and the sidebands’ eigenvalues, only pure solitons are remained. It provides a picture of the evolution of solitons without DW and demonstrates an alternative technique to eliminate sidebands, enhancing the quality of the seed source. Recently, Wang et al. applied NFT in fiber lasers to characterize the pure soliton evolution process without DW and sidebands [25,26]. They separate sideband eigenvalues in discrete spectrum and obtain pure soliton pulses after inverse NFT (INFT) operation. Pan et al. get pure cavity solitons in Kerr resonators by using the NFT-INFT processing [22]. However, only the eigenvalues were used in the above work, while the norming constants associated with each eigenvalue were ignored. The norming constants, according to NFT theory, provide phase information for the corresponding nonlinear components. By using NFT, we can not only separate the solitons and sidebands but also investigate their phase relationship, which can reveal the complicated evolution process of laser pulses.
In this paper, the phase calculation method based on NFT is validated first. Then we investigate the evolution of the pulse inside the fiber laser in the nonlinear spectrum. We note that the NFT is applied to the analysis of pulse signals instead of solving the evolution equation. The solitons and sidebands can be easily separated after applying NFT and their phase relationships are given by norming constants. We analyze the phase evolution of soliton and sideband components. The results show that NFT can be used to recover the phase of soliton and sidebands, which provides a unique perspective for investigating the pulse evolution.
2. NFT principle and Simulation model
The fiber laser is illustrated in Fig. 1. It consists of a 3 m long Erbium-doped fiber (EDF) with a group velocity dispersion (GVD) parameter of 23 ps2/km and a 6 m long standard single-mode fiber (SMF) with a GVD parameter of 21 ps2/km. A 15% fiber coupler is used to output the signals. We noted that the laser parameters are so selected that specific pulses, for example solitons with dip-type sidebands, could be achieved under appropriate operation. The pulse propagating in the cavity can be described by the complex nonlinear Ginzburg-Landau equation (CGLE) [29]:
The CGLE can be considered as a disturbed nonlinear Schrodinger equation. Despite it being a non-integrable equation, we can still formally calculate the nonlinear spectrum at each position in the cavity after obtaining the pulse evolution data. This goal is achieved by solving the Zakharov–Shabat problem (ZSP) [32]:
According to the spectral quantities, the nonlinear spectrum can be defined as:
The discrete spectrum specifies the soliton parameters: amplitude $2{\mathop{\rm Im}\nolimits} ({{\lambda_i}} )$ and frequency $- 2Re ({{\lambda_i}} )$, while the norming constant ${\tilde{Q}_d}$ associating with each eigenvalue provides phase information [24]:
3. Results and discussions
To verify the accuracy of our method, we calculated the phase transition of a soliton signal propagating in lossless fiber in both the temporal domain (based on Eq. (1) with g = 0) and the nonlinear domain (based on Eq. (4)–(9)). We define the phase at the profile peak as the soliton phase. The soliton pulse is expressed as:$q(t )= Asech (t/\tau )$. Since we have adopted the theoretical expression of soliton, there is no need to normalize the pulse before the NFT analysis. $A\tau $ is set to 1, 2, which represent the first- and second-order soliton. Additionally, we validate the case:$A\tau = 1.3$,which deviates from ideal solitons. The temporal evolutions are depicted in Figs. 2(a), 2(d) and 2(g). The corresponding nonlinear spectra, including eigenvalue distribution and continuous spectrum, are shown in Figs. 2(b), 2(e) and 2(h). In the eigenvalue distribution, the real part of the soliton eigenvalue is close to zero, whereas the imaginary part is nonzero and satisfies:${\mathop{\rm Im}\nolimits} ({{\lambda_i}} )= A\tau + 1/2 - n$,where n are positive integers satisfying: $A\tau + 1/2 - n > 0$ [34]. The continuous spectrum energy of first- and second-order soliton are shown in the color map of Figs. 2(b), and 2(h), which are nearly zero. The soliton takes up the majority of the energy and dominates the pulse. The first- and second-order soliton phase calculated from the time domain and norming constants (NC phase) are depicted in Figs. 2(c) and 2(i), respectively. There is virtually no distinction between them when $A\tau = 1$. The nonlinear spectrum contains two eigenvalues when $A\tau = 2$, the imaginary parts are 0.5 and 1.5. The phase change speeds of two eigenvalues differ by 9 times fitting in with theory [35], The NC phase corresponding to the eigenvalue whose imaginary part is 1.5, is in good agreement with time domain calculation results. There is a periodic deviation that is related to the soliton's periodic evolution. The maximum deviation is 0.1081π. If solitons have several values in the discrete spectrum, the deviation will be higher, so we normalized the laser pulse to have a single eigenvalue in this paper. Figures 2(d) and 2(e) show the pulse evolution and the corresponding nonlinear spectrum when $A\tau = 1.3$. The imaginary part of soliton eigenvalue is 0.8. Because $A\tau $ is no longer an integer, the pulse evolution in time domain exhibits oscillatory behavior and the continuous spectrum contains much energy with 3-order increasing. Periodic errors begin to arise as shown in Fig. 2(f). The period of error coincides with the period of pulse oscillation and the maximum calculation error is 0.0408π.
Fig. 2. Pulse evolution in time and nonlinear domain. (a),(d),(g) Time domain evolution of soliton pulse for $A\tau = 1$ $A\tau = 1.3, A\tau = 2$; (b),(e),(h) the corresponding evolution in nonlinear domain; (c),(f),(i) the corresponding phase calculated from temporal waveform and norming constants (NC phase).
When applying NFT to analyze laser pulses, normalization of input pulse data is necessary. In fact, NFT was developed based on integrable NLSE. Since the NFT spectrum undergoes a nonlinear change with the variation of time and amplitude, it is important to choose a proper form of normalization at the start of the NFT analysis. One natural thought is to scale amplitude and time with pulse energy and duration, which are obtained directly from experiments [28]. One can also normalize the intensities of laser pulses with respect to the ideal fundamental solitons [27]. There is another propagation coordinate-dependent normalization condition proved to produce an effective model of laser radiation [28]. For the sake of simplification, we scale the amplitude that the discrete spectrum has only a single soliton eigenvalue and there are only first-order sideband eigenvalues across the cavity:
where ${A_n} = 1.2$. The value of ${A_n}$ relies on the maximum and minimum amplitudes of the pulses. We note that the solitons may have multi-value and the sidebands may fade away in the discrete spectrum if the scale constants are inappropriate.
We set ${G_0} = 370{m^{ - 1}}$ (corresponding to gain of 30.45 dB) and obtain a stable pulse output. The evolution of a pulse within a roundtrip with a step of 0.1 m is shown in Fig. 3(a) and the corresponding spectrum is plotted in Fig. 3(b). The pulse propagation is affected by discrete components, including EDF, SA, and output couplers. Pulse amplitude decreases obviously after output coupler and SA. In EDF and SMF, solitons interact with DWs, and the pulse evolves continuously.
Fig. 3. (a) Temporal profile of pulse evolution in cavity; (b) the corresponding optical spectrum and (c) eigenvalue distribution;(d),(f) the spectrum of pulse at different position; (e),(g) the temporal profile of (d) and (f), the insert of (e),(g) is the corresponding eigenvalue distribution.
In a fiber laser, the pulse experiences discrete perturbations during the evolution in the cavity. The energy exchange between the soliton and the DW will change the pulse's characteristics, such as amplitude and temporal width. In the time domain, the soliton is embedded in the DW and co-evolves in the cavity, whereas in the nonlinear domain, soliton and low-order sidebands can be easily separated from the DW background, because they exist separately in discrete and continuous spectrum. We can obtain pulse evolution data at any position in the cavity through the split-step Fourier method. Then, the nonlinear spectrum can be calculated after the normalization process, which allows us to analyze the dynamic properties of soliton and sidebands in the nonlinear domain.
Figure 3(c) depicts the corresponding eigenvalue distribution of the pulse. The X axis is the real part of the eigenvalue, the Y axis represents intra-cavity position, and the Z axis is the imaginary part of the eigenvalue. In contrast to the theoretical soliton, the eigenvalue distribution of the fiber laser output contains more than soliton eigenvalues. According to NFT theory, the imaginary part of the eigenvalue represents the amplitude of the corresponding temporal component, while the real part represents the frequency. The eigenvalue with an almost zero real part and a much larger imaginary part represents the soliton component, because the soliton is located at the central frequency and has zero velocity relative to the group velocity reference frame. The sideband components have a non-zero real part because they deviate from the pulse center in the frequency domain. Only the first-order sidebands are depicted in Fig. 3(c), the higher-order sidebands are still present in the continuous spectrum because of their low intensity.
According to the various position within the cavity, our model exhibits dip-type sidebands in addition to the Kelly sidebands, as seen in Fig. 3(d) and Fig. 3 (f). The corresponding pulse profile is shown in Fig. 3(e) and Fig. 3(g). The insets in Fig. 3(e) and Fig. 3(g) represent the respective eigenvalue distributions. According to the eigenvalue distribution, there is little difference between the two kinds of sidebands. The eigenvalues of two kinds of sidebands have almost the same real and imaginary parts (the difference is about 0.1). The imaginary part of the soliton eigenvalue is a 1.45 larger when dip-type sidebands occur.
Figure 4(a) quantifies the corresponding evolution in the nonlinear domain: the black line depicts pulse energy obtained by time domain integration; the blue line depicts discrete the spectrum energy ${E_d} = \sum\limits_{k = 1}^N {4{\mathop{\rm Im}\nolimits} ({{\lambda_k}} )} $; the green line represents the continuous spectrum energy ${E_c} = \frac{1}{\pi }\int\limits_{ - \infty }^{ + \infty } {\log ({1 + {{|{{{\tilde{Q}}_c}(\lambda )} |}^2}} )} d\lambda $; the red line illustrates the nonlinear spectrum energy ${E_d} + {E_c}$. We normalize the pulse energy in an effort to more clearly observe the evolution. The pulse energy is concentrated in the discrete spectrum, and the proportion of continuous spectrum energy is less than 10%. Figure 4(b) and 4(c) quantify eigenvalues’ imaginary and real part evolution in Fig. 3(c). During the intracavity cycle, the real part of the eigenvalues basically does not change and the change of the imaginary part of the eigenvalue is consistent with the evolution of the amplitude. We note that there are two lines describing the sidebands’ imaginary part evolution in Fig. 4(b). The two lines almost coincide because the imaginary parts of the two sidebands’ eigenvalues are almost the same. Figure 4(d) is the continuous spectrum that characterizes the intensity of DWs. Figure 4 depicts the pulse's evolution in the nonlinear domain. The intensities of the soliton, sidebands, and DWs all increase when propagating in EDF. Then, the Soliton and the DW begin to interact extensively in the SMF. The Soliton width reduces, and amplitude increases. The imaginary part of the soliton eigenvalue grew when propagating in EDF, and the continuous spectrum is strengthened too. It again demonstrates the validity of NFT in translating time-domain pulse evolution into the nonlinear domain. The continuous spectrum reduced after SA, while the first-order sideband maintains its stability. In addition to the eigenvalues, the discrete spectrum contains norming constants, which can represent the phases of the corresponding components in time domain. The phases evolution of solitons and sidebands, respectively, are calculated using the norming constants, and the results are illustrated in Fig. 5.
Fig. 4. Nonlinear spectrum evolution of the pulse in the cavity: (a) the energy evolution of temporal profile (black solid line), discrete spectrum (blue solid line), continuous (green solid line) spectrum and nonlinear spectrum (red solid line); (b) the quantization of eigenvalue evolution of imaginary part; (c) the quantization of eigenvalue evolution of real part; (d) the continuous spectrum evolution across the cavity.
Fig. 5. (a) Phase evolution of soliton; (b) phase evolution of first-order sidebands; (c) phase difference between soliton and first-order sidebands over a roundtrip.
The solid black line in Fig. 5 (a) illustrates the phase evolution of the pulse peak in the cavity, which we consider as the soliton phase. The phase calculated from the pulse in the time domain (black line) is in good agreement with the phase calculated from the norming constants (purple line). The maximum deviation is 0.1125π. The fact that pulses produced by fiber lasers are dissipative solitons rather than theoretical solitons is the primary cause of the discrepancy. The phase evolution of first order sidebands calculated from the norming constants are shown in Fig. 5(b). At the position of SA, both the soliton and the sidebands phases jump since we consider the saturable absorption as a lumped one. The sideband phases are unstable at 3.5 m, this is due to the large gains when propagating in EDF. It is well known that the Kelly sidebands and the dip-type sidebands correspond to constructive and destructive interference, respectively. The formation of the two types of sidebands depends on different phase conditions. Figure 5(c) depicts the phase difference between the soliton and the sidebands. During a round trip in the cavity, the accumulated phase difference between the soliton and the sideband (of which the eigenvalue’s real part is greater than 0) is 1.9728π and the accumulated phase difference between the soliton and the sideband (of which the eigenvalue’s real part is less than 0) is 1.9743π. Both of them are close to the resonant phase requirement of 2π. The deviation might be resulted from the step accuracy of the split-step Fourier method. Here the computing step is 0.1 m. Given that the cavity length is 9 m and the phase change is constant for 2π, the calculation accuracy is about 0.022π. Therefore, sidebands are indeed a product of the phase resonance between the solitons and DW. The phase difference between the soliton and the first-order sideband is close to 0 at position about 5 m in cavity and it increases to π at position about 8 m. The corresponding time domain pulses and spectrum are shown in Fig. 3(d) - Fig. 3(g). At position about 5 m, it is the Kelly sideband presenting symmetric spectral peaks. The dip-type sidebands appear at position about 8 m, which satisfy the destructive interference condition. The abrupt reduction of phase difference in the SA is caused by the assumption of lumped saturable absorption. The pulse at position about 5 m in time domain as shown in Fig. 3(d) has a larger amplitude and a higher pedestal since the pulse was amplified during propagation in EDF.
4. Conclusions
In this paper, we verified the validity of obtaining the phase information via norming constants in NFT. Based on the NFT, the dynamics of the pulse evolution within a round trip are characterized. The evolution of phases of the soliton and first-order sidebands are calculated using norming constants in the nonlinear domain. The result shows that the accumulated phase difference over a round trip between the first-order sideband and the soliton is about 2π. Dip-type sidebands appear when phase differences satisfy destructive interference condition. Under specific parameter setting, dip-type sidebands can turn into peak-type sidebands when the phase difference between the first-order sideband and the soliton accumulates integer times of 2π. Our findings are consistent with the theoretical predictions. It is demonstrated that the NFT method can provide full-field information including phase information. Consequently, NFT can be used as a new but effective analysis tool to investigate laser pulses.
Funding
Fundamental Research Funds for the Central Universities (HUST2020kfyXJJS007); the Open Research Fund of China-Poland Belt and Road Joint Laboratory on Measurement and Control Technology, Huazhong University of Science and Technology (MCT202203); Fund from Shenzhen Science and Technology Program (JCYJ20220530160607016); the Protocol of the 38th Session of China-Poland Scientific and Technological Cooperation Committee (Grant No. 6); National Agency for Academic Exchange in Poland PPN/BCN/2019/1/00068.
Acknowledgment
This work was supported by Fundamental Research Funds for the Central Universities (HUST 2020kfyXJJS007); the Open Research Fund of China-Poland Belt and Road Joint Laboratory on Measurement and Control Technology, Huazhong University of Science and Technology (MCT202203); Fund from Shenzhen Science and Technology Program (JCYJ20220530160607016); the Protocol of the 38th Session of China-Poland Scientific and Technological Cooperation Committee (Grant No. 6); National Agency for Academic Exchange in Poland PPN/BCN/2019/1/00068.
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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