During the last decades, soliton fiber lasers have attracted great attention because of their excellent stability and compactness [1]. They have been widely used in remote sensing [2], optical communication [3], laser processing [4], and laser medical treatment [5]. The operating mechanism of a soliton laser can be understood by the use of the so-called average soliton model [6]. One of the most notable features of a soliton laser is that multiple sidebands can be resonantly generated if the cavity length is comparable to the dispersion length. Explaining the formation mechanism of these resonant sidebands aroused intense interest in the 1990s [7–11]. It can be explained well with the soliton perturbation theory [7]. The soliton that experiences periodical disturbances would simultaneously release the excess energy as dispersive waves during iterations inside the laser cavity. At certain wavelengths where the dispersive waves are phase-matched to the soliton, resonant sidebands are formed [7,8]. The resonant sidebands are typically distributed along both sides of the soliton spectrum, and the spectral intensities of the sidebands usually decrease exponentially as their central wavelengths shift away from the soliton.

Apart from manifesting themselves as notable features in the spectrum of soliton lasers, the resonant sidebands can have substantial influences on the soliton dynamics [12,13]. For instance, the resonant sidebands were shown to be able to induce bound states by acting as a long-range oscillating interaction potential [14], and lead to long-range soliton interactions from supramolecular structures in a mode-locked fiber laser [13]. From an application point of view, although the resonant sidebands were considered to be detrimental in some aspects such as leading to timing jitter [15] and imposing limitations on soliton amplification efficiency [16], they have been shown to be useful in many applications. For instance, the resonant sidebands have been used for dispersion measurement [10]. Recently, the resonant sideband has been leveraged for resonant coupling with a continuous-wave seed, which leads to the mitigation of supermodel noise [17]. It has also been shown that the resonant sidebands may find potential use in THz signal generation [18]. Moreover, since multiple resonant sidebands on both sides of the soliton spectrum are emitted simultaneously from the soliton, once their intensities are efficiently enhanced, they are inherently a good synchronized multiple-wavelength pulsed laser source. Therefore, generating multiple resonant sidebands with high spectral intensities is favored. However, an efficient way of achieving multiple resonant sidebands enhancement is still lacking.

In this paper, we tackle this problem through both numerical simulations and experiments. In particular, we demonstrate that by properly tailoring the gain profile and the nonlinear polarization evolution (NPE) transmittance, a number of resonant sidebands can be significantly enhanced. Both the spectral and temporal characteristics of the enhanced sidebands are numerically investigated. An explicit formula was derived by taking the third-order dispersion (TOD) effect into account, with which the wavelength of all resonant sidebands can be accurately predicted. In the experiment, we generated intense resonant sidebands whose maximum spectral intensity is more than 30 dB higher than that of the soliton, which is the highest yet reported.

We first use numerical modeling to illustrate the mechanism for the generation of unilaterally enhanced resonant sidebands. The schematic of the laser is shown in Fig. 1, which is a unidirectional ring cavity design. Segment AB consists of 2 m Er-doped fiber (EDF, LIEKKITM Er80-8/125). Segment BC consists of two inline polarization controllers (PCs) and a polarization-sensitive isolator (PS-ISO), which work as an artificial saturable absorber. 10% of intracavity power was extracted at point D via an output coupler. The EDF was pumped by a 980 nm laser diode (LD) via a 980/1550 nm wavelength-division multiplexer (WDM). Segments BC, CD, and DA consist of mainly Corning SMF-28 and have a length of 5.5 m, 1.21 m, and 1.51 m, respectively. The total cavity length is 10.22 m. The fiber laser is mode-locked by the nonlinear polarization evolution technique. The output pulse was diagnosed by an optical spectrum analyzer (Yokogawa, AQ6370D) and a digital oscilloscope (Keysight, DSOS404A).

We adopted a vector model to model the nonlinear polarization evolution of the pulse in the laser cavity. An iterative map was used with appropriate transfer functions for each cavity element [19–21]. The propagation of the two polarization components of the electric field was modeled by the coupled nonlinear Schrödinger equation [20,22]: $$\begin{gathered} ∂u∂z=+iΔβ02u−Δβ12∂u∂T−iβ22∂2u∂T2+β36∂3u∂T3+iγ(|u|2+23|v|2)u+iγ3v2u*+g^2u, \\ ∂v∂z=−iΔβ02v+Δβ12∂v∂T−iβ22∂2v∂T2+β36∂3v∂T3+iγ(|v|2+23|u|2)v+iγ3u2v*+g^2v, \end{gathered}$$(1)where u and v represent the electric field complex envelope along the two principal axes. The weak (bend-induced) birefringence in each segment is included via the parameter Δβ0=2π/LB, where LB is the beat length. Here a value of LB=2.0m was used for all segments, which is reasonable considering the bending of the fibers in the experiment [23,24]. The group index term was calculated from the approximation Δβ1≈Δβ0/ω0. The key components of the artificial saturable absorber were composed of two polarization controllers and a polarization-dependent isolator. They were modeled with the Jones matrix. For simplicity and without loss of generality, we simplify the two PCs, which have six paddles in total with four wave plates; thus the configuration is similar to that in Refs. [20,25,26]. In particular, the pulse propagation in the BC section was modeled by propagating the pulse through an equal length of the fiber and then sequentially passing through a quarter-wave plate (QWP1, α1), a half-wave plate (HWP1, α2), a polarizer (αp), a second half-wave plate (HWP2, α3), and a second quarter-wave plate (QWP2, α4). As will be shown below, numerical results calculated with such a simplified model show good agreement with the experiment, while the dimension of the parameter space searched was significantly reduced. The wave plate angles {α1,α2,α3,α4} are scanned to find different mode-locked solutions. We note that the angle of the transmission axis of the polarizer αp was set to zero in all the simulations [19].

The gain term g^(ω) is non-zero only in the EDF segment and is modeled by g^(ω)=g01+EEsatG(ω),(2)where g0 is the unsaturated small-signal gain, E=∫(|u|2+|v|2)dτ is the intracavity pulse energy, and Esat represents gain saturation energy parameter. The gain spectral profile was modeled with a superposition of two Gaussian functions given by G(ω)=c1exp(−(ω−c2)2c32)+c4exp(−(ω−c5)2c62),(3)where the coefficients ci, i=1,2,…,6 were varied to investigate the impact of the gain profiles, and their values will be specified in the following context. High-order nonlinear terms such as self-steepening and Raman effects are not considered in the model because of their negligible impact on the results shown in this paper. In the simulation, the central wavelength is 1613 nm. Since the dispersion and nonlinear parameters of the SMF and EDF are very similar, the same set of parameters was used for both types of fiber. In particular, we set β2=−25.4fs2/mm, β3=+130fs3/mm, and γ=0.0011 at 1613 nm. Another benefit of this simplification, as will be seen later, is that it is more straightforward to test the formula of the resonant wavelength derived in this paper. We used a sech pulse of 1 ps as the initial seed for searching for the stable mode-locked solution of the laser. In typical runs of our simulations, it takes 300–500 roundtrips for the iteration to well converge (the relative error of the pulse energy or pulse duration between consecutive roundtrips is below 10−4) to solutions with significant sideband enhancement.

In the next two subsections, we illustrate the impact of the saturable absorber and the gain profile, which are the dominant effects that lead to the multiple resonant sideband enhancement phenomenon.

We first illustrate the impact of the NPE transmittance on the resonant sideband generation. To calculate the transmittance (TSA) of the artificial saturable absorber (NPE mode-locker), the open-loop transfer function of the laser cavity was numerically calculated. In particular, a sech-shape signal pulse with a full width at half maximum (FWHM) duration of 10 ps was propagated for a single roundtrip. The propagation was initiated after the polarizer with a signal that polarized in parallel to the polarizer’s transmission axis. TSA was defined as the ratio of the output pulse energy to that of the input pulse. We note that the gain in the EDF was switched off to avoid the significant power variation caused by amplification, especially for a small signal input [19]. Although a sech pulse was used as the input signal, we have checked that a continuous wave input leads to similar transmission curves. Since TSA is a function of both the peak power P0s and central wavelength λ0s of the input signal, P0s and λ0s were scanned to investigate the linear and nonlinear transmission function of the laser cavity.

Three typical cases are considered. (i) The birefringence is switched off, which is equivalent to setting the beat length LB=+∞ in Eq. (1), and the wave plate angles are {2.372, 0.120, 0.035, 2.960} rad; in cases (ii) and (iii), the birefringence term is switched on with the beat length LB=2m. However, the wave plate angles are different for case (ii) and case (iii), which are {1.265, 0.5655, 1.508, 2.827} rad and {1.880, 0.1604, 0.7086, 2.787} rad, respectively.

The calculated transmittance curves for these three cases are shown in Fig. 2. The transmission spectra at both low (1 W) and high (450 W) input peak powers are calculated for each case. In particular, Figs. 2(a), 2(b), and 2(c) correspond to cases (i), (ii), and (iii), respectively. In each plot, the blue and green curves (left axis) correspond to the transmittance spectra at input peak powers of 1 W and 450 W, which correspond to the linear propagation of the dispersive waves (resonant sidebands) and nonlinear propagation of the soliton, respectively. The red filled curve (corresponding to the right vertical axis) represents the spectrum of the mode-locked soliton solution. We note that the gain of the EDF was switched on to calculate the mode-locked soliton with the saturation energy Esat=0.26nJ, 0.31 nJ, and 0.27 nJ for cases (i), (ii), and (iii), respectively. It was adjusted such that the average peak power of the soliton in each case is ∼450W. For results shown in Fig. 2, the gain spectrum is the same for all cases with a Gaussian profile centered at 1615 nm with a bandwidth of 40 nm. This corresponds to setting the coefficients in Eq. (2) as c1=0, c4=1.0, c5=0rad/s, c6=1.731×1013rad/s, and the small signal gain coefficient g0=1.03m−1.

It can be seen that the transmission spectrum is wavelength independent when the birefringence is turned off as shown in Fig. 2(a). As a result, the soliton spectrum is nearly symmetric. The slight asymmetry of the resonant sidebands on both sides of the soliton is attributed to the third-order dispersion (TOD) effect. We have checked that a perfectly symmetric spectrum can be obtained by switching the TOD off. In contrast, as can be seen from Figs. 2(b) and 2(c) that the transmission spectra become wavelength dependent when the birefringence is turned on. In case (ii), the transmittance decreases as the wavelength increases at both low and high input powers. However, the modulation depth (the difference between the transmittance at 450 W and 1 W) increases as the wavelength increases. In this case, the spectrum of the mode-locked pulse is highly asymmetric as shown in Fig. 2(b). In particular, the intensity of the resonant sideband on the blue side of the soliton is significantly enhanced. It is known that the resonant sidebands are low-power linear dispersive waves in the time domain, which can also be seen from Fig. 5 as will be shown later. Therefore, the lower linear transmission loss at shorter wavelength results in a higher spectral intensity of the resonant sideband therein. In contrast, the transmittance in case (iii) increases as the wavelength increases at both low and high input powers. Thus, the resonant sideband on the red side of the soliton is significantly enhanced as shown in Fig. 2(c). Therefore, it is possible to change the sideband asymmetry by changing the polarization controller while keeping the gain unchanged.

The transmittance as a function of the input power (saturable absorption curve) at three wavelengths (1600 nm, 1613 nm, and 1625 nm) was also calculated. Figures 2(d), 2(e), and 2(f) correspond to cases (i), (ii), and (iii), respectively. It can be seen that all saturable absorption curves generally show typical cosine profiles [19,23]. The transmittance increases as the input power increases and saturates at the power of ∼650W. When the power is further increased, an inverse saturable absorption may occur, which will lead to negative feedback [19]. As can be seen from Fig. 2(d), when the birefringence is switched off, the transmittance curves at different wavelengths are identical. This results in an identical modulation depth with q0=0.19 at each wavelength. In contrast, in case (ii), the linear transmittance (at ∼1W) is higher at a shorter wavelength, while the saturated transmittance (at ∼650W) is nearly the same for all wavelengths. Therefore, the modulation depth is lower at a shorter wavelength, with q0=0.09, 0.13, and 0.16 at 1600 nm, 1613 nm, and 1625 nm, respectively. As will be seen in the next subsection, this wavelength-dependent modulation depth is critical for stabilizing the soliton wavelength when an asymmetric gain profile is adopted, and thus it is key for multiple resonant sideband enhancement. The transmittance curves in case (iii) show similar properties, however, with the difference that the transmittance curve at a longer wavelength lies above that at a shorter wavelength as shown in Fig. 2(f). Therefore, the modulation depth is higher at shorter wavelengths, with q0=0.15, 0.11, and 0.08 at 1600 nm, 1613 nm, and 1625 nm, respectively.

The mode-locked solutions of a laser cavity result from a balance between loss and gain. Since the loss spectra can significantly influence the sideband intensity, it is reasonable to expect that a suitably tailored gain profile can lead to further enhancement of the sidebands.

Three different gain profiles are selected to illustrate their influences on the spectrum of the mode-locked pulse. The bandwidth and central wavelength of the two Gaussian components in Eq. (2) are fixed by setting the coefficients as c2=2.5×1013rad/s, c3=1.7556×1013rad/s, c5=0rad/s, and c6=1.731×1013rad/s. However, the ratio between the weights of the two Gaussian components is varied to change the overall gain landscape. The ratio c1/c4 is 0, 0.75, and 1.0 for the bottom, middle, and top gain profiles, respectively, as shown in Fig. 3(a). The gain profiles are normalized in each case. At first sight, one may expect that the intensity of the resonant sidebands can be easily enhanced by simply increasing the gain on either side of the soliton. However, this is not true as will be explained in the following text.

To discuss the impact of the gain profiles, the birefringence effects in case (i) and case (ii) shown in Fig. 2 are considered here. We note that the conclusion for case (iii) is similar to that for case (ii) despite that the resonant sidebands are enhanced on the longer wavelength side, so the gain impact for case (iii) is not discussed here. When the birefringence is switched off, the corresponding spectra of the mode-locked pulses with different gain profiles are shown in Fig. 3(b). It can be seen that as the gain at shorter wavelength increases, the central wavelength of the soliton also shifts to the shorter wavelength. However, the resonant sidebands on both sides of the soliton are still almost symmetric. We note that similar results were obtained when we used a scalar iterative map model with a simple power-dependent transmission function for the SA [27,28]. Therefore, we can conclude that the intensities of the resonant sidebands cannot be efficiently enhanced by simply tailoring the gain profile if the birefringence is not included in the model. In contrast, when the birefringence is switched on, the corresponding spectra of the mode-locked pulses are shown in Fig. 3(c). The settings of the wave plate angles are the same as that in case (ii) shown in Fig. 2. It can be seen that when the birefringence effect is contained, the central wavelength of the soliton is well stabilized around 1610 nm even when the gain at shorter wavelengths is significantly increased. This can be explained by considering the saturable absorption transmittance curves shown in Figs. 2(b) and 2(e). The modulation depth is higher at longer wavelengths, where mode-locking is more favorable. However, the gain decreases rapidly as the wavelength is above ∼1615nm. As a result, the soliton is stabilized at around 1610 nm, and the intensities of the resonant sidebands on the shorter wavelength side are significantly enhanced due to the higher gain therein.

It is worth noting that although the spectral asymmetry caused by birefringence has been shown in previous work [23], the combined effect of the birefringence and the gain profile has not been systematically investigated yet. Here we demonstrate for the first time, to the best of our knowledge, that by carefully tailoring the gain profile and properly adjusting the artificial saturable absorber transmittance, the intensities of multiple resonant sidebands can be significantly enhanced.

To gain more insights into the temporal and spectral characteristics of the enhanced sidebands, the temporal and spectral intracavity evolution dynamics are plotted and shown in Fig. 4. The simulation parameters are the same as that of the top subplot in Fig. 3(c). We note that although the pulse energy of one polarization component can be several times (which depends on the particular setting of the wave plate angles) larger than that of the other one, the temporal and spectral evolutions of the two polarization modes are very similar, which is because the birefringence inside the cavity is very weak. Here we plot the total intensity of both polarization components. The temporal intensity evolution is shown in Fig. 4(a) on a logarithmic scale. The most intense peak centered at ∼0ps corresponds to the main soliton with a peak power ranging from 200 W to 1050 W. The evolution of the main soliton (indicated in the dark dashed box) is shown as an inset on the top of Fig. 4(a). Note that it is shown on a linear scale in the inset for better visualization. The variation of the power is mainly due to the dissipative effect at the gain fiber, the polarizer, and the output coupler. The long leading tail that can be seen from the main plot in Fig. 4(a) corresponds to the resonant sidebands that shed from the soliton. We note that the fringes on the leading tail are due to interference, which is also known as the beat frequency between different orders of resonant sidebands, which is explained in more detail in Fig. 5. The corresponding spectral evolution is shown in Fig. 4(b) where the resonant sidebands can be more easily recognized. The sudden change of the temporal power and spectral intensity at 7.5 m (point C) is due to the polarization-dependent loss at the polarizer. The projected temporal profile and spectrum on the front vertical plane correspond to the pulse at the output coupler (point D).

Now we characterize the resonant sidebands in more detail. The spectrum at the output coupler (point D) is shown in Fig. 5(a) on a logarithmic scale. The inset corresponds to a linear scale visualization. The first four orders of sidebands are highlighted and denoted by S1, S2, S3, and S4. The angular frequencies of resonant sidebands can be derived from the phase-matching relationship and are given by [9,10] Ωn=±1T02nkp−1,(4)where Ωn=ω−ωs denotes the angular frequency offset from the soliton central angular frequency ωs, n denotes the order of the sideband, and the positive (negative) sign on the r.h.s of Eq. (4) corresponds to the blue (red) sidebands with respect to the soliton central wavelength. T0 is the duration of the soliton, kp=2πLD/Lc is the perturbation wave number normalized to the dispersion length, LD=T02/|β2| is the dispersion length, and Lc is the cavity length. The wavelength of the resonant sideband was commonly predicted with Eq. (4). Since the pulse duration varies along the cavity, a path-averaged duration was used, which is ∼220fs. The wavelength of the resonant sideband (on the blue side of the soliton) as a function of the sideband order is shown in Fig. 5(b). The dark solid curve denotes the wavelengths of the sidebands [peak positions in Fig. 5(a)] obtained directly from the simulated spectrum. The filled red circles denote the sideband positions predicted with Eq. (4). It can be seen that as the sideband order increases, the deviation between the predicted and the actual wavelength also increases. The prediction error with Eq. (4) is plotted in the inset with the filled red triangles, where the growing predicted error can be clearly seen. Note that the interval between two adjacent sidebands decreases with increasing order, which means that the prediction with Eq. (4) is even worse for higher-order sidebands (as an example, for n=13, the sideband interval is 1.33 nm, and the prediction error is 0.72 nm, which is larger than a half of the interval). This is because Eq. (4) was derived by considering only the second-order dispersion term. It works well for the first several orders of sidebands, while the prediction error increases significantly for higher (>5th) order sidebands on which the impact of third-order dispersion is significant.

To accurately predict the wavelengths of all enhanced high-order sidebands whose spectral intensities are still notable as shown in Fig. 5(a), we take the TOD effect into account. Although the TOD effect has already been discussed in previous literature [10], there is still no explicit formula for predicting high-order resonant wavelengths with the TOD effect included. Here we derived an explicit resonant wavelength formula that incorporates the TOD effect, which is given by Ωn=1T0(±(2nkp−1+52ϵ2(2nkp−1)32)+ϵ(2nkp−1)),(5)which is valid as long as the parameter ϵ=16β3|β2|1T0 is small (for details on the derivation and discussion on the range of ϵ over which Eq. (5) is valid, refer to Appendix A.1). The predicted sideband wavelengths with Eq. (5) are plotted in Fig. 5(b) with the filled blue circles, and the corresponding prediction error is shown in the inset with the filled blue triangles. It can be seen that when the high-order dispersion is taken into account, the formula Eq. (5) accurately predicts the resonant wavelength. We note that although the birefringence can also have an influence on the wavelengths of the resonant sidebands when the laser cavity contains significant birefringence [29], we have checked that it has negligible impact on the sideband positions in our case due to the very weak birefringence inside the cavity.

As shown in Fig. 4(a), a notable feature of the temporal profile is the long leading tail with an interference pattern. The temporal profile of the pulse at the output coupler (point D) is shown in Fig. 5(c). The inset shows an expanded view of the main soliton, and a hyperbolic secant soliton fit (dashed cyan curve) is for comparison. The soliton has a duration of T0=197fs. An expanded view of the low-amplitude leading tail indicated in the dashed box in Fig. 5(c) is shown in Fig. 5(d). It highlights a long-tail structure modulated by fringes that are actually the beats between resonant sidebands with different frequencies. To reveal the structure of each resonant sideband, we separately filtered out the first four orders of resonant sidebands, and their temporal profiles are shown in Fig. 5(e). Specifically, to obtain the temporal intensity of each resonant sideband, the complex spectral envelope of each of its two polarization components was multiplied by a narrow band Gaussian filter function centered at the frequency of the corresponding resonant sideband. Then the total temporal intensity of each filtered resonant sideband was calculated after taking Fourier transforms of its two filtered spectral polarization components. Since the bandwidth of the sidebands decreases with increasing sideband order as can be seen from Fig. 5(a), the bandwidths of the filters were adjusted to minimize the influence of the soliton during the filtering process. In particular, the bandwidths (FWHM) of the Gaussian filters are 0.5 nm, 0.25 nm, 0.17 nm, and 0.13 nm for the first four orders (in ascending order) of sidebands, respectively. It can be seen that a lower-order sideband has a higher peak power and a shorter pulse duration, which is in accord with the corresponding broader spectral bandwidth. An interesting temporal feature of the resonant sidebands is that the leading edge decays exponentially, which can be clearly seen even from the linear slope when visualized on a logarithmic scale [inset of Fig. 5(e)]. We note that the exponentially decaying leading edge of the first-order resonant sideband has been verified in an experiment [30] very recently. Here we demonstrate that high-order resonant sidebands show similar exponentially decaying edges, and the decay rate decreases as the sideband order increases. In particular, the intensity of the leading edge of the sideband of order n can be expressed as Inlead(t)=I0net−t0τn,(6)where I0n denotes the intensity at the reference time t0 (which is negative for the leading edge here), and τn represents the decay time constant of the sideband of order n. If the reference time t0 is chosen such that t0<−35ps, the leading edge of all four orders of sidebands can be very well modeled by Eq. (6). The decay time constants τn of the first four orders (in ascending order) of sidebands are 7.2 ps, 16.9 ps, 30.2 ps, and 52.0 ps, respectively. Note that although only the first four orders of sidebands are shown here, we have checked that higher-order sidebands exhibit similar exponentially decaying leading edges. To understand the reason for this phenomenon, it is useful to consider the gain and loss of the sidebands and the soliton more thoroughly. In a single roundtrip within the cavity, there are two sources for the gain of the sidebands. One originates from the gain fiber, and the other is from the soliton that continuously radiates energy to the sidebands (linear dispersive waves). To reach a steady mode-locked state, the gain must be balanced for both the sidebands and the soliton. We note that the linear losses from the splicing points and the coupler are the same for both the linear dispersive waves and the soliton. The main difference between the losses of the linear dispersive waves and the soliton comes from the polarizer. The reason is twofold as can be seen from Fig. 2: one is due to the wavelength-dependent linear loss caused by the birefringence, and the other is due to the saturable absorption caused by the nonlinear polarization evolution of the soliton. For the case of sideband enhancement solutions as shown in Fig. 5, the loss of most of the resonant sidebands (within gain bandwidth) is higher than that of the soliton at the polarizer. This higher loss experienced by the sidebands is mainly balanced by the gain from the soliton that continuously sheds energy to the sidebands during propagation. As the difference between the central frequency of the sideband and the soliton is larger for the higher-order sideband, it leads to a larger group velocity difference. In the time domain, after the sidebands are emitted from the soliton, they gradually drift away from the center of the soliton due to the mismatch of their group velocities. As the order of the sideband becomes higher, the mismatch between its group velocity and that of the soliton becomes larger. As a result, the higher-order sideband drifts more from the soliton within each cavity length. This explains why a higher-order sideband exhibits a longer decaying time constant. To confirm that the fringes in Fig. 5(d) are indeed coherent superpositions between different sidebands, we superposed the filtered electric field of the first three orders of sidebands, i.e., S1, S2, and S3. The superposed intensity profile is plotted in Fig. 5(d) with the red curve. It can be seen that the superposed field approximates the whole field (filled black curve) very well, and the minor difference can be eliminated by further superposing more higher-order sidebands. It is also useful to see how the superposed intensity profile (S1 + S2 + S3) evolved in the cavity. The evolution map is shown in Fig. 5(f) on a linear scale. It can be seen that the fringe separations keep constant, and the overall evolution of the intensity is similar to that of the main soliton shown in Fig. 4(a). Note that the slight spatial modulation of the intensity is due to the variation of the phase differences between the sidebands during propagation.

Multiple resonant sideband enhancement was also demonstrated in our experiment. The laser setup and cavity parameters are the same as that shown in Fig. 1. As shown in the above section, the gain profile is critical for the sideband enhancement. We adopted a highly doped erbium gain fiber (LIEKKI Er80-8-125) of which the gain profile can be easily tailored by changing the length of the fiber and the pump power. EDF of 2 m long was used, which at moderate pump powers (200–300 mW), shows a similar gain profile as used in the simulation [top curve in Fig. 3(a)]. The mode-locking threshold was ∼112mW. When the pump power was above the threshold, various mode-locking regimes could be obtained by adjusting the polarization controllers, which include the wavelength-tunable single soliton regime, soliton molecule regime, and dual-wavelength soliton regime. When the pump power was increased to ∼250mW, by carefully adjusting the two polarization controllers, a (dorsal-fin-like) spectrum with unilaterally enhanced resonant sidebands was observed. The measured average output power at the coupler is ∼0.76mW. The spectrum is shown in Fig. 6(a) on a logarithmic scale, and that on a linear scale is shown in Fig. 6(b). The central wavelength of the soliton is ∼1613nm. The inset in Fig. 6(b) shows a close-up view of the soliton, which is otherwise hard to recognize due to the much higher spectral intensities of the resonant sidebands. It can be seen that the peak spectral intensity of the sidebands is ∼30dB higher than that of the soliton. Note that a spectral difference as high as 32 dB, which is the highest yet reported, can be obtained by optimizing both the wave plate angles and the pump power (for more details, refer to Appendix A.2).

To gain more insights about how much the sidebands were enhanced, it is useful to estimate the ratio of the total energy of the sidebands to the energy of the whole spectrum, which is denoted by ηS. Specifically, the spectrum shown in Fig. 6(a) can be treated as a superposition of the spectrum of the soliton and that of the sidebands. The energy of the soliton was calculated from a sech2 fitting [10] of the central spectrum shown in Fig. 6(a), and then the total energy of the sidebands can be regarded as the difference between the energy of the whole spectrum and that of the soliton. As a result, we obtained ηS=0.74 for the spectrum in Fig. 6(a). It means that the energy of the sidebands is ∼3 times that of the soliton, which indicates a significant enhancement considering the fact that the energy of the sidebands is usually much less than that of the soliton [10]. One might find that there is a slight difference between the simulated spectrum in Fig. 5(a) and the experimental one in Fig. 6(a). In particular, for the spectrum shown in Fig. 5(a), the sidebands on the red side of the soliton manifest as low intensity peaks. In contrast, for the experimental spectrum in Fig. 6(a), most of the sidebands (order n>2) on the red side of the soliton manifest as shallow dips. We note that whether a sideband exhibits as a peak or dip on the spectrum depends on the phase difference (which is a function of the propagation distance) between the sideband and the soliton at the output position of the cavity [31]. We have checked that the simulated spectrum at other cavity positions [such as that at 5.37 m and 5.93 m shown in Fig. 4(b)] can exhibit a similar spectrum to that in Fig. 6(a). The difference between the simulated and experimental results may be attributed to the inaccurate birefringence used in the simulation, which leads to different relative phases between the soliton and the sidebands.

We also employed Eqs. (4) and (5) to predict the central wavelengths of the resonant sidebands. The vertical dashed lines in Fig. 6(a) highlight the central wavelengths predicted with Eq. (5), which agree well with the experiment. The wavelength of the resonant sideband (on the blue side of the soliton) as a function of the sideband order is shown in Fig. 6(c). The dark curve denotes the sideband positions obtained directly from the measured spectrum in Fig. 6(a). The filled red and blue circles denote the sideband positions calculated with Eq. (4) and Eq. (5), respectively. The inset represents the prediction error of the sideband positions with Eq. (4) (filled red triangles) and Eq. (5) (filled blue triangles). Again, Eq. (5), which takes the TOD effect into account, predicts all the resonant wavelengths with much higher accuracy. In the calculation with Eqs. (4) and (5), we used dispersion parameters of β2=−25.4fs2/mm and β3=+130fs3/mm at 1613 nm. Note that the second- and the third-order dispersions should be understood as average values of different types of fibers (EDF, SMF) in the current cavity, and were obtained by fitting the sideband positions from one experimental spectrum [10]. We note that once the dispersion parameters were obtained, their values were fixed when we predicted the sideband positions for all the other spectra. The duration of the soliton T0∼212fs can be deduced from the first two orders of sidebands with either Eq. (4) as in Refs. [1,32] or with Eq. (5) because the TOD impact on the very-low-order (n=1,2) sidebands is negligibly small for the current cavity parameters. As can be seen from the inset of Fig. 4(a), the duration of the soliton varies during propagation. Thus, the calculated T0 should also be understood as an average soliton duration along the cavity. It should be noted that for a larger perturbation wave number kp, the position of the first two orders of sidebands can be much further from the soliton center; then the impact of the TOD on the first two orders of sidebands can be notable, and the average duration T0 should be calculated with Eq. (5) instead of Eq. (4). Note that we did not plot the curves in Figs. 5(b) and 6(c) together because there are slight differences in the central wavelengths and pulse durations between the simulated and experimental results. However, it can be seen that the trends of the curves in Figs. 5(b) and 6(c) agree well. The output pulse train was measured with a 5 GHz photodiode (Thorlabs DET08CFC) and a 4 GHz oscilloscope as shown in Fig. 6(d). It can be seen that the uniform pulse train has a repetition rate of fr=19.95MHz (corresponding to a single roundtrip time of 50.1 ns), which is in accord with the measured cavity length.

The output spectrum can be tuned by rotating the wave plates and changing the pump power. Figure 7 shows three typical spectra obtained by adjusting the angle of a quarter-wave plate in the polarization controller. Figures 7(a)–7(c) correspond to the experimentally measured spectra. It is useful to see whether Eq. (5) still works well in predicting the sideband positions when the polarization states are changed as shown in Fig. 7. Similar to that shown in Fig. 6(c), the sideband position as a function of the sideband order is plotted in Figs. 7(d), 7(e), and 7(f) for spectra shown in Figs. 7(a), 7(b), and 7(c), respectively. The solid dark curve denotes the sideband wavelengths obtained directly from the measured spectrum. The filled red and blue circles denote the sideband positions calculated with Eq. (4) and Eq. (5), respectively. The inset in each plot represents the prediction error with Eq. (4) (filled red triangles) and Eq. (5) (filled blue triangles). Figures 7(g), 7(h), and 7(i) correspond to the simulated spectra obtained at three different wave plate angles of {1.243, 0.5655, 1.508, 2.827} rad, {1.268, 0.5655, 1.508, 2.827} rad, and {1.276, 0.5655, 1.508, 2.827} rad, respectively. The sideband position as a function of the sideband order is plotted in Figs. 7(j), 7(k), and 7(l) for the simulated spectra shown in Figs. 7(g), 7(h), and 7(i), respectively. The dark line denotes the sideband positions obtained directly from the numerical simulation. It can be seen that Eq. (5) predicts the sideband position precisely in all cases for both the experiment and the simulation. The gain profiles are also plotted on the tops of Figs. 7(c) and 7(i), which correspond to the right vertical axes. The experimentally measured gain profile was obtained by amplifying a continuous wave signal pumped by a 980 nm LD [33]. The wavelength of the signal was continuously tuned from 1524 nm to 1640 nm. The powers of the signal and the pump were fixed at 5 mW and 290 mW, respectively, which were chosen according to the typical operation powers of our laser. The gain profile used in the simulation was the same as the top profile in Fig. 3(a). It can be seen that as the wave plate angle continuously changes, the spectral intensity of the resonant sidebands around the peak wavelength of the gain profile (∼1570nm in the experiment, and ∼1580nm in the simulation) grows significantly. If the wave plate was further rotated, the resonant sidebands around the gain peak wavelength dominated the spectrum gradually, and finally extracted almost all the energy from the gain, which led to the vanishing of the soliton. We note that the spectral profiles of the resonant sidebands can also be adjusted by changing the pump power. The enhancement of the sidebands controlled by changing the pump power is shown in Appendix A.2. We note that only significant enhancement of multiple resonant sidebands on the blue side of the soliton was observed in our experiment. Although an asymmetric spectrum with a higher low-frequency sideband was also obtained, the enhancement was only limited to the first two or three orders. We attribute this to the limited gain bandwidth of the gain fiber we used, the gain of which rapidly drops for wavelengths longer than 1620 nm as shown in Fig. 7(c). The stability of the mode-locked states with significantly enhanced sidebands was also characterized by monitoring the spectrum and the output power for a long time (for details, refer to Appendix A.3). The laser shows good stability in terms of both the output power and the spectral intensity.

Although resonant sidebands are usually observed in cavities of anomalous dispersion (as is the case in this paper), they can also be generated in cavities of normal dispersion [11,29]. Moreover, fiber lasers can exhibit extra spectral sidebands, which are caused by the energy exchange between the two orthogonal polarization components of the vector solitons [34]. It will be beneficial to discuss whether the enhancement of the sidebands is possible for these cases. As shown in Section 2, the enhancement can be realized by carefully manipulating the gain profile and the wavelength-dependent NPE transmittance; it is reasonable to expect that the current proposed theory can also be applied to a normal dispersion laser that is mode-locked by the NPE technique. However, how to extend the current model to lasers mode-locked by a real saturable absorber such as SESAM needs further investigation.

In conclusion, we have demonstrated, numerically and experimentally, that multiple resonant sideband enhancement can be realized by carefully manipulating the gain profile and NPE transmittance. The peak spectral intensity of the enhanced resonant sidebands is ∼32dB higher than that of the soliton obtained in the experiment, which is the highest yet reported. To precisely predict the wavelengths of all the high-order resonant sidebands, we derived an explicit formula that takes the TOD effect into account, for the first time. Compared with the commonly used formula that only considers the GVD effect, our formula has higher accuracy, especially for the prediction of the high-order sidebands. In addition, we have investigated the temporal characteristics of the resonant sidebands by numerically filtering them out. We demonstrate that high-order resonant sidebands also show exponentially decaying leading edges, and the decay rate decreases as the sideband order increases. We have also demonstrated that the spectral profiles of the resonant sidebands can be tailored by adjusting the wave plate angle of the polarization controller and the pump power.