## Abstract

Period multiplication is a bifurcation phenomenon, which depicts a typical roadmap from periodic steady-state to chaos. There has been particular interest in studying period multiplication properties in mode-locked lasers, but the mechanism behind is still not well-studied. Here, we experimentally demonstrate a figure-of-9 fiber laser which can generate period-multiplied pulses with long-term stability. Further theoretical insight is obtained by numerical simulations, which clearly shows that the ratio between the distances of each bifurcation point is approaching the Feigenbaum constant. We believe that the laser design presented here could be a stable and reliable ultrafast optical platform to study bifurcation and chaos, which also has potential applications in optical frequency comb and all-optical modulation.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Fiber lasers are simple and robust solutions to generate ultrafast pulses. Compared with their solid-state counterparts, ultrafast fiber lasers benefit from a compact structure, cost effectiveness, easy operability and diffraction-limited beam quality [1]. However, these advantages come with a price, which brings special technical challenges to the design of ultrafast fiber lasers. Light in a fiber experiences tight confinement and long interaction lengths, resulting in serious nonlinearity. In fact, nonlinearity is the main barrier for the pulse energy scale-up of ultrafast fiber systems [2]. Yet, if we see this issue on the other side, nonlinearity can become friendly. Taking advantage of the high nonlinearity, ultrafast fiber lasers could be a quality platform to study various pulsing phenomena. Numerous pulsing phenomena have been reported in the ultrafast fiber systems, including harmonic mode-locking [3,4], soliton rain [5,6], bound states soliton [7,8] pulsating [9,10] and period multiplication [11–19].

When the train of output pulses exhibits periodic fluctuations, the pulse intensity is no longer uniform as we generally assumed. Such pulsing phenomenon is either called pulsating or period multiplication. They can be distinguished by the length of fluctuation period. The fluctuation period of pulsating could range from tens of to hundreds of round trip times. The one of period multiplied pulses, on the other hand, could be as short as several round trip times. Examples can be period-doubling [11] and period-quadrupling [12]. Compared with period multiplication, pulsating is easier to be reproduced and thus has received more extensive studies. Recently, time-stretch dispersive Fourier transform (TS-DFT) technique has been used to reveal both the temporal and spectral dynamics of the pulsating [9,10].

Period multiplication is a bifurcation phenomenon, which depicts a typical roadmap from periodic steady-state to chaos. It has been widely reported in mode-locked fiber lasers with different kinds of pulse shaping mechanisms, including conventional solitons [11,12,18,19], dissipative solitons [13] and dispersion-managed solitons [14,15]. It suggests that period multiplication is an intrinsic feature of mode-locked lasers, which is independent of the pulse property. Meanwhile, distinguished by types of saturable absorbers (SA), some of them were observed in vector soliton fiber lasers with material-based SAs [18,19]. Yet, the majority of them used nonlinear polarization rotation (NPR) technique as an artificial SA [11–17]. NPR-based period multiplication originates from overdriving of the artificial SA under high pump power [16]. If the pump power is too high that the peak power of the pulses in the cavity greatly exceeds the NPR switch threshold, its orientation of the polarization would be no longer aligned with the polarizer. The pulse would thus experience attenuation when it passes through the polarizer. Consequently, periodic modulation of the pulse intensity would be observed.

Period-multiplied mode-locking can generate new comb lines in the frequency domain, so they have been thought to be useful to develop novel optical frequency combs. Zhao et al. demonstrated a self-referenced f-to-2f beat note measurement in a period-doubled mode-locked Yb-fiber laser [16]. They showed that the carrier-envelope offset frequency was observed to shrink into half of the corresponding fundamentally mode-locked laser in the frequency domain. Wu et al. reported another optical frequency comb based on a period doubling mode-locked Er-fiber laser [17]. Their experimental results show that the new comb teeth generated in the period-doubling mode-locked fiber lasers are strongly correlated with the original teeth. Nevertheless, due to its working mechanism, NPR is rarely considered to be a good candidate to achieve long-term stability. Therefore, NPR-based period-multiplied fiber lasers have not yet to find widespread applications outside of labs, which greatly limit their impact.

For passively mode-locked fiber lasers, another widely used artificial SA is nonlinear amplified loop mirror (NALM). In contrast with NPR, NALM can be constructed into all-polarization maintaining (PM) structure, while still maintaining all-fiber advantage [20]. Therefore, NALM should be a promising solution to generate period-multiplied pulses with long-term stability. Recently, researchers have demonstrated a new-type NALM mode-locked fiber laser, which is named as figure-of-9 to distinguish it from the traditional figure-of-8 type [21,22]. By embedding a nonreciprocal phase shifter into the NALM loop, the figure-of-9 laser can achieve mode-locked operation under high repetition rates. Thanks to the all-PM configuration, such laser has been proved to be long-term stable and was used as a seed oscillator for the first optical frequency comb in space [23]. Nevertheless, the potential for such figure-of-9 fiber laser to generate period-multiplied pulses has not been reported by them.

Recently, we observed period-multiplied operation in an Yb-doped figure-of-9 fiber oscillator [24]. The NALM-based fiber laser constructed by all-PM fiber components can generate long-term stable, linearly polarized laser pulses with periodic non-uniform intensity. Here, a more in-depth study on the figure-of-9 period-multiplication is reported. We proved both experimentally and numerically that the period multiplication was originated from overdriving of the artificial SA under high pump power. Further theoretical insight is obtained by numerical simulations, which clearly shows that the ratio between the distances of each bifurcation point is approaching the Feigenbaum constant. We believe that the laser design presented here could be a stable and reliable ultrafast optical platform to study bifurcation and chaos.

## 2. Experiment setup

The design of our figure-of-9 mode-locked fiber laser is presented in Fig. 1. The cavity is constructed by entirely PM fibers (PANDA style) and PM fiber-pigtailed components. The central component of the structure is a 2×2 PM fiber coupler, which has a splitting ratio of 40/60% at 1064 nm. Two arms of the coupler were ringed up together to serve as the NALM loop. The 60% arm is connected to a 976/1064 nm wavelength division multiplexer (WDM), followed by a piece of 30 cm Yb-doped PM fiber (PM-YDF 5/130-VIII, Nufern) to serve as the gain medium. The 40% arm is connected to a nonreciprocal phase shifter to provide a linear π/2 phase bias. A piece of PM980 fiber (2.2 m) is inserted the NALM loop to set the position of the gain fiber away from the center of the loop. It is worth to emphasize that all the special design mentioned above, including the phase shifter, the extra PM980 fiber and the splitting ratio of the coupler, are to promote self-starting of the mode-locked operation. A chirped fiber Bragg grating (CFBG) with center wavelength at 1064 nm, ∼68% reflectivity, 0.5 nm bandwidth and 2 nm/cm chirp rate (providing a second order dispersion of -29 ps^{2}) is connected to the loop mirror to form the cavity. The overall fiber cavity length is ∼7.3 m, corresponding to a total second order dispersion of 0.175 ps^{2}. Therefore, the net cavity dispersion is -28.8 ps^{2}, which means the laser works in large anomalous dispersion regime.

## 3. Experiment results and discussion

Self-starting fundamental mode-locking was obtained when the pump power was greater than 170 mW. The pump power range for the fundamental mode-locking is from 150-215 mW. When the pump power was reduced to lower than 150 mW, the laser would switch to continuous wave mode. As shown in Fig. 1, the laser has two output ports. One is from the fiber coupler, which is named as output #I; the other is after the CFBG, which is named as output #II. Maximum average output powers of 11.7 mW and 13.3 mW was obtained at a pump power of 215 mW for output #I and #II respectively. The pulse repetition rate was measured to be at 27.4 MHz which gave the corresponding pulse energy of 0.43 nJ and 0.49 nJ. The temporal property of the pulse was measured with a Keysight 2.5 GHz oscilloscope (DSO-S 254A) through a 1 GHz InGaAs detector. The radio-frequency (RF) spectra were recorded by a Keysight RF spectrum analyzer (N9020A) with a bandwidth of 20 GHz. The measured pulse train and corresponding RF spectrum were shown in Fig. 2(a). The results clearly show a pulse spacing of 36.5 ns and a repetition rate of 27.4 MHz, which matches well with the 7.3 m long cavity length.

When the pump power was adjusted to slightly above 215 mW, the working state of the laser will switch into stable period-quadrupling mode-locked operation. The pump power range for such period-quadrupling operation is from 215-225 mW. Further increase of the pump power to over 225 mW will lead to pulse-splitting. The period-quadrupling operation could also be self-started, just like the case of fundamental mode-locking, which means period-multiplied pulsing can be established from noise. The pulse trains and corresponding RF spectra of the period-quadrupling operation were shown in Fig. 2(b, c) for output #I and #II, respectively. In contrast with the fundamental mode-locked state, the pulse intensity of the period-quadrupling is no longer uniform, but alternates between four different values, which indicates that steady-state period switches from 1 to 4 round trip times. Since the pulse spacing of the period-quadrupling state still stays at 36.5 ns, the modulation of the intensity would generate new frequency comb lines in the RF spectra. Strong intensity modulation will result in newly-generated comb lines with high energy, as shown in Fig. 2(b); Week intensity modulation will result in newly-generated comb lines with low energy, as shown in Fig. 2(c). The results agree with the principle of Fourier transform. For further in-depth study, the pulse trains from output #I and #II are measured simultaneously, and then marked with a group number, as shown in Fig. 2(d). In the measurement, the optical and electrical delay of these two channels are deliberately kept close so that the time delay between them would be in the level of tens of picoseconds. The group of pulses with highest intensity is marked as #1. The other three groups are marked as #2, #3 and #4 in order. It is worth to emphasis that multiplication with different period can be obtained by changing the cavity parameters such as the PM980 fiber length between the phase shifter and the gain fiber, or between the CFBG and the fiber coupler. Figure 3 is an example of period-tripling operation, obtained when extra 1 m PM980 fiber is added into the NALM loop. Here, we only take period-quadrupling as a general example for detailed analysis.

If the spectrum of the period-quadrupling pulses is measured by an optical spectrum analyzer, the result would be an average of the four groups of pulses. Therefore, in order to obtain the real spectrum of each group, a TS-DFT method was applied. The output pulses of port #I was fed into a commercial time-stretch spectrometer (Tachyonics Inc. / SPL Photonics), which has a high-speed photo-detectors with a bandwidth of 11 GHz inside. The measured data was captured by a real-time oscilloscope with a bandwidth of 33 GHz (Tektronix DPO75902SX). Together, they ensure a resolution of ∼0.055 nm for the TS-DFT measurements. Data sequences of 26 round trip times is recorded to study the spectral dynamics both in fundamental mode-locked and period-quadrupling mode-locked regime, as shown in Fig. 4(a) and 4(b) correspondingly. The spectral intensity profiles of the fundamental mode-locking stay constant with round trips. In contrast, the period-quadrupling mode-locking has a strong spectral intensity modulation with a period of 4 round trips. Pulses with higher temporal intensity have broader spectral width, due to self phase modulation.

Unlike material-based SA mode-locked fiber laser, whose period multiplication can be explained by the strong cross-phase modulation and coherent energy exchange between the two polarizations of the vector soliton [18], the figure-of-9 laser presented here has an all-PM structure and can only produce scalar soliton. Apart from the experimental observations, here we propose a simplified iteration model to understand the mechanism behind the NALM-based scalar soliton period multiplication. The transmission function of a material-based SA, such as SESAM and graphene, is a monotonic increasing function for the most likely case. However, as an artificial SA, NALM's transmission function is non-monotonic, which means that overdriving of the NALM under high pump power would lead to a considerable decrease of the transmission. A typical configuration of a NALM is shown in Fig. 5. Assuming that the input field is *|E _{IN}|^{2}* at port #1, the field of the transmitted light at port #1 and #2 is given by [25]:

*α*is the power-coupling ratio of the coupler (60%),

*n*is the nonlinear coefficient of the fiber (3×10

_{2}^{−16}cm

^{2}/W),

*L*is the length of the bias fiber in the loop (2.2 m),

*g*is the gain in the loop (3.2),

*λ*is the center wavelength (1064 nm) and

*θ*is the linear phase shift provided by the nonreciprocal phase shifter (0.5π).

For our figure-of-9 cavity, the transmission function is determined by Eq. (1), which can be generalized to a function as *y *= f(*x*), in which *y* represents the output field *|E _{OUT}|^{2}* and

*x*represents the input field

*|E*. Assuming that the reflectivity of the CFBG is

_{IN}|^{2}*R*, the output from the loop multiplies the

*R*would be the input for the next round trip. In other words, if

*y*=f(

_{1}*x*) represents the output field of the first round trip, the output of the second round trip can be expressed as

*y*=f(

_{2}*R*×

*y*)=f(

_{1}*R*×f(

*x*)), and so on, the output of the n

_{th}round trip can be expressed as

*y*=f(

_{n}*R*×

*y*). The black solid curve in the leftmost graph of Fig. 6 shows the NALM's transmission function of our laser. There exists a period-quadrupling solution, which is 197.2 W (point #1) –> 31.5 W (point #2) –> 52.7 W (point #3) –> 104.3 W (point #4) –> 197.2 W. These four points form a close-loop circle. The red curve (function of

_{n-1}*y*=

*x*/

*R*, with

*R*= 0.68), together with the black dash curve, represents the retrieval process of

*y*. The result numerically prove that overdriving of the NALM under high pump power, like the case of point #1 in the graph, could result in an n

_{n}_{th}close-loop solution, and thus lead to period-multiplied pulsing phenomenon.

Indeed the iteration model is over-simplified, only considering the NALM transmission at the peak power of the circulating pulses. The real physical process in the mode locked fiber laser is much more complicated, which will be numerically simulated in the next section. However, the fact that the simple model can phenomenally explain the observation indicates that pulse multiplication is a result of overdriving the artificial SA regardless of pulse formation mechanism.

We further designed a detailed experiment to verify the reliability of the iteration model. We used an acoustic-optic modulator as a pulse-picker (Gooch & Housego) to filter out the pulses with each group and then measured their autocorrelation traces using a commercial autocorrelator (APE PulseCheck, 150). The measured four groups’ autocorrelation traces at port #II are shown in Fig. 6. As we can tell from the results, the pulses of groups #2, #3 and #4 have a near-perfect Sech-function shape with pulse durations of 13.7 ps, 12.9 ps and 14.8 ps, respectively. Compared with them, the autocorrelation trace of groups #1 has an obvious pedestal, which makes it out of the Sech-function shape. These results can be explained by the former mentioned iteration model. According to the leftmost graph of Fig. 6, the pulses of groups #2, #3 and #4 are given close to linear transmission functions by the NALM. However, the pulses of groups #1 will suffer a nonlinear transmission function. Because it works at an overdriven point of the NALM, the top of the pulse, as well as the bottom of the pulse, will suffer from a low transmission. Meanwhile, the waist of the pulse will have a high transmission, which thus leads to the deformation of the pulse shape.

The four groups’ autocorrelation traces at port #I are measured and presented in Fig. 7. The transmission function of this output port can be determined by Eq. (2) and is plotted in the leftmost graph of Fig. 7. Interestingly, the results of the autocorrelation traces are completely opposite compared with the ones of port #II. This time the pulses of group #1 have a Sech-function shape with pulse durations of 7.3 ps; the pulses of groups #2, #3 and #4 have irregular pulse shapes with obvious pedestals. Such results can also be explained by the transmission function of port #I. For the pluses of group #1, the majority of the pulses will be given a close to linear transmission function after go through the NALM. Although the tail of the pulses suffers a nonlinear transmission function, its influence on the pulse-shaping can be ignored due to the high transmission contrast between the top and the tail of the pulses. However, for the pulses of groups #2, #3 and #4, such transmission contrast between the top and the tail is not that large, which consequently results in the deformation of the pulse shape. One can also compare the pulse widths of group #2, #3 and #4 in Fig. 6 with the one of group #1 in Fig. 7. The latter is almost half of the formers, which can also be explained by the transmission function in Fig. 6 and 7. The temporal filtering effect of the NALM to point #1 in Fig. 7 is twice as strong as the one to point #2, #3 and #4 in Fig. 6.

## 4. Numerical simulations and bifurcation analysis

Essentially, period-multiplied pulsing is a bifurcation phenomenon, which is related to the concept of chaos. Nonlinear system, whose output is periodic, may go through period bifurcation before its output becomes chaos. Examples could be as simple as droplets’ patterns of a water tap or as complex as discrete-time neural networks. As one of the nonlinear systems, ultrafast fiber lasers are known as an optical platform to investigate bifurcation and chaos.

In order to obtain deeper insight into the bifurcation working principle of our laser, numerical simulations based on the cubic Ginzburg–Landau equation and the split-step Fourier method were carried out. The detailed description about the simulation model can be found in our past works [26,27]. All the simulation parameters follow the experimental setup mentioned in the former section. It should be noted that no SA term was added deliberately into our simulation model; instead, a more authentic method was applied. We calculate the transmission states of the clockwise and counter-clockwise beam in the NALM loop separately. Due to the existence of nonlinear phase difference between these two beams, their interference would result in an intensity-dependent transmission. Such method can act like a SA term in our simulations, which exactly follows the working principle of the NALM.

The simulation results are plotted in Fig. 8, which presents the bifurcation route map from fundamental pulse operation to period-32 operation as a function of the CFBG's reflectivity. Period-4 pulsing was observed when the CFBG's reflectivity is set at 0.699, which indicates a good agreement between the experiment and numerical simulation. It is also interesting to notice that the working period could be adjusted by changing the value of the CFBG's reflectivity. The ranges of reflectivity for period-1, 2, 4, 8, 16, 32 are 0.44-0.594, 0.594-0.666, 0.666-0.699, 0.699-0.7054, 0.7054-0.70697, 0.70697-*n*, respectively. For period-32, *n* should be a value very close to 0.70697. Due to the large amount of calculation, we did not simulate the periods beyond 32. Because the working range of the reflectivity tends to shrink when the period increases, it is reasonable to predict that the working states of the system would eventually slide into chaos with a slight increase of the reflectivity over the value of 0.70697.

As discussed in the previous section, the mechanism of a figure-of-9 cavity could be characterized by the following function:

in which, function f(*x*) is the transmission function of NALM (Eq. (1)),

*x*is the peak power of the pulses coming into the NALM loop for n

_{n}_{th}round trip and

*R*is the CFBG's reflectivity. Other losses on the linear arm of the figure-of-9 cavity, like the loss and dispersion of the fiber, could be ignored, compared with the CFBG's reflectivity. Equation (3) is a typical Feigenbaum iterated function. Assuming

*R*is the reflectivity when the bifurcation from period-2

_{n}^{n-1}to period-2

^{n}happens, we can get the Feigenbaum constant by the following equation [28,29]:

*R*=0.594,

_{1}*R*=0.666,

_{2}*R*=0.699,

_{3}*R*=0.7054 and

_{4}*R*=0.70697. Therefore, we can calculate the first three

_{5}*δ*values as

*δ*=2.18,

_{1}*δ*=4.46 and

_{2}*δ*=4.71, which is approaching the Feigenbaum constant when the order increases.

_{3}It is worth to emphasize that the reported period-multiplied pulsing can be self-started and long-term stable. The laser was turned on and off for hundreds of times, and the period-quadrupling mode-locked operation can always be established from the noise with very repeatable performance. Such period-quadrupling operation could also be maintained after the laser was kept turning on for days. The result of 13 hours power stability test is shown in Fig. 9 when the laser is under period-quadrupling operation. A mean output power of 12.0 mW with ∼0.04 mW rms instabilities represents a 0.3% relative rms fluctuation, indicating the good power and mode-locking stability of the period-quadrupling operation. The excellent stability is result from the entirely PM-fiberized cavity design. Besides, the laser is constructed by merely four fiber components, which also helps to promote the system's robustness. Compared with the NPR-based period-multiplied fiber lasers, the figure-of-9 ultrafast fiber lasers can provide more reliable and robust optical platforms for the studies of bifurcation and chaos, which also has potential applications in optical frequency comb and all-optical modulation.

## 5. Conclusion

In summary, we have demonstrated a figure-of-9 fiber laser which can generate long-term stable period-multiplied pulses. Self-starting and stable period-quadrupling mode-locked operation is obtained with a fundamental repetition rate of 27.4 MHz. We studied the transmission function of the NALM in detail and proved both experimentally and numerically that the period multiplication was originated from overdriving of the artificial SA under high pump power. A comprehensive simulation based on the cubic Ginzburg–Landau equation was carried out, which proves that the working period could be adjusted by changing the CFBG's reflectivity. According to our simulation results, the ratio between the distances of each bifurcation point of our laser is approaching the Feigenbaum constant. We believe that the laser design presented here has a lot of potentials and could be a good ultrafast laser source for applications including studies of chaos, optical frequency comb and all-optical modulation.

## Funding

National Natural Science Foundation of China (61805262).

## Disclosures

The authors declare that there are no conflicts of interest related to this article.

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