## Abstract

we report three types of pulse generation in Yb-doped nonlinear polarization rotation mode-locked fiber lasers in all-normal-dispersion regime through simulation, including dissipative soliton, dissipative soliton resonance and noise-like pulse. We distinguish the different conditions of generating such different pulses by analyzing the transmission curve of saturable absorber, which plays a key role in pulse shaping.

© 2015 Optical Society of America

## 1. Introduction

Passively mode-locked fiber lasers, which can deliver ultrashort pulses down to femtosecond level, have attracted people’s attention for years. Comparing to crystal-based ultrafast solid-state lasers, mode-locked fiber lasers are associated with a number of advantages such as excellent pulse stability, compactness and low cost. The possibility of conventional soliton formation in optical fibers was suggested as early as 1973 [1], and solitons were observed in an experiment in 1980 [2]. Such kind of solitons are always generated in anomalous dispersion regime, and the single pulse energy is limited to 0.1 nJ due to area theory [3,4]. In 1993, people introduced some normal dispersion into the cavity to realize dispersion management and stretched soliton has been obtained. The single pulse energy can reach to nJ level, and the pulse width can be down to tens of fs [5]. After more than 10 years, similariton was realized in a mode-locked fiber laser as well [6]. The pulse has a parabolic shape and shows higher tolerance of nonlinear effect. This scheme leads to higher pulse energy more than 10 nJ [7]. Recently, people shift the dispersion to the normal regime and obtain dissipative soliton (DS) generation [8,9], pushing the single pulse energy up to 20 nJ.

Lots of work has been done on DS. Generally, DS implies a composite balance between dispersion, nonlinearity, gain and loss [10]. Although the DS is diversified, the typical pulse has large pulse energy and up chirps. Moreover, the pulse can be compressed to femtosecond level. The spectrum has typical steep edge due to the filter effect (or wavelength dependent linear loss or gain, in other words) [8]. But, it is not the only type of pulse in normal dispersion. In additional to typical DS, dissipative soliton resonance (DSR) [11] and noise-like pulse (NLP) [12] have been demonstrated as well.

The concept of DSR was proposed firstly in 2008 [11]. It has been shown that with certain cavity parameter selections, the energy of the formed dissipative solitons could increase infinitely, characterized as that the pulses increase their width while keeping their amplitude constant [13]. DSR can be modeled using complex cubic-quintic Ginzburg–Landau equation (CGLE), which adding cubic and quintic saturable absorption terms [14]. The two saturable absorption terms have been considered as two kinds of nonlinear loss, mostly introduced by saturable absorber (SA). And these saturable absorption terms lead to clamping effect on DSR pulse [15]. In practice, we consider these two terms as saturable absorption and reverse saturable absorption. That means the transmissivity of SA increases and decreases as intensity of light increases, respectively. As the nonlinear polarization rotation (NPR) and nonlinear optical loop mirror (NOLM) have both such opposite nonlinear absorptions, most reports on DSR generation based on these two mode-locking mechanisms.

The other type is NLP which was first mentioned in [16]. The NLP usually can be observed when the pump is strong. The typical characteristic is a narrow coherent spike on top of autocorrelation (AC) trace. In other words, there are a lot of random fine structures in the pulse. But the whole structures still stand together without walking off, combined as one pulse with relatively wide envelope. The NLP could be generated by introducing reverse saturable absorption, which was reported in [17].

It is shown that, DS, DSR and NLP can be realized in one and the same NPR mode-locked fiber laser. The cavity operation can be tuned from one type to another simply by tuning the polarization controller under appropriately tuning of the pump. So we think, in addition to the pump, the NPR device which works as a virtual SA, is the key to find out the relationship between these pulses. Here, we demonstrate the generation of such three types of pulses numerically by the analysis of transmission curve of SA in detail.

## 2. Theoretical modeling

The proposed mode-lock fiber ring laser is schematically shown in Fig. 1(a). The ring consists of a piece of ytterbium doped fiber (YDF) with a length of 1 m, a piece of single mode fiber (SMF) with a length of 10 m, a Gaussian shaped filer with a bandwidth of 6 nm, an output coupler (OC) with ratio of 50% and a saturable absorber (SA). All the fibers in the ring are assumed to have the same group velocity dispersion and nonlinear coefficient corresponding to *β*_{2} = 0.025 ps^{2}/m and γ = 4.66 W^{−1}km^{−1}, respectively. The saturable absorber here is assumed to be a nonlinear polarization rotation (NPR) device. The SA exhibits a sinusoidal shape transmissivity verse instantaneous power (Fig. 1(b)), and has a saturable absorption property within certain range of power (0~150 W and 300~450 W). The transmissivity increases as power increases. While in the other range (150~300 W), the SA demonstrates reverse saturable absorption. That is, the transmissivity decreases as power increases. The transmissivity function can be written as below:

*φ*is the phase delay which we set as zero in the whole journal for simplified; R

_{0}is the lowest transmissivity; dR is the modulation depth that controlling the difference of transmissivity values between the highest and the lowest; P

_{A}is a parameter which is similar to saturable power, determining the period of the curve (half period equals P

_{A}).

Pulse evolution starts from SA. To descript pulse propagation, we use generalized nonlinear Schrodinger equation instead of CGLE:

*A*denotes envelop of the optical pulse;

*α*is linear loss;

*β*

_{2}is second order dispersion; γ is third order nonlinear coefficient; $\tilde{A}(\omega )$ is the Fourier component of

*A*(

*t*);

*g*(

*ω*) is gain which depends on frequency

*ω*, and considering the gain saturation effect, there is:In Eq. (3),

*g*

_{0}is small signal gain at the central wavelength,

*E*is the gain saturation energy,

_{Sat}*E*is pulse energy.

_{pulse}## 3. Simulation results and analysis

The numerical model is solved by a split-step Fourier method. To accelerate the convergence of the calculation, the evolution starts from an initial 10 ps chirp-free pulse with 100 aJ pulse energy. The corresponding initial pulse peak power is ~9.4 μW. The spectrum of initial pulse centers at 1064 nm. The small signal gain is fixed at 6.9 m^{−1}, corresponding to 30 dB/m. And the bandwidth of the gain is 50 nm which has a Gaussian shape. We can realize three types of pulse generation by tuning the transmissivity curve of SA and gain saturation energy of gain fiber. The changing of gain saturation energy is equivalent to changing the pump power.

First, we set SA parameters as R_{0} = 0.2, dR = 0.2 and P_{A} = 150 W, corresponding to the transmission curve shown in Fig. 1(b). Under such SA parameters selection, we choose three different values of E_{Sat} as 0.2 nJ, 0.25 nJ and 0.3 nJ. Figure 2 is the simulation results which are chosen from the location before SA and after OC (other simulation results in this journal are from this location as well). Obviously, the cavity delivers DS with steep edges in spectrum, which is the symbol of typical DS. The pulse energy increases as E_{Sat} increases, up to 574.8 pJ, 699.2 pJ and 844.5 pJ, respectively. The peak power of the pulse increases to 95.6 W, 120.5 W and 140.3 W, respectively, and is approaching to the value of P_{A} = 150 W. As the peak power is increasing while the pulse width has little change (5.4 ps, 5.6 ps and 5.8 ps, respectively), it brings stronger nonlinear effect into cavity and broadens the bandwidth of spectrum which is 4.3 nm, 5.0 nm and 5.4nm, respectively.

Then, we continue increase the gain saturation energy to 0.4 nJ, 0.8 nJ and 1.2 nJ. In this range, cavity operates in a transition state (TS), demonstrating both DS and DSR properties (see Fig. 3). Pulse energy increases to 1.1 nJ, 2.1 nJ and 3.2 nJ, respectively. The peak power increases to 171.5 W, 229.0 W and 246.5 W, respectively. The speed of increasing of peak power has been slowed down, because all peak powers of the three pulses have exceeded the P_{A} = 150 W. In the range of reverse saturable absorption (150~300 W), transmissivity begins to decrease as power increases. In other words, higher instantaneous power part of pulse encounters larger loss induced by SA. So the pulse is starting to involve horizontally instead of vertically (pulse widths are 6.4 ps, 9.6 ps and 13.5 ps, respectively). Meanwhile, the 3dB spectrum width begins to decrease. But actually, the bandwidth between two steep edges at the bottom of spectrum doesn’t change too much. Because the central part of spectrum has larger gain than edges’, making the amplitude of the central spectrum much larger. And the shape of the spectrum is beginning to transfer from rectangular shape to bell-shaped. The result has agreement with [15]. So the 3dB bandwidth decreases naturally.

Further increasing the E_{Sat} to 2 nJ, 3 nJ and 4 nJ, the laser turns to generate DSR pulse completely (Fig. 4). The amplitude of the pulse has been clamped at a certain power (peak powers are 257.3 W, 261.7 W and 263.6 W, respectively), and the pulses are shaped into flat top pulses. The reverse saturable absorption has played a key role in generating DSR pulses. The pulse width increases fast, becomes wider and wider instead of taller (pulse widths are 21.6 ps, 31.7 ps and 41.8ps, respectively). The range of the chirp becomes larger, because the slope of the edges of pulse becomes larger (a trend of square shape transformation) as pump power increases. Meanwhile, the chirp within the central part of pulse becomes smaller. Because the top of pulse becomes more and more flattened, which introduces less self-phase modulation effect. The central part (the part of the pulse has a flat top in time domain, while edges don’t) of the pulse becomes wider, that is, most energy stores here where has little chirp (spectrum components distribute near the central wavelength). So most energy stores near the central wavelength, which forcing a narrower bandwidth of a DSR spectrum generation, comparing to a DS spectrum. Further increasing E_{Sat} to 6 nJ, the flat-top pulse continually broadens in pulse width and enlarges pulse energy, but little change in amplitude. And much more round trips are needed to make the simulation converge. So we didn’t increase the gain saturation energy anymore. It should be noted that the pulse energy could be improved dramatically due to DSR, but the peak power of pulse has been clamped to a certain level which is less than twice of the value of P_{A}. As the pump power increases, DSR pulse energy increases. In the condition of relative high energy, the pulse usually has a wide pulse width (even reach ns level), while the linear chirp is less and less. As the linear chirp of the pulse is so little, it is hard to compress the pulse obviously. We think the DSR is an efficient way to achieve high pulse energy, but it is not suitable for chasing high peak power.

The other type of pulse is Noise-like which can be realized with another different set of SA parameters. Here we change the dR and P_{A} to 0.6 and 200 W, respectively. When E_{Sat} is in a range of appropriate low value, we can still obtain DS generation. When E_{Sat} = 20 nJ, we can obtain the generation of NLP with pulse energy of 29 nJ as shown in Fig. 5 after operated for 1000 round trips. In the time domain, the pulse has a lot of fine structures which are different at every roundtrip. It has a typcial autocorrelation trace which has a narrow spike on top. The chirp of the pulse is a mess, which can explain that NLP can’t be compressed. The spectrum has plenty of fine structures as well, which is similar to the result in [9]. We have also demonstrated the evolution of NLP both in time and spectrum domains.

If we add some anomoalous dispersion fiber into cavity, it will be easier to generate NLP. Here, in the 10 m long SMF with normal dispersion, we change part of it into anomalous disperion with a length of 4 m. The absolute value of GVD remains unchange, which means *β*_{2} = −0.025 ps^{2}/m. The net dispersion of cavity is 0.075 ps^{2}.We set the gain saturation energy value as 0.2 nJ, which is much smaller than the former. Though the output pulse energy is only 3 nJ, the spectrum is wider obviously. The pulse has burst into several subpulses completely instead of lots random spikes growing on the same base. And the little spike on the top of AC trace is longer than the former (see Fig. 6).

In our case, we attribute the generation of NLP to reverse saturable absorption as well [17]. As the transmissivity of SA increases and decreases periodically. Different parts of the pulse, which have different instantaneous powers, encounter different amplitude modulation effect. As the NLP always occurs in the condition of relatively high gain, it is easy for the instantaneous power to exceed the P_{A} or even several times of P_{A}. We demonstrate this phenomenon qualitatively by simulation. We propagate a Gaussian pulse with ~940 W peak power and 10 ps pulse width through a SA. The SA has parameters as R_{0} = 0.2, dR = 0.6, P_{A} = 50 W and φ = 0. The peak power is nearly 19 time larger than P_{A}. After modulated by SA, the pulse has many fine structures and AC trace has a little spike on top (Fig. 7(a)). Futher propagating in a 5 m fiber with normal dispersion, the valley of fine structures becomes shallower, and the spike on top of AC trace is shorter(Fig. 7(b)). But if we change the fiber dispersion into anomalous, after repropagating the pulse, the valleys betweem the spikes are so deep that the pulse looks like been bursted into many subpulses. Each of the subpulses becomes narrower and higher due to the influence of anomalous dispersion. The spike on top of AC trace becomes longer (Fig. 7(c)). The results are in good agreement with the former simulations of NLP generation.

So both DSR and NLP can be formed by the existing of reverse saturable absorption effect, and the question is: when should DSR occur and when should NLP occur? After some simulation work, in our cavity configuration, we find the value of P_{A} is a decisive parameter. When P_{A} is large enough, cavity prefers NLP; otherwise is DSR. As in the former case, when SA has parameters as R_{0} = 0.2, dR = 0.6 and P_{A} = 200 W, the cavity has a NLP generation. But if we reduce the P_{A} into 40 W, the DSR can be realized. For further verification, we consider other conditions with different dR while fixing R_{0} = 0.2. After rough simulation work, we find the critical point of P_{A}. The cavity always prefers the DSR generation when P_{A} is less than the critical point (Fig. 8(a)). That is why most reported experimental DSR generation has a relative low peak power which is around several watts, though the pulse energies are remarkable, because P_{A} with a low value clamps the peak power in a low level. When modulation depth dR increases, the critical point decreases. Because large modulation depth leads to narrower and higher pulse, the DSR transformation needs reverse saturable absorption with steeper curve to clamp the peak power. So decreasing P_{A}, which is corresponding to tuning polarization controllers in experiment, should be an efficient way to obtain DSR pulse while avoiding NLP. Furthermore, when the value of P_{A} has been set higher than the critical point, there may be a mix state (MS) occurring (Fig. 8(b)). When R_{0} = 0.2, dR = 0.6 and P_{A} = 150 W, set the E_{Sat} = 7 nJ, the pulse has many fine structures within it and the AC trace still has a little spike on top which is very small. They are the typical characteristics of NLP. The envelope of pulse has approximate flat top shape, though there are obvious amplitude fluctuations. The AC trace is a triangle shape as a whole and the pulse width and energy increase as gain increases, which are properties of DSR phenomenon. As a result, we believe the DSR and NLP can coexist together.

The simulation work in earlier article is based on a saturable absorber with smooth and perfect sinusoidal transmission curve. For the sake of being closer to the practical conditions, we introduce some amplitude fluctuations in existing smooth sinusoidal transmission curve of SA. Five curves have been utilized as shown in Fig. 9(a). Curve 1, which has no amplitude fluctuations, has the same characteristics with the curve in Fig. 1(b). The fluctuations magnitudes of these five curves increase one by one. Curve 5 has the largest fluctuations. When the magnitudes of fluctuations are relatively low, such as curve 2 and 3, DSR pulses can still be obtained. But the flat-top of the pulse begins to bulge when fluctuation magnitude increases (Fig. 9(b)). The spectrum turns to a dual-wavelength generation (Fig. 9(c)). Larger magnitude of fluctuation will lead to NLP generation, such as in the cases of curve 4 and 5. But if we reduce the P_{A} of curve 5 from 150 W to 75 W, while keeping fluctuation magnitude remain the same, the clean DSR pulse can be obtained again (Fig. 10(e)). This further indicates that DSR pulse favors small P_{A} value.

If we increase the number of fluctuations of the curve 6 within the range of P_{A} (Fig. 9(d)), while keeping other parameters unchanged including the magnitude of fluctuation, the pulses become more and more unstable. When the fluctuation number within P_{A} is 1, clean DSR pulse with flat top as obtained (Figs. 10(a) and 10(e)); when the number is 2, there are always a little fine structures locating at the side of the pulse, but the AC trace doesn’t distort from triangular shape (Figs. 10(b) and 10(f)); when the number is 7, the pulse becomes much distorted and unstable, and looks like an old broken castle (Figs. 10(c) and 10(g)); if we increase the number of fluctuations to 34 as shown in Fig. 10(d), lots of fluctuations in amplitude of square-shaped pulse has occurred (Fig. 10(h)). This is the typical mixed state pulse as we mentioned earlier. The increasing numbers of fluctuations leading to instability of clean smooth DSR pulses, the top of the DSR pulse can’t keep being flattened anymore. Furthermore, we have tried to deduce the magnitude of fluctuations in transmission curves 7,8 and 9, as low as curve 2. The first two cases have clean DSR pulses generations again. In the last case, the fine structures in Fig. 10(h) have degenerated obviously. So if the magnitude of fluctuation in transmission curve is low enough, large numbers of fluctuations could be tolerated in clean DSR pulse generation.

## 4. Conclusion

In summary, we have numerically demonstrated three types of pulsed operation in a mode-locked fiber laser by simply tuning the transmission curve under appropriate pump level. When the gain saturation energy is low, the peak power doesn’t reach into the reverse saturable absorption power range, cavity always has a DS generation as DS really has a wide range of parameters to operate [10]. When the gain saturation energy is high enough, the reverse saturable absorption is activated, turning the laser operation to DSR or NLP. P_{A} plays big role in determining the final result. When P_{A} is small enough, the system prefers the DSR pulse generation; otherwise the NLP occurs. The mix state of DSR and NLP has been demonstrated as well, both properties of them can be observed in such state. Fluctuations have been introduced within transmission curve of SA. When the magnitude of fluctuation is relatively low, the system can remain the DSR pulse generation; while strong fluctuation will lead to NLP. Still, reducing the value of P_{A} can make laser favor DSR pulse generation under the condition of fluctuation. The number of fluctuation within power range of P_{A} has also considered. The DSR pulse exhibits more fine structures as the number increases, making the top of the share-shaped pulse not flatten anymore. Reducing the magnitude of the fluctuation of transmission curve will get rid of these fine structures.

In practice, mode-locking by NPR and NOLM should be easier way to observe the DSR phenomenon, because of their obvious reverse saturable absorption effect based on their mechanisms. But the peak power of DSR pulse is limited by P_{A}, and larger P_{A} value will lead to NLP generation. In addition, the high energy DSR pulse with wide pulse width has little linear chirp for compression. This makes DSR pulse less competitive for chasing high peak power.

## Acknowledgments

The authors acknowledge the financial support from the National Natural Science Foundation of China (NSFC, Nos. 61235010 and 61177048), the Beijing Municipal Education Commission (No. KZ2011100050011) and the Beijing University of Technology, China.

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