Bi-soliton generation and its properties in stretched pulse fiber ring laser

Xuesong Mao; Ying Xiong; Yajie Ma; Xiangcheng Chen; Akihiro Maruta

doi:10.1364/OE.22.028506

1. Introduction

Dispersion managed (DM) bi-soliton [1] was proposed as a candidate for carrying information in long haul and large capacity optical transmission line, since it is compact, free of pulse neighboring influences and Gaudon-Haus timing jitter, etc. Its transmission properties have been extensively studied within the framework of nonlinear Schrödinger equation (NLSE), especially the stability under the influence of high order dispersion [2]. To accomplish a soliton communication system, a light source used for generating uni-soliton and bi-soliton is most basic and important. However, due to the strict phase and chirp relationship between two neighboring pulses that form a bi-soliton, it is not feasible to request a device to bind two solitons directly to form a bi-soliton. In fact by applying averaging method [3], coupled Gaussian pulses can form a bi-soliton. However, averaging method is not practically realizable in optical field for obtaining bi-soliton because of complicated computation. A light source that can produce bi-soliton directly is then desirable. Since the transmission line is dispersion managed, it is natural to utilize a fiber laser that consists of a loop of fiber with the features of the same Kerr nonlinearity, anomalous/normal dispersion pattern, and distributed gain compensating locally for all losses. The stretched pulse fiber ring laser (SPFRL) applies a technique closely related to the principle of dispersion management in the optical transmission lines. For the reason, bi-soliton produced by SPFRL can propagate stably in DM transmission lines if the parameters are adjusted appropriately. Besides, pulses circulating in the ring compress and stretch periodically. This reduces some of the limits on pulse width and pulse energy imposed by nonlinear effects. With this type of laser, production of a compact, short and high energy pulse sequence is possible.

In the equation for modeling fiber ring laser, gain and loss are not regarded as perturbation terms anymore, since gain should be large to provide population inversion, and loss should also be large to characterize laser output. Thus the system equation of laser goes into the opposite side of NLSE, and is called the GL equation. Previous work [4] studied the soliton interaction starting from the two-soliton system in the framework of GL equation, and found that two solitons with a phase difference of π/2 are stable. We notice that, although some researchers [5–9] observed soliton pairs in the laser ring cavity or in the nonlinear accelerating beams, the work about how to generate bi-soliton in stretched pulse fiber ring laser has not been published yet. In addition, the previously observed bi-soliton withπ/2 phase difference in [4] is not preferred to the wavelength division multiplexing (WDM) transmission, where an anti-phase bi-soliton is required [1].

Reference [10] showed the process of generating harmonic mode locking pulses numerically when random noise is used as initial condition. The resulted pulse peak is relatively low compared with pulse full width at half maximum (FWHM). However, when pulse peak becomes large, nonlinear gain saturation should be taken into account to reduce the instability of the GL equation. In this paper, we show the possibility of generating a bi-soliton when the nonlinear gain saturation is added to the GL equation. In lasers, weak signal is often used as seed for producing high performance and high quality laser pulses. In our first effort, a weak single Gaussian pulse is used as seed pulse to guarantee the generation of bi-soliton. The results show that, after circulating a certain period unstably in the ring laser, the seed pulse converges into a bi-soliton for sure. On the other hand, when no seed pulse is used, in addition to the generation of bi-soliton, there are also possibilities of generating three stable pulses (Fig. 7 in [10]) even four in some cases.

In our equation, the term of high order nonlinear gain studied here is a novel consideration compared with Eq. (1) in [10]. Without the term, system equation becomes unstable if gain coefficient $ρ$ is large. Initial pulse diverges after it circulates the ring a few round trips. By using a low peak single Gaussian pulse as seed pulse, we study the effect of high order nonlinear gain by changing its coefficient μ and fixing all other coefficients in the regime where bi-soliton exists. μ is explicitly divided into two parts: symmetric anti-phase bi-soliton and asymmetric bi-soliton with unequal pulse peak and uncertain phase relation. The phenomenon that two pulses having unequal peak was also observed in harmonic mode locking situation [10]. Some phenomena such as pulse center shift in initial stage after bi-soliton is formed and phase difference between two pulses with unequal peak will be explained. When the coefficient μ is lower or higher than a particular level, irregular pulses circulates the ring unstably.

2. Basic equation

Considering the perturbations that induced by various imperfections of optical fiber, as well as some optical components introduced intentionally, the most basic form of the system equation to describe DM transmission is:

i \frac{\partial U}{\partial Z} - \frac{b (z)}{2} \frac{\partial^{2} U}{\partial T^{2}} + | U |^{2} U = p e r t u r b a t i o n t e r m s

Here,

U

is the normalized complex amplitude of optical pulse,

Z

and

T

stand for normalized propagation distance and retarded time, respectively.

b (z)

stands for fiber’s normalized group velocity dispersion (GVD), which is compensated periodically along the transmission line. Conversion from normalized parameters to real system parameters can be found in [1].

A similar technique to dispersion management described in Eq. (1) has found applications in the stretched pulse fiber ring laser [11]. Laser ring cavity, consisting of alternating fiber segments with negative and positive GVD, is similar to the alternating sequence of normal and anomalous dispersion fiber segments in DM line. Considering the gain properties (gain nonlinearity and bandwidth) of the solid state laser, Eq. (1) is modified as:

\begin{array}{l} i \frac{\partial U}{\partial Z} - \frac{b (z)}{2} \frac{\partial^{2} U}{\partial T^{2}} + | U |^{2} U = \\ - i α U + i {\frac{g (Z)}{1 + P_{0} (Z) / P_{s a t}} (1 + σ \frac{\partial^{2}}{\partial T^{2}}) + ρ {| U |}^{2} - μ {| U |}^{4}} U \end{array}

where perturbation terms are explicitly expressed.

g (Z)

and

P_{s a t}

stand for the medium gain and amplifier’s saturated power respectively.

P_{0} (Z)

stands for pulse energy, which is defined as:

P_{0} (Z) = \int_{- \infty}^{\infty} {| U (Z, T) |}^{2} d T

σ

accounts for gain bandwidth, which is related to spectral line broadening induced by inversed population exhaustion, atomic collision, and etc.

ρ

is the nonlinear gain coefficient, which is a key factor in the term of

ρ {| U |}^{2} U

that plays the role of fast saturable absorber to realize modelocking. The main differences between Eq. (2) and Eq. (1) in Ref [4]. are: (1) dispersion changes with the change of

Z

in Eq. (2), and (2) gain saturation is considered as well. In stretched pulse fiber ring laser, many approaches are proposed to achieve modelocking, such as additive pulse modelocking (APM) [12], Kerr-lens modelocking [13] and nonlinear polarization rotation [12].

μ

and

α

accounts for the nonlinear gain saturation and fiber loss, respectively. Perturbation terms in Eq. (1) which describes the optical transmission system can be designed, as a good approximation, to fit the terms at the right side of Eq. (2) [14,15]. Thus, bi-soliton generated by the laser can satisfy the requirement of transmitting bi-soliton stably in optical communication system directly. A design example of stretched pulse fiber ring laser can be found in [11].

Assume the values of GVD parameter $b (Z)$ and gain coefficient parameter $g (Z)$ keep unchanged within one fiber link and will change to other values for different fiber links, one can obtain a simple two-step model for $b (Z)$ and $g (Z)$ :

b (Z) = {\begin{cases} b_{1} (> 0) standard fiber \\ b_{2} (< 0) gain fiber \end{cases}

and

g (Z) = {\begin{cases} g_{1} (= 0) standard fiber \\ g_{2} (> 0) gain fiber \end{cases}

Since standard fiber with anomalous dispersion does not provide energy gain for laser generation, gain coefficient $g_{1}$ is set to be 0. A schematic gain and GVD distribution pattern are shown in Fig. 1. GVD distribution along the ring is important for DM soliton formation; no soliton solution can be found if it is not properly arranged [16].

Fig. 1 A schematic diagram of GVD and gain distribution along the fiber ring.

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In order to give a description on dispersion management property of the system, three parameters are defined:

\begin{array}{l} B = b_{1} l_{1} + b_{2} l_{2} the path-averaged GVD \\ S = - b_{1} l_{1} + b_{2} l_{2} the GVD strength \\ R = l_{1} the length ratio \end{array}

Here,

l_{1}

and

l_{2}

stand for the lengths of standard fiber and gain fiber, which satisfy

l_{1} + l_{2} = 1

and correspond to one round trip of the laser. It has been pointed out in [17] that cavity length and GVD distribution pattern determine laser pulse width. We fix three parameters

B

,

S

and

R

as

- 0.08

,

2.1

and

0.5

, respectively, since they are in the middle part of the subspace where DM bi-soliton exists as shown in [16]. Pulses propagating in the system with this set of parameters have larger tolerance to inhomogeneity of fiber parameters, which is induced by fabrication or installation environment.

In simulations of searching bi-soliton solution in section 3 of this paper, a weak single Gaussian pulse is chosen as seed pulse. Free propagation of the seed pulse in the laser ring cavity leads to the convergence from the seed pulse to a bi-soliton. The simulations are performed by using the well-known split-step Fourier method.

3. Bi-soliton in stretched pulse fiber ring laser

According to the study in [4], it is sure that GL equation has bi-soliton solutions. However, the problem is how to force initial pulse converge to a bi-soliton. When no seed pulse is injected, laser cavity cannot be guaranteed to generate bi-soliton. In fact, the possibility of generating bi-soliton from noise is very low by using the parameters we tested so far Thus, we consider of using a low peak single Gaussian pulse as injection, since it is easy to obtain by traditional methods. Reference [10] showed that perturbation terms are important in the formation process of harmonic mode locking pulses. In a similar way, we adjust parameters defined in Eq. (2) –pointwise and watch the resulted pulse shape till 20,000 periods of circulation around the laser cavity. In simulations, normalized round trip time is set to be 30. The first group of parameters we found, where well-shaped bi-soliton can be generated from a weak single Gaussian pulse, is:

g_{1} = 0 (in l_{1}), g_{2} = 0.04 (in l_{2}), σ = 7, ρ = 2.5, μ = 0.58, α = 0.55, P_{s a t} = 0.01

Figure 2 shows the resulted stable bi-soliton in a time window that the lower bound is 9 and upper bound is 15. In the initial stage of bi-soliton formation, pulse center shifts during every round-trip, and the speed of shifting gradually lowers down till the pulse center locates in a point at last. Such a phenomenon happens because pulse group velocity is changed during circulation. The first 4000 periods are not showed in the figure because pulse center randomly shifts out of the observing window.

Fig. 2 Bi-soliton circulating in the stretched pulse fiber ring laser.

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Bi-soliton formation process from a weak single Gaussian pulse is still mysterious. For watching bi-soliton formation process, the first 500 round-trips of the weak single Gaussian pulse are shown in Fig. 3. The initial pulse is so weak that it is even invisible in the figure. The Gaussian pulse splits into five pulses in the first stage. The five pulses circulate the ring as if they are one pulse. Then, five pulses suddenly emerge into one irregular pulse at Z = 250. And then the irregular pulse suddenly converges to a bi-soliton-like pulse packet at Z = 300. In the succeeding stage, the bi-soliton-like pulse circulates the ring and gradually converges to a bi-soliton.

Fig. 3 Bi-soliton formation process from a single Gaussian pulse. Initial pulse peak is so low that it is not visible in the figure.

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Bi-soliton generated by the laser has the same property as DM bi-soliton, which can be obtained by applying averaging method to coupled Gaussian pulses in DM transmission line [1]. These two pulses are closely grouped in a bi-soliton, and propagate as if they are one pulse. For this purpose, binding efficiency $ε = Δ T / τ$ is defined as a measure for describing the compaction of obtained bi-soliton, where $Δ T$ is pulse interval and $τ$ is pulse FWHM width. The pulse interval and FWHM width shown in Fig. 3 are 0.8372 and 0.5564, respectively, result in a binding efficiency of 1.5047, which is near to the value of anti-phase Schrödinger-bi-soliton. The pulse interval and FWHM width of the anti-phase bi-soliton are 1.5 and 1 respectively in [1]. Thus the binding efficiency is 1.5. The particulars of the pulse included in the low wing obviously present a symmetric wrinkle structure when the pulse is plotted in semi-log coordinates, which is shown in Fig. 4. The result shows that a seed pulse circulating the laser cavity converges to a bi-soliton with the same accuracy as generating a DM soliton by averaging method.

Fig. 4 Bi-soliton in semi-log scale coordinates. The center of pulse is adjusted to 0. Particulars included in the low wing present symmetric structure.

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All perturbation terms in Eq. (2) appear to be symmetric, which will not introduce the change of pulse group velocity. In the initial stage, the shift of bi-soliton center occurs, which is considered as the result of some asymmetric properties of pulse itself. We examined the spectrum of bi-soliton, and found that in the initial stage, the spectrum center does not locate at zero. It gradually shifts to the left side, and converges to 0 at last, as shown in Fig. 5. When spectrum center approaches to 0, pulse center stops shifting in time domain. In fact, it is easy to prove in mathematical way that a linear modulation of pulse by exp(jκT) in time domain leads to the change of group velocity. Such phenomenon can also be proved by comparing center shift shown in Fig. 6(a) and 6(b) for low μ situation.

Fig. 5 Pulse spectrum evolution. Pulse center spectrum gradually converges to 0.

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Fig. 6 Pulse propagation in time domain when μ = 0.54. Pulse spectrum center converges to nonzero, which leads to pulse linear shift in time domain, as shown in (a). Then, spectrum center is adjusted to 0 by multiply a factor exp(jκT)to pulse in time domain, pulse center ceases to shift, as shown in (b).

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When μ is low, difference can be seen from two figures in Fig. 6. In Fig. 6(a), pulse spectrum center does not converge to 0, but converge to a nonzero point. The bias of spectrum center when μ = 0.54 leads to a linear shift of pulse center in time domain. We adjusted spectrum center to 0 by multiplying a factor exp(-jκT) in the time domain, and checked how the spectrum center adjusts pulse propagation. The result is shown in Fig. 6(b). Comparison between Fig. 6(a) and 6(b) clearly shows that pulse center shift in time domain happens because of pulse spectrum center bias.

4. Effects of high order nonlinear gain coefficient μ

In order to obtain bi-soliton, nonlinear gain parameter ρ is set to be much greater (nearly 10 times) than that in harmonic mode locking situation studied in [10], which leads to pulse peak increasing rapidly as well as data overflow if high order nonlinear gain is not considered in Eq. (2). However, when real laser is used, this kind of phenomenon will not happen because of gain saturation effect, which is related to the inversed population consumption. Thus, when ρ is large, higher order nonlinear gain should be introduced to reduce errors between mathematical model and real laser.

By increasing nonlinear gain coefficient μ, amplification effect of the laser cavity at pulse peak is suppressed. When μ is less than 0.5, irregular unstable pulse is obtained due to instability introduced by nonlinearity. If we increase μ step by step while fixing other parameters with values shown in section 3, a nonlinear gain coefficient regime that can produce bi-soliton will be found. When μ is increased to 0.51, two stable pulses can be generated by the laser. However, the peaks and FWHM widths of these two pulses are not same. Gradually increasing μ leads two asymmetric pulses to converge to a bi-soliton. When μ is greater than 0.6, nonlinear gain effect is reduced by its high order term, resulting that mode locking effect is not strong enough to fix phase relations among oscillating modes. The obtained regime that can produce bi-soliton is divided into two parts: asymmetric bi-soliton with small μ and symmetric bi-soliton with large μ.

Figure 7 shows bi-soliton properties with respect to the high order nonlinear gain coefficient μ. As shown in Fig. 7(a), total pulse energy, which is defined by Eq. (3), decreases with the increase of μ. The phenomenon can be explained by the minus sign of μ in Eq. (2). Figure 7(b) shows that in asymmetric bi-soliton regime, pulse spacing increases with the increase of μ. When μ goes into symmetric bi-soliton regime, pulse spacing turns to decrease suddenly. The spacing between two pulses is defined as the time interval between two pulse peaks. Figure 7(c) and 7(d) show pulse peaks and FWHM widths with respect to μ, respectively. The upper branch of pulse peaks in pulse asymmetric regime shown in Fig. 7(c) corresponds to the lower branch of pulse FWHM widths shown in 7(d). From these two figures we can see that, in symmetric bi-soliton regime, when pulse peaks decreases pulse FWHM width increases. Such a phenomenon indicates that GL soliton keeps some properties of Schrödinger soliton. That is, in both situations, large pulse peak corresponds to low FWHM width (In Schrödinger soliton situation, multiplication of pulse peaks and FWHM widths keeps a constant [12]).

Fig. 7 Bi-soliton properties with respect to high order nonlinear gain coefficient μ. (a) pulse energy, (b) pulse peak power, (c) pulse spacing, (d) pulse FWHM width.

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In asymmetric bi-soliton regime, for each μ, there are two situations: i) left pulse peak is larger than right one, and ii) right pulse peak is larger than left one. We observed the two solutions and found that each spectrum center of the two situations locates at positive or negative side, as shown in Fig. 8. When the center locates at negative side, left pulse is larger than the right one, and vice versa.

Fig. 8 Pulse spectrum center with respect to high order nonlinear gain coefficient μ.

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Phase relation between two pulses of a bi-soliton is important in transmission system. Reference [1] demonstrated that anti-phase bi-soliton is more appropriate for transmission in WDM system since it is immune to pulse collision. Reference [4] showed the existence of bi-soliton solution with π/2 phase difference in fiber ring laser without dispersion management. However, how to determine the phase difference between two pulses of bi-soliton is not addressed at present.

For stretched pulse fiber ring laser studied here, phase relation between two pulses appears to be anti-phase in symmetric bi-soliton regime. While in asymmetric bi-soliton regime, phase difference between two pulses does not keep constant. Examples of phase relation for two situations are show in Fig. 9(a) and 9(b). In the field that pulse exists, Fig. 9(a) shows an exact anti-phase relation throughout the time field for symmetric bi-soliton, while Fig. 9(b) shows a point-wise variance in phase difference for asymmetric bi-soliton. For the later situation, amplitudes of corresponding part between two pulses do not equal each other. And the phase rotations introduced by fiber nonlinearity are not same either. The reason is that phase difference between two pulses does not keep a constant, even if initial asymmetric bi-soliton has exact anti-phase relation. Such an explanation can be proved by recording phase difference along transmission axis Z directly.

Fig. 9 Examples of phase difference for symmetric and asymmetric bi-soliton. (a) and (b) show pulse phase with respect to normalized time. (c) and (d) record phase difference between two peaks for symmetric and asymmetric bi-soliton within 20 periods propagation. (e) and (f) measure pulse spacing.

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Due to the varying phase difference for asymmetric bi-soliton along time axis, we only measure and record phase difference between two peaks along transmission axis for both symmetric and asymmetric bi-soliton. The result is shown in Fig. 9 (c) and 9(d), ignoring the probable initial unstable parts. For symmetric bi-soliton as shown in (c), a phase difference of π between two pulses is recorded along transmission axis, with only 1% error rate which due to the errors in pulse center positioning process introduced by large step size in time axis. However, for asymmetric bi-soliton, phase difference between two peaks varies periodically within the range of π and 1.2πas shown in Fig. 9(d). Although phase differences of different parts remarkably vary, the phase difference at two peaks is limited in a narrow range. The pulse asymmetry makes the phase difference at peaks of two pulses no longer be anti-phase.

Now, we can also explain the abrupt change of pulse spacing from asymmetric bi-soliton regime to symmetric bi-soliton regime showed in Fig. 7(b). For incoherent situation (many modes coexist in the laser cavity), the interaction between two pulses may change from attraction to repulsion and vice versa, which depends on their phase relation [18]. When neighboring pulses have exact anti-phase relation, attraction force occurs (see Fig. 4 in [18]) and increases with parameter μ. It gives the reason that pulse spacing gradually decreases with μ in symmetric bi-soliton regime. For asymmetric bi-soliton, phase difference between two pulses is not a constant, which leads to complicate force between two pulses, including both attraction and repulsion. Resultant attraction force decreases with μ, which explains the increase of pulse spacing in this regime. Pulse spacing also changes periodically within one dispersion map due to the varying GVD, but finally returns to the initial stage at the end of the period. Figure 9(e) and 9(f) show the measured pulse spacing for two situations.

5. Summarization

In this paper, we firstly discussed the relation between Schrödinger equation and GL equation, which are both suitable for describing fiber transmission system. GL equation is also appropriate for describing fiber ring laser approved by many experimental and theoretical researches. Transmission system described by Schrödinger equation can be designed to fit the propagation of GL-laser produced pulse.

Starting from a weak single Gaussian, bi-soliton generation is demonstrated within 500 periodical propagations. The result is very interesting although its origin still keeps mysterious. Since single Gaussian pulse generation from fiber ring laser is already a mature technique, direct bi-soliton generation from fiber ring laser will be possible if problems considered in this paper are solved in real system.

Then, we investigated the properties of observed bi-soliton and explained various phenomena against high order nonlinear gain. The high order nonlinear gain regime that can produce stable bi-soliton is divided into symmetric and asymmetric bi-soliton regimes. In symmetric bi-soliton regime, before spectrum center converges to 0, pulse center shifting in time domain is observed. Bi-solitons in this regime have exact anti-phase relation, which fits the WDM transmission requirements. In asymmetric bi-soliton regime, pulse spectrum center converges to a fixed nonzero point. There are two situations for one set of parameters: spectrum center is possible to locate at either positive side or negative side. The two solutions seem to be different but in fact same since one solution is exactly the reverse of the other. Bi-solitons in this regime have no fixed phase relation throughout time domain, even phase difference between two peaks fluctuates and deviates from anti-phase during propagation along Z axis. The deviation of phase difference explains the pulse spacing change against μ. Applications of these asymmetric bi-solitons are still unexplored since they are more complicated than those of the symmetric situation.

Acknowledgements

This work is supported by National Science Foundation of China project Grant No. 61104215, and the Scientific Research Foundation for the Returned Overseas Scholars, State Education Ministry.

References and links

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Bi-soliton generation and its properties in stretched pulse fiber ring laser

Abstract

1. Introduction

2. Basic equation

3. Bi-soliton in stretched pulse fiber ring laser

4. Effects of high order nonlinear gain coefficient μ

5. Summarization

Acknowledgements

References and links

Cited By

Figures (9)

Equations (7)

Optics Express