Simulation of the generation conditions and influence parameters of a self-mode-locked erbium-doped fiber laser

Zeqiu Hu; Zeqiu Hu; Duanduan Wu; Duanduan Wu; Duanduan Wu; Duanduan Wu; Wei Lin; Wei Lin; Zhe Li; Zhe Li; Shixun Dai; Shixun Dai; Shixun Dai; Shixun Dai

doi:10.1364/OE.477254

1. Introduction

As the low-cost and straightforward coherent ultrashort pulse sources, passively mode-locked fiber lasers have broad application prospects in fiber sensing, laser measurement, nonlinear optical properties of materials, spectroscopy, optical communications [1–4], and so on. The traditional mode-locked fiber lasers are realized with fast saturable absorbers (SAs) of materials, e.g. carbon nanotubes (CNTs), semiconductor saturable absorber mirrors (SESAMs), graphene, topological insulators (TIs), and so on [5–15]. However, these kinds of SAs always have limited damage thresholds, thus limiting the energy of their generated mode-locked pulses. For example, H. Zhang et al. used a graphene SA in an erbium-doped fiber laser (EDFL), obtaining a mode-locked pulse with the pulse energy of 7.3 nJ [16]. I.A. Litago et al. achieved a SESAM-based mode-locked EDFL with the pulse energy of 89 pJ [17]. M.A. Ismail et al. obtained the mode-locked pulse with 0.34 nJ pulse energy in an EDFL by using a CNT SA [18].

Recently, self-mode-locking technologies with high damage threshold characteristics have attracted much attention. In fiber lasers, self-mode-locking technologies mainly include inter-mode beat frequency [19], ion-pair quenching effect [20] and self-focusing effect [21]. The ion-pair quenching effect is achieved by using high concentration doped fibers as SAs, which is an additionally inserted doped fiber [22], or an overly long doped fiber in the laser acting as both gain medium and SA [23–27]. Therefore, the ion-pair quenching effect-based self-mode-locked lasers can possess simple structure and be without extra insertion lost. They have been widely studied in recent years [23–27]. For instance, A.F. El-Sherif et al. demonstrated a self-mode-locked heavily thulium-doped fiber laser with the mode-locking threshold of 2.5 W and the pulse energy of 3.75 µJ [23]. They qualitatively explained that the end-pumped geometry combined with the three-level nature of the Tm-ions, leads to the existence of a high absorption region at low pump levels, acting as a SA. C. Liu et al. used a high concentration thulium-doped fiber to achieve a self-mode-locked fiber laser with the mode-locking threshold of 2.05 W and the pulse energy of 32.7 nJ [24]. The generation of self-mode-locked pulses is considered to be attributed to the weak amplitude modulation of the intracavity laser and the weak saturable absorption of the thulium-doped fiber. M. Durán-Sánchez et al. used an Erbium-Ytterbium co-doped double-clad fiber to obtain self-mode-locked pulses with the mode-locking threshold of 4.89 W and the pulse energy of 50 nJ [25]. The self-mode-locking effect is explained as follows. The long length of active medium leads to a weakly pumped rear section. When the pump light is high enough, the rear section can reach bleaching through the saturable absorption mechanism. The above self-mode-locked fiber lasers are with anomalous dispersion conditions. Similar to other kinds of mode-locked fiber lasers, the self-mode-locked fiber lasers can also be achieved with normal dispersion conditions. For example, X. Li et al. observed the self-mode-locked pulses in a normal dispersion ytterbium-doped fibers with the mode-locking threshold of 56 mW [26]. The weak saturable absorption effect of ytterbium-doped fiber is considered to from the self-mode-locking effect. In this laser, the self-mode-locked pulses are initiated by carefully tuning the polarization controller (PC). Z. Feng et al. obtained the self-mode-locked pulses in a normal dispersion EDFL with the mode-locking threshold of 30 mW [27]. The weak saturable absorption effect of erbium-doped fiber (EDF) is used to explained the generation of the self-mode-locked pulses. The polarization state is mentioned to make great effect on the launch of the self-mode-locked operation. Based on the above literatures, the inner mechanism of self-mode-locked lasers is currently limited to qualitative analysis. And there are some differences among different explanations. Therefore, it is of great significance to quantitatively analyze the generation conditions and working principle of the self-mode-locked lasers. Moreover, one can notice that the performances of self-mode-locked lasers vary greatly under different dispersion conditions. For example, the anomalous dispersion self-mode-locked lasers possess high mode-locking thresholds. The normal dispersion ones have low mode-locking thresholds but sensitive polarization properties. So, a deep study of the laser cavity parameters on the self-mode-locked pulse performances is very necessary.

In this paper, the generation conditions and influence parameters of self-mode-locked fiber lasers are theoretically studied. By establishing a self-mode-locked EDFL simulation model with a high-concentration erbium-doped fiber-based SA, the effect of laser parameters on the mode-locked pulse generation and output performances are quantitatively analyzed.

2. Theory

2.1 High concentration doped fiber-based SA

The ion pairs effect can theoretically explain the observed self-pulsation generation in EDFLs [20,28–30]. It has been demonstrated that the EDFL operates continuously at low ion pair concentrations (x < 5%), where x is the proportion of ion pairs in the doped fiber [30]. But when the ion pair concentrations (x > 5%) are high, the laser turns to self-pulse operation. The physical mechanism of this ion pair-based self-pulsation is the quenching effect, which occurs between two adjacent Er ions at the $^4{I_{13/2}}$ energy level [31]. As shown in Fig. 1(a), if the pump light of 810 nm is used in an EDFL, the ground state ion will be transitioned to the $^4{I_{9/2}}$ energy level. Since the energy level lifetime of $^4{I_{9/2}}$ is about 7 µs, the ions at the $^4{I_{9/2}}$ energy level will quickly transition to the $^4{I_{13/2}}$ energy level [29]. Then, as the energy level lifetime of $^4{I_{13/2}}$ is about 10 ms [32], the ion at this energy level will transfer its energy to another ions, producing an upconverted $^4{I_{9/2}}$ ion and a ground state ion. At last, the upconverted ions rapidly decay to the $^4{I_{13/2}}$ energy level, as shown in Fig. 1(b). This is the so-called quenching effect. One should notice that the quenching effect is negligible in low doped concentrations EDF. It can only happen in EDF with high ion pair concentrations (i.e. high doped concentrations) [33]. The result of the quenching effect is that an excited Er ion is lost, and a ground state ion is gain [34]. Therefore, the laser radiation cannot transmit until the luminous flux is large enough to empty the ions at the ground state level. Moreover, when the upper energy state has a sufficient number of particles, the EDF becomes transparent to the laser radiation, resulting in a high transmittance, which is equivalent to a SA [35]. The following equation can be used to character the saturable absorption property of the high doped EDF-based SA [22]:

(1)$$T(I) = 1 - \Delta T \cdot \exp ( - \frac{I}{{{I_{sat}}}}) - {T_{ns}}$$

where $T(I)$ is the transmission, $\Delta T$ is the modulation depth, $I$ is the average input power, ${I_{sat}}$ is the saturated average power, and ${T_{ns}}$ is the unsaturated loss.

Fig. 1. (a) Energy level diagram of erbium ion and (b) quenching effect of erbium ion pairs [29].

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2.2 Model and parameters of the self-mode-locked fiber laser

By using an overly long high doped EDF as both gain medium and SA, a self-mode-locked EDFL is constructed. Fig. 2 shows the model of this self-mode-locked EDFL. The laser consists of an EDF, three single-mode fibers (SMFs), a PC, and an optical coupler (OC). The PC is used to control the laser polarization state. The OC outputs the laser from the laser cavity. The three single-mode fibers are pigtails of the PC and the OC. The intra-cavity laser is set to oscillate clockwise around the laser.

Fig. 2. Simulation model of the self-mode-locked EDFL.

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The light propagation in fiber section is described by the Ginzburg-Landau equation [36]:

(2)$$\frac{{\partial u}}{{\partial z}} = \frac{g}{2}u + (\frac{g}{{2\Omega _g^2}} - i\frac{{{\beta _2}}}{2})\frac{{{\partial ^2}u}}{{\partial {t^2}}} + i\gamma |u{|^2}\textrm{u}$$

where $u$ is the normalized electric field envelope, $z$ is the propagation distance, $t$ is the local time, $g$ is gain saturation, $g = {g_0}\exp ( - {E_{pulse}}/{E_{sat}})$, ${{g}_0}$ is the small-signal gain, which is equal to 0 for undoped fibers, ${E_{pulse}}$ is the pulse energy, ${E_{pulse}} = \int_{{{ - {T_R}} / 2}}^{{{{T_R}} / 2}} {|u{|^2}d\varepsilon }$, ${T_R}$ is the cavity round-trip time, ${E_{sat}}$ is the gain saturation energy, which usually represents pumping strength, ${\Omega _g}$ is the gain bandwidth, ${\beta _2}$ is the second-order dispersion coefficient, and $\gamma $ is the nonlinear coefficient. The items on the right of the formula correspond to gain, gain dispersion and second-order dispersion, and self-phase modulation, respectively.

Besides the fiber gain, dispersion and nonlinear effects, the polarization characteristic is also a key parameter, which can make great effect on the intra-cavity laser. Therefore, a PC is used to control the polarization characteristic of the intra-cavity laser. And the light transmission equation of the PC is considered during the simulation, which is expressed as follows [37]:

(3)$$T = {\sin ^2}(\theta ){\sin ^2}(\varphi ) + {\cos ^2}(\theta ){\cos ^2}(\varphi ) + 0.5\sin (2\theta )\sin (2\varphi )\cos (\Delta {\phi _L} + \Delta {\phi _{NL}})$$

where $\theta $ and $\varphi $ are the orientation angles of the polarizer and analyzer with respect to the fast axis of the fiber, $\Delta {\phi _L}$ is the linear phase delay, $\Delta {\phi _L} = \frac{{2\pi L}}{\lambda }\Delta {n_L} + {\phi _{PC}}$, $\Delta {n_L}$ is linear refractive index of fiber, ${\phi _{PC}}$ is phase difference between the fast axis of the fiber and the slow axis of the fiber, $\Delta {\phi _{NL}}$ is the nonlinear phase difference caused by self-phase modulation and cross-phase modulation, $\Delta {\phi _{NL}} ={-} \frac{{2\pi L{n_2}P}}{{3\lambda {A_{eff}}}}\cos (2\theta )$ [38], ${\textrm{n}_2}$ $\lambda $ is the pulse center wavelength, $L$ is the fiber length, ${n_2}$ is the nonlinear coefficient, ${A_{eff}}$ .is the effective mode field area, and $P$ is the optical power.

The simulation starts with a random weak noise signal. Then, the pulse evolution process is simulated by the standard split-step Fourier method. Table 1 shows the fiber parameters used for the simulation of the self-mode-locked EDFL. And other parameters are as follows: ${\Omega _g} = \textrm{ }40\textrm{ }nm$, $\Delta {\phi _L}\textrm{ } = \textrm{ }1.2\pi$, ${n_2}\textrm{ } = \textrm{ }2.6 \times {10^{ - 20}}\textrm{ }{m^2}/W$, $\lambda \textrm{ } = \textrm{ }1550\textrm{ }nm$, ${A_{eff}}\textrm{ } = \textrm{ }80\textrm{ }u{m^2}$, $\Delta T\textrm{ } = \textrm{ }0.18$, ${I_{sat}}\textrm{ } = \textrm{ }10\textrm{ }W$ and ${T_{ns}}\textrm{ } = \textrm{ }0.2$.

Table 1. Fiber parameters for simulation

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3. Simulation results

Since the self-mode-locked fiber lasers can operate under arbitrary dispersion condition, the simulation of the self-mode-locked EDFL is respectively executed with anomalous and normal dispersion. For the anomalous dispersion self-mode-locked EDFL, the initial cavity length is set to be 3.2 m. Its net cavity dispersion is calculated to be -0.0512 ps², which produces traditional soliton pulses. In the normal dispersion one, the initial cavity length is ∼14 m. The net cavity dispersion is calculated to be 0.0052 ps², which generates dissipative soliton pulses. During the simulation process, the gain saturation energy, the orientation angles of the polarizer and analyzer with respect to the fast axis of the fiber, and the laser coupling output ratio are chosen to study their effects on the generation and output performance of the self-mode-locked EDFL, respectively.

3.1 Simulation of the anomalous dispersion self-mode-locked EDFL

3.1.1 Effect of the gain saturation energy

Under the condition of 2.9 m total length of SMF, 10% laser coupling output ratio, and π⁄4 orientation angles of the polarizer and analyzer with respect to the fast axis of the fiber, the effect of the gain saturation energy on the self-mode-locked pulse property is firstly studied. As shown in Fig. 3(a), the mode-locked pulse appears at a gain saturation energy of 100 pJ. With further increasing the gain saturation energy, the intensity of the output mode-locked pulses increases. When the gain saturation energy reaches 500 pJ, the mode-locked pulses split. The peak power of the soliton is limited by the soliton area theory. So, the single pulse within the cavity breaks down into multiple pulses when the intracavity laser power is high enough [39]. Figure 3(b) shows the pulse duration and the pulse energy of the single mode-locked pulse as a function of the gain saturation energy. The pulse duration gradually decreases from 1.10 ps to 0.46 ps, and the pulse energy increases from 0.038 nJ to 0.156 nJ, as the increase of the gain saturation energy.

Fig. 3. (a) The self-mode-locked pulses under different gain saturation energy and (b) pulse duration and pulse energy versus the gain saturation energy.

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3.1.2 Effect of the orientation angles of the polarizer and analyzer with respect to the fast axis of the fiber

Under the condition of 300 pJ gain saturation energy, 10% laser coupling output ratio, and 2.9 m total length of SMF, the effect of the orientation angles of the polarizer and analyzer with respect to the fast axis of the fiber on the self-mode-locked pulse property is studied. As shown in Fig. 4, by changing the orientation angles of the polarizer and analyzer with respect to the fast axis of the fiber, the output pulses exhibit single-pulse and multi-pulse states. The nonlinear loss of different polarization light in the cavity is different. So, changing the polarization state is equivalent to change the pulse gain in the laser cavity [37]. When the pulse gain is small, the single pulse can stably transmit. But when the pulse gain is large, the pulse peak power can reach the limitation of soliton area theory. Then, the multiple pulses can be generated.

Fig. 4. The self-mode-locked pulses under different orientation angles of the polarizer and analyzer with respect to the fast axis of the fiber.

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3.1.3 Effect of the laser coupling output ratio

Under the condition of 2.9 m total length of SMF, 300 pJ gain saturation energy, and π⁄4 orientation angles of the polarizer and analyzer with respect to the fast axis of the fiber, the effect of the laser coupling output ratio on the self-mode-locked pulse property is studied. From Fig. 5(a), the mode-locking pulse cannot form when the laser coupling output ratio is 30%. It appears with the laser coupling output ratio of 25%. Continue to decrease the laser coupling output ratio, the intensity of the output pulse increases. The low cavity output ratio is conducive to the formation of mode-locked pulses. This is due to the low laser coupling output ratio makes the laser cavity possess little loss. In this case, more longitudinal modes will oscillate to help the mode-locked pulse generation. Then, the pulse duration and the pulse energy as a function of the laser coupling output ratio is show in Fig. 5(b). The pulse duration gradually increases from 0.39 ps to 0.90 ps as the laser coupling output ratio increases from 1% to 25%. At the same time, the pulse energy decreases from 0.167 nJ to 0.068 nJ.

Fig. 5. (a) The self-mode-locked pulses under different laser coupling output ratio and (b) pulse duration and pulse energy versus the laser coupling output ratio.

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3.2 Simulation of the normal dispersion self-mode-locked EDFL

For comparison, the normal dispersion self-mode-locked EDFL is also demonstrated. A section of dispersion compensation fiber (DCF) is used to control the net dispersion of the laser cavity of Fig. 2. The DCF is inserted behind the EDF. Table 2 shows the fiber parameters used for the simulation of the normal dispersion self-mode-locked EDFL.

Table 2. Fiber parameters for simulation

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3.2.1 Effect of the gain saturation energy

Under the condition of 7 m length of DCF, 10% laser coupling output ratio, and π⁄4 orientation angles of the polarizer and analyzer with respect to the fast axis of the fiber, the effect of the gain saturation energy on the self-mode-locked pulse property is firstly studied. As shown in Fig. 6(a), the gain saturation energy threshold of the mode-locked pulses is 10 pJ. As the gain saturation energy increases, the intensity of the mode-locking pulses increases. When the gain saturation energy reaches 100 pJ, the pulses start to split. The gain saturation energy thresholds of the self-mode-locked pulses generation and splitting under normal dispersion conditions are lower than those under anomalous one. This is due to the pulse shaping process of the net normal dispersion self-mode-locked EDFL is dissipative. The participation of dispersion and nonlinearity facilitates the balance of gain and loss during the dissipative process [40,41], which accelerates the soliton pulses evolution and enable the soliton pulses quick stabilization. Then, the single pulse duration and the single pulse energy as a function of the gain saturation energy is show in Fig. 6(b). The pulse duration gradually decreases from 0.90 ps to 0.49 ps. The pulse energy increases from 0.003 nJ to 0.049 nJ. These variation tendencies are similar to the anomalous ones.

Fig. 6. (a) The self-mode-locked pulses under different gain saturation energy and (b) pulse duration and pulse energy versus the gain saturation energy.

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3.2.2 Effect of the orientation angles of the polarizer and analyzer with respect to the fast axis of the fiber

Under the condition of 60 pJ gain saturation energy, 10% laser coupling output ratio, and 7 m length of DCF, the effect of the orientation angles of the polarizer and analyzer with respect to the fast axis of the fiber on the self-mode-locked pulse property is studied. Fig. 7 shows the self-mode-locked pulses under different orientation angles. Compared with the anomalous dispersion ones, these normal dispersion self-mode-locked pulses possess a no pulse state in addition to the single pulse and multiple pulse states. Therefore, the normal dispersion self-mode-locked EDFL is more sensitive to the polarization states, which has an obvious polarization-based pulse priming effect. Investigate its reason, the polarization rotation induced dynamic loss makes the normal dispersion self-mode-locked EDFL possess dynamic net gain. Meanwhile, this normal dispersion self-mode-locked lasers operates in a low requirement of gain energy. When the polarization loss is large, the laser may not get enough gain, thus cannot form a self-mode-locked pulse.

Fig. 7. The self-mode-locked pulses under different orientation angles of the polarizer and analyzer with respect to the fast axis of the fiber.

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As the effect of the normal dispersion self-mode-locked EDFL’s laser coupling output ratio on the self-mode-locked pulse property is similar to the anomalous dispersion one, we do not repeat it here. Besides, since the dispersion value plays a dominant role in mode-locked pulse formation and transmission, we independently investigate its effect on the self-mode-locked pulse properties in part 3.3.

3.3 Effect of dispersion value on the self-mode-locked pulse properties

Firstly, the anomalous and normal dispersion self-mode-locked EDFLs respectively set to work on the single pulse state. In the anomalous dispersion self-mode-locked EDFL, the gain saturation energy is 300 pJ. The laser coupling output ratio is 10%. The orientation angles of the polarizer and analyzer with respect to the fast axis of the fiber is π⁄4. For the normal dispersion one, the gain saturation energy is 60 pJ. The laser coupling output ratio and the orientation angles of the polarizer and analyzer with respect to the fast axis of the fiber are the same as the anomalous ones. Then, the pulse properties as a function of the dispersion value are investigated. As shown in Fig. 8, the single pulse state of these two lasers can respectively maintain within a certain range of dispersion value. After that, they turn to the multiple pulse state with increasing the dispersion value and also keep for a range of dispersion value. At last, when the dispersion value is too large, the self-mode-locked pulses become unstable. The explanations of the pulse state changes are as follows. On one hand, the exist of dispersion makes the front and rear edges of the self-mode-locked pulses transmit with different velocities [42]. When the velocity difference is large enough, the single pulse split into multiple pulses. With the increase of dispersion value, the pulses split more and more seriously. I.e. more pulses will generate. On the other hand, with the number of multiple pulses increasing, the interaction between the pulses enhances [43]. Finally, the multiple pulse state becomes unstable. Although the variation trends of pulse states of these two lasers are similar, the pulse states of the anomalous dispersion self-mode-locked EDFL can maintain for a wider range of dispersion value than that of the normal dispersion one. I.e. the anomalous dispersion self-mode-locked EDFL has a higher tolerance for dispersion value change. This maybe due to the formation of the normal dispersion self-mode-locked pulses involves more parameters than the anomalous dispersion ones. When the dispersion value changes, the normal dispersion EDFL requires not only the rebalance of dispersion and nonlinearity, but also the gain and loss. Therefore, its variable range of dispersion value will be narrow.

Fig. 8. Variation of the mode-locked states under different net-cavity dispersion (where A represents single-pulse state, B represents multi-pulse state, and C represents unstable pulse state).

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4. Conclusion

In summary, we have successfully demonstrated the simulation of a self-mode-locked EDFL based on a high-concentration erbium-doped fiber SA. The effect of gain saturation energy, orientation angles of the polarizer and analyzer with respect to the fast axis of the fiber, laser coupling output ratio, dispersion value and condition on the self-mode-locked pulse generation and performances has been quantitatively analyzed. This study has important reference value for the investigation of self-mode-locked fiber lasers: 1) a relatively low laser coupling output ratio can be conducive to activate the self-mode-locked pulses, 2) the anomalous dispersion self-mode-locked pulses need relatively high gain energy and have relatively large capability of dispersion value change, 3) the optimization of polarization is required for the initiation of the normal dispersion self-mode-locked pulses, and 4) the generation threshold of the normal dispersion self-mode-locked pulse is relatively low.

Funding

National Natural Science Foundation of China (61805124); Natural Science Foundation of Zhejiang Province (LY22F050008); 3315 Innovation Team in Ningbo City; K. C. Wong Magna Fund in Ningbo University.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Fiber type	$β_{2}$	$γ$	$L$	$g_{0}$
Fiber type	$p s^{2} / m$	$W^{- 1} m^{- 1}$	$m$	$d B / m$
SMF	-0.022	0.005	2.9	0
EDF	0.042	0.003	0.3	24

Fiber type	$β_{2}$	$γ$	$L$	$g_{0}$
Fiber type	$p s^{2} / m$	$W^{- 1} m^{- 1}$	$m$	$d B / m$
SMF	-0.022	0.005	6.7	0
DCF	0.020	0.003	7	0
EDF	0.042	0.003	0.3	24

Fiber type	$β_{2}$	$γ$	$L$	$g_{0}$
Fiber type	$p s^{2} / m$	$W^{- 1} m^{- 1}$	$m$	$d B / m$
SMF	-0.022	0.005	2.9	0
EDF	0.042	0.003	0.3	24

Fiber type	$β_{2}$	$γ$	$L$	$g_{0}$
Fiber type	$p s^{2} / m$	$W^{- 1} m^{- 1}$	$m$	$d B / m$
SMF	-0.022	0.005	6.7	0
DCF	0.020	0.003	7	0
EDF	0.042	0.003	0.3	24

Simulation of the generation conditions and influence parameters of a self-mode-locked erbium-doped fiber laser

Abstract

1. Introduction

2. Theory

2.1 High concentration doped fiber-based SA

2.2 Model and parameters of the self-mode-locked fiber laser

3. Simulation results

3.1 Simulation of the anomalous dispersion self-mode-locked EDFL

3.1.1 Effect of the gain saturation energy

3.1.2 Effect of the orientation angles of the polarizer and analyzer with respect to the fast axis of the fiber

3.1.3 Effect of the laser coupling output ratio

3.2 Simulation of the normal dispersion self-mode-locked EDFL

3.2.1 Effect of the gain saturation energy

3.2.2 Effect of the orientation angles of the polarizer and analyzer with respect to the fast axis of the fiber

3.3 Effect of dispersion value on the self-mode-locked pulse properties

4. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (8)

Tables (2)

Equations (3)

Optics Express