1.

Introduction

Single-wall carbon nanotubes (SWCNTs) have attracted much attention in the field of optical communications in recent years because of their ultrafast nonlinear optical properties in the near-infrared region arising from the saturation of excitonic transitions,^1–4 their interesting electronic and optical properties, and several applications in photonics, which include nanometer-scale devices for light generation, photodetection, photovoltaic applications, and application as saturable absorbers (SAs).^5,6 Recent developments in fiber lasers have led to a renewed interest in SAs. CNTs have emerged as a promising technology for the fabrication of SAs.⁷ Graphene^8–10and nanoscale graphite^11,12 materials based on CNTs have attracted much attention because of their high optical nonlinearity and fast recovery time when used as an SA in a mode-locked erbium-doped fiber laser (EDFL) for femtosecond/picosecond pulse generation.^13–18

Because SWCNTs possess subpicosecond recovery times and broad absorption spectra, active fibers doped with Yb:KLuW operating at a wavelength of 1000 nm,¹⁹ a praseodymium-doped fiber operating at 1300 nm,²⁰ a ytterbium-doped fiber operating at 1064 nm,²¹ and active fibers doped with ${\mathrm{Er}}^{+3}$ operating at 1500 nm have been mode-locked with SWCNT SAs.²²

Most recent studies on fiber lasers have focused only on the use of CNTs with erbium (Er) fiber lasers for generating short optical pulses at a 22 MHz repetition rate with a 50 kW peak power and a 1.1 ps pulse width^23,24 or a 39 MHz repetition rate with a 3.4 mW peak power and a 115 fs pulse width.²⁵

In addition, new SA materials, such as $\mathrm{TI}:{\mathrm{Bi}}_2{\mathrm{Se}}_3$ and $\mathrm{TI}:{\mathrm{Bi}}_2{\mathrm{Te}}_3$, have attracted much attention for mode locking of an Er-doped fiber laser and have yielded stable soliton pulses of 1.57 ps at 1564.6 nm and 1.21 ps at 1558 nm, respectively.^26,27 In 2014, Zhang developed an ${\mathrm{MoS}}_2$-based optical fiber SA device with an operation wavelength suitable for an ytterbium-doped fiber laser and experimentally generated nanosecond dissipative soliton pulses at 1054 nm.²⁸

Most previous studies have been experimental, and few theoretical analyses in this field have been presented. Analytical methods may be useful for enabling the study of a wide range of SA parameters. An analytical study involves performing a spatial analysis to obtain new information; a spatial analysis would be difficult to perform experimentally. The theoretical description of pulse shaping and propagation in a fiber ring-cavity laser is based on the generalized nonlinear Schrodinger (NLS) equation. The Schrodinger equation can be solved either analytically or numerically. Analytical methods are more rigorous and provide exact solutions, but they are difficult to use for complex problems. Many articles have been published on the exact solutions to nonlinear wave equations. These include studies of the Backlund transform,²⁹ the hyperbolic tangent expansion method,³⁰ the trial function method,³¹ the nonlinear transform method,³² transformed rational function method,³³ and exact 1-soliton solution of the complex modified Korteweg–de Vries equation method.³⁴

Numerical methods, such as the split-step Fourier method, have become popular with the development of computing capabilities, although they only give approximate solutions of the NLS equation. ³⁵ Previous studies of the exact solutions to nonlinear wave equations have not dealt with a range of SA parameters for CNTs.

In this paper, we describe and analyze a method for solving the NLS equation that involves the nonlinear effects of CNTs as an SA. The gain in the NLS equation is described by the saturation power of the gain medium, an average small-signal gain, and an average power over the cavity length. The rest of this paper is organized as follows. In Sec. 2, we derive the equations for representing the solitary wave. In Sec. 3.1, we discuss three cases for the chirp parameter and its effect on the generation of a soliton, and we investigate the effect of the SA parameter on the laser power, pulse phase, and pulse width. In Sec. 3.2, we focus on the chaotic behavior of the laser pulse. Section 3.3 considers the stability of the laser pulse; we focus on the phase of the pulse associated with the introduced transformations and present stationary solutions. In Sec. 3.4, we describe the effect of the SA parameter on the dynamic behavior of the chirped pulse.

2.

Model Algebraic Equations

In this section, we describe the optical pulse propagation in a fiber laser using the NLS equation for the pulse envelope $\psi (z,T)$ in the presence of mode locking with the use of an SA, including the gain dispersion, losses for the cavity and fiber, gain, group-velocity dispersion (GVD), self-phase modulation (SPM), and two-photon absorption (TPA). This equation can be written as⁵

When $\partial \psi /\partial z$ is calculated using Eq. (2), it is found to satisfy

The first part of Eq. (5) is given by

The second part of Eq. (5) is given by

We can write an equation for the second derivative of Eq. (2) with respect to $T$ after some algebra, as follows:

Let us assume the new parameters as

By calculating $\partial \mathrm{\Gamma }/\partial T$ and $\partial \mathrm{\Omega }/\partial T$, the result is given by

Substituting Eqs. (11) and (12) into Eq. (10), we obtain

We follow a similar procedure for finding the second derivative of Eq. (13):

Substituting Eqs. (2), (3), (4), (5), (6), (7), (13), and (14) into Eq. (1), after some algebra, we obtain the following equation:

Separating the real and imaginary parts of Eq. (15), we obtain the following two equations:

Equating the coefficient of ${\mathrm{sech}}^2(\sigma T)$ and other parameters in Eq. (16), we obtain the following equations:

Equation (17) can be separated into two equations in the form

3.

Results and Discussion

The laser configuration consists of a ring cavity. Two types of fibers are used in the cavity: a 6-m piece of Er-doped fiber (EDF) and a standard single-mode fiber (SMF28) with a total net anomalous GVD ${\beta }_2$ of $-0.012{\mathrm{ps}}^2/\mathrm{m}$.

The gain in our fiber laser comes from the EDF, which, when pumped with 980-nm light, exhibits spontaneous emission at 1550 nm. Continuously pumping the gain fiber results in the formation of an initial pulse by amplified spontaneous emission, and the pulse makes several passes through the resonant cavity.³⁶ It is important to fix the total length of the gain fiber so that the round trip time for a pulse through the resonant cavity basically equals the storage time of the gain fiber and the pulse will return on each successive pass to the gain fiber as the gain fiber has been pumped back up from a depleted state caused by the prior pass. A polarization-independent isolator was spliced into the cavity to force unidirectional operation of the ring. A CNT mode locker was placed between the EDF and the 10% fiber coupler, as shown in Fig. 1.

Fig. 1

Schematic of fiber laser components: Er-doped fiber (EDF), single-mode fiber, wavelength-division multiplexer, polarization controller, coupler, and optical isolator.

3.1.

Chirp Dynamics

To understand the dynamic behavior of the chirp, Eqs. (19) and (21) can be used to find an equation for the chirp.

Eq. (22)

$$[2\gamma g{T}_2^2-({\alpha }_2+\frac{{\delta }_{\mathrm{SA}}}{p_{\mathrm{sat}}^{\mathrm{CNT}}}-\frac{g}{p_{\mathrm{sat}}^{\mathrm{Er}}}\left){\beta }_2\right]c^2-3[2\gamma {\beta }_2+\frac{1}{2}g{T}_2^2({\alpha }_2+\frac{{\delta }_{\mathrm{SA}}}{p_{\mathrm{sat}}^{\mathrm{CNT}}}-\frac{g}{p_{\mathrm{sat}}^{\mathrm{Er}}}\left)\right]c-2[2\gamma g{T}_2^2-({\alpha }_2+\frac{{\delta }_{\mathrm{SA}}}{p_{\mathrm{sat}}^{\mathrm{CNT}}}-\frac{g}{p_{\mathrm{sat}}^{\mathrm{Er}}}\left){\beta }_2\right]=0.$$

After some algebra, we obtain the following equation:

Eq. (23)

$$c^2-3\frac{[2\gamma {\beta }_2+\frac{1}{2}g{T}_2^2({\alpha }_2+\frac{{\delta }_{\mathrm{SA}}}{p_{\mathrm{sat}}^{\mathrm{CNT}}}-\frac{g}{p_{\mathrm{sat}}^{\mathrm{Er}}}\left)\right]}{[2\gamma g{T}_2^2-({\alpha }_2+\frac{{\delta }_{\mathrm{SA}}}{p_{\mathrm{sat}}^{\mathrm{CNT}}}-\frac{g}{p_{\mathrm{sat}}^{\mathrm{Er}}}\left){\beta }_2\right]}c-2=0.$$

We can rewrite the above equation as

Eq. (24)

$$c^2-mc-n=0,$$

where the parameters $m$ and $n$ are given by

Eq. (25)

$$m=-3\left[\frac{2\gamma {\beta }_2+\frac{1}{2}g{T}_2^2({\alpha }_2+\frac{{\delta }_{\mathrm{SA}}}{p_{\mathrm{sat}}^{\mathrm{CNT}}}-\frac{g}{p_{\mathrm{sat}}^{\mathrm{Er}}})}{2\gamma g{T}_2^2-({\alpha }_2+\frac{{\delta }_{\mathrm{SA}}}{p_{\mathrm{sat}}^{\mathrm{CNT}}}-\frac{g}{p_{\mathrm{sat}}^{\mathrm{Er}}}){\beta }_2}\right],n=-2.$$

It is easy to show that the chirp parameter is

Eq. (26)

$$c=\frac{m}{2}\pm \sqrt{\frac{m^2-4n}{4}},$$

Eq. (27)

$$c=\frac{3}{2}\frac{[2\gamma {\beta }_2+\frac{1}{2}g{T}_2^2({\alpha }_2+\frac{{\delta }_{\mathrm{SA}}}{p_{\mathrm{sat}}^{\mathrm{CNT}}}-\frac{g}{p_{\mathrm{sat}}^{\mathrm{Er}}}\left)\right]}{[2\gamma g{T}_2^2-({\alpha }_2+\frac{{\delta }_{\mathrm{SA}}}{p_{\mathrm{sat}}^{\mathrm{CNT}}}-\frac{g}{p_{\mathrm{sat}}^{\mathrm{Er}}}\left){\beta }_2\right]}\pm \sqrt{\frac{9}{4}{\left[\frac{2\gamma {\beta }_2+\frac{1}{2}g{T}_2^2({\alpha }_2+\frac{{\delta }_{\mathrm{SA}}}{p_{\mathrm{sat}}^{\mathrm{CNT}}}-\frac{g}{p_{\mathrm{sat}}^{\mathrm{Er}}})}{2\gamma g{T}_2^2-({\alpha }_2+\frac{{\delta }_{\mathrm{SA}}}{p_{\mathrm{sat}}^{\mathrm{CNT}}}-\frac{g}{p_{\mathrm{sat}}^{\mathrm{Er}}}){\beta }_2}\right]}^2+2.}$$

We can use Eq. (27) to investigate the chirp parameter; we discuss the three cases separately.

• Case 1: The solution of Eq. (27) is called the symmetric state and exists only for $m^2-4n>0$. The chirp parameter is real and indicates that the pulse-formed soliton will propagate without changes in its shape; the pulse is stable.

• Case 2: When $m^2-4n=0$, the chirp is given by $c=m/2$, indicating that the frequency is always shifted (chirping always occurs) unless the pulse forms a soliton.

• Case 3: The solution of Eq. (27) represents an antisymmetric state and exists for all $m^2-4n<0$. In this case, the chirp parameter is purely imaginary, and it is impossible to obtain the soliton from a pulse because the antisymmetric states are unstable.

From the condition that the chirp parameter is real, we obtain

Eq. (28)

$$\frac{9}{4}{\left[\frac{2\gamma {\beta }_2+\frac{1}{2}g{T}_2^2({\alpha }_2+\frac{{\delta }_{\mathrm{SA}}}{p_{\mathrm{sat}}^{\mathrm{CNT}}}-\frac{g}{p_{\mathrm{sat}}^{\mathrm{Er}}})}{2\gamma g{T}_2^2-({\alpha }_2+\frac{{\delta }_{\mathrm{SA}}}{p_{\mathrm{sat}}^{\mathrm{CNT}}}-\frac{g}{p_{\mathrm{sat}}^{\mathrm{Er}}}){\beta }_2}\right]}^2+2\ge 0.$$

For a fiber laser employing CNTs, $g{T}_2^2$, ${\beta }_2$, ${\alpha }_2$, $g/p_{\mathrm{sat}}^{\mathrm{Er}}$, and $\gamma$ play important roles in the evolution of passive mode-locked pulses and should be included.

According to Eq. (28), we obtain

Eq. (29)

$${\delta }_{\mathrm{SA}}\ge \frac{2({\beta }_2+i\frac{4}{3}\sqrt{2}g{T}_2^2)}{(\frac{1}{2}g{T}_2^2+i\frac{2}{3}\sqrt{2}{\beta }_2)}\gamma p_{\mathrm{sat}}^{\mathrm{CNT}}-({\alpha }_2-\frac{g}{p_{\mathrm{sat}}^{\mathrm{Er}}})p_{\mathrm{sat}}^{\mathrm{CNT}}.$$

When a complex number (${\beta }_2+i(4/3)\sqrt{2}g{T}_2^2$) is multiplied by its conjugate, the result is

Eq. (30)

$${\delta }_{\mathrm{SA}}\ge \frac{\frac{41}{9}g{T}_2^2{\beta }_2}{(\frac{1}{4}{g}^2{T}_2^4+\frac{8}{9}{\beta }_2^2)}\gamma p_{\mathrm{sat}}^{\mathrm{CNT}}-i\frac{\frac{4}{3}\sqrt{2}({\beta }_2^2-g^2{T}_2^4)}{(\frac{1}{4}{g}^2{T}_2^4+\frac{8}{9}{\beta }_2^2)}\gamma p_{\mathrm{sat}}^{\mathrm{CNT}}-({\alpha }_2-\frac{g}{p_{\mathrm{sat}}^{\mathrm{Er}}})p_{\mathrm{sat}}^{\mathrm{CNT}}.$$

The SA parameter ${\delta }_{\mathrm{SA}}$ must be real; therefore, $4/3\sqrt{2}({\beta }_2^2-g^2{T}_2^4)=0$, and

Eq. (31)

$${\delta }_{\mathrm{SA}}\ge (4\gamma +\frac{g}{p_{\mathrm{sat}}^{\mathrm{Er}}}-{\alpha }_2)p_{\mathrm{sat}}^{\mathrm{CNT}}.$$

This equation shows that the condition that the saturable absorption parameter ${\delta }_{\mathrm{SA}}$ must yield a real chirp parameter $c$ can be related to the stable soliton.

According to Eq. (21) and the condition $|\xi {|}^2\ge 0$, we obtain

Eq. (32)

$${\alpha }_2+\frac{{\delta }_{\mathrm{SA}}}{p_{\mathrm{sat}}^{\mathrm{CNT}}}-\frac{g}{p_{\mathrm{sat}}^{\mathrm{Er}}}\le \frac{1}{T_{\mathrm{FWHM}}^2}(g{T}_2^2{c}^2-3{\beta }_{2}c-2).$$

This clearly illustrates that the nonlinear behavior is altered from saturable absorption to TPA as the intensity of the optical pulse increases. The transformation from saturable absorption to TPA suggests that another nonlinear process occurs and becomes dominant. This interesting effect can be used for optical pulse compression; the above behavior of the optical pulse depends on the values of the parameters $g{T}_2^2$, ${\beta }_2$, ${\alpha }_2$, ${\delta }_{\mathrm{SA}}/p_{\mathrm{sat}}^{\mathrm{CNT}}$, $g/p_{\mathrm{sat}}^{\mathrm{Er}}$, and $\gamma$.

We can use Eq. (27) to calculate the pulse chirp for ${\delta }_{\mathrm{SA}}$ values between 0.1 and 0.4. Figure 2 shows the changes in the chirp as a function of ${\delta }_{\mathrm{SA}}$; the chirp increases linearly with ${\delta }_{\mathrm{SA}}$.

Fig. 2

Chirp versus saturable absorption parameter ${\delta }_{\mathrm{SA}}$.

3.2.

Unstable Laser Pulse

Analytical simulations of pulses that grow through a 5-m fiber ring without a saturable absorption mode locker show that each input pulse develops an internal substructure consisting of many subpulses with widths on the order of femtoseconds. Figure 3 shows the evolution toward a parabolic shape when a “sech” pulse is amplified over the 5-m length of the fiber laser; the position and width of the subpulses change continuously in an apparently random manner. Changing the anomalous GVD values by varying the length of the SMF, therefore, will cause the total cavity dispersion to vary as well.³⁷ Figure 3 shows that each pulse still develops an internal substructure that depends on the frequencies of the photon laser through the ring resonator, and the pulse power also varies from pulse to pulse and exhibits chaotic behavior as the phase is varied in the cavity resonance.

Fig. 3

Evolution of substructure in a 5 m ring resonator when pulses grow from spontaneous emission with group-velocity dispersion: (a) ${\beta }_2=-0.014{\mathrm{ps}}^2/\mathrm{m}$, (b) ${\beta }_2=-0.012{\mathrm{ps}}^2/\mathrm{m}$. Other parameters are $\gamma =0.012{(\mathrm{Wm})}^{-1}$, ${\alpha }_2=0.3$, $T_2=0.047\mathrm{ps}$, $\alpha =0.17823{\mathrm{m}}^{-1}$, and $g=0.78275{\mathrm{m}}^{-1}$.

3.3.

Stable Pulse (Soliton)

The solution of Eq. (2) provides a shortcut to understanding the behavior of pulse laser propagation through an ${\mathrm{Er}}^{+3}$ ring cavity 6 m in length with a saturable absorption mode locker. In this section, we concentrate on the effects of the absorption parameter on the characteristics of the laser pulse; the absorption parameter is tuned in a wide range from 0.1 to 0.4 (Ref. 38) by varying the thickness of the CNTs. First, by joining Eqs. (20) and (27), we find the soliton width $\sigma$. Second, from Eq. (18), we obtain the wave number of the soliton, $k$. Third, from Eq. (21), we can obtain the soliton power. Fourth, we substitute the soliton width, soliton power, and wave number into Eqs. (3) and (4) and algebraically combine the resulting equations with Eq. (2) to obtain a solitary wave equation describing the dynamics and propagation of optical pulses in a 6-m-long EDFL.

Figure 4 shows the three-dimensional soliton pulse propagation in an EDFL, and Fig. 5 shows the sech pulse profile when the system parameters are $\gamma =0.012{(\mathrm{Wm})}^{-1}$, ${\beta }_2=-0.012{\mathrm{ps}}^2/\mathrm{m}$, ${\alpha }_2=0.3$, $T_2=0.047\mathrm{ps}$, ${\delta }_{\text{SA}}=\text{0.1}$, $\alpha =0.178{\mathrm{m}}^{-1}$, $p_{\mathrm{sat}}=210\mathrm{W}$, and $g=0.782{\mathrm{m}}^{-1}$.³⁸ The power of the soliton grows exponentially once the power of the pulse exceeds a threshold value, where, at $z=10\mathrm{cm}$, the power of the soliton is 0.007 W, and a picosecond optical pulse is amplified in the EDFL until the gain saturates at $z=4\mathrm{m}$ with a power of 1.2 W.

Fig. 4

Soliton growth from spontaneous emission with erbium-doped fiber laser parameters of $\gamma =0.012{(\mathrm{Wm})}^{-1}$, ${\beta }_2=-0.012{\mathrm{ps}}^2/\mathrm{m}$, ${\alpha }_2=0.3$, $T_2=0.047\mathrm{ps}$, $\alpha =0.17823{\mathrm{m}}^{-1}$, $g=0.78275{\mathrm{m}}^{-1}$, ${\delta }_{\mathrm{SA}}=0.1$ and $p_{\mathrm{sat}}=210\mathrm{W}$.

Fig. 5

Profiles of parabolic pulse at (a) $z=0.1\mathrm{m}$ and (b) $z=6\mathrm{m}$.

Figures 6(a) and 6(b) show the variation in the soliton phase with time and with the length of the fiber laser, respectively, according to the phase equation $\phi =kz-c\log [\cosh (\sigma T)]$, for a fixed length of $z=4\mathrm{m}$. The phase changes with time as a cosine wave with a range of 1 to $-1$, as shown in Fig. 6(a). The wave number $k$; in the phase equation depends on ${\beta }_2$, $g{T}_2^2$, $c$, and $T_{\mathrm{FWHM}}$, as shown in the following equation:

Eq. (33)

$$k=\frac{1}{2T_{\mathrm{FWHM}}^2}({\beta }_2{c}^2+2g{T}_2^{2}c-{\beta }_2),$$

where

Eq. (34)

$$T_{\mathrm{FWHM}}=\frac{1.76}{\sigma }.$$

Fig. 6

Soliton phase plotted as a function of (a) time and (b) fiber length.

Let us consider the case in which the soliton phase varies linearly along the fiber length, as shown in Fig. 6(b). At $z=0\mathrm{m}$, the soliton phase is $\phi =-0.7$, and the phase reaches a maximum value of $\phi =-0.57$ when $z=6\mathrm{m}$. Clearly, the values of the phase play an important role in building the soliton from spontaneous emission of the ${\mathrm{Er}}^{+3}$ fiber laser.

3.4.

Effect of SA on Soliton Behavior

First, we explain the control of the pulse width in a mode-locked fiber laser in which CNTs act as an SA by adjusting the laser’s intracavity SA parameter. Equation (35) shows that the SA parameter can vary with $T_{\mathrm{FWHM}}$.

Eq. (35)

$${\delta }_{\mathrm{SA}}=\frac{1}{T_{\mathrm{FWHM}}^2}(2{\beta }_{2}c+g{T}_2^2-g{T}_2^2{c}^2)+(g-\alpha ).$$

Figure 7(a) shows the soliton width (FWHM) as a function of ${\delta }_{\mathrm{SA}}$. The soliton width increases with ${\delta }_{\mathrm{SA}}$.

Fig. 7

(a) width of soliton and (b) output power versus ${\delta }_{\mathrm{SA}}$.

Second, we focus on the soliton power and SA parameter. Increasing ${\delta }_{\mathrm{SA}}$ reduces the soliton power, as can be inferred from Eq. (36). Furthermore, it is easy to show that ${\delta }_{\mathrm{SA}}$ depends on $p_{\mathrm{sat}}$ and $p$ and refers to the evolution of the building pulse toward a soliton as its width decreases and peak power increases, as shown in Fig. 7(b).

Eq. (36)

$${\delta }_{\mathrm{SA}}=\frac{p_{\mathrm{sat}}^{\mathrm{CNT}}}{p}\frac{T_2^2}{T_{\mathrm{FWHM}}^2}g(c^2-2)-3\frac{p_{\mathrm{sat}}^{\mathrm{CNT}}}{p}\frac{{\beta }_2}{T_{\mathrm{FWHM}}^2}c+(g-{\alpha }_2{p}_{\mathrm{sat}}^{\mathrm{CNT}}).$$

Figures 8(a) and 8(b) show the effect of changing the ${\delta }_{\mathrm{SA}}$ on the shape of the soliton; the soliton is repeatedly propagated around the laser cavity until a steady state is reached. The soliton power grows exponentially along the fiber length. SAs can also be used to filter sidebands associated with solitons because sidebands can be selectively transferred to the lossy core because of their low power level.

Fig. 8

Soliton growth from spontaneous emission: (a) ${\delta }_{\mathrm{SA}}=0.2$, (b) ${\delta }_{\mathrm{SA}}=0.3$.

The soliton power in the fiber resonators is evident from Eq. (37). Note that the power depends on the GVD ${\beta }_2$, gain dispersion $g{T}_2^2$, chirp $c$, SPM parameter $\gamma$, and soliton width $T_{\mathrm{FWHM}}$.

Eq. (37)

$${|\xi |}^2=\frac{1}{2\gamma T_{\mathrm{FWHM}}^2}(-{\beta }_2{c}^2+3g{T}_2^{2}c+{\beta }_2),$$

Eq. (38)

$$T_{\mathrm{FWHM}}=1.76\sqrt{\frac{-g{T}_2^2(1-c^2)-2{\beta }_{2}c}{(g-\alpha -{\delta }_{\mathrm{SA}})}}.$$

It is evident from Eqs. (37) and (38) that the parameters $\xi$ and $T_{\mathrm{FWHM}}$ can have real, positive values or imaginary values, depending on the value under the radical of Eq. (38). $\xi$ and $T_{\mathrm{FWHM}}$ play an important role in stable optical pulses; if $\xi$ and $T_{\mathrm{FWHM}}$ are real, the optical pulse forms a soliton. Figures 9(a) and 9(b) show that the output pulse forms a soliton and becomes broader with increasing ${\delta }_{\mathrm{SA}}$ and decreasing power.

Fig. 9

Profile of optical pulse that forms a soliton for (a) ${\delta }_{\mathrm{SA}}=0.2$ and (b) ${\delta }_{\mathrm{SA}}=0.3$.

4.

Conclusions

We highlighted four important properties: the power, width, chirp, and phase of the pulse laser on the basis of a mathematical model for fiber lasers with ring cavities and CNTs as an SA mode locker. For values of $\gamma =0.012{(\mathrm{Wm})}^{-1}$, ${\beta }_2=-0.012{\mathrm{ps}}^2/\mathrm{m}$, ${\alpha }_2=0.3$, $T_2=0.047\mathrm{ps}$, $\alpha =0.17823{\mathrm{m}}^{-1}$, $g=0.78275{\mathrm{m}}^{-1}$, and $p_{\mathrm{sat}}=210\mathrm{W}$, $\xi$ and $T_{\mathrm{FWHM}}$ must be positive and real to guarantee stable pulse propagation (a soliton). In this paper, we argued that ${\delta }_{\mathrm{SA}}=0.1$ is the best value for generating solitons at a power of $\sim 1.2\mathrm{W}$ and width of $\sim 0.5\mathrm{ps}$. The second major finding was that the soliton power decreases as the SA parameter ${\delta }_{\mathrm{SA}}$ increases, but that the soliton width increases with increasing ${\delta }_{\mathrm{SA}}$. Furthermore, different ${\delta }_{\mathrm{SA}}$ values yield different soliton characteristics; this provides a convenient method for tuning the properties of the generated soliton in ring-cavity fiber lasers, where the properties of the SA parameter depend on the size of the CNTs; therefore, the TPA from a thinner diameter of the CNTs could destroy the stability of mode-locked pulse formation. Consequently, a thicker diameter of the CNTs with less nonlinearity was identified as the mode locker to reduce the TPA, where both gain dispersion and two-photon absorption provide such a loss mechanism, and hence, the TPA plays an important role in establishing the soliton. These observations are in agreement with numerical simulations.³⁸ Moreover, our results for the parabolic shape of the pulse and the behavior of the laser pulse propagation through the fiber laser exhibit good agreement with previous studies.^39–43