Passive mode-locked (PML) fiber lasers have attracted much attention because of their advantages of high stability, higher beam quality, and compact structure^[1-5]. In PML lasers, saturable absorbers (SAs) are often used as passive mode-locking elements, crucial in forming optical solitons^[6-14]. Graphene has been chosen as an ideal saturable absorber because of its unique characteristics, such as ultrafast carrier dynamics, wide spectral bandwidth, and high optical damage threshold^[15]. In 2010, Sun et al. demonstrated the graphene film as a saturable absorber in the fiber laser to generate mode-locked pulses^[16]. Polarization plays a crucial role in formatting and stabilizing mode-locking pulses. Moreover, polarization serves to suppress noises in PML fiber lasers. Orthogonally polarized light significantly influences coherence, energy distribution, and pulse characteristics in PML fiber lasers. Graphene has advantages over traditional SAs like semiconductor saturable absorption mirrors (SESAMs); for example, the flexibility to cover the optical waveguide, a wide wavelength absorption spectra, and ultrafast carrier relaxation time for easily generating and controlling orthogonal polarization dissipative solitons. Therefore, it is necessary to study the evolutionary behavior and the impact of orthogonally polarized states on PML fiber lasers with graphene saturable absorbers.

Various soliton states in the graphene PML fiber laser have been studied extensively^[17-19]. Conventional solitons (CS) were discovered in fiber lasers with net negative cavity dispersion to obtain a natural balance between cavity dispersion and fiber nonlinearity^[20]. Other solitons, such as dissipative solitons (DSs), have been formed in graphene PML fiber lasers operating in the regime of net-normal cavity dispersion^[21]. DSs are coherent solutions of nonlinear wave equations that require an intricate and mutual interaction of nonlinearity, dispersion, loss, and gain^[22-24]. Their stable high-energy pulses make them ideal for high-energy laser machining^[25]. In 2021, by numerical simulations, Lee et al. demonstrated the optimal range of SA recovery time on the formation of dissipative solitons in Yb-doped fiber lasers^[26]. In 2022, Xu et al. achieved group-velocity-locked vector solitons switching between CS and DS by using a birefringent-related intracavity Lyot filter with an adjustable extinction ratio in the mode-locking fiber laser^[27]. However, in birefringent cavities, the effect of orthogonal polarization caused by polarization-dependent components on vector DS is hard to observe in intracavity and often overlooked. The vector DSs have significant potential for application in optics and quantum communication systems^[28].

In this paper, we build an orthogonally polarized numerical modeling of the graphene PML fiber laser with net-normal dispersion birefringent cavity for generating vector DSs. A polarization-dependent graphene saturable absorber is utilized to form a birefringent cavity. The effect of the recovery time of such polarization-dependent saturable absorbers on output DSs and optical spectra has been discussed and analyzed. This is of great significance for the development of vector DSs systems.

The schematic diagram of the graphene PML fiber laser is depicted in Fig. 1. The dispersion of the erbium-doped fiber (EDF) and single-mode fiber (SMF) employed in our laser are -46.25 ps/nm/km and 28 ps/nm/km, respectively, both possessing the same nonlinear coefficient of 0.004 W⁻¹·m⁻¹. An isolator (ISO) is implemented to enforce unidirectional operation within the fiber ring cavity. A 30:70 fiber optical coupler with a 30% port extracts the laser pulse output. Graphene is half-covered on the microfiber for polarization-dependent saturable absorber, called graphene half-covered microfiber SAs (GMSA). A bandpass filter with a bandwidth of 8 nm is utilized in our model to play a crucial role in the formation of DSs^[29]. A squeeze-typed polarization controller (PC) is employed to adjust the polarization state of pulses entering the GMSA. Due to the polarization-dependent property of the GMSA^[30], we first set the initial absorption rate of the saturable absorber (SA) to 0.56 for the X-polarization and 0.43 for the Y-polarization.

The total transmission T(I) of a SA consists of an unsaturated transmission (T₀) and modulation depth ($ \Delta T $), which is shown in the following:

$ T\left( I \right) = {T_0} + \left( {\Delta T - q\left( t \right)} \right) \quad,$ (1)

where q(t) is the saturation loss of an SA, which is expressed as:

$ \frac{{{\mathrm{d}}q\left( t \right)}}{{{\mathrm{d}}t}} = \frac{{q\left( t \right) - {q_0}}}{\tau } - \frac{{q\left( t \right)|A\left( t \right){|^2}}}{{{E_{{\mathrm{sat}}}}}}\quad, $ (2)

where q₀ is the initial absorption，$ \tau $ is the recovery time, and E_sat≈10 pJ is the saturable absorption energy of the SA. As the intensity of the incident pulse beam increases, the transmission also increases due to saturation absorption.

Here, orthogonally polarized states are considered. Thus, the coupled nonlinear Schrödinger equation (cNLSE) describes the evolution of orthogonally polarized pulses along the EDF or SMF^[31], neglecting higher-order dispersion. The expression of the coupled NLSE is as follows:

$\begin{split} & \frac{{\partial u}}{{\partial \textit{z}}} = - \frac{\alpha }{2}u - \delta \frac{{\partial u}}{{\partial T}} - i\frac{{{\beta _2}}}{2}\frac{{{\partial ^2}u}}{{\partial {T^2}}} + i\gamma \left( {|u{|^2} + \frac{2}{3}|v{|^2}} \right)u + \frac{g}{2}u + \frac{g}{{2{{\Omega }}_g^2}}\frac{{{\partial ^2}u}}{{\partial {T^2}}} \\ & \frac{{\partial v}}{{\partial\textit{z}}} = - \frac{\alpha }{2}v - \delta \frac{{\partial v}}{{\partial T}} - i\frac{{{\beta _2}}}{2}\frac{{{\partial ^2}v}}{{\partial {T^2}}} + i\gamma \left( {|v{|^2} + \frac{2}{3}|u{|^2}} \right)v + \frac{g}{2}v + \frac{g}{{2{{\Omega }}_g^2}}\frac{{{\partial ^2}v}}{{\partial {T^2}}} \quad, \end{split} $ (3)

where u and v represent the envelopes of the optical pulses along the two orthogonal polarization modes along the fiber, α is the loss coefficient of the fiber, δ is the group velocity difference between the two polarization modes, β₂ represents the fiber dispersion, γ is the third-order nonlinear coefficient associated with the refractive index of the medium, and Ω_g is the gain bandwidth of the laser, which is chosen to correspond to 50 nm. The variables T and z represent time and transmission distance, respectively. g is the net gain of the EDF, which is only nonzero in the gain fiber section, and it is modeled as g=g₀/(1+E_p/E_s). g₀=3 is the small-signal gain coefficient, E_s≈1 nJ is the gain saturation energy of the EDF^[32-33].

The split-step Fourier method is used to solve the equations iteratively and simulate the evolution from a chirped Gaussian pulse. The mathematical model of the squeeze-typed PC is represented by the Jones matrix^[34]:

$ J(\theta ,\gamma ) = \left( {\begin{array}{*{20}{c}} {\cos \left( {\dfrac{\gamma }{2}} \right) + i\cos (2\theta )\sin \left( {\dfrac{\gamma }{2}} \right)}&{i\sin (2\theta )\sin \left( {\dfrac{\gamma }{2}} \right)} \\ {i\sin (2\theta )\sin \left( {\dfrac{\gamma }{2}} \right)}&{\cos \left( {\dfrac{\gamma }{2}} \right) - i\cos (2\theta )\sin \left( {\dfrac{\gamma }{2}} \right)} \end{array}} \right)\quad. $ (4)

The force applied to the fiber is exerted on the fiber's cross-section, specifically in the direction that forms an angle θ with the horizontal plane, resulting in the compression of the fiber. $ \gamma =\delta L $, L denotes the length of the compressed fiber, which is set to 3 mm in our numerical simulations, $ \delta = KF/d $, F represents the pressure exerted on the fiber per unit length, and d represents the diameter of the compressed fiber, which is a constant value K=9.5×10⁻⁵ rad/m.

Compared to CS, DS’s formation mechanism is more complex^[35], especially for vector DS. Thus, we plan to conduct numerical simulations of all-normal-dispersion cavities from 0.01 to 0.1 ps², as shown in Table 1.

Table 1 illustrates different cavity lengths and their corresponding total net cavity dispersions. Fig. 2 (color online) depicts the corresponding output orthogonal DSs and optical spectra for various net-normal dispersion values.

Fig. 2 shows that larger cavity dispersion leads to large pulses energy, either X-polarized or Y-polarized. However, larger cavity dispersion could cause a wide spectrum shape for the X-polarized and Y-polarized. Table 2. illustrates the output pulse parameters extracted from Fig. 2. As the net-normal dispersion increases from 0.01 ps² to 0.1 ps², the 3 dB pulse widths of X-polarized and Y-polarized pulses increase ~2.42 ps and ~1.69 ps, respectively. The 3dB spectrum widths of X-polarized and Y-polarized pulses also increase by ~2.79 nm and ~2.52 nm, respectively. This means the effect of the dispersion on pulse and spectrum widths is minimal. Thus, we chose a net cavity dispersion value of 0.01 ps² in the subsequent simulation.

Fig. 3 (color online) presents the temporal evolution of orthogonally polarized dissipative solitons within the cavity. It is observed that the pulse gradually narrows and amplifies, eventually resulting in the output of stable orthogonally polarized dissipative solitons. The fiber laser model demonstrates excellent operational stability.

Interband transition relaxation time and intraband carrier scattering and recombination relaxation time are more important parameters for our proposed GMSA, where the intraband transition relaxation time is shorter than that of the interband^[36-37]. Graphene used to fabricate SAs has many fabrication methods, such as liquid phase exfoliation, chemical vapor deposition (CVD), reduced graphene oxide (rGO), micro-mechanical cleavage, etc. Different fabrication methods could cause the differing recovery times of graphene, which can vary from hundreds of femtoseconds to a few picoseconds^[38]. The short recovery time is related to the intraband transition, while the long is caused by the interband transition. Here, we choose recovery time ranging from 70 to 1700 fs, which is extracted from Ref [38] for covering common fabrication methods to make the GMSA.

The two polarizations’ pulse width and their piecewise fitted curves are shown in Fig. 4(a) (color online). At a recovery time of 70 fs, the pulse width was found to be ~7.76 ps for the X-polarized and ~8.71 ps for the Y-polarized. Pulse widths of two polarizations gradually decrease as the recovery time increases from 70 to 120 fs. The minimum pulse width is achieved at 120 fs, where the pulse width of the X-polarized is ~7.47 ps and the Y-polarized is ~8.06 ps. When the recovery time is changed from 120 to 1700 fs, the X-polarized and Y-polarized pulse widths are increased to ~11.9 ps and ~12.0 ps, respectively. It is observed that the pulse width of the Y-polarized is consistently larger than that of the X-polarized. This is because the polarization-dependent saturable absorption of our designed GMSA leads to insufficient absorption of the Y-polarized pulse sideband energy.

Additionally, the modulation depths of our GMSA for both polarizations are calculated by changing its recovery time, as shown in Fig. 4(b) (color online). With the recovery time increase, both modulation depths increase rapidly at first and then stabilize. The modulation depth of the X-polarized and Y-polarized varies from 0.17 to 0.45 and from 0.14 to 0.4, respectively. The pulse energy and chirp versus the recovery time also have been extracted from the output pulses, as shown in Fig. 4(c)−4(d) (color online). The pulse energy range of the X-polarized varies from 3.75 to 4.63 nJ, while for the Y-polarized, it varies from 3.86 to 4.65 nJ. The variation trend of X/Y polarized pulse energies is like that of the modulation depth of two polarizations. This means the pulse energy is related to the modulation depth of the GMSA. Additionally, this is also attributed to the slow response characteristics of the mode-locked fiber laser, where pulse energy accumulation "blocks" the subsequent pulse absorption. As a result, there is minimal absorption at the trailing edge of the pulse, which leads to an increase in pulse energy. With the increase of the recovery time, chirps for both polarizations decrease rapidly and then approach stability. Under the recovery time of 70 fs, both polarized pulses exhibit large chirp: 72 for the X-polarization and 66 for the Y-polarization. This could be attributed to the effect of the GMSA’s different recovery times on the pulse shape and energy, which may alter the interaction between the dispersion and nonlinearity and subsequently affect the manifestation of the chirp.

To explore the function of the recovery time of our GMSA on the mode-locking pulse generation, the formation mechanisms of the mode-locking pulses are discussed in Fig. 5 (color online). Here, the recovery time of such GMSA has been divided into three stages: 70 fs, 120 fs, and 1700 fs.

(1) Recovery time equals 70 fs

Fig. 5(a) shows the compression mechanism of polarized lights by the saturable absorber when the recovery time is 70 fs. The red and blue curves represent the saturation absorption of the X-polarized and the Y-polarized, where the initial saturation absorption of the X-polarized and the Y-polarized is fixed at 0.49 and 0.44, respectively. The modulation depth of the X-polarized and the Y-polarized are ~0.170 and ~0.147, respectively. The output pulse width of the X-polarized ~7.76 ps is slightly smaller than that of the Y-polarized ~8.71 ps. This is due to a larger modulation depth, thus resulting in an increased absorption of the low-energy sideband of the pulse. When the recovery time of the graphene is ~70 fs, the light causes the intraband absorption of the graphene as a fast absorber. When the low-energy sideband of the polarized pulse enters the graphene, it can be absorbed. With the increase in sideband energy, graphene can reach deep saturation and then release within an ultra-short time of 70 fs. Subsequently, the graphene always reaches deep saturation and 70 fs release until another low-energy sideband of the polarized pulse arrives. Thus, the absorption of the central pulse energy is lower than the sideband. Furthermore, larger modulation depth leads to larger energy absorption. Thus, the energy loss of the X-polarized (3.75 nJ) is lower than that of the Y-polarized (3.86 nJ).

(2) Recovery time equals 120 fs

From Fig. 4(a), at a recovery time of 120 fs, both polarization directions reach their minimum pulse widths. This is because too fast or too slow graphene recovery time cannot effectively absorb the output pulse sideband of the mode-locking laser^[32].

In Fig. 5(b), the red curve represents the GMSA with a modulation depth of 0.245 for the X-polarized, while the blue curve represents the GMSA with a modulation depth of 0.216 for the Y-polarized. The polarization-dependent modulation depth causes different pulse widths, where a larger modulation depth leads to a smaller pulse width; a larger modulation depth also leads to larger energy loss. The X-polarized and Y-polarized pulses’ energy is about 3.92 nJ and 4 nJ, respectively.

(3) Recovery time equals 1700 fs

The long recovery time of 1700 fs is related to the GMSA as a slow absorber. Moreover, As shown in Fig. 5(c), a long recovery time 1700 fs leads to a significant increase in the modulation depths for both polarization directions. The modulation depth for the X-polarized is calculated to be ~0.453, while the Y-polarized is ~0.406. The pulse widths of the X-polarized and Y-polarized pulses are 11.9 ps and 12.0 ps, respectively. These pulse widths are larger than that of fast recovery time. When the sideband of the pulse comes into the GMSA, the long recovery time and deep modulation depth cause the pulse sideband energy to be absorbed, and then the GMSA absorption reaches saturation. Subsequently, the GMSA should spend 1700 fs to recover. Thus, a large part of the pulse center enters the GMSA and cannot be absorbed. The overall energy of the pulse increases to broaden the pulse width. The output pulse energy for the X-polarized and Y-polarized pulses is about 4.63 and 4.65 nJ, respectively.

At a recovery time of 1700 fs, we observe the phenomenon of pulse splitting at the pulse peak. This can be attributed to the balance between nonlinearity, gain, loss, and dispersion. The fiber laser’s nonlinearity is related to the pulse power and dispersion. When the output pulse energy is high, optical nonlinear effects such as self-phase modulation and cross-phase modulation could affect the pulse shape and splitting.

An orthogonally polarized numerical modeling of the PML graphene fiber laser with a net-normal dispersion birefringent cavity has been established. The output of orthogonally polarized pulses with different normal net cavity dispersion has been discussed. The influence of the recovery time of the GMSA on the mode-locking performance is investigated by utilizing our modeling. It is found that the recovery time significantly affects the pulse characteristics of two orthogonally polarized pulses. GMSA with a recovery time of 120 fs could help to generate narrow dissipative soliton pulses with large chirps for two orthogonally polarized directions, about 7.47 ps and 8.06 ps.