## Abstract

It is demonstrated by numerical simulations that the CS_{2}-filled dual-core fiber coupler with appropriate parameters can provide single-mode operation, normal dispersions, low loss and high nonlinearities in 1550-nm and 2000-nm wavelength windows, which can contribute to a saturable absorber (SA) with short fiber length and low power needed for nonlinearity-induced saturation. The effects of stimulated Raman scattering (SRS) play a key role in the process of nonlinearity-induced saturation. The numerical results indicate that the SAs can be employed in the mode-locking fiber lasers with self-similar (SS) or dissipative soliton (DS) operations.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

For standard soliton mode-locked fiber laser with anomalous group velocity dispersion (GVD) components in the cavity, the output pulse energy is limited below about 10s pJ level due to the soliton energy quantization effect [1]. To achieve high pulse energies, the normal-dispersion mode-locked fiber lasers with net GVD near zero and large normal GVD, where the cavities consist of segments of normal and anomalous GVD, was experimentally demonstrated with three operating regimes [2], i.e., stretched dissipative solitons (SDS) fiber lasers [3,4], passive [5] and active [6,7] SS fiber lasers. Further increase in pulse energy can be generated in fiber lasers with all normal cavity dispersion, which can operate in the DS regime [8–10].

For passively mode-locked fiber lasers, the saturable absorber (SA) is a key component for initiating and stabilizing the pulses in the cavities. Generally, the SA can be divided into two categories, i.e., physical or artificial SA. The former includes semiconductor saturable-absorber mirrors (SESAMs), graphene and carbon nanotubes. The latter includes nonlinear optical loop mirror (NOLM) [11] and nonlinear polarization rotation (NPR) [12]. Recently, it was demonstrated that the dual-core fiber can also provide the characteristic of intensity discrimination due to the nonlinear mode coupling [13,14]. Furthermore, Mafi *et al*. proposed an all-fiber SA based on two-concentric-core chalcogenide glass optical fiber [15]. In this manuscript, we numerically investigate the SAs based on the CS_{2}-filled dual-core fiber couplers. The properties of the SA are presented in Section 2, where the influences of the effects of SRS and self-steepening on the transmission functions are also discussed. In Section 3, the applications of the proposed SA on the mode-locked fiber lasers are investigated by numerical simulations.

## 2. SA based on CS_{2}-filled dual-core fiber

#### 2.1 Properties of the CS_{2}-filled fiber

A conventional dual-core fiber, which contains two parallel identical single-mode cores, can be employed as directional coupler because optical power can periodically transfer between two fiber cores along the fiber [18]. When the input power is high enough that the nonlinear phase can’t be ignored, the fraction of the output power of each core depends not only the coupling coefficient but also the input power. In this case, the dual-core fibers with the kerr nonlinearity have attracted considerable attention because of their potential application to saturable absorber in passively mode-locked fiber lasers [19, 20]. Since weak Kerr nonlinearity of silica fiber leads to high switching power, the dual-core fibers made of high nonlinear material is desirable for lowering switching power [15]. In this paper, the design of the SA is based on a high nonlinearity dual-core fiber as shown in Fig. 1. The liquid CS_{2} is chosen as the filling material in two hollow cores of silica fiber because of its high nonlinearity and low absorption in the visible and near infrared region [21].

Since each single-core fiber supports a single-mode if the normalized frequency *V* is below 2.4048, the core diameters can be chosen to be 1.8 μm and 2.2 μm at the operating wavelengths of 1550 nm and 2000 nm as shown in Fig. 2(a), respectively. For the FM LP_{01} in single-core CS_{2}-filled fiber, the total loss, which takes into account the confinement loss and absorption loss, can be described in Fig. 2(b). It can be seen that the losses of the FM in fibers with *D* = 1.8 μm and 2.2 μm are much lower than 1 dB/m at most wavelengths ranging from 1500 nm to 2500 nm except for a loss peak near 2250 nm. Consequently, the loss of light can be reduced considerably for a CS_{2}-filled two-core fiber with a few centimeter lengths. As shown in Fig. 2(c), the fibers with *D* = 1.8, 2.0, 2.2, and 2.4 μm exhibit normal dispersion when the wavelength is < 2500 nm. Moreover, the nonlinear parameter *γ* at wavelengths < 2500 nm are higher than 0.3 (W·m)^{−1} in all cases as shown in Fig. 2(d), which is helpful to lower the power of the SA required for obtaining saturation.

When the optical power is launched into one of the two cores, the optical power can be transferred into the other core completely due to the linear coupling at the coupling length *L*_{C}, which can be given by

*κ*is the coupling coefficient [22]. As shown in Fig. 3, for given values of core diameters and wavelengths, the coupling lengths increase with an increase of core-to-core spacing

*d*due to a decrease of coupling coefficients. Consequently, if the SAs with 10s millimeters lengths are desirable, the values of

*d*should be limited to below 6 μm and 7 μm in the cases of

*D*= 1.8 μm and 2.2 μm, respectively.

#### 2.2 Properties of the SA based on the CS_{2}-filled two-core fiber

A CS_{2}-filled dual-core fiber with length *L*_{C} can be employed as the SA. As shown in Fig. 4, the core 1 as the input and output ports is located at the center of fiber for providing a convenient way of splicing with single-mode fibers.

In most cases of practical interest, mode-locked fiber lasers generate pulses with temporal width in the range from 10s picosecond to 100s femtosecond. Consequently, it is necessary to investigate the properties of the SA, including transmission functions, temporal shapes and spectra of output pulses. In this case, the coupling behavior can be described by [23]

*A*(

_{1}*z*,

*T*) and

*A*(

_{2}*z*,

*T*) represent the slowly varying pulse envelopes of the core 1 and core 2 in time domain, respectively.

*T*is the retarded time for a comoving frame at the envelope group velocity 1/

*β*

_{1},

*α*is the linear loss. The term

*κ*

_{1}=

*dκ / dω*is evaluated at

*ω*=

*ω*. The

_{0}*β*are the dispersion coefficients associated with the Taylor series expansion of the propagation constant

_{k}*β*(

*ω)*around the center frequency

*ω*. In the process of solving nonlinear coupled-mode equations, the dispersion operator in the frequency domain is applied through multiplication of the complex spectral envelope Ã(z,

_{0}*ω*) by the operator

*β*(

*ω*) - (

*ω*-

*ω*

_{0})β_{1}-

*β*

_{0}. The time derivative term on the right-hand side models the dispersion of the nonlinearity, which is associated with the effects of self-steepening and optical shock formation, characterized by a time scale

*τ*

_{shock}= 1 /

*ω*. The nonlinear response function

_{0}*R*(

*T*) = (1-

*f*

_{R})

*δ(*t) +

*f*

_{R}

*h*

_{R}(t) includes both instantaneous and delayed Raman contributions. The fractional contribution of the delayed Raman response to nonlinear polarization

*f*

_{R}is taken to be 0.89 for CS

_{2}. The function

*h*

_{R}(

*T*) is the Raman response in CS

_{2}[21]. In the following simulations, the initial pulse of the core 1 is set to be a Gaussian pulse ${A}_{1}(T)=\sqrt{{P}_{0}}\mathrm{exp}(-{T}^{2}/{T}_{0}^{2})$, where

_{${T}_{0}={T}_{FWHM}/\sqrt{2\mathrm{ln}2}$}, and

*T*is the full width at half-maximum (FWHM) pulse duration. The peak power

_{FWHM}*P*

_{0}can be obtained by

*P*

_{0}= 0.94

*E*

_{0}/

*T*

_{FWHM}, where

*E*

_{0}is the pulse energy.

Since the time-dependent terms in Eqs. (2) and (3) can be neglected for continuous-wave (CW) input, the behavior of the mode in each single-core waveguide can be described by the nonlinear coupled-mode equations, which are [24]

The results of equations are expressed as*cn*is a Jacobi elliptic function and the critical power

*P*

_{c}is defined as

According to the Eqs. (4) – (8), the output powers from the core 1 and core 2 as functions of wavelengths can be calculated for different input power, where the lengths of the SAs can be taken to be the coupling lengths *L*_{c} at 1550 nm and 2000 nm as shown in Figs. 5 and 6, respectively. In the former case, when the input powers are below 100 W, the output powers exhibit periodic oscillations induced by the linear coupling near the wavelength of 1550 nm, and become more sensitive to the wavelength at longer wavelengths because of an increase of the coupling coefficient. Furthermore, when the phase matching condition required for energy exchange between two cores can’t be satisfied due to the nonlinearity, most of the input power remains in the core 1 near 1550 nm as shown in Figs. 5(c) – 5(e), where the input powers are above 100 W. For the case of 2000 nm, the similar results can be seen in Fig. 6. Consequently, for the CW–input case, the SA can provide a high power transmission at an input-power level of 100s watts in the vicinity of the center wavelength.

Furthermore, when pulses with different temporal width input a SA, the transmission *T*_{r} can be defined as the ratio of output energy *E*_{out} to input energy *E*_{in}, which is given by

*T*

_{r}in the CW–input case is defined as

*P*

_{1}(

*L*

_{c})

**/**

*P*

_{1}(

*0*). When the input peak power is high enough, the nonlinear effects in the optical fibers detune the effective propagation constants of the modes, and thus break the phase-matching condition required for the energy exchange between two cores. As a result, a high transmission

*T*

_{r}and low efficiency of the energy exchange can be obtained for the SA. This behavior can be called nonlinearity-induced saturation in the dual-core fiber coupler. Figure 7 shows the transmission functions and output pulses of the SA in temporal and spectral domains for different

*T*

_{FWHM}of initial pulses. It can be seen that, when

*T*

_{FWHM}are relatively large (> 5 ps), the transmission curves become nearly identical to that of CW input (black curve). However, unlike the case of CW input, although the peak powers of input pulses are high enough, a part of the pulse energy from the core 1 is still transferred to the core 2 because the red components of the spectra have larger coupling coefficient with shorter coupling lengths than the center-frequency and blue components as shown in Fig. 3. Consequently, the influence of the nonlinear effects on suppressing the energy transfer efficiency for red components is weaker than those for the center-frequency and blue components. This effect can be clearly seen for the pulse with 1-ps

*T*

_{FWHM}because of its relatively broad spectral width (blue curves in Figs. 7(c) and (d)). For the same reason, the transmission for the 1-ps pulse is lower than those for input pulses with long widths as shown in Fig. 7(a). Moreover, the decrease in the peak power of the propagating pulse due to dispersion-induced pulse broadening can also weaken the influence of the nonlinearity, and enhance the energy transfer from the launched core to adjacent core due to the effect of linear coupling. As shown in Fig. 7(b), for a given peak power of initial pulses, the peak powers of the output pulses are reduced with a decrease of the input-pulse width because of the combined effects of dispersion-induced broadening and energy transfer.

The similar results for the SA with *D* = 1.8 μm and *d* = 5 μm can be seen in Figs. 8. Since the coupling coefficient decreases with an increase of the core-to-core spacing, the input peak powers required to reach saturation for the SA with *d* = 5 μm are much lower than those for the SA with *d* = 3 μm. Moreover, because of an increase of fiber length, the output pulses exhibit a further spectral and temporal broadening, especially for relatively short input pulses. In the 2000-nm wavelength window, the CS_{2}-filled dual-core fiber coupler as a SA show the similar results when D = 2.2 μm, *d* = 4 μm, and *D* = 2.2 μm, *d* = 6 μm, as shown in Figs. 9 and 10. Like the case of 1550-nm operating wavelength, the input peak power required for the saturation can be effectively reduced by increasing the core-to-core spacing from 3 μm to 6 μm. At the same time, the propagating pulses undergo more temporal and spectral broadening because of an increase of the coupling length, especially for relatively short pulses.

Since the liquid CS_{2} provides both high nonlinearity and a large fractional contribution of the Raman response to the total nonlinear polarization, it is necessary to investigate the influences of SRS and self-steepening on the transmission for the SA.

As shown in Fig. 11(a) and Fig. 12(a), when the effect of SRS is neglected, the curves of transmission *T _{r}* clearly deviate from the curves without neglecting SRS, and are nearly identical for different temporal widths of input pulses. The results imply that the effect of SRS has significant influence on the nonlinearity-induced saturation, especially for the input pulses with

*T*

_{FWHM}> 1 ps. At the same time, the output pulses for input pulses with

*T*

_{FWHM}< 1 ps exhibit clearly asymmetric shapes in the temporal domain due to the combined effects of SRS and normal dispersion as shown in Fig. 11(b) and Fig. 12(b).

As shown in Figs. 13 and 14, the influences of the self-steepening effect on the transmission curves and temporal shapes of the output pulses are relatively small compared with the results in Figs. 11 and 12. However, all the terms in the Eqs. (2) and (3) are needed in the following simulations for obtaining accurate results.

For the SA with a given core size *D*, the transmission curve of pulse energy is dependent on the parameters *d* and *T*_{FWHM} of the input pulse. The maximum transmission *T*_{r}^{max} occurs when the peak power is increased to *P _{0}^{I}*. Figure 15 shows

*T*

_{r}

^{max}and

*P*as functions of

_{0}^{I}*d*and

*T*

_{FWHM}in two cases of

*D*= 1.8 and 2.2 μm. It can be seen that the input pulses with

*T*

_{FWHM}< 1 ps exhibit relatively low

*T*

_{r}

^{max}and high

*P*compared with the cases of

_{0}^{I}*T*

_{FWHM}> 1 ps. The low

*T*

_{r}

^{max}can be explained by the linear-coupling-induced energy transfer mainly from the red components of input pulses. The reason of high

*P*is related to the fact that the critical peak power of the nonlinear fiber coupler can be dramatically reduced by the effect of SRS in CS

_{0}^{I}_{2}when the temporal width of the input pulse is much longer than the characteristic time

*t*

_{R}= 0.2 ps [25].

## 3. Applications of the SA on the mode-locked fiber lasers

The proposed SA can operate in both 1550 nm and 2000 nm wavelength windows. In the former case, Fig. 16 shows two configures of typical all-normal-dispersion mode-locked fiber lasers. The type-A schematic consists of a single mode fiber segment, a E_{r} fiber, a SA, a spectral filter, a fiber coupler, and an isolator in sequence as shown in Fig. 16. For the type-B schematic, the fiber laser has the same elements except for exchanging the positions of the spectral filter and fiber coupler. In the latter case, since most T_{m} fibers show anomalous dispersion in 2000-nm wavelength window, the ultrahigh numerical aperture (UHNA) fibers with large normal dispersion can be employed to increase the dispersion in the mode-locked T_{m} fiber lasers, which exhibit self-similar pulse evolutions with dissipative-soliton-like spectra [26]. Two configurations of T_{m} fiber laser are also shown in Fig. 17, where the proposed SA is introduced in the ring cavity.

The evolution of propagating pulse along the gain fiber can be described with the following nonlinear Schrödinger equation with the gain coefficient *g* [27, 28]:

*g*

_{0}is small-signal gain,

*E*is the pulse energy,

_{pulse}*E*

_{sat}is the gain saturation energy.

*Δω*is the gain bandwidth, which corresponds to 40 nm and 100 nm of wavelength bandwidth for Er fiber and Tm fiber, respectively. Additionally, the shape of the spectral filter is taken to be a Gaussian

_{g}*H*(

*ω*) = exp[-

*ω*/(2

^{2}*Δω*)], where

_{f}^{2}*Δω*represents the filter bandwidth.

_{f}The numerical model is solved with the split-step Fourier method and the simulation starts with white noise because other initial conditions without noise may result in misleading results [29]. The initial noise can be gradually shaped to the steady state pulse under the shaping effect of the gain, loss, nonlinearity, dispersion, filter and saturable absorption for every round trip. As shown in Fig. 18, the stable pulses can be obtained after tens of roundtrips for the Er fiber lasers. Figure 19 shows the steady-state output pulses in temporal and spectral domains after every component in the ring cavity for two configurations. The output pulse at the position 2 exhibits parabolic-like shape in the temporal domain and flat top in the spectral domain. Furthermore, as shown in Figs. 20(a) and 20(b), to investigate the pulse shaping, the kurtosis is used to analyze the temporal and spectral shapes of propagation pulses along the ring cavity [29]. Since the value of the kurtosis is −0.86 for an ideal parabola, one can see that both the spectral and temporal shapes of pulses are close to being parabolic at the output end of the gain fiber. As shown in Figs. 20(b) and 20(d), the pulse at position 2 has an expected parabolic shape with a linear frequency chirp, which is the characteristic of the self-similar solution.

Furthermore, Fig. 21 shows the pulse shapes with frequency chirps and spectra at the coupler output for different values of *d*. When *d* = 5.5, 5.0, 4.5, and 4.0 μm, the pulse energies are 0.41, 0.99, 1.34, 0.32 nJ for the type-A case, and 3.82, 6.03, 5.66, 1.44 nJ for the type-B case, respectively. Moreover, in two cases, *T*_{FWHM} of the output pulses are 0.45, 0.70, 0.92, 0.79 ps, and 6.10, 5.32, 5.10, 1.91 ps, respectively. It can be seen that the type-B schematic can output the pulses with higher energies, broader spectral and temporal widths compared with the type-A schematic, where the fiber coupler lies after the spectral filter. However, the output pulses in the type-A schematic exhibit symmetrical profiles and linear frequency chirps. For the type-B schematic, as show in Fig. 21(d), when *d* is increased from 4 μm to 5.5 μm, the 20-dB wavelength bandwidth increases from 50 nm (red) to 160 nm (blue) due to an increase of the SA fiber length. A noteworthy feature is that when the value of *d* is reduced below 4.5 μm, the temporal profiles of output pulses exhibit considerable degradations at the leading edge because of the effect of linear coupling. The results indicate that in practice the parameter *d* of the SA should be controlled above a level for avoiding linear-coupling induced distortion in the pulse shape.

For the case of the Tm fiber laser, Fig. 22 shows the evolutions of pulses along the ring cavity in temporal and spectral domains for both the type-A and type-B configurations. Furthermore, as shown in Figs. 23(a) and 23(b), the pulses at the UHNA4 output have linear frequency chirps and nearly parabolic temporal profiles except the leading and trailing edges mainly because of the anomalous-dispersion regimes in the Tm and SMF28e fibers. The spectra in Figs. 23(c) and 23(d) exhibit steep spectral edges and cat-ear features, which are the typical characteristics of DSs [27].

Furthermore, when varying the values of *d* of the SA, the detailed features of output pulses in temporal and spectral domains can be seen in Fig. 24. When *d* = 6.5, 6.0, 5.5, and 5.3 μm, the pulse energies are 4.48, 6.4, 7.73, 7.58 nJ for the type-A case, and 16.36, 17.25, 16.14, 14.72 nJ for the type-B case, respectively. In two cases, *T*_{FWHM} of the output pulses are 3.15, 4.37, 5.35, 5.43 ps, and 13.05, 13.38, 12.31, 11.66 ps, respectively. As shown in Fig. 24(d), the spectral bandwidth can reach up to about 100 nm at 20 dB level (the blue curve). For the SA, the similar conclusion to that of the Er fiber case is that the temporal distortions of pulses induced by the effect of linear coupling can be avoided by choosing an appropriate value of *d*.

## 4. Conclusion

The highly nonlinear coupler based on a CS_{2}-filled dual-core fiber is proposed and investigated numerically as the SA employed in the passively mode-locked fiber lasers. First, at the 1550 nm and 2000 nm operating wavelengths, the CS_{2}-filled silica fibers with *D* = 1.8 μm and 2.2 μm can operate in a single-mode regime with normal dispersions and large nonlinearities. Second, the coupling equations including the effects of SRS and self-steepening are employed to investigate the transmission properties of the SA. The comparisons between the CW-input and pulse-input cases with the same peak powers show that, the transmissions *T*_{r} of the latter are lower than those of the former, especially when the temporal width of the input pulse is < 1 ps. Furthermore, the effects of SRS play a significant role for the nonlinearity-induced saturation in the SA based on the CS_{2}-filled dual-core fiber. The influence of self-steepening on transmission function is much less than that of SRS. Finally, the mode-locked Er fiber and Tm fiber lasers with the proposed SA are numerically investigated. In the former case, the laser can operate in the SS regime due to all normal dispersion in the cavity. For the latter case, the laser can output pulses with approximately parabolic temporal shapes and square spectra. In both cases, the core-to-core spacing of the SA should be carefully chosen and above a certain level to avoid the temporal distortions of pulses induced by the linear coupling.

## Funding

National Natural Science Foundation of China (NSFC) (61575018).

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