Extra-Cavity Modulation of a Quartic Soliton with Negative Fourth-Order Dispersion

Abstract

Quartic solitons in ultrafast fibre lasers (intra-cavity optical fibre modulation systems) have been theoretically and experimentally analysed in recent years. However, there are few reports about extra-cavity modulating quartic solitons. In this situation, the purpose of this work is to investigate a quartic soliton’s extra-cavity modulation. In this paper, we theoretically simulate an extra-cavity-modulating quartic soliton with negative fourth-order dispersion at 1550 nm. The simulation relies on a physical model of a single-mode optical fibre system. Through controlling soliton parameters in an extra-cavity modulation system, a quartic soliton’s orthogonal polarisation modes will show unique characteristics depending on which kind of parameter is changed (seven parameters are considered for variation). For example, through the variation in the projection angle, only a horizontally polarised quartic soliton pulse is generated. These results explore the dynamics of quartic solitons in single-mode optical fibre modulation systems and are applicable to optical soliton transmission techniques in the field of optical fibre communication.

1. Introduction

Optical solitons have the characteristic of propagating over long distances without waveform distortions and exist in many types of nonlinear systems [1,2,3,4,5,6]. During past decades, the ultrafast fibre laser system was often selected as an ideal platforms with which to study optical soliton dynamics because it had the advantages of a compact and flexible structure, strong ability to resist electromagnetic interference, high heat dissipation efficiency, high laser beam quality, flexible parameter regulation and so on. Until now, optical solitons with specific properties (such as temporal pulse shape, optical spectrum and pulse chirp) have been theoretically studied and experimentally achieved, such as dispersion-managed solitons [7,8], dark solitons [9,10], dissipative solitons [11,12], noise-like pulses [13,14], dissipative soliton resonances [15,16], Kerr solitons [17,18], pure-quartic solitons [19,20,21,22,23,24,25] and so on. These research results effectively promote the exploration of optical solitons’ dynamic characteristics in optical fibre laser systems.

As a kind of intra-cavity modulation system, an ultrafast fibre laser can emit different optical solitons with appropriate intra-cavity parameter management. The controllable intra-cavity parameters include gain, loss, nonlinearity, dispersion, saturable energy, linear birefringence and some others. Gain can be varied through selecting a gain fibre with an appropriate length and absorption coefficient at a specific wavelength. Intra-cavity loss can be efficiently controlled by using an optical fibre attenuator. Nonlinearity is varied through controlling pump power or introducing a piece of optical fibre with high nonlinearity. Dispersion includes group-velocity dispersion and some other higher-order dispersions, which can be tuned with optical elements such as dispersion compensation fibres, chirped fibre Bragg grating, photonic crystal waveguide and spatial light modulators. Saturable energy depends on the saturable absorber that is adopted. The saturable absorber can be a real saturable absorber (such as SESAM, carbon nanotube, MXene, perovskite quantum dots and rare-earth-ion-doped fibre) or an artificial saturable absorber (such as nonlinear polarisation rotation). A change in linear birefringence in a fibre laser cavity can generate vector optical solitons when different intra-cavity parameters reach an equilibrium state. These vector solitons are group-velocity locked [26], polarisation locked [27] or polarisation rotated [28]. Different vector solitons have their own characteristics, including their orthogonal components’ central wavelengths, polarisation state, temporal pulse train and so on.

In addition to intra-cavity managing of optical solitons in ultrafast fibre laser cavities (oscillators or amplifiers), modulating optical solitons in extra-cavity optical fibre systems has also attracted some researchers’ attention [29,30,31,32]. For example, a dark bisoliton with a 1064 nm central wavelength was modulated in an extra-cavity manner based on an optical fibre system, and modulated orthogonal components showed unique properties in the time domain and frequency domain as different parameters varied [29]. Compared with intra-cavity modulation techniques, a larger tuning range in the time/frequency domain of optical solitons’ orthogonal components can be achieved with the adoption of extra-cavity modulation. So extra-cavity modulation systems can generate specific types of optical solitons, which is not easy for intra-cavtiy modulation systems. According to the structure, we classify extra-cavity modulation systems as single channel, dual-channel or multi-channel. In recent years, the extra-cavity modulation of optical solitons has already been studied by some researchers, and the relevant studies have potential applications in optical signal processing [33]. However, there are few reports about the extra-cavity modulation of quartic solitons. Recent studies indicate that quartic solitons can provide new ideas to develop high energy and high peak power laser pulses in ultrafast laser fields. For quartic solitons, fourth-order dispersion is introduced to compensate for higher-order nonlinear effects, which gives it the advantages of higher data rate and improved pulse quality in long-distance optical fibre transmission and high-power optical fibre transmission compared to conventional solitons. In this paper, we study the extra-cavity modulation of a quartic soliton with negative fourth-order dispersion at a 1550 nm wavelength regime. Through the control of several different extra-cavity parameters, the modulated quartic soliton can demonstrate different features in orthogonal polarisation direction and in the time domain and frequency domain. These results can provide meaningful values for developing fibre laser systems with quartic solitons.

2. Theoretical Physical Model

Figure 1 gives the physical model of our proposed modulation system. The fibre laser source can produce quartic soliton pulses. The quartic soliton pulses are modulated as follows: Firstly, the laser pulses are collimated by a collimator (COL1) and collected by another collimator (COL2). Through the mechanical rotation of three wave plates between the two COLs, the quartic soliton’s orthogonal phase difference can vary from zero to 2π, leading to the variation of the corresponding polarisation state. Secondly, the quartic soliton’s orthogonal electric fields are split by a polarisation beam splitter (PBS1) and combined by another polarisation beam splitter (PBS2). The two electric field amplitude modulators (AM1 and AM2) and two independent time delay lines (DL1 and DL2) are employed to independently change the orthogonal components’ amplitudes and orthogonal time delay, respectively. The fibres between PBS1 and PBS2 are polarisation-maintained (PM). Thirdly, the modulated quartic soliton’s polarisation state is further tuned by a polarisation controller (PC), then collimated by COL3 and split by PBS3. The orthogonally polarised components of modulated quartic solitons will demonstrate different characteristics as different parameters’ values change. It should be noted that the quartic soliton pulses can also be modulated directly in a single channel, but this method is usually used for modulating scalar soliton pulses.

The governing equation of a quartic soliton with negative fourth-order dispersion is:

$$i{\displaystyle \frac{\partial u}{\partial \xi }}+{\displaystyle \frac{1}{2}}{\displaystyle \frac{{\partial }^{2}u}{\partial {\tau }^2}}+{\left|u\right|}^{2}u={\delta }_4{\displaystyle \frac{{\partial }^{4}u}{\partial {\tau }^4}}.$$

(1)

In this equation, u is an electric field. ξ, τ and δ₄ can be expressed as following:

$$\xi ={\displaystyle \frac{z}{L_D}},$$

(2)

$$\tau ={\displaystyle \frac{t}{T_0}},$$

(3)

$${\delta }_4=-{\displaystyle \frac{{\beta }_4}{24\left|{\beta }_2\right|T_0^2}}.$$

(4)

In Equations (2)–(4), z is the position variable and L_D is the dispersion length ($T_0^2/\left|{\beta }_2\right|$). ξ is the normalized propagation distance. t is the time variable. T₀ is the initial pulse width. δ₄ is related to β₂, β₄ and T₀. β₂ and β₄ are second-order dispersion and fourth-order dispersion, respectively. In this work, β₄ is as a negative number to get the quartic soliton solution of the governing equation. β₄ can also be a positive number, but the relevant governing equation and soliton solution will be different.

The amplitudes after modulation and before the last PBS are expressed as follows:

$$u_1(z,\hspace{0.33em}t)=u_{10} {\mathrm{sech}}^2\left({\displaystyle \frac{t-\Delta T/2}{\sqrt{40{\delta }_4}{T}_0}}\right)\exp \left(i{\displaystyle \frac{2\pi ct}{\lambda }}+i{\displaystyle \frac{z_1}{25{\delta }_4{L}_{\mathrm{D}}}}\right),$$

(5)

$$u_2(z,\hspace{0.33em}t)=u_{20} {\mathrm{sech}}^2\left({\displaystyle \frac{t+\Delta T/2}{\sqrt{40{\delta }_4}{T}_0}}\right)\exp \left(i{\displaystyle \frac{2\pi ct}{\lambda }}+i{\displaystyle \frac{z_2}{25{\delta }_4{L}_{\mathrm{D}}}}+i\Delta \phi \right).$$

(6)

In Equations (5) and (6), u₁(z, t) and u₂(z, t) are the quartic soliton’s electric field components in orthogonal directions after changing amplitude ratio, time delay, phase difference, propagation distances and fourth-order dispersion. u₁₀ and u₂₀ are constant amplitudes. ∆T is the time delay between the two orthogonal electric fields. z₁ and z₂ are the propagation distances. ∆φ is the orthogonal phase difference. λ is the central wavelength. The orthogonal electric fields are calculated with the following equation:

$$\left[\begin{array}{c}{u}_x(z,\hspace{0.33em}t)\\ u_y(z,\hspace{0.33em}t)\end{array}\right]=\left[\begin{array}{cc}\cos \theta & \sin \theta \\ \sin \theta & -\cos \theta \end{array}\right]\left[\begin{array}{c}{u}_1(z,\hspace{0.33em}t)\\ u_2(z,\hspace{0.33em}t)\end{array}\right].$$

(7)

Here, θ is the modulated quartic soliton pulse’s projection angle on PBS3. u_x(z, t) and u_y(z, t) are the projected electric fields in a horizontal direction (x direction) and vertical direction (y direction). Modulated pulse shape intensities and optical spectral intensities are calculated with:

$$I_j(z,\hspace{0.33em}t)={\left|u_j(z,\hspace{0.33em}t)\right|}^2\hspace{1em}(j=x\hspace{0.33em}or\hspace{0.33em}y),$$

(8)

$$I_j(z,\hspace{0.33em}\omega )={\left|u_j(z,\hspace{0.33em}\omega )\right|}^2={\left|{\displaystyle {\int }_{-\infty }^{\infty }{u}_j(z,\hspace{0.33em}t)e^{-i\omega t}dt}\right|}^2\hspace{0.33em}(j=x\hspace{0.33em}or\hspace{0.33em}y).$$

(9)

In Equations (8) and (9), I_j(z, t) and I_j(z, ω) are pulse shape intensities and optical spectral intensities in orthogonal polarisation directions. ω is angular frequency, i represents the unit complex number. Here, u_j(z, ω) is a Fourier transform of u_j(z, t). In our theoretical simulation, β₂ and β₄ are two important parameters that can induce variations in temporal pulse widths in orthogonal polarisations, finally influencing the achievable modulation bandwidth and the fidelity of the encoded information.

3. Simulation

In this part, we present the simulation results under variations in different soliton parameters. In the simulation, we assume T₀ = 5 ps, λ = 1550 nm and β₂ = −21 ps²/km.

Figure 2 shows extra-cavity-modulated orthogonal pulse shapes and the corresponding optical spectra when u₂/u₁ = 1/1.26, 1/1.13, 1/1, 1.13/1 and 1.26/1, θ = 20°, ∆T = 0 ps, ∆φ = 0, z₁/L_D = z₂/L_D = 0.1 and β₄ = −1 ps⁴/km. In Figure 2a–e, as u₂/u₁ increases from 1/1.26 to 1/1, the horizontal/vertical pulse’s peak intensity decreases/increases. However, when u₂/u₁ increases from 1/1 to 1.26/1, both the orthogonal pulses’ peak intensities increase. These peak intensities’ variation tendencies are also applicable to corresponding optical spectra shown in Figure 2f–j. The relatively narrow (<1 nm) orthogonal optical spectrum bandwidths being shown in Figure 2f–j are accompanied by short temporal pulse widths (<1 ps). When u₂/u₁ = 1/1, orthogonal components still have different peak intensities in the time and frequency domains because θ ≠ 0°. Energy flow occurs between orthogonal directions in this situation, which comes from the projections of orthogonal electric fields on the axes of PBS3. It should be noted that the vertical component can have a higher peak intensity than the horizontal component in the time domain and frequency domain when u₂/u₁ increases further.

Figure 3 shows modulated quartic solitons’ temporal and frequency characteristics in output orthogonal polarisations when θ = 0°, 15°, 30°, 45° and 60°, u₂/u₁ = 1/1, ∆T = 0 ps, ∆φ = 0, z₁/L_D = z₂/L_D = 0.1 and β₄ = −1 ps⁴/km. From these results, we can see orthogonal pulses and optical spectra are the same when θ = 0°, and the horizontal/vertical pulse’s peak intensity will increase/decrease as θ increases from 0° to 45°. When θ = 45°, the vertical mode disappears and only the horizontal mode exists, meaning only horizontally polarised quartic soliton pulses are generated in the case of symmetrical projection. When θ further increases to 60°, the horizontal/vertical pulse peak intensity decreases/increases. The orthogonal optical spectra have the same variation trends compared with the corresponding orthogonal pulses. It should be noted that θ can be increased further, and orthogonal pulses and optical spectra will overlap again when θ = 90°. The peak intensities of a vertical pulse and optical spectrum can be larger than a horizontal component’s when θ is larger than 90°.

Figure 4 shows extra-cavity-modulated quartic solitons’ orthogonal pulse shapes and optical spectra as ∆T = 0.1 ps, 0.2 ps, 0.3 ps, 0.4 ps and 0.5 ps, u₂/u₁ = 1/1, θ = 10°, ∆φ = 0, z₁/L_D = z₂/L_D = 0.1 and β₄ = −1 ps⁴/km. In Figure 4a–e, the time delay between orthogonal pulses increases as ∆T increases, accompanied with amplitude variations. The peak intensity of a horizontally polarised pulse decreases with increased ∆T. And the situation is the contrary for a vertically polarised pulse. In Figure 4f–j, the orthogonal spectra-centering wavelengths are always centered at 1550 nm (see the zero point on the horizontal coordinate axis), with no obvious changes in peak intensities and 3 dB bandwidths as ∆T increases. The horizontal component always has a higher peak intensity than the vertical component either in time domain or frequency domain. When the time delay increases further, the vertical pulse’s peak intensity can be comparable to, but cannot exceed, the horizontal pulse’s peak intensity, while the vertical optical spectral peak intensity is always lower than the horizontal component’s.

Figure 5 shows the extra-cavity modulation results when ∆φ = π/9, 2π/9, π/3, 4π/9 and 5π/9, u₂/u₁ = 1/1, θ = 10°, ∆T = 0 ps, z₁/L_D = z₂/L_D = 0.1 and β₄ = −1 ps⁴/km. In Figure 5a–e, the peak intensity of the horizontal/vertical pulse decreases/increases with increased ∆φ. It should be noted that the horizontal/vertical pulse’s peak intensity decreases/increases as ∆φ increases from 0 to π, and the orthogonal pulses overlap when ∆φ = π/2. After that, horizontal/vertical pulse peak intensity will increase/decrease when ∆φ increases from π to 2π. Orthogonal pulses will overlap again when ∆φ = 3π/2. And in Figure 5f–j, the optical spectral peak intensities have the same variation trends in orthogonal directions compared with the corresponding pulse shapes’ peak intensities. We can see that the vertical component has higher temporal and optical spectral peak intensities than the horizontal component when ∆φ = 5π/9. Orthogonal pulses and optical spectra are always different when u₂/u₁ = 1/1, except for some specific ∆φ values.

Figure 6 shows the extra-cavity-modulated orthogonal components’ properties when z₁/L_D = z₂/L_D = 0.2, 0.3, 0.4, 0.5 and 0.6, u₂/u₁ = 1/1, θ = 10°, ∆T = 0 ps, ∆φ = π/9 and β₄ = −1 ps⁴/km. Figure 6 shows that, as z₁/L_D or z₂/L_D increases, both pulse shapes in Figure 6a–e and optical spectra in Figure 6f–j have no obvious changes. These results mean that modulated quartic solitons are robust or insensitive to changes in soliton propagation distances when z₁/L_D = z₂/L_D. If z₁/L_D or z₂/L_D further increases, orthogonal pulse shapes and optical spectra still maintain their stability, demonstrating that a quartic soliton can keep its temporal and spectral properties in long-distance propagation when z₁/L_D = z₂/L_D. The horizontal pulse/optical spectrum’s peak intensity is always higher than that for vertical component because θ ≠ 0°.

Figure 7 shows pulse shapes and corresponding optical spectral properties in an output terminal when z₁/L_D = 0.1, z₂/L_D = 0.2, 0.3, 0.4, 0.5 and 0.6., u₂/u₁ = 1/1, θ = 10°, ∆T = 0 ps, ∆φ = π/9 and β₄ = −1 ps⁴/km. Figure 7 shows that horizontal/vertical component time domain and frequency domain peak intensities decrease/increase as z₂/L_D increases, which is different from the simulation results shown in Figure 6. This variation trend in orthogonal polarisations means energy flow occurs between orthogonal components when orthogonal soliton components have different propagation distances. Horizontal component peak intensity is always at a higher level than that of vertical peak intensity, either in the time domain or frequency domain. If z₂/L_D increases further, orthogonal pulses and corresponding optical spectra will overlap when z₂/L_D = 1. And when z₂/L_D > 1, the vertical component’s pulse shape and optical spectrum have higher peak intensities than the horizontal peak intensities.

Figure 8 shows modulated quartic soliton properties in output orthogonal directions when β₄ = −1 ps⁴/km, −2 ps⁴/km, −3 ps⁴/km, −4 ps⁴/km and −5 ps⁴/km, u₂/u₁ = 1/1, θ = 10°, ∆T = 0 ps, ∆φ = π/10 and z₁/L_D = z₂/L_D = 0.1. β₄ can be varied with an intra-cavity spectral pulse shaper, which is located in the fibre laser source in Figure 1. In Figure 8a–e, orthogonal pulse peak intensities are almost unchanged when β₄ changes. However, orthogonal pulse widths will increase as β₄ varies from −1 ps⁴/km to −5 ps⁴/km, meaning that, in the design of long-distance optical fibre modulation system, β₄ should be carefully controlled to avoid temporal pulse broadening. In Figure 8f–j, orthogonal optical spectral peak intensities increase with the increased absolute value of β₄. The reason is that increased temporal pulse widths will induce decreased frequency domain bandwidths. According to the Parseval theorem, the total energy in the time domain and frequency domain are the same. Considering the total energy in one polarisation direction increases as $\left|{\beta }_4\right|$ increases, the corresponding peak intensity in Figure 8f–j will increase. And the horizontal component’s temporal peak intensity and optical spectral peak intensity are always higher than the vertical component’s, even if $\left|{\beta }_4\right|$ increases further.

4. Discussion

From the simulation results described above, we can see different soliton parameters will have different influences on the characteristics of pulse shapes and optical spectra in orthogonal polarisations. In previous works, studies about extra-cavity modulating optical fibre solitons focused on a normal or anomalous second-order dispersion regime, without consideration of higher-order dispersion [29,30,31,32]. In this manuscript, negative fourth-order dispersion is considered at an anomalous second-order dispersion regime. It is found that negative fourth-order dispersion can significantly influence the temporal pulse widths and optical spectral peak intensities of a modulated quartic soliton’s orthogonal components. In our future work, positive fourth-order dispersion will be considered to investigate extra-cavity-modulating self-similar quartic soliton pulses. Furthermore, the dual-channel structure can be replaced by a multi-channel structure to modulate soliton properties more flexibly, but it will also increase the complexity of the modulation system.

5. Conclusions

In the above-mentioned simulations, we mainly investigate extra-cavity-modulating a quartic soliton with negative fourth-order dispersion at 1550 nm wavelength regime. Different extra-cavity parameters were varied (including amplitude ratio, projection angle, time delay, orthogonal phase difference, propagation distances and fourth-order dispersion), to achieve the modulation of an input quartic soliton in a large adjustment range. The simulation results provided meaningful reference values for exploring quartic solitons’ dynamics in nonlinear optical fibre modulation systems. Additionally, research about extra-cavity modulation of a 1550 nm quartic soliton with negative fourth-order dispersion can find potential applications in C-band optical fibre soliton communication fields. These results also provide a new idea for modulating optical solitons with higher-order dispersions.