Phase shift, amplification, oscillation and attenuation of solitons in nonlinear optics.

In nonlinear optics, the soliton transmission in different forms can be described with the use of nonlinear Schrödinger (NLS) equations. Here, the soliton transmission is investigated by solving the NLS equation with the reciprocal of the group velocity β 1 ( z ) , the group velocity dispersion coefficient β 2 ( z ) and nonlinear coefficient γ ( z ) . Two-soliton solutions for the NLS equation are obtained through the Hirota method. According to the solutions obtained, β 1 ( z ) and γ ( z ) with different function forms are taken to study the characteristics of solitons. The effect of the phase shift on the soliton interaction is discussed, and the non-oscillating soliton amplification, which is transmitted in a bound state, is explored. Parabolic solitons with oscillations are analysed. Moreover, parabolic solitons can be reduced to dromion-like structures. Results indicate that the transmission of solitons can be adjusted with the group velocity dispersion and Kerr nonlinearity coefficients. The phase shift, amplification, oscillation and attenuation of solitons can also be controlled by other related parameters. This work accomplishes the theoretical study of transmission characteristics of optical solitons in spatially dependent inhomogeneous optical fibres. The conclusions of this research have theoretical guidance for the research of optical amplifier, all-optical switches and mode-locked lasers.

Introduction

Solitons have been investigated in such fields as mathematics and physics [1], [2], [3], [4], [5], [6]. They can propagate in a long distance without changes in their waveform, velocity and amplitude [7], [8], [9], [10]. The soliton phenomena are closely related to nonlinear evolution equation models [11], [12], [13], [14], [15], [16], [17]. Solitons are also applied in particle physics, fluid mechanics, Bose-Einstein condensation and nonlinear optics [18], [19], [20].

Some researchers have studied solitons by solving the nonlinear evolution equations and analysing their soliton solutions [21], [22], [23], [24], [25]. As one of the classic nonlinear evolution equations, the nonlinear Schrödinger (NLS) equation can be solved to obtain soliton solution, and has been widely investigated by using different methods [26], [27], [28], [29], [30], [31], [32]. With the differential quadrature method, the dynamic problems constructed by the NLS equation have been analysed [26]. The existence and stability of the standing wave solutions for the NLS equation in n-dimensional space have been studied [27]. Using the generalized exponential rational function, a new method to solve the exact special solutions of the NLS equation has been proposed [28]. The unified method has been used to acquire optical soliton solutions of the NLS equation [29]. Moreover, the stability of full dimensional KAM tori for the NLS equation has been proved [30]. Local Cauchy theory for the NLS equation has been discussed [31], and nonlinear instability of half-solitons has been analysed [32].

However, the soliton transmission process can be simulated more accurately with the variable coefficient NLS (vcNLS) equation, when the transmission medium or boundary condition is not uniform [33], [34], [35], [36], [37], [38], [39], [40]. For the vcNLS equation, dynamics of solitons have been explored [33], and soliton interactions have been discussed [34], [35]. In addition, the breather-to-soliton transitions for the vcNLS equation have been found [36], and nonautonomous multipeak solitons have been obtained [37].

The different transmission characteristics of solitons can be obtained by solving this vcNLS equation,where describes the temporal envelope of solitons; and represent the longitudinal coordinate and the time in the moving coordinate system, respectively; and , and are related to the reciprocal of the group velocity, group velocity dispersion (GVD) coefficient and the nonlinearity coefficient, respectively. If , Eq. (1) will become the standard vcNLS equation.

Eq. (1) can be used to describe the transmission of solitons. The discussion of and is helpful to the development of dispersion management communication systems [38]. In addition, the study of , and is of significance to the experimental and engineering application of mode-locked fiber lasers and nonlinear optics. The rogue wave solutions for Eq. (1) have been solved with similarity transformation [39], and the properties of oscillating solitons for Eq. (1) have been analysed [40].

However, parallel solitons, parabolic solitons and dromion-like solitons obtained by taking different functions for and have not been reported. The effect of the phase shift on the transmission of parallel solitons will be discussed. Here, the non-oscillating parallel soliton amplification, which transmits in bound states, will be studied. Parabolic solitons with oscillations will be analysed. The method of the parabolic soliton reducing to the dromion-like structure will be proposed by adjusting the corresponding parameters.

In Section ‘Material and methods’, the two-soliton solutions will be solved through use of the Hirota method. In Section ‘Results and discussion’, different soliton transmission characteristics are analysed with the variable coefficients taking different functions. And in Section ‘Conclusions’, the conclusions will be derived.

Material and methods

Firstly, the following independent variable transformation is introduced to solve the bilinear form according to the Hirota method [40],where and are the complex function and real function, respectively. Then the bilinear form is obtained,where is the complex conjugate. Moreover, the bilinear operators and are defined as follows

According to the Hirota method, the bilinear form can be solved by the following power series expansions of and ,

Here is a formal expansion parameter.

In order to obtain the two-soliton solutions, it can be assumed thatwhere

In addition,

Finally, one can obtain the solutions,and

Results and discussion

By analysing the preceding two-soliton solutions, we find that if , then the GVD coefficient is equal to the nonlinearity coefficient . Therefore, the different soliton transmission characteristics can be obtained by discussing the reciprocal of the group velocity and the nonlinearity coefficient .

In order to obtain the effect of the phase shift on parallel soliton transmission, one takes as the Gauss function, and as shown in Fig. 1. In Fig. 1(a) and (b), the phase shift of two solitons to the left influences the interaction between two solitons. In order to keep two solitons from interacting with each other after the phase shift, one can adjust the parameter . Comparing Fig. 1(b) with Fig. 1(c) or Fig. 1(a) with Fig. 1(d), one can find that the direction of the phase shift is related to the signs of and . When and are positive, two solitons shift to the left, while two solitons move in opposite directions when they are negative. The larger the value of or , the greater is the phase shift distance of the soliton. Therefore, one can adjust the value of as shown in Fig. 2 in order to separate the two interacting solitons after the phase shift. Moreover, the amplitude of the soliton increases as or increases.

Fig. 1

The phase shift has an effect on the transmission of parallel solitons. The parameters are , , , , and with (a): , , and ; (b): , , and ; (c): , , and ; (d): , , and .

Fig. 2

Amplitude comparison between two solitons at (red dotted line) and (green solid line) in (a): Fig. 1(a); (b): Fig. 1(b); (c): Fig. 1(c); (d): Fig. 1(d).

To observe whether soliton energy exchange occurs during the phase shift, we can measure the amplitude of the soliton before and after the phase shift as shown in Fig. 2. In Fig. 2(a) and (b), the amplitudes of two solitons before and after the phase shift have obvious changes. However, the amplitudes of two solitons before and after the phase shift are almost the same in Fig. 2(c) and (d). Therefore, when two solitons are simultaneously shifted to the left, the energy transfer between solitons occurs. When two solitons are in opposite phase shifts, they remain constant in energy. Moreover, it can be seen from Figs. 1(c) and 2(c) that two solitons retain shape and amplitude after the collision except for a certain phase shift, which indicates that the collision is elastic. These analyses can help eliminate the interaction between solitons and change the direction of the soliton transmission. This feature is useful for reducing the bit error rate, improving the quality of optical communication and studying all-optical switches.

Then, the effect of the phase shift on parallel solitons with the oscillation is discussed. In order to obtain the parallel solitons with oscillations, and are taken as Fig. 3 shows. In Fig. 3(a) and (b), one can find that the amplitude and phase shift of solitons can be changed by adjusting the parameters. The amplitude of the left soliton increases as or is increased. However, the amplitude of the right soliton increases with the decreasing of or the increasing of . The direction of the phase shift for two solitons is related to the sign of and . The phase shift distance increases as the values of and are increased. And when two solitons are in opposite phase shifts, elastic collision occurs during the phase shift. Because they have no change in amplitude and shape before and after the phase shift as shown in Fig. 4(a) and (b). From Fig. 3(c) and (d), one can see that not only are two solitons simultaneously shifted to the right, but also two solitons are shifted to the left at the same time. Further, in the process of the phase shift, the energy of two solitons is exchanged. Their amplitude changes before and after the phase shift, as shown in Fig. 4(c) and (d). These investigations can help weaken the energy exchange with the oscillatory parallel soliton during the phase shift. These studies can be helpful in the study of mode-locked fibre lasers.

Fig. 3

Phase shift affects the parallel soliton with oscillations. The parameters are , , , , and with (a): , , and ; (b): , , and ; (c): , , and ; (d): , , and .

Fig. 4

Amplitude comparison between two solitons at (red dotted line) and (green solid line) in (a): Fig. 3(a); (b): Fig. 3(b); (c): Fig. 3(c); (d): Fig. 3(d).

The effect of the phase shift on the parallel soliton transmission and energy exchange was investigated. Next, we discuss the non-oscillating parallel soliton amplification, which transmits in bound states. In order to study the transmission, one takes and as shown in Fig. 5. Near the point, it is mainly the sudden changes in and that lead to change in the interaction between two solitons. As a result, two solitons have exchanged energy.

Fig. 5

The parallel soliton amplification in bound states. The parameters are , , , , and with (a): , , and ; (b): , , and ; (c): , , and ; (d): , , and .

In Fig. 5(a), the left soliton amplitude increases and the right soliton amplitude decreases near the point. Energy is exchanged from the right soliton to the left. The situation in Fig. 5(a) and (d) is just the opposite. In Fig. 5(b), the amplitudes of two solitons are reduced at the same time near the point, while the amplitudes of two solitons are amplified in Fig. 5(c). And in Fig. 5(a) and (d), the interaction between two solitons has also changed near the point. This is because the inverse function taken by is changed suddenly around the point of . However, before and after the point in Fig. 5(b) and (c), the interaction between two solitons does not change. Moreover, the amplitude is also affected by .

Their changes in amplitude are shown in Fig. 6. The interaction between two solitons is inelastic. The energy is exchanged, and the amplitude changes between them without any outside influence. This can provide some help in the development of optical amplifiers.

Fig. 6

Amplitude comparison between two solitons at (red dotted line) and (green solid line) in (a): Fig. 5(a); (b): Fig. 5(b); (c): Fig. 5(c); (d): Fig. 5(d).

To investigate the effect of and on other types of soliton transmissions, one considers and . Then, a set of parabolic solitons with oscillations is obtained. In Fig. 7(a), two parabolic solitons maintain their original oscillation intensity due to the weak interaction. In Fig. 7(b), the interaction between two parabolic solitons causes the intensity of the inner parabolic soliton oscillations to become smooth. However, the interaction between parabolic solitons in Fig. 7(c) tends to smooth the intensity of the outer parabolic soliton oscillations. By analysing the parameters, one finds that affects the soliton oscillation intensity of the inner side, and the inner soliton oscillates are exacerbated as increases. The outer soliton oscillation intensity is controlled by and exacerbated with the increasing of . Moreover, the amplitude of the inner soliton is mainly controlled by , while the amplitude of the outer soliton is mainly influenced by . The amplitude of the inner and outer solitons is increased with increasing and , respectively. When or , the two solitons decay into a soliton as shown in Fig. 7(d). Those studies are of great help in controlling the oscillation intensity of parabolic solitons so that it can promote theoretical and experimental research on soliton transmission and collision.

Fig. 7

The interactions between parabolic solitons with oscillations. The parameters are , , , , and with (a): , , and ; (b): , , and ; (c): , , and ; (d): , , and .

In order to further explore the effect of and on other types of soliton transmissions, one takes and as shown in Fig. 8, so that the parabolic solitons can decay into two dromion-like structures. The method of how to adjust the parameters to achieve the suitable attenuation is proposed. The attenuation of solitons is mainly influenced by , and . The smaller the value, the faster the decay rate becomes. Only when and are positive can the parabolic solitons be attenuated to the dromion-like structures. Further, as and increase, the decay rate becomes faster. Moreover, the distance between two dromion-like structures can be adjusted by as shown in Fig. 8(a) and (b). Their opening directions are related to the sign of in . These parametric analyses are of great significance in the development of Bose-Einstein condensation.

Fig. 8

The parabolic solitons decay into two dromion-like structures. The parameters are , , , , and with (a): , , and ; (b): , , and .

Conclusions

Two soliton solutions of Eq. (1) were obtained through use of the Hirota method. By analysing the solution, different function forms for and were taken to obtain different types of soliton transmission. The effect of the phase shift on the transmission of smooth parallel solitons was discussed when and . If one takes and , the influence of the phase shift on parallel solitons was presented. In addition, the soliton amplification in a bound state has been explored when . When and , parabolic solitons with oscillations are demonstrated. Moreover, parabolic solitons have been reduced to dromion-like structures while and . Those presented results are applicable to theoretical analysis and experimental research of optical amplifier, all-optical switches and mode-locked lasers.

Conflict of interest

The authors declare they have no conflict of interest.

Compliance with Ethics Requirements

This article does not contain any studies with human or animal subjects.