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Abstract
We study the properties of dissipative solitons supported by a chirped lattice in a focusing Kerr medium with nonlinear loss and a transversal linear gain landscape consisting of single or three amplifying channels. The existence and stability of two types of dissipative solitons, including fundamental and three-peaked twisted solitons, have been explored. Stable fundamental solitons can only be found in a single-channel gain chirped lattice, and stable three-peaked twisted solitons can only be obtained in a three-channel gain chirped lattice. The instability of two types of dissipative solitons can be suppressed at a high chirp rate. Also, robust fundamental and three-peaked twisted nonlinear states can be obtained by excitation of Gaussian beams of arbitrary width in corresponding characteristic structures.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Periodic optical lattice can implement a spatial modulation (i.e., its period or strength) to become a new type of optical lattice, namely chirped optical lattice. It can significantly change the existence conditions of spatial solitons, because weak modulation can modify the band-gap lattice structure, while strong modulation can completely destroy it. In conservative systems, many interesting optical phenomena are found in chirped optical structures, such as plasmonic Bloch oscillations [1,2], families of power thresholdless surface waves [3–5], surface defect gap solitons [6], two-color localized nonlinear modes [7], asymmetric light propagation [8], rainbow trapping [9], self-deflecting plasmonic lattice solitons [10], and real-space imaging of a topologically protected edge state [11].
In contrast to conservative systems, gain and loss are usually present in nonconservative optical structures. A notable example is a parity-time symmetric lattice [12–19], in which the refractive index profile is an even symmetric function, and the distribution of linear gain and linear loss is odd symmetric. Stabilization of optical solitons [20], and diffraction management and soliton dynamics [21] have been explored in frequency-chirped parity-time symmetric lattices. Various concepts related to parity-time symmetry, including partially-parity-time symmetry [22–27] and non-parity-time symmetry [28–31], have also attracted great attentions. In such above nonconservative structures, it is very important to study the propagation dynamics of linear and nonlinear states (see reviews [32,33] and references therein). Another developed ramification of this topic is the study of trapped nonlinear modes in settings combining a spatially localized gain and nonlinear losses. In such settings, the properties of dissipative defect modes in both focusing and defocusing medium have been studied [34], dissipative surface solitons can be found at the interface between a semi-infinite periodic [35] (or chirped [36]) lattice and a homogeneous Kerr medium, edge and bulk dissipative solitons have been studied in modulated parity-time-symmetric waveguide arrays [37], symmetry breaking and multipeaked solitons in inhomogeneous gain landscapes have been addressed [38], and the existence of two-dimensional stable localized modes [39] and vortex solitons [40–43] supported by localized gain have been explored.
However, to the best of our knowledge, the properties of dissipative solitons supported by transversally single- or three-channel amplifying chirped lattices have not been exposed. In this article, we investigate the existence and stability of both fundamental and three-peaked twisted solitons in a focusing Kerr medium with nonlinear loss and a transversal linear gain landscape. We find that the localization degree of the field moduli of the dissipative solitons becomes obvious with the increase of linear gain coefficient or the decrease of nonlinear loss coefficient. In the characteristic structures, stable dissipative solitons can be found due to the balance of linear gain and nonlinear loss. And stable fundamental solitons can only exist in a single-channel gain chirped lattice, and stable three-peaked twisted solitons can only be obtained in a three-channel gain chirped lattice. We also discover that the instability of dissipative solitons can be suppressed by increasing the chirp rate of the lattice. And the excitations of the above two types of nonlinear states are also considered in their corresponding characteristic structures.
2. Theoretical model
We consider the propagation of a light beam along the $z$ axis in a focusing Kerr medium with chirped lattices, a single or several amplifying channels, and nonlinear losses, which can be described by the nonlinear Schrödinger equation for the dimensionless light field amplitude $\psi$:
Specifically, we set $V_r(x)=p_{\textrm {re}}\cos ^2(2x)[1-\tau |x|]$, where $p_{\textrm {re}}$ is the modulation depth, $\tau$ is a small constant and represents the chirp rate of the lattice. Similar to [34], we also assume that $V_i(x)\propto V_r(x)$ on the middle lattice channels, i.e., $V_i(x)=p_{\textrm {im}}\cos ^2(2x)[1-\tau |x|]$ for $|x|\leq n\pi /4$, and $V_i(x)\equiv 0$ otherwise, where $n$ is the number of lattice channels, $p\rm _{im}$ is positive and stands for the linear gain coefficient.
We search for stationary solutions of Eq. (1) in the form $\psi (x,z)=\phi (x)\exp (i\beta z)=[\phi _r(x)+i\phi _i(x)]\exp (i\beta z)$, where $\beta$ is the propagation constant, $\phi$, $\phi _r$ and $\phi _i$ are the complex function, real part and imaginary part of the profile. Upon substitution of the field in such a form into Eq. (1) one gets
In order to expound the stability of the stationary soliton solutions, we solve the eigenvalue problem by substituting the perturbed solutions $\psi =\{\phi +[f-g]\exp (\delta z)+[f+g]^*\exp (\delta ^*z)\}\exp (i\beta z)$ into Eq. (1), here $|f|, |g|\ll |\phi |$, $\delta =\delta _r+i\delta _i$, and $*$ represents the complex conjugation. The eigenvalue and eigenfunction of perturbation are determined by the following coupled equations:
3. Results and discussions
First, we address the properties of fundamental solitons supported by a central single-channel gain chirped lattice ($n=1$). Representative profiles of fundamental solitons are depicted in Figs. 1(a) and 1(b). Obviously, with an increase of $p_{\textrm {im}}$ or a decrease of $\kappa$, the peak value of fundamental soliton increases significantly, and the profile becomes more and more localized. The stationary solution is obtained due to the balance between linear gain and nonlinear loss.
Fig. 1. Profiles of fundamental solitons with single waveguide gain. $\kappa =0.8$, $p\rm _{im}=0.5$ (blue) and $4.0$ (red) in (a), and $p\rm _{im}=1.0$, $\kappa =0.4$ (blue) and $1.2$ (red) in (b). (c) Power $U$ and (d) propagation constant $\beta$ versus linear gain coefficient $p\rm _{im}$ at three different $\kappa$. $p\rm _{re}=5$, $\tau =0.05$, and $n=1$ in all cases.
The power of fundamental soliton, defined as $U=\int _{-\infty }^{+\infty }|\phi (x)|^2dx$, increases linearly with the growth of gain coefficient $p_{\textrm {im}}$ in the focusing Kerr medium. In addition, we find that for a fixed $p_{\textrm {im}}$ value, the power gradually decreases with the increase of the nonlinear coefficient $\kappa$ [Fig. 1(c)]. And the variation in the change of $U$ and $\beta$ is usually consistent [Fig. 1(d)].
We also study the profiles of three-peaked twisted solitons in the single-channel gain chirped lattice. We notice that, with an increase of $p_{\textrm {im}}$, the soliton power increases first and then decreases gradually [Fig. 2(a)]. In this situation, the profile of three-peaked twisted soliton is complicated, and the representative profiles are sketched in Figs. 2(b)–2(c).
Fig. 2. (a) Power $U$ of three-peaked twisted solitons versus linear gain coefficient $p\rm _{im}$ for three different $\kappa$. Profiles of three-peaked twisted solitons with single waveguide gain, which marked by plus sign in (a) are shown in (b), (c) and (d). $p\rm _{re}=5$, $\tau =0.05$, and $n=1$ in all cases.
For a smaller $p_{\textrm {im}}$, the field moduli of the solution is distributed in multiple lattice channels [Fig. 2(b)]. With increasing $p_{\textrm {im}}$, the main peaks of the soliton solutions are well confined in the three middle channels [Fig. 2(c)]. With the further increase of $p_{\textrm {im}}$, the peak in the middle of the soliton decreases, and the power is well localized in the two lattice channels adjacent to the central guide core [Fig. 2(d)].
The stability of both fundamental and three-peaked twisted solitons supported by the single-channel gain chirped lattice is an important issue. For small nonlinear loss coefficients, fundamental solitons are only stable in a narrow region near the lower cut-off values of $p_{\textrm {im}}$. As the nonlinear loss coefficient increases, the stable range of fundamental solitons increases obviously [Fig. 3(a)]. In addition, we find that the instability of two types of dissipative solitons can be suppressed by the increased chirp rate of the lattice [Figs. 3(b) and 3(c)]. However, the three-peaked twisted solitons are still unstable in their existence domains. This is because the power of field moduli of three-peaked twisted solitons localized outside the central lattice channel has no gain support to balance the nonlinear loss.
Fig. 3. Real part of instability growth rate $\delta _r$ for fundamental (a, b) and three-peaked twisted solitons (c) versus linear gain coefficient $p\rm _{im}$. $\tau =0.05$ in (a), $\kappa =0.4$ in (b) and $\kappa =0.7$ in (c). $p\rm _{re}=5$ and $n=1$ in all cases.
To verify the linear stability analysis of both fundamental and three-peaked twisted solitons we solved Eq. (1) numerically with input condition $\psi |_{z=0}=\phi$, where $\phi$ is the stationary soliton solution. When the maximum eigenvalue of the real part of the perturbation growth rate $\delta _r$ is equal to $0$, the fundamental soliton can always maintain its shape and phase during the propagation process [Fig. 4(a)]. Even if $\delta _r$ is a very small value, the propagation of fundamental solitons also shows unstable characteristics. The representative evolution is shown in Fig. 4(b). We can see that the field moduli of the soliton maintains its unique shape for a very long distance, but will produce small sidelobes after further distance. Interestingly, we find that when the value of $\delta _r$ is relatively large, the fundamental soliton can still maintain its field distribution for a certain distance, and then quickly disperse among chirped lattice channels. However, due to the presence of the central localized gain channel, the power corresponding to the field confined in the central channel is characterized by oscillation [Fig. 4(c)]. Also, we find that the three-peaked twisted solitons supported by single-channel gain are all unstable. It is clearly seen that the field outside the corresponding central channel disperses quickly, and the field in the central channel is quickly reconstructed and converted into a stable fundamental soliton [Fig. 4(d)].
Fig. 4. Stable, weak instable, and unstable propagation evolutions of fundamental solitons are shown in (a), (b) and (c), respectively. Unstable propagation of three-peaked twisted soliton is depicted in (d). $\kappa =0.4$, $p\rm _{im}=0.5$ in (a), $\kappa =0.4$, $p\rm _{im}=2.0$ in (b), $\kappa =0.4$, $p\rm _{im}=3.3$ in (c), $\kappa =0.7$, $p\rm _{im}=1.2$ in (d). $p\rm _{re}=5$, $\tau =0.05$, and $n=1$ in all cases.
Next, we study the properties of dissipative solitons supported by a three-channel gain chirped lattice ($n=3$). In such characteristic structures, we find that fundamental solitons are all unstable, while stable three-peaked twisted solitons can be found. Typical profiles of three-peaked twisted solitons are plotted in Figs. 5(a)–5(c). With the increase of the linear gain coefficient $p_{\textrm {im}}$, the field moduli of the three-peaked twisted soliton becomes more localized, and in the middle amplifying lattice channel, the amplitude of the imaginary part of the soliton increases while the amplitude of the real part decreases. Moreover, we find that for three-peaked twisted solitons, there exists a large stable region in the middle of their existence domain [Fig. 5(d)]. Intuitively, the stable region can be estimated by the transition point of the power curve slope, and it gradually increases with the growth of nonlinear coefficient.
Fig. 5. Profiles of three-peaked twisted solitons with triple waveguide gain, $\kappa =0.7$, $p\rm _{im}=0.5$ in (a), $\kappa =0.7$, $p\rm _{im}=2.0$ in (b), and $\kappa =0.7$, $p\rm _{im}=3.8$ in (c). (d) Power $U$ versus linear gain coefficient $p\rm _{im}$ for two different $\kappa$. $p\rm _{re}=5$, $\tau =0.05$, and $n=3$ in all cases.
We have also monitored the dynamic evolutions of three-peaked twisted solitons supported by the three-channel gain chirped lattice. The stable three-peaked twisted soliton will not be distorted even if it propagates for a very long distance [Fig. 6(a)]. However, for unstable three-peaked twisted ones, there exist three types of unstable evolution characteristics: (i) After the soliton solution enters the chirped lattices, the field distribution diffracts into multiple lattice channels, and the corresponding amplitude decreases. A reorganized field distribution with multipeaks satisfies the relationship ${\int _{-\infty }^{+\infty } (V_i|\phi |^2-\kappa |\phi |^4)dx}=0$ at $z\approx 400$, that is to say, the balanced linear gain and nonlinear loss are present again. Thus, an unstable three-peaked twisted soliton can be converted into a new multi-peaked twisted soliton [Fig. 6(b)]; (ii) For a large linear gain coefficient, the peak value of the field moduli is characterized by oscillation in the propagation process [Fig. 6(c)]; (iii) The input soliton solution disperses rapidly after entering the chirped lattice, and the field distribution is confusing because linear gain and nonlinear loss are not balanced [Fig. 6(d)].
Fig. 6. Stable (a) and unstable (b-d) propagation of three-peaked twisted solitons with triple waveguide gain. $\kappa =0.7$, $p\rm _{im}=2.0$ in (a), $\kappa =0.7$, $p\rm _{im}=0.5$ in (b), $\kappa =0.7$, $p\rm _{im}=3.8$ in (c), and $\kappa =0.4$, $p\rm _{im}=0.3$ in (d). $p\rm _{re}=5$, $\tau =0.05$, and $n=3$ in all cases.
Finally, we reveal that fundamental and three-peaked twisted solitons can be excited by an input Gaussian beam in the form of $\phi (x)=A\exp [-x^2/d^2]$, where $A$ and $d$ are the amplitude and width of the input beam, respectively. Obviously, for a large $d$, the power of outside of gain channel disperses quickly at the initial stage of propagation [Figs. 7(a) and 7(b)], while for a small $d$, the power in the central channel is coupled into the adjacent lattice channels [Figs. 7(c) and 7(d)]. Due to the balance between linear gain and nonlinear loss, we can find that robust fundamental [Figs. 7(a) and 7(c)] and three-peaked twisted [Figs. 7(b) and 7(d)] states are obtained in the corresponding characteristic amplifying chirped lattices.
Fig. 7. Excitation of fundamental (a, c) and three-peaked twisted solitons (b, d) by Gaussian beams with $A=2$, $\kappa =0.7$. $\omega =5\pi /4$ in (a, b) and $\omega =\pi /8$ in (c, d). (a, c) and (b, d) correspond to single and triple waveguide gain, respectively. $p\rm _{re}=5,~n=1$ in (a) and (c), $p\rm _{re}=5,~n=3$ in (b) and (d), $\tau =0.05$ in all cases.
4. Conclusion
In conclusion, we have investigated the existence and stability of both fundamental and three-peaked twisted solitons in a focusing Kerr medium with nonlinear loss and a transversal linear gain landscape consisting of a single or three amplifying channels. We have found that the width of the field moduli of the dissipative solitons becomes obviously narrow with the increase of linear gain coeffcient or the decrease of nonlinear loss coeffcient. Fundamental solitons supported by the single-channel amplified chirped lattice are stable near the lower cutoff. Stable three-peaked twisted solitons supported by the three-channel gain chirped lattices can also be found in large parts of their existence domain. Importantly, the stability regions for both fundamental and three-peaked twisted solitons increase with the growth of the nonlinear losses coefficient. And we predict that other stable multi-peaked twisted solitons can exist in other multi-channel amplified chirped lattices. Last but not least, we have disclosed that fundamental and three-peaked twisted solitons can be excited by Gaussian beams of arbitrary width in the corresponding characteristic gain lattice channel.
Funding
National Natural Science Foundation of China (11704339); Applied Basic Research Program of Shanxi Province (201901D211466); Natural Science Basic Research Plan in Shaanxi Province of China (2019JM-307, 2019JQ-089); Scientific and Technologial Innovation Programs of Higher Education Institutions in Shanxi (STIP) (2019L0896).
Disclosures
The authors declare no conflicts of interest.
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