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Abstract
A method for generating azimuthons in a nonlinear Kerr medium is presented. The findings indicate that azimuthons can be represented as a combination of two co-rotational azimuthally modulated degenerate linear eigenmodes, along with an additional shift in the propagation constant. Moreover, the conversions between azimuthons are showcased using longitudinal nonlinear modulation. The results reveal that, under the resonance condition, direct conversion between neighboring azimuthons is possible, leading to the emergence of Rabi oscillation. However, for non-neighboring azimuthons, direct conversion is less effective, requiring cascaded modulation for their conversion.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Self-trapped beams based on nonlinear Schrödinger equation have been extensively studied due to their ability to exhibit new physical phenomena and their potential applications in optical communication systems and photonic technologies [1–4]. Depending on the dimensionality of the system, various types of self-trapped beams, including bright solitons, dark solitons, and vortices, can exist [5–13]. Among these, azimuthons, which are a special type of azimuthally modulated vortex solitons, have received widespread attention since their initial proposal in the context of local nonlinear media [14]. Subsequently, theoretical studies in nonlocal nonlinear media and the first experimental observation were reported [15–23]. Investigations have shown that azimuthons exhibit a steady angular rotation during propagation. The rotational frequency and intensity profile of azimuthons can be uniquely determined by analyzing the eigenmodes of the linearized version of the nonlinear systems [19,21]. Moreover, azimuthons have also been reported in Bose-Einstein condensates [24–31].
Recently, Rabi oscillations of azimuthons in weakly nonlinear waveguides with weak longitudinally periodic modulation of the refractive index has been investigated [32]. In the field of optics, Rabi oscillations are manifested as resonant mode conversions. The longitudinally periodic modulation of the refractive index acts as an AC field, resulting in various phenomena such as parametric amplification, inhibition of light tunneling, and the realization of optical isolation [33–36]. Notably, there exists a mathematical analogy between the stimulated mode conversion process and Rabi flopping, which refers to periodic transitions between two stationary states of a quantum system driven by a resonant external field [37–40]. Rabi oscillations have been widely investigated in a variety of optical and photonic systems, including fibers [41,42], multimode waveguides [43–45], coupled waveguides [46], waveguide arrays [47,48], $\mathcal {PT}$ symmetry waveguides [49], nonlinear photonic waveguide [50], and two-dimensional (2D) modal structures [51,52]. Additionally, Rabi oscillations of topological edge states and resonant mode conversions in the fractional Schrödinger equation have been studied [53–55]. However, the conversion between azimuthons in nonlinear modulation has not been reported yet.
The objective of this study is to investigate the characterization of azimuthons in 2D super-Gaussian profile waveguides. Azimuthons in these waveguides can be represented as a combination of two co-rotational azimuthally modulated degenerate linear eigenmodes, with an additional shift in the propagation constant. Additionally, we aim to examine the conversion between azimuthons, which is achieved by introducing periodic longitudinal modulation of the Kerr-type nonlinearity.
The rest of this paper is organized as follows. In the next section, azimuthons induced by degenerate linear eigenmodes are constructed, which can be described as a superposition of two azimuthally modulated degenerate linear eigenmodes. Under nonlinear longitudinal modulation, the conversions between azimuthons with the same topological charge are demonstrated in Sec. III. The main results of the paper are summarized in Sec. IV
2. Model and azimuthons induced by degenerate linear eigenmodes
We consider the propagation of light in a multimode waveguide with longitudinal nonlinear modulation, which can be governed by the nonlinear Schrödinger equation
Firstly, we analyze the linear version of Eq. (1), which describes the linear eigenmodes supported by the waveguide potential $V(x,y)$. The linear eigenmodes with propagation constant $\beta$ are sought for in the polar form $\psi (x,y,z)=R(r)\exp (im\theta +i\beta z)$ with $R(r)$ satisfying the following linear eigenvalue equation
Fig. 1. (a) Values of the propagation constant $\beta _{mn_{r}}$. (b)-(e) Profiles of the radial wave functions $R_{mn_{r}}$ for $m=0,1,2,3$, respectively. Here, the system parameters are $V_{0}=200$ and $w=1$.
The real and imaginary parts of $R_{mn_{r}}\!(r)\exp (im\theta )$ provide two degenerate $(n_{r}+1)$th-order $2|m|$-pole linear eigenmodes with the propagation constant $\beta _{mn_{r}}$ as follows
Based on the degenerate linear eigenmodes provided in Eq. (3), the rotating azimuthon $\psi (r,\theta -\omega _{0}z)$ with a constant angular frequency $\omega _{0}$ in the Kerr nonlinear regime can be obtained by solving Eq. (1) with $\mu =0$ and initial input [32]
where $A$ is an amplitude factor, $B$ is a parameter related to the azimuthal modulation depth and $\omega _{0}$ is approximately given by [19,21] where $P=\iint r|\psi |^{2}drd\theta$ and $L_{z}=-i\iint r\psi ^{\ast } \frac {\partial \psi }{\partial \theta }drd\theta$ are the power and angular momentum, $P^{\prime }\!=\!\iint r|\frac {\partial \psi }{\partial \theta } |^{2}drd\theta$, $I=\iint r\psi ^{\ast }(\frac {1}{2}\nabla _{\perp }^{2}+V)\psi drd\theta$ and $I^{\prime }=i\iint r\frac {\partial \psi ^{\ast }}{\partial \theta }(\frac {1}{2}\nabla _{\perp }^{2}+V)\psi drd\theta$ are related to the diffraction mechanism and waveguide structure, whereas $N=\gamma \iint r|\psi |^{4}drd\theta$ and $N^{\prime }=i\gamma \iint r\frac {\partial \psi ^{\ast } }{\partial \theta }|\psi |^{2}\psi drd\theta$ are only involved in the nonlinearity.Figure 2 presents the numerical evolution result of the initial input (4) with $m=1$, $n_{r}=0$ and $A=1$, $B=0.4$. The initial input is a combination of two degenerate first-order dipole linear eigenmodes with azimuthal modulation. It is observed that, under the influence of Kerr nonlinearity, the evolution of the input exhibits a counterclockwise periodic rotation with an angular frequency $\omega _{0}\approx 0.2699$. This value is approximately equal to the theoretical value of $\omega _{0}=0.2562$, as given by Eq. (5). Thus, a first-order dipole azimuthon is fomed.
Fig. 2. Distributions of the first-order dipole azimuthon at the initial position, one-quarter period position, half period position, three-quarter period position and one period position, i.e., $z=0$, $5.82$, $11.64$, $17.46$, and $23.28$, respectively. Here, $m=1$, $n_{r}=0$, $A=1$, $B=0.4$, and the system parameters are $\gamma =1$, $\mu =0$, $V_{0} =200$, and $w=1$.
To gain a better understanding of the azimuthon induced by the initial input $\psi _{0}(r,\theta )$, let us express it as a superposition of all linear eigenmodes (3)
Fig. 3. Evolution of the weight functions in the first-order dipole azimuthon shown in Fig. 2. Here, the parameters are the same as in Fig. 2.
In general, in the absence of the nonlinear longitudinal modulation, i.e., $\mu =0$, the $(n_{r}+1)$th-order $2|m|$-pole azimuthon can be determined by two degenerate $(n_{r}+1)$th-order $2|m|$-pole linear eigenmodes. Therefore, the azimuthon induced by the initial input (3) can be approximately expressed as a superposition of two degenerate eigenmodes $u_{mn_{r}}(r,\theta )$ and $v_{mn_{r}}(r,\theta )$ in Eq. (3) as follows
Thus, Eq. (7) can be rewritten as
It should be noted that there are two special cases, which are $L_{z}=0$ and $L_{z}=mP$. In the former case, where $b_{1}=b_{2}=\sqrt {P}$ and $\omega _{0,mn_{r}}=0$, Eq. (11) takes the form of $\sqrt {P}u_{mn_{r}}(r,\theta )e^{i(\beta _{mn_{r}}+\beta _{mn_{r}}^{\prime })z}$. In the latter case, where $b_{1}=\sqrt {2P}$ and $b_{2}=0$, Eq. (11) can be written as $\psi (r,\theta,z)=\sqrt {P/2}R_{mn_{r}}\!(r)e^{im\theta +i(\beta _{mn_{r}}+\beta _{mn_{r}}^{\prime }-m\omega _{0,mn_{r}})z}$, which reduces to non-rotational stationary vortex. These two situations cannot form azimuthons.
Furthermore, comparing Eq. (4) and Eq. (11) with $z=0$, we have $A=(b_{1}+b_{2})/2$ and $B=(b_{1}-b_{2})/(b_{1}+b_{2})$. From which, it can be found that $P=A^{2}(1+B^{2})$ and $L_{z}=2mA^{2}B$. These imply that $0\leq B\leq 1$, where $B=0$ and $B=1$ correspond to the two especial cases of $L_{z}=0$ and $mP$, respectively. In this case, the rotational angular frequency and the propagation constant shift can be written as $\omega _{0,mn_{r}}=\gamma A^{2}BQ_{mn_{r}}/(2m)$ and $\beta _{mn_{r} }^{\prime }=3\gamma A^{2}(1+B^{2})Q_{mn_{r}}/4$, which increase with increasing of the parameters $A$ and $B$.
Figure 4 shows the numerical and analytical evolution plots of the second-order quadrupole azimuthon ($m=2$, $n_{r}=1$) as $A=1$ and $B=0.4$. One can see that the results from the numerical simulation are in agreement with those obtained through the analytical formula (11). It is worth noting that in Figs. 4(a) and 4(b), the intensity distributions are depicted at the initial position, one-third period position, two-thirds period position and one period position, respectively. Due to slight differences in rotational frequencies, which result in variations in the rotation period (see Fig. 4(c)), there are differences in the positions where their intensity distributions are displayed. In conclusion, the azimuthon can be described analytically as a combination of two co-rotational degenerate linear eigenmodes, incorporating an additional shift in the propagation constant. The linear version of the system determines the profile of the azimuthon and the nonlinearity results in the rotational angular frequency and the shift in the propagation constant.
Fig. 4. Intensity distributions of the second-order quadrupole azimuthon ($m=2$, $n_{r}=1$) at the initial position, one-third period position, two-thirds period position, and one period position. (a) Numerical result; (b) Analytical result, and (c) Evolutions of the corresponding weights $|c_{1}(z)|^{2}$ and $|c_{2}(z)|^{2}$, where the solid curves are the numerical result and the dashed curves are the analytical result given by Eq. (10). Here, the other parameters are the same as in Fig. 2.
The stability of the azimuthons can be addressed by numerically solving Eq. (1) with $\mu =0$ and the perturbed initial input. Figure 5 presents an example to demonstrate the evolution dynamics of a second-order quadrupole azimuthon perturbed by $20{\%}$ of noise. One can see that the characteristics of the azimuthon are largely retained, with the exception of some changes in intensity during the propagation, which can be recovered, as shown in Fig. 5(a).
Fig. 5. Evolution of a perturbed second-order quadrupole azimuthon, where a $20{\%}$ noise is introduced into the initial input (4). (a) Intensity distributions at the distinct positions, and (b) Evolutions of the respective weights $|c_{1}(z)|^{2}$ and $|c_{2}(z)|^{2}$. Here, the other parameters are the same as in Fig. 4.
3. Conversion between azimuthons under nonlinear modulation
In the presence of nonlinear longitudinal modulation, i.e., $\mu \neq 0$, the situation is different. To illustrate this difference, we present in Fig. 6 the evolution of the weights, $\rho _{kl_{r}} (z)=|c_{1,kl_{r}}(z)|^{2}+|c_{2,kl_{r}}(z)|^{2}$, for each pair of degenerate linear eigenmodes during the propagation of the initial input (4) for non-resonant and resonant cases. Here, $c_{1,kl_{r}}(z)$ and $c_{2,kl_{r}}(z)$ are calculated using the expressions given in Eq. (6). Comparing with Fig. 3, one find that under the nonlinear longitudinal modulation, in addition to degenerate eigenmodes present in the initial input, neighboring degenerate eigenmodes also participate in its evolution. Especially, when satisfying the resonance condition, the conversion between adjacent degenerate eigenmodes with the same topological charge is achieved, as shown in Fig. 6(b).
Fig. 6. Evolution of the weights for each pair of degenerate linear eigenmodes during the propagation of the initial input (4) under nonlinear longitudinal modulation with different frequency $\Omega$. (a) Non-resonant case, $\Omega =26.0059$, and (b) Resonant case, $\Omega =\beta _{20}-\beta _{21}=26.2309$, where the solid curves are the results obtained from numerical calculations and the dashed curves are the results from Eq. (13). Here, $m=2$, $n_{r}=0$, $A=0.89443$, $B=0.5$, $\mu =0.9$, and the other parameters are the same as in Fig. 2.
In this case, we consider a superposition of two sets of degenerate linear eigenmodes with the same topological charge and different node number as follows
Firstly, it can be shown that the total power and angular momentum of these azimuthons, i.e., $P=|c_{1}(z)|^{2}+|c_{2}(z)|^{2}+|c_{1}^{\prime } (z)|^{2}+|c_{2}^{\prime }(z)|^{2}$ and $L_{z}=im(c_{1}c_{2}^{\ast }-c_{2} c_{1}^{\ast }+c_{1}^{\prime }c_{2}^{\prime \ast }-c_{2}^{\prime }c_{1}^{\prime \ast })$, remain conversed quantities. Next, under the resonance condition, i.e., $\Omega =\beta _{mn_{r}}-\beta _{mn_{r}^{\prime }}$, and employing the rotating wave approximation, we can derive the following system of equations
Also, from Fig. 6(b), it can be seen that for the chosen parameters ($A=0.89443$, $B=0.5$ and $m=2$, $n_{r}=0$), $\rho (0)=A^{2} (1+B^{2})=1$ and $\rho ^{\prime }(0)=0$ according to Eq. (4), which specifies the initial first-order quadrupole azimuthon. With increasing of the propagation distance, $\rho (z)$ decreases and $\rho ^{\prime }(z)$ increases. At $z=z_{1}$, $\rho (z)$ reaches its minimum value $0.0010476$ and $\rho (z)$ reaches its maximum value $0.9989$, forming the second-order quadrupole azimuthon. This completes the conversion between the first-order and second-order quadrupole azimuthons with an efficiency of $99.89{\%}$. Subsequently, $\rho (z)$ increases while $\rho ^{\prime }(z)$ decreases until $z=z_{2}$, at which $\rho (z_{2})\approx 0.9999$ and $\rho ^{\prime } (z_{2})\approx 10^{-5}$, indicating the reappearance of the first-order quadrupole azimuthon. One cycle of Rabi oscillation is completed with an efficiency of $99.99{\%}$. The results indicate that under the resonance condition, the first-order quadrupole azimuthon can be converted to the second-order quadrupole azimuthon through the nonlinear longitudinal modulation, resulting in Rabi oscillation.
The intensity distributions at the initial position, conversion distance, and one Rabi oscillation cycle in the conversion process between the first-order and second-order quadrupole azimuthons are illustrated in Fig. 7(a). It can be observed that the initial first-order quadrupole azimuthon evolves into a second-order quadrupole azimuthon at $z=z_{1}$, and then reverts back to its original state at $z=z_{2}$. In addition, the conversions of the dipole, hexapole, and octapole azimuthons are shown in Figs. 7(b)–7(d), which depict their intensity distributions at the initial position, conversion distance, and one Rabi oscillation cycle, respectively.
Fig. 7. Intensity distributions at the initial position, conversion distance, and one Rabi oscillation cycle in the conversion (a) from first-order quadrupole azimuthon to second-order one; (b) from the second-order dipole azimuthon to third-order one; (c) from first-order hexapole azimuthon to second-order one, and (d) from the second-order octoxapole azimuthon to third-order one. Here $A=0.89443$, $B=0.5$, $\gamma =1$ and $\mu =0.9$, and the other parameters are the same as in Fig. 2 .
So far, we have analyzed and demonstrated the mode conversion between the azimuthons, which involves an important physical quantity: conversion distance $z_{L}$. Figure 8 illustrates the dependence of the conversion distance $z_{L}$ on the parameters $A$ and $B$ by employing Eq. (13). It is evident that increasing the values of $A$ and $B$ leads to a decrease in the conversion distance $z_{L}$. This is because higher values of $A$ and $B$ result in an increased rotational angular frequency of the azimuthons, facilitating faster conversion.
Fig. 8. Dependence of conversion distance $z_{L}$ on the parameters $A$ and $B$ for given $\mu$ and $\gamma$. (a) $m=2$, $n_{r}=0$; (b) $m=1$, $n_{r}=1$; (c) $m=3$, $n_{r}=0$, and (d) $m=4$, $n_{r}=1$. Here, $\gamma =1$, $\mu =0.9$, and the other parameters are the same as in Fig. 2.
Furthermore, cascade conversions between azimuthons can be achieved. Figure 9 provides an example demonstrating the cascade conversion from a first-order quadrupole azimuthon to a second-order quadrupole azimuthon and a third-order quadrupole azimuthon. Initially, by setting $\mu =0.89$ and under the resonance condition $\Omega =\beta _{20}-\beta _{21}$, an initial first-order quadrupole azimuthon evolves into a second-order quadrupole azimuthon at $z=z_{1}$. Subsequently, by adjusting $\mu =0.85$ and the resonance condition $\Omega =\beta _{21}-\beta _{22}$, the second-order quadrupole azimuthon converts to a third-order quadrupole azimuthon at $z=z_{2}$. Thus, the cascade conversion from the first-order quadrupole azimuthon to the third-order quadrupole azimuthon is achieved. Continuing to maintain these parameters, the third-order quadrupole azimuthon reverts to the second-order quadrupole azimuthon at $z=z_{3}$. Finally, by restoring the original parameters, $\mu =0.89$ and $\Omega =\beta _{20}-\beta _{21}$, the second-order quadrupole azimuthon returns to the initial first-order quadrupole azimuthon at $z=z_{4}$. Consequently, a cycle of Rabi oscillation from the first-order quadrupole azimuthon to the third-order quadrupole azimuthon is completed. After this cycle, a scenario without nonlinear modulation is set, i.e., $\mu =0$ as $z\,>\,z_{4}$, which shows the evolution of the first-order quadrupole azimuthon, as depicted in Fig. 9(b).
Fig. 9. Intensity distributions at the representative distances in the cascade conversion from first-order quadrupole azimuthon to second-order and third-order quadrupole azimuthon. Here $A=0.92848$, $B=0.4$, $\gamma =1$, and $\mu =0.89$, $\Omega =\beta _{20}-\beta _{21}=26.2309$ as $0\,<\,z\,<\,z_{1}$ and $z_{3}\,<\,z\,<\,z_{4}$; $\mu =0.85$, $\Omega =\beta _{21}-\beta _{22}=33.4565$ as $z_{1}\,<\,z\,<\,z_{3}$; and $\mu =0$ as $z>z_{4}$.
4. Conclusion
In conclusion, we analyzed and demonstrated the azimuthons induced by degenerate linear eigenmodes in nonlinear Kerr medium. Our results have shown that these azimuthons can be described analytically as a combination of two co-rotational degenerate linear eigenmodes, incorporating an additional shift in the propagation constant. The analytical values for the propagation constant shift and the rotational angular frequency were also determined. Furthermore, the conversions between azimuthons were demonstrated under the longitudinal nonlinear modulation, and the dependence of the conversion distance on the initial parameters was studied. It has been found that under the resonance condition, direct conversion between adjacent azimuthons is achievable, resulting in the formation of Rabi oscillation. However, for non-adjacent azimuthons, direct conversion is less efficient, necessitating cascading modulation for their conversion. This research provides a potential new avenue for the control of light in propagation.
Funding
National Natural Science Foundation of China (11705108); 111 Project (D18001); Hundred Talent Program of the Shanxi Province (2018).
Disclosures
The authors declare no conflicts of interest.
Data availability
Date that support the findings of this study are available from the corresponding author upon reasonable request.
References
1. Y. S. Kivshar and G. P. Agrawal, Optical solitons: from fibers to photonic crystals (Academic, 2003).
2. M. Segev and G. Stegeman, “Self-Trapping of Optical Beams: Spatial Solitons,” Phys. Today 51(8), 42–48 (1998). [CrossRef]
3. G. I. Stegeman and M. Segev, “Optical spatial solitons and their interactions: Universality and diversity,” Science 286(5444), 1518–1523 (1999). [CrossRef]
4. Z. Chen, M. Segev, and D. N. Christodoulides, “Optical spatial solitons: historical overview and recent advances,” Rep. Prog. Phys. 75(8), 086401 (2012). [CrossRef]
5. M. Segev, B. Crosignani, A. Yariv, and B. Fischer, “Spatial solitons in photorefractive media,” Phys. Rev. Lett. 68(7), 923–926 (1992). [CrossRef]
6. G. C. Duree, J. L. Shultz, G. J. Salamo, M. Segev, A. Yariv, B. Crosignani, P. Di Porto, E. J. Sharp, and R. R. Neurgaonkar, “Observation of self-trapping of an optical beam due to the photorefractive effect,” Phys. Rev. Lett. 71(4), 533–536 (1993). [CrossRef]
7. A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. II. Normal dispersion,” Appl. Phys. Lett. 23(4), 171–172 (1973). [CrossRef]
8. G. A. Swartzlander, D. R. Andersen, J. J. Regan, H. Yin, and A. E. Kaplan, “Spatial dark-soliton stripes and grids in self-defocusing materials,” Phys. Rev. Lett. 66(12), 1583–1586 (1991). [CrossRef]
9. Z. Chen, M. Segev, D. W. Wilson, R. E. Muller, and P. D. Maker, “Self-trapping of an optical vortex by use of the bulk photovoltaic effect,” Phys. Rev. Lett. 78(15), 2948–2951 (1997). [CrossRef]
10. Z. Chen, M. feng Shih, M. Segev, D. W. Wilson, R. E. Muller, and P. D. Maker, “Steady-state vortex-screening solitons formed in biased photorefractive media,” Opt. Lett. 22(23), 1751–1753 (1997). [CrossRef]
11. M. Soskin and M. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001). [CrossRef]
12. A. S. Desyatnikov, Y. S. Kivshar, and L. Torner, “Optical vortices and vortex solitons,” Prog. Opt. 47, 291–391 (2005). [CrossRef]
13. M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009). [CrossRef]
14. A. S. Desyatnikov, A. A. Sukhorukov, and Y. S. Kivshar, “Azimuthons: Spatially modulated vortex solitons,” Phys. Rev. Lett. 95(20), 203904 (2005). [CrossRef]
15. S. Lopez-Aguayo, A. S. Desyatnikov, Y. S. Kivshar, S. Skupin, W. Krolikowski, and O. Bang, “Stable rotating dipole solitons in nonlocal optical media,” Opt. Lett. 31(8), 1100–1102 (2006). [CrossRef]
16. S. Lopez-Aguayo, A. S. Desyatnikov, and Y. S. Kivshar, “Azimuthons in nonlocal nonlinear media,” Opt. Express 14(17), 7903–7908 (2006). [CrossRef]
17. S. Skupin, M. Saffman, and W. Królikowski, “Nonlocal stabilization of nonlinear beams in a self-focusing atomic vapor,” Phys. Rev. Lett. 98(26), 263902 (2007). [CrossRef]
18. D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Spiraling multivortex solitons in nonlocal nonlinear media,” Opt. Lett. 33(2), 198–200 (2008). [CrossRef]
19. S. Skupin, M. Grech, and W. Królikowski, “Rotating soliton solutions in nonlocal nonlinear media,” Opt. Express 16(12), 9118–9131 (2008). [CrossRef]
20. A. Minovich, D. N. Neshev, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Observation of optical azimuthons,” Opt. Express 17(26), 23610–23616 (2009). [CrossRef]
21. Y. Zhang, S. Skupin, F. Maucher, A. G. Pour, K. Lu, and W. Królikowski, “Azimuthons in weakly nonlinear waveguides of different symmetries,” Opt. Express 18(26), 27846–27857 (2010). [CrossRef]
22. C. Ruiz-Jiménez, H. Leblond, M. A. Porras, and B. A. Malomed, “Rotating azimuthons in dissipative Kerr media excited by superpositions of Bessel beams,” Phys. Rev. A 102(6), 063502 (2020). [CrossRef]
23. X. Huang, W. Tan, T. Jiang, S. Nan, Y. Bai, and X. Fu, “Propagation dynamics of modulated vortex beams in nonlocal nonlinear media,” Opt. Commun. 527, 128970 (2023). [CrossRef]
24. V. M. Lashkin, “Two-dimensional multisolitons and azimuthons in Bose-Einstein condensates,” Phys. Rev. A 77(2), 025602 (2008). [CrossRef]
25. V. M. Lashkin, “Stable three-dimensional spatially modulated vortex solitons in Bose-Einstein condensates,” Phys. Rev. A 78(3), 033603 (2008). [CrossRef]
26. V. M. Lashkin, E. A. Ostrovskaya, A. S. Desyatnikov, and Y. S. Kivshar, “Vector azimuthons in two-component Bose-Einstein condensates,” Phys. Rev. A 80(1), 013615 (2009). [CrossRef]
27. F. Maucher, S. Skupin, M. Shen, and W. Krolikowski, “Rotating three-dimensional solitons in Bose-Einstein condensates with gravitylike attractive nonlocal interaction,” Phys. Rev. A 81(6), 063617 (2010). [CrossRef]
28. V. M. Lashkin, A. S. Desyatnikov, E. A. Ostrovskaya, and Y. S. Kivshar, “Azimuthal vortex clusters in Bose-Einstein condensates,” Phys. Rev. A 85(1), 013620 (2012). [CrossRef]
29. G. Li, “Azimuthons and pattern formation in annularly confined exciton-polariton Bose-Einstein condensates,” Phys. Rev. A 93(1), 013837 (2016). [CrossRef]
30. A. C. White, Y. Zhang, and T. Busch, “Odd-petal-number states and persistent flows in spin-orbit-coupled Bose-Einstein condensates,” Phys. Rev. A 95(4), 041604 (2017). [CrossRef]
31. Y. V. Kartashov and D. A. Zezyulin, “Stable multiring and rotating solitons in two-dimensional spin-orbit-coupled Bose-Einstein condensates with a radially periodic potential,” Phys. Rev. Lett. 122(12), 123201 (2019). [CrossRef]
32. K. Jin, Y. Li, F. Li, M. R. Belic, Y. Zhang, and Y. Zhang, “Rabi oscillations of azimuthons in weakly nonlinear waveguides,” Adv. Photonics 2(04), 1 (2020). [CrossRef]
33. Y. V. Kartashov, L. Torner, and V. A. Vysloukh, “Parametric amplification of soliton steering in optical lattices,” Opt. Lett. 29(10), 1102–1104 (2004). [CrossRef]
34. Y. V. Kartashov, A. Szameit, V. A. Vysloukh, and L. Torner, “Light tunneling inhibition and anisotropic diffraction engineering in two-dimensional waveguide arrays,” Opt. Lett. 34(19), 2906–2908 (2009). [CrossRef]
35. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Enhancement and inhibition of light tunneling mediated by resonant mode conversion,” Opt. Lett. 39(4), 933–936 (2014). [CrossRef]
36. Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photonics 3(2), 91–94 (2009). [CrossRef]
37. S. Hughes, “Breakdown of the area theorem: Carrier-wave Rabi flopping of femtosecond optical pulses,” Phys. Rev. Lett. 81(16), 3363–3366 (1998). [CrossRef]
38. R. Robinett, “Quantum wave packet revivals,” Phys. Rep. 392(1-2), 1–119 (2004). [CrossRef]
39. I. L. Garanovich, S. Longhi, A. A. Sukhorukov, and Y. S. Kivshar, “Light propagation and localization in modulated photonic lattices and waveguides,” Phys. Rep. 518(1-2), 1–79 (2012). [CrossRef]
40. S. Longhi, “Quantum-optical analogies using photonic structures,” Laser Photonics Rev. 3(3), 243–261 (2009). [CrossRef]
41. K. O. Hill, B. Malo, K. Vineberg, F. Bilodeau, D. C. Johnson, and I. Skinner, “Efficient mode conversion in telecommunication fiber using externally written gratings,” in Optical Society of America Annual Meeting, (Optica Publishing Group, 1990), p. TuRR2.
42. K. S. Lee and T. Erdogan, “Fiber mode coupling in transmissive and reflective tilted fiber gratings,” Appl. Opt. 39(9), 1394–1404 (2000). [CrossRef]
43. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Resonant mode oscillations in modulated waveguiding structures,” Phys. Rev. Lett. 99(23), 233903 (2007). [CrossRef]
44. V. A. Vysloukh, Y. V. Kartashov, and K. Staliunas, “Efficient mode conversion in guiding structures with longitudinal modulation of nonlinearity,” Opt. Lett. 40(20), 4631–4634 (2015). [CrossRef]
45. X. Zhang, F. Ye, Y. V. Kartashov, and X. Chen, “Rabi oscillations and stimulated mode conversion on the subwavelength scale,” Opt. Express 23(5), 6731–6737 (2015). [CrossRef]
46. M. Ornigotti, G. D. Valle, T. T. Fernandez, A. Coppa, V. Foglietti, P. Laporta, and S. Longhi, “Visualization of two-photon Rabi oscillations in evanescently coupled optical waveguides,” J. Phys. B: At., Mol. Opt. Phys. 41(8), 085402 (2008). [CrossRef]
47. K. G. Makris, D. N. Christodoulides, O. Peleg, M. Segev, and D. Kip, “Optical transitions and Rabi oscillations in waveguide arrays,” Opt. Express 16(14), 10309–10314 (2008). [CrossRef]
48. K. Shandarova, C. E. Rüter, D. Kip, K. G. Makris, D. N. Christodoulides, O. Peleg, and M. Segev, “Experimental observation of Rabi oscillations in photonic lattices,” Phys. Rev. Lett. 102(12), 123905 (2009). [CrossRef]
49. V. A. Vysloukh and Y. V. Kartashov, “Resonant mode conversion in the waveguides with unbroken and broken $\mathcal {PT}$ symmetry,” Opt. Lett. 39(20), 5933–5936 (2014). [CrossRef]
50. J. Talukdar and D. Blume, “Undamped Rabi oscillations due to polaron-emitter hybrid states in a nonlinear photonic waveguide coupled to emitters,” Phys. Rev. A 106(1), 013722 (2022). [CrossRef]
51. G. K. L. Wong, M. S. Kang, H. W. Lee, F. Biancalana, C. Conti, T. Weiss, and P. S. J. Russell, “Excitation of orbital angular momentum resonances in helically twisted photonic crystal fiber,” Science 337(6093), 446–449 (2012). [CrossRef]
52. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Dynamics of topological light states in spiraling structures,” Opt. Lett. 38(17), 3414–3417 (2013). [CrossRef]
53. Y. Zhang, R. Wang, H. Zhong, J. Zhang, M. R. Belić, and Y. Zhang, “Resonant mode conversions and Rabi oscillations in a fractional Schrödinger equation,” Opt. Express 25(26), 32401–32410 (2017). [CrossRef]
54. Y. Zhang, Y. V. Kartashov, Y. Zhang, L. Torner, and D. V. Skryabin, “Resonant edge-state switching in polariton topological insulators,” Laser Photonics Rev. 12(8), 1700348 (2018). [CrossRef]
55. H. Zhong, Y. V. Kartashov, Y. Zhang, D. Song, Y. Zhang, F. Li, and Z. Chen, “Rabi-like oscillation of photonic topological valley Hall edge states,” Opt. Lett. 44(13), 3342–3345 (2019). [CrossRef]
