Rabi oscillations of azimuthons in weaklynonlinear waveguidesKaichao Jin,a,b Yongdong Li,a Feng Li,a Milivoj R. Belic,c Yanpeng Zhang,a and Yiqi Zhanga,b,*aXi’an Jiaotong University, Key Laboratory for Physical Electronics and Devices of the Ministry of Education, Xi’an, ChinabGuangdong Xi’an Jiaotong University Academy, Foshan, ChinacTexas A&M University at Qatar, Science Program, Doha, Qatar

Abstract. Rabi oscillation, an interband oscillation, describes periodic motion between two states that belongto different energy levels, in the presence of an oscillatory driving field. In photonics, Rabi oscillations can bemimicked by applying a weak longitudinal periodic modulation to the refractive index. However, the Rabioscillations of nonlinear states have yet to be introduced. We report the Rabi oscillations of azimuthons—spatially modulated vortex solitons—in weakly nonlinear waveguides with different symmetries. The periodof the Rabi oscillations can be determined by applying the coupled mode theory, which largely dependson the modulation strength. Whether the Rabi oscillations between two states can be obtained or not isdetermined by the spatial symmetry of the azimuthons and the modulating potential. Our results not onlydeepen the understanding of the Rabi oscillation phenomena, but also provide a new avenue in the studyof pattern formation and spatial field manipulation in nonlinear optical systems.

Keywords: Rabi oscillations; azimuthons; weak nonlinearity; longitudinally periodic modulation.

Received May 29, 2020; revised manuscript received Jun. 21, 2020; accepted for publication Jul. 6, 2020; published onlineAug. 5, 2020.

© The Authors. Published by SPIE and CLP under a Creative Commons Attribution 4.0 Unported License. Distribution orreproduction of this work in whole or in part requires full attribution of the original publication, including its DOI.

[DOI: 10.1117/1.AP.2.4.046002]

1 IntroductionRabi oscillations were introduced in quantum mechanics,1 butby now are widely investigated in a variety of optical and pho-tonic systems that include fibers,2,3 multimode waveguides,4–6

coupled waveguides,7 waveguide arrays,8–10 and two-dimensionalmodal structures.11,12 Recently, Rabi oscillations of topologicaledge states13,14 and modes in the fractional Schrödinger equation15

were also reported. Rabi oscillations are interband oscillationsthat require an AC field to be applied as an external periodicpotential. In optics, the longitudinal periodic modulation ofthe refractive index plays the role of an AC field in temporalquantum systems, and Rabi oscillations are indicated by theresonant mode conversion. As far as we know, the investigationof optical Rabi oscillations thus far has been limited to the linearregime only, and the Rabi oscillation in nonlinear systems isstill an open problem that needs to be explored.

The aim of this work is to investigate Rabi oscillations ofazimuthons in weakly nonlinear waveguides that is accomplished

by applying a weak longitudinally modulated periodic potential.Azimuthons are a special type of spatial soliton; they are azi-muthally modulated vortex beams that exhibit steady rotationupon propagation.16 Generally, azimuthons, especially the oneswith higher-order angular momentum structures, are unstablein media with local Kerr or saturable nonlinearities. To over-come the instability drawback, a nonlocal nonlinearity is intro-duced, and recently published reports demonstrate that the stablepropagation of azimuthons can indeed be obtained.17–19 In addi-tion, it was also reported that the spin-orbit-coupled Bose–Einstein condensates can support stable azimuthons as well.20

However, the treatment of nonlocal nonlinearity and spin-orbit-coupled Bose–Einstein condensates is challenging in both theo-retical modeling and experimental demonstration. Nevertheless, ithas been confirmed that weakly nonlinear waveguides21 representan ideal platform for the investigation of stable azimuthons,22,23

even with higher-order modal structures.Following this path of inquiry, we first investigate Rabi

oscillations of azimuthons in a circular waveguide and then ina square waveguide. Since in this nonlinear three-dimensionalwave propagation problem no analytical solutions are known,*Address all correspondence to Yiqi Zhang, E-mail: zhangyiqi@mail.xjtu.edu.cn

Research Article

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the mode of inquiry will necessarily be predominantly numericalwith some theoretical background. In the circular waveguide,the azimuthons will exhibit Rabi oscillation while rotatingduring propagation. In the square waveguide, the behavior ofthe azimuthons is different in two aspects:15 (i) azimuthons willrotate only if the corresponding Hamiltonian (energy) is biggerthan a certain threshold value and (ii) azimuthons will bedeformed during propagation. Hence, in this work, we chooseazimuthons with large enough energies to avoid wobblingmotions in the square waveguide during propagation.

2 Results

2.1 Theoretical Analysis

The propagation of a light beam in a photonic waveguide can bedescribed by the following Schrödinger-like paraxial waveequation:

i∂∂ZΨþ 1

2k0

� ∂2

∂X2þ ∂2

∂Y2

�Ψþ k0

n2nb

jΨj2Ψ

þ k0nðX; YÞ − nb

nb½1þ μ cosðδZÞ�Ψ

¼ 0; (1)

where ΨðX; Y; ZÞ is the complex amplitude of the light beam,the quantities ðX; YÞ and Z are the transverse and longitudinalcoordinates, and k0 ¼ 2πnb∕λ0 is the wavenumber in themedium, where λ0 is the wavelength in the vacuum. Other quan-tities in Eq. (1): μ ≪ 1 is the longitudinal modulation strength,δ is the longitudinal modulation frequency, n2 is the nonlinearKerr coefficient, nðX; YÞ is the linear refractive index distribu-tion, and nb is the ambient index. In Eq. (1), the refractive indexchange includes two parts, which are jn − nbj (the linear part)and n2jΨj2 (the nonlinear part). We would like to note thatweakly nonlinear waveguides demand not only the linear andnonlinear refractive index changes to be small in comparisonwith nb, but also the nonlinear part to be much smaller thanthe linear part. According to the relations x ¼ X∕r0, y ¼ Y∕r0,z ¼ Z∕ðk0r20Þ, d ¼ k0r20δ, and σ ¼ sgnðn2Þ, with r0 beingdetermined by the real beam width, Eq. (1) can be cast intoits dimensionless version:

i∂∂zψ þ 1

2

� ∂2

∂x2 þ∂2

∂y2�ψ þ σjψ j2ψ þ V½1þ μ cosðdzÞ�ψ ¼ 0;

(2)

where ψ¼k0r0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijn2j∕nb

pΨ and Vðx;yÞ¼k20r

20½nðx;yÞ−nb�∕nb.

Here we will consider propagation in a deep circular potentialVðx; yÞ ¼ V0 exp½−ðx2 þ y2Þ5∕w10�, where w characterizes thepotential width and V0 is the potential depth. Note that it ispossible to have a weak nonlinearity even when the potentialis deep. Thus the potential can be deep, but the potential modu-lation can still be shallow. The parameter σ ¼ 1 ðσ ¼ −1Þ cor-responds to the focusing (defocusing) nonlinearity. In this work,we consider the focusing nonlinearity, i.e., we take σ ¼ 1.

Different materials are used to produce waveguides, andsilica is one of the most popular, with the typical parametersnb ¼ 1.4, jn − nbj ≤ 9 × 10−3, and n2 ¼ 3 × 10−16 cm2∕Wfor light beams with the wavelengths ranging from the visible

to the near-infrared. Without loss of generality, we chooseλ0 ¼ 800 nm in this work. Therefore, if we choose V0 ¼ 500,the value of r0 ≈ 25.0 μm is obtained, according to the relationadopted in Eq. (2). Indeed, such a value is reasonable for amultimode fiber.24 According to the wavelength and r0, onelearns that the diffraction length is ∼7 mm. Considering thatthe group velocity dispersion coefficient is ∼35 fs2∕mm at thewavelength of λ0 ¼ 800 nm, the dispersion length is on theorder of kilometers for a picosecond light beam, which ismuch longer than the propagation distance taken in this work.As a result, it is safe to neglect temporal effects.

To start with, we consider the modes supported by the deeppotential alone; therefore, the nonlinear term and the longi-tudinal modulation in Eq. (2) are initially neglected. The corre-sponding solution of the reduced linear Eq. (2) can be writtenas ψðx; y; zÞ ¼ uðx; yÞ expðiβzÞ, where uðx; yÞ is the stationaryprofile of the mode and β is the propagation constant. Pluggingthis solution into the reduced Eq. (2), one obtains

βu ¼ 1

2

� ∂2

∂x2 þ∂2

∂y2�uþ Vu; (3)

which is the linear steady-state eigenvalue problem of Eq. (2)when σ and μ are set to zero. Equation (3) can be solved byutilizing the plane-wave expansion method, and the eigenstatessupported by the deep potential Vðx; yÞ can be easily obtained.In Fig. 1, the first-order as well as higher-order basic modes,degenerate dipole modes, degenerate quadrupole modes, degen-erate hexapole modes, and degenerate octopole modes that canexist in the potential are displayed. Here the term “degenerate”means that all modes feature the same propagation constants, inthe usual optical meaning. These linear modes will be used asthe input modes of the more general nonlinear and modulatedmodes of the complete Eq. (3).

Therefore, to seek approximate azimuthons in weakly non-linear waveguides, one takes the degenerate modes and makesa superposition of them as the initial wave

ψðx; yÞ ¼ A½u1ðx; yÞ þ iBu2ðx; yÞ� expðiβzÞ; (4)

where A is an amplitude factor, 1 − B is the azimuthal modu-lation depth, and u1;2ðx; yÞ are the degenerate linear modes(see examples in Fig. 1). Thus we set the transverse profile ofthe azimuthon at the initial place as

Uðx; y; z ¼ 0Þ ¼ A½u1ðx; yÞ þ iBu2ðx; yÞ� (5)

and numerically propagate it to obtain an output mode at an ar-bitrary z. We should note that the inputs depicted by Eq. (5) donot rotate in a linear medium, since the modes are degenerate.Rotation appears only when the nonlinearity is added to themodel.

In Fig. 2, we display such approximate azimuthons withA ¼ 0.4 and B ¼ 0.5. One finds that the phase of azimuthonsis nontrivial, displaying angular momentum and topologicalcharge. For dipole azimuthons the topological charge is �1,whereas for quadrupole, hexapole, and octopole azimuthonsthe values are �2, �3, and �4, respectively. As expected, theseazimuthons will rotate with a constant angular frequency ωduring propagation when the nonlinear term in Eq. (2) isincluded. Therefore, the wave Uðx; y; zÞ can be rewritten asUðr; θ − ωzÞ in polar coordinates, where r ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ y2

pand

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(a) (b) (c) (d) (e)

Fig. 1 (a) Basic modes, (b) degenerate dipole modes, (c) degenerate quadrupole modes,(d) degenerate hexapole modes, and (e) degenerate octopole modes. First row: first-order modes,second row: second-order modes, and third row: third-order modes. The panels are shown inthe window −2 ≤ x ≤ 2 and −2 ≤ y ≤ 2. Other parameters: V 0 ¼ 500 and w ¼ 1.

(a) (b) (c) (d)

(e) (f) (g) (h)

Fig. 2 Amplitude and phase of (a)–(d) the first-order and (e)–(h) the second-order azimuthonsconstructed from (a) and (e) the degenerate dipoles, (b) and (f) the quadrupoles, (c) and(g) the hexapoles, and (d) and (h) the octopoles. The panels are shown in the window−2 ≤ x ≤ 2 and −2 ≤ y ≤ 2. Other parameters: A ¼ 0.4 and B ¼ 0.5.

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θ is the azimuthal angle in the transverse plane ðx; yÞ. This factallows for a bit of theoretical analysis.

After plugging Eq. (4) into Eq. (2) with μ ¼ 0, multiplying byU� and ∂θU�, respectively, and integrating over the transversecoordinates, one ends up with a linear system of equations:

−βPþ ωLz þ I þ N ¼ 0;

− βLz þ ωP0 þ I0 þ N0 ¼ 0; (6)

where P¼ RR jUj2dxdy, Lz ¼ −iRR ð−y∂xUþ x∂yUÞU�dxdy,P0 ¼ RR j − y∂xU þ x∂yUj2dx dy, I ¼ RR

U�Δ⊥Udx dy, N ¼RR ðσjUj2 þ VÞjUj2dx dy, I0 ¼ iRR ð−y∂xU� þ∂yU�ÞΔ⊥Udxdy,

and N0 ¼ iRR ðσjUj2þVÞð−y∂xU�þx∂yU�ÞUdxdy. Obviously,

the quantities P and Lz stand for the power and angular momen-tum of the beam, and P0 is the norm of the state ∂θU. Theintegrals I and I0 are related to the diffraction mechanism ofthe system, whereas N and N0 account for the waveguide andnonlinearity. The angular frequency of the azimuthon duringpropagation can be obtained by directly solving Eq. (6), that is

ω ¼ PðI0 þ N0Þ − LzðI þ NÞL2z − PP0 : (7)

After these preliminaries, we are ready to address the Rabioscillation of azimuthons. To this end, we adopt the superposi-tion of two azimuthons Um;nðx; yÞ expðiβm;nzÞ as an input

ψ ¼ cmðzÞUmðx;yÞexpðiβmzÞþ cnðzÞUnðx;yÞexpðiβnzÞ; (8)

where cm;nðzÞ are the slowly varying complex amplitudes ofthe azimuthons and the modulation frequency is d ¼ βm − βn.Plugging Eq. (8) into Eq. (2), and without considering the non-linear term [but still on the level of analysis of Eq. (3)], oneobtains

i∂cm∂z Um expðidzÞ þ 1

2μcmVUm½1þ expð2idzÞ�

þ i∂cn∂z Un þ

1

2μcnVUn½expðidzÞ þ expð−idzÞ� ¼ 0: (9)

Since the azimuthons are constructed based on Eq. (5), theysatisfy the relation hUm;Uni ≠ 0 if m ¼ n and hUm;Uni ¼ 0 ifm ≠ n, thus forming a complete set of eigenstates. Here weborrowed the bra-ket notation from quantum mechanics. Notethat the orthogonality of azimuthon shapes is only valid in theweakly nonlinear regime. As a result, one obtains two coupledequations based on Eq. (9):

i∂cm∂z þ 1

2μ

DUmVUn

EDUmUm

E cn ¼ 0;

i∂cn∂z þ 1

2μ

DUnVUm

EDUnUn

E cm ¼ 0; (10)

where hUmVUni ¼RR

rU�mVUn dr dθ, with the asterisk repre-

senting the conjugate operation. Based on Eq. (10), the periodof the Rabi oscillation can be obtained as

zR ¼ π

jΩRj; (11)

where

ΩR ¼ μ

2

DUmVUn

EffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDUmUm

EDUnUn

Er : (12)

(a)

(b)

(c)

Fig. 3 Rabi transition of (a) a dipole and (b) a hexapole. In eachcase, the propagation is shown by the isosurface plot abovewhich amplitude distributions at selected distances are shown.In both cases, the weak longitudinally periodic modulation existsin the region 30 ≤ z ≤ 90, with d ≈ 25.2 and μ ≈ 0.031 in (a) andd ≈ 36.36 and μ ≈ 0.014 in (b). (c) The Rabi oscillation period zR

versus the frequency detuning l.

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We note that the azimuthon conversion happens at half of theperiod, i.e., at zR∕2. Note also that the Rabi spatial frequencydirectly depends on the modulation strength μ.

2.2 Circular Waveguide

We investigate the propagation of azimuthons in the circularweakly nonlinear waveguide by including the longitudinalmodulation, and the results are displayed in Fig. 3. Without lossof generality, we choose the dipole and hexapole azimuthons,which are shown in Figs. 3(a) and 3(b), respectively. By takingthe dipole azimuthon as an example [Fig. 3(a)], we want to seewhether the Rabi oscillation between the dipole azimuthon[Fig. 2(a)] and its corresponding second-order dipole azimuthon[Fig. 2(e)] can be established. So, to induce resonance, we setthe modulation frequency to be the difference between theeigenvalues of the two modes, which is d ≈ 25.2. As shown byEq. (12), the period of the Rabi oscillation is expected to dependon the modulation strength μ, and here we set it to be μ ≈ 0.031to make the period zR ∼ 60. As a consequence, one expects tosee the second-order dipole azimuthon at a distance ∼30 afterturning on the longitudinal modulation.

In Fig. 3(a), the propagation of the dipole azimuthon isexhibited as a three-dimensional (3-D) isosurface plot, in whichthe longitudinal modulation exists only in the interval30 ≤ z ≤ 90. When the propagation distance is smaller than

z ≤ 30, one in fact observes the stable rotational propagationof the dipole azimuthon. The selected amplitude distributionsat z ¼ 0 and z ¼ 30 are shown above the 3-D isosurface plots.In the interval 30 ≤ z ≤ 90, which is about one period of theRabi oscillation, the oscillation between the dipole azimuthonand the second-order azimuthon is displayed, in which thedipole azimuthon completely switches to the second-orderazimuthon at z ¼ 60. Indeed, the corresponding amplitude dis-tribution is the same as that in Fig. 2(e) except for rotation, andthe reason is quite obvious—azimuthons rotate steadily duringpropagation. When the propagation distance reaches z ¼ 90, thedipole azimuthon is recovered and the longitudinal modulationis also lifted at the same time. Therefore, one observes a stablerotating dipole azimuthon in the interval 90 ≤ z ≤ 120 and theamplitude distributions at z ¼ 90 and z ¼ 120, which are dipoleazimuthons explicitly, and are shown above the isosurface plot.The analogous propagation dynamics of the hexapole azimuthonis shown in Fig. 3(b), the setup of which is the same as that ofFig. 3(a); it also clearly displays the Rabi oscillation of a higher-order azimuthon.

Here we would like to note that the Rabi oscillation is notfeasible between two arbitrary azimuthons. Only azimuthonswith similar structures (e.g., the dipole and the higher-orderdipole azimuthons) can switch between each other, and azimu-thons with different symmetries (e.g., the dipole and thequadrupole azimuthons) will not, because the overlap integrals

(a) (b)

(c) (d)

(e) (f)

Fig. 4 (a) Rabi transition of a deformed dipole. (b) The amplitude and phase of the azimuthonbased on the dipole in (a). (c) The transition of a deformed hexapole. (d) The amplitude and phaseof the azimuthon based on the hexapole in (c). (e) The transition of a deformed higher-orderdipole. (f) The amplitude and phase of the azimuthon based on the higher-order dipole in (e).The panels are shown in the window −2 ≤ x ≤ 2 and −2 ≤ y ≤ 2. Other parameters: A ¼ 0.4and B ¼ 0.5.

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in general are zero, hUmVUni ¼ 0. Note also that the Rabioscillation between two modes with opposite symmetry is alsopossible if the potential is antisymmetrically modulated in thetransverse plane.25

Generally, there must exist some frequency detuningl ¼ d − d0 between the real modulation frequency d0 and theresonant frequency d. Therefore, it is reasonable to have a lookat the efficiency of the azimuthon conversion versus the detun-ing l. Unfortunately, one cannot obtain a direct measure of theefficiency via the projection of the field amplitude ψ on thetargeting azimuthons due to the rotation of azimuthons duringpropagation. However, the efficiency is reflected on the Rabioscillation period zR—the bigger the value of zR, the biggerthe efficiency.13–15 The dependence of zR on the frequency de-tuning l is shown in Fig. 3(c). As expected, one finds that the

efficiency of azimuthon conversion is the biggest at the resonantfrequency, and it reduces with the growth of the frequencydetuning l.

2.3 Square Waveguide

Now we investigate the azimuthon transition in a square wave-guide, which is generated by the potential in Eq. (2) in the formVðx; yÞ ¼ V0 exp½−ðx10 þ y10Þ∕w10�. Again, we solve for thelinear eigenmodes supported by the deep square waveguideusing the plane-wave expansion method. Connected with thegeometry of the potential, the amplitude distributions of thelinear modes are more complex than those in the regular circularwaveguide; therefore, we denote them as the deformed modes.In Fig. 4, we display three kinds of deformed modes and the

(a)

(b)

Fig. 5 (a) Transition between dipole and hexapole azimuthons with d ≈ 25.2 and μ ≈ 0.085.(b) Transition between dipole and hexapole azimuthons with d ≈ 22.1 and μ ≈ 0.034. The weaklongitudinally periodic modulation has to always exist during propagation.

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corresponding azimuthons, which will transform mutually be-cause of the relation hUmVUni ≠ 0.

Different from the azimuthons in circular waveguides, theazimuthons in square waveguides rotate conditionally. Due tothe symmetry of the square waveguide, a rotating azimuthonwill deform, i.e., its profile will change. Without consideringnonlinearity, the linear superposition of two degeneratemodes [e.g., dipoles in Fig. 4(a), labeled as u1;2ðx; yÞ] is alsoa dipole solution in the square potential, as u01;2ðx;yÞ ¼½u1ðx;yÞ�u2ðx;yÞ�∕

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRR ju1ðx;yÞ�u2ðx;yÞj2dxdyq

. We found

that an azimuthon in a square waveguide will rotate whenthe condition HðψÞ > Hðu01;2Þ is met. Here H represents theHamiltonian of the model.22 Even though the wobbling azi-muthons can be established in our numerical simulations,we are more interested in the rotating azimuthons, so we setagain A ¼ 0.4 and B ¼ 0.5 in this section to guarantee therotation of azimuthons during propagation.

Thus the azimuthons will be deformed during propagationbecause of the symmetry of the potential, hence to exhibitthe azimuthon conversion in a more clear way one has to prop-erly choose the value of μ to make the Rabi oscillation periodalmost equal to one half of the target azimuthon rotation period,since the modulation frequency d is determined beforehand.Numerical simulations reveal that the rotation periods of thehexapole azimuthon and the higher-order dipole azimuthonare ∼148.4 and ∼146.4, respectively. Therefore, according toEq. (12), the modulation strength μ for the two cases shouldbe ∼0.085 and ∼0.034, respectively. The numerical demonstra-tion of the Rabi oscillation is shown in Fig. 5. Different fromthe setting for the case of a circular waveguide, the longitudinalmodulation always accompanies the square waveguide.

As shown in Fig. 5(a), the hexapole azimuthon is obtained atz ∼ 37.1 (one half of the Rabi oscillation period and also a quar-ter of the rotation period of the hexapole azimuthon), whereasin Fig. 5(b) the higher-order dipole azimuthon is obtained atz ∼ 36.6. When the propagation distance reaches one Rabi os-cillation period, the dipole azimuthon is recovered with a smalldeformation. To show the azimuthon conversion more transpar-ently, we also display the corresponding phase distributions.Evidently, there is only one phase singularity in the phase ofthe dipole azimuthon (the topological charge is 1), five singu-larities for the hexapole azimuthon (the topological charge is 3),and nine for the higher-order dipole azimuthon (the topologicalcharge is again 1). As seen, the phase distributions are in accor-dance with the expectations and with those displayed in Fig. 4.

3 ConclusionWe investigated and demonstrated Rabi oscillations of azimu-thons in weakly nonlinear waveguides with weak longitudinallyperiodic modulations. Based on the coupled mode theory,we find the period of Rabi oscillation, which is affected bythe modulation strength and also by the spatial symmetry ofazimuthons. The analysis is feasible for both circular and squarewaveguides and can be extended to the waveguides of othersymmetries.

Based on the model considered in this work, the switchingbetween a vortex-carrying azimuthon and a multipole that isfree-of-vortex will not happen. The reason is that the initialazimuthon is composed of two degenerate modes u1;2 that willswitch into another two degenerate modes u3;4 during propaga-tion. So the output is a composition of u3;4, which also carries

a vortex. However, if the potential is modulated transversely ina proper manner, such a switch becomes possible once both thelongitudinal and transverse phase matching is satisfied.

Acknowledgments

This work was supported by Guangdong Basic and AppliedBasic Research Foundation (No. 2018A0303130057), theNational Natural Science Foundation of China (Nos. U1537210,11534008, and 11804267), and the Fundamental ResearchFunds for the Central Universities (Nos. xzy012019038 andxzy022019076). M. R. B. acknowledges support from theNPRP 11S-1126-170033 project from the Qatar NationalResearch Fund, whereas K. C. J. and Y. Q. Z. acknowledgethe computational resources provided by the HPC platform ofXi’an Jiaotong University. All authors acknowledge anonymousreviewers for their valuable comments. The authors declareno conflicts of interest.

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Kaichao Jin received his BS degree from Wuhan University ofTechnology in 2018. He is a master student working under the supervi-sion of Yiqi Zhang at the School of Electronic Science and Engineering ofXi’an Jiaotong University. Currently, he is working on topological andwaveguide array physics.

Yongdong Li received his PhD from Xi’an Jiaotong University, Xi’an,China, in 2005. He is currently a professor at the School of ElectronicScience and Engineering of Xi’an Jiaotong University. His research inter-ests include modeling and applications of plasmas.

Feng Li received his PhD from the University Nice Sophie Antipolis,France, in 2013. He is currently a professor at the School ofElectronic Science and Engineering of Xi’an Jiaotong University. Hisresearch interests include quantum and topological effects in photonicmicro/nanostructures.

Milivoj R. Belic received his PhD from The City College of New York in1980. He is a professor in physics at Texas A&M University at Qatar. Hisresearch interests include topological photonics, nonlinear optics, andnonlinear dynamics.

Yanpeng Zhang received his PhD from Xi’an Jiaotong University, Xi’an,China, in 2000. He is currently a professor at the School of ElectronicScience and Engineering of Xi’an Jiaotong University. His research inter-ests include quantum optics, topological photonics, and nonlinear optics.

Yiqi Zhang received his PhD from Xi’an Institute of Optics and PrecisionMechanics, Chinese Academy of Sciences, Xi’an, China, in 2011. He iscurrently an associate professor at the School of Electronic Science andEngineering of Xi’an Jiaotong University. His research interests includetopological photonics and nonlinear optics.

Jin et al.: Rabi oscillations of azimuthons in weakly nonlinear waveguides

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