Transformation of topological optical states via spiral modulation in fractional-diffraction systems

TL;DR

This paper introduces a method to manipulate topological optical states in fractional systems using spiral modulation, enabling mode conversion and exploring nonlinear effects, with potential applications in optical topological control.

Contribution

It presents a novel spiral modulation scheme for topological state manipulation in fractional optical systems, including analytical and simulation analyses of mode conversion mechanisms.

Findings

Supports direct mode conversion with topological-charge difference Δm=1

Requires intermediary mode for Δm=2 conversion with precise resonance tuning

Nonlinear effects cause resonance-frequency shifts affecting conversion efficiency

Abstract

We propose a scheme for manipulations of a variety of topological states in fractional optical systems through spiral modulation of the local refraction index. An analytical approximation, based on a truncated finite-mode system, and direct simulations reveal that the spiral modulation supports direct mutual conversion between eigenmodes with topological-charge difference \Delta m=1, driven by the resonant coupling between the modes and the underlying spiral modulation. The conversion between eigenmodes with \Delta m=2 requires involvement of an intermediary mode and precise tuning of the resonance condition. We further explore the conversion of degenerate modes under the action of azimuthal modulation. Modulated degenerate modes exhibit incomplete conversion, evolving into intermediate states with odd parity symmetry. Finally, we examine nonlinear effects on the spiral-modulation-induced mode conversion, identifying an essential nonlinearity-induced resonance-frequency shift that critically affects the conversion efficiency.

Equations

$$i\frac{\partial \psi }{\partial z}=\frac{1}{2}{\left(-\frac{{\partial }^2}{\partial x^2}-\frac{{\partial }^2}{\partial y^2}\right)}^{\alpha /2}\psi -V\left(x,y,z\right)\psi -\sigma \psi |\psi {|}^2.$$

$$V(x,y,z)=p\exp \{-[x+\eta \sin (\Omega z){]}^2-[y-\eta \cos (\Omega z){]}^2\}.$$

$$\psi (x,y,z)=U(x,y)\exp (i\beta z).$$

$$\beta U=-\frac{1}{2}{\left(-\frac{{\partial }^2}{\partial x^2}-\frac{{\partial }^2}{\partial y^2}\right)}^{\alpha /2}U+p{V}_0\left(x,y\right)U,$$

$$\psi (x,y,z)$$

$$+\left[c_1^{\prime }(z)u_{n^{\prime }{m}^{\prime }}(x,y)+c_2^{\prime }(z)v_{n^{\prime }{m}^{\prime }}(x,y)\right]e^{i{\beta }_{n^{\prime }{m}^{\prime }}z},$$

$$c_1(z)$$

$$c_2(z)$$

$$c_1^{\prime }(z)$$

$$c_2^{\prime }(z)$$

$$\frac{d{c}_1}{dz}$$

$$\frac{d{c}_2}{dz}$$

$$\frac{d{c}_1^{\prime }}{dz}$$

$$\frac{d{c}_2^{\prime }}{dz}$$

$$G(x,y,z)$$

$$=i\eta \left[\left(x-iy\right)e^{i\Omega z}-\left(x+iy\right)e^{-i\Omega z}\right]$$

$$-{\eta }^2-\frac{1}{2}{\eta }^2{\left[\left(x-iy\right)e^{i\Omega z}-\left(x+iy\right)e^{-i\Omega z}\right]}^2$$

$$+.....$$

$$\frac{d{c}_1}{dz}$$

$$\frac{d{c}_2}{dz}$$

$$\frac{d{c}_1^{\prime }}{dz}$$

$$\frac{d{c}_2^{\prime }}{dz}$$

$${\Omega }_R=2\eta p\left|D\right|,$$

$$\psi (x,y,0)=A\left[u_{nm}(x,y)+iB{v}_{nm}(x,y)\right],$$

$$\psi (x,y,z)$$

$$+\left[c_1^{\prime }(z)u_{n^{\prime }{m}^{\prime }}(x,y)+c_2^{\prime }(z)v_{n^{\prime }{m}^{\prime }}(x,y)\right]e^{i{\beta }_{n^{\prime }{m}^{\prime }}z}$$

$$+\left[c_1^{\prime \prime }(z)u_{n{}^{\prime \prime }{m}^{\prime \prime }}(x,y)+c_2^{\prime \prime }(z)v_{n{}^{\prime \prime }{m}^{\prime \prime }}(x,y)\right]e^{i{\beta }_{n^{\prime \prime }{m}^{\prime \prime }}z}$$

$$\frac{d{c}_1}{dz}$$

$$\frac{d{c}_2}{dz}$$

$$\frac{d{c}_1^{\prime }}{dz}$$

$$\frac{d{c}_2^{\prime }}{dz}$$

$$\frac{d{c}_1^{\prime \prime }}{dz}$$

$$\frac{d{c}_2^{\prime \prime }}{dz}$$

$$\frac{d{c}_1}{dz}$$

$$\frac{d{c}_2}{dz}$$

$$\frac{d{c}_1^{\prime }}{dz}$$

$$\frac{d{c}_2^{\prime }}{dz}$$

$${\beta }^{\prime }=\Omega -({\beta }_{nm}-{\beta }_{n^{\prime }{m}^{\prime }}),$$

Full text

I Introduction

The fractional Schrödinger equation (FSE), as a natural extension of the conventional Schrödinger equation, is the basis of fractional quantum mechanics [1, 2, 3]. The kinetic-energy operator in FSE is composed of fractional spatial derivatives with Lévy index $\alpha$, that usually takes values $1<\alpha\leq 2$, while conventional quantum mechanics corresponds to $\alpha=2$. The fractional operator leads to a profound change in the behavior of the wave function. The experimental realization of FSE requires to design a scheme for implementing the fractional derivatives. In particular, it was proposed to do that using one-dimensional Lévy crystals [4]. However, building such crystals remains a challenge. Alternatively, classical optical systems have emerged as a promising platform for the realization of fractional diffraction. In this connection, it was proposed by Longhi to implement FSE in optics, designing an aspherical optical cavity to emulate this equation [5]. This ground-breaking work has ushered in the realm of the beam dynamics realizing the FSE framework. In particular, the light-beam propagation in these models gives rise to a breather-like behavior [6, 7, 8].

A related topic of great interest is the nonlinear FSE, which relies upon the use of the Kerr nonlinearity of the optical material. It produces diverse families of optical fractional solitons, such as gap solitons [9], multipoles [10], trapped modes [11], Airy waves [12], as well as solitary vortices [13]. Setups supporting fractional solitons offer various applications for the design of photonic data-processing schemes [14, 15, 16]. Further, by combining FSE and a complex potential, the parity-time-symmetry and its breaking may be featured [17, 18, 19, 20, 21]. Recently, a research group has used programmable holograms and single-thermal measurement techniques to simulate Lévy waveguides and reconstruct the amplitude and phase of pulses, realizing the fractional Schrödinger equation in the time domain and observing typical phenomena such as temporal solitons and pulse splitting due to fractional group-velocity dispersion [22].

In the field of optics, Rabi oscillations provide the simplest realization of the resonant mode conversion. In this context, the longitudinally periodic modulation of the refractive index acts as an effective AC field which induces the Rabi coupling, resulting in various phenomena, such as the parametric amplification, inhibition of light tunneling, and the realization of optical isolation [23, 24, 25, 26]. Notably, there exists a similarity between the stimulated mode-conversion process and Rabi flopping, which refers to periodic transitions between two stationary states of a quantum system coupled by an external resonant AC field [27, 28, 29]. In recent years, Rabi oscillations have been widely investigated in a variety of optical and photonic systems, including fibers [30, 31], multimode [32, 33, 34], coupled [35] and arrayed [36, 37] waveguides. two-dimensional (2D) modal structures [38, 39], and waveguides with the parity-time ($\mathcal{PT}$) or partial $\mathcal{PT}$ symmetry [40, 41]. Rabi oscillations of topological edge states and resonant mode conversion in FSE have been studied too [42, 43, 44]. Recently, Rabi oscillations of azimuthons in weakly nonlinear waveguides with a weak longitudinally periodic modulation of the refractive index and nonlinearity have been investigated [45, 46].

The subject of the present work is resonant mode conversion by spiral longitudinal modulations in the framework of the nonlinear FSE. First, the characteristics of eigenmodes in the nonlinear FSE are addressed. Then, resonant mode conversion between modes with different topological charges is demonstrated under the action of the spiral modulation are demonstrated, with the help of the resonant coupled-mode theory. The results demonstrate the conversion between eigenmodes of all orders with topological-charge differences $1$ and $2$. Additionally, azimuthally-modulated degenerate superposition modes can be converted into multipole ones of odd orders. The nonlinearity results in strong asymmetrization of the resonant curves and a shift of the resonant frequencies.

The paper is organized as follows. In the next section, the model and its eigenmodes found in the linear regime are presented. Under the action of the spiral longitudinal modulation, the conversion between the modes with different topological charges is addressed in Sec. III. In Sec.  IV, the effect of the cubic nonlinearity on the mode conversion is studied. The paper is concluded in Sec. V.

II The model

We consider the light propagation in a multimode waveguide, modeled by the scaled nonlinear FSE for the field amplitude $\psi\left(x,y,z\right)$:

[TABLE]

Here, $x$ and $y$ are the transverse coordinates normalized by the characteristic scale $r_{0}$, and $z$ is the longitudinal coordinate normalized by $k_{0}r_{0}^{2}$, where the carrier wavenumber is $k_{0}=2\pi n_{b}/\lambda_{0}$, with wavelength $\lambda_{0}$ in vacuum and refractive index $n_{b}$. Further, $\alpha$ is the Lévy index and $\sigma$ (that may be positive or negative) is the coefficient of the Kerr nonlinearity. The spiral waveguide, characterized by strength $p$, potential rotation pitch $\eta\ll 1$, and spatial spiraling frequency $\Omega$, is represented by the effective potential

[TABLE]

First, we consider the linear version of Eq. (1) without the longitudinal modulation, i.e., $\sigma=0$ and $\eta=0$, which supports 2D linear eigenmodes with propagation constant $\beta$, sought for in the usual form:

[TABLE]

Here linear eigenmode $U(x,y)$ satisfies the stationary equation

[TABLE]

where $V_{0}\left(x,y\right)\equiv\exp(-x^{2}-y^{2})$. Numerical solution of Eq. (4) by means of the plane-wave expansion method produces 2D eigenmodes with propagation constant $\beta_{nm}$, as shown in Fig. 1. Here, $n$ and $m$ represent the radial node number and topological charge (vorticity) of these eigenmodes. Note that their degeneracy is two for $m\geq 1$, corresponding to cosine and sine modes $u_{nm}$ and $v_{nm}$, respectively [41, 46].

Further, Fig. 2 displays examples of the intensity distribution in the first- and higher-order zero-vorticity modes ($u_{00}$, $u_{10}$, and $u_{20}$), first- and second-order degenerate dipoles ($u_{01}$, $v_{01}$ and $u_{11}$, $v_{11}$), first- and second-order degenerate quadrupoles ($u_{02}$, $v_{02}$ and $u_{12}$, $v_{12}$), first-order degenerate hexapoles ($u_{03}$ and $v_{03}$), and first-order degenerate octupoles ($u_{04}$ and $v_{04}$).

III The conversion between linear eigenmodes

In this section, we consider the evolution of eigenmodes of the linearized version of Eq. (1) ($\sigma=0$) in the presence of spiral modulation ($\eta\neq 0$), and a possibility of the dynamical conversion between the eigenmodes with different topological charges. To this end, we consider a superposition of two sets of degenerate eigenmodes:

[TABLE]

where $\beta_{nm}$ and $\beta_{n^{\prime}m^{\prime}}$ are the above-mentioned propagation constants corresponding to the degenerate eigenmodes $u_{nm},v_{nm}$, and $u_{n^{\prime}m^{\prime}}$, $v_{n^{\prime}m^{\prime}}$, respectively, and weights $|c_{1}(z)|^{2}$, $|c_{2}(z)|^{2}$, $|c_{1}^{\prime}(z)|^{2}$, and $|c_{2}^{\prime}(z)|^{2}$ of the modes and can be calculated as

[TABLE]

Substituting expression. (5) in Eq. (1) with $\sigma=0$, multiplying both sides by $u_{nm}^{\ast}$, $v_{nm}^{\ast}$ and $u_{n^{\prime}m^{\prime}}^{\ast}$, $v_{n^{\prime}m^{\prime}}^{\ast}$, respectively, and integrating over the transverse coordinates, we derive the following system of evolution equations:

[TABLE]

where kernel $G$ may be expanded in powers of the small spiral-modulation pitch $\eta$:

[TABLE]

Adopting the resonance condition of the coincidence of the spiral-modulation frequency and a difference between two propagation constants found in the absence of the modulation,* viz*., $\Omega=\beta_{nm}-\beta_{n^{\prime}m^{\prime}}$, and the rotating-wave approximation (i.e., dropping rapidly oscillating terms), we reduce Eqs. (7) to the form of

[TABLE]

where $D_{1}={\iint}(x-iy)V_{0}u_{nm}^{\ast}u_{n^{\prime}m^{\prime}}dxdy$, $D_{2}={\iint}(x-iy)V_{0}u_{nm}^{\ast}v_{n^{\prime}m^{\prime}}dxdy$, $D_{3}={\iint}(x-iy)V_{0}v_{nm}^{\ast}u_{n^{\prime}m^{\prime}}dxdy$, and $D_{4}={\iint}(x-iy)V_{0}v_{nm}^{\ast}v_{n^{\prime}m^{\prime}}dxdy$.

Using Eq. (9), the frequency of the mode conversion between two mutually degenerate modes, coupled by the resonant modulation, is

[TABLE]

where $D\equiv D_{1}+D_{2}+D_{3}+D_{4}$ represents the collective coupling parameter. The mode conversion is governed by phase-matching conditions, which imply that if and only if the angular-momentum number difference satisfies condition $\Delta m\equiv|m-m^{\prime}|=1$, the collective coupling coefficient $D$ attains a nonzero value [39], which is written above. Under this crucial condition, the Rabi frequency $\Omega_{R}$ determines the efficiency of the inter-mode energy transfer. When $\Delta m\neq 1$, the symmetry-enforced destructive interference between the coupling terms annihilates the collective interaction, thereby completely suppressing the inter-mode energy conversion.

In particular, Eq. (9) predicts the conversion of the zero-vorticity mode $(m=0)$ to the dipole one $(m^{\prime}=1)$. In this case, $D_{3}=D_{4}=0$, as the zero-vorticity mode is a non-degenerate one. The mode conversion can occur between eigenmodes sharing identical radial node numbers $n$, as well as between those with different values of $n$.

Figure 3 shows the dependence of the conversion period $T=2\pi/\Omega_{R}$ [where $\Omega_{R}$ is given by Eq. ((10))] on parameter $\alpha$, revealing its crucial role in controlling the rate of the mode energy conversion. Figure 3(a) corresponds to the conversion period between modes with the same radial node number $n$ ($nm\leftrightarrow nm^{\prime}$), while Fig. 3(b) corresponds to that between modes with different radial node numbers $n$ ($nm\leftrightarrow n^{\prime}m^{\prime}$). As $\alpha$ decreases, the waveguide supports an expanded set of eigenmodes with higher radial/azimuthal quantum numbers (Fig. 1), dramatically increasing the number of phase-matched conversion pathways satisfying $\Delta m=1$. Overall, the conversion period in Fig. 3(b) is significantly larger than that in Fig. 3(a). Notably, the conversion period $T$ exhibits a non-monotonous trend, first decreasing and then increasing with the decrease of $\alpha$. This phenomenon stems from the dual effect of $\alpha$ on the collective coupling coefficient $D$: on the one hand, a decrease in $\alpha$ leads to more modes, which may enhance the coupling strength for certain mode pairs; on the other hand, very small $\alpha$ weakens mode confinement, causing the mode field distributions to broaden, thereby reducing the overlap integrals and ultimately weakening the coupling and increasing the conversion period $T$. This behavior indicates that, by optimizing $\alpha$, one can find an operating point that provides the minimum conversion period.

Figure 4 demonstrates the conversion between the third-order zero-vorticity mode $(u_{20})$ and first-order dipole ones $(u_{01},v_{01})$, with $\Delta m=1$. Under the action of the spiral modulation, the input mode $u_{20}$ undergoes a conversion into the degenerate superposed dipole, with the topological charge $1$, at $z=250$, and then returns to its original mode at $z=500$, thus completing a cycle of the Rabi oscillations(see Supplemental Material, Animation_Fig4 [47]). The numerical result for the period corroborates the analytical prediction given by Eq. (10), which yields $T=2\pi/\Omega_{R}=499.7$(Fig. 3(b)). Note that the two degenerate dipole modes are always represented by equal weights in the course of the conversion, i.e., $|c_{1}^{\prime}(z)|^{2}=|c_{2}^{\prime}(z)|^{2}$.

Next we discuss the evolution of the input represented by the single degenerate mode, corresponding to the initial conditions with $c_{2}(0)=1$ and $c_{1}(0)=c_{1}^{\prime}(0)=c_{2}^{\prime}(0)=0$. As shown in Fig. 5, at $z=0$, the input is the second-order dipole mode of the sine type, $v_{11}$, and we set the resonant frequency $\Omega=\beta_{11}-\beta_{12}$ to achieve the conversion between two modes of the same order. It is found that the dipole mode is converted to the tripolar one, with three peaks (instead of the quadrupole mode) at the half-Rabi-oscillation period, $z=55.582$, and returns to the dipole at the full Rabi oscillation period, $z=111.164$. However, unlike the original dipole sine mode $v_{11}$, it is the respective degenerate dipole-cosine mode $u_{11}$(see Supplemental Material, Animation_Fig5 [47]). Analyzing the evolution of the weight curve, it is found that the second-order tripole is actually formed by a superposition of four components of $u_{11},v_{11}$ and $u_{12},v_{12}$ with equal weights. During the full Rabi-oscillation period, $u_{12}$ and $v_{12}$ maintain equal weights, while phase rotation occurs between the degenerate sine $v_{11}$ and cosine $u_{11}$ modes [46]. This azimuthal pattern rotation, driven by their interconversion through the spiral modulation, preserves the dipole configuration while moving its angular orientation.

Continuing to consider the conversions between modes of different orders, Fig. 6 exhibits the evolution of the single second-order quadrupole mode $v_{12}$(see Supplemental Material, Animation_Fig6 [47]), under the resonance condition $\Omega=\beta_{03}-\beta_{12}$. The conversion creates a pentapole at the half-cycle. It is constructed as a superposition of two mutually degenerate second-order quadrupole modes $(u_{12},v_{12})$ and two mutually degenerate first-order hexapole ones $(u_{03},v_{03})$. In summary, we find that the action of the external spiral modulation on a single eigenmode sets it in the rotation in the transverse plane, simultaneously transforming it into an odd-pole mesostate, with half the conversion efficiency.

To investigate the rotational effect of the spiral modulation, we introduce azimuthons, which can be constructed from superimposed degenerate linear eigenmodes ($m\geq 1$) in the framework of the FSE by the following initial input [45, 46]:

[TABLE]

where $A$ is an amplitude factor and $B$ is a parameter which determines the depth of the azimuthal modulation.

Further, in Fig. 7 we use the input in the form of the first-order hexapole with the azimuthal-modulation intensity of $0.5$ and spiral-modulation frequency $\Omega=\beta_{03}-\beta_{04}$. In this case, the sign of the angular momentum of the input azimuthon hexapole agrees with the direction of the spiral modulation. Therefore, the azimuthon rotates in the same direction, exhibiting conversion into a heptapole, passing half the period, $z=41.735$, and returning to hexapole after the full period, $z=83.471$(see Supplemental Material, Animation_Fig7 [47]). In terms of the weight evolution, the conversion rate is greatly enhanced by the azimuthal modulation.

The observed minor discrepancy arises from the azimuthal energy redistribution under the spiral modulation: a non-uniform intensity distribution between the azimuthal lobes ($\theta$-directional peaks of the azimuthon) induces energy flow between adjacent lobes. This dynamical redistribution induces slight radiative losses in the course of the mode conversion, resulting in a measurable deviation between the predictions of the coupled-mode theory and numerical simulations.

Next, we use the coupled-mode theory to explore the dependence between the azimuthal-modulation depth and the azimuthon’s conversion rate. As shown in Fig. 8, $\rho_{04}=|c_{1}^{\prime}(z)|^{2}+|c_{2}^{\prime}(z)|^{2}$ denotes the weight of the octupole $(m=4)$ in the course of the conversion, and its maximum value represents the conversion rate. When $B>0$ in Eq. (11), the sign of the azimuthon’s angular momentum is the same as the sign of the spiral modulation, and the conversion rate grows with the increase of the azimuthal-modulation depth. When $B=1$, the equally weighted degenerate hexapole can be completely converted into the octupole. When $B=0$ , the wave function simplifies to a single constituent of the degenerate mode pair, achieving a $50\%$ conversion efficiency due to the absence of the inter-mode coupling. When $B<0$, the sign of the azimuthon’s angular momentum is opposite to that of the direction of the spiral modulation, resulting in a significant decrease in the conversion rate. The azimuthon cannot demonstrate complete conversion between modes with $\Delta m=1$ under the action of spiral modulation. Instead, it is converted into the odd-pole intermediate state, rotating under the action of the spiral modulation.

Finally, we consider the resonance-driven conversion between eigenmodes with $\Delta m=2$. To realize this transition, coupling to an intermediate mode needs to be introduced, therefore we approximate the dynamical state by the three-component form

[TABLE]

(cf. Eq. (5), where $\beta_{nm}>\beta_{n^{\prime}m^{\prime}}>\beta_{n^{\prime\prime}m^{\prime\prime}}$ are the propagation constants corresponding to the degenerate eigenmodes $\left(u_{nm},v_{nm}\right)$, $\left(u_{n^{\prime}m^{\prime}},v_{n^{\prime}m^{\prime}}\right)$, and $\left(u_{n^{\prime\prime}m^{\prime\prime}},v_{n^{\prime\prime}m^{\prime\prime}}\right)$, respectively.

Similar to the above consideration, we substituting ansatz (12) in Eq. (1) with $\sigma=0$, multiply both sides by $u_{nm}^{\ast}$, $v_{nm}^{\ast}$, $u_{n^{\prime}m^{\prime}}^{\ast}$, $v_{n^{\prime}m^{\prime}}^{\ast}$,$u_{n^{\prime\prime}m^{\prime\prime}}^{\ast}$ and $v_{n^{\prime\prime}m^{\prime\prime}}^{\ast}$, respectively, and integrate the result over the transverse coordinates, omitting rapidly oscillating terms. Thus we derive the truncated system:

[TABLE]

where $D_{5}={\iint}(x-iy)V_{0}u_{n^{\prime}m^{\prime}}^{\ast}u_{n^{\prime\prime}m^{\prime\prime}}dxdy$, $D_{6}={\iint}(x-iy)V_{0}u_{n^{\prime}m^{\prime}}^{\ast}v_{n^{\prime\prime}m^{\prime\prime}}dxdy$, $D_{7}\!\!=\!\!{\iint}\!(x-iy)V_{0}v_{n^{\prime}m^{\prime}}^{\ast}u_{n^{\prime\prime}m^{\prime\prime}}dxdy$, and $D_{8}={\iint}(x-iy)V_{0}v_{n^{\prime}m^{\prime}}^{\ast}v_{n^{\prime\prime}m^{\prime\prime}}dxdy$, $\delta=\beta_{nm}-\beta_{n^{\prime}m^{\prime}}-\Omega=\Omega-(\beta_{n^{\prime}m^{\prime}}-\beta_{n^{\prime\prime}m^{\prime\prime}})$. Modes $\left(nm,n^{\prime}m^{\prime}\right)$ and $\left(n^{\prime}m^{\prime}.n^{\prime\prime}m^{\prime\prime}\right)$ feature the resonance at $(\beta_{nm}-\beta_{n^{\prime}m^{\prime}})$ and at $(\beta_{n^{\prime}m^{\prime}}-\beta_{n^{\prime\prime}m^{\prime\prime}})$, respectively. To keep the respective frequency detuning $\delta$ as low as possible, we set the spiral-modulation spatial frequency to be $\Omega=[(\beta_{nm}-\beta_{n^{\prime}m^{\prime}})+(\beta_{n^{\prime}m^{\prime}}-\beta_{n^{\prime\prime}m^{\prime\prime}})]/2\equiv(\beta_{nm}-\beta_{n^{\prime\prime}m^{\prime\prime}})/2$. This way, the spiral modulation drives conversions with $\Delta m=|m-m^{\prime\prime}|=2$. Note that, to meet the above conditions, the mode order $n$ should be specially selected.

Figure 9 represents the conversion between the first-order zero-vorticity and quadrupole modes which $\Delta m=2$, the first-order dipole mode playing the role of the intermediary. As the zero-vorticity mode is nondegenerate, the weights of the three modes are $\rho_{b}=|c_{1}|^{2}$, $\rho_{d}=|c_{1}^{\prime}|^{2}+|c_{2}^{\prime}|^{2}$ and $\rho_{q}=|c_{1}^{\prime\prime}|^{2}+|c_{2}^{\prime\prime}|^{2}$, respectively. As shown in Fig. 9(a), mode $u_{00}$, which is the input at $z=0$, develops two holes under the action of the spiral modulation. With the increase of the transmission distance, the holes slowly move to the center and eventually merge, forming a large hole, so that the mode evolves into a ring-like structure(see Supplemental Material, Animation_Fig9 [47]). As concerns the phase structure, the two holes represent phase dislocations. In the course of the subsequent evolution, they fuse into a single phase singularity, which corresponds to the double topological charge. Eventually, the mode is converted into a quadrupole one.

Then, from the evolution of the weights observed in Fig. 9(b), we conclude that the weight curves exhibit Rabi oscillations. Roughly speaking, the evolving mode completes the transition from the first-order zero-vorticity one into the first-order quadrupole and returns to the zero-vorticity mode after one period of the oscillations. Note that all degenerate components keep equal weights during the transition. Throughout the entire process, in addition to the zero-vorticity mode and quadrupole, components of the dipole mode also appear, and the curves exhibits small oscillations with the resonance frequency, which causes inherent detuning in the course of the evolution. The dipole mode, acting as the intermediary, simultaneously couples to both the zero-vorticity and quadrupole modes, inducing the generation of the double topological charges and thus enabling the cross-level transformation.

We stress that the conversion method presented above is different from the known cascaded scheme for achieving inter-mode conversion [32, 46]. In the framework of that scheme, the conversion starts from the zero-vorticity mode, transforming it into the dipole one. After completing this stage, the driving frequency is changed to further convert the dipole mode into the quadrupole one In contrast, the present conversion method, based in the spiral modulation of the underlying medium, is different, as the entire process uses the single resonance modulation frequency. Accordingly, the different conversion techniques feature different characteristics. For the cascaded scheme, the conversion efficiency is high, and the stability is strong; however, the required adjustment method is complex, the admitted bandwidth of the resonance frequency is extremely narrow, and the resonance conditions are strict, leading to weak controllability. On the other hand, the spiral-modulation method, while not enabling complete conversion, requires solely adjusting the single resonance frequency to directly couple initial and target modes. By balancing the detuning across the interconnected transitions, this approach relaxes strict spectral-alignment constraints. Compared to the cumulative phase-matching limitations of the cascading method, it yields a broader operational bandwidth and makes the control of the setting significantly easier.

IV Effects of nonlinearity on the mode conversion

In this Section, we aim to demonstrate that efficient conversion between modes with different topological charges is possible in the nonlinear medium too, with a resonance shift induced by the nonlinearity. To address the nonlinear system, we substitute expression (5) in Eq. (1) with $\sigma\neq 0$, multiply both sides by $u_{nm}^{\ast}$, $v_{nm}^{\ast}$,  and $u_{n^{\prime}m^{\prime}}^{\ast}$, $v_{n^{\prime}m^{\prime}}^{\ast}$, respectively, and integrate the result over the transverse coordinates, retaining, as in Eq. (8), the first-order expansion term with respect to small spiral-modulation pitch $\eta$ and dropping rapidly oscillating terms. We thus derive the following system of nonlinear truncated-mode equations:

[TABLE]

where $Q_{11}=\frac{1}{3}\iint\left|u_{nm}\right|^{4}dxdy=\frac{1}{3}\iint\left|v_{nm}\right|^{4}dxdy=\iint\left|u_{nm}\right|^{2}\left|v_{nm}\right|^{2}dxdy$, $Q_{22}=\frac{1}{3}\iint\left|u_{n^{\prime}m^{\prime}}\right|^{4}dxdy=\frac{1}{3}\iint\left|v_{n^{\prime}m^{\prime}}\right|^{4}dxdy=\iint\left|u_{n^{\prime}m^{\prime}}\right|^{2}\left|v_{n^{\prime}m^{\prime}}\right|^{2}dxdy$, $Q_{12}=\iint\left|u_{nm}\right|^{2}\left|u_{n^{\prime}m^{\prime}}\right|^{2}dxdy=\iint\left|u_{nm}\right|^{2}\left|v_{n^{\prime}m^{\prime}}\right|^{2}dxdy=\iint\left|v_{nm}\right|^{2}\left|u_{n^{\prime}m^{\prime}}\right|^{2}dxdy=\iint\left|v_{nm}\right|^{2}\left|v_{n^{\prime}m^{\prime}}\right|^{2}dxdy$, the respective weights being $\rho_{nm}=|c_{1}|^{2}+|c_{2}|^{2}$ and $\rho_{n^{\prime}m^{\prime}}=|c_{1}^{\prime}|^{2}+|c_{2}^{\prime}|^{2}$. Here, $Q_{11}$ and $Q_{22}$ represent the nonlinear self-interaction of the two degenerate modes, while $Q_{12}$ governs the cross-interaction between them. The mutual conversion of the degenerate modes is predominantly driven by the cross-interaction $Q_{12}$, whereas the self-interactions $Q_{11}$ and $Q_{22}$ induce the rotational phase dynamics of the modal profiles [46]. As it follows from Eqs. (14), the nonlinearity break the resonance condition of the original linear system, making it necessary to introduce a correction to the resonance modulation frequency. Values of the correction, which is defined as

[TABLE]

are displayed below in Fig. 10.

We have also performed direct numerical simulations of the mode evolution in the framework of the full nonlinear FSE (1), as shown, in particular, in Fig. 10. The resonance frequency, including the nonlinearity-induced correction, is given by $\Omega=\beta_{00}-\beta_{01}+\beta^{\prime}$, see Eq. (15). Under the action of the spiral modulation, the zero-vorticity mode gradually transforms into the dipole one, and then returns to the zero-vorticity mode after the full cycle(see Supplemental Material, Animation_Fig10 [47]). The weight curve demonstrates that, due to the complexity of the nonlinear system, additional modes are coupled to the mode-transition process, leading to small oscillations and loss features on the curves, thereby reducing the conversion rate.

Finally, we use the coupled-mode theory to investigate the nonlinearity-induced resonance shift. As shown in Fig. 11, in the absence of the nonlinearity, the resonance curve exhibits the symmetric peak. When self-focusing/defocusing nonlinearity is introduced, with $\sigma>0/\sigma<0$, the peak is deformed in opposite directions. As the nonlinearity strength increases, the resulting resonance shift becomes more significant, while the corresponding conversion efficiency decreases conspicuously.

V Conclusion

We have elaborated the scheme for the resonant mode conversion in the optical waveguide, induced by the longitudinal spiral modulation added to the FSE (fractional Schrödinger equation). By adjusting the Lévy index $\alpha$, we enable the excitation of higher-order radial modes and vortex states with extended topological charges which are not stably supported in conventional nonlinear-Schrödinger waveguides, thereby expanding the accessible eigenmode space. Using the truncated coupled-mode theory, we have predicted that the longitudinal spiral modulation drives the resonant conversion between any two eigenmodes with the topological-charge difference $\Delta m=1$, where the conversion period $T$ can be minimized through the selection of the optimal value of $\alpha$. The resonant conversion between two eigenmodes of the same order with $\Delta m=2$ can be achieved with the help of the intermediary mode. This transition scheme elaborated here offers greater controllability in comparison to the known cascading mode-conversion scheme. For degenerate single-component modes, the longitudinal spiral modulation simultaneously drives both the rotation of the modes and their conversion, making complete mode conversion unachievable. However, introducing the azimuthal modulation in the direction identical to that of the spiral modulation, we can enhance the conversion efficiency of the azimuthally modulated vortex solitons. The nonlinearity induces a resonance-point shift, where the self-focusing $(\sigma>0)$ and defocusing $(\sigma<0)$ nonlinearities give rise to opposite directional deviations. This shift correlates with the marked reduction in the conversion efficiency, observed as the nonlinearity strengthens.

Acknowledgements.

This research was supported by the National National Natural Science Foundation of China (Grants No.11805141); Shanxi Province Basic Research Program (Grants No.202203021222250 and No. 202303021211185); Israel Science Foundation (Grant No.1695/22). R.L. was sponsored by the National Key Research and Development Program of China (Grant No.2022YFA1404900), National Natural Science Foundation of China (Grant No.12104353), and Fundamental Research Funds for the Central Universities (Grant No.QTZX25086).

Data Availability

The data that support the findings of this article are not publicly available upon publication because it is not technically feasible and/or the cost of preparing, depositing, and hosting the data would be prohibitive within the terms of this research project. The data are available from the authors upon reasonable request.