Stable propagation of the Poincaré polarization solitons in strongly nonlocal media

We report the first experimental observation of spatial solitons with complex polarization states, called the Poincaré polarization solitons (PPSs) in lead glass with strongly nonlocal nonlinearity. The formations of PPSs with topological charge of l = 1, including the cylindrical elliptical-polarization soliton (CEPS) and the angularly-hybrid polarization soliton (AHPS), were observed. We showed that the annular profiles and the complex polarization distributions of the first-order PPSs can be remained. Based on the linear stability analysis, we proved that the first-order PPSs are fully stable and the second-order PPS can survive only when one of the two component vortices dominates.


Introduction
A vector beam is the beam whose polarization states are inhomogenously distributed in space. The first proposed vector beam is cylindrical vector beam (CVB) which is composed of linearly-polarized light with cylindrical symmetry in polarization [1]. Due to their unique vectorial properties, CVB has attracted great interest in optical trapping [2,3], microscopic imaging [4,5], and optical communication [6,7]. In recent years, based on the superposition of two orthogonally-polarized phase vortices [8], there proposed a vector beam with complex vectorial nature, the Poincaré beam (PB) [9,10], of which the CVB is a particular case, where R and z are the radial and longitudinal coordinates, respectively, and ϕ is the polar angle. l 1,2 are the vortex topological charge, E 1,2 (R, ϕ, Z) are the spatial modes of the electric field corresponding to the topological charges, tan 2 θ and 2ϕ 0 are respectively the power ratio and the initial phase difference when ϕ = 0 between the two vortex fields, and e 1,2 are two transverse orthogonal unit polarization vectors (circular e 1 = e R = 1/ √ 2(e x − ie y ), e 2 = e L = 1/ √ 2(e x + ie y ) or linear e 1,2 = e x,y ). Generally, when the above parameters in equation (1) take different values, the distribution of the polarization of PB on the beam cross section shows different characteristics. Specifically, in the case of l 1 = −l 2 = l, when (e 1 , e 2 ) = (e R , e L ), PB can be named the cylindrical elliptical-polarization beam (CEPB). Its polarization state is characterized by polarization ellipses with uniform ellipticities determined by θ, but cylindrically-distributed optical orientations starting from the initial angle ϕ 0 at ϕ = 0. When (e 1 , e 2 ) = (e x , e y ), PB can be named the angularly-hybrid polarization beam (AHPB), whose polarization state is characterized by polarization ellipses with periodically-changed ellipticities and orientations along ϕ. It is worth mentioning that CEPB can be mapped onto the higher-order Poincaré sphere [11]. We collectively term the class of electric fields described by equation (1) the first-order PB in the case of l 1 = −l 2 = 1. Complex vector light fields are found to have novel properties, such as inducing curl force [12], carrying orbital angular momentum without phase vortices [13], and having complex multivortex fields under tight focussing [14].
In the nonlinear optical regime, most studies showed that vector beams cannot form stable solitons with spatially inhomogeneous polarization in Kerr media [15][16][17]. The collapse of the nonuniformly-polarized soliton can be arrested by using saturable nonlinearities [18], photorefractive crystals [19][20][21] or nonlocal materials [22,23]. The nonlocal properties of materials, either orientational [24] or thermal [25], can 'average' the angular modulation instability which makes the nonlocal spatial solitons more stable than local ones. Forms of the nonlocal spatial solitons include, such as, scalar vortex solitons [25][26][27], incoherent solitons [28], photon droplet [29], solitons based upon structured beam [30], and vector vortex solitons, which are composed of two incoherent linearly polarized vortex solitons [31][32][33][34][35][36]. Particularly, the nonlocality in lead glass has an infinite range but cut by its boundary [25], and more importantly, its thermal nature can reduce the sensitivity of the material response to the light polarization. This helps lead glass to support the stable propagation of two orthogonally-polarized vortex solitons [37,38]. Recently, the stable propagation of the first-order cylindrical vector vortex soliton is experimentally and theoretically demonstrated in lead glass [22]. From a point of view of the complexity of polarization distribution, cylindrical vector vortex soliton is the most basic nonuniformly-polarized soliton. The studies on stable Poincaré polarization solitons (PPSs) formed by PBs with complex vectorial nature are scare.
In this work, we experimentally demonstrate the stable propagation of first-order PPSs in lead glass, including the cylindrical elliptical-polarization soliton (CEPS) and the angularly-hybrid polarization soliton (AHPS). The results show that the soliton annular spot pattern and polarization distribution are well maintained on the exit surface of the lead glass. Based on the linear stability analysis method, we theoretically prove the stable propagation of first-order PPSs. In addition, we show that the second-order PPSs are generally unstable and will split after a short distance of propagation.

Experiments and results
A sketch of our experimental setup is shown in figure 1. The linearly polarized beam from the CW laser passes through a polarization beam splitter (PBS) to be split into two orthogonally polarized beams with tunable power ratio (tan 2 θ in equation (1)) changed by a half-wave plate (H 1 ) [32]. The vertically and horizontally polarized beams transmit in clockwise and counterclockwise directions, respectively, in a Sagnac interferometer [39] in which there inserted a first-order (l 1 = 1) vortex phase plate (VPP) and a Dove prism (DP). The DP here acts as an additional mirror (at its lower bottom) to allow different numbers of beam reflections [32] after the beams passing through the VPP in two opposite directions. After exiting PBS, the two vortex beams propagate coaxially with a combined electric field described by equation (1), carrying charges l 1 = 1 and l 2 = −1, respectively. After passing through a quarter wave plate (Q 1 ) and a convex lens F, they are focused onto the front face of the cylindrical lead glass sample with radius R = 7.5 mm and length L = 57.5 mm.
When the fast axis of Q 1 is π/4 relative to x-axis, the unit polarization vectors in equation (1) (e 1 , e 2 ) = (e R , e L ), and the initial phase difference | 2ϕ 0 |= 0, which can be changed by inserting a cascaded half-wave plates (H 2 -H 3 ) after Q 1 (right dashed square) [40]. When the fast axis of Q 1 is horizontal or vertical (along x-or y-axis), (e 1 , e 2 ) = (e x , e y ), and | 2ϕ 0 |= π/2, which can be changed by rotating Q 1 in the x-z or y-z planes to alter the corresponding optical distance. In the two above cases, we obtained first-order CEPB and AHPB respectively before the beams go into the lead glass. The beam profiles on the front and output face of the sample can be imaged by a CCD camera. Their polarization states can be detected by inserting a linear polarizer analyzer (LPA) (left dashed square) in front of the CCD, which is designed to cause the extinction of an arbitrary polarization state [10].
First, we study the formation process of the first-order PPS in a lead glass rod. Figure 2 represents the input and output beam images under the different input laser powers. The upper and lower rows are for CEPB and AHPB with θ = π/12, ϕ 0 = π/4, and θ = π/4, ϕ 0 = π/4, respectively. The initial ring radius (defined by the half of the maximum intensity distance [37]) of CEPB W max = 33.25 µm (figure 2(a)), and hence the normalized diffraction distance was z = L/Z 0 = 2.3 for our sample, where Z 0 = kW 2 0 is the Rayleigh length and W 0 is the Gaussian beam width of the Laguerre-Gauss profile. CEPB linearly diffracted at low input power of 10 mW (figure 2(b)), shrunk with the increasing input powers (figures 2(c), (d)), and finally approached their initial size with a ring radius of W max = 31.03 µm (figure 2(e)) at the critical power P c = 265 mW. Similarly, AHPB has an initial ring radius of 31.08 µm (figure 2(f)), output soliton radius of 30.51 µm (figure 2(j)) at the same critical power. It must be pointed out that the propagation distances of the first-order PPSs are relatively short in the experiment. This does not because they are inherently unstable when propagate longer, but because of the unavoidable loss, such as light scattering and absorption [22,25,34,37].  Next, we examined the beam polarization states by inserting a LPA in front of the CCD camera. We measured eight polarization states, each of which is theoretically expected to be located at two centrally-symmetric polar angles (ϕ, ϕ + π) as shown in figures 3(a) and (m) for CEPB and AHPB, respectively. The second and the fifth rows are respectively the experimental input beam images after the LPA for CEPB and AHPB. The third and the sixth rows are the corresponding output soliton images after the LPA. As well known, each polarization state can be mapped to a coordinate (α, β) on the basic Poincaré sphere. α and β represent the ellipticity and the azimuth for one polarization state and correspond, in our case, to the angles of the two half-wave plates H 4 and H 5 in the LPA, respectively. For specified values of (α, β) (three typical values, as examples, shown on the top or middle of figure 3 for CEPB and AHPB, respectively), we rotated H 4 and H 5 to the corresponding angles. The green dashed lines in the right three columns direct the polar angles, correspondingly depicted in the leftmost column, where the light has the polarization state described by (α, β) [10]. The theoretical beam images after the LPA are obtained by simulating the beams described by equation (1) with a first-order Laguerre-Gaussian profile passing through the Jones matrix of the LPA. We found that the measured angular deviation of the polarization state is not more than π/36 compared with the corresponding theoretical expectations figures 3(b)-(d) and (n)-(p).
Then we used a Glan prism (GP) for further polarization detection. The first and fourth rows of figure 4 are the theoretically expected beam images after the GP, whose orientation is shown with arrows, for the CEPB and AHPB, respectively. The second to third and the fifth to sixth rows of figure 4 are their corresponding input and output experimental results. For CEPB, owning cylindrical elliptical-polarization, there exist two places on the beam cross section after GP showing maximum intensities, and the line between them (not shown) rotate following the rotation of GP, while no extinction occurs except the zero intensity at the central singularity. As for AHPB, owning angularly-hybrid polarization, the beam images after GP display different characteristics when the direction of GP changes. When the direction of GP is vertical, the beam intensity after GP shows cylindrical feature (figures 4(l), (o), (r)), because only the 'e y ' component of AHPB contributes. When the direction of GP is perpendicular to the direction of a certain linear polarization on the beam cross section (refer to figure 3(m)), there will be 'black lines' on the beam images after GP in figures 4(k), (n), (q). This feature is the same as that in figures 3(s), (w) where LPA acts as a GP. Figures 3(m), (p) represent the 'intermediate state' between the above two cases, which neither has extinction nor the cylindrical feature. Combing the results of figures 3 and 4, we conclude that the measured polarization distributions of the first-order PPSs can be preserved during the propagation.

Theory and discussion
To model the basic physics behind the experimental results presented above, we theoretically investigated the PPS propagation in lead glass. The electric field E with an amplitude A can be expressed as where e 1 , e 2 are consistent with those in equation (1). The propagation of vector solitons in lead glass can be described by the coupled vector Helmholtz equation and the Poisson equation [22]. Taking account of the paraxial and slowly varying envelope approximations, we directly provide two coupled partial differential equations about the two components of the vector amplitude A in a dimensionless form: where a 1,2 = (αβk 2 W 4 0 /κn 0 ) 1/2 A 1,2 ; n = k 2 W 2 0 ∆n/n 0 is proportional to the nonlinear change ∆n of the linear refractive index n 0 (= 1.9); α = 0.07 cm −1 , β = 14 × 10 −6 K −1 , and κ = 0.7 W (mK) −1 are the absorption coefficient, the thermo-optical coefficient, and the thermal conductivity coefficient, respectively. r and z are the normalized radial and longitudinal coordinates, which are defined by R/W 0 and Z/kW 2 0 , respectively, with k being the wave number in lead glass.
The corresponding PPS solution of equation (3) is assumed to has the form a 1 = q(r) cos θe i(lϕ+ϕ0) e ibz , Here q(r) is the real soliton profile, b is the propagating constant. By inserting equation (4) we simplify equation (3) into coupled equations about q(r) It is notable that the above equation is independent of θ and ϕ 0 , which means all of the same order PPSs, whatever the polarization distributions they own, have the same soliton profiles. The light-induced refractive index distribution can be solved through the Green's function method. Under the circumstance of the radially symmetric excitation (circular cross section of the media, normal and central incidence, cylindrical beam spots), n = −´r 0 0 G 0 (r, ρ)q 2 (ρ)dρ, where the Green's function G 0 (r, ρ) is given by [27] and r 0 is the normalized sample radius. We refer to the Newton-iterative method used in [22] to find the solution of equation ( To analyze the stability of the PPSs, we performed the linear stability analysis of its solution with the azimuthal perturbation where u 1,2 and v 1,2 are the perturbation profiles, δ is the complex perturbation growth rate, and m is the azimuthal perturbation index. Substitution of the perturbed solution into equation (3) and linearization yields the eigenvalue equations which are similar to equations (9) and (10) in [22] but related to θ. The real part δ r of the perturbation growth rate represents the stability of the solitons. On the ground that δ r of the first-order PPSs is invariably no greater than zero (not shown), they are completely stable regardless of the value of θ. Figure 5(c) shows the change of the maximum value of δ r with θ when l = 2. We can see that there exist two stable regions for the second-order PPSs, 0 ⩽ θ ⩽ π/33 and (π/2 − π/33) ⩽ θ ⩽ π/2 (π/33 marked with a blue dot in figure 5(c)), where δ r is invariably zero. This is interesting that although the two second-order orthogonal vortices propagate stably alone [27], their incoherently superimposed PPS is generally unstable unless any one of the two vortices dominates. This result can also help to understand that the second-order cylindrical vector vortex soliton is unstable [22] because it is actually the PPS with θ = π/4, the central value in the unstable region in figure 5(c). The higher-order PPSs are completely unstable and their δ r is invariably greater than zero (not shown), regardless of the value of θ.
To verify the prediction of the stability analysis, we carry out the numerical simulations of propagation for the PPSs, based on equation (3). Figures 6(a) and (b) are the input beam images for first-order CEPS and AHPS, respectively and (c) and (d) are for second-order CEPBs. Their two output orthogonal linearly-and circularly-polarized vortex components are shown in figures 6(e)-(l). It is clear that the two vortex components of the first-order PPS and the second-order PPS located at the 'stable region' (first to third Figure 6. The numerical simulation of the input (first row) and the two orthogonal output (second and third rows) beam images. The polarization states are displayed with black lines, green and blue ellipses (circles) for linearly, right-handed and left-handed elliptically (circularly) polarized beams, respectively. The values of θ and ϕ0 of the first-order CEPB and AHPB (left two columns) are the same with those in figures 2-4. For second-order CEPBs, θ = π/36, ϕ0 = 0 in the third column, and θ = π/6, ϕ0 = 0 in the forth column (π/36 and π/6 are marked in figure 5(c)). The normalized propagation distances are shown in the respective figures. columns) can perfectly remain the donut profiles after relatively long propagation. This helps us to conclude that the corresponding composite PPSs propagate stably, with unchanged donut-profile and polarization distribution. While as shown in figures 6(h) and (l), the two circularly-polarized vortex components of second-order CEPB in the 'unstable region' undergo azimuthal breakup relatively soon. The two black dislocations in their corresponding output profiles indicate the influence of the second-order perturbation.

Conclusion
In conclusion, we experimentally observed the first-order PPSs in cylindrical lead glass, including two types of nonuniformly-polarized solitons in view of the polarization characteristic, CEPS and AHPS. The unchanged doughnut shapes of output soliton images were observed, and the cylindrical elliptical-and the angularly-hybrid polarizations were detected, for CEPS and AHPS respectively. The stable propagation of first-order PPS was proved in a rigorous sense based on the linear stability analysis method. According to the fact that, the change of any one of the parameters in equation (1) generates a new polarization distribution for PPS, our results indicates an infinite number of stable first-order PPSs with distinct polarization distribution. On the other hand, we proved that the second-order PPS survives only when one of the two orthogonal vortices dominates and the corresponding polarization is almost homogenous. This reveals that both the phase structure and the polarization complexity play important roles on the stability of a nonuniformly-polarized soliton. In addition, as expected, the higher-order PPSs are proved to be fully unstable. The numerical simulation verifies the prediction by monitoring the two orthogonally polarized vortex components of the PPS. This work guides further effort on more complicated PPS with |l 1 | ̸ = |l 2 | whose polarization distribution rotates with the propagation distance.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).