Polarization shaping for control of nonlinear propagation
We study the nonlinear optical propagation of two different classes of space-varying polarized light beams – radially symmetric vector beams and Poincaré beams with lemon and star topologies – in a rubidium vapour cell. Unlike Laguerre-Gauss and other types of beams that experience modulational instabilities, we observe that their propagation is not marked by beam breakup while still exhibiting traits such as nonlinear confinement and self-focusing. Our results suggest that by tailoring the spatial structure of the polarization, the effects of nonlinear propagation can be effectively controlled. These findings provide a novel approach to transport high-power light beams in nonlinear media with controllable distortions to their spatial structure and polarization properties.
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Introduction: Light beams that can propagate without significant change to their spatial profile are of interest for modern optical technologies and high-power laser systems. Self-trapped light filaments, or spatial solitons, are formed when their spreading due to linear diffraction is carefully balanced by a self-focusing (Kerr) nonlinearity that causes the beam to narrow. Due to their potential to carry an increased information content, there has been significant interest in the formation of spatial solitons carrying orbital angular momentum (OAM) firth:97 (); desyatnikov:01 (); musslimani:01 (); soljacic:01 (); bigelow:02 (); bigelow:04 (). OAM-carrying beams are characterized by an azimuthal phase dependence of the form allen:92 (), where the integer corresponds to the topological charge of the phase singularity present at the beam centre (e.g. see yao:11 () and references therein). Such beams include the Laguerre-Gauss (LG) modes barnett:07 (), which are solutions to the paraxial wave equation.
It is well known that in (2+1) dimensions, spatial solitons are unstable in homogeneous Kerr media rasmussen:86 (). One way to increase their stability is to use a saturable self-focusing medium to prevent the catastrophic collapse due to self-focusing. By using an intensity dependent nonlinear refractive index, , it becomes possible to balance out the effects of self-focusing and diffraction. However, even in the case of saturable self-focusing media, it is known that optical beams carrying OAM will fragment into several solitons possessing particle-like attributes desyatnikov:01 (); malomed:05 (). In particular, a scalar beam carrying an OAM value of is predicted to break up into daughter solitons firth:97 (); bigelow:04 ().
In comparison with scalar ring solitons that carry a definite non-zero OAM, it has been suggested that stability can be increased by using two beams with opposite OAM to produce a beam with a net zero OAM soljacic:01 (); bigelow:02 (). For example, “necklace” (petal) beams, which consist of a scalar superposition of two modes with equal and opposite OAM, have been shown to exhibit quasi-stable propagation in a self-focusing medium although they expand upon propagation soljacic:98 (). For vectorial superpositions of beams carrying OAM, however, the resulting vector solitons have been shown theoretically to exhibit quasi-stable propagation for much larger distances than the corresponding scalar vortex solitons and necklace beams desyatnikov:01 (); bigelow:02 ().
Vector vortex beams are fully correlated solutions to the vector paraxial wave equation that have space-varying polarization distributions. Cylindrical vector (CV) beams are a subclass of vector beams with an axially symmetric polarization profile about the beam’s propagation axis zhan:09 (); galvez:12 (). Examples, include radial, azimuthal and spiral polarization distributions. Another class of light beams with non-uniform polarization structure is that of full Poincaré beams beckley:10 (); galvez:12 (). These are of intrinsic interest because they carry polarization singularities. Such beams typically consist of a superposition of two orthogonally polarized LG modes of different orbital angular momenta beckley:10 (); galvez:12 () and thus carry a net value of angular momenta.
In this Letter, we demonstrate, both experimentally and numerically, the stable propagation of space-varying polarized light beams in a saturable self-focusing nonlinear medium. More specifically, our study focuses on vector vortex and Poincaré beams traveling through rubidium vapour. We compare the intensity and polarization distributions of the beams at the entrance and exit of the nonlinear cell. This allows us to see how beam-break up is affected both by the net OAM of the beam and by its polarization distribution.
Theory: Light beams with spatially inhomogeneous polarization distributions can be obtained by a superposition of two spatial transverse modes, and , with orthogonal polarizations
where and are orthonormal polarization vectors, and are the cylindrical coordinates. Here we adopt the circular polarization basis, i.e. and , and the LG basis for the spatial transverse modes beckley:10 (). If the two beams have equal but opposite OAM, the polarization of the beam varies along the azimuthal coordinate and, if the beams are equally weighted, spans the equator of the Poincaré sphere. These radially symmetric vector vortex beams zhan:09 () can have polarization distributions that are radial, azimuthal (see Fig. 1-(a), (b)) or spiral. If and carry a zero and a non-zero OAM value, respectively, the resulting full Poincaré beam beckley:10 () has a polarization that varies in both the angular and radial coordinates galvez:12 () and covers all polarization states on the Poincaré sphere beckley:10 (). The state of elliptic polarization varies with position dennis:09 () and polarization singularities occur at C-points where the azimuth is not defined and the polarization is circular nye:83b (), and along L-lines where the polarization is linear and its handedness is not defined nye:83a (). C-points can have three fundamental polarization topologies classified by the index and the number of polarization lines that terminate at the singularity: these topologies are known as “star” (, three lines), “lemon” (, one line) and “monstar” (, infinitely many, with three straight, lines) nye:99 (); dennis:09 (). Examples of lemon and star topologies are shown in Fig.1 (c) and (d).
We simulate propagation through the medium using a two-dimensional nonlinear Schrödinger equation with saturable self-focusing nonlinerity, derivable from the two-level model, under the slowly varying envelope and paraxial approximations and normalized to dimensionless quantities, and , where is the beam waist and is the Rayleigh range of the beam firth:97 (); bigelow:02 (); desyatnikov:01 (). As we are dealing with vector beams, our model consists of two coupled equations that interact through the cross phase modulation (XPM) term characterized by the parameter , which takes the value of for circularly polarized beams Agrawal ()
The parameters of importance are the nonlinear parameter, , and the saturation parameter, , given by:
where is the free-space wavenumber, and are the linear and nonlinear refractive indices ( for self-focusing), is the saturation intensity, and is the power of the incident laser beam.
In the simulations reported below, we have selected , that reproduce the experimental configuration of the natural rubidium (Rb) cell. We performed numerical integrations of the propagation Eqs. (2), using the split-step method with fast Fourier transforms and parameters corresponding to the experiments performed.
Experiment: We use a spatially filtered, linearly polarized, tunable CW single-mode diode laser (Toptica DL pro 780, 760 nm – 790 nm) together with a set of half- and quarter-wave plates to generate a Gaussian beam with an arbitrary polarization state, , where and are set by the orientation of the waveplates. This beam is converted into a space-varying polarized light beam using a -plate – a slab of patterned liquid crystal – that couples optical spin to OAM marrucci:06 (). The unitary action of a -plate in the circular polarization basis is described by
where is a half-integer number corresponding to the topological charge of the liquid crystal pattern, is the azimuthal orientation of the liquid crystal elements at (laboratory frame) and is the optical retardation of the -plate. The parameter can be experimentally adjusted by applying an electric field onto the plate in such a way that the resulting optical retardation corresponds to a half () or quarter () wavelength slussarenko:11 (). The generated beam is then focused by a -mm-focal-length lens into a -cm-long cell containing Rb atomic vapour. A detailed depiction of this experimental apparatus is provided in Fig. 2. The intensity of the incident beam and the temperature of the atomic vapour are set so that the medium exhibits saturable Kerr nonlinearities (powers in the vicinity of mW and a temperature of C). In order to observe beam breakup of OAM-carrying beams, the laser’s output wavelength was tuned near the transition line of Rb. Moreover, the laser was blue-detuned by less than GHz from the hyperfine transition.
The beam exiting the Rb cell is then imaged using a lens with a focal length of mm (not shown in the experimental setup) and its polarization distribution is reconstructed using polarization tomography. The tomography is performed using an appropriate sequence of a quarter-wave plate, a half-wave plate, a polarizer and a spatially resolving detector (CCD camera) set at an exposure time of s.
In order to generate vector vortex beams (radial, azimuthal and spiral), a linearly polarized Gaussian beam is sent to a tuned () -plate of topological charge cardano:12 (). Apart from a global phase, the generated beam will be given by , where depends on the orientation of the -plate with respect to the input polarization. The generated beams correspond to radial (), azimuthal () and spiral () vector vortex beams.
To generate the lemon and star Poincaré beams, a circularly polarized Gaussian beam is sent to a perfectly detuned () -plate with and , respectively cardano:13 (). Here, and does not affect the polarization topology but does cause a rotation in the polarization pattern. Thus, we choose , resulting in an output beam of the form , again omitting any global phase factors. Note also that monstar topologies cannot be readily generated in the laboratory. This scheme allows us to switch between different structured light beams without altering their intensities.
Analysis: It is well-known that azimuthal modulational instabilities associated with the helical phase structure found in beams carrying OAM result in their filamentation as they propagate through self-focusing media firth:97 (); desyatnikov:01 (); bigelow:02 (). It has been shown both analytically and numerically, however, that the dominant low-frequency perturbations that typically disrupt ring solitons are inhibited for vector solitons with no net OAM desyatnikov:01 (); bigelow:02 (). Here we confirm this experimentally and numerically by propagating a beam of the form through a self-focusing medium. When , the beam is a scalar LG mode of with left, right-hand circular polarization. For we have a vector vortex beam with linear polarization and for , a vector vortex beam of left, right-hand elliptical polarization, respectively, following the topology of the vector beam with . Note that each beam has the same total intensity.
A comparison of the experimental results with the simulations based on Eqs. (2) is shown in Fig. 3 for a vector beam defined by . From our results we can see that the nonlinearity counterbalances diffraction up until the point at which the beams fragment. As expected, we see that the scalar OAM-carrying beams () are starting to break-up at the exit of the nonlinear cell. The vector beam (), on the other hand, seems almost unperturbed, both in terms of its amplitude and polarization distribution. Indeed, we can see numerically that the fragmentation point occurs much later for vector beams, cm, than for scalar beams, cm. For the elliptically polarized vector beams () the stability length is between the cases of scalar and linear vector vortex beams. In this case the effect of the nonlinear propagation is also evident in the change of polarization distributions: the initial spiral distribution has now become almost azimuthal or radial, respectively (see Fig. 1). This suggests that vector beams allow for greater control over the nonlinearity.
In spite of the approximations used in the derivation of Eqs. (2), small differences in polarization distributions between the experiment and the numerical simulation are only really evident in the biased mode cases of , thus demonstrating the robustness of the vector vortex beam ().
It has been proposed that the increase in stability seen in vector beams is due to the fact that they carry no net OAM bigelow:02 (). We therefore repeated our analysis using lemon and star polarization topologies which have a net OAM.
The experimental and numerical results are shown in Fig. 4 where we have plotted the intensity and the polarization distributions for the lemon and star topologies after propagating through the Rb cell. These results again show that beam break-up has been inhibited when compared to scalar OAM-carrying beams. This result demonstrates that the increased stability is not simply due to a net OAM of zero. As in the case of linear vector vortex beams () above, the polarization distribution remains unaltered after nonlinear propagation and even preserves the number and kind of polarization singularities.
Conclusion: We have compared the propagation of scalar OAM-carrying beams with two different classes of beams with non-uniform transverse polarization distributions in a saturable self-focusing nonlinear medium of Rb vapour. With respect to scalar vortex beams (LG modes), we found that beam break up can be inhibited while nonlinear confinement, self-focusing and polarization distributions are not altered for specific cases of non-uniform spatial polarization, both with and without net OAM. This suggests that the spatial structure of the polarization plays an important role in preventing beam fragmentation. These findings provide a novel approach to transport high-power light beams in nonlinear media with controllable distortions to their spatial structure and polarization properties.
Acknowledgment: F.B., H.L., E.K. and R.W.B. acknowledge the support of the Canada Excellence Research Chairs (CERC) program. A. R. acknowledges funding from the European Union (FP7-PEOPLE-2012-CIG, PCIG12-GA-2012-326499-FOXIDUET). E.K. acknowledges the support of the Canada Research Chairs (CRC) program and Canada Foundation for Innovation (CFI).
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