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Vector cavity solitons in broad area Vertical-Cavity Surface-Emitting Lasers

We report the experimental observation of two-dimensional vector cavity solitons in a Vertical-Cavity Surface-Emitting Laser (VCSEL) under linearly polarized optical injection when varying optical injection linear polarization direction. The polarization of the cavity soliton is not the one of the optical injection as it acquires a distinct ellipticity. These experimental results are qualitatively reproduced by the spin-flip VCSEL model. Our findings open the road to polarization multiplexing when using cavity solitons in broad-area lasers as pixels in information technology.

Localized structures have been observed in various systems of nonlinear science such as chemistry, physics, plant ecology and optics [1][2][3][4][5][6][7][8] . Localized structures and localized patterns are stable structures that arise in a dissipative environment and belong to the class of dissipative structures found far from equilibrium 9,10 . The spontaneous emergence of dissipative structures arises from a principle of self-organization. A classic example of spatial self-organization is provided in the context of chemical reaction diffusion system referred to as Turing instability 11 .
In the framework of nonlinear optics and laser physics, two different kinds of dissipative structures are of particular interest. They are called temporal and spatial solitons. When they are generated inside a cavity they are referred to as Cavity Solitons (CSs). The Fresnel number of an optical cavity is defined as λ = /( ) F a L 2 with cavity width and length a and L, respectively and light wavelength λ. For small F, light propagates as well defined transverse modes and if the light pulses preserve their temporal shape during propagation, they are called temporal CSs. Typical examples are temporal CSs in fibre cavities, as they are a good example of one-dimensional systems 12 . When the Fresnel number F is sufficiently large, spatial CSs have been theoretically predicted 13,14 and experimentally observed 15,16 .
Cavity Solitons have always been linked with information technology because of their robustness and versatility. Their temporal version has been suggested to be used for optical storage in a fiber cavity (the fiber cavity is then used as a memory buffer) 12,17 and for use in information transmission 18 . Their spatial counterparts have been suggested as pixels 16 . The vectorial nature of optical waves can be used in the framework of CSs generation. When the CSs have a single polarization component, they are said to be scalar. Otherwise, these objects are called vector CSs. In the context of information technology, they are of great interest, as information can be stored in a three parameter space, instead of a single one. Scalar spatial CSs have been experimentally demonstrated in many optical systems, such as laser pumped sodium vapor 19 , laser with a saturable absorber 20 or photorefractive medium 21 , in liquid crystal light valve 22 , Kerr media 23,24 and Vertical-Cavity Surface-Emitting Lasers (VCSELs) 15 . VCSELs have several features that make them a choice material for studying vector CSs: their importance in the telecommunications market has allowed a mastering of the fabrication process, and hence of the desired properties of the VCSELs 25 . Their polarization dynamics have been studied extensively (see e.g. 26 for a review) and they have been proven to exhibit polarization chaos 27 . The CSs generated in a VCSEL with a saturable absorber have also been predicted to exhibit temporal chaos as well 28 .
Temporal vector solitons have been generated in erbium-doped fiber laser 29,30 , and in a small area (single mode) VCSEL, placed in an external cavity 31 . Their spatial counterparts still remains to be demonstrated, even though some work has been performed on the polarization behavior of CSs in VCSELs. In 32 , an orthogonally polarized CS is generated in a medium size VCSEL using a linearly polarized optical injection, orthogonal to the one the VCSEL spontaneously emits close to the lasing threshold. The theoretical model used to describe this experiment assumes locking of the VCSEL to the optical injection, and hence a polarization of the resulting field that is the same as the one of the optical injection. In 33 , CSs are generated in a monolithic system that does not have a preferred polarization direction: the VCSEL is submitted to frequency-selective feedback from a volume Bragg grating. In this system, the direction of the main axis of the polarization ellipse of the CSs is measured, but without any measurement of the ellipticity of these structures, and without providing a theoretical model.
In this letter, we report experimental and theoretical investigations of polarization properties of CSs generated in a VCSEL suject to optical injection. We demonstrate experimental evidence of two-dimensional vector CSs. We show that CSs are not linearly polarized. They are rather elliptically polarized. The experimental results are supported by theoretical investigations based on the spin-flip VCSEL model. Finally, we compare the experimental observations with numerical simulations of the vectorial nature of CSs.

Experiment
The VCSEL we use is a 80 μm diameter multiple quantum well bottom emitting device similar to the ones described in 34 . As it is pumped slightly above its lasing threshold, it emits linearly polarized light 35 . Our experimental setup is schematically shown in Fig. 1. The VCSEL is submitted to optical injection from an external cavity diode laser in Littrow configuration (Sacher Lasertechnik TEC100-960-60), isolated from the rest of the setup by an optical isolator. The direction of linear polarization of injected light is modified using a half-wave plate, before being purified with a Glan-Thompson prism. Optical injection power is then tuned by changing the orientation of the half wave plate. A 10/90% beam splitter is set between the polarizer and the VCSEL to allow a measurement of the optical injection power and to direct the light coming from the VCSEL towards the analysis branch. The analysis branch is made of an optical spectrum analyser (ANDO AQ6317-B), a photodiode, and a CCD camera, on which an image of the near field profile of the VCSEL is formed.
When the optical injection is linearly polarized and has its polarization direction matching the one that the VCSEL preferably emits when being pumped slightly above its lasing threshold, the situation is the same as in a previous communication 35 . Figure 2 shows an experimental bistability curve obtained, i.e. the VCSEL output power as a function of optical injection power hysteresis, for optical injection polarized in the direction of the one of the free-running VCSEL. The inset a) (resp. b)) represents the near field profile on the upper (resp. lower) branch of the hysteresis. On the lower branch of the hysteresis, the emission profile is rather homogeneous (see inset b)). As can be seen from Fig. 2, at P inj up , the profile of the VCSEL near field emission changes, and a bright dot appears at the surface of the VCSEL. The overall optical power emitted by the VCSEL also undergoes a sudden increase. Starting from that configuration, as the optical injection power is now attenuated, the bright dot stays in place down to when another threshold is reached: P inj down . At that point, the bright dot disappears, and the optical power emitted by the VCSEL decreases as strongly as it increased at first.
Such bistable spots have been obtained for different linear polarization directions of the optical injection. This bistability has been experimentally proved using the instruments contained in the box denoted (1) in Fig. 1. However, to measure the Stokes parameters of the CS, the setup needs to be modified. This is performed by replacing the box (1) by the box (2) as shown in Fig. 1. First, an iris is inserted in order to isolate the CS, and a quarter-wave plate, a polarizer and a photodiode are used to measure its Stokes parameters. They are experimentally determined by measuring a set of optical intensities optical injection. The results of this analysis are summarized in Fig. 3. The Stokes parameters , s 1 2 as a function of the angle Ψ between the horizontal and the linear polarization direction of the optical injection is plotted in Fig. 3a,b. The direction of the main axis of the polarization ellipse is obtained from the Stokes parameters , s 1 2 i.e., θ = ( / )/ s s arctan 2 2 1 . Its evolution as a function of Ψ is shown in Fig. 3c. The Stokes parameter s 3 as a function of Ψ is shown in Fig. 3d.
The presence of a nonzero , s 1 2 shown in Fig. 3a,b, accounts for the presence of a linear part in the CS polarization state. Moreover, the nonzero s 3 shown in Fig. 3d provides evidence of the presence of a circular part of the CS polarization state. From these two observations, we show that these CSs have two polarization components, which proves the vectorial nature of CSs. Vector CSs have been observed for a wide range of experimental parameters, as well as for a VCSEL biased below the lasing threshold.
We have carried out an additional set of experiments in order to provide further evidence of the vectorial nature of our cavity solitons. To this end, we fix the polarization angle Ψ of the holding beam at 20 degrees.
First, we have applied our procedure for Stokes parameter measurement on the pump and obtained the following Stokes parameters: = − . 3 . Indeed, this shows a clear evidence that the holding beam is well linearly polarized and that the amount of circularly polarized emission in the cavity soliton up to = . s 0 22 3 is not due to the holding beam emission but is inherent to the VCSEL polarization properties. We have also proved that the vectorial nature of the CS does not come from the background but it is inherent to the VCSEL. We first measure the s 3 parameter of a cavity soliton by inserting an iris in the detection setup to isolate the soliton. We obtain = .
, s 0 44 CS 3 . Notice that this value is twice as large as the one in Fig. 3, where the experiment has been performed for different conditions, such that the cavity soliton persists in the whole range of Ψ from 0 to 90 degrees. Then, we shift the iris transversely without changing its size and measure the s 3 parameter of the background obtaining = − . , s 0 07522 BG 3 . Clearly, while the background is almost linearly polarized with somewhat larger s 3 parameter than the holding beam, the CS acquires considerable amount of circularly polarized light, i.e. it is clearly not a scalar CS that simply follows the polarization of the holding beam.
Finally, we identify the principal and the orthogonal polarizations of the holding beam and acquire images of a cavity soliton in these two polarizations (see Fig. 4). Clearly, CS persists in the orthogonal polarization due to its vectorial nature.

Theory
Polarization properties of VCSELs are best described by the spin-flip VCSEL model 36 , where travelling wave and modulation instabilities leading to the formation of extended patterns have been reported. Another important study of the polarization state of VCSELs has been published in 37 . This paper best describes small area VCSELs, where transverse effects are neglected, i.e., diffraction is neglected. In what follows, we will adopt the same modelling procedure as in 37    where κ is the photon lifetime, γ a is the amplitude anisotropy of the VCSEL, γ p is the phase anisotropy, ω ∆ is the detuning parameter, α is the linewidth enhancement factor, E I is the optical injection amplitude, a is a scaling factor for diffraction, γ is the relaxation rate for the overall carrier density N , µ is the electric pumping and γ s is the spin-flip relaxation rate for the carrier density difference n 36,37 . Ψ is the angle between horizontal and the direction of the optical injection linear polarization, and "*" denotes a complex conjugate.
Homogeneous steady states (HSSs) of eqs 1-4 as a function of optical injection power E I is plotted in Fig. 5, in the bistable regime. Linear stability analysis of HSSs shows that there exists a symetry breaking instability that leads to the formation of periodic structures with an intrinsic wavelength. A portion of the lower HSS is stable, and coexists with a periodic pattern. In addition, the system exhibits a high degree of multistability in a finite range of injection values. More precisely, eq. (1) then admits an infinite set of odd and even CS's, i.e., a set of stationary solutions that exhibit different number of peaks. An evidence of the presence of vector CSs in the spin-flip VCSEL model is given in Fig. 6 where eqs 1-4 have been integrated in two-dimensional settings for the same parameters as in Fig. 5 and = E 1 I . From the Stokes parameters s 1 and s 2 we deduce that the CS has a nonzero linear polarization component. Furthermore, the CS acquires a certain ellipticity as evidenced by the nonzero s 3 Stokes parameter. The structures evidenced in Fig. 6 are hence vector CSs.
In order to investigate the dependence of the CS Stokes parameters on the polarization angle of the injection light, as has been done in the previous experimental section, we repeat these simulations in one dimension. The results of these simulations are presented in Fig. 7. The experimental results shown in Fig. 3 and the numerical results presented in Fig. 7 are in good agreement.

Conclusion
We have reported experimentally and numerically, for the first time, evidence of vector Cavity Solitons in a broad area VCSEL submitted to linearly polarized injection. We have characterized experimentally their polarization state by measuring their Stokes parameters. This analysis revealed that cavity solitons acquired ellipticity. The dependence of the linear polarization direction of the optical injection has been systematically investigated. To describe these experimental results, we used the spin-flip VCSEL model, and we have shown that this model admits Cavity Soliton as solutions. We have performed a systematic analysis of their polarization properties. These numerical results show good agreement with the experimental data. One may expect that, owing to its general character, the vector Cavity Soliton should be observed in other spatially extended systems. Using CSs in broad-area VCSELs that can be switched on and off independently as pixels for information processing 16 constitutes a bitmap. Considering the Stokes parameters of the vector CSs demonstrated here, one would potentially create a colormap instead of a bitmap, i.e. each pixel contains information in a 3-parameter space ( , , s s s 1 2 3 ) that easily can be translated into a color code. In such a way, the density of information can be dramatically increased.