Gaussian-beam-induced vortices in vertical-cavity surface-emitting lasers

We have triggered transverse pattern modifications by injecting a CW monochromatic (10 MHz) beam in a vertical-cavity surface-emitting laser (VCSEL). The injected Gaussian beam has a frequency close to that of the natural TEM₀₀ mode of the VCSEL (Fig. 1). The spot of the injection is not super-imposed with the original lasing spot (Fig. 2). However, it is close enough to lock the TEM₀₀ mode, as seen in the interferogram of Fig. 3. As a result we observe a system of two Gaussian beams locked in frequency and in phase, propagating on parallel axes, and with waists of different size and location. The interferogram in Fig. 3 shows two vortices of opposite topological charges located on both sides of the system of two Gaussian beams. It has been shown theoretically that if such a system had the two beams propagating along the same axis, a series of vortices would appear, with their trajectories along concentric rims normal to the axis of propagation. These vortices do not display the phase singularity behavior in the plane normal to the axis of propagation and cannot be observed interferometrically. However, when the beams' axes are not collinear, vortices appear with their trajectories along lines oriented in the same direction as the axes, and thus can be observed as phase singularities in the plane normal to the axes of propagation. In our experiment, the image plane coincided with the focal plane of the injected beam. In this plane, in a bipolar coordinate system (r₁, r₂) with poles coinciding with axes of the beams, the locations of vortices are given by the following formulas: r₁² = 2[1 + (m + 1/2 ) λ÷D] [D² + π² w₁⁴÷λ²], m = 0, ±1, ±2, . . . r₂² = w₂²[r₁÷w₁² + λ²D²/π²w₁² - ln A₁÷A₂], where index 1 stands for the laser beam, and index 2 is for the injected beam. D is the distance between focal planes of the two beams; w, are half-waists of the beams; A, are beam amplitudes; and λ is the wavelength. The pair of vortices observed in the experiment corresponds to the order m = - 15. A neighboring pair of vortices should be located outwards at a distance of more than 5 μm from the observed vortices. In our experimental setup, it is currently out of observation. The distance between the beams' axes is not a parameter in the above formulas. Hence, bringing the beams closer increases the distance between the vortices, and vice versa. This dependence was clearly confirmed in the experiment. Likewise, r₂ linearly depends on w₂. When w₂ increases, r₂ increases also, separating the two vortices further. These changes in positions of the vortices were also clearly seen in the experiment.