Natural shaping of the cylindrically polarized beams
We have experimentally and theoretically shown that the circularly polarized beam bearing a singly charged optical vortex propagating through a uniaxial crystal can be split after focusing into the radially and azimuthally polarized beams in the vicinity of the focal area provided that the polarization handedness and the vortex topological charge have opposite signs. Unique properties of the cylindrically polarized beams (i.e., the radially and azimuthally polarized ones) to produce an
... produce an extremely small focal spot  and to form an electric field with the only longitudinal component  under tight focusing have drawn intense attention for different applications, such as high-resolution microscopy , particle trapping devices , etc. There are many different ways to produce cylindrically polarized beams, e.g., intracavity laser devices , liquid-crystal phase modulators , and others. All these devices need especially accurate alignment of special optical gadgets. The question arises: does a simple, natural way (similar to natural focusing) exist to produce cylindrically polarized beams without losing the beam quality? As early as the beginning of 2000 we observed that the polarized beams propagating along the crystal optical axis form a complex polarized structure (see, e.g.,  and references therein) whose polarization states evenly fill all the Poincare sphere. Such a beam focused by a low-aperture lens can produce two focal spots with salient polarization distributions whose structures are defined by the initial state of the beam (namely, the vortex topological charge and the spin, i.e., handedness of the polarization state), the crystal, and the lens parameters  . The question is: how does one transform such a complex polarization structure into the beams with the desired polarization distribution? The aim of this Letter is twofold: (1) to form experimentally and theoretically cylindrically polarized beams via field focusing after a uniaxial crystal and (2) to estimate the quality of the polarization pattern. As a basis of our theoretical consideration, we assume the sketch of the experimental setup shown in Fig. 1 . The Gaussian beam generated by the laser is turned into the vortex beam by the computer-generated hologram CH (see, e.g.,  and references 12-14 therein). The polarizer Pol 1 and the phase retarder PR 1 transform it into the circularly polarized one that is focused by the lens with focal distance f 1 into a uniaxial crystal Cr whose optical axis is directed along the beam axis. The refractive indices for the beam complex amplitudes are n o , n e ¼ n 2 3 =n o , and n o , n 3 are the refractive indices along the crystallographic axes  . The lens with the focal distance f 2 focuses the beam after the crystal, whereas the diaphragm D truncates the wished field spot. The phase retarder PR 2 and the polarizer Pol 2 , together with computer processing, enable us to plot the polarization structure of the truncated beam field and estimate the quality of the field structure. The field structure of the beams after the crystal is sensitive to the signs of the spin s and the topological charge l of the initial vortex beam and can be radically transformed when changing the signs of s or l. We focus our attention only on the initial beam states with s ¼ 1, l ¼ −1 or s ¼ −1, l ¼ 1. The rest of the beam states are ineligible for our analysis. The transverse beam components can be presented in the circularly polarized basis with s ¼ where Ψ o;e ¼ ðz 0 r=q 2 o;e Þ exp½−ikn 1 r 2 =ð2q o;e Þ, r and φ are the radial and azimuthal coordinates, q o;e ¼ Zþ f 2 q ðo;eÞ 2 =ðf 2 − q ðo;eÞ 2 Þ, q 1 ¼ h þ d þ ðH þ iz 0 Þf 1 =ðf 1 þ Hþ iz 0 Þ, q ðo;eÞ 2 ¼ q 1 þ ðn 1 =n o;e Þz, z 0 ¼ kn 1 w 2 0 =2, w 0 is the radius of the initial beam waist at the laser's output, k ¼ 2π=λ, λ stands for the wavelength of the laser radiation, and n 1 is the refractive index outside the crystal. The distances h, d, z, H, and Z shown in Fig. 1 characterize the geometry of the optical system. In Eq. (1) we made use of the ABCD rule for the centered optical system. The beam state with s ¼ −1, l ¼ 1 can be obtained from Eq. (1) by replacing l → −l, E þ → E − , and E − → E þ . The complex amplitudes Ψ o and Ψ e are characterized by different refractive indices n o and n e ; therefore, the ordinary and extraordinary beams have different wave parameters, in particular, curvature radii of the wavefronts. It is the different curvature radii of these partial Fig. 1. (Color online) Sketch of the experimental setup. The longitudinal section of the focused beam is plotted for n o ¼ 1:654, n 3 ¼ 1:494 at the wavelength λ ¼ 0:634 μm, H ¼ 2:5 m, h ¼ 3:5 cm, d ¼ 4:2 cm, z ¼ 1 cm, f 1 ¼ −5 cm, f 2 ¼ 12:5 cm, and w 0 ¼ 1 mm.