Generalized $q$-plates and alternative kinds of vector and vortex beams

We took a generalization of the conventional concept of the $q$-plate, allowing in its definition nonlinear functions of the azimuthal coordinate, and simulated the resulting fields of applying this kind of element to uniformly polarized input beams, both in the near-field (Fresnel diffraction) and the far-field (Fraunhofer diffraction) approximations. In general terms, when working in the near-field regime, the chosen function defines the output polarization structure for linearly polarized input beams and the phase of the output field for circularly polarized input beams. In the far-field regime, it is obtained that when there are nonlinearities in the azimuthal variable, the central singularity of the polarization field of a vector or vortex beam may divide into several singularities of lower topological charge, preserving the total charge. Depending on the chosen $q$-plate function, different particular behaviors on the output beam are observed, which offers a whole range of possibilities for creating alternative kinds of vector and vortex beams, as well as polarization critical points and singularity distributions.

Figure 1

Argument phase function of the generalized $q$-plates determined by a polynomial growth in $\theta$. The first, second, and third columns show the cases for powers $p=1, p=2$, and $p=3$, respectively, while the first, second, and third rows show the cases for topological charges $q=1, q=3/2$, and $q=2$, respectively.

Figure 2

Polarization ellipses and their azimuths resulting from input vertical polarization for polynomial generalized $q$-plates. The first and third columns show intensity and polarization ellipses for $p=1$ and $p=2$, respectively. The second and fourth columns show polarization azimuths for $p=1$ and $p=2$, respectively. The first, second, and third rows show cases with topological charges $q=1/2, q=1$, and $q=3/2$, respectively. The output consists of vector beams in which the azimuth grows accordingly to the power $p$.

Figure 3

Polarization ellipses and phase distributions resulting from input left circular polarization for polynomial generalized $q$-plates. The first and third columns show intensity and polarization ellipses for $p=1$ and $p=2$, respectively. The second and fourth columns show the phase for $p=1$ and $p=2$, respectively. The first, second, and third rows show cases with topological charges $q=1/2, q=1$, and $q=3/2$, respectively. The output consists of uniformly right circular polarized beams in which the phase grows accordingly to the power $p$.

Figure 4

Polarization ellipses and their azimuths resulting from input vertical polarization for polynomial generalized $q$-plates, in the far-field regime. The first and third columns show the intensity distribution and the polarization structure for $p=1$ and $p=2$, respectively. The second and fourth columns show the azimuths of the polarization ellipses for $p=1$ and $p=2$, respectively. The first, second, and third rows show the cases for $q=1/2, q=1$, and $q=3/2$, respectively.

Figure 5

Zoom of one of the polarization singularities obtained in the far field for a polynomial generalized $q$-plate with $q=1/2$ and $p=2$, when the input polarization is vertical. (a) Intensity and polarization ellipses. (b) Azimuth of the polarization ellipses. (c) Form factor of the polarization ellipses.

Figure 6

Polarization ellipses and phase distribution resulting from an input beam with left circular polarization for polynomial generalized $q$-plates, in the far-field regime. The first and third columns show the intensity distribution and polarization structure for $p=1$ and $p=2$, respectively. Polarization is in every case uniform right circular. The second and fourth columns show the phase of the field for $p=1$ and $p=2$, respectively. The first, second, and third rows show the cases for $q=1/2, q=1$, and $q=3/2$, respectively.

Figure 7

Normalized weights $C_k^2=c_{s_k}^2+c_{p_k}^2$ of the terms in the Fourier transforms of the fields created with polynomial generalized $q$-plates with $q=1/2$ when the input beam is vertically polarized. (a) Linear case ($p=1$). (b) Nonlinear case ($p=2$).

Figure 8

Intensity and polarization patterns of Fourier transform terms of a field created with a polynomial generalized $q$-plate with $q=1/2$ and $p=1$, for vertically polarized input light. (a) Term with $k=-1$. (b) Term with $k=1$. (c) Sum of both terms.

Figure 9

Intensity and polarization patterns of Fourier transform terms of a field created with a polynomial generalized $q$-plate with $q=1/2$ and $p=2$, for vertically polarized input light. (a) Term with $k=0$. (b) Sum of terms with $k=-1$ and $k=1$. (c) Sum of terms with $k=-1, k=0$, and $k=1$.

Figure 10

Scheme of the optical device used for computing the beam evolution as it propagates from the generalized $q$-plate to the far-field regime.

Figure 11

Fresnel diffraction pattern at different planes resulting from a polynomial generalized $q$-plate with $q=1/2$ and $p=2$, when the input light is vertically polarized. The intensity distribution superposed to the polarization ellipses is shown in the first column. The azimuth and the form factor are depicted in the second and third columns, respectively. Rows show the evolution in the $z$ coordinate, for $z=0, z=\mathbf{f}-\mathbf{f}/2, z=\mathbf{f}-\mathbf{f}/8, z=\mathbf{f}-\mathbf{f}/32$, and $z=\mathbf{f}$, respectively.

Figure 12

Argument function $\left[2\mathrm{\Phi }\right(r,\theta \left)\right]$ for sinusoidal generalized $q$-plates with $q=1, q=2$, and $q=3$, respectively.

Figure 13

Results provided by sinusoidal generalized $q$-plates with $q=1$ (first row) and $q=2$ (second row). The first column shows the intensity and polarization distribution for the vertical input polarization and the second column shows its respective azimuth. The third column shows the intensity and polarization distribution for the left circular input polarization and the fourth column shows its respective phase.

Figure 14

Results for sinusoidal generalized $q$-plates in the far-field regime when the input polarization is vertical. The first column shows the polarization and intensity distribution, the second column shows the azimuth, and the third column shows the form factor. Rows show the cases for $q=1, q=2, q=3$, and $q=4$, respectively. Yellow lines demarcate minimum intensity areas.

Figure 15

Results for sinusoidal generalized $q$-plates in the far-field regime when the input polarization is left circular. The first column shows the intensity and polarization distribution, while the second column shows the phase distribution. Rows show the cases for $q=1, q=2, q=3$, and $q=4$, respectively. Yellow lines demarcate minimum intensity areas.

Figure 16

Experimental compact device proposed for emulating generalized $q$-plates using a reflective PA-LCoS.