Astigmatic tomography of orbital-angular-momentum superpositions

We use astigmatic transformations to characterize two-dimensional superpositions of orbital-angular-momentum states in laser beams. We propose two methods for doing this, both relying only on astigmatic transformations, viewed as rotations on the Poincaré sphere, followed by imaging. These methods can be used as a tomographic tool for communication protocols based on optical vortices.

Figure 1

(a) Poincaré sphere representation of the two-dimensional space of orbital-angular-momentum topological charge $\pm 1$. (b) The astigmatic transformations we apply correspond to rotations on the Poincaré sphere, by angle $-\pi /2$ around an axis on the equator. Here we picture one such rotation which does not map the initial point to the equator (red), and one which does (blue).

Figure 2

Experimental setup. The screen of the SLM was split in two sides, one for the preparation of the mode and the other to emulate the tilted lens. The screen connected to the CCD shows the three images used in method II; these images are not captured simultaneously.

Figure 3

Tilted lens orientation. A spherical lens (axis $\widehat{\mathbf{n}}$) implements the astigmatic mode conversion given by Eq. (3). The tilt angle $\xi$ is determined experimentally, so that a rotation of $-\pi /2$ radians on the Poincaré sphere is achieved. The axis of the rotation on the Poincaré sphere is on its equator, at an angle $\beta$ with the $x$ axis, determined by the angle of the projection of $\widehat{\mathbf{n}}$ with the transverse plane, as in the figure.

Figure 4

Poincaré sphere representation of the sample modes used for testing the astigmatic analyzer.

Figure 5

(a) Images of the input modes and the resulting outputs from the astigmatic analyzer for the sequence A–E. (b) Graphical representation of the measured orientations ${\alpha }_{\text{HG}}$ vs $\theta$. Blue dots are the experimental results and red solid line is the theoretical prediction given by Eq. (7).

Figure 6

(a) Images of the input modes and the resulting outputs from the astigmatic analyzer for the sequence D1–D8. (b) Graphical representation of the measured orientations ${\alpha }_{\text{HG}}$ vs $\varphi$. Blue dots are the experimental results and red solid line is the theoretical prediction given by Eq. (7).

Figure 7

(a) The spherical caps in blue, around the $x,y$, and $z$ axes, represent states for which one of the three images will have a mode with low visibility. Due to this, we will call theses caps polar blind spots. States in each cap will be determined from the two images which have modes with good visibility. (b) States near the three pictured great circles will result in an intersection between halves of great circles which corresponds to a very narrow triangle see Fig. 9. These will be obtained as the midpoint of the shortest triangle side [see Fig. 9].

Figure 8

(a) A state will be deemed near to the three problematic great circles of Fig. 7 when the three determined longitudes determine a narrow triangle (with shortest side smaller than half the next shortest side, in blue). In this case, we estimate the state as being at the half point of the shortest side. (b) For generic states, far from the polar caps of Fig. 7 and the great circles of Fig. 7, the state is deemed to be at the center of the spherical triangle (in blue).

Figure 9

Poincaré representation of the three-measurement scheme. Characterization of the angle that the line connecting the maxima makes with the horizontal gives information about the $z$ axis longitude (green). By previously rotating the Poincaré sphere via two astigmatic transformations, the same imaging technique gives us the $x$-axis longitude (blue) and $y$-axis longitude (red). In an idealized measurement, the three longitudes would intersect at the point representing the state.

Figure 10

Poincaré representation of the sample states used to test the three-measurement scheme.

Figure 11

The red x is the image center of mass, the green line is the horizontal, and the blue line is the one that we sum the intensity under.

Figure 12

Graphical representation of the summed intensity in arbitrary units vs $\eta$. The vertical error bar is very small and could not be represented within graphical precision.

Figure 13

The red line is the nodal line, the green line is the horizontal, the center of mass of each region is demarcated by the red x, and the green dashed line corresponds to the inclination of mode.