Superposition of two optical vortices with opposite integer or non-integer orbital angular momentum

This work develops a brief proposal to achieve the superposition of two opposite vortex beams, both with integer or non-integer mean value of the orbital angular momentum. The first part is about the generation of this kind of spatial light distributions through a modified Brown and Lohmann’s hologram. The inclusion of a simple mathematical expression into the pixelated grid’s transmittance function, based in Fourier domain properties, shifts the diffraction orders counterclockwise and clockwise to the same point and allows the addition of different modes. The strategy is theoretically and experimentally validated for the case of two opposite rotation helical wavefronts.


Introduction
Since Allen et al. (1992) explicitly demonstrated that light carries orbital angular momentum (OAM), and spin angular momentum (SAM), various methodologies have been proposed and developed to produce beams with this rotational dynamic in the wavefront.
Astigmatic converters, phase plates (Beijersbergen et al., 1993(Beijersbergen et al., , 1994) ) and computer-generated holograms (CGH) (Heckenberg, 1992;Janicijevic, 2008) are the techniques most often used to generate optical vortices with topological charge (TC) and therefore a beam with well defined OAM.However, based on the Díaz et al. (2012) review, CHG are selected as the most simple and feasible method to be implemented in a hybrid system, by the variety of modes that are generated and availability of equipment in the optics laboratory in Universidad Industrial de Santander (Bucaramanga/Colombia) for this kind of setup.
This optoelectronic system comprises a coherent light source of 532 nm wavelength, a collimator, one spatial light modulator (SLM), two computers and a CCD camera to record the intensity of the light.
The CGH that deploys over the SLM is a modified transmittance function; the original was exposed in the late sixties by Brown and Lohmann (1969).It allows to systems or circulars grids (Naidoo et al., 2011(Naidoo et al., , 2012;;Dudley, 2012), which attenuates the beams intensity and increases the number and length of optical paths for the mode to be propagated.
Below, a simple proposal to add wavefronts with a specified integer or non-integer mean value of the OAM is discussed.From the signal processing point of view, the modes superposition is made with a single propagation path and without additional hardware.

Optical vortices generation
All beams whose spatial light configuration is multiplied by an azimuthal phase term e − jmϕ ( x,y) have a wavefront with helical pattern and an optic vortex in the intensity, with a radius that varies according to the m value related with the OAM.The phase singularity is inserted on a Gaussian plane wavefront beam, were it is transmitted by a set of CGH displayed in a translucent matrix Holoeye, LC2002 model.The far field is composed of the different diffraction orders, each one with an optical vortex with different OAM mean value.The centered order at the optical axis has a null OAM and the vortices placed on the right and left sides are symmetrical with opposite signs.
However, there are various pixelated grids for that purpose.The Brown and Lohmann's blazed CGH (1969) Equation ( 1) adapts a specific angle for raising diffraction efficiency as much as possible, modifying the interfringe step d on the transmittance function T(x,y), with n equal to the number of orders to propagate.
The mathematical description above does not include an azimuthal mask.This transmittance function is unable to produce an optical vortex, and therefore a variant T 1 (x,y) in Equation ( 2) is proposed, inserting a phase singularity (3) with the step index m. (2) (3) The hologram from T 1 (x,y) presented in Figure 2 produces two optical vortices in the Fresnel-Fraunhofer space.The related simulations in Figures 3 and 4 are corresponding to the vortex profiles exposed by Leach (2004).The related signs of azimuthal index in the resulting spatial distribution of light indicate the direction of counterclockwise or clockwise rotation of the diffracted components (G.Gibson et al., 2004).The wavefront of each of the orders propagates with the same helical periodicity but opposite orientation.

Superposition of two optical vortices with opposite OAMs.
The process of superposition is mainly based in the use of two wavefronts generated with the physics property of interest, with the counterclockwise and clockwise rotation that is present in the far field.
The idea is movethe two spatial light configurations to the same propagation axis in the reciprocal space and without inserting other optical components.This is possible if these modes are convolved with two pulses located at the same point where the vorticess are placed.Refer to the Figure 7 for the particular case of superposition of two opposite rotation modes; the convolution between the impulse located in (−fy, −fx) and the vortex in (fy, fx), shift this to the optical axis.The same effect occurs with the convolution between the Dirac pulse placed in (fy, fx) and the beam centered at (−fy, −fx), as shown in Figure 8. Mathematically, in order to locate these Dirac pulses in the corresponding inverse space, multiply the transmittance function T 1 (x,y) by the cosine expression in the direct space; see Equation ( 4).This operation is equivalent to the inverse Fourier transform of two symmetrically shifted deltas Dirac distribution.
The intensity of the two overlapped diffraction orders is written as: From the last expression, it can be observed that resulting spatial light configuration must exhibit a number of constructive and destructive interference that is equal to twice the value of the TC of the modified CGH T 2 (x,y).
The theoretical and experimental results for the superposition of opposite modes with TC are presented below:     When extending the procedure described for optical vortices with non-integer OAM mean is obtained the next profiles.

Figure 1 .
Figure 1.Helical structure and intensity patterns with Bessel Gauss envelope and different values of m.

Figure 2 .
Figure 2. CGH generated from the new function with step index m = +5.

Figure 5 .Figure 6 .
Figure 5. Diffracted optical vortices with opposite orientation generated from incidence of fundamental mode on CGH.

Figure 8 .
Figure 8. Modes and impulses located in far field.

Figure 9 .
Figure 9. Simulation of the intensity profile of the diffraction orders superposition with m = −2 and m = +2.

Figure 10 .
Figure 10.Simulation of the intensity profile of the diffraction orders superposition with m = −3 and m = +3.

Figure 11 .
Figure 11.Simulation of the intensity profile of the diffraction orders superposition with m = −4 and m = +4.

Figure 12 .
Figure 12.Simulation of the intensity profile of the diffraction orders superposition with m = −5 and m = +5.

Figure 13 .
Figure 13.Experimental intensity profile of the diffraction orders superposition with m = −2 and m = +2.

Figure 14 .
Figure 14.Experimental intensity profile of the diffraction orders superposition with m = −3 and m = +3.

Figure 15 .
Figure 15.Experimental intensity profile of the diffraction orders superposition with m = −4 and m = +4.

Figure 16 .
Figure 16.Experimental intensity profile of the diffraction orders superposition with m = −5 and m = +5.