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We have experimentally created perfect optical vortices by the Fourier transformation of holographic masks with combination of axicons and spiral functions, which are displayed on a transmission liquid crystal spatial light modulator. We showed theoretically that the size of the annular vortex in the Fourier plane is independent of the spiral phase topological charge but it is dependent on the axicon. We also studied numerically and experimentally the free space diffraction of a perfect optical vortex after the Fourier back plane and we found that the size of the intensity pattern of a perfect optical vortex depends on the topological charge and the propagation distance.

As is well known, an optical vortex beam is an electromagnetic wave with a helical wavefront due to phase singularities [

To solve these requirements, Ostrovsk et al. have introduced the perfect optical vortex (POV) concept [

In this work, we present an experimental approach for generating perfect optical vortices by using of phase masks of three levels with shape of axicon and spiral phase functions, which are displayed in a transmission liquid crystal spatial light modulator. The intensity and vorticity of the POVs are measured with a CMOS camera and the far-field diffraction pattern through an equilateral triangular slit, respectively. We showed numerically and experimentally that the free space propagation diffraction patterns of the POVs are dependent on the distance and they are variant to ring-width size and to average ring-diameter.

As mentioned earlier, perfect optical vortices are beams whose diameter and width-ring are independent of its topological charge. POVs can be approximately generated by means of the Fourier transformation from Bessel-Gauss (BG) beams [

The field of a POV is obtained in the back focal plane of a converging lens by substituting (

Simulations of the intensity patterns and the phase structures for Bessel-Gauss beams and perfect optical vortices. (a)–(e) Intensity distributions of the BG beams with topological charges

(a) and (c) Theoretical line intensity profiles along the vortex centers of POVs with topological charges

Simulated intensity patterns for POVs and their respective Fraunhofer diffraction by an equilateral triangular aperture. (a)–(e) POVs with topological charges

Simulated intensity patterns for POVs in free space propagation with different TC (rows) and same distance (columns). (a)–(f) POV with topological charge

Figure

On the other hand

Finally, it is possible to analyze the comportment of a POV after the Fourier plane if it considers the Fresnel formula for the diffraction in free space given by [

Figure _{00} is spatially filtered and collimated. Subsequently, the beam is directed toward an experimental configuration composed of two lineal polarizers, a transmission liquid crystal spatial light modulator (TLC-SLM, Sony model LCX038ARA spatial resolution: 1024(H) × 768(V) pixels), and two quarter-wave plates. In order to create phase masks with combination of axicon and spiral functions, the TLC-SLM has been placed in the arrangement A1 composed by linear polarizer (P1), quarter-wave plate 1 (QWP1), TLC-SLM, linear analyzer (P2), and quarter-wave plate 2 (QWP2). In this arrangement the orientation of the linear polarizer, analyzer, and the waveplates in front of and behind the TLC-SLM was obtained through calibration previously of TLC-SLM in phase only modulation [

Schema of experimental setup for generated POVs: spatial filter and collimator; TLC-SLM: transmission liquid crystal spatial light modulator; P1: polarizer; QWP1: quarter-wave plate 1; P2: analyzer; QWP2: quarter-wave plate 2; P: pupil; L: Fourier lens with

The BG beams with the same axicon period

For verifying the nature of the perfect optical vortex generated from Fourier transform of a Bessel-Gauss beam we used an equilateral triangular aperture placed at the back focal plane of L1 within the experimental setup shown in Figure

Schema of experimental setup for verifying the POV nature: spatial filter and collimator; TLC-SLM: transmission liquid crystal spatial light modulator; P1: polarizer; QWP1: quarter-wave plate 1; P2: analyzer; QWP2: quarter-wave plate 2; L1: Fourier lens 1; AP; triangular aperture; L2, Fourier lens 2 with

Figure

Schema of experimental setup for finding the POV diffraction intensity pattern after the Fourier plane: spatial filter and collimator; TLC-SLM: transmission liquid crystal spatial light modulator; P1: polarizer; QWP1: quarter-wave plate 1; P2: analyzer; QWP2: quarter-wave plate 2; L: Fourier lens; FP: Fourier plane; PL1: plane 1; PL2: plane 2; PL3: plane 3;

Figure

Experimental POVs intensity pattern for topological charges

In this figure, one can see by naked eye that the intensity distributions are invariant with the topological charge. One can also observe that size of the intensity distributions changes with the axicon period according to theory; that is,

POVs intensity patterns and their respective experimental Fraunhofer diffraction. (a)–(e) POVs with topological charges

Finally, each column of Figure

POV experimental intensity patterns in free space propagation with different topological charge (columns) at the same distance (rows). (a)–(c) POVs at distance

We proved that an optical perfect vortex could be essentially created by the Fourier transformation of an adequate combination of an axicon function and a spiral function. We also showed that the size of the annular vortex on the back Fourier plane of the transforming lens is independent of topological charge of spiral phase function. Finally, numerical and experimental results for the free space diffraction propagation of perfect optical vortex, after the back Fourier plane, show that their intensity pattern depends on topological charge and propagation distance.

The authors declare that they have no conflicts of interest.

Thanks are due to Universidad Industrial de Santander for the support of Research and Services Vice-Chancellor VIE Funding Projects 5191/5803 and 5708, both from the institutional program for consolidation of research groups, years 2012 and 2013, respectively. The authors also acknowledge support from Colciencias under Project 110256934957, “Optics Devices for Quantum Key Distribution, High Dimensionality Systems Based in Orbital Angular Momentum of Light.” Funds were obtained from the National Call for the Bank of Projects in Science, Technology, and Innovation 2012.