Experimental generation and characterization of Devil’s vortex-lenses
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- Calatayud, A., Rodrigo, J.A., Remón, L. et al. Appl. Phys. B (2012) 106: 915. doi:10.1007/s00340-012-4913-0
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We propose the first experimental approach for both generation and characterization of high quality Devil’s vortex-lenses. These new type of lenses, able to produce a sequence of optical vortices, are addressed onto a programmable spatial light modulator (SLM) operating in phase-only modulation. The static aberrations arising by the lack of flatness of the SLM display are characterized and mostly compensated by using a Shack–Hartmann wavefront sensor. The analysis of the residual aberrations and their effect on the vortex-lens performance are studied.
Vortex lenses produce wavefronts with helical structure and zero axial intensity . These optical vortices are special optical traps having the ability to set the trapped particles into rotation due to the orbital angular momentum of light [2, 3]. The most common solution adopted to produce optical vortices are spiral phase plates [4, 5]. Inspired in fractal zone plates [6, 7] and in the Devil’s lenses , a new type of high efficient spiral fractal zone plates, the “Devil’s Vortex-Lenses” (DVLs)  have been recently proposed to generate a sequence of focused optical vortices. A DVL is designed using the “Devil’s staircase” or Cantor function, the distribution of the surface grooves in it producing a single main fractal focus composed of a sequence of discrete vortices. It was suggested  that the particular focal volume provided by DVLs could be profited as versatile and very efficient optical tweezers since, in optical trapping applications, in addition to rotating the trapped high index particles, the low-index particles can be trapped at center of the vortex. The relative angular velocity of the particles at the different traps can be modified by a parameter known as the topological charge, while the distances between the vortices can be modified by using different levels of the Cantor function.
In this paper DVLs are experimentally implemented in a programmable spatial light modulator (SLM). In order to observe the predicted focusing properties of different DVLs, an SLM operating in phase-only modulation was employed. In the experiment, the static aberrations arising from the proposed setup, mainly caused by the lack of flatness of the SLM display, are characterized and mostly compensated by using a Shack–Hartmann wavefront sensor. The effect of the residual aberrations in the vortex-lens performance is studied computing the intensity distribution along the optical axis and the transverse diffraction patterns along the propagation direction.
2 Devil’s vortex lenses
3 Experimental setup and its characterization
To implement the proposed lenses we use a programmable SLM operating in phase-only modulation . Specifically, we use a LCoS-SLM (Holoeye PLUTO, 8-bit gray-level, pixel pitch d=8 μm and 1920×1080 pixels) calibrated for a 2π phase-shift at λ=633 nm. In order to avoid efficiency degradation due to high time-fluctuations of the displayed phase, we addressed the so-called “5-5(543)” electrical signal scheme provided by the SLM manufacturer. This configuration gives better efficiency of the displayed phase even for phase-shift shorter than 2π, as demonstrated in . Notice that, according to the data provided by the manufacturer, the diffraction efficiency of such a SLM is more than 80% of the light reflected by the display, which has a reflectivity about 60%. Therefore a total light efficiency about 50% can be reached.
We underline that the 4-f setup is suitable for generating DVLs with different focal lengths by using an appropriate telescope-scaling factor M=f2/f1. To generate a sequence of optical vortices, we use a lens design given by (2). Specifically, for the considered scaling factor M=0.5, the generated lenses have a pupil radius of a=1 mm and the main focus is obtained at z=3.54 cm.
The focusing properties of Devil’s vortex-lenses have been experimentally analyzed for the first time in the literature. These lenses have been encoded onto an SLM operating in phase-only modulation with compensated aberrations. The good performance of the experimental setup is confirmed by the comparison between the experimental values of the irradiance along the optical axis and the theoretical predictions. It has been demonstrated that for multiple-plane optical trappings, the DVL can generate a light beam with axially distributed optical vortices. The transverse patterns appearing along the propagation distance present several concatenated doughnut modes. The particular focal volume provided by DVLs could be profited as versatile and efficient multiple-plane optical trap in the microscopic scale. Another potential application of DVL arises in X-ray microscopy  where the azimuthal component of the DVL acts as a Hilbert phase filter that can be used for detecting the phase component of objects with complex index of refraction. In the case of biological specimens, the additional phase sensitivity can provide enhanced contrast. Additionally, the multifocal nature of the lens, resulting from its fractal structure along the radial coordinate, could provide a high depth of field, especially with wideband sources [8, 10].
The financial support of the Spanish Ministry of Science and Innovation under projects DPI2008-02953, TRA2009-0215 and TEC2010-20307 is acknowledged. We also acknowledge the support from Generalitat Valenciana through the project PROMETEO2009-077. J.A.R. gratefully acknowledges a “Juan de la Cierva” grant and the financial assistance provided by the Universitat Politècnica de València (grants PAID-02-10 and PAID–5-11). L.R. acknowledges a fellowship of “Fundación Cajamurcia”, Spain.