Spectrally agile high-resolution and large-aperture self-engineered geometric-phase optical vortex generators

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1 Spectrally agile high-resolution and large-aperture self-engineered geometric-phase optical vortex generators refers to the threshold below which the liquid crystal orientation remains unperturbed. We attribute the presarxiv: v1 [physics.optics] 15 Mar 2018 Univ. Etienne Brasselet* Bordeaux, CNRS, LOMA, UMR 5798, F Talence, France (Dated: March 19, 2018) Artificially engineered liquid crystal films with space-variant molecular orientation are nowadays a widely used kind of geometric phase optical elements that may have tunable photonic functionalities owing to the sensitivity of liquid crystals to external fields. However, in the context of singular optics, a liquid crystal technology that combines high-resolution topological ordering with tunable reorientation magnitude remains elusive when dealing with handy macroscopic vectorial vortex masks. By using a magneto-electric external stimuli, here we report on the robust self-engineering of liquid crystal geometric phase vortex masks with electrically tunable operating wavelength over a broad range, centimeter-size clear aperture and high-quality topological ordering. Since Bhandari formulated two decades ago that space-variant anisotropic structures allow designing polarization-controlled optical elements [1] a variety of photonics technologies based on geometric phase have emerged. Geometric phase physics is a longstanding concept [2] that received a substantial attention in optics, especially when dealing with singular optics that is, quoting Soskin and Vasnetsov who introduced the terminology, this new branch of modern physical optics deals with a wide class of effects associated with phase singularities in wave fields, as well as with the topology of wave fronts [3]. Indeed, geometric phase optical elements are nowadays commonly used to produce optical vortices whose topological charge is controlled by the photon spin. The prototypical so-called q-plate design consists of a slab of uniaxial medium whose in-plane optical axis orientation angle is defined by ψ = qφ where q is a half-integer or an integer and φ is the polar angle. First fabricated using subwavelength gratings in the mid-infrared [4] it became broadly used once implemented in the visible domain using liquid crystals [5] and later extended to other media as well, such as liquid crystal polymers [6] or glasses [7]. To date, handy macroscopic q-plates all rely on machining techniques. In addition, there are independent strategies to tune the operating wavelength of these elements [8] or to fabricate high-resolution topological ordering of the optical anisotropy [9]. However, macroscopic standalone optical elements combining these assets are still missing and to address this issue is the purpose of the present work. Here we propose to use non-patterned nematic liquid crystal films under the influence of a magneto-electric external field to produce self-engineered liquid crystal q-plates with electrically tunable operating wavelength, centimeter-size clear aperture and high-quality topological ordering. Spectrally agile and pure optical vortex generation using collimated macroscopic laser beam is demonstrated while the high-resolution features are illustrated in the framework of spiral phase Fourier filtering. These results therefore extend previous attempts to tame Nature towards the generation of optical vortices using spontaneously formed liquid crystal topological defects under various kinds of environments and external fields [10 15]. However, these approaches have been restricted so far to small scale and this work fills a practical gap by offering easy manufacturing of high-tech spin-orbit photonic elements. Our experimental arrangement is shown in Fig. 1. The sample is a film of nematic liquid crystal with thickness L = 20 µm, birefringence δn = at 589 nm wavelength and negative dielectric anisotropy ɛ a = 6.4 at 1 khz frequency, both at 25 C. The liquid crystal slab is sandwiched between two glass substrates provided with transparent electrodes and perpendicular orientation for the liquid crystal molecules, Fig. 1. Under applied quasi-static voltage, a collection of umbilical defects with topological charges ±1 appears randomly and undergoes long-term collective annihilation dynamics [17], see Fig. 1(c). In presence of an external static magnetic field from an annular permanent magnet placed at a distance d from the liquid crystal layer as depicted in Fig. 1, a single defect with topological charge +1 can be obtained. The detailed principle of the method can be found in Ref. 18 in the case of a disk-shaped magnet. The main idea is that the transverse component of the magnetic field (with respect to the normal of the sample that defines the z axis) allows thresholdless axisymmetric realignment of the liquid crystal due to its diamagnetic properties. Then, the electric field amplifies the reorientation. However, in contrast to Ref. 18, the ring geometry for the magnet allows clear aperture, long-term stability and almost uniform birefringent phase retardation as discussed later. In the present case, we find that the distance d between magnet at the liquid crystal plays an important role in the nature of the steady topological structure. This is illustrated in Fig. 1(d) for d = 1, 2, 3 and 4 mm that show the sample observed between crossed linear polarizers 10 min after a square-waveform voltage at 2 khz and amplitude U = 1.2U th is switched on. There, U th 1.8 V

2 2 heavy electrodes. umbilic «+1»! ring magnet * (d) with magnet! = 1 mm! = 2 mm! = 3 mm! = 4 mm z 2'() (e) 2'+,- y x 4 mm FIG. 1: Geometry of the liquid crystal film under the action of magneto-electric stimulation. The magnetization of the Neodynium ring magnet points along the z axis. The magnet parameters are Rin = 4 mm, Rout = 7.5 mm and H = 6 mm, and it holds 5 kg. Lines of the director field, which refers to the local-averaged orientation of the liquid crystal molecules, of the sample at rest and of an umbilical defect with topological charge +1. Images of the sample observed between crossed linear polarizers (oriented along the x and y axes) using white light incoherent illumination 10 min after the voltage U/Uth = 1.2 is applied without (c) and with (d) the presence of the ring magnet placed at a distance d from the liquid crystal layer. (e) Sketch of the magnetic field lines for a ring magnet with axial magnetization taken from Ref. 19, where the star markers refer to magnetic field inversion points. ence of an umbilic necklace at lower values of d to the specific magnetic field lines that exhibit axial reversal, as illustrated in Fig. 1(e), which would ideally lead to an inversion wall for the liquid crystal orientation but breaks up in practice into a discrete set of point defects. At larger distance, the axisymmetry of the resulting structure is progressively lost, in agreement with the random reorientation without magnetic field shown in Fig. 1(c). In what follows we thus fix d = 2 mm and we note that the liquid crystal reorientation is hardly perceived between crossed linear polarizers at U = 0. Summarizing, the magnetic field acts as a topological orientational seed that grows under the effect of the electric field and lead to a macroscopic steady umbilic with topological charge +1 centered on-axis. From a general point of view, an umbilic is characterized by a molecular tilt angle with respect to the z axis that asymptotically converges to a constant value out- side from the core region, as sketched in Fig. 1. Consequently, discarding the core area, an umbilic behaves as an electrically tunable q-plate with q = +1 where the asymptotic birefringent phase retardation can be tuned between 0 and 2πδnL/λ, as shown in a previous work [12]. The explored range of retardation over the spectral range 400 nm < λ < 800 nm is summarized in Fig. 2, where the integer n refers to the multiple number of π, namely = nπ. Measurements are made by assessing the voltage values maximizing (n odd) and minimizing (n even) the transmitted power by placing the sample between crossed circular polarizers and illuminating it at normal incidence by a collimated supercontinuum laser beam with 2 mm diameter that propagates along the z axis. Data is collected for a set of nine wavelengths in steps of 50 nm, with 10 nm full-width at half-maximum transmission spectrum. Only the odd values of n are shown in Fig. 2 since pure optical vortex generation only happens when the liquid crystal film satisfies the half-wave retardance condition [5]. The demonstration of spectrally agile optical vortex generation at the macroscopic scale is illustrated in Fig. 3, where the far-field intensity profiles are shown in the particular case of an incident linear polarization state along the x axis. In that case, the output beam is a vectorial vortex that is the superposition of contracircularly polarized scalar vortex beams with opposite topological charges ±2 a so-called nonseparable optical state. As expected, annular-shaped total intensity profiles beams are obtained, see Fig. 3. In addition, the vectorial structure is resolved by placing a polarizer at the output of the sample, see Fig. 3. Maximally contrasted four-lobe patterns are observed, according to the expected inhomogeneous linearly polarization state whose azimuth rotates by 4π over a full turn. K=1 data1 K=3 data2 K=5 data3 K data4 =7 K data5 =9 4 th liquid crystal (c) without magnet ordering at rest 3 data7 data8 data9 n substrates UUJ//U U-@ z (nm) H6 (nm) FIG. 2: Spectral dependence of the voltage values Un that fulfill the half-wave retardance condition. Color markers refer to different values for n while the black curves are only guide for the eyes. The study is made at fixed d = 2 mm.

3 3 H = 400 nm H = 450 nm H = 500 nm H = 550 nm H = 600 nm H = 650 nm H = 700 nm H = 750 nm H = 800 nm FIG. 3: Optical characterization of the generation of spectrally agile optical vortex beams from a 2 mm diameter collimated supercontinuum laser beam filtered with a set of nine wavelengths from 400 nm to 800 nm by steps of 50 nm, in the case of an incident linear polarization state along the x axis. The study is performed for n = 1 at fixed d = 2 mm. Far-field total intensity profiles. Far-field intensity patterns by placing an output linear polarizer along the x axis. As a matter of fact, the proposed magneto-electric strategy enables the robust fabrication of a selfengineered vectorial vortex mask with large clear aperture that performs well over a broad spectral range without need of post-polarization filtering as is usually required for moderately pure vortex generator. Still, two structural features deserve additional comments. First, it is known that a purely electrically driven umbilic is associated a with characteristic core radius that depends on the applied voltage and scales according to rc [(U/Uth )2 1] 1/2 [17]. In the present case where the magnetic field plays only the role of imposing the axial symmetry, little change is expected. However, we stress that the characteristic distance r associated with less heavy nonuniform birefringent phase retardation does not scale Δ< = ; Δ< = 3; Δ< = 5; Δ< = 7; 50 µm U=2V U=6V U = 10 V U = 14 V 300 µm FIG. 4: Intensity profile nearby the defect (here located in the middle of each image) when observing the sample between crossed circular polarizers under white light incoherent illumination spectrally filtered at 532 nm wavelength, for n = 1, 3, 5 and 7. Observations under crossed linear polarizers using white light incoherent illumination for a representative set of voltage values. The study is made at fixed d = 2 mm. as rc and the Rapini model rapidly looses its validity as the reorientation magnitude increases. A qualitative experimental insight is thus eventually grasped by observing the sample between crossed circular polarizers, see Fig. 4. Indeed, r corresponds to the distance from the defect location below which the intensity distribution is nonuniform. It is found that r substantially decreases at large enough values of n, which gives a recipe to optimize the uniformity of the optical element whose topological ordering quality remains unquestionable whatever the voltage. The second issue is related to the twisted director field which is actually associated with ψ = φ + ϕ(r), see Fig. 1(d). It results from the fact that the initially imposed splay-type umbilic (i.e., with radial director field) imposed by the radial transverse magnetic field lines relaxes into a swirled umbilic due to the elastic anisotropy of the liquid crystal. More details on such a generic phenomenon can be found in Ref. 20 in the particular case of a liquid crystal light valve. Although this does not affect the generation of optical vortices with topological charges ±2 as such, this may be either a drawback [16] or an asset [21] when dealing with nonseparable spinorbit optical states. The trend of a decrease of the swirl magnitude as the voltage increases qualitatively exhibited in Fig. 4 is quantitatively summarized in Fig. 5 where the swirl profile ϕ(r) is shown. In particular, its value at the origin is fitted with the ad hoc function ϕ(0) = ϕ0 + ϕ1 exp[ (U Uth )/Uc ] with ϕ0 = 63.1, ϕ1 = 47.9 and Uc /Uth = 5.2. Such a behavior can therefore be exploited depending on the use sought after. Finally, we demonstrate that the fabricated geometric phase optical element can also be used when neat topological ordering at small scale, uniform half-wave birefringent phase retardation and large clear aperture are desirable at the same time. This is the case for optical imaging techniques relying on spiral phase Fourier filtering towards resolution and sensitivity enhancement. This

4 4 G (deg) input pupil exit pupil plane vortex mask U (V) G 0 (deg) D Camera f L 2f off-axis defect L f L on-axis defect F (µm) (c) U/U-@ FIG. 5: Voltage dependence of the swirl function ϕ(r) and its value ϕ(0) at the origin. Markers: experimental data. Solid curve: fit, see text for details. has been implemented in various standard and superresolution microscopy techniques or coronagraphy in astronomy, as reviewed in [22]. Here we illustrate it in a case corresponding to the basic scheme used in optical vortex coronagrahy where phase masks producing optical vortices with even topological charges are used [23]. In practice, this is done by placing the sample in the Fourier plane of a telescope and reimaging an input clear circular aperture according to the set-up sketched in Fig. 6. If the defect is not centered on the zeroth-order Fourier spot, the image of the aperture is almost unperturbed, as shown in Fig. 6. In contrast, when the defect is placed on-axis, the light intensity is redistributed at the periphery of the geometric image of the input pupil for odd n, as shown in Fig. 6(c). Therefore, the diffracted light can be totally suppressed by placing a diaphragm in the exit input pupil plane, which lead to the coronagraphic effect in the context of astronomical imaging of faint celestial objects nearby bright sources of light. We emphasize that no post polarization filtering is used, in contrast to previous attempts using self-engineered small-scale liquid crystal q-plates [24, 25]. In principle, as discussed above, the higher is the liquid crystal reorientation, the better it is. Indeed, higher values of n are associated with smaller birefringent core radius and smaller swirl. Optimization should also benefit from thinner liquid crystal layers since the core radius scales as L, though one should keep in mind that the topological seed effect arising from the magnetic field will be reduced since the reorienting magnetic torque scales as L 2. Nevertheless we recall that the quest for ever smaller umbilic core radius is not a necessary condition for potential applications. In particular, scalar vortex phase masks without phase singularity in the central area have been found useful to obtain shadow effects in spiral phase contrast imaging [26] and vectorial vortex phase masks without uniform birefringent phase retardation in the central area have been found useful to obtain optical vortex arrays [27]. FIG. 6: Experimental set-up used for the demonstration of spiral phase Fourier filtering at 700 nm wavelength using a so called 4f optical system made of a pair of lenses L with focal length 200 mm. This corresponds to an Airy spot radius r0 ' 40 µm (taken at the first zero of intensity) in the Fourier plane of the telescope. The input circular aperture has a diameter D = 4 mm. The self-engineered magneto-electric liquid crystal q-plate is placed in the Fourier plane. The input pupil is reimaged on a camera using a lens with focal lens 75 mm when the defect is displaced off-axis at a distance ' 10r0 and when it is placed on-axis (c), for n = 5 and d = 2 mm. Summarizing, the combined action of a static magnetic with a quasi-static electric field enables the robust self-engineering of handy macroscopic liquid crystal qplates. These are characterized by a large clear aperture of the order of 1 cm2, a high-quality topological structuring ensured by a nonsingular topological defect for the director field, and an operating wavelength that is electrically tunable over the full visible domain. Since liquid crystals remains highly birefringent and transparent up to terahertz frequencies, the extension to even larger wavelength range does not present fundamental difficulties. By combining large aperture while preserving high-resolution topological structuring at small scale, these technology-free and Nature-assisted versatile geometric phase optical elements invite to consider the implementation of soft-matter self-organization processes in spin-orbit photonics technologies. This work has been partially funded by the CNRS program De fi Instrumentation aux limites *etienne.brasselet@u-bordeaux.fr [1] R. Bhandari, Phys. Rep. 281, 1 64 (1997). [2] M. Berry, Phys. Today 43, (1990). [3] M. S. Soskin and M. V. Vasnetsov, Prog. Opt. 42, 219

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