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Abstract
A transmissive adaptable optical setup to generate a range of perfect vortex beams (PVBs) carrying different topological charges (TC) without using moving parts is presented. The setup is composed of an ad hoc transparent reconfigurable liquid crystal (LC) spiral phase plate (SPP), a refractive axicon and a convergent refractive lens. The LC SPP electrodes are manufactured ablating indium-tin oxide (ITO) glass substrates using direct laser writing (DLW) resulting in a very high fill factor device. In-house tailored electronics drive the 72 LC SPP electrodes giving rise to 72 different configurations with orbital angular momentum. In this work, the generation of PVBs with 36 positive or 36 negative TCs using this optical setup is accomplished.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The term optical vortices (OV) was first coined in 1989 by Coullet et al [1]. This term, also known as vortex beams [2] (VB), refers to light beams characterized by a helical wavefront. In 1992 Allen et al [3]. demonstrated that such beams carry an orbital angular momentum of light (OAM), whereby a laser beam can transfer a mechanical torque.
Since their discovery, OVs have aroused great interest in fields such as astrometry [4], microscopy [5], microfluidics [6], optical manipulation [7,8] direct laser writing polymerization [9] and optical communications [10]. The use of VBs in the latter field is of special interest since the OAM can be used as an additional degree of freedom to the currently used multiplexing schemes, allowing to drastically increase the capacity of communications systems [11,12].
VBs are characterized by having a phase singularity at the optical axis, giving rise to an annular pattern. Furthermore, VBs have an intrinsic property called topological charge (TC), directly related to the OAM carried by these light beams [13]. The TC describes the helical pitch in number of wavelengths, or equivalently the number of intertwined helical wavefronts.
In paraxial beams OVs are characterized by a central singularity with zero light intensity surrounded by a circular intensity pattern with a radius depending on the TC [14]. This radius dependence on the TC compromises applications like optical trapping and manipulation, where the exerted rotational force is preferably independent of the active area of actuation, and in optical communications where beams with different TC, all have to be focused into an optical fiber [15].
In 2013 Ostrovsky et al. proposed the perfect vortex beam (PVB) concept to avoid this dependence between the intensity pattern radius and the TC [16]. Thus, PVBs are light beams characterized by carrying different OAMs without modifying their intensity ring pattern radius [17].
Numerous ways of generating PVBs have been proposed, including spatial light modulators (SLMs) [16,18–20], digital micromirror devices (DMDs) [21,22], metasurfaces [23], polymer-based plates [24] or Pancharatnam-Berry (PB) elements [25]. In the case of devices based on polymer plates, PB elements or metasurfaces, their main drawback is the limited TC tuning of the generated beam. On the other hand, SLM-based systems have three major shortcomings: their relatively high cost, their aliasing limitations originating in the SLMs finite pixel size [15], and a more complex electronic driving requirement. Additionally, most of the previously reported configurations based on SLMs, metasurfaces and the ones using DMDs, are reflective setups, conditioning their implementation.
In this work, a bespoke electronically reconfigurable perfect vortex beam (PVB) system is presented, with a core element being a liquid crystal (LC) spiral phase plate (SPP). In this way, as opposed to previously reported approaches based on polymer plates or metasurfaces, the developed system enables the generation of PVB with a wide range of topological charges (TC) without the need of moving parts. Moreover, the system possesses an unpreceded simplicity, consisting of three main elements (SPP, Axicon and Lens) only, greatly reducing the cost and complexity compared to SLM-based approaches. Furthermore, in contrast with most of the previously reported systems based on SLMs or DMDs, the developed system is transmissive, thus suitable for applications where compactness is a crucial factor.
2. Theory
In 2015, Vaity and Rusch [18] demonstrated that PVBs could be generated using the Fourier transform property of a Bessel Beam. In their work, they used an SLM followed by a Fourier lens. Here, we propose an optical setup composed of a liquid crystal SPP, a refractive axicon and a refractive lens to generate PVBs.
An ideal SPP is an optical element with a continuous azimuthal phase delay [26], being used for generating vortex beams with different topological charges. The TC of the generated beam depends on the number of 2π phase delays imprinted by the SPP. Positive and negative values of the TC are defined by the handedness of the azimuthal phase delay, in this work positive TC values are assigned to the generation of a left-handed helical wavefront, which corresponds with a positive OAM.
In the optical setup, a tunable discrete SPP based on liquid crystal is used (Fig. 1). In this kind of devices, phase profile is divided into discrete steps [27,28] rather than generating a continuous azimuthal phase delay. To achieve a wide range of TCs in a thin device, 2π wrapping is employed resulting in a diffractive phase device. E.g. an SPP that introduces two phase-discontinuities of 2π in one revolution will produce a similar effect as an SPP that introduces only one phase-discontinuity of 4π. However, the SPP TC range is limited by its finite number of discrete electrodes, where the maximum TC is obtained for a binary phase profile consisting of electrodes with an alternating phase delay of 0 and π (Fig. 1(c)) as previously described [14,28].
Fig. 1. Optical vortex wavefront scheme, generated by a reconfigurable SPP with different topological charges (l). (a) l = -1. (b) l = -2. c) Required discrete SPP phase profiles to generate vortex beams carrying different topological charges.
The SPP transfers the OAM to the original laser beam converting an incident gaussian beam to a Laguerre-Gaussian (LG) beam [27], where the radius of the intensity ring increases with the TC absolute value. To avoid this dependence, the output beam generated by the SPP must be initially transformed by a refractive axicon.
Axicons are conical prisms first discovered by McLeod in 1954 [29]. The main characteristic of an axicon is that light rays are bent at the same angle, relative to the lens normal. In this way, an axicon converts a Gaussian beam into a zeroth-order Bessel beam, characterized by its long depth of focus (DOF) [30].
The Bessel beam zeroth-order is defined by the following equation:
where ${J_0}$ is the zeroth-order Bessel function, kz and kr are constants that represent the longitudinal and radial wavevectors, z represents the longitudinal component and r the radial component.In this work, the light incident onto the axicon is a Gaussian-shaped laser beam that has been transformed by the SPP. The combination of the phase profile of these two elements, performing a 2π wrapping, is shown in Fig. 2.
Fig. 2. Scheme of the SPP and axicon combination phase profile wrapped about 2π. On the top right corner, the phase profile using a discrete SPP is overlaid. The resultant phase profile is obtained for different SPP TCs (l): (a) l = 1. (b) l = 4.
This phase profile function is described as [24]:
where l is the topological charge, a is the axicon parameter, r is the distance to the optical axis and $\theta $ is the azimuthal angle. The axicon parameter is given by [25]: where $\beta $ is the axicon base angle and n is the refractive index. Hence, the resulting phase profile depends on the TC ($l$) and the axicon parameter (a), determined by the SPP configuration and the refractive axicon geometry respectively.Thus, when laser light passes the two devices, the LG beam generated by the SPP is converted by the refractive axicon into a nth-order Bessel-Gauss beam [25,31] (within the axicon DOF region):
When these Bessel-Gauss beams pass a focusing lens they produce perfect vortex beams (PVBs) [18,25], vortex beams with an intensity pattern radius independent of its TC [32]. Thus, by cascading a reconfigurable SPP, an axicon and a convergent lens, it is possible to generate PVBs carrying different TCs just modifying the SPP phase profile.
The PVB radius is determined by the radial wave vector [24], hence by changing the refractive axicon geometry, or using a tunable axicon [17], the PVB radius can be controlled. Moreover, using a setup composed of different lenses and an axicon can be used to modify the PVB radius, by changing the distance between optical elements [24,33].
The width of the PVB ring is inversely proportional to the diameter of the beam incident on the Axicon, thus by controlling the input beam waist the PVB ring width can be tuned [18].
3. Experimental
3.1 Liquid crystal cell manufacturing
The LC spiral phase plate (SPP) is made up of two indium tin oxide (ITO) coated glass substrates, in a sandwich-like configuration. One of the ITO substrates is used as a ground backplane, while the other is pixelated using direct laser writing (DLW), the electrodes design is described in the Supplement 1. A UV laser mounted over a CNC controlled XYZ-stage (Lasing SA) system is used to carry out the ablation process. This system allows movements of the substrate in the XY plane while maintaining the ablation focus in the Z axis with a closed loop feedback [34]. The interpixel size is of approximately 2 µm leading to a very high fill factor (>99%), outperforming the one that could have been obtained if an SLM had been used.
The SPP pixelated plane consists of 72 simultaneously and independently addressable electrodes, which have a pie-slice shape within the 2.54 cm diameter active area (Fig. 3). The pie-slices are separated by removing ITO radii using DLW. The 72 pixels enable the generation of 36 positive or 36 negative different topological charges. The electrodes shape eases a fan out to the periphery of the cell, where they are connected to an in-house developed electronic driver [35]. Connections between the electronic driver and the LC cell are carried out using a flex connector attached to the periphery of the substrate with an anisotropic conductive adhesive (Hitachi Chemical, Japan). The detailed electrode design is described in the Supplement 1.
Fig. 3. Manufactured SPP micrographs showing two different configurations under white illumination between crossed polarizers. (l): (a) l = 8. (b) l = 12.
Liquid crystal uniform homogenous alignment is conditioned using polyimide PIA-2304 (Chisso Lixon, Japan). The two ITO surfaces are coated with a polyimide precursor solution using a spin coating (30s @ 2500 rpm), and are polymerized at 180°C for 45 minutes. Finally, the alignment direction is defined by rubbing the polymerized polyimide with a velvet cloth. The LC cell thickness is set to 7.2 µm using cylindrical silica spacers. Finally, the cell is filled up with a high birefringence nematic liquid crystal, MDA-98-16002 (Merck GmbH, no = 1.52, ne = 1.78), and sealed using an epoxy adhesive.
The developed electronic driver generates 72 independent 12-bit pulse width modulation (PWM) outputs, determining the phase delay imposed to each pixel and therefore controlling the light beam TC. The PWM signal modulates a square 5Vpp, 1kHz signal generating the desired RMS voltage applied to apply to each pixel [17].
3.2 SPP calibration
To obtain the relationship between the applied RMS voltage signal (generated by the driver) and the light phase delay induced by the LC SPP, a calibration must be carried out. The cell was situated between crossed polarizers at +/- 45 degrees. The relation between the phase delay ($\delta $) and duty cycle (dc) has been empirically fitted with a pseudo-exponential function given by:
as previously described [14,17].3.3 Optical setup
The optical setup scheme is shown in Fig. 4. A He-Ne laser with a wavelength of 632.8 nm is used as light source. A spatial filter with a beam expander is attached to the laser, modifying the laser beam to obtain a collimated beam. A diaphragm reduces the diameter of the spot to 10 mm. Subsequently, a polarizer ensures that the light polarization is coincident with the rubbing alignment direction. This beam goes then through the LC SPP, followed by a 2° angle axicon (Thorlabs [36]). Finally, the light is focused using a refractive lens (20 cm focal length refractive lens) onto the camera sensor (Nikon D500 [37]).
Fig. 4. PVB optical setup scheme, where insets show beam propagation simulations, not taking the effect of the diaphragm into consideration.
4. Results
In this section, the generation of PVBs is experimentally demonstrated. The measurements of the transformations performed by each of the elements of the optical setup are shown below.
The intensity patterns generated by a spiral phase plate (SPP) for different topological charges are shown in Fig. 5. In the upper row, simulations of a continuous SPP and a discrete SPP composed of 72 different electrodes are shown. These simulations are based on scalar diffraction theory; in particular, the angular spectrum approach is used [38]. This method enables the calculus of the resulting electric field by knowing the electric field at a source plane (i.e. the SPP plane). The simulations are performed implementing the light propagation functions developed by Jason D. Schmidt [39]. In the bottom row, the experimental measurements recorded at the camera CMOS sensor are displayed. It is clear that the dark disc diameter increases with the topological charge.
Fig. 5. SPP Intensity outputs for different topological charges (l). Upper row contains simulations of a continuous ideal SPP as well as of a discrete one. Bottom row shows the measurements obtained from the LC SPP. The effect of the 10 mm diameter diaphragm giving rise to the Airy pattern has been included in the simulations.
The effect of the discrete SPP versus the ideal continuous SPP is appreciable in the simulations. As the topological charge is increased, less electrodes are used to recreate the ideal phase profile, giving rise to characteristic discrete diffraction patterns (Fig. 5) as previously shown [14]. Negative TCs can be also generated, producing dark discs with the same diameter as the equivalent positive TC. All the raw measurements can be found in the Supplement 1.
After the light impinges the SPP, it passes through the 2° angle refractive axicon located at a distance of 8 cm. Intensity measurements at the axicon output (at 14 cm) recorded with the camera CMOS sensor are displayed on Fig. 6. In this case, the characteristic high-order Bessel-Gauss beam patterns composed of concentric rings can be observed within the axicon DOF. As in the previous case, the dark disc present in the device optical axis has a size dependent on the generated beam TC. It should be noted that the obtained patterns are not perfectly homogeneous. The authors attribute this asymmetry to the fact that the laser beam does not have a homogeneous distribution, as well as to minor alignment errors. The higher order diffraction arising from the SPP discretization could also be observed as TC is increased, please find attached in the supplemental document the results obtained for the entire topological charges range.
Fig. 6. Axicon output (including SPP) measured intensity patterns for different topological charges (l).
Finally, the generated Bessel-Gauss beams are focused by a 20 cm focal length refractive lens placed within the axicon DOF (at 5.5 cm). The intensity patterns generated by the combination of the tunable SPP, the axicon and the refractive lens are displayed on Fig. 7 (recorded at the lens focal point). The generated dark hollow diameter is independent of the TCs imprinted by the SPP, thus perfect vortex beams are being generated. Using this optical setup, PVBs with TCs between -36 and 36 are generated. A full compilation of all the generated PVBs is presented in the supplementary file. As in the previous measurements the inhomogeneous intensity distribution can be appreciated. Furthermore, finite number and size of the electrodes leads to diffraction, which in turn leads to periodic oscillation of the light intensity in the PVBs. This intensity oscillation is inherent to discrete SPP and may be reduced by working with lower topological charges or increasing the number of pixels.
The optimal number of electrodes will depend on the specific application in which the system will be employed. A large number of electrodes may compromise the fill factor, and complicates the driving, while a small number of electrodes may lead to unacceptable high diffraction losses.
5. Conclusion
An electronically reconfigurable optical setup based on a liquid crystal spiral phase plate has been developed. In contrast with typical SLM configurations, the presented transmissive setup is able to generate perfect vortex having an unparalleled simple optical design. Using an in-house developed external electronic driver, the manufactured SPP can be set to 72 different configurations, where the topological charge (TC) of the impinging light beam is tuned up. The combination of this tunable SPP with a refractive axicon and a convergent lens enables the generation of PVBs, with 36 positive or 36 negative different TCs. Light transformations carried by each optical device have been measured. Experiments demonstrating the generation of these PVBs have been carried out, showing that the PVB topological charge can be tuned up without modifying the annular intensity size.
Funding
European Union’s Horizon 2020 research and innovation programme (Attract-IALL EU project G.A 101004462); European Space Agency (4000133048/20/NL/KML); Gobierno de España (BG20/00136, PDC2021-121370-C21, PID2020-114172RB-C22, PLEC2022-009381, TSI-063000-2021-83); Comunidad de Madrid (APOYO-JOVENES-21-9FOMOQ-22-0CNGFM, BEAGALINDO-21-QU81R4-7-0QQBF3, IND2020/TIC-17424, S2018/NMT-4326).
Acknowledgment
The authors are grateful to Professor J.M. Otón for creating the optical setup figure in Fig. 4.
Disclosures
The authors declare no conflicts of interest.
Author contributions. J.P.G. wrote the article, manufactured and characterized the device. In addition, J.P.G. carried out the device simulations. M.G.d.B. helped in the idea conceptualization and literature search. P.d.R.V. designed the SPP layout. X.Q.A., M.C.G. and M.A.G. designed the electronic driver. X.Q.A., M.C.G. and M.A.G. did the overall supervision. All authors reviewed the manuscript.
Data availability
All data in support of the findings of this paper are available within the article or as supplementary material. The simulations codes are available upon request to the authors.
Supplemental document
See Supplement 1 for supporting content.
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