Circular Airy beams realized via the photopatterning of liquid crystals [Invited] Download: 715次
An Airy beam is a propagation-invariant wave travelling along a curved parabolic trajectory while being resilient to perturbations. In other words, the Airy beam features non-diffraction, transverse acceleration, and self-healing characteristics. An ideal Airy beam consists of a main lobe and a series of adjacent side lobes trailing off to infinity whose intensity distribution follows the Airy function. These remarkable properties endow Airy beams with numerous attention, especially since their first experimental demonstration was realized by Siviloglou et al. via introducing an exponential truncation aperture[1]. Both the one- and two-dimensional Airy wave packets were derived from the Schrödinger equation in a Cartesian coordinate system. Afterwards, circular Airy beams (CABs) were proposed[2] and realized[3] based on cylindrical coordinates. The intrinsic lateral acceleration of the Airy beams during the diffraction-free propagation results in the peculiar autofocusing property of the CAB[2]. Namely, the CABs will keep a relatively low intensity before abrupt autofocusing with peak intensity increased by orders of magnitude. This feature is crucial for situations where high-energy lasers are needed inside transparent objects while the samples before the focus are required to be intact. Therefore, CABs show unique advantages in laser ablation and biomedical treatments like laser nanosurgery[3]. In addition, CABs have also been applied in microparticles’ manipulation[4] and the generation of nonlinear intense light bullets[5]. Moreover, they are excellent candidates for igniting nonlinear processes[6] such as optical filaments in gases, stimulated Raman scattering, multiphoton absorption, and so on.
The main strategy for generating CABs is based on the Fourier transform (FT) technique[3,7,8], i.e., modulating a Gaussian beam through the FT pattern of a CAB and then adopting a lens to perform the FT. Another way is utilizing holograms calculated by computing the interference of CABs and the plane wave[4,9]. Usually a system consisting of two lenses is demanded to filter the desired orders. For the design of the Fourier mask, a simpler and more convenient method is the superimposition of specific phase distributions[10]. In most of the above approaches, spatial light modulators (SLMs) are employed to display the phase patterns. However, the efficiency and quality of the output CABs are typically limited by the complex electrode matrices of the SLMs[11]. Besides, there must be beam expanders to match the size of the SLM chips, leading to a longer working distance and higher cost for the SLM-based optical systems.
To overcome the shortcomings mentioned above, we propose a liquid crystal (LC) geometric phase plate, LC circular Airy plate (CAP), to generate CABs. The LC-mediated geometric phase optical elements are remarkable candidates for efficient manipulation of light waves[12
Based on asymptotic analysis, the FT of a CAB is proportional to the zero-order Bessel function[7], whose argument contains a linear term and a cubic term. Therefore, the phase pattern for the CAB’s generation can be simply configured by superimposing the radial linear phase and the radial cubic phase, that is, where , is the linear period, and is a parameter relating to the cubic phase modulation. Figure
Fig. 1. Patterns of the (a) linear phase, (b) cubic phase, and (c) superimposed phase, where black to white indicates 0 to . (d) Micrograph of the LC CAP with an inset showing the zoom-in image of the central part marked by the dashed lines. The scale bar is 100 μm.
To carry out the above design, we take nematic LC E7 as the anisotropic medium and adopt sulfonic azo-dye SD1 as the alignment material. The SD1 molecules are sensitive to UV light and will orientate perpendicularly to the incident polarization direction. The orientation of SD1 will be further spread to the adjacent LCs by intermolecular interactions[19]. By means of the DMD ( pixels with pixel size )-based micro-lithography system and the eighteen-step five-time partly overlapping exposure process[20], the phase pattern in Fig.
The process of geometric phase modulation can be analyzed by the Jones calculus. The Jones matrix for an optical element with the optical axis orientating as can be expressed as where is the phase retardation, is the birefringence of the LC E7 at the incident wavelength , and is the LC cell gap. When the LC CAP is illuminated by a circularly polarized Gaussian beam, i.e., , where the sign of the spin eigenstate corresponds to left/right circular polarization (LCP/RCP), the output field can be written as
The first term of Eq. (
To verify the performance of our sample, we employ an optical setup, as shown in Fig.
Figure
Fig. 3. (a) Experimental and (b) simulated autofocusing dynamics of the CAB. Blue circles in (a) represent the measured radius or the transverse deflection of the CAB, and the red lines are parabolic fit curves. The inserted transverse profiles are detected at , 20, 30, 40, 50, 60, and 74 cm, respectively.
Furthermore, the incident polarization is adjusted to RCP to fully investigate the propagation process of the CAB modulated by the LC CAP. Figures
Fig. 4. (a)–(e) Intensity distributions of the defocusing CABs at , 20 40, 60, and 74 cm, (f) autofocusing CAB at by changing the incident polarization from RCP to LCP, and (g) transformed Bessel beam at . The insets in (f) and (g) are the zoom-in images of the central parts. (h) Detected OFF state of the CAB at when the applied voltage is tuned to meet the full-wave condition. The scale bars in (a) and the insets of (f)–(h) indicate 500 μm and 200 μm, respectively.
In conclusion, we have demonstrated a practical and convenient way to generate the CABs via a geometric phase element LC CAP. The LC sample is fabricated through the SD1-based photoaligning technique and the DMD-based dynamic photopatterning skill. A characteristic autofocusing process is observed, and the focal distance and propagation dynamics are in accordance with the calculations and simulations. Additionally, the autodefocusing state is also realized by switching the incident circular polarization, exhibiting the advantage of polarization controllability. The switching between ON and OFF states of the CAB can also be achieved by tuning the applied voltage. Besides, the LC CAP sample and the focal length of the CAB can be customized as required. Together with optical reconfigurability and electrical tunability, these merits will promote and expand the applications of CABs in optics and photonics, military field, medical sciences, and even interdisciplinary fields.
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Yuan Zhang, Bingyan Wei, Sheng Liu, Peng Li, Xin Chen, Yunlong Wu, Xian’an Dou, Jianlin Zhao. Circular Airy beams realized via the photopatterning of liquid crystals [Invited][J]. Chinese Optics Letters, 2020, 18(8): 080008.