Generation and three-dimensional characterization of complex nondiffracting optical beams

Nondiﬀracting optical beams play an important role in contemporary optics due to their special propagation characteristics, i.e., nondiﬀracting in a diﬀraction-free zone, shape recovering behind obstacles or self-healing property. Liquid crystal spatial light modulators (LC-SLM) are widely used for generating nondiﬀracting optical beams in virtue of programmable and dynamic features. In this paper, we propose a complex amplitude modulation technique that can encode any scalar complex ﬁelds for generating the complex nondiﬀracting beams. Before experiment, the phase modulation curve of the phase-only LC-SLM is optimized into being linear in a range of 0–2 (cid:25) by gamma correction in the way of variable binary phase gratings. Then, we experimentally generate the nonaccelerating beams, e.g., two zero-order Bessel beams with variable intensity distributions, and the nondiﬀracting petal-like beams generated by interfering with two coaxial Bessel beams. By scanning a reﬂection mirror near the focal region along the optical axis, a stack of two-dimensional images is acquired, and then a three-dimensional intensity proﬁle of the beam is reconstructed with a software. We also experimentally demonstrate a new kind of multi-main-lobe accelerating beam with parabolic accelerating trajectory by modifying the spatial spectrum of classical Airy beam. Compared with the so-called vectorial accelerating beam with multiple main lobes in spheroidal coordinates, our generated two-main-lobe accelerating beam has a very high energy eﬃciency. The self-healing property of the two-main-lobe accelerating beam is also demonstrated. The presented technique can generate a variety of complex nondiﬀracting optical beams rapidly and obtain their three-dimensional intensity distributions accurately, which has potential applications in the ﬁelds of optical microscope


Abstract
Nondiffracting optical beams play an important role in contemporary optics due to their special propagation characteristics, i.e., nondiffracting in a diffraction-free zone, shape recovering behind obstacles or self-healing property.
Liquid crystal spatial light modulators (LC-SLM) are widely used for generating nondiffracting optical beams in virtue of programmable and dynamic features.In this paper, we propose a complex amplitude modulation technique that can encode any scalar complex fields for generating the complex nondiffracting beams.Before experiment, the phase modulation curve of the phase-only LC-SLM is optimized into being linear in a range of 0-2π by gamma correction in the way of variable binary phase gratings.Then, we experimentally generate the nonaccelerating beams, e.g., two zero-order Bessel beams with variable intensity distributions, and the nondiffracting petal-like beams generated by interfering with two coaxial Bessel beams.By scanning a reflection mirror near the focal region along the optical axis, a stack of two-dimensional images is acquired, and then a three-dimensional intensity profile of the beam is reconstructed with a software.We also experimentally demonstrate a new kind of multi-main-lobe accelerating beam with parabolic accelerating trajectory by modifying the spatial spectrum of classical Airy beam.Compared with the so-called vectorial accelerating beam with multiple main lobes in spheroidal coordinates, our generated two-main-lobe accelerating beam has a very high energy efficiency.The self-healing property of the two-main-lobe accelerating beam is also demonstrated.
The presented technique can generate a variety of complex nondiffracting optical beams rapidly and obtain their threedimensional intensity distributions accurately, which has potential applications in the fields of optical microscope, optical date storage, optical trapping, optical micromachining, etc.

Fig. 1 .
Fig. 1. (color online) Optical setup for generation and measurement in three-dimension of nondiffracting optical beams: (a) The layout of optical configuration; (b) computer-generated-hologram loaded on the SLM; (c) intensity distribution of nondiffracting optical beams near the focal region of the objective lens.

Fig. 2 .
Fig. 2. (color online) Experimental results of gamma correction for the spatial light modulator in the way of variable binary phase gratings: (a) A series of binary gratings with different gray scales (corresponding to different phase depths) loaded on SLM for gamma correction; (b) diffraction patterns of Ronchi gratings before/after gamma correction; (c) diffraction patterns of blazed gratings before/after gamma correction.The experimental results indicate that the phase modulation of SLM is optimized to be linear in the range of 0-2π after gamma correction.

图 3 (
Fig. 3. (color online) Experimental results of two zero-order Bessel beams with controllable intensity distributions: (a) Computer-generated-hologram loaded on SLM; (b)-(f) intensity distributions of two Bessel beams with different peaks ratios.Inset curves are normalized intensity distributions.

Fig. 4 .
Fig. 4. (color online) Experimental results of generation and measurement of complex-mode Bessel beams: (a) Transversal intensity distribution of four zero-order Bessel beams in the focal region of objective; (b) 3D reconstruction of the measured four zero-order Bessel beams; (c) transversal intensity distribution of nondiffracting petal-like beams generated by interference of two coaxial propagation of ±6 orders Bessel beams in the focal region of objective; (d) 3D reconstruction of the measured nondiffracting petal-like beams.Scale bar: 3 µm.

图 5 (
Fig. 5. (color online) Experimental results of generation and measurement of accelerating beams: (a) Computergenerated-hologram loaded on SLM for the conventional Airy beam; (b) transversal intensity distribution of the conventional Airy beam; (c) 3D reconstruction of the measured Airy beam; (d) computer-generated-hologram for two-main-lobe accelerating beam; (e) transversal intensity distribution of the two-main-lobe accelerating beam; (f) 3D reconstruction of the measured two-main-lobe accelerating beam.Scale bar: 5 µm.

Fig. 6 .
Fig. 6. (color online) Experimental results of self-healing property of the two-main-lobe accelerating beam: (a)-(d) Transversal intensity distributions of the two-main-lobe accelerating beam in different axial positions without any obstacles; (e)-(h) transversal intensity distributions of the two-main-lobe accelerating beam with one main lobe blocked; (i)-(l) transversal intensity distribution of the two-main-lobe accelerating beam with both of main lobes blocked.Scale bar: 3 µm.