A novel method of generating qausi-non-diffracting Mahtieu beam based on axicon

Self-reconstruction properties of the Mathieu beam are studied theoretically and experimentally. By means of the Mathieu-Hankel wave's theory, the self-reconstruction mechanism of Mathieu beam is analyzed. Based on the Fresnel diffraction theory, an analytical expression for the optical field of a Mathieu beam partially blocked by a circular opaque obstacle is derived, and the self-reconstruction process of Mathieu beam is simulated based on the analytical expression. Using a cylindrical lens and an axicon system to generate Mathieu beam, the self-reconstruction properties of the Mathieu beam partially blocked by a circular opaque obstacle on axis and off axis are verified. Experimental results agree well with the theoretical analysis.

Theoretical and experimental studies on the self-reconstruction property of non-diffracting Mathieu beam

Wu¹,

Xiao-Xia²,

Sun³

2015

Odd version Mathieu-Gaussian beam based on Green function

Wu¹,

Zhijun²,

Lin-Yue³

et al. 2017

Like the theoretical pattern of non-diffracting Bessel beams, ideal non-diffracting Mathieu beams also carry infinite energy, but cannot be generated as a physically realizable entity. Mathieu-Gaussian beams can be experimentally generated by modulating ideal Mathieu beams with a Gaussian function, and thus they are a kind of pseudo-non-diffracting beams with finite energy and finite transverse extent. The research of Mathieu-Gaussian beam propagating characteristics in free space is of great significance. In order to analytically study the propagation of Mathieu-Gaussian beams, the Mathieu function is expanded into the superposition of a series of Bessel functions in polar coordinates based on the superposition principle of light waves. It means that the Mathieu-Gaussian beam can be converted into accumulation of the infinite terms of the Bessel beams with different orders. According to the properties of the Bessel function, the free-space propagation properties of Mathieu-Gaussian beams can be studied in the circular cylindrical coordinates. Thus, a group of virtual optical sources are introduced to generate the odd Mathieu-Gaussian beams of the first kind, i.e., (2n+2)th-order, which is a family of Mathieu-Gaussian beams. Using the virtual source technique and the Green function, we derive the rigorous integral formula for the odd Mathieu-Gaussian beams of the first kind. Taking for example the first three orders with non-paraxial corrections, the analytical solution of the on-axis field of odd Mathieu-Gaussian beams of the first kind is further obtained from the integral formula. The axial intensity distribution of the odd Mathieu-Gaussian beams of the first kind is numerically calculated by the integral formula. The simulation results show that the calculation results obtained with the paraxial theory and the rigorous integral expressions of non-paraxial Mathieu-Gaussian beams are obviously different when the propagation distance of the odd Mathieu-Gaussian beams of the first kind is small. The calculation results of the two methods are coming closer and closer with the increasing propagation distance. The results indicate that the correct results can be obtained with the paraxial theory when we study the propagation of Mathieu-Gaussian beams in the far-field, but the non-paraxial theory must be used to obtain correct results when we study the propagation of Mathieu-Gaussian beams in the near-field. Owing to the complexity of the non-paraxial theory, it is difficult to obtain the exact analytic solutions of Mathieu-Gaussian beams in the near-field with the classical diffraction theory. Based on the superposition principle of light waves, by introducing the virtual source technique and the Green function, the complex Mathieu-Gaussian function can be expanded into the superposition of a series of simple Bessel functions, and the axial intensity distributions of Mathieu-Gaussian beams in the far-field and the near-field can be studied well. It will also provide a feasible method to study other complex beams propagating in free space.

Generation and applications of non-diffraction beam

Liu¹,

Hu²,

Xia³

et al. 2018

In recent years, with the development of laser technology, various non-diffraction beams each with a central spot unchanged after a long distance propagation, have been generated, they being the Bessel beam, higher Bessel beam, Mathieu beam, higher Mathieu beam, cosine beam, parabolic beam, and Airy beam. Diffraction-free beams are widely used in laser drilling, laser precision alignment, optical precision control, optical micromanipulation, optical communication, plasma guidance, light bullet, synthesis of autofocusing beam, nonlinear optics, etc. In this paper, the expressions, generation methods and corresponding experimental results of the various non-diffraction beams are presented. There are many ways to generate the Bessel beam, they being circular slit, computed hologram, spherical aberration lens, resonant cavity, axicon, and metasurface. The main methods of generating the non-diffraction beams are summarized, and each method is analyzed in depth from the cost of the system, and then some suggestions for improving and perfecting are made. For the generation of non-diffraction beams, the passive methods are used most to convert other beams into corresponding non-diffraction beams by optical components. Due to the low damage threshold and high cost of optical components, the power, energy and beam quality of a non-diffracting beam will be limited. How to generate a high-power, high-beam quality non-diffracting beam will be a hot research spot. Diffractionless beams have attracted a great deal of interest due to their unique non-diffraction, transverse-accelerating (or self-bending) and self-healing property. Transverse-accelerating property refers to that non-diffraction beams propagate along a parabola trajectory. The diffractionless beams' propagation trajectory control method implemented by changing system parameters is simple and easily successful, but cannot reverse acceleration direction, and its controlling range is limited. The self-healing property means that the non-diffraction beam tends to reform during propagation in spite of severe perturbations imposed. Both the Airy beam and the Bessel beam exhibit self-healing properties during propagation. And non-diffraction beams have potential applications in many fields. In atmosphere, such as in optical communication, non-diffracting beam exhibits more resilience against perturbations. Finally, brief summary and outlook of non-diffraction beams playing important roles in future study, and their application prospects are presented. In addition to Airy beam and Bessel beam, for other non-diffraction beams due to the complexity of the beams themselves, by comparison, their applications are investigated very little, so the applications in Mathieu beam, cosine beam, and parabolic beam will be a hot research spot.