广义空间分数阶Bessel光束矢量性质的解析
Unraveling the vector nature of generalized space-fractional Bessel beams
摘要：
本文给出了柱坐标系下齐次空间分数阶Helmholtz方程的精确解析解。这个解称为矢量空间分数贝塞尔光束（SFBB），它是由洛伦兹规范条件和赫兹矢量变换建立的。我们执行标量和矢量波分析集中在电磁学的应用，特别是在光束的尺寸与其波长$(k_r \approx k)$相当的情况下。研究了光纤的衍射和自修复特性，着重研究了光纤的偏振态和横向传输模式。由于连续阶轨道角动量依赖性，该光束可以作为普通整数贝塞尔光束和分数贝塞尔光束之间的桥梁，因此，可以被认为是适用于整数维和分数维空间的空间分数波方程的广义解。所提出的SFBB提供对光束特性的更好控制，并且可以容易地使用数字微镜器件（DMD）、空间光调制器（SLM）、超表面或螺旋相位板来生成。我们的发现为电磁波传播提供了新的见解，从而为光镊、折射率传感、光阱和光通信等领域的新应用铺平了新的道路。
英文摘要：
We introduce an exact analytical solution of the homogeneous space-fractional Helmholtz equation in cylindrical coordinates. This solution, called vector Space-Fractional Bessel Beam (SFBB), has been established from the Lorenz' gauge condition and Hertz vector transformations. We perform scalar and vector wave analysis focusing on electromagnetics applications, especially in cases where the dimensions of the beam are comparable to its wavelength $(k_r \approx k)$. The propagation characteristics such as the diffraction and self-healing properties have been explored with particular emphasis on the polarization states and transverse propagation modes. Due to continuous order orbital angular momentum dependence, this beam can serve as a bridge between the ordinary integer Bessel beam and the fractional Bessel beam and, thus, can be considered as a generalized solution of the space-fractional wave equation that is applicable in both integer and fractional dimensional spaces. The proposed SFBBs provide better control over the beam characteristics and can be readily generated using Digital Micromirror Devices (DMDs), Spatial Light Modulators (SLMs), metasurfaces, or spiral phase plates. Our findings offer new insights on electromagnetic wave propagation, thus paving a new route towards novel applications in optical tweezers, refractive index sensing, optical trapping, and optical communications.
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