用傅里叶分析法研究艾里光束远场传播特性

Propagation Characteristics of Airy Beams in the Far-field with Fourier Analysis

为了从理论上深入分析新型无衍射光束艾里光束在有限能量条件下的远场传播特性,首先,从决定光波在自由空间传播的一维旁轴波动方程入手,采用傅里叶分析法,结合艾里函数的特殊性质,并利用经过指数衰减的有限能量初始条件,完整给出了有限能量条件下用于精确描述一维艾里光束在自由空间传播特性的波动方程解析解.然后,利用所得到解析解分别对一维和二维艾里光束在自由空间的传播特性进行了研究,重点分析了不同参量条件对艾里光束进行无衍射传播和横向自加速的影响.研究表明:当任意横向尺度为100μm,衰减系数为0.03、0.05、0.07、0.1、0.2时,二维艾里光束无衍射传播距离分别为1 014、624、455、338、193mm;当横向尺度保持不变时,衰减系数越小,艾里光束保持无衍射传播的距离越大;当衰减系数保持不变时,横向尺度越小,艾里光束横向自加速越大.所采用的研究方法也可用于研究艾里光束在介质中的传播特性.

In order to analyze the far-field propagation characteristics of the new non-diffractive Airy beams deeply in theory on the condition of finite energy,firstly,the one-dimensional paraxial wave equation which decides the propagation characteristics of light waves in free space was analyzed by taking advantage of the Fourier analysis method.Moreover,the special characteristics of Airy function and the initial condition of finite energy with exponential decay were also taken into consideration.As a result,the analytical solution of the wave equation can describe the propagation characteristics of onedimensional Airy beams accurately in free space under the condition of finite energy.Then,the propagation characteristics of one-dimensional and two-dimensional Airy beams were studied by utilizing the acquired analytical solution.What′s more,the influence of various parameters on nondiffractive propagation and transverse acceleration characteristics of Airy beams were studied in detail.Study results indicate the non-diffractive propagation distances of two-dimensional Airy beams are1014,624,455,338,193 mm separately when the arbitrary transverse scale is 100μm and the decay factors are 0.03,0.05,0.07,0.1,0.2,separately.The non-diffractive propagation distances will become longer if the decay factor becomes smaller when the arbitrary transverse scale keeps invariant.Moreover,the transverse acceleration of Airy beams will become larger with the decrease of arbitrary transverse scale when the decay factor keeps invariant.The adopted study method is also fit for analysis of Airy beams′propagation characteristics in the media with refractive index.