艾里光束是实验上观测到的第一类自加速光波，在艾里光束的基础上通过相位调制引入涡旋项可得到艾里涡旋光束。艾里涡旋光束具有无衍射、自愈以及横向自加速特性，并且携带轨道角动量(OAM)，其在光学捕获、光通信、表面等离激元、光子弹等方面有着广泛应用。本文针对艾里涡旋光束在介质中传输及相互作用的基础理论研究为艾里涡旋光束的光场调控及其在光通信、传感等领域的应用提供了理论支持。

本文采用矩阵光学法和矢量角谱表示法对艾里涡旋光束的波场进行描述，研究了不同类型艾里涡旋光束通过自由空间和手征介质的传输变换规律，以及艾里涡旋光束入射各向同性分层介质和手征介质的反射与透射特性，并分析了艾里涡旋光束入射石墨烯-氮化硼(hBN)异质结构产生的古斯-汉欣(GH)位移。主要研究成果如下：

1. 基于惠更斯-菲涅耳衍射积分公式，推导出傍轴条件下艾里涡旋光束在自由空间中传输的解析表达式。分析了不同模阶数的艾里涡旋光束在自由空间中的传输特性(包括光场强度和相位分布以及涡旋位置对光场强度分布的影响)。结果表明，在初始输入平面，受光学涡旋的影响，涡旋中心周围的光场呈暗中空分布；模阶数对光束的抛物线运动轨迹没有影响；在传输过程中，模阶数越大，光束保持无衍射传输的距离越长，维持艾里形状的能力越强。

2. 基于平面波角谱展开方法，研究了艾里涡旋光束斜入射介质分界面的反射与透射特性。在入射场模阶数为0时，推导出由角谱展开的高阶光场模组成的反射场和透射场的解析表达式。数值分析了不同波束入射角下几何光场模和高阶光场模对实际场的影响；以及衰减因子和光束宽度对光场强度分布和布鲁斯特角附近光束位移的影响。同时还讨论了光束在左手材料(LHM)中的成像问题。最后，对模阶数为1的艾里涡旋光束通过界面后幅度、相位以及轨道角动量态的变化进行了数值模拟。

3. 基于平面波角谱展开方法，研究了艾里涡旋光束斜入射电介质板的反射与透射特性。推导出模阶数为0的入射场通过介质板后反射场、内场和透射场的解析表达式。对光束通过介质板后幅度、相位变化以及介质板参数对光场的影响进行了模拟。不同于艾里光束斜入射介质分界面的情况，斜入射电介质板后角谱展开的各阶光场模对实际场的贡献与介质板折射率有关；由于光束在介质板内多次反射、透射，透射光束的相位相对于入射光束也发生了变化。反射场和透射场峰值强度随介质板光学厚度呈周期性变化。由于艾里光束是宽波束，不同于平面波，在临界角入射时能量不能被全部反射出去，会存在部分泄漏。最后，采用数值方法分析模阶数为1的艾里涡旋光束通过介质板后幅度、相位以及轨道角动量态的变化。研究结果可为光学薄膜的设计和优化提供理论支持。

4. 研究了模阶数为1的艾里涡旋光束在手征介质中的传输以及反射与透射特性。手征介质中电磁波以左旋圆偏振波和右旋圆偏振波的形式存在。基于矢量角谱表示方法，将艾里涡旋光束展开为沿不同方向传播的平面波叠加，结合电磁场边界条件，对艾里涡旋光束斜入射非手征-手征介质分界面的反射与透射场进行了计算。分析了不同介质参数下反射场与透射场的强度和相位分布。基于惠更斯-菲涅耳衍射积分公式，研究了傍轴条件下双曲余弦艾里涡旋(CAiV)光束在手征介质中的传输特性。数值分析了双曲余弦因子对光束强度分布、传输轨迹、坡印廷矢量和角动量密度的影响。双曲余弦因子的引入为艾里涡旋光束在手征介质中传输的操控提供了一个新的自由度，在不改变光束传输轨迹的条件下，可以通过改变双曲余弦因子控制光束的强度特征以满足实际应用需求，实现激光器的优化设计。

5. 基于平面波角谱展开方法，推导出艾里涡旋光束入射石墨烯-hBN异质结构反射产生的空间GH位移的解析解。通过模拟石墨烯-hBN异质结构在不同波长和入射角下的反射率，选取了合适的波长和入射角来获得较大的GH位移。数值分析了石墨烯-hBN异质结构的费米能级、弛豫时间、石墨烯层数、hBN厚度以及艾里涡旋光束的衰减因子和涡旋的位置对空间GH位移的影响。结果表明，布鲁斯特角的位置主要取决于石墨烯-hBN异质结构的弛豫时间和hBN厚度，在布鲁斯特角位置确定的情况下，通过调节费米能级可以实现对GH位移方向和幅度的有效控制。可控的GH位移可用于光开光、生物传感、表面等离子体共振成像探测、介质参数测量等应用。

Airy beams represent the first experimentally observed class of self-accelerating optical waves. Airy vortex beams can be obtained by introducing vortex terms through phase modulation. Airy vortex beams possess diffraction-free, self-healing and transverse self-accelerating properties, and carry orbital angular momentum (OAM). Airy vortex beams have been utilized in a variety of applications such as in optical trapping, optical communication, surface plasmon polariton and optical bullets. The study of propagation and interaction of Airy vortex beams in complex media provides theoretical supports for the application in the modulation of optical fields, optical communication and sensing.

 

The wave field descriptions of Airy vortex beams are obtained using matrix optics method and plane-wave angular spectrum representation. The propagation properties of different types of Airy vortex beams through free space and chiral medium are investigated. The reflection and transmission of Airy vortex beams incident on isotropic layered medium and chiral medium are analyzed, and the spatial GH shift of the Airy vortex beam impinging on graphene-hBN heterostructure is presented. The research results are as follows:

 

1. Under paraxial approximation, the analytical propagation expressions of the Airy vortex beams in free space are obtained by the Huygens-Fresnel diffraction integral formula and ABCD matrix. The propagation properties (including the intensity distribution, phase distribution, and the effect of vortex position on the intensity distribution) of Airy beams, first-order Airy vortex beams, and second-order Airy vortex beams are compared. It is found that with the increase of the propagation distance, the larger the topological charge, the stronger the ability of the beam to maintain shape, and the longer the propagation distance to keep diffraction-free.

 

2. Based on the plane-wave angular spectrum expansion, the reflection and transmission of an Airy vortex beam oblique impinging on a dielectric surface is presented. When the topological charge of the incident field is 0, the analytical expressions for the reflected and transmitted fields comprising of beam modes are derived. The contribution of the geometrically beam field, the first order beam mode field, the second order beam mode field to the actual beam field with different incident angles are simulated. The influences of decay factor and beam width on the intensity distributions of the incident, reflected, and transmitted beams and the beam shift of the reflected beam near the Brewster angle are studied. Meanwhile, an Airy beam incident from air to the left-handed material (LHM) is discussed. Finally, the amplitude, phase and orbital angular momentum states of an Airy vortex beam with topological charge l=1 passing through the interface are numerically simulated.

 

3. Based on the plane-wave angular spectrum expansion, the reflection and transmission of an Airy vortex beam incident in a dielectric slab are investigated. For the topological charge of the incident field is 0, the analytical expressions of the reflected field, internal field as well as transmitted field in each region are obtained. The change of intensity and phase distributions of the Airy beam passing through the slab and the influence of the parameters of the dielectric slab on the optical field are simulated. Different from the case of Airy beam oblique incident on the dielectric interface, the contribution of each order beam mode field to the actual beam field is related to the refractive index of the dielectric slab. Because the beam undergoes multiple reflection transmission in the dielectric slab, the phase of the transmitted beam is also changed with respect to the incident beam. The peak intensity of the reflected and transmitted beams varies periodically with the optical thickness of the dielectric slab. Different from the plane wave, the Airy beam has a finite width, so the energy cannot be reflected out completely at critical angle, and there will be partial leakage. Finally, the amplitude, phase and orbital angular momentum states of an Airy vortex beam with topological charge l=1 passing through the dielectric slab are analyzed numerically. The results can provide theoretical support for the design and optimization of optical films.

 

4. The propagation, reflection and transmission of an Airy vortex beam in a chiral medium is presented. Airy vortex beams split into the left circularly polarized vortex (LCPV) beams and the right circularly polarized vortex (RCPV) beams in the chiral medium. Based on the plane-wave angular spectrum expansion, Airy vortex beams are expanded into the superposition of plane waves with different transverse wave vectors. Combined with the boundary conditions, the reflection and transmission of the Airy vortex beams incident in an interface separating an isotropic achiral medium and an isotropic chiral medium are investigated. The influences of the parameters of the media on the intensity and phase distributions of the reflected and transmitted fields are analyzed. Based on the Huygens-Fresnel diffraction integral formula and ABCD transfer matrix, the paraxial propagation of the cosh-Airy vortex (CAiV) beams in a chiral medium is investigated. We numerically simulate the intensity distribution, propagation trajectory, peak intensity, main lobe’s intensity, Poynting vector, and angular momentum of the CAiV beams with different cosh parameters in a chiral medium. The introduction of the cosh parameters provides a new degree of freedom for manipulating the propagation of the CAiV beams in a chiral medium. Without the propagation trajectory changed, one can control intensity distribution, the main lobe’s intensity and peak intensity of the CAiV beams in the chiral medium by adjusting the cosh parameters to meet the actual usage and optimized design of lasers.

 

5. Based on the angular spectrum expansion, a full analytical theory for the spatial GH shifts of the Airy vortex beams impinging on graphene-hBN heterostructure is presented. By simulating the reflectance of the graphene-hBN heterostructure with different wavelength and incident angle, the appropriate wavelength and incident angle are obtained to enhance the GH shift. The influences of graphene-hBN heterostructure parameters (including Fermi energy, relaxation time, layer numbers of graphene, and hBN thickness) and incident Airy vortex beam parameters (including vortex position and decay factor) on the spatial GH shifts are analyzed in detail. The magnitude and sign of GH shifts at a certain Brewster angle can be controlled effectively by tuning the Fermi energy. The controlled GH shifts can be used in optical switch, biosensing, surface plasmon resonance imaging probing and precision measurement.

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