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为深入理解复杂光场在近场范围内的精确传输行为, 特别是在克服传统近轴近似限制下的传输行为, 本文发展了一种更为精确的理论框架, 全面揭示了广义抛物光束的传播机制及其能量传输模式. 基于光的独立传播与叠加原理, 利用虚点源方法, 结合韦伯积分公式及极坐标系下的傅里叶-贝塞尔变换方法, 严格推导出一套描述广义抛物光束近场传输的积分表达式. 这一表达式突破了传统近轴近似限制, 并涵盖了广义抛物光束的所有关键传输参数. 通过此积分表达式, 计算并分析了光束沿光轴的强度分布及相位特征, 从而揭示了其能量传输模式与相位特性. 基于推导所得的积分表达式, 运用数值模拟, 傍轴近似解和非傍轴修正解在远场模拟的结果展现出了良好的一致性, 验证了推导结果的正确性. 研究结果加深了对广义抛物光束近场传输机制的理解, 也为复杂光场在近场范围精确传输行为的计算奠定了理论基石.Generalized parabolic beams have various optical morphologies. They can be used in different research fields, such as component design, aero-optics, and microwave wireless power transmission. Studying the near-field transmission characteristics of these beams is important for improving utilization efficiency. We develop a more accurate theoretical framework to precisely understand the propagation behaviors of complex light fields in the near-field range, especially to break through the limitations of conventional near-axis approximation. This framework fully reveals the propagation mechanism of parabolic beams and their energy transmission modes. Here, based on the principle of independent propagation and the virtual source method, a group of virtual sources are introduced to analyze generalized parabolic beams. These beams can be expanded into the superposition of infinite continuous integer Bessel beams. Then, by combining the Weber integral formula and the Fourier Bessel transform, we rigorously derive an integral expression for generalized parabolic beams during near-field propagation. This expression breaks through the limitation of the traditional paraxial approximation and contains all the key propagation parameters of the family of beams. Based on this integral expression, the intensity distribution and phase characteristics of the generalized parabolic beam along the optical axis are further calculated and analyzed to reveal its energy transfer mode and phase characteristics. By comparing the paraxial approximate solution with the nonparaxial corrected solution for generalized parabolic beams, the far-field propagation of generalized parabolic beams is found to be the same when the propagation distance is sufficiently long. Such simulation results indirectly confirm the correctness of the obtained theoretical solution. The simple paraxial approximation theory can be used conveniently to calculate the far-field propagation of generalized parabolic beams. However, large errors exist when paraxial theory is used to calculate the near-field distribution of generalized parabolic beams. Although calculating nonparaxial propagation is especially complex, the nonparaxial correction solution is necessary when generalized parabolic beams are used in near-field research. Such research results not only deepen the understanding of the propagation mechanism of generalized parabolic beams but also lay a theoretical foundation for studying the precise propagation behaviors of other complex light fields in near-field optics.
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Keywords:
- generalized parabolic beams /
- virtual sources /
- paraxial approximation /
- superposition principle
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图 1 连续参数a = 0时的广义二阶抛物光束的轴上强度
Fig. 1. The on-axis intensity distribution of the Parabolic beams with continuous parameter a = 0.
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[1] Durnin J, Miceli J J, Eberly J H 1987 Phys. Rev. Lett. 58 1499
Google Scholar
[2] Durnin J 1987 J. Opt. Soc. Am. A 4 651
Google Scholar
[3] Gutiérrez-Vega J C, Iturbe-Castillo M D, Chávez-Cerda S 2000 Opt. Lett. 25 1493
Google Scholar
[4] Sosa-Sánchez C T, Silva-Ortigoza G, Juárez-Reyes S A, et al. 2017 J. Opt. 19 085604
Google Scholar
[5] Bandres M A, Gutirrez-Vega J C, Chvez-Cerda S 2004 Opt. Lett. 29 44
Google Scholar
[6] Khonina S N, Ustinov A V, Chávez-Cerda S 2018 J. Opt. Soc. Am. A 35 1511
Google Scholar
[7] Liang Y S, Yan S H, Wang Z J, Li R Z, Cai Y N, He M R, Yao B L, Lei M 2020 Rep. Prog. Phys. 83 032401
Google Scholar
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Google Scholar
[9] Deschamps G A 1971 Electron. Lett. 7 684
Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
[15] Gori F, Guattari G, Padovani C 1987 Opt. Commun. 64 491
Google Scholar
[16] Seshadri S R 2002 Opt. Lett. 27 1872
Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
Chen X M, Li H Y, Wu T, Meng X S, Li F X 2023 Acta Phys. Sin. 72 100302
Google Scholar
[21] 岳东宁, 董全力, 陈民, 赵耀, 耿盼飞, 远晓辉, 盛政明, 张杰 2023 物理学报 72 125201
Google Scholar
Yue D N, Dong Q L, Chen M, Zhao Y, Geng P F, Yuan X H, Sheng Z M, Zhang J 2023 Acta Phys. Sin. 72 125201
Google Scholar
[22] 尹培琪, 许博坪, 刘颖华, 王屹山, 赵卫, 汤洁 2024 物理学报 73 095202
Google Scholar
Yin P Q, Xu B P, Liu Y H, Wang Y S, Zhao W, Tang J 2024 Acta Phys. Sin. 73 095202
Google Scholar
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