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艾里变换是一种能实现高斯光束与艾里光束相互转换的神奇光学变换. 一阶艾里导数光束, 作为艾里光束的进阶型, 在经过艾里变换后会产生怎样的光束? 这就是本文所要研究的内容. 当艾里系数大于负的横向比例因子时, 一阶艾里导数光束的艾里变换在任意一个横向上的光场是偏心艾里光束和偏心一阶艾里导数光束之和. 当艾里系数等于负的横向比例因子时, 一阶艾里导数光束的艾里变换在任意一个横向上的光场是两个偏心优美厄米-高斯光束之和. 此外, 分别导出了一阶艾里导数光束经艾里变换后的质心和光束半宽在上述两种情形下的解析表达式. 最后, 实验上实现了一阶艾里导数光束的艾里变换, 并测量了艾里系数对光强分布、质心和光束半宽的影响. 一阶艾里导数光束艾里变换的研究拓宽了特殊形态分布光束的获取途径, 有望应用于光通信和分束技术等领域.As a remarkable optical transformation enabling mutual conversion between Gaussian and Airy beams, the Airy transformation raises intriguing questions when applied to Airyprime beam—an advanced variant of conventional Airy beam. To answer these questions, numerical simulations and experimental verification are combined in this study. The results show two different operation regimes: when the Airy coefficient exceeds the negative transverse scale factor, the Airy-transformed optical field of Airyprime beam in any transverse direction becomes equivalent to the superposition of eccentric Airy beam and eccentric Airyprime beam; when the Airy coefficient equals the negative transverse scale factor, the transformed optical field equivalently corresponds to the sum of two displaced elegant Hermite-Gaussian beams. Analytical expressions for centroid and beam half width under both regimes are rigorously derived and validated experimentally by using Airy transformation of Airyprime beams to systematically measure the influences of Airy coefficientson intensity distribution, centroid displacement, and beam half width. This investigation provides a novel method for generating complex beam profiles while enhancing the potential application value of such beams in optical communication and beam-splitting technology.
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Keywords:
- Airyprime beam /
- Airy transformation /
- Airy coefficients /
- centroid /
- beam half width
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图 2 一阶艾里导数光束经不同艾里变换后x方向上的归一化光强分布 (a) b = –0.50 mm; (b) b = –0.45 mm; (c) b = –0.40 mm; (d) b = –0.30 mm; (e) b = –0.20 mm; (f) b = –0.10 mm; (g) b = 0.00 mm; (h) b = 0.10 mm; (i) b = 0.20 mm
Fig. 2. Normalized light intensity distribution in the x-direction of an Airyprime beam after different Airy transformation: (a) b = –0.50 mm; (b) b = –0.45 mm; (c) b = –0.40 mm; (d) b = –0.30 mm; (e) b = –0.20 mm; (f) b = –0.10 mm; (g) b = 0.00 mm; (h) b = 0.10 mm; (i) b = 0.20 mm.
图 3 偏心艾里光束、偏心一阶艾里导数光束和交叉项在x方向上的光强分布 (a)—(c) b = –0.45 mm; (d)—(f) b = –0.20 mm; (g)—(i) b = 0.20 mm
Fig. 3. Light intensity distribution in the x–direction of the eccentric Airy beam, the eccentric Airyprime beam, and the cross term: (a)—(c) b = –0.45 mm; (d)—(f) b = –0.20 mm; (g)—(i) b = 0.20 mm.
图 4 一阶艾里导数光束经不同艾里变换后的二维归一化光强分布 (a) b = c = –0.50 mm; (b) b = c = –0.45 mm; (c) b = c = –0.40 mm; (d) b = c = –0.30 mm; (e) b = c = –0.20 mm; (f) b = c = –0.10 mm; (g) b = c = 0.00 mm; (h) b = c = 0.10 mm; (i) b = c = 0.20 mm
Fig. 4. Two-dimensional normalized intensity distribution of an Airyprime beam after different Airy transformation: (a) b = c = –0.50 mm; (b) b = c = –0.45 mm; (c) b = c = –0.40 mm; (d) b = c = –0.30 mm; (e) b = c = –0.20 mm; (f) b = c = –0.10 mm; (g) b = c = 0.00 mm; (h) b = c = 0.10 mm; (i) b = c = 0.20 mm.
图 5 艾里系数b对一阶艾里导数光束经艾里变换后x方向上的质心(a)和光束半宽(b)的影响
Fig. 5. Effect of the Airy coefficient b on the centroid (a) and the beam half width (b) in the x-direction of an Airyprime beam after Airy transformation.
图 6 一阶艾里导数光束艾里变换的实验装置示意图
Fig. 6. Experimental setup of the Airy transformation of an Airyprime beam.
图 7 (a) 用于产生一阶艾里导数光束的相位光栅; (b) 上传到SLM2的立方相位图
Fig. 7. (a) Phase grating for generation of an Airyprime beam; (b) cubic phase pattern uploaded onto SLM2.
图 8 一阶艾里导数光束经不同艾里变换后二维光强分布的实验记录 (a) b = c = –0.50 mm; (b) b = c = –0.45 mm; (c) b = c = –0.40 mm; (d) b = c = –0.30 mm; (e) b = c = –0.20 mm; (f) b = c = –0.10 mm; (g) b = c = 0.00 mm; (h) b = c = 0.10 mm; (i) b = c = 0.20 mm
Fig. 8. Experimental record of two-dimensional intensity profile of an Airyprime beam after different Airy transformation: (a) b = c = –0.50 mm; (b) b = c = –0.45 mm; (c) b = c = –0.40 mm; (d) b = c = –0.30 mm; (e) b = c = –0.20 mm; (f) b = c = –0.10 mm; (g) b = c = 0.00 mm; (h) b = c = 0.10 mm; (i) b = c = 0.20 mm.
图 9 艾里系数b对一阶艾里导数光束经艾里变换后x方向上的质心(a)和光束半宽(b)影响的实验测量
Fig. 9. Experimental measurement of the effect of the Airy coefficient b on the centroid (a) and the beam half width (b) in the x-direction of an Airyprime beam after Airy transformation.
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