Hermite–Gaussian beams with sinusoidal vortex phase modulation

Guanxue Wang; Yue Li; Xinzhi Shan; Yu Miao; Xiumin Gao

doi:10.3788/COL202018.042601

Journals >Chinese Optics Letters >Volume 18 >Issue 4 >Page 042601 > Article

Chinese Optics Letters
Vol. 18, Issue 4, 042601 (2020)

Hermite–Gaussian beams with sinusoidal vortex phase modulation

Guanxue Wang, Yue Li, Xinzhi Shan, Yu Miao, and Xiumin Gao^*

Author Affiliations

School of Optical-Electrical and Computer Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China

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DOI: 10.3788/COL202018.042601 Cite this Article Set citation alerts

Guanxue Wang, Yue Li, Xinzhi Shan, Yu Miao, Xiumin Gao. Hermite–Gaussian beams with sinusoidal vortex phase modulation[J]. Chinese Optics Letters, 2020, 18(4): 042601 Copy Citation Text

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Abstract

In this Letter, vortex phase and sinusoidal phase modulations of Hermite–Gaussian beams are studied theoretically and experimentally. The coding method of the experiment is introduced in detail, and the evolution law of focus under different beam order (m, n) and topological charge (l) is given. In order to verify the accuracy of the generation experiment, the optical field distribution under sinusoidal vortex modulation is analyzed deeply. The relevant analysis and methods provided in this Letter have certain practical significance for the development of laser mode analysis, optical communication, and other fields.

Keywords

focus mode Hermite–Gaussian beams phase modulation sinusoidal vortex phase modulation spatial light modulator

HG (x, y, z) = \frac{1}{ω (z)} {(\frac{2^{1 - n - m}}{π n! m!})}^{1 / 2} H_{n} [\frac{\sqrt{x}}{ω (z)}] H_{m} [\frac{\sqrt{y}}{ω (z)}] \times \exp [i (n + m + 1) ξ (z)] \exp {- {[\frac{p}{ω (z)}]}^{2}} \times \exp (- \frac{i k p^{2}}{2 R}) \exp (- i k z) \exp (i l \sin ϕ) .

(1)

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R (z) = z \sqrt{1 + {(\frac{z_{R}}{z})}^{2}} .

(2)

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p = \sqrt{x^{2} + y^{2}},

(3)

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ξ (z) = \arctan (\frac{z}{z_{R}}) .

(4)

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A_{HG} (x, y, z) = \frac{1}{ω (z)} \sqrt{\frac{2^{1 - n - m}}{π n! m!}} H_{n} [\frac{\sqrt{x}}{ω (z)}] H_{m} [\frac{\sqrt{y}}{ω (z)}] \exp [{- \frac{p}{ω (z)}]}^{2} .

(5)

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ϕ_{HG} (x, y, z) = \exp [i (1 + n + m) ξ (z)] \exp (- \frac{i k p^{2}}{2 R}) \times \exp (- i k z) \exp (i m \sin ϕ) .

(6)

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h (x, y) = \exp {i ψ [A_{HG} (x, y, z), ϕ_{HG} (x, y, z)]},

(7)

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h (x, y) = \sum_{- \infty}^{\infty} c_{q}^{A_{HG} (x, y, z)} \exp [i q ϕ_{HG} (x, y, z)],

(8)

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ψ [A_{HG} (x, y, z), ϕ_{HG} (x, y, z)] = f_{HG} \sin [ϕ_{HG} (x, y, z)] .

(9)

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\exp {{if}_{HG} \sin [ϕ_{HG} (x, y, z)]} = \sum_{m = - \infty}^{\infty} J_{m} f_{HG} \exp [im ϕ_{HG} (x, y, z)] .

(10)

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A_{HG} (x, y, z) = J_{1} f_{HG} .

(11)

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Φ_{SLM} = f_{HG} \sin (J_{HG} + G_{X} X + G_{Y} Y),

(12)

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Guanxue Wang, Yue Li, Xinzhi Shan, Yu Miao, Xiumin Gao. Hermite–Gaussian beams with sinusoidal vortex phase modulation[J]. Chinese Optics Letters, 2020, 18(4): 042601

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