Researching | Experimental determination of the azimuthal and radial mode orders of a partially coherent LGpl beam (Invited Paper)

Journals >Chinese Optics Letters >Volume 15 >Issue 3 >Page 030002 > Article

Chinese Optics Letters
Vol. 15, Issue 3, 030002 (2017)

Experimental determination of the azimuthal and radial mode orders of a partially coherent LG_pl beam (Invited Paper)

Xianlong Liu^1、2, Tengfei Wu^1、2, Lin Liu^1、2, Chengliang Zhao^1、2, and Yangjian Cai^1、2、*

Author Affiliations

¹College of Physics, Optoelectronics and Energy & Collaborative Innovation Center of Suzhou Nano Science and Technology, Soochow University, Suzhou 215006, China

²Key Lab of Advanced Optical Manufacturing Technologies of Jiangsu Province and Key Lab of Modern Optical Technologies of Education Ministry of China, Soochow University, Suzhou 215006, China

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DOI: 10.3788/COL201715.030002 Cite this Article Set citation alerts

Xianlong Liu, Tengfei Wu, Lin Liu, Chengliang Zhao, Yangjian Cai. Experimental determination of the azimuthal and radial mode orders of a partially coherent LG_pl beam (Invited Paper)[J]. Chinese Optics Letters, 2017, 15(3): 030002 Copy Citation Text

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Abstract

It is known that one can determine the mode orders (i.e., the azimuthal order and radial order) of a partially coherent

{LG}_{p l}

beam (i.e., a partially coherent vortex beam) based on the measurement of the cross-correlation function (CCF) and the double correlation function (DCF) together. The technique for measuring the CCF is known. In this Letter, we propose a method for measuring the DCF. Based on the proposed method, the determination of the mode orders of a partially coherent

{LG}_{p l}

beam is demonstrated experimentally.

Γ (ρ_{1,} ρ_{2}) = {(\frac{\sqrt{2} ρ_{1}}{ω_{0}})}^{l} {(\frac{\sqrt{2} ρ_{2}}{ω_{0}})}^{l} L_{p}^{l} (\frac{2 ρ_{1}^{2}}{ω_{0}^{2}}) L_{p}^{l} (\frac{2 ρ_{2}^{2}}{ω_{0}^{2}}) \exp [i l (φ_{1} - φ_{2})] \times \exp (- \frac{ρ_{1}^{2} + ρ_{2}^{2}}{ω_{0}^{2}}) \exp [- \frac{ρ_{1}^{2} + ρ_{2}^{2} - 2 ρ_{1} ρ_{2} \cos (φ_{1} - φ_{2})}{2 δ_{g}^{2}}],

(1)

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μ (r_{1}, r_{2}) = \frac{Γ (r_{1}, r_{2})}{\sqrt{〈 I (r_{1}) 〉 〈 I (r_{2}) 〉}} .

(2)

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g^{(2)} (r_{1}, r_{2}) = \frac{〈 I (r_{1}) I (r_{2}) 〉}{〈 I (r_{1}) 〉 〈 I (r_{2}) 〉} = 1 + {| μ (r_{1}, r_{2}) |}^{2},

(3)

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〈 I (r) 〉 = \sum_{n}^{N} I (r) / N,

(4)

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〈 I (r_{1}) I (r_{2}) 〉 = \sum_{n}^{N} I_{n} (r_{1}) I_{n} (r_{2}) / N .

(5)

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{| μ (r_{1}, r_{2}) |}^{2} = \frac{N \sum_{n}^{N} I_{n} (r_{1}) I_{n} (r_{2})}{\sum_{n}^{N} I_{n} (r_{1}) \sum_{n}^{N} I_{n} (r_{2})} - 1 .

(6)

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I_{n}^{'} (x^{'}, y^{'}) = (\begin{matrix} (x_{- m}^{'}, y_{m}^{'}) & (x_{- m + 1}^{'}, y_{m}^{'}) & \dots & (x_{m}^{'}, y_{m}^{'}) \\ (x_{- m}^{'}, y_{m - 1}^{'}) & (x_{- m + 1}^{'}, y_{m - 1}^{'}) & \dots & \dots \\ \dots & \dots & \dots & \dots \\ (x_{- m}^{'}, y_{- m}^{'}) & \dots & \dots & (x_{m}^{'}, y_{- m}^{'}) \end{matrix}) = (\begin{matrix} (x_{- m}, y_{m}) & (x_{- m + 1}, y_{m}) & \dots & (x_{m}, y_{m}) \\ (x_{- m}, y_{m - 1}) & (x_{- m + 1}, y_{m - 1}) & \dots & \dots \\ \dots & \dots & \dots & \dots \\ (x_{- m}, y_{- m}) & \dots & \dots & (x_{m}, y_{- m}) \end{matrix}) .

(7)

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I_{n}^{'} (2 x^{'}, 2 y^{'}) = I_{n}^{''} (x^{''}, y^{''}) = (\begin{matrix} (x_{- m}^{''}, y_{m}^{''}) & (x_{- m + 1}^{''}, y_{m}^{''}) & \dots & (x_{m}^{''}, y_{m}^{''}) \\ (x_{- m}^{''}, y_{m - 1}^{''}) & (x_{- m + 1}^{''}, y_{m - 1}^{''}) & \dots & \dots \\ \dots & \dots & \dots & \dots \\ (x_{- m}^{''}, y_{- m}^{''}) & \dots & \dots & (x_{m}^{''}, y_{- m}^{''}) \end{matrix}) = (\begin{matrix} (x_{- 2 m}, y_{2 m}) & (x_{- 2 m + 2}, y_{2 m}) & \dots & (x_{2 m}, y_{2 m}) \\ (x_{- 2 m}, y_{2 m - 2}) & (x_{- 2 m + 2}, y_{2 m - 2}) & \dots & \dots \\ \dots & \dots & \dots & \dots \\ (x_{- 2 m}, y_{- 2 m}) & \dots & \dots & (x_{2 m}, y_{- 2 m}) \end{matrix}),

(8)

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{| μ (r^{'}, 2 r^{'}) |}^{2} = {| μ (x^{'}, y^{'}; 2 x^{'}, 2 y^{'}) |}^{2} = \frac{N \sum_{n}^{N} I_{n} (x^{'}, y^{'}) I_{n} (2 x^{'}, 2 y^{'})}{\sum_{n}^{N} I_{n} (x^{'}, y^{'}) \sum_{n}^{N} I_{n} (2 x^{'}, 2 y^{'})} - 1 .

(9)

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Xianlong Liu, Tengfei Wu, Lin Liu, Chengliang Zhao, Yangjian Cai. Experimental determination of the azimuthal and radial mode orders of a partially coherent LG_pl beam (Invited Paper)[J]. Chinese Optics Letters, 2017, 15(3): 030002

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