Determination of the focused beam waist of lasers with weak measurements

Linguo Xie; Xiaodong Qiu; Jiangdong Qiu; Zhiyou Zhang; Jinglei Du; Fuhua Gao

doi:10.3788/COL201513.112401

Journals >Chinese Optics Letters >Volume 13 >Issue 11 >Page 112401 > Article

Chinese Optics Letters
Vol. 13, Issue 11, 112401 (2015)

Determination of the focused beam waist of lasers with weak measurements

Linguo Xie¹, Xiaodong Qiu¹, Jiangdong Qiu¹, Zhiyou Zhang^1,2,*..., Jinglei Du^1,2 and Fuhua Gao^1,2|Show fewer author(s)

Author Affiliations

¹Key Laboratory of High Energy Density Physics and Technology of Ministry of Education, Sichuan University, Chengdu 610064, China

²College of Physical Science and Technology, Sichuan University, Chengdu 610064, China

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DOI: 10.3788/COL201513.112401 Cite this Article Set citation alerts

Linguo Xie, Xiaodong Qiu, Jiangdong Qiu, Zhiyou Zhang, Jinglei Du, Fuhua Gao, "Determination of the focused beam waist of lasers with weak measurements," Chin. Opt. Lett. 13, 112401 (2015) Copy Citation Text

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Abstract

In this Letter, a method for detecting the focused beam waist of lasers is proposed by using weak measurements based on the so-called weak-value amplification. We establish a propagation model to describe the quantitative relation between the beam waist and the amplified shift of the spin Hall effect of light (SHEL), which is sensitive to the variation of the beam waist. We experimentally measure the amplified shift corresponding to a different beam waist and the experimental data agrees well with theoretical calculation. These results confirm the rationality and feasibility of our method.

{\tilde{E}}_{i}^{H} = \frac{ω_{0}}{\sqrt{2 π}} \exp [- \frac{ω_{0}^{2} (k_{i x}^{2} + k_{i y}^{2})}{4}] {e⃗}_{i x},

(1)

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[\begin{matrix} {\tilde{E}}_{r}^{H} \\ {\tilde{E}}_{r}^{V} \end{matrix}] = [\begin{matrix} r_{p} & \frac{k_{r y} (r_{p} + r_{s}) \cot θ_{i}}{k_{0}} \\ - \frac{k_{r y} (r_{p} + r_{s}) \cot θ_{i}}{k_{0}} & r_{s} \end{matrix}] [\begin{matrix} {\tilde{E}}_{i}^{H} \\ {\tilde{E}}_{i}^{V} \end{matrix}],

(2)

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{\tilde{E}}_{r | σ 〉} = \frac{ω_{0} r_{p}}{2 \sqrt{π}} \exp [- \frac{ω_{0}^{2} (k_{r x}^{2} + k_{r y}^{2})}{4}] \exp ({i σ k}_{r y} δ),

(3)

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E_{r | σ 〉} = \iint {\tilde{E}}_{r | σ 〉} \exp [i (k_{r x} x_{r} + k_{r y} y_{r} + k_{r z} z_{r})] {d k}_{r x} {d k}_{r y},

(4)

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δ_{r | σ 〉} = \frac{\iint y_{r} I_{r | σ 〉} {d x}_{r} {d y}_{r}}{\iint I_{r | σ 〉} {d x}_{r} {d y}_{r}},

(5)

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{\tilde{E}}_{post} = {\tilde{E}}_{r}^{H} \sin Δ + {\tilde{E}}_{r}^{V} \cos Δ,

(6)

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Δ_{H} = \frac{2 z_{r} δ_{r} \tan Δ}{δ_{r}^{2} k_{0} + 2 R_{0} \tan^{2} Δ},

(7)

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ω_{0} = \frac{f ω}{\sqrt{R^{2} + (d - f^{2})}},

(8)

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ω_{0} = \sqrt{(\frac{2 z_{r}}{k_{0} Δ_{H}} - \frac{δ_{r}}{\tan Δ}) \frac{δ_{r}}{\tan Δ}} .

(9)

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Linguo Xie, Xiaodong Qiu, Jiangdong Qiu, Zhiyou Zhang, Jinglei Du, Fuhua Gao, "Determination of the focused beam waist of lasers with weak measurements," Chin. Opt. Lett. 13, 112401 (2015)

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