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Journals >Photonics Research >Volume 11 >Issue 9 >Page 1535 > Article

Photonics Research
Vol. 11, Issue 9, 1535 (2023)

Partially coherent beam generation with metasurfaces | Spotlight on Optics

Roman Calpe^1,*, Atri Halder¹, Meilan Luo^1,2, Matias Koivurova^3,4, and Jari Turunen¹

Author Affiliations

¹Center for Photonics Sciences, University of Eastern Finland, FI-80101 Joensuu, Finland

²Department of Physics and Synergetic Innovation Center for Quantum Effects and Applications, Hunan Normal University, Changsha 410081, China

³Tampere Institute for Advanced Study, Tampere University, 33100 Tampere, Finland

⁴Faculty of Engineering and Natural Sciences, Tampere University, FI-33720 Tampere, Finland

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DOI: 10.1364/PRJ.492233 Cite this Article Set citation alerts

Roman Calpe, Atri Halder, Meilan Luo, Matias Koivurova, Jari Turunen, "Partially coherent beam generation with metasurfaces," Photonics Res. 11, 1535 (2023) Copy Citation Text

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Abstract

An optical system for the generation of partially coherent beams with genuine cross-spectral density functions from spatially modulated globally incoherent sources is presented. The spatial intensity modulation of the incoherent source is achieved by quasi-planar metasurfaces based on spatial-frequency modulation of binary Bragg surface-relief diffraction gratings. Two types of beams are demonstrated experimentally: (i) azimuthally periodic, radially quasi-periodic beams and (ii) rotationally symmetric Bessel-correlated beams with annular far-zone radiation patterns.

W (ρ_{1}, ρ_{2}) = \int_{- \infty}^{\infty} p (v) H^{*} (ρ_{1}, v) H (ρ_{2}, v) d^{2} v .

(1)

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H (ρ, v) = H (ρ) \exp (- i 2 π ρ \cdot v),

(2)

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W (ρ_{1}, ρ_{2}) = H^{*} (ρ_{1}) H (ρ_{2}) \int_{- \infty}^{\infty} p (v) \exp (- i 2 π Δ ρ \cdot v) d^{2} v,

(3)

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W^{'} (ρ_{1}, ρ_{2}) = t^{*} (ρ_{1}) t (ρ_{2}) W (ρ_{1}, ρ_{2})

(4)

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p (v, ϕ) = p_{m} \exp (- 2 π^{2} σ_{0}^{2} v^{2}) [1 + C_{m} J_{m} (2 π ρ_{0} v) \cos (m ϕ)]

(5)

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p_{m} = 2 π σ_{0}^{2} {\begin{cases} {1 + C_{m} \exp [- ρ_{0}^{2} / (2 σ_{0}^{2})]}^{- 1} & when m = 0, \\ 1 & when m \neq 0 . \end{cases}

(6)

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S (ρ) = \exp (- 2 ρ^{2} / w_{0}^{2}) .

(7)

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μ (Δ ρ) = \exp (- Δ ρ^{2} / 2 σ_{0}^{2}) \times [1 + C_{m} {(- i)}^{m} \cos (m Δ φ) \exp (- \frac{ρ_{s}^{2}}{2}) I_{m} (\frac{ρ_{s} Δ ρ}{σ_{0}})],

(8)

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S (ρ, z) = \frac{w_{0}^{2}}{w^{2} (Δ z)} \exp [- \frac{2 ρ^{2}}{w^{2} (Δ z)}] {1 + C_{m} {(- 1)}^{m} \cos (m φ) \times \exp [- \frac{R^{2}}{2 q^{2}} c (Δ z)] J_{m} [\frac{2 R}{q \sqrt{1 + q^{2}}} \frac{Δ z / z_{R}}{1 + {(Δ z / z_{R})}^{2}} \frac{ρ}{w_{0}}]},

(9)

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w (Δ z) = w_{0} {[1 + {(Δ z / z_{R})}^{2}]}^{1 / 2}

(10)

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c (Δ z) = \frac{1 + {(Δ z / z_{G})}^{2}}{1 + {(Δ z / z_{R})}^{2}},

(11)

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z_{R} = z_{G} {(1 + q^{- 2})}^{- 1 / 2}

(12)

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p (v, ϕ) = p_{0} \exp (- 2 π^{2} σ_{0}^{2} v^{2}) [1 \pm J_{0} (2 π ρ_{0} v)],

(13)

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B_{0} = {[1 \pm \exp (- ρ_{s}^{2} / 2)]}^{- 1} .

(14)

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μ (Δ ρ) = B_{0} \exp (- Δ ρ^{2} / 2 σ_{0}^{2}) \times [1 \pm \exp (- \frac{ρ_{s}^{2}}{2}) I_{0} (\frac{ρ_{s} Δ ρ}{σ_{0}})] .

(15)

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S (ρ, z) = B_{0} \frac{w_{0}^{2}}{w^{2} (Δ z)} \exp [- \frac{2 ρ^{2}}{w^{2} (Δ z)}] \times {1 \pm \exp [- \frac{R^{2}}{2 q^{2}} c (Δ z)] J_{0} [\frac{2 R}{q \sqrt{1 + q^{2}}} \frac{Δ z / z_{R}}{1 + {(Δ z / z_{R})}^{2}} \frac{ρ}{w_{0}}]},

(16)

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μ (Δ ρ) = \exp (- \frac{Δ ρ^{2}}{2 σ_{0}^{2}}) (1 - \frac{Δ ρ^{2}}{2 σ_{0}^{2}}),

(17)

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S (ρ, z) = \frac{w_{0}^{2}}{w^{2} (Δ z)} \exp [- \frac{2 ρ^{2}}{w^{2} (Δ z)}] \times [c (Δ z) + \frac{2 Δ z^{2} z_{R}^{2} ρ^{2}}{(w_{0}^{2} + σ_{0}^{2}) {(Δ z^{2} + z_{R}^{2})}^{2}}] .

(18)

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\sin θ_{B} = \frac{λ_{0}}{2 nd} .

(19)

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M (v, ϕ) = M_{0} [1 + J_{m} (2 π ρ_{0} v) \cos (m ϕ)],

(20)

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M (v, ϕ) = M_{0} [1 - J_{0} (2 π ρ_{0} v)],

(21)

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References (31)

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Roman Calpe, Atri Halder, Meilan Luo, Matias Koivurova, Jari Turunen, "Partially coherent beam generation with metasurfaces," Photonics Res. 11, 1535 (2023)

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