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Advanced Photonics
Vol. 1, Issue 1, 016005 (2019)

Noniterative spatially partially coherent diffractive imaging using pinhole array mask

Xingyuan Lu¹, Yifeng Shao², Chengliang Zhao^1、2、*, Sander Konijnenberg², Xinlei Zhu¹, Ying Tang², Yangjian Cai^1、3, and H. Paul Urbach²

Author Affiliations

¹Soochow University, School of Physical Science and Technology, Suzhou, China

²Delft University of Technology, Optics Research Group, Delft, Netherlands

³Shandong Normal University, School of Physics and Electronics, Center of Light Manipulation and Application, Jinan, China

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DOI: 10.1117/1.AP.1.1.016005 Cite this Article Set citation alerts

Xingyuan Lu, Yifeng Shao, Chengliang Zhao, Sander Konijnenberg, Xinlei Zhu, Ying Tang, Yangjian Cai, H. Paul Urbach. Noniterative spatially partially coherent diffractive imaging using pinhole array mask[J]. Advanced Photonics, 2019, 1(1): 016005 Copy Citation Text

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Abstract

We propose and experimentally demonstrate a noniterative diffractive imaging method for reconstructing the complex-valued transmission function of an object illuminated by spatially partially coherent light from the far-field diffraction pattern. Our method is based on a pinhole array mask, which is specially designed such that the correlation function in the mask plane can be obtained directly by inverse Fourier transforming the diffraction pattern. Compared to the traditional iterative diffractive imaging methods using spatially partially coherent illumination, our method is noniterative and robust to the degradation of the spatial coherence of the illumination. In addition to diffractive imaging, the proposed method can also be applied to spatial coherence property characterization, e.g., free-space optical communication and optical coherence singularity measurement.

Keywords

coherent diffractive imaging noniterative method partial coherence pinhole array mask

M (r) = δ (r - 0) + \sum_{m, n} δ (r - r_{m, n}),

(1)

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I (k) = \iint M (r_{1}) M {(r_{2})}^{*} W (r_{1}, r_{2}) \exp [- i 2 π k \cdot (r_{1} - r_{2})] d^{2} r_{1} d^{2} r_{2} = W (0, 0) + \sum_{m_{1}, n_{1}} \sum_{m_{2}, n_{2}} W (r_{m_{1}, n_{1}}, r_{m_{2}, n_{2}}) \exp [- i 2 π k \cdot (r_{m_{1}, n_{1}} - r_{m_{2}, n_{2}})] + \sum_{m_{1}, n_{1}} W (r_{m_{1}, n_{1}}, 0) \exp (- i 2 π k \cdot r_{m_{1}, n_{1}}) + \sum_{m_{2}, n_{2}} W (0, r_{m_{2}, n_{2}}) \exp (+ i 2 π k \cdot r_{m_{2}, n_{2}}),

(2)

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F^{- 1} [I (k)] (r) = W (0, 0) δ (r - 0) + \sum_{m_{1}, n_{1}} \sum_{m_{2}, n_{2}} W (r_{m_{1}, n_{1}}, r_{m_{2}, n_{2}}) δ [r - (r_{m_{1}, n_{1}} - r_{m_{2}, n_{2}})] + \sum_{m_{1}, n_{1}} W (r_{m_{1}, n_{1}}, 0) δ (r - r_{m_{1}, n_{1}}) + \sum_{m_{2}, n_{2}} W (0, r_{m_{2}, n_{2}}) δ (r + r_{m_{2}, n_{2}}),

(3)

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M^{'} (r) = \sum_{m, n} δ (r - r_{m, n}) .

(4)

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W (r_{1}, r_{2}) = \iint O (ρ_{1}) O {(ρ_{2})}^{*} W_{0} (ρ_{1}, ρ_{2}) \exp [- i 2 π (ρ_{1} \cdot r_{1} - ρ_{2} \cdot r_{2})] d^{2} ρ_{1} d^{2} ρ_{2},

(5)

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W (r_{m, n}, 0) = \iint O (ρ_{1}) O (ρ_{2})^{*} W_{0} (ρ_{1}, ρ_{2}) \exp (- i 2 π ρ_{1} \cdot r_{m, n}) d^{2} ρ_{1} d^{2} ρ_{2} .

(6)

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W (r_{m, n}, 0) = \int T (ρ_{1}) O (ρ_{1}) \exp (- i 2 π ρ_{1} \cdot r_{m, n}) d^{2} ρ_{1},

(7)

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T (ρ_{1}) = \int O (ρ_{2})^{*} W_{0} (ρ_{1}, ρ_{2}) d^{2} ρ_{2} .

(8)

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W_{δ} (r_{m, n}, 0) = \iint [O (ρ_{1}) + C δ (ρ_{1} - ρ_{0})] [O (ρ_{2}) + C δ (ρ_{2} - ρ_{0})]^{*} \times W_{0} (ρ_{1}, ρ_{2}) \exp (- i 2 π ρ_{1} \cdot r_{m, n}) d^{2} ρ_{1} d^{2} ρ_{2} .

(9)

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W_{δ} (r_{m, n}, 0) = W (r_{m, n}, 0) + {| C |}^{2} W_{0} (ρ_{0}, ρ_{0}) + \iint C δ (ρ_{1} - ρ_{0}) O (ρ_{2})^{*} W_{0} (ρ_{1}, ρ_{2}) \exp (- i 2 π ρ_{1} \cdot r_{m, n}) d^{2} ρ_{1} d^{2} ρ_{2} + \iint O (ρ_{1}) C^{*} δ (ρ_{2} - ρ_{0}) W_{0} (ρ_{1}, ρ_{2}) \exp (- i 2 π ρ_{1} \cdot r_{m, n}) d^{2} ρ_{1} d^{2} ρ_{2} .

(10)

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W_{δ} (r_{m, n}, 0) = W (r_{m, n}, 0) + {| C |}^{2} W_{0} (ρ_{0}, ρ_{0}) + [\int C \cdot O (ρ_{2})^{*} W_{0} (ρ_{0}, ρ_{2}) d^{2} ρ_{2}] \exp (- i 2 π ρ_{0} \cdot r_{m, n}) + \int O (ρ_{1}) C^{*} W_{0} (ρ_{1}, ρ_{0}) \exp (- i 2 π ρ_{1} \cdot r_{m, n}) d^{2} ρ_{1} .

(11)

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F^{- 1} [W_{δ} (r_{m, n}, 0) - W (r_{m, n}, 0)] = O (ρ_{1}) C^{*} W_{0} (ρ_{1}, ρ_{0}) + {| C |}^{2} W_{0} (ρ_{0}, ρ_{0}) δ (ρ - 0) + [\int C \cdot O {(ρ_{2})}^{*} W_{0} (ρ_{0}, ρ_{2}) d^{2} ρ_{2}] δ (ρ - ρ_{0}) .

(12)

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W_{0} (ρ_{1}, ρ_{2}) = \exp (- \frac{ρ_{1}^{2} + ρ_{2}^{2}}{w_{0}^{2}}) \exp [- \frac{{(ρ_{1} - ρ_{2})}^{2}}{2 σ^{2}}],

(13)

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W_{0} (ρ_{1}, ρ_{2}) = \exp (- \frac{ρ_{1}^{2} + ρ_{2}^{2}}{w_{0}^{2}}) \exp [- \frac{{(ρ_{1} - ρ_{2})}^{2}}{2 σ^{2}}] L_{n}^{0} [\frac{{(ρ_{1} - ρ_{2})}^{2}}{2 σ^{2}}],

(14)

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Xingyuan Lu, Yifeng Shao, Chengliang Zhao, Sander Konijnenberg, Xinlei Zhu, Ying Tang, Yangjian Cai, H. Paul Urbach. Noniterative spatially partially coherent diffractive imaging using pinhole array mask[J]. Advanced Photonics, 2019, 1(1): 016005

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