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Chinese Optics Letters
Vol. 19, Issue 5, 052601 (2021)

Optimizing illumination’s complex coherence state for overcoming Rayleigh’s resolution limit

Chunhao Liang¹, Yashar E. Monfared², Xin Liu¹, Baoxin Qi¹, Fei Wang^3、*, Olga Korotkova^4、**, and Yangjian Cai^1、3、***

Author Affiliations

¹Shandong Provincial Engineering and Technical Center of Light Manipulations & Shandong Provincial Key Laboratory of Optics and Photonic Device, School of Physics and Electronics, Shandong Normal University, Jinan 250014, China

²Department of Chemistry, Dalhousie University, Halifax, NS B3H 4R2, Canada

³School of Physical Science and Technology & Collaborative Innovation Center of Suzhou Nano Science and Technology, Soochow University, Suzhou 215006, China

⁴Department of Physics, University of Miami, Coral Gables, Florida 33146, USA

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DOI: 10.3788/COL202119.052601 Cite this Article Set citation alerts

Chunhao Liang, Yashar E. Monfared, Xin Liu, Baoxin Qi, Fei Wang, Olga Korotkova, Yangjian Cai. Optimizing illumination’s complex coherence state for overcoming Rayleigh’s resolution limit[J]. Chinese Optics Letters, 2021, 19(5): 052601 Copy Citation Text

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Abstract

We suggest tailoring of the illumination’s complex degree of coherence for imaging specific two- and three-point objects with resolution far exceeding the Rayleigh limit. We first derive a formula for the image intensity via the pseudo-mode decomposition and the fast Fourier transform valid for any partially coherent illumination (Schell-like, non-uniformly correlated, twisted) and then show how it can be used for numerical image manipulations. Further, for Schell-model sources, we show the improvement of the two- and three-point resolution to 20% and 40% of the classic Rayleigh distance, respectively.

Keywords

imaging light manipulation optical coherence

W_{0}^{'} (r_{1}, r_{2}) = O^{*} (r_{1}) O (r_{2}) W_{0} (r_{1}, r_{2}) .

(1)

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W_{im} (ρ_{1}, ρ_{2}) = \int W_{0}^{'} (r_{1}, r_{2}) K^{*} (r_{1}, ρ_{1}) K (r_{2}, ρ_{2}) d^{2} r_{1} d^{2} r_{2},

(2)

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K (r, ρ) = - \frac{1}{λ^{2} f^{2}} \int P (ξ) \exp [- \frac{i k}{f} ξ \cdot (r + ρ)] d^{2} ξ,

(3)

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W_{0} (r_{1}, r_{2}) = τ^{*} (r_{1}) τ (r_{2}) \int p (v) H_{0}^{*} (v, r_{1}) H_{0} (v, r_{2}) d^{2} v,

(4)

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S_{im} (ρ) = \int d^{2} v p (v) {| \int τ (r) O (r) H_{0} (v, r) K (r, ρ) d^{2} r |}^{2} .

(5)

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S_{im} (ρ) = \int d^{2} v p (v) {| \int F (v, r) K (- r - ρ) d^{2} r |}^{2} = \int d^{2} v p (v) {| IFT [{\tilde{F}}_{1} (v, f) \tilde{K} (f)] {- ρ} |}^{2},

(6)

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S_{im} (ρ) = \sum_{i}^{N} \sum_{j}^{N} p (v_{x i}, v_{y j}) M (v_{x i}, v_{y j}, - ρ),

(7)

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S_{im} (ρ) = \int d^{2} v p (v) {| IFT [{\tilde{F}}_{1} (f - v) \tilde{K} (f)] {- ρ} |}^{2},

(8)

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O (r) = δ (x - d / 2, y) + δ (x + d / 2, y)

(9)

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S_{im} (ρ) = \exp (- d^{2} / 8 σ_{0}^{2}) \int d^{2} v \times p (v) {{| S_{+} |}^{2} + {| S_{-} |}^{2} + 2 Re [S_{+} S_{-}^{*} μ (d, 0)]},

(10)

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μ (Δ r) = \frac{W (r_{1}, r_{2})}{\sqrt{W (r_{1}, r_{1}) W (r_{2}, r_{2})}} = \frac{\int d^{2} v p (v) \exp (2 π i v \cdot Δ r)}{\int d^{2} v p (v)}

(11)

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S_{\pm} = \frac{2 π R^{2}}{λ^{2} f^{2}} \frac{J_{1} [2 π R \sqrt{{(ρ_{x} \pm d / 2)}^{2} + ρ_{y}^{2}} / λ f]}{2 π R \sqrt{{(ρ_{x} \pm d / 2)}^{2} + ρ_{y}^{2}} / λ f},

(12)

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p (v) = e^{- a^{2} [{(v_{x} - b / 2)}^{2} + v_{y}^{2}]} + e^{- a^{2} [{(v_{x} + b / 2)}^{2} + v_{y}^{2}]},

(13)

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μ (Δ r) = e^{- π^{2} Δ r^{2} / a^{2}} \cos (π Δ x / b),

(14)

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O_{1} (r) = δ (x, y - d / \sqrt{3}) + δ (x - d / 2, y + \sqrt{3} d / 6) + δ (x + d / 2, y + \sqrt{3} d / 6) .

(15)

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S_{im} (ρ) = \exp (- \frac{d^{2}}{6 σ^{2}}) {{| S_{1} |}^{2} + {| S_{2} |}^{2} + {| S_{3} |}^{2} + 2 Re [S_{1} S_{2}^{*} \times μ (- \frac{d}{2}, \frac{\sqrt{3} d}{2}) + S_{1}^{*} S_{2} μ (- \frac{d}{2}, - \frac{\sqrt{3} d}{2}) + S_{2} S_{3}^{*} μ (d, 0)]},

(16)

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p (v) = e^{- a^{2} [{(v_{x} - \frac{1}{\sqrt{3} b})}^{2} + v_{y}^{2}]} + e^{- a^{2} [{(v_{x} + \frac{1}{2 \sqrt{3} b})}^{2} + {(v_{y} + \frac{1}{2 b})}^{2}]} + e^{- a^{2} [{(v_{x} + \frac{1}{2 \sqrt{3} b})}^{2} + {(v_{y} - \frac{1}{2 b})}^{2}]} .

(17)

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μ (Δ r) = \frac{1}{3} e^{- \frac{π^{2} Δ r^{2}}{a^{2}}} [2 \cos (\frac{π Δ y}{b}) e^{- \frac{i π Δ x}{\sqrt{3} b}} + e^{i \frac{2 π Δ x}{\sqrt{3} b}}],

(18)

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Chunhao Liang, Yashar E. Monfared, Xin Liu, Baoxin Qi, Fei Wang, Olga Korotkova, Yangjian Cai. Optimizing illumination’s complex coherence state for overcoming Rayleigh’s resolution limit[J]. Chinese Optics Letters, 2021, 19(5): 052601

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