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Chinese Optics Letters
Vol. 16, Issue 9, 092701 (2018)

Improving the resolution in quantum and classical temporal imaging | Editors' Pick

Junheng Shi^1、2、3, Giuseppe Patera³, Youzhen Gui¹, Mikhail I. Kolobov^3、*, Dmitri B. Horoshko^3、4, and Shensheng Han^1、**

Author Affiliations

¹Key Laboratory for Quantum Optics, Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai 201800, China

²University of Chinese Academy of Sciences, Beijing 100049, China

³Univ. Lille, CNRS, UMR 8523—PhLAM—Physique des Lasers Atomes et Molécules, F-59000 Lille, France

⁴B I. Stepanov Institute of Physics, NASB, Minsk 220072, Belarus

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DOI: 10.3788/COL201816.092701 Cite this Article Set citation alerts

Junheng Shi, Giuseppe Patera, Youzhen Gui, Mikhail I. Kolobov, Dmitri B. Horoshko, Shensheng Han. Improving the resolution in quantum and classical temporal imaging[J]. Chinese Optics Letters, 2018, 16(9): 092701 Copy Citation Text

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Abstract

The point-spread function of an optical system determines its optical resolution for both spatial and temporal imaging. For spatial imaging, it is given by a Fourier transform of the pupil function of the system. For temporal imaging based on nonlinear optical processes, such as sum-frequency generation or four-wave mixing, the point-spread function is related to the waveform of the pump wave by a nonlinear transformation. We compare the point-spread functions of three temporal imaging schemes: sum-frequency generation, co-propagating four-wave mixing, and counter-propagating four-wave mixing, and demonstrate that the last scheme provides the best temporal resolution. Our results are valid for both quantum and classical temporal imaging.

k_{μ} (ω) \approx k_{μ} (ω_{μ}) + β_{μ}^{(1)} Ω + β_{μ}^{(2)} Ω^{2} / 2,

(1)

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{\hat{E}}_{μ}^{(+)} (t, z) = E_{μ} e^{i (k_{μ} z - ω_{μ} t)} {\hat{A}}_{μ} (τ, z),

(2)

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{\hat{A}}_{μ} (τ, z) = \frac{1}{2 π} \int_{- \infty}^{\infty} {\hat{ϵ}}_{μ} (Ω, z) e^{i β_{μ}^{(2)} Ω^{2} (z - z_{0}) / 2 - i Ω τ} d Ω,

(3)

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{\hat{A}}_{s} (τ, z_{2}) = c (τ) {\hat{A}}_{s} (τ, z_{1}) - s (τ) e^{- i ϕ (τ)} {\hat{A}}_{i} (τ, z_{1}), {\hat{A}}_{i} (τ, z_{2}) = s (τ) e^{i ϕ (τ)} {\hat{A}}_{s} (τ, z_{1}) + c (τ) {\hat{A}}_{i} (τ, z_{1}),

(4)

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c (τ) = \cos [g A_{p} (τ) L], s (τ) = \sin [g A_{p} (τ) L] .

(5)

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{| c (τ) |}^{2} + {| s (τ) |}^{2} = 1.

(6)

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{\hat{A}}_{out} (τ) = \frac{i}{\sqrt{| M |}} \exp (- \frac{i τ^{2}}{2 | M | D_{f}}) \times [\int_{- \infty}^{\infty} \tilde{p} (τ, τ^{'}) {\hat{A}}_{in} (\frac{τ^{'}}{M}) d τ^{'} + \int_{- \infty}^{\infty} \tilde{q} (τ, τ^{'}) {\hat{B}}_{in} (\frac{τ^{'}}{M}) d τ^{'}],

(7)

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\tilde{p} (τ, τ^{'}) = p (τ - τ^{'}) e^{i θ (τ, τ^{'})},

(8)

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\tilde{q} (τ, τ^{'}) = q (τ - τ^{'}) e^{i θ (τ, τ^{'})},

(9)

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p (τ) = \frac{1}{2 π} \int_{- \infty}^{\infty} d Ω e^{i τ Ω} s (D_{out} Ω),

(10)

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q (τ) = \frac{1}{2 π} \int_{- \infty}^{\infty} d Ω e^{i τ Ω} c^{'} (D_{out} Ω),

(11)

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θ (τ, τ^{'}) = \frac{τ^{2} - τ^{' 2}}{2 | M | D_{out}} .

(12)

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\frac{1}{D_{in}} + \frac{1}{D_{out}} = \frac{1}{D_{f}},

(13)

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\int_{- \infty}^{\infty} \tilde{p} (τ, s) {\tilde{p}}^{*} (τ^{'}, s) d s + \int_{- \infty}^{\infty} \tilde{q} (τ, s) {\tilde{q}}^{*} (τ^{'}, s) d s = δ (τ - τ^{'}),

(14)

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{\hat{A}}_{out} (τ) = \frac{i}{\sqrt{| M |}} \exp (- \frac{i τ^{2}}{2 | M | D_{f}}) \times [\int_{- \infty}^{\infty} p (τ - τ^{'}) {\hat{A}}_{in} (\frac{τ^{'}}{M}) d τ^{'} + \int_{- \infty}^{\infty} q (τ - τ^{'}) {\hat{B}}_{in} (\frac{τ^{'}}{M}) d τ^{'}] .

(15)

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{\hat{A}}_{out} (τ) = \frac{i}{\sqrt{| M |}} \exp (- \frac{i τ^{2}}{2 | M | D_{f}}) {\hat{A}}_{in} (\frac{τ}{M}),

(16)

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R = \frac{Δ}{| M |} .

(17)

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A_{p} (τ) = A_{p} \exp [- \frac{τ^{2}}{2 {(D_{f} δ ω_{p})}^{2}}] .

(18)

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s (τ) = - \sin {θ_{0} \exp [- \frac{τ^{2}}{2 {(D_{f} δ ω_{p})}^{2}}]} .

(19)

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p (τ) = \sum_{n = 1}^{\infty} \frac{{(- 1)}^{n} θ_{0}^{2 n - 1}}{(2 n - 1)!} \exp [- \frac{(2 n - 1) τ^{2}}{2 {(D_{f} δ ω_{p})}^{2}}] .

(20)

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p (τ) = \sum_{n = 1}^{\infty} \frac{{(- 1)}^{n} θ_{0}^{2 n - 1}}{(2 n - 1)!} \frac{δ ω_{p}}{\sqrt{2 n - 1}} \times \exp [- \frac{{(τ δ ω_{p})}^{2}}{2 (2 n - 1) M^{2}}] .

(21)

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R = 2 \sqrt{2 \ln 2} \frac{| M |}{δ ω_{p}} / | M | \approx \frac{2.35}{δ ω_{p}} .

(22)

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p_{3} (τ) = δ ω_{p} θ_{0} {\exp [- \frac{{(τ δ ω_{p})}^{2}}{2 M^{2}}] - \frac{θ_{0}^{2}}{3! \sqrt{3}} \exp [- \frac{{(τ δ ω_{p})}^{2}}{6 M^{2}}] + \frac{θ_{0}^{4}}{5! \sqrt{5}} \exp [- \frac{{(τ δ ω_{p})}^{2}}{10 M^{2}}]} .

(23)

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R \approx \frac{2.12}{δ ω_{p}} .

(24)

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α_{1} (τ) = A_{1} (τ) e^{i ϕ_{1} (τ)}, α_{2} (τ) = A_{2} (τ) e^{i ϕ_{2} (τ)} .

(25)

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{\hat{A}}_{s} (τ, z_{2}) = c^{'} (τ) {\hat{A}}_{s} (τ, z_{1}) + i s^{'} (τ) e^{- i ϕ (τ)} {\hat{A}}_{i} (τ, z_{1}), {\hat{A}}_{i} (τ, z_{2}) = i s^{'} (τ) e^{i ϕ (τ)} {\hat{A}}_{s} (τ, z_{1}) + c^{'} (τ) {\hat{A}}_{i} (τ, z_{1}),

(26)

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c^{'} (τ) = \cos [g A_{1} (τ) A_{2} (τ) L], s^{'} (τ) = \sin [g A_{1} (τ) A_{2} (τ) L] .

(27)

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s^{'} (τ) = \sin [g A_{1} (τ) A_{2} (τ) L] .

(28)

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A_{1} (τ) = A_{2} (τ) = A_{p} \exp [- \frac{τ^{2}}{2 {(2 D_{f} δ ω_{p})}^{2}}],

(29)

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ϕ_{1} (τ) = - ϕ_{2} (τ) = \frac{τ^{2}}{4 D_{f}} .

(30)

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p^{'} (τ) = \sqrt{2} θ_{0^{'}} δ ω_{p} \exp [- \frac{{(τ δ ω_{p})}^{2}}{4 M^{2}}] .

(31)

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R \approx \frac{1.67}{δ ω_{p}} .

(32)

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R \approx \frac{1.50}{δ ω_{p}} .

(33)

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{\hat{A}}_{s} (τ, z_{2}) = c^{''} (τ) {\hat{A}}_{s} (τ, z_{1}) + i s^{''} (τ) e^{- i ϕ (τ)} {\hat{A}}_{i} (τ, z_{1}), {\hat{A}}_{i} (τ, z_{2}) = i s^{''} (τ) e^{i ϕ (τ)} {\hat{A}}_{s} (τ, z_{1}) + c^{''} (τ) {\hat{A}}_{i} (τ, z_{1}),

(34)

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c^{''} (τ) = \frac{1}{\cosh [γ A_{1} (τ) A_{2} (τ) L]}, s^{''} (τ) = \tanh [γ A_{1} (τ) A_{2} (τ) L] .

(35)

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s^{''} (τ) = \tanh {θ_{0}^{'} \exp [- \frac{τ^{2}}{2 {(\sqrt{2} D_{f} δ ω_{p})}^{2}}]} .

(36)

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R \approx \frac{1.47}{δ ω_{p}} .

(37)

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R \approx \frac{0.93}{δ ω_{p}},

(38)

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Junheng Shi, Giuseppe Patera, Youzhen Gui, Mikhail I. Kolobov, Dmitri B. Horoshko, Shensheng Han. Improving the resolution in quantum and classical temporal imaging[J]. Chinese Optics Letters, 2018, 16(9): 092701

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