• Chinese Optics Letters
  • Vol. 17, Issue 9, 091101 (2019)
Zhentao Liu, Xia Shen, Honglin Liu, Hong Yu, and Shensheng Han*
Author Affiliations
  • Key Laboratory for Quantum Optics, Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai 201800, China
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    DOI: 10.3788/COL201917.091101 Cite this Article Set citation alerts
    Zhentao Liu, Xia Shen, Honglin Liu, Hong Yu, Shensheng Han. Lensless Wiener–Khinchin telescope based on second-order spatial autocorrelation of thermal light[J]. Chinese Optics Letters, 2019, 17(9): 091101 Copy Citation Text show less

    Abstract

    The resolution of a conventional imaging system based on first-order field correlation can be directly obtained from the optical transfer function. However, it is challenging to determine the resolution of an imaging system through random media, including imaging through scattering media and imaging through randomly inhomogeneous media, since the point-to-point correspondence between the object and the image plane in these systems cannot be established by the first-order field correlation anymore. In this Letter, from the perspective of ghost imaging, we demonstrate for the first time, to the best of our knowledge, that the point-to-point correspondence in these imaging systems can be quantitatively recovered from the second-order correlation of light fields, and the imaging capability, such as resolution, of such imaging schemes can thus be derived by analyzing second-order autocorrelation of the optical transfer function. Based on this theoretical analysis, we propose a lensless Wiener–Khinchin telescope based on second-order spatial autocorrelation of thermal light, which can acquire the image of an object by a snapshot via using a spatial random phase modulator. As an incoherent imaging approach illuminated by thermal light, the lensless Wiener–Khinchin telescope can be applied in many fields such as X-ray astronomical observations.
    It(r)=E0(r0,t)E0*(r0,t)thE(r;r0)hE*(r;r0)dr0dr0,(1)

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    E0(r0,t)E0*(r0,t)t=κI0(r0)δ(r0r0),(2)

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    It(r)=κI0(r0)hI(r;r0)dr0,(3)

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    GIt(2)(r+Δr,r)=Et*(r+Δr)Et*(r)Et(r)Et(r+Δr)r={Et*(r+Δr)Et*(r)Et(r)Et(r+Δr)}s¯={It(r)It(r+Δr)}¯s,(4)

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    GIt(2)(r+Δr,r)=Gh(2)(r+Δr,r0+Δr0;r,r0)×I0(r0+Δr0)I0(r0)dr0dΔr0,(5)

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    Gh(2)(r+Δr,r0+Δr0;r,r0)={hE*(r+Δr;r0+Δr0)hE*(r;r0)hE(r;r0)hE(r+Δr;r0+Δr0)}s¯(6)

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    Gh(2)(r+Δr,r0+Δr0;r,r0)=B[1+gh(2)(r+Δr,r0+Δr0;r,r0)](7)

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    gh(2)(r+Δr,r0+Δr0;r,r0)=|{hE*(r+Δr;r0+Δr0)hE(r;r0)}¯s|2B(8)

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    hE(r;r0)=exp[j2π(z1+z2)/λ]λ2z1z2exp[jπ(rr0)2λ(z1+z2)]P(rm)t(rm)exp[jπ(z1+z2)λz1z2(rmz1r+z2r0z1+z2)2]drm,(9)

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    gh(2)(r+Δr,r0+Δr0;r,r0)1B|P(rm)P*(rm){t(rm)t*(rm)}¯s=exp[jπ(z1+z2)λz1z2(rmz1r+z2r0z1+z2)2]exp{jπ(z1+z2)λz1z2[rmz1(r+Δr)+z2(r0+Δr0)z1+z2]2}drmdrm|2.(10)

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    Rη(rm,rm)={η(rm)η(rm)}¯s=ω2exp[(rmrmζ)2]=Rη(Δrm),Δrm=rmrm,(11)

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    gh(2)(r+Δr,r0+Δr0;r,r0)|{exp{2[2π(n1)λ]2[ω2Rη(2λz1z2z1+z2ν)]}{|P(μ)|2}μν}z1Δr+z2Δr02λz1z2|2=gh(2)(z1Δr+z2Δr02λz1z2),(12)

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    GIt(2)(r+Δr,r)B{[1+gh(2)(Δr02λz1)]GI0(2)(r0+Δr0,r0)}z1z2Δr,(13)

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    GI0(2)(r0+Δr0,r0)=I0(r0+Δr0)I0(r0)r0=I0(r0)I0(r0+Δr0)dr0(14)

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    gh(2)(Δr02λz1)=|{exp{2[2π(n1)λ]2[ω2Rη(2λz1z2z1+z2ν)]}{|P(μ)|2}μν}Δr02λz1|2.(15)

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    GI0(2)(r0+Δr0,r0)=1{|{I0(r0)}r0f0|2}f0Δr0.(16)

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    GIt(2)(r+Δr,r){[1+gh(2)(Δr02λz1)]1{|{I0(r0)}r0f0|2}f0Δr0}z1z2Δr.(17)

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    gθ(2)(Δθ)=exp{[2πωλ(n2sin2(Δθi)n)]2}exp{[πnωλsin2(Δθ)]2},(18)

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    FOV<Lz2(19)

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    gh(2)(Δr02λz1)|{|P(μ)|2}μΔr02λz1|2.(20)

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    gh(2)(Δr02λz1)[J1(2πDΔr0z1λ)2πDΔr0z1λ]2.(21)

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    gh(2)(Δr02λz1)exp{4[2π(n1)ωz2Δr0λ(z1+z2)ζ]2}(22)

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    PCCD<z2Mz1gh(2)(Δr02λz1),(23)

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    gh(2)(Δr02λz1)|{exp{2[2π(n1)λ]2[ω2Rη(2λz2ν)]}{|P(μ)|2}μν}Δr02λz1|2.(24)

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    Zhentao Liu, Xia Shen, Honglin Liu, Hong Yu, Shensheng Han. Lensless Wiener–Khinchin telescope based on second-order spatial autocorrelation of thermal light[J]. Chinese Optics Letters, 2019, 17(9): 091101
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