• Infrared and Laser Engineering
  • Vol. 50, Issue 12, 20211059 (2021)
Zhentao Liu1, Chenyu Hu1,2, Zhishen Tong1, Chunyan Chu3, and Shensheng Han1,2,*
Author Affiliations
  • 1Key Laboratory for Quantum Optics of CAS, Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai 201800, China
  • 2Hangzhou Institute for Advanced Study, University of Chinese Academy of Sciences, Hangzhou 310024, China
  • 3Beijing Key Laboratory for Precision Optoelectronic Measurement Instrument and Technology, School of Optics and Photonics, Beijing Institute of Technology, Beijing 100081, China
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    DOI: 10.3788/IRLA20211059 Cite this Article
    Zhentao Liu, Chenyu Hu, Zhishen Tong, Chunyan Chu, Shensheng Han. Some research progress on the theoretical study of ghost imaging in Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences (Invited)[J]. Infrared and Laser Engineering, 2021, 50(12): 20211059 Copy Citation Text show less

    Abstract

    Compared with traditional imaging that uses the first-order correlation of light fields to realize the one-to-one correspondence between the object and image spaces, ghost imaging realizes the correspondence based on the second-order correlation of light fields and thereby obtains object’s image information. By introducing light field fluctuation modulation and computational reconstruction, ghost imaging not only has higher information acquisition efficiency, but also improves the flexibility of image information acquisition modes, thus having imaging capabilities that traditional imaging does not have. The further developments of ghost imaging in system optimization and technical applications bring new demands and challenges to ghost imaging theory. In this paper, the recent research progress of ghost imaging theory in Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences was introduced from three aspects: the physical nature, the image information acquisition theory and theoretical resolution. And the future theoretical research on ghost imaging was prospected.
    $ I(\boldsymbolr)=I(\boldsymbolr,t)t=E(\boldsymbolr,t)E(\boldsymbolr,t)t $(1)

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    $ I(\boldsymbolr)=Eo(\boldsymbolr,t)Eo(\boldsymbolr,t)t+Er(\boldsymbolr,t)Er(\boldsymbolr,t)t+Eo(\boldsymbolr,t)Er(\boldsymbolr,t)t+Er(\boldsymbolr,t)Eo(\boldsymbolr,t)t $(2)

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    $I1(\boldsymbolr1,t)=E1(\boldsymbolr1,t)E1(\boldsymbolr1,t)I2(\boldsymbolr2,t+Δt)=E2(\boldsymbolr2,t+Δt)E2(\boldsymbolr2,t+Δt) $(3)

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    $ ΔI1(\boldsymbolr1,t)ΔI2(\boldsymbolr2,t+Δt)=|ΓE(\boldsymbolr1,\boldsymbolr2;Δt)|2 $(4)

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    $ ΓE(\boldsymbolr1,\boldsymbolr2;Δt)=E1(\boldsymbolr1,t)E2(\boldsymbolr2,t+Δt) $(5)

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    $ΔIt(\boldsymbolrt)ΔIr(\boldsymbolrr)|d\boldsymbolr0d\boldsymbolr0'ht(\boldsymbolrt,\boldsymbolr0)hr(\boldsymbolrr,\boldsymbolr0')E0(\boldsymbolr0)E0(\boldsymbolr0')|2$(6)

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    $ ΔIt(\boldsymbolrt,t)ΔIr(\boldsymbolrr,t)t|d\boldsymbolr0ht(\boldsymbolrt,\boldsymbolr0)hr(\boldsymbolrr,\boldsymbolr0')|2 $(7)

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    $ ΔIt(t)ΔIr(\boldsymbolrr,t)t|T(\boldsymbolrr)|2 $(8)

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    $ ΔIt(\boldsymbolrt,t)ΔIr(\boldsymbolrr,t)t|F{T}\boldsymbolrt\boldsymbolrrλz2|2 $(9)

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    $ΔIt(\boldsymbolrt)ΔIr(\boldsymbolrt;\boldsymbolr0,k)\boldsymbolrtT(\boldsymbolr0,k) $(10)

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    $ x^=1Ni=1N(It(i)It)(Ir(i)(\boldsymbolrr)Ir(\boldsymbolrr)) $(11)

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    $ x^1Ni=1N(It(i)It)(Ir(i)(\boldsymbolrr)Ir(\boldsymbolrr))1Ni=1Nx(i) $(12)

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    $ E{x^}=E{x} $(13)

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    $ V{x^}=1Ni=1NV{x} $(14)

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    $ It=αd\boldsymbolr0Ir(\boldsymbolr0)T(\boldsymbolr0)+n $(15)

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    $Ir(\boldsymbolr0)Ir $(16)

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    $ΔIr(\boldsymbolrr)Δn=ΔIr(\boldsymbolrr)Δn=0 $(17)

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    $ E{x}=E{ΔItΔIr(\boldsymbolrr)}αIr2ScT(\boldsymbolrr) $(18)

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    $ V{x}=E{x2}E{x}2α2Ir4[Scd\boldsymbolr0T2(\boldsymbolr0)+7Sc2T2(\boldsymbolrr)+σn2α2Ir2] $(19)

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    $ E{x^}T(\boldsymbolrr) $(20)

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    $ V{x^}1N[1Scd\boldsymbolr0T2(\boldsymbolr0)+7T2(\boldsymbolrr)+σn2α2Sc2Ir2]=1N[1Scd\boldsymbolr0T2(\boldsymbolr0)+7T2(\boldsymbolrr)+ς2SNR2] $(21)

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    $ It(\boldsymbolrt)=It(0)(\boldsymbolrt)+n $(22)

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    $ E{x}=Δ[It(0)(\boldsymbolrt)+n]ΔIr(\boldsymbolrr)=ΔIt(0)(\boldsymbolrt)ΔIr(\boldsymbolrr)+ΔnΔIr(\boldsymbolrr) $(23)

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    $ E{x}=ΔIt(0)(\boldsymbolrt)ΔIr(\boldsymbolrr)=It(0)Ir|ΓE(\boldsymbolrt,\boldsymbolrr)|2|F{T}\boldsymbolrt\boldsymbolrrλz2|2 $(24)

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    $ |ΓE(\boldsymbolrt,\boldsymbolrr)|2=z2z22STSs|F{T}\boldsymbolrt\boldsymbolrrλz2|2 $(25)

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    $ V{x}It(0)2Ir2[1+3|ΓE(\boldsymbolrt,\boldsymbolrr)|4+4|ΓE(\boldsymbolrt,\boldsymbolrr)|2+σn2It(0)2] $(26)

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    $E{x^}|ΓE(\boldsymbolrt,\boldsymbolrr)|2 $(27)

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    $ V{x^}1N[1+3|ΓE(\boldsymbolrt,\boldsymbolrr)|4+4|ΓE(\boldsymbolrt,\boldsymbolrr)|2+σn2It(0)2]=1N[1+3|ΓE(\boldsymbolrt,\boldsymbolrr)|4+4|ΓE(\boldsymbolrt,\boldsymbolrr)|2+1SNR2] $(28)

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    $ \boldsymboly=\boldsymbolΦ\boldsymbolx $(29)

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    $ \boldsymboly=β0\boldsymbolA\boldsymbolx+\boldsymboln $(30)

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    $ \boldsymbolaj=[Ir(\boldsymbolr0,j,t1),Ir(\boldsymbolr0,j,t2),,Ir(\boldsymbolr0,j,ti),,Ir(\boldsymbolr0,j,tN)]T\boldsymbolx=[T(\boldsymbolr0,1),T(\boldsymbolr0,2),,T(\boldsymbolr0,j),,T(\boldsymbolr0,M)]Tj=1,,M\boldsymboly=[It(t1),It(t2),,It(ti),,It(tN)]Ti=1,,N $(31)

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    $ p(\boldsymboly,\boldsymbolA;\boldsymbolx)=p(\boldsymboly|\boldsymbolA;\boldsymbolx)p(\boldsymbolA)=1(2πσn)Nexp{12σn2(\boldsymbolyβ0\boldsymbolA\boldsymbolx)T(\boldsymbolyβ0\boldsymbolA\boldsymbolx)}p(\boldsymbolA) $(32)

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    $lnp(\boldsymboly,\boldsymbolA;\boldsymbolx)=Nln2πσn12σn2(\boldsymbolyβ0\boldsymbolA\boldsymbolx)T(\boldsymbolyβ0\boldsymbolA\boldsymbolx)+lnp(\boldsymbolA) $(33)

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    $ lnp(\boldsymboly,\boldsymbolA;\boldsymbolx)\boldsymbolxi=β0σn2[\boldsymbolAT(\boldsymbolyβ0\boldsymbolA\boldsymbolx)]i $(34)

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    $lnp(\boldsymboly,\boldsymbolA;\boldsymbolx)\boldsymbolxi\boldsymbolxj=β02σn2[\boldsymbolAT\boldsymbolA]ij $(35)

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    $ [I(\boldsymbolx)]ij=E\boldsymboly,\boldsymbolA{lnp(\boldsymboly,\boldsymbolA;\boldsymbolx)\boldsymbolxi\boldsymbolxj}=β02σn2E\boldsymbolA[\boldsymbolAT\boldsymbolA]ij $(36)

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    $ E\boldsymbolA[\boldsymbolAT\boldsymbolA]ijNIr2(1+δij) $(37)

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    $ [I(\boldsymbolx)]ijβ02NIr2σn2(1+δij) $(38)

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    $ V{\boldsymbolxi}=[I1(\boldsymbolx)]iiσn2β02NIr2MM+1$(39)

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    $ V{\boldsymbolxi}σn2β02NIr2=(STSc)2T2NSNR2=ς2NSNR2 $(40)

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    $ It(\boldsymbolrt)=Et(\boldsymbolrt)Et(\boldsymbolrt)=αλ4z24d\boldsymbolrrd\boldsymbolrr'Er(\boldsymbolrr)Er(\boldsymbolrr')×exp{jπλz2(\boldsymbolrr2\boldsymbolrr'2)}F{T}\boldsymbolrt\boldsymbolrrλz2F{T}\boldsymbolrt\boldsymbolrr'λz2 $(41)

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    $Ir(\boldsymbolrr)=Er(\boldsymbolrr)Er(\boldsymbolrr')κ1δ(\boldsymbolrr\boldsymbolrr') $(42)

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    $ It'(\boldsymbolrt)=ακ1λ4z24d\boldsymbolrrIr(\boldsymbolrr)|F{T}\boldsymbolrt\boldsymbolrrλz2|2 $(43)

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    $E{It'(\boldsymbolrt)}=E{It(\boldsymbolrt)} $(44)

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    $ It(\boldsymbolrt)=It'(\boldsymbolrt)+Δ=ακ1λ4z24d\boldsymbolrrIr(\boldsymbolrr)|F{T}\boldsymbolrt\boldsymbolrrλz2|2+Δ=αSTλ2z22d\boldsymbolrrIr(\boldsymbolrr)|ΓE(\boldsymbolrt,\boldsymbolrr)|2+Δ $(45)

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    $ Δ=It(\boldsymbolrt)It'(\boldsymbolrt) $(46)

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    $ E{Δ}=0 $(47)

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    $ V{Δ}=E{Δ2}E{Δ}2=E{It2(\boldsymbolrt)}+E{It'2(\boldsymbolrt)}2E{It(\boldsymbolrt)It'(\boldsymbolrt)}=It2{1κ1d\boldsymbolrr|F{T}\boldsymbolrt\boldsymbolrrλz2|4[d\boldsymbolrr|F{T}\boldsymbolrt\boldsymbolrrλz2|2]2} $(48)

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    $ d\boldsymbolrr|F{T}\boldsymbolrt\boldsymbolrrλz2|4[d\boldsymbolrr|F{T}\boldsymbolrt\boldsymbolrrλz2|2]2 $(49)

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    $V{Δ}It2 $(50)

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    $ \boldsymbol{y}=\beta \boldsymbol{Ax}+{\boldsymbol{\varepsilon}} $(51)

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    $ xj=|ΓE(\boldsymbolrt,\boldsymbolrr,j)|2j=1,,Myi=It(\boldsymbolrt,ti)i=1,,N\boldsymbolaj=[Ir(\boldsymbolrr,j,t1),Ir(\boldsymbolrr,j,t2),,Ir(\boldsymbolrr,j,ti),,Ir(\boldsymbolrr,j,tN)] $(52)

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    $ \boldsymbol{\varepsilon} =\left\{\boldsymbol{n}+\mathrm{\Delta }\right\} \sim N\left(0,{\sigma }^{2}\right),\ {\sigma }^{2}=\mathbb{V}\left\{\boldsymbol{n}\right\}+\mathbb{V}\left\{\mathrm{\Delta }\right\} ={\sigma }_{n}^{2}+\mathbb{V}\left\{\mathrm{\Delta }\right\}$(53)

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    $ V{\boldsymbolxi}σ2β2NIr2MM+1 $(54)

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    $ V{\boldsymbolxi}σn2+It2β2NIr2MM+1 $(55)

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    $ V{\boldsymbolxi}σn2+It2β2NIr2=1N(1+1SNR2)$(56)

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    $ p(Ir,It;μ2)=1IrIt(1μ2)×exp{IrIt+ItIrIrIt(1μ2)}J0{2μ1μ2IrItIrIt} $(57)

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    $ p(Ir,It|ρ1It<It<ρ2It)=1eρ1eρ21IrIt(1μ2)×exp{IrIt+ItIrIrIt(1μ2)}J0{2μ1μ2IrItIrIt} $(58)

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    $ I(μ2|ρ1It<It<ρ2It)=ρ2Itρ1ItdIt0dIrp(Ir,It|ρ1It<It<ρ2It)×[lnp(Ir,It|ρ1It<It<ρ2It)μ2]2 $(59)

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    $ lnp(Ir,It|ρ1It<It<ρ2It)μ2=11μ2(IrIr+ItIt)11μ2+J1{2μ1μ2IrItIrIt}J0{2μ1μ2IrItIrIt}IrItIrItμ+1μ(1μ2)2 $(60)

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    $ p(Ir|It>ρiIt)=ρiItdItp(Ir,It)p(It>ρiIt)=eρiρiItdItp(Ir,It) $(61)

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    $ R+=0dIrp(Ir|It>ρiIt)Ir=Ir(1+ρiμ2) $(62)

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    $ R=0dIrp(Ir|It<ρjIt)Ir=Ir(1eρjρj1eρjμ2) $(63)

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    $ ΔR=R+R=Irρi(ρiρj)eρj1eρjμ2 $(64)

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    $ ΔR=Irρ1eρμ2 $(65)

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    $ μ(\boldsymbolΦ):=max1i<jM|\boldsymbolΦiT\boldsymbolΦj|\boldsymbolΦi2\boldsymbolΦj2 $(66)

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    $ μ(\boldsymbolΦ)=max1i<jM|g(2)(i,j)| $(67)

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    $ max1i<jM|g(2)(i,j)|<12K1 $(68)

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    $ exp[(2π(np1)ωΔλminλ2)2][2J1(πD|Δ\boldsymbolrmin|2λf)πD|Δ\boldsymbolrmin|2λf]2<12K1 $(69)

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    Zhentao Liu, Chenyu Hu, Zhishen Tong, Chunyan Chu, Shensheng Han. Some research progress on the theoretical study of ghost imaging in Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences (Invited)[J]. Infrared and Laser Engineering, 2021, 50(12): 20211059
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