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    Journals >Chinese Optics Letters >Volume 22 >Issue 6 >Page 060011 > Article
    • Chinese Optics Letters
    • Vol. 22, Issue 6, 060011 (2024)
    Ghost imaging—its physics and application [Invited]
    Yanhua Shih*
    Author Affiliations
  • Department of Physics, University of Maryland, Baltimore County, Baltimore, MD 21250, USA
    • show less
    DOI: 10.3788/COL202422.060011 Cite this Article Set citation alerts
    Yanhua Shih, "Ghost imaging—its physics and application [Invited]," Chin. Opt. Lett. 22, 060011 (2024) Copy Citation Text show less

    Abstract

    Since its first experimental demonstration, “ghost imaging” has attracted much attention, perhaps not only because of its interesting physics but also because of its attractive application. This review article discusses the physics and application of ghost imaging: (1) emphasizes the nonlocal two-photon interference nature of ghost imaging, including detailed analysis and calculations; (2) introduces three types of applications with unique advantages of ghost imaging, including a light detection and ranging device with imaging ability, namely, an Imaging Lidar or ILidar system; a turbulence-resistant, or turbulence-free, imaging technology; and a vibration-resistant X-ray microscope of high resolving capability. This article is prepared for a Special Issue of Chinese Optics Letters, intended for general audiences, especially young researchers and students who are interested in ghost imaging.

    Keywords

    1si+1so=1f,

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    ⟨I(ρ→i)⟩=⟨|∫objdρ→oA(ρ→o)δ(ρ→o+ρ→iμ)|2⟩=∫objdρ→o|A(ρ→o)|2δ(ρ→o+ρ→iμ)=|A(ρ→o)|2⊗δ(ρ→o+ρ→iμ)≃|A(ρ→i/μ)|2,

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    ⟨I(ρ→i)⟩=⟨|∫objdρ→oA(ρ→o)somb(Rsoωc|ρ→o+ρ→iμ|)|2⟩=⟨|A(ρ→o)⊗somb(Rsoωc|ρ→o+ρ→iμ|)|2⟩=|A(ρ→o)|2⊗somb2(Rsoωc|ρ→o+ρ→iμ|),

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    E(ρ→i,so+si,ti)=E(ρ→o)gρ→o(ρ→i,zi,ti)=∫lensdρ→lE(ρ→o)[gρ→o(ρ→l,zl,tl)](glens)[gρ→l(ρ→i,zi,ti)],

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    gρ→o(ρ→i,zi,ti)=∫lensdρ→l[−iω2πceiωcsosoG(|ρ→l−ρ→o|,ωcso)]×[G(|ρ→l|,−ωcf)][−iω2πceiωcsisiG(|ρ→i−ρ→l|,ωcsi)],

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    gρ→o(ρ→i,zi,ti)=−ω2(2πc)2sosieiωc(so+si)G(|ρ→i|,ωcsi)G(|ρ→o|,ωcso)×∫lensdρ→lG[|ρ→l|,ωc(1so+1si−1f)]e−iωc(ρ→oso+ρ→isi)·ρ→l.

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    somb(x)=2J1(x)xwith x=Rsoωc|ρ→o+ρiμ|.

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    gρ→o(ρ→i,zi,ti)=−ω2(2πc)2sosieiωc(so+si)G(|ρ→i|,ωcsi)G(|ρ→o|,ωcso)×somb(Rsoωc|ρ→o+ρ→iμ|).

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    gρ→o(ρ→i,zi,ti)=−ω2eiφsi(2πc)2sosiδ(|ρ→o+ρ→iμ|),

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    ⟨|E(ρ→o)gρ→o(ρ→i,zi,ti)|2⟩∝somb2(Rsoωc|ρ→o+ρ→iμ|),

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    ⟨|E(ρ→o)gρ→o(ρ→i,zi,ti)|2⟩∝δ(|ρ→o+ρ→iμ|),

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    ⟨I(ρ→i,zi,ti)⟩=⟨|E(ρ→i,so+si)|2⟩=⟨|∫objdρ→oE(ρ→o)gρ→o(ρ→i,zi,ti)|2⟩=⟨|∫objdρ→oA(ρ→o)gρ→o(ρ→l,zl,tl)glens∫lensdρ→lgρ→l(ρ→i,zi,ti)|2⟩=⟨|∫objdρ→oA(ρ→o)∫lensdρ→lgρ→o(ρ→l,zl,tl)glensgρ→l(ρ→i,zi,ti)|2⟩.

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    E(ρ→i,so+si)=∫objdρ→o∫lensdρ→l[A(ρ→o)][−iω2πceiωcsosoG(|ρ→l−ρ→o|,ωcso)]×[G(|ρ→l|,−ωcf)][−iω2πceiωcsisiG(|ρ→i−ρ→l|,ωcsi)],

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    E(ρ→i,z=so+si)=−ω2(2πc)2sosieiωc(so+si)G(|ρ→i|,ωcsi)∫objdρ→oA(ρ→o)G(|ρ→o|,ωcso)×∫lensdρ→lG[|ρ→l|,ωc(1so+1si−1f)]e−iωc(ρ→oso+ρ→isi)·ρ→l.

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    E(ρ→i,z=so+si)=−ω2(2πc)2sosieiωc(so+si)G(|ρ→i|,ωcsi)∫objdρ→o[A(ρ→o)G(|ρ→o|,ωcso)]×somb(Rsoωc|ρ→o+ρ→iμ|),

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    ⟨I(ρ→i)⟩∝⟨|∫objdρ→oA(ρ→o)somb(Rsoωc|ρ→o+ρ→iμ|)|2⟩=∫objdρ→o|A(ρ→o)|2somb2(Rsoωc|ρ→o+ρ→iμ|)≃|A(ρ→o)|2⊗δ(|ρ→o+ρ→i/μ|)=|A(ρ→i/μ)|2,

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    G(2))(r1,t1;r2,t2)=⟨Ψ|E^(−)(r1,t1)E^(−)(r2,t2)E^(+)(r2,t2)⁢E^(+)(r1,t1)|Ψ⟩,

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    |Ψ⟩=Ψ0∑s,iδ(ωs+ωi−ωp)δ(ks+ki−kp)as†(ks)ai†(ki)|0⟩,

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    E^(+)(r,t)=∑kE^(+)(k,rs,ts)g(k,r−rs,t−ts).

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    g(κ→s,ωs;ρ→o,zo=d1+so)=eiωsczo∫lensdρ→l∫sourcedρ→S[−iωs2πcd1eiκ→s·ρ→SG(|ρ→S−ρ→l|,ωscd1)]×[G(|ρ→l|,ωscf)][−iωs2πcsoG(|ρ→l−ρ→o|,ωscso)],

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    g(κ→i,ωi;ρ→2,z2=d2)=−iωi2πcd2eiωicd2∫sourcedρ→S′G(|ρ→S′−ρ→2|,ωicd2)eiκ→i·ρ→S′,

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    G(2)(r1,t1;r2,t2)=|⟨0|E^(+)(r2,t2)E^(+)(r1,t1)|Ψ⟩|2≡|Ψ(r1,t1;r2,t2)|2,

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    Ψ(ρ→1,z1,t1;ρ→2,z2,t2)=⟨0|∫dω′dκ→′g(κ→′,ω′;ρ→2,z2)e−iω′t2a^(ω′,κ→′)×∫dω′′dκ→′′g(κ→′′,ω′′;ρ→1,z1)e−iω′′t1a^(ω′′,κ→′′)×∫dνdκ→sdκ→if(ν)δ(κ→s+κ→i)as†(ωs0+ν,κ→s)⁢ai†(ωi0−ν,κ→i)|0⟩.

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    Ψ(ρ→o,ρ→2)∝∫dκ→sdκ→iδ(κ→s+κ→i)g(κ→s,ω;ρ→o,zo)g(κ→i,ω;ρ→2,z2)∝eiωc(s0+si)∫dκ→sdκ→iδ(κ→s+κ→i)×∫lensdρ→l∫sourcedρ→Seiκ→s·ρ→SG(|ρ→S−ρ→l|,ωcd1)×G(|ρ→l|,ωcf)G(|ρ→l−ρ→o|,ωcso)×∫sourcedρ→S′eiκ→i·ρ→S′G(|ρ→S′−ρ→2|,ωcd2),

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    ∫dκ→sdκ→iδ(κ→s+κ→i)eiκ→s·ρ→Seiκ→i·ρ→S′∼δ(ρ→S−ρ→S′),

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    Ψ(ρ→o,ρ→2)∝∫lensdρ→l∫sourcedρ→SG(|ρ→2−ρ→S|,ωcd2)⁢G(|ρ→S−ρ→l|,ωcd1)×G(|ρ→l|,ωcf)G(|ρ→l−ρ→o|,ωcso).

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    Ψ(ρ→o,ρ→2)∝∫lensdρ→lG(|ρ→2−ρ→l|,ωcsi)G(|ρ→l|,ωcf)⁢G(|ρ→l−ρ→o|,ωcso).

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    ∫lensdρ→lG[|ρ→l|,ωc(1so+1si−1f)]e−iωc(ρ→oso+ρ→isi)·ρ→l∝δ(ρ→o+ρ→im),

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    ∫lensdρ→le−iωc(ρ→oso+ρ→isi)·ρ→l∝somb[Rsoωc(ρ→o+ρ→im)],

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    Ψ(ρ→o,ρ→i)∝δ(ρ→o+ρ→im),

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    G(2)(ρ→o,ρ→i)=|Ψ(ρ→o,ρ→i)|2∝|δ(ρ→o+ρ→im)|2.

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    R1,2∝|∫objdρ→oA(ρ→o)Ψ(ρ→o,ρ→i)|2≃|A(ρ→o)⊗δ(ρ→o+ρ→2m)|2=|A(−ρ→2m)|2,

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    g(κ→s,ωs;ρ→1,z1=d1+so+fcoll)=eiωscz1∫objdρ→o∫lensdρ→l∫sourcedρ→S[−iωs2πcd1eiκ→s·ρ→S⁢G(|ρ→S−ρ→l|,ωscd1)]×G(|ρ→l|,ωscf)[−iωs2πcsoG(|ρ→l−ρ→o|,ωscso)]A(ρ→o)×G(|ρ→o|,ωscfcoll)[−iωs2πcfcollG(|ρ→o−ρ→1|,ωscfcoll)],

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    Ψ(ρ→1,ρ→2)∝∫objdρ→oA(ρ→o)δ(ρ→o+ρ→2m)=A(ρ→o)⊗δ(ρ→o+ρ→2m),

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    R1,2∝G(2)(ρ→1,ρ→2)∝|A(ρ→o)⊗δ(ρ→o+ρ→2m)|2=|A(−ρ→2m)|2,

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    ⟨n(r1,t1)n(r2,t2)⟩=⟨n(r1,t1)⟩⟨n(r2,t2)⟩+⟨Δn(r1,t1)Δn(r2,t2)⟩,⟨I(r1,t1)I(r2,t2)⟩=⟨I(r1,t1)⟩⟨I(r2,t2)⟩+⟨ΔI(r1,t1)ΔI(r2,t2)⟩,

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    E(r,t)=∑m=1MEm(r,t),

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    ⟨n(r1,t1)n(r2,t2)⟩∝⟨I(r1,t1)I(r2,t2)⟩=⟨E*(r1,t1)E(r1,t1)E*(r2,t2)E(r2,t2)⟩=⟨∑m,n,p,qEm*(r1,t1)En(r1,t1)Ep*(r2,t2)Eq(r2,t2)⟩=∑mEm*(r1,t1)Em(r1,t1)∑nEn*(r2,t2)En(r2,t2)+∑m≠nEm*(r1,t1)En(r1,t1)En*(r2,t2)Em(r2,t2)=∑m,n|12[Em(r1,t1)En(r,t2)+En(r1,t1)Em(r2,t2)]|2,

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    ⟨n(r1,t1)⟩⟨n(r2,t2)⟩∝⟨I(r1,t1)⟩⟨I(r2,t2)⟩=∑mEm*(r1,t1)Em(r1,t1)∑nEn*(r2,t2)En(r2,t2)⟨Δn(r1,t1)Δn(r2,t2)⟩∝⟨ΔI(r1,t1)ΔI(r2,t2)⟩=∑m≠nEm*(r1,t1)En(r1,t1)En*(r2,t2)Em(r2,t2).

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    |12[Em(r1,t1)En(r,t2)+En(r1,t1)Em(r2,t2)]|2,

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    ⟨Δn(ρ→1,z1)Δn2⟩=∑m,n[Em*gm*(ρ→1,z1)][Engn(ρ→1,z1)]×[∫dρ→oEn*A*(ρ→o)gn*(ρ→o,zo)]⁢[∫dρ→′oEmA(ρ→′o)gm(ρ→′o,zo)]=|∑m[Em*gm*(ρ→1,z1)][∫dρ→oEmA(ρ→o)gm(ρ→o,zo)]|2∝|∫dρ→s[gρ→s*(ρ→1,z1)][∫dρ→oA(ρ→o)gρ→s(ρ→o,zo)]|2,

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    gm(ω;ρ→j,zj)=cozje−iωτjeiω2czj|ρ→j−ρ→m|2,

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    ⟨Δn(ρ→1,z1)Δn2⟩|z1=zo∝|∫dρ→s[gρ→s*(ω;ρ→1,z1)][∫dρ→oA(ρ→o)gρ→s(ω;ρ→o,zo)]|2≃|∫dρ→oA(ρ→o)sombπΔθsλ|ρ→1−ρ→o||2=|A(ρ→o)⊗sombπΔθsλ|ρ→1−ρ→o||2,

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    ⟨Δn(ρ→1,z1)Δn2⟩|z1=zo≃|∫dρ→oA(ρ→o)δ(|ρ→1−ρ→o|)|2=|A(ρ→o)⊗δ(|ρ→1−ρ→o|)|2=|A(ρ→1)|2.

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    ⟨Δn(ρ→o,zo)Δn(ρ→1,z1)⟩∝|∫dρ→s[gρ→s(ω;ρ→1,z1)][gρ→s*(ω;ρ→o,zo)]|2=|∫dρ→s{[gρ→s(ω;ρ→gi,zgi)][∫dρ→gigρ→gi(ω;ρ→L,zL)](glens)×[∫dρ→LgL(ω;ρ→1,z1)]}[gρ→s*(ω;ρ→o,zo)]|2≃|∫dρ→giδ(|ρ→o−ρ→gi|)sombπDsoλ|ρ→gi−ρ→1/m||2≃somb2πDsoλ|ρ→o−ρ→1/m|,

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    ⟨Δn(ρ→1,z1)Δn2⟩∝|∫dρ→s[gρ→s(ω;ρ→1,z1)][∫dρ→oA(ρ→o)gρ→s*(ω;ρ→o,zo)]|2=|∫dρ→s{[gρ→s(ω;ρ→gi,zGI)][∫dρ→gigρ→gi(ω;ρ→L,zL)](glens)×[∫dρ→LgL(ω;ρ→1,z1)]}{∫dρ→oA(ρ→o)[gρ→s*(ω;ρ→o,zo)]}|2≃|∫dρ→oA(ρ→o)∫dρ→giδ(|ρ→o−ρ→gi|)⁢sombπDsoλ|ρ→gi−ρ→1/m||2≃|∫dρ→oA(ρ→o)sombπDsoλ|ρ→o−ρ→1/m||2.

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    ⟨ΔI(ρ→1,z1)ΔI2⟩≃|∫dρ→oA(ρ→o)δ(|ρ→o−ρ→1/m|)|2=|A(ρ→o)⊗δ(|ρ→o−ρ→1/m|)|2=|A(ρ→1/m)|2,

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    γ(ρ→1,ρ→2)≃δ(ρ→s−ρ→1/m)δ(ρ→s−ρ→2/m),

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    |Ψ⟩≃Ψ0∑s,iδ(ωs+ωi−ωp)as†(ωs)ai†(ωi)|0⟩.

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    Ψ(τ1,τ2)=⟨0|E^(+)(r2,t2)E^(+)(r1,t1)|Ψ⟩≃Ψ0∑s,iδ(ωs+ωi−ωp)e−iωsτ1e−iωiτ2=Ψ0∑s,iΨs,i(ωs,ωi),

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    Ψ(τ1;τ2)≅Ψ0e−i(ωs0τ1+ωi0τ2)∫dνf(ν)e−iν(τ1−τ2)=[Ψ0e−i2(ωs0+ωi0)(τ1+τ2)]{Fτ1−τ2[f(ν)]e−i2(ωs0−ωi0)(τ1−τ2)},

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    G(2)(τ1−τ2)∝|Fτ1−τ2[f(ν)]|2.

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    G(2)(τ1−τ2)∝sinc2[Δω(τ1−τ2)2π].

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    ΨT(τ1,τ2)≃Ψ0∑s,iδ(ωs+ωi−ωp)[e−iδφ(r1)]e−iωsτ1e−iωiτ2=[Ψ0e−iδφ(r1)]Ψ(τ1,τ2),

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    G(2)(τ1−τ2)=|ΨT(τ1,τ2)|2=|Ψ(τ1,τ2)|2

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    G(2)(t˜1−t˜2)=∫tcdt1dt2G(2)(t1−t2)D(t˜1−t1)D(t˜2−t2),

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    GTurb(2)(ρ→1,z1;ρ→2,z2)=∫dκ→dκ→′|12[g2(ρ→2,z2;κ→)eiδφ2g1(ρ→1,z1;κ→′)eiδφ1+g2(ρ→2,z2;κ→′)eiδφ2g1(ρ→1,z1;κ→)eiδφ1]|2=G(2)(ρ→1,z1;ρ→2,z2).

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    gm(ρ→j,zj)=co|zj−zm|e−iωτjeiω2c|zj−zm||ρ→j−ρ→m|2,

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    ⟨ΔI(ρ→gi,zgi)ΔI(ρ→o,zo)⟩|zgi=zo∝⟨|∫dρ→s[gs*(ρ→gi,zgi)][gs(ρ→o,zo)]|2⟩∝somb2πΔθsλ|ρ→gi−ρ→o|,

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    ⟨ΔI(ρ→gi,zgi)ΔI(ρ→o,zo)⟩∝δ(zgi−zo)δ(|ρ→gi−ρ→o|).

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    A(ρ→o)≃∫dzoA(ρ→o,zo).

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    ⟨ΔI(ρ→1,z1)ΔI2⟩=|∫dρ→o[∫dzoA(ρ→o,zo)]⁢[δ(z1−zo)sombπΔθsλ|ρ→1−ρ→o|]|2≃|∫dρ→o[∫dzoA(ρ→o,zo)][δ(z1−zo)δ(|ρ→1−ρ→o|)]|2≃|A(ρ→1=ρ→o,z1=zo)|2,

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    ⟨ΔI(ρ→i,zi)ΔI(ρ→o,zo)⟩∝|∫dρ→sgs*(ρ→o,zo)gs(ρ→i,zi)|2=|∫dρ→sgρ→s*(ρ→o,zo){gs(ρ→gi,zgi)×[∫dρ→giggi(ρ→L,zL)](glens)⁢[∫dρ→LgL(ρ→i,zi)]}|2,

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    ⟨ΔI(ρ→i,zi)ΔI(ρ→o,zo)⟩∝⟨|∫dρ→giδ(|ρ→gi−ρ→o|)sombπDsoλ|ρ→gi−ρ→i/μ||2⟩∝somb2πDsoλ|ρ→o−ρ→i/μ|,

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    ⟨ΔI(ρ→1,z1)ΔI2⟩∝⟨|∫dρ→oA˜(ρ→o,zo=zgi)sombπDsoλ|ρ→o−ρ→1/μ||2⟩∝∫dρ→o|A˜(ρ→o,zo=zgi)|2somb2πDsoλ|ρ→o−ρ→1/μ|≃|A˜(ρ→1/μ,zo=zgi)|2.

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    ΔI(ρ→i,zi)=∑m≠nEm*(ρ→i,zi)En(ρ→i,zi)=∑m≠n∫lensdρ→lEm*(ρ→gi)[gρ→gi*(ρ→l,zl)](glens)[gρ→l*(ρ→i,zi)]×∫lensdρ→l′En(ρ→gi)[ggi(ρ→l′,zl′)](glens)[gl′(ρ→i,zi)]=∑m≠nEm*(ρ→gi)En(ρ→gi)somb2πDsoλv|ρ→o+ρ→i/μ|=ΔI(ρ→gi,zgi)somb2πDsoλv|ρ→o+ρ→i/μ|.

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    ⟨ΔI(ρ→1,z1)ΔI2⟩∝⟨|∫dρ→oA˜(ρ→o,zo=zgi)sombπDsoλv|ρ→o−ρ→1/μ||2⟩∝∫dρ→o|A˜(ρ→o,zo=zgi)|2somb2πDsoλv|ρ→o−ρ→1/μ|.

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    Yanhua Shih, "Ghost imaging—its physics and application [Invited]," Chin. Opt. Lett. 22, 060011 (2024)
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