Abstract

The interdependence between the temporal and the longitudinal and transverse spatial coherence is analyzed. The analysis concerns both the derivation of the exact analytical propagation equation of the mutual coherence in free space from a planar and spherical boundary surface and the numerical analysis of a specific model. This model assumes the mutual coherence function to be spatially incoherent at a planar surface. The temporal coherence of the mutual coherence function on the planar or spherical boundary surface is not quasi-monochromatic but much more general and given by Eq. (41). Two theorems are derived that are a generalization of the van Cittert–Zernike theorem.

© 2012 Optical Society of America

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  1. H. Liu and S. Han, “Spatial longitudinal coherence length of a thermal source and its influence on lensless ghost imaging,” Opt. Lett. 33, 824–826 (2008).
    [CrossRef]
  2. F. Ferri, D. Magatti, V. Sala, and A. Gatti, “Longitudinal coherence in ghost imaging,” Appl. Phys. Lett. 92, 261109 (2008).
    [CrossRef]
  3. L. Helseth, “Strongly focused polarized light pulse,” Phys. Rev. E 72, 047602 (2005).
    [CrossRef]
  4. D. Lyaking and V. Ryabukho, “Changes in longitudinal spatial coherence length of optical field in image space,” Tech. Phys. Lett. 37, 45–48 (2011).
    [CrossRef]
  5. Y. Cai and S.-Y. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E 71, 056607 (2005).
    [CrossRef]
  6. M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975), Chap. 10.7.
  7. E. Wolf, A. Devaney, and F. Gori, “Relationship between spectral properties and spatial coherence properties in one-dimensional free fields,” Opt. Commun. 46, 45–48 (1983).
    [CrossRef]
  8. L. Sereda, M. Bertolotti, and A. Ferrari, “Coherence properties of nonstationary light wave fields,” J. Opt. Soc. Am. A 15, 695–705 (1998).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  11. M. Bertolotti, A. Ferrari, and L. Sereda, “Coherence properties of nonstationary polychromatic light sources,” J. Opt. Soc. Am. B 12, 341–347 (1995).
    [CrossRef]
  12. P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204, 53–58 (2002).
    [CrossRef]
  13. M. Born and E. Wolf, Principle of Optics, 7th ed. (Pergamon, 1999).
  14. H. Lajunen and P. Vahimaa, “Theory of spatially and spectrally partially coherent pulses,” J. Opt. Soc. Am. A 22, 1536–1545 (2005).
    [CrossRef]
  15. I. Christov, “Propagation of partially coherent light pulses,” Opt. Acta 33, 63–72 (1986).
    [CrossRef]
  16. L. Wang, Q. Lin, H. Chen, and Z. Zhu, “Propagation of partially coherent pulsed beams in the spatiotemporal domain,” Phys. Rev. E 67, 056613 (2003).
    [CrossRef]
  17. P. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953), Vol. I, Section 7.3, p. 865.
  18. The phrase “uniform strength” means also “uniformly” illuminated.
  19. Equation (25) is derived by repeated inversion of the operator (24) occurring in Eq. (23).
  20. A. Forsyth, Theory of Differential Equations part 4 (Cambridge University, 1906), Vol. 6, Chap. 23.
  21. E. Copson, Partial Differential Equations (Cambridge University, 1975), Chap. 6.1.

2011

D. Lyaking and V. Ryabukho, “Changes in longitudinal spatial coherence length of optical field in image space,” Tech. Phys. Lett. 37, 45–48 (2011).
[CrossRef]

2008

F. Ferri, D. Magatti, V. Sala, and A. Gatti, “Longitudinal coherence in ghost imaging,” Appl. Phys. Lett. 92, 261109 (2008).
[CrossRef]

H. Liu and S. Han, “Spatial longitudinal coherence length of a thermal source and its influence on lensless ghost imaging,” Opt. Lett. 33, 824–826 (2008).
[CrossRef]

2005

H. Lajunen and P. Vahimaa, “Theory of spatially and spectrally partially coherent pulses,” J. Opt. Soc. Am. A 22, 1536–1545 (2005).
[CrossRef]

L. Helseth, “Strongly focused polarized light pulse,” Phys. Rev. E 72, 047602 (2005).
[CrossRef]

Y. Cai and S.-Y. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E 71, 056607 (2005).
[CrossRef]

2003

L. Wang, Q. Lin, H. Chen, and Z. Zhu, “Propagation of partially coherent pulsed beams in the spatiotemporal domain,” Phys. Rev. E 67, 056613 (2003).
[CrossRef]

2002

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204, 53–58 (2002).
[CrossRef]

1998

1996

1995

1986

I. Christov, “Propagation of partially coherent light pulses,” Opt. Acta 33, 63–72 (1986).
[CrossRef]

1983

E. Wolf, A. Devaney, and F. Gori, “Relationship between spectral properties and spatial coherence properties in one-dimensional free fields,” Opt. Commun. 46, 45–48 (1983).
[CrossRef]

Bertolotti, M.

Born, M.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975), Chap. 10.7.

M. Born and E. Wolf, Principle of Optics, 7th ed. (Pergamon, 1999).

Cai, Y.

Y. Cai and S.-Y. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E 71, 056607 (2005).
[CrossRef]

Chen, H.

L. Wang, Q. Lin, H. Chen, and Z. Zhu, “Propagation of partially coherent pulsed beams in the spatiotemporal domain,” Phys. Rev. E 67, 056613 (2003).
[CrossRef]

Christov, I.

I. Christov, “Propagation of partially coherent light pulses,” Opt. Acta 33, 63–72 (1986).
[CrossRef]

Copson, E.

E. Copson, Partial Differential Equations (Cambridge University, 1975), Chap. 6.1.

Devaney, A.

E. Wolf, A. Devaney, and F. Gori, “Relationship between spectral properties and spatial coherence properties in one-dimensional free fields,” Opt. Commun. 46, 45–48 (1983).
[CrossRef]

Ferrari, A.

Ferri, F.

F. Ferri, D. Magatti, V. Sala, and A. Gatti, “Longitudinal coherence in ghost imaging,” Appl. Phys. Lett. 92, 261109 (2008).
[CrossRef]

Feshbach, H.

P. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953), Vol. I, Section 7.3, p. 865.

Forsyth, A.

A. Forsyth, Theory of Differential Equations part 4 (Cambridge University, 1906), Vol. 6, Chap. 23.

Friberg, A. T.

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204, 53–58 (2002).
[CrossRef]

Gatti, A.

F. Ferri, D. Magatti, V. Sala, and A. Gatti, “Longitudinal coherence in ghost imaging,” Appl. Phys. Lett. 92, 261109 (2008).
[CrossRef]

Gori, F.

E. Wolf, A. Devaney, and F. Gori, “Relationship between spectral properties and spatial coherence properties in one-dimensional free fields,” Opt. Commun. 46, 45–48 (1983).
[CrossRef]

Han, S.

Helseth, L.

L. Helseth, “Strongly focused polarized light pulse,” Phys. Rev. E 72, 047602 (2005).
[CrossRef]

Lajunen, H.

Lin, Q.

L. Wang, Q. Lin, H. Chen, and Z. Zhu, “Propagation of partially coherent pulsed beams in the spatiotemporal domain,” Phys. Rev. E 67, 056613 (2003).
[CrossRef]

Liu, H.

Lyaking, D.

D. Lyaking and V. Ryabukho, “Changes in longitudinal spatial coherence length of optical field in image space,” Tech. Phys. Lett. 37, 45–48 (2011).
[CrossRef]

Magatti, D.

F. Ferri, D. Magatti, V. Sala, and A. Gatti, “Longitudinal coherence in ghost imaging,” Appl. Phys. Lett. 92, 261109 (2008).
[CrossRef]

Morse, P.

P. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953), Vol. I, Section 7.3, p. 865.

Pääkkönen, P.

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204, 53–58 (2002).
[CrossRef]

Ryabukho, V.

D. Lyaking and V. Ryabukho, “Changes in longitudinal spatial coherence length of optical field in image space,” Tech. Phys. Lett. 37, 45–48 (2011).
[CrossRef]

Sala, V.

F. Ferri, D. Magatti, V. Sala, and A. Gatti, “Longitudinal coherence in ghost imaging,” Appl. Phys. Lett. 92, 261109 (2008).
[CrossRef]

Sereda, L.

Turunen, J.

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204, 53–58 (2002).
[CrossRef]

Vahimaa, P.

H. Lajunen and P. Vahimaa, “Theory of spatially and spectrally partially coherent pulses,” J. Opt. Soc. Am. A 22, 1536–1545 (2005).
[CrossRef]

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204, 53–58 (2002).
[CrossRef]

Wang, L.

L. Wang, Q. Lin, H. Chen, and Z. Zhu, “Propagation of partially coherent pulsed beams in the spatiotemporal domain,” Phys. Rev. E 67, 056613 (2003).
[CrossRef]

Wolf, E.

E. Wolf, A. Devaney, and F. Gori, “Relationship between spectral properties and spatial coherence properties in one-dimensional free fields,” Opt. Commun. 46, 45–48 (1983).
[CrossRef]

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975), Chap. 10.7.

M. Born and E. Wolf, Principle of Optics, 7th ed. (Pergamon, 1999).

Wyrowski, F.

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204, 53–58 (2002).
[CrossRef]

Zhu, S.-Y.

Y. Cai and S.-Y. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E 71, 056607 (2005).
[CrossRef]

Zhu, Z.

L. Wang, Q. Lin, H. Chen, and Z. Zhu, “Propagation of partially coherent pulsed beams in the spatiotemporal domain,” Phys. Rev. E 67, 056613 (2003).
[CrossRef]

Appl. Phys. Lett.

F. Ferri, D. Magatti, V. Sala, and A. Gatti, “Longitudinal coherence in ghost imaging,” Appl. Phys. Lett. 92, 261109 (2008).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Acta

I. Christov, “Propagation of partially coherent light pulses,” Opt. Acta 33, 63–72 (1986).
[CrossRef]

Opt. Commun.

E. Wolf, A. Devaney, and F. Gori, “Relationship between spectral properties and spatial coherence properties in one-dimensional free fields,” Opt. Commun. 46, 45–48 (1983).
[CrossRef]

P. Pääkkönen, J. Turunen, P. Vahimaa, A. T. Friberg, and F. Wyrowski, “Partially coherent Gaussian pulses,” Opt. Commun. 204, 53–58 (2002).
[CrossRef]

Opt. Lett.

Phys. Rev. E

L. Wang, Q. Lin, H. Chen, and Z. Zhu, “Propagation of partially coherent pulsed beams in the spatiotemporal domain,” Phys. Rev. E 67, 056613 (2003).
[CrossRef]

L. Helseth, “Strongly focused polarized light pulse,” Phys. Rev. E 72, 047602 (2005).
[CrossRef]

Y. Cai and S.-Y. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E 71, 056607 (2005).
[CrossRef]

Tech. Phys. Lett.

D. Lyaking and V. Ryabukho, “Changes in longitudinal spatial coherence length of optical field in image space,” Tech. Phys. Lett. 37, 45–48 (2011).
[CrossRef]

Other

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975), Chap. 10.7.

P. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953), Vol. I, Section 7.3, p. 865.

The phrase “uniform strength” means also “uniformly” illuminated.

Equation (25) is derived by repeated inversion of the operator (24) occurring in Eq. (23).

A. Forsyth, Theory of Differential Equations part 4 (Cambridge University, 1906), Vol. 6, Chap. 23.

E. Copson, Partial Differential Equations (Cambridge University, 1975), Chap. 6.1.

M. Born and E. Wolf, Principle of Optics, 7th ed. (Pergamon, 1999).

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Figures (7)

Fig. 1.
Fig. 1.

Geometry of the system.

Fig. 2.
Fig. 2.

Spatial coherence properties of the normalized mutual coherence function along the z axis.

Fig. 3.
Fig. 3.

Spatial coherence properties of the normalized mutual coherence function along the z axis for different values of the temporal coherence time Tc.

Fig. 4.
Fig. 4.

Spatial coherence properties of the normalized mutual coherence function along the z axis for different values of t1 and t2, and in the absence of temporal correlations, viz. Tc.

Fig. 5.
Fig. 5.

Spatial coherence properties of the normalized mutual coherence function in the planes y1=y2=0 for five different values of x1 and as a function of x2.

Fig. 6.
Fig. 6.

Spatial coherence properties of the normalized mutual coherence function in the planes x1=y1=y2=0 for three values of Tc as a function of x2.

Fig. 7.
Fig. 7.

Temporal coherence properties of the normalized mutual coherence function in the planes x1=x2=y1=y2=0 for different values of Tc as a function of t2.

Equations (55)

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ψ(r1;t1)=14πS0dS·{R1R13ψ(rS;t1)|(t1=t1R1c)R1cR12t1ψ(rS;t1)|(t1=t1R1c)}.
R1=|r1rS|,
R1=r1rS,
rSsurfaceS0,
r1is the observation point.
Γ(r1,r2;t1,t2)=ψ(r1;t1)ψ*(r2;t2)
Γ(r1,r2;t1,t2)=14πS1dS1·{R1R13ψ(rS1;t1)|(t1=t1R1c)R1cR12t1ψ(rS1;t1)|(t1=t1R1c)}×14πS2dS2·{R2R23ψ*(rS2;t2)|(t2=t2R2c)R2cR22t2ψ*(rS2;t2)|(t2=t2R2c)}.
Γ(ret)(r1,r2;t1,t2)Γ(r1,r2;t1,t2)|t1=t1R1c,t2=t2R2c.
Γ(r1,r2;t1,t2)=116π2S1S2dS1·dS2·[R1R13R2R23Γ(ret)(rS1,rS2;t1,t2)R1R13R2cR22t2Γ(ret)(rS1,rS2;t1,t2)R1cR12R2R23t1Γ(ret)(rS1,rS2;t1,t2)+R1cR12R2cR22t1t2Γ(ret)(rS1,rS2;t1,t2)].
ψ(r1;t1)=14π0t1+dt1S1dS1·S1(δ(t1t1+R1c)R1)ψ(rS1;t1).
Γ(r1,r2;t1,t2)=116π20t1+0t2+dt1dt2S1S2dS1·dS2·[Γ(rS1,rS2;t1,t2)×S1(δ(t1t1+R1c)R1)S2(δ(t2t2+R2c)R2)].
ψ(rSj;tj)Sj(δ(tjtj+Rjc)Rj)ψ(rSj;tj)(RjRj3RjcRj2tj)(δ(tjtj+Rjc)Rj),j=1,2,.
Γ(r1,r2,r3,r4;t1,t2,t3,t4)=1256π40t1+0t2+0t3+0t4+dt1dt2dt3dt4×S1S2S3S4dS1·dS2·dS3·dS4·Γ(rS1,rS2,rS3,rS4;t1,t2,t3,t4)j=1j=4Sj(δ(tjtj+Rjc)Rj).
Γ(r1,r2,r3,r4;t1,t2,t3,t4)=S1S2S3S4dS1·dS2·dS3·dS4·×1256π4j=1j=4(RjRj3RjcRj2tj)Γ(rS1,rS2,rS3,rS4;t1,t2,t3,t4)|tj=tjRjc.
Γ(S1)(rS1,rS2;t1,t2)=δ(rS1rS2)Γ(S1)(t1,t2).
Γ(ret)(rS1,rS2;t1,t2)=Γ(S1)(rS1,rS1;t1,t2)=δ(rS1rS2)Γ(S1)(t1,t2).
δ(t1t1+R1c)R1δ(t2t2+R2c)R2.
δ(t1t1R1c)R1=dkdωexp(ik·(r1r1)iω(t1t1))c2|k|2ω2,
whereR1=|r1r1|.
δ(t1t1+R1c)R1δ(t2t2+R2c)R2=dk1dω1dk2dω2×exp(ik1.(rS1+rn1r1)+ik2·(rS2+rn2r2)iω1(t1t1)iω2(t2t2))(c2|k1|2ω12)(c2|k2|2ω22),
rj=rSj+rnj,j=1,2.
Γ(S1)(rS1,rS1;t1,t2)=δ(rS1rS2)Γ(S1)(t1,t2).
Iprop(r1,r2;t1,t2)=dS1dk1dω1dk2dω2×exp(ik1·(rS1+rn1r1)+ik2·(rS1+rn2r2)iω1(t1t1)iω2(t2t2))(c2|k1|2ω12)(c2|k2|2ω22)Γ(S1)(t1,t2).
δ(xx)=12π+dfexp(if(xx)),
I(prop)(r1,r2;t1,t2)=dk1zdk2dω1dω2exp(ik2.(rn2r2)+i(k1zz1+k2xx1+k2yy1k1zz1)iω1(t1t1)iω2(t2t2))(ω12c2(k2x2+k2y2+k1z2))(ω22c2|k22|).
X1=x2x1,
Y1=y2y1,
(2X12+2Y12+2z121c22t12)(2X12+2Y12+2z221c22t22)I(prop)(r1,r2;t1,t2)=δ(X1)δ(Y1)δ(z1z1)δ(z2z2)δ(t1t1)δ(t1t2).
O(X1,Y2,z1,z2;t1,t2)=(2X12+2Y12+2z121c22t12)(2X12+2Y12+2z221c22t22)
I(prop)(r1,r2;t1,t2)=dX1dY1δ(t1t1+|R1(entangl)R1(entangl)|c)|R1(entangl)R1(entangl)|δ(t2t2+|R2(entangl)|c)|R2(entangl)|,
R1(entangl)=(X1,Y1,z1),R1(entangl)=(X1,Y1,z1),
R2(entangl)=(X1,Y1,z2z2).
Γ(r1,r2;t1,t2)=dt1dt2zS1zS2Γ(t1,t2)I(prop)(r1,r2;t1,t2).
Γ(rS1,rS2;t1,t2)=Γ(S1,2)(t1,t2)δ(ϕ1ϕ2)δ(cos(θ1)cos(θ2)).
G(r1,r1;t1,t2)=|k1|dk1dω1exp(iω1(t1t1))(c2|k1|2ω12)l=0m=lm=+lhl1(k1r1)Ylm(θ1,ϕ1)Ylm*(θ1,ϕ1)hl1(k1a).
δ(t1t1)δ(ϕ1ϕ2)δ(cos(θ1)cos(θ2))
Γ(r1,r2;t1,t2)=0t1+0t2+dt1dt2dS1dS2dk1dω1dk2dω2exp(iω1(t1t1)iω2(t2t2))(c2|k1|2ω12)(c2|k2|2ω22)l=0m=lm=+lhl1(k1r1)Ylm(θ1,ϕ1)Ylm*(θ1,ϕ1)hl1(k1a)l=0m=lm=+lhl1(k2r2)Ylm(θ2,ϕ2)Ylm*(θ2,ϕ2)hl1(k2a)Γ(rS1,rS2;t1,t2).
Γ(r1,r2;t1,t2)=0t1+0t2+dt1dt2dS1dS2dk1dω1dk2dω2exp(iω1(t1t1)iω2(t2t2))(c2|k1|2ω12)(c2|k2|2ω22)l=0m=lm=+lhl1(k1r1)Ylm(θ1,ϕ1)Ylm*(θ1,ϕ1)hl1(k1a)×l=0m=lm=+lhl1(k2r2)Ylm(θ2,ϕ2)Ylm*(θ2,ϕ2)hl1(k2a)Γ(S1,2)(t1,t2)δ(ϕ1ϕ2)(δ(cosθ1)δ(cosθ2)).
dΩYlm(θ1,ϕ1)Ylm*(θ1,ϕ1)=δl,lδm,m
Ylm*(θ1,ϕ1)=Ylm(θ1,ϕ1).
kjx=|kj|sin(θj)cos(ϕj),kjy=|kj|sin(θj)sin(ϕj),kjz=|kj|cos(θj),j=1,2.
Γ(r1,r2;t1,t2)=0t1+0t2+dt1dt2dS1dS2dω1dω2k12dk1k22dk2exp(iω1(t1t1)iω2(t2t2))(c2|k1|2ω12)(c2|k2|2ω22)Γ(S1,2)(t1,t2)×{l=0m=lm=+lhl1(k1r1)Ylm(θ1,ϕ1)Ylm*(θ1,ϕ1)hl1(k1a)l=0m=lm=+lhl1(k2r2)Ylm(θ2,ϕ2)Ylm*(θ1,ϕ1)hl1(k2a)}.
Γ(r1,r2;t1,t2)=0t1+0t2+dt1dt2dS1dω1dω2k12dk1k22dk2Γ(S1,2)(t1,t2)×exp(iω1(t1t1)iω2(t2t2))(c2|k1|2ω12)(c2|k2|2ω22)l=0m=lm=+lhl1(k1r1)Ylm(θ1,ϕ1)hl1(k2r2)Ylm(θ2,ϕ2)hl1(k1a)hl1(k2a).
Pl(cosθ)=4π2n+1m=lm=lYlm(θ1,ϕ1)Yl*m(θ2,ϕ2),
Γ(r1,r2;t1,t2)=0t1+0t2+dt1dt2dS1dω1dω2k12dk1k22dk2Γ(S1,2)(t1,t2)×exp(iω1(t1t1)iω2(t2t2))(c2|k1|2ω12)(c2|k2|2ω22)l=02n+14πhl1(k1r1)hl1(k2r2)hl1(k1a)hl1(k2a)Pl(cosθ).
μ(ω1,ω2)=w0exp[(ω1ω0)2+(ω2ω0)22Ω2]exp[(ω1ω2)22Ωc2].
Γ(t1,t2)=I0exp{(t1t2)2+iω(t1t2)2Tc2t12+t222T2}
T=(1Ω2+1Ωc2)1/2,
Tc=2ΩcΩ(1Ω2+1Ωc2)1/2.
ψ(t|rr|c)|rr|,
Γ(r1S,r2S;t1,t2)=Γ(t1,t2).
a2Γ(t1|r1|c+ac,t2|r2|c+ac)|r1||r2|.
Γ(r1,r2;t1,t2)=I0exp((t1|r1|ct2+|r2|c)2+iω(t1|r1|ct2+|r2|c)2Tc2)×exp((t1|r1|c)2+(t2|r2|c)22T2).
Γ(r1,r2;t1,t2)=116π2S1S2dS1·dS2·[R1R13R2R23Γ(ret)(rS1,rS2;t1,t2)R1R13R2cR22t2Γ(ret)(rS1,rS2;t1,t2)R1cR12R2R23t1Γ(ret)(rS1,rS2;t1,t2)+R1cR12R2cR22t1t2Γ(ret)(rS1,rS2;t1,t2)]
Γ(ret)(rS1,rS2;t1,t2)=δ(rS1rS2)exp(|rS1|2+|rS2|2σ2)exp((t1t2)2Tc2)exp(t12t22τ2),

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