Abstract

A spectral approach to the theory of partial coherence of nonstationary light [M. Bertolotti et al., Pure Appl. Opt. 6, 153 (1997)] is applied to the study of spatial coherence properties of nonstationary wave fields on propagation in free space. Three types of nonstationary light sources are considered—a point source, a spatially incoherent source, and a uniformly coherent source, the last being perfectly coherent in both space–time and space–frequency domains. An example of the diffraction pattern from a slit with nonstationary partially coherent illumination is used to illustrate the general discussion.

© 1998 Optical Society of America

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  1. M. Bertolotti, L. Sereda, A. Ferrari, “Application of the spectral representation of stochastic processes to the study of nonstationary light radiation: a tutorial,” Pure Appl. Opt. 6, 153–171 (1997).
    [CrossRef]
  2. M. Bertolotti, A. Ferrari, L. Sereda, “Coherence properties of nonstationary polychromatic light sources,” J. Opt. Soc. Am. B 12, 341–347 (1995).
    [CrossRef]
  3. M. Bertolotti, A. Ferrari, L. Sereda, “Far-zone diffraction of polychromatic and nonstationary plane waves from a slit,” J. Opt. Soc. Am. B 12, 1519–1526 (1995).
    [CrossRef]
  4. L. Sereda, A. Ferrari, M. Bertolotti, “On the partial coherence theory for nonstationary light,” in Coherence and Quantum Optics VII: J. H. Eberly, L. Mandel, E. Wolf, eds. (Plenum, New York, 1996), pp. 689–690.
  5. L. Sereda, A. Ferrari, M. Bertolotti, “Spectral and time evolution in diffraction from a slit of polychromatic and nonstationary plane waves,” J. Opt. Soc. Am. B 13, 1394–1402 (1996).
    [CrossRef]
  6. L. Sereda, A. Ferrari, M. Bertolotti, “Diffraction of a time Gaussian-shaped pulsed plane wave from a slit,” Pure Appl. Opt. 5, 349–353 (1996).
    [CrossRef]
  7. L. Sereda, A. Ferrari, M. Bertolotti, “Spectral and time evolution of nonstationary plane waves in diffraction from a slit,” in Coherence and Quantum Optics VII: J. H. Eberly, L. Mandel, E. Wolf, eds. (Plenum, New York, 1996), pp. 693–694.
  8. L. Sereda, A. Ferrari, M. Bertolotti, “Spectral and time evolution of non-stationary plane waves in a two-beam interference experiment,” J. Mod. Opt. 43, 2503–2522 (1996).
    [CrossRef]
  9. L. Sereda, A. Ferrari, M. Bertolotti, “Diffraction of a pulsed plane wave from an amplitude diffraction grating,” J. Mod. Opt. 44, 1321–1343 (1997).
    [CrossRef]
  10. L. Sereda, “The spectral theory of coherence, interference and diffraction of nonstationary light,” Ph.D. thesis (University of Rome “La Sapienza,”Rome, 1997).
  11. About the definitions of the source spectral composition and time intensity see also Refs. 1-3.
  12. L. Mandel, E. Wolf, “Some properties of coherent light,” J. Opt. Soc. Am. 51, 815–819 (1961).
    [CrossRef]
  13. L. Mandel, E. Wolf, “Complete coherence in the space–frequency domain,” Opt. Commun. 36, 247–249 (1981).
    [CrossRef]
  14. W. H. Carter, “Difference in the definitions of coherence in the space–time domain and in the space–frequency domain,” J. Mod. Opt. 39, 1461–1470 (1992).
    [CrossRef]
  15. As is shown in Refs. 5 and 6, the energy of a pulsed source is equal to the sum of intensities of the source monochromatic components—in our case given by Eq. (2.3), multiplied by the factor 2π, which constitutes a generalization of the Parceval theorem to pulsed stochastic processes of finite energy.
  16. L. Mandel, E. Wolf, “Spectral coherence and the concept of cross-spectral purity,” J. Opt. Soc. Am. 66, 529–535 (1976).
    [CrossRef]
  17. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).
  18. E. Wolf, “New spectral representation of random sources and of the partially coherent fields that they generate,” Opt. Commun. 38, 3–6 (1981).
    [CrossRef]
  19. An example of such a situation for a source, consisting of two partially correlated point sources, is given in Ref. 1.
  20. R. J. Adler, The Geometry of Random Fields (Wiley, New York, 1981).
  21. It means also that if a thin opaque screen with a small aperture is illuminated from a point source, the spatial coherence of light will not change across the aperture, forming in this way a secondary uniformly coherent source, as is defined by Eqs. (2.1) and (2.2).
  22. Considering the far-zone wave field, it is more exact to say that the normalized spectrum, time intensity, and energy remain unchanged in certain directions of observation, changing if this direction changes, as is shown in Refs. 3-10.
  23. M. Planck, The Theory of Heat Radiation (Dover, New York, 1959).

1997

M. Bertolotti, L. Sereda, A. Ferrari, “Application of the spectral representation of stochastic processes to the study of nonstationary light radiation: a tutorial,” Pure Appl. Opt. 6, 153–171 (1997).
[CrossRef]

L. Sereda, A. Ferrari, M. Bertolotti, “Diffraction of a pulsed plane wave from an amplitude diffraction grating,” J. Mod. Opt. 44, 1321–1343 (1997).
[CrossRef]

1996

L. Sereda, A. Ferrari, M. Bertolotti, “Diffraction of a time Gaussian-shaped pulsed plane wave from a slit,” Pure Appl. Opt. 5, 349–353 (1996).
[CrossRef]

L. Sereda, A. Ferrari, M. Bertolotti, “Spectral and time evolution of non-stationary plane waves in a two-beam interference experiment,” J. Mod. Opt. 43, 2503–2522 (1996).
[CrossRef]

L. Sereda, A. Ferrari, M. Bertolotti, “Spectral and time evolution in diffraction from a slit of polychromatic and nonstationary plane waves,” J. Opt. Soc. Am. B 13, 1394–1402 (1996).
[CrossRef]

1995

1992

W. H. Carter, “Difference in the definitions of coherence in the space–time domain and in the space–frequency domain,” J. Mod. Opt. 39, 1461–1470 (1992).
[CrossRef]

1981

E. Wolf, “New spectral representation of random sources and of the partially coherent fields that they generate,” Opt. Commun. 38, 3–6 (1981).
[CrossRef]

L. Mandel, E. Wolf, “Complete coherence in the space–frequency domain,” Opt. Commun. 36, 247–249 (1981).
[CrossRef]

1976

1961

Adler, R. J.

R. J. Adler, The Geometry of Random Fields (Wiley, New York, 1981).

Bertolotti, M.

L. Sereda, A. Ferrari, M. Bertolotti, “Diffraction of a pulsed plane wave from an amplitude diffraction grating,” J. Mod. Opt. 44, 1321–1343 (1997).
[CrossRef]

M. Bertolotti, L. Sereda, A. Ferrari, “Application of the spectral representation of stochastic processes to the study of nonstationary light radiation: a tutorial,” Pure Appl. Opt. 6, 153–171 (1997).
[CrossRef]

L. Sereda, A. Ferrari, M. Bertolotti, “Spectral and time evolution in diffraction from a slit of polychromatic and nonstationary plane waves,” J. Opt. Soc. Am. B 13, 1394–1402 (1996).
[CrossRef]

L. Sereda, A. Ferrari, M. Bertolotti, “Diffraction of a time Gaussian-shaped pulsed plane wave from a slit,” Pure Appl. Opt. 5, 349–353 (1996).
[CrossRef]

L. Sereda, A. Ferrari, M. Bertolotti, “Spectral and time evolution of non-stationary plane waves in a two-beam interference experiment,” J. Mod. Opt. 43, 2503–2522 (1996).
[CrossRef]

M. Bertolotti, A. Ferrari, L. Sereda, “Coherence properties of nonstationary polychromatic light sources,” J. Opt. Soc. Am. B 12, 341–347 (1995).
[CrossRef]

M. Bertolotti, A. Ferrari, L. Sereda, “Far-zone diffraction of polychromatic and nonstationary plane waves from a slit,” J. Opt. Soc. Am. B 12, 1519–1526 (1995).
[CrossRef]

L. Sereda, A. Ferrari, M. Bertolotti, “Spectral and time evolution of nonstationary plane waves in diffraction from a slit,” in Coherence and Quantum Optics VII: J. H. Eberly, L. Mandel, E. Wolf, eds. (Plenum, New York, 1996), pp. 693–694.

L. Sereda, A. Ferrari, M. Bertolotti, “On the partial coherence theory for nonstationary light,” in Coherence and Quantum Optics VII: J. H. Eberly, L. Mandel, E. Wolf, eds. (Plenum, New York, 1996), pp. 689–690.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

Carter, W. H.

W. H. Carter, “Difference in the definitions of coherence in the space–time domain and in the space–frequency domain,” J. Mod. Opt. 39, 1461–1470 (1992).
[CrossRef]

Ferrari, A.

M. Bertolotti, L. Sereda, A. Ferrari, “Application of the spectral representation of stochastic processes to the study of nonstationary light radiation: a tutorial,” Pure Appl. Opt. 6, 153–171 (1997).
[CrossRef]

L. Sereda, A. Ferrari, M. Bertolotti, “Diffraction of a pulsed plane wave from an amplitude diffraction grating,” J. Mod. Opt. 44, 1321–1343 (1997).
[CrossRef]

L. Sereda, A. Ferrari, M. Bertolotti, “Spectral and time evolution of non-stationary plane waves in a two-beam interference experiment,” J. Mod. Opt. 43, 2503–2522 (1996).
[CrossRef]

L. Sereda, A. Ferrari, M. Bertolotti, “Diffraction of a time Gaussian-shaped pulsed plane wave from a slit,” Pure Appl. Opt. 5, 349–353 (1996).
[CrossRef]

L. Sereda, A. Ferrari, M. Bertolotti, “Spectral and time evolution in diffraction from a slit of polychromatic and nonstationary plane waves,” J. Opt. Soc. Am. B 13, 1394–1402 (1996).
[CrossRef]

M. Bertolotti, A. Ferrari, L. Sereda, “Far-zone diffraction of polychromatic and nonstationary plane waves from a slit,” J. Opt. Soc. Am. B 12, 1519–1526 (1995).
[CrossRef]

M. Bertolotti, A. Ferrari, L. Sereda, “Coherence properties of nonstationary polychromatic light sources,” J. Opt. Soc. Am. B 12, 341–347 (1995).
[CrossRef]

L. Sereda, A. Ferrari, M. Bertolotti, “Spectral and time evolution of nonstationary plane waves in diffraction from a slit,” in Coherence and Quantum Optics VII: J. H. Eberly, L. Mandel, E. Wolf, eds. (Plenum, New York, 1996), pp. 693–694.

L. Sereda, A. Ferrari, M. Bertolotti, “On the partial coherence theory for nonstationary light,” in Coherence and Quantum Optics VII: J. H. Eberly, L. Mandel, E. Wolf, eds. (Plenum, New York, 1996), pp. 689–690.

Mandel, L.

Planck, M.

M. Planck, The Theory of Heat Radiation (Dover, New York, 1959).

Sereda, L.

L. Sereda, A. Ferrari, M. Bertolotti, “Diffraction of a pulsed plane wave from an amplitude diffraction grating,” J. Mod. Opt. 44, 1321–1343 (1997).
[CrossRef]

M. Bertolotti, L. Sereda, A. Ferrari, “Application of the spectral representation of stochastic processes to the study of nonstationary light radiation: a tutorial,” Pure Appl. Opt. 6, 153–171 (1997).
[CrossRef]

L. Sereda, A. Ferrari, M. Bertolotti, “Spectral and time evolution in diffraction from a slit of polychromatic and nonstationary plane waves,” J. Opt. Soc. Am. B 13, 1394–1402 (1996).
[CrossRef]

L. Sereda, A. Ferrari, M. Bertolotti, “Diffraction of a time Gaussian-shaped pulsed plane wave from a slit,” Pure Appl. Opt. 5, 349–353 (1996).
[CrossRef]

L. Sereda, A. Ferrari, M. Bertolotti, “Spectral and time evolution of non-stationary plane waves in a two-beam interference experiment,” J. Mod. Opt. 43, 2503–2522 (1996).
[CrossRef]

M. Bertolotti, A. Ferrari, L. Sereda, “Coherence properties of nonstationary polychromatic light sources,” J. Opt. Soc. Am. B 12, 341–347 (1995).
[CrossRef]

M. Bertolotti, A. Ferrari, L. Sereda, “Far-zone diffraction of polychromatic and nonstationary plane waves from a slit,” J. Opt. Soc. Am. B 12, 1519–1526 (1995).
[CrossRef]

L. Sereda, A. Ferrari, M. Bertolotti, “Spectral and time evolution of nonstationary plane waves in diffraction from a slit,” in Coherence and Quantum Optics VII: J. H. Eberly, L. Mandel, E. Wolf, eds. (Plenum, New York, 1996), pp. 693–694.

L. Sereda, A. Ferrari, M. Bertolotti, “On the partial coherence theory for nonstationary light,” in Coherence and Quantum Optics VII: J. H. Eberly, L. Mandel, E. Wolf, eds. (Plenum, New York, 1996), pp. 689–690.

L. Sereda, “The spectral theory of coherence, interference and diffraction of nonstationary light,” Ph.D. thesis (University of Rome “La Sapienza,”Rome, 1997).

Wolf, E.

L. Mandel, E. Wolf, “Complete coherence in the space–frequency domain,” Opt. Commun. 36, 247–249 (1981).
[CrossRef]

E. Wolf, “New spectral representation of random sources and of the partially coherent fields that they generate,” Opt. Commun. 38, 3–6 (1981).
[CrossRef]

L. Mandel, E. Wolf, “Spectral coherence and the concept of cross-spectral purity,” J. Opt. Soc. Am. 66, 529–535 (1976).
[CrossRef]

L. Mandel, E. Wolf, “Some properties of coherent light,” J. Opt. Soc. Am. 51, 815–819 (1961).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

J. Mod. Opt.

L. Sereda, A. Ferrari, M. Bertolotti, “Spectral and time evolution of non-stationary plane waves in a two-beam interference experiment,” J. Mod. Opt. 43, 2503–2522 (1996).
[CrossRef]

L. Sereda, A. Ferrari, M. Bertolotti, “Diffraction of a pulsed plane wave from an amplitude diffraction grating,” J. Mod. Opt. 44, 1321–1343 (1997).
[CrossRef]

W. H. Carter, “Difference in the definitions of coherence in the space–time domain and in the space–frequency domain,” J. Mod. Opt. 39, 1461–1470 (1992).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. B

Opt. Commun.

E. Wolf, “New spectral representation of random sources and of the partially coherent fields that they generate,” Opt. Commun. 38, 3–6 (1981).
[CrossRef]

L. Mandel, E. Wolf, “Complete coherence in the space–frequency domain,” Opt. Commun. 36, 247–249 (1981).
[CrossRef]

Pure Appl. Opt.

M. Bertolotti, L. Sereda, A. Ferrari, “Application of the spectral representation of stochastic processes to the study of nonstationary light radiation: a tutorial,” Pure Appl. Opt. 6, 153–171 (1997).
[CrossRef]

L. Sereda, A. Ferrari, M. Bertolotti, “Diffraction of a time Gaussian-shaped pulsed plane wave from a slit,” Pure Appl. Opt. 5, 349–353 (1996).
[CrossRef]

Other

L. Sereda, A. Ferrari, M. Bertolotti, “Spectral and time evolution of nonstationary plane waves in diffraction from a slit,” in Coherence and Quantum Optics VII: J. H. Eberly, L. Mandel, E. Wolf, eds. (Plenum, New York, 1996), pp. 693–694.

L. Sereda, A. Ferrari, M. Bertolotti, “On the partial coherence theory for nonstationary light,” in Coherence and Quantum Optics VII: J. H. Eberly, L. Mandel, E. Wolf, eds. (Plenum, New York, 1996), pp. 689–690.

L. Sereda, “The spectral theory of coherence, interference and diffraction of nonstationary light,” Ph.D. thesis (University of Rome “La Sapienza,”Rome, 1997).

About the definitions of the source spectral composition and time intensity see also Refs. 1-3.

An example of such a situation for a source, consisting of two partially correlated point sources, is given in Ref. 1.

R. J. Adler, The Geometry of Random Fields (Wiley, New York, 1981).

It means also that if a thin opaque screen with a small aperture is illuminated from a point source, the spatial coherence of light will not change across the aperture, forming in this way a secondary uniformly coherent source, as is defined by Eqs. (2.1) and (2.2).

Considering the far-zone wave field, it is more exact to say that the normalized spectrum, time intensity, and energy remain unchanged in certain directions of observation, changing if this direction changes, as is shown in Refs. 3-10.

M. Planck, The Theory of Heat Radiation (Dover, New York, 1959).

As is shown in Refs. 5 and 6, the energy of a pulsed source is equal to the sum of intensities of the source monochromatic components—in our case given by Eq. (2.3), multiplied by the factor 2π, which constitutes a generalization of the Parceval theorem to pulsed stochastic processes of finite energy.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

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Figures (4)

Fig. 1
Fig. 1

Illustration to Eqs. (3.2) and (4.3). D is the source domain.

Fig. 2
Fig. 2

Geometry of the diffraction of light from an extended source. ρ0 is an arbitrary fixed point of the source domain D; dmax is the maximum linear dimension of the source (maximum linear distance between point ρ0 and any other source point ρD); Ψmax is the maximum angular dimension of the source, visible from point r of the wave field; and s=|r-ρ|, s0=|r-ρ0|, Δs=s-s0, and d=|ρ-ρ0|.

Fig. 3
Fig. 3

Illustration to Eqs. (5.1)–(5.4). D is the source domain.

Fig. 4
Fig. 4

Geometry of the diffraction of partially coherent light from a slit. W is an incident wave, A is a plane of a slit of width 2a, and S is a cylindrical surface of observation of radius R0.

Equations (129)

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Q(ρ, t)=- exp(iωt)uQ(ρ, ω)dω,
sQ(ρ1, ρ2; ω1, ω2)=uQ(ρ1, ω1)uQ*(ρ2, ω2),
ΓQ(ρ1, ρ2; t1, t2)=Q(ρ1, t1)Q*(ρ2, t2)=-- exp[i(ω1t1-ω2t2)]×sQ(ρ1, ρ2; ω1, ω2)dω1dω2,
γQ(ρ1, ρ2; t1, t2)=ΓQ(ρ1, ρ2; t1, t2)[IQ(ρ1, t1)IQ(ρ2, t2)]1/2
μQ(ρ1, ρ2; ω1, ω2)=sQ(ρ1, ρ2; ω1, ω2)[IQ(ρ1, ω1)IQ(ρ2, ω2)]1/2
IQ(ρ, t)=|Q(ρ, t)|2=ΓQ(ρ, ρ; t, t)
IQ(ρ, ω)=|uQ(ρ, ω)|2=sQ(ρ, ρ; ω, ω)
sQ(ρ1, ρ2; ω1, ω2)=sQ(ω1, ω2)
foranypairρ1, ρ2D.
ΓQ(ρ1, ρ2; t1, t2)=ΓQ(t1, t2)
foranypairρ1, ρ2D;
IQ(ρ, ω)=sQ(ω, ω)=IQ(ω)foranyρD,
IQ(ρ, t)=ΓQ(t, t)=IQ(t)foranyρD.
EQ=-IQ(t)dt=2π-IQ(ω)dω.
γQ(ρ1, ρ2; t1, t2)=γQ(t1, t2)=ΓQ(t1, t2)[IQ(t1)IQ(t2)]1/2,
μQ(ρ1, ρ2; ω1, ω2)=μQ(ω1, ω2)=sQ(ω1, ω2)[IQ(ω1)IQ(ω2)]1/2.
γQ(t1, t2)=γQ(τ)=ΓQ(τ)/IQ,τ=t1-t2,
μQ(ω1, ω2)=μQ(ω)1,ω=ω1=ω2,
Q(ρ, t)=η(ρ) exp[iω0t-12(t/τ)2],
IQ(ρ, t)=|Q(ρ, t)|2=I(ρ) exp[-(t/τ)2],
I(ρ)=|η(ρ)|2
uQ(ρ, ω)=η(ρ)τ(2π)-1/2 exp{-12[(ω-ω0)τ]2},
IQ(ρ, ω)=|uQ(ρ, ω)|2=I(ρ)(τ2/2π) exp{-[(ω-ω0)τ]2}.
sQ(ρ1, ρ2; ω1, ω2)=η(ρ1)η*(ρ2)(τ2/2π) exp{-12[(ω1-ω0)+(ω2-ω0)]2τ2},
μQ(ρ1, ρ2; ω1, ω2)=μQ(ρ1, ρ2)=η(ρ1)η*(ρ2)[I(ρ1)I(ρ2)]1/2.
ΓQ(ρ1, ρ2; t1, t2)=η(ρ1)η*(ρ2) exp-12t12+t22τ2+iω0(t1-t2),
γQ(ρ1, ρ2; t1, t2)=μQ(ρ1, ρ2) exp[iω0(t1-t2)].
γQ(ρ1, ρ2)=|γQ(ρ1, ρ2; t1, t2)|=μQ(ρ1, ρ2).
Q(ρ, t)=η(ρ)Q(t).
Q(ρ, t)=δ(ρ)Q(t).
ΓQ(ρ1, ρ2; t1, t2)=δ(ρ)ΓQ(t1, t2),
sQ(ρ1, ρ2; ω1, ω2)=δ(ρ)sQ(ω1, ω2),
sQ(ρ1, ρ2; ω1, ω2)=δ(ρ)sQ(ρ; ω1, ω2),
ρ=ρ1-ρ2,
sQ(ρ; ω1, ω2)=sQ(ρ, ρ; ω1, ω2)
ΓQ(ρ1, ρ2; t1, t2)=δ(ρ)ΓQ(ρ; t1, t2),ρ=ρ1-ρ2,
ΓQ(ρ; t1, t2)=ΓQ(ρ, ρ; t1, t2)
V(r, t)=- exp(iωt)uF(r, ω)dω,rR3,
uF(r, ω)=ρD exp(-i(ω/v)s)suQ(ρ, ω)dρ,
V(r, t)=ρD 1sQρ, t-svdρ,rR3.
V(r, t)=1sQt-sv,rR3,
s=|r|
uF(r, ω) exp(iωt)=uQ(ω) 1sexpiωt-sv,
uF(r, ω)=exp-i ωvssuQ(ω).
s=s0+Δs
Δss0,orΔs/s01.
Δs<d+2s0 sin(Ψmax/2)
Δsmax<dmax+2s0 sin(Ψmax/2)
dmaxs0+2 sinΨmax21.
V(r, t)1s0ρDQρ, t-s0v-Δsvdρ.
Q(t)=ρDQ(ρ, t)dρ,
V(r, t)1s0Qt-s0v,
sF(r1, r2; ω1, ω2)=uF(r1, ω1)uF*(r2, ω2),
ΓF(r1, r2; t1, t2)=V(r1, t1)V*(r2, t2)=--exp[i(ω1t1-ω2t2)]×sF(r1, r2; ω1, ω2)dω1dω2.
sF(r1, r2; ω1, ω2)=ρ1D ρ2D ×exp[-(i/v)(ω1s11-ω2s22)]s11s12×sQ(ρ1, ρ2; ω1, ω2)dρ1dρ2,
ΓF(r1, r2; t1, t2)=ρ1D ρ2D 1s11s22ΓQρ1, ρ2; t1-s11v, t2-s22vdρ1dρ2,
sF(r1, r2; ω1, ω2)=exp[-(i/v)(ω1s1-ω2s2)]s1s2×sQ(ω1, ω2),
ΓF(r1, r2; t1, t2)=1s1s2ΓQt1-s1v, t2-s2v.
s1=|r1|,s2=|r2|
ΓF(r1, r2; t1, t2)=ΓF(|r1|, |r2|; t1, t2).
γF(r1, r2; t1, t2)=ΓF(r1, r2; t1, t2)[IF(r1, t1)IF(r2, t2)]1/2,
IF(r, t)=|V(r, t)|2=ΓF(r, r; t, t)
μF(r1, r2; ω1, ω2)=sF(r1, r2; ω1, ω2)[IF(r1, ω1)IF(r2, ω2)]1/2,
IF(r, ω)=|uF(r, ω)|2=sF(r, r; ω, ω)
γF(r1, r2; t1, t2)=γQt1-s1v, t2-s2v,
μF(r1, r2; ω1, ω2)=exp-iv(ω1s1-ω2s2)×μQ(ω1, ω2).
Q(ρ, t)=δ(ρ)η exp[-12(t/τ)2+iω0t].
μQ(ρ1, ρ2; ω1, ω2)=μQ(ω1, ω2)1,
γQ(ρ1, ρ2; t1, t2)=γQ(t1, t2)=exp[iω0(t1-t2)].
γF(r1, r2; t1, t2)=expiω0t1-t2+s2-s1v,
μF(r1, r2; ω1, ω2)=exp-iv(ω1s1-ω2s2).
|γF(r1, r2; t1, t2)|=|μF(r1, r2; ω1, ω2)|=1
foranyr1andr2.
Q(ρ, t)=δ(ρ)Q(t).
sF(r1, r2; ω1, ω2)=ρD exp[-(i/v)(ω1s1-ω2s2)]s1s2×sQ(ρ; ω1, ω2)dρ,
ΓF(r1, r2; t1, t2)=ρD 1s1s2×ΓQρ; t1-s1v, t2-s2vdρ,
s=|r1|=|r2|,
sF(r1, r2; ω1, ω2)=1s02exp-i s0v(ω1-ω2)×ρD expivω2Δs21×sQ(ρ; ω1, ω2)dρ,
ΓF(r1, r2; t1, t2)=1s02ρDΓQρ; t1-s0v, t2-s0+Δs21vdρ.
Δs21=s2-s1.
sF(r1, r2; ω, ω)=1s02ρD expivωΔs21IQ(ρ, ω)dρ,
ΓF(r1, r2; τ)=1s02ρDΓQρ, τ+Δs21vdρ,
τ=t1-t2,
sF(r1, r2; ω, ω)=IQ(ω) 1s02ρD expivωΔs21dρ,
ΓF(r1, r2; τ)=1s02ρDΓQτ+Δs21vdρ.
IF(r, ω)=ρ1D ρ2D exp[-i(ω/v)(s1-s2)]s1s2×sQ(ρ1, ρ2; ω, ω)dρ1dρ2.
IF(r, t)=ρ1D ρ2D 1s1s2ΓQρ1, ρ2;t-s1v, t-s2vdρ1dρ2.
EF(r)=-IF(r, t)dt=2π-IF(r, ω)dω,
EF(r)=2πω=- ρ1D ρ2D exp[-i(ω/v)(s1-s2)]s1s2×sQ(ρ1, ρ2; ω, ω)dωdρ1dρ2.
IF(r, ω)=1s2IQ(ω).
IQ(ω)=|uQ(ω)|2
IF(r, t)=1s2IQt-sv,
IQ(t)=|Q(t)|2
EF(r, ω)=1s2E0,
E0=2π-IQ(ω)dω.
IF(r, ω)=sF(r, r; ω, ω)=ρD 1s2IQ(ρ; ω),
IF(r, t)=ρD 1s2IQρ, t-svdρ.
EF(r)=ρD 1s2EQ(ρ)dρ,
EQ(ρ)=2π-IQ(ρ, ω)dω
IF(r, ω)=1s02ρDIQ(ρ, ω),
IF(r, t)=1s02ρDIQρ, t-svdρ,
EF(r)=1s02ρDEQ(ρ)dρ.
IF(r, ω)=1s02AQIQ(ω),
IF(r, t)=1s02AQIQt-s0v,
EF(r)=1s02AQEQ.
AQ=ρDdρ
IF(r, ω)=1s02ρ1D ρ2D exp-i ωvΔs12×sQ(ρ1, ρ2; ω, ω)dρ1dρ2.
IF(r, t)=1s02ρ1D ρ2DΓQρ1, ρ2; t-s0v, t-s0-Δs12vdρ1dρ2,
EF(r)=2πs02ω=- ρ1D ρ2D×exp-i ωvΔs12sQ(ρ1, ρ2; ω, ω)dρ1dρ2.
Δs12=s1-s2
IF(r, ω)=1s02IQ(ω)ρ1D ρ2D exp-i ωvΔs12dρ1dρ2,
IF(r, t)=1s02ρ1D ρ2DΓQt-s0v, t-s0-Δs12v×dρ1dρ2,
EF(r)=2πs02ω=-IQ(ω)×ρ1D ρ2D exp-i ωvΔs12dρ1dρ2dω.
sQ(ρ1, ρ2; ω1, ω2)=sQ(x1, x2; ω1, ω2)=uQ(x1, ω1)uQ*(x2, ω2),
uF(r, ω)=-aa exp-i ωvssuQ(x, ω)dx.
s=R0-x sin Θ,
uF(r, ω)=1R0exp-i ωvR0×-aa expi ωvx sin ΘuQ(x, ω)dx.
sF(r1, r2; ω1, ω2)=1R02exp-i ω1-ω2vR0×-aa -aaexpiv(ω1x1 sin Θ1-ω2x2 sin Θ2)×sQ(x1, x2; ω1, ω2)dx1dx2.
ΓF(r1, r2; t1, t2)=1R02-aa -aaΓQx1, x2;t1-R0-x1 sin Θ1v,t2-R0+x2 sin Θ2vdx1dx2.
sF(r1, r2; ω1, ω2)=1R02exp-i ω1-ω2vR0×-aaexpi xv(ω1 sin Θ1-ω2 sin Θ2)sQ(x; ω1, ω2)dx,
ΓF(r1, r2; t1, t2)=1R02-aaΓQ×x; t1-R0-x sin Θ1v,t2-R0+x sin Θ2vdx.
IF(r, ω)=sF(r, r; ω, ω)=1R02-aaIQ(x, ω)dx,
IF(r, t)=ΓF(r, r; t, t)=1R02-aaIQx, t-R0vdx.
ωmaxva dmaxR0=2π aλmindmaxR00
sothatωmax avsin Θ1, ωmax avsin Θ20,
sF(r1, r2; ω1, ω2)=1R02sQ(ω1, ω2)×exp-i ω1-ω2vR0×-aa -aaexpiv(ω1x1 sin Θ1-ω2x2 sin Θ2)dx1dx2=2aR02sQ(ω1, ω2)×exp-i ω1-ω2vR0×sincω1va sin Θ1×sincω2va sin Θ2,
ΓF(r1, r2; t1, t2)=1R02-aa -aaΓQ×t1-R0-x1 sin Θ1v, t2-R0+x2 sin Θ2vdx1dx2,
sF(r1, r2; ω1, ω2)2aR02sQ(ω1, ω2)×exp-i ω1-ω2vR0,
ΓF(r1, r2; t1, t2)2aR02ΓQt1-R0v, t2-R0v.

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