Abstract

Formulas are derived for determining low-pass filtered versions of the two constituent factors of the two-point spatial correlation function of a quasi-homogeneous random medium from knowledge of the cross-spectral density of the scattered field in the far zone. The results are illustrated by an example. When the scale lengths of variation of these quantities are greater than approximately half of the mean wavelength of the incident light, the reconstructions are found to be practically exact.

© 1994 Optical Society of America

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  1. A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, New York, 1988).
  2. W. Swindell, H. H. Barrett, “Computerized tomography: taking sectional x-rays,” Phys. Today 30(12), 32–41 (1977).
    [Crossref]
  3. E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
    [Crossref]
  4. A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imag. 4, 336–350 (1982).
  5. A. J. Devaney, “A computer simulation study of diffraction tomography,” IEEE Trans. Biomed. Eng. BE-30, 377–386 (1983).
    [Crossref]
  6. E. Wolf, R. P. Porter, “On the physical contents of some integral equations for inverse scattering from inhomogeneous objects,” Rad. Sci. 21, 627–634 (1986).
    [Crossref]
  7. E. Wolf, “Determination of the amplitude and the phase of scattered fields by holography,” J. Opt. Soc. Am. 60, 18–20 (1970).
    [Crossref]
  8. A. J. Devaney, “Structure determination from intensity measurements in scattering experiments,” Phys. Rev. Lett. 62, 2385–2388 (1989).
    [Crossref] [PubMed]
  9. M. H. Maleki, A. J. Devaney, A. Schatzberg, “Tomographic reconstruction from optical scattered intensities,” J. Opt. Soc. Am. A 9, 1356–1363 (1992).
    [Crossref]
  10. P. Debye, A. M. Bueche, “Scattering by an inhomogeneous solid,” J. Appl. Phys. 20, 518–525 (1949).
    [Crossref]
  11. R. A. Silverman, “Scattering of plane waves by locally homogeneous dielectric noise,” Proc. Cambr. Phil. Soc. 54, 530–537 (1958).
    [Crossref]
  12. G. Ross, “Light scattering in amorphous media: coherence of the radiation field and its effects,” Opt. Acta 16, 611–628 (1969).
    [Crossref]
  13. G. Ross, “Light scattering in amorphous media: the object wave and its coherence,” Opt. Acta 25, 57–66 (1978).
    [Crossref]
  14. W H. Carter, E. Wolf, “Scattering from quasi-homogeneous media,” Opt. Commun. 67, 85–90 (1988).
    [Crossref]
  15. E. Wolf, J. T. Foley, “Scattering of electromagnetic fields of any state of coherence from space–time fluctuations,” Phys. Rev. A 40, 579–587 (1989).
    [Crossref] [PubMed]
  16. E. Wolf, J. T. Foley, F. Gori, “Frequency shifts of spectral lines produced by scattering from spatially random media,” J. Opt. Soc. Am. A 6, 1142–1149 (1989);J. Opt. Soc. Am. A 7, 173 (1990).
    [Crossref]
  17. J. Howard, W. A. Peebles, D. L. Brower, S. K. Kim, N. C. Luhmann, “Application of diffraction tomography to plasma density measurements,” Rev. Sci. Instrum. 61, 2829–2833 (1990).
    [Crossref]
  18. D. Rouseff, R. P. Porter, “Diffraction tomography and the stochastic inverse scattering problem,” J. Acoust. Soc. Am. 89, 1599–1605 (1991).
    [Crossref]
  19. E. Wolf, “New theory of partial coherence in the space-frequency domain. Part I: Spectra and cross-spectra of steady-state sources,” J. Opt. Soc. Am. 72, 343–351 (1982);“Part II. Steady-state fields and higher-order correlations,” J. Opt. Soc. Am. A 3, 76–85 (1986).
    [Crossref]
  20. R. W. James, The Optical Principles of the Diffraction of X-Rays (Bell, London, 1948), pp. 14–15.
  21. W. H. Carter, E. Wolf, “Correlation theory of wavefields generated by fluctuating three-dimensional scalar sources. II. Radiation from isotropic model sources,” Opt. Acta 28, 245–259 (1981).
    [Crossref]
  22. M. Kowarz, “Non-interferometric reconstructions of wave-field correlations,” Phys. Rev. E (to be published).
  23. E. Wolf, “Two inverse problems in spectroscopy with partially coherent sources and the scaling law,” J. Mod. Opt. 39, 9–20 (1992).
    [Crossref]
  24. The usual form for the frequency dependence of a refractive index that is modeled as a single resonance at ω1with width n2(ω)=1+ωp2[ω12−ω2+iΓ1ω]−1, where ωp is the plasma frequency.

1992 (2)

M. H. Maleki, A. J. Devaney, A. Schatzberg, “Tomographic reconstruction from optical scattered intensities,” J. Opt. Soc. Am. A 9, 1356–1363 (1992).
[Crossref]

E. Wolf, “Two inverse problems in spectroscopy with partially coherent sources and the scaling law,” J. Mod. Opt. 39, 9–20 (1992).
[Crossref]

1991 (1)

D. Rouseff, R. P. Porter, “Diffraction tomography and the stochastic inverse scattering problem,” J. Acoust. Soc. Am. 89, 1599–1605 (1991).
[Crossref]

1990 (1)

J. Howard, W. A. Peebles, D. L. Brower, S. K. Kim, N. C. Luhmann, “Application of diffraction tomography to plasma density measurements,” Rev. Sci. Instrum. 61, 2829–2833 (1990).
[Crossref]

1989 (3)

E. Wolf, J. T. Foley, “Scattering of electromagnetic fields of any state of coherence from space–time fluctuations,” Phys. Rev. A 40, 579–587 (1989).
[Crossref] [PubMed]

E. Wolf, J. T. Foley, F. Gori, “Frequency shifts of spectral lines produced by scattering from spatially random media,” J. Opt. Soc. Am. A 6, 1142–1149 (1989);J. Opt. Soc. Am. A 7, 173 (1990).
[Crossref]

A. J. Devaney, “Structure determination from intensity measurements in scattering experiments,” Phys. Rev. Lett. 62, 2385–2388 (1989).
[Crossref] [PubMed]

1988 (1)

W H. Carter, E. Wolf, “Scattering from quasi-homogeneous media,” Opt. Commun. 67, 85–90 (1988).
[Crossref]

1986 (1)

E. Wolf, R. P. Porter, “On the physical contents of some integral equations for inverse scattering from inhomogeneous objects,” Rad. Sci. 21, 627–634 (1986).
[Crossref]

1983 (1)

A. J. Devaney, “A computer simulation study of diffraction tomography,” IEEE Trans. Biomed. Eng. BE-30, 377–386 (1983).
[Crossref]

1982 (2)

1981 (1)

W. H. Carter, E. Wolf, “Correlation theory of wavefields generated by fluctuating three-dimensional scalar sources. II. Radiation from isotropic model sources,” Opt. Acta 28, 245–259 (1981).
[Crossref]

1978 (1)

G. Ross, “Light scattering in amorphous media: the object wave and its coherence,” Opt. Acta 25, 57–66 (1978).
[Crossref]

1977 (1)

W. Swindell, H. H. Barrett, “Computerized tomography: taking sectional x-rays,” Phys. Today 30(12), 32–41 (1977).
[Crossref]

1970 (1)

1969 (2)

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[Crossref]

G. Ross, “Light scattering in amorphous media: coherence of the radiation field and its effects,” Opt. Acta 16, 611–628 (1969).
[Crossref]

1958 (1)

R. A. Silverman, “Scattering of plane waves by locally homogeneous dielectric noise,” Proc. Cambr. Phil. Soc. 54, 530–537 (1958).
[Crossref]

1949 (1)

P. Debye, A. M. Bueche, “Scattering by an inhomogeneous solid,” J. Appl. Phys. 20, 518–525 (1949).
[Crossref]

Barrett, H. H.

W. Swindell, H. H. Barrett, “Computerized tomography: taking sectional x-rays,” Phys. Today 30(12), 32–41 (1977).
[Crossref]

Brower, D. L.

J. Howard, W. A. Peebles, D. L. Brower, S. K. Kim, N. C. Luhmann, “Application of diffraction tomography to plasma density measurements,” Rev. Sci. Instrum. 61, 2829–2833 (1990).
[Crossref]

Bueche, A. M.

P. Debye, A. M. Bueche, “Scattering by an inhomogeneous solid,” J. Appl. Phys. 20, 518–525 (1949).
[Crossref]

Carter, W H.

W H. Carter, E. Wolf, “Scattering from quasi-homogeneous media,” Opt. Commun. 67, 85–90 (1988).
[Crossref]

Carter, W. H.

W. H. Carter, E. Wolf, “Correlation theory of wavefields generated by fluctuating three-dimensional scalar sources. II. Radiation from isotropic model sources,” Opt. Acta 28, 245–259 (1981).
[Crossref]

Debye, P.

P. Debye, A. M. Bueche, “Scattering by an inhomogeneous solid,” J. Appl. Phys. 20, 518–525 (1949).
[Crossref]

Devaney, A. J.

M. H. Maleki, A. J. Devaney, A. Schatzberg, “Tomographic reconstruction from optical scattered intensities,” J. Opt. Soc. Am. A 9, 1356–1363 (1992).
[Crossref]

A. J. Devaney, “Structure determination from intensity measurements in scattering experiments,” Phys. Rev. Lett. 62, 2385–2388 (1989).
[Crossref] [PubMed]

A. J. Devaney, “A computer simulation study of diffraction tomography,” IEEE Trans. Biomed. Eng. BE-30, 377–386 (1983).
[Crossref]

A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imag. 4, 336–350 (1982).

Foley, J. T.

E. Wolf, J. T. Foley, “Scattering of electromagnetic fields of any state of coherence from space–time fluctuations,” Phys. Rev. A 40, 579–587 (1989).
[Crossref] [PubMed]

E. Wolf, J. T. Foley, F. Gori, “Frequency shifts of spectral lines produced by scattering from spatially random media,” J. Opt. Soc. Am. A 6, 1142–1149 (1989);J. Opt. Soc. Am. A 7, 173 (1990).
[Crossref]

Gori, F.

Howard, J.

J. Howard, W. A. Peebles, D. L. Brower, S. K. Kim, N. C. Luhmann, “Application of diffraction tomography to plasma density measurements,” Rev. Sci. Instrum. 61, 2829–2833 (1990).
[Crossref]

James, R. W.

R. W. James, The Optical Principles of the Diffraction of X-Rays (Bell, London, 1948), pp. 14–15.

Kak, A. C.

A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, New York, 1988).

Kim, S. K.

J. Howard, W. A. Peebles, D. L. Brower, S. K. Kim, N. C. Luhmann, “Application of diffraction tomography to plasma density measurements,” Rev. Sci. Instrum. 61, 2829–2833 (1990).
[Crossref]

Kowarz, M.

M. Kowarz, “Non-interferometric reconstructions of wave-field correlations,” Phys. Rev. E (to be published).

Luhmann, N. C.

J. Howard, W. A. Peebles, D. L. Brower, S. K. Kim, N. C. Luhmann, “Application of diffraction tomography to plasma density measurements,” Rev. Sci. Instrum. 61, 2829–2833 (1990).
[Crossref]

Maleki, M. H.

Peebles, W. A.

J. Howard, W. A. Peebles, D. L. Brower, S. K. Kim, N. C. Luhmann, “Application of diffraction tomography to plasma density measurements,” Rev. Sci. Instrum. 61, 2829–2833 (1990).
[Crossref]

Porter, R. P.

D. Rouseff, R. P. Porter, “Diffraction tomography and the stochastic inverse scattering problem,” J. Acoust. Soc. Am. 89, 1599–1605 (1991).
[Crossref]

E. Wolf, R. P. Porter, “On the physical contents of some integral equations for inverse scattering from inhomogeneous objects,” Rad. Sci. 21, 627–634 (1986).
[Crossref]

Ross, G.

G. Ross, “Light scattering in amorphous media: the object wave and its coherence,” Opt. Acta 25, 57–66 (1978).
[Crossref]

G. Ross, “Light scattering in amorphous media: coherence of the radiation field and its effects,” Opt. Acta 16, 611–628 (1969).
[Crossref]

Rouseff, D.

D. Rouseff, R. P. Porter, “Diffraction tomography and the stochastic inverse scattering problem,” J. Acoust. Soc. Am. 89, 1599–1605 (1991).
[Crossref]

Schatzberg, A.

Silverman, R. A.

R. A. Silverman, “Scattering of plane waves by locally homogeneous dielectric noise,” Proc. Cambr. Phil. Soc. 54, 530–537 (1958).
[Crossref]

Slaney, M.

A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, New York, 1988).

Swindell, W.

W. Swindell, H. H. Barrett, “Computerized tomography: taking sectional x-rays,” Phys. Today 30(12), 32–41 (1977).
[Crossref]

Wolf, E.

E. Wolf, “Two inverse problems in spectroscopy with partially coherent sources and the scaling law,” J. Mod. Opt. 39, 9–20 (1992).
[Crossref]

E. Wolf, J. T. Foley, F. Gori, “Frequency shifts of spectral lines produced by scattering from spatially random media,” J. Opt. Soc. Am. A 6, 1142–1149 (1989);J. Opt. Soc. Am. A 7, 173 (1990).
[Crossref]

E. Wolf, J. T. Foley, “Scattering of electromagnetic fields of any state of coherence from space–time fluctuations,” Phys. Rev. A 40, 579–587 (1989).
[Crossref] [PubMed]

W H. Carter, E. Wolf, “Scattering from quasi-homogeneous media,” Opt. Commun. 67, 85–90 (1988).
[Crossref]

E. Wolf, R. P. Porter, “On the physical contents of some integral equations for inverse scattering from inhomogeneous objects,” Rad. Sci. 21, 627–634 (1986).
[Crossref]

E. Wolf, “New theory of partial coherence in the space-frequency domain. Part I: Spectra and cross-spectra of steady-state sources,” J. Opt. Soc. Am. 72, 343–351 (1982);“Part II. Steady-state fields and higher-order correlations,” J. Opt. Soc. Am. A 3, 76–85 (1986).
[Crossref]

W. H. Carter, E. Wolf, “Correlation theory of wavefields generated by fluctuating three-dimensional scalar sources. II. Radiation from isotropic model sources,” Opt. Acta 28, 245–259 (1981).
[Crossref]

E. Wolf, “Determination of the amplitude and the phase of scattered fields by holography,” J. Opt. Soc. Am. 60, 18–20 (1970).
[Crossref]

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[Crossref]

IEEE Trans. Biomed. Eng. (1)

A. J. Devaney, “A computer simulation study of diffraction tomography,” IEEE Trans. Biomed. Eng. BE-30, 377–386 (1983).
[Crossref]

J. Acoust. Soc. Am. (1)

D. Rouseff, R. P. Porter, “Diffraction tomography and the stochastic inverse scattering problem,” J. Acoust. Soc. Am. 89, 1599–1605 (1991).
[Crossref]

J. Appl. Phys. (1)

P. Debye, A. M. Bueche, “Scattering by an inhomogeneous solid,” J. Appl. Phys. 20, 518–525 (1949).
[Crossref]

J. Mod. Opt. (1)

E. Wolf, “Two inverse problems in spectroscopy with partially coherent sources and the scaling law,” J. Mod. Opt. 39, 9–20 (1992).
[Crossref]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (2)

Opt. Acta (3)

W. H. Carter, E. Wolf, “Correlation theory of wavefields generated by fluctuating three-dimensional scalar sources. II. Radiation from isotropic model sources,” Opt. Acta 28, 245–259 (1981).
[Crossref]

G. Ross, “Light scattering in amorphous media: coherence of the radiation field and its effects,” Opt. Acta 16, 611–628 (1969).
[Crossref]

G. Ross, “Light scattering in amorphous media: the object wave and its coherence,” Opt. Acta 25, 57–66 (1978).
[Crossref]

Opt. Commun. (2)

W H. Carter, E. Wolf, “Scattering from quasi-homogeneous media,” Opt. Commun. 67, 85–90 (1988).
[Crossref]

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[Crossref]

Phys. Rev. A (1)

E. Wolf, J. T. Foley, “Scattering of electromagnetic fields of any state of coherence from space–time fluctuations,” Phys. Rev. A 40, 579–587 (1989).
[Crossref] [PubMed]

Phys. Rev. Lett. (1)

A. J. Devaney, “Structure determination from intensity measurements in scattering experiments,” Phys. Rev. Lett. 62, 2385–2388 (1989).
[Crossref] [PubMed]

Phys. Today (1)

W. Swindell, H. H. Barrett, “Computerized tomography: taking sectional x-rays,” Phys. Today 30(12), 32–41 (1977).
[Crossref]

Proc. Cambr. Phil. Soc. (1)

R. A. Silverman, “Scattering of plane waves by locally homogeneous dielectric noise,” Proc. Cambr. Phil. Soc. 54, 530–537 (1958).
[Crossref]

Rad. Sci. (1)

E. Wolf, R. P. Porter, “On the physical contents of some integral equations for inverse scattering from inhomogeneous objects,” Rad. Sci. 21, 627–634 (1986).
[Crossref]

Rev. Sci. Instrum. (1)

J. Howard, W. A. Peebles, D. L. Brower, S. K. Kim, N. C. Luhmann, “Application of diffraction tomography to plasma density measurements,” Rev. Sci. Instrum. 61, 2829–2833 (1990).
[Crossref]

Ultrason. Imag. (1)

A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imag. 4, 336–350 (1982).

Other (4)

A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, New York, 1988).

M. Kowarz, “Non-interferometric reconstructions of wave-field correlations,” Phys. Rev. E (to be published).

R. W. James, The Optical Principles of the Diffraction of X-Rays (Bell, London, 1948), pp. 14–15.

The usual form for the frequency dependence of a refractive index that is modeled as a single resonance at ω1with width n2(ω)=1+ωp2[ω12−ω2+iΓ1ω]−1, where ωp is the plasma frequency.

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Figures (6)

Fig. 1
Fig. 1

Illustrating notation relating to measurement of the scattered field in the far zone. s0 and s are unit vectors in the directions of incidence and scattering, respectively.

Fig. 2
Fig. 2

Height factor H(θ), calculated from Eq. (4.20). The spectrum of the incident field was taken to be Gaussian [cf. Eq. (4.1)] with center frequency ω0 = 1×1010 Hz and width Γ0 = ω0/5 (S0 = 1). The degree of spatial correlation was assumed to have a Gaussian profile [Eq. (4.6)], with σg = 0.5λ0. The frequency-dependent factor iF (ω) was taken to have the form of Eq. (4.5), with center frequency ω1 = 1010 Hz and width Γ1 = ω1/10.

Fig. 3
Fig. 3

Normalized far-zone spectrum of the scattered field s U ( s ) ( r s , s 0 ; ω ) = S U ( s ) ( r s , s 0 ; ω ) / 0 S U ( s ) ( r s , s 0 ; ω ) d ω ,in units of [Γ0]−1, generated by the scattering of a plane wave on a quasi-homogeneous medium, for several values of the scattering angle θ, under the same circumstances as relating to Fig. 2.

Fig. 4
Fig. 4

Reconstructed degree of spatial correlation [gF(r′)]rec of the scattering potential, calculated from far-zone spectra by the use of formulas (3.8) and (3.9). The parameters are the same as those for Fig. 2, except that the width σg of the degree of spatial correlation is varied.

Fig. 5
Fig. 5

Reconstructed frequency-dependent factor [iF(ω)]rec in units of [Γ1]−1 calculated from far-zone spectra by the use of Eq. (3.34). The parameters are the same as those for Fig. 2, except that the width σg of the degree of spatial correlation is varied.

Fig. 6
Fig. 6

Reconstructed relative space-dependent factor [ N ¯ ( r ) ] rec in units of [ σ I 3 ] 1, calculated from the far-zone spectral degree of coherence by the use of Eq. (3.38). The space-dependent factor N(r) was assumed to have a Gaussian profile [Eq. (4.3)] of width σI. The remaining parameters are the same as those for Fig. 2.

Equations (80)

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U ( i ) ( r , ω ) = a ( ω ) exp ( i k s 0 · r ) ,
k = ω / c
U ( s ) ( r s , s 0 ; ω ) = a ( ω ) exp ( ikr ) r D F ( r , ω ) × exp [ i k ( s s 0 ) · r ] d 3 r ,
F ( r , ω ) = k 2 4 π [ n 2 ( r , ω ) 1 ]
W U ( s ) ( r s 1 , r s 2 ; s 0 ; ω ) = U ( s ) * ( r s 1 , s 0 ; ω ) U ( s ) ( r s 2 , s 0 ; ω ) ,
S U ( s ) ( r s , s 0 ; ω ) W U ( s ) ( r s , r s , s 0 ; ω ) = U ( s ) * ( r s , s 0 ; ω ) U ( s ) ( r s , s 0 ; ω ) .
C F ( r 1 , r 2 , ω ) = F * ( r 1 , ω ) F ( r 2 , ω ) n ,
μ F ( r 1 , r 2 , ω ) = C F ( r 1 , r 2 , ω ) [ C F ( r 1 , r 1 , ω ) ] 1 / 2 [ C F ( r 2 , r 2 , ω ) ] 1 / 2
I F ( r , ω ) = F * ( r , ω ) F ( r , ω ) n .
μ F ( r 1 , r 2 , ω ) = C F ( r 1 , r 2 , ω ) [ I F ( r 1 , ω ) ] 1 / 2 [ I F ( r 2 , ω ) ] 1 / 2 ,
C F ( r 1 , r 2 , ω ) = [ I F ( r 1 , ω ) ] 1 / 2 [ I F ( r 2 , ω ) ] 1 / 2 μ F ( r 1 , r 2 , ω ) .
μ F ( r 1 , r 2 , ω ) = g F ( r 2 r 1 , ω ) .
C F ( r 1 , r 2 , ω ) I F [ ( r 1 + r 2 ) / 2 , ω ] g F ( r 2 r 1 , ω ) .
I F ( r 1 , ω ) I F ( r 2 , ω ) I F [ ( r 1 + r 2 ) / 2 , ω ] ,
W U ( s ) ( r s 1 , r s 2 ; s 0 ; ω ) = 1 r 2 S ( i ) ( ω ) I F [ k ( s 2 s 1 ) , ω ] × g F { k [ ( s 1 + s 2 ) / 2 s 0 ] , ω } ,
I F ( K , ω ) = D I F ( r , ω ) exp ( i K · r ) d 3 r ,
g F ( K , ω ) = D g F ( r , ω ) exp ( i K · r ) d 3 r
S ( i ) ( ω ) = a * ( ω ) a ( ω )
S U ( s ) ( r s , s 0 ; ω ) = 1 r 2 S ( i ) ( ω ) I F ( 0 , ω ) g F [ k ( s s 0 ) , ω ] .
M S ( s s 0 , ω ) = r 2 S U ( s ) ( r s , s 0 ; ω ) .
M S ( s s 0 , ω ) = S ( i ) ( ω ) I F ( 0 , ω ) g F [ k ( s s 0 ) , ω ] .
ξ = s s 0
[ g F ( r , ω ) ] LP = ( k 2 π ) 3 1 S ( i ) ( ω ) I F ( 0 , ω ) × | ξ | 2 M S ( ξ , ω ) exp ( i k ξ · r ) d 3 ξ ,
[ g F ( r , ω ) ] LP = 1 ( 2 π ) 3 | ξ | 2 g F ( k ξ , ω ) exp ( i k ξ · r ) d 3 ( k ξ ) .
1 ( k 2 π ) 3 1 S ( i ) ( ω ) I F ( 0 , ω ) | ξ | 2 M S ( ξ , ω ) d 3 ξ ,
I F ( 0 , ω ) ( k 2 π ) 3 1 S ( i ) ( ω ) | ξ | 2 M S ( ξ , ω ) d 3 ξ .
[ g F ( r , ω ) ] LP | ξ | 2 M S ( ξ , ω ) exp ( i k ξ · r ) d 3 ξ | ξ | 2 M S ( ξ , ω ) d 3 ξ .
[ g F ( r , ω ) ] rec [ g F ( r , ω ) ] LP .
μ U ( s ) ( r s 1 , r s 2 ; s 0 ; ω ) = W U ( s ) ( r s 1 , r s 2 ; s 0 ; ω ) [ W U ( s ) ( r s 1 , r s 1 ; s 0 ; ω ) ] 1 / 2 [ W U ( s ) ( r s 2 , r s 2 ; s 0 ; ω ) ] 1 / 2 .
μ U ( s ) ( r s 1 , r s 2 ; s 0 ; ω ) = I F [ k ( s 2 s 1 ) , ω ] I F ( 0 , ω ) × G F ( k s 1 , k s 2 ; k s 0 ; ω ) ,
G F ( k s 1 , k s 2 ; k s 0 ; ω ) = g F { k [ ( s 1 + s 2 ) / 2 s 0 ] , ω } { g F [ k ( s 1 s 0 ) , ω ] } 1 / 2 { g F [ k ( s 2 s 0 ) , ω ] } 1 / 2 .
G F ( k s 1 , k s 2 ; k s 0 ; ω ) G F [ k ( s 1 + s 2 ) / 2 , k ( s 1 + s 2 ) / 2 ; k s 0 ; ω ] = 1 ,
μ U ( s ) ( r s 1 , r s 2 ; s 0 ; ω ) I F [ k ( s 2 s 1 ) , ω ] I F ( 0 , ω ) .
M μ ( s 2 s 1 , ω ) = μ U ( s ) ( r s 1 , r s 2 ; s 0 ; ω ) .
M μ ( s 2 s 1 , ω ) = I F [ k ( s 2 s 1 ) , ω ] I F ( 0 , ω ) ,
M μ ( η , ω ) = I F ( k η , ω ) I F ( 0 , ω ) .
[ I F ( r , ω ) ] LP = ( k 2 π ) 3 I F ( 0 , ω ) × | η | 2 M μ ( η , ω ) exp ( i k η · r ) d 3 η ,
[ I F ( r , ω ) ] LP = 1 ( 2 π ) 3 | η | 2 I F ( k η , ω ) exp ( i k η · r ) d 3 η .
[ I F ( r , ω ) ] LP ( k 2 π ) 6 1 S ( i ) ( ω ) | ξ | 2 M S ( ξ , ω ) d 3 ξ × | η | 2 M μ ( η , ω ) exp ( i k η · r ) d 3 η .
[ I F ( r , ω ) ] rec [ I F ( r , ω ) ] LP .
I F ( r , ω ) = N ( r ) i F ( ω ) ,
0 S ( i ) ( ω ) i F ( ω ) d ω = 1 .
N ( r ) = 0 S ( i ) ( ω ) I F ( r , ω ) d ω .
( 2 + k 2 ) U ( s ) ( r , ω ) = 4 π F ( r , ω ) U ( i ) ( r , ω ) .
S sec ( 0 ) ( r , ω ) ρ sec * ( r , ω ) ρ sec ( r , ω ) = S U ( i ) ( ω ) I F ( r , ω ) .
s sec ( 0 ) ( r , ω ) S U ( i ) ( ω ) I F ( r , ω ) 0 S U ( i ) ( ω ) I F ( r , ω ) d ω
s sec ( 0 ) ( r , ω ) = S U ( i ) ( ω ) i F ( ω ) .
I F ( K , ω ) = N ( K ) i F ( ω ) ,
N ( K ) = D N ( r ) exp ( i K · r ) d 3 r .
i F ( ω ) ( k 2 π ) 3 1 N ( 0 ) S ( i ) ( ω ) | ξ | 2 M S ( ξ , ω ) d 3 ξ .
1 ( 2 π ) 3 N ( 0 ) 0 d ω k 3 | ξ | 2 M S ( ξ , ω ) d 3 ξ 1 ,
N ( 0 ) 1 ( 2 π ) 3 0 d ω k 3 | ξ | 2 M S ( ξ , ω ) d 3 ξ .
[ i F ( ω ) ] rec = ω 3 S ( i ) ( ω ) | ξ | 2 M S ( ξ , ω ) d 3 ξ 0 d ω ω 3 | ξ | 2 M S ( ξ , ω ) d 3 ξ .
[ N ( r ) ] LP = ( k 2 π ) 3 N ( 0 ) | η | 2 M μ ( η , ω ) exp ( i k η · r ) d 3 η .
[ N ( r ) ] LP = k 3 ( 2 π ) 6 0 d ω k 3 | ξ | 2 M S ( ξ , ω ) d 3 ξ × | η | 2 M μ ( η , ω ) exp ( i k η · r ) d 3 η .
N ¯ ( r ) N ( r ) D N ( r ) d 3 r = N ( r ) N ( 0 ) ,
[ N ¯ ( r ) ] LP = ( k 2 π ) 3 | η | 2 M μ ( η , ω ) exp ( i k η · r ) d 3 η .
S ( i ) ( ω ) = S 0 exp [ ( ω ω 0 ) 2 / 2 Γ 0 2 ] ,
I F ( r , ω ) = I 0 exp ( | r | 2 / 2 σ I 2 ) ω 4 ( ω 1 2 ω 2 ) 2 + Γ 1 2 ω 2 ,
N ( r ) = C I 0 exp ( | r | 2 / 2 σ I 2 ) ,
C = S 0 0 exp [ ( ω ω 0 ) 2 / 2 Γ 0 2 ] ω 4 ( ω 1 2 ω 2 ) 2 + Γ 1 2 ω 2 d ω
i F ( ω ) = 1 C ω 4 ( ω 1 2 ω 2 ) 2 + Γ 1 2 ω 2 .
g F ( r ) = exp ( | r | 2 / 2 σ g 2 ) ,
σ I σ g .
I F ( 0 , ω ) = I 0 ( 2 π σ I 2 ) 3 / 2 ω 4 ( ω 1 2 ω 2 ) 2 + Γ 1 2 ω 2 ,
g F ( K ) = ( 2 π σ g 2 ) 3 / 2 exp ( σ g 2 | K | 2 / 2 ) .
S U ( s ) ( r s , s 0 ; ω ) 1 r 2 M S ( s s 0 , ω ) ,
M S ( s s 0 , ω ) = A exp [ ( ω ω 0 ) 2 / 2 Γ 0 2 ] × ω 4 ( ω 1 2 ω 2 ) 2 + Γ 1 2 ω 2 × exp ( σ g 2 k 2 | s s 0 | 2 / 2 ,
A = S 0 I 0 ( 2 π σ I σ g ) 3 .
| s s 0 | = 2 sin ( θ / 2 ) ,
M S ( s s 0 , ω ) = A exp [ ( ω ω 0 ) 2 / 2 Γ 0 2 ] × ω 4 ( ω 1 2 ω 2 ) 2 + Γ 1 2 ω 2 × exp [ 2 σ g 2 k 2 sin ( θ / 2 ) ] .
exp [ ( ω ω 0 ) 2 / 2 Γ 0 2 ] exp [ 2 σ g 2 k 2 sin 2 ( θ / 2 ) ] = exp { ω 0 2 / 2 [ Γ 0 2 + α 2 ( θ ) ] } × exp { [ ω ω ( θ ) ] 2 / 2 Γ 2 ( θ ) } ,
ω ( θ ) = α 2 ( θ ) Γ 0 2 + α 2 ( θ ) ω 0 ,
1 Γ 2 ( θ ) = 1 Γ 0 2 + 1 α 2 ( θ ) ,
α ( θ ) = c 2 σ g sin ( θ / 2 ) .
S U ( s ) ( r s , s 0 ; ω ) = A r 2 H ( θ ) exp { [ ω ω ( θ ) ] 2 / 2 Γ 2 ( θ ) } × ω 4 ( ω 1 2 ω 2 ) 2 + Γ 1 2 ω 2 ,
H ( θ ) = exp { ( ω 0 2 / 2 [ Γ 0 2 + α 2 ( θ ) ] } .
I F ( K , ω ) = I 0 ( 2 π σ I 2 ) 3 / 2 ω 4 ( ω 1 2 ω 2 ) 2 + Γ 1 2 ω 2 exp ( σ I 2 | K | 2 / 2 ) .
μ U ( s ) ( r s 1 , r s 2 ; s 0 ; ω ) = exp ( σ I 2 k 2 | s 2 s 1 | 2 / 2 ) .
s U ( s ) ( r s , s 0 ; ω ) = S U ( s ) ( r s , s 0 ; ω ) / 0 S U ( s ) ( r s , s 0 ; ω ) d ω ,

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