Abstract

In Part I of this investigation [ E. Wolf, J. Opt. Soc. Am. 72. 343 ( 1982)] new representations were introduced for the cross-spectral density of a steady-state source of any state of coherence. The central concept in that formulation was the notion of a coherent source mode (a natural mode of oscillation). In the present paper the theory is developed further and new representations are obtained for the cross-spectral densities of all orders, both of the source and of the field that the source generates. These representations involve only the previously introduced coherent source modes and the moments of certain random coefficients that characterize the statistical properties of the source. The results provide a new mathematical framework for analyzing coherence properties of all orders of stationary sources and of stationary fields. Some potential applications of the theory are mentioned.

© 1986 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. 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). There is a misprint in Eq. (5.5) of this reference. The factor [λn(ω)]2/3 should be replaced by [λn(ω)]1/2.
    [CrossRef]
  2. E. Wolf, “A new description of second-order coherence phenomena in the space-frequency domain,”AIP Conf. Proc. 65, 42–48 (1981).
    [CrossRef]
  3. E. Wolf, “New spectral representation of random sources and of the partially coherent fields that they generate,” Opt. Commun. 38, 3–6 (1981).
    [CrossRef]
  4. A. Starikov, E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and of their radiation fields,”J. Opt. Soc. Am. 72, 923–928 (1982).
    [CrossRef]
  5. A. Starikov, “Effective number of degrees of freedom of partially coherent sources,”J. Opt. Soc. Am. 72, 1538–1544 (1982).
    [CrossRef]
  6. F. Gori, “Mode propagation of the field generated by Collett–Wolf Schell-model sources,” Opt. Commun. 46, 149–154 (1983).
    [CrossRef]
  7. F. Gori, R. Grella, “Shape invariant propagation of polychromatic fields,” Opt. Commun. 49, 173–177 (1984).
    [CrossRef]
  8. E. Wolf, G. S. Agarwal, “Coherence theory of laser resonator modes,” J. Opt. Soc. Am. A 1, 541–546 (1984).
    [CrossRef]
  9. R. Martinez-Herrero, P. M. Mejias, “Radiometric definitions for partially coherent sources,” J. Opt. Soc. Am. A 1, 556–558 (1984); J. T. Foley, M. Nieto-Vesperinas, “Radiance functions that depend nonlinearly on the cross-spectral density,” J. Opt. Soc. Am. A 2, 1446–1447 (1985).
    [CrossRef]
  10. I. J. LaHaie, “Inverse source problem for three-dimensional partially coherent sources and fields,” J. Opt. Soc. Am. A 2, 35–45 (1985).
    [CrossRef]
  11. L. Mandel, E. Wolf, “Spectral coherence and the concept of cross-spectral purity,”J. Opt. Soc. Am. 66, 529–535 (1976), Sec. II. A relationship between the degree of spectral coherence μ(r1, r2, ω) and the more familiar complex degree of coherence γ(r1, r2, τ) in the space–time domain, as well as a method for measuring μ(r1, r2, ω), is discussed in E. Wolf, “Young’s interference fringes with narrow-band light,” Opt. Lett. 8, 250–252 (1983).
    [CrossRef] [PubMed]
  12. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Sec. 10.3.1.
  13. L. Mandel, E. Wolf, “Coherence properties of optical fields,” Rev. Mod. Phys. 37, 231–287 (1965), Sec. 3.1.
    [CrossRef]
  14. C. H. Papas, Theory of Electromagnetic Wave Propagation (McGraw-Hill, New York, 1965), Sec. 2.1.
  15. If Q(r, t) is the complex analytic signal representation (Ref. 10, Sec. 10.2) of a real source distribution, V(r, t) will also be an analytic signal. Consequently ΓQ(r1, r2, τ) and ΓV(r1, r2, τ) will not contain any negative frequency components (Ref. 12, Sec. 10.3.2), and the lower limits in the integrals in Eqs. (3.17) and (3.18) can then be replaced by zero.
  16. Rayleigh, The Theory of Sound (reprinted by Dover, New York, 1945), Vol. II, Sec. 278 [with a modification appropriate to time dependence exp(−iωt) used in the present paper].
  17. Field correlations of arbitrary order appear to have been discussed first in the framework of the classical theory of stationary fields by L. Mandel in Ref. 18. See also E. Wolf, “Basic concepts of optical coherence theory,” in Proceedings of the Symposium on Optical Masers, J. Fox, ed. (Wiley, New York, 1963), pp. 29–42.
  18. L. Mandel, “Some coherence properties of non-Gaussian light,” in Quantum Electronics III, P. Grivet, N. Bloembergen, eds. (Columbia U. Press, New York, 1964), Vol. I, pp. 101–109.
  19. It is implicitly assumed here that M> 1. When M= 1 one readily finds that the first product term (with the index j) in Eq. (4.10), and in all subsequent formulas where it appears, must be replaced by unity.
  20. Cross-spectral density functions of arbitrary orders in classical wave fields were first considered by L. Mandel in Ref. 18. See also Refs. 21–23.
  21. E. Wolf, “Light fluctuations as a new spectroscopic tool,” Jpn. J. Appl. Phys. Suppl. 1 4, 1–14, (1965).
  22. C. L. Mehta, L. Mandel, “Some properties of higher order coherence functions,” in Electromagnetic Wave Theory, J. Brown, ed. (Pergamon, New York, 1967), Part 2, pp. 1069–1075.
  23. L. Mandel, “Photoelectric correlations and fourth-order coherence properties of optical fields,”AIP Conf. Proc. 65, 178–195 (1981).
    [CrossRef]
  24. Similar remarks apply here to those made in note 19 in connection with Eq. (4.10).
  25. A preliminary account of this application of our theory was presented at the 1984 Annual Meeting of the Optical Society of America [J. Opt. Soc. Am. A 1, 1311 (A) (1984)].
  26. Equation (A13) was obtained previously by a somewhat less rigorous argument by W. H. Carter, E. Wolf in “Correlation theory of wave-fields generated by fluctuating, three-dimensional, primary, scalar sources. I: General theory,” Opt. Acta 28, 227–244 (1981), Sec. 2.
    [CrossRef]

1985 (1)

1984 (4)

1983 (1)

F. Gori, “Mode propagation of the field generated by Collett–Wolf Schell-model sources,” Opt. Commun. 46, 149–154 (1983).
[CrossRef]

1982 (3)

1981 (4)

E. Wolf, “A new description of second-order coherence phenomena in the space-frequency domain,”AIP Conf. Proc. 65, 42–48 (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, “Photoelectric correlations and fourth-order coherence properties of optical fields,”AIP Conf. Proc. 65, 178–195 (1981).
[CrossRef]

Equation (A13) was obtained previously by a somewhat less rigorous argument by W. H. Carter, E. Wolf in “Correlation theory of wave-fields generated by fluctuating, three-dimensional, primary, scalar sources. I: General theory,” Opt. Acta 28, 227–244 (1981), Sec. 2.
[CrossRef]

1976 (1)

1965 (2)

L. Mandel, E. Wolf, “Coherence properties of optical fields,” Rev. Mod. Phys. 37, 231–287 (1965), Sec. 3.1.
[CrossRef]

E. Wolf, “Light fluctuations as a new spectroscopic tool,” Jpn. J. Appl. Phys. Suppl. 1 4, 1–14, (1965).

Agarwal, G. S.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Sec. 10.3.1.

Carter, W. H.

Equation (A13) was obtained previously by a somewhat less rigorous argument by W. H. Carter, E. Wolf in “Correlation theory of wave-fields generated by fluctuating, three-dimensional, primary, scalar sources. I: General theory,” Opt. Acta 28, 227–244 (1981), Sec. 2.
[CrossRef]

Gori, F.

F. Gori, R. Grella, “Shape invariant propagation of polychromatic fields,” Opt. Commun. 49, 173–177 (1984).
[CrossRef]

F. Gori, “Mode propagation of the field generated by Collett–Wolf Schell-model sources,” Opt. Commun. 46, 149–154 (1983).
[CrossRef]

Grella, R.

F. Gori, R. Grella, “Shape invariant propagation of polychromatic fields,” Opt. Commun. 49, 173–177 (1984).
[CrossRef]

LaHaie, I. J.

Mandel, L.

L. Mandel, “Photoelectric correlations and fourth-order coherence properties of optical fields,”AIP Conf. Proc. 65, 178–195 (1981).
[CrossRef]

L. Mandel, E. Wolf, “Spectral coherence and the concept of cross-spectral purity,”J. Opt. Soc. Am. 66, 529–535 (1976), Sec. II. A relationship between the degree of spectral coherence μ(r1, r2, ω) and the more familiar complex degree of coherence γ(r1, r2, τ) in the space–time domain, as well as a method for measuring μ(r1, r2, ω), is discussed in E. Wolf, “Young’s interference fringes with narrow-band light,” Opt. Lett. 8, 250–252 (1983).
[CrossRef] [PubMed]

L. Mandel, E. Wolf, “Coherence properties of optical fields,” Rev. Mod. Phys. 37, 231–287 (1965), Sec. 3.1.
[CrossRef]

C. L. Mehta, L. Mandel, “Some properties of higher order coherence functions,” in Electromagnetic Wave Theory, J. Brown, ed. (Pergamon, New York, 1967), Part 2, pp. 1069–1075.

L. Mandel, “Some coherence properties of non-Gaussian light,” in Quantum Electronics III, P. Grivet, N. Bloembergen, eds. (Columbia U. Press, New York, 1964), Vol. I, pp. 101–109.

Martinez-Herrero, R.

Mehta, C. L.

C. L. Mehta, L. Mandel, “Some properties of higher order coherence functions,” in Electromagnetic Wave Theory, J. Brown, ed. (Pergamon, New York, 1967), Part 2, pp. 1069–1075.

Mejias, P. M.

Papas, C. H.

C. H. Papas, Theory of Electromagnetic Wave Propagation (McGraw-Hill, New York, 1965), Sec. 2.1.

Rayleigh,

Rayleigh, The Theory of Sound (reprinted by Dover, New York, 1945), Vol. II, Sec. 278 [with a modification appropriate to time dependence exp(−iωt) used in the present paper].

Starikov, A.

Wolf, E.

E. Wolf, G. S. Agarwal, “Coherence theory of laser resonator modes,” J. Opt. Soc. Am. A 1, 541–546 (1984).
[CrossRef]

A. Starikov, E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and of their radiation fields,”J. Opt. Soc. Am. 72, 923–928 (1982).
[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). There is a misprint in Eq. (5.5) of this reference. The factor [λn(ω)]2/3 should be replaced by [λn(ω)]1/2.
[CrossRef]

E. Wolf, “A new description of second-order coherence phenomena in the space-frequency domain,”AIP Conf. Proc. 65, 42–48 (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]

Equation (A13) was obtained previously by a somewhat less rigorous argument by W. H. Carter, E. Wolf in “Correlation theory of wave-fields generated by fluctuating, three-dimensional, primary, scalar sources. I: General theory,” Opt. Acta 28, 227–244 (1981), Sec. 2.
[CrossRef]

L. Mandel, E. Wolf, “Spectral coherence and the concept of cross-spectral purity,”J. Opt. Soc. Am. 66, 529–535 (1976), Sec. II. A relationship between the degree of spectral coherence μ(r1, r2, ω) and the more familiar complex degree of coherence γ(r1, r2, τ) in the space–time domain, as well as a method for measuring μ(r1, r2, ω), is discussed in E. Wolf, “Young’s interference fringes with narrow-band light,” Opt. Lett. 8, 250–252 (1983).
[CrossRef] [PubMed]

L. Mandel, E. Wolf, “Coherence properties of optical fields,” Rev. Mod. Phys. 37, 231–287 (1965), Sec. 3.1.
[CrossRef]

E. Wolf, “Light fluctuations as a new spectroscopic tool,” Jpn. J. Appl. Phys. Suppl. 1 4, 1–14, (1965).

Field correlations of arbitrary order appear to have been discussed first in the framework of the classical theory of stationary fields by L. Mandel in Ref. 18. See also E. Wolf, “Basic concepts of optical coherence theory,” in Proceedings of the Symposium on Optical Masers, J. Fox, ed. (Wiley, New York, 1963), pp. 29–42.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Sec. 10.3.1.

AIP Conf. Proc. (2)

E. Wolf, “A new description of second-order coherence phenomena in the space-frequency domain,”AIP Conf. Proc. 65, 42–48 (1981).
[CrossRef]

L. Mandel, “Photoelectric correlations and fourth-order coherence properties of optical fields,”AIP Conf. Proc. 65, 178–195 (1981).
[CrossRef]

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (4)

Jpn. J. Appl. Phys. Suppl. 1 (1)

E. Wolf, “Light fluctuations as a new spectroscopic tool,” Jpn. J. Appl. Phys. Suppl. 1 4, 1–14, (1965).

Opt. Acta (1)

Equation (A13) was obtained previously by a somewhat less rigorous argument by W. H. Carter, E. Wolf in “Correlation theory of wave-fields generated by fluctuating, three-dimensional, primary, scalar sources. I: General theory,” Opt. Acta 28, 227–244 (1981), Sec. 2.
[CrossRef]

Opt. Commun. (3)

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

F. Gori, “Mode propagation of the field generated by Collett–Wolf Schell-model sources,” Opt. Commun. 46, 149–154 (1983).
[CrossRef]

F. Gori, R. Grella, “Shape invariant propagation of polychromatic fields,” Opt. Commun. 49, 173–177 (1984).
[CrossRef]

Rev. Mod. Phys. (1)

L. Mandel, E. Wolf, “Coherence properties of optical fields,” Rev. Mod. Phys. 37, 231–287 (1965), Sec. 3.1.
[CrossRef]

Other (10)

C. H. Papas, Theory of Electromagnetic Wave Propagation (McGraw-Hill, New York, 1965), Sec. 2.1.

If Q(r, t) is the complex analytic signal representation (Ref. 10, Sec. 10.2) of a real source distribution, V(r, t) will also be an analytic signal. Consequently ΓQ(r1, r2, τ) and ΓV(r1, r2, τ) will not contain any negative frequency components (Ref. 12, Sec. 10.3.2), and the lower limits in the integrals in Eqs. (3.17) and (3.18) can then be replaced by zero.

Rayleigh, The Theory of Sound (reprinted by Dover, New York, 1945), Vol. II, Sec. 278 [with a modification appropriate to time dependence exp(−iωt) used in the present paper].

Field correlations of arbitrary order appear to have been discussed first in the framework of the classical theory of stationary fields by L. Mandel in Ref. 18. See also E. Wolf, “Basic concepts of optical coherence theory,” in Proceedings of the Symposium on Optical Masers, J. Fox, ed. (Wiley, New York, 1963), pp. 29–42.

L. Mandel, “Some coherence properties of non-Gaussian light,” in Quantum Electronics III, P. Grivet, N. Bloembergen, eds. (Columbia U. Press, New York, 1964), Vol. I, pp. 101–109.

It is implicitly assumed here that M> 1. When M= 1 one readily finds that the first product term (with the index j) in Eq. (4.10), and in all subsequent formulas where it appears, must be replaced by unity.

Cross-spectral density functions of arbitrary orders in classical wave fields were first considered by L. Mandel in Ref. 18. See also Refs. 21–23.

Similar remarks apply here to those made in note 19 in connection with Eq. (4.10).

C. L. Mehta, L. Mandel, “Some properties of higher order coherence functions,” in Electromagnetic Wave Theory, J. Brown, ed. (Pergamon, New York, 1967), Part 2, pp. 1069–1075.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Sec. 10.3.1.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Equations (102)

Equations on this page are rendered with MathJax. Learn more.

Γ Q ( r 1 , r 2 , τ ) = Q * ( r 1 , t ) Q ( r 2 , t + τ )
W Q ( r 1 , r 2 , ω ) = 1 2 π - Γ Q ( r 1 , r 2 , τ ) e i ω τ d τ
W Q ( r 1 , r 2 , ω ) = n λ n ( ω ) ϕ n * ( r 1 , ω ) ϕ n ( r 2 , ω ) ,
D W Q ( r 1 , r 2 , ω ) ϕ n ( r 1 , ω ) d 3 r 1 = λ n ( ω ) ϕ n ( r 2 , ω ) .
D ϕ n ( r , ω ) ϕ m ( r , ω ) d 3 r = δ n m ,
λ n ( ω ) > 0.
μ ( r 1 , r 2 , ω ) = W ( r 1 , r 2 , ω ) [ W ( r 1 , r 1 , ω ) ] 1 / 2 [ W ( r 2 , r 2 , ω ) ] 1 / 2 ,
W Q ( n ) ( r 1 , r 2 , ω ) = λ n ( ω ) ϕ n * ( r 1 , ω ) ϕ n ( r 2 , ω )
μ Q ( n ) ( r 1 , r 2 , ω ) = W Q ( n ) ( r 1 , r 2 , ω ) [ W Q ( n ) ( r 1 , r 1 , ω ) ] 1 / 2 [ W Q ( n ) ( r 2 , r 2 , ω ) ] 1 / 2
W Q ( r 1 , r 2 , ω ) = U Q * ( r 1 , ω ) U Q ( r 2 , ω ) ω ,
U Q ( r , ω ) = n a n ( ω ) ϕ n ( r , ω ) ,
a n * ( ω ) a m ( ω ) ω = λ n ( ω ) δ n m .
S Q ( r , ω ) = n λ n ( ω ) ϕ n ( r , ω ) 2
S Q ( r , ω ) = U Q ( r , ω ) 2 ω .
( 2 - 1 c 2 2 t 2 ) V ( r , t ) = - 4 π Q ( r , t ) ,
Γ V ( r 1 , r 2 , τ ) = V * ( r 1 , t ) V ( r 2 , t + τ )
W V ( r 1 , r 2 , ω ) = 1 2 π - Γ V ( r 1 , r 2 , τ ) e i ω t d τ
M 2 M 1 W V ( r 1 , r 2 , ω ) = ( 4 π ) 2 W Q ( r 1 , r 2 , ω ) ,
M s s 2 + k 2             ( s = 1 , 2 ) ;
M ψ n ( r , ω ) = - 4 π ϕ n ( r , ω ) .
ψ n ( r , ω ) = D ϕ n ( r , ω ) exp ( i k r - r ) r - r d 3 r .
U V ( r , ω ) = n a n ( ω ) ψ n ( r , ω ) ,
M U V ( r , ω ) = - 4 π U Q ( r , ω ) .
M 2 M 1 U V * ( r 1 , ω ) U V ( r 2 , ω ) ω = ( 4 π ) 2 W Q ( r 1 , r 2 , ω ) ,
W V ( r 1 , r 2 , ω ) = U V * ( r 1 , ω ) U V ( r 2 , ω ) ω .
W V ( r 1 , r 2 , ω ) = n λ n ( ω ) ψ n * ( r 1 , ω ) ψ n ( r 2 , ω ) .
W V * ( r 1 , r 2 , ω ) = λ n ( ω ) ψ n * ( r 1 , ω ) ψ n ( r 2 , ω )
μ V ( n ) ( r 1 , r 2 , ω ) = W V ( n ) ( r 1 , r 2 , ω ) [ W V ( n ) ( r 1 , r 1 , ω ) ] 1 / 2 [ W V ( n ) ( r 2 , r 2 , ω ) ] 1 / 2
S V ( r , ω ) = n λ n ( ω ) ψ n ( r , ω ) 2 ,
S V ( r , ω ) = U V ( r , ω ) 2 ω ,
Γ V ( r 1 , r 2 , τ ) = - U V ( r 1 , ω ) U V ( r 2 , ω ) ω e - i ω τ d ω
Γ V ( r 1 , r 2 , τ ) = n - λ n ( ω ) ψ n * ( r 1 , ω ) ψ n ( r 2 , ω ) e - i ω τ d ω .
Ψ n ( r , ω ) = D ϕ n ( r , ω ) G ( r , r , ω ) d 3 r .
Ψ n ( r , ω ) = 1 2 π S ϕ n ( r , ω ) z [ exp ( i k r - r ) r - r ] d 2 r ,
S W V ( 0 ) ( r 1 , r 2 , ω ) ϕ n ( r 1 , ω ) d 2 r 1 = λ n ( ω ) ϕ n ( r 2 , ω )
G Q ( M , N ) ( r 1 , r 2 , r M + N ; t 1 , t 2 , t M + N ) = Q * ( r 1 , t 1 ) Q * ( r 2 , t 2 ) Q * ( r M , t M ) × Q ( r M + 1 , t M + 1 ) Q ( r M + 2 , t M + 2 ) Q ( r M + N , t M + N ) .
G Q ( M , N ) ( r 1 , r 2 , r M + N ; t 1 , t 2 , t M + N ) = j = 1 M Q * ( r j , t j ) k = M + 1 M + N Q ( r k , t k ) ,
Q ( r , t ) = - q ( r , ω ) e - i ω t d ω .
q ( r , ω ) = 1 2 π - Q ( r , t ) e i ω t d t .
G Q ( M , N ) ( r 1 , r 2 , r M + N ; t 1 , t 2 , t M + N ) = - d ω 1 - d ω 2 , - d ω M + N × Φ Q ( M , N ) ( r 1 , r 2 , , r M + N ; ω 1 , ω 2 , , ω M + N ) × j = 1 M exp ( i ω j t j ) k = M + 1 M + N exp ( - i ω k t k ) ,
Φ Q ( M , N ) ( r 1 , r 2 , , r M + N ; ω 1 , ω 2 , , ω M + N ) = j = 1 M q * ( r j , ω j ) k = M + 1 M + N q ( r k , ω k ) .
Φ Q ( M , N ) ( r 1 , r 2 , r M + N ; ω 1 , ω 2 , ω M + N ) = 1 ( 2 π ) M + N - d t 1 - d t 2 , - d t M + N × G Q ( M , N ) ( r 1 , r 2 , , r M + N ; t 1 , t 2 , , t M + N ) × j = 1 M exp ( - i ω j t j ) k = M + 1 M + N exp ( i ω k t k ) .
t l = t 1 + τ l             ( l = 2 , 3 M + N ) ,
G Q ( M , N ) ( r 1 , r 2 , r M + N ; t 1 , t 2 , t M + N ) Γ Q ( M , N ) ( r 1 , r 2 , , r M + N ; τ 1 , τ 2 , , τ M + N ) .
Φ Q ( M , N ) ( r 1 , r 2 , r M + N ; ω 1 , ω 2 , ω M + N ) = δ ( ω 1 + ω 2 + ω M - ω M + 1 - ω M + 2 - ω M + N ) × W Q ( M , N ) ( r 1 , r 2 , , r M + N ; ω 2 , ω 3 , , ω M + N ) ,
W Q ( M , N ) ( r 1 , r 2 , r M + N ; ω 1 , ω 2 , ω M + N ) = 2 ( 2 π ) M + N - 1 - d τ 2 - d τ 3 - d τ M + N × Γ Q ( M , N ) ( r 1 , r 2 , , r M + N ; τ 2 , τ 3 , , τ M + N ) × j = 2 M exp ( - i ω j t j ) k = M + 1 M + N exp ( i ω k t k ) .
Φ Q ( M , N ) ( r 1 , r 2 , r M + N ; ω 1 , ω 2 , ω M + N ) = 0
ω 1 + ω 2 + ω M - ω M + 1 - ω M + 2 - ω M + N = 0 ;
W Q ( 1 , 1 ) ( r 1 , r 2 ; ω 2 ) = 1 2 π - Γ Q ( 1 , 1 ) ( r 1 , r 2 ; τ 2 ) exp ( i ω 2 τ 2 ) d τ 2 ,
Γ Q ( 1 , 1 ) ( r 1 , r 2 ; τ 2 ) = Q * ( r 1 , t 1 ) Q ( r 2 , t 1 + τ 2 ) .
W Q ( M , N ) ( r 1 , r 2 , , r M + N ; ω 2 , ω 3 , , ω M + N ) = σ ^ ( M , N ) j = 1 M U Q * ( r j ; ω j ) k = M + 1 M + N U Q ( r k , ω k ) ω .
U Q ( r , ω ) = n a n ( ω ) ϕ n ( r , ω ) ,
a n 1 * ( ω ) a n 2 ( ω ) ω = λ n 1 ( ω ) δ n 1 n 2 ,
ω 1 = - ω 2 - ω 3 - ω M + ω M + 1 + ω M + 2 + ω M + N
W Q ( M , N ) ( r 1 r 2 r M + N ; ω 2 ω 3 , ω M + N ) = n 1 n 2 n M + N σ ^ ( M , N ) M n 1 n 2 n M + N ( M , N ) ( ω 1 , ω 2 , ω M + N ) × σ ^ ( M , N ) j = 1 M ϕ n j * ( r j , ω j ) k = M + 1 M + N ϕ n k ( r k , ω k ) ,
M n 1 n 2 n M + N ( M , N ) ( ω 1 , ω 2 , ω M + N ) = j = 1 M a n j * ( ω j ) k = M + 1 M + N a n k ( ω k ) ω
σ ^ ( M , N ) j = 1 M ϕ m j ( r j , ω j ) k = M + 1 M + N ϕ m k * ( r k , ω k )
σ ^ ( M , N ) M n 1 n 2 n M + N ( M , N ) ( ω 1 , ω 2 , ω M + N ) = D d 3 r 1 D d 3 r 2 D d 3 r M + N × W Q ( M + N ) ( r 1 r 2 , r M + N ; ω 2 , ω 3 , ω M + N ) × σ ^ ( M , N ) j = 1 M ϕ n j ( r j , ω j ) k = M + 1 M + N ϕ n k * ( r k , ω k ) .
σ ^ ( M , N ) M n 1 n 2 n M + N ( M , N ) ( ω 1 , ω 2 , ω M + N ) = 1 ( 2 π ) M + N - 1 - d τ 2 - d τ 3 - d τ M + N × n 1 n 2 n M + N ( M , N ) ( τ 2 , τ 3 , τ M + N ; ω 2 , ω 3 ω M + N ) × j = 2 M exp ( - i ω j τ j ) k = M + 1 M + N exp ( i ω k τ k ) ,
n 1 n 2 n M + N ( M , N ) ( τ 2 , τ 3 τ M + N ; ω 2 , ω 3 , ω M + N ) = D d 3 r 1 D d 3 r 2 D d 3 r M + N × Γ Q ( M , N ) ( r 1 , r 2 , r M + N ; τ 2 , τ 3 τ M + N ) × σ ^ ( M , N ) j = 1 M ϕ n j ( r j , ω j ) k = M + 1 M + N ϕ n k * ( r k , ω k ) .
M n 1 n 2 n M + N M , N ( ω 1 , ω 2 ω M + N ) = 0 ,
ω 1 + ω 2 + ω M - ω M + 1 - ω M + 2 - - ω M + N 0.
Ψ Q ( M , N ) ( r 1 , r 2 , r M + N ; ω 1 , ω 2 ω M + N ) = j = 1 M U Q * ( r j , ω j ) k = M + 1 M + N U Q ( r k , ω k ) ω .
Ψ Q ( M , N ) ( r 1 , r 2 , r M + N ; ω 1 , ω 2 ω M + N ) = n 1 n 2 n M + N M n 1 n 2 n M + N M + N ( ω 1 , ω 2 , , ω M + N ) × j = 1 M ϕ n j * ( r j , ω j ) k = M + 1 M + N ϕ n k ( r k , ω k ) .
Ψ Q ( M , N ) ( r 1 , r 2 , r M + N ; ω 1 , ω 2 ω M + N ) = W Q ( M , N ) ( r 1 , r 2 , r M + N ; ω 2 , ω 3 ω M + N )
ω 1 + ω 2 + ω M - ω M + 1 - ω M + 2 - - ω M + N = 0
Ψ Q ( M , N ) ( r 1 , r 2 , r M + N ; ω 1 , ω 2 ω M + N ) = 0
ω 1 + ω 2 + ω M - ω M + 1 - ω M + 2 - - ω M + N 0.
G V ( M , N ) ( r 1 , r 2 , r M + N ; t 1 , t 2 t M + N ) = j = 1 M V * ( r j , t j ) k = M + 1 M + N V ( r k , t k ) .
v ( r , ω ) = 1 2 π - V ( r , t ) e i ω t d t ,
Φ V ( M , N ) ( r 1 , r 2 , r M + N ; ω 1 , ω 2 ω M + N ) = j = 1 M v * ( r j ; ω j ) k = M + 1 M + N v ( r k , ω k )
G V ( M , N ) ( r 1 , r 2 , r M + N ; t 1 , t 2 t M + N ) = Γ V ( M , N ) r 1 , r 2 , r M + N ; τ 2 , τ 3 τ M + N ) .
Φ V ( M , N ) ( r 1 , r 2 , r M + N ; ω 1 , ω 2 ω M + N ) = δ ( ω 1 + ω 2 + ω M - ω M + 1 - ω M + 2 - - ω M + N ) × W V ( M , N ) ( r 1 , r 2 , r M + N ; ω 2 , ω 3 ω M + N ) ,
W V ( M , N ) ( r 1 , r 2 , r M + N ; ω 2 , ω 3 ω M + N ) = 1 ( 2 π ) M + N - 1 - d τ 2 - d τ 3 - d τ M + N × Γ V ( M , N ) ( r 1 , r 2 , r M + N ; τ 1 , τ 2 τ M + N ) × j = 2 M exp ( - i ω j τ j ) k = M + 1 M + N exp ( i ω k τ k ) .
σ ^ ( M , N ) s = 1 M + N M s W V ( M , N ) ( r 1 , r 2 , r M + N ; ω 2 , ω 3 ω M + N ) = ( - 4 π ) M + N W Q ( M , N ) ( r 1 , r 2 , r M + N ; ω 2 , ω 3 ω M + N ) ,
M s = s 2 + k s 2             ( s = 1 , 2 M + N )
k s = ω s c             ( s = 1 , 2 M + N ) ,
M s ψ s ( r , ω s ) = - 4 π ϕ s ( r , ω s ) ,
U V ( r , ω ) = n a n ( ω ) ψ n ( r , ω ) ,
W V ( M , N ) ( r 1 , r 2 , r M + N ; ω 2 , ω 3 ω M + N ) × σ ^ ( M , N ) j = 1 M U V * ( r j ; ω j ) k = M + 1 M + N U V ( r k , ω k ) ω ,
W V ( M , N ) ( r 1 , r 2 , r M + N ; ω 2 , ω 3 ω M + N ) = n 1 n 2 n M + N σ ^ ( M , N ) M n 1 , n 2 n M + N ( M , N ) ( ω 1 , ω 2 , ω M + N ) × σ ^ ( M , N ) j = 1 M ψ n j * ( r j ; ω j ) k = M + 1 M + N ψ n k ( r k , ω k ) .
L V ( r , t ) = - 4 π Q ( r , t ) ,
L 2 - 1 c 2 2 t 2
L 1 V * ( r 1 , t 1 ) L 2 V ( r 2 , t 2 ) = ( 4 π ) 2 Q * ( r 1 , t 1 ) Q ( r 2 , t 2 ) ,
L 2 L 1 V * ( r 1 , t 1 ) V ( r 2 , t 2 ) = ( 4 π ) 2 Q * ( r 1 , t 1 ) Q ( r 2 , t 2 ) .
L 2 L 1 Γ V ( r 1 , r 2 , t 2 - t 1 ) = ( 4 π ) 2 Γ Q ( r 1 , r 2 , t 2 - t 1 ) .
t 2 - t 1 = τ ,
L 2 L 1 Γ V ( r 1 , r 2 , τ ) = ( 4 π ) 2 Γ Q ( r 1 , r 2 , τ ) ,
L j = j 2 - 1 c 2 2 τ 2             ( j = 1 , 2 ) ,
Γ V ( r 1 , r 2 , τ ) = - W V ( r 1 , r 2 , ω ) e - i ω τ d ω ,
Γ Q ( r 1 , r 2 , τ ) = - W Q ( r 1 , r 2 , ω ) e - i ω τ d ω .
- M 2 M 1 W V ( r 1 , r 2 , ω ) e - i ω τ d ω = ( 4 π ) 2 - W Q ( r 1 , r 2 , ω ) e - i ω τ d ω ,
M j = j 2 + k 2             ( j = 1 , 2 )
M 2 M 1 W V ( r 1 , r 2 , ω ) = ( 4 π ) 2 W Q ( r 1 , r 2 , ω ) ,
s = 1 M + N L s G V ( M , N ) ( r 1 , r 2 , , r M + N ; t 1 , t 2 , t M + N ) = ( - 4 π ) M + N G Q ( M , N ) ( r 1 , r 2 , , r M + N ; t 1 , t 2 , t M + N ) .
L s s 2 - 1 c 2 2 t s 2 ,
s = 1 M + N M s Φ V ( M , N ) ( r 1 , r 2 , , r M + N ; ω 1 , ω 2 , ω M + N ) = ( - 4 π ) ( M + N ) Φ Q ( M + N ) ( r 1 , r 2 , , r M + N ; t 1 , t 2 , t M + N )
M s s 2 + k s 2             ( s = 1 , 2 , M + N )
k s = ω s c             ( s = 1 , 2 , M + N ) ,
σ ^ ( M , N ) s = 1 M + N M s W V ( M , N ) ( r 1 , r 2 , , r M + N ; ω 2 , ω 3 , ω M + N ) = ( - 4 π ) M + N W Q ( M , N ) ( r 1 , r 2 , , r M + N ; ω 2 , ω 3 , ω M + N ) .
( 1 2 + k 2 2 ) ( 2 2 + k 2 2 ) W V ( 1 , 1 ) ( r 1 , r 2 , ω 2 ) = ( 4 π ) 2 W Q ( 1 , 1 ) ( r 1 , r 2 , ω 2 )
[ 1 2 + ( k 3 - k 2 ) 2 ] ( 2 2 + k 2 2 ) ( 3 2 + k 3 2 ) × W V ( 2 , 1 ) ( r 1 , r 2 , r 3 ; ω 2 , ω 3 ) = ( 4 π ) 3 W Q ( 2 , 1 ) ( r 1 , r 2 , r 3 ; ω 2 , ω 3 ) ,

Metrics