Abstract

An analysis of transverse laser resonator modes is presented, based on a recently developed coherence theory in the space-frequency domain. The modes are introduced by means of solutions of an integral equation that expresses a steady-state condition for a second-order correlation function of the field across a mirror of the laser cavity. All solutions of this integral equation are found to be expressible as quadratic forms involving the Fox–Li modes of the conventional theory. If there is no degeneracy, each mode is shown to be necessarily completely spatially coherent, at each frequency, within the framework of second-order correlation theory. It is also shown that, if several transverse modes are excited, the output cannot be completely spatially coherent.

© 1984 Optical Society of America

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References

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  1. A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).
  2. E. Wolf, “Spatial coherence of resonant modes in a maser interferometer,” Phys. Lett. 3, 166–168 (1963).
    [CrossRef]
  3. W. Streifer, “Spatial coherence in periodic systems,”J. Opt. Soc. Am., 56, 1481–1489 (1966).
    [CrossRef]
  4. L. Allen, S. Gatehouse, D. G. C. Jones, “Enhancement of optical coherence during light propagation in bounded media,” Opt. Commun. 4, 169–171 (1971).
    [CrossRef]
  5. F. Gori, “Propagation of the mutual intensity through a periodic structure,” Atti Fond. Giorgio Ronchi 35, 434–447 (1980).
  6. M. Bertolotti, B. Daino, F. Gori, D. Sette, “Coherence properties of a laser beam,” Nuovo Cimento 38, 1505–1514 (1965).
    [CrossRef]
  7. For definition of the analytic signal and for explanation of the concepts of coherence theory used in this paper, see M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Chap. X, or L. Mandel, E. Wolf, “Coherence properties of optical fields,” Rev. Mod. Phys. 37, 231–287 (1965).
    [CrossRef]
  8. 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). Although this paper deals with stationary primary sources, strictly similar results hold also for stationary secondary sources and for stationary fields. In this connection, see Refs. 9 and 10.
    [CrossRef]
  9. E. Wolf, “A new description of second-order coherence phenomena in the space-frequency domain,” in Optics in Four Dimensions—1980, M. A. Machado, L. M. Narducci, eds.,AIP Conf. Proc.65, 42–48 (1981).
  10. E. Wolf, “New spectral representation of random sources and the partially coherent fields that they generate,” Opt. Commun. 38, 3–6 (1981).
    [CrossRef]
  11. Because we assumed the domain D to be three-dimensional, the index n stands for an ordered triplet of integers (n1, n2, n3). We will later need the analogs of some of the formulas for fields in two-dimensional domains. In general, if the domain containing the field is N dimensional, the index n will stand for N ordered integers (n1, n2, … nN), and the integrals in Eqs. (2.4) and (2.5) will, of course, extend over the N-dimensional domain.
  12. L. Mandel, E. Wolf, “Spectral coherence and the concept of cross-spectral purity,”J. Opt. Soc. Am. 66, 529–535 (1976), Sec. II.
    [CrossRef]
  13. E. Wolf, “Coherence and radiometry,”J. Opt. Soc. Am. 68, 6–17 (1978).
    [CrossRef]
  14. G. D. Boyd, H. Kogelnik, “Generalized confocal resonator theory,” Bell Syst. Tech. J. 44, 1347–1369 (1962).
  15. P. M. Morse, H. Feshbach, Methods of Mathematical Physics (McGraw-Hill, New York (1953)), Part I, pp. 919–920; see also pp. 884–886. Unfortunately neither in this reference nor in any of the well-known texts on integral equations conditions are given for the validity of the biorthogonal expansion [Eq. (4.5) below]. However, L(ρ1, ρ2, ω) belongs to a class of well-behaved kernels to which the important relations (4.3) and (4.4) apply. Moreover, it is known that, for laser cavities of the usual geometries, L(ρ1, ρ2, ω) has a complete set of eigenfunctions in the Hilbert space of square-integrable functions. These facts suggest that for such cavities L(ρ1, ρ2, ω) admits of such an expansion.
  16. For proofs of one-dimensional versions of these results, see F. Smithies, Integral Equations (Cambridge U. Press, Cambridge, 1970), theorems 6.7.3 and 6.7.4.
  17. W. Streifer, H. Gamo, “On the Schmidt expansion for optical resonator modes,” in Proceedings of the Symposium on Quasi-Optics (Polytechnic, Brooklyn, N.Y., 1964), pp. 351–365.
  18. W. Streifer, “Optical resonator modes—rectangular reflectors of spherical curvature,”J. Opt. Soc. Am. 55, 868–877 (1965).
    [CrossRef]
  19. J. C. Heurtley, W. Streifer, “Optical resonator modes—circular reflectors of spherical curvature,”J. Opt. Soc. Am. 55, 1472–1479 (1965).
    [CrossRef]
  20. It seems worthwhile to stress that the subscript that labels W has here a different meaning from that in Section 3. It now distinguishes the different eigenfunctions of Eq. (3.6), whereas in Section 3 it indicated the number of complete cycles of propagation of light between the two mirrors.
  21. L. Mandel, E. Wolf, “Complete coherence in the space-frequency domain,” Opt. Commun. 36, 247–249 (1981).
    [CrossRef]

1982 (1)

1981 (2)

E. Wolf, “New spectral representation of random sources and 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]

1980 (1)

F. Gori, “Propagation of the mutual intensity through a periodic structure,” Atti Fond. Giorgio Ronchi 35, 434–447 (1980).

1978 (1)

1976 (1)

1971 (1)

L. Allen, S. Gatehouse, D. G. C. Jones, “Enhancement of optical coherence during light propagation in bounded media,” Opt. Commun. 4, 169–171 (1971).
[CrossRef]

1966 (1)

1965 (3)

1963 (1)

E. Wolf, “Spatial coherence of resonant modes in a maser interferometer,” Phys. Lett. 3, 166–168 (1963).
[CrossRef]

1962 (1)

G. D. Boyd, H. Kogelnik, “Generalized confocal resonator theory,” Bell Syst. Tech. J. 44, 1347–1369 (1962).

1961 (1)

A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).

Allen, L.

L. Allen, S. Gatehouse, D. G. C. Jones, “Enhancement of optical coherence during light propagation in bounded media,” Opt. Commun. 4, 169–171 (1971).
[CrossRef]

Bertolotti, M.

M. Bertolotti, B. Daino, F. Gori, D. Sette, “Coherence properties of a laser beam,” Nuovo Cimento 38, 1505–1514 (1965).
[CrossRef]

Born, M.

For definition of the analytic signal and for explanation of the concepts of coherence theory used in this paper, see M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Chap. X, or L. Mandel, E. Wolf, “Coherence properties of optical fields,” Rev. Mod. Phys. 37, 231–287 (1965).
[CrossRef]

Boyd, G. D.

G. D. Boyd, H. Kogelnik, “Generalized confocal resonator theory,” Bell Syst. Tech. J. 44, 1347–1369 (1962).

Daino, B.

M. Bertolotti, B. Daino, F. Gori, D. Sette, “Coherence properties of a laser beam,” Nuovo Cimento 38, 1505–1514 (1965).
[CrossRef]

Feshbach, H.

P. M. Morse, H. Feshbach, Methods of Mathematical Physics (McGraw-Hill, New York (1953)), Part I, pp. 919–920; see also pp. 884–886. Unfortunately neither in this reference nor in any of the well-known texts on integral equations conditions are given for the validity of the biorthogonal expansion [Eq. (4.5) below]. However, L(ρ1, ρ2, ω) belongs to a class of well-behaved kernels to which the important relations (4.3) and (4.4) apply. Moreover, it is known that, for laser cavities of the usual geometries, L(ρ1, ρ2, ω) has a complete set of eigenfunctions in the Hilbert space of square-integrable functions. These facts suggest that for such cavities L(ρ1, ρ2, ω) admits of such an expansion.

Fox, A. G.

A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).

Gamo, H.

W. Streifer, H. Gamo, “On the Schmidt expansion for optical resonator modes,” in Proceedings of the Symposium on Quasi-Optics (Polytechnic, Brooklyn, N.Y., 1964), pp. 351–365.

Gatehouse, S.

L. Allen, S. Gatehouse, D. G. C. Jones, “Enhancement of optical coherence during light propagation in bounded media,” Opt. Commun. 4, 169–171 (1971).
[CrossRef]

Gori, F.

F. Gori, “Propagation of the mutual intensity through a periodic structure,” Atti Fond. Giorgio Ronchi 35, 434–447 (1980).

M. Bertolotti, B. Daino, F. Gori, D. Sette, “Coherence properties of a laser beam,” Nuovo Cimento 38, 1505–1514 (1965).
[CrossRef]

Heurtley, J. C.

Jones, D. G. C.

L. Allen, S. Gatehouse, D. G. C. Jones, “Enhancement of optical coherence during light propagation in bounded media,” Opt. Commun. 4, 169–171 (1971).
[CrossRef]

Kogelnik, H.

G. D. Boyd, H. Kogelnik, “Generalized confocal resonator theory,” Bell Syst. Tech. J. 44, 1347–1369 (1962).

Li, T.

A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).

Mandel, L.

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

Morse, P. M.

P. M. Morse, H. Feshbach, Methods of Mathematical Physics (McGraw-Hill, New York (1953)), Part I, pp. 919–920; see also pp. 884–886. Unfortunately neither in this reference nor in any of the well-known texts on integral equations conditions are given for the validity of the biorthogonal expansion [Eq. (4.5) below]. However, L(ρ1, ρ2, ω) belongs to a class of well-behaved kernels to which the important relations (4.3) and (4.4) apply. Moreover, it is known that, for laser cavities of the usual geometries, L(ρ1, ρ2, ω) has a complete set of eigenfunctions in the Hilbert space of square-integrable functions. These facts suggest that for such cavities L(ρ1, ρ2, ω) admits of such an expansion.

Sette, D.

M. Bertolotti, B. Daino, F. Gori, D. Sette, “Coherence properties of a laser beam,” Nuovo Cimento 38, 1505–1514 (1965).
[CrossRef]

Smithies, F.

For proofs of one-dimensional versions of these results, see F. Smithies, Integral Equations (Cambridge U. Press, Cambridge, 1970), theorems 6.7.3 and 6.7.4.

Streifer, W.

Wolf, E.

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). Although this paper deals with stationary primary sources, strictly similar results hold also for stationary secondary sources and for stationary fields. In this connection, see Refs. 9 and 10.
[CrossRef]

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 the partially coherent fields that they generate,” Opt. Commun. 38, 3–6 (1981).
[CrossRef]

E. Wolf, “Coherence and radiometry,”J. Opt. Soc. Am. 68, 6–17 (1978).
[CrossRef]

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

E. Wolf, “Spatial coherence of resonant modes in a maser interferometer,” Phys. Lett. 3, 166–168 (1963).
[CrossRef]

E. Wolf, “A new description of second-order coherence phenomena in the space-frequency domain,” in Optics in Four Dimensions—1980, M. A. Machado, L. M. Narducci, eds.,AIP Conf. Proc.65, 42–48 (1981).

For definition of the analytic signal and for explanation of the concepts of coherence theory used in this paper, see M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Chap. X, or L. Mandel, E. Wolf, “Coherence properties of optical fields,” Rev. Mod. Phys. 37, 231–287 (1965).
[CrossRef]

Atti Fond. Giorgio Ronchi (1)

F. Gori, “Propagation of the mutual intensity through a periodic structure,” Atti Fond. Giorgio Ronchi 35, 434–447 (1980).

Bell Syst. Tech. J. (2)

G. D. Boyd, H. Kogelnik, “Generalized confocal resonator theory,” Bell Syst. Tech. J. 44, 1347–1369 (1962).

A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).

J. Opt. Soc. Am. (6)

Nuovo Cimento (1)

M. Bertolotti, B. Daino, F. Gori, D. Sette, “Coherence properties of a laser beam,” Nuovo Cimento 38, 1505–1514 (1965).
[CrossRef]

Opt. Commun. (3)

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

L. Allen, S. Gatehouse, D. G. C. Jones, “Enhancement of optical coherence during light propagation in bounded media,” Opt. Commun. 4, 169–171 (1971).
[CrossRef]

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

Phys. Lett. (1)

E. Wolf, “Spatial coherence of resonant modes in a maser interferometer,” Phys. Lett. 3, 166–168 (1963).
[CrossRef]

Other (7)

It seems worthwhile to stress that the subscript that labels W has here a different meaning from that in Section 3. It now distinguishes the different eigenfunctions of Eq. (3.6), whereas in Section 3 it indicated the number of complete cycles of propagation of light between the two mirrors.

Because we assumed the domain D to be three-dimensional, the index n stands for an ordered triplet of integers (n1, n2, n3). We will later need the analogs of some of the formulas for fields in two-dimensional domains. In general, if the domain containing the field is N dimensional, the index n will stand for N ordered integers (n1, n2, … nN), and the integrals in Eqs. (2.4) and (2.5) will, of course, extend over the N-dimensional domain.

For definition of the analytic signal and for explanation of the concepts of coherence theory used in this paper, see M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Chap. X, or L. Mandel, E. Wolf, “Coherence properties of optical fields,” Rev. Mod. Phys. 37, 231–287 (1965).
[CrossRef]

E. Wolf, “A new description of second-order coherence phenomena in the space-frequency domain,” in Optics in Four Dimensions—1980, M. A. Machado, L. M. Narducci, eds.,AIP Conf. Proc.65, 42–48 (1981).

P. M. Morse, H. Feshbach, Methods of Mathematical Physics (McGraw-Hill, New York (1953)), Part I, pp. 919–920; see also pp. 884–886. Unfortunately neither in this reference nor in any of the well-known texts on integral equations conditions are given for the validity of the biorthogonal expansion [Eq. (4.5) below]. However, L(ρ1, ρ2, ω) belongs to a class of well-behaved kernels to which the important relations (4.3) and (4.4) apply. Moreover, it is known that, for laser cavities of the usual geometries, L(ρ1, ρ2, ω) has a complete set of eigenfunctions in the Hilbert space of square-integrable functions. These facts suggest that for such cavities L(ρ1, ρ2, ω) admits of such an expansion.

For proofs of one-dimensional versions of these results, see F. Smithies, Integral Equations (Cambridge U. Press, Cambridge, 1970), theorems 6.7.3 and 6.7.4.

W. Streifer, H. Gamo, “On the Schmidt expansion for optical resonator modes,” in Proceedings of the Symposium on Quasi-Optics (Polytechnic, Brooklyn, N.Y., 1964), pp. 351–365.

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Equations (59)

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Γ ( r 1 , r 2 , τ ) = V * ( r 1 , t ) V ( r 2 , t + τ )
W ( r 1 , r 2 , ω ) = 1 2 π - Γ ( r 1 , r 2 , τ ) e i ω τ d τ
W ( r 1 , r 2 , ω ) = n λ n ( ω ) ψ n * ( r 1 , ω ) ψ n ( r 2 , ω ) .
D W ( r 1 , r 2 , ω ) ψ n ( r 1 , ω ) d 3 r 1 = λ n ( ω ) ψ n ( r 2 , ω ) .
D ψ n * ( r , ω ) ψ m ( r , ω ) d 3 r = δ n m ,
W ( n ) ( r 1 , r 2 , ω ) = ψ n * ( r 1 , ω ) ψ n ( r 2 , ω )
μ ( n ) ( r 1 , r 2 , ω ) = W ( n ) ( r 1 , r 2 , ω ) [ W ( n ) ( r 1 , r 1 , ω ) ] 1 / 2 [ W ( n ) ( r 2 , r 2 , ω ) ] 1 / 2 ,
μ ( n ) ( r 1 , r 2 , ω ) = 1.
j 2 W ( n ) ( r 1 , r 2 , ω ) + k 2 W ( n ) ( r 1 , r 2 , ω ) = 0             ( j = 1 , 2 ) ,
k = ω / c ,
W ( r 1 , r 2 , ω ) = U * ( r 1 , ω ) U ( r 2 , ω ) ω ,
U ( r , ω ) = a n ( ω ) ψ n ( r , ω ) ,
a * n ( ω ) a m ( ω ) ω = λ n ( ω ) δ n m .
2 U ( r , ω ) + k 2 U ( r , ω ) = 0
Γ ( r 1 , r 2 , τ ) = n Γ ( n ) ( r 1 , r 2 , τ ) ,
Γ ( n ) ( r 1 , r 2 , τ ) = 0 λ n ( ω ) ψ n * ( r 1 , ω ) ψ n ( r 2 , ω ) e - i ω τ d ω .
γ ( n ) ( r 1 , r 2 , τ ) = Γ ( n ) ( r 1 , r 2 , τ ) [ Γ ( n ) ( r 1 , r 1 , 0 ) ] 1 / 2 [ Γ ( n ) ( r 2 , r 2 , 0 ) ] 1 / 2
W j ( ρ 1 , ρ 2 , ω ) = U j * ( ρ 1 , ω ) U j ( ρ 2 , ω ) ω ,
U j + 1 ( ρ , ω ) = A L ( ρ , ρ , ω ) U j ( ρ , ω ) d 2 ρ
W j + 1 ( ρ 1 , ρ 2 , ω ) = A A L * ( ρ 1 , ρ 1 , ω ) L ( ρ 2 , ρ 2 , ω ) × W j ( ρ 1 , ρ 2 , ω ) d 2 ρ 1 d 2 ρ 2 .
W j + 1 ( ρ 1 , ρ 2 , ω ) = σ ( ω ) W j ( ρ 1 , ρ 2 , ω ) .
σ ( ω ) > 0.
A A W ( ρ 1 , ρ 2 , ω ) L * ( ρ 1 , ρ 1 , ω ) L ( ρ 2 , ρ 2 , ω ) d 2 ρ 1 d 2 ρ 2 = σ ( ω ) W ( ρ 1 , ρ 2 , ω ) .
A L ( ρ 1 , ρ 2 , ω ) ϕ n ( ρ 2 , ω ) d 2 ρ 2 = α n ( ω ) ϕ n ( ρ 1 , ω ) ,
A L * ( ρ 2 , ρ 1 , ω ) χ n ( ρ 2 , ω ) d 2 ρ 2 = β n ( ω ) χ n ( ρ 1 , ω ) .
β n = α n * .
A ϕ n * ( ρ , ω ) χ m ( ρ , ω ) d 2 ρ = δ n m .
L ( ρ 1 , ρ 2 , ω ) = n α n ( ω ) ϕ n ( ρ 1 , ω ) χ n * ( ρ 2 , ω ) .
n m α n * ( ω ) α m ( ω ) w n m ( ω ) ϕ n * ( ρ 1 , ω ) ϕ m ( ρ 2 , ω ) = σ ( ω ) W ( ρ 1 , ρ 2 , ω ) ,
w n m = A A W ( ρ 1 , ρ 2 , ω ) χ n ( ρ 1 , ω ) × χ m * ( ρ 2 , ω ) d 2 ρ 1 d 2 ρ 2 .
n m α n * ( ω ) α m ( ω ) w n m ( ω ) δ n N δ m M = σ ( ω ) w N M ( ω ) ,
[ σ ( ω ) - α N * ( ω ) α M ( ω ) ] w N M = 0             ( no summation ) .
σ N M ( ω ) = α N * ( ω ) α M ( ω ) .
α N * ( ω ) α M ( ω ) = α N * ( ω ) α M ( ω ) .
σ k l ( ω ) = α k * ( ω ) α l ( ω )
W ( ρ 1 , ρ 2 , ω ) = w k l ( ω ) ϕ k * ( ρ 1 ) ϕ l ( ρ 2 ) .
W ( ρ 2 , ρ 1 , ω ) = W * ( ρ 1 , ρ 2 , ω ) ,
w k l ϕ k * ( ρ 2 , ω ) ϕ l ( ρ 1 , ω ) = w k l * ϕ k ( ρ 1 , ω ) ϕ l * ( ρ 2 , ω ) ,
ϕ l ( ρ 1 , ω ) ϕ k ( ρ 1 , ω ) = w k l * w k l ϕ l * ( ρ 2 , ω ) ϕ k * ( ρ 2 , ω ) .
ϕ l ( ρ , ω ) = γ k l ( ω ) ϕ k ( ρ , ω ) .
W ( ρ 1 , ρ 2 , ω ) = w k l ( ω ) γ k l ( ω ) ϕ k * ( ρ 1 , ω ) ϕ k ( ρ 2 , ω ) .
w n m = w k l γ k l A ϕ k * ( ρ 1 , ω ) χ n ( ρ 1 , ω ) d 2 ρ 1 × A ϕ k ( ρ 2 , ω ) χ m * ( ρ 2 ω ) d 2 ρ 2 .
w n m ( ω ) = w k l ( ω ) γ k l δ k n δ k m ,
w n m ( ω ) = 0             unless             n = m = k
w k k ( ω ) = w k l ( ω ) γ k l ( ω ) .
W k ( ρ 1 , ρ 2 , ω ) = w k k ( ω ) ϕ k * ( ρ 1 , ω ) ϕ k ( ρ 2 , ω ) .
σ k ( ω ) = α k * ( ω ) α k ( ω ) .
A ϕ k ( ρ , ω ) 2 d 2 ρ = 1.
ϕ k ( ρ , ω ) ψ k ( ρ , ω ) ,
w k k ( ω ) = λ k ( ω ) .
A W k ( ρ , ρ , ω ) d 2 ρ = λ k ( ω ) .
W k ( ρ 1 , ρ 2 , ω ) = λ k ( ω ) ϕ k * ( ρ 1 , ω ) ϕ k ( ρ 2 , ω )
W ( ρ 1 , ρ 2 , ω ) = k l c k l ( ω ) ϕ k * ( ρ 1 , ω ) ϕ l ( ρ 2 , ω ) ,
W ( ρ 1 , ρ 2 , ω ) = k Λ k ( ω ) f k * ( ρ 1 , ω ) f k ( ρ 2 , ω ) ,
f k ( ρ 1 , ω ) = l u k l ( ω ) ϕ l ( ρ , ω ) .
Γ k ( ρ 1 , ρ 2 , τ ) = 0 λ k ( ω ) ϕ k * ( ρ 1 , ω ) ϕ k ( ρ 2 , ω ) e - i ω τ d ω ,
Γ ( ρ 1 , ρ 2 , τ ) = k 0 Λ k ( ω ) f * k ( ρ 1 , ω ) f k ( ρ 2 , ω ) e - i ω τ d ω .
S k ( ρ , ω ) W k ( ρ , ρ , ω ) = λ k ( ω ) ϕ k ( ρ , ω ) 2
S ( ρ , ω ) = k λ k ( ω ) f k ( ρ , ω ) 2

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