Abstract

The purpose of this paper is to suggest a possible approach to the recovery of the spectral density function g(ω) through a knowledge of the first few measured complex zeros of the complex degree of coherence γ(τ). The assumption that γ(τ) is band-limited allows us to express the sums of inverse powers of the complex zeros of γ(τ) in terms of the moments of g(ω). Only the lowest-order moments can be evaluated in this manner with any accuracy for reasons discussed in the text. We use two estimation-type solutions that utilize lower-order moments: beta distribution model and the Shannon maximum entropy model to estimate g(CD). Representative numerical calculations are discussed.

© 1980 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon, New York, 1965).
  2. J. Perina, Coherence of Light (Van Nostrand Reinhold, London, 1972), p. 55.
  3. E. Wolf, “Is a complete determination of the energy spectrum of light possible from measurements of the degree of coherence?,” Proc. Phys. Soc. London 80, 1269–1272 (1962).
    [Crossref]
  4. A. Walther, “The question of phase retrieval in optics,” Opt. Acta 10, 41–49 (1963).
    [Crossref]
  5. M. L. Goldberger, H. W. Lewis, and K. M. Watson, “Use of intensity correlations to determine the phase of a scattering amplitude,” Phys. Rev. 1322764–2787 (1963).
    [Crossref]
  6. H. Nussenzveig, “Phase problem in coherence theory,” J. Math. Phys. 8, 561–572 (1967).
    [Crossref]
  7. R. A. Gonsalves, “Phase retrieval from modulus data,” J. Opt. Soc. Am. 66, 961–964 (1976).
    [Crossref]
  8. J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27–29 (1978).
    [Crossref] [PubMed]
  9. R. E. Burge, M. A. Fiddy, M. Nieto-Vesperinas, and M. W. L. Wheeler, “The phase problem in scattering theory: the zeros of entire functions and their significance,” Proc. R. Soc. London Ser. A 360, 25–45 (1978).
    [Crossref]
  10. W. O. Saxton, Computer Techniques for Image Processing in Electron Microscopy (Academic, New York, 1978).
  11. Inverse Source Problems in Optics, edited by H. P. Baltes, (Springer-Verlag, Berlin, 1978).
    [Crossref]
  12. E. J. Akutowicz, “On the determination of the phase of a Fourier integral, I.” Trans. Am. Math. Soc. 83, 179–192 (1956).
  13. E. J. Akutowicz, “On the determination of the phase of a Fourier integral, II.” Proc. Am. Math. Soc. 8, 234–238 (1957).
  14. P. J. Napier, “The brightness temperature distributions defined by a measured intensity interferogram,” N. Z. J. Sci. 15, 342–355 (1972).
  15. F. J. Ynduráin, “The moment problem and applications,” in Padé Approximants, edited by P. Graves-Morris (Institute of Physics, London, 1973),p. 45.
  16. J. A. Shotat and J. D. Tamarkin, The Problem of Moments (American Mathematical Society, Providence, 1950).
  17. R. P. Boas, Entire Functions (Academic, New York, 1954),p. 103.
  18. E. C. Titchmarsh, “The zeros of certain integral functions,” Proc. London Math. Soc. 25, 283–302 (1926).
    [Crossref]
  19. B. Ya. Levin, Distribution of Zeros of Entire Functions (American Mathematical Society, Providence, 1964).
  20. P. Henrici, Applied and Computational Complex Analysis (Wiley-Interscience, New York, 1974), Vols. 1 and 2.
  21. R. Fortet, Elements of Probability Theory (Gordon and Breach, New York, 1977), p. 271.
  22. W. Elderton and N. Johnson, Systems of Frequency Curves (Cambridge University, Cambridge, 1969), Chap. 4.
    [Crossref]
  23. G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge University, Cambridge, 1944).
  24. Handbook of Mathematical Functions, edited by H. Abramowitz and I. A. Stegun, Natl. Bur. Std., U.S. Appl. Math Ser. (U.S. Government Printing Office, Washington, D. C., 1964).
  25. Lord Rayleigh, “Note on the numerical calculation of the roots of fluctuating functions,” Proc. London Math. Soc. 5, 119–124 (1874).
  26. I. N. Sneddon, Special Functions of Mathematical Physics and Chemistry (Interscience, New York, 1961).
  27. L. M. Delves and J. N. Lyness, “A numerical method for locating the zeros of an analytic function,” Math. Comp. 21, 543–560 (1967).
    [Crossref]
  28. C. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423 (1948);Bell Syst. Tech. J. 27, 623–656 (1948).
    [Crossref]
  29. E. T. Jaynes, “Information theory and statistical mechanics,” Phys. Rev. 106, 620–630 (1957);Phys. Rev. 107, 17–190 (1957).
    [Crossref]
  30. L. Ronkin, Introduction to the Theory of Entire Functions of Several Variables (American Mathematical Society, Providence, 1974), Chap. 4.

1978 (2)

J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27–29 (1978).
[Crossref] [PubMed]

R. E. Burge, M. A. Fiddy, M. Nieto-Vesperinas, and M. W. L. Wheeler, “The phase problem in scattering theory: the zeros of entire functions and their significance,” Proc. R. Soc. London Ser. A 360, 25–45 (1978).
[Crossref]

1976 (1)

1972 (1)

P. J. Napier, “The brightness temperature distributions defined by a measured intensity interferogram,” N. Z. J. Sci. 15, 342–355 (1972).

1967 (2)

L. M. Delves and J. N. Lyness, “A numerical method for locating the zeros of an analytic function,” Math. Comp. 21, 543–560 (1967).
[Crossref]

H. Nussenzveig, “Phase problem in coherence theory,” J. Math. Phys. 8, 561–572 (1967).
[Crossref]

1963 (2)

A. Walther, “The question of phase retrieval in optics,” Opt. Acta 10, 41–49 (1963).
[Crossref]

M. L. Goldberger, H. W. Lewis, and K. M. Watson, “Use of intensity correlations to determine the phase of a scattering amplitude,” Phys. Rev. 1322764–2787 (1963).
[Crossref]

1962 (1)

E. Wolf, “Is a complete determination of the energy spectrum of light possible from measurements of the degree of coherence?,” Proc. Phys. Soc. London 80, 1269–1272 (1962).
[Crossref]

1957 (2)

E. J. Akutowicz, “On the determination of the phase of a Fourier integral, II.” Proc. Am. Math. Soc. 8, 234–238 (1957).

E. T. Jaynes, “Information theory and statistical mechanics,” Phys. Rev. 106, 620–630 (1957);Phys. Rev. 107, 17–190 (1957).
[Crossref]

1956 (1)

E. J. Akutowicz, “On the determination of the phase of a Fourier integral, I.” Trans. Am. Math. Soc. 83, 179–192 (1956).

1948 (1)

C. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423 (1948);Bell Syst. Tech. J. 27, 623–656 (1948).
[Crossref]

1926 (1)

E. C. Titchmarsh, “The zeros of certain integral functions,” Proc. London Math. Soc. 25, 283–302 (1926).
[Crossref]

1874 (1)

Lord Rayleigh, “Note on the numerical calculation of the roots of fluctuating functions,” Proc. London Math. Soc. 5, 119–124 (1874).

Akutowicz, E. J.

E. J. Akutowicz, “On the determination of the phase of a Fourier integral, II.” Proc. Am. Math. Soc. 8, 234–238 (1957).

E. J. Akutowicz, “On the determination of the phase of a Fourier integral, I.” Trans. Am. Math. Soc. 83, 179–192 (1956).

Boas, R. P.

R. P. Boas, Entire Functions (Academic, New York, 1954),p. 103.

Born, M.

M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon, New York, 1965).

Burge, R. E.

R. E. Burge, M. A. Fiddy, M. Nieto-Vesperinas, and M. W. L. Wheeler, “The phase problem in scattering theory: the zeros of entire functions and their significance,” Proc. R. Soc. London Ser. A 360, 25–45 (1978).
[Crossref]

Delves, L. M.

L. M. Delves and J. N. Lyness, “A numerical method for locating the zeros of an analytic function,” Math. Comp. 21, 543–560 (1967).
[Crossref]

Elderton, W.

W. Elderton and N. Johnson, Systems of Frequency Curves (Cambridge University, Cambridge, 1969), Chap. 4.
[Crossref]

Fiddy, M. A.

R. E. Burge, M. A. Fiddy, M. Nieto-Vesperinas, and M. W. L. Wheeler, “The phase problem in scattering theory: the zeros of entire functions and their significance,” Proc. R. Soc. London Ser. A 360, 25–45 (1978).
[Crossref]

Fienup, J. R.

Fortet, R.

R. Fortet, Elements of Probability Theory (Gordon and Breach, New York, 1977), p. 271.

Goldberger, M. L.

M. L. Goldberger, H. W. Lewis, and K. M. Watson, “Use of intensity correlations to determine the phase of a scattering amplitude,” Phys. Rev. 1322764–2787 (1963).
[Crossref]

Gonsalves, R. A.

Henrici, P.

P. Henrici, Applied and Computational Complex Analysis (Wiley-Interscience, New York, 1974), Vols. 1 and 2.

Jaynes, E. T.

E. T. Jaynes, “Information theory and statistical mechanics,” Phys. Rev. 106, 620–630 (1957);Phys. Rev. 107, 17–190 (1957).
[Crossref]

Johnson, N.

W. Elderton and N. Johnson, Systems of Frequency Curves (Cambridge University, Cambridge, 1969), Chap. 4.
[Crossref]

Levin, B. Ya.

B. Ya. Levin, Distribution of Zeros of Entire Functions (American Mathematical Society, Providence, 1964).

Lewis, H. W.

M. L. Goldberger, H. W. Lewis, and K. M. Watson, “Use of intensity correlations to determine the phase of a scattering amplitude,” Phys. Rev. 1322764–2787 (1963).
[Crossref]

Lyness, J. N.

L. M. Delves and J. N. Lyness, “A numerical method for locating the zeros of an analytic function,” Math. Comp. 21, 543–560 (1967).
[Crossref]

Napier, P. J.

P. J. Napier, “The brightness temperature distributions defined by a measured intensity interferogram,” N. Z. J. Sci. 15, 342–355 (1972).

Nieto-Vesperinas, M.

R. E. Burge, M. A. Fiddy, M. Nieto-Vesperinas, and M. W. L. Wheeler, “The phase problem in scattering theory: the zeros of entire functions and their significance,” Proc. R. Soc. London Ser. A 360, 25–45 (1978).
[Crossref]

Nussenzveig, H.

H. Nussenzveig, “Phase problem in coherence theory,” J. Math. Phys. 8, 561–572 (1967).
[Crossref]

Perina, J.

J. Perina, Coherence of Light (Van Nostrand Reinhold, London, 1972), p. 55.

Rayleigh, Lord

Lord Rayleigh, “Note on the numerical calculation of the roots of fluctuating functions,” Proc. London Math. Soc. 5, 119–124 (1874).

Ronkin, L.

L. Ronkin, Introduction to the Theory of Entire Functions of Several Variables (American Mathematical Society, Providence, 1974), Chap. 4.

Saxton, W. O.

W. O. Saxton, Computer Techniques for Image Processing in Electron Microscopy (Academic, New York, 1978).

Shannon, C.

C. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423 (1948);Bell Syst. Tech. J. 27, 623–656 (1948).
[Crossref]

Shotat, J. A.

J. A. Shotat and J. D. Tamarkin, The Problem of Moments (American Mathematical Society, Providence, 1950).

Sneddon, I. N.

I. N. Sneddon, Special Functions of Mathematical Physics and Chemistry (Interscience, New York, 1961).

Tamarkin, J. D.

J. A. Shotat and J. D. Tamarkin, The Problem of Moments (American Mathematical Society, Providence, 1950).

Titchmarsh, E. C.

E. C. Titchmarsh, “The zeros of certain integral functions,” Proc. London Math. Soc. 25, 283–302 (1926).
[Crossref]

Walther, A.

A. Walther, “The question of phase retrieval in optics,” Opt. Acta 10, 41–49 (1963).
[Crossref]

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge University, Cambridge, 1944).

Watson, K. M.

M. L. Goldberger, H. W. Lewis, and K. M. Watson, “Use of intensity correlations to determine the phase of a scattering amplitude,” Phys. Rev. 1322764–2787 (1963).
[Crossref]

Wheeler, M. W. L.

R. E. Burge, M. A. Fiddy, M. Nieto-Vesperinas, and M. W. L. Wheeler, “The phase problem in scattering theory: the zeros of entire functions and their significance,” Proc. R. Soc. London Ser. A 360, 25–45 (1978).
[Crossref]

Wolf, E.

E. Wolf, “Is a complete determination of the energy spectrum of light possible from measurements of the degree of coherence?,” Proc. Phys. Soc. London 80, 1269–1272 (1962).
[Crossref]

M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon, New York, 1965).

Ynduráin, F. J.

F. J. Ynduráin, “The moment problem and applications,” in Padé Approximants, edited by P. Graves-Morris (Institute of Physics, London, 1973),p. 45.

Bell Syst. Tech. J. (1)

C. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423 (1948);Bell Syst. Tech. J. 27, 623–656 (1948).
[Crossref]

J. Math. Phys. (1)

H. Nussenzveig, “Phase problem in coherence theory,” J. Math. Phys. 8, 561–572 (1967).
[Crossref]

J. Opt. Soc. Am. (1)

Math. Comp. (1)

L. M. Delves and J. N. Lyness, “A numerical method for locating the zeros of an analytic function,” Math. Comp. 21, 543–560 (1967).
[Crossref]

N. Z. J. Sci. (1)

P. J. Napier, “The brightness temperature distributions defined by a measured intensity interferogram,” N. Z. J. Sci. 15, 342–355 (1972).

Opt. Acta (1)

A. Walther, “The question of phase retrieval in optics,” Opt. Acta 10, 41–49 (1963).
[Crossref]

Opt. Lett. (1)

Phys. Rev. (2)

M. L. Goldberger, H. W. Lewis, and K. M. Watson, “Use of intensity correlations to determine the phase of a scattering amplitude,” Phys. Rev. 1322764–2787 (1963).
[Crossref]

E. T. Jaynes, “Information theory and statistical mechanics,” Phys. Rev. 106, 620–630 (1957);Phys. Rev. 107, 17–190 (1957).
[Crossref]

Proc. Am. Math. Soc. (1)

E. J. Akutowicz, “On the determination of the phase of a Fourier integral, II.” Proc. Am. Math. Soc. 8, 234–238 (1957).

Proc. London Math. Soc. (2)

Lord Rayleigh, “Note on the numerical calculation of the roots of fluctuating functions,” Proc. London Math. Soc. 5, 119–124 (1874).

E. C. Titchmarsh, “The zeros of certain integral functions,” Proc. London Math. Soc. 25, 283–302 (1926).
[Crossref]

Proc. Phys. Soc. London (1)

E. Wolf, “Is a complete determination of the energy spectrum of light possible from measurements of the degree of coherence?,” Proc. Phys. Soc. London 80, 1269–1272 (1962).
[Crossref]

Proc. R. Soc. London Ser. A (1)

R. E. Burge, M. A. Fiddy, M. Nieto-Vesperinas, and M. W. L. Wheeler, “The phase problem in scattering theory: the zeros of entire functions and their significance,” Proc. R. Soc. London Ser. A 360, 25–45 (1978).
[Crossref]

Trans. Am. Math. Soc. (1)

E. J. Akutowicz, “On the determination of the phase of a Fourier integral, I.” Trans. Am. Math. Soc. 83, 179–192 (1956).

Other (15)

I. N. Sneddon, Special Functions of Mathematical Physics and Chemistry (Interscience, New York, 1961).

L. Ronkin, Introduction to the Theory of Entire Functions of Several Variables (American Mathematical Society, Providence, 1974), Chap. 4.

W. O. Saxton, Computer Techniques for Image Processing in Electron Microscopy (Academic, New York, 1978).

Inverse Source Problems in Optics, edited by H. P. Baltes, (Springer-Verlag, Berlin, 1978).
[Crossref]

M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon, New York, 1965).

J. Perina, Coherence of Light (Van Nostrand Reinhold, London, 1972), p. 55.

B. Ya. Levin, Distribution of Zeros of Entire Functions (American Mathematical Society, Providence, 1964).

P. Henrici, Applied and Computational Complex Analysis (Wiley-Interscience, New York, 1974), Vols. 1 and 2.

R. Fortet, Elements of Probability Theory (Gordon and Breach, New York, 1977), p. 271.

W. Elderton and N. Johnson, Systems of Frequency Curves (Cambridge University, Cambridge, 1969), Chap. 4.
[Crossref]

G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge University, Cambridge, 1944).

Handbook of Mathematical Functions, edited by H. Abramowitz and I. A. Stegun, Natl. Bur. Std., U.S. Appl. Math Ser. (U.S. Government Printing Office, Washington, D. C., 1964).

F. J. Ynduráin, “The moment problem and applications,” in Padé Approximants, edited by P. Graves-Morris (Institute of Physics, London, 1973),p. 45.

J. A. Shotat and J. D. Tamarkin, The Problem of Moments (American Mathematical Society, Providence, 1950).

R. P. Boas, Entire Functions (Academic, New York, 1954),p. 103.

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Figures (4)

FIG. 1
FIG. 1

Beta spectrum with ν = 3 and estimates: ——exact spectrum; –·– N = 10; --- N = 20.

FIG. 2
FIG. 2

Beta spectrum with ν = 3: —— exact spectrum; --- modified estimate with N = 10.

FIG. 3
FIG. 3

Nonsymmetric beta spectrum and estimates: —— exact spectrum; –·– N = 10; --- N = 20.

FIG. 4
FIG. 4

Nonsymmetric beta spectrum: —— exact spectrum; --- maximum entropy estimate given exact moments.

Tables (2)

Tables Icon

TABLE I Values of cumulants K2 and K4 in terms of first 2N zeros of J3(τ).

Tables Icon

TABLE II Values of cumulants in terms of first 2N zeros of M (2,3,− ).

Equations (52)

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γ ( τ ) = 0 g ( ω ) e i τ ω d ω < τ < .
g ( ω ) 0 ,
0 g ( ω ) d ω = 1 = γ ( 0 ) .
γ ( τ ) = a b g ( ω ) e i τ ω d ω 0 | τ | < .
γ ( τ ) = e ( 1 / 2 ) ( b + a ) i τ n = 1 ( 1 τ / τ n ) ,
γ ( τ ) = e γ ( 0 ) τ n = 1 [ ( 1 τ / τ n ) e τ / τ n ] .
Log γ = Log | γ | + i ϕ ,
Log γ ( τ ) = ( ½ ) ( b + a ) ( i τ ) + n = 1 Log ( 1 τ / τ n ) .
Log ( 1 τ τ n ) = l = 1 1 l ( τ τ n ) l | τ | < | τ 1 | ,
Log γ ( τ ) = n = 1 1 n ! K n ( i τ ) n | τ | < | τ 1 | .
K 1 = ( ½ ) ( b + a ) −; i l ( τ l ) 1 ,
K n = i n ( n 1 ) ! l ( τ l ) n n 2 ,
γ ( τ ) = n = 0 1 n ! ω n ( i τ ) n ,
ω n = a b ω n g ( ω ) d ω n = 0 , 1 ,
n = 1 1 n ! K n ( i τ ) n = L o g [ 1 + n = 1 1 n ! ω n ( i τ ) n ] ,
K 1 = ω , ω K 1 + K 2 = ω 2 , ( ½ ) ω 2 K 1 + ω K 2 + ( ½ ) K 3 = ( ½ ) ω 3 .
ω = K 1 , ω 2 K 2 + K 1 2 , ω 3 = K 3 + 3 K 2 K 1 + K 1 3 , ω 4 = K 4 + 4 K 3 K 1 + 3 K 2 2 + 6 K 2 K 1 2 + K 1 4 .
i l ( τ l ) 1 = ( ½ ) ( b + a ) ω
K n ~ i n ( n 1 ) ! τ 1 n n 1 .
g ( ω ) = ( ω a ) p 1 ( b ω ) q 1 B ( p , q ) ( b a ) p + q 1 a ω b .
B ( p , q ) = Γ ( p ) Γ ( q ) / Γ ( p + q ) .
mode = ( p 1 ) b + ( 1 1 ) a p + q 2 .
ω = 1 B ( p , q ) l = 0 n ( n l ) ( b n ) 1 l a l B ( p + l , q ) ,
g ( ω ) = ( 1 + ω ) ν 1 / 2 ( 1 ω ) ν 1 / 2 2 ν B ( ν + ½ , ν + ½ ) 1 ω 1 ,
γ ( τ ) = Γ ( 1 + ν ) J ν ( τ ) / ( τ / 2 ) ν γ ( 0 ) = 1 ,
K 2 n + 1 = ( 2 n ) ! i 2 n + 1 l = 1 [ ( j ν , l ) ( 2 n + 1 ) + ( j ν , l ) ( 2 n + 1 ) ] 0 ,
K 2 n = ( 2 n 1 ) ! i 2 n 2 l = 1 ( j ν , l ) 2 n = 2 ( 2 n 1 ) i 2 n σ ν ( n ) ,
σ ν ( n ) l = 1 ( j ν , l ) 2 n
σ ν ( 1 ) = 1 2 2 ( 1 + ν ) , σ ν ( 2 ) = 1 2 4 ( 1 + ν ) 2 ( 2 + ν ) ,
K 2 = 1 2 ν + 2 , K 4 = 3 4 ( ν + 1 ) 2 ( ν + 2 ) .
g ( ω ) = ( 1 ω 2 ) 1.224 B ( 0.5 , 2.224 ) ν = 1.724 ;
g ( ω ) = ( 1 ω 2 ) 1.784 B ( 0.5 , 2.784 ) ν = 2.284 ;
g ( ω ) = ( 1 ω 2 ) 2.5 B ( 0.5 , 3.5 ) ν = 3.000 .
g ( ω ) = ( 1 ω 2 ) 2.110 B ( 0.5 , 3.110 ) ν = 2.610 .
g ( ω ) = ω p = 1 ( 1 ω ) q 1 / B ( p , q ) 0 ω 1 .
γ ( τ ) = M ( p , q + q , i τ ) ,
γ ( τ ) = M ( 2 , 3 , i τ ) = 2 ( i τ ) 2 0 i τ x e x d x ,
γ ( τ ) = 2 ( i τ ) 2 [ 1 ( i τ ) e i τ e i τ ] .
K 1 = 2 / 5 , K 2 = 1 / 25 , K 3 = 2 / 875 , K 4 = 9 / 8750 .
g ( ω ) = ω 0.241 ( 1 ω ) 0.887 B ( 1.241 , 1.887 ) ;
g ( ω ) = ω 1.040 ( 1 ω ) 1.828 B ( 2.040 , 2.828 ) .
g ( ω ) = ω 1.042 ( 1 ω ) 2.103 B ( 1.042 , 2.103 ) .
H = a b g ( ω ) log [ g ( ω ) ] d ω
ω n a b ω n g ( ω ) d ω n = 1 , , N ,
g ( ω ) = g 0 exp ( n = 1 N λ n ω n ) ,
γ ( τ 1 , τ 2 ) = a b g ( ω 1 , ω w ) e i τ 1 ω 1 i τ 2 ω 2 d ω 1 d ω 2 .
β 1 = ( ω ω ) 3 2 / ( ω ω ) 2 3 ,
β 2 = ( ω ω ) 4 / ( ω ω ) 2 2 .
β 1 = K 3 2 / K 2 3 ,
β 2 = ( K 4 + 3 K 2 2 ) / K 2 2 .
r = 6 ( β 2 β 1 1 ) / ( 6 + 3 β 1 2 β 2 ) .
p , q = ½ [ ( r 2 ) ± r ( r + 2 ) × ( β 1 β 1 2 ( r + 2 ) 2 + 16 ( r + 1 ) ) 1 / 2 ] ,