Abstract

A calculational procedure for obtaining the complete set of scattering coefficients for a multilayered sphere is proposed. The procedure is based on the utilization of a prescription which relates the coefficients for an r-layered sphere to those for an (r − 1)-layered sphere. The prescription is derived directly from the determinantal form of the scattering amplitudes for a multilayered sphere. The complete set of coefficients considered includes the coefficients required to describe the fields within the various regions of the multilayered sphere.

© 1985 Optical Society of America

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References

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  1. P. Chylek, R. Bhandari, “Specific Absorption of Graphitic Carbon within a Double-Layered Sphere,” Appl Opt., submitted.
  2. A. L. Aden, “Electromagnetic Scattering from Spheres with Sizes Comparable to the Wavelength,” J. Appl. Phys. 22, 601 (1951).
    [CrossRef]
  3. J. V. Dave, “Subroutines for Computing the Parameters of the Electromagnetic Radiation Scattered by a Sphere,” Report 320-3237, IBM Scientific Center, Palo Alto, Calif. (1968).
  4. J. V. Dave, “Scattering of Visible Light by Large Water Spheres,” Appl. Opt. 8, 155 (1969).
    [CrossRef] [PubMed]
  5. G. A. Kattawar, D. A. Hood, “Electromagnetic Scattering from a Spherical Polydispersion of Coated Spheres,” Appl. Opt. 15, 1996 (1976).
    [CrossRef] [PubMed]
  6. O. B. Toon, T. P. Ackerman, “Algorithms for the Calculation of Scattering by Stratified Spheres,” Appl. Opt. 20, 3657 (1981).
    [CrossRef] [PubMed]
  7. J. R. Wait, “Electromagnetic Scattering from a Radially Inhomogeneous Sphere,” Appl. Sci. Res. Sect. B 10, 441 (1963).
    [CrossRef]
  8. H. E. Bussey, J. H. Richmond, “Scattering by a Lossy Dielectric Circular Cylindrical Multilayer, Numerical Values,” IEEE Trans.Antennas Propag. AP-23, 723 (1975).
    [CrossRef]
  9. Reference 7 (and also Ref. 8) came to the author's attention after the work reported in this paper had been carried out. The prescription for the multilayered sphere, as given in Ref. 7, is not in a form suitable for numerical calculation, but it can be transformed into a calculational form along lines similar to ours.
  10. R. Bhandari, “Scattering by a Double-Layered Sphere in the Limit of a Thin Inner Shell,” Appl. Opt., to be submitted.
  11. R. Bhandari, “Tiny Core or Thin Layer as a Perturbation in Scattering by a Single-Layered Sphere,” J. Opt. Soc. Am. A, submitted.
  12. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).
  13. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).
  14. G. W. Kattawar, G. N. Plass, “Electromagnetic Scattering from Absorbing Spheres,” Appl. Opt. 6, 1377 (1967).
    [CrossRef] [PubMed]

1981 (1)

1976 (1)

1975 (1)

H. E. Bussey, J. H. Richmond, “Scattering by a Lossy Dielectric Circular Cylindrical Multilayer, Numerical Values,” IEEE Trans.Antennas Propag. AP-23, 723 (1975).
[CrossRef]

1969 (1)

1967 (1)

1963 (1)

J. R. Wait, “Electromagnetic Scattering from a Radially Inhomogeneous Sphere,” Appl. Sci. Res. Sect. B 10, 441 (1963).
[CrossRef]

1951 (1)

A. L. Aden, “Electromagnetic Scattering from Spheres with Sizes Comparable to the Wavelength,” J. Appl. Phys. 22, 601 (1951).
[CrossRef]

Ackerman, T. P.

Aden, A. L.

A. L. Aden, “Electromagnetic Scattering from Spheres with Sizes Comparable to the Wavelength,” J. Appl. Phys. 22, 601 (1951).
[CrossRef]

Bhandari, R.

P. Chylek, R. Bhandari, “Specific Absorption of Graphitic Carbon within a Double-Layered Sphere,” Appl Opt., submitted.

R. Bhandari, “Scattering by a Double-Layered Sphere in the Limit of a Thin Inner Shell,” Appl. Opt., to be submitted.

R. Bhandari, “Tiny Core or Thin Layer as a Perturbation in Scattering by a Single-Layered Sphere,” J. Opt. Soc. Am. A, submitted.

Bussey, H. E.

H. E. Bussey, J. H. Richmond, “Scattering by a Lossy Dielectric Circular Cylindrical Multilayer, Numerical Values,” IEEE Trans.Antennas Propag. AP-23, 723 (1975).
[CrossRef]

Chylek, P.

P. Chylek, R. Bhandari, “Specific Absorption of Graphitic Carbon within a Double-Layered Sphere,” Appl Opt., submitted.

Dave, J. V.

J. V. Dave, “Scattering of Visible Light by Large Water Spheres,” Appl. Opt. 8, 155 (1969).
[CrossRef] [PubMed]

J. V. Dave, “Subroutines for Computing the Parameters of the Electromagnetic Radiation Scattered by a Sphere,” Report 320-3237, IBM Scientific Center, Palo Alto, Calif. (1968).

Hood, D. A.

Kattawar, G. A.

Kattawar, G. W.

Kerker, M.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

Plass, G. N.

Richmond, J. H.

H. E. Bussey, J. H. Richmond, “Scattering by a Lossy Dielectric Circular Cylindrical Multilayer, Numerical Values,” IEEE Trans.Antennas Propag. AP-23, 723 (1975).
[CrossRef]

Toon, O. B.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

Wait, J. R.

J. R. Wait, “Electromagnetic Scattering from a Radially Inhomogeneous Sphere,” Appl. Sci. Res. Sect. B 10, 441 (1963).
[CrossRef]

Appl. Opt. (4)

Appl. Sci. Res. Sect. B (1)

J. R. Wait, “Electromagnetic Scattering from a Radially Inhomogeneous Sphere,” Appl. Sci. Res. Sect. B 10, 441 (1963).
[CrossRef]

IEEE Trans.Antennas Propag. (1)

H. E. Bussey, J. H. Richmond, “Scattering by a Lossy Dielectric Circular Cylindrical Multilayer, Numerical Values,” IEEE Trans.Antennas Propag. AP-23, 723 (1975).
[CrossRef]

J. Appl. Phys. (1)

A. L. Aden, “Electromagnetic Scattering from Spheres with Sizes Comparable to the Wavelength,” J. Appl. Phys. 22, 601 (1951).
[CrossRef]

Other (7)

J. V. Dave, “Subroutines for Computing the Parameters of the Electromagnetic Radiation Scattered by a Sphere,” Report 320-3237, IBM Scientific Center, Palo Alto, Calif. (1968).

P. Chylek, R. Bhandari, “Specific Absorption of Graphitic Carbon within a Double-Layered Sphere,” Appl Opt., submitted.

Reference 7 (and also Ref. 8) came to the author's attention after the work reported in this paper had been carried out. The prescription for the multilayered sphere, as given in Ref. 7, is not in a form suitable for numerical calculation, but it can be transformed into a calculational form along lines similar to ours.

R. Bhandari, “Scattering by a Double-Layered Sphere in the Limit of a Thin Inner Shell,” Appl. Opt., to be submitted.

R. Bhandari, “Tiny Core or Thin Layer as a Perturbation in Scattering by a Single-Layered Sphere,” J. Opt. Soc. Am. A, submitted.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

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Figures (2)

Fig. 1
Fig. 1

Multilayered sphere which scatters an incident plane wave.12mi,xi are the refractive index and the size parameter for the ith region.

Fig. 2
Fig. 2

Homogeneous sphere, a single-layered sphere, and a double-layered sphere.

Equations (57)

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u ( r , j ) = ( 1 ) r j exp ( i ω t ) cos φ n = 1 m j ( i ) n 2 n + 1 n ( n + 1 ) P n 1 ( cos θ ) × { [ a n ( r , j ) δ j , r + 1 ] j n ( m j x ) c n ( r , j ) n n ( m j x ) } ,
υ ( r , j ) = ( 1 ) r j exp ( i ω t ) sin φ n = 1 m j ( i ) n 2 n + 1 n ( n + 1 ) P n 1 ( cos θ ) × { [ b n ( r , j ) δ j , r + 1 ] j n ( m j x ) d n ( r , j ) n n ( m j x ) } ( 1 j r + 1 ) .
σ ext = λ 2 2 π n = 1 ( 2 n + 1 ) Re [ a n ( r , r + 1 ) + b n ( r , r + 1 ) ] ,
σ sc = λ 2 2 π n = 1 ( 2 n + 1 ) [ | a n ( r , r + 1 ) | 2 + | b n ( r , r + 1 ) | 2 ] ,
σ abs = σ ext σ sc = λ 2 2 π n = 1 ( 2 n + 1 ) { Re [ a n ( r , r + 1 ) + b n ( r , r + 1 ) ] | a n ( r , r + 1 ) | 2 | b n ( r , r + 1 ) | 2 } .
a n ( r , r + 1 ) = N ( r , r + 1 ) / D ( r , r + 1 ) ,
N ( r , r + 1 ) = | m r + 1 ψ n ( m r + 1 x r ) m r ψ n ( m r x r ) m r χ n ( m r x r ) 0 0 0 0 ψ n ( m r + 1 x r ) ψ n ( m r x r ) χ n ( m r x r ) 0 0 0 0 0 m r ψ n ( m r x r 1 ) m r χ n ( m r x r 1 ) m r 1 ψ n ( m r 1 x r 1 ) m r 1 χ n ( m r 1 x r 1 ) 0 0 0 ψ n ( m r x r 1 ) χ n ( m r x r 1 ) ψ n ( m r 1 x r 1 ) χ n ( m r 1 x r 1 ) 0 0 0 0 0 m r 1 ψ n ( m r 1 x r 2 ) m r 1 χ n ( m r 1 x r 2 ) 0 0 0 0 ψ n ( m r 1 x r 2 ) χ n ( m r 1 x r 2 ) 0 : : : 0 0 : : : : : : m 1 ψ n ( m 1 x 1 ) 0 0 0 0 0 ψ n ( m 1 x 1 ) | .
N ( r + 1 , r + 2 ) = | m r + 2 ψ n ( m r + 2 x r + 1 ) m r + 1 ψ n ( m r + 1 x r + 1 ) m r + 1 χ n ( m r + 1 x r + 1 ) 0 0 0 0 ψ n ( m r + 2 x r + 1 ) ψ n ( m r + 1 x r + 1 ) χ n ( m r + 1 x r + 1 ) 0 0 0 0 0 m r + 1 ψ n ( m r + 1 x r ) m r + 1 χ n ( m r + 1 x r ) m r ψ n ( m r x r ) m r χ n ( m r x r ) 0 0 0 ψ n ( m r + 1 x r ) χ n ( m r + 1 x r ) ψ n ( m r x r ) χ n ( m r x r ) 0 0 0 0 0 m r ψ n ( m r x r 1 ) m r χ n ( m r x r 1 ) 0 0 0 0 ψ n ( m r x r 1 ) χ n ( m r x r 1 ) 0 : : : 0 0 : : : : : : m 1 ψ n ( m 1 x 1 ) 0 0 0 0 0 ψ n ( m 1 x 1 ) | .
N ( r + 1 , r + 2 ) = | | | 0 | | | 0 | | | m r + 1 χ n ( m r + 1 x r ) | | | χ n ( m r + 1 x r ) | | | 0 | | | | | | 0 | | + | | | m r + 1 χ n ( m r + 1 x r + 1 ) | | | χ n ( m r + 1 x r + 1 ) | | | 0 | | | 0 | | | 0 | | | | | | 0 | | .
N ( r + 1 , r + 2 ) = n H Ñ m ( r , r + 1 ) ñ H N m ( r , r + 1 ) ,
n H = n H ( m r + 2 , m r + 1 , x r + 1 ) = m r + 2 ψ n ( m r + 2 , x r + 1 ) ψ n ( m r + 1 x r + 1 ) m r + 1 ψ n ( m r + 2 x r + 1 ) ψ n ( m r + 1 x r + 1 ) , m r + 2 = 1 ,
ñ H = m r + 2 ψ n ( m r + 2 x r + 1 ) χ n ( m r + 1 x r + 1 ) m r + 1 ψ n ( m r + 2 x r + 1 ) χ n ( m r + 1 x r + 1 ) , m r + 2 = 1 ,
Ñ m ( r , r + 1 ) = N m ( r , r + 1 ) ,
N ( r , r + 1 ) = N ( 1 , 2 ) = n H ( m 2 , m 1 , x 1 ) = m 2 ψ n ( m 2 x 1 ) ψ n ( m 1 x 1 ) m 1 ψ n ( m 2 x 1 ) ψ n ( m 1 x 1 ) = W 1 ( m 2 = 1 ) .
N ( r , r + 1 ) = N ( 2 , 3 ) = n H ( m 3 , m 2 , x 2 ) Ñ m ( 1 , 2 ) ñ H ( m 3 , m 2 , x 2 ) N m ( 1 , 2 ) = W 2 W 6 W 7 W 1 ,
W 7 = m 3 ψ n ( m 3 x 2 ) χ n ( m 2 x 2 ) m 2 ψ n ( m 3 x 2 ) χ n ( m 2 x 2 ) , W 6 = m 2 χ n ( m 2 x 1 ) ψ n ( m 1 x 1 ) m 1 χ n ( m 2 x 1 ) ψ n ( m 1 x 1 ) , W 2 = m 3 ψ n ( m 3 x 2 ) ψ n ( m 2 x 2 ) m 2 ψ n ( m 3 x 2 ) ψ n ( m 2 x 2 ) , ( m 3 = 1 ) ,
N ( r , r + 1 ) = N ( 3 , 4 ) = n H ( m 4 , m 3 , x 3 ) Ñ ( 2 , 3 ) ñ H ( m 4 , m 3 , x 3 ) N m ( 2 , 3 ) = W 3 ( W 5 W 6 W 8 W 1 ) W 4 ( W 2 W 6 W 7 W 1 ) ,
W 3 = m 4 ψ n ( m 4 x 3 ) ψ n ( m 3 x 3 ) m 3 ψ n ( m 4 x 3 ) ψ n ( m 3 x 3 ) , W 4 = m 4 ψ n ( m 4 x 3 ) χ n ( m 3 x 3 ) m 3 ψ n ( m 4 x 3 ) ψ n ( m 3 x 3 ) , W 5 = m 3 χ n ( m 3 x 2 ) ψ n ( m 2 x 2 ) m 2 χ n ( m 3 x 2 ) ψ n ( m 2 x 2 ) , W 8 = m 3 χ n ( m 3 x 2 ) χ n ( m 2 x 2 ) m 2 χ n ( m 3 x 2 ) χ n ( m 2 x 2 ) , ( m 4 = 1 ) ,
a n ( r , r ) = N ( r , r ) / D ( r , r + 1 ) ,
N ( r , r ) = ( i ) Ñ m ( r 1 , r ) / D ( r , r + 1 ) .
a n ( r , j ) = ( 1 ) r j ( m 1 m 2 m r ) ( i ) Ñ m ( j 1 , j ) ( m 1 m 2 m j ) D ( r , r + 1 ) ( 1 j r ) .
c n ( r , j ) = ( 1 ) r j 1 ( m 1 m 2 m r ) ( i ) N m ( j 1 , j ) ( m 1 m 2 m j ) D ( r , r + 1 ) ( 2 j r ) , c n ( r , 1 ) = 0 , as pointed out earlier .
b n ( 2 , 3 ) = N ( 2 , 3 ) / D ( 2 , 3 ) ,
N ( 2 , 3 ) = W 2 W 6 W 7 W 1
W 1 = m 1 ψ n ( m 2 x 1 ) ψ n ( m 1 x 1 ) m 2 ψ n ( m 2 x 1 ) ψ n ( m 1 x 1 ) , W 7 = m 2 ψ n ( m 3 x 2 ) χ n ( m 2 x 2 ) m 3 ψ n ( m 3 x 2 ) χ n ( m 2 x 2 ) , W 6 = m 1 χ n ( m 2 x 1 ) ψ n ( m 1 x 1 ) m 2 χ n ( m 2 x 1 ) ψ n ( m 1 x 1 ) , W 2 = m 2 ψ n ( m 3 x 2 ) ψ n ( m 2 x 2 ) m 3 ψ n ( m 3 x 2 ) χ n ( m 2 x 2 ) , ( m 3 = 1 ) .
N ( 2 , 3 ) = | ψ n ( x 2 ) m 2 ψ n ( m 2 x 2 ) m 2 χ n ( m 2 x 2 ) 0 ψ n ( x 2 ) ψ n ( m 2 x 2 ) χ n ( m 2 x 2 ) 0 0 m 2 ψ n ( m 2 x 1 ) m 2 χ n ( m 2 x 1 ) m 1 ψ n ( m 1 x 1 ) 0 ψ n ( m 2 x 1 ) χ n ( m 2 x 1 ) ψ n ( m 1 x 1 ) | .
lim x 1 0 N ( 2 , 3 ) = | ψ n ( x 2 ) m 2 ψ n ( m 2 x 2 ) ψ n ( x 2 ) ψ n ( m 2 x 2 ) | | m 2 χ n ( m 2 x 1 ) m 1 ψ n ( m 2 x 1 ) χ n ( m 2 x 1 ) ψ n ( m 2 x 1 ) | .
D ( 2 , 3 ) = | ζ n ( x 2 ) m 2 ψ n ( m 2 x 2 ) ζ n ( x 2 ) ψ n ( m 2 x 2 ) | | m 2 χ n ( m 2 x 1 ) m 1 ψ n ( m 1 x 1 ) χ n ( m 2 x 1 ) ψ n ( m 1 x 1 ) | .
N ( r + 1 , r + 2 ) = ψ n ( m r + 2 x r + 1 ) ψ n ( m r + 1 x r + 1 ) ζ n ( m r + 1 x r ) ψ n ( m r x r ) × ζ n ( m r x r 1 ) ζ n ( m 2 x 1 ) ψ n ( m 1 x 1 ) × N R ( r + 1 , r + 2 ) ,
n H = ψ n ( m r + 2 x r + 1 ) ψ n ( m r + 1 x r + 1 ) × [ m r + 2 ψ n ( m r + 1 x r + 1 ) / ψ n ( m r + 1 x r + 1 ) m r + 1 ψ n ( m r + 2 x r + 1 ) / ψ n ( m r + 2 x r + 1 ) ] .
Ñ m ( r , r + 1 ) = ζ n ( m r + 1 x r ) ψ n ( m r x r ) ζ n ( m r x r 1 ) ψ n ( m r 1 x r 1 ) ζ n ( m 2 x 1 ) ψ n ( m 1 x 1 ) Ñ mR ( r , r + 1 ) ,
ñ H = ψ n ( m r + 2 x r + 1 ) ζ n ( m r + 1 x r ) [ m r + 2 ζ n ' ( m r + 1 x r + 1 ) ζ n ( m r + 1 x r + 1 ) ζ n ( m r + 1 x r + 1 ) ζ n ( m r + 1 x r ) m r + 1 ζ n ( m r + 1 x r + 1 ) ζ n ( m r + 1 x r ) ψ n ( m r + 2 x r + 1 ) ψ n ( m r + 2 x r + 1 ) ] .
N m ( r , r + 1 ) = ψ n ( m r + 1 x r ) ψ n ( m r x r ) ζ n ( m r x r 1 ) ζ n ( m 2 x 1 ) × ψ n ( m 1 x 1 ) N mR ( r , r + 1 ) = ψ n ( m r + 1 x r + 1 ) ψ n ( m r x r ) ζ n ( m r x r 1 ) ζ n ( m 2 x 1 ) × ψ n ( m 1 x 1 ) [ ψ n ( m r + 1 x r ) ψ n ( m r + 1 x r + 1 ) ] N mR ( r , r + 1 ) .
N R ( r + 1 , r + 2 ) = [ n HR Ñ mR ( r , r + 1 ) ζ n ( m r + 1 x r + 1 ) ζ n ( m r + 1 x r ) ψ n ( m r + 1 x r ) ψ n ( m r + 1 x r + 1 ) ñ HR N mR ( r , r + 1 ) ] ,
n HR = m r + 2 ψ n ( m r + 1 x r + 1 ) ψ n ( m r + 1 x r + 1 ) m r + 1 ψ n ( m r + 2 x r + 1 ) ψ n ( m r + 2 x r + 1 ) ,
ñ HR = m r + 2 ζ n ' ( m r + 1 x r + 1 ) ζ n ( m r + 1 x r + 1 ) m r + 1 ψ n ' ( m r + 2 x r + 1 ) ψ n ( m r + 2 x r + 1 ) .
D ( r + 1 , r + 2 ) = ζ n ( m r + 2 x r + 1 ) ψ n ( m r + 1 x r + 1 ) × ζ n ( m r + 1 x r ) ζ n ( m 2 x 1 ) ψ n ( m 1 x 1 ) D R ( r + 1 , r + 2 ) ,
a n ( r + 1 , r + 2 ) = N ( r + 1 , r + 2 ) / D ( r + 1 , r + 2 ) = ψ n ( m r + 2 x r + 1 ) ζ n ( m r + 2 x r + 1 ) N R ( r + 1 , r + 2 ) / D R ( r + 1 , r + 2 ) .
N R ( 1 , 2 ) = n HR = m 2 ψ n ( m 1 x 1 ) ψ n ( m 1 x 1 ) m 1 ψ n ( m 2 x 1 ) ψ n ( m 2 x 1 ) .
A n 1 ( z ) = n / z 1 / [ A n ( z ) + n / z ]
A N ( z ) = 0.0 + i 0.0 , N > n max ,
F n ( z ) = n / z + 1 / [ n / z F n 1 ( z ) ] ,
F 0 ( z ) = i .
ψ n ( m i x j ) ψ n ( m i x j + 1 ) = ψ n 1 ( m i x j ) ψ n 1 ( m i x j + 1 ) ( x j x j + 1 ) [ m i x j + 1 A n ( m i x j + 1 ) + n ] [ m i x j A n ( m i x j ) + n ] ,
ψ o ( m i x j ) ψ o ( m i x j + 1 )
ψ o ( m i x j ) ψ o ( m i x j + 1 ) = exp ( i a 1 ) exp ( i a 1 ) exp ( 2 b 1 ) exp ( i a 2 ) exp ( i a 2 ) exp ( 2 b 2 ) exp ( b 1 b 2 ) ,
ζ n ( m i x j + 1 ) ζ n ( m i x j ) = ζ n 1 ( m i x j + 1 ) ζ n 1 ( m i x j ) ( x j x j + 1 ) [ n x j + 1 F n 1 ( m i x j + 1 ) n x j F n 1 ( m i x j ) ] .
ζ n ( z ) = ( n / z ) ζ n 1 ( z ) ζ n 1 ( z ) .
ζ o ( m i x j + 1 ) / ζ o ( m i x j ) = exp [ i ( a 2 a 1 ) ] exp [ ( b 1 b 2 ) ] ,
N R ( 1 , 2 ) = G 1 ( m 2 = 1 ) ;
N R ( 2 , 3 ) = G 2 G 6 S n Q n G 7 G 1 ( m 3 = 1 ) ;
N R ( 3 , 4 ) = G 3 ( G 5 G 6 S n Q n G 8 G 1 ) R n P n G 4 ( G 2 G 6 S n Q n G 7 G 1 ) ( m 4 = 1 ) .
G 1 = m 2 A n ( z 1 ) m 1 A n ( z 2 ) , G 2 = m 3 A n ( z 3 ) m 2 A n ( z 4 ) , G 6 = m 2 A n ( z 1 ) m 1 F n ( z 2 ) , G 7 = m 3 F n ( z 3 ) m 2 A n ( z 4 ) , G 3 = m 4 A n ( z 5 ) m 3 A n ( z 6 ) , G 5 = m 3 A n ( z 3 ) m 2 F n ( z 4 ) , G 8 = m 3 F n ( z 3 ) m 2 F n ( z 4 ) , G 4 = m 4 F n ( z 5 ) m 3 A n ( z 6 ) , }
z 1 = m 1 x 1 , z 2 = m 2 x 1 , z 3 = m 2 x 2 , z 4 = m 3 x 2 , z 5 = m 3 x 3 , z 6 = m 4 x 3 ,
A n ( z i ) = ψ n ( z i ) / ψ n ( z i ) , F n ( z i ) = ζ n ( z i ) / ζ n ( z i ) ,
S n = ζ n ( z 3 ) / ζ n ( z 2 ) , R n = ζ n ( z 5 ) / ζ n ( z 4 ) ,
Q n = ψ n ( z 2 ) / ψ n ( z 3 ) , P n = ψ n ( z 4 ) / ψ n ( z 5 ) .

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