Abstract

Optical systems produce canonical transformations on phase space that are nonlinear. When a power expansion of the coordinates is performed around a chosen optical axis, the linear part is the paraxial approximation, and the nonlinear part is the ideal of aberrations. When the optical system has axial symmetry, its linear part is the symplectic group Sp(2, R) represented by 2 × 2 matrices. It is used to provide a classification of aberrations into multiplets of spin that are irreducible under the group, in complete analogy with the quantum harmonic-oscillator states. The “magnetic” axis of the latter may be chosen to adapt to magnifying systems or to optical fiberlike media. There seems to be a significant computational advantage in using the symplectic classification of aberrations.

© 1988 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. Sánchez-Mondragón, K. B. Wolf, eds., Proceedings of the CIFMO-CIO Workshop on Lie Methods in Optics, Vol. 250 of Lecture Notes in Physics (Springer-Verlag, Heidelberg, 1986).
    [CrossRef]
  2. A. J. Dragt, “Lie algebraic theory of geometrical optics and optical aberrations,”J. Opt. Soc. Am. 72, 372–379 (1982).
    [CrossRef]
  3. V. Guillemin, S. Sternberg, Symplectic Techniques in Physics (Cambridge U. Press, Cambridge, 1984).
  4. A. J. Dragt, E. Forest, K. B. Wolf, in Proceedings of the CIFMO-CIO Workshop on Lie Methods in Optics, J. Sánchez-Mondragón, K. B. Wolf, eds., Vol. 250 of Lecture Notes in Physics (Springer-Verlag, Heidelberg, 1986), pp. 105–157.
    [CrossRef]
  5. T. Sekiguchi, K. B. Wolf, “The Hamiltonian formulation of optics,” Am. J. Phys. 55, 830–835 (1987).
    [CrossRef]
  6. H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1959).
  7. J. R. Klauder, E. C. G. Sudarshan, Fundamentals of Quantum Optics (Benjamin, New York, 1968).
  8. A. J. Dragt, “Lectures on Nonlinear Orbit Dynamics,”AIP Conf. Proc. 87 (1982).
    [CrossRef]
  9. A. J. Dragt, J. Finn, “Lie series and invariant functions for analytic symplectic maps,”J. Math. Phys. 17, 2215–2227 (1976); see also A. J. Dragt, E. Forest, “Computation of nonlinear behavior of Hamiltonian systems using Lie algebraic methods,”J. Math. Phys. 24, 2734–2744 (1983).
    [CrossRef]
  10. K. B. Wolf, “Symmetry in Lie optics,” Ann. Phys. 172, 1–25 (1986).
    [CrossRef]
  11. K. B. Wolf, “On time-dependent quadratic quantum Hamiltonians,” SIAM J. Appl. Math. 40, 419–431 (1981).
    [CrossRef]
  12. S. Steinberg, in Proceedings of the CIFMO-CIO Workshop on Lie Methods in Optics, J. Sánchez-Mondragón, K. B. Wolf, eds., Vol. 250 of Lecture Notes in Physics (Springer-Verlag, Heidelberg, 1986), pp. 45–103.
    [CrossRef]
  13. M. Navarro-Saad, K. B. Wolf, “Factorization of the phase-space transformation produced by an arbitrary refracting surface,” J. Opt. Soc. Am. A 3, 340–346 (1986).
    [CrossRef]
  14. See Ref. 1. Add. A.
  15. H. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge U. Press, Cambridge, 1970).
  16. This aberration coefficient appears as p6, the missing coefficient, in Ref. 11, Sec. 121, and is treated in Sec. 125 with some incidental remarks on duality (i.e., Fourier conjugation).
  17. M. Navarro-Saad, K. B. Wolf, “The group-theoretical treatment of aberrating systems. I. Aligned lens systems in third aberration order,”J. Math. Phys. 27, 1449–1457 (1986).
    [CrossRef]
  18. L. C. Biedenharn, J. D. Louck, Angular Momentum in Quantum Physics, Vol. 8 of Encyclopedia of Mathematics, G.-C. Rota, ed. (Addison-Wesley, Reading, Mass., 1981), Sec. 6.16.
  19. K. B. Wolf, “The group-theoretical treatment of aberrating systems. II. Axis-symmetric inhomogeneous systems and fiber optics in third aberration order,”J. Math. Phys. 27, 1458–1465 (1986).
    [CrossRef]
  20. V. Bargmann, “On a Hubert space of analytic functions and an associated integral transform,” Commun. Pure Appl. Math. 14, 187–214 (1961).
    [CrossRef]
  21. K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, New York, 1979), Chap. 9.
  22. M. Navarro-Saad, K. B. Wolf, “Applications of a factorization theorem for ninth-order aberration optics,”J. Symbolic Comp. 1, 235–239 (1985).
    [CrossRef]
  23. A. J. Dragt, “Elementary and advanced Lie algebraic methods with applications to accelerator design, electron microscopes and light optics,” Nucl. Instrum. Meth. Phys. Res. A 258, 339–354 (1987).
    [CrossRef]
  24. K. B. Wolf, “The group-theoretical treatment of aberrating systems. III. The classification of asymmetric aberrations,”J. Math. Phys. 28, 2498–2507 (1987).
    [CrossRef]

1987 (3)

T. Sekiguchi, K. B. Wolf, “The Hamiltonian formulation of optics,” Am. J. Phys. 55, 830–835 (1987).
[CrossRef]

A. J. Dragt, “Elementary and advanced Lie algebraic methods with applications to accelerator design, electron microscopes and light optics,” Nucl. Instrum. Meth. Phys. Res. A 258, 339–354 (1987).
[CrossRef]

K. B. Wolf, “The group-theoretical treatment of aberrating systems. III. The classification of asymmetric aberrations,”J. Math. Phys. 28, 2498–2507 (1987).
[CrossRef]

1986 (4)

K. B. Wolf, “Symmetry in Lie optics,” Ann. Phys. 172, 1–25 (1986).
[CrossRef]

M. Navarro-Saad, K. B. Wolf, “Factorization of the phase-space transformation produced by an arbitrary refracting surface,” J. Opt. Soc. Am. A 3, 340–346 (1986).
[CrossRef]

M. Navarro-Saad, K. B. Wolf, “The group-theoretical treatment of aberrating systems. I. Aligned lens systems in third aberration order,”J. Math. Phys. 27, 1449–1457 (1986).
[CrossRef]

K. B. Wolf, “The group-theoretical treatment of aberrating systems. II. Axis-symmetric inhomogeneous systems and fiber optics in third aberration order,”J. Math. Phys. 27, 1458–1465 (1986).
[CrossRef]

1985 (1)

M. Navarro-Saad, K. B. Wolf, “Applications of a factorization theorem for ninth-order aberration optics,”J. Symbolic Comp. 1, 235–239 (1985).
[CrossRef]

1982 (2)

1981 (1)

K. B. Wolf, “On time-dependent quadratic quantum Hamiltonians,” SIAM J. Appl. Math. 40, 419–431 (1981).
[CrossRef]

1976 (1)

A. J. Dragt, J. Finn, “Lie series and invariant functions for analytic symplectic maps,”J. Math. Phys. 17, 2215–2227 (1976); see also A. J. Dragt, E. Forest, “Computation of nonlinear behavior of Hamiltonian systems using Lie algebraic methods,”J. Math. Phys. 24, 2734–2744 (1983).
[CrossRef]

1961 (1)

V. Bargmann, “On a Hubert space of analytic functions and an associated integral transform,” Commun. Pure Appl. Math. 14, 187–214 (1961).
[CrossRef]

Bargmann, V.

V. Bargmann, “On a Hubert space of analytic functions and an associated integral transform,” Commun. Pure Appl. Math. 14, 187–214 (1961).
[CrossRef]

Biedenharn, L. C.

L. C. Biedenharn, J. D. Louck, Angular Momentum in Quantum Physics, Vol. 8 of Encyclopedia of Mathematics, G.-C. Rota, ed. (Addison-Wesley, Reading, Mass., 1981), Sec. 6.16.

Buchdahl, H.

H. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge U. Press, Cambridge, 1970).

Dragt, A. J.

A. J. Dragt, “Elementary and advanced Lie algebraic methods with applications to accelerator design, electron microscopes and light optics,” Nucl. Instrum. Meth. Phys. Res. A 258, 339–354 (1987).
[CrossRef]

A. J. Dragt, “Lie algebraic theory of geometrical optics and optical aberrations,”J. Opt. Soc. Am. 72, 372–379 (1982).
[CrossRef]

A. J. Dragt, “Lectures on Nonlinear Orbit Dynamics,”AIP Conf. Proc. 87 (1982).
[CrossRef]

A. J. Dragt, J. Finn, “Lie series and invariant functions for analytic symplectic maps,”J. Math. Phys. 17, 2215–2227 (1976); see also A. J. Dragt, E. Forest, “Computation of nonlinear behavior of Hamiltonian systems using Lie algebraic methods,”J. Math. Phys. 24, 2734–2744 (1983).
[CrossRef]

A. J. Dragt, E. Forest, K. B. Wolf, in Proceedings of the CIFMO-CIO Workshop on Lie Methods in Optics, J. Sánchez-Mondragón, K. B. Wolf, eds., Vol. 250 of Lecture Notes in Physics (Springer-Verlag, Heidelberg, 1986), pp. 105–157.
[CrossRef]

Finn, J.

A. J. Dragt, J. Finn, “Lie series and invariant functions for analytic symplectic maps,”J. Math. Phys. 17, 2215–2227 (1976); see also A. J. Dragt, E. Forest, “Computation of nonlinear behavior of Hamiltonian systems using Lie algebraic methods,”J. Math. Phys. 24, 2734–2744 (1983).
[CrossRef]

Forest, E.

A. J. Dragt, E. Forest, K. B. Wolf, in Proceedings of the CIFMO-CIO Workshop on Lie Methods in Optics, J. Sánchez-Mondragón, K. B. Wolf, eds., Vol. 250 of Lecture Notes in Physics (Springer-Verlag, Heidelberg, 1986), pp. 105–157.
[CrossRef]

Goldstein, H.

H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1959).

Guillemin, V.

V. Guillemin, S. Sternberg, Symplectic Techniques in Physics (Cambridge U. Press, Cambridge, 1984).

Klauder, J. R.

J. R. Klauder, E. C. G. Sudarshan, Fundamentals of Quantum Optics (Benjamin, New York, 1968).

Louck, J. D.

L. C. Biedenharn, J. D. Louck, Angular Momentum in Quantum Physics, Vol. 8 of Encyclopedia of Mathematics, G.-C. Rota, ed. (Addison-Wesley, Reading, Mass., 1981), Sec. 6.16.

Navarro-Saad, M.

M. Navarro-Saad, K. B. Wolf, “The group-theoretical treatment of aberrating systems. I. Aligned lens systems in third aberration order,”J. Math. Phys. 27, 1449–1457 (1986).
[CrossRef]

M. Navarro-Saad, K. B. Wolf, “Factorization of the phase-space transformation produced by an arbitrary refracting surface,” J. Opt. Soc. Am. A 3, 340–346 (1986).
[CrossRef]

M. Navarro-Saad, K. B. Wolf, “Applications of a factorization theorem for ninth-order aberration optics,”J. Symbolic Comp. 1, 235–239 (1985).
[CrossRef]

Sekiguchi, T.

T. Sekiguchi, K. B. Wolf, “The Hamiltonian formulation of optics,” Am. J. Phys. 55, 830–835 (1987).
[CrossRef]

Steinberg, S.

S. Steinberg, in Proceedings of the CIFMO-CIO Workshop on Lie Methods in Optics, J. Sánchez-Mondragón, K. B. Wolf, eds., Vol. 250 of Lecture Notes in Physics (Springer-Verlag, Heidelberg, 1986), pp. 45–103.
[CrossRef]

Sternberg, S.

V. Guillemin, S. Sternberg, Symplectic Techniques in Physics (Cambridge U. Press, Cambridge, 1984).

Sudarshan, E. C. G.

J. R. Klauder, E. C. G. Sudarshan, Fundamentals of Quantum Optics (Benjamin, New York, 1968).

Wolf, K. B.

T. Sekiguchi, K. B. Wolf, “The Hamiltonian formulation of optics,” Am. J. Phys. 55, 830–835 (1987).
[CrossRef]

K. B. Wolf, “The group-theoretical treatment of aberrating systems. III. The classification of asymmetric aberrations,”J. Math. Phys. 28, 2498–2507 (1987).
[CrossRef]

M. Navarro-Saad, K. B. Wolf, “Factorization of the phase-space transformation produced by an arbitrary refracting surface,” J. Opt. Soc. Am. A 3, 340–346 (1986).
[CrossRef]

M. Navarro-Saad, K. B. Wolf, “The group-theoretical treatment of aberrating systems. I. Aligned lens systems in third aberration order,”J. Math. Phys. 27, 1449–1457 (1986).
[CrossRef]

K. B. Wolf, “The group-theoretical treatment of aberrating systems. II. Axis-symmetric inhomogeneous systems and fiber optics in third aberration order,”J. Math. Phys. 27, 1458–1465 (1986).
[CrossRef]

K. B. Wolf, “Symmetry in Lie optics,” Ann. Phys. 172, 1–25 (1986).
[CrossRef]

M. Navarro-Saad, K. B. Wolf, “Applications of a factorization theorem for ninth-order aberration optics,”J. Symbolic Comp. 1, 235–239 (1985).
[CrossRef]

K. B. Wolf, “On time-dependent quadratic quantum Hamiltonians,” SIAM J. Appl. Math. 40, 419–431 (1981).
[CrossRef]

A. J. Dragt, E. Forest, K. B. Wolf, in Proceedings of the CIFMO-CIO Workshop on Lie Methods in Optics, J. Sánchez-Mondragón, K. B. Wolf, eds., Vol. 250 of Lecture Notes in Physics (Springer-Verlag, Heidelberg, 1986), pp. 105–157.
[CrossRef]

K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, New York, 1979), Chap. 9.

AIP Conf. Proc. (1)

A. J. Dragt, “Lectures on Nonlinear Orbit Dynamics,”AIP Conf. Proc. 87 (1982).
[CrossRef]

Am. J. Phys. (1)

T. Sekiguchi, K. B. Wolf, “The Hamiltonian formulation of optics,” Am. J. Phys. 55, 830–835 (1987).
[CrossRef]

Ann. Phys. (1)

K. B. Wolf, “Symmetry in Lie optics,” Ann. Phys. 172, 1–25 (1986).
[CrossRef]

Commun. Pure Appl. Math. (1)

V. Bargmann, “On a Hubert space of analytic functions and an associated integral transform,” Commun. Pure Appl. Math. 14, 187–214 (1961).
[CrossRef]

J. Math. Phys. (4)

K. B. Wolf, “The group-theoretical treatment of aberrating systems. III. The classification of asymmetric aberrations,”J. Math. Phys. 28, 2498–2507 (1987).
[CrossRef]

A. J. Dragt, J. Finn, “Lie series and invariant functions for analytic symplectic maps,”J. Math. Phys. 17, 2215–2227 (1976); see also A. J. Dragt, E. Forest, “Computation of nonlinear behavior of Hamiltonian systems using Lie algebraic methods,”J. Math. Phys. 24, 2734–2744 (1983).
[CrossRef]

M. Navarro-Saad, K. B. Wolf, “The group-theoretical treatment of aberrating systems. I. Aligned lens systems in third aberration order,”J. Math. Phys. 27, 1449–1457 (1986).
[CrossRef]

K. B. Wolf, “The group-theoretical treatment of aberrating systems. II. Axis-symmetric inhomogeneous systems and fiber optics in third aberration order,”J. Math. Phys. 27, 1458–1465 (1986).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

J. Symbolic Comp. (1)

M. Navarro-Saad, K. B. Wolf, “Applications of a factorization theorem for ninth-order aberration optics,”J. Symbolic Comp. 1, 235–239 (1985).
[CrossRef]

Nucl. Instrum. Meth. Phys. Res. A (1)

A. J. Dragt, “Elementary and advanced Lie algebraic methods with applications to accelerator design, electron microscopes and light optics,” Nucl. Instrum. Meth. Phys. Res. A 258, 339–354 (1987).
[CrossRef]

SIAM J. Appl. Math. (1)

K. B. Wolf, “On time-dependent quadratic quantum Hamiltonians,” SIAM J. Appl. Math. 40, 419–431 (1981).
[CrossRef]

Other (11)

S. Steinberg, in Proceedings of the CIFMO-CIO Workshop on Lie Methods in Optics, J. Sánchez-Mondragón, K. B. Wolf, eds., Vol. 250 of Lecture Notes in Physics (Springer-Verlag, Heidelberg, 1986), pp. 45–103.
[CrossRef]

V. Guillemin, S. Sternberg, Symplectic Techniques in Physics (Cambridge U. Press, Cambridge, 1984).

A. J. Dragt, E. Forest, K. B. Wolf, in Proceedings of the CIFMO-CIO Workshop on Lie Methods in Optics, J. Sánchez-Mondragón, K. B. Wolf, eds., Vol. 250 of Lecture Notes in Physics (Springer-Verlag, Heidelberg, 1986), pp. 105–157.
[CrossRef]

H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1959).

J. R. Klauder, E. C. G. Sudarshan, Fundamentals of Quantum Optics (Benjamin, New York, 1968).

See Ref. 1. Add. A.

H. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge U. Press, Cambridge, 1970).

This aberration coefficient appears as p6, the missing coefficient, in Ref. 11, Sec. 121, and is treated in Sec. 125 with some incidental remarks on duality (i.e., Fourier conjugation).

J. Sánchez-Mondragón, K. B. Wolf, eds., Proceedings of the CIFMO-CIO Workshop on Lie Methods in Optics, Vol. 250 of Lecture Notes in Physics (Springer-Verlag, Heidelberg, 1986).
[CrossRef]

L. C. Biedenharn, J. D. Louck, Angular Momentum in Quantum Physics, Vol. 8 of Encyclopedia of Mathematics, G.-C. Rota, ed. (Addison-Wesley, Reading, Mass., 1981), Sec. 6.16.

K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, New York, 1979), Chap. 9.

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.


Figures (2)

Fig. 1
Fig. 1

The sextet of third-order aberrations in the monomial (Seidel) classification: S, spherical aberration; C, coma; A, astigmatism; F, curvature of field; D, distortion; P, pocus. Fourier conjugation reflects across the AF line.

Fig. 2
Fig. 2

Harmonic-oscillator, symplectic classification of aberrations. The k = 2 level contains the third-order aberrations: a singlet (j = 0) and a quintuplet (j = 2). Both aberration multiplets transform irreducibly under the paraxial subgroup. The Seidel magnetic-number classification shown is that of pure magnifiers.

Equations (50)

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

w = ( p q ) , p = ( p x p y ) , q = ( q x q y ) .
{ f , g } ( p , q ) = f q · g p f p · g q = ( : f : g ) ( p , q ) .
[ exp ( z : f : ) g ] ( p , q ) = [ m = 0 1 m ! ( z : f : ) m g ] ( p , q ) = m = 0 z m m ! { f , { f , { { f , g } } } } ( p , q ) = g ( p , q ; z ) .
d d z g ( p , q ; z ) = : f : g ( p , q ; z ) , g ( p , q ; 0 ) = g ( p , q ) .
H ( p , q ) = [ n ( q ) 2 p 2 ] 1 / 2 ,
[ exp ( : x · p + y · q : ) g ] ( p , q ) = g ( p + y , q x ) .
exp ( : α p 2 + β p · q + γ q 2 : ) ( p q ) = [ cos u + β sinc u 2 γ sinc u 2 α sinc u cos u β sinc u ] ( p q ) ,
u = ± ( 4 α γ β 2 ) 1 / 2 , sinc x = x 1 sin x .
m = exp : f 5 : exp : f 4 : exp : f 3 : exp : f 2 :
p 2 = 2 ξ + = ( ξ 1 + i ξ 2 ) ,
p · q = ξ 0 = ξ 3 ,
q 2 = 2 ξ = ξ 1 i ξ 2 ,
m f = exp : f 8 : exp : f 6 : exp : f 4 :
exp : f 2 n : exp : f 2 m : = exp : g 2 ( n + m 1 ) : exp : f 2 m : exp : f 2 n : .
G ( f ; M ) = G ( f 2 k , f 2 k 2 , , f 4 ; M ) = G ( f ; 1 ) exp : f 2 : .
G ( f ; M ) G ( g ; N ) = G ( f # [ exp : f 2 : g ] ; MN ) .
f 2 k ( p , q ) = k + + k 0 + k = k υ k + , k 0 , k ( k ) M k + , k 0 , k ( ξ ) ,
M k + , k 0 , k ( ξ ) = ( p 2 ) k + ( p · q ) k 0 ( q 2 ) k .
exp : f 2 k : q = q υ k + k 0 k ( k ) [ 2 M k + 1 , k 0 , k p + M k + , k 0 1 , k q ] + .
υ k , 0,0 , υ k 1,1,0 , υ k 2,2,0 , υ k 1,0,1 , υ k 3,3,0 , υ k 2,1,1 , υ 0 k 0 , υ 1 , k 2,1 , , υ 0,3 , k 3 , υ 1,1 , k 2 , υ 0,2 , k 2 , υ 1,0 , k 1 , υ 0,1 , k 1 , υ 0,0 , k spherical aberration ( S ) , circular coma ( C ) , oblique spherical aberration , ( nameless ) , υ k / 2,0 , k / 2 or υ ( k 1 ) / 2,1 , ( k 1 ) / 2 , elliptical coma , curvature of field ( F ) , astigmatism ( A ) , distortion ( D ) , ( nameless ) ( P ) .
ξ 2 = ξ 1 2 + ξ 2 2 + ξ 3 2 = ξ 0 2 2 ξ + ξ = ( p · q ) 2 p 2 q 2 = ( p × q ) 2 ,
exp ( α : p · q : ) ( p q ) = [ e α 0 0 e α ] ( p q ) ,
exp ( α : ξ 0 : ) g ( ξ + , ξ 0 , ξ ) = g ( e 2 α ξ + , ξ 0 , e 2 α ξ ) .
χ j m j ( ξ ) = [ 4 π ( 2 j + 1 ) ( j + m ) ! ( j m ) ! ( 2 j 1 ) ! ! ] 1 / 2 y m i ( ξ ) = ( j + m ) ! ( j m ) ! 2 m / 2 ( 2 j 1 ) ! ! × ν 1 2 ν ξ + m + ν ( m + ν ) ! ξ 0 j m 2 ν ( j m 2 ν ) ! ξ ν ν ! ,
χ k m j ( ξ ) = ( ξ 2 ) ( k j ) / 2 χ j m j ( ξ )
f 2 k ( p , q ) = j = k , ( 2 ) 1 or 0 m = + j j x k m j χ k m j ( ξ ) .
χ 2 0 0 ( ξ ) = 2 ξ + ξ ξ 0 2 = p 2 q 2 ( p · q ) 2 = ( p × q ) 2 ,
χ 0 2 ( ξ ) = 1 3 ξ 2 + ξ 0 2 = 2 3 ( ξ + ξ + ξ 0 2 ) = 1 3 p 2 q 2 + 2 3 ( p · q ) 2 .
H osc = 1 2 ( p 2 + q 2 ) = 2 ( ξ + + ξ ) = i ξ 2 ,
B = exp ( 1 8 i π : p 2 q 2 : ) .
B ( p q ) = 1 2 [ 1 i i 1 ] ( p q ) = ( ( p + i q ) / 2 ( q + i p ) / 2 ) = ( η ζ )
B ( ξ + ξ 0 ξ ) = [ 1 2 i / 2 1 2 i / 2 0 i / 2 1 2 i / 2 1 2 ] ( ξ + ξ 0 ξ ) = ( σ + σ 0 σ ) ,
σ 1 = ( σ σ + ) / 2 = ξ 1 = 1 2 ( p 2 q 2 ) ,
σ 2 = i ( σ + σ + ) / 2 = ξ 3 = p · q ,
σ 3 = σ 0 = ξ 2 = i ( 1 2 ) ( p 2 + q 2 ) .
B χ k m j ( ξ ) = m = j j B m m j k χ m j ( ξ ) = χ k m j ( σ ) ,
B m m j = ( j + m ) ! ( j m ) ! ( 2 ) j i m + m × ν ( 1 ) ν ( ν m m ) ( j + m ν ) ! ( j + m ν ) ! ν ! .
f 2 k ( p , q ) = j = k , ( 2 ) 1 or 0 m = j j s k m j χ k m j ( σ ) ,
s k m j = m x k m j B m m j ,
x k m j = m s k m j B m m j * .
n ( q ) = [ n 0 2 q 2 β ( q 2 ) 2 + ] 1 / 2 .
H f = n 0 + 1 2 n 0 ( p 2 + q 2 ) + 1 8 n 0 3 ( p 2 + q 2 ) 2 + β 2 n 0 ( q 2 ) 2 = n 0 i n 0 χ 1 0 1 ( σ ) + β 8 n 0 [ χ 2 2 2 ( σ ) + χ 2 2 2 ( σ ) ] + i β 2 n 0 [ χ 2 1 2 ( σ ) χ 2 1 2 ( σ ) ] ( 1 2 n 0 3 + 3 β 4 n 0 ) χ 2 0 2 ( σ ) + 1 6 n 0 3 χ 2 0 0 ( σ ) .
exp ( : ω χ 0 1 + a · χ 2 + b χ 0 0 : ) exp ( : a · χ 2 + b χ 0 0 : ) × exp ( : ω χ 0 1 : ) ,
a m = a m e 2 m ω 1 2 m ω , m = 2,1,0 , 1 , 2 ( a 0 = a 0 ) ,
s 2 2 = 1 32 i β [ 1 exp ( 4 i z / n 0 ) ] = s 2 2 * ,
s 1 2 = 1 4 β [ 1 exp ( 2 i z / n 0 ) ] = s 1 2 * ,
s 0 2 = ( 1 2 n 0 3 + 3 β 4 n 0 ) z , s 0 0 = 1 6 n 0 3 z .
exp ( : z H f : ) ζ exp ( i z / n 0 ) [ ζ + ( 4 s 2 2 η 2 2 s 1 2 η · ζ { 2 3 s 0 2 + 1 2 s 0 0 } ζ 2 ) η + ( s 1 2 η 2 { 4 3 s 0 2 1 2 s 0 0 } η · ζ s 1 2 ζ 2 ) ζ ] .
{ M a b c , M a b c } = 4 ( c a a c ) M a + a 1 , b + b + 1 , c + c 1 + 2 ( b a a b + c b b c ) × M a + a , b + b 1 , c + c .
{ χ k m j , χ k m j } = j = | m 1 + m 2 | k + k + j odd j + j 1 S m m j j j X m + m j k + k 1 .

Metrics