Abstract

In the present paper we have given the theory of the fifth-order aberrations of rotationally symmetric optical systems. Explicit formulas have been derived for the aberrations of an inhomogeneous medium. We have used these formulas to make an explicit evaluation of the variation of aberrations of meridional and skew rays with the angle of injection of the ray with the z axis. The results have been compared with those obtained by numerically solving the ray equation. We have also shown that for the values of parameters found in experimentally prepared graded-index media the fifth-order contribution is significant and must be taken into consideration while calculating the image patterns formed by these media.

© 1976 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, and H. Matsumura, IEEE J. Quantum Electron. QE-6, 606 (1970).
    [Crossref]
  2. A. D. Pearson, W. G. French, and E. G. Rawson, Appl. Phys. Lett. 15, 76 (1969).
    [Crossref]
  3. E. W. Marchand, in Progress in Optics, edited by E. Wolf, (North Holland, Amsterdam, 1973), Vol. XI, p. 307.
  4. P. J. Sands, J. Opt. Soc. Am. 61, 1086 (1970).
    [Crossref]
  5. D. T. Moore, J. Opt. Soc. Am. 61, 1195 (1971).
    [Crossref]
  6. E. G. Rawson, D. R. Herriott, and J. Mckenna, Appl. Opt. 9, 753 (1970).
    [Crossref] [PubMed]
  7. K. Thyagarajan and A. K. Ghatak, Optik 44, 329 (1976).
  8. W. Streifer and K. B. Paxton, Appl. Opt. 10, 769 (1971).
    [Crossref] [PubMed]
  9. K. B. Paxton and W. Streifer, Appl. Opt. 10, 1164 (1971).
    [Crossref] [PubMed]
  10. K. B. Paxton and W. Streifer, Appl. Opt. 10, 2090, (1971).
    [Crossref] [PubMed]
  11. D. T. Moore, J. Opt. Soc. Am. 61, 886 (1971).
    [Crossref]
  12. E. W. Marchand and D. J. Janeczko, J. Opt. Soc. Am. 64, 846 (1974).
    [Crossref]
  13. H. A. Buchdahl, Optical Aberration Coefficients (Dover, New York, 1968).
  14. R. K. Luneburg, Mathematical Theory of Optics (University of California, Berkeley, 1964).
  15. H. A. Buchdahl, Introduction to Hamiltonian Optics (Cambridge U.P., Cambridge, England, 1970).
  16. E. G. Rawson and R. G. Murray, IEEE J. Quantum Electron. QE-9, 1114 (1973).
    [Crossref]

1976 (1)

K. Thyagarajan and A. K. Ghatak, Optik 44, 329 (1976).

1974 (1)

1973 (1)

E. G. Rawson and R. G. Murray, IEEE J. Quantum Electron. QE-9, 1114 (1973).
[Crossref]

1971 (5)

1970 (3)

E. G. Rawson, D. R. Herriott, and J. Mckenna, Appl. Opt. 9, 753 (1970).
[Crossref] [PubMed]

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, and H. Matsumura, IEEE J. Quantum Electron. QE-6, 606 (1970).
[Crossref]

P. J. Sands, J. Opt. Soc. Am. 61, 1086 (1970).
[Crossref]

1969 (1)

A. D. Pearson, W. G. French, and E. G. Rawson, Appl. Phys. Lett. 15, 76 (1969).
[Crossref]

Buchdahl, H. A.

H. A. Buchdahl, Optical Aberration Coefficients (Dover, New York, 1968).

H. A. Buchdahl, Introduction to Hamiltonian Optics (Cambridge U.P., Cambridge, England, 1970).

French, W. G.

A. D. Pearson, W. G. French, and E. G. Rawson, Appl. Phys. Lett. 15, 76 (1969).
[Crossref]

Furukawa, M.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, and H. Matsumura, IEEE J. Quantum Electron. QE-6, 606 (1970).
[Crossref]

Ghatak, A. K.

K. Thyagarajan and A. K. Ghatak, Optik 44, 329 (1976).

Herriott, D. R.

Janeczko, D. J.

Kitano, I.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, and H. Matsumura, IEEE J. Quantum Electron. QE-6, 606 (1970).
[Crossref]

Koizumi, K.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, and H. Matsumura, IEEE J. Quantum Electron. QE-6, 606 (1970).
[Crossref]

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (University of California, Berkeley, 1964).

Marchand, E. W.

E. W. Marchand and D. J. Janeczko, J. Opt. Soc. Am. 64, 846 (1974).
[Crossref]

E. W. Marchand, in Progress in Optics, edited by E. Wolf, (North Holland, Amsterdam, 1973), Vol. XI, p. 307.

Matsumura, H.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, and H. Matsumura, IEEE J. Quantum Electron. QE-6, 606 (1970).
[Crossref]

Mckenna, J.

Moore, D. T.

Murray, R. G.

E. G. Rawson and R. G. Murray, IEEE J. Quantum Electron. QE-9, 1114 (1973).
[Crossref]

Paxton, K. B.

Pearson, A. D.

A. D. Pearson, W. G. French, and E. G. Rawson, Appl. Phys. Lett. 15, 76 (1969).
[Crossref]

Rawson, E. G.

E. G. Rawson and R. G. Murray, IEEE J. Quantum Electron. QE-9, 1114 (1973).
[Crossref]

E. G. Rawson, D. R. Herriott, and J. Mckenna, Appl. Opt. 9, 753 (1970).
[Crossref] [PubMed]

A. D. Pearson, W. G. French, and E. G. Rawson, Appl. Phys. Lett. 15, 76 (1969).
[Crossref]

Sands, P. J.

Streifer, W.

Thyagarajan, K.

K. Thyagarajan and A. K. Ghatak, Optik 44, 329 (1976).

Uchida, T.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, and H. Matsumura, IEEE J. Quantum Electron. QE-6, 606 (1970).
[Crossref]

Appl. Opt. (4)

Appl. Phys. Lett. (1)

A. D. Pearson, W. G. French, and E. G. Rawson, Appl. Phys. Lett. 15, 76 (1969).
[Crossref]

IEEE J. Quantum Electron. (2)

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, and H. Matsumura, IEEE J. Quantum Electron. QE-6, 606 (1970).
[Crossref]

E. G. Rawson and R. G. Murray, IEEE J. Quantum Electron. QE-9, 1114 (1973).
[Crossref]

J. Opt. Soc. Am. (4)

Optik (1)

K. Thyagarajan and A. K. Ghatak, Optik 44, 329 (1976).

Other (4)

E. W. Marchand, in Progress in Optics, edited by E. Wolf, (North Holland, Amsterdam, 1973), Vol. XI, p. 307.

H. A. Buchdahl, Optical Aberration Coefficients (Dover, New York, 1968).

R. K. Luneburg, Mathematical Theory of Optics (University of California, Berkeley, 1964).

H. A. Buchdahl, Introduction to Hamiltonian Optics (Cambridge U.P., Cambridge, England, 1970).

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 (5)

FIG. 1
FIG. 1

The above figure shows the variation of the meridional ray aberration (in units of z1) as a function of γ0 (the initial angle made by the ray with the z axis) for an axial object point. The dashed lines show the third-order aberrations (calculated using the formulas developed in Ref. 7) and the solid lines show the sum of the third- and fifth-order aberrations. The labeling M, H, P, and E refer, respectively, to the refractive-index distribution of meridionally exact, helically exact, parabolic and experimental fibers. Notice that the fifth-order contribution of experimental fibers is significant. The curves corresponding to the meridionally exact distribution coincide with the abcissa.

FIG. 2
FIG. 2

The meridional ray aberrations (in units of z1) as a function of γ0 for a nonaxial object point situated at x0 = 0.095 z1. The labeling of the curves is the same as in Fig. 1. The curves corresponding to the meridionally exact distribution coincide with the abcissa.

FIG. 3
FIG. 3

The skew ray aberrations (in units of z1) as a function of γ0 for a nonaxial object point situated at x0 = 0.095 z1. The labeling of the curves is the same as in Fig. 1.

FIG. 4
FIG. 4

The meridional ray aberrations (z1 = 100 units) as a function of γ0 for a nonaxial object point situated at x0 = 4.75 units. The figure shows a comparison between our results (G3 and G5) and those obtained by Paxton and Streifer (PS) and Marchand (M) (Ref. 10). The labeling G3 and G5 refer, respectively, to the third-order aberrations and the sum of the third-, and fifth-order aberrations.

FIG. 5
FIG. 5

The skew ray aberrations (z1 = 100 units) as a function of γ0 for a nonaxial object point situated at x0 = 4.75 units. The labeling of the curves is the same as in Fig. 4.

Equations (73)

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

N 2 ( r ) = N 0 2 [ 1 - δ ( r r 0 ) 2 + α 2 δ 2 ( r r 0 ) 4 + α 3 δ 3 ( r r 0 ) 6 + ] .
H = - [ n 2 ( U , z ) - V ] 1 / 2 ,
d X d z = 2 P H V ,
d P d z = - 2 X H U .
X = X 1 + X 3 + X 5 + ,
P = P 1 + P 3 + P 5 + ,
H = H 0 + H 1 U + H 2 V + 1 2 ( H 11 U 2 + 2 H 12 U V + H 22 V 2 ) + 1 6 ( H 111 U 3 + 3 H 112 U 2 V + 3 H 122 U V 2 + H 222 V 3 ) + ,
X ˙ 1 = 2 H 2 P 1 ,
X ˙ 3 = 2 H 2 P 3 + 2 ( H 12 U 1 + H 22 V 1 ) P 1 ,
X ˙ 5 = 2 H 2 P 5 + 2 ( H 12 U 1 + H 22 V 1 ) P 3 + 4 ( H 12 U 13 + H 22 V 13 ) P 1 + ( H 112 U 1 2 + 2 H 122 U 1 V 1 + H 222 V 1 2 ) P 1 ,
P ˙ 1 = - 2 H 1 X 1 ,
P ˙ 3 = - 2 H 1 X 3 - 2 ( H 11 U 1 + H 12 V 1 ) X 1 ,
P ˙ 5 = 2 H 1 X 5 - 2 ( H 11 U 1 + H 12 V 1 ) X 3 - 4 ( H 11 U 13 + H 12 V 13 ) X 1 - ( H 111 U 1 2 + 2 H 112 U 1 V 1 + H 122 V 1 2 ) X 1 ,
h ( z 0 ) = 0 ,             h ( ζ ) = 1 ,             H ( z 0 ) = 1 ,             H ( ζ ) = 0.
x = x 0 H ( z ) + ξ h ( z ) ,             p = x 0 θ ( z ) + ξ ϑ ( z ) , y = y 0 H ( z ) + η h ( z ) ,             q = y 0 θ ( z ) + η ϑ ( z ) .
X 3 ( z 0 ) = X 5 ( z 0 ) = Y 3 ( z 0 ) = Y 5 ( z 0 ) = 0.
d d z ( X 3 ϑ - h P 3 ) = 2 [ ( H 12 U 1 + H 22 V 1 ) p ϑ + ( H 11 U 1 + H 12 V 1 ) x h ] .
P 3 ( z ) = ( ϑ / h ) X 3 ( z ) - ( 2 / h ) I 1 ( z ) ,
I 1 ( z ) = z 0 z [ H 12 U 1 + H 22 V 1 ) p ϑ + ( H 11 U 1 + H 12 V 1 ) x h ] d z .
X ˙ 3 - ( h ˙ / h ) X 3 = 2 ( H 22 V 1 + H 12 U 1 ) p - 2 ( h ˙ / h ϑ ) I 1 ( z ) ,
X 3 ( z ) = 2 h ζ z ( H 22 V 1 + H 12 U 1 ) ( p / h ) d z - 2 h ζ z ( h ˙ / h 2 ϑ ) I 1 ( z ) d z .
h ˙ = 2 H 2 ϑ ,             ϑ ˙ = - 2 H 1 h .
d d z ( X 5 ϑ - h P 5 ) = 2 [ ( H 12 U 1 + H 22 V 1 ) P 3 ϑ + ( H 11 U 1 + H 12 V 1 ) X 3 h + 2 ( H 12 U 13 + H 22 V 13 ) p ϑ + 2 ( H 11 U 13 + H 12 V 13 ) x h + 1 2 ( H 112 U 1 2 + 2 H 122 U 1 V 1 + H 222 V 1 2 ) p ϑ + 1 2 ( H 111 U 1 2 + 2 H 112 U 1 V 1 + H 122 V 1 2 ) x h ] .
X 5 ( z 1 ) = 2 ϑ ( z 1 ) z 0 z 1 [ ( H 12 U 1 + H 22 V 1 ) P 3 ϑ + ( H 11 U 1 + H 12 V 1 ) X 3 h + 2 ( H 12 U 13 + H 22 V 13 ) p ϑ + 2 ( H 11 U 13 + H 12 V 13 ) x h + 1 2 ( H 112 U 1 2 + 2 H 122 U 1 V 1 + H 222 V 1 2 ) p ϑ + 1 2 ( H 111 U 1 2 + 2 H 112 U 1 V 1 + H 122 V 1 2 ) x h ] d z .
X 5 ( z 1 ) = A ρ 5 cos θ + ( B 1 + B 2 cos 2 θ ) x 0 ρ 4 + x 0 2 ρ 3 ( C 1 + C 2 cos 2 θ ) cos θ + x 0 3 ρ 2 ( D 1 + D 2 cos 2 θ ) + x 0 4 ρ cos θ E + F x 0 5 ,
A = 2 ϑ ( z 1 ) z 0 z 1 [ 3 ( H 12 h 2 + H 22 ϑ 2 ) ϑ I D + 3 ( H 11 h 2 + H 12 ϑ 2 ) h I D + 1 2 H 111 h 6 + 3 2 H 112 h 4 ϑ 2 + 3 2 H 122 h 2 ϑ 4 + 1 2 H 222 ϑ 6 ] d z ,
B 1 = 2 ϑ ( z 1 ) z 0 z 1 [ ( 3 H 12 θ ϑ h + 4 H 11 H h 2 + H 12 ϑ 2 H ) I D + ( 4 H 22 θ ϑ 2 + 3 H 12 H h ϑ + H 12 h 2 θ ) I D + ( H 12 h 2 + H 22 ϑ 2 ) ϑ ( 2 I E + 3 I H ) + ( H 11 h 2 + H 12 ϑ 2 ) h ( 2 I E + 3 I H ) + 3 2 H 111 H h 5 + 3 H 112 ( 1 2 h 4 θ ϑ + H h 3 ϑ 2 ) + 3 H 122 ( θ ϑ 3 h 2 + 1 2 H h ϑ 4 ) + 3 2 H 222 θ ϑ 5 ] d z ,
B 2 = 2 ϑ ( z 1 ) z 0 z 1 [ ( 2 H 11 H h 2 + H 12 θ ϑ h + H 12 ϑ 2 H ) I D + ( H 12 H h ϑ + H 12 h 2 θ + 2 H 22 θ ϑ 2 ) I D + ( H 12 ϑ 2 + H 11 h 2 ) h ( 3 I H + I E ) + ( H 22 ϑ 2 + H 12 h 2 ) ϑ ( 3 I H + I E ) + H 111 H h 5 + H 112 ( 2 H h 3 ϑ 2 + θ ϑ h 4 ) + H 122 ( 2 θ ϑ 3 h 2 + H h ϑ 4 ) + H 222 θ ϑ 5 ] d z ,
C 1 = 2 ϑ ( z 1 ) z 0 z 1 [ ( 3 H 11 H 2 h + H 12 θ 2 h + 2 H 12 θ ϑ H ) I D + ( H 12 H 2 ϑ + 2 H 12 H h θ + 3 H 22 θ 2 ϑ ) I D + 4 H 12 θ ϑ h ( I E + I H ) + 4 H 12 H h ϑ ( I E + I H ) + 2 H 11 H h 2 ( 3 I E + 2 I H ) + 2 H 22 θ ϑ 2 ( 3 I E + 2 I H ) + ( H 11 h 2 + H 12 ϑ 2 ) h ( 2 I F + 3 I G ) + ( H 12 h 2 + H 22 ϑ 2 ) ϑ ( 2 I F + 3 I G ) + 2 H 12 ϑ 2 H I E + 2 H 12 h 2 θ I E + 3 H 111 H 2 h 4 + H 112 ( 4 H h 3 θ ϑ + 4 H 2 h 2 ϑ 2 + h 4 θ 2 ) + H 122 ( 4 θ 2 ϑ 2 h 2 + 4 H h θ ϑ 3 + H 2 ϑ 4 ) + 3 H 222 θ 2 ϑ 4 ] d z ,
C 2 = 2 ϑ ( z 1 ) z 0 z 1 [ 4 ( H 12 ϑ 2 + H 11 h 2 ) h I F + 4 ( H 12 h 2 + H 22 ϑ 2 ) ϑ I F + 4 ( 2 H 11 H h 2 + H 12 θ ϑ h + H 12 ϑ 2 H ) I H + 4 ( H 12 H h ϑ + H 12 h 2 θ + 2 H 22 θ ϑ 2 ) I H + 2 H 111 H 2 h 4 + 2 H 112 ( H ϑ + 2 h θ ) H h 2 ϑ + 2 H 122 ( 2 H ϑ + h θ ) h θ ϑ 2 + 2 H 222 θ 2 ϑ 4 ] d z ,
D 1 = 2 ϑ ( z 1 ) z 0 z 1 [ 2 h ( H 11 h 2 + H 12 ϑ 2 ) I A + 2 ϑ ( H 12 h 2 + H 22 ϑ 2 ) I A + ( H 12 θ 2 h + 3 H 11 H 2 h + 2 H 12 θ ϑ H ) ( I E + I H ) + ( H 12 H 2 ϑ + 2 H 12 H h θ + 3 H 22 θ 2 ϑ ) ( I E + I H ) + 2 H 11 H h 2 ( 3 I F + 2 I G ) + H 12 H ϑ 2 ( 2 I F + I G ) + H 12 h 2 θ ( 2 I F + I G ) + H 12 θ ϑ h ( 4 I F + 3 I G ) + H 12 H h ϑ ( 4 I F + 3 I G ) + 2 H 22 θ ϑ 2 ( 3 I F + 2 I G ) + ( H 112 H h + H 122 θ ϑ ) ( 5 H h θ ϑ + 2 H 2 ϑ 2 + 2 h 2 θ 2 ) + 3 H 111 H 3 h 3 + 3 H 222 θ 3 ϑ 3 ] d z ,
D 2 = 2 ϑ ( z 1 ) z 0 z 1 [ ( H 12 ϑ 2 + H 11 h 2 ) h I A + ( H 12 h 2 + H 22 ϑ 2 ) ϑ I A + ( 2 H 12 θ ϑ h + 2 H 11 H h 2 ) I F + ( 2 H 22 θ ϑ 2 + 2 H 12 H h ϑ ) I F + ( 2 H 11 H h 2 + H 12 θ ϑ h + H 12 ϑ 2 H ) ( 2 I F + I G ) + ( H 12 H h ϑ + H 12 h 2 θ + 2 H 22 θ ϑ 2 ) ( 2 I F + I G ) + ( 3 H 11 H 2 h + H 12 θ 2 h + 2 H 12 θ ϑ H ) I H + ( H 12 H 2 ϑ + 2 H 12 H h θ + 3 H 22 θ 2 ϑ ) I H + 2 H 111 H 3 h 3 + ( H 112 H h + H 122 θ ϑ ) ( 4 H h θ ϑ + H 2 ϑ 2 + h 2 θ 2 ) + 2 H 222 θ 3 ϑ 3 ] d z ,
E = 2 ϑ ( z 1 ) z 0 z 1 [ ( 6 H 11 H h 2 + 4 H 12 θ ϑ h + 2 H 12 ϑ 2 H ) I A + ( 4 H 12 H h ϑ + 2 H 12 h 2 θ + 6 H 22 θ ϑ 2 ) I A + ( 3 H 11 H 2 h + H 12 θ 2 h + 2 H 12 θ ϑ H ) ( 2 I F + I G ) + ( H 12 H 2 ϑ + 2 H 12 H h θ + 3 H 22 θ 2 ϑ ) ( 2 I F + I G ) + 5 2 H 111 H 4 h 2 + H 112 ( 4 H h θ ϑ + 1 2 ϑ 2 H 2 + 3 h 2 θ 2 ) H 2 + H 122 ( 4 H h θ ϑ + 1 2 h 2 θ 2 + 3 H 2 ϑ 2 ) θ 2 + 5 2 H 222 θ 4 ϑ 2 ] d z ,
F = 2 ϑ ( z 1 ) z 0 z 1 [ ( H 12 θ 2 h + 3 H 11 H 2 h + 2 H 12 θ ϑ H ) I A + ( H 12 H 2 ϑ + 2 H 12 H h θ + 3 H 22 θ 2 ϑ ) I A + 1 2 H 111 H 5 h + H 112 ( 1 2 H ϑ + h θ ) H 3 θ + H 122 ( 1 2 h θ + H ϑ ) H θ 3 + 1 2 H 222 θ 5 ϑ ] d z .
Y 5 ( z 1 ) = A ρ 5 sin θ + x 0 ρ 4 G sin 2 θ + x 0 2 ρ 3 sin θ ( H 1 + H 2 cos 2 θ ) + x 0 3 ρ 2 P sin 2 θ + x 0 4 ρ Q sin θ ,
G = 2 ϑ ( z l ) z 0 z 1 [ ( 2 H 11 H h 2 + H 12 H ϑ 2 + H 12 θ ϑ h ) I D + ( H 12 H h ϑ + H 12 θ h 2 + 2 H 22 θ ϑ 2 ) I D + ( H 12 h ϑ 2 + H 11 h 3 ) I E + ( H 12 ϑ h 2 + H 22 ϑ 3 ) I E + ( 3 H 11 h 3 + 3 H 12 h ϑ 2 ) I H + ( 3 H 12 ϑ h 2 + 3 H 22 ϑ 3 ) I H + H 111 H h 5 + ( 2 ϑ 2 H h 3 + θ ϑ h 4 ) H 112 + ( 2 θ ϑ 3 h 2 + H h ϑ 4 ) H 122 + H 222 θ ϑ 5 ] d z ,
H 1 = 2 ϑ ( z 1 ) z 0 z 1 [ ( H 11 H 2 h + H 12 θ 2 h ) I D + ( H 12 H 2 ϑ + H 22 θ 2 ϑ ) I D + ( H 11 h 2 H + H 12 ϑ 2 H ) 2 I E + ( H 12 h 2 θ + H 22 ϑ 2 θ ) 2 I E + ( H 11 h 3 + H 12 ϑ 2 h ) 3 I G + ( H 12 h 2 ϑ + H 22 ϑ 3 ) 3 I G + H 111 H 2 h 4 + ( 2 H 2 h 2 ϑ 2 + h 4 θ 2 ) H 112 + ( 2 h 2 θ 2 ϑ 2 + H 2 ϑ 4 ) H 122 + H 222 θ 2 ϑ 4 ] d z ,
H 2 = 2 ϑ ( z 1 ) z 0 z 1 [ ( H 11 h 3 + H 12 ϑ 2 h ) 4 I F + ( H 12 h 2 ϑ + H 22 ϑ 3 ) 4 I F + ( 2 H 11 H h 2 + H 12 θ ϑ h + H 12 ϑ 2 H ) 4 I H + ( H 12 H h ϑ + H 12 h 2 θ + 2 H 22 θ ϑ 2 ) 4 I H + 2 H 111 H 2 h 4 + ( H 2 h 2 ϑ 2 + 2 H h 3 θ ϑ ) 2 H 112 + ( h 2 θ 2 ϑ 2 + 2 H h θ ϑ 3 ) 2 H 122 + 2 H 222 θ 2 ϑ 4 ] d z ,
P = 2 ϑ ( z 1 ) z 0 z 1 [ ( H 11 h 3 + H 12 ϑ 2 h ) I A + ( H 12 h 2 ϑ + H 22 ϑ 3 ) I A + ( H 11 h 2 H + H 12 ϑ 2 H ) 2 I F + ( H 12 h 2 θ + H 22 θ ϑ 2 ) 2 I F + ( 2 H 11 H h 2 + H 12 θ ϑ h + H 12 ϑ 2 H ) I G + ( H 12 H h ϑ + H 12 h 2 θ + 2 H 22 θ ϑ 2 ) I G + ( H 11 H 2 h + H 12 θ 2 h ) I H + ( H 12 H 2 ϑ + H 22 θ 2 ϑ ) I H + H 111 H 3 h 3 + ( H 3 h ϑ 2 + H 2 θ ϑ h 2 + H h 3 θ 2 ) H 112 + ( H 2 θ ϑ 3 + H h θ 2 ϑ 2 + θ 3 ϑ h 2 ) H 122 + H 222 θ 3 ϑ 3 ] d z ,
Q = 2 ϑ ( z 1 ) z 0 z 1 [ ( H 11 h 2 H + H 12 ϑ 2 H ) 2 I A + ( H 12 h 2 θ + H 22 ϑ 2 θ ) 2 I A + ( H 11 H 2 h + H 12 θ 2 h ) I G + ( H 12 H 2 ϑ + H 22 θ 2 ϑ ) I G + 1 2 H 111 H 4 h 2 + ( 1 2 ϑ 2 H 4 + H 2 θ 2 h 2 ) H 112 + ( 1 2 θ 4 h 2 + H 2 θ 2 ϑ 2 ) H 122 + 1 2 H 222 θ 4 ϑ 2 ] d z .
I A = 2 h [ ζ z ( H 22 θ 3 h + H 12 H 2 θ h ) d z - ζ z h ˙ h 2 ϑ I a d z ] ,
I D = 2 h [ ζ z ( H 22 ϑ 3 h + H 12 h ϑ ) d z - ζ z h ˙ h 2 ϑ I d d z ] ,
I E = 2 h [ ζ z ( H 22 ϑ 2 θ h + H 12 h θ ) d z - ζ z h ˙ h 2 ϑ I e d z ] ,
I F = 2 h [ ζ z ( H 22 θ 2 ϑ h + H 12 H θ ) d z - ζ z h ˙ h 2 ϑ I f d z ] ,
I G = 2 h [ ζ z ( H 22 θ 2 ϑ h + H 12 H 2 ϑ h ) d z - ζ z h ˙ h 2 ϑ I g d z ] ,
I H = 2 h [ ζ z ( H 22 θ ϑ 2 h + H 12 H ϑ ) d z - ζ z h ˙ h 2 ϑ I h d z ] ,
I a = z 0 z ( H 12 H 2 θ ϑ + H 22 θ 3 ϑ + H 11 H 3 h + H 12 θ 2 H h ) d z ,
I d = z 0 z ( 2 H 12 h 2 ϑ 2 + H 22 ϑ 4 + H 11 h 4 ) d z ,
I e = z 0 z ( H 12 h 2 θ ϑ + H 22 θ ϑ 3 + H 11 H h 3 + H 12 H h ϑ 2 ) d z ,
I f = z 0 z ( 2 H 12 H h θ ϑ + H 22 θ 2 ϑ 2 + H 11 H 2 h 2 ) d z ,
I g = z 0 z ( H 12 H 2 ϑ 2 + H 22 θ 2 ϑ 2 + H 11 H 2 h 2 + H 12 h 2 θ 2 ) d z ,
I h = z 0 z ( H 12 H h ϑ 2 + H 22 θ ϑ 3 + H 11 H h 3 + H 12 θ ϑ h 2 ) d z .
I X = ( ϑ / h ) I X - ( 2 / h ) I x ,
n ( U , z ) = n 0 ( 1 - 1 2 α 2 U + 1 2 β α 4 U 2 + γ α 6 U 3 ) ,
h ( z ) = sin α z , ϑ ( z ) = n 0 α cos α z , H ( z ) = cos α z , θ ( z ) = - n 0 α sin α z ,
α z 1 = 2 m π , α ζ = ( 2 m - 1 ) π / 2 ,             m = 1 , 2 , 3 , .
H 11 = - n 0 β α 4 ,             H 12 = α 2 / 4 n 0 ,             H 22 = 1 / 4 n 0 3 , H 111 = - 6 n 0 γ α 6 ,             H 112 = ( α 4 / 2 n 0 ) ( 1 2 - β ) , H 122 = 3 α 2 / 8 n 0 3 ,             H 222 = 3 / 8 n 0 5 .
A = 1 8 z 1 α 5 ( 159 128 - 15 γ - 69 16 β - 33 8 β 2 ) ,
B 1 = 2 α 6 z 1 [ 1 64 z 1 ( - 3 β 2 + 13 2 β - 35 16 ) + 1 2 ζ ( 15 β 2 - 58 4 β + 55 16 ) ] ,
B 2 = 2 α 6 z 1 [ 1 64 z 1 ( β - 6 β 2 + 5 8 ) + 1 8 ζ ( 3 β 2 - 2 β + 5 16 ) ] ,
C 1 = 1 8 α 5 z 1 ( 189 64 - 18 γ - 63 8 β - 3 4 β 2 ) ,
C 2 = 1 8 α 5 z 1 ( - 55 32 - 12 γ - 11 4 β + 9 2 β 2 ) ,
D 1 = 2 α 6 z 1 [ 1 64 z 1 ( - 41 β 2 + 79 4 β - 2 ) + 1 4 ζ ( 3 β 2 - 11 4 β + 5 8 ) ] ,
D 2 = 2 α 6 z 1 [ 1 16 z 1 ( - β 2 - 1 4 β + 1 2 ) + 1 8 β ζ ( 3 β - 5 4 ) ] ,
E = 1 8 α 5 z 1 ( 79 128 - 15 γ - 170 32 β + 15 8 β 2 ) ,
F = 1 32 α 6 z 1 ( 3 β - 5 4 ) 2 ( 2 ζ - z 1 ) ,
G = ( 3 β - 5 4 ) 2 α 6 z 1 [ 1 8 ( β - 1 4 ) ζ - 1 32 z 1 ( β + 1 4 ) ] ,
H 1 = 2 α 5 z 1 ( - 5 64 β 2 - 114 256 β - 3 8 γ + 251 1024 ) ,
H 2 = 2 α 5 z 1 ( 9 32 β 2 - 11 64 β - 3 4 γ - 55 512 ) ,
P = 2 α 6 z 1 [ z 1 ( - 3 32 β 2 + 1 64 β + 5 512 ) + ζ ( 5 16 β 2 - 9 32 β + 9 256 ) ] ,
Q = 2 α 5 z 1 ( 11 128 β 2 - 73 256 β - 3 16 γ + 203 2048 ) .
n = n 0 ( 1 - 1 2 α 2 U + 5 24 α 4 U 2 - 61 720 α 6 U 3 ) .
n = n 0 sech ( α U ) .