Abstract

The fifth-order aberration coefficients for gradient-index lenses are presented. A procedure for calculation is outlined.

© 1983 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. P. J. Sands, “Third-order aberrations of inhomogeneous lenses,” J. Opt. Soc. Am. 60, 1436–1443 (1970).
    [Crossref]
  2. S. D. Fantone, “Fifth-order aberration theory of gradient-index optics—examples,” J. Opt. Soc. Am. 73, 1162–1164 (1983).
    [Crossref]
  3. H. A. Buchdahl, Optical Aberration Coefficients (Dover, New York, 1968).
  4. S. D. Fantone, “Design, engineering, and manufacturing aspects of gradient index optical components,” Ph.D. Thesis (University of Rochester, Rochester, New York, 1979).
  5. M. P. Rimmer, “Optical aberration coefficients,” M.S. Thesis (University of Rochester, Rochester, New York, 1963).

1983 (1)

1970 (1)

Buchdahl, H. A.

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

Fantone, S. D.

S. D. Fantone, “Fifth-order aberration theory of gradient-index optics—examples,” J. Opt. Soc. Am. 73, 1162–1164 (1983).
[Crossref]

S. D. Fantone, “Design, engineering, and manufacturing aspects of gradient index optical components,” Ph.D. Thesis (University of Rochester, Rochester, New York, 1979).

Rimmer, M. P.

M. P. Rimmer, “Optical aberration coefficients,” M.S. Thesis (University of Rochester, Rochester, New York, 1963).

Sands, P. J.

J. Opt. Soc. Am. (2)

Other (3)

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

S. D. Fantone, “Design, engineering, and manufacturing aspects of gradient index optical components,” Ph.D. Thesis (University of Rochester, Rochester, New York, 1979).

M. P. Rimmer, “Optical aberration coefficients,” M.S. Thesis (University of Rochester, Rochester, New York, 1963).

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

Aberration contributions in a gradient lens system.

Fig. 2
Fig. 2

Real ray path in a lens.

Tables (4)

Tables Icon

Table 1 Expansion and Coefficients of the Fifth-Order Quasi-Invariant in Terms of ξ, η, ζ, S, and Ta

Tables Icon

Table 2 Relationship of μ and ` s Coefficientsa

Tables Icon

Table 3 Iteration Table for Surface Contributions

Tables Icon

Table 4 Iteration Table for Transfer Contributions

Equations (159)

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

Spherical aberration :             a 1 = a ˆ + κ y a 4 ,
Coma :             a 2 = q a ˆ + κ y a 3 y b ,
Astigmatism :             a 3 = q 2 a ˆ + κ y a 2 y b 2 ,
Petzval :             a 4 = 1 2 λ 2 c Δ ( 1 / N 0 ) ,
Distortion :             a 5 = q 3 a ˆ + κ y a y b 3 ,
Δ Λ = - ( N 1 ξ - 1 2 N o ζ ) y a V [ y a ( 4 N 2 ξ + N 1 ζ ) Y + v a ( N 1 ξ - 1 2 N 0 ζ ) V ] d x + 0 ( 5 ) .
a 1 * = 1 2 ( N 0 y a v a 3 ) + ( 4 N 2 y a 4 + 2 N 1 y a 2 v a 2 - 1 2 N 0 v a 4 ) d x ,
a 2 * = 1 2 ( N 0 y a v a 2 v b ) + [ 4 N 2 y a 3 y b + N 1 y a v a ( y a v b + y b v a ) - 1 2 N 0 v a 3 v b ] d x ,
a 3 * = 1 2 ( N 0 y a v a v b 2 ) + ( 4 N 2 y a 2 y b 2 + 2 N 1 y a y b v a v b - 1 2 N 0 v a 2 v b 2 ) d x ,
a 4 * = λ 2 ( N 1 / N 0 2 ) d x ,
a 5 * = 1 2 ( N 0 y a v b 3 ) + [ 4 N 2 y a y b 3 + N 1 y b v b ( y a v b + y b v a ) - 1 2 N 0 v a v b 3 ] d x .
y = σ 1 ρ 3 cos θ + σ 2 ρ 2 H ( 2 + cos 2 θ ) + ( 3 σ 3 + σ 4 ) ρ H 2 cos θ + σ 5 H 3 ,
z = σ 1 ρ 3 sin θ + σ 2 ρ 2 H sin 2 θ + ( σ 3 + σ 4 ) ρ H 2 sin θ ,
σ i = μ j = 1 k ( a i j + a * i j ) ,
σ 1 = μ j = 1 k ( a 1 j + a * 1 j ) .
ˆ = j = 1 k Δ Λ j = A k ξ S + Ā k ξ T + B k η S + B ¯ k η T + C k ζ S + C ¯ k ζ T ,
A k = j = 1 k a j , Ā k = j = 1 k ā j , B k = j = 1 k b j , B ¯ k = j = 1 k b ¯ j , C k = j = 1 k c j , C ¯ k = j = 1 k c ¯ j .
Δ Λ = 1 2 N 0 { y a I [ 2 V ( k - 1 ) I + ( k - 1 ) 2 I 2 ] + c i a ξ ( k - 1 ) I } .
Δ Λ = 1 2 N 0 ( k - 1 ) ( i a S + i b T ) { y a ξ 1 ( i a + v a ) i a + 2 η 1 y a ( i a + v a ) i b + ζ 1 [ y a ( i a + v a ) i b - c y b ] λ / N } .
Δ Λ a = ( a a h ξ 1 + b a h η 1 + c a h ζ 1 ) S + ( ā a h ξ 1 + b ¯ a h η 1 + c ¯ a h ζ 1 ) T ,
a a h = 1 2 N 0 ( k - 1 ) y a i a 2 ( i a + v a ) ,
ā a h = 1 2 N 0 ( k - 1 ) y a i a i b ( i a + v a ) ,
b a h = N 0 ( k - 1 ) i a i b y a ( i a + v a ) ,
b ¯ a h = N 0 ( k - 1 ) i b 2 y a ( i a + v a ) ,
c a h = 1 2 N 0 ( k - 1 ) i a [ y a i b ( i a + v b ) - c y b λ / N ] ,
c ¯ a h = 1 2 N 0 ( k - 1 ) i b [ y a i b ( i a + v b ) - c y b λ / N ] .
c ¯ b h = 1 2 N 0 ( k - 1 ) y b i b 2 ( i b + v a ) = a q ,
c b h = a q × i a / i b ,
b ¯ a h = 2 a q × i a / i b = 2 ā q ,
b b h = 2 a q ( i a / i b ) 2 ,
ā b h = 1 2 N 0 ( k - 1 ) i b [ y b i a ( i a + v a ) + c y a λ / N ] ,
a b h = 1 2 N 0 ( k - 1 ) i a [ y b i a ( i a + v a ) + c y a λ / N ] .
Δ Λ b = ( c b h ξ 1 + b b h η 1 + a b h ζ 1 ) T + ( c ¯ b h ξ 1 + b ¯ b h η 1 + ā b h ζ 1 ) S .
Δ Λ g = - y a Δ ( N 1 ξ V ) + κ y a Y ξ ,
Δ Λ g = - y a ξ [ V Δ N 1 + N 1 ( k - 1 ) I ] + κ y a ξ Y ,
Δ Λ g = y a ξ [ κ ( y a S + y b T ) + ( v a S + v b T ) N 1 + N 1 ( k - 1 ) ( i a S + i b T ) ] .
a a g = y a 4 κ + y a 3 [ v a Δ N 1 + N 1 ( k - 1 ) i a ] ,
ā a g = y a 3 y b κ + y a 3 [ v b Δ N 1 + N 1 ( k - 1 ) i b ] ,
b a g = 2 y a 3 y b κ + 2 y a 2 y b [ v a Δ N 1 + N 1 ( k - 1 ) i a ] ,
b ¯ a g = 2 y a 2 y b 2 κ + 2 y a 2 y b [ v b Δ N 1 + N 1 ( k - 1 ) i b ] ,
c a g = y a 2 y b 2 κ + y a y b 2 [ v a Δ N 1 + N 1 ( k - 1 ) i a ] ,
c ¯ a g = y a y b 3 κ + y a y b 2 [ v b Δ N 1 + N 1 ( k - 1 ) i b ] ,
c ¯ b g = y b 4 κ + y b 3 [ v b Δ N 1 + N 1 ( k - 1 ) i b ] ,
c b g = y b 3 y a κ + y b 3 [ v a Δ N 1 + N 1 ( k - 1 ) i a ] ,
b ¯ b g = 2 y a y b 3 κ + 2 y b 2 y a [ v b Δ N 1 + N 1 ( k - 1 ) i b ] ,
b b g = 2 y a 2 y b 2 κ + 2 y b 2 y a [ v a Δ N 1 + N 1 ( k - 1 ) i a ] ,
ā b g = y a 2 y b 2 κ + y b y a 2 [ v b Δ N 1 + N 1 ( k - 1 ) i b ] ,
a b g = y a 3 y b κ + y b y a 2 [ v a Δ N 1 + N 1 ( k - 1 ) i a ] .
Λ = - ( N 1 ξ - 1 2 N 0 ζ ) y a V + [ y a ( 4 N 2 ξ + N 1 ζ ) Y + v a ( N 1 ξ - 1 2 N 0 ζ ) V ] d x .
a a m = - ( N 1 y a 2 - 1 2 N 0 v a 2 ) y a v a + [ y a 2 ( 4 N 2 y a 2 + N 1 v a 2 ) + v a v b ( N 1 y a 2 - 1 2 N 0 v a 2 ) ] d x ,
ā a m = - ( N 1 y a 2 - 1 2 N 0 v a 2 ) y a v b + [ y a y b ( 4 N 2 y a 2 + N 1 v a 2 ) + v a v b ( N 1 y a 2 - 1 2 N 0 v a 2 ) ] d x ,
b a m = - ( 2 N 1 y a y b - N 0 v a v b ) y a v a + [ y a 2 ( 8 N 2 y a y b + 2 N 1 v a v b ) + v a 2 ( 2 N 1 y a y b - N 0 v a v b ) ] d x ,
b ¯ a m = - ( 2 N 1 y a y b - N 0 v a v b ) y a v b + [ y a y b ( 8 N 2 y a y b + 2 N 1 v a v b ) + v a v b ( 2 N 1 y a y b - N 0 v a v b ) ] d x ,
c a m = - ( N 1 y b 2 - 1 2 N 0 v b 2 ) y a v a + [ y a 2 ( 4 N 2 y b 2 + N 1 v b 2 ) + v a 2 ( N 1 y b 2 - 1 2 N 0 v b 2 ) ] d x ,
c ¯ a m = - ( N 1 y b 2 - 1 2 N 0 v b 2 ) y a v b + [ y a y b ( 4 N 2 y b 2 + N 1 v b 2 ) + v a v b ( N 1 y b 2 - 1 2 N 0 v b 2 ) ] d x ,
c ¯ b m = - ( N 1 y b 2 - 1 2 N 0 v b 2 ) y b v b + [ y b 2 ( 4 N 2 y b 2 + N 1 v b 2 ) + v b 2 ( N 1 y b 2 - 1 2 N 0 v b 2 ) ] d x ,
c b m = - ( N 1 y b 2 - 1 2 N 0 v b 2 ) y b v a + [ y a y b ( 4 N 2 y b 2 + N 1 v b 2 ) + v b v a ( N 1 y b 2 - 1 2 N 0 v b 2 ) ] d x ,
b ¯ b m = - ( 2 N 1 y a y b - N 0 v a v b ) y b v b + [ y b 2 ( 8 N 2 y b y a + 2 N 1 v a v b ) + v b 2 ( 2 N 1 y a y b - N 0 v a v b ) ] d x ,
b b m = - ( 2 N 1 y b y a - N 0 v a v b ) y b v b + [ y a y b ( 8 N 2 y a y b + 2 N 1 v a v b ) + v a v b ( 2 N 1 y a y b - N 0 v a v b ) ] d x ,
ā b m = - ( N 1 y a 2 - 1 2 N 0 v a 2 ) y b v b + [ y b 2 ( 4 N 2 y a 2 + N 1 v a 2 ) + v b 2 ( N 1 y a 2 - 1 2 N 0 v a 2 ) ] d x ,
a b m = - ( N 1 y a 2 - 1 2 N 0 v a 2 ) y b v a + [ y a y b ( 4 N 2 y a 2 + N 1 v b 2 ) + v a v b ( N 1 y a 2 - 1 2 N 0 v a 2 ) ] d x .
Δ Λ a = ( a a m ξ 1 + b a m η 1 + c a m ζ 1 ) S + ( ā a m ξ 1 + b ¯ a m η 1 + c ¯ a m ζ 1 ) T ,
Δ Λ b = ( a b m ζ 1 + b b m η 1 + c c m ξ 1 ) S + ( a b m ζ 1 + b b m η 1 + c b m ξ 1 ) T .
a a , j = a a h , j + a a g , j + a a m , j ,
A a = A a h + A a g + A a m ,
A a , k = j = 1 k a a h , j + j = 1 k a a g , j + j = 1 k a a m , j ,
A a , k = A a h , k + A a g , k + A a m , k .
A a h , k = j = 1 k - 1 a a h , j , A a h , k = j = 1 k a a h , j .
σ 1 = μ A a , k = μ ( A a h , k + A a g , k + A a m , k ) ,
σ 2 = μ Ā a , k = μ ( Ā a h , k + Ā a g , k + Ā a m , k ) ,
σ 3 = 1 2 μ B ¯ a , k = 1 2 μ ( B ¯ a h , k + B ¯ a g , k + B ¯ a m , k ) ,
σ 4 = μ ( C a , k - 1 2 B ¯ a , k ) = μ [ ( C a h , k - 1 2 B ¯ a h , k ) + ( C a g , k - 1 2 B ¯ a g , k ) + ( C a m , k - 1 2 B a m , k ) ] ,
σ 5 = μ C ¯ a , k = μ ( C ¯ a h , k + C ¯ a g , k + C ¯ a m , k ) ,
y = σ 1 ρ 3 cos θ + σ 2 [ + cos ( 2 θ ) ] ρ 2 H + ( 3 σ 3 + σ 4 ) ρ H 2 cos θ + σ 5 H 3 ,
z = σ 1 ρ 3 sin θ + σ 2 sin ( 2 θ ) ρ 2 H + ( σ 3 + σ 4 ) sin θ ρ H 2 .
5 = μ j Δ Λ 5 , j = μ ( ` s 1 ξ 1 2 S + ` s ¯ 1 ξ ¯ 1 2 T ) + ` s 2 ξ 1 η 1 S + ` s ¯ 2 ξ 1 η 1 T + ` s 2 ξ 1 ζ 1 S + ` s ¯ 3 ξ 1 ζ 1 T + ` s 4 η 1 2 S + ` s ¯ 4 η 1 2 T + ` s 5 η 1 ζ 1 S + ` s ¯ 5 η 1 ζ 1 T + ` s 6 ζ 1 2 S + ` s ¯ 6 ζ 1 2 T ) ,
S y ρ cos θ , S z ρ sin θ , T y H , T z 0 , ξ 1 ρ 2 , η 1 ρ H cos θ , ζ 1 H 2 , ξ 1 2 ρ 4 , ξ 1 η 1 ρ 3 H cos θ , ξ 1 ζ 1 ρ 2 H 2 , η 1 2 ρ 2 H 2 cos 2 θ , η 1 ζ 1 ρ H 3 cos θ , ζ 1 2 H 4 .
5 y = μ { ` s 1 ρ 5 cos θ + ( ` s ¯ 1 + ` s 2 cos 2 θ ) ρ 4 H + ( ` s ¯ 2 + ` s 3 + ` s 4 cos 2 θ ) ρ 3 H 2 cos θ + [ ` s ¯ 3 + ( ` s ¯ 4 + ` s 5 ) cos 2 θ ] ρ 2 H 3 + ( ` s ¯ 5 + ` s 6 ) ρ H 4 cos θ + ` s ¯ 6 H 5 } ,
5 x = μ [ ` s 1 ρ 5 sin θ + ` s 2 ρ 4 H sin θ cos θ + ( ` s 3 + ` s 4 cos 2 θ ) ρ 3 H 2 sin θ + ` s 5 ρ 2 H 3 sin θ cos θ + ` s 6 H 4 ρ sin θ ] .
5 y = μ 1 ρ 5 cos θ + ( μ 2 + μ 3 cos 2 θ ) ρ 4 H + cos θ ( μ 4 + μ 6 cos 2 θ ) ρ 3 H 2 + ( μ 7 + μ 8 cos 2 θ ) ρ 2 H 3 + μ 10 cos θ ρ H 4 + μ 12 H 5 ,
5 x = μ 1 ρ 5 sin θ + μ 3 sin 2 θ ρ 4 H + ( μ 5 + μ 6 cos 2 θ ) sin θ ρ 3 H 2 + μ 9 sin 2 θ ρ 2 H 3 + μ 11 sin θ ρ H 4 .
Δ Λ g = y a Δ ( N 0 I ) g - N 0 i a Δ Y g .
Δ Λ g 5 = y a N 0 I ξ { ξ ( k - 1 ) c 2 ( k N 1 N 0 + κ ) + η [ k ( k - 1 ) ( c N 1 N 0 - N ˙ 0 2 N 0 c 2 ) + c ( 2 k - 1 ) κ + ( k - 1 ) c N 1 N 0 ] + ζ [ c N ˙ 0 2 N 0 ( k 2 + 1 ) + k κ ] } + y a ( k - 1 ) ξ [ N 1 I + c ( N 1 + c 2 N ˙ 0 ) Y ] × [ c 2 ξ ( k - 1 ) + 2 k c η + ( k + 1 ) ζ ] / 2 - y a ξ 2 { c Y [ Δ ( 3 N 2 + 3 4 c N ˙ 1 + c 2 8 N ¨ 0 + c 3 N ˙ 0 8 ) + ( k - 1 ) ( N 2 + c 2 N 1 4 + c 3 N ˙ 0 8 ) ] + V [ Δ ( N 2 + c 2 4 N 1 + c 3 N ˙ 0 8 ) + ( k - 1 ) ( N 2 + c 2 4 N 1 + c 3 N ˙ 0 8 ) ] } - y a ξ [ c - Y η Δ ( c N ˙ 0 2 + N 1 ) ] - y a ξ ( c Y { ζ Δ N 1 + N 1 ( k - 1 ) [ ( k - 1 ) c 2 ξ + 2 k c η + k ζ ] } - 1 2 c ζ V Δ N ˙ 0 - 1 2 c N ˙ 0 ( k - 1 ) I × [ ( k - 1 ) 2 c 2 ξ + ( 2 k 2 - k - 1 ) c η + ( k 2 + k + 1 ) ζ ] ) + N 0 i a c ξ 2 2 { c Y × [ ( k - 1 ) N 1 N 0 + κ - 1 2 Δ ( N 1 N 0 ) + 1 4 c N ˙ 0 N 0 ( k - 1 ) ] + V [ c ( 1 - k ) N ˙ 0 2 N 0 + κ + c 4 Δ ( N ˙ 0 N 0 ) + 1 4 c N ˙ 0 N 0 ( k - 1 ) ] } + y a ξ 2 c 2 ( 2 N 1 + c 2 N ˙ 0 ) ( k - 1 ) I 2 - y a ξ N 1 [ ( k - 1 ) I G ( 1 ) + κ I ξ + α I ξ ( k - 1 ) + ( 1 - k ) c η I - ( k - 1 ) 2 c Y ( c 2 ξ + 2 c η + ζ ) ] .
Δ Λ g 5 = ( A ξ 2 + B ξ η + C ξ ζ ) S + ( Ā ξ 2 + B ¯ ξ η + C ¯ ξ ζ ) T ,
A = y a { N 0 i a ( k - 1 ) c 2 ( k N 1 N 0 + κ ) + ( k - 1 ) [ N 1 i a + c ( N 1 + c 2 N ˙ 0 ) y b ] c 2 ( k - 1 ) - c y a [ Δ ( 3 N 2 + 3 4 c N ˙ 1 + c 2 8 N ¨ 0 + c 3 N 0 8 ) + ( k - 1 ) ( N 2 + c 4 N 1 + c 3 8 N ˙ 0 ) ] - v a [ Δ ( N 2 + c 2 4 N 1 + c 3 8 N ˙ 0 ) + ( k - 1 ) ( N 2 + c 2 4 N 1 + c 3 8 N ˙ 0 ) ] - y 3 a N 1 ( k - 1 ) 2 + 1 2 c 3 N ˙ 0 ( k - 1 ) 3 i a + c 2 ( 2 N 1 + c 2 N ˙ 0 ) ( k - 1 ) 2 i a - N 1 [ ( k - 1 ) i a c 2 2 ( k 2 - k + 1 ) + κ i a + c N 0 ( N 1 y a - 1 2 N ˙ 0 v a ) ( k - 1 ) - ( k - 1 ) 2 c y 3 a ] } + N 0 i a c 2 { c y a [ ( k - 1 ) N 1 N 0 + κ - 1 2 Δ ( N 1 N 0 ) + 1 4 c N ˙ 0 N 0 ( k - 1 ) ] + v a [ c ( 1 - k ) 2 N 0 N ˙ 0 + κ + c 4 Δ ( N ˙ 0 N 0 ) + 1 4 c N ˙ 0 N 0 ( k - 1 ) ] } ,
B = y a { N 0 i a [ k ( k - 1 ) ( c N 1 N 0 - N ˙ 0 2 N 0 c 2 ) + c ( 2 k - 1 ) κ + ( k - 1 ) c N 1 N 0 ] + ( k - 1 ) [ N 1 i a + c ( N 1 + c 2 N ˙ 0 ) y a ] k c - c 2 y a Δ ( c N ˙ 0 2 + N 1 ) - 2 c y a ( k - 1 ) k N 1 + c 2 2 N ˙ 0 ( k - 1 ) i a · ( 2 k 2 - k - 1 ) - N 1 [ ( k - 1 ) i a k 2 c + ( 1 - k ) c i a - 2 ( k - 1 ) 2 c 2 y a ] } ,
C = y a { N 0 i a [ c N ˙ 0 2 N 0 ( 1 - k 2 ) + k κ ] + ( k - 1 ) [ N 1 i a + c ( N 1 + c 2 N ˙ 0 ) y a ] ( k + 1 ) / 2 - c y a [ Δ N 1 + N 1 ( k - 1 ) k ] + 1 2 c v a Δ N ˙ 0 + 1 2 c N ˙ 0 ( k - 1 ) ( k 2 + k + 1 ) i a - N 1 [ k ( k 2 - 1 ) i a - ( k - 1 ) 2 c y a ] } ,
Ā = y a { N 0 i b ( k - 1 ) c 2 ( k N 1 N 0 + κ ) + ( k - 1 ) [ N 1 i b + c ( N 1 + c 2 N ˙ 0 ) y b ] · c 2 ( k - 1 ) - c y b [ Δ ( 3 N 2 + 3 4 c N ˙ 1 + c 2 8 N ¨ 0 + c 3 N 0 8 ) + ( k - 1 ) ( N 2 + c 4 N 1 + c 3 8 N ˙ 0 ) ] - v b [ Δ ( N 2 + c 2 4 N 1 + c 3 8 N ˙ 0 ) + ( k - 1 ) ( N 2 + c 2 4 N 1 + c 3 8 N ˙ 0 ) ] - c 3 y b N 1 ( k - 1 ) 2 + 1 2 c 3 N ˙ 0 ( k - 1 ) 2 i b + c 2 ( 2 N 1 + c 2 N ˙ 0 ) ( k - 1 ) 2 i b - N 1 [ ( k - 1 ) i b c 2 2 ( k 2 - k + 1 ) + κ i b + c N 0 ( N 1 y b - 1 2 N ˙ 0 v b ) ( k - 1 ) - ( k - 1 ) 2 c 3 y b ] } + N 0 i a c 2 { c y b [ ( k - 1 ) N 1 N 0 + κ - 1 2 Δ ( N 1 N 0 ) + 1 4 c N ˙ 0 N 0 ( k - 1 ) ] + v b [ c ( 1 - k ) 2 N 0 N ˙ 0 + κ + c 4 Δ ( N ˙ 0 N 0 ) + 1 4 c N ˙ 0 N 0 ( k - 1 ) ] } ,
B ¯ = y a { N 0 i b [ k ( k - 1 ) ( c N 1 N 0 - N ˙ 0 2 N 0 c 2 ) + c ( 2 k - 1 ) κ + ( k - 1 ) c N 1 N 0 ] + ( k - 1 ) [ N 1 i b + c ( N 1 + c 2 N ˙ 0 ) y b ] k c - c 2 y b Δ ( c N 0 2 + N 1 ) - 2 c y b ( k - 1 ) k N 1 + c 2 2 N ˙ 0 ( k - 1 ) i b · ( 2 k 2 - k - 1 ) - N 1 × [ ( k - 1 ) i b k 2 c + ( 1 - k ) c i b - 2 ( k - 1 ) 2 c 2 y b ] } ,
C ¯ = y a { N 0 i b [ c N ˙ 0 2 N 0 ( 1 - k 2 ) + k κ ] + ( k - 1 ) [ N 1 i b + c ( N 1 + c 2 N ˙ 0 ) y b ] ( k + 1 ) / 2 - c y b [ Δ N 1 + N 1 ( k - 1 ) k ] + 1 2 c v b Δ N ˙ 0 + 1 2 c N ˙ 0 ( k - 1 ) ( k 2 + k + 1 ) i b - N 1 [ k ( k 2 - 1 ) i b - ( k - 1 ) 2 c y b ] } .
ξ = y a 2 ξ 1 + 2 y a y b η 1 + y b 2 ζ 1 ,
η = y a v a ξ 1 + ( y a v b + y b v a ) η 1 + y b v b ζ 1 ,
ζ = v a 2 ξ 1 + 2 v a v b η 1 + v b 2 ζ 1 ,
ξ 2 = y a 4 ξ 1 2 + 4 y a 3 y b ξ 1 η 1 + 2 y a 2 y b 2 ξ 1 ζ 1 + 4 y a 2 y b 2 η 1 2 + 4 y a y b 3 η 1 ζ 1 + y b 4 ζ 1 2 ,
ξ η = y a 3 v a ξ 1 2 + ( 2 y a 2 y b v a + y a 3 v b + y a 3 v b + y b y a 2 v a ) ξ 1 η 1 + 2 y a y b ( y a v b + y b v a ) η 1 2 + η 1 ζ 1 [ 2 y b 2 v b y a + y b 2 ( y a v b + y b v a ) ] + y b 3 v b ζ 1 2 ,
ξ ζ = y a 2 v a 2 ξ 1 2 + ξ 1 η 1 ( 2 y a 2 v a v b + 2 y a y b v a 2 ) + ξ 1 ζ 1 ( y a 2 v b 2 + y b 2 v a 2 ) + η 1 2 ( 4 v a v b y a y b ) + η 1 ζ 1 2 ( v a v b y b 2 + y a y b v b 2 ) + ζ 1 2 v b 2 y b 2 ,
η 2 = y a 2 v a 2 ξ 1 2 + 2 y a v a ( y a v b + y b v a ) ξ 1 η 1 + ξ 1 ζ 1 2 ( y a v a y b v b ) + η 1 2 ( y a 2 v b 2 + y b 2 v a 2 + 2 y a y b v a v b ) + η 1 ζ 1 2 y b v b ( y a v b + y b v a ) + y b 2 v b 2 ζ 1 2 ,
η ζ = y a v a 3 ξ 1 2 + ξ 1 η 1 ( 3 v a 2 y a v b + v a 3 y b ) + ξ 1 ζ 1 ( y a v a v b 2 + y b v b v a 2 ) + 2 η 1 2 ( y a v a v b 2 + y b v a 2 v b ) + η 1 ζ 1 ( 3 y b v a v b 2 + y a v b 3 ) + ζ 1 2 v b 3 y b ,
ζ 2 = v a 4 ξ 1 2 + 4 v a 3 v b ξ 1 η 1 + 2 v a 2 v b 2 ξ 1 ζ 1 + 4 v a 2 v b 2 η 2 + 4 v a v b 3 η 1 ζ 1 + v b 4 ζ 1 2 .
ξ 2 = y a 4 ξ 1 2 + 4 y a 3 y b ξ 1 η 1 + 2 y a 2 y b 2 ξ 1 ζ 1 + 4 y a 2 y b 2 η 1 2 + 4 y a y b 3 η 1 ζ 1 + y b 4 ζ 1 2 .
co ( ξ 2 , ξ 1 η 1 ) = 4 y a 3 y b ,
s 1 g = A co ( ξ 2 , ξ 1 2 ) + B co ( ξ η , ξ 1 2 ) + C co ( ξ ζ , ξ 1 2 ) ,
s ¯ 1 g = Ā co ( ξ 2 , ξ 1 2 ) + B ¯ co ( ξ η , ξ 1 2 ) + C ¯ co ( ξ ζ , ξ 1 2 ) ,
s 2 g = A co ( ξ 2 , ξ 1 η 1 ) + B co ( ξ η , ξ 1 η 1 ) + C co ( ξ ζ , ξ 1 ζ 1 ) ,
s ¯ 2 g = Ā co ( ξ 2 , ζ 1 η 1 ) + B ¯ co ( ξ η , ξ 1 η 1 ) + C ¯ co ( ξ ζ , ξ 1 ζ 1 ) ,
s 3 g = A co ( ξ 2 , ξ 1 ζ 1 ) + B co ( ξ η , ξ 1 ζ 1 ) + C co ( ξ ζ , ξ 1 ζ 1 ) ,
s ¯ 3 g = Ā co ( ξ 2 , ξ 1 ζ 1 ) + B ¯ co ( ξ η , ξ 1 ζ 1 ) + C ¯ co ( ξ ζ , ξ 1 ζ 1 ) ,
s 4 g = A co ( ξ 2 , η 1 2 ) + B co ( ξ η , η 1 2 ) + C co ( ξ ζ , η 1 2 ) ,
s ¯ 4 g = Ā co ( ξ 2 , η 1 2 ) + B ¯ co ( ξ η , η 1 2 ) + C ¯ co ( ξ ζ , η 1 2 ) ,
s 5 g = A co ( ξ 2 , η 1 ζ 1 ) + B co ( ξ η , η 1 ζ 1 ) + C co ( ξ ζ , η 1 ζ 1 ) ,
s ¯ 5 g = Ā co ( ξ 2 , η 1 ζ 1 ) + B ¯ co ( ξ η , η 1 ζ 1 ) + C co ( ξ ζ , η 1 ζ 1 ) ,
s 6 g = A co ( ξ 2 , ζ 1 2 ) + C co ( ξ η , ζ 1 2 ) + C co ( ξ ζ , ζ 1 2 ) ,
s ¯ 6 g = Ā co ( ξ 2 , ζ 1 2 ) + C ¯ co ( ξ η , ζ 1 2 ) + C ¯ co ( ξ ζ , ζ 1 2 ) .
s 1 g = y a N 0 i a y a 2 { y a 2 ( k - 1 ) c 2 ( k N 1 N 0 + κ ) + y a v a [ k ( k - 1 ) × ( c N 1 N 0 - N ˙ 0 2 N 0 c 2 ) + c ( 2 k - 1 ) κ + ( k - 1 ) c N 1 N 0 ] + v a 2 [ c N ˙ 0 2 N 0 ( 1 - k 2 ) + k κ ] } + y a ( k - 1 ) y a 2 × [ N 1 i a + c ( N 1 + c 2 N ˙ 0 ) y a ] × [ c 2 y a 2 ( k - 1 ) + 2 c k y a v a + ( k + 1 ) v a 2 ] / 2 - y a 3 { c y a [ Δ ( 3 N 2 + 3 4 c N ˙ 1 + c 2 8 N ¨ 0 + c 3 N 0 8 ) + ( k - 1 ) ( N 2 + c 2 N 1 4 + c 3 N ˙ 0 8 ) ] + v a [ Δ ( N 2 + c 2 N 1 4 + c 3 N ˙ 0 8 ) + ( k - 1 ) ( N 2 + c 2 4 N 1 + c 3 N ˙ 0 8 ) ] } - y a 3 { c 2 y a [ y a v a Δ ( c N ˙ 0 2 + N 1 ) ] } - y a 3 { c y a [ v a 2 Δ N 1 + N 1 ( k - 1 ) ] [ ( k - 1 ) c 2 y a 2 + 2 k c y a v a + k v a 2 ] - 1 2 c v a 3 Δ N ˙ 0 - 1 2 c N ˙ 0 ( k - 1 ) i a [ ( k - 1 ) 2 c 2 y a 2 + 2 c ( k 2 - k - 1 ) y a v a + ( k 2 + k + 1 ) ζ ] } + N 0 i a c y a 4 2 { c y a [ ( k - 1 ) N 1 N 0 + κ - 1 2 Δ ( N 1 N 0 ) + 1 4 c N ˙ 0 N 0 ( k - 1 ) ] + v a [ c ( 1 - k ) N ˙ 0 2 N 0 + κ + c 4 Δ ( N ˙ 0 N 0 ) + 1 4 c N ˙ 0 N 0 ( k - 1 ) ] } + y a 5 c 2 ( 2 N 1 + c 2 N ˙ 0 ) ( K - 1 ) 2 i a - y a 3 N 1 { ( k - 1 ) i a [ 1 2 ( k 2 - k + 1 ) × c 2 y a 2 + y a v a k 2 c + 1 2 k ( k + 1 ) v a 2 ] + κ i a y a 2 + c N 0 ( N 1 y a - 1 2 N ˙ 0 v a ) y a 2 ( k - 1 ) + ( 1 - k ) c y a v a i a - 1 2 ( k - 1 ) 2 c y a ( c 2 y a 2 + 2 c y a v a + v a 2 ) } .
L = N [ x , ξ ( x ) ] ( 1 + ζ ) 1 / 2 ,             ζ = V 2 + W 2 ,
= L 0 + L 1 + L 2 + L 3 + ,
L n = m = 0 n ( 1 / 2 m ) N n - m ( x ) ξ n - m ζ m ,
N [ x , ξ ( x ) ] = N 0 ( x ) + N 1 ( x ) ξ + N 2 ( x ) ξ 2 + N 3 ( x ) ξ 3 + ,
( 1 + ζ ) 1 / 2 = 1 + ζ / 2 - ζ 2 8 + 1 16 ζ 3 + .
L = N 0 + ( N 1 ξ + N 0 ξ / 2 ) + ( N 2 ξ 2 + N 1 ξ ζ 2 - 1 8 N 0 ξ 2 ) + ( N 3 ξ 3 + N 2 ξ 2 ζ 2 ) - ( 1 8 N 1 ξ ζ 2 + 1 16 N 0 ζ 3 ) + ,
L 0 = N 0 ,
L 1 = N 1 ξ + N 0 ζ / 2 ,
L 2 = N 2 ξ 2 + ( N 1 ξ ζ ) / 2 - 1 8 N 0 ζ 2 ,
L 3 = N 3 ξ 3 + N 2 ( ξ 2 ζ ) / 2 - 1 8 ( N 1 ξ ζ 2 + 1 16 N 0 ζ 3 ) .
F = L * Y - d d x ( L * V ) .
F = L 3 Y - d d x ( L 3 V ) .
F 3 = 6 N 3 Y ξ 2 + 2 N 2 Y ξ ζ - 1 4 N 1 Y ζ 2 - d d x ( N 2 ξ 2 V - 1 2 N 1 ξ ζ V + 3 8 N 0 ζ 2 V ) .
d Λ d x = y a ( x ) F ( x ) .
Λ ( x ) = x j x j + 1 y a ( x ) F ( x ) d x .
Λ g 5 = x j x j + 1 y a ( x ) { ( 6 N 3 ξ 2 + 2 N 2 ξ ζ - 1 4 N 1 ζ 2 ) Y - d d x [ ( N 2 ξ 2 - 1 2 N 1 ξ ζ + 3 8 N 0 ζ 2 ) V ] } d x .
Λ g 5 = - [ y a ( N 2 ξ 2 - 1 2 N 1 ξ ζ + 3 8 N 0 ζ 2 ) V ] + [ y a ( 6 N 3 ξ 2 + 2 N 2 ξ ζ - 1 4 N 1 ζ 2 ) Y + v a ( N 2 ξ 2 - 1 2 N 1 ξ ζ + 3 8 N 0 ζ 2 ) V ] d x + 0 ( 7 ) .
s 1 m = - [ y a ( N 2 y a 4 - 1 2 N 1 y a 2 v a 2 + 3 8 N 0 v a 4 ) v a ] + [ y a ( 6 N 3 y a 4 + 2 N 2 y a 2 v a 2 - 1 4 N 1 v a 4 ) y a + v a ( N 2 y a 4 - 1 2 N 1 y a 2 v a 2 + 3 8 N 0 v a 4 ) v a ] d x ,
s 2 m = - [ y a ( 4 N 2 y a 3 y b - N 1 ( y a v b + y a v a ) y a v a + 3 2 N 0 v a 3 v b ) v a ] + [ Y a ( 24 N 3 y a 3 y b + 4 N 2 y a v a ( y a v b + y b v a ) - 4 N 1 v a 3 v b ) y a + v a ( 4 N 2 y a 3 y b - N 1 y a v a ( y a v b + y b v a ) + 3 2 N 0 v a 3 v b ) v a ] d x ,
s 3 m = - [ y 2 ( 2 N 2 y a 2 y b 2 - 1 2 N 1 ( y a 2 v b 2 + y b 2 v a 2 ) + 3 4 N 0 v a 2 v b 2 ) v a ] + [ y a ( 12 N 3 y a 2 y b 2 + 2 N 2 y a 2 v b 2 + y b 2 v a 2 ) - 1 2 N 1 v a 2 v b 2 ) y a + v a ( 2 N 2 y a 2 y b 2 - 1 2 N 1 ( y a 2 v b 2 + y b 2 v b 2 ) ) + 3 4 N 0 v a 2 v b 2 ) v a ] d x ,
s 4 m = - [ y a ( 4 y a 2 y b 2 N 2 - 2 N 1 v a y a v b y b + 3 2 N 0 v a 2 v b 2 ) v a ] + [ y a ( 24 N 3 y a 2 y b 2 + 8 n 2 v a y a v b y b - N 1 v a 2 v b 2 ) y a - v a ( 4 y a 2 y b 2 N 2 - 2 N 1 v a y a v b y b + 3 2 N 0 v a 2 v b 2 ) v a ] d x ,
s 5 m = - [ y a ( 4 N 2 y a y b 3 - N 1 ( v a y b + v b y a ) y b v b + 3 2 N 0 v a v b 3 ) v a ) ] + [ y a ( 24 N 3 y a y b 3 - 4 N 2 ( v a y b + v b y a ) y b v b - N 1 v a v b 3 ) y a ) - v a ( 4 N 2 y a y b 3 - N 1 ( v a y b + v b y a ) y b v b + 3 2 N 0 v a v b 3 ) v a ] d x ,
s 6 m = - [ y a ( y b 4 N 2 - 1 2 N 1 v b 2 y b 2 + 3 8 N 0 v b 4 ) v a ] + [ y a ( 6 N 3 y b 4 + 2 N 2 v b 2 y b 2 - 1 4 N 1 v b 4 ) y a + v a ( N 2 y b 4 - 1 2 N 1 v b 2 y b 2 + 3 8 N 0 v b 4 ) v a ] d x ,
s ¯ 1 m = - [ y a ( N 2 y a 4 - 1 2 N 1 y a 2 v a 2 + 3 8 N 0 v a 4 ) v b ] + [ y a ( 6 N 3 y a 4 + 2 N 2 y a 2 v a 2 - 1 4 N 1 v a 4 ) y b + v a ( N 2 y a 4 - 1 2 N 1 y a 2 v a 2 + 3 8 N 0 v a 4 ) v b ] d x ,
s ¯ 2 m = - [ y a ( 4 N 2 y a 3 y b - N 1 ( y a v b + y b v a ) y a v a + 3 2 N 0 v a 3 v b ) v b ] + [ Y a ( 24 N 3 y a 3 y b + 4 N 2 y a v a ( y a v b + y b v a ) - 4 N 1 v a 3 v b ) y b + v a ( 4 N 2 y a 3 y b - N 1 y a v a ( y a v b + y b v a ) + 3 2 N 0 v a 3 v b ) v b ] d x ,
s ¯ 3 m = - [ y a ( 2 N 2 y a 2 y b 2 - 1 2 N 1 ( y a 2 v b 2 + y b 2 v α 2 ) + 3 4 N 0 v a 2 v b 2 ) v b ] + [ y a ( 12 N 3 y a 2 y b 2 + 2 N 2 y a 2 v b 2 + y b 2 v a 2 ) - 1 2 N 1 v a 2 v b 2 ) y b + v a ( 2 N 2 y a 2 y b 2 - 1 2 N 1 ( y a 2 v b 2 + y b 2 v a 2 ) + 3 4 N 0 v a 2 v b 2 ) v b ] d x ,
s ¯ 4 m = - [ y a ( 4 y a 2 y b 2 N 2 - 2 N 1 v a y a v b y b + 3 2 n 0 v a 2 v b 2 ) v b ] + [ y a ( 24 N 3 y a 2 y b 2 + 8 N 2 v a y a v b y b - N 1 v a 2 v b 2 ) y b - v a ( 4 y a 2 y b 2 N 2 - 2 N 1 v a y a v b y b + 3 2 N 0 v a 2 v b 2 ) v b ] d x ,
s ¯ 5 m = - [ y a ( 4 N 2 y a y b 3 - N 1 ( v a y b + v b y a ) y b v b + 3 2 N 0 v a v b 3 ) v b ) ] + [ y a ( 24 N 3 y a y b 3 - 4 N 2 ( v a y b + v b y a ) y b v b - N 1 v a v b 3 ) y b ) - v a ( 4 N 2 y a y b 3 - N 1 ( v a y b + v b y a ) y b v b + 3 2 N 0 v a v b 3 ) v b ] d x ,
s ¯ 6 m = - [ y a ( y b 4 N 2 - 1 2 N 1 v b 2 y b 2 + 3 8 N 0 v b 4 ) v b ] + [ y a ( 6 N 3 y b 4 + 2 N 2 v b 2 y b 2 - 1 4 N 1 v b 4 ) y b + v a ( N 2 y b 4 - 1 2 N 1 v b 2 y b 2 + 3 8 N 0 v b 4 ) v b ] d x .
- ( y a N 2 ξ 2 V ) .
Λ = - ( N 1 ξ - 1 2 N 0 ζ ) y a V + [ y a ( 4 N 2 ξ + N 1 ζ ) Y + v a ( N 1 ξ - 1 2 N 0 ζ ) v ] d x .
qc ξ = y a 2 ξ 1 * + 2 y a y b η 1 * + y b 2 ζ 1 * , ζ = v a 2 ξ 1 * + 2 v a v b η 1 * + v b 2 ζ 1 * .
Λ = - { [ ( N 1 y a 2 - 1 2 N 0 v a 2 ) ξ 1 * + 2 ( N 1 y a y b - 1 2 N 0 v a v b ) η 1 * + ( N 1 y b 2 - 1 2 N 0 v b 2 ) ζ 1 * ] y a V } + { y a [ ( 4 N 2 y a 2 + N 1 v a 2 ) ξ 1 * + 2 ( 4 N 2 y a y b + N 1 v a v b ) η 1 * + ( 4 N 2 y b 2 + N 1 v b 2 ) ζ 1 * ] Y + v a [ ( N 1 y a 2 - 1 2 N 0 v a 2 ) ξ 1 * + 2 ( N 1 y a y b - 1 2 N 0 v a v b ) η 1 * + ( N 1 v a 2 - 1 2 N 0 v b 2 ) ζ 1 * ) V ] d x .
Y y a ( S + δ S ) + y b ( T + δ T ) , V v a ( S + δ S ) + v b ( T + δ T ) , ξ 1 * ξ 1 + 2 S δ S + 0 ( δ 2 ) , η 1 * η 1 + ( S δ T + T δ S ) + 0 ( δ 2 ) , ζ 1 * ζ 1 + 2 T δ T + 0 ( δ 2 ) ,
δ S j = - i = 1 j - 1 Δ Λ b j , δ S j = - 1 λ [ ( ` A b j ξ 1 + ` B b j η 1 + ` C b j ζ 1 ) S + ( ` Ā b j ξ 1 + ` B ¯ b j η 1 + ` C ¯ b j ζ 1 ) T ] ,
δ T j = + i = 1 j - 1 Δ Λ a j , δ T j = ( ` A a j ξ 1 + ` B a j η 1 + ` C a j ζ 1 ) S + ( ` Ā a j ξ 1 + ` B ¯ a j η 1 + ` C ¯ a j ζ 1 ) T .
s 1 T m = - [ ( N 1 y a 3 - 1 2 N 0 v a 2 y a ) ( - 3 v a ` A b + v b ` A a ) + 2 ( N 1 y a 2 y b - N 0 v a v b y a ) v a ` A a ] + { y a [ ( 4 N 2 y a 2 + N 1 v a 2 ) ( - 3 y a ` A b + y b ` A a ) + 2 ( 4 N 2 y a y b + N 1 v a v b ) y b ` A a ] + v a [ ( N 1 y a 2 - 1 2 N 0 v a 2 ) ( - 3 v a ` A b + v b ` A a ) + 2 ( N 1 y a y b - 1 2 N 0 v a v b ) v a ` A a ] } d x .
a a m = - a 1 a m + a ˙ 2 a m d x , b a m = - b 1 a m + b ˙ 2 a m d x , , c ¯ a m = - c ¯ 1 a m + c ¯ ˙ 2 a m d x .
s 1 T m = - [ - 3 ` A b a 1 a m + ` A a ( ā 1 a m + b 1 a m ) ] + [ - 2 ` A b a ˙ 2 a m + ` A a ( ā ˙ 2 a m + b ˙ 2 a m ) ] d x .
{ A a a } = - ( ` A a a 1 a m ) + ` A a a ˙ 2 a m d x .
s 1 T m = { - 3 A b a + ` A a ( ā + b ) } .
Δ Λ = ( S ¯ 6 b ξ 1 2 + S ¯ 5 b ξ 1 η 1 + S ¯ 4 b ξ 1 ζ 1 + S ¯ 2 b η 1 ζ 1 + S ¯ 1 b ζ 1 2 ) S + ( S 6 b ξ 1 2 + S 5 b ξ 1 η 1 + S 4 b ξ 1 ζ 1 + S 3 b η 1 2 + S 2 b η 1 ζ 1 + S 1 b ζ 1 2 ) T .
ˆ y b = S 1 b H 5 .
S y H , S z 0 , T y ρ cos θ , T z ρ sin θ , ξ 1 ρ 2 , η 1 ρ H sin θ , ζ 1 H 2 ,
ˆ y b = [ S 1 b ρ 5 cos θ + ( S ¯ 1 b + S 2 b cos 2 θ ) ρ 4 H ] .