Abstract

Analytic solutions are found for skew ray paths in inhomogeneous cylindrical systems such as GRIN rods and gas lenses, which focus rays by virtue of a gradual radial decrease in refractive index. The solutions are obtained by employing digital computers to carry out analytic, as opposed to numerical, calculations.

© 1971 Optical Society of America

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References

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  1. A. D. Pearson, W. G. French, E. G. Rawson, Appl. Phys. Lett. 15, 76 (1969); see also F. P. Kapron, J. Opt. Soc. Amer. 60, 1433 (1970), P. J. Sands, J. Opt. Soc. Amer. 60, 1436 (1970), D. T. Moore, M. S. thesis, Institute of Optics, University of Rochester (1970) and the papers cited in Ref. 4.
    [CrossRef]
  2. T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matsumura, paper presented at the IEEE/Opt. Soc. Amer. Joint Conf. Laser Eng. Appl., May 1969, Washington, D. C.
  3. D. Marcuse, S. E. Miller, Bell Sys. Tech. J. 43, 1759 (1964).
  4. W. Streifer, K. B. Paxton, Appl. Opt. 10, 769 (1971).
    [CrossRef] [PubMed]
  5. N. N. Bogoliubov, Y. A. Mitropolsky, Asymptotic Methods in the Theory of Non-Linear Oscillations (Gordon and Breach, New York, 1961).
  6. M. Herzberger, Modern Geometrical Optics (Interscience, New York, 1958), p. 30.

1971 (1)

1969 (1)

A. D. Pearson, W. G. French, E. G. Rawson, Appl. Phys. Lett. 15, 76 (1969); see also F. P. Kapron, J. Opt. Soc. Amer. 60, 1433 (1970), P. J. Sands, J. Opt. Soc. Amer. 60, 1436 (1970), D. T. Moore, M. S. thesis, Institute of Optics, University of Rochester (1970) and the papers cited in Ref. 4.
[CrossRef]

1964 (1)

D. Marcuse, S. E. Miller, Bell Sys. Tech. J. 43, 1759 (1964).

Bogoliubov, N. N.

N. N. Bogoliubov, Y. A. Mitropolsky, Asymptotic Methods in the Theory of Non-Linear Oscillations (Gordon and Breach, New York, 1961).

French, W. G.

A. D. Pearson, W. G. French, E. G. Rawson, Appl. Phys. Lett. 15, 76 (1969); see also F. P. Kapron, J. Opt. Soc. Amer. 60, 1433 (1970), P. J. Sands, J. Opt. Soc. Amer. 60, 1436 (1970), D. T. Moore, M. S. thesis, Institute of Optics, University of Rochester (1970) and the papers cited in Ref. 4.
[CrossRef]

Furukawa, M.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matsumura, paper presented at the IEEE/Opt. Soc. Amer. Joint Conf. Laser Eng. Appl., May 1969, Washington, D. C.

Herzberger, M.

M. Herzberger, Modern Geometrical Optics (Interscience, New York, 1958), p. 30.

Kitano, I.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matsumura, paper presented at the IEEE/Opt. Soc. Amer. Joint Conf. Laser Eng. Appl., May 1969, Washington, D. C.

Koizumi, K.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matsumura, paper presented at the IEEE/Opt. Soc. Amer. Joint Conf. Laser Eng. Appl., May 1969, Washington, D. C.

Marcuse, D.

D. Marcuse, S. E. Miller, Bell Sys. Tech. J. 43, 1759 (1964).

Matsumura, H.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matsumura, paper presented at the IEEE/Opt. Soc. Amer. Joint Conf. Laser Eng. Appl., May 1969, Washington, D. C.

Miller, S. E.

D. Marcuse, S. E. Miller, Bell Sys. Tech. J. 43, 1759 (1964).

Mitropolsky, Y. A.

N. N. Bogoliubov, Y. A. Mitropolsky, Asymptotic Methods in the Theory of Non-Linear Oscillations (Gordon and Breach, New York, 1961).

Paxton, K. B.

Pearson, A. D.

A. D. Pearson, W. G. French, E. G. Rawson, Appl. Phys. Lett. 15, 76 (1969); see also F. P. Kapron, J. Opt. Soc. Amer. 60, 1433 (1970), P. J. Sands, J. Opt. Soc. Amer. 60, 1436 (1970), D. T. Moore, M. S. thesis, Institute of Optics, University of Rochester (1970) and the papers cited in Ref. 4.
[CrossRef]

Rawson, E. G.

A. D. Pearson, W. G. French, E. G. Rawson, Appl. Phys. Lett. 15, 76 (1969); see also F. P. Kapron, J. Opt. Soc. Amer. 60, 1433 (1970), P. J. Sands, J. Opt. Soc. Amer. 60, 1436 (1970), D. T. Moore, M. S. thesis, Institute of Optics, University of Rochester (1970) and the papers cited in Ref. 4.
[CrossRef]

Streifer, W.

Uchida, T.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matsumura, paper presented at the IEEE/Opt. Soc. Amer. Joint Conf. Laser Eng. Appl., May 1969, Washington, D. C.

Appl. Opt. (1)

Appl. Phys. Lett. (1)

A. D. Pearson, W. G. French, E. G. Rawson, Appl. Phys. Lett. 15, 76 (1969); see also F. P. Kapron, J. Opt. Soc. Amer. 60, 1433 (1970), P. J. Sands, J. Opt. Soc. Amer. 60, 1436 (1970), D. T. Moore, M. S. thesis, Institute of Optics, University of Rochester (1970) and the papers cited in Ref. 4.
[CrossRef]

Bell Sys. Tech. J. (1)

D. Marcuse, S. E. Miller, Bell Sys. Tech. J. 43, 1759 (1964).

Other (3)

N. N. Bogoliubov, Y. A. Mitropolsky, Asymptotic Methods in the Theory of Non-Linear Oscillations (Gordon and Breach, New York, 1961).

M. Herzberger, Modern Geometrical Optics (Interscience, New York, 1958), p. 30.

T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, H. Matsumura, paper presented at the IEEE/Opt. Soc. Amer. Joint Conf. Laser Eng. Appl., May 1969, Washington, D. C.

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Figures (2)

Fig. 1
Fig. 1

aθ phase plane for ai = ρi/√2.

Fig. 2
Fig. 2

aθ phase plane for ai = ρi/2.

Equations (119)

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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 ) 4 + ] ,
x ¨ + x = x j = 2 j α j δ j - 1 r 2 ( j - 1 )
and             y ¨ + y = y j = 2 j α j δ j - 1 r 2 ( j - 1 ) ,
x = ξ / r 0 ,             y = η / r 0 ,             r = r / r 0 ,
z = [ ( δ 1 2 / r 0 ) ( N 0 / N i ) sec γ i ] ζ .
x = a cos ψ + δ x 1 ( a , b , ψ , ϕ ) + δ 2 x 2 ( a , b , ψ , ϕ ) + ,
y = b cos ϕ + δ y 1 ( a , b , ψ , ϕ ) + δ 2 y 2 ( a , b , ψ , ϕ ) + ,
where             a ˙ = δ A 1 ( a , b , θ ) + δ 2 A 2 ( a , b , θ ) + ,
b ˙ = δ B 1 ( a , b , θ ) + δ 2 B 2 ( a , b , θ ) + ,
ψ ˙ = 1 + δ C 1 ( a , b , θ ) + δ 2 C 2 ( a , b , θ ) + ,
and             ϕ ˙ = 1 + δ D 1 ( a , b , θ ) + δ 2 D 2 ( a , b , θ ) + .
θ = ψ - ϕ
θ ˙ = δ [ C 1 ( a , b , θ ) - D 1 ( a , b , θ ) ] + δ 2 [ C 2 ( a , b , θ ) - D 2 ( a , b , θ ) ] + .
a b ,             ψ ϕ ,             θ - θ .
a ˙ = δ A 1 ( a , b , θ ) = - [ ( δ b 2 a ) / 4 ] α 2 sin ( 2 θ ) ,
ψ ˙ = 1 + δ C 1 ( a , b , θ ) = 1 - [ ( δ α 2 ) / 4 ] ( 3 a 2 + 2 b 2 ) - [ ( δ b 2 ) / 4 ] α 2 cos ( 2 θ ) ,
x 1 ( a , b , ψ , ϕ ) = - [ ( b 2 a ) / 16 ] α 2 cos ( ψ + 2 ϕ ) - ( a 3 / 16 ) α 2 cos ( 3 ψ ) ,
a ˙ = δ A 1 ( a , b , θ ) = - [ ( δ b 2 a ) / 4 ] α 2 sin ( 2 θ ) ,
b ˙ = δ B 1 ( a , b , θ ) = [ ( δ a 2 b ) / 4 ] α 2 sin ( 2 θ ) ,
and θ ˙ = δ [ C 1 ( a , b , θ ) - D 1 ( a , b , θ ) ] = - { [ δ ( a 2 - b 2 ) α 2 ] / 2 } sin 2 θ .
a 2 + b 2 = a i 2 + b i 2 = ρ i 2 ,
d a d θ = a ˙ θ ˙ = - a ρ i 2 - a 2 ρ i 2 - 2 a 2 cot ( θ ) .
a 2 ( θ ) = ρ i 2 2 ± [ ρ i 4 4 - a i 2 ( ρ i 2 - a i 2 ) sin 2 θ i sin 2 θ ] 1 2 ,
or             a 2 ( θ ) = ρ i 2 2 ± [ ρ i 4 4 - a i 2 b i 2 sin 2 θ i sin 2 θ ] 1 2 ,
θ m = sin - 1 [ 2 a i b i ρ i 2 sin θ i ] ,
b 2 ( θ ) = ρ i 2 - a 2 ( θ ) = ρ i 2 2 [ ρ i 4 4 - a i 2 b i 2 sin 2 θ i sin 2 θ ] 1 2 ,
θ ˙ ( θ ) = δ α 2 sin 2 θ [ ρ i 4 4 - a i 2 b i 2 sin 2 θ i sin 2 θ ] 1 2 .
κ 1 = δ α 2 a i b i sin θ i
and             τ i = cos - 1 [ ( tan θ m ) / ( tan θ i ) ] ,
θ ( z ) = sin - 1 [ sin 2 θ m 1 - cos 2 θ m sin 2 ( κ 1 z - τ i ) ] 1 2 ,
a ( z ) = ( ρ i / 2 1 2 ) [ 1 - cos θ m sin ( κ 1 z - τ i ) ] 1 2
and             b ( z ) = ( ρ i / 2 1 2 ) [ 1 + cos θ m sin ( κ 1 z - τ i ) ] 1 2 ,
ψ ( z ) = ( 1 - δ α 2 3 4 ρ i 2 ) z + tan - 1 { csc θ m tan [ ( κ 1 z - τ i ) / 2 ] - cot θ m } - tan - 1 [ csc θ m tan ( - τ i / 2 ) - cot θ m ] + ψ i .
ϕ ( z ) = ( 1 - δ α 2 3 4 ρ i 2 ) z + tan - 1 { csc θ m tan [ ( κ 1 z - τ i ) / 2 ] + cot θ m } - tan - 1 [ csc θ m tan ( - τ i / 2 ) + cot θ m ] + ϕ i ,
x ( z ) = a ( z ) cos ψ ( z ) + δ x 1 [ a ( z ) , b ( z ) , ψ ( z ) , ϕ ( z ) ] , and             y ( z ) = b ( z ) cos ϕ ( z ) + δ y 1 [ a ( z ) , b ( z ) , ψ ( z ) , φ ( z ) ] ,
a i ( x i , y i , x ˙ i , y ˙ i ) = a 0 ( x i , y i , x ˙ i , y ˙ i ) + δ a i ( x i , y i , x ˙ i , y ˙ i ) + ,
ψ i ( x i , y i , x ˙ i , y ˙ i ) = ψ 0 ( x i , y i , x ˙ i , y ˙ i ) + δ ψ 1 ( x i , y i , x ˙ i , y ˙ i ) + ,
a 0 = ( x i 2 + x ˙ i 2 ) 1 2
and             ψ 0 = - tan - 1 ( x ˙ i / x i ) .
a 1 = n m C 1 ( n , m ) cos ( n ψ 0 + m ϕ 0 )
and             ψ 1 = n m S 1 ( n , m ) sin ( n ψ 0 + m ϕ 0 ) ,
C 1 ( 0 , 0 ) = [ ( a 0 α 2 ) / 8 ] ( 3 a 0 2 + 2 b 0 2 ) ,
C 1 ( 2 , - 2 ) = [ ( a 0 α 2 ) / 8 ] b 0 2 ,
C 1 ( 2 , 0 ) = - [ ( a 0 α 2 ) / 4 ] ( a 0 2 + b 0 2 ) ,
C 1 ( 2 , 2 ) = - [ ( a 0 α 2 ) / 16 ] b 0 2 ,
C 1 ( 4 , 0 ) = - [ ( a 0 3 α 2 ) / 16 ] ,
and             S 1 ( 0 , 2 ) = [ ( b 0 2 α 2 ) / 4 ] ,
S 1 ( 2 , - 2 ) = - [ ( b 0 2 α 2 ) / 4 ] ,
S 1 ( 2 , 0 ) = ( α 2 / 4 ) ( 2 a 0 2 + b 0 2 ) ,
S 1 ( 2 , 2 ) = [ ( b 0 2 / α 2 ) / 16 ] ,
S 1 ( 4 , 0 ) = [ ( a 0 2 α 2 ) / 16 ] .
ψ 1 = ψ 2 = ϕ 1 = ϕ 2 = 0 ,
a 1 = [ ( a 0 α 2 ) / 2 4 ] ( a 0 2 - b 0 2 ) ,
b 1 = [ ( b 0 α 2 ) / 2 4 ] ( 7 a 0 2 + 9 b 0 2 ) ,
a 2 = - [ ( a 0 α 3 ) / 2 4 ] ( b 0 4 + a 0 2 b 0 2 - 2 a 0 4 ) - [ ( a 0 α 2 2 ) / 2 8 ] ( 37 b 0 4 + 18 a 0 2 b 0 2 - 23 a 0 4 ) ,
b 2 = [ ( b 0 α 3 ) / 2 4 ] ( 7 a 0 4 + 7 a 0 2 b 0 2 + 10 b 0 4 ) , + [ ( b 0 α 2 2 ) / 2 8 ] ( 43 a 0 4 + 190 a 0 2 b 0 2 + 55 b 0 4 ) .
x 1 0 ,             x 2 0 ,             and             x 3 0 ,
ψ ˙ = 1 - δ α 2 ( ρ i 2 / 2 ) - δ 2 ( ρ i 4 / 2 3 ) ( α 2 2 + 3 α 3 ) - δ 3 ( ρ i 6 / 2 4 ) ( α 2 3 + 3 α 2 α 3 + 4 α 4 ) ,
ϕ ˙ = ψ ˙ ,
and             ϕ = ψ ± π / 2.
y ˙ = - b ϕ ˙ sin ϕ = - b ψ ˙ cos ψ and             y ˙ i = - ( ρ i / 2 ) ψ ˙ ,
ψ ˙ z = δ 1 2 r 0 N 0 N i ( sec γ i ) ψ ˙ ζ = ( δ 1 2 / r 0 ) ψ ˙ ζ [ ( N i / N 0 ) 2 - δ ρ i 2 2 ( ψ ˙ ) 2 ] 1 2 ,
ψ ˙ z = [ 1 + δ ρ i 2 2 ( 1 - α 2 ) + δ 2 ρ i 4 2 3 ( 3 - 5 α 2 - α 2 2 - 3 α 3 ) + δ 3 ρ i 6 2 4 ( 2 - 6 α 2 - α 2 2 - 7 α 3 - α 2 3 - 3 α 2 α 3 - 4 α 4 ) ] ( δ 1 2 / r 0 ) ζ .
N 2 = N 0 2 [ 1 + δ ( r / r 0 ) 2 ] - 1
ψ ˙ z = ( δ 1 2 / r 0 ) ζ ,
x ¨ + x = 4 α 4 δ 3 x r 6
and             y ¨ + y = 4 α 4 δ 3 y r 6 .
x ¨ + x = δ x r 6
and             y ¨ + y = δ y r 6 ,
r ˙ 2 + r 2 + c 3 2 r 2 - j = 2 α j δ j - 1 r 2 j = c 2 ,
where             c 2 + x ˙ i 2 + y ˙ i 2 + r i 2 - j = 2 α j δ j - 1 r i 2 j ,
and             c 3 = x i y ˙ i - y i x ˙ i ,
δ = 0.01 , r 0 = 1.0 , x i = 0.596 902 6 , y i = 0.0 , x ˙ i = 0.0 , y ˙ i = 0.174 214 0 , α 2 = 1.0 , α j = 0.0 ; j = 3 , 4 , , γ i = 1.0 ° ,
r min = 0.174 551 9             and r max = 0.596 902 6.
r min = 0. 174 551 5             and r max = 0. 596 902 6.
x ¨ + x = δ x f ( r 2 , δ ) ,
and             y ¨ + y = δ y f ( r 2 , δ ) ,
where             f ( r 2 , δ ) = j = 2 j α j δ j - 2 r 2 ( j - 1 ) ,
x = a cos ψ + δ x 1 ( a , b , ψ , ϕ ) + ,
y = b cos ϕ + δ y 1 ( a , b , ψ , ϕ ) + ,
with             a ˙ = δ A 1 ( a , b , θ ) + ,
b ˙ = δ B 1 ( a , b , θ ) + ,
ψ ˙ = 1 + δ C 1 ( a , b , θ ) + ,
and             ϕ ˙ = 1 + δ D 1 ( a , b , θ ) + .
x ( 0 ) = a cos ψ ,
y ( 0 ) = b cos ϕ ,
δ x f ( r 2 , δ ) = 2 δ α 2 a cos ψ ( a 2 cos 2 ψ + b 2 cos 2 ϕ ) + 0 ( δ 2 ) .
x ˙ = a ˙ cos ψ - a ψ ˙ sin ψ + δ ( a ˙ x 1 a + b ˙ x 1 b + ψ ˙ x 1 ψ + ϕ ˙ x 1 ϕ ) ,
x ˙ = - a sin ψ + δ [ A 1 ( a , b , θ ) cos ψ - a C 1 ( a , b , θ ) sin ψ + x 1 ψ + x 1 ϕ ] + 0 ( δ 2 ) .
x ¨ = - a cos ψ + δ [ - 2 A 1 sin ψ - 2 a C 1 cos ψ + x 1 ψ ψ + 2 x 1 ψ ϕ + x 1 ϕ ϕ ] + 0 ( δ 2 ) ,
x 1 ψ ψ + 2 x 1 ψ ϕ + x 1 ϕ ϕ - 2 A 1 ( a , b , θ ) sin ψ - 2 a C 1 ( a , b , θ ) cos ψ + x 1 = 2 α 2 a cos ψ ( a 2 cos 2 ψ + b 2 cos 2 ϕ ) .
n , m h m n ( a , b ) exp [ i ( n ψ + m ϕ ) ] ,
- 2 A 1 ( a , b , θ ) sin ψ - 2 a C 1 ( a , b , θ ) cos ψ = n , m n + m = ± 1 h m m ( a , b ) exp [ i ( n ψ + m ϕ ) ] ,
x 1 ψ ψ + 2 x 1 ψ ϕ + x 1 ϕ ϕ + x 1 = n , m n + m ± 1 h n m ( a , b ) exp [ i ( n ψ + m ϕ ) ] .
exp [ i ( n ψ + m ϕ ) ] = exp { i [ ( n + m ) ψ - m θ ] } = exp ( ± i ψ ) exp ( - i m θ ) = exp ( - i m θ ) [ cos ψ ± i sin ψ ] .
- 2 A 1 ( a , b , θ ) sin ψ - 2 a C 1 ( a , b , θ ) cos ψ = ( cos ψ ± i sin ψ ) n , m n + m = ± 1 h n m ( a , b ) exp ( - i m θ )
A 1 ( a , b , θ ) = - i 2 n , m n + m = ± 1 ( n + m ) h n m ( a , b ) exp ( - i m θ ) ,
and             C 1 ( a , b , θ ) = - 1 2 a n , m n + m = ± 1 h n m ( a , b ) exp ( - i m θ ) .
x 1 ( a , b , ψ , ϕ ) = n , m n + m 1 ± g n m ( a , b ) exp [ i ( n ψ + m ϕ ) ] ,
n , m n + m ± 1 [ 1 - ( n + m ) 2 ] g n m ( a , b ) exp [ i ( n ψ + m ϕ ) ] = n , m n + m ± 1 h n m ( a , b ) exp [ i ( n ψ + m ϕ ) ] .
g n m ( a , b ) = h n m ( a , b ) [ 1 - ( n + m ) 2 ] ,             n + m ± 1.
a ˙ = δ A 1 + δ 2 A 2 + δ 3 A 3 , A 1 = - [ ( b 2 a ) / 4 ] α 2 sin 2 α , A 2 = - [ ( b 2 a ) / 2 5 ] ρ i 2 ( 5 α 2 2 + 12 α 3 ) sin 2 θ , A 3 = A 3 ( 2 ) sin 2 θ + A 3 ( 4 ) sin 4 θ , A 3 ( 2 ) = - [ ( b 2 a ) / 2 10 ] ( 139 α 2 3 + 384 α 2 α 3 + 480 α 4 ) - [ ( b 4 a 3 ) / 2 10 ] ( 304 α 2 3 + 768 α 2 α 3 + 768 α 4 ) - [ ( b 2 a 5 ) / 2 10 ] ( 139 α 2 3 + 384 α 2 α 3 + 480 α 4 ) , A 3 ( 4 ) = [ ( b 4 a 3 ) / 2 10 ] ( 13 α 2 3 - 96 α 4 ) .
ψ ˙ = 1 + δ C 1 + δ 2 C 2 + δ 3 C 3 C 1 = - ( α 2 / 4 ) ( 3 a 2 + 2 b 2 ) - [ ( b 2 α 2 ) / 4 ] cos 2 θ , C 2 = C 2 ( 0 ) + C 2 ( 2 ) cos 2 θ , C 2 ( 0 ) = - ( b 4 / 2 6 ) ( 5 α 2 2 + 36 α 3 ) - [ ( b 2 a 2 ) / 2 6 ] ( 26 α 2 2 + 72 α 3 ) - ( a 4 / 2 6 ) ( 15 α 2 2 + 60 α 3 ) , C 2 ( 2 ) = - ( b 4 / 2 5 ) ( 5 α 2 2 + 12 α 3 ) - [ ( b 2 a 2 ) / 2 5 ] ( 2 α 2 2 + 24 α 3 ) , C 3 = C 3 ( 0 ) + C 3 ( 2 ) cos 2 θ + C 3 ( 4 ) cos 4 θ , C 3 ( 0 ) = - ( b 6 / 2 10 ) ( - 16 α 2 3 + 96 α 2 α 3 + 640 α 4 ) - [ ( b 4 a 2 ) / 2 10 ] ( 238 α 2 3 + 864 α 2 α 3 + 1728 α 4 ) - [ ( b 2 a 4 ) / 2 10 ] ( 416 α 2 3 + 1248 α 2 α 3 + 1920 α 4 ) - ( a 6 / 2 10 ) ( 123 α 2 3 + 480 α 2 α 3 + 1120 α 4 ) , C 3 ( 2 ) = - ( b 2 / 2 10 ) ( 139 α 2 3 + 384 α 2 α 3 + 480 α 4 ) - [ ( b 4 a 2 ) / 2 10 ] ( 144 α 2 3 + 576 α 2 α 3 + 1536 α 4 ) - [ ( b 2 a 4 ) / 2 10 ] ( - 47 α 2 3 + 192 α 2 α 3 + 1440 α 4 ) , C 3 ( 4 ) = - [ ( b 4 a 2 ) / 2 10 ] ( - 13 α 2 3 + 96 α 4 ) .
x = a cos ψ + δ x 1 + δ 2 x 2 + δ 2 x 3 ,
x i = Σ n Σ m x i ( n , m ) cos ( n ψ + m ϕ ) ,             i = 1 , 2 , 3.
x 1 ( 1 , 2 ) = - [ ( b 2 a ) / 2 4 ] α 2 , x 1 ( 3 , 0 ) = - ( a 3 / 2 4 ) α 2 , x 2 ( 1 , - 4 ) = - [ ( b 4 a ) / 2 8 ] ( α 2 2 + 6 α 3 ) , x 2 ( 1 , 2 ) = - [ ( b 4 a ) / 2 8 ] ( 20 α 2 2 + 24 α 3 ) - [ ( b 2 a 3 ) / 2 8 ] ( 22 α 2 2 + 36 α 3 ) , x 2 ( 1 , 4 ) = [ ( b 4 a ) / 2 8 ] ( α 2 2 - 2 α 3 ) , x 2 ( 3 , 0 ) = - [ ( b 2 a 3 ) / 2 8 ] ( 20 α 2 2 + 24 α 3 ) - ( a 5 / 2 8 ) ( 21 α 2 2 + 30 α 3 ) , x 2 ( 3 , 2 ) = [ ( b 2 a 3 ) / 2 8 ] ( 2 α 2 2 - 4 α 3 ) , x 2 ( 5 , 0 ) = ( a 5 / 2 8 ) ( α 2 2 - 2 α 3 ) , x 3 ( 1 , - 6 ) = [ ( b 6 a ) / 2 12 ] [ α 2 3 + 8 α 2 α 3 - ( 32 / 3 ) α 4 ] , x 3 ( 1 , - 4 ) = - [ ( b 6 a ) / 2 12 ] ( 42 α 2 3 + 176 α 2 α 3 + 192 α 4 ) - [ ( b 4 a 3 ) / 2 12 ] ( 13 α 2 3 + 216 α 2 α 3 + 288 α 4 ) , x 3 ( 1 , 2 ) = - [ ( b 6 a ) / 2 12 ] ( 375 α 2 3 + 808 α 2 α 3 + 480 α 4 ) - [ ( b 4 a 3 ) / 2 12 ] ( 892 α 2 3 + 1888 α 2 α 3 + 1152 α 4 ) - [ ( b 2 a 5 ) / 2 12 ] ( 430 α 2 3 + 1200 α 2 α 3 + 960 α 4 ) , x 3 ( 1 , 4 ) = [ ( b 6 a ) / 2 12 ] ( 42 α 2 3 + 16 α 2 α 3 - 64 α 4 ) + [ ( b 4 a 3 ) / 2 12 ] ( 45 α 2 3 + 40 α 2 α 3 - 96 α 4 ) , x 3 ( 1 , 6 ) = - [ ( b 6 a ) / 2 12 ] [ α 2 3 - 6 α 2 α 3 + ( 16 / 3 ) α 4 ] , x 3 ( 3 , 0 ) = - [ ( b 4 a 3 ) / 2 12 ] ( 346 α 2 3 + 848 α 2 α 3 + 576 α 4 ) - [ ( b 2 a 5 ) / 2 12 ] ( 850 α 2 3 + 1712 α 2 α 3 + 960 α 4 ) - ( a 7 / 2 12 ) ( 417 α 2 3 + 984 α 2 α 3 + 672 α 4 ) , x 3 ( 3 , 2 ) = [ ( b 4 a 3 ) / 2 12 ] ( 84 α 2 3 + 32 α 2 α 3 - 128 α 4 ) + [ ( b 2 a 5 ) / 2 12 ] ( 87 α 2 3 + 56 α 2 α 3 - 160 α 4 ) , x 3 ( 3 , 4 ) = - [ ( b 4 a 3 ) / 2 12 ] ( 3 α 2 3 - 18 α 2 α 3 + 16 α 4 ) , x 3 ( 5 , - 2 ) = [ ( b 2 a 5 ) / 2 12 ] ( 29 α 2 3 - 40 α 2 α 3 - 96 α 4 ) , x 3 ( 5 , 0 ) = [ ( b 2 a 5 ) / 2 12 ] ( 42 α 2 2 + 16 α 2 α 3 - 64 α 4 ) + ( a 7 / 2 12 ) [ 43 α 2 3 + 24 α 2 α 3 - ( 224 / 3 ) α 4 ] , x 3 ( 5 , 2 ) = - [ ( b 2 a 5 ) / 2 12 ] ( 3 α 2 3 - 18 α 2 α 3 + 16 α 4 ) , x 3 ( 7 , 0 ) = - ( a 7 / 2 12 ) [ α 2 3 - 6 α 2 α 3 + ( 16 / 3 ) α 4 ] .
a 2 + b 2 = a i 2 + b i 2 = ρ i 2 .
d a d θ = - a ρ i 2 - a 2 ρ i 2 - 2 a 2 cot θ ,
a 2 ( θ ) = ρ i 2 2 ± { ρ i 4 2 - a i 2 b i 2 sin 2 θ i sin 2 θ } 1 2 .
b 2 ( θ ) = ρ i 2 2 { ρ i 4 4 - a i 2 b i 2 sin 2 θ i sin 2 θ } 1 2 ,
a b sin θ = a i b i sin θ i ,
θ ˙ = [ ( δ α 2 ) / 2 ] ( ρ i 2 - 2 a 2 ) sin 2 θ ( κ 3 / κ 1 ) ,
κ 3 = δ a i b i sin θ i { α 2 + δ 8 ρ i 2 ( 5 α 2 2 + 12 α 3 ) + δ 2 [ ρ i 4 2 8 ( 139 α 2 3 + 384 α 2 α 3 + 480 α 4 ) + a i 2 b i 2 sin 2 θ i 2 6 ( 13 α 2 3 - 96 α 4 ) ] } ,
ψ ˙ = 1 + δ L 1 + δ 2 L 2 + δ 3 L 3 + [ ( δ α 2 ) / 2 ] ( κ 3 / κ 1 ) b 2 sin 2 θ ,
L 1 = - 3 4 α 2 ρ i 2 ,
L 2 = - 15 2 6 ρ i 4 ( α 2 2 + 4 α 3 ) + a i 2 b i 2 sin 2 θ i 2 4 ( - 3 α 2 2 + 12 α 3 ) ,
and             L 3 = - ρ i 6 2 10 ( 123 α 2 3 + 480 α 2 α 3 + 1120 α 4 ) - a i 2 b i 2 sin 2 θ i 2 8 ρ i 2 ( 93 α 2 3 + 96 α 2 α 3 - 480 α 4 ) .
ψ ( z ) = ( 1 + δ L 1 + δ 2 L 2 + δ 3 L 3 ) z + tan - 1 { csc θ m tan [ ( κ 3 z - τ i ) / 2 ] - cot θ m } - tan - 1 [ csc θ m tan ( - τ i / 2 ) - cot θ m ] + ψ i ,
ϕ ( z ) = ( 1 + δ L 1 + δ 2 L 2 + δ 3 L 3 ) z + tan - 1 { csc θ m tan [ ( κ 3 z - τ i ) / 2 ] + cot θ m } - tan - 1 [ csc θ m tan ( - τ i / 2 ) + cot θ m ] + ϕ i .

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