Abstract

A set of simple analytic expressions has been developed for calculating the third-order aberration coefficients in weak or nearly thin axial and radial gradient-index lenses.

© 1990 Optical Society of America

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References

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  1. W. J. Smith, Modern Optical Engineering (McGrawHill, New York, 1966), p. 281.
  2. P. J. Sands, “Third-Order Aberrations of Inhomogeneous Lenses,” J. Opt. Soc. Am. 60, 1436–1443 (1970).
    [CrossRef]
  3. E. W. Marchand, Gradient Index Optics (Academic, New York, 1978), Chap. 8.
  4. P. J. Sands, “Inhomogeneous Lenses. IV: Aberrations of Lenses with Axial Index Distributions,” J. Opt. Soc. Am. 61, 1086–1091 (1971).
    [CrossRef]
  5. D. T. Moore, P. J. Sands, “Third-Order Aberrations of Inhomogeneous Lenses with Cylindrical Index Distributions,” J. Opt. Soc. Am. 61, 1195–1201 (1971).
    [CrossRef]
  6. D. T. Moore, “Aberration Correction Using Index Gradients,” M.S. Thesis, U. Rochester (1970).
  7. J. B. Caldwell et al., “Gradient-Index Binocular Objective Design,” Appl. Opt. 25, 3345–3350 (1986).
    [CrossRef] [PubMed]
  8. J. P. Bowen et al., “Radial Gradient-Index Eyepiece Design,” Appl. Opt. 27, 3170–3176 (1988).
    [CrossRef] [PubMed]
  9. P. J. Sands, “Inhomogeneous Lenses. III: Paraxial Optics,” J. Opt. Soc. Am. 61, 879–885 (1971).
    [CrossRef]
  10. R. W. Wood, Physical Optics (Macmillan, New York, 1934), p. 88.

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Tables (6)

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Table I Axial GRIN Singlet with a N01 Coefficient: Design Data

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Table II Axial GRIN Singlet with a N01 Coefficient: Third-Order Transverse Aberration Data

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Table III Wood Singlet with a N10 Coefficient: Design Data

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Table IV Wood Singlet with a N10 Coefficient: Third-Order Transverse Aberration Data

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Table V Radial GRIN Singlet with a N10 Coefficient: Design Data

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Table VI Radial GRIN Singlet with a N10 Coefficient: Third-Order Transverse Aberration Data

Equations (46)

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N ( z , r ) z [ 1 + ( d Y d z ) 2 ] d Y d z + N ( z , r ) d 2 Y d z 2 - [ 1 + ( d Y d z ) 2 ] N ( z , r ) Y = 0 ,
N ( z ) = i = 0 N 0 i z i ,
N ( r ) = i = 0 N i 0 r 2 i ,
y a , b ( z ) = y a , b ( 0 ) + N 00 u a , b ( 0 ) 0 t d z N ( z ) ,
u a , b ( z ) = N 00 u a , b ( 0 ) / N ( z ) ,
y a , b ( z ) = [ y a , b ( 0 ) 2 + u a , b ( 0 ) 4 α ] × exp ( 2 α z ) + [ y a , b ( 0 ) 2 - u a , b ( 0 ) 4 α ] × exp ( - 2 α z ) ,
u a , b ( z ) = 2 α × { [ y a , b ( 0 ) 2 + u a , b ( 0 ) 4 α ] × exp ( 2 α z ) - [ y a , b ( 0 ) 2 - u a , b ( 0 ) 4 α ] × exp ( 2 α z ) } ,
α = N 10 2 N 00 .
y a , b ( z ) y a , b ( 0 ) + N 00 u a , b ( 0 ) ( z N 00 - N 01 z 2 2 N 00 2 ) ,
u a , b ( z ) = N 00 u a , b ( 0 ) / ( N 00 + N 01 z ) .
y a , b ( z ) y a , b ( 0 ) + u a , b ( 0 ) z ,
u a , b ( z ) = N 00 u a , b ( 0 ) / ( N 00 + N 02 z 2 ) .
y a , b ( z ) ( 1 + 2 α 2 z 2 ) y a , b ( 0 ) + u a , b ( 0 ) z ,
u a , b ( z ) 4 α 2 y a , b ( 0 ) z + ( 1 + 2 α 2 z 2 ) u a , b ( 0 ) .
σ i = - 1 n k u a K ( j = 1 k a i j + j = 1 k a i j * )             i = 1 , , 5 ,
a 1 = a + κ y a 4 ,             a = N 0 2 ( N 0 N 0 - 1 ) y a i a 2 ( i a + u a ) , a 2 = q a + κ y a 3 y b ,             κ = - C Δ [ 2 N 10 + 1 2 C ( d N 0 d z ) ] , a 3 = q 2 a + κ y a 2 y b 2 ,             H = N 0 ( y b u a - y a u b ) , a 4 = 1 2 H 2 C Δ ( 1 N 0 ) , a 5 = q 3 a + q a 4 + κ y a y b 3 ,
a 1 * = 1 2 ( N 0 y a u a 3 ) + 0 t ( 4 N 20 y a 4 + 2 N 10 y a 2 u a 2 - 1 2 N 0 u a 4 ) d z , a 2 * = 1 2 ( N 0 y a u a 2 u b ) + 0 t [ 4 N 20 y a 3 y b + N 10 y a u a ( y a u b + y b u a ) - 1 2 N 0 u a 3 u b ] d z , a 3 * = 1 2 ( N 0 y a u a u b 2 ) + 0 t ( 4 N 20 y a 2 y b 2 + N 10 y a y b u a u b - 1 2 N 0 u a 2 u b 2 ) d z , a 4 * = H 2 0 t ( N 10 / N 0 2 ) d z , a 5 * = 1 2 ( N 0 y a u b 3 ) + 0 t [ 4 N 20 y a y b 3 + N 10 y b u b ( y a u b + y b u a ) - 1 2 N 0 u a u b 3 ] d z ,
a 1 a 1 , H + Δ N y a 1 u a 1 2 [ 2 N 00 2 C 2 2 y a 1 2 + ( 3 N 00 2 + 2 ) C 2 y a 1 2 u a 1 + 2 u a 1 2 ] + N 00 ( N 00 2 - 1 ) u a 1 3 t 2 ( 4 C 2 y a 1 + u a 1 ) , a 2 a 2 , H * + Δ N y a 1 u a 1 2 { C 2 [ ( N 00 2 + 2 ) y a 1 u b 1 + 2 y b 1 u a 1 ] + 2 u a 1 u b 1 } + N 00 ( N 00 2 - 1 ) u a 1 3 u b 1 t 2 , a 3 a 3 , H * + Δ N y a 1 u b 1 2 { 2 N 00 2 C 2 2 y b 1 + 2 u a 1 u b 1 + C 2 [ N 00 2 y a 1 u b 1 + 2 ( N 00 2 + 1 ) y b 1 u a 1 } + N 00 ( N 00 2 - 1 ) u a 1 u b 1 t 2 [ 2 C 2 ( y a 1 u b 1 + y b 1 u a 1 ) + u a 1 u b 1 ] , a 4 - H 2 2 N 00 ( Φ H - C 2 Δ N N 00 ) , a 5 a 5 , H * + Δ N y b 1 u b 1 2 { C 2 [ 2 y a 1 u b 1 + ( N 00 2 + 2 ) y b 1 u a 1 ] + 2 u a 1 u b 1 } + N 00 ( N 00 2 - 1 ) u a 1 u b 1 3 t 2 - H u b 1 { N 00 C 2 [ ( N 00 - 1 ) u b 1 t + y b 1 Δ N ] + Δ N u b 1 N 00 ,
a 1 a 1 , H + ( 3 N 00 - 2 ) N 00 y a 1 4 C 2 3 Δ N 2 , a 2 a 2 , H * + N 00 C 2 2 y a 1 3 2 × { Δ N [ ( 3 N 00 - 2 ) y b 1 C 2 + ( 2 N 00 - 1 ) u b 1 ] + ( N 00 - 1 ) C 2 N 00 u b 1 t } , a 3 a 3 , H * + N 00 C 2 y a 1 2 2 × [ ( u b 1 + y b 1 C 2 ) 2 N 00 Δ N + 2 ( N 00 - 1 ) C 2 u b 1 ( y b 1 Δ N + N 00 u b 1 t ] , a 5 a 5 , H * + C 2 Δ N y a 1 y b 1 u b 1 2 - H u b 1 { N 00 C 2 [ ( N 00 - 1 ) u b 1 t + y a 1 Δ N ] + Δ N u b 1 N 00 } .
a 2 a 2 , H - Δ N H u a 1 2 N 00 [ ( N 00 2 + 2 ) C 2 y a 1 + 2 u a 1 ] + N 00 ( N 00 2 - 1 ) u a 1 3 u b 1 t 2 , a 3 = - Φ H H 2 2 - Δ N H 2 N 00 ( - N 00 H C 2 + 2 u a 1 u b 1 ) + ( N 00 2 - 1 ) u a 1 u b 1 t 2 ( - 2 C 2 H + N 00 u a 1 u b 1 ) , a 5 N 00 ( N 00 2 - 1 ) u a 1 u b 1 3 t 2 - H u b 1 2 [ N 00 ( N 00 - 1 ) C 2 t + Δ N N 00 ] .
a 2 a 2 , H - C 2 2 y a 1 2 H 2 [ Δ N ( 2 N 00 - 1 ) + ( N 00 - 1 ) C 2 N 00 t ] , a 3 - H 2 2 [ Φ H - C 2 [ Δ N + ( N 00 - 1 ) ( 2 C 2 t + C 2 2 t 2 ) ] } , a 5 - H u b 1 2 2 { ( N 00 - 1 ) C 2 t [ 3 N 00 ( 1 + C 2 t ) + 1 ] + 2 Δ N N 00 ( 1 + C 2 t ) } .
a 1 a 1 , H - Φ g y a 1 2 ( N 00 - 1 ) ( C 2 y a 1 + u a 1 ) 2 N 00 × [ ( N 00 + 1 ) ( - 2 Φ g y a 1 + 3 N 00 u a 1 ) + N 00 ( 3 N 00 + 1 ) C 2 y a 1 ] , a 2 a 2 , H * - ( N 00 - 1 ) Φ g y a 1 2 × { C 2 2 y a 1 2 y b 1 ( 3 N 00 + 1 ) + ( N 00 + 1 ) u a 1 ( 2 y a 1 u b 1 + y b 1 u a 1 ) + C 2 y a 1 [ y a 1 u b 1 ( 2 N 00 + 1 ) + y b 1 u a 1 ( 4 N 00 + 3 ) ] } , a 3 a 3 , H * - Φ g y a 1 ( N 00 - 1 ) ( C 2 y b 1 + u b 1 ) 2 N 00 × { ( N 00 + 1 ) [ y a 1 ( - 2 Φ g y b 1 + N 00 u b 1 ) + 2 N 00 y b 1 u a 1 ] + N 00 ( 3 N 00 + 1 ) C 2 y a 1 y b 1 } , a 4 = - Φ H H 2 2 N 00 , a 5 a 5 , H * - ( N 00 - 1 ) Φ g y b 1 2 × { C 2 2 y a 1 y b 1 2 ( 3 N 00 + 1 ) + ( N 00 + 1 ) u b 1 ( y a 1 u b 1 + 2 y b 1 u a 1 ) + C 2 y b 1 [ y a 1 u b 1 ( 4 N 00 + 3 ) + y b 1 u a 1 ( 2 N 00 + 1 ) ] } - ( N 00 - 1 ) H 2 × [ ( N 00 + 1 ) Φ g y b 1 N 00 2 ( Φ g y b 1 - 2 N 00 u b 1 ) + 2 C 2 ( - Φ g y b 1 2 + N 00 u b 1 2 t ) ] ,
a 1 a 1 , H - Φ g y a 1 2 ( N 00 - 1 ) C 2 2 N 00 × [ - 2 Φ g ( N 00 + 1 ) + N 00 ( 3 N 00 + 1 ) C 2 ] , a 2 a 2 , H * - ( N 00 - 1 ) Φ g C 2 y a 1 3 2 × [ ( 3 N 00 + 1 ) C 2 y b 1 + ( 2 N 00 + 1 ) u b 1 ] , a 3 a 3 , H * - Φ g y a 1 2 ( N 00 - 1 ) ( C 2 y b 1 + u b 1 ) 2 N 00 × [ ( N 00 + 1 ) ( - 2 Φ g y b 1 + N 00 u b 1 ) + N 00 ( 3 N 00 + 1 ) C 2 y b 1 ] , a 5 a 5 , H * - ( N 00 - 1 ) Φ g y a 1 y b 1 2 × [ ( 3 N 00 + 1 ) C 2 2 y b 1 2 + ( 4 N 00 + 3 ) C 2 y b 1 u b 1 + ( N 00 + 1 ) u b 1 2 ] - ( N 00 - 1 ) H 2 × [ ( N 00 + 1 ) Φ g y b 1 N 00 2 ( Φ g y b 1 - 2 N 00 u b 1 ) + 2 C 2 ( - Φ g y b 1 2 + N 00 u b 1 2 t ) ] .
a 2 a 2 , H + ( N 00 - 1 ) Φ g y a 1 H 2 N 00 [ ( 2 N 00 + 1 ) C 2 y a 1 + 2 ( N 00 + 1 ) u a 1 ] , a 3 - H 2 2 [ Φ H + Φ g ( 1 - 1 N 00 2 ) ] + ( N 00 2 - 1 ) u a 1 u b 1 t 2 ( - 2 H C 2 + N 00 u a 1 u b 1 , a 5 ( N 00 - 1 ) u b 1 2 t 2 N 00 { ( N 00 + 1 ) [ H ( 2 Φ g - N 00 C 2 ) + N 00 2 u a 1 u b 1 } - 2 H N 00 2 C 2 }
a 2 a 2 , H + ( N 00 - 1 ) C 2 H y a 1 2 2 [ ( 2 N 00 + 1 ) Φ g N 00 - N 00 2 C 2 2 t ] , a 3 - H 2 2 [ Φ H + Φ g ( 1 - 1 N 00 2 ) ] + ( N 00 - 1 ) H 2 C 2 2 t , a 5 ( N 00 - 1 ) H u b 1 2 t 2 N 00 [ 2 ( N 00 + 1 ) Φ g - N 00 ( 3 N 00 + 1 ) C 2 ] .
a 1 Φ g y a 1 2 ( N 00 2 - 1 ) u a 1 2 N 00 ( - 2 Φ g y a 1 + 3 N 00 u a 1 ) , a 2 - ( N 00 2 - 1 ) Φ g y a 1 u a 1 2 ( 2 y a 1 u b 1 + y b 1 u a 1 ) , a 3 - ( N 00 2 - 1 ) Φ g y a 1 u b 1 2 N 00 × [ y a 1 ( - 2 Φ g y b 1 + N 00 u b 1 ) + 2 N 00 y b 1 u a 1 ) , a 4 = 0 , a 5 - ( N 00 2 - 1 ) Φ g y b 1 u b 1 2 × ( y a 1 u b 1 + 2 y b 1 u a 1 ) - ( N 00 2 - 1 ) H Φ g y b 1 2 N 00 2 × ( Φ g y b 1 - 2 N 00 u b 1 ) .
a 1 - Φ g 3 y a 1 4 ( N 00 2 - 1 ) 2 N 00 2 ( 1 - Φ g t 2 N 00 ) , a 2 Φ g 2 y a 1 3 ( N 00 2 - 1 ) 2 N 00 ( - Φ g y b 1 N 00 + u b 1 ) , a 3 - ( N 00 2 - 1 ) Φ g y a 1 2 u b 1 2 N 00 ( - 2 Φ g y b 1 + N 00 u b 1 ) , a 5 - ( N 00 2 - 1 ) Φ g y b 1 2 N 00 2 [ N 00 2 y a 1 u a 1 2 - H ( Φ g y b 1 - 2 N 00 u b 1 ) ] .
a 2 ( N 00 2 - 1 ) Φ g y a 1 u a 1 H N 00 , a 3 - H 2 Φ g 2 ( 1 - 1 N 00 2 ) + N 00 ( N 00 2 - 1 ) u a 1 2 u b 1 2 t 2 , a 5 u b 1 2 t ( N 00 2 - 1 ) 2 N 00 ( 2 Φ g H + N 00 2 u a 1 u b 1 ) .
a 2 - Φ g 2 y a 1 2 H ( N 00 2 - 1 ) 2 N 00 2 ( 1 - Φ g t N 00 ) , a 3 - H 2 2 Φ g ( 1 - 1 N 00 2 ) ( 1 - 3 Φ g t 2 N 00 ) , a 5 u b 1 2 Φ g H t ( N 00 2 - 1 ) N 00 .
a 1 a 1 , H + N 00 ( N 00 2 - 1 ) u a 1 3 t 2 ( 4 C 2 y a 1 + u a 1 ) , a 2 a 2 , H * + N 00 ( N 00 2 - 1 ) u a 1 2 t 2 [ C 2 ( 3 y a 1 u b 1 + y b 1 u a 1 ) + u a 1 u b 1 ] , a 3 a 3 , H * + N 00 ( N 00 2 - 1 ) u a 1 u b 1 t 2 [ 2 C 2 ( y a 1 u b 1 + y b 1 u a 1 ) + u a 1 u b 1 ] , a 4 = - Φ H H 2 2 N 00 , a 5 a 5 , H * + N 00 ( N 00 2 - 1 ) u b 1 2 t 2 × { C 2 [ y a 1 ( C 2 y b 1 + u b 1 ) + 3 y b 1 u a 1 ] + u a 1 u b 1 } - ( N 00 - 1 ) C 2 H u b 1 t × [ ( N 00 - 1 ) C 2 y b 1 + N 00 u b 1 * ] .
a 1 a 1 , H , a 2 a 2 , H * + N 00 2 ( N 00 - 1 ) C 2 3 y a 1 3 u b 1 t 2 , a 2 a 3 , H * + N 00 2 ( N 00 - 1 ) y a 1 2 C 2 t 2 × { C 2 [ C 2 ( 2 y b 1 u b 1 + u b 1 2 t ) + 2 u b 1 2 ] } , a 5 a 5 , H * + N 00 ( N 00 2 - 1 ) u b 1 2 t 2 × [ C 2 y a 1 ( C 2 y b 1 + u b 1 ) ] - ( N 00 - 1 ) C 2 H u b 1 t [ ( N 00 - 1 ) C 2 y b 1 + N 00 u b 1 ] .
a 2 a 2 , H + ( N 00 2 - 1 ) u a 1 2 t 2 ( - 3 H C 2 + N 00 u a 1 u b 1 ] , a 3 - Φ H H 2 2 + ( N 00 2 - 1 ) u a 1 u b 1 t 2 ( - 2 H C 2 + N 00 u a 1 u b 1 ) , a 5 ( N 00 - 1 ) u b t 2 t 2 [ ( N 00 + 1 ) ( - H C 2 + N 00 u a 1 u b 1 ) - 2 N 00 C 2 H ] .
a 2 a 2 , H - N 00 ( N 00 - 1 ) C 2 3 H y a 1 2 t 2 , a 3 - Φ H H 2 2 + ( N 00 - 1 ) H 2 C 2 2 t 2 ( 2 + C 2 t ) , a 5 - ( N 00 - 1 ) ( 3 N 00 + 1 ) C 2 H u b 1 2 t 2 .
a 1 + y a 1 4 N 01 2 ( C 2 2 - C 1 2 ) + 2 C y 2 y a 1 3 u a 1 Δ N , a 2 + y a 1 3 y b 1 N 01 2 ( C 2 2 - C 1 2 ) + C 2 2 y a 1 2 Δ N 2 ( y a 1 u b 1 + 3 y b 1 u a 1 ) , a 3 + y a 1 2 y b 1 2 N 01 2 ( C 2 2 - C 1 2 ) + C 2 2 y a 1 y b 1 Δ N ( y a 1 u b 1 + y b 1 u b 1 ) , a 4 + = 0 , a 5 + y a 1 y b 1 3 N 01 2 ( C 2 2 - C 1 2 ) + C 2 2 y b 1 2 Δ N 2 ( 3 y a 1 u b 1 + y b 1 u a 1 ) ,
a 2 + - C 2 2 H y a 1 Δ N 2 N 00 ( y a 1 + 3 u a 1 t ) , a 3 + - C 2 2 H Δ N t 2 N 00 ( - H N 00 + 2 u a 1 u b 1 t ) , a 5 + C 2 2 u b 1 2 Δ N t 2 2 ( - H N 00 + u a 1 u b 1 t ) .
a 1 + y a 1 4 C 2 2 N 02 t + 4 C 2 2 y a 1 3 u a 1 Δ N , a 2 + y a 1 3 y b 1 C 2 2 N 02 t + C 2 2 y a 1 2 Δ N ( y a 1 u b 1 + 3 y b 1 u a 1 ) , a 3 + y a 1 2 y b 1 2 C 2 2 N 02 t + 2 C 2 2 y a 1 y b 1 Δ N ( y a 1 u b 1 + y b 1 u a 1 ) , a 4 + = 0 , a 5 + y a 1 y b 1 3 C 2 2 N 02 t + C 2 2 y b 1 2 Δ N ( 3 y a 1 u b 1 + y b 1 u a 1 ) ,
a 2 + - C 2 2 H y a 1 Δ N N 00 ( y a 1 + 3 u a 1 t ) , a 3 + - C 2 2 H Δ N t N 00 ( - H N 00 + 2 u a 1 u b 1 t ) , a 5 + C 2 2 u b 1 2 Δ N t 2 ( - H N 00 + u a 1 u b 1 t ) .
a 1 + - 2 N 10 Φ H y a 1 4 N 00 - 1 - 4 C 2 Φ g y a 1 3 u a 1 , a 2 + - 2 N 10 Φ H y a 1 3 y b 1 N 00 - 1 - C 2 Φ g y a 1 2 ( y a 1 u b 1 + 3 y b 1 u a 1 ) , a 3 + - 2 N 10 Φ H y a 1 2 y b 1 2 N 00 - 1 - 2 C 2 Φ g y b 1 ( y a 1 u b 1 + y b 1 u a 1 ) , a 4 + = 0 , a 5 + - 2 N 10 Φ H y a 1 y b 1 3 N 00 - 1 - C 2 Φ g y b 1 2 ( 3 y a 1 u b 1 + y b 1 u a 1 ) .
a 2 + C 2 H y a 1 Φ g N 00 ( y a 1 + 3 u a 1 t ) , a 3 + C 2 H Φ g t N 00 ( - H N 00 + 2 u a 1 u b 1 t ) , a 5 + - C 2 u b 1 2 Φ g t 2 ( - H N 00 + u a 1 u b 1 t ) .
a 1 * - y a 1 u a 1 3 Δ N , a 2 * - y a 1 u a 1 2 u b 1 Δ N , a 3 * - y a 1 u a 1 u b 1 2 Δ N , a 4 * = 0 , a 5 * - y a 1 u b 1 Δ N ,
a 1 * 5 Φ g y a 1 2 u a 1 2 ( Φ g y a 1 N 00 - u a 1 ) , a 2 * y a 1 Φ g [ Φ g y a 1 4 N 00 ( 3 y a 1 u b 1 + 7 y b 1 u a 1 ) - u a 1 2 ( 3 y a 1 u b 1 + 2 y b 1 u a 1 ) ] , a 3 * y a 1 Φ g [ Φ g y b 1 2 N 00 ( 3 y a 1 u b 1 + 2 y b 1 u a 1 ) - u b 1 2 ( y a 1 u b 1 + 4 y b 1 u a 1 ) ] , a 4 * - H 2 Φ g 2 N 00 2 , a 5 * y b 1 Φ g [ Φ g y b 1 4 N 00 ( 9 y a 1 u b 1 + y b 1 u a 1 ) - u b 1 2 ( 4 y a 1 u b 1 + y b 1 u a 1 ) ] .
a 1 * - 5 Φ g 3 y a 1 4 6 N 00 2 , a 2 * 3 Φ g 2 y a 1 3 u b 1 4 N 00 , a 3 * y a 1 2 u b 1 Φ g 2 N 00 ( 3 Φ g y b 1 - N 00 u b 1 ) , a 5 * y a 1 y b 1 u b 1 Φ g ( 9 Φ g y b 1 4 N 00 - 2 u b 1 ) .
a 2 * - 3 Φ g y a 1 H 4 N 00 ( Φ g y a 1 N 00 - 2 u a 1 ) , a 3 * Φ g H 4 N 00 ( - 2 H N 00 + 3 u a 1 u b 1 t ) , a 5 * Φ g H u b 1 2 t N 00 ,
a 2 * - 3 Φ g 2 y a 1 2 H 4 N 00 2 , a 3 * - H 2 Φ g 2 N 00 2 , a 5 * Φ g H u b 1 2 t N 00 .
a 1 * 4 N 20 y a 1 3 t ( y a 1 + 2 u a 1 t ) , a 2 * 4 N 20 y a 1 2 t [ y a 1 ( y b 1 + 1 2 u b 1 t ) ) + 3 2 y b 1 u a 1 t ] , a 3 * 4 N 20 y a 1 y b 1 t [ y a 1 ( y b 1 + u b 1 t ) + y b 1 u a 1 t ] , a 4 * = 0 , a 5 * 4 N 20 y b 1 2 t [ y b 1 ( y b 1 + 3 2 u b 1 t ) + 1 2 y b 1 u a 1 t ] .
a 2 * - 4 N 20 H y a 1 N 00 t 2 ( 1 2 y a 1 + u a 1 t ) , a 3 * - 4 N 20 H t 3 N 00 ( - H 3 N 00 + 1 2 u a 1 u b 1 t ) , a 5 * 4 N 20 u b 1 2 t 4 ( - H 4 N 00 + 1 5 u a 1 u b 1 t ) .

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