Abstract

Mathematical expressions derived for the primary aberrations of a thin, flat Fresnel lens (grooved on both surfaces, in general) as functions of the constructional parameters are convenient for the analytic design of optical systems containing one or more Fresnel lenses. In general. such systems have five primary monochromatic aberrations of the Seidel type plus a sixth that is identically zero for ordinary lenses. The new aberration (called line coma to distinguish it from ordinary circular coma) bears the same relation to sagittal and tangential coma that Petzval curvature bears to sagittal and tangential astigmatism. Moreover line coma is independent of stop position, whereas Petzval curvature varies with stop position unless the line coma is corrected. The primary chromatic aberrations and the aberration contributions of Fresnel aspherics are the same as for ordinary lenses. Specific applications of the theory are discussed.

© 1974 Optical Society of America

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References

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  1. C. Hofmann and R. Tiedeken, Jenaer Jahrb. 1964, 109 (1964).
  2. C. Hofmann, Jenaer Jahrb. 1966, 89 (1966).
  3. C. Hofmann, Jenaer Jahrb. 1967, 7 (1967).
  4. C. Hofmann, C. Z. Jena, and J. Klebe, Optik 25, 389 (1967).
  5. C. Hofmann and J. Neumann, Jenaer Jahrb. 1968, 7 (1968).
  6. C. Hofmann and J. Neumann, Jenaer Jahrb. 1969/70, 69 (1969/70).
  7. O. E. Miller, J. H. McLeod, and W. T. Sherwood, J. Opt. Soc. Am. 41, 807 (1951).
    [Crossref]
  8. E. Barkan and R. J. Kapash, J. Opt. Soc. Am. 61, 686A (1971).
  9. H. H. Hopkins, Wave Theory of Aberrations (Oxford U. P., London, 1950).
  10. See Ref. 9, Ch. IX.
  11. See Ref. 9, Ch. VII.
  12. See Ref. 9, p. 134, Eq. (191).
  13. E. Delano, Appl. Opt. 2, 1251 (1963). See Appendix.
    [Crossref]
  14. See Ref. 9, p. 153, Eqs. (216) and (217).
  15. See Ref. 9, p. 135, Eqs. (192).

1971 (1)

E. Barkan and R. J. Kapash, J. Opt. Soc. Am. 61, 686A (1971).

1968 (1)

C. Hofmann and J. Neumann, Jenaer Jahrb. 1968, 7 (1968).

1967 (2)

C. Hofmann, Jenaer Jahrb. 1967, 7 (1967).

C. Hofmann, C. Z. Jena, and J. Klebe, Optik 25, 389 (1967).

1966 (1)

C. Hofmann, Jenaer Jahrb. 1966, 89 (1966).

1964 (1)

C. Hofmann and R. Tiedeken, Jenaer Jahrb. 1964, 109 (1964).

1963 (1)

1951 (1)

Barkan, E.

E. Barkan and R. J. Kapash, J. Opt. Soc. Am. 61, 686A (1971).

Delano, E.

Hofmann, C.

C. Hofmann and J. Neumann, Jenaer Jahrb. 1969/70, 69 (1969/70).

C. Hofmann and J. Neumann, Jenaer Jahrb. 1968, 7 (1968).

C. Hofmann, Jenaer Jahrb. 1967, 7 (1967).

C. Hofmann, C. Z. Jena, and J. Klebe, Optik 25, 389 (1967).

C. Hofmann, Jenaer Jahrb. 1966, 89 (1966).

C. Hofmann and R. Tiedeken, Jenaer Jahrb. 1964, 109 (1964).

Hopkins, H. H.

H. H. Hopkins, Wave Theory of Aberrations (Oxford U. P., London, 1950).

Jena, C. Z.

C. Hofmann, C. Z. Jena, and J. Klebe, Optik 25, 389 (1967).

Kapash, R. J.

E. Barkan and R. J. Kapash, J. Opt. Soc. Am. 61, 686A (1971).

Klebe, J.

C. Hofmann, C. Z. Jena, and J. Klebe, Optik 25, 389 (1967).

McLeod, J. H.

Miller, O. E.

Neumann, J.

C. Hofmann and J. Neumann, Jenaer Jahrb. 1969/70, 69 (1969/70).

C. Hofmann and J. Neumann, Jenaer Jahrb. 1968, 7 (1968).

Sherwood, W. T.

Tiedeken, R.

C. Hofmann and R. Tiedeken, Jenaer Jahrb. 1964, 109 (1964).

Appl. Opt. (1)

J. Opt. Soc. Am. (2)

O. E. Miller, J. H. McLeod, and W. T. Sherwood, J. Opt. Soc. Am. 41, 807 (1951).
[Crossref]

E. Barkan and R. J. Kapash, J. Opt. Soc. Am. 61, 686A (1971).

Jenaer Jahrb. (5)

C. Hofmann and R. Tiedeken, Jenaer Jahrb. 1964, 109 (1964).

C. Hofmann, Jenaer Jahrb. 1966, 89 (1966).

C. Hofmann, Jenaer Jahrb. 1967, 7 (1967).

C. Hofmann and J. Neumann, Jenaer Jahrb. 1968, 7 (1968).

C. Hofmann and J. Neumann, Jenaer Jahrb. 1969/70, 69 (1969/70).

Optik (1)

C. Hofmann, C. Z. Jena, and J. Klebe, Optik 25, 389 (1967).

Other (6)

H. H. Hopkins, Wave Theory of Aberrations (Oxford U. P., London, 1950).

See Ref. 9, Ch. IX.

See Ref. 9, Ch. VII.

See Ref. 9, p. 134, Eq. (191).

See Ref. 9, p. 153, Eqs. (216) and (217).

See Ref. 9, p. 135, Eqs. (192).

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Figures (5)

Fig. 1
Fig. 1

Bending a thin, flat Fresnel lens with grooves on both surfaces. The lens may be regarded as having a shape determined by the shape of the ordinary lens from which it is derived, and shown directly above it. The power is redistributed among the two Fresnel surfaces, while keeping the power of the lens constant.

Fig. 2
Fig. 2

An arbitrary skew ray from the point P, incident upon a Fresnel lens lying in the y,z plane. The distance l and the angle Ū are negative as shown.

Fig. 3
Fig. 3

The relation between Comas, ComaT′T, and ComaL measured in the gaussian image plane. Point P′ is the ideal image point; points S and T are the sagittal and tangential foci, respectively. All other aberrations are assumed to be zero.

Fig. 4
Fig. 4

As the pupil is shifted, its size is changed in such a way that the marginal ray P0C from the axial object point P0 is unchanged. Because the object height h is also unchanged, the value of the Lagrange invariant Q for any ray also remains the same.

Fig. 5
Fig. 5

A keplerian telescope consisting of three Fresnel lenses. It can be corrected for all the primary aberrations except longitudinal color.

Equations (91)

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â = ( N / N ) â + Γ n ˆ , where Γ = [ 1 - ( N / N ) 2 ( â × n ˆ ) 2 ] 1 2 - ( N / N ) â · n ˆ ,
â · n ˆ = α μ + β ν + γ λ = 1 - 1 2 [ ( β - ν ) 2 + ( γ - λ ) 2 ] + O ( 4 ) , ( â × n ˆ ) 2 = ( β λ - γ ν ) 2 + ( γ μ - α λ ) 2 + ( α ν - β μ ) 2 = ( β - ν ) 2 + ( γ - λ ) 2 + O ( 4 ) ,
Γ = [ 1 - ( N / N ) ] { 1 + 1 2 ( N / N ) × [ ( β - ν ) 2 + ( γ - λ ) 2 ] } + O ( 4 ) .
â 1 = ( N 1 / N 1 ) â 1 + Γ 1 n ˆ 1 ,             â 2 = ( N 2 / N 2 ) â 2 + Γ 2 n ˆ 2 ,
â 2 = â 1 + n Γ 1 n ˆ 1 + Γ 2 n ˆ 2 .
β = β + ( n - 1 ) ( ν 1 - ν 2 ) + 1 2 ( 1 - n - 1 ) [ { ( β - ν 1 ) 2 + ( γ - λ 1 ) 2 } ν 1 - { [ β - ν 1 + n ( ν 1 - ν 2 ) ] 2 + [ γ - λ 1 + n ( λ 1 - λ 2 ) ] 2 } ν 2 ] + O ( 5 ) , γ = γ + ( n - 1 ) ( λ 1 - λ 2 ) + 1 2 ( 1 - n - 1 ) [ { ( β - ν 1 ) 2 + ( γ - λ 1 ) 2 } λ 1 - { [ β - ν 1 + n ( ν 1 - ν 2 ) ] 2 + [ γ - λ 1 + n ( λ 1 - λ 2 ) ] 2 } λ 2 ] + O ( 5 ) ,
u = - ( y - h ) / l ,             v = - z / l .
D = [ l 2 + ( y - h ) 2 + z 2 ] 1 2 = l [ 1 + 1 2 u 2 + 1 2 v 2 ] + O ( 4 ) .
β = - ( y - h ) / D ,             γ = - z / D ,
β = u [ 1 - 1 2 ( u 2 + v 2 ) ] + O ( 5 ) , γ = v [ 1 - 1 2 ( u 2 + v 2 ) ] + O ( 5 ) .
u = u + ( n - 1 ) ( ν 1 - ν 2 ) + 1 2 [ u ( u 2 + v 2 ) - u ( u 2 + v 2 ) ] + 1 2 ( 1 - n - 1 ) [ ν 1 { ( u - ν 1 ) 2 + ( v - λ 1 ) 2 } - ν 2 { [ u - ν 1 + n ( ν 1 - ν 2 ) ] 2 + [ v - λ 1 + n ( λ 1 - λ 2 ) ] 2 } ] + O ( 5 ) , v = v + ( n - 1 ) ( λ 1 - λ 2 ) + 1 2 [ v ( u 2 + v 2 ) - v ( u 2 + v 2 ) ] + 1 2 ( 1 - n - 1 ) [ λ 1 { ( u - ν 1 ) 2 + ( v - λ 1 ) 2 } - λ 2 { [ u - ν 1 + n ( ν 1 - ν 2 ) ] 2 + [ v - λ 1 + n ( λ 1 - λ 2 ) ] 2 } ] + O ( 5 ) .
ν = - y c + 1 2 c 3 y ( y 2 + z 2 ) + O ( 5 ) , λ = - z c + 1 2 c 3 z ( y 2 + z 2 ) + O ( 5 ) ,
( n - 1 ) ( ν 1 - ν 2 ) = - K y + 1 2 ( n - 1 ) × ( c 1 3 - c 2 3 ) y ( y 2 + z 2 ) + O ( 5 ) , ( n - 1 ) ( λ 1 - λ 2 ) = - K z + 1 2 ( n - 1 ) × ( c 1 3 - c 2 3 ) z ( y 2 + z 2 ) + O ( 5 ) ,
u = u - K y + δ u ,             v = v - K z + δ v ,
δ Y / l = 1 2 ( n - 1 ) ( c 1 3 - c 2 3 ) y ( y 2 + z 2 ) + 1 2 [ u ( u 2 + v 2 ) - u ( u 2 + v 2 ) ] + 1 2 ( 1 - n - 1 ) [ ν 1 { ( u - ν 1 ) 2 + ( ν - λ 1 ) 2 } - ν 2 { [ u - ν 1 + n ( ν 1 - ν 2 ) ] 2 + [ v - λ 1 + n ( λ 1 - λ 2 ) ] 2 } ] + O ( 5 ) , δ Z / l = 1 2 ( n - 1 ) ( c 1 3 - c 2 3 ) z ( y 2 + z 2 ) + 1 2 [ v ( u 2 + v 2 ) - v ( u 2 + v 2 ) ] + 1 2 ( 1 - n - 1 ) [ λ 1 { ( u - ν 1 ) 2 + ( ν - λ 1 ) 2 } - λ 2 { [ u - ν 1 + n ( ν 1 - ν 2 ) ] 2 + [ v - λ 1 + n ( λ 1 - λ 2 ) ] 2 } ] + O ( 5 ) ,
c 1 = 1 2 K [ S + ( n - 1 ) - 1 ] ,             c 2 = 1 2 K [ S - ( n - 1 ) - 1 ] .
v = 1 2 K z ( T + 1 ) ,             v = 1 2 K z ( T - 1 ) .
u = - ( y - h ) / l = - y / l + h / l ,
u = 1 2 K y ( T + 1 ) + ū ,             u = 1 2 K y ( T - 1 ) + ū ,
δ Y / l = - 1 8 K 3 y ( y 2 + z 2 ) [ ( ω - 1 ) S 2 + 2 ( 1 + ω ) S T + ( 3 + ω ) T 2 + ( 1 - ω ) - 1 ] - 1 2 K 2 y 2 ū [ ( 1 + ω ) S + ( 2 + ω ) T ] - 1 2 K 2 ( y 2 + z 2 ) ū T - 1 2 K y ū 2 ( 3 + ω ) + O ( 5 ) , δ Z / l = - 1 8 K 3 z ( y 2 + z 2 ) [ ( ω - 1 ) S 2 + 2 ( 1 + ω ) S T + ( 3 + ω ) T 2 + ( 1 - ω ) - 1 ] - 1 2 K 2 y z ū [ ( 1 + ω ) S + ( 2 + ω ) T ] - 1 2 K z ū 2 ( 1 + ω ) + O ( 5 ) ,
η = y / r m ,             ζ = z / r m ,             σ = ū / ū m ,
δ Y = a 1 η ( η 2 + ζ 2 ) + a 2 η 2 σ + a 3 ( η 2 + ζ 2 ) σ + a 4 η σ 2 + O ( 5 ) , δ Z = a 1 ζ ( η 2 + ζ 2 ) + a 2 η ζ σ + a 5 ζ σ 2 + O ( 5 ) ,
a 1 = - 1 8 l K 3 r m 3 [ ( ω - 1 ) S 2 + 2 ( 1 + ω ) S T + ( 3 + ω ) T 2 + ( 1 - ω ) - 1 ] , a 2 = - 1 2 l K 2 r m 2 ū m [ ( 1 + ω ) S + ( 2 + ω ) T ] , a 3 = - 1 2 l K 2 r m 2 ū m T , a 4 = - 1 2 l K r m ū m 2 ( 3 + ω ) , a 5 = - 1 2 l K r m ū m 2 ( 1 + ω ) .
δ Y = b 1 ρ 3 cos θ + b 2 c ρ 2 σ ( 2 + cos 2 θ ) + b 2 l ρ 2 σ + ( 3 b 3 + b 4 ) ρ σ 2 cos θ + O ( 5 ) , δ Z = b 1 ρ 3 sin θ + b 2 c ρ 2 σ sin 2 θ + ( b 3 + b 4 ) ρ σ 2 sin θ + O ( 5 ) ,
b 1 = a 1 ,             b 2 c = 1 2 a 2 ,             b 2 l = a 3 - 1 2 a 2 , b 3 = 1 2 ( a 4 - a 5 ) ,             b 4 = 1 2 ( 3 a 5 - a 4 ) .
Coma T = ( 3 b 2 c + b 2 l ) ρ 2 σ ;
Coma S = ( b 2 c + b 2 l ) ρ 2 σ ,
Coma L = b 2 l ρ 2 σ ,
Coma T - Coma L = 3 ( Coma S - Coma L ) .
X T - X P = 3 ( X S - X P ) ,
ψ = sin - 1 [ b 2 c / ( 2 b 2 c + b 2 l ) ] .
( y * - h ) / ( y - h ) = z * / z = d * / d .
η = η * + p σ ,             ζ = ζ * , where p = r ¯ m / r m ,             r ¯ m = h m ( d * - d ) / d * .
δ Y = a 1 * η * ( η * 2 + ζ * 2 ) + a 2 * η * 2 σ + a 3 * ( η * 2 + ζ * 2 ) σ + a 4 * η * σ 2 + a 6 * σ 3 + O ( 5 ) , δ Z = a 1 * ζ * ( η * 2 + ζ * 2 ) + a 2 * η * ζ * σ + a 5 * ζ * σ 2 + O ( 5 ) ,
a 1 * = a 1 , a 2 * = a 2 + 2 p a 1 , a 3 * = a 3 + p a 1 , a 4 * = a 4 + 2 p ( a 2 + a 3 ) + 3 p 2 a 1 , a 5 * = a 5 + p a 2 + p 2 a 1 , a 6 * = p a 4 + p 2 ( a 2 + a 3 ) + p 3 a 1 .
δ Y = b 1 * ρ * 3 cos θ * + b 2 c * ρ * 2 σ ( 2 + cos 2 θ * ) + b 2 l * ρ * 2 σ + ( 3 b 3 * + b 4 * ) ρ * σ 2 cos θ * + b 5 * σ 3 + O ( 5 ) , δ Z = b 1 * ρ * 3 sin θ * + b 2 c * ρ * 2 σ sin 2 θ * + ( b 3 * + b 4 * ) ρ * σ 2 sin θ * + O ( 5 ) ,
b 1 * = b 1 , b 2 c * = b 2 c + p b 1 , b 2 l * = b 2 l , b 3 * = b 3 + p ( 2 b 2 c + b 2 l ) + p 2 b 1 , b 4 * = b 4 - p b 2 l             ( N . B . a minus sign here ! ) b 5 * = a 6 * = b 5 + p ( 3 b 3 + b 4 ) + p 2 ( 3 b 2 c + b 2 l ) + p 3 b 1 ,
δ Y tot = j m j k δ Y j + O ( 5 ) , δ Z tot = j m j k δ Z j + O ( 5 ) ,
E j = - 2 N j v j δ Y j ,             F j = - 2 N j v j δ Z j .
E tot = - 2 N k v k δ Y tot = j E j + O ( 5 ) , F tot = - 2 N k v k δ Z tot = j F j + O ( 5 ) ,
v = - z / l = ( - r m / l ) ( z / r m ) = v m ζ = v m ρ sin θ ,
E = [ S 1 * ρ * 3 cos θ * + S 2 c * ρ * 2 σ ( 2 + cos 2 θ * ) + S 2 l * ρ * 2 σ + ( 3 S 3 * + S 4 * ) ρ * σ 2 cos θ * + S 5 * σ 3 ] ρ * sin θ * + O ( 5 ) , F = [ S 1 * ρ * 3 sin θ * + S 2 c * ρ * 2 σ sin 2 θ * + ( S 3 * + S 4 * ) ρ * σ 2 sin θ * ] ρ * sin θ * + O ( 5 ) ,
( S i ) tot = j ( S i * ) j .
S 1 = - 1 4 r m 4 K 3 [ ( ω - 1 ) S 2 + 2 ( 1 + ω ) S T + ( 3 + ω ) T 2 + ( 1 - ω ) - 1 ] , S 2 c = - 1 2 Q m r m 2 K 2 [ ( 1 + ω ) S + ( 2 + ω ) T ] , S 2 l = + 1 2 Q m r m 2 K 2 [ ( 1 + ω ) S + ω T ]             ( N . B . the plus sign ) , S 3 = - Q m 2 K , S 4 = - Q m 2 K ω , S 5 = 0.
v m * = r m * / d * = r m / d = v m .
S 1 * = S 1 , S 2 c * = S 2 c + p S 1 , S 2 l * = S 2 l , S 3 * = S 3 + p ( 2 S 2 c + S 2 l ) + p 2 S 1 , S 4 * = S 4 - p S 2 l , S 5 * = S 5 + p ( 3 S 3 + S 4 ) + p 2 ( 3 S 2 c + S 2 l ) + p 3 S 1 ,
ū * = 1 2 K y ¯ ( T ¯ + 1 ) ,             ū * = 1 2 K y ¯ ( T ¯ - 1 ) ,
Q m = 1 2 K r m r ¯ m ( T ¯ - T ) .
S 1 * = - 1 4 r m 4 K 3 [ ( ω - 1 ) S 2 + 2 ( 1 + ω ) S T + ( 3 + ω ) T 2 + ( 1 - ω ) - 1 ] , S 2 c * = - 1 4 r m 3 r ¯ m K 3 [ ( ω - 1 ) S 2 + ( 1 + ω ) S ( T + T ¯ ) + T 2 + ( 2 + ω ) T T ¯ + ( 1 - ω ) - 1 ] , S 2 l * = + 1 2 Q m r m 2 K 2 [ ( 1 + ω ) S + ω T ] , S 3 * = - 1 4 r m 2 r ¯ m 2 K 3 [ ( ω - 1 ) S 2 + ( 1 + ω ) S ( T + T ¯ ) + T ¯ 2 + ( 2 + ω ) T T ¯ + ( 1 - ω ) - 1 ] , S 4 * = - 1 2 Q m r m r ¯ m K 2 [ ( 1 + ω ) S + ω T ¯ ] , S 5 * = - 1 4 r m r ¯ m 3 K 3 [ ( ω - 1 ) S 2 + 2 ( 1 + ω ) S T ¯ + ( 3 + ω ) T ¯ 2 + ( 1 - ω ) - 1 ] .
x = 1 2 c ( y 2 + z 2 ) + e ( y 2 + z 2 ) 2 + O ( 6 ) ,
ν = - y c + 1 2 ( c 3 - 8 e ) y ( y 2 + z 2 ) + O ( 5 ) , λ = - z c + 1 2 ( c 3 - 8 e ) z ( y 2 + z 2 ) + O ( 5 ) .
Δ S 1 = - 2 v m Δ b 1 = - 8 r m 4 ( n - 1 ) ( e 1 - e 2 ) ,
Δ S 1 * = Δ S 1 , Δ S 2 c * = p Δ S 1 , Δ S 2 l * = 0 , Δ S 3 * = p 2 Δ S 1 , Δ S 4 * = 0 , Δ S 5 * = p 3 Δ S 1 ,
e 1 = 1 8 c 1 3 = 1 64 K 3 [ S 3 + 3 S 2 ( n - 1 ) - 1 + 3 S ( n - 1 ) - 2 + ( n - 1 ) - 3 ] , e 2 = 1 8 c 2 3 = 1 64 K 3 [ S 3 - 3 S 2 ( n - 1 ) - 1 + 3 S ( n - 1 ) - 2 - ( n - 1 ) - 3 ] ,
Δ S 1 = - 1 4 r m 4 K 3 [ 3 S 2 + ω 2 ( 1 - ω ) - 2 ] .
S 1 + Δ S 1 = - 1 4 r m 4 K 3 [ ( 2 + ω ) S 2 + 2 ( 1 + ω ) S T + ( 3 + ω ) T 2 + 1 + ω ( 1 - ω ) - 2 ]
u = u - K y ,             v = v - K z .
δ u = - y δ K = - y K / ν ,             δ v = - z δ K = - z K / ν ,
δ Y = a 7 η ,             δ Z = a 7 ζ ,
a 7 = - l r m K / ν .
δ Y = a 7 * η * + a 8 * σ ,             δ Z = a 7 * ζ * ,
δ Y = b 7 * ρ * cos θ * + b 8 * σ ,             δ Z = b 7 * ρ * sin θ * ,
E = ( C 1 * ρ * cos θ * + C 2 * σ ) ρ * sin θ * , F = C 1 * ρ * 2 sin 2 θ * ,
C 1 = - 2 r m 2 K / ν ,             C 2 = 0.
C 1 * = C 1 = - 2 r m 2 K / ν ,             C 2 * = C 2 + p C 1 = - 2 r m r ¯ m K / ν .
T = ( v m + v m ) / ( v m - v m ) 2 v m ( K r m ) - 1
K a = 1 3 ,             K b = 1 ,             K c = 2 ,             A = 2 ,             B = 1 ,             C = 1 3 ,
u a = 0 ,             u a = - 1 ,             u b = - 2 ,             u c = 0 , y a = 3 ,             y b = 1 ,             y c = - 1 , ū a = 1 3 ,             ū a = 1 3 ,             ū b = - 1 3 ,             ū c = - 1 , y ¯ a = 0 ,             y ¯ b = 2 3 ,             y ¯ c = 1 3 ,
T a = - 1 , T b = - 3 , T c = 1 , T ¯ a = , T ¯ b = 0 , T ¯ c = - 2.
( S 1 ) tot = - 1 4 y a 4 K a 3 [ γ a + 2 β a T a + ( 3 + ω ) T a 2 ] - 1 4 y b 4 K b 3 [ γ b + 2 β b T b + ( 3 + ω ) T b 2 ] - 1 4 y c 4 K c 3 [ γ c + 2 β c T c + ( 3 + ω ) T c 2 ] = 0 , ( S 2 c ) tot = - 1 2 Q y a 2 K a 2 [ β a + ( 2 + ω ) T a ] - 1 4 y b 3 y ¯ b K b 3 [ γ b + β b ( T b + T ¯ b ) + T b 2 + ( 2 + ω ) T b T ¯ b ] - 1 4 y c 3 y ¯ c K c 3 [ γ c + β c ( T c + T ¯ c ) + T c 2 + ( 2 + ω ) T c T ¯ c ] = 0 , ( S 2 l ) tot = 1 2 Q y a 2 K a 2 ( β a + ω T a ) + 1 2 Q y b 2 K b 2 ( β b + ω T b ) + 1 2 Q y c 2 K c 2 ( β c + ω T c ) = 0 , ( S 3 ) tot = - Q 2 K a - 1 4 y b 2 y ¯ b 2 K b 3 [ γ b + β b ( T b + T ¯ b ) + T ¯ b 2 + ( 2 + ω ) T b T ¯ b ] - 1 4 y c 2 y ¯ c 2 K c 3 [ γ c + β c ( T c + T ¯ c ) + T ¯ c 2 + ( 2 + ω ) T c T ¯ c ] = 0 , ( S 4 ) tot = - Q 2 K a ω - 1 2 Q y b y ¯ b K b 2 ( β b + ω T ¯ b ) - 1 2 Q y c y ¯ c K c 2 ( β c + ω T ¯ c ) = 0 , ( S 5 ) tot = - 1 4 y b y ¯ b 3 K b 3 [ γ b + 2 β b T ¯ b + ( 3 + ω ) T ¯ b 2 ] - 1 4 y c y ¯ c 3 K c 3 [ γ c + 2 β c T ¯ c + ( 3 + ω ) T ¯ c 2 ] = 0 ,
β a = ( 1 + ω ) S a , γ a = ( ω - 1 ) S a 2 + ( 1 - ω ) - 1 - 4 y a - 4 K a - 3 ( Δ S 1 ) a ,
β a = - 29 / 3 , β b = 1 , β c = 13 / 6 , γ a = - 169 / 3 , γ b = 19 / 3 , γ c = 1 / 3.
S a = - 5.8 , S b = 0.6 , S c = 1.3 , ( Δ S 1 ) a = 36.09 , ( Δ S 1 ) b = - 0.8633 , ( Δ S 1 ) c = 4.2067.
â = A â + Γ n ˆ ,
â × n ˆ = A â × n ˆ ,             since n ˆ × n ˆ = 0.
1 = ( A â + Γ n ˆ ) · ( A â + Γ n ˆ ) = A 2 + 2 A Γ â · n ˆ + Γ 2 .
Γ = - A â · n ˆ ± [ ( A â · n ˆ ) 2 - ( A 2 - 1 ) ] 1 2 = ± { 1 - A 2 [ 1 - ( â · n ˆ ) 2 ] } 1 2 - A â · n ˆ .
x = 1 2 c ( y 2 + z 2 ) + e ( y 2 + z 2 ) 2 ,
ϕ ( x , y , z ) = x - 1 2 c ( y 2 + z 2 ) - e ( y 2 + z 2 ) 2 = 0 ,
μ = κ ϕ / x ,             ν = - κ ϕ / y ,             λ = κ ϕ / z , where κ = [ ( ϕ / x ) 2 + ( ϕ / y ) 2 + ( ϕ / z ) 2 ] - 1 2 ,
ν = - y c + 1 2 ( c 3 - 8 e ) y ( y 2 + z 2 ) + O ( 5 ) , λ = - z c + 1 2 ( c 3 - 8 e ) z ( y 2 + z 2 ) + O ( 5 ) .
( n - 1 ) ( c 1 3 - c 2 3 ) = 1 4 K 3 [ 3 S 2 + ( n - 1 ) - 2 ] .
u ( u 2 + v 2 ) - u ( u 2 + v 2 ) = - K y [ 1 4 K 2 y 2 ( 3 T 2 + 1 ) + 3 K y T ū + 3 ū 2 ] - 1 2 K 2 z 2 [ 1 2 K y ( 3 T 2 + 1 ) + 2 T ū ] + O ( 5 ) .
A ν 1 [ ( u - ν 1 ) 1 + ( v - λ 1 ) 2 ] = - 1 2 K y [ S + ( n - 1 ) - 1 ] [ { 1 2 K y [ S + T + n ( n - 1 ) - 1 ] + ū } 2 + { 1 2 K z [ S + T + n ( n - 1 ) - 1 ] } 2 ] + O ( 5 ) ,
B ν 2 { [ u - ν 1 + n ( ν 1 - ν 2 ) ] 2 + [ v - λ 1 + n ( λ 1 - λ 2 ) ] 2 } = - 1 2 K y [ S - ( n - 1 ) - 1 ] [ { 1 2 K y [ S + T - n ( n - 1 ) - 1 ] + ū } 2 + { 1 2 K z [ S + T - n ( n - 1 ) - 1 ] } 2 ] + O ( 5 ) .
A - B = - K y S ( 1 - n - 1 ) - 1 [ 1 2 K 2 ( y 2 + z 2 ) ( S + T ) + K y ū ] - K y ( n - 1 ) - 1 { 1 4 K 2 ( y 2 + z 2 ) [ ( S + T ) 2 + ( 1 - n - 1 ) - 2 ] + K y ū ( S + T ) + ū 2 } + O ( 5 ) .
v ( u 2 + v 2 ) - v ( u 2 + v 2 ) = - K z [ 1 4 K 2 y 2 ( 3 T 2 + 1 ) + 2 K y T ū + ū 2 ] - 1 4 K 3 z 3 ( 3 T 2 + 1 ) + O ( 5 ) .
Q m = r m ū m * - r ¯ m v m .
Q m = 1 2 r m K r ¯ m ( T ¯ + 1 ) - 1 2 r ¯ m K r m ( T + 1 ) = 1 2 K r m r ¯ m ( T ¯ - T ) ,
S 1 = - 1 4 r m 4 K 3 [ ( 1 + 2 ω ) S 2 + 4 ( 1 + ω ) S T + ( 3 + 2 ω ) T 2 + ( 1 - ω ) - 2 ] , S 2 c = - 1 2 Q m r m 2 K 2 [ ( 1 + ω ) S + ( 2 + ω ) T ] , S 2 l = 0 , S 3 = - Q m 2 K , S 4 = - Q m 2 K ω , S 5 = 0 , C 1 = - 2 r m 2 K / ν , C 2 = 0.
S 1 * = - 1 4 r m 4 K 3 [ ( 1 + 2 ω ) S 2 + 4 ( 1 + ω ) S T + ( 3 + 2 ω ) T 2 + ( 1 - ω ) - 2 ] S 2 c * = - 1 4 r m 3 r ¯ m K 3 [ ( 1 + 2 ω ) S 2 + ( 1 + ω ) S ( 3 T + T ¯ ) + ( 1 + ω ) T 2 + ( 2 + ω ) T T ¯ + ( 1 - ω ) - 2 ] , S 2 l * = 0 , S 3 * = - 1 4 r m 2 r ¯ m 2 K 3 [ ( 1 + 2 ω ) S 2 + 2 ( 1 + ω ) S ( T + T ¯ ) + 2 ( 1 + ω ) T T ¯ + T ¯ 2 + ( 1 - ω ) - 2 ] , S 4 * = - Q m 2 K ω , S 5 * = - 1 4 r m r ¯ m 3 K 3 [ ( 1 + 2 ω ) S 2 + ( 1 + ω ) S ( T + 3 T ¯ ) + ω T T ¯ + ( 3 + ω ) T ¯ 2 + ( 1 - ω ) - 2 ] , C 1 * = - 2 r m 2 K / ν , C 2 * = - 2 r m r ¯ m K / ν .