Abstract

Short and accurate analytic formulas for the Seidel coefficients of gradient-index lenses with arbitrary radial refractive-index distributions have been obtained. Starting from analytic ray-tracing formulas, we have developed a technique for decomposing the two components of the transverse aberration of an arbitrary skew ray in surface and inhomogeneous transfer contributions and a technique for shortening the large expressions for the third-order terms resulting from the transfer contributions. Unlike previously known derivation methods, our method delivers simple algebraic expressions for all Seidel coefficients.

© 1994 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. P. J. Sands, “Aberrations of lenses with axial index distributions,” J. Opt. Soc. Am. 61, 1086–1091 (1971).
    [CrossRef]
  3. 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]
  4. S. Minami, “Third-order aberration theory in GRIN lenses,” in Digest of Topical Meeting on Gradient-Index Optical Imaging Systems (Optical Society of America, Washington, D.C., 1984), paper FE-A1-1-4.
  5. D. Wang, D. T. Moore, “Third-order aberration theory for weak gradient-index lenses,” Appl. Opt. 29, 4016–4025 (1990).
    [CrossRef] [PubMed]
  6. H. A. Buchdahl, Optical Aberration Coefficients (Dover, New York, 1968), p. 305.
  7. F. Bociort, J. Kross, “New ray-tracing method for radial gradient-index lenses,” in Lens and Optical Systems Design, H. Zuegge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1780, 216–225 (1993).
  8. W. T. Welford, Aberrations of Optical Systems (Hilger, Bristol, UK, 1986), p. 139.
  9. A. Cox, A System of Optical Design (Focal, London, 1964), p. 144.
  10. E. W. Marchand, Gradient Index Optics (Academic, New York, 1978), p. 72.
  11. E. W. Marchand, “Rapid ray tracing in radial gradients,” Appl. Opt. 27, 465–467 (1988).
    [CrossRef] [PubMed]
  12. H. H. Hopkins, “The nature of the paraxial approximation,” J. Mod. Opt. 38, 427–472 (1991).
    [CrossRef]
  13. S. Wolfram, mathematica, A System for Doing Mathematics by Computer (Addison-Wesley, New York, 1991).

1991 (1)

H. H. Hopkins, “The nature of the paraxial approximation,” J. Mod. Opt. 38, 427–472 (1991).
[CrossRef]

1990 (1)

1988 (1)

1971 (2)

1970 (1)

Bociort, F.

F. Bociort, J. Kross, “New ray-tracing method for radial gradient-index lenses,” in Lens and Optical Systems Design, H. Zuegge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1780, 216–225 (1993).

Buchdahl, H. A.

H. A. Buchdahl, Optical Aberration Coefficients (Dover, New York, 1968), p. 305.

Cox, A.

A. Cox, A System of Optical Design (Focal, London, 1964), p. 144.

Hopkins, H. H.

H. H. Hopkins, “The nature of the paraxial approximation,” J. Mod. Opt. 38, 427–472 (1991).
[CrossRef]

Kross, J.

F. Bociort, J. Kross, “New ray-tracing method for radial gradient-index lenses,” in Lens and Optical Systems Design, H. Zuegge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1780, 216–225 (1993).

Marchand, E. W.

Minami, S.

S. Minami, “Third-order aberration theory in GRIN lenses,” in Digest of Topical Meeting on Gradient-Index Optical Imaging Systems (Optical Society of America, Washington, D.C., 1984), paper FE-A1-1-4.

Moore, D. T.

Sands, P. J.

Wang, D.

Welford, W. T.

W. T. Welford, Aberrations of Optical Systems (Hilger, Bristol, UK, 1986), p. 139.

Wolfram, S.

S. Wolfram, mathematica, A System for Doing Mathematics by Computer (Addison-Wesley, New York, 1991).

Appl. Opt. (2)

J. Mod. Opt. (1)

H. H. Hopkins, “The nature of the paraxial approximation,” J. Mod. Opt. 38, 427–472 (1991).
[CrossRef]

J. Opt. Soc. Am. (3)

Other (7)

S. Minami, “Third-order aberration theory in GRIN lenses,” in Digest of Topical Meeting on Gradient-Index Optical Imaging Systems (Optical Society of America, Washington, D.C., 1984), paper FE-A1-1-4.

H. A. Buchdahl, Optical Aberration Coefficients (Dover, New York, 1968), p. 305.

F. Bociort, J. Kross, “New ray-tracing method for radial gradient-index lenses,” in Lens and Optical Systems Design, H. Zuegge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1780, 216–225 (1993).

W. T. Welford, Aberrations of Optical Systems (Hilger, Bristol, UK, 1986), p. 139.

A. Cox, A System of Optical Design (Focal, London, 1964), p. 144.

E. W. Marchand, Gradient Index Optics (Academic, New York, 1978), p. 72.

S. Wolfram, mathematica, A System for Doing Mathematics by Computer (Addison-Wesley, New York, 1991).

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

Plane A defined by the finite ray EF at the first surface of the system.

Fig. 2
Fig. 2

Definition of the barred marginal ray data. The medium after the first surface of the system was assumed to be homogeneous.

Tables (1)

Tables Icon

Table 1 Confirmation by Ray Tracing of the Correctness of the Seidel Aberration Formulas

Equations (83)

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

n 2 ( r 2 ) = n 0 2 ( 1 - k r 2 + N 4 k 2 r 4 ) + O ( 6 ) ,
Ξ x , 3 = [ Γ 1 ( σ x 2 + σ y 2 ) + 2 Γ 2 ( σ x τ x + σ y τ y ) + ( Γ 3 + P ) ( τ x 2 + τ y 2 ) ] σ x + [ Γ 2 ( σ x 2 + σ y 2 ) + 2 Γ 3 ( σ x τ x + σ y τ y ) + Γ 4 ( τ x 2 + τ y 2 ) ] τ x , Ξ y , 3 = [ Γ 1 ( σ x 2 + σ y 2 ) + 2 Γ 2 ( σ x τ x + σ y τ y ) + ( Γ 3 + P ) ( τ x 2 + τ y 2 ) ] σ y + [ Γ 2 ( σ x 2 + σ y 2 ) + 2 Γ 3 ( σ x τ x + σ y τ y ) + Γ 4 ( τ x 2 + τ y 2 ) ] τ y .
Γ p = - 1 2 n Q u Q [ surfaces ( S p + S p * ) + GRIN media T p ] ,             p = 1 , 2 , 3 , 4 , P = - 1 2 n Q u Q ( surfaces P S + GRIN media P T ) .
n 0 i = n 0 h ρ - n 0 u , n 0 j = n 0 m ρ - n 0 w .
H = m n 0 u - h n 0 w .
S 1 = ( n 0 i ) 2 h Δ ( u / n 0 ) , S 2 = n 0 i n 0 j h Δ ( u / n 0 ) , S 3 = ( n 0 j ) 2 h Δ ( u / n 0 ) , P S = - ρ H 2 Δ ( 1 / n 0 ) , S 4 = ( n 0 j ) 2 m Δ ( u / n 0 ) + n 0 j H Δ ( w / n 0 ) .
S 1 * = - 2 h 4 ρ Δ ( n 0 k ) , S 2 * = - 2 h 3 m ρ Δ ( n 0 k ) , S 3 * = - 2 h 2 m 2 ρ Δ ( n 0 k ) , S 4 * = - 2 h m 3 ρ Δ ( n 0 k ) .
δ z = ½ ρ ( x 2 + y 2 ) + O ( 4 ) ,
z = d + δ z - δ z .
t = n 0 ζ g z .
x = x ϕ c ( t ) + ξ n 0 g ϕ s ( t ) , y = y ϕ c ( t ) + η n 0 g ϕ s ( t ) , ξ = n 0 g x ϕ ˙ c ( t ) + ξ ϕ ˙ s ( t ) , η = n 0 g y ϕ ˙ c ( t ) + η ϕ ˙ s ( t ) ,
μ = ½ ( 1 - ζ 2 / n 0 2 ) .
ϕ c = ϕ c 0 + μ N 4 ϕ c 11 + , ϕ s = ϕ s 0 + μ N 4 ϕ s 11 + .
ϕ c 0 = cos t , ϕ ˙ c 0 = - sin t , ϕ s 0 = sin t , ϕ ˙ s 0 = cos t ,
ϕ c 11 = - 1 8 a cos 3 t - 1 8 b sin 3 t - 1 2 b t cos t + 1 8 a cos t + 1 2 ( a + 2 ) t sin t + 7 8 b sin t , ϕ ˙ c 11 = - 3 8 b cos 3 t + 3 8 a sin 3 t + 1 2 ( a + 2 ) t cos t + 3 8 b cos t + 1 2 b t sin t + 1 8 ( 3 a + 8 ) sin t , ϕ s 11 = 1 8 b cos 3 t - 1 8 a sin 3 t + 1 2 ( a - 2 ) t cos t - 1 8 b cos t + 1 2 b t sin t - 1 8 ( a - 8 ) sin t , ϕ ˙ s 11 = - 3 8 a cos 3 t - 3 8 b sin 3 t + 1 2 b t cos t + 3 8 a cos t - 1 2 ( a - 2 ) t sin t + 5 8 b sin t ,
a = 1 μ g 2 ( x 2 + y 2 ) - 1 ,             b = g n 0 μ ( x ξ + y η ) .
x ˜ = m τ x + h σ x , ξ ˜ = - n 0 w τ x - n 0 u σ x , y ˜ = m τ y + h σ y , n ˜ = - n 0 w τ y - n 0 u σ y .
ξ 2 + η 2 + ζ 2 = n 2 ( r 2 ) ,
ζ 2 = n 0 2 [ 1 - ( g r ) 2 + N 4 ( g r ) 4 ] - ξ 2 - η 2 ,
t ˜ = g d .
μ ˜ = ½ [ 1 n 0 2 ( ξ ˜ 2 + η ˜ 2 ) + g 2 ( x ˜ 2 + y ˜ 2 ) ] .
x ˜ = x ˜ cos g d + ξ ˜ / ( n 0 g ) sin g d , ξ ˜ = ξ ˜ cos g d - n 0 g x ˜ sin g d .
x ˜ = h ,             x ˜ = h ,             ξ ˜ = - n 0 u ,             ξ ˜ = - n 0 u ,
u = u cos g d + h g sin g d , h = - u / g sin g d + h cos g d .
w = w cos g d + m g sin g d , m = - w / g sin g d + m cos g d .
sin i x = i sinh x , cos i x = cosh x .
u = u cosh g ^ d - h g ^ sinh g ^ d , h = - u / g ^ sinh g ^ d + h cosh g ^ d , w = w cosh g ^ d - m g ^ sinh g ^ d , m = - w / g ^ sinh g ^ d + m cosh g ^ d .
u = u , h = h - u d , w = w , m = m - w d .
μ ˜ = ½ [ ( k h 2 + u 2 ) ( σ x 2 + σ y 2 ) + 2 ( k h m + u w ) × ( σ x τ x + σ y τ y ) + ( k m 2 + w 2 ) ( τ x 2 + τ y 2 ) ]
e 1 = k h 2 + u 2 , e 2 = k h m + u w , e 3 = k m 2 + w 2
e 1 e 3 = e 2 2 + k n 0 2 H 2 .
Ξ x = x Q - x ˜ Q ,             Ξ y = y Q - y ˜ Q .
Λ ˜ x = n 0 u x ˜ + h ξ ˜ .
Λ x P = n P u P x P ,             Λ x Q = n Q u Q x Q .
Λ ˜ x Q = Λ ˜ x P = Λ x P .
n Q u Q ( x Q - x ˜ Q ) = n Q u Q x Q - n Q u Q x ˜ Q = n Q u Q x Q - n P u P x P ,
Ξ x = Λ x Q - Λ x P n Q u Q .
Λ x = n 0 F ( u ) x + F ( h ) f ( ξ ) ,
F ˜ ( u ) = u ,             F ˜ ( h ) = h ,             f ˜ ( ξ ) = ξ ˜ ,
Λ x Q - Λ x P = Δ Λ x .
F ( u ) = u ,             F ( h ) = h
f ( ξ ) = n ζ ξ ,
Δ x = ξ ζ z ,             Δ h = - u z ,
Δ Λ x = n u Δ x + n ξ ζ Δ h = 0.
f ( ξ ) = n ( r 2 ) ζ ξ ,
u ¯ = u ,             h ¯ = h - u δ z ,             u ¯ = u ,             h ¯ = h - u δ z .
F ( u ) = u ¯ ,             F ( h ) = h ¯ ,
Λ x = n 0 u ¯ x + h ¯ n ( r 2 ) ξ ζ .
Λ x Q - Λ x P = surfaces Δ Λ x + GRIN media Δ Λ x .
x = x 1 + x 3 + O ( 5 ) , y = y 1 + y 3 + O ( 5 ) , ξ = ξ 1 + ξ 3 + O ( 5 ) , η = η 1 + η 3 + O ( 5 ) .
ϕ c = ϕ c , 0 + ϕ c , 2 + O ( 4 ) ,             ϕ s = ϕ s , 0 + ϕ s , 2 + O ( 4 ) ,
x p + 1 = x ϕ c , p + ξ n 0 g ϕ s , p , ξ p + 1 = n 0 g x ϕ ˙ c , p + ξ ϕ ˙ s , p ,             p = 0 , 2 , .
t = n 0 ζ g z = g z / ( 1 - 2 μ ) 1 / 2 = g z ( 1 + μ ) + O ( 4 ) .
cos t = cos g z - μ g z sin g z + O ( 4 ) , sin t = sin g z + μ g z cos g z + O ( 4 ) .
ϕ c , 0 = cos g z ,             ϕ s , 0 = sin g z , ϕ ˙ c , 0 = - sin g z ,             ϕ ˙ s , 0 = cos g z , ϕ c , 2 = - μ g z sin g z + μ N 4 ϕ c 11 , ϕ ˙ c , 2 = - μ g z cos g z + μ N 4 ϕ ˙ c 11 , ϕ s , 2 = μ g z cos g z + μ N 4 ϕ s 11 , ϕ ˙ s , 2 = - μ g z sin g z + μ N 4 ϕ ˙ s 11 .
x 1 = x cos g z + ξ / ( n 0 g ) sin g z , ξ 1 = - n 0 g x sin g z + ξ cos g z ,
x 3 = μ z ξ 1 n 0 + μ N 4 ( x ϕ c 11 + ξ n 0 g ϕ s 11 ) , ξ 3 = - μ n 0 g 2 z x 1 + μ N 4 ( n 0 g x ϕ ˙ c 11 + ξ ϕ ˙ s 11 ) .
n ζ = ( n ζ ) 0 + ( n ζ ) 2 + O ( 4 )
n ζ = n ( n 2 - ξ 2 - η 2 ) 1 / 2 = 1 [ 1 - 1 n 2 ( ξ 2 + η 2 ) ] 1 / 2 = 1 + 1 2 n 0 2 ( ξ 2 + η 2 ) + O ( 4 ) .
Λ x = n 0 u ¯ x + h ¯ ξ + 1 2 n 0 2 h ¯ ξ ( ξ 2 + η 2 ) + O ( 5 )
Λ x = n 0 u ¯ ( x 1 + x 3 ) + h ¯ [ ( n ζ ) 0 + ( n ζ ) 2 ] × ( ξ 1 + ξ 3 ) + O ( 5 ) = n 0 u ¯ x 1 + h ¯ ξ 1 + n 0 u ¯ x 3 + h ¯ ξ 3 + 1 2 n 0 2 h ¯ ξ 1 ( ξ 1 2 + η 1 2 ) + O ( 5 ) .
n 0 u ¯ x + h ¯ ξ = n 0 u ¯ x 1 + h ¯ ξ 1 .
Λ x - Λ x = n 0 u ¯ x 3 + h ¯ ξ 3 + 1 2 n 0 2 × [ h ¯ ξ 1 ( ξ 1 2 + η 1 2 ) - h ¯ ξ ( ξ 2 + η 2 ) ] + O ( 5 ) .
( Δ Λ x ) 3 = n 0 u x ˜ 3 + h ξ ˜ 3 + 1 2 n 0 2 Δ [ h ξ ˜ ( ξ ˜ 2 + η ˜ 2 ) ] ,
x ˜ 3 = μ ˜ d ξ ˜ n 0 + μ ˜ N 4 ( x ˜ ϕ ˜ c 11 + ξ ˜ n 0 g ϕ ˜ s 11 ) , ξ ˜ 3 = - μ ˜ n 0 g 2 d x ˜ + μ ˜ N 4 ( n 0 g x ˜ ϕ ˙ ˜ c 11 + ξ ˜ ϕ ˙ ˜ s 11 ) .
( Δ Λ x ) 3 = [ c 1 ( σ x 2 + σ y 2 ) + c 2 ( σ x τ x + σ y τ y ) + c 3 ( τ x 2 + τ y 2 ) ] × σ x + [ c 4 ( σ x 2 + σ y 2 ) + c 5 ( σ x τ x + σ y τ y ) + c 6 ( τ x 2 + τ y 2 ) ] τ x .
- 2 ( Δ Λ x ) 3 = [ T 1 ( σ x 2 + σ y 2 ) + 2 T 2 ( σ x τ x + σ y τ y ) + ( T 3 + P T ) ( τ x 2 + τ y 2 ) ] σ x + [ T 2 ( σ x 2 + σ y 2 ) + 2 T 3 ( σ x τ x + σ y τ y ) + T 4 ( τ x 2 + τ y 2 ) ] τ x .
T 1 = - 2 c 1 ,             T 2 = - 2 c 4 = - c 2 ,             T 3 = - c 5 , P T = - 2 c 3 + c 5 ,             T 4 = - 2 c 6 .
P T = k d H 2 / n 0 ,
k d H 2 , d e p e q ,             p , q = 1 , 2 , 3.
H Δ ( u 2 ) ,             H Δ ( u w ) ,             H Δ ( w 2 ) , e p Δ ( h u ) ,             e p Δ ( m u ) ,             e p Δ ( m w ) ,             p = 1 , 2 , 3.
e p Δ ( h w ) = e p Δ ( m u ) , H k Δ ( h 2 ) = - H Δ ( u 2 ) , H k Δ ( h m ) = - H Δ ( u w ) , H k Δ ( m 2 ) = - H Δ ( w 2 ) .
Δ ( h u 3 ) , Δ ( h u 2 w ) , Δ ( h u w 2 ) , Δ ( h w 3 ) , Δ ( m u 3 ) , Δ ( m u 2 w ) , Δ ( m u w 2 ) , Δ ( m w 3 ) .
T 1 :             MRD 4 , T 2 :             MRD 3 CRD , T 3 :             MRD 2 CRD 2 , T 4 :             MRD ( CRD 3 ) ,
T 1 = n 0 [ C 1 d e 1 2 + C 2 e 1 Δ ( h u ) + C 3 Δ ( h u 3 ) ] .
- n 0 ( C 3 - N 4 - 1 ) [ 1 4 h u ( g 2 h 2 - u 2 ) cos 4 g d + 1 16 g ( g 4 h 4 - 6 g 2 h 2 u 2 + u 4 ) sin 4 g d ] + n 0 ( 2 C 2 + C 3 + 4 N 4 - 1 ) [ 1 4 h u ( g 2 h 2 + u 2 ) × cos 2 g d + 1 8 g ( g 4 h 4 - u 4 ) sin 2 g d ] + n 0 d 4 ( 2 C 1 + 3 N 4 - 2 ) ( g 2 h 2 + u 2 ) 2 - n 0 4 ( 2 C 2 + 5 N 4 ) g 2 h 3 u - n 0 4 ( 2 C 2 + 2 C 3 + 3 N 4 - 2 ) h u 3 = 0.
C 3 - N 4 - 1 = 0 , 2 C 2 + C 3 + 4 N 4 - 1 = 0 , 2 C 1 + 3 N 4 - 2 = 0 , 2 C 2 + 5 N 4 = 0 , 2 C 2 + 2 C 3 + 3 N 4 - 2 = 0.
C 1 = 1 - 3 N 4 / 2 ,             C 2 = - 5 N 4 / 2 ,             C 3 = 1 + N 4 ,
T 1 = n 0 d e 1 2 ( 1 - 3 N 4 / 2 ) + n 0 ( 1 + N 4 ) Δ ( h u 3 ) - 5 n 0 N 4 e 1 Δ ( h u ) / 2 , T 2 = n 0 d e 1 e 2 ( 1 - 3 N 4 / 2 ) + n 0 ( 1 + N 4 ) Δ ( h u 2 w ) - 5 n 0 N 4 e 2 Δ ( h u ) / 2 - N 4 H Δ ( u 2 ) , T 3 = n 0 d e 2 2 ( 1 - 3 N 4 / 2 ) + n 0 ( 1 + N 4 ) Δ ( h u w 2 ) - 5 n 0 N 4 e 3 Δ ( h u ) / 2 - 2 N 4 H Δ ( u w ) - N 4 P T / 2 , T 4 = n 0 d e 2 e 3 ( 1 - 3 N 4 / 2 ) + n 0 ( 1 + N 4 ) Δ ( h w 3 ) - 5 n 0 N 4 e 3 Δ ( m u ) / 2 - N 4 H Δ ( w 2 ) / 2.
n 2 ( r 2 ) = n 0 2 ( 1 + r 4 ) + O ( 6 ) .
P T = 0 ,
T 1 = - 4 5 d n 0 ( h 4 + h 3 h + h 2 h 2 + h h 3 + h 4 ) , T 2 = - 1 5 d n 0 ( 4 h 3 m + 3 h 2 h m + 2 h h 2 m + h 3 m + h 3 m + 2 h 2 h m + 3 h h 2 m + 4 h 3 m ) , T 3 = - 2 15 d n 0 ( 6 h 2 m 2 + 3 h h m 2 + h 2 m 2 + 3 h 2 m m + 4 h h m m + 3 h 2 m m + h 2 m 2 + 3 h h m 2 + 6 h 2 m 2 ) , T 4 = - 1 5 d n 0 ( 4 h m 3 + h m 3 + 3 h m 2 m + 2 h m 2 m + 2 h m m 2 + 3 h m m 2 + h m 3 + 4 h m 3 ) .
n 0 = 1.637 ,             k = 0.108796 ,             N 4 = 0.694658 , ρ 1 = - ρ 2 = - 0.783032 ,             d = 9.169290.

Metrics