Abstract

A simple derivation of analytic expressions for the chromatic paraxial aberration coefficients of radial gradient-index lenses is presented. By decomposing the transverse chromatic aberration vector of an arbitrary paraxial ray in contributions from refraction at the surfaces and from transfer through the inhomogeneous media of the system, remarkably short formulas for the contributions of transfer through the gradient medium to the axial and lateral color coefficients are obtained. In the thin-lens approximation these expressions lead to well-known results for the total chromatic aberrations of a radial gradient-index lens.

© 1996 Optical Society of America

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References

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  1. L. G. Atkinson, S. N. Houde-Walter, D. T. Moore, D. P. Ryan, J. M. Stagaman, “Design of a gradient-index photographic objective,” Appl. Opt. 21, 993–998 (1982).
    [Crossref] [PubMed]
  2. H. Tsuchida, T. Nagaoka, K. Yamamoto, “On the design of optical systems using GRIN materials,” in Lens and Optical Systems Design, H. Zuegge, ed., Proc. SPIE1780, 456–463 (1993).
  3. D. P. Ryan-Howard, D. T. Moore, “Model for the chromatic properties of graident-index glass,” Appl. Opt. 24, 4356–4366 (1985).
    [Crossref] [PubMed]
  4. P. J. Sands, “Inhomogeneous lenses. II. Chromatic paraxial aberrations,”J. Opt. Soc. Am. 61, 777–783 (1971).
    [Crossref] [PubMed]
  5. P. J. Sands, “Inhomogeneous lenses. V. Chromatic paraxial aberrations of lenses with axial or cylindrical index distributions,”J. Opt. Soc. Am. 61, 1495–1500 (1971).
    [Crossref]
  6. When in Ref. 5 the second of Eqs. (21) was calculated according to Eq. (20), a coefficient 1/2 was lost. This minor omission, affecting one term in a sum for each chromatic transfer contribution, has, however, a harmful effect on the final results.
  7. P. O. McLaughlin, J. J. Miceli, D. T. Moore, D. P. Ryan, J. M. Stagaman, “Design of a gradient-index binocular objective,” in 1980 International Lens Design Conference, R. E. Fischer, ed., Proc. SPIE237, 369–379 (1980).
  8. F. Bociort, J. Kross, “Seidel aberration coefficients for radial gradient-index lenses,” J. Opt. Soc. Am. A 11, 2647–2656 (1994).
    [Crossref]
  9. H. H. Hopkins, “The nature of the paraxial approximation,” J. Mod. Opt. 38, 427–472 (1991).
    [Crossref]
  10. E. W. Marchand, Gradient Index Optics (Academic, New York, 1978).
  11. H. A. Buchdahl, Optical Aberration Coefficients (Dover, New York, 1968).
  12. A. Cox, A System of Optical Design (Focal, London, 1964).

1994 (1)

1991 (1)

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

1985 (1)

1982 (1)

1971 (2)

Atkinson, L. G.

Bociort, F.

Buchdahl, H. A.

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

Cox, A.

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

Hopkins, H. H.

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

Houde-Walter, S. N.

Kross, J.

Marchand, E. W.

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

McLaughlin, P. O.

P. O. McLaughlin, J. J. Miceli, D. T. Moore, D. P. Ryan, J. M. Stagaman, “Design of a gradient-index binocular objective,” in 1980 International Lens Design Conference, R. E. Fischer, ed., Proc. SPIE237, 369–379 (1980).

Miceli, J. J.

P. O. McLaughlin, J. J. Miceli, D. T. Moore, D. P. Ryan, J. M. Stagaman, “Design of a gradient-index binocular objective,” in 1980 International Lens Design Conference, R. E. Fischer, ed., Proc. SPIE237, 369–379 (1980).

Moore, D. T.

D. P. Ryan-Howard, D. T. Moore, “Model for the chromatic properties of graident-index glass,” Appl. Opt. 24, 4356–4366 (1985).
[Crossref] [PubMed]

L. G. Atkinson, S. N. Houde-Walter, D. T. Moore, D. P. Ryan, J. M. Stagaman, “Design of a gradient-index photographic objective,” Appl. Opt. 21, 993–998 (1982).
[Crossref] [PubMed]

P. O. McLaughlin, J. J. Miceli, D. T. Moore, D. P. Ryan, J. M. Stagaman, “Design of a gradient-index binocular objective,” in 1980 International Lens Design Conference, R. E. Fischer, ed., Proc. SPIE237, 369–379 (1980).

Nagaoka, T.

H. Tsuchida, T. Nagaoka, K. Yamamoto, “On the design of optical systems using GRIN materials,” in Lens and Optical Systems Design, H. Zuegge, ed., Proc. SPIE1780, 456–463 (1993).

Ryan, D. P.

L. G. Atkinson, S. N. Houde-Walter, D. T. Moore, D. P. Ryan, J. M. Stagaman, “Design of a gradient-index photographic objective,” Appl. Opt. 21, 993–998 (1982).
[Crossref] [PubMed]

P. O. McLaughlin, J. J. Miceli, D. T. Moore, D. P. Ryan, J. M. Stagaman, “Design of a gradient-index binocular objective,” in 1980 International Lens Design Conference, R. E. Fischer, ed., Proc. SPIE237, 369–379 (1980).

Ryan-Howard, D. P.

Sands, P. J.

Stagaman, J. M.

L. G. Atkinson, S. N. Houde-Walter, D. T. Moore, D. P. Ryan, J. M. Stagaman, “Design of a gradient-index photographic objective,” Appl. Opt. 21, 993–998 (1982).
[Crossref] [PubMed]

P. O. McLaughlin, J. J. Miceli, D. T. Moore, D. P. Ryan, J. M. Stagaman, “Design of a gradient-index binocular objective,” in 1980 International Lens Design Conference, R. E. Fischer, ed., Proc. SPIE237, 369–379 (1980).

Tsuchida, H.

H. Tsuchida, T. Nagaoka, K. Yamamoto, “On the design of optical systems using GRIN materials,” in Lens and Optical Systems Design, H. Zuegge, ed., Proc. SPIE1780, 456–463 (1993).

Yamamoto, K.

H. Tsuchida, T. Nagaoka, K. Yamamoto, “On the design of optical systems using GRIN materials,” in Lens and Optical Systems Design, H. Zuegge, ed., Proc. SPIE1780, 456–463 (1993).

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. (2)

J. Opt. Soc. Am. A (1)

Other (6)

H. Tsuchida, T. Nagaoka, K. Yamamoto, “On the design of optical systems using GRIN materials,” in Lens and Optical Systems Design, H. Zuegge, ed., Proc. SPIE1780, 456–463 (1993).

When in Ref. 5 the second of Eqs. (21) was calculated according to Eq. (20), a coefficient 1/2 was lost. This minor omission, affecting one term in a sum for each chromatic transfer contribution, has, however, a harmful effect on the final results.

P. O. McLaughlin, J. J. Miceli, D. T. Moore, D. P. Ryan, J. M. Stagaman, “Design of a gradient-index binocular objective,” in 1980 International Lens Design Conference, R. E. Fischer, ed., Proc. SPIE237, 369–379 (1980).

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

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

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

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Figures (2)

Fig. 1
Fig. 1

Ray parameters of the marginal ray OB and the chief ray AP at the first surface of the system.

Fig. 2
Fig. 2

Focal length of a thin Wood lens (see text).

Equations (85)

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n 2 ( r 2 ) = n 0 2 ( 1 k r 2 + ) ,
x P = r P τ x , y P = r P τ y
x E P = r E P σ x , y E P = r E P σ y .
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 , η ˜ = n 0 w τ y n 0 u σ y .
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 .
u = u cosh g ˆ d h g ˆ sinh g ˆ d , h = u g ˆ sinh g ˆ d + h cosh g ˆ d , w = w cosh g ˆ d m g ˆ sinh g ˆ d , m = w g ˆ sinh g ˆ d + m cosh g ˆ d .
E 1 ( k d 2 ) = cos g d , E 2 ( k d 2 ) = sin g d g d ,
u = u E 1 ( k d 2 ) + h k d E 2 ( k d 2 ) , h = u d E 2 ( k d 2 ) + h E 1 ( k d 2 ) .
E 1 ( k d 2 ) = 1 1 2 ! k d 2 + 1 4 ! ( k d 2 ) 2 , E 2 ( k d 2 ) = 1 1 3 ! k d 2 + 1 5 ! ( k d 2 ) 2 .
u = u , h = h u d .
H = m n 0 u h n 0 w ;
Λ ˜ x = n 0 u x ˜ + h ξ ˜ ,
e 1 = k h 2 + u 2 , e 2 = k h m + u w , e 3 = k m 2 + w 2 .
Ξ ˜ λ x = x ˜ λ Q x ˜ Q , Ξ ˜ λ y = y ˜ λ Q y ˜ Q
Λ ˜ λ x P = n P u P x ˜ λ P , Λ ˜ λ x Q = n Q u Q x ˜ λ Q .
n Q u Q ( x ˜ λ Q x ˜ Q ) = Λ ˜ λ x Q Λ ˜ x Q .
Λ ˜ x Q = Λ ˜ x P = Λ ˜ λ x P .
n Q u Q Ξ ˜ λ x = Λ ˜ λ x Q Λ ˜ λ x P .
n Q u Q Ξ ˜ λ x = Δ Λ ˜ λ x .
Δ x ˜ λ = ξ ˜ λ d n λ , Δ h = u d .
Λ ˜ λ x = n u x ˜ λ + n n λ h ξ ˜ λ
Δ Λ ˜ λ x = n u Δ x ˜ λ + n n λ ξ ˜ λ Δ h = 0.
Λ ˜ λ x = n 0 u x ˜ λ + n 0 n 0 λ h ξ ˜ λ .
x ˜ λ = m λ τ x + h λ σ x , ξ ˜ λ = n 0 λ w λ τ x n 0 λ u λ σ x ,
n Q u Q Ξ ˜ λ x = Δ Λ ˜ λ x = Γ λ 1 σ x + Γ λ 2 τ x .
Γ λ 1 = n Q u Q h λ Q .
δ λ s = h λ Q / u λ Q
δ λ s = Γ λ 1 n Q u Q 2 .
Γ λ 2 = n Q u Q ( m λ Q m Q ) .
δ λ X = X λ X ,
δ λ m Q m Q = Γ λ 2 H .
Δ Λ ˜ λ x = Δ ( n 0 u x ˜ λ + n 0 n 0 λ h ξ ˜ λ ) = Δ [ n 0 u ( x ˜ λ x ˜ ) + h ( n 0 n 0 λ ξ ˜ λ ξ ˜ ) ] .
Δ Λ ˜ λ x = Δ [ n 0 u ( δ λ h σ x + δ λ m τ x ) n 0 h ( δ λ u σ x + δ λ w τ x ) ] .
Δ Λ ˜ λ x = Δ [ n 0 ( h δ λ u u δ λ h ) ] σ x + Δ [ n 0 ( h δ λ w u δ λ m ) ] τ x .
Δ Λ ˜ λ x = S λ 1 σ x + S λ 2 τ x ,
Δ Λ ˜ λ x = T λ 1 σ x + T λ 2 τ x .
Γ λ p = surfaces S λ p + RGRIN media T λ p , p = 1,2 ,
S λ 1 = h n 0 i Δ ( δ λ n 0 n 0 ) , S λ 2 = h n 0 j Δ ( δ λ n 0 n 0 ) .
n 0 i = n 0 h ρ n 0 u , n 0 j = n 0 m ρ n 0 w ,
T λ 1 = n 0 [ ( h δ λ u u δ λ h ) ( h δ λ u u δ λ h ) ] , T λ 2 = n 0 [ ( h δ λ w u δ λ m ) ( h δ λ w u δ λ m ) ] .
δ λ u = A 1 + A 2 + A 3 , δ λ h = B 1 + B 2 + B 3 ,
A 1 = δ λ u cos g d + δ λ h g sin g d , A 2 = ( u sin g d + h g cos g d ) d δ λ g , A 3 = h sin g d δ λ g ,
B 1 = δ λ u g sin g d + δ λ h cos g d , B 2 = ( u g cos g d h sin g d ) d δ λ g , B 3 = 1 g 2 u sin g d δ λ g .
( h δ λ u u δ λ h ) ( h δ λ u u δ λ h ) = C 1 + C 2 + C 3 ,
C 1 = h A 1 u B 1 ( h δ λ u u δ λ h ) , C 2 = h A 2 u B 2 , C 3 = h A 3 u B 3 .
A 2 = h d g δ λ g , B 2 = u d δ λ g g .
C 2 = ( h 2 d g 2 + u 2 d ) ( δ λ g g ) = d e 1 δ λ g g ,
A 3 = ( u u cos g d ) δ λ g g ,
C 3 = δ λ g g [ h u u ( h cos g d + 1 g u sin g d ) ] .
h cos g d + 1 g u sin g d = h ,
C 3 = δ λ g g ( h u h u ) .
δ λ g g = 1 2 δ λ k k ,
T λ 1 = n 0 2 δ λ k k [ d e 1 + Δ ( h u ) ] .
T λ 2 = n 0 2 δ λ k k [ d e 2 + Δ ( h w ) ] .
d | f W | .
| k | d 2 1.
u = u + k d h ,
h = h u d .
Δ h = 0.
h / f W = u Q = n 0 u = n 0 k d h .
f W = ( n 0 k d ) 1 ,
ϕ g = n 0 k d .
ϕ h = ( n 0 1 ) ( ρ 1 ρ 2 )
d e 1 = k h 2 d + u 2 d = h Δ u u Δ h = h Δ u .
d e 1 + Δ ( h u ) = 2 h Δ u = 2 h 2 ϕ g n 0 ,
T λ 1 = δ λ k k h 2 ϕ g .
Γ λ 1 = S λ 1,1 + T λ 1 + S λ 1,2 .
S λ 1,1 + S λ 1,2 = h δ λ n 0 n 0 [ ( n 0 i ) 1 ( n 0 i ) 2 ] ,
( n 0 i ) 1 ( n 0 i ) 2 = n 0 h ρ 1 n 0 u ( n 0 h ρ 2 n 0 u ) = n 0 h ( ρ 1 ρ 2 ) + n 0 k d h = n 0 h n 0 1 ϕ h + h ϕ g .
δ λ n 0 n 0 + δ λ k k = δ λ ( n 0 k ) n 0 k ,
Γ λ 1 = h 2 [ δ λ n 0 n 0 1 ϕ h + δ λ ( n 0 k ) n 0 k ϕ g ] .
Γ λ 2 = h m [ δ λ n 0 n 0 1 ϕ h + δ λ ( n 0 k ) n 0 k ϕ g ] .
Γ λ 1 = h 2 ( ϕ λ ϕ ) .
ϕ λ ϕ = δ λ ϕ = δ λ ϕ h + δ λ ϕ g = ( ρ 1 ρ 2 ) δ λ n 0 + d δ λ ( n 0 k ) = ϕ h δ λ n 0 n 0 1 + ϕ g δ λ ( n 0 k ) n 0 k ,
n 0 1 δ λ n 0 = ν h = n 0 d 1 n 0 F n 0 C ,
n 0 k δ λ ( n 0 k ) = ν g = ( n 0 k ) d ( n 0 k ) F ( n 0 k ) C ,
Γ λ 1 = h 2 ( ϕ h ν h + ϕ g ν g ) .
n ( r ) = N 00 + N 10 r 2 + .
N 00 = n 0 , N 10 = n 0 k / 2 .
ν g = N 10 d N 10 F N 10 C .
δ λ n 0 n 0 = n 0 1 n 0 ν h ,
δ λ k k = ν g 1 n 0 1 n 0 ν h .

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