Abstract

Chromatic effects of radial gradient-index materials have been analyzed, and several important conclusions have been derived in terms of material dispersion data. The use of Buchdahl dispersion data, both for base glass materials and ion-exchange pairs, provides some simple relationships for chromatic aberration and helps in selecting suitable materials for producing achromatic radial gradient-index lenses.

© 1996 Optical Society of America

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References

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  1. M. L. Huggins, K. H. Sun, D. O. Davis, “The dispersion of silicate glasses as a function of composition. II,” J. Opt. Soc. Am.. 32, 635–650 (1942).
  2. S. D. Fantone, “Refractive index and spectral models for gradient index materials,”Appl. Opt. 22, 432–440 (1983).
  3. D. P. Ryan-Howard, D. T. Moore, “Model for the chromatic properties of gradient index glass,”Appl. Opt. 24, 4356–4366 (1985).
  4. K. Siva, Rama Krishna, A. Sharma, “Model for optical properties of silicate glasses,” Appl. Opt. 33, 8030–8035 (1994).
  5. K. Siva, Rama Krishna, A. Sharma, “Evaluation of optical glass composition by optimization models,” Appl. Opt. 34, 5628–5634 (1995).
  6. P. J. Sands, “Inhomogeneous lenses. II. Chromatic par-axial aberrations,” J. Opt. Soc. Am. 61, 777–783 (1971).
  7. 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).
  8. P. N. Robb, R. I. Mercado, “Calculation of refractive indices using Buchdahl’s chromatic coordinate,” Appl. Opt. 22, 1198–1215 (1983).
  9. D. T. Moore, “Design of singlets with continuously varying indices of refraction,” J. Opt. Soc. Am. 61, 886–894 (1971).
  10. D. T. Moore, P. J. Sands, “Third-order aberrations of inhomogeneous lenses with cylindrical index distributions,” J. Opt. Soc. Am. 61, 1195–1201 (1971).

1995 (1)

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1985 (1)

1983 (2)

1971 (4)

1942 (1)

M. L. Huggins, K. H. Sun, D. O. Davis, “The dispersion of silicate glasses as a function of composition. II,” J. Opt. Soc. Am.. 32, 635–650 (1942).

Davis, D. O.

M. L. Huggins, K. H. Sun, D. O. Davis, “The dispersion of silicate glasses as a function of composition. II,” J. Opt. Soc. Am.. 32, 635–650 (1942).

Fantone, S. D.

Huggins, M. L.

M. L. Huggins, K. H. Sun, D. O. Davis, “The dispersion of silicate glasses as a function of composition. II,” J. Opt. Soc. Am.. 32, 635–650 (1942).

Krishna, Rama

Mercado, R. I.

Moore, D. T.

Robb, P. N.

Ryan-Howard, D. P.

Sands, P. J.

Sharma, A.

Siva, K.

Sun, K. H.

M. L. Huggins, K. H. Sun, D. O. Davis, “The dispersion of silicate glasses as a function of composition. II,” J. Opt. Soc. Am.. 32, 635–650 (1942).

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Tables (1)

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Table 1 Invariant Exchange Index and Dispersion Values of Some Univalent Ion-Exchange Pairs

Equations (43)

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n 0 ( ω λ ) = n 00 λ0 + v 01 ω λ + v 02 ω λ 2 + ,
ω λ = ( λ−λ 0 ) 1 + 2.5 ( λ λ 0 )
V 00 λ 0 , F C = ( n 00 λ 0 1 ) / [ n 0 ( ω F ) n 0 ( ω C ) ] ,
V 00 λ0, F C = n 00 λ 0 1 v 01 ( ω F ω C ) [ 1 ( v 02 / v 01 ) ( ω F + ω C ) ] ,
V 00 λ 0 , F C n 00 λ 0 1 v 01 ( ω F ω C ) .
a i λ = K i λ 0 + L i ω λ + M i ω λ 2 ,
n ( r , ω λ ) = n 0 ( ω λ ) + n 1 ( ω λ ) r 2 + ,
n 0 ( ω λ ) = n 00 λ 0 + v 01 ω λ + v 02 ω λ 2 + = n 00 λ 0 + m v 0 m ω λ m , n 1 ( ω λ ) = n 10 λ 0 + v 11 ω λ + v 12 ω λ 2 + = n 10 λ 0 + m v 1 m ω λ m ,
n ( r , ω λ ) = n λ 0 ( r ) + m = 1 v m ( r ) ω λ m ,
n λ 0 ( r ) = n 00 λ 0 + n 10 λ 0 r 2 + n 20 λ 0 r 4 + , v m ( r ) = v 0 m + v 1 m r 2 + v 2 m r 4 + .
T L ch C = μ   m = 1 F a m ( ω F m ω C m ) ,
T ch C = μ   m = 1 F ¯ a m ( ω F m ω C m ) ,
μ = 1 / ( n 00 λ0, k υ a , k ) , F a m = j f a m + j f a m * , F ¯ a m = j f ¯ a m + j f ¯ a m * .
T L ch C = μ ( f a 1 + f a 1 * ) ( ω F ω C ) ,
T ch C = μ ( f ¯ a 1 + f ¯ a 1 * ) ( ω F ω C ) ,
f a 1 = n 00 λ 0 y a i a Δ ( v 01 n 00 λ0 ) ,
f ¯ a 1 = n 00 λ 0 y a i b Δ ( v 01 n 00 λ0 ) ,
f a 1 * = 2 n 10 λ0 Ψ 1 [ y a 2 g 0 + y a υ a g 1 + υ a 2 g 2 ] ,
f ¯ a 1 * = 2 n 10 λ 0 Ψ 1 [ y a y b g 0 + 0.5 ( y a υ b + y b υ a ) × g 1 + υ a υ b g 2 ] ,
g 0 = 0.5 ( t + C S ) , g 1 = S 2 g 2 = ( t C S ) 2 N ¯ 1 , C = cos ( α t )   for   n 10 λ0 < 0 = cosh ( α t )   for   n 10 λ0 > 0 , S = sin ( α t ) α   for   n 10 λ0 < 0 = sinh ( α t ) α   for   n 10 λ0 > 0 , N ¯ 1 = α 2 = 2 n 10 λ 0 n 00 λ 0 , Δ ( v 01 n 00 λ0 ) = ( v 01 n 00 λ0 v 01 n 00 λ0 ) , Ψ 1 = ( v 11 n 10 λ0 v 01 n 00 λ0 ) .
L ch C = l F l C = T L ch C υ a , k .
n ( r , ω λ ) = n 0 ( ω λ ) + n 1 ( ω λ ) r 2 , = n 0 ( ω λ ) [ 1 + β ( ω λ ) r 2 ] ,
Δ n ( ω λ ) = n ( r , ω λ ) n 0 ( ω λ ) = n 0 ( ω λ ) β ( ω λ ) r 2 .
β ( ω λ ) = 1 r 2 [ Δ n ( ω λ ) n 0 ( ω λ ) ] .
β ( ω λ ) = 1 r 2 ( Δ n λ 0 + Δ v 1 ω λ + Δ v 2 ω λ 2 n 00 λ 0 + v 01 ω λ + v 02 ω λ 2 ) .
β ( ω λ ) = Δ n λ0 n 00 λ 0 r 2 [ 1 + Ψ 1 ω λ + ( Ψ 2 v 01 n 00 λ 0 Ψ 1 ) ω λ 2 ] ,
Ψ 1 = ( Δ v 1 Δ n λ0 v 01 n 00 λ 0 ) , Ψ 2 = ( Δ v 2 Δ n λ0 v 02 n 00 λ 0 )
n λ = 1 + i R i λ V 0 = 1 + i a i λ N i V 0 ,
n λ ( χ ) = 1 + R 0 λ + χΔ R λ V 0 + χΔV ,
R 0 λ = [ n λ ( χ = 0 ) 1 ] V 0 , Δ R λ = ( γ a dfλ a exλ ) N ex0 , V 0 = k + V i + V ex0 , Δ V = ( γ c df c ex ) N ex0 , γ= valence of exchance ion valence of diffusing ion .
Δ n λ ( χ ) = R 0 λ + χΔ R λ V 0 + χΔ V R 0 λ V 0 , ( Δ R λ Δ V R 0 λ V 0 ) Δ V V 0 χ, = { Δ R λ Δ V [ n λ ( χ = 0 ) 1 ] } Δ V V 0 χ.
Δ n λ ( χ ) = [ n I λ , df ex n λ ( χ = 0 ) ] Δ V V 0 χ,
n I λ , df ex = 1 + γ a dfλ a exλ γ c df c ex .
n λ ( χ = 0 ) = n λ ( r , ω λ  ) = n λ0 ( r ) + v 1 ( r ) ω λ + v 2 ( r ) ω λ 2 , = n 00 λ 0 + v 01 ω λ + v 02 ω λ 2 , a dfλ = K dfλ 0 + L df ω λ + M df ω λ 2 , a exλ = K exλ 0 + L ex ω λ + M ex ω λ 2 .
Δ n λ 0 ( χ ) = ( 1 + γK dfλ 0 K exλ 0 γ c df c ex n 00 λ0 ) Δ V V 0 χ, Δ v 1 ( χ ) = ( γ L df L ex γ c df c ex v 01 ) Δ V V 0 χ, Δ v 2 ( χ ) = ( γ M df M ex γ c df c ex v 02 ) Δ V V 0 χ.
n I λ 0 , df ex = 1 + γ K dfλ0 K exλ 0 γ c df c ex ,
v 1 I , df ex = γ L df L ex γ c df c ex ,
Δ n λ0 ( χ ) = ( n I λ0,df ex n 00 λ 0 ) Δ V V 0 χ,
Δ v 1 ( χ ) = ( v 1 I , df ex v 01 ) Δ V V 0 χ.
Δ v 1 Δ n λ0 = v 01 n 00 λ 0 .
v 1 I , df ex n I λ 0 , df ex = v 01 n 00 λ 0 .
Ψ 1 = v 1 I , df ex n I λ 0 , df ex v 01 n 00 λ 0 ,
= ( v 11 n 10 λ 0 v 01 n 00 λ 0 ) .

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