Abstract

A model has been developed which predicts the chromatic properties of gradient-index materials based on the composition of the glass. It yields a gradient Abbe number and a gradient partial dispersion which can be used by the lens designer in the design of achromatic gradient-index lens systems. Over 100 glasses with various ion exchange pairs have been investigated. The result is a gradient Abbe number ranging from −2000 to +5000 and a gradient partial dispersion ranging from −5 to +7. This wide range of gradient Abbe numbers and gradient partial dispersions can be further extended by using a glass which has two exchange ions with one diffusing ion or a glass which has one exchange ion with two diffusing ions. In addition to an extended range, the designer is afforded the luxury of a continuously varying Abbe number rather than the discrete Abbe number of conventional materials.

© 1985 Optical Society of America

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References

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  1. M. L. Huggins, “The Dispersion of Silicate Glasses as a Function of Composition,” J. Opt. Soc. Am. 30, 514 (1940).
    [CrossRef]
  2. 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 (1942).
    [CrossRef]
  3. M. L. Huggins, K. H. Sun, “Calculation of Density and Optical Constants of a Glass From its Composition in Weight Percentage,” J. Opt. Soc. Am. 26, 4 (1936).

1942

1940

1936

M. L. Huggins, K. H. Sun, “Calculation of Density and Optical Constants of a Glass From its Composition in Weight Percentage,” J. Opt. Soc. Am. 26, 4 (1936).

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Figures (16)

Fig. 1
Fig. 1

Invariant lines with quadrants for nd and V10.

Fig. 2
Fig. 2

Invariant lines with quadrants for V10 and P10.

Fig. 3
Fig. 3

Graphical representation of V10 and P10—monovalent ions, full exchange.

Fig. 4
Fig. 4

Graphical representation of V10 and P10—50% exchange.

Fig. 5
Fig. 5

Index change as a function of V10—monovalent ions, conventional glass.

Fig. 6
Fig. 6

Index change as a function of V10—divalent ions, conventional glass.

Fig. 7
Fig. 7

Single diffusant (Li) with two exchange ions (K and Na) in SF64.

Fig. 8
Fig. 8

Single diffusant (Li) with two exchange ions (K and Na) in BaK1.

Fig. 9
Fig. 9

Single diffusant (Ag) with two exchange ions (K and Na) in SF64.

Fig. 10
Fig. 10

Mixed bath (Ag and Li) with one exchange ion (Na) and BL glass.

Fig. 11
Fig. 11

Mixed bath (Ag and Li) with one exchange ion (K) in KzF1.

Fig. 12
Fig. 12

Mixed bath (K and Li) with one exchange ion (Na) in BL glass.

Fig. 13
Fig. 13

Mixed hath (Na and Li) with one exchange ion (K) in KzF1.

Fig. 14
Fig. 14

Percent error in index and density using old silicon constants.

Fig. 15
Fig. 15

Percent error in index and density using revised silicon constants.

Fig. 16
Fig. 16

Volume and refraction constant for KHF2.

Tables (9)

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Table I New Constants for Constituents

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Table II Summary of Revised Silicon Constants

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Table III Constants for Constituents

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Table IV Invariant Lines

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Table V Calculated Values for Sample Glasses and Ion Exchange Pairs

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Table VI Effect of Degree of Exchange on V10 and P10

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Table VII Diffusion Coefficient Ratio (Na/K)

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Table VIII Ranges of V10 for Sample Glasses with Mixed Bath and One Exchange Ion

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Table IX Summary of Old Silicon Constants

Equations (25)

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n λ ( χ ) = 1 + [ a M λ N M ( χ ) ] / V ( χ ) ,
a M λ = d M [ 1 / ( g M 1 / λ 2 ) 4.8 × 10 5 λ 2 ] ,
N M = ( m M f M / W M ) / A ,
A = n M f M / W M ,
V ( χ ) = k + b M + c M N M ( χ ) ,
ρ = 1 / V 0 A ,
a M λ = d M 0 [ 1 / ( g M 0 1 / λ 2 ) 4.8 × 10 5 λ 2 ] + d M [ 1 / ( g M 1 / λ 2 ) ] N M .
a M λ = d M 0 [ 1 / ( g M 0 1 / λ 2 ) 4.8 × 10 5 λ 2 ] + d M [ 1 / ( g M 1 / λ 2 ) ] N M 2 .
N B = N B + N B ,
N B = 2 4 N Si 3 N B and N B = 4 ( N Si + N B ) 2 if ( N Si + N B ) > 0.5 , N B = N B , and N B = 0 if ( N Si + N B ) < 0.5 .
N Si B = N Si + N B for N B / N Si < 0.55 , N Si B = N Si + 0.85 N B for N B / N Si 0.55 . }
N λ ( r , z ) = N 00 λ + N 10 λ r 2 + N 20 λ r 4 + + N 01 λ z + N 02 λ z 2 + N 03 λ z 3 + ,
V ( r ) = V 00 + V 10 r 2 + V 20 r 4 + ,
V i j = N i j d / ( N i j F N i j C ) except for i = j = 0 ,
V 00 = ( N 00 d 1 ) / ( N 00 F N 00 C ) .
P A C = 1 N 00 d , k u a k i = 1 k ( y a i 2 ϕ h d , i V 00 , i + y a i 2 ϕ W d , i V 10 , i ) ,
V ( r , z ) = V i j r 2 i z j
P ( r , z ) = P i j r 2 i z j ,
V 00 = ( N 00 d 1 ) / ( N 00 F N 00 C ) , V i j = N i j d / ( N i j F N i j C ) , P i j = ( N i j d N i j C ) / ( N i j F N i j C ) .
n d , I = 1 + γ a f , d a e , d γ c f c e , n F C , I = γ a f , F C a e , F C γ c f c e , n d C , I = γ a f , d C a e , d C γ c f c e ,
a Ca , d = 12.66 + 0.15 N Ca , a Ba , d = 19.80 + 5.00 N Ba , a Pb , d = 28.40 + 22.74 N Pb 2 .
N B = N B + N B ,
N B = 2 4 N Si 3 N B and N B = 4 ( N Si + N B ) 2 if ( N B + N Si ) < 0.5 , N B = N B and N B = 0 if ( N B + N Si ) > 0.5 .
N Si B = N Si + N B for N B / N Si < 0.55 ,
N Si B = N Si + 0.85 N B for N B / N Si 0.55 .

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