Abstract

Models are presented which describe the refractive index and spectral variation of gradient-index materials. Classes of gradient which exhibit little or no variation with wavelength are predicted. These are termed achromatic gradients. Experimental verification of the models is presented in a following paper.

© 1983 Optical Society of America

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References

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  1. R. W. Wood, Physical Optics (Macmillan, New York, 1905), pp. 86–91.
  2. J. C. Maxwell, The Scientific Papers of James Clerk Maxwell, W. D. Niven, Ed. (Dover, New York, 1965).
  3. R. K. Luneberg, Mathematical Theory of Optics (U. California Press, Berkeley, 1966).
  4. D. T. Moore, “Gradient Index Optics: Aspects of Design, Testing, Tolerancing and Fabrication,” Ph.D. Thesis, The Institute of Optics, U. Rochester, New York (1974).
  5. M. L. Huggins, J. Opt. Soc. Am. 30, 420 (1940).
    [CrossRef]
  6. M. L. Huggins, J. Opt. Soc. Am. 30, 495 (1940).
    [CrossRef]
  7. M. L. Huggins, J. Opt. Soc. Am. 30, 514 (1940).
    [CrossRef]
  8. M. L. Huggins, J. Opt. Soc. Am. 32, 635 (1942).
    [CrossRef]
  9. K. H. Sun, J. Am. Ceram. Soc. 30, 277 (1947).
    [CrossRef]
  10. S. D. Fantone, “Design, Engineering, and Manufacturing Aspects of Gradient Index Optical Components,” Ph.D. Thesis, The Institute of Optics, U. Rochester, New York (1979).
  11. G. W. Morey, The Properties of Glass (Reinhold, New York, 1954).

1947

K. H. Sun, J. Am. Ceram. Soc. 30, 277 (1947).
[CrossRef]

1942

1940

Fantone, S. D.

S. D. Fantone, “Design, Engineering, and Manufacturing Aspects of Gradient Index Optical Components,” Ph.D. Thesis, The Institute of Optics, U. Rochester, New York (1979).

Huggins, M. L.

Luneberg, R. K.

R. K. Luneberg, Mathematical Theory of Optics (U. California Press, Berkeley, 1966).

Maxwell, J. C.

J. C. Maxwell, The Scientific Papers of James Clerk Maxwell, W. D. Niven, Ed. (Dover, New York, 1965).

Moore, D. T.

D. T. Moore, “Gradient Index Optics: Aspects of Design, Testing, Tolerancing and Fabrication,” Ph.D. Thesis, The Institute of Optics, U. Rochester, New York (1974).

Morey, G. W.

G. W. Morey, The Properties of Glass (Reinhold, New York, 1954).

Sun, K. H.

K. H. Sun, J. Am. Ceram. Soc. 30, 277 (1947).
[CrossRef]

Wood, R. W.

R. W. Wood, Physical Optics (Macmillan, New York, 1905), pp. 86–91.

J. Am. Ceram. Soc.

K. H. Sun, J. Am. Ceram. Soc. 30, 277 (1947).
[CrossRef]

J. Opt. Soc. Am.

Other

R. W. Wood, Physical Optics (Macmillan, New York, 1905), pp. 86–91.

J. C. Maxwell, The Scientific Papers of James Clerk Maxwell, W. D. Niven, Ed. (Dover, New York, 1965).

R. K. Luneberg, Mathematical Theory of Optics (U. California Press, Berkeley, 1966).

D. T. Moore, “Gradient Index Optics: Aspects of Design, Testing, Tolerancing and Fabrication,” Ph.D. Thesis, The Institute of Optics, U. Rochester, New York (1974).

S. D. Fantone, “Design, Engineering, and Manufacturing Aspects of Gradient Index Optical Components,” Ph.D. Thesis, The Institute of Optics, U. Rochester, New York (1979).

G. W. Morey, The Properties of Glass (Reinhold, New York, 1954).

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Figures (3)

Fig. 1
Fig. 1

Invariant exchange index lines.

Fig. 2
Fig. 2

Invariant exchange dispersion lines.

Fig. 3
Fig. 3

Invariant exchange β lines.

Tables (9)

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Table I Volume Constants for Silica

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Table II Volume Constants for Glass Constituents

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Table III Compositional Data for SF-2

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Table IV Renormalization of SF-2 Compositional Data

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Table V Refraction Data for Constituents

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Table VI Invariant Exchange Index

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Table VII Composition of BK7

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Table VIII Invariant Exchange Dispersion

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Table IX Invariant Exchange Slope for Fibers

Equations (82)

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A = i n i f i W i ,
B = m M f M W M = number of moles of positive ion M gram of glass ,
N M = B M / A = number of moles of positive ion M number of moles of oxygen ( O ) .
N M = i m M f M W M n i f i W i ,
N M = i m M f ¯ M n i f ¯ i .
V 0 = 1 ρ A ,
V 0 = k + b Si + c Si N Si + M c M N M ,
V 0 = b Si + c oxide N oxide , or V 0 = 5.69 + c N .
V 0 = 1 ρ n W = W n ρ ;
c M = ( V 0 5.69 ) n m .
A = n M f M W M = 1.69533 , V 0 = 15.104 + k , ρ = 1 A V 0 , and if k = 0 , then ρ = 3.905 , or if k = 0.05 , then ρ = 3.892.
A = n M f M W M = 1.7006 , N Si = 0.4061 , N Na = 0.0197 , N K = 0.0858 , N Pb = 0.1326 , and N As = 0.0016.
A = n M f M W M = 1.70383 , V 0 = k + 15.104.
N B = N B + N B .
N B = 2 4 N Si 3 N B ,
N B = 4 ( N Si + N B ) 2.
V 0 = k + b Si + c Si N Si + c B N B + c B N B + M c m N M .
N Li = χ N Na and N Na = ( 1 χ ) N Na .
V = k + b Si + c Si N Si + N Na c Na + χ N Na ( C Li C Na ) + c M N M .
V = V 0 + χ Δ V ,
N C = N A ( b c a d ) χ , and
N A = ( 1 χ ) N A .
Δ V = N A ( b c a d c C c A ) , or
Δ V = N A ( γ c C c A ) ,
γ = valence of the ion A valence of the ion C .
N A = N A ( 1 u υ ) ,
N C = N A x · c b a d = N A u γ C ,
N E = N A y · e b a f = N A υ γ E ,
γ C = c b a d , γ E = e b a f .
Δ V = ( c C γ C c A ) N A 1 + α u + ( c E γ E c A ) N A 1 + α α u ,
n d = 1 + M R M V = 1 + M a M N M V ,
n d = 1 + M a M N M k + b Si + c Si N Si + M c M N M .
n d = 1 + 9.842 k + 15.1039 .
n d ( χ ) = 1 + a M N M ( χ ) k + b Si + c Si N Si ( χ ) + c M N M ( χ ) .
n d ( χ ) = 1 + a Si N Si + ( a Pb + a Pb N Pb 2 ) N Pb + a As + N As + a K 0 N K 0 + χ ( a Li a K ) N K 0 V 0 + χ Δ V ,
n d ( χ ) = 1 + R 0 + χ Δ R V 0 + χ Δ V ,
R 0 = [ n d ( 0 ) 1 ] V 0 , Δ R = ( a Li a K ) N K 0 , V 0 = k + b Si + c Si N Si + c Pb N Pb + c K 0 N K 0 + c As N As , Δ V = ( c Li c K ) N K 0 .
Δ n = R 0 + χ Δ R V 0 + χ Δ V R 0 V 0 , Δ n 1 V 0 ( R 0 + χ Δ R ) ( 1 χ V 0 Δ V ) R 0 V 0 ,
Δ n χ V 0 ( Δ R R 0 Δ V V 0 ) .
Δ n max = Δ R V 0 { [ n d ( 0 ) 1 ] Δ V V 0 } .
n d χ | χ = 0 = χ ( 1 + R 0 + χ Δ R V 0 + χ Δ V ) | χ = 0 = V 0 Δ R R 0 Δ V V 0 2 = 1 V 0 { Δ R [ n d ( 0 ) 1 ] Δ V } .
n d ( 0 ) 1 = γ a d a e γ c d c e ,
n I , d = 1 + 6.0179 9.5387 8.7 15.5 = 1.518.
Δ n d = n d χ χ = 0 Δ χ and Δ χ = 1 .
Δ n d = Δ V V 0 [ Δ R d Δ V ( n d 1 ) ] .
Δ n d = ( n I , d n d ) Δ V V 0 .
Δ n n I , d n d 15 ( γ c d c e ) N e .
Δ n d = 0.001 15 ( 8.7 15.5 ) 0.1 = 4.5 × 10 5 .
n d ( χ ) = 1 + R d + Δ R d χ V 0 = n d ( 0 ) + Δ R d V 0 χ , Δ n d = n d ( 1 ) n d ( 0 ) = Δ R d V 0 = ( γ a d a e ) N e V 0 .
a M ( χ ) = d M ( 1 g M 1 λ 2 4.8 × 10 5 λ 2 ) ,
n ( λ ) = 1 + a M ( λ ) N M V ,
n F n C = n F C = N m [ a M ( λ F = 0.48613 μ m ) a M ( λ C = 0.65627 μ m ) ] V ,
n F C = a M F C N M V = R F C V .
a M ( χ ) = d M o ( 1 g M o 1 λ 2 4.8 × 10 5 λ 2 ) + d M ( 1 g M 1 λ 2 ) N M .
a Pb ( λ ) = d Pb o ( 1 g Pb o 1 λ 2 4.8 × 10 5 λ 2 ) + d Pb ( 1 g Pb 1 λ 2 ) N Pb 2 .
V # = n d 1 n F n C = 34.71.
n F C ( χ ) = a M F C N M ( χ ) V ( χ ) , n F C ( χ ) = R F C + χ Δ R F C V 0 + χ Δ V ,
R F C = a M , F C N M , Δ R F C = N a ( γ a b , F C a a , F C ) , V 0 = k + b Si + N Si + c M N M , Δ V = ( γ c b c a ) N a , γ = valence of ion a valence of ion b .
N A = N A ( 1 u υ ) ,
N C = N A u γ C ,
N E = N A υ γ E .
n ( u , υ ) = 1 + a M ( λ ) N M + a A ( λ ) N A + a C ( λ ) N C + a E ( λ ) N E V 0 + u Δ V C + υ Δ V E .
n d ( u ) = 1 + a M ( λ ) N M + a A ( λ ) N A ( 1 u ) + N A γ C a C ( λ ) + N A α u γ E a E ( λ ) V 0 + u Δ V .
n F C ( χ ) χ | χ = 0 = 0.
n F C ( χ ) χ | χ = 0 = ( V 0 + χ Δ F ) Δ R F C Δ V ( R F C + χ Δ R F C ) ( V 0 + χ Δ V ) 2 | χ = 0 = V 0 Δ R F C V Δ R F C V 0 2 .
n F C ( χ ) χ | χ = 0 = 1 V 0 [ Δ R F C Δ V n F C ( 0 ) ] = 0.
Δ R F C = Δ V n F C ( 0 ) , n F C ( 0 ) = Δ R F C Δ V = ( γ a b F C a a F C ) ( γ c b c a ) = n I , F C ( a , b ) .
Δ n F C = Δ V V ( n I , F C n F C ) or Δ n F C = ( γ c b c a ) V N a ( n I , F C n F C ) .
Δ n F C = Δ R F C V 0 = ( γ a b , F C a a , F C ) N a V 0 .
n ( r , λ ) = n 0 ( λ ) [ 1 + σ ( λ ) χ ( r ) ] ,
χ ( r ) = ρ r 2 .
n ( r , λ ) = n 0 ( λ ) [ 1 + σ ( λ ) ρ r 2 ] = n 0 ( λ ) [ 1 + β ( λ ) r 2 ] ,
n ( 1 , λ ) = n 0 ( λ ) + n 0 ( λ ) β ( λ ) or = n 0 ( λ ) + Δ n ( λ ) .
β ( λ ) = Δ n ( λ ) n 0 ( λ ) β 0 + β 1 ( λ ) Δ λ,
n 0 ( λ ) n 0 ( λ 0 ) + ( n F n C ) ( λ F λ C ) Δ λ, Δ n ( λ ) Δ n ( λ 0 ) + Δ ( n F n C ) ( λ F λ C ) Δ λ .
n 0 ( λ ) n d + n F n C ( λ F λ C ) Δ λ, Δ n ( λ ) Δ n d + Δ ( n F n C ) ( λ F λ C ) Δ λ,
β ( λ ) Δ n d + Δ ( n F n C ) λ F n C Δ λ n d + ( n F n C ) ( λ F λ C ) Δ λ , β ( λ ) Δ n d n 1 ι + Δ λ n d ( λ F λ C ) [ Δ ( n F n C ) Δ n d n d ( n F n c ) ] Δ n F C n F C n d 2 ( λ F λ C ) 2 Δ λ 2 .
Δ ( n F n C ) = Δ n d n d ( n F n C ) , or = Δ n d V # ( 1 1 / n d ) .
Δ ( n F n C ) = N ex V 0 [ γ a i , F C a ex , F C n F C ( γ c i c ex ) ] , Δ n d = N ex V 0 [ ( γ c i c ex ) ( n d 1 ) + ( γ a i a ex ) ] .
n d n F n C = ( γ a i a ex ) + ( γ c i c ex ) ( γ a i , F C a ex , F C ) = S T ,
Δ ( n F n C ) = N ex V 0 [ ( γ 1 y 1 a d 1 + γ 2 y 2 a d 2 ) a ex n F C ( γ 1 y 1 c d 1 + γ 2 y 2 c d 2 c ex ) ] ,
n d n F n C = ( y 1 γ 1 a d 1 + γ 2 y 2 a d 2 a ex ) + ( y 1 γ 1 c d 1 + y 2 γ 2 c d 2 c ex ) ( y 1 γ 1 a d 1 , F C + y 2 γ 2 c d 2 , F C a ex , F C ) ,

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