Abstract

In gradient-index glass the variation of glass composition can also cause a variation in the thermal properties across the gradient region. The difference in thermal expansion and dn/dT across the gradient-index region has been modeled. Several examples illustrate how the models can be utilized to select gradient-index glass compositions that will have constant thermal properties within the gradient region. Experimental measurement of the variations of αL and dn/dT in gradient-index glass is presented in a separate paper.

© 1985 Optical Society of America

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References

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  1. M. L. Huggins, “The Density of Silicate Glasses as a Function of Composition,” J. Opt. Soc. Am. 30, 420 (1940).
    [CrossRef]
  2. A. A. Appen, “Versuch zur Klassifizierung von Komponenten nach ihrem Einfluβ auf die Oberflächenspannung von Silikatschmelzen,” Silikattechnik 5, 113 (1954). Also, H. Scholze, Glas Natur Strukur, und Eigenschaften (Springer-Verlag, Berlin, 1977), pp. 148–149.
  3. K. Takahashi, “Thermal Expansion Coefficients and the Structure of Glass. Part I,” J. Soc. Glass Technol. 37, 3N (1953).
  4. 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]
  5. T. Baak, “Thermal Coefficient of Refractive Index,” J. Opt. Soc. Am. 59, 851 (1969). Through personal communications with Baak it was confirmed that the values for ki and γi reported in Table IV of his paper are in error by a constant multiplicative term, n/(n − 1). Table VII (this paper) shows that good agreement between the model and catalog data is obtained when the recalculated values for ki and γi from Table VI of this paper are used.
    [CrossRef]
  6. G. N. Ramachandran, “Thermo-Optic Behavior of Solids. VI. Optical Glasses,” Proc. Indian Acad. Sci. Sect. A 25, 498 (1947).
  7. S. D. Fantone, “Refractive Index and Spectral Models for Gradient-Index Materials,” Appl. Opt. 22, 432 (1983).
    [CrossRef] [PubMed]
  8. 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).

1983 (1)

1969 (1)

1954 (1)

A. A. Appen, “Versuch zur Klassifizierung von Komponenten nach ihrem Einfluβ auf die Oberflächenspannung von Silikatschmelzen,” Silikattechnik 5, 113 (1954). Also, H. Scholze, Glas Natur Strukur, und Eigenschaften (Springer-Verlag, Berlin, 1977), pp. 148–149.

1953 (1)

K. Takahashi, “Thermal Expansion Coefficients and the Structure of Glass. Part I,” J. Soc. Glass Technol. 37, 3N (1953).

1947 (1)

G. N. Ramachandran, “Thermo-Optic Behavior of Solids. VI. Optical Glasses,” Proc. Indian Acad. Sci. Sect. A 25, 498 (1947).

1942 (1)

1940 (1)

Appen, A. A.

A. A. Appen, “Versuch zur Klassifizierung von Komponenten nach ihrem Einfluβ auf die Oberflächenspannung von Silikatschmelzen,” Silikattechnik 5, 113 (1954). Also, H. Scholze, Glas Natur Strukur, und Eigenschaften (Springer-Verlag, Berlin, 1977), pp. 148–149.

Baak, T.

Davis, D. O.

Fantone, S. D.

S. D. Fantone, “Refractive Index and Spectral Models for Gradient-Index Materials,” Appl. Opt. 22, 432 (1983).
[CrossRef] [PubMed]

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.

Ramachandran, G. N.

G. N. Ramachandran, “Thermo-Optic Behavior of Solids. VI. Optical Glasses,” Proc. Indian Acad. Sci. Sect. A 25, 498 (1947).

Sun, K. H.

Takahashi, K.

K. Takahashi, “Thermal Expansion Coefficients and the Structure of Glass. Part I,” J. Soc. Glass Technol. 37, 3N (1953).

Appl. Opt. (1)

J. Opt. Soc. Am. (3)

J. Soc. Glass Technol. (1)

K. Takahashi, “Thermal Expansion Coefficients and the Structure of Glass. Part I,” J. Soc. Glass Technol. 37, 3N (1953).

Proc. Indian Acad. Sci. Sect. A (1)

G. N. Ramachandran, “Thermo-Optic Behavior of Solids. VI. Optical Glasses,” Proc. Indian Acad. Sci. Sect. A 25, 498 (1947).

Silikattechnik (1)

A. A. Appen, “Versuch zur Klassifizierung von Komponenten nach ihrem Einfluβ auf die Oberflächenspannung von Silikatschmelzen,” Silikattechnik 5, 113 (1954). Also, H. Scholze, Glas Natur Strukur, und Eigenschaften (Springer-Verlag, Berlin, 1977), pp. 148–149.

Other (1)

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).

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Figures (2)

Fig. 1
Fig. 1

Thermal expansion factors vs D2/2Z.

Fig. 2
Fig. 2

Additive factors vs D2/2Z at λC.

Tables (9)

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Table I Partial Linear Expansion Factors According to Appen

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Table II Partial Linear Expansion Factors According to Appen

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Table III Calculated and Experimental Thermal Expansion Coefficients for Some Commercially Available Homogeneous Glasses

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Table IV Calculated Thermal Expansion Coefficients for Gradient-Index Glass

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Table V Calculated Thermal Expansion Coefficients for Gradient-Index Glass

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Table VI Additive Factors for Calculating dn/dT

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Table VII Comparison of Calculated and Experimental Values of dn/dT at Two Wavelengths

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Table VIII Calculated Temperature Coefficients of Refractive Index for Gradient-Index Glass

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Table IX Calculated Temperature Coefficients of Refractive Index for Gradient-Index Glass

Equations (14)

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α L = i α i p i ,
α L ( x , y , z ) = i α i p i ( x , y , z ) ,
p ex = p ex ( 1 u υ ) , p b = p ex u , p c = p ex υ , 0 u 1 0 υ 1 0 u + υ 1 , }
α L ( u , υ ) = i α i p i + [ α ex ( 1 u υ ) + α b u + α c υ ] p ex i ex ,
α L ( u , υ ) = α L ( 0,0 ) + [ ( α b α ex ) u + ( α c α ex ) υ ] p ex ,
Case 1 u 0 , α c = α ex or υ 0 , α b = α ex ,
Case 2 u υ = ( α c α ex ) ( α b α ex ) = constant .
n ( λ , T , x , y , z ) = n ( λ , T 0 , x , y , z ) + d n ( λ , T , x , y , z ) d T Δ T ,
d n ( λ ) d T = [ n ( λ ) 1 ] i p i ( k i λ 2 + γ i ) ,
n ( λ ) = 1 + R ( λ ) V = 1 + i R i ( λ ) i V i ,
d n ( λ , x , y , z ) d T = [ n ( λ , x , y , z ) 1 ] i p i ( x , y , z ) ( k i λ 2 + γ i ) .
d n ( λ , u , υ ) d T = [ n ( λ , u , υ ) 1 ] [ i p i ( k i λ 2 + γ i ) ] + u p ex ( k b k ex λ 2 + γ b γ ex ) + υ p ex ( k c k ex λ 2 + γ c γ ex ) ] ,
d n ( λ , u , υ ) d T = n ( λ , u , υ ) 1 n ( λ ,0,0 ) 1 { d n ( λ ,0,0 ) d T + [ n ( λ ,0,0 ) 1 ] p ex [ u ( k b k ex λ 2 + γ b γ ex ) + υ ( k c k ex λ 2 + γ b γ ex ) ] } ,
n ( λ , u , υ ) 1 n ( λ ,0,0 ) 1 ,

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