Abstract

The authors suggest that the optical qualities of a transparent material are given with the help of three graphs, one a plot of the reciprocal dispersion 1/(nFnC) against the ν-value, the other two giving two data ρA and ρh signifying the deviation of the dispersion of the glass toward the red and violet ends of the spectrum from the regular standard. A universal dispersion formula is given which permits one (a) to calculate the refractive index for any given wave-length if the above data are given; and (b) to compute the above data if the refractive index is given for four or more values.

The dispersion formula is linear with respect to the given data.

The graphs so designed permit one to estimate immediately the powers of a doublet corrected for two wave-lengths and the amount of the secondary spectrum in the red and violet ends for any two given glasses.

Testing these formulas for different types of glass and other optically important materials (fluorite, quartz, lithium fluoride, salt, etc.) shows an agreement with the measurements throughout the spectrum which is sufficient for optical calculation (±2×10−5).

© 1949 Optical Society of America

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References

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  1. M. Herzberger, “The dispersion of optical glass,” J. Opt. Soc. Am. 32, 70–77 (1942).
    [Crossref]

1942 (1)

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Figures (4)

G I
G I

Reciprocal dispersion plotted against ν-value. The inclined lines designate the index of refraction.

G II
G II

ν-value plotted against residual partial dispersion for red end of spectrum. Inclined lines designate partial dispersion.

G III
G III

ν-value plotted against residual partial dispersion for violet end of spectrum. Inclined lines designate partial dispersion.

F. 1
F. 1

Graphic illustration to show how to compute the powers of an achromatic doublet for two glasses chosen from Graph I.

Tables (4)

Equations (38)

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μ = μ 0 + μ 1 λ 2 + μ 2 / ( λ 2 0.035 ) + μ 3 / ( λ 2 0.035 ) 2 .
K 1 = ( 1 / r 2 1 / r 1 ) 1 K 2 = ( 1 / r 2 1 / r 1 ) 2
Φ = ( n 1 1 ) K 1 + ( n 1 1 ) K 2 = μ 1 K 1 + μ 2 K 2 .
Φ D = μ 1 D K 1 + μ 2 D K 2 Φ F Φ C = ( μ 1 F μ 1 C ) K 1 + ( μ 2 F μ 2 C ) K 2 Φ λ Φ D = ( μ 1 λ μ 1 D ) K 1 + ( μ 2 λ μ 2 D ) K 2 ,
ν = μ D / ( μ F μ C ) ,
P λ = ( μ λ μ D ) / ( μ F μ C ) .
Φ D = μ 1 D K 1 + μ 2 D K 2 Φ F Φ C = ( μ 1 D K 1 / ν 1 ) + ( μ 2 D K 2 / ν 2 ) Φ λ Φ D = ( μ 1 D K 1 / ν 1 ) P 1 λ + ( μ 2 D K 2 / ν 2 ) P 2 λ .
| Φ D 1 1 Φ F Φ C 1 / ν 1 1 / ν 2 Φ λ Φ D P 1 / ν 1 P 2 / ν 2 | = 0
Φ D ( P 2 P 1 ) ( Φ F Φ C ) ( P 2 ν 1 P 1 ν 2 ) + ( Φ λ Φ D ) ( ν 1 ν 2 ) = 0 .
Φ λ Φ D = ( P 2 P 1 / ν 2 ν 1 ) Φ D .
P 2 = P 1 .
Φ D = μ 1 D K 1 + μ 2 D K 2 0 = ( μ 1 D K 1 / ν 1 ) + ( μ 1 D K 2 / ν 2 )
K 1 = Φ D ν 1 / μ 1 ν 1 ν 2 = ( 1 / n F n C ) 1 ν 1 ν 2 Φ D = N 1 ν 1 ν 2 Φ D K 2 = Φ D ν 2 / μ 2 ν 2 ν 1 = ( 1 / n F n C ) 2 ν 2 ν 1 Φ D = N 2 ν 2 ν 1 Φ D ,
P λ = A 1 ν + A 2 ,
P λ = A 1 ν + A 2 + ρ λ ,
ρ λ = A 3 ρ A + A 4 ρ h ,
P λ = A 1 ν + A 2 + A 3 ρ A + A 4 ρ h
P A = A 1 A ν + A 2 A + ρ A P h = A 1 h ν + A 2 h + ρ h .
N μ λ = [ ( A 1 + 1 ) ν D + A 2 + A 3 ρ A + A 4 ρ h ] ,
A i = α 0 + α 1 λ 2 + α 2 / ( λ 2 0.035 ) + α 3 / ( λ 2 0.035 ) 2 ,
μ λ μ F = B 1 ( μ C μ F ) + B 2 ( μ A μ F ) + B 3 ( μ h μ F )
μ λ = B 1 μ c + B 2 μ A + B 3 μ h + B 4 μ F
B 1 + B 2 + B 3 + B 4 = 1 .
μ λ = C 1 μ D + C 2 ( μ F μ C ) + C 3 μ A + C 4 μ h ,
μ D = B 1 D μ c + B 2 D μ A + B 3 D μ h + B 4 D μ F μ F μ C = μ C + μ F μ A = μ A μ h = μ h .
μ λ = ( C 1 B 1 D C 2 ) μ c + ( C 1 B 2 D + C 3 ) μ A + ( C 1 B 3 D + C 4 ) μ h + ( C 1 B 4 D + C 2 ) μ F .
B 1 = C 1 B 1 D C 2 B 2 = C 1 B 2 D + C 3 B 3 = C 1 B 3 D + C 4 B 4 = C 1 B 4 D + C 2
C 1 = ( B 1 + B 4 ) / ( B 1 D + B 4 D ) C 2 = B 1 + C 1 B 1 D C 3 = B 2 C 1 B 2 D C 4 = B 3 C 1 B 3 D ,
μ λ = ( A 1 + 1 ) μ D + A 2 ( μ F μ C ) + A 3 r A + A 3 r A + A 4 r h ,
r A = ρ A ( μ F μ C ) r h = ρ h ( μ F μ C ) ,
μ A = ( A 1 A + 1 ) μ D + A 2 A ( μ F μ C ) + r A μ h = ( A 1 h + 1 ) μ D + A 2 h ( μ F μ C ) + r h .
A 1 = C 1 + C 3 ( A 1 A + 1 ) + C 4 ( A 1 h + 1 ) 1 A 2 = C 2 + C 3 A 2 A + C 4 A 2 h A 3 = C 3 A 4 = C 4 .
μ λ = ( A 1 + 1 ) μ D + A 2 ( μ F μ C ) .
N μ λ = ( A 1 + 1 ) ν + A 2 + A 3 ρ A + A 4 ρ h ,
tan α = N 2 / ( ν 2 ν 1 ) tan β = N 1 / ( ν 1 ν 2 ) .
K 1 = Φ tan β , K 2 = Φ tan α ,
0 = P 1 P 2 = A 1 ( ν 1 ν 2 ) + ( ρ 1 ρ 2 ) ,
A 1 = 0.001746 , for λ = λ A A 1 = 0.004568 , for λ = λ h .