Abstract

It is now well known that a system of interference fringes photographed on a photographic plate has very interesting properties when used as a diffraction grating. This paper considers two cases. One is when the photographic plate is a plane surface; the aberration properties of this are worked out as a function of wavelength. There are three positions for which the spherical aberration is zero. Of these, one is of little interest as this is simply zero-order position. Another case considered was that of a photographic plate in the form of a concave spherical surface. In this case, there are three positions at which the spherical aberration is zero. The nature of variation of spherical aberration between these zero-aberration positions is presented in the form of curves computed from the theory of these gratings.

© 1971 Optical Society of America

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References

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  1. M. V. R. K. Murty, J. Opt. Soc. Am. 50, 923 (1960).
    [Crossref]
  2. M. V. R. K. Murty, J. Opt. Soc. Am. 52, 768 (1962).
    [Crossref]

1962 (1)

1960 (1)

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Figures (16)

Fig. 1
Fig. 1

Diagram showing the construction of the plane grating.

Fig. 2
Fig. 2

Diagram showing the plane grating with a polychromatic source at A. The image in wavelength λ0 is focused at B. E is the paraxial focus of any wavelength λ and PD is an exactly diffracted ray for the same wavelength.

Fig. 3
Fig. 3

Plot of angular aberration AA′ of plane grating vs λ/λ0 for various values of the aperture variable β(4°, 5°, and 6°). m = 0.

Fig. 4
Fig. 4

Same as Fig. 3 except that m = 0.5. Note the merging of the two zero-aberration positions at λ/λ0 = 0.

Fig. 5
Fig. 5

Same as Fig. 3 except that m = 1.0.

Fig. 6
Fig. 6

Same as Fig. 3 except that m = 2.0. Note again the merging of two zero-aberration positions at λ/λ0 = 1.0. Here the aperture variable is α.

Fig. 7
Fig. 7

Same as Fig. 6 except that m = ∞. Note in Figs. 37 the progressive shift of one zero-aberration position from λ/λ0 = −1.0 to λ/λ0 = 2.0 as m varies from 0 to ∞. The other two zero-aberration positions remain fixed at λ/λ0 = 0 and λ/λ0 = 1.0 for all values of m.

Fig. 8
Fig. 8

Diagram showing the construction of a spherical grating. The points A and B are inverse points with respect to the center of the sphere.

Fig. 9
Fig. 9

Diagram showing the spherical grating with a polychromatic source at A. The image in wavelength λ0 is focused at B. E is the paraxial focus of wavelength λ and PD is the exactly diffracted ray for the same wavelength.

Fig. 10
Fig. 10

Plots of angular aberration AA′ of concave spherical grating vs λ/λ0 for different values of the aperture variable ψ (4°, 5°, and 6°). The value of m = 0. Note that two zero-aberration positions have merged at λ/λ0 = 1.0.

Fig. 11
Fig. 11

Same as Fig. 10 except that m = 0.5. Note the small amount of aberration between the zero positions.

Fig. 12
Fig. 12

Same as Fig. 10 except that m = 1.0. Note again the merging of two zero-aberration positions at λ/λ0 = 1.0.

Fig. 13
Fig. 13

Same as Fig. 10 except that m = 2.0. Note the small amount of aberration between the zero positions.

Fig. 14
Fig. 14

Same as Fig. 10 except that m = ∞. Note again the merging of the two zero-aberration positions at λ/λ0 = 0. Note also that one zero-aberration position is constant at λ/λ0 = 1.0 for all values of m. The other two zero positions are variable as can be seen from Figs. 1014.

Fig. 15
Fig. 15

Diagram showing the construction of the spherical grating with the two coherent sources at O and A.

Fig. 16
Fig. 16

Diagram showing the construction of the spherical grating with the two coherent sources at O and B.

Tables (3)

Tables Icon

Table I Aberration-free wavelengths imaged at O, A, and B when a polychromatic source is located at O, A, and B. The grating is constructed with wavelength λ0 and the two coherent sources are located at A and B.

Tables Icon

Table II Aberration-free wavelengths imaged at O, A, and B when a polychromatic source is located at O, A, and B. The grating is constructed with wavelength λ0 and the two coherent sources are located at O and A.

Tables Icon

Table III Aberration-free wavelengths imaged at O, A, and B when a polychromatic source is located at O, A, and B. The grating is constructed with wavelength λ0 and the two coherent sources are located at O and B.

Equations (38)

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tan α = m tan β .
sin α + sin β = λ 0 / σ ,
1 / σ = λ 0 - 1 [ sin β + m tan β ( 1 + m 2 tan 2 β ) - 1 2 ] .
sin α + sin ϕ = λ / σ .
sin ϕ = ( λ / λ 0 ) sin β + [ ( λ / λ 0 ) - 1 ] m tan β ( 1 + m 2 tan 2 β ) - 1 2 .
ϕ = ( λ / λ 0 ) β + [ ( λ / λ 0 ) - 1 ] m β ,
1 / l E = [ ( m + 1 ) λ / λ 0 - m ] ( 1 / l B ) .
tan ϕ 0 = [ ( m + 1 ) λ / λ 0 - m ] tan β .
A A = ( ϕ 0 - ϕ ) .
sin ϕ = [ ( λ / λ 0 ) - 1 ] sin α + ( λ / λ 0 ) k tan α ( 1 + k 2 tan 2 α ) - 1 2 , tan ϕ 0 = [ ( 1 + k ) λ / λ 0 - 1 ] tan α ,
or             λ / λ 0 = ( 2 m - 1 ) / ( m + 1 ) λ / λ 0 = ( 2 - k ) / ( 1 + k ) .
sin ϕ = ( 2 m - 1 m + 1 ) tan β ( 1 + tan 2 β ) - 1 2 + ( m - 2 m + 1 ) m tan β ( 1 + m 2 tan 2 β ) - 1 2
sin ϕ 0 = ( m - 1 ) tan β [ 1 + ( m - 1 ) 2 tan 2 β ] - 1 2 .
sin ϕ 0 = ( m - 1 ) tan β - ( m - 1 ) 3 2 tan 3 β + 3 8 ( m - 1 ) 5 tan 5 β
sin ϕ = ( m - 1 ) tan β - ( m - 1 ) 3 2 tan 3 β + 3 8 ( m 5 - 3 m 4 + 3 m 3 - 3 m 2 + 3 m - 1 ) tan 5 β + .
sin ϕ 0 = ( 1 - k ) tan α - ( 1 - k ) 3 2 tan 3 α + 3 8 ( 1 - k ) 5 tan 5 α sin ϕ = ( 1 - k ) tan α - ( 1 - k ) 3 2 tan 3 α + 3 8 ( 1 - 3 k + 3 k 2 - 3 k 3 + 3 k 4 - k 5 ) tan 5 α + .
and             sin α + sin β = λ 0 / σ sin β + sin ϕ = λ / σ ,
( sin α / sin β ) = constant = m ( say ) .
1 / σ = ( m + 1 ) sin β / λ 0
sin ϕ = [ ( m + 1 ) λ / λ 0 - 1 ] sin β .
L = ( r sin ϕ ) / sin ( α + β - ϕ ) .
l = r ϕ / ( α + β - ϕ ) .
α = m β
ϕ = [ ( m + 1 ) λ / λ 0 - 1 ] β .
l / r = [ ( m + 1 ) λ / λ 0 - 1 ] / { ( m + 1 ) - [ ( m + 1 ) λ / λ 0 - 1 ] } = A .
l / r = sin ϕ 0 / sin ( α + β - ϕ 0 ) .
tan ϕ 0 = A sin ( α + β ) / [ 1 + A cos ( α + β ) ] .
sin α + sin β = λ 0 / σ
sin β = [ λ 0 / ( m + 1 ) ] ( 1 / σ )
2 sin β = [ 2 λ 0 / ( m + 1 ) ] ( 1 / σ ) .
sin α = [ m λ 0 / ( m + 1 ) ] ( 1 / σ )
2 sin α = [ 2 m λ 0 / ( m + 1 ) ] ( 1 / σ ) .
sin β = λ 0 / σ .
sin α = m λ 0 / σ .
sin α + sin β = ( m + 1 ) λ 0 / σ .
sin α = λ 0 / σ .
sin β = ( λ 0 / m ) ( 1 / σ ) ;
sin α + sin β = { [ ( m + 1 ) λ 0 ] / m } ( 1 / σ ) .