Abstract

The properties of a plane grating when it is illuminated by a convergent or divergent wave front are investigated. The investigation shows that the image principally suffers an unsymmetric aberration of coma type in the direction of dispersion. Remedies have been indicated to correct this aberration.

© 1962 Optical Society of America

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References

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  1. H. E. Ives, J. Opt. Soc. Am. 1, 172 (1917).
    [Crossref]
  2. G. S. Monk, J. Opt. Soc. Am. 17, 358 (1928).
    [Crossref]
  3. H. T. Smyth, J. Opt. Soc. Am. 25, 312 (1935).
    [Crossref]
  4. A. H. C. P. Gillieson, J. Sci. Instr. 26, 335 (1949).
    [Crossref]
  5. E. W. T. Richards, A. R. Thomas, and W. Weinstein, Reports A.E.R.E. c/r 2152 and c/r 2163 (1957).
  6. G. R. Rosendahl, “Contributions to the optics of mirror systems and gratings with oblique incidence,” Ball Brothers Research Rept., April20, 1959.
  7. H. G. Beutler, J. Opt. Soc. Am. 35, 311 (1945).
    [Crossref]
  8. T. Namioka, J. Opt. Soc. Am. 49, 446 (1959).
    [Crossref]
  9. J. P. C. Southall, The Principles and Methods of Geometrical Optics (The Macmillan Company, New York, 1910), Chap. IV.
  10. H. Yoshinaga, B. Okazaki, and S. Tatsuoka, J. Opt. Soc. Am. 50, 437 (1960).
    [Crossref]
  11. G. R. Rosendahl, J. Opt. Soc. Am. 51, 1 (1961).
    [Crossref]
  12. G. H. Spencer and M. V. R. K. Murty, J. Opt. Soc. Am. 52, 672 (1962).
    [Crossref]

1962 (1)

1961 (1)

1960 (1)

1959 (1)

1949 (1)

A. H. C. P. Gillieson, J. Sci. Instr. 26, 335 (1949).
[Crossref]

1945 (1)

1935 (1)

1928 (1)

1917 (1)

Beutler, H. G.

Gillieson, A. H. C. P.

A. H. C. P. Gillieson, J. Sci. Instr. 26, 335 (1949).
[Crossref]

Ives, H. E.

Monk, G. S.

Murty, M. V. R. K.

Namioka, T.

Okazaki, B.

Richards, E. W. T.

E. W. T. Richards, A. R. Thomas, and W. Weinstein, Reports A.E.R.E. c/r 2152 and c/r 2163 (1957).

Rosendahl, G. R.

G. R. Rosendahl, J. Opt. Soc. Am. 51, 1 (1961).
[Crossref]

G. R. Rosendahl, “Contributions to the optics of mirror systems and gratings with oblique incidence,” Ball Brothers Research Rept., April20, 1959.

Smyth, H. T.

Southall, J. P. C.

J. P. C. Southall, The Principles and Methods of Geometrical Optics (The Macmillan Company, New York, 1910), Chap. IV.

Spencer, G. H.

Tatsuoka, S.

Thomas, A. R.

E. W. T. Richards, A. R. Thomas, and W. Weinstein, Reports A.E.R.E. c/r 2152 and c/r 2163 (1957).

Weinstein, W.

E. W. T. Richards, A. R. Thomas, and W. Weinstein, Reports A.E.R.E. c/r 2152 and c/r 2163 (1957).

Yoshinaga, H.

J. Opt. Soc. Am. (8)

J. Sci. Instr. (1)

A. H. C. P. Gillieson, J. Sci. Instr. 26, 335 (1949).
[Crossref]

Other (3)

E. W. T. Richards, A. R. Thomas, and W. Weinstein, Reports A.E.R.E. c/r 2152 and c/r 2163 (1957).

G. R. Rosendahl, “Contributions to the optics of mirror systems and gratings with oblique incidence,” Ball Brothers Research Rept., April20, 1959.

J. P. C. Southall, The Principles and Methods of Geometrical Optics (The Macmillan Company, New York, 1910), Chap. IV.

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Figures (11)

Fig. 1
Fig. 1

Diagram showing the loci of tangential foci for angles of incidence 0°, 15°, 30°, and 45° for a convergent beam incident on a plane grating. The curve marked SF is the locus of sagittal foci for all angles of incidence.

Fig. 2
Fig. 2

Diagram showing the variables for derivation of the optical-path function F from A to B via P on the plane grating.

Fig. 3
Fig. 3

Diagram used for showing that hyperbolic rulings on a plane are needed to obtain an aberration-free image of the point A into A.

Fig. 4
Fig. 4

Schematic diagram showing how the hyperbolic rulings on the grating could be obtained by photographing interference pattern of two co-coherent monochromatic point sources.

Fig. 5
Fig. 5

Schematic diagram of an objective spectrometer using a plane grating in the convergent beam. M-paraboloidal mirror. G-plane grating with hyperbolic rulings. S-spectrum.

Fig. 6
Fig. 6

Schematic diagram of a monochromator using a plane grating in convergent light. M-ellipsoidal mirror with foci at S1 and S2. G-plane grating with hyperbolic rulings. S1-entrance slit. S2-exit slit. S2-second focus of the mirror towards which the convergent beam is directed.

Fig. 7
Fig. 7

Schematic diagram of an objective spectrometer in which a plane grating of conventional type is used in the convergent light. In order to correct the coma of the plane grating, the paraboloidal mirror is used off-axis. M-paraboloidal mirror. G-plane grating of conventional type. S-spectrum.

Fig. 8
Fig. 8

Schematic diagram of a monochromator in which a plane grating of conventional type is used in the convergent light. In order to correct the coma of the plane grating, the ellipsoidal mirror is used off-axis. M-ellipsoidal mirror. G-plane grating of conventional type. S1-entrance slit. S2-exit slit. S2-point towards which the convergent beam is directed.

Fig. 9
Fig. 9

Schematic diagram showing the path of meridional rays of a convergent beam after refraction through a thin prism.

Fig. 10
Fig. 10

Schematic diagram of an objective spectromator in which the plane grating and a thin prism are used in convergent light to correct for coma. M-paraboloidal mirror. G-plane grating of conventional type. P-thin prism. S-spectrum.

Fig. 11
Fig. 11

Schematic diagram of a monochromator in which the plane grating and a thin prism are used in convergent light to correct for coma. M-ellipsoidal mirror with foci at S1 and S2. G-plane grating of conventional type. P-thin prism. S1-entrance slit. S2-exit slit. S2-second focus of the mirror towards which the convergent beam is directed.

Equations (35)

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( cos 2 α r cos α R ) + ( cos 2 β r t cos β R ) = 0
( 1 r cos α R ) + ( 1 r s cos β R ) = 0 ,
( cos 2 α / r ) + ( cos 2 β / r t ) = 0 ,
( 1 / r ) + ( 1 / r s ) = 0.
F = AP + BP + mw λ / σ ,
F = r + r w ( sin α + sin β ) + m w λ σ + w 2 2 ( cos 2 α r + cos 2 β r ) + l 2 2 ( 1 r + 1 r ) + w 3 2 ( sin α cos 2 α r 2 + sin β cos 2 β r 2 ) + w l 2 2 ( sin α r 2 + sin β r 2 ) + higher terms .
sin α + sin β = m λ / σ , ( cos 2 α / r ) + ( cos 2 β / r ) = 0 , ( 1 / r ) + ( 1 / r ) = 0.
( W 3 / 4 r 2 ) sin α cos 2 α 1 4 λ .
( W 3 / 2 r 2 ) cos 2 α σ ,
W / 2 ( f number ) 2 σ .
x 2 ( 1 2 n λ ) 2 y 2 + z 2 r 2 ( 1 2 n λ ) 2 = 1.
y sin α + x cos α = 0.
x = x cos α y sin α , y = y cos α + x sin α .
w 2 [ r 2 sin 2 α ( 1 2 n λ ) 2 ( 1 2 n λ ) 2 { r 2 ( 1 2 n λ ) 2 } ] l 2 { r 2 ( 1 2 n λ ) 2 } = 1.
w 2 = ( 1 2 n λ ) 2 [ r 2 ( 1 2 n λ ) 2 r 2 sin 2 α ( 1 2 n λ ) 2 ] .
w = n σ 0 [ ( 1 n 2 σ 0 2 sin 2 α r 2 ) 1 2 ( 1 n 2 σ 0 2 r 2 ) 1 2 ] .
w = n σ 0 [ 1 + n 2 σ 0 2 cos 2 α / 2 r 2 ]
σ = σ 0 [ 1 + 3 2 ( n 2 σ 0 2 cos 2 α / r 2 ) ] .
σ = σ 0 [ 1 + 3 2 ( w 2 cos 2 α / r 2 ) ] ,
( z 2 / 4 f ) tan θ = l 2 tan α / r .
tan θ = 2 tan α .
M 0 = ( 1 + e ) / ( 1 e ) ,
M = ( a + e x ) / ( a e x ) ,
x = a ( 1 z 2 / b 2 ) 1 2 ,
( M 0 M ) = [ e / ( 1 e ) 4 a 2 ] z 2
e z 2 H / ( 1 e ) 4 a 2 = l 2 tan α / r .
H = ( 2 / 9 ) a tan α .
coma T = y 2 sin δ / ρ
coma S = z 2 sin δ / 3 ρ .
2 z 2 sin δ / 3 ρ = l 2 tan α / r .
sin δ = ( 3 2 ) tan α .
A = [ 3 / 2 ( μ 1 ) ] tan α .
OPD = ( L 3 / 4 r 2 ) ( tan α tan α * ) ,
sin α sin α * = ( λ λ * ) / 2 σ
OPD = L 3 8 r 2 ( λ λ * σ )