Abstract

The optical properties of plane-grating monochromators such as the Ebert and Czerny–Turner types are examined by geometrical optics. The grating equation and the expression for the focal surface are given; by using them, expressions are found for the image form; the optical system that minimizes wavelength impurity is discussed; and such residual aberrations as coma and astigmatism are examined collectively. In any monochromator, the focal surface depends upon the grating position and the off-axial properties of the collimator. In an Ebert-type monochromator with a circular-arc-shaped entrance slit the image form is a circular arc only in a symmetric arrangement; it is elliptic in other arrangements; the light through a circular-arc-shaped exit slit is free of wavelength error; and even if the slit is long, the image will not be much blurred by residual aberrations. Therefore, it is advisable to use long circular-arc slits if the system is of the Ebert type. In the Ebert- or Czerny–Turner-type monochromator with a straight entrance slit the image is parabolic. In the Czerny–Turner type both the image form and focal surface are affected by its nonconcentric character.

© 1965 Optical Society of America

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References

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1964 (2)

1963 (1)

1961 (2)

1960 (1)

K. Kudo, Sci. Light 9, 1 (1960).

1959 (2)

1957 (1)

M. A. Ford and et al., J. Sci. Instr. 35, 55 (1957).
[CrossRef]

1956 (1)

W. Leo, Z. Angew. Phys. 8, 196 (1956).

1952 (1)

1945 (1)

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Figures (11)

Fig. 1
Fig. 1

Various kinds of plane-grating monochromators with spherical mirrors as collimators. (a) Ebert type, (b) Czerny–Turner type.

Fig. 2
Fig. 2

Coordinates of optical elements in two-mirror system.

Fig. 3
Fig. 3

Schematic diagram showing the angular relation of the incident and diffracted central rays.

Fig. 4
Fig. 4

Angular distribution of isochromatic central rays (A′,A″,A‴,⋯) diffracted by a grating when the central rays (A) that are incident to the grating center are along a generating line of a conical surface with its vertex on the grating. d=10 μ, n=1, γ=10°. For (a) θ8μ=23.965°, (b) θ7μ=20.818°, (c) θ6μ=17.736°.

Fig. 5
Fig. 5

Angular distribution of isochromatic central rays (A′,A″,A‴,⋯) diffracted by a grating when the central rays (A) that are incident on the grating center are in a vertical plane. d=10 u, n=1, γ=10°. For (a) θ8μ=23.965°, (b) θ7μ=20.818°, (c) θ6μ=17.736°.

Fig. 6
Fig. 6

Angular distribution of isochromatic central rays (A′,A″,A‴,⋯) diffracted by a grating when the central rays (A) that are incident on the grating center are in y=0 plane. d=10 μ, n=1, γ=0°, φ20=10°. For (a) θ8μ=23.965°, (b) θ7μ=20.818°, (c) θ6μ=17.736°.

Fig. 7
Fig. 7

Schematic diagram showing the image formation and the image form in the Ebert monochromator in symmetric arrangement with a circular-arc-shaped entrance slit.

Fig. 8
Fig. 8

Another schematic diagram of image formation of the Ebert monochromator with a circular-arc-shaped entrance slit. S1, object point; S1′, image point; M1, M2, portions of the collimating mirror; C1, collimator; E, Ebert circle; C2, sub-Ebert circle.

Fig. 9
Fig. 9

Arrangement of a modified Ebert type.

Fig. 10
Fig. 10

Schematic diagram showing the image formation and the image form in Ebert horizontal-type monochromator in symmetric arrangement with a straight entrance slit.

Fig. 11
Fig. 11

Calculated values of total effect of the off-axial aberrations such as astigmatism and coma. (a) Ebert type with circular arc slit, (b) horizontal Ebert type with straight slit, (c) vertical Ebert type with straight slit. In (a) and (b): R=100 cm, W=5 cm, L0=4 cm, h0=50 cm, y10=10 cm. In (c) R, W, L0 same as in (a) and (b), h 0 = R / 3, y10=0, φ20=l0°.

Equations (101)

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s 1 = x 1 - x 0 s 2 = y 1 - y 0 z 1 = z 1 } ,             x = x 2 - x 0 y = y 2 - y 0 z = z 2 } , G 1 = ξ - x 0 = h 0 - ω sin θ G 2 = η - y 0 = - y 0 + w cos θ         l = l } ,
h 0 = c - x 0 .
L 1 = x R - 1 M 1 = y R - 1 N 1 = z R - 1 } ,             λ 1 = ( x - s 1 ) ρ 1 - 1 μ 1 = ( y - s 2 ) ρ 1 - 1 ν 1 = ( z - z 1 ) ρ 1 - 1 } ,             λ 2 = ( G 1 - x ) ρ 2 - 1 μ 2 = ( G 2 - y ) ρ 2 - 1 ν 2 = ( l - z ) ρ 2 - 1 } ,
R 2 = x 2 + y 2 + z 2 ,
ρ 1 2 = R 2 + s 2 - 2 ( s 1 x + s 2 y + z 1 z ) ,
ρ 2 2 = R 2 + g 2 - 2 ( G 1 x + G 2 y + l z ) ,
s 2 = s 1 2 + s 2 2 + z 1 2 ,
g 2 = G 1 2 + G 2 2 + l 2 = g 0 2 + g 1 + g 2 ,
g 0 2 = h 0 2 + y 0 2 g 1 = - 2 ( h 0 sin θ + y 0 cos θ ) w g 2 = w 2 + l 2 } .
λ 1 - λ 2 L = μ 1 - μ 2 M = ν 1 - ν 2 N = 2 ( L λ 1 + M μ 1 + N ν 1 ) .
x = ( G 1 + α s 1 ) a - 1 y = ( G 2 + α s 2 ) a - 1 z = ( l + α z 1 ) a - 1 } ,
a = 1 - ( ρ 1 2 - s 2 ) α R - 2 ,
α = ρ 2 ρ 1 - 1 .
α 2 s 2 + 2 α k 2 + g 2 - R 2 a 2 = 0 ,
k 2 = s 1 G 1 + s 2 G 2 + z 1 l = k 0 2 + k 1 ,
k 0 2 = h 0 s 1 - y 0 s 2 k 1 = ( - s 1 sin θ + s 2 cos θ ) w + z 1 l } .
α = [ R s a ( 1 - m a - 2 ) 1 2 - k 2 ] s - 2 ,
m = ( s 2 g 2 - k 4 ) ( R s ) - 2 .
ρ 1 2 = R 2 + s 2 - 2 R s ( 1 - m a - 2 ) 1 2 ,
ρ 2 2 = R 2 + g 2 - 2 R 2 m a - 1 - 2 R s - 1 k 2 ( 1 - m a - 2 ) 1 2 ,
ρ 2 = [ R s a ( 1 - m a - 2 ) 1 2 - k 2 ] ρ 1 s - 2 .
a = 1 - [ R - 2 s ( 1 - m a - 2 ) 1 2 ] × [ R s ( 1 - m a - 2 ) 1 2 - k 2 ] R - 1 s - 2 .
a = 1 - a .
a = 1 - ( e 0 + e 1 a + e 2 a 2 + ) ,
a = 1 - p q ( 1 + e 10 + e 10 2 + e 10 3 + ) - ( p q ) 2 e 20 ( 1 + 2 e 10 + ) - ( p q ) 3 e 30 + ,
p = ( R s β 1 1 2 - k 2 ) R - 1 s - 1 , q = ( R - 2 s β 1 1 2 ) s - 1 , e 10 = ( 2 p m - q ) β 1 - 1 2 , e 20 = [ 3 p - 2 β 1 - 1 2 + p m β 1 - 1 - 1 2 q β 1 - 1 ] m β 1 - 1 2 , e 30 = [ p ( 4 + 3 m β 1 - 1 + m 2 β 1 - 2 ) - ( 3 - m ) β 1 - 3 2 - 1 2 q β 1 - 2 ] m β 1 - 1 2 ,
β 1 = 1 - m .
ρ 1 + ρ 2 = ψ 0 + ψ 1 + ψ 2 + ψ 3 + , ψ 0 = ρ 10 ( 1 + G 0 s - 2 ) , ψ 1 = ( ψ 10 sin θ + ψ 11 cos θ ) w + ψ 12 l ,
ψ 10 = - h 0 ρ 20 - 1 ( 1 - a 0 - 1 ) + s 1 ρ 10 - 1 a 0 - 1 ,
ψ 11 = - y 0 ρ 20 - 1 ( 1 - a 0 - 1 ) - s 2 ρ 10 - 1 a 0 - 1 ,
ψ 12 = - z 1 ρ 10 - 1 a 0 - 1 ,
ρ 10 2 = s 2 + R Q 0 ρ 20 = ρ 10 G 0 s - 2 } ,
Q 0 = R - 2 s β 0 1 2 ,             G 0 = R s a 0 β 0 1 2 - k 0 2 ,             β 0 = 1 - m 0 a 0 - 2 ,
a 0 = 1 - G 0 Q 0 R - 1 s - 2 = 1 - α 0 Q 0 R - 1 = 1 - p 0 q 0 - ( 2 m 0 p 0 - q 0 ) p 0 q 0 β 10 - 1 2 - , α 0 = ρ 20 ρ 10 - 1 = G 0 s - 2 ,             p 0 = ( R s β 10 1 2 - k 0 2 ) R - 1 s - 1 ,             q 0 = ( R - 2 s β 10 1 2 ) s - 1 , }
β 10 = 1 - m 0 . ψ 2 = [ ( ψ 20 + ψ 200 ) sin 2 θ + 2 ψ 21 sin θ cos θ + ( ψ 22 + ψ 220 ) cos 2 θ ] w 2 + 2 ( ψ 23 sin θ + ψ 24 cos θ ) w l + ψ 25 l 2 ,
ψ 20 = - C 1 [ ( s 2 2 + z 1 2 ) ( m 0 D 1 + R - 1 Q 0 a 0 2 ) + R - 2 Q 0 2 a 0 2 ( h 0 2 - G 0 ) + Δ 20 ] ,
ψ 200 = C 1 ( R - 2 D 1 y 0 2 z 1 2 - 2 R - 1 Q 0 a 0 2 G 0 - 1 s 1 2 s 2 y 0 ) ,
ψ 21 = - C 1 [ s 1 s 2 ( m 0 D 1 + R - 1 Q 0 a 0 2 - s - 2 Q 0 2 a 0 2 ) + R - 2 D 1 h 0 y 0 z 1 2 + R - 1 Q 0 a 0 2 G 0 - 1 s 2 s 1 y 0 + Δ 21 ] ,
ψ 22 = - C 1 [ ( s 1 2 + z 1 2 ) ( m 0 D 1 + R - 1 Q 0 a 0 2 ) + R - 2 Q 0 2 a 0 2 ( R 2 - G 0 ) + Δ 22 ] ,
ψ 220 = C 1 ( R - 2 D 1 h 0 2 z 1 2 - R - 1 Q 0 a 0 2 s 2 y 0 ) ,
ψ 23 = - C 1 z 1 [ s 1 ( m 0 D 1 + R - 1 Q 0 a 0 2 - s - 2 Q 0 2 a 0 2 ) - R - 2 D 1 y 0 ( h 0 s 2 + y 0 s 1 ) + Δ 23 ] ,
ψ 24 = C 1 z 1 [ s 2 ( m 0 D 1 + R - 1 Q 0 a 0 2 - s - 2 Q 0 2 a 0 2 ) - R - 2 D 1 h 0 ( h 0 s 2 + y 0 s 1 ) + R - 1 Q 0 a 0 2 G 0 - 1 s 2 y 0 + Δ 24 ] ,
ψ 25 = - C 1 { R - 1 Q 0 s 2 a 0 2 + Q 0 2 ( 1 - R - 2 G 0 ) + z 1 2 [ ( R - 2 g 0 2 - m 0 ) D 1 - 1 4 R - 1 G 0 Q 0 a 0 2 ( R 2 - G 0 ) + Δ 25 ] } ,
C 1 = 1 2 ρ 10 - 3 a 0 - 4 , D 1 = 1 + 2 m 0 + 4 m 0 2 + m 0 = ( s 2 g 0 2 - k 0 4 ) ( R s ) - 2 = [ h 0 2 ( s 2 2 + z 1 2 ) + 2 h 0 y 0 s 1 s 2 + y 0 2 ( s 1 2 + z 1 2 ) ] ( R s ) - 2 } ,
ψ 3 = ( ψ 30 sin 3 θ + ψ 31 sin 2 θ cos θ + ψ 32 sin θ cos 2 θ + ψ 33 cos 3 θ ) w 3 + ( ψ 34 sin 2 θ + 2 ψ 35 sin θ cos θ + ψ 36 cos 2 θ ) w 2 l + ( ψ 37 sin θ + ψ 38 cos θ ) w l 2 + ψ 39 l 3 ,
ψ 30 = A 1 ρ 20 - 2 ( ψ 20 + ψ 200 ) - 1 4 C 2 B 1 s - 1 ( s 2 2 + z 1 2 ) + Δ 30 ,
ψ 31 = ρ 20 - 2 [ 2 A 1 ψ 21 + A 2 ( ψ 20 + ψ 200 ) ] + ,
ψ 32 = ρ 20 - 2 [ A 1 ( ψ 22 + ψ 220 ) + 2 A 2 ψ 21 ] - 1 4 C 2 s - 1 [ B 1 ( s 1 2 + z 1 2 ) - 4 B 2 s 1 s 2 ] + Δ 32 ,
ψ 33 = A 2 ρ 20 - 2 ( ψ 22 + ψ 220 ) + ,
ψ 34 = ρ 20 - 2 [ 2 A 1 ψ 23 + A 3 ( ψ 20 + ψ 200 ) ] + ,
ψ 35 = ρ 20 - 2 ( A 1 ψ 24 + A 2 ψ 23 + A 3 ψ 21 ) + 1 2 C 2 s - 1 ( B 1 s 2 z 1 + B 2 s 1 z 1 + B 3 s 1 s 2 - B 4 s z 1 ) + Δ 35 ,
ψ 36 = ρ 20 - 2 [ 2 A 2 ψ 24 + A 3 ( ψ 22 + ψ 220 ) ] + ,
ψ 37 = ρ 20 - 2 ( A 1 ψ 25 + 2 A 3 ψ 23 ) - 1 4 C 2 [ B 5 s ( 1 - s - 2 z 1 2 ) + 2 B 6 z 1 a 0 - 3 β 0 - 1 2 × ( R - 2 g 0 2 - m 0 ) - 4 B 3 s - 1 s 1 z 1 ] + Δ 37 ,
ψ 38 = ρ 20 - 2 ( A 2 ψ 25 + 2 A 3 ψ 24 ) + ,
ψ 39 = A 3 ρ 20 - 2 ψ 25 + 1 2 C 2 B 3 s ( 1 - s - 2 z 1 2 ) + ,
λ 1 = λ 2 - 2 L K ,             μ 1 = μ 2 - 2 M K ,             ν 1 = ν 2 - 2 N K ,
K = L λ 2 + M μ 2 + N ν 2 .
K = - ( L λ 1 + M μ 1 + N ν 1 ) .
F = ρ 1 + ρ 2 + ρ 2 + ρ 1 + ( n λ / d ) w ,
F / w = 0 = ( F 1 + F 1 ) + 2 ( F 2 + F 2 ) w + ( F 31 + F 31 ) ,
F / l = 0 = ( ψ 12 + ψ 12 ) + ( F 32 + F 32 ) ,
F 1 = ψ 10 sin θ + ψ 11 cos θ ,
F 2 = ψ 20 sin 2 θ + ψ 22 cos 2 θ ,
F 31 = 2 ( ψ 200 sin 2 θ + 2 ψ 21 sin θ cos θ + ψ 220 cos 2 θ ) w + 2 ( ψ 23 sin θ + ψ 24 cos θ ) l + 3 ( ψ 30 sin 3 θ + ψ 31 sin 2 θ cos θ + ψ 32 sin θ cos 2 θ + ψ 33 cos 3 θ ) w 2 + 2 ( ψ 34 sin 2 θ + 2 ψ 35 sin θ cos θ + ψ 36 cos 2 θ ) w l + ( ψ 37 sin θ + ψ 38 cos θ ) l 2 + ,
F 32 = 2 ( ψ 23 sin θ + ψ 24 cos θ ) w + 2 ψ 25 l + ( ψ 34 sin 2 θ + 2 ψ 35 sin θ cos θ + ψ 36 cos 2 θ ) w 2 + 2 ( ψ 37 sin θ + ψ 38 cos θ ) w l + 3 ψ 39 l 2 + .
x 20 = a 0 - 1 ( h 0 + α 0 s 1 ) ,             y 20 = a 0 - 1 ( - y 0 + α 0 s 2 ) , z 20 = a 0 - 1 α 0 z 1 .
λ 20 = ρ 20 - 1 h 0 ( 1 - a 0 - 1 ) - ρ 10 - 1 s 1 a 0 - 1 = - ψ 10 = - cos φ 2 cos ϕ 2 μ 20 = - ρ 20 - 1 y 0 ( 1 - a 0 - 1 ) - ρ 10 - 1 s 2 a 0 - 1 = ψ 11 = - cos φ 2 sin ϕ 2 ν 20 = - ρ 10 - 1 z 1 a 0 - 1 = ψ 12 = - sin φ 2 } .
( cos φ 2 cos ϕ 2 + cos φ 2 cos ϕ 2 ) sin θ - ( cos φ 2 sin ϕ 2 + cos φ 2 sin ϕ 2 ) cos θ + n λ / d = 0 ,
φ 2 = - φ 2 .
cos φ 2 [ sin ( θ - ϕ 2 ) + sin ( θ - ϕ 2 ) ] + n λ / d = 0 ,
( F 2 + F 2 ) w = 0 ,
( ψ 20 + ψ 20 ) sin 2 θ + ( ψ 22 + ψ 22 ) cos 2 θ = 0.
ψ 2 i + ψ 2 i = 0             ( i = 0 , 2 ) .
ψ 2 i = 0             and             ψ 2 i = 0
m 0 D 1 + R - 1 Q 0 + = 0.
R - 2 s β 10 3 2 R - 2 s ( 1 - 3 m 0 ) 1 2 = 0.
s 1 = 1 2 R - ( R 2 - 3 h 0 2 ) ( s 2 2 + z 1 2 ) R 3 - ( R 2 - 3 h 0 2 ) 2 ( s 2 2 + z 1 2 ) 2 R 7 - + 3 h 0 s 2 y 0 R 2 + [ 3 4 R + 3 ( s 2 2 + z 1 2 ) R 3 - 3 ( R 2 - 3 h 0 2 ) ( 3 s 2 2 + z 1 2 ) 2 R 5 ] y 0 2 + 9 h 0 s 2 R 4 y 0 3 + .
s 1 = s 1 ( R R , s 2 s 2 , z 1 z 1 , y 0 y 0 ) .
ϕ 2 = 2 θ - ϕ 2 ,
( φ 2 ) z 1 = 0 = ( - φ 2 ) z 1 = 0 = 0.
sin ( θ - γ ) + sin ( θ - ϕ 20 ) + n λ / d = 0 ,
2 cos γ sin θ + n λ / d = 0.
cos φ 2 [ sin θ + sin ( θ - ϕ 2 ) ] + n λ / d = 0.
2 cos φ 20 sin θ + n λ / d = 0.
x 20 = x 20
y 20 2 + z 20 2 = α 0 2 a 0 - 2 ( y 1 2 + z 1 2 ) = α 0 2 a 0 - 2 ( y 1 2 + z 1 2 ) .
y 1 2 + z 1 2 = y 1 2 + z 1 2 = y 10 2 .
2 cos φ 2 cos ϕ 2 sin θ + n λ / d = 0.
2 cos γ sin θ = cos φ 2 [ sin ( θ - γ ) + sin ( β + Δ β ) ] ,
[ ( 2 cos γ sin θ / cos φ 2 ) - sin ( θ - γ ) ] 2 ( 1 + tan 2 Δ β ) = ( sin β + cos β tan Δ β ) 2 .
y 10 + y 1 = - ρ 10 tan Δ β / cos             ( y 1 < 0 ) .
y 10 + y 10 = - 2 z 1 2 tan θ R cos ( 1 - tan γ tan θ ) × [ 1 - 2 h 0 2 ( 3 R - 4 h 0 ) ( y 10 2 + z 1 2 ) R 5 + 4 h 0 2 y 10 y 0 R 4 - 3 R 2 + 4 h 0 2 2 R 4 y 0 2 + ] × [ 1 + 2 z 1 2 R 2 ( 3 2 + cos γ sin θ sin ( θ + γ ) cos 2 ( θ + γ ) ) + ] .
σ 0 c = - R 4 cot θ cos ( 1 - tan γ tan θ ) × [ 1 + 2 h 0 2 y 10 2 ( 3 R - 4 h 0 ) R 5 - 4 h 0 2 y 10 y 0 R 4 + ( 3 R 2 + 4 h 0 2 ) y 0 2 2 R 4 + ] ,
y 10 + y 1 = - R tan θ ( 1 - 2 z 10 2 3 R 2 + 94 z 10 4 9 R 4 ) ( Y - 1 ) × [ Y + 1 cos 2 θ Y 2 ( Y - 1 ) ( 1 - 2 cos 2 θ + ) + ] ,
Y = cos φ 20 cos φ 2 = 1 + 2 R 2 ( z 1 2 - z 10 2 ) × [ 1 - 0.9211 R 2 ( z 1 2 + z 10 2 ) + ] × [ 1 + 3 R 2 ( z 1 2 + z 10 2 4 ) + ] .
Δ y = ( ρ 10 / cos ) ( F 31 + F 31 ) Δ z = ( ρ 10 / cos ) ( F 32 + F 32 ) } ,
Δ a c = ( 1 / σ ) ( y ¯ Δ y + z ¯ 1 Δ z ) .
Δ a c = 2 z 1 ρ 10 - 2 ( H 1 w + H 2 l ) + z 1 ρ 10 - 3 ρ 20 - 1 sin θ ( H 3 w 2 + H 4 w l + H 5 l 2 ) + y ¯ 1 s 1 ρ 10 - 3 ρ 20 - 1 sin θ H 6 ,
H 1 = σ - 1 R - 2 [ h 0 2 ( y ¯ 1 z 1 cos 2 θ - s 2 z ¯ 1 cos θ ) + Δ 1 ] , H 2 = - σ - 1 R - 2 { h 0 2 [ s 2 y ¯ 1 cos θ - ( z ¯ 1 / z 1 ) s 2 2 ] + Δ 2 } , H 3 = σ - 1 R - 2 { - 3 h 0 2 s 1 y ¯ 1 z 1 cos 2 θ + 2 z ¯ 1 R 2 s 1 G 0 [ m 11 + ( m 12 / z 1 ) s 2 ] cos θ + Δ 3 } , H 4 = 2 s 1 σ - 1 R - 2 { 2 R 2 G 0 y ¯ 1 cos θ [ m 11 + ( m 12 / z 1 ) s 2 ] + ( z ¯ 1 / z 1 ) [ 1 2 R 2 s [ R m 0 + 2 ( s / s 1 ) G m m 10 ] - h 0 2 s 2 2 - 1 4 h 0 2 z 1 2 ] + Δ 4 } , H 5 = 2 σ - 1 R - 2 ( h 0 2 s 1 y ¯ 1 z 1 + Δ 5 ) , H 6 = σ - 1 ( ( 6 s 2 G 0 m 11 cos 2 θ + ) w 2 + { ( s / 2 ) [ R m 0 + 2 ( s / s 1 ) G 0 m 10 ] - ( h 0 2 / R 2 ) s 2 2 } l 2 + Δ 6 ) ,
Δ R = ρ 10 λ / 2 W cos ( θ + γ ) .
s 20 3 W 2 < R 4 λ / 48 sin θ cos 2 θ cos ( θ + γ ) .