Abstract

A new Eagle-type monochromator mounting for ruled concave gratings at off-plane angles of 45° is presented. The monochromator operates at an angle of incidence α range of ∼0–15°, where α is the angle between the grating normal and the projection of the incident central ray in the Rowland plane. Spherical, cylindrical, ellipsoidal, or toroidal gratings can be used. With toroidal or ellipsoidal grating surfaces one stigmatic image point is obtained, and the amount of astigmatism is strongly reduced in the whole spectral range. In the presence of astigmatism there is no comatic broadening of the line image. With toroidal and ellipsoidal gratings the comatic aberration is small being proportional to the angle α and depending only on the groove length. The width-dependent spherical aberration is proportional to α2 and is therefore small. The new monochromator is expected to have applications in the far UV region as the low wavelength limit is expected to be reduced by a factor of √2 compared to in-plane or near in-plane mountings.

© 1980 Optical Society of America

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References

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  1. W. Werner, “Imaging Properties of Diffraction Gratings,” Thesis, Delft, The Netherlands.
  2. W. Werner, Appl. Opt. 16, 2078 (1977).
    [CrossRef] [PubMed]
  3. X. Da Silva, U.S. Patent4,068,954 (1978).
  4. A. Danielsson, P. Lindblom, Optik 41, 441 (1974).
  5. A. Danielsson, P. Lindblom, Optik 41, 465 (1975).
  6. J. Dahlbacka, P. Lindblom, A. Danielsson, Optik 43, 213 (1975).
  7. R. A. Sawyer, Experimental Spectroscopy (Prentice-Hall, New York, 1951), p. 311.
  8. D. O. Landon, Appl. Opt. 4, 450 (1963).
    [CrossRef]

1977 (1)

1975 (2)

A. Danielsson, P. Lindblom, Optik 41, 465 (1975).

J. Dahlbacka, P. Lindblom, A. Danielsson, Optik 43, 213 (1975).

1974 (1)

A. Danielsson, P. Lindblom, Optik 41, 441 (1974).

1963 (1)

D. O. Landon, Appl. Opt. 4, 450 (1963).
[CrossRef]

Da Silva, X.

X. Da Silva, U.S. Patent4,068,954 (1978).

Dahlbacka, J.

J. Dahlbacka, P. Lindblom, A. Danielsson, Optik 43, 213 (1975).

Danielsson, A.

J. Dahlbacka, P. Lindblom, A. Danielsson, Optik 43, 213 (1975).

A. Danielsson, P. Lindblom, Optik 41, 465 (1975).

A. Danielsson, P. Lindblom, Optik 41, 441 (1974).

Landon, D. O.

D. O. Landon, Appl. Opt. 4, 450 (1963).
[CrossRef]

Lindblom, P.

A. Danielsson, P. Lindblom, Optik 41, 465 (1975).

J. Dahlbacka, P. Lindblom, A. Danielsson, Optik 43, 213 (1975).

A. Danielsson, P. Lindblom, Optik 41, 441 (1974).

Sawyer, R. A.

R. A. Sawyer, Experimental Spectroscopy (Prentice-Hall, New York, 1951), p. 311.

Werner, W.

W. Werner, Appl. Opt. 16, 2078 (1977).
[CrossRef] [PubMed]

W. Werner, “Imaging Properties of Diffraction Gratings,” Thesis, Delft, The Netherlands.

Appl. Opt. (2)

Optik (3)

A. Danielsson, P. Lindblom, Optik 41, 441 (1974).

A. Danielsson, P. Lindblom, Optik 41, 465 (1975).

J. Dahlbacka, P. Lindblom, A. Danielsson, Optik 43, 213 (1975).

Other (3)

R. A. Sawyer, Experimental Spectroscopy (Prentice-Hall, New York, 1951), p. 311.

X. Da Silva, U.S. Patent4,068,954 (1978).

W. Werner, “Imaging Properties of Diffraction Gratings,” Thesis, Delft, The Netherlands.

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Figures (5)

Fig. 1
Fig. 1

Projections of focal curves in Rowland plane for monochromator mounting with α = β0 at different off-plane angles. Note how the curves approach circle r = a−1 when off-plane angle approaches ϕ = 45°.

Fig. 2
Fig. 2

Quantity (a−1r)/a−l of the focal curves plotted as function of the incidence angle α for different off-plane angles. Thus the curves show departure of focal curves from circle r = a−1.

Fig. 3
Fig. 3

Slit orientation and schematic scanning mechanism for off-plane monochromator. Line BO is line of intersection between Rowland plane and plane containing centers F and G of entrance and exit slits, respectively, and center O of grating surface. Scanning is made by rotating grating around z axis. Arm ED mounted orthogonally to line BO is coupled to arm DO, which coincides with grating normal. Distance EO is chosen to be a−1 sin2ϕ or ≅ ½ a−1 for ϕ ≅ 45°. Thus length of arm ED is a−1 tanα sin2ϕ. Arm AB is mounted similarly to arm ED but in opposite direction. Elongation of arm ED is transferred to arm AB through arm AD so that AB = −ED. Common focal plane for the slits is the plane through point B orthogonal to line BO, which plane thus contains arm AB. Lines AF and AG thus coincide with entrance and exit slits, and tilt angles AFB and AGB are given by Eqs. (4) and (5) as BF = −BG = a−1 tanϕ. Slits thus rotate in the focal plane synchronously with the grating.

Fig. 4
Fig. 4

Spot images of a point entrance aperture produced by spherical and a cylindrical grating in monochromator mounting. Both gratings have a−1 = 1000 mm. Off-plane angle is ϕ = 44.8°, and angles of incidence are α = 3.9° [Fig. 4(a)], α = 8.2° [Fig. 4(b)], and α = 11.0° [Fig. 4(c)]. For a 2400-g/mm grating these correspond to wavelengths 40.2, 84.3, and 112.8 nm, respectively. Widths of the gratings are 80 mm and heights 32 mm. Note that astigmatic line images for the two gratings are very similar. Each spot image has been expanded in the direction orthogonal to astigmatic line [Eq. (4)] to show widths of images. Tilt of astigmatic lines are 1.95, 4.12, and 5.55°, respectively, as calculated from Eq. (4).

Fig. 5
Fig. 5

Images of toroidal grating used in monochromator. Data are same as Fig. 4. Astigmatism has been eliminated at αs = 8.25° from Eq. (12) using αmin = 3.9 and αmax = 11.0°. For a−1 = 1000 mm Eq. (10) gives b−1 = 1965.791 mm. Note that astigmatism is strongly reduced for all incidence angles in comparison with Fig. 4.

Tables (1)

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Table I Comparison of Results

Equations (20)

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1 r ( cos 2 α + tan 2 ϕ ) = a cos α cos 2 ϕ ,
r = a 1 { 1 + ½ cos 2 ϕ ( tan 2 ϕ 1 ) α 2 + 1 24 cos 2 ϕ ( 5 tan 2 ϕ + 1 ) α 4 + ... } .
r = a 1 ( 1 + α 4 + ... ) .
tan γ x = r Δ β Δ z = Δ β Δ ϕ cos 2 ϕ = sin ϕ cos ϕ tan α ,
tan γ I = sin ϕ cos ϕ tan α .
l ast = [ ( Δ z ) 2 + ( r Δ β ) 2 ] 1 / 2 | l = 1 / 2 L = 2 L r | Q | ( 1 + sin 2 ϕ cos 2 ϕ tan 2 α ) 1 / 2 ,
Q = 1 r b cos α cos 2 ϕ
l ast sph = 2 L | 1 2 cos α | ( 1 + ¼ tan 2 ϕ ) 1 / 2
l ast cyl = 2 L ( 1 + ¼ tan 2 α ) 1 / 2
l ast sph l ast cyl 2 L .
1 cos 2 ϕ = a b + sin 2 α s , 1 r s = cos α s ( a + b sin 2 α s ) = b cos α s cos 2 ϕ = a cos α s 1 cos 2 ϕ sin 2 α s ,
l ast = L | α 2 α s 2 |
α s = 1 2 ( α max 2 + α min 2 ) 1 / 2 .
Δ Z = 2 l r ( a 2 b cos α ) + 4 b l 2 ( 1 2 b a ) w l α ( a + 2 b ) , r Δ β = ½ tan α · ( Δ Z ) + ¼ l 2 α 1 a ( a 2 10 a b + 8 b 2 ) ,
Δ Z = 2 l ( 1 2 cos α ) 4 l 2 R 3 w l R α , r Δ β = 1 2 tan α · ( Δ Z ) l 2 4 R α ,
Δ Z = 2 l [ w l / R ] α , r Δ β = 1 2 tan α · ( Δ Z ) + l 2 4 R α .
Δ Z = 2 w l α s , r Δ β = ½ a l 2 α s .
r Δ β | l = 0 = w ( 2 α 2 Δ + 3 α 2 Δ 2 + ¼ α 4 ) 3 w 2 2 r α 2 Δ + w 2 2 r 2 α 2 ,
r Δ β | l = 0 = w ( 2 α 2 Δ + ¼ α 4 ) + w 3 2 r 2 α 2 .
sin θ = λ e c 1 ( n π m ) 1 / 2 ,

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