Abstract

The second-order focusing properties of a grating spectrograph with fixed entrance slit and a circular image field are investigated analytically. It is shown that the Rowland mount is the only circular field spectrograph mount which allows perfect tangential focusing (to second order) over a predetermined wavelength range, while two classes of mounts exist, one of which is nontrivial, for which-astigmatism is corrected exactly on a circular arc.

© 1990 Optical Society of America

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References

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  1. T. Namioka, “Theory of the Concave Grating,” J. Opt. Soc. Am. 49, 446–460 (1958).
    [CrossRef]
  2. C. Palmer, “Absolute Astigmatism Correction for Flat Field Spectrographs,” Appl. Opt. 28, 1605–1607 (1989).
    [CrossRef] [PubMed]
  3. H. A. Rowland, “Preliminary Notice of the Results Accomplished in the Manufacture and Theory of Gratings for Optical Purposes,” Philos. Mag. 13, 469–474 (1882); “On Concave Gratings for Optical Purposes,” Philos. Mag. 16, 197–210 (1883).
    [CrossRef]
  4. C. Palmer, “Limitations of Aberration Correction in Spectrometer Imaging,” Proc. Soc. Photo-Opt. Instrum. Eng. 1055, 359–369 (1989).

1989 (2)

C. Palmer, “Limitations of Aberration Correction in Spectrometer Imaging,” Proc. Soc. Photo-Opt. Instrum. Eng. 1055, 359–369 (1989).

C. Palmer, “Absolute Astigmatism Correction for Flat Field Spectrographs,” Appl. Opt. 28, 1605–1607 (1989).
[CrossRef] [PubMed]

1958 (1)

1882 (1)

H. A. Rowland, “Preliminary Notice of the Results Accomplished in the Manufacture and Theory of Gratings for Optical Purposes,” Philos. Mag. 13, 469–474 (1882); “On Concave Gratings for Optical Purposes,” Philos. Mag. 16, 197–210 (1883).
[CrossRef]

Namioka, T.

Palmer, C.

C. Palmer, “Absolute Astigmatism Correction for Flat Field Spectrographs,” Appl. Opt. 28, 1605–1607 (1989).
[CrossRef] [PubMed]

C. Palmer, “Limitations of Aberration Correction in Spectrometer Imaging,” Proc. Soc. Photo-Opt. Instrum. Eng. 1055, 359–369 (1989).

Rowland, H. A.

H. A. Rowland, “Preliminary Notice of the Results Accomplished in the Manufacture and Theory of Gratings for Optical Purposes,” Philos. Mag. 13, 469–474 (1882); “On Concave Gratings for Optical Purposes,” Philos. Mag. 16, 197–210 (1883).
[CrossRef]

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

Philos. Mag. (1)

H. A. Rowland, “Preliminary Notice of the Results Accomplished in the Manufacture and Theory of Gratings for Optical Purposes,” Philos. Mag. 13, 469–474 (1882); “On Concave Gratings for Optical Purposes,” Philos. Mag. 16, 197–210 (1883).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

C. Palmer, “Limitations of Aberration Correction in Spectrometer Imaging,” Proc. Soc. Photo-Opt. Instrum. Eng. 1055, 359–369 (1989).

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Figures (3)

Fig. 1
Fig. 1

General circular field spectrometer geometry. Light from point A is diffracted in order m by the grating of major radius R centered at O; the ideal image points of the spectrum {λSL} are located along a circular arc of radius R′ centered at point S′(x0,y0). An intermediate wavelength λ is diffracted to an ideal image point located at the point B. The center of curvature of the grating is situated at point S(R,0).

Fig. 2
Fig. 2

Rowland circle spectrometer geometry. Light from point A is diffracted in order m by the grating centered at O; the ideal image points of the spectrum {λSL} are located along an arc which lies on the Rowland circle, which itself is tangent to the concave grating surface at its vertex O and has radius R′ = R/2, where R is the major radius of the grating. An intermediate wavelength λ is diffracted to an ideal image point located at the point B. Both A and B lie on the Rowland circle.

Fig. 3
Fig. 3

Astigmatism-free circular field spectrometer geometry. The sagittal focal curve lies on a circle which intersects the concave grating surface at its vertex O and has radius R′ given by Eq. (18). The center of this circle is located at point S′(x0,y0), whose Cartesian coordinates are given by Eq. (19). A and B are the entrance slit and image points.

Tables (2)

Tables Icon

Table I Circular Field Solutions of F20 = 0 for Values of A, B, C, and Q

Tables Icon

Table II Circular Field Solutions of F02 = 0 for Values of D, E, F, and Q

Equations (36)

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( x - x 0 ) 2 + ( y - y 0 ) 2 = R 2 ,
F 20 = cos 2 α 2 r + cos 2 β 2 r - a 20 ( cos α + cos β ) + m λ λ 0 H 20 = 0 ,
x ( y , z ) = i = 0 j = 0 a i j y i z j .
r ( β ) = cos 2 β A + B cos β + C sin β ,
A = B cos α + C sin α - cos 2 α r ,
B = 2 a 20 ,
C = - 2 d H 20 λ 0 .
r ( β ) = x 0 cos β + y 0 sin β ± [ x 0 cos β + y 0 sin β ] 2 + R 2 - r 0 2 ,
( K 0 2 - L 0 2 ) + 2 ( K 0 K 1 - L 0 L 1 ) sin β + ( K 1 2 + 2 K 0 K 2 - K 0 2 - L 1 2 - 2 L 0 L 2 ) sin 2 β + ( 2 K 0 K 3 + 2 K 1 K 2 - 2 K 0 K 1 - 2 L 0 L 3 - 2 L 1 L 2 ) sin 3 β + ( K 2 2 + 2 K 1 K 3 - K 1 2 - 2 K 0 K 2 - L 2 2 - 2 L 0 L 4 - 2 L 1 L 3 ) sin 4 β + ( 2 K 2 K 3 - 2 K 0 K 3 - 2 K 1 K 2 - 2 L 1 L 4 - 2 L 2 L 3 ) sin 5 β + ( K 3 2 - K 2 2 - 2 K 1 K 2 - L 3 2 - 2 L 2 L 4 ) sin 6 β - ( 2 K 2 K 3 + 2 L 3 L 4 ) sin 7 β - ( K 3 2 + L 4 2 ) sin 8 β = 0 ,
K 0 = 2 Q A B + 2 x 0 A ,
K 1 = 2 Q B C + 2 y 0 B + 2 x 0 C ,
K 2 = - 2 x 0 A ,
K 3 = - 2 y 0 B - 2 x 0 C ,
L 0 = 1 - 2 x 0 B - Q A 2 - Q B 2 ,
L 1 = - 2 y 0 A - 2 Q A C ,
L 2 = - 2 + 4 x 0 B - 2 y 0 C + Q B 2 - Q C 2 ,
L 3 = 2 y 0 A ,
L 4 = 1 - 2 x 0 B + 2 y 0 C ,
Q = R 2 - r 0 2 .
n = 0 8 c n sin n β = 0 ,
K 0 = K 1 = K 2 = K 3 = L 0 = L 1 = L 2 = L 3 = L 4 = 0.
F 02 = 1 2 r + 1 2 r - a 02 ( cos α + cos β ) + m λ λ 0 H 02 = 0 ,
r ( β ) = 1 D + E cos β + F sin β ,
D = E cos α + F sin α - 1 r ,
E = 2 a 02 ,
F = - 2 d H 02 λ 0 .
( M 0 2 - N 0 2 ) + 2 ( M 0 M 1 - N 0 N 1 ) sin β + ( M 1 2 - M 0 2 - N 1 2 - 2 N 0 N 2 ) sin 2 β - ( 2 M 0 M 1 + 2 N 1 N 2 ) sin 3 β - ( M 1 2 + N 2 2 ) sin 4 β = 0 ,
M 0 = 2 Q D E + 2 x 0 D ,
M 1 = 2 Q E F + 2 y 0 E + 2 x 0 F ,
N 0 = 1 - 2 x 0 E - Q D 2 - Q E 2 ,
N 1 = - 2 y 0 D - 2 Q D F ,
N 2 = 2 x 0 E - 2 y 0 F + Q E 2 - Q F 2 .
M 0 = M 1 = N 0 = N 1 = N 2 = 0.
R = r 0 = 1 4 1 a 02 2 + λ 0 2 ( d H 02 ) 2 ,
( x 0 , y 0 ) = ( 1 4 a 02 , - λ 0 4 d H 02 ) .
r = 1 2 [ a 02 cos α - d H 02 λ 0 sin α ] - 1 .

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