Abstract

The analytic theory of aberrations has been used to derive an expression for the magnitude of the coma width of the image in the meridional plane in a Czerny–Turner spectrograph with unequal mirror radii. The calculated properties of a 4-m spectrograph with equal radii and a recently constructed 3.34-m spectrograph with unequal radii are compared with the results obtained by tracing individual rays. The agreement is excellent, in contrast to the results of Chandler [ J. Opt. Soc. Am. 58, 895 ( 1968)]. The lateral position of the grating for complete elimination of coma found experimentally with the 3.34-m instrument is in fair agreement with the theory. A correction to the 3 longitudinal grating position is given for a Czerny–Turner spectrograph which results in a flatter focal surface.

© 1969 Optical Society of America

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References

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  1. M. Czerny and A. F. Turner, Z. Physik 61, 792 (1930).
    [CrossRef]
  2. G. Chandler, J. Opt. Soc. Am. 58, 895 (1968).
    [CrossRef]
  3. A. Shafer, L. Megill, and L. Droppleman, J. Opt. Soc. Am. 54, 879 (1964).
    [CrossRef]
  4. H. G. Beutler, J. Opt. Soc. Am. 35, 311 (1945).
    [CrossRef]
  5. G. Rosendahl, J. Opt. Soc. Am. 52, 412 (1962).
    [CrossRef]
  6. W. G. Fastie, U. S. Patent3,011,391 (1961).
  7. S. A. Khrshanovskii, Opt. Spectry. (USSR) 9, 207 (1960).
  8. K. Mielenz, J. Res. Natl. Bur. Std. (U. S.) 68C, 205 (1964).
    [CrossRef]
  9. The diameter of the camera mirror is determined by the requirement that the entire grating be visible from all points of the plate holder. This gives D>W cosβ+2L(1 − m/r).
  10. This change was necessitated by a statement in the Megill program which sets the direction cosines of the central ray relative to the x, y, and z axes equal to −1, 0, and 0, respectively. Thus, the central ray is assumed by the program to be parallel to the x axis. If the geometry is not changed, the calculated spot diagram and image width do not refer to rays which actually form the image. This is equivalent to tracing rays through a system that is different from the one specified by the input parameters.
  11. W. G. Fastie, private communication.

1968 (1)

1964 (2)

K. Mielenz, J. Res. Natl. Bur. Std. (U. S.) 68C, 205 (1964).
[CrossRef]

A. Shafer, L. Megill, and L. Droppleman, J. Opt. Soc. Am. 54, 879 (1964).
[CrossRef]

1962 (1)

1960 (1)

S. A. Khrshanovskii, Opt. Spectry. (USSR) 9, 207 (1960).

1945 (1)

1930 (1)

M. Czerny and A. F. Turner, Z. Physik 61, 792 (1930).
[CrossRef]

Beutler, H. G.

Chandler, G.

Czerny, M.

M. Czerny and A. F. Turner, Z. Physik 61, 792 (1930).
[CrossRef]

Droppleman, L.

Fastie, W. G.

W. G. Fastie, private communication.

W. G. Fastie, U. S. Patent3,011,391 (1961).

Khrshanovskii, S. A.

S. A. Khrshanovskii, Opt. Spectry. (USSR) 9, 207 (1960).

Megill, L.

Mielenz, K.

K. Mielenz, J. Res. Natl. Bur. Std. (U. S.) 68C, 205 (1964).
[CrossRef]

Rosendahl, G.

Shafer, A.

Turner, A. F.

M. Czerny and A. F. Turner, Z. Physik 61, 792 (1930).
[CrossRef]

J. Opt. Soc. Am. (4)

J. Res. Natl. Bur. Std. (U. S.) (1)

K. Mielenz, J. Res. Natl. Bur. Std. (U. S.) 68C, 205 (1964).
[CrossRef]

Opt. Spectry. (USSR) (1)

S. A. Khrshanovskii, Opt. Spectry. (USSR) 9, 207 (1960).

Z. Physik (1)

M. Czerny and A. F. Turner, Z. Physik 61, 792 (1930).
[CrossRef]

Other (4)

W. G. Fastie, U. S. Patent3,011,391 (1961).

The diameter of the camera mirror is determined by the requirement that the entire grating be visible from all points of the plate holder. This gives D>W cosβ+2L(1 − m/r).

This change was necessitated by a statement in the Megill program which sets the direction cosines of the central ray relative to the x, y, and z axes equal to −1, 0, and 0, respectively. Thus, the central ray is assumed by the program to be parallel to the x axis. If the geometry is not changed, the calculated spot diagram and image width do not refer to rays which actually form the image. This is equivalent to tracing rays through a system that is different from the one specified by the input parameters.

W. G. Fastie, private communication.

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Figures (6)

Fig. 1
Fig. 1

Explanation of symbols for light-path function for spherical mirror in two dimensions.

Fig. 2
Fig. 2

Geometry of a Czerny–Turner spectrograph for plate-curvature considerations. The y axis is located at a distance R2/2 from the center of curvature of the camera mirror.

Fig. 3
Fig. 3

Focal curves for a Czerny–Turner spectrograph with the grating at (a) flat-field position; (b) corrected-flat-field position, = 15 mm; (c) super-flat-field position, = 39 mm. The meaning of x and θ are given in Fig. 2.

Fig. 4
Fig. 4

Image widths for an optimized 4-m Czerny–Turner spectrograph with fixed slit, grating, and plate positions found in present work (solid line) compared with results of Chandler2 (broken line).

Fig. 5
Fig. 5

Calculated and experimental transverse grating positions, zg (see appendix), for a coma-free image at the center of the plate for the present 3.34-m Czerny–Turner spectrograph. Experimental points have error bars.

Fig. 6
Fig. 6

General geometry of a Czerny–Turner spectrograph.

Tables (2)

Tables Icon

Table I Image quality for a 4-m Czerny–Turner spectrograph with fixed slit, grating, and plate positions. System optimized for θN = 62°. Successive points across plate are separated by 6.25 cm. All entries in table are in units of 0.001 mm. The ray-tracing width is the maximum separation between rays in the meridional plane found by ray tracing. The calculated coma is the coma width found from Eq. (1). A plus sign indicates a flare away from the slit; a minus sign indicates a flare toward the slit. Where no sign is given for the ray tracing width, the meridional rays were found to be symmetric about the central ray. The ray-tracing length (astigmatism) is the extension of the image perpendicular to the meridional plane, as found by ray tracing.

Tables Icon

Table II Image quality for a 3.34-m Czerny–Turner spectrograph with fixed slit, grating, and plate positions. System optimized for θN = 62°. Successive points across plate are separated by 6.25 cm. All entries in table are in units of 0.001 mm. The ray-tracing width is the maximum separation between rays in the meridional plane, found by ray tracing. The calculated coma is the coma width found from Eq. (1). A plus sign indicates a flare away from the slit; a minus sign indicates a flare toward the slit. Where no sign is given for the ray tracing width, the meridional rays were found to be symmetric about the central ray. The ray-tracing length (astigmatism) is the extension of the image perpendicular to the meridional plane, as found by ray tracing.

Equations (24)

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F = r + r - w ( sin α + sin α ) + w 2 2 [ ( cos 2 α r - cos α R 1 ) + ( cos 2 α r - cos α R 1 ) ] + 1 2 w 3 [ sin α r ( cos 2 α r - cos α R 1 ) + sin α r ( cos 2 α r - cos α R 1 ) ] .
F w ( for r = ) = - 1 cos α ( sin α + sin α ) + w cos 2 α [ cos 2 α r - cos α R 1 - cos α R 1 ] + 3 w 2 2 cos 3 α [ sin α r ( cos 2 α r - cos α R 1 ) ] = ? 0.
( F / w ) ( for r = , α = - α , r = 1 2 R 1 cos α , w = 1 2 W cos α g ) = 3 W 2 cos 2 α g sin α 4 R 1 2 cos 3 α .
δ β g = - cos α g cos β g δ α g = - 3 W 2 sin α cos 3 α g 4 R 1 2 cos 3 α cos β g .
Δ α = 1 2 R 2 cos β δ β g = - 3 W 2 R 2 sin α cos β cos 3 α g 8 R 1 2 cos 3 α cos β g ,
( F / w ) ( for r = , β = - β , r = 1 2 R 2 cos β , w = 1 2 W cos β g ) = 3 W 2 cos 2 β g sin β 4 R 2 2 cos 3 β .
Δ β = 1 2 R 2 cos β F w = 3 W 2 cos β g sin β 8 R 2 cos 2 β .
Δ = Δ α + Δ β = 3 W 2 R 2 cos 2 α g cos β 8 cos 3 α × [ sin β R 2 2 · cos 2 β g cos 3 α cos 2 α g cos 3 β - sin α cos α g R 1 2 cos β g ] .
sin β sin α = R 2 2 R 1 2 cos 3 β cos 3 α cos 3 α g cos 3 β g ,
β / α = cos 3 α g / cos 3 β g ,
x R 2 = - 1 4 [ 1 - 3 ( m R 2 ) 2 ] θ 2 + 1 48 [ 1 - 30 ( m R 2 ) 2 + 48 ( m R 2 ) 3 - 27 ( m R 2 ) 4 ] θ 4 .
x ( , θ min ) = x ( , θ max )
2 x ( , θ center ) = x ( , θ min ) + x ( , θ max )
θ min = 3 β center - 3 L / 2 R 2 , θ center = 3 β center ,
θ max = 3 β center + 3 L / 2 R 2 ,
α = z g / 2 x g .
β = ( E 1 + h Δ β g - z g ) / 2 l + ( E 2 + 1 2 R 2 Δ β g - E 1 - h Δ β g ) / R 2 ,
Δ β g = β g - β g ( center of plate ) l = [ h 2 - ( E 1 - z g ) 2 ] 1 2 .
γ 1 = z g R 1 / 2 x g γ 2 = E 2 - R 2 ( E 1 - z g ) / 2 l .
δ 1 = [ R 1 2 - γ 1 2 ] - R 1 δ 2 = x g - l - R 2 + [ R 2 2 - ( E 1 - γ 2 ) 2 ] 1 2 .
m = { ( l - [ R 2 2 - ( E 1 - γ 2 ) 2 ] 1 2 ) 2 + ( z g - γ 2 ) 2 } 1 2 .
θ = arcsin [ ( R 2 / m ) sin β ] .
Slope = z g - γ 2 [ R 2 2 - ( E 1 - γ 2 ) 2 ] 1 2 - l .
β g ( center of plate ) - α g = ( E 1 - z g ) / l + z g / x g .