Abstract

The Ebert spectrograph has recently been used extensively as a monochromator. In view of the spherical mirror being used as a collimator-camera mirror and the plane grating as the dispersing element, the instrument suffers mainly from spherical aberration if the f-number of the system is smaller than about 9 for a 1-m focal length. Also, if many exit slits are needed over the entire spectrum of interest, the coma cannot be neglected. In order to obtain a fast spectrometer in which many exit slits could be used, the two surfaces in the system must be aspherized. The theory of such a system is presented in this paper. Specifically, a numerical example is worked out for an f/5 system with 40-in. focal length and 1200 lines/mm rulings on the grating. The spectrum of interest is 1000 to 4000 Å in first order.

© 1962 Optical Society of America

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References

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  1. H. Ebert, Wiedemanns Ann. 38, 489 (1889).
  2. W. G. Fastie, J. Opt. Soc. Am. 42, 641 (1952).
    [Crossref]
  3. Hilger Journal VI, No. 2, 29 (1960).
  4. R. F. Jarrell, J. Opt. Soc. Am. 45, 259 (1955).
    [Crossref]
  5. P. C. von Planta, J. Opt. Soc. Am. 47, 629 (1957).
    [Crossref]
  6. W. G. Fastie, H. M. Crosswhite, and P. Gloersen, J. Opt. Soc. Am. 48, 106 (1958).
    [Crossref]
  7. D. W. Robinson, J. Opt. Soc. Am. 49, 966 (1959).
    [Crossref]
  8. F. B. Wright, Publs. Astron. Soc. Pacific 47, 300 (1935).
    [Crossref]
  9. G. H. Spencer and M. V. R. K. Murty (to be published).

1960 (1)

Hilger Journal VI, No. 2, 29 (1960).

1959 (1)

1958 (1)

1957 (1)

1955 (1)

1952 (1)

1935 (1)

F. B. Wright, Publs. Astron. Soc. Pacific 47, 300 (1935).
[Crossref]

1889 (1)

H. Ebert, Wiedemanns Ann. 38, 489 (1889).

Crosswhite, H. M.

Ebert, H.

H. Ebert, Wiedemanns Ann. 38, 489 (1889).

Fastie, W. G.

Gloersen, P.

Jarrell, R. F.

Murty, M. V. R. K.

G. H. Spencer and M. V. R. K. Murty (to be published).

Robinson, D. W.

Spencer, G. H.

G. H. Spencer and M. V. R. K. Murty (to be published).

von Planta, P. C.

Wright, F. B.

F. B. Wright, Publs. Astron. Soc. Pacific 47, 300 (1935).
[Crossref]

Hilger Journal (1)

Hilger Journal VI, No. 2, 29 (1960).

J. Opt. Soc. Am. (5)

Publs. Astron. Soc. Pacific (1)

F. B. Wright, Publs. Astron. Soc. Pacific 47, 300 (1935).
[Crossref]

Wiedemanns Ann. (1)

H. Ebert, Wiedemanns Ann. 38, 489 (1889).

Other (1)

G. H. Spencer and M. V. R. K. Murty (to be published).

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Figures (10)

Fig. 1
Fig. 1

Schematic diagram of the Ebert spectrometer: M—spherical mirror, G—plane grating with rulings perpendicular to the plane of the paper, S1 and S2—entrance and exit slits.

Fig. 2
Fig. 2

Unfolded schematic diagram of the Ebert spectrometer. M1 and M2 are the same mirror M shown in Fig. 1. M1 is the collimating part and M2 the focusing part. S1 and S2 are the entrance and exit slits.

Fig. 3
Fig. 3

This is the same as Fig. 2 except that S1 and S2 are brought on axis to emphasize the fact that spherical aberration of the system with the spherical mirror is twice that of a single spherical mirror in parallel light.

Fig. 4
Fig. 4

Unfolded schematic diagram showing how the spectrum is corrected for spherical aberration and coma. M1 and M2 are the same physically and G is the plane grating in transmission. S is the entrance slit at the paraxial focus of M1. S.P. is a Schmidt plate to correct the spherical aberration. This is purely a fictitious one and in the actual system it is the aspheric correction on the reflection grating surface. Sp is the spectrum spread.

Fig. 5
Fig. 5

Schematic diagram of the aspheric spectrometer in which the grating has a central slot to allow for the light to pass through. S is the entrance slit at the paraxial focus of the collimator-camera mirror M. The mirror M is an ellipsoid of revolution about its minor axis. The grating G is slightly aspherized and is rotationally symmetric. The aspheric term on the grating is of the form α(x2 + y2)2. The spectrum Sp is spread with 1000 and 4000 Å wavelengths at either end.

Fig. 6
Fig. 6

Schematic diagram of the aspheric system in which the grating is free of the slot. This has been achieved by designing a faster system in the direction parallel to the rulings and then rejecting one-half of the system above. The diagram is shown in elevation only because the plan is the same as shown in Fig. 5. The aspherics on the mirror M and the grating G are off-axis elements of rotationally symmetric surfaces.

Fig. 7
Fig. 7

Diagram showing the aberration curves for the basic system shown in Fig. 5 for the three wavelengths 1000, 2500, and 4000 A for the central fan of rays on the grating in the plane of dispersion. The wavelength 2500 Å being imaged on axis is free from spherical aberration and coma. The two wavelengths 1000 and 4000 Å are imaged off-axis and the images are a little under and overcorrected for spherical aberration. The coma, however, is well corrected.

Fig. 8
Fig. 8

The spot diagrams for the system shown in Fig. 5, for the three wavelengths 1000, 2500, and 4000 Å. Eighty-one rays are traced from the center of the entrance slit and intersecting the grating surface at equal intervals over its area of 8 × 8-in.

Fig. 9
Fig. 9

Diagram showing the aberration curves for the system of Fig. 6, for the three wavelengths 1000, 2500, and 4000 Å for the central fan of rays on the grating in the plane of dispersion. The curves are of the same type as shown in Fig. 7.

Fig. 10
Fig. 10

The spot diagrams for the system shown in Fig. 6 for the three wavelengths 1000, 2500, and 4000 Å. Eighty-one rays are traced from the center of the entrance slit and intersecting the grating surface at equal intervals over its area of 8 × 8 in.

Equations (23)

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Δ = ( y 4 / 2 r 3 ) ( 2 y 2 δ / r 2 ) .
δ = ( y 2 max / 4 r ) ,
Δ max = y 4 max / 8 r 3 .
f number 1 4 ( f / λ ) 1 4 .
Δ h s = ( y 2 / 4 f ) ( h / f ) .
Δ h T = 3 y 2 h / 4 f 2 .
coma T = ( y 3 h / 4 f 3 ) = 2 y 3 h / r 3 ,
coma T = 2 y 3 y p / r 3 .
coma T = 8 A y 3 y p .
A = 1 / 4 r 3 .
A y 4 = [ ( 1 / 4 r 3 ) ( 1 / 8 r 3 ) ] y 4 = ( 1 / 8 r 3 ) y 4 .
Z = c ( x 2 + y 2 ) 1 + [ 1 c 2 ( x 2 + y 2 ) ] 1 2 ( x 2 + y 2 ) 2 8 r 3 ,
z = ( x 2 + y 2 ) 2 / 2 r 3 .
h = y p [ r / ( r 2 a ) ] .
z = c ( x 2 + y 2 ) 1 + [ 1 c 2 ( x 2 + y 2 ) ] 1 2 k ( x 2 + y 2 ) 2 8 r 3 ,
z = c ( x 2 + y 2 ) 1 + [ 1 c 2 ( 1 + k ) ( x 2 + y 2 ) ] 1 2 .
z = [ ( 1 + k ) / 4 r 3 ] ( x 2 + y 2 ) 2 .
2 sin θ = λ / σ ,
z = [ ( 1 + k ) / 4 r 3 ] ( x 2 cos 2 θ + y 2 ) 2 ( 1 / cos θ ) .
z = [ ( 1 + k ) / 4 r 3 ] cos 3 θ ( x 2 + y 2 ) 2 .
z = 0.10486549 × 10 5 ( x 2 + y 2 ) 2 ,
z = 0.10567 × 10 5 ( x 2 + y 2 ) 2 .
z = 0.10923 × 10 5 ( x 2 + y 2 ) 2 ,