Abstract

Tandem use of two classical concave gratings makes it possible to design a double-dispersion spectrograph that is essentially free of astigmatism and coma over a large wavelength range near normal incidence. The first grating is used as a Wadsworth collimator. Light of different wavelengths is dispersed by the Wadsworth collimator so that it illuminates different portions of the second grating. The second grating is placed so that it acts as a Wadsworth camera, in which the light bundle of a certain wavelength illuminating a particular section of the second grating is diffracted along the local normal of that section. In this way, the Wadsworth condition for stigmatic and coma-free imaging is almost fulfilled for all wavelengths. Only two reflecting surfaces are needed. The instrument is a double-dispersion spectrograph with additive dispersion. It does not use an intermediate slit, but has the stray-light-suppression characteristics of such a mount. A comparison of its imaging capabilities with other stigmatic concave-grating spectrographs is presented.

© 1975 Optical Society of America

Full Article  |  PDF Article

Corrections

John-David F. Bartoe and Guenter E. Brueckner, "Erratum," J. Opt. Soc. Am. 65, 617-617 (1975)
https://www.osapublishing.org/josa/abstract.cfm?uri=josa-65-5-617

References

  • View by:
  • |
  • |
  • |

  1. A classical concave grating has grooves arranged on a segment of a spherical surface such that the projection of the grooves onto a plane tangent to the vertex of the surface forms straight, equally spaced lines.
  2. F. L. O. Wadsworth, Astrophys. J. 3, 54 (1896).
  3. T. Namioka, J. Opt. Soc. Am. 51, 4 (1961).
    [CrossRef]
  4. Y. Sakayanagi, Science of Light 16, 129 (1967).
  5. A. Labeyrie and J. Flamand, Opt. Spectra 50, No. 6 (1969).
  6. A. E. Douglas and G. Herzberg, J. Opt. Soc. Am. 47, 625 (1957).
    [CrossRef]
  7. W. A. Rense and T. Violett, J. Opt. Soc. Am. 49, 139 (1959).
    [CrossRef]
  8. Yu P. Shchepetkin, Opt. Spectrosk. 6, 538 (1959) [Opt. Spectrosc. 6, 822 (1959)].
  9. H. E. Blackwell, G. S. Shipp, M. Ogawa, and G. L. Weissler, J. Opt. Soc. Am. 56, 665 (1966).
    [CrossRef]
  10. R. M. Bonnet, J. E. Blamont, and P. Gildwarg, Astrophys. J. 148, L115 (1967).
    [CrossRef]
  11. P. Lemaire, Astrophys. Lett. 3, 43 (1969).
  12. C. R. Detwiler, D. L. Garrett, J. D. Purcell, and R. Tousey, Ann. de Géophysique 17, 9 (1961).
  13. H. G. Beutler, J. Opt. Soc. Am. 35, 349 (1945).
    [CrossRef]
  14. Equations (5) are based upon the assumption that the ruling density of the second grating G2is constant along the circle R. In practice, this is not the case for a classical concave grating, for which the ruling density is constant along a straight line tangent to the vertex of the sphere. Therefore, for a large grating with a small radius of curvature, the image of a particular wavelength will not be located exactly on the local normal. This second-order effect results in a very small amount of astigmatism, which has not been included in our analytical treatment.
  15. H. G. Beutler, J. Opt. Soc. Am. 35, 319 (1945).
    [CrossRef]
  16. J-D. F. Bartoe, An Analysis of Some Stigmatic Concave Grating Spectrographs, M.S. thesis (Georgetown University, Washington, D.C., 1973).
  17. J. Cordelle, J. Flamand, G. Pieuchard, and A. Labeyrie, in Proceedings of Eighth Congress of the International Commission for Optics, Reading, England, 1969.
  18. The coma of a Wadsworth mount using a spherical collimating mirror is given byΔp=3W28Rcosα[tanβ0cosβ(cosα+cosβ0)cosβ0(cosα+cosβ)]×[1+tan2β(1+cosαcosα+cosβ)]1/2,where Δp is the linear size of the coma measured along the horizontal focal plane, W is the grating width, R is the grating radius, α and β are the angles of incidence and diffraction for the grating, and β0is the angle of diffraction at which the coma is zero. If β0= 0, then the source point must be on the normal of the mirror and the coma computed from the equation for Δp is due to the grating only. Under this condition, the equation for Δp also gives the coma for the tandem Wads-worth mount, because the source point is on the normal of the first grating. The angular size of the coma is then given by θc≅Δp/γ2h′.M. Seya and T. Namioka, Science of Light 16, 167 (1967).
  19. Warren J. Smith, Modern Optical Engineering (McGraw-Hill, New York, 1966), p. 387.
  20. The laser wavelength presently used for production of type III holographic gratings is 4880 Å. Thus m= 0.297.

1969 (2)

A. Labeyrie and J. Flamand, Opt. Spectra 50, No. 6 (1969).

P. Lemaire, Astrophys. Lett. 3, 43 (1969).

1967 (3)

R. M. Bonnet, J. E. Blamont, and P. Gildwarg, Astrophys. J. 148, L115 (1967).
[CrossRef]

The coma of a Wadsworth mount using a spherical collimating mirror is given byΔp=3W28Rcosα[tanβ0cosβ(cosα+cosβ0)cosβ0(cosα+cosβ)]×[1+tan2β(1+cosαcosα+cosβ)]1/2,where Δp is the linear size of the coma measured along the horizontal focal plane, W is the grating width, R is the grating radius, α and β are the angles of incidence and diffraction for the grating, and β0is the angle of diffraction at which the coma is zero. If β0= 0, then the source point must be on the normal of the mirror and the coma computed from the equation for Δp is due to the grating only. Under this condition, the equation for Δp also gives the coma for the tandem Wads-worth mount, because the source point is on the normal of the first grating. The angular size of the coma is then given by θc≅Δp/γ2h′.M. Seya and T. Namioka, Science of Light 16, 167 (1967).

Y. Sakayanagi, Science of Light 16, 129 (1967).

1966 (1)

1961 (2)

T. Namioka, J. Opt. Soc. Am. 51, 4 (1961).
[CrossRef]

C. R. Detwiler, D. L. Garrett, J. D. Purcell, and R. Tousey, Ann. de Géophysique 17, 9 (1961).

1959 (2)

W. A. Rense and T. Violett, J. Opt. Soc. Am. 49, 139 (1959).
[CrossRef]

Yu P. Shchepetkin, Opt. Spectrosk. 6, 538 (1959) [Opt. Spectrosc. 6, 822 (1959)].

1957 (1)

1945 (2)

H. G. Beutler, J. Opt. Soc. Am. 35, 349 (1945).
[CrossRef]

H. G. Beutler, J. Opt. Soc. Am. 35, 319 (1945).
[CrossRef]

1896 (1)

F. L. O. Wadsworth, Astrophys. J. 3, 54 (1896).

Bartoe, J-D. F.

J-D. F. Bartoe, An Analysis of Some Stigmatic Concave Grating Spectrographs, M.S. thesis (Georgetown University, Washington, D.C., 1973).

Beutler, H. G.

H. G. Beutler, J. Opt. Soc. Am. 35, 349 (1945).
[CrossRef]

H. G. Beutler, J. Opt. Soc. Am. 35, 319 (1945).
[CrossRef]

Blackwell, H. E.

Blamont, J. E.

R. M. Bonnet, J. E. Blamont, and P. Gildwarg, Astrophys. J. 148, L115 (1967).
[CrossRef]

Bonnet, R. M.

R. M. Bonnet, J. E. Blamont, and P. Gildwarg, Astrophys. J. 148, L115 (1967).
[CrossRef]

Cordelle, J.

J. Cordelle, J. Flamand, G. Pieuchard, and A. Labeyrie, in Proceedings of Eighth Congress of the International Commission for Optics, Reading, England, 1969.

Detwiler, C. R.

C. R. Detwiler, D. L. Garrett, J. D. Purcell, and R. Tousey, Ann. de Géophysique 17, 9 (1961).

Douglas, A. E.

Flamand, J.

A. Labeyrie and J. Flamand, Opt. Spectra 50, No. 6 (1969).

J. Cordelle, J. Flamand, G. Pieuchard, and A. Labeyrie, in Proceedings of Eighth Congress of the International Commission for Optics, Reading, England, 1969.

Garrett, D. L.

C. R. Detwiler, D. L. Garrett, J. D. Purcell, and R. Tousey, Ann. de Géophysique 17, 9 (1961).

Gildwarg, P.

R. M. Bonnet, J. E. Blamont, and P. Gildwarg, Astrophys. J. 148, L115 (1967).
[CrossRef]

Herzberg, G.

Labeyrie, A.

A. Labeyrie and J. Flamand, Opt. Spectra 50, No. 6 (1969).

J. Cordelle, J. Flamand, G. Pieuchard, and A. Labeyrie, in Proceedings of Eighth Congress of the International Commission for Optics, Reading, England, 1969.

Lemaire, P.

P. Lemaire, Astrophys. Lett. 3, 43 (1969).

Namioka, T.

The coma of a Wadsworth mount using a spherical collimating mirror is given byΔp=3W28Rcosα[tanβ0cosβ(cosα+cosβ0)cosβ0(cosα+cosβ)]×[1+tan2β(1+cosαcosα+cosβ)]1/2,where Δp is the linear size of the coma measured along the horizontal focal plane, W is the grating width, R is the grating radius, α and β are the angles of incidence and diffraction for the grating, and β0is the angle of diffraction at which the coma is zero. If β0= 0, then the source point must be on the normal of the mirror and the coma computed from the equation for Δp is due to the grating only. Under this condition, the equation for Δp also gives the coma for the tandem Wads-worth mount, because the source point is on the normal of the first grating. The angular size of the coma is then given by θc≅Δp/γ2h′.M. Seya and T. Namioka, Science of Light 16, 167 (1967).

T. Namioka, J. Opt. Soc. Am. 51, 4 (1961).
[CrossRef]

Ogawa, M.

Pieuchard, G.

J. Cordelle, J. Flamand, G. Pieuchard, and A. Labeyrie, in Proceedings of Eighth Congress of the International Commission for Optics, Reading, England, 1969.

Purcell, J. D.

C. R. Detwiler, D. L. Garrett, J. D. Purcell, and R. Tousey, Ann. de Géophysique 17, 9 (1961).

Rense, W. A.

Sakayanagi, Y.

Y. Sakayanagi, Science of Light 16, 129 (1967).

Seya, M.

The coma of a Wadsworth mount using a spherical collimating mirror is given byΔp=3W28Rcosα[tanβ0cosβ(cosα+cosβ0)cosβ0(cosα+cosβ)]×[1+tan2β(1+cosαcosα+cosβ)]1/2,where Δp is the linear size of the coma measured along the horizontal focal plane, W is the grating width, R is the grating radius, α and β are the angles of incidence and diffraction for the grating, and β0is the angle of diffraction at which the coma is zero. If β0= 0, then the source point must be on the normal of the mirror and the coma computed from the equation for Δp is due to the grating only. Under this condition, the equation for Δp also gives the coma for the tandem Wads-worth mount, because the source point is on the normal of the first grating. The angular size of the coma is then given by θc≅Δp/γ2h′.M. Seya and T. Namioka, Science of Light 16, 167 (1967).

Shchepetkin, Yu P.

Yu P. Shchepetkin, Opt. Spectrosk. 6, 538 (1959) [Opt. Spectrosc. 6, 822 (1959)].

Shipp, G. S.

Smith, Warren J.

Warren J. Smith, Modern Optical Engineering (McGraw-Hill, New York, 1966), p. 387.

Tousey, R.

C. R. Detwiler, D. L. Garrett, J. D. Purcell, and R. Tousey, Ann. de Géophysique 17, 9 (1961).

Violett, T.

Wadsworth, F. L. O.

F. L. O. Wadsworth, Astrophys. J. 3, 54 (1896).

Weissler, G. L.

Ann. de Géophysique (1)

C. R. Detwiler, D. L. Garrett, J. D. Purcell, and R. Tousey, Ann. de Géophysique 17, 9 (1961).

Astrophys. J. (2)

R. M. Bonnet, J. E. Blamont, and P. Gildwarg, Astrophys. J. 148, L115 (1967).
[CrossRef]

F. L. O. Wadsworth, Astrophys. J. 3, 54 (1896).

Astrophys. Lett. (1)

P. Lemaire, Astrophys. Lett. 3, 43 (1969).

J. Opt. Soc. Am. (6)

Opt. Spectra (1)

A. Labeyrie and J. Flamand, Opt. Spectra 50, No. 6 (1969).

Opt. Spectrosk. (1)

Yu P. Shchepetkin, Opt. Spectrosk. 6, 538 (1959) [Opt. Spectrosc. 6, 822 (1959)].

Science of Light (2)

The coma of a Wadsworth mount using a spherical collimating mirror is given byΔp=3W28Rcosα[tanβ0cosβ(cosα+cosβ0)cosβ0(cosα+cosβ)]×[1+tan2β(1+cosαcosα+cosβ)]1/2,where Δp is the linear size of the coma measured along the horizontal focal plane, W is the grating width, R is the grating radius, α and β are the angles of incidence and diffraction for the grating, and β0is the angle of diffraction at which the coma is zero. If β0= 0, then the source point must be on the normal of the mirror and the coma computed from the equation for Δp is due to the grating only. Under this condition, the equation for Δp also gives the coma for the tandem Wads-worth mount, because the source point is on the normal of the first grating. The angular size of the coma is then given by θc≅Δp/γ2h′.M. Seya and T. Namioka, Science of Light 16, 167 (1967).

Y. Sakayanagi, Science of Light 16, 129 (1967).

Other (6)

Warren J. Smith, Modern Optical Engineering (McGraw-Hill, New York, 1966), p. 387.

The laser wavelength presently used for production of type III holographic gratings is 4880 Å. Thus m= 0.297.

J-D. F. Bartoe, An Analysis of Some Stigmatic Concave Grating Spectrographs, M.S. thesis (Georgetown University, Washington, D.C., 1973).

J. Cordelle, J. Flamand, G. Pieuchard, and A. Labeyrie, in Proceedings of Eighth Congress of the International Commission for Optics, Reading, England, 1969.

A classical concave grating has grooves arranged on a segment of a spherical surface such that the projection of the grooves onto a plane tangent to the vertex of the surface forms straight, equally spaced lines.

Equations (5) are based upon the assumption that the ruling density of the second grating G2is constant along the circle R. In practice, this is not the case for a classical concave grating, for which the ruling density is constant along a straight line tangent to the vertex of the sphere. Therefore, for a large grating with a small radius of curvature, the image of a particular wavelength will not be located exactly on the local normal. This second-order effect results in a very small amount of astigmatism, which has not been included in our analytical treatment.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (14)

FIG. 1
FIG. 1

The Wadsworth mount. Parallel light is incident on the grating, G, which has a radius of curvature, R, at an angle α with respect to the grating normal, υ and h are the vertical and horizontal focal planes, γ υ and γ h are the vertical and horizontal focal distances from the grating.

FIG. 2
FIG. 2

The tandem Wadsworth mount. G1 is the Wadsworth collimator grating. G2 is the Wadsworth-camera grating, having a radius R centered at M. N1 is the normal of G1 and N2 is the local normal of G2. υ and h are the vertical and horizontal focal planes of G2. β and α2 are the diffraction angles of G1 and G2. γ1s is the distance between the vertex of G1 and the entrance slit, S. d is the distance between the vertex of Gx and the local vertex of G2. γ 2 υ and γ 2 h are the vertical and horizontal focal distances from the local vertex to G2 to the focal planes υ and h.

FIG. 3
FIG. 3

Special tandem Wadsworth arrangements. S is the slit. G1 and G2 are the gratings. P is an off-axis parabola. F is the focal plane. R is the radius of the second grating. (A) Both gratings have the same ruling density. (B) Ruling densities and radii of both gratings are equal. (C) Wadsworth collimator replaced by plane grating, ruling densities of both gratings are equal. (d) Tandem Wadsworth spectrograph with off-axis paraboloidal mirror.

FIG. 4
FIG. 4

Astigmatism of the symmetric tandem Wadsworth mount. For the case illustrated in Fig. 3B, |A| is the relative astigmatic difference and λ/λs is the normalized wavelength.

FIG. 5
FIG. 5

Astigmatism of different stigmatic concave-grating spectrographs; diffraction angle βs = 10°. TW = symmetric tandem Wadsworth mount; CW = crossed double Wadsworth mount; W = conventional Wadsworth mount (in this case, αs = 10° and βs = 0); E = Rowland mount with an ellipsoidal grating; V = Rowland mount with a varying spacing grating; T = Rowland mount with a toroidal correcting mirror; H = holographic type III grating.

FIG. 6
FIG. 6

Astigmatism of different stigmatic concave-grating spectrographs; diffraction angle βs = 20°.

FIG. 7
FIG. 7

Astigmatism of different stigmatic concave-grating spectrographs; diffraction angle βs = 30°.

FIG. 8
FIG. 8

Astigmatism of different stigmatic concave-grating spectrographs; diffraction angle βs = 40°.

FIG. 9
FIG. 9

Astigmatism of a Rowland-mount spectrograph with a type III holographic grating and astigmatism of a tandem Wads-worth-mount spectrograph, m is the ratio of the stigmatic wavelength to the wavelength of the laser that produced the grating.

FIG. 10
FIG. 10

The changing ruling density of a classical concave grating. N2 is the normal at the center of the grating G2. N2 is the local normal of G2 at angle θ with respect to N2. C is the center of curvature of G2. α2 and β2 are the angles of incidence and diffraction with respect to the local normal, d and d are the ruling spacings along lines perpendicular to N2 and N 2 , respectively.

FIG. 11
FIG. 11

Coma of the symmetric tandem Wadsworth mount (for the case illustrated in Fig. 3B). θcN2 is the angular size of the coma multiplied by the square of the f number and λ/λs is the normalized wavelength.

FIG. 12
FIG. 12

Coma of the Wadsworth mount and symmetric tandem Wadsworth mount. TW = symmetric tandem Wadsworth mount; W = conventional Wadsworth mount.

FIG. 13
FIG. 13

Total aberration of the symmetric tandem Wadsworth mount for a f/15 system. The solid lines are from Eq. (42) and the broken lines are from the ray-tracing program.

FIG. 14
FIG. 14

Optical layout of a high-resolution telescope and spectrograph (HRTS) using the symmetric tandem Wadsworth mount. M1 and M2 are the mirrors of the Cassegrain telescope, S is the spectrograph entrance slit, G1 and G2 are concave gratings in the symmetric tandem Wadsworth arrangement, M3 is a folding mirror, and F is the final focal plane. The broken line represents the rocket-payload skin.

Equations (49)

Equations on this page are rendered with MathJax. Learn more.

γ h = R cos 2 β cos α + cos β
γ υ = R cos α + cos β ,
| A | = | γ υ γ h γ av | = 2 | γ υ γ h γ υ + γ h | = 2 [ 1 cos 2 β 1 + cos 2 β ] .
sin β R = sin α 2 d 1 .
sin β = λ l 1 for G 1 , sin α 2 = λ l 2 for G 2 ,
l 1 / l 2 = R / d 1 .
γ 1 = R 1 1 + cos β ,
sin β s = l 1 λ s ,
γ 1 s = R 1 1 + cos β s .
γ h = [ cos α + cos β R cos 2 α γ ] 1 cos 2 β ,
γ υ = [ cos α + cos β R 1 γ ] 1 ,
γ h = R 1 cos 2 β cos β cos β s
γ υ = R 1 cos β cos β s .
γ h = d γ h ,
γ υ = d γ υ ,
d = d 1 cos β + ( R 2 d 1 2 sin 2 β ) 1 / 2 .
γ 2 h = R [ 1 + cos α 2 R ( cos β cos β s ) cos 2 α 2 d ( cos β cos β s ) R 1 cos 2 β ] 1 .
γ 2 υ = R [ 1 + cos α 2 R ( cos β cos β s ) d ( cos β cos β s ) R 1 ] 1 ,
cos α 2 = [ 1 d 1 2 R 2 sin 2 β ] 1 / 2 .
| A | = | 2 γ 2 υ γ 2 h γ 2 υ + γ 2 h | .
γ 2 h = R [ 1 + cos β R cos β ( cos β cos β s ) 2 R ( cos β cos β s ) R 1 cos β ] 1 ,
γ 2 υ = R [ 1 + cos β R ( cos β cos β s ) 2 R cos β ( cos β cos β s ) R 1 ] 1 .
γ 2 h = R [ 1 + cos β cos β ( cos β cos β s ) 2 ( cos β cos β s ) cos β ] 1 ,
γ 2 υ = R [ 1 + cos β cos β cos β s 2 cos β ( cos β cos β s ) 1 ] 1 .
γ 2 h = γ 2 υ = R ( 1 + cos β ) 1 .
( d λ d s ) TW = 1 + cos β s 2 l R
( d λ d s ) CW = ( 2 l R ) 1 / 2 .
( d λ d s ) W = 2 l R .
( d λ d s ) E = cos β s l R .
( d λ d s ) V = cos 3 β s l R .
( d λ d s ) T = cos β s l R .
( d λ d s ) H = cos β s 2 l R .
l λ s = ( sin β s + sin α ) ,
W 2 W 1 cos β cos α 2 + L ( 1 + 1 cos α ) ,
W 2 W 1 + 2 L .
θ c 3 8 1 N 2 ( γ 1 s ) 2 R γ 2 h tan β 2 cos α 2 [ 1 + tan 2 β 2 ( 1 + cos α 2 cos β 2 + cos α 2 ) ] 1 / 2 ,
sin β = λ l 1 for G 1 ,
sin α 2 + sin β 2 = λ l 2 for G 2 ,
d = d / cos θ ,
θ = ( β s + α 2 s ) ( β + α 2 ) ,
l = 1 / d = l cos 2 ( β s β ) .
sin β 2 = λ λ s [ cos 2 ( β s α 2 ) 1 ] sin β s ,
sin α 2 = λ λ s sin β s .
θ = 1 128 1 N 3 ,
θ s = 1 128 ( 1 N 1 3 + 1 N 2 3 ) ,
θ s = 1 64 1 N 3 .
θ A = 1 2 | A / N | ,
θ T = θ s + θ c + θ A .
Δp=3W28Rcosα[tanβ0cosβ(cosα+cosβ0)cosβ0(cosα+cosβ)]×[1+tan2β(1+cosαcosα+cosβ)]1/2,