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.

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

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

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

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

A. E. Douglas and G. Herzberg, J. Opt. Soc. Am. 47, 625 (1957).

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

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

H. E. Blackwell, G. S. Shipp, M. Ogawa, and G. L. Weissler, J. Opt. Soc. Am. 56, 665 (1966).

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

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

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

Equations (5) are based upon the assumption that the ruling density of the second grating *G*_{2} is 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.

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

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.

The coma of a Wadsworth mount using a spherical collimating mirror is given by [Equations] where Δ*p* 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 β_{0} is the angle of diffraction at which the coma is zero. If β_{0} = 0, then the source point must be on the normal of the mirror and the coma computed from the equation for Δ*P* is due to the grating only. Under this condition, the equation for Δ*p* also gives the coma for the tandem Wadsworth mount, because the source point is on the normal of the first grating. The angular size of the coma is then given by θ_{c} ≅ Δ*p*/γ′_{2h}. M. Seya and T. Namioka, Science of Light 16, 167 (1967).

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.