## Theory of second-generation holographic diffraction gratings

JOSA A, Vol. 6, Issue 8, pp. 1175-1188 (1989)

http://dx.doi.org/10.1364/JOSAA.6.001175

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### Abstract

The geometric theory of imaging for spectrometer systems is extended to include those with diffraction gratings produced holographically by the intersection of two sets of spherical wave fronts that have been rendered aspheric through reflection from concave mirrors. Aberration coefficients are derived that reduce to those of conventional holographic grating systems in the appropriate limit, and degrees of freedom for aberration correction are discussed.

© 1989 Optical Society of America

**History**

Original Manuscript: October 11, 1988

Manuscript Accepted: March 9, 1989

Published: August 1, 1989

**Citation**

Christopher Palmer, "Theory of second-generation holographic diffraction gratings," J. Opt. Soc. Am. A **6**, 1175-1188 (1989)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-6-8-1175

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### References

- M. C. Hutley, Diffraction Gratings (Academic, London, 1982), p. 10.
- J. M. Burch, “Photographic production of scales for moiré fringe applications,” in Optics in Metrology, P. Mollet, ed. (Pergamon, New York, 1960), pp. 361–368.
- D. Rudolph, G. Schmahl, “Verhehren zur Herstellung von Röntgenlinsen und Beugungsgittern,” Umschau Wiss. Tech. 67, 225 (1967); “Über ein Verfehren zur Herstellung von Beugungsgittern und Zonenplatten,” Mitt. Astron. Ges. 23, 46 (1967).
- A. Labeyrie, J. Flamand, “Spectroscopic performance of holographically made diffraction gratings,” Opt. Commun. 1, 5–8 (1969). [CrossRef]
- M. Chrisp, “Aberration-corrected holographic gratings and their mountings,” in Applied Optics and Optical Engineering (Academic, New York, 1987), Vol. 10, Chap. 7; T. Namioka, M. Seya, H. Noda, “Design and performance of holographic concave gratings,” Jpn. J. Appl. Phys. 15, 1181–1197 (1976). [CrossRef]
- M. Koike, Y. Harada, H. Noda, “New blazed holographic gratings fabricated by using an aspherical recording with an ion-etching method,” in Application and Theory of Periodic Structures, Diffraction Gratings, and Moiré Phenomena III, J. M. Lerner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.815, 96–101 (1987). [CrossRef]
- See, for instance, T. Namioka, “Theory of the concave grating. I.,” J. Opt. Soc. Am. 49, 446–460 (1959). [CrossRef]
- H. Noda, T. Namioka, M. Seya, “Geometric theory of the grating,”J. Opt. Soc. Am. 64, 1031–1048 (1974). [CrossRef]
- M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), p. 115.
- With regard to classical and first-generation holographic gratings, Namioka7 considered a more general case, in which the entrance and exit slits (and the recording source points) may be out of plane (that is, they may have nonzero zcoordinates). For second-generation holographic gratings, though, enough degrees of freedom should be found to permit significant aberration correction while leaving the optics in the principal plane.
- The analysis that follows is not restricted to spherical mirrors but may apply to any reflecting surfaces of revolution. At this point in the development, though, it is more instructive to consider a spherical mirror, which has one center of curvature. Generally the center of curvature for that slice of the mirror lying in the principal plane and the center of the mirror surface define the axis of symmetry of the surface of revolution; we require this particular line to be parallel to the grating normal.
- H. Noda, Ph.D. dissertation (Tokyo Kyoiku University, Tokyo, Japan, 1974); see also M. R. Howells, “Some geometrical considerations concerning grazing incidence reflectors,” (Brookhaven National Laboratory, Upton, N.Y., 1980).
- H. Beutler, “The theory of the concave grating,”J. Opt. Soc. Am. 35, 311–350 (1945). [CrossRef]
- T. Namioka, Tohoku University, 2-1-1 Katahira, Sendai 980, Japan (personal communication, 1986). Namioka also allowed the source and ideal image points A and B to lie outside the principal plane; setting z= z′ = 0 in his expressions produces agreement with the terms given in Ref. 15.
- W. R. McKinney, C. Palmer, “Numerical design method for aberration-reduced concave grating spectrometers,” Appl. Opt. 26, 3108–3118 (1987). [CrossRef] [PubMed]
- Strictly speaking, F10= F01= 0 only for the principal ray, but the power-series expansion of Ψ [Eq. (25)] provides aberration coefficients Fijthat are independent of the pupil coordinates yand z.
- F01= 0 identically, since we have imposed a plane of symmetry (the xyplane); for optical systems without a plane of symmetry, the F01term provides the familiar law of reflection in the plane perpendicular to the principal plane (viz., parallel to the grooves).
- M. Abramowitz, I. Stegun, eds., Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1972), p. 15, Eq. (3.6.11).
- A further restriction, namely, lC= rC, provides Y= Z= −1/2; this can readily be seen to be valid when it is recalled that Yand Zare the derivatives ∂η/∂yand ∂ζ/∂z, respectively, and for a planar mirror parallel to the grating tangent plane and equidistant from the source and the grating center, a shift dηin the mirror coordinate will produce twice the shift in the grating coordinate. The negative sign is an artifact of our coordinate systems: ηincreases as ydecreases, and vice versa.
- R. P. Gillespie, Partial Differentiation (Oliver and Boyd, Edinburgh, 1951), pp. 57–61.
- D. J. Schroeder, Astronomical Optics (Academic, San Diego, California, 1987), pp. 79–81.
- This can be seen from Eq. (51), in which the groove spacing dfor a first-generation holographic grating is nonzero only if the recording angles γand δare distinct.

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