Abstract

We present a theory of the aberrations of gratings that are recorded holographically. Stigmatic imaging at one or two wavelengths is possible with such a grating. We give a summary of the cases of stigmatism, including some newly found. The general principles of the correction of aberrations of holographic recording are described. The theory is applied to the design of Rowland, Seya-Namioka, Wadsworth, and Eagle mountings. We show that the extra freedom of design given by holographic recording can be used to obtain a simpler scanning mechanism or better correction of aberrations, or both.

© 1977 Optical Society of America

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References

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  1. C. H. F. Velzel, J. Opt. Soc. Am. 66, 346 (1976).
    [Crossref]
  2. A. Labeyrie and J. Flamand, Opt. Commun. 1, 5 (1969).
    [Crossref]
  3. J. Cordelle, J. Flamand, and G. Pieuchard, in Optical Instruments and Techniques, edited by J. Home Dickson (Oriel, Newcastle, 1970), p. 117.
  4. E. W. Palmer, M. C. Hutley, A. Franks, J. F. Verrill, and B. Gale, Rep. Prog. Phys. 38, 975 (1975).
    [Crossref]
  5. Jobin Yvon, Diffraction Gratings, Ruled and Holographic (Company Publication, City, 1973).
  6. R. J. Speer, J. Opt. Soc. Am. 65, 1214 (1975).
  7. W. T. Welford, Opt. Acta 9, 389 (1962).
    [Crossref]
  8. H. Noda, T. Namioka, and M. Seya, J. Opt. Soc. Am. 64, 1037 (1974).
    [Crossref]
  9. M. Pouey, J. Opt. Soc. Am. 64, 1616 (1974).
    [Crossref]
  10. H. Noda, T. Namioka, and M. Seya, J. Opt. Soc. Am. 64, 1043 (1974).
    [Crossref]
  11. M. Seya, Sci. Light 2, 8 (1952).
  12. H. Greiner and E. Schäffer, Optik 14, 263 (1957);Optik 15, 51 (1958).

1976 (1)

1975 (2)

E. W. Palmer, M. C. Hutley, A. Franks, J. F. Verrill, and B. Gale, Rep. Prog. Phys. 38, 975 (1975).
[Crossref]

R. J. Speer, J. Opt. Soc. Am. 65, 1214 (1975).

1974 (3)

1969 (1)

A. Labeyrie and J. Flamand, Opt. Commun. 1, 5 (1969).
[Crossref]

1962 (1)

W. T. Welford, Opt. Acta 9, 389 (1962).
[Crossref]

1957 (1)

H. Greiner and E. Schäffer, Optik 14, 263 (1957);Optik 15, 51 (1958).

1952 (1)

M. Seya, Sci. Light 2, 8 (1952).

Cordelle, J.

J. Cordelle, J. Flamand, and G. Pieuchard, in Optical Instruments and Techniques, edited by J. Home Dickson (Oriel, Newcastle, 1970), p. 117.

Flamand, J.

A. Labeyrie and J. Flamand, Opt. Commun. 1, 5 (1969).
[Crossref]

J. Cordelle, J. Flamand, and G. Pieuchard, in Optical Instruments and Techniques, edited by J. Home Dickson (Oriel, Newcastle, 1970), p. 117.

Franks, A.

E. W. Palmer, M. C. Hutley, A. Franks, J. F. Verrill, and B. Gale, Rep. Prog. Phys. 38, 975 (1975).
[Crossref]

Gale, B.

E. W. Palmer, M. C. Hutley, A. Franks, J. F. Verrill, and B. Gale, Rep. Prog. Phys. 38, 975 (1975).
[Crossref]

Greiner, H.

H. Greiner and E. Schäffer, Optik 14, 263 (1957);Optik 15, 51 (1958).

Hutley, M. C.

E. W. Palmer, M. C. Hutley, A. Franks, J. F. Verrill, and B. Gale, Rep. Prog. Phys. 38, 975 (1975).
[Crossref]

Labeyrie, A.

A. Labeyrie and J. Flamand, Opt. Commun. 1, 5 (1969).
[Crossref]

Namioka, T.

Noda, H.

Palmer, E. W.

E. W. Palmer, M. C. Hutley, A. Franks, J. F. Verrill, and B. Gale, Rep. Prog. Phys. 38, 975 (1975).
[Crossref]

Pieuchard, G.

J. Cordelle, J. Flamand, and G. Pieuchard, in Optical Instruments and Techniques, edited by J. Home Dickson (Oriel, Newcastle, 1970), p. 117.

Pouey, M.

Schäffer, E.

H. Greiner and E. Schäffer, Optik 14, 263 (1957);Optik 15, 51 (1958).

Seya, M.

Speer, R. J.

R. J. Speer, J. Opt. Soc. Am. 65, 1214 (1975).

Velzel, C. H. F.

Verrill, J. F.

E. W. Palmer, M. C. Hutley, A. Franks, J. F. Verrill, and B. Gale, Rep. Prog. Phys. 38, 975 (1975).
[Crossref]

Welford, W. T.

W. T. Welford, Opt. Acta 9, 389 (1962).
[Crossref]

Yvon, Jobin

Jobin Yvon, Diffraction Gratings, Ruled and Holographic (Company Publication, City, 1973).

J. Opt. Soc. Am. (5)

Opt. Acta (1)

W. T. Welford, Opt. Acta 9, 389 (1962).
[Crossref]

Opt. Commun. (1)

A. Labeyrie and J. Flamand, Opt. Commun. 1, 5 (1969).
[Crossref]

Optik (1)

H. Greiner and E. Schäffer, Optik 14, 263 (1957);Optik 15, 51 (1958).

Rep. Prog. Phys. (1)

E. W. Palmer, M. C. Hutley, A. Franks, J. F. Verrill, and B. Gale, Rep. Prog. Phys. 38, 975 (1975).
[Crossref]

Sci. Light (1)

M. Seya, Sci. Light 2, 8 (1952).

Other (2)

Jobin Yvon, Diffraction Gratings, Ruled and Holographic (Company Publication, City, 1973).

J. Cordelle, J. Flamand, and G. Pieuchard, in Optical Instruments and Techniques, edited by J. Home Dickson (Oriel, Newcastle, 1970), p. 117.

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Figures (7)

FIG. 1
FIG. 1

Geometry for holographic recording. S is the spherical substrate of radius r with center of curvature M. C and D are coherent point sources; in the case of D we have indicated the two possibilities of convergence and divergence. In practice, a convergent wave can be generated by launching a wave from a point that is the image of D by the grating substrate; but one has to correct for the aberrations associated with the imaging.

FIG. 2
FIG. 2

Coordinate system for a grating with the entrance slit at A and the exit slit at B. Object and image points are given in the XYZ coordinate system. The grating surface is given in the ξ, η coordinate system that coincides with the XYZ system; ξ and η are referred to as pupil coordinates.

FIG. 3
FIG. 3

Spherical grating substrate with center of curvature M. We have indicated the possibility of harmonic conjugation. When CM·HM = r2 the optical paths HP and CP have the ratio CP/CH = CM/r for every point P of the grating surface.

FIG. 4
FIG. 4

Recording geometry with sources C and D for which CM = DM. The entrance and exit slits A and B are situated so that likewise AM = BM. Moreover M, C, and A are in line as well as M, D, and B, and AM·CM = r2.

FIG. 5
FIG. 5

Rowland curves for r/t = 0, 1 4, 1 2, and 1. For r/t = 0 the curve is equal to the classical Rowland circle. For r/t = 1 the part of the curve with positive values of L is nearly a straight line. The mountings with r/t = 1 are therefore well corrected for astigmatism.

FIG. 6
FIG. 6

(a) Length of the focal line l as a function of L1 for L = 3 8 and r/t = 0, 1 4, 1 2, 3 4, and 1. For r/t = 1 the ratio 1/b is under 0.02 for values of mλ/p = L + L1 from 1 4 to 1. (b) Length of the focal line for L = 1 2 and r/t = 0, 1 4, 1 2, 3 4, and 1. For r/t the ratio 1/6 is smaller than 0.02 for values of mλ/p roughly between 3 8 and 9 8.

FIG. 7
FIG. 7

Curve a gives the value of (r2/b2(3)λ/λ as a function of L1 Curve b gives the value of (r3/ab2(4)λ/λ. Both curves are valid for L = 3 8 and r/t = 1.

Tables (3)

Tables Icon

TABLE I One of the recording sources at the center of curvature.

Tables Icon

TABLE II The recording sources coincide.

Tables Icon

TABLE III The recording sources are harmonically conjugated.

Equations (30)

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φ = ( 2 π / λ 0 ) ( C P ± D P ) ,
F = A P + P B m ( λ / λ 0 ) ( C P ± D P )
α k l = k + 1 ξ k + 1 l η l { F } , β k l = k ξ k l + 1 η l + 1 { F } ( ξ = η = 0 )
α k l = A k l + A k l m ( λ / λ 0 ) ( A k l C ± A k l D ) , β k l = B k l + B k l m ( λ / λ 0 ) ( B k l C ± B k l D ) ,
s = r cos i 1 r tan i / t ,
m λ / λ = 1 + L 1 / L .
i = i + α .
α 10 = N 2 s N r + L t + N 1 2 s N 1 r + L 1 t ,
1 t = 1 L C + L D ( N C 2 s C N C r + N D 2 s D N D r )
N 2 s + N 1 2 s + L 1 + L t = N 1 + N r , 2 L N s 2 L 1 N 1 s + N 1 + N t = L 1 + L r , 2 ( L 2 N 2 ) s + 2 ( L 1 2 N 1 2 ) s L 1 + L t = N 1 + N r , 8 L N s + 8 L 1 N 1 s N 1 + N t = L 1 + L r ,
L 1 N = L N 1 or sin α = 0 , L L 1 + N N 1 = 1 or cos α = 1 , 2 L L 1 = N N 1 or tan α = ( 1 + sin 2 i ) / sin i cos i .
i = 24 ° 2 8 , i = 47 ° 4 2 α = 72 ° 1 0 .
s / r = 0.714 , s / r = 0.936 , t / r = 4.82 .
α 10 = β 01 = 1 s 1 + N r m λ λ 0 ( 1 s D 1 ± N C r ) .
1 / s D = ( 1 ± N C ) / r .
α 30 = α 12 = β 21 = β 03 = ( 1 + N ) N 2 2 r 3 + ( m λ / λ 0 ) ( 1 ± N C ) N C 2 2 r 3 .
α 10 = 2 ( 1 L 2 ) s + 2 L t 2 N r ,
β 01 = 2 ( 1 M 2 ) s + 2 L u 2 N r ,
1 u = 1 L C + L D [ 1 M C 2 s C N C r ± ( 1 M D 2 s D N D r ) ] .
1 L 2 s + L t = N r ,
2 L N s + N t = L r ,
2 ( L 2 N 2 ) s L t = N r .
1 L 2 = 2 ( L 2 N 2 ) .
α 20 = 3 ( L / s ) [ ( L 2 L 0 2 ) / s ( N N 0 ) / r ] , α 02 = 3 ( L / s ) ( N N 0 ) / r .
1 2
1 2 p
1 2
1 2 ( p + 1 )
1 2 p
1 2 ( p + 1 )