Abstract

We investigated the ability of aberration-corrected concave holographic gratings used in the Rowland mount at normal diffraction to provide high spectral resolution in the far-ultraviolet region. By assuming that astigmatism and spherical aberration are geometrically corrected by an ellipsoid, we show that holography can be used to correct the remaining prominent second-type coma. Stigmatic sources require a laser wavelength that is too far in the ultraviolet for current recording technology. However, at 3336 Å a simple compact symmetric mount, which involves two spherical mirrors, can generate aberrated wave fronts that can be used to record a coma-corrected holographic grating. When compared with the equivalent equally spaced straight-groove grating, which requires a modified ellipsoid substrate, holography cancels the additional asymmetrical term of deformation that permits the use of a simpler surface for the substrate. Some areas of potential difficulty in the holographic mounting are briefly analyzed.

© 1991 Optical Society of America

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References

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  1. H. Noda, T. Namioka, M. Seya, “Geometric theory of the grating,” J. Opt. Soc. Am. 64, 1031–1048 (1974).
    [CrossRef]
  2. B. J. Brown, I. J. Wilson, “Holographic grating aberration correction for a rowland circle mount,” Opt. Acta 28, 1587–1599 (1981).
    [CrossRef]
  3. M. C. E. Huber, J. G. Timothy, J. S. Morgan, G. Lemaître, G. Tondello, E. Janniti, P. Scarin, “Imaging extreme ultraviolet spectrometer employing a single toroidal diffraction grating: the initial evaluation,” Appl. Opt. 27, 3503–3510 (1988).
    [CrossRef] [PubMed]
  4. H. Noda, Y. Harada, M. Koike, “Holographic grating recorded using aspheric wavefronts for a Seya–Namioka monochromator,” Appl. Opt. 28, 4375–4380 (1989).
    [CrossRef] [PubMed]
  5. W. C. Cash, “Aspheric concave grating spectrographs,” Appl. Opt. 23, 4518–4522 (1984).
    [CrossRef] [PubMed]
  6. W. Moos, (principal investigator), lyman, Phase A Study Final Report (Goddard Space Flight Center, NASA, Greenbelt, Md., 1989), Vol. II, App. E.
  7. C. Palmer, “Theory of second-generation holographic diffraction gratings,” J. Opt. Soc. Am. A 6, 1175–1188 (1989).
    [CrossRef]
  8. 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).
  9. W. T. Welford, Aberrations of Optical Systems (Hilger, Bristol, 1986), pp. 75–78.
  10. E. G. Loewen, “Diffraction gratings, ruled and holographic,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds. (Academic, London, 1983), Vol. IX, pp. 33–71.

1989 (2)

1988 (1)

1984 (1)

1981 (1)

B. J. Brown, I. J. Wilson, “Holographic grating aberration correction for a rowland circle mount,” Opt. Acta 28, 1587–1599 (1981).
[CrossRef]

1974 (1)

Brown, B. J.

B. J. Brown, I. J. Wilson, “Holographic grating aberration correction for a rowland circle mount,” Opt. Acta 28, 1587–1599 (1981).
[CrossRef]

Cash, W. C.

Harada, Y.

H. Noda, Y. Harada, M. Koike, “Holographic grating recorded using aspheric wavefronts for a Seya–Namioka monochromator,” Appl. Opt. 28, 4375–4380 (1989).
[CrossRef] [PubMed]

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).

Huber, M. C. E.

Janniti, E.

Koike, M.

H. Noda, Y. Harada, M. Koike, “Holographic grating recorded using aspheric wavefronts for a Seya–Namioka monochromator,” Appl. Opt. 28, 4375–4380 (1989).
[CrossRef] [PubMed]

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).

Lemaître, G.

Loewen, E. G.

E. G. Loewen, “Diffraction gratings, ruled and holographic,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds. (Academic, London, 1983), Vol. IX, pp. 33–71.

Moos, W.

W. Moos, (principal investigator), lyman, Phase A Study Final Report (Goddard Space Flight Center, NASA, Greenbelt, Md., 1989), Vol. II, App. E.

Morgan, J. S.

Namioka, T.

Noda, H.

H. Noda, Y. Harada, M. Koike, “Holographic grating recorded using aspheric wavefronts for a Seya–Namioka monochromator,” Appl. Opt. 28, 4375–4380 (1989).
[CrossRef] [PubMed]

H. Noda, T. Namioka, M. Seya, “Geometric theory of the grating,” J. Opt. Soc. Am. 64, 1031–1048 (1974).
[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).

Palmer, C.

Scarin, P.

Seya, M.

Timothy, J. G.

Tondello, G.

Welford, W. T.

W. T. Welford, Aberrations of Optical Systems (Hilger, Bristol, 1986), pp. 75–78.

Wilson, I. J.

B. J. Brown, I. J. Wilson, “Holographic grating aberration correction for a rowland circle mount,” Opt. Acta 28, 1587–1599 (1981).
[CrossRef]

Appl. Opt. (3)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Opt. Acta (1)

B. J. Brown, I. J. Wilson, “Holographic grating aberration correction for a rowland circle mount,” Opt. Acta 28, 1587–1599 (1981).
[CrossRef]

Other (4)

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).

W. T. Welford, Aberrations of Optical Systems (Hilger, Bristol, 1986), pp. 75–78.

E. G. Loewen, “Diffraction gratings, ruled and holographic,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds. (Academic, London, 1983), Vol. IX, pp. 33–71.

W. Moos, (principal investigator), lyman, Phase A Study Final Report (Goddard Space Flight Center, NASA, Greenbelt, Md., 1989), Vol. II, App. E.

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Figures (5)

Fig. 1
Fig. 1

Schematic diagram of the Rowland circle mount. The laser sources (C, D) are also on the Rowland circle. Conventions are taken from Noda et al.1 Numerical values are listed Table I.

Fig. 2
Fig. 2

Holography with aberrated sources: geometry and conventions. The virtual aberrated sources are in the dispersion plane. Normals at the mirrors' centers are not parallel.

Fig. 3
Fig. 3

Dependence of the holographic coma coefficients (H30 and H12) on the distance of the stigmatic sources from the grating. It is assumed that λ0 = 3336 Å, rc = rd, and β = −γ. The dashed line represents the required value of H12 for correction of the second-type coma. The final solution, which uses two spherical mirrors, is reached at a large distance, rc ∼20 m. H12 meets the required value at ∼2.7 m, but H30 is too large and cannot be compensated for with a simple optics.

Fig. 4
Fig. 4

Schematic diagram of the proposed antisymmetric recording geometry. Laser sources at 3336 Å are imaged by two spherical concave mirrors into aberrated virtual images located outside the Rowland circle. The groove pattern at the substrate surface contains only coma terms. The plane of symmetry is the dispersion plane. The plane perpendicular to the dispersion plane, including the substrate normal, is also a plane of symmetry. The characteristics are listed in column c of Table II. The linear scale does not apply to the size of the optical components.

Fig. 5
Fig. 5

Spot diagrams for five wavelengths in the spectrum: (a) the equispaced straight grooves and modified ellipsoid, (b) a grating recorded with stigmatic sources (λ0 = 1940 Å), (c) a grating recorded with aberrated sources (λ0 = 3336 Å). The second-type comacorrected holographic gratings (b) and (c) provide an image quality that compares closely with that given by the classical equivalent (a). Slight changes between (a) and (b) or (c) reflect primarily differences in the choice of a precise wavelength to minimize an aberration. Astigmatism and second-type coma are arbitrarily set here to zero for the central wavelength.

Tables (2)

Tables Icon

Table I Mounting Parameters, where aij Are the Coefficients of the Polynomial Expansion for an Ellipsoidal Blank

Tables Icon

Table II Recording Parameters

Equations (16)

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F = AP + PB + nm λ ,
n λ 0 = ( CP DP ) ( CO DO ) ,
x ( y , z ) = a 20 y 2 + a 02 z 2 + a 30 y 3 + a 12 y z 2 + a 40 y 4 + a 22 y 2 z 2 + a 04 z 4 .
F ( y , z ) = F 00 + y F 10 + 1 / 2 y 2 F 20 + 1 / 2 z 2 F 02 + 1 / 2 y 3 f 30 + 1 / 2 y z 2 F 12 + 1 / 8 y 4 F 40 + 1 / 4 y 2 z 2 F 22 + 1 / 8 z 4 F 04 .
F i , j = M i , j + m λ λ 0 H i , j .
σ = λ 0 / ( sin δ sin γ ) .
H 02 = 1 R cos γ 1 R cos δ 1 ρ ( cos γ cos δ ) = 0 ,
F 12 = M 12 + m λ λ 0 H 12 = 0 ,
σ = λ 0 2 sin ( | δ | ) ,
M 12 = [ 1 R cos α cos α ρ ] sin α R cos α = 4.53 × 10 8 ( mm 2 ) , H 12 = 2 sin δ R cos δ ( 1 R cos δ cos δ ρ ) .
cos 2 δ = m λ σ R 1 M 12 R + m λ / σ ρ
H 12 = S c sin γ r c S d sin δ r d + C 2 cos γ D 2 cos δ ,
S c = r c r c 2 cos γ ρ , S d = r d r d 2 cos γ ρ .
C 2 = sin η c sec η c [ S c 1 p c S c 2 q c r c ] ( q c r c 1 ) ( 1 q c r c ) 2 , D 2 = sin η d sec η d [ S d 1 p d S d 2 q d r d ] ( q d r d 1 ) ( 1 q d r d ) 2 ,
S c 1 = 1 p c cos η c R c , S c 2 = q c r c ( q c r c ) 2 cos η c R c , S d 1 = 1 p d cos η d R d , S d 2 = q d r d ( q d r d ) 2 cos η d R d .
F 12 = M 12 + m λ λ 0 H 12 = 0 ; hence H 12 = 1.56 × 10 7 ( mm 2 ) .

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