Abstract

Analytic formulas are developed for the semiaxes and for the deformation coefficients of a deformed ellipsoidal grating that minimize aberrations over a given spectral range. These include the Rowland circle mount and quasi-Rowland circle mounts, as required for systems that combine spectra from multiple gratings on a common detector. It is also shown that the necessary condition for a holographic grating to give a better performance over a conventional grating is the use of a convergent beam and a divergent beam for the recording. Examples applicable to the Far-Ultraviolet Spectroscopic Explorer are given. The light-path function for a deformed ellipsoidal holographic grating is also presented.

© 1993 Optical Society of America

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References

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  1. W. C. Cash, “Aspheric concave grating spectrographs,” Appl. Opt. 23, 4518–4522 (1984).
    [CrossRef] [PubMed]
  2. D. Content, C. Trout, P. Davila, M. Wilson, “Aberration corrected aspheric gratings for far ultraviolet spectrographs: conventional approach,” Appl. Opt. 30, 801–806 (1991).
    [CrossRef] [PubMed]
  3. C. Trout, D. Content, P. Davila, “Aberration-corrected aspheric grating designs for the Lyman/Far-Ultraviolet Spectroscopic Explorer high-resolution spectrograph: a comparison,” Appl. Opt. 31, 943–948 (1992).
    [CrossRef] [PubMed]
  4. T. Namioka, M. Koike, “Design of compact high-resolution far-ultraviolet spectrographs equipped with a spherical grating having variable spacing and curved grooves,” presented at the Tenth International Colloquium on UV and X-Ray Spectroscopy of Astrophysical and Laboratory Plasmas, Berkeley, Calif., 3–5 February 1992.
  5. P. Davila, D. Content, C. Trout, “Aberration-corrected aspheric gratings for far-ultraviolet spectrographs: holographic approach,” Appl. Opt. 31, 949–954 (1992).
    [CrossRef] [PubMed]
  6. R. Grange, M. Laget, “Holographic diffraction gratings generated by aberrated wave fronts: application to a high-resolution far-ultraviolet spectrograph,” Appl. Opt. 30, 3598–3603 (1991).
    [CrossRef] [PubMed]
  7. M. Duban, “Some reflections about a high-resolution spectrograph,” Appl. Opt. 31, 443–445 (1992).
    [CrossRef] [PubMed]
  8. M. Duban, “Third-generation Rowland holographic mounting,” Appl. Opt. 30, 4019–4025 (1991).
    [CrossRef] [PubMed]
  9. R. Grange, “Aberration-reduced holographic spherical gratings for Rowland circle spectrographs,” Appl. Opt. 31, 3744–3749 (1992).
    [CrossRef] [PubMed]
  10. H. W. Moos, ed., LYMAN, the Far Ultraviolet Spectroscopic Explorer, Phase A Study Final Report (NASA Goddard Space Flight Center, Greenbelt, Md., 1989).
  11. D. A. Content, P. M. Davila, J. F. Osantowski, T. T. Saha, M. E. Wilson, “Optical design of Lyman/FUSE,” in Instrumentation in Astronomy II, D. L. Crawford, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1235, 943–952 (1990).
  12. H. Noda, T. Namioka, M. Seya, “Geometric theory of the grating,” J. Opt. Soc. Am. 64, 1031–1036 (1974).
    [CrossRef]
  13. T. Namioka, M. Koike, “Analytical representation of spot diagrams and its application to the design of monochromators,” Nucl. Instrum. Methods A 319, 219–227 (1992).
    [CrossRef]
  14. A. Takahashi, “Optical transfer function-based merit functions for automatic diffraction grating system design,” J. Mod. Opt. 36, 675–684 (1989).
    [CrossRef]
  15. T. Namioka, “Theory of the ellipsoidal concave grating. I,” J. Opt. Soc. Am. 51, 4–12 (1961); “Theory of the ellipsoidal concave grating. II. Application of the theory to the specific grating mountings,” J. Opt. Soc. Am. 51, 13–16 (1961).
    [CrossRef]
  16. M. Koike, Y. Harada, H. Noda, “New blazed holographic grating fabricated by using an aspherical recording with an ion-etching method,” in Application and Theory of Periodic Structures, Diffraction Gratings, and Moire Phenomena III, J. M. Lerner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.815, 96–101 (1988).
  17. It is assumed here that the imaging of the FUSE telescope will not be drastically better than the current 1-arcsec specification; if it were, the spectral resolving power of the system would improve correspondingly.

1992 (5)

1991 (3)

1989 (1)

A. Takahashi, “Optical transfer function-based merit functions for automatic diffraction grating system design,” J. Mod. Opt. 36, 675–684 (1989).
[CrossRef]

1984 (1)

1974 (1)

1961 (1)

Cash, W. C.

Content, D.

Content, D. A.

D. A. Content, P. M. Davila, J. F. Osantowski, T. T. Saha, M. E. Wilson, “Optical design of Lyman/FUSE,” in Instrumentation in Astronomy II, D. L. Crawford, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1235, 943–952 (1990).

Davila, P.

Davila, P. M.

D. A. Content, P. M. Davila, J. F. Osantowski, T. T. Saha, M. E. Wilson, “Optical design of Lyman/FUSE,” in Instrumentation in Astronomy II, D. L. Crawford, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1235, 943–952 (1990).

Duban, M.

Grange, R.

Harada, Y.

M. Koike, Y. Harada, H. Noda, “New blazed holographic grating fabricated by using an aspherical recording with an ion-etching method,” in Application and Theory of Periodic Structures, Diffraction Gratings, and Moire Phenomena III, J. M. Lerner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.815, 96–101 (1988).

Koike, M.

T. Namioka, M. Koike, “Analytical representation of spot diagrams and its application to the design of monochromators,” Nucl. Instrum. Methods A 319, 219–227 (1992).
[CrossRef]

M. Koike, Y. Harada, H. Noda, “New blazed holographic grating fabricated by using an aspherical recording with an ion-etching method,” in Application and Theory of Periodic Structures, Diffraction Gratings, and Moire Phenomena III, J. M. Lerner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.815, 96–101 (1988).

T. Namioka, M. Koike, “Design of compact high-resolution far-ultraviolet spectrographs equipped with a spherical grating having variable spacing and curved grooves,” presented at the Tenth International Colloquium on UV and X-Ray Spectroscopy of Astrophysical and Laboratory Plasmas, Berkeley, Calif., 3–5 February 1992.

Laget, M.

Namioka, T.

T. Namioka, M. Koike, “Analytical representation of spot diagrams and its application to the design of monochromators,” Nucl. Instrum. Methods A 319, 219–227 (1992).
[CrossRef]

H. Noda, T. Namioka, M. Seya, “Geometric theory of the grating,” J. Opt. Soc. Am. 64, 1031–1036 (1974).
[CrossRef]

T. Namioka, “Theory of the ellipsoidal concave grating. I,” J. Opt. Soc. Am. 51, 4–12 (1961); “Theory of the ellipsoidal concave grating. II. Application of the theory to the specific grating mountings,” J. Opt. Soc. Am. 51, 13–16 (1961).
[CrossRef]

T. Namioka, M. Koike, “Design of compact high-resolution far-ultraviolet spectrographs equipped with a spherical grating having variable spacing and curved grooves,” presented at the Tenth International Colloquium on UV and X-Ray Spectroscopy of Astrophysical and Laboratory Plasmas, Berkeley, Calif., 3–5 February 1992.

Noda, H.

H. Noda, T. Namioka, M. Seya, “Geometric theory of the grating,” J. Opt. Soc. Am. 64, 1031–1036 (1974).
[CrossRef]

M. Koike, Y. Harada, H. Noda, “New blazed holographic grating fabricated by using an aspherical recording with an ion-etching method,” in Application and Theory of Periodic Structures, Diffraction Gratings, and Moire Phenomena III, J. M. Lerner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.815, 96–101 (1988).

Osantowski, J. F.

D. A. Content, P. M. Davila, J. F. Osantowski, T. T. Saha, M. E. Wilson, “Optical design of Lyman/FUSE,” in Instrumentation in Astronomy II, D. L. Crawford, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1235, 943–952 (1990).

Saha, T. T.

D. A. Content, P. M. Davila, J. F. Osantowski, T. T. Saha, M. E. Wilson, “Optical design of Lyman/FUSE,” in Instrumentation in Astronomy II, D. L. Crawford, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1235, 943–952 (1990).

Seya, M.

Takahashi, A.

A. Takahashi, “Optical transfer function-based merit functions for automatic diffraction grating system design,” J. Mod. Opt. 36, 675–684 (1989).
[CrossRef]

Trout, C.

Wilson, M.

Wilson, M. E.

D. A. Content, P. M. Davila, J. F. Osantowski, T. T. Saha, M. E. Wilson, “Optical design of Lyman/FUSE,” in Instrumentation in Astronomy II, D. L. Crawford, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1235, 943–952 (1990).

Appl. Opt. (8)

J. Mod. Opt. (1)

A. Takahashi, “Optical transfer function-based merit functions for automatic diffraction grating system design,” J. Mod. Opt. 36, 675–684 (1989).
[CrossRef]

J. Opt. Soc. Am. (2)

Nucl. Instrum. Methods A (1)

T. Namioka, M. Koike, “Analytical representation of spot diagrams and its application to the design of monochromators,” Nucl. Instrum. Methods A 319, 219–227 (1992).
[CrossRef]

Other (5)

M. Koike, Y. Harada, H. Noda, “New blazed holographic grating fabricated by using an aspherical recording with an ion-etching method,” in Application and Theory of Periodic Structures, Diffraction Gratings, and Moire Phenomena III, J. M. Lerner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.815, 96–101 (1988).

It is assumed here that the imaging of the FUSE telescope will not be drastically better than the current 1-arcsec specification; if it were, the spectral resolving power of the system would improve correspondingly.

H. W. Moos, ed., LYMAN, the Far Ultraviolet Spectroscopic Explorer, Phase A Study Final Report (NASA Goddard Space Flight Center, Greenbelt, Md., 1989).

D. A. Content, P. M. Davila, J. F. Osantowski, T. T. Saha, M. E. Wilson, “Optical design of Lyman/FUSE,” in Instrumentation in Astronomy II, D. L. Crawford, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1235, 943–952 (1990).

T. Namioka, M. Koike, “Design of compact high-resolution far-ultraviolet spectrographs equipped with a spherical grating having variable spacing and curved grooves,” presented at the Tenth International Colloquium on UV and X-Ray Spectroscopy of Astrophysical and Laboratory Plasmas, Berkeley, Calif., 3–5 February 1992.

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Figures (7)

Fig. 1
Fig. 1

Schematic layout diagram of the Rowland circle mount and quasi-Rowland circle mounts. The vertex O of the grating is taken as the origin with the grating normal as the x axis and the plane of symmetry of the grating as the xy plane. The quasi-Rowland circle mount places the detector on the Rowland circle and the entrance slit at a point on line AT, where A is the slit position of the Rowland circle mount and T is the intersection of line OA and the tangent to the Rowland circle at point N, which is the diametrical point of O. When the slit is placed at point T, the mount is referred to as the tangential Mount. Points C and D represent the holographic recording point sources (in the figure they are located on the Rowland circle).

Fig. 2
Fig. 2

Spectral resolving power versus wavelength for the three differing cases in Table 1 (columns labeled 1, 3, and 4). A slit width of 1 arcsec was assumed in the ray tracing. The Rowland circle mount (solid curve) has essentially ideal performance. The tangential mount (dotted curve) and the quasi-Rowland circle mount with rA = R (dashed curve) show degradation away from the center of the spectral range.

Fig. 3
Fig. 3

Spectral rms spot size versus wavelength for the three cases of Fig. 2. Spectral imaging is primarily limited by astigmatic curvature M120 and also by coma M300 in the quasi-Rowland circle mounts.

Fig. 4
Fig. 4

Spatial rms spot size versus wavelength for the three cases of Fig. 2. The spatial spot size (measured perpendicular to the plane of the Rowland circle) is controlled by the ellipsoidal correction of astigmatism and so is not strongly affected by the mount chosen.

Fig. 5
Fig. 5

Spot diagrams for a point source on axis at 91, 94, 97, 100, and 103 nm (from left to right) for the Rowland circle mount design. A 1 arcsec × 3 arcsec box is shown for scale. The spots indicate cancellation of astigmatism near 94 and 100 nm and high spectral resolution throughout the bandpass.

Fig. 6
Fig. 6

Spot diagrams for a point source on axis at 91, 94, 97, 100, and 103 nm (from left to right) for the tangential mount design. A 1 arcsec × 3 arcsec box is shown for scale.

Fig. 7
Fig. 7

Spot diagrams for a point source on axis at 91, 94, 97, 100, and 103 nm (from left to right) for the quasi-Rowland circle mount design with rA = R. A 1 arcsec × 3 arcsec box is shown for scale.

Tables (1)

Tables Icon

Table 1 Grating Designs for the FUSE 91–103-nm Bandpass

Equations (52)

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u = a a [ 1 ( w 2 b 2 + l 2 c 2 ) ] 1 / 2 + 12 w l 2 + 30 w 3 + 40 w 4 + 22 w 2 l 2 + 04 l 4 ,
F = A P + P B + n m λ = r A + r B + w [ ( m λ / d ) ( sin α + sin β ) ] l [ ( z A / r A ) + ( z B / r B ) ] + 1 2 w 2 F 200 + 1 2 l 2 F 020 + 1 2 z A 2 F 002 + 1 2 w 3 F 300 + 1 2 w l 2 F 120 + w l z A F 111 + 1 2 w z A 2 F 102 + 1 8 w 4 F 400 + 1 4 w 2 l 2 F 220 + 1 8 l 4 F 040 + 1 4 w 2 z A 2 F 202 + 1 2 l 3 z A F 031 + 1 4 w 2 z A 2 F 022 + 1 2 l z A 3 F 013 + 1 2 w 2 l z A F 211 + 1 8 z A 4 F 004 + ,
F i j k = M i j k + ( m λ / λ 0 ) H i j k .
p [ β 1 β 2 ( F i j 0 ) 2 d β ] = 0 or p [ β 1 β 2 ( M i j 0 ) 2 d β ] = 0 .
b 2 c 2 = 1 D [ 4 β m + 2 sec α sin β m + cos α log ( 1 + sin β m 1 sin β m ) ] = sec α sec β a ,
D = β m ( 1 + 2 cos 2 α ) + sin β m ( cos β m + 4 cos α )
12 = 1 R 2 D [ tan α ( sec α a R c 2 cos α ) ( β m cos α + sin β m ) ] ,
40 = 1 8 R 2 ( a 2 c 2 a c 2 ) ,
22 = b 2 4 R 3 c 2 ( 2 R a ) + 1 4 R 3 D { [ 2 sec α ( 2 tan 2 α b 2 c 2 ) 8 12 R 2 sin α ] ( β m cos α + sin β m ) 2 β m ( 2 + b 2 c 2 ) + 2 tan β m ( 2 + cos α sec β m ) cos α ( 1 + b 2 c 2 ) log ( 1 + sin β m 1 sin β m ) } ,
04 = b 4 8 R 3 c 4 ( 1 + R a ) + 1 8 R 3 D × [ 2 sec α ( b 4 c 4 + 2 b 2 c 2 sec 2 α ) ( β m cos α + sin β m ) + 2 β m ( b 4 c 4 + 2 b 2 c 2 ) tan β m ( 2 + cos α sec β m ) + cos α ( b 4 c 4 + 2 b 2 c 2 1 2 ) log ( 1 + sin β m 1 sin β m ) ] .
a = c = R D 4 β m + 2 sec α sin β m + cos α log ( 1 + sin β m 1 sin β m ) , b = R a .
H 200 = ( cos 2 γ r c cos γ R ) ( cos 2 δ r D cos δ R ) = 0 ,
r c = R cos γ , r D = R cos δ .
H 200 = H 020 = H 300 = H 400 = H 040 = 0 , H 120 = ( 2 S D / R ) tan δ , H 220 = ( 8 12 / R ) sin δ .
12 = 1 R 2 D sin α ( β m cos α + sin β m ) ( sec 2 α sec 2 δ ) ,
22 = [ Eq . ( 9 ) ] 12 = 0 .
[ M 020 ] conv [ F 020 ] holo = [ M 400 ] conv [ F 400 ] hoIo = [ M 040 ] conv [ F 040 ] holo = 0 , [ M 120 ] conv [ F 120 ] holo = E R 2 [ sec 2 δ ( b 2 / c 2 ) ] , [ M 220 ] conv [ F 220 ] holo = 4 E R [ 12 ] conv ,
E = sin α + sin β 2 D sin α ( cos α + cos β ) × ( β m cos α + sin β m ) .
H 300 = T C ( sin γ r c sin δ r D ) = T D ( sin γ r C sin δ r D ) = 0 .
r D r C = sin δ sin γ < 0 if γ < 0 , > 0 if γ > 0 .
a R b 2 σ = 1 D { β m ( 1 + 2 ρ A cos 2 α ) + sin β m [ cos β m + 2 cos α ( 1 + ρ A ) ] } ,
a R c 2 τ = 1 D [ 2 β m ( 1 + ρ A ) + 2 ρ A sec α sin β m + cos α log ( 1 + sin β m 1 sin β m ) ] ,
ρ A R cos α / r A .
a = c = R / τ , b = R / στ .
12 = 1 R 2 D × [ ρ A tan α ( β m cos α + sin β m ) ( ρ A sec α τ cos α ) ] ,
30 = 1 R 2 D [ ρ A sin α ( β m cos α + sin β m ) ( ρ A σ ) ] ,
40 = a 8 b 4 + 1 8 R 3 D [ 2 A 40 ( β m cos α + sin β m ) + β m ( σ 2 2 σ + 3 ) ( 5 σ ) ( 1 σ ) × sin β m ( cos β m + 2 cos α ) + ( 2 σ ) 2 cos α log ( 1 + sin β m 1 sin β m ) ] ,
22 = τ 4 R 3 ( 2 R 2 b 2 ) + 1 4 R 3 D [ 2 A 22 ( β m cos α + sin β m ) β m ( 3 σ ) ( 2 + τ ) + 2 tan β m ( 2 + cos α sec β m ) + τ ( 1 σ ) sin β m ( 2 cos α + cos β m ) ( 1 + τ ) ( 2 σ ) cos α log ( 1 + sin β m 1 sin β m ) ] ,
04 = τ 2 8 R 3 ( 1 + R a ) + 1 8 R 3 D [ 2 A 04 ( β m cos α + sin β m ) + 2 β m τ ( τ + 2 ) tan β m ( 2 + cos α sec β m ) + ( τ 2 + 2 τ 1 2 ) cos α log ( 1 + sin β m 1 sin β m ) ] ,
A 40 = 4 T A R ρ A 2 tan 2 α + ( σ 2 R 2 T A 2 ) ρ A sec α 8 30 R 2 ρ A sin α ,
A 22 = 2 ρ A 2 R S ¯ A tan 2 α + ρ A sec α ( στ R 2 T A S ¯ A ) 2 τ cos α 4 ρ A 12 R 2 sin α ,
A 04 = τ 2 cos α ( 1 + ρ A tan 2 α ) + 2 τρ A 2 sec α ρ A 3 sec 3 α .
M 200 = T A + T B ,
M 200 = S ¯ A + S ¯ B ,
M 300 = T A r A sin α + T B r B sin β 2 30 ( cos α + cos β ) ,
M 120 = S ¯ A r A sin α + S ¯ B r B sin β 2 12 ( cos α + cos β ) ,
M 400 = 4 T A r A 2 sin 2 α T A 2 r A + 4 T B r B 2 sin 2 β T B 2 r B + a 2 b 4 × ( S A + S B ) 8 30 ( 1 r A sin α cos α + 1 r B sin β cos β ) 8 40 ( cos α + cos β ) ,
M 220 = 2 S ¯ A r A 2 sin 2 α T A S ¯ A r A + 2 S ¯ B r B 2 sin 2 β T B S ¯ B r B + a 2 b 2 c 2 ( S A + S B ) 4 12 ( 1 r A sin α cos α + 1 r B sin β cos β ) 4 22 ( cos α + cos β ) ,
M 040 = a 2 c 4 ( S A + S B ) S ¯ A 2 r A S ¯ B 2 r B 8 04 ( cos α + cos β ) ,
T A = cos 2 α r A a cos α b 2 , T B = cos 2 β r B a cos β b 2 ,
S A = 1 r A cos α a , S B = 1 r B cos β a ,
S ¯ A = 1 r A a cos α c 2 , S ¯ B = 1 r B a cos β c 2 .
H 200 = T C T D ,
H 020 = S ¯ C S ¯ D ,
H 300 = T C r C sin γ T D r D sin δ 2 30 ( cos γ cos δ ) ,
H 120 = S ¯ C r C sin γ S ¯ D r D sin δ 2 12 ( cos γ cos δ ) ,
H 400 = 4 T C r C 2 sin 2 γ T C 2 r C 4 T D r D 2 sin 2 δ + T D 2 r D + a 2 b 4 × ( S C S D ) 8 30 ( 1 r C sin γ cos γ 1 r D sin δ cos δ ) 8 40 ( cos γ cos δ ) ,
H 220 = 2 S ¯ C r C 2 sin 2 γ T C S ¯ C r C 2 S ¯ D r D 2 sin 2 δ + T D S ¯ D r D + a 2 b 2 c 2 × ( S C S D ) 4 12 ( 1 r C sin γ cos γ 1 r D sin δ cos δ ) 4 22 ( cos γ cos δ ) ,
H 040 = a 2 c 4 ( S C S D ) S ¯ C 2 r C + S ¯ D 2 r D 8 04 ( cos γ cos δ ) ,
T C = cos 2 γ r C a cos γ b 2 , T D = cos 2 δ r D a cos δ b 2 ,
S C = 1 r C cos γ a , S D = 1 r D cos δ a ,
S ¯ C = 1 r C a cos γ c 2 , S ¯ D = 1 r D a cos δ c 2 .

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