Abstract

Self-supporting transmission gratings suitable for the soft x-ray and extreme ultraviolet must be reinforced by support structures which, if regularly placed, tend to produce undesirable artifacts in the diffraction plane. Because these artifacts appear in the neighborhood of the principal maxima and can be substantial in magnitude, they may confuse the spectrum. Methods are described whereby these unwanted diffraction effects are much reduced or eliminated. In one method, the members of the support structure parallel to the grating slits are placed in a random pattern so that on the average the coherence of the support structure is drastically reduced everywhere. In a second method, these support structure members are distributed pseudorandomly in such a manner that the diffraction pattern of the support structure is completely removed from the diffraction plane. A third method is investigated in which the pseudorandom placement of supports is organized into a repeated pattern, which may be easier to manufacture than the configurations of the first two methods.

© 1989 Optical Society of America

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References

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  1. K. S. Wood et al., “The HEAO A-1 X-Ray Source Catalog,” Astrophys. J. Suppl. 56, 507 (1984).
    [CrossRef]
  2. H. Gursky, T. Zehnpfennig, “An Image-Forming Slitless Spectrometer for Soft X-Ray Astronomy,” Appl. Opt. 5, 875 (1966).
    [PubMed]
  3. K. P. Beuermann, H. Brauninger, J. Trumper, “Aberrations of a Facet-Type Transmission Grating for Cosmic X-Ray and XUV Spectroscopy,” Appl. Opt. 17, 2304 (1978).
    [CrossRef] [PubMed]
  4. A. C. Brinkman, J. H. Dijkstra, W. F. P. A. L. Geerlings, F. A. van Rooijen, C. Timmerman, P. A. J. de Korte, “Efficiency and Resolution Measurements of X-Ray Transmission Gratings Between 7.1 and 304 Å,” Appl. Opt. 19, 1601 (1980).
    [CrossRef] [PubMed]
  5. F. D. Seward et al., “Calibration and Efficiency of the Einstein Objective Grating Spectrometer,” Appl. Opt. 21, 2012 (1982).
    [CrossRef] [PubMed]
  6. H. Brauninger, H. Kraus, H. Dangschat, K. P. Beuermann, P. Predehl, J. Trumper, “Fabrication of Transmission Gratings for Use in Cosmic X-Ray and XUV Astronomy,” Appl. Opt. 18, 3502 (1979).
    [CrossRef] [PubMed]
  7. J. A. Hogbom, “Aperture Synthesis with a Non-Regular Distribution of Interferometer Baselines,” Astron. Astrophys. Suppl. 15, 417 (1974).

1984 (1)

K. S. Wood et al., “The HEAO A-1 X-Ray Source Catalog,” Astrophys. J. Suppl. 56, 507 (1984).
[CrossRef]

1982 (1)

1980 (1)

1979 (1)

1978 (1)

1974 (1)

J. A. Hogbom, “Aperture Synthesis with a Non-Regular Distribution of Interferometer Baselines,” Astron. Astrophys. Suppl. 15, 417 (1974).

1966 (1)

Beuermann, K. P.

Brauninger, H.

Brinkman, A. C.

Dangschat, H.

de Korte, P. A. J.

Dijkstra, J. H.

Geerlings, W. F. P. A. L.

Gursky, H.

Hogbom, J. A.

J. A. Hogbom, “Aperture Synthesis with a Non-Regular Distribution of Interferometer Baselines,” Astron. Astrophys. Suppl. 15, 417 (1974).

Kraus, H.

Predehl, P.

Seward, F. D.

Timmerman, C.

Trumper, J.

van Rooijen, F. A.

Wood, K. S.

K. S. Wood et al., “The HEAO A-1 X-Ray Source Catalog,” Astrophys. J. Suppl. 56, 507 (1984).
[CrossRef]

Zehnpfennig, T.

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Figures (8)

Fig. 1
Fig. 1

Schematic of the orientation of the facet grating and the relationship of the vectors R, r, and x. The vector x lies in the plane of the grating.

Fig. 2
Fig. 2

Schematic diagram of a single grating chip. The rectangular slits are oriented parallel to the x axis, have dimensions Δx and Δy, and are placed at periodic intervals of 2Δy. The chip has dimensions ξΔx and (2L + ηy, where L is the number of slits. In order that the slit spacing is maintained from one chip to the next, η is an integer multiple of 2Δy.

Fig. 3
Fig. 3

Contour plot of the intensity of the diffracted radiation for a grating with a regularly spaced support structure, and where L = 5, N = 7, J = 5, ξ = 1.1, and K = 98 (see text). The uppermost contour is one-tenth of the zero-order intensity. Both axes are labeled in units of π radians.

Fig. 4
Fig. 4

Relative intensity of the diffracted radiation in the diffraction plane for the case given in Fig. 3.

Fig. 5
Fig. 5

Example of random placement of the supports which are parallel to the slits.

Fig. 6
Fig. 6

Contour plots of the intensity of the diffracted radiation from a grating in which the supports parallel to the slits are randomly placed, and where L = 5, N = 7, J = 5, ξ = 1.1, and K = 98 (see text). (a) The uppermost contour is one-tenth of the zero-order intensity. (b) The contour plot in (a) has been redrawn to emphasize weak features. The uppermost contour is 1/100 of the zero-order intensity.

Fig. 7
Fig. 7

Relative intensity of the diffracted radiation in the diffraction plane for the random case given in Fig. 6.

Fig. 8
Fig. 8

Contour plot of the intensity of the diffracted radiation from a grating in which the supports parallel to the slits are pseudorandomly placed, and where L = 5, N = 7, J = 5, ξ = 1.1, and K = 98 (see text). The uppermost contour is 1/100 of the zero-order intensity.

Equations (28)

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R = R ( sin θ cos ϕ e ^ x + sin θ sin ϕ e ^ y + cos θ e ^ z ) ,
r R x sin θ cos ϕ y sin θ sin ϕ .
exp ( i k x + i k r ) exp [ 2 π i ( R + u y + υ x ) / λ ] ,
A ( slit ) = Q exp [ 2 π i ( u Y + υ X ) ] ( sin α / α ) ( sin β / β ) ,
P ( slit ) = | A ( slit ) | 2 , = ( sin α / α ) ( sin β / β ) 2 ,
α = 0 , α = tan α ; α ± ( m + 3 / 2 ) π , β = 0 , β = tan β ; β ± ( m + 3 / 2 ) π ,
l = 0 L 1 exp ( 4 π i l u Δ y / λ ) = [ exp ( 4 π i L u Δ y / λ ) 1 ] [ exp ( 4 π i u Δ y / λ ) 1 ] = Q sin ( 2 L β ) / sin ( 2 β ) .
A ( chip ) = Q exp [ 2 π i ( u Y + υ X ) ] ( sin α / α ) × [ sin ( 2 L β ) / sin ( 2 β ) ] ( sin β / β ) ,
β = n π / 2 , tan ( 2 L β ) = L tan ( 2 β ) ; β ± ( m + 3 / 2 ) π / 2 L ,
X = k ξ Δ x , Y = 2 j N Δ y ,
j = 0 j 1 exp ( 4 π i u j N Δ y / λ ) = [ exp ( 4 π i u J N Δ y / λ ) 1 ] [ exp ( 4 π i u N Δ y / λ ) 1 ] = Q sin ( 2 J N β ) / sin ( 2 N β ) ,
k = 0 K 1 exp ( 2 π i υ k ξ Δ x / λ ) = [ exp ( 2 π i υ K ξ Δ x / λ ) 1 ] [ exp ( 2 π i υ ξ Δ x / λ ) 1 ] = Q sin ( K ξ α ) / sin ( ξ α ) ,
A (total) = Q [ sin ( K ξ α ) / sin ( ξ α ) ] [ sin α / α ] × [ sin ( 2 N J β ) / sin ( 2 N β ) ] × [ sin ( 2 L β ) / sin ( 2 β ) ] [ sin β / β ] .
P ( total ) = | A ( total ) | 2 ,
α = 0 , α ± ( m + 3 / 2 ) π , α = k π / ξ , tan ( K ξ α ) = K tan ( ξ α ) ; α ± ( m + 3 / 2 ) π / K ξ , β = 0 , β ± ( m + 3 / 2 ) π , β = n π / 2 , β ± ( m + 3 / 2 ) π / 2 L , β = k π / 2 N , tan ( 2 J N β ) = J tan ( 2 N β ) ; β ± ( m + 3 / 2 ) π / 2 N L ,
k = 0 K 1 exp { 2 π i [ υ k ξ Δ x + 2 u f ( k ) Δ y ] / λ } = E ( 2 π i υ ξ Δ x / λ ) RF ( 4 π i u Δ y / λ ) ,
A ( total ) = Q E ( 2 π i υ ξ Δ x / λ ) RF ( 4 π i u Δ y / λ ) [ sin α / α ] × [ sin ( 2 N J β ) / sin 2 N β ) ] × [ sin ( 2 L β ) / sin ( 2 β ) ] [ sin β / β ] ,
A = ( 1 / M ) m = 1 M E ( 2 π i υ ξ Δ x / λ ) R m F ( 4 π i u Δ y / λ ) .
A = ( 1 / N ) E ( 2 π i υ ξ Δ x / λ ) R sum F  ( 4 π i u Δ y / λ ) ,
A = ( 1 / N ) [ k = 0 K 1 exp ( 2 π i k υ ξ Δ x / λ ) ] [ m = 0 N 1 exp ( 4 π i u m Δ y / λ ) ] = ( 1 / N ) Q [ sin ( K ξ α ) / sin ( ξ α ) ] [ sin ( 2 N β ) / sin ( 2 β ) ] .
A ( total ) = ( 1 / N ) Q [ sin ( K ξ α ) / sin ( ξ α ) ] [ sin α / α ] × [ sin ( 2 N J β ) / sin ( 2 β ) ] × [ sin ( 2 L β ) / sin ( 2 β ) ] [ sin β / β ] .
P = ( K / N ) [ sin ( 2 N J β ) / sin ( 2 N β ) ] [ sin ( 2 L β ) / sin ( 2 β ) ] [ sin β / β ] 2 ,
β = 0 , β ± ( m + 3 / 2 ) π , β = n π / 2 , β ± ( m + 3 / 2 ) π / 2 L , β ± ( m + 3 / 2 ) π / 2 N J .
A ( total ) = Q T [ sin α / α ] [ sin ( 2 N J β ) / sin ( 2 N β ) ] × [ sin ( 2 L β ) / sin ( 2 β ) ] [ sin β / β ] ,
T = k = 0 K 1 exp [ 2 i k α ξ + 4 i β f ( k ) ] ,
T ( ξ α = j π ) = ( K / N ) Q [ sin ( 2 β N ) / sin ( 2 β ) ] ,
T = Q [ sin ( K ξ α / 2 N ) / sin ( ξ α ) ] k = 0 2 N 1 exp [ 2 i k α ξ + 4 i β f ( k ) ] .
T = 2 Q [ sin ( K ξ α / 2 N ) / sin ( ξ α ) ] k = 0 N 1 exp [ 4 i β f ( k ) ] × cos [ ξ α ( 2 k N + 1 ) ] .

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