Abstract

A method for calculating the image distribution for one-dimensional lobster-eye optics is presented. This method gives the image distribution exactly for certain cases and offers improved speed over the method of ray tracing. Examples of the use of the algorithm are given. We show that the algorithm gives the same results as detailed ray-tracing codes. Extension of the method to the two-dimensional case is discussed.

© 1998 Optical Society of America

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References

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  1. W. C. Priedhorsky, A. G. Peele, K. A. Nugent, “X-ray all-sky monitor with extraordinary sensitivity,” Mon. Not. R. Astron. Soc. 279, 733–750 (1996).
    [CrossRef]
  2. G. W. Fraser, A. N. Brunton, J. E. Lees, J. F. Pearson, W. B. Feller, “X-ray focusing using square-pore microchannel plates. First observation of cruxiform image structure,” Nucl. Instrum. Methods A 324, 404–407 (1993).
    [CrossRef]
  3. A. G. Peele, K. A. Nugent, A. Rode, K. Gabel, M. C. Richardson, R. Strack, W. Siegmund, “X-ray focusing with lobster-eye optics: a comparison of theory with experiment,” Appl. Opt. 35, 4420–4425 (1996).
    [CrossRef] [PubMed]
  4. H. N. Chapman, K. A. Nugent, S. W. Wilkins, “X-ray focusing using square-channel capillary arrays,” Rev. Sci. Instrum. 62, 1542–1561 (1991).
    [CrossRef]
  5. W. K. H. Schmidt, “A proposed x-ray focusing device with wide field of view for use in x-ray astronomy,” Nucl. Instrum. Methods 127, 285–292 (1975).
    [CrossRef]
  6. J. R. P. Angel, “Lobster eyes as x-ray telescopes,” Astrophys. J. 233, 364–373 (1979).
    [CrossRef]
  7. A. G. Peele, “X-ray optics of square channel arrays,” Ph.D. dissertation (The University of Melbourne, Melbourne, Australia, 1996).
  8. D. X. Balaic, K. A. Nugent, “X-ray optics of tapered capillaries,” Appl. Opt. 34, 7263–7272 (1995).
    [CrossRef] [PubMed]
  9. H. N. Chapman, A. R. Rode, “Geometric optics of arrays of reflective surfaces,” Appl. Opt. 33, 2419–2436 (1994).
    [CrossRef] [PubMed]

1996 (2)

1995 (1)

1994 (1)

1993 (1)

G. W. Fraser, A. N. Brunton, J. E. Lees, J. F. Pearson, W. B. Feller, “X-ray focusing using square-pore microchannel plates. First observation of cruxiform image structure,” Nucl. Instrum. Methods A 324, 404–407 (1993).
[CrossRef]

1991 (1)

H. N. Chapman, K. A. Nugent, S. W. Wilkins, “X-ray focusing using square-channel capillary arrays,” Rev. Sci. Instrum. 62, 1542–1561 (1991).
[CrossRef]

1979 (1)

J. R. P. Angel, “Lobster eyes as x-ray telescopes,” Astrophys. J. 233, 364–373 (1979).
[CrossRef]

1975 (1)

W. K. H. Schmidt, “A proposed x-ray focusing device with wide field of view for use in x-ray astronomy,” Nucl. Instrum. Methods 127, 285–292 (1975).
[CrossRef]

Angel, J. R. P.

J. R. P. Angel, “Lobster eyes as x-ray telescopes,” Astrophys. J. 233, 364–373 (1979).
[CrossRef]

Balaic, D. X.

Brunton, A. N.

G. W. Fraser, A. N. Brunton, J. E. Lees, J. F. Pearson, W. B. Feller, “X-ray focusing using square-pore microchannel plates. First observation of cruxiform image structure,” Nucl. Instrum. Methods A 324, 404–407 (1993).
[CrossRef]

Chapman, H. N.

H. N. Chapman, A. R. Rode, “Geometric optics of arrays of reflective surfaces,” Appl. Opt. 33, 2419–2436 (1994).
[CrossRef] [PubMed]

H. N. Chapman, K. A. Nugent, S. W. Wilkins, “X-ray focusing using square-channel capillary arrays,” Rev. Sci. Instrum. 62, 1542–1561 (1991).
[CrossRef]

Feller, W. B.

G. W. Fraser, A. N. Brunton, J. E. Lees, J. F. Pearson, W. B. Feller, “X-ray focusing using square-pore microchannel plates. First observation of cruxiform image structure,” Nucl. Instrum. Methods A 324, 404–407 (1993).
[CrossRef]

Fraser, G. W.

G. W. Fraser, A. N. Brunton, J. E. Lees, J. F. Pearson, W. B. Feller, “X-ray focusing using square-pore microchannel plates. First observation of cruxiform image structure,” Nucl. Instrum. Methods A 324, 404–407 (1993).
[CrossRef]

Gabel, K.

Lees, J. E.

G. W. Fraser, A. N. Brunton, J. E. Lees, J. F. Pearson, W. B. Feller, “X-ray focusing using square-pore microchannel plates. First observation of cruxiform image structure,” Nucl. Instrum. Methods A 324, 404–407 (1993).
[CrossRef]

Nugent, K. A.

A. G. Peele, K. A. Nugent, A. Rode, K. Gabel, M. C. Richardson, R. Strack, W. Siegmund, “X-ray focusing with lobster-eye optics: a comparison of theory with experiment,” Appl. Opt. 35, 4420–4425 (1996).
[CrossRef] [PubMed]

W. C. Priedhorsky, A. G. Peele, K. A. Nugent, “X-ray all-sky monitor with extraordinary sensitivity,” Mon. Not. R. Astron. Soc. 279, 733–750 (1996).
[CrossRef]

D. X. Balaic, K. A. Nugent, “X-ray optics of tapered capillaries,” Appl. Opt. 34, 7263–7272 (1995).
[CrossRef] [PubMed]

H. N. Chapman, K. A. Nugent, S. W. Wilkins, “X-ray focusing using square-channel capillary arrays,” Rev. Sci. Instrum. 62, 1542–1561 (1991).
[CrossRef]

Pearson, J. F.

G. W. Fraser, A. N. Brunton, J. E. Lees, J. F. Pearson, W. B. Feller, “X-ray focusing using square-pore microchannel plates. First observation of cruxiform image structure,” Nucl. Instrum. Methods A 324, 404–407 (1993).
[CrossRef]

Peele, A. G.

A. G. Peele, K. A. Nugent, A. Rode, K. Gabel, M. C. Richardson, R. Strack, W. Siegmund, “X-ray focusing with lobster-eye optics: a comparison of theory with experiment,” Appl. Opt. 35, 4420–4425 (1996).
[CrossRef] [PubMed]

W. C. Priedhorsky, A. G. Peele, K. A. Nugent, “X-ray all-sky monitor with extraordinary sensitivity,” Mon. Not. R. Astron. Soc. 279, 733–750 (1996).
[CrossRef]

A. G. Peele, “X-ray optics of square channel arrays,” Ph.D. dissertation (The University of Melbourne, Melbourne, Australia, 1996).

Priedhorsky, W. C.

W. C. Priedhorsky, A. G. Peele, K. A. Nugent, “X-ray all-sky monitor with extraordinary sensitivity,” Mon. Not. R. Astron. Soc. 279, 733–750 (1996).
[CrossRef]

Richardson, M. C.

Rode, A.

Rode, A. R.

Schmidt, W. K. H.

W. K. H. Schmidt, “A proposed x-ray focusing device with wide field of view for use in x-ray astronomy,” Nucl. Instrum. Methods 127, 285–292 (1975).
[CrossRef]

Siegmund, W.

Strack, R.

Wilkins, S. W.

H. N. Chapman, K. A. Nugent, S. W. Wilkins, “X-ray focusing using square-channel capillary arrays,” Rev. Sci. Instrum. 62, 1542–1561 (1991).
[CrossRef]

Appl. Opt. (3)

Astrophys. J. (1)

J. R. P. Angel, “Lobster eyes as x-ray telescopes,” Astrophys. J. 233, 364–373 (1979).
[CrossRef]

Mon. Not. R. Astron. Soc. (1)

W. C. Priedhorsky, A. G. Peele, K. A. Nugent, “X-ray all-sky monitor with extraordinary sensitivity,” Mon. Not. R. Astron. Soc. 279, 733–750 (1996).
[CrossRef]

Nucl. Instrum. Methods (1)

W. K. H. Schmidt, “A proposed x-ray focusing device with wide field of view for use in x-ray astronomy,” Nucl. Instrum. Methods 127, 285–292 (1975).
[CrossRef]

Nucl. Instrum. Methods A (1)

G. W. Fraser, A. N. Brunton, J. E. Lees, J. F. Pearson, W. B. Feller, “X-ray focusing using square-pore microchannel plates. First observation of cruxiform image structure,” Nucl. Instrum. Methods A 324, 404–407 (1993).
[CrossRef]

Rev. Sci. Instrum. (1)

H. N. Chapman, K. A. Nugent, S. W. Wilkins, “X-ray focusing using square-channel capillary arrays,” Rev. Sci. Instrum. 62, 1542–1561 (1991).
[CrossRef]

Other (1)

A. G. Peele, “X-ray optics of square channel arrays,” Ph.D. dissertation (The University of Melbourne, Melbourne, Australia, 1996).

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Figures (15)

Fig. 1
Fig. 1

Cylindrical array viewed along the x axis. The radius of curvature, R, and array thickness, t, are shown.

Fig. 2
Fig. 2

Acceptance widths for zero, one, two, and three reflections.

Fig. 3
Fig. 3

Projected view of general channel rotated to align with the optic axis. The angle a ray makes with the wall of the channel is γ n ; the z and the y intersections for the ray corresponding to the nth reflection are y n and z n . The angle made with the optic axis is θ n , and the gradient of the nth ray is m n . The width of a channel wall w and the interior ε and exterior ε f channel taper angles are also shown.

Fig. 4
Fig. 4

Schematic showing parallel-sided reflectors with effective taper angle and radius of curvature.

Fig. 5
Fig. 5

Reflection end points for locating acceptance widths. The maximum and the minimum number of reflections for a channel are found when the incident ray enters the channel at the upper and the lower surface as in (a) and (b), respectively. The height of the entry point of the rays in (c) and (d) locates one end of the acceptance width. In (c), the entry point gives the lower end of δ nmax, which is also the upper end of δ nmax-1. In (d), the entry point gives the upper end of δ nmin, which is also the lower end of δ nmin+1.

Fig. 6
Fig. 6

Calculated acceptance widths for an array with R = 0.5 m, d = 500 μm, and t = 0.1 m. The solid curve is the approximate calculation with the method of either Schmidt5 or Chapman et al.4 The dashed curves are calculated with the exact method described in the text. The first half-triangle is the efficiency for zero reflections, the next triangle is for one reflection, and so on.

Fig. 7
Fig. 7

Intensity distribution for one-dimensional array.

Fig. 8
Fig. 8

Transverse scan across the focal line of Fig. 7. The dashed curve is the prediction made by Chapman et al.4 All parameters are as for Fig. 6. The full width of the line focus for the exact case is indicated by the dotted-dash lines.

Fig. 9
Fig. 9

Projected acceptance widths for the 7th, the 8th, and the 9th channels above and below the optic axis for one reflection and the sum of those projections.

Fig. 10
Fig. 10

Transverse scans across the focal line with the Fresnel reflectivity (solid curve) and unit reflectivity (dashed curve) for Schott 8092 glass at a 1.5-keV energy. All other parameters are as for Fig. 8. The full-width of the line focus for the Fresnel reflectivity case is indicated by the dotted-dash lines.

Fig. 11
Fig. 11

Relative alignment of channels in a spherical array. Sizes of the channels and the curvatures are greatly exaggerated.

Fig. 12
Fig. 12

Intensity distribution for two-dimensional array with Fresnel reflectivity and the parameters in Fig. 10. The high-intensity central focus and the two-arm foci are clear, whereas the low-intensity checkerboard pattern background should be just observable.

Fig. 13
Fig. 13

Comparison between the result shown in Fig. 10 (solid curve) and the ray-trace simulation for an array and a detector with the same parameters (plotted points). Within the noise level of the simulation, the two are in exact agreement.

Fig. 14
Fig. 14

Flux in a 400-μm collector as a function of array thickness. The solid curve was generated by use of the algorithm; the data points were generated by use of the ray trace. The array parameters are described in the text.

Fig. 15
Fig. 15

Comparison of (a) exact algorithm result and (b) ray-trace result.

Equations (18)

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z + y tan θ y ± ε 2 = 0 ,
γ n = γ 1 + n - 1 ε .
θ n = γ n - ε 2 = ψ y + θ y + n - 1 ε ,
m n = - - 1 n tan ( ψ y + θ y + n - 1 ε ) .
y n = - 1 n tan ε 2 z n .
y n = m n z n - z n - 1 + y n - 1 .
z n = z n - 1 sin ψ y + θ y + n - 2 ε sin ψ y + θ y + n - 1 ε ,
z n = z 1 sin ψ y + θ y sin ψ y + θ y + n - 1 ε .
δ n = R + t 2 sin ψ y + θ y - R - t 2 sin ψ y + θ y + n - 1 ε - i = n + 1 n   max   δ i .
y image = - - 1 n + 1 tan ψ y + θ y - ε 2 + n ε + - 1 n + 1 × ψ y + θ y - ε 2 Z - z n + y n .
z n = - R + t 2 sin ψ y + θ y sin ψ y + θ y + n - 1 ε cos ψ y + θ y - ε 2 - - 1 n ε 2 - R - t 2 cos ψ y + θ y - ε 2 - - 1 n + 1 ε 2 - R - t 2 cos ψ y + θ y - ε 2 - - 1 n ε 2 - R + t 2 sin ψ y + θ y - ε sin ψ y + θ y + n - 1 ε cos ψ y + θ y - ε 2 - - 1 n ε 2 n = n   max ,   upper n   max > n n   min ,   upper n   max n > n   min ,   lower n = n   min ,   lower , y n = R + t 2 sin ψ y + θ y sin ψ y + θ y + n - 1 ε sin ψ y + θ y - ε 2 - - 1 n ε 2 R - t 2 sin ψ y + θ y - ε 2 - - 1 n + 1 ε 2 R - t 2 sin ψ y + θ y - ε 2 - - 1 n ε 2 R + t 2 sin ψ y + θ y - ε sin ψ y + θ y + n - 1 ε sin ψ y + θ y - ε 2 - - 1 n ε 2 n = n   max ,   upper n   max > n n   min ,   upper n   max n > n   min ,   lower n = n   min ,   lower .
I y = k = 0 N j = n   min n   max i = 0 j   R i ψ y + i - 1 2 ε + k ε f × y n u k ,   y n l k ,
sin   ψ x = sin   ψ x / sin cos - 1 sin   2 θ y + ε / 2 cos   ψ x ,
z + x tan θ x ± ε 2 = 0 ,
y image = - - 1 n + 1 tan   α l s - z n + y n ,
tan   α = k d + 2 w + d / 2 l s - t / 2 k d + 2 w + d / 2 + nd l s + t / 2 k d + 2 w + d / 2 + n - 1 d l s + t / 2 k - 1 d + 2 w + d / 2 l s - t / 2 n = n max ,   upper n max > n n min ,   upper n max n > n min ,   lower n = n min ,   lower .
z n = - t 2 + n - 1 d tan   α t 2 t 2 - t 2 + n   d tan   α n = n max ,   upper n max > n n min ,   upper n max n > n min ,   lower n = n min ,   lower .
y n = k d + 2 w - - 1 n d 2 k d + 2 w - - 1 n + 1 d 2 k d + 2 w - - 1 n d 2 k d + 2 w - - 1 n d 2 n = n max ,   upper n max > n n min ,   upper n max n > n min ,   lower n = n min ,   lower .

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