Abstract

Nested flat plates bent into the Kirkpatrick-Baez parabolic geometry and supported at discrete points are one means of obtaining large area grazing incidence x-ray optics with good angular resolution. A method for optimizing the on-axis resolution combining finite element and ray trace techniques with selective masking is presented. The optimally determined supported point location technique can lead to greatly reduced in situ labor costs.

© 1981 Optical Society of America

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References

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  1. P. Gorenstein, “X-ray Optics for the LAMAR Facility, An Overview,” Proc. Soc. Photo-Opt. Instrum. Eng. 184, 63 (1979).
  2. P. Kirkpatrick, A. V. Baez, J. Opt. Soc. Am. 38, 766 (1948).
    [CrossRef] [PubMed]
  3. There are other known methods of supporting K/B mirror elements including that of Underwood.4 However, the described support system has previously been utilized5 and is the system chosen as the reference system.
  4. J. H. Underwood, Space Sci. Instrum. 3, 259 (1977).
  5. P. Gorenstein, A. DeCaprio, R. Chase, B. Harris, Rev. Sci. Instrum. 44, 539 (1973).
    [CrossRef]
  6. The problem of determining the deformations, strains, and stresses of a thin rectangular plate arbitrarily loaded and supported is one of continuum mechanics. To derive the proper differential equations along with associated boundary and loading conditions is not an easy task. The underlying philosophy of the FEM is that we can construct a numerical model of the continuous structure which will result in a good approximation to the desired solution. To establish this solution, the numerical model is constructed of small individual elements whose solutions within their domain we can accurately approximate. The boundary conditions of this numerical model will also be approximated by some set of generalized deformations. The errors associated with the solution are dependent upon the size of the elements and the degrees of freedom (allowable deformations) assigned to each element. The finer the mesh, the smaller the error.
  7. L. P. VanSpeybroeck, R. C. Chase, T. F. Zehnpfennig, Appl. Opt. 10, 945 (1971).
    [CrossRef] [PubMed]

1979 (1)

P. Gorenstein, “X-ray Optics for the LAMAR Facility, An Overview,” Proc. Soc. Photo-Opt. Instrum. Eng. 184, 63 (1979).

1977 (1)

J. H. Underwood, Space Sci. Instrum. 3, 259 (1977).

1973 (1)

P. Gorenstein, A. DeCaprio, R. Chase, B. Harris, Rev. Sci. Instrum. 44, 539 (1973).
[CrossRef]

1971 (1)

1948 (1)

Baez, A. V.

Chase, R.

P. Gorenstein, A. DeCaprio, R. Chase, B. Harris, Rev. Sci. Instrum. 44, 539 (1973).
[CrossRef]

Chase, R. C.

DeCaprio, A.

P. Gorenstein, A. DeCaprio, R. Chase, B. Harris, Rev. Sci. Instrum. 44, 539 (1973).
[CrossRef]

Gorenstein, P.

P. Gorenstein, “X-ray Optics for the LAMAR Facility, An Overview,” Proc. Soc. Photo-Opt. Instrum. Eng. 184, 63 (1979).

P. Gorenstein, A. DeCaprio, R. Chase, B. Harris, Rev. Sci. Instrum. 44, 539 (1973).
[CrossRef]

Harris, B.

P. Gorenstein, A. DeCaprio, R. Chase, B. Harris, Rev. Sci. Instrum. 44, 539 (1973).
[CrossRef]

Kirkpatrick, P.

Underwood, J. H.

J. H. Underwood, Space Sci. Instrum. 3, 259 (1977).

VanSpeybroeck, L. P.

Zehnpfennig, T. F.

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

P. Gorenstein, “X-ray Optics for the LAMAR Facility, An Overview,” Proc. Soc. Photo-Opt. Instrum. Eng. 184, 63 (1979).

Rev. Sci. Instrum. (1)

P. Gorenstein, A. DeCaprio, R. Chase, B. Harris, Rev. Sci. Instrum. 44, 539 (1973).
[CrossRef]

Space Sci. Instrum. (1)

J. H. Underwood, Space Sci. Instrum. 3, 259 (1977).

Other (2)

There are other known methods of supporting K/B mirror elements including that of Underwood.4 However, the described support system has previously been utilized5 and is the system chosen as the reference system.

The problem of determining the deformations, strains, and stresses of a thin rectangular plate arbitrarily loaded and supported is one of continuum mechanics. To derive the proper differential equations along with associated boundary and loading conditions is not an easy task. The underlying philosophy of the FEM is that we can construct a numerical model of the continuous structure which will result in a good approximation to the desired solution. To establish this solution, the numerical model is constructed of small individual elements whose solutions within their domain we can accurately approximate. The boundary conditions of this numerical model will also be approximated by some set of generalized deformations. The errors associated with the solution are dependent upon the size of the elements and the degrees of freedom (allowable deformations) assigned to each element. The finer the mesh, the smaller the error.

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Figures (6)

Fig. 1
Fig. 1

Upper: single orthogonal pair of K/B reflectors. Lower: nested set of orthogonal K/B reflectors.

Fig. 2
Fig. 2

K/B mirror element deformed shape (each edge held at five discrete points located on a parabolic surface).

Fig. 3
Fig. 3

Typical K/B mirror element geometry.

Fig. 4
Fig. 4

K/B mirror element finite element computer model geometry.

Fig. 5
Fig. 5

Lamar-SAO.

Fig. 6
Fig. 6

Masked area of a K/B mirror element.

Tables (1)

Tables Icon

Table I Optical Performance of Various Cases

Equations (5)

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S theo = [ p ( 2 · p · x i + p 2 ) 1 / 2 ] ,
E rror = 1 n ( S theo S act ) 2 n .
n E rror 2 = 1 n ( S theo S act ) 2 .
S i = S i , base + j = 1 j = 5 Δ S i , j δ j ,
n E rror 2 = i = 1 n [ S i , theo ( S i , base + j = 1 j = 5 Δ S i , j δ j ) ] 2 · W i 2 .

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