Abstract

By using Fourier techniques and linear systems theory we have derived an analytic expression for a generalized transfer function for grazing incidence optical systems operating at ultraviolet and x-ray wavelengths that includes the effects of optical fabrication errors over the entire range of relevant spatial frequencies. The Fourier transform of this transfer function yields the image distribution (or point spread function) from which encircled energy characteristics or other image quality criteria can be obtained. This transfer function characterization of grazing incidence optical systems allows parametric trade studies and sensitivity analyses to be performed as well as the derivation of fabrication tolerances necessary to satisfy a given image quality requirement.

© 1988 Optical Society of America

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References

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  1. P. M. Duffieux “L’Integrale de Fourier et Ses Applications a L’Optique,” (Besancon, 1946).
  2. O. Schade, J. Soc. Motion Pict. Telev. Eng. 56, 137 (1951); J. Soc. Motion Pict. Telev. Eng. 58, 182 (1952); J. Soc. Motion Pict. Telev. Eng. 61, 98 (1953); J. Soc. Motion Pict. Telev. Eng. 64, 593 (1955).
  3. R. V. Shack, “Outline of Practical Characteristics of an Image-Forming System,” J. Opt. Soc. Am. 46, 755 (1956).
    [CrossRef]
  4. E. Ingelstam, “Nomenclature for Fourier Transforms of Spread Functions,” J. Opt. Soc. Am. 51, 1441 (1961).
  5. F. D. Smith, “Optical Image Evaluation and the Transfer Function,” Appl. Opt. 2, 335 (1963).
    [CrossRef]
  6. M. D. Rosenau, “Image-Motion Modulation Transfer Functions,” Symposium on the Practical Application of Modulation Transfer Functions, Society of Photographic Scientists and Engineers, Inc. (1965).
  7. W. Swindell, “A Noncoherent Optical Analog Image Processor,” Appl. Opt. 9, 2459 (1970).
    [CrossRef] [PubMed]
  8. J. E. Harvey, “Transfer Function Characterization of Scattering, Surfaces,” J. Opt. Soc. Am. 66, 1136 (1976).
  9. J. E. Harvey, “Light-Scattering Characteristics of Optical Surfaces,” Ph.D. Dissertation, U. Arizona (1976).
  10. J. C. Stover, “Roughness Characterization of Smooth Machined Surfaces by Light Scattering,” Appl. Opt. 14, 1796 (1975).
    [CrossRef] [PubMed]
  11. E. L. Church, H. A. Jenkinson, J. M. Zavada, “Relationship Between Surface Scattering and Microtopographic Features,” Opt. Eng. 18, 125 (1979).
    [CrossRef]
  12. R. J. Noll, “Effect of Mid and High Spatial Frequencies on Optical Performance,” Opt. Eng. 18, 137 (1979).
    [CrossRef]
  13. A. Slomba, R. Babish, P. Glenn, “Mirror Surface Metrology and Polishing for AXAF/TMA,” Proc. Soc. Photo-Opt. Instrum. Eng. 597, 40 (1986).
  14. P. Glenn, A. Slomba, “Derivation of Requirements for Surface Quality and Metrology Instrumentation for AXAF/TMA,” Proc. Soc. Photo-Opt. Instrum. Eng. 597, 55 (1986).
  15. L. Van Speybroeck, “Grazing Incidence Optics for United States High Resolution X-Ray Astronomy Program,” Proc. Soc. Photo-Opt. Instrum. Eng.830, in press (1987).
  16. J. E. Harvey, “Diffraction-Limited Performance of Grazing Incidence Optical Systems,” Proc. Soc. Photo-Opt. Instrum. Eng. 640, 2 (1986).
  17. M. V. Zombeck, “Advanced X-Ray Astrophysics Facility (AXAF)—Performance Requirements and Design Considerations,” Proc. Soc. Photo-Opt. Instrum. Eng. 184, 50 (1979).
  18. R. Giacconi et al., “The Einstein (HEAO-2) X-Ray Observatory,” Astrophys. J. 230, 540 (1979).
    [CrossRef]

1986 (3)

A. Slomba, R. Babish, P. Glenn, “Mirror Surface Metrology and Polishing for AXAF/TMA,” Proc. Soc. Photo-Opt. Instrum. Eng. 597, 40 (1986).

P. Glenn, A. Slomba, “Derivation of Requirements for Surface Quality and Metrology Instrumentation for AXAF/TMA,” Proc. Soc. Photo-Opt. Instrum. Eng. 597, 55 (1986).

J. E. Harvey, “Diffraction-Limited Performance of Grazing Incidence Optical Systems,” Proc. Soc. Photo-Opt. Instrum. Eng. 640, 2 (1986).

1979 (4)

M. V. Zombeck, “Advanced X-Ray Astrophysics Facility (AXAF)—Performance Requirements and Design Considerations,” Proc. Soc. Photo-Opt. Instrum. Eng. 184, 50 (1979).

R. Giacconi et al., “The Einstein (HEAO-2) X-Ray Observatory,” Astrophys. J. 230, 540 (1979).
[CrossRef]

E. L. Church, H. A. Jenkinson, J. M. Zavada, “Relationship Between Surface Scattering and Microtopographic Features,” Opt. Eng. 18, 125 (1979).
[CrossRef]

R. J. Noll, “Effect of Mid and High Spatial Frequencies on Optical Performance,” Opt. Eng. 18, 137 (1979).
[CrossRef]

1976 (1)

J. E. Harvey, “Transfer Function Characterization of Scattering, Surfaces,” J. Opt. Soc. Am. 66, 1136 (1976).

1975 (1)

1970 (1)

1963 (1)

1961 (1)

E. Ingelstam, “Nomenclature for Fourier Transforms of Spread Functions,” J. Opt. Soc. Am. 51, 1441 (1961).

1956 (1)

1951 (1)

O. Schade, J. Soc. Motion Pict. Telev. Eng. 56, 137 (1951); J. Soc. Motion Pict. Telev. Eng. 58, 182 (1952); J. Soc. Motion Pict. Telev. Eng. 61, 98 (1953); J. Soc. Motion Pict. Telev. Eng. 64, 593 (1955).

Babish, R.

A. Slomba, R. Babish, P. Glenn, “Mirror Surface Metrology and Polishing for AXAF/TMA,” Proc. Soc. Photo-Opt. Instrum. Eng. 597, 40 (1986).

Church, E. L.

E. L. Church, H. A. Jenkinson, J. M. Zavada, “Relationship Between Surface Scattering and Microtopographic Features,” Opt. Eng. 18, 125 (1979).
[CrossRef]

Duffieux, P. M.

P. M. Duffieux “L’Integrale de Fourier et Ses Applications a L’Optique,” (Besancon, 1946).

Giacconi, R.

R. Giacconi et al., “The Einstein (HEAO-2) X-Ray Observatory,” Astrophys. J. 230, 540 (1979).
[CrossRef]

Glenn, P.

A. Slomba, R. Babish, P. Glenn, “Mirror Surface Metrology and Polishing for AXAF/TMA,” Proc. Soc. Photo-Opt. Instrum. Eng. 597, 40 (1986).

P. Glenn, A. Slomba, “Derivation of Requirements for Surface Quality and Metrology Instrumentation for AXAF/TMA,” Proc. Soc. Photo-Opt. Instrum. Eng. 597, 55 (1986).

Harvey, J. E.

J. E. Harvey, “Diffraction-Limited Performance of Grazing Incidence Optical Systems,” Proc. Soc. Photo-Opt. Instrum. Eng. 640, 2 (1986).

J. E. Harvey, “Transfer Function Characterization of Scattering, Surfaces,” J. Opt. Soc. Am. 66, 1136 (1976).

J. E. Harvey, “Light-Scattering Characteristics of Optical Surfaces,” Ph.D. Dissertation, U. Arizona (1976).

Ingelstam, E.

E. Ingelstam, “Nomenclature for Fourier Transforms of Spread Functions,” J. Opt. Soc. Am. 51, 1441 (1961).

Jenkinson, H. A.

E. L. Church, H. A. Jenkinson, J. M. Zavada, “Relationship Between Surface Scattering and Microtopographic Features,” Opt. Eng. 18, 125 (1979).
[CrossRef]

Noll, R. J.

R. J. Noll, “Effect of Mid and High Spatial Frequencies on Optical Performance,” Opt. Eng. 18, 137 (1979).
[CrossRef]

Rosenau, M. D.

M. D. Rosenau, “Image-Motion Modulation Transfer Functions,” Symposium on the Practical Application of Modulation Transfer Functions, Society of Photographic Scientists and Engineers, Inc. (1965).

Schade, O.

O. Schade, J. Soc. Motion Pict. Telev. Eng. 56, 137 (1951); J. Soc. Motion Pict. Telev. Eng. 58, 182 (1952); J. Soc. Motion Pict. Telev. Eng. 61, 98 (1953); J. Soc. Motion Pict. Telev. Eng. 64, 593 (1955).

Shack, R. V.

Slomba, A.

A. Slomba, R. Babish, P. Glenn, “Mirror Surface Metrology and Polishing for AXAF/TMA,” Proc. Soc. Photo-Opt. Instrum. Eng. 597, 40 (1986).

P. Glenn, A. Slomba, “Derivation of Requirements for Surface Quality and Metrology Instrumentation for AXAF/TMA,” Proc. Soc. Photo-Opt. Instrum. Eng. 597, 55 (1986).

Smith, F. D.

Stover, J. C.

Swindell, W.

Van Speybroeck, L.

L. Van Speybroeck, “Grazing Incidence Optics for United States High Resolution X-Ray Astronomy Program,” Proc. Soc. Photo-Opt. Instrum. Eng.830, in press (1987).

Zavada, J. M.

E. L. Church, H. A. Jenkinson, J. M. Zavada, “Relationship Between Surface Scattering and Microtopographic Features,” Opt. Eng. 18, 125 (1979).
[CrossRef]

Zombeck, M. V.

M. V. Zombeck, “Advanced X-Ray Astrophysics Facility (AXAF)—Performance Requirements and Design Considerations,” Proc. Soc. Photo-Opt. Instrum. Eng. 184, 50 (1979).

Appl. Opt. (3)

Astrophys. J. (1)

R. Giacconi et al., “The Einstein (HEAO-2) X-Ray Observatory,” Astrophys. J. 230, 540 (1979).
[CrossRef]

J. Opt. Soc. Am. (3)

J. E. Harvey, “Transfer Function Characterization of Scattering, Surfaces,” J. Opt. Soc. Am. 66, 1136 (1976).

R. V. Shack, “Outline of Practical Characteristics of an Image-Forming System,” J. Opt. Soc. Am. 46, 755 (1956).
[CrossRef]

E. Ingelstam, “Nomenclature for Fourier Transforms of Spread Functions,” J. Opt. Soc. Am. 51, 1441 (1961).

J. Soc. Motion Pict. Telev. Eng. (1)

O. Schade, J. Soc. Motion Pict. Telev. Eng. 56, 137 (1951); J. Soc. Motion Pict. Telev. Eng. 58, 182 (1952); J. Soc. Motion Pict. Telev. Eng. 61, 98 (1953); J. Soc. Motion Pict. Telev. Eng. 64, 593 (1955).

Opt. Eng. (2)

E. L. Church, H. A. Jenkinson, J. M. Zavada, “Relationship Between Surface Scattering and Microtopographic Features,” Opt. Eng. 18, 125 (1979).
[CrossRef]

R. J. Noll, “Effect of Mid and High Spatial Frequencies on Optical Performance,” Opt. Eng. 18, 137 (1979).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng. (4)

A. Slomba, R. Babish, P. Glenn, “Mirror Surface Metrology and Polishing for AXAF/TMA,” Proc. Soc. Photo-Opt. Instrum. Eng. 597, 40 (1986).

P. Glenn, A. Slomba, “Derivation of Requirements for Surface Quality and Metrology Instrumentation for AXAF/TMA,” Proc. Soc. Photo-Opt. Instrum. Eng. 597, 55 (1986).

J. E. Harvey, “Diffraction-Limited Performance of Grazing Incidence Optical Systems,” Proc. Soc. Photo-Opt. Instrum. Eng. 640, 2 (1986).

M. V. Zombeck, “Advanced X-Ray Astrophysics Facility (AXAF)—Performance Requirements and Design Considerations,” Proc. Soc. Photo-Opt. Instrum. Eng. 184, 50 (1979).

Other (4)

L. Van Speybroeck, “Grazing Incidence Optics for United States High Resolution X-Ray Astronomy Program,” Proc. Soc. Photo-Opt. Instrum. Eng.830, in press (1987).

P. M. Duffieux “L’Integrale de Fourier et Ses Applications a L’Optique,” (Besancon, 1946).

J. E. Harvey, “Light-Scattering Characteristics of Optical Surfaces,” Ph.D. Dissertation, U. Arizona (1976).

M. D. Rosenau, “Image-Motion Modulation Transfer Functions,” Symposium on the Practical Application of Modulation Transfer Functions, Society of Photographic Scientists and Engineers, Inc. (1965).

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Figures (13)

Fig. 1
Fig. 1

Geometric configuration for scattering measurements.

Fig. 2
Fig. 2

(a) Scattered intensity vs scattering angle; (b) scattered radiance in direction cosine space.

Fig. 3
Fig. 3

Illustration of surface height variations and associated statistical parameters.

Fig. 4
Fig. 4

Surface transfer function and the associated angle spread function are related by the Fourier transform operation just as are the optical transfer function (OTF) and the point spread function (PSF) of modern image formation theory.

Fig. 5
Fig. 5

Transfer function characterization of individual error sources greatly facilitates the prediction of system performance.

Fig. 6
Fig. 6

Grazing incidence reduces the rms wavefront error induced by surface irregularities and foreshortens the wavefront features in the plane of incidence. An isoplanatic surface thus produces an elliptical wavefront autocovariance function.

Fig. 7
Fig. 7

(a) Optical surface profile composed of low spatial frequency figure errors, midspatial frequency surface irregularities, and high spatial frequency microroughness; (b) composite surface autocovariance function; (c) image intensity distribution consisting of a narrow image core, a small-angle scatter distribution, and a wide-angle scattered halo.

Fig. 8
Fig. 8

Point spread function of an imaging system is related to the complex pupil function in precisely the same way that the surface PSD is related to the surface profile.

Fig. 9
Fig. 9

Schematic representation of the scattering effects of a Wolter I x-ray telescope.

Fig. 10
Fig. 10

(a) Elongated intensity distribution due to a foreshortened wavefront autocovariance function. (b) Entire annular aperture produces the same fractional encircled energy as the azimuthal subaperture element.

Fig. 11
Fig. 11

These four families of curves illustrate the sensitivity of the image quality of the above Wolter I telescope to variations in the relevant optical surface parameters about their nominal values given in Eqs. (31) and (32).

Fig. 12
Fig. 12

Schematic illustration of five nested Wolter I x-ray telescopes with focal length F, radius ri, and length of paraboloids and hyperboloids L.

Fig. 13
Fig. 13

Effective collecting area of the individual shells and the composite nested Wolter I x-ray telescope. These data are necessary to calculate the effective area weighted average encircled energy of such telescopes.

Equations (36)

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H s ( x ˆ , y ˆ ) = surface transfer function = exp { ( 4 π σ ˆ s ) 2 [ 1 C s ( x ˆ l ˆ , y ˆ l ˆ ) / σ s 2 ] } .
H s ( x ˆ , y ˆ ) = A + B Q ( x ˆ , y ˆ ) ,
A = exp [ ( 4 π σ ˆ s ) 2 ] ,
B = 1 A = 1 exp [ ( 4 π σ ˆ s ) 2 ] ,
Q ( x ˆ , y ˆ ) = exp [ ( 4 π σ ˆ s ) 2 C ( x ˆ l ˆ , y ˆ l ˆ ) / σ s 2 ] 1 exp ( 4 π σ ˆ s ) 2 1 .
A 1 ( 4 π σ ˆ s ) 2 ,
B ( 4 π σ ˆ s ) 2 ,
Q ( x ˆ , y ˆ ) C ( x ˆ l ˆ , y ˆ l ˆ ) / σ s 2 .
S ( α , β ) = scattering function = B F { Q ( x ˆ , y ˆ ) } ( 4 π / λ ) 2 PSD .
σ w = 2 σ s sin ϕ ,
C w ( x ˆ , y ˆ ) = 4 sin 2 ϕ C s ( x ˆ l ˆ , y ˆ l ˆ sin ϕ ) .
H s ( x ˆ , y ˆ ) = exp { ( 4 π sin ϕ σ ˆ s ) 2 [ 1 C ( x ˆ l ˆ y ˆ l ˆ sin ϕ ) / σ s 2 ] } ,
H s ( x ˆ , y ˆ ) = A + B Q ( x ˆ , y ˆ ) .
A = exp [ ( 4 π sin ϕ σ ˆ s ) 2 ] ,
B = 1 A = 1 exp [ ( 4 π sin ϕ σ ˆ s ) 2 ] ,
Q ( x ˆ , y ˆ ) = exp [ ( 4 π sin ϕ σ ˆ s ) 2 C ( x ˆ l ˆ , y ˆ l ˆ sin ) / σ s ] 1 exp [ ( 4 π sin ϕ σ ˆ s ) 2 ] 1 .
H fab = H L H M H H ,
H fab = exp { ( 4 π sin ϕ σ ˆ L ) 2 [ 1 C L / σ L 2 ] } × exp { ( 4 π sin ϕ σ ˆ M ) 2 [ 1 C M / σ M 2 ] } × exp { ( 4 π sin ϕ σ ˆ N ) 2 [ 1 C N / σ H 2 ] } .
C = C L + C M + C H .
σ 2 = σ L 2 + σ M 2 + σ H 2 ,
H fab = exp { ( 4 π sin ϕ σ ) 2 [ 1 C / σ 2 ] } .
P ( x ˆ , y ˆ ) = A ( x ˆ , y ˆ ) exp [ 2 π W ˆ ( x ˆ , y ˆ ) ] , W ˆ ( x ˆ , y ˆ ) = 2 sin ϕ h ˆ ( x ˆ , y ˆ ) .
H ( x ˆ , y ˆ ) = H c ( x ˆ , y ˆ ) H s ( x ˆ , y ˆ ) .
I ( α , β ) = I c ( α , β ) * S ( α , β ) .
S ( α , β ) = A δ ( α , β ) + B F { Q ( x ˆ , y ˆ ) } ,
B = B 1 + B 2 = B 1 ( 1 + A 1 ) = ( 1 A 1 ) ( 1 + A 1 ) = 1 A 1 2 ,
B = 1 exp [ ( 4 π 2 sin ϕ σ ˆ s ) 2 ] .
H s ( x ˆ , y ˆ ) = exp { ( 4 π 2 sin ϕ σ ˆ s ) 2 [ 1 C s ( x ˆ l ˆ , y ˆ l ˆ sin ϕ ) / σ s 2 ] } .
I barrel stave ( α , β ) = I core ( α , β ) * S ( α , β ) .
I ( α , β ) = ψ = 0 2 π I barrel stave ( α , β ) d ψ .
C M = σ M 2 exp [ r / l M ] , σ M = 25 Å , l M = 20 mm ,
C H = σ H 2 exp [ r / l H ] , σ H = 10 Å , l H = 20 μ m .
E E ( λ ) = 1 A T ( λ ) i = 1 N A i ( λ ) E E i ( λ ) .
A i ( λ ) = G i R 2 ( λ , α i ) .
G i = 2 π r i L tan α i ,
α i = 1 4 tan 1 ( r i F ) .

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