Abstract

We describe methods of predicting the degradation of the performance of a simple imaging system in terms of the statistics of the shape errors of the focusing element and, conversely, of specifying those statistics in terms of requirements on image quality. Results are illustrated for normal-incidence, x-ray mirrors with figure errors plus conventional and/or fractal finish errors. It is emphasized that the imaging properties of a surface with fractal errors are well behaved even though fractal-power spectra diverge at low spatial frequencies.

© 1993 Optical Society of America

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References

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  1. E. L. Church, P. Z. Takacs, “Prediction of mirror performance from laboratory measurements,” in X-Ray/EUV Optics for Astronomy and Microscopy, R. B. Hoover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1160, 323–336 (1989). This reference discusses the system-limited behavior of glancing-incidence mirrors in terms of their one-dimensional or profile power spectra.
  2. E. L. Church, P. Z. Takacs, “Specification of the surface figure and finish of optical elements in terms of system performance,” in Specification and Measurement of Optical Systems, L. R. Baker, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1781, 118–130 (1992). This reference extends the research in Ref. 1 to normal-incidence optics and expresses the results in terms of the two-dimensional or area power spectra.
  3. E. L. Church, P. Z. Takacs, “Specifying the surface finish of x-ray mirrors,” in Soft-X-Ray Projection Lithography, Vol. 18 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1993). This reference extends the research in Ref. 2 to diffraction-limited optics.
  4. The Gaussian forms of Eq. (15), (31), (32), and (51) follow from the assumption that the error fluctuations are members of a Gaussian random process with a Gaussian height distribution. In the more general case of a kth-order gamma distribution, Eq. (15) is replaced by OTF(τ) = [1 + (4π/λ)2D(τ)/2k]−k. In the smooth-surface limit, however, results are independent of the form of the height distribution.
  5. The statistical stability of the results is ensured by the smallness of the parameter (W/D0)2.
  6. S1(fx) and S2(f) are the cosine and zeroth-order Hankel transforms of a common correlation function and form, then, an Abel-transform pair. That is, S1 is the two-to-one-dimensional stereological projection or half-integral of S2. See also Ref. 9.
  7. E. L. Church, P. Z. Takacs, “Instrumental effects in surface finish measurements,” in Surface Measurement and Characterization, J. M. Bennett, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1009, 46–55 (1988).
  8. E. L. Church, P. Z. Takacs, “The optimal estimation of finish parameters,” in Optical Scatter: Applications, Measurement, and Theory, J. C. Stover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1530, 71–86 (1991).
  9. E. L. Church, H. A. Jenkinson, J. M. Zavada, “Relationship between surface scattering and microtopographic features,” Opt. Eng. 18, 125–136 (1979).
  10. E. L. Church, “Fractal surface finish,” Appl. Opt. 27, 1518–1526 (1988). The coefficient 45 in Eq. (A5) of this reference should read 74.
    [CrossRef] [PubMed]
  11. The bars over the two-dimensional power spectra in Eqs. (33) and (38) denote their convolution with the system spread function, Eq. (13). This has a negligible effect for polished surfaces but would, for example, determine the nonvanishing widths of the spectral lines caused by periodic tool marks in machined surfaces.
  12. Statistical scattering theory depends on the structure function of the surface-height fluctuations and not their autocovariance function. The structure function exists when the fluctuations have statistically stationary first differences, while the autocovariance function exists only in the more restrictive condition that the magnitudes themselves are also stationary. Fractal surfaces have stationary differences but nonstationary magnitudes, while conventional surfaces have stationary differences and magnitudes. A telltale difference is whether the surface in question has a finite intrinsic rms roughness. Conventional surfaces do; fractal surfaces do not.
  13. More precisely, the limits 0 and ∞ in Eqs. (44) and (45) are 2/D0 and 2/λ.
  14. Melles Griot Industrie (France) advertises x-ray mirrors with rms slopes of < 5 μrad and rms roughnesses of < 5 Å. An rms slope of 5 μrad corresponds to an rms gradient of 7 μrad.
  15. H. H. Hopkins, “The aberration permissible in optical systems,” Proc. Phys. Soc. London Sec. B 52, 449–470 (1957).
    [CrossRef]
  16. P. Beckmann, A. Spizzichino, The Scattering of Electroma-gentci Waves from Rough Surfaces (Pergamon, New York, 1963).
  17. S. K. Sinha, E. B. Sirota, S. Garoff, H. B. Stanley, “X-ray and neutron scattering from rough surfaces,” Phys. Rev. B 38, 2297–2311 (1988).
    [CrossRef]
  18. D. G. Stearns, “X-ray scattering from nonideal multilayer structures,” J. Appl. Phys. 65, 491–506 (1989).
    [CrossRef]
  19. D. G. Stearns, “X-ray scattering from interfacial roughness in multilayer structures,” J. Appl. Phys. 71, 4286–4298 (1992).
    [CrossRef]

1992

D. G. Stearns, “X-ray scattering from interfacial roughness in multilayer structures,” J. Appl. Phys. 71, 4286–4298 (1992).
[CrossRef]

1989

D. G. Stearns, “X-ray scattering from nonideal multilayer structures,” J. Appl. Phys. 65, 491–506 (1989).
[CrossRef]

1988

E. L. Church, “Fractal surface finish,” Appl. Opt. 27, 1518–1526 (1988). The coefficient 45 in Eq. (A5) of this reference should read 74.
[CrossRef] [PubMed]

S. K. Sinha, E. B. Sirota, S. Garoff, H. B. Stanley, “X-ray and neutron scattering from rough surfaces,” Phys. Rev. B 38, 2297–2311 (1988).
[CrossRef]

1979

E. L. Church, H. A. Jenkinson, J. M. Zavada, “Relationship between surface scattering and microtopographic features,” Opt. Eng. 18, 125–136 (1979).

1957

H. H. Hopkins, “The aberration permissible in optical systems,” Proc. Phys. Soc. London Sec. B 52, 449–470 (1957).
[CrossRef]

Beckmann, P.

P. Beckmann, A. Spizzichino, The Scattering of Electroma-gentci Waves from Rough Surfaces (Pergamon, New York, 1963).

Church, E. L.

E. L. Church, “Fractal surface finish,” Appl. Opt. 27, 1518–1526 (1988). The coefficient 45 in Eq. (A5) of this reference should read 74.
[CrossRef] [PubMed]

E. L. Church, H. A. Jenkinson, J. M. Zavada, “Relationship between surface scattering and microtopographic features,” Opt. Eng. 18, 125–136 (1979).

E. L. Church, P. Z. Takacs, “Instrumental effects in surface finish measurements,” in Surface Measurement and Characterization, J. M. Bennett, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1009, 46–55 (1988).

E. L. Church, P. Z. Takacs, “Specifying the surface finish of x-ray mirrors,” in Soft-X-Ray Projection Lithography, Vol. 18 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1993). This reference extends the research in Ref. 2 to diffraction-limited optics.

E. L. Church, P. Z. Takacs, “Prediction of mirror performance from laboratory measurements,” in X-Ray/EUV Optics for Astronomy and Microscopy, R. B. Hoover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1160, 323–336 (1989). This reference discusses the system-limited behavior of glancing-incidence mirrors in terms of their one-dimensional or profile power spectra.

E. L. Church, P. Z. Takacs, “The optimal estimation of finish parameters,” in Optical Scatter: Applications, Measurement, and Theory, J. C. Stover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1530, 71–86 (1991).

E. L. Church, P. Z. Takacs, “Specification of the surface figure and finish of optical elements in terms of system performance,” in Specification and Measurement of Optical Systems, L. R. Baker, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1781, 118–130 (1992). This reference extends the research in Ref. 1 to normal-incidence optics and expresses the results in terms of the two-dimensional or area power spectra.

Garoff, S.

S. K. Sinha, E. B. Sirota, S. Garoff, H. B. Stanley, “X-ray and neutron scattering from rough surfaces,” Phys. Rev. B 38, 2297–2311 (1988).
[CrossRef]

Hopkins, H. H.

H. H. Hopkins, “The aberration permissible in optical systems,” Proc. Phys. Soc. London Sec. B 52, 449–470 (1957).
[CrossRef]

Jenkinson, H. A.

E. L. Church, H. A. Jenkinson, J. M. Zavada, “Relationship between surface scattering and microtopographic features,” Opt. Eng. 18, 125–136 (1979).

Sinha, S. K.

S. K. Sinha, E. B. Sirota, S. Garoff, H. B. Stanley, “X-ray and neutron scattering from rough surfaces,” Phys. Rev. B 38, 2297–2311 (1988).
[CrossRef]

Sirota, E. B.

S. K. Sinha, E. B. Sirota, S. Garoff, H. B. Stanley, “X-ray and neutron scattering from rough surfaces,” Phys. Rev. B 38, 2297–2311 (1988).
[CrossRef]

Spizzichino, A.

P. Beckmann, A. Spizzichino, The Scattering of Electroma-gentci Waves from Rough Surfaces (Pergamon, New York, 1963).

Stanley, H. B.

S. K. Sinha, E. B. Sirota, S. Garoff, H. B. Stanley, “X-ray and neutron scattering from rough surfaces,” Phys. Rev. B 38, 2297–2311 (1988).
[CrossRef]

Stearns, D. G.

D. G. Stearns, “X-ray scattering from interfacial roughness in multilayer structures,” J. Appl. Phys. 71, 4286–4298 (1992).
[CrossRef]

D. G. Stearns, “X-ray scattering from nonideal multilayer structures,” J. Appl. Phys. 65, 491–506 (1989).
[CrossRef]

Takacs, P. Z.

E. L. Church, P. Z. Takacs, “Prediction of mirror performance from laboratory measurements,” in X-Ray/EUV Optics for Astronomy and Microscopy, R. B. Hoover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1160, 323–336 (1989). This reference discusses the system-limited behavior of glancing-incidence mirrors in terms of their one-dimensional or profile power spectra.

E. L. Church, P. Z. Takacs, “Instrumental effects in surface finish measurements,” in Surface Measurement and Characterization, J. M. Bennett, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1009, 46–55 (1988).

E. L. Church, P. Z. Takacs, “Specifying the surface finish of x-ray mirrors,” in Soft-X-Ray Projection Lithography, Vol. 18 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1993). This reference extends the research in Ref. 2 to diffraction-limited optics.

E. L. Church, P. Z. Takacs, “The optimal estimation of finish parameters,” in Optical Scatter: Applications, Measurement, and Theory, J. C. Stover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1530, 71–86 (1991).

E. L. Church, P. Z. Takacs, “Specification of the surface figure and finish of optical elements in terms of system performance,” in Specification and Measurement of Optical Systems, L. R. Baker, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1781, 118–130 (1992). This reference extends the research in Ref. 1 to normal-incidence optics and expresses the results in terms of the two-dimensional or area power spectra.

Zavada, J. M.

E. L. Church, H. A. Jenkinson, J. M. Zavada, “Relationship between surface scattering and microtopographic features,” Opt. Eng. 18, 125–136 (1979).

Appl. Opt.

J. Appl. Phys.

D. G. Stearns, “X-ray scattering from nonideal multilayer structures,” J. Appl. Phys. 65, 491–506 (1989).
[CrossRef]

D. G. Stearns, “X-ray scattering from interfacial roughness in multilayer structures,” J. Appl. Phys. 71, 4286–4298 (1992).
[CrossRef]

Opt. Eng.

E. L. Church, H. A. Jenkinson, J. M. Zavada, “Relationship between surface scattering and microtopographic features,” Opt. Eng. 18, 125–136 (1979).

Phys. Rev. B

S. K. Sinha, E. B. Sirota, S. Garoff, H. B. Stanley, “X-ray and neutron scattering from rough surfaces,” Phys. Rev. B 38, 2297–2311 (1988).
[CrossRef]

Proc. Phys. Soc. London Sec. B

H. H. Hopkins, “The aberration permissible in optical systems,” Proc. Phys. Soc. London Sec. B 52, 449–470 (1957).
[CrossRef]

Other

P. Beckmann, A. Spizzichino, The Scattering of Electroma-gentci Waves from Rough Surfaces (Pergamon, New York, 1963).

The bars over the two-dimensional power spectra in Eqs. (33) and (38) denote their convolution with the system spread function, Eq. (13). This has a negligible effect for polished surfaces but would, for example, determine the nonvanishing widths of the spectral lines caused by periodic tool marks in machined surfaces.

Statistical scattering theory depends on the structure function of the surface-height fluctuations and not their autocovariance function. The structure function exists when the fluctuations have statistically stationary first differences, while the autocovariance function exists only in the more restrictive condition that the magnitudes themselves are also stationary. Fractal surfaces have stationary differences but nonstationary magnitudes, while conventional surfaces have stationary differences and magnitudes. A telltale difference is whether the surface in question has a finite intrinsic rms roughness. Conventional surfaces do; fractal surfaces do not.

More precisely, the limits 0 and ∞ in Eqs. (44) and (45) are 2/D0 and 2/λ.

Melles Griot Industrie (France) advertises x-ray mirrors with rms slopes of < 5 μrad and rms roughnesses of < 5 Å. An rms slope of 5 μrad corresponds to an rms gradient of 7 μrad.

E. L. Church, P. Z. Takacs, “Prediction of mirror performance from laboratory measurements,” in X-Ray/EUV Optics for Astronomy and Microscopy, R. B. Hoover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1160, 323–336 (1989). This reference discusses the system-limited behavior of glancing-incidence mirrors in terms of their one-dimensional or profile power spectra.

E. L. Church, P. Z. Takacs, “Specification of the surface figure and finish of optical elements in terms of system performance,” in Specification and Measurement of Optical Systems, L. R. Baker, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1781, 118–130 (1992). This reference extends the research in Ref. 1 to normal-incidence optics and expresses the results in terms of the two-dimensional or area power spectra.

E. L. Church, P. Z. Takacs, “Specifying the surface finish of x-ray mirrors,” in Soft-X-Ray Projection Lithography, Vol. 18 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1993). This reference extends the research in Ref. 2 to diffraction-limited optics.

The Gaussian forms of Eq. (15), (31), (32), and (51) follow from the assumption that the error fluctuations are members of a Gaussian random process with a Gaussian height distribution. In the more general case of a kth-order gamma distribution, Eq. (15) is replaced by OTF(τ) = [1 + (4π/λ)2D(τ)/2k]−k. In the smooth-surface limit, however, results are independent of the form of the height distribution.

The statistical stability of the results is ensured by the smallness of the parameter (W/D0)2.

S1(fx) and S2(f) are the cosine and zeroth-order Hankel transforms of a common correlation function and form, then, an Abel-transform pair. That is, S1 is the two-to-one-dimensional stereological projection or half-integral of S2. See also Ref. 9.

E. L. Church, P. Z. Takacs, “Instrumental effects in surface finish measurements,” in Surface Measurement and Characterization, J. M. Bennett, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1009, 46–55 (1988).

E. L. Church, P. Z. Takacs, “The optimal estimation of finish parameters,” in Optical Scatter: Applications, Measurement, and Theory, J. C. Stover, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1530, 71–86 (1991).

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Figures (6)

Fig. 1
Fig. 1

General form of the structure function of a randomly rough surface.

Fig. 2
Fig. 2

Profile-power spectral density of a silicon x-ray synchrotron-radiation mirror plotted on log–log scales. Two types of measuring instrument were used to cover the range of spatial frequencies shown.

Fig. 3
Fig. 3

Image-intensity distribution for a conventional surface calculated from Eq. (33) for θ0 = λ/W = 100 μrad, λ = 140 Å; μ0 = 0, σ0 = 5 Å, and l0 = 1 μm. In this case the Strehl factor multiplying the image core is 0.799.

Fig. 4
Fig. 4

Image-intensity distribution for a fractal surface calculated from Eq. (37) for the same system and figure parameters as in Fig. 3, but n = 4/3 and K = 1 × 10−8 μm5/3. In this case the on-axis Strehl factor is 0.828.

Fig. 5
Fig. 5

Function K0 given by Eq. (41), where q = Wfx = θ/θ0.

Fig. 6
Fig. 6

Function K2 given by Eq. (48), where q = Wfx = θ/θ0.

Equations (52)

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I ( θ ) = 1 I i R d I d θ = 1 λ 2 d τ exp ( i 2 π f · τ ) OTF ( τ ) ,
f = ( f x f y ) = sin θ λ ( cos φ sin φ ) θ λ .
OTF ( τ ) = OTF s ( τ ) × OTF e ( τ ) .
OTF s ( τ ) = A 1 ( τ ) × A 2 ( τ ) × A 3 ( τ ) ,
A 1 ( τ ) = 2 π { arccos ( τ D 0 ) τ D 0 [ 1 ( τ D 0 ) 2 ] 1 / 2 } ,
I 1 ( θ ) = π 4 ( D 0 λ ) 2 [ J 1 ( π D 0 θ / λ ) π D 0 θ / 2 λ ] 2 .
A 2 ( τ ) = exp [ ( π θ S τ / λ ) 2 ] ,
I 2 ( θ ) = 1 π θ S 2 exp [ ( θ / θ s ) 2 ] .
A 3 ( τ ) = J 1 ( π θ D τ / λ ) π θ D τ / 2 λ ,
I 3 ( θ ) = 4 π θ D 2 Cyl ( θ / θ D ) ,
I s ( θ ) = I 1 ( θ ) * I 2 ( θ ) * I 3 ( θ ) .
OTF s ( τ ) = exp [ ( π τ / W ) 2 ] ,
I s ( θ ) = 1 π ( W λ ) 2 exp [ ( W θ / λ ) 2 ] ,
width I s ( θ ) = λ W Δ ̅ θ 0 .
OTF e ( τ ) = exp [ 1 2 ( 4 π λ ) 2 D ( τ ) ] ,
D ( τ ) = [ Z ( x + τ ) Z ( x ) ] 2 .
S 1 ( f x ) = lim L 2 L | L / 2 + L / 2 d x exp ( i 2 π f x x ) Z ( x ) | 2 ,
D ( τ ) = 4 0 d f x S 1 ( f x ) sin 2 ( π f x τ ) .
D ( τ ) = ½ μ 0 2 τ 2 ,
μ 0 2 = 2 0 d f x S 1 ( f x ) ( 2 π f x ) 2 ,
D ( τ ) = 2 σ 0 2 [ 1 exp ( τ / l 0 ) ] ,
S 1 ( f x ) = 4 σ 0 2 l 0 [ 1 + ( 2 π l 0 f x ) 2 ] 1 .
D ( τ ) = T 2 ( τ / T ) n 1 ,
Hausdorff Besicovitch dimension of surface = ( 7 n ) / 2 ,
S 1 ( f x ) = K / f x n ,
K = π ( 1 / 2 ) n Γ ( n 2 ) Γ ( 1 n 2 ) T 3 n .
I ( θ ) = 1 λ 2 d τ exp ( i 2 π f · τ ) exp [ ( τ / W ) 2 ] × exp [ ( 2 π μ 0 τ / λ ) 2 ] exp [ ( 4 π / λ ) 2 D ( τ ) / 2 ] .
W = W [ 1 + 4 ( W λ ) 2 μ 0 2 ] 1 / 2 .
I ( 0 ) I s ( 0 ) = ( W W ) 2 ,
width I ( θ ) width I s ( θ ) = ( W W ) 1 .
I ( θ ) = 1 4 π μ 0 2 exp [ ( θ / 2 μ 0 ) 2 ] * I ( θ ) ,
I ( θ ) = exp [ ( 4 π σ 0 / λ ) 2 ] × I ( θ ) + c ,
I ( θ ) = I s ( θ ) { 1 4 θ 0 2 μ 0 2 [ 1 ( θ θ 0 ) 2 ] ( 4 π σ 0 λ ) 2 } + 16 π 2 λ 4 S ̅ 2 ( f ) .
S 2 ( f ) = d τ exp ( i 2 π f · τ ) C ( τ ) ,
C ( τ ) = σ 0 2 ½ D ( τ )
S 2 ( f ) = 2 π σ 0 2 l 0 2 [ 1 + ( 2 π l 0 f ) 2 ] 3 / 2 .
I ( θ ) = I s ( θ ) { 1 4 θ 0 2 μ 0 2 [ 1 ( θ θ 0 ) 2 ] } + 16 π 2 λ 4 [ Γ ( 1 + n 2 ) Γ ( 1 n 2 ) 2 π Γ ( n / 2 ) K ] W n + 1 × 1 F 1 [ 1 + n 2 ; 1 ; ( θ θ 0 ) 2 ] ,
I ( θ ) 16 π 2 λ 4 S 2 ( f ) ,
S 2 ( f ) = Γ ( 1 + n 2 ) 2 π Γ ( n / 2 ) × K f n + 1 ,
I ( 0 ) I s ( 0 ) = 1 ( 4 π λ ) 2 0 d f x S 1 ( f x ) K 0 ( f x ) ,
K 0 ( f x ) = 1 1 F 1 [ 1 ; ½ ; ( W f x ) 2 ] .
K 0 ( f x ) { 2 ( W f x ) 2 0 f x 1 / 2 W 1 otherwise ,
I ( 0 ) I s ( 0 ) 1 4 θ 0 2 ( μ 0 2 + μ W 2 ) ( 4 π λ ) 2 σ W 2 ,
μ W 2 = 2 0 1 / 2 W d f x S 1 ( f x ) ( 2 π f x ) 2 ,
σ W 2 = 1 / 2 W d f x S 1 ( f x )
width I ( θ ) Δ ̅ [ 1 2 d 2 d θ 2 { I ( θ ) I ( 0 ) } 0 ] 1 / 2 .
width I ( θ ) width I s ( θ ) = 1 + ( 4 π λ ) 2 0 d f x S 1 ( f x ) K 2 ( f x ) ,
K 2 ( f x ) = 2 ( W f x ) 2 F 1 1 [ 2 ; 3 / 2 ; ( W f x ) 2 ]
K 2 ( f x ) { 2 ( W f x ) 2 0 f x 1 / 2 W 0 otherwise .
width I ( θ ) width I s ( θ ) 1 + 2 θ 0 2 ( μ 0 2 + μ W 2 ) ,
OTF ( g ) = OTF s ( g ) × exp [ 1 2 ( 4 π λ ) 2 D ( g ) ] ,
H ( g ) = OTF ( g ) OTF s ( g ) ,

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