Abstract

Image analysis in the presence of surface scatter due to residual optical fabrication errors is often perceived to be complicated, nonintuitive, and achieved only by computationally intensive nonsequential ray tracing with commercial optical analysis codes such as ASAP, Zemax, Code V, TracePro, or FRED. However, we show that surface scatter can be treated very similarly to conventional wavefront aberrations. For multielement imaging systems degraded by both surface scatter and aberrations, the composite point spread function is obtained in explicit analytic form in terms of convolutions of the geometrical point spread function and scaled bidirectional scattering distribution functions of the individual surfaces of the imaging system. The approximations and assumptions in this formulation are discussed, and the result is compared to the irradiance distribution obtained using commercial software for the case of a two-mirror telescope operating at an extreme ultraviolet wavelength. The two results are virtually identical.

© 2012 Optical Society of America

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References

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  1. G. L. Peterson, “Analytic expressions for in-field stray light irradiance in imaging systems,” Master’s report (University of Arizona, 2003).
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    [CrossRef]
  3. J. E. Harvey, N. Choi, A. Krywonos, G. Peterson, and M. Bruner, “Image degradation due to scattering effects in two-mirror telescopes,” Opt. Eng. 49, 063202 (2010).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  7. J. E. Harvey, M. Atanassova, and A. Krywonos, “Systems engineering analysis of five “as-manufactured” SXI telescopes,” Proc. SPIE 5867, 58670F (2005).
    [CrossRef]
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2011 (1)

2010 (5)

2009 (1)

J. E. Harvey, N. Choi, A. Krywonos, and J. Marcen, “Calculating BRDFs from surface PSDs for moderately rough surfaces,” Proc. SPIE 7426, 64260I (2009).

2007 (1)

J. E. Harvey, A. Krywonos, M. Atanassova, and P. L. Thompson, “The Solar X-ray Imager (SXI) on GOES-13: design, analysis, and on-orbit performance,” Proc. SPIE 6689, 66890I (2007).
[CrossRef]

2005 (1)

J. E. Harvey, M. Atanassova, and A. Krywonos, “Systems engineering analysis of five “as-manufactured” SXI telescopes,” Proc. SPIE 5867, 58670F (2005).
[CrossRef]

2004 (1)

G. L. Peterson, “Analytic expressions for in-field scattered light distribution,” Proc. SPIE 5178, 184–193 (2004).
[CrossRef]

2000 (1)

J. E. Harvey and A. Krywonos, “A systems engineering analysis of image quality,” Proc. SPIE 4093, 379–388 (2000).
[CrossRef]

1989 (1)

J. E. Harvey, “Surface scatter phenomena: a linear, shift-invariant process,” Proc. SPIE 1165, 87–99 (1989).

1988 (1)

1982 (1)

1970 (1)

Akram, M. N.

Arfken, G. B.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 5th ed. (Academic, 2001).

Atanassova, M.

J. E. Harvey, A. Krywonos, M. Atanassova, and P. L. Thompson, “The Solar X-ray Imager (SXI) on GOES-13: design, analysis, and on-orbit performance,” Proc. SPIE 6689, 66890I (2007).
[CrossRef]

J. E. Harvey, M. Atanassova, and A. Krywonos, “Systems engineering analysis of five “as-manufactured” SXI telescopes,” Proc. SPIE 5867, 58670F (2005).
[CrossRef]

Barbastathis, G.

Barton, J. K.

Beckmann, P.

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, 1963).

Bruner, M.

J. E. Harvey, N. Choi, A. Krywonos, G. Peterson, and M. Bruner, “Image degradation due to scattering effects in two-mirror telescopes,” Opt. Eng. 49, 063202 (2010).
[CrossRef]

Castro, J.

Choi, N.

A. Krywonos, J. E. Harvey, and N. Choi, “Linear systems formulation of scattering theory for rough surfaces with arbitrary incident and scattering angles,” J. Opt. Soc. Am. A 28, 1121–1138 (2011).
[CrossRef]

J. E. Harvey, N. Choi, A. Krywonos, G. Peterson, and M. Bruner, “Image degradation due to scattering effects in two-mirror telescopes,” Opt. Eng. 49, 063202 (2010).
[CrossRef]

J. E. Harvey, N. Choi, A. Krywonos, and J. Marcen, “Calculating BRDFs from surface PSDs for moderately rough surfaces,” Proc. SPIE 7426, 64260I (2009).

Church, E. L.

Coluccelli, N.

Glenn, P.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

Harvey, J. E.

A. Krywonos, J. E. Harvey, and N. Choi, “Linear systems formulation of scattering theory for rough surfaces with arbitrary incident and scattering angles,” J. Opt. Soc. Am. A 28, 1121–1138 (2011).
[CrossRef]

J. E. Harvey, N. Choi, A. Krywonos, G. Peterson, and M. Bruner, “Image degradation due to scattering effects in two-mirror telescopes,” Opt. Eng. 49, 063202 (2010).
[CrossRef]

J. E. Harvey, N. Choi, A. Krywonos, and J. Marcen, “Calculating BRDFs from surface PSDs for moderately rough surfaces,” Proc. SPIE 7426, 64260I (2009).

J. E. Harvey, A. Krywonos, M. Atanassova, and P. L. Thompson, “The Solar X-ray Imager (SXI) on GOES-13: design, analysis, and on-orbit performance,” Proc. SPIE 6689, 66890I (2007).
[CrossRef]

J. E. Harvey, M. Atanassova, and A. Krywonos, “Systems engineering analysis of five “as-manufactured” SXI telescopes,” Proc. SPIE 5867, 58670F (2005).
[CrossRef]

J. E. Harvey and A. Krywonos, “A systems engineering analysis of image quality,” Proc. SPIE 4093, 379–388 (2000).
[CrossRef]

J. E. Harvey, “Surface scatter phenomena: a linear, shift-invariant process,” Proc. SPIE 1165, 87–99 (1989).

J. E. Harvey, “Light-scattering characteristics of optical surfaces,” Ph.D. dissertation (University of Arizona, 1976).

Kostuk, R. K.

Krywonos, A.

A. Krywonos, J. E. Harvey, and N. Choi, “Linear systems formulation of scattering theory for rough surfaces with arbitrary incident and scattering angles,” J. Opt. Soc. Am. A 28, 1121–1138 (2011).
[CrossRef]

J. E. Harvey, N. Choi, A. Krywonos, G. Peterson, and M. Bruner, “Image degradation due to scattering effects in two-mirror telescopes,” Opt. Eng. 49, 063202 (2010).
[CrossRef]

J. E. Harvey, N. Choi, A. Krywonos, and J. Marcen, “Calculating BRDFs from surface PSDs for moderately rough surfaces,” Proc. SPIE 7426, 64260I (2009).

J. E. Harvey, A. Krywonos, M. Atanassova, and P. L. Thompson, “The Solar X-ray Imager (SXI) on GOES-13: design, analysis, and on-orbit performance,” Proc. SPIE 6689, 66890I (2007).
[CrossRef]

J. E. Harvey, M. Atanassova, and A. Krywonos, “Systems engineering analysis of five “as-manufactured” SXI telescopes,” Proc. SPIE 5867, 58670F (2005).
[CrossRef]

J. E. Harvey and A. Krywonos, “A systems engineering analysis of image quality,” Proc. SPIE 4093, 379–388 (2000).
[CrossRef]

Lin, P. D.

Liu, C.

Luo, Y.

Mahajan, V. N.

V. N. Mahajan, Optical Imaging and Aberrations: Part I (SPIE, 1998), pp. 206–208, 281–287.

Marcen, J.

J. E. Harvey, N. Choi, A. Krywonos, and J. Marcen, “Calculating BRDFs from surface PSDs for moderately rough surfaces,” Proc. SPIE 7426, 64260I (2009).

Nicodemus, F. E.

Noll, R. J.

Peterson, G.

J. E. Harvey, N. Choi, A. Krywonos, G. Peterson, and M. Bruner, “Image degradation due to scattering effects in two-mirror telescopes,” Opt. Eng. 49, 063202 (2010).
[CrossRef]

Peterson, G. L.

G. L. Peterson, “Analytic expressions for in-field scattered light distribution,” Proc. SPIE 5178, 184–193 (2004).
[CrossRef]

G. L. Peterson, “Analytic expressions for in-field stray light irradiance in imaging systems,” Master’s report (University of Arizona, 2003).

Smith, W. J.

W. J. Smith, Modern Optical Engineering, 3rd and international ed. (McGraw-Hill, 2000), pp. 360–361, 372.

Spizzichino, A.

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, 1963).

Stover, J. C.

J. C. Stover, Optical Scattering, Measurement and Analysis, 2nd ed. (SPIE, 1995).

Thompson, P. L.

J. E. Harvey, A. Krywonos, M. Atanassova, and P. L. Thompson, “The Solar X-ray Imager (SXI) on GOES-13: design, analysis, and on-orbit performance,” Proc. SPIE 6689, 66890I (2007).
[CrossRef]

Weber, H. J.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 5th ed. (Academic, 2001).

Welford, W. T.

W. T. Welford, Aberrations of Optical Systems (Hilger, 1991), pp. 113–114.

Appl. Opt. (6)

J. Opt. Soc. Am. A (1)

Opt. Eng. (1)

J. E. Harvey, N. Choi, A. Krywonos, G. Peterson, and M. Bruner, “Image degradation due to scattering effects in two-mirror telescopes,” Opt. Eng. 49, 063202 (2010).
[CrossRef]

Opt. Express (1)

Proc. SPIE (6)

J. E. Harvey, “Surface scatter phenomena: a linear, shift-invariant process,” Proc. SPIE 1165, 87–99 (1989).

J. E. Harvey, N. Choi, A. Krywonos, and J. Marcen, “Calculating BRDFs from surface PSDs for moderately rough surfaces,” Proc. SPIE 7426, 64260I (2009).

G. L. Peterson, “Analytic expressions for in-field scattered light distribution,” Proc. SPIE 5178, 184–193 (2004).
[CrossRef]

J. E. Harvey and A. Krywonos, “A systems engineering analysis of image quality,” Proc. SPIE 4093, 379–388 (2000).
[CrossRef]

J. E. Harvey, A. Krywonos, M. Atanassova, and P. L. Thompson, “The Solar X-ray Imager (SXI) on GOES-13: design, analysis, and on-orbit performance,” Proc. SPIE 6689, 66890I (2007).
[CrossRef]

J. E. Harvey, M. Atanassova, and A. Krywonos, “Systems engineering analysis of five “as-manufactured” SXI telescopes,” Proc. SPIE 5867, 58670F (2005).
[CrossRef]

Other (10)

W. J. Smith, Modern Optical Engineering, 3rd and international ed. (McGraw-Hill, 2000), pp. 360–361, 372.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 5th ed. (Academic, 2001).

V. N. Mahajan, Optical Imaging and Aberrations: Part I (SPIE, 1998), pp. 206–208, 281–287.

J. E. Harvey, “Light-scattering characteristics of optical surfaces,” Ph.D. dissertation (University of Arizona, 1976).

G. L. Peterson, “Analytic expressions for in-field stray light irradiance in imaging systems,” Master’s report (University of Arizona, 2003).

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

Zemax Development Corporation, Zemax User Manual, 3001 112th Avenue NE, Suite 202, Bellevue, Washington 98004-8017, USA (2009).

W. T. Welford, Aberrations of Optical Systems (Hilger, 1991), pp. 113–114.

P. Beckmann and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, 1963).

J. C. Stover, Optical Scattering, Measurement and Analysis, 2nd ed. (SPIE, 1995).

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Figures (11)

Fig. 1.
Fig. 1.

Schematic layout of an optical imaging system consisting of a series of coaxial optical surfaces.

Fig. 2.
Fig. 2.

Schematic layout of a single-surface optical imaging system.

Fig. 3.
Fig. 3.

Schematic layout of a two-surface optical imaging system.

Fig. 4.
Fig. 4.

Ray aberration due to scattering in a single surface.

Fig. 5.
Fig. 5.

Schematic layout of a Cassegrain-type two-mirror telescope. The marginal ray heights at the primary and secondary mirrors are designated by h 1 and h 2 , respectively.

Fig. 6.
Fig. 6.

Composite surface PSD function determined from four different metrology instruments. A K -correlation function has been fit to the experimental data to characterize the surface.

Fig. 7.
Fig. 7.

Predicted BSDF profiles for three small angles of incidence ( θ i ) and small angle of scattering for 9.4 nm wavelength of light in (a) direction cosine space and (b) shifted direction cosine space.

Fig. 8.
Fig. 8.

PSF by scattering and aberrations for 0.5° field angle by (a) the convolution method and (b) Zemax; (c) contour map of (a); (d) contour map of (b). Numbers in the axis denote axial distance from the Gaussian image point in millimeters.

Fig. 9.
Fig. 9.

Illustration of the contour map of the composite PSF due to both scattering and aberrations for a 0.5° field angle as calculated by (a) the convolution method, (b) Zemax. Numbers in the axis denote axial distance from the Gaussian image point in millimeters.

Fig. 10.
Fig. 10.

En-squared energy for 0° and 0.5° field angle for the convolution method (solid line), Zemax (asterisks), and aberration-free case (dotted line) centered on the Gaussian image point.

Fig. 11.
Fig. 11.

Two infinitesimal areas in (a) pupil plane and (b) image plane. The infinitesimal area in (a) is mapped into infinitesimal square in (b).

Tables (1)

Tables Icon

Table 1. Defocus and Seidel Aberration Coefficients of the Two-Mirror Telescope ( λ = 9.4 nm )

Equations (54)

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Ψ output ( η output ) = d η input K ( η output , η input ) · Ψ input ( η input ) ,
W j ( ξ j | x j 1 ) = W 040 p j 2 + W 131 p j c j + W 222 c j 2 + W 220 p j q j + W 311 q j c j ,
W ( ξ n | x 0 ) = j = 1 n W j ( ξ n / m j n ( p ) | m 1 m 2 m j 1 x 0 ) .
ε j ( ξ j | x j 1 ) = r j j W j ( ξ j | x j 1 ) ,
ε ( ξ n | x 0 ) = r n n W ( ξ n | x 0 ) = j = 1 n r n n W j ( ξ n / m j n ( p ) | m 1 m 2 m j 1 x 0 ) = j = 1 n [ m j + 1 m n r j j W j ( ξ j | x j 1 ) ] = j = 1 n [ m j + 1 m n ε j ( ξ j | x j 1 ) ] .
δ ( x x 1 ) = δ ( x z 1 z 1 x 0 ) .
E ( x 1 ) = P inc d 2 x 0 1 z 1 2 δ ( x 1 z 1 x 0 z 1 ) E 0 ( x 0 ) = P inc δ ( x 1 m 1 x obj ) ,
δ ( x x 1 ) = δ ( x z 1 z 1 x 0 ε 1 ( ξ 1 | x 0 ) ) .
E ( x 1 ) = E p d 2 x 0 d 2 ξ 1 1 z 1 2 δ ( x 1 z 1 x 0 z 1 ε 1 ( ξ 1 | x 0 ) z 1 ) E 0 ( x 0 ) .
δ ( x a b + c ) = d 2 x δ ( x a b ) δ ( x x + a c ) ,
E ( x 1 ) = d 2 x 1 [ 1 z 1 2 δ ( x 1 z 1 x obj z 1 ) ] [ E p d 2 ξ 1 δ ( x 1 x 1 ε 1 ( ξ 1 | x obj ) ) ] .
E ( x 1 ) = PSF G ( x 1 | x obj ) δ ( x 1 m 1 x obj ) ,
PSF G ( x 1 | x obj ) E p d 2 ξ 1 δ ( x 1 + r 1 1 W ( ξ 1 | x obj ) ) .
δ ( x x 2 ) = d 2 x 1 δ ( x 1 z 1 z 1 x 0 ε 1 ( ξ 1 | x 0 ) ) δ ( x z 2 z 2 x 1 ε 2 ( ξ 2 | x 1 ) ) ,
δ ( x x 2 ) d 2 x 1 δ ( x 1 z 1 z 1 x 0 ) δ ( x z 2 z 2 x 1 z 2 z 2 ε 1 ( ξ 1 | x 0 ) ε 2 ( ξ 2 | z 1 z 1 x 0 ) ) .
ε 2 ( ξ 2 | m 1 x 0 + ε 1 ( ξ 1 | x 0 ) ) ε 2 ( ξ 2 | m 1 x 0 ) .
E ( x 2 ) = E p d 2 x 0 d 2 x 1 d 2 ξ [ j = 1 2 1 z j 2 δ ( x j z j x j 1 z j ε j z j ) ] E 0 ( x 0 ) ,
E ( x 2 ) = d 2 x 1 d 2 x 1 d 2 x 2 [ j = 1 2 1 z j 2 δ ( x j z j x j 1 z j ) ] [ G ( x 1 , x 2 , x 1 , x 2 | x obj ) ] ,
G ( x 1 , x 2 , x 1 , x 2 | x obj ) = E p d 2 ξ δ ( x 1 x 1 ε 1 ) δ ( x 2 x 2 ε 2 ) δ ( x 1 x 1 ) PSF G ( x 2 x 2 | x obj )
PSF G ( x 2 | x obj ) E p d 2 ξ 2 δ ( x 2 + r 2 2 W ( ξ 2 | x obj ) ) .
E ( x 2 ) PSF G ( x 2 | x obj ) δ ( x 2 m 2 m 1 x obj ) ,
E ( x n ) = E p d 2 x 0 d 2 x n 1 d 2 ξ [ j = 1 n 1 z j 2 δ ( x j z j x j 1 z j ε j z j ) ] E 0 ( x 0 ) .
E ( x n ) = d 2 x 0 d 2 x n 1 d 2 x 1 d 2 x n [ j = 1 n 1 z j 2 δ ( x j z j x j 1 z j ) ] [ G ( x 1 , , x n , x 1 , , x n | x 0 ) ] E 0 ( x 0 ) ,
G ( x 1 , , x n , x 1 , , x n | x 0 ) = E p d 2 ξ [ j = 1 n δ ( x j x j ε j ) ] .
G ( x 1 , , x n , x 1 , , x n | x 0 ) [ j = 1 n 1 δ ( x j x j ) ] PSF G ( x n x n | x 0 ) ,
PSF G ( x n | x obj ) = E p d 2 ξ n δ ( x n + r n n W ( ξ n | x obj ) ) .
E ( x n ) = PSF G ( x n | x obj ) δ ( x n m x obj ) .
BSDF ( α s , α 0 ) = d L s ( α s , α 0 ) d E i ,
BSDF ( α s α 0 ) Q · [ A · δ ( α s α 0 ) + S ( α s α 0 ) ] ,
W ( x 0 , ξ , α ) = W ( x 0 , ξ ) + W s ( α , ξ ) ,
W ( x 0 , ξ , α ) = W ( x 0 , ξ ) α · ξ ,
δ ( x x j ) = δ ( x z j z j x j 1 r j α j ) .
d P = BSDF j ( α j ) d α j = r j 2 z j 2 BSDF j ( r j z j α j ) d α j .
E ( x n ) = P inc d 2 x 0 d 2 x n 1 d 2 α 1 d 2 α j j = 1 n r j 2 z j 2 BSDF j ( r j z j α j ) 1 z j 2 δ ( x j z j x j 1 z j r j z j α j ) E 0 ( x 0 ) = P inc d 2 x 0 d 2 x n 1 [ j = 1 n T j ( x j | x j 1 ) ] E 0 ( x 0 ) ,
T j ( x j | x j 1 ) = d 2 α j r j 2 z j 2 BSDF j ( r j z j α j ) 1 z j 2 δ ( x j z j x j 1 z j r j z j α j ) = 1 z j 2 BSDF j ( x j z j x j 1 z j ) .
x j z j x j 1 z j α s α 0 .
E ( x n ) = P inc 1 d n 2 BSDF n ( x n d n ) 1 d 1 2 BSDF 1 ( x n d 1 ) δ ( x n m x obj ) P inc PSF S ( x n ) δ ( x n m x obj ) ,
PSF S ( x ) = P inc 1 d j 2 BSDF j ( x d j ) ,
δ ( x x j ) = δ ( x z j z j x j 1 r j α j ε j ) .
E ( x n ) = d 2 x 0 d 2 x n 1 K ( x 0 , , x n ) E 0 ( x 0 ) ,
K = E p d 2 α 1 d 2 α n d 2 ξ [ j = 1 n r j 2 z j 2 BSDF j ( r j z j α j ) 1 z j 2 δ ( x j z j x j 1 z j r j α j z j ε j z j ) ] .
ε j + 1 ( ξ j + 1 | m j x j 1 + r j α j + ε j ) ε j + 1 ( ξ j + 1 | m j x j 1 ) ,
K = d 2 x 1 d 2 x n [ j = 1 n T j ( x j | x j 1 ) ] [ G ( x 1 , , x n , x 1 , , x n | x 0 ) ] ,
E ( x n ) = PSF G ( x n | x obj ) PSF S ( x n ) δ ( x n m x obj ) PSF ( x n ) δ ( x n m x obj ) ,
ε j + 1 ( T ) m j + 1 ( ε j + z j α j ) + ( ε j + 1 + z j + 1 α j + 1 ) ,
PSF S ( x ) = 1 f 2 ( h 2 / h 1 ) 2 BSDF 2 ( x f h 2 / h 1 ) 1 f 2 BSDF 1 ( x f ) .
ε ( ξ ) = T [ W ( ξ ) ] ,
E ( x c ) = E p d S p d S i ,
E ( x ) = E p S p N m = 1 N δ ( x T [ W ( ξ m ) ] ) ,
E ( x c ) = 1 Δ S i Δ S i d 2 x E ( x ) = E p S p N 1 Δ S i Δ S i d 2 x m = 1 N δ ( x T [ W ( ξ m ) ] ) = E p S p N 1 Δ S i d 2 x m = 1 M δ ( x ) = E p S p N 1 Δ S i M = E p 1 Δ S i ( S p M N ) ,
Lim Δ S i 0 ( Lim N E ) = Lim Δ S i 0 ( E p Δ S p Δ S i ) = E p d S p d S i ,
τ ( x ) = Lim Δ S i 0 1 Δ S i , Δ x 2 < x < Δ x 2 , Δ x 2 < y < Δ x 2 , = 0 , elsewhere ,
E ( x ) = E p d 2 ξ τ ( x T [ W ( ξ ) ] ) .
E ( x c ) = Lim Δ S i 0 E p Δ S p d 2 ξ 1 Δ S i = Lim Δ S i 0 E p Δ S p Δ S i = E p d S p d S i ,

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