Abstract

Numerical modeling of bright-field and dark-field imaging with spatially partially coherent light is considered. The illuminating field is expressed as a superposition of transversely shifted fully coherent elementary fields of identical form. Examples of imaging under variable coherence conditions demonstrate the computational feasibility of the model even when the coherence area of the illumination is in the wavelength scale.

© 2015 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Elementary-field analysis of partially coherent beam shaping

Manisha Singh, Jani Tervo, and Jari Turunen
J. Opt. Soc. Am. A 30(12) 2611-2617 (2013)

Independent-elementary-field model for three-dimensional spatially partially coherent sources

Jari Turunen and Pasi Vahimaa
Opt. Express 16(9) 6433-6442 (2008)

Shifted-elementary-mode representation for partially coherent vectorial fields

Jani Tervo, Jari Turunen, Pasi Vahimaa, and Frank Wyrowski
J. Opt. Soc. Am. A 27(9) 2004-2014 (2010)

References

  • View by:
  • |
  • |
  • |

  1. H. H. Hopkins, “The concept of partial coherence in optics,” Proc. Royal Soc. 208, 263–277 (1951).
    [Crossref]
  2. J. W. Goodman, Statistical Optics (Wiley, 2000).
  3. E. C. Kintner, “Method for the calculation of partially coherent imagery,” Appl. Opt. 17, 2747–2753 (1978).
    [Crossref] [PubMed]
  4. J. van der Gracht, “Simulation of partially coherent imaging by outer-product expansion,” Appl. Opt. 17, 3725–3731 (1994).
    [Crossref]
  5. K. Yamazoe, “Computation theory of partially coherent imaging by stacked pupil shift matrix,” J. Opt. Soc. Am. A 25, 3111–3119 (2008).
    [Crossref]
  6. S. B. Mehta and C. J. R. Sheppard, “Phase-space representation of partially coherent imaging using the Cohen class distribution,” Opt. Lett. 35, 348–350 (2010).
    [Crossref] [PubMed]
  7. K. Yamazoe, “Two models for partially coherent imaging,” J. Opt. Soc. Am. A 29, 2591–2597 (2012).
    [Crossref]
  8. A. E. Rosenbluth, S. J. Bukofsky, M. Hibbs, K. Lai, R. N. Singh, and A. K. K. Wong, “Optimum mask and source patterns to print a given shape,” J. Micro/Nanolith., Microfab., Microsyst. 1, 13–30 (2002).
  9. R. L. Gordon and A. E. Rosenbluth, “Lithographic Image Simulation for the 21st Century with 19th-Century Tools,” Proc. SPIE 5182, 73–87 (2003).
    [Crossref]
  10. R. L. Gordon, “Exact Computation of Scalar, 2D Aerial Imagery,” Proc. SPIE 4692, 517–531 (2002).
    [Crossref]
  11. E. Wolf, “New theory of partial coherence in the space-frequency domain. Part I: Spectra and cross-spectra of steady-state sources,” J. Opt. Soc. Am. 72, 343–351 (1982).
    [Crossref]
  12. B. E. A. Saleh and M. Rabbani, “Simulation of partially coherent imagery in the space and frequency domains and by modal expansion,” Appl. Opt. 21, 2770–2777 (1982).
    [Crossref] [PubMed]
  13. A. S. Ostrovsky, O. Ramos-Romero, and M. V. Rodríguez-Solís, “Coherent-mode representation of partially coherent imagery,” Opt. Rev. 3, 492–496 (1996).
    [Crossref]
  14. A. Burvall, A. Smith, and C. Dainty, “Elementary functions: propagation of partially coherent light,” J. Opt. Soc. Am. A 26, 1721–1729 (2009).
    [Crossref]
  15. F. Gori and C. Palma, “Partially coherent sources which give rise to highly directional laser beams,” Opt. Commun. 27, 185–188 (1978).
    [Crossref]
  16. P. Vahimaa and J. Turunen, “Finite-elementary-source model for partially coherent radiation,” Opt. Express 14, 1376–1381 (2006).
    [Crossref] [PubMed]
  17. J. Tervo, J. Turunen, P. Vahimaa, and F. Wyrowski, “Shifted-elementary-mode representation for partially coherent vectorial fields,” J. Opt. Soc. Am. A 27, 2004–2014 (2010).
    [Crossref]
  18. F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32, 3531–3533 (2007).
    [Crossref] [PubMed]
  19. R. Martínez-Herrero, P. M. Mejías, and F. Gori, “Genuine cross-spectral densities and pseudo-modal expansions,” Opt. Lett. 34, 1399–1401 (2009).
    [Crossref] [PubMed]
  20. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999), Sect. 10.6.2.
    [Crossref]
  21. C. J. R. Sheppard and T. Wilson, “The theory of the direct-view confocal microscope,” J. Microsc. 124, 107–117 (1981).
    [Crossref] [PubMed]
  22. J. W. Goodman, Fourier Optics, 3rd ed. (Roberts & Company, 2005), Chap. 6.
  23. R. E. Kinzly, “Images of coherently illuminated edged objects formed by scanning optical systems,” J. Opt. Soc. Soc. Am. 56, 9–11 (1966).
    [Crossref]
  24. B. Möller, “Imaging of a straight edge in the partially coherent illumination in the presence of spherical aberration,” Opt. Acta 15, 223–236 (1968).
    [Crossref]
  25. B. M. Watrasiewicz, “Theoretical calculations of images of straight edges in partially coherent illumination,” Opt. Acta: Int. J. of Opt. 12, 391–400 (1965).
    [Crossref]
  26. D. S. Goodman and A. E. Rosenbluth, “Condenser aberrations in Köhler illumination,” Proc. SPIE 922, 108–134 (1988).
    [Crossref]
  27. S. Subramanian, “Rapid calculation of defocused partially coherent images,” Appl. Opt. 20, 1854–1857 (1981).
    [Crossref] [PubMed]
  28. L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University., 2006).
    [Crossref]
  29. H. Kim, J. Park, and B. Lee, Fourier Modal Method and Its Applications in Computational Nanophotonics (CRC Press, 2012).

2012 (1)

2010 (2)

2009 (2)

2008 (1)

2007 (1)

2006 (1)

2003 (1)

R. L. Gordon and A. E. Rosenbluth, “Lithographic Image Simulation for the 21st Century with 19th-Century Tools,” Proc. SPIE 5182, 73–87 (2003).
[Crossref]

2002 (2)

R. L. Gordon, “Exact Computation of Scalar, 2D Aerial Imagery,” Proc. SPIE 4692, 517–531 (2002).
[Crossref]

A. E. Rosenbluth, S. J. Bukofsky, M. Hibbs, K. Lai, R. N. Singh, and A. K. K. Wong, “Optimum mask and source patterns to print a given shape,” J. Micro/Nanolith., Microfab., Microsyst. 1, 13–30 (2002).

1996 (1)

A. S. Ostrovsky, O. Ramos-Romero, and M. V. Rodríguez-Solís, “Coherent-mode representation of partially coherent imagery,” Opt. Rev. 3, 492–496 (1996).
[Crossref]

1994 (1)

J. van der Gracht, “Simulation of partially coherent imaging by outer-product expansion,” Appl. Opt. 17, 3725–3731 (1994).
[Crossref]

1988 (1)

D. S. Goodman and A. E. Rosenbluth, “Condenser aberrations in Köhler illumination,” Proc. SPIE 922, 108–134 (1988).
[Crossref]

1982 (2)

1981 (2)

S. Subramanian, “Rapid calculation of defocused partially coherent images,” Appl. Opt. 20, 1854–1857 (1981).
[Crossref] [PubMed]

C. J. R. Sheppard and T. Wilson, “The theory of the direct-view confocal microscope,” J. Microsc. 124, 107–117 (1981).
[Crossref] [PubMed]

1978 (2)

E. C. Kintner, “Method for the calculation of partially coherent imagery,” Appl. Opt. 17, 2747–2753 (1978).
[Crossref] [PubMed]

F. Gori and C. Palma, “Partially coherent sources which give rise to highly directional laser beams,” Opt. Commun. 27, 185–188 (1978).
[Crossref]

1968 (1)

B. Möller, “Imaging of a straight edge in the partially coherent illumination in the presence of spherical aberration,” Opt. Acta 15, 223–236 (1968).
[Crossref]

1966 (1)

R. E. Kinzly, “Images of coherently illuminated edged objects formed by scanning optical systems,” J. Opt. Soc. Soc. Am. 56, 9–11 (1966).
[Crossref]

1965 (1)

B. M. Watrasiewicz, “Theoretical calculations of images of straight edges in partially coherent illumination,” Opt. Acta: Int. J. of Opt. 12, 391–400 (1965).
[Crossref]

1951 (1)

H. H. Hopkins, “The concept of partial coherence in optics,” Proc. Royal Soc. 208, 263–277 (1951).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999), Sect. 10.6.2.
[Crossref]

Bukofsky, S. J.

A. E. Rosenbluth, S. J. Bukofsky, M. Hibbs, K. Lai, R. N. Singh, and A. K. K. Wong, “Optimum mask and source patterns to print a given shape,” J. Micro/Nanolith., Microfab., Microsyst. 1, 13–30 (2002).

Burvall, A.

Dainty, C.

Goodman, D. S.

D. S. Goodman and A. E. Rosenbluth, “Condenser aberrations in Köhler illumination,” Proc. SPIE 922, 108–134 (1988).
[Crossref]

Goodman, J. W.

J. W. Goodman, Fourier Optics, 3rd ed. (Roberts & Company, 2005), Chap. 6.

J. W. Goodman, Statistical Optics (Wiley, 2000).

Gordon, R. L.

R. L. Gordon and A. E. Rosenbluth, “Lithographic Image Simulation for the 21st Century with 19th-Century Tools,” Proc. SPIE 5182, 73–87 (2003).
[Crossref]

R. L. Gordon, “Exact Computation of Scalar, 2D Aerial Imagery,” Proc. SPIE 4692, 517–531 (2002).
[Crossref]

Gori, F.

Hecht, B.

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University., 2006).
[Crossref]

Hibbs, M.

A. E. Rosenbluth, S. J. Bukofsky, M. Hibbs, K. Lai, R. N. Singh, and A. K. K. Wong, “Optimum mask and source patterns to print a given shape,” J. Micro/Nanolith., Microfab., Microsyst. 1, 13–30 (2002).

Hopkins, H. H.

H. H. Hopkins, “The concept of partial coherence in optics,” Proc. Royal Soc. 208, 263–277 (1951).
[Crossref]

Kim, H.

H. Kim, J. Park, and B. Lee, Fourier Modal Method and Its Applications in Computational Nanophotonics (CRC Press, 2012).

Kintner, E. C.

Kinzly, R. E.

R. E. Kinzly, “Images of coherently illuminated edged objects formed by scanning optical systems,” J. Opt. Soc. Soc. Am. 56, 9–11 (1966).
[Crossref]

Lai, K.

A. E. Rosenbluth, S. J. Bukofsky, M. Hibbs, K. Lai, R. N. Singh, and A. K. K. Wong, “Optimum mask and source patterns to print a given shape,” J. Micro/Nanolith., Microfab., Microsyst. 1, 13–30 (2002).

Lee, B.

H. Kim, J. Park, and B. Lee, Fourier Modal Method and Its Applications in Computational Nanophotonics (CRC Press, 2012).

Martínez-Herrero, R.

Mehta, S. B.

Mejías, P. M.

Möller, B.

B. Möller, “Imaging of a straight edge in the partially coherent illumination in the presence of spherical aberration,” Opt. Acta 15, 223–236 (1968).
[Crossref]

Novotny, L.

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University., 2006).
[Crossref]

Ostrovsky, A. S.

A. S. Ostrovsky, O. Ramos-Romero, and M. V. Rodríguez-Solís, “Coherent-mode representation of partially coherent imagery,” Opt. Rev. 3, 492–496 (1996).
[Crossref]

Palma, C.

F. Gori and C. Palma, “Partially coherent sources which give rise to highly directional laser beams,” Opt. Commun. 27, 185–188 (1978).
[Crossref]

Park, J.

H. Kim, J. Park, and B. Lee, Fourier Modal Method and Its Applications in Computational Nanophotonics (CRC Press, 2012).

Rabbani, M.

Ramos-Romero, O.

A. S. Ostrovsky, O. Ramos-Romero, and M. V. Rodríguez-Solís, “Coherent-mode representation of partially coherent imagery,” Opt. Rev. 3, 492–496 (1996).
[Crossref]

Rodríguez-Solís, M. V.

A. S. Ostrovsky, O. Ramos-Romero, and M. V. Rodríguez-Solís, “Coherent-mode representation of partially coherent imagery,” Opt. Rev. 3, 492–496 (1996).
[Crossref]

Rosenbluth, A. E.

R. L. Gordon and A. E. Rosenbluth, “Lithographic Image Simulation for the 21st Century with 19th-Century Tools,” Proc. SPIE 5182, 73–87 (2003).
[Crossref]

A. E. Rosenbluth, S. J. Bukofsky, M. Hibbs, K. Lai, R. N. Singh, and A. K. K. Wong, “Optimum mask and source patterns to print a given shape,” J. Micro/Nanolith., Microfab., Microsyst. 1, 13–30 (2002).

D. S. Goodman and A. E. Rosenbluth, “Condenser aberrations in Köhler illumination,” Proc. SPIE 922, 108–134 (1988).
[Crossref]

Saleh, B. E. A.

Santarsiero, M.

Sheppard, C. J. R.

Singh, R. N.

A. E. Rosenbluth, S. J. Bukofsky, M. Hibbs, K. Lai, R. N. Singh, and A. K. K. Wong, “Optimum mask and source patterns to print a given shape,” J. Micro/Nanolith., Microfab., Microsyst. 1, 13–30 (2002).

Smith, A.

Subramanian, S.

Tervo, J.

Turunen, J.

Vahimaa, P.

van der Gracht, J.

J. van der Gracht, “Simulation of partially coherent imaging by outer-product expansion,” Appl. Opt. 17, 3725–3731 (1994).
[Crossref]

Watrasiewicz, B. M.

B. M. Watrasiewicz, “Theoretical calculations of images of straight edges in partially coherent illumination,” Opt. Acta: Int. J. of Opt. 12, 391–400 (1965).
[Crossref]

Wilson, T.

C. J. R. Sheppard and T. Wilson, “The theory of the direct-view confocal microscope,” J. Microsc. 124, 107–117 (1981).
[Crossref] [PubMed]

Wolf, E.

Wong, A. K. K.

A. E. Rosenbluth, S. J. Bukofsky, M. Hibbs, K. Lai, R. N. Singh, and A. K. K. Wong, “Optimum mask and source patterns to print a given shape,” J. Micro/Nanolith., Microfab., Microsyst. 1, 13–30 (2002).

Wyrowski, F.

Yamazoe, K.

Appl. Opt. (4)

J. Micro/Nanolith., Microfab., Microsyst. (1)

A. E. Rosenbluth, S. J. Bukofsky, M. Hibbs, K. Lai, R. N. Singh, and A. K. K. Wong, “Optimum mask and source patterns to print a given shape,” J. Micro/Nanolith., Microfab., Microsyst. 1, 13–30 (2002).

J. Microsc. (1)

C. J. R. Sheppard and T. Wilson, “The theory of the direct-view confocal microscope,” J. Microsc. 124, 107–117 (1981).
[Crossref] [PubMed]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (4)

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

R. E. Kinzly, “Images of coherently illuminated edged objects formed by scanning optical systems,” J. Opt. Soc. Soc. Am. 56, 9–11 (1966).
[Crossref]

Opt. Acta (1)

B. Möller, “Imaging of a straight edge in the partially coherent illumination in the presence of spherical aberration,” Opt. Acta 15, 223–236 (1968).
[Crossref]

Opt. Acta: Int. J. of Opt. (1)

B. M. Watrasiewicz, “Theoretical calculations of images of straight edges in partially coherent illumination,” Opt. Acta: Int. J. of Opt. 12, 391–400 (1965).
[Crossref]

Opt. Commun. (1)

F. Gori and C. Palma, “Partially coherent sources which give rise to highly directional laser beams,” Opt. Commun. 27, 185–188 (1978).
[Crossref]

Opt. Express (1)

Opt. Lett. (3)

Opt. Rev. (1)

A. S. Ostrovsky, O. Ramos-Romero, and M. V. Rodríguez-Solís, “Coherent-mode representation of partially coherent imagery,” Opt. Rev. 3, 492–496 (1996).
[Crossref]

Proc. Royal Soc. (1)

H. H. Hopkins, “The concept of partial coherence in optics,” Proc. Royal Soc. 208, 263–277 (1951).
[Crossref]

Proc. SPIE (3)

R. L. Gordon and A. E. Rosenbluth, “Lithographic Image Simulation for the 21st Century with 19th-Century Tools,” Proc. SPIE 5182, 73–87 (2003).
[Crossref]

R. L. Gordon, “Exact Computation of Scalar, 2D Aerial Imagery,” Proc. SPIE 4692, 517–531 (2002).
[Crossref]

D. S. Goodman and A. E. Rosenbluth, “Condenser aberrations in Köhler illumination,” Proc. SPIE 922, 108–134 (1988).
[Crossref]

Other (5)

J. W. Goodman, Fourier Optics, 3rd ed. (Roberts & Company, 2005), Chap. 6.

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University., 2006).
[Crossref]

H. Kim, J. Park, and B. Lee, Fourier Modal Method and Its Applications in Computational Nanophotonics (CRC Press, 2012).

J. W. Goodman, Statistical Optics (Wiley, 2000).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999), Sect. 10.6.2.
[Crossref]

Supplementary Material (4)

NameDescription
» Visualization 1: MPEG (288 KB)      Visualization 1 demonstrates the diffraction image when K2 increases from 0.1P to 2P
» Visualization 2: MPEG (209 KB)      Visualization 2 shows the image when K1 = P and K2 varies from P to 2P.
» Visualization 3: MPEG (191 KB)      Visualization 3 demonstrates the effect of changing the value of a from zero to -2.
» Visualization 4: MPEG (333 KB)      Visualization 4 shows the effect of changing a from zero to -2

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1 Köhler illumination of an object O with condenser LC, followed by a telecentric imaging system consisting of lenses (or lens systems) L1 and L2 and an aperture A. Though marked here in the S plane, the symbols R1 and R2 represent the limiting spatial frequencies generated by the annular primary source in the condenser system. Correspondingly, R denotes the maximum spatial frequency passed by the circular aperture of the imaging system.
Fig. 2
Fig. 2 Edge imaging in partially coherent illumination.
Fig. 3
Fig. 3 The 2D resolution target used as the object in numerical simulations. The periods of the three four-bar gratings are r, 2r, and 5r, respectively, where r = 0.61/R is the resolution limit of the setup. The window (with the frame) contains 156 × 573 sampling points separated by a distance r/10.
Fig. 4
Fig. 4 Bright-field imaging (R1 = 0). Top: R2 = 1.5R. Middle: R2 = 0.7R. Bottom: R2 = 0.1R. Visualization 1 demonstrates the diffraction image when R2 increases from 0.1R to 2R.
Fig. 5
Fig. 5 Dark-field imaging (R1 = R). Top: R2 = 2R. Middle: R2 = 1.6P. Bottom: R2 = 1.1R. Visualization 2 shows the image when R1 = R and R2 varies from R to 2R.
Fig. 6
Fig. 6 Bright-field imaging in the presence of spherical aberration and defocus (R1 =0 and R2 = 0.7R). Top: a = 0. Middle: a = −0.75. Bottom: a = −1. Visualization 3 demonstrates the effect of changing the value of a from zero to −2.
Fig. 7
Fig. 7 Dark-field imaging in the presence of spherical aberration and defocus (R1 = R and R2 = 1.6R). Top: a = 0. Middle: a = 0.5. Bottom: a = 1. Visualization 4 shows the effect of changing a from zero to 2.

Tables (3)

Tables Icon

Table 1 Convergence of the method when the distance (characterized by Δx, a dimensionless number) between the adjacent shifted elementary fields is reduced.

Tables Icon

Table 2 Convergence of the method when the size (characterized by L, a dimensionless number) of the elementary-field window is increased.

Tables Icon

Table 3 Simulation parameters for Figs. 4 and 5. Size is the number N2 of sampling points of the elementary field and Sep. is the separation of their center points (the same in x and y directions) in pixel units 0.61/R, Num. is the total number of elementary fields, and Time is the simulation time on a desktop computer (Intel Core i7-2600K Processor).

Equations (17)

Equations on this page are rendered with MathJax. Learn more.

W 0 ( ρ 1 , ρ 2 ) = t * ( ρ 1 ) t ( ρ 2 ) W 0 ( ρ 1 , ρ 2 ) .
W ( ρ 1 , ρ 2 ) W 0 ( ρ 1 , ρ 2 ) K * ( ρ 1 , ρ 1 ) K ( ρ 2 , ρ 2 ) d 2 ρ 1 d 2 ρ 1 .
W ( ρ 1 , ρ 2 ) = p ( ρ ¯ ) H * ( ρ 1 , ρ ¯ ) H ( ρ 2 , ρ ¯ ) d 2 ρ ¯ ,
H ( ρ , ρ ¯ ) = t ( ρ ) H 0 ( ρ , ρ ¯ ) K ( ρ , ρ ) d ρ .
W 0 ( ρ 1 , ρ 2 ) = 1 ( 2 π ) 2 T ( κ ) exp ( i Δ ρ κ ) d 2 κ ,
μ 0 ( ρ 1 , ρ 2 ) = W 0 ( ρ 1 , ρ 2 ) S 0 ( ρ 1 ) S 0 ( ρ 2 ) = T ( κ ) exp ( i Δ ρ κ ) d 2 κ T ( κ ) d 2 κ
W 0 ( ρ 1 , ρ 2 ) = e 0 * ( ρ 1 ρ ¯ ) e 0 ( ρ 2 ρ ¯ ) d 2 ρ ¯ ,
e 0 ( ρ ) = 1 ( 2 π ) 2 T ( κ ) exp ( i ρ κ ) d 2 κ
W ( ρ 1 , ρ 2 ) = e * ( ρ 1 ρ ¯ ) e ( ρ 2 ρ ¯ ) d 2 ρ ¯ ,
e ( ρ ρ ¯ ) = t ( ρ ) e 0 ( ρ ρ ¯ ) K ( ρ ρ ) d 2 ρ .
S ( ρ ) = W ( ρ , ρ ) = | e ( ρ ρ ¯ ) | 2 d 2 ρ ¯ ,
g ( ξ , ρ ¯ ) = t ( ρ ) e 0 ( ρ ρ ¯ ) exp ( i ξ ρ ) d 2 ρ
P ( ξ ) = K ( ρ ) exp ( i ξ ρ ) d 2 ρ .
e ( ρ ρ ¯ ) = 1 ( 2 π ) 2 P ( ξ ) g ( ξ , ρ ¯ ) exp ( i ξ ρ ) d 2 ξ .
μ 0 ( Δ ρ ) = 2 [ R 2 J 1 ( R 2 | Δ ρ | ) R 1 J 1 ( R 1 | Δ ρ | ) ] ( R 2 2 R 1 2 ) | Δ ρ | ,
e 0 ( ρ ) = 2 [ R 2 J 1 ( R 2 | ρ | ) R 1 J 1 ( R 1 | ρ | ) ] ( R 2 2 R 1 2 ) | ρ | .
w ( ξ ) = a ( | ξ | / R ) 2 + b ( | ξ | / R ) 4 ,

Metrics