Abstract

The elementary function method is an approximate method for propagation calculations in spatially, partially coherent light in two dimensions. In this paper, we present the numerical application of this method to a 248 nm UV excimer laser source. We present experimental results of the measurement of the degree of spatial coherence and the beam profile of this source. The elementary function method is then applied to the real beam data and used to simulate the effects of imaging an opaque edge with a source of varying degrees of spatial coherence. The effect of spatial coherence on beam homogenization is also presented.

© 2013 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  9. 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]
  10. A. Starikov and E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and their radiation fields,” J. Opt. Soc. Am. 72, 923–928 (1982).
    [CrossRef]
  11. M. Unser, “Sampling—50 years after Shannon,” Proc. IEEE 88, 569–587 (2000).
    [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]
  13. R. Castañeda and F. F. Medina, “Partially coherent imaging with Schell-model beams,” Opt. Laser Technol. 29, 165–170 (1997).
    [CrossRef]
  14. F. Gori, M. Santarsiero, R. Simon, G. Piquero, R. Borghi, and G. Guattari, “Coherent-mode decomposition of partially polarized, partially coherent sources,” J. Opt. Soc. Am. A 20, 78–84 (2003).
    [CrossRef]
  15. Y. Cai and Q. Lin, “Partially coherent flat-topped multi-Gaussian-Schell-model beam and its propagation,” Opt. Commun. 239, 33–41 (2004).
    [CrossRef]
  16. J. Turunen and P. Vahimaa, “Independent-elementary-field model for three-dimensional spatially partially coherent sources,” Opt. Express 16, 6433–6442 (2008).
    [CrossRef]
  17. F. Zernike, “The concept of degree of coherence and its applications to optical problems,” Physica 5, 785–795 (1938).
    [CrossRef]
  18. E. Tervonen, J. Turunen, and A. T. Friberg, “Transverse laser-mode structure determination from spatial coherence measurements: experimental results,” Appl. Phys. B 49, 409–414 (1989).
    [CrossRef]
  19. K. A. Nugent and J. E. Trebes, “Coherence measurement technique for short-wavelength light sources,” Rev. Sci. Instrum. 63, 2146–2151 (1992).
    [CrossRef]
  20. R. Castañeda and Z. Jaroszewicz, “Determination of the spatial coherence of Schell-model beams with diffraction gratings,” Opt. Commun. 173, 115–121 (2000).
    [CrossRef]
  21. J. Garcia-Sucerquia, and R. Castañeda, “Full retrieving of the complex degree of spatial coherence: theoretical analysis,” Opt. Commun. 228, 9–19 (2003).
    [CrossRef]
  22. Y. Mejía, and A. I. González, “Measuring spatial coherence by using a mask with multiple apertures,” Opt. Commun. 273, 428–434 (2007).
    [CrossRef]
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  26. J. Turunen, P. Paakkonen, M. Kuittinen, P. Laakkonen, J. Simonen, T. Kajava, and M. Kaivola, “Diffractive shaping of excimer laser beams,” J. Mod. Opt. 47, 2467–2475 (2000).
  27. E. T. Kana, S. Bollanti, P. D. Lazzaro, and D. Murra, “Beam homogenization: theory, modelling and application to an excimer laser beam,” Proc. SPIE 5777, 716–724 (2005).
    [CrossRef]
  28. E. T. Kana, S. Bollanti, P. D. Lazzaro, D. Murra, O. Bouba, and M. B. Onana, “Laser beam homogenization: modelling and comparison with experimental results,” Opt. Commun. 264, 187–192 (2006).
    [CrossRef]
  29. K. Mann, A. Hopfmüller, P. Gorzellik, R. Schild, W. Stöffler, H. Wagner, and G. Wolbold, “Monitoring and shaping of excimer laser beam profiles,” Proc. SPIE 1834, 184–194 (1992).
    [CrossRef]
  30. A. Büttner, and U. D. Zeitner, “Wave optical analysis of light-emitting diode beam shaping using microlens arrays,” Opt. Eng. 41, 2393–2401 (2002).
    [CrossRef]
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    [CrossRef]
  32. M. Zimmermann, N. Lindlein, R. Voelkel, and K. J. Weible, “Microlens laser beam homogenizer—from theory to application,” Proc. SPIE 6663, 6663021 (2007).
    [CrossRef]
  33. E. Hecht, Optics (Addison Wesley, 2002).

2009 (1)

2008 (1)

2007 (2)

M. Zimmermann, N. Lindlein, R. Voelkel, and K. J. Weible, “Microlens laser beam homogenizer—from theory to application,” Proc. SPIE 6663, 6663021 (2007).
[CrossRef]

Y. Mejía, and A. I. González, “Measuring spatial coherence by using a mask with multiple apertures,” Opt. Commun. 273, 428–434 (2007).
[CrossRef]

2006 (1)

E. T. Kana, S. Bollanti, P. D. Lazzaro, D. Murra, O. Bouba, and M. B. Onana, “Laser beam homogenization: modelling and comparison with experimental results,” Opt. Commun. 264, 187–192 (2006).
[CrossRef]

2005 (2)

M. Wald, M. Burkhardt, A. Pesch, H. Gross, and J. Greif, “Design of a microscopy illumination using a partial coherent light source,” Proc. SPIE 5962, 420–429 (2005).
[CrossRef]

E. T. Kana, S. Bollanti, P. D. Lazzaro, and D. Murra, “Beam homogenization: theory, modelling and application to an excimer laser beam,” Proc. SPIE 5777, 716–724 (2005).
[CrossRef]

2004 (1)

Y. Cai and Q. Lin, “Partially coherent flat-topped multi-Gaussian-Schell-model beam and its propagation,” Opt. Commun. 239, 33–41 (2004).
[CrossRef]

2003 (2)

J. Garcia-Sucerquia, and R. Castañeda, “Full retrieving of the complex degree of spatial coherence: theoretical analysis,” Opt. Commun. 228, 9–19 (2003).
[CrossRef]

F. Gori, M. Santarsiero, R. Simon, G. Piquero, R. Borghi, and G. Guattari, “Coherent-mode decomposition of partially polarized, partially coherent sources,” J. Opt. Soc. Am. A 20, 78–84 (2003).
[CrossRef]

2002 (1)

A. Büttner, and U. D. Zeitner, “Wave optical analysis of light-emitting diode beam shaping using microlens arrays,” Opt. Eng. 41, 2393–2401 (2002).
[CrossRef]

2001 (1)

2000 (3)

R. Castañeda and Z. Jaroszewicz, “Determination of the spatial coherence of Schell-model beams with diffraction gratings,” Opt. Commun. 173, 115–121 (2000).
[CrossRef]

J. Turunen, P. Paakkonen, M. Kuittinen, P. Laakkonen, J. Simonen, T. Kajava, and M. Kaivola, “Diffractive shaping of excimer laser beams,” J. Mod. Opt. 47, 2467–2475 (2000).

M. Unser, “Sampling—50 years after Shannon,” Proc. IEEE 88, 569–587 (2000).
[CrossRef]

1997 (1)

R. Castañeda and F. F. Medina, “Partially coherent imaging with Schell-model beams,” Opt. Laser Technol. 29, 165–170 (1997).
[CrossRef]

1992 (2)

K. A. Nugent and J. E. Trebes, “Coherence measurement technique for short-wavelength light sources,” Rev. Sci. Instrum. 63, 2146–2151 (1992).
[CrossRef]

K. Mann, A. Hopfmüller, P. Gorzellik, R. Schild, W. Stöffler, H. Wagner, and G. Wolbold, “Monitoring and shaping of excimer laser beam profiles,” Proc. SPIE 1834, 184–194 (1992).
[CrossRef]

1989 (1)

E. Tervonen, J. Turunen, and A. T. Friberg, “Transverse laser-mode structure determination from spatial coherence measurements: experimental results,” Appl. Phys. B 49, 409–414 (1989).
[CrossRef]

1982 (3)

1981 (1)

E. Wolf, “New spectral representation of random sources and of the partially coherent fields that they generate,” Opt. Commun. 38, 3–6 (1981).
[CrossRef]

1980 (1)

F. Gori, “Directionality and spatial coherence,” Optica Acta 27, 1025–1034 (1980).
[CrossRef]

1971 (1)

1957 (1)

1954 (1)

E. Wolf, “A macroscopic theory of interference and diffraction of light from finite sources,” Proc. R. Soc. Lond. A 225, 96–111 (1954).
[CrossRef]

1951 (1)

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

1938 (1)

F. Zernike, “The concept of degree of coherence and its applications to optical problems,” Physica 5, 785–795 (1938).
[CrossRef]

Bollanti, S.

E. T. Kana, S. Bollanti, P. D. Lazzaro, D. Murra, O. Bouba, and M. B. Onana, “Laser beam homogenization: modelling and comparison with experimental results,” Opt. Commun. 264, 187–192 (2006).
[CrossRef]

E. T. Kana, S. Bollanti, P. D. Lazzaro, and D. Murra, “Beam homogenization: theory, modelling and application to an excimer laser beam,” Proc. SPIE 5777, 716–724 (2005).
[CrossRef]

Borghi, R.

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

Bouba, O.

E. T. Kana, S. Bollanti, P. D. Lazzaro, D. Murra, O. Bouba, and M. B. Onana, “Laser beam homogenization: modelling and comparison with experimental results,” Opt. Commun. 264, 187–192 (2006).
[CrossRef]

Buck, J.

Burkhardt, M.

M. Wald, M. Burkhardt, A. Pesch, H. Gross, and J. Greif, “Design of a microscopy illumination using a partial coherent light source,” Proc. SPIE 5962, 420–429 (2005).
[CrossRef]

Burvall, A.

Büttner, A.

A. Büttner, and U. D. Zeitner, “Wave optical analysis of light-emitting diode beam shaping using microlens arrays,” Opt. Eng. 41, 2393–2401 (2002).
[CrossRef]

Cai, Y.

Y. Cai and Q. Lin, “Partially coherent flat-topped multi-Gaussian-Schell-model beam and its propagation,” Opt. Commun. 239, 33–41 (2004).
[CrossRef]

Castañeda, R.

J. Garcia-Sucerquia, and R. Castañeda, “Full retrieving of the complex degree of spatial coherence: theoretical analysis,” Opt. Commun. 228, 9–19 (2003).
[CrossRef]

R. Castañeda and Z. Jaroszewicz, “Determination of the spatial coherence of Schell-model beams with diffraction gratings,” Opt. Commun. 173, 115–121 (2000).
[CrossRef]

R. Castañeda and F. F. Medina, “Partially coherent imaging with Schell-model beams,” Opt. Laser Technol. 29, 165–170 (1997).
[CrossRef]

Dainty, C.

Friberg, A. T.

E. Tervonen, J. Turunen, and A. T. Friberg, “Transverse laser-mode structure determination from spatial coherence measurements: experimental results,” Appl. Phys. B 49, 409–414 (1989).
[CrossRef]

Garcia-Sucerquia, J.

J. Garcia-Sucerquia, and R. Castañeda, “Full retrieving of the complex degree of spatial coherence: theoretical analysis,” Opt. Commun. 228, 9–19 (2003).
[CrossRef]

Golay, M. J. E.

González, A. I.

Y. Mejía, and A. I. González, “Measuring spatial coherence by using a mask with multiple apertures,” Opt. Commun. 273, 428–434 (2007).
[CrossRef]

Gori, F.

Gorzellik, P.

K. Mann, A. Hopfmüller, P. Gorzellik, R. Schild, W. Stöffler, H. Wagner, and G. Wolbold, “Monitoring and shaping of excimer laser beam profiles,” Proc. SPIE 1834, 184–194 (1992).
[CrossRef]

Greif, J.

M. Wald, M. Burkhardt, A. Pesch, H. Gross, and J. Greif, “Design of a microscopy illumination using a partial coherent light source,” Proc. SPIE 5962, 420–429 (2005).
[CrossRef]

Gross, H.

M. Wald, M. Burkhardt, A. Pesch, H. Gross, and J. Greif, “Design of a microscopy illumination using a partial coherent light source,” Proc. SPIE 5962, 420–429 (2005).
[CrossRef]

Guattari, G.

Hecht, E.

E. Hecht, Optics (Addison Wesley, 2002).

Hopfmüller, A.

K. Mann, A. Hopfmüller, P. Gorzellik, R. Schild, W. Stöffler, H. Wagner, and G. Wolbold, “Monitoring and shaping of excimer laser beam profiles,” Proc. SPIE 1834, 184–194 (1992).
[CrossRef]

Hopkins, H. H.

H. H. Hopkins, “Applications of coherence theory in microscopy and interferometry,” J. Opt. Soc. Am. 47, 508–526 (1957).
[CrossRef]

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

Jaroszewicz, Z.

R. Castañeda and Z. Jaroszewicz, “Determination of the spatial coherence of Schell-model beams with diffraction gratings,” Opt. Commun. 173, 115–121 (2000).
[CrossRef]

Kaivola, M.

J. Turunen, P. Paakkonen, M. Kuittinen, P. Laakkonen, J. Simonen, T. Kajava, and M. Kaivola, “Diffractive shaping of excimer laser beams,” J. Mod. Opt. 47, 2467–2475 (2000).

Kajava, T.

J. Turunen, P. Paakkonen, M. Kuittinen, P. Laakkonen, J. Simonen, T. Kajava, and M. Kaivola, “Diffractive shaping of excimer laser beams,” J. Mod. Opt. 47, 2467–2475 (2000).

Kana, E. T.

E. T. Kana, S. Bollanti, P. D. Lazzaro, D. Murra, O. Bouba, and M. B. Onana, “Laser beam homogenization: modelling and comparison with experimental results,” Opt. Commun. 264, 187–192 (2006).
[CrossRef]

E. T. Kana, S. Bollanti, P. D. Lazzaro, and D. Murra, “Beam homogenization: theory, modelling and application to an excimer laser beam,” Proc. SPIE 5777, 716–724 (2005).
[CrossRef]

Kuittinen, M.

J. Turunen, P. Paakkonen, M. Kuittinen, P. Laakkonen, J. Simonen, T. Kajava, and M. Kaivola, “Diffractive shaping of excimer laser beams,” J. Mod. Opt. 47, 2467–2475 (2000).

Laakkonen, P.

J. Turunen, P. Paakkonen, M. Kuittinen, P. Laakkonen, J. Simonen, T. Kajava, and M. Kaivola, “Diffractive shaping of excimer laser beams,” J. Mod. Opt. 47, 2467–2475 (2000).

Lawrence, G. N.

Lazzaro, P. D.

E. T. Kana, S. Bollanti, P. D. Lazzaro, D. Murra, O. Bouba, and M. B. Onana, “Laser beam homogenization: modelling and comparison with experimental results,” Opt. Commun. 264, 187–192 (2006).
[CrossRef]

E. T. Kana, S. Bollanti, P. D. Lazzaro, and D. Murra, “Beam homogenization: theory, modelling and application to an excimer laser beam,” Proc. SPIE 5777, 716–724 (2005).
[CrossRef]

Lin, Q.

Y. Cai and Q. Lin, “Partially coherent flat-topped multi-Gaussian-Schell-model beam and its propagation,” Opt. Commun. 239, 33–41 (2004).
[CrossRef]

Lin, Y.

Lindlein, N.

M. Zimmermann, N. Lindlein, R. Voelkel, and K. J. Weible, “Microlens laser beam homogenizer—from theory to application,” Proc. SPIE 6663, 6663021 (2007).
[CrossRef]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Mann, K.

K. Mann, A. Hopfmüller, P. Gorzellik, R. Schild, W. Stöffler, H. Wagner, and G. Wolbold, “Monitoring and shaping of excimer laser beam profiles,” Proc. SPIE 1834, 184–194 (1992).
[CrossRef]

Medina, F. F.

R. Castañeda and F. F. Medina, “Partially coherent imaging with Schell-model beams,” Opt. Laser Technol. 29, 165–170 (1997).
[CrossRef]

Mejía, Y.

Y. Mejía, and A. I. González, “Measuring spatial coherence by using a mask with multiple apertures,” Opt. Commun. 273, 428–434 (2007).
[CrossRef]

Murra, D.

E. T. Kana, S. Bollanti, P. D. Lazzaro, D. Murra, O. Bouba, and M. B. Onana, “Laser beam homogenization: modelling and comparison with experimental results,” Opt. Commun. 264, 187–192 (2006).
[CrossRef]

E. T. Kana, S. Bollanti, P. D. Lazzaro, and D. Murra, “Beam homogenization: theory, modelling and application to an excimer laser beam,” Proc. SPIE 5777, 716–724 (2005).
[CrossRef]

Nugent, K. A.

K. A. Nugent and J. E. Trebes, “Coherence measurement technique for short-wavelength light sources,” Rev. Sci. Instrum. 63, 2146–2151 (1992).
[CrossRef]

Onana, M. B.

E. T. Kana, S. Bollanti, P. D. Lazzaro, D. Murra, O. Bouba, and M. B. Onana, “Laser beam homogenization: modelling and comparison with experimental results,” Opt. Commun. 264, 187–192 (2006).
[CrossRef]

Paakkonen, P.

J. Turunen, P. Paakkonen, M. Kuittinen, P. Laakkonen, J. Simonen, T. Kajava, and M. Kaivola, “Diffractive shaping of excimer laser beams,” J. Mod. Opt. 47, 2467–2475 (2000).

Pesch, A.

M. Wald, M. Burkhardt, A. Pesch, H. Gross, and J. Greif, “Design of a microscopy illumination using a partial coherent light source,” Proc. SPIE 5962, 420–429 (2005).
[CrossRef]

Piquero, G.

Rabbani, M.

Saleh, B. E. A.

Santarsiero, M.

Schild, R.

K. Mann, A. Hopfmüller, P. Gorzellik, R. Schild, W. Stöffler, H. Wagner, and G. Wolbold, “Monitoring and shaping of excimer laser beam profiles,” Proc. SPIE 1834, 184–194 (1992).
[CrossRef]

Simon, R.

Simonen, J.

J. Turunen, P. Paakkonen, M. Kuittinen, P. Laakkonen, J. Simonen, T. Kajava, and M. Kaivola, “Diffractive shaping of excimer laser beams,” J. Mod. Opt. 47, 2467–2475 (2000).

Smith, A.

Starikov, A.

Stöffler, W.

K. Mann, A. Hopfmüller, P. Gorzellik, R. Schild, W. Stöffler, H. Wagner, and G. Wolbold, “Monitoring and shaping of excimer laser beam profiles,” Proc. SPIE 1834, 184–194 (1992).
[CrossRef]

Tervonen, E.

E. Tervonen, J. Turunen, and A. T. Friberg, “Transverse laser-mode structure determination from spatial coherence measurements: experimental results,” Appl. Phys. B 49, 409–414 (1989).
[CrossRef]

Trebes, J. E.

K. A. Nugent and J. E. Trebes, “Coherence measurement technique for short-wavelength light sources,” Rev. Sci. Instrum. 63, 2146–2151 (1992).
[CrossRef]

Turunen, J.

J. Turunen and P. Vahimaa, “Independent-elementary-field model for three-dimensional spatially partially coherent sources,” Opt. Express 16, 6433–6442 (2008).
[CrossRef]

J. Turunen, P. Paakkonen, M. Kuittinen, P. Laakkonen, J. Simonen, T. Kajava, and M. Kaivola, “Diffractive shaping of excimer laser beams,” J. Mod. Opt. 47, 2467–2475 (2000).

E. Tervonen, J. Turunen, and A. T. Friberg, “Transverse laser-mode structure determination from spatial coherence measurements: experimental results,” Appl. Phys. B 49, 409–414 (1989).
[CrossRef]

Unser, M.

M. Unser, “Sampling—50 years after Shannon,” Proc. IEEE 88, 569–587 (2000).
[CrossRef]

Vahimaa, P.

Voelkel, R.

M. Zimmermann, N. Lindlein, R. Voelkel, and K. J. Weible, “Microlens laser beam homogenizer—from theory to application,” Proc. SPIE 6663, 6663021 (2007).
[CrossRef]

Wagner, H.

K. Mann, A. Hopfmüller, P. Gorzellik, R. Schild, W. Stöffler, H. Wagner, and G. Wolbold, “Monitoring and shaping of excimer laser beam profiles,” Proc. SPIE 1834, 184–194 (1992).
[CrossRef]

Wald, M.

M. Wald, M. Burkhardt, A. Pesch, H. Gross, and J. Greif, “Design of a microscopy illumination using a partial coherent light source,” Proc. SPIE 5962, 420–429 (2005).
[CrossRef]

Weible, K. J.

M. Zimmermann, N. Lindlein, R. Voelkel, and K. J. Weible, “Microlens laser beam homogenizer—from theory to application,” Proc. SPIE 6663, 6663021 (2007).
[CrossRef]

Wolbold, G.

K. Mann, A. Hopfmüller, P. Gorzellik, R. Schild, W. Stöffler, H. Wagner, and G. Wolbold, “Monitoring and shaping of excimer laser beam profiles,” Proc. SPIE 1834, 184–194 (1992).
[CrossRef]

Wolf, E.

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]

A. Starikov and E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and their radiation fields,” J. Opt. Soc. Am. 72, 923–928 (1982).
[CrossRef]

E. Wolf, “New spectral representation of random sources and of the partially coherent fields that they generate,” Opt. Commun. 38, 3–6 (1981).
[CrossRef]

E. Wolf, “A macroscopic theory of interference and diffraction of light from finite sources,” Proc. R. Soc. Lond. A 225, 96–111 (1954).
[CrossRef]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

E. Wolf, Introduction to the Theory of Coherence and Polarisation of Light (Cambridge University, 2007).

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

Zeitner, U. D.

A. Büttner, and U. D. Zeitner, “Wave optical analysis of light-emitting diode beam shaping using microlens arrays,” Opt. Eng. 41, 2393–2401 (2002).
[CrossRef]

Zernike, F.

F. Zernike, “The concept of degree of coherence and its applications to optical problems,” Physica 5, 785–795 (1938).
[CrossRef]

Zimmermann, M.

M. Zimmermann, N. Lindlein, R. Voelkel, and K. J. Weible, “Microlens laser beam homogenizer—from theory to application,” Proc. SPIE 6663, 6663021 (2007).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. B (1)

E. Tervonen, J. Turunen, and A. T. Friberg, “Transverse laser-mode structure determination from spatial coherence measurements: experimental results,” Appl. Phys. B 49, 409–414 (1989).
[CrossRef]

J. Mod. Opt. (1)

J. Turunen, P. Paakkonen, M. Kuittinen, P. Laakkonen, J. Simonen, T. Kajava, and M. Kaivola, “Diffractive shaping of excimer laser beams,” J. Mod. Opt. 47, 2467–2475 (2000).

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (2)

Opt. Commun. (6)

Y. Cai and Q. Lin, “Partially coherent flat-topped multi-Gaussian-Schell-model beam and its propagation,” Opt. Commun. 239, 33–41 (2004).
[CrossRef]

E. T. Kana, S. Bollanti, P. D. Lazzaro, D. Murra, O. Bouba, and M. B. Onana, “Laser beam homogenization: modelling and comparison with experimental results,” Opt. Commun. 264, 187–192 (2006).
[CrossRef]

R. Castañeda and Z. Jaroszewicz, “Determination of the spatial coherence of Schell-model beams with diffraction gratings,” Opt. Commun. 173, 115–121 (2000).
[CrossRef]

J. Garcia-Sucerquia, and R. Castañeda, “Full retrieving of the complex degree of spatial coherence: theoretical analysis,” Opt. Commun. 228, 9–19 (2003).
[CrossRef]

Y. Mejía, and A. I. González, “Measuring spatial coherence by using a mask with multiple apertures,” Opt. Commun. 273, 428–434 (2007).
[CrossRef]

E. Wolf, “New spectral representation of random sources and of the partially coherent fields that they generate,” Opt. Commun. 38, 3–6 (1981).
[CrossRef]

Opt. Eng. (1)

A. Büttner, and U. D. Zeitner, “Wave optical analysis of light-emitting diode beam shaping using microlens arrays,” Opt. Eng. 41, 2393–2401 (2002).
[CrossRef]

Opt. Express (1)

Opt. Laser Technol. (1)

R. Castañeda and F. F. Medina, “Partially coherent imaging with Schell-model beams,” Opt. Laser Technol. 29, 165–170 (1997).
[CrossRef]

Optica Acta (1)

F. Gori, “Directionality and spatial coherence,” Optica Acta 27, 1025–1034 (1980).
[CrossRef]

Physica (1)

F. Zernike, “The concept of degree of coherence and its applications to optical problems,” Physica 5, 785–795 (1938).
[CrossRef]

Proc. IEEE (1)

M. Unser, “Sampling—50 years after Shannon,” Proc. IEEE 88, 569–587 (2000).
[CrossRef]

Proc. R. Soc. Lond. A (2)

E. Wolf, “A macroscopic theory of interference and diffraction of light from finite sources,” Proc. R. Soc. Lond. A 225, 96–111 (1954).
[CrossRef]

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

Proc. SPIE (4)

M. Wald, M. Burkhardt, A. Pesch, H. Gross, and J. Greif, “Design of a microscopy illumination using a partial coherent light source,” Proc. SPIE 5962, 420–429 (2005).
[CrossRef]

M. Zimmermann, N. Lindlein, R. Voelkel, and K. J. Weible, “Microlens laser beam homogenizer—from theory to application,” Proc. SPIE 6663, 6663021 (2007).
[CrossRef]

K. Mann, A. Hopfmüller, P. Gorzellik, R. Schild, W. Stöffler, H. Wagner, and G. Wolbold, “Monitoring and shaping of excimer laser beam profiles,” Proc. SPIE 1834, 184–194 (1992).
[CrossRef]

E. T. Kana, S. Bollanti, P. D. Lazzaro, and D. Murra, “Beam homogenization: theory, modelling and application to an excimer laser beam,” Proc. SPIE 5777, 716–724 (2005).
[CrossRef]

Rev. Sci. Instrum. (1)

K. A. Nugent and J. E. Trebes, “Coherence measurement technique for short-wavelength light sources,” Rev. Sci. Instrum. 63, 2146–2151 (1992).
[CrossRef]

Other (4)

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

E. Wolf, Introduction to the Theory of Coherence and Polarisation of Light (Cambridge University, 2007).

E. Hecht, Optics (Addison Wesley, 2002).

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Figures (15)

Fig. 1.
Fig. 1.

Plot of nine shifted and weighted elementary functions for a GSM source. Only five functions contribute significantly to the reconstructed intensity envelope: the other four functions have a low weighting and are not visible on this plot.

Fig. 2.
Fig. 2.

Comparison of analytical and numerical results for the elementary function method applied to a GSM beam. (a)–(c) Intensity (thick blue line) and reconstructed intensity (red line) are shown for one-dimensional (1D) GSM beams of different degrees of coherence, along with the scaled and shifted squares of the elementary functions (thin lines), as calculated numerically. (d)–(f) Intensity and reconstructed intensity (thick line) are shown for 1D GSM beams of different degrees of coherence, along with the scaled and shifted squares of the elementary functions (thin blue lines), as calculated analytically and previously presented in [1]. The intensity distribution is the same for all beams, with σI=0.01m, while the coherence varies from high to low as (a), (d) σg=0.03m with seven elementary functions required, (b), (e) σg=0.01m with 13 elementary functions required, and (c), (f) σg=0.003m with 39 elementary functions required.

Fig. 3.
Fig. 3.

Finished design of mask with arrays of five pinholes. The pinhole diameters are A=5μm, B=10μm, and C=15μm. The slashes represent the pinholes and the pinhole separations are shown to scale. The gridlines are used to provide scale and position information for the five sets of pinholes.

Fig. 4.
Fig. 4.

(a) Simulated intensity cross section of the interference pattern produced by five pinholes illuminated with spatially coherent light. (b) Normalized Fourier magnitude spectrum of (a). Each delta function has a magnitude equal to one fifth of the normalized central maximum, except the class 5 which is doubled as it occurs twice in the pinhole mask.

Fig. 5.
Fig. 5.

Interference pattern generated by 10 μm pinholes orientated (a) along the short axis and (b) along the long axis of the 248 nm KrF laser source. (c) Intensity cross section taken through the center of (a). Similarly, (d) shows the intensity cross section of (b). (e) 1D Fourier transform of an integrated sample taken along the short axis of the beam [using data from (a)]. (f) 1D Fourier transform of an integrated sample taken along the long axis of the beam [using data from (b)].

Fig. 6.
Fig. 6.

Plot of average visibility (%) as a function of pinhole separation (μm) for the long and short axis of a 248 nm source. The pinholes used in the measurements were 10 μm and 15 μm in diameter.

Fig. 7.
Fig. 7.

Best Gaussian fit of coherence distribution data across the (a) short and (b) long axis of the 248 nm source.

Fig. 8.
Fig. 8.

Comparison of the original intensity distribution and the noise-reduced reconstructed intensity distribution calculated for the (a) short and (b) long axis of the 248 nm Braggstar source.

Fig. 9.
Fig. 9.

Intensity distribution in the image of a straight opaque edge illuminated by coherent, incoherent, and partially coherent light with a Gaussian intensity distribution. The intensity width is the same in all cases, σI=1000μm, and the coherence width varies with (a) σg=100μm, requiring 115 elementary functions, (b) σg=10μm, requiring 1145 elementary functions, (c) σg=1μm, requiring 11,459 elementary functions, and (d) σg=0.1μm, requiring 11,4591 elementary functions. In (a)–(c), the data points representing the partially coherent data coincide with the points representing the coherent data.

Fig. 10.
Fig. 10.

Intensity distribution in the image of a straight opaque edge illuminated by coherent, incoherent, and partially coherent light. (a) Corresponds to the measured data for the short edge of the 248 nm beam with the intensity width σI=0.003m, and the coherence width σg=0.0007m. (b) Corresponds to the measured data for the long edge of the 248 nm beam, with an intensity width σI=0.0085m, and the coherence width σg=0.0002m. As seen in Fig. 10, the data points representing coherent illumination coincide with the data points representing partially coherent illumination.

Fig. 11.
Fig. 11.

Special case of multiple beam interference to imitate the effect of a lenslet array. The aperture (slit) diameter is equal to the aperture separation. The beamlet corresponding to the central lenslet is unaffected. A phase shift, dependant on the distance from the axis, is applied to all other beamlets. Beamlets experiencing the same phase shift will interfere with each other, e.g., beamlet 2 and 2 will interfere with each other, as will 3 and 3.

Fig. 12.
Fig. 12.

Simulated intensity in the image plane of an imaging homogenizer illuminated by GSM sources of increasing intensity width. In each case, the intensity width is the same (σI=0.1mm) and the coherence width (σg) is varied. (a) σg=0.1mm, requiring 13 elementary functions, (b) σg=0.05mm, requiring 23 elementary functions, (c) σg=0.025mm, requiring 47 elementary functions, (d) σg=0.01mm, requiring 115 elementary functions, (e) σg=0.005mm, requiring 229 elementary functions, and (f) σg=0.003mm, requiring 287 elementary functions.

Fig. 13.
Fig. 13.

Simulated intensity in the image plane of an imaging homogenizer illuminated by an incoherent source with the same intensity width as the (a) short axis and (b) long axis 248 nm beam. The intensity width in (a) is 3.27 mm and in (b) the intensity width is 8.964 mm.

Fig. 14.
Fig. 14.

Simulated intensity in the image plane of an imaging homogenizer illuminated by a GSM source with the same intensity and coherence widths are the (a) short axis and (b) long axis of the 248 nm excimer source. In (a) the intensity and coherence widths are 3.27 and 0.67 mm, respectively, corresponding to a ratio σg/σI of approximately 0.2, which is a very coherent source. In (b), the intensity and coherence widths are 8.946 and 0.214 mm, respectively, to give a ratio σg/σI of approximately 0.025, which is very incoherent.

Fig. 15.
Fig. 15.

Simulated intensity in the image plane of an imaging homogenizer illuminated by the (a) short axis and (b) long axis of the 248 nm excimer source. The intensity and coherence widths in (a) are 3.27 and 0.67 mm, respectively, and in (b) the intensity and coherence widths are 8.946 and 0.214 mm, respectively.

Equations (23)

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W(r1,r2)=d2r1d2r2h*(r1,r1)h(r2,r2)W0(r1,r2),
W0(r1,r2)=d2ra(r)f(r1r)f(r2r),
f^(u)W^0(u,u),
a^(u)=I^0(u)f^2(u),
W0(r1,r2)=nmamnf(r1rmn)f(r2rmn).
W^0(u1,u2)=d2r1d2r2mnamnf(r1rmn)×f(r2rmn)exp(2πiu1·r1)exp(2πiu2·r2).
W^0(u1,u2)=f^(u1)f^(u2)mnamnexp(2πi(u1+u2)·rmn).
W^0(u,u)=f^2(u)mnamn,
f^(u)W^0(u,u).
W(r1,r2)=I(r1)I(r2)γ(r1r2).
W0(r1,r2)=I0(r1)I0(r2)γ(r1r2).
W^0(u,u)=d2uU^2(uu)γ^(u),
f^(u)W^0(u,u).
I0(r)=W0(r,r)=mnamnf2(rrmn),
φmn(r˜)=Df2(Δx(x˜m),Δy(y˜n)).
Δx=πσIσg(lnc)(σg2+4σI2),
mam=ΔxU^0(0)f^2(0).
Ur(r)=f2(r)mam.
I(2)(Q)=I(1)(Q)+I(2)(Q)+2I(1)(Q)I(2)(Q)|j12|cos[β12δ].
ΔS|=|s2s1|=λ¯2πδλ¯2Δλ,
γ12(τ)||j12|,
I(3)(Q)=3I(Q)+2I(Q)|j12|cos[β12δ]+2I(Q)|j23|cos[β23δ]+2I(Q)|j13|cos[β13δ].
I˜(3)(u)=3I(Q)[δ(u)+|j12|3(δ(uu1)+δ(u+u1))+|j23|3(δ(uu2)+δ(u+u2))+|j13|3(δ(uu3)+δ(u+u3))].

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