Abstract

We consider spatial shaping of partially coherent fields in two types of optical systems: a 2F Fourier-transforming system with the beam shaping element in the input plane and a 4F imaging system with the element in the intermediate Fourier plane. Different representations of the spatially partially coherent field in terms of fully coherent fields are examined to permit reduction of the dimensionality of the propagation integrals. The standard Mercer-type coherent-mode representation of the incident cross-spectral density (CSD) function is compared to expansions of CSD in either spatially or angularly shifted elementary field modes, all sharing the same spatial profile. In Fourier-transforming systems, the angular elementary-field representation proves computationally superior, while in imaging systems the spatially shifted elementary-field expansion is the best choice. Considering the Fourier-plane element as a generalized pupil, the latter leads to the concept’s generalized amplitude associated with the elementary field and to a generalized transfer function of the system. These concepts reduce to the standard point spread function and the optical transfer function in the limit of spatial incoherence at the object plane. Examples of the effects of partial coherence in spatial beam shaping are given.

© 2013 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2013 (1)

2011 (1)

J. Turunen, “Elementary-field representations in partially coherent optics,” J. Mod. Opt. 58, 509–527 (2011).
[CrossRef]

2009 (2)

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A 11, 085706 (2009).
[CrossRef]

R. Martínez-Herrero and P. M. Mejías, “Elementary-field expansions of genuine cross-spectral density matrices,” Opt. Lett. 34, 2303–2305 (2009).
[CrossRef]

2007 (1)

2004 (1)

2003 (1)

2000 (1)

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

1999 (2)

S. R. Seshadri, “Partially coherent Gaussian Schell-model electromagnetic beams,” J. Opt. Soc. Am. A 16, 1373–1380 (1999).
[CrossRef]

P. Laakkonen, J. Lautanen, V. Kettunen, J. Turunen, and M. Schirmer, “Analog diffractive elements in SiO2 by electron beam lithography and proportional etching with negative resist,” J. Mod. Opt. 46, 1295–1307 (1999).
[CrossRef]

1982 (3)

1980 (2)

F. Gori, “Collett-Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[CrossRef]

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

1978 (1)

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

1974 (1)

1967 (1)

A. C. Schell, “A technique for the determination of the radiation pattern of a partially coherent aperture,” IEEE Trans. Antennas Propag. 15, 187–188 (1967).
[CrossRef]

1961 (1)

1953 (1)

H. H. Hopkins, “On the diffraction theory of optical images,” Proc. R. Soc. Lond. Ser. A 217, 408–432 (1953).
[CrossRef]

Borghi, R.

Bryngdahl, O.

Friberg, A. T.

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, 2000), Chap. 7.

Gori, F.

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A 11, 085706 (2009).
[CrossRef]

F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32, 3531–3533 (2007).
[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]

F. Gori, “Collett-Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[CrossRef]

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

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

Guattari, G.

Hopkins, H. H.

H. H. Hopkins, “On the diffraction theory of optical images,” Proc. R. Soc. Lond. Ser. A 217, 408–432 (1953).
[CrossRef]

Kaivola, M.

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

Kajava, T.

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

Kettunen, V.

P. Laakkonen, J. Lautanen, V. Kettunen, J. Turunen, and M. Schirmer, “Analog diffractive elements in SiO2 by electron beam lithography and proportional etching with negative resist,” J. Mod. Opt. 46, 1295–1307 (1999).
[CrossRef]

Kuittinen, M.

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

Laakkonen, P.

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

P. Laakkonen, J. Lautanen, V. Kettunen, J. Turunen, and M. Schirmer, “Analog diffractive elements in SiO2 by electron beam lithography and proportional etching with negative resist,” J. Mod. Opt. 46, 1295–1307 (1999).
[CrossRef]

Lautanen, J.

P. Laakkonen, J. Lautanen, V. Kettunen, J. Turunen, and M. Schirmer, “Analog diffractive elements in SiO2 by electron beam lithography and proportional etching with negative resist,” J. Mod. Opt. 46, 1295–1307 (1999).
[CrossRef]

Lee, W.-H.

W.-H. Lee, “Computer-generated holograms: techniques and applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, 1978), Vol. 16, Chap. 3, pp. 119–223.

Mandel, L.

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

Martínez-Herrero, R.

Mejías, P. M.

Miyamoto, K.

Pääkkönen, P.

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

Partanen, H.

Piquero, G.

Ramírez-Sánchez, V.

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A 11, 085706 (2009).
[CrossRef]

Santarsiero, M.

Schell, A. C.

A. C. Schell, “A technique for the determination of the radiation pattern of a partially coherent aperture,” IEEE Trans. Antennas Propag. 15, 187–188 (1967).
[CrossRef]

A. C. Schell, “The multiple plate antenna,” Ph.D. dissertation (Massachusetts Institute of Technology, 1961).

Schirmer, M.

P. Laakkonen, J. Lautanen, V. Kettunen, J. Turunen, and M. Schirmer, “Analog diffractive elements in SiO2 by electron beam lithography and proportional etching with negative resist,” J. Mod. Opt. 46, 1295–1307 (1999).
[CrossRef]

Seshadri, S. R.

Setälä, T.

Shirai, T.

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A 11, 085706 (2009).
[CrossRef]

Simon, R.

Simonen, J.

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

Starikov, A.

Tervo, J.

Turunen, J.

H. Partanen, J. Tervo, and J. Turunen, “Spatial coherence of broad-area laser diodes,” Appl. Opt. 52, 3221–3228 (2013).
[CrossRef]

J. Turunen, “Elementary-field representations in partially coherent optics,” J. Mod. Opt. 58, 509–527 (2011).
[CrossRef]

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

P. Laakkonen, J. Lautanen, V. Kettunen, J. Turunen, and M. Schirmer, “Analog diffractive elements in SiO2 by electron beam lithography and proportional etching with negative resist,” J. Mod. Opt. 46, 1295–1307 (1999).
[CrossRef]

Wolf, E.

Appl. Opt. (1)

IEEE Trans. Antennas Propag. (1)

A. C. Schell, “A technique for the determination of the radiation pattern of a partially coherent aperture,” IEEE Trans. Antennas Propag. 15, 187–188 (1967).
[CrossRef]

J. Mod. Opt. (3)

P. Laakkonen, J. Lautanen, V. Kettunen, J. Turunen, and M. Schirmer, “Analog diffractive elements in SiO2 by electron beam lithography and proportional etching with negative resist,” J. Mod. Opt. 46, 1295–1307 (1999).
[CrossRef]

J. Turunen, “Elementary-field representations in partially coherent optics,” J. Mod. Opt. 58, 509–527 (2011).
[CrossRef]

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

J. Opt. A (1)

F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A 11, 085706 (2009).
[CrossRef]

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (3)

Opt. Acta (1)

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

Opt. Commun. (2)

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

F. Gori, “Collett-Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[CrossRef]

Opt. Lett. (2)

Proc. R. Soc. Lond. Ser. A (1)

H. H. Hopkins, “On the diffraction theory of optical images,” Proc. R. Soc. Lond. Ser. A 217, 408–432 (1953).
[CrossRef]

Other (4)

J. W. Goodman, Statistical Optics (Wiley, 2000), Chap. 7.

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

A. C. Schell, “The multiple plate antenna,” Ph.D. dissertation (Massachusetts Institute of Technology, 1961).

W.-H. Lee, “Computer-generated holograms: techniques and applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, 1978), Vol. 16, Chap. 3, pp. 119–223.

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