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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  1. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  2. H. H. Hopkins, “On the diffraction theory of optical images,” Proc. R. Soc. Lond. Ser. A 217, 408–432 (1953).
    [CrossRef]
  3. J. W. Goodman, Statistical Optics (Wiley, 2000), Chap. 7.
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    [CrossRef]
  5. 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]
  6. 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]
  7. J. Tervo, T. Setälä, and A. T. Friberg, “Theory of partially coherent electromagnetic fields in the space–frequency domain,” J. Opt. Soc. Am. A 21, 2205–2215 (2004).
    [CrossRef]
  8. A. Starikov and E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and of their radiation fields,” J. Opt. Soc. Am. 72, 923–928 (1982).
    [CrossRef]
  9. A. Starikov, “Effective number of degrees of freedom of partially coherent sources,” J. Opt. Soc. Am. 72, 1538–1544 (1982).
    [CrossRef]
  10. F. Gori and C. Palma, “Partially coherent sources which give rise to highly directional laser beams,” Opt. Commun. 27, 185–188 (1978).
    [CrossRef]
  11. F. Gori, “Directionality and partial coherence,” Opt. Acta 27, 1025–1034 (1980).
    [CrossRef]
  12. F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32, 3531–3533 (2007).
    [CrossRef]
  13. F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A 11, 085706 (2009).
    [CrossRef]
  14. R. Martínez-Herrero and P. M. Mejías, “Elementary-field expansions of genuine cross-spectral density matrices,” Opt. Lett. 34, 2303–2305 (2009).
    [CrossRef]
  15. J. Turunen, “Elementary-field representations in partially coherent optics,” J. Mod. Opt. 58, 509–527 (2011).
    [CrossRef]
  16. 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]
  17. H. Partanen, J. Tervo, and J. Turunen, “Spatial coherence of broad-area laser diodes,” Appl. Opt. 52, 3221–3228 (2013).
    [CrossRef]
  18. A. C. Schell, “The multiple plate antenna,” Ph.D. dissertation (Massachusetts Institute of Technology, 1961).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  23. 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]
  24. 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.

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)

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]

S. R. Seshadri, “Partially coherent Gaussian Schell-model electromagnetic beams,” J. Opt. Soc. Am. A 16, 1373–1380 (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, “Collett-Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (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]

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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Figures (6)

Fig. 1.
Fig. 1.

Illustration of the optical system considered in this paper and the propagation of two independent fully coherent fields (an axial and an off-axis field in the object plane) through the system.

Fig. 2.
Fig. 2.

Fourier-plane intensity profiles generated by some lowest-order Mercer modes of a Gaussian Schell-model field with σ=0.2w, and the superposition of all significant modal contributions, which gives rise to a nearly flat-top intensity profile.

Fig. 3.
Fig. 3.

Fourier-plane intensity profiles generated by some laterally displaced elementary field modes when σ=0.2w. The dashed red line is the profile obtained with 10 field modes and the black solid line is the converged result with 50 field modes.

Fig. 4.
Fig. 4.

Effect of varying degree of spatial coherence in the shape of the flat-top profile. The curves have been scaled such that I(0) is the same for all values of σ.

Fig. 5.
Fig. 5.

Effect of fabrication error of a diffractive beam shaping element in flat-top profiles for different degrees of coherence of illumination when (a) the profile is slightly too shallow and (b) when it is slightly too deep.

Fig. 6.
Fig. 6.

Transformation of a discrete array of mutually uncorrelated Gaussian sources into a flat-top illumination profile in the image plane of a 4F system using a beam shaping element in the Fourier plane; effect of the individual target half-width a of the beam shaping element in the image-plane intensity profile.

Equations (48)

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W(ρ1,ρ2)=t*(ρ1)t(ρ2)W(0)(ρ1,ρ2),
W(0)(κ1,κ2)=1(Fλ)2W(ρ1,ρ2)×exp[i2πFλ(κ1·ρ1κ2·ρ2)]d2ρ1d2ρ2.
W(κ1,κ2)=t*(κ1)t(κ2)W(0)(κ1,κ2)
W(ρ1,ρ2)=1(Fλ)2W(κ1,κ2)×exp[i2πFλ(ρ1·κ1ρ2·κ2)]d2κ1d2κ2,
W(0)(ρ1,ρ2)=m=0αmψm(0)*(ρ1)ψm(0)(ρ2),
ψm(ρ)=t(ρ)ψm(0)(ρ).
W(0)(ρ1,ρ2)=g(0)*(ρ1)g(0)(ρ2)q(Δρ),
S(0)(ρ)=W(0)(ρ,ρ)=|g(0)(ρ)|2
μ(0)(ρ1,ρ2)=W(0)(ρ1,ρ2)S(0)(ρ1)S(0)(ρ2)
μ(0)(ρ1,ρ2)=q(Δρ)exp[iΦ(ρ1,ρ2)]
Φ(ρ1,ρ2)=arg[g(0)(ρ2)]arg[g(0)(ρ1)],
g(ρ)=t(ρ)g(0)(ρ)
W(ρ1,ρ2)=g*(ρ1)g(ρ2)q(Δρ).
F(u)=F[f(v)]=1(2π)2f(v)exp(iu·v)d2v
f(v)=F1[F(u)]=F(u)exp(iv·u)d2u.
W(0)(κ1,κ2)=(2π)4(Fλ)2Q(χ)×G*(2πFλκ1χ)G(2πFλκ2χ)d2χ,
S(0)(κ)=(2π)4(Fλ)2Q(χ)|G(2πFλκχ)|2d2χ,
W(0)(ρ1,ρ2)=p(ρe)f(0)*(ρ1ρe)×f(0)(ρ2ρe)d2ρe,
W(0)(κ1,κ2)=(2π)6(Fλ)2P(2πFλΔκ)×F(0)*(2πFλκ1)F(0)(2πFλκ2),
S(0)(κ)=(2π)6(Fλ)2P(0)|F(0)(2πFλκ)|2.
R(2πFλκ,ρe)=1Fλt(ρ)f(0)(ρρe)×exp(2πFλκ·ρ)d2ρ,
W(0)(κ1,κ2)=p(ρe)×R*(2πFλκ1,ρe)R(2πFλκ2,ρe)d2ρe
S(0)(κ)=p(ρe)|R(2πFλκ,ρe)|2d2ρe.
W(ρ1,ρ2)=p(ρe)f*(ρ1ρe)f(ρ2ρe)d2ρe
S(ρ)=p(ρe)|f(ρρe)|2d2ρe.
f(ρ)=F(χ)exp(iρ·χ)d2χ
F(χ)=t(χ)F(0)(χ)
F(0)(χ)=1(2π)2f(ρ)exp(iχ·ρ)d2ρ
F[S(ρ)]=P(χ)K(χ),
K(χ)=|f(ρ)|2exp(iχ·ρ)d2ρ
S(ρ)=F1[P(χ)K(χ)]
W(0)(x1,x2)=g(0)(x1)g(0)(x2)q(Δx)
g(0)(x)=exp(x2w2),
q(Δx)=exp(Δx22σ2),
ψm(0)(x)=(2/π)1/42mm!wcHm(2xwc)exp(x2wc2)
wc=wβ,
β=[1+(w/σ)2]1/2.
αm=2πw1+1/β(1β1+β)m.
p(xe)=p0exp(2xe2wp2)
f(0)(x)=exp(x2we2)
wp=w1β2
we=wβ.
ϕ(x)=2πaFλ{w2π[exp(2x2w2)1]+xerf(2xw)},
N=12(1+1β),
NlnCln(1β)/(1+β)lnC2wσ,
N4/β4w/σ,
t(x)=q=Gqexp[iqϕ(x)].
Gq=sinc(αq)exp[iπ(αq)],

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