Abstract

We investigate the spatial coherence properties in the focal region of a converging, spatially partially coherent wave field. In particular, we find that, depending on the effective coherence length of the field in the aperture, the longitudinal and transverse coherence lengths in the focal region can be either larger or smaller than the corresponding width of the intensity distribution. Also, the correlation function is shown to exhibit phase singularities.

© 2004 Optical Society of America

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References

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  1. W. Wang, A. T. Friberg, E. Wolf, “Focusing of partially coherent light in systems of large Fresnel number,” J. Opt. Soc. Am. A 14, 491–496 (1997).
    [CrossRef]
  2. A. T. Friberg, T. D. Visser, W. Wang, E. Wolf, “Focal shifts of converging diffracted waves of any state of spatial coherence,” Opt. Commun. 196, 1–7 (2001).
    [CrossRef]
  3. B. Lü, B. Zhang, B. Cai, “Focusing of a Gaussian Schell-model beam through a circular lens,” J. Mod. Opt. 42, 289–298 (1995).
    [CrossRef]
  4. T. D. Visser, G. Gbur, E. Wolf, “Effect of the state of coherence on the three-dimensional spectral intensity distribution near focus,” Opt. Commun. 213, 13–19 (2002).
    [CrossRef]
  5. M. Born, E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, Cambridge, UK, 1999).
  6. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).
  7. M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1965), Sec. 9.6.16.
  8. H. F. Schouten, G. Gbur, T. D. Visser, E. Wolf, “Phase singularities of the coherence functions in Young’s interference pattern,” Opt. Lett. 28, 968–970 (2003).
    [CrossRef] [PubMed]
  9. G. Gbur, T. D. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222, 117–125 (2003).
    [CrossRef]
  10. G. Gbur, T. D. Visser, E. Wolf, “Hidden’ singularities in partially coherent wavefields,” J. Opt. A Pure Appl. Opt. 6, 5239–5242 (2004).
    [CrossRef]

2004 (1)

G. Gbur, T. D. Visser, E. Wolf, “Hidden’ singularities in partially coherent wavefields,” J. Opt. A Pure Appl. Opt. 6, 5239–5242 (2004).
[CrossRef]

2003 (2)

2002 (1)

T. D. Visser, G. Gbur, E. Wolf, “Effect of the state of coherence on the three-dimensional spectral intensity distribution near focus,” Opt. Commun. 213, 13–19 (2002).
[CrossRef]

2001 (1)

A. T. Friberg, T. D. Visser, W. Wang, E. Wolf, “Focal shifts of converging diffracted waves of any state of spatial coherence,” Opt. Commun. 196, 1–7 (2001).
[CrossRef]

1997 (1)

1995 (1)

B. Lü, B. Zhang, B. Cai, “Focusing of a Gaussian Schell-model beam through a circular lens,” J. Mod. Opt. 42, 289–298 (1995).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, Cambridge, UK, 1999).

Cai, B.

B. Lü, B. Zhang, B. Cai, “Focusing of a Gaussian Schell-model beam through a circular lens,” J. Mod. Opt. 42, 289–298 (1995).
[CrossRef]

Friberg, A. T.

A. T. Friberg, T. D. Visser, W. Wang, E. Wolf, “Focal shifts of converging diffracted waves of any state of spatial coherence,” Opt. Commun. 196, 1–7 (2001).
[CrossRef]

W. Wang, A. T. Friberg, E. Wolf, “Focusing of partially coherent light in systems of large Fresnel number,” J. Opt. Soc. Am. A 14, 491–496 (1997).
[CrossRef]

Gbur, G.

G. Gbur, T. D. Visser, E. Wolf, “Hidden’ singularities in partially coherent wavefields,” J. Opt. A Pure Appl. Opt. 6, 5239–5242 (2004).
[CrossRef]

H. F. Schouten, G. Gbur, T. D. Visser, E. Wolf, “Phase singularities of the coherence functions in Young’s interference pattern,” Opt. Lett. 28, 968–970 (2003).
[CrossRef] [PubMed]

G. Gbur, T. D. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222, 117–125 (2003).
[CrossRef]

T. D. Visser, G. Gbur, E. Wolf, “Effect of the state of coherence on the three-dimensional spectral intensity distribution near focus,” Opt. Commun. 213, 13–19 (2002).
[CrossRef]

Lü, B.

B. Lü, B. Zhang, B. Cai, “Focusing of a Gaussian Schell-model beam through a circular lens,” J. Mod. Opt. 42, 289–298 (1995).
[CrossRef]

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).

Schouten, H. F.

Visser, T. D.

G. Gbur, T. D. Visser, E. Wolf, “Hidden’ singularities in partially coherent wavefields,” J. Opt. A Pure Appl. Opt. 6, 5239–5242 (2004).
[CrossRef]

H. F. Schouten, G. Gbur, T. D. Visser, E. Wolf, “Phase singularities of the coherence functions in Young’s interference pattern,” Opt. Lett. 28, 968–970 (2003).
[CrossRef] [PubMed]

G. Gbur, T. D. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222, 117–125 (2003).
[CrossRef]

T. D. Visser, G. Gbur, E. Wolf, “Effect of the state of coherence on the three-dimensional spectral intensity distribution near focus,” Opt. Commun. 213, 13–19 (2002).
[CrossRef]

A. T. Friberg, T. D. Visser, W. Wang, E. Wolf, “Focal shifts of converging diffracted waves of any state of spatial coherence,” Opt. Commun. 196, 1–7 (2001).
[CrossRef]

Wang, W.

A. T. Friberg, T. D. Visser, W. Wang, E. Wolf, “Focal shifts of converging diffracted waves of any state of spatial coherence,” Opt. Commun. 196, 1–7 (2001).
[CrossRef]

W. Wang, A. T. Friberg, E. Wolf, “Focusing of partially coherent light in systems of large Fresnel number,” J. Opt. Soc. Am. A 14, 491–496 (1997).
[CrossRef]

Wolf, E.

G. Gbur, T. D. Visser, E. Wolf, “Hidden’ singularities in partially coherent wavefields,” J. Opt. A Pure Appl. Opt. 6, 5239–5242 (2004).
[CrossRef]

H. F. Schouten, G. Gbur, T. D. Visser, E. Wolf, “Phase singularities of the coherence functions in Young’s interference pattern,” Opt. Lett. 28, 968–970 (2003).
[CrossRef] [PubMed]

T. D. Visser, G. Gbur, E. Wolf, “Effect of the state of coherence on the three-dimensional spectral intensity distribution near focus,” Opt. Commun. 213, 13–19 (2002).
[CrossRef]

A. T. Friberg, T. D. Visser, W. Wang, E. Wolf, “Focal shifts of converging diffracted waves of any state of spatial coherence,” Opt. Commun. 196, 1–7 (2001).
[CrossRef]

W. Wang, A. T. Friberg, E. Wolf, “Focusing of partially coherent light in systems of large Fresnel number,” J. Opt. Soc. Am. A 14, 491–496 (1997).
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, Cambridge, UK, 1999).

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).

Zhang, B.

B. Lü, B. Zhang, B. Cai, “Focusing of a Gaussian Schell-model beam through a circular lens,” J. Mod. Opt. 42, 289–298 (1995).
[CrossRef]

J. Mod. Opt. (1)

B. Lü, B. Zhang, B. Cai, “Focusing of a Gaussian Schell-model beam through a circular lens,” J. Mod. Opt. 42, 289–298 (1995).
[CrossRef]

J. Opt. A Pure Appl. Opt. (1)

G. Gbur, T. D. Visser, E. Wolf, “Hidden’ singularities in partially coherent wavefields,” J. Opt. A Pure Appl. Opt. 6, 5239–5242 (2004).
[CrossRef]

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

Opt. Commun. (3)

A. T. Friberg, T. D. Visser, W. Wang, E. Wolf, “Focal shifts of converging diffracted waves of any state of spatial coherence,” Opt. Commun. 196, 1–7 (2001).
[CrossRef]

G. Gbur, T. D. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222, 117–125 (2003).
[CrossRef]

T. D. Visser, G. Gbur, E. Wolf, “Effect of the state of coherence on the three-dimensional spectral intensity distribution near focus,” Opt. Commun. 213, 13–19 (2002).
[CrossRef]

Opt. Lett. (1)

Other (3)

M. Born, E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, Cambridge, UK, 1999).

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).

M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1965), Sec. 9.6.16.

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Figures (7)

Fig. 1
Fig. 1

Illustration of the notation.

Fig. 2
Fig. 2

Real part (solid curve) and imaginary part (dashed curve) of the spectral degree of coherence μ(0, 0, 0; 0, 0, z). In this example a=1 cm, f=2 cm, σg=0.5 cm, and λ=0.6328 μm.

Fig. 3
Fig. 3

Modulus of the spectral degree of coherence, |μ(0, 0, z1; 0, 0, z2)|. In this example a=1 cm, f=2 cm, σg=0.4 cm, and λ=0.6328 μm.

Fig. 4
Fig. 4

Modulus of the spectral degree of coherence, |μ(0, 0, -z; 0, 0, z)| for several values of the scaled coherence length σg/a. In this example a=1 cm, f=2 cm, and λ=0.6328 μm.

Fig. 5
Fig. 5

Modulus of the spectral degree of coherence, |μ(0, 0, -z; 0, 0, z)| and the normalized axial spectral density S(0, 0, z)/S(0, 0, 0) for different values of σg/a. (a) σg/a=2, (b) σg/a=1, (c) σg/a=0.6, (d) σg/a=0.4, (e) σg/a=0.2, (f) σg/a=0.1. In all examples a=1 cm, f=2 cm, and λ=0.6328 μm.

Fig. 6
Fig. 6

Spectral degree of coherence, μ(0, 0, 0; x, 0, 0) for several values of the scaled coherence length σg/a. Notice that μ(0, 0, 0; x, 0, 0) is strictly real. In all examples a=1 cm, f=2 cm, and λ=0.6328 μm.

Fig. 7
Fig. 7

Spectral degree of coherence, μ(0, 0, 0; x, 0, 0) and the normalized spectral density S(x, 0, 0)/S(0, 0, 0) for different values of σg/a. (a) σg/a=5, (b) σg/a=0.8, (c) σg/a=0.6, (d) σg/a=0.4, (e) σg/a=0.2. In all examples a=1 cm, f=2 cm, and λ=0.6328 μm.

Equations (41)

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U(r, ω)=-iλSU(0)(r, ω) exp(iks)sd2r,
W(0)(r1, r2, ω)=U(0)*(r1, ω)U(0)(r2, ω).
W(r1, r2, ω)=U*(r1, ω)U(r2, ω)
W(r1, r2, ω)=1λ2SSW(0)(r, r, ω)× exp[ik(s2-s1)]s1s2d2rd2r,
s1=|r1-r|,
s2=|r2-r|.
W(0)(r, r)=W(0)(ρ, ρ)=exp[-(ρ-ρ)2/2σg2],
W(r1, r2)=1(λf)2SSexp[-(ρ-ρ)2/2σg2]×exp[ik(s2-s1)]d2rd2r.
s1f-qr1,
s2f-qr2,
W(r1, r2)=1(λf)2SSexp[-(ρ-ρ)2/2σg2]×exp[ik(qr1-qr2)]d2rd2r.
μ(r1, r2)=W(r1, r2)[S(r1)S(r2)]1/2,
S(ri)=W(ri, ri).
r1=(0, 0, z1),
r2=(0, 0, z2).
qr1-z1(1-ρ2/2f2),
qr2-z2(1-ρ2/2f2),
W(0, 0, z1; 0, 0, z2)=1λf202π0a02π0a×exp{-[ρ2+ρ2-2ρρcos(ϕ-ϕ)]/2σg2}×exp{ik[-z1(1-ρ2/2f2)+z2(1-ρ2/2f2)]}×ρρdϕdρdϕdρ,
02π02πexp[ρρcos(ϕ-ϕ)/σg2]dϕdϕ
=4π2I0ρρσg2,
W(0, 0, z1; 0, 0, z2)=2πλf20a0aexp[-(ρ2+ρ2)/2σg2]I0ρρσg2×exp{ik[-z1(1-ρ2/2f2)+z2(1-ρ2/2f2)]}×ρρdρdρ.
S(0, 0, z)=W(0, 0, z; 0, 0, z)
=2πλf20a0aexp[-(ρ2+ρ2)/2σg2]I0ρρσg2×exp{ik[z(ρ2-ρ2)/2f2]}×ρρdρdρ.
S(0, 0, z)=2πλf20a0aexp[-(ρ2+ρ2)/2σg2]I0ρρσg2×cos[kz(ρ2-ρ2)/2f2]ρρdρdρ.
W(0, 0, -z1; 0, 0, -z2)=W(0, 0, z1; 0, 0, z2)*,
S(0, 0, -z)=S(0, 0, z).
μ(0, 0, z1; 0, 0, z2)=μ(0, 0, -z1; 0, 0, -z2)*.
z01=±2λ(f/a)2,
z02=±4λ(f/a)2,
r1=(0, 0, 0),
r2=(x, 0, 0).
s1=f.
q=[ρcos ϕ/f, ρsin ϕ/f, -(1-ρ2/f2)1/2],
s2f-ρx cos ϕ/f.
W(0, 0, 0; x, 0, 0)
=1(λf)202π0a02π0aexp{-[ρ2+ρ2-2ρρcos(ϕ-ϕ)]/2σg2}×exp[-ik(ρx cos ϕ)/f]ρρdϕdρdϕdρ
=2π(λf)20a02π0aexp[-(ρ2+ρ2)/2σg2]I0ρρσg2×exp[-ik(ρx cos ϕ)/f]ρρdρdϕdρ,
=2πλf20a0aexp[-(ρ2+ρ2)/2σg2]I0ρρσg2×J0kρxfρρdρdρ,
W(x, 0, 0; x, 0, 0)
=1(λf)202π0a02π0aexp{-[ρ2+ρ2-2ρρ×cos(ϕ-ϕ)]/2σg2}×exp[ikx(ρcos ϕ-ρcos ϕ)/f]ρρdϕdρdϕdρ
=1(λf)202π0a02π0aexp{-[ρ2+ρ2-2ρρcos(ϕ-ϕ)]/2σg2}×cos[kx(ρcos ϕ-ρcos ϕ)/f]×ρρdϕdρdϕdρ,

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