Abstract

We investigate the spatial coherence properties of a twisted, partially coherent field in the focal region of diffractive axicons. We demonstrate that the focused field is a combination of an infinite number of weighted, mutually uncorrelated, helical components, whose weights depend on both the coherence width and the twist strength, and the total helicity of the field inverts its handedness depending on the twist handedness and vanishes at the nontwist limit. Depending on the variances of whichever the effective coherence width, the twist strength, the twist handedness of the illumination, or the shape of the axicon phase function, substantive changes will intervene on the distribution of the spatial coherence degree of the focused field. In particular, the twist strength of the illumination influences both the phase and amplitude of the spatial degree of coherence, while the twist handedness just inverts its phase. In addition, the spectral degree of coherence of the focused field possesses phase singularities, and their locations and shapes are affected by the coherence and twist properties of the illumination and the shape of the axicon phase function.

© 2012 Optical Society of America

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References

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  1. W. Wang, A. T. Friberg, and E. Wolf, “Focusing of partially coherent light in systems of large Fresnel numbers,” J. Opt. Soc. Am. A 14, 491–497 (1997).
    [CrossRef]
  2. A. T. Friberg, T. D. Visser, W. Wang, and E. Wolf, “Focal shifts of converging diffracted waves of any state of spatial coherence,” Opt. Commun. 196, 1–7 (2001).
    [CrossRef]
  3. H. F. Schouten, G. Gbur, T. D. Visser, and E. Wolf, “Phase singularities of the coherence functions in Young’s interference pattern,” Opt. Lett. 28, 968–970 (2003).
    [CrossRef]
  4. G. Gbur and T. D. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222, 117–125 (2003).
    [CrossRef]
  5. D. G. Fischer and T. D. Visser, “Spatial correlation properties of focused partially coherent light,” J. Opt. Soc. Am. A 21, 2097–2102 (2004).
    [CrossRef]
  6. R. Simon and N. Mukunda, “Twisted Gaussian Schellmodel beams,” J. Opt. Soc. Am. A 10, 95–109 (1993).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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  11. A. T. Friberg, “Stationary-phase analysis of generalized axicons,” J. Opt. Soc. Am. A 13, 743–750 (1996).
    [CrossRef]
  12. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).
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    [CrossRef]
  14. Abdu. A. Alkelly, M. A. Shukri, and Y. S. Alarify, “Influences of twist phenomenon of partially coherent field with uniform-intensity diffractive axicons,” J. Opt. Soc. Am. A 29, 417–425 (2012).
    [CrossRef]
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    [CrossRef]
  16. S. M. Barnett and L. Allen, “Orbital angular momentum and nonparaxial light-beams,” Opt. Commun. 110, 670–678(1994).
    [CrossRef]
  17. K. Volke-Sepulveda, V. Garces-Chavez, S. Chavez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B Quantum Semiclass. Opt. 4, S82–S89 (2002).
    [CrossRef]
  18. L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
    [CrossRef]
  19. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  20. J. Sochacki, Z. Jaroszewicz, L. R. Staroński, and A. Kołodziejczyk, “Annular-aperture logarithmic axicon,” J. Opt. Soc. Am. A 10, 1765–1768 (1993).
    [CrossRef]
  21. M. R. Perrone, A. Piegari, and S. Scaglione, “On the super-Gaussian unstable resonators for high-gain short-pulse laser media,” IEEE J. Quantum Electron. 29, 1423–1487 (1993).
    [CrossRef]
  22. S. Yu. Popov, A. T. Friberg, M. Honkanen, J. Lautanen, J. Turunen, and B. Schnabel, “Apodized annular-aperture diffractive axicons fabricated by continuous-path-control electron beam lithography,” Opt. Commun. 154, 359–367 (1998).
    [CrossRef]

2012

2006

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
[CrossRef]

H. Kim and B. Lee, “Analytic design of an anamorphic optical system for generating anisotropic partially coherent GSM beams,” Opt. Commun. 260, 383–397 (2006).
[CrossRef]

2004

2003

2002

K. Volke-Sepulveda, V. Garces-Chavez, S. Chavez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B Quantum Semiclass. Opt. 4, S82–S89 (2002).
[CrossRef]

2001

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

1998

F. Gori, M. Santarsiero, R. Borghi, and S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 45, 539–554 (1998).
[CrossRef]

S. Yu. Popov, A. T. Friberg, M. Honkanen, J. Lautanen, J. Turunen, and B. Schnabel, “Apodized annular-aperture diffractive axicons fabricated by continuous-path-control electron beam lithography,” Opt. Commun. 154, 359–367 (1998).
[CrossRef]

1997

1996

1994

1993

1992

1991

Alarify, Y. S.

Alkelly, Abdu. A.

Allen, L.

S. M. Barnett and L. Allen, “Orbital angular momentum and nonparaxial light-beams,” Opt. Commun. 110, 670–678(1994).
[CrossRef]

Arlt, J.

K. Volke-Sepulveda, V. Garces-Chavez, S. Chavez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B Quantum Semiclass. Opt. 4, S82–S89 (2002).
[CrossRef]

Barnett, S. M.

S. M. Barnett and L. Allen, “Orbital angular momentum and nonparaxial light-beams,” Opt. Commun. 110, 670–678(1994).
[CrossRef]

Borghi, R.

F. Gori, M. Santarsiero, R. Borghi, and S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 45, 539–554 (1998).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

Chavez-Cerda, S.

K. Volke-Sepulveda, V. Garces-Chavez, S. Chavez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B Quantum Semiclass. Opt. 4, S82–S89 (2002).
[CrossRef]

Davidson, N.

Dholakia, K.

K. Volke-Sepulveda, V. Garces-Chavez, S. Chavez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B Quantum Semiclass. Opt. 4, S82–S89 (2002).
[CrossRef]

Fischer, D. G.

Friberg, A. T.

Friesem, A. A.

Garces-Chavez, V.

K. Volke-Sepulveda, V. Garces-Chavez, S. Chavez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B Quantum Semiclass. Opt. 4, S82–S89 (2002).
[CrossRef]

Gbur, G.

Gori, F.

F. Gori, M. Santarsiero, R. Borghi, and S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 45, 539–554 (1998).
[CrossRef]

Hasman, E.

Honkanen, M.

S. Yu. Popov, A. T. Friberg, M. Honkanen, J. Lautanen, J. Turunen, and B. Schnabel, “Apodized annular-aperture diffractive axicons fabricated by continuous-path-control electron beam lithography,” Opt. Commun. 154, 359–367 (1998).
[CrossRef]

Jaroszewicz, Z.

Kim, H.

H. Kim and B. Lee, “Analytic design of an anamorphic optical system for generating anisotropic partially coherent GSM beams,” Opt. Commun. 260, 383–397 (2006).
[CrossRef]

Kolodziejczyk, A.

Lautanen, J.

S. Yu. Popov, A. T. Friberg, M. Honkanen, J. Lautanen, J. Turunen, and B. Schnabel, “Apodized annular-aperture diffractive axicons fabricated by continuous-path-control electron beam lithography,” Opt. Commun. 154, 359–367 (1998).
[CrossRef]

Lee, B.

H. Kim and B. Lee, “Analytic design of an anamorphic optical system for generating anisotropic partially coherent GSM beams,” Opt. Commun. 260, 383–397 (2006).
[CrossRef]

Mandel, L.

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

Manzo, C.

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
[CrossRef]

Marrucci, L.

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
[CrossRef]

Mukunda, N.

Paparo, D.

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
[CrossRef]

Perrone, M. R.

M. R. Perrone, A. Piegari, and S. Scaglione, “On the super-Gaussian unstable resonators for high-gain short-pulse laser media,” IEEE J. Quantum Electron. 29, 1423–1487 (1993).
[CrossRef]

Piegari, A.

M. R. Perrone, A. Piegari, and S. Scaglione, “On the super-Gaussian unstable resonators for high-gain short-pulse laser media,” IEEE J. Quantum Electron. 29, 1423–1487 (1993).
[CrossRef]

Popov, S. Yu.

S. Yu. Popov, A. T. Friberg, M. Honkanen, J. Lautanen, J. Turunen, and B. Schnabel, “Apodized annular-aperture diffractive axicons fabricated by continuous-path-control electron beam lithography,” Opt. Commun. 154, 359–367 (1998).
[CrossRef]

A. T. Friberg and S. Yu. Popov, “Radiometric description of intensity and coherence in generalized holographic axicon images,” Appl. Opt. 35, 3039–3046 (1996).
[CrossRef]

Santarsiero, M.

F. Gori, M. Santarsiero, R. Borghi, and S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 45, 539–554 (1998).
[CrossRef]

Scaglione, S.

M. R. Perrone, A. Piegari, and S. Scaglione, “On the super-Gaussian unstable resonators for high-gain short-pulse laser media,” IEEE J. Quantum Electron. 29, 1423–1487 (1993).
[CrossRef]

Schnabel, B.

S. Yu. Popov, A. T. Friberg, M. Honkanen, J. Lautanen, J. Turunen, and B. Schnabel, “Apodized annular-aperture diffractive axicons fabricated by continuous-path-control electron beam lithography,” Opt. Commun. 154, 359–367 (1998).
[CrossRef]

Schouten, H. F.

Shukri, M. A.

Simon, R.

Sochacki, J.

Staronski, L. R.

Tervonen, E.

Turunen, J.

S. Yu. Popov, A. T. Friberg, M. Honkanen, J. Lautanen, J. Turunen, and B. Schnabel, “Apodized annular-aperture diffractive axicons fabricated by continuous-path-control electron beam lithography,” Opt. Commun. 154, 359–367 (1998).
[CrossRef]

A. T. Friberg, E. Tervonen, and J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11, 1818–1826 (1994).
[CrossRef]

Vicalvi, S.

F. Gori, M. Santarsiero, R. Borghi, and S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 45, 539–554 (1998).
[CrossRef]

Visser, T. D.

D. G. Fischer and T. D. Visser, “Spatial correlation properties of focused partially coherent light,” J. Opt. Soc. Am. A 21, 2097–2102 (2004).
[CrossRef]

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

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

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

Volke-Sepulveda, K.

K. Volke-Sepulveda, V. Garces-Chavez, S. Chavez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B Quantum Semiclass. Opt. 4, S82–S89 (2002).
[CrossRef]

Wang, W.

A. T. Friberg, T. D. Visser, W. Wang, and 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, and E. Wolf, “Focusing of partially coherent light in systems of large Fresnel numbers,” J. Opt. Soc. Am. A 14, 491–497 (1997).
[CrossRef]

Wolf, E.

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

A. T. Friberg, T. D. Visser, W. Wang, and 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, and E. Wolf, “Focusing of partially coherent light in systems of large Fresnel numbers,” J. Opt. Soc. Am. A 14, 491–497 (1997).
[CrossRef]

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

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

Appl. Opt.

IEEE J. Quantum Electron.

M. R. Perrone, A. Piegari, and S. Scaglione, “On the super-Gaussian unstable resonators for high-gain short-pulse laser media,” IEEE J. Quantum Electron. 29, 1423–1487 (1993).
[CrossRef]

J. Mod. Opt.

F. Gori, M. Santarsiero, R. Borghi, and S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 45, 539–554 (1998).
[CrossRef]

J. Opt. B Quantum Semiclass. Opt.

K. Volke-Sepulveda, V. Garces-Chavez, S. Chavez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B Quantum Semiclass. Opt. 4, S82–S89 (2002).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

S. Yu. Popov, A. T. Friberg, M. Honkanen, J. Lautanen, J. Turunen, and B. Schnabel, “Apodized annular-aperture diffractive axicons fabricated by continuous-path-control electron beam lithography,” Opt. Commun. 154, 359–367 (1998).
[CrossRef]

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

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

H. Kim and B. Lee, “Analytic design of an anamorphic optical system for generating anisotropic partially coherent GSM beams,” Opt. Commun. 260, 383–397 (2006).
[CrossRef]

S. M. Barnett and L. Allen, “Orbital angular momentum and nonparaxial light-beams,” Opt. Commun. 110, 670–678(1994).
[CrossRef]

Opt. Lett.

Phys. Rev. Lett.

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
[CrossRef]

Other

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

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

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Figures (8)

Fig. 1.
Fig. 1.

Illustrating the notation relating to the propagation of a partially coherent beam.

Fig. 2.
Fig. 2.

On-axis distributions of the magnitude of the complex degree of spatial coherence |μ(0,0,z1;0,0,z2)| when one of the field points locates at the center of focal line (z1=150mm), and the other point varies throughout the image section for different values of σμ/R2 and twist parameter η.

Fig. 3.
Fig. 3.

Spectral degree of coherence μ(0,0,z;ρ2,0,z) (solid curves) and associated normalized spectral density S(ρ2,z)/S(0,z) (dashed curves) for various values of σμ/R2 and twist parameter η.

Fig. 4.
Fig. 4.

Spectral degree of coherence μ(0,ϕ,z;ρ2,ϕ,z) of partially coherent beam propagating through axicon at the different propagation distances for twist parameter η=0 with effective coherence width σμ/R2 of (a) 0.2; (b) 0.6.

Fig. 5.
Fig. 5.

Amplitude of the spectral degree of coherence |μ(ρ1,z;ρ2,z)| within the plane z=150mm [the first point is located at (0.05, 0.0, 150) mm in Cartesian coordinates (x1,y1,z), and the latter point varies throughout the plane z=150mm], for effective coherence width σμ/R2=0.2 and twist parameter η=0. The system parameters are the same as in Fig. 2

Fig. 6.
Fig. 6.

Same as in Fig. 5, but for first point located at r1=(x1,y1,z)=(0.05,0.05,150)mm and effective coherence width σμ/R2 of (a) 0.05; (b) 0.1; (c) 0.2; (d) 0.4.

Fig. 7.
Fig. 7.

Schematic illustration of surfaces on which the observation points r1 and r2 are located for which the spectral degree of coherence μ(r1;r2) is evaluated in this section.

Fig. 8.
Fig. 8.

Same as in Fig. 6, but for (a) σμ/R2=0.2, η=0.4; (b) σμ/R2=0.2, η=0.8; (c) σμ/R2=0.1, η=0.4; (d) σμ/R2=0.1, η=0.8.

Equations (28)

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

W(r1;r2;ν)=(k2π)2AW0(r1;r2;ν)t(r1)t(r2)exp{ik[φ(r1)φ(r2)]}exp[ik(s1s2)]s1s2d2r1d2r2,
W0(r1;r2;ν)=S0exp[(ρ12+ρ22)/4σI2]×exp[(ρ1ρ2)2/2σμ2]exp[ikuρ1.ρ2],
sj=|rjrj|zj+ρj22zj+ρj22zjρjρjzjcos(ϕiϕi).
W(r1;r2)=S0(k2π)2P*(ρ1,z1)P(ρ2,z2)z1z2aCT(ρ1,ρ2;r1;r2;σI,σμ,u)t(ρ1)t(ρ2)×exp{ik[ϕ(ρ1)ϕ(ρ2)]}exp[ik(ρ12/2z2ρ22/2z2)]ρ1ρ2dρ1dρ2,
CT(ρ1,ρ2;r1;r2;σI,σμ,u)=exp[(ρ12+ρ22)/δ2]02πexp[ρ1ρ2cos(ϕ1ϕ2)/σμ2]×exp[ikuρ1ρ2sin(ϕ1ϕ2)]×exp{ik[ρ1ρ1cos(ϕ1ϕ1)/z1ρ2ρ2cos(ϕ2ϕ2)/z2]}dϕ1dϕ2,
1δ2=14σI2+12σμ2.
CT(ρ1,ρ2;r1;r2;σI,σμ,η)=(2π)2exp(ρ12+ρ22δ2)m=(1+η1η)m2exp[im(ϕ1ϕ2)]×Im(ρ1ρ21η2/σμ2)Jm(kρ1ρ1/z1)Jm(kρ2ρ2/z2),
W(r1;r2)=S0k2P*(ρ1,z1)P(ρ2,z2)z1z2m=exp[im(ϕ1ϕ2)]Tm(η)Gm(ρ1,z1;ρ2,z2;σI,σμ,η),
Gm(ρ1,z1;ρ2,z2;σI,σμ,η)=at(ρ1)t(ρ2)exp{ik[ψ(ρ1,z1)ψ(ρ2,z2)]}×exp[(ρ12+ρ22)(14σI2+12σμ2)]Im(1η2σμ2ρ1ρ2)×Jm(kρ1z1ρ1)Jm(kρ2z2ρ2)ρ1ρ2dρ1dρ2,
Tm(η)=[(1+η)/(1η)]m2,
Gm(ρ1,z1;ρ2,z2;σI,σμ,η)=Gm(ρ1,z1;ρ2,z2;σI,σμ,η),
Gm(ρ1,z1;ρ2,z2;σI,σμ,η)=Gm(ρ1,z1;ρ2,z2;σI,σμ,η).
W(r2;r1)*=W(r1;r2).
S(r)=S0k2z2m=Tm(η)Gm(ρ,z;ρ,z;σI,σμ,η).
S(ρ,z)=S(ρ,z).
μ(r1;r2)=W(r1;r2)S(r1)S(r2).
μ(r2;r1)*=μ(r1;r2).
φ(ρ)=12a¯ln[1+a¯(ρ2R12)/d1],
t(ρ)=exp[(ρR¯ϖ)n],
W(0,0,z1;0,0,z2)=S0k2exp[ik(z1z2)]z1z2G0(0,z1;0,z2;σI,σμ,η).
W(0,ϕ,z;ρ2,ϕ,z)=S0k2exp[ikρ22/2z]z2G0(0,z;ρ2,z;σI,σμ,η).
Ω(ρ1,ρ2,Δϕ;σμ,η)=μ(ρ1,ϕ1,z;ρ2,ϕ1+Δϕ,z).
Ω(ρ1,ρ2,Δϕ;σμ,0)=Ω(ρ1,ρ2,Δϕ;σμ,0),
Ω(ρ,ρ,Δϕ;σμ,η)=Ω(ρ,ρ,Δϕ;σμ,η)*,
|Ω(ρ1,ρ2,Δϕ;σμ,η)|=|Ω(ρ1,ρ2,Δϕ;σμ,η)|.
Ω(ρ1,ρ2,Δϕ;σμ,η)=Ω(ρ1,ρ2,Δϕ;σμ,η).
Ω(ρ,ρ,Δϕ;σμ,η)=Ω(ρ,ρ,Δϕ;σμ,η)*,
|Ω(ρ1,ρ2,Δϕ;σμ,η)|=|Ω(ρ1,ρ2,Δϕ;σμ,η)|.

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