Abstract

Spiral phase plates for the generation of Laguerre–Gaussian (LG) beam with non-null radial index were designed and fabricated by electron beam lithography on polymethylmethacrylate over glass substrates. The optical response of these phase optical elements was theoretically considered and experimentally measured, and the purity of the experimental beams was investigated in terms of LG modes contributions. The far-field intensity pattern was compared with theoretical models and numerical simulations, whereas interferometric analyses confirmed the expected phase features of the generated beams. The high quality of the output beams confirms the applicability of these phase plates for the generation of high-order LG beams.

© 2014 Optical Society of America

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    [CrossRef]
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2013 (1)

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, Science 340, 1545 (2013).
[CrossRef]

2010 (2)

2009 (3)

D. Wildanger, J. Buckers, V. Westphal, S. W. Hell, and L. Kastrup, Opt. Express 17, 16100 (2009).
[CrossRef]

G. Anzolin, F. Tamburini, A. Bianchini, and C. Barbieri, Phys. Rev. A 79, 033845 (2009).
[CrossRef]

M. Prasciolu, F. Tamburini, G. Anzolin, E. Mari, M. Melli, A. Carpentiero, C. Barbieri, and F. Romanato, Microelectron. Eng. 86, 1103 (2009).
[CrossRef]

2008 (2)

C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, J. Microsc. 230, 134 (2008).
[CrossRef]

N. Matsumoto, T. Ando, T. Inoue, Y. Ohtake, N. Fukuchi, and T. Hara, J. Opt. Soc. Am. A 25, 1642 (2008).
[CrossRef]

2006 (1)

2004 (1)

2001 (2)

A. T. O’Neil and M. J. Padgett, Opt. Commun. 193, 45 (2001).
[CrossRef]

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 313 (2001).
[CrossRef]

1998 (2)

Z. S. Sacks, D. Rozas, and G. A. Swartzlander, J. Opt. Soc. Am. B 15, 2226 (1998).
[CrossRef]

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, J. Mod. Opt. 45, 1231 (1998).
[CrossRef]

1997 (1)

E. B. Kley, Microelectron. Eng. 34, 261 (1997).
[CrossRef]

1996 (1)

G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, Opt. Commun. 127, 183 (1996).
[CrossRef]

1995 (1)

M. J. Padgett and L. Allen, Opt. Commun. 121, 36 (1995).
[CrossRef]

1993 (1)

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, Opt. Commun. 96, 123 (1993).
[CrossRef]

1992 (2)

N. R. Heckenberg, R. Mc Duff, C. P. Smith, H. Rubinsztein-Dunlop, and M. J. Wegener, Opt. Quantum Electron. 24, S951 (1992).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef]

1989 (1)

P. Coullet, L. Gil, and F. Rocca, Opt. Commun. 73, 403 (1989).
[CrossRef]

1974 (1)

J. F. Nye and M. V. Berry, Proc. R. Soc. A 336, 165 (1974).
[CrossRef]

’t Hooft, G. W.

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).

Allen, L.

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, J. Mod. Opt. 45, 1231 (1998).
[CrossRef]

G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, Opt. Commun. 127, 183 (1996).
[CrossRef]

M. J. Padgett and L. Allen, Opt. Commun. 121, 36 (1995).
[CrossRef]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, Opt. Commun. 96, 123 (1993).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef]

Ando, T.

Anzolin, G.

E. Mari, G. Anzolin, F. Tamburini, M. Prasciolu, G. Umbriaco, A. Bianchini, C. Barbieri, and F. Romanato, Opt. Express 18, 2339 (2010).
[CrossRef]

M. Prasciolu, F. Tamburini, G. Anzolin, E. Mari, M. Melli, A. Carpentiero, C. Barbieri, and F. Romanato, Microelectron. Eng. 86, 1103 (2009).
[CrossRef]

G. Anzolin, F. Tamburini, A. Bianchini, and C. Barbieri, Phys. Rev. A 79, 033845 (2009).
[CrossRef]

Arlt, J.

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, J. Mod. Opt. 45, 1231 (1998).
[CrossRef]

Barbieri, C.

E. Mari, G. Anzolin, F. Tamburini, M. Prasciolu, G. Umbriaco, A. Bianchini, C. Barbieri, and F. Romanato, Opt. Express 18, 2339 (2010).
[CrossRef]

M. Prasciolu, F. Tamburini, G. Anzolin, E. Mari, M. Melli, A. Carpentiero, C. Barbieri, and F. Romanato, Microelectron. Eng. 86, 1103 (2009).
[CrossRef]

G. Anzolin, F. Tamburini, A. Bianchini, and C. Barbieri, Phys. Rev. A 79, 033845 (2009).
[CrossRef]

Barsuglia, M.

M. Granata, C. Buy, R. Ward, and M. Barsuglia, Phys. Rev. Lett. 105, 231102 (2010).
[CrossRef]

Beijersbergen, M. W.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, Opt. Commun. 96, 123 (1993).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef]

Bernet, S.

C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, J. Microsc. 230, 134 (2008).
[CrossRef]

Berry, M. V.

J. F. Nye and M. V. Berry, Proc. R. Soc. A 336, 165 (1974).
[CrossRef]

Bianchini, A.

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1986).

Bozinovic, N.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, Science 340, 1545 (2013).
[CrossRef]

Buckers, J.

Buy, C.

M. Granata, C. Buy, R. Ward, and M. Barsuglia, Phys. Rev. Lett. 105, 231102 (2010).
[CrossRef]

Carpentiero, A.

M. Prasciolu, F. Tamburini, G. Anzolin, E. Mari, M. Melli, A. Carpentiero, C. Barbieri, and F. Romanato, Microelectron. Eng. 86, 1103 (2009).
[CrossRef]

Coullet, P.

P. Coullet, L. Gil, and F. Rocca, Opt. Commun. 73, 403 (1989).
[CrossRef]

Dholakia, K.

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, J. Mod. Opt. 45, 1231 (1998).
[CrossRef]

Eliel, E. R.

Fukuchi, N.

Furhapter, S.

C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, J. Microsc. 230, 134 (2008).
[CrossRef]

Gil, L.

P. Coullet, L. Gil, and F. Rocca, Opt. Commun. 73, 403 (1989).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

Granata, M.

M. Granata, C. Buy, R. Ward, and M. Barsuglia, Phys. Rev. Lett. 105, 231102 (2010).
[CrossRef]

Hara, T.

Heckenberg, N. R.

N. R. Heckenberg, R. Mc Duff, C. P. Smith, H. Rubinsztein-Dunlop, and M. J. Wegener, Opt. Quantum Electron. 24, S951 (1992).
[CrossRef]

Hell, S. W.

Huang, H.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, Science 340, 1545 (2013).
[CrossRef]

Inoue, T.

Jesacher, A.

C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, J. Microsc. 230, 134 (2008).
[CrossRef]

Kastrup, L.

Kley, E. B.

E. B. Kley, Microelectron. Eng. 34, 261 (1997).
[CrossRef]

Kloosterboer, J. G.

Kristensen, P.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, Science 340, 1545 (2013).
[CrossRef]

Mair, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 313 (2001).
[CrossRef]

Mari, E.

E. Mari, G. Anzolin, F. Tamburini, M. Prasciolu, G. Umbriaco, A. Bianchini, C. Barbieri, and F. Romanato, Opt. Express 18, 2339 (2010).
[CrossRef]

M. Prasciolu, F. Tamburini, G. Anzolin, E. Mari, M. Melli, A. Carpentiero, C. Barbieri, and F. Romanato, Microelectron. Eng. 86, 1103 (2009).
[CrossRef]

Matsumoto, N.

Maurer, C.

C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, J. Microsc. 230, 134 (2008).
[CrossRef]

Mc Duff, R.

N. R. Heckenberg, R. Mc Duff, C. P. Smith, H. Rubinsztein-Dunlop, and M. J. Wegener, Opt. Quantum Electron. 24, S951 (1992).
[CrossRef]

Melli, M.

M. Prasciolu, F. Tamburini, G. Anzolin, E. Mari, M. Melli, A. Carpentiero, C. Barbieri, and F. Romanato, Microelectron. Eng. 86, 1103 (2009).
[CrossRef]

Nye, J. F.

J. F. Nye and M. V. Berry, Proc. R. Soc. A 336, 165 (1974).
[CrossRef]

O’Neil, A. T.

A. T. O’Neil and M. J. Padgett, Opt. Commun. 193, 45 (2001).
[CrossRef]

Oemrawsingh, S. S. R.

Ohtake, Y.

Padgett, M. J.

A. T. O’Neil and M. J. Padgett, Opt. Commun. 193, 45 (2001).
[CrossRef]

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, J. Mod. Opt. 45, 1231 (1998).
[CrossRef]

G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, Opt. Commun. 127, 183 (1996).
[CrossRef]

M. J. Padgett and L. Allen, Opt. Commun. 121, 36 (1995).
[CrossRef]

Prasciolu, M.

E. Mari, G. Anzolin, F. Tamburini, M. Prasciolu, G. Umbriaco, A. Bianchini, C. Barbieri, and F. Romanato, Opt. Express 18, 2339 (2010).
[CrossRef]

M. Prasciolu, F. Tamburini, G. Anzolin, E. Mari, M. Melli, A. Carpentiero, C. Barbieri, and F. Romanato, Microelectron. Eng. 86, 1103 (2009).
[CrossRef]

Ramachandran, S.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, Science 340, 1545 (2013).
[CrossRef]

Ren, Y.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, Science 340, 1545 (2013).
[CrossRef]

Ritsch-Marte, M.

C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, J. Microsc. 230, 134 (2008).
[CrossRef]

Robertson, D. A.

G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, Opt. Commun. 127, 183 (1996).
[CrossRef]

Rocca, F.

P. Coullet, L. Gil, and F. Rocca, Opt. Commun. 73, 403 (1989).
[CrossRef]

Romanato, F.

E. Mari, G. Anzolin, F. Tamburini, M. Prasciolu, G. Umbriaco, A. Bianchini, C. Barbieri, and F. Romanato, Opt. Express 18, 2339 (2010).
[CrossRef]

M. Prasciolu, F. Tamburini, G. Anzolin, E. Mari, M. Melli, A. Carpentiero, C. Barbieri, and F. Romanato, Microelectron. Eng. 86, 1103 (2009).
[CrossRef]

Rozas, D.

Rubinsztein-Dunlop, H.

N. R. Heckenberg, R. Mc Duff, C. P. Smith, H. Rubinsztein-Dunlop, and M. J. Wegener, Opt. Quantum Electron. 24, S951 (1992).
[CrossRef]

Sacks, Z. S.

Smith, C. P.

N. R. Heckenberg, R. Mc Duff, C. P. Smith, H. Rubinsztein-Dunlop, and M. J. Wegener, Opt. Quantum Electron. 24, S951 (1992).
[CrossRef]

Smith, G. M.

G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, Opt. Commun. 127, 183 (1996).
[CrossRef]

Soifer, V. A.

V. A. Soifer, Methods for Computer Design of Diffractive Optical Elements (Wiley, 2002).

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).

Swartlander, G. A.

Swartzlander, G. A.

Tamburini, F.

E. Mari, G. Anzolin, F. Tamburini, M. Prasciolu, G. Umbriaco, A. Bianchini, C. Barbieri, and F. Romanato, Opt. Express 18, 2339 (2010).
[CrossRef]

M. Prasciolu, F. Tamburini, G. Anzolin, E. Mari, M. Melli, A. Carpentiero, C. Barbieri, and F. Romanato, Microelectron. Eng. 86, 1103 (2009).
[CrossRef]

G. Anzolin, F. Tamburini, A. Bianchini, and C. Barbieri, Phys. Rev. A 79, 033845 (2009).
[CrossRef]

Torner, L.

J. P. Torres and L. Torner, Twisted Photons (Wiley, 2011).

Torres, J. P.

J. P. Torres and L. Torner, Twisted Photons (Wiley, 2011).

Tur, M.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, Science 340, 1545 (2013).
[CrossRef]

Turnbull, G. A.

G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, Opt. Commun. 127, 183 (1996).
[CrossRef]

Umbriaco, G.

van der Veen, H. E. L. O.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, Opt. Commun. 96, 123 (1993).
[CrossRef]

van Houwelingen, J. A. W.

Vaziri, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 313 (2001).
[CrossRef]

Verstegen, E. J. K.

Ward, R.

M. Granata, C. Buy, R. Ward, and M. Barsuglia, Phys. Rev. Lett. 105, 231102 (2010).
[CrossRef]

Wegener, M. J.

N. R. Heckenberg, R. Mc Duff, C. P. Smith, H. Rubinsztein-Dunlop, and M. J. Wegener, Opt. Quantum Electron. 24, S951 (1992).
[CrossRef]

Weihs, G.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 313 (2001).
[CrossRef]

Westphal, V.

Wildanger, D.

Willner, A. E.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, Science 340, 1545 (2013).
[CrossRef]

Woerdman, J. P.

S. S. R. Oemrawsingh, J. A. W. van Houwelingen, E. R. Eliel, J. P. Woerdman, E. J. K. Verstegen, J. G. Kloosterboer, and G. W. ’t Hooft, Appl. Opt. 43, 688 (2004).
[CrossRef]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, Opt. Commun. 96, 123 (1993).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1986).

Yue, Y.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, Science 340, 1545 (2013).
[CrossRef]

Zeilinger, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 313 (2001).
[CrossRef]

Appl. Opt. (1)

J. Microsc. (1)

C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, J. Microsc. 230, 134 (2008).
[CrossRef]

J. Mod. Opt. (1)

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, J. Mod. Opt. 45, 1231 (1998).
[CrossRef]

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

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

Microelectron. Eng. (2)

E. B. Kley, Microelectron. Eng. 34, 261 (1997).
[CrossRef]

M. Prasciolu, F. Tamburini, G. Anzolin, E. Mari, M. Melli, A. Carpentiero, C. Barbieri, and F. Romanato, Microelectron. Eng. 86, 1103 (2009).
[CrossRef]

Nature (1)

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 313 (2001).
[CrossRef]

Opt. Commun. (5)

G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, Opt. Commun. 127, 183 (1996).
[CrossRef]

M. J. Padgett and L. Allen, Opt. Commun. 121, 36 (1995).
[CrossRef]

A. T. O’Neil and M. J. Padgett, Opt. Commun. 193, 45 (2001).
[CrossRef]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, Opt. Commun. 96, 123 (1993).
[CrossRef]

P. Coullet, L. Gil, and F. Rocca, Opt. Commun. 73, 403 (1989).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Opt. Quantum Electron. (1)

N. R. Heckenberg, R. Mc Duff, C. P. Smith, H. Rubinsztein-Dunlop, and M. J. Wegener, Opt. Quantum Electron. 24, S951 (1992).
[CrossRef]

Phys. Rev. A (2)

G. Anzolin, F. Tamburini, A. Bianchini, and C. Barbieri, Phys. Rev. A 79, 033845 (2009).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef]

Phys. Rev. Lett. (1)

M. Granata, C. Buy, R. Ward, and M. Barsuglia, Phys. Rev. Lett. 105, 231102 (2010).
[CrossRef]

Proc. R. Soc. A (1)

J. F. Nye and M. V. Berry, Proc. R. Soc. A 336, 165 (1974).
[CrossRef]

Science (1)

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, Science 340, 1545 (2013).
[CrossRef]

Other (5)

M. Born and E. Wolf, Principles of Optics (Pergamon, 1986).

V. A. Soifer, Methods for Computer Design of Diffractive Optical Elements (Wiley, 2002).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

J. P. Torres and L. Torner, Twisted Photons (Wiley, 2011).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).

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Figures (5)

Fig. 1.
Fig. 1.

SEM micrograph of PMMA SSP for the generation of LG51 beam.

Fig. 2.
Fig. 2.

Generation of LG31 beam. (a) Phase pattern of the corresponding SSP. (b) Simulation of the far-field. Brightness and colors refer, respectively, to intensity and phase of the field. (c) Comparison between a cross section of the experimental far-field intensity (blue dots), numerical computation of the far-field intensity (blue solid line) and theoretical profile of a LG31 beam (red solid line). (d) Experimental far-field intensity, normalized to the maximal value.

Fig. 3.
Fig. 3.

Efficiency γp of LGp generation for different SSPs encoding LG beams with radial number pSPP from 0 to 5, fixed azimuth number =1. Diagonal efficiency terms: γ01=0.93, γ11=0.91, γ21=0.88, γ31=0.86, γ41=0.84, γ51=0.81.

Fig. 4.
Fig. 4.

Experimental data of the first-ring radius RM, normalized to w0/2 (blue dots), and of the corresponding value 1/RM2 (red squares), as a function of the radial number p for fixed azimuth number =1. Comparison with the theoretical trends, given by interpolation of data points calculated on LG beams (solid lines). In particular, the fit with a linear trend 1/RM2=k·(p+1)+q gives k=1.18, q=2e2.

Fig. 5.
Fig. 5.

Experimental results. (a) Intensity of the pattern obtained from interference of a reference wave with a generated LG31 beam and (b) details of phase dislocations. (c) Detail of the interference pattern for an LG32 beam. Central dislocations confirm the presence of a phase singularity along the axis of the beam. Shifts of half-period between the interference fringes of consecutive rings confirm the π phase-shift because of the change of sign in the associated Laguerre polynomial.

Equations (8)

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ψp(r,φ,z)=Cplw(2rw)||er2w2Lp||(2r2w2)eiφeikz+ikr22Rei(2p+||+1)ζ(z),
2π=2πλ(nSPP1)h,
Up(r,φ)=eiφ·sign[Lp||(2r2w02)].
χ(ρ,θ)=FT[c0u(i)Upl]=c002πdφ0+u(i)Upleirρcos(θφ)rdr,
χ(ρ,θ)=FT[c0u(i)eiφ]*FT[sign[Lp||(2r2w02)]],
FT[c0u(i)eiφ]=c0iπ3/22w12eiθeη2/2η[I12(η22)I+12(η22)],
FT[sign[Lp||(2r2w02)]]=k=1p2rk(1)kJ1(rkρ)ρδ(ρ)2π,
γp=ψp|χexpχexp|χexp.

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