Abstract

We derive general coupled-mode equations describing the nonlinear interaction of electromagnetic modes in periodic media with loss and gain. Our approach is rigorously based on the Lorentz reciprocity theorem, and it can be applied to a broad range of metal–dielectric photonic structures, including plasmonic waveguides and metamaterials. We verify that our general results agree with the previous analysis of particular cases, and predict novel effects on self- and cross-phase modulation in multilayer nonlinear fishnet metamaterials.

© 2014 Optical Society of America

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  1. C. Denz, S. Flach, and Y. S. Kivshar, eds., Nonlinearities in Periodic Structures and Metamaterials, Vol. 150 (Springer-Verlag, 2009).
  2. I. V. Shadrivov, A. A. Zharov, and Y. S. Kivshar, J. Opt. Soc. Am. B 23, 529 (2006).
    [CrossRef]
  3. A. I. Maimistov and I. R. Gabitov, Eur. J. Phys. Special Topics 147, 265 (2007).
    [CrossRef]
  4. A. K. Popov, S. A. Myslivets, and V. M. Shalaev, Physica B 405, 2999 (2010).
    [CrossRef]
  5. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, 1983).
  6. D. Michaelis, U. Peschel, C. Wachter, and A. Brauer, Phys. Rev. E 68, 065601 (2003).
    [CrossRef]
  7. R. Iliew, C. Etrich, T. Pertsch, and F. Lederer, Phys. Rev. B 77, 115124 (2008).
    [CrossRef]
  8. N. C. Panoiu, J. F. McMillan, and C. W. Wong, IEEE J. Sel. Top. Quantum Electron. 16, 257 (2010).
    [CrossRef]
  9. A. V. Shahraam and T. M. Monro, Opt. Express 17, 2298 (2009).
    [CrossRef]
  10. A. Rose, S. Larouche, E. Poutrina, and D. R. Smith, Phys. Rev. A 86, 033816 (2012).
    [CrossRef]
  11. G. Lecamp, J. P. Hugonin, and P. Lalanne, Opt. Express 15, 11042 (2007).
    [CrossRef]
  12. E. Feigenbaum and M. Orenstein, Opt. Lett. 32, 674 (2007).
    [CrossRef]
  13. A. R. Davoyan, I. V. Shadrivov, and Y. S. Kivshar, Opt. Express 17, 21732 (2009).
    [CrossRef]
  14. A. Marini and D. V. Skryabin, Phys. Rev. A 81, 033850 (2010).
    [CrossRef]
  15. A. Marini, R. Hartley, A. V. Gorbach, and D. V. Skryabin, Phys. Rev. A 84, 063839 (2011).
    [CrossRef]
  16. Z. C. Ruan, G. Veronis, K. L. Vodopyanov, M. M. Fejer, and S. F. Fan, Opt. Express 17, 13502 (2009).
    [CrossRef]
  17. P. Y. Chen, R. C. McPhedran, C. M. de Sterke, C. G. Poulton, A. A. Asatryan, L. C. Botten, and M. J. Steel, Phys. Rev. A 82, 053825 (2010).
    [CrossRef]
  18. A. Andryieuski, S. Ha, A. A. Sukhorukov, Y. S. Kivshar, and A. V. Lavrinenko, Phys. Rev. B 86, 035127 (2012).
    [CrossRef]
  19. R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic, 2008), Sec. 4.2.
  20. J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, Nature 455, 376 (2008).
    [CrossRef]
  21. R. Todorov, D. Tsankov, J. Pirov, and K. Petkov, J. Phys. D 44, 305401 (2011).
    [CrossRef]
  22. P. B. Johnson and R. W. Christy, Phys. Rev. B 6, 4370 (1972).
    [CrossRef]
  23. G. K. M. Thutupalli and S. G. Tomlin, J. Phys. C 10, 467 (1977).
    [CrossRef]
  24. D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, Phys. Rev. B 65, 195104 (2002).
    [CrossRef]
  25. M. Soljacic and J. D. Joannopoulos, Nat. Mater. 3, 211 (2004).
    [CrossRef]
  26. H. Suchowski, K. O’Brien, Z. J. Wong, A. Salandrino, X. Yin, and X. Zhang, Science 342, 1223 (2013).

2013 (1)

H. Suchowski, K. O’Brien, Z. J. Wong, A. Salandrino, X. Yin, and X. Zhang, Science 342, 1223 (2013).

2012 (2)

A. Rose, S. Larouche, E. Poutrina, and D. R. Smith, Phys. Rev. A 86, 033816 (2012).
[CrossRef]

A. Andryieuski, S. Ha, A. A. Sukhorukov, Y. S. Kivshar, and A. V. Lavrinenko, Phys. Rev. B 86, 035127 (2012).
[CrossRef]

2011 (2)

R. Todorov, D. Tsankov, J. Pirov, and K. Petkov, J. Phys. D 44, 305401 (2011).
[CrossRef]

A. Marini, R. Hartley, A. V. Gorbach, and D. V. Skryabin, Phys. Rev. A 84, 063839 (2011).
[CrossRef]

2010 (4)

N. C. Panoiu, J. F. McMillan, and C. W. Wong, IEEE J. Sel. Top. Quantum Electron. 16, 257 (2010).
[CrossRef]

A. Marini and D. V. Skryabin, Phys. Rev. A 81, 033850 (2010).
[CrossRef]

P. Y. Chen, R. C. McPhedran, C. M. de Sterke, C. G. Poulton, A. A. Asatryan, L. C. Botten, and M. J. Steel, Phys. Rev. A 82, 053825 (2010).
[CrossRef]

A. K. Popov, S. A. Myslivets, and V. M. Shalaev, Physica B 405, 2999 (2010).
[CrossRef]

2009 (3)

2008 (2)

R. Iliew, C. Etrich, T. Pertsch, and F. Lederer, Phys. Rev. B 77, 115124 (2008).
[CrossRef]

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, Nature 455, 376 (2008).
[CrossRef]

2007 (3)

2006 (1)

2004 (1)

M. Soljacic and J. D. Joannopoulos, Nat. Mater. 3, 211 (2004).
[CrossRef]

2003 (1)

D. Michaelis, U. Peschel, C. Wachter, and A. Brauer, Phys. Rev. E 68, 065601 (2003).
[CrossRef]

2002 (1)

D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, Phys. Rev. B 65, 195104 (2002).
[CrossRef]

1977 (1)

G. K. M. Thutupalli and S. G. Tomlin, J. Phys. C 10, 467 (1977).
[CrossRef]

1972 (1)

P. B. Johnson and R. W. Christy, Phys. Rev. B 6, 4370 (1972).
[CrossRef]

Andryieuski, A.

A. Andryieuski, S. Ha, A. A. Sukhorukov, Y. S. Kivshar, and A. V. Lavrinenko, Phys. Rev. B 86, 035127 (2012).
[CrossRef]

Asatryan, A. A.

P. Y. Chen, R. C. McPhedran, C. M. de Sterke, C. G. Poulton, A. A. Asatryan, L. C. Botten, and M. J. Steel, Phys. Rev. A 82, 053825 (2010).
[CrossRef]

Bartal, G.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, Nature 455, 376 (2008).
[CrossRef]

Botten, L. C.

P. Y. Chen, R. C. McPhedran, C. M. de Sterke, C. G. Poulton, A. A. Asatryan, L. C. Botten, and M. J. Steel, Phys. Rev. A 82, 053825 (2010).
[CrossRef]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic, 2008), Sec. 4.2.

Brauer, A.

D. Michaelis, U. Peschel, C. Wachter, and A. Brauer, Phys. Rev. E 68, 065601 (2003).
[CrossRef]

Chen, P. Y.

P. Y. Chen, R. C. McPhedran, C. M. de Sterke, C. G. Poulton, A. A. Asatryan, L. C. Botten, and M. J. Steel, Phys. Rev. A 82, 053825 (2010).
[CrossRef]

Christy, R. W.

P. B. Johnson and R. W. Christy, Phys. Rev. B 6, 4370 (1972).
[CrossRef]

Davoyan, A. R.

de Sterke, C. M.

P. Y. Chen, R. C. McPhedran, C. M. de Sterke, C. G. Poulton, A. A. Asatryan, L. C. Botten, and M. J. Steel, Phys. Rev. A 82, 053825 (2010).
[CrossRef]

Etrich, C.

R. Iliew, C. Etrich, T. Pertsch, and F. Lederer, Phys. Rev. B 77, 115124 (2008).
[CrossRef]

Fan, S. F.

Feigenbaum, E.

Fejer, M. M.

Gabitov, I. R.

A. I. Maimistov and I. R. Gabitov, Eur. J. Phys. Special Topics 147, 265 (2007).
[CrossRef]

Genov, D. A.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, Nature 455, 376 (2008).
[CrossRef]

Gorbach, A. V.

A. Marini, R. Hartley, A. V. Gorbach, and D. V. Skryabin, Phys. Rev. A 84, 063839 (2011).
[CrossRef]

Ha, S.

A. Andryieuski, S. Ha, A. A. Sukhorukov, Y. S. Kivshar, and A. V. Lavrinenko, Phys. Rev. B 86, 035127 (2012).
[CrossRef]

Hartley, R.

A. Marini, R. Hartley, A. V. Gorbach, and D. V. Skryabin, Phys. Rev. A 84, 063839 (2011).
[CrossRef]

Hugonin, J. P.

Iliew, R.

R. Iliew, C. Etrich, T. Pertsch, and F. Lederer, Phys. Rev. B 77, 115124 (2008).
[CrossRef]

Joannopoulos, J. D.

M. Soljacic and J. D. Joannopoulos, Nat. Mater. 3, 211 (2004).
[CrossRef]

Johnson, P. B.

P. B. Johnson and R. W. Christy, Phys. Rev. B 6, 4370 (1972).
[CrossRef]

Kivshar, Y. S.

Lalanne, P.

Larouche, S.

A. Rose, S. Larouche, E. Poutrina, and D. R. Smith, Phys. Rev. A 86, 033816 (2012).
[CrossRef]

Lavrinenko, A. V.

A. Andryieuski, S. Ha, A. A. Sukhorukov, Y. S. Kivshar, and A. V. Lavrinenko, Phys. Rev. B 86, 035127 (2012).
[CrossRef]

Lecamp, G.

Lederer, F.

R. Iliew, C. Etrich, T. Pertsch, and F. Lederer, Phys. Rev. B 77, 115124 (2008).
[CrossRef]

Love, J. D.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, 1983).

Maimistov, A. I.

A. I. Maimistov and I. R. Gabitov, Eur. J. Phys. Special Topics 147, 265 (2007).
[CrossRef]

Marini, A.

A. Marini, R. Hartley, A. V. Gorbach, and D. V. Skryabin, Phys. Rev. A 84, 063839 (2011).
[CrossRef]

A. Marini and D. V. Skryabin, Phys. Rev. A 81, 033850 (2010).
[CrossRef]

Markos, P.

D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, Phys. Rev. B 65, 195104 (2002).
[CrossRef]

McMillan, J. F.

N. C. Panoiu, J. F. McMillan, and C. W. Wong, IEEE J. Sel. Top. Quantum Electron. 16, 257 (2010).
[CrossRef]

McPhedran, R. C.

P. Y. Chen, R. C. McPhedran, C. M. de Sterke, C. G. Poulton, A. A. Asatryan, L. C. Botten, and M. J. Steel, Phys. Rev. A 82, 053825 (2010).
[CrossRef]

Michaelis, D.

D. Michaelis, U. Peschel, C. Wachter, and A. Brauer, Phys. Rev. E 68, 065601 (2003).
[CrossRef]

Monro, T. M.

Myslivets, S. A.

A. K. Popov, S. A. Myslivets, and V. M. Shalaev, Physica B 405, 2999 (2010).
[CrossRef]

O’Brien, K.

H. Suchowski, K. O’Brien, Z. J. Wong, A. Salandrino, X. Yin, and X. Zhang, Science 342, 1223 (2013).

Orenstein, M.

Panoiu, N. C.

N. C. Panoiu, J. F. McMillan, and C. W. Wong, IEEE J. Sel. Top. Quantum Electron. 16, 257 (2010).
[CrossRef]

Pertsch, T.

R. Iliew, C. Etrich, T. Pertsch, and F. Lederer, Phys. Rev. B 77, 115124 (2008).
[CrossRef]

Peschel, U.

D. Michaelis, U. Peschel, C. Wachter, and A. Brauer, Phys. Rev. E 68, 065601 (2003).
[CrossRef]

Petkov, K.

R. Todorov, D. Tsankov, J. Pirov, and K. Petkov, J. Phys. D 44, 305401 (2011).
[CrossRef]

Pirov, J.

R. Todorov, D. Tsankov, J. Pirov, and K. Petkov, J. Phys. D 44, 305401 (2011).
[CrossRef]

Popov, A. K.

A. K. Popov, S. A. Myslivets, and V. M. Shalaev, Physica B 405, 2999 (2010).
[CrossRef]

Poulton, C. G.

P. Y. Chen, R. C. McPhedran, C. M. de Sterke, C. G. Poulton, A. A. Asatryan, L. C. Botten, and M. J. Steel, Phys. Rev. A 82, 053825 (2010).
[CrossRef]

Poutrina, E.

A. Rose, S. Larouche, E. Poutrina, and D. R. Smith, Phys. Rev. A 86, 033816 (2012).
[CrossRef]

Rose, A.

A. Rose, S. Larouche, E. Poutrina, and D. R. Smith, Phys. Rev. A 86, 033816 (2012).
[CrossRef]

Ruan, Z. C.

Salandrino, A.

H. Suchowski, K. O’Brien, Z. J. Wong, A. Salandrino, X. Yin, and X. Zhang, Science 342, 1223 (2013).

Schultz, S.

D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, Phys. Rev. B 65, 195104 (2002).
[CrossRef]

Shadrivov, I. V.

Shahraam, A. V.

Shalaev, V. M.

A. K. Popov, S. A. Myslivets, and V. M. Shalaev, Physica B 405, 2999 (2010).
[CrossRef]

Skryabin, D. V.

A. Marini, R. Hartley, A. V. Gorbach, and D. V. Skryabin, Phys. Rev. A 84, 063839 (2011).
[CrossRef]

A. Marini and D. V. Skryabin, Phys. Rev. A 81, 033850 (2010).
[CrossRef]

Smith, D. R.

A. Rose, S. Larouche, E. Poutrina, and D. R. Smith, Phys. Rev. A 86, 033816 (2012).
[CrossRef]

D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, Phys. Rev. B 65, 195104 (2002).
[CrossRef]

Snyder, A. W.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, 1983).

Soljacic, M.

M. Soljacic and J. D. Joannopoulos, Nat. Mater. 3, 211 (2004).
[CrossRef]

Soukoulis, C. M.

D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, Phys. Rev. B 65, 195104 (2002).
[CrossRef]

Steel, M. J.

P. Y. Chen, R. C. McPhedran, C. M. de Sterke, C. G. Poulton, A. A. Asatryan, L. C. Botten, and M. J. Steel, Phys. Rev. A 82, 053825 (2010).
[CrossRef]

Suchowski, H.

H. Suchowski, K. O’Brien, Z. J. Wong, A. Salandrino, X. Yin, and X. Zhang, Science 342, 1223 (2013).

Sukhorukov, A. A.

A. Andryieuski, S. Ha, A. A. Sukhorukov, Y. S. Kivshar, and A. V. Lavrinenko, Phys. Rev. B 86, 035127 (2012).
[CrossRef]

Thutupalli, G. K. M.

G. K. M. Thutupalli and S. G. Tomlin, J. Phys. C 10, 467 (1977).
[CrossRef]

Todorov, R.

R. Todorov, D. Tsankov, J. Pirov, and K. Petkov, J. Phys. D 44, 305401 (2011).
[CrossRef]

Tomlin, S. G.

G. K. M. Thutupalli and S. G. Tomlin, J. Phys. C 10, 467 (1977).
[CrossRef]

Tsankov, D.

R. Todorov, D. Tsankov, J. Pirov, and K. Petkov, J. Phys. D 44, 305401 (2011).
[CrossRef]

Ulin-Avila, E.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, Nature 455, 376 (2008).
[CrossRef]

Valentine, J.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, Nature 455, 376 (2008).
[CrossRef]

Veronis, G.

Vodopyanov, K. L.

Wachter, C.

D. Michaelis, U. Peschel, C. Wachter, and A. Brauer, Phys. Rev. E 68, 065601 (2003).
[CrossRef]

Wong, C. W.

N. C. Panoiu, J. F. McMillan, and C. W. Wong, IEEE J. Sel. Top. Quantum Electron. 16, 257 (2010).
[CrossRef]

Wong, Z. J.

H. Suchowski, K. O’Brien, Z. J. Wong, A. Salandrino, X. Yin, and X. Zhang, Science 342, 1223 (2013).

Yin, X.

H. Suchowski, K. O’Brien, Z. J. Wong, A. Salandrino, X. Yin, and X. Zhang, Science 342, 1223 (2013).

Zentgraf, T.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, Nature 455, 376 (2008).
[CrossRef]

Zhang, S.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, Nature 455, 376 (2008).
[CrossRef]

Zhang, X.

H. Suchowski, K. O’Brien, Z. J. Wong, A. Salandrino, X. Yin, and X. Zhang, Science 342, 1223 (2013).

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, Nature 455, 376 (2008).
[CrossRef]

Zharov, A. A.

Eur. J. Phys. Special Topics (1)

A. I. Maimistov and I. R. Gabitov, Eur. J. Phys. Special Topics 147, 265 (2007).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

N. C. Panoiu, J. F. McMillan, and C. W. Wong, IEEE J. Sel. Top. Quantum Electron. 16, 257 (2010).
[CrossRef]

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

J. Phys. C (1)

G. K. M. Thutupalli and S. G. Tomlin, J. Phys. C 10, 467 (1977).
[CrossRef]

J. Phys. D (1)

R. Todorov, D. Tsankov, J. Pirov, and K. Petkov, J. Phys. D 44, 305401 (2011).
[CrossRef]

Nat. Mater. (1)

M. Soljacic and J. D. Joannopoulos, Nat. Mater. 3, 211 (2004).
[CrossRef]

Nature (1)

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, Nature 455, 376 (2008).
[CrossRef]

Opt. Express (4)

Opt. Lett. (1)

Phys. Rev. A (4)

A. Marini and D. V. Skryabin, Phys. Rev. A 81, 033850 (2010).
[CrossRef]

A. Marini, R. Hartley, A. V. Gorbach, and D. V. Skryabin, Phys. Rev. A 84, 063839 (2011).
[CrossRef]

A. Rose, S. Larouche, E. Poutrina, and D. R. Smith, Phys. Rev. A 86, 033816 (2012).
[CrossRef]

P. Y. Chen, R. C. McPhedran, C. M. de Sterke, C. G. Poulton, A. A. Asatryan, L. C. Botten, and M. J. Steel, Phys. Rev. A 82, 053825 (2010).
[CrossRef]

Phys. Rev. B (4)

A. Andryieuski, S. Ha, A. A. Sukhorukov, Y. S. Kivshar, and A. V. Lavrinenko, Phys. Rev. B 86, 035127 (2012).
[CrossRef]

R. Iliew, C. Etrich, T. Pertsch, and F. Lederer, Phys. Rev. B 77, 115124 (2008).
[CrossRef]

D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, Phys. Rev. B 65, 195104 (2002).
[CrossRef]

P. B. Johnson and R. W. Christy, Phys. Rev. B 6, 4370 (1972).
[CrossRef]

Phys. Rev. E (1)

D. Michaelis, U. Peschel, C. Wachter, and A. Brauer, Phys. Rev. E 68, 065601 (2003).
[CrossRef]

Physica B (1)

A. K. Popov, S. A. Myslivets, and V. M. Shalaev, Physica B 405, 2999 (2010).
[CrossRef]

Science (1)

H. Suchowski, K. O’Brien, Z. J. Wong, A. Salandrino, X. Yin, and X. Zhang, Science 342, 1223 (2013).

Other (3)

C. Denz, S. Flach, and Y. S. Kivshar, eds., Nonlinearities in Periodic Structures and Metamaterials, Vol. 150 (Springer-Verlag, 2009).

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, 1983).

R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic, 2008), Sec. 4.2.

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Figures (2)

Fig. 1.
Fig. 1.

(a) Scheme of a multilayer fishnet metamaterial consisting of 45 nm thick silver layers separated by 30 nm thick silicon layers (d=75nm). The 110×190nm holes are filled with chalcogenide glass, and positioned with periodicity of 270 nm in both lateral directions (x,y). (b) Effective linear refractive index: real (black) and imaginary (red) parts. Lines are calculated using Bloch modes formalism [18], dots—by inverted-Fresnel formulas [24].

Fig. 2.
Fig. 2.

Absolute values of the electric fields in a (y,z) cross section and aligned with a hole center for linear Bloch modes at different optical wavelengths: (a), (b) 1550 nm; (c), (d) 1250 nm; and (e), (f) 1047 nm. Shown are the [(a), (c), (e)] y-components and [(b), (d), (f)] z-components. The fields are normalized to unity energy flow across one period in the (x,y) cross section.

Tables (1)

Tables Icon

Table 1. Calculated Parameters for Fishnet Metamaterial at Three Different Wavelengths λ: Effective Refractive Index neff, Propagation Constant β, and SPM Coefficient Normalized to Its Bulk Value

Equations (5)

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

SFq·dA=iωVEq·PnldV,
{E,H}(x,y,z)=mam(z){e,h}m(x,y,z).
damdziβmam=iωsmddxdyd/2d/2dξeiβmξem(x,y,z+ξ)·P˜nl(x,y,z,ξ),
da+/dziβa+=i(γSPM|a+|2+γXPM|a|2)a+,da/dz+iβa=i(γSPM|a|2+γXPM|a+|2)a,
γSPM=ωs+ddxdyd/2d/2dze(x,y,z)·[χA(e+·e+*)e++χB(e+·e+)e+*/2],γXPM=ωs+ddxdyd/2d/2dze(x,y,z)·[χA(e·e*)e++χA(e+·e*)e+χB(e+·e)e*].

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