Abstract

We present a study a new type of optical slow-light structure comprising a serpentine shaped waveguide were the loops are coupled. The dispersion relation, group velocity and GVD are studied analytically using a transfer matrix method and numerically using finite difference time domain simulations. The structure exhibits zero group velocity points at the ends of the Brillouin zone, but also within the zone. The position of mid-zone zero group velocity point can be tuned by modifying the coupling coefficient between adjacent loops. Closed-form analytic expressions for the dispersion relations, group velocity and the mid-zone zero vg points are found and presented.

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2009 (1)

2008 (3)

2007 (4)

2006 (2)

J. Scheuer and A. Yariv, “Sagnac effect in coupled-resonator slow-light waveguide structures,” Phys. Rev. Lett. 96(5), 053901 (2006).
[CrossRef] [PubMed]

X. Fengnian, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics 1, 65–71 (2006).

2005 (3)

B. Z. Steinberg, “Rotating photonic crystals: a medium for compact optical gyroscopes,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(5), 056621 (2005).
[CrossRef] [PubMed]

J. Scheuer, G. T. Paloczi, J. K. S. Poon, and A. Yariv, “Coupled resonator optical waveguides: towards slowing and storing of light,” Opt. Photon. News 16(2), 36–40 (2005).
[CrossRef]

M. Sumetsky, “Uniform coil optical resonator and waveguide: transmission spectrum, eigenmodes, and dispersion relation,” Opt. Express 13(11), 4331–4340 (2005).
[CrossRef] [PubMed]

2004 (3)

J. Heebner, P. Chak, S. Pereira, J. Sipe, and R. Boyd, “Distributed and localized feedback in microresonator sequences for linear and nonlinear optics,” J. Opt. Soc. Am. B 21(10), 1818–1832 (2004).
[CrossRef]

M. F. Yanik and S. H. Fan, “Stopping light all optically,” Phys. Rev. Lett. 92(8), 083901 (2004).
[CrossRef] [PubMed]

S. Mookherjea, “Semiconductor coupled-resonator optical waveguide laser,” Appl. Phys. Lett. 84(17), 3265–3267 (2004).
[CrossRef]

2003 (2)

T. A. Ibrahim, W. Cao, Y. Kim, J. Li, J. Goldhar, P.-T. Ho, and C. H. Lee, “All-optical switching in a laterally coupled microring resonator by carrier injection,” IEEE Photon. Technol. Lett. 15(1), 36–38 (2003).
[CrossRef]

A. Melloni, F. Morichetti, and M. Martinelli, “Linear and nonlinear pulse propagation in coupled resonator slow-wave optical structures,” Opt. Quantum Electron. 35(4/5), 365–379 (2003).
[CrossRef]

2002 (2)

J. E. Heebner and R. W. Boyd, “'Slow’ and 'fast' light in resonator-coupled waveguides,” J. Mod. Opt. 49(14), 2629–2636 (2002).
[CrossRef]

J. Heebner, R. Boyd, and Q. Park, “SCISSOR solitons and other novel propagation effects in microresonator-modified waveguides,” J. Opt. Soc. Am. B 19(4), 722–731 (2002).
[CrossRef]

2000 (2)

R. C. Polson, G. Levina, and Z. V. Vardeny, “Spectral analysis of polymer microring lasers,” Appl. Phys. Lett. 76(26), 3858–3860 (2000).
[CrossRef]

J. V. Hryniewicz, P. P. Absil, B. E. Little, R. A. Wilson, and P. T. Ho, “Higher order filter response in coupled microring resonators,” IEEE Photon. Technol. Lett. 12(3), 320–322 (2000).
[CrossRef]

1999 (3)

1998 (1)

S. Nojima, “Enhancement of optical gain in two dimensional photonic crystal with active lattice points,” Jpn. J. Appl. Phys. 37(Part 2, No. 5B), L565–L567 (1998).
[CrossRef]

1997 (1)

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[CrossRef]

1994 (1)

J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band edge laser: a new approach to gain enhancement,” J. Appl. Phys. 75(4), 1896–1899 (1994).
[CrossRef]

Absil, P. P.

J. V. Hryniewicz, P. P. Absil, B. E. Little, R. A. Wilson, and P. T. Ho, “Higher order filter response in coupled microring resonators,” IEEE Photon. Technol. Lett. 12(3), 320–322 (2000).
[CrossRef]

Behroozi, C. H.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397(6720), 594–598 (1999).
[CrossRef]

Bertino, M. F.

Bloemer, M. J.

J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band edge laser: a new approach to gain enhancement,” J. Appl. Phys. 75(4), 1896–1899 (1994).
[CrossRef]

Boag, A.

Botten, L. C.

Bowden, C. M.

J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band edge laser: a new approach to gain enhancement,” J. Appl. Phys. 75(4), 1896–1899 (1994).
[CrossRef]

Boyd, R.

Boyd, R. W.

J. E. Heebner and R. W. Boyd, “'Slow’ and 'fast' light in resonator-coupled waveguides,” J. Mod. Opt. 49(14), 2629–2636 (2002).
[CrossRef]

Cao, W.

T. A. Ibrahim, W. Cao, Y. Kim, J. Li, J. Goldhar, P.-T. Ho, and C. H. Lee, “All-optical switching in a laterally coupled microring resonator by carrier injection,” IEEE Photon. Technol. Lett. 15(1), 36–38 (2003).
[CrossRef]

Chak, P.

Chigrin, D. N.

Chu, S. T.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[CrossRef]

Dossou, K. B.

Dowling, J. P.

J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band edge laser: a new approach to gain enhancement,” J. Appl. Phys. 75(4), 1896–1899 (1994).
[CrossRef]

Dutton, Z.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397(6720), 594–598 (1999).
[CrossRef]

Fan, S. H.

M. F. Yanik and S. H. Fan, “Stopping light all optically,” Phys. Rev. Lett. 92(8), 083901 (2004).
[CrossRef] [PubMed]

Fengnian, X.

X. Fengnian, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics 1, 65–71 (2006).

Foresi, J.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[CrossRef]

Goldhar, J.

T. A. Ibrahim, W. Cao, Y. Kim, J. Li, J. Goldhar, P.-T. Ho, and C. H. Lee, “All-optical switching in a laterally coupled microring resonator by carrier injection,” IEEE Photon. Technol. Lett. 15(1), 36–38 (2003).
[CrossRef]

Ha, S.

Harris, S. E.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397(6720), 594–598 (1999).
[CrossRef]

Hau, L. V.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397(6720), 594–598 (1999).
[CrossRef]

Haus, H. A.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[CrossRef]

Heebner, J.

Heebner, J. E.

J. E. Heebner and R. W. Boyd, “'Slow’ and 'fast' light in resonator-coupled waveguides,” J. Mod. Opt. 49(14), 2629–2636 (2002).
[CrossRef]

Herrera, M. R.

Ho, P. T.

J. V. Hryniewicz, P. P. Absil, B. E. Little, R. A. Wilson, and P. T. Ho, “Higher order filter response in coupled microring resonators,” IEEE Photon. Technol. Lett. 12(3), 320–322 (2000).
[CrossRef]

Ho, P.-T.

T. A. Ibrahim, W. Cao, Y. Kim, J. Li, J. Goldhar, P.-T. Ho, and C. H. Lee, “All-optical switching in a laterally coupled microring resonator by carrier injection,” IEEE Photon. Technol. Lett. 15(1), 36–38 (2003).
[CrossRef]

Hryniewicz, J. V.

J. V. Hryniewicz, P. P. Absil, B. E. Little, R. A. Wilson, and P. T. Ho, “Higher order filter response in coupled microring resonators,” IEEE Photon. Technol. Lett. 12(3), 320–322 (2000).
[CrossRef]

Ibrahim, T. A.

T. A. Ibrahim, W. Cao, Y. Kim, J. Li, J. Goldhar, P.-T. Ho, and C. H. Lee, “All-optical switching in a laterally coupled microring resonator by carrier injection,” IEEE Photon. Technol. Lett. 15(1), 36–38 (2003).
[CrossRef]

Khurgin, J. B.

Kim, Y.

T. A. Ibrahim, W. Cao, Y. Kim, J. Li, J. Goldhar, P.-T. Ho, and C. H. Lee, “All-optical switching in a laterally coupled microring resonator by carrier injection,” IEEE Photon. Technol. Lett. 15(1), 36–38 (2003).
[CrossRef]

Kivshar, Y. S.

Laine, J. P.

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[CrossRef]

Lavrinenko, A. V.

Lee, C. H.

T. A. Ibrahim, W. Cao, Y. Kim, J. Li, J. Goldhar, P.-T. Ho, and C. H. Lee, “All-optical switching in a laterally coupled microring resonator by carrier injection,” IEEE Photon. Technol. Lett. 15(1), 36–38 (2003).
[CrossRef]

Lee, R. K.

Levina, G.

R. C. Polson, G. Levina, and Z. V. Vardeny, “Spectral analysis of polymer microring lasers,” Appl. Phys. Lett. 76(26), 3858–3860 (2000).
[CrossRef]

Li, J.

T. A. Ibrahim, W. Cao, Y. Kim, J. Li, J. Goldhar, P.-T. Ho, and C. H. Lee, “All-optical switching in a laterally coupled microring resonator by carrier injection,” IEEE Photon. Technol. Lett. 15(1), 36–38 (2003).
[CrossRef]

Little, B. E.

J. V. Hryniewicz, P. P. Absil, B. E. Little, R. A. Wilson, and P. T. Ho, “Higher order filter response in coupled microring resonators,” IEEE Photon. Technol. Lett. 12(3), 320–322 (2000).
[CrossRef]

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[CrossRef]

Martinelli, M.

A. Melloni, F. Morichetti, and M. Martinelli, “Linear and nonlinear pulse propagation in coupled resonator slow-wave optical structures,” Opt. Quantum Electron. 35(4/5), 365–379 (2003).
[CrossRef]

Melloni, A.

A. Melloni, F. Morichetti, and M. Martinelli, “Linear and nonlinear pulse propagation in coupled resonator slow-wave optical structures,” Opt. Quantum Electron. 35(4/5), 365–379 (2003).
[CrossRef]

Miller, D.

Mookherjea, S.

S. Mookherjea, “Semiconductor coupled-resonator optical waveguide laser,” Appl. Phys. Lett. 84(17), 3265–3267 (2004).
[CrossRef]

Morichetti, F.

A. Melloni, F. Morichetti, and M. Martinelli, “Linear and nonlinear pulse propagation in coupled resonator slow-wave optical structures,” Opt. Quantum Electron. 35(4/5), 365–379 (2003).
[CrossRef]

Nojima, S.

S. Nojima, “Enhancement of optical gain in two dimensional photonic crystal with active lattice points,” Jpn. J. Appl. Phys. 37(Part 2, No. 5B), L565–L567 (1998).
[CrossRef]

Paloczi, G. T.

J. Scheuer, G. T. Paloczi, J. K. S. Poon, and A. Yariv, “Coupled resonator optical waveguides: towards slowing and storing of light,” Opt. Photon. News 16(2), 36–40 (2005).
[CrossRef]

Park, Q.

Pelinovsky, D. E.

Pereira, S.

Polson, R. C.

R. C. Polson, G. Levina, and Z. V. Vardeny, “Spectral analysis of polymer microring lasers,” Appl. Phys. Lett. 76(26), 3858–3860 (2000).
[CrossRef]

Poon, J. K.

Poon, J. K. S.

J. Scheuer, G. T. Paloczi, J. K. S. Poon, and A. Yariv, “Coupled resonator optical waveguides: towards slowing and storing of light,” Opt. Photon. News 16(2), 36–40 (2005).
[CrossRef]

Sakoda, K.

Scalora, M.

J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band edge laser: a new approach to gain enhancement,” J. Appl. Phys. 75(4), 1896–1899 (1994).
[CrossRef]

Scherer, A.

Scheuer, J.

O. Weiss and J. Scheuer, “Side coupled adjacent resonators CROW--formation of mid-band zero group velocity,” Opt. Express 17(17), 14817–14824 (2009).
[CrossRef] [PubMed]

B. Z. Steinberg, J. Scheuer, and A. Boag, “Rotation-induced superstructure in slow-light waveguides with mode-degeneracy: optical gyroscopes with exponential sensitivity,” J. Opt. Soc. Am. B 24(5), 1216–1224 (2007).
[CrossRef]

J. Scheuer and A. Yariv, “Sagnac effect in coupled-resonator slow-light waveguide structures,” Phys. Rev. Lett. 96(5), 053901 (2006).
[CrossRef] [PubMed]

J. Scheuer, G. T. Paloczi, J. K. S. Poon, and A. Yariv, “Coupled resonator optical waveguides: towards slowing and storing of light,” Opt. Photon. News 16(2), 36–40 (2005).
[CrossRef]

Sekaric, L.

X. Fengnian, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics 1, 65–71 (2006).

Sipe, J.

Steinberg, B. Z.

Sukhorukov, A. A.

Sumetsky, M.

Vardeny, Z. V.

R. C. Polson, G. Levina, and Z. V. Vardeny, “Spectral analysis of polymer microring lasers,” Appl. Phys. Lett. 76(26), 3858–3860 (2000).
[CrossRef]

Vlasov, Y.

X. Fengnian, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics 1, 65–71 (2006).

Weiss, O.

Wilson, R. A.

J. V. Hryniewicz, P. P. Absil, B. E. Little, R. A. Wilson, and P. T. Ho, “Higher order filter response in coupled microring resonators,” IEEE Photon. Technol. Lett. 12(3), 320–322 (2000).
[CrossRef]

Xu, Y.

Yamilov, A. G.

Yanik, M. F.

M. F. Yanik and S. H. Fan, “Stopping light all optically,” Phys. Rev. Lett. 92(8), 083901 (2004).
[CrossRef] [PubMed]

Yariv, A.

P. Chak, J. K. Poon, and A. Yariv, “Optical bright and dark states in side-coupled resonator structures,” Opt. Lett. 32(13), 1785–1787 (2007).
[CrossRef] [PubMed]

J. Scheuer and A. Yariv, “Sagnac effect in coupled-resonator slow-light waveguide structures,” Phys. Rev. Lett. 96(5), 053901 (2006).
[CrossRef] [PubMed]

J. Scheuer, G. T. Paloczi, J. K. S. Poon, and A. Yariv, “Coupled resonator optical waveguides: towards slowing and storing of light,” Opt. Photon. News 16(2), 36–40 (2005).
[CrossRef]

A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24(11), 711–713 (1999).
[CrossRef]

Appl. Phys. Lett. (2)

R. C. Polson, G. Levina, and Z. V. Vardeny, “Spectral analysis of polymer microring lasers,” Appl. Phys. Lett. 76(26), 3858–3860 (2000).
[CrossRef]

S. Mookherjea, “Semiconductor coupled-resonator optical waveguide laser,” Appl. Phys. Lett. 84(17), 3265–3267 (2004).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

J. V. Hryniewicz, P. P. Absil, B. E. Little, R. A. Wilson, and P. T. Ho, “Higher order filter response in coupled microring resonators,” IEEE Photon. Technol. Lett. 12(3), 320–322 (2000).
[CrossRef]

T. A. Ibrahim, W. Cao, Y. Kim, J. Li, J. Goldhar, P.-T. Ho, and C. H. Lee, “All-optical switching in a laterally coupled microring resonator by carrier injection,” IEEE Photon. Technol. Lett. 15(1), 36–38 (2003).
[CrossRef]

J. Appl. Phys. (1)

J. P. Dowling, M. Scalora, M. J. Bloemer, and C. M. Bowden, “The photonic band edge laser: a new approach to gain enhancement,” J. Appl. Phys. 75(4), 1896–1899 (1994).
[CrossRef]

J. Lightwave Technol. (1)

B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997).
[CrossRef]

J. Mod. Opt. (1)

J. E. Heebner and R. W. Boyd, “'Slow’ and 'fast' light in resonator-coupled waveguides,” J. Mod. Opt. 49(14), 2629–2636 (2002).
[CrossRef]

J. Opt. Soc. Am. B (6)

Jpn. J. Appl. Phys. (1)

S. Nojima, “Enhancement of optical gain in two dimensional photonic crystal with active lattice points,” Jpn. J. Appl. Phys. 37(Part 2, No. 5B), L565–L567 (1998).
[CrossRef]

Nat. Photonics (1)

X. Fengnian, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics 1, 65–71 (2006).

Nature (1)

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397(6720), 594–598 (1999).
[CrossRef]

Opt. Express (4)

Opt. Lett. (3)

Opt. Photon. News (1)

J. Scheuer, G. T. Paloczi, J. K. S. Poon, and A. Yariv, “Coupled resonator optical waveguides: towards slowing and storing of light,” Opt. Photon. News 16(2), 36–40 (2005).
[CrossRef]

Opt. Quantum Electron. (1)

A. Melloni, F. Morichetti, and M. Martinelli, “Linear and nonlinear pulse propagation in coupled resonator slow-wave optical structures,” Opt. Quantum Electron. 35(4/5), 365–379 (2003).
[CrossRef]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

B. Z. Steinberg, “Rotating photonic crystals: a medium for compact optical gyroscopes,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(5), 056621 (2005).
[CrossRef] [PubMed]

Phys. Rev. Lett. (2)

J. Scheuer and A. Yariv, “Sagnac effect in coupled-resonator slow-light waveguide structures,” Phys. Rev. Lett. 96(5), 053901 (2006).
[CrossRef] [PubMed]

M. F. Yanik and S. H. Fan, “Stopping light all optically,” Phys. Rev. Lett. 92(8), 083901 (2004).
[CrossRef] [PubMed]

Supplementary Material (1)

» Media 1: MOV (13939 KB)     

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Fig. 1
Fig. 1

The SOW structure: Unit cell and parameter definitions.

Fig. 3
Fig. 3

Dispersion relation (a) and normalized group velocity (b) for the marked band. The structure parameters are defined in the text.

Fig. 2
Fig. 2

The mode profile of the waveguide.

Fig. 4
Fig. 4

The dependence of the dispersion curve on the coupling coefficient.

Fig. 5
Fig. 5

The dependence of the mid Brillouin zone zero group velocity point on the coupling coefficient.

Fig. 6
Fig. 6

The group velocity of SOW structure with κ = 0.1 vs. the wavenumber (a) and the frequency (b). Note the formation of the inflection point at K = 0.

Fig. 7
Fig. 7

Comparison between the numerically (blue stars) and the theoretically (solid green) dispersion relations. The structure parameters are defined in the text.

Fig. 8
Fig. 8

Transmission spectral response as a function of the coupling coefficient for a 5 unit-cells long SOW. The rest of the parameters are as in Fig. 4. Insets – transmission function for various coupling levels.

Fig. 9
Fig. 9

Transmission spectral response of a 5 unit-cell SOW with κ = 0.9. The rest of the parameters are as in Fig. 4.

Fig. 10
Fig. 10

Transmission spectral response as a function of the coupling coefficient for a 7 unit-cells long SOW with no loss (a) and −1dB/cm propagation loss (b).

Fig. 11
Fig. 11

Propagation of a 30 ps pulse along a SOW comprising 300 unit-cells. The pulse profile is shown at 100ps snapshots. The group velocity is approximately 0.35 unit-cells per ps corresponding to vg≈0.14 × c. media 1 – animation of the pulse propagation (Media 1).

Fig. 12
Fig. 12

Phase (a) and amplitude (b) responses of a 7 unit-cell long SOW. The average time delay is ~40ps yielding group velocity of approximately 0.2 unit-cells per ps.

Equations (7)

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E _ n + 1 = [ E 1 ... E 6 ] n + 1   T = M [ E 1 ... E 6 ] n   T
M = ( 0 i 1 κ κ i κ e i ( φ A + φ B ) 0 0 0 i 1 κ κ 0 0 i κ e i ( φ A + φ B ) 0 0 0 1 κ κ e i ( φ A + φ B ) 1 κ κ 0 i κ e 2 i φ A 0 1 κ κ e i ( φ A + φ B ) 0 0 1 κ κ 0 i κ e 2 i φ A 1 κ e i ( φ A + φ B ) 0 0 1 κ κ 0 i 1 κ κ e 2 i φ A 0 1 κ e i ( φ A + φ B ) 1 κ κ 0 i 1 κ κ e 2 i φ A 0 )
| M e i K Λ I | = 0
cos ( 4 φ A + 2 φ B ) = 4 κ 2 cos 3 ( K Λ ) 4 κ ( 1 κ ) cos 2 ( K Λ ) + ( 1 4 κ ) cos ( K Λ )
v g = Λ c n e f f sin ( K Λ ) n e f f ( 4 L A + 2 L B ) 12 κ 2 cos 3 ( K Λ ) 8 κ ( 1 κ ) cos ( K Λ ) + ( 1 4 κ ) sin [ k 0 n e f f ( 4 L A + 2 L B ) ]
cos ( K Λ ) | v g = 0 = ( 1 4 κ ) / 6 κ
κ 0.1

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