Abstract

This study investigates the transversely propagating waves in a body-centered tetragonal photonic crystal based on a holographic polymer-dispersed liquid crystal film. Rotating the film reveals three different transverse propagating waves. Degeneracy of optical Bloch waves from reciprocal lattice vectors explains their symmetrical distribution.

© 2011 OSA

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References

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  1. G. Y. Dong, X. L. Yang, and L. Z. Cai, “Anomalous refractive effects in honeycomb lattice photonic crystals formed by holographic lithography,” Opt. Express 18(16), 16302–16308 (2010).
    [CrossRef] [PubMed]
  2. D. Bernier, X. Le Roux, A. Lupu, D. Marris-Morini, L. Vivien, and E. Cassan, “Compact, low cross-talk CWDM demultiplexer using photonic crystal superprism,” Opt. Express 16(22), 17209–17214 (2008).
    [CrossRef] [PubMed]
  3. R. J. Liu, Z. Y. Li, F. Zhou, and D. Z. Zhang, “Waveguide coupler in three-dimensional photonic crystal,” Opt. Express 16(8), 5681–5688 (2008).
    [CrossRef] [PubMed]
  4. S. T. Wu, M. S. Li, and A. Y. G. Fuh, “Unusual refractions in photonic crystals based on polymer-dispersed liquid crystal films,” Appl. Phys. Lett. 91(25), 251117 (2007).
    [CrossRef]
  5. Y. C. Hsu and L. W. Chen, “Bloch surface wave excitation based on coupling from photonic crystal waveguide,” J. Opt. 12(9), 095709 (2010).
    [CrossRef]
  6. P. Zhang, C. Lou, S. Liu, J. Zhao, J. Xu, and Z. Chen, “Tuning of Bloch modes, diffraction, and refraction by two-dimensional lattice reconfiguration,” Opt. Lett. 35(6), 892–894 (2010).
    [CrossRef] [PubMed]
  7. E. Cassan, D. Bernier, G. Maire, D. M. Morini, and L. Vivien, “Bloch wave decomposition for prediction of strong light coupling efficiency into extended planar photonic crystals,” J. Opt. Soc. Am. B 24(5), 1211–1215 (2007).
    [CrossRef]
  8. B. Lombardet, L. A. Dunbar, R. Ferrini, and R. Houdre, “Bloch wave propagation in two-dimensional photonic crystals: Influence of the polarization,” Opt. Quantum Electron. 37(1–3), 293–307 (2005).
    [CrossRef]
  9. M. S. Li, S. T. Wu, and A. Y.-G. Fuh, “Superprism phenomenon based on holographic polymer dispersed liquid crystal films,” Appl. Phys. Lett. 88(9), 091109 (2006).
    [CrossRef]
  10. M.-S. Li, S.-Y. Huang, S.-T. Wu, H.-C. Lin, and A. Y.-G. Fuh, “Optical and electro-optical properties of photonic crystals based on polymer-dispersed liquid crystals,” Appl. Phys. B 101(1–2), 245–252 (2010).
    [CrossRef]
  11. W. Hergert and M. Däne, “Group theoretical investigations of photonic band structures,” Phys. Status Solidi 197(3), 620–634 (2003).
    [CrossRef]
  12. K. Sakoda, “Group-theoretical classification of eigenmodes in three-dimensional photonic lattices,” Phys. Rev. B 55(23), 15345–15348 (1997).
    [CrossRef]
  13. P. Zhang, N. K. Efremidis, A. Miller, Y. Hu, and Z. Chen, “Observation of coherent destruction of tunneling and unusual beam dynamics due to negative coupling in three-dimensional photonic lattices,” Opt. Lett. 35(19), 3252–3254 (2010).
    [CrossRef] [PubMed]
  14. S. Foteinopoulou and C. M. Soukoulis, “Electromagnetic wave propagation in two-dimensional photonic crystals: A study of anomalous refractive effects,” Phys. Rev. B 72(16), 165112 (2005).
    [CrossRef]
  15. Z. Peng, T. De-Xing, Z. Jian-Lin, S. Kun, Z. Jian-Bo, L. Bi-Li, and Y. Dong-Sheng, “Light-Induced Array of Three-Dimensional Waveguides in Lithium Niobate by Employing Two-Beam Interference Field,” Chin. Phys. Lett. 21(8), 1558–1561 (2004).
    [CrossRef]
  16. Y. Liu, S. Liu, and X. Zhang, “Fabrication of three-dimensional photonic crystals with two-beam holographic lithography,” Appl. Opt. 45(3), 480–483 (2006).
    [CrossRef] [PubMed]
  17. N. D. Lai, W. P. Liang, J. H. Lin, C. C. Hsu, and C. H. Lin, “Fabrication of two- and three-dimensional periodic structures by multi-exposure of two-beam interference technique,” Opt. Express 13(23), 9605–9611 (2005).
    [CrossRef] [PubMed]
  18. R. Gajić, R. Meisels, F. Kuchar, and K. Hingerl, “Refraction and rightness in photonic crystals,” Opt. Express 13(21), 8596–8605 (2005).
    [CrossRef] [PubMed]
  19. S. Mahmoodian, A. A. Sukhorukov, S. Ha, A. V. Lavrinenko, C. G. Poulton, K. B. Dossou, L. C. Botten, R. C. McPhedran, and C. M. de Sterke, “Paired modes of heterostructure cavities in photonic crystal waveguides with split band edges,” Opt. Express 18(25), 25693–25701 (2010).
    [CrossRef] [PubMed]
  20. O. Painter and K. Srinivasan, “Localized defect states in two-dimensional photonic crystal slab waveguides: A simple model based upon symmetry analysis,” Phys. Rev. B 68(3), 035110 (2003).
    [CrossRef]
  21. G. Sun and A. G. Kirk, “On the relationship between Bloch modes and phase-related refractive index of photonic crystals,” Opt. Express 15(20), 13149–13154 (2007).
    [CrossRef] [PubMed]

2010

2008

2007

2006

M. S. Li, S. T. Wu, and A. Y.-G. Fuh, “Superprism phenomenon based on holographic polymer dispersed liquid crystal films,” Appl. Phys. Lett. 88(9), 091109 (2006).
[CrossRef]

Y. Liu, S. Liu, and X. Zhang, “Fabrication of three-dimensional photonic crystals with two-beam holographic lithography,” Appl. Opt. 45(3), 480–483 (2006).
[CrossRef] [PubMed]

2005

N. D. Lai, W. P. Liang, J. H. Lin, C. C. Hsu, and C. H. Lin, “Fabrication of two- and three-dimensional periodic structures by multi-exposure of two-beam interference technique,” Opt. Express 13(23), 9605–9611 (2005).
[CrossRef] [PubMed]

R. Gajić, R. Meisels, F. Kuchar, and K. Hingerl, “Refraction and rightness in photonic crystals,” Opt. Express 13(21), 8596–8605 (2005).
[CrossRef] [PubMed]

S. Foteinopoulou and C. M. Soukoulis, “Electromagnetic wave propagation in two-dimensional photonic crystals: A study of anomalous refractive effects,” Phys. Rev. B 72(16), 165112 (2005).
[CrossRef]

B. Lombardet, L. A. Dunbar, R. Ferrini, and R. Houdre, “Bloch wave propagation in two-dimensional photonic crystals: Influence of the polarization,” Opt. Quantum Electron. 37(1–3), 293–307 (2005).
[CrossRef]

2004

Z. Peng, T. De-Xing, Z. Jian-Lin, S. Kun, Z. Jian-Bo, L. Bi-Li, and Y. Dong-Sheng, “Light-Induced Array of Three-Dimensional Waveguides in Lithium Niobate by Employing Two-Beam Interference Field,” Chin. Phys. Lett. 21(8), 1558–1561 (2004).
[CrossRef]

2003

W. Hergert and M. Däne, “Group theoretical investigations of photonic band structures,” Phys. Status Solidi 197(3), 620–634 (2003).
[CrossRef]

O. Painter and K. Srinivasan, “Localized defect states in two-dimensional photonic crystal slab waveguides: A simple model based upon symmetry analysis,” Phys. Rev. B 68(3), 035110 (2003).
[CrossRef]

1997

K. Sakoda, “Group-theoretical classification of eigenmodes in three-dimensional photonic lattices,” Phys. Rev. B 55(23), 15345–15348 (1997).
[CrossRef]

Bernier, D.

Bi-Li, L.

Z. Peng, T. De-Xing, Z. Jian-Lin, S. Kun, Z. Jian-Bo, L. Bi-Li, and Y. Dong-Sheng, “Light-Induced Array of Three-Dimensional Waveguides in Lithium Niobate by Employing Two-Beam Interference Field,” Chin. Phys. Lett. 21(8), 1558–1561 (2004).
[CrossRef]

Botten, L. C.

Cai, L. Z.

Cassan, E.

Chen, L. W.

Y. C. Hsu and L. W. Chen, “Bloch surface wave excitation based on coupling from photonic crystal waveguide,” J. Opt. 12(9), 095709 (2010).
[CrossRef]

Chen, Z.

Däne, M.

W. Hergert and M. Däne, “Group theoretical investigations of photonic band structures,” Phys. Status Solidi 197(3), 620–634 (2003).
[CrossRef]

de Sterke, C. M.

De-Xing, T.

Z. Peng, T. De-Xing, Z. Jian-Lin, S. Kun, Z. Jian-Bo, L. Bi-Li, and Y. Dong-Sheng, “Light-Induced Array of Three-Dimensional Waveguides in Lithium Niobate by Employing Two-Beam Interference Field,” Chin. Phys. Lett. 21(8), 1558–1561 (2004).
[CrossRef]

Dong, G. Y.

Dong-Sheng, Y.

Z. Peng, T. De-Xing, Z. Jian-Lin, S. Kun, Z. Jian-Bo, L. Bi-Li, and Y. Dong-Sheng, “Light-Induced Array of Three-Dimensional Waveguides in Lithium Niobate by Employing Two-Beam Interference Field,” Chin. Phys. Lett. 21(8), 1558–1561 (2004).
[CrossRef]

Dossou, K. B.

Dunbar, L. A.

B. Lombardet, L. A. Dunbar, R. Ferrini, and R. Houdre, “Bloch wave propagation in two-dimensional photonic crystals: Influence of the polarization,” Opt. Quantum Electron. 37(1–3), 293–307 (2005).
[CrossRef]

Efremidis, N. K.

Ferrini, R.

B. Lombardet, L. A. Dunbar, R. Ferrini, and R. Houdre, “Bloch wave propagation in two-dimensional photonic crystals: Influence of the polarization,” Opt. Quantum Electron. 37(1–3), 293–307 (2005).
[CrossRef]

Foteinopoulou, S.

S. Foteinopoulou and C. M. Soukoulis, “Electromagnetic wave propagation in two-dimensional photonic crystals: A study of anomalous refractive effects,” Phys. Rev. B 72(16), 165112 (2005).
[CrossRef]

Fuh, A. Y. G.

S. T. Wu, M. S. Li, and A. Y. G. Fuh, “Unusual refractions in photonic crystals based on polymer-dispersed liquid crystal films,” Appl. Phys. Lett. 91(25), 251117 (2007).
[CrossRef]

Fuh, A. Y.-G.

M.-S. Li, S.-Y. Huang, S.-T. Wu, H.-C. Lin, and A. Y.-G. Fuh, “Optical and electro-optical properties of photonic crystals based on polymer-dispersed liquid crystals,” Appl. Phys. B 101(1–2), 245–252 (2010).
[CrossRef]

M. S. Li, S. T. Wu, and A. Y.-G. Fuh, “Superprism phenomenon based on holographic polymer dispersed liquid crystal films,” Appl. Phys. Lett. 88(9), 091109 (2006).
[CrossRef]

Gajic, R.

Ha, S.

Hergert, W.

W. Hergert and M. Däne, “Group theoretical investigations of photonic band structures,” Phys. Status Solidi 197(3), 620–634 (2003).
[CrossRef]

Hingerl, K.

Houdre, R.

B. Lombardet, L. A. Dunbar, R. Ferrini, and R. Houdre, “Bloch wave propagation in two-dimensional photonic crystals: Influence of the polarization,” Opt. Quantum Electron. 37(1–3), 293–307 (2005).
[CrossRef]

Hsu, C. C.

Hsu, Y. C.

Y. C. Hsu and L. W. Chen, “Bloch surface wave excitation based on coupling from photonic crystal waveguide,” J. Opt. 12(9), 095709 (2010).
[CrossRef]

Hu, Y.

Huang, S.-Y.

M.-S. Li, S.-Y. Huang, S.-T. Wu, H.-C. Lin, and A. Y.-G. Fuh, “Optical and electro-optical properties of photonic crystals based on polymer-dispersed liquid crystals,” Appl. Phys. B 101(1–2), 245–252 (2010).
[CrossRef]

Jian-Bo, Z.

Z. Peng, T. De-Xing, Z. Jian-Lin, S. Kun, Z. Jian-Bo, L. Bi-Li, and Y. Dong-Sheng, “Light-Induced Array of Three-Dimensional Waveguides in Lithium Niobate by Employing Two-Beam Interference Field,” Chin. Phys. Lett. 21(8), 1558–1561 (2004).
[CrossRef]

Jian-Lin, Z.

Z. Peng, T. De-Xing, Z. Jian-Lin, S. Kun, Z. Jian-Bo, L. Bi-Li, and Y. Dong-Sheng, “Light-Induced Array of Three-Dimensional Waveguides in Lithium Niobate by Employing Two-Beam Interference Field,” Chin. Phys. Lett. 21(8), 1558–1561 (2004).
[CrossRef]

Kirk, A. G.

Kuchar, F.

Kun, S.

Z. Peng, T. De-Xing, Z. Jian-Lin, S. Kun, Z. Jian-Bo, L. Bi-Li, and Y. Dong-Sheng, “Light-Induced Array of Three-Dimensional Waveguides in Lithium Niobate by Employing Two-Beam Interference Field,” Chin. Phys. Lett. 21(8), 1558–1561 (2004).
[CrossRef]

Lai, N. D.

Lavrinenko, A. V.

Le Roux, X.

Li, M. S.

S. T. Wu, M. S. Li, and A. Y. G. Fuh, “Unusual refractions in photonic crystals based on polymer-dispersed liquid crystal films,” Appl. Phys. Lett. 91(25), 251117 (2007).
[CrossRef]

M. S. Li, S. T. Wu, and A. Y.-G. Fuh, “Superprism phenomenon based on holographic polymer dispersed liquid crystal films,” Appl. Phys. Lett. 88(9), 091109 (2006).
[CrossRef]

Li, M.-S.

M.-S. Li, S.-Y. Huang, S.-T. Wu, H.-C. Lin, and A. Y.-G. Fuh, “Optical and electro-optical properties of photonic crystals based on polymer-dispersed liquid crystals,” Appl. Phys. B 101(1–2), 245–252 (2010).
[CrossRef]

Li, Z. Y.

Liang, W. P.

Lin, C. H.

Lin, H.-C.

M.-S. Li, S.-Y. Huang, S.-T. Wu, H.-C. Lin, and A. Y.-G. Fuh, “Optical and electro-optical properties of photonic crystals based on polymer-dispersed liquid crystals,” Appl. Phys. B 101(1–2), 245–252 (2010).
[CrossRef]

Lin, J. H.

Liu, R. J.

Liu, S.

Liu, Y.

Lombardet, B.

B. Lombardet, L. A. Dunbar, R. Ferrini, and R. Houdre, “Bloch wave propagation in two-dimensional photonic crystals: Influence of the polarization,” Opt. Quantum Electron. 37(1–3), 293–307 (2005).
[CrossRef]

Lou, C.

Lupu, A.

Mahmoodian, S.

Maire, G.

Marris-Morini, D.

McPhedran, R. C.

Meisels, R.

Miller, A.

Morini, D. M.

Painter, O.

O. Painter and K. Srinivasan, “Localized defect states in two-dimensional photonic crystal slab waveguides: A simple model based upon symmetry analysis,” Phys. Rev. B 68(3), 035110 (2003).
[CrossRef]

Peng, Z.

Z. Peng, T. De-Xing, Z. Jian-Lin, S. Kun, Z. Jian-Bo, L. Bi-Li, and Y. Dong-Sheng, “Light-Induced Array of Three-Dimensional Waveguides in Lithium Niobate by Employing Two-Beam Interference Field,” Chin. Phys. Lett. 21(8), 1558–1561 (2004).
[CrossRef]

Poulton, C. G.

Sakoda, K.

K. Sakoda, “Group-theoretical classification of eigenmodes in three-dimensional photonic lattices,” Phys. Rev. B 55(23), 15345–15348 (1997).
[CrossRef]

Soukoulis, C. M.

S. Foteinopoulou and C. M. Soukoulis, “Electromagnetic wave propagation in two-dimensional photonic crystals: A study of anomalous refractive effects,” Phys. Rev. B 72(16), 165112 (2005).
[CrossRef]

Srinivasan, K.

O. Painter and K. Srinivasan, “Localized defect states in two-dimensional photonic crystal slab waveguides: A simple model based upon symmetry analysis,” Phys. Rev. B 68(3), 035110 (2003).
[CrossRef]

Sukhorukov, A. A.

Sun, G.

Vivien, L.

Wu, S. T.

S. T. Wu, M. S. Li, and A. Y. G. Fuh, “Unusual refractions in photonic crystals based on polymer-dispersed liquid crystal films,” Appl. Phys. Lett. 91(25), 251117 (2007).
[CrossRef]

M. S. Li, S. T. Wu, and A. Y.-G. Fuh, “Superprism phenomenon based on holographic polymer dispersed liquid crystal films,” Appl. Phys. Lett. 88(9), 091109 (2006).
[CrossRef]

Wu, S.-T.

M.-S. Li, S.-Y. Huang, S.-T. Wu, H.-C. Lin, and A. Y.-G. Fuh, “Optical and electro-optical properties of photonic crystals based on polymer-dispersed liquid crystals,” Appl. Phys. B 101(1–2), 245–252 (2010).
[CrossRef]

Xu, J.

Yang, X. L.

Zhang, D. Z.

Zhang, P.

Zhang, X.

Zhao, J.

Zhou, F.

Appl. Opt.

Appl. Phys. B

M.-S. Li, S.-Y. Huang, S.-T. Wu, H.-C. Lin, and A. Y.-G. Fuh, “Optical and electro-optical properties of photonic crystals based on polymer-dispersed liquid crystals,” Appl. Phys. B 101(1–2), 245–252 (2010).
[CrossRef]

Appl. Phys. Lett.

S. T. Wu, M. S. Li, and A. Y. G. Fuh, “Unusual refractions in photonic crystals based on polymer-dispersed liquid crystal films,” Appl. Phys. Lett. 91(25), 251117 (2007).
[CrossRef]

M. S. Li, S. T. Wu, and A. Y.-G. Fuh, “Superprism phenomenon based on holographic polymer dispersed liquid crystal films,” Appl. Phys. Lett. 88(9), 091109 (2006).
[CrossRef]

Chin. Phys. Lett.

Z. Peng, T. De-Xing, Z. Jian-Lin, S. Kun, Z. Jian-Bo, L. Bi-Li, and Y. Dong-Sheng, “Light-Induced Array of Three-Dimensional Waveguides in Lithium Niobate by Employing Two-Beam Interference Field,” Chin. Phys. Lett. 21(8), 1558–1561 (2004).
[CrossRef]

J. Opt.

Y. C. Hsu and L. W. Chen, “Bloch surface wave excitation based on coupling from photonic crystal waveguide,” J. Opt. 12(9), 095709 (2010).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Express

G. Y. Dong, X. L. Yang, and L. Z. Cai, “Anomalous refractive effects in honeycomb lattice photonic crystals formed by holographic lithography,” Opt. Express 18(16), 16302–16308 (2010).
[CrossRef] [PubMed]

D. Bernier, X. Le Roux, A. Lupu, D. Marris-Morini, L. Vivien, and E. Cassan, “Compact, low cross-talk CWDM demultiplexer using photonic crystal superprism,” Opt. Express 16(22), 17209–17214 (2008).
[CrossRef] [PubMed]

R. J. Liu, Z. Y. Li, F. Zhou, and D. Z. Zhang, “Waveguide coupler in three-dimensional photonic crystal,” Opt. Express 16(8), 5681–5688 (2008).
[CrossRef] [PubMed]

N. D. Lai, W. P. Liang, J. H. Lin, C. C. Hsu, and C. H. Lin, “Fabrication of two- and three-dimensional periodic structures by multi-exposure of two-beam interference technique,” Opt. Express 13(23), 9605–9611 (2005).
[CrossRef] [PubMed]

R. Gajić, R. Meisels, F. Kuchar, and K. Hingerl, “Refraction and rightness in photonic crystals,” Opt. Express 13(21), 8596–8605 (2005).
[CrossRef] [PubMed]

S. Mahmoodian, A. A. Sukhorukov, S. Ha, A. V. Lavrinenko, C. G. Poulton, K. B. Dossou, L. C. Botten, R. C. McPhedran, and C. M. de Sterke, “Paired modes of heterostructure cavities in photonic crystal waveguides with split band edges,” Opt. Express 18(25), 25693–25701 (2010).
[CrossRef] [PubMed]

G. Sun and A. G. Kirk, “On the relationship between Bloch modes and phase-related refractive index of photonic crystals,” Opt. Express 15(20), 13149–13154 (2007).
[CrossRef] [PubMed]

Opt. Lett.

Opt. Quantum Electron.

B. Lombardet, L. A. Dunbar, R. Ferrini, and R. Houdre, “Bloch wave propagation in two-dimensional photonic crystals: Influence of the polarization,” Opt. Quantum Electron. 37(1–3), 293–307 (2005).
[CrossRef]

Phys. Rev. B

S. Foteinopoulou and C. M. Soukoulis, “Electromagnetic wave propagation in two-dimensional photonic crystals: A study of anomalous refractive effects,” Phys. Rev. B 72(16), 165112 (2005).
[CrossRef]

K. Sakoda, “Group-theoretical classification of eigenmodes in three-dimensional photonic lattices,” Phys. Rev. B 55(23), 15345–15348 (1997).
[CrossRef]

O. Painter and K. Srinivasan, “Localized defect states in two-dimensional photonic crystal slab waveguides: A simple model based upon symmetry analysis,” Phys. Rev. B 68(3), 035110 (2003).
[CrossRef]

Phys. Status Solidi

W. Hergert and M. Däne, “Group theoretical investigations of photonic band structures,” Phys. Status Solidi 197(3), 620–634 (2003).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Experimental setup used to fabricate body-centered tetragonal (BCT) photonic crystals (PCs). The intersection angle (θ) between the two beams is ~39°. M and B.S. represent mirror and beam splitter, respectively.

Fig. 2
Fig. 2

(a) Simulated intensity profile in interference region, (b) illustration of unit cell, (c) top-view SEM image of polymer profile and (d) side-view SEM image of polymer profile.

Fig. 3
Fig. 3

Photos taken with sample rotated along the Y-axis with the normal film making angles of (a) 0°, (b) 20°, (c) 40°and (d) 50° with respect to the incident light. Notably, the CCD camera is fixed (i.e., not rotated with the sample).

Fig. 4
Fig. 4

Illustration of BCT PC alignment in FDTD simulation. (a) The XZ-, YZ- and XY-plane in the simulation are shown in yellow, pink and blue, respectively; (b) The cut-plane of the BCT structure with respect to the laboratory coordinates.

Fig. 5
Fig. 5

FDTD simulation results for light wave transfer on the X-Z and Y-Z planes of the film. The (a)-(c) and (d)-(f) are the results from the X-Z, Y-Z plane at Y=0 μm and X=0 μm, respectively; the intersect angles are 0°, 20°, and 50° from left to right.

Fig. 6
Fig. 6

(a) Projection of Gl,m on the X- Y- plane, (b) the intensity profile of transversely propagating beams for a normal incident light. The X-Y plane FDTD simulation of light wave at Z= −3 μm in the film with incident angles of (c) 20°, (d) 40°, and (e) 50°.

Equations (2)

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

I = n I n = n = 1 4 E n E 0 exp [ i ( k n k 0 ) r ] = n = 1 4 E n E 0 exp [ i ( G n , 0 ) r ] ,
u ( r ) = n , m = 0 u n , m ( r ) e j ( k + G n m ) r = n , m = 0 u n , m ( r ) e j ( k + ( G n , 0 G m , 0 ) ) r ,

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