Abstract

We study propagation of light beams in two-dimensional photonic lattices created by periodically curved waveguide arrays. We demonstrate that by designing the waveguide bending, one can control not only the strength and sign of the beam diffraction, but also to engineer the effective geometry and even dimensionality of the two-dimensional photonic lattice. We reveal that diffraction of different spectral components of polychromatic light can display completely different patterns in the same periodically modulated structure, e.g. one-dimensional, hexagonal, or rectangular. Our results suggest novel opportunities for efficient self-collimation, focusing, and reshaping of light beams in two-dimensional photonic structures.

© 2007 Optical Society of America

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    [CrossRef]
  28. P. G. Kevrekidis, B. A. Malomed, and Y. B. Gaididei, "Solitons in triangular and honeycomb dynamical lattices with the cubic nonlinearity," Phys. Rev. E 66,016609-10 (2002).
    [CrossRef]
  29. A. Szameit, T. Pertsch, F. Dreisow, S. Nolte, A. Tunnermann, U. Peschel, and F. Lederer, "Light evolution in arbitrary two-dimensional waveguide arrays," Phys. Rev. A 75,053814-14 (2007).
    [CrossRef]
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    [CrossRef] [PubMed]
  31. I. L. Garanovich, A. A. Sukhorukov, and Yu. S. Kivshar, "Soliton control in modulated optically-induced photonic lattices," Opt. Express 13,5704-5710 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-15-5704.
    [CrossRef] [PubMed]
  32. C. R. Rosberg, I. L. Garanovich, A. A. Sukhorukov, D. N. Neshev,W. Krolikowski, and Yu. S. Kivshar, "Demonstration of all-optical beam steering in modulated photonic lattices," Opt. Lett. 31,1498-1500 (2006).
    [CrossRef] [PubMed]
  33. C. Zener, Proc. R. Soc. London A 145,523-529 (1934).

2007 (3)

2006 (9)

C. R. Rosberg, I. L. Garanovich, A. A. Sukhorukov, D. N. Neshev,W. Krolikowski, and Yu. S. Kivshar, "Demonstration of all-optical beam steering in modulated photonic lattices," Opt. Lett. 31,1498-1500 (2006).
[CrossRef] [PubMed]

P. T. Rakich, M. S. Dahlem, S. Tandon, M. Ibanescu, M. Soljacic, G. S. Petrich, J. D. Joannopoulos, L. A. Kolodziejski, and E. P. Ippen, "Achieving centimetre-scale supercollimation in a large-area two-dimensional photonic crystal," Nat. Mater. 5,93-96 (2006).
[CrossRef] [PubMed]

A. Szameit, D. Blomer, J. Burghoff, T. Pertsch, S. Nolte, and A. Tunnermann, "Hexagonal waveguide arrays written with fs-laser pulses," Appl. Phys. B 82,507-512 (2006).
[CrossRef]

I. L. Garanovich, A. A. Sukhorukov, and Yu. S. Kivshar, "Broadband diffraction management and selfcollimation of white light in photonic lattices," Phys. Rev. E 74,066609-4 (2006).
[CrossRef]

N. Chiodo, G. Della Valle, R. Osellame, S. Longhi, G. Cerullo, R. Ramponi, P. Laporta, and U. Morgner, "Imaging of Bloch oscillations in erbium-doped curved waveguide arrays," Opt. Lett. 31,1651-1653 (2006).
[CrossRef] [PubMed]

H. Trompeter, T. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Brauer, and U. Peschel, "Visual observation of Zener tunneling," Phys. Rev. Lett. 96,023901-4 (2006).
[CrossRef] [PubMed]

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Yu. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, "Bloch oscillations and Zener tunneling in two-dimensional photonic lattices," Phys. Rev. Lett. 96,053903-4 (2006).
[CrossRef] [PubMed]

A. Fratalocchi, G. Assanto, K. A. Brzdakiewicz, and M. A. Karpierz, "Optically induced Zener tunneling in one-dimensional lattices," Opt. Lett. 31,790-792 (2006).
[CrossRef] [PubMed]

S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, "Observation of dynamic localization in periodically curved waveguide arrays," Phys. Rev. Lett. 96,243901-4 (2006).
[CrossRef] [PubMed]

2005 (6)

2004 (1)

2003 (4)

S. Nolte, M. Will, J. Burghoff, and A. Tuennermann, "Femtosecond waveguide writing: a new avenue to three-dimensional integrated optics," Appl. Phys. A 77,109-111 (2003).
[CrossRef]

D. N. Chigrin, S. Enoch, C. M. S. Torres, and G. Tayeb, "Self-guiding in two-dimensional photonic crystals," Opt. Express 11,1203-1211 (2003), http://www.opticsinfobase.org/abstract.cfm?URI=oe-11-10-1203.
[CrossRef] [PubMed]

G. Lenz, R. Parker, M. C. Wanke, and C. M. de Sterke, "Dynamical localization and AC Bloch oscillations in periodic optical waveguide arrays," Opt. Commun. 218,87-92 (2003).
[CrossRef]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, "Discretizing light behaviour in linear and nonlinear waveguide lattices," Nature 424,817-823 (2003).
[CrossRef] [PubMed]

2002 (1)

P. G. Kevrekidis, B. A. Malomed, and Y. B. Gaididei, "Solitons in triangular and honeycomb dynamical lattices with the cubic nonlinearity," Phys. Rev. E 66,016609-10 (2002).
[CrossRef]

2000 (1)

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, "Diffraction management," Phys. Rev. Lett. 85,1863-1866 (2000).
[CrossRef] [PubMed]

1999 (4)

G. Lenz, I. Talanina, and C. M. de Sterke, "Bloch oscillations in an array of curved optical waveguides," Phys. Rev. Lett. 83,963-966 (1999).
[CrossRef]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Self-collimating phenomena in photonic crystals," Appl. Phys. Lett. 74,1212-1214 (1999).
[CrossRef]

T. Pertsch, P. Dannberg, W. Elflein, A. Brauer, and F. Lederer, "Optical Bloch oscillations in temperature tuned waveguide arrays," Phys. Rev. Lett. 83,4752-4755 (1999).
[CrossRef]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and K. Silberberg, "Experimental observation of linear and nonlinear optical Bloch oscillations," Phys. Rev. Lett. 83,4756-4759 (1999).
[CrossRef]

1973 (1)

S. Somekh, E. Garmire, A. Yariv, H. L. Garvin, and R. G. Hunsperger, "Channel optical waveguide directional couplers," Appl. Phys. Lett. 22,46-47 (1973).
[CrossRef]

1965 (1)

1934 (1)

C. Zener, Proc. R. Soc. London A 145,523-529 (1934).

Aitchison, J. S.

R. Iyer, J. S. Aitchison, J. Wan, M. M. Dignam, and C. M. de Sterke, "Exact dynamic localization in curved AlGaAs optical waveguide arrays," Opt. Express 15,3212-3223 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-6-3212.
[CrossRef] [PubMed]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, "Diffraction management," Phys. Rev. Lett. 85,1863-1866 (2000).
[CrossRef] [PubMed]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and K. Silberberg, "Experimental observation of linear and nonlinear optical Bloch oscillations," Phys. Rev. Lett. 83,4756-4759 (1999).
[CrossRef]

Assanto, G.

Bartelt, H.

Blomer, D.

Brauer, A.

H. Trompeter, T. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Brauer, and U. Peschel, "Visual observation of Zener tunneling," Phys. Rev. Lett. 96,023901-4 (2006).
[CrossRef] [PubMed]

T. Pertsch, P. Dannberg, W. Elflein, A. Brauer, and F. Lederer, "Optical Bloch oscillations in temperature tuned waveguide arrays," Phys. Rev. Lett. 83,4752-4755 (1999).
[CrossRef]

Brzdakiewicz, K. A.

Burghoff, J.

Cerullo, G.

Chigrin, D. N.

Chiodo, N.

Christodoulides, D. N.

Y. V. Kartashov, L. Torner, and D. N. Christodoulides, "Soliton dragging by dynamic optical lattices," Opt. Lett. 30,1378-1380 (2005).
[CrossRef] [PubMed]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, "Discretizing light behaviour in linear and nonlinear waveguide lattices," Nature 424,817-823 (2003).
[CrossRef] [PubMed]

Cianci, E.

S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, "Observation of dynamic localization in periodically curved waveguide arrays," Phys. Rev. Lett. 96,243901-4 (2006).
[CrossRef] [PubMed]

Dahlem, M. S.

P. T. Rakich, M. S. Dahlem, S. Tandon, M. Ibanescu, M. Soljacic, G. S. Petrich, J. D. Joannopoulos, L. A. Kolodziejski, and E. P. Ippen, "Achieving centimetre-scale supercollimation in a large-area two-dimensional photonic crystal," Nat. Mater. 5,93-96 (2006).
[CrossRef] [PubMed]

Dannberg, P.

T. Pertsch, P. Dannberg, W. Elflein, A. Brauer, and F. Lederer, "Optical Bloch oscillations in temperature tuned waveguide arrays," Phys. Rev. Lett. 83,4752-4755 (1999).
[CrossRef]

de Sterke, C. M.

R. Iyer, J. S. Aitchison, J. Wan, M. M. Dignam, and C. M. de Sterke, "Exact dynamic localization in curved AlGaAs optical waveguide arrays," Opt. Express 15,3212-3223 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-6-3212.
[CrossRef] [PubMed]

J. Wan, M. Laforest, C. M. de Sterke, and M. M. Dignam, "Optical filters based on dynamic localization in curved coupled optical waveguides," Opt. Commun. 247,353-365 (2005).
[CrossRef]

G. Lenz, R. Parker, M. C. Wanke, and C. M. de Sterke, "Dynamical localization and AC Bloch oscillations in periodic optical waveguide arrays," Opt. Commun. 218,87-92 (2003).
[CrossRef]

G. Lenz, I. Talanina, and C. M. de Sterke, "Bloch oscillations in an array of curved optical waveguides," Phys. Rev. Lett. 83,963-966 (1999).
[CrossRef]

Della Valle, G.

Desyatnikov, A. S.

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Yu. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, "Bloch oscillations and Zener tunneling in two-dimensional photonic lattices," Phys. Rev. Lett. 96,053903-4 (2006).
[CrossRef] [PubMed]

Dignam, M. M.

R. Iyer, J. S. Aitchison, J. Wan, M. M. Dignam, and C. M. de Sterke, "Exact dynamic localization in curved AlGaAs optical waveguide arrays," Opt. Express 15,3212-3223 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-6-3212.
[CrossRef] [PubMed]

J. Wan, M. Laforest, C. M. de Sterke, and M. M. Dignam, "Optical filters based on dynamic localization in curved coupled optical waveguides," Opt. Commun. 247,353-365 (2005).
[CrossRef]

Dreisow, F.

A. Szameit, T. Pertsch, F. Dreisow, S. Nolte, A. Tunnermann, U. Peschel, and F. Lederer, "Light evolution in arbitrary two-dimensional waveguide arrays," Phys. Rev. A 75,053814-14 (2007).
[CrossRef]

Eisenberg, H. S.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, "Diffraction management," Phys. Rev. Lett. 85,1863-1866 (2000).
[CrossRef] [PubMed]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and K. Silberberg, "Experimental observation of linear and nonlinear optical Bloch oscillations," Phys. Rev. Lett. 83,4756-4759 (1999).
[CrossRef]

Elflein, W.

T. Pertsch, P. Dannberg, W. Elflein, A. Brauer, and F. Lederer, "Optical Bloch oscillations in temperature tuned waveguide arrays," Phys. Rev. Lett. 83,4752-4755 (1999).
[CrossRef]

Enoch, S.

Foglietti, V.

S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, "Observation of dynamic localization in periodically curved waveguide arrays," Phys. Rev. Lett. 96,243901-4 (2006).
[CrossRef] [PubMed]

Fratalocchi, A.

Gaididei, Y. B.

P. G. Kevrekidis, B. A. Malomed, and Y. B. Gaididei, "Solitons in triangular and honeycomb dynamical lattices with the cubic nonlinearity," Phys. Rev. E 66,016609-10 (2002).
[CrossRef]

Garanovich, I. L.

Garmire, E.

S. Somekh, E. Garmire, A. Yariv, H. L. Garvin, and R. G. Hunsperger, "Channel optical waveguide directional couplers," Appl. Phys. Lett. 22,46-47 (1973).
[CrossRef]

Garvin, H. L.

S. Somekh, E. Garmire, A. Yariv, H. L. Garvin, and R. G. Hunsperger, "Channel optical waveguide directional couplers," Appl. Phys. Lett. 22,46-47 (1973).
[CrossRef]

Han, Y. G.

Hunsperger, R. G.

S. Somekh, E. Garmire, A. Yariv, H. L. Garvin, and R. G. Hunsperger, "Channel optical waveguide directional couplers," Appl. Phys. Lett. 22,46-47 (1973).
[CrossRef]

Ibanescu, M.

P. T. Rakich, M. S. Dahlem, S. Tandon, M. Ibanescu, M. Soljacic, G. S. Petrich, J. D. Joannopoulos, L. A. Kolodziejski, and E. P. Ippen, "Achieving centimetre-scale supercollimation in a large-area two-dimensional photonic crystal," Nat. Mater. 5,93-96 (2006).
[CrossRef] [PubMed]

Ippen, E. P.

P. T. Rakich, M. S. Dahlem, S. Tandon, M. Ibanescu, M. Soljacic, G. S. Petrich, J. D. Joannopoulos, L. A. Kolodziejski, and E. P. Ippen, "Achieving centimetre-scale supercollimation in a large-area two-dimensional photonic crystal," Nat. Mater. 5,93-96 (2006).
[CrossRef] [PubMed]

Iyer, R.

Joannopoulos, J. D.

P. T. Rakich, M. S. Dahlem, S. Tandon, M. Ibanescu, M. Soljacic, G. S. Petrich, J. D. Joannopoulos, L. A. Kolodziejski, and E. P. Ippen, "Achieving centimetre-scale supercollimation in a large-area two-dimensional photonic crystal," Nat. Mater. 5,93-96 (2006).
[CrossRef] [PubMed]

Jones, A. L.

Karpierz, M. A.

Kartashov, Y. V.

Kawakami, S.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Self-collimating phenomena in photonic crystals," Appl. Phys. Lett. 74,1212-1214 (1999).
[CrossRef]

Kawashima, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Self-collimating phenomena in photonic crystals," Appl. Phys. Lett. 74,1212-1214 (1999).
[CrossRef]

Kevrekidis, P. G.

P. G. Kevrekidis, B. A. Malomed, and Y. B. Gaididei, "Solitons in triangular and honeycomb dynamical lattices with the cubic nonlinearity," Phys. Rev. E 66,016609-10 (2002).
[CrossRef]

Kivshar, Yu. S.

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Yu. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, "Bloch oscillations and Zener tunneling in two-dimensional photonic lattices," Phys. Rev. Lett. 96,053903-4 (2006).
[CrossRef] [PubMed]

I. L. Garanovich, A. A. Sukhorukov, and Yu. S. Kivshar, "Broadband diffraction management and selfcollimation of white light in photonic lattices," Phys. Rev. E 74,066609-4 (2006).
[CrossRef]

C. R. Rosberg, I. L. Garanovich, A. A. Sukhorukov, D. N. Neshev,W. Krolikowski, and Yu. S. Kivshar, "Demonstration of all-optical beam steering in modulated photonic lattices," Opt. Lett. 31,1498-1500 (2006).
[CrossRef] [PubMed]

I. L. Garanovich, A. A. Sukhorukov, and Yu. S. Kivshar, "Soliton control in modulated optically-induced photonic lattices," Opt. Express 13,5704-5710 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-15-5704.
[CrossRef] [PubMed]

Kobelke, J.

Kolodziejski, L. A.

P. T. Rakich, M. S. Dahlem, S. Tandon, M. Ibanescu, M. Soljacic, G. S. Petrich, J. D. Joannopoulos, L. A. Kolodziejski, and E. P. Ippen, "Achieving centimetre-scale supercollimation in a large-area two-dimensional photonic crystal," Nat. Mater. 5,93-96 (2006).
[CrossRef] [PubMed]

Kosaka, H.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Self-collimating phenomena in photonic crystals," Appl. Phys. Lett. 74,1212-1214 (1999).
[CrossRef]

Krolikowski, W.

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Yu. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, "Bloch oscillations and Zener tunneling in two-dimensional photonic lattices," Phys. Rev. Lett. 96,053903-4 (2006).
[CrossRef] [PubMed]

C. R. Rosberg, I. L. Garanovich, A. A. Sukhorukov, D. N. Neshev,W. Krolikowski, and Yu. S. Kivshar, "Demonstration of all-optical beam steering in modulated photonic lattices," Opt. Lett. 31,1498-1500 (2006).
[CrossRef] [PubMed]

Laforest, M.

J. Wan, M. Laforest, C. M. de Sterke, and M. M. Dignam, "Optical filters based on dynamic localization in curved coupled optical waveguides," Opt. Commun. 247,353-365 (2005).
[CrossRef]

Laporta, P.

S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, "Observation of dynamic localization in periodically curved waveguide arrays," Phys. Rev. Lett. 96,243901-4 (2006).
[CrossRef] [PubMed]

N. Chiodo, G. Della Valle, R. Osellame, S. Longhi, G. Cerullo, R. Ramponi, P. Laporta, and U. Morgner, "Imaging of Bloch oscillations in erbium-doped curved waveguide arrays," Opt. Lett. 31,1651-1653 (2006).
[CrossRef] [PubMed]

Lederer, F.

A. Szameit, T. Pertsch, F. Dreisow, S. Nolte, A. Tunnermann, U. Peschel, and F. Lederer, "Light evolution in arbitrary two-dimensional waveguide arrays," Phys. Rev. A 75,053814-14 (2007).
[CrossRef]

H. Trompeter, T. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Brauer, and U. Peschel, "Visual observation of Zener tunneling," Phys. Rev. Lett. 96,023901-4 (2006).
[CrossRef] [PubMed]

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Yu. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, "Bloch oscillations and Zener tunneling in two-dimensional photonic lattices," Phys. Rev. Lett. 96,053903-4 (2006).
[CrossRef] [PubMed]

A. Szameit, D. Blomer, J. Burghoff, T. Schreiber, T. Pertsch, S. Nolte, A. Tunnermann, and F. Lederer, "Discrete nonlinear localization in femtosecond laser written waveguides in fused silica," Opt. Express 13,10552-10557 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-26-10552.
[CrossRef] [PubMed]

T. Pertsch, U. Peschel, F. Lederer, J. Burghoff, M. Will, S. Nolte, and A. Tunnermann, "Discrete diffraction in two-dimensional arrays of coupled waveguides in silica," Opt. Lett. 29,468-470 (2004).
[CrossRef] [PubMed]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, "Discretizing light behaviour in linear and nonlinear waveguide lattices," Nature 424,817-823 (2003).
[CrossRef] [PubMed]

T. Pertsch, P. Dannberg, W. Elflein, A. Brauer, and F. Lederer, "Optical Bloch oscillations in temperature tuned waveguide arrays," Phys. Rev. Lett. 83,4752-4755 (1999).
[CrossRef]

Lee, S. B.

Lenz, G.

G. Lenz, R. Parker, M. C. Wanke, and C. M. de Sterke, "Dynamical localization and AC Bloch oscillations in periodic optical waveguide arrays," Opt. Commun. 218,87-92 (2003).
[CrossRef]

G. Lenz, I. Talanina, and C. M. de Sterke, "Bloch oscillations in an array of curved optical waveguides," Phys. Rev. Lett. 83,963-966 (1999).
[CrossRef]

Lobino, M.

S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, "Observation of dynamic localization in periodically curved waveguide arrays," Phys. Rev. Lett. 96,243901-4 (2006).
[CrossRef] [PubMed]

Longhi, S.

Malomed, B. A.

P. G. Kevrekidis, B. A. Malomed, and Y. B. Gaididei, "Solitons in triangular and honeycomb dynamical lattices with the cubic nonlinearity," Phys. Rev. E 66,016609-10 (2002).
[CrossRef]

Marangoni, M.

S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, "Observation of dynamic localization in periodically curved waveguide arrays," Phys. Rev. Lett. 96,243901-4 (2006).
[CrossRef] [PubMed]

Michaelis, D.

H. Trompeter, T. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Brauer, and U. Peschel, "Visual observation of Zener tunneling," Phys. Rev. Lett. 96,023901-4 (2006).
[CrossRef] [PubMed]

Morandotti, R.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, "Diffraction management," Phys. Rev. Lett. 85,1863-1866 (2000).
[CrossRef] [PubMed]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and K. Silberberg, "Experimental observation of linear and nonlinear optical Bloch oscillations," Phys. Rev. Lett. 83,4756-4759 (1999).
[CrossRef]

Morgner, U.

Neshev, D. N.

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Yu. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, "Bloch oscillations and Zener tunneling in two-dimensional photonic lattices," Phys. Rev. Lett. 96,053903-4 (2006).
[CrossRef] [PubMed]

C. R. Rosberg, I. L. Garanovich, A. A. Sukhorukov, D. N. Neshev,W. Krolikowski, and Yu. S. Kivshar, "Demonstration of all-optical beam steering in modulated photonic lattices," Opt. Lett. 31,1498-1500 (2006).
[CrossRef] [PubMed]

Nolte, S.

A. Szameit, T. Pertsch, F. Dreisow, S. Nolte, A. Tunnermann, U. Peschel, and F. Lederer, "Light evolution in arbitrary two-dimensional waveguide arrays," Phys. Rev. A 75,053814-14 (2007).
[CrossRef]

A. Szameit, D. Blomer, J. Burghoff, T. Pertsch, S. Nolte, and A. Tunnermann, "Hexagonal waveguide arrays written with fs-laser pulses," Appl. Phys. B 82,507-512 (2006).
[CrossRef]

A. Szameit, D. Blomer, J. Burghoff, T. Schreiber, T. Pertsch, S. Nolte, A. Tunnermann, and F. Lederer, "Discrete nonlinear localization in femtosecond laser written waveguides in fused silica," Opt. Express 13,10552-10557 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-26-10552.
[CrossRef] [PubMed]

T. Pertsch, U. Peschel, F. Lederer, J. Burghoff, M. Will, S. Nolte, and A. Tunnermann, "Discrete diffraction in two-dimensional arrays of coupled waveguides in silica," Opt. Lett. 29,468-470 (2004).
[CrossRef] [PubMed]

S. Nolte, M. Will, J. Burghoff, and A. Tuennermann, "Femtosecond waveguide writing: a new avenue to three-dimensional integrated optics," Appl. Phys. A 77,109-111 (2003).
[CrossRef]

Notomi, M.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Self-collimating phenomena in photonic crystals," Appl. Phys. Lett. 74,1212-1214 (1999).
[CrossRef]

Osellame, R.

Parker, R.

G. Lenz, R. Parker, M. C. Wanke, and C. M. de Sterke, "Dynamical localization and AC Bloch oscillations in periodic optical waveguide arrays," Opt. Commun. 218,87-92 (2003).
[CrossRef]

Pertsch, T.

A. Szameit, T. Pertsch, F. Dreisow, S. Nolte, A. Tunnermann, U. Peschel, and F. Lederer, "Light evolution in arbitrary two-dimensional waveguide arrays," Phys. Rev. A 75,053814-14 (2007).
[CrossRef]

A. Szameit, D. Blomer, J. Burghoff, T. Pertsch, S. Nolte, and A. Tunnermann, "Hexagonal waveguide arrays written with fs-laser pulses," Appl. Phys. B 82,507-512 (2006).
[CrossRef]

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Yu. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, "Bloch oscillations and Zener tunneling in two-dimensional photonic lattices," Phys. Rev. Lett. 96,053903-4 (2006).
[CrossRef] [PubMed]

H. Trompeter, T. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Brauer, and U. Peschel, "Visual observation of Zener tunneling," Phys. Rev. Lett. 96,023901-4 (2006).
[CrossRef] [PubMed]

A. Szameit, D. Blomer, J. Burghoff, T. Schreiber, T. Pertsch, S. Nolte, A. Tunnermann, and F. Lederer, "Discrete nonlinear localization in femtosecond laser written waveguides in fused silica," Opt. Express 13,10552-10557 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-26-10552.
[CrossRef] [PubMed]

T. Pertsch, U. Peschel, F. Lederer, J. Burghoff, M. Will, S. Nolte, and A. Tunnermann, "Discrete diffraction in two-dimensional arrays of coupled waveguides in silica," Opt. Lett. 29,468-470 (2004).
[CrossRef] [PubMed]

T. Pertsch, P. Dannberg, W. Elflein, A. Brauer, and F. Lederer, "Optical Bloch oscillations in temperature tuned waveguide arrays," Phys. Rev. Lett. 83,4752-4755 (1999).
[CrossRef]

Peschel, U.

A. Szameit, T. Pertsch, F. Dreisow, S. Nolte, A. Tunnermann, U. Peschel, and F. Lederer, "Light evolution in arbitrary two-dimensional waveguide arrays," Phys. Rev. A 75,053814-14 (2007).
[CrossRef]

H. Trompeter, T. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Brauer, and U. Peschel, "Visual observation of Zener tunneling," Phys. Rev. Lett. 96,023901-4 (2006).
[CrossRef] [PubMed]

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Yu. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, "Bloch oscillations and Zener tunneling in two-dimensional photonic lattices," Phys. Rev. Lett. 96,053903-4 (2006).
[CrossRef] [PubMed]

T. Pertsch, U. Peschel, F. Lederer, J. Burghoff, M. Will, S. Nolte, and A. Tunnermann, "Discrete diffraction in two-dimensional arrays of coupled waveguides in silica," Opt. Lett. 29,468-470 (2004).
[CrossRef] [PubMed]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and K. Silberberg, "Experimental observation of linear and nonlinear optical Bloch oscillations," Phys. Rev. Lett. 83,4756-4759 (1999).
[CrossRef]

Petrich, G. S.

P. T. Rakich, M. S. Dahlem, S. Tandon, M. Ibanescu, M. Soljacic, G. S. Petrich, J. D. Joannopoulos, L. A. Kolodziejski, and E. P. Ippen, "Achieving centimetre-scale supercollimation in a large-area two-dimensional photonic crystal," Nat. Mater. 5,93-96 (2006).
[CrossRef] [PubMed]

Rakich, P. T.

P. T. Rakich, M. S. Dahlem, S. Tandon, M. Ibanescu, M. Soljacic, G. S. Petrich, J. D. Joannopoulos, L. A. Kolodziejski, and E. P. Ippen, "Achieving centimetre-scale supercollimation in a large-area two-dimensional photonic crystal," Nat. Mater. 5,93-96 (2006).
[CrossRef] [PubMed]

Ramponi, R.

N. Chiodo, G. Della Valle, R. Osellame, S. Longhi, G. Cerullo, R. Ramponi, P. Laporta, and U. Morgner, "Imaging of Bloch oscillations in erbium-doped curved waveguide arrays," Opt. Lett. 31,1651-1653 (2006).
[CrossRef] [PubMed]

S. Longhi, M. Marangoni, M. Lobino, R. Ramponi, P. Laporta, E. Cianci, and V. Foglietti, "Observation of dynamic localization in periodically curved waveguide arrays," Phys. Rev. Lett. 96,243901-4 (2006).
[CrossRef] [PubMed]

Ropke, U.

Rosberg, C. R.

Sato, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Self-collimating phenomena in photonic crystals," Appl. Phys. Lett. 74,1212-1214 (1999).
[CrossRef]

Schreiber, T.

Schuster, K.

Silberberg, K.

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and K. Silberberg, "Experimental observation of linear and nonlinear optical Bloch oscillations," Phys. Rev. Lett. 83,4756-4759 (1999).
[CrossRef]

Silberberg, Y.

D. N. Christodoulides, F. Lederer, and Y. Silberberg, "Discretizing light behaviour in linear and nonlinear waveguide lattices," Nature 424,817-823 (2003).
[CrossRef] [PubMed]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, "Diffraction management," Phys. Rev. Lett. 85,1863-1866 (2000).
[CrossRef] [PubMed]

Soljacic, M.

P. T. Rakich, M. S. Dahlem, S. Tandon, M. Ibanescu, M. Soljacic, G. S. Petrich, J. D. Joannopoulos, L. A. Kolodziejski, and E. P. Ippen, "Achieving centimetre-scale supercollimation in a large-area two-dimensional photonic crystal," Nat. Mater. 5,93-96 (2006).
[CrossRef] [PubMed]

Somekh, S.

S. Somekh, E. Garmire, A. Yariv, H. L. Garvin, and R. G. Hunsperger, "Channel optical waveguide directional couplers," Appl. Phys. Lett. 22,46-47 (1973).
[CrossRef]

Streppel, U.

H. Trompeter, T. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Brauer, and U. Peschel, "Visual observation of Zener tunneling," Phys. Rev. Lett. 96,023901-4 (2006).
[CrossRef] [PubMed]

Sukhorukov, A. A.

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Yu. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, "Bloch oscillations and Zener tunneling in two-dimensional photonic lattices," Phys. Rev. Lett. 96,053903-4 (2006).
[CrossRef] [PubMed]

I. L. Garanovich, A. A. Sukhorukov, and Yu. S. Kivshar, "Broadband diffraction management and selfcollimation of white light in photonic lattices," Phys. Rev. E 74,066609-4 (2006).
[CrossRef]

C. R. Rosberg, I. L. Garanovich, A. A. Sukhorukov, D. N. Neshev,W. Krolikowski, and Yu. S. Kivshar, "Demonstration of all-optical beam steering in modulated photonic lattices," Opt. Lett. 31,1498-1500 (2006).
[CrossRef] [PubMed]

I. L. Garanovich, A. A. Sukhorukov, and Yu. S. Kivshar, "Soliton control in modulated optically-induced photonic lattices," Opt. Express 13,5704-5710 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-15-5704.
[CrossRef] [PubMed]

Szameit, A.

A. Szameit, T. Pertsch, F. Dreisow, S. Nolte, A. Tunnermann, U. Peschel, and F. Lederer, "Light evolution in arbitrary two-dimensional waveguide arrays," Phys. Rev. A 75,053814-14 (2007).
[CrossRef]

A. Szameit, D. Blomer, J. Burghoff, T. Pertsch, S. Nolte, and A. Tunnermann, "Hexagonal waveguide arrays written with fs-laser pulses," Appl. Phys. B 82,507-512 (2006).
[CrossRef]

A. Szameit, D. Blomer, J. Burghoff, T. Schreiber, T. Pertsch, S. Nolte, A. Tunnermann, and F. Lederer, "Discrete nonlinear localization in femtosecond laser written waveguides in fused silica," Opt. Express 13,10552-10557 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-26-10552.
[CrossRef] [PubMed]

Talanina, I.

G. Lenz, I. Talanina, and C. M. de Sterke, "Bloch oscillations in an array of curved optical waveguides," Phys. Rev. Lett. 83,963-966 (1999).
[CrossRef]

Tamamura, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Self-collimating phenomena in photonic crystals," Appl. Phys. Lett. 74,1212-1214 (1999).
[CrossRef]

Tandon, S.

P. T. Rakich, M. S. Dahlem, S. Tandon, M. Ibanescu, M. Soljacic, G. S. Petrich, J. D. Joannopoulos, L. A. Kolodziejski, and E. P. Ippen, "Achieving centimetre-scale supercollimation in a large-area two-dimensional photonic crystal," Nat. Mater. 5,93-96 (2006).
[CrossRef] [PubMed]

Tayeb, G.

Tomita, A.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Self-collimating phenomena in photonic crystals," Appl. Phys. Lett. 74,1212-1214 (1999).
[CrossRef]

Torner, L.

Torres, C. M. S.

Trompeter, H.

H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Yu. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, "Bloch oscillations and Zener tunneling in two-dimensional photonic lattices," Phys. Rev. Lett. 96,053903-4 (2006).
[CrossRef] [PubMed]

H. Trompeter, T. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Brauer, and U. Peschel, "Visual observation of Zener tunneling," Phys. Rev. Lett. 96,023901-4 (2006).
[CrossRef] [PubMed]

Tuennermann, A.

S. Nolte, M. Will, J. Burghoff, and A. Tuennermann, "Femtosecond waveguide writing: a new avenue to three-dimensional integrated optics," Appl. Phys. A 77,109-111 (2003).
[CrossRef]

Tunnermann, A.

Unger, S.

Wan, J.

R. Iyer, J. S. Aitchison, J. Wan, M. M. Dignam, and C. M. de Sterke, "Exact dynamic localization in curved AlGaAs optical waveguide arrays," Opt. Express 15,3212-3223 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-6-3212.
[CrossRef] [PubMed]

J. Wan, M. Laforest, C. M. de Sterke, and M. M. Dignam, "Optical filters based on dynamic localization in curved coupled optical waveguides," Opt. Commun. 247,353-365 (2005).
[CrossRef]

Wanke, M. C.

G. Lenz, R. Parker, M. C. Wanke, and C. M. de Sterke, "Dynamical localization and AC Bloch oscillations in periodic optical waveguide arrays," Opt. Commun. 218,87-92 (2003).
[CrossRef]

Will, M.

T. Pertsch, U. Peschel, F. Lederer, J. Burghoff, M. Will, S. Nolte, and A. Tunnermann, "Discrete diffraction in two-dimensional arrays of coupled waveguides in silica," Opt. Lett. 29,468-470 (2004).
[CrossRef] [PubMed]

S. Nolte, M. Will, J. Burghoff, and A. Tuennermann, "Femtosecond waveguide writing: a new avenue to three-dimensional integrated optics," Appl. Phys. A 77,109-111 (2003).
[CrossRef]

Yariv, A.

S. Somekh, E. Garmire, A. Yariv, H. L. Garvin, and R. G. Hunsperger, "Channel optical waveguide directional couplers," Appl. Phys. Lett. 22,46-47 (1973).
[CrossRef]

Zener, C.

C. Zener, Proc. R. Soc. London A 145,523-529 (1934).

Appl. Phys. A (1)

S. Nolte, M. Will, J. Burghoff, and A. Tuennermann, "Femtosecond waveguide writing: a new avenue to three-dimensional integrated optics," Appl. Phys. A 77,109-111 (2003).
[CrossRef]

Appl. Phys. B (1)

A. Szameit, D. Blomer, J. Burghoff, T. Pertsch, S. Nolte, and A. Tunnermann, "Hexagonal waveguide arrays written with fs-laser pulses," Appl. Phys. B 82,507-512 (2006).
[CrossRef]

Appl. Phys. Lett. (2)

S. Somekh, E. Garmire, A. Yariv, H. L. Garvin, and R. G. Hunsperger, "Channel optical waveguide directional couplers," Appl. Phys. Lett. 22,46-47 (1973).
[CrossRef]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Self-collimating phenomena in photonic crystals," Appl. Phys. Lett. 74,1212-1214 (1999).
[CrossRef]

J. Opt. Soc. Am. (1)

Nat. Mater. (1)

P. T. Rakich, M. S. Dahlem, S. Tandon, M. Ibanescu, M. Soljacic, G. S. Petrich, J. D. Joannopoulos, L. A. Kolodziejski, and E. P. Ippen, "Achieving centimetre-scale supercollimation in a large-area two-dimensional photonic crystal," Nat. Mater. 5,93-96 (2006).
[CrossRef] [PubMed]

Nature (1)

D. N. Christodoulides, F. Lederer, and Y. Silberberg, "Discretizing light behaviour in linear and nonlinear waveguide lattices," Nature 424,817-823 (2003).
[CrossRef] [PubMed]

Opt. Commun. (2)

J. Wan, M. Laforest, C. M. de Sterke, and M. M. Dignam, "Optical filters based on dynamic localization in curved coupled optical waveguides," Opt. Commun. 247,353-365 (2005).
[CrossRef]

G. Lenz, R. Parker, M. C. Wanke, and C. M. de Sterke, "Dynamical localization and AC Bloch oscillations in periodic optical waveguide arrays," Opt. Commun. 218,87-92 (2003).
[CrossRef]

Opt. Express (6)

R. Iyer, J. S. Aitchison, J. Wan, M. M. Dignam, and C. M. de Sterke, "Exact dynamic localization in curved AlGaAs optical waveguide arrays," Opt. Express 15,3212-3223 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-6-3212.
[CrossRef] [PubMed]

D. N. Chigrin, S. Enoch, C. M. S. Torres, and G. Tayeb, "Self-guiding in two-dimensional photonic crystals," Opt. Express 11,1203-1211 (2003), http://www.opticsinfobase.org/abstract.cfm?URI=oe-11-10-1203.
[CrossRef] [PubMed]

U. Ropke, H. Bartelt, S. Unger, K. Schuster, and J. Kobelke, "Two-dimensional high-precision fiber waveguide arrays for coherent light propagation," Opt. Express 15,6894-6899 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-11-6894.
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Y. G. Han and S. B. Lee, "Tunable dispersion compensator based on uniform fiber Bragg grating and its application to tunable pulse repetition-rate multiplication," Opt. Express 13,9224-9229 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-23-9224.
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A. Szameit, D. Blomer, J. Burghoff, T. Schreiber, T. Pertsch, S. Nolte, A. Tunnermann, and F. Lederer, "Discrete nonlinear localization in femtosecond laser written waveguides in fused silica," Opt. Express 13,10552-10557 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-26-10552.
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Opt. Lett. (6)

Phys. Rev. A (1)

A. Szameit, T. Pertsch, F. Dreisow, S. Nolte, A. Tunnermann, U. Peschel, and F. Lederer, "Light evolution in arbitrary two-dimensional waveguide arrays," Phys. Rev. A 75,053814-14 (2007).
[CrossRef]

Phys. Rev. E (2)

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[CrossRef]

I. L. Garanovich, A. A. Sukhorukov, and Yu. S. Kivshar, "Broadband diffraction management and selfcollimation of white light in photonic lattices," Phys. Rev. E 74,066609-4 (2006).
[CrossRef]

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Supplementary Material (8)

» Media 1: GIF (1194 KB)     
» Media 2: GIF (2482 KB)     
» Media 3: GIF (2392 KB)     
» Media 4: GIF (2357 KB)     
» Media 5: GIF (2528 KB)     
» Media 6: GIF (2265 KB)     
» Media 7: GIF (2375 KB)     
» Media 8: GIF (2324 KB)     

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Figures (6)

Fig. 1.
Fig. 1.

(1.2MB) Sketch of modulated hexagonal photonic lattice. (a) Two-dimensional cross-section. Shading marks the unit cell, each lattice site has six nearest neighbors. (b) Schematic of the individual waveguide with the axis periodically curved in the z-direction. (c) Couplings between the nearest neighbours in the hexagonal lattice. Lattice sites are numbered along the n and m-axes.

Fig. 2.
Fig. 2.

(1.2MB) Discrete diffraction in a hexagonal lattice of straight waveguides. Animation shows the patterns of the light-field distribution as the beam propagates over the distance 4L=84 mm. Wavelength is λ=633 nm.

Fig. 3.
Fig. 3.

(2.4MB) Self-collimation in a modulated hexagonal lattice. The field distribution is shown at the input, and after the propagation for two and four bending periods. Animation shows the beam propagation in a fixed viewpoint. Wavelength is λ=633 nm.

Fig. 4.
Fig. 4.

One-dimensional diffraction in a modulated hexagonal lattice. The field distribution is shown at the input, and after the propagation for two and four bending periods. (a) (2.3MB) Lattice shift occurs in the y-direction. (b) (2.3MB) Lattice shift occurs in the x-direction. Animations show the propagation dynamics in a fixed viewport. Wavelength is λ=633 nm.

Fig. 5.
Fig. 5.

(2.5MB) Rectangular diffraction in a modulated hexagonal lattice. The field distribution is shown at the input, and after the propagation for two and four bending periods. Animation shows beam propagation dynamics in a fixed viewport. Wavelength is λ=633 nm.

Fig. 6.
Fig. 6.

Examples of different diffraction patterns in the same modulated hexagonal lattice. (a) (2.2MB) One-dimensional diffraction for the wavelength λ=633 nm. (b) (2.3MB) Hexagonal diffraction for the wavelength λ=583 nm. (c) (2.3MB) Rectangular diffraction for the wavelength λ=550 nm. Animations show the beam propagation dynamics in a fixed viewport. The bending period and amplitude are L=10.5 mm and A 1=35 µm, respectively.

Equations (14)

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i E z + λ 4 π n 0 ( 2 E x 2 + 2 E y 2 ) + 2 π λ ν [ x x 0 ( z ) , y y 0 ( z ) ] E = 0 .
E = n , m Ψ n , m ( z ) E 0 [ x x n , m x 0 ( z ) , y y n , m y 0 ( z ) , z ]
exp { 2 i p x ˙ 0 ( z ) [ x x n , m x 0 ( z ) ] + 2 i p y ˙ 0 ( z ) [ y y n , m y 0 ( z ) ] }
exp { ip 0 z [ x ˙ 0 ( ξ ) ] 2 d ξ + ip 0 z [ y ˙ 0 ( ξ ) ] 2 d ξ } ,
i d Ψ n , m dz + n , m n , m C ˜ n , n , m , m Ψ n , m = 0 ,
i d Ψ n , m dz + C ˜ 1 * Ψ n 1 , m + C ˜ 1 Ψ n + 1 , m + C ˜ 2 * Ψ n , m 1 + C ˜ 2 Ψ n , m + 1 + C ˜ 3 * Ψ n 1 , m + 1 + C ˜ 3 Ψ n + 1 , m 1 = 0 ,
C 1 eff = C 1 L 1 0 L cos [ ω x ˙ 0 ( ζ ) ] d ζ ,
C 2 eff = C 2 L 1 0 L cos [ ω 2 x ˙ 0 ( ζ ) + 3 2 ω y ˙ 0 ( ζ ) ] d ζ ,
C 3 eff = C 3 L 1 0 L cos [ ω 2 x ˙ 0 ( ζ ) 3 2 ω y ˙ 0 ( ζ ) ] d ζ .
x 0 ( z ) = { A 1 { cos [ 2 π z z 0 ] 1 } , if 0 z z 0 A 2 { cos [ 2 π ( z z 0 ) ( L 2 z 0 ) ] 1 } , if z 0 z L 2
y 0 ( z ) = { B 1 { cos [ 2 π z z 0 ] 1 } , if 0 z z 0 B 2 { cos [ 2 π ( z z 0 ) ( L 2 z 0 ) ] 1 } , if z 0 z L 2,
C 1 eff = 2 C 1 L 1 [ z 0 J 0 ( ξ ˜ 1 ) + ( L 2 z 0 ) J 0 ( ξ ˜ 2 ) ] ,
C 2 eff = 2 C 2 L 1 [ z 0 J 0 ( ξ ˜ 1 2 + 3 2 ξ ¯ 1 ) + ( L 2 z 0 ) J 0 ( ξ ˜ 2 2 + 3 2 ξ ¯ 2 ) ] ,
C 3 eff = 2 C 3 L 1 [ z 0 J 0 ( ξ ˜ 1 2 3 2 ξ ¯ 1 ) + ( L 2 z 0 ) J 0 ( ξ ˜ 2 2 3 2 ξ ¯ 2 ) ] ,

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