Abstract

We investigate the evanescent coupling of femtosecond laser written waveguides with elliptical and circular shape. A directional tuning of the coupling properties is realized in a cubic array by tilting the elliptical waveguides. This allows to specifically pronounce diagonal coupling. In contrast, directional insensitive coupling is demonstrated in a circular waveguide array based on circular waveguides.

© 2007 Optical Society of America

Full Article  |  PDF Article

Corrections

Alexander Szameit, Felix Dreisow, Thomas Pertsch, Stefan Nolte, and Andreas Tünnermann, "Control of directional evanescent coupling in fs laser written waveguides: Erratum," Opt. Express 15, 8492-8492 (2007)
https://www.osapublishing.org/oe/abstract.cfm?uri=oe-15-13-8492

References

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  1. H. Haus and L. Molter-Orr, "Coupled multiple waveguide systems," IEEE J. Quantum Electron. 19, 840-844 (1983).
    [CrossRef]
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    [CrossRef]
  3. D. Christodoulides and R. Joseph, "Discrete self-focusing in nonlinear arrays of coupled waveguides," Opt. Lett. 13, 794-796 (1988).
    [CrossRef] [PubMed]
  4. H. Eisenberg, Y. Silberberg, R. Morandotti, A. Boyd, and J. Aitchison, "Discrete spatial optical solitons in waveguide arrays," Phys. Rev. Lett. 81, 3383-3386 (1998).
    [CrossRef]
  5. A. Szameit, J. Burghoff, T. Pertsch, S. Nolte, A. Tuennermann, and F. Lederer, "Discrete Nonlinear Localization in Femtosecond Laser Written Waveguides in Fused Silica," Opt. Exp. 13, 10,552-10,557 (2005).
    [CrossRef]
  6. A. Fratalocchi, G. Assanto, K. Brzdakiewicz, and M. Karpierz, "Discrete propagation and spatial solitons in nematic liquid crystals," Opt. Lett. 29, 1530-1532 (2004).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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  13. D. Bloemer, A. Szameit, F. Dreisow, J. Burghoff, T. Schreiber, T. Pertsch, S. Nolte, F. Lederer, and A. Tuennermann, "Measurement of the nonlinear refractive index of fs-laser-written waveguides in fused silica," Opt. Exp. 14, 2151-2157 (2006).
    [CrossRef]
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    [CrossRef]
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  16. M. Ams, G. Marshall, D. Spence, and M. Withford, "Slit-beam shaping method for femtosecond laser direct-write fabrication of symmetric waveguides in bulk glasses," Opt. Exp. 13, 5676-5681 (2005).
    [CrossRef]
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2006 (4)

A. Szameit, D. Bloemer, J. Burghoff, T. Pertsch, S. Nolte, F. Lederer, and A. Tuennermann, "Two-dimensional soliton in cubic fs laser written waveguide arrays in fused silica," Opt. Exp. 14, 6055-6062 (2006).
[CrossRef]

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

K. Itoh, W. Watanabe, S. Nolte, and C. Schaffer, "Ultrafast Processes for Bulk Modification of Transparent Materials," MRS Bulletin 31, 620-625 (2006).
[CrossRef]

D. Bloemer, A. Szameit, F. Dreisow, J. Burghoff, T. Schreiber, T. Pertsch, S. Nolte, F. Lederer, and A. Tuennermann, "Measurement of the nonlinear refractive index of fs-laser-written waveguides in fused silica," Opt. Exp. 14, 2151-2157 (2006).
[CrossRef]

2005 (2)

A. Szameit, J. Burghoff, T. Pertsch, S. Nolte, A. Tuennermann, and F. Lederer, "Discrete Nonlinear Localization in Femtosecond Laser Written Waveguides in Fused Silica," Opt. Exp. 13, 10,552-10,557 (2005).
[CrossRef]

M. Ams, G. Marshall, D. Spence, and M. Withford, "Slit-beam shaping method for femtosecond laser direct-write fabrication of symmetric waveguides in bulk glasses," Opt. Exp. 13, 5676-5681 (2005).
[CrossRef]

2004 (2)

2003 (1)

J. Fleischer, M. Segev, N. Efremidis, and D. Christodoulides, "Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices," Nature 422, 147-150 (2003).
[CrossRef] [PubMed]

1998 (1)

H. Eisenberg, Y. Silberberg, R. Morandotti, A. Boyd, and J. Aitchison, "Discrete spatial optical solitons in waveguide arrays," Phys. Rev. Lett. 81, 3383-3386 (1998).
[CrossRef]

1996 (1)

I. Mansour and F. Caccavale, "An improved procedure to calculate the refractive index profile from the measured near-field intensity," J. Lightwave Technol. 14, 423-428 (1996).
[CrossRef]

1994 (1)

1988 (1)

1983 (1)

H. Haus and L. Molter-Orr, "Coupled multiple waveguide systems," IEEE J. Quantum Electron. 19, 840-844 (1983).
[CrossRef]

Aitchison, J.

H. Eisenberg, Y. Silberberg, R. Morandotti, A. Boyd, and J. Aitchison, "Discrete spatial optical solitons in waveguide arrays," Phys. Rev. Lett. 81, 3383-3386 (1998).
[CrossRef]

Ams, M.

M. Ams, G. Marshall, D. Spence, and M. Withford, "Slit-beam shaping method for femtosecond laser direct-write fabrication of symmetric waveguides in bulk glasses," Opt. Exp. 13, 5676-5681 (2005).
[CrossRef]

Assanto, G.

Bloemer, D.

A. Szameit, D. Bloemer, J. Burghoff, T. Pertsch, S. Nolte, F. Lederer, and A. Tuennermann, "Two-dimensional soliton in cubic fs laser written waveguide arrays in fused silica," Opt. Exp. 14, 6055-6062 (2006).
[CrossRef]

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

D. Bloemer, A. Szameit, F. Dreisow, J. Burghoff, T. Schreiber, T. Pertsch, S. Nolte, F. Lederer, and A. Tuennermann, "Measurement of the nonlinear refractive index of fs-laser-written waveguides in fused silica," Opt. Exp. 14, 2151-2157 (2006).
[CrossRef]

Boyd, A.

H. Eisenberg, Y. Silberberg, R. Morandotti, A. Boyd, and J. Aitchison, "Discrete spatial optical solitons in waveguide arrays," Phys. Rev. Lett. 81, 3383-3386 (1998).
[CrossRef]

Brzdakiewicz, K.

Burghoff, J.

A. Szameit, D. Bloemer, J. Burghoff, T. Pertsch, S. Nolte, F. Lederer, and A. Tuennermann, "Two-dimensional soliton in cubic fs laser written waveguide arrays in fused silica," Opt. Exp. 14, 6055-6062 (2006).
[CrossRef]

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

D. Bloemer, A. Szameit, F. Dreisow, J. Burghoff, T. Schreiber, T. Pertsch, S. Nolte, F. Lederer, and A. Tuennermann, "Measurement of the nonlinear refractive index of fs-laser-written waveguides in fused silica," Opt. Exp. 14, 2151-2157 (2006).
[CrossRef]

A. Szameit, J. Burghoff, T. Pertsch, S. Nolte, A. Tuennermann, and F. Lederer, "Discrete Nonlinear Localization in Femtosecond Laser Written Waveguides in Fused Silica," Opt. Exp. 13, 10,552-10,557 (2005).
[CrossRef]

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

Caccavale, F.

I. Mansour and F. Caccavale, "An improved procedure to calculate the refractive index profile from the measured near-field intensity," J. Lightwave Technol. 14, 423-428 (1996).
[CrossRef]

Christodoulides, D.

J. Fleischer, M. Segev, N. Efremidis, and D. Christodoulides, "Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices," Nature 422, 147-150 (2003).
[CrossRef] [PubMed]

D. Christodoulides and R. Joseph, "Discrete self-focusing in nonlinear arrays of coupled waveguides," Opt. Lett. 13, 794-796 (1988).
[CrossRef] [PubMed]

Cronin-Golomb, M.

Dreisow, F.

D. Bloemer, A. Szameit, F. Dreisow, J. Burghoff, T. Schreiber, T. Pertsch, S. Nolte, F. Lederer, and A. Tuennermann, "Measurement of the nonlinear refractive index of fs-laser-written waveguides in fused silica," Opt. Exp. 14, 2151-2157 (2006).
[CrossRef]

Efremidis, N.

J. Fleischer, M. Segev, N. Efremidis, and D. Christodoulides, "Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices," Nature 422, 147-150 (2003).
[CrossRef] [PubMed]

Eisenberg, H.

H. Eisenberg, Y. Silberberg, R. Morandotti, A. Boyd, and J. Aitchison, "Discrete spatial optical solitons in waveguide arrays," Phys. Rev. Lett. 81, 3383-3386 (1998).
[CrossRef]

Fleischer, J.

J. Fleischer, M. Segev, N. Efremidis, and D. Christodoulides, "Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices," Nature 422, 147-150 (2003).
[CrossRef] [PubMed]

Fratalocchi, A.

Haus, H.

H. Haus and L. Molter-Orr, "Coupled multiple waveguide systems," IEEE J. Quantum Electron. 19, 840-844 (1983).
[CrossRef]

Itoh, K.

K. Itoh, W. Watanabe, S. Nolte, and C. Schaffer, "Ultrafast Processes for Bulk Modification of Transparent Materials," MRS Bulletin 31, 620-625 (2006).
[CrossRef]

Joseph, R.

Karpierz, M.

Krolikowski, W.

Lederer, F.

D. Bloemer, A. Szameit, F. Dreisow, J. Burghoff, T. Schreiber, T. Pertsch, S. Nolte, F. Lederer, and A. Tuennermann, "Measurement of the nonlinear refractive index of fs-laser-written waveguides in fused silica," Opt. Exp. 14, 2151-2157 (2006).
[CrossRef]

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

A. Szameit, D. Bloemer, J. Burghoff, T. Pertsch, S. Nolte, F. Lederer, and A. Tuennermann, "Two-dimensional soliton in cubic fs laser written waveguide arrays in fused silica," Opt. Exp. 14, 6055-6062 (2006).
[CrossRef]

A. Szameit, J. Burghoff, T. Pertsch, S. Nolte, A. Tuennermann, and F. Lederer, "Discrete Nonlinear Localization in Femtosecond Laser Written Waveguides in Fused Silica," Opt. Exp. 13, 10,552-10,557 (2005).
[CrossRef]

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

Mansour, I.

I. Mansour and F. Caccavale, "An improved procedure to calculate the refractive index profile from the measured near-field intensity," J. Lightwave Technol. 14, 423-428 (1996).
[CrossRef]

Marshall, G.

M. Ams, G. Marshall, D. Spence, and M. Withford, "Slit-beam shaping method for femtosecond laser direct-write fabrication of symmetric waveguides in bulk glasses," Opt. Exp. 13, 5676-5681 (2005).
[CrossRef]

Molter-Orr, L.

H. Haus and L. Molter-Orr, "Coupled multiple waveguide systems," IEEE J. Quantum Electron. 19, 840-844 (1983).
[CrossRef]

Morandotti, R.

H. Eisenberg, Y. Silberberg, R. Morandotti, A. Boyd, and J. Aitchison, "Discrete spatial optical solitons in waveguide arrays," Phys. Rev. Lett. 81, 3383-3386 (1998).
[CrossRef]

Nolte, S.

D. Bloemer, A. Szameit, F. Dreisow, J. Burghoff, T. Schreiber, T. Pertsch, S. Nolte, F. Lederer, and A. Tuennermann, "Measurement of the nonlinear refractive index of fs-laser-written waveguides in fused silica," Opt. Exp. 14, 2151-2157 (2006).
[CrossRef]

A. Szameit, D. Bloemer, J. Burghoff, T. Pertsch, S. Nolte, F. Lederer, and A. Tuennermann, "Two-dimensional soliton in cubic fs laser written waveguide arrays in fused silica," Opt. Exp. 14, 6055-6062 (2006).
[CrossRef]

K. Itoh, W. Watanabe, S. Nolte, and C. Schaffer, "Ultrafast Processes for Bulk Modification of Transparent Materials," MRS Bulletin 31, 620-625 (2006).
[CrossRef]

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

A. Szameit, J. Burghoff, T. Pertsch, S. Nolte, A. Tuennermann, and F. Lederer, "Discrete Nonlinear Localization in Femtosecond Laser Written Waveguides in Fused Silica," Opt. Exp. 13, 10,552-10,557 (2005).
[CrossRef]

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

Pertsch, T.

A. Szameit, D. Bloemer, J. Burghoff, T. Pertsch, S. Nolte, F. Lederer, and A. Tuennermann, "Two-dimensional soliton in cubic fs laser written waveguide arrays in fused silica," Opt. Exp. 14, 6055-6062 (2006).
[CrossRef]

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

D. Bloemer, A. Szameit, F. Dreisow, J. Burghoff, T. Schreiber, T. Pertsch, S. Nolte, F. Lederer, and A. Tuennermann, "Measurement of the nonlinear refractive index of fs-laser-written waveguides in fused silica," Opt. Exp. 14, 2151-2157 (2006).
[CrossRef]

A. Szameit, J. Burghoff, T. Pertsch, S. Nolte, A. Tuennermann, and F. Lederer, "Discrete Nonlinear Localization in Femtosecond Laser Written Waveguides in Fused Silica," Opt. Exp. 13, 10,552-10,557 (2005).
[CrossRef]

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

Peschel, U.

Schaffer, C.

K. Itoh, W. Watanabe, S. Nolte, and C. Schaffer, "Ultrafast Processes for Bulk Modification of Transparent Materials," MRS Bulletin 31, 620-625 (2006).
[CrossRef]

Schmidt-Hattenberger, C.

Schreiber, T.

D. Bloemer, A. Szameit, F. Dreisow, J. Burghoff, T. Schreiber, T. Pertsch, S. Nolte, F. Lederer, and A. Tuennermann, "Measurement of the nonlinear refractive index of fs-laser-written waveguides in fused silica," Opt. Exp. 14, 2151-2157 (2006).
[CrossRef]

Segev, M.

J. Fleischer, M. Segev, N. Efremidis, and D. Christodoulides, "Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices," Nature 422, 147-150 (2003).
[CrossRef] [PubMed]

Silberberg, Y.

H. Eisenberg, Y. Silberberg, R. Morandotti, A. Boyd, and J. Aitchison, "Discrete spatial optical solitons in waveguide arrays," Phys. Rev. Lett. 81, 3383-3386 (1998).
[CrossRef]

Spence, D.

M. Ams, G. Marshall, D. Spence, and M. Withford, "Slit-beam shaping method for femtosecond laser direct-write fabrication of symmetric waveguides in bulk glasses," Opt. Exp. 13, 5676-5681 (2005).
[CrossRef]

Szameit, A.

D. Bloemer, A. Szameit, F. Dreisow, J. Burghoff, T. Schreiber, T. Pertsch, S. Nolte, F. Lederer, and A. Tuennermann, "Measurement of the nonlinear refractive index of fs-laser-written waveguides in fused silica," Opt. Exp. 14, 2151-2157 (2006).
[CrossRef]

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

A. Szameit, D. Bloemer, J. Burghoff, T. Pertsch, S. Nolte, F. Lederer, and A. Tuennermann, "Two-dimensional soliton in cubic fs laser written waveguide arrays in fused silica," Opt. Exp. 14, 6055-6062 (2006).
[CrossRef]

A. Szameit, J. Burghoff, T. Pertsch, S. Nolte, A. Tuennermann, and F. Lederer, "Discrete Nonlinear Localization in Femtosecond Laser Written Waveguides in Fused Silica," Opt. Exp. 13, 10,552-10,557 (2005).
[CrossRef]

Trutschel, U.

Tuennermann, A.

D. Bloemer, A. Szameit, F. Dreisow, J. Burghoff, T. Schreiber, T. Pertsch, S. Nolte, F. Lederer, and A. Tuennermann, "Measurement of the nonlinear refractive index of fs-laser-written waveguides in fused silica," Opt. Exp. 14, 2151-2157 (2006).
[CrossRef]

A. Szameit, D. Bloemer, J. Burghoff, T. Pertsch, S. Nolte, F. Lederer, and A. Tuennermann, "Two-dimensional soliton in cubic fs laser written waveguide arrays in fused silica," Opt. Exp. 14, 6055-6062 (2006).
[CrossRef]

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

A. Szameit, J. Burghoff, T. Pertsch, S. Nolte, A. Tuennermann, and F. Lederer, "Discrete Nonlinear Localization in Femtosecond Laser Written Waveguides in Fused Silica," Opt. Exp. 13, 10,552-10,557 (2005).
[CrossRef]

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

Watanabe, W.

K. Itoh, W. Watanabe, S. Nolte, and C. Schaffer, "Ultrafast Processes for Bulk Modification of Transparent Materials," MRS Bulletin 31, 620-625 (2006).
[CrossRef]

Will, M.

Withford, M.

M. Ams, G. Marshall, D. Spence, and M. Withford, "Slit-beam shaping method for femtosecond laser direct-write fabrication of symmetric waveguides in bulk glasses," Opt. Exp. 13, 5676-5681 (2005).
[CrossRef]

Appl. Phys. B. (1)

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

IEEE J. Quantum Electron. (1)

H. Haus and L. Molter-Orr, "Coupled multiple waveguide systems," IEEE J. Quantum Electron. 19, 840-844 (1983).
[CrossRef]

J. Lightwave Technol. (1)

I. Mansour and F. Caccavale, "An improved procedure to calculate the refractive index profile from the measured near-field intensity," J. Lightwave Technol. 14, 423-428 (1996).
[CrossRef]

MRS Bulletin (1)

K. Itoh, W. Watanabe, S. Nolte, and C. Schaffer, "Ultrafast Processes for Bulk Modification of Transparent Materials," MRS Bulletin 31, 620-625 (2006).
[CrossRef]

Nature (1)

J. Fleischer, M. Segev, N. Efremidis, and D. Christodoulides, "Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices," Nature 422, 147-150 (2003).
[CrossRef] [PubMed]

Opt. Exp. (4)

A. Szameit, D. Bloemer, J. Burghoff, T. Pertsch, S. Nolte, F. Lederer, and A. Tuennermann, "Two-dimensional soliton in cubic fs laser written waveguide arrays in fused silica," Opt. Exp. 14, 6055-6062 (2006).
[CrossRef]

A. Szameit, J. Burghoff, T. Pertsch, S. Nolte, A. Tuennermann, and F. Lederer, "Discrete Nonlinear Localization in Femtosecond Laser Written Waveguides in Fused Silica," Opt. Exp. 13, 10,552-10,557 (2005).
[CrossRef]

D. Bloemer, A. Szameit, F. Dreisow, J. Burghoff, T. Schreiber, T. Pertsch, S. Nolte, F. Lederer, and A. Tuennermann, "Measurement of the nonlinear refractive index of fs-laser-written waveguides in fused silica," Opt. Exp. 14, 2151-2157 (2006).
[CrossRef]

M. Ams, G. Marshall, D. Spence, and M. Withford, "Slit-beam shaping method for femtosecond laser direct-write fabrication of symmetric waveguides in bulk glasses," Opt. Exp. 13, 5676-5681 (2005).
[CrossRef]

Opt. Lett. (4)

Phys. Rev. Lett. (1)

H. Eisenberg, Y. Silberberg, R. Morandotti, A. Boyd, and J. Aitchison, "Discrete spatial optical solitons in waveguide arrays," Phys. Rev. Lett. 81, 3383-3386 (1998).
[CrossRef]

Other (3)

T. Pertsch, T. Zentgraf, U. Peschel, A. Braeuer, and F. Lederer, "Anomalous refraction and diffraction in discrete optical systems," Phys. Rev. Lett. 88, 0939,011-4 (2002).
[CrossRef]

A. Yariv, Optical Electronics, 4th ed. (Saunders College Publ., 1991).

H. Trompeter, W. Krolikowski, D. Neshev, A. Desyatnikov, A. Sukhorukov, Y. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, "Bloch Oscillations and Zener Tunneling in Two-Dimensional Photonic Lattices," Phys. Rev. Lett.  96, 0539,031-4 (2006).
[CrossRef]

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Figures (8)

Fig. 1.
Fig. 1.

Scheme of the writing process in a transparent bulk material by use of fs laser pulses. Inset: Measured profile of refractive index change.

Fig. 2.
Fig. 2.

a) Microscope image of waveguide pairs with a separation of 20 μm written in different orientations for the measurement of the coupling constant. b) Resulting output patterns at 633 nm for a propagation length of 10 mm. The excited waveguide is marked by a white circle.

Fig. 3.
Fig. 3.

(a) Coupling constant for different waveguide separations and different coupling angles at λ = 800 nm. (b) Coupling constant for different wavelengths and different waveguide separations at a orientation angle of 45°.

Fig. 4.
Fig. 4.

(a) Microscope images of square waveguide arrays with a waveguide separation of 18 μm exhibiting orientation angles 0°, 15°, 30° and 45°. (b-e) Corresponding (rotated) output patterns for excitation of the center waveguide at λ = 614 nm. (f-i) Corresponding calculated output patterns. The excited waveguide is marked by a white circle.

Fig. 5.
Fig. 5.

Left: Setup for the fabrication of cylindrical waveguides. Right: Comparison of elliptical and circular waveguides.

Fig. 6.
Fig. 6.

(a-e) Mode profile of a circular waveguide for different input wavelengths. Also the measured profile of the refractive index change at λ = 633 nm is shown.

Fig. 7.
Fig. 7.

(a) Coupling constant for different waveguide separations and different coupling angles for cylindrical waveguides at λ = 800 nm. (b) Comparison of the dependence of the coupling on the direction at 18 μm waveguide separation.

Fig. 8.
Fig. 8.

(a) Microscope image of a circular waveguide array with a diameter of 286 μm. (b) Measured intensity output pattern for λ = 633 nm and (c) corresponding theoretical calculation assuming isotropic coupling. The excited waveguide is marked by a white circle.

Equations (12)

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w 0 = M 2 λ π NA
b = M 2 λ πN A 2 .
n 2 ( x , y ) = n eff 2 λ 2 4 π 2 Δ A ( x , y ) A ( x , y ) ,
i d dz a n ( z ) + c ( a n + 1 ( z ) + a n 1 ( z ) ) = 0
c ~ ω Δ ε n + 1 E n ( x , y ) E n + 1 ( x , y ) dxdy ,
i d dz A ( z ) + cB ( z ) = 0
i d dz B ( z ) + cA ( z ) = 0 .
A ( z ) = cos cz
B ( z ) = i sin cz .
c = 1 z arctan I B I A
l c = π 2 c .
R = p 1 ε cos φ

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