Abstract

We describe experimental and theoretical results of research on a new type of waveguide, the so-called grating-mediated waveguide (GMW) recently reported by our group. This waveguide structure relies on Bragg diffractions from a 1D grating giving rise to wave guiding in the direction normal to the grating wave vector. The structure consists of a shallow 1D grating having a bell- or trough-shaped amplitude in the confinement direction. We provide the theoretical analysis of the underlying wave-guiding mechanism along with experimental evidence for both the bell- and the trough-shaped waveguides. We investigate the robustness of grating-mediated wave guiding and suggest more elaborate, 2D structures, such as a GMW superlattice and a grating-mediated ring waveguide. Finally we discuss the relation between grating-mediated wave guiding and holographic solitons, which are the beams that are self-trapped solely by virtue of their jointly induced grating.

© 2005 Optical Society of America

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  1. G. P. Agrawal, Fiber-Optic Communication Systems, 2nd ed. (Wiley, 1997).
  2. V. I. Kopp, V. M. Churikov, J. Singer, N. Chao, D. Neugroschl, and A. Z. Genack, "Chiral fiber gratings," Science 305, 74-75 (2004).
    [CrossRef] [PubMed]
  3. J. C. Knight, J. Broeng, T. A. Birks, and P. St. J. Russell, "Photonic band gap guidance in optical fibers," Science 282, 1476-1478 (1998).
    [CrossRef] [PubMed]
  4. D. Collodon and J. Babinet, Comptes Rendes 15, 800 (1842).
  5. A. Yariv, Optical Electronics in Modern Communications (Oxford, 1997).
  6. P. Yeh and A. Yariv, "Bragg reflection waveguides," Opt. Commun. 19, 427-430 (1976).
    [CrossRef]
  7. M. Ibanescu, Y. Fink, S. Fan, E. L. Thomas, and J. D. Joannopoulos, "An all-dielectric coaxial waveguide," Science 289, 415-419 (2000).
    [CrossRef] [PubMed]
  8. P. St. J. Russell, "Photonic crystal fibers," Science 299, 358-362 (2003).
    [CrossRef] [PubMed]
  9. A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, "Coupled-resonator optical waveguide: a proposal and analysis," Opt. Lett. 24, 711-713 (1999).
    [CrossRef]
  10. O. Cohen, B. Freedman, J. W. Fleischer, M. Segev, and D. N. Christodoulides, "Grating-mediated waveguiding," Phys. Rev. Lett. 93, 103902 (2004).
    [CrossRef] [PubMed]
  11. N. W. Ashcroft and N. D. Mermin, Solid State Physics, (Saunders, 1976).
  12. In general, the grating modes at the edge of the first Brillouin zone that belong to the odd bands can be written as an infinite sum Sum=0∞amcos[(2m+1)pix/d]. Here the grating is shallow; hence we neglect the coupling to higher bands and approximate the grating mode of the first band as cos(pix/d). Similarly, we approximate the grating mode of the second band at the edge of the Brillouin zone as sin(pix/d).
  13. A. L. Jones, "Coupling of optical fibers and scattering in fibers," J. Opt. Soc. Am. 55, 261-271 (1965).
    [CrossRef]
  14. S. Somekh, E. Garmire, A. Yariv, H. L. Garvin, and R. G. Hunsperger, "Channel optical waveguide directional couplers," Appl. Phys. Lett. 22, 46-48 (1973).
    [CrossRef]
  15. H. Eisenberg, Y. Silberberg, R. Morandotti, and J. Aitchison, "Diffraction management," Phys. Rev. Lett. 85, 1863-1866 (2000).
    [CrossRef] [PubMed]
  16. M. J. Ablowitz and Z. H. Musslimani, "Discrete diffraction managed spatial solitons," Phys. Rev. Lett. 87, 254102 (2001).
    [CrossRef] [PubMed]
  17. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, "Rotary solitons in Bessel optical lattices," Phys. Rev. Lett. 93, 093940 (2004).
  18. D. N. Christodoulides and R. I. Joseph, "Discrete self-focusing in nonlinear arrays of coupled waveguides," Opt. Lett. 13, 794-796 (1988).
    [CrossRef] [PubMed]
  19. D. N. Christodoulides, F. Lederer, and Y. Silberberg, "Discretizing light behavior in linear and nonlinear waveguide lattices," Nature 424, 817-823 (2003).
    [CrossRef] [PubMed]
  20. 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]
  21. J. Fleischer, M. Segev, N. Efremidis, and D. N. Christodoulides, "Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices," Nature 422, 147-150 (2003).
    [CrossRef] [PubMed]
  22. O. Cohen, T. Carmon, M. Segev, and S. Odoulov, "Holographic solitons," Opt. Lett. 27, 2031-2033 (2002).
    [CrossRef]
  23. M. Vaupel, C. Seror, and R. Dykstra, "Self-focusing in photorefractive two-wave mixing," Opt. Lett. 22, 1470-1472 (1997).
    [CrossRef]
  24. R. J. D. Miller, M. Pierre, T. S. Rose, and M. D. Fayer, "A coherent photoacoustic approach to excited-state-excited-state absorption spectroscopy: application to the investigation of a near-resonant contribution to ultrasonic diffraction," J. Phys. Chem. 88, 3021-3025 (1984).
    [CrossRef]
  25. M. Greiner, O. Mandel, T. W. Hansch, and I. Bloch, "Collapse and revival of the matter wave field of a Bose-Einstein condensate," Nature 419, 51-54 (2002).
    [CrossRef] [PubMed]

2004 (3)

V. I. Kopp, V. M. Churikov, J. Singer, N. Chao, D. Neugroschl, and A. Z. Genack, "Chiral fiber gratings," Science 305, 74-75 (2004).
[CrossRef] [PubMed]

O. Cohen, B. Freedman, J. W. Fleischer, M. Segev, and D. N. Christodoulides, "Grating-mediated waveguiding," Phys. Rev. Lett. 93, 103902 (2004).
[CrossRef] [PubMed]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, "Rotary solitons in Bessel optical lattices," Phys. Rev. Lett. 93, 093940 (2004).

2003 (3)

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

P. St. J. Russell, "Photonic crystal fibers," Science 299, 358-362 (2003).
[CrossRef] [PubMed]

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

2002 (2)

M. Greiner, O. Mandel, T. W. Hansch, and I. Bloch, "Collapse and revival of the matter wave field of a Bose-Einstein condensate," Nature 419, 51-54 (2002).
[CrossRef] [PubMed]

O. Cohen, T. Carmon, M. Segev, and S. Odoulov, "Holographic solitons," Opt. Lett. 27, 2031-2033 (2002).
[CrossRef]

2001 (1)

M. J. Ablowitz and Z. H. Musslimani, "Discrete diffraction managed spatial solitons," Phys. Rev. Lett. 87, 254102 (2001).
[CrossRef] [PubMed]

2000 (2)

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

M. Ibanescu, Y. Fink, S. Fan, E. L. Thomas, and J. D. Joannopoulos, "An all-dielectric coaxial waveguide," Science 289, 415-419 (2000).
[CrossRef] [PubMed]

1999 (1)

1998 (2)

J. C. Knight, J. Broeng, T. A. Birks, and P. St. J. Russell, "Photonic band gap guidance in optical fibers," Science 282, 1476-1478 (1998).
[CrossRef] [PubMed]

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]

1997 (1)

1988 (1)

1984 (1)

R. J. D. Miller, M. Pierre, T. S. Rose, and M. D. Fayer, "A coherent photoacoustic approach to excited-state-excited-state absorption spectroscopy: application to the investigation of a near-resonant contribution to ultrasonic diffraction," J. Phys. Chem. 88, 3021-3025 (1984).
[CrossRef]

1976 (1)

P. Yeh and A. Yariv, "Bragg reflection waveguides," Opt. Commun. 19, 427-430 (1976).
[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, 46-48 (1973).
[CrossRef]

1965 (1)

1842 (1)

D. Collodon and J. Babinet, Comptes Rendes 15, 800 (1842).

Ablowitz, M. J.

M. J. Ablowitz and Z. H. Musslimani, "Discrete diffraction managed spatial solitons," Phys. Rev. Lett. 87, 254102 (2001).
[CrossRef] [PubMed]

Agrawal, G. P.

G. P. Agrawal, Fiber-Optic Communication Systems, 2nd ed. (Wiley, 1997).

Aitchison, J.

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

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]

Ashcroft, N. W.

N. W. Ashcroft and N. D. Mermin, Solid State Physics, (Saunders, 1976).

Babinet, J.

D. Collodon and J. Babinet, Comptes Rendes 15, 800 (1842).

Birks, T. A.

J. C. Knight, J. Broeng, T. A. Birks, and P. St. J. Russell, "Photonic band gap guidance in optical fibers," Science 282, 1476-1478 (1998).
[CrossRef] [PubMed]

Bloch, I.

M. Greiner, O. Mandel, T. W. Hansch, and I. Bloch, "Collapse and revival of the matter wave field of a Bose-Einstein condensate," Nature 419, 51-54 (2002).
[CrossRef] [PubMed]

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]

Broeng, J.

J. C. Knight, J. Broeng, T. A. Birks, and P. St. J. Russell, "Photonic band gap guidance in optical fibers," Science 282, 1476-1478 (1998).
[CrossRef] [PubMed]

Carmon, T.

Chao, N.

V. I. Kopp, V. M. Churikov, J. Singer, N. Chao, D. Neugroschl, and A. Z. Genack, "Chiral fiber gratings," Science 305, 74-75 (2004).
[CrossRef] [PubMed]

Christodoulides, D. N.

O. Cohen, B. Freedman, J. W. Fleischer, M. Segev, and D. N. Christodoulides, "Grating-mediated waveguiding," Phys. Rev. Lett. 93, 103902 (2004).
[CrossRef] [PubMed]

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

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

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

Churikov, V. M.

V. I. Kopp, V. M. Churikov, J. Singer, N. Chao, D. Neugroschl, and A. Z. Genack, "Chiral fiber gratings," Science 305, 74-75 (2004).
[CrossRef] [PubMed]

Cohen, O.

O. Cohen, B. Freedman, J. W. Fleischer, M. Segev, and D. N. Christodoulides, "Grating-mediated waveguiding," Phys. Rev. Lett. 93, 103902 (2004).
[CrossRef] [PubMed]

O. Cohen, T. Carmon, M. Segev, and S. Odoulov, "Holographic solitons," Opt. Lett. 27, 2031-2033 (2002).
[CrossRef]

Collodon, D.

D. Collodon and J. Babinet, Comptes Rendes 15, 800 (1842).

Dykstra, R.

Efremidis, N.

J. Fleischer, M. Segev, N. Efremidis, and D. N. 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, and J. Aitchison, "Diffraction management," Phys. Rev. Lett. 85, 1863-1866 (2000).
[CrossRef] [PubMed]

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]

Fan, S.

M. Ibanescu, Y. Fink, S. Fan, E. L. Thomas, and J. D. Joannopoulos, "An all-dielectric coaxial waveguide," Science 289, 415-419 (2000).
[CrossRef] [PubMed]

Fayer, M. D.

R. J. D. Miller, M. Pierre, T. S. Rose, and M. D. Fayer, "A coherent photoacoustic approach to excited-state-excited-state absorption spectroscopy: application to the investigation of a near-resonant contribution to ultrasonic diffraction," J. Phys. Chem. 88, 3021-3025 (1984).
[CrossRef]

Fink, Y.

M. Ibanescu, Y. Fink, S. Fan, E. L. Thomas, and J. D. Joannopoulos, "An all-dielectric coaxial waveguide," Science 289, 415-419 (2000).
[CrossRef] [PubMed]

Fleischer, J.

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

Fleischer, J. W.

O. Cohen, B. Freedman, J. W. Fleischer, M. Segev, and D. N. Christodoulides, "Grating-mediated waveguiding," Phys. Rev. Lett. 93, 103902 (2004).
[CrossRef] [PubMed]

Freedman, B.

O. Cohen, B. Freedman, J. W. Fleischer, M. Segev, and D. N. Christodoulides, "Grating-mediated waveguiding," Phys. Rev. Lett. 93, 103902 (2004).
[CrossRef] [PubMed]

Garmire, E.

S. Somekh, E. Garmire, A. Yariv, H. L. Garvin, and R. G. Hunsperger, "Channel optical waveguide directional couplers," Appl. Phys. Lett. 22, 46-48 (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, 46-48 (1973).
[CrossRef]

Genack, A. Z.

V. I. Kopp, V. M. Churikov, J. Singer, N. Chao, D. Neugroschl, and A. Z. Genack, "Chiral fiber gratings," Science 305, 74-75 (2004).
[CrossRef] [PubMed]

Greiner, M.

M. Greiner, O. Mandel, T. W. Hansch, and I. Bloch, "Collapse and revival of the matter wave field of a Bose-Einstein condensate," Nature 419, 51-54 (2002).
[CrossRef] [PubMed]

Hansch, T. W.

M. Greiner, O. Mandel, T. W. Hansch, and I. Bloch, "Collapse and revival of the matter wave field of a Bose-Einstein condensate," Nature 419, 51-54 (2002).
[CrossRef] [PubMed]

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, 46-48 (1973).
[CrossRef]

Ibanescu, M.

M. Ibanescu, Y. Fink, S. Fan, E. L. Thomas, and J. D. Joannopoulos, "An all-dielectric coaxial waveguide," Science 289, 415-419 (2000).
[CrossRef] [PubMed]

Joannopoulos, J. D.

M. Ibanescu, Y. Fink, S. Fan, E. L. Thomas, and J. D. Joannopoulos, "An all-dielectric coaxial waveguide," Science 289, 415-419 (2000).
[CrossRef] [PubMed]

Jones, A. L.

Joseph, R. I.

Kartashov, Y. V.

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, "Rotary solitons in Bessel optical lattices," Phys. Rev. Lett. 93, 093940 (2004).

Knight, J. C.

J. C. Knight, J. Broeng, T. A. Birks, and P. St. J. Russell, "Photonic band gap guidance in optical fibers," Science 282, 1476-1478 (1998).
[CrossRef] [PubMed]

Kopp, V. I.

V. I. Kopp, V. M. Churikov, J. Singer, N. Chao, D. Neugroschl, and A. Z. Genack, "Chiral fiber gratings," Science 305, 74-75 (2004).
[CrossRef] [PubMed]

Lederer, F.

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

Lee, R. K.

Mandel, O.

M. Greiner, O. Mandel, T. W. Hansch, and I. Bloch, "Collapse and revival of the matter wave field of a Bose-Einstein condensate," Nature 419, 51-54 (2002).
[CrossRef] [PubMed]

Mermin, N. D.

N. W. Ashcroft and N. D. Mermin, Solid State Physics, (Saunders, 1976).

Miller, R. J. D.

R. J. D. Miller, M. Pierre, T. S. Rose, and M. D. Fayer, "A coherent photoacoustic approach to excited-state-excited-state absorption spectroscopy: application to the investigation of a near-resonant contribution to ultrasonic diffraction," J. Phys. Chem. 88, 3021-3025 (1984).
[CrossRef]

Morandotti, R.

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

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]

Musslimani, Z. H.

M. J. Ablowitz and Z. H. Musslimani, "Discrete diffraction managed spatial solitons," Phys. Rev. Lett. 87, 254102 (2001).
[CrossRef] [PubMed]

Neugroschl, D.

V. I. Kopp, V. M. Churikov, J. Singer, N. Chao, D. Neugroschl, and A. Z. Genack, "Chiral fiber gratings," Science 305, 74-75 (2004).
[CrossRef] [PubMed]

Odoulov, S.

Pierre, M.

R. J. D. Miller, M. Pierre, T. S. Rose, and M. D. Fayer, "A coherent photoacoustic approach to excited-state-excited-state absorption spectroscopy: application to the investigation of a near-resonant contribution to ultrasonic diffraction," J. Phys. Chem. 88, 3021-3025 (1984).
[CrossRef]

Rose, T. S.

R. J. D. Miller, M. Pierre, T. S. Rose, and M. D. Fayer, "A coherent photoacoustic approach to excited-state-excited-state absorption spectroscopy: application to the investigation of a near-resonant contribution to ultrasonic diffraction," J. Phys. Chem. 88, 3021-3025 (1984).
[CrossRef]

Russell, P. St. J.

P. St. J. Russell, "Photonic crystal fibers," Science 299, 358-362 (2003).
[CrossRef] [PubMed]

J. C. Knight, J. Broeng, T. A. Birks, and P. St. J. Russell, "Photonic band gap guidance in optical fibers," Science 282, 1476-1478 (1998).
[CrossRef] [PubMed]

Scherer, A.

Segev, M.

O. Cohen, B. Freedman, J. W. Fleischer, M. Segev, and D. N. Christodoulides, "Grating-mediated waveguiding," Phys. Rev. Lett. 93, 103902 (2004).
[CrossRef] [PubMed]

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

O. Cohen, T. Carmon, M. Segev, and S. Odoulov, "Holographic solitons," Opt. Lett. 27, 2031-2033 (2002).
[CrossRef]

Seror, C.

Silberberg, Y.

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

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

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]

Singer, J.

V. I. Kopp, V. M. Churikov, J. Singer, N. Chao, D. Neugroschl, and A. Z. Genack, "Chiral fiber gratings," Science 305, 74-75 (2004).
[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, 46-48 (1973).
[CrossRef]

Thomas, E. L.

M. Ibanescu, Y. Fink, S. Fan, E. L. Thomas, and J. D. Joannopoulos, "An all-dielectric coaxial waveguide," Science 289, 415-419 (2000).
[CrossRef] [PubMed]

Torner, L.

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, "Rotary solitons in Bessel optical lattices," Phys. Rev. Lett. 93, 093940 (2004).

Vaupel, M.

Vysloukh, V. A.

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, "Rotary solitons in Bessel optical lattices," Phys. Rev. Lett. 93, 093940 (2004).

Xu, Y.

Yariv, A.

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

P. Yeh and A. Yariv, "Bragg reflection waveguides," Opt. Commun. 19, 427-430 (1976).
[CrossRef]

S. Somekh, E. Garmire, A. Yariv, H. L. Garvin, and R. G. Hunsperger, "Channel optical waveguide directional couplers," Appl. Phys. Lett. 22, 46-48 (1973).
[CrossRef]

A. Yariv, Optical Electronics in Modern Communications (Oxford, 1997).

Yeh, P.

P. Yeh and A. Yariv, "Bragg reflection waveguides," Opt. Commun. 19, 427-430 (1976).
[CrossRef]

Appl. Phys. Lett. (1)

S. Somekh, E. Garmire, A. Yariv, H. L. Garvin, and R. G. Hunsperger, "Channel optical waveguide directional couplers," Appl. Phys. Lett. 22, 46-48 (1973).
[CrossRef]

Comptes Rendes (1)

D. Collodon and J. Babinet, Comptes Rendes 15, 800 (1842).

J. Opt. Soc. Am. (1)

J. Phys. Chem. (1)

R. J. D. Miller, M. Pierre, T. S. Rose, and M. D. Fayer, "A coherent photoacoustic approach to excited-state-excited-state absorption spectroscopy: application to the investigation of a near-resonant contribution to ultrasonic diffraction," J. Phys. Chem. 88, 3021-3025 (1984).
[CrossRef]

Nature (3)

M. Greiner, O. Mandel, T. W. Hansch, and I. Bloch, "Collapse and revival of the matter wave field of a Bose-Einstein condensate," Nature 419, 51-54 (2002).
[CrossRef] [PubMed]

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

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

Opt. Commun. (1)

P. Yeh and A. Yariv, "Bragg reflection waveguides," Opt. Commun. 19, 427-430 (1976).
[CrossRef]

Opt. Lett. (4)

Phys. Rev. Lett. (5)

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]

O. Cohen, B. Freedman, J. W. Fleischer, M. Segev, and D. N. Christodoulides, "Grating-mediated waveguiding," Phys. Rev. Lett. 93, 103902 (2004).
[CrossRef] [PubMed]

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

M. J. Ablowitz and Z. H. Musslimani, "Discrete diffraction managed spatial solitons," Phys. Rev. Lett. 87, 254102 (2001).
[CrossRef] [PubMed]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, "Rotary solitons in Bessel optical lattices," Phys. Rev. Lett. 93, 093940 (2004).

Science (4)

M. Ibanescu, Y. Fink, S. Fan, E. L. Thomas, and J. D. Joannopoulos, "An all-dielectric coaxial waveguide," Science 289, 415-419 (2000).
[CrossRef] [PubMed]

P. St. J. Russell, "Photonic crystal fibers," Science 299, 358-362 (2003).
[CrossRef] [PubMed]

V. I. Kopp, V. M. Churikov, J. Singer, N. Chao, D. Neugroschl, and A. Z. Genack, "Chiral fiber gratings," Science 305, 74-75 (2004).
[CrossRef] [PubMed]

J. C. Knight, J. Broeng, T. A. Birks, and P. St. J. Russell, "Photonic band gap guidance in optical fibers," Science 282, 1476-1478 (1998).
[CrossRef] [PubMed]

Other (4)

A. Yariv, Optical Electronics in Modern Communications (Oxford, 1997).

N. W. Ashcroft and N. D. Mermin, Solid State Physics, (Saunders, 1976).

In general, the grating modes at the edge of the first Brillouin zone that belong to the odd bands can be written as an infinite sum Sum=0∞amcos[(2m+1)pix/d]. Here the grating is shallow; hence we neglect the coupling to higher bands and approximate the grating mode of the first band as cos(pix/d). Similarly, we approximate the grating mode of the second band at the edge of the Brillouin zone as sin(pix/d).

G. P. Agrawal, Fiber-Optic Communication Systems, 2nd ed. (Wiley, 1997).

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Figures (8)

Fig. 1
Fig. 1

Schematic index structure of (a) Type I (bell-shaped) and (b) Type II (trough-shaped) GMWs.

Fig. 2
Fig. 2

(a) Index grating with three different amplitudes corresponding to three different y planes (“layers”) in GMWs. (b) The dispersion curves (propagation constant β versus the transverse momentum k x ) near the edge of the first Brillouin zone for the gratings shown in (a). The inset shows the dispersion curves at the edge of the first Brillouin zone. Typical grating index amplitudes of (c) Type I and (d) Type II GMWs. (e) The effective waveguide structure in y; Type I beams need a cos ( π x d ) dependence, Type II beams need a sin ( π x d ) dependence to experience Bragg-mediated wave guiding.

Fig. 3
Fig. 3

Propagation of different beams in a Type I (bell-shaped) GMW, each with a y profile of the first guided mode. Shown are the intensities at the propagation planes z = ( a ) 0 , (b) 2 3 cm , and (c) 2 cm for a beam uniform in x, (d)–(f) a beam with cos ( π x d ) dependence, and (g)–(i) a beam with exp ( i π x d ) dependence. Note that for the exponential profile, the sine component radiates while the cosine component is trapped.

Fig. 4
Fig. 4

Propagation of different beams in a Type II (trough-shaped) GMW, each with a y profile of the first guided mode. Shown are the intensities at the propagation planes z = ( a ) 0 , (b) 2 3 cm , and (c) 2 cm for a beam uniform in x (d)–(f), a beam with sin ( π x d ) dependence, and (g)–(i) a beam with cos ( π x d ) dependence.

Fig. 5
Fig. 5

(a) Schematic of the optically-induced technique we use to obtain a Type II GMW. Photographs of a probe beam at the (b) input and (c) output of the waveguide providing experimental proof of concept for GMW. Diffraction in unguided conditions: (d) a homogeneous medium, (e) when the beams are not Bragg-matched with the index grating, (f) when the beams are Bragg-matched with the grating but have the “wrong” phase relative to the grating. In (b)–(f) the intensity in each figure is normalized to its own peak intensity.

Fig. 6
Fig. 6

Same as Fig. 5 but for a Type I GMW.

Fig. 7
Fig. 7

Detuning in x direction of a probe entering a GMW. (a) Shifting of each beam by an equal opposite angle δ, (b) shifting both beams together by an angle Δ in one direction relative to the waveguide grating. Numerical results showing the guided modes for (c) Type I and (d) Type II GMW with detuning of 2.5%, both for a type of detuning described in (b).

Fig. 8
Fig. 8

Discrete diffraction in a simulated 2D GMW superlattice. The phase of the probe itself relative to the initial superlattice layer determines whether that particular layer of the superlattice will act as a waveguide.

Equations (6)

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n ( x , y ) = n 0 [ 1 + ε A ( y ) cos ( 2 π x d ) ] .
2 Ψ x 2 + 2 Ψ y 2 + ( k 0 2 n 2 β 2 ) Ψ = 0 ,
cos ( π x d ) sin ( π x d ) { Φ + Φ [ k 0 2 n 0 2 ( π d ) 2 β 2 + 2 ε A ( y ) k 0 2 n 0 2 cos ( 2 π x d ) ] } = 0 .
Φ + Φ [ k 0 2 n 0 2 ( π d ) 2 β 2 ± ε k 0 2 n 0 2 A ( y ) ] = 0 ,
Δ n I 1 cos ( 2 π x d + π 2 ) [ I 1 + I 2 exp ( y 2 y 0 2 ) ] .
Δ n I 1 cos ( 2 π x d + π 2 ) [ I 1 + I 2 tanh 2 ( y 2 y 0 2 ) ] .

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