Abstract

We present an approach for the realization of reflectionless potentials in linear coupled-resonator optical waveguides or optical waveguide arrays with modulated coupling. We show that soliton solutions of nonlinear Ablowitz–Ladik equations can be used to define analytically reflectionless modulations for linear photonic structures. Such structures fully transmit incident waves and simultaneously support a pair of localized modes. These results suggest new possibilities for pulse and beam shaping, including a realization of cavities that do not reflect incident light.

© 2010 Optical Society of America

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2008 (2)

M. Notomi, E. Kuramochi, and T. Tanabe, Nat. Photonics 2, 741 (2008).
[CrossRef]

M. I. Molina and Yu. S. Kivshar, Opt. Lett. 33, 917 (2008).
[CrossRef] [PubMed]

2007 (1)

2006 (2)

A. A. Sukhorukov, Phys. Rev. Lett. 96, 113902 (2006).
[CrossRef] [PubMed]

A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications, 6th ed. (Oxford U. Press, 2006).

2005 (1)

2003 (3)

2002 (1)

2001 (1)

A. A. Sukhorukov and Yu. S. Kivshar, Phys. Rev. Lett. 87, 083901 (2001).
[CrossRef] [PubMed]

1999 (2)

M. J. Ablowitz, Y. Ohta, and A. D. Trubatch, Phys. Lett. A 253, 287 (1999).
[CrossRef]

A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, Opt. Lett. 24, 711 (1999).
[CrossRef]

1996 (2)

W. Krolikowski and Yu. S. Kivshar, J. Opt. Soc. Am. B 13, 876 (1996).
[CrossRef]

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, Phys. Rev. E 53, 1172 (1996).
[CrossRef]

1956 (1)

I. Kay and H. E. Moses, J. Appl. Phys. 27, 1503 (1956).
[CrossRef]

Ablowitz, M. J.

M. J. Ablowitz, Y. Ohta, and A. D. Trubatch, Phys. Lett. A 253, 287 (1999).
[CrossRef]

Aceves, A. B.

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, Phys. Rev. E 53, 1172 (1996).
[CrossRef]

Agarwal, G. S.

Aitchison, J. S.

Brauer, A.

Chen, Z. G.

Christodoulides, D. N.

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
[CrossRef] [PubMed]

D. N. Christodoulides and N. K. Efremidis, Opt. Lett. 27, 568 (2002).
[CrossRef]

De Angelis, C.

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, Phys. Rev. E 53, 1172 (1996).
[CrossRef]

Efremidis, N. K.

Eisenberg, H. S.

Fedele, F.

Gupta, S. D.

Kay, I.

I. Kay and H. E. Moses, J. Appl. Phys. 27, 1503 (1956).
[CrossRef]

Kivshar, Yu. S.

Krolikowski, W.

Kuramochi, E.

M. Notomi, E. Kuramochi, and T. Tanabe, Nat. Photonics 2, 741 (2008).
[CrossRef]

Lederer, F.

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
[CrossRef] [PubMed]

H. Trompeter, U. Peschel, T. Pertsch, F. Lederer, U. Streppel, D. Michaelis, and A. Brauer, Opt. Express 11, 3404 (2003).
[CrossRef] [PubMed]

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, Phys. Rev. E 53, 1172 (1996).
[CrossRef]

Lee, R. K.

Mandelik, D.

Michaelis, D.

Modotto, D.

Molina, M. I.

Morandotti, R.

Moses, H. E.

I. Kay and H. E. Moses, J. Appl. Phys. 27, 1503 (1956).
[CrossRef]

Muschall, R.

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, Phys. Rev. E 53, 1172 (1996).
[CrossRef]

Notomi, M.

M. Notomi, E. Kuramochi, and T. Tanabe, Nat. Photonics 2, 741 (2008).
[CrossRef]

Ohta, Y.

M. J. Ablowitz, Y. Ohta, and A. D. Trubatch, Phys. Lett. A 253, 287 (1999).
[CrossRef]

Pertsch, T.

Peschel, T.

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, Phys. Rev. E 53, 1172 (1996).
[CrossRef]

Peschel, U.

Scherer, A.

Silberberg, Y.

Sorel, M.

Stanley, C. R.

Streppel, U.

Sukhorukov, A. A.

A. A. Sukhorukov, Phys. Rev. Lett. 96, 113902 (2006).
[CrossRef] [PubMed]

A. A. Sukhorukov and Yu. S. Kivshar, Phys. Rev. Lett. 87, 083901 (2001).
[CrossRef] [PubMed]

Tanabe, T.

M. Notomi, E. Kuramochi, and T. Tanabe, Nat. Photonics 2, 741 (2008).
[CrossRef]

Trillo, S.

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, Phys. Rev. E 53, 1172 (1996).
[CrossRef]

Trompeter, H.

Trubatch, A. D.

M. J. Ablowitz, Y. Ohta, and A. D. Trubatch, Phys. Lett. A 253, 287 (1999).
[CrossRef]

Wabnitz, S.

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, Phys. Rev. E 53, 1172 (1996).
[CrossRef]

Xu, Y.

Yang, J. K.

Yariv, A.

A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications, 6th ed. (Oxford U. Press, 2006).

A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, Opt. Lett. 24, 711 (1999).
[CrossRef]

Yeh, P.

A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications, 6th ed. (Oxford U. Press, 2006).

J. Appl. Phys. (1)

I. Kay and H. E. Moses, J. Appl. Phys. 27, 1503 (1956).
[CrossRef]

J. Opt. Soc. Am. B (1)

Nat. Photonics (1)

M. Notomi, E. Kuramochi, and T. Tanabe, Nat. Photonics 2, 741 (2008).
[CrossRef]

Nature (1)

D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature 424, 817 (2003).
[CrossRef] [PubMed]

Opt. Express (2)

Opt. Lett. (5)

Phys. Lett. A (1)

M. J. Ablowitz, Y. Ohta, and A. D. Trubatch, Phys. Lett. A 253, 287 (1999).
[CrossRef]

Phys. Rev. E (1)

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, Phys. Rev. E 53, 1172 (1996).
[CrossRef]

Phys. Rev. Lett. (2)

A. A. Sukhorukov and Yu. S. Kivshar, Phys. Rev. Lett. 87, 083901 (2001).
[CrossRef] [PubMed]

A. A. Sukhorukov, Phys. Rev. Lett. 96, 113902 (2006).
[CrossRef] [PubMed]

Other (1)

A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications, 6th ed. (Oxford U. Press, 2006).

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Figures (3)

Fig. 1
Fig. 1

Top row, schematic of coupled resonators with (a) homogeneous arrangement and (b), (c) modulated coupling: (c) reflectionless modulation based on Eqs. (2, 3) with C = 1 , μ = 1 , and n 0 = 0.5 ; (b) nonreflectionless modulation based on same equations and parameters as (c) but with the values of ψ n reduced by a factor of 2, such that the modulation is weaker compared with (c). Second row, corresponding dependencies on wavenumber of the amplitude and phase of transmission coefficient T ( k ) . Bottom row, propagation of tilled beams through the waveguide arrays.

Fig. 2
Fig. 2

Top, examples of reflectionless modulations corresponding to different values of μ as indicated in the legend. Bottom, corresponding phases of transmission coefficients versus the wavenumber. For all the plots C = 1 and n 0 = 0.5 .

Fig. 3
Fig. 3

Top, schematic of coupled wave guides with different modulation positions: (a) n 0 = 0 and (b) n 0 = 0.5 . Second row, corresponding profiles of two localized modes. Third row, beam propagation for an initial excitation of waveguide number n = 0 .

Equations (4)

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i E n z + C n , n 1 E n 1 + C n , n + 1 E n + 1 + V n E n = 0 .
C n , n + 1 = C n + 1 , n = ( C + | ψ n | 2 ) ( C + | ψ n + 1 | 2 ) .
ψ n ( z ) = C sinh ( μ ) sech [ μ ( n n 0 ) ] exp ( i β z ) ,
E n = ψ n C + | ψ n | 2

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