Abstract

It is shown that asymmetric waveguides with gain and loss can support a stable propagation of optical beams. This means that the propagation constants of modes of the corresponding complex optical potential are real. A class of such waveguides is found from a relation between two spectral problems. A particular example of an asymmetric waveguide, described by the hyperbolic functions, is analyzed. The existence and stability of linear modes and of continuous families of nonlinear modes are demonstrated.

© 2014 Optical Society of America

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  1. Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, Phys. Rev. Lett. 100, 030402 (2008).
    [CrossRef]
  2. A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, Phys. Rev. Lett. 103, 093902 (2009).
    [CrossRef]
  3. C. M. Bender, Rep. Prog. Phys. 70, 947 (2007).
    [CrossRef]
  4. M. Wadati, J. Phys. Soc. Jpn. 77, 074005 (2008).
    [CrossRef]
  5. C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192 (2010).
    [CrossRef]
  6. F. Kh. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zezyulin, Phys. Rev. A 83, 041805(R) (2011).
    [CrossRef]
  7. E. N. Tsoy, S. Sh. Tadjimuratov, and F. Kh. Abdullaev, Opt. Commun. 285, 3441 (2012).
    [CrossRef]
  8. Z. Lin, J. Schindler, F. M. Ellis, and T. Kottos, Phys. Rev. A 85, 050101(R) (2012).
    [CrossRef]
  9. M.-A. Miri, M. Heinrich, and D. N. Christodoulides, Phys. Rev. A 87, 043819 (2013).
    [CrossRef]
  10. J. Yang, Phys. Lett. A 378, 367 (2014).
    [CrossRef]
  11. F. Cooper, A. Khare, and U. Sukhatme, Phys. Rep. 251, 267 (1995).
    [CrossRef]
  12. A. A. Andrianov, M. V. Ioffe, F. Cannata, and J. P. Dedonder, Int. J. Mod. Phys. A 14, 2675 (1999).
    [CrossRef]
  13. G. Lévai and M. Znojil, J. Phys. A 35, 8793 (2002).
    [CrossRef]
  14. Yu. S. Kivshar and G. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003).
  15. S. P. Novikov, S. V. Manakov, L. P. Pitaevskii, and V. E. Zakharov, Theory of Solitons: The Inverse Scattering Method (Springer-Verlag, 1984).
  16. G. L. Lamb, Elements of Soliton Theory (Wiley, 1980).
  17. S. V. Manakov, Zh. Eksp. Teor. Fiz. 65, 1392 (1973) [Sov. Phys. JETP 38, 693 (1974)].
  18. J. Satsuma and N. Yajima, Suppl. Prog. Theor. Phys. 55, 284 (1974).
    [CrossRef]
  19. Z. Ahmed, Phys. Lett. A 282, 343 (2001).
    [CrossRef]
  20. M. Klaus and J. K. Shaw, Phys. Rev. E 65, 036607 (2002).
    [CrossRef]
  21. E. N. Tsoy and F. Kh. Abdullaev, Phys. Rev. E 67, 056610 (2003).
    [CrossRef]
  22. T. Pang, An Introduction to Computational Physics (Cambridge University, 2006).
  23. N. Akhmediev and A. Ankiewicz, eds., Lecture Notes in Physics: Dissipative Solitons (Springer, 2005).

2014 (1)

J. Yang, Phys. Lett. A 378, 367 (2014).
[CrossRef]

2013 (1)

M.-A. Miri, M. Heinrich, and D. N. Christodoulides, Phys. Rev. A 87, 043819 (2013).
[CrossRef]

2012 (2)

E. N. Tsoy, S. Sh. Tadjimuratov, and F. Kh. Abdullaev, Opt. Commun. 285, 3441 (2012).
[CrossRef]

Z. Lin, J. Schindler, F. M. Ellis, and T. Kottos, Phys. Rev. A 85, 050101(R) (2012).
[CrossRef]

2011 (1)

F. Kh. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zezyulin, Phys. Rev. A 83, 041805(R) (2011).
[CrossRef]

2010 (1)

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192 (2010).
[CrossRef]

2009 (1)

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, Phys. Rev. Lett. 103, 093902 (2009).
[CrossRef]

2008 (2)

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, Phys. Rev. Lett. 100, 030402 (2008).
[CrossRef]

M. Wadati, J. Phys. Soc. Jpn. 77, 074005 (2008).
[CrossRef]

2007 (1)

C. M. Bender, Rep. Prog. Phys. 70, 947 (2007).
[CrossRef]

2003 (1)

E. N. Tsoy and F. Kh. Abdullaev, Phys. Rev. E 67, 056610 (2003).
[CrossRef]

2002 (2)

M. Klaus and J. K. Shaw, Phys. Rev. E 65, 036607 (2002).
[CrossRef]

G. Lévai and M. Znojil, J. Phys. A 35, 8793 (2002).
[CrossRef]

2001 (1)

Z. Ahmed, Phys. Lett. A 282, 343 (2001).
[CrossRef]

1999 (1)

A. A. Andrianov, M. V. Ioffe, F. Cannata, and J. P. Dedonder, Int. J. Mod. Phys. A 14, 2675 (1999).
[CrossRef]

1995 (1)

F. Cooper, A. Khare, and U. Sukhatme, Phys. Rep. 251, 267 (1995).
[CrossRef]

1974 (1)

J. Satsuma and N. Yajima, Suppl. Prog. Theor. Phys. 55, 284 (1974).
[CrossRef]

1973 (1)

S. V. Manakov, Zh. Eksp. Teor. Fiz. 65, 1392 (1973) [Sov. Phys. JETP 38, 693 (1974)].

Abdullaev, F. Kh.

E. N. Tsoy, S. Sh. Tadjimuratov, and F. Kh. Abdullaev, Opt. Commun. 285, 3441 (2012).
[CrossRef]

F. Kh. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zezyulin, Phys. Rev. A 83, 041805(R) (2011).
[CrossRef]

E. N. Tsoy and F. Kh. Abdullaev, Phys. Rev. E 67, 056610 (2003).
[CrossRef]

Agrawal, G.

Yu. S. Kivshar and G. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003).

Ahmed, Z.

Z. Ahmed, Phys. Lett. A 282, 343 (2001).
[CrossRef]

Aimez, V.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, Phys. Rev. Lett. 103, 093902 (2009).
[CrossRef]

Andrianov, A. A.

A. A. Andrianov, M. V. Ioffe, F. Cannata, and J. P. Dedonder, Int. J. Mod. Phys. A 14, 2675 (1999).
[CrossRef]

Bender, C. M.

C. M. Bender, Rep. Prog. Phys. 70, 947 (2007).
[CrossRef]

Cannata, F.

A. A. Andrianov, M. V. Ioffe, F. Cannata, and J. P. Dedonder, Int. J. Mod. Phys. A 14, 2675 (1999).
[CrossRef]

Christodoulides, D. N.

M.-A. Miri, M. Heinrich, and D. N. Christodoulides, Phys. Rev. A 87, 043819 (2013).
[CrossRef]

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192 (2010).
[CrossRef]

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, Phys. Rev. Lett. 103, 093902 (2009).
[CrossRef]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, Phys. Rev. Lett. 100, 030402 (2008).
[CrossRef]

Cooper, F.

F. Cooper, A. Khare, and U. Sukhatme, Phys. Rep. 251, 267 (1995).
[CrossRef]

Dedonder, J. P.

A. A. Andrianov, M. V. Ioffe, F. Cannata, and J. P. Dedonder, Int. J. Mod. Phys. A 14, 2675 (1999).
[CrossRef]

Duchesne, D.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, Phys. Rev. Lett. 103, 093902 (2009).
[CrossRef]

El-Ganainy, R.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192 (2010).
[CrossRef]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, Phys. Rev. Lett. 100, 030402 (2008).
[CrossRef]

Ellis, F. M.

Z. Lin, J. Schindler, F. M. Ellis, and T. Kottos, Phys. Rev. A 85, 050101(R) (2012).
[CrossRef]

Guo, A.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, Phys. Rev. Lett. 103, 093902 (2009).
[CrossRef]

Heinrich, M.

M.-A. Miri, M. Heinrich, and D. N. Christodoulides, Phys. Rev. A 87, 043819 (2013).
[CrossRef]

Ioffe, M. V.

A. A. Andrianov, M. V. Ioffe, F. Cannata, and J. P. Dedonder, Int. J. Mod. Phys. A 14, 2675 (1999).
[CrossRef]

Kartashov, Y. V.

F. Kh. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zezyulin, Phys. Rev. A 83, 041805(R) (2011).
[CrossRef]

Khare, A.

F. Cooper, A. Khare, and U. Sukhatme, Phys. Rep. 251, 267 (1995).
[CrossRef]

Kip, D.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192 (2010).
[CrossRef]

Kivshar, Yu. S.

Yu. S. Kivshar and G. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003).

Klaus, M.

M. Klaus and J. K. Shaw, Phys. Rev. E 65, 036607 (2002).
[CrossRef]

Konotop, V. V.

F. Kh. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zezyulin, Phys. Rev. A 83, 041805(R) (2011).
[CrossRef]

Kottos, T.

Z. Lin, J. Schindler, F. M. Ellis, and T. Kottos, Phys. Rev. A 85, 050101(R) (2012).
[CrossRef]

Lamb, G. L.

G. L. Lamb, Elements of Soliton Theory (Wiley, 1980).

Lévai, G.

G. Lévai and M. Znojil, J. Phys. A 35, 8793 (2002).
[CrossRef]

Lin, Z.

Z. Lin, J. Schindler, F. M. Ellis, and T. Kottos, Phys. Rev. A 85, 050101(R) (2012).
[CrossRef]

Makris, K. G.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192 (2010).
[CrossRef]

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, Phys. Rev. Lett. 100, 030402 (2008).
[CrossRef]

Manakov, S. V.

S. V. Manakov, Zh. Eksp. Teor. Fiz. 65, 1392 (1973) [Sov. Phys. JETP 38, 693 (1974)].

S. P. Novikov, S. V. Manakov, L. P. Pitaevskii, and V. E. Zakharov, Theory of Solitons: The Inverse Scattering Method (Springer-Verlag, 1984).

Miri, M.-A.

M.-A. Miri, M. Heinrich, and D. N. Christodoulides, Phys. Rev. A 87, 043819 (2013).
[CrossRef]

Morandotti, R.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, Phys. Rev. Lett. 103, 093902 (2009).
[CrossRef]

Musslimani, Z. H.

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, Phys. Rev. Lett. 100, 030402 (2008).
[CrossRef]

Novikov, S. P.

S. P. Novikov, S. V. Manakov, L. P. Pitaevskii, and V. E. Zakharov, Theory of Solitons: The Inverse Scattering Method (Springer-Verlag, 1984).

Pang, T.

T. Pang, An Introduction to Computational Physics (Cambridge University, 2006).

Pitaevskii, L. P.

S. P. Novikov, S. V. Manakov, L. P. Pitaevskii, and V. E. Zakharov, Theory of Solitons: The Inverse Scattering Method (Springer-Verlag, 1984).

Rüter, C. E.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192 (2010).
[CrossRef]

Salamo, G. J.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, Phys. Rev. Lett. 103, 093902 (2009).
[CrossRef]

Satsuma, J.

J. Satsuma and N. Yajima, Suppl. Prog. Theor. Phys. 55, 284 (1974).
[CrossRef]

Schindler, J.

Z. Lin, J. Schindler, F. M. Ellis, and T. Kottos, Phys. Rev. A 85, 050101(R) (2012).
[CrossRef]

Segev, M.

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192 (2010).
[CrossRef]

Shaw, J. K.

M. Klaus and J. K. Shaw, Phys. Rev. E 65, 036607 (2002).
[CrossRef]

Siviloglou, G. A.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, Phys. Rev. Lett. 103, 093902 (2009).
[CrossRef]

Sukhatme, U.

F. Cooper, A. Khare, and U. Sukhatme, Phys. Rep. 251, 267 (1995).
[CrossRef]

Tadjimuratov, S. Sh.

E. N. Tsoy, S. Sh. Tadjimuratov, and F. Kh. Abdullaev, Opt. Commun. 285, 3441 (2012).
[CrossRef]

Tsoy, E. N.

E. N. Tsoy, S. Sh. Tadjimuratov, and F. Kh. Abdullaev, Opt. Commun. 285, 3441 (2012).
[CrossRef]

E. N. Tsoy and F. Kh. Abdullaev, Phys. Rev. E 67, 056610 (2003).
[CrossRef]

Volatier-Ravat, M.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, Phys. Rev. Lett. 103, 093902 (2009).
[CrossRef]

Wadati, M.

M. Wadati, J. Phys. Soc. Jpn. 77, 074005 (2008).
[CrossRef]

Yajima, N.

J. Satsuma and N. Yajima, Suppl. Prog. Theor. Phys. 55, 284 (1974).
[CrossRef]

Yang, J.

J. Yang, Phys. Lett. A 378, 367 (2014).
[CrossRef]

Zakharov, V. E.

S. P. Novikov, S. V. Manakov, L. P. Pitaevskii, and V. E. Zakharov, Theory of Solitons: The Inverse Scattering Method (Springer-Verlag, 1984).

Zezyulin, D. A.

F. Kh. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zezyulin, Phys. Rev. A 83, 041805(R) (2011).
[CrossRef]

Znojil, M.

G. Lévai and M. Znojil, J. Phys. A 35, 8793 (2002).
[CrossRef]

Int. J. Mod. Phys. A (1)

A. A. Andrianov, M. V. Ioffe, F. Cannata, and J. P. Dedonder, Int. J. Mod. Phys. A 14, 2675 (1999).
[CrossRef]

J. Phys. A (1)

G. Lévai and M. Znojil, J. Phys. A 35, 8793 (2002).
[CrossRef]

J. Phys. Soc. Jpn. (1)

M. Wadati, J. Phys. Soc. Jpn. 77, 074005 (2008).
[CrossRef]

Nat. Phys. (1)

C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192 (2010).
[CrossRef]

Opt. Commun. (1)

E. N. Tsoy, S. Sh. Tadjimuratov, and F. Kh. Abdullaev, Opt. Commun. 285, 3441 (2012).
[CrossRef]

Phys. Lett. A (2)

J. Yang, Phys. Lett. A 378, 367 (2014).
[CrossRef]

Z. Ahmed, Phys. Lett. A 282, 343 (2001).
[CrossRef]

Phys. Rep. (1)

F. Cooper, A. Khare, and U. Sukhatme, Phys. Rep. 251, 267 (1995).
[CrossRef]

Phys. Rev. A (3)

Z. Lin, J. Schindler, F. M. Ellis, and T. Kottos, Phys. Rev. A 85, 050101(R) (2012).
[CrossRef]

M.-A. Miri, M. Heinrich, and D. N. Christodoulides, Phys. Rev. A 87, 043819 (2013).
[CrossRef]

F. Kh. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zezyulin, Phys. Rev. A 83, 041805(R) (2011).
[CrossRef]

Phys. Rev. E (2)

M. Klaus and J. K. Shaw, Phys. Rev. E 65, 036607 (2002).
[CrossRef]

E. N. Tsoy and F. Kh. Abdullaev, Phys. Rev. E 67, 056610 (2003).
[CrossRef]

Phys. Rev. Lett. (2)

Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, Phys. Rev. Lett. 100, 030402 (2008).
[CrossRef]

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, Phys. Rev. Lett. 103, 093902 (2009).
[CrossRef]

Rep. Prog. Phys. (1)

C. M. Bender, Rep. Prog. Phys. 70, 947 (2007).
[CrossRef]

Suppl. Prog. Theor. Phys. (1)

J. Satsuma and N. Yajima, Suppl. Prog. Theor. Phys. 55, 284 (1974).
[CrossRef]

Zh. Eksp. Teor. Fiz. (1)

S. V. Manakov, Zh. Eksp. Teor. Fiz. 65, 1392 (1973) [Sov. Phys. JETP 38, 693 (1974)].

Other (5)

T. Pang, An Introduction to Computational Physics (Cambridge University, 2006).

N. Akhmediev and A. Ankiewicz, eds., Lecture Notes in Physics: Dissipative Solitons (Springer, 2005).

Yu. S. Kivshar and G. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003).

S. P. Novikov, S. V. Manakov, L. P. Pitaevskii, and V. E. Zakharov, Theory of Solitons: The Inverse Scattering Method (Springer-Verlag, 1984).

G. L. Lamb, Elements of Soliton Theory (Wiley, 1980).

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Figures (4)

Fig. 1.
Fig. 1.

Stable dynamics of waveguide modes for (a) γ=0, and (b) γ=1 and P=2. The other parameters are v0=2, w1=1, and w2=0.5.

Fig. 2.
Fig. 2.

Dependence of EVs (propagation constant) μj on v0 for potential (7), w1=1 and w2=0.5. Solid (dashed and dotted) lines are for the linear (nonlinear) modes, γ=0 (P=2, γ=1 and γ=1). The inset shows the real part (solid line) and the imaginary part (dashed line) of the fundamental eigenmode of a linear waveguide for v0=2.

Fig. 3.
Fig. 3.

Dependence of μ on P for γ=1 (solid line) and for γ=1 (dashed line). The parameters of potential (7) are v0=2, w1=1, and w2=0.5.

Fig. 4.
Fig. 4.

Dependence of the amplitude A (solid lines) and of the FWHM a (dashed lines) of the nonlinear fundamental mode on v0 for potential (7), w1=1, w2=0.5 and P=2.

Equations (10)

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

iψz+ψxx/2+V(x)ψ+γ|ψ|2ψ=0,
uxx/2+V(x)u+γ|u|2u=μu.
V(x)=[v2(x)±ivx(x)]/2,
iϕ1,xiv(x)ϕ2=λϕ1,iϕ2,xiv*(x)ϕ1=λϕ2.
u=ϕ1+iϕ2,r=i(ϕ1iϕ2),
μ=λ2/2.
V(x)=12[ηv02sech2(x/w)iv0wsech(x/w)tanh(x/w)],
w=w1forx<0,andw2forx0,
Pz=Im[ψxxψ*]dx2VI|ψ|2dxI1I2.
iqz+qxx/2+|q|2q=0.

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