Abstract

We study the effect of lifting the degeneracy of vortex modes with a parity time (PT) symmetric defect, using discrete vortices in a circular array of nonlinear waveguides as an example. When the defect is introduced, the degenerate linear vortex modes spontaneously break PT symmetry and acquire complex eigenvalues, but nonlinear propagating modes with real propagation constants can still exist. The stability of nonlinear modes depends on both the magnitude and the sign of the vortex charge; thus PT symmetric systems offer new mechanisms to control discrete vortices.

© 2013 Optical Society of America

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  1. F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
    [CrossRef]
  2. A. S. Desyatnikov, Yu. S. Kivshar, and L. Torner, Prog. Opt. 47, 291 (2005).
    [CrossRef]
  3. B. A. Malomed and P. G. Kevrekidis, Phys. Rev. E 64, 026601 (2001).
    [CrossRef]
  4. T. J. Alexander, A. A. Sukhorukov, and Yu. S. Kivshar, Phys. Rev. Lett. 93, 063901 (2004).
    [CrossRef]
  5. A. Bezryadina, E. Eugenieva, and Z. Chen, Opt. Lett. 31, 2456 (2006).
    [CrossRef]
  6. A. S. Desyatnikov, M. R. Dennis, and A. Ferrando, Phys. Rev. A 83, 063822 (2011).
    [CrossRef]
  7. C. M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998).
    [CrossRef]
  8. S. V. Dmitriev, A. A. Sukhorukov, and Y. S. Kivshar, Opt. Lett. 35, 2976 (2010).
    [CrossRef]
  9. C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192 (2010).
    [CrossRef]
  10. K. Li and P. G. Kevrekidis, Phys. Rev. E 83, 066608 (2011).
    [CrossRef]
  11. D. A. Zezyulin and V. V. Konotop, Phys. Rev. Lett. 108, 213906 (2012).
    [CrossRef]
  12. A. A. Sukhorukov, S. V. Dmitriev, S. S. Suchkov, and Yu. S. Kivshar, Opt. Lett. 37, 2148 (2012).
    [CrossRef]
  13. D. D. Scott and Y. N. Joglekar, Phys. Rev. A 85, 062105 (2012).
    [CrossRef]
  14. E. Caliceti, F. Cannata, and S. Graffi, J. Phys. A 39, 10019 (2006).
    [CrossRef]

2012 (3)

D. A. Zezyulin and V. V. Konotop, Phys. Rev. Lett. 108, 213906 (2012).
[CrossRef]

A. A. Sukhorukov, S. V. Dmitriev, S. S. Suchkov, and Yu. S. Kivshar, Opt. Lett. 37, 2148 (2012).
[CrossRef]

D. D. Scott and Y. N. Joglekar, Phys. Rev. A 85, 062105 (2012).
[CrossRef]

2011 (2)

K. Li and P. G. Kevrekidis, Phys. Rev. E 83, 066608 (2011).
[CrossRef]

A. S. Desyatnikov, M. R. Dennis, and A. Ferrando, Phys. Rev. A 83, 063822 (2011).
[CrossRef]

2010 (2)

S. V. Dmitriev, A. A. Sukhorukov, and Y. S. Kivshar, Opt. Lett. 35, 2976 (2010).
[CrossRef]

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

2008 (1)

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

2006 (2)

A. Bezryadina, E. Eugenieva, and Z. Chen, Opt. Lett. 31, 2456 (2006).
[CrossRef]

E. Caliceti, F. Cannata, and S. Graffi, J. Phys. A 39, 10019 (2006).
[CrossRef]

2005 (1)

A. S. Desyatnikov, Yu. S. Kivshar, and L. Torner, Prog. Opt. 47, 291 (2005).
[CrossRef]

2004 (1)

T. J. Alexander, A. A. Sukhorukov, and Yu. S. Kivshar, Phys. Rev. Lett. 93, 063901 (2004).
[CrossRef]

2001 (1)

B. A. Malomed and P. G. Kevrekidis, Phys. Rev. E 64, 026601 (2001).
[CrossRef]

1998 (1)

C. M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998).
[CrossRef]

Alexander, T. J.

T. J. Alexander, A. A. Sukhorukov, and Yu. S. Kivshar, Phys. Rev. Lett. 93, 063901 (2004).
[CrossRef]

Assanto, G.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

Bender, C. M.

C. M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998).
[CrossRef]

Bezryadina, A.

Boettcher, S.

C. M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998).
[CrossRef]

Caliceti, E.

E. Caliceti, F. Cannata, and S. Graffi, J. Phys. A 39, 10019 (2006).
[CrossRef]

Cannata, F.

E. Caliceti, F. Cannata, and S. Graffi, J. Phys. A 39, 10019 (2006).
[CrossRef]

Chen, Z.

Christodoulides, D. N.

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

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

Dennis, M. R.

A. S. Desyatnikov, M. R. Dennis, and A. Ferrando, Phys. Rev. A 83, 063822 (2011).
[CrossRef]

Desyatnikov, A. S.

A. S. Desyatnikov, M. R. Dennis, and A. Ferrando, Phys. Rev. A 83, 063822 (2011).
[CrossRef]

A. S. Desyatnikov, Yu. S. Kivshar, and L. Torner, Prog. Opt. 47, 291 (2005).
[CrossRef]

Dmitriev, S. V.

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]

Eugenieva, E.

Ferrando, A.

A. S. Desyatnikov, M. R. Dennis, and A. Ferrando, Phys. Rev. A 83, 063822 (2011).
[CrossRef]

Graffi, S.

E. Caliceti, F. Cannata, and S. Graffi, J. Phys. A 39, 10019 (2006).
[CrossRef]

Joglekar, Y. N.

D. D. Scott and Y. N. Joglekar, Phys. Rev. A 85, 062105 (2012).
[CrossRef]

Kevrekidis, P. G.

K. Li and P. G. Kevrekidis, Phys. Rev. E 83, 066608 (2011).
[CrossRef]

B. A. Malomed and P. G. Kevrekidis, Phys. Rev. E 64, 026601 (2001).
[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, Y. S.

Kivshar, Yu. S.

A. A. Sukhorukov, S. V. Dmitriev, S. S. Suchkov, and Yu. S. Kivshar, Opt. Lett. 37, 2148 (2012).
[CrossRef]

A. S. Desyatnikov, Yu. S. Kivshar, and L. Torner, Prog. Opt. 47, 291 (2005).
[CrossRef]

T. J. Alexander, A. A. Sukhorukov, and Yu. S. Kivshar, Phys. Rev. Lett. 93, 063901 (2004).
[CrossRef]

Konotop, V. V.

D. A. Zezyulin and V. V. Konotop, Phys. Rev. Lett. 108, 213906 (2012).
[CrossRef]

Lederer, F.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

Li, K.

K. Li and P. G. Kevrekidis, Phys. Rev. E 83, 066608 (2011).
[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]

Malomed, B. A.

B. A. Malomed and P. G. Kevrekidis, Phys. Rev. E 64, 026601 (2001).
[CrossRef]

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]

Scott, D. D.

D. D. Scott and Y. N. Joglekar, Phys. Rev. A 85, 062105 (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]

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

Silberberg, Y.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

Stegeman, G. I.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

Suchkov, S. S.

Sukhorukov, A. A.

Torner, L.

A. S. Desyatnikov, Yu. S. Kivshar, and L. Torner, Prog. Opt. 47, 291 (2005).
[CrossRef]

Zezyulin, D. A.

D. A. Zezyulin and V. V. Konotop, Phys. Rev. Lett. 108, 213906 (2012).
[CrossRef]

J. Phys. A (1)

E. Caliceti, F. Cannata, and S. Graffi, J. Phys. A 39, 10019 (2006).
[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. Lett. (3)

Phys. Rep. (1)

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[CrossRef]

Phys. Rev. A (2)

A. S. Desyatnikov, M. R. Dennis, and A. Ferrando, Phys. Rev. A 83, 063822 (2011).
[CrossRef]

D. D. Scott and Y. N. Joglekar, Phys. Rev. A 85, 062105 (2012).
[CrossRef]

Phys. Rev. E (2)

K. Li and P. G. Kevrekidis, Phys. Rev. E 83, 066608 (2011).
[CrossRef]

B. A. Malomed and P. G. Kevrekidis, Phys. Rev. E 64, 026601 (2001).
[CrossRef]

Phys. Rev. Lett. (3)

T. J. Alexander, A. A. Sukhorukov, and Yu. S. Kivshar, Phys. Rev. Lett. 93, 063901 (2004).
[CrossRef]

D. A. Zezyulin and V. V. Konotop, Phys. Rev. Lett. 108, 213906 (2012).
[CrossRef]

C. M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998).
[CrossRef]

Prog. Opt. (1)

A. S. Desyatnikov, Yu. S. Kivshar, and L. Torner, Prog. Opt. 47, 291 (2005).
[CrossRef]

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Figures (3)

Fig. 1.
Fig. 1.

(a) Schematic of N-site ring with gain (+) and loss () at waveguides 1 and N. Phase circulation direction of vortices with m>0 and m<0 is indicated by anticlockwise (red) and clockwise (blue) arrows. (b) β versus C for a conservative (γ=0) N=4 ring. Degenerate vortex modes only occur at the intersection marked by the black circle. (c) and (d) Linear spectrum for N=3 and N=4 rings with γ=0.2. Modes with TCs +1 (1) are shown in red (blue).

Fig. 2.
Fig. 2.

Stable (unstable) nonlinear modes of N=3 ring are shown with solid (dashed) lines. TCs are indicated next to the curves, m=+1 in red (purple), m=1 in blue (brown), and m=0 in black (gray). Parameters are: (a) C=0.6, γ=0.2<γth; (b) C=0.6, γ=0.6>γth; (c) C=1.3, γ=0.2<γth; and (d) C=1.3, γ=0.6>γth.

Fig. 3.
Fig. 3.

Charge selectivity in an array with N=6, C=1, γ=0.2, and δ=1. The (a) total power and (b) TC are shown versus propagation distance z for m=+2 (black, dashed) and m=2 (gray, solid) inputs. Vortex (red) and antivortex (blue) lines are shown in (c) for input m=+2 and in (d) for m=2.

Equations (2)

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izE1+CEN+E2iγE1+δ|E1|2E1=0,izEn+En1+En+1+δ|En|2En=0,izEN+CE1+EN1+iγEN+δ|EN|2EN=0.
βE+HE+δF(|E|2)E=0,H=H0+iγH1,

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