Abstract

Multidimensional complex optical potentials with partial parity-time (PT) symmetry are proposed. The usual PT symmetry requires that the potential is invariant under complex conjugation and simultaneous reflection in all spatial directions. However, we show that if the potential is only partially PT symmetric, i.e., it is invariant under complex conjugation and reflection in a single spatial direction, then it can also possess all-real spectra and continuous families of solitons. These results are established analytically and corroborated numerically.

© 2014 Optical Society of America

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    [CrossRef]
  18. K. Li, P. G. Kevrekidis, Phys. Rev. E 83, 066608 (2011).
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  19. R. Driben, B. A. Malomed, Opt. Lett. 36, 4323 (2011).
    [CrossRef]
  20. Y. He, X. Zhu, D. Mihalache, J. Liu, Z. Chen, Phys. Rev. A 85, 013831 (2012).
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  26. Y. V. Kartashov, Opt. Lett. 38, 2600 (2013).
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2014

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

2013

S. Nixon, J. Yang, Opt. Lett. 38, 1933 (2013).
[CrossRef]

Y. V. Kartashov, Opt. Lett. 38, 2600 (2013).
[CrossRef]

I. V. Barashenkov, L. Baker, N. V. Alexeeva, Phys. Rev. A 87, 033819 (2013).
[CrossRef]

P. G. Kevrekidis, D. E. Pelinovsky, D. Y. Tyugin, SIAM J. Appl. Dyn. Syst. 12, 1210 (2013).
[CrossRef]

C. Huang, C. Li, L. Dong, Opt. Express 21, 3917 (2013).
[CrossRef]

2012

Y. He, X. Zhu, D. Mihalache, J. Liu, Z. Chen, Phys. Rev. A 85, 013831 (2012).
[CrossRef]

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

S. Nixon, Y. Zhu, J. Yang, Opt. Lett. 37, 4874 (2012).
[CrossRef]

I. V. Barashenkov, S. V. Suchkov, A. A. Sukhorukov, S. V. Dmitriev, Y. S. Kivshar, Phys. Rev. A 86, 053809 (2012).
[CrossRef]

V. V. Konotop, D. E. Pelinovsky, D. A. Zezyulin, Euro. Phys. Lett. 100, 56006 (2012).
[CrossRef]

S. Nixon, L. Ge, J. Yang, Phys. Rev. A 85, 023822 (2012).
[CrossRef]

D. A. Zezyulin, V. V. Konotop, Phys. Rev. A 85, 043840 (2012).
[CrossRef]

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, U. Peschel, Nature 488, 167 (2012).
[CrossRef]

2011

H. Wang, J. Wang, Opt. Express 19, 4030 (2011).
[CrossRef]

Z. Lu, Z. Zhang, Opt. Express 19, 11457 (2011).
[CrossRef]

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, D. N. Christodoulides, Phys. Rev. Lett. 106, 213901 (2011).
[CrossRef]

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

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

R. Driben, B. A. Malomed, Opt. Lett. 36, 4323 (2011).
[CrossRef]

2010

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

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, Z. H. Musslimani, Phys. Rev. A 81, 063807 (2010).
[CrossRef]

2009

S. Longhi, Phys. Rev. Lett. 103, 123601 (2009).
[CrossRef]

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

2008

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

2007

2005

A. Ruschhaupt, F. Delgado, J. G. Muga, J. Phys. A 38, L171 (2005).
[CrossRef]

1998

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

Abdullaev, F. K.

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

Agrawal, G. P.

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

Aimez, V.

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

Alexeeva, N. V.

I. V. Barashenkov, L. Baker, N. V. Alexeeva, Phys. Rev. A 87, 033819 (2013).
[CrossRef]

Baker, L.

I. V. Barashenkov, L. Baker, N. V. Alexeeva, Phys. Rev. A 87, 033819 (2013).
[CrossRef]

Barashenkov, I. V.

I. V. Barashenkov, L. Baker, N. V. Alexeeva, Phys. Rev. A 87, 033819 (2013).
[CrossRef]

I. V. Barashenkov, S. V. Suchkov, A. A. Sukhorukov, S. V. Dmitriev, Y. S. Kivshar, Phys. Rev. A 86, 053809 (2012).
[CrossRef]

Bender, C.

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

Bersch, C.

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, U. Peschel, Nature 488, 167 (2012).
[CrossRef]

Boettcher, S.

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

Cao, H.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, D. N. Christodoulides, Phys. Rev. Lett. 106, 213901 (2011).
[CrossRef]

Chen, Z.

Y. He, X. Zhu, D. Mihalache, J. Liu, Z. Chen, Phys. Rev. A 85, 013831 (2012).
[CrossRef]

Christodoulides, D. N.

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, U. Peschel, Nature 488, 167 (2012).
[CrossRef]

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, D. N. Christodoulides, Phys. Rev. Lett. 106, 213901 (2011).
[CrossRef]

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, Z. H. Musslimani, Phys. Rev. A 81, 063807 (2010).
[CrossRef]

C. E. Rueter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, 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, D. N. Christodoulides, Phys. Rev. Lett. 103, 093902 (2009).
[CrossRef]

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

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, Z. H. Musslimani, Opt. Lett. 32, 2632 (2007).
[CrossRef]

Delgado, F.

A. Ruschhaupt, F. Delgado, J. G. Muga, J. Phys. A 38, L171 (2005).
[CrossRef]

Dmitriev, S. V.

I. V. Barashenkov, S. V. Suchkov, A. A. Sukhorukov, S. V. Dmitriev, Y. S. Kivshar, Phys. Rev. A 86, 053809 (2012).
[CrossRef]

Dong, L.

Driben, R.

Duchesne, D.

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

Eichelkraut, T.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, D. N. Christodoulides, Phys. Rev. Lett. 106, 213901 (2011).
[CrossRef]

El-Ganainy, R.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, Z. H. Musslimani, Phys. Rev. A 81, 063807 (2010).
[CrossRef]

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

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

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, Z. H. Musslimani, Opt. Lett. 32, 2632 (2007).
[CrossRef]

Ge, L.

S. Nixon, L. Ge, J. Yang, Phys. Rev. A 85, 023822 (2012).
[CrossRef]

Guo, A.

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

He, Y.

Y. He, X. Zhu, D. Mihalache, J. Liu, Z. Chen, Phys. Rev. A 85, 013831 (2012).
[CrossRef]

Huang, C.

Kartashov, Y. V.

Y. V. Kartashov, Opt. Lett. 38, 2600 (2013).
[CrossRef]

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

Kevrekidis, P. G.

P. G. Kevrekidis, D. E. Pelinovsky, D. Y. Tyugin, SIAM J. Appl. Dyn. Syst. 12, 1210 (2013).
[CrossRef]

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

Kip, D.

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

Kivshar, Y. S.

I. V. Barashenkov, S. V. Suchkov, A. A. Sukhorukov, S. V. Dmitriev, Y. S. Kivshar, Phys. Rev. A 86, 053809 (2012).
[CrossRef]

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

Konotop, V. V.

D. A. Zezyulin, V. V. Konotop, Phys. Rev. A 85, 043840 (2012).
[CrossRef]

V. V. Konotop, D. E. Pelinovsky, D. A. Zezyulin, Euro. Phys. Lett. 100, 56006 (2012).
[CrossRef]

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

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

Kottos, T.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, D. N. Christodoulides, Phys. Rev. Lett. 106, 213901 (2011).
[CrossRef]

Li, C.

Li, K.

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

Lin, Z.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, D. N. Christodoulides, Phys. Rev. Lett. 106, 213901 (2011).
[CrossRef]

Liu, J.

Y. He, X. Zhu, D. Mihalache, J. Liu, Z. Chen, Phys. Rev. A 85, 013831 (2012).
[CrossRef]

Longhi, S.

S. Longhi, Phys. Rev. Lett. 103, 123601 (2009).
[CrossRef]

Lu, Z.

Makris, K. G.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, Z. H. Musslimani, Phys. Rev. A 81, 063807 (2010).
[CrossRef]

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

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

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, Z. H. Musslimani, Opt. Lett. 32, 2632 (2007).
[CrossRef]

Malomed, B. A.

Mihalache, D.

Y. He, X. Zhu, D. Mihalache, J. Liu, Z. Chen, Phys. Rev. A 85, 013831 (2012).
[CrossRef]

Miri, M. A.

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, U. Peschel, Nature 488, 167 (2012).
[CrossRef]

Morandotti, R.

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

Muga, J. G.

A. Ruschhaupt, F. Delgado, J. G. Muga, J. Phys. A 38, L171 (2005).
[CrossRef]

Musslimani, Z. H.

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, Z. H. Musslimani, Phys. Rev. A 81, 063807 (2010).
[CrossRef]

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

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, Z. H. Musslimani, Opt. Lett. 32, 2632 (2007).
[CrossRef]

Nixon, S.

Onishchukov, G.

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, U. Peschel, Nature 488, 167 (2012).
[CrossRef]

Pelinovsky, D. E.

P. G. Kevrekidis, D. E. Pelinovsky, D. Y. Tyugin, SIAM J. Appl. Dyn. Syst. 12, 1210 (2013).
[CrossRef]

V. V. Konotop, D. E. Pelinovsky, D. A. Zezyulin, Euro. Phys. Lett. 100, 56006 (2012).
[CrossRef]

Peschel, U.

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, U. Peschel, Nature 488, 167 (2012).
[CrossRef]

Ramezani, H.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, D. N. Christodoulides, Phys. Rev. Lett. 106, 213901 (2011).
[CrossRef]

Regensburger, A.

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, U. Peschel, Nature 488, 167 (2012).
[CrossRef]

Rueter, C. E.

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

Ruschhaupt, A.

A. Ruschhaupt, F. Delgado, J. G. Muga, J. Phys. A 38, L171 (2005).
[CrossRef]

Salamo, G. J.

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

Segev, M.

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

Siviloglou, G. A.

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

Suchkov, S. V.

I. V. Barashenkov, S. V. Suchkov, A. A. Sukhorukov, S. V. Dmitriev, Y. S. Kivshar, Phys. Rev. A 86, 053809 (2012).
[CrossRef]

Sukhorukov, A. A.

I. V. Barashenkov, S. V. Suchkov, A. A. Sukhorukov, S. V. Dmitriev, Y. S. Kivshar, Phys. Rev. A 86, 053809 (2012).
[CrossRef]

Tyugin, D. Y.

P. G. Kevrekidis, D. E. Pelinovsky, D. Y. Tyugin, SIAM J. Appl. Dyn. Syst. 12, 1210 (2013).
[CrossRef]

Volatier-Ravat, M.

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

Wang, H.

Wang, J.

Yang, J.

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

S. Nixon, J. Yang, Opt. Lett. 38, 1933 (2013).
[CrossRef]

S. Nixon, Y. Zhu, J. Yang, Opt. Lett. 37, 4874 (2012).
[CrossRef]

S. Nixon, L. Ge, J. Yang, Phys. Rev. A 85, 023822 (2012).
[CrossRef]

Zezyulin, D. A.

D. A. Zezyulin, V. V. Konotop, Phys. Rev. A 85, 043840 (2012).
[CrossRef]

V. V. Konotop, D. E. Pelinovsky, D. A. Zezyulin, Euro. Phys. Lett. 100, 56006 (2012).
[CrossRef]

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

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

Zhang, Z.

Zhu, X.

Y. He, X. Zhu, D. Mihalache, J. Liu, Z. Chen, Phys. Rev. A 85, 013831 (2012).
[CrossRef]

Zhu, Y.

Euro. Phys. Lett.

V. V. Konotop, D. E. Pelinovsky, D. A. Zezyulin, Euro. Phys. Lett. 100, 56006 (2012).
[CrossRef]

J. Phys. A

A. Ruschhaupt, F. Delgado, J. G. Muga, J. Phys. A 38, L171 (2005).
[CrossRef]

Nat. Phys.

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

Nature

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, U. Peschel, Nature 488, 167 (2012).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Lett. A

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

Phys. Rev. A

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, Z. H. Musslimani, Phys. Rev. A 81, 063807 (2010).
[CrossRef]

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

S. Nixon, L. Ge, J. Yang, Phys. Rev. A 85, 023822 (2012).
[CrossRef]

D. A. Zezyulin, V. V. Konotop, Phys. Rev. A 85, 043840 (2012).
[CrossRef]

I. V. Barashenkov, S. V. Suchkov, A. A. Sukhorukov, S. V. Dmitriev, Y. S. Kivshar, Phys. Rev. A 86, 053809 (2012).
[CrossRef]

Y. He, X. Zhu, D. Mihalache, J. Liu, Z. Chen, Phys. Rev. A 85, 013831 (2012).
[CrossRef]

I. V. Barashenkov, L. Baker, N. V. Alexeeva, Phys. Rev. A 87, 033819 (2013).
[CrossRef]

Phys. Rev. E

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

Phys. Rev. Lett.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, D. N. Christodoulides, Phys. Rev. Lett. 106, 213901 (2011).
[CrossRef]

S. Longhi, Phys. Rev. Lett. 103, 123601 (2009).
[CrossRef]

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

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Figures (3)

Fig. 1.
Fig. 1.

(a), (b) Real and imaginary parts of the partially PT symmetric potential (7) for β=0.1 and (c), (d) spectrum of this potential for β=0.1 and 0.3, respectively.

Fig. 2.
Fig. 2.

Upper left: power diagram of the soliton family in potential (7) with β=0.1 and σ=1 (solid blue are stable and dashed red unstable); upper and lower right: amplitude fields of solitons (|ψ|) at points “b,c” of the power curve; lower left: phase field of the soliton at “b”.

Fig. 3.
Fig. 3.

Linear-stability spectra of the four solitons marked by letters “a,b,c,d” on the power curve of Fig. 2.

Equations (22)

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iΨz+2Ψ+V(x,y)Ψ+σ|Ψ|2Ψ=0,
V*(x,y)=V(x,y).
(2+V)ψ=λψ,
V(x,y)=V1(x)+V2(y).
V1*(x)=V1(x),V2*(y)=V2(y).
λ=Λ1+Λ2
[xx+V1(x)]Ψ1(x)=Λ1Ψ1(x),[yy+V2(y)]Ψ2(y)=Λ2Ψ2(y).
V(x,y)=V0(x,y)+ϵVp(x,y),
λ=λ0+ϵλ1+ϵ2λ2+,ψ=ψ0+ϵψ1+ϵ2ψ2+.
Lψ1=(λ1Vp)ψ0,
λ1=ψ0*,Vpψ0ψ0*,ψ0,
f,g=f*(x,y)g(x,y)dxdy.
f,g*=f*,g*=f(x,y),g(x,y)=f,g.
V(x,y)=3(e(xx0)2(yy0)2+e(x+x0)2(yy0)2)+2(e(xx0)2(y+y0)2+e(x+x0)2(y+y0)2)+iβ[2(e(xx0)2(yy0)2e(x+x0)2(yy0)2)+(e(xx0)2(y+y0)2e(x+x0)2(y+y0)2)],
V(x,y)=(1+iγxy)ex2y2,
Ψ(x,y,t)=ψ(x,y)eiμz,
2ψ+V(x,y)ψ+σ|ψ|2ψ=μψ
ψ(x,y;μ)=ϵ1/2[c0ψ0+ϵψ1+ϵ2ψ2+],
Lψ1=c0(ρψ0σ|c0|2|ψ0|2ψ0),
|c0|2=ρψ0*,ψ0σψ0*,|ψ0|2ψ0.
Ψ(x,y,z)=eiμz[ψ(x,y)+f(x,y)eλz+g*(x,y)eλ*z],
i[M1M2M2*M1*][fg]=λ[fg],

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