Abstract

We demonstrate that light propagation in waveguide arrays that include PT-symmetric structures can exhibit strongly nonlocal sensitivity to topology of the array at fixed other parameters. We consider an array composed of lossless waveguides, that includes a pair of PT-symmetric waveguides with balanced gain and loss, and reveal that PT-symmetry breaking thresholds are different for planar and circular array configurations. These results demonstrate that PT-symmetric structures can offer new regimes for optical beam shaping compared to conservative structures.

© 2012 Optical Society of America

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  1. Y. J. Chen, A. W. Snyder, and D. N. Payne, IEEE J. Quantum Electron. 28, 239 (1992).
    [CrossRef]
  2. A. Ruschhaupt, F. Delgado, and J. G. Muga, J. Phys. A 38, L171 (2005).
    [CrossRef]
  3. R. El-Ganainy, K. G. Makris, D. N. Christodoulides, and Z. H. Musslimani, Opt. Lett. 32, 2632 (2007).
    [CrossRef]
  4. S. Longhi, Phys. Rev. Lett. 103, 123601 (2009).
    [CrossRef]
  5. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, Phys. Rev. A 81, 063807 (2010).
    [CrossRef]
  6. C. E. Ruter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192 (2010).
    [CrossRef]
  7. 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]
  8. H. F. Jones, Phys. Rev. D 76, 125003 (2007).
    [CrossRef]
  9. M. Znojil, Phys. Rev. D 80, 045009 (2009).
    [CrossRef]
  10. S. V. Dmitriev, A. A. Sukhorukov, and Yu. S. Kivshar, Opt. Lett. 35, 2976 (2010).
    [CrossRef]
  11. M. C. Zheng, D. N. Christodoulides, R. Fleischmann, and T. Kottos, Phys. Rev. A 82, 010103 (2010).
    [CrossRef]
  12. S. Longhi, F. Cannata, and A. Ventura, Phys. Rev. B 84, 235131 (2011).
    [CrossRef]
  13. S. V. Dmitriev, S. V. Suchkov, A. A. Sukhorukov, and Yu. S. Kivshar, Phys. Rev. A 84, 013833 (2011).
    [CrossRef]

2011 (2)

S. Longhi, F. Cannata, and A. Ventura, Phys. Rev. B 84, 235131 (2011).
[CrossRef]

S. V. Dmitriev, S. V. Suchkov, A. A. Sukhorukov, and Yu. S. Kivshar, Phys. Rev. A 84, 013833 (2011).
[CrossRef]

2010 (4)

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

M. C. Zheng, D. N. Christodoulides, R. Fleischmann, and T. Kottos, Phys. Rev. A 82, 010103 (2010).
[CrossRef]

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

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

2009 (3)

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]

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

M. Znojil, Phys. Rev. D 80, 045009 (2009).
[CrossRef]

2007 (2)

2005 (1)

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

1992 (1)

Y. J. Chen, A. W. Snyder, and D. N. Payne, IEEE J. Quantum Electron. 28, 239 (1992).
[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]

Cannata, F.

S. Longhi, F. Cannata, and A. Ventura, Phys. Rev. B 84, 235131 (2011).
[CrossRef]

Chen, Y. J.

Y. J. Chen, A. W. Snyder, and D. N. Payne, IEEE J. Quantum Electron. 28, 239 (1992).
[CrossRef]

Christodoulides, D. N.

M. C. Zheng, D. N. Christodoulides, R. Fleischmann, and T. Kottos, Phys. Rev. A 82, 010103 (2010).
[CrossRef]

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

C. E. Ruter, 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]

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

Delgado, F.

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

Dmitriev, S. V.

S. V. Dmitriev, S. V. Suchkov, A. A. Sukhorukov, and Yu. S. Kivshar, Phys. Rev. A 84, 013833 (2011).
[CrossRef]

S. V. Dmitriev, A. A. Sukhorukov, and Yu. S. Kivshar, Opt. Lett. 35, 2976 (2010).
[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. Ruter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192 (2010).
[CrossRef]

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

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

Fleischmann, R.

M. C. Zheng, D. N. Christodoulides, R. Fleischmann, and T. Kottos, Phys. Rev. A 82, 010103 (2010).
[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]

Jones, H. F.

H. F. Jones, Phys. Rev. D 76, 125003 (2007).
[CrossRef]

Kip, D.

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

Kivshar, Yu. S.

S. V. Dmitriev, S. V. Suchkov, A. A. Sukhorukov, and Yu. S. Kivshar, Phys. Rev. A 84, 013833 (2011).
[CrossRef]

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

Kottos, T.

M. C. Zheng, D. N. Christodoulides, R. Fleischmann, and T. Kottos, Phys. Rev. A 82, 010103 (2010).
[CrossRef]

Longhi, S.

S. Longhi, F. Cannata, and A. Ventura, Phys. Rev. B 84, 235131 (2011).
[CrossRef]

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

Makris, K. G.

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

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

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

Muga, J. G.

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

Musslimani, Z. H.

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

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

Payne, D. N.

Y. J. Chen, A. W. Snyder, and D. N. Payne, IEEE J. Quantum Electron. 28, 239 (1992).
[CrossRef]

Ruschhaupt, A.

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

Ruter, C. E.

C. E. Ruter, 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]

Segev, M.

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

Snyder, A. W.

Y. J. Chen, A. W. Snyder, and D. N. Payne, IEEE J. Quantum Electron. 28, 239 (1992).
[CrossRef]

Suchkov, S. V.

S. V. Dmitriev, S. V. Suchkov, A. A. Sukhorukov, and Yu. S. Kivshar, Phys. Rev. A 84, 013833 (2011).
[CrossRef]

Sukhorukov, A. A.

S. V. Dmitriev, S. V. Suchkov, A. A. Sukhorukov, and Yu. S. Kivshar, Phys. Rev. A 84, 013833 (2011).
[CrossRef]

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

Ventura, A.

S. Longhi, F. Cannata, and A. Ventura, Phys. Rev. B 84, 235131 (2011).
[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]

Zheng, M. C.

M. C. Zheng, D. N. Christodoulides, R. Fleischmann, and T. Kottos, Phys. Rev. A 82, 010103 (2010).
[CrossRef]

Znojil, M.

M. Znojil, Phys. Rev. D 80, 045009 (2009).
[CrossRef]

IEEE J. Quantum Electron. (1)

Y. J. Chen, A. W. Snyder, and D. N. Payne, IEEE J. Quantum Electron. 28, 239 (1992).
[CrossRef]

J. Phys. A (1)

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

Nat. Phys. (1)

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

Opt. Lett. (2)

Phys. Rev. A (3)

M. C. Zheng, D. N. Christodoulides, R. Fleischmann, and T. Kottos, Phys. Rev. A 82, 010103 (2010).
[CrossRef]

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

S. V. Dmitriev, S. V. Suchkov, A. A. Sukhorukov, and Yu. S. Kivshar, Phys. Rev. A 84, 013833 (2011).
[CrossRef]

Phys. Rev. B (1)

S. Longhi, F. Cannata, and A. Ventura, Phys. Rev. B 84, 235131 (2011).
[CrossRef]

Phys. Rev. D (2)

H. F. Jones, Phys. Rev. D 76, 125003 (2007).
[CrossRef]

M. Znojil, Phys. Rev. D 80, 045009 (2009).
[CrossRef]

Phys. Rev. Lett. (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]

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

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Figures (3)

Fig. 1.
Fig. 1.

Schematic of (a) planar and (b) circular waveguide array with a pair of PT-symmetric waveguides at sites n=0, 1 with balanced gain and loss.

Fig. 2.
Fig. 2.

Fastest mode amplification rate in parameter plane |ρ/C2| and |C1/C2| for (a), (c) planar and (b), (d) circular waveguide arrays of sizes (a), (b) N=20 and (c), (d) N=100. Solid lines show analytical instability threshold according to (a), (c) Eq. (6) and (b), (d) Eq. (8). Dashed lines show asymptotic instability threshold for infinitely large structures according to Eq. (11).

Fig. 3.
Fig. 3.

Optical beam dynamics in (a) planar and (b) circular waveguide arrays for ρ/C2=0.8, and (c) planar or circular (practically identical figures) for ρ/C2=2. Left plots: the modulus of amplitudes at the waveguides |an| along the propagation distance z. Right: calculated total power versus distance (solid line) and power trend for the fastest growing eigenmode (dashed blue line). For all the plots, C1/C2=1.5 and N=20.

Equations (11)

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

idandz+C1an1+C1an+1=0,n0,1,
ida0dz+iρa0+C1a1+C2a1=0,
ida1dziρa1+C2a0+C1a2=0,
aN+20,aN10,
aN+2aN,aN1aN+1.
|ρ|>ρp=|C2|,
C1|an|[sin(ϕn+1ϕn)|an+1|+sin(ϕn1ϕn)|an1|]=0.
|ρ|>ρc=||C1||C2||,
an=F±eikn+iβz+B±eikn+iβz,
|J|2=C12C22+ρ22C1[ρ2C2Im(J)]sin(k)C12C22+ρ2+2C1ρsin(k).
|ρ|>ρinf=C12+C22.

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