Abstract

Nonlinear dynamics of wave packets in two-dimensional parity-time-symmetric optical lattices near the phase transition point are analytically studied. A fourth-order equation is derived for the envelope of these wave packets. A pyramid diffraction pattern is demonstrated in both the linear and nonlinear regimes. Blow-up is also possible in the nonlinear regime for both focusing and defocusing nonlinearities.

© 2013 Optical Society of America

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  1. C. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998).
    [CrossRef]
  2. A. Ruschhaupt, F. Delgado, and J. G. Muga, J. Phys. A: Math. Gen. 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. 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]
  5. C. E. Rueter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, 192 (2010).
    [CrossRef]
  6. A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, Nature 488, 167 (2012).
    [CrossRef]
  7. R. Driben and B. A. Malomed, Opt. Lett. 36, 4323 (2011).
    [CrossRef]
  8. I. V. Barashenkov, S. V. Suchkov, A. A. Sukhorukov, S. V. Dmitriev, and Y. S. Kivshar, Phys. Rev. A 86, 053809 (2012).
    [CrossRef]
  9. Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, Phys. Rev. Lett. 100, 030402 (2008).
    [CrossRef]
  10. S. Longhi, Phys. Rev. Lett. 103, 123601 (2009).
    [CrossRef]
  11. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, Phys. Rev. A 81, 063807 (2010).
    [CrossRef]
  12. Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, Phys. Rev. Lett. 106, 213901 (2011).
    [CrossRef]
  13. S. Nixon, L. Ge, and J. Yang, Phys. Rev. A 85, 023822 (2012).
    [CrossRef]
  14. S. Nixon, Y. Zhu, and J. Yang, Opt. Lett. 37, 4874 (2012).
    [CrossRef]

2012 (4)

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

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

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

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

2011 (2)

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

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

2010 (2)

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

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

2009 (2)

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, and D. N. Christodoulides, Phys. Rev. Lett. 103, 093902 (2009).
[CrossRef]

2008 (1)

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

2007 (1)

2005 (1)

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

1998 (1)

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

Barashenkov, I. V.

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

Bender, C.

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

Bersch, C.

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

Boettcher, S.

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

Cao, H.

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

Christodoulides, D. N.

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

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

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

C. E. Rueter, 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]

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: Math. Gen. 38, L171 (2005).
[CrossRef]

Dmitriev, S. V.

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

Driben, R.

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]

Eichelkraut, T.

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

El-Ganainy, R.

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

C. E. Rueter, 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]

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

Ge, L.

S. Nixon, L. Ge, and 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, and D. N. Christodoulides, Phys. Rev. Lett. 103, 093902 (2009).
[CrossRef]

Kip, D.

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

Kivshar, Y. S.

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

Kottos, T.

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

Lin, Z.

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

Longhi, S.

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. Rueter, 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]

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

Malomed, B. A.

Miri, M. A.

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and 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, 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: Math. Gen. 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]

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

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

Nixon, S.

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

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

Onishchukov, G.

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

Peschel, U.

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

Ramezani, H.

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

Regensburger, A.

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

Rueter, C. E.

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

Ruschhaupt, A.

A. Ruschhaupt, F. Delgado, and J. G. Muga, J. Phys. A: Math. Gen. 38, L171 (2005).
[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. Rueter, 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]

Suchkov, S. V.

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

Yang, J.

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

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

Zhu, Y.

J. Phys. A: Math. Gen. (1)

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

Nat. Phys. (1)

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

Nature (1)

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

Opt. Lett. (3)

Phys. Rev. A (3)

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

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

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

Phys. Rev. Lett. (5)

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

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

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

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

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Figures (3)

Fig. 1.
Fig. 1.

(Upper left) Diffraction relation near intersection point (kx,ky,μ)=(1,1,2), marked by red dot. (Upper right) Linear diffraction of initial Gaussian envelope at phase transition in envelope Eq. (11). (Lower row) Linear diffraction of initial Gaussian wave packet at phase transition in full Eq. (1).

Fig. 2.
Fig. 2.

Pyramid diffraction of a Gaussian wavepacket in the linear equation below phase transition. (Upper row) Diffraction in the envelope Eq. (11). (Lower row) Diffraction in full Eq. (1).

Fig. 3.
Fig. 3.

Nonlinear dynamics of wave packets below phase transition. (Upper row) Envelope solutions in Eq. (11) at Z2 for three values of A0 in (15). (Lower row) Solutions of Eq. (1) for the initial wavepacket with A0=6 (left) at later distances under focusing (middle) and defocusing (right) nonlinearities.

Equations (19)

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

iΨz+2Ψ+V(x,y)Ψ+σ|Ψ|2Ψ=0,
V˜(x)=V02[cos(2x)+iW0sin(2x)],
ϕe1(x)=I˜n1(V0eix),ϕe2(y)=I˜n2(V0eiy),
ϕ01(x,y)=ϕe1(x)ϕe2(y),
ϕ02(x,y)=ϕe1(x)ϕg2(y)ϕg1(x)ϕe2(y).
ϕ11(x,y)=[ϕe1(x)ϕg2(y)+ϕg1(x)ϕe2(y)]/2,
ϕ21(x,y)=ϕg1(x)ϕg2(y)/2,
Ψ=ε32ψeiμt,ψ=ψ0+εψ1+,
(L+μ)ψ=iεψZ2ε(ψxX+ψyY)ε2(X2+Y2)ψ+iηε2[sin(2x)+sin(2y)]ψε3σ|ψ|2ψ.
ψ1=iAZϕ112AXϕa2AYϕb+Bϕ02,
iBZ=AZXAZY+2AXX2AYY,
Z3A8(X2+Y2)ZA8(X2Y2)(XY)A+8(X2Y2)(iB)+αZA+iσ˜|A|2A=0,
α=2V02η,σ˜=iσ02π02π|ϕ01|2ϕ01ϕ21dxdy02π02πϕ01ϕ21dxdy.
Z4A8(X2+Y2)Z2A+16(X2Y2)2A+αZ2A+iσ˜Z(|A|2A)=0.
Ψε32(Aϕ01+ϵBϕ02+ϵCϕ11+ϵ2Dϕ21),
C=i(AZ+2AX+2AY),
D=AZZ2AZX2AZYα2A+4iBX4iBY.
A=A0e(X2+Y2),AZ=AZZ=0,Z3A=iσ˜|A|2A,
A=A1(X2Z,Y2Z)+A2(X2Z,Y+2Z)+A3(X+2Z,Y2Z)+A4(X+2Z,Y+2Z)

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