Abstract

We address the existence and properties of one-dimensional solitons maintained by localized parametric gain in focusing and defocusing lossy nonlinear media. Localized parametric gain supports both fundamental and multipole solitons. We found that the family of fundamental solitons is partly stable in focusing nonlinear medium, and completely stable in defocusing medium, while all higher-order solitons are unstable. In addition to numerical results, the existence threshold for the solitons and a particular stable exact solution are obtained in an exact analytical form.

© 2013 Optical Society of America

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2012 (1)

B. A. Malomed, E. Ding, K. W. Chow, and S. K. Lai, Phys. Rev. E 86, 036608 (2012).
[CrossRef]

2011 (2)

2010 (4)

2009 (2)

E. Kenig, B. A. Malomed, M. C. Cross, and R. Lifshitz, Phys. Rev. E 80, 046202 (2009).
[CrossRef]

C.-K. Lam, B. A. Malomed, K. W. Chow, and P. K. A. Wai, Eur. Phys. J. Spec. Top. 173, 233 (2009).
[CrossRef]

2006 (1)

M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta, Nature 441, 960 (2006).
[CrossRef]

2003 (2)

I. V. Barashenkov, S. Cross, and B. A. Malomed, Phys. Rev. E 68, 056605 (2003).
[CrossRef]

W. C. K. Mak, B. A. Malomed, and P. L. Chu, Phys. Rev. E 67, 026608 (2003).
[CrossRef]

1999 (1)

I. V. Barashenkov and E. V. Zemlyanaya, Phys. Rev. Lett. 83, 2568 (1999).
[CrossRef]

1998 (1)

1997 (1)

1995 (1)

M. Bondila, I. V. Barashenkov, and M. M. Bogdan, Physica D 87, 314 (1995).
[CrossRef]

1991 (1)

I. V. Barashenkov, M. M. Bogdan, and V. I. Korobov, Europhys. Lett. 15, 113 (1991).
[CrossRef]

Agarwal, A.

Agrawal, G.

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

Barashenkov, I. V.

I. V. Barashenkov, S. Cross, and B. A. Malomed, Phys. Rev. E 68, 056605 (2003).
[CrossRef]

I. V. Barashenkov and E. V. Zemlyanaya, Phys. Rev. Lett. 83, 2568 (1999).
[CrossRef]

M. Bondila, I. V. Barashenkov, and M. M. Bogdan, Physica D 87, 314 (1995).
[CrossRef]

I. V. Barashenkov, M. M. Bogdan, and V. I. Korobov, Europhys. Lett. 15, 113 (1991).
[CrossRef]

Bartolini, G. D.

Bogdan, M. M.

M. Bondila, I. V. Barashenkov, and M. M. Bogdan, Physica D 87, 314 (1995).
[CrossRef]

I. V. Barashenkov, M. M. Bogdan, and V. I. Korobov, Europhys. Lett. 15, 113 (1991).
[CrossRef]

Bondila, M.

M. Bondila, I. V. Barashenkov, and M. M. Bogdan, Physica D 87, 314 (1995).
[CrossRef]

Chow, K. W.

B. A. Malomed, E. Ding, K. W. Chow, and S. K. Lai, Phys. Rev. E 86, 036608 (2012).
[CrossRef]

C.-K. Lam, B. A. Malomed, K. W. Chow, and P. K. A. Wai, Eur. Phys. J. Spec. Top. 173, 233 (2009).
[CrossRef]

Chu, P. L.

W. C. K. Mak, B. A. Malomed, and P. L. Chu, Phys. Rev. E 67, 026608 (2003).
[CrossRef]

Clerk, A. A.

A. A. Clerk, M. H. Devoret, S. Girvin, F. Marquardt, and R. J. Schoelkopf, Rev. Mod. Phys. 82, 1155 (2010).
[CrossRef]

Cross, M. C.

E. Kenig, B. A. Malomed, M. C. Cross, and R. Lifshitz, Phys. Rev. E 80, 046202 (2009).
[CrossRef]

Cross, S.

I. V. Barashenkov, S. Cross, and B. A. Malomed, Phys. Rev. E 68, 056605 (2003).
[CrossRef]

Devoret, M. H.

A. A. Clerk, M. H. Devoret, S. Girvin, F. Marquardt, and R. J. Schoelkopf, Rev. Mod. Phys. 82, 1155 (2010).
[CrossRef]

Ding, E.

B. A. Malomed, E. Ding, K. W. Chow, and S. K. Lai, Phys. Rev. E 86, 036608 (2012).
[CrossRef]

Foster, M. A.

M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta, Nature 441, 960 (2006).
[CrossRef]

Gaeta, A. L.

M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta, Nature 441, 960 (2006).
[CrossRef]

Girvin, S.

A. A. Clerk, M. H. Devoret, S. Girvin, F. Marquardt, and R. J. Schoelkopf, Rev. Mod. Phys. 82, 1155 (2010).
[CrossRef]

Haelterman, M.

Kartashov, Y. V.

Kath, W. L.

Kenig, E.

E. Kenig, B. A. Malomed, M. C. Cross, and R. Lifshitz, Phys. Rev. E 80, 046202 (2009).
[CrossRef]

Kivshar, Y. S.

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

Konotop, V. V.

Korobov, V. I.

I. V. Barashenkov, M. M. Bogdan, and V. I. Korobov, Europhys. Lett. 15, 113 (1991).
[CrossRef]

Kumar, P.

Kutz, J. N.

Lai, S. K.

B. A. Malomed, E. Ding, K. W. Chow, and S. K. Lai, Phys. Rev. E 86, 036608 (2012).
[CrossRef]

Lam, C.-K.

C.-K. Lam, B. A. Malomed, K. W. Chow, and P. K. A. Wai, Eur. Phys. J. Spec. Top. 173, 233 (2009).
[CrossRef]

Lifshitz, R.

E. Kenig, B. A. Malomed, M. C. Cross, and R. Lifshitz, Phys. Rev. E 80, 046202 (2009).
[CrossRef]

Lipson, M.

M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta, Nature 441, 960 (2006).
[CrossRef]

Mak, W. C. K.

W. C. K. Mak, B. A. Malomed, and P. L. Chu, Phys. Rev. E 67, 026608 (2003).
[CrossRef]

Malomed, B. A.

B. A. Malomed, E. Ding, K. W. Chow, and S. K. Lai, Phys. Rev. E 86, 036608 (2012).
[CrossRef]

C.-K. Lam, B. A. Malomed, K. W. Chow, and P. K. A. Wai, Eur. Phys. J. Spec. Top. 173, 233 (2009).
[CrossRef]

E. Kenig, B. A. Malomed, M. C. Cross, and R. Lifshitz, Phys. Rev. E 80, 046202 (2009).
[CrossRef]

I. V. Barashenkov, S. Cross, and B. A. Malomed, Phys. Rev. E 68, 056605 (2003).
[CrossRef]

W. C. K. Mak, B. A. Malomed, and P. L. Chu, Phys. Rev. E 67, 026608 (2003).
[CrossRef]

Marquardt, F.

A. A. Clerk, M. H. Devoret, S. Girvin, F. Marquardt, and R. J. Schoelkopf, Rev. Mod. Phys. 82, 1155 (2010).
[CrossRef]

McGrath, C. W.

Rosanov, N. N.

N. N. Rosanov, Spatial Hysteresis and Optical Patterns (Springer, 2002).

Schmidt, B. S.

M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta, Nature 441, 960 (2006).
[CrossRef]

Schoelkopf, R. J.

A. A. Clerk, M. H. Devoret, S. Girvin, F. Marquardt, and R. J. Schoelkopf, Rev. Mod. Phys. 82, 1155 (2010).
[CrossRef]

Serkland, D. K.

Sharping, J. E.

M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta, Nature 441, 960 (2006).
[CrossRef]

Sheppard, A.

Torner, L.

Trillo, S.

Turner, A. C.

M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta, Nature 441, 960 (2006).
[CrossRef]

Vysloukh, V. A.

Wai, P. K. A.

C.-K. Lam, B. A. Malomed, K. W. Chow, and P. K. A. Wai, Eur. Phys. J. Spec. Top. 173, 233 (2009).
[CrossRef]

Williams, M. O.

Zemlyanaya, E. V.

I. V. Barashenkov and E. V. Zemlyanaya, Phys. Rev. Lett. 83, 2568 (1999).
[CrossRef]

Zezyulin, D. A.

Eur. Phys. J. Spec. Top. (1)

C.-K. Lam, B. A. Malomed, K. W. Chow, and P. K. A. Wai, Eur. Phys. J. Spec. Top. 173, 233 (2009).
[CrossRef]

Europhys. Lett. (1)

I. V. Barashenkov, M. M. Bogdan, and V. I. Korobov, Europhys. Lett. 15, 113 (1991).
[CrossRef]

Nature (1)

M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta, Nature 441, 960 (2006).
[CrossRef]

Opt. Express (1)

Opt. Lett. (6)

Phys. Rev. E (4)

W. C. K. Mak, B. A. Malomed, and P. L. Chu, Phys. Rev. E 67, 026608 (2003).
[CrossRef]

B. A. Malomed, E. Ding, K. W. Chow, and S. K. Lai, Phys. Rev. E 86, 036608 (2012).
[CrossRef]

I. V. Barashenkov, S. Cross, and B. A. Malomed, Phys. Rev. E 68, 056605 (2003).
[CrossRef]

E. Kenig, B. A. Malomed, M. C. Cross, and R. Lifshitz, Phys. Rev. E 80, 046202 (2009).
[CrossRef]

Phys. Rev. Lett. (1)

I. V. Barashenkov and E. V. Zemlyanaya, Phys. Rev. Lett. 83, 2568 (1999).
[CrossRef]

Physica D (1)

M. Bondila, I. V. Barashenkov, and M. M. Bogdan, Physica D 87, 314 (1995).
[CrossRef]

Rev. Mod. Phys. (1)

A. A. Clerk, M. H. Devoret, S. Girvin, F. Marquardt, and R. J. Schoelkopf, Rev. Mod. Phys. 82, 1155 (2010).
[CrossRef]

Other (3)

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

N. Akhmediev and A. Ankiewicz, eds., Dissipative Solitons: From Optics to Biology and Medicine (Springer, 2008).

N. N. Rosanov, Spatial Hysteresis and Optical Patterns (Springer, 2002).

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Figures (5)

Fig. 1.
Fig. 1.

Distributions of the field’s absolute value u and current j in fundamental solitons pinned to the hot spot in the focusing medium with (a) a=2, (b) a=4, and (c) a=6. (d) The same in the defocusing medium with a=4. The soliton in (c) is unstable, the others being stable. For the sake of the presentation, the current in panels (b) and (c) is multiplied by 0.3 and 0.1, respectively. These and all other examples are shown for γ=1.

Fig. 2.
Fig. 2.

Same as in Fig. 1, but for unstable dipole solitons in the (a) focusing and (b) defocusing media with a=4. The current in (a) is multiplied by 0.1.

Fig. 3.
Fig. 3.

Energy flow versus the gain for fundamental (Uf) and dipole (Ud) solitons in the focusing (a) and defocusing (b) media. Stable and unstable parts of the fundamental soliton branch are black and red, respectively. Dipoles are completely unstable. Circles in (a) and (b) correspond to the solitons shown in Figs. 1(a)1(c) and Figs. 1(d), 2(b), respectively. (c) The integral width versus gain for fundamental solitons in the focusing and defocusing media (black and green curves, respectively). The point at the junction of the curves represents the gain-guided linear mode, existing at a=ag.

Fig. 4.
Fig. 4.

(a) Critical, threshold, and initial (corresponding to the linear limit) values of the gain, acr, ath, and ag, versus linear loss γ for fundamental solitons in the focusing medium. (b) The real part of the instability growth rate versus a at γ=1.

Fig. 5.
Fig. 5.

(a) Stable and (b) unstable propagation of perturbed fundamental solitons from the upper branch in Fig. 3(a) (in the focusing medium) at a=3 and a=6, respectively. (c) The transformation of an unstable dipole in the case of focusing medium into a stable fundamental soliton at a=3.5. In all cases γ=1.

Equations (5)

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

iqξ=122qη2(k+iγ)q+(σiα)|q|2qaeη2/d2q*,
exp(η2/d2)π1/2dδ(η).
ag2(k)=(2/πd2)(k2+γ2)1/2
q(η)=(γ/2)1/2exp(iπ/8)[sinh(γ1/2(|η|+κ))]1+i,
q(η)=(3γ/2α)1/2exp(iδ)[sinh(γ1/2(|η|+κ)/μ1/2)]1+iμ,

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