## Defect superlattice solitons

Optics Express, Vol. 15, Issue 22, pp. 14498-14503 (2007)

http://dx.doi.org/10.1364/OE.15.014498

Acrobat PDF (507 KB)

### Abstract

We reveal theoretically that defect superlattice solitons (DSSs) exist at the defect site in one-dimensional optical superlattices with focusing saturable nonlinearity. Solitons with some unique properties exist in superlattices with defects. For a positive defect, solitons exist at the semi-infinite gap, and solitons are stable at low power but unstable at high power. For a negative defect, most solitons exist in the first finite gap and can propagate stably. In particular, it is found that the solitons can be divided into two equal parts upon propagation in a certain regime of parameters.

© 2007 Optical Society of America

## 1. Introduction

01. D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behavior in linear and nonlinear waveguide lattices,” Nature **424**, 817–823 (2003). [CrossRef] [PubMed]

02. D. K. Campbell, S. Flach, and Y. S. Kivshar, “Localizing energy through nonlinearity and discreteness,” Phys. Today **57**, 43–49 (2004). [CrossRef]

03. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Rotary solitons in Bessel optical lattices,” Phys. Rev. Lett. **93**, 093904 (2004). [CrossRef] [PubMed]

04. Y. J. He and H. Z. Wang, “(1+1)-dimensional dipole solitons supported by optical lattice,” Opt. Express **14**, 9832–9837 (2006) http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-21-9832 [CrossRef] [PubMed]

05. P. J. Y. Louis, E. A. Ostrovskaya, and Y. S. Kivshar, “Dispersion control for matter waves and gap solitons in optical superlattices,” Phys. Rev. A **71**, 023612 (2005). [CrossRef]

06. M. A. Porter, P.G. Kevrekidis, R. Carretero-González, and D. J. Frantzeskakis, “Dynamics and manipulation of matter-wave solitons in optical superlattices,” Phys. Lett. A **352**, 210–215 (2006). [CrossRef]

07. N. G. R. Broderick and C. M. de Sterke, “Theory of grating superstructures,” Phys. Rev. E **55**, 3634–3646 (2006). [CrossRef]

08. K. Yagasaki, I. M. Merhasin, B. A. Malomed, T. Wagenknecht, and A. R. Champneys, “Gap solitons in Bragg gratings with a harmonic superlattice,” Europhys. Lett. **74**, 1006–1012 (2006). [CrossRef]

05. P. J. Y. Louis, E. A. Ostrovskaya, and Y. S. Kivshar, “Dispersion control for matter waves and gap solitons in optical superlattices,” Phys. Rev. A **71**, 023612 (2005). [CrossRef]

06. M. A. Porter, P.G. Kevrekidis, R. Carretero-González, and D. J. Frantzeskakis, “Dynamics and manipulation of matter-wave solitons in optical superlattices,” Phys. Lett. A **352**, 210–215 (2006). [CrossRef]

09. J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature **422**, 147–150 (2003). [CrossRef] [PubMed]

10. Z. Chen and K. McCarthy, “Spatial soliton pixels from partially incoherent light,” Opt. Lett. **27**, 2019–2021 (2002). [CrossRef]

11. F. Fedele, J. Yang, and Z. Chen, “Defect modes in one-dimensional photonic lattices,” Opt. Lett. **30**, 1506–1508 (2005). [CrossRef] [PubMed]

12. A. A. Sukhorukov and Y. S. Kivshar, “Nonlinear localized waves in a periodic medium,” Phys. Rev. Lett. **87**, 083901 (2001). [CrossRef] [PubMed]

13. J. Yang and Z. Chen, “Defect solitons in photonic lattices,” Phys. Rev. E **73**, 026609 (2006). [CrossRef]

## 2. The model

14. H. Martin, E. D. Eugenieva, Z. Chen, and D. N. Christodoulides, “Discrete solitons and soliton-induced dislocations in partially coherent photonic lattices,” Phys. Rev. Lett. **92**, 123902 (2004). [CrossRef] [PubMed]

13. J. Yang and Z. Chen, “Defect solitons in photonic lattices,” Phys. Rev. E **73**, 026609 (2006). [CrossRef]

13. J. Yang and Z. Chen, “Defect solitons in photonic lattices,” Phys. Rev. E **73**, 026609 (2006). [CrossRef]

15. B. A. Malomed, T. Mayteevarunyoo, E. A. Ostrovskaya, and Y. S. Kivshar, “Coupled-mode theory for spatial gap solitons in optically induced lattices,” Phys. Rev. E **71**, 056616 (2005). [CrossRef]

*q*is the slowly varying amplitude of the probe beam, z is the propagation distance (in units of 2

*k*

_{1}

*D*

^{2}/

*π*

^{2}),

*k*

_{1}=

*k*

_{0}

*n*,

_{e}*n*is the unperturbed refractive index,

_{e}*k*

_{0}= 2

*π*/

*λ*

_{0}is the wave number (

*λ*

_{0}is the wavelength in a vacuum), D is the lattice spacing,

*x*is the transverse distance (in units of

*D*/

*π*),

*E*

_{0}is the applied dc field [in units of

*π*

^{2}/(

*k*

^{2}

_{0}

*n*

^{4}

_{e}*D*

^{2}

*γ*

_{33})], and

*γ*

_{33}is the electro-optic coefficient of the crystal.

*I*is the intensity profile of the optical lattice described by

_{L}*I*is the peak intensity without modulating the uniform photonic lattice (

_{0}*i.e.*

*ε*

_{1}=1).

*ε*

_{1}represents the modulation parameter of superlattice peak intensity. Therefore, the peak intensity of superlattices is determined by the

*I*and

_{0}*ε*

_{1}.

*ε*

_{2}represents the modulation parameter of the peak intensity of defect. The constant

*c*is introduced to make the peak intensity of the defect at

*ε*

_{2}=0 approximate to that of superlattices at a given value of

*ε*

_{1}. We take

*c*= 0.85 when

*ε*

_{1}=0.3. We consider the intensity of defect with a single oscillation of a sine function, and the defect locates at the center of superlattices and is modulated by

*ε*

_{2}, as shown in Figs. 1(a)-1(c). In Fig. 1, different defect optical superlattice profiles with

*I*=3 and

_{0}*ε*

_{1}=0.3 are shown at (a)

*ε*

_{2}=0.45, (b)

*ε*

_{2}= -0.45, and (c)

*ε*

_{2}=0. The superlattice potential given by Eq. (2) can be induced optically by launching a beam into the amplitude mask whose intensity distribution of transmission light is the same as the superlattice potential.

*D*= 30

*μm*,

*ε*

_{1}=0.3,

*λ*=0.5

*μm*,

*n*=2.3,

_{e}*γ*

_{33}=280

*pm*/

*V*, then

*x*=1,

*z*=1, and

*E*=1 correspond to 9.55

_{0}*μ*m, 5.3 mm, and 8.86 V/mm, respectively. We take

*I*=3 and

_{0}*E*=6, which are typical in experimental conditions as shown in Ref. [16].

_{0}*q*(

*x*,

*z*) =

*f*(

*x*) exp(

*ikx*+

*iμz*) to the linear version of Eq. (1) where

*μ*is the real propagation constant,

*k*is the Bloch wave number, and

*f*(

*x*) is the complex periodic function [here

*f*(

*x*) =

*f*(

*x*+

*π*)]. The substitution of the light field in such form yields the eigenvalue problem [17]

*μ*(

*E*) of the (infinite) superlattice as shown in Fig. 1(d), which shows the bandgap structure in the uniform (defect-free) superlattice.

_{0}*q*(

*x*,

*z*) =

*f*(

*x*)exp(

*iμz*/

*z*), where

*f*(

*x*) is the real function satisfying the equation

*P*of a soliton is defined as

*P*= ∫

^{+∞}

_{-∞}

*f*

^{2}(

*x*)

*dx*. By numerically solving Eq. (4) using the shooting method, we get the soliton profiles in Section 3 as in Figs. 2(c)-2(e). To indicate the stability of DSSs, we search for the perturbed solution of Eq. (1) in the form

*q*(

*x*,

*z*) = [

*f*(

*x*) +

*h*(

*x*,

*z*) +

*ie*(

*x*,

*z*)]exp(

*iμz*), where

*h*(

*x*,

*z*), and

*e*(

*x*,

*z*) are the real and imaginary parts of perturbation that can grow with complex rate δ upon propagation. Omit the neglectable nonlinear terms in Eq. (1), the eigenmodes of coupled equations as follows:

*δ*).

## 3. Numerical results

*x*)], where

*ρ*(

*x*) is a Gaussian random function with <

*ρ*>=0 and <

*ρ*

^{2}> =

*δ*

^{2}(we choose that δ is equal to 10% of the input soliton amplitude).

*ε*

_{2}= 0.45 as a typical case for the positive defect. Figure 2(a) shows that the power of DSSs increase with the increase of propagation constant

*μ*. Figures 2(c)-2(e) show the profiles of DSSs with different propagation constants

*μ*=-2.35, -1.8, and -1.5, respectively. Figures 2(f)-2(h) show the solitons’ propagations corresponding to Figs. 2(c)-2(e), respectively. In the range of propagation constant -2.35≤

*μ*≤-1.8, the DSSs can stably propagate, but the propagations of DSSs are unstable when the propagation constant is

*μ*>-1.8 (corresponding to higher power). This phenomenon is similar to defect solitons in a regular lattice [13

**73**, 026609 (2006). [CrossRef]

**73**, 026609 (2006). [CrossRef]

*ε*

_{2}= 0, DSSs exist in the semi-infinite gap and their stability is similar to the positive defect except in the case of the lower power. Figure 3(a) shows that the power of DSSs increases with the increase of the propagation constant

*μ*. Figures 3(c)-3(e) show the profiles of DSSs with different propagation constants

*μ*=-2.4, -1.9, and -2.55, respectively, and Figs. 3(f)-3(h) show their propagations. In the range of propagation constant -2.52≤

*μ*≤-1.85, the DSSs can be stably propagated, but they are unstable with propagation constants

*μ*>-1.85 (corresponding to higher power) and

*μ*<-2.52 (corresponding to lower power). The power

*P*increases with the increase of propagation constant

*μ*when

*μ*≥-2.52. At high power, the propagations of solitons are unstable, which is different from the usual Vakhitov-Kolokolov criterion [13

**73**, 026609 (2006). [CrossRef]

*μ*≤-1.85, the propagations of solitons are stable because of

*dP*/

*dμ*> 0, but unstable when

*μ*<-2.52 (such as near point A) because of

*dP*/

*dμ*< 0. The stability of DSSs is analyzed by solving Eq. (5) to get the growth rate Re(δ), as shown in Fig. 3(b), which is in agreement with the above analytic result.

^{st}and 2

^{nd}bands). As a typical case, we consider

*ε*

_{2}= -0.45. Figure 4(a) shows that the power of DSSs increases with the increase of

*μ*. Figures 4(b) and 4(c) show the profile of DSSs for

*μ*= -3.31 and

*μ*= -3.85, respectively. For the negative defect, most of DSSs exist in the first finite gap and can be stable in propagation, which is the same as that of uniform photonic lattices [13

**73**, 026609 (2006). [CrossRef]

*ε*

_{2}=-0.2; i.e., for

*ε*

_{2}>-0.2, the DSSs exist in the semi-infinite gap, while for

*ε*

_{2}<-0.2, DSSs exist in the first finite gap. For

*ε*

_{2}=-0.2, DSSs exist not only in the semi-infinite gap but also in the first finite gap, as shown in Figs. 5(a) and (b). We give the DSSs’ stable/unstable domains according to the relation defect of the modulation parameter to the propagation constant in Fig. 5, where the stable domains are shown in gray.

*ε*

_{2}from zero to negative when

*ε*

_{1}is changed from high to low. As an example, we consider the cases of

*ε*

_{1}=0.1 and

*ε*

_{2}=-0.2. Figures 6(a) and 6(b) show the profiles of DSSs for

*μ*=-2.67 and

*μ*=-2.7, respectively, and Figs. 6(c) and 6(d) show that propagation splits. It is worthy of discussion as to whether the defect of superlattices can be used as the Y waveguide by this phenomenon. Table 1 shows the regions of parameters in which DSSs split upon propagation.

## 4. Summary

18. W. H. Chen, Y. J. He, and H. Z. Wang, “Surface defect gap solitons,” Opt. Express **14**, 11271–11276 (2006). [CrossRef] [PubMed]

## Acknowledgments

## References and links

01. | D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behavior in linear and nonlinear waveguide lattices,” Nature |

02. | D. K. Campbell, S. Flach, and Y. S. Kivshar, “Localizing energy through nonlinearity and discreteness,” Phys. Today |

03. | Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Rotary solitons in Bessel optical lattices,” Phys. Rev. Lett. |

04. | Y. J. He and H. Z. Wang, “(1+1)-dimensional dipole solitons supported by optical lattice,” Opt. Express |

05. | P. J. Y. Louis, E. A. Ostrovskaya, and Y. S. Kivshar, “Dispersion control for matter waves and gap solitons in optical superlattices,” Phys. Rev. A |

06. | M. A. Porter, P.G. Kevrekidis, R. Carretero-González, and D. J. Frantzeskakis, “Dynamics and manipulation of matter-wave solitons in optical superlattices,” Phys. Lett. A |

07. | N. G. R. Broderick and C. M. de Sterke, “Theory of grating superstructures,” Phys. Rev. E |

08. | K. Yagasaki, I. M. Merhasin, B. A. Malomed, T. Wagenknecht, and A. R. Champneys, “Gap solitons in Bragg gratings with a harmonic superlattice,” Europhys. Lett. |

09. | J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature |

10. | Z. Chen and K. McCarthy, “Spatial soliton pixels from partially incoherent light,” Opt. Lett. |

11. | F. Fedele, J. Yang, and Z. Chen, “Defect modes in one-dimensional photonic lattices,” Opt. Lett. |

12. | A. A. Sukhorukov and Y. S. Kivshar, “Nonlinear localized waves in a periodic medium,” Phys. Rev. Lett. |

13. | J. Yang and Z. Chen, “Defect solitons in photonic lattices,” Phys. Rev. E |

14. | H. Martin, E. D. Eugenieva, Z. Chen, and D. N. Christodoulides, “Discrete solitons and soliton-induced dislocations in partially coherent photonic lattices,” Phys. Rev. Lett. |

15. | B. A. Malomed, T. Mayteevarunyoo, E. A. Ostrovskaya, and Y. S. Kivshar, “Coupled-mode theory for spatial gap solitons in optically induced lattices,” Phys. Rev. E |

16. | I. Makasyuk, Z. Chen, and J. Yang, “Observation of light confinement by defects in optically-induced photonic lattices,” in |

17. | Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Surface gap solitons,” Phys. Rev. Lett. |

18. | W. H. Chen, Y. J. He, and H. Z. Wang, “Surface defect gap solitons,” Opt. Express |

19. | Y. J. He, W. H. Chen, H. Z. Wang, and B. A. Malomed, “Surface superlattice gap solitons,” Opt. Lett. |

20. | W. H. Chen, Y. J. He, and H. Z. Wang, “Surface defect superlattice solitons,” J. Opt. Soc. Am. B |

**OCIS Codes**

(190.0190) Nonlinear optics : Nonlinear optics

(190.5530) Nonlinear optics : Pulse propagation and temporal solitons

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: July 19, 2007

Revised Manuscript: September 3, 2007

Manuscript Accepted: September 19, 2007

Published: October 19, 2007

**Citation**

W. H. Chen, Y. J. He, and H. Z. Wang, "Defect superlattice solitons," Opt. Express **15**, 14498-14503 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-22-14498

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### References

- D. N. Christodoulides, F. Lederer, and Y. Silberberg, "Discretizing light behavior in linear and nonlinear waveguide lattices," Nature 424, 817-823 (2003). [CrossRef] [PubMed]
- D. K. Campbell, S. Flach, and Y. S. Kivshar, "Localizing energy through nonlinearity and discreteness," Phys. Today 57, 43-49 (2004). [CrossRef]
- Y. V. Kartashov, V. A. Vysloukh, and L. Torner, "Rotary solitons in Bessel optical lattices," Phys. Rev. Lett. 93, 093904 (2004). [CrossRef] [PubMed]
- Y. J. He and H. Z. Wang, "(1+1)-dimensional dipole solitons supported by optical lattice," Opt. Express 14, 9832-9837 (2006) http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-21-9832> [CrossRef] [PubMed]
- P. J. Y. Louis, E. A. Ostrovskaya, and Y. S. Kivshar, "Dispersion control for matter waves and gap solitons in optical superlattices," Phys. Rev. A 71, 023612 (2005). [CrossRef]
- M. A. Porter, P.G. Kevrekidis, R. Carretero-González, D. J. Frantzeskakis, "Dynamics and manipulation of matter-wave solitons in optical superlattices," Phys. Lett. A 352, 210-215 (2006). [CrossRef]
- N. G. R. Broderick and C. M. de Sterke, "Theory of grating superstructures," Phys. Rev. E 55, 3634-3646 (2006). [CrossRef]
- K. Yagasaki I. M. Merhasin, B. A. Malomed, T. Wagenknecht, and A. R. Champneys, "Gap solitons in Bragg gratings with a harmonic superlattice," Europhys. Lett. 74, 1006-1012 (2006). [CrossRef]
- J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, "Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices," Nature 422, 147-150 (2003). [CrossRef] [PubMed]
- Z. Chen and K. McCarthy, "Spatial soliton pixels from partially incoherent light," Opt. Lett. 27, 2019-2021 (2002). [CrossRef]
- F. Fedele, J. Yang, and Z. Chen, "Defect modes in one-dimensional photonic lattices," Opt. Lett. 30, 1506-1508 (2005). [CrossRef] [PubMed]
- A. A. Sukhorukov and Y. S. Kivshar, "Nonlinear localized waves in a periodic medium," Phys. Rev. Lett. 87, 083901 (2001). [CrossRef] [PubMed]
- J. Yang and Z. Chen, "Defect solitons in photonic lattices," Phys. Rev. E 73, 026609 (2006). [CrossRef]
- H. Martin, E. D. Eugenieva, Z. Chen, and D. N. Christodoulides, "Discrete solitons and soliton-induced dislocations in partially coherent photonic lattices," Phys. Rev. Lett. 92, 123902 (2004). [CrossRef] [PubMed]
- B. A. Malomed, T. Mayteevarunyoo, E. A. Ostrovskaya, and Y. S. Kivshar, "Coupled-mode theory for spatial gap solitons in optically induced lattices," Phys. Rev. E 71, 056616 (2005). [CrossRef]
- I. Makasyuk, Z. Chen, and J. Yang, "Observation of light confinement by defects in optically-induced photonic lattices," in Nonlinear Guided Waves and Their Applications, Technical Digest (CD) (Optical Society of America, 2005), paper TuC8.
- Y. V. Kartashov, V. A. Vysloukh, and L. Torner, "Surface gap solitons," Phys. Rev. Lett. 96, 073901 (2006). [CrossRef] [PubMed]
- W. H. Chen, Y. J. He, and H. Z. Wang, "Surface defect gap solitons," Opt. Express 14, 11271-11276 (2006). [CrossRef] [PubMed]
- Y. J. He, W. H. Chen, H. Z. Wang, and B. A. Malomed, "Surface superlattice gap solitons," Opt. Lett. 32, 1390-1392 (2007). [CrossRef] [PubMed]
- W. H. Chen, Y. J. He, and H. Z. Wang, "Surface defect superlattice solitons," J. Opt. Soc. Am. B 24, 2584-2588 (2007) [CrossRef]

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