## Three-dimensional structures in nonlinear cavities containing left-handed materials

Optics Express, Vol. 14, Issue 20, pp. 9338-9343 (2006)

http://dx.doi.org/10.1364/OE.14.009338

Acrobat PDF (304 KB)

### Abstract

We study the coupling between negative diffraction and direct dispersion in a nonlinear ring cavity containing slabs of Kerr nonlinear right-handed and left-handed materials. Within the mean field approximation, we show that a portion of the homogeneous response curve is affected by a three-dimensional modulational instability. We show numerically that the light distribution evolves through a sequence of three-dimensional dissipative structures with different lattice symmetry. These structures are unstable with respect to the upswitching process, leading to a premature transition to the upper branch in the homogeneous hysteresis cycle.

© 2006 Optical Society of America

## 1. Introduction

4. N. Akhmediev and A. Ankiewicz, *Dissipative Solitons* (Springer, Berlin, 2005). [CrossRef]

6. M. Tlidi, M. Haelterman, and P. Mandel, “Three-dimensional structures in diffractive and dispersive nonlinear ring cavities,” Europhys. Lett. **42**, 505–509 (1998). [CrossRef]

7. K. Staliunas, “Three-Dimensional Turing Structures and Spatial Solitons in Optical Parametric Oscillators,” Phys. Rev. Lett. **81**, 81–84 (1998). [CrossRef]

8. M. Tlidi and P. Mandel, “Three-Dimensional Optical Crystals and Localized Structures in Cavity Second Harmonic Generation,” Phys. Rev. Lett. **83**, 4995–4998 (1999). [CrossRef]

9. S. V. Fedorov, N. N. Rosanov, A. N. Shatsev, N. A. Veretenov, and A. G. Vladimirov, “Topologically multicharged and multihumped rotating solitons in wide-aperture lasers with a saturable absorber,” IEEE J. Quantum Electron. **39**, 197–205 (2003). [CrossRef]

10. M. Brambilla, T. Maggipinto, G. Patera, and L. Columbo, “Cavity Light Bullets: Three-Dimensional Localized Structures in a Nonlinear Optical Resonator,” Phys. Rev. Lett. **93**, 203901 (2004). [CrossRef] [PubMed]

11. D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and Negative Refractive Index,” Science **305**, 788–792 (2004). [CrossRef] [PubMed]

12. V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of *ε* and *µ*,” Sov. Phys. Usp. **10**, 509–514 (1968). [CrossRef]

13. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental Verification of a Negative Index of Refraction,” Science **292**, 77–79 (2001). [CrossRef] [PubMed]

14. A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, “Nanofabricated media with negative permeability at visible frequencies,” Nature **438**, 335–338 (2006). [CrossRef]

15. J. B. Pendry, “Negative Refraction Makes a Perfect Lens,” Phys. Rev. Lett. **85**, 39663969 (2000). [CrossRef]

16. A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear Properties of Left-Handed Metamaterials,” Phys. Rev. Lett. **91**, 037401 (2003). [CrossRef] [PubMed]

17. N. Lazarides and G. P. Tsironis, “Coupled nonlinear Schrödinger field equations for electromagnetic wave propagation in nonlinear left-handed materials,” Phys. Rev. E **71**, 036614 (2005). [CrossRef]

18. V. M. Agranovich, Y. R. Shen, R. H. Baughman, and A. A. Zakhidov, “Linear and nonlinear wave propagation in negative refraction metamaterials,” Phys. Rev. B **69**, 165112 (2004). [CrossRef]

## 2. Model equations and modulational instability

*A*of the electric field during propagation in the RHM can be described by the standard nonlinear Schrödinger equation (NLSE) and it has been shown that the NLSE also applies to LHMs [19

19. M. Scalora, M. S. Syrchin, N. Akozbek, E. Y. Poliakov, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, and A. M. Zheltikov, “Generalized Nonlinear Schrödinger Equation for Dispersive Susceptibility and Permeability,” Phys. Rev. Lett. **95**, 013902 (2005). [CrossRef] [PubMed]

*z*,

*x*and

*y*are the longitudinal and transverse space coordinates and τ is the time coordinate in the frame moving with the group velocity. Subscripts R and L denote parameters in the RHM and LHM. The strength of the nonlinearity is determined by the coefficients

*γ*

_{R,L}and the diffraction term is inversely proportional to the wavenumber k R,L. This means that the diffraction coefficient is always negative in a LHM [19

19. M. Scalora, M. S. Syrchin, N. Akozbek, E. Y. Poliakov, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, and A. M. Zheltikov, “Generalized Nonlinear Schrödinger Equation for Dispersive Susceptibility and Permeability,” Phys. Rev. Lett. **95**, 013902 (2005). [CrossRef] [PubMed]

19. M. Scalora, M. S. Syrchin, N. Akozbek, E. Y. Poliakov, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, and A. M. Zheltikov, “Generalized Nonlinear Schrödinger Equation for Dispersive Susceptibility and Permeability,” Phys. Rev. Lett. **95**, 013902 (2005). [CrossRef] [PubMed]

*β*has been calculated based on a Drude model describing the dispersive properties of

*ε*(

*ω*) and

*µ*(

*ω*), which has been verified to a good extent for the microwave LHMs. There, it is demonstrated that

*β*can be positive or negative depending on the ratio of the working frequency to the resonance frequency of the LHM. The stability of the NLSE for LHM has been discussed in Ref. [21

21. M. Marklund, P. K. Shukla, and L. Stenflo, “Ultrashort solitons and kinetic effects in nonlinear metamaterials,” Phys. Rev. E **73**, 037601 (2006). [CrossRef]

*ω*

_{0}/2

*π*. The intracavity field

*A*undergoes a coherent superposition with this input field at the input mirror:

*ρ*is the product of the amplitude reflection coefficients of the input and output mirrors,

*θ*is the amplitude transmission coefficient at the input mirror.

*A*

_{m}to a single partial differential equation using the well-known mean field approach. This method is based on three assumptions: (1) reflections at the interfaces of the layers can be neglected; (2) the dissipative Fresnel number is large; (3) the cavity is shorter than the diffraction, dispersion and nonlinearity space scales. The first of these approximations needs some more explanation as such reflections and the associated counterpropagating beam could alter the dynamics considerably. Having two independent material parameters (

*ε*and

*µ*), the structure of the LHM can be tuned to have an impedance

*T*is introduced, where

*T*=

*πt*/(

*𝓕t*

_{r}), with

*t*

_{r}the cavity roundtrip time and 𝓕 its finesse. The slow time evolution of the envelope

*A*(

*x*,

*y*,τ,

*T*) is then governed by

6. M. Tlidi, M. Haelterman, and P. Mandel, “Three-dimensional structures in diffractive and dispersive nonlinear ring cavities,” Europhys. Lett. **42**, 505–509 (1998). [CrossRef]

*β*are an arithmetic average of the respective coefficients in the NLSE weighted by the lengths of the layers and, consequently, that they can be tuned between the values that they would have if the cavity were filled with either of both materials. The inclusion of the left-handed element in the ring cavity thus allows to engineer the strength of the diffraction coefficient D and even to change its sign when

*l*

_{L}>-

*l*

_{R}

*k*

_{L}/

*k*

_{R}. Furthermore, it is possible to obtain both signs for

*β*, since both the RHM and the LHM can come with normal or anomalous dispersion. As we will see, these changes of sign have an important consequence in the 3D pattern formation process. We will consider the case of

*𝒟*<0 and

*β*<0 in the numerical simulations presented below.

*A*. Additionally, the combination of the RHM and the LHM allows to satisfy the resonance condition with cavities that are much thinner than λ=

*c*/

*ν*[22

22. N. Engheta, “An Idea for Thin Subwavelength Cavity Resonators Using Metamaterials With Negative Permittivity and Permeability,” IEEE Ant. Wireless Prop. Lett. **1**, 10–13 (2002). [CrossRef]

*A*

_{s}of Eq. (3) are

*𝓔*=[1+i(Δ-Γ|

*A*

_{s}|

^{2})]

*A*

_{s}(Fig. 2). |

*A*

_{s}|

^{2}as a function of |

*𝓔*|

^{2}is single-valued for Δ<√3 and multiple-valued for Δ>√3. We have performed the stability analysis of these HSS. With periodic boundaries, the deviation from the steady state is taken proportional to exp (i

**k**·

**r**+

*λT*) with

**k**=(

*k*

_{x},

*k*

_{y},

*k*

_{τ}) and

**r**=(

*x*,y,

*τ*). The threshold associated with modulational instability is |

*A*

_{c}|=1/√Γ and

*𝓔*(

*k*

_{x},

*k*

_{y},

*k*

_{τ}) if 𝒟 and

*β*have equal sign, and an hyperboloid for opposite signs (see Fig. 3).

## 3. Three-dimensional structures and up-switching process

*k*

_{x},

*k*

_{y},

*k*

_{τ}), as given by the linear stability analysis, which means that they are of constant magnitude, but arbitrarily directed in the Fourier space. Therefore, the modulational instability will tend to impose periodicity in the Euclidean space (

*x*,

*y*,τ). Due to the nonlinearity, the unstable modes will interact, resulting in the formation of dissipative structures. However, in the same input intensity range, stable homogeneous states still exist on the upper branch. A second possible dynamical process is thus the up-switching phenomenon, which will tend to restore uniformity in the transverse plane of the cavity.

*𝒟*<0 and

*β*<0. The input is chosen in the unstable part of the lower steady state branch, i.e., between the threshold for modulational instability and the normal up-switching point (red dotted branch in Fig. 2). We have discretized Eq. (3) by applying a 6-point Crank-Nicholson scheme with periodic boundary conditions in all directions (number of mesh points is 40×40×40 in the cube 0<

*x*,

*y*,τ<16) and as initial condition the homogeneous field

*A*

_{s}to which a small noise term (10

^{-5}) is added.

_{num}=5.3) compares well to the value obtained from the linear stability analysis (Γ

_{c}=2

*π*/

*k*

_{c}=

*π*); the small difference is due to the periodic boundary conditions. Subsequently, the bullets on diagonal planes of the cubic structure merge to form plates of interconnected bullets [Fig. 4(b)]. The formation of dissipative structures thus seems to dominate the dynamics at small times. However, the patterns further evolve to a homogeneous state (note the contracting scale in the movie of Fig. 4). We conclude therefore that all dissipative structures are unstable and that the intracavity field switches to the upper homogeneous state. Note the dramatic difference with the positive diffraction case, where stable light bullets are formed [6

6. M. Tlidi, M. Haelterman, and P. Mandel, “Three-dimensional structures in diffractive and dispersive nonlinear ring cavities,” Europhys. Lett. **42**, 505–509 (1998). [CrossRef]

*A*

_{c}|

^{2}=1/Γ rather than at the limit point

## 4. Conclusions

## Acknowledgments

## References and links

1. | N. N. Rosanov, |

2. | K. Staliunas and J. V. Sanchez-Morcillo, |

3. | P. Mandel and M. Tlidi, “Transverse dynamics in cavity nonlinear optics (2000–2003),” J. Opt. B: Quantum Semiclass. Opt. |

4. | N. Akhmediev and A. Ankiewicz, |

5. | Y. S. Kivshar and G. P. Agrawal, |

6. | M. Tlidi, M. Haelterman, and P. Mandel, “Three-dimensional structures in diffractive and dispersive nonlinear ring cavities,” Europhys. Lett. |

7. | K. Staliunas, “Three-Dimensional Turing Structures and Spatial Solitons in Optical Parametric Oscillators,” Phys. Rev. Lett. |

8. | M. Tlidi and P. Mandel, “Three-Dimensional Optical Crystals and Localized Structures in Cavity Second Harmonic Generation,” Phys. Rev. Lett. |

9. | S. V. Fedorov, N. N. Rosanov, A. N. Shatsev, N. A. Veretenov, and A. G. Vladimirov, “Topologically multicharged and multihumped rotating solitons in wide-aperture lasers with a saturable absorber,” IEEE J. Quantum Electron. |

10. | M. Brambilla, T. Maggipinto, G. Patera, and L. Columbo, “Cavity Light Bullets: Three-Dimensional Localized Structures in a Nonlinear Optical Resonator,” Phys. Rev. Lett. |

11. | D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and Negative Refractive Index,” Science |

12. | V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of |

13. | R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental Verification of a Negative Index of Refraction,” Science |

14. | A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, “Nanofabricated media with negative permeability at visible frequencies,” Nature |

15. | J. B. Pendry, “Negative Refraction Makes a Perfect Lens,” Phys. Rev. Lett. |

16. | A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear Properties of Left-Handed Metamaterials,” Phys. Rev. Lett. |

17. | N. Lazarides and G. P. Tsironis, “Coupled nonlinear Schrödinger field equations for electromagnetic wave propagation in nonlinear left-handed materials,” Phys. Rev. E |

18. | V. M. Agranovich, Y. R. Shen, R. H. Baughman, and A. A. Zakhidov, “Linear and nonlinear wave propagation in negative refraction metamaterials,” Phys. Rev. B |

19. | M. Scalora, M. S. Syrchin, N. Akozbek, E. Y. Poliakov, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, and A. M. Zheltikov, “Generalized Nonlinear Schrödinger Equation for Dispersive Susceptibility and Permeability,” Phys. Rev. Lett. |

20. | P. Tassin, G. Van der Sande, I. Veretennicoff, M. Tlidi, and P. Kockaert, “Analytical model for the optical propagation in a nonlinear left-handed material,” in |

21. | M. Marklund, P. K. Shukla, and L. Stenflo, “Ultrashort solitons and kinetic effects in nonlinear metamaterials,” Phys. Rev. E |

22. | N. Engheta, “An Idea for Thin Subwavelength Cavity Resonators Using Metamaterials With Negative Permittivity and Permeability,” IEEE Ant. Wireless Prop. Lett. |

**OCIS Codes**

(190.4420) Nonlinear optics : Nonlinear optics, transverse effects in

**ToC Category:**

Metamaterials

**History**

Original Manuscript: May 31, 2006

Revised Manuscript: September 5, 2006

Manuscript Accepted: September 8, 2006

Published: October 2, 2006

**Citation**

Philippe Tassin, Guy Van der Sande, Nikolay Veretenov, Pascal Kockaert, Irina Veretennicoff, and Mustapha Tlidi, "Three-dimensional structures in nonlinear cavities containing left-handed materials," Opt. Express **14**, 9338-9343 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-20-9338

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### References

- N. N. Rosanov, Spatial Hysteresis and Optical Patterns (Springer-Verlag, Berlin, 2002).
- K. Staliunas and J. V. Sanchez-Morcillo, Transverse Patterns in Nonlinear Optical Resonators (Springer-Verlag, Berlin, 2003).
- P. Mandel and M. Tlidi, "Transverse dynamics in cavity nonlinear optics (2000-2003)," J. Opt. B: Quantum Semiclass. Opt. 6, R60-R75 (2004). [CrossRef]
- N. Akhmediev and A. Ankiewicz, Dissipative Solitons (Springer, Berlin, 2005). [CrossRef]
- Y. S. Kivshar and G. P. Agrawal, Optical Solitons (Academic Press, San Diego, 2003).
- M. Tlidi, M. Haelterman, and P. Mandel, "Three-dimensional structures in diffractive and dispersive nonlinear ring cavities," Europhys. Lett. 42, 505-509 (1998). [CrossRef]
- K. Staliunas, "Three-Dimensional Turing Structures and Spatial Solitons in Optical Parametric Oscillators," Phys. Rev. Lett. 81, 81-84 (1998). [CrossRef]
- M. Tlidi and P. Mandel, "Three-Dimensional Optical Crystals and Localized Structures in Cavity Second Harmonic Generation," Phys. Rev. Lett. 83, 4995-4998 (1999). [CrossRef]
- S. V. Fedorov, N. N. Rosanov, A. N. Shatsev, N. A. Veretenov, and A. G. Vladimirov, "Topologically multicharged and multihumped rotating solitons in wide-aperture lasers with a saturable absorber," IEEE J. Quantum Electron. 39, 197-205 (2003). [CrossRef]
- M. Brambilla, T. Maggipinto, G. Patera, and L. Columbo, "Cavity Light Bullets: Three-Dimensional Localized Structures in a Nonlinear Optical Resonator," Phys. Rev. Lett. 93, 203901 (2004). [CrossRef] [PubMed]
- D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, "Metamaterials and Negative Refractive Index," Science 305, 788-792 (2004). [CrossRef] [PubMed]
- V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of ε and μ," Sov. Phys. Usp. 10, 509-514 (1968). [CrossRef]
- R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental Verification of a Negative Index of Refraction," Science 292, 77-79 (2001). [CrossRef] [PubMed]
- A. N. Grigorenko, A. K. Geim, H. F. Gleeson, Y. Zhang, A. A. Firsov, I. Y. Khrushchev, and J. Petrovic, "Nanofabricated media with negative permeability at visible frequencies," Nature 438, 335-338 (2006). [CrossRef]
- J. B. Pendry, "Negative Refraction Makes a Perfect Lens," Phys. Rev. Lett. 85, 39663969 (2000). [CrossRef]
- A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, "Nonlinear Properties of Left-Handed Metamaterials," Phys. Rev. Lett. 91, 037401 (2003). [CrossRef] [PubMed]
- N. Lazarides and G. P. Tsironis, "Coupled nonlinear Schr¨odinger field equations for electromagnetic wave propagation in nonlinear left-handed materials, " Phys. Rev. E 71, 036614 (2005). [CrossRef]
- V. M. Agranovich, Y. R. Shen, R. H. Baughman, and A. A. Zakhidov, "Linear and nonlinear wave propagation in negative refraction metamaterials," Phys. Rev. B 69, 165112 (2004). [CrossRef]
- M. Scalora, M. S. Syrchin, N. Akozbek, E. Y. Poliakov, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, and A. M. Zheltikov, "Generalized Nonlinear Schr¨odinger Equation for Dispersive Susceptibility and Permeability," Phys. Rev. Lett. 95, 013902 (2005). [CrossRef] [PubMed]
- P. Tassin, G. Van der Sande, I. Veretennicoff, M. Tlidi, and P. Kockaert, "Analytical model for the optical propagation in a nonlinear left-handed material," in Metamaterials, T. Szoplik, E. ¨ Ozbay, C.M. Soukoulis, N. I. Zheludev, eds., Proc. SPIE 5955, 595500X (2005).
- M. Marklund, P. K. Shukla, and L. Stenflo, "Ultrashort solitons and kinetic effects in nonlinear metamaterials," Phys. Rev. E 73, 037601 (2006). [CrossRef]
- N. Engheta, "An Idea for Thin Subwavelength Cavity Resonators Using Metamaterials With Negative Permittivity and Permeability," IEEE Ant. Wireless Prop. Lett. 1, 10-13 (2002). [CrossRef]

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