Abstract

We demonstrate the existence of spatial soliton solutions in photonic crystal fibers (PCF’s). These guided localized nonlinear waves appear as a result of the balance between the linear and nonlinear diffraction properties of the inhomogeneous photonic crystal cladding. The spatial soliton is realized self-consistently as the fundamental mode of the effective fiber defined simultaneously by the PCF linear and the self-induced nonlinear refractive indices. It is also shown that the photonic crystal cladding is able to stabilize these solutions, which would be unstable otherwise if the medium was entirely homogeneous.

© 2003 Optical Society of America

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References

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Appl. Phys. Lett.

H. A. Haus, �??Higher order trapped light beam solutions,�?? Appl. Phys. Lett. 8, 128�??129 (1966).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Lett.

Phys. Rev. E

A. Ferrando, M. Zacares, and P. F. de Cordoba, �??Ansatz-independent solution of a soliton in a strong dispersion-management system,�?? Phys. Rev. E 62, 7320�??7329 (2000).
[CrossRef]

P. Xie, Z.-Q. Zhang, and X. Zhang, �??Gap solitons and soliton trains in .nite-sized two-dimensional periodic and quasiperiodic photonic crystals,�?? Phys. Rev. E 67, 026607-1�?? 026607-5 (2003).
[CrossRef]

S. F.Mingaleev, Y. S. Kivshar, and R. A. Sammut, �??Long-range interaction and nonlinear localized modes in photonic crystal waveguides,�?? Phys. Rev. E 62, 5777�??5782 (2000).
[CrossRef]

Phys. Rev. Lett.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, �??Discrete spatial optical solitons in waveguide arrays,�?? Phys. Rev. Lett. 81, 3383�??3387 (1998).
[CrossRef]

S. John and N. Akozbek, �??Nonlinear optical solitary waves in a photonic band gap,�?? Phys. Rev. Lett. 71, 1168�??1171 (1993).
[CrossRef] [PubMed]

S. F. Mingaleev and Y. S. Kivshar, �??Self-trapping and stable localized modes in nonlinear photonic crystals,�?? Phys. Rev. Lett. 86, 5474�??5477 (2001).
[CrossRef] [PubMed]

R. Y. Chiao, E. Garmire, and C. H. Townes, �??Self-trapping of optical beams,�?? Phys. Rev. Lett. 13, 479�??482 (1964).
[CrossRef]

Supplementary Material (1)

» Media 1: GIF (2611 KB)     

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Figures (5)

Fig. 1.
Fig. 1.

(a) Schematic representation of the transverse section of a PCF. (b) The box with dimensions D×D corresponds to the unit cell used to implement periodic boundary conditions. In our simulations we have chosen D=7Λ, Λ being the spatial periodicity, or pitch, of the photonic crystal cladding.

Fig. 2.
Fig. 2.

Intensity distribution of different solutions of Eq. (1) for a PCF with pitch Λ=23µm, radius a=4µm, λ=1.55 µm and different nonlinear couplings: (a), linear mode (γ=0); (b) and (c), spatial soliton solutions for γ=0.0010 and γ=0.0015, respectively;(d), unstable nonlinear solution of the homogeneous medium (γ=γc=0.0017); and (e), self-focusing instability (γ>γc).

Fig. 3.
Fig. 3.

(a) Dependence of the gap function Δ on the nonlinear coupling γ for different hole sizes. As in Fig. 2, Λ=23µm and λ=1.55µm. (b) Diagram of existence of solutions for a nonlinear PCF. The shaded region is the nonlinear soliton phase. The other region corresponds to the homogeneous-instability phase. The inter-phase is given by the γc(a) curve. As before, Λ=23 µm and λ=1.55 µm.

Fig. 4.
Fig. 4.

(2.55 MB) Evolution of the field amplitude in z for a large-scale PCF with Λ=23µm, a=8µm and λ=1.55µm. We show the transient from an initial Gaussian profile towards an asymptotic spatial soliton solution.

Fig. 5.
Fig. 5.

Typical evolution behavior in z of 〈ϕ|L(ϕ)|ϕ〉 for a large-scale PCF with Λ=23µm, a=8µm and λ=1.55µm.

Equations (2)

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[ t 2 + k 0 2 ( n 0 2 ( x ) + n 2 2 ( x ) ϕ 2 ) ] ϕ = β 2 ϕ ,
[ t 2 + k 0 2 ( n 0 2 ( x ) + n 2 2 ( x ) ϕ 2 ) ] ϕ = 2 ϕ z 2 .

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