## Spatial soliton formation in photonic crystal fibers

Optics Express, Vol. 11, Issue 5, pp. 452-459 (2003)

http://dx.doi.org/10.1364/OE.11.000452

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### 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

## 1. Introduction

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

2. H. A. Haus, “Higher order trapped light beam solutions,” Appl. Phys. Lett. **8**, 128–129 (1966). [CrossRef]

3. S. John and N. Aközbek, “Nonlinear optical solitary waves in a photonic band gap,” Phys. Rev. Lett. **71**, 1168–1171 (1993). [CrossRef] [PubMed]

4. 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]

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

6. 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]

7. 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]

3. S. John and N. Aközbek, “Nonlinear optical solitary waves in a photonic band gap,” Phys. Rev. Lett. **71**, 1168–1171 (1993). [CrossRef] [PubMed]

4. 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]

8. A. Ferrando, E. Silvestre, J. J. Miret, P. Andrés, and M. V. Andrés, “Donor and acceptor guided modes in photonic crystal fibers,” Opt. Lett. **25**, 1238–1330 (2000). [CrossRef]

## 2. Description of the method

*k*

_{0}the vacuum wavenumber. We search for electric field solutions with well-defined constant polarization

**E**(

*x⃗*)=

**u**ϕ(

*x⃗*). The linear refractive index profile function

*n*

_{0}(

*x⃗*) is one in the air-holes and equals

*n*

_{silica}in silica, whereas the nonlinear index profile function

*n*

_{2}(

*x⃗*) is different from zero only in silica (

_{0}

*cn*

_{0(silica)}). We approach this problem using a generalization of our modal method to describe linear propagation in PCF’s [9

9. A. Ferrando, E. Silvestre, J. J. Miret, P. Andrés, and M. V. Andrés, “Full-vector analysis of a realistic photonic crystal fiber,” Opt. Lett. **24**, 276–278 (1999). [CrossRef]

10. A. Ferrando, E. Silvestre, J. J. Miret, P. Andrés, and M. V. Andrés, “Vector description of higherorder modes in photonic crystal fibers,” J. Opt. Soc. Am. A **17**, 1333–1340 (2000). [CrossRef]

*x⃗*)). Despite the complicated PCF geometry, it is also possible here to evaluate the fourth-order tensor analytically. This nonlinear eigenvalue equation is formally identical to that found for a soliton solution in a modal approach to dispersion-management systems [11

11. A. Ferrando, M. Zacarés, and P. F. de Córdoba, “Ansatz-independent solution of a soliton in a strong dispersion-management system,” Phys. Rev. E **62**, 7320–7329 (2000). [CrossRef]

11. A. Ferrando, M. Zacarés, and P. F. de Córdoba, “Ansatz-independent solution of a soliton in a strong dispersion-management system,” Phys. Rev. E **62**, 7320–7329 (2000). [CrossRef]

## 3. Results

*A*

_{0}=1), which is equivalent to define a power-dependent dimensionless nonlinear coupling γ=

*A*

_{0}, where

*P*is the total power carried by the unnormalized field and

*A*

_{0}is a magnitude with dimensions of area characterizing the core size (here we choose

*A*

_{0}=π(Λ/2)

^{2}, Λ being the spatial periodicity—or pitch—of the PCF). Solutions of the nonlinear eigenvalue equation are found for different values of the PCF parameters and the nonlinear coupling. In all cases, the solution we find correspond to the eigenvector with highest β

^{2}of the nonlinear eigenvalue problem. The solution can be envisaged as the fundamental mode of the effective fiber generated by the combined effect of the PCF refractive index and the nonlinear index induced by the solution amplitude itself (

*n*

^{2}(

*x⃗*)=

*x⃗*)+

*x⃗*)|ϕ

_{sol}(

*x⃗*)|

^{2}). Since the fundamental mode of the linear theory is also a localized solution, it is convenient to provide an efficient criterion to establish a distinction between localized nonlinear solutions and the fundamental mode of a linear PCF. We use as a measure of distinction the difference, or gap, (normalized to the given vacuum wavenumber

*k*

_{0}) Δ≡(β

_{sol}-β

_{fund})/

*k*

_{0}, where β

_{sol}and β

_{fund}are the propagation constants of the solution of the nonlinear equation and the fundamental mode of the linear fiber, respectively. The generation of a gap implies that the nonlinear solution has a different shape than the fundamental linear mode.

_{max}≈10

^{-3}). For these reasons, it is more realistic to consider PCF structures that possess larger cores. One simple way of generating such structures is by magnifying the conventional PCF parameters, i.e., the hole radius

*a*and the pitch Λ, (

*a*→

*Ma*, Λ

*M*Λ). The core size area

*A*

_{0}is then scaled by a

*M*

^{2}factor and γ by 1/

*M*

^{2}. In our simulation, we choose

*M*=10 with respect to an ordinary PCF configuration characterized by Λ=2.3 µm and

*a*ranging from 0.2 to 1.0 microns. That is, we simulate large-scale PCF’s with Λ=23 µm and

*a*=2–10 µm. In this way, we guarantee that all solutions we find will lie below the breakdown intensity threshold.

*a*, at which the soliton acquires a critical shape and propagation constant (Fig. 2(d)). At the critical coupling, our calculations recover the rotationally invariant nonlinear solution obtained in an homogeneous medium (also known as Townes soliton) [1

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

^{2}(Fig. 2(e)). The physical interpretation of these solutions is clear. As we increase the nonlinear coupling, the localized mode is more confined and it feels the photonic crystal cladding less and less. At the critical coupling, it stops seeing the cladding completely. Thus, its behavior corresponds to a nonlinear mode in an homogeneous medium. That is why we recover the rotationally invariant solution (no discrete symmetry is left) of an homogeneous medium. This argument also applies above the critical coupling, where we expect to recover the same physics as in an homogeneous medium. It is known that the Townes soliton in an homogeneous medium is unstable under power perturbations. In particular, if power is increased the solution collapses experimenting a process of filamentation induced by a self-focusing instability and generating an infinitely narrow solution with an infinite β

^{2}[12

12. G. Fibich and A. L. Gaeta, “Critical power for self-focusing in bulk media and in hollow waveguides,” Opt. Lett. **25**335–337 (2000). [CrossRef]

*a*→0), which corresponds to the homogeneous medium case. The differences and similarities with respect to the PCF case are enlightening. Below the critical coupling, the method does not detect any nontrivial solution and only finds the trivial zero solution of Eq. (1). This is also consistent with known results for an homogeneous medium, since no stationary nonlinear solutions are found below the critical power that corresponds to the Townes soliton [1

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

12. G. Fibich and A. L. Gaeta, “Critical power for self-focusing in bulk media and in hollow waveguides,” Opt. Lett. **25**335–337 (2000). [CrossRef]

_{c}(

*a*) are unstable since we recognized them as Townes solitons of an homogeneous medium and instability is a known feature of them. Stability is a dynamical issue which implies the study of the evolution equation associated to Eq. (1):

*z*of the expectation value of the differential operator in Eq.(1)—let us denote it by

*L*(ϕ)—with respect the field amplitude ϕ(

*x*,

*y*,

*z*). This field verifies the 3D evolution equation (2), which in terms of the

*L*(ϕ)-operator is expressed as

*L*(ϕ)ϕ=-∂

^{2}ϕ/∂

*z*

^{2}. Thus, the expectation value of the

*L*(ϕ)-operator, 〈ϕ|

*L*|ϕ〉, is

*z*-dependent unless it represents an stationary solution of the nonlinear eigenvalue equation (1): ϕ(

*x*,

*y*,

*z*)=ϕ

_{s}(

*x*,

*y*)

*e*

^{iβz}.

*L*|ϕ〉=β

^{2}. In Fig. 5 we show a typical behavior of the evolution of 〈ϕ|

*L*|ϕ〉 (

*z*) where the stabilization mechanism becomes evident: the expectation value evolves (oscillating) until it asymptotically reaches a plateau indicating the existence of a spatial soliton solution. The asymptotic β

^{2}value corresponds to the spatial soliton solution of Eq. (1) for the same PCF parameters with a γ given by the asymptotic power. Notice that power is not conserved because of radiation and decreases during propagation, so that the asymptotic solution possesses a smaller power than any other intermediate solution. It is worth mentioning that we have also checked that these spatial solitons are stable under both small transverse displacements relative to the photonic crystal cladding and launching with small transverse momentum (slight offaxis illumination). Analysis of the evolution of the expectation value 〈ϕ|

*L*|ϕ〉 (

*z*) shows in both cases a similar asymptotic behavior as for the Gaussian profile in Fig. 5: damped oscillations tending towards a spatial soliton plateau.

## 4. Conclusions

## References and links

1. | R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. |

2. | H. A. Haus, “Higher order trapped light beam solutions,” Appl. Phys. Lett. |

3. | S. John and N. Aközbek, “Nonlinear optical solitary waves in a photonic band gap,” Phys. Rev. Lett. |

4. | S. F. Mingaleev and Y. S. Kivshar, “Self-trapping and stable localized modes in nonlinear photonic crystals,” Phys. Rev. Lett. |

5. | P. Xie, Z.-Q. Zhang, and X. Zhang, “Gap solitons and soliton trains in finite-sized two-dimensional periodic and quasiperiodic photonic crystals,” Phys. Rev. E |

6. | S. F. Mingaleev, Y. S. Kivshar, and R. A. Sammut, “Long-range interaction and nonlinear localized modes in photonic crystal waveguides,” Phys. Rev. E |

7. | H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete spatial optical solitons in waveguide arrays,” Phys. Rev. Lett. |

8. | A. Ferrando, E. Silvestre, J. J. Miret, P. Andrés, and M. V. Andrés, “Donor and acceptor guided modes in photonic crystal fibers,” Opt. Lett. |

9. | A. Ferrando, E. Silvestre, J. J. Miret, P. Andrés, and M. V. Andrés, “Full-vector analysis of a realistic photonic crystal fiber,” Opt. Lett. |

10. | A. Ferrando, E. Silvestre, J. J. Miret, P. Andrés, and M. V. Andrés, “Vector description of higherorder modes in photonic crystal fibers,” J. Opt. Soc. Am. A |

11. | A. Ferrando, M. Zacarés, and P. F. de Córdoba, “Ansatz-independent solution of a soliton in a strong dispersion-management system,” Phys. Rev. E |

12. | G. Fibich and A. L. Gaeta, “Critical power for self-focusing in bulk media and in hollow waveguides,” Opt. Lett. |

13. | R. W. Boyd, Nonlinear Optics (Academic Press, 1992). |

**OCIS Codes**

(060.4370) Fiber optics and optical communications : Nonlinear optics, fibers

(190.4370) Nonlinear optics : Nonlinear optics, fibers

**ToC Category:**

Research Papers

**History**

Original Manuscript: January 30, 2003

Revised Manuscript: February 26, 2003

Published: March 10, 2003

**Citation**

Albert Ferrando, Mario Zacares, Pedro Fernandez de Cordoba, Daniele Binosi, and Juan Monsoriu, "Spatial soliton formation in photonic crystal fibers," Opt. Express **11**, 452-459 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-5-452

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### References

- R. Y. Chiao, E. Garmire, and C. H. Townes, �??Self-trapping of optical beams,�?? Phys. Rev. Lett. 13, 479�??482 (1964). [CrossRef]
- H. A. Haus, �??Higher order trapped light beam solutions,�?? Appl. Phys. Lett. 8, 128�??129 (1966). [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]
- 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]
- 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]
- A. Ferrando, E. Silvestre, J. J. Miret, P. Andres, and M. V. Andres, �??Donor and acceptor guided modes in photonic crystal fibers,�?? Opt. Lett. 25, 1238�??1330 (2000). [CrossRef]
- A. Ferrando, E. Silvestre, J. J. Miret, P. Andres, and M. V. Andres, �??Full-vector analysis of a realistic photonic crystal fiber,�?? Opt. Lett. 24, 276�??278 (1999). [CrossRef]
- A. Ferrando, E. Silvestre, J. J. Miret, P. Andres, and M. V. Andres, �??Vector description of higherorder modes in photonic crystal fibers,�?? J. Opt. Soc. Am. A 17, 1333�??1340 (2000). [CrossRef]
- 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]
- G. Fibich and A. L. Gaeta, �??Critical power for self-focusing in bulk media and in hollow waveguides,�?? Opt. Lett. 25 335�??337 (2000). [CrossRef]

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