# FDTD analysis of modal dispersive properties of nonlinear photonic crystal fibers

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## Abstract

This paper presents a full-wave electromagnetic analysis of soft-glass photonic crystal fibers developed for the generation of supercontinuum based on third-order nonlinearity. It is shown that a two-dimensional finite-difference time-domain method for guided problems provides results very similar to the measurement data of real fiber structures, enabling the reduction of costly hardware prototyping, thus, opening the way for the application of FDTD to the modeling of nonlinear optical processes.

## Keywords

FDTD Photonic crystal fibers Supercontinuum Dispersion## 1 Introduction

Photonic crystal fibers (PCFs) are a new class of fibers applicable in areas such as nonlinear optics (Knight and Skryabin 2007), generation of supercontinuum (Dudley et al. 2006), biomedical measurements or sensors (Konorov et al. 2005). Several numerical methods, dedicated to the electromagnetic (EM) analysis of PCFs, have been developed (Szpulak et al. 2006). The most commonly used are the finite element method (Rahman et al. 2007), plane wave expansion method, (Ho et al. 1990), the polar Fourier decomposition method, (Issa and Poladian 2003), finite-difference time-domain (FDTD) method, (Taflove and Hagness 2005), and finite-difference frequency-domain (FDFD) method, (Zhu and Brown 2002). Since dispersive properties of PCFs, critical for nonlinear phenomena, are very sensitive to geometrical properties of a structure, it is essential to have access to a highly reliable EM simulation method in order to reduce costly hardware prototyping to the necessary minimum.

In this paper, a two-dimensional FDTD method developed for guided problems (Gwarek et al. 1993) is applied in a design cycle of PCFs (Salski et al. 2010). The performance of the method is validated against the FDFD method as another widely applicable tool in computational electromagnetics. Both approaches are compared in terms of computational effort and accuracy of obtained results. Eventually, the results are verified against measurement data obtained for microstructured optical fibers (Joannopoulos 2008).

## 2 Electromagnetic analysis of PCF

Sellmeier and Lorentz coefficients

Model type | Parameters | Glass type | |
---|---|---|---|

TWNN16 | PBS517 | ||

Sellmeier | \(B_{1}\) | 2.496654501 | 1.5289 |

\(B_{2}\) | 0.799980033 | 0.7282 | |

\(B_{3}\) | 1.602289511 | 0.9981 | |

\(C_{1}\) | 0.016999569 | 0.0219 | |

\(C_{2}\) | 0.076752705 | 0.0219 | |

\(C_{3}\) | 149.808796932 | 100.0000 | |

Lorentz | \(\varepsilon _\infty \) | 1.0 | 1.0 |

\(\varepsilon _s \) | 3.4967 | 2.5289 | |

\(f_{p1}\) (THz) | 2,299.3 | 2,025.8 | |

\(f_{p2}\) (THz) | 1,082.1 | 29.979 | |

\(f_{p3}\) (THz) | 24.494 | – | |

\(A_{1}\) | 1.0 | 1.0 | |

\(A_{2}\) | 0.3204 | 1.1291 | |

\(A_{3}\) | 0.6418 | – |

Once the material properties are properly represented, total fiber dispersion can be determined. Total fiber dispersion is comprised of material dispersion and waveguide dispersion. For proper modeling of the fiber both parameters have to be taken into account. Several works previously studied this task (Zhu and Brown 2002; Koshiba and Saitoh 2001; Ferrando et al. 1999).

## 3 Ideal PCF structure

Electromagnetic analysis is undertaken on the Intel Core i5-2410M CPU 2.30 GHz platform. Computing time of full range characteristics with FDFD is around 2 h, while in the case of the FDTD model, computation of the whole characteristic takes ca. 1.5 h. It can be noticed that both methods provide very similar results with a regular discrepancy, although FDFD gives slightly lower values of the effective refractive index (see Fig. 3). The discrepancy increases with the filling factor \(d/a\) and with the wavelength \(\lambda \). It can be also noticed that the application of TWNN16 glass results in more flattened dispersion \(D\) in the range from \(\lambda = 1~\upmu \)m up to \(\lambda =2~\upmu \)m. However, changing the geometry of the photonic crystal cladding, in terms of the filling factor \(d/a\), allows wider adjustment of the dispersion amplitude in the aforementioned spectrum range, which is essential to the efficient supercontinuum generation (compare Fig. 3b with Fig. 4b).

In the next Section, real PCF geometry will be considered and the corresponding computational results will be validated against measurement data, thus, enabling the validation of the applied EM modeling methods.

## 4 Real PCF structure

As it can be seen in Fig. 5c, the measured and calculated dispersion characteristics are in decent agreement, indicating that both numerical methods give reasonable results. However, if the zero dispersion wavelength (ZDW) has to be determined, the FDTD method gives a result much closer to the experimental data. In this case, ZDW is 1,356 and 1,492 nm according to results of the FDTD and FDFD method, respectively, while the measurements indicate that ZDW = 1,382 nm. It means that discrepancy is at the level of about 1.9 and 8.0 % for FDTD and FDFD solutions, respectively. Since determination of ZDW is critical for supercontinuum generation in nonlinear PCFs (Buczynski et al. 2011), the use of the FDFD method can bring a meaningful inaccuracy. Discrepancy between experimental and modeled result is mostly related to credibility of SEM image of the fiber. Since fiber is covers with layer of gold/palladium before the SEM image is taken, diameters of air holes might be slightly different on SEM image and in real fiber. Although the difference is very small it can noticeably influence the dispersion curvature, since dispersion is proportional to the second derivative of effective refractive index (see Eq. 2).

## 5 Conclusions

Application of the FDTD method to the computation of modal dispersive properties of photonic crystal fibers has been addressed. The accuracy of the proposed method was verified against the FDFD method and against the experimental results. Due to a frequency-domain approach, the extraction of the effective refractive index with FDFD is based on tracing of a selected mode versus wavelength. On the contrary, FDTD allows tracing all the modes simultaneously, which can be advantageous when modes coupling becomes of our interest. In general, both methods give similar results and can be successfully applied to the modeling of photonic crystal fibers.

## Notes

### Acknowledgments

This work was supported by the project TEAM/2012-9/1 operated within the Foundation for Polish Science Team Programme co-financed by the European Regional Development Fund, Operational Program Innovative Economy 2007–2013. The authors thank to Dr. R Stepien and D. Pysz from the Institute of Electronic Materials Technology for providing the PCF and Dr. T. Martynkien from Wroclaw University of Technology for the measurement of a dispersion characteristic of the fiber.

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