## Leakage properties of photonic crystal fibers

Optics Express, Vol. 10, Issue 23, pp. 1314-1319 (2002)

http://dx.doi.org/10.1364/OE.10.001314

Acrobat PDF (427 KB)

### Abstract

An analysis of the confinement losses in photonic crystal fibers due to the finite numbers of air holes is performed by means of the finite element method. The high flexibility of the numerical method allows to consider fibers with regular lattices, like the triangular and the honeycomb ones, and circular holes, but also fibers with more complicated cross sections like the cobweb fiber. Numerical results show that by increasing the number of air hole rings the attenuation constant decreases. This dependence is very strong for triangular and cobweb fibers, whereas it is very weak for the honeycomb one.

© 2002 Optical Society of America

## 1. Introduction

## 2 The Finite Element Method Formulation

6. S. Selleri, L. Vincetti, A. Cucinotta, and M. Zoboli, “Complex FEM modal solver of optical waveguides with PML boundary conditions,” Opt. Quantum Electron. **33**, 359–371(2001). [CrossRef]

*h̄*is the magnetic field,

*ε̿*and

_{r}*μ̿*are the relative dielectric permittivity and magnetic permeability complex tensors, respectively, and

_{r}*k*

_{0}= 2

*π*/

*λ*the wave number in the vacuum,

*λ*being the wavelength. The magnetic field of the modal solution is expressed as

*h̄*=

*H̄e*

^{-γz}where

*H̄*is the field distribution on the transverse plane and

*γ*=

*α*+

*jβ*is the complex propagation constant with

*α*the attenuation constant and

*β*the phase constant. By applying the variational finite element procedure, the full vector equations (1) yields the algebraic problem [6

6. S. Selleri, L. Vincetti, A. Cucinotta, and M. Zoboli, “Complex FEM modal solver of optical waveguides with PML boundary conditions,” Opt. Quantum Electron. **33**, 359–371(2001). [CrossRef]

*H*} and the eigenvalue

## 3 Results

### 3.1 Triangular fiber

*d*= 0.69

*μm*and pitch Λ = 2.3

*μm*. Notice that due to fiber symmetry just a quarter of the cross section can be considered for the analysis. The hexagonal symmetry as well as the leakage due to the interruption of the lattice are evident. The field confinement and its decay rate play a fundamental role in the leakage properties. They depend on the hole diameter, on their pitch and on the number of rings. Figure 2 shows the losses versus the diameter

*d*normalized to the pitch value Λ = 2.3

*μm*. As expected, the losses quickly decrease by increasing the numbers of rings and the hole diameter. Also the slope increases with these parameters. In fact, with

*d*= 0.69

*μm*, that is

*d*/Λ = 0.3, going from a ring to eight rings losses decrease from 10

^{5}

*dB*/

*m*to a value a little lower than 10

*dB*/

*m*. Notice that this corresponds to pass from six to two hundred and sixteen holes. By increasing the hole diameter to

*d*= 0.9

*μm*= 0.4Λ, the upper value of the range goes to 0.5 × 10

^{5}

*dB*/

*m*, whereas the lower value goes to 10

^{-5}which is negligible with respect to all the other causes of losses, like to absorption and the Rayleigh scattering. Figure 2(b) shows that a loss reduction is also obtained by fixing the ratio

*d*/Λ and by increasing the pitch Λ. The pitch and the hole diameter are changed of the same scale factor, consequently greater pitches correspond to greater core size and thus more confined fields. Specific dispersion profile can be obtained playing on ratio

*d*/Λ and on the scale factor [9

9. A. Ferrando, E. Silvestre, P. Andrés, J. J. Miret, and M. V. Andrés, “Designing the properties of dispersion-flattend photonic crystal fibers,” Opt. Express **9**, 687–697 (2001). http: //www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-687 [CrossRef] [PubMed]

*nm*to 1700

*nm*with one ring, the losses are almost wavelength-independent, conversely, with five rings, the losses increase of about two orders of magnitude. This variation decreases of one order going from a pitch Λ = 2.3

*μm*to a pitch Λ = 4.6

*μm*. In fact greater pitches correspond to larger cores and thus more confined fields. As a consequence, a wavelength increment corresponds to a little increment of the confinement with a little reduction of the losses.

### 3.2 Honeycomb fiber

*dB*/

*m*. The figure also shows another important difference with respect to the triangular fiber. Higher ratios

*d*/Λ exhibit lower losses only up to six rings, after that the losses slightly increase. This can be ascribed to the fiber guidance mechanism. In fact the lattice of the fiber determines a truly photonic band gap only if the number of rings is very high. Also the wavelength dependence, shown in figure 5(b), is weaker than in the triangular one. It slightly increases with ring numbers and it reaches a order of magnitude with nine rings, over the wavelength range here considered.

### 3.3 Cobweb fiber

7. J.C. Knight, J. Arriaga, T.A. Birks, A. Ortigosa-Blanch, W.J. Wadsworth, and P. St. Russell, “Anomalous dispersion in photonic crystal fiber,” IEEE Photon. Technol. Lett. , **12**, 807–809 (2000). [CrossRef]

5. A. Cucinotta, S. Selleri, L. Vincetti, and M. Zoboli, “Holey fiber analysis through the finite element method,” IEEE Photon. Technol. Lett. **14**, 1530–15322002. [CrossRef]

*nm*. This low value combined with very large holes assures very low losses as showed in figure 6(b). The wavelength dependence is stronger than in the previous fibers and it increases by increasing the number of rings. For two rings, passing from 1300

*nm*to 1700

*nm*of wavelength, the losses change of six orders of magnitude. However, for all wavelengths here considered, with just three rings the losses are lower than 10

^{-5}

*dB*/

*m*. The losses are due mainly to the field which passes through the silica bridges.

## 4 Conclusion

*dB*/

*m*, values which could affect their applications. Several aspects, like the role of the bridge thickness in the cobweb fibers, the dimension of the central defect in honeycomb fibers or the influence of the confinement losses on dispersion parameters, must be still investigated and will be the object of future works.

## References and links

1. | T. P. White, R. C. McPhedram, C. M. de Sterke, L. C. Botten, and M. J. Steel, “Confinement losses in microstructured optical fibers,” Opt. Lett. |

2. | V. Finazzi, T. M. Monro, and D. J. Richardson, “Confinement losses in highly nonlinear holey optical fibers,” in Optical Fiber Communication 2002, vol. 70 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C.2002), paper ThS4. |

3. | K. Saitoh and M. Koshiba, “Full-vectorial imaginary-distance beam propagation method based on a finite element scheme: application to photonic crystal fibers,” IEEE J. Quantum Electron. |

4. | A. Cucinotta, S. Selleri, L. Vincetti, and M. Zoboli, “Perturbation analysis of dispersion properties in photonic crystal fibers through the finite element method,” J. Lightwave Technol. |

5. | A. Cucinotta, S. Selleri, L. Vincetti, and M. Zoboli, “Holey fiber analysis through the finite element method,” IEEE Photon. Technol. Lett. |

6. | S. Selleri, L. Vincetti, A. Cucinotta, and M. Zoboli, “Complex FEM modal solver of optical waveguides with PML boundary conditions,” Opt. Quantum Electron. |

7. | J.C. Knight, J. Arriaga, T.A. Birks, A. Ortigosa-Blanch, W.J. Wadsworth, and P. St. Russell, “Anomalous dispersion in photonic crystal fiber,” IEEE Photon. Technol. Lett. , |

8. | S. E. Barkou, J. Broeng, and A. Bjarklev, “Dispersion properties of photonic bandgap guiding fibers,” in Optical Fiber Communication Conference , OSA Technical Digest (Optical Society of America, Washington DC, 1998), FG5. |

9. | A. Ferrando, E. Silvestre, P. Andrés, J. J. Miret, and M. V. Andrés, “Designing the properties of dispersion-flattend photonic crystal fibers,” Opt. Express |

**OCIS Codes**

(060.2280) Fiber optics and optical communications : Fiber design and fabrication

(060.2400) Fiber optics and optical communications : Fiber properties

**ToC Category:**

Research Papers

**History**

Original Manuscript: September 23, 2002

Revised Manuscript: October 30, 2002

Published: November 18, 2002

**Citation**

D. Ferrarini, Luca Vincetti, M. Zoboli, A. Cucinotta, and S. Selleri, "Leakage properties of photonic crystal fibers," Opt. Express **10**, 1314-1319 (2002)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-10-23-1314

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### References

- T. P. White, R. C. McPhedram, C. M. de Sterke, L. C. Botten, and M. J. Steel, �??Confinement losses in microstructured optical fibers,�?? Opt. Lett. 26, 1660-1662 (2001). [CrossRef]
- V. Finazzi, T. M. Monro, and D. J. Richardson, �??Confinement losses in highly nonlinear holey optical fibers,�?? in Optical Fiber Communication 2002, vol. 70 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C. 2002), paper ThS4.
- K. Saitoh, and M. Koshiba, �??Full-vectorial imaginary-distance beam propagation method based on a finite element scheme: application to photonic crystal fibers,�?? IEEE J. Quantum Electron. 38, 927-933 (2002). [CrossRef]
- A. Cucinotta, S. Selleri, L. Vincetti, and M. Zoboli, �??Perturbation analysis of dispersion properties in photonic crystal fibers through the finite element method,�?? J. Lightwave Technol. 20, (2002). [CrossRef]
- A. Cucinotta, S. Selleri, L. Vincetti, and M. Zoboli, �??Holey fiber analysis through the finite element method,�?? IEEE Photon. Technol. Lett. 14, 1530-1532 2002. [CrossRef]
- S. Selleri, L. Vincetti, A. Cucinotta, and M. Zoboli, �??Complex FEM modal solver of optical waveguides with PML boundary conditions,�?? Opt. Quantum Electron. 33, 359-371(2001). [CrossRef]
- J. C. Knight, J. Arriaga, T. A. Birks, A. Ortigosa-Blanch, W. J. Wadsworth and P. St. Russell, �??Anomalous dispersion in photonic crystal fiber,�?? IEEE Photon. Technol. Lett. 12, 807-809 (2000). [CrossRef]
- S. E. Barkou, J. Broeng, and A. Bjarklev, �??Dispersion properties of photonic bandgap guiding fibers,�?? in Optical Fiber Communication Conference , OSA Technical Digest (Optical Society of America, Washington DC, 1998), FG5.
- A. Ferrando, E. Silvestre, P.Andres, J. J. Miret, and M. V. Andres, �??Designing the properties of dispersion-.attend photonic crystal fibers,�?? Opt. Express 9, 687-697 (2001). <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-687">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-687</a> [CrossRef] [PubMed]

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