## Dispersion properties of square-lattice photonic crystal fibers

Optics Express, Vol. 12, Issue 5, pp. 941-946 (2004)

http://dx.doi.org/10.1364/OPEX.12.000941

Acrobat PDF (1190 KB)

### Abstract

In this paper the guiding properties of photonic crystal fibers with a square lattice of air-holes in a silica matrix have been studied for the first time. The dispersion curves of fibers with different hole-to-hole spacing and air-hole diameter have been accurately calculated. Negative values of the dispersion parameter and the dispersion slope have been obtained with a hole-to-hole spacing of 1 *µm*. A comparison between fibers with square and triangular lattice has been also performed, taking into account the dispersion properties and the effective area in the wavelength range between 1200 *nm* and 1600 *nm*.

© 2004 Optical Society of America

## 1. Introduction

10. P. St. J. Russell, E. Marin, A. Díez, S. Guenneau, and A. B. Movchan, “Sonic band gaps in PCF preforms: enhancing the interaction of sound and light,” Opt. Express **11**, 2555–2560 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-20-2555. [CrossRef] [PubMed]

10. P. St. J. Russell, E. Marin, A. Díez, S. Guenneau, and A. B. Movchan, “Sonic band gaps in PCF preforms: enhancing the interaction of sound and light,” Opt. Express **11**, 2555–2560 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-20-2555. [CrossRef] [PubMed]

*d*of the air-hole in the fiber cross-section (Fig. 1(a)). Simulation results have shown that the dispersion parameter values increase in the wavelength range considered, from 1200

*nm*to 1600

*nm*, for a fixed hole-to-hole spacing, chosen between 1

*µm*and 3

*µm*, as the air-holes become bigger. Moreover, it has been demonstrated that the square-lattice PCFs with the smallest pitch, that is 1

*µm*, have negative dispersion parameter in the wavelength range around 1550

*nm*. The square-lattice PCFs with small pitch and large air-hole diameter, whose dispersion slope is negative too, can be used, as done with the triangular ones [4

4. F. Poli, A. Cucinotta, M. Fuochi, S. Selleri, and L. Vincetti, “Characterization of microstructured optical fibers for wideband dispersion compensation,” J. Opt. Soc. Am. A **20**, 1958–1962 (2003). [CrossRef]

## 2. Dispersion properties of the square-lattice PCFs

*d*values, showing a lower average value of the refractive index around the core in the triangular PCF. In fact, in this case the first ring comprises six air-holes whose distance from the core center is equal to Λ, thus resulting in a stronger field confinement.

4. F. Poli, A. Cucinotta, M. Fuochi, S. Selleri, and L. Vincetti, “Characterization of microstructured optical fibers for wideband dispersion compensation,” J. Opt. Soc. Am. A **20**, 1958–1962 (2003). [CrossRef]

11. 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]

12. A. Cucinotta, F. Poli, S. Selleri, L. Vincetti, and M. Zoboli, “Amplification Properties of Er^{3+}-Doped Photonic Crystal Fibers,” J. Lightwave Technol. **21**, 782–788 (2003). [CrossRef]

*n*for the different wavelengths in the range between 1200

_{eff}*nm*and 1600

*nm*has been obtained by the FEM approach [4

4. F. Poli, A. Cucinotta, M. Fuochi, S. Selleri, and L. Vincetti, “Characterization of microstructured optical fibers for wideband dispersion compensation,” J. Opt. Soc. Am. A **20**, 1958–1962 (2003). [CrossRef]

8. F. Poli, A. Cucinotta, S. Selleri, and A. H. Bouk, “Tailoring of flattened dispersion in highly nonlinear photonic crystal fibers,” IEEE Photon. Technol. Lett., to be published (2004). [CrossRef]

**20**, 1958–1962 (2003). [CrossRef]

8. F. Poli, A. Cucinotta, S. Selleri, and A. H. Bouk, “Tailoring of flattened dispersion in highly nonlinear photonic crystal fibers,” IEEE Photon. Technol. Lett., to be published (2004). [CrossRef]

*n*values versus the wavelength, the dispersion parameter has been derived using simple finite difference formulas [4

_{eff}**20**, 1958–1962 (2003). [CrossRef]

8. F. Poli, A. Cucinotta, S. Selleri, and A. H. Bouk, “Tailoring of flattened dispersion in highly nonlinear photonic crystal fibers,” IEEE Photon. Technol. Lett., to be published (2004). [CrossRef]

12. A. Cucinotta, F. Poli, S. Selleri, L. Vincetti, and M. Zoboli, “Amplification Properties of Er^{3+}-Doped Photonic Crystal Fibers,” J. Lightwave Technol. **21**, 782–788 (2003). [CrossRef]

*d*/Λ has been investigated considering PCFs with five rings of air-holes in the cross-section. For fibers with low Λ values, a FEM solver with Perfectly Matched Layers as boundary conditions which accounts for leakage losses has been used not to affect the dispersion results. Five values of the hole-to-hole spacing Λ, that is 1, 1.5, 2, 2.5 and 3

*µm*, have been chosen, and

*d*/Λ has been varied in the range 0.5÷0.9.

*D*(

*λ*) of the square-lattice PCFs with different

*d*/Λ values and Λ=1

*µm*, Λ=2

*µm*and Λ=3

*µm*, respectively, for the wavelengths between 1200

*nm*and 1600

*nm*. All the square-lattice PCFs with the smallest pitch, 1

*µm*, have negative dispersion parameter in the C band, around 1550

*nm*(Fig. 2(a)), since the core dimension is very little and the waveguide dispersion dominates on the material one [6

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

7. B. T. Kuhlmey, G. Renversez, and D. Maystre, “Chromatic dispersion and losses of microstructured optical fibers,” Appl. Opt. OT **42**, 634–639 (2003). [CrossRef]

*nm*, -277

*ps/km·nm*, is obtained with the PCF characterized by Λ=1

*µm*and

*d*/Λ=0.6. Notice that

*D*values increase with the air-hole diameter, so only the PCFs with

*d*/Λ≤0.7, that is with small air-holes, have negative dispersion parameter in all the wavelength range here considered. The fiber with the smallest air-holes, that is the one with

*d*/Λ=0.5, has a dispersion curve with a minimum, about -248

*ps/km·nm*, around 1550

*nm*and a positive dispersion slope for the longer wavelengths. The other square-lattice PCFs, with

*d*/Λ≥0.6, have negative dispersion slope, so they could be used as dispersion compensating fibers. In fact, as it has been already demonstrated for the triangular PCFs [6

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

7. B. T. Kuhlmey, G. Renversez, and D. Maystre, “Chromatic dispersion and losses of microstructured optical fibers,” Appl. Opt. OT **42**, 634–639 (2003). [CrossRef]

*µm*, the dispersion parameter of all the PCFs becomes positive, independently from the air-hole dimension, that is from the

*d*/Λ value. It is interesting to notice that, as the pitch increases, the dispersion slope of the curves becomes more positive. Moreover, a change of

*d*/Λ value causes a smaller difference in the dispersion parameter values, of about 8

*ps/km·nm*, for the PCFs with the higher Λ, that is 3

*µm*. Notice that the dispersion curve of the square-lattice PCF with

*d*/Λ=0.5 and Λ=2

*µm*is quite flat, around the value of 53

*ps/km·nm*, from 1425

*nm*to 1550

*nm*(Fig. 2(b)).

*d*/Λ value. In this case

*d*/Λ=0.9 has been chosen, but the results are almost the same for different air-hole dimension. Notice that an increase of 0.5

*µm*in the pitch value, that is from 1

*µm*to 1.5

*µm*, causes a significant change in the dispersion curve. In fact, there is a great difference between the dispersion parameter values of the two PCFs, which increases with the wavelength, being about 56

*ps/km·nm*at 1250

*nm*and about 310

*ps/km·nm*at 1600

*nm*. Moreover, the dispersion slope, which is negative for the PCF with Λ=1

*µm*in all the wavelength range considered, becomes almost null, in the wavelength range between 1200

*nm*and 1450

*nm*, and positive, for the longer wavelengths for the PCF with Λ=1.5

*µm*. When Λ≥2

*µm*, the slope of the dispersion curves is always positive. Finally, the dispersion parameter values, which are all greater than 50

*ps/km·nm*for these PCFs, decrease as the pitch Λ increases from 2

*µm*to 3

*µm*.

## 3. Comparison between the square-lattice and the triangular PCFs

*d*/Λ, that is 1

*µm*and 0.9, respectively. It has been already demonstrated that the triangular PCF with these geometric parameters has negative dispersion and dispersion slope, and can be successfully used as a dispersion compensating fiber for a NZDF [4

**20**, 1958–1962 (2003). [CrossRef]

*nm D*=-293

*ps/km·nm*for the triangular PCF and

*D*=-157

*ps/km·nm*for the square-lattice one. The last PCF could better compensate the positive dispersion of a NZDF in a wide wavelength range, since its dispersion slope is lower around 1550

*nm*. In fact, its compensation ratio values at 1500

*nm*and 1600

*nm*, being 0.964 and 1.007 respectively, are closer to the optimum value, that is 1, than those for the triangular PCF, which are 0.913 at 1500

*nm*and 1.047 at 1600

*nm*[4

**20**, 1958–1962 (2003). [CrossRef]

*nm*, due to its lower negative dispersion parameter value at this wavelength.

*µm*

^{2}in all the wavelength range considered, but they are higher than those of the triangular PCF. As an example, the square-lattice PCF has an effective area at 1550

*nm*which is 18% larger than that of the fiber with the triangular lattice. This difference can be explained by considering that the square lattice is characterized by a lower air-filling fraction

*f*=(

*π*/4)(

*d*/Λ)

^{2}, which is almost 86% of the one for the triangular lattice, that is

*f*=(

*π*/2√3)(

*d*/Λ)

^{2}. As a consequence, the square-lattice PCFs provide higher values of the average refractive index of the cladding, that is a lower step index which results in a lower field confinement.

*d*/Λ value, that is 0.5, has been chosen for the two fibers, so that the triangular PCF is singlemode in all the wavelength range considered also for the largest pitch Λ=3

*µm*[13

13. B. T. Kuhlmey, R. C McPhedran, C. M. de Sterke, P. A. Robinson, G. Renversez, and D. Maystre, “Microstructured optical fibers: where’s the edge?,” Opt. Express **10**, 1285–1290 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-22-1285. [CrossRef] [PubMed]

14. N. A. Mortensen, J. R. Folkenberg, M. D. Nielsen, and K. P. Hansen, “Modal cutoff and the V parameter in photonic crystal fibers,” Opt. Lett. **28**, 1879–1881 (2003). [CrossRef] [PubMed]

*µm*, and lower

*D*values when the hole-to-hole distance is large, that is Λ=3

*µm*(Fig. 5(a)). The dispersion slope is only slightly influenced by the geometric characteristics of the lattice, being similar for the two PCFs. The PCFs with the square lattice have larger effective area for both the pitch values here considered (Fig. 5(b)). In particular, there is a quite greater difference between the

*A*values of the two kind of PCFs if the pitch is large, that is Λ=3

_{e f f}*µm*. Finally, notice the tight field confinement obtained in both the PCFs with

*d*/Λ=0.5 and Λ=3

*µm*, due to the large core diameter (Fig. 6). Since the fundamental component of the magnetic field is all confined inside the first air-hole ring, its shape clearly underlines the differences in the position of the air-holes belonging to the inner ring and, as a consequence, the different geometric characteristics of the two lattices, that is the square and the triangular one. Notice that, due to their field shape, square-lattice PCFs could be useful if applied as pig-tail fibers of integrated optical devices with a rectangular or a square transverse section.

## 4. Conclusion

*nm*. Moreover, a comparison has been made between square-lattice and triangular PCFs with the same Λ and

*d*/Λ, taking into account the dispersion parameter and the effective area values in the wavelength range between 1200

*nm*and 1600

*nm*.

## References and links

1. | T.M. Monro, “Tutorial - Holey fibers: fundamentals and applications,” Optical Fiber Communication Conference 2002, TuD. |

2. | A. Bjarklev, “Photonic Crystal Fibers and their Applications,” European Conference on Optical Communication 2003, We3.3. |

3. | P. St. J. Russell, J. C. Knight, T. A. Birks, P. J. Roberts, and H. Sabert, “Photonic crystal fibres: mastering the flow of light,” European Conference on Optical Communication 2003, We1.7.1. |

4. | F. Poli, A. Cucinotta, M. Fuochi, S. Selleri, and L. Vincetti, “Characterization of microstructured optical fibers for wideband dispersion compensation,” J. Opt. Soc. Am. A |

5. | Lin-Ping Shen, Wei-Ping Huang, and Shui-Sheng Jian, “Design of Photonic Crystal Fibers for Dispersion-Related Applications,” J. Lightwave Technol. |

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

7. | B. T. Kuhlmey, G. Renversez, and D. Maystre, “Chromatic dispersion and losses of microstructured optical fibers,” Appl. Opt. OT |

8. | F. Poli, A. Cucinotta, S. Selleri, and A. H. Bouk, “Tailoring of flattened dispersion in highly nonlinear photonic crystal fibers,” IEEE Photon. Technol. Lett., to be published (2004). [CrossRef] |

9. | K. Saitoh and M. Koshiba, “Chromatic dispersion control in photonic crystal fibers: application to ultra-flattened dispersion,” Opt. Express |

10. | P. St. J. Russell, E. Marin, A. Díez, S. Guenneau, and A. B. Movchan, “Sonic band gaps in PCF preforms: enhancing the interaction of sound and light,” Opt. Express |

11. | A. Cucinotta, S. Selleri, L. Vincetti, and M. Zoboli, “Holey Fiber Analysis Through the Finite-Element Method,” IEEE Photon. Technol. Lett. |

12. | A. Cucinotta, F. Poli, S. Selleri, L. Vincetti, and M. Zoboli, “Amplification Properties of Er |

13. | B. T. Kuhlmey, R. C McPhedran, C. M. de Sterke, P. A. Robinson, G. Renversez, and D. Maystre, “Microstructured optical fibers: where’s the edge?,” Opt. Express |

14. | N. A. Mortensen, J. R. Folkenberg, M. D. Nielsen, and K. P. Hansen, “Modal cutoff and the V parameter in photonic crystal fibers,” Opt. Lett. |

**OCIS Codes**

(060.2310) Fiber optics and optical communications : Fiber optics

(060.2340) Fiber optics and optical communications : Fiber optics components

(060.2400) Fiber optics and optical communications : Fiber properties

**ToC Category:**

Research Papers

**History**

Original Manuscript: February 3, 2004

Revised Manuscript: March 1, 2004

Published: March 8, 2004

**Citation**

A. Bouk, A. Cucinotta, F. Poli, and S. Selleri, "Dispersion properties of square-lattice photonic crystal fibers," Opt. Express **12**, 941-946 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-5-941

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### References

- T.M. Monro, ???Tutorial - Holey fibers: fundamentals and applications,??? Optical Fiber Communication Conference 2002, TuD.
- A. Bjarklev, ???Photonic Crystal Fibers and their Applications,??? European Conference on Optical Communication 2003, We3.3.
- P. St. J. Russell, J. C. Knight, T. A. Birks, P. J. Roberts and H. Sabert, ???Photonic crystal fibres: mastering the flow of light,??? European Conference on Optical Communication 2003, We1.7.1
- F. Poli, A. Cucinotta, M. Fuochi, S. Selleri and L. Vincetti, ???Characterization of microstructured optical fibers for wideband dispersion compensation,??? J. Opt. Soc. Am. A 20, 1958???1962 (2003). [CrossRef]
- Lin-Ping Shen,Wei-Ping Huang and Shui-Sheng Jian, ???Design of Photonic Crystal Fibers for Dispersion-Related Applications,??? J. Lightwave Technol. 21, 1644???1651 (2003 [CrossRef]
- A. Ferrando, E. Silvestre, P. Andrés, J. J. Miret and M. V. Andrés, ???Designing the properties of dispersion-flattened 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]
- B. T. Kuhlmey, G. Renversez and D. Maystre, ???Chromatic dispersion and losses of microstructured optical fibers,??? Appl. Opt. 42, 634???639 (2003). [CrossRef]
- F. Poli, A. Cucinotta, S. Selleri and A. H. Bouk, ???Tailoring of flattened dispersion in highly nonlinear photonic crystal fibers,??? IEEE Photon. Technol. Lett., to be published (2004). [CrossRef]
- K. Saitoh and M. Koshiba, ???Chromatic dispersion control in photonic crystal fibers: application to ultra-flattened dispersion,??? Opt. Express 11, 843???852 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-8-843">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-8-843</a> [CrossRef] [PubMed]
- P. St. J. Russell, E. Marin, A. Díez, S. Guenneau and A. B. Movchan, ???Sonic band gaps in PCF preforms: enhancing the interaction of sound and light,??? Opt. Express 11, 2555???2560 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-20-2555">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-20-2555</a> [CrossRef] [PubMed]
- 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]
- A. Cucinotta, F. Poli, S. Selleri, L. Vincetti and M. Zoboli, ???Amplification Properties of Er3+-Doped Photonic Crystal Fibers,??? J. Lightwave Technol. 21, 782???788 (2003). [CrossRef]
- B. T. Kuhlmey, R. C McPhedran, C. M. de Sterke, P. A. Robinson, G. Renversez and D. Maystre, ???Microstructured optical fibers: where???s the edge?,??? Opt. Express 10, 1285???1290 (2002), <a href= "http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-22-1285">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-22-1285</a> [CrossRef] [PubMed]
- N. A. Mortensen, J. R. Folkenberg, M. D. Nielsen and K. P. Hansen, ???Modal cutoff and the V parameter in photonic crystal fibers,??? Opt. Lett. 28, 1879???1881 (2003). [CrossRef] [PubMed]

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