## Analysis of photonic crystal fibers: Scalar solution and polarization correction

Optics Express, Vol. 14, Issue 24, pp. 11848-11854 (2006)

http://dx.doi.org/10.1364/OE.14.011848

Acrobat PDF (377 KB)

### Abstract

A numerical approach based on the scalar finite element method is applied to analyse the modal properties of photonic crystal fibers having a solid core and a cladding region with either circular or non-circular microstructured holes. A correction which accounts for the polarization effects due to the large refractive index difference between silica materials and air holes is included in the analysis. Numerical results show that the proposed technique is an efficient and accurate alternative to vector ones.

© 2006 Optical Society of America

## 1. Introduction

1. J. C. Knight, T. A. Birks, P. St. J. Russell, and D. M. Aktin, “All silica single mode optical fiber with photonic crystal cladding,” Opt. Lett. **21**, 1547–1549 (1999). [CrossRef]

2. T. A. Birks, J. C. Knight, and P. St. J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. **22**, 961–963 (1997). [CrossRef] [PubMed]

3. W. J. Wadsworth, J. C. Knight, A. Ortigosa-Blanch, J. Arriaga, E. Silvestre, and P. St. J. Russell, “Soliton effects in photonic crystal fibers at 850 nm,” Electron. Lett. **36**, 53–55 (2000). [CrossRef]

4. K. Furusawa, A. N. Malinowski, J. H. V. Price, T. M. Monro, J. K. Sahu, J. Nilsson, and D. J. Richardson, “Cladding pumped Ytterbium-doped fiber laser with holey inner and outer cladding,” Opt. Express **9**, 714–720 (2001). [CrossRef] [PubMed]

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

8. 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]

9. T. M. Monro, D. J. Richardson, N. G. R. Broderick, and P. J. Bennett, “Modeling large air fraction holey optical fibers,” J. Lightwave Technol. **18**, 50–56 (2000). [CrossRef]

10. F. Fogli, L. Saccomandi, P. Bassi, G. Bellanca, and S. Trillo, “Full vectorial BPM modeling of indexguiding photonic crystal fibers and couplers,” Opt. Express **10**, 54–59 (2002). [PubMed]

11. K. Saitoh and M. Koshiba, “Full-vectorial imaginary-distance beam propagation method based on a finite element scheme: application to photonic crystal fibers,” J. Quantum Electron. **38**, 927–933 (2002). [CrossRef]

12. T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. Martijn de Sterke, and L. C. Botten, “Multipole method for microstructured optical fibers. I. Formulation,” J. Opt. Soc. Am. B **19**, 2322–2330 (2002). [CrossRef]

13. S. Campbell, R. C. McPhedran, C. Martijn de Sterke, and L. C. Botten, “Differential multipole method for microstructured optical fibers,” J. Opt. Soc. Am. B **21**, 1919–1928 (2004). [CrossRef]

14. Z. Zhu and T. G. Brown, “Full-vectorial finite-difference analysis of microstructured optical fibers, “Opt. Express **10**, 853–864 (2002). [PubMed]

15. C. P. Yu and H. C. Chang, “Applications of the finite difference mode solution method to photonic crystal structures,” Opt. Quantum Electron. **36**, 145–163 (2004). [CrossRef]

16. M. Qiu, “Analysis of guided modes in photonic crystal fibers using the finite-difference time-domain method,” Microwave Opt. Technol. Lett. **30**, 327–330 (2001). [CrossRef]

17. K. Saitoh and M. Koshiba, “Numerical modeling of photonic crystal fibers,” J. Lightwave Technol. **23**, 3580–3580 (2005). [CrossRef]

2. T. A. Birks, J. C. Knight, and P. St. J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. **22**, 961–963 (1997). [CrossRef] [PubMed]

18. T. N. Monro, D. J. Richardson, N. G. R. Broderick, and P. J. Bennett, “Holey optical fibers: An efficient modal model,” J. Lightwave Technol. **17**, 1093–1102 (1999). [CrossRef]

## 2. Analysis method

### 2.1 Scalar solution

*β*is the propagation constant,

*ẑ*is the unit vector parallel to the waveguide axis and

*=*

**E**_{t}*E*

_{x}

*x̂*+

*E*

_{y}

*ŷ*and

*E*

_{z}are transversal and longitudinal components of the electric field, respectively. If we work with the fields of Eq. (1) in the full vectorial wave equation, it is easy to demonstrate that the transverse modal electric field satisfies the vector wave equation

*n*=

*n*(

*x*,

*y*) is the refractive-index profile, and

*k*

_{0}=2π/λ the wave number in the vacuum, λ being the wavelength.

*and*

**Ẽ**_{t}*are the scalar field and its corresponding propagating constant, respectively. This approximation is valid when coupling between orthogonal field components become negligible.*β ˜

*e*} contains the values of the electric field at the vertices of the triangular elements used for discretization.

2. T. A. Birks, J. C. Knight, and P. St. J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. **22**, 961–963 (1997). [CrossRef] [PubMed]

20. J. Riishede, N. A. Mortensen, and J. N. Lægsgaard, “A ‘poor man’s approach’ to modelling microstructured optical fibres,” J. Opt. A: Pure Appl. Opt. **5**, 534–538 (2003). [CrossRef]

23. N. A. Mortensen, “Semianalytical approach to short-wavelength dispersion and modal properties of photonic crystal fibers,” Opt. Lett. **30**, 1455–1457 (2005). [CrossRef] [PubMed]

### 2.2. Polarization correction

*and*β ˜

*β*as described in Ref. [24]

*A*is the fiber cross section.

*δβ*

^{2}exactly we would have to solve the vector wave Eq. (2). However, using simple perturbations methods [24], we have that the polarization correction to

*x*- or

*y*-polarized mode, to first order, reduces to

*i*and

*x*

_{i}are equal to

*x*or

*y*for

*x*and

*y*polarized mode, respectively. This term is always negative and therefore tends to shift down the propagation constant obtained from the scalar analysis.

## 3. Results and discussion

*n*

_{eff}=

*β*/

*k*

_{0}of the fundamental mode for different PCFs with vector results reported in literature. A single-polarization single mode (SPSM) fiber is also included in our analysis, as a case to predict sensitive vector quantities such as birefringence. All results were obtained by using large computational domains with Dirichlet boundary conditions.

### 3.1. Fiber with triangular lattice cladding

*d*is the hole diameter. The PCF symmetry allows just one quarter of the structure to be considered for the numerical simulation. The scalar and the corrected scalar approaches were tested by comparing the computed curves of the effective index for the H

11. K. Saitoh and M. Koshiba, “Full-vectorial imaginary-distance beam propagation method based on a finite element scheme: application to photonic crystal fibers,” J. Quantum Electron. **38**, 927–933 (2002). [CrossRef]

*d*/Λ is taken as a parameter, over a wide range of wavelength from λ=0.4 µm to λ=2.0 µm. As it was expected, the scalar solution is valid in the shorter wavelength region (λ<0.5 µm), and begins to break down with increasing air filling fraction. On the other hand, the results that include the polarization correction show an overall good agreement with vectorial ones, hence justifying the proposed calculation scheme. For large hole size

*d*/Λ=0.7, where the polarization effects are important, the maximum relative error at longer wavelength of the studied range is 0.12%, which underlines the relevance of our approximation.

### 3.2. Annular-shaped holes fiber

25. H. P. Uranus and H. J. W. M. Hoekstr, “Modelling of microstructured waveguides using a finite-elementbased vectorial mode solver with transparent boundary conditions,” Opt. Express **12**, 2795–2809 (2004). [CrossRef] [PubMed]

*r*

_{1}=1 µm and outer radius

*r*

_{2}=2 µm and angular width of 108°. In this case, we use a half-circle with radius

*r*=3.0 µm as computational window. The computed effective-index curve of the HE

_{11}-like mode of this structure is shown in Fig. 2(b) with the curve reported in Ref. [25

25. H. P. Uranus and H. J. W. M. Hoekstr, “Modelling of microstructured waveguides using a finite-elementbased vectorial mode solver with transparent boundary conditions,” Opt. Express **12**, 2795–2809 (2004). [CrossRef] [PubMed]

### 3.3. Cobweb fiber

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

26. 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]

### 3.4. SPSM fiber

27. K. Saitoh and M. Koshiba, “Single-polarization single-mode photonic crystal fibers,” IEEE Photon. Technol. Lett. **15**, 1384–1386 (2003). [CrossRef]

^{-3}at λ=1.45 µm and is in good agreement with the reported value of 3.27×10

^{-3}obtained with a full-vector FEM [27

27. K. Saitoh and M. Koshiba, “Single-polarization single-mode photonic crystal fibers,” IEEE Photon. Technol. Lett. **15**, 1384–1386 (2003). [CrossRef]

## 4. Conclusion

## Acknowledgments

## References and links

1. | J. C. Knight, T. A. Birks, P. St. J. Russell, and D. M. Aktin, “All silica single mode optical fiber with photonic crystal cladding,” Opt. Lett. |

2. | T. A. Birks, J. C. Knight, and P. St. J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. |

3. | W. J. Wadsworth, J. C. Knight, A. Ortigosa-Blanch, J. Arriaga, E. Silvestre, and P. St. J. Russell, “Soliton effects in photonic crystal fibers at 850 nm,” Electron. Lett. |

4. | K. Furusawa, A. N. Malinowski, J. H. V. Price, T. M. Monro, J. K. Sahu, J. Nilsson, and D. J. Richardson, “Cladding pumped Ytterbium-doped fiber laser with holey inner and outer cladding,” Opt. Express |

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

6. | W. H. Reeves, J. C. Knight, P. St. J. Russell, and P. J. Roberts, “Demonstration of ultra-flattened dispersion in photonic crystal fibers,” Opt. Express |

7. | K. Saitoh, M. Koshiba, T. Hasegawa, and E. Sasaoka, “Chromatic dispersion control in photonic crystal fiber: application to ultra-flattened dispersion,” Opt. Express |

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

9. | T. M. Monro, D. J. Richardson, N. G. R. Broderick, and P. J. Bennett, “Modeling large air fraction holey optical fibers,” J. Lightwave Technol. |

10. | F. Fogli, L. Saccomandi, P. Bassi, G. Bellanca, and S. Trillo, “Full vectorial BPM modeling of indexguiding photonic crystal fibers and couplers,” Opt. Express |

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

12. | T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. Martijn de Sterke, and L. C. Botten, “Multipole method for microstructured optical fibers. I. Formulation,” J. Opt. Soc. Am. B |

13. | S. Campbell, R. C. McPhedran, C. Martijn de Sterke, and L. C. Botten, “Differential multipole method for microstructured optical fibers,” J. Opt. Soc. Am. B |

14. | Z. Zhu and T. G. Brown, “Full-vectorial finite-difference analysis of microstructured optical fibers, “Opt. Express |

15. | C. P. Yu and H. C. Chang, “Applications of the finite difference mode solution method to photonic crystal structures,” Opt. Quantum Electron. |

16. | M. Qiu, “Analysis of guided modes in photonic crystal fibers using the finite-difference time-domain method,” Microwave Opt. Technol. Lett. |

17. | K. Saitoh and M. Koshiba, “Numerical modeling of photonic crystal fibers,” J. Lightwave Technol. |

18. | T. N. Monro, D. J. Richardson, N. G. R. Broderick, and P. J. Bennett, “Holey optical fibers: An efficient modal model,” J. Lightwave Technol. |

19. | C. E. Kerbage, B. J. Eggleton, P. S. Westbrook, and R. S. Windeler, “Experimental and scalar beam propagation analysis of an air-silica microstructure fiber,” Opt. Express |

20. | J. Riishede, N. A. Mortensen, and J. N. Lægsgaard, “A ‘poor man’s approach’ to modelling microstructured optical fibres,” J. Opt. A: Pure Appl. Opt. |

21. | V. H. Aristizabal, F. J. Vélez, and P. Torres, “Modelling of photonic crystal fibers with the scalar finite element method,” in |

22. | T. A. Birks, D. M. Bird, T. D. Hedley, J. M. Pottage, and P. St. J. Russell, “Scaling laws and vector effects in bandgap-guiding fibres,” Opt. Express |

23. | N. A. Mortensen, “Semianalytical approach to short-wavelength dispersion and modal properties of photonic crystal fibers,” Opt. Lett. |

24. | A. W. Snyder and J. D. Love, |

25. | H. P. Uranus and H. J. W. M. Hoekstr, “Modelling of microstructured waveguides using a finite-elementbased vectorial mode solver with transparent boundary conditions,” Opt. Express |

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

27. | K. Saitoh and M. Koshiba, “Single-polarization single-mode photonic crystal fibers,” IEEE Photon. Technol. Lett. |

**OCIS Codes**

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

(060.2400) Fiber optics and optical communications : Fiber properties

(060.2430) Fiber optics and optical communications : Fibers, single-mode

**ToC Category:**

Photonic Crystal Fibers

**History**

Original Manuscript: September 18, 2006

Revised Manuscript: November 8, 2006

Manuscript Accepted: November 11, 2006

Published: November 27, 2006

**Citation**

Víctor Hugo Aristizabal, Francisco Javier Vélez, and Pedro Torres, "Analysis of photonic crystal fibers: Scalar solution and polarization correction," Opt. Express **14**, 11848-11854 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-24-11848

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### References

- J. C. Knight, T. A. Birks, P. St. J. Russell, and D. M. Aktin, "All silica single mode optical fiber with photonic crystal cladding," Opt. Lett. 21, 1547-1549 (1999). [CrossRef]
- T. A. Birks, J. C. Knight, P. St. J. Russell, "Endlessly single-mode photonic crystal fiber," Opt. Lett. 22, 961-963 (1997). [CrossRef] [PubMed]
- W. J. Wadsworth, J. C. Knight, A. Ortigosa-Blanch, J. Arriaga, E. Silvestre, and P. St. J. Russell, "Soliton effects in photonic crystal fibers at 850 nm," Electron. Lett. 36, 53-55 (2000). [CrossRef]
- K. Furusawa, A. N. Malinowski, J. H. V. Price, T. M. Monro, J. K. Sahu, J. Nilsson, and D. J. Richardson, "Cladding pumped Ytterbium-doped fiber laser with holey inner and outer cladding," Opt. Express 9, 714-720 (2001). [CrossRef] [PubMed]
- J. C. Knight, J. Arriaga, T. A. Birks, A. Ortigosa-Blanch, W. J. Wadsworth, and P. St. J. Russell, "Anomalous dispersion in photonic crystal fiber," IEEE Photon. Technol. Lett. 12, 807-809 (2000). [CrossRef]
- W. H. Reeves, J. C. Knight, P. St. J. Russell, and P. J. Roberts, "Demonstration of ultra-flattened dispersion in photonic crystal fibers," Opt. Express 10, 609-613 (2002). [PubMed]
- K. Saitoh, M. Koshiba, T. Hasegawa, and E. Sasaoka, "Chromatic dispersion control in photonic crystal fiber: application to ultra-flattened dispersion," Opt. Express 11, 843-852 (2003). [CrossRef] [PubMed]
- A. Ferrando, E. Silvestre, J. J. Miret, P. Andrés, M. V. Andrés, "Full-vector analysis of a realistic photonic crystal fiber," Opt. Lett. 24, 276-278 (1999). [CrossRef]
- T. M. Monro, D. J. Richardson, N. G. R. Broderick, and P. J. Bennett, "Modeling large air fraction holey optical fibers," J. Lightwave Technol. 18, 50-56 (2000). [CrossRef]
- F. Fogli, L. Saccomandi, P. Bassi, G. Bellanca, and S. Trillo, "Full vectorial BPM modeling of index-guiding photonic crystal fibers and couplers," Opt. Express 10, 54-59 (2002). [PubMed]
- K. Saitoh and M. Koshiba, "Full-vectorial imaginary-distance beam propagation method based on a finite element scheme: application to photonic crystal fibers," J. Quantum Electron. 38, 927-933 (2002). [CrossRef]
- T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. Martijn de Sterke, and L. C. Botten, "Multipole method for microstructured optical fibers. I. Formulation," J. Opt. Soc. Am. B 19, 2322-2330 (2002). [CrossRef]
- S. Campbell, R. C. McPhedran, C. Martijn de Sterke, and L. C. Botten, "Differential multipole method for microstructured optical fibers," J. Opt. Soc. Am. B 21, 1919-1928 (2004). [CrossRef]
- Z. Zhu and T. G. Brown, "Full-vectorial finite-difference analysis of microstructured optical fibers, "Opt. Express 10, 853-864 (2002). [PubMed]
- C. P. Yu and H. C. Chang, "Applications of the finite difference mode solution method to photonic crystal structures," Opt. Quantum Electron. 36, 145-163 (2004). [CrossRef]
- M. Qiu, "Analysis of guided modes in photonic crystal fibers using the finite-difference time-domain method," Microwave Opt. Technol. Lett. 30, 327-330 (2001). [CrossRef]
- K. Saitoh and M. Koshiba, "Numerical modeling of photonic crystal fibers," J. Lightwave Technol. 23, 3580-3580 (2005). [CrossRef]
- T. N. Monro, D. J. Richardson, N. G. R. Broderick, and P. J. Bennett, "Holey optical fibers: An efficient modal model," J. Lightwave Technol. 17, 1093-1102 (1999). [CrossRef]
- C. E. Kerbage, B. J. Eggleton, P. S. Westbrook, and R. S. Windeler, "Experimental and scalar beam propagation analysis of an air-silica microstructure fiber," Opt. Express 7, 113-122 (2000). [CrossRef] [PubMed]
- J. Riishede, N. A. Mortensen, and J. N. Lægsgaard, "A ‘poor man’s approach’ to modelling micro-structured optical fibres," J. Opt. A: Pure Appl. Opt. 5, 534-538 (2003). [CrossRef]
- V. H. Aristizabal, F. J. Vélez, and P. Torres, "Modelling of photonic crystal fibers with the scalar finite element method," in 5th Iberoamerican Meeting on Optics and 8th Latin American Meeting on Optics, Laser and their Applications, A. Marcano and J. L. Paz, eds., Proc. SPIE 5622, 849-854 (2004). [CrossRef]
- T. A. Birks, D. M. Bird, T. D. Hedley, J. M. Pottage, and P. St. J. Russell, "Scaling laws and vector effects in bandgap-guiding fibres," Opt. Express 12, 69-74 (2004). [CrossRef] [PubMed]
- N. A. Mortensen, "Semianalytical approach to short-wavelength dispersion and modal properties of photonic crystal fibers," Opt. Lett. 30, 1455-1457 (2005). [CrossRef] [PubMed]
- A. W. Snyder and J. D. Love, Optical Waveguide Theory (Kluwer Academic, 2000).
- H. P. Uranus and H. J. W. M. Hoekstr, "Modelling of microstructured waveguides using a finite-element-based vectorial mode solver with transparent boundary conditions," Opt. Express 12, 2795-2809 (2004). [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]
- K. Saitoh and M. Koshiba, "Single-polarization single-mode photonic crystal fibers," IEEE Photon. Technol. Lett. 15, 1384-1386 (2003). [CrossRef]

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