## Birefringence induced by irregular structure in photonic crystal fiber

Optics Express, Vol. 11, Issue 22, pp. 2799-2806 (2003)

http://dx.doi.org/10.1364/OE.11.002799

Acrobat PDF (157 KB)

### Abstract

The unintentional birefringence induced by the irregular structure in photonic crystal fibers is analyzed numerically using the plane wave expansion method. The statistical correlations between the birefringence and the various irregularities are obtained. The birefringence is found to be largely dependent on the fiber design parameters as well as the degree of the irregularity. And the large pitch and the small air hole make the fiber less sensitive to the structural irregularity, which is successfully explained by the simple perturbation theory. The accuracy of our analyses is confirmed by the detailed investigation of computational errors. This study provides the essential information for the characterization and the design of low birefringence photonic crystal fibers.

© 2003 Optical Society of America

## 1. Introduction

1. 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]

2. 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]

3. J. C. Knight, T. A. Birks, R. F. Cregan, P. St. J. Russel, and J. -P. deSandro, “Large mode area photonic crystal fiber,” Electron. Lett. **34**, 1347–1348 (1998). [CrossRef]

4. A. Ortigosa-Blanch, J. C. Knight, W. J. Wadsworth, J. Arriaga, B. J. Mangan, T. A. Birks, and P. St. J. Russell, “Highly birefringent photonic crystal fibers,” Opt. Lett. **25**1325–1327 (2000). [CrossRef]

5. Theis P. Hansen, Jes Broeng, Stig E. B. Libori, Erik Knudsen, Anders Bjarklev, Jacob Riis Jensen, and Harald Simonsen, “Highly birefringent index-guiding photonic crystal fibers,” IEEE Photon. Technol. Lett. **13**, 588–590 (2001). [CrossRef]

7. A. Peyrilloux, T. Chartier, A. Hideur, L. Berthelot, G. Mélin, S. Lempereur, D. Pagnoux, and P. Roy, “Theoretical and experimental study of the birefringence of a photonic crystal fiber,” J. Lightwave Technol. **21**, 536–539 (2003). [CrossRef]

8. M. J. Steel, T. P. White, C. Martijn de Sterke, R. C. McPhedran, and L C. Botten, “Symmetry and degeneracy in microstructured optical fibers,” Opt. Lett. **26**, 488–490 (2001). [CrossRef]

9. Se-Heon Kim and Yong-Hee Lee, “Symmetry Relations of Two-Dimensional Photonic Crystal Cavity Modes,” IEEE J. Quantum Electron. **39**, 1081–1086 (2003). [CrossRef]

13. Masanori Koshiba and Kunimasa Saitoh, “Numerical verification of degeneracy in hexagonal photonic crystal fibers,” IEEE Photon. Technol. Lett. **13**, 1313–1315 (2001). [CrossRef]

## 2. Numerical birefringence of the perfect PCF

*d*/Λ and the scale factor Λ/

*λ*. In this paper,

*d*/Λ=0.48 is selected for initial studies. It enables the single mode operation over a wide range of Λ/

*λ*, at least between 1 and 11, and the relatively large hole size provides strong guidance of the optical mode against the fiber bend [1

1. 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]

11. S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express **8**, 173–190 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173 [CrossRef] [PubMed]

12. Yong-Jae Lee, Dae-Sung Song, Se-Heon Kim, Jun Huh, and Yong-Hee Lee, “Modal characteristics of photonic crystal fibers,” J. Opt. Soc. Korea **7**, 47–52 (2003). [CrossRef]

*n*=10

^{-4}~10

^{-7}), it is important to guarantee the computation error to be below this level. The calculation accuracy in the plane wave expansion method is influenced by several parameters such as the number of plane waves, the size of the supercell, and the tolerance. The tolerance is the numerical criterion of zero to be reached through iterations. Here we try to find the optimum parameter set by repeating the calculation for the perfectly symmetric PCF while varying these computational parameters.

13. Masanori Koshiba and Kunimasa Saitoh, “Numerical verification of degeneracy in hexagonal photonic crystal fibers,” IEEE Photon. Technol. Lett. **13**, 1313–1315 (2001). [CrossRef]

*λ*. This amount of numerical birefringence in the perfect PCF comes from the finiteness of the computational model, and limits the calculation accuracy of the actual birefringence. It should be noted that the numerical birefringence error is bigger at a longer wavelength (

*λ*) or for a smaller fiber structure (Λ).

^{-7}. In most calculations, the resolution of 64 is used, which corresponds to the number of plane waves of (7×64)

^{2}. Only when the total birefringence of a PCF sample is close to or below the numerical error level, that sample is recalculated with resolution of 256 to confirm the accuracy of the result.

8. M. J. Steel, T. P. White, C. Martijn de Sterke, R. C. McPhedran, and L C. Botten, “Symmetry and degeneracy in microstructured optical fibers,” Opt. Lett. **26**, 488–490 (2001). [CrossRef]

9. Se-Heon Kim and Yong-Hee Lee, “Symmetry Relations of Two-Dimensional Photonic Crystal Cavity Modes,” IEEE J. Quantum Electron. **39**, 1081–1086 (2003). [CrossRef]

*d*/Λ decreases. However, the fluctuation of the error due to the variation of the hole size is still well below the real birefringence studied in this work.

*x*or

*y*. When this supercell is resolved into the same number of grids along two sides of the rhombus, the grids have different densities (or resolutions) along

*x*- and

*y*-directions. The numerical error due to this uneven resolution also decreases with the dimension of the grid.

*x*and

*y*directions are different by only ~1%. Figure 3 shows the calculated numerical birefringence at different resolutions. At the resolutions of 128 and 256, the numerical errors are much smaller than those shown in Fig. 2(b), which proves the effect of the supercell shape. In the following calculations, however, the rhombus shape is used for the supercell, since no improvement is observed for the rectangular supercell at the resolution of 64.

## 3. Birefringence of the PCFs with irregular structures

*δd/d*) serves as the degree of irregularity. First, with

*δd/d*=0.2, 20 PCF samples are randomly generated and their birefringence are calculated using the above method. Figure 4(b) shows the birefringence distribution of the 20 samples at Λ/

*λ*=6.21. Two insets show the fiber structures of two samples with the smallest and largest birefringence. The lower one has relatively uniform holes at the innermost ring, by accident, while the upper one shows the severe asymmetry. This confirms that the symmetry of innermost holes plays the major role for the birefringence of PCFs.

^{-6}and 3.0×10

^{-6}, respectively, in this case.

*δd/d*, and the results are summarized in Fig. 4(c). The mean and the standard deviation of each distribution of the birefringence are plotted as dots and error bars, respectively. The data for Λ/

*λ*=6.21 correspond to the birefringence, for example, at 1550 nm in a PCF with Λ=9.3

*µm*and

*d*=4.3

*µm*. One interesting result is that the birefringence strongly depends on the scale factor Λ/

*λ*[10]. Increasing the size of the fiber structure as well as suppressing the hole size variation will have significant effects on the reduction of the birefringence. The origin of this scale dependence will be discussed later.

^{-6}, at Λ/

*λ*=6.21. Or, once they have knowledge about the structural quality of PCFs, their birefringence distribution can be estimated from the plot.

*A*and

*B*are the fitting parameters. Table 1 shows

*A*and

*B*values at each scale factor.

*A*is close to unity at the small value of Λ/

*λ*, which means the magnitude of the birefringence is proportional to

*δd/d*in Eq. (2). However, for large values of Λ/

*λ, A*is smaller than 1 and the birefringence grows slower than the linear case.

*q*, has the Gaussian probability distribution with a standard deviation,

*δq*, and the direction of the displacement of each hole is random. The hole size is fixed at

*d*/Λ=0.46 and the degree of the hole position variation is defined as

*δq*/Λ. 20 random samples are generated for each degree of the position variation. The mean and the standard deviation of the birefringence are plotted in Fig. 6. Again, the linear relation is found in the log-log plot, and its coefficients are shown in Table 2. The overall behaviors, including the dependence of the coefficient

*A*on the scale factor, are found to be very similar between the two different types of the irregularities.

*S*is the hole area. In case of the hole position variation,

*δq*is the offset from the original position. In both cases, the perturbed area

*δS*is proportional to the degree of irregularity, in agreement with the semi-linear relations between the birefringence and (

*δd/d*) or (

*δq*/Λ). The coefficient 2

*d*Λ in Eq. (5) is bigger than the coefficient (

*π*/2)

*d*

^{2}in Eq. (4) by factor of ~2.8. It explains why the position-dependent birefringence is bigger than the size-dependent birefringence at the same degree of the irregularity. (Compare the

*B*values in Table 1 and 2.)

*y*-polarized) as a function of

*x*at

*y*=0 in Fig. 1. The vertical lines denote the boundary of innermost air holes, and the markers indicate the optical intensity at the first boundary. Note that the optical power is better confined in the silica for the larger structure or at the short wavelength, leaving little power in and around the holes. The intensity at the hole boundary for Λ/

*λ*=2.09 is more than 10 times stronger than that for Λ/

*λ*=10.35. This observation is consistent with the suppression of the birefringence at the larger scale factor by factor of 30 ~100 as observed in Fig. 4(c) and 5(b).

*d*/Λ=0.46. To investigate the behavior at different hole sizes, the same process is repeated at

*d*/Λ=0.30 with the hole size variation of 10%. The obtained birefringence is lower than that shown in Fig. 4(c) for all Λ/

*λ*, and it is equivalent to ~3% hole size variation at

*d*/Λ=0.46. This indicates that the PCF with the small holes is advantageous in suppressing the birefringence as long as

*δd/d*is constant.

*d*

^{2}. The reduction of the hole diameter from 0.46 to 0.30 results in the reduction of

*dS*by factor of 2.4, which explains the reduction of the birefringence.

_{d}## 4. Design for low birefringence fibers

*λ*and/or a small hole size

*d*/Λ is desirable for the lower birefringence assuming the degrees of the regularities (

*δd/d*) and (

*δq*/Λ) are determined regardless of

*d*and Λ. Especially, increasing the scale factor is very effective. For example, the increase of the scale factor by a factor of 2 brings the equivalent effect of reducing the degree of regularity by a factor of ~10. (See Fig. 4(c) and 5(b).)

*d*), so it is approximately proportional to the scale factor. The bend loss, in general, depends on the effect index difference between the core and the cladding regions, which decreases at the larger scale factor and the smaller hole size, resulting in the higher bend losses [1

1. 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]

*λ*, it is desirable to maximize the pitch Λ, and minimize the hole size

*d*as long as the mode size and the bend loss allow.

## 5. Conclusion

## Acknowledgments

## References and links

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

2. | 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. |

3. | J. C. Knight, T. A. Birks, R. F. Cregan, P. St. J. Russel, and J. -P. deSandro, “Large mode area photonic crystal fiber,” Electron. Lett. |

4. | A. Ortigosa-Blanch, J. C. Knight, W. J. Wadsworth, J. Arriaga, B. J. Mangan, T. A. Birks, and P. St. J. Russell, “Highly birefringent photonic crystal fibers,” Opt. Lett. |

5. | Theis P. Hansen, Jes Broeng, Stig E. B. Libori, Erik Knudsen, Anders Bjarklev, Jacob Riis Jensen, and Harald Simonsen, “Highly birefringent index-guiding photonic crystal fibers,” IEEE Photon. Technol. Lett. |

6. | T. Niemi, H. Ludvigsen, F. Scholder, M. Legré, M. Wegmuller, N. Gisin, J. R. Jensen, A. Petersson, and P. M. W. Skovgaard, “Polarization properties of single-moded, large-mode area photonic crystal fibers,” in |

7. | A. Peyrilloux, T. Chartier, A. Hideur, L. Berthelot, G. Mélin, S. Lempereur, D. Pagnoux, and P. Roy, “Theoretical and experimental study of the birefringence of a photonic crystal fiber,” J. Lightwave Technol. |

8. | M. J. Steel, T. P. White, C. Martijn de Sterke, R. C. McPhedran, and L C. Botten, “Symmetry and degeneracy in microstructured optical fibers,” Opt. Lett. |

9. | Se-Heon Kim and Yong-Hee Lee, “Symmetry Relations of Two-Dimensional Photonic Crystal Cavity Modes,” IEEE J. Quantum Electron. |

10. | S. B. Libori, J. Broeng, E. Knudsen, A. Bjarklev, and H. R. Simonsen, “High-birefringent photonic crystal fiber,” in |

11. | S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express |

12. | Yong-Jae Lee, Dae-Sung Song, Se-Heon Kim, Jun Huh, and Yong-Hee Lee, “Modal characteristics of photonic crystal fibers,” J. Opt. Soc. Korea |

13. | Masanori Koshiba and Kunimasa Saitoh, “Numerical verification of degeneracy in hexagonal photonic crystal fibers,” IEEE Photon. Technol. Lett. |

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

**OCIS Codes**

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

(060.2400) Fiber optics and optical communications : Fiber properties

(260.1440) Physical optics : Birefringence

(260.5430) Physical optics : Polarization

**ToC Category:**

Research Papers

**History**

Original Manuscript: September 23, 2003

Revised Manuscript: October 15, 2003

Published: November 3, 2003

**Citation**

In-Kag Hwang, Yong-Jae Lee, and Yong-Hee Lee, "Birefringence induced by irregular structure in photonic crystal fiber," Opt. Express **11**, 2799-2806 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-22-2799

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### References

- 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]
- 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]
- J. C. Knight, T. A. Birks, R. F. Cregan, P. St. J. Russell, and J. �??P. de Sandro, �??Large mode area photonic crystal fiber,�?? Electron. Lett. 34, 1347-1348 (1998). [CrossRef]
- A. Ortigosa-Blanch, J. C. Knight, W. J. Wadsworth, J. Arriaga, B. J. Mangan, T. A. Birks, and P. St. J. Russell, �??Highly birefringent photonic crystal fibers,�?? Opt. Lett. 25 1325-1327 (2000). [CrossRef]
- Theis P. Hansen, Jes Broeng, Stig E. B. Libori, Erik Knudsen, Anders Bjarklev, Jacob Riis Jensen, and Harald Simonsen, �??Highly birefringent index-guiding photonic crystal fibers,�?? IEEE Photon. Technol. Lett. 13, 588-590 (2001). [CrossRef]
- T. Niemi, H. Ludvigsen, F. Scholder, M. Legré, M. Wegmuller, N. Gisin, J. R. Jensen, A. Petersson, and P. M. W. Skovgaard, �??Polarization properties of single-moded, large-mode area photonic crystal fibers,�?? in Proc. European Conference on Optical Communication 2002, paper M.S1.09.
- A. Peyrilloux, T. Chartier, A. Hideur, L. Berthelot, G. Mélin, S. Lempereur, D. Pagnoux, and P. Roy, �??Theoretical and experimental study of the birefringence of a photonic crystal fiber,�?? J. Lightwave Technol. 21, 536-539 (2003). [CrossRef]
- M. J. Steel, T. P. White, C. Martijn de Sterke, and R. C. McPhedran, and L C. Botten, �??Symmetry and degeneracy in microstructured optical fibers,�?? Opt. Lett. 26, 488-490 (2001). [CrossRef]
- Se-Heon Kim and Yong-Hee Lee, �??Symmetry Relations of Two-Dimensional Photonic Crystal Cavity Modes,�?? IEEE J. Quantum Electron. 39, 1081-1086 (2003). [CrossRef]
- S. B. Libori, J. Broeng, E. Knudsen, A. Bjarklev, and H. R. Simonsen, �??High-birefringent photonic crystal fiber,�?? in Proc. Optical Fiber Communication Conference 2001, Vol. 54 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1900), pp. TuM2-1 -TuM2-3.
- S. G. Johnson and J. D. Joannopoulos, "Block-iterative frequency-domain methods for Maxwell's equations in a planewave basis," Opt. Express 8, 173-190 (2001), <a href=" http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173</a>. [CrossRef] [PubMed]
- Yong-Jae Lee, Dae-Sung Song, Se-Heon Kim, Jun Huh, and Yong-Hee Lee, �??Modal characteristics of photonic crystal fibers,�?? J. Opt. Soc. Korea 7, 47-52 (2003). [CrossRef]
- Masanori Koshiba, and Kunimasa Saitoh, �??Numerical verification of degeneracy in hexagonal photonic crystal fibers,�?? IEEE Photon. Technol. Lett. 13, 1313-1315 (2001). [CrossRef]
- A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, New York, 1983), Chap. 18.

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