## Structural dependence of effective area and mode field diameter for holey fibers

Optics Express, Vol. 11, Issue 15, pp. 1746-1756 (2003)

http://dx.doi.org/10.1364/OE.11.001746

Acrobat PDF (174 KB)

### Abstract

A rigorous full-vector finite element method is effectively applied to evaluating the effective area *A*_{eff}
and the mode field diameter (MFD) of holey fibers (HFs) with finite cross sections. The effective modal spot size (a half of MFD), *w*_{eff}
, is defined with the help of the second
moment of the optical intensity distribution. The influence of hole diameter, hole pitch, operating wavelength, and number of rings of air holes on *A*_{eff}
and *w*_{eff}
is investigated in detail. As a result, it is shown that *A*_{eff}
and *w*_{eff}
are almost independent of the number of hole rings and that the relation *A*_{eff}
=π*A*_{eff}
and *w*_{eff}
obtained here, the beam divergence and the nonlinear phase shift are calculated and are compared with the earlier experimental results.

© 2003 Optical Society of America

## 1. Introduction

1. J. Broeng, D. Mogilevstev, S.E. Barkou, and A. Bjarklev, “Photonic crystal fibers: A new class of optical waveguides,” Opt. Fiber Technol. **5**, 305–330 (1999). [CrossRef]

6. M. Koshiba and Y. Tsuji,“Curvilinear hybrid edge/nodal elements with triangular shape for guided-wave problems,” J. Lightwave Technol. **18**, 737–743, (2000). [CrossRef]

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

*A*

_{eff}and MFD of HFs with finite cross sections are evaluated. In order to treat axially nonsymmetrical fibers such as HFs with air holes arranged in a triangular lattice in the cladding region, the second moment of the optical intensity distribution is introduced as the definition of

*w*

_{eff}. The influence of hole diameter, hole pitch, operating wavelength, and number of hole rings on

*A*

_{eff}and

*w*

_{eff}is investigated in detail, including confinement losses arising from the leaky nature of HFs with finite cross sections. As a result, it is shown that in contrast with the confinement losses which is strongly dependent of the number of hole rings [8

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

*A*

_{eff}and

*w*

_{eff}are almost independent of it and that the relation

*A*

_{eff}=π

*A*

_{eff}and

*w*

_{eff}obtained here, the beam divergence and the nonlinear phase shift are calculated and are compared with the earlier experimental results [11

11. M.J. Gander, R. McBride, J.D.C. Jones, T.A. Birks, J.C. Knight, P.St.J. Russell, P.M. Blanchard, J.G. Burnett, and A.H. Greenaway, “Measurement of the wavelength dependence of beam divergence for photonic crystal fiber,” Opt. Lett. **24**, 1017–1019 (1999). [CrossRef]

13. J.H. Lee, P.C. Teh, Z. Yusoff, M. Ibsen, W. Belardi, T.M. Monro, and D.J. Richardson, “A holey fiber-based nonlinear thresholding devices for optical CDMA receiver performance enhancement,” IEEE Photon. Technol. Lett. **14**, 876–878 (2002). [CrossRef]

## 2. Analysis method

*x*and

*y*are the transverse directions,

*z*is the propagation direction,

*d*is the hole diameter, and Λ is the hole pitch. Anisotropic perfectly matched layers (PMLs) [7

7. K. Saitoh and M. Koshiba, “Full-vectorial finite element beam propagation method with perfectly matched layers for anisotropic optical waveguides,” J. Lightwave Technol. **19**, 405–413 (2001). [CrossRef]

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

**38**, 927–933 (2002). [CrossRef]

*is the electric field vector,*

**E***n*is the refractive index, [

*s*] is the PML matrix, and [

*s*]

^{-1}is an inverse matrix of [

^{s}].

6. M. Koshiba and Y. Tsuji,“Curvilinear hybrid edge/nodal elements with triangular shape for guided-wave problems,” J. Lightwave Technol. **18**, 737–743, (2000). [CrossRef]

*E*

_{z}, a nodal element with six variables,

*E*

_{z}

_{1}to

*E*

_{z}

_{6}, is employed, while for the transverse fields,

*E*

_{x}and

*E*

_{y}, an edge element with eight variables,

*E*

_{t}

_{1}to

*E*

_{t}

_{8}, is employed, resulting in significantly fast convergence of solutions [6

6. M. Koshiba and Y. Tsuji,“Curvilinear hybrid edge/nodal elements with triangular shape for guided-wave problems,” J. Lightwave Technol. **18**, 737–743, (2000). [CrossRef]

*E*

_{t}}

_{e}and {

*E*

_{z}}

_{e}are, respectively, the edge and the nodal variables for each element

*e*, {

*U*} and {

*V*} are the shape function vectors for edge elements, {

*N*} is the shape function vector for nodal elements, and

*T*denotes a transpose.

*E*} is the global electric field vector and the finite element matrices [

*K*] and [

*M*] are given in [8

**38**, 927–933 (2002). [CrossRef]

*K*] and [

*M*], Eq. (3) is solved with the multifrontal method [14

14. J.W.H. Liu, “The multifrontal method for sparse matrix solutions: theory and practice,” SIAM Rev. **34**, 82–109 (1992). [CrossRef]

*is the transverse electric field vector and*

**E**_{t}*S*denotes the whole fiber cross section. Using the eigenvector {

*E*} corresponding to the eigenvalue β

^{2}obtained from Eq. (3) and noting the relation of Eq. (2),

*A*

_{eff}is calculated as

_{e}extends over all different elements and the optical intensity distribution for each element,

*I*

_{e}, is given by

*x*) direction and the other in the vertical (

*y*) direction, and these two fundamental modes are degenerate [17

17. 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]

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

16. K. Petermann, “Constraints for fundamental-mode spot size for broadband dispersion-compensated single-mode fibers,” Electron. Lett. **19**, 712–714 (1983). [CrossRef]

19. K. Hayata, M. Koshiba, and M. Suzuki, “Modal spot size of axially nonsymmetrical fibers,” Electron. Lett. **22**, 127–129 (1986). [CrossRef]

*x*and

*y*directions,

*w*

_{x}and

*w*

_{y}, are given by

*x*

_{c}and

*y*

_{c}are, respectively, the

*x*and

*y*coordinates of the center of the field distribution, which are calculated as

*x*and

*y*, are, in general, approximated with quadratic polynomials using the local coordinates,

*L*

_{1},

*L*

_{2}, and

*L*

_{3}, which are also called the area coordinates for the rectilinear elements with triangular shape [6

**18**, 737–743, (2000). [CrossRef]

*U*}, {

*V*}, and {

*N*}, are also given as the functions of

*L*

_{1},

*L*

_{2}, and

*L*

_{3}, Eqs. (6) and (8) to Eq. (11) can be easily and accurately evaluated by using the well-established numerical integration formulas (see Eq. (16) in [6

**18**, 737–743, (2000). [CrossRef]

*x*and

*y*directions,

*w*

_{x}and

*w*

_{y}, are, in general, different from each other, and so we define the following effective modal spot size

*w*

_{eff}:

*w*

_{eff}remains the 1/

*e*

^{2}intensity radius (namely, the 1/

*e*field radius) for a Gaussian profile. A similar definition has been introduced to the measurement of the far-field intensity profile with a Gaussian beam approximation [11

11. M.J. Gander, R. McBride, J.D.C. Jones, T.A. Birks, J.C. Knight, P.St.J. Russell, P.M. Blanchard, J.G. Burnett, and A.H. Greenaway, “Measurement of the wavelength dependence of beam divergence for photonic crystal fiber,” Opt. Lett. **24**, 1017–1019 (1999). [CrossRef]

## 3. Effective area and confinement loss

*d*/Λ, as a parameter, where

*N*is the number of rings of air holes, the background silica index

*n*is assumed to be 1.45, and the normalized confinement loss in dB,

*L*

_{c}Λ, is defined as

*L*

_{c}Λ to be 10

^{-8}dB, the confinement loss

*L*

_{c}becomes 10 dB/km for Λ=1 µm and 1 dB/km for Λ=10 µm.

*et al.*[5]. This seems to be due to the fact that even if

*A*

_{eff}/Λ

^{2}≅ π, the spread of effective area is in the order of Λ corresponding to the position of the first ring. In Fig. 3(f), the results of a rigorous full-vector multipole method (MM) [5] are also plotted. Our results agree well with those of MM, showing the reliability of a full-vector FEM with PMLs.

## 4. Effective area and mode field diameter

*A*

_{eff}and mode field diameter (MFD) as a function of wavelength for different hole sizes, where the MFD is expressed in the form of π

*w*

_{eff}(a half of MFD) being the effective modal spot size given in Eq. (12). Roughly speaking, the results for

*A*

_{eff}and π

*A*

_{eff}=π

*d*/Λ=0.3 at λ/Λ=0.2, where the intensity counters are spaced by 1 dB. The left and right profiles shows the principal field components of the horizontally polarized and vertically polarized modes, respectively, which are degenerate [17

17. 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]

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

*A*

_{eff}=π

*A*

_{eff}.

*w*

_{eff},

*w*

_{x}, and

*w*

_{y}, as a function of wavelength. For the vertically polarized mode, the values of

*w*

_{x}and

*w*

_{y}are interchanged. The results of the effective modal spot size

*w*

_{eff}are the same as those in Fig. 4. The modal spot sizes in the

*x*and

*y*directions,

*w*

_{x}and

*w*

_{y}, are, respectively, obtained from Eqs. (8) and (9). It is worth noting that for HFs with large air holes operating at longer wavelengths, these values are significantly different from each other. This suggests that the mode field in these cases may be considerably deviated from that in ordinary axially symmetrical fibers. Figure 7 shows the intensity profiles of the horizontally polarized and vertically polarized modes in a HF with

*d*/Λ=0.9 at λ/Λ=1.2, where the intensity contours are spaced by 1 dB. Surprisingly, in contrast to those in Fig. 5, these intensity profiles are entirely different from each other, even though these two linearly polarized modes are degenerate. They have no longer six-fold rotational symmetry. Such curious mode shapes have not been reported so far, to our knowledge. Traditional methods, which rely on Gaussian optics to estimate the splice loss, source-to-fiber coupling efficiency, and so on, would fail in these cases, and therefore, more rigorous numerical methods may be required.

## 5. Beam divergence and nonlinear phase shift

11. M.J. Gander, R. McBride, J.D.C. Jones, T.A. Birks, J.C. Knight, P.St.J. Russell, P.M. Blanchard, J.G. Burnett, and A.H. Greenaway, “Measurement of the wavelength dependence of beam divergence for photonic crystal fiber,” Opt. Lett. **24**, 1017–1019 (1999). [CrossRef]

13. J.H. Lee, P.C. Teh, Z. Yusoff, M. Ibsen, W. Belardi, T.M. Monro, and D.J. Richardson, “A holey fiber-based nonlinear thresholding devices for optical CDMA receiver performance enhancement,” IEEE Photon. Technol. Lett. **14**, 876–878 (2002). [CrossRef]

*d*/Λ=0.26 and with Λ=7.2 µm and

*d*/Λ=0.53, where the beam divergence θ is given by

*w*

_{eff}being the effective modal spot size in (12

12. N.A. Mortensen, J.R. Folken, P.M.W. Skovgaard, and J. Broeng, “Numerical aperture of single-mode photonic crystal fibers,” IEEE Photon. Technol. Lett. **14**, 1094–1096 (2002). [CrossRef]

*A*

_{eff}=π

*d*/Λ=0.53 and λ/Λ < 0.14, the relation

*A*

_{eff}=π

*A*

_{eff}is very similar to that obtained from

*w*

_{eff}. For a considerably small air-hole case,

*d*/Λ=0.26 and λ/Λ < 0.42, on the other hand, the relation

*A*

_{eff}=π

*w*

_{eff}, in other word, MFD, are in good quantitative agreement with the experimental ones [11

**24**, 1017–1019 (1999). [CrossRef]

12. N.A. Mortensen, J.R. Folken, P.M.W. Skovgaard, and J. Broeng, “Numerical aperture of single-mode photonic crystal fibers,” IEEE Photon. Technol. Lett. **14**, 1094–1096 (2002). [CrossRef]

13. J.H. Lee, P.C. Teh, Z. Yusoff, M. Ibsen, W. Belardi, T.M. Monro, and D.J. Richardson, “A holey fiber-based nonlinear thresholding devices for optical CDMA receiver performance enhancement,” IEEE Photon. Technol. Lett. **14**, 876–878 (2002). [CrossRef]

*A*

_{eff}and MFD of such an actual fiber, the real-model simulation [20

20. M. Koshiba and K. Saitoh, “Polarization-dependent losses in actual holey fibers,” IEEE Photon. Technol. Lett. **15**, 691–693 (2003). [CrossRef]

17. 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]

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

*A*

_{eff}and MFD are independent of the polarization states. In the actual fiber structure, on the other hand, they depend on the polarization states. Figure 10 shows the values of

*A*

_{eff}and π

^{-3}. The calculated

*A*

_{eff}and the modal birefringence are in good agreement with the experimental ones [13

**14**, 876–878 (2002). [CrossRef]

*A*

_{eff}=π

*n*

_{2}is the nonlinear-index coefficient for pure silica,

*P*is the signal power, and the operating wavelength is taken as 1.553 µm. The calculated ϕ

_{SPM}for the slow-axis mode agree well with the experimental one [13

**14**, 876–878 (2002). [CrossRef]

## 6. Conclusion

*A*

_{eff}and the mode field diameter (MFD) of holey fibers (HFs) with finite cross sections were evaluated. In order to treat axially nonsymmetrical fibers such HFs with air holes arranged in a triangular lattice in the cladding region, the second moment of the optical intensity distribution was introduced as the definition of the effective modal spot size

*w*

_{eff}(a half of MFD). We have shown that

*A*

_{eff}and

*w*

_{eff}are almost independent of the number of rings of air holes and that the relation

*A*

_{eff}=π

*A*

_{eff}and MFD obtained here were effectively applied to evaluating the beam divergence and the nonlinear phase shift of HFs. The applicability of

*A*

_{eff}and MFD to the splice loss, bending loss, and source-to-fiber coupling efficiency are now under consideration.

## References and links

1. | J. Broeng, D. Mogilevstev, S.E. Barkou, and A. Bjarklev, “Photonic crystal fibers: A new class of optical waveguides,” Opt. Fiber Technol. |

2. | T.A. Birks, J.C. Knight, B.J. Mangan, and P.St.J. Russell, “Photonic crystal fibers: An endless variety,” IEICE Trans. Electron. |

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

4. | N.A. Mortensen, “Effective area of photonic crystal fibers,” Opt. Express |

5. | T.M. Monro, V. Finazzi, W. Belardi, K.M. Kiang, J.H. Lee, and D.J. Richardson, “Highly nonlinear holey optical fibers: design, manufacture and device applications,” Proc. European Conf. Opt. Commun., Symposium 1.5. (2002). |

6. | M. Koshiba and Y. Tsuji,“Curvilinear hybrid edge/nodal elements with triangular shape for guided-wave problems,” J. Lightwave Technol. |

7. | K. Saitoh and M. Koshiba, “Full-vectorial finite element beam propagation method with perfectly matched layers for anisotropic optical waveguides,” J. Lightwave Technol. |

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

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

10. | V. Finazzi, T.M. Monro, and D.J. Richardson, “Confinement loss in highly nonlinear holey optical fibers,” Proc. Optical Fiber Commun. Conf. ThS4. (2002). |

11. | M.J. Gander, R. McBride, J.D.C. Jones, T.A. Birks, J.C. Knight, P.St.J. Russell, P.M. Blanchard, J.G. Burnett, and A.H. Greenaway, “Measurement of the wavelength dependence of beam divergence for photonic crystal fiber,” Opt. Lett. |

12. | N.A. Mortensen, J.R. Folken, P.M.W. Skovgaard, and J. Broeng, “Numerical aperture of single-mode photonic crystal fibers,” IEEE Photon. Technol. Lett. |

13. | J.H. Lee, P.C. Teh, Z. Yusoff, M. Ibsen, W. Belardi, T.M. Monro, and D.J. Richardson, “A holey fiber-based nonlinear thresholding devices for optical CDMA receiver performance enhancement,” IEEE Photon. Technol. Lett. |

14. | J.W.H. Liu, “The multifrontal method for sparse matrix solutions: theory and practice,” SIAM Rev. |

15. | G. Agrawal, Nonlinear Fiber Optics, Academic Press (San Diego, CA), 2dn Edition (1995). |

16. | K. Petermann, “Constraints for fundamental-mode spot size for broadband dispersion-compensated single-mode fibers,” Electron. Lett. |

17. | 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. |

18. | M. Koshiba and K. Saitoh, “Numerical verification of degeneracy in hexagonal photonic crystal fibers,” IEEE Photon. Technol. Lett. |

19. | K. Hayata, M. Koshiba, and M. Suzuki, “Modal spot size of axially nonsymmetrical fibers,” Electron. Lett. |

20. | M. Koshiba and K. Saitoh, “Polarization-dependent losses in actual holey fibers,” IEEE Photon. Technol. Lett. |

**OCIS Codes**

(060.2270) Fiber optics and optical communications : Fiber characterization

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

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

**ToC Category:**

Research Papers

**History**

Original Manuscript: June 26, 2003

Revised Manuscript: July 16, 2003

Published: July 28, 2003

**Citation**

M. Koshiba and K. Saitoh, "Structural dependence of effective area and mode field diameter for holey fibers," Opt. Express **11**, 1746-1756 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-15-1746

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### References

- J. Broeng, D. Mogilevstev, S.E. Barkou, and A. Bjarklev, �??Photonic crystal fibers: A new class of optical waveguides,�?? Opt. Fiber Technol. 5, 305-330, (1999). [CrossRef]
- T.A. Birks, J.C. Knight, B.J. Mangan, and P.St.J. Russell, �??Photonic crystal fibers: An endless variety,�?? IEICE Trans. Electron. E84-C, 585-592, (2001).
- 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]
- N.A. Mortensen, �??Effective area of photonic crystal fibers,�?? Opt. Express 10, 341-348, (2002), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-7-341">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-7-341</a> [CrossRef] [PubMed]
- T.M. Monro, V. Finazzi, W. Belardi, K.M. Kiang, J.H. Lee, and D.J. Richardson, �??Highly nonlinear holey optical fibers: design, manufacture and device applications,�?? Proc. European Conf. Opt. Commun., Symposium 1.5. (2002).
- M. Koshiba and Y. Tsuji,�??Curvilinear hybrid edge/nodal elements with triangular shape for guided-wave problems,�?? J. Lightwave Technol. 18, 737-743, (2000). [CrossRef]
- K. Saitoh and M. Koshiba, �??Full-vectorial finite element beam propagation method with perfectly matched layers for anisotropic optical waveguides,�?? J. Lightwave Technol. 19, 405-413, (2001). [CrossRef]
- K. Saitoh and M. Koshiba, �??Full-vectorial imaginary-distance beam propagation method based on finite element scheme: Application to photonic crystal fibers,�?? IEEE J. Quantum Electron. 38, 927-933, (2002). [CrossRef]
- T.P. White, R.C. McPhedran, 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 loss in highly nonlinear holey optical fibers,�?? Proc. Optical Fiber Commun. Conf. ThS4. (2002).
- M.J. Gander, R. McBride, J.D.C. Jones, T.A. Birks, J.C. Knight, P.St.J. Russell, P.M. Blanchard, J.G. Burnett, and A.H. Greenaway, �??Measurement of the wavelength dependence of beam divergence for photonic crystal fiber,�?? Opt. Lett. 24, 1017-1019, (1999). [CrossRef]
- N.A. Mortensen, J.R. Folken, P.M.W. Skovgaard, and J. Broeng, �??Numerical aperture of single-mode photonic crystal fibers,�?? IEEE Photon. Technol. Lett. 14, 1094-1096, (2002). [CrossRef]
- J.H. Lee, P.C. Teh, Z. Yusoff, M. Ibsen, W. Belardi, T.M. Monro, and D.J. Richardson, �??A holey fiberbased nonlinear thresholding devices for optical CDMA receiver performance enhancement,�?? IEEE Photon. Technol. Lett. 14, 876-878, (2002). [CrossRef]
- J.W.H. Liu, �??The multifrontal method for sparse matrix solutions: theory and practice,�?? SIAM Rev. 34, 82- 109, (1992). [CrossRef]
- G. Agrawal, Nonlinear Fiber Optics, Academic Press (San Diego, CA), 2dn Edition (1995).
- K. Petermann, �??Constraints for fundamental-mode spot size for broadband dispersion-compensated singlemode fibers,�?? Electron. Lett. 19, 712-714, (1983). [CrossRef]
- 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]
- M. Koshiba and K. Saitoh, �??Numerical verification of degeneracy in hexagonal photonic crystal fibers,�?? IEEE Photon. Technol. Lett. 13, 1313-1315, (2001). [CrossRef]
- K. Hayata, M. Koshiba, and M. Suzuki, �??Modal spot size of axially nonsymmetrical fibers,�?? Electron. Lett. 22, 127-129, (1986). [CrossRef]
- M. Koshiba and K. Saitoh, �??Polarization-dependent losses in actual holey fibers,�?? IEEE Photon. Technol. Lett. 15, 691-693, (2003). [CrossRef]

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