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Optics Express

  • Editor: Michael Duncan
  • Vol. 11, Iss. 15 — Jul. 28, 2003
  • pp: 1746–1756
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Structural dependence of effective area and mode field diameter for holey fibers

M. Koshiba and K. Saitoh  »View Author Affiliations


Optics Express, Vol. 11, Issue 15, pp. 1746-1756 (2003)
http://dx.doi.org/10.1364/OE.11.001746


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Abstract

A rigorous full-vector finite element method is effectively applied to evaluating the effective area Aeff and the mode field diameter (MFD) of holey fibers (HFs) with finite cross sections. The effective modal spot size (a half of MFD), weff , 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 Aeff and weff is investigated in detail. As a result, it is shown that Aeff and weff are almost independent of the number of hole rings and that the relation Aeffweff2, which is frequently utilized in the conventional optical fibers, does not always hold, especially in smaller air-filling fraction and/or longer wavelength regions. In addition, we find that for HFs with large air holes operating at longer wavelengths, the mode profiles of the two linearly polarized fundamental modes are significantly different from each other, even though they are degenerate. Using the values of Aeff and weff 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

Photonic crystal fibers, also called holey fibers (HFs), have been one of the most interesting developments in recent fiber optics [1

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]

],[2

2. 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).

].

The effective area Aeff is a quantity of great importance in fiber optics. It was originally introduced as a measure of nonlinearities; a small effective area would be useful for enhancing nonlinear effects. Often a large effective area is desired, and then high powers can be transmitted without introducing any unwanted nonlinear effects in the fiber. Although several authors have reported theoretical studies of effective area for HFs [3

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. 18, 50–56 (2000). [CrossRef]

],[4

4. N.A. Mortensen, “Effective area of photonic crystal fibers,” Opt. Express 10, 341–348 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-7-341 [CrossRef] [PubMed]

], the influence of the extent of the confining air hole region on effective area has not been investigated. More recently, Monro et al. have reported a pioneering work regarding the effective area of HFs with a few rings of air holes and have claimed that Aeff is effectively independent of the number of hole rings [5

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).

]. However, in [5

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).

] the number of rings of air holes has been limited 2 to 4 and the operating wavelength has been fixed to 1550 nm.

In this paper, using a rigorous full-vector finite element method [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]

]–[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]

], Aeff 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 weff . The influence of hole diameter, hole pitch, operating wavelength, and number of hole rings on Aeff and weff 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]

]–[10

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).

], Aeff and weff are almost independent of it and that the relation Aeffweff2 does not always hold, especially in smaller air-filling fraction and/or longer wavelength regions. In addition, we find that for HFs with large air holes operating at longer wavelengths, the mode profiles of the horizontally polarized and vertically polarized fundamental modes are significantly different from each other, even though these two linearly polarized modes are degenerate in strict sense. Using the values of Aeff and weff 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

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

We consider a HF with finite cross section as shown in Fig. 1, where 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

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]

] are incorporated as absorbing boundary conditions to evaluate confinement losses. From Maxwell’s equations the following vector wave equation is derived [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]

]:

×([s]1×E)k02n2[s]E=0
(1)

where E is the electric field vector, n is the refractive index, [s] is the PML matrix, and [s]-1 is an inverse matrix of [s].

When applying a full-vector FEM to HFs, a curvilinear hybrid edge/nodal element [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]

] as shown in Fig. 2 is very useful for avoiding spurious solutions and for accurately modeling curved boundaries of circular air holes. For the axial field, Ez , a nodal element with six variables, Ez 1 to Ez 6, is employed, while for the transverse fields, Ex and Ey , an edge element with eight variables, Et 1 to Et 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]

].

Fig. 1. Holey fiber with finite cross section.
Fig. 2. Curvilinear hybrid edge/nodal element.

Dividing the fiber cross section into curvilinear hybrid edge/nodal elements, we expand the transverse and axial fields in each element as

E=[ExEyEz]=[{U(x,y)}T{Et}eexp(jβz){V(x,y)}T{Et}eexp(jβz)jβ{N(x,y)}T{Ez}eexp(jβz)]
(2)

where β is the propagation constant, {Et } e and {Ez } 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.

Applying the standard finite element technique to Eq. (1), we can obtain the following eigenvalue equation:

[K]{E}=β2[M]{E}
(3)

with

{E}=[{Et}{Ez}]
(4)

where {E} is the global electric field vector and the finite element matrices [K] and [M] are given in [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]

]. Utilizing sparse nature of [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]

].

Aeff=(SEt2dxdy)2SEt4dxdy
(5)

where Et is the transverse electric field vector and S denotes the whole fiber cross section. Using the eigenvector {E} corresponding to the eigenvalue β2 obtained from Eq. (3) and noting the relation of Eq. (2), Aeff is calculated as

Aeff=[eeIe(x,y)dxdy]2eeIe2(x,y)dxdy
(6)

where Σ e extends over all different elements and the optical intensity distribution for each element, Ie , is given by

Ie(x,y)={U(x,y)}T{Et}e2+{V(x,y)}T{Et}e2.
(7)

wx2=4S(xxc)2Et2dxdySEt2dxdy=4ee(xxc)2Ie(x,y)dxdyeeIe(x,y)dxdy
(8)
wy2=4S(yyc)2Et2dxdySEt2dxdy=4ee(yyc)2Ie(x,y)dxdyeeIe(x,y)dxdy
(9)

where xc and yc are, respectively, the x and y coordinates of the center of the field distribution, which are calculated as

xc=SxEt2dxdySEt2dxdy=eexIe(x,y)dxdyeeIe(x,y)dxdy
(10)
yc=SyEt2dxdySEt2dxdy=eeyIe(x,y)dxdyeeIe(x,y)dxdy.
(11)

In the curvilinear elements with triangular shape the Cartesian coordinates, 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

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]

]. Noting that all the shape function vectors, {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

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]

]).

weff2=wx2+wy22.
(12)

With this definition, weff 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

HFs are usually made from pure silica, and so the guided modes are inherently leaky because the core index is the same as the index of the outer cladding region without air holes. Figure 3 shows the normalized effective area and the normalized confinement loss of the fundamental mode in HFs with finite cross sections, taking the ratio of hole diameter to pitch, 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, Lc Λ, is defined as

LcΛ=8.686Im[βΛ].
(13)

Here Im stands for the imaginary part. For example, assuming the value of Lc Λ to be 10-8 dB, the confinement loss Lc becomes 10 dB/km for Λ=1 µm and 1 dB/km for Λ=10 µm.

As expected, increasing the air-hole size, the mode becomes more confined, and thus, the effective area and the confinement loss are both reduced. Also, increasing the number of rings of air holes, the confinement loss is significantly reduced. On the other hand, the effective area is almost independent of the number of hole rings, as claimed by Monro et al. [5

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).

]. This seems to be due to the fact that even if Aeff2 ≅ 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

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).

] are also plotted. Our results agree well with those of MM, showing the reliability of a full-vector FEM with PMLs.

Figure 3 is a general “map” of the effective area which can be used for designing various HFs, such as ultra-low nonlinearity HFs and high nonlinearity HFs with desired confinement loss properties. If, for example, the operating wavelength is given, the effective area and the confinement loss can to a large extent be tailored via the choice of hole pitch, hole size, and number of hole rings.

Fig. 3. Effective area and confinement loss of holey fibers operating at the wavelength of (a) λ/Λ=0.2, (b) λ/Λ=0.4, (c) λ/Λ=0.6, (d) λ/Λ=0.8, (e) λ/Λ=1.0, and (f) λ/Λ=1.29.

4. Effective area and mode field diameter

Fig. 4. Relationship between effective area and mode field diameter as a function of wavelength.
Fig. 5. Intensity profile of (a) horizontally polarized and (b) vertically polarized modes in a holey fiber with d/Λ=0.3 at λ/Λ=0.2, where |Ex |2 in (a) and |Ey |2 in (b) are expressed in the intensity contours spaced by 1 dB.

Fig. 6. Relationship among three kinds of modal spot sizes as a function of wavelength.
Fig. 7. Intensity profile of (a) horizontally polarized and (b) vertically polarized modes in a holey fiber with d/Λ=0.9 at λ/Λ=1.2, where |Ex |2 in (a) and |Ey |2 in (b) are expressed in the intensity contours spaced by 1 dB.

5. Beam divergence and nonlinear phase shift

In this section, using the values of effective area and MFD obtained here, the beam divergence and the nonlinear phase shift of HFs 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

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]

].

Figure 8 shows the beam divergences of HFs with hole pitch Λ=2.39 µm and hole diameter d/Λ=0.26 and with Λ=7.2 µm and d/Λ=0.53, where the beam divergence θ is given by

θ=tan1λπweff
(14)

Fig. 8. Beam divergences of holey fibers.

ϕSPM=4πn2PλAeff
(15)

where 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 ϕSPM for the slow-axis mode agree well with the experimental one [13

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]

].

Fig. 9. Highly nonlinear holey fiber [13].
Fig. 10. Effective area, modal spot size, and birefringence of a holey fiber shown in Fig. 9.
Fig. 11. Nonlinear phase shift of a holey fiber shown in Fig. 9.

6. Conclusion

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. 5, 305–330 (1999). [CrossRef]

2.

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).

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. 18, 50–56 (2000). [CrossRef]

4.

N.A. Mortensen, “Effective area of photonic crystal fibers,” Opt. Express 10, 341–348 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-7-341 [CrossRef] [PubMed]

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. 18, 737–743, (2000). [CrossRef]

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]

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. 26, 1660–1662 (2001). [CrossRef]

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. 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]

14.

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

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. 19, 712–714 (1983). [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]

19.

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

20.

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

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

  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]
  2. 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).
  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. 18, 50-56, (2000). [CrossRef]
  4. 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]
  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. 18, 737-743, (2000). [CrossRef]
  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]
  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. 26, 1660-1662, (2001). [CrossRef]
  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. 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 fiberbased nonlinear thresholding devices for optical CDMA receiver performance enhancement,�?? IEEE Photon. Technol. Lett. 14, 876-878, (2002). [CrossRef]
  14. J.W.H. Liu, �??The multifrontal method for sparse matrix solutions: theory and practice,�?? SIAM Rev. 34, 82- 109, (1992). [CrossRef]
  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 singlemode fibers,�?? Electron. Lett. 19, 712-714, (1983). [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]
  19. K. Hayata, M. Koshiba, and M. Suzuki, �??Modal spot size of axially nonsymmetrical fibers,�?? Electron. Lett. 22, 127-129, (1986). [CrossRef]
  20. 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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