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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 11, Iss. 23 — Nov. 17, 2003
  • pp: 3100–3109
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Leakage loss and group velocity dispersion in air-core photonic bandgap fibers

Kunimasa Saitoh and Masanori Koshiba  »View Author Affiliations


Optics Express, Vol. 11, Issue 23, pp. 3100-3109 (2003)
http://dx.doi.org/10.1364/OE.11.003100


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Abstract

The wavelength dependence and the structural dependence of leakage loss and group velocity dispersion (GVD) in air-core photonic bandgap fibers (PBGFs) are numerically investigated by using a full-vector finite element method. It is shown that at least seventeen rings of arrays of air holes are required in the cladding region to reduce the leakage losses to a level of 0.1 dB/km in 1.55-µm wavelength range even if using large air holes of the diameter to pitch ratio of 0.9 and that the leakage losses in air-core PBGFs decrease drastically with increasing the hole diameter to pitch ratio. Moreover, it is shown that the waveguide GVD and dispersion slope of air-core PBGFs are much larger than those of conventional silica fibers and that the shape of air-core region greatly affects the leakage losses and the dispersion properties.

© 2003 Optical Society of America

1. Introduction

Optical fibers with silica-air microstructures called photonic crystal fibers (PCFs) [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).

] have attracted a considerable amount of attention recently, because of their unique properties that are not realised in conventional optical fibers. PCFs, which are also called holey fibers or microstructured fibers, are divided into two different kinds of fibers. The first one guides light by total internal reflection between a solid core and a cladding region with multiple air-holes [3

3. J.C. Knight, T.A. Birks, P.St.J. Russell, and D.M. Atkin, “All-silica single-mode optical fiber with photonic crystal cladding,” Opt. Lett. 21, 1547–1549 (1996). [CrossRef] [PubMed]

, 4

4. 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]

]. On the other hand, the second one uses a perfectly periodic structure exhibiting a photonic bandgap (PBG) effect at the operating wavelength to guide light in a low index core-region [5

5. J.C. Knight, J. Broeng, T.A. Birks, and P.St.J. Russell, “Photonic band gap guidance in optical fiber,” Science 282, 1476–1478 (1998). [CrossRef] [PubMed]

, 6

6. R.F. Cregan, B.J. Mangan, J.C. Knight, T.A. Birks, P.St.J. Russell, P.J. Roberts, and D.C. Allan,“Single-mode photonic band gap guidance of light in air,” Science 285, 1537–1539 (1999). [CrossRef] [PubMed]

], which is also called photonic bandgap fiber (PBGF).

Especially, air-core/hollow-core PBGFs [6

6. R.F. Cregan, B.J. Mangan, J.C. Knight, T.A. Birks, P.St.J. Russell, P.J. Roberts, and D.C. Allan,“Single-mode photonic band gap guidance of light in air,” Science 285, 1537–1539 (1999). [CrossRef] [PubMed]

, 7

7. J. Broeng, S.E. Barkou, T. Søndergaard, and A. Bjarklev, “Analysis of air-guiding photonic bandgap fibers,” Opt. Lett. 25, 96–98 (2000). [CrossRef]

] have possibilities of extremely low-loss transmission, high-power delivery with low nonlinearity, and many applications such as enhanced Raman scattering [8

8. F. Benabid, J.C. Knight, G. Antonopoulos, and P.St.J. Russell, “Stimulated Raman scattering in hydrogen-filled hollow-core photonic crystal fiber,” Science 298, 399–402 (2002). [CrossRef] [PubMed]

], soliton transmission [9

9. D.G. Ouzouno, F.R. Ahmad, D. Müller, N. Venkataraman, M.T. Gallagher, M.G. Thomas, J. Silcox, K.W. Koch, and A.L. Gaeta, “Generation of megawatt optical solitons in hollow-core photonic bang-gap fibers,” Science 301, 1702–1704 (2003). [CrossRef]

], and so on. It is very important to know the leakage-loss and dispersion properties of PBGFs for practical applications. A number of studies so far have been made on leakage losses, chromatic dispersion, and imperfection analysis in index-guiding PCFs [10

10. 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]

15

15. G. Renversez, B. Kuhlmey, and R. McPhedran, “Dispersion management with microstructured optical fibers: ultraflattened chromatic dispersion with low losses,” Opt. Lett. 28, 989–991 (2003). [CrossRef] [PubMed]

], PBGFs [16

16. K. Saitoh and M. Koshiba, “Confinement losses in air-guiding photonic bandgap fibers,” IEEE Photon. Technol. Lett. 15, 236–238 (2003). [CrossRef]

, 17

17. N.A. Issa, A. Argyros, M.A. van Eijkelenborg, and J. Zagari, “Identifying hollow waveguide guidance in air-cored microstructured optical fibers,” Opt. Express 9, 996–1001 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-996 [CrossRef]

], and OmniGuide fibers [18

18. S.G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, T.D. Engeness, M. Soljačić, S.A. Jacobs, J.D. Joannopoulos, and Y. Fink, “Low-loss asymptotically single-mode propagation in large-core OmniGuide fibers,” Opt. Express 9, 748–779 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-748 [CrossRef] [PubMed]

21

21. T.D. Engeness, M. Ibanescu, S.G. Johnson, O. Weisberg, M. Skorobogatiy, S. Jacobs, and Y. Fink, “Dispersion tailoring and compensation by modal interactions in OmniGuide fibers,” Opt. Express 11, 1175–1196 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-10-1175 [CrossRef] [PubMed]

]. However, the structural dependence of leakage losses and group velocity dispersion (GVD) characteristics in air-core PBGFs with a finite number of air-hole rings has not been reported in the literature.

In this paper, the wavelength dependence and the structural dependence of leakage loss and GVD in air-core PBGFs with a finite number of air-hole rings are numerically investigated by using a full-vector finite element method (FEM). Here, anisotropic perfectly matched layers (PMLs) [22

22. 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 into a full-vector FEM with curvilinear hybrid edge/nodal elements [23

23. 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]

] to evaluate leakage losses. It is shown that at least seventeen rings of arrays of air holes are required in the cladding region to reduce the leakage losses to a level of 0.1 dB/km in 1.55-µm wavelength range even if using large air holes of the diameter to pitch ratio of 0.9 and that the leakage losses in air-core PBGFs decrease drastically with increasing the hole diameter to pitch ratio. Moreover, it is shown that the waveguide GVD and dispersion slope of air-core PBGFs are much larger than those of conventional silica fibers and that the shape of air-core region greatly affects the leakage losses and the dispersion properties.

2. Full-vector finite element method formulation

We consider a PCF with finite cross section in the xy (transverse) plane and assume that the structure is uniform along the propagation direction (z axis), the cross section of which is shown in Fig. 1. In order to enclose the computational domain without affecting the numerical solution and to evaluate leakage losses, anisotropic PMLs [22

22. 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 placed before the boundary. From Maxwell’s equations the following vector wave equation is derived [22

22. 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, k 0 is the free space wavenumber, n is the refractive index, [s] is the PML matrix, and [s]-1 is an inverse matrix of [s]. Because of the uniformity of the fiber, we can write the electric field E as

E(x,y,z)=e(x,y)exp(γz)
(2)

with

γ=α+jβ
(3)

where γ is the complex propagation constant, α is the attenuation constant, and β is the phase constant, respectively.

In the present formulation curvilinear hybrid edge/nodal elements [23

23. 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]

] have been used for accurately modeling curved boundaries [24

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

]. Dividing the fiber cross section into curvilinear hybrid edge/nodal elements and applying the variational finite element procedure, we can obtain the following eigenvalue equation:

[K]{E}=γ2[M]{E}
(4)

where {E} is the discretized electric field vector and the finite element matrices [K] and [M] are given in [22

22. 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]

].

The leakage loss, which is also called confinement loss, is an important parameter to design a PCF with a finite number of air holes. In air-core PBGFs with an infinite number of air holes, the light is confined to the air-core region by a full two-dimensional PBG and leakage losses do not occur. In fabricated PBGFs, however, the number of air holes in the cladding is finite, and so the modes of such fibers are inherently leaky. The leakage loss, Lc, is deduced from the value of α as

Lc=8.686α
(5)

in decibels per meter. The waveguide GVD, Dw, is easily calculated from the computed wavelength dependence of the effective index neff=β/k 0 as

Dw=λcd2neffdλ2
(6)

where λ is the wavelength, c is the velocity of light in a vacuum, and the wavelength dependence of the index of silica is neglected. Of course, the material dispersion should be taken into account to evaluate total wavelength dispersion of PCFs.

Fig. 1. Photonic crystal fiber with finite cross section.

3. Leakage loss and group velocity dispersion

Fig.2. Air-core PBGF with ten rings of arrays of air holes.
Fig. 3. Modal dispersion curve as a function of normalized wavelength for the air-core PBGF with ten rings of air holes in Fig. 2, where d/Λ=0.9.
Fig. 4. Intensity profile of horizontally polarized fundamental mode in an air-core PBGF with d/Λ=0.9 and Λ=2.32 µm at λ=1.55 µm, where |Ex|2 is expressed in the intensity contours spaced by 1 dB.
Fig. 5. Leakage loss as a function of wavelength for an air-core PBGF with a finite number of air holes. The hole pitch Λ=2.32 µm and d/Λ=0.9.
Fig. 6. Leakage loss as a function of the number of rings. The hole pitch Λ=2.32 µm, d/Λ=0.9, and λ=1.55 µm.
Fig. 7. Waveguide group velocity dispersion of an air-core PBGF with a finite number of air holes. The hole pitch Λ=2.32 µm and d/Λ=0.9.

Fig. 8. PBG boundaries and modal dispersion curves of the fundamental modes for two values of d/Λ=0.9 and d/Λ=0.95 as a function of normalized wavelength.
Fig. 9. Normalized leakage loss as a function of the normalized wavelength in an air-core PBGF with ten rings of arrays of air holes. The hole diameter to pitch ratio d/Λ is taken as a parameter.
Fig. 10. Normalized waveguide GVD for the fundamental mode of the air-core PBGF with ten rings of air holes in Fig. 2, where the hole diameter to pitch ratio d/Λ is taken as a parameter.

Aeff=(E2dxdy)2E4dxdy.
(7)

Fig. 11. Schematics of air-core PBGF cross sections for (a) type-1, (b) type-2, and (c) type-3 PBGFs.
Fig. 12. Intensity profiles of horizontally polarized fundamental modes for air-core PBGFs in Fig. 11, where d/Λ=0.9, λ/Λ=0.67, and |Ex|2 is expressed in the intensity contours spaced by 1 dB.
Fig. 13. (a) Modal dispersion curves and (b) normalized waveguide GVD as a function of normalized wavelength for the three types of air-core PBGFs as shown in Fig. 11, where d/Λ=0.9 and the number of air-hole rings is ten.
Fig. 14. (a) Normalized leakage loss and (b) normalized effective mode area as a function of normalized wavelength for the three types of air-core PBGFs as shown in Fig. 11, where d/Λ=0.9 and the number of air-hole rings is ten.

4. Conclusions

The wavelength dependence and the structural dependence of leakage loss and GVD in air-core PBGFs have been numerically investigated by using a full-vector FEM. It was shown that at least seventeen rings of arrays of air holes are required in the cladding region to reduce the leakage losses to a level of 0.1 dB/km in 1.55-µm wavelength range even if using large air holes of the diameter to pitch ratio of 0.9 and that the leakage losses in air-core PBGFs decrease drastically with increasing the hole diameter to pitch ratio. Moreover, it was shown that the waveguide GVD and dispersion slope of air-core PBGFs are much larger than those of conventional silica fibers and that the shape of air-core region greatly affects the leakage losses and the dispersion properties.

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.

J.C. Knight, T.A. Birks, P.St.J. Russell, and D.M. Atkin, “All-silica single-mode optical fiber with photonic crystal cladding,” Opt. Lett. 21, 1547–1549 (1996). [CrossRef] [PubMed]

4.

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]

5.

J.C. Knight, J. Broeng, T.A. Birks, and P.St.J. Russell, “Photonic band gap guidance in optical fiber,” Science 282, 1476–1478 (1998). [CrossRef] [PubMed]

6.

R.F. Cregan, B.J. Mangan, J.C. Knight, T.A. Birks, P.St.J. Russell, P.J. Roberts, and D.C. Allan,“Single-mode photonic band gap guidance of light in air,” Science 285, 1537–1539 (1999). [CrossRef] [PubMed]

7.

J. Broeng, S.E. Barkou, T. Søndergaard, and A. Bjarklev, “Analysis of air-guiding photonic bandgap fibers,” Opt. Lett. 25, 96–98 (2000). [CrossRef]

8.

F. Benabid, J.C. Knight, G. Antonopoulos, and P.St.J. Russell, “Stimulated Raman scattering in hydrogen-filled hollow-core photonic crystal fiber,” Science 298, 399–402 (2002). [CrossRef] [PubMed]

9.

D.G. Ouzouno, F.R. Ahmad, D. Müller, N. Venkataraman, M.T. Gallagher, M.G. Thomas, J. Silcox, K.W. Koch, and A.L. Gaeta, “Generation of megawatt optical solitons in hollow-core photonic bang-gap fibers,” Science 301, 1702–1704 (2003). [CrossRef]

10.

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]

11.

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]

12.

D. Ferrarini, L. Vincetti, M. Zoboli, A. Cucinotta, and S. Selleri, “Leakage properties of photonic crystal fibers,” Opt. Express 10, 1314–1319 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-23-1314 [CrossRef] [PubMed]

13.

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

14.

K. Saitoh, M. Koshiba, T. Hasegawa, and E. Sasaoka, “Chromatic dispersion control in photonic crystal fibers: application to ultra-flattened dispersion,” Opt. Express 11, 843–852 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-8-843 [CrossRef] [PubMed]

15.

G. Renversez, B. Kuhlmey, and R. McPhedran, “Dispersion management with microstructured optical fibers: ultraflattened chromatic dispersion with low losses,” Opt. Lett. 28, 989–991 (2003). [CrossRef] [PubMed]

16.

K. Saitoh and M. Koshiba, “Confinement losses in air-guiding photonic bandgap fibers,” IEEE Photon. Technol. Lett. 15, 236–238 (2003). [CrossRef]

17.

N.A. Issa, A. Argyros, M.A. van Eijkelenborg, and J. Zagari, “Identifying hollow waveguide guidance in air-cored microstructured optical fibers,” Opt. Express 9, 996–1001 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-996 [CrossRef]

18.

S.G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, T.D. Engeness, M. Soljačić, S.A. Jacobs, J.D. Joannopoulos, and Y. Fink, “Low-loss asymptotically single-mode propagation in large-core OmniGuide fibers,” Opt. Express 9, 748–779 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-748 [CrossRef] [PubMed]

19.

S.G. Johnson, M. Ibanescu, M.A. Skorobogatiy, O. Weisberg, J.D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E65, 066611 (2002). [CrossRef]

20.

M. Skorobogatiy, S.A. Jacobs, S.G. Johnson, and Y. Fink, “Geometric variations in high index-contrast waveguides, coupled mode theory in curvilinear coordinates,” Opt. Express 10, 1227–1243 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-21-1227 [CrossRef] [PubMed]

21.

T.D. Engeness, M. Ibanescu, S.G. Johnson, O. Weisberg, M. Skorobogatiy, S. Jacobs, and Y. Fink, “Dispersion tailoring and compensation by modal interactions in OmniGuide fibers,” Opt. Express 11, 1175–1196 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-10-1175 [CrossRef] [PubMed]

22.

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]

23.

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]

24.

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

25.

J.A. Weat, N. Venkataraman, C.M. Smith, and M.T. Gallagher, “Photonic crystal fibers,” Proc. European Conf. Opt. Commun., Th.A.2.2 (2001).

26.

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

OCIS Codes
(060.2270) Fiber optics and optical communications : Fiber characterization
(060.2280) Fiber optics and optical communications : Fiber design and fabrication

ToC Category:
Research Papers

History
Original Manuscript: October 6, 2003
Revised Manuscript: November 4, 2003
Published: November 17, 2003

Citation
Kunimasa Saitoh and Masanori Koshiba, "Leakage loss and group velocity dispersion in air-core photonic bandgap fibers," Opt. Express 11, 3100-3109 (2003)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-23-3100


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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. J.C. Knight, T.A. Birks, P.St.J. Russell, and D.M. Atkin, �??All-silica single-mode optical fiber with photonic crystal cladding,�?? Opt. Lett. 21, 1547-1549 (1996). [CrossRef] [PubMed]
  4. 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]
  5. J.C. Knight, J. Broeng, T.A. Birks, and P.St.J. Russell, �??Photonic band gap guidance in optical fiber,�?? Science 282, 1476-1478 (1998). [CrossRef] [PubMed]
  6. R.F. Cregan, B.J. Mangan, J.C. Knight, T.A. Birks, P.St.J. Russell, P.J. Roberts, and D.C. Allan, �??Single-mode photonic band gap guidance of light in air,�?? Science 285, 1537-1539 (1999). [CrossRef] [PubMed]
  7. J. Broeng, S.E. Barkou, T. Sondergaard, and A. Bjarklev, �??Analysis of air-guiding photonic bandgap fibers,�?? Opt. Lett. 25, 96-98 (2000) [CrossRef]
  8. F. Benabid, J.C. Knight, G. Antonopoulos, and P.St.J. Russell, �??Stimulated Raman scattering in hydrogen-filled hollow-core photonic crystal fiber,�?? Science 298, 399-402 (2002). [CrossRef] [PubMed]
  9. D.G. Ouzouno, F.R. Ahmad, D. Müller, N. Venkataraman, M.T. Gallagher, M.G. Thomas, J. Silcox, K.W. Koch, A.L. Gaeta, �??Generation of megawatt optical solitons in hollow-core photonic bang-gap fibers,�?? Science 301, 1702-1704 (2003). [CrossRef]
  10. 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]
  11. 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]
  12. D. Ferrarini, L. Vincetti, M. Zoboli, A. Cucinotta, and S. Selleri, �??Leakage properties of photonic crystal fibers,�?? Opt. Express 10, 1314-1319 (2002), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-23-1314">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-23-1314</a>. [CrossRef] [PubMed]
  13. B. Kuhlmey, G. Renversez, and D. Maystre, �??Chromatic dispersion and losses of microstructured optical fibers,�?? Appl. Opt. 42, 634-639 (2003). [CrossRef] [PubMed]
  14. K. Saitoh, M. Koshiba, T. Hasegawa, and E. Sasaoka, �??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]
  15. G. Renversez, B. Kuhlmey, and R. McPhedran, �??Dispersion management with microstructured optical fibers: ultraflattened chromatic dispersion with low losses,�?? Opt. Lett. 28, 989-991 (2003). [CrossRef] [PubMed]
  16. K. Saitoh and M. Koshiba, �??Confinement losses in air-guiding photonic bandgap fibers,�?? IEEE Photon. Technol. Lett. 15, 236-238 (2003). [CrossRef]
  17. N.A. Issa, A. Argyros, M.A. van Eijkelenborg, and J. Zagari, �??Identifying hollow waveguide guidance in air-cored microstructured optical fibers,�?? Opt. Express 9, 996-1001 (2003),<a href=" http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-996"> http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-996</a> [CrossRef]
  18. S.G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, T.D. Engeness, M. Soljacic, S.A. Jacobs, J.D. Joannopoulos, and Y. Fink, �??Low-loss asymptotically single-mode propagation in large-core OmniGuide fibers,�?? Opt. Express 9, 748-779 (2001), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-748"> http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-748</a> [CrossRef] [PubMed]
  19. S.G. Johnson, M. Ibanescu, M.A. Skorobogatiy, O. Weisberg, J.D. Joannopoulos, and Y. Fink, �??Perturbation theory for Maxwell�??s equations with shifting material boundaries,�?? Phys. Rev. E 65, 066611 (2002) [CrossRef]
  20. M. Skorobogatiy, S.A. Jacobs, S.G. Johnson, and Y. Fink, �??Geometric variations in high index-contrast waveguides, coupled mode theory in curvilinear coordinates,�?? Opt. Express 10, 1227-1243 (2002), <a href= " http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-21-1227">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-21-1227</a> [CrossRef] [PubMed]
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