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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 17 — Aug. 18, 2008
  • pp: 13175–13187
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Meshed index profile method for photonic crystal fibers with arbitrary structures

Kwang No Park and Kyung Shik Lee  »View Author Affiliations


Optics Express, Vol. 16, Issue 17, pp. 13175-13187 (2008)
http://dx.doi.org/10.1364/OE.16.013175


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Abstract

A meshed index profile method, which is based on the localized function method, is demonstrated for analyzing modal characteristics of photonic crystal fibers with arbitrary air-hole structures. The index profile of PCF, which is expressed as a sum of meshed unit matrix, is substituted to full wave equation. By solving this full wave equation, we obtain the modal characteristics of the PCF such as the mode field distribution, the birefringence and the waveguide dispersion. The accuracy of the proposed meshed index profile method (MIPM) is demonstrated by examining the effective index and the birefringence of the two degenerate fundamental modes in the PCF with a triangular air-hole lattice. The MIPM is not restricted to the PCF structure and will be useful in designing various PCF devices.

© 2008 Optical Society of America

1. Introduction

2. Electric field decomposition

Et=(x,y)=Ex(x,y)x̂+Ey(x,y)ŷ,
(1)

where Ex(x,y) and Ey(x,y) are the field distributions of the modes linearly polarized along the x-axis and y-axis, respectively. And the mode field equations as sums of the localized basis functions can be expressed as,

Ex(x,y)=m=0Sn=0Semnxψm(x)ψn(y),
(2)
Ey(x,y)=m=0Sn=0Semnyψm(x)ψn(y),
(3)

where (S+1) is the number of basis functions used to express the transverse electric field distribution. Ψm(x) and Ψn(x) are the Hermite-Gaussian basis functions used to express the mode field distribution in the LFM and are expressed as,

ψi(s)=2i2π14i!wexp(s22w2)Hi(sw),
i=m,ns=x,y
(4)

[2k2βx2k2+n2(x,y)]Ex=1k2x[Exlnn2(x,y)x+Eylnn2(x,y)y],
(5)
[2k2βy2k2+n2(x,y)]Ey=1k2y[Exlnn2(x,y)x+Eylnn2(x,y)y],
(6)

3. Meshed index profile of the PCF and eigenvalue matrix

The transverse index profile of the PCF has recently been determined by using the SOM [13

13. W. Zhi, R Guobin, and L. Shuqin, “A Novel Supercell Overlapping Method for Different Photonic Crystal Fibers,” J. Lightwave Technol. 22, 903–916 (2004). [CrossRef]

] and the orthogonal function method [22

22. T. M. 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]

]. The similarity between the SOM and the orthogonal function method is that they use a combination of several virtual photonic crystal structures to determine the index profile of the PCF. In order to describe the index profile of the PCF, two two-dimensional virtual photonic crystals were used in the SOM [13

13. W. Zhi, R Guobin, and L. Shuqin, “A Novel Supercell Overlapping Method for Different Photonic Crystal Fibers,” J. Lightwave Technol. 22, 903–916 (2004). [CrossRef]

, 22

22. T. M. 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]

]. Several different two two-dimensional photonic crystal structures were also proposed and demonstrated in reference 22. These methods are simple and efficient in determining the index profile of the PCF, but restricted to the PCF with a triangular air-hole lattices. If the PCF has more complex structures with non-symmetric air-hole lattice, elliptical air-holes or different sized air-holes, many virtual photonic crystals should be used to fully express the index profile of the PCF. The number of virtual photonic crystals used to express the index profile is strongly correlated with the computation time. Therefore, these methods are not efficient for analyzing a PCF with non-symmetric air-hole lattice, elliptical air-holes or different sized air-holes.

n2(x,y)=n2(x,yN)+n2(x,y(N1))++n02(x,y0)++n2(x,yN1)+n2(x,yN)
=nN2(x)+n(N1)2(x)++n02(x)++nN12(x)+nN2(x)=i=NNni2(x),or
n2(x,y)=n2(xN,y)+n2(x(N1),y)++n02(x0,y)++n2(xN1,y)+n2(xN,y)
=nN2(y)+n(N1)2(y)++n02(y)++nN12(y)+nN2(y)=i=NNni2(y).
(7)

where 2N+1 is the number of unit row (n 2 i(x)) or column (n 2 i(y)) matrix. In this way, an index profile function (n 2(x,y)) is deduced directly and easily from the index profile of the PCF. Therefore, the index composition method used in the MIPM is efficient and very useful in expressing the index profile of the PCF.

Fig. 1. (a) Meshed transverse dielectric structure of a PCF and (b) the detailed definition of index profile function.

Mabcdxex=(Iabcd(1)+k2Iabcd(2)+Iabcd(3)x)ex=βx2ex,
(8)
Mabcdyey=(Iabcd(1)+k2Iabcd(2)+Iabcd(3)y)ey=βy2ey,
(9)

Iabcd(1)=ψa(x)ψb(y)2[ψc(x)ψd(y)]dxdy,
(10)
Iabcd(2)=n2(x,y)ψa(x)ψb(y)ψc(x)ψd(y)dxdy,
(11)
Iabcd(3)x=ψa(x)ψb(y)x[ψc(x)ψd(y)lnn2(x,y)x]dxdy,
(12)
Iabcd(3)y=ψa(x)ψb(y)y[ψc(x)ψd(y)lnn2(x,y)y]dxdy,
(13)

where Ψa(x), Ψb(y), Ψc(x) and Ψd(y) are the Hermite-Gaussian basis functions.

Next, the meshed index profile function n 2(x,y) expressed by (7) is substituted into the overlap integrals. The overlap integrals can be calculated with the orthonormality of the Hermite-Gaussian basis functions and are rewritten as [24

24. I. S. Gradshtein and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1994).

, 25

25. I. Kimel and L. R. Elias, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quant. Electron. 29, 2562–2567 (1993). [CrossRef]

],

Iabcd(1)=2a+2b+1w2δacδbd+2b(2b+1)2w2δacδb1,d+2a(2a+1)2w2δa1,cδbd
+2(b+1)(2b+1)2w2δacδb,d1+2(a+1)(2a+1)2w2δa,c1δbd,
(14)
Iabcd(2)=i=NNΔyni2(x)ψa(x)ψb(y)ψc(x)ψd(y)dxdy,
=i=NN[Δyψb(yi)ψd(yi)dyni2(x)ψa(x)ψc(x)dx]
=i=NN[ψb(yi)ψd(yi)Δyni2(x)ψa(x)ψc(x)dx],
(15)
Iabcd(3)x=ψa(x)ψb(y)x[ψc(x)ψd(y)i=NNlnni2(x)x]dxdy
=i=NN[Δyψb(y)ψd(y)dylnni2(x)xψa(x)x[ψc(x)]dx]
=i=NN[ψb(yi)ψd(yi)Δylnni2(x)xψa(x)x[ψc(x)]dx],
(16)
Iabcd(3)y=ψa(x)ψb(y)y[ψc(x)ψd(y)i=NNlnni2(y)y]dxdy
=i=NN[Δxψa(x)ψc(x)dxlnni2(y)yψc(y)y[ψd(y)]dy]
=i=NN[ψa(xi)ψc(xi)Δxlnni2(y)yψc(y)y[ψd(y)]dy].
(17)

4. Modal electric field distribution, dispersion and birefringence

Fig. 2 x-polarized fundamental mode field intensities (|Ex|2) of PCFs with different air-hole periods Λ and d/Λ=0.2.
Fig. 3 x-polarized contour plots of the PCFs with the same parameters shown in Fig. 2. The contour plots are spaced by 2dB(a,b) and 1dB(c,d) from -30dB.

Figure 2 shows the x-polarized fundamental mode field intensities (|Ex|2) at two different wavelengths 0.633µm and 1.55µm. Figure 3 displays the corresponding contour plots shown in Fig. 2. From the contour plots shown in Fig. 3, we establish that the mode field intensity distribution at the short wavelengths has greater localization over the core region. Meanwhile, at longer wavelengths, mode field intensity distribution spreads over the cladding region as predicted.

Figure 4(a) and Fig. 4(b) are the x-polarized mode field intensity distribution and its contour plot of the PCF with an elliptical air-hole lattice, respectively. In this simulation, Λ of 2.3µm and the elliptical ratio of the air-hole of 0.5 were used, where the major width of the elliptical air-hole is 0.4Λ and the minor is 0.2Λ. We expect that the mode field intensity distribution is affected by the shape and location of the air-hole. Figure 4 shows that the mode field intensity distribution of the PCF with the elliptical air-hole lattice is elliptical and this result matches with our expectation.

Fig. 4 x-polarized mode intensity distribution and contour plot of the PCF with an elliptical air-hole lattice. The contour plots are spaced by 2dB from -40dB.
Fig. 5. Transverse structure of the PM-PCF with Λ=4.4µm, D1=4.5µm and D2=2.2µm.
Fig. 6. x-polarized mode intensity distribution and contour plot of the PM-PCF shown in Fig. 5. The contour plots are spaced by 2dB from -30dB

Fig. 7. Birefringence of a PM-PCF as a function of wavelength.

The total dispersion of the PCF is defined as the sum of the material dispersion (Dm) and the waveguide dispersion (Dw). Because the PCF is mainly made of pure silica, Dm is same for all silica PCFs with different structures. It means that the total dispersion of the PCF is dominantly affected by the waveguide dispersion. The waveguide dispersions of the PCFs with different parameters are calculated by the MIPM. And the obtained waveguide dispersion of the PCF should satisfy the condition of scaling transformation property of the Maxwell’s equations which are expressed as [27

27. A. Ferrando, E. Silvestre, P. Andres, and J. J. Miret, “Designing the properties of dispersion-flattened photonic crystal fibers,” Opt. Express 9, 687–697 (2001) , http://www.opticsinfobase.org/abstract.cfm?URI=oe-9-13-687. [CrossRef] [PubMed]

],

Dw(λ,MΛ,f)=1MDw(λM,Λ,f),
(18)

where M is the scaling factor and f is air filling fraction of the PCF. The scaling property of the waveguide dispersion expressed in Eq (18) is well known and very useful in designing the waveguide dispersion of PCF.

Fig. 8. Waveguide dispersions of PCFs with different parameters.

5. Accuracy verification of the MIPM

Fig. 9. Rms error of the effective index as a function of wavelength in the range between 0.5µm and 1.8µm.

To check the accuracy of the MIPM, we compute the rms error defined as the root-mean-square(rms) values of deviation of the effective indices evaluated by the MIPM from those computed by the MPM. Fig. 9 shows the rms errors of the effective indices calculated by using the MIPM of the PCF with Λ=4.6µm and d/Λ=0.2. The rms error decreases as the number of unit row or column matrix, used to express the index profile of the PCF, increases. Note that the rms error is reduced less than 5×10-6 and saturated by increasing the number of unit matrix above 1200 as shown in the solid boxed region of the Fig. 9. From these results, one can say that the accuracy of the MIPM can be improved by increasing the number of unit matrix above 1200. However, increasing the number of unit matrix and the number of basis functions require additional computation time.

Meanwhile, the fundamental modes of the fibers with sixfold rotational symmetry should be degenerate [28

28. M. J. Steel, P. T. P. White, C. M. Sterke, R. C. McPhedran, and L. C. Bottn, “Symmerty and degeneracy in microstructured optical fibers,” Opt. Lett. 26, 488–490 (2001). [CrossRef]

]. This means that the fundamental modes (HE11x and HE11y) in the PCF with a triangular air-hole lattice, which means sixfold rotational symmetry, should be degenerate modes. Therefore, the PCF with a triangular air-hole lattice should exhibit very small birefringence.

Fig. 10. Birefringence of a PCF in the wavelength range between 0.5µm and 1.8µm

6. Discussion and conclusion

Acknowledgments

This paper was supported by the 63 Research Fund, from the Sungkyunkwan University, in 2006 and by the Second Stage of Brain Korea 21 Project.

References and links

1.

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

2.

S. K. Varshney, M. P. Singh, and R. K. Sinha, “Propagation Characteristics of Photonic Crystal Fibers,” J. Opt. Commun. 24, 192–198 (2003).

3.

M. Midrio, M. P. Singh, and C. G. Someda, “The Space Filling Mode of Holey Fibers : An Analytical Vectorial Soultion,” J. Lightwave Technol. 18, 1031–1037 (2000). [CrossRef]

4.

Y. Li, Y. Yao, M. Hu, L. Chai, and C. Wang, “Improved fully vectorial effective index method for photonic crystal fibers: evaluation and enhancement,” Appl. Opt. 47, 399–406 (2008). [CrossRef] [PubMed]

5.

H. Li, A. Mafi, A. Schülzgen, L. Li, V. L. Temyanko, N. Peyghambarian, and J. V. Moloney, “Analysis and Design of Photonic Crystal Fibers Based on an Improved Effective-Index Method,” J. Lightwave Technol. 25, 1224–1230 (2007). [CrossRef]

6.

Y. Li, C. Wang, N. Zhang, C. Wang, and Q. Xing, “Analysis and design of terahertz photonic crystal fibers by an effective-index method,” Appl. Opt. 45, 8462–8465 (2006). [CrossRef] [PubMed]

7.

B. T. Kuhlmey, H. C. Nguyen, M. J. Steel, and B. J. Eggleton, “Confinement loss in adiabatic photonic crystal fiber tapers,” J. Opt. Soc. Am. B 23, 1965–1974 (2006). [CrossRef]

8.

T. P. Whiteet al, “Calculations of air-guided modes in photonic crystal fibers using the multipole method,” Opt. Express 11, 721–732 (2001) , http://www.opticsinfobase.org/abstract.cfm?URI=oe-9-13-721. [CrossRef]

9.

A. Cucinotta, S. Selleri, L. Vincetti, and M. Zoboli, “Perturbation Analysis of Dispersion Properties in Photonic Crystal Fibers Through the Finite Element Method,” J. Lightwave Technol. 20, 1433–1442 (2002). [CrossRef]

10.

F. Brechetet al, “Complete analysis of the propagation characteristics into photonic crystal fibers, be the finite element method,” J. Optical Fiber Technol. 6, 181–201 (2001). [CrossRef]

11.

M. A. R. Franco, H. T. Hattori, F. Sircilli, A. Passaro, and N. M. Abe, “Finite Element Analysis of Photonic Crystal Fiber,” in PROC IEEE MTT-S IMOC, 5–7 (2001).

12.

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]

13.

W. Zhi, R Guobin, and L. Shuqin, “A Novel Supercell Overlapping Method for Different Photonic Crystal Fibers,” J. Lightwave Technol. 22, 903–916 (2004). [CrossRef]

14.

D. Mogilevtsev, T. A. Birks, and P. St. J. Russell, “Localized Function Method for Modeling Defect Modes in 2-D Photonic Crystals,” J. Lightwave Technol. 17, 2078–2081 (1999). [CrossRef]

15.

D. Mogilevtsev, T. A. Birks, and P. St. J. Russell, “Group-velocity dispersion in photonic crystal fibers,” Opt. Lett. 23, 1662–1664 (1998). [CrossRef]

16.

T. M. Monro, D. J. Richardson, N. G. R. Broderick, and P. J. Bennett, “Holey optical fibers : An efficient modal model,” J. Lightwave Technol. 17, 1093–1101 (1999). [CrossRef]

17.

Z. Zhang and S. Satpathy, “Electromagnetic wave propagation in structures : Bloch wave solution of Maxwell’s equation,” Phys. Rev. Lett. 65, 2650–2653 (1990). [CrossRef] [PubMed]

18.

S. Guo and S. Albin, “Simple plane wave implementation for photonic crystal calculations,” Opt. Express 11, 167–175 (2003) , http://www.opticsinfobase.org/abstract.cfm?URI=oe-11-2-167. [CrossRef] [PubMed]

19.

Z. Zhu and T. G. Brown, “Full-vectorial finite-difference analysis of microstructured optical fiber,” Opt. Express 10, 853–864 (2002) , http://www.opticsinfobase.org/abstract.cfm?URI=oe-10-17-853. [PubMed]

20.

K. N. Park and K. S. Lee, “Improved effective index method for analysis of photonic cyrstal fibers,” Opt. Lett. 30, 958–960 (2005). [CrossRef] [PubMed]

21.

W. Zhi, R. Guobin, L. Shuqin, and J. Shuisheng, “Supercell lattice method for photonic crystal fibers,” Opt. Express 11, 980–991 (2003) , http://www.opticsinfobase.org/abstract.cfm?URI=oe-11-9-980. [CrossRef] [PubMed]

22.

T. M. 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]

23.

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]

24.

I. S. Gradshtein and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1994).

25.

I. Kimel and L. R. Elias, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quant. Electron. 29, 2562–2567 (1993). [CrossRef]

26.

T.-L. Wu and C.-H. Chao, “Photonic Crystal Fiber Analysis Through the Vector Boundary-Element Method : Effect of Elliptical Air Hole,” IEEE Photon. Tech. Lett. 16, 126–128 (2004). [CrossRef]

27.

A. Ferrando, E. Silvestre, P. Andres, and J. J. Miret, “Designing the properties of dispersion-flattened photonic crystal fibers,” Opt. Express 9, 687–697 (2001) , http://www.opticsinfobase.org/abstract.cfm?URI=oe-9-13-687. [CrossRef] [PubMed]

28.

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

OCIS Codes
(060.2270) Fiber optics and optical communications : Fiber characterization
(350.3950) Other areas of optics : Micro-optics
(060.5295) Fiber optics and optical communications : Photonic crystal fibers

ToC Category:
Photonic Crystal Fibers

History
Original Manuscript: May 27, 2008
Revised Manuscript: July 18, 2008
Manuscript Accepted: August 10, 2008
Published: August 13, 2008

Citation
Kwang No Park and Kyung Shik Lee, "Meshed index profile method for photonic crystal fibers with arbitrary structures," Opt. Express 16, 13175-13187 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-17-13175


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References

  1. T. A. Birks, J. C. Knight, and P. St. J. Russel, "Endlessly single mode photonic crystal fiber," Opt. Lett. 22, 961-963 (1997). [CrossRef] [PubMed]
  2. S. K. Varshney, M. P. Singh, and R. K. Sinha, "Propagation Characteristics of Photonic Crystal Fibers," J. Opt. Commun. 24, 192-198 (2003).
  3. M. Midrio, M. P. Singh, and C. G. Someda, "The Space Filling Mode of Holey Fibers : An Analytical Vectorial Soultion," J. Lightwave Technol. 18, 1031-1037 (2000). [CrossRef]
  4. Y. Li, Y. Yao, M. Hu, L. Chai, and C. Wang, "Improved fully vectorial effective index method for photonic crystal fibers: evaluation and enhancement," Appl. Opt. 47, 399-406 (2008). [CrossRef] [PubMed]
  5. H. Li, A. Mafi, A. Schülzgen, L. Li, V. L. Temyanko, N. Peyghambarian, and J. V. Moloney, "Analysis and Design of Photonic Crystal Fibers Based on an Improved Effective-Index Method," J. Lightwave Technol. 25, 1224-1230 (2007). [CrossRef]
  6. Y. Li, C. Wang, N. Zhang, C. Wang, and Q. Xing, "Analysis and design of terahertz photonic crystal fibers by an effective-index method," Appl. Opt. 45, 8462-8465 (2006). [CrossRef] [PubMed]
  7. B. T. Kuhlmey, H. C. Nguyen, M. J. Steel, and B. J. Eggleton, "Confinement loss in adiabatic photonic crystal fiber tapers," J. Opt. Soc. Am. B 23, 1965-1974 (2006). [CrossRef]
  8. T. P. White et al, "Calculations of air-guided modes in photonic crystal fibers using the multipole method," Opt. Express 11, 721-732 (2001), http://www.opticsinfobase.org/abstract.cfm?URI=oe-9-13-721. [CrossRef]
  9. A. Cucinotta, S. Selleri, L. Vincetti, and M. Zoboli, "Perturbation Analysis of Dispersion Properties in Photonic Crystal Fibers Through the Finite Element Method," J. Lightwave Technol. 20, 1433-1442 (2002). [CrossRef]
  10. F. Brechet et al, "Complete analysis of the propagation characteristics into photonic crystal fibers, be the finite element method," J. Optical Fiber Technol. 6, 181-201 (2001). [CrossRef]
  11. M. A. R. Franco, H. T. Hattori, F. Sircilli, A. Passaro, and N. M. Abe, "Finite Element Analysis of Photonic Crystal Fiber," in PROC IEEE MTT-S IMOC, 5-7 (2001).
  12. 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]
  13. W. Zhi, R, Guobin, and L. Shuqin, "A Novel Supercell Overlapping Method for Different Photonic Crystal Fibers," J. Lightwave Technol. 22, 903-916 (2004). [CrossRef]
  14. D. Mogilevtsev, T. A. Birks, and P. St. J. Russell, "Localized Function Method for Modeling Defect Modes in 2-D Photonic Crystals," J. Lightwave Technol. 17, 2078-2081 (1999). [CrossRef]
  15. D. Mogilevtsev, T. A. Birks, and P. St. J. Russell, "Group-velocity dispersion in photonic crystal fibers," Opt. Lett. 23, 1662-1664 (1998). [CrossRef]
  16. T. M. Monro, D. J. Richardson, N. G. R. Broderick, and P. J. Bennett, "Holey optical fibers : An efficient modal model," J. Lightwave Technol. 17, 1093-1101 (1999). [CrossRef]
  17. Z. Zhang and S. Satpathy, "Electromagnetic wave propagation in structures: Bloch wave solution of Maxwell's equation," Phys. Rev. Lett. 65, 2650-2653 (1990). [CrossRef] [PubMed]
  18. S. Guo and S. Albin, "Simple plane wave implementation for photonic crystal calculations," Opt. Express 11, 167-175 (2003), http://www.opticsinfobase.org/abstract.cfm?URI=oe-11-2-167. [CrossRef] [PubMed]
  19. Z. Zhu and T. G. Brown, "Full-vectorial finite-difference analysis of microstructured optical fiber," Opt. Express 10, 853-864 (2002), http://www.opticsinfobase.org/abstract.cfm?URI=oe-10-17-853">http://www.opticsinfobase.org/abstract.cfm?URI=oe-10-17-853. [PubMed]
  20. K. N. Park and K. S. Lee, "Improved effective index method for analysis of photonic cyrstal fibers," Opt. Lett. 30, 958-960 (2005). [CrossRef] [PubMed]
  21. W. Zhi, R. Guobin, L. Shuqin, and J. Shuisheng, "Supercell lattice method for photonic crystal fibers," Opt. Express 11, 980-991 (2003), http://www.opticsinfobase.org/abstract.cfm?URI=oe-11-9-980. [CrossRef] [PubMed]
  22. T. M. 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]
  23. 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]
  24. I. S. Gradshtein and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1994).
  25. I. Kimel and L. R. Elias, "Relations between Hermite and Laguerre Gaussian modes," IEEE J. Quant. Electron. 29, 2562-2567 (1993). [CrossRef]
  26. T.-L. Wu and C.-H. Chao, "Photonic Crystal Fiber Analysis Through the Vector Boundary-Element Method : Effect of Elliptical Air Hole," IEEE Photon. Tech. Lett. 16, 126-128 (2004). [CrossRef]
  27. A. Ferrando, E. Silvestre, P. Andres, and J. J. Miret, "Designing the properties of dispersion-�?attened photonic crystal �?bers," Opt. Express 9, 687-697 (2001), http://www.opticsinfobase.org/abstract.cfm?URI=oe-9-13-687. [CrossRef] [PubMed]
  28. M. J. Steel, P. T. P. White, C. M. Sterke, R. C. McPhedran, and L. C. Bottn, "Symmerty and degeneracy in microstructured optical fibers," Opt. Lett. 26, 488-490 (2001). [CrossRef]

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