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Optics Express

  • Editor: Michael Duncan
  • Vol. 11, Iss. 17 — Aug. 25, 2003
  • pp: 1966–1979
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Dependence of mode characteristics on the central defect in elliptical hole photonic crystal fibers

Wang Zhi, Ren Guobin, Lou Shuqin, and Jian Shuisheng  »View Author Affiliations


Optics Express, Vol. 11, Issue 17, pp. 1966-1979 (2003)
http://dx.doi.org/10.1364/OE.11.001966


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Abstract

Some mode characteristics are obtained by the full vector supercell overlap method that has been developed to model triangular lattice elliptical hole photonic crystal fibers regardless of whether the light is guided by total internal reflection or a photonic bandgap mechanism. When the central defect hole is large enough, the modes are disordered. Birefringence (Δn) dependence on the central defect is discussed in detail by numerical analysis.

© 2003 Optical Society of America

1. Introduction

Intensive research has been generated by the development of photonic crystal fibers (PCFs), which are silica optical fibers that consist of a central defect region in a regular lattice of air holes. According to the mechanism used to guide light, PCFs can be divided into two general classes [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]

]: a total internal reflection (TIR) PCF and a photonic bandgap (PBG) PCF. These fibers exhibit some unusual properties including endlessly single mode [2

2. T. A. Birks, D. Mogilevtsev, J. C. Knight, P. St. J. Russell, J. Broeng, P. J. Roberts, J. A. West, D. C. Allan, and J. C. Fajardo, “The analogy between photonic crystal fibers and step index fibers,” in Optical Fiber Communication Conference, Vol. 2 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), paper FG4, pp. 114–116.

,3

3. 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]

], very large or very small mode area [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]

6

6. J. M. Dudley and S. Coen, “Numerical simulations and coherence properties of supercontinuum generation in photonic crystal and tapered optical fibers,” IEEE J. Sel. Top. Quantum Electron. 8, 651–659 (2002). [CrossRef]

], zero dispersion wavelength shifting into the visible spectrum [7

7. S. E. Barkou, J. Broeng, and A. Bjarklev, “Dispersion properties of photonic bandgap guiding fibers,” in Optical Fiber Communication Conference, Vol. 2 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), paper FG5, pp. 117–119.

10

10. R. G. Bin, L. S. Qin, W. Zhi, and S. S. Jian, “Study on dispersion properties of photonic crystal fiber by effective-index model,” Acta Opt. Sin. 24, (2004), in Chinese (to be published).

], and guidance in air [11

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

13

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

]. Several authors have considered high birefringence PCFs, in which the birefringence is introduced either by a local change in location or size of a few holes [14

14. S. E. B. Libori, J. Broeng, E. Knudsen, A. Bjarklev, and H. Simonsen, “High-birefringent photonic crystal fibers,” in Optical Fiber Communication Conference, Vol. 54 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2001), paper TuM2.

,15

15. T. P. Hansen, J. Broeng, S. E. B. Libori, E. Knudsen, A. Bjarklev, J. R. Jensen, and H. Simonsen, “Highly birefringent index-guiding photonic crystal fibers,” IEEE Photon. Technol. Lett. 13, 588–590 (2001). [CrossRef]

] or by use of elliptical holes [16

16. J. Broeng, D. Mogilevtsev, S. E. B. Libori, and A. Bjarklev, “Polarization-preserving holey fibers,” in Optical Fiber Communication Conference, Vol. 54 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2001), paper MA1.3.

18

18. M. J. Steel and R. M. Osgood, “Polarization and dispersive properties of elliptical-hole photonic crystal fibers,” J. Lightwave Technol. 19, 495–503 (2001). [CrossRef]

].

Together with the technological advancement in the fabrication [19

19. K. Tajima, J. Zhou, K. Nakajima, and K. Sato, “Ultra low loss and long length photonic crystal fiber,” in Optical Fiber Communication Conference, Vol. 86 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2003), pp. PD1-1–PD1-3.

] of PCFs, powerful theoretical tools have been developed to model the guidance properties. Some of these are the effective index method [2

2. T. A. Birks, D. Mogilevtsev, J. C. Knight, P. St. J. Russell, J. Broeng, P. J. Roberts, J. A. West, D. C. Allan, and J. C. Fajardo, “The analogy between photonic crystal fibers and step index fibers,” in Optical Fiber Communication Conference, Vol. 2 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), paper FG4, pp. 114–116.

,10

10. R. G. Bin, L. S. Qin, W. Zhi, and S. S. Jian, “Study on dispersion properties of photonic crystal fiber by effective-index model,” Acta Opt. Sin. 24, (2004), in Chinese (to be published).

,20

20. R. G. Bin, W. Zhi, L. S. Qin, and S. S. Jian, “Study on dispersion properties of photonic crystal fiber by effective-index model,” accepted by OECC’2003, Shanghai. (to be published by the Chinese Optical Society)

], the plane wave expansion method [21

21. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding theFlow of Light (Princeton University, Princeton, N.J., 1995).

23

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

], the localized basis function method [24

24. 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]

28

28. D. Mogilevtsev, T. A. Birks, and P. St. J. Russell, “Localized function method for modeling defect mode in 2-d photonic crystal,” J. Lightwave Technol. 17, 2078–2081 (1999). [CrossRef]

], the finite-element method [29

29. M. Koshiba, “Full vector analysis of photonic crystal fibers using the finite element method,” IEICE Electron. E85-C, 881–888 (2002).

,30

30. Z. Zhu and T. G. Brown, “Full-vectorial finite difference analysis of microstructured optical fibers,” Opt. Express 10, 853–864 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-8-853. [CrossRef] [PubMed]

], and more recently the beam propagation method [31

31. F. Fogli, L. Saccomandi, P. Bassi, G. Bellanca, and S. Trillo, “Full vectorial BPM modeling of indexguiding photonic crystal fibers and couplers,” Opt. Express 10, 54–59 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-1-54. [CrossRef] [PubMed]

], and the multipole method [32

32. T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. M. de Sterke, and L. C. Botten, “Multipole method for microstructured optical fibers. I. Formulation,” J. Opt. Soc. Am. B 19, 2322–2330 (2002). [CrossRef]

,33

33. B. T. Kuhlmey, T. P. White, G. Renversez, D. Maystre, L. C. Botten, C. M. de Sterke, and R. C. McPhedran, “Multipole method for microstructured optical fibers. II. Implementation and results,” J. Opt. Soc. Am. B 19, 2331–2340 (2002). [CrossRef]

]. When higher-order modes or polarization properties are considered, a full vector approach is crucial to assess the true behavior of the electromagnetic wave in a complex inhomogeneous structure such as a PCF.

The polarization and dispersive properties of elliptical hole PCFs (EHPCFs) have been investigated with the plane wave method [18

18. M. J. Steel and R. M. Osgood, “Polarization and dispersive properties of elliptical-hole photonic crystal fibers,” J. Lightwave Technol. 19, 495–503 (2001). [CrossRef]

] in detail. The results show that a number of important quantities, including birefringence and walk-off parameters, exhibit frequency dependence quite unlike conventional birefringent systems. It was found that the presence of a central defect, a smaller hole in the core region, gives rise to larger birefringence than in the case of a missing hole [16

16. J. Broeng, D. Mogilevtsev, S. E. B. Libori, and A. Bjarklev, “Polarization-preserving holey fibers,” in Optical Fiber Communication Conference, Vol. 54 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2001), paper MA1.3.

]. To our knowledge, there is nothing in the literature that describes how central defects affect mode properties.

2. Full Vector Model

2.1 Eigenequation

We believe that the PCF is lossless and uniform in the propagation (z) direction, so our main task is to investigate the transverse mode field distribution, which satisfies a pair of coupled wave equations [25

25. 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]

,34

34. A.W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

]. When the transverse mode field is expanded as the decomposition of an orthogonal set of Hermite-Gaussian basis functions, the eigenvalue equation, shown as Eq. (1), will be obtained by use of the orthonormality of the Hermite-Gaussian basis functions [35

35. R. Guobin, W. Zhi, L. Shuqin, and J. Shuisheng, “Mode classification and degeneracy in photonic crystal fiber,” Opt. Express 11, 1310–1321 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-1310. [CrossRef] [PubMed]

].

L[exey][Iabcd(1)+k2Iabcd(2)+Iabcd(3)xIabcd(4)xIabcd(4)yIabcd(1)+k2Iabcd(2)+Iabcd(3)y][exey]=βj2[exey],
(1)

Iabcd(1)=+ψa(x)ψb(y)t2[ψc(x)ψd(y)]dxdy,
(2a)
Iabcd(2)=+εψa(x)ψb(y)ψc(x)ψd(y)dxdy,
(2b)
Iabcd(3)x=+ψa(x)ψb(y)x[ψc(x)ψd(y)lnεx]dxdy,
(2c)
Iabcd(3)y=+ψa(x)ψb(y)y[ψc(x)ψd(y)lnεy]dxdy,
(2d)
Iabcd(4)x=+ψa(x)ψb(y)x[ψc(x)ψd(y)lnεy]dxdy,
(2e)
Iabcd(4)y=+ψa(x)ψb(y)y[ψc(x)ψd(y)lnεx]dxdy,
(2f)

where ε is the dielectric constant distribution ε(x,y), ψi (s) (I=a,b,c,d, s=x,y) are elements of the orthogonal set of Hermite-Gaussian basis functions, and ψi(s)=2i2π14i!ωexp(s22ω2)Hi(sω), where Hi (s/ω) is the ith-order Hermite polynomial and ω is the characteristic width of the basis set.

2.2 Dielectric Constant Profile

We use the supercell overlap method [27

27. W. Zhi, R.G. Bin, L. S. Qin, and S. S. Jian, “Supercell lattice method for photonic crystal fibers,” Opt. Express 11, 980–991 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-980. [CrossRef] [PubMed]

,35

35. R. Guobin, W. Zhi, L. Shuqin, and J. Shuisheng, “Mode classification and degeneracy in photonic crystal fiber,” Opt. Express 11, 1310–1321 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-1310. [CrossRef] [PubMed]

,36

36. W. Zhi, R. G. Bin, L. S. Qin, and S. S. Jian, “Novel supercell overlapping method for different photonic crystal fibers,” J. Lightwave Technol. submitted.

] to describe the transverse dielectric structure of PCF. In this method, we introduce two two-dimensional (2-D) virtually perfect photonic crystals (PC1 and PC2) that can be combined to reconstruct the dielectric structure of a supercell lattice PCF. The dielectric structure of the PCF can be expressed as the sum of the decomposition of both PC1 and PC2. Figure 1 shows an example to illustrate the way in which the transverse dielectric structure is constructed, and Table 1 lists the parameters of both virtual PCs.

Fig. 1. Schematic of the way in which the transverse dielectric structure is constructed. PC1 is a perfect triangular lattice constructed by the blue elliptical holes, the dielectric constant is εair in the hole and εsi outside the hole. PC2 is another perfect triangular lattice constructed by the coaxial elliptical rings with which the inner major axis is dc and the outer major axis is d, the dielectric constant is 0 in dc or outside d and εsi -εair between dc and d.

Table 1. Parameters of Virtual PCs for Triangular Lattice PCFs

table-icon
View This Table

Decomposing the dielectric structure of both PC1 and PC2 by use of cosine functions, the dielectric constant of the supercell lattice PCF can be expressed as

ε(x,y)=ε|PC1+ε|PC2,lnε=(lnε)|PC1+(lnε)|PC2,
(3)

where

ε|PC1=a,b=0P11P1abcos2πaxl1xcos2πbyl1y,
(4a)
εPC2=a,b=0P21P2abcos2πaxl2xcos2πbyl2y,
(4b)
(lnε)PC1=a,b=0P11P1ablncos2πaxl1xcos2πbyl1y,
(4c)
(lnε)PC2=a,b=0P21P2ablncos2πaxl2xcos2πbyl2y,
(4d)

and where P 1 and P 2 are the number of decomposition terms of PC1 and PC2, respectively; and P 1ab, P 2ab, P1abln, and P2abln are the coefficients that can be obtained through the Fourier transform. P 1ab and P1abln can be obtained as described in Refs. [27

27. W. Zhi, R.G. Bin, L. S. Qin, and S. S. Jian, “Supercell lattice method for photonic crystal fibers,” Opt. Express 11, 980–991 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-980. [CrossRef] [PubMed]

, 36

36. W. Zhi, R. G. Bin, L. S. Qin, and S. S. Jian, “Novel supercell overlapping method for different photonic crystal fibers,” J. Lightwave Technol. submitted.

]; the approach to evaluate P 2ab and P2abln is simply described as follows.

For the EHPCF, the central defect elliptical hole size is dc along the y axis, elliptical ratio η is the same as the cladding holes. There are two fractional parameters, fc and f 2, of PC2 that are defined as follows:

fc=πdc2(2ηl2xl2y),f2=πd2(2ηl2xl2y),
(5)

where η is defined as η=d/e, and d and e represent the length of the elliptical holes along the y and x axes, respectively, as shown in Fig. 1.

When the dielectric structure of PC2 is expressed in the form of a Fourier transform as

ε(x,y)PC2=m,n=(P21)P21F2(Kmn)cos(k1x)cos(k2y),
(6)

the Fourier transform coefficient will be

F2(Kmn)=2(nsi2nair2)[f.J1(Kmnd2)Kmnd2fc.J1(Kmndc2)Kmndc2],Kmn0,
(7a)
F2(0)=(ffc)(nsi2nair2),
(7b)

where J 1 is the first-order Bessel function, k 1 and k 2 are the real and imaginary parts of Kmn , i.e., Km n =k 1+ik 2 (i is the imaginary unit). Kmn depends on the reciprocal lattice vector [21

21. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding theFlow of Light (Princeton University, Princeton, N.J., 1995).

,22

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

] of PCFs and can be expressed as

Kmn=(m+n)kxi(mn)ky,kx=2πl2xη,ky=2πl2y.
(8)

P 2ab can be obtained when F 1 is replaced with F 2 in Eq. (9) in Ref. 28

28. D. Mogilevtsev, T. A. Birks, and P. St. J. Russell, “Localized function method for modeling defect mode in 2-d photonic crystal,” J. Lightwave Technol. 17, 2078–2081 (1999). [CrossRef]

. P2abln can be analytically obtained in the same way as P 2ab, but only when the dielectric constant parameters are replaced with their logarithm as shown in Table 1.

Figure 2 shows the simulation results of the triangular lattice EHPCF, with a structure parameter hole pitch of D=2.3 µm, a cladding elliptical hole size along the y axis of d=0.8D, a central defect elliptical hole size along the y axis of dc =0.4D, and an elliptical ratio of both the cladding hole and the central defect η=2. To show the dielectric structure clearly, the supercell parameter N is set at 4 in Fig. 2, whereas N=10 in the following simulations. Figures 2(a), 2(b), and 2(c) are the periodic dielectric profiles of the reconstructed PCF, the virtual PC1, and the virtual PC2, respectively. The unit cell in Fig. 2(c) is a coaxial elliptical ring with which the inner major axis is dc and the outer major axis is d, the dielectric constant is 0 in dc or outside d, and εsi -εair between dc and d.

Fig. 2. Simulation results of the dielectric constant profiles of the PCF and both virtually perfect PCs with parameters of D=2.3 µm, d=0.8D, dc =0.4D, η=2, and N=4.

2.3 Overlap Integrals

Substituting decomposition Eq. (4) into Eq. (2), the overlap integrals can be calculated by use of the orthonormality of the Hermite-Gaussian basis functions and the standard integration by parts technique combined with some of the definite integrals available in collections [37

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

] along with some identities [38

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

]. The fact that all these integrals can be performed analytically is such a major advantage that the numerical precision and efficiency can be significantly improved, especially when a significant number of terms are needed in the expansions. The expressions for overlap integrals I (1), I (2), and I (3) can be found in Refs. [27

27. W. Zhi, R.G. Bin, L. S. Qin, and S. S. Jian, “Supercell lattice method for photonic crystal fibers,” Opt. Express 11, 980–991 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-980. [CrossRef] [PubMed]

, 36

36. W. Zhi, R. G. Bin, L. S. Qin, and S. S. Jian, “Novel supercell overlapping method for different photonic crystal fibers,” J. Lightwave Technol. submitted.

]; I (4)x and I (4)y are shown here:

Iabcd(4)x=f,g=0P11P1fglnIfac(42)xIgbd(41)yf,g=0P21P2fglnINfac(42)xINgbd(41)y,
(9a)
Iabcd(4)y=f,g=0P11P1fglnIfac(41)xIgbd(42)yf,g=0P21P2fglnINfac(41)xINgbd(42)y,
(9b)

where

Ii1i2i3(41)s=+cos(2πi1sl1s)sψi2(s)ψi3(s)ds,
(9c)
Ii1i2i3(42)s=+ψi3(s)cos(2πi1sl1s)sψi2(s)ds.
(9d)

When l 1s is replaced with l 2s, IN (41)s and IN (42)s can be evaluated in the same way.

3. Mode Symmetry and Classification

The modes of a waveguide with C symmetry either exhibit the full waveguide symmetry (C ) and are nondegenerate or occur in degenerate pairs that support this symmetry only in combination [36

36. W. Zhi, R. G. Bin, L. S. Qin, and S. S. Jian, “Novel supercell overlapping method for different photonic crystal fibers,” J. Lightwave Technol. submitted.

,39

39. P. R. McIsaac, “Symmetry-induced modal characteristics of uniform waveguides-I: summary of results,” IEEE Trans. Microwave Theory Tech. MTT-23, 421–429 (1975). [CrossRef]

41

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

]. Because the cross section of the triangular lattice EHPCF studied here possesses C 2ν symmetry, all the modes can be classified into four classes. These classes are nondegenerate and exhibit the full waveguide symmetry individually [39

39. P. R. McIsaac, “Symmetry-induced modal characteristics of uniform waveguides-I: summary of results,” IEEE Trans. Microwave Theory Tech. MTT-23, 421–429 (1975). [CrossRef]

,40

40. P. R. McIsaac, “Symmetry-induced modal characteristics of uniform waveguides- II: theory,” IEEE Trans. Microwave Theory Tech. MTT-23, 429–433 (1975). [CrossRef]

].

Fig. 3. Minimum sectors for waveguides with C 2ν symmetry. The waveguide modes are classified into four classes (p=1,2,3,4). The solid lines indicate short-circuit boundary conditions, the dashed lines indicate open-circuit boundary conditions.

By applying our approach to the triangular lattice EHPCF with structure parameters: D=2.3 µm, d/D=0.8, dc =D, and η=2, we extract the first 12 eigenvalues βj/k (modal index) at wavelength λ=D/0.8 from the eigenequation and plot them in Fig. 4, and the electric field vector of the first six modes is shown in Fig. 5. We found that all modes are nondegenerate, with modes 1–6 belonging to mode class p=4, 1, 2, 3, 3, 1, respectively.

Fig. 4. Modal index of the first 12 modes of the triangular lattice EHPCF with the parameters D=2. 3 µm, d=0.8D, dc =D, η=2, λ=D/0.8.
Fig. 5. Electric field vector of the first six modes in the PCF with parameters as in Fig. 4.

There is one important fact, that is, the electric field of modes 1 and 4 looks like the HE11 mode in step-index fibers, which we refer to as HE11y and HE11x. The doublet components of the fundamental mode, as we know, are degenerate in conventional standard fibers and in circular hole triangular lattice PCFs [35

35. R. Guobin, W. Zhi, L. Shuqin, and J. Shuisheng, “Mode classification and degeneracy in photonic crystal fiber,” Opt. Express 11, 1310–1321 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-1310. [CrossRef] [PubMed]

,39

39. P. R. McIsaac, “Symmetry-induced modal characteristics of uniform waveguides-I: summary of results,” IEEE Trans. Microwave Theory Tech. MTT-23, 421–429 (1975). [CrossRef]

41

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

]. However, when the air holes are elliptical, the degeneracy splits significantly, one is mode 1, and the other is mode 4. Regardless of whether both modes are confined or leaky, the birefringence will be defined as Δn=n 1-n 4 rather than Δn=n 1-n 2, which will be discussed in detail in the following text. The reason for this notation is that, when the central defect is large, here dc =D, the light propagates in the PCF guided by the PBG mechanism, not TIR. In air-silica PBG fibers, it is true that higher-order modes are bounded whereas lower-order modes are leaky [11

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

,18

18. M. J. Steel and R. M. Osgood, “Polarization and dispersive properties of elliptical-hole photonic crystal fibers,” J. Lightwave Technol. 19, 495–503 (2001). [CrossRef]

]. Here we contribute to the mode disorder and the birefringence properties regardless of whether the light is TIR guided or PBG guided.

Fig. 6. Modal index of the first 12 modes of the triangular lattice EHPCF with parameters D=2.3 µm, d=0.8D, dc =0.2D, η=2, λ=D/0.8.
Fig. 7. Electric field vector of the first three modes in the PCF with parameters as in Fig. 6.

4. Mode Disorder in Electronic Hole Photonic Crystal Fiber

At a certain wavelength, all the eigenvalue-eigenvector pairs can be labeled in the decreasing order of the propagation constant, whereas the mode with the greatest β is labeled as mode 1. Consequently, all the modes can be arranged in descending order of β and an increasing order of labels. Conventionally, the two lowest-order modes are the HE11x and HE11y, which are called the fundamental modes. When the fiber structure is complicated, the order of modes is changed, and all the modes are rearranged in different orders, which is called mode disorder.

What is the physical reason for mode disorder in EHPCFs? The propagation constant, as is well known, is the longitudinal (z axis of fibers) component of the wave vector. The term mode means the transverse electric field pattern, which can be considered as the interference pattern of electromagnetic waves in the cross section. In polar coordinates, a wave vector can be decomposed into three orthogonal vectors along three different basis vectors as k=k r+k φ+k z, where k z is just the propagation constant β. Standing cylinder waves along the r direction with wave vector k r and standing waves along the φ direction with wave vector k φ will interfere to produce the mode pattern. All the modes are labeled with two numbers m and n, which are the numbers of modes of the two standing waves along the φ and r directions, respectively, such as mode HEmn or mode EHmn. At wavelength λ, the amplitude of the wave vector is |k|=2π/λ, which can be rewritten in the form of |k|2=|k r|2+|k φ|2+|k z|2. One can easily deduce that two or more different (k r, k φ) pairs could exist with the same k z at a fixed wavelength. We can also state that there are two or more modes with the same propagation constants (β) at this wavelength. Modes with the same β at a wavelength will exchange their mode order near that wavelength, at which point the modes are then disordered and rearranged.

4.1. Mode Disorder between the Fundamental and the Higher-Order Modes

In Fig. 5, where the air holes are elliptical and the central hole is large enough, the HE11x mode is the fourth mode, the degeneracy between the fundamental doublet is significantly split. It can be said that the mode disorder occurs between the fundamental and the higher-order modes.

Fig. 8. Mode order of HE11y and HE11x of an EHPCF. The parameters are shown at top right.

The property of mode disorder depends on many parameters such as the structure of PCFs and the operating wavelength. To demonstrate mode disorder between the fundamental and the higher-order modes clearly, a triangular lattice EHPCF with parameters D=2.3µm, d=0.8D, dc =1.2D, and η=2 is considered. The mode order of both HE11y and HE11x varies with the operating wavelength, which is shown in Fig. 8. It is obvious that the HE11y mode keeps the lowest mode within a wide wavelength range, and the HE11x mode keeps the second lowest mode within a narrower wavelength range. As the wavelength decreases, the mode disorder of both doublets increases, i.e., there are some other modes that become the first or the second mode.

4.2. Mode Disorder between the Fundamental Doublets

In Fig. 8, the order of HE11y is always lower than HE11x, and mode disorder does not exist between the fundamental doublets. When the size of the cladding air hole varies, some novelty emerges. Figure 9(a) illustrates the mode order of the fundamental doublets in an EHPCF, which has the structure parameters D=2.3µm, d=0.4D, dc =1.2D, and η=2. Figure 9 also shows that both modes are the lowest two, but the mode orders exchange with each other when the wavelength is nearly 0.8 µm.

For clarity, Δn=ny -nx is introduced and demonstrated as the solid blue curve in Fig. 9(b). It was found that the mode order of both doublets exchanges at point A where Δn is zero. When the wavelength is longer than λ A, the HE11y mode is the lowest, its mode index is greater than HE11x, hence the slower axis of the EHPCF is along the y axis. When the wavelength is shorter than λ A, it is inverse, and the y axis becomes the faster axis of the EHPCF. The fact that Δn increases as λ decreases at a much shorter wavelength needs to be researched, so it is not discussed in this contribution.

At point A, the mode indices of both modes are exactly the same, so they degenerate at wavelength λ A. How does this happen and what is the mechanism of the degeneracy? Figure 10 provides a qualitative understanding of the answer to the question. Figure 10(a) is an EHPCF with a central defect air hole, and only one ring of cladding air holes is plotted for simplicity. It can be considered as the composition of two structures: one is an EHPCF with the same photonic crystal cladding and without the central air hole, the other is a fiber with a central elliptical air hole and a homogeneous cladding, all of which are illustrated in Figs. 10(b) and 10(c), respectively, and labeled as fibers a, b, and c in the following text.

Fig. 9. (a) Mode orders of HE11y and HE11x of an EHPCF with the parameters shown at right. (b) Relationship between Δn and λ of fibers a, b, and c.
Fig. 10. Schematic of fibers a, b, and c, which is helpful to understand the mode disorder between the fundamental doublets.

With regard to fiber b, the polarization property of this kind of EHPCF has been investigated by a few researchers [17

17. M. J. Steel and P. M. Osgood, Jr, “Elliptical-hole photonic crystal fibers,” Opt. Lett. 26, 229–231 (2001). [CrossRef]

,18

18. M. J. Steel and R. M. Osgood, “Polarization and dispersive properties of elliptical-hole photonic crystal fibers,” J. Lightwave Technol. 19, 495–503 (2001). [CrossRef]

], who all reached the same conclusion that the mode index of HE11y is greater than that of HE11x. The birefringence of the fundamental doublets can be analyzed and is shown as the dotted green curve in Fig. 9(b). The birefringence of fiber c is also plotted as the dash-dot red curve in Fig. 9(b).

Δn of fiber b increases monotonically when λ increases. At high frequencies, the vector terms in the wave equation [i.e., I (3) and I (4) in the eigenequation Eq. (1)] become negligible, and the birefringence must vanish in the scalar regime at short wavelengths. When λ increases, the effective mode area increases [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]

,5

5. 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]

,27

27. W. Zhi, R.G. Bin, L. S. Qin, and S. S. Jian, “Supercell lattice method for photonic crystal fibers,” Opt. Express 11, 980–991 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-980. [CrossRef] [PubMed]

] because the electric field penetrates into the cladding area more easily, and the more elliptical holes will be sampled by the modes. The long-wavelength behavior of the birefringence is due to the cladding region, which acts as a birefringent material.

In contrast, birefringence Δn of fiber c, which can be regarded as an inversion of the conventional standard elliptical core fiber, tends to zero at both high and low frequencies. At high frequencies, the zero birefringence exists for the same reason as in fiber b, whereas at low frequencies the bulk of the mode energy is distributed far from the core-cladding interface, which induces birefringence. The maximum value at intermediate frequencies scales with the relative refractive-index difference between the fiber core and the cladding in the weak guidance limit [18

18. M. J. Steel and R. M. Osgood, “Polarization and dispersive properties of elliptical-hole photonic crystal fibers,” J. Lightwave Technol. 19, 495–503 (2001). [CrossRef]

].

In Fig. 9(b), the a curve can be approximately understood as the sum of curve b and curve c, i.e., the birefringence of fiber a can be considered as the sum of both fibers b and c. When the birefringence of fibers b and c are equal and have opposite sign, fiber a will have a zero Δn. As the wavelength decreases, the mode field becomes more confined, so the birefringence property can be approximate to fiber c, which has only nonsymmetrical geometry in the core region. As the wavelength increases, the case is reversed. The mode field penetrates more into the cladding area and fills more elliptical holes that have nonsymmetrical geometry everywhere, so fiber a can be approximate to fiber b at long wavelengths.

5. Birefringence

Δn increases monotonically when λ/D increases for a certain PCF (fixed dc /D), which can also be seen in Fig. 12, which shows the relationship between Δn and D/λ at different dc /D. The birefringence profile is the same as curve b in Fig. 9(b), which has been discussed above. In the long-wavelength regime, a majority of the mode energy is located beyond the first ring of air holes, and the field attempts to approach the distribution of the fundamental space filling mode (FSM) as closely as possible [18

18. M. J. Steel and R. M. Osgood, “Polarization and dispersive properties of elliptical-hole photonic crystal fibers,” J. Lightwave Technol. 19, 495–503 (2001). [CrossRef]

]. It can thus be concluded that the birefringence between the FSMy and the FSMx modes must be the maximum value at a fixed wavelength for a certain PCF.

Fig. 11. Relationship between Δn and central defect size dc /D at different wavelengths. The structure parameters of the PCF are D=2.3 µm, d=0.8D, and η=2.
Fig. 12. Relationship between Δn and normalized frequency D/λ in the PCFs with different central defect sizes. The parameters are D=2.3 µm, d=0.8D, and η=2.

At a fixed d and λ/D, when the central defect diameter dc /D increases from 0 (without the central air hole), Δn increases. Once the central defect increases to the same size as the cladding holes, the PCF becomes a perfect 2D photonic crystal, hence the fundamental mode is the FSM mode. We thus expect that Δn will reach its peak value at point dc /D=d/D. It is fortunate that the conclusion is almost validated by the numerical results shown in Fig. 11, where the cladding hole size is d/D=0.8. When dc /D is greater than 0.8, Δn then decreases, which is not shown in Fig. 12 because the mode disorder is too complicated to obtain an accurate Δn.

Fig. 13. Relationship between Δn and normalized frequency D/λ in PCFs with different elliptical ratios. The parameters are D=2.3 µm and d=0.8D. The major axis of the elliptical hole is along the (a) y axis or (b) x axis. The major axis length is dc =0.4D.

Figure 13 shows the relationship between Δn and the normalized frequency D/λ at different elliptical ratios η with structure parameters D=2.3 µm and d=0.8D. In Fig. 13(a) the major axis of the elliptical hole is along the y axis and is maintained at dc =0.4D. In Fig. 13(b) the major axis is along the x axis and is also maintained at 0.4D, which is the same as in Fig.11 or Fig. 12 in which birefringence Δn increases as D/λ decreases. As η decreases, Δn increases gradually when the major axis is along the y axis, and Δn increases significantly when the major axis is along the x axis. By comparing Figs. 13(a) and 13(b) one will find that the birefringence of the x major axis is greater than that of the y major axis, a fact that can be easily understood from the waveguide structure shown in the Fig. 13 inset, which has been discussed in detail in Refs. 18

18. M. J. Steel and R. M. Osgood, “Polarization and dispersive properties of elliptical-hole photonic crystal fibers,” J. Lightwave Technol. 19, 495–503 (2001). [CrossRef]

and 42

42. R. G. Bin, W. Zhi, L. S. Qin, and S. S. Jian, “Birefringence properties of elliptical-hole photonic crystal fiber,” Chin. J. Lasers 31 (2004), in Chinese (to be published)

and will not be repeated here.

6. Discussion and Conclusion

Some modes are cut off as the frequency decreases. From the electric field distribution it can be found that all the modes are cut off in Fig. 5 and only the first two modes are bounded in Fig. 7. When we discussed the mode disorder properties, only the electric field distribution was considered. For example, in Fig. 5 the first mode is HE11y and the fourth mode is HE11x according to the vector distribution of the transverse electric field, no matter whether it is bounded or leaky or whether it is guided by a PBG or TIR.

We thank Wu Chongqing for beneficial discussions about the physical basis of mode disorder as well as information about dielectric waveguides.

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, D. Mogilevtsev, J. C. Knight, P. St. J. Russell, J. Broeng, P. J. Roberts, J. A. West, D. C. Allan, and J. C. Fajardo, “The analogy between photonic crystal fibers and step index fibers,” in Optical Fiber Communication Conference, Vol. 2 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), paper FG4, pp. 114–116.

3.

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]

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.

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]

6.

J. M. Dudley and S. Coen, “Numerical simulations and coherence properties of supercontinuum generation in photonic crystal and tapered optical fibers,” IEEE J. Sel. Top. Quantum Electron. 8, 651–659 (2002). [CrossRef]

7.

S. E. Barkou, J. Broeng, and A. Bjarklev, “Dispersion properties of photonic bandgap guiding fibers,” in Optical Fiber Communication Conference, Vol. 2 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), paper FG5, pp. 117–119.

8.

T. A. Birk, D. Mogilevtsev, J. C. Knight, and P. St. J. Russell, “Single material fibers for dispersion compensation,” in Optical Fiber Communication Conference (Optical Society of America, Washington, D.C., 1999), paper FG2, pp. 108–110.

9.

A. Bjarklev, J. Broeng, K. Dridi, and S. E. Barkou, “Dispersion properties of photonic crystal fibres,” in European Conference on Optical Communication (Madrid, Spain, 1998), pp. 135–136.

10.

R. G. Bin, L. S. Qin, W. Zhi, and S. S. Jian, “Study on dispersion properties of photonic crystal fiber by effective-index model,” Acta Opt. Sin. 24, (2004), in Chinese (to be published).

11.

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

12.

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

13.

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

14.

S. E. B. Libori, J. Broeng, E. Knudsen, A. Bjarklev, and H. Simonsen, “High-birefringent photonic crystal fibers,” in Optical Fiber Communication Conference, Vol. 54 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2001), paper TuM2.

15.

T. P. Hansen, J. Broeng, S. E. B. Libori, E. Knudsen, A. Bjarklev, J. R. Jensen, and H. Simonsen, “Highly birefringent index-guiding photonic crystal fibers,” IEEE Photon. Technol. Lett. 13, 588–590 (2001). [CrossRef]

16.

J. Broeng, D. Mogilevtsev, S. E. B. Libori, and A. Bjarklev, “Polarization-preserving holey fibers,” in Optical Fiber Communication Conference, Vol. 54 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2001), paper MA1.3.

17.

M. J. Steel and P. M. Osgood, Jr, “Elliptical-hole photonic crystal fibers,” Opt. Lett. 26, 229–231 (2001). [CrossRef]

18.

M. J. Steel and R. M. Osgood, “Polarization and dispersive properties of elliptical-hole photonic crystal fibers,” J. Lightwave Technol. 19, 495–503 (2001). [CrossRef]

19.

K. Tajima, J. Zhou, K. Nakajima, and K. Sato, “Ultra low loss and long length photonic crystal fiber,” in Optical Fiber Communication Conference, Vol. 86 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2003), pp. PD1-1–PD1-3.

20.

R. G. Bin, W. Zhi, L. S. Qin, and S. S. Jian, “Study on dispersion properties of photonic crystal fiber by effective-index model,” accepted by OECC’2003, Shanghai. (to be published by the Chinese Optical Society)

21.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding theFlow of Light (Princeton University, Princeton, N.J., 1995).

22.

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

23.

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

24.

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]

25.

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]

26.

T. M. Monro, D. J. Richardson, and N. G. R. Broderick, “Efficient modeling of holey fibers,” in Optical Fiber Communication Conference (Optical Society of America, Washington, D.C., 1999), paper FG3, pp. 111–113.

27.

W. Zhi, R.G. Bin, L. S. Qin, and S. S. Jian, “Supercell lattice method for photonic crystal fibers,” Opt. Express 11, 980–991 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-980. [CrossRef] [PubMed]

28.

D. Mogilevtsev, T. A. Birks, and P. St. J. Russell, “Localized function method for modeling defect mode in 2-d photonic crystal,” J. Lightwave Technol. 17, 2078–2081 (1999). [CrossRef]

29.

M. Koshiba, “Full vector analysis of photonic crystal fibers using the finite element method,” IEICE Electron. E85-C, 881–888 (2002).

30.

Z. Zhu and T. G. Brown, “Full-vectorial finite difference analysis of microstructured optical fibers,” Opt. Express 10, 853–864 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-8-853. [CrossRef] [PubMed]

31.

F. Fogli, L. Saccomandi, P. Bassi, G. Bellanca, and S. Trillo, “Full vectorial BPM modeling of indexguiding photonic crystal fibers and couplers,” Opt. Express 10, 54–59 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-1-54. [CrossRef] [PubMed]

32.

T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. M. de Sterke, and L. C. Botten, “Multipole method for microstructured optical fibers. I. Formulation,” J. Opt. Soc. Am. B 19, 2322–2330 (2002). [CrossRef]

33.

B. T. Kuhlmey, T. P. White, G. Renversez, D. Maystre, L. C. Botten, C. M. de Sterke, and R. C. McPhedran, “Multipole method for microstructured optical fibers. II. Implementation and results,” J. Opt. Soc. Am. B 19, 2331–2340 (2002). [CrossRef]

34.

A.W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

35.

R. Guobin, W. Zhi, L. Shuqin, and J. Shuisheng, “Mode classification and degeneracy in photonic crystal fiber,” Opt. Express 11, 1310–1321 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-1310. [CrossRef] [PubMed]

36.

W. Zhi, R. G. Bin, L. S. Qin, and S. S. Jian, “Novel supercell overlapping method for different photonic crystal fibers,” J. Lightwave Technol. submitted.

37.

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

38.

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

39.

P. R. McIsaac, “Symmetry-induced modal characteristics of uniform waveguides-I: summary of results,” IEEE Trans. Microwave Theory Tech. MTT-23, 421–429 (1975). [CrossRef]

40.

P. R. McIsaac, “Symmetry-induced modal characteristics of uniform waveguides- II: theory,” IEEE Trans. Microwave Theory Tech. MTT-23, 429–433 (1975). [CrossRef]

41.

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

42.

R. G. Bin, W. Zhi, L. S. Qin, and S. S. Jian, “Birefringence properties of elliptical-hole photonic crystal fiber,” Chin. J. Lasers 31 (2004), in Chinese (to be published)

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

ToC Category:
Research Papers

History
Original Manuscript: June 20, 2003
Revised Manuscript: August 8, 2003
Published: August 25, 2003

Citation
Wang Zhi, Ren Guobin, Lou Shuqin, and Jian Shuisheng, "Dependence of mode characteristics on the central defect in elliptical hole photonic crystal fibers," Opt. Express 11, 1966-1979 (2003)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-17-1966


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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, D. Mogilevtsev, J. C. Knight, P. St. J. Russell, J. Broeng, P. J. Roberts, J. A. West, D. C. Allan, and J. C. Fajardo, �??The analogy between photonic crystal fibers and step index fibers,�?? in Optical Fiber Communication Conference, Vol. 2 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), paper FG4, pp. 114�??116.
  3. 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]
  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. 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]
  6. J. M. Dudley and S. Coen, �??Numerical simulations and coherence properties of supercontinuum generation in photonic crystal and tapered optical fibers,�?? IEEE J. Sel. Top. Quantum Electron. 8, 651�??659 (2002). [CrossRef]
  7. S. E. Barkou, J. Broeng, and A. Bjarklev, �??Dispersion properties of photonic bandgap guiding fibers,�?? in Optical Fiber Communication Conference, Vol. 2 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), paper FG5, pp. 117�??119.
  8. T. A. Birk, D. Mogilevtsev, J. C. Knight, and P. St. J. Russell, �??Single material fibers for dispersion compensation,�?? in Optical Fiber Communication Conference (Optical Society of America, Washington, D.C., 1999), paper FG2, pp. 108�??110.
  9. A. Bjarklev, J. Broeng, K. Dridi, and S. E. Barkou, �??Dispersion properties of photonic crystal fibres,�?? in European Conference on Optical Communication (Madrid, Spain, 1998), pp. 135�??136.
  10. R. G. Bin, L. S. Qin, W. Zhi, and S. S. Jian, �??Study on dispersion properties of photonic crystal fiber by effective-index model,�?? Acta Opt. Sin. 24, (2004), in Chinese (to be published).
  11. J. Broeng, S. E. B. Libori, T. Sondergaard, and A. Bjarklev, �??Analysis of air-guiding photonic bandgap fibers,�?? Opt. Lett. 25, 96�??98 (2000). [CrossRef]
  12. J. C. Knight, J. Broeng, T. A. Birks, and P. St. J. Russell, �??Photonic band gap guidance in optical fibers,�?? Science 282, 1476�??1478 (1998). [CrossRef] [PubMed]
  13. 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]
  14. S. E. B. Libori, J. Broeng, E. Knudsen, A. Bjarklev, and H. Simonsen, �??High-birefringent photonic crystal fibers,�?? in Optical Fiber Communication Conference, Vol. 54 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2001), paper TuM2.
  15. T. P. Hansen, J. Broeng, S. E. B. Libori, E. Knudsen, A. Bjarklev, J. R. Jensen, and H. Simonsen, �??Highly birefringent index-guiding photonic crystal fibers,�?? IEEE Photon. Technol. Lett. 13, 588�??590 (2001). [CrossRef]
  16. J. Broeng, D. Mogilevtsev, S. E. B. Libori, and A. Bjarklev, �??Polarization-preserving holey fibers,�?? in Optical Fiber Communication Conference, Vol. 54 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2001), paper MA1.3.
  17. M. J. Steel and P. M. Osgood, Jr, �??Elliptical-hole photonic crystal fibers,�?? Opt. Lett. 26, 229�??231 (2001). [CrossRef]
  18. M. J. Steel and R. M. Osgood, �??Polarization and dispersive properties of elliptical-hole photonic crystal fibers,�?? J. Lightwave Technol. 19, 495�??503 (2001). [CrossRef]
  19. K. Tajima, J. Zhou, K. Nakajima, and K. Sato, �??Ultra low loss and long length photonic crystal fiber,�?? in Optical Fiber Communication Conference, Vol. 86 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2003), pp. PD1-1�??PD1-3.
  20. R. G. Bin, W. Zhi, L. S. Qin, and S. S. Jian, �??Study on dispersion properties of photonic crystal fiber by effective-index model,�?? accepted by OECC�??2003, Shanghai. (to be published by the Chinese Optical Society).
  21. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding theFlow of Light ( Princeton University, Princeton, N.J., 1995).
  22. S. Guo and S. Albin, �??Simple plane wave implementation for photonic crystal calculations,�?? Opt. Express 11, 167�??175 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-2-167">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-2-167</a>. [CrossRef] [PubMed]
  23. S. G. Johnson and J. D. Joannopoulos, �??Block-iterative frequency-domain methods for Maxwell's equations in a plane wave basis,�?? Opt. Express 8, 173�??190 (2001), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173.">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173.</a> [CrossRef] [PubMed]
  24. 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]
  25. 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]
  26. T. M. Monro, D. J. Richardson, and N. G. R. Broderick, �??Efficient modeling of holey fibers,�?? in Optical Fiber Communication Conference (Optical Society of America, Washington, D.C., 1999), paper FG3, pp. 111�??113.
  27. W. Zhi, R.G. Bin, L. S. Qin, and S. S. Jian, �??Supercell lattice method for photonic crystal fibers,�?? Opt. Express 11, 980�??991 (2003),<a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-980">http://www.opticsexpress.org/abstract.cfm? URI=OPEX-11-9-980</a>. [CrossRef] [PubMed]
  28. D. Mogilevtsev, T. A. Birks, and P. St. J. Russell, �??Localized function method for modeling defect mode in 2-d photonic crystal,�?? J. Lightwave Technol. 17, 2078�??2081 (1999). [CrossRef]
  29. M. Koshiba, �??Full vector analysis of photonic crystal fibers using the finite element method,�?? IEICE Electron. E85-C, 881�??888 (2002).
  30. Z. Zhu and T. G. Brown, �??Full-vectorial finite difference analysis of microstructured optical fibers,�?? Opt. Express 10, 853�??864 (2002), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-17-853">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-17-853</a>. [CrossRef] [PubMed]
  31. F. Fogli, L. Saccomandi, P. Bassi, G. Bellanca, and S. Trillo, �??Full vectorial BPM modeling of indexguiding photonic crystal fibers and couplers,�?? Opt. Express 10, 54�??59 (2002), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-1-54">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-1-54</a>. [CrossRef] [PubMed]
  32. T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. M. de Sterke, and L. C. Botten, �??Multipole method for microstructured optical fibers. I. Formulation,�?? J. Opt. Soc. Am. B 19, 2322�??2330 (2002). [CrossRef]
  33. B. T. Kuhlmey, T. P. White, G. Renversez, D. Maystre, L. C. Botten, C. M. de Sterke, and R. C. McPhedran, �??Multipole method for microstructured optical fibers. II. Implementation and results,�?? J. Opt. Soc. Am. B 19, 2331�??2340 (2002). [CrossRef]
  34. A.W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).
  35. R. Guobin, W. Zhi, L. Shuqin, and J. Shuisheng, �??Mode classification and degeneracy in photonic crystal fiber,�?? Opt. Express 11, 1310�??1321 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-11-1310">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-11-1310</a>. [CrossRef] [PubMed]
  36. W. Zhi, R. G. Bin, L. S. Qin, and S. S. Jian, �??Novel supercell overlapping method for different photonic crystal fibers,�?? J. Lightwave Technol. submitted.
  37. I. S. Gradshtein, and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1994).
  38. I. Kimel and L. R. Elias, �??Relations between Hermite and Laguerre Gaussian modes,�?? IEEE J. Quantum Electron. 29, 2562�??2567 (1993). [CrossRef]
  39. P. R. McIsaac, �??Symmetry-induced modal characteristics of uniform waveguides-I: summary of results,�?? IEEE Trans. Microwave Theory Tech. MTT-23, 421�??429 (1975). [CrossRef]
  40. P. R. McIsaac, �??Symmetry-induced modal characteristics of uniform waveguides- II: theory,�?? IEEE Trans. Microwave Theory Tech. MTT-23, 429�??433 (1975). [CrossRef]
  41. M. J. Steel, T. P. White, C. M. Sterke, R. C. McPhedran, and L. C. Botten, �??Symmetry and degeneracy in microstructured optical fibers,�?? Opt. Lett. 26, 488�??490 (2001). [CrossRef]
  42. R. G. Bin, W. Zhi, L. S. Qin, and S. S. Jian, �??Birefringence properties of elliptical-hole photonic crystal fiber,�?? Chin. J. Lasers 31 (2004), in Chinese (to be published).

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