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Optics Express

  • Editor: Michael Duncan
  • Vol. 11, Iss. 9 — May. 5, 2003
  • pp: 980–991
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Supercell lattice method for photonic crystal fibers

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


Optics Express, Vol. 11, Issue 9, pp. 980-991 (2003)
http://dx.doi.org/10.1364/OE.11.000980


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Abstract

A supercell lattice method, believed to be novel, deduced from the plane-wave expansion method and the localized basis function method, is presented for analyzing photonic crystal fibers (PCFs). The electric field is decomposed by use of Hermite—Gaussian functions, and the dielectric constant of PCFs missing a central air hole is considered as the sum of two virtual different periodic dielectric structures of perfect photonic crystals (PCs). The structures of both virtual PCs are expanded in cosine functions. From the wave equation and the orthonormality of the Hermite—Gaussian functions, the propagation characteristics of the PCFs, such as the mode field distribution, the effective area, and the dispersion property, are obtained. The accuracy of the novel method is demonstrated as we obtain the same results when the dielectric constant is split into two virtual ideal PCs in different ways.

© 2003 Optical Society of America

1. Introduction

The PWM [2

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

4

4. 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]

] solves the full vector wave equation for the magnetic field and, as the name implies, is based on a plane-wave expansion of the field and the position-dependent dielectric constant, which is equal to the Fourier transform. It allows one to calculate the mode field distribution and the photonic band structure of the PCFs and thereby the possible existence, width, and positioning of any PBGs. The plane-wave expansion of the periodic dielectric constant is simple, and the coefficients of all terms are analytical, but it will cost too much time because it requires a large number of terms for expanding the field and the dielectric constant for certain accuracy.

There are many other traditional algorithms for analyzing the electromagnetic field, such as the FEM [14

14. M. Koshiba, “Full vector analysis of photonic crystal fibers using the finite element method,” IEICE Electron, E85-C , 4, 881–888 (2002).

] and the finite-difference time domain (FDTD) [15

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

], which can also be used to investigate the PCFs with very high accuracy. But again the time requirement will be great because of the complicated structure of PCFs. In fact, the LFM model is more practical because it is almost at the same precision level as the FEM or the FDTD [16

16. L. P. Shen, C. L. Xu, and W. P. Huang are preparing a manuscript to be called “Modal characteristics of index-guiding photonic crystal fibers: a comparison between scalar and vector analysis.”

].

2. Supercell lattice

In fact, the split up into PC1 and PC2 could be made in different ways as demonstrated by the two gray lines of gray color Table 1. It is assumed that x, y, z and z’ are selected as desired; then the following identities must be satisfied in order to reconstruct the missing hole region.

z+(nsi2x)=nair2,z+(lnnsi2y)=0.
(1)
Fig. 1. Schematic of the way in which the transverse dielectric structure is constructed. (a) supercell lattice PCF, (b) PC1, (c) PC2.

Regardless of the dielectric constant parameters (x, y, z and z’) of PC1 are selected, the results will not change. For the case of simplicity, they are set as the last two lines in Table 1. Here x and y of PC1 are selected as nsi2 and ln nsi2; consequently the dielectric constant and the logarithm of the background of PC2 are both zero. This choice seems to be incorrect, but we use only the sum of both PC1 and PC2 in this novel method and do not investigate them individually. When the sum of both PC1 and PC2 is fixed, the results will not be affected by the way in which the PCF is split.

Table 1. Structure Parameters of Two Perfect Photonic Crystals (n si and n air are the refractive index of pure silica and air at the operation wavelength)

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)=εPC1+εPC2,lnε=(lnε)PC1+(lnε)PC2,
(2)

where

εPC1=a,b=0P1P1abcos2πaxDcos2πby3D,
(3a)
εPC2=a,b=0P2P2abcos2πaxNDcos2πby3ND,
(3b)
(lnε)PC1=a,b=0P1P1ablncos2πaxDcos2πby3D,
(3c)
(lnε)PC2=a,b=0P2P2ablncos2πaxNDcos2πby3ND,
(3d)

and where (P 1+1) and (P 2+1) are the number of decomposition terms of PC1 and PC2, respectively, and P 1ab, P 2ab, P1ab1n, and P2ab1n present the coefficients that can be obtained through the Fourier transform. To show how to obtain the coefficients, P 1ab is illustrated as an example.

The dielectric constant of PC1 can be expressed in the form of a Fourier transform as

ε(x,y)PC1=m,n=P1P1F(Kmn)cos(kxx)cos(kyy),
(4)

where F(K mn), the 2-D Fourier transform coefficients, can be written as [4

4. 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]

]

F(Kmn)=nair2δ(Kmn)+2(nair2nsi2)fJ1(KmnR)KmnR,
F(0)=nair2+f(nair2nsi2)
(5)

where J 1 is the first-order Bessel function and

Kmn=(m+n)kxi(mn)ky(iistheimaginaryunit),kx=2πD,ky=2π(3D).
(6)

k x and k y depend on the reciprocal lattice vector of the triangular lattice [3

3. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, New York, 1995).

,4

4. 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]

]. f, a fraction parameter, is defined as the ratio of the hole volume to the cell volume:

f=2πR23D2,(R=d2,theholeradius).
(7)

When we transform the subscripts m and n from [-P 1, P 1] to [0, 2P 1], Eq. (4) will become Eq. (8).

ε(x,y)PC1=m,n=02P1F(Kmn)cos2π(m+n2P1)xDcos2π(mn)y3D.
(8)

When we compare Eq. (8) with Eq. (3a), P 1ab can be analytically expressed as the composition of F(K mn) shown as Eq. (9).

P1ab=F(Ka+b+2k2,ab+2k2)+F(Kab+2k2,a+b+2k2)
+F(Ka+b+2k2,ab+2k2)+F(Kab+2k2,a+b+2k2),
(9a)
P1ab=F(Ka+b+2k2,ab+2k2)+F(Kab+2k2,a+b+2k2),whena=0orb=0,
(9b)
P100=F(Kk,k)whena=0andb=0.
(9c)

Other coefficients can also be evaluated analytically in the same way. Figure 2 demonstrates the simulation results along the y=0 axis of the dielectric reconstructions for a PCF with the structure parameters D=2.3 µm, d=0.69 µm, and P 1=50, N=10, P 2=500. Obviously, it can describe the supercell lattice efficiently and accurately. In fact, when P 2=P 1*N, the reconstructed dielectric structure will be the most accurate according to the Fourier transform.

Fig. 2. Cross sections along the y=0 axis of the dielectric structure of PC1, PC2, and PCF.

3. Electric field and eigenvalue system

It is assumed that the PCF is lossless and uniform in the propagation (z) direction, so a main task is to investigate the transverse mode field distribution et (x,y), which can be divided into two polarization components along the x or y axis:

et(x,y)=ex(x,y)x̂+ey(x,y)ŷ.
(10)

ex(x,y)=a,b=0Fεabxψa(x)ψb(y),ey(x,y)=a,b=0Fεabyψa(x)ψb(y),
(11)

where (F+1) is the number of terms retained in this expansion, ψi(s) (i=a,b, s=x,y) are elements of the orthogonal set of Hermite—Gaussian basis functions:

ψi(s)=2iπ14(2i)!ωexp(s22ω2)H2i(sω),
(12)

where H 2i(s/ω) is the 2ith-order Hermite polynomial and ω is the characteristic width of the basis set. As discussed by Monro et al. [11

11. 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]

], only even-order Hermite polynomials are used because the observed fundamental mode profiles in these fibers are even.

When the decompositions in Eq. (11) are substituted into the wave equation, the eigenvalue equation will be obtained and shown as Eq. (13) [10

10. 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]

,11

11. 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]

].

Msεs=βs2εs,(s=x,ycorrespondingtoex,ey),
(13)

Iabcd(1)=+ψa(x)ψb(y)t2[ψc(x)ψd(y)]dxdy,
Iabcd(2)=+εψa(x)ψb(y)ψc(x)ψd(y)dxdy,
Iabcd(3)x=+ψa(x)ψb(y)x[ψc(x)ψd(y)lnεx]dxdy,
Iabcd(3)y=+ψa(x)ψb(y)y[ψc(x)ψd(y)lnεy]dxdy.
(14)

When we substitute the decomposition equation Eq. (3) into Eq. (14), the overlap integrals can be calculated with 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 [18

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

] along with some identities [19

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

]. The fact that all these integrals can be performed analytically is a significant advantage, especially when a great number of terms are needed in the expansions. Then Eq. (14) can be written as

Iabcd(1)=2a+2b+1ϖ2δacδbd+2b(2b+1)2ϖ2δacδb1,d+2a(2a+1)2ϖ2δa1,cδbd
+2(b+1)(2b+1)2ϖ2δacδb,d1+2(a+1)(2a+1)2ϖ2δa,c1δbd,
(15a)
Iabcd(2)=f,g=0P1P1fgIfac(21)xIgbd(21)y+f,g=0P2P2fgINfac(21)xINgbd(21)y,
(15b)
Iabcd(3)x=f,g=0P1P1fglnIfac(32)xIgbd(21)yf,g=0P2P2fglnINfac(32)xINgbd(21)y,
(15c)
Iabcd(3)y=f,g=0P1P1fglnIfac(21)xIgbd(32)yf,g=0P2P2fglnINfac(21)xINgbd(32)y,
(15d)

where

Ii1i2i3(21)s=+cos(2πi1sls)ψi2(s)ψi3(s)ds,
(16a)
INi1i2i3(21)s=+cos(2πi1sNls)ψi2(s)ψi3(s)ds,
(16b)
Ii1i2i3(32)s=+cos(2πi1sls)sψi2(s)sψi3(s)ds,
(16c)
INi1i2i3(32)s=+cos(2πi1sNls)sψi2(s)sψi3(s)ds,(s=x,y)
(16d)

where l x=D and l y=sqrt(3)D. Both parts of the overlap integrals I (2), I (3)x, and I (3)y depend on the decomposition coefficients P 1ab, P 2ab, P1ab1n, and P2ab1n, and this is the only way that the division of PCF into both virtual PC1 and PC2 is reflected in the eigenvalue equation. When the division satisfies Eq. (1) and Table 1, the overlap integrals I (2), I (3)x, and I (3)y will stay unchanged no matter how the PCF is split.

Through the subscript transform as in Eq. (17), M s and εs can be transferred into an (F+1)2×(F+1)2 2-D matrix and a vector with (F+1)2 elements, which are still written as M s and εs for compactness.

MabcdsMs[(a1)F+b,(c1)F+d].
(17)

4. Results

4.1 Modal electric field distribution

Fig. 3. x-polarized mode intensity (|Ex |2) distribution of the triangular lattice circular-hole PCF, with structural parameters D=2.3 µm; d=0.69 µm; and P1 =50, N=10, P2 =500.
Fig. 4. Contour lines of the x-polarized mode intensity (|Ex |2) of the PCF; the structural parameters are the same as in Fig. 3. The dielectric constant profile is superimposed. The intensity contours are spaced by 2 dB from -30 dB.

4.2 Effective modal area Aeff

A eff is defined as Eq. (18) [17

17. A. W. Snyder, Optical Waveguide Theory (Chapman and Hall, New York, 1983).

].

Aeff=[++E(x,y)2dxdy]2++E(x,y)4dxdy.
(18)

When we substitute the decomposition equation Eq. (11) into Eq. (18), and again use the orthogonality of the basis set and the Feldhiem’s identity, A eff can be expressed as

Aeff=2[πωa1,a2=0F1εa1,a22]2a1,a2,a3,a4=0b1,b2,b3,b4=0F1εa1,b1εa2,b2εa3,b3εa4,b4ξ·ζa·ζb,
(19)

where

ξ=(2a1)!(2a2)!(2a3)!(2a4)!(2b1)!(2b2)!(2b3)!(2b4)!,
(20a)
ζs=t1=0min(2s1,2s2)t2=0min(2s3,2s4)(1)κsΓ(κs)t1!(2s1t1)!(2s2t1)!t2!(2s3t2)!(2s4t2)!,
(20b)
κs=s1+s2+s3+s4t1t2,(s=a,b).
(20c)

Γ(κs) is gamma function. Using the evaluated eigenvector from the eigenvalue Eq. (13), A eff can be calculated efficiently and accurately for a given PCF at any wavelength.

Fig. 5. Effective modal area for triangular-lattice circular-hole PCFs with different D and d/D.

We should note that A eff of this paper is twice that shown in the results of Monro et al. in Refs. [10

10. 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]

] and [11

11. 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]

] because of the factor 2 in Eq. (19).

4.3 Dispersion

The total dispersion coefficient D t could be written as the sum of the waveguide dispersion D w and the material dispersion D m, and D m can be calculated by application of the Sellmeyer formula. Because PCF is made of silica, the material dispersion is the same for PCFs with different structural parameters, and the total dispersion coefficient will be dominated by D w. When we take account of the scaling transformation property of Maxwell’s equations, there is a scaling property of the waveguide dispersion expressed as Eq. (21) [24

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

].

Dw(λ;MD,f)=1MDw(λM;D,f).
(21)

Equation (21) tells us that if the scale of PCF is magnified by a factor M, the waveguide dispersion coefficient is reduced to 1/M, and at the same time the corresponding wavelength shifts to Mλ. This equation is useful for analyzing the waveguide dispersion of PCFs.

Fig. 6. Dispersion profile of x-polarized mode of the triangular-lattice circular-hole PCFs. All the solid lines in different colors are the waveguide dispersions; the dotted curves are the material dispersion.

5. Accuracy and efficiency

The novelty of this method is in how it splits the PCF into two virtual perfect photonic crystals. We performed this method in different ways to split the PCF, and it is shown that regardless of the dielectric constant parameters of both virtual photonic crystals selected, the results will not change when the sum of both PC1 and PC2 is fixed. In fact, the results shown if Fig. 6(a) are obtained by selection of the structural parameters z=1/2 and z’=0.

Fig. 7. Modal birefringence Δn=n x-n y.

6. Discussion and conclusion

The supercell lattice method can be used to analyze many other structures of PCFs, such as the elliptical-hole PCF, the square-lattice PCF, and the PBG PCF. For example, when the fraction parameters f and k x in Eq. (4) are divided by the ellipticity ratio η=b/a, where b and a are the lengths of the major and minor axes of the elliptical holes, the SLM can be used to investigate the elliptical hole PCF [28

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

30

30. 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]

] with a straightforward extension of the approach described in this paper. Figure 8 shows the dielectric structure of an elliptical hole PCF with the parameters D=2.3 µm, b/D=0.8, η=3, P1 =30, N=10, P2 =300.

Fig. 8. Simulated dielectric structure of an elliptical-hole PCF.

References and links

1.

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

2.

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

3.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, New York, 1995).

4.

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]

5.

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 fibres and step index fibres,” in Optical Fiber Communication Conference, (Optical Society of America, Washington, D.C., 1998), FG4, pp. 114–116.

6.

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]

7.

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), FG2, pp. 108–110.

8.

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.

9.

R. Guobin, L. Shuqin, W. Zhi, and J. Shuisheng are preparing a manuscript to be called “Study on dispersion properties of photonic crystal fiber by effective-index model” (in Chinese).

10.

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]

11.

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]

12.

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), FG3, 111–113.

13.

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

14.

M. Koshiba, “Full vector analysis of photonic crystal fibers using the finite element method,” IEICE Electron, E85-C , 4, 881–888 (2002).

15.

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

16.

L. P. Shen, C. L. Xu, and W. P. Huang are preparing a manuscript to be called “Modal characteristics of index-guiding photonic crystal fibers: a comparison between scalar and vector analysis.”

17.

A. W. Snyder, Optical Waveguide Theory (Chapman and Hall, New York, 1983).

18.

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

19.

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

20.

W. Zhi, R. Guobin, L. Shuqin, L. Weijun, and J. Shuisheng, “the mode characteristics of the photonic crystal fibers,” to be published by ACTA OPTICA SINICA (in Chinese).

21.

J.M. Dudley and S. Coen, “Numerical simulations and coherence properties of supercontinuum Generation in Photonic Crystal and Tapered Optical Fibers,” IEEE J. Selected Topics in Quantum. Electron. 8, 651–659 (2002). [CrossRef]

22.

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

23.

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]

24.

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

25.

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]

26.

R. G. Bin, W. Zhi, L. S. Qin, and S. S. Jian are preparing a manuscript to be called “Study on dispersion properties of photonic crystal fiber by effective-index model.”

27.

L. P. Shen, W. P. Huang, and S. S. Jian, “Design of photonic crystal fibers for dispersion-related applications” IEEE Photon. Tech. Lett. (to be published).

28.

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

29.

J. Broeng, D. Mogilevtsev, S. E. B. Libori, and A. Bjarklev, “Polarization-preserving holey fibers,” in Optical Fiber Communication Conference (Optical Society of America, Washington, D.C., 2001), MA1.3, pp. 6–7.

30.

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]

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: March 31, 2003
Revised Manuscript: April 14, 2003
Published: May 5, 2003

Citation
Wang Zhi, Ren Guobin, Lou Shuqin, and Jian Shuisheng, "Supercell lattice method for photonic crystal fibers," Opt. Express 11, 980-991 (2003)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-9-980


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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,�?? Optical Fiber Technol. 5, 305-330 (1999). [CrossRef]
  2. S. E. Barkou, J. Broeng, and A. Bjarklev, �??Dispersion properties of photonic bandgap guiding fibers,�?? in Optical Fiber Communication Conference (Optical Society of America, Washington, D.C., 1998), FG5, pp. 117-119.
  3. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, New York, 1995).
  4. 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]
  5. 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 fibres and step index fibres,�?? in Optical Fiber Communication Conference, (Optical Society of America, Washington, D.C., 1998), FG4, pp. 114-116.
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