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Optics Express

  • Editor: Michael Duncan
  • Vol. 10, Iss. 17 — Aug. 26, 2002
  • pp: 853–864
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Full-vectorial finite-difference analysis of microstructured optical fibers

Zhaoming Zhu and Thomas G. Brown  »View Author Affiliations


Optics Express, Vol. 10, Issue 17, pp. 853-864 (2002)
http://dx.doi.org/10.1364/OE.10.000853


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Abstract

In this paper we present a full-vectorial finite-difference analysis of microstructured optical fibers. A new mode solver is described which uses Yee’s 2-D mesh and an index averaging technique. The modal characteristics are calculated for both conventional optical fibers and microstructured optical fibers. Comparison with previous finite difference mode solvers and other numerical methods is made and excellent agreement is achieved.

© 2002 Optical Society of America

1. Introduction

Among optical waveguides, only a few structures such as slab waveguides and step-index optical fibers can be solved analytically. For more complex waveguide structures, rigorous numerical methods have been proposed including both finite element methods and finite difference (FD) methods [1

1. K. S. Chiang, “Review of numerical and approximate methods for the modal analysis of general optical dielectric waveguides,” Opt. Quantum Electron. 26, s113–s134 (1994). [CrossRef]

,2

2. C. Vassallo, “1993-1995 optical mode solvers,” Opt. Quantum Electron. 29, 95–114 (1997). [CrossRef]

].

In the FDFD mode solvers, two discretization schemes have been used. One is that first proposed by Stern [22

22. M. S. Stern, “Semivectorial polarized finite difference method for optical waveguides with arbitrary index profiles,” IEE Proc. J. Optoelectron. 135, 56–63 (1988). [CrossRef]

] in which possible discontinuities lie between two adjacent mesh grids and every grid point corresponds to a unique refractive index. The wave equation in terms of transverse electric field E t (or magnetic field H t )

(t2+k02εr)Et+t(εr1tεr·Et)=β2Et

or

(t2+k02εr)Ht+εr1tεr×(t×Ht)=β2Ht
(1)

where k 0=2π/λ is the wave number in free space, ε r is the waveguide dielectric constant, and β is the propagation constant, is directly discretized by finite difference [23

23. W. P. Huang and C. L. Xu, “Simulation of three-dimensional optical waveguides by a full-vector beam propagation method,” IEEE J. Quantum Electron. 29, 2639–2649 (1993). [CrossRef]

]. This discretization scheme is usually used in the context of the beam propagation method [23

23. W. P. Huang and C. L. Xu, “Simulation of three-dimensional optical waveguides by a full-vector beam propagation method,” IEEE J. Quantum Electron. 29, 2639–2649 (1993). [CrossRef]

,24

24. W. P. Huang, C. L. Xu, S. T. Chu, and S. K. Chaudhuri, “The finite-difference vector beam propagation method. Analysis and Assessment,” J. Lightwave Technol. 10, 295–305 (1992). [CrossRef]

], which can also be used as a mode solver. Another discretization scheme is that first proposed by Bierwirth et al. [25

25. K. Bierwirth, N. Schulz, and F. Arndt, “Finite-difference analysis of rectangular dielectric waveguide structures,” IEEE Trans. Microwave Theory Tech. 34, 1104–1113 (1986). [CrossRef]

]. In this scheme, possible discontinuities lie on the mesh grids, so that any grid point can be associated with up to four different refractive indices. The transverse magnetic components are usually used in deriving the discretization matrix [26

26. H. Dong, A. Chronopoulos, J. Zou, and A. Gopinath, “Vectorial integrated finite-difference analysis of dielectric waveguides,” J. Lightwave Technol. 11, 1559–1563 (1993). [CrossRef]

,27

27. P. Lüsse, P. Stuwe, J. Schüle, and H. G. Unger, “Analysis of vectorial mode fields in optical waveguides by a new finite difference method,” J. Lightwave Technol. 12, 487–493 (1994). [CrossRef]

].

In this paper, we present a full-vector finite difference mode solver that is based on discretization scheme first proposed by Yee [28

28. K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propagat. 14, 302–307 (1966). [CrossRef]

]. Yee’s mesh is widely used in the FDTD analysis [29

29. K. S. Kunz and R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics, (CRC, Boca Raton, 1993).

]. Here we use Yee’s two-dimensional mesh in our frequency domain mode solver for complex optical waveguides. In Section 2, the formulation of the method is described in detail. In order to improve the staircase approximation for curved interface, an index averaging technique is used for the cells across interfaces. Numerical results for both conventional optical fibers and MOFs are presented in Section 3. The comparison between our results and those by other methods are also made. Section 4 concludes the paper.

Fig. 1. (a) Yee’s 2-D mesh; (b) Mesh cells across a curved interface.

2. Formulation

The Yee’s 2-D mesh is illustrated in Fig. 1(a). The mesh grids for electric fields lie on possible dielectric discontinuities. Since all the transverse field components are tangential to the unit cell boundaries, the continuity conditions are automatically satisfied.

Assume the fields have dependence of exp[i(βz-ωt)]. From Maxwell’s curl equations (∇×E = -∂B/∂t, ∇×H = ∂D/∂t), after scaling E by the free space impedance Z0=μ0ε0, we have

ik0Hx=EzyEy,
(2a)
ik0Hy=ExEzx,
(2b)
ik0Hz=EyxExy,
(2c)
ik0εrEx=HzyHy,
(3a)
ik0εrEy=HxHzx,
(3b)
ik0εrEz=HyxHxy.
(3c)

Discretizing Eqs. (2) and (3) yields

ik0Hx(j,l)=[Ez(j,l+1)Ez(j,l)]ΔyEy(j,l),
(4a)
ik0Hy(j,l)=Ex(j,l)[Ez(j+1,l)Ez(j,l)]Δx,
(4b)
ik0Hz(j,l)=[Ey(j+1,l)Ey(j,l)]Δx[Ex(j,l+1)Ex(j,l)]Δy,
(4c)
ik0εrx(j,l)Ex(j,l)=[Hz(j,l)Hz(j,l1)]ΔyHy(j,l),
(5a)
ik0εry(j,l)Ey(j,l)=Hx(j,l)[Hz(j,l)Hz(j1,l)]Δx,
(5b)
ik0εrz(j,l)Ez(j,l)=[Hy(j,l)Hy(j1,l)]Δx[Hx(j,l)Hx(j,l1)]Δy,
(5c)

where

εrx(j,l)=[εr(j,l)+εr(j,l1)]2,
(6a)
εry(j,l)=[εr(j,l)+εr(j1,l)]2,
(6b)
εrz(j,l)=[εr(j,l)+εr(j1,l1)+εr(j,l1)+εr(j1,l)]4.
(6c)

In writing Eqs. (6a)~(6c), we have approximated the refractive indices by averaging the refractive indices of adjacent cells.

Eqs. (4) and (5) can be written in matrix form as

ik0[HxHyHz]=[0IUyI0UxUyUx0][ExEyEz],
(7)
ik0[εrx000εry000εrz][ExEyEz]=[0IVyI0VxVyVx0][HxHyHz],
(8)

where I is a square identity matrix, ε rx , ε ry and ε rz are diagonal matrices determined by Eqs. (6). U x , U y , V x and V y are square matrices and depend on the boundary conditions of the rectangular computation window. For example, when the zero-value boundary condition is chosen for the computation window edges, we have

Ux=1Δx[1111111],Uy=1Δy[111111],
Vx=1Δx[11111111],Vy=1Δy[111111].
(9)

From Eqs. (7) and (8), after some algebra, we obtain an eigenvalue equation in terms of transverse electric fields:

P[ExEy]=[PxxPxyPyxPyy][ExEy]=β2[ExEy],
(10)

where

Pxx=k02Uxεrz1VyVxUy+(k02I+Uxεrz1Vx)(εrx+k02VyUy),
Pyy=k02Uyεrz1VxVyUx+(k02I+Uyεrz1Vy)(εry+k02VxUx),
Pxy=Uxεrz1Vy(εry+k02VxUx)k02(k02I+Uxεrz1Vx)VyUx,
Pyx=Uyεrz1Vx(εrx+k02VyUy)k02(k02I+Uyεrz1Vy)VxUy.
(11)

Alternatively, we can obtain an eigenvalue equation in terms of transverse magnetic fields:

Q[HxHy]=[QxxQxyQyxQyy][HxHy]=β2[HxHy],
(12)

with

Qxx=k02VxUyUxεrz1Vy+(εry+k02VxUx)(k02I+Uyεrz1Vy),
Qyy=k02VyUxUyεrz1Vx+(εrx+k02VyUy)(k02I+Uxεrz1Vx),
Qxy=(εry+k02VxUx)Uyεrz1Vx+k02VxUy(k02I+Uxεrz1Vx),
Qyx=(εrx+k02VyUy)Uxεrz1Vy+k02VyUx(k02I+Uyεrz1Vy).
(13)

Under proper boundary conditions (zero value boundary, for example), one can get V x =-UxT and V y =-UyT , where the superscript T denotes transpose operation. Then it is clear from Eqs. (11) and (13) that

Qxx=PyyT,Qyy=PxxT,Qxy=PxyT,Qyx=PyxT.
(14)

Other FD mode solvers [23

23. W. P. Huang and C. L. Xu, “Simulation of three-dimensional optical waveguides by a full-vector beam propagation method,” IEEE J. Quantum Electron. 29, 2639–2649 (1993). [CrossRef]

,27

27. P. Lüsse, P. Stuwe, J. Schüle, and H. G. Unger, “Analysis of vectorial mode fields in optical waveguides by a new finite difference method,” J. Lightwave Technol. 12, 487–493 (1994). [CrossRef]

] obtain eigenvalue equations similar to Eq. (10) or (12

23. W. P. Huang and C. L. Xu, “Simulation of three-dimensional optical waveguides by a full-vector beam propagation method,” IEEE J. Quantum Electron. 29, 2639–2649 (1993). [CrossRef]

), but the coefficient matrices P and Q are different. One advantage of the present approach is that, once E t or H t is obtained, all other field components are readily obtained through Eqs. (7) and (8).

3. Numerical Results

In this section, we first use the proposed FD mode solver to calculate the fundamental effective index of a step-index optical fiber and compare them with other FD mode solvers [23

23. W. P. Huang and C. L. Xu, “Simulation of three-dimensional optical waveguides by a full-vector beam propagation method,” IEEE J. Quantum Electron. 29, 2639–2649 (1993). [CrossRef]

,27

27. P. Lüsse, P. Stuwe, J. Schüle, and H. G. Unger, “Analysis of vectorial mode fields in optical waveguides by a new finite difference method,” J. Lightwave Technol. 12, 487–493 (1994). [CrossRef]

] and the analytical solution. Then we apply the mode solver for the characterization of MOFs, including air-hole assisted optical fibers, and, finally holey fibers with both circular air hole and elliptical air holes. Comparisons with other approaches are also given.

3.1. Step-Index Fiber

The step-index circular fiber we considered has core diameter of 6 μm, core refractive index of 1.45 at wavelength λ=1.5 μm, and air cladding with unity refractive index. The analytical solution gives the fundamental mode index neff = 1.438604. We choose such a high index difference fiber in order to illustrate the applicability of the FD mode solver in modeling waveguides with curved high-index difference interfaces. In Fig.2 we show the relative error of the calculated fundamental mode index neff from the three FD solvers (both with and without index averaging) when the number of grids along the x-axis is varied. As can be clearly seen, the index averaging technique can significantly stabilize and improve the accuracy for the three FD mode solvers. In Table 1, we list the calculated results from different approaches. Uniform meshes are used in all the numerical calculations.

We also carried out calculations on fibers with different parameters. All the comparisons reveal that the index averaging technique can significantly improve modeling accuracy. The proposed FD mode solver and the mode solver by Huang et al. [23

23. W. P. Huang and C. L. Xu, “Simulation of three-dimensional optical waveguides by a full-vector beam propagation method,” IEEE J. Quantum Electron. 29, 2639–2649 (1993). [CrossRef]

] both give better results than that from the approach by Lüsse et al. [27

27. P. Lüsse, P. Stuwe, J. Schüle, and H. G. Unger, “Analysis of vectorial mode fields in optical waveguides by a new finite difference method,” J. Lightwave Technol. 12, 487–493 (1994). [CrossRef]

] when few grids are used. Compared with that by Huang et al., the current mode solver appears to get better results even with very sparse grids.

Table 1. Calculated fundamental mode indices of a step-index fiber from different FD mode solvers. The fiber parameters are the same as in Fig. 2. The analytical solution for the fundamental mode index is 1.438604.

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Fig. 2. Relative error in the calculated fundamental mode index neff of a step-index circular optical fiber. The fiber has a core diameter 6 μm, a core refractive index of 1.45 at wavelength 1.5 μm, and air cladding with unity refractive index. The calculation window is chosen to be the first quadrant of the fiber cross section with a computation window size of 6μm by 6 μm. The left boundary is magnetic wall, the bottom is electric wall; all others are zero-value boundaries.

3.2. Microstructured Optical Fiber (MOF)

With the demonstrated validity of the FD mode solvers in modeling conventional optical fiber, in this sub-section we apply them in the characterization of two types of MOFs: air-hole assisted optical fibers and holey optical fibers.

Table 2. Calculated fundamental mode indices for air-hole-assisted optical fiber shown in Fig. 3(a). Core index 1.45, silica cladding index 1.42, r 0=2μm, r a=2μm, Λ=5μm, wavelength=1.5 μm, first quadrant window size 8μm by 8μm, C1: magnetic wall, C2~C4: electric wall. The multipole analysis [32] gives neff =1.4353607.

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Fig. 3. (a) Schematic of an air-hole assisted optical fiber; (b) Relative errors of calculated fundamental mode index from different mode solvers when the number of grids along the x-axis is varied (fiber parameters are the same as in Table 2); (c) Magnetic field plots of the fundamental mode (y-polarization); (d) Calculated GVD curves (material dispersion not considered).
Fig. 4. (a) Calculation window of the holey fiber; (b) Comparison between different FD mode solvers (Λ=2.3 μm, r a=0.5 μm, silica refractive index of 1.45 is assumed at λ=1.5 μm, calculation window of 3Λ by 3Λ is the first quadrant bounded by C1~C4. C1: magnetic wall, C2~C4: electric wall); (c) Field plots of the y-polarized fundamental mode; (d) Calculated effective modal index β/k 0, nFSM and GVD (Silica refractive index of 1.45 is assumed at all wavelengths).

Table 3. Calculated fundamental mode indices for the holey fiber shown in Fig. 4(a) from different methods. The fiber parameters are same as in Fig. 4(b). The number of grids along both x-axis and y-axis is 120.

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Fig. 5. Convergence of modal birefringence of the fundamental modes in (a) air-hole assisted optical fiber and in (b) holey optical fiber. The parameters for the two fibers are listed in Table 2 and Fig. 4(b), respectively.

Finally, we apply this method to the calculation of the fundamental mode and FSM indices, and the GVD of a holey fiber with elliptical air-holes. The structure analyzed is similar to that of Fig. 4(a), but with elliptical holes (eccentricity e=0.375) in the place of the circular holes. Steel et al. [37

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

] analyzed these structures using the plane wave expansion method, and identified possible applications. Fig. 6 shows the numerical results for the two lowest order modes. As expected, the degeneracy of the fundamental mode is lifted, with a predicted birefringence of B=6.5×10-4 at λ=1.5 μm. The indices of the x-polarized and y-polarized FSM are also calculated and show much larger birefringence that the fundamental modes. This has important implications for the design of polarization-maintaining MOFs because it can affect both the single-mode cutoff condition and the differential loss (e.g. bending loss or leakage). As before, the material dispersion is not considered in the calculations.

Fig. 6. (a) Calculated effective index and GVD of the x- and y-polarized fundamental modes for a holey fiber with elliptical air holes. The air holes have their major axis along the y-direction. The semimajor axis is 0.8 μm; the semiminor axis is 0.5 μm. Other calculation parameters are same as in Fig. 4. The effective indices of x-polarized and y-polarized FSM are also shown. (b) Contour plots of major magnetic field components for x-polarized (left) and y-polarized (right) fundamental modes at wavelength of 1.5 μm.

4. Discussion and Conclusion

Acknowledgement

This work is supported in part by the Laboratory for Laser Energetics of the University of Rochester and by the National Science Foundation, grant number ECS-9816251.

References and Links

1.

K. S. Chiang, “Review of numerical and approximate methods for the modal analysis of general optical dielectric waveguides,” Opt. Quantum Electron. 26, s113–s134 (1994). [CrossRef]

2.

C. Vassallo, “1993-1995 optical mode solvers,” Opt. Quantum Electron. 29, 95–114 (1997). [CrossRef]

3.

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

4.

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

5.

J. C. Knight, J. Arriaga, T. A. Birks, A. Ortigosa-Blanch, W. J. Wadsworth, and P. St. J. Russell, “Anomalous dispersion in photonic crystal fiber,” IEEE Photon. Technol. Lett. 12, 807–809 (2000). [CrossRef]

6.

A. Ferrando, E. Silvestre, J. J. Miret, and P. Andres, “Nearly zero ultraflattened dispersion in photonic crystal fibers,” Opt. Lett. 25, 790–792 (2000). [CrossRef]

7.

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]

8.

N. G. R. Broderick, T. M. Monro, P. J. Bennett, and D. J. Richardson, “Nonlinearity in holey optical fibers: measurement and future opportunities,” Opt. Lett. 24, 1395–1397 (1999). [CrossRef]

9.

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]

10.

J. K. Ranka, R. S. Windeler, and A. J. Stentz, “Visible continuum generation in air silica microstructure optical fibers with anomalous dispersion at 800nm,” Opt. Lett. 25, 25–27 (2000). [CrossRef]

11.

B. J. Eggleton, C. Kerbage, P. Westbrook, R. S. Windeler, and A. Hale, “Microstructured optical fiber devices,” Opt. Express 9, 698–713 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-698 [CrossRef] [PubMed]

12.

J. C. Knight, T. A. Birks, P. St. J. Russell, and J. P. de Sandro, “Properties of photonic crystal fiber and the effective index model,” J. Opt. Soc. Am. A 15, 748–752 (1998). [CrossRef]

13.

A. Ferrando, E. Silvestre, J. J. Miret, P. Andres, and M. V. Andres, “Full-vector analysis of a realistic photonic crystal fiber,” Opt. Lett. 24, 276–278 (1999). [CrossRef]

14.

A. Ferrando, E. Silvestre, J. J. Miret, and P. Andres, “Vector description of higher-order modes in photonic crystal fibers,” J. Opt. Soc. Am. A 17, 1333–1340 (2000). [CrossRef]

15.

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

16.

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]

17.

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]

18.

F. Brechet, J. Marcou, D. Pagnoux, and P. Roy, “Complete analysis of the characteristics of propagation into photonic crystal fibers by the finite element method,” Opt. Fiber Technol. 6, 181–191 (2000). [CrossRef]

19.

G. E. Town and J. T. Lizer, “Tapered holey fibers for spot size and numerical-aperture conversion,” Opt. Lett. 26, 1042–1044 (2001). [CrossRef]

20.

M. Qiu, “Analysis of guided modes in photonic crystal fibers using the finite-difference time-domain method,” Microwave Opt. Technol. Lett. 30, 327–330 (2001). [CrossRef]

21.

T. P. White, R. C. McPhedran, C. M. de Sterke, L. C. Botten, and M. J. Steel, “Confinement losses in microstructured optical fibers,” Opt. Lett. 26, 1660–1662 (2001). [CrossRef]

22.

M. S. Stern, “Semivectorial polarized finite difference method for optical waveguides with arbitrary index profiles,” IEE Proc. J. Optoelectron. 135, 56–63 (1988). [CrossRef]

23.

W. P. Huang and C. L. Xu, “Simulation of three-dimensional optical waveguides by a full-vector beam propagation method,” IEEE J. Quantum Electron. 29, 2639–2649 (1993). [CrossRef]

24.

W. P. Huang, C. L. Xu, S. T. Chu, and S. K. Chaudhuri, “The finite-difference vector beam propagation method. Analysis and Assessment,” J. Lightwave Technol. 10, 295–305 (1992). [CrossRef]

25.

K. Bierwirth, N. Schulz, and F. Arndt, “Finite-difference analysis of rectangular dielectric waveguide structures,” IEEE Trans. Microwave Theory Tech. 34, 1104–1113 (1986). [CrossRef]

26.

H. Dong, A. Chronopoulos, J. Zou, and A. Gopinath, “Vectorial integrated finite-difference analysis of dielectric waveguides,” J. Lightwave Technol. 11, 1559–1563 (1993). [CrossRef]

27.

P. Lüsse, P. Stuwe, J. Schüle, and H. G. Unger, “Analysis of vectorial mode fields in optical waveguides by a new finite difference method,” J. Lightwave Technol. 12, 487–493 (1994). [CrossRef]

28.

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propagat. 14, 302–307 (1966). [CrossRef]

29.

K. S. Kunz and R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics, (CRC, Boca Raton, 1993).

30.

S. Dey and R. Mittra, “A conformal finite-difference time-domain technique for modeling cylindrical dielectric resonators,” IEEE Trans. Microwave Theory Tech. 47, 1717–1739 (1999).

31.

T. Hasegawa, E. Sasaoka, M. Onishi, M. Nishimura, Y. Tsuji, and M. Koshiba, “Hole-assisted lightguide fiber for large anomalous dispersion and low optical loss,” Opt. Express 9, 681–686 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-681 [CrossRef] [PubMed]

32.

Z. Zhu and T. G. Brown, “Multipole analysis of hole-assisted optical fibers”, Opt. Commun. 206, 333–339 (2002). [CrossRef]

33.

M. Midrio, M. P. Singh, and C. G. Someda, “The space filling mode of holey fibers: an analytical vectorial solution,” J. Lightwave Technol. 18, 1031–1037 (2000). [CrossRef]

34.

Z. Zhu and T. G. Brown, “Analysis of the space filling modes of photonic crystal fibers,” Opt. Express 8, 547–554 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-10-547 [CrossRef] [PubMed]

35.

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

36.

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

37.

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

OCIS Codes
(060.2270) Fiber optics and optical communications : Fiber characterization
(060.2400) Fiber optics and optical communications : Fiber properties
(230.3990) Optical devices : Micro-optical devices

ToC Category:
Research Papers

History
Original Manuscript: June 26, 2002
Revised Manuscript: July 29, 2002
Published: August 26, 2002

Citation
Zhaoming Zhu and Thomas Brown, "Full-vectorial finite-difference analysis of microstructured optical fibers," Opt. Express 10, 853-864 (2002)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-10-17-853


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References

  1. K. S. Chiang, �??Review of numerical and approximate methods for the modal analysis of general optical dielectric waveguides,�?? Opt. Quantum Electron. 26, s113-s134 (1994). [CrossRef]
  2. C. Vassallo, �??1993-1995 optical mode solvers,�?? Opt. Quantum Electron. 29, 95-114 (1997). [CrossRef]
  3. J. C. Knight, T. A. Birks, P. St. J. Russell, and D. M. Atkin, �??All-silica single mode optical fiber with photonic crystal cladding,�?? Opt. Lett. 21, 1547-1549 (1996). [CrossRef] [PubMed]
  4. T. A. Birks, J. C. Knight, and P. St. J. Russell, �??Endlessly single-mode photonic crystal fiber,�?? Opt. Lett. 22, 961-963 (1997). [CrossRef] [PubMed]
  5. J. C. Knight, J. Arriaga, T. A. Birks, A. Ortigosa-Blanch, W. J. Wadsworth, and P. St. J. Russell, �??Anomalous dispersion in photonic crystal fiber,�?? IEEE Photon. Technol. Lett. 12, 807-809 (2000). [CrossRef]
  6. A. Ferrando, E. Silvestre, J. J. Miret, and P. Andres, �??Nearly zero ultraflattened dispersion in photonic crystal fibers,�?? Opt. Lett. 25, 790-792 (2000). [CrossRef]
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