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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 13 — Jun. 20, 2011
  • pp: 12582–12593
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A vector boundary matching technique for efficient and accurate determination of photonic bandgaps in photonic bandgap fibers

Liang Dong  »View Author Affiliations


Optics Express, Vol. 19, Issue 13, pp. 12582-12593 (2011)
http://dx.doi.org/10.1364/OE.19.012582


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Abstract

A vector boundary matching technique has been proposed and demonstrated for finding photonic bandgaps in photonic bandgap fibers with circular nodes. Much improved accuracy, comparing to earlier works, comes mostly from using more accurate cell boundaries for each mode at the upper and lower edges of the band of modes. It is recognized that the unit cell boundary used for finding each mode at band edges of the 2D cladding lattice is not only dependent on whether it is a mode at upper or lower band edge, but also on the azimuthal mode number and lattice arrangements. Unit cell boundaries for these modes are determined by mode symmetries which are governed by the azimuthal mode number as well as lattice arrangement due to mostly geometrical constrains. Unit cell boundaries are determined for modes at both upper and lower edges of bands of modes dominated by m = 1 and m = 2 terms in their longitudinal field Fourier-Bessel expansion series, equivalent to LP0s and LP1s modes in the approximate LP mode representations, for hexagonal lattice to illustrate the technique. The novel technique is also implemented in vector form and incorporates a transfer matrix algorithm for the consideration of nodes with arbitrary refractive index profiles. Both are desired new capabilities for further explorations of advanced new designs of photonic bandgap fibers.

© 2011 OSA

1. Introduction

Plane-wave decomposition technique has been proven to be a very useful tool for solving this infinite lattice problem [7

7. J. M. Stone, G. J. Pearce, F. Luan, T. A. Birks, J. C. Knight, A. K. George, and D. M. Bird, “An improved photonic bandgap fiber based on an array of rings,” Opt. Express 14(13), 6291–6296 (2006). [CrossRef] [PubMed]

], with infinite lattice being already implies in its algorithm and implementation. It is, however, very inefficient computationally for the lattices of concern in photonic bandgap fibers. The small physical size of the high index nodes and their steep refractive index boundaries require a broad spatial spectral coverage including very high spatial frequencies (see Fig. 1
Fig. 1 Illustration of a photonic lattice.
for an illustration of a lattice). Recently a new technique using a curvilinear space has been proposed to ease this problem somewhat [8

8. G. J. Pearce, T. D. Hedley, and D. M. Bird, “Adaptive curvilinear coordinates in a plane-wave solution of Maxwell’s equations in photonic crystals,” Phys. Rev. B 71(19), 195108 (2005). [CrossRef]

]. Additionally, photonic bandgaps of lattices with nodes of complex refractive index profiles have not been demonstrated so far. There are increasing needs for considering complex refractive index distributions of the nodes. All practical fibers have smoothed-out refractive index distributions, instead of step index profiles. More advanced designs also require the optimizations of refractive index profiles of the nodes. Some recent work on photonic lattice of rings has also pointed in this direction [7

7. J. M. Stone, G. J. Pearce, F. Luan, T. A. Birks, J. C. Knight, A. K. George, and D. M. Bird, “An improved photonic bandgap fiber based on an array of rings,” Opt. Express 14(13), 6291–6296 (2006). [CrossRef] [PubMed]

].

Mode solvers based on multipole algorithm are very accurate and highly efficient for solving discrete circular boundary problems [9

9. 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(10), 2322–2330 (2002). [CrossRef]

,10

10. 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(10), 2331–2340 (2002). [CrossRef]

]. This is largely due to the fact that the Fourier-Bessel expansion series represents very well the scattered fields around a circular scatter. The algorithm has been very successful in determining leakage losses of modes in finite cladding structures with circular scatters [9

9. 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(10), 2322–2330 (2002). [CrossRef]

,10

10. 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(10), 2331–2340 (2002). [CrossRef]

]. The plane-wave decomposition technique cannot find leakage loss due to the implied infinity in the cladding lattice. In theory, multipole algorithm with a very large number of nodes in the cladding can be used to approximate the case of infinite cladding. In practice, this is very costly, because the computational efficiency goes down very quickly with an increased number of nodes. A propagation model based on Bloch theorem and lattice sum has been proposed to find photonic bandgaps in photonic bandgap fibers [11

11. T. White, R. McPhedran, L. Botten, G. Smith, and C. M. de Sterke, “Calculations of air-guided modes in photonic crystal fibers using the multipole method,” Opt. Express 9(13), 721–732 (2001). [CrossRef] [PubMed]

]. It is, however, fundamentally ill-suited for this eigenvalue problem.

Recently, a very interesting approach has been proposed for a quick determination of photonic bandgaps in a 2D photonic lattice [12

12. T. A. Birks, G. J. Pearce, and D. M. Bird, “Approximate band structure calculation for photonic bandgap fibres,” Opt. Express 14(20), 9483–9490 (2006). [CrossRef] [PubMed]

]. This technique is based on recognizing that only the modes at the edges of the photonic bandgaps need to be determined to know where the bands are. Furthermore, the authors recognized that the edge modes have some well-defined characteristics, namely the transverse fields of the modes determining the upper edge of bands of modes are likely in-phase among neighboring nodes and the fields of the modes determining lower edge of bands of modes are likely anti-phase among the neighboring nodes. The original implementation of this idea used a circular boundary for each unit cell, a very rough approximation. This can introduce significant errors especially in cases of strong couplings among neighboring nodes.

2. Fields around a circular node with a complex refractive index profile

The complex refractive index of each circular node is represented by N concentric rings (not necessarily equally spaced) of uniform indexes. Using the Fourier-Bessel series for field representations, component of the electric and magnetic fields along the propagation direction z in layer p can be written as [9

9. 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(10), 2322–2330 (2002). [CrossRef]

]
ezp=m=+[ampJm(kpr)+bmpHm1(kpr)]eimθ
(1)
Hzp=m=+[CmpJm(kpr)+DmpHm1(kpr)]eimθ
(2)
where a, b, C, D are field coefficients to be determined for each layer; Jm is the Bessel function of the first kind; H1 m is the Hankel function of the first kind, r is radial coordination centered at the node center in the transverse plane; and θ is the corresponding azimuthal coordination. The first term of each field representation in Eq. (1) and Eq. (2) can be viewed as incoming wave originated from outside of the node and the second term as outgoing scattered wave from the node as described in [9]. We also have
kp2=k2np2β2
(3)
where k is vacuum wave vector; np is refractive index of layer p; and β is the propagation constant of the mode. It is worth noting that we have allowed β>knp in Eq. (3), resulting in imaginary kpr for Jm and H1 m in Eq. (1) and (2) in this case. The fields in Eq. (1) and (2) are still valid in this case considering the transformation Km(-iz) = (π/2)im + 1H1 m(z) and Im(-iz) = i˗mJm(z) where Im and Km are modified Bessel function of the first and second kind respectively. H in Eq. (2) is a normalized magnetic field component for easier representations. It is connected to regular magnetic field h by

H=μ0ε0h
(4)

Other field components can be determined from the z field components [13

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

]. Boundary conditions require the continuity of all tangential field components, i.e. z and θ components, at each ring boundary. This produces four equations to be satisfied at each boundary by z and θ components of electric and magnetic fields. This gives us the following relationship deriving all field coefficients of one layer from an adjacent layer.

[amp+1bmp+1Cmp+1Dmp+1]=Mmp[ampbmpCmpDmp]
(5)

The innermost circle is denoted by p = 0 and the outmost region is denoted by p = N in our denominations. The details of the transfer matrix are given in the appendix. The first thing to note is that there is no coupling between each set of m-field coefficients. This is due to the circular boundary between layers which does not couple between the m terms. The second thing to note is that there is generally some coupling between electric and magnetic fields at each boundary. It is then easy to see that we have the following relationship between fields outside the node and fields at the center of the node
[amNbmNCmNDmN]=Mm[am0bm0Cm0Dm0]
(6)
where

Mm=p=0N1Mmp
(7)

Bearing in mind that scattering fields are only generated at the discontinuous boundaries, there is no outgoing wave over the innermost circle, i.e. bm = Dm = 0 for p = 0. It can now be understood that all the four field coefficients for the outmost region, i.e. am, bm, Cm and Dm for p = N, can be determined by the only two non-zero components am and Cm of the innermost circle for each m. Since our following analysis is going to mostly involve the field in the outmost region, we will write them down.

ezN=m=+[amNJm(kpr)+bmNHm1(kpr)]eimθ
(8)
HzN=m=+[CmNJm(kpr)+DmNHm1(kpr)]eimθ
(9)

It needs to be emphasized that there are only two independent coefficients in these field representations in Eq. (8) and Eq. (9). Other field components can be derived from these. The transverse field can then be derived for the region around the node

exN=erNcos(θ)eθNsin(θ)
(10)
eyN=erNsin(θ)+eθNcos(θ)
(11)

Corresponding magnetic fields can be similarly written.

To implement the boundary matching technique for the edge modes, symmetries of the mode field distributions at the upper and lower edges have to be determined first for each band of modes. This can be done using either a mode solver or alternatively by a mode symmetry analysis. It is worth noting mode symmetry analysis can be complicated, especially considering that fields over adjacent nodes can be dominated by different m-terms in the Fourier-Bessel expansion series and the specific geometric constraints of the lattice. A Multipole mode solver is heavily relied on in this work to be sure of the mode symmetry. It needs to be noted that these edge mode symmetries need to be determined only once for each band of modes for each lattice arrangement. This is because these edge mode symmetries are only dependent on the dominant azimuthal mode number and lattice symmetry. Therefore, as it will be discussed in more details later, each band needs to be determined separately as the mode symmetry is expected to be dependent on the dominant azimuthal mode number.

3. Vector boundary matching technique for the first band of modes

In this section, the upper and lower edge modes of the first band of modes, which are derivatives of the fundamental mode of the node, are used to illustrate the boundary matching technique. Although hexagonal lattice is used throughout this work, this technique can be similarly implemented for any other lattice arrangements.

Symmetries of the edge modes of the first band of modes is characterized by the dominance of m = 1 terms in ez and hz fields. Unlike in the scalar mode approximations, other m terms are non-zero, but smaller in comparison. The z components are dominated by two anti-phase symmetric lobes over the node (see Fig. 2
Fig. 2 Amplitudes of ez (a), hz (b), and et (c) for a lattice with node V = 0.9 and d/Λ = 0.19. This is a mode located at upper edge of the first band.
and Fig. 3
Fig. 3 Amplitudes of ez (a), hz (b), and et (c) for a lattice with node V = 0.8 and d/Λ = 0.17. This is a mode located at lower edge of the first band.
). This gives a transverse field over the node resembles the quasi-Gaussian mode with which we are familiar. The orientations of each z field components need to be such so that the transverse fields are in-phase between adjacent nodes for modes determining the upper edge of this band of modes. This can be easily achieved for this mode as shown in Fig. 2.

The mode in Fig. 2 is determined by a multipole mode solver with a finite number of nodes. The field amplitudes of the ez, hz and transverse electric field et are given in Fig. 2 for a hexagonal lattice with d/Λ = 0.19 where d is node diameter and Λ is node spacing. The node normalized frequency V is 0.9 in this case, where normalized frequency V is defined as for conventional fibers. In this case, this upper edge mode has an identical distribution over each node. The hexagonal boundary of each unit cell is apparent (see the white cell boundary in Fig. 2). Each corner of this unit cell is located at the center of the equilateral triangle formed by the three adjacent nodes. The transverse field is in-phase between any adjacent nodes. The hexagonal boundary of the center unit cell is illustrated in the et plot along with radial lines from center of coordination of this cell. In this case, there are 24 radial lines equally spaced by π/12 degree angle. At the hexagonal cell boundary, we should have ∂\et\/∂r = 0 and ∂\ht\/∂r = 0, due to the symmetry of this mode. This allows us to write a total of 48 boundary equations in this case, two at each intersection point for each radial line with the cell boundary. Bearing in mind that the field parameters around the node can be derived from am and Cm at the center of the node for each m from Eq. (6). We can assemble the following cell boundary matching equations.

M[aM0...aM0bM0...bM0]=0
(12)

Each two rows of M are from ∂\et\/∂r = 0 and ∂\ht\/∂r = 0 respectively at each intersection points, making a total 48 rows. We used ∂\ex\/∂r = 0 and ∂\hx\/∂r = 0 in our implementations. This is adequate to constrain all the field coefficients, as ey and hy are similarly constrained. The m is chosen from –mmax to mmax, making a total of 2(2mmax + 1) variables. It is worth noting that we are looking for modes with a minimum, i.e. trench, along the cell boundary. ∂\ex\/∂r = 0 and ∂\hx\/∂r = 0 are valid even though most of the radial lines are not crossing the cell boundary at a right angle.

For the lower edge of the first band, a simple symmetry analysis shows that it is impossible to find a field distribution which allows anti-phase between any adjacent nodes in a hexagonal lattice while maintaining a similar power distribution at each node. The field amplitudes given by multipole mode solver for a lower edge mode are given in Fig. 3. It can be seen that there are nodes with zero fields so that the remaining nodes can have anti-phase field between any adjacent nodes with non-zero fields. The unit cell boundary in this case is triangular (see Fig. 3) and its three corners are at the centers of the three nodes with zero field. Due to the field symmetries and the anti-phase nature of field between adjacent nodes, we have et = 0 and ht = 0 on the triangular unit cell boundary of the transverse electric field. We again used 24 radial lines in our case with mmax = 12 to set up a matrix M with a dimension of 48 × 50 to find this band edge.

Since we are expecting the unit cell for edge modes is only dependent on the dominant azimuthal mode number and lattice arrangement, we can then apply the hexagonal and triangular unit cell obtained here for upper and lower band edges respectively for all first band of modes for hexagonal lattices.

4. Determination of higher order bandgaps

The amplitudes of ez, hz and et of a mode on the upper edge of the second band of modes are given in Fig. 4
Fig. 4 Amplitudes of ez (a), hz (b), and et (c) for a lattice with node V = 2.6 and d/Λ = 0.46. This is a mode located at upper edge of the second band.
for a hexagonal lattice. This mode symmetry is complicated by the geometry constrains of the fields dominated by m = 2 terms. The resulting transverse fields have alternating field distribution from one node to an adjacent node. This makes precise determination of the boundary cell very difficult for this case. We resort to use a similar hexagonal unit cell as that for the upper edge of the first band of modes in this case. The hexagonal unit cell is illustrated in Fig. 4. On this unit cell boundary, we have ∂\et\/∂r = 0 and ∂\ht\/∂r = 0. A total number of 24 equally spaced radial lines (not illustrated) similar to that in last section is used to generate 48 boundary equations in this study.

The amplitudes of ez, hz and et of a mode on the lower edge of the second band of modes are given in Fig. 5
Fig. 5 Amplitudes of ez (a), hz (b), and et (c) for a lattice with node V = 2.65 and d/Λ = 0.47. This is a mode located at lower edge of the second band.
for a hexagonal lattice of nodes. In this case, the field symmetry is straight forward, resulting in a hexagonal unit cell (see Fig. 5). Transverse fields should be zero over the boundary of this unit cell due to the anti-phase symmetries. A total number of 24 equally spaced radial lines (not illustrated) similar to that in last section is used to generate 48 boundary equations in this study. The mode symmetry of the analysis in this section equally applied to all other modes dominated by m = 2 terms in the field summations as it has been pointed out earlier in the discussion of modes dominated by m = 1 terms.

5. Impact of unit cell boundaries of the edge modes

Since a great deal of efforts has been spent so far in determining the unit cell of the edge modes, some discussion of the impact of the choice of unit cell on the precision in determination of the bandgap edges is warranted. A triangular and hexagonal unit cell have been determined in the last section for the lower edge modes for modes dominated by m = 1 and m = 2 terms respectively in the z field summations. In order to determine the impact of the choice of unit cell on the band edges, the lower band edges of modes dominated by all m-terms for kΛ<180 are determined using both triangular and hexagonal unit cells for the fiber simulated in [12]. The results are shown in Fig. 6
Fig. 6 Lower band edges determined using the triangular unit cell (black dots) and hexagonal unit cell (red circles) for the fiber in [12] (d/Λ = 0.41, node index n0 = 1.48716 and background index nb = 1.458.).
, where the edges determined by the triangular unit cell are given in black dots and those by hexagonal unit cell red circles.

The two unit cells provide very close results for the upper part of the edges, i.e. regime of low coupling among adjacent nodes. This part of the band edge happens to be an area of strong interests. This is where most relevant bandgaps are for typical photonic bandgap fibers. There are strong deviations especially for lower part of the edges where there is strong coupling among adjacent nodes.

6. Full bandgap simulations

A comparison with the bandgap diagram in [12

12. T. A. Birks, G. J. Pearce, and D. M. Bird, “Approximate band structure calculation for photonic bandgap fibres,” Opt. Express 14(20), 9483–9490 (2006). [CrossRef] [PubMed]

] simulated with plane wave technique for the same fiber indicates a good agreement and an improved accuracy over the approximate method used in [12

12. T. A. Birks, G. J. Pearce, and D. M. Bird, “Approximate band structure calculation for photonic bandgap fibres,” Opt. Express 14(20), 9483–9490 (2006). [CrossRef] [PubMed]

]. A precise comparison is difficult due to the complexity of the bandgap diagram. The bandgap diagram from the plane wave technique is a mode density plot and edges do not always show up clearly. The location of the bandgaps is, however, reproduced with excellent accuracy.

7. Bandgap of lattice with nodes of graded index

8. Summary

9. Appendix

The elements for the transfer matrix can be obtained by first deriving other field terms from the z field components for layer p [13

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

].
eθp=m={mβrkp2[ampJm(kpr)+bmpHm1(kpr)]ikkp[CmpJm1(kpr)Jm+1(kpr)2+DmpHm1(kpr)Hm+11(kpr)2]}eimθ
(13)
Hθp=m={mβrkp2[CmpJm(kpr)+DmpHm1(kpr)]+iknp2kp[ampJm1(kpr)Jm+1(kpr)2+bmpHm1(kpr)Hm+11(kpr)2]}eimθ
(14)
ez, Hz, eθ and Hθ are continuous at each boundary p, resulting in four equations for adjacent layers p and p + 1 for each m. The transfer matrix can then be obtained. The following matrix elements are for a specific m; but subscript m is omitted for simplicity.
M11p=1M0{Jm(kpρp)[Hm11(kp+1ρp)Hm+11(kp+1ρp)]np2kp+1np+12kpHm1(kp+1ρp)[Jm1(kpρp)Jm+1(kpρp)]}
M12p=1M0{Hm1(kpρp)[Hm11(kp+1ρp)Hm+11(kp+1ρp)]np2kp+1np+12kpHm1(kp+1ρp)[Hm11(kpρp)Hm+11(kpρp)]}
M13p=1M0i2mβkp+1ρpnp+12k(1kp21kp+12)Hm1(kp+1ρp)Jm(kpρp)
M14p=1M0i2mβkp+1ρpnp+12k(1kp21kp+12)Hm1(kp+1ρp)Hm1(kpρp)
M21p=1M0{Jm(kpρp)[Jm1(kp+1ρp)Jm+1(kp+1ρp)]np2kp+1np+12kpJm(kp+1ρp)[Jm1(kpρp)Jm+1(kpρp)]}
M22p=1M0{Hm1(kpρp)[Jm1(kp+1ρp)Jm+1(kp+1ρp)]np2kp+1np+12kpJm(kp+1ρp)[Hm11(kpρp)Hm+11(kpρp)]}
M23p=1M0i2mβkp+1ρpnp+12k(1kp21kp+12)Jm(kp+1ρp)Jm(kpρp)
M24p=1M0i2mβkp+1ρpnp+12k(1kp21kp+12)Jm(kp+1ρp)Hm1(kpρp)
M31p=1M0i2mβkp+1ρpk(1kp21kp+12)Hm1(kp+1ρp)Jm(kpρp)
M32p=1M0i2mβkp+1ρpk(1kp21kp+12)Hm1(kp+1ρp)Hm1(kpρp)
M33p=1M0{Jm(kpρp)[Hm11(kp+1ρp)Hm+11(kp+1ρp)]kp+1kpHm1(kp+1ρp)[Jm1(kpρp)Jm+1(kpρp)]}
M34p=1M0{Hm1(kpρp)[Hm11(kp+1ρp)Hm+11(kp+1ρp)]kp+1kpHm1(kp+1ρp)[Hm11(kpρp)Hm+11(kpρp)]}
M41p=1M0i2mβkp+1ρpk(1kp21kp+12)Jm(kp+1ρp)Jm(kpρp)
M42p=1M0i2mβkp+1ρpk(1kp21kp+12)Jm(kp+1ρp)Hm1(kpρp)
M43p=1M0{Jm(kpρp)[Jm1(kp+1ρp)Jm+1(kp+1ρp)]kp+1kpJm(kp+1ρp)[Jm1(kpρp)Jm+1(kpρp)]}
M44p=1M0{Hm1(kpρp)[Jm1(kp+1ρp)Jm+1(kp+1ρp)]kp+1kpJm(kp+1ρp)[Hm11(kpρp)Hm+11(kpρp)]}
(15)
where

M0=Jm(kp+1ρp)[Hm11(kp+1ρp)Hm+11(kp+1ρp)]Hm1(kp+1ρp)[Jm1(kp+1ρp)Jm1(kp+1ρp)]
(16)

The eθ and Hθ for p = N describe the azimuthal fields around the node and can be derived similarly to these in Eq. (14). Radial field components er and Hr for p = N can be obtained through the following equations.

erN=m={mkrkb2[CmNJm(kbr)+DmNHm1(kbr)]+iβkb[amNJm1(kbr)Jm+1(kbr)2+bmNHm1(kbr)Hm+11(kbr)2]}eimθ
(17)
HrN=m={mknb2rkb2[amNJm(kbr)+bmNHm1(kbr)]+iβkb[CmNJm1(kbr)Jm+1(kbr)2+DmNHm1(kbr)Hm+11(kbr)2]}eimθ
(18)

The transverse fields around the node can then be obtained using Eq. (10) and Eq. (11). The boundary conditions at each intersection of the radial lines with the unit cell boundary can be obtained so that matrix M for the linear equations in Eq. (12) can be set up.

References and links

1.

C. B. Olausson, C. I. Falk, J. K. Lyngsø, B. B. Jensen, K. T. Therkildsen, J. W. Thomsen, K. P. Hansen, A. Bjarklev, and J. Broeng, “Amplification and ASE suppression in a polarization-maintaining ytterbium-doped all-solid photonic bandgap fibre,” Opt. Express 16(18), 13657–13662 (2008). [CrossRef] [PubMed]

2.

A. Shirakawa, C. B. Olausson, H. Maruyama, K. Ueda, J. K. Lyngsø, and J. Broeng, “High power ytterbium fiber lasers at extremely long wavelengths by photonic bandgap fiber technology,” Opt. Fiber Technol. 16(6), 449–457 (2010). [CrossRef]

3.

F. Luan, A. K. George, T. D. Hedley, G. J. Pearce, D. M. Bird, J. C. Knight, and P. St. J. Russell, “All-solid photonic bandgap fiber,” Opt. Lett. 29(20), 2369–2371 (2004). [CrossRef] [PubMed]

4.

A. Argyros, T. A. Birks, S. G. Leon-Saval, C. M. B. Cordeiro, F. Luan, and P. St. J. Russell, “Photonic bandgap with an index step of one percent,” Opt. Express 13(1), 309–314 (2005). [CrossRef] [PubMed]

5.

A. Argyros, T. A. Birks, S. G. Leon-Saval, C. M. B. Cordeiro, and P. St J Russell, “Guidance properties of low-contrast photonic bandgap fibres,” Opt. Express 13(7), 2503–2511 (2005). [CrossRef] [PubMed]

6.

G. Bouwmans, L. Bigot, Y. Quiquempois, F. Lopez, L. Provino, and M. Douay, “Fabrication and characterization of an all-solid 2D photonic bandgap fiber with a low-loss region (< 20 dB/km) around 1550 nm,” Opt. Express 13(21), 8452–8459 (2005). [CrossRef] [PubMed]

7.

J. M. Stone, G. J. Pearce, F. Luan, T. A. Birks, J. C. Knight, A. K. George, and D. M. Bird, “An improved photonic bandgap fiber based on an array of rings,” Opt. Express 14(13), 6291–6296 (2006). [CrossRef] [PubMed]

8.

G. J. Pearce, T. D. Hedley, and D. M. Bird, “Adaptive curvilinear coordinates in a plane-wave solution of Maxwell’s equations in photonic crystals,” Phys. Rev. B 71(19), 195108 (2005). [CrossRef]

9.

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(10), 2322–2330 (2002). [CrossRef]

10.

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(10), 2331–2340 (2002). [CrossRef]

11.

T. White, R. McPhedran, L. Botten, G. Smith, and C. M. de Sterke, “Calculations of air-guided modes in photonic crystal fibers using the multipole method,” Opt. Express 9(13), 721–732 (2001). [CrossRef] [PubMed]

12.

T. A. Birks, G. J. Pearce, and D. M. Bird, “Approximate band structure calculation for photonic bandgap fibres,” Opt. Express 14(20), 9483–9490 (2006). [CrossRef] [PubMed]

13.

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

OCIS Codes
(060.0060) Fiber optics and optical communications : Fiber optics and optical communications
(060.2280) Fiber optics and optical communications : Fiber design and fabrication
(060.5295) Fiber optics and optical communications : Photonic crystal fibers

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: April 18, 2011
Revised Manuscript: May 31, 2011
Manuscript Accepted: June 7, 2011
Published: June 14, 2011

Citation
Liang Dong, "A vector boundary matching technique for efficient and accurate determination of photonic bandgaps in photonic bandgap fibers," Opt. Express 19, 12582-12593 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-13-12582


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References

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