## Full-vectorial analysis of complex refractive-index photonic crystal fibers

Optics Express, Vol. 12, Issue 6, pp. 1126-1135 (2004)

http://dx.doi.org/10.1364/OPEX.12.001126

Acrobat PDF (365 KB)

### Abstract

We investigated the modal properties of complex refractive-index core photonic crystal fibers (PCFs) with the supercell model. The validity of the approach is shown when we compare our results with those reported earlier on a step complex refractive-index profile. The imaginary part of the electric field results in wave-front distortion in the complex refractive-index profile PCFs, which means that the power flows out or into the doped region according to the sign of the imaginary part of the refractive index. A simple formula is proposed for calculating the gain or loss coefficients of these fibers. The numerical results obtained by the approximation formula agree well with the full-vectorial results.

© 2004 Optical Society of America

## 1. Introduction

1. A. Reisinger, “Characteristics of optical guided modes in lossy waveguides,” Appl. Opt. **12**, 1015–1025 (1973). [CrossRef] [PubMed]

5. R. Singh Sunanda and E. Khular Sharma, “Propagation characteristics of single-mode optical fibers with arbitrary complex index profiles: a direct numerical approach,” IEEE J. Quantum Electron. **37**, 635–640 (2001). [CrossRef]

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. M. Ibanescu, Y. Fink, S. Fan, E. L. Thomas, and J. D. Joannopoulos, “An all-dielectric coaxial waveguide,” Science **289**, 415–419(2000). [CrossRef] [PubMed]

*et al*. [8

8. W. J. Wadsworth, J.C. Knight, W. H. Reeves, P.S.J. Russell, and J. Arriaga, “Yb^{3+}-doped photonic crystal fibre laser, ” Electronics Lett. **36**, 1452–1454 (2000) [CrossRef]

9. K. G. Hougaard, J. Broeng, and A. Bjarklev, “Low pump power photonic crystal fibre amplifiers,” Electron. Lett. , **39**, 599–600 (2003) [CrossRef]

10. J. Limpert, T. Schreiber, S. Nolte, H. Zellmer, T. Tunnermann, R. Iliew, F. Lederer, J. Broeng, G. Vienne, A. Petersson, and C. Jakobsen, “High-power air-clad large-mode-area photonic crystal fiber laser,” Opt. Express **11**, 818–823 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-7-818 [CrossRef] [PubMed]

## 2. Simulation method

*z*) direction, so our main task here is to investigate the transverse modal field distribution

*e⃗*

_{t}(

*x y*) which can be divided into two polarization components along the

*x*and

*y*axes:

*e⃗*

_{t}(

*x*,

*y*)=

*x̂e*

_{x}(

*x*,

*y*)+

*ŷe*

_{y}(

*x*,

*y*). For a dielectric waveguide, when its dielectric constant profile

*ε*(

*x*,

*y*) has

*x*and

*y*axial symmetry, i.e., it is an even function of both

*x*and

*y*, it can be proved that

*e*

_{x}(

*x*,

*y*) and

*e*

_{y}(

*x*,

*y*) always have opposite parities in the

*x*and

*y*directions for each eigenmode [11

11. Yu-Li Hsueh, E. S.T. Hu, M. E. Marhic, and G. Kazovsky, “Opposite-parity orthonormal function expansion for efficient full-vectorial modeling of holey optical fibers,” Opt. Lett. **28**, 1188–1190 (2003). [CrossRef] [PubMed]

*m*and

*n*are introduced to express the opposite parities of the mode electric field, which have the logical value 0 or 1, and are used to describe the symmetry of the

*x*component

*e*

_{x}(

*x,y*) as

*e*

_{x}(-

*x,y*)=(-1)

^{m}

*e*

_{x}(

*x,y*) and

*e*

_{x}(

*x*,-

*y*)=(-1)

^{n}

*e*

_{x}(

*x,y*). All the compositions of

*mn*are [00, 01, 10, 11], which can completely express the symmetry of the mode electric field about both axes. To improve computational efficiency, the transverse electric field can be expanded with the localized orthonormal Hermite-Gaussian basis functions as follows:

*mn*indicates that there are four sets of (

*e*

_{x},

*e*

_{y}) with different parity.

*F*is the number of expansion terms,

*εab*

^{s}(

*s*=

*x*,

*y*) are the expansion coefficients, and

*ψ*

_{i}(

*s*) is the ith-order orthonormal Hermite-Gaussian function [12

12. 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]

13. W. Zhi, R.G. Bin, L.S. Qin, and J. S. Sheng, “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]

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

*x*and

*y*, i.e.,

*ε*(-

*x*,

*y*)=

*ε*(

*x*,-

*y*)=

*ε*(

*x*,

*y*), the expression can be given as a sum of the cosine functions as

*P*is the number of expansion items and

*P*

_{ab},

*D*

_{x}(

*D*

_{y}) is the characteristic period in the

*x*(

*y*) direction. When fibers with a complex refractive-index profile are examined, the expansion coefficients

*P*

_{ab},

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

15. W. Zhi, R. Guobin, L. Shuqin, L. Weijun, and S. Guo, “Compact supercell method based on opposite parity for Bragg fibers,” Opt. Express **11**, 3542–3549 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-26-3542 [CrossRef] [PubMed]

*β*is the propagation constant corresponding to the mode field distribution (

*e*

_{x},

*e*

_{y}). If the complex dielectric structure

*ε*(

*x*,

*y*) is considered, the corresponding propagation constant is thus complex and can be written as

*β*=

*β*

_{r}+

*iβ*

_{i},

*β*

_{r},

*β*

_{i}are the phase item and the gain (loss) item, respectively. The gain or loss of the modal field is hence 8.686

*β*

_{i}in decibels per meter.

*L*

_{mn}is a four-dimensional matrix, and

*M*

_{1},

*M*

_{2},

*M*

_{3}, and

*M*

_{4}are the overlapping integrals, which are a four-dimensional

*F*×

*F*×

*F*×

*F*matrix. These overlapping integrals can be calculated analytically. All the overlapping integrals are not shown here because of their complicated form, which we have discussed previously [15

15. W. Zhi, R. Guobin, L. Shuqin, L. Weijun, and S. Guo, “Compact supercell method based on opposite parity for Bragg fibers,” Opt. Express **11**, 3542–3549 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-26-3542 [CrossRef] [PubMed]

*L*

_{mn}and

*ε*

^{s}can be transferred into a [2×

*F*

^{2}]×[2×

*F*

^{2}] two-dimensional complex matrix and a vector with 2×

*F*

^{2}elements, with which the eigensystem Eq. (4) can be solved by a complex solver. When the real parts of the eigenvalues at the wavelength

*λ*are labeled in decreasing order, the modal electric fields from the fundamental to higher order can be obtained. If the vector items

*M*

_{3x},

*M*

_{3y},

*M*

_{4x}, and

*M*

_{4y}are neglected, the eigenvalue problems transform into a scalar approximation.

*n*(

*x*,

*y*)=

*n*

_{r}(

*x*,

*y*)+

*i*

*n*

_{i}(

*x*,

*y*). The positive (negative) sign of the imaginary part indicates a gain (lossy) media. If we take the complex index as

*n*

^{*}, where the asterisk represents a complex conjugate operation, it can be proved that the matrix

*L*

_{mn}will be

*n*results in the complex conjugate of the propagation constant and the modal field.

## 3. Propagation properties of complex index PCFs

### 3.1 Validity of the method

5. R. Singh Sunanda and E. Khular Sharma, “Propagation characteristics of single-mode optical fibers with arbitrary complex index profiles: a direct numerical approach,” IEEE J. Quantum Electron. **37**, 635–640 (2001). [CrossRef]

*a*=2.2 µm (core radius),

*n*

_{core}=1.475+

*in*

_{i},

*n*

_{clad}=1.458, at

*λ*=1.55 µm;

*n*

_{i}is the imaginary part of the refractive index in fiber core. The numbers of the expansion items in Eqs. (1)–(3) are taken as

*F*=18,

*P*=600. The values of real and imaginary parts of mode index

*n*

_{e}=

*n*

_{er}+

*in*

_{ei}obtained by our method are listed in Table 1. As seen from the table, the scalar method gives quite an accurate modal index. A larger

*F*gives more-accurate results, but there is a trade-off between the accuracy and computation time. The mode indices obtained by the vectorial method are also shown in table. We note that the vectorial results are a little smaller than the scalar results; this is because the vector items

*M*

_{3x},

*M*

_{3y}and the coupling items

*M*

_{4x},

*M*

_{4y}are taken into account in Eq. (4).

### 3.2 Propagation properties of complex index core PCF

*Λ*and relative hole size

*d/Λ*are used to define the structure of the PCF (shown in Fig. 1). We assume that the fiber core is doped with core radius

*R*and that its refractive index has an imaginary part. We take the fiber parameters as

*Λ*=4.4 µm,

*R*=2.2 µm and the relative hole size to be

*d/Λ*=0.3. The refractive index of the fiber core is taken as

*n*

_{core}=1.475+i

*n*

_{i}, where

*n*

_{i}=10

^{-3}, the index of the cladding matrix is

*n*

_{matrix}=1.458, and the index of air is

*n*

_{air}=1. The complex refractive-index profile can be considered as a function of radial distance, pump and signal wavelength, and dopant profiles for active medium [4

4. Sunanda and E. K. Sharma, “Field variational analysis for modal gain in erbium-doped fiber amplifiers,” J. Opt. Soc. Am. B , **16**, 1344–1347 (1999). [CrossRef]

_{11}) and second-order modes (HE

_{21}, TM

_{01}and TE

_{01}) exist at wavelength

*λ*=1550 nm. The expression

*E*(

*x*,

*y*)=

*E*

_{r}(

*x*,

*y*)+

*iE*

_{i}(

*x*,

*y*) is used to represent the modal field, where

*E*

_{r}(

*x*,

*y*) and

*E*

_{i}(

*x*,

*y*) are the real and imaginary parts of the modal field, respectively. Figures 2 and 3 show |

*E*

_{r}|

^{2}and |

*E*

_{i}|

^{2}for the HE

_{11}and TM

_{01}modes. It is shown that both the real and the imaginary parts of modal field reflect the symmetry of the dielectric structure; the imaginary part of the modal field is much smaller than the real part because of the small

*n*

_{i}. An apparent characteristic of the imaginary part of the modal field is that the modal field reaches the minimum near the edge of the fiber core.

*n*

_{i}—the imaginary part of the refractive index in the fiber core. Figure 5 shows the phase distribution along the

*x*=0 axis with different

*n*

_{i}for

*d/Λ*=0.3,

*λ*=1550nm. In Fig. 5, the phase is multiplied by 10 for

*n*

_{i}=10

^{-4}and 100 for

*n*

_{i}=10

^{-5}. It is obvious that these curves are overlapped; therefore it can be concluded that the phase distribution

*P*(

*x*,

*y*) satisfies

*P*(

*x*,

*y*)∝

*n*

_{i}. For different

*n*

_{i}, the line types of these phase distributions are the same except for the amplitude.

*d/Λ*is an important parameter for determining the modal characteristics of PCF. Figure 6 shows the dependence of mode index

*n*

_{e}=

*β*/

*k*

_{0}=

*n*

_{er}+

*in*

_{ei}of the fundamental mode on

*d/Λ*at wavelengths of 980, 1310, and 1550 nm.

*k*

_{0}=2

*π/λ*is the wave number of the vacuum. At a certain wavelength, the real part of mode index

*n*

_{er}decreases with increasing air hole; whereas the imaginary part of mode index

*n*

_{ei}increases as

*d/Λ*increases. On the other hand, the mode index (including

*n*

_{er}and

*n*

_{ei}) increases as the wavelength decreases because more field energy is confined in the high-index region (core). The introduction of air hole into the fiber cladding will enhance the gain of the fiber efficiently. It is shown that the use of erbium- or ytterbinm-doped PCF as fiber lasers or amplifiers allows for improved features of amplification properties with respect to standard step-index fibers [9

9. K. G. Hougaard, J. Broeng, and A. Bjarklev, “Low pump power photonic crystal fibre amplifiers,” Electron. Lett. , **39**, 599–600 (2003) [CrossRef]

*d/Λ*is less than 0.406 [17

17. B. T. Kuhlmey, R. C. McPhedran, C. M. de Sterke, P. A. Robinson, G. Renversez, and D. Maystre,“Microstructured optical fibers: where’s the edge?” Opt. Express **10**, 1285–1290 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-22-1285 [CrossRef] [PubMed]

*d/Λ*=0.3) is multimode (including the fundamental mode and second-order modes) at wavelength

*λ*=1550 nm. Because the fiber core is up-doped, it is reasonable that the critical value of

*d/Λ*for endlessly single-mode operation will decrease and depend on the refractive index of the core. To extend the available parameter space, the refractive index of the fiber core should be depressed, for example, by codoping with fluorine, to compensate the refractive-index increase.

## 4. Approximate formula for calculation gain or loss coefficient

*β*

_{i}is the imaginary part of propagation constant and

*n*

_{r}(

*x*,

*y*),

*n*

_{i}(

*x*,

*y*) denote the real and imaginary parts of the refractive-index profile. When the waveguide is only slightly absorbing (amplifying), as is normally the case in practice, we can replace the modal fields with their corresponding modal fields of nonabsorbing waveguide (the imaginary part of the refractive-index profile is neglected).

*r*denote the real values. In the weakly guiding approximation, we have

*k*denotes the

*k*th layer from the fiber core to the out-cladding,

*n*

_{kr}(

*n*

_{ki}) is the real (imaginary) part of refractive index in the

*k*th layer, and

*Γ*

_{k}is the optical power confinement factor in the

*k*th layer. For the complex index core PCF discussed in this paper, we have

*n*

_{corer}(

*n*

_{corei}) is the real (imaginary) part of the refractive index in the fiber core and

*Γ*is the fraction of modal power in the core. The imaginary part of the propagation constant (i.e., gain or loss coefficient) is

*β*

_{i}=

*k*

_{0}

*n*

_{ei}. According to Eq. (11),

*n*

_{ei}is determined by the confinement factor and the mode index

*n*

_{er}when the refractive-index profile of the fiber is fixed. The configuration of PCF can provide more flexibility to control modal properties and then the gain (loss) coefficient.

*n*

_{e}and the power confinement factor

*Γ*between the complex refractive-index core PCF and the corresponding fibers, in which the imaginary part of the imaginary refractive index is neglected. For clarity, fiber A represents PCF with

*n*

_{core}=1.475+i10

^{-3}and the refractive index of the cladding matrix is

*n*

_{matrix}=1.458. Fiber B represents PCF with

*n*

_{core}=1.475 and

*n*

_{matrix}=1.458. Figure 7 shows the mode indices (for fiber A, only the real part of mode index are demonstrated) and the confinement factors of fiber A and B as a function of

*d/Λ*. The differences of mode index and confinement factor are also given in the figure. We note that as the relative air hole size

*d/Λ*increases, the differences of two types of fiber decrease. The restraining assumption of the stand method of perturbation is that the gain or loss exhibited by the waveguide does not alter the field significantly [1

1. A. Reisinger, “Characteristics of optical guided modes in lossy waveguides,” Appl. Opt. **12**, 1015–1025 (1973). [CrossRef] [PubMed]

^{-5}for mode index, and 10

^{-4}for power confinement factor. Hence for practical fibers, the perturbation method is also applicable.

*d/Λ*at wavelengths of 980, 1310, and 1550 nm; in Fig. 8(b), the imaginary parts of the effective mode indices are shown as a function of wavelength for different relative air hole size

*d/Λ*(0.1, 0.3 0.5 0.7). As shown in Fig. 8, Eq. (11) agrees well with the full-vectorial method. We find that the difference of the two groups of curves is less than 10

^{-6}. It is observed that the difference between the approximate formula and the full-vectorial method is small at short wavelengths but increases for long wavelengths. Further, the difference increases as the air hole size

*d/Λ*increases. This is understood to occur because the weakly guiding approximation used in Eq. (11) brings more error for the large air hole and long wavelength.

## 5. Conclusion

## Acknowledgments

## References and links

1. | A. Reisinger, “Characteristics of optical guided modes in lossy waveguides,” Appl. Opt. |

2. | J. E. Sader, “Method for analysis of complex refractive index profile fibers,” Opt. Lett. |

3. | E. K. Sharma, Mukesh. P. Singh, and A. Sharma, “Variational analysis of optical fibers with loss or gain,” Opt. Lett. |

4. | Sunanda and E. K. Sharma, “Field variational analysis for modal gain in erbium-doped fiber amplifiers,” J. Opt. Soc. Am. B , |

5. | R. Singh Sunanda and E. Khular Sharma, “Propagation characteristics of single-mode optical fibers with arbitrary complex index profiles: a direct numerical approach,” IEEE J. Quantum Electron. |

6. | T. A. Birks, J. C. Knight, and P. St. J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. |

7. | M. Ibanescu, Y. Fink, S. Fan, E. L. Thomas, and J. D. Joannopoulos, “An all-dielectric coaxial waveguide,” Science |

8. | W. J. Wadsworth, J.C. Knight, W. H. Reeves, P.S.J. Russell, and J. Arriaga, “Yb |

9. | K. G. Hougaard, J. Broeng, and A. Bjarklev, “Low pump power photonic crystal fibre amplifiers,” Electron. Lett. , |

10. | J. Limpert, T. Schreiber, S. Nolte, H. Zellmer, T. Tunnermann, R. Iliew, F. Lederer, J. Broeng, G. Vienne, A. Petersson, and C. Jakobsen, “High-power air-clad large-mode-area photonic crystal fiber laser,” Opt. Express |

11. | Yu-Li Hsueh, E. S.T. Hu, M. E. Marhic, and G. Kazovsky, “Opposite-parity orthonormal function expansion for efficient full-vectorial modeling of holey optical fibers,” Opt. Lett. |

12. | T.M. Monro, D.J. Richardson, N.G.R. Broderick, and P. J. Bennett, “Holey optical fibers: an efficient modal model,” J. Lightwave Technol. |

13. | W. Zhi, R.G. Bin, L.S. Qin, and J. S. Sheng, “Supercell lattice method for photonic crystal fibers,” Opt. Express |

14. | R. Guobin, W. Zhi, L. Shuqin, and J. Shuisheng, “Mode classification and degeneracy in photonic crystal fibers,” Opt. Express |

15. | W. Zhi, R. Guobin, L. Shuqin, L. Weijun, and S. Guo, “Compact supercell method based on opposite parity for Bragg fibers,” Opt. Express |

16. | A.W. Snyder and J.D. Love, |

17. | B. T. Kuhlmey, R. C. McPhedran, C. M. de Sterke, P. A. Robinson, G. Renversez, and D. Maystre,“Microstructured optical fibers: where’s the edge?” Opt. Express |

**OCIS Codes**

(060.2280) Fiber optics and optical communications : Fiber design and fabrication

(060.2310) Fiber optics and optical communications : Fiber optics

(230.7370) Optical devices : Waveguides

**ToC Category:**

Research Papers

**History**

Original Manuscript: January 16, 2004

Revised Manuscript: February 25, 2004

Published: March 22, 2004

**Citation**

Ren Guobin, Wang Zhi, Lou Shuqin, Liu Yan, and Jian Shuisheng, "Full-vectorial analysis of complex refractive index photonic crystal fibers," Opt. Express **12**, 1126-1135 (2004)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-6-1126

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### References

- A. Reisinger, �??Characteristics of optical guided modes in lossy waveguides,�?? Appl. Opt. 12, 1015-1025 (1973). [CrossRef] [PubMed]
- J.E. Sader, �??�??Method for analysis of complex refractive index profile fibers,�??�?? Opt. Lett. 15, 107-109 (1990). [CrossRef]
- E. K. Sharma, Mukesh. P. Singh, and A. Sharma, �??Variational analysis of optical fibers with loss or gain,�?? Opt. Lett. 18, 2096-2098 (1993). [CrossRef] [PubMed]
- Sunanda and E. K. Sharma, �??Field variational analysis for modal gain in erbium-doped fiber amplifiers,�?? J. Opt. Soc. Am. B, 16, 1344�??1347 (1999). [CrossRef]
- R. Singh Sunanda, and E. Khular Sharma, �??Propagation characteristics of single-mode optical fibers with arbitrary complex index profiles: a direct numerical approach,�?? IEEE J. Quantum Electron. 37, 635-640 (2001). [CrossRef]
- 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]
- M. Ibanescu, Y. Fink, S. Fan, E. L. Thomas, and J. D. Joannopoulos, �??An all-dielectric coaxial waveguide,�?? Science 289, 415-419(2000). [CrossRef] [PubMed]
- W. J. Wadsworth, J.C. Knight, W. H. Reeves, P.S.J. Russell, and J Arriaga, "Yb3+-doped photonic crystal fibre laser," Electronics Lett. 36, 1452 �?? 1454 (2000) [CrossRef]
- K. G. Hougaard, J. Broeng, and A. Bjarklev, �??Low pump power photonic crystal fibre amplifiers,�?? Electron. Lett., 39, 599-600 (2003) [CrossRef]
- J. Limpert, T. Schreiber, S. Nolte, H. Zellmer, T. Tunnermann, R. Iliew, F. Lederer, J. Broeng, G. Vienne, A. Petersson, and C. Jakobsen, "High-power air-clad large-mode-area photonic crystal fiber laser," Opt. Express 11, 818-823 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-7-818">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-7-818</a> [CrossRef] [PubMed]
- Yu-Li Hsueh, E. S.T. Hu, M. E. Marhic, and G. Kazovsky, �??Opposite- arity orthonormal function expansion for efficient full-vectorial modeling of holey optical fibers,�?? Opt. Lett. 28, 1188-1190 (2003). [CrossRef] [PubMed]
- 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]
- W. Zhi, R.G. Bin, L.S. Qin, and J. S. Sheng, �??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]
- R. Guobin, W. Zhi, L. Shuqin, and J. Shuisheng, "Mode classification and degeneracy in photonic crystal fibers," 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]
- W. Zhi, R. Guobin, L. Shuqin, L. Weijun, and S. Guo, "Compact supercell method based on opposite parity for Bragg fibers," Opt. Express 11, 3542-3549 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-26-3542">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-26-3542</a> [CrossRef] [PubMed]
- A.W. Snyder and J.D. Love, Optical Waveguide Theory (Chapman and Hall, New York, 1983).
- B. T. Kuhlmey, R. C. McPhedran, C. M. de Sterke, P. A. Robinson, G. Renversez, and D. Maystre, "Microstructured optical fibers: where�??s the edge?" Opt. Express 10, 1285-1290 (2002), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-22-1285">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-22-1285</a> [CrossRef] [PubMed]

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