## Meshed index profile method for photonic crystal fibers with arbitrary structures

Optics Express, Vol. 16, Issue 17, pp. 13175-13187 (2008)

http://dx.doi.org/10.1364/OE.16.013175

Acrobat PDF (2506 KB)

### Abstract

A meshed index profile method, which is based on the localized function method, is demonstrated for analyzing modal characteristics of photonic crystal fibers with arbitrary air-hole structures. The index profile of PCF, which is expressed as a sum of meshed unit matrix, is substituted to full wave equation. By solving this full wave equation, we obtain the modal characteristics of the PCF such as the mode field distribution, the birefringence and the waveguide dispersion. The accuracy of the proposed meshed index profile method (MIPM) is demonstrated by examining the effective index and the birefringence of the two degenerate fundamental modes in the PCF with a triangular air-hole lattice. The MIPM is not restricted to the PCF structure and will be useful in designing various PCF devices.

© 2008 Optical Society of America

## 1. Introduction

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

8. T. P. Whiteet al, “Calculations of air-guided modes in photonic crystal fibers using the multipole method,” Opt. Express **11**, 721–732 (2001)
, http://www.opticsinfobase.org/abstract.cfm?URI=oe-9-13-721. [CrossRef]

9. A. Cucinotta, S. Selleri, L. Vincetti, and M. Zoboli, “Perturbation Analysis of Dispersion Properties in Photonic Crystal Fibers Through the Finite Element Method,” J. Lightwave Technol. **20**, 1433–1442 (2002). [CrossRef]

13. W. Zhi, R Guobin, and L. Shuqin, “A Novel Supercell Overlapping Method for Different Photonic Crystal Fibers,” J. Lightwave Technol. **22**, 903–916 (2004). [CrossRef]

14. 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. Z. Zhang and S. Satpathy, “Electromagnetic wave propagation in structures : Bloch wave solution of Maxwell’s equation,” Phys. Rev. Lett. **65**, 2650–2653 (1990). [CrossRef] [PubMed]

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

19. Z. Zhu and T. G. Brown, “Full-vectorial finite-difference analysis of microstructured optical fiber,” Opt. Express **10**, 853–864 (2002)
, http://www.opticsinfobase.org/abstract.cfm?URI=oe-10-17-853. [PubMed]

20. K. N. Park and K. S. Lee, “Improved effective index method for analysis of photonic cyrstal fibers,” Opt. Lett. **30**, 958–960 (2005). [CrossRef] [PubMed]

13. W. Zhi, R Guobin, and L. Shuqin, “A Novel Supercell Overlapping Method for Different Photonic Crystal Fibers,” J. Lightwave Technol. **22**, 903–916 (2004). [CrossRef]

21. W. Zhi, R. Guobin, L. Shuqin, and J. Shuisheng, “Supercell lattice method for photonic crystal fibers,” Opt. Express **11**, 980–991 (2003)
, http://www.opticsinfobase.org/abstract.cfm?URI=oe-11-9-980. [CrossRef] [PubMed]

22. 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]

## 2. Electric field decomposition

*E*(

_{x}*x*,

*y*) and

*E*(

_{y}*x*,

*y*) are the field distributions of the modes linearly polarized along the x-axis and y-axis, respectively. And the mode field equations as sums of the localized basis functions can be expressed as,

*S*+1) is the number of basis functions used to express the transverse electric field distribution.

*Ψ*(

_{m}*x*) and

*Ψ*(

_{n}*x*) are the Hermite-Gaussian basis functions used to express the mode field distribution in the LFM and are expressed as,

*H*(

_{i}*s*/

*w*) is the

*i*th order Hermite polynomial and

*w*is the characteristics width of the basis function.

*e*and

^{x}_{mn}*e*are the coefficients to express the mode field distributions and will be calculated by using the MIPM in the next section. This indicates that the method of selecting the basis functions, the characteristic width of the basis function, and the number of basis functions of the expansion can determine the numerical results. We can express more exact mode field distribution by increasing the number of basis functions. However, if the number of terms is increased, considerably more time is required. Inserting

^{y}_{mn}*E*(

_{x}*x*,

*y*) and

*E*(

_{y}*x*,

*y*) into the full vector wave equation [22

22. 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]

23. 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]

*k*=2

*π*/λ is the wave number,

*β*and

_{x}*β*are the propagation constants of the

_{y}*x*and

*y*polarized modes, respectively. By solving these coupled wave equations, we can then analyze the modal characteristics of the PCF including the effective index, the mode field distribution and the dispersion. If we express the transverse index profile (

*n*

^{2}(

*x*,

*y*)) in a simple form in the coupled wave equations, the coupled wave equations can be easily solved by using the overlap integrals of the Hermite-Gaussian basis functions and the some integrals with specific identities. A two-dimensional Fourier transform has been used to express the transverse index profile (

*n*

^{2}(

*x*,

*y*)) in the earlier simulation methods such as the SOM [13

13. W. Zhi, R Guobin, and L. Shuqin, “A Novel Supercell Overlapping Method for Different Photonic Crystal Fibers,” J. Lightwave Technol. **22**, 903–916 (2004). [CrossRef]

22. 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]

## 3. Meshed index profile of the PCF and eigenvalue matrix

*n*

^{2}(

*x*,

*y*)) from the transverse structure of the PCF. The transverse plane of the PCF is divided into (

*2N*+

*1*)×(

*2N*+

*1*) unit cells and then the total index profile of the PCF is expressed as the sum of unit row matrix functions (

*n*(

_{i}*x*)) or as the sum of column matrix functions (

*n*(

_{i}*y*)). Because

*x*and

_{i}*y*are the constants in a respective same row matrix and column matrix, we can define the unit row and column matrix functions as

_{i}*n*(

*x*,

*y*)=

_{i}*n*(

_{i}*x*) and

*n*(

*x*,

_{i}*y*)=

*n*(

_{i}*y*), respectively. Then, the total index profile of the PCF can be expressed as,

*2N*+

*1*is the number of unit row (

*n*

^{2}

_{i}(

*x*)) or column (

*n*

^{2}

_{i}(

*y*)) matrix. In this way, an index profile function (

*n*

^{2}(

*x*,

*y*)) is deduced directly and easily from the index profile of the PCF. Therefore, the index composition method used in the MIPM is efficient and very useful in expressing the index profile of the PCF.

*n*

^{2}(

*x*,

*y*)) into the full vector wave equations (5) and (6). Then the two eigenvalue equations are obtained and given by,

*k*=

*2π*/

*λ*is the wave number,

*β*and

_{x}*β*are the propagation constants of the modal electric fields

_{y}*E*and

_{x}*E*, respectively. Here,

_{y}*M*and

^{x}_{abcd}*M*are

^{y}_{abcd}*(S+1)*×

*(S+1)*×

*(S+1)*×

*(S+1)*-order four-dimensional matrices which are used in calculating the modal characteristics of the PCF. The overlap integrals

*I*,

^{(1)}_{abcd}*I*,

^{(2)}_{abcd}*I*and

^{(3)x}_{abcd}*I*in (8) and (9) are defined as follows:

^{(3)y}_{abcd}*Ψ*(

_{a}*x*),

*Ψ*(

_{b}*y*),

*Ψ*(

_{c}*x*) and

*Ψ*(

_{d}*y*) are the Hermite-Gaussian basis functions.

*n*

^{2}(

*x*,

*y*) expressed by (7) is substituted into the overlap integrals. The overlap integrals can be calculated with the orthonormality of the Hermite-Gaussian basis functions and are rewritten as [24, 25

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

*M*,

^{x}_{abcd}*M*to analyze the modal characteristics of the PCF by substituting Eqs. (14)–(17) into Eq. (8) and (9). The four-dimensional matrices

^{y}_{abcd}*M*,

^{x}_{abcd}*M*can be transferred into the (

^{y}_{abcd}*S*+

*1*)

^{2}×(

*S*+

*1*)

^{2}two-dimensional matrices[21

21. W. Zhi, R. Guobin, L. Shuqin, and J. Shuisheng, “Supercell lattice method for photonic crystal fibers,” Opt. Express **11**, 980–991 (2003)
, http://www.opticsinfobase.org/abstract.cfm?URI=oe-11-9-980. [CrossRef] [PubMed]

*M*,

^{x}_{abcd}*M*become a vector with (

^{y}_{abcd}*S*+

*1*)

^{2}elements. The Propagation constants of the guided modes

*β*and

_{x}*β*are the eigenvalues of

_{y}*M*and

^{x}_{abcd}*M*, respectively. The eigenvectors corresponding to the eigenvalues

^{y}_{abcd}*β*and

_{x}*β*are used as the mode field distribution coefficients (

_{y}*e*,

^{x}_{mn}*e*) in Eqs. (2) and (3), respectively.

^{y}_{mn}## 4. Modal electric field distribution, dispersion and birefringence

*Λ*of 2.3µm and 4.6µm, respectively. And

*d*/

*Λ*for the PCF is 0.2, where d is the air-hole diameter of the PCF. The dimension of index profile matrix used in the MIPM is 1201×1201 and the number of basis functions (

*S*+

*1*) is 16.

*E*|

_{x}^{2}) at two different wavelengths 0.633µm and 1.55µm. Figure 3 displays the corresponding contour plots shown in Fig. 2. From the contour plots shown in Fig. 3, we establish that the mode field intensity distribution at the short wavelengths has greater localization over the core region. Meanwhile, at longer wavelengths, mode field intensity distribution spreads over the cladding region as predicted.

*Λ*of 2.3µm and the elliptical ratio of the air-hole of 0.5 were used, where the major width of the elliptical air-hole is 0.4

*Λ*and the minor is 0.2

*Λ*. We expect that the mode field intensity distribution is affected by the shape and location of the air-hole. Figure 4 shows that the mode field intensity distribution of the PCF with the elliptical air-hole lattice is elliptical and this result matches with our expectation.

*Λ*=4.4µm, large air-hole diameter D

_{1}=4.5µm and small air-hole diameter D

_{2}=2.2µm. In order to calculate the effective index, birefringence and fundamental mode field intensity distribution of this PM-PCF, first, the transverse structure of the PM-PCF is used to derive the meshed index profile. The meshed index profile derived from the transverse structure of the PM-PCF is then substituted into the eigenvalue equation and the modal characteristics of the PM-PCF are obtained by solving this eigenvalue equation. If the PCF has an arbitrary structure, these processes for analyzing the modal characteristics of the PCF are identical. Figure 6(a) and Fig. 6(b) are the x-polarized fundamental mode intensity distribution and the contour plot of the PM-PCF in Fig. 5, respectively. Figure 7 is the birefringence (

*Δn*=|

*n*-

_{x}*n*|) of the PM-PCF shown in Fig. 5. If the PCF has the symmetric air-hole lattice, the two polarized fundamental modes are degenerate and the mode field intensity distributions are nearly identical. Then the modal birefringence becomes very small for the fundamental mode in the PCF with a symmetrically triangular air-hole lattice. But the PM-PCF has two air-holes of which size are different from the other air-holes in the triangular lattice as shown in Fig. 5. In that case the birefringence of the PM-PCF becomes larger than the birefringence of the PCF with a symmetrically triangular air-hole lattice. Figure 7 shows the birefringence of the PM-PCF computed by the MIPM, which increases with wavelength as predicted [26

_{y}26. T.-L. Wu and C.-H. Chao, “Photonic Crystal Fiber Analysis Through the Vector Boundary-Element Method : Effect of Elliptical Air Hole,” IEEE Photon. Tech. Lett. **16**, 126–128 (2004). [CrossRef]

*D*) and the waveguide dispersion (

_{m}*D*). Because the PCF is mainly made of pure silica,

_{w}*D*is same for all silica PCFs with different structures. It means that the total dispersion of the PCF is dominantly affected by the waveguide dispersion. The waveguide dispersions of the PCFs with different parameters are calculated by the MIPM. And the obtained waveguide dispersion of the PCF should satisfy the condition of scaling transformation property of the Maxwell’s equations which are expressed as [27

_{m}27. A. Ferrando, E. Silvestre, P. Andres, and J. J. Miret, “Designing the properties of dispersion-flattened photonic crystal fibers,” Opt. Express **9**, 687–697 (2001)
, http://www.opticsinfobase.org/abstract.cfm?URI=oe-9-13-687. [CrossRef] [PubMed]

*M*is the scaling factor and

*f*is air filling fraction of the PCF. The scaling property of the waveguide dispersion expressed in Eq (18) is well known and very useful in designing the waveguide dispersion of PCF.

## 5. Accuracy verification of the MIPM

*Δn*=|

*n*-

_{x}*n*| (

_{y}*n*and

_{x}*n*are effective indices of

_{y}*HE*and

_{11x}*HE*mode, respectively).

_{11y}*Λ*=4.6µm and

*d*/

*Λ*=0.2. The rms error decreases as the number of unit row or column matrix, used to express the index profile of the PCF, increases. Note that the rms error is reduced less than 5×10

^{-6}and saturated by increasing the number of unit matrix above 1200 as shown in the solid boxed region of the Fig. 9. From these results, one can say that the accuracy of the MIPM can be improved by increasing the number of unit matrix above 1200. However, increasing the number of unit matrix and the number of basis functions require additional computation time.

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

*HE*and

_{11x}*HE*) in the PCF with a triangular air-hole lattice, which means sixfold rotational symmetry, should be degenerate modes. Therefore, the PCF with a triangular air-hole lattice should exhibit very small birefringence.

_{11y}*Λ*=2.3µm,

*d*/

*Λ*=0.2) and the PCF (

*Λ*=4.6µm,

*d*/

*Λ*=0.2) by the MIPM and are less than 6×10

^{-5}and 1.3×10

^{-5}, respectively, over the wavelength range between 0.5µm and 1.8µm as shown in Fig. 10. This very small value of the birefringence <6×10

^{-5}over the wide wavelength range implies that the MIPM is accurate [28

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

## 6. Discussion and conclusion

^{-5}, indicating that the MIPM is reasonably accurate. The proposed MIPM has high accuracy like other simulation methods such as FEM, MPM and FDM. The eigenvalue matrix calculated by the MIPM used to analyze the modal property of the PCF is independent of wavelength. Even though the operating wavelength is changed, we can calculate the modal property without reconstructing the eigenvalue matrix. Therefore, MIPM is very efficient for designing the PCF with ultra-flattened dispersion or the ultra-wideband birefringent PCF. In conclusion, we are convinced that the MIPM is an efficient method for analyzing modal characteristics of the PCF with a complex index profile, because an arbitrary index profile of the PCF and its modal characteristics can be effectively obtained.

## Acknowledgments

## References and links

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

2. | S. K. Varshney, M. P. Singh, and R. K. Sinha, “Propagation Characteristics of Photonic Crystal Fibers,” J. Opt. Commun. |

3. | M. Midrio, M. P. Singh, and C. G. Someda, “The Space Filling Mode of Holey Fibers : An Analytical Vectorial Soultion,” J. Lightwave Technol. |

4. | Y. Li, Y. Yao, M. Hu, L. Chai, and C. Wang, “Improved fully vectorial effective index method for photonic crystal fibers: evaluation and enhancement,” Appl. Opt. |

5. | H. Li, A. Mafi, A. Schülzgen, L. Li, V. L. Temyanko, N. Peyghambarian, and J. V. Moloney, “Analysis and Design of Photonic Crystal Fibers Based on an Improved Effective-Index Method,” J. Lightwave Technol. |

6. | Y. Li, C. Wang, N. Zhang, C. Wang, and Q. Xing, “Analysis and design of terahertz photonic crystal fibers by an effective-index method,” Appl. Opt. |

7. | B. T. Kuhlmey, H. C. Nguyen, M. J. Steel, and B. J. Eggleton, “Confinement loss in adiabatic photonic crystal fiber tapers,” J. Opt. Soc. Am. B |

8. | T. P. Whiteet al, “Calculations of air-guided modes in photonic crystal fibers using the multipole method,” Opt. Express |

9. | A. Cucinotta, S. Selleri, L. Vincetti, and M. Zoboli, “Perturbation Analysis of Dispersion Properties in Photonic Crystal Fibers Through the Finite Element Method,” J. Lightwave Technol. |

10. | F. Brechetet al, “Complete analysis of the propagation characteristics into photonic crystal fibers, be the finite element method,” J. Optical Fiber Technol. |

11. | M. A. R. Franco, H. T. Hattori, F. Sircilli, A. Passaro, and N. M. Abe, “Finite Element Analysis of Photonic Crystal Fiber,” in PROC IEEE MTT-S IMOC, 5–7 (2001). |

12. | K. Saitoh and M. Koshiba, “Full-Vectorial Imaginary-Distance Beam Propagation Method Based on a Finite Element Scheme : Application to Photonic Crystal Fibers,” J. Quantum Electron. |

13. | W. Zhi, R Guobin, and L. Shuqin, “A Novel Supercell Overlapping Method for Different Photonic Crystal Fibers,” J. Lightwave Technol. |

14. | 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. |

15. | D. Mogilevtsev, T. A. Birks, and P. St. J. Russell, “Group-velocity dispersion in photonic crystal fibers,” Opt. Lett. |

16. | 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. | Z. Zhang and S. Satpathy, “Electromagnetic wave propagation in structures : Bloch wave solution of Maxwell’s equation,” Phys. Rev. Lett. |

18. | S. Guo and S. Albin, “Simple plane wave implementation for photonic crystal calculations,” Opt. Express |

19. | Z. Zhu and T. G. Brown, “Full-vectorial finite-difference analysis of microstructured optical fiber,” Opt. Express |

20. | K. N. Park and K. S. Lee, “Improved effective index method for analysis of photonic cyrstal fibers,” Opt. Lett. |

21. | W. Zhi, R. Guobin, L. Shuqin, and J. Shuisheng, “Supercell lattice method for photonic crystal fibers,” Opt. Express |

22. | T. M. Monro, D. J. Richardson, N. G. R. Broderick, and P. J. Bennett, “Holey Optical Fibers : An Efficient Modal Model,” J. Lightwave Technol. |

23. | T. M. Monro, D. J. Richardson, N. G. R. Broderick, and P. J. Bennett, “Modeling large air fraction holey optical fibers,” J. Lightwave Technol. |

24. | I. S. Gradshtein and I. M. Ryzhik, |

25. | I. Kimel and L. R. Elias, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quant. Electron. |

26. | T.-L. Wu and C.-H. Chao, “Photonic Crystal Fiber Analysis Through the Vector Boundary-Element Method : Effect of Elliptical Air Hole,” IEEE Photon. Tech. Lett. |

27. | A. Ferrando, E. Silvestre, P. Andres, and J. J. Miret, “Designing the properties of dispersion-flattened photonic crystal fibers,” Opt. Express |

28. | M. J. Steel, P. T. P. White, C. M. Sterke, R. C. McPhedran, and L. C. Bottn, “Symmerty and degeneracy in microstructured optical fibers,” Opt. Lett. |

**OCIS Codes**

(060.2270) Fiber optics and optical communications : Fiber characterization

(350.3950) Other areas of optics : Micro-optics

(060.5295) Fiber optics and optical communications : Photonic crystal fibers

**ToC Category:**

Photonic Crystal Fibers

**History**

Original Manuscript: May 27, 2008

Revised Manuscript: July 18, 2008

Manuscript Accepted: August 10, 2008

Published: August 13, 2008

**Citation**

Kwang No Park and Kyung Shik Lee, "Meshed index profile method for photonic crystal fibers with arbitrary structures," Opt. Express **16**, 13175-13187 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-17-13175

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### References

- T. A. Birks, J. C. Knight, and P. St. J. Russel, "Endlessly single mode photonic crystal fiber," Opt. Lett. 22, 961-963 (1997). [CrossRef] [PubMed]
- S. K. Varshney, M. P. Singh, and R. K. Sinha, "Propagation Characteristics of Photonic Crystal Fibers," J. Opt. Commun. 24, 192-198 (2003).
- M. Midrio, M. P. Singh, and C. G. Someda, "The Space Filling Mode of Holey Fibers : An Analytical Vectorial Soultion," J. Lightwave Technol. 18, 1031-1037 (2000). [CrossRef]
- Y. Li, Y. Yao, M. Hu, L. Chai, and C. Wang, "Improved fully vectorial effective index method for photonic crystal fibers: evaluation and enhancement," Appl. Opt. 47, 399-406 (2008). [CrossRef] [PubMed]
- H. Li, A. Mafi, A. Schülzgen, L. Li, V. L. Temyanko, N. Peyghambarian, and J. V. Moloney, "Analysis and Design of Photonic Crystal Fibers Based on an Improved Effective-Index Method," J. Lightwave Technol. 25, 1224-1230 (2007). [CrossRef]
- Y. Li, C. Wang, N. Zhang, C. Wang, and Q. Xing, "Analysis and design of terahertz photonic crystal fibers by an effective-index method," Appl. Opt. 45, 8462-8465 (2006). [CrossRef] [PubMed]
- B. T. Kuhlmey, H. C. Nguyen, M. J. Steel, and B. J. Eggleton, "Confinement loss in adiabatic photonic crystal fiber tapers," J. Opt. Soc. Am. B 23, 1965-1974 (2006). [CrossRef]
- T. P. White et al, "Calculations of air-guided modes in photonic crystal fibers using the multipole method," Opt. Express 11, 721-732 (2001), http://www.opticsinfobase.org/abstract.cfm?URI=oe-9-13-721. [CrossRef]
- A. Cucinotta, S. Selleri, L. Vincetti, and M. Zoboli, "Perturbation Analysis of Dispersion Properties in Photonic Crystal Fibers Through the Finite Element Method," J. Lightwave Technol. 20, 1433-1442 (2002). [CrossRef]
- F. Brechet et al, "Complete analysis of the propagation characteristics into photonic crystal fibers, be the finite element method," J. Optical Fiber Technol. 6, 181-201 (2001). [CrossRef]
- M. A. R. Franco, H. T. Hattori, F. Sircilli, A. Passaro, and N. M. Abe, "Finite Element Analysis of Photonic Crystal Fiber," in PROC IEEE MTT-S IMOC, 5-7 (2001).
- K. Saitoh and M. Koshiba, "Full-Vectorial Imaginary-Distance Beam Propagation Method Based on a Finite Element Scheme: Application to Photonic Crystal Fibers," J. Quantum Electron. 38, 927-933 (2002). [CrossRef]
- W. Zhi, R, Guobin, and L. Shuqin, "A Novel Supercell Overlapping Method for Different Photonic Crystal Fibers," J. Lightwave Technol. 22, 903-916 (2004). [CrossRef]
- 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]
- D. Mogilevtsev, T. A. Birks, and P. St. J. Russell, "Group-velocity dispersion in photonic crystal fibers," Opt. Lett. 23, 1662-1664 (1998). [CrossRef]
- 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, 1093-1101 (1999). [CrossRef]
- Z. Zhang and S. Satpathy, "Electromagnetic wave propagation in structures: Bloch wave solution of Maxwell's equation," Phys. Rev. Lett. 65, 2650-2653 (1990). [CrossRef] [PubMed]
- S. Guo and S. Albin, "Simple plane wave implementation for photonic crystal calculations," Opt. Express 11, 167-175 (2003), http://www.opticsinfobase.org/abstract.cfm?URI=oe-11-2-167. [CrossRef] [PubMed]
- Z. Zhu and T. G. Brown, "Full-vectorial finite-difference analysis of microstructured optical fiber," Opt. Express 10, 853-864 (2002), http://www.opticsinfobase.org/abstract.cfm?URI=oe-10-17-853">http://www.opticsinfobase.org/abstract.cfm?URI=oe-10-17-853. [PubMed]
- K. N. Park and K. S. Lee, "Improved effective index method for analysis of photonic cyrstal fibers," Opt. Lett. 30, 958-960 (2005). [CrossRef] [PubMed]
- W. Zhi, R. Guobin, L. Shuqin, and J. Shuisheng, "Supercell lattice method for photonic crystal fibers," Opt. Express 11, 980-991 (2003), http://www.opticsinfobase.org/abstract.cfm?URI=oe-11-9-980. [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]
- 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]
- I. S. Gradshtein and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1994).
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