## Supercell lattice method for photonic crystal fibers

Optics Express, Vol. 11, Issue 9, pp. 980-991 (2003)

http://dx.doi.org/10.1364/OE.11.000980

Acrobat PDF (1139 KB)

### Abstract

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

© 2003 Optical Society of America

## 1. Introduction

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

**E**or

**H**.

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

10. T. M. Monro, D. J. Richardson, N. G. R. Broderick, and P. J. Bennett, “Modeling large air fraction holey optical fibers,” J. Lightwave Technol. **18**, 50–56 (2000). [CrossRef]

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

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

## 2. Supercell lattice

*D*, the hole diameter

*d*, and the supercell lattice period

*ND*. To describe the dielectric structure of Fig. 1(a), two 2-D perfect PCs (PC1 and PC2) are introduced, as shown in Fig. 1(b) and Fig. 1(c), with the parameters shown in Table 1. Adding the dielectric structures of both PC1 [Fig. 1(b)] and PC2 [Fig. 1(c)] will form the dielectric structure of the supercell lattice PCF [Fig. 1(a)].

*x, y, z*and

*z*’ are selected as desired; then the following identities must be satisfied in order to reconstruct the missing hole region.

*x, y, z*and

*z*’) of PC1 are selected, the results will not change. For the case of simplicity, they are set as the last two lines in Table 1. Here

*x*and

*y*of PC1 are selected as

*P*

_{1}+1) and (

*P*

_{2}+1) are the number of decomposition terms of PC1 and PC2, respectively, and

*P*

_{1ab},

*P*

_{2ab},

*P*

_{1ab}is illustrated as an example.

*F*(

*K*

_{mn}), the 2-D Fourier transform coefficients, can be written as [4

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

*J*

_{1}is the first-order Bessel function and

*k*

_{x}and

*k*

_{y}depend on the reciprocal lattice vector of the triangular lattice [3,4

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

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

*m*and

*n*from [-

*P*

_{1},

*P*

_{1}] to [0, 2

*P*

_{1}], Eq. (4) will become Eq. (8).

*P*

_{1ab}can be analytically expressed as the composition of

*F*(

*K*

_{mn}) shown as Eq. (9).

*y*=0 axis of the dielectric reconstructions for a PCF with the structure parameters

*D*=2.3 µm,

*d*=0.69 µm, and

*P*

_{1}=50,

*N*=10,

*P*

_{2}=500. Obviously, it can describe the supercell lattice efficiently and accurately. In fact, when

*P*

_{2}=

*P*

_{1}*

*N*, the reconstructed dielectric structure will be the most accurate according to the Fourier transform.

## 3. Electric field and eigenvalue system

*z*) direction, so a main task is to investigate the transverse mode field distribution

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

*x*or

*y*axis:

*e*

_{x}(

*x,y*) and

*e*

_{y}(

*x,y*) satisfy a pair of coupled wave equations [10

10. T. M. Monro, D. J. Richardson, N. G. R. Broderick, and P. J. Bennett, “Modeling large air fraction holey optical fibers,” J. Lightwave Technol. **18**, 50–56 (2000). [CrossRef]

*F*+1) is the number of terms retained in this expansion, ψ

_{i}(

*s*) (

*i*=

*a,b, s*=

*x,y*) are elements of the orthogonal set of Hermite—Gaussian basis functions:

*H*

_{2i}(

*s*/ω) is the 2

*i*th-order Hermite polynomial and ω is the characteristic width of the basis set. As discussed by Monro

*et al*. [11

11. T. M. Monro, D. J. Richardson, N. G. R. Broderick, and P. J. Bennett, “Holey optical fibers: an efficient modal model,” J. Lightwave Technol. **17**, 1093–1102 (1999). [CrossRef]

10. T. M. Monro, D. J. Richardson, N. G. R. Broderick, and P. J. Bennett, “Modeling large air fraction holey optical fibers,” J. Lightwave Technol. **18**, 50–56 (2000). [CrossRef]

11. T. M. Monro, D. J. Richardson, N. G. R. Broderick, and P. J. Bennett, “Holey optical fibers: an efficient modal model,” J. Lightwave Technol. **17**, 1093–1102 (1999). [CrossRef]

*M*

^{s}=

*I*

^{(1)}+

*k*

^{2}

*I*

^{(2)}+

*I*

^{(3)s}is an (

*F*+1)×(

*F*+1)×(

*F*+1)×(

*F*+1)-order four-dimensional (4D) matrix; β

_{s}is the propagation constant of

*e*

_{x}or

*e*

_{y}; ε

^{s}is an (

*F*+1)×(

*F*+1)-order 2-D matrix;

*k*=2π/λ is the wave number; and

*I*

^{(1)},

*I*

^{(2)}, and

*I*

^{(3)}are overlap integrals of the modal functions, which are defined as [10

**18**, 50–56 (2000). [CrossRef]

11. T. M. Monro, D. J. Richardson, N. G. R. Broderick, and P. J. Bennett, “Holey optical fibers: an efficient modal model,” J. Lightwave Technol. **17**, 1093–1102 (1999). [CrossRef]

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

*l*

_{x}=

*D*and

*l*

_{y}=sqrt(3)

*D*. Both parts of the overlap integrals

*I*

^{(2)},

*I*

^{(3)x}, and

*I*

^{(3)y}depend on the decomposition coefficients

*P*

_{1ab},

*P*

_{2ab},

*I*

^{(2)},

*I*

^{(3)x}, and

*I*

^{(3)y}will stay unchanged no matter how the PCF is split.

*M*

^{s}and ε

^{s}can be transferred into an (

*F*+1)

^{2}×(

*F*+1)

^{2}2-D matrix and a vector with (

*F*+1)

^{2}elements, which are still written as

*M*

^{s}and ε

^{s}for compactness.

## 4. Results

### 4.1 Modal electric field distribution

*x*-polarized fundamental modal intensity profiles (|

*E*

_{x}|

^{2}) at 0.633 and 1.55 µm are shown in Figs. 3(a) and 3(b), respectively. Figure 4 demonstrates the corresponding contour lines, and the dielectric constant profile is superimposed. The intensity contours are spaced by 2 dB from -30 dB. Obviously, the wavelength is one of the key factors in determining the localization extent of the transverse mode. At shorter wavelengths, the field is limited in the high-index core, and at longer wavelengths, the field penetrates further into the periodic cladding region.

### 4.2 Effective modal area A_{eff}

*D*and the hole size

*d*. Much current research focuses on designing fibers with very large or very small effective modal areas [21

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

23. N.A. Mortensen, “Effective area of photonic crystal fibers,” Opt. Express **10**, 341–348 (2002), http://www.opticsexpress.org/abstract.cfm?URI = OPEX-10-7-341. [CrossRef] [PubMed]

*D*or

*d*during the fabrication process by means of drawing the fiber under different conditions.

*A*

_{eff}can be expressed as

_{s}) is gamma function. Using the evaluated eigenvector from the eigenvalue Eq. (13),

*A*

_{eff}can be calculated efficiently and accurately for a given PCF at any wavelength.

*d*/

*D*, whereas the hole spacing remains constant at

*D*=2.3 µm; and Fig. 5(b) shows

*A*

_{eff}with different hole spacing

*D*, whereas

*d*/

*D*=0.3.

*A*

_{eff}of the PCFs can be several times less than or larger than that of conventional fibers, even extremely less or larger with extreme structure parameters. Hence more attention is paid to the nonlinearity characteristics of PCFs. When

*d*/

*D*is greater, the mode field will be confined more in the central area of the PCF. When

*D*, the distance between the nearest holes, is greater, it is natural that the mode field will extend outside. When the wavelength increases, the same as with conventional fibers, the electric field will penetrate into the outer cladding more easily. This can also be seen from Figs. 3 and 4. In general, when

*D*increases, or

*d*/

*D*decreases, or the wavelength increases,

*A*

_{eff}will increase.

*A*

_{eff}of this paper is twice that shown in the results of Monro

*et al*. in Refs. [10

**18**, 50–56 (2000). [CrossRef]

**17**, 1093–1102 (1999). [CrossRef]

### 4.3 Dispersion

*D*

_{t}is proportional to the second-order derivative of the modal effective index (or the propagation constant) with respect to the wavelength λ.

*D*

_{t}could be written as the sum of the waveguide dispersion

*D*

_{w}and the material dispersion

*D*

_{m}, and

*D*

_{m}can be calculated by application of the Sellmeyer formula. Because PCF is made of silica, the material dispersion is the same for PCFs with different structural parameters, and the total dispersion coefficient will be dominated by

*D*

_{w}. When we take account of the scaling transformation property of Maxwell’s equations, there is a scaling property of the waveguide dispersion expressed as Eq. (21) [24

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

*M*, the waveguide dispersion coefficient is reduced to 1/

*M*, and at the same time the corresponding wavelength shifts to

*M*λ. This equation is useful for analyzing the waveguide dispersion of PCFs.

## 5. Accuracy and efficiency

*z*=1/2 and

*z*’=0.

*et al*. [25

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

_{11}-like modes, so they must be degenerate. As a result, the observation of birefringence must be a result of asymmetry in the structure or the error produced by the numerical calculations. When an ideal PCF is examined, the modal birefringence Δ

*n*=

*n*

_{x}-

*n*

_{y}can be used to scale the accuracy of the algorithm. The lower the value of |Δ

*n*|, the more accurate the numerical method. Figure 7 shows the modal birefringence evaluated from the SLM model. Obviously, the result is accurate, since |Δ

*n*| is less than 5×10

^{-5}over the wavelength range of 0.9–1.7 µm. The accuracy can be improved while increasing

*P*

_{1},

*N*, and

*P*

_{2}for a certain PCF, but this will cost more time.

## 6. Discussion and conclusion

*f*and

*k*

_{x}in Eq. (4) are divided by the ellipticity ratio η=

*b*/

*a*, where

*b*and

*a*are the lengths of the major and minor axes of the elliptical holes, the SLM can be used to investigate the elliptical hole PCF [28

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

30. M. J. Steel and R. M. Osgood, “Polarization and dispersive properties of elliptical-hole photonic crystal fibers,” J. Lightwave Technol. **19**, 495–503 (2001). [CrossRef]

*D*=2.3 µm, b/

*D*=0.8, η=3,

*P*

_{1}=30,

*N*=10,

*P*

_{2}=300.

## References and links

1. | J. Broeng, D. Mogilevstev, S. E. Barkou, and A. Bjarklev, “Photonic crystal fibers: a new class of optical waveguides,” Optical Fiber Technol. |

2. | S. E. Barkou, J. Broeng, and A. Bjarklev, “Dispersion properties of photonic bandgap guiding fibers,” in |

3. | J. D. Joannopoulos, R. D. Meade, and J. N. Winn, |

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

5. | T. A. Birks, D. Mogilevtsev, J. C. Knight, P. St. J. Russell, J. Broeng, P. J. Roberts, J. A. West, D. C. Allan, and J. C. Fajardo, “The analogy between photonic crystal fibres and step index fibres,” in |

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

7. | T. A. Birk, D. Mogilevtsev, J. C. Knight, and P. St. J. Russell, “Single material fibers for dispersion compensation,” in |

8. | A. Bjarklev, J. Broeng, K. Dridi, and S. E. Barkou, “Dispersion properties of photonic crystal fibres,” in |

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

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

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

12. | T.M. Monro, D.J. Richardson, and N.G.R. Broderick, “Efficient modeling of holey fibers,” in |

13. | D. Mogilevtsev, T. A. Birks, and P. St. Russell, “Localized function method for modeling defect mode in 2-d photonic crystal,” J. Lightwave Technol. |

14. | M. Koshiba, “Full vector analysis of photonic crystal fibers using the finite element method,” IEICE Electron, E85-C , |

15. | Z. Zhu and T. G. Brown, “Full-vectorial finite-difference analysis of microstructured optical fibersOpt. Express |

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

17. | A. W. Snyder, |

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

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

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

21. | J.M. Dudley and S. Coen, “Numerical simulations and coherence properties of supercontinuum Generation in Photonic Crystal and Tapered Optical Fibers,” IEEE J. Selected Topics in Quantum. Electron. |

22. | N.A. Mortensen, J.R. Folken, P.M.W. Skovgaard, and J. Broeng, “Numerical aperture of single-mode photonic crystal fibers,” IEEE Photon. Tech. Lett. , |

23. | N.A. Mortensen, “Effective area of photonic crystal fibers,” Opt. Express |

24. | A. Ferrando, E. Silvestre, P. Andres, J.J. Miret, and M.V. Andres, “Designing the properties of dispersionflattened photonic crystal fibers,” Opt. Express |

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

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

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

28. | M. J. Steel and P. M. Osgood, Jr, “Ellipitical-hole photonic crystal fibers,” Opt. Lett. |

29. | J. Broeng, D. Mogilevtsev, S. E. B. Libori, and A. Bjarklev, “Polarization-preserving holey fibers,” in |

30. | M. J. Steel and R. M. Osgood, “Polarization and dispersive properties of elliptical-hole photonic crystal fibers,” J. Lightwave Technol. |

**OCIS Codes**

(060.2270) Fiber optics and optical communications : Fiber characterization

(350.3950) Other areas of optics : Micro-optics

**ToC Category:**

Research Papers

**History**

Original Manuscript: March 31, 2003

Revised Manuscript: April 14, 2003

Published: May 5, 2003

**Citation**

Wang Zhi, Ren Guobin, Lou Shuqin, and Jian Shuisheng, "Supercell lattice method for photonic crystal fibers," Opt. Express **11**, 980-991 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-9-980

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### References

- J. Broeng, D. Mogilevstev, S. E. Barkou, and A. Bjarklev, �??Photonic crystal fibers: a new class of optical waveguides,�?? Optical Fiber Technol. 5, 305-330 (1999). [CrossRef]
- S. E. Barkou, J. Broeng, and A. Bjarklev, �??Dispersion properties of photonic bandgap guiding fibers,�?? in Optical Fiber Communication Conference (Optical Society of America, Washington, D.C., 1998), FG5, pp. 117-119.
- J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, New York, 1995).
- S. Guo and S. Albin, �??Simple plane wave implementation for photonic crystal calculations,�?? Opt. Express 11, 167-175 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI = OPEX-11-2-167">http://www.opticsexpress.org/abstract.cfm?URI = OPEX-11-2-167</a> [CrossRef] [PubMed]
- T. A. Birks, D. Mogilevtsev, J. C. Knight, P. St. J. Russell, J. Broeng, P. J. Roberts, J. A. West, D. C. Allan, and J. C. Fajardo, �??The analogy between photonic crystal fibres and step index fibres,�?? in Optical Fiber Communication Conference, (Optical Society of America, Washington, D.C., 1998), FG4, pp. 114-116.
- 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]
- T. A. Birk, D. Mogilevtsev, J. C. Knight, and P. St. J. Russell, �??Single material fibers for dispersion compensation,�?? in Optical Fiber Communication Conference (Optical Society of America, Washington, D.C., 1999), FG2, pp. 108-110.
- A. Bjarklev, J. Broeng, K. Dridi, and S. E. Barkou, �??Dispersion properties of photonic crystal fibres,�?? in European Conference on Optical Communication, (Madrid, Spain, 1998), pp. 135-136.
- R. Guobin, L. Shuqin, W. Zhi, and J. Shuisheng are preparing a manuscript to be called �??Study on dispersion properties of photonic crystal fiber by effective-index model�?? (in Chinese).
- 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]
- 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, �??Efficient modeling of holey fibers,�?? in Optical Fiber Communication Conference (Optical Society of America, Washington, D.C., 1999), FG3, 111-113.
- D. Mogilevtsev, T. A. Birks, P. St. Russell, �??Localized function method for modeling defect mode in 2-d photonic crystal,�?? J. Lightwave Technol. 17, 2078-2081 (1999). [CrossRef]
- M. Koshiba, �??Full vector analysis of photonic crystal fibers using the finite element method,�?? IEICE Electron, E85-C, 4, 881-888 (2002). (C) 2003 OSA 5 May 2003 / Vol. 11, No. 9 / OPTICS EXPRESS 980 #2290 - $15.00 US Received March 31, 2003; Revised April 15, 2003
- Z. Zhu and T. G. Brown, �??Full-vectorial finite-difference analysis of microstructured optical fibers Opt. Express 10, 853-864 (2002), <a href="http://www.opticsexpress.org/abstract.cfm?URI = OPEX-10-17-853">http://www.opticsexpress.org/abstract.cfm?URI = OPEX-10-17-853</a> [CrossRef] [PubMed]
- L. P. Shen, C. L. Xu, and W. P. Huang are preparing a manuscript to be called �??Modal characteristics of index-guiding photonic crystal fibers: a comparison between scalar and vector analysis.�??
- A. W. Snyder, Optical Waveguide Theory (Chapman and Hall, New York, 1983).
- I.S. Gradshtein and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1994).
- I. Kimel and L. R. Elias, �??Relations between Hermite and Laguerre Gaussian modes,�?? IEEE J. Quant. Electron. 29, 2562-2567 (1993). [CrossRef]
- W. Zhi, R. Guobin, L. Shuqin, L. Weijun, and J. Shuisheng, �??the mode characteristics of the photonic crystal fibers,�?? to be published by ACTA OPTICA SINICA (in Chinese).
- J.M. Dudley, and S. Coen, �??Numerical simulations and coherence properties of supercontinuum Generation in Photonic Crystal and Tapered Optical Fibers,�?? IEEE J. Selected Topics in Quantum. Electron. 8, 651-659 (2002). [CrossRef]
- N.A. Mortensen, J.R. Folken, P.M.W. Skovgaard, and J. Broeng, �??Numerical aperture of single-mode photonic crystal fibers,�?? IEEE Photon. Tech. Lett., 14, 1094-1096 (2002). [CrossRef]
- N.A. Mortensen, �??Effective area of photonic crystal fibers,�?? Opt. Express 10, 341-348 (2002), <a href="http://www. opticsexpress.org/abstract.cfm?URI = OPEX-10-7-341.">http://www. opticsexpress.org/abstract.cfm?URI = OPEX-10-7-341</a> [CrossRef] [PubMed]
- A. Ferrando, E. Silvestre, P. Andres, J.J. Miret, and M.V. Andres, �??Designing the properties of dispersion- flattened photonic crystal fibers,�?? Opt. Express 9, 687-697 (2001), <a href="http://www.opticsexpress.org/abstract.cfm?URI = OPEX-9-13-687">http://www.opticsexpress.org/abstract.cfm?URI = OPEX-9-13-687</a> [CrossRef] [PubMed]
- M. J. Steel, T. P. White, C. M. Sterke, R. C. McPhedran, and L. C. Botten, �??Symmetry and degeneracy in microstructured optical fibers,�?? Opt. Lett. 26, 488-490 (2001). [CrossRef]
- R. G. Bin, W. Zhi, L. S. Qin, and S. S. Jian are preparing a manuscript to be called �??Study on dispersion properties of photonic crystal fiber by effective-index model.�??
- P. Shen, W. P. Huang, and S. S. Jian, �??Design of photonic crystal fibers for dispersion-related applications�?? IEEE Photon. Tech. Lett. (to be published).
- M. J. Steel and P. M. Osgood, Jr, �??Ellipitical-hole photonic crystal fibers,�?? Opt. Lett. 26, 229-231 (2001). [CrossRef]
- J. Broeng, D. Mogilevtsev, S. E. B. Libori, and A. Bjarklev, �??Polarization-preserving holey fibers,�?? in Optical Fiber Communication Conference (Optical Society of America, Washington, D.C., 2001), MA1.3, pp. 6-7.
- M. J. Steel and R. M. Osgood, �??Polarization and dispersive properties of elliptical-hole photonic crystal fibers,�?? J. Lightwave Technol. 19, 495-503 (2001). [CrossRef]

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