## Compact supercell method based on opposite parity for Bragg fibers

Optics Express, Vol. 11, Issue 26, pp. 3542-3549 (2003)

http://dx.doi.org/10.1364/OE.11.003542

Acrobat PDF (940 KB)

### Abstract

The supercell- based orthonormal basis method is proposed to investigate the modal properties of the Bragg fibers. A square lattice is constructed by the whole Bragg fiber which is considered as a supercell, and the periodical dielectric structure of the square lattice is decomposed using periodic functions (cosine). The modal electric field is expanded as the sum of the orthonormal set of Hermite-Gaussian basis functions based on the opposite parity of the transverse electric field. The propagation characteristics of Bragg fibers can be obtained after recasting the wave equation into an eigenvalue system. This method is implemented with very high efficiency and accuracy.

© 2003 Optical Society of America

## 1. Introduction

1. P. Yeh, A. Yariv, and E. Marom, “Theory of Bragg fibers,” J. Opt. Soc. Am. , **68**, 1196–1201(1978). [CrossRef]

2. Y. Fink, J.N. Winn, S.H. Fan, C. Chen, J. Michel, J.D. Joannopoulos, and E.L. Thomas, “A dielectric omnidirectional reflector,” Science **282**, 1679–1682(1998). [CrossRef] [PubMed]

5. S.D. Hart, G.R. Maskaly, B. Temelkuran, P.H. Prideaux, J.D. Joannopoulos, and Y. Fink, “External reflection from omnidirectional dielectric mirror fibers,” Science **296**, 510–513(2002). [CrossRef] [PubMed]

6. P. Russell, “Photonic crystal fibers,” Science **299**, 358–362 (2003). [CrossRef] [PubMed]

8. Jes Broeng, Stig E. B. Libori, T. Sondergaard, and Anders Bjarklev, “Analysis of air-guiding photonic bandgap fibers,” Opt. Lett. **25**, 96–98 (2000). [CrossRef]

2. Y. Fink, J.N. Winn, S.H. Fan, C. Chen, J. Michel, J.D. Joannopoulos, and E.L. Thomas, “A dielectric omnidirectional reflector,” Science **282**, 1679–1682(1998). [CrossRef] [PubMed]

5. S.D. Hart, G.R. Maskaly, B. Temelkuran, P.H. Prideaux, J.D. Joannopoulos, and Y. Fink, “External reflection from omnidirectional dielectric mirror fibers,” Science **296**, 510–513(2002). [CrossRef] [PubMed]

4. 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]

9. M. Ibanescu, S.G. Johnson, M. Soljacic, J.D. Joannopoulos, and Y. Fink, “Analysis of mode structure in hollow dielectric waveguide fibers,” Phys. Rev. E **67**, 046608-1-8(2003). [CrossRef]

11. S.G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, T.D. Engeness, M. Soljacic, S.A. Jacobs, J.D. Joannopoulos, and Y. Fink, “Low-loss asymptotically single-mode propagation in large-core omniguide fibers,” Opt. Express **9**, 748–779 (2001), http://www.opticsexpress.org/ abstract. cfm? URI =OPEX-9-13-748. [CrossRef] [PubMed]

11. S.G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, T.D. Engeness, M. Soljacic, S.A. Jacobs, J.D. Joannopoulos, and Y. Fink, “Low-loss asymptotically single-mode propagation in large-core omniguide fibers,” Opt. Express **9**, 748–779 (2001), http://www.opticsexpress.org/ abstract. cfm? URI =OPEX-9-13-748. [CrossRef] [PubMed]

12. G. Ourang, Yong Xu, and A. Yariv, “Theoretical study on dispersion compensation in air-core Bragg fibers,” Opt. Express **10**, 899–908 (2002), http://www.opticsexpress.org/ abstract. cfm? URI =OPEX-10-17-899. [CrossRef]

14. J.A. Mosoriu, E. Silvestre, A. Ferrando, P. Andres, and J.J. Miret, “High-index-core Bragg fibers: dispersion properties,” Opt. Express **11**, 1400–1405 (2003), http://www.opticsexpress.org/ abstract. cfm? URI =OPEX-11-12-1400. [CrossRef]

4. 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]

5. S.D. Hart, G.R. Maskaly, B. Temelkuran, P.H. Prideaux, J.D. Joannopoulos, and Y. Fink, “External reflection from omnidirectional dielectric mirror fibers,” Science **296**, 510–513(2002). [CrossRef] [PubMed]

1. P. Yeh, A. Yariv, and E. Marom, “Theory of Bragg fibers,” J. Opt. Soc. Am. , **68**, 1196–1201(1978). [CrossRef]

11. S.G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, T.D. Engeness, M. Soljacic, S.A. Jacobs, J.D. Joannopoulos, and Y. Fink, “Low-loss asymptotically single-mode propagation in large-core omniguide fibers,” Opt. Express **9**, 748–779 (2001), http://www.opticsexpress.org/ abstract. cfm? URI =OPEX-9-13-748. [CrossRef] [PubMed]

12. G. Ourang, Yong Xu, and A. Yariv, “Theoretical study on dispersion compensation in air-core Bragg fibers,” Opt. Express **10**, 899–908 (2002), http://www.opticsexpress.org/ abstract. cfm? URI =OPEX-10-17-899. [CrossRef]

14. J.A. Mosoriu, E. Silvestre, A. Ferrando, P. Andres, and J.J. Miret, “High-index-core Bragg fibers: dispersion properties,” Opt. Express **11**, 1400–1405 (2003), http://www.opticsexpress.org/ abstract. cfm? URI =OPEX-11-12-1400. [CrossRef]

15. E. Silvestre, M.V. Andrés, and P. Andrés, “Biorthonormal-basis method for the vector description of optical-fiber mode,” J. Lightwave Technol. **16**, 923–928 (1998). [CrossRef]

16. R. D. Meade, A.M. Rappe, K.D. Brommer, J.D. Joannopoulos, and O.L. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B **48**, 8434–8437 (1993). [CrossRef]

*x*- and

*y*- direction. In Section 3, the transverse electric field is decomposed into Hermite-Gaussian basis based on the opposite parity of the transverse electric field, and an eigenvalue system is deduced from the full-vectorial coupling wave equations. In Section 4, a Bragg fiber, which has the same structure as in Ref. [11

**9**, 748–779 (2001), http://www.opticsexpress.org/ abstract. cfm? URI =OPEX-9-13-748. [CrossRef] [PubMed]

## 2. Supercell of the dielectric structure

*Λ*, and the low-index-layer thichness,

*a*, as illustrated in Fig. 1(a). The dielectric constants are alternately

*ε*

_{1}and

*ε*

_{2}, where

*ε*

_{2}<

*ε*

_{1}. The dielectric constant of the hollow core is

*ε*

_{3}, and its radius is

*R*. In order to analyze this fiber, a square lattice is constructed by the whole transverse profile of the Bragg fiber that is considered as a supercell, the lattice constant of the square lattice is

*D*, as illustrated in Fig. 1(b).

19. Shangping Guo and Sacharia 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]

*D*in Fig. 1(b), the Fourier transform of the dielectric constant is:

**r**=(

*x, y*) is the space position,

*A*=

*D*

^{2}is the area of the supercell,

**k**=(

*k*

_{x}

*, k*

_{y}), the vector in the reciprocal space of the supercell lattice, is linearly combined by the primitive reciprocal lattice vectors (2π/

*D*, 2π/

*D*),

*ε*(

**r**) is the dielectric structure of the supercell, which can be expressed in different regions as follows:

*ε*

_{i}and

*r*

_{i}are the dielectric constant and the outer radius of the

*i*th layer, and

*r*

_{0}is set zero,

*m*is the total number of the layers,

*ε*

_{b}is the background dielectric constant. Eq. (1) will be analytically expressed as [18]:

*x*- and

*y*- axial symmetric, i.e.,

*ε*(-

*x,y*)=

*ε*(

*x,-y*)=

*ε*(

*x,y*), it can be expressed as a sum of the cosine functions as

*P*+1) is the number of the expansion items,

*P*

_{ab},

*ε*

_{F}(

**k**) in Eq. (3) and expressed as Eq. (5).

**k**

_{m,n}=(

*k*

_{x}

*, k*

_{y})=2

*π*/

*D*×(

*m, n*).

*ε*(

**r**).

**9**, 748–779 (2001), http://www.opticsexpress.org/ abstract. cfm? URI =OPEX-9-13-748. [CrossRef] [PubMed]

*R*=30

*Λ*, and

*Λ*=0.434µm. The dielectric constant of the periodical cladding are

*ε*

_{1}=4.6

^{2}and

*ε*

_{2}=1.6

^{2}, the background is

*ε*

_{2}too. There are 17 layers, and the lower-index layer thichness is

*a*=0.78

*Λ*. The supercell lattice constant

*D*is set as 1.2(2

*R*+18

*Λ*), the expansion parameter

*P*=1200.

## 3. Eigenvalue system

### 3.1 opposite parity of the mode field

*z*-axis), both components

*e*

_{x}(

*x,y*) and

*e*

_{y}(

*x,y*) of the transverse electric field must satisfy the full-vectorial coupling wave equations [20,21

21. 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]

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

*e*

_{x}

*, e*

_{y}),

*k*

_{0}=2

*π*/

*λ*is the wave number of the vacuum.

*ε*(

*x, y*) is an even function about both the x and y directions, 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 guided eigenmode [22

22. 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]

*e*

_{x}(

*x,y*) is an even function about

*x*,

*e*

_{y}(

*x,y*) will be an odd function about

*x*, i.e.,

*e*

_{x}(-

*x,y*)=

*e*

_{x}(

*x,y*),

*e*

_{y}(-

*x,y*)=-

*e*

_{y}(

*x,y*).

### 3.2 subscripts

*m*and

*n*are introduced to express the opposite parity 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. For example, the subscript ‘10’ means that

*e*

_{x}(

*x,y*) is an odd function about

*x*and an even function about

*y, e*

_{y}(

*x,y*) is an even function about

*x*and an odd function about

*y*.

*mn*’ means that there are four sets of (

*e*

_{x}

*, e*

_{y}) with different parity. In Eq. (7),

*F*is the number of the expansion terms,

*s*=

*x,y*) are the expansion coefficients,

*ψ*

_{i}(

*s*) is the

*i*th order orthonormal Hermite-Gaussian function which is defined as:

*H*

_{i}(

*s*/

*ϖ*

_{s}) is the

*i*th Hermite function which has the parity of

*H*

_{i}(-

*s*/

*ϖ*

_{s})=(-1)

^{i}

*H*

_{i}(

*s*/

*ϖ*

_{s}),

*ϖ*

_{s}is the character width [21

21. 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]

22. 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]

### 3.3 eigen equations

*L*

_{mn}is a four-dimensional matrix,

*I*

^{(1)},

*I*

^{(2)},

*I*

^{(3)}and

*I*

^{(4)}are the overlapping integrals which are four-dimensional

*F*×

*F*×

*F*×

*F*matrix and expressed as follows.

23. 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]

24. R.G. Bin, W. Zhi, L.S. Qin, and J.S. Sheng, “Mode Classification and Degeneracy in Photonic Crystal Fiber,” Opt. Express **11**, 1310–1321 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-11-1310. [CrossRef]

*L*

_{mn}and

*ε*

^{s}can be transferred into a [2×

*F*

^{2}]×[2×

*F*

^{2}] 2-D matrix and a vector with 2×

*F*

^{2}elements, with which the eigen system Eq. (9) can be solved directly, the modes and the corresponding propagation constant

*β*can be calculated. There are 2×

*F*

^{2}eigenvalue- eigenvector pairs for every ‘

*mn*’, so the total results must be 4×2×

*F*

^{2}eigenvalue- eigenvector pairs after solving eigen equations corresponding to 4 different ‘

*mn*’. When the eigen values at the wavelength

*λ*are labeled in the decreasing order, the electric fields of the modes from the fundamental to higher-order can be obtained.

*F*

^{2}eigenvalue- eigenvector pairs, the decomposition terms must be 2

*F*, and the eigen equation will be a [2×(2

*F*)

^{2}]×[2×(2

*F*)

^{2}] 2-D matrix in which the element number exhibits an increase factor of 4×4=16 than that discussed above. The computation time is mainly dominated by three mechanisms: (1) constructing the matrix by analytically calculating definite integrals of Hermite-Gaussian functions, (2) the localization property of the Hermite-Gaussian functions increasing the efficiency and the accuracy for the confined modes and (3) the eigen systems with much less matrix elements greatly reducing the computation time for solving eigenvalue problems when standard algorithms are employed.

*x*- and

*y*- components; the coupling property is included in the coupled terms

*I*

^{(4)x}and

*I*

^{(4)y}; the fiber with complex dielectric structure can be investigated while it is made of the dielectric constant with negative or positive imaginary part (gain or absorption); the degeneracy property can be obtained from the full vectorial coupling wave equation.

## 4. Numerical results

*F*is set as 15. The transverse electric field of the lowest-order mode, which is a degenerated pair of modes HE

_{11x}and HE

_{11y}, and the second order modes are quivered in Fig. 3 at wavelength 1550nm. It can be found that light is strongly confined in the low index air core by the PBG due to the outside annular Bragg reflector. The critical property of TE

_{0m}(especially TE

_{01}) modes is that they have a node in their electric field (

*E*

_{φ}) near

*r*=

*R*. The fractional |

**E**|

^{2}in the cladding, hence the radiation loss, scales as 1/R

^{3}and 1/R for TE

_{0m}and other modes (TM, HE, EH), respectively. Because of the substantial discrimination between the single lowest-loss mode TE

_{01}and other higher-order guided modes, it allows even a highly multimode omniguide fiber to operate in an effectively single-mode fashion [9

9. M. Ibanescu, S.G. Johnson, M. Soljacic, J.D. Joannopoulos, and Y. Fink, “Analysis of mode structure in hollow dielectric waveguide fibers,” Phys. Rev. E **67**, 046608-1-8(2003). [CrossRef]

**9**, 748–779 (2001), http://www.opticsexpress.org/ abstract. cfm? URI =OPEX-9-13-748. [CrossRef] [PubMed]

_{01}is shown in Fig. 4(a), in which the solid line is obtained by the supercell method, and the cross is computed by the transfer matrix method (TMM). The difference between both approaches is demonstrated in Fig. 4(b). In our computation for the case with

*Λ*=0.434µm,

*λ*=1550nm, i.e.

*ω*=0.28×2π

*c*/

*Λ*, which is same as Ref. [11

**9**, 748–779 (2001), http://www.opticsexpress.org/ abstract. cfm? URI =OPEX-9-13-748. [CrossRef] [PubMed]

*Λ*, which has very good agreement with the value

*β*=0.27926×2π/

*Λ*in Ref. [11

**9**, 748–779 (2001), http://www.opticsexpress.org/ abstract. cfm? URI =OPEX-9-13-748. [CrossRef] [PubMed]

## 5. Conclusion

**9**, 748–779 (2001), http://www.opticsexpress.org/ abstract. cfm? URI =OPEX-9-13-748. [CrossRef] [PubMed]

## References and links

1. | P. Yeh, A. Yariv, and E. Marom, “Theory of Bragg fibers,” J. Opt. Soc. Am. , |

2. | Y. Fink, J.N. Winn, S.H. Fan, C. Chen, J. Michel, J.D. Joannopoulos, and E.L. Thomas, “A dielectric omnidirectional reflector,” Science |

3. | Y. Fink, D.J. Ripin, S. Fan, C. Chen, J.D. Joannopoulos, and E.L. Thomas, “Guiding optical light in air using an all-dielcetric structure,” J. Lightwave Technol. |

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

5. | S.D. Hart, G.R. Maskaly, B. Temelkuran, P.H. Prideaux, J.D. Joannopoulos, and Y. Fink, “External reflection from omnidirectional dielectric mirror fibers,” Science |

6. | P. Russell, “Photonic crystal fibers,” Science |

7. | J.C. Knight and P. St. Russell, “New ways to guide light,” Science |

8. | Jes Broeng, Stig E. B. Libori, T. Sondergaard, and Anders Bjarklev, “Analysis of air-guiding photonic bandgap fibers,” Opt. Lett. |

9. | M. Ibanescu, S.G. Johnson, M. Soljacic, J.D. Joannopoulos, and Y. Fink, “Analysis of mode structure in hollow dielectric waveguide fibers,” Phys. Rev. E |

10. | I.M. Bassett and A. Argyros, “Elimination of polarization degeneracy in round waveguides,” Opt. Express |

11. | S.G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, T.D. Engeness, M. Soljacic, S.A. Jacobs, J.D. Joannopoulos, and Y. Fink, “Low-loss asymptotically single-mode propagation in large-core omniguide fibers,” Opt. Express |

12. | G. Ourang, Yong Xu, and A. Yariv, “Theoretical study on dispersion compensation in air-core Bragg fibers,” Opt. Express |

13. | T.D. Engeness, M. Ibanescu, S.G. Johnson, O. Weisberg, M. Skorobogatiy, S. Jacobs, and Y. Fink, “Dispersion tailoring and compensation by modal interactions in omniguide fibers,” Opt. Express |

14. | J.A. Mosoriu, E. Silvestre, A. Ferrando, P. Andres, and J.J. Miret, “High-index-core Bragg fibers: dispersion properties,” Opt. Express |

15. | E. Silvestre, M.V. Andrés, and P. Andrés, “Biorthonormal-basis method for the vector description of optical-fiber mode,” J. Lightwave Technol. |

16. | R. D. Meade, A.M. Rappe, K.D. Brommer, J.D. Joannopoulos, and O.L. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B |

17. | Shangping Guo, Feng Wu, Khalid Ikram, and Sacharia Albin, “Analysis of circular fibers with arbitrary index profile by Galerkin method,” to be published by Opt. Lett.. |

18. | J.D. Joannopoulos, R.D. Meade, and J.N. Winn, “Photonic crystals: molding the flow of light,” (New York, Princeton university press, 1995). |

19. | Shangping Guo and Sacharia Albin, “Simple plane wave implementation for photonic crystal calculations,” Opt. Express |

20. | A.W. Snyder and J.D. Love, Optical waveguide theory, (New York: Chapman and Hall, 1983). |

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

22. | 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. |

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

24. | R.G. Bin, W. Zhi, L.S. Qin, and J.S. Sheng, “Mode Classification and Degeneracy in Photonic Crystal Fiber,” 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: October 30, 2003

Revised Manuscript: December 10, 2003

Published: December 29, 2003

**Citation**

Wang Zhi, Ren Guobin, Lou Shuqin, Liang Weijun, and Shangping Guo, "Compact supercell method based on opposite parity for Bragg fibers," Opt. Express **11**, 3542-3549 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-26-3542

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### References

- P. Yeh, A. Yariv, E. Marom, �??Theory of Bragg fibers,�?? J. Opt. Soc. Am. 68, 1196-1201 (1978). [CrossRef]
- Y. Fink, J.N. Winn, S.H. Fan, C.Chen, J. Michel, J.D. Joannopoulos, E.L. Thomas, �??A dielectric omnidirectional reflector,�?? Science 282, 1679-1682 (1998). [CrossRef] [PubMed]
- Y.Fink, D.J. Ripin, S.Fan, C. Chen, J.D. Joannopoulos, E.L. Thomas, �??Guiding optical light in air using an all-dielcetric structure,�?? J. Lightwave Technol. 17, 2039-2041 (1999). [CrossRef]
- M. Ibanescu, Y. Fink, S. Fan, E.L. Thomas, J.D. Joannopoulos, �??An all-dielectric coaxial waveguide,�?? Science 289, 415-419 (2000). [CrossRef] [PubMed]
- S.D. Hart, G.R.Maskaly, B.Temelkuran, P.H. Prideaux, J.D. Joannopoulos, Y.Fink, �??External reflection from omnidirectional dielectric mirror fibers,�?? Science 296, 510-513 (2002). [CrossRef] [PubMed]
- P. Russell, �??Photonic crystal fibers,�?? Science 299, 358-362 (2003). [CrossRef] [PubMed]
- J.C. Knight, P. St. Russell, �??New ways to guide light,�?? Science 296, 276-277 (2002). [CrossRef] [PubMed]
- Jes Broeng, Stig E. B. Libori, T. Sondergaard, and Anders Bjarklev, �??Analysis of air-guiding photonic bandgap fibers,�?? Opt. Lett. 25, 96-98 (2000). [CrossRef]
- M. Ibanescu, S.G. Johnson, M. Soljacic, J.D. Joannopoulos, Y.Fink, �??Analysis of mode structure in hollow dielectric waveguide fibers,�?? Phys. Rev. E 67, 046608-1-8 (2003). [CrossRef]
- I.M. Bassett, A. Argyros, �??Elimination of polarization degeneracy in round waveguides,�?? Opt. Express 10, 1342-1346 (2002), <a href="http://www.opticsexpress.org/ abstract. cfm? URI =OPEX-10�??23-1342">http://www.opticsexpress.org/ abstract. cfm? URI =OPEX-10�??23-1342</a>. [CrossRef] [PubMed]
- S.G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, T.D. Engeness, M. Soljacic, S.A. Jacobs, J.D. Joannopoulos, Y.Fink, �??Low-loss asymptotically single-mode propagation in large-core omniguide fibers,�?? Opt. Express 9, 748-779 (2001), <a href="http://www.opticsexpress.org/ abstract. cfm? URI =OPEX-9�??13-748">http://www.opticsexpress.org/ abstract. cfm? URI =OPEX-9�??13-748</a>. [CrossRef] [PubMed]
- G. Ourang, Yong Xu, A. Yariv, �??Theoretical study on dispersion compensation in air-core Bragg fibers,�?? Opt. Express 10, 899-908 (2002), <a href="http://www.opticsexpress.org/ abstract. cfm? URI =OPEX-10�??17-899">http://www.opticsexpress.org/ abstract. cfm? URI =OPEX-10�??17-899</a>. [CrossRef]
- T.D. Engeness, M. Ibanescu, S.G. Johnson, O.Weisberg, M.Skorobogatiy, S.Jacobs, Y.Fink, �??Dispersion tailoring and compensation by modal interactions in omniguide fibers,�?? Opt. Express 11, 1175-1196 (2003), <a href="http://www.opticsexpress.org/ abstract. cfm? URI =OPEX-11�??10-1175">http://www.opticsexpress.org/ abstract. cfm? URI =OPEX-11�??10-1175</a>. [CrossRef] [PubMed]
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