## Analysis of the space filling modes of photonic crystal fibers

Optics Express, Vol. 8, Issue 10, pp. 547-554 (2001)

http://dx.doi.org/10.1364/OE.8.000547

Acrobat PDF (617 KB)

### Abstract

We study the cladding modes of photonic crystal fibers (PCFs) using a fully vectorial method. This approach enables us to analyze the modes and incorporate material dispersion in a straightforward fashion. We find the field flow lines, intensity distribution and polarization properties of these modes. The effective cladding indices of different PCFs are investigated in detail.

© Optical Society of America

## 1. Introduction

1. D. Mogilevtsev, T. A. Birks, and P. St. J. Russell, “Group-velocity dispersion in photonic crystal fibers,” Opt. Lett. **23**, 1662–1664 (1998). [CrossRef]

2. 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. 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]

4. J. C. Knight, J. Boreng, T. A. Birks, and P. St. J. Russell, “Photonic band gap guidance in optical fibers,” Science **282**, 1476–1478 (1998). [CrossRef] [PubMed]

5. R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allen, “Single-mode photonic band gap guidance of light in air,” Science **285**, 1537–1539 (1999). [CrossRef] [PubMed]

6. M. Midrio, M. P. Singh, and C. G. Someda, “The space filling mode of holey fibers: an analytical vectorial solution,” J. Lightwave Technol. **18**, 1031 (2000). [CrossRef]

7. A. A. Maradudin and A. R. McGurn, “Out of plane propagation of electromagnetic waves in two-dimensional periodic dielectric medium,” J. Mod. Opt. **41**, 275–284 (1994). [CrossRef]

10. A. Ferrando, E. Silvestre, J. J. Miret, P. Andres, and M. V. Andres, “Full-vector analysis of a realistic photonic crystal fiber,” Opt. Lett. **24**, 276–278 (1999). [CrossRef]

3. 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]

6. M. Midrio, M. P. Singh, and C. G. Someda, “The space filling mode of holey fibers: an analytical vectorial solution,” J. Lightwave Technol. **18**, 1031 (2000). [CrossRef]

7. A. A. Maradudin and A. R. McGurn, “Out of plane propagation of electromagnetic waves in two-dimensional periodic dielectric medium,” J. Mod. Opt. **41**, 275–284 (1994). [CrossRef]

8. J. Broeng, T. Sondergaard, S. E. Barkou, P. M. Barbeito, and A. Bjarklev, “Waveguidance by the photonic bandgap effect in optical fiber,” J. Opt. A—Pure Appl. Opt. **1**, 477–482 (1999). [CrossRef]

3. 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]

6. M. Midrio, M. P. Singh, and C. G. Someda, “The space filling mode of holey fibers: an analytical vectorial solution,” J. Lightwave Technol. **18**, 1031 (2000). [CrossRef]

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

10. A. Ferrando, E. Silvestre, J. J. Miret, P. Andres, and M. V. Andres, “Full-vector analysis of a realistic photonic crystal fiber,” Opt. Lett. **24**, 276–278 (1999). [CrossRef]

## 2. The Method

*z*) direction, and write the modal magnetic field in the form

**H**

_{t}and

*H*

_{z}and are the transverse and longitudinal components of the modal magnetic field, respectively. Substituting Eq. (1) into the wave equation

**H**

_{t}:

*k*=

*ω*/

*c*=2

*π*/

*λ*is the wavevector and

*ε*=

*ε*(

*x*,

*y*) is the transverse dielectric constant profile. We write

**H**

_{t}as a column vector

*H*

_{x}and

*H*

_{y}, we expand

*ε*(

*x*,

*y*), ln

*ε*(

*x*,

*y*) and

*H*

_{t}(

*x*,

*y*) as

**x**

_{t}=

**x̂**

*x*+

**ŷ**

*y*, and

**G**(

*l*)=

*l*

_{x}

**b**

_{x}+

*l*

_{y}

**b**

_{y}is a vector in the reciprocal space. Here

*l*

_{x}and

*l*

_{y}are any two integers that we denote collectively by

*l*,

**b**

_{x}and

**b**

_{y}are the primitive vector of the reciprocal lattice. When these expansions are substituted into Eq. (3), it becomes the algebraic eigenvalue problem

**L**in the plane-wave basis.

*L*can be written as

*(*ε ^

**G**)and

*(*κ ^

**G**)in Eq. (4) as

*f*=

*πR*

^{2}/

*A*

_{c}is the air filling ratio,

*R*is the radius of the air holes,

*A*

_{c}is the area of the primitive cell.

*ε*

_{a}and

*ε*

_{b}are the dielectric constants of air and silica, respectively.

## 3. Cladding modes

*R*is the radius of the air-holes. The two vectors of the primitive cell are

*εb*is calculated using the Sellmeier equation with the parameters for fused silica as given in Ref. 11. The dielectric constant of air

*ε*

_{a}is assumed to be 1.0 at all wavelengths considered.

10. A. Ferrando, E. Silvestre, J. J. Miret, P. Andres, and M. V. Andres, “Full-vector analysis of a realistic photonic crystal fiber,” Opt. Lett. **24**, 276–278 (1999). [CrossRef]

*R*=0.3 µm. The fundamental cladding mode is a two-fold degenerate mode. Fig. 3 shows the transverse magnetic fields for these two degenerate modes at λ=1.5 µm. We can see that they are essentially linearly polarized, with minute deviation from linear polarization near silica-air interfaces. In Fig. 4 we show the transverse magnetic field intensity of the x-polarized fundamental mode. As expected, the field is strongly concentrated in the high dielectric region (silica).

2. 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]

*n*

_{eff,cl}is a function of wavelength λ, dielectric constant of silica

*ε*

_{b}, lattice pitch Λ, and air hole radius

*R*. Due to fact that Maxwell’s equations have no fundamental length scale,

*n*

_{eff,cl}is in fact a function of

*ε*

_{b}and normalized parameters λ/Λ and

*R*/Λ. In Fig. 6 we show a surface plot of

*n*

_{eff,cl}. The material dispersion of silica is considered in the calculations.

**22**, 961–963 (1997). [CrossRef] [PubMed]

*V*=

*k*Λ(

^{1/2}gives a good estimation of the number of guided modes inside the PCF. Fig. 7 shows how

*V*varies with wavelength for PCFs of different lattice pitches and air-filling ratios. We can see that the

*V*is well below the red dashed line (which corresponds to the single-mode cutoff value for traditional step-index optical fibers) for PCFs with small air holes. It is clearly that PCFs with relatively small air holes can be single mode over a very large spectral range. To appreciate the difference between the vectorial and scalar methods, we also show the results from the scalar approximation in Fig. 7(a). The relative difference between the two approaches reaches about 10% in the results shown. So, in order to accurately predict the single-mode behavior of PCFs (especially near the single-mode cutoff), it is advisable to use the vectorial method.

*R*=0.8 µm. More plane waves are needed to keep the accuracy for increasing wavelength λ and air-hole radius

*R*.

## 4. Discussion and Conclusion

2. 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]

12. 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);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]

## References and links

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

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

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

4. | J. C. Knight, J. Boreng, T. A. Birks, and P. St. J. Russell, “Photonic band gap guidance in optical fibers,” Science |

5. | R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allen, “Single-mode photonic band gap guidance of light in air,” Science |

6. | M. Midrio, M. P. Singh, and C. G. Someda, “The space filling mode of holey fibers: an analytical vectorial solution,” J. Lightwave Technol. |

7. | A. A. Maradudin and A. R. McGurn, “Out of plane propagation of electromagnetic waves in two-dimensional periodic dielectric medium,” J. Mod. Opt. |

8. | J. Broeng, T. Sondergaard, S. E. Barkou, P. M. Barbeito, and A. Bjarklev, “Waveguidance by the photonic bandgap effect in optical fiber,” J. Opt. A—Pure Appl. Opt. |

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

10. | A. Ferrando, E. Silvestre, J. J. Miret, P. Andres, and M. V. Andres, “Full-vector analysis of a realistic photonic crystal fiber,” Opt. Lett. |

11. | G. Agrawal, |

12. | 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);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] |

**OCIS Codes**

(060.2270) Fiber optics and optical communications : Fiber characterization

(060.2400) Fiber optics and optical communications : Fiber properties

**ToC Category:**

Research Papers

**History**

Original Manuscript: March 22, 2001

Published: May 7, 2001

**Citation**

Zhaoming Zhu and Thomas Brown, "Analysis of the space filling modes of photonic crystal fibers," Opt. Express **8**, 547-554 (2001)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-8-10-547

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### References

- 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, 1093-1102 (1999). [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]
- J. C. Knight, J. Boreng, T. A. Birks, and P. St. J. Russell, "Photonic band gap guidance in optical fibers," Science 282, 1476-1478 (1998). [CrossRef] [PubMed]
- R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allen, "Single-mode photonic band gap guidance of light in air," Science 285, 1537-1539 (1999). [CrossRef] [PubMed]
- M. Midrio, M. P. Singh, and C. G. Someda, "The space filling mode of holey fibers: an analytical vectorial solution," J. Lightwave Technol. 18, 1031 (2000). [CrossRef]
- A. A. Maradudin and A. R. McGurn, "Out of plane propagation of electromagnetic waves in two-dimensional periodic dielectric medium," J. Mod. Opt. 41, 275-284 (1994). [CrossRef]
- J. Broeng, T. Sondergaard, S. E. Barkou, P. M. Barbeito, and A. Bjarklev, "Waveguidance by the photonic bandgap effect in optical fiber," J. Opt. A - Pure Appl. Opt. 1, 477-482 (1999). [CrossRef]
- E. Silvestre, M. V. Andres, and P. Andres, "Biorthonormal-basis method for the vector description of optical fiber modes," J. Lightwave Technol. 16, 923-928 (1998). [CrossRef]
- A. Ferrando, E. Silvestre, J. J. Miret, P. Andres, and M. V. Andres, "Full-vector analysis of a realistic photonic crystal fiber," Opt. Lett. 24, 276-278 (1999). [CrossRef]
- G. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1995).
- 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); 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]

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