## Numerical analysis of directional coupling in dual-core microstructured optical fibers

Optics Express, Vol. 17, Issue 18, pp. 15778-15789 (2009)

http://dx.doi.org/10.1364/OE.17.015778

Acrobat PDF (407 KB)

### Abstract

In this paper, we present a numerical analysis of the coupling coefficients in dual-core air-silica microstructured optical fibers with π/6 symmetry. The calculations are based on an especially fitted application of the coupled mode theory for microstructured optical fibers. This method is compared with three other techniques, the supermode method, the beam propagation method and the equivalent fiber model, and is shown to be very computationally efficient. Our studies enable us to derive a formula linking the coupling coefficients to core separation according to the wavelength, the pitch and the hole diameter of the fiber structure.

© 2009 Optical Society of America

## 1. Introduction

1. J. C. Knight, T. A. Birks, P. St. J. Russell, and D. M. Atkin, “All-silica single-mode optical fiber with photonic crystal cladding,” Opt. Lett. **21**, 1547–1549 (1996).
[CrossRef] [PubMed]

2. J. C. Knight, J. Arriaga, T. A. Birks, A. Ortigosa-Blanch, W. J. Wadsworth, and P. St. J. Russell, “Anomalous dispersion in photonic crystal fiber,” IEEE Photon. Technol. Lett. **12**, 807–809 (2000).
[CrossRef]

3. B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, and A. H. Greenaway, “Experimental study of dual-core photonic crystal fibre,” Electron. Lett. **36**, 1358–1359 (2000).
[CrossRef]

4. I. Velchev and J. Toulouse, “Directional coupling and switching in multi-core microstructure fibers,” in *Conference on Lasers and Electro-Optics* Technical digest(CD) (Optical Society of America, 2004), paper CTuV1, http://www.opticsinfobase.org/abstract.cfm?URI=CLEO-2004-CTuV1.

5. M. Zghal, R. Cherif, and F. Bahloul, “Improving triangular-lattice photonic-crystal-fiber couplers by introducing geometric nonuniformities,” Opt. Eng.46, n°095004 (2007). [CrossRef]

6. K. Saitoh, Y. Sato, and M. Koshiba, “Coupling characteristics of dual-core photonic crystal fiber couplers,” Opt. Express **11**, 3188–3195(2003), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-11-24-3188.
[CrossRef] [PubMed]

5. M. Zghal, R. Cherif, and F. Bahloul, “Improving triangular-lattice photonic-crystal-fiber couplers by introducing geometric nonuniformities,” Opt. Eng.46, n°095004 (2007). [CrossRef]

7. D. M. Taylor, C. R. Bennet, T. J. Shepherd, L. F. Michaille, M. D. Nielsen, and H. R. Simonsen, “Demonstration of multi-core photonic crystal fibre in an optical interconnect,” Electron. Lett. **42**, 331–332 (2006).
[CrossRef]

8. K. L. Reichenbach and C. Xu, “Independent core propagation in two-core photonic crystal fibers resulting from structural nonuniformities,” Opt. Express **13**, 10336–10348 (2005). http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-13-25-10336 [CrossRef] [PubMed]

9. Y. Yan, J. Toulouse, I. Velchev, and S. V. Rotkin, ≪ Decoupling and asymmetric coupling in triplecore photonic crystal fibers, » J. Opt. Soc. Am. B **25**, 1488–1495 (2008).
[CrossRef]

10. F. Fogli, L. Saccomandi, P. Rossi, G. Bellanca, and S. Trillo, “Full vectorial BPM modeling of Index-Guiding Photonic Crystal Fibers and Couplers,” Opt. Express **10**, 54–59 (2002), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-10-1-54 [PubMed]

6. K. Saitoh, Y. Sato, and M. Koshiba, “Coupling characteristics of dual-core photonic crystal fiber couplers,” Opt. Express **11**, 3188–3195(2003), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-11-24-3188.
[CrossRef] [PubMed]

11. N. Florous, K. Saitoh, and M. Koshiba, “A novel approach for designing photonic crystal fiber splitters with polarization independent propagation characteristics,” Opt. Express **13**, 7365–7373 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-13-19-7365 [CrossRef] [PubMed]

12. X. Yu, M. Liu, Y. Chung, M. Yan, and P. Shum, “Coupling coefficient of two-core microstructured optical fiber,” Opt. Commun. **260**, 164–169 (2006).
[CrossRef]

## 2. Studied microstructure definition

^{2}which are involved in many MOF guiding properties. The cladding lattice is a triangular arrangement of air holes in silica background material. The guiding properties are due to modified total internal reflection between the solid core and the holey cladding.

^{th}hole next to the first core, denoted h

_{m}. Thus, the distance between cores S is defined relatively to the pitch size of the microstructure Λ by the relation S=mΛ with m≥2 in geometry #1 or S=2mΛ cos(π/6) with m≥1 in geometry #2.

## 3. Numerical methods for coupling coefficient calculations

### 3.1 Supermode method (SM)

^{+}

_{x}, E

^{-}

_{x}the symmetric and antisymmetric x-polarized fields, and E

^{+}

_{y}and E

^{-}

_{y}the y-polarized ones, respectively.

^{+}

_{x/y}and β

^{-}

_{x/y}are different, they propagate at different phase velocities. These phase mismatches induce along the fiber an evolution of the global field distribution. This can be interpreted as power coupling between the two cores as well as the spatial beating of the two corresponding supermodes. The coupling coefficients C

_{x}and C

_{y}corresponding to each polarization can be inferred from [13] as

14. T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. Martijn de Sterke, and L. G. Botten, “Multipole method for microstructured optical fibers. I. Formulation,” J. Opt. Soc. Am. B **19**, 2322–2330 (2002).
[CrossRef]

15. T. P. White, R. C. McPhedran, L. G. Botten, G. Smith, and C. Martijn de Sterke, “Calculation of air-guided modes in photonic crystal fibers using the multipole method,” Opt. Express **9**, 721–732 (2001).
[CrossRef] [PubMed]

16. P. J. Roberts and T. J. Shepherd, “The guidance properties of multi-core photonic crystal fibres,” J. Opt. A, Pure Appl. Opt. **3**, 133–140 (2001).
[CrossRef]

### 3.2 Beam Propagation Method (BPM)

_{x/y}is the coupling length for the x-axis or the y-axis polarized modes respectively, i.e. the length over which 100% of the energy is transferred from one core to another.

### 3.3 Equivalent Fiber Model (EFM)

12. X. Yu, M. Liu, Y. Chung, M. Yan, and P. Shum, “Coupling coefficient of two-core microstructured optical fiber,” Opt. Commun. **260**, 164–169 (2006).
[CrossRef]

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

### 3.4 Coupled Mode Theory (CMT)

_{1}(x, y) and e⃗

_{2}(x,y) are the transverse electric fields of the guided modes of each optically independent core (i.e. the unperturbed MOF) constituting the dual-core MOF (i.e. the perturbed MOF). Thus, the transverse field e⃗

_{2}(x,y) in the second core exhibits the same spatial distribution as e⃗

_{1}(x, y) but is translated with respect to e⃗

_{1}(x, y) by the value of the core separation S (Fig. 1) along the x-axis, leading to e⃗

_{2}(x, y)=e⃗

_{1}(x-S,y).

_{perturbation}) because the difference between a two-core MOF and a single-core MOF defined by the expression Δn

^{2}(x,y)=n

^{2}(x,y)-n̄

^{2}(x,y) is non-null only in the air-hole corresponding to the second core position (Fig. 4).

_{1}(x, y)). In our case, the field distribution e⃗

_{1}of the guided mode in a single-core MOF is computed using a FEM based software (Fig. 5) but all other methods (FDM, MM, PWEM) giving field distribution can be utilized. One can note that the electric field distribution in the different holes (m=2 to 4) are very similar to each other while the amplitude decreases as m is increased.

_{perturbation}is the hole area). The main advantage of this technique is that the calculation time is independent of C even for very low coupling coefficient values for which the coupling length is very long thus requiring long computational time up to several days with beam propagation methods. However an upper limit exists since this method is based on the assumption that the field of a two-core MOF can be decomposed as the sum of two modes evolving in independent single-core MOFs which becomes incorrect in the case of very close cores.

^{th}hole corresponding to the position of the second core. Using our model, it is straightforward to determine the coupling coefficients in two-core microstructured optical fibers in both geometries and for the two polarization states.

7. D. M. Taylor, C. R. Bennet, T. J. Shepherd, L. F. Michaille, M. D. Nielsen, and H. R. Simonsen, “Demonstration of multi-core photonic crystal fibre in an optical interconnect,” Electron. Lett. **42**, 331–332 (2006).
[CrossRef]

## 4. Comparison of the different methods

## 5. Simplified model

_{1}(x, y) in these holes are very similar to each other (Fig. 5). This is confirmed by the results of the field overlap integral calculations between the electric fields present in the different air holes (Table 2) that are all very close to the unity. There are no more than 5% residual errors, from the first hole neighboring the core (m=1) to the fourth one (m=4).

_{1}(x,y)|

_{m}in one particular hole hm and its immediately following one h

_{m+1}(field e⃗

_{1}(x,y)|

_{m+1}) is shown to be constant whatever the hole under consideration in the studied domain (m=1 to 4). This can be explained by the fact that such a microstructured optical fiber guides the light by modified total internal reflection and consequently, the electric field in the silica cladding has a decreasing exponential form corresponding to an evanescent field. An illustration is given in Fig. 9 showing the field evolution of the guided mode in a logarithmic scale (dB) as a function of the horizontal distance along the x-axis. A linear decreasing of the amplitude of the field at the different air-hole position is clearly visible. Obviously, the slope depends on the ratio d/Λ, as well as on the pitch Λ and on the wavelength λ.

_{1}and m

_{2}are integer values representing the positions of the two considered air holes in the microstructured cladding.

_{1}=0) and a second one placed at the m

^{th}air-hole position (m

_{2}=m):

_{1}(x, y)|

_{m}with the field in the second hole e⃗

_{1}(x, y)|

_{2}and γ:

_{2}is the coupling coefficient between two cores separated by one hole (m=2).

_{m}between one core and a second one placed at the m

^{th}hole position can simply be obtained from the knowledge of γ and C

_{2}. Such a result is confirmed by the results presented in Fig. 7 on which we can verify that for a particular ratio d/Λ the distance between the curves of identical polarization remains constant.

## 6. Conclusion

^{-10}m

^{-1}) while the other methods present precision or computing time limitations below 0.1 m

^{-1}values. The influence of the ratio d/Λ, of the microstructured geometry, of the polarization state and of the core separation was presented.

## Acknowledgments

## References and links

1. | J. C. Knight, T. A. Birks, P. St. J. Russell, and D. M. Atkin, “All-silica single-mode optical fiber with photonic crystal cladding,” Opt. Lett. |

2. | J. C. Knight, J. Arriaga, T. A. Birks, A. Ortigosa-Blanch, W. J. Wadsworth, and P. St. J. Russell, “Anomalous dispersion in photonic crystal fiber,” IEEE Photon. Technol. Lett. |

3. | B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, and A. H. Greenaway, “Experimental study of dual-core photonic crystal fibre,” Electron. Lett. |

4. | I. Velchev and J. Toulouse, “Directional coupling and switching in multi-core microstructure fibers,” in |

5. | M. Zghal, R. Cherif, and F. Bahloul, “Improving triangular-lattice photonic-crystal-fiber couplers by introducing geometric nonuniformities,” Opt. Eng.46, n°095004 (2007). [CrossRef] |

6. | K. Saitoh, Y. Sato, and M. Koshiba, “Coupling characteristics of dual-core photonic crystal fiber couplers,” Opt. Express |

7. | D. M. Taylor, C. R. Bennet, T. J. Shepherd, L. F. Michaille, M. D. Nielsen, and H. R. Simonsen, “Demonstration of multi-core photonic crystal fibre in an optical interconnect,” Electron. Lett. |

8. | K. L. Reichenbach and C. Xu, “Independent core propagation in two-core photonic crystal fibers resulting from structural nonuniformities,” Opt. Express |

9. | Y. Yan, J. Toulouse, I. Velchev, and S. V. Rotkin, ≪ Decoupling and asymmetric coupling in triplecore photonic crystal fibers, » J. Opt. Soc. Am. B |

10. | F. Fogli, L. Saccomandi, P. Rossi, G. Bellanca, and S. Trillo, “Full vectorial BPM modeling of Index-Guiding Photonic Crystal Fibers and Couplers,” Opt. Express |

11. | N. Florous, K. Saitoh, and M. Koshiba, “A novel approach for designing photonic crystal fiber splitters with polarization independent propagation characteristics,” Opt. Express |

12. | X. Yu, M. Liu, Y. Chung, M. Yan, and P. Shum, “Coupling coefficient of two-core microstructured optical fiber,” Opt. Commun. |

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

14. | T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. Martijn de Sterke, and L. G. Botten, “Multipole method for microstructured optical fibers. I. Formulation,” J. Opt. Soc. Am. B |

15. | T. P. White, R. C. McPhedran, L. G. Botten, G. Smith, and C. Martijn de Sterke, “Calculation of air-guided modes in photonic crystal fibers using the multipole method,” Opt. Express |

16. | P. J. Roberts and T. J. Shepherd, “The guidance properties of multi-core photonic crystal fibres,” J. Opt. A, Pure Appl. Opt. |

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

18. | D. Marcuse, |

19. | K. P. L. Reichenbach, “Numerical analysis and experimental study of fiber bundles and multi-core photonic crystal fibers for use in endoscopes,” PhD dissertation, Cornell University (January 2007). |

**OCIS Codes**

(060.2270) Fiber optics and optical communications : Fiber characterization

(060.2330) Fiber optics and optical communications : Fiber optics communications

(060.2400) Fiber optics and optical communications : Fiber properties

(060.4005) Fiber optics and optical communications : Microstructured fibers

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: March 10, 2009

Revised Manuscript: May 12, 2009

Manuscript Accepted: June 29, 2009

Published: August 21, 2009

**Citation**

Philippe Di Bin and Nicolas Mothe, "Numerical analysis of directional coupling in dual-core microstructured optical fibers," Opt. Express **17**, 15778-15789 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-18-15778

Sort: Year | Journal | Reset

### References

- J. C. Knight, T. A. Birks, P. St. J. Russell, and D. M. Atkin, "All-silica single-mode optical fiber with photonic crystal cladding," Opt. Lett. 21, 1547-1549 (1996). [CrossRef] [PubMed]
- J. C. Knight, J. Arriaga, T. A. Birks, A. Ortigosa-Blanch, W. J. Wadsworth, and P. St. J. Russell, "Anomalous dispersion in photonic crystal fiber," IEEE Photon. Technol. Lett. 12, 807-809 (2000). [CrossRef]
- B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, and A. H. Greenaway, "Experimental study of dual-core photonic crystal fibre," Electron. Lett. 36, 1358-1359 (2000). [CrossRef]
- I. Velchev and J. Toulouse, "Directional coupling and switching in multi-core microstructure fibers," in Conference on Lasers and Electro-Optics Technical digest(CD) (Optical Society of America, 2004), paper CTuV1, http://www.opticsinfobase.org/abstract.cfm?URI=CLEO-2004-CTuV1.
- M. Zghal, R. Cherif, and F. Bahloul, "Improving triangular-lattice photonic-crystal-fiber couplers by introducing geometric nonuniformities," Opt. Eng. 46, 095004 (2007). [CrossRef]
- K. Saitoh, Y. Sato, and M. Koshiba, "Coupling characteristics of dual-core photonic crystal fiber couplers," Opt. Express 11, 3188-3195(2003), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-11-24-3188. [CrossRef] [PubMed]
- D. M. Taylor, C. R. Bennet, T. J. Shepherd, L. F. Michaille, M. D. Nielsen, and H. R. Simonsen, "Demonstration of multi-core photonic crystal fibre in an optical interconnect," Electron. Lett. 42, 331-332 (2006). [CrossRef]
- K. L. Reichenbach and C. Xu, "Independent core propagation in two-core photonic crystal fibers resulting from structural nonuniformities," Opt. Express 13, 10336-10348 (2005). http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-13-25-10336 [CrossRef] [PubMed]
- Y. Yan, J. Toulouse, I. Velchev, and S. V. Rotkin, "Decoupling and asymmetric coupling in triplecore photonic crystal fibers," J. Opt. Soc. Am. B 25, 1488-1495 (2008). [CrossRef]
- F. Fogli, L. Saccomandi, P. Rossi, G. Bellanca, and S. Trillo, "Full vectorial BPM modeling of Index-Guiding Photonic Crystal Fibers and Couplers," Opt. Express 10, 54-59 (2002), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-10-1-54 [PubMed]
- N. Florous, K. Saitoh, and M. Koshiba, "A novel approach for designing photonic crystal fiber splitters with polarization independent propagation characteristics," Opt. Express 13, 7365-7373 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-13-19-7365 [CrossRef] [PubMed]
- X. Yu, M. Liu, Y. Chung, M. Yan, and P. Shum, "Coupling coefficient of two-core microstructured optical fiber," Opt. Commun. 260, 164-169 (2006). [CrossRef]
- A. W. Snyder, and J. D. Love, Optical waveguide theory (Kluwer Academic Publishers, 2000).
- T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. Martijn de Sterke, and L. G. Botten, "Multipole method for microstructured optical fibers. I. Formulation," J. Opt. Soc. Am. B 19, 2322-2330 (2002). [CrossRef]
- T. P. White, R. C. McPhedran, L. G. Botten, G. Smith, and C. Martijn de Sterke, "Calculation of air-guided modes in photonic crystal fibers using the multipole method," Opt. Express 9, 721-732 (2001). [CrossRef] [PubMed]
- P. J. Roberts and T. J. Shepherd, "The guidance properties of multi-core photonic crystal fibres," J. Opt. A, Pure Appl. Opt. 3, 133-140 (2001). [CrossRef]
- K. N. Park, and K. S. Lee, "Improved effective-index method for analysis of photonic crystal fibers," Opt. Lett. 30, 958-960 (2005). [CrossRef] [PubMed]
- D. Marcuse, Theory of dielectric optical waveguides, Y.-H. Pao and P. Kelley, ed. (Academic Press, New York, 1974).
- K. P. L. Reichenbach, "Numerical analysis and experimental study of fiber bundles and multi-core photonic crystal fibers for use in endoscopes," PhD dissertation, Cornell University (January 2007).

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.