## Non-proximity resonant tunneling in multi-core photonic band gap fibers: An efficient mechanism for engineering highly-selective ultra-narrow band pass splitters

Optics Express, Vol. 14, Issue 11, pp. 4861-4872 (2006)

http://dx.doi.org/10.1364/OE.14.004861

Acrobat PDF (6021 KB)

### Abstract

The objective of the present investigation is to demonstrate the possibility of designing compact ultra-narrow band-pass filters based on the phenomenon of non-proximity resonant tunneling in multi-core photonic band gap fibers (PBGFs). The proposed PBGF consists of three identical air-cores separated by two defected air-holes which act as highly-selective resonators. With a fine adjustment of the design parameters associated with the resonant-air-holes, phase matching at two distinct wavelengths can be achieved, thus enabling very narrow-band resonant directional coupling between the input and the two output cores. The validation of the proposed design is ensured with an accurate PBGF analysis based on finite element modal and beam propagation algorithms. Typical characteristics of the proposed device over a single polarization are: reasonable short coupling length of 2.7 mm, dual bandpass transmission response at wavelengths of 1.339 and 1.357 µm, with corresponding full width at half maximum bandwidths of 1.2 nm and 1.1 nm respectively, and a relatively high transmission of 95% at the exact resonance wavelengths. The proposed ultra-narrow band-pass filter can be employed in various applications such as all-fiber bandpass/bandstop filtering and resonant sensors.

© 2006 Optical Society of America

## 1. Introduction

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

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. W. N. MacPherson, J. D. C. Jones, B. J. Mangan, J. C. Knight, and P. St. J. Russell, “Two-core photonic crystal fiber for Doppler difference velocimetry,” Opt. Commun. **233**, 375–380 (2003). [CrossRef]

5. K. Kitayama and Y. Ishida, “Wavelength-selective coupling of two-core optical fiber: application and design,” J. Opt. Soc. Am. A **2**, 90–94 (1985). [CrossRef]

7. E. Eisenmann and E. Weidel, “Single-mode fused biconical couplers for wavelength division multiplexing with channel spacing between 100–300 nm,” J. Lightwave Technol. **LT-6**, 113–119 (1988). [CrossRef]

8. K. Thyagarajan, S.D. Seshadri, and A.K. Ghatak, “Waveguide polarizer based on resonant tunneling,” J. Lightwave Technol. **9**, 315–317 (1991). [CrossRef]

9. K. Saitoh, N. Florous, M. Koshiba, and M. Skorobogatiy, “Design of narrow band-pass filters based on the resonant-tunneling phenomenon in multi-core photonic crystal fibers,” Opt. Express **13**, 10327–10335 (2005). [CrossRef] [PubMed]

11. K. Saitoh and M. Koshiba, “Full-vectorial imaginary-distance beam propagation method based on a finite element scheme: application to photonic crystal fibers,” IEEE J. Quantum Electron. **38**, 927–933 (2002). [CrossRef]

12. K. Saitoh and M. Koshiba, “Full-vectorial finite element beam propagation method with perfectly matched layers for anisotropic optical waveguides,” J. Lightwave Technol. **19**, 405–413 (2001). [CrossRef]

## 2. Schematic representation and design guidelines for engineering MC-PBGF splitters

*n*=1.45, by removing two rows of tubes and smoothing the resulting core edges. The pitch constant is chosen to be Λ=2 µm, while the air-hole diameters in the cladding of the fiber is

*d*/Λ=0.9, with a total of six hole layers in the cladding. Fundamental band gap where the core guided modes are found, extends between 1.29 µm < λ < 1.40 µm [13

13. K. Saitoh and M. Koshiba, “Leakage loss and group velocity dispersion in air-core photonic bandgap fibers,” Opt. Express **11**, 3100–3109 (2003). [CrossRef] [PubMed]

*N*=8 periods apart from each other, counting from the center of each core (along the

*x*axis) as shown in Fig. 1. Two dissimilar transverse resonators with

*d*

_{1}/Λ and

*d*

_{2}/Λ are then introduced by reducing (high index defects) the diameters of the air-holes in the middle of the line joining the cores. By an accurate modal analysis performed using an accurate FEM solver [11

11. K. Saitoh and M. Koshiba, “Full-vectorial imaginary-distance beam propagation method based on a finite element scheme: application to photonic crystal fibers,” IEEE J. Quantum Electron. **38**, 927–933 (2002). [CrossRef]

*x*-polarized (horizontally polarized) fundamental (blue solid curve), as well as the x-polarized excited odd resonant modes (red dashed curves), as a function of the operating wavelength and for several incremental values of the resonator’s normalized diameter

*d*

_{r}/Λ, ranged from 0.6 to 0.8. The reason for showing the odd excited modes in the graph is the fact that only these modes can be coupled to the fundamental

*x*-polarized mode, while the even excited modes can not be coupled at all. Note that the computation of the effective refractive index of the fundamental mode was performed assuming the core to be isolated; while the resonant excited modes have been calculated assuming these resonators isolated from the cores. This approximation was confirmed to give accurate results when comparing with the results associated with the coupled system’s performance (cores plus resonators). We can clearly see that the effective index of the fundamental mode is being crossed at certain wavelengths by the excited resonant modes. The physical interpretation of this crossing is that the excited modes at wavelengths of λ

_{1}, λ

_{2}, λ

_{3}, λ

_{n}, corresponding to different normalized resonator’s diameters

*d*

_{r}/Λ, can be effectively transferred via resonant coupling through the dissimilar resonators, from the central input core-A into the output cores-B and C. This simply means that at a given normalized resonator’s diameter

*d*

_{r}/Λ, there exists a wavelength where resonant tunneling can be achieved between the input (core-A) and output (core B or C) through the resonator. To identify the evolution of the resonance wavelengths as the resonator’s diameter changes, in Fig. 2(b) we plot the resonance wavelength as a function of the resonator’s normalized diameter

*d*

_{r}/Λ. From the results in Fig. 2(b) we can observe that as soon as the resonator’s diameter decreases the resonance wavelength increases. Due to the fact that the resonance wavelength must lie within the photonic band gap (PBG) region (shown by the grey boundaries), the possible values of the resonator’s diameters are in the range: 0.62<

*d*

_{r}/Λ<0.8, for the

*x*-polarized state. The same calculations are repeated in Figs. 3(a) and (b) for the

*y*-polarized (vertically polarized) fundamental mode and for the y-polarized excited even resonant modes. In the second case of y-polarization, where the odd-type excited

*y*-polarized modes can not be coupled to the fundamental

*y*-polarized resonant mode, we can observe a significant difference in the behavior of the evolution curve of the resonance wavelengths. By comparing the results in Figs. 2(b) and 3(b) we can see a great difference between the two polarizations. From Fig. 3(b) it is evident that the range of allowable values of the normalized resonator’s diameters for the

*y*-polarized mode which will result in resonance wavelengths within the PBG of the structure is significantly larger: 0<

*d*

_{r}/Λ<0.86. In addition a very interesting phenomenon occurs. This phenomenon is the insensitivity of the resonance wavelength for the following range of the normalized resonator’s diameters 0.1<dr/Λ<0.4, since at this range the resonance wavelength appears almost constant at a value of λ

_{res}=1.372 µm. The physical explanation for this drastic difference among the two polarization states can be given in terms of the feature of each polarization. This means that while the

*y*-polarized resonant state has an even-type profile, the

*x*-polarized resonant state significantly differs because its profile is anti-symmetric (odd), as will be demonstrated qualitatively later on. Therefore as a conclusion of this section we have described the basic principle of operation for this type of multi-core PBGF splitter and we have identified the impact of each polarization state to the resonance wavelength of the coupled system consisting of the input core and the resonator. In addition we have identified the allowable range that the resonators can give a resonance wavelength within the PBG capabilities of the structure.

## 3. Numerical results and device performance

*y*-polarization), and two anti-symmetric and one symmetric (for

*x*-polarization), with corresponding fields defined as ϕ

_{1}, ϕ

_{2}, ϕ

_{3}and qualitatively shown in Fig. 4. Note that the shape of the supermodes considered here are the envelopes of the supermodes, calculated by the modal analysis of the complete system. This was done only for qualitative purposes and has no particular meaning.

*n*

_{eff,1},

*n*

_{eff,2}, and

*n*

_{eff,3}represent the effective refractive indices of the supermodes corresponding to the fields ϕ

_{1}, ϕ

_{2}, ϕ

_{3}, respectively, for each of the polarization states. Then assuming that initially all the energy is in the input core-A, this will correspond to the excitation of a supermode combination of the following type:

*z*, this excitation pattern will evolve into:

_{i}=2π

*n*

_{eff,i}/λ

_{0}(

*i*=1, 2, 3) and λ

_{0}is operating wavelength. If we design the PBGF so that the effective refractive indexes of its supermodes satisfy the condition:

_{0}by choosing mode propagation length

*z=L*

_{c}with

_{1}and

_{,3}denote the effective indexes of the supermodes, for

*x*-polarization or

*y*-polarization respectively. Assuming the following geometrical parameters of the PBGF for the

*y*-polarized state: Λ=2 µm,

*d*/Λ=0.9,

*d*

_{1}/Λ=0.7 and

*d*

_{2}/Λ=0.6, from the results in Fig. 3(a) we can clearly see that the excited resonant modes cross the effective index curve of the fundamental mode, at two distinct wavelengths of λ

_{1},

*y*=1.339 µm and λ

_{2, y}=1.357 µm. In case of

*x*-polarization, by choosing the resonators sizes to be

*d*

_{1}/Λ=0.74 and

*d*

_{2}/Λ=0.70, the resonant wavelengths will be different: λ

_{1,x}=1.331 µm and λ

_{2,x}=1.356 µm. Using the relation in Eq. (5) combined with an accurate modal solver based on the FEM computational algorithm, in Fig. 5 we plot the calculated coupling lengths as a function of the normalized resonator’s diameter

*d*

_{r}/Λ for (a)

*x*-polarization and (b)

*y*-polarization, respectively. From the results in Fig. 5 we can at first observe a significant difference of the coupling lengths evolution for the two different polarization states. Particularly from Fig. 5(a) we can see that the coupling length changes linearly as the resonator’s diameter increases, while a different rate of change occurs in Fig. 5(b), where we can clearly see the asymptotic behavior of the coupling length as the resonator’s diameter approaches lower values. The partial coupling lengths in this case are:

_{1}=1.331 µm)=7.3 mm,

_{2}=1.356 µm)=6.4 mm,

_{1}=1.339 µm)=2.9 mm, and

_{2}=1.357 µm)=2.6 mm. In Fig. 6 we plot the obtained spectral characteristics of this novel type of PBGF splitter with total fiber length of 2.7 mm, for the y-polarized state, by using an accurate analysis based on the BPM algorithm [12

12. K. Saitoh and M. Koshiba, “Full-vectorial finite element beam propagation method with perfectly matched layers for anisotropic optical waveguides,” J. Lightwave Technol. **19**, 405–413 (2001). [CrossRef]

_{1, y}=1.339 µm and λ

_{2, y}=1.357 µm. The highly selectivity in the filter’s response we could obtain in this case, indicates the potential capability of the non-proximity resonator’s states to synthesize highly selective resonant coupling characteristics. The full width at half maximum (FWHM) bandwidths of this filter are 1.2 nm and 1.1 nm for the y-polarized state at wavelengths of λ1, y=1.339 µm and λ2, y=1.357 µm, respectively, while for the

*x*-polarized state the FWHM bandwidths were found to be a bit smaller. In both cases a transmission of about 95 % at the resonant wavelengths of λ

_{1}and λ

_{2}could be achieved. The difference in the values of the FWHM bandwidths for the two different polarization states, is associated with the larger coupling length of the

*x*-polarization in comparison to the

*y*-polarization (see Fig. 5), a fact which in general will result in a weaker coupling between the input core-A and the output core-B or C for the

*x*-polarization, thus resulting in a lower FWHM.

12. K. Saitoh and M. Koshiba, “Full-vectorial finite element beam propagation method with perfectly matched layers for anisotropic optical waveguides,” J. Lightwave Technol. **19**, 405–413 (2001). [CrossRef]

*E*

_{y}), for (a) λ

_{1, y}=1.339 µm, (b) λ

_{2, y}=1.357 µm, calculated at a distance of

*z*=0 mm, (c) λ

_{1, y}=1.339 µm (d) λ

_{2, y}=1.357 µm, at a distance of

*z*=1.0 mm, (e) λ

_{1, y}=1.339 µm, and (f) λ

*2,*

_{y}=1.357 µm, at a distance of

*z*=1.5 mm, (g) λ

_{1, y}=1.339 µm, and (h) λ

_{2, y}=1.357 µm, at a distance of

*z*=2.0 mm, and finally for (i) λ

_{1, y}=1.339 µm, (j) λ

_{2, y}=1.357 µm, calculated at the coupling length of

*z*=

*L*

_{c}=2.7 mm. We can clearly observe that at the coupling length of

*L*

_{c}=2.7 mm the two different wavelengths were splitted in the output cores B and C within a power decrement of 5 % from the targeted level of 100 %, associated with the slightly difference between the partial coupling lengths corresponding to the two different operating wavelengths.

## 4. Realization of polarization-insensitive PBGF splitters operating at a single wavelength

14. 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). [CrossRef] [PubMed]

15. S. K. Varshney, N. Florous, K. Saitoh, and M. Koshiba, “The impact of elliptical deformations for optimizing the performance of dual-core fluorine-doped photonic crystal fiber couplers,” Opt. Express **14**, 1982–1995 (2006). [CrossRef] [PubMed]

*d*

_{r}/Λ, for

*x*-polarization (blue curve) and

*y*-polarization (red curve). These two curves cross each other at a unique point corresponding to

*d*

_{r}/Λ=0.75 with resonance wavelength of λ

_{res}=1.325 µm. This unique selection of the resonator’s diameter will lead to polarizationindependent propagation characteristics, for the structure shown in Fig. 9, at the prescribed single resonance wavelength. So by choosing the resonator’s diameter as

*d*

_{r}/Λ=0.75, the resulting structure will operate as an effectively 100 % coupler from core-A into core-B, with polarization-independent propagation characteristics, operating at a single wavelength. The corresponding coupling length in this case was calculated by the modal analysis to be

*L*

_{c}=22.3 mm, a relatively short coupling length, acceptable for most practical applications. In order to verify the exact coupling length, in Fig. 10 we perform a simulation of the normalized power propagation along the PBGF splitter, using an accurate BPM algorithm [12

**19**, 405–413 (2001). [CrossRef]

_{res}=1.325 µm, for

*x*-polarization (blue curve) and y-polarization (red curve), in the input core-A as a function of the propagating distance in mm. The same simulation is shown in Fig. 10(b) for the output core-B. From these results we can see that at the coupling length of Lc=22.3 mm, the power is transferred from the input core-A to the output core-B with a transmission of more than 90 %, independent of the polarization state of the input signal. The main conclusion arising from this section is that the proposed MC-PBGF technology has indeed the potential capabilities of realizing polarization-independent devices by a judicious choice of the design parameters.

## 5. Conclusions

## Acknowledgments

## References and links

1. | P. St. J. Russell, “Photonic crystal fibers,” Science |

2. | S. Kawanishi, T. Yamamoto, H. Kubota, M. Tanaka, and S. Yamaguchi, “Dispersion controlled and polarization maintaining photonic crystal fibers for high performance network systems,” IEICE Trans. Electron. |

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. | W. N. MacPherson, J. D. C. Jones, B. J. Mangan, J. C. Knight, and P. St. J. Russell, “Two-core photonic crystal fiber for Doppler difference velocimetry,” Opt. Commun. |

5. | K. Kitayama and Y. Ishida, “Wavelength-selective coupling of two-core optical fiber: application and design,” J. Opt. Soc. Am. A |

6. | R. Zengerle and O. G. Leminger, “Narrow-band wavelength-selective directional couplers made of dissimilar single-mode fibers,” J. Lightwave Technol. |

7. | E. Eisenmann and E. Weidel, “Single-mode fused biconical couplers for wavelength division multiplexing with channel spacing between 100–300 nm,” J. Lightwave Technol. |

8. | K. Thyagarajan, S.D. Seshadri, and A.K. Ghatak, “Waveguide polarizer based on resonant tunneling,” J. Lightwave Technol. |

9. | K. Saitoh, N. Florous, M. Koshiba, and M. Skorobogatiy, “Design of narrow band-pass filters based on the resonant-tunneling phenomenon in multi-core photonic crystal fibers,” Opt. Express |

10. | M. Skorobogatiy, K. Saitoh, and M. Koshiba, “Transverse light guides in microstructured optical fibers,” Opt. Lett. |

11. | K. Saitoh and M. Koshiba, “Full-vectorial imaginary-distance beam propagation method based on a finite element scheme: application to photonic crystal fibers,” IEEE J. Quantum Electron. |

12. | K. Saitoh and M. Koshiba, “Full-vectorial finite element beam propagation method with perfectly matched layers for anisotropic optical waveguides,” J. Lightwave Technol. |

13. | K. Saitoh and M. Koshiba, “Leakage loss and group velocity dispersion in air-core photonic bandgap fibers,” Opt. Express |

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

15. | S. K. Varshney, N. Florous, K. Saitoh, and M. Koshiba, “The impact of elliptical deformations for optimizing the performance of dual-core fluorine-doped photonic crystal fiber couplers,” Opt. Express |

16. | T. Tjugiarto, G. D. Peng, and P. L. Chu, “Bandpass filtering effect in tapered asymmetrical twin-core optical fibers,” Electron. Lett. |

17. | B. Wu and P. L. Chu, “Narrow-bandpass filter with gain by use of twin-core rare-earth-doped fiber,” Opt. Lett. |

18. | B. Ortega and L. Dong, “Accurate tuning of mismatched twin-core fiber filters,” Opt. Lett. |

**OCIS Codes**

(060.2270) Fiber optics and optical communications : Fiber characterization

(060.2430) Fiber optics and optical communications : Fibers, single-mode

**ToC Category:**

Photonic Crystal Fibers

**History**

Original Manuscript: March 22, 2006

Revised Manuscript: May 1, 2006

Manuscript Accepted: May 10, 2006

Published: May 29, 2006

**Citation**

Nikolaos J. Florous, Kunimasa Saitoh, Tadashi Murao, Masanori Koshiba, and Maksim Skorobogatiy, "Non-proximity resonant tunneling in multi-core photonic band gap fibers: An efficient mechanism for engineering highly-selective ultra-narrow band pass splitters," Opt. Express **14**, 4861-4872 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-11-4861

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### References

- P. St. J. Russell, "Photonic crystal fibers," Science 299, 358-362 (2003). [CrossRef] [PubMed]
- S. Kawanishi, T. Yamamoto, H. Kubota, M. Tanaka, and S. Yamaguchi, "Dispersion controlled and polarization maintaining photonic crystal fibers for high performance network systems," IEICE Trans. Electron. E87-C, 336-342 (2004).
- 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]
- W. N. MacPherson, J. D. C. Jones, B. J. Mangan, J. C. Knight, and P. St. J. Russell, "Two-core photonic crystal fiber for Doppler difference velocimetry," Opt. Commun. 233, 375-380 (2003). [CrossRef]
- K. Kitayama and Y. Ishida, "Wavelength-selective coupling of two-core optical fiber: application and design," J. Opt. Soc. Am. A 2, 90-94 (1985). [CrossRef]
- R. Zengerle and O. G. Leminger, "Narrow-band wavelength-selective directional couplers made of dissimilar single-mode fibers," J. Lightwave Technol. LT-5, 1196-1198 (1987). [CrossRef]
- E. Eisenmann and E. Weidel, "Single-mode fused biconical couplers for wavelength division multiplexing with channel spacing between 100-300 nm," J. Lightwave Technol. LT-6, 113-119 (1988). [CrossRef]
- K. Thyagarajan, S.D. Seshadri, and A.K. Ghatak, "Waveguide polarizer based on resonant tunneling," J. Lightwave Technol. 9, 315-317 (1991). [CrossRef]
- K. Saitoh, N. Florous, M. Koshiba, and M. Skorobogatiy, "Design of narrow band-pass filters based on the resonant-tunneling phenomenon in multi-core photonic crystal fibers," Opt. Express 13, 10327-10335 (2005). [CrossRef] [PubMed]
- M. Skorobogatiy, K. Saitoh, and M. Koshiba, "Transverse light guides in microstructured optical fibers," Opt. Lett. 31, 314-316 (2006). [CrossRef] [PubMed]
- K. Saitoh and M. Koshiba, "Full-vectorial imaginary-distance beam propagation method based on a finite element scheme: application to photonic crystal fibers," IEEE J. Quantum Electron. 38, 927-933 (2002). [CrossRef]
- K. Saitoh and M. Koshiba, "Full-vectorial finite element beam propagation method with perfectly matched layers for anisotropic optical waveguides," J. Lightwave Technol. 19, 405-413 (2001). [CrossRef]
- K. Saitoh and M. Koshiba, "Leakage loss and group velocity dispersion in air-core photonic bandgap fibers," Opt. Express 11, 3100-3109 (2003). [CrossRef] [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). [CrossRef] [PubMed]
- S. K. Varshney, N. Florous, K. Saitoh, and M. Koshiba, "The impact of elliptical deformations for optimizing the performance of dual-core fluorine-doped photonic crystal fiber couplers," Opt. Express 14, 1982-1995 (2006). [CrossRef] [PubMed]
- T. Tjugiarto, G. D. Peng, and P. L. Chu, "Bandpass filtering effect in tapered asymmetrical twin-core optical fibers," Electron. Lett. 29, 1077-1078 (1993). [CrossRef]
- B. Wu and P. L. Chu, "Narrow-bandpass filter with gain by use of twin-core rare-earth-doped fiber," Opt. Lett. 18, 1913-1915 (1993). [CrossRef] [PubMed]
- B. Ortega and L. Dong, "Accurate tuning of mismatched twin-core fiber filters," Opt. Lett. 23, 1277-1279 (1998). [CrossRef]

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