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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 10, Iss. 23 — Nov. 18, 2002
  • pp: 1314–1319
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Leakage properties of photonic crystal fibers

D. Ferrarini, L. Vincetti, M. Zoboli, A. Cucinotta, and S. Selleri  »View Author Affiliations


Optics Express, Vol. 10, Issue 23, pp. 1314-1319 (2002)
http://dx.doi.org/10.1364/OE.10.001314


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Abstract

An analysis of the confinement losses in photonic crystal fibers due to the finite numbers of air holes is performed by means of the finite element method. The high flexibility of the numerical method allows to consider fibers with regular lattices, like the triangular and the honeycomb ones, and circular holes, but also fibers with more complicated cross sections like the cobweb fiber. Numerical results show that by increasing the number of air hole rings the attenuation constant decreases. This dependence is very strong for triangular and cobweb fibers, whereas it is very weak for the honeycomb one.

© 2002 Optical Society of America

1. Introduction

2 The Finite Element Method Formulation

Both electric field and magnetic field based formulations can be developed. Here for the sake of simplicity a magnetic field formulation is presented. The reader can find more details in [6

6. S. Selleri, L. Vincetti, A. Cucinotta, and M. Zoboli, “Complex FEM modal solver of optical waveguides with PML boundary conditions,” Opt. Quantum Electron. 33, 359–371(2001). [CrossRef]

]. The formulation starts from the curl-curl equation obtained decoupling the Maxwell equations:

¯×(ε̿r1¯×h¯)k02μ̿ rh¯=0
(1)

([A](γk0)2[B]){H}=0

where the eigenvector {H} and the eigenvalue (γk0)2 provide, respectively, the full vectorial magnetic field distribution and the effective index of the mode. In the present formulation triangular high order edge elements have been used. In order to enclose the computational domain without affecting the numerical solution, anisotropic perfectly matched layers are placed before the outer boundary. This formulation is able to deal with anisotropic material both in terms of dielectric permittivity and magnetic permeability allowing anisotropic PMLs to be directly implemented. Furthermore fiber symmetry can be used to reduce the computational domain and consequently both time and memory required.

3 Results

3.1 Triangular fiber

The PCF first considered in the analysis consists of a triangular lattice of air holes. An example of its cross section is reported in figure 1(a) in the case of four rings. The different colors show different rings. Figure 1(b) shows the magnetic field main component of one of the two polarizations of the fundamental mode. The fiber considered consists of two rings of holes having diameter d = 0.69μm and pitch Λ = 2.3μm. Notice that due to fiber symmetry just a quarter of the cross section can be considered for the analysis. The hexagonal symmetry as well as the leakage due to the interruption of the lattice are evident. The field confinement and its decay rate play a fundamental role in the leakage properties. They depend on the hole diameter, on their pitch and on the number of rings. Figure 2 shows the losses versus the diameter d normalized to the pitch value Λ = 2.3μm. As expected, the losses quickly decrease by increasing the numbers of rings and the hole diameter. Also the slope increases with these parameters. In fact, with d = 0.69μm, that is d/Λ = 0.3, going from a ring to eight rings losses decrease from 105 dB/m to a value a little lower than 10dB/m. Notice that this corresponds to pass from six to two hundred and sixteen holes. By increasing the hole diameter to d = 0.9μm = 0.4Λ, the upper value of the range goes to 0.5 × 105 dB/m, whereas the lower value goes to 10-5 which is negligible with respect to all the other causes of losses, like to absorption and the Rayleigh scattering. Figure 2(b) shows that a loss reduction is also obtained by fixing the ratio d/Λ and by increasing the pitch Λ. The pitch and the hole diameter are changed of the same scale factor, consequently greater pitches correspond to greater core size and thus more confined fields. Specific dispersion profile can be obtained playing on ratio d/Λ and on the scale factor [9

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

]. Thus, the dependence on these parameters is very important to design fibers with a good trade-off between the dispersion and loss properties. In this frame, another important aspect to investigate is the wavelength dependence. Figure 3 shows the results of this analysis. As expected, the losses increase with the wavelength because the field confinement decreases. The numbers of rings still affects this dependence. For few rings the dependence is weak, whereas for many rings it becomes stronger. Passing from 1300nm to 1700nm with one ring, the losses are almost wavelength-independent, conversely, with five rings, the losses increase of about two orders of magnitude. This variation decreases of one order going from a pitch Λ = 2.3μm to a pitch Λ = 4.6μm. In fact greater pitches correspond to larger cores and thus more confined fields. As a consequence, a wavelength increment corresponds to a little increment of the confinement with a little reduction of the losses.

Fig. 1. (a): triangular fiber with four hole rings. The lines with different colors show different rings: black first ring, red second one, yellow third one, and blue fourth one. The dashed lines shows the quarter of the structure considered in the analysis. (b): main component of the magnetic field for a fiber having two rings, d/Λ = 0.3 and Λ = 2.3μm.
Fig. 2. (a): confinement loss as a function of hole diameter d normalized to pitch Λ = 2.3μm for different numbers of rings. (b): confinement loss as a function of pitch Λ for different ratios d/Λ. In both cases a wavelength λ = 1.55μm is assumed

3.2 Honeycomb fiber

It is very interesting also to investigate the loss phenomena in fibers where guiding can be ascribed to photonic band gap as, for example, the honeycomb fibers. The cross section of the honeycomb fiber considered in the analysis is reported in figure 4(a). The defect consists on a extra hole with the same diameter as that of the lattice holes. The holes belonging to different rings are highlighted with different colors. Notice that in this kind of fiber the number of holes increases with the number of rings more quickly than in the triangular one. Passing from two to three rings in a triangular fiber the numbers of holes pass from eighteen to thirty-six while in a honeycomb they pass from fifty-four to ninety-six. However, due to the lower air filling factor, that is the ratio between the areas of the air holes and the unitary cell, the mode is less confined as shown in figure 4(b). This causes higher losses and above all a weaker dependence on the number of rings. Figure 5(a) shows this aspect. Comparing the results of figure 5 and 2, for few rings, the losses are a little bit higher than those of the triangular fiber, but by increasing the rings the difference grows reaching six orders of magnitude for eight rings. As a consequence in the considered fibers, the losses are always higher than 10dB/m. The figure also shows another important difference with respect to the triangular fiber. Higher ratios d/Λ exhibit lower losses only up to six rings, after that the losses slightly increase. This can be ascribed to the fiber guidance mechanism. In fact the lattice of the fiber determines a truly photonic band gap only if the number of rings is very high. Also the wavelength dependence, shown in figure 5(b), is weaker than in the triangular one. It slightly increases with ring numbers and it reaches a order of magnitude with nine rings, over the wavelength range here considered.

Fig. 3. Confinement loss as a function of the wavelength λ for different numbers of rings and d/Λ = 0.5. (a) Λ = 2.3μm; (b) Λ = 4.6μm,
Fig. 4. (a): honeycomb fiber with three hole rings. The circles with different colors show different rings: red first ring and black second one. The dashed lines shows the quarter of the structure considered in the analysis. (b): fundamental mode profile of a honeycomb fiber having three rings and d/Λ = 0.41 at the wavelength λ = 1.55μm.

3.3 Cobweb fiber

Fig. 5. (a): confinement loss as a function of numbers of rings at the wavelength λ = 1.55μm and for d/Λ = 0.41 red line and for d/Λ = 0.55 green line. (b): confinement loss as a function of the wavelength λ for different numbers of rings and d/Λ = 0.41. In both cases Λ = 1.62μm is assumed.
Fig. 6. (a): cross sections of the cobweb fiber with one (top) and three (bottom) rings. (b): confinement loss as a function of the wavelength for different ring numbers.

4 Conclusion

References and links

1.

T. P. White, R. C. McPhedram, C. M. de Sterke, L. C. Botten, and M. J. Steel, “Confinement losses in microstructured optical fibers,” Opt. Lett. 26, 1660–1662 (2001). [CrossRef]

2.

V. Finazzi, T. M. Monro, and D. J. Richardson, “Confinement losses in highly nonlinear holey optical fibers,” in Optical Fiber Communication 2002, vol. 70 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C.2002), paper ThS4.

3.

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]

4.

A. Cucinotta, S. Selleri, L. Vincetti, and M. Zoboli, “Perturbation analysis of dispersion properties in photonic crystal fibers through the finite element method,” J. Lightwave Technol. 20,(2002). [CrossRef]

5.

A. Cucinotta, S. Selleri, L. Vincetti, and M. Zoboli, “Holey fiber analysis through the finite element method,” IEEE Photon. Technol. Lett. 14, 1530–15322002. [CrossRef]

6.

S. Selleri, L. Vincetti, A. Cucinotta, and M. Zoboli, “Complex FEM modal solver of optical waveguides with PML boundary conditions,” Opt. Quantum Electron. 33, 359–371(2001). [CrossRef]

7.

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

8.

S. E. Barkou, J. Broeng, and A. Bjarklev, “Dispersion properties of photonic bandgap guiding fibers,” in Optical Fiber Communication Conference , OSA Technical Digest (Optical Society of America, Washington DC, 1998), FG5.

9.

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

OCIS Codes
(060.2280) Fiber optics and optical communications : Fiber design and fabrication
(060.2400) Fiber optics and optical communications : Fiber properties

ToC Category:
Research Papers

History
Original Manuscript: September 23, 2002
Revised Manuscript: October 30, 2002
Published: November 18, 2002

Citation
D. Ferrarini, Luca Vincetti, M. Zoboli, A. Cucinotta, and S. Selleri, "Leakage properties of photonic crystal fibers," Opt. Express 10, 1314-1319 (2002)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-10-23-1314


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References

  1. T. P. White, R. C. McPhedram, C. M. de Sterke, L. C. Botten, and M. J. Steel, �??Confinement losses in microstructured optical fibers,�?? Opt. Lett. 26, 1660-1662 (2001). [CrossRef]
  2. V. Finazzi, T. M. Monro, and D. J. Richardson, �??Confinement losses in highly nonlinear holey optical fibers,�?? in Optical Fiber Communication 2002, vol. 70 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C. 2002), paper ThS4.
  3. 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]
  4. A. Cucinotta, S. Selleri, L. Vincetti, and M. Zoboli, �??Perturbation analysis of dispersion properties in photonic crystal fibers through the finite element method,�?? J. Lightwave Technol. 20, (2002). [CrossRef]
  5. A. Cucinotta, S. Selleri, L. Vincetti, and M. Zoboli, �??Holey fiber analysis through the finite element method,�?? IEEE Photon. Technol. Lett. 14, 1530-1532 2002. [CrossRef]
  6. S. Selleri, L. Vincetti, A. Cucinotta, and M. Zoboli, �??Complex FEM modal solver of optical waveguides with PML boundary conditions,�?? Opt. Quantum Electron. 33, 359-371(2001). [CrossRef]
  7. J. C. Knight, J. Arriaga, T. A. Birks, A. Ortigosa-Blanch, W. J. Wadsworth and P. St. Russell, �??Anomalous dispersion in photonic crystal fiber,�?? IEEE Photon. Technol. Lett. 12, 807-809 (2000). [CrossRef]
  8. S. E. Barkou, J. Broeng, and A. Bjarklev, �??Dispersion properties of photonic bandgap guiding fibers,�?? in Optical Fiber Communication Conference , OSA Technical Digest (Optical Society of America, Washington DC, 1998), FG5.
  9. A. Ferrando, E. Silvestre, P.Andres, J. J. Miret, and M. V. Andres, �??Designing the properties of dispersion-.attend 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]

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