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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 10, Iss. 1 — Jan. 14, 2002
  • pp: 54–59
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Full vectorial BPM modeling of Index-Guiding Photonic Crystal Fibers and Couplers

Fabrizio Fogli, Luca Saccomandi, Paolo Bassi, Gaetano Bellanca, and Stefano Trillo  »View Author Affiliations


Optics Express, Vol. 10, Issue 1, pp. 54-59 (2002)
http://dx.doi.org/10.1364/OE.10.000054


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Abstract

A 3D full-vectorial Beam Propagation Method is successfully applied to compute both the propagation constants and the modal profiles in high-contrast silica-air index-guiding Photonic Crystal Fibers. The approach is intrinsically suited to investigate longitudinally varying structures or propagation and polarization effects, which are of practical interest for advanced optical applications. As an example we model a dual-core coupler, showing that efficient polarization preserving coupling can be expected.

© Optical Society of America

1. Introduction

2. Dispersion features of the modes

For the sake of brevity, we refer the reader to Ref. [18

18. F. Fogli, G. Bellanca, P. Bassi, I. Madden, and W. Johnstone, “Highly Efficient Full-Vectorial 3-D BPM Modeling of Fiber to Planar Waveguide Couplers,” IEEE J. Lightwave Tecnol. 17, 136–143 (1999). [CrossRef]

] for the details of our finite difference implementation of BPM with Perfectly Matched Layers (PML) boundary conditions used in the calculations.

First, in order to validate our BPM code, the propagation in a PCF structure described in Ref. [8

8. A. Ferrando, E. Silvestre, J. J. Miret, P. Andrés, and M. V. Andrés, “Vector description of higher-order modes in photonic crystal fibers,” J. Opt. Soc. Am. A 17, 1333–1340 (2000). [CrossRef]

] has been simulated. In this way, the output of our code could be directly compared with results already obtained in the literature by means of a completely different method.

The structure consists of an air-filled silica (n = 1.46) index-guiding PCF characterized by an hexagonal distribution of air holes of radius a = 0.6 μm and pitch ʌ = 2.3 μm with a central silica-filled hole which represents a high index defect. A transverse window of 19 μm × 14.2 μm in the x and y direction respectively, surrounded by a 1 μm thick PML boundary, has been used in the simulations. The mesh for the finite-difference computation has been generated using a spatial sampling step Δx = Δy = Δz =0.1 μm

Fig. 1. Real part of the transverse profiles of the launched field (the imaginary parts are set to zero). Left frame: fundamental mode excitation. Right frame: upper mode excitation.

As far as mode degeneracy is concerned [9–11

9. T. M. Monro, D. J. Richardson, N. G. R. Broderick, and P. J. Bennet, “Modeling large air fraction holey optical fibers,” IEEE J. Lightwave Tecnol. 18, 50–56 (2000). [CrossRef]

], we have launched all the possible polarization states which compose the doublet and the multiplet, and found that maximum differences of effective index stabilize around 10-7 and 10-5 respectively, supporting the argument of degeneracy, within the estimated numerical precision of the method.

The three electric field components of the fundamental mode distribution are shown in Fig. 3 for a normalized frequency ʌ/λ = 5. They have been obtained by propagating the input gaussian field of Fig. 1 (left frame), until it reaches a stable configuration over the transverse section. The layout of the investigated structure is superimposed as well, for better readability of the results.

Fig. 2. Modal effective index vs. normalized frequency ʌ/λ (λ varies from 300 to 1600 nm) for the fundamental mode and the next higher-order one of a PCF with a = 0.6 and ʌ = 2.3 μm. The upper (red) and the lower (blue) curves refer to the fundamental and the higher order modes respectively. The open circles stand for BPM calculations whereas the solid lines show for comparison the data taken from Fig. 3 of Ref. [7].
Fig. 3. Moduli of the x, y and z electric field components of the fundamental mode at the output section of the PCF for a normalized frequency ʌ/λ = 5.

If both the x-polarized modes of the upper multiplet are excited by launching a 45 degrees oriented antisymmetric field distribution for the x-component, they reach the stable pattern shown in Fig. 4.

Fig. 4. Moduli of the x, y and z electric field components of the first upper mode at the output section of the fiber for a normalized frequency ʌ/λ = 5.

Remarkably, the most effective x-directed field component, shown in Figs. 3 and 4, are very similar to the experimental near-field map presented in Ref. [5

5. J. C. Knight, T. A. Birks, P. St. J. Russel, and J. P. de Sandro, “Properties of photonic crystal fiber and the effective index model,” J. Opt. Soc. Am. A 15, 748–752 (1998). [CrossRef]

]. Moreover, the less significant one, directed along y, still exhibits a very regular pattern, despite its very low values, confirming the good reliability of our BPM algorithm.

3. The two-core structure

Once the BPM code has been successfully validated, we have applied it to study a dual-core PCF coupler, obtained by filling two holes of a single-mode PCF (a = 0.3 μm, ʌ = 2.3 μm [8

8. A. Ferrando, E. Silvestre, J. J. Miret, P. Andrés, and M. V. Andrés, “Vector description of higher-order modes in photonic crystal fibers,” J. Opt. Soc. Am. A 17, 1333–1340 (2000). [CrossRef]

]) with an inter-core separation of 4.6 μm (twice the pitch). The x-polarized single fiber mode has been preliminarily calculated using the same procedure described before at a normalized frequency ʌ/λ, = 2.7. Then, this field has been launched along the left channel of the coupler, as shown in the left frame of Fig. 5 for the most significant field component.

Fig. 5. Moduli of the x electric field components in the dual-core PCF Coupler for a normalized frequency ʌ/λ = 2.7 overlaid with the investigated coupler geometry. Left: launched field in the starting section. Right: field distribution after a propagation distance of nearly 0.715 mm.

The right frame of Fig. 5 displays the same field component after a propagation distance of 0.7 mm. As shown, almost all the launched power is coupled into the right channel (around the right-end defect) of the coupler. Although our BPM code encompasses the possibility to have a dynamical evolution of the polarization, no significant polarization coupling is observed, meaning that the structure can operate in polarization preserving mode. The whole longitudinal field evolution along the structure has been recorded and post-processed to extract the coupling length of the device, which turns out to be Lc=0.7335 mm. Interestingly enough, by launching the y-polarization state of the doublet, we end up with the different value Lc=0.7505 mm. Such small difference in the coupling strength can be attributed to slight but significant differences in the x- and y-polarized field profiles of the doublet, which become detectable mainly at the air-silica interfaces around the main hole ring (surrounding the defect), where the field continuity plays a major role.

4. Conclusions

The 3D full-vectorial finite-difference BPM with PML boundary conditions has been proposed and validated to investigate the propagation properties of holey fibers. The method proves efficient in spite of the high-contrast of silica-air structures. The propagating fields and the relevant dispersion curves for both the fundamental and the higher-order modes have been successfully compared with previously published results obtained by means of completely different approaches. However, the method has greater flexibility as compared with mode-finder techniques, since it permits the study of propagation phenomena in PCF structures. To show this we have demonstrated that a dual-defect silica-air PCF can operate as a polarization preserving directional coupler. All the successfully performed tests are encouraging towards the applicability of this technique to study more complex directional coupling phenomena in longitudinally-invariant or adiabatically varying structures and devices such as those recently proposed [12–14

12. X. Liu, C. Xu, W. Knox, J. K. Chandalia, B. J. Eggleton, S. G. Kosinski, and R. S. Windeler, “Soliton self-frequency shift in short tapered air-silica microstructure fiber,” Opt. Lett. 26, 358–400 (2001). [CrossRef]

].

Acknowledgem ents

Funding from CNR (MADESS II project) and MURST is gratefully acknowledged.

References and links

1.

J. C. Knight, J. Broeng, T. A. Birks, and P. St. J. Russel, “Single mode photonic band gap guidance of light in air,” Science 282, 1476–1478 (1998). [CrossRef] [PubMed]

2.

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

3.

T.A. Birks and P. St. J. Russel, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. 22, 961–963 (1997). [CrossRef] [PubMed]

4.

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

5.

J. C. Knight, T. A. Birks, P. St. J. Russel, and J. P. de Sandro, “Properties of photonic crystal fiber and the effective index model,” J. Opt. Soc. Am. A 15, 748–752 (1998). [CrossRef]

6.

E. Silvestre, M. V. Andrés, and P. Andrés, “Biorthonormal-basis method for the vector description of optical-fiber modes,” IEEE J. Lightwave Tecnol. 23, 923928 (1998). [CrossRef]

7.

A. Ferrando, E. Silvestre, J. J. Miret, P. Andrés, and M. V. Andrés, “Full-vector analysis of a realistic photonic crystal fiber modes,” Opt. Lett. 24, 276278 (1999). [CrossRef]

8.

A. Ferrando, E. Silvestre, J. J. Miret, P. Andrés, and M. V. Andrés, “Vector description of higher-order modes in photonic crystal fibers,” J. Opt. Soc. Am. A 17, 1333–1340 (2000). [CrossRef]

9.

T. M. Monro, D. J. Richardson, N. G. R. Broderick, and P. J. Bennet, “Modeling large air fraction holey optical fibers,” IEEE J. Lightwave Tecnol. 18, 50–56 (2000). [CrossRef]

10.

J. Broeng, S. E. Barkou, T. Sondergaard, and A. Bjarklev, “Analysis of air-guiding photonic bandgap fibers,” Opt. Lett. 25, 96–98 (2000). [CrossRef]

11.

M. J. Steel, T. P. White, C. M. de Sterke, R. C. McPhedran, and L. C. Botten, “Symmetry and degeneracy in microstructured optical fibers,” Opt. Lett. 26, 488–490 (2001). [CrossRef]

12.

X. Liu, C. Xu, W. Knox, J. K. Chandalia, B. J. Eggleton, S. G. Kosinski, and R. S. Windeler, “Soliton self-frequency shift in short tapered air-silica microstructure fiber,” Opt. Lett. 26, 358–400 (2001). [CrossRef]

13.

T. Sorensen, J. Broeng, A. Bjarklev, E. Knudsen, and S. E. Libori Barkou, “Macro-bending loss properties of photonic crystal fibre,” Electron. Lett. 37, 287–289 (2001). [CrossRef]

14.

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

15.

M.D. Feit and J. A. Fleck, “Computation of mode properties in optical fiber waveguides by a propagating beam method,” Appl. Opt. 19, 1154–1164 (1980). [CrossRef] [PubMed]

16.

B. J. Eggleton, P. S. Westbrook, C. A. White, C. Kerbage, R. S. Windeler, and G. L. Burdge, “Claddding-mode-resonances in air-silica microstructure optical fibers,” IEEE J. Lightwave Tecnol. 18, 1084–1100 (2000). [CrossRef]

17.

C. E. Kerbage, B. J. Eggleton, P. S. Westbrook, and R. S. Windeler, “Experimental and scalar beam propagation analysis of an air-silica microstructure fiber,” Opt. Expr. 7, 13–122 (2000). [CrossRef]

18.

F. Fogli, G. Bellanca, P. Bassi, I. Madden, and W. Johnstone, “Highly Efficient Full-Vectorial 3-D BPM Modeling of Fiber to Planar Waveguide Couplers,” IEEE J. Lightwave Tecnol. 17, 136–143 (1999). [CrossRef]

OCIS Codes
(060.2270) Fiber optics and optical communications : Fiber characterization
(060.2310) Fiber optics and optical communications : Fiber optics
(060.2340) Fiber optics and optical communications : Fiber optics components
(060.2400) Fiber optics and optical communications : Fiber properties

ToC Category:
Research Papers

History
Original Manuscript: December 13, 2001
Published: January 14, 2002

Citation
Fabrizio Fogli, Luca Saccomandi, Paolo Bassi, Gaetano Bellanca, and Stefano 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


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References

  1. J. C. Knight, J. Broeng, T. A. Birks, and P. St. J. Russel, "Single mode photonic band gap guidance of light in air," Science 282, 1476-1478 (1998). [CrossRef] [PubMed]
  2. R. F. Cregan, B. J. Mangan J. C. Knight, T. A. Birks, P. St. J. Russel, P. J. Roberts, andD. C. Allan, "Single mode photonic band gap guidance of light in air," Science 285, 1537-1539 (1999). [CrossRef] [PubMed]
  3. T.A. Birks, and P. St. J. Russel, "Endlessly single-mode photonic crystal fiber," Opt. Lett. 22, 961-963 (1997). [CrossRef] [PubMed]
  4. D. Mogilevtsev, T. A. Birks, and P. St. J. Russel, "Group-velocity dispersion in photonic crystal fibers," Opt. Lett. 23, 1662-1664 (1998). [CrossRef]
  5. J. C. Knight, T. A. Birks, P. St. J. Russel, and J. P. de Sandro, "Properties of photonic crystal fiber and the effective index model," J. Opt. Soc. Am. A 15, 748-752 (1998). [CrossRef]
  6. E. Silvestre, M. V. Andr?s, and P. Andr?s, "Biorthonormal-basis method for the vector description of optical-fiber modes," IEEE J. Lightwave Tecnol. 23, 923-928 (1998). [CrossRef]
  7. A. Ferrando, E. Silvestre, J. J. Miret, P. Andr?s, and M. V. Andr?s, "Full-vector analysis of a realistic photonic crystal fiber modes," Opt. Lett. 24, 276-278 (1999). [CrossRef]
  8. A. Ferrando, E. Silvestre, J. J. Miret, P. Andr?s, and M. V. Andr?s, "Vector description of higher-order modes in photonic crystal fibers," J. Opt. Soc. Am. A 17, 1333-1340 (2000). [CrossRef]
  9. T. M. Monro, D. J. Richardson, N. G. R. Broderick, and P. J. Bennet, "Modeling large air fraction holey optical fibers," IEEE J. Lightwave Tecnol. 18, 50-56 (2000). [CrossRef]
  10. J. Broeng, S. E. Barkou, T. Sondergaard, and A. Bjarklev, "Analysis of air-guiding photonic bandgap fibers," Opt. Lett. 25, 96-98 (2000). [CrossRef]
  11. M. J. Steel, T. P. White, C. M. de Sterke, R. C.McPhedran, and L. C. Botten, "Symmetry and degeneracy in microstructured optical fibers," Opt. Lett. 26, 488-490 (2001). [CrossRef]
  12. X. Liu, C. Xu, W. Knox, J. K. Chandalia, B. J. Eggleton, S. G. Kosinski, and R. S. Windeler, "Soliton selffrequency shift in short tapered air-silica microstructure fiber," Opt. Lett. 26, 358-400 (2001). [CrossRef]
  13. T. Sorensen, J. Broeng, A. Bjarklev, E. Knudsen, and S. E. Barkou Libori, "Macro-bending loss properties of photonic crystal fibre," Electron. Lett. 37, 287-289 (2001). [CrossRef]
  14. B.J. Mangan, J.C. Knight, T. A. Birks, T.A. Roberts, P. St. J. Russel, and A. H. Greenaway, "Experimental study of dual-core photonic crystal fibre," Electron. Lett. 36, 1358-1359 (2000). [CrossRef]
  15. M.D. Feit, and J. A. Fleck, "Computation of mode properties in optical fiber waveguides by a propagating beam method," Appl. Opt. 19, 1154-1164 (1980). [CrossRef] [PubMed]
  16. B. J. Eggleton, P. S. Westbrook, C. A. White, C. Kerbage, R. S. Windeler and G. L. Burdge, "Cladddingmode-resonances in air-silica microstructure optical fibers," IEEE J. Lightwave Tecnol. 18, 1084-1100 (2000). [CrossRef]
  17. C. E. Kerbage, B. J. Eggleton, P. S. Westbrook, R. S. Windeler, "Experimental and scalar beam propagation analysis of an air-silica microstructure fiber," Opt. Express 7, 13-122 (2000). [CrossRef]
  18. F. Fogli, G. Bellanca, P. Bassi, I. Madden, andW. Johnstone, "Highly Efficient Full-Vectorial 3-D BPM Modeling of Fiber to Planar Waveguide Couplers," IEEE J. Lightwave Tecnol. 17, 136-143 (1999). [CrossRef]

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