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

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  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, 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]
  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]
  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]
  13. A. W. Snyder, and J. D. Love, Optical waveguide theory (Kluwer Academic Publishers, 2000).
  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]
  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]
  18. D. Marcuse, Theory of dielectric optical waveguides, Y.-H. Pao and P. Kelley, ed. (Academic Press, New York, 1974).
  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).

2008 (1)

2007 (1)

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

2006 (2)

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]

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]

2005 (3)

2003 (1)

2002 (2)

2001 (2)

2000 (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]

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]

1996 (1)

Arriaga, J.

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]

Atkin, D. M.

Bahloul, F.

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

Bellanca, G.

Bennet, C. R.

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]

Birks, T. A.

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]

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]

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]

Botten, L. G.

Cherif, R.

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

Chung, Y.

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]

Florous, N.

Fogli, F.

Greenaway, A. H.

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]

Knight, J. C.

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]

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]

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]

Koshiba, M.

Kuhlmey, B. T.

Lee, K. S.

Liu, M.

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]

Mangan, B. J.

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]

Martijn de Sterke, C.

Maystre, D.

McPhedran, R. C.

Michaille, L. F.

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]

Nielsen, M. D.

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]

Ortigosa-Blanch, A.

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]

Park, K. N.

Reichenbach, K. L.

Renversez, G.

Roberts, P. J.

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]

Rossi, P.

Rotkin, S. V.

Russell, P. St. J.

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]

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]

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]

Saccomandi, L.

Saitoh, K.

Sato, Y.

Shepherd, T. J.

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]

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]

Shum, P.

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]

Simonsen, H. R.

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]

Smith, G.

Taylor, D. M.

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]

Toulouse, J.

Trillo, S.

Velchev, I.

Wadsworth, W. J.

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]

White, T. P.

Xu, C.

Yan, M.

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]

Yan, Y.

Yu, X.

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]

Zghal, M.

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

Electron. Lett. (2)

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]

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]

IEEE Photon. Technol. Lett. (1)

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]

J. Opt. A, Pure Appl. Opt. (1)

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]

J. Opt. Soc. Am. B (2)

Opt. Commun. (1)

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]

Opt. Eng. (1)

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

Opt. Express (5)

Opt. Lett. (2)

Other (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.

A. W. Snyder, and J. D. Love, Optical waveguide theory (Kluwer Academic Publishers, 2000).

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).

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Figures (9)

Fig. 1.
Fig. 1.

Description of the two DC-MOFs configurations (m=2): a) Geometry #1; b) Geometry #2.

Fig. 2.
Fig. 2.

x- polarized symmetric and antisymmetric supermodes of a typical DC-MOF (geometry #1, m=4).

Fig. 3.
Fig. 3.

Index profile (geometry #1, m=4, Λ=4µm, d=1µm,, λ=1.55µm) under simulation (left) and corresponding scalar BPM simulation results (right). Representation of the optical power distribution in the DC-MOF. The z-axis is the light propagation direction.

Fig. 4.
Fig. 4.

A DC-MOF is a single-core MOF perturbed by a silica rod filling one air-hole in place of the second core (e.g. geometry #1, m=2).

Fig .5.
Fig .5.

Example of typical electric field distribution (x-polarized mode) in a single-core MOF. Enlarged images of the fields e⃗1 in different holes of the cladding (m=2, 3, 4 respectively). All the images are amplitude normalized to their local maximum to optimize the color scale and improve their readability.

Fig. 6.
Fig. 6.

Graph of the coupling coefficients for both polarizations and geometries. Cx and Cy are the coupling coefficients for electric field polarized along x-axis and y-axis, respectively (m=2, Λ=4 µm, λ=1.55 µm).

Fig. 7.
Fig. 7.

Coupling coefficients calculated as a function of the ratio d/Λ for different core separations (geometry #1, Λ=4 µm, λ=1.55 µm).

Fig. 8.
Fig. 8.

Convergence study for weak coupling coefficients in geometry #1 with cores separated by 3 holes (m=4, Λ=4µm, λ=1.55µm, x-polarized).

Fig. 9.
Fig. 9.

Examples of cross-sectional view of the electric field x-component in a MOF (x-polarized mode) presented with a log-scale and showing the linear slope from one hole to the other.

Tables (3)

Tables Icon

Table 1. Results of the convergence test carried out for an x-polarized mode (geometry #1, m=2, Λ=4µm, λ=1.55µm).

Tables Icon

Table 2. Overlap integrals between different air hole fields as a function of their respective rank m in the structure

Tables Icon

Table 3. Comparisons between some CMT and simplified model results for the calculation of the coupling coefficient CX (x-polarized) for different positions of the second core (m=2 to 5) for geometry #1. Other fixed parameters are λ=1.55µm and Λ=4µm.

Equations (9)

Equations on this page are rendered with MathJax. Learn more.

Cxy = βxy+βxy2
Cxy = π2Lcxy
C = 14(N1N2)12 (ε0μ0)12 k (n2(x,y)n̅2(x,y)) Aperturbatione1*(x,y)e2*(x,y)dA
C = 14(N1N2)12 (ε0μ0)12 k Δn2(x,y) Aperturbatione1*(x,y).e1(xS,y)dA
γ = e1(x,y)m+1e1(x,y)m
e1 (x,y)m2=γ(m2m1)e1(x,y)m1
Cm = 14(N1N2)12 (ε0μ0)12 k Δ n2 (x,y)hme1*(x,y)m.e2(x,y)dxdy
Cm = γm2 [14(N1N2)12(ε0μ0)12kΔn2(x,y)h2e1*(x,y)2.e1(xS,y)dxdy]
Cm = γm2 C2

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