Abstract

We present a study of the large numerical aperture and high capture efficiency in a class of microstructured optical fibers, also called ‘air-clad’ fibers. We employ a recently developed method where the leaky modes supported by a waveguide are used to determine the far-field angular intensity distributions. These distributions are subsequently used to calculate the capture efficiency and numerical aperture. Their dependence on length, wavelength, bridge thickness and number of layers is presented. Based on the physical insights provided by the analysis, two simplified heuristic models are presented which are valid for either single layer or multiple layer fibers. They show good agreement with the full numerical calculations.

© 2004 Optical Society of America

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  1. J.K. Sahu, C.C. Renaud, K. Furusawa, R. Selvas, J.A. Alvarez-Chavez, D.J. Richardson, and J. Nilsson, “Jacketed air-clad cladding pumped Ytterbium-doped fiber laser with wide tuning range,” Electron. Lett. 371116–1117 (2001).
    [CrossRef]
  2. W.J. Wadsworth, R.M. Percival, G. Bouwmans, J.C. Knight, and P.St.J. Russell, “High power air-clad photonic crystal fibre laser,” Opt. Express. 1148–53 (2003). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-1-48
    [CrossRef] [PubMed]
  3. G. Bouwmans, R.M. Percival, W.J. Wadsworth, J.C. Knight, and P.St.J. Russell, “High-power Er:Yb fiber laser with very high numerical aperture pump-cladding waveguide,” App. Phys. Lett. 83817–818 (2003).
    [CrossRef]
  4. K. Furusawa, A. Malinowski, J.H.V. Price, T.M. Monro, J.K. Sahu, J. Nilsson, and D.J. Richardson, “Caldding pumped Ytterbium-doped fiber laser with holey inner and outer cladding,” Opt. Express 9714–720 (2001). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-714.
    [CrossRef] [PubMed]
  5. J. Limpert, T. Schreiber, A. Liem, S. Nolte, H. Zellmer, T. Peschel, V. Guyenot, and A. Tünnermann, “Thermo-optical properties of air-clad photonic crystal fiber lasers in high power operation,” Opt. Express. 112982–90 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-22-2982
    [CrossRef] [PubMed]
  6. D. Feuermann, J.M. Gordon, and M. Huleihil, “Light leakage in optical fibers: Experimental results, modelling and the consequences for solar collectors,” Solar Evergy 72195–204 (2002).
    [CrossRef]
  7. N.A. Issa, “High numerical aperture in multimode microstructured optical fibers,” submitted to Applied Optics. Temporarily available at http://www.oftc.usyd.edu.au/docs/ext/HighNAinMultimodeMOFs.pdf
  8. C.P. Achenbach and J.H. Cobb, “Computational studies of light acceptance and propagation in straight and curved multimode active fibers,” J. Opt. A 5239–249 (2003).
    [CrossRef]
  9. R.J. Potter, “Transmission properties of Optical Fibers,” J. Opt. Soc. Am. 511079–89 (1961).
    [CrossRef]
  10. W.J. Wadsworth, R.M. Percival, G. Bouwmans, J.C. Knight, T.A. Birks, T.D. Hedley, and P.St.J. Russell, “Very high numerical aperture fibers,” IEEE. Phot. Tech. Lett. 16843–5 (2004).
    [CrossRef]
  11. N.A. Issa and L. Poladian, “Vector wave expansion method for leaky modes of microstructured optical fibres,” J. Lightwave Tech. 211005–12 (2003).
    [CrossRef]
  12. K. Sakoda, “Optical properties of photonic crystals,” (Springer, Berlin, Germany, 2001), pp. 21–23.
  13. A.W. Snyder and J.D. Love, Optical waveguide theory, (Chapman and Hall, New York, 1983), Chapters 4, 12 and 20.

2004 (1)

W.J. Wadsworth, R.M. Percival, G. Bouwmans, J.C. Knight, T.A. Birks, T.D. Hedley, and P.St.J. Russell, “Very high numerical aperture fibers,” IEEE. Phot. Tech. Lett. 16843–5 (2004).
[CrossRef]

2003 (5)

N.A. Issa and L. Poladian, “Vector wave expansion method for leaky modes of microstructured optical fibres,” J. Lightwave Tech. 211005–12 (2003).
[CrossRef]

W.J. Wadsworth, R.M. Percival, G. Bouwmans, J.C. Knight, and P.St.J. Russell, “High power air-clad photonic crystal fibre laser,” Opt. Express. 1148–53 (2003). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-1-48
[CrossRef] [PubMed]

G. Bouwmans, R.M. Percival, W.J. Wadsworth, J.C. Knight, and P.St.J. Russell, “High-power Er:Yb fiber laser with very high numerical aperture pump-cladding waveguide,” App. Phys. Lett. 83817–818 (2003).
[CrossRef]

J. Limpert, T. Schreiber, A. Liem, S. Nolte, H. Zellmer, T. Peschel, V. Guyenot, and A. Tünnermann, “Thermo-optical properties of air-clad photonic crystal fiber lasers in high power operation,” Opt. Express. 112982–90 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-22-2982
[CrossRef] [PubMed]

C.P. Achenbach and J.H. Cobb, “Computational studies of light acceptance and propagation in straight and curved multimode active fibers,” J. Opt. A 5239–249 (2003).
[CrossRef]

2002 (1)

D. Feuermann, J.M. Gordon, and M. Huleihil, “Light leakage in optical fibers: Experimental results, modelling and the consequences for solar collectors,” Solar Evergy 72195–204 (2002).
[CrossRef]

2001 (2)

K. Furusawa, A. Malinowski, J.H.V. Price, T.M. Monro, J.K. Sahu, J. Nilsson, and D.J. Richardson, “Caldding pumped Ytterbium-doped fiber laser with holey inner and outer cladding,” Opt. Express 9714–720 (2001). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-714.
[CrossRef] [PubMed]

J.K. Sahu, C.C. Renaud, K. Furusawa, R. Selvas, J.A. Alvarez-Chavez, D.J. Richardson, and J. Nilsson, “Jacketed air-clad cladding pumped Ytterbium-doped fiber laser with wide tuning range,” Electron. Lett. 371116–1117 (2001).
[CrossRef]

1961 (1)

Achenbach, C.P.

C.P. Achenbach and J.H. Cobb, “Computational studies of light acceptance and propagation in straight and curved multimode active fibers,” J. Opt. A 5239–249 (2003).
[CrossRef]

Alvarez-Chavez, J.A.

J.K. Sahu, C.C. Renaud, K. Furusawa, R. Selvas, J.A. Alvarez-Chavez, D.J. Richardson, and J. Nilsson, “Jacketed air-clad cladding pumped Ytterbium-doped fiber laser with wide tuning range,” Electron. Lett. 371116–1117 (2001).
[CrossRef]

Birks, T.A.

W.J. Wadsworth, R.M. Percival, G. Bouwmans, J.C. Knight, T.A. Birks, T.D. Hedley, and P.St.J. Russell, “Very high numerical aperture fibers,” IEEE. Phot. Tech. Lett. 16843–5 (2004).
[CrossRef]

Bouwmans, G.

W.J. Wadsworth, R.M. Percival, G. Bouwmans, J.C. Knight, T.A. Birks, T.D. Hedley, and P.St.J. Russell, “Very high numerical aperture fibers,” IEEE. Phot. Tech. Lett. 16843–5 (2004).
[CrossRef]

W.J. Wadsworth, R.M. Percival, G. Bouwmans, J.C. Knight, and P.St.J. Russell, “High power air-clad photonic crystal fibre laser,” Opt. Express. 1148–53 (2003). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-1-48
[CrossRef] [PubMed]

G. Bouwmans, R.M. Percival, W.J. Wadsworth, J.C. Knight, and P.St.J. Russell, “High-power Er:Yb fiber laser with very high numerical aperture pump-cladding waveguide,” App. Phys. Lett. 83817–818 (2003).
[CrossRef]

Cobb, J.H.

C.P. Achenbach and J.H. Cobb, “Computational studies of light acceptance and propagation in straight and curved multimode active fibers,” J. Opt. A 5239–249 (2003).
[CrossRef]

Feuermann, D.

D. Feuermann, J.M. Gordon, and M. Huleihil, “Light leakage in optical fibers: Experimental results, modelling and the consequences for solar collectors,” Solar Evergy 72195–204 (2002).
[CrossRef]

Furusawa, K.

J.K. Sahu, C.C. Renaud, K. Furusawa, R. Selvas, J.A. Alvarez-Chavez, D.J. Richardson, and J. Nilsson, “Jacketed air-clad cladding pumped Ytterbium-doped fiber laser with wide tuning range,” Electron. Lett. 371116–1117 (2001).
[CrossRef]

K. Furusawa, A. Malinowski, J.H.V. Price, T.M. Monro, J.K. Sahu, J. Nilsson, and D.J. Richardson, “Caldding pumped Ytterbium-doped fiber laser with holey inner and outer cladding,” Opt. Express 9714–720 (2001). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-714.
[CrossRef] [PubMed]

Gordon, J.M.

D. Feuermann, J.M. Gordon, and M. Huleihil, “Light leakage in optical fibers: Experimental results, modelling and the consequences for solar collectors,” Solar Evergy 72195–204 (2002).
[CrossRef]

Guyenot, V.

J. Limpert, T. Schreiber, A. Liem, S. Nolte, H. Zellmer, T. Peschel, V. Guyenot, and A. Tünnermann, “Thermo-optical properties of air-clad photonic crystal fiber lasers in high power operation,” Opt. Express. 112982–90 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-22-2982
[CrossRef] [PubMed]

Hedley, T.D.

W.J. Wadsworth, R.M. Percival, G. Bouwmans, J.C. Knight, T.A. Birks, T.D. Hedley, and P.St.J. Russell, “Very high numerical aperture fibers,” IEEE. Phot. Tech. Lett. 16843–5 (2004).
[CrossRef]

Huleihil, M.

D. Feuermann, J.M. Gordon, and M. Huleihil, “Light leakage in optical fibers: Experimental results, modelling and the consequences for solar collectors,” Solar Evergy 72195–204 (2002).
[CrossRef]

Issa, N.A.

N.A. Issa and L. Poladian, “Vector wave expansion method for leaky modes of microstructured optical fibres,” J. Lightwave Tech. 211005–12 (2003).
[CrossRef]

N.A. Issa, “High numerical aperture in multimode microstructured optical fibers,” submitted to Applied Optics. Temporarily available at http://www.oftc.usyd.edu.au/docs/ext/HighNAinMultimodeMOFs.pdf

Knight, J.C.

W.J. Wadsworth, R.M. Percival, G. Bouwmans, J.C. Knight, T.A. Birks, T.D. Hedley, and P.St.J. Russell, “Very high numerical aperture fibers,” IEEE. Phot. Tech. Lett. 16843–5 (2004).
[CrossRef]

W.J. Wadsworth, R.M. Percival, G. Bouwmans, J.C. Knight, and P.St.J. Russell, “High power air-clad photonic crystal fibre laser,” Opt. Express. 1148–53 (2003). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-1-48
[CrossRef] [PubMed]

G. Bouwmans, R.M. Percival, W.J. Wadsworth, J.C. Knight, and P.St.J. Russell, “High-power Er:Yb fiber laser with very high numerical aperture pump-cladding waveguide,” App. Phys. Lett. 83817–818 (2003).
[CrossRef]

Liem, A.

J. Limpert, T. Schreiber, A. Liem, S. Nolte, H. Zellmer, T. Peschel, V. Guyenot, and A. Tünnermann, “Thermo-optical properties of air-clad photonic crystal fiber lasers in high power operation,” Opt. Express. 112982–90 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-22-2982
[CrossRef] [PubMed]

Limpert, J.

J. Limpert, T. Schreiber, A. Liem, S. Nolte, H. Zellmer, T. Peschel, V. Guyenot, and A. Tünnermann, “Thermo-optical properties of air-clad photonic crystal fiber lasers in high power operation,” Opt. Express. 112982–90 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-22-2982
[CrossRef] [PubMed]

Love, J.D.

A.W. Snyder and J.D. Love, Optical waveguide theory, (Chapman and Hall, New York, 1983), Chapters 4, 12 and 20.

Malinowski, A.

Monro, T.M.

Nilsson, J.

K. Furusawa, A. Malinowski, J.H.V. Price, T.M. Monro, J.K. Sahu, J. Nilsson, and D.J. Richardson, “Caldding pumped Ytterbium-doped fiber laser with holey inner and outer cladding,” Opt. Express 9714–720 (2001). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-714.
[CrossRef] [PubMed]

J.K. Sahu, C.C. Renaud, K. Furusawa, R. Selvas, J.A. Alvarez-Chavez, D.J. Richardson, and J. Nilsson, “Jacketed air-clad cladding pumped Ytterbium-doped fiber laser with wide tuning range,” Electron. Lett. 371116–1117 (2001).
[CrossRef]

Nolte, S.

J. Limpert, T. Schreiber, A. Liem, S. Nolte, H. Zellmer, T. Peschel, V. Guyenot, and A. Tünnermann, “Thermo-optical properties of air-clad photonic crystal fiber lasers in high power operation,” Opt. Express. 112982–90 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-22-2982
[CrossRef] [PubMed]

Percival, R.M.

W.J. Wadsworth, R.M. Percival, G. Bouwmans, J.C. Knight, T.A. Birks, T.D. Hedley, and P.St.J. Russell, “Very high numerical aperture fibers,” IEEE. Phot. Tech. Lett. 16843–5 (2004).
[CrossRef]

W.J. Wadsworth, R.M. Percival, G. Bouwmans, J.C. Knight, and P.St.J. Russell, “High power air-clad photonic crystal fibre laser,” Opt. Express. 1148–53 (2003). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-1-48
[CrossRef] [PubMed]

G. Bouwmans, R.M. Percival, W.J. Wadsworth, J.C. Knight, and P.St.J. Russell, “High-power Er:Yb fiber laser with very high numerical aperture pump-cladding waveguide,” App. Phys. Lett. 83817–818 (2003).
[CrossRef]

Peschel, T.

J. Limpert, T. Schreiber, A. Liem, S. Nolte, H. Zellmer, T. Peschel, V. Guyenot, and A. Tünnermann, “Thermo-optical properties of air-clad photonic crystal fiber lasers in high power operation,” Opt. Express. 112982–90 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-22-2982
[CrossRef] [PubMed]

Poladian, L.

N.A. Issa and L. Poladian, “Vector wave expansion method for leaky modes of microstructured optical fibres,” J. Lightwave Tech. 211005–12 (2003).
[CrossRef]

Potter, R.J.

Price, J.H.V.

Renaud, C.C.

J.K. Sahu, C.C. Renaud, K. Furusawa, R. Selvas, J.A. Alvarez-Chavez, D.J. Richardson, and J. Nilsson, “Jacketed air-clad cladding pumped Ytterbium-doped fiber laser with wide tuning range,” Electron. Lett. 371116–1117 (2001).
[CrossRef]

Richardson, D.J.

J.K. Sahu, C.C. Renaud, K. Furusawa, R. Selvas, J.A. Alvarez-Chavez, D.J. Richardson, and J. Nilsson, “Jacketed air-clad cladding pumped Ytterbium-doped fiber laser with wide tuning range,” Electron. Lett. 371116–1117 (2001).
[CrossRef]

K. Furusawa, A. Malinowski, J.H.V. Price, T.M. Monro, J.K. Sahu, J. Nilsson, and D.J. Richardson, “Caldding pumped Ytterbium-doped fiber laser with holey inner and outer cladding,” Opt. Express 9714–720 (2001). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-714.
[CrossRef] [PubMed]

Russell, P.St.J.

W.J. Wadsworth, R.M. Percival, G. Bouwmans, J.C. Knight, T.A. Birks, T.D. Hedley, and P.St.J. Russell, “Very high numerical aperture fibers,” IEEE. Phot. Tech. Lett. 16843–5 (2004).
[CrossRef]

W.J. Wadsworth, R.M. Percival, G. Bouwmans, J.C. Knight, and P.St.J. Russell, “High power air-clad photonic crystal fibre laser,” Opt. Express. 1148–53 (2003). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-1-48
[CrossRef] [PubMed]

G. Bouwmans, R.M. Percival, W.J. Wadsworth, J.C. Knight, and P.St.J. Russell, “High-power Er:Yb fiber laser with very high numerical aperture pump-cladding waveguide,” App. Phys. Lett. 83817–818 (2003).
[CrossRef]

Sahu, J.K.

J.K. Sahu, C.C. Renaud, K. Furusawa, R. Selvas, J.A. Alvarez-Chavez, D.J. Richardson, and J. Nilsson, “Jacketed air-clad cladding pumped Ytterbium-doped fiber laser with wide tuning range,” Electron. Lett. 371116–1117 (2001).
[CrossRef]

K. Furusawa, A. Malinowski, J.H.V. Price, T.M. Monro, J.K. Sahu, J. Nilsson, and D.J. Richardson, “Caldding pumped Ytterbium-doped fiber laser with holey inner and outer cladding,” Opt. Express 9714–720 (2001). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-714.
[CrossRef] [PubMed]

Sakoda, K.

K. Sakoda, “Optical properties of photonic crystals,” (Springer, Berlin, Germany, 2001), pp. 21–23.

Schreiber, T.

J. Limpert, T. Schreiber, A. Liem, S. Nolte, H. Zellmer, T. Peschel, V. Guyenot, and A. Tünnermann, “Thermo-optical properties of air-clad photonic crystal fiber lasers in high power operation,” Opt. Express. 112982–90 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-22-2982
[CrossRef] [PubMed]

Selvas, R.

J.K. Sahu, C.C. Renaud, K. Furusawa, R. Selvas, J.A. Alvarez-Chavez, D.J. Richardson, and J. Nilsson, “Jacketed air-clad cladding pumped Ytterbium-doped fiber laser with wide tuning range,” Electron. Lett. 371116–1117 (2001).
[CrossRef]

Snyder, A.W.

A.W. Snyder and J.D. Love, Optical waveguide theory, (Chapman and Hall, New York, 1983), Chapters 4, 12 and 20.

Tünnermann, A.

J. Limpert, T. Schreiber, A. Liem, S. Nolte, H. Zellmer, T. Peschel, V. Guyenot, and A. Tünnermann, “Thermo-optical properties of air-clad photonic crystal fiber lasers in high power operation,” Opt. Express. 112982–90 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-22-2982
[CrossRef] [PubMed]

Wadsworth, W.J.

W.J. Wadsworth, R.M. Percival, G. Bouwmans, J.C. Knight, T.A. Birks, T.D. Hedley, and P.St.J. Russell, “Very high numerical aperture fibers,” IEEE. Phot. Tech. Lett. 16843–5 (2004).
[CrossRef]

W.J. Wadsworth, R.M. Percival, G. Bouwmans, J.C. Knight, and P.St.J. Russell, “High power air-clad photonic crystal fibre laser,” Opt. Express. 1148–53 (2003). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-1-48
[CrossRef] [PubMed]

G. Bouwmans, R.M. Percival, W.J. Wadsworth, J.C. Knight, and P.St.J. Russell, “High-power Er:Yb fiber laser with very high numerical aperture pump-cladding waveguide,” App. Phys. Lett. 83817–818 (2003).
[CrossRef]

Zellmer, H.

J. Limpert, T. Schreiber, A. Liem, S. Nolte, H. Zellmer, T. Peschel, V. Guyenot, and A. Tünnermann, “Thermo-optical properties of air-clad photonic crystal fiber lasers in high power operation,” Opt. Express. 112982–90 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-22-2982
[CrossRef] [PubMed]

App. Phys. Lett. (1)

G. Bouwmans, R.M. Percival, W.J. Wadsworth, J.C. Knight, and P.St.J. Russell, “High-power Er:Yb fiber laser with very high numerical aperture pump-cladding waveguide,” App. Phys. Lett. 83817–818 (2003).
[CrossRef]

Electron. Lett. (1)

J.K. Sahu, C.C. Renaud, K. Furusawa, R. Selvas, J.A. Alvarez-Chavez, D.J. Richardson, and J. Nilsson, “Jacketed air-clad cladding pumped Ytterbium-doped fiber laser with wide tuning range,” Electron. Lett. 371116–1117 (2001).
[CrossRef]

IEEE. Phot. Tech. Lett. (1)

W.J. Wadsworth, R.M. Percival, G. Bouwmans, J.C. Knight, T.A. Birks, T.D. Hedley, and P.St.J. Russell, “Very high numerical aperture fibers,” IEEE. Phot. Tech. Lett. 16843–5 (2004).
[CrossRef]

J. Lightwave Tech. (1)

N.A. Issa and L. Poladian, “Vector wave expansion method for leaky modes of microstructured optical fibres,” J. Lightwave Tech. 211005–12 (2003).
[CrossRef]

J. Opt. A (1)

C.P. Achenbach and J.H. Cobb, “Computational studies of light acceptance and propagation in straight and curved multimode active fibers,” J. Opt. A 5239–249 (2003).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Express (1)

Opt. Express. (2)

W.J. Wadsworth, R.M. Percival, G. Bouwmans, J.C. Knight, and P.St.J. Russell, “High power air-clad photonic crystal fibre laser,” Opt. Express. 1148–53 (2003). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-1-48
[CrossRef] [PubMed]

J. Limpert, T. Schreiber, A. Liem, S. Nolte, H. Zellmer, T. Peschel, V. Guyenot, and A. Tünnermann, “Thermo-optical properties of air-clad photonic crystal fiber lasers in high power operation,” Opt. Express. 112982–90 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-22-2982
[CrossRef] [PubMed]

Solar Evergy (1)

D. Feuermann, J.M. Gordon, and M. Huleihil, “Light leakage in optical fibers: Experimental results, modelling and the consequences for solar collectors,” Solar Evergy 72195–204 (2002).
[CrossRef]

Other (3)

N.A. Issa, “High numerical aperture in multimode microstructured optical fibers,” submitted to Applied Optics. Temporarily available at http://www.oftc.usyd.edu.au/docs/ext/HighNAinMultimodeMOFs.pdf

K. Sakoda, “Optical properties of photonic crystals,” (Springer, Berlin, Germany, 2001), pp. 21–23.

A.W. Snyder and J.D. Love, Optical waveguide theory, (Chapman and Hall, New York, 1983), Chapters 4, 12 and 20.

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Figures (8)

Fig. 1.
Fig. 1.

Cross section along optical fiber axis. The core refractive index, n co=1.45, is assumed equal to the index of the infinite jacket. Both are lossless.

Fig. 2.
Fig. 2.

Schematic illustrating the class of air-clad fiber geometry studied here. All bridges have equal thickness δ and width w. N denotes the number of bridges in each layer while R denotes the number of layers.

Fig. 3.
Fig. 3.

Example external angular transmission distribution used to determine NA, showing dependence on the number of rings. Increasing the number of rings results in a shift in the transition from near θbTE to near θbTM, where θbTE/TM=sin-1(NAbTE/TM). The values δ/λ=0.3, N=100, w/λ=8 and L/D=20m/D o were used in the calculation.

Fig. 4.
Fig. 4.

Total loss, -10log[T(θ)], for three mode classes clearly showing a shift in the loss knee for TE and HE+EH mode classes as the number of rings is increased. The vertical lines indicate nbTE and nbTM, which are the effective indices of the TE fundamental and TM second order bridge modes. They are calculated for the equivalent slab waveguide of thickness 0.3 and the polarizations are prescribed with respect to the orientation of radial bridges. The values δ/λ=0.3, N=100, w/λ=8 and L/D=20m/D o were used in the calculation.

Fig. 5.
Fig. 5.

Example intensity plots of sample TE and TM-like modes in regions I, II (neffr≃1.3>nbTE), III, IV (neffr≃1.172 between nbTE and nbTM), and V, VI (neffr≃1.104<nbTM). The extent of light penetration is shown to correlate with confinement loss which is similarly strongly polarization dependent.

Fig. 6.
Fig. 6.

Numerical aperture as a function of δ/λ and number of layers, full numerical results (data points) are compared to the heuristic estimates. Largely independent of source intensity distribution and weakly length dependent. L/D=10m/Do . Inset: Percentage increase in NA by adding multiple layers, 100×(NAbTM-NAbTE)/NAbTE.

Fig. 7.
Fig. 7.

Capture efficiency as a function of δ/λ and number of layers for an isotropic source. L/D=10m/Do . Inset: Percentage increase in capture efficiency by adding multiple layers, 100×(εbTM-εbTE)/εbTE.

Fig. 8.
Fig. 8.

Relative capture efficiency as a function of δ/λ, L/D and number of rings.

Equations (13)

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T in ( θ in ) I ( θ in ) α ( θ in ) ,
α ( θ in ) = 1 M m = 1 M exp ( 2 { n eff i } m k L ) ,
T ( θ ) 1 n co d θ in d θ I ( θ in ) ( F TE ( θ ) + F TM ( θ ) 2 ) α ( θ in ) ,
NA = sin ( θ max ) ,
ε = 1 P o ϕ = 0 2 π θ in = 0 π 2 T in ( θ in ) sin ( θ in ) d θ in d ϕ ,
N A L n co ε L ,
N A i n co ε i ( 2 ε i ) ,
ε L ε i ( 2 ε i ) .
N A b TE = n co 2 n b TE 2 ,
ε b TE = 1 n b TE n co ,
N A b TM = n co 2 n b TM 2 ,
ε b TM = 1 n b TM n co .
Δ ε = ε ε b TE ,

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