Abstract

Wave chaos is a concept which has already proved its practical usefulness in design of double-clad fibers for cladding-pumped fiber lasers and fiber amplifiers. In general, classically chaotic geometries will favor strong pump absorption and we address the extent of chaotic wave dynamics in typical air-clad geometries. While air-clad structures supporting sup-wavelength convex air-glass interfaces (viewed from the high-index side) will promote chaotic dynamics we find guidance of regular whispering-gallery modes in air-clad structures resembling an overall cylindrical symmetry. Highly symmetric air-clad structures may thus suppress the pump-absorption efficiency η below the ergodic scaling law η ∝ Ac/A cl, where Ac and A cl are the areas of the rare-earth doped core and the cladding, respectively.

© 2007 Optical Society of America

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  1. A. Tünnermann, T. Schreiber, F. Röser, A. Liem, S. Höfer, H. Zellmer, S. Nolte, and J. Limpert, "The renaissance and bright future of fibre lasers," J. Phys. B 38, S681 - S693 (2005).
    [CrossRef]
  2. H.-R. Müller, J. Kirchhof, V. Reichel, and S. Unger, "Fibers for high-power lasers and amplifiers," C. R. Phys. 7, 154 - 162 (2006).
    [CrossRef]
  3. J. Canning, "Fibre lasers and related technologies," Opt. Lasers Eng. 44, 647 - 676 (2006).
    [CrossRef]
  4. J. Limpert, F. Röser, T. Schreiber, I. Manek-Hönninger, F. Salin, and A. Tünnermann, "Ultrafast high power fiber laser systems," C. R. Phys. 7, 187 - 197 (2006).
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  10. J. Limpert, T. Schreiber, S. Nolte, H. Zellmer, A. Tünnermann, R. Iliew, F. Lederer, J. Broeng, G. Vienne, A. Petersson, and C. Jakobsen, "High-power air-clad large-mode-area photonic crystal fiber laser," Opt. Express 11, 818 - 823 (2003).
    [CrossRef] [PubMed]
  11. G. Bouwmans, R. M. Percival, W. J. Wadsworth, J. C. Knight, and P. S. J. Russell, "High-power Er : Yb fiber laser with very high numerical aperture pump-cladding waveguide," Appl. Phys. Lett. 83, 817 - 818 (2003).
    [CrossRef]
  12. J. Limpert, N. D. Robin, I. Manek-Hönninger, F. Salin, F. Röser, A. Liem, T. Schreiber, S. Nolte, H. Zellmer, A. Tünnermann, J. Broeng, A. Petersson, and C. Jakobsen, "High-power rod-type photonic crystal fiber laser," Opt. Express 13, 1055 - 1058 (2005).
    [CrossRef] [PubMed]
  13. N. A. Issa, "High numerical aperture in multimode microstructured optical fibers," Appl. Optics 43, 6191 - 6197 (2004).
    [CrossRef]
  14. M. Aslund, S. D. Jackson, J. Canning, A. Teixeira, and K. Lyytikainen-Digweed, "The influence of skew rays on angular losses in air-clad fibres," Opt. Commun. 262, 77 - 81 (2006).
    [CrossRef]
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    [CrossRef]
  16. J. Broeng, G. Vienne, A. Petersson, P. M. W. Skovgaard, J. R. Folkenberg, M. D. Nielsen, C. Jakobsen, H. Simonsen, and N. A. Mortensen, "Air-clad photonic crystal fibers for high-power single-mode lasers," Proc. SPIE 5335, 192 - 201 (2004).
    [CrossRef]
  17. S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, "Perturbation theory for Maxwell’s equations with shifting material boundaries," Phys. Rev. E 65, 066611 (2002).
    [CrossRef]
  18. N. A. Mortensen, M. Stach, J. Broeng, A. Petersson, H. R. Simonsen, and R. Michalzik, "Multi-mode photonic crystal fibers for VCSEL based data transmission," Opt. Express 11, 1953 - 1959 (2003).
    [CrossRef] [PubMed]
  19. V. Doya, O. Legrand, F. Mortessagne, and C. Miniatura, "Speckle statistics in a chaotic multimode fiber," Phys. Rev. E 65, 056223 (2002).
    [CrossRef]
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    [CrossRef]
  21. D. Kouznetsov and J. V. Moloney, "Efficiency of pump absorption in double-clad fiber amplifiers. II. Broken circular symmetry," J. Opt. Soc. Am. B 19, 1259 - 1263 (2002).
    [CrossRef]
  22. D. Kouznetsov and J. V. Moloney, "Efficiency of pump absorption in double-clad fiber amplifiers. III. Calculation of modes," J. Opt. Soc. Am. B 19, 1304 - 1309 (2002).
    [CrossRef]
  23. J. Limpert, O. Schmidt, J. Rothhardt, F. Röser, T. Schreiber, A. Tünnermann, S. Ermeneux, P. Yvernault, and F. Salin, "Extended single-mode photonic crystal fiber lasers," Opt. Express 14, 2715 - 2720 (2006).
    [CrossRef] [PubMed]

2006 (5)

H.-R. Müller, J. Kirchhof, V. Reichel, and S. Unger, "Fibers for high-power lasers and amplifiers," C. R. Phys. 7, 154 - 162 (2006).
[CrossRef]

J. Canning, "Fibre lasers and related technologies," Opt. Lasers Eng. 44, 647 - 676 (2006).
[CrossRef]

J. Limpert, F. Röser, T. Schreiber, I. Manek-Hönninger, F. Salin, and A. Tünnermann, "Ultrafast high power fiber laser systems," C. R. Phys. 7, 187 - 197 (2006).
[CrossRef]

M. Aslund, S. D. Jackson, J. Canning, A. Teixeira, and K. Lyytikainen-Digweed, "The influence of skew rays on angular losses in air-clad fibres," Opt. Commun. 262, 77 - 81 (2006).
[CrossRef]

J. Limpert, O. Schmidt, J. Rothhardt, F. Röser, T. Schreiber, A. Tünnermann, S. Ermeneux, P. Yvernault, and F. Salin, "Extended single-mode photonic crystal fiber lasers," Opt. Express 14, 2715 - 2720 (2006).
[CrossRef] [PubMed]

2005 (2)

2004 (2)

N. A. Issa, "High numerical aperture in multimode microstructured optical fibers," Appl. Optics 43, 6191 - 6197 (2004).
[CrossRef]

J. Broeng, G. Vienne, A. Petersson, P. M. W. Skovgaard, J. R. Folkenberg, M. D. Nielsen, C. Jakobsen, H. Simonsen, and N. A. Mortensen, "Air-clad photonic crystal fibers for high-power single-mode lasers," Proc. SPIE 5335, 192 - 201 (2004).
[CrossRef]

2003 (4)

2002 (4)

V. Doya, O. Legrand, F. Mortessagne, and C. Miniatura, "Speckle statistics in a chaotic multimode fiber," Phys. Rev. E 65, 056223 (2002).
[CrossRef]

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, "Perturbation theory for Maxwell’s equations with shifting material boundaries," Phys. Rev. E 65, 066611 (2002).
[CrossRef]

D. Kouznetsov and J. V. Moloney, "Efficiency of pump absorption in double-clad fiber amplifiers. II. Broken circular symmetry," J. Opt. Soc. Am. B 19, 1259 - 1263 (2002).
[CrossRef]

D. Kouznetsov and J. V. Moloney, "Efficiency of pump absorption in double-clad fiber amplifiers. III. Calculation of modes," J. Opt. Soc. Am. B 19, 1304 - 1309 (2002).
[CrossRef]

2001 (3)

2000 (1)

W. J. Wadsworth, J. C. Knight, W. H. Reeves, P. S. J. Russell, and J. Arriaga, "Yb3+-doped photonic crystal fibre laser," Electron. Lett. 36, 1452 - 1454 (2000).
[CrossRef]

1989 (1)

E. Snitzer, "Rare-earth fiber lasers," J. Less-Common Met. 148, 45 - 58 (1989).
[CrossRef]

Arriaga, J.

W. J. Wadsworth, J. C. Knight, W. H. Reeves, P. S. J. Russell, and J. Arriaga, "Yb3+-doped photonic crystal fibre laser," Electron. Lett. 36, 1452 - 1454 (2000).
[CrossRef]

Aslund, M.

M. Aslund, S. D. Jackson, J. Canning, A. Teixeira, and K. Lyytikainen-Digweed, "The influence of skew rays on angular losses in air-clad fibres," Opt. Commun. 262, 77 - 81 (2006).
[CrossRef]

Bouwmans, G.

G. Bouwmans, R. M. Percival, W. J. Wadsworth, J. C. Knight, and P. S. J. Russell, "High-power Er : Yb fiber laser with very high numerical aperture pump-cladding waveguide," Appl. Phys. Lett. 83, 817 - 818 (2003).
[CrossRef]

W. J. Wadsworth, R. M. Percival, G. Bouwmans, J. C. Knight, and P. S. J. Russel, "High power air-clad photonic crystal fibre laser," Opt. Express 11, 48 - 53 (2003).
[CrossRef] [PubMed]

Broeng, J.

Canning, J.

M. Aslund, S. D. Jackson, J. Canning, A. Teixeira, and K. Lyytikainen-Digweed, "The influence of skew rays on angular losses in air-clad fibres," Opt. Commun. 262, 77 - 81 (2006).
[CrossRef]

J. Canning, "Fibre lasers and related technologies," Opt. Lasers Eng. 44, 647 - 676 (2006).
[CrossRef]

Doya, V.

V. Doya, O. Legrand, F. Mortessagne, and C. Miniatura, "Speckle statistics in a chaotic multimode fiber," Phys. Rev. E 65, 056223 (2002).
[CrossRef]

V. Doya, O. Legrand, and F. Mortessagne, "Optimized absorption in a chaotic double-clad fiber amplifier," Opt. Lett. 26, 872 - 874 (2001).
[CrossRef]

Ermeneux, S.

Fink, Y.

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, "Perturbation theory for Maxwell’s equations with shifting material boundaries," Phys. Rev. E 65, 066611 (2002).
[CrossRef]

Folkenberg, J. R.

J. Broeng, G. Vienne, A. Petersson, P. M. W. Skovgaard, J. R. Folkenberg, M. D. Nielsen, C. Jakobsen, H. Simonsen, and N. A. Mortensen, "Air-clad photonic crystal fibers for high-power single-mode lasers," Proc. SPIE 5335, 192 - 201 (2004).
[CrossRef]

Furusawa, K.

Höfer, S.

A. Tünnermann, T. Schreiber, F. Röser, A. Liem, S. Höfer, H. Zellmer, S. Nolte, and J. Limpert, "The renaissance and bright future of fibre lasers," J. Phys. B 38, S681 - S693 (2005).
[CrossRef]

Ibanescu, M.

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, "Perturbation theory for Maxwell’s equations with shifting material boundaries," Phys. Rev. E 65, 066611 (2002).
[CrossRef]

Iliew, R.

Issa, N. A.

N. A. Issa, "High numerical aperture in multimode microstructured optical fibers," Appl. Optics 43, 6191 - 6197 (2004).
[CrossRef]

Jackson, S. D.

M. Aslund, S. D. Jackson, J. Canning, A. Teixeira, and K. Lyytikainen-Digweed, "The influence of skew rays on angular losses in air-clad fibres," Opt. Commun. 262, 77 - 81 (2006).
[CrossRef]

Jakobsen, C.

Joannopoulos, J. D.

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, "Perturbation theory for Maxwell’s equations with shifting material boundaries," Phys. Rev. E 65, 066611 (2002).
[CrossRef]

Johnson, S. G.

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, "Perturbation theory for Maxwell’s equations with shifting material boundaries," Phys. Rev. E 65, 066611 (2002).
[CrossRef]

Kirchhof, J.

H.-R. Müller, J. Kirchhof, V. Reichel, and S. Unger, "Fibers for high-power lasers and amplifiers," C. R. Phys. 7, 154 - 162 (2006).
[CrossRef]

Knight, J. C.

W. J. Wadsworth, R. M. Percival, G. Bouwmans, J. C. Knight, and P. S. J. Russel, "High power air-clad photonic crystal fibre laser," Opt. Express 11, 48 - 53 (2003).
[CrossRef] [PubMed]

G. Bouwmans, R. M. Percival, W. J. Wadsworth, J. C. Knight, and P. S. J. Russell, "High-power Er : Yb fiber laser with very high numerical aperture pump-cladding waveguide," Appl. Phys. Lett. 83, 817 - 818 (2003).
[CrossRef]

W. J. Wadsworth, J. C. Knight, W. H. Reeves, P. S. J. Russell, and J. Arriaga, "Yb3+-doped photonic crystal fibre laser," Electron. Lett. 36, 1452 - 1454 (2000).
[CrossRef]

Kouznetsov, D.

Lederer, F.

Legrand, O.

V. Doya, O. Legrand, F. Mortessagne, and C. Miniatura, "Speckle statistics in a chaotic multimode fiber," Phys. Rev. E 65, 056223 (2002).
[CrossRef]

V. Doya, O. Legrand, and F. Mortessagne, "Optimized absorption in a chaotic double-clad fiber amplifier," Opt. Lett. 26, 872 - 874 (2001).
[CrossRef]

Liem, A.

Limpert, J.

Lyytikainen-Digweed, K.

M. Aslund, S. D. Jackson, J. Canning, A. Teixeira, and K. Lyytikainen-Digweed, "The influence of skew rays on angular losses in air-clad fibres," Opt. Commun. 262, 77 - 81 (2006).
[CrossRef]

Malinowski, A.

Manek-Hönninger, I.

Michalzik, R.

Miniatura, C.

V. Doya, O. Legrand, F. Mortessagne, and C. Miniatura, "Speckle statistics in a chaotic multimode fiber," Phys. Rev. E 65, 056223 (2002).
[CrossRef]

Moloney, J. V.

Monro, T.M.

Mortensen, N. A.

J. Broeng, G. Vienne, A. Petersson, P. M. W. Skovgaard, J. R. Folkenberg, M. D. Nielsen, C. Jakobsen, H. Simonsen, and N. A. Mortensen, "Air-clad photonic crystal fibers for high-power single-mode lasers," Proc. SPIE 5335, 192 - 201 (2004).
[CrossRef]

N. A. Mortensen, M. Stach, J. Broeng, A. Petersson, H. R. Simonsen, and R. Michalzik, "Multi-mode photonic crystal fibers for VCSEL based data transmission," Opt. Express 11, 1953 - 1959 (2003).
[CrossRef] [PubMed]

Mortessagne, F.

V. Doya, O. Legrand, F. Mortessagne, and C. Miniatura, "Speckle statistics in a chaotic multimode fiber," Phys. Rev. E 65, 056223 (2002).
[CrossRef]

V. Doya, O. Legrand, and F. Mortessagne, "Optimized absorption in a chaotic double-clad fiber amplifier," Opt. Lett. 26, 872 - 874 (2001).
[CrossRef]

Müller, H.-R.

H.-R. Müller, J. Kirchhof, V. Reichel, and S. Unger, "Fibers for high-power lasers and amplifiers," C. R. Phys. 7, 154 - 162 (2006).
[CrossRef]

Nielsen, M. D.

J. Broeng, G. Vienne, A. Petersson, P. M. W. Skovgaard, J. R. Folkenberg, M. D. Nielsen, C. Jakobsen, H. Simonsen, and N. A. Mortensen, "Air-clad photonic crystal fibers for high-power single-mode lasers," Proc. SPIE 5335, 192 - 201 (2004).
[CrossRef]

Nilsson, J.

Nolte, S.

Percival, R. M.

G. Bouwmans, R. M. Percival, W. J. Wadsworth, J. C. Knight, and P. S. J. Russell, "High-power Er : Yb fiber laser with very high numerical aperture pump-cladding waveguide," Appl. Phys. Lett. 83, 817 - 818 (2003).
[CrossRef]

W. J. Wadsworth, R. M. Percival, G. Bouwmans, J. C. Knight, and P. S. J. Russel, "High power air-clad photonic crystal fibre laser," Opt. Express 11, 48 - 53 (2003).
[CrossRef] [PubMed]

Petersson, A.

Price, J. H. V.

Reeves, W. H.

W. J. Wadsworth, J. C. Knight, W. H. Reeves, P. S. J. Russell, and J. Arriaga, "Yb3+-doped photonic crystal fibre laser," Electron. Lett. 36, 1452 - 1454 (2000).
[CrossRef]

Reichel, V.

H.-R. Müller, J. Kirchhof, V. Reichel, and S. Unger, "Fibers for high-power lasers and amplifiers," C. R. Phys. 7, 154 - 162 (2006).
[CrossRef]

Richardson, D. J.

Robin, N. D.

Röser, F.

J. Limpert, F. Röser, T. Schreiber, I. Manek-Hönninger, F. Salin, and A. Tünnermann, "Ultrafast high power fiber laser systems," C. R. Phys. 7, 187 - 197 (2006).
[CrossRef]

J. Limpert, O. Schmidt, J. Rothhardt, F. Röser, T. Schreiber, A. Tünnermann, S. Ermeneux, P. Yvernault, and F. Salin, "Extended single-mode photonic crystal fiber lasers," Opt. Express 14, 2715 - 2720 (2006).
[CrossRef] [PubMed]

A. Tünnermann, T. Schreiber, F. Röser, A. Liem, S. Höfer, H. Zellmer, S. Nolte, and J. Limpert, "The renaissance and bright future of fibre lasers," J. Phys. B 38, S681 - S693 (2005).
[CrossRef]

J. Limpert, N. D. Robin, I. Manek-Hönninger, F. Salin, F. Röser, A. Liem, T. Schreiber, S. Nolte, H. Zellmer, A. Tünnermann, J. Broeng, A. Petersson, and C. Jakobsen, "High-power rod-type photonic crystal fiber laser," Opt. Express 13, 1055 - 1058 (2005).
[CrossRef] [PubMed]

Rothhardt, J.

Russel, P. S. J.

Russell, P. S. J.

G. Bouwmans, R. M. Percival, W. J. Wadsworth, J. C. Knight, and P. S. J. Russell, "High-power Er : Yb fiber laser with very high numerical aperture pump-cladding waveguide," Appl. Phys. Lett. 83, 817 - 818 (2003).
[CrossRef]

W. J. Wadsworth, J. C. Knight, W. H. Reeves, P. S. J. Russell, and J. Arriaga, "Yb3+-doped photonic crystal fibre laser," Electron. Lett. 36, 1452 - 1454 (2000).
[CrossRef]

Sahu, J. K.

Salin, F.

Schmidt, O.

Schreiber, T.

Simonsen, H.

J. Broeng, G. Vienne, A. Petersson, P. M. W. Skovgaard, J. R. Folkenberg, M. D. Nielsen, C. Jakobsen, H. Simonsen, and N. A. Mortensen, "Air-clad photonic crystal fibers for high-power single-mode lasers," Proc. SPIE 5335, 192 - 201 (2004).
[CrossRef]

Simonsen, H. R.

Skorobogatiy, M. A.

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, "Perturbation theory for Maxwell’s equations with shifting material boundaries," Phys. Rev. E 65, 066611 (2002).
[CrossRef]

Skovgaard, P. M. W.

J. Broeng, G. Vienne, A. Petersson, P. M. W. Skovgaard, J. R. Folkenberg, M. D. Nielsen, C. Jakobsen, H. Simonsen, and N. A. Mortensen, "Air-clad photonic crystal fibers for high-power single-mode lasers," Proc. SPIE 5335, 192 - 201 (2004).
[CrossRef]

Snitzer, E.

E. Snitzer, "Rare-earth fiber lasers," J. Less-Common Met. 148, 45 - 58 (1989).
[CrossRef]

Stach, M.

Teixeira, A.

M. Aslund, S. D. Jackson, J. Canning, A. Teixeira, and K. Lyytikainen-Digweed, "The influence of skew rays on angular losses in air-clad fibres," Opt. Commun. 262, 77 - 81 (2006).
[CrossRef]

Tünnermann, A.

Unger, S.

H.-R. Müller, J. Kirchhof, V. Reichel, and S. Unger, "Fibers for high-power lasers and amplifiers," C. R. Phys. 7, 154 - 162 (2006).
[CrossRef]

Vienne, G.

J. Broeng, G. Vienne, A. Petersson, P. M. W. Skovgaard, J. R. Folkenberg, M. D. Nielsen, C. Jakobsen, H. Simonsen, and N. A. Mortensen, "Air-clad photonic crystal fibers for high-power single-mode lasers," Proc. SPIE 5335, 192 - 201 (2004).
[CrossRef]

J. Limpert, T. Schreiber, S. Nolte, H. Zellmer, A. Tünnermann, R. Iliew, F. Lederer, J. Broeng, G. Vienne, A. Petersson, and C. Jakobsen, "High-power air-clad large-mode-area photonic crystal fiber laser," Opt. Express 11, 818 - 823 (2003).
[CrossRef] [PubMed]

Wadsworth, W. J.

G. Bouwmans, R. M. Percival, W. J. Wadsworth, J. C. Knight, and P. S. J. Russell, "High-power Er : Yb fiber laser with very high numerical aperture pump-cladding waveguide," Appl. Phys. Lett. 83, 817 - 818 (2003).
[CrossRef]

W. J. Wadsworth, R. M. Percival, G. Bouwmans, J. C. Knight, and P. S. J. Russel, "High power air-clad photonic crystal fibre laser," Opt. Express 11, 48 - 53 (2003).
[CrossRef] [PubMed]

W. J. Wadsworth, J. C. Knight, W. H. Reeves, P. S. J. Russell, and J. Arriaga, "Yb3+-doped photonic crystal fibre laser," Electron. Lett. 36, 1452 - 1454 (2000).
[CrossRef]

Weisberg, O.

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, "Perturbation theory for Maxwell’s equations with shifting material boundaries," Phys. Rev. E 65, 066611 (2002).
[CrossRef]

Wright, E. M.

Yvernault, P.

Zellmer, H.

Appl. Optics (1)

N. A. Issa, "High numerical aperture in multimode microstructured optical fibers," Appl. Optics 43, 6191 - 6197 (2004).
[CrossRef]

Appl. Phys. Lett. (1)

G. Bouwmans, R. M. Percival, W. J. Wadsworth, J. C. Knight, and P. S. J. Russell, "High-power Er : Yb fiber laser with very high numerical aperture pump-cladding waveguide," Appl. Phys. Lett. 83, 817 - 818 (2003).
[CrossRef]

C. R. Phys. (2)

H.-R. Müller, J. Kirchhof, V. Reichel, and S. Unger, "Fibers for high-power lasers and amplifiers," C. R. Phys. 7, 154 - 162 (2006).
[CrossRef]

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Supplementary Material (1)

» Media 1: GIF (3328 KB)     

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Figures (6)

Fig. 1.
Fig. 1.

Plot of the pump-absorption efficiency η(L). The solid line shows the ergodic limit, Eq. (7), while the dashed line indicates the initial dynamics and the critical length L *, Eq. (11). The inset illustrates a setup employing a double-clad fiber for conversion of multi-mode light with a low beam quality into single-mode light with a high beam quality.

Fig. 2.
Fig. 2.

Panel (a) shows a micrograph of an air-clad photonic crystal fiber made in silica (courtesy Crystal Fibre A/S). The pump light is corralled by an air clad consisting of 42 air holes (dark regions) separated by sub-micron silica bridges. The diameter of the inner cladding is of the order 125μm. Panels (b) to (h) show time-dependent finite-element simulations of a scalar wave equation at different times (click panel to view animation, 3.3 Mbyte) for the initial Gaussian excitation shown in panel (b). [Media 1]

Fig. 3.
Fig. 3.

Ray-tracing dynamics illustrating the different response of integrable geometries, panels (a,b), and non-integrable chaotic geometries, panels (c,d). Panels (a) shows initial ray trajectories in the transverse section of an integrable geometry corresponding to an idealized multi-mode fibers having perfect circular symmetry. Panel (b) shows the regular whispering-gallery like pattern that results after propagating the initial rays in panel (a) for a long time (> 103 round trips). Panel (c) shows corresponding results for a non-integrable chaotic geometry including a circular scatterer. Note that the ray pattern results from the same initial conditions as for panel (b). Panel (d) shows a corresponding ‘scarred’ wave function obtained from a finite-element solution of a scalar wave equation.

Fig. 4.
Fig. 4.

Distribution functions for a regular integrable and non-integrable chaotic geometry. The dashed lines shows numerical fits to Eq. (12) with 〈f〉/f 0 ~ 9% and 〈f〉/f 0 ~ 100% for the circular and D-shape geometries, respectively.

Fig. 5.
Fig. 5.

Qualitative illustration of the distribution function for an air-clad structure displaying partly integrable (λ δ) and partly chaotic dynamics (λ δ). The insets illustrate the two classes of modes representing a vanishing overlap f ~ 0 and a finite overlap f ~ f 0, respectively.

Fig. 6.
Fig. 6.

Panel (a) shows a whispering-gallery mode in a ring-shaped air-clad geometry while panel (b) shows a corresponding mode for a D-shaped air-clad geometry.

Equations (13)

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ε ( r ) = ε′ ( r ) + i ε″ ( r ) = { ε c + i ε c , r Ω c , ε cl , r Ω cl , 1 , r Ω air ,
× × E m = ε ( ω m c ) 2 E m , m = 1 , 2 , 3 ,
Δ β m = ( ω m β ) 1 ω m 2 E m Δ ε E m E m ε E m
α m = 2 β m = f m × ω ν g , m ε c ε c f m × α c
f m = E m ε E m c E m ε E m c + cl = E m D m c E m D m c + cl
η ( L ) = 0 d α P ( α ) [ 1 exp ( α L ) ] 0 d f P ( f ) [ 1 exp ( f × α c L ) ] .
η ( L ) = f α c L 1 2 f 2 ( α c L ) 2 + 𝓞 ( [ α c L ] 3 ) .
L * f 1 × α c 1
f 0 = ε c A c ε c A c + ε cl A cl = A c A cl + 𝓞 ( [ A c A cl ] 2 ) + 𝓞 ( ε c ε cl )
η ( L ) 1 exp ( f 0 α c L ) ( ergodic limit )
L * f 0 1 × α c 1 A cl A c × α c 1
P fit ( f ) = { f 1 exp ( f f ) , f f 0 ~ 9 % , ( circular ) , 4 f f 2 exp ( 2 f f ) , f f 0 ~ 100 % , ( D shape ) ,
λ λ n cl 2 n eff 2

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