Abstract

We derive a sufficient condition for the existence of indexguided modes in a very general class of dielectric waveguides, including photonic-crystal fibers (arbitrary periodic claddings, such as “holey fibers”), anisotropic materials, and waveguides with periodicity along the propagation direction. This condition provides a rigorous guarantee of cutoff-free index-guided modes in any such structure where the core is formed by increasing the index of refraction (e.g. removing a hole). It also provides a weaker guarantee of guidance in cases where the refractive index is increased “on average” (precisely defined). The proof is based on a simple variational method, inspired by analogous proofs of localization for two-dimensional attractive potentials in quantum mechanics.

© 2008 Optical Society of America

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References

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  1. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983).
  2. P. Russell, “Photonic crystal fibers,” Science 299, 358–362 (2003).
    [Crossref] [PubMed]
  3. A. Bjarklev, J. Broeng, and A. S. Bjarklev, Photonic Crystal Fibres (Springer, New York, 2003).
    [Crossref]
  4. F. Zolla, G. Renversez, A. Nicolet, B. Kuhlmey, S. Guenneau, and D. Felbacq, Foundations of Photonic Crystal Fibres (Imperial College Press, London, 2005).
    [Crossref]
  5. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton Univ. Press, 2008).
  6. R. Ramaswami and K. N. Sivarajan, Optical Networks: A Practical Perspective (Academic Press, London, 1998).
  7. C. Elachi, “Waves in active and passive periodic structures: A review,” Proc. IEEE 64, 1666–1698 (1976).
    [Crossref]
  8. S. Fan, J. N. Winn, A. Devenyi, J. C. Chen, R. D. Meade, and J. D. Joannopoulos, “Guided and defect modes in periodic dielectric waveguides,” J. Opt. Soc. Am. B 12, 1267–1272 (1995).
    [Crossref]
  9. A. Bamberger and A. S. Bonnet, “Mathematical analysis of the guided modes of an optical fiber,” SIAM J. Math. Anal. 21, 1487–1510 (1990).
    [Crossref]
  10. H. P. Urbach, “Analysis of the domain integral operator for anisotropic dielectric waveguides,” Journal on Mathematical Analysis 27 (1996).
  11. K. Yang and M. de Llano, “Simple variational proof that any two-dimensional potential well supports at least one bound state,” Am. J. Phys. 57, 85–86 (1989).
    [Crossref]
  12. R. G. Hunsperger, Integrated Optics: Theory and Technology (Springer-Verlag, 1982).
  13. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991).
    [Crossref]
  14. C.-L. Chen, Foundations for Guided-Wave Optics (Wiley, 2006).
    [Crossref]
  15. B. T. Kuhlmey, R. C. McPhedran, C. M. de Sterke, P. A. Robinson, G. Renversez, and D. Maystre, “Microstructured optical fibers: where’s the edge?” Opt. Express 10, 1285–1290 (2002).
    [PubMed]
  16. S. Wilcox, L. Botten, C. M. de Sterke, B. Kuhlmey, R. McPhedran, D. Fussell, and S. Tomljenovic-Hanic, “Long wavelength behavior of the fundamental mode in microstructured optical fibers,” Opt. Express 13 (2005).
    [Crossref] [PubMed]
  17. S. Kawakami and S. Nishida, “Characteristics of a doubly clad optical fiber with a low-index inner cladding,” IEEE J. Quantum Electron. 10, 879–887 (1974).
    [Crossref]
  18. T. Okoshi and K. Oyamoda, “Single-polarization single-mode optical fibre with refractive-index pits on both sides of core,” Electron. Lett. 16, 712–713 (80).
    [Crossref]
  19. W. Eickhoff, “Stress-induced single-polarization single-mode fiber,” Opt. Lett. 7 (1982).
    [Crossref] [PubMed]
  20. J. R. Simpson, R. H. Stolen, F. M. Sears, W. Pleibel, J. B. Macchesney, and R. E. Howard, “A single-polarization fiber,” IEEE J. Lightwave Technol. 1, 370–374 (1983).
    [Crossref]
  21. M. J. Messerly, J. R. Onstott, and R. C. Mikkelson, “A broad-band single polarization optical fiber,” IEEE J. Lightwave Technol. 9, 817–820 (1991).
    [Crossref]
  22. H. Kubota, S. Kawanishi, S. Koyanagi, M. Tanaka, and S. Yamaguchi, “Absolutely single polarization photonic crystal fiber,” IEEE Photon. Tech. Lett. 16, 182–184 (2004).
    [Crossref]
  23. M.-J. Li, X. Chen, D. A. Nolan, G. E. Berkey, J. Wang, W. A. Wood, and L. A. Zenteno, “High bandwidth single polarization fiber with elliptical central air hole,” IEEE J. Lightwave Technol. 23, 3454–3460 (2005).
    [Crossref]
  24. P. Kuchment, “The Mathematics of Photonic Crystals,” in Mathematical Modeling in Optical Science, G. Bao, L. Cowsar, and W. Masters, eds., Frontiers in Applied Mathematics, pp. 207–272 (SIAM, Philadelphia, 2001).
  25. D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous materials,” Phys. Rev. E 71, 036,617 (2005).
    [Crossref]
  26. P. Kuchment and B. Ong, “On guided waves in photonic crystal waveguides,” in Waves in Periodic and Random Media, vol. 339 of Contemporary Mathematics, pp. 105–115 (AMS, Providence, RI, 2003).
  27. B. Simon, “The bound state of weakly coupled Schrödinger operators in one and two dimensions,” Ann. Phys. 97, 279–288 (1976).
    [Crossref]
  28. L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Addison-Wesley, 1977).
  29. H. Picq, “Détermination et calcul numérique de la première valeur propre d’opérateurs de Schrödinger dans le plan,” Ph.D. thesis, Université de Nice, Nice, France (1982).
  30. E. N. Economou, Green’s functions in quantum physics (Springer, 2006).
  31. C. Cohen-Tannoudji, B. Din, and F. Laloë, Quantum Mechanics (Hermann, Paris, 1977).
  32. D. ter Haar, Selected Problems in Quantum Mechanics (Academic Press, New York, 1964).
  33. E. Hewitt and K. Stromberg, Real and Abstract Analysis (Springer, 1965).
  34. J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1998).
  35. J. A. Kong, Electromagnetic Wave Theory (Wiley, New York, 1975).
  36. S. G. Johnson, M. L. Povinelli, M. Soljačić, A. Karalis, S. Jacobs, and J. D. Joannopoulos, “Roughness losses and volume-current methods in photonic-crystal waveguides,” Appl. Phys. B 81, 283–293 (2005).
    [Crossref]
  37. P. Yeh, A. Yariv, and E. Marom, “Theory of Bragg fiber,” J. Opt. Soc. Am. 68, 1196–1201 (1978).
    [Crossref]

2005 (4)

S. Wilcox, L. Botten, C. M. de Sterke, B. Kuhlmey, R. McPhedran, D. Fussell, and S. Tomljenovic-Hanic, “Long wavelength behavior of the fundamental mode in microstructured optical fibers,” Opt. Express 13 (2005).
[Crossref] [PubMed]

M.-J. Li, X. Chen, D. A. Nolan, G. E. Berkey, J. Wang, W. A. Wood, and L. A. Zenteno, “High bandwidth single polarization fiber with elliptical central air hole,” IEEE J. Lightwave Technol. 23, 3454–3460 (2005).
[Crossref]

D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous materials,” Phys. Rev. E 71, 036,617 (2005).
[Crossref]

S. G. Johnson, M. L. Povinelli, M. Soljačić, A. Karalis, S. Jacobs, and J. D. Joannopoulos, “Roughness losses and volume-current methods in photonic-crystal waveguides,” Appl. Phys. B 81, 283–293 (2005).
[Crossref]

2004 (1)

H. Kubota, S. Kawanishi, S. Koyanagi, M. Tanaka, and S. Yamaguchi, “Absolutely single polarization photonic crystal fiber,” IEEE Photon. Tech. Lett. 16, 182–184 (2004).
[Crossref]

2003 (1)

P. Russell, “Photonic crystal fibers,” Science 299, 358–362 (2003).
[Crossref] [PubMed]

2002 (1)

1996 (1)

H. P. Urbach, “Analysis of the domain integral operator for anisotropic dielectric waveguides,” Journal on Mathematical Analysis 27 (1996).

1995 (1)

1991 (1)

M. J. Messerly, J. R. Onstott, and R. C. Mikkelson, “A broad-band single polarization optical fiber,” IEEE J. Lightwave Technol. 9, 817–820 (1991).
[Crossref]

1990 (1)

A. Bamberger and A. S. Bonnet, “Mathematical analysis of the guided modes of an optical fiber,” SIAM J. Math. Anal. 21, 1487–1510 (1990).
[Crossref]

1989 (1)

K. Yang and M. de Llano, “Simple variational proof that any two-dimensional potential well supports at least one bound state,” Am. J. Phys. 57, 85–86 (1989).
[Crossref]

1983 (1)

J. R. Simpson, R. H. Stolen, F. M. Sears, W. Pleibel, J. B. Macchesney, and R. E. Howard, “A single-polarization fiber,” IEEE J. Lightwave Technol. 1, 370–374 (1983).
[Crossref]

1982 (1)

W. Eickhoff, “Stress-induced single-polarization single-mode fiber,” Opt. Lett. 7 (1982).
[Crossref] [PubMed]

1978 (1)

1976 (2)

C. Elachi, “Waves in active and passive periodic structures: A review,” Proc. IEEE 64, 1666–1698 (1976).
[Crossref]

B. Simon, “The bound state of weakly coupled Schrödinger operators in one and two dimensions,” Ann. Phys. 97, 279–288 (1976).
[Crossref]

1974 (1)

S. Kawakami and S. Nishida, “Characteristics of a doubly clad optical fiber with a low-index inner cladding,” IEEE J. Quantum Electron. 10, 879–887 (1974).
[Crossref]

Bamberger, A.

A. Bamberger and A. S. Bonnet, “Mathematical analysis of the guided modes of an optical fiber,” SIAM J. Math. Anal. 21, 1487–1510 (1990).
[Crossref]

Berkey, G. E.

M.-J. Li, X. Chen, D. A. Nolan, G. E. Berkey, J. Wang, W. A. Wood, and L. A. Zenteno, “High bandwidth single polarization fiber with elliptical central air hole,” IEEE J. Lightwave Technol. 23, 3454–3460 (2005).
[Crossref]

Bjarklev, A.

A. Bjarklev, J. Broeng, and A. S. Bjarklev, Photonic Crystal Fibres (Springer, New York, 2003).
[Crossref]

Bjarklev, A. S.

A. Bjarklev, J. Broeng, and A. S. Bjarklev, Photonic Crystal Fibres (Springer, New York, 2003).
[Crossref]

Bonnet, A. S.

A. Bamberger and A. S. Bonnet, “Mathematical analysis of the guided modes of an optical fiber,” SIAM J. Math. Anal. 21, 1487–1510 (1990).
[Crossref]

Botten, L.

S. Wilcox, L. Botten, C. M. de Sterke, B. Kuhlmey, R. McPhedran, D. Fussell, and S. Tomljenovic-Hanic, “Long wavelength behavior of the fundamental mode in microstructured optical fibers,” Opt. Express 13 (2005).
[Crossref] [PubMed]

Broeng, J.

A. Bjarklev, J. Broeng, and A. S. Bjarklev, Photonic Crystal Fibres (Springer, New York, 2003).
[Crossref]

Chen, C.-L.

C.-L. Chen, Foundations for Guided-Wave Optics (Wiley, 2006).
[Crossref]

Chen, J. C.

Chen, X.

M.-J. Li, X. Chen, D. A. Nolan, G. E. Berkey, J. Wang, W. A. Wood, and L. A. Zenteno, “High bandwidth single polarization fiber with elliptical central air hole,” IEEE J. Lightwave Technol. 23, 3454–3460 (2005).
[Crossref]

Cohen-Tannoudji, C.

C. Cohen-Tannoudji, B. Din, and F. Laloë, Quantum Mechanics (Hermann, Paris, 1977).

de Llano, M.

K. Yang and M. de Llano, “Simple variational proof that any two-dimensional potential well supports at least one bound state,” Am. J. Phys. 57, 85–86 (1989).
[Crossref]

de Sterke, C. M.

S. Wilcox, L. Botten, C. M. de Sterke, B. Kuhlmey, R. McPhedran, D. Fussell, and S. Tomljenovic-Hanic, “Long wavelength behavior of the fundamental mode in microstructured optical fibers,” Opt. Express 13 (2005).
[Crossref] [PubMed]

B. T. Kuhlmey, R. C. McPhedran, C. M. de Sterke, P. A. Robinson, G. Renversez, and D. Maystre, “Microstructured optical fibers: where’s the edge?” Opt. Express 10, 1285–1290 (2002).
[PubMed]

Devenyi, A.

Din, B.

C. Cohen-Tannoudji, B. Din, and F. Laloë, Quantum Mechanics (Hermann, Paris, 1977).

Economou, E. N.

E. N. Economou, Green’s functions in quantum physics (Springer, 2006).

Eickhoff, W.

W. Eickhoff, “Stress-induced single-polarization single-mode fiber,” Opt. Lett. 7 (1982).
[Crossref] [PubMed]

Elachi, C.

C. Elachi, “Waves in active and passive periodic structures: A review,” Proc. IEEE 64, 1666–1698 (1976).
[Crossref]

Fan, S.

Felbacq, D.

F. Zolla, G. Renversez, A. Nicolet, B. Kuhlmey, S. Guenneau, and D. Felbacq, Foundations of Photonic Crystal Fibres (Imperial College Press, London, 2005).
[Crossref]

Fussell, D.

S. Wilcox, L. Botten, C. M. de Sterke, B. Kuhlmey, R. McPhedran, D. Fussell, and S. Tomljenovic-Hanic, “Long wavelength behavior of the fundamental mode in microstructured optical fibers,” Opt. Express 13 (2005).
[Crossref] [PubMed]

Guenneau, S.

F. Zolla, G. Renversez, A. Nicolet, B. Kuhlmey, S. Guenneau, and D. Felbacq, Foundations of Photonic Crystal Fibres (Imperial College Press, London, 2005).
[Crossref]

Hewitt, E.

E. Hewitt and K. Stromberg, Real and Abstract Analysis (Springer, 1965).

Howard, R. E.

J. R. Simpson, R. H. Stolen, F. M. Sears, W. Pleibel, J. B. Macchesney, and R. E. Howard, “A single-polarization fiber,” IEEE J. Lightwave Technol. 1, 370–374 (1983).
[Crossref]

Hunsperger, R. G.

R. G. Hunsperger, Integrated Optics: Theory and Technology (Springer-Verlag, 1982).

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, New York, 1998).

Jacobs, S.

S. G. Johnson, M. L. Povinelli, M. Soljačić, A. Karalis, S. Jacobs, and J. D. Joannopoulos, “Roughness losses and volume-current methods in photonic-crystal waveguides,” Appl. Phys. B 81, 283–293 (2005).
[Crossref]

Joannopoulos, J. D.

S. G. Johnson, M. L. Povinelli, M. Soljačić, A. Karalis, S. Jacobs, and J. D. Joannopoulos, “Roughness losses and volume-current methods in photonic-crystal waveguides,” Appl. Phys. B 81, 283–293 (2005).
[Crossref]

S. Fan, J. N. Winn, A. Devenyi, J. C. Chen, R. D. Meade, and J. D. Joannopoulos, “Guided and defect modes in periodic dielectric waveguides,” J. Opt. Soc. Am. B 12, 1267–1272 (1995).
[Crossref]

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton Univ. Press, 2008).

Johnson, S. G.

S. G. Johnson, M. L. Povinelli, M. Soljačić, A. Karalis, S. Jacobs, and J. D. Joannopoulos, “Roughness losses and volume-current methods in photonic-crystal waveguides,” Appl. Phys. B 81, 283–293 (2005).
[Crossref]

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton Univ. Press, 2008).

Karalis, A.

S. G. Johnson, M. L. Povinelli, M. Soljačić, A. Karalis, S. Jacobs, and J. D. Joannopoulos, “Roughness losses and volume-current methods in photonic-crystal waveguides,” Appl. Phys. B 81, 283–293 (2005).
[Crossref]

Kawakami, S.

S. Kawakami and S. Nishida, “Characteristics of a doubly clad optical fiber with a low-index inner cladding,” IEEE J. Quantum Electron. 10, 879–887 (1974).
[Crossref]

Kawanishi, S.

H. Kubota, S. Kawanishi, S. Koyanagi, M. Tanaka, and S. Yamaguchi, “Absolutely single polarization photonic crystal fiber,” IEEE Photon. Tech. Lett. 16, 182–184 (2004).
[Crossref]

Kong, J. A.

J. A. Kong, Electromagnetic Wave Theory (Wiley, New York, 1975).

Koschny, T.

D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous materials,” Phys. Rev. E 71, 036,617 (2005).
[Crossref]

Koyanagi, S.

H. Kubota, S. Kawanishi, S. Koyanagi, M. Tanaka, and S. Yamaguchi, “Absolutely single polarization photonic crystal fiber,” IEEE Photon. Tech. Lett. 16, 182–184 (2004).
[Crossref]

Kubota, H.

H. Kubota, S. Kawanishi, S. Koyanagi, M. Tanaka, and S. Yamaguchi, “Absolutely single polarization photonic crystal fiber,” IEEE Photon. Tech. Lett. 16, 182–184 (2004).
[Crossref]

Kuchment, P.

P. Kuchment, “The Mathematics of Photonic Crystals,” in Mathematical Modeling in Optical Science, G. Bao, L. Cowsar, and W. Masters, eds., Frontiers in Applied Mathematics, pp. 207–272 (SIAM, Philadelphia, 2001).

P. Kuchment and B. Ong, “On guided waves in photonic crystal waveguides,” in Waves in Periodic and Random Media, vol. 339 of Contemporary Mathematics, pp. 105–115 (AMS, Providence, RI, 2003).

Kuhlmey, B.

S. Wilcox, L. Botten, C. M. de Sterke, B. Kuhlmey, R. McPhedran, D. Fussell, and S. Tomljenovic-Hanic, “Long wavelength behavior of the fundamental mode in microstructured optical fibers,” Opt. Express 13 (2005).
[Crossref] [PubMed]

F. Zolla, G. Renversez, A. Nicolet, B. Kuhlmey, S. Guenneau, and D. Felbacq, Foundations of Photonic Crystal Fibres (Imperial College Press, London, 2005).
[Crossref]

Kuhlmey, B. T.

Laloë, F.

C. Cohen-Tannoudji, B. Din, and F. Laloë, Quantum Mechanics (Hermann, Paris, 1977).

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Addison-Wesley, 1977).

Li, M.-J.

M.-J. Li, X. Chen, D. A. Nolan, G. E. Berkey, J. Wang, W. A. Wood, and L. A. Zenteno, “High bandwidth single polarization fiber with elliptical central air hole,” IEEE J. Lightwave Technol. 23, 3454–3460 (2005).
[Crossref]

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Addison-Wesley, 1977).

Love, J. D.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983).

Macchesney, J. B.

J. R. Simpson, R. H. Stolen, F. M. Sears, W. Pleibel, J. B. Macchesney, and R. E. Howard, “A single-polarization fiber,” IEEE J. Lightwave Technol. 1, 370–374 (1983).
[Crossref]

Marom, E.

Maystre, D.

McPhedran, R.

S. Wilcox, L. Botten, C. M. de Sterke, B. Kuhlmey, R. McPhedran, D. Fussell, and S. Tomljenovic-Hanic, “Long wavelength behavior of the fundamental mode in microstructured optical fibers,” Opt. Express 13 (2005).
[Crossref] [PubMed]

McPhedran, R. C.

Meade, R. D.

S. Fan, J. N. Winn, A. Devenyi, J. C. Chen, R. D. Meade, and J. D. Joannopoulos, “Guided and defect modes in periodic dielectric waveguides,” J. Opt. Soc. Am. B 12, 1267–1272 (1995).
[Crossref]

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton Univ. Press, 2008).

Messerly, M. J.

M. J. Messerly, J. R. Onstott, and R. C. Mikkelson, “A broad-band single polarization optical fiber,” IEEE J. Lightwave Technol. 9, 817–820 (1991).
[Crossref]

Mikkelson, R. C.

M. J. Messerly, J. R. Onstott, and R. C. Mikkelson, “A broad-band single polarization optical fiber,” IEEE J. Lightwave Technol. 9, 817–820 (1991).
[Crossref]

Nicolet, A.

F. Zolla, G. Renversez, A. Nicolet, B. Kuhlmey, S. Guenneau, and D. Felbacq, Foundations of Photonic Crystal Fibres (Imperial College Press, London, 2005).
[Crossref]

Nishida, S.

S. Kawakami and S. Nishida, “Characteristics of a doubly clad optical fiber with a low-index inner cladding,” IEEE J. Quantum Electron. 10, 879–887 (1974).
[Crossref]

Nolan, D. A.

M.-J. Li, X. Chen, D. A. Nolan, G. E. Berkey, J. Wang, W. A. Wood, and L. A. Zenteno, “High bandwidth single polarization fiber with elliptical central air hole,” IEEE J. Lightwave Technol. 23, 3454–3460 (2005).
[Crossref]

Okoshi, T.

T. Okoshi and K. Oyamoda, “Single-polarization single-mode optical fibre with refractive-index pits on both sides of core,” Electron. Lett. 16, 712–713 (80).
[Crossref]

Ong, B.

P. Kuchment and B. Ong, “On guided waves in photonic crystal waveguides,” in Waves in Periodic and Random Media, vol. 339 of Contemporary Mathematics, pp. 105–115 (AMS, Providence, RI, 2003).

Onstott, J. R.

M. J. Messerly, J. R. Onstott, and R. C. Mikkelson, “A broad-band single polarization optical fiber,” IEEE J. Lightwave Technol. 9, 817–820 (1991).
[Crossref]

Oyamoda, K.

T. Okoshi and K. Oyamoda, “Single-polarization single-mode optical fibre with refractive-index pits on both sides of core,” Electron. Lett. 16, 712–713 (80).
[Crossref]

Picq, H.

H. Picq, “Détermination et calcul numérique de la première valeur propre d’opérateurs de Schrödinger dans le plan,” Ph.D. thesis, Université de Nice, Nice, France (1982).

Pleibel, W.

J. R. Simpson, R. H. Stolen, F. M. Sears, W. Pleibel, J. B. Macchesney, and R. E. Howard, “A single-polarization fiber,” IEEE J. Lightwave Technol. 1, 370–374 (1983).
[Crossref]

Povinelli, M. L.

S. G. Johnson, M. L. Povinelli, M. Soljačić, A. Karalis, S. Jacobs, and J. D. Joannopoulos, “Roughness losses and volume-current methods in photonic-crystal waveguides,” Appl. Phys. B 81, 283–293 (2005).
[Crossref]

Ramaswami, R.

R. Ramaswami and K. N. Sivarajan, Optical Networks: A Practical Perspective (Academic Press, London, 1998).

Renversez, G.

B. T. Kuhlmey, R. C. McPhedran, C. M. de Sterke, P. A. Robinson, G. Renversez, and D. Maystre, “Microstructured optical fibers: where’s the edge?” Opt. Express 10, 1285–1290 (2002).
[PubMed]

F. Zolla, G. Renversez, A. Nicolet, B. Kuhlmey, S. Guenneau, and D. Felbacq, Foundations of Photonic Crystal Fibres (Imperial College Press, London, 2005).
[Crossref]

Robinson, P. A.

Russell, P.

P. Russell, “Photonic crystal fibers,” Science 299, 358–362 (2003).
[Crossref] [PubMed]

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991).
[Crossref]

Sears, F. M.

J. R. Simpson, R. H. Stolen, F. M. Sears, W. Pleibel, J. B. Macchesney, and R. E. Howard, “A single-polarization fiber,” IEEE J. Lightwave Technol. 1, 370–374 (1983).
[Crossref]

Simon, B.

B. Simon, “The bound state of weakly coupled Schrödinger operators in one and two dimensions,” Ann. Phys. 97, 279–288 (1976).
[Crossref]

Simpson, J. R.

J. R. Simpson, R. H. Stolen, F. M. Sears, W. Pleibel, J. B. Macchesney, and R. E. Howard, “A single-polarization fiber,” IEEE J. Lightwave Technol. 1, 370–374 (1983).
[Crossref]

Sivarajan, K. N.

R. Ramaswami and K. N. Sivarajan, Optical Networks: A Practical Perspective (Academic Press, London, 1998).

Smith, D. R.

D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous materials,” Phys. Rev. E 71, 036,617 (2005).
[Crossref]

Snyder, A. W.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983).

Soljacic, M.

S. G. Johnson, M. L. Povinelli, M. Soljačić, A. Karalis, S. Jacobs, and J. D. Joannopoulos, “Roughness losses and volume-current methods in photonic-crystal waveguides,” Appl. Phys. B 81, 283–293 (2005).
[Crossref]

Soukoulis, C. M.

D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous materials,” Phys. Rev. E 71, 036,617 (2005).
[Crossref]

Stolen, R. H.

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H. Kubota, S. Kawanishi, S. Koyanagi, M. Tanaka, and S. Yamaguchi, “Absolutely single polarization photonic crystal fiber,” IEEE Photon. Tech. Lett. 16, 182–184 (2004).
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B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991).
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H. P. Urbach, “Analysis of the domain integral operator for anisotropic dielectric waveguides,” Journal on Mathematical Analysis 27 (1996).

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D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous materials,” Phys. Rev. E 71, 036,617 (2005).
[Crossref]

Wang, J.

M.-J. Li, X. Chen, D. A. Nolan, G. E. Berkey, J. Wang, W. A. Wood, and L. A. Zenteno, “High bandwidth single polarization fiber with elliptical central air hole,” IEEE J. Lightwave Technol. 23, 3454–3460 (2005).
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S. Wilcox, L. Botten, C. M. de Sterke, B. Kuhlmey, R. McPhedran, D. Fussell, and S. Tomljenovic-Hanic, “Long wavelength behavior of the fundamental mode in microstructured optical fibers,” Opt. Express 13 (2005).
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S. Fan, J. N. Winn, A. Devenyi, J. C. Chen, R. D. Meade, and J. D. Joannopoulos, “Guided and defect modes in periodic dielectric waveguides,” J. Opt. Soc. Am. B 12, 1267–1272 (1995).
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J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton Univ. Press, 2008).

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M.-J. Li, X. Chen, D. A. Nolan, G. E. Berkey, J. Wang, W. A. Wood, and L. A. Zenteno, “High bandwidth single polarization fiber with elliptical central air hole,” IEEE J. Lightwave Technol. 23, 3454–3460 (2005).
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H. Kubota, S. Kawanishi, S. Koyanagi, M. Tanaka, and S. Yamaguchi, “Absolutely single polarization photonic crystal fiber,” IEEE Photon. Tech. Lett. 16, 182–184 (2004).
[Crossref]

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M.-J. Li, X. Chen, D. A. Nolan, G. E. Berkey, J. Wang, W. A. Wood, and L. A. Zenteno, “High bandwidth single polarization fiber with elliptical central air hole,” IEEE J. Lightwave Technol. 23, 3454–3460 (2005).
[Crossref]

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F. Zolla, G. Renversez, A. Nicolet, B. Kuhlmey, S. Guenneau, and D. Felbacq, Foundations of Photonic Crystal Fibres (Imperial College Press, London, 2005).
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Am. J. Phys. (1)

K. Yang and M. de Llano, “Simple variational proof that any two-dimensional potential well supports at least one bound state,” Am. J. Phys. 57, 85–86 (1989).
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S. G. Johnson, M. L. Povinelli, M. Soljačić, A. Karalis, S. Jacobs, and J. D. Joannopoulos, “Roughness losses and volume-current methods in photonic-crystal waveguides,” Appl. Phys. B 81, 283–293 (2005).
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[Crossref]

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H. Kubota, S. Kawanishi, S. Koyanagi, M. Tanaka, and S. Yamaguchi, “Absolutely single polarization photonic crystal fiber,” IEEE Photon. Tech. Lett. 16, 182–184 (2004).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (1)

Journal on Mathematical Analysis (1)

H. P. Urbach, “Analysis of the domain integral operator for anisotropic dielectric waveguides,” Journal on Mathematical Analysis 27 (1996).

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S. Wilcox, L. Botten, C. M. de Sterke, B. Kuhlmey, R. McPhedran, D. Fussell, and S. Tomljenovic-Hanic, “Long wavelength behavior of the fundamental mode in microstructured optical fibers,” Opt. Express 13 (2005).
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B. T. Kuhlmey, R. C. McPhedran, C. M. de Sterke, P. A. Robinson, G. Renversez, and D. Maystre, “Microstructured optical fibers: where’s the edge?” Opt. Express 10, 1285–1290 (2002).
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B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991).
[Crossref]

C.-L. Chen, Foundations for Guided-Wave Optics (Wiley, 2006).
[Crossref]

A. Bjarklev, J. Broeng, and A. S. Bjarklev, Photonic Crystal Fibres (Springer, New York, 2003).
[Crossref]

F. Zolla, G. Renversez, A. Nicolet, B. Kuhlmey, S. Guenneau, and D. Felbacq, Foundations of Photonic Crystal Fibres (Imperial College Press, London, 2005).
[Crossref]

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton Univ. Press, 2008).

R. Ramaswami and K. N. Sivarajan, Optical Networks: A Practical Perspective (Academic Press, London, 1998).

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A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983).

P. Kuchment, “The Mathematics of Photonic Crystals,” in Mathematical Modeling in Optical Science, G. Bao, L. Cowsar, and W. Masters, eds., Frontiers in Applied Mathematics, pp. 207–272 (SIAM, Philadelphia, 2001).

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C. Cohen-Tannoudji, B. Din, and F. Laloë, Quantum Mechanics (Hermann, Paris, 1977).

D. ter Haar, Selected Problems in Quantum Mechanics (Academic Press, New York, 1964).

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J. A. Kong, Electromagnetic Wave Theory (Wiley, New York, 1975).

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Figures (3)

Fig. 1.
Fig. 1.

Schematics of various types of dielectric waveguides in which our theorem is applicable. Light propagates in the z direction (along which the structure is either uniform or periodic) and is confined in the xy direction by a higher-index core compared to the surrounding (homogeneous or periodic) cladding.

Fig. 2.
Fig. 2.

Example dispersion relation of a simple 2d dielectric waveguide in air (inset) for the TM polarization (electric field out of the plane), showing the light cone, the light line, the fundamental (cutoff-free) guided mode, and a higher-order guided mode with a cutoff.

Fig. 3.
Fig. 3.

Plot of γ [eq. (11)], γ′/α [eq. (12)], and γ″/α [eq. (13)] versus r for α=0.1. All three functions go to zero for r → ∞, with no extrema other than those shown.

Equations (59)

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D c * · ( ε 1 ε c 1 ) D c < 0 ,
ε ( x , y ) = ε c + Δε ( x , y ) ,
Δ ( x , y ) ε 1 ε c 1 .
Δ ( x , y ) dxdy < 0 ,
β × 1 ε β × H = Θ ̂ β H = ω 2 c 2 H ,
β + z ̂ ,
β · H = 0
ω min 2 ( β ) c 2 = inf β · H = 0 H * · Θ ̂ β H dxdy H * · H dxdy
ε c H * · Θ ̂ β H dxdy β 2 H * · H dxdy
= ε c 1 ε β × H 2 dxdy β 2 H 2 dxdy < 0 ,
H = r ̂ γ cos ϕ ϕ ̂ ( r γ ) sin ϕ ,
γ = γ ( r ) = e 1 ( r 2 + 1 ) α
γ = 2 α r ( r 2 + 1 ) α 1 γ ,
γ = 2 α ( r 2 + 1 ) α 1 γ [ 1 + 2 α r 2 ( r 2 + 1 ) α 1 + 2 ( 1 α ) r 2 ( r 2 + 1 ) 1 ] ,
ε c 1 ε ( r ) ( + i β z ̂ ) × H ( r , ϕ ) 2 d 2 r β 2 H 2 d 2 r
= ε c ( 1 ε c + Δ ( r ) ) z ̂ 1 r [ r ( r H ϕ ) H r ϕ ] + i β z ̂ × H 2 d 2 r β 2 H 2 d 2 r
= ε c ( 1 ε c + Δ ( r ) ) ( sin 2 ϕ r 2 { [ r ( r γ ) ] γ } 2 + β 2 H 2 ) d 2 r β 2 H 2 d 2 r
= ε c ( 1 ε c + Δ ( r ) ) sin 2 ϕ r 2 ( 3 r γ + r 2 γ ) 2 d 2 r ε c β 2 Δ ( r ) H 2 d 2 r
lim α 0 β 2 Δ ( r ) H 2 d 2 r = β 2 Δ ( r ) d 2 r < 0 .
lim α 0 0 0 2 π sin 2 ϕ r 2 ( 3 r γ + r 2 γ ) 2 rdrd ϕ
= 16 π lim α 0 0 α 2 r 3 ( r 2 + 1 ) 2 α 2 γ 2 [ 2 + α r 2 ( r 2 + 1 ) α 1 + ( 1 α ) r 2 ( r 2 + 1 ) 1 ] 2 dr
16 π lim α 0 0 α 2 r ( r 2 + 1 ) 2 α 1 γ 2 [ 2 + α ( r 2 + 1 ) α + ( 1 α ) ] 2 dr
= 16 π lim α 0 1 α 2 t 4 α 1 e 2 2 t 2 α [ ( 3 α ) + α t 2 α ] 2 dt
8 π lim α 0 0 α u e 2 2 u [ ( 3 α ) + α u ] 2 du
= 8 π e 2 lim α 0 α [ 3 8 α 2 + 1 2 α ( 3 α ) + 1 4 ( 3 α ) 2 ] = 0 .
ε ( x , y , z ) = ε c ( x , y , z ) + Δ ε ( x , y , z ) ,
ω min 2 ( β ) c 2 = inf β · H = 0 H * · Θ ̂ β H H * · H .
H * · Θ ̂ β H ω c 2 ( β ) c 2 H * · H < 0 .
H = β × ( γ A c ) = γ H c + γ × A c .
lim α 0 H * · ( β × Δ β × H )
= H c * · ( β × Δ β × H c )
= ( β × H c ) * · Δ ( β × H c )
= ω c 2 c 2 D c * · Δ D c
H * · ( β × ε c 1 γ β × H c )
= H * · ( γ β × ε c 1 β × H c + ( γ ) × ε c 1 β × H c )
= H * · γ ω c 2 c 2 H c + H * · ( γ × ε c 1 β × H c )
= H * · ω c 2 c 2 H H * · ω c 2 c 2 γ × A c + H * · ( γ × ε c 1 β × H c )
H * · ( β × ε c 1 β × H )
= H * · ( β × ε c 1 γ β × H c ) + H * · ( β × ε c 1 γ × H c )
+ H * · ( β × ε c 1 β × ( γ × A c ) ) .
ω c 2 c 2 H * · ( γ × A c ) + H * · [ γ × ( i ω c c E c ) ]
+ H * · β × ε c 1 [ γ × H c + β × ( γ × A c ) ]
= ω c 2 c 2 [ γ H c + γ × A c ] * · ( γ × A c ) i ω c c γ γ · ( E c × H c * )
i ω c c ( γ × A c ) * · ( γ × E c ) + ( γ β × H c ) * · ε c 1 [ γ × H c + β × ( γ × A c ) ]
+ ( γ × H c ) * · ε c 1 [ γ × H c + β × ( γ × A c ) ]
+ ( β × γ × A c ) * · ε c 1 [ γ × H c + β × ( γ × A c ) ]
= ω c 2 c 2 γ H c * · ( γ × A c ) ω c 2 c 2 γ × A c 2 i ω c c γ γ · ( E c × H c * )
i ω c c ( γ × A c ) * · ( γ × E c ) + i ω c c γ E c * · ( γ × H c )
+ i ω c c γ ( β × E c ) * · ( γ × A c ) + i ω c c ( γ × E c ) * · ( γ × A c )
+ ( γ × H c ) * · ε c 1 ( γ × H c )
+ ( ( γ × H c ) * · ε c 1 [ β × ( γ × A c ) ] + c . c . )
+ ( β × ( γ × A c ) ) * · ε c 1 ( β × ( γ × A c ) ) .
= ω c 2 c 2 γ H c * · ( γ × A c ) ω c 2 c 2 γ × A c 2 2 i ω c c γ γ · { E c × H c * }
+ ( i ω c c ( γ × A c ) · ( γ × E c ) * + c . c . ) + ω c 2 c 2 γ H c * · ( γ × A c )
+ ( γ × H c ) * · ε c 1 ( γ × H c ) + ( ( γ × H c ) * · ε c 1 [ β × ( γ × A c ) ] + c . c . )
+ ( β × ( γ × A c ) ) * · ε c 1 ( β × ( γ × A c ) )
2 i ω c c γ γ · { E c × H c * }
= 2 i ω c 2 c γ 2 · { E c × H c * }
= i ω c c · ( γ 2 { E c × H c * } ) + i ω c c γ 2 · ( { E c × H c * } )

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