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

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  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," J. Math. Anal. 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 premiére 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. Lalo¨e, 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. Solja??i??, 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. Solja??i??, 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," J. Math. Anal. 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)

B. Simon, "The bound state of weakly coupled Schrödinger operators in one and two dimensions," Ann. Phys. 97, 279-288 (1976).
[CrossRef]

C. Elachi, "Waves in active and passive periodic structures: A review," Proc. IEEE 64, 1666-1698 (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]

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]

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]

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.

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.

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]

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]

Jacobs, S.

S. G. Johnson, M. L. Povinelli, M. Solja??i??, 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. Solja??i??, 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]

Johnson, S. G.

S. G. Johnson, M. L. Povinelli, M. Solja??i??, 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]

Karalis, A.

S. G. Johnson, M. L. Povinelli, M. Solja??i??, 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]

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]

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]

Kuhlmey, B. T.

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]

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.

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]

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]

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]

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. Solja??i??, 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]

Renversez, G.

Robinson, P. A.

Russell, P.

P. Russell, "Photonic crystal fibers," Science 299, 358-362 (2003).
[CrossRef] [PubMed]

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]

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]

Solja??i??, M.

S. G. Johnson, M. L. Povinelli, M. Solja??i??, 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.

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]

Tanaka, M.

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]

Tomljenovic-Hanic, S.

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]

Urbach, H. P.

H. P. Urbach, "Analysis of the domain integral operator for anisotropic dielectric waveguides," J. Math. Anal. 27, (1996).

Vier, D. C.

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).
[CrossRef]

Wilcox, S.

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]

Winn, J. N.

Wood, W. 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]

Yamaguchi, 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]

Yang, K.

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]

Yariv, A.

Yeh, P.

Zenteno, L. 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]

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).
[CrossRef]

Ann. Phys. (1)

B. Simon, "The bound state of weakly coupled Schrödinger operators in one and two dimensions," Ann. Phys. 97, 279-288 (1976).
[CrossRef]

Appl. Phys. B (1)

S. G. Johnson, M. L. Povinelli, M. Solja??i??, 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]

Electron. Lett (1)

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]

IEEE J. Lightwave Technol. (3)

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]

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]

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]

IEEE J. Quantum Electron. (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]

IEEE Photon. Tech. Lett. (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]

J. Math. Anal. (1)

H. P. Urbach, "Analysis of the domain integral operator for anisotropic dielectric waveguides," J. Math. Anal. 27, (1996).

J. Opt. Soc. Am. (1)

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

Opt. Express (2)

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]

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]

Opt. Lett. (1)

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

Phys. Rev. E (1)

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]

Proc. IEEE (1)

C. Elachi, "Waves in active and passive periodic structures: A review," Proc. IEEE 64, 1666-1698 (1976).
[CrossRef]

Science (1)

P. Russell, "Photonic crystal fibers," Science 299, 358-362 (2003).
[CrossRef] [PubMed]

SIAM J. Math. Anal. (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]

Other (18)

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

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).

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

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]

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).

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).

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

H. Picq, "Détermination et calcul numérique de la premiére valeur propre d???opérateurs de Schrödinger dans le plan," Ph.D. thesis, Université de Nice, Nice, France (1982).

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

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

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

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

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

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

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


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)

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

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 * } )

Metrics