Abstract

We present a formal analogy between the eigenvalue problem for guided scalar modes in a low-contrast photonic bandgap fiber and quasi-stationary TM modes of a two-dimensional (2D) photonic structure. Using this analogy, we numerically study the confinement losses of disordered microstructured fibers through the leakage rate of an open 2D system with high refractive index inclusions. Our results show that for large values of the disorder, the confinement losses increase. However, they also suggest that losses might be improved in strongly disordered fibers by exploring ranges of physical parameters where Anderson localization sets in.

© 2009 Optical Society of America

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References

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  1. V. Pureur, G. Bouwmans, M. Perrin, Y. Quiquempois, and M. Douay, “Impact of transversal defects on confinement loss of an all-solid 2-D photonic-bandgap fiber, ” J. Lightwave Technol. 25, 3589-3596 (2007).
    [CrossRef]
  2. A. Argyros, T. Birks, S. Leon-Saval, C. M. B. Cordeiro, and P. St. J. Russell, “Guidance properties of low-contrast photonic bandgap fibres,” Opt. Express 13, 2503-2511(2005).
    [CrossRef] [PubMed]
  3. M. M. Sigalas, C. M. Soukoulis, C.-T. Chan, and D. Turner, “Localization of electromagnetic waves in two-dimensional disordered systems,” Phys. Rev. B 53, 8340-8348 (1996).
    [CrossRef]
  4. C. Rockstuhl, U. Peschel, and F. Lederer, “Correlation between single-cylinder properties and bandgap formation in photonic structures,” Opt. Lett. 31, 1741-1743 (2006).
    [CrossRef] [PubMed]
  5. A. W. Snyder, “Coupled-mode theory for optical fibers,” J. Opt. Soc. Am. 62, 1267-1277 (1972).
    [CrossRef]
  6. D. Marcuse, “Coupled-mode theory of round optical fibers,” Bell Syst. Tech. J. 52, 817-842 (1973).
  7. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, 1983).
  8. D. Laurent, O. Legrand, P. Sebbah, C. Vanneste, and F. Mortessagne, “Localized modes in a finite-size open disordered microwave Cavity,” Phys. Rev. Lett. 99, 253902(2007).
    [CrossRef]
  9. E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, “Scaling theory of localization: absence of quantum diffusion in two dimensions,” Phys. Rev. Lett. 42, 673-676 (1979).
    [CrossRef]
  10. H. C. Van de Hulst, Light Scattering by Small Particles (Dover, 1981).
  11. A. Derode, A. Tourin, and M. Fink, “Random multiple scattering of ultrasound. I. Coherent and ballistic waves,” Phys. Rev. E 64, 036605 (2001).
    [CrossRef]

2007

V. Pureur, G. Bouwmans, M. Perrin, Y. Quiquempois, and M. Douay, “Impact of transversal defects on confinement loss of an all-solid 2-D photonic-bandgap fiber, ” J. Lightwave Technol. 25, 3589-3596 (2007).
[CrossRef]

D. Laurent, O. Legrand, P. Sebbah, C. Vanneste, and F. Mortessagne, “Localized modes in a finite-size open disordered microwave Cavity,” Phys. Rev. Lett. 99, 253902(2007).
[CrossRef]

2006

2005

2001

A. Derode, A. Tourin, and M. Fink, “Random multiple scattering of ultrasound. I. Coherent and ballistic waves,” Phys. Rev. E 64, 036605 (2001).
[CrossRef]

1996

M. M. Sigalas, C. M. Soukoulis, C.-T. Chan, and D. Turner, “Localization of electromagnetic waves in two-dimensional disordered systems,” Phys. Rev. B 53, 8340-8348 (1996).
[CrossRef]

1979

E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, “Scaling theory of localization: absence of quantum diffusion in two dimensions,” Phys. Rev. Lett. 42, 673-676 (1979).
[CrossRef]

1973

D. Marcuse, “Coupled-mode theory of round optical fibers,” Bell Syst. Tech. J. 52, 817-842 (1973).

1972

Abrahams, E.

E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, “Scaling theory of localization: absence of quantum diffusion in two dimensions,” Phys. Rev. Lett. 42, 673-676 (1979).
[CrossRef]

Anderson, P. W.

E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, “Scaling theory of localization: absence of quantum diffusion in two dimensions,” Phys. Rev. Lett. 42, 673-676 (1979).
[CrossRef]

Argyros, A.

Birks, T.

Bouwmans, G.

Chan, C.-T.

M. M. Sigalas, C. M. Soukoulis, C.-T. Chan, and D. Turner, “Localization of electromagnetic waves in two-dimensional disordered systems,” Phys. Rev. B 53, 8340-8348 (1996).
[CrossRef]

Cordeiro, C. M. B.

Derode, A.

A. Derode, A. Tourin, and M. Fink, “Random multiple scattering of ultrasound. I. Coherent and ballistic waves,” Phys. Rev. E 64, 036605 (2001).
[CrossRef]

Douay, M.

Fink, M.

A. Derode, A. Tourin, and M. Fink, “Random multiple scattering of ultrasound. I. Coherent and ballistic waves,” Phys. Rev. E 64, 036605 (2001).
[CrossRef]

Laurent, D.

D. Laurent, O. Legrand, P. Sebbah, C. Vanneste, and F. Mortessagne, “Localized modes in a finite-size open disordered microwave Cavity,” Phys. Rev. Lett. 99, 253902(2007).
[CrossRef]

Lederer, F.

Legrand, O.

D. Laurent, O. Legrand, P. Sebbah, C. Vanneste, and F. Mortessagne, “Localized modes in a finite-size open disordered microwave Cavity,” Phys. Rev. Lett. 99, 253902(2007).
[CrossRef]

Leon-Saval, S.

Licciardello, D. C.

E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, “Scaling theory of localization: absence of quantum diffusion in two dimensions,” Phys. Rev. Lett. 42, 673-676 (1979).
[CrossRef]

Love, J. D.

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

Marcuse, D.

D. Marcuse, “Coupled-mode theory of round optical fibers,” Bell Syst. Tech. J. 52, 817-842 (1973).

Mortessagne, F.

D. Laurent, O. Legrand, P. Sebbah, C. Vanneste, and F. Mortessagne, “Localized modes in a finite-size open disordered microwave Cavity,” Phys. Rev. Lett. 99, 253902(2007).
[CrossRef]

Perrin, M.

Peschel, U.

Pureur, V.

Quiquempois, Y.

Ramakrishnan, T. V.

E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, “Scaling theory of localization: absence of quantum diffusion in two dimensions,” Phys. Rev. Lett. 42, 673-676 (1979).
[CrossRef]

Rockstuhl, C.

Russell, P. St. J.

Sebbah, P.

D. Laurent, O. Legrand, P. Sebbah, C. Vanneste, and F. Mortessagne, “Localized modes in a finite-size open disordered microwave Cavity,” Phys. Rev. Lett. 99, 253902(2007).
[CrossRef]

Sigalas, M. M.

M. M. Sigalas, C. M. Soukoulis, C.-T. Chan, and D. Turner, “Localization of electromagnetic waves in two-dimensional disordered systems,” Phys. Rev. B 53, 8340-8348 (1996).
[CrossRef]

Snyder, A. W.

A. W. Snyder, “Coupled-mode theory for optical fibers,” J. Opt. Soc. Am. 62, 1267-1277 (1972).
[CrossRef]

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

Soukoulis, C. M.

M. M. Sigalas, C. M. Soukoulis, C.-T. Chan, and D. Turner, “Localization of electromagnetic waves in two-dimensional disordered systems,” Phys. Rev. B 53, 8340-8348 (1996).
[CrossRef]

Tourin, A.

A. Derode, A. Tourin, and M. Fink, “Random multiple scattering of ultrasound. I. Coherent and ballistic waves,” Phys. Rev. E 64, 036605 (2001).
[CrossRef]

Turner, D.

M. M. Sigalas, C. M. Soukoulis, C.-T. Chan, and D. Turner, “Localization of electromagnetic waves in two-dimensional disordered systems,” Phys. Rev. B 53, 8340-8348 (1996).
[CrossRef]

Van de Hulst, H. C.

H. C. Van de Hulst, Light Scattering by Small Particles (Dover, 1981).

Vanneste, C.

D. Laurent, O. Legrand, P. Sebbah, C. Vanneste, and F. Mortessagne, “Localized modes in a finite-size open disordered microwave Cavity,” Phys. Rev. Lett. 99, 253902(2007).
[CrossRef]

Bell Syst. Tech. J.

D. Marcuse, “Coupled-mode theory of round optical fibers,” Bell Syst. Tech. J. 52, 817-842 (1973).

J. Lightwave Technol.

J. Opt. Soc. Am.

Opt. Express

Opt. Lett.

Phys. Rev. B

M. M. Sigalas, C. M. Soukoulis, C.-T. Chan, and D. Turner, “Localization of electromagnetic waves in two-dimensional disordered systems,” Phys. Rev. B 53, 8340-8348 (1996).
[CrossRef]

Phys. Rev. E

A. Derode, A. Tourin, and M. Fink, “Random multiple scattering of ultrasound. I. Coherent and ballistic waves,” Phys. Rev. E 64, 036605 (2001).
[CrossRef]

Phys. Rev. Lett.

D. Laurent, O. Legrand, P. Sebbah, C. Vanneste, and F. Mortessagne, “Localized modes in a finite-size open disordered microwave Cavity,” Phys. Rev. Lett. 99, 253902(2007).
[CrossRef]

E. Abrahams, P. W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, “Scaling theory of localization: absence of quantum diffusion in two dimensions,” Phys. Rev. Lett. 42, 673-676 (1979).
[CrossRef]

Other

H. C. Van de Hulst, Light Scattering by Small Particles (Dover, 1981).

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

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Figures (4)

Fig. 1
Fig. 1

Index profile and the corresponding potential profile.

Fig. 2
Fig. 2

Periodic system and an example of a disordered system: the map of a mode is shown together with its amplitude (logarithmic scale) along a section marked on the map.

Fig. 3
Fig. 3

Attenuation rates of fiber modes 1 and 2 as a function of the amount of disorder σ.

Fig. 4
Fig. 4

Localization length as a function of the V parameter for scatterers of radius 5 μm . The different curves, labeled by n = 3.0 , 4.0 , 5.0 , 6.0 , correspond to different values of the refractive index of the scatterers.

Equations (9)

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

( Δ + N 2 ( r ) K p 2 ) Φ p = 0.
[ Δ + ( 1 N 2 ( r ) ) K p 2 ] Φ p ( r ) = K p 2 Φ p ( r ) ,
( Δ k 2 ( n p eff ) 2 ) ϕ p + n 2 ( r ) k 2 ϕ p = 0 ,
Δ ϕ p + [ n 0 2 n 2 ( r ) ] k 2 ϕ p = [ n 0 2 ( n p eff ) 2 ] k 2 ϕ p ,
0 < n 0 2 ( n p eff ) 2 1.
[ n 0 2 ( n p eff ) 2 ] k 2 with     K p 2 ,
( n 0 2 n 1 2 ) k 2 with     ( 1 N scat 2 ) K p 2 ,
k 2 Re n eff Im n eff = Re K Im K = Ω 2 c 2 Γ ,
V = 2 π r ( n 1 2 n 0 2 ) 1 / 2 λ ,

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