Abstract

A perturbation theory is developed that treats a localised mode embedded within a continuum of states. The method is applied to a model rectangular hollow-core photonic crystal fibre structure, where the basic modes are derived from an ideal, scalar model and the perturbation terms include vector effects and structural difference between the ideal and realistic structures. An expression for the attenuation of the fundamental mode due to interactions with cladding modes is derived, and results are presented for a rectangular photonic crystal fibre structure. Attenuations calculated in this way are in good agreement with numerical simulations. The origin of the guidance in our model structure is explained through this quantitative analysis. Further perspectives are obtained through investigating the influence of fibre parameters on the attenuation.

© 2011 OSA

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References

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  1. L. Chen, G. J. Pearce, T. A. Birks, and D. M. Bird, “Guidance in Kagome-like photonic crystal fibres I: analysis of an ideal fibre structure,” Submitted to Opt. Express (2011).
    [PubMed]
  2. P. W. Anderson, “Localized magnetic states in metals,” Phys. Rev. 124, 41–53 (1961).
    [CrossRef]
  3. L. Chen, “Modelling of photonic crystal fibres,” Ph.D. thesis, University of Bath (2009).
  4. S. Davison and M. Stesliska, Basic Theory of Surface States (Oxford U. Press, 1992).
  5. E. Economou, Green’s Functions in Quantum Physics (Springer-Verlag, 1990).
  6. F. Couny, F. Benabid, and P. S. Light, “Large-pitch Kagome-structured hollow-core photonic crystal fiber,” Opt. Lett. 31, 3574–3576 (2006).
    [CrossRef] [PubMed]
  7. F. Couny, F. Benabid, P. J. Roberts, P. S. Light, and M. G. Raymer, “Generation and Photonic Guidance of Multi-Octave Optical-Frequency Combs,” Science 318, 1118–1121 (2007).
    [CrossRef] [PubMed]
  8. A. Argyros and J. Pla, “Hollow-core polymer fibres with a kagome lattice: potential for transmission in the infrared,” Opt. Express 15, 7713–7719 (2007).
    [CrossRef] [PubMed]
  9. F. Couny, P. J. Roberts, T. A. Birks, and F. Benabid, “Square-lattice large-pitch hollow-core photonic crystal fiber,” Opt. Express 16, 20626–20636 (2008).
    [CrossRef] [PubMed]
  10. N. Guan, S. Habu, K. Takenaga, K. Himeno, and A. Wada, “Boundary element method for analysis of holey optical fibers,” J. Lightwave Technol. 21, 1787–1792 (2003).
    [CrossRef]
  11. T.-L. Wu and C.-H. Chao, “Photonic crystal fiber analysis through the vector boundary-element method: effect of elliptical air hole,” IEEE Photon. Technol. Lett. 16, 126–128 (2004).
    [CrossRef]
  12. X. Wang, J. Lou, C. Lu, C.-L. Zhao, and W. Ang, “Modeling of pcf with multiple reciprocity boundary element method,” Opt. Express 12, 961–966 (2004).
    [CrossRef] [PubMed]
  13. Y. Wang, F. Couny, P. Roberts, and F. Benabid, “Low loss broadband transmission in optimized core-shape kagome hollow-core pcf,” in “Lasers and Electro-Optics (CLEO) and Quantum Electronics and Laser Science Conference (QELS), 2010 Conference on,” (2010), pp. 1–2.

2008

2007

A. Argyros and J. Pla, “Hollow-core polymer fibres with a kagome lattice: potential for transmission in the infrared,” Opt. Express 15, 7713–7719 (2007).
[CrossRef] [PubMed]

F. Couny, F. Benabid, P. J. Roberts, P. S. Light, and M. G. Raymer, “Generation and Photonic Guidance of Multi-Octave Optical-Frequency Combs,” Science 318, 1118–1121 (2007).
[CrossRef] [PubMed]

2006

2004

T.-L. Wu and C.-H. Chao, “Photonic crystal fiber analysis through the vector boundary-element method: effect of elliptical air hole,” IEEE Photon. Technol. Lett. 16, 126–128 (2004).
[CrossRef]

X. Wang, J. Lou, C. Lu, C.-L. Zhao, and W. Ang, “Modeling of pcf with multiple reciprocity boundary element method,” Opt. Express 12, 961–966 (2004).
[CrossRef] [PubMed]

2003

1961

P. W. Anderson, “Localized magnetic states in metals,” Phys. Rev. 124, 41–53 (1961).
[CrossRef]

Anderson, P. W.

P. W. Anderson, “Localized magnetic states in metals,” Phys. Rev. 124, 41–53 (1961).
[CrossRef]

Ang, W.

Argyros, A.

Benabid, F.

Birks, T. A.

Chao, C.-H.

T.-L. Wu and C.-H. Chao, “Photonic crystal fiber analysis through the vector boundary-element method: effect of elliptical air hole,” IEEE Photon. Technol. Lett. 16, 126–128 (2004).
[CrossRef]

Couny, F.

Davison, S.

S. Davison and M. Stesliska, Basic Theory of Surface States (Oxford U. Press, 1992).

Economou, E.

E. Economou, Green’s Functions in Quantum Physics (Springer-Verlag, 1990).

Guan, N.

Habu, S.

Himeno, K.

Light, P. S.

F. Couny, F. Benabid, P. J. Roberts, P. S. Light, and M. G. Raymer, “Generation and Photonic Guidance of Multi-Octave Optical-Frequency Combs,” Science 318, 1118–1121 (2007).
[CrossRef] [PubMed]

F. Couny, F. Benabid, and P. S. Light, “Large-pitch Kagome-structured hollow-core photonic crystal fiber,” Opt. Lett. 31, 3574–3576 (2006).
[CrossRef] [PubMed]

Lou, J.

Lu, C.

Pla, J.

Raymer, M. G.

F. Couny, F. Benabid, P. J. Roberts, P. S. Light, and M. G. Raymer, “Generation and Photonic Guidance of Multi-Octave Optical-Frequency Combs,” Science 318, 1118–1121 (2007).
[CrossRef] [PubMed]

Roberts, P. J.

F. Couny, P. J. Roberts, T. A. Birks, and F. Benabid, “Square-lattice large-pitch hollow-core photonic crystal fiber,” Opt. Express 16, 20626–20636 (2008).
[CrossRef] [PubMed]

F. Couny, F. Benabid, P. J. Roberts, P. S. Light, and M. G. Raymer, “Generation and Photonic Guidance of Multi-Octave Optical-Frequency Combs,” Science 318, 1118–1121 (2007).
[CrossRef] [PubMed]

Stesliska, M.

S. Davison and M. Stesliska, Basic Theory of Surface States (Oxford U. Press, 1992).

Takenaga, K.

Wada, A.

Wang, X.

Wu, T.-L.

T.-L. Wu and C.-H. Chao, “Photonic crystal fiber analysis through the vector boundary-element method: effect of elliptical air hole,” IEEE Photon. Technol. Lett. 16, 126–128 (2004).
[CrossRef]

Zhao, C.-L.

IEEE Photon. Technol. Lett.

T.-L. Wu and C.-H. Chao, “Photonic crystal fiber analysis through the vector boundary-element method: effect of elliptical air hole,” IEEE Photon. Technol. Lett. 16, 126–128 (2004).
[CrossRef]

J. Lightwave Technol.

Opt. Express

Opt. Lett.

Phys. Rev.

P. W. Anderson, “Localized magnetic states in metals,” Phys. Rev. 124, 41–53 (1961).
[CrossRef]

Science

F. Couny, F. Benabid, P. J. Roberts, P. S. Light, and M. G. Raymer, “Generation and Photonic Guidance of Multi-Octave Optical-Frequency Combs,” Science 318, 1118–1121 (2007).
[CrossRef] [PubMed]

Other

L. Chen, G. J. Pearce, T. A. Birks, and D. M. Bird, “Guidance in Kagome-like photonic crystal fibres I: analysis of an ideal fibre structure,” Submitted to Opt. Express (2011).
[PubMed]

Y. Wang, F. Couny, P. Roberts, and F. Benabid, “Low loss broadband transmission in optimized core-shape kagome hollow-core pcf,” in “Lasers and Electro-Optics (CLEO) and Quantum Electronics and Laser Science Conference (QELS), 2010 Conference on,” (2010), pp. 1–2.

L. Chen, “Modelling of photonic crystal fibres,” Ph.D. thesis, University of Bath (2009).

S. Davison and M. Stesliska, Basic Theory of Surface States (Oxford U. Press, 1992).

E. Economou, Green’s Functions in Quantum Physics (Springer-Verlag, 1990).

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Figures (4)

Fig. 1
Fig. 1

Variation of the imaginary part of Δ(β Λ)2 for the fundamental mode with smoothing width σ for different sizes of supercells calculated using Eq. (28). The left and right panels show contributions from air-guided and glass-guided cladding modes, respectively. The black arrows at (β Λ)2 = 1587.463 indicate the location of the shifted fundamental guided mode.

Fig. 2
Fig. 2

Imaginary part of the propagation constant of the fundamental mode over a range of frequencies in the first transmission window. (a) Comparison between perturbation and boundary element methods. The inset shows the schematic model PCF structure calculated by the boundary element method. (b) The effect of the perturbation terms, where the red line indicates the full perturbation result; the green and blue lines are results that include only the vector and high-index perturbation terms, respectively.

Fig. 3
Fig. 3

Imaginary part of the propagation constant as a function of the glass strip thickness in the first transmission window when k 0Λ = 40. The full, vector and high-index perturbation results are shown in red, green and blue, respectively.

Fig. 4
Fig. 4

(a) Dependence of attenuation on the size of the central defect. Full, vector and high-index perturbation results are shown in red, green and blue, respectively. (b) Attenuations for the two sizes of central defect as a function of frequency. The red and green lines respectively correspond to D = 1.20Λ and D = 1.11Λ.

Equations (28)

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L 0 h n = β 0 n 2 h n ,
L 0 H + δ L ( H ) = β 2 H ,
δ L ( H ) = ( t × H ) × t ln n r 2 + Δ n 2 k 0 2 H ,
L 0 ( n a n h n ) + δ L ( n a n h n ) = β 2 n a n h n .
n ( L mn 0 + δ L mn ) a n = β 2 a m ,
L mn 0 = δ mn β 0 m 2
δ L mn = h m * · δ L ( h n ) d A .
δ L mn a = h m · [ ( t × ( h n ) ) × t ln n r 2 ] d A
δ L mn b = h m · ( Δ n 2 k 0 2 ) h n d A .
L mm 0 + δ L mm = β 0 m 2 ,
v ( L mm 0 δ mv + δ L mm δ mv + δ L mv ) G v n = δ mn .
( L mm 0 + δ L mm ) G mm 0 = 1 ,
G mm 0 = ( β 2 β 0 m 2 + i ɛ ) 1 ,
G a b = G a a 0 δ a b + p , q G a p 0 δ L p q G q b ,
G m 0 x m 0 x = G m 0 x m 0 x 0 + G m 0 x m 0 x 0 m δ L m 0 x m G mm 0 x ,
G m 0 y m 0 x = G m 0 y m 0 y 0 m δ L m 0 y m G mm 0 x ,
G mm 0 x = G mm 0 δ L mm 0 x G m 0 x m 0 x + G mm 0 δ L mm 0 y G m 0 y m 0 x ,
G m 0 x m 0 x = G m 0 x m 0 x 0 + G m 0 x m 0 x 0 G m 0 x m 0 x V m 0 x m 0 x + G m 0 x m 0 x 0 G m 0 y m 0 x V m 0 x m 0 y ,
G m 0 y m 0 x = G m 0 y m 0 y 0 G m 0 x m 0 x V m 0 y m 0 x 1 G m 0 y m 0 y 0 V m 0 y m 0 y ,
V a b = m δ L a m δ L mb β 2 β 0 m 2 + i ɛ .
G m 0 x m 0 x = G m 0 x m 0 x 0 1 G m 0 x m 0 x 0 V m 0 x m 0 X G m 0 x m 0 x 0 G m 0 y m 0 y 0 V m 0 x m 0 y V m 0 y m 0 x 1 G m 0 y m 0 y 0 V m 0 y m 0 y .
G m 0 x m 0 x = ( ( G m 0 x m 0 x 0 ) 1 V m 0 x m 0 x V m 0 x m 0 y 2 ( G m 0 x m 0 x 0 ) 1 V m 0 x m 0 x ) 1 .
G m 0 x m 0 x = ( β 2 β 0 m 0 x 2 + i ɛ m δ L m 0 x m δ L mm 0 x β 2 β 0 m 2 + i ɛ ) 1 ,
lim y 0 1 x + i y = P 1 x i π δ ( x ) ,
Δ β m 0 x [ Imag ] 2 = π m δ L m 0 x m δ L mm 0 x δ ( β 2 β 0 m 2 ) .
β m 0 x [ Cmplx ] 2 = β 0 m 0 x [ Real ] 2 + i Δ β m 0 x [ Imag ] 2 .
β m 0 x [ Imag ] = Δ β m 0 x [ Imag ] 2 / ( 2 β 0 m 0 x [ Real ] ) .
Δ β m 0 x [ Imag ] 2 = π m 1 2 π σ exp { [ β 2 β 0 m 2 / σ ] 2 2 } δ L m 0 x m δ L mm 0 x ,

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