Abstract

In the hollow core photonic bandgap fibers, modal losses are strongly differentiated, potentially enabling effectively single mode guidance. However, in the presence of macro-bending, due to mode coupling, power in the low-loss mode launched into a bend is partially transferred into the modes with higher losses, thus resulting in increased propagation loss, and degradation of the beam quality. We show that coupled mode theory formulated in the curvilinear coordinates associated with a bend can describe correctly both the bending induced loss and beam degradation. Suggested approach works both in absorption dominated regime in which fiber modes are square integrable over the fiber crossection, as well as in radiation dominated regime in which leaky modes are not square integrable. It is important to stress that for multimode fibers, full-vectorial coupled mode theory developed in this work is not a simple approximation, but it is on par with such “exact” numerical approaches as finite element and finite difference methods for prediction of macro-bending induced losses.

© 2008 Optical Society of America

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References

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  1. P. Russell, “Photonic crystal fibers,” Science 299, 358 (2003).
    [Crossref] [PubMed]
  2. C.M. Smith, N. Venkataraman, M. T. Gallagher, D. Muller, J. A. West, N. F. Borrelli, D. C. Allan, and K. W. Koch, “Low-loss hollow-core silica/air photonic bandgap fibre,” Nature 424, 657 (2003).
    [Crossref] [PubMed]
  3. B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, and Y. Fink, “Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO2 laser transmission,” Nature 420, 650 (2002).
    [Crossref] [PubMed]
  4. S.G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weiseberg, T. D. Engeness, M. Soljacic, S. A. Jacobs, J. D. Joannopoulos, and Y. Fink, “Low-loss asymptotically single-mode propagation in large core OmniGuide fibers,” Opt. Express 9, 748 (2001).
    [Crossref] [PubMed]
  5. T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. Martijn de Sterke, and L. C. Botten “Multipole method for microstructured optical fibers. I. Formulation,” JOSA B 19, 2322 (2002).
    [Crossref]
  6. E. Pone, A. Hassani, S. Lacroix, A. Kabashin, and M. Skorobogatiy, “Boundary integral method for the challenging problems in bandgap guiding, plasmonics and sensing,” Opt. Express 15, 10231 (2007).
    [Crossref] [PubMed]
  7. K. Saitoh and M. Koshiba, “Full-vectorial imaginary-distance beam propagation method based on a finite element scheme: application to Photonic Crystal fibers,” IEEE J. Quantum Electron. 38, 297 (2002).
    [Crossref]
  8. D.M. Shyroki, J. Lgsgaard, and O. Bang, “Finite-difference modeling of Bragg Fibers with ultrathin cladding layers via adaptive coordinate transformation,” Proc. of SPIE 6728, 672830 (2007).
  9. M. Skorobogatiy, S. A. Jacobs, S.G. Johnson, and Y. Fink, “Geometric variations in high index-contrast waveguides, coupled mode theory in curvilinear coordinates,” Opt. Express 10, 1227 (2002).
    [PubMed]
  10. M. Skorobogatiy, M. Ibanescu, S. G. Johnson, O. Weiseberg, T. D. Engeness, M. Soljacic, S. A. Jacobs, and Y. Fink, “Analysis of general geometric scaling perturbations in a transmitting waveguide,” JOSA B 19, 2867 (2002).
    [Crossref]

2007 (2)

E. Pone, A. Hassani, S. Lacroix, A. Kabashin, and M. Skorobogatiy, “Boundary integral method for the challenging problems in bandgap guiding, plasmonics and sensing,” Opt. Express 15, 10231 (2007).
[Crossref] [PubMed]

D.M. Shyroki, J. Lgsgaard, and O. Bang, “Finite-difference modeling of Bragg Fibers with ultrathin cladding layers via adaptive coordinate transformation,” Proc. of SPIE 6728, 672830 (2007).

2003 (2)

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

C.M. Smith, N. Venkataraman, M. T. Gallagher, D. Muller, J. A. West, N. F. Borrelli, D. C. Allan, and K. W. Koch, “Low-loss hollow-core silica/air photonic bandgap fibre,” Nature 424, 657 (2003).
[Crossref] [PubMed]

2002 (5)

B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, and Y. Fink, “Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO2 laser transmission,” Nature 420, 650 (2002).
[Crossref] [PubMed]

M. Skorobogatiy, S. A. Jacobs, S.G. Johnson, and Y. Fink, “Geometric variations in high index-contrast waveguides, coupled mode theory in curvilinear coordinates,” Opt. Express 10, 1227 (2002).
[PubMed]

M. Skorobogatiy, M. Ibanescu, S. G. Johnson, O. Weiseberg, T. D. Engeness, M. Soljacic, S. A. Jacobs, and Y. Fink, “Analysis of general geometric scaling perturbations in a transmitting waveguide,” JOSA B 19, 2867 (2002).
[Crossref]

K. Saitoh and M. Koshiba, “Full-vectorial imaginary-distance beam propagation method based on a finite element scheme: application to Photonic Crystal fibers,” IEEE J. Quantum Electron. 38, 297 (2002).
[Crossref]

T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. Martijn de Sterke, and L. C. Botten “Multipole method for microstructured optical fibers. I. Formulation,” JOSA B 19, 2322 (2002).
[Crossref]

2001 (1)

Allan, D. C.

C.M. Smith, N. Venkataraman, M. T. Gallagher, D. Muller, J. A. West, N. F. Borrelli, D. C. Allan, and K. W. Koch, “Low-loss hollow-core silica/air photonic bandgap fibre,” Nature 424, 657 (2003).
[Crossref] [PubMed]

Bang, O.

D.M. Shyroki, J. Lgsgaard, and O. Bang, “Finite-difference modeling of Bragg Fibers with ultrathin cladding layers via adaptive coordinate transformation,” Proc. of SPIE 6728, 672830 (2007).

Benoit, G.

B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, and Y. Fink, “Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO2 laser transmission,” Nature 420, 650 (2002).
[Crossref] [PubMed]

Borrelli, N. F.

C.M. Smith, N. Venkataraman, M. T. Gallagher, D. Muller, J. A. West, N. F. Borrelli, D. C. Allan, and K. W. Koch, “Low-loss hollow-core silica/air photonic bandgap fibre,” Nature 424, 657 (2003).
[Crossref] [PubMed]

Botten, L. C.

T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. Martijn de Sterke, and L. C. Botten “Multipole method for microstructured optical fibers. I. Formulation,” JOSA B 19, 2322 (2002).
[Crossref]

Engeness, T. D.

M. Skorobogatiy, M. Ibanescu, S. G. Johnson, O. Weiseberg, T. D. Engeness, M. Soljacic, S. A. Jacobs, and Y. Fink, “Analysis of general geometric scaling perturbations in a transmitting waveguide,” JOSA B 19, 2867 (2002).
[Crossref]

S.G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weiseberg, T. D. Engeness, M. Soljacic, S. A. Jacobs, J. D. Joannopoulos, and Y. Fink, “Low-loss asymptotically single-mode propagation in large core OmniGuide fibers,” Opt. Express 9, 748 (2001).
[Crossref] [PubMed]

Fink, Y.

M. Skorobogatiy, S. A. Jacobs, S.G. Johnson, and Y. Fink, “Geometric variations in high index-contrast waveguides, coupled mode theory in curvilinear coordinates,” Opt. Express 10, 1227 (2002).
[PubMed]

M. Skorobogatiy, M. Ibanescu, S. G. Johnson, O. Weiseberg, T. D. Engeness, M. Soljacic, S. A. Jacobs, and Y. Fink, “Analysis of general geometric scaling perturbations in a transmitting waveguide,” JOSA B 19, 2867 (2002).
[Crossref]

B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, and Y. Fink, “Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO2 laser transmission,” Nature 420, 650 (2002).
[Crossref] [PubMed]

S.G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weiseberg, T. D. Engeness, M. Soljacic, S. A. Jacobs, J. D. Joannopoulos, and Y. Fink, “Low-loss asymptotically single-mode propagation in large core OmniGuide fibers,” Opt. Express 9, 748 (2001).
[Crossref] [PubMed]

Gallagher, M. T.

C.M. Smith, N. Venkataraman, M. T. Gallagher, D. Muller, J. A. West, N. F. Borrelli, D. C. Allan, and K. W. Koch, “Low-loss hollow-core silica/air photonic bandgap fibre,” Nature 424, 657 (2003).
[Crossref] [PubMed]

Hart, S. D.

B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, and Y. Fink, “Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO2 laser transmission,” Nature 420, 650 (2002).
[Crossref] [PubMed]

Hassani, A.

Ibanescu, M.

M. Skorobogatiy, M. Ibanescu, S. G. Johnson, O. Weiseberg, T. D. Engeness, M. Soljacic, S. A. Jacobs, and Y. Fink, “Analysis of general geometric scaling perturbations in a transmitting waveguide,” JOSA B 19, 2867 (2002).
[Crossref]

S.G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weiseberg, T. D. Engeness, M. Soljacic, S. A. Jacobs, J. D. Joannopoulos, and Y. Fink, “Low-loss asymptotically single-mode propagation in large core OmniGuide fibers,” Opt. Express 9, 748 (2001).
[Crossref] [PubMed]

Jacobs, S. A.

Joannopoulos, J. D.

B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, and Y. Fink, “Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO2 laser transmission,” Nature 420, 650 (2002).
[Crossref] [PubMed]

S.G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weiseberg, T. D. Engeness, M. Soljacic, S. A. Jacobs, J. D. Joannopoulos, and Y. Fink, “Low-loss asymptotically single-mode propagation in large core OmniGuide fibers,” Opt. Express 9, 748 (2001).
[Crossref] [PubMed]

Johnson, S. G.

M. Skorobogatiy, M. Ibanescu, S. G. Johnson, O. Weiseberg, T. D. Engeness, M. Soljacic, S. A. Jacobs, and Y. Fink, “Analysis of general geometric scaling perturbations in a transmitting waveguide,” JOSA B 19, 2867 (2002).
[Crossref]

Johnson, S.G.

Kabashin, A.

Koch, K. W.

C.M. Smith, N. Venkataraman, M. T. Gallagher, D. Muller, J. A. West, N. F. Borrelli, D. C. Allan, and K. W. Koch, “Low-loss hollow-core silica/air photonic bandgap fibre,” Nature 424, 657 (2003).
[Crossref] [PubMed]

Koshiba, M.

K. Saitoh and M. Koshiba, “Full-vectorial imaginary-distance beam propagation method based on a finite element scheme: application to Photonic Crystal fibers,” IEEE J. Quantum Electron. 38, 297 (2002).
[Crossref]

Kuhlmey, B. T.

T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. Martijn de Sterke, and L. C. Botten “Multipole method for microstructured optical fibers. I. Formulation,” JOSA B 19, 2322 (2002).
[Crossref]

Lacroix, S.

Lgsgaard, J.

D.M. Shyroki, J. Lgsgaard, and O. Bang, “Finite-difference modeling of Bragg Fibers with ultrathin cladding layers via adaptive coordinate transformation,” Proc. of SPIE 6728, 672830 (2007).

Martijn de Sterke, C.

T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. Martijn de Sterke, and L. C. Botten “Multipole method for microstructured optical fibers. I. Formulation,” JOSA B 19, 2322 (2002).
[Crossref]

Maystre, D.

T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. Martijn de Sterke, and L. C. Botten “Multipole method for microstructured optical fibers. I. Formulation,” JOSA B 19, 2322 (2002).
[Crossref]

McPhedran, R. C.

T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. Martijn de Sterke, and L. C. Botten “Multipole method for microstructured optical fibers. I. Formulation,” JOSA B 19, 2322 (2002).
[Crossref]

Muller, D.

C.M. Smith, N. Venkataraman, M. T. Gallagher, D. Muller, J. A. West, N. F. Borrelli, D. C. Allan, and K. W. Koch, “Low-loss hollow-core silica/air photonic bandgap fibre,” Nature 424, 657 (2003).
[Crossref] [PubMed]

Pone, E.

Renversez, G.

T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. Martijn de Sterke, and L. C. Botten “Multipole method for microstructured optical fibers. I. Formulation,” JOSA B 19, 2322 (2002).
[Crossref]

Russell, P.

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

Saitoh, K.

K. Saitoh and M. Koshiba, “Full-vectorial imaginary-distance beam propagation method based on a finite element scheme: application to Photonic Crystal fibers,” IEEE J. Quantum Electron. 38, 297 (2002).
[Crossref]

Shyroki, D.M.

D.M. Shyroki, J. Lgsgaard, and O. Bang, “Finite-difference modeling of Bragg Fibers with ultrathin cladding layers via adaptive coordinate transformation,” Proc. of SPIE 6728, 672830 (2007).

Skorobogatiy, M.

Smith, C.M.

C.M. Smith, N. Venkataraman, M. T. Gallagher, D. Muller, J. A. West, N. F. Borrelli, D. C. Allan, and K. W. Koch, “Low-loss hollow-core silica/air photonic bandgap fibre,” Nature 424, 657 (2003).
[Crossref] [PubMed]

Soljacic, M.

M. Skorobogatiy, M. Ibanescu, S. G. Johnson, O. Weiseberg, T. D. Engeness, M. Soljacic, S. A. Jacobs, and Y. Fink, “Analysis of general geometric scaling perturbations in a transmitting waveguide,” JOSA B 19, 2867 (2002).
[Crossref]

S.G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weiseberg, T. D. Engeness, M. Soljacic, S. A. Jacobs, J. D. Joannopoulos, and Y. Fink, “Low-loss asymptotically single-mode propagation in large core OmniGuide fibers,” Opt. Express 9, 748 (2001).
[Crossref] [PubMed]

Temelkuran, B.

B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, and Y. Fink, “Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO2 laser transmission,” Nature 420, 650 (2002).
[Crossref] [PubMed]

Venkataraman, N.

C.M. Smith, N. Venkataraman, M. T. Gallagher, D. Muller, J. A. West, N. F. Borrelli, D. C. Allan, and K. W. Koch, “Low-loss hollow-core silica/air photonic bandgap fibre,” Nature 424, 657 (2003).
[Crossref] [PubMed]

Weiseberg, O.

M. Skorobogatiy, M. Ibanescu, S. G. Johnson, O. Weiseberg, T. D. Engeness, M. Soljacic, S. A. Jacobs, and Y. Fink, “Analysis of general geometric scaling perturbations in a transmitting waveguide,” JOSA B 19, 2867 (2002).
[Crossref]

S.G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weiseberg, T. D. Engeness, M. Soljacic, S. A. Jacobs, J. D. Joannopoulos, and Y. Fink, “Low-loss asymptotically single-mode propagation in large core OmniGuide fibers,” Opt. Express 9, 748 (2001).
[Crossref] [PubMed]

West, J. A.

C.M. Smith, N. Venkataraman, M. T. Gallagher, D. Muller, J. A. West, N. F. Borrelli, D. C. Allan, and K. W. Koch, “Low-loss hollow-core silica/air photonic bandgap fibre,” Nature 424, 657 (2003).
[Crossref] [PubMed]

White, T. P.

T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. Martijn de Sterke, and L. C. Botten “Multipole method for microstructured optical fibers. I. Formulation,” JOSA B 19, 2322 (2002).
[Crossref]

IEEE J. Quantum Electron. (1)

K. Saitoh and M. Koshiba, “Full-vectorial imaginary-distance beam propagation method based on a finite element scheme: application to Photonic Crystal fibers,” IEEE J. Quantum Electron. 38, 297 (2002).
[Crossref]

JOSA B (2)

T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. Martijn de Sterke, and L. C. Botten “Multipole method for microstructured optical fibers. I. Formulation,” JOSA B 19, 2322 (2002).
[Crossref]

M. Skorobogatiy, M. Ibanescu, S. G. Johnson, O. Weiseberg, T. D. Engeness, M. Soljacic, S. A. Jacobs, and Y. Fink, “Analysis of general geometric scaling perturbations in a transmitting waveguide,” JOSA B 19, 2867 (2002).
[Crossref]

Nature (2)

C.M. Smith, N. Venkataraman, M. T. Gallagher, D. Muller, J. A. West, N. F. Borrelli, D. C. Allan, and K. W. Koch, “Low-loss hollow-core silica/air photonic bandgap fibre,” Nature 424, 657 (2003).
[Crossref] [PubMed]

B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, and Y. Fink, “Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO2 laser transmission,” Nature 420, 650 (2002).
[Crossref] [PubMed]

Opt. Express (3)

Proc. of SPIE (1)

D.M. Shyroki, J. Lgsgaard, and O. Bang, “Finite-difference modeling of Bragg Fibers with ultrathin cladding layers via adaptive coordinate transformation,” Proc. of SPIE 6728, 672830 (2007).

Science (1)

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

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Figures (3)

Fig. 1.
Fig. 1.

(a) Schematic of a hollow core Bragg fiber. Sz is the transverse distribution of the longitudinal energy flux component for the Gaussian-like HE 11 core guided mode. (b) Spectrum of the modal propagation constants for the hollow core Bragg fiber shown in (a), evaluated at λ = 10.6µm.

Fig. 2.
Fig. 2.

(a) Schematic of a fiber bend. (b) Losses of the eigen modes of a bent hollow core Bragg fiber evaluated at λ = 10.6µm as a function of the bending radius. Material absorbtion dominated regime. (c) Losses of an HE 11-like mode of a bend as a function of the wavelength of operation for various values of the bending radius.

Fig. 3.
Fig. 3.

(a) Bending losses of a hollow core Bragg fiber under HE 11 launching condition for various values of bending radius (absorption dominated regime). (b) Plots of intensity distribution at the bend output, under HE 11 launching conditions for various values of bending radius. (c) Bending losses of a hollow core Bragg fiber under HE 11 launching condition for various values of bending radius (radiation dominated regime).

Equations (11)

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x = R b ( R b x ) cos ( s R ) ,
z = ( R b x ) sin ( s R )
F t β b ( x , y , s ) = exp ( i β b s ) β r C β r β b F t β r ( x , y ) ,
B β r , β r = s ̂ · d x d y ( E t β r × H t β r + E t β r × H t β r ) ,
Δ M β r , β r = ω R b d x d y [ ( H s β r H s β r + H t β r · H t β r ) + ε ( x , y ) ( E s β r E s β r + E t β r · E t β r ) ] · x .
β b B C β b = ( B B r + Δ M ) C β b ,
Out r = C b exp ( i B b R b θ b ) C b 1 In r ,
F t out ( x , y ) = β r Out β r r F t β r ( x , y ) .
P out = s ̂ · d x d y S s out ( x , y ) = β r , β r Out β r r Out β r r * s ̂ · d x d y Re ( E t β r ( x , y ) × H t β r * ( x , y ) ) 2 .
α bend [ dB m ] = 10 log 10 ( P out P in ) ( θ b R b ) .
β b β r = β r β r Δ M β r , β r 2 B β r , β r B β r , β r 1 β r β r ~ R b 2 .

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