Abstract

Using an eigenmode decomposition technique, we numerically determine the backreflection coefficient of the modes of air-core photonic bandgap fibers for flat terminations. This coefficient is found to be very small for the fundamental air-guided mode, of the order of 10-5 to 10-6, in contrast with the surface and bulk modes, which exhibit significantly higher reflections, by about three to four orders of magnitude. For the Crystal Fibre HC-1550-2 fiber, we find a reflection coefficient of 1.9×10-6 for an air termination, and approximately 3.3% for a silica termination. We also find that the Fresnel approximation is ill suited for the determination of the modal reflection coefficient, and instead propose a more accurate new formula based on an averaged modal index.

© 2007 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. Broeng, D. Mogilevstev, S. E. Barkou, and A. Bjarklev, "Photonic crystal fibers: A new class of optical waveguides," Opt. Fiber Technol. 5, 305-330 (1999).
    [CrossRef]
  2. J. C. Knight, T. A. Birks, R. F. Cregan, P. St. J. Russell, and J. P. Sandro, "Photonic crystals as optical fibers-physics and applications," Opt. Mater. 11, 143-151 (1999).
    [CrossRef]
  3. R. S. Windeler, J. L. Wagener, and D. J. Giovanni, "Silica-air microstructured fibers: Properties and applications," Optical Fiber Communications Conference, San Diego, CA, USA (1999).
  4. V. Dangui, H. K. Kim, M. J. F. Digonnet, and G. S. Kino, "Phase sensitivity to temperature of the fundamental mode in air-guiding photonic-bandgap fibers," Opt. Express 13, 6669-6684 (2005).H
    [CrossRef] [PubMed]
  5. D. M. Dagenais, K. P. Koo, and F. Bucholtz, "Fiber interferometry limitations due to parasitic optical cavities," Lasers and Electro-Optics Society Conference, San Jose, CA, USA (1993).
  6. J. Corbett, A. Dabirian, T. Butterley, N. A. Mortensen, and J. R. Allington-Smith, "The coupling performance of photonic crystal fibres in fibre stellar interferometry," Mon. Not. Roy. Astron. Soc. 368, 203-210 (2006).
  7. Crystal Fibre website, http://www.crystal-fibre.com
  8. D. Khalil, "Reflection at the end of strongly guiding dielectric waveguide," Opto-Electronics and Communications Conference, Yokohama, Japan, 1, 352-353 (2002).
  9. P. Gerard, P. Benech, D. Khalil, R. Rimet, and S. Tedjini, "Towards a full vectorial and modal technique for the analysis of integrated optics structures: The Radiation Spectrum Method (RSM)," Opt. Commun. 140, 128-145 (1997).
    [CrossRef]
  10. D. Marcuse, Theory of dielectric optical waveguides, (Academic Press, 1974).
  11. V. Dangui, M. J. F. Digonnet, and G. S. Kino, "A fast and accurate numerical tool to model the modal properties of photonic-bandgap fibers," Opt. Express 14, 2979-2993 (2006).H
    [CrossRef] [PubMed]
  12. H. . van der Vorst, "BI-CGSTAB: A fast and smoothly converging variant of BI-CG for the solution of nonsymmetric linear systems," SIAM J. Sci. Stat. Comput. 13, 631-644 (1992).
    [CrossRef]
  13. H. K. Kim, J. Shin, S. H. Fan, M. J. F. Digonnet, and G. S. Kino, "Designing air-core photonic-bandgap fibers free of surface modes," IEEE J. Quantum Eectron. 40, 551-556 (2004).
    [CrossRef]
  14. D. C. Allan, N. F. Borrelli, M. T. Gallagher, D. Müller, C. M. Smith, N. Venkataraman, J. A. West, P. Zhang, and K. W. Koch, "Surface modes and loss in air-core photonic band-gap fibers," in Photonic Crystal Materials and Devices, A. Adibi, A. Scherer, S. Yu Lin, eds., Proc. SPIE 5000,161-174 (2003).
    [CrossRef]
  15. K. Saitoh, N. A. Mortensen, and M. Koshiba, "Air-core photonic band-gap fibers: the impact of surface modes," Opt. Express 12, 394-400 (2004).
    [CrossRef] [PubMed]
  16. J. A. West, C. M. Smith, N. F. Borrelli, D. C. Allan, and K. W. Koch, "Surface modes in air-core photonic band-gap fibers," Opt. Express 12, 1485-1496 (2004).
    [CrossRef] [PubMed]

2006 (2)

J. Corbett, A. Dabirian, T. Butterley, N. A. Mortensen, and J. R. Allington-Smith, "The coupling performance of photonic crystal fibres in fibre stellar interferometry," Mon. Not. Roy. Astron. Soc. 368, 203-210 (2006).

V. Dangui, M. J. F. Digonnet, and G. S. Kino, "A fast and accurate numerical tool to model the modal properties of photonic-bandgap fibers," Opt. Express 14, 2979-2993 (2006).H
[CrossRef] [PubMed]

2005 (1)

2004 (3)

1999 (2)

J. Broeng, D. Mogilevstev, S. E. Barkou, and A. Bjarklev, "Photonic crystal fibers: A new class of optical waveguides," Opt. Fiber Technol. 5, 305-330 (1999).
[CrossRef]

J. C. Knight, T. A. Birks, R. F. Cregan, P. St. J. Russell, and J. P. Sandro, "Photonic crystals as optical fibers-physics and applications," Opt. Mater. 11, 143-151 (1999).
[CrossRef]

1997 (1)

P. Gerard, P. Benech, D. Khalil, R. Rimet, and S. Tedjini, "Towards a full vectorial and modal technique for the analysis of integrated optics structures: The Radiation Spectrum Method (RSM)," Opt. Commun. 140, 128-145 (1997).
[CrossRef]

1992 (1)

H. . van der Vorst, "BI-CGSTAB: A fast and smoothly converging variant of BI-CG for the solution of nonsymmetric linear systems," SIAM J. Sci. Stat. Comput. 13, 631-644 (1992).
[CrossRef]

Allan, D. C.

Allington-Smith, J. R.

J. Corbett, A. Dabirian, T. Butterley, N. A. Mortensen, and J. R. Allington-Smith, "The coupling performance of photonic crystal fibres in fibre stellar interferometry," Mon. Not. Roy. Astron. Soc. 368, 203-210 (2006).

Barkou, S. E.

J. Broeng, D. Mogilevstev, S. E. Barkou, and A. Bjarklev, "Photonic crystal fibers: A new class of optical waveguides," Opt. Fiber Technol. 5, 305-330 (1999).
[CrossRef]

Benech, P.

P. Gerard, P. Benech, D. Khalil, R. Rimet, and S. Tedjini, "Towards a full vectorial and modal technique for the analysis of integrated optics structures: The Radiation Spectrum Method (RSM)," Opt. Commun. 140, 128-145 (1997).
[CrossRef]

Birks, T. A.

J. C. Knight, T. A. Birks, R. F. Cregan, P. St. J. Russell, and J. P. Sandro, "Photonic crystals as optical fibers-physics and applications," Opt. Mater. 11, 143-151 (1999).
[CrossRef]

Bjarklev, A.

J. Broeng, D. Mogilevstev, S. E. Barkou, and A. Bjarklev, "Photonic crystal fibers: A new class of optical waveguides," Opt. Fiber Technol. 5, 305-330 (1999).
[CrossRef]

Borrelli, N. F.

Broeng, J.

J. Broeng, D. Mogilevstev, S. E. Barkou, and A. Bjarklev, "Photonic crystal fibers: A new class of optical waveguides," Opt. Fiber Technol. 5, 305-330 (1999).
[CrossRef]

Butterley, T.

J. Corbett, A. Dabirian, T. Butterley, N. A. Mortensen, and J. R. Allington-Smith, "The coupling performance of photonic crystal fibres in fibre stellar interferometry," Mon. Not. Roy. Astron. Soc. 368, 203-210 (2006).

Corbett, J.

J. Corbett, A. Dabirian, T. Butterley, N. A. Mortensen, and J. R. Allington-Smith, "The coupling performance of photonic crystal fibres in fibre stellar interferometry," Mon. Not. Roy. Astron. Soc. 368, 203-210 (2006).

Cregan, R. F.

J. C. Knight, T. A. Birks, R. F. Cregan, P. St. J. Russell, and J. P. Sandro, "Photonic crystals as optical fibers-physics and applications," Opt. Mater. 11, 143-151 (1999).
[CrossRef]

Dabirian, A.

J. Corbett, A. Dabirian, T. Butterley, N. A. Mortensen, and J. R. Allington-Smith, "The coupling performance of photonic crystal fibres in fibre stellar interferometry," Mon. Not. Roy. Astron. Soc. 368, 203-210 (2006).

Dangui, V.

Digonnet, M. J. F.

Fan, S. H.

H. K. Kim, J. Shin, S. H. Fan, M. J. F. Digonnet, and G. S. Kino, "Designing air-core photonic-bandgap fibers free of surface modes," IEEE J. Quantum Eectron. 40, 551-556 (2004).
[CrossRef]

Gerard, P.

P. Gerard, P. Benech, D. Khalil, R. Rimet, and S. Tedjini, "Towards a full vectorial and modal technique for the analysis of integrated optics structures: The Radiation Spectrum Method (RSM)," Opt. Commun. 140, 128-145 (1997).
[CrossRef]

Khalil, D.

P. Gerard, P. Benech, D. Khalil, R. Rimet, and S. Tedjini, "Towards a full vectorial and modal technique for the analysis of integrated optics structures: The Radiation Spectrum Method (RSM)," Opt. Commun. 140, 128-145 (1997).
[CrossRef]

Kim, H. K.

V. Dangui, H. K. Kim, M. J. F. Digonnet, and G. S. Kino, "Phase sensitivity to temperature of the fundamental mode in air-guiding photonic-bandgap fibers," Opt. Express 13, 6669-6684 (2005).H
[CrossRef] [PubMed]

H. K. Kim, J. Shin, S. H. Fan, M. J. F. Digonnet, and G. S. Kino, "Designing air-core photonic-bandgap fibers free of surface modes," IEEE J. Quantum Eectron. 40, 551-556 (2004).
[CrossRef]

Kino, G. S.

Knight, J. C.

J. C. Knight, T. A. Birks, R. F. Cregan, P. St. J. Russell, and J. P. Sandro, "Photonic crystals as optical fibers-physics and applications," Opt. Mater. 11, 143-151 (1999).
[CrossRef]

Koch, K. W.

Koshiba, M.

Mogilevstev, D.

J. Broeng, D. Mogilevstev, S. E. Barkou, and A. Bjarklev, "Photonic crystal fibers: A new class of optical waveguides," Opt. Fiber Technol. 5, 305-330 (1999).
[CrossRef]

Mortensen, N. A.

J. Corbett, A. Dabirian, T. Butterley, N. A. Mortensen, and J. R. Allington-Smith, "The coupling performance of photonic crystal fibres in fibre stellar interferometry," Mon. Not. Roy. Astron. Soc. 368, 203-210 (2006).

K. Saitoh, N. A. Mortensen, and M. Koshiba, "Air-core photonic band-gap fibers: the impact of surface modes," Opt. Express 12, 394-400 (2004).
[CrossRef] [PubMed]

Rimet, R.

P. Gerard, P. Benech, D. Khalil, R. Rimet, and S. Tedjini, "Towards a full vectorial and modal technique for the analysis of integrated optics structures: The Radiation Spectrum Method (RSM)," Opt. Commun. 140, 128-145 (1997).
[CrossRef]

Russell, P. St. J.

J. C. Knight, T. A. Birks, R. F. Cregan, P. St. J. Russell, and J. P. Sandro, "Photonic crystals as optical fibers-physics and applications," Opt. Mater. 11, 143-151 (1999).
[CrossRef]

Saitoh, K.

Sandro, J. P.

J. C. Knight, T. A. Birks, R. F. Cregan, P. St. J. Russell, and J. P. Sandro, "Photonic crystals as optical fibers-physics and applications," Opt. Mater. 11, 143-151 (1999).
[CrossRef]

Shin, J.

H. K. Kim, J. Shin, S. H. Fan, M. J. F. Digonnet, and G. S. Kino, "Designing air-core photonic-bandgap fibers free of surface modes," IEEE J. Quantum Eectron. 40, 551-556 (2004).
[CrossRef]

Smith, C. M.

Tedjini, S.

P. Gerard, P. Benech, D. Khalil, R. Rimet, and S. Tedjini, "Towards a full vectorial and modal technique for the analysis of integrated optics structures: The Radiation Spectrum Method (RSM)," Opt. Commun. 140, 128-145 (1997).
[CrossRef]

van der Vorst, H.

H. . van der Vorst, "BI-CGSTAB: A fast and smoothly converging variant of BI-CG for the solution of nonsymmetric linear systems," SIAM J. Sci. Stat. Comput. 13, 631-644 (1992).
[CrossRef]

West, J. A.

IEEE J. Quantum Eectron. (1)

H. K. Kim, J. Shin, S. H. Fan, M. J. F. Digonnet, and G. S. Kino, "Designing air-core photonic-bandgap fibers free of surface modes," IEEE J. Quantum Eectron. 40, 551-556 (2004).
[CrossRef]

Mon. Not. Roy. Astron. Soc. (1)

J. Corbett, A. Dabirian, T. Butterley, N. A. Mortensen, and J. R. Allington-Smith, "The coupling performance of photonic crystal fibres in fibre stellar interferometry," Mon. Not. Roy. Astron. Soc. 368, 203-210 (2006).

Opt. Commun. (1)

P. Gerard, P. Benech, D. Khalil, R. Rimet, and S. Tedjini, "Towards a full vectorial and modal technique for the analysis of integrated optics structures: The Radiation Spectrum Method (RSM)," Opt. Commun. 140, 128-145 (1997).
[CrossRef]

Opt. Express (4)

Opt. Fiber Technol. (1)

J. Broeng, D. Mogilevstev, S. E. Barkou, and A. Bjarklev, "Photonic crystal fibers: A new class of optical waveguides," Opt. Fiber Technol. 5, 305-330 (1999).
[CrossRef]

Opt. Mater. (1)

J. C. Knight, T. A. Birks, R. F. Cregan, P. St. J. Russell, and J. P. Sandro, "Photonic crystals as optical fibers-physics and applications," Opt. Mater. 11, 143-151 (1999).
[CrossRef]

SIAM J. Sci. Stat. Comput. (1)

H. . van der Vorst, "BI-CGSTAB: A fast and smoothly converging variant of BI-CG for the solution of nonsymmetric linear systems," SIAM J. Sci. Stat. Comput. 13, 631-644 (1992).
[CrossRef]

Other (6)

D. C. Allan, N. F. Borrelli, M. T. Gallagher, D. Müller, C. M. Smith, N. Venkataraman, J. A. West, P. Zhang, and K. W. Koch, "Surface modes and loss in air-core photonic band-gap fibers," in Photonic Crystal Materials and Devices, A. Adibi, A. Scherer, S. Yu Lin, eds., Proc. SPIE 5000,161-174 (2003).
[CrossRef]

R. S. Windeler, J. L. Wagener, and D. J. Giovanni, "Silica-air microstructured fibers: Properties and applications," Optical Fiber Communications Conference, San Diego, CA, USA (1999).

D. M. Dagenais, K. P. Koo, and F. Bucholtz, "Fiber interferometry limitations due to parasitic optical cavities," Lasers and Electro-Optics Society Conference, San Jose, CA, USA (1993).

D. Marcuse, Theory of dielectric optical waveguides, (Academic Press, 1974).

Crystal Fibre website, http://www.crystal-fibre.com

D. Khalil, "Reflection at the end of strongly guiding dielectric waveguide," Opto-Electronics and Communications Conference, Yokohama, Japan, 1, 352-353 (2002).

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 (18)

Fig. 1.
Fig. 1.

Configuration studied for the computation of the modal reflection coefficient of a PBF terminated in a uniform medium. The boundary between regions I and II is marked by the red dotted line, and defines the transverse plane.

Fig. 2.
Fig. 2.

Typical matrix A used for computation of the modal backreflection.

Fig. 3.
Fig. 3.

Refractive index profile of the PBF modeled in this section, for which ρ=0.47Λ and R=0.8Λ. The gray areas indicate air (refractive index equal to 1); the black areas indicate silica (refractive index chosen equal to 1.45).

Fig. 4.
Fig. 4.

Bandgap and dispersion curve of the PBF shown in Fig. 3. The fundamental modes are depicted in red, and the bandgap limits are shown in black.

Fig. 5.
Fig. 5.

Fundamental mode longitudinal Poynting vector profile, at a wavelength of λ = 0.6Λ, for the PBF of Fig. 3.

Fig. 6.
Fig. 6.

Fourier transform intensity, in logarithmic scale, of the transverse electric field of the mode plotted in Fig. 5. The black circle marks all wavevectors such that k 2 x +k 2 y k 2 0, corresponding to plane waves that can be excited in air.

Fig. 7.
Fig. 7.

Fundamental mode reflection coefficient for an air termination, for the PBF of Fig. 3, across the bandgap, as a function of the normalized wavelength λ/Λ. The incident field is the mostly x-polarized fundamental mode.

Fig. 8.
Fig. 8.

Surface mode electric field intensity profile for a PBF with R = 2Λ and ρ = 0.47Λ at a wavelength λ = 0.6Λ.

Fig. 9.
Fig. 9.

Comparison between the computed reflection coefficients and the Fresnel approximation predictions for a few surface and core modes at a wavelength of λ = 0.6Λ, for a PBF with ρ = 0.47Λ and R = 2Λ.

Fig. 10.
Fig. 10.

Variation of the fundamental mode reflection with PBF core radius, for a PBF cladding with ρ = 0.47Λ, at a wavelength of λ = 0.6Λ.

Fig. 11.
Fig. 11.

Variation of the reflection coefficient with termination medium refractive index, for a PBF structure with ρ = 0.47Λ, R = 0.8Λ, at a wavelength λ = 0.6Λ.

Fig. 12.
Fig. 12.

Plot of the averaged modal index of the fundamental mode as a function of normalized core radius for a PBF with ρ = 0.47Λ, at a wavelength of λ = 0.6Λ.

Fig. 13.
Fig. 13.

Variation of the calculated reflection (blue diamonds) and reflection predicted by the averaged modal index approximation (red squares) of the fundamental mode as a function of the core radius for a PBF with ρ = 0.47Λ, at a wavelength λ = 0.6Λ.

Fig. 14.
Fig. 14.

Scanning electron microscope picture of the cross-section of the HC-1550-2 PBF by Crystal Fibre. [7]

Fig. 15.
Fig. 15.

Refractive index profile cross-section of the PBF model used to approximate the fiber HC-1550-2. Grey areas mark air (refractive index 1) and black areas mark silica (refractive index chosen equal to 1.45).

Fig. 16.
Fig. 16.

Normalized plot of the longitudinal Poynting vector component for the y-polarized fundamental air-guided mode of the PBF HC-1550-2 model, at a wavelength of λ = 0.4Λ.

Fig. 17.
Fig. 17.

Fourier transform intensity, in logarithmic scale, of the transverse electric field of the fundamental mode plotted in Fig. 16. The black circle marks all wavevectors such that k 2 x + k 2 y k 2 0, corresponding to plane waves that can be excited in air.

Fig. 18.
Fig. 18.

Dependence of the reflection coefficient on termination medium refractive index, calculated with two different models. The PBF refractive index profile is shown in Fig. 15.

Equations (26)

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

R = ( n eff n 0 n eff + n 0 ) 2 ,
E R ( r ) = m = 0 N M a m E m ( r ) ,
R = a 0 2 ,
E I + E R = E T ,
H I + H R = H T ,
n mod al = n ( r ) E T ( r ) 2 dS E T ( r ) 2 dS ,
R = ( n mod al n 0 n mod al + n 0 ) 2 ,
E R ( r ) = m = 0 N M a m E m ( r ) ,
R = a 0 2 ,
E T ( r ) = k Ω ( b k 1 E W , k 1 + b k 2 E W , k 2 ) e i k r S ,
k 2 = k x 2 + k y 2 + k z 2 k 0 2 = ( 2 π λ ) 2 ,
H I ( r ) = H 0 ( r ) ,
H R ( r ) = a m m = 0 N M H m ( r ) ,
H T ( r ) = k Ω ( b k 1 H W , k 1 + b k 2 H W , k 2 ) e i k r S ,
E I + E R = E T
E 0 ( r ) + m = 0 N M a m E m ( r ) = k Ω ( b k 1 E W , k 1 + b k 2 E W , k 2 ) e i k r S ,
H I + H R = H T
H 0 ( r ) + m = 0 N M a m H m ( r ) = k Ω ( b k 1 H W , k 1 + b k 2 H W , k 2 ) e i k r S ,
A B = 2 A * ( r ) B ( r ) dS ,
E 0 2 ( 1 + a 0 ) = k∊Ω ( b k 1 E 0 E w , k 1 e ik r s + b k 2 E 0 E w , k 2 e ik r s ) ,
E m 2 a m = k∊Ω ( b k 1 E m E w , k 1 e ik r s + b k 2 E m E w , k 2 e ik r s ) , m 1 ,
H w , k 1 e ik r s 2 b k 1 = ( 1 a 0 ) H w , k 1 e ik r s H 0 m = 1 N M a m H w , k 1 e ik r s H m ,
H w , k 2 e ik r s 2 b k 2 = ( 1 a 0 ) H w , k 2 e ik r s H 0 m = 1 N M a m H w , k 2 e ik r s H m ,
A ν = w ,
( LU ) 1 A ν = ( LU ) 1 w ,
A ν 0 w w = ε 10 12 ,

Metrics