Abstract

We present a detailed description of the role of surface modes in photonic band-gap fibers (PBGFs). A model is developed that connects the experimental observations of high losses in the middle of the transmission spectrum to the presence of surface modes supported at the core-cladding interface. Furthermore, a new PBGF design is proposed that avoids these surface modes and produces single-mode operation.

© 2004 Optical Society of America

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References

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  1. N. Venkataraman, M.T. Gallagher, D. Müller, Charlene M. Smith, J. A. West, K. W. Koch, and J. C. Fajardo, �??Low-Loss (13 dB/km) Air-Core Photonic Band-Gap Fibre�??, Proceedings of ECOC 2002 (Copenhagen, Denmark, 2002) PD1.1.
  2. C. M. Smith, N. Venkataraman, M. T. Gallagher, D. Müller, J. A. West, N. F. Borrelli, D. C. Allan and K. W. Koch, "Low-loss hollow-core silica/air photonic bandgap fibre," Nature 424, 657-659 (2003).
    [CrossRef] [PubMed]
  3. K. Saitoh and M. Koshiba, "Leakage loss and group velocity dispersion in air-core photonic bandgap fibers," Opt. Express 11, 3100-3109 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-23-3100">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-23-3100</a>.
    [CrossRef] [PubMed]
  4. 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), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-3-394">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-3-394</a>.
    [CrossRef] [PubMed]
  5. D. C. Allan, et al., Photonic Crystals and Light Localization in the 21st Century, C. M. Soukoulis (ed.), (Kluwer Academic Press, The Netherlands, 2001), pp. 305-320.
    [CrossRef]
  6. D. C. Allan, N. F. Borrelli, M. T. Gallagher, D. Müller, C. M. Smith, N. Venkataraman, J. A. West, P. Zhang, K.W. Koch, �??Surface modes and loss in air-core photonic bandgap fibers,�?? in Photonic Crystal Materials and Devices, Ali Adibi, Axel Scherer, and Shawn Yu Lin;, eds. Proc. SPIE 5000, p. 161-174 (2003).
  7. J. D. Joannopoulos, R. D. Meade and J. N. Winn, Photonic Crystals (Princeton University Press, Princeton, N.J., 1995), pp. 73-76.
  8. P. Yeh, Optical Waves in Layered Media, (John Wiley & Sons, New York, N.Y., 1988) pp. 337-345.
  9. D. C. Allan, N.F. Borrelli, J. C. Fajardo, K. W. Koch, and J. A. West. Corning Incorporated �??Optimized defects in band-gap waveguides,�?? U.S. Pat. Appl. 20020136516-A1. February 4 2002.
  10. S. G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, �??Perturbation theory for Maxwell�??s equations with shifting material boundaries,�?? Phys. Rev. E 65, 66611 (2002).
    [CrossRef]
  11. 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, 1227-1243 (2002), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-21-1227">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-21-1227</a>.
    [CrossRef] [PubMed]
  12. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Kluwer Academic Publishers, Boston, MA, 2000), Eq. 31-50a.
  13. S. G. Johnson and J. D. Joannopoulos, "Block-iterative frequency-domain methods for Maxwell's equations in a planewave basis," Opt. Express 8, 173-190 (2001), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173</a>.
    [CrossRef] [PubMed]
  14. B. J. Mangan, L. Farr, A. Langford, P. J. Roberts, D. P. Williams, F. Couny, M. Lawman, M. Mason, S. Coupland, R. Flea and H. Sabert, �??Low loss (1.7 dB/km) hollow core photonic bandgap fiber,�?? Proceedings of OFC 2004, (OSA, Los Angeles, CA, 2004) PDP24.

ECOC 2002 (1)

N. Venkataraman, M.T. Gallagher, D. Müller, Charlene M. Smith, J. A. West, K. W. Koch, and J. C. Fajardo, �??Low-Loss (13 dB/km) Air-Core Photonic Band-Gap Fibre�??, Proceedings of ECOC 2002 (Copenhagen, Denmark, 2002) PD1.1.

Nature (1)

C. M. Smith, N. Venkataraman, M. T. Gallagher, D. Müller, J. A. West, N. F. Borrelli, D. C. Allan and K. W. Koch, "Low-loss hollow-core silica/air photonic bandgap fibre," Nature 424, 657-659 (2003).
[CrossRef] [PubMed]

OFC 2004 (1)

B. J. Mangan, L. Farr, A. Langford, P. J. Roberts, D. P. Williams, F. Couny, M. Lawman, M. Mason, S. Coupland, R. Flea and H. Sabert, �??Low loss (1.7 dB/km) hollow core photonic bandgap fiber,�?? Proceedings of OFC 2004, (OSA, Los Angeles, CA, 2004) PDP24.

Opt. Express (4)

Phys. Rev. E (1)

S. G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, �??Perturbation theory for Maxwell�??s equations with shifting material boundaries,�?? Phys. Rev. E 65, 66611 (2002).
[CrossRef]

Proc. SPIE (1)

D. C. Allan, N. F. Borrelli, M. T. Gallagher, D. Müller, C. M. Smith, N. Venkataraman, J. A. West, P. Zhang, K.W. Koch, �??Surface modes and loss in air-core photonic bandgap fibers,�?? in Photonic Crystal Materials and Devices, Ali Adibi, Axel Scherer, and Shawn Yu Lin;, eds. Proc. SPIE 5000, p. 161-174 (2003).

Other (5)

J. D. Joannopoulos, R. D. Meade and J. N. Winn, Photonic Crystals (Princeton University Press, Princeton, N.J., 1995), pp. 73-76.

P. Yeh, Optical Waves in Layered Media, (John Wiley & Sons, New York, N.Y., 1988) pp. 337-345.

D. C. Allan, N.F. Borrelli, J. C. Fajardo, K. W. Koch, and J. A. West. Corning Incorporated �??Optimized defects in band-gap waveguides,�?? U.S. Pat. Appl. 20020136516-A1. February 4 2002.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Kluwer Academic Publishers, Boston, MA, 2000), Eq. 31-50a.

D. C. Allan, et al., Photonic Crystals and Light Localization in the 21st Century, C. M. Soukoulis (ed.), (Kluwer Academic Press, The Netherlands, 2001), pp. 305-320.
[CrossRef]

Supplementary Material (2)

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Figures (11)

Fig. 1.
Fig. 1.

(a) Optical attenuation as a function of wavelength for the 65-m long air-core PBGF using a conventional cut-back technique [2]. The loss feature between 1550 and 1650 nm is attributed to surface modes. (b) SEM of the air-silica PBGF profile, truncated to show central core defect and first rings of holes of microstructured cladding. The parameters are described in Ref. [2].

Fig. 2.
Fig. 2.

Modal plots of 7 surface modes, and 2 core modes that lie within the band gap of an undistorted silica structure (shown at lower right) with defect radius of Rd=1.15Λ and air-hole radius of r=0.45Λ. Note that the core mode has some spatial overlap with the surface modes.

Fig. 3.
Fig. 3.

(a) Calculated dispersion of air-core modes and band-gap modes for the fiber profile in Fig. 1(b). The black symbols represent modes of the cladding structure. The blue symbols represent modes whose intensity is predominantly on the surface of the air-core defect. The red symbols represent the modes with a majority of their intensity within the air-core defect. The fundamental air-core modes have a higher effective index than the higher-order air-core modes. The shaded region indicates the continuum regions surrounding the band gap. (b) Expanded view of core-surface-mode avoided crossing near 1655 nm. Red indicates core-like modes, blue indicates surface-like modes and modes which cannot be accurately approximated by either a core or surface mode are colored green. The dashed lines show the approximate core and surface modes used in the supermode theory. The numbers are used in Figs. 6 and 7 to refer to the modes.

Fig. 4.
Fig. 4.

(a) Experimental (black line with points) and calculated attenuation spectra of the fiber shown in Fig. 1. The solid blue curve assumes no change in the fiber scale with γ=4250 km-1; the red curve models a 4% variation in the fiber scale over the 65-meter length with γ=1750 km-1. The band edges are approximated by Lorentzians to illustrate the possible impact of the band edges on attenuation. (b) Individual loss contributions αij (thin, multicolored lines) compared to the overall loss coefficient α (heavy red line) for a fiber with no scale variation.

Fig. 5.
Fig. 5.

(a) Fraction of transmitted light in the two fundamental core modes (red) or the four higher-order modes (blue) relative to the total light transmitted calculated using the same model as the red curve in Fig. 4(a). Between 1530 and 1630 nm, almost 100% of the light transmitted is in the higher order modes. (b) (750 Kb) Movie of a spectrally resolved experimental measurement of the spatial distribution of modes of the fiber in Fig. 1. Note the “surface-mode” character of the mode between 1500 and 1650 nm and also that the light is transmitted in a higher-order mode between 1570 and 1630 nm in agreement with the predictions of the surface-mode theory.

Fig. 6.
Fig. 6.

(a) Coupling coefficients for a 2% change in air-filling fraction (AFF) for the modes shown in Fig. 3(b). The mode labels are given in Fig. 3(b). (b) The diagonal coupling coefficients indicate the impact of the perturbation on the propagation constants of the modes. The color scheme is the same as in Fig. 3(b).

Fig. 7.
Fig. 7.

(a) For the modes shown in Fig. 3(b), we plot the fraction of the mode energy contained in the glass, maintaining the color scheme of Fig. 3(b). (b) For the modes shown in Fig 3(a) we color-code the modes based on the fraction of the modal energy in the glass: >20%, blue; <2%, red; all others, green.

Fig. 8.
Fig. 8.

(1 Mb) Movie of band-gap for varying core radii, Rd, showing the surface modes (blue) entering the gap and interacting with the core-guided modes (black). The continuum modes and modes whose character is neither core nor surface are colored red.

Fig. 9.
Fig. 9.

Contour plot of fraction of core-confined energy vs. the propagation constant and the scaled defect radius Rd.

Fig. 10.
Fig. 10.

(a) The red curve is the fraction of core-confined energy vs. defect radius calculated for the most highly confined mode at each Rd in the contour plot from Fig. 9. The blue curve shows the βz at which the mode appears. (b) A log-log plot of the fractional energy outside the core radius (red) and the fractional energy in the glass (blue) vs. defect radius for the same modes as shown in the left-hand plot. The dashed lines indicate that the while the energy outside of the core radius Rd scales inversely as the cube of Rd, the scaling for the energy in the glass is closer to Rd2.5.

Fig. 11.
Fig. 11.

(a) Index profile of an improved PBGF design with r=0.47Λ and Rd=1.00Λ with no surface modes in the band gap. (b) Corresponding core mode at a scaled propagation constant of βz=1.67.

Equations (9)

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dA i dz = i κ ij A j e i Δ β ij z
dA j dz = i κ ij * A i e i Δ β ij z γ j A j
α ij ( dB km ) 20 ln ( 10 ) γ j κ ij 2 ( Δ β ij ) 2
( Δ β ij ) 2 = ( 2 π λ ) 2 ( Δ n eff , ij ) 2
Δ n eff , ij = Δ n min , ij + Λ s ij ( λ λ min , ij ) 2
s ij = dn eff , i d ω ˜ dn eff , j d ω ˜ ,
α ( λ ) = 10 L * log 10 ( i I i ( 0 ) 10 j α ij ( λ ) L i I i ( 0 ) )
λ min , ij ( z ) = λ min , ij ( 0 ) ( 1 + z x L )
C 12 A ( ε ε ˜ ) { E t 1 * · E t 2 + ε ˜ ε E z 1 * E z 2 } dA

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