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Analysis of ring-structured Bragg fibres for single TE mode guidance

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Abstract

Ring-structured Bragg fibres that support a single TE-polarisation mode are investigated. The fibre designs consist of a hollow core and rings of holes concentric with the core, which form the low-index layers of the Bragg reflector in the cladding. The effects of varying the air fraction in each ring of holes on the transmission properties of the fibres are analysed and an approximate model based on homogenisation is explored. Surface modes and transitions thereof are also discussed.

©2004 Optical Society of America

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Supplementary Material (1)

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Figures (8)

Fig. 1.
Fig. 1. Schematic of the ring-structured Bragg fibre (a) with some of the parameters indicated in (b). (c) The 2D lattice of holes that forms at large distances from the core. The wavevector is indicated by k, its longitudinal component by β and its transverse component by κ.
Fig. 2.
Fig. 2. (a) The loss of the TE01 mode (at the lowest loss wavelength) as a function of the number of rings for the different values of the air fraction. (b) The loss of the TE01 mode as a function of wavelength for the different values of f with 9 rings of holes, showing the loss and the wavelength of lowest loss increasing with decreasing air fraction. (c) The loss of the HE11 and TM01 modes for f=0.65 (Λi=0.403 µm) with 9 rings of holes. (d) The loss of the HE11 and TM01 modes for f=0.43 (Λi=0.607 µm) with 9 rings of holes. The loss of the hollow waveguide HE11 and TM01 modes is also shown in (c) and (d) for comparison. The mechanism for isolating the TE01 mode is the large difference in the loss of the modes [9]. The HE11 mode has a loss of 103 to 104 times larger than that of the TE01 mode, depending on the value of Λi.
Fig. 3.
Fig. 3. Band diagrams for the highest and lowest air fraction of each ring of holes used for the TE and TM polarisations with the band gap regions denoted by white. The dispersion lines of the TE01 modes are shown on the TE diagrams, and of the HE11 mode on the TM diagrams (mostly indistinguishable from the light line on these graphs). The point where the TM band gap closes falls very close to the light line. This was achieved by incorporating the Brewster angle in these fibre designs [9].
Fig. 4.
Fig. 4. The average refractive index profile n(r) obtained for one ring of holes with f=0.65 (Λi=0.403 µm) using the various methods discussed in the text.
Fig. 5.
Fig. 5. Loss at the lowest loss wavelength for the ring-structured fibres and the various homogenisation approaches used. Only the curve for x=1 is plotted for Eq. (5) as it gave the best approximation compared to other values of x as described in the text.
Fig. 6.
Fig. 6. (a) Schematic of Re{n eff} as a function of wavelength indicating the effect of the avoided crossings and the surface mode “s”, core mode “c” and transition stage “t” of each mode. The Re{n eff} (b) and loss (c) as a function of wavelength for f=0.43 (Λi=0.607 µm) and 9 rings of holes for modes of TE and HE polarisation from the middle of the primary band gap to longer wavelengths. The number of lobes in the intensity profile Sz of each mode is indicated. The intensity profile for the HE mode with 8 lobes is shown in Fig. 7 (for the wavelengths indicated) and in Fig. 8.
Fig. 7.
Fig. 7. The z-component of the Poynting vector Sz across the fibre for a HE polarisation mode in the surface mode stage (a), in the core mode stage (b), and in the transition stage (c and d).
Fig. 8.
Fig. 8. The z-component of the Poynting vector Sz across the fibre for a HE polarisation mode as the wavelength is increased from 1.18 to 1.57 µm, showing how the mode changes from a surface mode to a core mode and the transitions thereafter. Red indicates large intensities and violet low intensities. (animation -1.61 MB)

Equations (6)

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κ d ¯ = k 2 ( n 2 Re { n eff } 2 ) d ¯ = π 2 ,
n ( r ) = 1 2 π 0 2 π n ( r , ϕ ) d ϕ ,
n av = π d 4 Λ i + ( 1 π d 4 Λ i ) n 1 = f + ( 1 f ) n 1 ,
n av = [ π d 4 Λ i + ( 1 π d 4 Λ i ) n 1 x ] 1 x = [ f + ( 1 f ) n 1 x ] 1 x , 2 x 2 ,
n ( r ) = [ 1 2 π 0 2 π n x ( r , ϕ ) d ϕ ] 1 x , 2 x 2 .
n ( r ) = e 1 2 π 0 2 π ln [ n ( r , ϕ ) ] d ϕ ,
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