The properties of the out-of-plane bandgaps [105] that allow light confinement in PBGFs are quite different from those obtained when the component of the wavevector parallel to the periodic stack is equal to zero [106, 107]. Examples of the latter case are the in-plane propagation in a 2D photonic crystal or the fibre Bragg grating. In both cases the period of the lattice Λ must be of the order of half the wavelength, requiring sub- micrometre features for operation in the visible or near-IR. In contrast, for out-of-plane propagation the requirement is typically relaxed: in the most common configuration, the propagation constant β1 is several times larger than the wave number in the transverse
plane k⊥ =
√ n2k2
0−β2. As a result the bandgap can operate on shorter wavelengths
than in the analogous in-plane version, and a more practical Λ of several µm can be employed to create a PBG at visible or telecom wavelengths.
An additional advantage of the out-of-plane propagation is that a lower refractive index contrast is needed to open a bandgap. For a silica matrix with a triangular lattice of circular holes there is no complete photonic bandgap in the transverse propagation plane, due to an insufficient index contrast between the two regions [108]. Forβ 6= 0 however, a large contrast in transverse rather than absolute wave vectors is needed, and this can be made arbitrarily large, even if the contrast between the refractive indices itself is small [35]. This has allowed the use of pure silica to fabricate hollow core, air-guiding PBGFs, such as the one shown in Figure 1.11 and analysed theoretically for the first time by Broeng et al.[109].
1Although in practice also the modes of PBGFs are leaky (Section 1.2.5) and therefore their prop- agation constantβ is complex, for simplicity in this section we will neglect this aspect and refer toβ
The existence of an out-of-plane bandgap for a silica-air structure is clearly shown by the band diagram in Figure 1.12, where for a periodic cladding similar to those achievable in practice [110] and forβΛ = 15 (note that, in comparison, Bragg gratings operate at a much smaller βΛ = π) a gap of no allowable frequencies appears around kΛ = 15. The specific value ofβΛ in this example was chosen to demonstrate that a bandgap can exist for an effective index neff= β/k around the index of air (=1). This is the main
requirement for modal confinement in air.
In analogy to the electronic density of states, we can also introduce a photonic density of states (PDOS, or simply DOS),ρ(ω, β), defined as the number of electromagnetic states existing in the frequency interval between ω and δω and for a propagation constant betweenβ and δβ: ρ(ω, β) =X m Z 1BZ δ[ωm(k⊥, β)−ω]dk⊥ (1.10) Here m indexes the cladding eigenmodes with eigenvalues ωm, δ[ω] is the Dirac delta function and the integral is calculated for the transverse wave vectork⊥ scanning over the irreducible Brillouin zone. The DOS plot shown in Figure 1.12 for βΛ = 15 clearly confirms the indication that electromagnetic states can be completely inhibited for cer- tain wavelength ranges.
Figure 1.12: Band diagrams showing the normalised frequency kΛ versusk⊥ calcu-
lated along the contour of the irreducible Brillouin zone and forβΛ = 15. The periodic dielectric structure consists of hexagonal holes with rounded corners and it is shown in the inset of Figure 1.13. The filled curve on the right shows the corresponding density of states (DOS). From both images a bandgap around the air index (β/k= 1) is visible.
In general the position and width of the bandgaps depend on the propagation constant
β, and Figure 1.13 shows the results of changing β. The green patched areas represents bandgaps, where light propagation in the cladding is prohibited, and the dashed black line represents the air-line. As can be seen, several complete bandgaps exist for the studied lattice, and some of them are crossed by the air line.
Figure 1.13: Plot of neff=β/k versus λ/Λ for the realistic air-silica cladding shown
in the inset and presenting d/Λ=0.96 and hexagons with rounded holes. The patched green regions indicate the bandgaps and the dashed line shows the air-line.
If a defect suitable to sustain one or mode modes is introduced, the structure can guide in air. Creganet al. suggested, by employing a modification of a method often used in quantum electrodynamics to determine the density of modes [111], that the number of guided modes in the fibre is approximately given by:
NP BGF = ρ 2(β2
H−βL2)
4 (1.11)
whereβH andβLare the upper and lower edges of the PBG at a givenλandρis the core radius [3]. By employing typical values for a PBGF, it can be shown that a defect having nearly the size of Λ is insufficiently large to guide a mode. For a 7-cell defect however, not only a degenerate pair of fundamental modes, but also a group of additional 4 higher order modes, can be guided [109]. The defect modes will, to first approximation, follow the air-line, and their bandgap edges will thus be roughly given by the intersection of the air-line with the bandgap edges of the cladding shown in Figure 1.13.
The dependence of the central wavelength and of the edges of the bandgap as a function of the air-filling fractionf has been studied by Mortensenet al. [110], who showed that by increasingf the centre of the bandgap shifts towards shorter wavelengths, and at the same time the relative bandwidth increases (almost exponentially withf2).