The Photonic Band Structure

The photonic band structure of photonic-crystal slabs is characterized by the light-line problem, which discriminates between guided modes and quasi-guided modes. This is one of the main novelties with respect to conventional photonic crystals. Moreover, it has been mentioned that there can exist Bloch waves with a cut-off, which depends on the waveguide geometry. To get more insight on these concepts and have a better understanding of the photonic band structure, it is useful to study a simple case, namely a self-standing dielectric membrane in air, completely etched with a one-dimensional lattice like a Bragg reflector. The formation of the band structure is explained starting from the guided modes of the effective waveguide, which corresponds to the empty-lattice picture seen for two-dimensional photonic crystals. The core layer is chosen to have

²= 12, the cladding²= 1. The filling factor isf = 30% and the waveguide thickness isd= 0.5a.

The effective waveguide, which defines the basis states, is an homogenous self-standing membrane with dielectric constant ²= 8.7, given by Eq. (1.26).

The left panel of Fig. 1.10 displays the dispersion relation of the H-like modes for the effective waveguide, folded in the Brillouin zone. The light lines are drawn with dashed lines. The dark gray region is where no solutions can exist. The white region, between the light lines, contains the guided modes. The light gray area corresponds to leaky modes. These guided modes represent the empty-lattice states that are used as basis set in the numerical method. Actually, in order to form a complete basis set, the expansion should include all the waveguide modes that are found for a given Bloch vector k. In this case, the number of states would be infinite, because of the leaky modes continuum, and the expansion would be numerically untractable.

This picture is similar to Fig. 1.2a, where plane waves have been replaced by guided modes, to account for the vertical confinement. There are two main differences with respect to Fig. 1.2a:

there exist modes characterized by a cut-off energy that do not reach the long-wavelength limit, and the presence of a continuum spectrum (leaky modes). The long-wavelength limit corresponds to the fundamental waveguide mode, while the states with cut-off are higher-order modes.

The form of the dielectric matrix [[²]] is the origin of propagation losses in photonic crystal slabs.

When a guided mode is folded, it crosses the air light line and enters the leaky mode region.

However, the mode remains truly guided, because the coupling with leaky modes is null, since the dielectric tensor of the effective waveguide is diagonal. Indeed, the photonic band picture of the effective waveguide is equivalent to the conventional unfolded dispersion relation, where guided modes never cross the light line. When the effective waveguide is replaced by the photonic crystal slab, the numerical method outputs a discrete spectrum above and below the light line, which is shown in the right panel of Fig. 1.10. Below the light line, the photonic band structure is made of guided Bloch states, which may form a photonic band gap likewise in a one dimensional photonic crystal, see Fig. 1.2b. Once the Bloch mode has crossed the light line, even if it is calculated as a state with zero line-width, in fact, it becomes a resonance, due to the non-zero off-diagonal elements of the dielectric matrix [[²]], which couple the Bloch mode to the external field. Therefore, these states are subject to intrinsic propagation losses. The physical process that causes losses, is thus

diffraction, since states with different G vectors are coupled by the off-diagonal elements of [[²]], and the origin of diffraction is the periodicity of the dielectric function.

The spectrum is thus a continuum of states and the photonic band picture seems to break down.

However, assuming that above the light line, the dispersion relation of photonic crystal slabs is not a mere continuum of states, but it is organized in resonances, with central frequency and width well defined, the photonic band picture is still valid. In summary, the photonic bands lying below the light line represent the dispersion of guided modes, while those above the light line represent the dispersion of resonances. The numerical method does not provide the width of the resonances, so that, in principle, it is not known if the above statement holds. This issue will be addressed in the next two chapters, to show that the photonic band picture indeed is valid also for modes above the light line, provided that the structure is properly designed [Krauss T. F.,et al. (1996)].

Also the concept of photonic band gap requires some clarifications. Looking at band structure in the right panel of Fig. 1.10, notice that the band gaps are not characterized by a null density of states. In fact, in Eq. (1.17) the sum is performed over the whole Brillouin zone, which includes the leaky mode region. Considering that above the light line the states are organized in resonances, it is correct to assume that far from the resonances the density of states is almost zero, or at least very small. In this sense, the concept of photonic band gap is reformulated as follows: the spectral region[ω₁, ω₂]for which∀ω ∈[ω₁, ω₂],@(k, n) :ω= ˜ω_n(k), whereω˜_n(k)is either a guided mode or a resonance, is called the photonic band gap. Therefore, photonic-crystal slabs represent a trade-off also in the sense that the photonic band gap does not exactly imply a null density of states.

Another important feature of the band structure of photonic crystal slabs is that the band gap can be closed also by the onset of a higher order mode, as shown in Fig. 1.10 for the band gap at ωa/2πc ∼ 0.4. For this reason, it is favorable to work in the frequency region where only the fundamental mode can exist.

As a final remark, it is worth to mention that the choice of the effective waveguide is by no means unique. The effective waveguide determines the guided modes used as basis set. Since the set is not complete, changing the effective waveguide is not irrelevant. However, calculations performed taking different values for the average dielectric functions gave very similar results, apart the cut-off energies of the higher order modes, which strongly depend on the effective waveguide. That is why

0.0

Figure 1.11 Photonic bands for the air-bridge structure of Fig. 1.9d, with hole radius r = 0.24a. (a) Waveguide thicknessd= 0.3a; (b) waveguide thicknessd= 0.6a; (c) ideal 2D case. Solid (dashed) lines represent modes that are even (odd) with respect to the xy mirror plane. The dotted lines in (a) and (b) refer to the light lines in air and in the effective waveguide material.

the cut-off energies have to be treated with particular care.