Numerical Method

The basic idea underneath this method is the expansion of the magnetic field in terms of the guided modes of an “effective” waveguide, in place of a super-cell with plane-waves. Each basis mode is then folded in the two-dimensional Brillouin zone, where it is coupled by the inverse dielectric tensor η_G,G⁰, just like plane waves in a two-dimensional photonic crystal. The eigenstates falling below the light line are identified as guided modes, whereas those above the light line represent the guided resonances.

The importance of using guided modes instead of plane waves is readily explained. Below the light

line, the spectrum is discrete, ω_n(k), and the modes exhibit exponential decay in the cladding;

above it is continuum, ω_ρ(k), and the modes have an oscillatory profile in the cladding. If the spectrum is discrete, an expansion withN plane waves yieldsN eigenstates with increasing energy:

ω₁(k), ω₂(k), . . . , ω_N(k). Also, if the super-cell is large enough, the overlap among guided modes of nearest super-cells is negligible, because the mode profile decays exponentially in the cladding.

When the spectrum becomes continuum, there are two issues with the super-cell method. First of all, since the mode profile is not evanescent in the cladding, there will be interaction among modes of neighboring super-cells, causing an error in the evaluation of the eigenfrequencies. Secondly, the diagonalization of the hamiltonian will give only the states of the continuum that are just above the light line. In fact, if ω_LL(k) is the frequency of the light line for Bloch vectork, ∀² >0,∃ω_ρ(k) : ω_LL(k)< ω_ρ(k)< ω_LL(k) +². Thus, given any number N of plane waves, all the eigenfrequencies above the light line will lie in an infinitesimal around of the accumulation point ω_LL(k). In other words, the plane-wave expansion does not pick up the resonances of the continuum, but it yields the first states, without exception, starting from the light line and within an infinitesimal interval ofω_LL(k). That is why the plane-wave expansion is valid only for modes below the light line, where the spectrum is discrete, and not for the resonances.

As already mentioned, the numerical method proposed by Andreani, L. C. (2002) avoids the use of a super-cell and expands the electromagnetic field in an orthonormal set of guided modes of an effective waveguide. The great advantage is that the method is able to calculate the dispersion relation of the resonances. Within this approach, both modes above and below the light line are considered as truly guided, so that the spectrum is discrete, formally. This avoids the problem encountered with the plane-wave expansion. Indeed, while plane waves form a complete basis set, the expansion on guided modes is not such, because leaky modes are missing. Therefore, the two

“hamiltonian” differ: ifHis the “hamiltonian” in the plane-wave expansion andω_n(k), ω_ρ(k) is the spectrum (discrete and continuum), the “hamiltonian” in the guided-modes expansion will be He with spectrum ˜ω_n(k) only discrete. Below the light line, the assumption is that ˜ω_n(k)∼ω_n(k) with good approximation. The eigenfrequencies above the light line simply represent the frequencies of the resonances, ˜ω_n(k), with zero line-width. The method is not exact, in principle, and its validity relies on the contribution of leaky modes to the eigenfrequencies. The assumption is that

the frequency shift due to coupling with leaky modes of the effective waveguide is small, if not negligible, for both guided and quasi-guided modes. As to the spectrum below the light line, very good agreement has been found in comparison with the super-cell method. Concerning the modes above the light line, the method has been tested with results in the literature [Ochiai, T., et al.

A (2001)] and, indirectly, with the scattering-matrix method [Andreani, L. C. (2002)]. Also for this case the agreement is very good.

The implementation of this method is not much different from the plane-wave expansion method.

The basis set is chosen to consist of the guided modes of an effective planar waveguide, where each layer j has the homogenous dielectric function given by the spatial average of ²_j(x) within the unit cell; i.e. the diagonal elements of the dielectric matrix ²_j,G,G, see Eq. (1.26). Therefore, in Eq. (1.20), the plane waves are replaced by the guided modes, and the magnetic field is

H_k(r) = X

G∈G

X

α

c_α(k+G)ˆf_α(z)e^ı(k+G)·x, (1.29)

whereˆf_α(z)e^ı(k+G)·x represents a guided mode with wave-vectork+G;α is a discrete index that labels the guided mode,k is the Bloch vector andGare reciprocal vectors of the two-dimensional lattice. Likewise for the plane-wave expansion method, the G vectors are limited by a cut-off K and the master equation is transformed into a linear eigenvalue problem

X

G⁰,α⁰

H^α,α_G,G⁰0c_α⁰(k+G⁰) = ω²

c²c_α(k+G), (1.30)

where the “hamiltonian” matrix is given by H^α,α_G,G⁰0 =

The matrix elements H^α,α_G,G⁰0 of Eq. (1.31) can be calculated by noting that the dxintegral in each layer j yields the matrix Fourier transform [[²⁻¹_j ]] of the inverse dielectric function. This is the same quantity that appears in the two-dimensional case and is computed using the inverse method [[²⁻¹_j ]]⇒[[η_j]].

Besides a wave-vector cut-off K, it is also convenient to specify a maximum number of guided modes of the effective waveguide. 109 plane waves are found to give stable photonic bands up to ωa/2πc ∼0.7 in the whole range of hole radii from zero to the close-packing condition r = 0.5a.

Figure 1.10 Left (right) panel: empty-lattice (photonic) bands for a self-standing dielectric membrane, patterned with a 1D lattice with air filling ratio f = 30%. The width of the waveguide is d/a = 0.5 and the core dielectric function is ² = 12. The dashed lines delimit the guided-mode region (white area) and refer to the dispersion of light in air and in the effective core material (² = 8.7). Light gray is for the leaky mode region, dark gray is where no solutions can exist.

For symmetric waveguides, keeping four guided modes in each parity sector gives also very good convergence, except when the waveguide is so thick that several higher-order modes are required.