Gap maps, diffraction losses, and exciton–polaritons in photonic crystal slabs

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Abstract

A theory of photonic crystal (PhC) slabs is described, which relies on an expansion in the basis of guided modes of an effective homogeneous waveguide and on treating the coupling to radiative modes and the resulting losses by perturbation theory. The following applications are discussed for the case of a high-index membrane: gap maps for photonic lattices in a waveguide; exciton–polariton states, when the PhC slab contains a quantum well with an excitonic resonance; propagation losses of line-defect modes in W1 waveguides, also in the presence of disorder; the quality factors of photonic nanocavities. In particular, we predict that disorder-induced losses below 0.2 dB/mm can be achieved in state-of-the-art samples by increasing the channel width of W1 waveguides.

Introduction

Photonic crystals embedded in planar waveguides, also known as photonic crystal (PhC) slabs, can lead to a full control of light propagation because of the two-dimensional (2D) photonic lattice in the slab plane combined with dielectric confinement in the vertical direction [1], [2]. Electromagnetic eigenmodes in PhC slabs can be either truly guided (if their frequency lies below the light line of the cladding material) or quasi-guided (if the frequency lies above the light line). Quasi-guided modes are subject to intrinsic radiative losses because of diffraction out of the slab plane. Truly guided modes are only subject to extrinsic losses due to disorder. Diffraction losses represent a crucial problem for prospective applications of PhC slabs to integrated photonic devices.

In this work, we present a theory of photonic modes and of radiation-matter interaction in PhC slabs that relies on an expansion of the magnetic field on the basis of guided modes of an effective homogeneous waveguide. Coupling to radiative modes of the effective waveguide is taken into account by perturbation theory and leads to a determination of diffraction losses. The theory allows calculation of the following quantities: photonic band dispersion and gap maps for 1D and 2D lattices embedded in a waveguide; the intrinsic losses of quasi-guided modes, due to the non-separable form of the dielectric modulation; the extrinsic losses of truly guided modes, by means of a Gaussian model of disorder; exciton–polariton states, when the PhC slab contains a quantum well with an excitonic resonance. Throughout this paper we consider the self-standing membrane (air bridge) as a prototype of a PhC slab with a strong refractive index contrast. In particular, we calculate propagation losses of line defect modes in W1 waveguides in the presence of structural disorder (modelled as random variations of hole radii in the triangular lattice) and Q-factors of photonic nanocavities.

Section snippets

Method

In order to formulate the method, we start from the second-order equation for the magnetic field, ∇×1ɛ(r)∇×H=ω2c2H,where ɛ(r) is the spatially-dependent dielectric constant. If the magnetic field is expanded in an orthonormal set of basis states as H(r)=∑μcμHμ(r), then Eq. (1) is transformed into a linear eigenvalue problem, ∑νĤμνcν=ω2c2cμ,where the matrix Ĥμν (which is the analog of a quantum Hamiltonian for an electronic problem) is given by Ĥμν=∫1ɛ(r)(∇×Hμ*(r))⋅(∇×Hν(r))dr.For the case of a

Photonic bands and gap maps

A notable feature of the present method is that it leads to a determination of the photonic mode dispersion both below and above the light line. This allows calculation of gap maps for photonic lattices embedded in a waveguide. As an example, in Fig. 1 we show the gap map of a 1D lattice of air stripes in a dielectric slab with ɛ=12 and core thickness d/a=0.4. Transverse electric (TE) and transverse magnetic (TM) polarizations are defined with respect to a vertical plane perpendicular to the

Exciton–polaritons in photonic crystal slabs

Exciton–polaritons are the mixed modes of the electromagnetic field and an excitonic resonance, and are a well established concept in solid state physics [8]. Polaritonic effects in PhC slabs infiltrated with organic materials have been already shown experimentally[9] and studied theoretically[10], [11] on a classical basis.

Here we describe a quantum theory of exciton polaritons in semiconductor-based PhC slabs. The method described in Section 2 allows calculation of the coefficients of the

Propagation losses and effect of disorder in W1 waveguides

We consider a line defect consisting of a missing row of holes along the ΓK direction of the triangular lattice: this is called a W1 waveguide. The structure is shown in Fig. 4a (inset). The channel width w equals w0≡3a, if the positions of the surroundings holes are those of the triangular lattice, but waveguides with reduced or increased channel widths have also been realized [15], [16]. The W1 waveguide supports defect modes in the photonic gap, which opens only for states even with respect

Photonic cavities

Point defects in PhC slabs behave as 0D cavities and support localized modes in the photonic gap. Cavity modes are always subject to intrinsic losses, as they have no wave vector and are coupled to the continuum of leaky slab modes by the dielectric modulation. Still, photonic cavities with large quality factor Q and small mode volumes can be defined. The quality factor can be increased by a momentum-space design, which allows reducing the radiative component of the confined photonic mode [22],

Conclusions

The approach to PhC slabs consisting on an expansion in the basis of guided modes and on treating coupling to leaky modes by perturbation theory proves to be an efficient method for calculating photonic bands, gap maps, and diffraction losses for modes below and above the light cone. This approach is especially suited for PhC slabs with strong refractive index contrast. Line and point defects are treated by introducing a supercell, like in the usual plane-wave calculations. The formulation

Acknowledgments

This work was supported by MIUR through the Cofin program and by INFM through PRA PHOTONIC.

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