Large-bandwidth planar photonic crystal waveguides

Abstract

A general design principle is presented for making finite-height photonic crystal waveguides that support leakage-free guidance of light over large frequency intervals. The large bandwidth waveguides are designed by introducing line defects in photonic crystal slabs, where the material in the line defect has appropriate dispersion properties relative to the photonic crystal slab material surrounding the line defect. A three-dimensional theoretical analysis is given for large-bandwidth waveguide designs based on a silicon-air photonic crystal slab suspended in air. In one example, the leakage-free single-mode guidance is found for a large frequency interval covering 60% of the photonic bandgap.

Introduction

The discovery of photonic bandgap materials or photonic crystals [1], [2] has opened new possibilities for making novel optical components. These materials are periodic dielectric structures with a periodicity being comparable to the wavelength of light. Photonic crystals can be designed to exhibit frequency intervals, photonic bandgaps, where light cannot propagate in the material for certain directions and polarizations. By making a structure being periodic in three dimensions, it is even possible to inhibit propagation of light within certain frequency ranges for any propagation direction and any polarization.

Photonic crystals become particularly useful when properly designed defects are introduced in the periodic structure, since this allows making e.g. novel cavities and waveguides where light is confined to a point-like defect or a line defect by the photonic bandgap of the crystal structure surrounding the defect [3], [4]. A two-dimensional analysis of photonic crystal waveguides (PCWs), where the design is based on introducing a line defect in a photonic crystal material, has previously been given by a number of authors [5], [6], [7], [8], [9], [10]. The dispersion properties of these waveguides may depend strongly on design parameters such as the size of air-holes in a dielectric material, the width of a waveguide and the type of waveguide. It is important to understand how PCWs can be designed to have desired dispersion properties. The aim in this paper is to develop the ideas and theory necessary for making PCWs that support leakage-free bandgap guidance of light over large frequency intervals. A two-dimensional analysis is sufficient for understanding some of the physics of planar PCWs. Even the properties of finite-height waveguides can be addressed when two-dimensional calculations are compared with dispersion relations for the media above and below the finite-height waveguides [8]. The two-dimensional approach does not, however, allow the calculation of exact dispersion properties for finite-height structures. The effect of the height of the structure on bandgaps and guided modes is difficult to address by two-dimensional calculations. In this paper, we will provide a full three-dimensional analysis of large-bandwidth planar PCWs. The three-dimensional analysis allows the calculation of the dispersion properties of a type of realistic finite-height PCW which is within reach with todays fabrication technology. Previous three-dimensional analysis of PCWs that did not focus on obtaining a large bandwidth can be found in [11], [12], [13], [14], [15], [16], [17].

Obtaining leakage-free guidance of light in planar PCWs requires first of all a high vertical index contrast between a photonic crystal slab and the material above and below the slab. The highest vertical index contrast is achieved when a photonic crystal slab is suspended in air. This is the approach that will give the largest frequency range with leakage-free bandgap guidance of light in a planar PCW. In this paper, we focus on the case of a photonic crystal slab suspended in air. Photonic crystal slabs suspended in air have been investigated experimentally both for making lasers [18] and for making waveguides [15], [19], [20], [21], [22].

A sufficient vertical index contrast can also be achieved by using the semiconductor-on-insulator materials system [13], [16], [17], [23], [24], [25]. Due to the smaller vertical index contrast the photonic bandgap region in which leakage-free bandgap guidance of light is possible is smaller compared to the case of a photonic crystal slab suspended in air. On the other hand the waveguide design is more robust.

Researchers also explore the possibility of making waveguides with a relatively low vertical index contrast [26], [27], [28], [29]. In that case leakage-free bandgap guidance of light is not possible, but the losses can be low. Experimentally losses as low as 11 and 20 dB/mm have been reported [27], [28]. In this paper, we will be concerned with waveguides having a high vertical index contrast and focus on leakage-free guidance of light.

The motivation for this work can be understood from the features of a typical banddiagram for finite height PCWs based on holes in a high-index material. An example of such a diagram is shown in Fig. 1. All banddiagrams presented in this paper have been calculated by plane–wave-expansion theory and a variational principle [30]. First, we will explain what is seen in the banddiagram and then explain the physics that can be read off the diagram for the waveguide in concern in terms of frequency ranges with guidance and group velocity of guided modes. The waveguide considered is shown as an inset, and consists of a photonic crystal structure with air-holes arranged periodically on a triangular lattice in a silicon slab. The diameter of the air-holes is D=0.7Λ, where Λ is the photonic crystal lattice constant. This choice of hole-diameter gives a large photonic bandgap [9]. By removing one row of air-holes, a line defect or waveguide has been formed. The height of the slab is h=0.76Λ, and we assume that the slab is suspended in air. Due to the periodic nature of the waveguide the solutions for the electromagnetic fields can be expanded in Bloch modes on the form

$$\text{E}{}_{\mathrm{<mtext>k</mtext>}}\text{(}\text{r}\text{)=}\text{U}{}_{\mathrm{<mtext>k</mtext>}}\text{(}\text{r}\text{)}\text{exp}\text{(}\text{i}\text{kx),}$$

where $\text{U}{}_{\mathrm{<mtext>k</mtext>}}\text{(}\text{r}\text{)}$ is a function which is periodic with the same periodicity as the PCW, k is the Bloch wave number, and x is the position coordinate along the direction of the straight waveguide. The banddiagram in Fig. 1 shows the allowed combinations of normalized frequencies Λ/λ and normalized Bloch wave numbers kΛ/2π, where λ is the free space wavelength. Because the structure is vertically symmetric, it is possible to divide the Bloch mode solutions into two classes of modes that are sometimes referred to as even modes and odd modes [13]. In the banddiagrams in this paper, we consider only the even modes where a complete photonic bandgap exists in the slab modes of the considered photonic crystal slabs.

The continuum regions in Fig. 1 correspond to those Bloch modes that are allowed in the photonic crystal slab surrounding the waveguide and to the modes that are allowed in free space above and below the waveguide. These are not guided modes, and only these modes would be present if a linedefect had not been introduced. The discrete bands, however, correspond to electromagnetic modes that are localized to the region of the linedefect. These modes are leakage-free guided modes. Notice the gap in (Λ/λ,kΛ/2π)-space in the continuum of modes above the frequency Λ/λ=0.26. In this gap, we find a number of discrete dispersion curves corresponding to photonic bandgap guided modes. These bands only cover rather limited frequency intervals and they are flat. The slope of the bands is proportional to the group velocity or energy velocity [31] of the guided modes, and thereby the flatness of the bands can be interpreted as a very low group velocity. The features concerning the flat bands and the narrow bandwidth of the guided modes that we observe in Fig. 1 are common to PCWs with high-index linedefects. For some applications, e.g. filtering, it may be convenient to have guidance over only narrow frequency intervals. A narrow bandwidth can be achieved to an even larger extent by using coupled cavity PCWs [32], [33], [34], [35]. The aim of this paper, however, is to design PCWs that support leakage-free guidance of light over large frequency intervals, and preferably single-mode guidance is desired over a large part of the gap.

Previously, waveguides based on removing a row of dielectric pillars along a line in a periodic structure of dielectric pillars have also been studied using two-dimensional calculations [7], [31], [36]. In a two-dimensional analysis where the pillars are assumed infinitely high, and the light is assumed to propagate in the plane of the crystal, a large bandwidth can be obtained. In that case, however, a three-dimensional analysis shows that the waveguides tend to be leaky waveguides. This has been a major reason for instead considering designs such as the one shown in Fig. 1 because the high refractive index contrast between the waveguide slab and the media above and below the slab (in this case free space) is sufficient for providing a guidance mechanism vertically (vertically light is confined by total internal reflection). But as it has become clear, these waveguides have very different dispersion properties. The designs that will be presented in this paper posses both the vertical confinement of light and the large bandwidth.

In Section 2 a general design principle is given for making large bandwidth PCWs, and an example is given that illustrates the theoretical idea. In Section 3, we consider a photonic crystal slab similar to the one in Fig. 1 where the line defect will consist of one or more rows of air-holes with various sizes. These waveguide designs are within reach with todays technology.