**2. State-of-the-art in multi-channel laser generation techniques**

A phase-locked operation of multi-channel waveguide devices supporting propagation of lateral modes (also known as supermodes) was studied mostly in 80's. The main goal of theoretical and experimental research was to achieve higher power density of the coherent laser beams generated in semiconductors. Phased array lasers, consisting of *N* single-mode waveguides, can guide, in total, *N* array modes. In practice, the most likely excited mode is of the highest order (Yariv, 1997). Consequently, relatively broad far-field patterns as well as broad spectral linewidths are obtained. To solve or at least alleviate that disadvantageous property, it is essential to distinguish appropriate supermodes.

Fig.3 shows near-field patterns of five supermodes supported by index-guided arrays consisting of five identical and non-identical channels. The supermode patterns were calculated with a numerical solver of Maxwell's equations (Kapon et al., 1984a). In particular, the excitation of a fundamental supermode results in a single-lobe radiation beam aligned with the array channels. However, as it has been shown in Fig.3a, in uniform arrays with identical channels, intensity patterns of the fundamental and the highest order supermodes are similar to each other, so their discrimination becomes difficult. Moreover, as it has been shown in (Kapon et al., 1984a), since inter-channel regions are usually lossy, the highest order supermode, with a two-lobe far field pattern, is often favoured over the other modes. Subsequently, variation in the channels' width (known as chirped arrays)

 Fig. 2. A perspective view of dielectric membranes with square (left) and triangular (right)

**2. State-of-the-art in multi-channel laser generation techniques** 

property, it is essential to distinguish appropriate supermodes.

In the next Section, a brief overview of the developments of mutli-channel laser generation techniques is given, especially in the context of so-called supermode multi-channel

A phase-locked operation of multi-channel waveguide devices supporting propagation of lateral modes (also known as supermodes) was studied mostly in 80's. The main goal of theoretical and experimental research was to achieve higher power density of the coherent laser beams generated in semiconductors. Phased array lasers, consisting of *N* single-mode waveguides, can guide, in total, *N* array modes. In practice, the most likely excited mode is of the highest order (Yariv, 1997). Consequently, relatively broad far-field patterns as well as broad spectral linewidths are obtained. To solve or at least alleviate that disadvantageous

Fig.3 shows near-field patterns of five supermodes supported by index-guided arrays consisting of five identical and non-identical channels. The supermode patterns were calculated with a numerical solver of Maxwell's equations (Kapon et al., 1984a). In particular, the excitation of a fundamental supermode results in a single-lobe radiation beam aligned with the array channels. However, as it has been shown in Fig.3a, in uniform arrays with identical channels, intensity patterns of the fundamental and the highest order supermodes are similar to each other, so their discrimination becomes difficult. Moreover, as it has been shown in (Kapon et al., 1984a), since inter-channel regions are usually lossy, the highest order supermode, with a two-lobe far field pattern, is often favoured over the other modes. Subsequently, variation in the channels' width (known as chirped arrays)

PhC air-hole lattices.

propagation.

Fig. 1. The definition of square (left) and triangular (right) air-hole lattices.

results in significantly different near field envelope patterns of the fundamental and higherorder supermodes, in contrast to the case of uniform arrays (Kapon et al., 1984a). In such arrays, higher order supermodes can be suppressed by employing a proper gain distribution.

Fig. 3. Near-field patterns of the supermodes in a five-element a) uniform array b) inverted-V chirped array, c) linearly chirped array (Kapon et al., 1984a).

Next, in an inverted-V chirped array (Fig.3b), the power of the fundamental supermode is concentrated in central channels, whereas the higher order supermodes are more localized in the outermost channels. Since gain in the active region is larger when the laser channels are wider, the fundamental supermode is expected to have a higher modal gain (near threshold) and, in consequence, is more likely to oscillate (Kapon et al., 1984b).

In (Kapon et al., 1986), a buried ridge array has been proposed. In such arrays, a small refractive index contrast between the channels and inter-channel regions is applied. That soft index profile ensures effective coupling between the adjacent array laser channels via their evanescent optical fields. Since the channels in those arrays are defined by a built-in distribution of the refractive index, it is possible to achieve a uniform gain distribution across the array, while maintaining the channel definition. Such an approach makes the buried ridge arrays different from the gain guided arrays, in which inter-channel regions are inherently more lossy. Moreover, buried ridge arrays operate mainly with the fundamental supermode, thus, producing a single-lobe radiation beam.

On the Applicability of Photonic Crystal Membranes to Multi-Channel Propagation 101

tools applied. First, dispersive properties of an infinitely large PhC membrane with no defects are investigated to exemplify general rules for the photonic bandgap (PBG) generation as a function of PhC membrane geometry and incident light wavelength. Once a PBG dispersion diagram is achieved, a defect channel is processed in the PhC membrane and dispersive properties of such an optical waveguide are considered. For the purpose of this Chapter, propagation of transverse–electric (TE) modes in defect PhC membrane channels based on the square lattice type is studied only. However, the introduced methodology may be easily extended to other lattice types with either TE or TM polarisation. The obtained PBG diagrams will help detecting the supermodes within a photonic bandgap. Eventually, electric field patterns of those modes are computed to assess their applicability to the laser beam generation. As it is shown in Section 4.3, those field distributions are useful to calculate laser characteristics of the single- and multi-channel

Two common lattice types processed in a photonic crystal membrane are investigated, namely square and triangular (see Fig.4). The lattices are cut with air holes in an indium gallium arsenide phosphide (InGaAsP) layer with a refractive index of *n* = 3.4. At this stage, the goal is to specify design rules for the photonic bandgap generation as a function of the most critical parameters of those structures, that is, a membrane's thickness *d*, a lattice

Numerical computations are performed using a full-wave electromagnetic approach with a finite-difference time-domain (FDTD) method implemented in a QuickWave-3D simulator (Taflove & Hagness, 2005; QWED). Since the structure is periodic in two dimensions, the computation with FDTD is enhanced with the Floquet's theorem (Collin, 1960), also known as the Bloch's one, which allows us to reduce a computational domain to a single period of the lattice (Salski, 2010), as exemplified in Fig.4. Considering periodicity along the *z*-axis, the following periodic boundary conditions (PBCs), derived from the Floquet's theorem, are

,, , ,,, *<sup>j</sup> E xyz Lt E xyzt e*

,,, ,, , *<sup>j</sup> H xyzt H xyz Lt e*

where *L* is the period of the structure along the *z*-axis, denotes the components transverse

As it has been shown in (Celuch-Marcysiak & Gwarek, 1995; Salski, 2010), incorporation of the Floquet's theorem into FDTD schemes results in a complex notation of time-domain electromagnetic fields with the real and imaginary FDTD grids computed simultaneously at the same structure's mesh and coupled via PBCs in each iteration cycle. The method is known as Complex-Looped FDTD (CL-FDTD) and is implemented in the QuickWave-3D simulator (QWED). Additionally, due to conformal meshing implemented in QuickWave-3D (Gwarek, 1985), curvature of the air holes, as shown in Fig.4, is accurately represented on

(1)

(2)

is a fundamental Floquet phase shift

photonic crystal membrane lasers.

constant *a* and an air holes' radius *r*.

enforced at periodic faces of the structure:

to periodicity (in this case *x*- and *y*- components), and

per period *L* understood as a user-defined parameter.

**3.1 Bandgaps in photonic crystal membranes** 

The first analytical interpretation of supermodes behaviour in the phased array lasers was proposed in (Scifres et al., 1979). Experimental data was interpreted by considering a diffraction pattern of a structure with equally-spaced slits corresponding to individual laser array elements. Such an approach is usually known as a simple diffraction theory. Although the simple diffraction theory has been proved useful to interpret some experimental results (Scifres et al., 1979; AcMey & Engelmann, 1981; van der Ziel et al., 1984), it provides no means to describe the allowed oscillating modes in the array of coupled emitters.

In the early 70's, an alternative method, known as a coupled mode theory, was intensively investigated (Yariv, 1973; Yariv & Taylor, 1981; Kogelnik, 1979). It has been successfully applied to the modelling and analysis of various guided-wave optoelectronic and fibre optical devices, such as optical directional couplers (Taylor, 1973; Kogelnik & Schmidt, 1976), optical fibres (Digonnet & Shaw, 1982; Zhang & Garmire, 1987), phase-locked laser arrays (Kapon et al., 1984c; Mukai et al., 1984; Hardy et al., 1988), distributed feedback lasers (Kogelnik & Shank, 1972) and distributed Bragg reflectors (Schmidt et al., 1974).

One of major assumptions made in the conventional coupled mode theory is that the modes of uncoupled systems are orthogonal to each other. In coupled systems, however, one often chooses the modes of isolated systems as the basis for the mode expansion and these modes may not be orthogonal. Therefore, the orthogonal coupled mode theory (OCMT) is not suitable for the description of the mode-coupling process in that case. Non-orthogonality of modes in optical couplers, due to crosstalk between the waveguide modes, was first recognized in (Chen & Wang, 1984). Later on, several formulations of the non-orthogonal coupled mode theory (NCMT) were developed by several authors (Hardy & Streifer, 1985; Chuang, 1987a; Chuang, 1987b; Chuang, 1987c). It has been shown that NCMT yields more accurate dispersion characteristics and field patterns for the modes in the coupled waveguides. Better accuracy is even more essential to the modelling of coupling between non-identical waveguides. It is evident for weak coupling, though the new formulation extends the applicability of the coupled mode theory to geometries with more strongly coupled waveguides. However, NCMT becomes inaccurate when considering very strongly coupled waveguide modes (Hardy & Streifer, 1985).

To the best of authors' knowledge, edge-emitting multi-channel membrane lasers have not been manufactured so far, although single-channel membrane lasers processed on a GaAs photonic crystal membrane were already presented (Yang, et al., 2005; Yang, et al., 2007; Lu, et al. 2009). One of the major reasons lies in technological challenges in achieving acceptable repeatability of the photonic crystal structure manufacturing process (Massaro, et al., 2008). However, with the advent of new technology nodes those challenges will likely be overcome or at least substantially alleviated, opening a wide range of applications to the methodology addressed below.

In this Chapter, a complete design cycle of a new type of phased array laser structures processed in photonic crystal membranes is presented. Due to a very strong coupling between the adjacent channels in the array, a non-orthogonal coupled mode theory was applied in order to maintain the rigidity of the analysis.
