**3. Numerical modelling of optical channels in photonic crystal membranes**

A complete electromagnetic design cycle of single- and multi-channel optical propagation in PhC membranes is presented in this Section, together with the computational methods and 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 photonic crystal membrane lasers.

#### **3.1 Bandgaps in photonic crystal membranes**

100 Photonic Crystals – Innovative Systems, Lasers and Waveguides

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

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

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

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

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

**3. Numerical modelling of optical channels in photonic crystal membranes**  A complete electromagnetic design cycle of single- and multi-channel optical propagation in PhC membranes is presented in this Section, together with the computational methods and

means to describe the allowed oscillating modes in the array of coupled emitters.

(Kogelnik & Shank, 1972) and distributed Bragg reflectors (Schmidt et al., 1974).

coupled waveguide modes (Hardy & Streifer, 1985).

applied in order to maintain the rigidity of the analysis.

methodology addressed below.

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

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 enforced at periodic faces of the structure:

$$
\vec{E}\_{\perp} \left( \mathbf{x}\_{\prime} y\_{\prime} z + \mathbf{L}\_{\prime} t \right) = \vec{E}\_{\perp} \left( \mathbf{x}\_{\prime} y\_{\prime} z\_{\prime} t \right) e^{j\chi} \tag{1}
$$

$$
\vec{H}\_{\perp} \left( \mathbf{x}, \mathbf{y}, z, t \right) = \vec{H}\_{\perp} \left( \mathbf{x}, \mathbf{y}, z + L\_{\prime} t \right) e^{-j\chi t} \tag{2}
$$

where *L* is the period of the structure along the *z*-axis, denotes the components transverse to periodicity (in this case *x*- and *y*- components), and is a fundamental Floquet phase shift per period *L* understood as a user-defined parameter.

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

On the Applicability of Photonic Crystal Membranes to Multi-Channel Propagation 103

Fig. 6. A TE mode photonic bandgap diagram for an air-hole square lattice cut in

Owing to the wideband spectral properties of the CL-FDTD method, a single simulation provides information about all the modal frequencies within the spectrum of our interest satisfying the imposed Floquet's phase shifts. Thus, the simulator has to be invoked as many times as the number of wave vector points chosen to collect a PBG diagram along a whole contour of an irreducible Brillouin zone (Salski, 2010). In this case, a single simulation of the model consisting of 18 400 FDTD cells (ca. 4MB RAM) takes 11 seconds on Intel Core i7 CPU 950 with the speed of 1785 iter/sec. An FDTD cell size is set to *a*/20, leading to at least 40 FDTD cells per wavelength in free space and ca. 12 in the membrane. Calculation of the whole PBG diagram shown in Fig.6 with 55 wave vector points takes, in total, ca. 55 x 11

In the case of an air-hole square lattice cut in an InGaAsP membrane, the PBG diagram of which is shown in Fig.6, an 8.8% wide indirect *X*-*M* TE bandgap for the normalised

bandgap is not present in that spectrum range, what may be considered as a potential disadvantage in applications when the precise control of beam propagation is necessary. It can be solved using a triangular lattice, where both TE and TM bandgaps may coincide within the same spectrum range. However, this issue extends beyond the scope of the

Consider now the impact of geometrical settings on a TE bandgap in the investigated square PhC membrane. Fig.7 shows the modal dispersion as a function of the membrane's thickness *d/a*. It can be seen that, the TE bandgap decreases with the increasing membrane's thickness *d/a*, while covering the spectrum width of 8.8%, 9.2% and 10.4% for *d/a* = 0.4, 0.5 and 0.6, respectively. It shows that the membrane's thickness *d/a* has a relatively minor impact on the photonic bandgap width. Next, Fig.8 presents the computation results for a variable airholes' radius *r/a,* and it is evident that the radius, in contrast to the membrane's thickness, has a substantial impact on the bandgap width, which reaches zero below ca. *r/a* = 0.25. Later in this paper, those PhC structures are applied for single- and multi-channel

= 0.348 … 0.380 is found. Although it is not exemplified in Fig.6, a TM

an InGaAsP membrane (*n* = 3.4, *r/a* = 0.4, *d/a* = 0.4).

sec = 10 minutes.

Chapter and is not considered here.

propagation of optical pulses.

frequency *a/*

the FDTD mesh with no deteriorating effect on memory storage and computing time. A vertical cross-section of a unit membrane lattice cell sketched in Fig.4 (right) indicates that the structure is situated in air which, in order to reduce the computational volume, is truncated with absorbing boundary conditions, usually known as Mur superabsorption (Mei et al., 1992).

Fig. 4. A horizontal cross-section view of FDTD models of a unit cell of square (left) and triangular (centre) air-hole lattices and a vertical cross-section (right) with absorbing boundary conditions truncating the air regions below and above the membrane.

In each simulation run, for a particular set of Floquet's phase shifts per *y*- and *z*- periods, a point excitation located somewhere inside a unit cell is driven with a wideband pulse (e.g. a Dirac's delta), injecting energy into the structure. As the simulation continues, the Fourier transform is iteratively calculated until a convergent state is achieved. Fig.5 shows an exemplary spectrum of an injected electric current for the lattice shown in Fig.4 (left). Resonances indicate the eigenvalues (frequencies) of the detected modes satisfying Floquet's phase shifts imposed by periodic boundary conditions.

Fig. 5. The spectrum of an electric current injected into a unit cell of a square PhC membrane as shown in Fig.4 (left) with the imposed Floquet's phase shifts along *y*- and *z*-axis *y* = *<sup>z</sup>* = 0 radians (*n* = 3.4, *r/a* = 0.4, *d/a* = 0.4).

the FDTD mesh with no deteriorating effect on memory storage and computing time. A vertical cross-section of a unit membrane lattice cell sketched in Fig.4 (right) indicates that the structure is situated in air which, in order to reduce the computational volume, is truncated with absorbing boundary conditions, usually known as Mur superabsorption

phase shifts imposed by periodic boundary conditions.

*<sup>z</sup>* = 0 radians (*n* = 3.4, *r/a* = 0.4, *d/a* = 0.4).

Fig. 4. A horizontal cross-section view of FDTD models of a unit cell of square (left) and triangular (centre) air-hole lattices and a vertical cross-section (right) with absorbing boundary conditions truncating the air regions below and above the membrane.

In each simulation run, for a particular set of Floquet's phase shifts per *y*- and *z*- periods, a point excitation located somewhere inside a unit cell is driven with a wideband pulse (e.g. a Dirac's delta), injecting energy into the structure. As the simulation continues, the Fourier transform is iteratively calculated until a convergent state is achieved. Fig.5 shows an exemplary spectrum of an injected electric current for the lattice shown in Fig.4 (left). Resonances indicate the eigenvalues (frequencies) of the detected modes satisfying Floquet's

Fig. 5. The spectrum of an electric current injected into a unit cell of a square PhC membrane

as shown in Fig.4 (left) with the imposed Floquet's phase shifts along *y*- and *z*-axis

(Mei et al., 1992).

*y* = 

Fig. 6. A TE mode photonic bandgap diagram for an air-hole square lattice cut in an InGaAsP membrane (*n* = 3.4, *r/a* = 0.4, *d/a* = 0.4).

Owing to the wideband spectral properties of the CL-FDTD method, a single simulation provides information about all the modal frequencies within the spectrum of our interest satisfying the imposed Floquet's phase shifts. Thus, the simulator has to be invoked as many times as the number of wave vector points chosen to collect a PBG diagram along a whole contour of an irreducible Brillouin zone (Salski, 2010). In this case, a single simulation of the model consisting of 18 400 FDTD cells (ca. 4MB RAM) takes 11 seconds on Intel Core i7 CPU 950 with the speed of 1785 iter/sec. An FDTD cell size is set to *a*/20, leading to at least 40 FDTD cells per wavelength in free space and ca. 12 in the membrane. Calculation of the whole PBG diagram shown in Fig.6 with 55 wave vector points takes, in total, ca. 55 x 11 sec = 10 minutes.

In the case of an air-hole square lattice cut in an InGaAsP membrane, the PBG diagram of which is shown in Fig.6, an 8.8% wide indirect *X*-*M* TE bandgap for the normalised frequency *a/* = 0.348 … 0.380 is found. Although it is not exemplified in Fig.6, a TM bandgap is not present in that spectrum range, what may be considered as a potential disadvantage in applications when the precise control of beam propagation is necessary. It can be solved using a triangular lattice, where both TE and TM bandgaps may coincide within the same spectrum range. However, this issue extends beyond the scope of the Chapter and is not considered here.

Consider now the impact of geometrical settings on a TE bandgap in the investigated square PhC membrane. Fig.7 shows the modal dispersion as a function of the membrane's thickness *d/a*. It can be seen that, the TE bandgap decreases with the increasing membrane's thickness *d/a*, while covering the spectrum width of 8.8%, 9.2% and 10.4% for *d/a* = 0.4, 0.5 and 0.6, respectively. It shows that the membrane's thickness *d/a* has a relatively minor impact on the photonic bandgap width. Next, Fig.8 presents the computation results for a variable airholes' radius *r/a,* and it is evident that the radius, in contrast to the membrane's thickness, has a substantial impact on the bandgap width, which reaches zero below ca. *r/a* = 0.25. Later in this paper, those PhC structures are applied for single- and multi-channel propagation of optical pulses.

On the Applicability of Photonic Crystal Membranes to Multi-Channel Propagation 105

Fig. 9. TE mode photonic bandgap diagrams for air-hole triangular lattices cut in an InGaAsP membrane in function of a membrane's thickness *d/a* (*n* = 3.4, *r/a* = 0.35).

Fig. 10. TE mode photonic bandgap diagrams for air-hole triangular lattices cut in an InGaAsP membrane in function of an air-holes' radius *r/a* (*n* = 3.4, *d/a* = 0.4).

dielectric membranes exhibit the following properties:

lattices,

used,

radius *r/a* for both square and triangular air-hole lattices,

Concluding, it may be noticed from the investigation given in this Section and from the literature as well, that two-dimensional air-hole photonic crystals processed in thin




Fig. 7. TE mode photonic bandgap diagrams for air-hole square lattices cut in an InGaAsP membrane in function of a membrane's thickness *d/a* (*n* = 3.4, *r/a* = 0.4).

Fig. 8. TE mode photonic bandgap diagrams for air-hole square lattices cut in an InGaAsP membrane in function of an air-holes' radius *r/a* (*n* = 3.4, *d/a* = 0.4).

Similar computations were carried out for a membrane with a triangular air-hole lattice. First of all, as it may be inferred from Fig.9, a direct TE bandgap is achieved at a critical point. Secondly, the achieved TE bandgap spectra are much wider when compared to their counterparts computed for the square lattice. Fig.9 shows that the spectrum width amounts to 40.7%, 42.6% and 43.8% for *d/a* = 0.4, 0.5 and 0.6, respectively. Next, Fig.10 depicts the impact of the air-holes' radius *r/a* on the TE bandgap spectrum width, which amounts to 16.6%, 40.7% and 36.9% for *r/a* = 0.3, 0.35 and 0.4, respectively. The simulations show that if a narrow bandgap is favourable in a considered application, the square lattice is a better option, while the triangular one allows creation of a wider bandgap.

Fig. 7. TE mode photonic bandgap diagrams for air-hole square lattices cut in an InGaAsP

Fig. 8. TE mode photonic bandgap diagrams for air-hole square lattices cut in an InGaAsP

Similar computations were carried out for a membrane with a triangular air-hole lattice.

point. Secondly, the achieved TE bandgap spectra are much wider when compared to their counterparts computed for the square lattice. Fig.9 shows that the spectrum width amounts to 40.7%, 42.6% and 43.8% for *d/a* = 0.4, 0.5 and 0.6, respectively. Next, Fig.10 depicts the impact of the air-holes' radius *r/a* on the TE bandgap spectrum width, which amounts to 16.6%, 40.7% and 36.9% for *r/a* = 0.3, 0.35 and 0.4, respectively. The simulations show that if a narrow bandgap is favourable in a considered application, the square lattice is a better

critical

First of all, as it may be inferred from Fig.9, a direct TE bandgap is achieved at a

membrane in function of an air-holes' radius *r/a* (*n* = 3.4, *d/a* = 0.4).

option, while the triangular one allows creation of a wider bandgap.

membrane in function of a membrane's thickness *d/a* (*n* = 3.4, *r/a* = 0.4).

Fig. 9. TE mode photonic bandgap diagrams for air-hole triangular lattices cut in an InGaAsP membrane in function of a membrane's thickness *d/a* (*n* = 3.4, *r/a* = 0.35).

Fig. 10. TE mode photonic bandgap diagrams for air-hole triangular lattices cut in an InGaAsP membrane in function of an air-holes' radius *r/a* (*n* = 3.4, *d/a* = 0.4).

Concluding, it may be noticed from the investigation given in this Section and from the literature as well, that two-dimensional air-hole photonic crystals processed in thin dielectric membranes exhibit the following properties:


On the Applicability of Photonic Crystal Membranes to Multi-Channel Propagation 107

Fig. 12. A TE mode photonic bandgap diagram for a single-channel in an air-hole square lattice cut in an InGaAsP membrane (*n* = 3.4, *r/a* = 0.4, *d/a* = 0.4, *b/a* = 0.3). The light cone is

Fig.13 shows a PBG diagram for TE polarisation computed for a square dual-channel (*n* = 3.4, *r/a* = 0.4, *d/a* = 0.6, *b/a* = 0.3). Two supermodes are distinguished with blue and green colours. However, only the 1st order supermode (green) has a uniquely defined phase constant within a photonic bandgap (red semi-transparent zone), additionally reduced by a

shown in Fig.14 with those in Fig.13, it can be seen that an increase in the channel's width *b/a* from 0.3 to 0.4 results in a decrease of the supermode's frequency. Most of the 1st order supermode's unique phase constant range shown in Fig.14 is within the photonic bandgap (red semi-transparent zone). Unfortunately, the light cone limits the choice to the *a/*

0.318… 0.323 spectrum range (1.5% wide). In this case, however, the allowed spectrum is more distant from the 2nd order supermode (blue), reducing the risk of its unintended oscillation. On the other hand, excitation of the modes in the photonic crystal surrounding the channel is more likely to happen. Concluding, it can be seen that an appropriate adjustment of the light cone, photonic bandgap and channel's width gives a lot of possibilities to modify the allowed supermode's spectrum range (Lesniewska-Matys, 2011). In the next Section, electric field distributions of a few exemplary supermodes obtained in

**3.3 Electromagnetic field distribution in photonic crystal membrane channels**

The calculation of laser characteristics of above-threshold generation in the considered PhC membrane channels requires quantitative knowledge of a field distribution of an undisturbed travelling wave propagating along the channel at one of selected modes (see Section 4.3). Therefore, envelopes of electric field components within a unit row of the photonic crystal waveguides have to be computed. For that purpose, an FDTD computational model as shown in Fig.15 is used to generate a travelling wave in the channel(s), which may be then integrated in time to obtain the envelopes. The photonic crystal is equipped on the left with an additional input section, where an appropriate mode

= 0.337… 0.342 spectrum range (1.5% wide). Comparing the results

=

shown with a blue semi-transparent colour.

photonic crystal membrane channels are given.

light cone to the *a/*

