**3.2 Bandgaps in defect channels processed in photonic crystal membranes**

Spectral properties of TE modes propagating in defect channels, as exemplified in Fig.11, processed in PhC membranes are investigated below. PBG diagrams are computed with the aid of the QuickWave-3D electromagnetic FDTD simulator (QWED), in the same way as in the case of non-defect PhC membranes as shown in Fig.4. This time, however, an FDTD model consists of a single PhC row, as marked with a red dashed line in Fig.11. Since it is assumed that the waveguide is infinitely long, the Floquet's periodic boundary conditions are enforced only along the channel's axis, while lateral dimensions are truncated with the absorbing boundary conditions (Mei, 1992). PBG diagrams for the square lattice channels are computed for phase shifts within a range designated by and *X* critical points of the first irreducible Brillouin zone of the corresponding non-defect PhC membranes.

Fig.12 depicts the modes computed for a single square 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). Black curves indicate the propagating modes with one distinguished by a green colour, while the red curves depict the modes of the non-defect PhC membrane surrounding the channel. It can be seen that a single defect mode is achieved (green) within a photonic bandgap (red semi-transparent zone) of the surrounding PhC. The mode has a uniquely defined phase constant within the *a/* = 0.356… 0.369 spectrum range, that is 3.6% wide, although a light cone additionally limits the allowed spectrum range to *a/* = 0.356… 0.365 (2.5% wide). Since single-mode propagation is achievable in that spectral range, it may be useful to design edge-emitting lasers based on 2D photonic crystal membranes.

Fig. 11. The definition of square single- (left) and dual-channel (right) air-hole lattices with regions chosen for an FDTD simulation (see red dashed line).

A single simulation of a model consisting of 220 320 FDTD cells (ca. 26MB RAM) takes 140 seconds on Intel Core i7 CPU 950 with the speed of 220 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. Thus, calculation of the whole PBG diagram with the step of /10 takes about 19 x 140 sec = 45 minutes.


Spectral properties of TE modes propagating in defect channels, as exemplified in Fig.11, processed in PhC membranes are investigated below. PBG diagrams are computed with the aid of the QuickWave-3D electromagnetic FDTD simulator (QWED), in the same way as in the case of non-defect PhC membranes as shown in Fig.4. This time, however, an FDTD model consists of a single PhC row, as marked with a red dashed line in Fig.11. Since it is assumed that the waveguide is infinitely long, the Floquet's periodic boundary conditions are enforced only along the channel's axis, while lateral dimensions are truncated with the absorbing boundary conditions (Mei, 1992). PBG diagrams for the square lattice channels are computed for phase shifts within a range designated by

*X* critical points of the first irreducible Brillouin zone of the corresponding non-defect

Fig.12 depicts the modes computed for a single square 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). Black curves indicate the propagating modes with one distinguished by a green colour, while the red curves depict the modes of the non-defect PhC membrane surrounding the channel. It can be seen that a single defect mode is achieved (green) within a photonic bandgap (red semi-transparent zone) of the surrounding PhC. The mode has a uniquely defined phase constant within the

= 0.356… 0.369 spectrum range, that is 3.6% wide, although a light cone additionally

propagation is achievable in that spectral range, it may be useful to design edge-emitting

 Fig. 11. The definition of square single- (left) and dual-channel (right) air-hole lattices with

A single simulation of a model consisting of 220 320 FDTD cells (ca. 26MB RAM) takes 140 seconds on Intel Core i7 CPU 950 with the speed of 220 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

= 0.356… 0.365 (2.5% wide). Since single-mode

/10 takes about 19

and

the membrane is thick enough (Joannopoulos et al., 2008).

PhC membranes.

x 140 sec = 45 minutes.

limits the allowed spectrum range to *a/*

lasers based on 2D photonic crystal membranes.

regions chosen for an FDTD simulation (see red dashed line).

membrane. Thus, calculation of the whole PBG diagram with the step of

*a/*

**3.2 Bandgaps in defect channels processed in photonic crystal membranes**

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 shown with a blue semi-transparent colour.

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 light cone to the *a/* = 0.337… 0.342 spectrum range (1.5% wide). Comparing the results 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 photonic crystal membrane channels are given.

#### **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

On the Applicability of Photonic Crystal Membranes to Multi-Channel Propagation 109

(right) for the single-channel (left) and dual-channel (right) waveguides, the PBG diagrams

For instance, Fig.16 shows the distribution of an instantaneous electric field vector in the

within the channel near the hole and has its minimum in the middle between the rows, where the longitudinal electric component dominates over the transverse one. Similarly, Fig.17 presents the distribution of an instantaneous electric field vector in the dual-channel

channels oscillate in-phase or not. A thorough look onto the picture reveals that vectors in the adjacent channels have the same direction prompting the conclusion that both modes creating the supermode are in-phase polarised. Thus, the gain of a far-field radiation pattern

Instantaneous electric field distributions like those shown in Fig.16 are, afterwards, integrated in time and in a whole volume of a single row of a channel. As it is shown in the subsequent Section, those envelopes are used to compute gain characteristics of phased

Taking advantage of the already computed PBG diagrams of the photonic crystal membranes with one and two waveguide channels (see Section 3), the phase constant of the supermodes may be easily determined. That knowledge is essential to build an equivalent effective waveguide model, which enables an approximate analytical representation of a field distribution of the guided modes in passive structures (Lesniewska-Matys, 2011). As it can be seen in Fig.17, in the proposed model, a photonic crystal waveguide is replaced with a two dimensional planar one with the same membrane's thickness but the channel's width

**4. Supermode laser generation in photonic crystal membranes** 

= 0.321. This time, it is crucial to determine whether the fields in both

= 0.356. It can be seen that the field is mostly concentrated

= 0.356 (left) and for *a/*

= 0.321

Fig. 16. Vector views of an instantaneous electric field for *a/*

of which are shown in Fig.12 and Fig.14, respectively.

single-channel waveguide for *a/*

increases leading to higher laser beam intensity.

array lasers based on photonic crystal membranes.

**4.1 The model of an effective planar waveguide**

adjusted so as to obtain the same phase constant.

waveguide for *a/*

Fig. 13. A TE mode photonic bandgap diagram for a dual-channel in an air-hole square lattice cut in an InGaAsP membrane (*n* = 3.4, *r/a* = 0.4, *d/a* = 0.6, *b/a* = 0.3). The light cone is shown with a blue semi-transparent colour.

Fig. 14. A TE mode photonic bandgap diagram for a dual-channel in an air-hole square lattice cut in an InGaAsP membrane (*n* = 3.4, *r/a* = 0.4, *d/a* = 0.6, *b/a* = 0.4). The light cone is shown with a blue semi-transparent colour.

is excited using a mode template generation technique (Celuch-Marcysiak et al., 1996). The end of the waveguide on the right is truncated with a perfectly matched layer (PML) (Berenger, 1994) to avoid any reflections that would disturb the travelling wave.

Fig. 15. The view of an FDTD model of a photonic crystal waveguide.

Fig. 13. A TE mode photonic bandgap diagram for a dual-channel in an air-hole square lattice cut in an InGaAsP membrane (*n* = 3.4, *r/a* = 0.4, *d/a* = 0.6, *b/a* = 0.3). The light cone is

Fig. 14. A TE mode photonic bandgap diagram for a dual-channel in an air-hole square lattice cut in an InGaAsP membrane (*n* = 3.4, *r/a* = 0.4, *d/a* = 0.6, *b/a* = 0.4). The light cone is

(Berenger, 1994) to avoid any reflections that would disturb the travelling wave.

Fig. 15. The view of an FDTD model of a photonic crystal waveguide.

is excited using a mode template generation technique (Celuch-Marcysiak et al., 1996). The end of the waveguide on the right is truncated with a perfectly matched layer (PML)

shown with a blue semi-transparent colour.

shown with a blue semi-transparent colour.

Fig. 16. Vector views of an instantaneous electric field for *a/* = 0.356 (left) and for *a/* = 0.321 (right) for the single-channel (left) and dual-channel (right) waveguides, the PBG diagrams of which are shown in Fig.12 and Fig.14, respectively.

For instance, Fig.16 shows the distribution of an instantaneous electric field vector in the single-channel waveguide for *a/* = 0.356. It can be seen that the field is mostly concentrated within the channel near the hole and has its minimum in the middle between the rows, where the longitudinal electric component dominates over the transverse one. Similarly, Fig.17 presents the distribution of an instantaneous electric field vector in the dual-channel waveguide for *a/* = 0.321. This time, it is crucial to determine whether the fields in both channels oscillate in-phase or not. A thorough look onto the picture reveals that vectors in the adjacent channels have the same direction prompting the conclusion that both modes creating the supermode are in-phase polarised. Thus, the gain of a far-field radiation pattern increases leading to higher laser beam intensity.

Instantaneous electric field distributions like those shown in Fig.16 are, afterwards, integrated in time and in a whole volume of a single row of a channel. As it is shown in the subsequent Section, those envelopes are used to compute gain characteristics of phased array lasers based on photonic crystal membranes.
