**3.2.1 Photonic crystal slow-light waveguides**

The band diagram of a photonic crystal waveguide on a two dimensional photonic crystal is shown in Fig.9. The periodicity has been broken in the *y* direction, due to the introduction of the linear defect, and thence only *kx* is conserved. Therefore, the modes are not represented anymore over the irreducible Brillouin zone. Their projection on *kx* is depicted instead. One can create a photonic crystal waveguide by simply removing a single row of holes. Such a waveguide, normally referred as W1, results in the appearance of a number of defect modes within the photonic band gap. As monomode behaviour is desirable, narrowing the waveguide is normally required. By reducing the width of W1 by a factor of 0.7, one gets a single defect mode within the band gap, the red-coloured band in Fig.9. Lateral confinement of the defect mode within the waveguide is ensured by the photonic bandgap, as the electric field density simulation in Fig.10 proves. Vertical confinement of light within the slab is achieved by total internal reflection at the interface between the high-dielectric constant slab and the surrounding air. The blue region in Fig. 9 represents the so called light cone, i.e. those extended modes propagating in air and not confined within the slab.

Fig. 9. Band diagram of a photonic crystal waveguide on a high-dielectric constant slab.

Unmodified photonic crystal structures, as the ones in the previous section, present a series of applications originating from the existence of a photonic band gap, as wavelength selective mirrors or stop-band filters. However, to fully exploit photonic crystal capacities for modelling electromagnetic propagation, one has to resort to the creation of defects within the otherwise perfectly periodic structure. Punctual and linear defects within the photonic crystal allow for tight confinement of light, at band gap frequencies, inside cavities

The band diagram of a photonic crystal waveguide on a two dimensional photonic crystal is shown in Fig.9. The periodicity has been broken in the *y* direction, due to the introduction of the linear defect, and thence only *kx* is conserved. Therefore, the modes are not represented anymore over the irreducible Brillouin zone. Their projection on *kx* is depicted instead. One can create a photonic crystal waveguide by simply removing a single row of holes. Such a waveguide, normally referred as W1, results in the appearance of a number of defect modes within the photonic band gap. As monomode behaviour is desirable, narrowing the waveguide is normally required. By reducing the width of W1 by a factor of 0.7, one gets a single defect mode within the band gap, the red-coloured band in Fig.9. Lateral confinement of the defect mode within the waveguide is ensured by the photonic bandgap, as the electric field density simulation in Fig.10 proves. Vertical confinement of light within the slab is achieved by total internal reflection at the interface between the high-dielectric constant slab and the surrounding air. The blue region in Fig. 9 represents the so called light cone, i.e.

those extended modes propagating in air and not confined within the slab.

Fig. 9. Band diagram of a photonic crystal waveguide on a high-dielectric constant slab.

**3.2 Slow light photonic crystal waveguides and cavities** 

**3.2.1 Photonic crystal slow-light waveguides** 

and waveguides.

Fig. 10. Electric field energy density confined within the photonic crystal waveguide.

Group velocity is given by Eq. 1, i.e. by the slope of the modes appearing in the dispersion diagram. Computed group velocity for light pulses coupled to the defect mode is depicted in Fig. 11. The top graphic represents *vg* as a function of the longitudinal wave vector *kx* for the guided mode. It is noticeable that the group velocity is zero at the edges of the band and for certain wavelengths. Unfortunately, these working regions are not desirable in practice. At the band edges, any fluctuation in the structure, due to fabrication imperfections, causes oscillations between guided and not guided states. Moreover, the operational bandwidth for this ultra-low velocity is very small due to group velocity dispersion and higher-order dispersion. Special designs have been proposed to minimize dispersion and enable higher bandwidths (Baba, 2008; O'Faolain, 2009). The bottom graphic at Fig. 11 shows guided mode *vg* as a function of wavelength. By setting the lattice period *a* equal to 500nm the band gap is located around the third communications window. Since we are interested in guided modes and not in modes extended in air, we must only consider those frequencies out of the Light cone. These correspond to wavelengths superior to 1.4 μm.

Our group is working on optimized waveguide designs trying to achieve a balance between reduced group velocity and bandwidth. The waveguide depicted in Fig.12 has been created by diminishing the radii of a row of holes and filling it with a material of ε=7. Furthermore, to achieve monomode behaviour the waveguide width has been reduced by a factor of 0.64 (Andonegui et al., 2011). Group velocities of c/100 are achieved for a 33% of the k-vector space, achieving a good balance between low information velocity and bandwidth.

#### **3.2.2 Photonic crystal cavities**

The group velocity-bandwidth trade-off of photonic crystal waveguides can also be addressed by coupling a series of punctual defects (cavities) within the photonic band gap material. High quality factor (Q) photonic crystal cavities are capable to store photons for a relative long time in an extremely small volume. A high-Q photonic crystal cavity is the basic block of the so called coupled-resonator optical waveguides (CROWs). Remarkable achievements have been done in this field, as in (Notomi et al., 2008), where more than 100 high-Q cavities were coupled, achieving *vg* of c/170 in pulse propagation experiments and notable storage capacity.

Photonic Band Gap Engineered Materials for Controlling the Group Velocity of Light 67

light at certain resonant frequency can be coupled from the waveguide to the cavity. This light will be coupled back from the waveguide to the cavity after a time, tstorage, proportional to Q.

Fig. 13. Band diagram of a photonic crystal cavity on a high-dielectric constant slab.

Fig. 14. Amplitude of the Hy component of the optical field confined within the cavity.

Fast and fine device tunability is a requirement for many applications of slow light, e.g. for optical buffer memories. Fast reconfiguration of the photonic crystal device can be achieved by a variety of effects: by using thermo-optical effect, electro-optic effect, or carrier injection among others. Along this section, we show how photonic crystal waveguides and cavities

Lithium niobate (LiNbO3) is an anisotropic crystalline material, i.e. its refractive index depends on the crystal axis direction. Consequently its response to the electro-optic effect is given by a matrix of coefficients giving the electro-optic response for each direction. LiNbO3 index response to an applied field along the z axis, depicted in Fig.16, is given by the expression

**3.2.3 Tunability of photonic crystal slow light structures** 

can be fast and efficiently tuned by exploiting Pockels effect.

Fig. 11. Group velocity of light pulses coupled to the defect mode confined within the waveguide as a function of wave vector *kx* (top) and as a function of wavelength (bottom)

Fig. 12. Optimized photonic crystal waveguide on a triangular lattice of air holes on silicon with d1=0.51a, d2=0.76a, W=0.64W (left) and group velocity of the defect mode (right)

By introducing a punctual defect the periodicity in both *x* and *y* direction is broken and therefore k vector is not conserved in any direction. Consequently, the band structure of a cavity is naïve and it is not usually represented. Nevertheless, it is very instructive to visualize the shape of the defect mode confined in the cavity, as in Fig. 13. Notice that the defect mode is parallel to the *k*-axis, giving useful information: a single resonant frequency remains confined into the cavity with zero group velocity, as illustrated in Fig.14.

A more sophisticated cavity consisting on a missing hole and a gradual change of surrounding holes radii is shown in the design of Fig.15. The cavity is adjacent to a waveguide so that the

Fig. 11. Group velocity of light pulses coupled to the defect mode confined within the waveguide as a function of wave vector *kx* (top) and as a function of wavelength (bottom)

Fig. 12. Optimized photonic crystal waveguide on a triangular lattice of air holes on silicon with d1=0.51a, d2=0.76a, W=0.64W (left) and group velocity of the defect mode (right)

By introducing a punctual defect the periodicity in both *x* and *y* direction is broken and therefore k vector is not conserved in any direction. Consequently, the band structure of a cavity is naïve and it is not usually represented. Nevertheless, it is very instructive to visualize the shape of the defect mode confined in the cavity, as in Fig. 13. Notice that the defect mode is parallel to the *k*-axis, giving useful information: a single resonant frequency

A more sophisticated cavity consisting on a missing hole and a gradual change of surrounding holes radii is shown in the design of Fig.15. The cavity is adjacent to a waveguide so that the

remains confined into the cavity with zero group velocity, as illustrated in Fig.14.

light at certain resonant frequency can be coupled from the waveguide to the cavity. This light will be coupled back from the waveguide to the cavity after a time, tstorage, proportional to Q.

Fig. 13. Band diagram of a photonic crystal cavity on a high-dielectric constant slab.

Fig. 14. Amplitude of the Hy component of the optical field confined within the cavity.
