**3. Photonic band gap engineered materials for slow light applications**

Dielectric materials having periodic dielectric constant present singular properties for electromagnetic waves propagation. The most remarkable of these properties is the appearance of the so called photonic band gap, a range of frequencies unable to propagate through the material. Many profitable effects stem from the existence of photonic band gaps: from the possibility of tight light confinement to the opportunity of achieving extremely low group velocities.

Photonic crystals are periodic, artificial, dielectric materials, which under certain conditions, present a photonic band gap. Biological photonic crystals are found in nature: in butterflies' wings, in peacocks' feathers, in comb-jellyfishes... The characteristics of photonics crystals confer these species their peculiar and outstanding colouration. Engineered or artificial photonic crystals can be designed with tailored electromagnetic and propagation properties, giving rise to a huge range of possibilities and practical applications. The next points give a flavour on the principles of design of photonic crystal devices with tailored group velocity.

#### **3.1 Principles of photonic crystals**

A photonic band gap engineered material, or a photonic crystal possessing a band gap, can be built from the periodic arrangement of different dielectric media in one, two or three dimensions. It has to be noticed that, even under the appropriate conditions of index contrast, lattice period and geometry, the photonic band gap will only appear in the plane of periodicity. Therefore, only a three-dimensional photonic crystal will be able to localize light in three dimensions by means of the photonic band gap, while light modes in one- or twodimensional arrangements will only be localized in one or two dimensions respectively. Examples of one-, two-, and three-dimensional photonic crystals are shown in Fig. 6.

To understand how light propagates within a photonic crystal one has to resort to the macroscopic Maxwell equations and specialise to the particular case of mixed dielectric media in the absence of free currents or charges and in which the structure does not vary with time. This development has already been done in several excellent books and we refer the reader to (Joannopoulus et al., 2008) or (Sakoda, 2005), just to cite two of them, for a comprehensive understanding. In the following, the main results of this development,

Fig. 5. Illustration of an optical biosensor based on a slow light photonic crystal waveguide.

Dielectric materials having periodic dielectric constant present singular properties for electromagnetic waves propagation. The most remarkable of these properties is the appearance of the so called photonic band gap, a range of frequencies unable to propagate through the material. Many profitable effects stem from the existence of photonic band gaps: from the possibility of tight light confinement to the opportunity of achieving extremely low

Photonic crystals are periodic, artificial, dielectric materials, which under certain conditions, present a photonic band gap. Biological photonic crystals are found in nature: in butterflies' wings, in peacocks' feathers, in comb-jellyfishes... The characteristics of photonics crystals confer these species their peculiar and outstanding colouration. Engineered or artificial photonic crystals can be designed with tailored electromagnetic and propagation properties, giving rise to a huge range of possibilities and practical applications. The next points give a flavour on the principles of design of photonic crystal

A photonic band gap engineered material, or a photonic crystal possessing a band gap, can be built from the periodic arrangement of different dielectric media in one, two or three dimensions. It has to be noticed that, even under the appropriate conditions of index contrast, lattice period and geometry, the photonic band gap will only appear in the plane of periodicity. Therefore, only a three-dimensional photonic crystal will be able to localize light in three dimensions by means of the photonic band gap, while light modes in one- or twodimensional arrangements will only be localized in one or two dimensions respectively.

Examples of one-, two-, and three-dimensional photonic crystals are shown in Fig. 6.

To understand how light propagates within a photonic crystal one has to resort to the macroscopic Maxwell equations and specialise to the particular case of mixed dielectric media in the absence of free currents or charges and in which the structure does not vary with time. This development has already been done in several excellent books and we refer the reader to (Joannopoulus et al., 2008) or (Sakoda, 2005), just to cite two of them, for a comprehensive understanding. In the following, the main results of this development,

**3. Photonic band gap engineered materials for slow light applications** 

group velocities.

devices with tailored group velocity.

**3.1 Principles of photonic crystals** 

Fig. 6. Examples of one-, two- and three-dimensional photonic crystals.

indispensable to understand the basic principles of photonic crystals, are summarized. The resultant master equation is the following:

$$\nabla \times \left(\frac{1}{\varepsilon(r)} \nabla \times H(r)\right) = \left(\frac{\alpha}{c}\right)^2 H(r) \tag{4}$$

Given a known spatial dielectric constant arrangement, *ε(r)*, one can find the magnetic field spatial profile of the modes allowed by the structure, *H(r)*, and their corresponding frequencies, ω, by solving Eq. 4 subject to the following transversality requirements:

$$\nabla \cdot H(r) = 0\tag{5}$$

$$\nabla \cdot \left( \varepsilon(r) E(r) \right) = 0 \tag{6}$$

Subsequently one can obtain the electric field by using the following expression:

Photonic Band Gap Engineered Materials for Controlling the Group Velocity of Light 63

Modes propagating parallel to the periodicity plane have kz=0 and posess mirror symmetry through that plane. This allows us to classify modes in transverse electric (TE) and transverse magnetic (TM), having the electric field (E) in plane and the magnetic field (H)

In Fig.8 the computed TE band structure for a two-dimensional photonic crystal made of air holes on silicon substrate is represented, accompanied by its transmission diagram. It is noticeable that transmission is zero at the photonic band gap frequencies. This type of arrangement, under certain conditions on index contrast and holes radii, presents a photonic band gap for TE and TM polarizations. Nonetheless, typically only TE polarized light is used since TE band gap is larger and therefore confinement will be

Several concepts must be clarified to be able to interpret this diagram correctly. First, notice that the wave vector in the plane of periodicity, *k*||, is indexed in four points (Γ, M, K, Γ), representing the limits of the irreducible Brillouin zone for this particular structure. A complete description of Brillouin zone is given in the Appendix B of (Joannopoulus et al., 2008). Here we will just mention that the Brillouin zone is a primitive cell in the reciprocal lattice, the Fourier transform of the spatial function of the original lattice. The irreducible Brillouin zone is the first Brillouin zone reduced by all symmetries in the point group of the lattice, i.e. all periodicity- and symmetry-imposed redundancies in *k* are avoided. Secondly, the frequency is given in dimensionless units ωa/2πc, or equivalently a/λ. By normalizing all magnitudes, the frequency and the holes radii in this case, by the lattice period *a* it is possible to explode the inherent scalability of photonic crystals. If the given structure had a=1μm, the middle of the band gap ωm=0.3 would be at a wavelength λ=1/0.3= 3.33 μm. If of one wanted to set the middle of the band gap at the third window of communications, λ=1550nm, it would suffice with setting

Fig. 8. Band structure and transmission diagram of a 2D photonic crystal made of air holes on silicon. The lattice period is *a* , the dielectric constant is ε=11,9716 and the hole radii r=0.36a

polarized in z and vice versa.

stronger.

a=0,31550=465nm.

$$E(r) = \frac{i}{a\varkappa\_0 \varepsilon(r)} \nabla \times H(r) \tag{7}$$

The solutions to the master equation *H(r)* can be written as Bloch states, i.e. as a product of a plane wave and a periodic function, with a period equal to the lattice period of the photonic crystal:

$$H\_k(r) = e^{ikr} u\_k(r) \tag{8}$$

All information about the spatial profile is given by the wave vector, *k,* and the periodic function, uk(r). For a given k, an infinite family of modes with discretely spaced frequencies ωn(k) satisfy the master equation in Eq. 4 in particular these functions ωn(k), represented in the *band structure* diagram, provide us with most of the valuable information about the optical properties of a given photonic crystal.

From this moment on, we will restrict our attention to two-dimensional structures, mainly because the state of the art fabrication techniques do not allow for reliable and costaffordable realization of three-dimensional photonic crystals. Additionally, 2D structures are suitable for photonics on-chip integration. 3D light confinement can be obtained in 2D photonic crystals by using the photonic band gap effect to confine light laterally and total internal reflection for vertical confinement.

An ideal 2D photonic crystal is homogeneous in *z* (real 2D photonic crystal devices will be patterned in slabs or membranes to ensure vertical confinement). Modes will oscillate in that direction with its wave vector *kz* unrestricted. We can particularize Eq. 8 for this case, expressing the Bloch modes of a two-dimensional photonic crystal structure as a function of the mode index, *n*, the in-plane and off-plane wave vectors, *k||* and *kz*, and the projection of *r* in the *xy* plane, *ρ*:

$$H\_{\{\mathbf{n},k\_z,\mathbf{k}\_{\parallel\perp}\}}(r) = e^{i\mathbf{\bar{k}}\_{\parallel}\cdot\rho} e^{i\mathbf{\bar{k}}\_x r} \mathbf{u}\_{\{\mathbf{n},k\_z,\mathbf{k}\_{\parallel\perp}\}}(\rho) \tag{9}$$

Fig. 7. Modes propagating parallel to the plane of periodicity xy (kz=0), can be classified in transverse electric (TE) and transverse magnetic (TM) in two dimensional photonic crystals

(7)

() () *ikr Hr eur k k* (8)

0 ( ) ( ) ( ) *<sup>i</sup> E r H r r*

crystal:

optical properties of a given photonic crystal.

internal reflection for vertical confinement.

in the *xy* plane, *ρ*:

The solutions to the master equation *H(r)* can be written as Bloch states, i.e. as a product of a plane wave and a periodic function, with a period equal to the lattice period of the photonic

All information about the spatial profile is given by the wave vector, *k,* and the periodic function, uk(r). For a given k, an infinite family of modes with discretely spaced frequencies ωn(k) satisfy the master equation in Eq. 4 in particular these functions ωn(k), represented in the *band structure* diagram, provide us with most of the valuable information about the

From this moment on, we will restrict our attention to two-dimensional structures, mainly because the state of the art fabrication techniques do not allow for reliable and costaffordable realization of three-dimensional photonic crystals. Additionally, 2D structures are suitable for photonics on-chip integration. 3D light confinement can be obtained in 2D photonic crystals by using the photonic band gap effect to confine light laterally and total

An ideal 2D photonic crystal is homogeneous in *z* (real 2D photonic crystal devices will be patterned in slabs or membranes to ensure vertical confinement). Modes will oscillate in that direction with its wave vector *kz* unrestricted. We can particularize Eq. 8 for this case, expressing the Bloch modes of a two-dimensional photonic crystal structure as a function of the mode index, *n*, the in-plane and off-plane wave vectors, *k||* and *kz*, and the projection of *r*

> || z || || k k ( , ,k ) ( , ,k ) ( ) ( ) *z z <sup>i</sup> i r H re eu n k n k*

> > **k||**

**kz**

Fig. 7. Modes propagating parallel to the plane of periodicity xy (kz=0), can be classified in transverse electric (TE) and transverse magnetic (TM) in two dimensional photonic crystals

(9)

Modes propagating parallel to the periodicity plane have kz=0 and posess mirror symmetry through that plane. This allows us to classify modes in transverse electric (TE) and transverse magnetic (TM), having the electric field (E) in plane and the magnetic field (H) polarized in z and vice versa.

In Fig.8 the computed TE band structure for a two-dimensional photonic crystal made of air holes on silicon substrate is represented, accompanied by its transmission diagram. It is noticeable that transmission is zero at the photonic band gap frequencies. This type of arrangement, under certain conditions on index contrast and holes radii, presents a photonic band gap for TE and TM polarizations. Nonetheless, typically only TE polarized light is used since TE band gap is larger and therefore confinement will be stronger.

Several concepts must be clarified to be able to interpret this diagram correctly. First, notice that the wave vector in the plane of periodicity, *k*||, is indexed in four points (Γ, M, K, Γ), representing the limits of the irreducible Brillouin zone for this particular structure. A complete description of Brillouin zone is given in the Appendix B of (Joannopoulus et al., 2008). Here we will just mention that the Brillouin zone is a primitive cell in the reciprocal lattice, the Fourier transform of the spatial function of the original lattice. The irreducible Brillouin zone is the first Brillouin zone reduced by all symmetries in the point group of the lattice, i.e. all periodicity- and symmetry-imposed redundancies in *k* are avoided. Secondly, the frequency is given in dimensionless units ωa/2πc, or equivalently a/λ. By normalizing all magnitudes, the frequency and the holes radii in this case, by the lattice period *a* it is possible to explode the inherent scalability of photonic crystals. If the given structure had a=1μm, the middle of the band gap ωm=0.3 would be at a wavelength λ=1/0.3= 3.33 μm. If of one wanted to set the middle of the band gap at the third window of communications, λ=1550nm, it would suffice with setting a=0,31550=465nm.

Fig. 8. Band structure and transmission diagram of a 2D photonic crystal made of air holes on silicon. The lattice period is *a* , the dielectric constant is ε=11,9716 and the hole radii r=0.36a

Photonic Band Gap Engineered Materials for Controlling the Group Velocity of Light 65

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

cone. These correspond to wavelengths superior to 1.4 μm.

**3.2.2 Photonic crystal cavities** 

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

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

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.

space, achieving a good balance between low information velocity and bandwidth.

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

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 and waveguides.
