**1. Introduction**

12 Will-be-set-by-IN-TECH

52 Advanced Photonic Sciences

Wieczorek, W., Krischek, R., Kiesel, N., Michelberger, P., Tóth, G. & Weinfurter, H. (2009).

Yuan, Z., Chen, Y., Zhao, B., Chen, S., Schmiedmayer, J. & Pan, J. (2008). Experimental demonstration of a bdcz quantum repeater node, *Nature* 454(7208): 1098–1101. Zhang, Q., Goebel, A., Wagenknecht, C., Chen, Y., Zhao, B., Yang, T., Mair, A., Schmiedmayer,

Zukowski, M., Zeilinger, A., Horne, M. A. & Ekert, A. K. (1993). "Event-ready-detectors" Bell experiment via entanglement swapping, *Phys. Rev. Lett.* 71(26): 4287–4290.

*Letters* 103(2): 020504.

system, *Nature Physics* 2(10): 678–682.

Experimental entanglement of a six-photon symmetric dicke state, *Physical Review*

J. & Pan, J. (2006). Experimental quantum teleportation of a two-qubit composite

The power of creating engineered materials in which one can customize the propagation of light unleashes a huge range of fascinating applications. Commonly, these materials are based on the appearance of a forbidden frequency range, after which they are named photonic band gap materials or, more generally, photonic crystals. Photonic crystals are artificial, periodic arrangements of dielectric media with sub-wavelength periodicity lengths. As a consequence of the periodic dielectric function, in analogy with natural crystals where atoms or molecules are periodically spaced, a range of forbidden propagating energies may appear under the proper conditions, giving rise to a photonic band gap. Exploiting the effects associated to the photonic band gap, one can dramatically alter the flow of electro-magnetic waves (Joannopoulos et al., 2008).

One of the optical properties that can be radically modified in such materials is the group velocity of light pulses. Usually, the unmatched velocity of light (in vacuum c=299.792.458 m/s) is an advantage. Thanks to this fact, light confined within the core of an optical fiber could give a complete round to the equatorial circumference of Earth in 200 milliseconds (light in glass travels at c/1.5). In other cases, where processing the information carried by light pulses is needed, the huge speed of light entails a problem. In quantum optics, for instance, the quantum state of light can be stored for longer if light is slowed down. As another example, buffering and storing information on a photonic chip is crucial in all-optical communications routers and optical computing, the same as electronic RAMs are essential in conventional networks and computers. Furthermore, countless applications arise from the increase of energy density, and consequent stronger light-matter interaction, due to slower light propagation. Higher energy density implies enhanced nonlinearities, and therefore more efficient, compact, and low-power consuming nonlinear optical schemes, as Raman amplification or optical regenerators (Krauss, 2008). Stronger light-matter interaction is also beneficial for sensing applications, especially for biosening, and more efficient photovoltaic cells.

Unluckily, light is inherently difficult to control, not to mention how hard it is to store it. This is not only due to its speed, but also to the very nature of its basic unit, the photon, being massless and having no electric charge. A number of ingenious and diverse schemes to diminish the speed of light in a controllable fashion has been proposed, giving rise to the intriguing field of slow light. All slow light schemes are based in the same basic principle, in

Photonic Band Gap Engineered Materials for Controlling the Group Velocity of Light 55

It can be seen that for media in which *ω* varies linearly with *k* the phase and the group velocity coincide. However, in general, for dispersive media, phase and group velocities

There is one last concept that it is worth to mention, the front velocity. The front velocity is the speed of the pulse leading edge, i.e. the velocity of the earliest part of a pulse. This is truly the velocity of the signal, i.e. of the information carried by the light wave. According to the principle of Einstein causality, information can never travel faster than light in vacuum and therefore the signal velocity it is always slower than *c*. This concept has led to discrepancies among the scientific community in the past, when some groups claimed measured superluminal *signal* velocity (Heitmann & Nimtz, 1994; Steinberg et al., 1993), menacing the validity of the special theory of relativity. In those experiments, the reported measured velocity corresponded to that of the peak of the photon's wave and that of the envelope of the signal, but not to the signal velocity itself. This is explained by the reshaping of the pulse along its propagation, so that its peak moves towards its leading edge. The velocity of the peak may be, in this case, faster than the velocity of light in vacuum, but the signal (front) velocity will remain bellow c. In other words, even if the group velocity

Fig. 1. Representation of the points of measurement for the group, phase and front velocities

Coming back to the point of slow light, it has been defined as the phenomenon of light propagation at reduced group velocity. Bearing in mind Eq. 1, it is straightforward to see

The refractive index of the material *n* varies with frequency and can be slightly altered by making use of a plurality of phenomena, such as electro-optic or thermo-optic effects. Nonetheless, when aiming at a significant change in the group index, it is the second term in Eq. 3, dn/d*ω*, the one that dominates, i.e. the dispersive behaviour of the media. This is the

By making use of different approaches, that will be briefly reviewed in next section, extremely low group velocities have been achieved, as low as 17 ms-1 (Hau et al., 1999).

*dn*

(3)

*d* 

that slow light relies on increasing the group index of the medium, which is given by

*g*

reason why sharp spectral resonances are always behind slow light techniques.

*n n*

differ and both vary with frequency.

appears to be higher than c, causality still holds.

spite of their interdisciplinary nature: the existence of a single sharp resonance or multiple resonances (Khurgin & Tucker, 2009). As it will be justified, photonic crystal devices are, to our understanding, the most promising approach for the development of practical photonic devices based on slow light.

This chapter begins by reviewing the motivation and principles of slow light, followed by a description and assessment of the different techniques to diminish the speed of light and the applications of this phenomenon. In a subsequent section the particularly promising approach of slow-light photonic crystal arrangements will be addressed. In that section, together with the fundamentals of photonic band gap materials, their simulation methods and fabrication techniques, simulations results and analysis on our designs of slow-light photonic crystal arrangements will be presented. The chapter will be concluded with an assessment of photonic band gap engineered materials as a technology for developing slowlight based devices for telecommunications and microwave photonics applications.

#### **2. Slow light: From theory to practice**

Being able of governing something as essentially free as light is captivating in itself. Along this section, the physical principles behind the drastic reduction of group velocity and the techniques to achieve it will be presented. However, what is more appealing about slow light is its significant number of applications. In practice, unfortunately, slow light entails severe difficulties and a noteworthy research effort is being done by many groups to cover the gap between theory and practical applications.

#### **2.1 Principles of slow light**

The term slow lights refers to the phenomenon of light propagation in media with reduced group velocity. It is thence convenient to throw some light on the concepts of group velocity, phase velocity, and front velocity. Fig. 1 serves as a reference for those concepts.

The group velocity is the speed at which the wave envelope, the overall shape of an amplitude varying wave, propagates. This is frequently considered to be the velocity of the information transported by light signals, which is correct for most cases. Nevertheless, as it will be discussed bellow, there are some exceptions to this statement. The group velocity of a wave in a certain medium corresponds to the velocity of light in vacuum, *c*, divided by the group index of the medium, *ng*, and it is given by the equation

$$
\sigma\_{\mathcal{S}} = \frac{c}{n\_{\mathcal{S}}} = \frac{dco}{dk} \tag{1}
$$

where *ω* is the angular frequency and *k* is the angular wave number. The variation *ω* with *k* is called the dispersion relation, and thence, the group velocity is given by the slope of the dispersion relation.

The phase velocity is the speed at which the phase of every particular frequency component of the wave travels and it is given by

$$w\_p = \frac{\alpha}{k} \tag{2}$$

spite of their interdisciplinary nature: the existence of a single sharp resonance or multiple resonances (Khurgin & Tucker, 2009). As it will be justified, photonic crystal devices are, to our understanding, the most promising approach for the development of practical photonic

This chapter begins by reviewing the motivation and principles of slow light, followed by a description and assessment of the different techniques to diminish the speed of light and the applications of this phenomenon. In a subsequent section the particularly promising approach of slow-light photonic crystal arrangements will be addressed. In that section, together with the fundamentals of photonic band gap materials, their simulation methods and fabrication techniques, simulations results and analysis on our designs of slow-light photonic crystal arrangements will be presented. The chapter will be concluded with an assessment of photonic band gap engineered materials as a technology for developing slow-

Being able of governing something as essentially free as light is captivating in itself. Along this section, the physical principles behind the drastic reduction of group velocity and the techniques to achieve it will be presented. However, what is more appealing about slow light is its significant number of applications. In practice, unfortunately, slow light entails severe difficulties and a noteworthy research effort is being done by many groups to cover

The term slow lights refers to the phenomenon of light propagation in media with reduced group velocity. It is thence convenient to throw some light on the concepts of group velocity, phase velocity, and front velocity. Fig. 1 serves as a reference for those concepts.

The group velocity is the speed at which the wave envelope, the overall shape of an amplitude varying wave, propagates. This is frequently considered to be the velocity of the information transported by light signals, which is correct for most cases. Nevertheless, as it will be discussed bellow, there are some exceptions to this statement. The group velocity of a wave in a certain medium corresponds to the velocity of light in vacuum, *c*, divided by the

> *g c d*

where *ω* is the angular frequency and *k* is the angular wave number. The variation *ω* with *k* is called the dispersion relation, and thence, the group velocity is given by the slope of the

The phase velocity is the speed at which the phase of every particular frequency component

*k* 

*p v*

*n dk* 

(1)

(2)

*g*

*v*

light based devices for telecommunications and microwave photonics applications.

devices based on slow light.

**2.1 Principles of slow light** 

dispersion relation.

of the wave travels and it is given by

**2. Slow light: From theory to practice** 

the gap between theory and practical applications.

group index of the medium, *ng*, and it is given by the equation

It can be seen that for media in which *ω* varies linearly with *k* the phase and the group velocity coincide. However, in general, for dispersive media, phase and group velocities differ and both vary with frequency.

There is one last concept that it is worth to mention, the front velocity. The front velocity is the speed of the pulse leading edge, i.e. the velocity of the earliest part of a pulse. This is truly the velocity of the signal, i.e. of the information carried by the light wave. According to the principle of Einstein causality, information can never travel faster than light in vacuum and therefore the signal velocity it is always slower than *c*. This concept has led to discrepancies among the scientific community in the past, when some groups claimed measured superluminal *signal* velocity (Heitmann & Nimtz, 1994; Steinberg et al., 1993), menacing the validity of the special theory of relativity. In those experiments, the reported measured velocity corresponded to that of the peak of the photon's wave and that of the envelope of the signal, but not to the signal velocity itself. This is explained by the reshaping of the pulse along its propagation, so that its peak moves towards its leading edge. The velocity of the peak may be, in this case, faster than the velocity of light in vacuum, but the signal (front) velocity will remain bellow c. In other words, even if the group velocity appears to be higher than c, causality still holds.

Fig. 1. Representation of the points of measurement for the group, phase and front velocities

Coming back to the point of slow light, it has been defined as the phenomenon of light propagation at reduced group velocity. Bearing in mind Eq. 1, it is straightforward to see that slow light relies on increasing the group index of the medium, which is given by

$$m\_{\mathcal{S}} = n + o \frac{dn}{d\alpha} \tag{3}$$

The refractive index of the material *n* varies with frequency and can be slightly altered by making use of a plurality of phenomena, such as electro-optic or thermo-optic effects. Nonetheless, when aiming at a significant change in the group index, it is the second term in Eq. 3, dn/d*ω*, the one that dominates, i.e. the dispersive behaviour of the media. This is the reason why sharp spectral resonances are always behind slow light techniques.

By making use of different approaches, that will be briefly reviewed in next section, extremely low group velocities have been achieved, as low as 17 ms-1 (Hau et al., 1999).

Photonic Band Gap Engineered Materials for Controlling the Group Velocity of Light 57

suited for realizing this kind of systems, as in electromagnetically induced transparency (EIT), and coherent population oscillations (CPO). EIT and CPO have been used to create narrow transparency windows in absorbing materials. These techniques, not only allow making controllable the degree of slowing (or speeding), but they also alleviate other important issues related to strong atomic resonances: excessive absorption and dispersion. The electromagnetically induced transparency is a quantum effect that permits the propagation of light through a medium otherwise opaque. In (Hau et al., 1999), the extremely low speed of light of 17 ms-1 was achieved by exploiting this technique. The biggest disadvantages of EIT arise because the method has a highly limited operation bandwidth owing to its narrow transparency window and to higher-order dispersion effects. Moreover EIT relies on delicate interference between two quantum amplitudes and thus the presence of collisions or any other dephasing effect can destroy the interference. On the face of it, EIT requires cryogenic temperatures and atomic media, preventing its

The quantum coherence technique of coherent population oscillations is studied as an alternative to EIT, due to its larger bandwidth and because it is highly insensitive to dephasing effects. The CPO method relies on creating a spectral hole due to population oscillations. Slow light propagation with a group velocity as low as 57.5m/s was observed employing CPO at room temperature in a ruby crystal (Bigelow et al., 2003). However,

More recently, stimulated Brillouin and Raman scattering have also been proposed as material-based methods for slowing light. Brillouin scattering arises from the interaction of light with propagating density waves or acoustic phonons. Raman scattering arises from the interaction of light with the vibrational or rotational modes of the molecules in the scattering medium. The main advantage of these techniques is that they make use of optical fibers, which is an unmatched medium in terms of low attenuation levels. They are very adequate to provide practical moderate delays as in (Gonzalez-Herraez et al., 2005). Unluckily, Brillouin presents a limited bandwidth due to its narrow gain linewidth while Raman

The innovative realm of metamaterials has also been explored for developing photonic buffers. Metamaterials are artificial composites with dramatically different electromagnetic properties. The most noticeable approach to storing light in metamaterials is presented in (Tsakmakidis & Hess, 2007). So far, this theoretical approach is unfeasible due to the

Recently, novel approaches to slow light based on plasmonics have been proposed, as in (Søndergaard & Bozhevolnyi, 2007). The main argument for using plasmonics is that it overcomes the diffraction limit affecting photonics, i.e. surface plasmon polaritons (SPPs) enable focusing light in nanoscale while photonics is size-limited to a wavelength scale. The main limitation to plasmonics today is that plasmons tend to dissipate after only a few millimetres. For sending data longer distances, the technology would need a great breakthrough. SPPs are also very sensitive to surface roughness and they are difficult to excite efficiently, although some "tricks" (e.g. Kretschmann geometry) have been already

significant delays can only be achieved for signals of a few kb/s.

material losses, and, difficulty of fabrication in a light wavelength scale.

presents reduced slow-light efficiency.

proposed for their excitation.

practical use.

Nonetheless, from the point of view of practical applications, a more useful figure of merit (FOM) than the absolute value of the group velocity itself is the number of pulse widths that can be delayed by the system. This FOM is given by the delay-bandwidth product, being the bandwidth inversely proportional to the duration of the pulse, and it gives an idea of the information storage capacity of the system (Boyd & Narum, 2007). The delay-bandwidth product is limited by two major impediments: the pulse distortion, mainly caused by group velocity dispersion, and the propagation loss. These two factors present different origin and significance for each slow-light approach but it is a common feature to all of them that higher delays come at the expense of bandwidth reductions and, usually, of higher loss.
