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

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 can be fast and efficiently tuned by exploiting Pockels effect.

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

Photonic Band Gap Engineered Materials for Controlling the Group Velocity of Light 69

defect mode frequency shift as a response to a refractive index change originating from the supplied voltage. When light at a proper wavelength proceeding from a narrow-width optical source (laser) is launched into the slow light device, it couples to the guided mode.

Fig. 16. Lithium niobate refractive index response to an electric field applied due to Pockels effect. The inset shows how to exploit this effect by applying a voltage to the electrodes

Depending on the source frequency (or wavelength) the signal will propagate with a given group velocity as given by the graphics in Fig.11. At this point, if a voltage is applied to the device electrodes, the mode shifts in frequency but the injected frequency remains the same. In consequence, the signal is switched to another velocity regime or even from a guided- to a non-guided regime. This simple mechanism enables a plurality of device applications from

Fig. 17. Frequency shift of a photonic crystal waveguide guided mode as a function of

placed on the lateral sides of the waveguide.

tunable delay lines to transistors, modulators or tunable filters.

refractive index change stimulated by Pockels effect.

Fig. 15. a) Dielectric constant structure of a cavity coupled to a waveguide on a 2D triangular lattice of holes in lithium niobate; b) DFT of H┴; c) Transmission diagram over wavelength and detail of the resonance and surrounding wavelengths.

$$
\Delta \mathbf{n} = \left(\frac{\mathbf{n}^3}{2}\right) \cdot \mathbf{r}\_{33} \left(\frac{\mathbf{V}}{\mathbf{d}}\right) \tag{10}
$$

where *r33* is one of the matrix component and *d* is the electrode width. A refractive index change of Δn=1x10-3is given by an electric field of ΔE0= -6.45 V/μm. Taking into account that electrode width will be of the order of half a micron, small electric fields will be needed to achieve tunability.

We subsequently consider the appliance of an incremental electric field, by steps of |ΔE0|= 64.5 mV/μm, to a LiNbO3 slab patterned with a 2D photonic crystal. This generates refractive index changes from 10-5 to 10-1, later we will see how only a range of these index changes is achievable in practice. This is achieved by a means of a voltage supplied to electrodes placed on the surface of the photonic crystal device, as in the inset of Fig.15.

First, we pay attention to waveguides reconfigurability. The red line in the band diagram of Fig.9 represented the frequencies of the defect mode guided within the waveguide as a function of the longitudinal wave vector. The group velocity of the guided mode was also computed and represented in Fig.11. Now, we explore how the group velocity of the signals coupled to this mode can be dynamically switched. The graphic in Fig.17 is evidence for the

**a) b)** 

**c)**  Fig. 15. a) Dielectric constant structure of a cavity coupled to a waveguide on a 2D triangular lattice of holes in lithium niobate; b) DFT of H┴; c) Transmission diagram over wavelength

3

where *r33* is one of the matrix component and *d* is the electrode width. A refractive index change of Δn=1x10-3is given by an electric field of ΔE0= -6.45 V/μm. Taking into account that electrode width will be of the order of half a micron, small electric fields will be needed

We subsequently consider the appliance of an incremental electric field, by steps of |ΔE0|= 64.5 mV/μm, to a LiNbO3 slab patterned with a 2D photonic crystal. This generates refractive index changes from 10-5 to 10-1, later we will see how only a range of these index changes is achievable in practice. This is achieved by a means of a voltage supplied to electrodes placed on the surface of the photonic crystal device, as in the inset of Fig.15.

First, we pay attention to waveguides reconfigurability. The red line in the band diagram of Fig.9 represented the frequencies of the defect mode guided within the waveguide as a function of the longitudinal wave vector. The group velocity of the guided mode was also computed and represented in Fig.11. Now, we explore how the group velocity of the signals coupled to this mode can be dynamically switched. The graphic in Fig.17 is evidence for the

n r

33 n V

(10)

2 d 

and detail of the resonance and surrounding wavelengths.

to achieve tunability.

defect mode frequency shift as a response to a refractive index change originating from the supplied voltage. When light at a proper wavelength proceeding from a narrow-width optical source (laser) is launched into the slow light device, it couples to the guided mode.

Fig. 16. Lithium niobate refractive index response to an electric field applied due to Pockels effect. The inset shows how to exploit this effect by applying a voltage to the electrodes placed on the lateral sides of the waveguide.

Depending on the source frequency (or wavelength) the signal will propagate with a given group velocity as given by the graphics in Fig.11. At this point, if a voltage is applied to the device electrodes, the mode shifts in frequency but the injected frequency remains the same. In consequence, the signal is switched to another velocity regime or even from a guided- to a non-guided regime. This simple mechanism enables a plurality of device applications from tunable delay lines to transistors, modulators or tunable filters.

Fig. 17. Frequency shift of a photonic crystal waveguide guided mode as a function of refractive index change stimulated by Pockels effect.

Photonic Band Gap Engineered Materials for Controlling the Group Velocity of Light 71

Substantial work has been done to provide numerical solution of Maxwell equations. In this subsection a rough idea on the different computational methods to solve photonic crystal problems is given, for a broader notion we refer the reader to (Joannopoulus et al., 2008).

As a first approach, one can divide computational methods in frequency domain methods and time-domain methods, each of them useful for solving different problems typologies. Frequency domain methods are used to solve problems such as the computation of band diagrams and stationary mode profiles. On the other side, time-domain methods are better suited to perform computations involving time evolution of fields, such as transmission and reflection spectra or resonant cavities decay in time. Numerical methods can be alternatively classified on the basis of the used discretization schemes in: finite differences, finite

Several commercial and open-source software packages implementing different numerical methods are available for computational photonics. Just to cite a few of these free-software products we will mention MPB (using plane wave expansion frequency-domain method) and Meep (implementing finite differences in time domain) MIT's packages and CAMFR (based in eigenmode expansion and advanced boundary conditions like perfectly matched

Next, a coarse notion on the fabrication techniques to synthesize photonic crystals is given. A good set of references about 3D and 2D photonic crystal fabrication techniques is given in (Skorobogatiy & Yang, 2009). It has to be noticed that, in spite of its outstanding potential, photonic crystals mostly remain at research stage and this is mostly due to current technological limitations of fabrication techniques. Focused Ion Beam (FIB) and electron beam (e-beam) lithography combined with reactive ion-etching (RIE) are two the methods used in laboratories for high accuracy and high-resolution fabrication of planar 2D photonic crystals. However, it is necessary to start moving photonic crystal technology out of the laboratory and onto the production floor for building photonic devices for practical applications. Recent advances in nano-imprint lithography are fulfilling this goal (Kreindl et

Photonic band gap materials are a powerful tool for tailoring light propagation properties. This Chapter has put the emphasis on the control over the signal group velocity given the wide range of applications enabled by slow light. All slow light techniques reviewed rely on slowing down the information or the energy transported by light signals, more than on

Among the plurality of foreseen slow-light applications we have mainly highlighted two due to its high technological and societal expected impact. The development of optical buffer memories is of utmost importance for the deployment of all-optical networks. Slow light is a promising approach for fast, scalable and low-power consuming on-chip optical buffers. Given the state of the art of the technology, nowadays, such devices cannot replace electronic RAMs in their current functions; however major breakthroughs are still expected in this field. Concerning slow light application to biosensing, not so bandwidth demanding

and less affected by losses, near future commercial prospects are encouraging.

**3.3 Computational methods and fabrication techniques** 

elements, boundary-element and spectral methods.

layers).

al., 2010).

**4. Conclusion** 

slowing the photons themselves.

The reconfiguration possibilities of waveguides are somehow limited when compared with that of the cavities. The reconfiguration capabilities of the photonic crystal cavity proposed in Fig. 15 via electro-optic effects are presented next. Recall that the cavity had a resonant frequency at around 1500nm, precisely at 1499.1nm. We alter the refractive index of the photonic crystal material from 0 to 0.1 in steps of 10-5 by exploiting Pockels. The overlap of transmission spectra for different refractive index values proves the resonant wavelength redshift. This diagram resulting from our FDTD simulations is presented in Fig. 18. Subsequently we have focused in index changes achievable for reasonable values of applied electric field in normal applications, i.e. from 0.1 to 10MV/ μm. This reduces the range of achievable index changes to values up to 1.5510-3. In Fig.19 we represent the computed cavity resonant wavelengths as a function of the index change generated by Pockels effect using realistic electric field values. This simulation results are the proof of the concept of fast (switching time bellow 1 ns) and efficient (every |ΔE0|= 64.5 mV/μm implies Δλ= 6pm) reconfigurability.

Fig. 18. Overlapped transmission spectrum for refractive index increments of Δn=(0, 0.001, 0.01, 0.1)

Fig. 19. Resonant wavelength versus refractive index change for the photonic crystal cavity shown in Fig.15.
