## Abstract

A comparison between several 1D periodic structures designed to enhance non-linear effects for high-speed all-optical applications is presented. These structures allow for a small group velocity of the propagating waves, so the light-matter interaction is increased, making the non-linear process to be more efficient. In addition, the propagating wave is compressed, making the field intensity to be higher in the non-linear material. Thus, a significant reduction in both the structure length and the input power needed to induce a particular phase shift is achieved. The selected 1D periodic structures are compared by means of properties such as the modal effective volume, coupling efficiency, mode bandwidth, group velocity dispersion, and easiness of fabrication, in order to determine the optimum configuration in terms of non-linear enhancement.

©2008 Optical Society of America

## 1. Introduction

It has been demonstrated that a low group velocity (*v _{g}*) can enhance the efficiency of non-linear processes in optical materials [1]. Since the interaction between field and matter takes a longer time, shorter structures and lower power consumption are needed to induce the same non-linear phase. In addition, a small

*v*results in higher electric field intensities for the same total optical power. The improvement in the refraction index change induced by a third-order non-linearity has been quantified as (

_{g}*v*/

_{g,WG}*v*)

_{g,SW}^{2}, where

*v*and

_{g,WG}*v*are the group velocities of a conventional dielectric optical waveguide (no slow-wave structure is introduced) and a slow-wave optical waveguide, respectively [2].

_{g,SW}Periodic photonic structures, where propagation is in the form of Bloch modes, are one of
the means to achieve slow-wave behavior [3]. When a structure is periodic along the direction of propagation, wave vectors of the propagating mode are folded in the first Brillouin zone (BZ). The dispersion relationship of guided modes flattens at the edge of this BZ, providing regions with low *v _{g}* values (theoretically zero just at the band edge). In this work, different types of one-dimensional (1D) periodic structures created by inserting a strong periodic modulation in a high-index material with a third-order non-linearity are compared. Three-dimensional periodic structures are not considered because our interest is centered in planar photonic structures, whose fabrication is quite simple; two-dimensional periodic structures are not considered either because periodicity is only required in one direction to achieve the slow-wave behavior. Moreover, 1D periodic structures are expected to be easier to design and fabricate than 2D structures.

These 1D periodic structures are very promising as slow-wave non-linear elements in
high-speed all-optical functionalities (switching, routing, gating,…). Thus, some crucial
aspects such as bandwidth or dispersion have to be considered in addition to the small *v _{g}* giving rise to the enhancement of non-linear effects. Since values of

*v*as low as desired can be achieved for any periodic structure just by getting closer to the 1

_{g}^{st}BZ edge, other parameters of the guided mode are obtained and analyzed in order to make a comparison between the structures and choose the most suitable for non-linear applications. The application of these 1D periodic structures in a functional device (a XOR all-optical logic gate) is also commented at the end of this text.

## 2. Benefits of low group velocity in non-linear applications

A reduction of the group velocity results in an increased matter-field interaction owing to two main mechanisms:

- for a given power flow, the energy is temporarily compressed and the field amplitude is increased as the group velocity decreases,
- longer time is necessary for an electromagnetic wave to travel through a given length if the group velocity is small.

These phenomena give rise to an enhancement of any physical effect (amplification,
absorption, loss, phase shift,…) experimented by a signal that travels through a slow-wave
structure. It has been theoretically predicted that the non-linear phase shift resulting from the cross-phase modulation (XPM) due to the Kerr effect in a waveguide with fixed length and for a given input power is enhanced by a factor (*v _{g,WG}*/

*v*)

_{g,SW}^{2}when a slow-wave structure is used [2].

For example, a medium that possesses a Kerr non-linearity experiments a certain change
in the index of refraction Δ*n* when an optical signal propagates along the structure. If we consider an optical waveguide structure of length *L* made of this material, the propagating signal will experiment an additional non-linear phase shift Δ*ϕ* that can be well approximated by the following expression [2]:

where *σ* is the fraction of energy of the guided mode inside the non-linear material. Eq. (1) indicates that the non-linear phase shift depends on the quantity of the field that is confined in the non-linear material of the structure and is inversely proportional to the group velocity of the guided mode. In addition, since Δ*n* is caused by a Kerr effect Δ*n*=*n*
_{2}
*I*, where *n*
_{2} is the Kerr coefficient and *I* is the optical intensity, which is proportional to |*E*|^{2}. Due to the energy temporal compression in the slow-wave structure, it can be predicted that |*E*|^{2} is inversely proportional to *v _{g}* [2]:

This means that the non-linear phase shift induced by a given structure is inversely proportional to the square of *v _{g}*. So, the phase shift induced by a slow-wave section is (

*v*/

_{g,WG}*v*)

_{g,SW}^{2}higher than that of a conventional waveguide:

In conclusion, the benefits of the low group velocity in non-linear applications can be exploited in two different ways:

- a reduction of the total length of the structure needed to achieve a particular phase shift. This is mandatory to achieve large scale integration of photonic devices at low cost,
- the total power required to achieve a particular phase shift can be significantly reduced due to the field enhancement in the slow-wave structures, which results in power saving.

## 3. Criteria for the comparison between structures

As previously said, group velocities as low as desired can be achieved for any periodic
structure. Therefore, in order to characterise the selected structures, a reasonably low value of v_{g} is selected and other parameters/properties of the guided modes are compared.

**normalized effective volume parameter,**(effective volume per period): This parameter characterise how the energy of the mode is concentrated in the non-linear region. The lower*V*_{eff,n}*V*value, the higher the concentration of the energy in the non-linear region. This parameter is an equivalent of the effective area [4] for structures with discrete translational symmetry (the effective area is only valid for structures with continuous translational symmetry)._{eff,n}*V*is numerically calculated from the field profile of the guided mode as:_{eff,n}The optical intensity in the waveguide can be expressed as

*I*=*P*/*V*, where_{eff,n}*P*is the optical power, so the induced index change Δ*n*will be inversely proportional to*V*._{eff,n}**coupling efficiency:**Since the induced non-linearity is proportional to the optical power in this region, the more efficiently the light is coupled into the slow-waveguide, the less power will be required at the input of the structure in order to induce the same index change. Group index*n*of the guided mode in the periodic structure is very high when working close to the band edge, what provokes a great mode mismatch with external waveguides and increases coupling losses. The coupling efficiency for analyzed structures is estimated by means of 2D-FDTD calculations using an effective index for the material of the waveguide to take into account the vertical confinement. Since mode mismatch between access and slow-wave structures is mainly due to in-plane geometry variations, results obtained from 2D calculations are equivalent to those for 3D structures. Coupling techniques such as tapering or adiabatic transitions are evaluated for some structures in order to increase coupling efficiency, as it can be seen in Fig. 1._{g}**bandwidth, BW**Time limited signals (i.e., pulses) are considered to be transmitted along the slow-wave structure, so the guided mode must have enough bandwidth to allocate the spectrum of these signals. The distance between the wavelength with the selected group velocity and the edge of the band is defined to be BW_{max}:_{max}/2. This is a limiting factor for the structures, since those with a small bandwidth will not be able to propagate signals at determined bit rates. For example, for bit rates of 40 Gbit/s, signal energy has a spectral bandwidth (at 95%) around 0.8 nm, so structures with a BW_{max}higher than this value are required.**dispersion parameter,**: Pulse shape is desired to remain unperturbed after crossing the slow-wave structure, with only a phase variation due to the non-linear index change. Unfortunately, dispersion in periodic structures is extremely high when we get close to the band edge of the guided mode [3].*D**D*is obtained at the working wavelength for the periodic structures analyzed, looking for low values of this parameter.*D*is expressed here in ps/(mm·nm) since very high values are obtained and sizes are small (*D*is usually expressed in ps/(km·nm) in fibre-optics communications systems [6]).Dispersion compensation mechanisms consisting in cascading two stages of slow-wave structures with opposite values of GVD are considered. In a 1D periodic structure this can be done easily since the 1

^{st}and the 2^{nd}guided bands have opposite GVDs near their respective band edges. Fig. 2 shows schematically a usual band diagram of a generic 1D periodic structure where this fact can be seen, and how two cascaded structures can be used to compensate the GVD accumulated along the first periodic structure, if required.**easiness of fabrication**: This is a more subjective property of the structures but equally important. Propagation losses in photonic structures are mainly due to imperfections in the fabrication process (e.g., surface roughness or scatterers in the structures). A reduction in the group velocity of the mode also “enhances” the interaction between the field and the imperfections of the structure, increasing propagation losses. Some works state that losses scale with (*c*/*v*) or (_{g}*c*/*v*)_{g}^{2}[7, 8], what makes them to grow dramatically in the small group velocity region. But we think that this relation is not so simple and it will greatly depend on the concrete structure [9]. Hence, we have only centred our work in the enhancement of non-linear effects due to the reduction of*v*. Therefore, a very accurate fabrication process is mandatory in order to use slow-wave structures in non-linear applications, without the problem of losses, independently of their relation with_{g}*v*. Hence, the simpler the designed structure is, the more accurate the fabrication could be done._{g}

The characterization of the proposed structures is divided in the following steps. First, the dimensions of the structure are designed in order to allocate the band edge of the guided modes around 1550 nm. Both modes with negative and positive *D* will be selected in order to perform dispersion compensation if needed. Then, band diagrams for each configuration are calculated using the plane wave expansion (PWE) method [10] and the parameters of the mode are obtained. A group velocity 0.07*c* is selected (any other value could have been selected), and the distance to the band edge (used to obtain BW_{max}) and the dispersion parameter *D* are obtained for the wavelength with this *v _{g}*. Then, the normalized effective volume

*V*of the guided mode is determined from the electric field distribution in the structure (obtained from PWE calculations). Finally, the effective index of the core material for the selected band is obtained in order to perform 2D FDTD simulations to study transmission performance and coupling efficiencies.

_{eff,n}## 4. 1D periodic structures analyzed and results

Fig. 3 shows the 1D periodic structures under analysis. All the structures are considered to be made of CdTe (*n*=2.74) as non-linear material, with a SiO_{2} (*n*=1.45) lowercladding and surrounded by air. CdTe is chosen as non-linear material for the calculations since it presents high Kerr effect at wavelengths around 1550 nm (*n*
_{2}=5.23·10^{-13} cm^{2}/W) and small losses due to two-photon absorption (TPA) effect (*β*=18.3 mm/GW) [11]. Fabrication process is under development for this material, but conclusions of this work can be used for any other high-index non-linear material (e.g., GaAs - *n*=3.34/*n*
_{2}=2.93·10^{-13} cm^{2}/W/*β*=103 mm/GW) [11].

First, an access waveguide (a strip high-index contrast waveguide) is designed. Dimensions for the waveguide are *w*=550 nm (width) and *h*=350 nm (height) in order to be single mode and to have low effective area. These parameters have been used for the design of the slow-wave structures. It is seen in Fig. 3 that analyzed structures vary from structures with isolated defects ((a) and (b)) to structures where a periodicity is introduced in the previously designed strip waveguide ((c), (d), (e) and (f)).

Calculations and results for each configuration are presented in following subsections. Three main considerations for the design process have been taken into account:

- height of the structures is set to
*h*=350 nm, the same height as for the single mode waveguide designed, - in order to avoid leaky propagation for the guided modes, only those located below the light line of the silica (the cladding with higher refraction index) are considered. This condition determines the maximum period
*a*allowable:*a*<(*k*·_{edge}*n*)·_{cladd}*λ*→_{work}*a*<535 nm, - the minimum size of elements and the minimum distance between them is set to 100 nm in order to avoid problems in a future fabrication process. For example, in the rods configurations the minimum radius of them is set to 50 nm, while their maximum radius will be determined by the minimum distance between elements (100 nm) and the period selected.

## 4.1 CdTe rods chain with n=1

This structure is shown in Fig. 3(a). In this structure light is laterally confined inside the rods by total internal reflection (TIR) and propagates in the longitudinal direction due to the coupling of the tail of the field between nanopillars. Several rows of nanopillars can be placed in parallel with a spacing *d* between adjacent rows, as shown in Fig. 3(b). The use of several rows provides the advantage of a larger lateral size, which in principle would be advantageous for a more efficient coupling from/to external waveguides. Previous works [12, 13] concluded that the formation of large photonic band gaps (PBG) occurs for TM-polarized modes.

A design process aimed at locating the edge of the guided modes (slow-wave region)
around 1550 nm is carried out. Since the columns height *h* is fixed, a sweep for the lattice constant *a* and the pillar radius *r* is made, obtaining the wavelengths of the modes at the edge of the BZ with 3D PWE calculations. Calculations are made for lattice constant values between 400 nm and 535 nm, and radii values between 50 nm and 210 nm, due to dimension restrictions previously commented. All the combinations of these parameters for this configuration have their frequencies above the light line of the silica, being leaky modes. A higher percentage of dielectric material (CdTe) would be needed in order to shift modes below the light line, but fabrication constraints (mainly the minimum distance between pillars) as well as the small height of the columns prevent the modes to be non-leaky. Hence, no designs for the 1D periodic structure with *n*=1 column are selected because all the configurations give leaky guided modes.

## 4.2 CdTe rods chain with n=3

This structure is shown in Fig. 3(b), where it can be seen that two rows of adjacent rods are added to the previously analyzed structure. In this case, the lateral distance between
nanopillars, *d*, is set to be equal to the lattice constant of the structure. Again, the limitations in the maximum and minimum values of the parameters are considered in the calculations.

Fig. 4 shows a wavelength map for the first guided band of the structure. A color map is
used to represent the wavelength of the guided mode at the edge of the BZ for each combination of parameters (*a*, *r*). The grey shaded area denotes the configurations where the distance between adjacent columns is smaller than 100 nm (dimension restrictions are not accomplished). Configurations giving guided modes with their band edges located at 1550 nm are depicted by a black solid line. Solid triangles indicate the sign of the group velocity and polarization of the calculated mode for each parameters combination. Triangles pointing upwards indicate positive group velocity, while triangles pointing downwards indicate negative group velocity. The polarization of the mode is indicated by the filling color of the triangles: TE – red and TM – blue. It can be seen that the polarization of the first mode changes when modifying structure parameters, because the relative position between modes changes (the first TE mode can be above the first TM mode for some configuration and below for others). These wavelength maps are useful to determine the appropriate dimensions of the structure to allocate the edge of each band at 1550 nm.

As it can be seen in Fig. 4, guided modes with their edge located at 1550 nm and below the light line can be achieved. However, the dimensions of the structure are very close to the physical constraints imposed by the hypothetic fabrication process, what could easily make the modes to be slightly leaky. Moreover, only modes with positive group velocity (negative dispersion parameter) can be achieved, so no dispersion compensation mechanisms could be performed. Another problem of this configuration is that modes are barely confined in the non-linear region, so that the induced index change will be very slight.

## 4.3 CdTe waveguide with air holes

This kind of 1D periodic structure is created by periodically introducing several air holes inside the strip waveguide. Propagation mechanism in this structure is very similar to that in the nanopillars structure: the space between the holes is the place where the EM field is mainly confined, which act as cavities. But now, continuity between high index regions makes a wider PBG to appear for TE modes [12]. Again, periodicity causes the bands to become flat at the edge of the BZ, giving low values of group velocity. Since this structure is directly created inside the access strip waveguide, the coupling efficiency of light from/to it is expected to be very high. Coupling mechanisms to enhance this issue are also studied below.

Fabrication of air holes inside the CdTe strip waveguide is not expected to be so difficult, since etching depths required are not very high (350 nm) and only reasonable aspect ratios would be required.

A sweep for lattice constant values between 400 nm and 535 nm, and holes radii values between 50 nm and 175 nm has been carried out. Wavelength maps in Fig. 5 show the wavelength of each guided mode of the band diagram at the edge of the BZ for each
combination of parameters (*a, r*). The Grey shaded area indicates the combinations (*a, r*) where the distance between holes is smaller than 100 nm. White area in the third and forth bands
indicate combinations where the frequency of the mode is above the light line of the silica (leaky mode). It can be seen now that appropriate combinations of parameters can be obtained in order to achieve guided modes with slow-wave behaviour near 1550 nm and with both signs for the group velocity. Configurations to obtain proper TE and TM modes with both signs of group velocity are selected and their parameters are shown in Table 1 and Table 2. These tables show the parameters (*a, r*) selected for each configuration, the order of the mode in the band diagram, the sign of the group velocity, the bandwidth allowable when working at the wavelength with *v _{g}*=0.07

*c*, the dispersion parameter

*D*at this wavelength, the volume of the non-linear region in the basic cell

*V*, the normalized effective volume

_{NL,n}*V*, and the percentage of electromagnetic energy in the non-linear region, α.

_{eff,n}As expected from results in [12], the best results are obtained for TE-polarized modes.
The curvature of the bands is very smooth, obtaining modes with extremely high bandwidths
(~ 5 nm) and low dispersion parameters (|*D*| ~10 ps/(mm·nm)), so propagated signal will only be slightly distorted along propagation. Moreover, modes with opposite sign of dispersion and very similar absolute values are obtained, which eases dispersion compensation if needed. Furthermore, energy is highly confined in the non-linear region, that gives normalized effective volume parameters with very low values (*V _{eff,n}* ~0.5-1.5 µm

^{2}). However, these low values of

*V*are a little deceptive since they are greatly influenced by the low volume of the non-linear material of the structure. The insertion of air holes in the strip CdTe waveguide significantly reduces the region where non-linearities can be induced. It can be seen that effective volumes are one order of magnitude higher than non-linear volumes of the structures.

_{eff,n}Worse but still good results are obtained for TM-polarized modes. A bandwidth enough to propagate 40 Gbps signals is obtained (>0.8 nm), but worse values of the parameters of dispersion (|*D*| ~35–50 ps/(mm·nm)) and effective volume (*V _{eff,n}* ~2–4 µm

^{2}) are achieved.

The coupling issue is analyzed only for TE-polarized modes. First, the case of coupling to
the 2^{nd} band (*a*=500 nm, *r*=100.6 nm, *n _{eff}*=2.28) is considered. Figure 6(a) shows the
obtained transmission spectra. The blue curve shows the response for the holes waveguide
without any coupling section. It can be appreciated that a strong ripple appears due to both the modal and the group velocity mismatching between the waveguide mode and the mode in the holes structure, which results in a typical Fabry-Perot-like response. Adiabatic transitions have been implemented at both ends of the holes chain by linearly diminishing the radius of the holes in N steps (N=3, 10 and 20). This adiabatic coupling technique is schematically shown in Fig. 6(b) for the input side of the chain of holes. The objective is to improve the coupling by an adiabatic transition and reduce the Fabry-Perot ripple. The period is kept constant, so a reduction of the radius is equivalent to increasing the index of the structure and the second band is shifted up to higher wavelengths, which results in an improved coupling. It can be thought that an almost perfect coupling could be obtained with a sufficiently long tapering structure. Nevertheless, a linear reduction of the radius of the holes seems to be difficult to achieve in the fabrication process. In fact, holes with radius smaller than 100 nm might be drilled.

An equivalent technique to adiabatically increase the radius of the air holes is to decrease the width of the strip waveguide. This alternative is shown in Fig. 7(a). It can be seen from spectra in Fig. 7(b) that transmission response is enhanced by this technique, but results not as good as before are obtained (insertion losses are slightly higher). The problem with this configuration is that the access waveguide has to be wider than the one used to the 1D periodic section.

The enhancement of the coupling for the first band is a harder issue. Now, instead of increasing the percentage of high-index material in the transition section (as it was done before by reducing the radius of the holes or by increasing the width of the waveguide), the percentage of this material has to be reduced. The ways to do that are either to increase the radius of the air holes or reduce the width of the CdTe waveguide in the transition section, but this makes the elements to get very close and dimension restrictions previously commented are not accomplished.

As a general conclusion for the air holes structure, extremely good results in terms of bandwidth and dispersion are obtained. Very low effective volumes are also achieved, but they are mainly due to the low percentage of non-linear material in the structure. Therefore, induced index change will not be as high as desired. Moreover, enhancement of the coupling efficiency from/to access strip waveguides seems to be a difficult issue, since elements are needed to be very small or very close between them.

## 4.4 CdTe waveguide with adjacent CdTe rods

Some problems were encountered when analyzing the previously proposed structures. For the chain of nanopillars, modes are very close or even above the light line of the silica, avoiding non-leaky modes. For the holes structure, very good results for the analyzed configurations were obtained, since it is directly based on the CdTe strip waveguide. But the dimensions of the holes are too big and CdTe region is greatly reduced, what is not desirable for a non-linear application. The low values for the effective volume parameter were influenced by the reduction of the non-linear region when creating the holes.

In order to avoid the problems commented above for the analyzed periodic structures, a structure where a periodicity is introduced near an “intact” CdTe strip waveguide is proposed. Two variations are considered, with one or two chains of rods placed besides the strip waveguide, as shown in Figs. 3(d) and 3(e). In these structures, light is mainly guided by the strip waveguide and a chain of rods (or two chains symmetrically placed) is placed adjacent to the waveguide in order to create a periodicity. So, slow-wave behavior will appear at the edge of the BZ.

However, it has been seen that modes near the band edge given by these configurations
are not valid for our purpose. The electromagnetic field is highly confined in the strip
waveguide (see Fig. 8(a) for the configuration with *n*=1 adjacent row of rods) what makes
the influence of the adjacent rods to be almost negligible over the total response. An example of band diagram for this kind of structure is depicted in Fig. 8(b), where it can be seen the very slight splitting of the mode at the edge of the BZ. These bands have an almost negligible curvature, so low group velocities would be only obtained for frequencies very close to the band edge, having very small bandwidths and very high dispersion. This behavior has been obtained for all possible configurations of this structure (with *n*=1 and *n*=2 rows of rods), so this scheme of 1D periodic structure must be discarded for non-linear enhancing purposes.

## 4.5 CdTe corrugated waveguide

This structure is also based on introducing a periodicity in the CdTe strip waveguide, similar to what was made for the previous structure. In this case, a group of CdTe transversal elements (corrugations) separated by a period *a* are introduced along the waveguide in order to create the periodicity [5]. This structure is shown in Fig. 3(f). Now, the periodicity is created by straight elements similar to waveguides, which will be easier to fabricate than rods or holes. Moreover, they are completely joint to the main CdTe strip waveguide, so their influence will be higher than that of the rods in the other structure. The expected advantages are the same as those expected for the waveguide with adjacent rods: a greater index change due to the non-linear effect (the field is almost completely confined in the strip waveguide) and a highly efficient coupling of light from/to the access strip waveguide. Moreover, optimal structures to enhance the coupling of light for this structure are also proposed in [5]. The main parameters of this structure are: width w and height *h* of strip waveguide (fixed from design of the single mode waveguide), period *a*, and width *w _{i}* and length

*w*of the transversal elements.

_{e}This structure is expected to be much easier to fabricate than all the previous ones since
all elements are straight lines similar to waveguides. Problems could appear if transversal elements width wi or distance between them (*a - w _{i}*) were too small. This could be solved by using an appropriate hard mask or lift-off technique in order to achieve a higher resolution.

The same design process as for the other structures is used for the CdTe corrugated
waveguide, but now the values of three parameters (*a, w _{i}, w_{e}*) need to be determined in order to design the structure to work around 1550 nm. This fact hugely increases the number of calculations to be done in order to determine these parameters. Therefore, elements length

*w*is fixed to a value of 2 µm. This value is expected to be enough to give proper results, since it is four times greater than the width of the waveguide

_{e}*w*and it will be theoretically around ten times greater than the width of the transversal elements

*w*.

_{i}Before sweeping for the other two parameters (*a, w _{i}*) and obtaining the wavelength maps for the modes of the waveguide, a band diagram of a corrugated waveguide is calculated. Fig. 9(a) shows the band diagram for both polarizations of a corrugated waveguide with

*a*=363.2 nm and

*w*=150 nm. Symmetries in the in-plane direction are also depicted. It can be seen that modes with several polarizations and symmetries are obtained, overlapping each other. Nevertheless, if this structure is accessed from the designed single mode waveguide, TE-polarized modes (solid line) with an odd symmetry in the transversal direction (crosses) are excited. Therefore, only modes with these symmetries must be taken into account, as depicted in Fig. 9(b). Moreover, it can be seen that the first two modes have positive and negative slopes at the edge of the BZ, what makes them suitable for dispersion compensation purposes.

_{i}Fig. 9(b) also shows that single mode behavior is not achieved for the first two modes at the edge of the BZ. It can be seen that the uprising part of the second band overlaps with the frequencies of both modes at the edge of the BZ, so as not be single mode. In order to overcome this, this uprising part must be located over the light line of the silica. In this way, only the mode excited at the edge of the BZ will be truly guided, while the mode excited for this uprising part over the light line will be leaky. In order to increase the frequency of the bands, the quantity of high index material in the structure must be reduced. Hence, the width of the transversal elements wi is set to the minimum value, 100 nm.

So, the only parameter to be determined is the period a to center the first and the second
bands on 1550 nm. For this purpose, periods *a*=371.7 nm and *a*=390.1 nm need to be taken. Parameters of these two modes are shown in Table 3.
Fig. 10 shows the band diagrams for these two configurations. It can be seen that only one mode is excited when working at the edge of the band for these two configurations (uprising part of the second band is over the light line of the silica).

It can be seen from Table 3 that the bandwidth of these two modes is enough to transmit a
40 Gbps signal (bandwidth of 0.8 was commented). Dispersion values are not as low as for
the holes configuration, but magnitude is almost the same for the first and the second bands (|*D*| ~45 ps/(mm·nm)). Therefore, dispersion compensation would be easily performed by using two sections of approximately the same length and equal but opposite *D*. But the main advantage of this configuration is the high concentration of the EM field in the non-linear region. Concentration values around 90 % are obtained for both modes, what gives extremely low values of normalized effective volume (around 0.34 µm^{2}). These values are very close to the effective area of the CdTe strip waveguide, that has been calculated to be 0.27 µm^{2}. Therefore, a huge enhancement of the non-linear effect will be produced, which is its main advantage. Contrary to what occurred for the air holes configuration, the volume of non-linear material is very high, so the low values of *V _{eff,n}* are due to the high concentration of the EM field (as desired, notice that they are of the same order of magnitude than the non-linear volume).

If values of Table 3 are used in Eq. (1), an input power of *P*≈0.1 W would be needed to induce a π-phase-shift in a corrugated waveguide of length *L*=500 µm for any of the two selected modes. By using the same equation, we obtain that an input power of ~1.8 W would be needed to induce the same phase shift in the CdTe strip waveguide. Concerning dispersion, if pulses width of the 40 Gbps signal is 7.5 ps, *D* values for this structure will cause a slight broadening of pulses to ~8.4 ps for a corrugated waveguide of this length (*L*=500 µm) (classical broadening expression given in [6] has been used for the calculation).

The coupling issue has been analyzed by means of 2D-FDTD calculations with effective index for the CdTe. Fig. 11 shows the transmission spectrum of the corrugated waveguide with parameters *a*=371.7 nm and *w _{i}*=100 nm, and length of N=50 transversal elements, when coupling from/to it with the CdTe strip waveguide. It can be seen that transmission efficiencies for the first band (wavelengths higher than 1550 nm) around 75 % are obtained
when the corrugated waveguide is directly connected to the strip CdTe waveguide. The second band (wavelengths lower than 1500 nm) provides efficiencies around 65 %, but large
Fabry-Perot resonances appear due to the reflections at the interfaces between both waveguides. Efficiencies are highly enhanced if a simple linear taper of 5 elements is
introduced to connect both waveguides (see inset in Fig. 11). Efficiency values are now
obtained near to 100 % for the first band and around 85 % for the second. It can be also
observed that Fabry-Perot resonances are reduced due to smaller reflections at the interfaces of the waveguides. Contrary to what happens for other structures such as the chain of holes, a taper can be easily fabricated, since it is only based on reducing the length of the transversal elements and fabrication constraints are always accomplished. In [5] it is shown how coupling efficiency can be enhanced even more by using a variable-rate taper. Instead of using a linear rate for the taper, an n-order polynomial is used to describe the shape of the taper, obtaining an almost perfect coupling between structures.

Therefore, as a conclusion for the corrugated waveguide, it seems to be the best alternative for the purpose of non-linearity enhancement. The corrugated waveguide provides extremely high concentrations of the EM field in the non-linear region and extremely high coupling efficiencies from/to the access CdTe strip waveguide. These coupling efficiencies can be easily enhanced by using a simple linear taper. Concerning propagation properties of the guided modes, bandwidth enough to transmit 40 Gbit/s signal is obtained. Moreover, even though dispersion values for these modes are not very low, first and second bands have almost the same absolute values of dispersion with different signs, easing the application of compensation techniques.

## 5. Application of non-linear enhancement for optical processing

Fig. 12 shows an example of structure where non-linear enhancement by slow-wave elements can be applied. It consists on an all-optical XOR logic gate based on a Mach-Zehnder Interferometer (MZI), where two slow-wave structures are created in its arms in order to increase the efficiency of the non-linearity, similar to what was studied in [2] for photonic crystals. When an input data signal is present, it induces a phase shift over the control signal due to the XPM mechanism. This phase shift is increased by the presence of the slow-wave structure, what reduces the group velocity and increases the field amplitude, giving shorter sections to achieve a desired phase shift.

The operation of the XOR logic gate can be briefly explained as follows. In absence or presence of the two input data signals, the same phase shift is induced over the control signal in both arms of the MZI and they interfere destructively at the output due to the π-phase shifter element (it can be implemented by only changing the optical length of one MZI arm). That is, no signal is obtained at the output of the MZI (0-bit) when both data signals are the same (two 0-bits or two 1-bits). If only one of the two input signals is present (two different data signals, one 1-bit and one 0-bit), the control signal in one of the MZI arms experiments a higher phase shift, giving a non-zero signal at the output of the MZI (1-bit). In the optimum case, the difference of the phase induced in both MZI arms is π, what gives a completely constructive interaction between the signals in both arms, achieving then the maximum amplitude. Experimental demonstration of this kind of structure as an optical correlator has already been demonstrated with discrete optical elements and semiconductor optical amplifiers (SOAs) as non-linear elements [14].

By using these 1D periodic structures, the length of the structure needed to achieve a desired phase shift is reduced, making the total device smaller. This is a need to achieve higher and higher integration of photonic devices. Other possibilities to increase the efficiency of non-linearities have been proposed. One of them is the use of ring resonators [15]. This alternative has some drawbacks as:

- non-linear enhancement is only achieved for the wavelength of resonance of the ring and the bandwidth provided is very narrow. This makes the operation of the structure very sensitive to any fabrication deviation,
- the complete MZI structure is wider (ring radius around 5 µm-10 µm for high index materials) than the one obtained using the 1D periodic structures proposed in this text (in the worst case, the corrugated waveguide, the width of the structure is increased until 2 µm). Hence, the capacity of integration is reduced.

## 6. Conclusions

Several 1D periodic structures to enhance non-linearities by slow-wave propagation have been analyzed in this work. Some of these configurations can be directly discarded because of their poor transmission properties. CdTe rods configurations give modes that are very close or even above the light line of the silica, what makes the modes to radiate into the lowercladding (leaky modes). CdTe waveguide with adjacent rods give bands with a small split at the edge of the BZ and with high curvature at this region, due to the slight influence of the periodic rods over the strip waveguide. This makes the modes have extremely high dispersions and extremely low bandwidths.

Good results are obtained for TE modes in the holes structure (extremely high bandwidths and extremely low dispersions). Low effective volumes are obtained, but they are influenced by the low percentage of non-linear material in the final structure (non-linear material is removed by introducing the air holes). Good coupling efficiencies are obtained, but adiabatic transitions to reduce Fabry-Perot resonances are difficult to perform because they require very small elements or very small distances between them.

The corrugated waveguide seems to be the best option for non-linear enhancement purposes. Although bandwidth and dispersion parameters are not as good as those for the holes structure, they are enough to properly transmit the data signals at high bit rates (up to 40 Gbps). Moreover, the dispersion can be easily compensated by combining two sections with negative and positive dispersion parameter. On the positive side, energy is extremely confined in the CdTe non-linear region (very small normalized effective volumes are obtained) and coupling from/to the access CdTe strip waveguide is greatly enhanced when using a simple linear taper. Therefore, energy will be very efficiently used in this configuration, diminishing total power requirements to induce the required index change.

## Acknowledgments

Funding by EC under project PHOLOGIC - FP6 - 017158 and Spanish MCyT and EU-FEDER under contract TEC2005-07830 is acknowledged.

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