## Abstract

We modelled strong slow wave modulation enhancement in a rib corrugated waveguide with respect to a conventional rib waveguide, both embedded in a reverse biased pn junction. This enhancement is characterized in terms of effective index change versus reverse bias variations from 0V to −10V and for moderate group velocities varying in the range 0.02c to 0.15c. Interaction lengths and insertion losses below 750 μm and 3dB are respectively found for voltage variations in the range −8V to −10V. Furthermore, the device electrical modulation bandwidth is expected to be higher than 10 GHz.

©2009 Optical Society of America

## 1. Introduction

The slow wave phenomenon has attracted numerous researchers in the last few years due to the deeper control on light it offers. Indeed, slow wave structures, due to their periodic nature, present the advantage of slowing down the propagation of light. Slow light has been observed experimentally in SOI dielectric structures such as PhCs [1,2] and ring resonators [3–5]. The brilliant demonstration made by F. Xia et al. [5] in the field of optical delay lines marked a major milestone in optical buffering. Furthermore, the strong enhancement of non linear interactions due to local energy compression occurring in the slow light regime, was predicted by M. Soljačić et al. [6]. In the latter, the potential reduction of the interaction length and power consumption in all- and electro-optical devices was underlined and triggered a series of publications on switches [2,7], delay lines [8], non-linear enhancement in waveguides [9], and electro-optical modulators [10,11], opening up the way to much tinier devices operated under lower power. In the latter, the enhancing passive structures were chosen to be 2D Photonic crystals (PhCs) slab, specially tailored to enable light to propagate with low group velocity at a wavelength rounding 1550 nm. The slow light is then modulated at GHz frequency by carrier injection in a p-i-n diode. Though widely studied and now well know, conventional 2D PhCs slab with air holes is somewhat sensitive to fabrication variations, which can therefore be an issue if one wants to maintain their slow light features in the right wavelength range. This paper aims at showing that a similar enhancement can be also achieved using simpler planar geometry fully compatible with CMOS fabrication processes. The selected electro-optical structure is a 1D PhC rib shaped corrugated waveguide embedded in a pn junction, which, to the best of our knowledge, has not been explored yet.

## 2. Phase shifter description

The passive optical structure shown in Fig. 0.1 consists of a *W* = 300 nm x *H* = 220 nm single mode silicon rib waveguide with an *h* = 50 nm thick slab and transversal elements of dimensions *W _{e}* and

*W*repeated periodically by

_{i}*a*= 330nm. The periodic repetition of identical transversal elements gives rise to the slow light behavior at the edge of the Brillouin zone in the reciprocal space. Thus, light propagates with lower group velocity in a certain range of frequencies, depending upon the width (

*W*) and length (

_{e}*W*) of the transversal elements. The proposed electrical structure, an asymmetrical pn junction (Fig. 1 ) already demonstrated in [12], turned out to be an efficient modulation scheme and has therefore been employed in the electrical simulation to show the improvement that such corrugated slow wave structure is capable to bring with respect to conventional rib waveguides.

_{i}The corrugated waveguide is splitted into both *n-* and *p-*type doped regions, which respective net doping concentrations are 6.10^{17} cm^{−3} and 2.10^{17} cm^{−3}, respectively. Both *n +* and *p +* doping regions are implanted with a concentration of 1.10^{20} cm^{−3} to ensure ohmic contacts and are positioned at a distance *S _{wide}* = 1μm away from the thin section of the waveguide. The parameter

*S*is to be determined below, depending on the dimensions of both

_{thin}*W*and

_{e}*W*. The passive device design guideline is orientated by the doping concentrations into the waveguide as the optical properties of the corrugated structure are slightly modified by the presence of impurities. Therefore, the doping concentration distributions along the waveguide cross section, altering the refractive index distribution with respect to a non doped waveguide have been taken into account in the calculations. The refractive index distribution for a doped waveguide is derived from Soref’s expression [13] at

_{i}*λ*= 1.55 μm with no bias applied:

The density of electrons *N _{e}* and holes

*N*are calculated with ATLAS , a semiconductor CAD software from Silvaco©. The refractive index of undoped Silicon is

_{h}*n*= 3.47641 The determination of

_{Si}*W*, and

_{e}*W*has been made as a results of plane wave expansion calculations using BANDSOLVE form Rsoft© including the altered refractive index distribution calculated above in Eq. (1). The two dimensional scan was carried out in order for wavelengths surrounding 1550 nm to match the edge of the Brillouin zone, i.e. the slow wave region. Figure 2 below illustrates the results of such calculations.

_{i}From Fig. 2, the basic couple of parameters *(W _{i},W_{e})* can be extracted to fix the corrugation dimensions . The multiple choices offered by Fig. 2 are oriented by three primary fabrication concerns:

- 1. The minimum separation
*(a-W*between two consecutive transversal elements should be 100nm as well as the minimum length_{i})*W*should not be lower than 100nm (Minimum resolution of deep UV lithography)._{i} - 2. The corrugations should not be closer than 500 nm from the
*p +*and*n +*highly doped regions, i.e.*S*>500nm to avoid high optical losses._{wide} - 3. The wavelength sensitivity to impurities must be as low as possible.

Among many others suitable, the selected dimensions for this enhanced modulator to work at *λ* = 1.55 μm in the slow light regime are fixed to *W _{e}* = 650 nm and

*W*= 100nm, although further device optimizations and experimental tests could adjust slightly these values. The following section shows how the slow group velocity can enhance the modulator performances in terms of refractive index change and interaction length.

_{i}## 3. Slow wave modulation enhancement

The modulation enhancement is due to the intensity of the interaction between the free carriers and the slow optical mode propagating through the corrugated waveguide. Intuitively, the slower the mode propagates, the greater the interaction with carriers. Hence, the interaction length can be greatly reduced with respect to a conventional rib waveguide, resulting in a shorter phase shifter with lower power requirements. In what follows, the phase shifter is operated under reverse bias from *V _{ini}* = 0V up to

*V*= −10V, i.e.

_{fin}*ΔV*=

*V*-

_{fin}*V*< 0. As a result, the space charge region extends as carriers are depleted from the junction. The effect of the applied voltage on the real refractive index distribution is calculated in 2D with ATLAS using again Soref’s expression at

_{ini}*λ*= 1.55 μm, as shown in Eq. (2).

The real refractive index change *Δn* = *n(V _{fin})-n(V_{ini})* distribution has been worked out in 2D for both thin and wide waveguide sections and represented below on Fig. 3
for different static voltages

These real refractive index distributions have been then included into the plane wave expansion simulator BANDSOLVE ** ^{TM}** by means of a in-house conversion program, in order to work out the effective index change as a function of the applied reverse bias variations. Figure 4
below shows the refractive index change versus applied reverse bias while varying the group velocity

*V*. The desired low group velocities values have been chosen by shifting slightly the wavelength around

_{G,opt}*λ*= 1.55 μm. From a practical point of view and due to fabrication imperfections, the main issue here would be for the desired wavelength,

*λ*= 1.55 μm, to match the right group velocity. This could be achieved for instance by thermal tuning. However, in these simulations, we considered that once the desired group velocity is found (by shifting slightly the wavelength) the device is assumed to be operated in this regime at

*λ*= 1.55 μm. The effective index change enhancement can be quantified by calculating the ratio between the effective index change achieved in a corrugated waveguide and a conventional rib waveguide of dimensions 300 nm x 220 nm with a 50 nm thick slab. This figure of merit is shown in Eq. (3) and represented on Fig. 4.

The first observation that can be made is the strong effective index variation enhancement in the slow light regime compared to the conventional rib waveguide. The information that can be directly extracted from Fig. 4 is the size reduction with respect to conventional rib waveguides. As an example, the refractive change *Δn _{eff}* = 2.912.10

^{−4}or an applied voltage variation of −10 V in a rib waveguide, leads to an interaction length

*L*= λ/2

_{π,rib}*Δn*= 2661.4 μm to achieve a π-phase shift. Figure 4 shows that this refractive index change is more than one order of magnitude greater (10.94) with a group velocity

_{eff}*V*= 0.02c at −10V. The resulting length to achieve the same phase shift in the slow wave device is then only

_{G,opt}*L*= 243.3 μm. The very advantageous feature of such configuration becomes straightforward and opens the way to very compact silicon active devices. Moreover, after [6] the power consumption scales as the device length does, which is perhaps one of the main achievement for reliable photonic integration. However, such enhancement can hide two major challenges due to low group velocity, which are the propagation losses and bandwidth limitations. The following sections are devoted to the study of both potential issues.

_{π,corr}## 4. Handling slow wave induced propagation losses

Propagation losses have always been mentioned as an issue in PhC based devices. The insertion losses in a conventional modulator are essentially due to charges carrier absorption and sidewall roughness. In 2D planar PhC, it has been recently mentioned in [14] that out of plane scattering is the dominant loss mechanism for moderately low group velocities, i.e. *V _{G}* down to 0.03c (and probably lower) and increases sub linearly with inverse group velocity. In the case of the corrugated waveguide, out of plane scattering losses are intuitively not the dominant mechanism (in the vertical direction the waveguide has rather smooth texture). On the contrary, the major sources of losses are expected to be caused by the sidewall roughness as in conventional rib waveguide. Figure 1 shows that the slow mode is mainly confined into the narrow part of the waveguide. Therefore, we have considered a simple model where the losses may be mainly due to charge carrier absorption and larger interaction of the optical mode with sidewall roughness. The scaling factor is assumed to be 1/V

_{G}as the selected group velocities lies in a moderately low (V

_{G}down to 0.02c) group velocity range. The insertion losses in dB are then given by the following Equation (Eq. (4).

The carrier absorption at a fixed voltage has been calculated through Soref’s expression Eq. (5).

_{roughness}is then introduced into Eq. (4). Figure 5 below illustrates the total insertion losses as well as the required interaction length L

_{π}to achieve a π-phase shift as a function of the applied voltage variations.

Figure 5 (a) shows that in order to maintain the insertion losses below 3 dB, the minimum applied voltage variation should lie between −7V and –8V depending upon the selected group velocity, except for the case of the rib waveguide where the 1/V_{G} dependency has not been considered in Eq. (4). The interaction length versus the applied voltage variations has been represented in the range of voltages from −7V down to −10V. For applied voltage variations between −8V and −10V and group velocities being in the range [0.02c, 0.07c], the interaction lengths lie below 750 μm.

## 5. Bandwidth limitations

Modulation speed is indeed a key aspect to evaluate the phase shifter performances. Such bandwidth limitation is generally due to the group velocity mismatch between the electrical and optical signal. This mismatch can be compensated by a convenient travelling wave design, enabling both waves to propagate with the same group velocities, i.e. *V _{G, opt} = V_{G,}*

_{el}. However, such complex architecture can be avoided by sufficiently reducing the phase shifter length, removing the need for these group velocity values to be equal. Indeed, considering an open circuit configuration (no terminating resistances) and following the reasoning of [15], the limiting modulation frequency is given by:

*L*is the interaction length, i.e. the required length to achieve a π-phase shift.

_{π}Assuming that the electrical wave travels faster than the optical wave, i.e. *V _{G,opt} << V_{G,el}*
Eq. (6) can be reduced to:

In this case, the limiting modulation frequency depends both upon the group velocity of the optical wave and the interaction length. This cut-off frequency dependence against both group and velocity applied voltage variations is represented on Fig. 6

A first observation on Fig. 6 is the strong cut-off frequency dependence on the applied voltage variations, which is basically due to the poorer efficiency of the modulator at lower voltage variations, leading to larger interaction lengths and lower cut-off frequencies according to Eq. (7). Another observation on Fig. 6 shows that lower group velocity can limit to some extent the modulation frequency in our device, although it remains well beyond 10 GHz for voltage variations between −8V and −10V. Here we assumed that *V _{G,opt} << V_{G,el}* holds for group velocities up to 0.15c. However, the frontier up to where this inequality remains valid is not well defined since the group velocity of the electrical signal will mainly depend on the coplanar microwave electrode design. Ideally, and according to [15], if a careful travelling wave electrode design with terminating impedances is made, then the electrical modulation bandwidth would be theoretically infinite. Indeed, in this configuration, the 3dB cut-off frequency is given by Eq. (8) below:

The travelling wave design is made such that *V _{G,opt} = V_{G,el}*, leading to an infinite bandwidth. However, if such design option is not selected for practical reasons, then Eq. (6) holds and the only way to avoid strong bandwidth limitations due to the factor

*V*, is having

_{G,opt}/V_{G,el}*V*, which only occurs if the optical group velocity sufficiently decreases with respect to the electrical group velocity. This challenge can be overcomed by the slow wave structures.

_{G,opt}<< V_{G,el}## 6. Conclusion

Overall, this paper has shown the modulation enhancement that a 1D PhC rib shaped corrugated waveguide embedded in a pn junction can potentially bring with respect to a conventional rib waveguide embedded in the same electrical configuration. Such enhancement is due to the reduced group velocity in the passive structure, strengthening the interaction between charge carriers and the slow optical mode. The slow wave phase shifter exhibits theoretical enhanced performances within a moderately low group velocity range [0.02c, 0.07c] under −8V to −10V applied voltage variations by showing interaction lengths and insertion losses below 750 μm and 3 dB, respectively. Moreover, the device electrical modulation bandwidth is expected to be higher than 10 GHz.

## Acknowledgements

The authors gratefully acknowledge financial support from FP6-IST 004525 ePIXnet and FP7-ICT 224312 HELIOS as well as TEC2008-06360 DEMOTEC national project.

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