Abstract
Silicon photonics, considered as a major photonic platform for optical communications in data centers, is today also developed for others applications including quantum photonics and sensing. Advanced silicon functionalities based on optical nonlinearities are then required. As the presence of inversion symmetry in the Si crystal structure prevents the exploitation of secondorder optical nonlinearities, the generation of strain gradients in Si by a stressed material can be considered. However, due to the semiconductor nature of silicon with the presence of carriers, no clear evidence of secondorder nonlinearities have been reported yet. Here we report an experimental demonstration of highspeed Pockels effect in silicon waveguides at 1550 nm. Additionally, a theoretical model is developed to describe its frequency behavior. A secondorder nonlinear susceptibility \(\chi _{xxy}^{(2)}\) of −1.8 ± 0.2 pm V^{−1} is then experimentally determined. These results pave the way for the development of fast linear electrooptic effect for advanced silicon photonics devices.
Introduction
The development of silicon photonics has the purpose of improving data communications by combining photonics and electronics in the same integrated circuits^{1}. By taking advantage of the high index contrast, provided by the silicononinsulator (SOI) platform, compact devices can be fabricated with the wellestablished complementary metaloxidesemiconductor (CMOS) fabrication techniques used in microelectronics while reducing fabrication costs. Besides, silicon has proven to be an excellent material for the realization of a wide range of photonic devices. However, the inversion symmetry present in its crystal structure^{2} prevents the exploitation of secondorder optical nonlinearities, a set of effects that would be of great interest for silicon photonics.
Indeed, noncentrosymmetric materials are well known to exhibit secondorder electric susceptibility (χ^{(2)}) properties including secondharmonic generation and a linear electrooptic effect, i.e., Pockels effect for a large range of applications: optical communication systems, quantum photonics, security, aerospace, biology…^{3,4,5,6} Among all, lithium niobate (LiNbO_{3}) is the most frequently used due to its large χ^{(2)} coefficients, providing inter alia, efficient, and highspeed electrooptic modulation. However, for the purpose of data communication applications, lower costs, higher compactness, and higher fabrication yields on chip are required, as permitted with silicon.
The inversion symmetry limitation may be overcomed by straining the silicon lattice. Indeed, a nonuniform strain applied to the crystal induces displacements of the silicon atoms, which break its centrosymmetry. The new atomic arrangement plays an important role in the generated secondorder polarization, which in turn, directly intervenes in the components of the χ^{(2)} tensor. This technique to generate χ^{(2)} in silicon through the use of strain has been explored theoretically^{7,8,9,10,11,12,13}, as well as experimentally^{14,15,16,17,18,19} in the past years.
Indeed, several experimental demonstrations have been reported, attempting the estimations of the straininduced χ^{(2)} through the Pockels effect. This gave rise to high values of secondorder electric susceptibility, in many cases competing to those typically achievable in lithium niobate^{15,16,17}. However, without considering the inherent semiconductor nature of silicon, such analysis of a linear electrooptic effect could not account for realistic Pockels coefficients. Recently, it has been demonstrated that free carriers play a critical role on the electrooptic effect in strained silicon waveguides. By applying an electric field to a silicon waveguide, the freecarrier concentration and distribution are indeed changed, which results in a variation of the refractive index due to the plasmadispersion effect^{20,21}. Furthermore, it has been reported that the linear electrooptic effect detected in strained silicon waveguides was induced by charging centers in the SiN stress overlayer, which moves the silicon waveguide directly into an inversion regime, where the electron concentration grows linearly with the voltage^{22,23,24}. This result has deep consequences for the interpretation of the electrooptic effect in strained silicon waveguides, particularly in the conclusion that there are two main electrooptic (EO) effects that jointly contribute to the total change in effective mode index (Δn_{eff}) in a strained silicon waveguide. These effects are: the plasmadispersion effect \(( {{\mathrm{\Delta }}n_{{\mathrm{eff}}_{\mathrm{c}}}} )\) induced by carriers and the straininduced Pockels effect \(( {{\mathrm{\Delta }}n_{{\mathrm{eff}}_{\mathrm{p}}}} )\). Therefore, the overall electrooptic effect may be written as follows:
The inclusion of the freecarrier effect into the electrooptic description of strained silicon waveguides has led to the conclusion that the experimental results previously reported on the straininduced χ^{(2)} in silicon have been erroneously interpreted in the characterization of strained silicon waveguides^{22}. Consequently, one of the recent challenges is to experimentally demonstrate the straininduced Pockels effect by isolating both electrooptic effects, alongside distinctive validation of the existence of Pockels effect in strained silicon waveguides. It is wellknown that an electric field (induced by a voltage) applied into the Si waveguide automatically affects the carrier distribution, inducing a direct change in the effective index of the guided mode. However, these intrinsic changes in carrier distribution also modify the distribution of the electrical field inside the silicon waveguide, thereby significantly affecting the electrooptic efficiency induced by Pockels effect^{25}. The carrier evolution under a bias voltage induces a screening effect, which makes the probing of the Pockels effect and a subsequent estimation of χ^{(2)} coefficient in strained silicon waveguides vastly challenging.
Nevertheless, the two effects that come into play have very distinct origins, which results in different physical properties that can be advantageously exploited during the experimental characterizations. It is wellknown that the Pockels effect is an ultrafast electrooptic effect, with a response time of the order of femtoseconds^{26}, while the freecarrier effect involved in the waveguiding system (i.e., into Si waveguide, at the interfaces, and into the stress layer) is typically characterized by a comparatively lower response time of nanoseconds, corresponding to the carriers recombination time^{27,28}. Hence, when increasing the frequency in the microwave domain, the contribution of carriers to modulation becomes negligible, thereby leaving the Pockels effect to be solely responsible for overall electrooptic modulation.
In this work, we demonstrate fast electrooptic properties based on the Pockels effect at a wavelength of about 1550 nm. Its efficiency according to the direction of light propagation with respect to the crystal lattice is also experimentally and theoretically studied. Highspeed electrooptic responses are demonstrated using differentoriented travelingwave Mach–Zehnder interferometers (TWMZI) in order to dissociate both fundamental contributions, i.e., carrier and Pockels effects. Alongside the highspeed Pockelsbased electrooptic characterizations, a model is developed to fit the results obtained and retrieve critical parameters needed to describe the secondorder nonlinear susceptibility in a strained silicon waveguide.
Results
Characteristics of devices
A set of 10 asymmetric Mach–Zehnder interferometers (MZI), of length L_{arm} = 2 mm and a delay ΔL = 60 μm between the two arms, with different orientation angles φ ranging between 0° and 45° (with 5° step) according to the wafer orientation notch along the [01̄1] axis, was fabricated (Fig. 1). The different angled waveguides were designed to work in a singlemode condition with a transverse electric (TE) polarization. A crosssectional view of the electric field intensity of the fundamental TE mode is shown in Fig. 2. The sample was covered by a silicon nitride (SiN) stressed thin film, with a level of internal stress σ_{i} = −1.3 GPa to induce strain gradients in silicon waveguides. Coplanar electrodes were placed on top of the MZI to generate an electric field under an applied voltage (See Methods for fabrication details).
Frequency domain characterizations
The efficiency of a straininduced Pockels effect in silicon was determined in a highspeed regime by taking the advantage of the traveling wave electrodes placed above the MZI in order to propagate an AC electric signal at microwave frequency (see Methods for highspeed measurements details). In this frequency range, it is necessary to take into account the waving behavior of the driving electric signal and properly optimize its interaction with the optical wave. The electrodes were designed to satisfy the impedance matching and low transmission loss conditions to ensure the probing of Pockels effect. For each angled MZI, the electrooptic spectra at a quadrature point λ_{opt} (3 dB below the maximum transmission where the MZI transfer function is linearized) have been measured and analyzed. In addition, a DC bias V_{DC} between −15V and +15 V was applied to investigate the influence of carrier distribution from depletion to inversion regimes in frequency domain.
Varying the DC bias from +15 V to −15 V, a noticeable variation of the optical responses between 100 MHz and about 3 GHz was measured, while the electrical response remained unchanged at higher frequencies (beyond 3 GHz), as shown in Fig. 3 with the 40° MZI. Similar behaviors were also observed at all other angles.
The variation of the electrooptic signal in the lowfrequency regime with respect to the DC bias is attributed to variable modulation efficiencies in view of the different regimes of carriers in silicon. As a matter of fact, for reverse bias, free carriers progressively enter to the depletion regime, in which their redistribution is more easily affected under an AC voltage, as shown in Fig. 4. As a consequence, this leads to stronger variations of the electric field within the silicon waveguide, as shown in Fig. 5.
Conversely, for forward bias, the inversion regime is reached and the redistribution of free carriers is less affected by an AC voltage. This reduces the electric field redistribution in the silicon waveguide and leads to lower RF responses. Furthermore, at the given lowcarrier distribution change obtained, plasmadispersion effect has a weak contribution on the electrooptic response. Besides at highfrequency regime, the electrooptic response (with respect to the DC bias) is unchanged. This can be explained by the fact that at such frequencies free carriers can no longer keep up the fast voltage variations, and their distributions in the waveguide remain unchanged. Consequently, the measured responses from about 3 GHz is attributed to the Pockels effect. Finally, the decreasing of the signal beyond 10 GHz is explained by the group velocity mismatch between the microwave and the optical signals (as will be explained later).
Silicon being a crystal, different χ^{(2)} values are expected along its crystallographic axis. Then, at a given forward DC bias (+15 V) to reduce the impact of carriers on the electrooptic behaviors at low frequency, the effect of the orientation angle of the MZI according to the crystallographic directions φ has been studied (Fig. 1). This angle variation allowed studying the evolution of the Pockels effect for different strain gradient conditions. Fig. 6 presents the electrooptic response as a function of the frequency for 10° and 40° TWMZI. For similar electrical and optical input signals, we noticed a difference of the response level, while the overall behavior remained identical.
This difference of the response level with respect to the orientation angle φ, is illustrated in Fig. 7 at 15 GHz. A clear increase in Pockels response was observed when the orientation angle increased, proving that the crystallographic directions play an important role in secondorder nonlinear process efficiency.
Theoretical analysis of the Pockels effect
Due to silicon mechanical anisotropy (see Methods), higher strain gradients and then higher χ^{(2)} coefficients, are obtained when the angle φ increases. In addition, it has been previously shown^{13} that any \(\chi _{ijk}^{(2)}\) component can be expressed as a linear combination of the strain gradient components \(\eta _{mnl} = \frac{{d\varepsilon _{mn}}}{{dl}}\) (i.e., the variation of the deformation ε_{mn} along the direction l in the xyz coordinate system shown in Fig. 1):
where the coefficients Γ_{ijk,mnl} depend only on the nature of the Si crystal^{13}. These Γ coefficients are of paramount importance, as they allow the determination of χ^{(2)} components in every scenario, whatever the stress applied to silicon. However, they can only be determined from robust experiments, which were not available until now due to carrier contributions. This is a critical issue for proper optimizations of secondorder nonlinear processes in silicon waveguides. In view of our set up configuration (light polarization and electric field direction), only the \(\chi _{xxy}^{(2)}\) component played a major role on the detected Pockels responses, and the main two strain gradient components are η_{xxy} and η_{yyy}, when applying SiN stress layer on the top of Si waveguides. Thus, Eq. (2) applied to \(\chi _{xxy}^{(2)}\) can be simplified into:
In our approach, the linear electrooptic effect (i.e., Pockels effect) in silicon was determined in highspeed regime. In this way, the frequency evolution including carriers and Pockels was taken into account. Then, an estimation of Γ coefficients could be performed by extending the model developed in ref.^{25} to the microwave domain. The model hereafter consists the description of the S_{21} electrical signal of the electrooptic responses measured, as shown in Fig. 6, in order to give an accurate value of χ^{(2)} for strained Si photonics structures. These electrical signals, R_{m}, in frequency domain are defined as:
where \(P_{{\mathrm{out}}}^{{\mathrm{AC}}}\) is the optical power at the output of the TWMZI and detected by the highspeed photodiode, \(V_{{\mathrm{in}}}^{{\mathrm{AC}}}\) is the AC peak voltage at the input of the microwave chain, and Z_{0} is the 50 Ω reference impedance. The optical power \(P_{{\mathrm{out}}}^{{\mathrm{AC}}}\) was expressed as a function of the optical input power P_{in} and the AC optical phase shift Δϕ^{AC} (See Methods). The expression shown in Eq. (4) includes each physical effect that play a role in the electrooptic effect, including both freecarrier plasma dispersion and Pockels effects. To extract the straininduced Pockels effect, we consider the response from 3 GHz to 30 GHz, as it is the predominant effect in the considered frequency domain.
Using the approximation obtained in Eq. (3), the Pockels AC phase shift \({\mathrm{\Delta }}\phi _{{\mathrm{P}}}^{{\mathrm{AC}}}\) can then be written as (see Methods):
where λ_{opt} is the optical wavelength, V_{DC} the DC voltage, V_{AC} the AC voltage, f_{m} the microwave frequency, and \(\widetilde{\widehat{\eta_{mnl}^{ijk}}}\) the harmonic effective strain gradients. The calculation of the term P^{(1)}(f_{m}) was performed by describing the AC voltage, at any position z along the TWMZI, as follows^{29,30,31}:
where \(V_{{\mathrm{AC}}_0}\) is the AC voltage at the input of the TWMZI, f_{m} the microwave frequency, a_{m} the microwave loss, and \(\delta _{{\mathrm{mo}}}\left({f_{\mathrm{m}}}\right) = n_{{\mathrm{g}}_{\mathrm{m}}}\left( {f_{\mathrm{m}}} \right)  n_{{\mathrm{g}}_{\mathrm{o}}}( {\lambda _{{\mathrm{opt}}}} )\) is the group velocity mismatch between the microwave and the optical signals. The behavior of \(V_{{\mathrm{AC}}_0}\) was determined from Sparameter measurements (see Methods). The microwave loss, a_{m}, was retrieved using different coplanar waveguide lengths, and the microwave group index was estimated at about 2.15, between 3 GHz and 30 GHz, due to a low dispersion of the coplanar waveguide in this frequency range. Hence, P^{(1)}(f_{m}) mainly drives the shape of the electrooptic responses and explains the drop in modulation efficiency after 10 GHz by the group velocity mismatch between the two copropagating waves. Thus, knowing the propagative efficiency P^{(1)} as a function of the microwave frequency, it is possible to fit the highspeed electrooptic results via the angle dependence and estimate both unknown coefficients Γ_{xxy,xxy} and Γ_{xxy,yyy}.
To determine the carrier regime behavior in the silicon waveguide, highfrequency capacitance–voltage (CV) and conductance–voltage (GV) measurements were performed in order to retrieve carrier densities via the estimation of the surface fixedcharge Q_{f} and the interface traps D_{it} densities (see Methods). Results have shown that Q_{f} ~ 10^{12} cm^{−2} and D_{it} ~ 10^{12} cm^{−2}. Semiconductor simulations in DC regime can be considered to describe the electric field distribution within the silicon waveguide for any DC bias applied, including the influence of Q_{f} and D_{it}. Furthermore, by performing harmonic perturbationbased simulations at high frequencies around a DC operating point, the distribution of the electric field in AC regime is calculated. The simulations of optical modes, strain gradients, carriers, and electric field distributions were performed using COMSOL Multiphysics software. Then, the variation of the different \(\widetilde{\widehat{\eta_{mnl}^{ijk}}}\) with respect to the AC voltage V_{AC} is determined. Consequently, it is possible to calculate the slopes \(\frac{\partial \widetilde{\widehat{\eta_{mnl}^{ijk}}}}{{\partial V}}\) around every DC operating point and for each orientation angles φ (see Methods). Finally, by fitting experimental results from Fig. 7 with this described model, we found:
These values are comparable with the theoretical ones reported in^{13}, indicating that our experimental approach is appropriate to describe these coefficients in strained silicon. Furthermore, we underline that both Γ_{xxy,xxy} and Γ_{xxy,yyy} are unique for silicon and do not depend on the stress applied to silicon nor its orientation. Moreover a good agreement between experimental results and the model developed in this work was obtained at all frequency, as shown in Fig. 6.
Finally, the distribution in a silicon waveguide of any χ^{(2)} components is deduced using the strain gradient tensor. In Fig. 8, we determined the distribution of \(\chi _{xxy}^{(2)}\) in the silicon waveguide for different crystal orientations.
Discussion
First of all, it must be noted that higher \(\chi _{xxy}^{(2)}\) values are achieved at 45° orientation angle, corresponding here to the [01̄0] crystallographic direction. This is in line with the fact that stronger strain gradients and higher Γ_{ijk,mnl} coefficients are achieved for <100> equivalent directions. Furthermore, we notice that for a given orientation angle, the highest values of \(\chi _{xxy}^{(2)}\) (few pm V^{−1}) are only achieved at the corners of the waveguide where the strain gradients are the strongest, but their overlap with the optical mode the lowest. Due to the heterogenous behavior of \(\chi _{xxy}^{(2)}\) into the waveguide, an effective value of \(\chi _{xxy}^{(2)}\), which makes sense when describing the secondorder nonlinear effects in silicon, must then be determined. This effective value is obtained by computing the overlap integral of the TE propagating mode at a wavelength of 1550 nm and the strain gradients. Hence, an effective \(\chi _{xxy}^{(2)}\) coefficient of −1.8 ± 0.2 pm V^{−1} was then determined at the 45° orientation angle. In comparison, lithium niobate (LiNbO_{3}) and lithium tantalate (LiTaO_{3}) have, respectively, a nonlinear susceptibility \(\chi _{xxy}^{(2)}\) of 72 and −2 pm V^{−1}, which is either comparable or one order of magnitude higher than strained silicon for this specific χ^{(2)} component. The small value achieved for strained silicon is explained by the presence of sign inversion within \(\chi _{xxy}^{(2)}\) distribution and a poor overlap with the mode that significantly decreases its effective value, thereby reducing the overall efficiency of secondorder nonlinear processes involved. However, it is worth noting that several solutions can be developed to enhance the nonlinear efficiency in strained silicon photonics devices. The first solution consists in increasing the strain gradients by means of higher stress (greater than 2 GPa) applied to silicon using other straining materials than SiN, such as aluminum nitride (AlN), alumina (Al_{2}O_{3}), or functional oxides. Then, an optimal designed waveguide can also be considered in order to increase the overlaps between the optical mode and the strain gradients region. For instance, a strip waveguide including on its top shallow etched grooves and narrow slot may improve the strain gradient distribution and provide better overlaps with the optical mode. Asymmetric waveguides can also be a solution to better improve the strain gradient distribution according to the optical mode profile. On the other hand, increased nonlinear performances can also be obtained, depositing structured stressed layer on top of the silicon waveguide. Finally, the increase of the electric field using PIN diode in silicon could also allow achieving strong linear electrooptic effect.
In conclusion, we determined with no ambiguity a linear electrooptic effect in silicon, a wellknown centrosymmetric material. A clear distinction between carrier effect and Pockels effect was achieved for the first time in highspeed regime. Besides, we characterized the straininduced χ^{(2)} tensor in silicon through the measurements of Pockels responses in the microwave regime. A multiphysics model has been developed in order to accurately describe these frequency responses and extract critical coefficients (i.e., the Γ_{ijk,mnl} coefficients) necessary for the complete identification of each χ^{(2)} components with respect to the strain gradient distributions induced in a silicon waveguide. Hence, this model allowed the calculation of an effective value of −1.8 ± 0.2 pm V^{−1} for the \(\chi _{xxy}^{(2)}\) component for a conventional silicon strip waveguide strained by SiN. Finally, we want to highlight that the knowledge of the unique Γ_{ijk,mnl} coefficients paves the way for the optimization of siliconbased photonic structures. Considering a stressed layer of 5 GPa, which is not the maximum value achievable in silicon, and an optimized strain gradient distribution into the silicon waveguide, more than 100 pm V^{−1} may be reached, suitable for efficient light modulation.
Methods
Sample fabrication
The device was fabricated using a ptype highresistivity SOI wafer with a 260nmthick silicon film on a 2μmthick silicon dioxide. A first electron beam lithography was performed to define the waveguides in a ZEP ebeam resist. Then, fully etched waveguides were obtained using an inductively coupled plasma etching technique. A 700nmthick highly stressed (σ_{i} = −1.3 GPa) silicon nitride was deposited by plasma enhanced chemical vapor deposition (PECVD) technique in order to strain the silicon waveguides. A second lithography level was performed to define the coplanar electrodes. A 500nmthick gold layer was evaporated all over the sample and liftedoff away to make the travelingwave electrodes. Finally, the samples were cleaved to enable the optical edge coupling measurements.
Highspeed measurements setup
Highspeed measurements were carried out to obtain the electrooptic response of TWMZI as a function of the RF/microwave electric signal frequency.
A Sparameters test set combined with a lightwave component analyzer (LCA) was used to generate a RF electric signal, with a frequency variation from 100 MHz to 40 GHz. This electric signal was subsequently amplified using a RF modulator driver working between 40 kHz and 38 GHz before being injected into a travelingwave coplanar electrode via GSGbased RF probes. Two bias tees were added, one between the output of the RF modulator driver and the input GSGbased RF probes, and the other after the output GSGbased RF probes. A DC bias between −15 V and +15 V could be superimposed with the RF signal in order to modify the carrier regime within the silicon waveguide. The RF signal propagated through the circuit until a 50 Ω load, while the DC bias was applied in an open circuit to avoid high electric current along the electrodes.
In parallel, a continuous wavelength tunable laser working in the Cband was employed to inject a TEpolarized optical signal through the use of a lensed optical fiber. A total of 90% of the output power was then coupled to the LCA, while the remaining 10% was detected by an optical component power meter to retrieve the optical transmission of the Mach–Zehnder interferometers.
The modulated optical signal was detected by a highspeed photodiode included inside the LCA and converted into an electrical signal. The later was then normalized by the input electrical signal generated from the Sparameters test set and the result was plotted on a vector network analyzer to read and characterize the electrooptic response of the modulator.
where \(P_{{\mathrm{out}}}^{{\mathrm{AC}}}\) is the optical power at the output of the TWMZI and detected by the highspeed photodiode, \(I_{{\mathrm{in}}}^{{\mathrm{AC}}}\) is the AC peak intensity, \(V_{{\mathrm{in}}}^{{\mathrm{AC}}}\) is the AC peak voltage at the input of the microwave chain, and Z_{0} is the 50 Ω reference impedance.
All the RF components were separately characterized between 100 MHz and 40 GHz with a Sparameters test set. The calibration was performed using a SOLT (short–open–loadthru) tool kit.
Characterization of surface fixedcharges and interface traps densities through CV and GV measurements
The offload silicon electronic configuration is known via the characterization of the surface fixedcharges Q_{f} and interface traps D_{it} densities. Indeed, positive charges are present in the SiN cladding around the silicon waveguide and consequently a constant electric field is applied to the waveguide, moving carriers to the inversion regime. Furthermore, interface traps, caused by defects at the Si/SiN interface, creating new energy levels in the silicon band gap, alter the redistribution of carriers under a DC bias. Highfrequency capacitancevoltage (CV) and conductance–voltage (GV) measurements of stressed SiN insulator layers on silicon substrate were performed to calculate Q_{f} and D_{it}, using methods described in ref.^{32}.
Silicon mechanical anisotropy
Silicon being an anisotropic mechanical material, its Young’s modulus and Poisson’s ratio are varying according to the crystal orientation^{33}, as shown in Fig. 9. When the angle of the waveguide increases in reference of the [01̄1], Young’s modulus and Poisson’s ratio are, respectively, decreasing and increasing. This means that silicon becomes more easily strained and deformed by a given level of external stress.
Expression of the optical power at the output of the DUT using the MZI transfer function
The MZI transfert function is expressed as:
where P_{out} is the output optical power of the MZI, P_{in} the input optical power, and Δϕ is the optical phase shift between the two arms. The later can be decomposed as the sum of a DC component Δϕ^{DC} (that also takes into account the initial asymmetry of the MZI) and an AC component Δϕ^{AC}. Indeed in our experiment, the voltage applied to the electrodes is a combination of a DC bias and an AC signal, and both of them induced electrooptic effects. Because every measurements were performed at quadrature operating points (where the transfer function is linearized), Eq. (9) is simplified into:
Expression of Γ coefficients
In ref.^{13}, it has been reported that the Γ_{ijk,mnl} coefficients can be expressed as a function of two unknown parameters α and β. In our experiments, only Γ_{xxy,xxy} and Γ_{xxy,yyy} are needed to describe the main component responsible for Pockels responses, i.e., \(\chi _{xxy}^{(2)}\). They are written hereafter:
with d = 0.235 nm the distance between the atoms in the silicon lattice, K = −1.18 × 10^{29} C^{3}m^{−3}eV^{−3}^{13} and φ the TWMZI orientation angle.
Modeling description of optical phase shift induced by Pockels effect in AC regime
The optical phase shift Δϕ_{P} induced by Pockels effect along the arm of one φangled MZI is expressed as:
The effective index change \({\mathrm{\Delta }}n_{{\mathrm{eff}}_{\mathrm{P}}}\) was described using the method employed in ref.^{25}, which combined the perturbation theory for waveguides with the description of straininduced χ^{(2)}. Hence, under an arbitrary voltage V(z) we have
where \(\widehat {\eta _{mnl}^{ijk}}\) are the effective strain gradients, describing the overlap between the mode profile, each strain gradient tensor components η_{mnl} and the electric field induced by the electrodes. Such overlaps are written
where the parameter N is given by
and corresponds to the active power of the optical mode.
During the highspeed experiments performed, the voltage applied at a position z along the electrode was a combination of a DC bias and a microwave AC voltage with frequency f_{m}
The DC bias V_{DC} allowed the modification of the carriers regime and in turn the electric field within the silicon waveguide while the AC component V_{AC} was used to probe the response of Pockels effect at high frequencies around that DC operating point. A firstorder Taylor expansion can be performed around the DC operating point such that the effective strain gradients can be decomposed into a DC and an AC term as follows
where \(\widetilde {\widehat {\eta _{mnl}^{ijk}}}\) is the harmonic effective strain gradient calculated considering V_{AC} as a harmonic perturbation. By combining Eqs. (13) and (17) the Pockelsbased effective index change can be expressed as the sum of a DC bias \({\mathrm{\Delta }}n_{{\mathrm{eff}}_{\mathrm{P}}}^{{\mathrm{DC}}}\) and an AC component \({\mathrm{\Delta }}n_{{\mathrm{eff}}_{\mathrm{P}}}^{{\mathrm{AC}}}\). The later is expressed
By integrating equation (18) along the interaction zone of the TWMZI, as showed by equation (12), we may write the phase change induced through Pockels effect \({\mathrm{\Delta }}\phi _{\mathrm{P}}^{{\mathrm{AC}}}\) as
\(T_{\alpha ,\beta }^{(1)}\) and P^{(1)} represent the transverse Pockels efficiency and the propagative Pockels efficiency respectively. Both terms depict how the microwave modulation is transfered to the optical wave.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (ERC POPSTAR  grant agreement No 647342). Authors thank JeanMarc Fedeli from CEA/LETI, France for its technical help, and Frederic Boeuf and Charles Baudot from STMicroelectronics for their support in strained silicon photonics developments. The fabrication of the devices was performed at the technological nanocenter of C2N, which was partially funded by the Conseil Général de l’Essonne. This work was supported by the French RENATECH network. D.M.M. acknowledges support by the Institut Universitaire de France.
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X.L.R. and M.B. fabricated the samples and developed the Si strained process with the help of P.D. and G.M. D.B. helped M.B. with the optical coupling of light in the samples. P.C., D.M.M., and M.B. performed microwave electrical characterizations of the coplanar waveguides and RF devices. Electrooptic measurements were carried out by M.B. with the help of D.P.G., C.A.R., and L.V. M.B. developed the highspeed model with the help of P.C. and E.C. L.V. initiated and supervised the research project. All authors discussed the results and contributed to writing the manuscript.
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Correspondence to Mathias Berciano.
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Berciano, M., Marcaud, G., Damas, P. et al. Fast linear electrooptic effect in a centrosymmetric semiconductor. Commun Phys 1, 64 (2018). https://doi.org/10.1038/s420050180064x
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