Integrated electro-optic (EO) modulators are a key component in systems such as classical and quantum optical communications, photonics-based quantum computing, and sensing. These systems target high-speed and low power consumption optical modulators. Silicon (Si) modulators, which rely primarily on the plasma dispersion effect [1], are intrinsically limited in speed due to their high RC constant [2]. Si modulators relying on the Pockels effect could overcome these limitations to produce a fast and pure phase modulation. Since silicon does not have a natural χ(2) due to its centrosymmetric structure, such modulation cannot be achieved directly except by straining the crystal lattice [3] leading to a low resulting Pockels coefficient. The integration of high-χ(2) materials on the Si platform has been widely considered. These include doped polymers, barium titanate (BTO) [4], lead zirconate titanate (PZT) [4], and lithium niobate (LN) [4]. These approaches require the development of hybrid or heterogeneous integration processes that increase the technology complexity. An EO modulation in Si can also be achieved through the DC Kerr effect that electrically induces an effective χ(2), which can be, hence, exploited to vary the refractive index by applying an electrical modulation superimposed to a static field. The DC Kerr effect has been studied in bulk silica [5], bulk silicon [6,7], silicon interface [8], bulk antiferromagnetic NiO [9], and in integrated platforms including silicon–organic hybrid [10], silicon-rich nitride [11], silicon rich carbide [12], and silicon nitride [13]. It has also been studied in the silicon platform for electric field-induced (EFI) second-harmonic generation (EFISHG) [14], EO modulation (EOM) [15,16], slow light regime [17], and cryogenic experiments [18]. However, the high-speed EOM in Refs. [15–17] using PN junctions led to a plasma dispersion effect that has a higher contribution to the modulation than the DC Kerr effect. While the DC Kerr effect has been well studied in the DC regime, no assessment discriminating the contribution of the DC Kerr effect and plasma dispersion modulation in the dynamic regime has been reported to our knowledge. This paper presents a comprehensive analysis of the DC Kerr effect induced in a PIN diode inserted in a silicon Mach–Zehnder interferometer (MZI) in both static and dynamic regimes. Data transmission has been analyzed up to 100 Gbit/s in non-return-to-zero (NRZ) format. An experimental method has been developed to assess the relative contribution of plasma dispersion from the Kerr effect in the dynamic regime.

The DC Kerr effect, also known as EFI Pockels effect, originates from the third-order nonlinear susceptibility tensor χ(3) in presence of a static electric field. The refractive index change induced by the Kerr effect when a static electric field FDC and an RF field FRFcosΩt are applied to the PIN junction is given by [10]Δn(t)=3χ(3)2nsi(FDC2+12FRF2+2FDCFRFcosΩt+12FRF2cos2Ωt),(1)with Ω=2πf, f being the RF frequency, nsi=3.48 being the silicon refractive index, and χ(3)=2.8×10−19m2·V−2 at λ=1.55μm for a silicon waveguide with a cross section oriented along the crystallographic axis [110] [19,20]. Equation (1) exhibits three kinds of dependencies. The first one corresponds to the static refractive index growing with the square of the field amplitudes that will be called later the DC Kerr effect concerning FDC. The second one relies on an index modulation at an angular frequency Ω, which has its amplitude growing with the product of the DC and RF field amplitudes. It will be called later the EFI linear EO effect. At last an index modulation at a 2Ω component exhibits an amplitude growing with the square of the RF field amplitude alone. It will be called later the quadratic EO effect.

Static and dynamic studies are conducted to distinguish Kerr effects from plasma dispersion on the index variation in three different unbalanced Mach–Zehnder modulators (MZMs). They consist of either PN or PIN junctions named PN, PIN2, PIN3, and their respective intrinsic region widths are w=0, 0.33, and 1.05 μm (Fig. 1). Each junction waveguide has the same cross-sectional design with a 450 nm width, a 220 nm height, and a 100 nm slab thickness, suitable for the propagation of a single TE polarization mode (Appendix G, Fig. 10). The unbalancing of the MZMs is realized by a length difference ΔL=200μm between the arms leading to a passive phase shift Δθ=2π/λngΔL with ng=3.6, the group index of our waveguide. The operating point of the MZM can, thus, be adjusted at the quadrature (Δθ=π/2) without the need of heaters by only tuning the laser wavelength around 1550 nm.

The first experiments focus on the comparison between the three junctions in MZMs under a DC bias voltage only. The variation of the effective index of the guided mode (ΔnDC) as a function of the reverse DC voltage (VDC) applied to the junction is obtained by measuring the shift of the resonance wavelength Δλr, ΔnDC(VDC)=λrΔλr(VDC)FSR(λr)L,(2)with λr being the resonance wavelength, FSR(λr) the free spectral range of the MZM, and L the length of the electrodes all along the junctions. See Appendix A. Optical and EO simulations taking into account the DC Kerr and plasma dispersion effects were performed to design the three different PN/PIN waveguides. The measured and simulated variations of the effective index of the three junctions are presented in Fig. 1. Total refractive index modulations are in good agreement with the simulations. By increasing the width of the intrinsic region of the junction to 1.05 μm, the contribution of the plasma dispersion effect is significantly reduced to become minor compared to the DC Kerr effect, while it is dominant for the PN junction waveguide. The DC Kerr effect can, thus, contribute up to 82% of the total index change in the PIN3 junction waveguide.

The study of the EFI linear EO effect in the Ω angular frequency modulation focuses on the PIN3 junction, which shows a dominant contribution of the DC Kerr effect in the effective index change (4 times greater than the contribution from plasma dispersion) as seen in Fig. 1(f). A common DC bias voltage is applied to both arms of the MZM and a sinusoidal RF signal (f=5GHz) is split with two opposite phases to be applied in push–pull configuration. The optical wavelength is chosen to operate at the quadrature point. A simplified schematic view of the experimental setup to characterize the EOM is provided in Fig. 2(a). It is worthwhile to notice that the push–pull configuration of the MZM driving leads to assessing the index variation versus voltage as an equivalent efficiency of a single path because the measured index variation is twice the index variation in each arm while the considered voltage is twice of what is applied to each arm. The RF analysis in the push–pull configuration leads moreover to the cancellation of DC shift terms from Eq. (1) of the index variation in the MZM output measurements because the shift is the same in each arm.

The transfer function of the MZM as a function of the phase shift Δϕ(t) is P(t)P0=12{1+cos[Δϕ(t)+Δθ]},(3)with P0 being the maximum output power of the MZM. The EOM response at the Ω angular frequency can be approximated at the quadrature point (Δθ=π/2) as PΩ(t)=P0Δϕ(t)/2 with Δϕ(t)=mΩcosΩt, mΩ being the modulation index, mΩ,k being the EFI linear EO modulation index, and mΩ,c being the carrier modulation index, mΩ=mΩ,k+mΩ,c,(4)mΩ,k=Γ2πλLeff13χ(3)nsiFDCFRF,(5)with the mode overlap Γ=0.87 in the Si waveguide, the effective length Leff1=[1−exp(−αRFL)]/αRF, and the RF field loss αRF=4.3dB⋅cm−1. See Appendices B and C for more details. Both the EFI linear EO effect and the plasma dispersion effect are expected to increase linearly with the RF amplitude. Only the EFI linear EO effect is expected to increase with the applied reverse DC bias following the Eq. (5). The dynamic carrier modulation is expected to decrease with VDC considering a small signal approximation on its static response.

For a 6 mm long junction, a linear behavior of the effective index change ΔnΩ=mΩλ/(2πLeff1) as a function of the applied reverse DC bias and RF amplitude is observed in Figs. 2(b) and 2(c), respectively. This is a clear signature of the EFI linear EO effect. In Fig. 2(b), the non-zero intersection and nonlinear behavior of ΔnΩ around VDC=0V indicates that carrier modulation is dominant relative to the EFI linear EO effect at low reverse DC voltages. The slope of the curve allows us to determine the χ(3) coefficient (χ(3)=1.0×10−19m2·V−2). See Appendices D and E for more information. This value is slightly underestimated (Appendix E) due to the carrier contribution having a negative evolution with VDC. However, it remains relatively close to the χ(3) values found in the literature.

The quadratic EO effect at the angular frequency of 2Ω can only be observed in a single-drive configuration, as it is proportional to the square of the electric field. We studied the transfer function at angular frequencies of Ω and 2Ω to separate the modulation behavior resulting from the distortion produced by the nonlinear transfer function of the MZM [Eq. (3)] and the quadratic EO effect. A bandpass RF filter centered at Ω was placed at the signal generator output ensuring a very high rejection at 2Ω. We considered the PIN3 junction where distortion due to the carrier absorption modulation is negligible.

The phase shift induced by the plasma dispersion and the Kerr effects can then be written as Δϕ(t)=mΩcosΩt+m2Ωcos2Ωt,(6)where m2Ω is the modulation index associated with the quadratic EO effect, m2Ω=Γ2πλLeff23χ(3)4nsiFRF2,(7)and Leff2=[1−exp(−2αRFL)]/(2αRF) is the effective length for the 2Ω component.

The Ω and 2Ω components of the MZI spectral response can be written—after inserting the phase shift Δϕ(t) [Eq. (6)] in the MZM transfer function P(t)/P0 [Eq. (3)], performing a Jacobi–Anger expansion, and neglecting intermodulations—as follows: PΩ(t)P0=sin(Δθ)J1(mΩ)cosΩt,(8)P2Ω(t)P0=[−cos(Δθ)J2(mΩ)+sin(Δθ)J0(mΩ)J1(m2Ω)]cos2Ωt,(9)where Jn(mΩ) represents the Bessel functions of the first kind.

The modulation indices mΩ and m2Ω are determined by fitting the DC transmission and the spectral responses using Eqs. (8) and (9) at fixed reverse DC and RF voltages. See Appendix F.

The measurements performed for a 5 mm long PIN3 junction [Fig. 3(a)] show that the 2Ω component is induced by the quadratic EO effect and not the signal distortion (the modulation operates at quadrature). Then, we can extract the corresponding modulation index m2Ω from the response of the PIN3 junction. We can notice that it is, however, not possible to extract the m2Ω modulation index from the responses of the PN and PIN2 junctions because the distortion induced by carriers is too important (see Appendix F, Fig. 7). The modulation indices mΩ and m2Ω are accurately extracted at different reverse DC and RF bias voltages for the PIN3 junction using this method.

Figure 3(b) shows the linear variation of the refractive index change Δn2Ω=m2Ωλ/(2πLeff2) as a function of the square RF voltage (i.e., Δn2Ω quadratically increases with the RF voltage). This variation is independent of the applied reverse DC voltage, as expected with a quadratic EO effect. In addition, a linear fit of Δn2Ω with respect to FRF2 is performed to extract the χ(3) coefficient (χ(3)=1.5×10−19m2·V−2). This value is close to the average value from the literature and is consistent with the value found in the previous section.

Moreover, the measurements of the Ω and 2Ω components of the spectral response can be used to calculate the EFI linear EOM contribution to the modulation at Ω using Eqs. (5) and (7), mΩ,k=4FDCLeff1FRFLeff2m2Ω.(10)

The DC electric field inside the PIN junction is estimated using FDC=(VDC+Vbi)/w with Vbi the built-in voltage and w the width of the intrinsic region [16]. See Appendix D. The RF field is estimated from the small signal approximation FRF≈VRFdFDC/dVDC.

The contributions of the EFI linear EOM (mΩ,k/mΩ) and carrier modulation [(mΩ−mΩ,k)/mΩ] in the Ω spectral response are reported in Fig. 3(c) showing that, above VDC=5V and at a modulation frequency of 5 GHz, the EFI linear EO effect contribution to the modulation becomes greater than the carrier modulation and reaches more than a factor of 3 at 15 V. A good agreement with simulations from Fig. 1(f) is obtained.

The data transmission characteristics of EO modulators based on the DC Kerr effect using PIN3 diode have been analyzed. The DATA and DATA‾ signals from an SHF bit pattern generator were amplified and transmitted to the respective arms of the MZM in push–pull configuration. A schematic view of the setup is shown in Fig. 4(a).

First, optical eye diagrams were acquired at 10 Gbit/s on a digital communication analyzer (DCA) from a 6 mm long modulator with each arm driven at 4Vpp and at different reverse DC bias voltages. The extinction ratio (ER) and the signal-to-noise ratio (SNR) of the modulated optical signal were computed by the DCA. The ER is greatly improved by reverse bias VDC [Fig. 4(b)]. Indeed, for a VDC varying from 2 to 30 V, the measured ER increases from 1.5 to 3.7 dB, and the SNR increases from 8.9 to 15.6. These results are entirely explainable with the EFI linear EO modulation behavior studied in Fig. 2. (More eye diagrams as a function of VDC are presented in Appendix G, Fig. 8.)

At higher data rate, the DC Kerr effect improves the transmission capability, reaching a maximum data rate of 40 Gbit/s for the same 6 mm long PIN3 modulator with each arm driven at 4Vpp [Appendix G Fig. 9(b)]. Its speed is limited by the RF electrode bandwidth, which can be further improved by redesigning the traveling wave electrodes to achieve an expected EO bandwidth of about 40 GHz for 1 cm propagation length [21].

Then, the bandwidth limitation of DC Kerr modulators for higher speed optical modulation was investigated on shorter modulators with 1 mm long PIN3 modulator with each arm driven at 2Vpp. The obtained speed limit shows a closing of the eye diagram around 80 Gbit/s [Fig. 5(a)] imputed to the limited traveling wave electrodes bandwidth, which is the same as the achieved speed limit of 1 mm long conventional depletion modulation under same test setup [22]. At 100 Gbit/s, the use of numerical six taps feed-forward equalization (FFE) has led to the open eye diagram [Fig. 5(b)] showing that such a DC Kerr modulator associated with the proper equalizing equipment could be promising to achieve very high speed modulation.

The speed of silicon DC Kerr modulators can be improved by designing better traveling wave electrode or by removing the substrate to increase the electrode bandwidth by reducing the high-frequency electrode losses and the mismatch between the optical and RF group indices [23]. The doping level in PIN junctions does not limit their high-frequency response as their intrinsic RC bandwidth is greater due to the reduced loaded capacitance. Additionally, simulation results show the PIN modulator leads to a low doping-induced insertion loss of 0.11 dB/mm to be compared to about 1.5 dB/mm [24] for normal PN junction waveguides with VπL1.6V⋅cm. The VπL for PIN phase shifter is estimated at 22V⋅cm for the current device working at 10 Gbit/s with 26VDC–30VDC. The low optical loss of PIN phase shifters at high reverse DC biases [25] allows to build high Q-factor silicon ring resonators based on the DC Kerr effect to improve frequency comb generation quality of ring modulators [26]. Such implementation of the DC Kerr ring modulator would reduce the footprint compared to MZI, and the speed limit will only be limited by the cavity quality factor. Because the maximum refractive index shift achievable using the DC Kerr effect is obtained when the electrical field inside the junction is maximum (close to the PIN junction avalanche) [27], a trade-off needs to be made among the high DC bias necessary to reach the maximum field, the carrier contribution, and the desired EO bandwidth limited by the RC constant, all increasing with the intrinsic region width [27]. Further improvements of the DC Kerr effect could be driven by optimizing the overlap between the optical and electric fields, i.e., the optical confinement in the waveguide and by carefully selecting the doping region to maximize the electric field location inside the waveguide.

The EFI Pockels effect (i.e., DC Kerr effect) has been observed in a Si PIN junction-based MZM. The refractive index variations as a function of both reverse DC bias voltage and RF amplitude have been measured in the dynamic regime showing a linear response with the DC bias voltage at a fixed RF amplitude. The refractive index modulations at angular frequencies Ω and 2Ω resulting from an applied RF signal at the angular frequency Ω have been extracted to quantify the EFI linear EO effect contribution to the modulation. We have shown that the DC Kerr effect is the main reason for the high-speed modulation above 5 V DC bias voltages in comparison with plasma dispersion effect. Furthermore, optical modulation has been demonstrated up to 100 Gbit/s for a 1 mm long MZM. Silicon modulators based on the EFI linear EO modulation show promising characteristics for high-speed optical communications when using in short MZMs or in ring modulators. But they can also be used in quantum and LIDAR applications, where the low loss and efficiency characteristics are more important than the high-speed one, using longer phase shifters thanks to the low loss of PIN junctions and pure phase modulation of Kerr effect.

Acknowledgment. The authors acknowledge CORNERSTONE team of University of Southampton for the device fabrication. J. Peltier acknowledge Victor Turpaud for fruitful discussions, and Quentin Chateiller and Bruno Garbin for the development of the Python package Autolab used in his experiments. D. J. Thomson acknowledges funding from the Royal Society for his University Research Fellowship.