1. Introduction

Broadband efficient electro-optic (EO) modulators are key elements for a wide variety of applications, ranging from high-speed optical communications [1–6] to ultra-fast signal processing [7,8] and further to optical metrology and sensing [9,10]. Among the various technology platforms that are available today, silicon photonics (SiP) represents a particularly promising option, exploiting mature CMOS fabrication processes for cost-efficient mass production of densely integrated photonic circuits [11–13]. When it comes to EO modulators, hybrid combinations of SiP slot waveguides with highly efficient organic EO cladding materials are an attractive approach to overcome the intrinsic lack of second-order optical nonlinearities in bulk silicon [14]. Exploiting theory-guided optimization of organic EO materials on a molecular level [15], these silicon-organic hybrid (SOH) devices primarily stand out due to small voltage-length products down to 0.32 Vmm [16] — more than an order of magnitude below that of conventional depletion-type SiP modulators [17]. SOH devices have been demonstrated to offer attractive performance parameters such as EO bandwidths of 100 GHz [18] as well as line rates of up to 120 Gbit/s for intensity modulation [5,19] and up to 400 Gbit/s for coherent 16QAM signaling [6], while supporting driver-less operation with sub-1 V drive signals obtained from binary outputs of a standard CMOS FPGA [20]. However, the EO bandwidth of these devices is still subject to large uncertainties. In addition, most of the aforementioned experiments [5,6,18,19,21] relied on a so-called gate voltage that is applied across the buried oxide (BOX) of the underlying silicon-on-insulator (SOI) substrate to induce a highly conductive electron accumulation layer that reduces the RC time constant of the slot waveguide [22]. Moreover, an experimentally verified model describing the EO frequency response of SOH modulators is still lacking, which represents a major obstacle towards a targeted design of the modulator structures and of the associated drive circuitry.

In this paper, we formulate and experimentally verify a reliable quantitative model of slot-waveguide modulators that accounts for the various bandwidth-limiting effects. We build upon an equivalent-circuit approach proposed by Witzens et al. [23], which relies on a distributed-element description of the slot waveguide structure and of the associated radio-frequency (RF) transmission line. We refine this model and verify it quantitatively using experimental data obtained from SOH devices. We find that the model accurately predicts the EO frequency response of a slot-waveguide modulator based on purely electrical RF characteristics of the device, which are easily accessible by measurements. Our model allows to identify non-optimum doping profiles as the main reason for a limited EO bandwidth of our current device generation. Based on these findings, we extract design rules that lead to an optimum trade-off between optical loss due to free-carrier absorption (FCA), EO bandwidth, and π-voltage. Optimized device designs should lead to sub-1 mm Mach-Zehnder modulators (MZM) with π-voltages smaller than 1 V, electro-optic-electric (EOE) bandwidths larger than 100 GHz, and FCA-induced optical losses below 0.1 dB. To the best of our knowledge, this work represents the first experimental validation of a general model of slot-waveguide modulators. We believe that quantitatively reliable models represent a key step towards fast and reliable design and efficient wafer-level characterization of slot-waveguide modulators which are embedded into complex optical systems and RF drive circuits.

The paper is structured as follows: In Section 2, we introduce the concept of SOH modulators and formulate a model based on distributed elements. In Section 3, we discuss S-parameter measurements of SOH modulators, extract the characteristic impedance and the RF propagation parameter, and determine the parameters of the equivalent RF circuit by a least-squares fit. In Section 4, we validate the model by comparing the measured EO frequency response to the model prediction using parameters extracted from purely electrical measurements. In Section 5, we summarize the findings and discuss design guidelines and achievable performance parameters of SOH EO modulators.

2. Device principle and model based on distributed-element equivalent circuit

The concept of a slot-waveguide SOH Mach-Zehnder modulator (MZM) is depicted in Fig. 1 [24–26]. The MZM consists of two parallel phase shifter sections, which typically have a length $\ell $ between 0.5 mm and 1.5 mm, see Fig. 1(a). The electric drive signal is applied via a coplanar RF transmission line in ground-signal-ground (GSG) configuration. The device is realized as a travelling-wave structure, in which the modulating electric mode co-propagates with the optical mode. Each phase shifter comprises a silicon slot waveguide with a slot width typically between 60 nm and 160 nm, which is filled with a highly efficient organic electro-optic (EO) material, that may offer in-device EO coefficients in excess of 300 pm/V [16], see cross-section in Fig. 1(b). To activate the macroscopic EO activity in the slot, chromophores in the organic EO material have to be aligned in a one-time poling process, see [24,27] for details. The rails of the slot waveguide are connected to the metal RF transmission line via thin, doped silicon slabs, such that the externally applied drive voltage predominantly drops across the narrow slot. At the same time, the optical mode is highly confined to the slot region, leading to a very strong interaction with the modulating electric field. The EO frequency response of SOH modulators strongly depends on the RC time constant that is associated with the slot capacitance and the slab resistance. A two-level doping profile as illustrated in Fig. 1(b) can help to increase the bandwidth while maintaining acceptable optical loss [28,29]: In the rails and near the slot, the doping concentration is kept low (light blue) to avoid excessive free-carrier absorption, while a higher doping concentration (dark blue) is used further away from the optical waveguide to increase the electric conductivity. The impact of increased slab conductivity can also be studied by applying a DC “gate” voltage U_gate between the substrate and the slabs. This induces a highly conductive electron accumulation layer at the interface between the Si slabs and the BOX [22].

 

Fig. 1. Device concept and equivalent-circuit model. (a) Schematic top view of the Mach-Zehnder modulator (MZM) with metal electrodes (yellow) that form a coplanar ground-signal-ground electrical transmission line of length $\ell $. The device is equipped with contact pads at both ends. The optical waveguide depicted in blue is split into two arms to realize the Mach-Zehnder interferometer. (b) Cross-section of the MZM along the line A-A’ indicated in Subfigure (a). Each arm comprises a SiP slot waveguide filled with an organic electro-optic (EO) material. The metal electrodes are connected to the rails of the slot waveguide via thin, doped Si slabs. A modulating voltage U_mod, applied to the signal electrode, drops mainly across the narrow slot. This leads to a strong electric field in the slot that changes the refractive index of the organic EO material and thus modulates the phase of the optical wave. The Si slabs are doped to increase the conductivity. A low doping concentration (light blue) is used near the slot to keep optical losses low. A higher doping concentration (dark blue) is used further away from the slot to achieve a low resistance of the slabs and to thus decrease the RC time constant of the modulator. In addition, a DC “gate” voltage U_gate can be applied between the substrate and the ground electrodes. This induces a highly conductive electron accumulation layer at the interface between the Si slabs and the buried oxide layer (BOX) and thus further decreases the RC time constant. (c) Enlarged cross-sectional view of the MZM along with the associated elements of an equivalent-circuit representation of the modulator. The capacitance of the slot is represented by a capacitor with capacitance C_S/2. The finite conductivity of each slab is represented by two parallel conductances G_S,bulk and G_S,acc that account for the conductance of the bulk Si slab and of the accumulation layer formed by the gate voltage, respectively. The metal electrodes have a total capacitance C_TL. (d) Equivalent-circuit model of an infinitesimally short element of the electrical transmission line formed by the MZM. The distributed elements shown in yellow correspond to the metal electrodes. The distributed elements shown in blue represent the additional RC shunt load formed by the slot waveguide. (e) Telegrapher-type equivalent-circuit representation of an MZM transmission-line element [23]. This representation is obtained by transforming the circuit in Subfigure (d), using a frequency-dependent shunt conductance ${\bar{G}_\textrm{S}}$ and a frequency-dependent shunt capacitance ${\bar{C}_\textrm{S}},$ see Eq. (2) and Eq. (3).

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For a quantitative description of the electric behavior of the SOH modulator, we adapt the model suggested in [23], which has also been adapted to model depletion-type pn-modulators [30–36]. Reference [23] is primarily focused on the design of phase-matched travelling-wave transmission lines, which is key to achieve broadband operation of MZM with rather large interaction lengths $\ell $ of several millimeters. Over the previous years, however, substantial improvements of organic EO materials and in-device EO efficiencies [16] have allowed to significantly reduce the interaction length. This renders phase-matching less crucial and requires adapted design considerations that focus on the RC time constant of the slab/slot configuration and on the associated trade-off with respect to optical loss.

In our analysis, we rely on a distributed-element model of the underlying RF transmission line, see Fig. 1(c) for a cross-sectional view with elements of an equivalent circuit representing an infinitesimally short transmission-line section of length $\Delta z$. The full equivalent circuit is shown in Fig. 1(d). First, we consider the circuit elements depicted in yellow in Fig. 1(d) that describe the coplanar GSG transmission line formed by the metal electrodes only, without the optical slot-waveguide structure. These circuits elements comprise a series resistance ${R^{\prime}_{\textrm{TL}}}\Delta z$, a series inductance ${L^{\prime}_{\textrm{TL}}}\Delta z$, and a shunt capacitance ${C^{\prime}_{\textrm{TL}}}\Delta z$, where ${R^{\prime}_{\textrm{TL}}},$ ${L^{\prime}_{\textrm{TL}}},$ and ${C^{\prime}_{\textrm{TL}}}$ are differential quantities that describe the resistance, inductance, and capacitance of the GSG transmission line per length in Ω/m, Η/m, and F/m, respectively. This corresponds to the classical telegrapher circuit with negligible shunt conductance. In a second step, we expand the equivalent circuit by the electrical representation of the Si structure (blue) forming the optical slot waveguide. For practical doping concentrations, the conductivity of the slabs is much smaller than that of the metal electrodes. As a consequence, the longitudinal currents can be assumed to be confined to the metal electrodes, whereas the currents in the silicon slabs flow predominantly in the transverse (x) direction between the electrodes and the slot. These assumptions are in accordance with findings from finite-element simulations [23] and allow us to model the electrical behavior of the optical slot waveguide by simply adding an RC shunt to the equivalent circuit [23,30,37]. The associated elements are the differential capacitance ${C^{\prime}_\textrm{S}}$ of the pair of slots, the differential conductance ${G^{\prime}_{\textrm{S,bulk}}}$ of the resistive Si slabs to both sides of each of the slot waveguides, and the differential conductance ${G^{\prime}_{\textrm{S,acc}}} = {g^{\prime}_{\textrm{S,acc}}}{U_{\textrm{gate}}}$, which accounts for the increased conductance of an accumulation layer that may be induced by applying a gate voltage U_gate. Note that the capacitance between center signal electrode and each of the ground electrodes amounts to ${{{C_{\textrm{TL}}}} \mathord{\left/ {\vphantom {{{C_{\textrm{TL}}}} 2}} \right.} 2}$, and that the overall shunt capacitance of the metal electrodes forming the coplanar GSG transmission line is hence ${C_{\textrm{TL}}} = {C^{\prime}_{\textrm{TL}}}\Delta z$. Similarly, ${{{C_\textrm{S}}} \mathord{\left/ {\vphantom {{{C_\textrm{S}}} 2}} \right.} 2}$ is the capacitance of an individual slot, and ${C_\textrm{S}} = {C^{\prime}_\textrm{S}}\Delta z$ represents the combination of both slots. The quantities ${G_{\textrm{S,bulk}}} = {G^{\prime}_{\textrm{S,bulk}}}\Delta z$ and ${G_{\textrm{S,acc}}} = {G^{\prime}_{\textrm{S,acc}}}\Delta z$ refer to the slab conductance to either side of each slot, and the equivalent circuit-element in Fig. 1(d) represents the combined effect of four slabs. The total differential conductance is ${G^{\prime}_\textrm{S}} = {G^{\prime}_{\textrm{S,bulk}}} + {G^{\prime}_{\textrm{S,acc}}}.$ The two slot waveguides thus create a differential shunt admittance ${\underline{Y} _S}^\prime $ given by

The overall circuit shown in Fig. 1(d) can be transformed into the circuit in Fig. 1(e) by introducing the frequency-dependent differential conductance

For analyzing the EO interaction, we now consider a travelling-wave EO modulator with characteristic impedance Z₀, complex RF propagation parameter γ, optical group refractive index n_g,opt in the phase shifter, and active phase-shifter length $\ell $. The device is connected to an RF source with impedance Z_src and terminated by an impedance Z_trm. Due to the Pockels effect, the local optical phase shift increases in proportion to the local modulation voltage. If a sinusoidal voltage with angular frequency ${\omega _{\textrm{RF}}}$ is applied to the modulator’s signal electrode, the total phase shift $\varphi (t) = \Re \{{\underline{\hat{\varphi }} ({\omega_{\textrm{RF}}}){\textrm{e}^{\textrm{j}{\omega_{\textrm{RF}}}t}}} \}$ in each arm can then be obtained from an integral of the modulation voltage seen by the optical signal along the phase shifter length $\ell $, see Ref. [39], Eq. (2). In this relation, the complex amplitude $\underline{\hat{\varphi }} ({\omega _{\textrm{RF}}})$ contains both the frequency-dependent amplitude and the phase of the sinusoidally varying phase shift $\varphi (t)$. The complex frequency response ${\underline m _{\textrm{PH}}}({\omega _{\textrm{RF}}})$ of such a phase modulator is given by the phase amplitude $\underline{\hat{\varphi }} ({\omega _{\textrm{RF}}})$, normalized to its value that would be obtained for an angular frequency ${\omega _{\textrm{RF}}} = 0$ for a source and a termination impedance that are both perfectly matched to the modulator’s characteristic transmission-line impedance ${\underline{Z} _0}$ ([39], Eq. (3)) . This leads to

To apply the model developed here to a given device design, the complex propagation parameter γ and the characteristic impedance Z₀ can be directly determined from S-parameter measurements or with numerical mode solvers. The equivalent-circuit parameters ${R^{\prime}_{\textrm{TL}}},{L^{\prime}_{\textrm{TL}}},{C^{\prime}_{\textrm{TL}}},{G^{\prime}_\textrm{S}},$ ${C^{\prime}_\textrm{S}}$ can then be obtained by fitting Eq. (4) and Eq. (5) to the frequency-dependence of γ and Z₀, see Section 3. If, in addition, n_g,opt is known, Eq. (6) allows to predict the EO response of the modulator.

3. Electrical characterization and fitting procedure

To experimentally verify the model introduced in Section 2, we determine the frequency-dependent characteristic impedance Z₀ and propagation parameter γ of a typical SOH modulator. These parameters fully describe the electrical behavior of the device and build the base for determining the frequency response $|{{{\underline{m} }_{\textrm{EOE}}}({\omega_{\textrm{RF}}})} |$ of the electro-optic-electric (EOE) conversion, see Section 4. To ensure comparability with the measurement of the EOE device characteristics, we used a modulator with the organic EO material already applied to the slot waveguides by means of a dispensing technique. Direct-current (DC) measurements confirm that the organic material behaves as a dielectric with negligible parasitic currents flowing through the slot. For the devices used here, the thickness of the BOX layer was 3 µm, the device layer thickness was 220 nm, and the slot width was 120 nm. To de-embed the characteristics of the devices-under-test (DUT) from those of the RF contact-pad parasitics, we investigate two MZM with nominally identical cross sections and different active lengths ${\ell _1} = 500$ µm and ${\ell _2} = 750$ µm of the phase shifters. In a first step, we measure S-parameters of the embedded devices using a vector network analyzer (VNA) and microwave contact probes. Figure 2(c) shows the measured (embedded) transmission S₂₁ and reflection S₁₁ of the two modulators for a gate voltage U_gate = 0 V. The reference plane was moved to the tip of the probes using a short-open-load-through (SOLT) calibration routine with an impedance-standard substrate. The phase-shifter sections of both devices are thus only embedded into identical on-chip contact circuits that are formed by the contact pads and subsequent transitions on both ends of the modulator, see Fig. 2(a), and that are referred to as contact fixtures in the following. Each of these fixtures is modeled by a series impedance Z and a shunt admittance Y as shown in Fig. 2(b), which can be represented by electrical transfer matrices (T-matrices [38]) T_L and T_R for the left and the right fixture, respectively, see Appendix A, Eq. (22) and Eq. (23). These T-matrices can be calculated from the measured S-parameters S₂₁ and S₁₁ using standard microwave theory, see Appendix A. To de-embed the T-matrices T_DUT1 and T_DUT2 of the phase shifters with lengths ${\ell _1}$ and ${\ell _2}$, we first extract the measured T-matrix of the embedded modulator T_m1 = T_LT_DUT1T_R and T_m2 = T_LT_DUT2T_R from the measured S-parameters. Introducing the T-matrix T_DUT0 of an ${\ell _0} = 250$ µm-long de-embedded modulator section and exploiting that ${\ell _1} = 2{\ell _0}$ and ${\ell _2} = 3{\ell _0},$ we can rewrite T_DUT1 and T_DUT2 as T_DUT1 = T_DUT0T_DUT0 and T_DUT2 = T_DUT0T_DUT0T_DUT0. The T-matrix of the virtual through connection shown in Fig. 2(b) is defined as T_thru= T_LT_R and can then be calculated from the measured matrices as ${\underline T _{\textrm{thru}}}\; = \underline T _{\textrm{m1}}^{}\underline T _{\textrm{m2}}^{ - 1}\underline T _{\textrm{m1}}^{}\underline T _{\textrm{m2}}^{ - 1}\underline T _{\textrm{m1}}^{}$. This allows to determine T_L and T_R directly from the measured transfer matrices T_m1 and T_m2 and to de-embed the modulator response by calculating ${\underline T _{\textrm{DUT1}}} = \underline T _\textrm{L}^{ - 1}{\underline T _{\textrm{m1}}}\underline T _\textrm{R}^{ - 1}$ and ${\underline T _{\textrm{DUT2}}} = \underline T _\textrm{L}^{ - 1}{\underline T _{\textrm{m2}}}\underline T _\textrm{R}^{ - 1}$. Using Eq. (28) of Appendix A, we can then extract the frequency-dependent characteristic impedance Z₀ and the frequency-dependent propagation parameter γ from the transfer matrices T_DUT1 and T_DUT2 of the de-embedded modulator, see black traces in Figs. 2(d-g). We find that the results obtained from T_DUT1 and T_DUT2 show very good agreement, thereby confirming the validity of our de-embedding approach. For comparison, we also extract Z₀ and γ by applying Eqs. (24–28) directly to the measured T-matrices T_m1 and T_m2 of the embedded modulators, thereby ignoring the impact of the contact fixtures. The resulting frequency characteristics of Z₀ and γ are depicted as blue and red traces in Figs. 2(d-g) and show clear differences, thereby underlining the importance of proper de-embedding.

 

Fig. 2. (a) Simplified electrical structure of the SOH modulators. The phase-shifter sections with lengths ${\ell _1}$ or ${\ell _2}$ and T-matrices T_DUT1 or T_DUT2 are electrically embedded into contact fixtures, modeled by T-matrices T_L and T_R. (b) The contact fixtures are modelled by a series impedance Z and a shunt admittance Y. (c) Amplitude and phase of measured S-parameters for embedded MZM with lengths ${\ell _1}\; = \; 500$ µm (red) and ${\ell _2}\; = \; 750$ µm (blue). (d) Real part and (e) imaginary part of the characteristic impedance Z₀, along with (f) real part and (g) imaginary part of the propagation constant γ. Black curves indicate values as extracted from de-embedded modulators of lengths ${\ell _1}\;$ and ${\ell _2}\;$. These results show good agreement and hence confirm the validity of our de-embedding approach. In contrast to that, the red and blue curves correspond to values extracted from the embedded modulators by ignoring the impact of the contact fixtures. Consequently, the results obtained for the different device lengths are not in agreement.

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Our results indicated that Z₀ has a negligible imaginary part and amounts to approximately 50 Ω – except for the low-frequency region, where Z₀ deviates from its ideal low-loss and high-frequency approximation, see Eq. (29) in Appendix B. This ensures good impedance matching to typical feeding networks and drivers. The real part α of γ, associated with RF propagation loss, increases with frequency due to loss in the metal stripes and in the resistive slabs, see the following paragraphs for a detailed discussion of the loss mechanisms. The imaginary part β of γ shows a linear increase indicating a low dispersion of the RF transmission line.

The measurement of S-parameters and subsequent de-embedding and extraction of γ and Z₀ was repeated for a range of gate voltages U_gate between 0 V and 300 V. Figure 3 shows the numerical values obtained from measurements in blue. To determine the six equivalent-circuit parameters ${R^{\prime}_{\textrm{TL}}},{L^{\prime}_{\textrm{TL}}},{C^{\prime}_{\textrm{TL}}},{G^{\prime}_{\textrm{S,bulk}}},{G^{\prime}_{\textrm{S,acc}}}$, and ${C^{\prime}_\textrm{S}}$, the following approach was used: First, Eq. (4) and Eq. (5) were used to calculate ${R^{\prime}_{\textrm{TL}}} = \Re \{{{{\underline Z }_0}\underline \gamma } \}$ and ${L^{\prime}_{\textrm{TL}}} = {{{\Im \{{{{\underline Z }_0}\underline \gamma } \}} \mathord{\left/ {\vphantom {{\Im \{{{{\underline Z }_0}\underline \gamma } \}} \omega }} \right.} \omega }_{\textrm{RF}}}$ directly from γ and Z₀. Using the fact that both ${R^{\prime}_{\textrm{TL}}}$ and ${L^{\prime}_{\textrm{TL}}}$ are independent of frequency and of the applied gate voltage, the two values were calculated for all frequencies and gate voltages and then averaged. In a second step, a multi-dimensional multi-parameter least-squares fit was used to obtain ${C^{\prime}_{\textrm{TL}}},{G^{\prime}_{\textrm{S,bulk}}},{G^{\prime}_{\textrm{S,acc}}}$ and ${C^{\prime}_\textrm{S}}$ by fitting Eq. (4) and Eq. (5) to the measured frequency-dependent and gate-voltage-dependent values of γ and Z₀, see Table 1 for the resulting numerical values. We estimate, e.g., a differential capacitance of the transmission line of 130 fF/mm, which compares well to the numerical result of 151 fF/mm obtained for a similar device geometry in [23]. The fitted curves are shown in red in Fig. 3 and show good agreement with the measured data. For increasing frequencies, the characteristic impedance quickly approaches its ideal high-frequency real value ${\underline Z _0} = \sqrt {{{L^{\prime}_{TL}}}/({{\bar{C}^{\prime}_\textrm{S}}} + {{C^{\prime}_{\textrm{TL}}}})} \approx 50{\kern 1pt} \,\Omega $, which can be obtained from Eq. (5) using ${R^{\prime}_{\textrm{TL}}} \ll {\omega _{\textrm{RF}}}{L^{\prime}_{\textrm{TL}}}$ and ${\bar{G}^{\prime}_\textrm{S}} \ll {\omega _{\textrm{RF}}}({\bar{C}^{\prime}_\textrm{S}} + {C^{\prime}_{TL}}).$ The characteristic impedance Z₀ as well as the imaginary part of γ are rather insensitive to the gate voltage.

 

Fig. 3. Measured values (blue) and fitted curves (red) according to Eq. (4) and (5) as a function of both RF frequency and gate voltage U_gate for (a) the real part of the characteristic impedance Z₀, (b) the imaginary part of the characteristic impedance Z₀, (c) the real part of the complex propagation parameter γ associated with loss, and (d) the imaginary part of the complex propagation parameter γ.

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Table 1. Circuit parameters obtained by fitting Eq. (4) and (5) to the measured frequency-dependent and gate-voltage-dependent values of γ and Z₀, see Fig. 3.

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In contrast to this, the losses show a strong dependence on the conductivity of the slabs and thus on U_gate. To explain this finding, we first consider the losses in the metal transmission line represented by ${R^{\prime}_{\textrm{TL}}},{L^{\prime}_{\textrm{TL}}}$ and ${C^{\prime}_{\textrm{TL}}}$ separately from the losses in the RC element which represents the slab and the slot and which is formed by ${G^{\prime}_{\textrm{S,bulk}}},{G^{\prime}_{\textrm{S,acc}}}$ and ${C^{\prime}_\textrm{S}}$. The different loss contributions are plotted as a function of frequency in Fig. 4 for three different gate voltages. Note that the separation and the independent analysis of the two loss contributions represents an approximation and might thus lead to small deviations from the mathematically correct overall loss that can be obtained from the real part $\alpha $ of the complex propagation parameter $\underline \gamma $ according to Eq. (4).

 

Fig. 4. Detailed representation of RF loss mechanisms in the slot-waveguide modulator. Blue data points represent measured data, and red traces correspond to the loss obtained by calculating the real part of the complex propagation parameter γ, Eq. (4), for the circuit parameters obtained by the fit. The three panels correspond to three values of the gate voltage U_gate and represent slices of the 3D plot in Fig. 3(c). Besides the gate voltage U_gate, the associated slab conductance $G{^{\prime}_\textrm{S}}$ is specified. The black dotted traces depict the loss contribution α_TL of the transmission line formed by the metal electrodes only, see Eq. (15), and the black dashed traces show the loss contribution α_RC of the RC element formed by the slab and the slot, see Eq. (18). The solid black line shows the sum of α_TL and α_RC, which is in good agreement with the loss obtained from the full model represented by the red solid trace. (a) U_gate = 0 V: Except for very low frequencies, see inset, the loss is constant with frequency. (b) U_gate = 150 V: The quadratic frequency-dependence of α_RC for frequencies below the corner frequency ${f_{\textrm{RC}}} = {\omega _{\textrm{RC}}}/(2\pi )$ of the RC element, see Eq. (19), leads to a strong increase of the total loss with frequency. At frequencies ${f_{\textrm{RF}}} = {{{\omega _{\textrm{RF}}}} \mathord{\left/ {\vphantom {{{\omega_{\textrm{RF}}}} {(2\pi )}}} \right.} {(2\pi )}} \gg {f_{\textrm{RC}}}$ the total loss approaches a constant value, see Eq. (17) and Eq. (20). (c) U_gate = 300 V: The corner frequency f_RC is further increased compared to (b), and the total loss no longer reaches its constant high-frequency value in the considered frequency range.

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For the contribution of the bare transmission line, we neglect ${G^{\prime}_{\textrm{S,bulk}}},{G^{\prime}_{\textrm{S,acc}}}$ and ${C^{\prime}_\textrm{S}}$ in Eq. (4), such that the loss can be written as

For the loss contribution of the RC-element that is formed by the slot capacitance ${C^{\prime}_\textrm{S}}$ and the slab conductance ${G^{\prime}_\textrm{S}}\; = \; {G^{\prime}_{\textrm{S,acc}}} + {G^{\prime}_{\textrm{S,bulk}}}$, we derive a relationship in Appendix C,

The overall loss $\alpha $ can be estimated by adding ${\alpha _{\textrm{TL}}}$ and ${\alpha _{\textrm{RC}}}$, leading to the solid black lines in Fig. 4. These estimated losses are in good agreement with the loss obtained from directly using $\alpha = \Re \{{\underline \gamma } \}$ according to Eq. (4), see red lines in Fig. 4.

The three panels for Fig. 4(a, b, c) correspond to different gate voltages of 0 V, 150 V, and 300 V, respectively. For the samples used in our experiments, the doping of the slabs is rather low, and hence the resistance $G^{\prime -1}_\textrm{S}$ of the slabs is rather high in case no gate voltage is applied. This leads to a small corner frequency ${\omega _{\textrm{RC}}}$, illustrated by a dashed vertical line in Fig. 4(a), and the loss contribution ${\alpha _{\textrm{RC}}}$ of the RC-element quickly reaches the frequency-independent value given by Eq. (20). For low frequencies, the total loss is dominated by the contribution ${\alpha _{\textrm{TL}}}$ of the bare transmission line, Eq. (16), and increases in proportion to $\sqrt {{\omega _{\textrm{RF}}}} $. Small discrepancies between measured and predicted curves can be observed for small values of U_gate and for the frequency range beyond 40 GHz, where the measurement data become noisy. These discrepancies may be reduced by introducing additional free parameters into the model, thereby improving the fit to measured data, but sacrificing the clear correlation between model parameters and physical device parameters such as geometry and material properties, see Section 4 for a more detailed discussion. If U_gate is increased, Fig. 4(b) and 4(c), ${\omega _{\textrm{RC}}}$ becomes larger and exceeds ${\omega _{\textrm{TL}}}$, which leads to a strongly frequency-dependent increase of the overall loss which is dominated by ${\alpha _{\textrm{RC}}} \sim \omega _{\textrm{RF}}^2$ for frequencies ${\omega _{\textrm{RF}}}$ between ${\omega _{\textrm{TL}}}$ and ${\omega _{\textrm{RC}}}$. At even higher frequencies (${\omega _{\textrm{RF}}} \gg {\omega _{\textrm{RC}}}$), the total losses are again frequency-independent, see Eq. (17) and Eq. (20). For a gate voltage of 300 V, Fig. 4(c), the plateau, where the overall loss becomes frequency-independent, is no longer reached in the considered range of RF frequencies. For the discussion of design trade-offs (Section 5) it is important to note that in range between 5 GHz and 50 GHz, increasing the slab conductivity, in the range considered here by changing U_gate, brings the modulator from a regime with weakly frequency-dependent loss to a regime with strongly frequency-dependent loss and may hence limit the bandwidth of the device.

4. Measured and modelled electro-optic response

With the knowledge of the six equivalent-circuit parameters ${R^{\prime}_{TL}},{L^{\prime}_{\textrm{TL}}},{C^{\prime}_{\textrm{TL}}},{G^{\prime}_{\textrm{S,bulk}}},{G^{\prime}_{\textrm{S,acc}}}$, ${C^{\prime}_\textrm{S}}$ and of the optical group velocity v_g,opt in the slot waveguide, Eq. (4–11) can be used to predict the EO frequency response of the SOH modulator. In the following we assume an operating wavelength of 1.55 µm. Note that SOH modulators can be adapted to other wavelengths such as 1300 nm, where the electro-optic coefficient of the underlying materials should be even higher [43] The optical group velocity v_g,opt = c/n_g,opt at 1.55 µm is obtained by calculating the group refractive index n_g,_opt = 3.2 by means of a numerical mode solver. In Fig. 5(a), the red lines show the EO frequency response as predicted by the model for a 750 µm-long modulator terminated with a 50 Ω impedance under the influence of gate voltages between 0 V to 300 V. These predictions are based on the circuit parameters listed in Table 1, as obtained from the purely electrical measurements of the frequency-dependent RF scattering parameters S₂₁ and S₁₁. Increasing the gate voltage increases the bandwidth, which is mainly due to a reduction of the RC time-constant of the slab-loaded slot waveguides. The predicted frequency response is compared against measured values obtained by using a VNA and a calibrated optical receiver [44], see blue traces in Fig. 5(a). In these measurements, the swept-source microwave stimulus of the VNA is applied to the modulator via microwave probes. The modulator is biased at its 3 dB point and the microwave signals amplitude is chosen small enough for the modulator to be operated under small-signal condition. The optical signal at a center wavelength of 1.55 µm is subsequently amplified using an erbium-doped fiber amplifier (EDFA), filtered by a band-pass filter to remove out-of-band amplified-spontaneous-emission noise (ASE), and converted back to the electrical domain using a photodiode that is connected to the receiver port of the VNA. The frequency-dependent decay of the EOE frequency response $|{\underline m _{\textrm{EOE}}}({{\omega_{\textrm{RF}}}} )|$ according to Eq. (14) and Eq. (6) can be derived from the VNA measurement by the relation

 

Fig. 5. Electro-optic-electric (EOE) frequency response $|{{{\underline m }_{\textrm{EOE}}}({\omega_{\textrm{RF}}})} |$ according to Eq. (6) and Eq. (14) of a 750 µm-long SOH modulator terminated with a 50 Ω impedance. (a) EOE response for different gate voltages U_gate of 300 V, 200 V, 150 V, 100 V, 75 V, 50 V, 40 V, 30 V, 20 V, 10 V, and 0 V (from top to bottom). The red curve represents the prediction of the model according to Eq. (4) to Eq. (11), and the blue line corresponds to the measurement results, where the different shades indicate three independent measurements of the same device. With increasing gate voltage, the frequency response flattens. (b) 3 dB EOE bandwidth f_3dB,EOE (red solid line: model; blue circles: measurement) and 6 dB EOE bandwidth f_6dB,EOE (red broken line: model; blue diamonds: measurement) as extracted from the curves in (a) as a function of applied gate voltage U_gate. The predicted bandwidths agree well with their measured counterparts, thus confirming that the proposed method allows to accurately predict the bandwidth of an SOH modulator based on the knowledge of purely electrical characteristics.

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The different shades of blue in Fig. 5(a) indicate three independent measurements of the same device with small statistical variations. Overall, we find a good agreement of the measurements with the model predictions (red). In Fig. 5(b), both the 3 dB and the 6 dB EOE bandwidths of the modulator are shown for different gate voltages. For the 6 dB EOE bandwidth f_6dB,EOE, blue diamonds correspond to the measured values, and the dotted red line represents the bandwidths predicted by the model. Similarly, blue circles and the solid red line indicated measurements and model predictions of the 3 dB EOE bandwidth f_3dB,EOE. The good agreement of the predicted bandwidths with their measured counterparts validates the model introduced in Section 2, i.e., the proposed method allows to accurately predict the EOE frequency response of an SOH modulator based on the knowledge of purely electrical characteristics along with the group velocity ${v_{\textrm{g,opt}}}$ of the optical signal in the slot waveguide. Again, some discrepancies between measured and predicted curves can be observed for small values of U_gate. We believe that our model can be further refined by considering additional effects and/or by including additional equivalent-circuit elements, thereby reducing the discrepancies to the measured data. As an example, we assume a simple linear relationship between the differential conductance ${G^{\prime}_{\textrm{S,acc}}} = {g^{\prime}_{\textrm{S,acc}}}{U_{\textrm{gate}}}$ and the gate voltage U_gate without any threshold behavior. This is to be expected for an ideal accumulation layer of majority carriers in the slab, but the physical situation in the device might be more complex. The benefit of our fairly simple model is the clear correlation between model parameters and physical device parameters such as geometry and material properties. This correlation is essential for deriving guidelines for future device designs. The electrical characteristics are easily accessible by either measurements in the RF domain, by simulations of the electrical RF device characteristics, or by estimating the distributed-element parameters from the modulator geometry and material properties. We believe that high-throughput wafer-level testing protocols might greatly benefit from a quantitatively verified model-based prediction of the EOE frequency response based on easily accessible electrical scattering parameters.

5. Design guidelines

5.1 Overcoming bandwidth limitations

Having validated the model of the SOH modulator, various design parameters can be tested for their impact on device performance. In the following, we assume a device with transmission line parameters ${R^{\prime}_{\textrm{TL}}},{L^{\prime}_{\textrm{TL}}}$, and ${C^{\prime}_{\textrm{TL}}}$ as specified in Table 1 and investigate the impact of variations of the slot capacitance ${C^{\prime}_\textrm{S}}$ and of the slab conductance ${G^{\prime}_\textrm{S}}$ on the EO bandwidth. This leads to the following considerations:

 

Fig. 6. 6 dB EOE bandwidth f_6dB,EOE of a modulator only limited by the RC low-pass formed by the slot capacitance ${C^{\prime}_\textrm{S}}$ and the slab conductance ${G^{\prime}_\textrm{S}}$. Both a reduction of ${C^{\prime}_\textrm{S}}$ and an increase of ${G^{\prime}_\textrm{S}}$ linearly increases the bandwidth. The parameters of the devices used in this paper for experimental verification of the model, see Table 1, are indicated by a green star for the case of a gate voltage of 0 V and by a magenta star for a gate voltage of 300 V. For the parameters indicated by Ⓐ,Ⓑ, and Ⓒ, the frequency response of a 750 µm-long modulator is exemplarily shown in Fig. 8.

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Next, we consider the impact of RF loss, impedance mismatch, and velocity mismatch on the modulator bandwidth. Figure 7(a) shows f_6dB,EOE of a 750 µm-long modulator limited only by RF loss, impedance mismatch, and velocity mismatch, explicitly neglecting RC limitations. The response of such a theoretical device is given by Eq. (6) for ${\underline H _{\textrm{RC}}}({\omega _{\textrm{RF}}}) = \textrm{const}\textrm{.}$ For all slot capacitances ${C^{\prime}_\textrm{S}}$, an increase in the slab conductance ${G^{\prime}_\textrm{S}}$ tends to reduce the bandwidth, see Fig. 7(a). This can be understood by considering that increasing the slab conductance ${G^{\prime}_\textrm{S}}$ initially brings the modulator from a regime with low and frequency-independent RF loss to a regime with high RF loss that increases with frequency, see Fig. 4. If the slab conductance ${G^{\prime}_\textrm{S}}$ is increased further, the RF loss and its frequency dependence are reduced again, and the impact on the bandwidth decreases. In practical devices, however, ${G^{\prime}_\textrm{S}}$ is constrained due to optical loss caused by FCA in the highly doped slab regions. This trade-off can be mitigated by optimized doping profiles, see Section 5.2.

 

Fig. 7. (a) 6 dB EOE bandwidth f_6dB,EOE of a 750 µm-long modulator limited only by RF loss, impedance mismatch, and velocity mismatch, explicitly neglecting the impact of the RC-low-pass formed by $C{^{\prime}_\textrm{S}}$ and ${G^{\prime}_\textrm{S}}$. For a large $C{^{\prime}_\textrm{S}}$ and a low ${G^{\prime}_\textrm{S}}$, an increase of ${G^{\prime}_\textrm{S}}$ leads to a strong frequency-dependent increase of the RF loss and thus to a reduced bandwidth. For the parameters indicated by Ⓐ,Ⓑ, and Ⓒ the frequency response is exemplarily shown in Fig. 8. (b) RF loss for a 750 µm-long modulator at 50 GHz. The RF loss initially increases with increasing ${G^{\prime}_\textrm{S}}$, see Fig. 4(a) and Fig. 4(b), but decreases again for very large values of ${G^{\prime}_\textrm{S}}$, see Eq. (19). (c) Characteristic impedance Z₀ at 50 GHz. The impedance remains close to the system impedance of 50 Ω. (d) Effective RF refractive index n_eff,RF . For values of ${G^{\prime}_\textrm{S}} \approx 50$ S/m and ${C^{\prime}_\textrm{S}} \approx 130$ pF/m, n_eff,RF is close to the effective optical group refractive index of 3.2 Increasing the slot capacitance slows down the RF wave and hence increases the RF refractive index. The detailed analysis in Fig. 8 shows that the bandwidth is not limited by velocity mismatch for practical values of $C{^{\prime}_\textrm{S}}$ for the considered length of the modulator. The parameters of the devices (see Table 1) used in this paper for experimental verification of the model are indicated by a green star for the case of 0 V gate voltage applied and by a magenta star for 300 V gate voltage applied.

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We further investigate the overall RF loss α of the modulator, including both the loss in the metal electrodes and in the RC element formed by the slot waveguide and the slabs, as well as the associated real-part Z₀ of the characteristic impedance Z₀ and the effective RF index n_eff,RF. The numerical values for α, Z₀, and n_eff,RF are shown exemplarily for an RF frequency of 50 GHz in Fig. 7(b), Fig. 7(c), and Fig. 7(d), respectively. While the RF loss, Fig. 7(b), initially increases substantially with the slab conductance ${G^{\prime}_\textrm{S}}$, the characteristic impedance Z₀, Fig. 7(c), stays close to 50 Ω. For the effective microwave index n_eff,RF, we observe a strong initial increase with the slab conductance ${G^{\prime}_\textrm{S}}$, Fig. 7(d). This can be understood by considering that, for a low slab conductivity ${G^{\prime}_\textrm{S}}$, the RC element is operated above its corner-frequency ${\omega _{\textrm{RC}}} < 50\,\textrm{GHz}$. In this regime, the slot capacitance ${C^{\prime}_\textrm{S}}$ is only partially charged during one cycle. If the slab conductance ${G^{\prime}_\textrm{S}}$ is increased, the slot capacitance becomes increasingly effective, and the RF wave is slowed down. For larger values of the slab conductance ${G^{\prime}_\textrm{S}}$, the RC element is operated above its corner frequency and the Capacitor is fully charged and de-charged during each cycle, leading to an RF index n_eff,RF that essentially independent of the slab conductance. Following the same argument, it can also be understood why an increase of the slot capacitance ${C^{\prime}_\textrm{S}}$ therefore leads to an increase of n_eff,RF.

To investigate the frequency dependence of the RF loss, of the characteristic impedance, of the effective RF index, and of the RC low-pass as well as the associated impact on the modulator bandwidth, we consider three exemplary ${C^{\prime}_\textrm{S}}$-${G^{\prime}_\textrm{S}}$-value-pairs for a 750 µm-long device. The chosen value pairs are marked by Ⓐ,Ⓑ, and Ⓒ in Fig. 6 and Fig. 7. The slot capacitance is fixed to ${C^{\prime}_\textrm{S}} = 100\;{{\textrm{pF}} \mathord{\left/ {\vphantom {{\textrm{pF}} \textrm{m}}} \right.} \textrm{m}}$, while the slab conductance ${G^{\prime}_\textrm{S}}$ is increased from ${G^{\prime}_\textrm{S}} = 2.76\;{\textrm{S} \mathord{\left/ {\vphantom {\textrm{S} \textrm{m}}} \right.} \textrm{m}}$ (Ⓐ), which corresponds to the value extracted from the device used in our experiment, to ${G^{\prime}_\textrm{S}} = 40\;{\textrm{S} \mathord{\left/ {\vphantom {\textrm{S} \textrm{m}}} \right.} \textrm{m}}$ (Ⓑ) and further to ${G^{\prime}_\textrm{S}} = 80\;{\textrm{S} \mathord{\left/ {\vphantom {\textrm{S} \textrm{m}}} \right.} \textrm{m}}$(Ⓒ). Figure 8(a) shows, for the weakly conducting slab with ${G^{\prime}_\textrm{S}} = 2.76$ S/m, the EOE frequency response $|{{m_{\textrm{EOE}}}({\omega_{\textrm{RF}}})} |$, the RF loss α, the characteristic impedance Z₀, and the effective RF refractive index n_eff,RF, all as a function of RF frequency up to 150 GHz. The solid red line depicts the total frequency response. Broken lines in blue, magenta, green, and black depict the frequency response by the sole contribution of the RC low-pass, the RF loss, the impedance mismatch, and the velocity mismatch, respectively. For a weakly conducting slab with ${G^{\prime}_\textrm{S}} = 2.76$ S/m, the total bandwidth (red line) is limited almost exclusively by the RC low-pass (blue broken line), whereas the impact of RF loss (red magenta broken line), impedance-mismatch (green broken line) and of velocity mismatch (black broken line) are negligible, because α, Z₀, and n_eff,RF show little frequency dependence. If the slab conductivity is increased to, e.g., 40 S/m, Fig. 8(b), or 80 S/m, Fig. 8(c), the RC corner frequency increases. However, frequency-dependent RF loss starts to become significant and limits the overall bandwidth of the device to values smaller than the RC corner frequency. For the increased slab conductance ${G^{\prime}_\textrm{S}} = 40\;{\textrm{S} \mathord{\left/ {\vphantom {\textrm{S} \textrm{m}}} \right.} \textrm{m}}$ and ${G^{\prime}_\textrm{S}} = 80\;{\textrm{S} \mathord{\left/ {\vphantom {\textrm{S} \textrm{m}}} \right.} \textrm{m}},$ impedance mismatch and velocity mismatch do not significantly affect the bandwidth either. This confirms that RF loss is the most important bandwidth-limiting effect besides the RC time constant of the slab-slot configuration.

 

Fig. 8. Electro-optic bandwidth and RF properties for a 750 µm-long modulator with ${C^{\prime}_\textrm{S}} = 100\;{{\textrm{pF}} \mathord{\left/ {\vphantom {{\textrm{pF}} \textrm{m}}} \right.} \textrm{m}}$ and (a) ${G^{\prime}_S} = 2.76$ S/m, (b)${G^{\prime}_S} = 40$ S/m, and (c)${G^{\prime}_S} = 80$ S/m, corresponding to the points marked by Ⓐ,Ⓑ, and Ⓒ, respectively, in Fig. 6, Fig. 7(a), and Fig. 9(b). The first row shows the overall frequency response $|{{{\underline m }_{\textrm{EOE}}}({\omega_{\textrm{RF}}})} |$ of the modulator (solid red line), as well as the frequency responses that correspond to the sole impact of RC limitations (blue dashed line), RF loss (magenta dashed line), impedance mismatch (green dashed line), and velocity mismatch (black dashed line). The second row shows the total RF loss α. In (a) the RF loss is low and frequency-independent, see Eq. (17). In (b) and (c) losses in the resistive slab are increased and the losses becomes strongly frequency-dependent, see Eq. (19), thereby reducing the bandwidth to values below the RC limit. The third row shows the characteristic impedance Z₀, which stays close to the system impedance of 50 Ω., except for low frequencies, where the Z₀ deviates from its ideal high frequency, low loss value, see Appendix B. The last row shows the effective RF refractive index n_eff,RF. In (a), the RC corner frequency ${\omega _{\textrm{RC}}}$ is small and the slot capacitance is not fully charged and de-charged as the RF wave propagates along the modulator. If the resistance of the slabs is reduced, see (b) and (c), ${\omega _{\textrm{RC}}}$ increases and the slot capacitance has a stronger impact on the RF wave, leading to a reduced velocity. The effective RF refractive index n_eff,RF then becomes close to the optical group index n_g,opt = 3.2. Overall, impedance and velocity-mismatch are negligible for all of the three considered cases. In (a), the device is limited by the RC time constant of the slab-slot configuration. In (b) and (c) the RC time constant is reduced and the associated bandwidth is increased accordingly. At the same time, the increased RC corner frequency leads to an increased frequency-dependence of the RF loss, which causes additional bandwidth limitations.

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Finally, we consider the 6 dB EOE bandwidth ${f_{\textrm{6dB,EOE}}}$ obtained from the full model as a function of the slab conductance ${G^{\prime}_\textrm{S}}$ and the slot capacitance ${C^{\prime}_\textrm{S}}$, taking into account all bandwidth-limiting effects, i.e., the RF loss, the impedance mismatch, the velocity mismatch and the RC low-pass. Figure 9 shows ${f_{\textrm{6dB,EOE}}}$ as obtained by the full model for four different lengths $\ell $ of the modulator. For long devices, the frequency dependence of the RF loss is more significant, and increasing the slab conductivity ${G^{\prime}_\textrm{S}}$ has a smaller impact on the overall bandwidth than for short devices, see Fig. 9(a). Specifically, if $\ell $ is reduced to 500 µm, Fig. 9(c), and further to 200 µm, Fig. 9(d), the bandwidth approaches the RC-limited case shown in Fig. 6. A combination of reduced device length, reduced slab capacitance ${C^{\prime}_\textrm{S}}$, and increased slab conductance ${G^{\prime}_\textrm{S}}$ thus enables bandwidths well beyond 100 GHz.

 

Fig. 9. 6 dB EOE bandwidth f_6dB,EOE for modulators of different lengths $\ell $ based on the full model considering all bandwidth-limiting effects. The color scale is the same as in Fig. 6. If the length is reduced, the bandwidth of the modulator approaches the RC-limited case depicted in Fig. 6. The frequency response of the 750-µm-long modulator is shown in Fig. 8 for the ${C^{\prime}_\textrm{S}}$-$G{^{\prime}_\textrm{S}}$-values indicated by Ⓐ,Ⓑ, and Ⓒ in (b). The parameters of the devices used in this paper for experimental verification of the model, see Table 1, are indicated by a green star for the case of a gate voltage of 0 V and by a magenta star for a gate voltage of 300 V.

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5.2 Optimized doping profiles for low-loss modulators

To increase the bandwidth of the modulators, highly conductive slab regions are desirable. Increasing the doping concentration near the optical slot waveguide, however, leads to optical loss due to free-carrier absorption (FCA). This effect can be mitigated by a two-step doping profile, see Fig. 1(b) [23,28,29]. Based on this approach, we suggest the following design procedure for SOH modulators:

 

Fig. 10. Optimization of the doping profile in the slab-waveguide region to minimize optical loss. (a) Cross section of the slot waveguide and the adjacent slab: A low doping concentration is chosen for the rails of the slot waveguide and for the adjacent region of the slab (depicted in green), which interact strongly with the guided light. The doping is increased in the slab region that is further away from the rails (depicted in blue). We consider a waveguide with rails having a width of w_rail = 240 nm and a height of h_rail = 220 nm. The slabs have a height of h_slab = 70 nm and a width of w_slab = 1.66 µm. The width of the slot is 200 nm. For a fixed resistivity of the weakly doped silicon (here ρ_weak = 5.5×10⁻⁴ Ωm), w_weak and ρ_heavy can be varied to realize the target conductivity of the slab (here ${G^{\prime}_\textrm{S}} = 102$ S/m). (b) Bulk resistivity ρ_heavy in the heavily doped slab regions that is required to reach the target conductivity, specified as a function of w_weak for fixed values of ρ_weak = 5.5×10⁻⁴ Ωm and w_slab = 1.66 µm. (c) Corresponding material absorption coefficient in the doped silicon. Note that α_Si refers to the power attenuation coefficient for the optical wave, whereas the symbol α in Eq. (4) refers to the amplitude attenuation coefficient of the corresponding RF wave. (d) Power loss coefficient experienced by the guided optical mode, obtained from the numerically calculated optical mode field and its overlap with the doped silicon regions. An optimum in terms of optical loss is found for an intermediate extension of the heavily doped section in combination with a medium resistivity.

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For simplicity, we choose a fixed resistivity ρ_weak of the weakly doped section in a first step, here ρ_weak= 5.5×10⁻⁴ Ωm. For a target conductance ${G^{\prime}_\textrm{S}} = 102$ S/m of the slab, Fig. 10(b) shows the required bulk resistivity ρ_heavy of the heavily doped region as a function of its width w_heavy. To reach the overall target conductance of ${G^{\prime}_\textrm{S}} = 102$ S/m, the width w_weak of the heavily doped region must stay below 1.16 µm. The bulk resistivity in the weakly and heavily doped regions can be translated into optical loss parameters using the experimentally obtained relations published in [47]. Figure 10(c) shows the bulk power absorption coefficient $\alpha _{\textrm{Si}}^{}$ in the heavily and weakly doped silicon. Note that $\alpha _{\textrm{Si}}^{}$ refers to the optical power – in contrast to the RF attenuation coefficient $\alpha $, specified in Eq. (4), which refers to the amplitude. The underlying model for the optical power absorption coefficient assumes a linear increase of the optical loss with the electrical conductivity in Si [47]. This assumption is only valid as long as the electrical conductivity is not limited by electron-impurity-scattering [48]. To translate the bulk absorption coefficient into a waveguide loss parameter, we compute the optical mode in the slot waveguide and the overlap with the doped silicon region [24] using a vectorial mode solver. The resulting power absorption coefficient $\alpha _{\textrm{WG}}^{}$ for the optical waveguide mode is shown in Fig. 10(d). If the weakly doped region is narrow, the heavily doped section comes close to the slot region leading to high optical loss. If the heavily doped section is far away from the slot waveguide, the conductivity has to be increased to maintain a given overall conductance ${G^{\prime}_\textrm{S}}$, which again increases the optical loss. A minimum excess loss of approximately 0.048 mm^-1 (0.2 dB/mm) is found for a weakly doped region with an intermediate width of w_weak = 1 µm and a medium resistivity of the heavily doped section of ρ_heavy= 1.43×10⁻⁴ Ωm. Note that this doping-related excess loss does not significantly change the overall propagation loss of practically used slot waveguides, which is of the order of 0.5 dB/mm [49]. Note that these loss figures are still dominated by the imperfections of the silicon slot waveguides and not by the organic EO cladding. The bulk absorption of the EO material is of the order of 0.1 dB/mm, and only a fraction of the optical mode propagates in the slot such that the contribution to the overall loss should be even smaller.

So far, we have only considered a fixed resistivity ρ_weak= 5.5×10⁻⁴ Ωm of the weakly doped section. Varying this resistivity allows to further increase the bandwidth of the device or decrease its optical loss. This degree of freedom becomes particularly important when the length of the modulator is varied. For a quantitative analysis, we have repeated the optimization exemplarily described in the previous paragraph for devices of different lengths $\ell = 0.200\;\textrm{mm,}$ 0.5 mm, and 1.0 mm and for different resistivities ρ_weak = 5.5×10⁻⁴ Ωm, 16.5×10⁻⁴ Ωm, and 49.5×10⁻⁴ Ωm of the weakly doped region. Figure 11 shows the resulting optical excess loss caused by absorption in the doped silicon regions for modulators of different lengths $\ell $ as a function of their total bandwidth f_6dB,EOE. In this plot, the trade-off between the modulator figures of merits ${U_\pi } \propto {\ell ^{ - 1}}$, bandwidth and insertion loss becomes evident. The numerical values presented in Fig. 11 suggest that it will be possible to realize SOH devices with a length of, e.g., 0.5 mm, having a bandwidth of f_6dB,EOE = 100 GHz while keeping the doping-related optical excess loss below 0.1 dB. Using highly efficient EO polymers [16], such a device would have a π-voltage of less than 1 V. The detailed device parameters leading to the results shown in Fig. 11 are depicted in Fig. 12. Figure 12(a) shows the relationship between the device bandwidth and the overall slab conductance ${G^{\prime}_\textrm{S}}$ for the three different modulator lengths $\ell $ considered in Fig. 11. The graphs are obtained from the full model according to Eq. (6) in combination with a slot capacitance of ${C^{\prime}_\textrm{S}} = 100\;{{\textrm{pF}} \mathord{\left/ {\vphantom {{\textrm{pF}} \textrm{m}}} \right.} \textrm{m}}$ and the transmission line parameters ${R^{\prime}_{\textrm{TL}}},{L^{\prime}_{\textrm{TL}}}$, and ${C^{\prime}_{\textrm{TL}}}$ as specified in Table 1. Figure 12(b) specifies the optimized resistivity ρ_heavy of the heavily doped slab region along with the associated width w_weak of the weakly doped region, see Fig. 12(c), and the associated power absorption coefficient $\alpha _{\textrm{WG}}^{}$ for the optical waveguide mode, see Fig. 12(d), considering three fixed values of ρ_weak = 5.5×10⁻⁴ Ωm, 16.5×10⁻⁴ Ωm, and 49.5×10⁻⁴ Ωm.

 

Fig. 11. Optical excess loss caused by absorption in the doped silicon regions for modulators of length $\ell $ = 0.2 mm, 0.5 mm, and 1 mm as a function of the total 6 dB electro-optic-electric (EOE) bandwidth f_6dB,EOE. For each length, we consider three different resistivities ρ_weak of 5.5×10⁻⁴ Ωm, 16.5×10⁻⁴ Ωm, and 49.5×10⁻⁴ Ωm. The parameters ρ_heavy and w_weak are then optimized according to the procedure described in Fig. 9. For a given length of the modulator, there is a trade-off between electro-optic bandwidth and optical insertion loss. For a given bandwidth, optimizing the doping profile can lead to a reduction of optical loss. Data points are connected by interpolated lines as guide to the eye.

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Fig. 12. Details of the optimized SOH modulator design for the modulators of length $\ell $ = 0.2 mm, 0.5 mm, and 1 mm considered in Fig. 11. For each length, we consider three different resistivities ρ_weak of 5.5×10⁻⁴ Ωm, 16.5×10⁻⁴ Ωm, and 49.5×10⁻⁴ Ωm. The parameters ρ_heavy and w_weak are then optimized according to the procedure described in Fig. 9. (a) Relationship between device bandwidth f_6dB,EOE and overall slab conductance ${G^{\prime}_\textrm{S}}$ for three different modulator lengths. (b) Optimized width w_weak of the weakly doped region, (c) the associated resistivity ρ_heavy of the heavily doped slab region, and (d) the associated power absorption coefficient for the optical mode in the doped region of the silicon, all as a function of slab conductance ${G^{\prime}_\textrm{S}}$, considering the three fixed values of ρ_weak. Data points in all subfigures are connected by interpolated lines as guide to the eye.

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6. Summary

We formulate and experimentally validate an equivalent-circuit model of silicon-organic hybrid (SOH) slot-waveguide modulators based on distributed elements that describe the electric properties of the device. The model parameters can be extracted from purely electrical measurements, that provide the frequency-dependent scattering parameters of the travelling-wave device. With these parameters, we can accurately predict the small-signal electro-optic (EO) frequency response of the modulators, as confirmed by direct measurements of the EO frequency response. We further formulate design guidelines that lead to an optimum trade-off between EO bandwidth and optical insertion loss of SOH slot-waveguide modulators. We find that proper choice of doping concentrations and geometrical device parameters can enable SOH modulators with lengths of 0.5 mm and 6 dB electro-optic-electric (EOE) bandwidths of more than f_6dB,EOE = 100 GHz while maintaining π-voltages of less than 1 V and while keeping the optical excess loss caused by free-carrier absorption in the doped silicon regions below 0.1 dB.