Results

\begin{table}[hp] \centering \begin{tabularx}{\textwidth}{@{}lX@{}} \toprule Section & Key Result \ \midrule 4.1 & Rectlinear modulators have a limited MD/IL ratio of $f-1$. \ 4.1.1 & A Fermi level standard deviation of \qty{0.1}{eV} deteriorates the switching performance similar to a high scattering rate of \gamma=\num{3e13} at room temperature. \ 4.2 & Modulators resulting from inverse design only diverge in MD for high IL. \ 4.3 & To enhance EA-modulation it is beneficial to use the overcoupled regime, compared to the undercoupled regime. \ 4.3.1 & MD and IL are related as depicted in figure . \ 4.3.2 & The use of resonant modulators (without changing $f$) only yields marginal imrovements of less than \qty{5}{%} in links limited by thermal noise (+ realistic RIN). \ 4.3.3 & Only approximately \qty{25}{%} of the optical bandwidth are useable for resonantly enhanced modulation. \ 4.3.4 & The degradation due to additional unmodulated loss in the resonator probably overshadows the marginal improvements found in subsection 4.3.2 \ 4.4 & Resonant structures can shrink the required graphene dimensions, which can lead to an improvement of $f$ \ 4.4.1 & For locally correlated doping variations with an exponential autocorrelation, the influence of those variation is mitigated for devices smaller than the autocorrelation length $L_{corr}$ \ 4.4.2 & Extracting $L_{corr}$ from spatially resolved Raman maps suggests it to be approximately \qty{450}{nm}. The measurement results exhibit features destinct from those that would arrise if $L_{corr}$ was an artifact from the finite beam size ao the measurement system. However, further investigation is deemed neccessary. \ 4.4.3 & Theoretically optimal resonator dimensions are introduced, however these strongly depend on fabrication nonidealities (i.e. scattering loss). To minimize the graphene size in arbitrary resonators it is beneficial to optimize $Q/V$, which has been studied extensively in literature. \ 4.5 & A mask design and measurement plan is provided to verify the optimal resonator geometries, and to characterize the fabrication-induced losses for given processing capabilities. \ \bottomrule \end{tabularx} \caption{Overview of the results of this work.} \label{tab:results} \end{table}

Rectlinear Modulator¶

In the following, the modulation behavior of a rectlinear graphene modulator is analyzed with respect to MD, IL, and their relation. The switching factor $f$ is introduced, which plays a significant role in the analysis of the resonant modulators. Lastly, the increase in laser power due to an imperfect modulator in a noisy link will be discussed.

As seen in section Graphene Modeling it is not possible to switch graphene to a fully transparent state due to defect-activated intraband transitions in the Pauli-blocking regime. Additionally, a limited gate voltage swing impedes the contrast between the transparent ($\alpha_{on}$) and opaque state ($\alpha_{off}$). To quantify this deficiency, the switching factor $f>0$ is defined such that $\alpha_{off} = f\alpha_{on}$ and equivalently $\Im(\delta \varepsilon_{r,off}) = f\Im(\delta \varepsilon_{r,on})$. This results in a transmission of $p_{off}=p_{on}^f$.

MD and IL can be evaluated for the rectlinear modulator according to their definition (in \si{dB}):

Thus, these opposing figures of merits can neither be adjusted independently by the modulator length nor by the overlap $\Gamma$. This locks the ratio of MD and IL for a given $f$ to:

Furthermore, the non-overlapping sections of graphene used to contact the modulator capacitor, the proximity of the metal pads to the optical mode, optically active residues/contaminants introduced by the graphene transfer, and a second graphene layer used as an electrostatic gate can lead to additional non-modulated loss, degrading $\mathrm{MD}/\mathrm{IL}$ to below $f-1$. As overlap and pad distance also affect the RC limited bandwidth, a trade-off has to be made when determining these parameters, which is however out of the scope of this work.

The different techniques presented in Photonic Modeling were compared using an exemplary width sweep. The effective permittivity method was observed to require much higher mesh resolution compared to the other methods (see appendix \ref{ap:fde_convergence}). The access graphene regions are not doped electrostatically and can therefore be considered absorptive in both modulator states. Figure Figure 3 (b) compares the MD/IL ratio with and without considering this fact. As expected it is found that the ratio does not depend on the geometry when all graphene is modulated in tandem. For the realistic case of unmodulated access graphene, the ratio is found to be maximized for some waveguide width. This waveguide width minimizes the lateral extent of the modal \efield. For smaller width, the mode increasingly leaks out of the waveguide core, while for larger widths the mode widens alongside the waveguide.

Required Laser Power¶

From the optical link model introduced in section it becomes apparent, that the detection limit (sensitivity) of the receiver in conjunction with the losses along the link determines the necessary laser power. Laser power is a major contribution to the overall power budget.

The nonideal modulator behavior makes it necessary to increase the laser power by some factor $\delta P_{l}$. Besides MD and IL, the magnitude of $\delta P_{l}$ is dependent on the types and amount of noise present in the link. It is derived in appendix \ref{ap:thermal_noise} that the laser power has to be increased by a factor of $\delta P_{l, thermal}$ due to the nonideal modulation behavior in a thermal-noise-limited link:

Which is equivalent to the inverse ROMA (dashed line in figure (a)). As expected the maximum ROMA increases with $f$. Similarly, in the case of a shot-noise-limited link $\delta P_{l, thermal}$ is found as:

Introducing the EROMA, analogous to the thermal-noise-limited link. Figure compares the effects of the modulator for the shot and thermal-noise-limited link.

\begin{figure}[t] \centering \includegraphics[width=\textwidth]{thermal_and_shot.pdf} \caption{Effect of nonideal modulation on the laser power for different dominant noise sources (thermal and shot noise). (a) EROMA for different dominant noise terms (see equations \ref{eq:rOMA_max_straight} and \ref{eq:eroma}). The dashed line indicates the position and magnitude of the maximum as predicted by equations \ref{eq:rect_roma_pos_max} and \ref{eq:rOMA_max_straight} respectively. The dotted line indicates the IL limit. (b) Comparison of the laser power increase due to a nonideal rectlinear modulator, comparing a link limited by shot noise and thermal noise. The left graph is the zoom of the right graph indicated by the grey-shaded region.} \label{fig:compare_thermal_shot} \end{figure} The last source of noise investigated here is RIN, which scales linearly with the optical power. Therefore $Q$ is not dependent on $P_{in}$ when considering RIN as the only source of noise:

Thus in this case there is no incentive to increase the laser power for worse modulators. However, $Q$ and $C_{rin}$ directly determine the minimum $md$ as:

With a $C_{rin}$ close to the limit set by equation \ref{eq:rin_md_req} however, additional thermal noise will quickly lower Q beyond the targeted value. Suppressing the thermal noise requires a strongly increased power budget.

The resulting equations and derivation of $\delta P_{l, rin, th}$ are presented in appendix \ref{ap:rin_power}. It is noted that the expression for $\delta P_{l, rin, th}$ (equation \ref{eq:rin_th_p_laser}) reduces to the expression for $\delta P_{l, thermal}$ (equation \ref{eq:p_laser_th}) in the limit of low $C_{rin}$ as expected.

Local Doping Variations¶

Wafer-scale CVD growth and transfer of graphene is the subject of ongoing research Liu et al., 2022Wang et al., 2022Deokar et al., 2015Xu et al., 2022Li, 2022. Macro-Smith et al., 2017, micro-Whelan et al., 2021 and nanoscopicMartin et al., 2008 variation in the properties of the graphene film are observed. In the following, it will be investigated, to what extent such variations lead to a degradation in switching performance. Such degradation will be described by the change in $\mu_c$ necessary to tune $\Re{\sigma}$ from \qty{90}{%} to \qty{10}{%} of its maximum value at $\mu_c=0$ (see fig. (a)). For reference this quantity, denoted as $\Delta\mu_{c,80\

\begin{figure}[h] \centering \includegraphics[width=\textwidth]{widening.pdf} \caption{Temperature and scattering dependent widening of the switching behavior of graphene. (a) Definition of $\Delta \mu_{80\

\label{fig:widening}

\end{figure} As shown in appendix \ref{ap:dop_var}, the switching width due to doping variations evaluates to $\Delta\mu_{c,80\

Considering \qty{0.1}{eV} as a representative Fermi level standard deviation thus yields a $\Delta\mu_{c,80\

Goos Modulator¶

\begin{figure}[h] \centering \includegraphics[width=\textwidth]{problem_mod.png} \caption{Results of initial optimizations: The geometries in the on and off state were optimized independently (which is unphysical), allowing them to reach very high performance. In the on state, the geometry converges to a structure resembling a waveguide (a) for the transmissive and a reflection grating (b) for the reflective modulator. The off state (b+d) has a less recognizable structure, which however effectively avoids any power in the output waveguide.} \label{fig:goos_problem} \end{figure} Dynamically steering a resonator into critical coupling can lead to full extinction Yariv, 2002. In such cases, MD diverges to infinity. Modulating the resonator out of critical coupling by any amount will result in finite (but small) transmission. Thus IL is finite too, leading to a divergence of the MD/IL ratio. It is therefore clear that equation \ref{eq:md_il_f} does not hold for resonant modulators. It was concluded, that it should be possible to optimize the modulator performance by selecting a suitable geometry of the waveguide layer. Photonic inverse design was leveraged, to determine such geometry (see section Inverse Design with Goos). Initial optimizations performed here yielded highly promising performance improvements. Subsequently, time domain simulations were performed to verify the results reported by the optimization algorithm. These reproduced the expected on-state transmission. However, the transmission in the off state was much higher than expected from the FDFD simulations. At first, possible mismatches in the simulation mesh or the way the graphene absorption was modeled were suspected and ruled out. It was found eventually that the geometries in the on and off state were optimized independently. This lead to geometries ideally suited for the requirement of the respective state (transmission/reflection or suppression). Figure demonstrates the distribution of dielectric and the resulting field distribution found for both states of a transmissive and reflective modulator. While both geometries nicely fulfill the task at hand in the single state it is entirely impossible to switch between the different geometries during modulation. It was investigated how the optimization could be constrained to the same geometry in both modulator states. The method described in Separate Simulation of On and Off State was devised and implemented. \begin{figure}[t] \centering \includegraphics[width=0.6\textwidth]{MD_IL_invdes.pdf} \caption{Modulator devices found by inverse design. The bold markers indicate the performance after the optimization has terminated, while faint markers indicate the performance at previous iterations (marker size increases with iteration step). The desired performance is located at the top left of the diagram. All modulators are operated in the transmissive mode. Different colors indicate different widths of the waveguide and design region. slim: \qty{500}{nm}, \qty{2000}{nm}; med: \qty{1000}{nm}, \qty{1000}{nm}; wide: \qty{2000}{nm}, \qty{2000}{nm}. Med and wide were intended to be used as in figure f). The black line indicates the rectlinear resonator limit for $f=7.068$ which was also used for the inversely designed devices. More then \num{11e3} distinct device geometries are displayed. The weighted MD/IL objective function was used with varying weights.} \label{fig:inv_des_md_il} \end{figure} Subsequently, the optimizer struggled to find appropriate geometries. The different architectures proposed in Inverse Design with Goos were tested varying the size of the optimization region, the type of parametrization (grating/half-grating and cubic spline density based) and the objective functions listed in table \ref{tab:obj_functions}. The MD/IL ratio would only exceed the rectlinear modulator limit in the regime of high IL (see figure ). \begin{figure}[t] \centering \includegraphics[width=\textwidth]{obj_compare.pdf} \caption{Comparison of the device performance reached by inverse design with different objective functions: frac (a), rOMA (b), and MD target (c). The black line indicates the rectlinear modulator limit for the same $f$ as used in the inverse design. The dashed line indicates the performance limit of a singly coupled resonator, which will be derived in section Single Resonator Modulator. The color coding is the same as in figure . The marker shape indicates whether a reflective (circle) or transmissive (cross) architecture was used}

\label{fig:inv_des_obj}

\end{figure} Furthermore, it was observed that by artificially increasing the LMI of the graphene, this regime was reached more frequently. This resulted in an improved capability of the optimizer to minimize the objective function. This behavior indicates that a strong LMI eases the inverse design process, as it allows the consideration of relatively small devices. Comparing different symmetry constraints (and the lack thereof) did not yield conclusive results, as the statistical variability of the devices found here was substantial, depending on the random seed used to initialize the design region. This variability also hints at the optimization problem not being well conditioned depending on the objective function (see figure ). It was observed specifically, that objective functions, that do not discriminate simultaneously high (or low) MD and IL generated a large spread in the final device performance. The MD target objective function was specifically devised to counteract such behavior. It was however found, that the performance would not increase beyond the straight modulator limit under this objective function. This is attributed to an imbalance in penalty/benefit from optimizing MD to be closer to its target value and decreasing IL. It remains an unsolved challenge to deterministically generate highly performant resonant modulator devices using inverse design. It was found that the inverse design algorithm was generally able to perform better for reflective modulators (example in figure (a)). A photonic circuit was devised, to extract the reflected light (see appendix \ref{ap:refl_conversion}).

Single Resonator Modulator¶

\begin{figure}[h] \centering \includegraphics[width=\textwidth]{Tmin.pdf} \caption{Transmission of a singly-coupled resonator as governed by equation \ref{eq:tmin}. (a) Transmission is plotted against roundtrip transmission $a$ (or coupler transmission $\tau$) for a fixed $\tau$ ($a$) of $\sqrt{0.75}$ and $\sqrt{0.5}$ in the solid and dashed lines respectively (similar to Phare et al., 2015). With the x-axis representing $a$ the grey region marks the overcoupled regime for values $a$ larger than the critical coupling point (the inverse holds when considering $\tau$). (b) Transmission for fixed $\tau$ (values as in (a)) with $\xi$ as defined in equation \ref{eq:xi}. The grey lines have been mirrored around critical coupling, expressed by $T_{min}(\xi^{-1})$. They serve as a guide to the eye, indicating lower IL in the overcoupled regime when $a$ is switched.} \label{fig:crit_coup} \end{figure} It has been suggested in the literature, that strong modulation at moderate drive voltage swings can be achieved by tuning a resonator in and out of critical coupling Phare et al., 2015Midrio et al., 2012Yanna et al., 2015Wang et al., 2021. In this section, the critical coupling modulation is investigated exemplarily for a ring resonator, focusing on the effects of a limited switching factor $f$. Due to the analogy between ring resonators and Fabry-Pérot-like cavities, however, this description holds for all resonators, which can be coupled to by a single channel.

It is shown, that for a singly coupled resonator the relation between MD and IL is limited to $(f-1)$ (as is the case for rectlinear modulators) in the limit of low IL, while for higher IL this constraint is lifted. It is also shown, that a doubly coupled resonator (separate input and output channels) is not capable of surpassing the $(f-1)$ limit.

At critical coupling all light present in the bus couples to the resonator, where it gets dissipated/radiated by the intrinsic loss mechanisms of the resonator. This leads to perfect extinction and no light present at the output port. Therefore, critical coupling seems highly suitable for absorption modulation, as it results in infinite MD. However steering the resonator away from critical coupling is not instantaneous, leading to large IL for small deflections from the critical coupling point. As seen from equation \ref{eq:crit_coup} two options exist to steer a resonator in and out of critical coupling: By changing either the ring transmission $a$ or the coupling between bus and resonator $\kappa$ (expressed in terms of transmission $\tau$ for convenience) the outcoupled light can be modulated (see fig. ). In this section, the intrinsic resonator loss is considered to be solely graphene loss, while an analysis of the impact of additional loss mechanisms will be given in subsection Domination Factor. To match the switching behavior found for the rectlinear modulator $\xi$ will be used to represent the deflection from critical coupling. $\xi$ will be defined such that:

Thus $\xi=f$, if critical coupling is reached in the off state of the modulator.

The transmission at resonance and corresponding MD and IL were evaluated for on and off states separated by $f=1.5$ and $4$, assuming the more transmissive state to be the on state (see figure ). Similar to the observations made during inverse design the MD diverges only in regions of high IL. Figure shows that the IL is lower in the overcoupled regime. The magnitude of this effect increases for low $\tau$, which agrees well with figure . \begin{figure}[h] \centering \includegraphics[width=\textwidth]{MD_IL_xi_f4_0.pdf} \caption{Switching behavior of a singly coupled resonator. The round trip loss is modulated by the switching factor $f$ as introduced in section Rectlinear Modulator starting from $\xi$ ((a) $f=1.5$ and (b) $f=4$). MD and IL are calculated as defined in section Performance Metrics. An exemplary IL limit is indicated by the dashed line. Note how MD has two singularities. One is located at $\xi=1$ and one at $\xi=1/f$, which arise from critical coupling in the original and switched state respectively.} \label{fig:MD_IL_xi} \end{figure}

Semianalytical Model¶

\begin{figure}[h] \centering \includegraphics[width=\textwidth]{md_straight_vs_resonant.pdf} \caption{Limitation of MD found by evaluating the analytical expression in \ref{eq:il_contraint}, finding the corresponding on state roundtrip transmission from the definition of $f$. The right panel compares the MD accessible by the resonant modulator for a given rectlinear MD. Minor ticks on the colorbar indicate the used $f$.} \label{fig:MD_IL_limit} \end{figure} The acceptable IL is oftentimes limited. It is thus of interest how much MD can be reached under such IL limit (dashed line in indicates exemplary limit of \qty{3}{dB}) and a certain available $f$. As IL is determined by $T_{res}$ a minimum (maximum) $a$ can be found for the overcoupled (undercoupled) case, so that a given IL constraint is met, by solving equation \ref{eq:tmin} for a:

Where $il$ is IL expressed as a scalar power factor. From this constrained round trip transmission in the transmitting state of the modulator $a$, the round trip transmission in the blocking state can be found by $a^f$ or $a^{1/f}$ depending on the mode of operation. MD can then be evaluated according to its definition. As soon as critical coupling can be reached the available MD diverges to infinity and stays there for higher acceptable IL also, as $f$ can intentionally be reduced to fit (e.g. by reducing the voltage swing). The resulting modulation performance is displayed and compared to the straight modulator in figure . Cascading resonators is clearly not beneficial, as it reduces the acceptable IL per device moving it further away from the MD divergence. The corresponding limit was also added to the inverse design results in figure . The best-performing devices approximate and are limited to the optimal performance of a single resonator modulator. The inverse design algorithm is in principle free to generate an arbitrary number of arbitrarily coupled resonators (given a sufficient extent of the design region and a sufficiently small minimum feature size). The inability of the inversely designed modulators to surpass the single resonator limit thus suggests that this limit holds for any resonant EA-modulator.

Influence on Laser Power¶

\begin{figure}[h] \centering \includegraphics[width=\textwidth]{resonant_compare.pdf} \caption{Comparison of the laser power penalty of a rectlinear (dashed) and resonant modulator (solid) for links limited by different noise sources. Due to the strong increase of MD close to critical coupling the optimum is shifted towards higher IL. It is visible that this shift for the shot noise-limited link (b) is more substantial compared to the thermal noise-limited link (a). The different colors represent different values of $f$. The same palette is used as in figure . The lines for the resonant modulator end where MD diverges to infinity. From this point on the curve follow the IL limit (dotted line).} \label{fig:resonant_compare} \end{figure} To evaluate the reduction in required laser power that a resonant modulator provides to a thermal and shot noise limited link, the ROMA and effective ROMA respectively are compared to the rectlinear modulator. The semianalytical expression developed above can be directly used with the expressions for ROMA (see figure ) and EROMA. The improvement in the shot noise limited case is observed to be more substantial, which is caused by the zero-level shot noise being eliminated by high MD which is favorable for the resonant modulator. Comparing the resonant and rectlinear modulators one finds a maximum improvement of \qty{4.63}{%} and \qty{84.72}{%} for the thermal- and shot-noise-dominated links respectively. As seen in figure the largest improvement is available when the rectlinear $f$ approaches \num{1}. \begin{figure}[h] \centering \includegraphics[width=\textwidth]{shot_rin_thermal_limited.pdf} \caption{Improvement of the required laser power of a resonant graphene modulator relative to the rectlinear modulator for a link limited by (left panel) thermal noise (dashed), RIN and thermal noise (colored) and (right panel) shot noise} \label{fig:resonant_improvement} \end{figure} As described in section Rectlinear Modulator the laser power penalty by RIN is only meaningful for links limited by additional noise terms.

The laser power penalty was compared for resonant and rectlinear modulators for a link affected by RIN and thermal noise (figure ). As seen the improvement diverges for $f$ approaching one, which can be explained by the proximity to the cutoff of equation \ref{eq:rin_md_req}. It is further noted that for low $C_{rin}$ and/or high $f$ the improvement converges to the thermal noise limited case as expected.

Off-Resonant Modulation¶

For wavelengths detuned from resonance significantly less IL is introduced in the overcoupled regime (see fig. Figure 5). In the following section, it will be shown, that it is nevertheless not beneficial to the MD/IL ratio to use a resonant modulator at off-resonant wavelengths. A meaningful measure of detuning from resonance can be found by considering $\Delta \phi$ normalized by the width of the resonance dip. The HWHM in terms of $\phi$ at critical coupling can be evaluated according to appendix \ref{ap:hwhm}.

\begin{figure}[h] \centering \includegraphics[width=\textwidth]{offres.pdf} \caption{Semi analytical evaluation of the relation between IL and MD when detuned from resonance by $\phi_{norm}$ normalized to the $\mathrm{HWHM}$ for coupler transmission $\tau$ for f=\num{1.5}. (b) Numerically extracted improvement in the MD/IL ratio (upper panel) and ROMA (lower panel) based on equations \ref{eq:off_res_il_analytical} and \ref{eq:md_ring}. The values of $\phi_{norm}$ not present in the colorbar are indicated directly in (a).} \label{fig:MD_offres_analytical} \end{figure} With the IL in scalar units (power) $il$ equivalent to the transmission of the resonator at an offset from resonance $\phi$ as given by equation \ref{eq:t_allpass} solving for $a$ one finds:

MD is then evaluated at the given $a$ from (in scalar form):

By comparing the different branches and switching directions (i.e. over and undercoupled) MD is found to be maximized for the $+$ branch of $a$ and $f>1$ (the overcoupled regime). This branch is depicted in figure (a) for different detuning from resonance and different $\tau$. It is observed, that the influence of $\tau$ only becomes significant for very strong detuning. The region relevant for resonant modulation enhancement is located in the upper right of the diagram, where MD diverges when at resonance. It is found, that the enhancement in MD/IL ratio decays with detuning, with the divergence to infinity only present perfectly at resonance. The maximum relative improvement of the MD/IL ratio is extracted numerically for different detuning and is depicted in figure (b). Similarly the ROMA is determined for such MD and IL. IL is chosen to maximize ROMA and the corresponding ROMA is recorded for different detuning. It is observed that despite the improvement in MD/IL decreasing rapidly an improvement in ROMA is sustained for more than \qty{20}{%} of the resonance width.

Domination Factor¶

The round trip transmission in the different modulator states is related by the graphene switching behavior $a_{on} = a_{off}^f$ only if the switched graphene is the only source of loss^[1]. Considering additional round trip loss $a_{add}$ however one finds a reduced switching factor $f^\prime$:

Introducing the domination factor $d$ such that $a_{add}^d = a_{on}$, indicating the extent by which the overall loss is dominated by the contributions of the graphene, one finds (derivation in \ref{ap:domination}):

The effects of this degradation are shown in figure Domination Factor. It should be noted, that even for high^[2] $d$, a deterioration of some percent is observed. This shows that the ROMA improvement by introducing a resonator (<\qty{5}{\%}) is marginal and likely overshadowed by variations in the unmodulated loss. \begin{figure}[h] \centering \includegraphics[width=\textwidth]{domination_influence.pdf} \caption{Influence of unswitched loss expressed by the domination factor $d$ on the effective switching factor $f^\prime$. The left panel shows the absolute value of $f^\prime$ while the right panel shows the relative change from $f$ to $f^\prime$.} \label{fig:domination} \end{figure}

Minimizing Graphene Footprint¶

To this point resonant enhancement of the switching behavior of graphene has proven limited potential to overcome the MD vs. IL trade-off, assuming that $f$ of the graphene conductivity stays unchanged irrespective of the device geometry.

It is shown, that for locally correlated doping variations, with a correlation length similar to the device dimensions, the average deterioration in $f$ due to these variations can be reduced by decreasing the device size. Therefore depending on the different contributions to $f$, reducing the device size can lead to a significant improvement in switching performance. Furthermore, a reduced graphene capacitance would positively affect the energy per bit (see equation \ref{eq:e_per_bit}). Despite not being reflected by the cross-sectional bandwidth model, literature suggests that reducing the device length has a positive effect on bandwidth Agarwal et al., 2021 (see also appendix \ref{ap:bw_experimental}). Thus it is of increased interest to reduce the modulator footprint. It is shown that a minimum $d$ limits the required graphene length in resonant modulators. The resulting optimum device dimensions will be evaluated for ring resonators and Fabry P'{e}rot resonators.

Device Size with locally correlated Doping¶

\begin{figure}[h] \centering \includegraphics[width=.53\textwidth]{full.png} \includegraphics[width=.45\textwidth]{colorbar_advanced.pdf} \caption{Numerically generated doping surface. Three-dimensional representation of a doping surface with $L_{corr}=164$ grid points on a 4096x4096 grid. The vertical displacement represents the deviation of the local chemical potential from its average value. The corresponding probability density function displays small deviations from the expected gaussian shape (dashed line), which were observed to be less pronounced for surfaces with a higher ratio between the total grid points in each direction compared to their $L_{corr}$} \label{fig:doping_surface} \end{figure} \begin{figure}[h!] \centering \includegraphics[width=\textwidth]{all.pdf} \caption{Modulation penalty due to doping variations for different device sizes with respect to the autocorrelation length $L_{corr}$ of the variations. (a) to (e) with the exception of (b) show the penalty imposed on the required modulation voltage swing normalized by the penalty in the large device size limit (see section Local Doping Variations) for different device widths. The quantiles marked by different colors are consistent across the subfigures. (b) depicts the underlying raw data (0.01 quantile and mean) of (a). The dotted black circle marks a small numerical inaccuracy introduced by the device width being in the order of the grid quantization. This inaccuracy is however minimized in the subsequent averaging to generate the stitched plots (see (a)).}

\label{fig:device_size}

\end{figure}

Considering a spatially slowly varying doping distribution, it is hypothesized, that selecting a patch that is small compared to these variations should yield a more homogeneous distribution in the selected patch. Long-lengthscale variations should be knocked out, as only the local and thus strongly correlated doping levels are considered. In order to quantify this effect for technologically relevant doping variations the approach introduced in section was used to numerically generate doping maps of defined mean and standard deviation and with an exponential radial ACF (see fig. ). Evaluating modulation performance for small and large devices simultaneously requires prohibitively large doping maps. Square maps with a side length of only \num{4096} pixels were generated for different $L_{corr}$ (see figure ), to avoid large computational expense. The results were merged together in a post-processing step (see figure a) and b)).

Using the simplifying assumption that the standard deviation of the Dirac voltage $\sigma(V_d)$ relates linearly to the peak-to-peak modulator voltage $V_{mod}$, the influence of device length (along the direction of propagation in the waveguide) was evaluated for multiple device widths. The results are displayed in figure . It is seen, that for all widths the mean and median $V_{mod}$ reduce for shorter devices, which supports the hypothesis stated above. It is however observed, that the graphene width strongly influences the magnitude of mentioned decrease. As expected smaller-width devices show a more pronounced knock-out effect. It is observed, that the variations in $V_{mod}$ decrease for long devices independent of their width. For short devices, a similar decrease of variability is observed, which is however limited to devices with a width smaller than $L_{corr}$. For all but the widest device the $V_{mod}$ variability has a maximum around $5L_{corr}$. The spread towards higher $V_{mod}$ is broader compared to the downward direction, which is also evident from the mean exceeding the median for all configurations.

Correlation Length of Doping Variations¶

It was found that device scaling can knock out negative influences of doping variability. In the following it will be evaluated, whether it is possible to decrease the graphene footprint in modulators to an extent, that has a significant and positive influence on $V_{mod}$ (or $f$ for a fixed $V_{mod}$. The doping length scales are characterized by spatially resolved Raman measurements in the first step. In the next subsection, it will be discussed to what minimum dimensions the graphene patches can be scaled while maintaining modulation performance. \begin{figure}[t] \centering \includegraphics[width=\textwidth]{L_corr/doping_maps.pdf} \caption{Spatially resolved distribution of the chemical potential in as transferred graphene on \qty{90}{nm} $SiO_2$. The spectroscopic maps of increasing resolution were recorded by confocal Raman microscopy and $\mu_c$ extracted according to section Raman Spectroscopy. (a) \qtyproduct{20x20}{\micro\m} at \qty{333}{nm} resolution, (b) \qtyproduct{5x5}{\micro\m} at \qty{100}{nm} resolution, (c) \qtyproduct{2x2}{\micro\m} at \qty{20}{nm} resolution. The black lines in (a) correspond to the position of wrinkles visible by optical microscopy, while the optical inspection of areas in (b) and (c) did not detect obvious defects in the graphene layer. White dots indicate points at which reliably fitting both Raman peaks failed.} \label{fig:doping_maps} \end{figure} The doping of graphene has been observed experimentally to be non-uniform. It rather exhibits electron-hole puddles on nanoscopic scales which have been investigated extensively Martin et al., 2008Zhang et al., 2009Samaddar et al., 2016Singh & Gupta, 2018. The characteristic puddle size has been observed to be less than \qty{20}{nm}. It has been found to depend on the average doping. Charged impurities in the dielectric environment of the graphene layer have been identified as the predominant cause of such puddles Samaddar et al., 2016Singh & Gupta, 2018. In graphene grown by CVD on copper substrates and transferred to foreign substrates using the PMMA method, a zoo of large-scale imperfections exists, that can alter the electrical properties. These include grains up to multiple hundreds of $\mu m$ in size and their boundaries, wrinkles, folds, add layers, polymeric residues and other contaminants Bousa et al., 2016Sinterhauf et al., 2020Tyagi et al., 2022Lee et al., 2012. Extraction of $L_{corr}$ according to section was initially performed on a \qtyproduct{20x20}{\micro\m} map, which is shown in figure (a). The extraction was repeated for the higher resolution maps of (b) \qtyproduct{5x5}{\micro\m} and (c) \qtyproduct{2x2}{\micro\m}, to ensure that the spatial quantization does not distort the ACF at small $r$. The radial profile of the autocorrelation function $\mathrm{ACF}(r)$ for all three maps is shown in figure . As indicated the measurement corresponds well to an exponential autocorrelation of L_{corr}\approx\qty{450}{nm} except for the longest $r$ per measurement, which is however attributed to the limited spatial extent of the recorded maps, which was chosen to keep the measurement duration acceptable. It is further noted that the horizontal autocorrelation (fast scanning axis) is consistently higher compared to the vertical autocorrelation (slow axis). This effect is assumed to originate from the scanning behavior of the confocal microscope. An alternative explanation could involve some astigmatism in the microscope optics and a resulting elliptical beam shape. The $\mathrm{ACF}(r=0)$ per definition takes a value of \num{1}. However, the exponential fit takes a value of \num{0.92} at $r=0$. As demonstrated in figure , this effect was faithfully modeled using additive white (uncorrelated) measurement noise. \begin{figure}[t] \centering \includegraphics[width=0.6\textwidth]{L_corr/correlation_length_extraction.pdf} \caption{Radial profile of the two-dimensional autocorrelation of the three $\mu_c$ surfaces shown in figure . The solid lines indicate the radial average of the $\mathrm{ACF}$ while the differently shaded areas correspond to the horizontal and vertical cut. Least squares fitting was highly sensitive to the variations at long length scales. Therefore an approximate fit to an exponential was performed manually, yielding a L_{corr}\approx\qty{450}{nm} indicated by the dashed line. The dash-dotted line shows an exemplary Gaussian profile.} \label{fig:acf_extract} \end{figure} The extracted $L_{corr}$ is on the same order of magnitude as the beam diameter (see section Raman Spectroscopy). Therefore the spatial resolution of the imaging optics might determine the measured $L_{corr}$. The effects of a finite beam diameter on recorded Raman signatures were modeled assuming that the illuminated area can be divided into infinitesimal patches, which contribute to the extracted peak position proportionally to the illumination intensity. This intensity was assumed to have a two-dimensional (round) Gaussian distribution with a beam diameter $d_{beam}$ (FWHM). The measurement is modeled as a convolution with the Gaussian beam.

The $\mathrm{ACF}(r)$ was evaluated for beams of different size relative to the original $L_{corr}$ (see figure ). \begin{figure}[t] \centering \includegraphics[width=\textwidth]{L_corr/gaussian_filtering_behavior.pdf} \caption{Effect of convolution with a Gaussian beam of diameter $d_{beam}$ on the $\mathrm{ACF}(r)$ of a map with original correlation length $L_{corr}$. The right inset shows the resulting new correlation length (extracted by least squares fit) in blue. Additionally, the $R^2$ parameter is indicated for fitting with an exponential and Gaussian.} \label{fig:gaussian_convolution} \end{figure} As expected the convolution increasingly introduces additional long-scale autocorrelation with broader beams. However, the shape of $\mathrm{ACF}(r)$ changes from an exponential to a Gaussian. This change is clearly observed from the coefficient of determination $R^2$ of a least squares Gaussian and exponential fit indicated in the right panel of figure . Conversely, the measured autocorrelation profiles fit more closely to an exponential shape than a Gaussian. Furthermore, the convolution with a Gaussian has been observed to narrow the standard deviation $\sigma_{\mu_c}$ of $\mu_c$ in the resulting extraction by a factor of $\delta\sigma_{\mu_c}$ (see figure ). The measured standard deviation would thus need to be multiplied by the inverse of $\delta\sigma_{\mu_c}$ to find the deviation physically present. As the measured $\sigma_{\mu_c}$ are already in the range of \qty{0.1}{eV} this scenario is considered unreasonable. \begin{figure}[t] \centering \includegraphics[width=\textwidth]{L_corr/Beam_statistics.pdf} \caption{Influence of the convolution with a Gaussian on the statistical distribution of $\mu_c$. The left panel shows a mean-free histogram of the extracted chemical potentials $\mu_c$ normalized by the standard deviation of the original doping distribution. The right panel shows the relative change in the standard deviation due to the convolution, expressed as a multiplicative factor. The datapoints for large original $L_{corr}$ and large beam diameter are greyed out and should not be considered, as the beam size approached the total size of the map, leading to border effects in the convolution.} \label{fig:beam_statistics} \end{figure} In summary, a correlation length of approximately \qty{450}{nm} has been identified based on three measurements of different grid spacing and extent. It was investigated, whether the extracted $L_{corr}$ could be an artifact of the beam diameter of the measurement system. This, however, does not seem to be the case. The experimental data set, this analysis is based on, is very sparse. Therefore further investigation on the extent of mesoscopic variations in the doping of graphene should be conducted. Moreover, the origins of such variations and the potential influence of cleaning or annealing steps and substrates should be investigated systematically.

Limitation of Graphene Length¶

The required length of graphene is determined by the round trip loss $a$, which in turn has to be matched correctly to $\tau$ of the coupling mechanism.

Increasing $\tau$ shrinks the required graphene length $l_{gr}$. Narrowing the optical linewidth can be avoided by simultaneously reducing the device length (see figure Figure 1).

To allow for a decreased graphene-induced loss without harming $d$, it is advantageous to minimize $a$ to reach the smallest possible graphene footprint:

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l_{gr} = \frac{d \cdot 20\log_{10}(\tau a)}{\alpha_{\mathrm{trans}}[\mathrm{dB}/\unit{\micro \m}]}

Where $\tau$ and $a$ are the coupler and round-trip transmission respectively. $\alpha_{\mathrm{trans}}$ is the graphene-induced loss in the transparent state.

Ring Resonator¶

The round trip transmission $a$ in an unloaded ring resonator is determined by a combination of radiation loss decreasing exponentially with bending radius and loss mechanisms like material absorption, roughness scattering, etc., which are also present in straight waveguides. Thus there should exist a radius that minimizes $a$. Published bend loss simulations Xu et al., 2008Midrio et al., 2012Chrostowski & Hochberg, 2015 were considered and additional FDE simulations were performed, to identify the optimal radius $R_{opt}$. It was found that the results presented in Midrio et al., 2012 coincide with the calculated losses from Chrostowski & Hochberg, 2015 including mode mismatch loss (see fig. (a)). Mode mismatch losses are relevant whenever a transition between sections of waveguide of different radius are present (e.g. bent to straight), which is the case for racetrack resonators. As Midrio et al. (2012) analyses a circular ring, mode mismatch losses should not be considered.^[3]

The radiative bend loss disregarding mode mismatch loss was simulated using FDE. FDTD simulations of small silicon microrings with slightly larger waveguide height of \qty{250}{nm} (\qty{220}{nm} used here) are used for comparison Xu et al., 2008. The required length of graphene was calculated for the different simulated bend losses assuming a graphene absorption of \qty{0.01}{dB\per\micro\m} in the transparent state and a required domination factor of $d=5$ (see figure (b)). \begin{figure}[t] \centering \includegraphics[width=\textwidth]{graphenelength.pdf} \caption{Left: Comparison of the simulated bend loss from Midrio et al., 2012 (Midrio et al.) and Chrostowski & Hochberg, 2015 (Chrostowski et al.). Chrostowski & Hochberg (2015) considers a right-angle waveguide bend connecting two straight waveguides. Thus a mode mismatch at both transitions leads to additional loss. The displayed loss is scaled by a factor of \num{4} to account for a full roundtrip. Midrio et al. (2012) considers a circular ring resonator. Mode mismatch losses should therefore be neglected, which seems to not be the case with both curves showing similar behavior. Adding the influence of straight waveguide loss (\qty{4}{dB \per \cm} dash-dotted line) one finds a minimum roundtrip loss around \qty{10}{\mu m}. Right: The interaction of different loss mechanisms in ring resonators leads to the roundtrip loss being minimized for $R\approx$\qty{1.7}{\micro \m} depending on the remnant loss of a straight waveguide. The radiation loss was determined from FDE simulations and FDTD simulations of the full resonator performed by Xu et al. (2008). The required graphene length reaches a minimum of less than \qty{3}{\micro \m}} \label{fig:bend_loss_compare} \end{figure} The power propagation in a bent waveguide is shifted radially outward increasingly with decreasing bending radius. Typically the waveguide sidewall roughness is the primary source of scattering loss. Therefore it is reasonable to assume that the scattering loss increases in tight bends, as the mode moves towards the rough sidewall surface Vlasov & McNab, 2004Roberts et al., 2022. This effect has only recently been modeled by a modification to the Payne-Lacey model Payne & Lacey, 1994 for waveguide bending loss Roberts et al., 2022. No accurate description for tightly bent narrow high index contrast waveguides is known to the author as of today. An experiment was designed to obtain such description if desired (see section Mask design).

Fabry-Pérot Resonator¶

Minimizing the round trip loss in a Fabry-Pérot resonator involves minimizing the loss of the reflectors and the propagating section. The latter is achieved most effectively by reducing the length of the propagating section such that the operating wavelength resonates as the fundamental mode (DFB operation). Sagnac reflectors rely on waveguides, that have to be bent more tightly for the same roundtrip length compared to a ring resonator. Thus these are in any case outperformed by ring resonators.

Scattering parameter simulations assuming Fresnel coefficients at the interfaces of waveguide sections were performed to obtain the penetration depth and reflectivity of DBRs (see figure (a)). The penetration depth is defined as the length into the reflector at which the field has decayed to $1/e$. Increased propagation losses were observed to limit the maximum reachable reflectivity. The influence of propagation loss can be mitigated by introducing a stronger contrast between the $n_{eff}$ and thus shorter $d_p$. It is assumed, that more light is scattered out per length in a DBR compared to a regular waveguide, due to increased roughness and high field intensities at the surfaces normal to the direction of propagation. This effect is pronounced for DBRs with large width variation. It is therefore not immediately clear what amount of width variation is preferable. To evaluate the influence of variations in the length of segments and their $n_{eff}$ (e.g. by width variations) Monte Carlo simulations with varying variability in those parameters was performed (see figure (b)). It was observed, that the reflectivity was reduced for DBR of a given length. By elongating the DBR however, it was possible to recover the reflectivity to its original value (neglecting propagation loss). It is concluded that the DBR performance is highly fabrication dependent and should thus be evaluated experimentally. For this purpose, a mask including corresponding test structures was generated. \begin{figure}[h] \centering \includegraphics[width=\textwidth]{dbr.pdf} \caption{Ability of DBR to reflect light based on scattering parameter simulations assuming Fresnel coefficients. (a) shows the penetration dept $d_p$ at which the light decayed to $1/e$ is highly dependent on the contrast of $n_{eff}$ between the wide and narrow sections. (b) depicts a Monte Carlo simulation of the reflectivity $r$ of DBRs of different length for a nominal $\Delta n_{eff} = 0.3$.} \label{fig:dbr} \end{figure}

Arbitrarily shaped Cavities¶

For resonators of arbitrary shape, it is not well defined what constitutes a roundtrip. It is thus of interest to identify a more general property of resonators that determines to what size the graphene patch can be shrunk. The absorption is proportional to the intensity of the \efield parallel to the graphene. From the prerequisite, that almost all loss in the modulator should stem from the graphene becomes apparent, that to minimize the graphene area it is required to increase the absorption per area. In a resonant modulator, this increase in absorption is provided by the enhancement of the \efield intensity. For a critically coupled ring the intensity enhancement is found by the transfer matrix method to be:

Where the nominator is dependent on the position around the ring (in this case directly before the coupler). $\chi_{crit}^{-1}$ is approximately proportional to expression \ref{eq:graphene_length} for $\tau$ close to \num{1}.

At steady state the power dissipated in a resonator and the power added to the resonator have to be equal yielding with the definition of $Q$:

$W_{res}$ being the energy stored in the resonator. The energy densities of the electric field ($U_E$) and the magnetic field ($U_H$) are equal to $\varepsilon |\vec{E}|^2$ and $\mu |\vec{H}|^2$ respectively Wolski, 2011. Thus one has to evaluate $Q/V$ to obtain a measure for the average intensity enhancement in the resonator.

The ratio of $Q/V$ is proportional to the Purcell factor describing the increase in spontaneous emission of a single emitter strongly coupled to a cavity in CQED Agio & Cano, 2013Romeira & Fiore, 2018:

Due to its relevance in quantum optics maximizing the Purcell enhancement has received considerable attention Robinson et al., 2005Lu et al., 2013Vučković, 2017Choi et al., 2017. Inverse design has been shown to be applicable to such optimizations Wang et al., 2018. Using the description of the ring modulator it is possible to obtain the proportionality constant $\eta$ relating $Q/V$ and $l_{gr}$. Under the conditions presented in appendix \ref{ap:qv_eta} it is found to be \eta\approx\qty{2e17}{m^-2}. This indicates, that cavities presented in literature are capable to reduce $l_{gr}$ well below \qty{1}{\micro \m} Lu et al., 2013Seidler et al., 2013.^[4] It should be noted that this analysis disregards possible changes in the coupling of the graphene to the resonator. Going forward such effects should be analyzed in further detail.

Mask design¶

Layouts of different types and geometries of photonic resonators were compiled into a mask, to experimentally evaluate to what extent $Q/V$ is degraded in the fabrication process. The generated mask is intended for a single EBEAM lithography and subsequent silicon etch on a \qty{220}{nm} SOI wafer. Grating couplers are used, to couple light into and from the chip. The dimensions of the gratings have been designed by a co-worker in such a way, that they can be defined lithographically and etched in the same step as all other structures. Besides the cavity test structures, the mask contains cutback blocks for the waveguides and special cutback blocks for \ang{90} and \ang{180} bends. Determining the loss of these different bends allows decomposing the mode mismatch loss and the loss bend angle. By extracting the bend loss for different radii of curvature the influence of a radius-dependent scattering loss as described in Roberts et al., 2022 can be evaluated. If required the model used to determine $R_{opt}$ for the ring resonators can be adjusted. As described by Xu et al., 2008 a small ring radius results in a significant phase mismatch between bus and ring of equal width. A reduced bus width fulfilling phase matching was thus calculated from $n_{eff}$ extracted from FDE simulations. In addition to single-ring test structures, arrays of five rings coupled to the same bus have been designed so that their resonances should be spaced equally within one FSR. These test structures are intended to investigate the fabrication sensitivity of the relative alignments of these rings placed closely on the mask. It is expected, that the large FSR and relatively low $Q$ resulting from the small ring radius will help to stabilize (relative to the FSR) the resonance grid. Similarly, different distributed feedback cavities are present in the mask to evaluate experimentally the achievable round-trip transmission.

Fabrication and Characterization of Teststructures¶

The optical measurements of the test structures should be evaluated in the following order: First, the waveguide loss and coupler loss should be extracted from the cutback block. Similarly, the loss of a single circular \ang{90} and \ang{180} bend should be extracted separately for the different bend radii. From there the radiation loss in the bent waveguide and the mode mismatch loss can be separated and compared to the predictions presented here. The transmission spectra of the ring resonators should be recorded and fitted by a Lorentzian or according to equation \ref{eq:t_allpass}. A variation of the coupler gap is included, to avoid ambiguity of the coupling state. The different coupling strengths enable identifying the over or undercoupled state from the direction in which the maximum extinction at resonance varies for increasing coupling. Consequently the loaded and intrinsic $Q$-factors can be extracted from the FWHM and by finding the relation between $a$ and $\tau$ via equation \ref{eq:tmin} respectively. As the mode volume $V$ is assumed to be mostly unaffected by fabrication variations, roughness, etc., the $Q/V$ ratio can be identified using this approach. $Q/V$ is related to the required graphene length as described in appendix \ref{ap:qv_eta}. Similarly, the other resonators should be analyzed. However, the coupling variations either coincide with a variation in roundtrip loss (DFB resonators) or are not present in the mask. The coupling state of these devices can be identified by introducing additional loss to the resonator in a second measurement (e.g. by covering it with graphene).