1. INTRODUCTION

Thermo-optical phase shifters (TOPSs), or optical-waveguide heaters, are used to locally modify the phase of light, allowing for optical operations through dynamically controlled constructive or destructive interference. As a result, these devices have found numerous applications in photonic integrated circuits (PICs) [1,2], nanophotonic phased arrays [3], optical neural networks [1,4], programmable photonic circuits [5,6], and quantum photonic devices [7–9]. As shown in Fig. 1(a), a TOPS is a device consisting of an optical waveguide and a resistive heater. The heater, usually made of a metal [10–13] or a doped semiconductor [10,14,15], will generate heat, which will change the refractive index of the waveguide. While Fig. 1(a) shows a 3D depiction of a TOPS, we work with extruded 2D heater geometries since the TOPS setups we are interested in tailoring consist of heating translationally invariant waveguides. Thus, we can exploit the significantly reduced computational cost associated with the numerical simulations of the 2D physics problem. In this case, and under suitable simplifying assumptions [16], the phase-shift of the mode propagating through the waveguide can be expressed as

 

Fig. 1. (a) Sketch of the working principle of the TOPS and (b) optical loss mitigation example via PT-symmetry breaking. Temperature and electric field intensity are shown in max-normalized units.

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There are many approaches to building an efficient TOPS. In this work we focus on designing cross-sections of compact TOPS devices with low optical losses on a silicon on insulator (SOI) platform where the heater is a highly lossy metallic material, in this case, titanium nitride (TiN) [10]. In many conventional TOPS designs, the heater is placed far away from the waveguide to mitigate optical losses, which often results in poor power performance of the device. However, in several studies it has been demonstrated that through the use of ENZ materials [18], or the use of parity–time (PT)-symmetry breaking [19], it is possible to limit the optical losses while having the heater in direct contact with the optical waveguide. In particular, PT-symmetry breaking has widely been employed to change the spatial distribution of gain/loss properties in waveguides and resonators [20–22]. In Fig. 1(b) we show an example of a PT symmetry-breaking system, where the propagating waveguide mode has low optical intensity near the metal, resulting in low optical losses.

In this work, we propose a general framework, based on inverse design by topology optimization, to design more efficient TOPS for problem-specific input parameters. As an application, we use the framework to design a TOPS starting from an initially uniform starting guess and compare it with a reference device based on PT-symmetry breaking [19]. The reference design has the advantage that a metallic heater is positioned directly on top of the waveguide while simultaneously reducing losses by exploiting PT-symmetry breaking, allowing for good heat-conducting low-loss devices. TO is an inverse design [23] tool that has been successfully used in the photonics community in many applications, such as waveguide design [24–26], cavity design [27–30], photonic demultiplexer design [31], metasurface design [32,33], microresonators [34], topological insulator design [35], etc. While a series of impressive results has been achieved by employing TO for photonic devices, previous optics applications were for single physics or uncoupled multiphysics problems; see, e.g., [36,37]. While coupled multiphysics TO applications have been demonstrated in, e.g., MEMS design [38], we here derive the first coupled multiphysics TO formulation in the nano-optics literature and demonstrate how TO can be applied to designing metallic heaters in low-optical-loss TOPS. Additionally, we provide the detailed derivation of the adjoint sensitivities for multiphysics coupled problems required to implement our approach, and showcase the various features of the framework, such as modifying the maximum amount of metal allowed, constraining the input power of the heater, and accounting for manufacturability issues.

2. TOPS: MODEL PROBLEM AND REFERENCE DESIGN

In order to model the reference TOPS device in a two-dimensional setting, we consider the simulation domain illustrated in Fig. 2(a). This domain is a representation of the cross-section of a waveguide [see, e.g., Fig. 1(b)], where we have a silicon (Si) optical waveguide in the region ${\Omega _1}$ and a silica (${{\rm SiO}_2}$) cladding in ${\Omega _2}$ with width ${w_{{\Omega _2}}} = 14\;\unicode{x00B5}{\rm m}$ and height ${h_{{\Omega _2}}} = 4\;\unicode{x00B5}{\rm m}$, which together form the simulation volume $\Omega$. Additionally, on top of the waveguide, we have a metal (TiN) with variable width. For more information on the material properties and system parameters, consult Appendix C.

 

Fig. 2. PT-symmetry breaking TOPS and its thermal and max-normalized optical field in the two-dimensional model.

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To accurately model the physics of TOPS, we consider a weakly coupled multiphysics problem involving both heat and electromagnetic field propagation. The heat problem can be modeled using Fourier’s law

The optical problem is modeled using Maxwell’s equations assuming time-harmonic excitation, which in the frequency domain can be written as a single vector-values wave equation of the form

Using the physics model described above, obtaining the fields illustrated in Fig. 2(b), we are able to reproduce the physical response of current state-of-the-art devices, such as the ones introduced in [19]. These devices harness passive PT-symmetry breaking with a single SOI waveguide with metal directly deposited on top, which enhances the heat conduction of TOPS while limiting the optical losses of the system [19]. Accordingly, we use this system as a reference to be compared to optimized designs.

To quantify the propagation of an electric mode in a translationally invariant waveguide, one may use the following approaches: analyzing the system’s response when excited at the frequencies of interest in the frequency domain or solving an eigenvalue problem to determine the eigenmodes of a cross-section of the waveguide. In our modeling, we first compute the temperature field finding the stationary solution of Eq. (2) to determine the refractive index of the waveguide under both heated and unheated conditions. Then, we solve Eq. (2) with an analysis in the frequency domain by performing a frequency sweep, as illustrated in Fig. 3. By plotting the maximum intensity of the electric field against the effective index, we characterize the modes based on their central effective index ${n_{{\rm eff}}}$ and losses $\alpha$. To characterize the optical modes using the frequency domain model, we adopt a methodology commonly employed for characterizing optical cavities [41]. This involves fitting a Lorentzian peak to each mode, enabling us to extract the central effective index of the mode ${n_{{\rm eff}}}$ and the full-width-half-maximum (FWHM) values, where the latter is equivalent to the optical loss of the mode $\alpha$. Alternatively, and for validation purposes, we solve an optical eigenproblem, following a similar approach to [19]. The system’s complex eigenvalues, denoted as $\Lambda = {n_{{\rm eff}}} + {\rm i}\alpha$, provide insights into the mode properties. Calculating the mode properties with the frequency domain approach and validating it with the eigensolution yields a relative error, defined as $\Delta _x^{A \to B}[\%] = 100 \cdot ({x_B} - {x_A})/{x_A}$ for the quantity $x$ with cases $A$ and $B$, of ${\lt}1\%$ between the frequency domain and eigenvalue approaches. For more information on model validation, consult Appendix D.

 

Fig. 3. Optical response of the modes in the PT-symmetry breaking system for two different metal widths as a function of the effective index ${n_{{\rm eff}}}$ for an input power of ${P_{{\rm in}}} = 20\;{\rm mW} $.

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Figure 3 illustrates the frequency domain procedure for estimating the losses and index shift as an effect of heating the device, using two simple heater setups with different metal widths: ${w_{{\rm metal}}} = 4\;\unicode{x00B5}{\rm m}$ (shown in red) and ${w_{{\rm metal}}} = 2\;\unicode{x00B5}{\rm m}$ (shown in blue). The plot shows the behavior of the fundamental mode in the unheated and heated conditions, exhibiting a clearly visible shift in the effective index $\Delta {n_{{\rm eff}}}$, and an increase in losses (lowering of the peak) is observed due to the heating for both setups. By studying the figure, one may observe that there are two loss mechanisms at play: the inherent loss due to the introduction of the lossy metallic heater, and the non-uniform heating of the waveguide, which results in a non-uniform refractive index profile of the optical waveguide. In the unheated conditions, both mode peaks show a broadening due to the losses introduced by the presence of the metal. Since we have less metal volume for ${w_{{\rm metal}}} = 2\;\unicode{x00B5}{\rm m}$, the peak is higher and shows less broadening than the peak for ${w_{{\rm metal}}} = 4\;\unicode{x00B5}{\rm m}$. When both waveguides are heated, we observe an increase in the loss factor caused by the inhomogeneous heat distribution over the waveguide [see, e.g., Fig. 2(b)]. Specifically, the temperature change is most significant in the vicinity of the metal region. As a result, the sections of the waveguide closer to the metal experience a larger shift in refractive index. Consequently, the guiding capability of the mode, which relies on a uniform index, becomes compromised, leading to increased losses. It is important, as it is possibly non-intuitive, to note that this effect becomes more substantial when the metal width is smaller. With the same power applied, a smaller metal width generates more heat. Consequently in the heated scenario, the shift in refractive index and the effect of the non-uniform heating become more significant, resulting in higher losses.

Finally, we relate the loss coefficient $\alpha$ to the loss per unit-length in the physical waveguide as

For our reference devices, using the eigenvalue solution of $\alpha$, we determine that the unheated case yields ${{\rm loss}_{{\rm unheated}}} = \def\LDeqbreak{}44\;{\rm dB/cm}$ for the ${w_{{\rm metal}}} = 4\;\unicode{x00B5}{\rm m}$ reference, while the heated case yields ${{\rm loss}_{{\rm heated}}} = 47\;{\rm dB/cm}$. Regarding the ${w_{{\rm metal}}} = 2\;\unicode{x00B5}{\rm m}$ reference, the unheated case shows ${{\rm loss}_{{\rm unheated}}} = 16\;{\rm dB/cm}$, whereas the heated case shows ${{\rm loss}_{{\rm heated}}} = 18\;{\rm dB/cm}$.

3. THERMO-OPTICAL DESIGN PROBLEM

In order to design extreme performance, low-loss TOPS, we employ inverse design by topology optimization. To maximize the TOPS performance, we seek to minimize the losses incurred due to the introduction of the metal in the optical problem at the desired wave numbers, yielding the targeted phase-shift. In essence, since the waveguide acts as a resonator, we aim to narrow the resonant peaks, similar to those illustrated in Fig. 3. As we have discussed earlier, these peaks can be understood as Lorentzian resonances. Therefore, by maximizing the maximum value of these peaks, we can expect a decrease in the FWHM, which directly corresponds to reduced losses [42]. Thus, we choose to maximize a figure of merit (FOM) related to the optical field intensity inside the waveguide, which is equivalent to maximizing the response at a particular effective index. This can be expressed as

The design procedure for maximizing the FOM in Eq. (9) is based on the configuration in Fig. 2(a) where we have a simulation domain $\Omega$ consisting of a cladding and an optical waveguide, similar to the one presented in the previous section. The cladding in ${\Omega _2}$ acts as the design domain, where we seek to identify the optimal distribution of a metal (TiN) and an insulator (${{\rm SiO}_2}$) that gives rise to the best performing TOPS. We identify the optimized material distribution by formulating the TOPS design problem as an optimization problem, where the material distribution is controlled by a set of design variables through the field $\xi$. This problem is then solved iteratively using the gradient-based MMA optimization algorithm [43]. Note that for all the starting guesses in the optimization problem, we choose a homogeneous design ($\xi = {\xi _0}$), leaving the optimizer free to tailor the device geometry without biasing the result by prescribing any geometrical features in the initial guess.

Like in many other TO formulations, we adopt a filtering and thresholding scheme similar to the one described in [44]. For the filter, we use a convolution-based filter:

To obtain the material parameters using the filtered and thresholded design field ${\bar {\tilde \xi}}$, we employ two different material interpolations. For the optical problem, we use the non-linear material interpolation based on [45]:

For a given material configuration, the electric field intensity must be optimized inside the waveguide for both the heated and the unheated device for the FOM in Eq. (9). Therefore, we use the maxmin formulation [46] for the optimization problem where the two FOMs are reformulated as constraints. In addition, and in order to regularize the problem, we include a volume constraint. Recasting the non-differentiable maxmin formulation into a so-called bound formulation using constraints, the optimization problem being solved can be stated as

 

Fig. 4. Optimized device and its optical response for the ${w_{{\rm metal}}} = 4\;\unicode{x00B5}{\rm m}$ base case.

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Fig. 5. Optimized device and its optical response for the ${w_{{\rm metal}}} = 2\;\unicode{x00B5}{\rm m}$ base case.

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4. OPTIMIZATION RESULTS

As our first example, we seek to design a device under similar conditions to those for the reference device from [19], considering the cases of ${w_{{\rm metal}}} = 4\;\unicode{x00B5}{\rm m}$ and ${w_{{\rm metal}}} = 2\;\unicode{x00B5}{\rm m}$. Accordingly, we optimize the design of the metal heater to maximize the FOM in Eq. (9) for the effective indexes ${n_{{\rm eff}}}$ where we found the resonances in Fig. 3. Therefore, we will target different effective indexes depending on the metal width. Note that the starting guess for our inverse design problem is not the reference device but a “gray design,” which is given by ${\xi _0} = 0.5$ and represents a non-physical mixture of the two different materials (TiN, and ${{\rm SiO}_2}$). In the inverse design process, this non-physical mixture of materials is gradually pushed to a final black and white design consisting solely of regions with TiN and ${{\rm SiO}_2}$. We also choose $f = 0.3$, which is a volume much larger than the reference. For the rest of the parameters used in the optimization, see Appendix C.

With these optimization settings, we find the designs (framed in purple) in Figs. 4 and 5. Although not prescribed a priori, a feature reminiscent of the PT-symmetry breaking reference device emerges on the left side of the waveguide, which serves to confine the mode to the right side of the waveguide, while minimizing optical losses. This is shown in Fig. 6, where we plot the thermal and optical fields for the optimized design from Fig. 4. The thermal field generates a homogeneous temperature field along the waveguide, more homogeneous than the temperature field generated by the reference design in Fig. 2. This results in the optical mode not getting pulled into the vicinity of the metal and staying in the dielectric, low-loss, region. Consequently, the optimized design leads to a device with lower optical losses. Similar heat and optical profiles arise from the design in Fig. 5, which also leads to decreased optical losses.

 

Fig. 6. Thermal and optical field simulation for the optimized design in Fig. 4 for ${P_{{\rm in}}} = 20\;{\rm mW} $.

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We can also study the phase modulation of the optimized phase shifters, which is related to the shift in effective refractive index and the length of the waveguide $L$, defined in Table 2. Using the expression in Eq. (1), we find that for both devices, we obtain a phase-shift per length of $\Delta \Phi /L = 2 \cdot {10^{- 3}}\pi /\unicode{x00B5}{\rm m}$, which for the length of our device yields a total phase-shift of $\Delta \Phi = 0.4\pi$. This is enough to create interference effects with the parameters chosen in Table 2, and it could be further tailored by modifying the TOPS length $L$ and the input power ${P_{{\rm in}}}$. Accordingly, through this framework, it is possible to tailor the phase of light while having a device with a compact cross-section, and a length along the waveguide comparable to state-of-the-art devices [10–12,14,47].

Evaluating the optimized designs in terms of loss per cm, using Eq. (8) we find that the optimized design based on the reference with ${w_{{\rm metal}}} = 4\;\unicode{x00B5}{\rm m}$ achieves a ${\rm loss} = 29\;{\rm dB/cm}$ for both the unheated and heated cases, which is a remarkable reduction in optical losses of around 18 and 15 dB/cm, respectively. For the design optimized based on the reference with ${w_{{\rm metal}}} = 2\;\unicode{x00B5}{\rm m}$, we obtain ${\rm loss} = 12\;{\rm dB/cm}$ for the unheated and heated cases, resulting in an improvement of 4 and 6 dB/cm, respectively. Alternatively, we can compare the reduction in loss $\alpha$ using the relative error metric introduced in Section 2. Here, we find a reduction in losses $\Delta _{\alpha ,{\rm unheated}}^{{\rm ref}{\rm .} \to {\rm opt}{\rm .}} = - 31\%$ in the unheated condition and $\Delta _{\alpha ,{\rm heated}}^{{\rm ref}{\rm .} \to {\rm opt}{\rm .}} = - 33\%$ in the heated condition for the ${w_{{\rm metal}}} = 4\;\unicode{x00B5}{\rm m}$ baseline. For the ${w_{{\rm metal}}} = 2\;\unicode{x00B5}{\rm m}$ baseline, we find a reduction $\Delta _{\alpha ,{\rm unheated}}^{{\rm ref}{\rm .} \to {\rm opt}{\rm .}} = - 21\%$ for the unheated condition and $\Delta _{\alpha ,{\rm heated}}^{{\rm ref}{\rm .} \to {\rm opt}{\rm .}} = - 24\%$ for the heated condition.

A. Effects of Constraining the Maximum Metal Volume

Comparing the cases in the previous section to the PT-symmetry breaking reference in Fig. 3, much more metal is used. It is therefore relevant to systematically explore the effect of constraining the allowed fraction of metal in the device design. This is done by using the volume constraint introduced in the optimization problem [see Eq. (14)]:

With a starting guess of ${\xi _0} = 0.3$, we find the designs and performances in Fig. 7. Interestingly, we observe that the design optimized under the tightest volume constraint of $f = 0.0125$ (shown in black) converges to a similar design to the reference with the important alteration to the device geometry of an extra symmetric heater under the waveguide, with an associated performance increase of $\Delta _\Phi ^{{\rm ref}{\rm .} \to {\rm opt}{\rm .}} = 10\%$. This mirrored metal allows the waveguide to confine the mode more tightly while heating the waveguide more homogeneously in the vertical direction. If we relax the volume constraint to $f = 0.05$ (shown in yellow) we observe a difference in the topology of the design, with a new feature emerging above and further away from the waveguide. This new feature acts like a heater, which helps provide an additional homogeneous refractive index shift throughout the waveguide, resulting in further enhanced performance of $\Delta _\Phi ^{{\rm ref}{\rm .} \to {\rm opt}{\rm .}} = 25\%$. For the rest of the designs, we find similar heater features that encapsulate the top of the waveguide without disturbing the optical mode, providing further homogeneous heating and enhanced performance. We find the best performance to saturate at $f = 0.1$ or $f = 0.2$, with even a slight deterioration for the higher values likely due to the optimizer becoming stuck in local optima for larger volumes.

 

Fig. 7. Optical performance of designs optimized under different volume constraints compared to the reference baseline with ${w_{{\rm metal}}} = 4\;\unicode{x00B5}{\rm m}$. Under each design, we note the response time estimated through the transient analysis.

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Fig. 8. Optical performance and optimization cross-validation of optimized devices for different input powers. The solid line shows the envelope of the performance for different input powers.

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The introduction of the volume constraint also allows us to address another important property of TOPS: the response time $\tau$, which is inversely proportional to the modulation speed. The response time is given by the expression $\tau = H/(G \cdot A)$, where $H$ is the heat capacity of the metal, $G$ is the thermal conductance between the waveguide and the heater, and $A$ is the waveguide cross-section area [10]. Since the overall device heat capacity decreases with decreasing volume, for a fixed thermal conductance, smaller devices should lead to increased modulation speed (decreased response time). This is also true for the metal fraction alone; thus for several applications, it will be beneficial to design devices for smaller metal volume fractions to achieve a faster response even at the expense of some optical performance degradation. We show this in Fig. 7, where we calculate the response time by running a time-dependent analysis until the temperature variation in the system is smaller than ${10^{- 4}}\;{\rm K}$, considering it the steady-state temperature ${T_s}$. Then, we estimate the time constant by finding the time for the temperature ${T_s}(1 - 1/e)$. For the reference design with ${w_{{\rm metal}}} = 4\;\unicode{x00B5}{\rm m}$ in Fig. 4, we find ${\tau _{{\rm ref}{\rm .}}} \approx 5.3\;\unicode{x00B5} {\rm s}$. For the optimized designs in Fig. 7, the results show how the time constant decreases with decreasing volume, as expected. It is interesting to note that the designs for $f = 0.3$ (shown in blue) and $f = 0.4$ (shown in green) have a similar time constant; despite having a different metal volume, the device with $f = 0.4$ has a larger metal encapsulation, which results in a higher thermal conductance. Remarkably, the design with the tightest volume constraint of $f = 0.0125$ (shown in black) is slightly faster than the reference while also having enhanced optical performance. For more information on the time-dependent analysis, see Appendix C.

B. Impact of Power on Optical Performance

As stated in the introduction, it is beneficial to optimize TOPS to achieve lower power consumption. The optimized designs in the previous section indicate that it is beneficial to have the metal in direct contact with the waveguide, as this results in better heat conduction from the heater to the waveguide. As a consequence, one can intuitively expect better power consumption in such devices, compared to where the metal is far away from the waveguide. Leaving aside intuition for a moment, it might be possible to achieve even better performance by optimizing for lower power consumption directly. To study this, we use one of the volume fractions performing the best in Fig. 7, namely, we choose $f = 0.2$, which should allow the optimizer to find high-performance devices while allowing for more design flexibility than the design optimized for $f = 0.1$. With this setup, in Fig. 8 we optimize a set of TOPS designs for four different input powers and check their optical performances. To solve the optimization problem, we reuse the same optimization formulation in Eq. (14), and we vary the thermal excitation ${\textbf{F}_T}$ by modifying the input power ${P_{{\rm in}}}$. For more on the definition of the thermal excitation, see Appendix A.

First, we cross-validate that each optimized design achieves the highest performance for their respective targeted input power. Reading Fig. 8 from right to left, we observe a logarithmic decrease in the performance of optimized devices with a linear decrease in power. This happens because as we go lower in input power, we need to decrease the metal volume to generate a similar temperature profile. The optimizer then removes metal from the top, but mostly the bottom of the waveguide, which, while present, supports homogeneous heating of the waveguide, and as a result, eliminating it leads to more optical losses. These results tell us that one could fix a maximum acceptable optical loss and tailor the TOPS design for minimum power usage. Vice versa, one could also, given a power constraint, select the device that gives the lowest optical loss.

 

Fig. 9. Topology optimized design for a simulation domain with a larger cladding volume.

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C. Optimizing Larger Domains: Cladded Device

Many devices implemented in practice are embedded in a larger cladding volume to create more heat insulation. Accordingly, and in contrast to the previous designs that had a more compact cross-section, we consider a device embedded in a larger cladding volume, similar to the one considered in [10]. Thus, we consider a simulation domain with ${w_{{\Omega _2}}} = 40\;\unicode{x00B5}{\rm m}$ and ${h_{{\Omega _2}}} = 20\;\unicode{x00B5}{\rm m}$. We solve the heat problem in this larger domain, while still solving the optical problem in the smaller domain employed throughout the previous sections. This means that our design domain effectively remains the same. To achieve a similar effective index shift as in the reference in Section 2, in this case, we use a power of ${P_{{\rm in}}} = 5\;{\rm mW} $. This means that due to the nature of our simulation domain, we will be working with less compact but more power-efficient devices.

Using this new setup, we optimize our design based on the reference with ${w_{{\rm metal}}} = 4\;\unicode{x00B5}{\rm m}$ and find the optimized design in Fig. 9. Since the model domain in this problem is much larger than before, the thermal BCs are, in practical terms, far away from the optical problem. This means that the design features responsible for homogeneous heat control disappear since the cladding now acts as a heat regulator. On that account, the optimizer distributes the metal to heat the waveguide in its vicinity in an almost symmetric manner, while conserving the curved features around the optical mode. Since the metal heater on top of the waveguide is no longer needed for uniform heat distribution, the optimizer distributes more metal close to the waveguide to ensure that the device reaches a similar temperature, resulting in thicker metal features compared to the more compact case. Comparing the performance of the optimized design in Fig. 9 to the reference, using the relative error metric from Section 2, reveals a reduction in losses of around $\Delta _\alpha ^{{\rm ref}{\rm .} \to {\rm opt}{\rm .}} = - 32\%$ both for the unheated and the heated cases.

In this new setup, we also check the time constant for the reference and optimized designs for the larger cladding volume and we find ${\tau _{{\rm ref}{\rm .}}} \approx 80\;\unicode{x00B5} {\rm s}$ and ${\tau _{{\rm opt}{\rm .}}} \approx 84\;\unicode{x00B5} {\rm s}$, respectively. The increase in the time constant of the optimized design with respect to the reference is a direct result of increased metal volume in the design. As we can observe, the aforementioned time constants are much larger than the response times calculated in the volume study in Fig. 7. This is due to the larger setup, where the heat needs more time to propagate in the simulation domain and reach the steady state. Therefore, based on this study and as also pointed out in [10], when designing TOPSs, there is a clear trade-off between building power-efficient and fast-response devices.

5. TOWARDS MANUFACTURABLE DEVICES

While the results in the previous sections have demonstrated that it is possible to achieve significant improvements for TOPS designs by employing TO, the optimized designs studied thus far would be hard to manufacture using current state-of-the-art fabrication processes. Therefore, in this section, we introduce two ways of moving towards optimized design blueprints that account for some fabrication limitations: adding length-scale control constraints on top of the optimization procedure and designing devices based on simple polygonal approximations of the optimized designs. Using this framework, one can introduce other fabrication constraints that further bridge the gap between optimized theoretical design geometries and manufacturable geometries; for instance, one can consider linking the design variables in the vertical direction to obtain designs fabricable in a single lithographic step, as detailed in [44], or impose an out-of-plane sidewall slant angle constraint for metallic nanostructures as done in [48].

A. Length-Scale Control

One of the main difficulties for the practical implementation of the previously presented optimized designs is the appearance of small features that might be out of reach of current fabrication capabilities. Therefore, similar to other nanophotonic TO problems [29,49], we implement two geometric length-scale control constraints, detailed in [50]:

 

Fig. 10. Optimized designs for the reference with ${w_{{\rm metal}}} = 4\;\unicode{x00B5}{\rm m}$ with tunable length-scale constraint.

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Adding these constraints to the optimization problem, and using a similar setup to Subsection 4.B, we achieve the results in Fig. 10 for two different filter radii, where we have introduced the length-scale constraint as part of the continuation scheme (see Appendix C). As previously explained, by using different filter radii, we can control the length-scale of the designs. Using the same filter radius as in the previous sections, we obtain the design in Fig. 10(a). If we compare the losses with the relative error metric introduced in Section 2, we find only $\Delta _\alpha ^{{\rm TO} \to {\rm LS}} = 1\%$ higher losses, where LS denotes the length-scale optimized design. This is a promising achievement since it means that we can achieve very similar optical performance while introducing a fixed length-scale in the designs. The new design eliminates several of the small features, and the measured minimum length-scale for the device is around ${l_{{\rm meas}}} \approx 0.11\;\unicode{x00B5}{\rm m}$. In Fig. 10(b) we illustrate the design obtained by choosing a larger filter radius ${r_f} = 0.5\;\unicode{x00B5}{\rm m}$. Since we impose a larger feature size in this device, the heaters merge with each other, and we see an increase in losses of $\Delta _\alpha ^{{\rm TO} \to {\rm LS}} = 3\%$. The length-scale constraint ensures that only larger features remain, and we measure a minimum length-scale for the device around ${l_{{\rm meas}}} \approx 0.245\;\unicode{x00B5}{\rm m}$.

B. TO-Inspired Design

Another avenue for exploiting the knowledge gained about the optimal geometry for the TOPS from the topology optimized devices is to simplify the designs by means of extruding simplified designs described by simple geometrical objects. In this section, we present a TO-inspired (TOI) design that can be achieved by composing a set of rectangles. Inspired by the volume constraint study in Subsection 4.A, we seek to achieve better optical performance by curving the metal close to the optical mode. We approximate the curved surface by layering metal rectangles on top of each other on the top left of the waveguide. Moreover, to achieve lower losses through uniform heating of the device, we add a rectangular heater further away from the waveguide, aligned to the center of the propagating mode, with the placement and width inspired by the design with $f = 0.1$ (shown in red) in Fig. 7. The resulting design is shown in Fig. 11.

 

Fig. 11. TO-inspired (TOI) design generated through the use of simple rectangles.

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If we once more use the relative error in Section 2 to compare this design to the ${w_{{\rm metal}}} = 4\;\unicode{x00B5}{\rm m}$ reference, we observe that this design decreases losses around $\Delta _{\alpha ,{\rm unheated}}^{{\rm ref}{\rm .} \to {\rm opt}{\rm .}} = - 20\%$ and $\Delta _{\alpha ,{\rm heated}}^{{\rm ref}{\rm .} \to {\rm opt}{\rm .}} = - 24\%$ for the unheated and heated devices, respectively. As would be expected from a non-optimal design, the TOI design performs slightly worse than the TO-optimized design for $f = 0.1$; specifically, we observe an increase in losses of $\Delta _{\alpha ,{\rm unheated}}^{{\rm TO} \to {\rm TOI}} = 19\%$ and $\Delta _{\alpha ,{\rm heated}}^{{\rm TO} \to {\rm TOI}} = 20\%$ for the unheated and heated cases with respect to the optimized design for $f = 0.1$ (shown in red) in Fig. 7. Nevertheless, quite remarkably, the TOI design still enhances the performance while accounting for manufacturability issues. In particular, if we calculate the losses using Eq. (4), we find ${\rm loss} = 35\;{\rm dB/cm}$ for the unheated and heated cases, meaning that this design is able to reduce the losses by 9 and 12 dB/cm with respect to the reference, i.e., more than an order of magnitude.

6. CONCLUSION

In this work, we have derived a topology optimization framework to solve coupled multiphysics thermal and optical inverse design problems. This is achieved by optimizing the design of metallic heaters in TOPSs to minimize optical losses, a first in the nano-optics community. The adjoint sensitivity analysis provided in Appendix A allows for the treatment of a range of weakly coupled multiphysics problems by introducing minor, problem-specific modifications. Thus, the framework derived, demonstrated, and used in this work may be translated to study a range of other coupled multiphysics problems in photonics, hereby paving the way to apply topology optimization to a wider class of inverse design problems in nano-optics.

We present and analyze the designs of cross-sectionally compact TOPSs, where the heater (metal) is in direct contact with the waveguide, demonstrating that losses may be decreased by up to 33% compared to a state-of-the-art reference device, which is equivalent to reducing losses by 18 dB/cm. Notably, this is obtained without providing an initial highly performing design geometry (only a uniform initial guess is used), and our design procedure finds a highly efficient design that partially resembles the conventional reference design. We show that the framework is not limited to minimizing optical losses but can also be used to design devices under volume constraints and study the optimized design performance against input power. This allows tailoring of the designs based on material and energy limitations, and specifically, in the case of the volume constraint, it allows taking into account the response time of the system.

Additionally, and in order to facilitate the experimental realization of the devices, we have shown that it is possible to include fabrication constraints by means of a length-scale constraint or TOI designs. Interestingly, the TOI design leads to a considerable reduction in optical losses, specifically 12 dB/cm, which is performance comparable to the TO-optimized designs while having a significantly simpler geometry, describable solely using rectangles.

In addition to the TOPS design problems studied in this work, one could imagine several other avenues open for further investigation. One example is to directly account for the time response of the system in the optimization problem formulation as well as directly optimize for minimum power consumption. Another simple extension is to apply the proposed design formulation to other systems consisting of other materials and waveguide/device geometries. One could also apply a three-material interpolation [51] to design the waveguide and heater geometries simultaneously, which could lead to devices with even higher performance. Lastly, note that our work does not consider the insertion losses associated with coupling from an input waveguide to the TOPS along the waveguide direction. The framework presented here could be expanded to account for this issue by including the design of optical couplers between the homogeneous waveguide and the TOPS device, resulting in low-loss mode conversion between the waveguide and the TOPS, as, e.g., demonstrated for pure mode conversion in prior work [37].

APPENDIX A: ADJOINT SENSITIVITY ANALYSIS

In this appendix we calculate the sensitivities of the figure of merit ($\Phi$) that was introduced in Eq. (9) with respect to design field variations. For that we need two vectors of Lagrange multipliers to account for the complex field, similar to the approach presented in [44]. In this case, we have to add an additional vector accounting for the heat problem, which contributes to the multi-physical coupling. Based on the optimization problem defined in Eq. (14), it is possible then to rewrite the expression as

(A1)$$\begin{split}\tilde \Phi &= \Phi + \lambda _{{\rm opt}}^ \top ({\textbf{S}_{{\rm opt}}}(\varepsilon (T))\textbf{E} - {\textbf{F}_{{\rm opt}}})\\&\quad + \lambda _{{\rm opt}}^\dagger (\textbf{S}_{{\rm opt}}^*(\varepsilon (T)){\textbf{E}^*} - \textbf{F}_{{\rm opt}}^*) + \lambda _T^ \top ({\textbf{S}_T}\textbf{T} - {\textbf{F}_T}) ,\end{split}$$

where opt denotes the optical problem, $T$ denotes the heat problem, $\textbf{S}$ is the system matrix, $\textbf{E}$ is the electric field vector, $\textbf{T}$ is the temperature field vector, $\lambda$ is a Lagrange multiplier vector, and $\textbf{F}$ is the forcing term. Using the previous definitions, we can take the derivatives, using the chain rule, to calculate the sensitivities:

(A2)$$\begin{split}\frac{{\partial \tilde \Phi}}{{\partial {{{\bar {\tilde \xi}}}_k}}} &= \frac{{\partial \Phi}}{{\partial {\textbf{E}_\Re}}}\frac{{\partial {\textbf{E}_\Re}}}{{\partial \varepsilon}}\left({\frac{{\partial \varepsilon}}{{\partial {{{\bar {\tilde \xi}}}_k}}} + \frac{{\partial \varepsilon}}{{\partial T}}\frac{{\partial \textbf{T}}}{{\partial {{{\bar {\tilde \xi}}}_k}}}} \right) + \frac{{\partial \Phi}}{{\partial {\textbf{E}_\Im}}}\frac{{\partial {\textbf{E}_\Im}}}{{\partial \varepsilon}}\left({\frac{{\partial \varepsilon}}{{\partial {{{\bar {\tilde \xi}}}_k}}} + \frac{{\partial \varepsilon}}{{\partial T}}\frac{{\partial \textbf{T}}}{{\partial {{{\bar {\tilde \xi}}}_k}}}} \right) \\ &\quad+\lambda _{{\rm opt}}^ \top \left[{\frac{{\partial {\textbf{S}_{{\rm opt}}}}}{{\partial \varepsilon}}\left({\frac{{\partial \varepsilon}}{{\partial {{{\bar {\tilde \xi}}}_k}}} + \frac{{\partial \varepsilon}}{{\partial T}}\frac{{\partial \textbf{T}}}{{\partial {{{\bar {\tilde \xi}}}_k}}}} \right)\textbf{E} + {\textbf{S}_{{\rm opt}}}\left({\frac{{\partial {\textbf{E}_\Re}}}{{\partial \varepsilon}} + {\rm i}\frac{{\partial {\textbf{E}_\Im}}}{{\partial \varepsilon}}} \right)\left({\frac{{\partial \varepsilon}}{{\partial {{{\bar {\tilde \xi}}}_k}}} + \frac{{\partial \varepsilon}}{{\partial T}}\frac{{\partial \textbf{T}}}{{\partial {{{\bar {\tilde \xi}}}_k}}}} \right)} \right] \\&\quad +\lambda _{{\rm opt}}^\dagger \left[{\frac{{\partial \textbf{S}_{{\rm opt}}^*}}{{\partial \varepsilon}}\left({\frac{{\partial \varepsilon}}{{\partial {{{\bar {\tilde \xi}}}_k}}} + \frac{{\partial \varepsilon}}{{\partial T}}\frac{{\partial \textbf{T}}}{{\partial {{{\bar {\tilde \xi}}}_k}}}} \right){\textbf{E}^*} + \textbf{S}_{{\rm opt}}^*\left({\frac{{\partial {\textbf{E}_\Re}}}{{\partial \varepsilon}} - {\rm i}\frac{{\partial {\textbf{E}_\Im}}}{{\partial \varepsilon}}} \right)\left({\frac{{\partial \varepsilon}}{{\partial {{{\bar {\tilde \xi}}}_k}}} + \frac{{\partial \varepsilon}}{{\partial T}}\frac{{\partial \textbf{T}}}{{\partial {{{\bar {\tilde \xi}}}_k}}}} \right)} \right] + \lambda _T^ \top \left[{\frac{{\partial {\textbf{S}_T}}}{{\partial {{{\bar {\tilde \xi}}}_k}}}\textbf{T} + {\textbf{S}_T}\frac{{\partial \textbf{T}}}{{\partial {{{\bar {\tilde \xi}}}_k}}} - \frac{{\partial {\textbf{F}_T}}}{{\partial {{{\bar {\tilde \xi}}}_k}}}} \right] ,\end{split}$$

where $k$ and $i$ denote element numbers. One can then rearrange the terms as

(A3)$$\begin{split}\frac{{\partial \tilde \Phi}}{{\partial {{{\bar {\tilde \xi}}}_k}}} &= \frac{{\partial {\textbf{E}_\Re}}}{{\partial \varepsilon}}\frac{{\partial \varepsilon}}{{\partial {{{\bar {\tilde \xi}}}_k}}}\left({\frac{{\partial \Phi}}{{\partial {\textbf{E}_\Re}}} + \lambda _{{\rm opt}}^ \top {\textbf{S}_{{\rm opt}}} + \lambda _{{\rm opt}}^\dagger \textbf{S}_{{\rm opt}}^*} \right) + \frac{{\partial {\textbf{E}_\Im}}}{{\partial \varepsilon}}\frac{{\partial \varepsilon}}{{\partial {{{\bar {\tilde \xi}}}_k}}}\left({\frac{{\partial \Phi}}{{\partial {\textbf{E}_\Im}}} + {\rm i}\lambda _{{\rm opt}}^ \top {\textbf{S}_{{\rm opt}}} - {\rm i}\lambda _{{\rm opt}}^\dagger \textbf{S}_{{\rm opt}}^*} \right) \\&\quad +\frac{{\partial \textbf{T}}}{{\partial {{{\bar {\tilde \xi}}}_k}}}\left[\frac{{\partial {\textbf{E}_\Re}}}{{\partial \varepsilon}}\frac{{\partial \varepsilon}}{{\partial T}}\left({\frac{{\partial \Phi}}{{\partial {\textbf{E}_\Re}}} + \lambda _{{\rm opt}}^ \top {\textbf{S}_{{\rm opt}}} + \lambda _{{\rm opt}}^\dagger \textbf{S}_{{\rm opt}}^*} \right) + \frac{{\partial {\textbf{E}_\Im}}}{{\partial \varepsilon}}\frac{{\partial \varepsilon}}{{\partial T}}\left({\frac{{\partial \Phi}}{{\partial {\textbf{E}_\Im}}} + {\rm i}\lambda _{{\rm opt}}^ \top {\textbf{S}_{{\rm opt}}} - {\rm i}\lambda _{{\rm opt}}^\dagger \textbf{S}_{{\rm opt}}^*} \right) \right.\\ &\quad+2\Re \left.\left\{{\lambda _{{\rm opt}}^ \top \frac{{\partial {\textbf{S}_{{\rm opt}}}}}{{\partial \varepsilon}}\frac{{\partial \varepsilon}}{{\partial T}}\textbf{E}} \right\} + \lambda _T^ \top {\textbf{S}_T}\right] + 2\Re \left\{{\lambda _{{\rm opt}}^ \top \frac{{\partial {\textbf{S}_{{\rm opt}}}}}{{\partial \varepsilon}}\frac{{\partial \varepsilon}}{{\partial {{{\bar {\tilde \xi}}}_k}}}\textbf{E}} \right\} + \lambda _T^ \top \left({\frac{{\partial {\textbf{S}_T}}}{{\partial {{{\bar {\tilde \xi}}}_k}}}\textbf{T} - \frac{{\partial {\textbf{F}_T}}}{{\partial {{{\bar {\tilde \xi}}}_k}}}} \right) .\end{split}$$

To eliminate the field derivatives $\partial \textbf{T}/\partial {{\bar {\tilde \xi}} _k}$, $\partial {\textbf{E}_\Re}/\partial \varepsilon$ and $\partial {\textbf{E}_\Im}/\partial \varepsilon$, we solve the adjoint equations

(A4)$$\textbf{S}_{{\rm opt}}^ \top {\lambda _{{\rm opt}}} = - \frac{1}{2}{\left({\frac{{\partial \Phi}}{{\partial {\textbf{E}_\Re}}} - {\rm i}\frac{{\partial \Phi}}{{\partial {\textbf{E}_\Im}}}} \right)^ \top} $$

and

(A5)$$\textbf{S}_T^ \top {\lambda _T} = - 2\Re \left\{{{\textbf{E}^ \top}\frac{{\partial \textbf{S}_{{\rm opt}}^ \top}}{{\partial T}}{\lambda _{{\rm opt}}}\;} \right\} .$$

Note that when we solve the forward thermo-optical problem in Eq. (14), we solve for the thermal field $\textbf{T}$ and insert it to solve the electric field $\textbf{E}$. For the adjoint problem, we observe that we first need to solve the optical problem in Eq. (A4) and use its solution ${\lambda _{{\rm opt}}}$ to solve the thermal adjoint problem. Using the Lagrange multipliers, the sensitivity calculation can be simplified to

(A6)$$\frac{{\partial \tilde \Phi}}{{\partial {{{\bar {\tilde \xi}}}_k}}} = 2\Re \left\{{\lambda _{{\rm opt}}^ \top \frac{{\partial {\textbf{S}_{{\rm opt}}}}}{{\partial \varepsilon}}\frac{{\partial \varepsilon}}{{\partial {{{\bar {\tilde \xi}}}_k}}}\textbf{E}} \right\} + \lambda _T^ \top \left({\frac{{\partial {\textbf{S}_T}}}{{\partial {{{\bar {\tilde \xi}}}_k}}}\textbf{T} - \frac{{\partial {\textbf{F}_T}}}{{\partial {{{\bar {\tilde \xi}}}_k}}}} \right) .$$

It is possible to calculate these quantities by having access to the element matrices, and the material interpolation. However, the excitation term in the conductive heat equation ${\textbf{F}_T}$ remains to be defined. In our case, since we want a design-variable-dependent heat source defined by a heat rate given by the input power, we have taken

(A7)$${\textbf{F}_{T,i}}({\bar {\tilde \xi}}) = {P_{{\rm in}}}\frac{{{\bar {\tilde \xi}} _i^p}}{{L{a_e}\sum\limits_j {{{\bar {\tilde \xi}}}_j}}}\iint_{{\Omega ^e}} N_i^e {\rm d}\Omega ,$$

where $L$ is the out-of-plane length of the element, $p$ is a penalization factor set to $p = 3$ to avoid grayscale in our designs, ${a_e}$ is the area of the element, and $N_i^e$ is the interpolation function of the ${i}$th element.

Lastly, to recover the final sensitivities, we need to take into account that we have filtered and thresholded design variables obtained using Eqs. (10) and (11). Accordingly, we need to apply the chain rule

(A8)$$\frac{{d\Phi}}{{d{\xi _h}}} = \sum\limits_{k \in {{\cal B}_{e,h}}} \frac{{\partial {{\tilde \xi}_k}}}{{\partial {\xi _h}}}\frac{{\partial {{{\bar {\tilde \xi}}}_k}}}{{\partial {{\tilde \xi}_k}}}\frac{{d\Phi}}{{d{{{\bar {\tilde \xi}}}_k}}},$$

where ${{\cal B}_{e,h}}$ denotes the $h$th set of finite elements whose center point is within the filter radius ${r_f}$ of the $h$th element.

APPENDIX B: GEOMETRIC LENGTH-SCALE SENSITIVITIES

In this appendix, we show the sensitivity derivations of the geometric length-scale constraint from Eqs. (16) and (17). To simplify our equations, we consider only the case where ${\tilde \xi _i} - {\eta _e} \lt 0$ or ${\eta _d} - {\tilde \xi _i} \lt 0$, for the solid and void constraints, respectively; otherwise, the result will give zero. The sensitivities for the solid constraint are

(B1)$$\frac{{\partial {g^s}}}{{\partial {\xi _j}}} = \frac{1}{n}\left({\sum\limits_{i \in \mathbb{N}} \frac{{\partial I_i^s}}{{\partial {\xi _j}}}{{({{\tilde \xi}_i} - {\eta _e})}^2} + 2I_i^s({{\tilde \xi}_i} - {\eta _e})\frac{{\partial {{\tilde \xi}_i}}}{{\partial {\xi _j}}}{\delta _{\textit{ij}}}} \right) .$$

Since the design variables should not be correlated to each other, we introduce the Kronecker delta ${\delta _{\textit{ij}}} = \partial {\xi _i}/\partial {\xi _j}$. Now we take the sensitivity of the indicator function:

(B2)$$\frac{{\partial I_i^s}}{{\partial {\xi _j}}} = {{\rm e}^{- c \cdot |\nabla {{\tilde \xi}_i}{|^2}}}\;\left({\frac{{\partial {{{\bar {\tilde \xi}}}_i}}}{{\partial {{\tilde \xi}_j}}}\frac{{\partial {{\tilde \xi}_j}}}{{\partial {\xi _j}}}{\delta _{\textit{ij}}} - 2c{{{\bar {\tilde \xi}}}_i}|\nabla {{\tilde \xi}_i}|\frac{{\partial |\nabla {{\tilde \xi}_i}|}}{{\partial {\xi _j}}}{\delta _{\textit{ij}}}} \right) .$$

In a similar fashion, the void constraint sensitivities are

(B3)$$\frac{{\partial {g^v}}}{{\partial {\xi _j}}} = \frac{1}{n}\left({\sum\limits_{i \in \mathbb{N}} \frac{{\partial I_i^v}}{{\partial {\xi _j}}}{{({\eta _d} - {{\tilde \xi}_i})}^2} - 2I_i^v({{\tilde \xi}_i} - {\eta _e})\frac{{\partial {{\tilde \xi}_i}}}{{\partial {\xi _j}}}{\delta _{\textit{ij}}}} \right) ,$$

and the sensitivities for the void indicator function are

(B4)$$\frac{{\partial I_i^v}}{{\partial {\xi _j}}} = {{\rm e}^{- c \cdot |\nabla {{\tilde \xi}_i}{|^2}}}\!\left({- \frac{{\partial {{{\bar {\tilde \xi}}}_i}}}{{\partial {{\tilde \xi}_j}}}\frac{{\partial {{\tilde \xi}_j}}}{{\partial {\xi _j}}}{\delta _{\textit{ij}}} - 2c(1 - {{{\bar {\tilde \xi}}}_i})|\nabla {{\tilde \xi}_i}|\frac{{\partial |\nabla {{\tilde \xi}_i}|}}{{\partial {\xi _j}}}{\delta _{\textit{ij}}}} \right)\! .$$

The sensitivities are thus analytically found without the need for solving an adjoint problem.

APPENDIX C: SIMULATION PARAMETERS

In this appendix, we provide the parameters used to reproduce the different results shown in the previous sections. Namely, we provide the material, system, simulation, and optimization parameters.

1. Material Parameters

In our study, we consider linear, static, homogeneous, isotropic, non-dispersive, and non-magnetic materials. These materials can be characterized with the parameters used in the material interpolations in Eqs. (12) and (13), which are the real part of the refractive index $n$, the extinction coefficient $\kappa$, and the thermal conductivity $k$. All the data are gathered in Table 1, with some based on the values in [10].

Table 1. Material Parameters Used in the Material Interpolations, Given at $\lambda = 1.55\;\unicode{x00B5}{\rm m}$ and $T = 300 \;{\rm K}$

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Table 2. System Parameters Used in the Different Studies

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2. System Parameters

The system parameters represented in Fig. 2(a) and used in the different studies include the width of the waveguide ${w_{\text{wg}}}$, the height of the waveguide ${h_{\text{wg}}}$, the width of the region ${\Omega _2}\;{w_{{\Omega _2}}}$, the height of the region ${\Omega _2}\;{h_{{\Omega _2}}}$, the height of the metal ${h_{{\rm metal}}}$, the out-of-plane length of the waveguide $L$, the convective coefficient $h$ acting on the boundary ${\Gamma _{{\rm top}}}$ in Fig. 2(a), and the constant temperature ${T_{{\rm BC}}}$ acting on the boundary ${\Gamma _{{\rm bot}}}$. These are all gathered in Table 2. Note that for the larger domain used in Subsection 4.C, we have used ${w_{{\Omega _2}}} = 40\;\unicode{x00B5}{\rm m}$ and ${h_{{\Omega _2}}} = 20\;\unicode{x00B5}{\rm m}$ to solve the thermal problem.

Table 3. Simulation Parameters for the Reference Study in Section 2^a

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Table 4. Optimization Parameters for Studies in Different Sections

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3. Simulation Parameters

In Table 3 we show the parameters of the frequency domain and eigenvalue simulations in Section 2 where ${{\rm nEl}_x}$ are the number of elements in the X direction, ${{\rm nEl}_y}$ are the number of elements in the Y direction, Exc. point denotes the excitation point of the system in the spatial coordinates, Exc. value denotes the value of the excitation, and $n_{{\rm eff}{\rm .}}^{{\rm guess}}$ and $n_{{\rm eff}{\rm .,T}}^{{\rm guess}}$ are the guesses for the unheated and heated effective indexes, respectively. Note that in Table 3 there are some empty sites because the frequency domain solver requires an excitation, which is not the case for the eigensolver that finds the closest eigenvalue to an initial guess.

The setup for the $4\;\unicode{x00B5}{\rm m}$ reference in Table 3 is applied to solve the finite-element problem for the rest of the systems in this study, with the exception of the cladded device in Subsection 4.C, where the power is ${P_{{\rm in}}} = 5\;{\rm mW} $.

Finally, for the volume constraint study in Subsection 4.A and for the larger thermal domain in Subsection 4.C, we use COMSOL [52] to run a time-dependent study to determine the time constant of each design. For the volume constraint study in Fig. 7, we use the same parameters as in the frequency domain solution, with the addition of outputting the temperature each $\Delta t = 0.05\;\unicode{x00B5} {\rm s}$ and of running the simulation for a total simulation time of ${T_s} = 200\;\unicode{x00B5} {\rm s}$ to reach steady state. Similarly, for the larger thermal domain study, we output the temperature each $\Delta t = 1\;\unicode{x00B5} {\rm s}$ and run for a total simulation time of ${T_s} = 3\;{\rm ms} $. Thus, we account for a discretization error of ${\pm}0.05\;\unicode{x00B5} {\rm s}$ in the volume constraint study and of ${\pm}1\;\unicode{x00B5} {\rm s}$ in the larger domain study.

4. Optimization Parameters

The optimization parameters for the different substudies have been split up into their respective sections and subsections. In Table 4 we show the optimization parameters for the different studies, where ${n_{{\rm eff}{\rm .}}}$ refers to the selected effective index for the unheated device, ${n_{{\rm eff}{\rm .,T}}}$ refers to the selected effective index for the heated device, ${n_{{\rm it}}}$ refers to the total number of iterations, and ${\xi _0}$ corresponds to the initial value of the design variables in the homogeneous initial guess. Since in our optimization procedure we are using a continuation scheme, we increase the threshold sharpness $\beta$, with initial value ${\beta _0}$, and the scaling factor $\alpha$ in Eq. (12) for every ${n_{{\rm it,cont}}}$ iterations. Namely, we increase $\beta$ by a factor of 1.5, while for $\alpha$ we start with $\alpha = 0$, increase to $\alpha = 1.5$ on iteration ${n_{{\rm it,cont}}}$, and then further increase by a factor of 1.5 every ${n_{{\rm it,cont}}}$ iterations. In addition, for the length-scale study in Subsection 5.A, we include the geometric length-scale constraints in Eqs. (16) and (17) in the continuation scheme. We do so by introducing them on iteration $2{n_{{\rm it,cont}}}$ with $\epsilon ={10^{- 4}}$, decreasing the error to $\epsilon = {10^{- 6}}$ on iteration 5 ${n_{{\rm it,cont}}}$, and then decreasing one order of magnitude every ${n_{{\rm it,cont}}}$ iterations.

APPENDIX D: MODEL VALIDATION

In Section 2 we solve the reference problem with the frequency domain and eigenvalue methods. We note that these methods can calculate the effective refractive index ${n_{{\rm eff}}}$ and the losses $\alpha$ of the different modes. The eigenvalue solution gives a more accurate solution to the problem, but since the TO problem is tackled using a frequency domain solver, it is necessary to quantify the error between both approaches. To validate the accuracy of the frequency domain solution, using Table 5 we calculate the difference between the effective index and the losses for the ${w_{{\rm metal}}} = 4\;\unicode{x00B5}{\rm m}$ unheated reference case. We do this by calculating the relative error $\Delta$ with the expression introduced in Section 2:

(D1)$$\Delta _x^{A \to B}[\%] = 100 \cdot ({x_B} - {x_A})/{x_A}$$

for the quantity $x$ with cases $A$ and $B$. With this metric, we find a relative error $|\Delta _{{{\rm n}_{{\rm eff}}}}^{{\rm freq}{\rm .} \to {\rm eig}{\rm .}}| \simeq 3 \cdot {10^{- 6}} \%$ in the effective index and a relative error of $|\Delta _\alpha ^{{\rm freq}{\rm .} \to {\rm eig}{\rm .}}| \simeq 1 \%$ in the losses of the system, where eig. denotes the eigenvalue solution and freq. denotes the frequency domain solution. The error is higher in the case of the losses because it is more difficult to capture the width of a peak in a multimodal frequency response, where multiple peaks might be overlapping with each other. All in all, the obtained errors are low enough to capture the essential physics of the problem, and we observe discrepancies of the same order of magnitude for the heated case, the ${w_{{\rm metal}}} = 2\;\unicode{x00B5}{\rm m}$ reference case, and the posterior analysis and optimization.

Table 5. Comparison of Fitted Frequency Domain Solution and Eigenvalue Solution for the ${w_{{\rm metal}}} = 4\;\unicode{x00B5}{\rm m}$ Reference Case^a

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Funding

Danmarks Grundforskningsfond (DNRF147).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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