1. Introduction

The adjoint method is an effective and powerful tool for optimization problems in high dimensional parameter spaces [1,2]. By calculating the gradients of one or more Figure-of-Merit (FoM) functions in an efficient manner, the adjoint method makes possible the exploration of large parameter spaces that are otherwise intractable. Despite its effectiveness, the adjoint method has limitations. First, response surfaces of the FoM functions with respect to the design parameters are in general not convex, and gradient-based minimum-searching algorithms can be trapped at local minima and result in suboptimal designs. This issue can be partly alleviated by expanding the coverage of the gradient descendent algorithm, with techniques such as basin hopping or perturbation of initial conditions [3] at the cost of time and computational resources. However, it is still difficult to reach the global optima or to even validate the claim that a design is at the global optimum. Second, to migrate to a new process that has different silicon thickness, etch depth, or minimum trench width, the adjoint optimization has to be rerun for the new process, which may take hours or even days for complex problems such as grating couplers, barring convergence issues. One potential solution is to extract physics-based, process-independent knowledge from the adjoint optimized design, which can be used to reduce the parameters under consideration for the forward optimization problem and to expedite the migration of the grating design between different processes with a minimum of effort. As will be exemplified by the wideband grating coupler in the following of this paper, such physics-based, process-independent knowledge can be the grating architecture that uses two stagger-tuned beams by dividing the grating into two or more regions. As technology changes, the specific design parameters will be lost, but the architecture can be preserved and transferred to the new technology. Forward optimization after dimensionality reduction can be finished within similar or shorter amount of time than adjoint optimization without the convergence issues, at the cost of potential performance penalty as compared to the adjoint optimization. It is worth noting that with forward optimization, one still has to rerun simulations to identify parameter windows for the new process. Third, process assumptions made in the electromagnetic simulations differ from actual fabrication processes and the differences result in deviations of the optical performances from the expected value. Design-of-Experiments (DOEs) that consist of variants of the adjoint optimized design have to be taped out and tested on multiple dies and wafers in Design Verification Tests (DVT) to account for the deviation. The total number of devices that can be accommodated within one reticle at commercial CMOS foundries, and that can be tested on an automated fiber optics probestation within a reasonable amount of time, are limited to several thousands. For the problem of grating couplers, geometry optimization consists of a parameter set of about 50 width values, and dimensionality of the parameter space has to be reduced to be efficiently explored by DOEs. In this paper, we tackle the challenges with a different design paradigm. Instead of relying on the adjoint method to determine the geometry of the grating coupler, we first strive to understand the rationale behind the adjoint optimized designs, and then re-parameterize the designs in manners that are more viable for validation by tapeouts and testing of DOEs.

We demonstrated this design paradigm with an example of practical importance, that is, a grating coupler for coupling to 8 degree angle-polished single-mode fibers with sufficiently large 1dB bandwidth to cover the entire Coarse Wavelength-Division-Multiplexing (CWDM) O-band, which has center wavelengths ranging from 1270 nm to 1330 nm. Grating couplers have intrinsically limited bandwidths as an attribute of the wavelength-sensitive phase matching conditions. Well-designed grating couplers on SOI platform in general have a 1-dB bandwidth of 20∼30 nm [4,5]. Larger bandwidths can be achieved at the cost of lower coupling efficiency and/or higher design and fabrication complexity, but it is difficult for grating couplers to have sufficient 1-dB bandwidth to cover the entire CWDM O-band [6,7]. Instead, edge couplers are used as optical interfaces of CWDM systems for their large bandwidths, at the cost of high complexity in process integration, low throughput in optical sorting and packaging, and low density of optical ports [8–11]. Conversely, grating couplers enable easy process integration, high throughput optical sorting and packaging, and high density of optical ports, and it is therefore highly desirable to develop wideband grating couplers as the optical interfaces for CWDM systems.

2. Adjoint method optimization

Consider grating couplers on an SOI platform, which has a 270 nm-thick silicon layer on top of a 2 um-thick buried oxide layer. A set of 140 nm-deep trenches are etched into the top silicon layer to form a grating coupler. Since it is not immediately clear which architecture is required for the gratings to achieve sufficient 1dB bandwidth to cover the entire CWDM O-band, a bottom-up approach is to allow all the trench and pitch widths to vary freely while the GC coupling efficiency is being evaluated. Here the pitch width is defined as the spacing from one trench’s starting edge to the next trench’s starting edge. A grating coupler on SOI that couples to a single-mode fiber consists of about 25 periods, and therefore the optimization problem needs to sweep 50 free parameters, namely 25 trench widths and 25 pitch widths, to explore the entire parameter space for best coupling efficiency. Optimization within such a large parameter space is infeasible, if not impossible at all, without the help of the adjoint method.

We use the Python-based open-source adjoint method toolkit EMopt [12], which has a built-in 2D FDFD electromagnetic solver. The FDFD solver needs to perform one forward simulation for every individual wavelength in the Figure-of-Merit (FoM), and the total simulation time grows approximately linearly with the number of wavelengths in the FoM. We intend to keep the FoM simple as we will only use the adjoint optimized design as a starting point to obtain physics-based parameterization that allows us to transfer the design among different processes with a minimum of effort. In this study, the FoM of the optimization problem is simply defined as the mean of the coupling efficiencies in linear scale at 1270 nm and 1330 nm respectively. We did not include all 4 band centers of CWDM O-band at 1270/1290/1310/1330 nm in the FoM, which would take twice as long for forward simulations. Note that adjoint optimization tools relying on FDTD solvers with broadband pulse sources do not have this scaling bottleneck, as the fields at different wavelengths can be evaluated by a single time-domain simulation [13]. Without prior knowledge of the architecture of an ultra-wide-band grating coupler, one naïve choice of the initial condition for the optimization would be a grating coupler that is optimized for maximum coupling efficiency at 1300 nm, which sits at the center of the CWDM O-band. In this study, the trench width of the initial condition design starts at 60 nm and ramps up linearly at a rate of 10 nm; the pitch width starts at 464 nm and ramps up linearly at a rate of 2 nm; the initial condition design has a peak coupling efficiency of 0.9 dB at 1300 nm. After 50 iterations of minimum-searching using the Conjugate Gradient method, the optimized grating coupler achieves a FoM of about 0.5. We also tested the BFGS method for the minimization, which resulted in a similar design with slightly higher FoM. Again the choice of the minimization algorithm is not critical, and we do not intend to use the adjoint optimized grating as is. Instead we will learn from the adjoint optimized design and parameterize it accordingly. Figure 1 shows the optimization status dashboard of EMopt, which consist of the electric field intensity at 1270 nm, the refractive index profile of the grating, and the FoM plotted versus the iteration number.

 

Fig. 1. Screenshot of the EMopt simulation status dashboard after 50 iterations. Top left: electric field at 1270 nm overlaid on the grating structure. Bottom left: refractive index profile of the grating. Right: FoM plotted versus the iteration number.

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Since the 2D FDFD solver only computes the coupling efficiency at two wavelengths of 1270 nm and 1330 nm, the geometry of the optimized grating design is exported and validated in a commercial 2D FDTD solver (Lumerical FDTDSolutions) for the full coupling efficiency spectrum, which is shown in the blue curve (ratio = 1) in Fig. 2. It can be seen that the adjoint method design does not immediately satisfy the requirements to cover the entire CWDM band. Instead, peak loss of 1.5 dB is achieved with a 1-dB bandwidth of about 36 nm, which spans from 1260 nm to 1296 nm. Also the coupling efficiency spectrum is skewed with much lower efficiency at 1330 nm than that at 1270 nm, despite the same weight in calculating the FoM. This asymmetry in the coupling efficiency spectrum is primarily attributed to the construction of the FoM, which only seeks to minimize the mean of the two efficiencies in linear scale without constraining the imbalance between them. One way to equalize the coupling efficiencies at the two wavelengths is to assign different weights to the two wavelengths in the calculation of the FoM. In this case, more weight should be assigned to the 1330 nm efficiency than that of the 1270 nm to generate a more leveled coupling efficiency spectrum. Figure 2 shows the overlaid coupling efficiency spectra of 6 adjoint method optimized designs, which are obtained from the same initial conditions, but the 1330 nm efficiency is weighted more than that of the 1270 nm by a ratio of 1.0, 1.02, 1.03, 1.036, 1.038, and 1.04 respectively. It can be seen that as the ratio of weight increases, the 1270 nm efficiency is penalized while the 1330 nm efficiency is improved.

 

Fig. 2. Coupling efficiency spectra of adjoint optimized designs with 1330 nm weight of 1, 1.02, 1.03, 1.036, 1.038, 1.04 respectively with reference to 1270 nm.

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It is worth noting that in this work, we choose the widely adopted 1 dB bandwidth as a FoM, such that the interband loss penalty (i.e., the maximum difference between the coupling efficiencies at the 4 CWDM center wavelengths) does not exceed 1 dB. The 1 dB bandwidth is increased at the cost of lower efficiencies within the CWDM band. For example as the weight ratio increases from 1.0 to 1.038, the best efficiency (around 1270 nm for ratio = 1.0 and 1330 nm for ratio = 1.038) is penalized by about 1 dB while the efficiencies at the 4 CWDM center wavelengths become more balanced. The choice of performance parameters for a grating coupler depends largely on the CWDM system’s architecture and requirements, and in this work we choose to maximize the peak efficiency and 1 dB bandwidth simultaneously, at the cost of lower peak efficiency in the CWDM band. Note that this is by no mean the only choice of performance parameters for this type of grating couplers, and others may find it more favorable to trade off the interband loss penalty for better efficiencies.

Note that none of the 6 designs shown in Fig. 2 achieved sufficient 1-dB bandwidth to cover the entire CWDM O-band, despite fine tuning of the weight ratio to the third digit after the point. Also note that the spectral imbalance does not change monotonically with the weight ratio, which implies that some or all of the designs are local minima. In principle, eventually we could achieve the bandwidth goal by further tweaking the ratio of the weights or adding more wavelength points to the calculation of the FoM, at the cost of simulation time and resources. Instead, we stop with the most promising design of the adjoint method, which has the weight ratio = 1.038, and take a different approach forward by trying to understand the rationale behind the design.

First we examine the far field patterns of the adjoint optimized design. Figure 3 shows the electric field intensity of the far field of the adjoint optimized design at 1271 nm (blue) and 1323 nm (red) respectively, where the coupling efficiency spectrum peaks. The tilt angle of the fiber mode, which is 8 degree in this study, is also marked as the green line in Fig. 3. It can be seen that at each wavelength, the far field pattern consists of two peaks, which are stagger-tuned with a spacing of approximately 5 degree. At 1271 nm, the two peaks are at 8.4 degree and 13.5 degree respectively, the former of which matches the 8 degree fiber mode. As wavelength redshifts to 1323 nm, both peaks shift toward smaller beam angles of 2.0 degree and 8.1 degree respectively, the latter of which matches the 8 degree fiber mode. It is also evident from the far field patterns in Fig. 3 that the 8 degree peak at 1323 nm is higher in intensity than the 8 degree peak at 1271 nm, which is consistent with the asymmetry of the efficiency spectrum. It is worth noting that grating couplers with this architecture are expected to have coupling losses of about 3 dB as the upward scattering is approximately equally distributed between the two stagger-tuned beams, the angle difference between which is so large that the off-axis beam does not contribute to the coupling efficiency significantly.

 

Fig. 3. Far field E intensity of the optimized grating at 1271 nm (blue) and 1323 nm (red).

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Next, we examine the near-field phase matching of the adjoint optimized grating. The phase match condition of grating couplers requires that the wavevector of the grating waveguide minus the perturbation wavevector of the scattering elements is equal to the projection of the exit beam’s wavevector on the chip surface. Conversely, given trench and pitch widths of each period of the grating, the exit beam angle can be calculated at each individual period for all periods across the grating, which help dissect the operation of the grating coupler. The relation between the exit beam angle and the grating geometry within each period is shown in Eq. (1), where θ is the exit beam angle, t the trench width, p the pitch width of the unit cell, n_etched and n_unetched the slab effective indices of the etched and unetched regions respectively, n_clad the refractive index of the cladding, and λ the operation wavelength. It should be noted that the calculated beam angle for each individual period should not be interpreted as that of a standalone trench, where pitch cannot be defined. Also note that for simplicity, the grating waveguide’s effective index is assumed to be the weighted average index of the etched and unetched widths. Numerical validation shows that this method overestimates the grating waveguide’s effective index by approximately 3%. Accordingly, the exit beam angle calculated by Eq. (1) overestimates the actual beam angle by about 2.5 degree for 8 degree nominal beam angle.

 

Fig. 4. (a) Duty cycle versus period index of the adjoint optimized grating. (b) Exit beam angle versus period index at 1271 nm. (c) Exit beam angle versus period index at 1323 nm.

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Evolution of the grating geometry as the optimization progresses also provides valuable insights on the adjoint optimized grating design. Figures 5(a)–5(c) show the coupling efficiency spectra, trench width profile, and pitch width profile respectively, at 6 steps of the optimization procedure, where iteration 1 is the initial condition, and iteration 51 is the final step. From iteration 1 to iteration 11, the efficiency spectrum is redshifted by about 25 nm as all the pitch widths increase. Then at iterations 21∼51, the coupling efficiency at 1270 nm is boosted by decreasing the pitch widths of the last 6 periods, which emit 8deg beam at shorter wavelength. Figure 5(b) shows that trench widths in the first 7 periods are apodized with a parabolic profile to ramp up the scattering intensity adiabatically for better overlap with the Gaussian mode of optical fibers. As the duty cycle reaches ∼50% after 7 periods, the trench width stops ramping up and remains constant for 4 periods at 50% duty cycle to generate maximum scattering intensity, and then regresses to the initial conditions toward the end of the grating where the impact of scattering intensity diminishes. The geometry and the efficiency spectrum converge after about 30 iterations. Note that we did not implement any fabrication constraints in the adjoint optimization, so the first few trenches of the optimized design are narrower than 60 nm despite the minimum trench width of 60 nm in the initial condition. We do not intend to introduce constraints on minimum trench width in the adjoint optimization, as that can be simply set as a design parameter in the parameterization scheme.

 

Fig. 5. (a) Coupling efficiency spectra at various iterations. (b) Trench width versus period index at various iterations. (c) Pitch width versus period index at various iterations.

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Lastly, we would like to understand the abrupt changes in the pitch width profile of the adjoint optimized designs, namely the dip at period #6, and the larger pitch widths of the 4 periods of #13∼16. For the 4 periods of #13∼16, we decrease the pitch widths as a whole by 10 nm, 20 nm, and 30 nm respectively while preserving all other geometric parameters, and examine the change in coupling efficiency spectra, which are shown in Fig. 6. It can be seen that as the 4 pitch widths decrease, both the 1271 nm and the 1323 nm peaks blueshift by approximately 0.75 nm/nm, which is only one third of the expected blueshift if all pitch widths were to be decreased together. Also the 1323 nm coupling efficiency is significantly improved while the 1271 nm coupling efficiency is penalized, which further implies that the 1323 nm peak is contributed collectively by the first and the second regions.

 

Fig. 6. Pitch width versus period index (a), and coupling efficiency spectra (b), with the middle 4 periods decreased by 10 nm, 20 nm, and 30 nm, respectively.

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For the dip in pitch width at period #6, we increase its width by 10 nm, 20 nm, and 30 nm respectively while preserving all other geometric parameters. The coupling efficiency spectra of the 4 designs are shown in Fig. 7. As the pitch width increases, the 1323 nm peak is higher while the 1271 nm peak lower, so pitch #6 works with the last few periods in the tail region as a cohort to emit 8 degree beam at 1271 nm. This non-intuitive design implies that the adjoint optimized grating may be at a local minimum instead of a global minimum. Nonetheless, we do not seek to escape local minima, and a fortiori, to approach global minima. Instead, we will simply disregard trivial features like the dip in pitch #6 and focus on significant features like the 3-section stagger-tuning structures.

 

Fig. 7. Pitch width versus period index (a), and coupling efficiency spectra (b), with the 6th pitch width increased by 10 nm, 20 nm, and 30 nm, respectively.

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3. Parameterization and validation

The adjoint-optimized design in the previous section does not immediately meet the bandwidth requirements to cover the entire CWDM O-band. To achieve the bandwidth goal, one can invest more time and computational resources to fine tune the weights in the FoM or construct a new FoM to eliminate the imbalance, but the gradient-based minimum-searching algorithm cannot guarantee convergence to global minima given the non-convexity of the response surface. In this study, we take a different route by parameterizing the adjoint optimized design, even though it has yet to achieve the bandwidth goal. As explained earlier, it is difficult to explore a parameter space with 50 degrees of freedom. Instead, a more viable approach is to learn from the adjoint optimized design, disregard trivial features that do not contribute significantly to the coupling efficiency, and capture important features of the grating with a reasonable number of parameters based on physics. The parameterization scheme should seek to explore as much relevant parameter space as possible, without exceeding the capacity of tapeout and testing. For simple grating test structures and full reticle at commercial CMOS foundries, the total number of devices that can be taped out and tested on an automated fiber probestation within a reasonable amount of time is in general on the order of several thousands.

In this work, DOEs are employed to study parameter-parameter and parameter-process relationships under the constraint of limited tapeout and testing resources. Sweeps of individual parameters may help understand simple parameter-process relationships, but are not sufficient for more sophisticated designs such as the stagger-tuned grating couplers in this paper. For example, to match the amplitude of the grating mode to the fiber’s Gaussian mode, the trench width ramps up at the beginning of the grating such that the scattering intensity is lower at the beginning and higher at the beam center. To match the grating’s phase front to the planar phase front of the fiber beam, the pitch width of the grating also needs to ramp up at a rate that is related to the trench width’s ramp rate, and the phase difference between the two stagger-tuned beams needs to be adjusted accordingly. It can be seen that as design complexity increases, the dependence of the scattered mode on the design variables and process becomes less intuitive, such that DOEs are required to study the relationships between the parameters and the process. In physics-based parameterization schemes, there is a higher likelihood that more parameters will be strongly correlated with fabrication effects. For example, the trench ramp rate controls the beam size along the grating, while the pitch ramp rate controls the slope of the phase front, both of which have stronger correlation with fabrication effects than individual trench or pitch widths. From the perspective of photonics library development, it saves cost and time-to-market to cover as much relevant parameter space with DOEs on each tapeout, since increased coverage of relevant parameter space with DOEs can significantly improve the odds of having a better design on silicon with as few tapeouts as possible, which are both expensive and slow.

One scheme to parameterize the adjoint optimized grating is to divide the trench and pitch width profiles into 3 regions, as shown schematically in Fig. 8. In the head region, the trench widths ramp up parabolically while the pitch widths ramp up linearly; in the waist region, both the trench and pitch widths remain constant; in the tail region, both the trench and pitch widths ramp up linearly. The offsets in pitch widths between the three regions are vital parameters to stagger tune the two peaks to achieve wide band operation. The offset in trench width between the second and third regions is an artifact of the selection of initial condition and plays a less important role. Note that we need not to constrain the minimal trench width in the initial adjoint method optimization design phase, since we can simply set the minimum trench width in the parameterized scheme according to the lithography technology node. In this paper we set the first and minimum trench width to be 60 nm, which can be achieved at 65 nm CMOS nodes with immersion lithography.

 

Fig. 8. Schematic drawing of trench and pitch widths versus period index of the proposed parameterized grating design, which is overlaid on top of the adjoint optimized design.

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The adjoint optimized design provides a near-optimal starting point for parameter sweeping. Preliminary parameter sweeps show that the trench width parameters do not significantly affect the grating performance at the vicinity of the adjoint optimized geometry with the exception of the parabolic ramp rate of trench width in the first region, so we focus on the pitch width parameters. The parameterization scheme is then reduced to 6 independent parameters of the pitch width: first pitch width, ramp rates of the first and the third regions, the two offsets between the three regions, and the numbers of periods for the first and second regions. For validation by tapeout and testing, smaller DOEs of individual trench parameters crossed with pitch parameters can be constructed to confirm the weaker correlation between parameters. A full factorial DOE that varies each parameter by 3 values consists of 729 designs, which are reasonable for validation by simulations, tapeout at commercial CMOS foundries, and testing on an automated fiber-optics probe station. Simulation results of the 6-parameter full factorial DOE, with trench parameters also individually varied in 10 different configurations, are shown in the multivariate scatterplot of Fig. 9, where the peak coupling efficiency in linear scale, 1dB bandwidth, and center wavelength of the 1dB band are shown. There are 7290 designs in total in the scatter plots. Wideband, low loss designs are designated by the upper convex hull (highlighted by the green curve) in the efficiency-bandwidth scatterplot, where 1-dB bandwidth of around 70 nm can be achieved with a 3 dB peak loss.

 

Fig. 9. Multivariate scatterplot of coupling efficiency, 1-dB bandwidth, and center wavelength of the 1-dB band for the 7290 simulated designs. Note that all the 6 blobs are projections of the same set of data points onto the 6 different pairs of axis combinations. The upper convex hull of the efficiency-bandwidth scatterplot is highlighted by a green curve.

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As more than one design can achieve the 1dB bandwidth goal, we have the freedom to further select those designs that meet the bandwidth requirements with process variation. Based on our experience at commercial CMOS foundries with 65 nm technology node, the etch depth may vary by approximately 10 nm within a 300 mm SOI wafer [14], and we will evaluate the performance sensitivity to etch depth variation of ± 5 nm. Coupling efficiency spectra of one selected design at nominal etch as well as etch depth ± 5 nm, are shown in Fig. 10(a). The efficiency spectrum blue shifts with deeper etch by approximately 1.3 nm/nm, with 1270 nm efficiency improved and 1330nm efficiency penalized. Coupling efficiency spectra of the selected design at nominal trench widths as well as trench width ± 5 nm for all periods, are shown in Fig. 10(b). The coupling efficiency spectrum does not shift significantly with trench width bias, but the 1330 nm peak is modulated more than the 1270 nm peak. For both etch depth and trench width CD corners considered in this study, the 1dB band is sufficient to cover the entire CWDM O-band from 1270 nm to 1330 nm, although it becomes marginal at the etch corners.

 

Fig. 10. (a) Coupling efficiency spectra with nominal and ± 5 nm etch depth. (b) Coupling efficiency spectra with nominal and ± 5 nm trench width CD.

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Figure 11 shows the spectra of the total power (blue) reflected into the monitor behind the waveguide source and the TE0 component of the reflection (red). The reflection spectra clearly show two stagger-tuned dips, which correspond to the efficiency peaks of the out-coupling to fibers. Within the CWDM O-band from 1270 nm to 1330 nm, the worst case reflection of TE0 component is about -13.5 dB, which is on par with that of a normal grating coupler. Reflection of grating couplers designed for tilted fibers can be greatly reduced without redesigning the grating couplers, by simply tilting the grating trenches at an angle [15].

 

Fig. 11. Reflection spectra of the total power (blue) into the monitor behind the waveguide source, and the TE0 component (red).

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4. Discussion

Note that the main aim of the parameterization procedure is to facilitate DOE construction, not to improve the fabrication robustness. Since the parameterization simply tries to mimic the adjoint-optimized design by capturing physics based features, we expect to see similar process sensitivities between the adjoint-optimized design and its parameterized version. Simulations show that the efficiency spectrum of the adjoint-optimized design blueshifts with deeper etch by 1.2 nm/nm, which is similar to the process sensitivity of its parameterized version. Also the adjoint optimized grating and its parameterized version have similar wavelength and loss sensitivities to the etch depth as those of a conventional grating, for example, the linearly ramped initial design that is centered at 1300 nm. The adjoint optimized designs do not have superior tolerance to process variation since it is not formulated in the construction of the FoM at all. If the etch depth variation on one wafer exceeds the 10 nm range, or when wafer-to-wafer and lot-to-lot variations are superimposed on the wafer-level etch depth variation, the 1-dB bandwidth will start to fall short on the extreme corners and cause yield loss. It is possible to formulate the adjoint optimization to take into consideration the etch depth corners, but the simulation time will scale up accordingly and the resulted designs will likely comprise more subtle features to capture and parameterize.

Intuitively, a grating coupler with two stagger-tuned beams can be constructed with only two sections at the minimum, one for each of the beam angles. The three-section architecture of the adjoint optimized design could be due to two reasons: either an additional third section is required for phase transition or beam shaping between the two main sections, or the adjoint optimization may have converged to a local minimum. Our simulations using different initial conditions, and simulations on a different substrate all show similar three-section architectures, which confirmed that the additional third section indeed serves a purpose of phase transition or beam shaping.

The selected design’s coupling efficiency varies by about 0.6 dB among the 4 center wavelengths of CWDM O-band, namely 1270 nm, 1290 nm, 1310 nm and 1330 nm. Potentially the inter-band penalty can be improved by adding more wavelength points to the FoM at which the coupling efficiency is evaluated. Again this will add to the simulation time and complexity of the parameterization, and the better inter-band uniformity is likely achieved at the cost of lower efficiency. One can also look at this problem from another angle. The wideband operation of the adjoint optimized grating is achieved by stagger tuning two beams with approximately 5 degree spacing in between, and in principle better inter-band uniformity or even larger bandwidth can be achieved by stagger tuning more beams with smaller spacing. However, more stagger-tuned beams also mean that the finite number of grating periods need to be divided into more groups of shorter segments, each of which have worse mode matching to a fiber’s Gaussian mode at 8 degree.

In this paper, the proposed design paradigm is to do preliminary adjoint optimization, understand the physics behind the adjoint optimized design, and parameterize the design with DOEs that are suited for tapeouts at commercial CMOS foundries and wafer-level testing on fiber probe stations. This design paradigm has been exemplified by grating couplers, but one should be able to apply it to certain other photonic devices without fundamental difficulties, especially for complex devices with many parameters. Our expectation is that for problems where there is already a significant body of physical intuition and deep insights, such as grating couplers [16] and directional couplers, the adjoint method can be an organic part of the design flow and an aid to inspire designers, instead of a simple black box procedure.