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Peer Review File

Reviewers' comments

Reviewer #1 (Remarks to the Author)

Dual

waveband silicon grating coupler was optimized by a fast integral method. The authors

considered the foundry fabrication constra

ints and got nice measurement results. Though the

functionality has been demonstrated before, the optimization method is of interests to the

community. The manuscript is well organized and the write

up is good. I only have a few

comments listed below

1. C

ompared with brute

force parameter searching via FEM or FDTD, the optimization process is

speed

up by adjoint method and the integral equation method. How is the adjoint method

implemented in the grating design

2. The authors mentioned computation time co

st is two or more orders of magnitude less

compared with conventional approaches. Are there any comparison results

3. From Fig. 3, the directionality of the coupler is not high which may limit the maximum

efficiency. Is there any back reflection problem f

or vertical coupling? I suggest to discuss the loss

mechanisms of the coupler

4. The coupling is very sensitive to the illumination position. It would be useful to investigate the

sensitivity to the coupling angle error as well

5. How much improvement ca

n be achieved if the minimum feature size constraints are relaxed

Reviewer #2 (Remarks to the Author)

The authors present simulation and experimental results on design of a grating coupler

demultiplexer for 1300 nm and 1550 nm optical communication ban

ds. They exploit the well

known method of inverse design for which they have used the fast integral method for solving

Maxwell’s equations. The results are sound, timely, and of importance to the community. The

paper is well written and easy to follow and

understand. In general, the claims are supported by

evidence, however further work needs to be done. Therefore, before I can recommend this for

publication, I have listed my comments/suggestions for improvement, and I would very much like

to see clarificat

ions from the authors on these points, both in their reply and by taking appropriate

measures in their revisions

Comments

1) In view of this work impact, it is important for such devices to also be capable of working as

multiplexers. Can the authors c

omment on the ability of their device for such purposes? Moreover,

will the back reflections in the reverse operation become an issue? Any numbers

2) Fabrication robustness needs to be supported, at the very least, by simulation data and

preferably by pr

oviding experimental statistics. The authors rather use the agreement between

theory and experiment (which itself needs further clarification in this work), and therefore it is

weak

3) Following the previous comment, in Fig 6, experimental results are co

nsiderably higher than

simulated, which while a bonus, this needs to be explained and clarified. Are these actually the

same devices in terms of features sizes? In other words, was the optimized design, re

visited after

fabrication to better find a match b

etween theory and experiment

4) Line 68, the authors say that “... lithography, and therefore required E

beam lithography for

fabrication.” What about technologies such as immersion deep

5) Line 138, the authors mention use of windowing functions for

truncation of infinite waveguide

domains. This needs further elaboration and clarification to the reader in terms of how it works, so

a proper judgment can be made from method/numerical perspective

6) The authors argue that their Maxwell solver is more e

fficient compared to FDTD and FEM. This

however needs to be supported and quantified by data and numbers, for this work (aside from

referencing previous work). For example, 2D simulations of grating couplers (a single simulation)

can be efficiently done in

20 seconds or less with Lumerical solver and a similar workstation as

used by the authors. Therefore, it is very important to make this statement more accurate and

more supported

7) Line 149, the authors need to either provide references on their implem

entation of graded

meshing, or explain it in more details in the supplementary information. Judging this as it is

written now, is very difficult

8) I am wondering why the material index is kept constant in this optimization

Reviewer #3 (Remarks to the

Author)

In the manuscript "Foundry

Fabricated Grating Coupler Demultiplexer Inverse

Designed via Fast

Integral Methods" the authors present the design and experimental characterization of a

wavelength demultiplexer based on a surface grating in the silic

insulator platform. The

device couples light from a standard SMF 28 fiber placed vertically on top of the chip and splits the

O and C optical communication bands into the two output waveguides pointing in opposite

directions. The design of the gratin

g was optimized exploiting the L

BFGS gradient

based

algorithm along with adjoint simulations for gradient estimation and an efficient integral equation

based Maxwell solver. Owing to a large critical dimension of 160 nm, the device has been realized

throu

gh a standard multi

project waver run at the A*STAR silicon photonics foundry with 193 nm

UV lithography and partial etch. The subsequent characterization shows a good agreement with

designed performance and according to the authors' claims some of the bes

t performance reported

so far in the literature for this type of devices

The paper is easy to follow even if some information should be added to the main text, as I detail

in my comments below. I believe that the most interesting and timely aspect of this work is the

report of an inverse designed device with high performance an

d also compatible with commercial

UV lithography, a critical aspect to allow for the widespread of these innovative design approaches

However, in my view, the authors fall short in providing convincing and solid arguments for the

novelty of their work. T

he use of the L

BFGS algorithm and adjoint method have become the de

facto standard for inverse design and are even provided in commercial software packages. The use

of a more efficient solver rather than FDTD or FEM is of absolute interest but already pre

sented in

ref 14, as largely commented also by the authors. The fact of being applied to a dual

band grating

instead of a single wavelength devices as in ref 14 can be regarded as quite incremental unless

this poses additional difficulties in the optimizat

ion process which are not on the other hand

discussed in the manuscript. The device concept is not new but the authors claim quite strong

improvements in performance compared to published results, in particular on insertion losses and

power unbalance. Thes

e results represent the core of the paper novelty but I believe their solidity

should be better discussed. Insertion loss performance heavily depends on the applied

normalizations, which becomes hence a critical point. In figure 6a, experimental results ar

e largely

better than those expected by simulation (purple and red curves) which is very surprising. Yet, this

aspect is not discussed. If this is indeed the case then one could assume that the optimizer did not

finished its job since a better design was i

ndeed found during fabrication! Normalizations for

directional couplers and PIN modulators are based on FDTD simulations that may not grasp all the

source losses in a fabricated device or loss differences between side by side devices. It is not

specified i

f the four directional couplers in the rest device are identical or not. It is not clear if the

responsivity of the photodiodes was measured on dedicated devices realized on the same chip or

just represents an average value for the foundry. I think clarify

ing these aspect is crucial to

convince the reader that the presented device has indeed best

class performance

Some other minor aspect should be considered. Regarding parametrization, while it is true that

adjoint simulations are often applied to topo

logy optimization which optimizes material distribution

in a volume, as described at the end of page 5, there are many examples in literature where the

same is used for parametrized devices without the use of any threshold or level set function,

exactly as

done here. This advantage is hence independent on the specific solver and is not unique

to the tool chosen by the authors

Moreover, it would be interesting if the authors could comment on the reflections generated by

their device since this is the real

difficulty in designing perfectly vertical couplers and the one that

normally forces the critical dimension to few tens on nanometers

Since the authors claim, for example at the end of the introduction, that they were able to find a

better device compare

d for example to ref 16, if think they should discuss the reason for this. Is it

for an optimization algorithm that can be better explore the design space (which is however hard

to claim since they are using a local optimizer)? Is it for a better initial g

uess? Is it because they

are penalizing other aspect, such as reflections

Lastly, I feel in some points the text should be strengthened

In the abstract, it is not clear to me what "high

order accuracy" of the solver means

In the introduction, page

3, I don't think the sentence "The majority of grating couplers are

designed to be efficient at coupling in a narrow band around a single wavelength of interest" being

true. Many devices have been reported working on a single band but on very large bandwid

ths, up

to several hundreds of nanometers

In the same page, the authors discuss manual design of grating couplers but cite also ref 16,

which is indeed on inverse design

Also stating that "it appears completely impossible to even attempt to use such

approaches in

the design of multi

wavelength grating splitters" is incorrect, as proved by refs 15 and 17

Insertion losses are reported as 3 dB and 4.95 dB for the two bands in the introduction, page 4,

and as 2.96 dB and 5 dB in table 1

Page 4, it i

s not clear to me the wording "engineering designs". There is also a typo in the same

line

It is not clear what is an "auto

generated" initial guess. Does it mean it has been chosen

randomly? Initial guess is key for local optimizers

At the beginning

of page 5, "sufficiently optimized design" is very vague. Does it mean the

algorithm has converged or it has simply been stopped at an arbitrary iteration

I feel that information on silicon thickness, number of periods in the grating and fiber mode

dia

meter should be included in the main text, not in the supplementary since they are quite

fundamental piece of data (e.g. choosing a different fiber would heavily affect the achievable

efficiency). The authors did not mention which polarization they are usi

ng. Reflections from the

silicon substrate were considered in the simulation/optimization

Page 8, how is it defined the adjoint problem? Power is lunched simultaneously in the two

waveguides

Page 9, the abbreviation "resp." is not clear to me

Figu

re 2, results are obtained with FDTD or their simulator

In commenting eq 2, larger splitting ratios means more power into the correct output only for the

left port, for the right port eq. 2 should be flipped, I think

On line 220, page 12, why the ste

p size if 50 nm in one direction and 40 times larger in the other

one

Line 222 and subsequent, including table 1, what is the isolation? Does it refer to eq 2

I could not find a call

out for figure 7, which does not seem to be commented

Table 1, P

igott 2014 should be ref 16 not 1

Replies to Reviewers

October 16, 2021

We thank all reviewers for their valuable comments, which have led to significant improvements of

our manuscript and major expansion of our Supplementary Information. In what follows we describe

the changes introduced, and we reply to each one of the reviewers’ queries.

All literature citations

refer to the bibliography included in the primary manuscript.

Replies to Queries by Reviewer 1

1. Compared with brute-force parameter searching via FEM or FDTD, the optimization process

is speed-up by adjoint method and the integral equation method. How is the adjoint method

implemented in the grating design?

We have expanded significantly on the integral equation approach that we developed and used to

design the grating splitter, as well as our implementation of the adjoint method in this context.

Due to space limittions, this has primarily been incorporated in the Supplementary Information

in sections 1.1, 1.2, and 1.3.

2. The authors mentioned computation time cost is two or more orders of magnitude less compared

with conventional approaches. Are there any comparison results?

We have performed a comparison against a state-of-the-art commercial FDTD solver for photonic

devices. In the right image of Figure 1 in the Supplementary Information, we plot the amount

of time required to achieve a desired error for both our integral equation formulation and the

FDTD solver to produce the solution for the GCWD structure under consideration. As discussed

more in Section 1.2 of the Supplementary Information, we suggest that a simulation error of

better than 1% is necessary to not become a dominant source of performance degradation of the

fabricated device in view of approximately 10% fabrication error expected. In order to achieve

1% simulation accuracy, our integral equation method and the FDTD solver require 10 seconds

and 2000 seconds respectively, which is more than 2 orders of magnitude faster, as suggested in

the main body of the paper.

3. From Fig. 3, the directionality of the coupler is not high which may limit the maximum efficiency.

Is there any back reflection problem for vertical coupling? I suggest to discuss the loss mechanisms

of the coupler.

Thank you for the suggestion to analyze the loss mechanisms of the coupler. We have done

so and included a brief discussion in the main text reporting that 24

7% and 13

7% of the

power is reflected back upwards at 1.33

m and 1.55

m respectively, and that the remaining

power which does not couple into the output waveguides is transmitted downwards through

the substrate. Since the measured coupling efficiency into the desired outputs exceeds that of

previously reported work, we expect that the losses should be less than those of other reported

vertical-coupling grating splitters. We believe that the relatively low measured insertion losses

with respect to the desired coupling into the output waveguide makes this device suitable for a

number of wavelength-demultiplexing applications.

4. The coupling is very sensitive to the illumination position. It would be useful to investigate the

sensitivity to the coupling angle error as well.

We agree that this important and useful information. Figure. 7, which was already present in

the original submission, plots coupling efficiency variation with respect to both incident fiber

position and angle variation. We have added an extra line in the text to call attention to this

figure, and we have also performed additional 3D simulations where the angle of the incident

excitation is swept from -10 degrees to 10 degrees and plotted the changes in the insertion loss and

splitting ratio performance metrics at each of the two wavelengths. These results are presented in

Supplementary Section 3.2 alongside a relevant discussion. Briefly, the device performance only

moderately degrades due to angular variations of up to several degrees in the incident coupling

angle.

5. How much improvement can be achieved if the minimum feature size constraints are relaxed?

In order to properly answer this question, we used our inverse design framework to optimize a new

grating coupler wavelength demultiplexer with relaxed 50nm minimum feature size constraints.

In order to be able to keep the size of the grating the same as the GCWD presented here while

allowing for smaller features without requiring correspondingly much larger features to span the

whole length, we increased the number of widths and spacings from 39 to 159. As expected, the

resulting simulated performance of the new grating splitter with relaxed size constraints improved

appreciably. The coupling efficiency into the right and left output waveguides at 1.55

m and

1.3

m are 52.1% and 66.7% respectively, compared to 37.6% and 27.5% for the grating presented

with 160nm minimum feature size constraints. The splitting ratios of the 50nm relaxed constraint

coupler are 23.5dB and 21.6dB at 1.55

m and 1.3

m respectively, also showed an improvement,

albeit a lesser one, over the 160nm constrained design which had 19dB and 21dB isolation at

1.55

m and 1.3

m respectively. Although the performance of the feature size constrained splitter

is better, it may still be advantageous to design gratings with larger minimum feature sizes for

two reasons: 1. E-beam lithography would be required to fabricate structures with 50nm size

features, which is costly, time-consuming, and does not scale to mass production. 2. Process

variation during fabrication is expected to be more deleterious for gratings which rely on smaller

feature sizes. These results have been included in Section 4 of the Supplementary Information.

Replies to Queries by Reviewer 2

1. In view of this work impact, it is important for such devices to also be capable of working

as multiplexers. Can the authors comment on the ability of their device for such purposes?

Moreover, will the back reflections in the reverse operation become an issue? Any numbers?

We agree that a wavelength mutliplexing grating used as a transmit, rather than receive, coupler

also has many important applications. Although the grating that we report on in this work was

designed and optimized specifically as a receive coupler and a wavelength demultiplexer, since

the coupling efficiencies and splitting ratios were quite favorable, due to reciprocity, we might

expect that it could also work well as a multiplexing transmit coupler. In order to determine

whether this is indeed the case, we performed two extra simulations: one in which we excited

the left waveguide with the fundamental mode at 1.3

m, and the second in which we excited

the right waveguide with the fundamental mode at 1.55

m. We then measured the amounts of

power that couples through to the opposing waveguide, the power that is back-reflected, and

the power that is radiated upwards in the desired direction, and finally the amount of power

coupling downwards through the silicon substrate. We found that the power coupling through

to the opposing waveguide, as well as the power back-reflecting into the excitation waveguide,

were neligible in both simulations. Importantly, 63.4% (resp. 56.6%) of the power is radiated

upwards for the 1.3

m (resp. 1.55

m) wavelength excitation, resulting in a front-to-back ratio

of 69% (resp. 65%). This confirms that our device, despite not being designed to be operated as

a wavelength multiplexer, can indeed be used successfully for such a purpose. We included the

results of these simulations and a corresponding discussion in Section 5 of the Supplementary

Information.

2. Fabrication robustness needs to be supported, at the very least, by simulation data and preferably

by providing experimental statistics. The authors rather use the agreement between theory and

experiment (which itself needs further clarification in this work), and therefore it is weak.

We have verified robustness by performing Monte Carlo simulations with our integral equation

solver. We generated and simulated 855,893 independently distributed random samples with

10nm standard deviation from the nominal dimensions in the grating wall position parameters.

The results are plotted and discussed in Section 3.1 of the Supplementary Information. Briefly,

at 2

$

sigma variation (95% of the samples), the maximum deviations in coupling efficiencies

of the right and left outputs at 1.55

m and 1.3

m respectively are

±

3% and

±

7%, and

the maximum deviation in the splitting ratio is

±

8dB for both wavelengths. Based on these

results, the proposed GCWD design is expected to be robust to process variation, which is also

corroborated by our measurement results.

3. Following the previous comment, in Fig 6, experimental results are considerably higher than

simulated, which while a bonus, this needs to be explained and clarified. Are these actually the

same devices in terms of features sizes? In other words, was the optimized design, re-visited after

fabrication to better find a match between theory and experiment?

We fabricated the same design that we designed and simulated. We believe that any variations

in performance are due to process variation and fabrication defects. Although the measured

coupling efficiency was indeed better than the simulated results, the measured splitting ratio

was worse. As a result, the objective function (eq. 1) of the measured device, which considers

a weighted sum of the coupling efficiency as well as the splitting ratio, was actually worse than

that of the simulated device. We have added the following text in the revised manuscript to

clarify this:

“In fact, for both left and right transmission, Fig. 6 shows experimental power values higher

than the numerical values obtained for the simulated device; the departure is significantly more

noticeable for the power transmitted into the left fiber. But the experimental data does not

contradict the predictions, since the objective function consists not only of the power transmitted

into the desired waveguide, but it also includes a penalty for power transmitted into the undesired

waveguide. When both powers are taken into account the apparent contradiction is eliminated:

the value of the objective function for the combined left and right power, for both frequencies,

lies below the optimal predicted value.”

4. Line 68, the authors say that ”... lithography, and therefore required E-beam lithography for

fabrication.” What about technologies such as immersion deep-UV?

Immersion deep-UV lithography can indeed achieve smaller feature sizes; however, to the best of

our knowledge, the majority of silicon photonics foundry processes available today do not support

this technology. More specifically, the foundries which we have worked with in the past (including

A*STAR AMF and TowerJazz) have minimum achievable feature sizes of approximately 160nm

as reported in our manuscript. However, should such a technology node become accessible for

mass fabrication of silicon photonics devices, our proposed optimization approach can readily

be used to optimize gratings which fully take advantage of the reduced minimum feature sizes

available. In fact, in response to question #5 from Reviewer #1, we optimized a new grating

splitter design with 50nm minimum width features. Indeed, the resulting grating shows significant

improvements in coupling efficiency. Specifically, the coupling efficiency into the right and left

output waveguides at 1.55

m and 1.3

m are 52.1% and 66.7% respectively, compared to 37.6%

and 27.5% for the grating presented with 160nm minimum feature size constraints. These results

have been included in Section 4 of the Supplementary Information.

5. Line 138, the authors mention use of windowing functions for truncation of infinite waveguide

domains. This needs further elaboration and clarification to the reader in terms of how it works,

so a proper judgment can be made from method/numerical perspective.

The new Section 1 in the Supplementary Information, provides details on the integral equation

method used along with the windowed Green function (WGF) approach for truncating infinite

waveguides. Specifically, Section 1.1 describes the WGF technique, Section 1.2 describes the

adjoint optimization method and its specific efficient implementation in the context of optimizing

gratings via integral equation methods, and Section 1.3 compares our solver’s accuracy as well

as speed with a well-know, state-of-the-art commercial FDTD solver.

6. The authors argue that their Maxwell solver is more efficient compared to FDTD and FEM. This

however needs to be supported and quantified by data and numbers, for this work (aside from

referencing previous work). For example, 2D simulations of grating couplers (a single simulation)

can be efficiently done in 20 seconds or less with Lumerical solver and a similar workstation as

used by the authors. Therefore, it is very important to make this statement more accurate and

more supported.

The new Section 1.3 in the Supplementary Information presents a comparison between the accu-

racy and speed of our integral equation based simulation approach to a well-known, commercial

state-of-the-art FDTD solver. As noted by the reviewer, a 20 second calculation by an FDTD

solver yields a degree of accuracy. However, according to our convergence tests, this is sufficient

for achieving a solution with at best 10% error, which as argued in the discussion included in

Section 1.3, is not accurate enough for performing optimization under an experimental error level

of 10%. Assuming a 10% fabrication error, the simulation error should be bounded below 1% in

order to have a negligible effect on the overall departure from the design performance caused by

fabrication error. Based on our simulations, it takes the commercial FDTD solver 2000 seconds

to achieve a solution with no larger than 1% error, whereas our solver only requires 10 seconds

to achieve such an error.

7. Line 149, the authors need to either provide references on their implementation of graded mesh-

ing, or explain it in more details in the supplementary information. Judging this as it is written

now, is very difficult.

We included a reference which explains the graded meshing approach that we used for the corners.

Specifically, we added the following lines to the main manuscript text:

“Furthermore, to accurately resolve the field singularities present at the corners, a graded mesh,

which is induced via a reparametrization, is used to properly cluster points at such corners, as

described on pages 83-84 in reference [23]—so as to give rise to high-order convergence. Additional

details concerning our integral equation implementation are presented in Sections 1.1 and 1.2 in

the Supplementary Information. Supplementary Section 1.3 compares the performance of the

proposed solver against that of a commercial state-of-the-art FDTD solver in terms of accuracy

vs. computing time as well as mesh resolution, clearly demonstrating the advantages of the

proposed approach.”

8. I am wondering why the material index is kept constant in this optimization.

Since only the boundaries are changed there is no need to allow for refractive index variations.

The boundary integral method we use is applicable only to situations, like the one considered in

the paper, in which the material index is constant by pieces. Brief mention of this matter with a

reference to previous work is made at the beginning of the new Section 1.3 in the Supplementary

Information.

Replies to Queries by Reviewer 3

In the manuscript ”Foundry-Fabricated Grating Coupler Demultiplexer Inverse-Designed via Fast In-

tegral Methods” the authors present the design and experimental characterization of a wavelength

demultiplexer based on a surface grating in the silicon-on-insulator platform. The device couples light

from a standard SMF 28 fiber placed vertically on top of the chip and splits the O and C optical

communication bands into the two output waveguides pointing in opposite directions. The design of

the grating was optimized exploiting the L-BFGS gradient-based algorithm along with adjoint simula-

tions for gradient estimation and an efficient integral equation-based Maxwell solver. Owing to a large

critical dimension of 160 nm, the device has been realized through a standard multi-project waver

run at the A*STAR silicon photonics foundry with 193 nm UV lithography and partial etch. The

subsequent characterization shows a good agreement with designed performance and according to the

authors’ claims some of the best performance reported so far in the literature for this type of devices.

The paper is easy to follow even if some information should be added to the main text, as I detail

in my comments below. I believe that the most interesting and timely aspect of this work is the

report of an inverse designed device with high performance and also compatible with commercial UV

lithography, a critical aspect to allow for the widespread of these innovative design approaches.

Thank you for the very useful and constructive feedback. We have made changes in the manuscript

accordingly and significantly expanded the Supplementary Information to address this as well as the

remarks by the other reviewers. We have included our responses to each of the remarks individually

below.

1. However, in my view, the authors fall short in providing convincing and solid arguments for

the novelty of their work. The use of the L-BFGS algorithm and adjoint method have become

the de-facto standard for inverse design and are even provided in commercial software packages.

The use of a more efficient solver rather than FDTD or FEM is of absolute interest but already

presented in ref 14, as largely commented also by the authors. The fact of being applied to a

dual-band grating instead of a single wavelength devices as in ref 14 can be regarded as quite

incremental unless this poses additional difficulties in the optimization process which are not

on the other hand discussed in the manuscript. The device concept is not new but the authors

claim quite strong improvements in performance compared to published results, in particular on

insertion losses and power unbalance. These results represent the core of the paper novelty but

I believe their solidity should be better discussed.

Although L-BFGS and the adjoint method are indeed used quite often for inverse design of

nanophotonic devices, the majority of prior work (and any commercial packages) which use these

optimization methods rely on the Finite Difference Time Domain (FDTD) method for simulation.

Deriving the adjoint method and applying it to integral equation methods, as done here and in

our previous work (ref [14]) is not as straightforward as its application to the FDTD method

and requires careful treatment in order to maintain high efficiency. We have included extra

details specific to the simulation and optimization of the grating coupler wavelength demultiplexer

presented in this work, beyond what is covered in ref [14] which is a general introduction to

nanophotonic optimization using integral methods, in Sections 1.1 and 1.2 of the Supplementary

Information.

In Section 1.3, we also performed a comparison of the time it takes to achieve a desired accu-

racy with our integral equation based simulation approach versus a commercial state-of-the-art

FDTD solver. In order to demonstrate the advantages resulting from use of the new approach

in the design of this particular device, we present performance comparison results against the

best-known commercial FDTD solver used for photonics simulations and include a discussion on

the accuracy level required for optimization purposes—which can only be produced by the com-

mercial solver at a significantly higher expense (200 times slower as indicated in Supplementary

Section 1.3) than that required by integral solver we use. The validity of the solutions obtained

is confirmed in Figure 2 in the main body of the paper, which presents 2D and 3D simulations

based on the commercial solver, displaying optimal coupling efficiencies in clear agreement with

the design produced by means of the integral solver.

2. Insertion loss performance heavily depends on the applied normalizations, which becomes hence

a critical point. In figure 6a, experimental results are largely better than those expected by

simulation (purple and red curves) which is very surprising. Yet, this aspect is not discussed. If

this is indeed the case then one could assume that the optimizer did not finished its job since a

better design was indeed found during fabrication!

A similar comment was provided by Reviewer #2. Briefly, although the measured coupling

efficiency was indeed better than the simulated results, the measured splitting ratio was worse,

thus resulting in a value of the objective function which, as expected, is worse for the fabricated

device than predicted by the numerical simulation. We have added the following text in the

revised manuscript to clarify this:

“In fact, for both left and right transmission, Fig. 6 shows experimental power values higher

than the numerical values obtained for the simulated device; the departure is significantly more

noticeable for the power transmitted into the left fiber. But the experimental data does not

contradict the predictions, since the objective function consists not only of the power transmitted

into the desired waveguide, but it also includes a penalty for power transmitted into the undesired

waveguide. When both powers are taken into account the apparent contradiction is eliminated:

the value of the objective function for the combined left and right power, for both frequencies,

lies below the optimal predicted value.”

3. Normalizations for directional couplers and PIN modulators are based on FDTD simulations

that may not grasp all the source losses in a fabricated device or loss differences between side

by side devices. It is not specified if the four directional couplers in the rest device are identical

or not. It is not clear if the responsivity of the photodiodes was measured on dedicated devices

realized on the same chip or just represents an average value for the foundry. I think clarifying

these aspect is crucial to convince the reader that the presented device has indeed best-in-class

performance.

All of the directional couplers are identical. The responsivity of the photodiodes were measured

as being 0.5 A/W using a separate test structure on the same chip, which is also in agreement with

the average value reported by the foundry. The losses of the directional couplers and PIN diodes

were simulated due to not having test structures for characterizing them on this chip; however,

the same designs have been previously characterized and agreed well with simulation on prior chip

designs by the group. We agree that there could be some variations in the losses of the directional

couplers and PIN modulators used, due to process variation; however, our simulated results

should represent conservative estimates. If indeed the actual fabricated directional couplers

and/or PIN modulators had more loss than simulated, they would account for a larger fraction

of the total measured insertion loss and the actual measured grating splitter efficiency would be

better than what we reported.

4. Some other minor aspect should be considered. Regarding parametrization, while it is true

that adjoint simulations are often applied to topology optimization which optimizes material

distribution in a volume, as described at the end of page 5, there are many examples in literature

where the same is used for parametrized devices without the use of any threshold or level set

function, exactly as done here. This advantage is hence independent on the specific solver and

is not unique to the tool chosen by the authors.

We do not agree that the boundary perturbation based optimization technique is independent

of the specific solver—the boundary integral formulation naturally lends itself to optimization

techniques which optimize structures by directly perturbing their boundary curves. The only

other approach that we are familiar with which optimizes parametrized boundaries that is not

based on level sets is the work presented in reference (13). This approach still relies on expensive

volumetric simulation methods (FDTD) and requires material averaging in order to translate

small boundary perturbations into changes in the volumetric dielectric distribution. Aside from

leading to potentially inaccurate representations of the actual boundary due to non-physical

material averaging on a Cartesian mesh, the volumetric solver based approach is significantly

more expensive as well. In our previous work (14), we optimize a 2D taper structure with the

same material parameters and size as that in (13) and we found that our integral equation based

simulation and optimization approach took only 15 minutes to complete the whole optimization,

compared to 35.7 hrs reported in (13).

5. Moreover, it would be interesting if the authors could comment on the reflections generated by

their device since this is the real difficulty in designing perfectly vertical couplers and the one

that normally forces the critical dimension to few tens on nanometers.

We have previously addressed this matter in response to question number 3 by reviewer #1.

In particular, we have included a brief discussion in the main text reporting that 24

7% and

7% of the power is reflected back upwards at 1.33

m and 1.55

m respectively, and that the

remaining power which does not couple into the output waveguides is transmitted downwards

through the substrate. Since the measured coupling efficiency into the desired outputs exceeds

that of previously reported work, we expect that the losses should be less than those of other

reported vertical-coupling grating splitters. We believe that the relatively low measured insertion

losses with respect to the desired coupling into the output waveguide makes this device suitable

for a number of wavelength-demultiplexing applications.

6. Since the authors claim, for example at the end of the introduction, that they were able to find

a better device compared for example to ref 16, if think they should discuss the reason for this.

Is it for an optimization algorithm that can be better explore the design space (which is however

hard to claim since they are using a local optimizer)? Is it for a better initial guess? Is it because

they are penalizing other aspect, such as reflections?

We believe that this is, on one hand, due to lower accuracy of the computational representation of

the problem, including the quality of the FDTD solution in presence of a level-set approximation

for domain partition, and, on the other hand, caused by effects arising from the thresholding

method that is an integral part of the optimization approach presented in ref 16. Note, in

particular that Figure 4 in reference 16 indicates a very significant discrepancy between the

predicted simulation results and measured fabrication results—including, for example, a 15dB

relative coupling difference between the two wavelengths in measurement vs. the simulations.

7. Lastly, I feel in some points the text should be strengthened.

- In the abstract, it is not clear to me what ”high-order accuracy” of the solver means.

We replaced the words ”high-order accuracy” were replaced by ”extremely rapid convergence”

in the abstract and the introduction; the latter occurrence now refers to the new Figure 1 left in

the Supplementary Information.

- In the introduction, page 3, I don’t think the sentence ”The majority of grating couplers are

designed to be efficient at coupling in a narrow band around a single wavelength of interest” being

true. Many devices have been reported working on a single band but on very large bandwidths,

up to several hundreds of nanometers.

We removed the word “narrow” from the sentence to make it more accurate and better convey

the statement that the majority of grating couplers are designed to be single, rather than multi

band.

- In the same page, the authors discuss manual design of grating couplers but cite also ref 16,

which is indeed on inverse design.

We do not cite reference 16 as an example of manual design. The sentence: “Although it may

be possible to design a

single-wavelength

grating coupler with reasonable efficiency via trial and

error or sequences of parametric sweeps (a challenging and time-consuming approach which may

additionally lead to suboptimal designs), it appears completely impossible to even attempt to

use such approaches in the design of multi-wavelength grating splitters.” which discusses the

difficulty of manual design does not cite reference 16. Reference 16 is cited as an example of a

previous implementation of a grating splitter without mention of manual design.

- Also stating that ”it appears completely impossible to even attempt to use such approaches in

the design of multi-wavelength grating splitters” is incorrect, as proved by refs 15 and 17.

Thank you for bringing this up as a point which needs clarification. References 15 and 17

present two dimensional grating couplers, which in essence couple a single wavelength (or polar-

ization) in each dimension. We have added the word “one dimensional” to “multi-wavelength

one-dimensional

grating splitters” to clarify and qualify that we are not aware of any manual

design approach for multi-wavelength coupling in single dimensional gratings such as the device

we present in this manuscript.

- Insertion losses are reported as 3 dB and 4.95 dB for the two bands in the introduction, page

4, and as 2.96 dB and 5 dB in table 1.

Thank you for noticing this. We have adjusted the table and introduction so that the significant

figures are consistent in both sections.

- Page 4, it is not clear to me the wording ”engineering designs”. There is also a typo in the

same line.

We have changed the sentence to read “Inverse design is a computational technique that enables

automated design of photonic devices.”

- It is not clear what is an ”auto-generated” initial guess. Does it mean it has been chosen

randomly? Initial guess is key for local optimizers.

We have changed the text to the following to clarify: “Starting from an initial guess (which is

typically selected randomly, as we do in this paper, but which could otherwise be chosen on

the basis of prior insight), an iterative optimization process is subsequently used which leads to

increasingly more efficient structures, until the desired structural response has been achieved.”

- At the beginning of page 5, ”sufficiently optimized design” is very vague. Does it mean the

algorithm has converged or it has simply been stopped at an arbitrary iteration?

We have changed “sufficienty optimized design” to “until convergence to a minimum” to clarify

that indeed we let the algorithm converge.

- I feel that information on silicon thickness, number of periods in the grating and fiber mode

diameter should be included in the main text, not in the supplementary since they are quite

fundamental piece of data (e.g. choosing a different fiber would heavily affect the achievable

efficiency). The authors did not mention which polarization they are using. Reflections from the

silicon substrate were considered in the simulation/optimization?

We have incorporated all of this information into the main text, as part of the discussion of

Figure 1, as follows:

“This figure demonstrates some of the characteristics of the design, including the top oxide

passivation layer (

= 1

= 2

m), the grating Silicon device layer (

= 3

48,

= 0

m),

the buried oxide layer (

= 1

44,

= 2

m), and the silicon substrate handle (

= 3

48). A

TM-polarized, vertically incident Gaussian beam excitation of 10

m waist diameter at a height

of 0.22

m above the passivation layer was used to simulate the fundamental mode of the SMF-28

optical fiber vertically incident over the GCWD device.”

For ease of reference, this information is also contained in the Supplementary Information—as

part of the text referring to the optimized grating widths and spacings.

The objective function does not explicitly consider substrate reflections, although they are im-

plicitly minimized by maximizing the coupling efficiency from the input fiber to the desired

output waveguide. The simulations are full-wave solutions of Maxwell’s equations and therefore

consider the whole physics of the problem, including any scattering and reflections.

- Page 8, how is it defined the adjoint problem? Power is lunched simultaneously in the two

waveguides?

We use the discrete adjoint method. The right hand side excitation of the adjoint problem

is calculated based on the objective function vector multiplied by the derivative of the right

hand side. This should indeed have the effect of placing sources inside each the waveguides in

the ratios determined by the objective function. We have added more details on our choice of

parametrization for enabling fast implementation of the adjoint method to the Supplementary

Information, and we also refer the reader to our previous work in ref [14] for a detailed derivation

of the discrete adjoint method for integral equation solvers.

- Page 9, the abbreviation ”resp.” is not clear to me.

We have expanded the first couple instances to “respectively” to be more clear; however, for the

sake of brevity we left the remaining instances intact as we believe it should be clear since they

are used in the same manner and style.

- Figure 2, results are obtained with FDTD or their simulator?

The results in Figure 2 are obtained with FDTD for comparison and to show agreement with

our solver’s predicted performance. We have added clarification to both the caption of Figure 2

and to the sentence in the manuscript referring to it:

“The simulated insertion losses from the incident optical fiber to the left and right output ports

of the grating are plotted as a function of the wavelength in Figure 2 using both 2D as well as

full-wave 3D

FDTD

simulations for comparison.”

We thank the reviewer for their detailed suggestions and aside from these changes have also

significantly expanded the supplementary information to provide a fuller description of the opti-

mization methods used and their specific application towards the design of the presented grating

coupler wavelength demultiplexer. More specifically, we added Sections 1.1, 1.2, and 1.3 in the

Supplementary which focus on the boundary integral methods and our adaptation of the adjoint

method to them for rapid gradient computation. Section 2 presents the optimized parameters

of the fabricated GCWD design. Sections 3, 4, and 5 present a number of additional simulation

results of the presented grating splitter, including studies concerning back-reflections, reverse op-

eration as a transmit grating multiplexer, angular sensitivity, and comparison to another grating

design using our same algorithm but with 50nm minimum feature sizes rather than 160nm.

REVIEWERS' COMMENTS

Reviewer #1 (Remarks to the Author)

The authors have addressed all the comments with satisfactory. The revised manuscript and

supplementary information are in much better quality

Reviewer #3 (

Remarks to the Author)

The authors convincingly addressed most of my previous comments, in particular regarding the

novelty of their work. I think the manuscript can be considered for publication after a few last

minor points are addressed

In particula

r, I could not find a comment to some of my previous suggestions that I propose again

here (page and line numbers refer to the previous version of the manuscript)

In commenting eq 2, larger splitting ratios means more power into the correct output only

for the

left port, for the right port eq. 2 should be flipped, I think

On line 220, page 12, why the step size if 50 nm in one direction and 40 times larger in the other

one

Line 222 and subsequent, including table 1, what is the isolation? Does it

refer to eq 2

Table 1, Pigott 2014 should be ref 16 not 1

Additionally, on page 9 of the revised manuscript, I think "top oxide passivation layer (n = 1.0, t =

2.78μm)" should be " top oxide passivation layer (n = 1.44, t = 2.78μm)" according to figur

e 1

Rebuttal Letter

We would

to thank all the reviewers

for their insightful feedback and

suggestions throughout the review process, which have helped us significantly

improve the quality of our manuscript.

Reviewer #1 (Remarks to the Author):

The authors have addressed all the comments with satisfactory. The revised manuscript

and supplementary information are in much better quality.

Reviewer #3 (Remarks to the Author):

The authors convincingly address

ed most of my previous comments, in particular

regarding the novelty of their work. I think the manuscript can be considered for

publication after a few last minor points are addressed.

In particular, I could not find a comment to some of my previous suggestions that I

propose again here (page and line numbers refer to the previous version of the

manuscript):

- In commenting eq 2, larger splitting ratios means more power into the correct output

only for the left port, for the right port eq. 2 should be flipped, I think.

- On line 220, page 12, why the step size if 50 nm in one direction and 40 times larger in

the other one?

- Line 222 and subsequent, including table 1, what is the isolation? Does it refer to eq 2?

- Table 1, Pigott 2014 should be ref 16 not 12

Additionally, on page 9 of the revised manuscript, I think "top oxide passivation layer (n

= 1.0, t =

2.78μm)" should be " top oxide passivation layer (n = 1.44, t = 2.78μm)"

according to figure 1.

We have incorporated all of th

ese changes.