Foundry-fabricated grating coupler demultiplexer inverse-designed via fast integral methods

ARTICLE

Foundry-fabricated grating coupler demultiplexer

inverse-designed via fast integral methods

Constantine Sideris

✉

, Aroutin Khachaturian

, Alexander D. White

, Oscar P. Bruno

4,5

& Ali Hajimiri

2,5

Silicon photonics is an emerging technology which, enabling nanoscale manipulation of light

on chips, impacts areas as diverse as communications, computing, and sensing. Wavelength

division multiplexing is commonly used to maximize throughput over a single optical channel

by modulating multiple data streams on different wavelengths concurrently. Traditionally,

wavelength (de)multiplexers are implemented as monolithic devices, separate from the

grating coupler, used to couple light into the chip. This paper describes the design and

measurement of a grating coupler demultiplexer

—

a single device which combines both light

coupling and demultiplexing capabilities. The device was designed by means of a custom

inverse design algorithm which leverages boundary integral Maxwell solvers of extremely

rapid convergence as the mesh is re

fi

ned. To the best of our knowledge, the fabricated device

enjoys the lowest insertion loss reported for grating demultiplexers, small size, high splitting

ratio, and low coupling-ef

fi

ciency imbalance between ports, while meeting the fabricability

constraints of a standard UV lithography process.

https://doi.org/10.1038/s42005-022-00839-w

OPEN

University of Southern California, Ming Hsieh Department of Electrical and Computer Engineering, Los Angeles, CA 90089, USA.

California Institute of

Technology, Electrical Engineering, Pasadena, CA 91125, USA.

Stanford University, Department of Electrical Engineering and Ginzton Laboratory, Stanford,

CA 94305, USA.

California Institute of Technology, Computational and Mathematical Sciences, Pasadena, CA 91125, USA.

These authors contributed

equally: Oscar P. Bruno, Ali Hajimiri.

✉

email:

csideris@usc.edu

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1234567890():,;

S

ilicon photonics is a rapidly expanding industry due to

its potential for positively impacting a broad range of

important application domains, including high-speed

interconnects, optical phased array transceivers

, one- and two-

dimensional lens-less cameras

, nanophotonic lens-less projec-

tion systems

, integrated LIDAR

, biosensors

, nanophotonic

gyroscopes

, etc.

This paper presents a photonic dual-band grating coupler

wavelength demultiplexer (GCWD) for the bands most com-

monly used in telecommunication applications, namely, the

O-band (centered at 1310 nm) and the C-band (centered at

1550 nm) under vertical incidence. A grating coupler couples

light into and out of on-chip waveguides via an external optical

fi

ber. The grating, which consists of many etched

“

teeth

”

which

scatter the incident

fi

eld, must be designed in such a way that the

sum of the scattered contributions interferes coherently in the

desired mode. Use of high-ef

fi

ciency grating couplers, which

could allow for reductions in power requirements of the laser

source, can be greatly bene

fi

cial. The majority of grating couplers

are designed to be ef

fi

cient at coupling in a band around a single

wavelength of interest. However, there exist many important

applications, such as wavelength division multiplexing (WDM)

for optical interconnects where data is encoded on multiple

wavelengths and sent concurrently through the same optical

fi

ber

link; other applications include spectroscopy, biosensing, and

nonlinear harmonic extraction. In such situations, it is useful to

utilize a single grating coupler device which can both couple as

well as split the signal for the relevant wavelengths incident from

the

fi

ber into separate on-chip waveguides; the

fi

rst such devices

were introduced by Xu et al. and other authors

–

. Although it

may be possible to design a single-wavelength grating coupler

with reasonable ef

fi

ciency 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 dual-wavelength one-dimensional

grating demultiplexers.

Silicon photonic chips are comprised of nanophotonic struc-

tures of large electrical size which typically contain complex sub-

wavelength features, and, thus, the design and optimization of

photonic devices amounts to a highly challenging computational

problem. Recently, inverse design, which is an optimization

approach that can automatically design structures under

given desired speci

fi

cations and fabrication constraints, has

emerged as a powerful methodology for the development of high-

performance photonic devices

. A brief description of the

general inverse-design methodology is presented at the beginning

of the Results section. Signi

fi

cant efforts have been devoted over

the last few years to the numerical simulation and topology

optimization of such structures

–

The proposed GCWD design was obtained on the basis of a

fast integral-equation electromagnetic inverse design framework

introduced in Sideris et al.

, which is outlined, demonstrated,

and compared to other approaches in Supplementary Note 1.

Notably, the designed device meets all of the design rule con-

straints of, and was fabricated using, a standard UV-lithography

silicon photonics foundry process; Supplementary Notes 3 and 4

explore the robustness of the design and the tradeoffs that result

when the foundry-prescribed 160 nm minimum feature size

constraint is relaxed to 50 nm. Additionally, the potential of the

GCWD to be used in reverse, as a wavelength combiner, is

established in Supplementary Note 5. Finally, the fabricated

device measurements are in close agreement with the computa-

tional predictions. To the best of our knowledge, the measured

device achieves the lowest insertion loss ever reported for a

GCWD device (4.95 and 2.96 dB at 1310 and 1550 nm

respectively), in addition to a number of other important

advantages listed in Table

and described in the

“

Discussion

”

section.

Results

Grating coupler inverse design via fast integral methods

Inverse design is a computational technique that enables the

automated design of photonic devices. 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 ef

fi

cient structures, until the

desired structural response has been achieved. The optimization

step proceeds by evaluation of the sensitivity (gradient) of the

underlying objective function (which quanti

fi

es the quality of the

device and its performance) to perturbations of the parameters

describing the structure. The gradient points in the direction of

maximum improvement. A complete design is then obtained by

iteratively recomputing the gradient and adjusting the design

parameters at each step, as be

fi

ts minimization/maximization of

nonlinear functions, until convergence to a minimum is achieved.

The gradient evaluation would naively require a number of

solutions of the

fi

eld equations: one solution per design para-

meter. The cost of the procedure would be prohibitive for most

photonic applications, including the one considered in this paper.

To tackle this problem we rely on a powerful alternative com-

monly used in modern inverse design methods, namely, the

“

adjoint method" for gradient evaluation, which produces the

same gradient, without approximations, by relying on only two

simulations

–

It was recently shown that the use of integral equation methods

can provide certain advantages in the areas of photonic inverse

design

. Indeed, integral methods, which are based on Green

’

functions and only require discretization of material interfaces

can effectively be used to represent complex photonic designs.

This is in contrast with the

fi

nite element method (FEM) and the

fi

nite-difference time-domain (FDTD) which require meshing the

complete domain while adequately respecting the material

interfaces and satisfying Maxwell

’

s boundary conditions on those

interfaces. Aside from signi

fi

cantly reducing the number of

unknowns required to solve the forward problem, the integral

equation method also leads to a natural boundary-based

formulation which is particularly valuable for purposes of

optimization, since it allows for modi

fi

cation of the material

interfaces without requiring remeshing of any kind. Traditional

FEM or FDTD-based inverse design methods, on the other hand,

either rely on optimization of a continuous volumetric dielectric

distribution and subsequent thresholding

or use of level-set

methods to simulate boundaries, which must then be mapped to

and from the volumetric dielectric distribution at each iteration

at signi

fi

cant accuracy loss and computational cost, as discussed

in Sideris et al.

and Supplementary Note 1.3. Since the integral

equation based approach

only uses the interfaces to discretize

the problem, the optimization parameters directly control the

shape of the boundary and the equations that need to be solved,

and thus allow the device boundaries to be optimized within a

computational context that is well adapted to the evaluation of

Maxwell solutions under arbitrary variation of the interface

boundaries.

More speci

fi

cally, the proposed GCWD design was obtained by

parametrizing the boundaries of the grating as shown in Fig.

The unknowns to be optimized are the grating tooth widths

and tooth spacings

fi

xed etch depth of 130 nm was chosen

and the minimum width and spacing were constrained to be

greater than or equal to 160 nm in order to meet the required

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foundry speci

fi

cations

. An objective function was subsequently

fi

ned whose minimization corresponds to joint optimization of

the coupling ef

fi

ciencies for both wavelengths into the designated

output waveguides while concurrently maintaining a high

splitting ratio between the two outputs for each frequency.

In detail, a given candidate grating structure of the type

depicted in Fig.

is considered in what follows, whose speci

fi

dimensions are characterized by a parameter vector

=

(

...

) of spacings and widths, with the

fi

rst spacing

fi

ned relative to a prescribed,

fi

xed point along the

horizontal. Note that the

fi

rst spacing,

, allows the complete

grating to shift left or right to align with the center of the incident

excitation. The quality of the design is evaluated by means of the

“

objective function

”

Þ¼

1550

1300

αη

1300

1550

;

whose minimization yields ef

fi

cient GCWD designs. Here,

;

1300

;

1550

denotes the coupling ef

fi

ciency into the guide at the

wavelength indicated by the subscript and measured either at the

left or right output waveguide, as indicated by the superscript (cf.

eq. (

)). The positive, scalar parameter

controls the emphasis of

the objective function on maximizing the coupling ef

fi

ciency at

each wavelength vs. minimizing coupling of each wavelength into

the unintended port. Note that the design was optimized for

center wavelengths of 1300 and 1550 nm as can be seen in the

simulated coupling ef

fi

ciency results (Fig.

), although 1310 and

1550 nm wavelengths were used for the experimental measure-

ments due to the limitations of the lasers available in the

laboratory. For a given grating structure (that is, for a given

parameter vector

), these coupling ef

fi

ciencies are calculated on

the basis of electromagnetic simulation

and subsequent

evaluation of mode overlap integrals across the two output

waveguide cross-sections.

Like the Method of Moments

, the electromagnetic solver we

used utilizes Green

’

s functions, discretizations of surface currents,

and integral equations. The actual integral formulation

used in

this contribution is based on point-based

“

Nyström

”

discretiza-

tion of the boundary and the use of windowing functions for

truncation of in

fi

nite waveguide domains. The approach is

designed to produce high accuracy in short computing times by

exploiting numerical techniques that yield extremely rapid

convergence as the mesh is re

fi

ned; see e.g., the left portion of

Supplementary Fig. 1. The high speed of the method has

signi

fi

cantly facilitated the design process. For example, Fig.

the contribution Sideris et al.

compares the integral solver

under consideration to other methods often used in engineering

practice, demonstrating improvements in design times of the

order of two or more orders of magnitude over corresponding

design times required by FDTD- and FEM- based approaches; cf.

Supplementary Note 1.3.

As mentioned above, this numerical method only requires

discretization of the material interfaces. In the structure under

consideration all interfaces are composed of horizontal or

vertical planar segments, as can be seen in Fig.

.Eachoneof

these segments is discretized using six points per wavelength but

no less than a total of twenty points per segment. Furthermore,

to accurately resolve the

fi

eld 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

–

84 in the book by Colton et al.

—

so as to give rise to high-

order convergence. Additional details concerning our integral

equation implementation are presented in Supplementary

Notes 1.1 and 1.2. Supplementary Note 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. Once the integral-equation

solution is found, the

fi

eld values at any point within the

structure can be obtained in terms of certain interface integrals

involving both Green functions and the values of the surface

unknowns obtained as part of the integral-equation solution

process. Using such

fi

eld values, the aforementioned mode-

overlap integrals are then computed at approximately two

wavelengths away from the ends of the grating design on the

left and right sides by evaluating the

fi

elds for integration along

vertical lines perpendicular to the waveguide cross-section. The

optimization process was tackled by means of a quasi-Newton

Fig. 1 Diagram of the proposed grating coupler wavelength demultiplexer

geometry (GCWD).

The individual tooth widths and spacings (shown to be

uniform in the

fi

gure, for simplicity) are parametrized by the variables

and

respectively. These variables are selected by the inverse design

algorithm to optimize the demultiplexing objective function. The input

excitation is modeled as a radiative-type Gaussian beam which closely

matches the free-space

fi

eld emanating from an optical

fi

ber. The quantities

inc

, and

denote the refractive indices, the incident beam

excitation, and the power coupled into the left and right silicon waveguides,

respectively.

Fig. 2 2D and 3D

fi

nite difference time domain (FDTD) coupling

fi

ciency simulation.

Simulated coupling ef

fi

ciency vs. wavelength for left

and right output ports. As intended, the simulated coupling ef

fi

ciency peaks

for 1300 nm on the left port (at 27.5%, corresponding to 5.6 dB insertion

loss) and for 1550 nm on the right port (at 35.5%, corresponding to 4.5 dB

insertion loss). Dotted black lines are full-wave 3D simulations of the

fi

nal,

fabricated grating structure. The close match between the insertion losses

obtained via 2D and 3D FDTD simulations justi

fi

es using our 2D integral

equation simulation approach during the optimization phase.

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method, namely, the L-BFGS algorithm

(Limited-memory

Broyden

–

Fletcher

–

Goldfarb

–

Shanno) in the variables con-

tained in the vector

, to minimize the function (

). As

mentioned above, the necessary gradient is obtained by means of

the adjoint integral approach introduced by Sideris et al.

, which

requires a solution of a certain adjoint integral problem that can

be discretized and solved in the same manner as the original

forward electromagnetic problem on the waveguide. For the types

of problems under consideration in this paper, the minimization

process was achieved in approximately

fi

fty to two hundred

L-BFGS iterations.

Demultiplexer simulation and design

. The design under con-

sideration, for a given vector

of widths

and spacings

,is

depicted in Fig.

. This

fi

gure demonstrates some of the char-

acteristics of the design, including the top oxide passivation layer

(

=

1.44,

=

2.78

μ

m), the grating Silicon device layer (

=

3.48,

=

0.22

μ

m), the buried oxide layer (

=

1.44,

=

2.0

μ

m), and

the silicon substrate handle (

=

3.48). A TM-polarized (Trans-

verse Magnetic), vertically incident Gaussian beam excitation of

μ

m waist diameter at a height of 0.22

μ

m above the passivation

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

optical

fi

ber vertically incident over the GCWD device.

The simplicity of the width and spacing parameters for

geometry characterization leads to a straightforward implementa-

tion of the optimization problem on the basis of the L-BFGS

algorithm

—

by using L-BFGS box constraints requiring that each

width and spacing meets the foundry fabrication constraints, as

lower bounds, and an arbitrary suf

fi

ciently large spacing value

(taken to equal 1

μ

m for this design) as upper bounds. Thus,

minimum width and spacing restrictions of 160 nm, as required

by the A

*

STAR AMF silicon photonics foundry

that was used

to fabricate the structure, were adopted in our design. The

optimization process was completed in 122 L-BFGS iterations,

resulting in a total computing time of approximately 130 single-

core minutes on a Xeon E5-2630 v3 2.4GHz CPU. The simulated

insertion losses from the incident optical

fi

ber to the left and right

output ports of the grating are plotted as a function of the

wavelength in Fig.

using both 2D as well as full-wave 3D FDTD

simulations for comparison. The close agreement in insertion

losses produced by the 2D and 3D simulations justi

fi

a-posteriori the use of a 2D solver as part of the optimization

process.

Note, in particular, that, as intended, the simulated coupling

fi

ciency peaks for 1300 nm on the left port (at 27.5%,

corresponding to 5.6 dB insertion loss) and for 1550 nm on the

right port (at 35.5%, corresponding to 4.5 dB insertion loss).

Furthermore, the coupling ef

fi

ciency on the right output port at

1550 nm (resp. left output port at 1300 nm) is small, at only

0.29% (resp. 0.33%), implying simulated splitting-ratio values of

21 and 19 dB on the left and right ports respectively. The

corresponding optimized design,

fi

nally, is depicted in Fig.

Figure

a (resp.

b) demonstrates coupling to the left (resp. right)

port due to incident illumination at the 1300 nm (resp. 1550 nm)

wavelength

—

clearly demonstrating the splitting capability of the

proposed device, whose experimental realization, as discussed in

the following section, achieves a lower measured insertion loss

(Table

) than previously reported grating splitters. For reference,

24.7 and 13.7% of the incident power is back-re

fl

ected upwards at

1300 and 1550 nm respectively, and the remaining power is either

transmitted through one of the two output waveguides as

indicated above or transmitted downwards through the substrate.

The parameter vector

listing the tooth widths and spacings of

the optimized GCWD is presented in Supplementary Note 2

(Supplementary Table 1).

Experimental results

. As mentioned above, the grating coupler

demultiplexer presented in this paper has demonstrated favorable

fi

ciencies and wavelength demultiplexing. In order to quanti-

tatively evaluate the device performance, we present the experi-

mentally observed values of the insertion losses and splitting

ratios at each wavelength. Thus, using the outgoing power

parameters

left

and

right

—

which equal, for each of the two

wavelengths

considered, the power obtained at the left and right

ports, respectively

—

, following

we de

fi

ne the splitting ratio at

wavelength

10 log

left

right

:

Note that, on account of the absolute value used in this expres-

sion,

yields the logarithmic ratio between the amounts of power

going into the intended port and the opposite port for both

wavelengths

=

1300 nm and

=

1550 nm. The splitting ratio is

an important metric for GCWD devices, and should be max-

imized in order to avoid cross-talk between information in the

two bands, but it is not the only important metric in this context.

Indeed, the insertion loss from the input incident

fi

ber to each

output and the corresponding absolute coupling ef

fi

ciency at each

wavelength are also very important parameters for the assessment

of the device performance. The absolute coupling ef

fi

ciency and

the insertion loss in dB at the left (L) and right (R) ports for a

given incident wavelength

are de

fi

ned by

;

;

inc

;

;

10 log

;

;

respectively. A GCWD used for optical interconnect applications,

for example, must feature both low insertion loss and low mis-

match between the insertion losses at the two wavelengths in

order to minimize the incident power required for a desired

Fig. 3 Simulation of vertical incident

fi

eld coupling into the left and right waveguides.

Electric

fi

eld magnitudes in the GCWD (grating coupler

wavelength demultiplexer) for 1300 and 1550 nm incident excitations.

-component

of the electric

fi

eld at 1300 nm.

fi

eld component at 1550 nm.

The white lines in these

fi

gures correspond to interfaces separating the various material regions (Air, Passivation, Silicon, Buried Oxide, and Silicon

Substrate) as illustrated in Fig.

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signal-to-noise ratio. Although Piggott et al.

do not report

absolute coupling ef

fi

ciency measurements, Fig.

b in that paper

does show that the relative insertion loss of the signal at 1550 nm

is at least 10 dB lower than that at 1310 nm.

Figure

b depicts a 3D rendering of the designed GCWD

device alongside a scanning electron micrograph (c) of the actual

fabricated device, and the measurement setup (a) used to

characterize it. The coupling losses and alignment tolerances for

the demultiplexer were characterized in both the O-band

(1310 nm) and the C-band (1550 nm). The structure was

vertically illuminated with a 2D scanning

fi

ber and the coupled

power at the two ports was recorded for both wavelengths as

illustrated in Fig.

a. The coupled power at the two ports was

measured using integrated Germanium positive-intrinsic-

negative (PIN) photodiodes at both wavelengths. The outgoing

power at the two output ports of the structure was measured via

the photodiodes as the

fi

ber position was swept at normal

incidence over a 50

μ

m×50

μ

m area centered at the nominal

design position with a 50 nm × 2

μ

m step size; Fig.

displays the

power collected at each port for each one of the two wavelengths

for each position considered in the sweep. Note that a

signi

fi

cantly

fi

ner step size was used for the sweep in the

direction (the axis of propagation of the GCWD) compared to

the transverse

-direction axis, on account of the essentially

uniform character of the design, and corresponding slow

fi

eld

variation, in the

-direction.

The experimentally measured splitting ratio

between the two

ports is 9.16 dB at 1310 nm and 8.18 dB at 1550 nm for the input

fi

ber at position (

=

μ

m) (centered over the GCWD). The

measured absolute coupling ef

fi

ciency at each port and

corresponding wavelength, at the cross-section

=

μ

m, are

extracted from Fig.

and plotted in Fig.

alongside simulation

results. It can be seen that there is good agreement between the

measured and simulated results. In fact, for both left and right

transmission, Fig.

shows experimental power values higher than

the numerical values obtained for the simulated device; the

departure is signi

fi

cantly more noticeable for the power

transmitted into the left

fi

ber. 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. The simulated

sensitivity of the device with respect to variations in position and

angle of the incident

fi

ber is plotted in Fig.

. Additional details

and analysis concerning sensitivity to angular variation are

presented in Supplementary Note 3.2.

Directional coupler and PIN modulator devices were included

in-line with the output waveguides in order to provide added

fl

exibility in the experimental characterization of the GCWD. The

test devices introduce an extra insertion loss which must be

subtracted to accurately characterize the performance of the

GCWD itself. To do this, the insertion losses of the directional

couplers and PIN modulators were characterized using FDTD

simulation. It was found that the PIN modulators have an

insertion loss of 0.75 dB at 1310 nm and 0.05 dB at 1550 nm. The

directional couplers result in an additional loss of 0.12 dB at

1310 nm and 2.49 dB at 1550 nm. The responsivity of the

photodiodes was measured at 0.5 A/W at both wavelengths.

Finally, as predicted by the aforementioned results presented in

Fig.

, the peak collected power for the wavelength demultiplexer

corresponds to zero degree incident angle and after de-

embedding the responsivity of the photodiodes, the directional

couplers, and the modulators, the adjusted insertion losses of the

measured wavelength demultiplexer in isolation were found to be

4.95 dB at 1310 nm and 2.96 dB at 1550 nm, as reported in

Table

Discussion

This paper demonstrated the design, fabrication, and measure-

ment of a grating coupler wavelength demultiplexer design with

state-of-the-art performance for optical interconnect applications.

The GCWD was designed, without any human interaction aside

from speci

fi

cation of desired performance criteria and fabrication

constraints, by means of a high-speed, inverse-design approach

based on the boundary-integral Maxwell solver introduced in

Sideris et al.

. The tested device was fabricated by A

*

STAR

AMF

and the measurements results were found to agree well

with the predicted simulations indicating a clear potential for

applicability in practical settings. Most importantly, minimum

feature size constraints were incorporated in the design process

and the grating was designed for operation under vertical input

incidence (cf. Fig.

)

—

which makes the device fabricable in a

standard SOI foundry process and easy to package, respectively.

Both of these characteristics are crucial for reducing cost and

increasing yield in a commercial setting.

A comparison between the measured performance of the

proposed device and related prior work is presented in Table

As can be seen from the table, the proposed device provides

signi

fi

cant advantages in several of the performance categories

listed: among the devices considered, it exhibits the lowest

insertion loss as well as high splitting ratio between the ports, low

insertion-loss imbalance between the two wavelengths, small

total device size, and it was fabricated in a standard UV

lithography-based silicon photonics foundry process which was

possible in view of its large minimum feature size. For reference,

the contribution Piggott et al.

,wherethe1DGCWDconcept

was

fi

rst demonstrated, presents a design with minimum feature

size of 61.1 nm, which is incompatible with foundry UV litho-

graphy, and therefore required E-beam lithography for fabrica-

tion. Further, an imbalance of ~14 dB in the coupling ef

fi

ciency

of the two wavelengths can be seen in Fig.

b in Piggott et al.

compared to the 1.95 dB imbalance in the present design.

Fig. 4 Wavelength demultiplexer design and structures used for

characterization. a

On-chip structures used to measure and characterize

the wavelength demultiplexer. The coupled power from the wavelength

demultiplexer device at both output ports is measured simultaneously by

using two photodiodes (PD1 and PD2). Positive-intrinsic-negative (PIN)

modulator devices are displayed next to PD1 and PD2. Directional couplers

act as sniffers to measure the coupled power at both wavelengths.

Wavelength Demultiplexing structure rendering (Port 1: 1310 nm, Port 2:

1550 nm).

Scanning electron microscope (SEM) image of the fabricated

wavelength demultiplexer structure.

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Fig. 5 Coupled output power measurement.

Coupled power

PD1

and

PD2

(in dB) measured by the photodiodes PD1 and PD2 displayed in Fig.

at the

corresponding output ports of the wavelength demultiplexer and for the corresponding wavelengths, 1310 and 1550 nm. The measurement was obtained by

laterally scanning the incident

fi

ber above the device as shown in Fig.

Table 1 Performance comparison of various GCWD designs.

This work

Piggott et al. 2014

Streshinsky et al. 2013

Xu et al. 2011

Test wavelengths:

λ

λ

1310 nm, 1550 nm

1310 nm, 1540 nm

1310 nm, 1550 nm

1480 nm, 1530 nm

Insertion loss at

λ

4.95 dB

N.R.

8.2 dB

6.5 dB

Insertion loss at

λ

2.96 dB

N.R.

7.1 dB

5.8 dB

Insertion loss imbalance

2.04 dB

∼

14 dB

1.1 dB

1.3 dB

λ

Splitting ratio (

→

→

)

9.16 dB

17 dB

8.4 dB

N.R.

λ

Splitting ratio (

→

→

)

8.18 dB

12 dB

24 dB

N.R.

Incident angle

∘

N.R.

Total device size

12 × 13

μ

μ

m × N.R.

25 × 25

μ

9×9

μ

Minimum feature size

160 nm

61.1 nm

236 nm

360 nm

Foundry fabricated/no E-beam lithography

Yes

N.R.: data not reported in the corresponding publication.

With respect to the normal angle.

Approximated from

fi

gures.

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Additionally, like Xu et al.

and Piggott et al.

,theproposed

device functions under a direct vertical input incidence which

greatly simpli

fi

es the packaging and alignment requirements in a

practical setting. And,

fi

nally, although the design presented by

Streshinsky et al.

enjoys a high splitting ratio at 1550 nm, this

comes at the expense of non-vertical

fi

ber incidence (23

∘

inci-

dence angle is used in that reference), high insertion loss, and

very large device size.

In all, we suggest that both, the proposed GCWD itself and the

underlying rapid boundary integral equation method used in its

design, have signi

fi

cant potential for improving the performance

and reducing the size of next generation silicon photonic systems.

Methods

Device simulation and optimization

. The simulation and inverse design approach

used in this paper for modeling and optimization of the grating coupler wavelength

demultiplexer is described in detail in Sideris et al.

. Brie

fl

y, the 2D simulation and

optimization proceeds by means of a custom fast boundary integral solver which,

based on the use of the Maxwell Green function and like the method of moments,

does not rely on volumetric discretization, and it results in highly ef

fi

cient solution

of the Maxwell equations. The device optimization, in turn, utilizes the L-BFGS

quasi-Newton gradient-descent algorithm

together with the integral-equation

based version of the adjoint method for gradient computation. Both 2D and 3D

FDTD simulations were additionally used to independently validate the coupling

fi

ciency vs, wavelength at each of the two outputs of the GCWD and to con

fi

the correctness of the 2D optimization results.

Device fabrication

.The

fi

nalized GCWD design was laid out using the freely

available K-Layout CAD software

using the Process Design Kit (PDK) provided by

*

STAR AMF

. The design was exported in the GDSII

fi

le format and sent to the

foundry electronically for processing and fabrication. The chip was fabricated by AMF

using their standard 193 nm UV lithography multi-project-wafer (MPW) process.

Device measurement

. The system was measured and characterized using IFA-600

active alignment system with Keysight 81657A DFB laser module in O-band

(1312.1 nm) and C-Band (1546.9 nm). Input polarization was adjusted using an

external

fi

ber polarization controller. The two output photodiode currents mea-

sured from the integrated Germanium detectors were used in an automated loop to

align the illumination

fi

ber and characterize the alignment tolerance. An alignment

tolerance of the system better than 0.1 dB was obtained.

Data availability

The design parameters that characterize the proposed device as well as an alternate

device requiring smaller minimum feature sizes, are included in the Supplementary

Notes 2 and 4. The data used to produce the

fi

gures can be obtained upon reasonable

request from the corresponding author.

Code availability

The computer codes used for simulation and design are available from the corresponding

author upon reasonable request.

Received: 5 May 2021; Accepted: 21 February 2022;

Fig. 6 Comparison of simulation and measured results for 0

∘

(vertical) incident angle on the left (L) and right (R) ports.

Measured absolute coupling

fi

ciency at

Port 1 (1310 nm) and

Port 2 (1500 nm), at the cross-section

=

μ

m as a function of the

fi

ber position in the

(propagation) direction,

extracted from Fig.

and plotted alongside simulation results.

Fig. 7 Coupling ef

fi

ciencies at the two ports vs.

fi

ber position and orientation.

Simulated GCWD (grating coupler wavelength demultiplexer) coupling

fi

ciencies at the two ports as functions of the input

fi

ber position and incidence angle.

Port 1 (1310 nm) and

Port 2 (1500 nm).

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7

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Acknowledgements

O.P.B. gratefully acknowledges suppo

rt from NSF under contracts DMS-1714169

and DMS-2109831, from AFOSR under contract FA9550-21-1-0373, from

DARPA under contract HR00111720035, and f

rom the NSSEFF Vannevar Bush Fel-

lowship under contract number N00014-16-1-2808. C.S. gratefully acknowledges

support by the National Science Foundati

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2047433, and AFOSR under contract FA9550-20-1-0087. The authors gratefully

acknowledge

fi

nancial support from Caltech RI2 Program under contract CIT-

2021RI2-1.

Author contributions

C.S. and O.P.B. developed the inverse design framework used, and utilized it to obtain

the proposed design. C.S., A.K., A.D.W., and S.A.H. developed the testing methodology

including incorporating additional test structures for decoupling the losses of the device

itself from those of the measurement setup. A.K., A.D.W., and C.S. characterized the

fabricated design experimentally. C.S., A.K., and O.P.B. analyzed the measurement

results and compared them against numerical simulations. All authors contributed to the

preparation and review of the manuscript.

Competing interests

The authors declare no competing interests.

Additional information

Supplementary information

The online version contains supplementary material

available at

https://doi.org/10.1038/s42005-022-00839-w

Correspondence

and requests for materials should be addressed to Constantine Sideris.

Peer review information

Communications Physics

thanks Ke Xu, Daniele Melati, and the

other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer

reviewer reports are available.

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