1
Large-Scale Crosstalk-Corrected Thermo-Optic
Phase Shifter Arrays in Silicon Photonics
B. Volkan Gurses,
Student Member, IEEE,
Reza Fatemi,
Member, IEEE,
Aroutin Khachaturian,
Member, IEEE,
and Ali Hajimiri,
Fellow, IEEE
Abstract
—We introduce a thermo-optic phase shifter (TOPS)
array architecture with independent phase control of each
phase shifter for large-scale and high-density photonic integrated
circuits with two different control schemes: pulse amplitude mod-
ulation (PAM) and pulse width modulation (PWM). We realize
a compact spiral TOPS and a 288-element high-density row-
column TOPS array with this architecture and drive TOPS with
waveforms of both control schemes and of different array sizes.
We present a thermal excitation model and a finite difference
method-based simulation to simulate large-scale TOPS arrays
and compare both schemes experimentally and theoretically. We
also analyze the effects of thermal crosstalk in the realized TOPS
array and implement a thermal crosstalk correction algorithm
with the developed model. The high-density TOPS array architec-
ture and the thermal crosstalk correction algorithm pave the way
for high-density TOPS arrays with independent phase control in
large-scale photonic integrated circuits interfaced with electronics
limited in voltage swing and bandwidth.
Index Terms
—optical phase shifters, thermo-optic effects, time
division multiplexing, crosstalk, large-scale circuits, integrated
optoelectronics, silicon
I. I
NTRODUCTION
I
NTEGRATED photonics allows numerous bulk optic com-
ponents such as lenses, modulators, and fiber-optics to
be integrated on a thin substrate to realize millimeter-scale,
energy-efficient optical systems with better manufacturability
and cost [1]. The advent of large-scale photonic integrated
circuits (PICs) largely driven by silicon photonics enabled
high-performing systems driving practical applications in sens-
ing, communications, and computing [2].
Silicon photonics leverage the high refractive index of
silicon to enable the realization of optical waveguides with
tight mode confinement. This allows compact nanophotonic
structures to be implemented on chip at a high volume. Com-
bined with its CMOS compatibility, silicon photonics enable
the largest scale PICs with tens of thousands of components
realized on a single chip [3]. Consequently, many large-scale
systems such as optical phased arrays [4]–[6], programmable
photonic circuits [7], optical neural networks [8], and quantum
photonic processors [9] have been implemented on silicon
photonics platform.
B. Volkan Gurses, Reza Fatemi, Aroutin Khachaturian and Ali Hajimiri
are with the Department of Electrical Engineering, California Institute of
Technology, Pasadena, CA,91125 USA e-mail: gurses@caltech.edu .
© 2022 IEEE. Personal use of this material is permitted. Permission from
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PICs ubiquitously utilize phase shifters for modulation,
tuning, and correction of random phases along the signal path
[10]. As the circuits scale, more phase shifters are needed for
signal manipulation. Thermo-optic phase shifters (TOPS) are a
good candidate for high-density silicon PICs because of their
low optical loss and small form factor [11], [12]. However,
high-density TOPS arrays suffer from thermal crosstalk [13].
In high-density integration of TOPS, unless photonics and
electronics are both monolithically integrated, electrical con-
nections required to drive each phase shifter in a direct ad-
dressing scheme can pose a problem for scaling. Furthermore,
if each phase shifter requires its own driver, the complexity
of the control electronics and their power consumption scale
rapidly [14], becoming a limiting factor for scalability.
To overcome these challenges, row-column or matrix ad-
dressing schemes have been implemented [4]–[6], [15]. In
these schemes, the number of electrical connections scales as
N
with
N
2
phase shifters. This reduces the complexity of the
electronic interface as opposed to a direct addressing scheme
where each phase shifter has a separate connection and where
the number of electrical connections scales as
N
. In the row-
column architecture, each phase shifter is connected between a
column and row node in series with a diode. The diode serves
as a switch for the current to flow between a column and
row node and modulate the phase shifter. In such a scheme,
columns determine which diodes will be in forward bias, while
rows determine the amplitude of the electrical signal. Electrical
signals used to modulate the TOPS are time multiplexed across
multiple columns, by which each TOPS receives an average
electrical power proportional to the duty cycle as shown in
Fig. 1. The average electrical power received by each TOPS
can then be controlled by either pulse-amplitude modulation
(PAM) or pulse-width modulation (PWM) of the row voltages.
In this paper, we realize a compact TOPS design and a large-
scale 288-element row-column TOPS array with independent
phase control using a folded row-column architecture [4] along
with integrated Mach-Zehnder interferometers (MZIs) and
row-column photodiodes (PDs) to probe the TOPS phase shifts
and calibrate the array. Driving a TOPS+MZI test structure
with PAM and PWM waveforms of different array sizes, we
compare PAM and PWM driving schemes. We introduce a
thermal excitation model accompanied with a finite difference
method-based simulation to predict the experimental results
and analyze both schemes. With the row-column array, we
also showcase the effects of thermal crosstalk between phase
shifters and implement a thermal crosstalk correction algo-
rithm to completely cancel the crosstalk in the array.
arXiv:2206.04525v1 [physics.optics] 5 Jun 2022
2
Fig. 1: Temporal multiplexing of TOPS arrays showing the
AC averaging for direct modulation and thermal crosstalk.
II. T
HERMO
-O
PTIC
P
HASE
S
HIFTER
TOPS utilizes the waveguide refractive index dependence
on temperature. Phase shift,
Φ
, due to refractive index,
n
, is
Φ = 2
πL
n
λ
0
(1)
where
λ
0
is the wavelength in free space and
L
is the length
of the heated waveguide. Then, phase shift in the TOPS is
∆Φ =
2
πL
λ
0
∆
n
=
2
πL
λ
0
γ
∆
T
(2)
where
∆
T
is the temperature difference, and
γ
is the temper-
ature coefficient of refractive index. Then,
∆
T
π
, temperature
difference for
π
phase shift is
∆
T
π
=
λ
0
2
Lγ
(3)
A single TOPS can be modeled as a simple RC circuit, with
P
π
, power required for
π
phase shift, defined as
P
π
=
Ak
∆
T
π
=
Ak
λ
0
2
Lγ
(4)
where
A
is the effective surface area and
k
is the effective
thermal conductivity between the waveguide and the heat sink.
In the RC model,
G
= 1
/R
=
Ak
is the thermal conductance
of TOPS to the heat sink.
Another parameter to characterize a TOPS is its time
constant,
τ
, similarly defined with the first-order RC approx-
imation as
τ
=
C
Ak
(5)
where
C
is the heat capacity of TOPS.
A. Thermal Excitation Model
We now develop a time-varying lumped model starting with
a 3D analysis of TOPS seen in Fig. 2, which shows the cross-
section of the realized TOPS design and array. Without loss of
generality, we consider a heater on one side of the waveguide
in an XYZ space. Here, we analyze the dynamics between four
Fig. 2: Cross-section of TOPS array showing the interactions
between the four main structures.
structures: heater, waveguide, substrate, and the environment
acting as a perfect heat sink. We now assume uniform material
for all structures, uniform temperature distribution within each
structure and in X and Z directions. We define a time-
varying temperature function,
T
(
y,t
)
, that characterizes the
local instantaneous temperature. When some instantaneous
electrical power,
P
(
t
)
, is applied to the resistive heater, this
power is expended through four mechanisms (ignoring power
dissipated in forms other than Joule heating):
1) Heat used to increase the heater temperature (
q
H
=
ρ
H
c
H
dT
(
y,t
)
dV
) where
ρ
H
,
c
H
and
V
are the heater
mass density, specific heat capacity and volume.)
2) Heat exchange with the waveguide (
q
W
=
−
k
WH
dtd
~
A
·
∇
T
(
y,t
)
where
k
WH
is the effective thermal conduc-
tivity between the waveguide and the heater,
d
~
A
is
the differential surface area vector, and
∇
T
(
y,t
)
is the
instantaneous temperature gradient.)
3) Heat exchange with the substrate (
q
S
=
−
k
SH
dtd
~
A
·
∇
T
(
y,t
)
where
k
SH
is the effective thermal conductiv-
ity between the substrate and the heater.)
4) Heat exchange with surrounding environment other than
the waveguide and substrate (
q
E
=
−
k
EH
dtd
~
A
·
∇
T
(
y,t
)
where
k
EH
is the effective thermal conduc-
tivity between the environment and the heater.)
1) No Crosstalk:
First, we neglect the crosstalk between
phase shifters. Then,
dP
(
t
)
dt
=
q
H
+
q
W
+
q
S
+
q
E
. Going
through the derivation in the Supplementary Material, we have
the matrix ODE for the lumped model of the system.
d
dt
T
H
T
W
T
S
=
−
D
H
C
H
G
WH
C
H
G
SH
C
H
G
WH
C
W
−
D
W
C
W
G
SW
C
W
G
SH
C
S
G
SW
C
S
−
D
S
C
S
T
H
T
W
T
S
+
P
+
G
EH
T
E
C
H
G
EW
T
E
C
W
G
ES
T
E
C
S
(6)
where we defined
G
ij
= 1
/R
ij
=
A
ij
k
ij
. Since we assumed
uniform temperature distribution within each structure,
k
ij
re-
duces to effective thermal contact conductivity between the re-
spective structures. Furthermore,
D
H
=
G
WH
+
G
SH
+
G
EH
,
D
W
=
G
WH
+
G
SW
+
G
EW
, and
D
S
=
G
SH
+
G
SW
+
G
ES
are the sum of thermal conductances for the matrix diagonal