Femtojoule femtosecond all-optical
switching in lithium niobate nanophotonics
In the format provided by the
authors and unedited
Supplementary information
https://doi.org/10.1038/s41566-022-01044-5
Supplementary Information for “Femtojoule, femtosecond all-optical switching in
lithium niobate nanophotonics”
Qiushi Guo
1
,
∗
, Ryoto Sekine
1
,
∗
, Luis Ledezma
1
,
2
,
∗
, Rajveer Nehra
1
, Devin J.
Dean
3
, Arkadev Roy
1
, Robert M. Gray
1
, Saman Jahani
1
and Alireza Marandi
1
,
†
1
Department of Electrical Engineering, California Institute of Technology, Pasadena, California 91125, USA.
2
Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California 91109, USA.
3
Department of Applied Physics, Cornell University, Ithaca, New York 14850, USA.
∗
These authors contributed equally to this work.
†
Email: marandi@caltech.edu
CONTENTS
I. Comparison with nonlinear directional coupler presented in Ref.[1]
2
II. Design of the adiabatically tapered directional coupler
4
III. Characterization of input pulses
6
IV. Single envelope simulation
7
V. Deterministically finding the quasi-phase-matching condition
8
VI. Calibration of the input/output coupling loss
9
VII. Device fabrication variation and device operating wavelength tuning
10
VIII. Nonlinear dynamics
11
A. Nonlinear dynamics in the SHG region
11
B. Nonlinear dynamics in the entire device
11
C. Operating regime for the all-optical switching
13
IX. Performance comparison of on-chip all-optical switches
15
X. Toward higher switching contrast
16
XI. Dispersion engineering at telecommunication wavelengths
17
XII. Applications of femotosecond, femotojoule all-optical switch
18
A. Terabit/s (Tbps) information processing
18
B. On-chip generation of ultrashort light pulses
20
References
22
2
I. COMPARISON WITH NONLINEAR DIRECTIONAL COUPLER PRESENTED IN REF.[1]
Here, we want to emphasize that both the design and operating principle of our all-optical switch are fundamen-
tally different from those reported by Schiek et al. [1]. We also explain why our device operating principle leads to
significant improvement in the device performance compared this work.
FIG. S1. Comparison of device design and operating principle
As shown in Fig. S1, the nonlinear directional coupler (NDC) device presented by R. Schiek et al. has two
neighboring PPLN waveguide with a very long coupling region. Because the coupling region is long, the input FH can
be coupled back and forth between the two neighboring waveguides. Both of the PPLN waveguides operate in large
phase-mismatched (
∆
k
∼
16
π
) second harmonic generation (or effective
χ
(3)
) regime. (
∆
k
= 1
.
8
π
×
(219
−
T/
◦
C))
and the device is operating at 210
◦
C
(see page 2 of Ref.[1]). The phase-mismatched SHG produces an effective
χ
(3)
nonlinearity and self-phase modulation on the input FH pulse. The intensity-dependent self-phase modulation of the
FH further perturbs the evanescent coupling ratio of the FH pulse, thus leading to the switching behavior.
In sharp contrast, in our device as shown in Fig. S1, we employed phase-matched SHG and DOPA processes
separated by the “poling defect” in the waveguide. In the first section of our device, energy flows monotonically from
the input pulse towards its second harmonic, while on the second section of our device, energy flows monotonically
from the second harmonic towards the fundamental. Therefore, there is no self-phase modulation of the FH pulse
in our device. The wavelength selective directional coupler still operates as a linear splitter over its length (i.e. we
do not perturb its operation due to the propagation of the modes). Instead, the switching behavior of our device is
enabled by the phase-matched frequency conversion and frequency dependent output coupling. Here, we explain why
our device operating principle leads to significantly better performance.
Power efficiency:
the fundamental difference in operating principle described above leads to dramatic perfor-
mance difference between our device and the NDC by Schiek et al. Compared to phase-matched
χ
(2)
processes
presented in our work, the all-optical switch based phase-mismatched
χ
(3)
nonlinearities requires much higher input
power. We want to emphasize that even if we implement the NLDC approach in nanophotonic LN platform with the
same device length and waveguide geometry as our case, the device performance will still be much inferior to our device.
To ensure a fair comparison, we compare the peak power requirements if both designs are implemented on the
same nanophotonic PPLN platform and operate at 2090 nm. The NDC demonstrated by Schiek et al. requires more
than 50 W of peak power in 50-mm-long diffused waveguides at 1550 nm. In previous work, Schiek et al. report
a normalized nonlinear efficiency (
η
) of 60
%
/W/cm
2
for their waveguides at 1550 nm [2], which is typical of those
large-mode waveguides. To estimate the required power of their design in our PPLN platform and at our wavelength,
we first consider the wavelength difference. Since
η
scales as
ω
4
[3], at 2090 nm
η
will be reduced to 18
%
/W/cm
2
for
the Ti-diffused waveguides, while our nanophotonic platform provides
η
∼
1100
%/W/cm
2
at these wavelengths[4].
Therefore, implementing such a NDC in LN nanophotonic platform would benefit from a 61-fold larger normalized
3
nonlinear efficiency at the same wavelength.
Second, we consider the length difference. The NDC mechanism is based on self-phase-modulation due to phase-
mismatched (2) nonlinearities. In the limit of large phase mismatch, with the same amount of
∆
kL
, the phase
shift scales linearly with normalized nonlinear efficiency and power, and quadratically with the device length, i.e.
Φ
SPM
∝
ηPL
2
(Equation 6 in Ref.[5]). By implementing NDC on our nanophotonic PPLN platform with the same
length as our device (
∼
6 mm),
L
2
is smaller by a factor of 69. Therefore, to realize the same amount of phase shift
with the same amount of
∆
kL
, the NDC design demonstrated by Schiek et al. still requires a peak power level of
above 56 W if it is implemented in our nanophotonic PPLN platform. In our device, however, we have experimentally
demonstrated all-optical switching with only
∼
1
.
5
W of peak power (80 fJ of 46 fs pulse) in a 6-mm-long waveguide.
This illustrates that just the operating principle of our device leads to more than an order of magnitude improvement
in the power requirement compared to the NDCs design. The fundamental reason behind this is that the phase-
matched interactions are more efficient than the phase-mismatched interactions employed in NDCs.
FIG. S2. Comparison of input pulse envelop (grey) and the nonlinear response (blue) for phase-matched
χ
(2)
process in
dispersion-engineered LN waveguides (the operation principle of our design).
Bandwidth:
Another important disadvantage of using phase-mismatched
χ
(2)
nonlinearity is the attainable band-
widths. As highlighted in Page 3 of Ref.[1]: “. . . the wavelength dependence of the phase-mismatch yields different
effective nonlinearities for the different spectral components which introduce a kind of nonlinear dispersion.” In other
words, to enable large bandwidths (and femtosecond switching), the nonlinear effect should have a flat response
over the entire pulse bandwidth. In our work, we achieve this through dispersion engineering and operating in the
phase-matched case with low group velocity mismatch between the pump and the second harmonic. As shown in Fig.
S2, we can achieve a flat phase-matching spectrum over the pump bandwidth even for a 46-fs input pulses and hence
all-optical switching with minimal distortion.
On the other hand, Fig. S3 shows the phase matching plot for the Ti-diffused waveguide device described in Ref.[1],
which operates far from phase matching in order to obtain self-phase modulation of the input pulses. Even though
the pulse is 9-ps long and the entire pulse bandwidth is far from the phase matching peak, there is still considerable
dispersion, with different parts of the pulse spectrum experiencing different effective
χ
(3)
values. This leads to the
pulse distortions and asymmetries described in Ref.[1] even for 9-ps pulses.
These fundamental spectral dependences in phase-mismatched effective
χ
(3)
processes indicate that it may not be
even possible to achieve distortion-free all-optical switching with femtosecond pulses using the NDC design. By using
the same mismatch (
∆
kL
= 16
π
) as Ref.[1], we compare in Fig. S4 the cases of a 9-ps pulse and a 46-fs pulse in
dispersion-engineered nanophotonic waveguides. In this scenario, the pulse spectrum is closer to the phase matching
peak, so the effective
χ
(3)
function is even more dispersive than the previously reported experiment (indicating that
even a 9-ps pulse would undergo more severe distortions for an NDC implemented in our nanophotonic platform
compared to the old platform). This results in even more distortion for 46-fs pulses, as expected. The simple solution
is to operate even farther away from phase-matching, which will decrease the effective
χ
(3)
coefficient. Therefore, even
in the presence of dispersion engineering, it is not clear (and likely impossible) how to achieve femtosecond, femtojoule
performance with the NDC design implemented in state-of-the-art nonlinear photonic platforms.
4
FIG. S3. Comparison of input pulse envelop (grey) and the nonlinear response (blue) for phase-mismatched (2) process leading
to effective (3) (the operation principle of NDC).
FIG. S4. Comparison of input pulse envelop (grey) and the nonlinear response (blue) for phase-mismatched
χ
(2)
process in
dispersion engineered nanophotonic LN waveguides.
II. DESIGN OF THE ADIABATICALLY TAPERED DIRECTIONAL COUPLER
During the device fabrication process, the waveguide width, height, and coupling gap can vary. As a result, the
effective index
n
eff
of the waveguide, as well as the coupling strength between the waveguides, will change. The
coupling efficiency of the conventional directional coupler, which has neighboring waveguides of identical size, usually
suffers from poor tolerance to fabrication errors and can hardly be used as broadband component[6]. Here we adopt
an adiabatically tapered directional coupler design[6], which ensures broadband operation and good tolerance to
fabrication errors. Figure S5 shows the design of the directional coupler, which is composed of a tapered top waveguide
(linearly tapered from
W
1
to
W
2
and a non-tapered bottom waveguide with a width of
W
0
. The dimensions, including
W
1
,
W
2
,
W
0
, the coupling length
L
and the coupling gap are labeled in the figure. The gap is fixed throughout
the tapered region. Note that the etched LN waveguide has
∼
60
◦
slant angle, the geometry shown in Figure S5
corresponds to the top surface of the etched ridge waveguide. For an adiabatic coupler shown in Figure S5, the power
coupling efficiency
ζ
can be estimated by the Landau-Zener formula[6, 7]:
ζ
= 1
−
exp
−
2
πg
2
∂n
eff
/∂z
2
π
λ
(1)
where the coupling strength
g
equals to the half of the
n
eff
index difference between the even mode and the odd
mode at the center of coupler, the
∂n
eff
/∂z
is the changing rate of the
n
eff
when varying the waveguide width along
the propagation direction z, and
λ
is the wavelength. For very small
g
, light can hardly be coupled to neighboring
waveguide since
ζ
≈
0
, while for large
g
the coupling efficiency
ζ
≈
1
.
Based on Eq. 1, in Fig. S6 we plot the coupling efficiency
ζ
of the adiabatically tapered directional coupler with the
design parameters shown in Fig. S5. When the coupling length is 70
μ
m, the coupling efficiency for the fundamental
5
L=70 μm
W
0
=1.65 μm
W
1
=1.55 μm
gap=0.65 μm
W
1
=1.75 μm
FIG. S5. Design of the adiabatically tapered directional coupler. The directional coupler is composed two neighboring waveg-
uides. The top width of the top waveguide is linearly tapered from
W
1
= 1
.
55
μ
m to
W
2
= 1
.
75
μ
m, whereas the bottom
waveguide is not tapered, with a constant top width of
W
0
= 1
.
65
μ
m. The coupling length
L
is 70
μ
m.
0
50
100
150
200
250
0
20
40
60
80
100
Coupling ratio (%)
Coupling Length (
m
m)
2090 nm
1045 nm
85%
5%
FIG. S6. Coupling efficiency as a function of coupling length for 2090 nm and 1045 nm light. When the coupling length is is
70
μ
m, the coupling efficiency for 2090 nm and 1045 nm are 85
%
and 5
%
, respectively.
TE mode at 2090 nm and the fundamental TE mode at 1045 nm are 85
%
and 5
%
, respectively. By performing a
frequency dependent analysis (Fig. 2c in the main text), we have verified that the coupler is broadband around 2090
nm, and has a low coupling ratio for light at 1000 nm.
6
III. CHARACTERIZATION OF INPUT PULSES
The quantitatively analysis of our switching device including the nonlinear dynamics, the input/output coupling
loss, the switching time and energy necessitates an accurate measurement of the input pulses. In Fig. S7a, we plot
the measured spectrum of input pulses (red solid line). By comparing it with the spectra of 30-fs, 35-fs and 40-fs
pulses centered at 2.09
μ
m, we found that the 35-fs pulse has the best agreement with our experimental spectrum.
We also performed the interferometric autocorrelation measurement of the input pulses, as shown in Fig. S7b. The
Gaussian fitting of the peaks of the autocorrelation has a FWHM of 65.2 fs, indicating that the actual pulse length
is close to 46.2 fs. The slightly longer pulse length obtained from the autocorrelation measurement indicates that the
input pulse is chirped, presumably due to the dispersive elements in our setup such as the pellicle, the long-pass filter
and the neutral density (ND) filter. The relation between temporal profile of the pulse before (
a
(
t
)
) and after (
a
′′
(
t
)
)
the dispersive element is given by[8]
a
′′
(
t
)
≈
1 +
j
β
2
L
2
d
2
dt
2
a
(
t
)
(2)
where
L
is length of the dispersive medium and
β
2
is the group velocity dispersion (GVD) of the dispersive medium.
Based on the results in Fig. S1a and b, we can estimate a total group dispersion delay (GDD) of
β
2
L
=
±
362
fs
2
.
We determine the sign of GDD in section V.
-300
-200
-100
0
100
200
300
0
1
2
3
4
5
6
7
8
9
10
Inteferometric autocorrelation
Gaussian fitting (FWHM=65.2 ± 1.81 fs)
Delay (fs)
Second harmonic Intensity (a.u.)
1600
1800
2000
2200
2400
-80
-60
-40
Power (dBm)
Wavelength (nm)
35 fs sech
2
30 fs sech
2
40 fs sech
2
Measured spectrum
a
b
FIG. S7.
a
, Measured spectrum of input 2.09
μ
m pulses (red) compared with the spectra of 30- (green), 35- (blue) and
40-fs (orange) pulses.
b
, autocorrelation of the input pulses. The Gaussian fit (black) of the envelop of the interferometric
autocorrelation has a FWHM of 65.2 fs.
7
IV. SINGLE ENVELOPE SIMULATION
We used a method similar to that described in [9] to simulate quadratic interactions over a large bandwidth using
a single envelope in the frequency domain. We write a spectral component of the electric field propagating in the
z
-direction on a single waveguide mode as:
E
(
x,y,ω
) =
A
(
z,
Ω)
e
(
x,y,ω
)
e
−
i
(
β
0
−
ω
0
/v
ref
)
z
,
(3)
where
ω
and
Ω =
ω
−
ω
0
are the optical and envelope angular frequencies,
ω
0
is the simulation center frequency,
β
0
is the waveguide propagation constant at
ω
0
,
v
ref
is the simulation reference frame velocity,
x,y
are the transversal
waveguide coordinates,
e
(
x,y,ω
)
is the mode transversal field distribution, and
A
(
z,ω
)
is the complex amplitude of
the field that evolves during propagation. Note that
A
(
z,ω
)
is a rapidly-varying envelope, i.e. it includes the phase
factor
e
−
iβ
(
ω
)
z
acquired during linear propagation. Furthermore,
A
(
z,ω
)
is an analytic signal, i.e., it only contains
positive frequencies (
A
(
z,ω <
0) = 0
).
We obtained an equation of motion for
A
(
z,
Ω)
by ignoring counter-propagating terms (which are usually phase
mismatched), and assuming a constant nonlinear coefficient and mode overlap integral, both of which are weak
functions of frequency away from any material resonances. No limitations are placed upon the maximum spectral
bandwidth of the simulation. The resulting propagation equation is,
∂A
∂z
=
−
i
β
(
ω
)
−
β
0
−
Ω
v
ref
−
i
α
2
A
−
iωε
0
X
0
8
d
(
z
)
F
Ω
n
a
2
(
z,t
)
e
jφ
(
z,t
)
+ 2
a
(
z,t
)
a
∗
(
z,t
)
e
−
jφ
(
z,t
)
o
,
(4)
where
d
(
z
) =
±
1
is the sign of the quadratic nonlinear coefficient that is modulated in quasi-phase matching,
a
(
z,t
)
is the time domain representation of
A
(
z,
Ω)
,
φ
(
z,t
) =
ω
0
t
−
(
β
0
−
ω
0
/v
ref
)
z
,
F
Ω
is the Fourier transform in the
Ω
variable. The effective nonlinear coefficient
X
0
is defined as:
X
0
=
X
ijk
χ
(2)
ijk
Z
e
∗
i
(
ω
1
)
e
j
(
ω
2
)
e
k
(
ω
1
−
ω
2
) d
S ,
(5)
where
χ
(2)
ijk
is the quadratic nonlinear susceptibility tensor,
j,k,l
are Cartesian components of the corresponding
vectors, and
ω
1
and
ω
2
are two suitable chosen frequencies, e.g., the signal and pump frequencies in our case.
The time domain terms inside the Fourier transform of Eq. (4) represent the processes of sum frequency generation