Supplemental Document
Intense optical parametric amplification in
dispersion-engineered nanophotonic lithium
niobate waveguides: supplement
L
UIS
L
EDEZMA
,
1,2,†
R
YOTO
S
EKINE
,
1,†
Q
IUSHI
G
UO
,
1,†
R
AJVEER
N
EHRA
,
1
S
AMAN
J
AHANI
,
1
AND
A
LIREZA
M
ARANDI
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
†
These authors contributed equally to this work.
∗
Corresponding author:
marandi@caltech.edu
This supplement published with Optica Publishing Group on 14 March 2022 by The Authors
under the terms of the Creative Commons Attribution 4.0 License in the format provided by the
authors and unedited. Further distribution of this work must maintain attribution to the author(s)
and the published article’s title, journal citation, and DOI.
Supplement DOI: https://doi.org/10.6084/m9.figshare.19274822
Parent Article DOI: https://doi.org/10.1364/OPTICA.442332
Intense optical parametric
amplification in
dispersion-engineered nanophotonic
lithium niobate waveguides:
supplemental document
On this supplementary document, we include background theory on CW parametric amplification
in waveguides as well as details on the nonlinear single envelope simulations for OPA and OPG.
We also include additional fabrication and characterization details and a companion table for Fig.
4 of the main text.
1. CW PARAMETRIC AMPLIFICATION THEORY
We briefly review here the CW theory of degenerate parametric amplification, in which a strong
pump field at frequency
ω
p
interacts with a signal field at frequency
ω
s
=
ω
p
/
2. In the limit of
no loss, the coupled wave equations are
∂
A
p
∂
z
=
−
i
κ
A
2
s
e
i
∆
kz
,
(S1)
∂
A
s
∂
z
=
−
i
κ
A
∗
s
A
p
e
−
i
∆
kz
,
(S2)
where
A
p
and
A
s
are the pump and signal complex envelopes, normalized such that
|
A
i
|
2
is the
power carried by the pulses,
∆
k
=
β
p
−
2
β
s
−
2
π
/
Λ
is the phase mismatch, and
κ
is the nonlinear
coupling coefficient, which is related to the second harmonic normalized efficiency
η
by
η
=
κ
2
=
2
ω
2
s
d
2
eff
n
2
ω
s
n
ω
p
e
0
c
3
A
eff
.
(S3)
Here,
d
eff
= (
2
/
π
)
d
33
for periodic quasi-phase matching with 50% duty cycle,
d
33
≈
20.5
pm/V
for a pump centered at 1045 nm, and
A
eff
is the effective area of the nonlinear interaction, given
by [1],
A
eff
=
(
∫
(
e
ω
s
×
h
∗
ω
s
)
·
dS
)
2
∫
(
e
ω
p
×
h
∗
ω
p
)
·
dS
∣
∣
∣
∫
∑
i
,
j
,
k
̄
d
ijk
e
∗
i
,
ω
p
e
j
,
ω
s
e
k
,
ω
s
dxdy
∣
∣
∣
2
,
(S4)
where
e
μ
are dimensionless transverse mode profiles scaled such that the peak value of
(
e
μ
×
h
∗
μ
)
·
z
is unity, and
̄
d
ijk
is the
χ
(
2
)
tensor normalized with respect to
d
33
. Note that a large overlap
integral (denominator of
A
eff
) leads to a small effective area and a stronger nonlinear interaction
(large
η
).
Assuming no pump depletion, the signal equation of motion reduces to
∂
A
s
∂
z
=
−
i
κ
A
p
A
∗
s
e
−
i
∆
kz
=
γ
A
∗
s
e
−
i
∆
kz
,
(S5)
where
γ
≡ −
i
κ
A
p
can be made real by an appropriate definition of the pump absolute phase.
This equation can be solved by separating the signal into its real and imaginary parts (i.e. its
quadratures),
A
s
=
A
1
+
iA
2
, yielding for the real part:
A
1
(
z
) =
A
1
(
0
)
[
cosh
(
gz
)
+
γ
g
sinh
(
gz
)
]
exp
(
i
∆
kz
/2
)
+
A
2
(
0
)
∆
k
2
g
sinh
(
gz
)
exp
(
i
∆
kz
/2
)
,
(S6)
where we have introduced the parametric gain parameter
g
=
√
|
γ
|
2
−
(
∆
k
/2
)
2
=
√
η
P
p
−
(
∆
k
/2
)
2
.
In the large gain regime,
η
P
p
(
∆
k
/
2
)
2
,
γ
/
g
≈
1 and
∆
k
/
2
g
≈
0, so the power of the real
quadrature grows as
|
A
1
(
z
)
|
2
=
G
|
A
1
(
0
)
|
2
=
|
A
1
(
0
)
|
2
exp
(
2
gz
)
≈
|
A
1
(
0
)
|
2
exp
(
2
√
η
P
p
z
)
,
(S7)
with a power gain
G
≈
exp
(
2
√
η
P
p
z
)
.
For the case of non-degenerate operation, if only the pump and the signal are present at the
input, then an idler will be generated with the right phase to produce signal amplification. In that
case the amplification is not phase-sensitive and the gain is given by:
G
=
γ
2
g
2
sinh
2
(
gz
)
,
(S8)
where
g
and
γ
are the same as in the degenerate case, but the phase mismatch is
∆
k
=
β
p
−
β
s
−
β
i
−
2
π
/
Λ
. This gain expression was used to generate Fig. 3e on the main paper.
2. SINGLE ENVELOPE SIMULATION
We used a method similar to that described in [2] 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
,
(S9)
where
ω
and
Ω
=
ω
−
ω
0
are the optical and envelope angular frequencies,
ω
0
is the simula-
tion 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
−
i
β
(
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
−
i
ωe
0
X
0
8
d
(
z
)
F
Ω
{
a
2
(
z
,
t
)
e
j
φ
(
z
,
t
)
+
2
a
(
z
,
t
)
a
∗
(
z
,
t
)
e
−
j
φ
(
z
,
t
)
}
,
(S10)
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
=
∑
ijk
χ
(
2
)
ijk
∫
e
∗
i
(
ω
1
)
e
j
(
ω
2
)
e
k
(
ω
1
−
ω
2
)
d
S
,
(S11)
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. (S10) represent the processes of
sum frequency generation
(
∝
a
(
t
)
2
)
and difference frequency generation
(
∝
a
(
t
)
a
(
t
)
∗
)
, which
combined can predict all classical second order interactions, such as second harmonic generation
and parametric amplification. Since
A
(
z
)
is fast varying, carrier dynamics can be resolved. In
particular, phase mismatch is automatically included and the term
d
(
z
)
can be used to accurately
simulate different quasi-phase matching gratings. This also means that the spatial domain needs
to be sampled finely enough to resolve these dynamics. We solve the evolution equation (S10)
with the split-step Fourier technique using the fourth-order Runge-Kutta method for the nonlinear
step.
We used a combination of commercial software (Lumerical Inc.) and custom built effective-
index routines to solve for the waveguide modes and generate the dispersion characteristics
shown in Fig. 1c,d of the main text.
2
3. SMALL-SIGNAL OPA SIMULATION
We simulated optical parametric amplification for the dispersion-engineered 2.5-mm-long waveg-
uide described in the main text, including quasi-phase matching through a periodic modulation
of
d
(
z
)
. As the input, we used a 100-fs-long pump pulse centered at 1045 nm, and a 35-fs-long
signal pulse centered at 2090 nm, both with a hyperbolic secant profile. The output power spectral
density (PSD) for three pump power levels, with a fixed input signal level, are shown in Fig. S1a.
For the largest pump pulse energy shown, significant spectral broadening is observed at both,
the pump and signal wavelengths, revealing a strong non-linear regime similar to [
1
]. Figure
S1b shows the gain as a function of the pump energy for three different input signal levels as
well as the CW theory and measured data. Pump depletion and spectral broadening effects make
the gain deviate from the theoretical CW prediction at large pump and signal power levels (to
calculate the gain, the output signal energy is integrated from 1,600 nm to 3,000 nm). At low pump
power levels, the pulsed gain curve follows the CW theory as expected for dispersion-engineered
waveguides according to the argument provided in the main paper. Note that a low signal level
is necessary in order to extract the maximum gain from a strong pump. It is for this reason that to
characterize the maximum gain of our waveguides we performed measurements on the OPG
regime, with no input signal except for vacuum fluctuations.
1000
1200
1400
1600
1800
2000
2200
2400
2600
Wavelength (nm)
100
80
60
40
20
0
PSD (dBm/THz)
Ein = 5 fJ
Epump = 0.1 pJ
Epump = 1 pJ
Epump = 10 pJ
Input Signal
10
2
10
1
10
0
10
1
Pump Energy (pJ)
0
50
100
150
200
Gain (dB/cm)
CW Theory
Ein = 0.5 fJ
Ein = 5 fJ
Ein = 50 fJ
Measured
Fig. S1.
Numerical simulation of small-signal optical parametric amplification. a
, Power
spectral density at the output of a 2.5-mm-long, dispersion-engineered waveguide showing
low distortion parametric amplification for pump energies of 0.1 pJ and 1 pJ, and spectral
broadening for a pump energy of 10 pJ. The 100-fs-long pump is centered at 1045 nm and the
35-fs-long signal is centered at 2090 nm. The input signal energy is fixed at 5 fJ.
b
, Simulated
gain as a function of pump power for three different input signal levels along with the predic-
tion from the CW theory and the measured values (from main paper Fig. 2c). It is evident that
the CW theory is valid for this dispersion-engineered waveguide before gain saturation and
spectral broadening effects start dominating at high pump power levels. The maximum mea-
sured gain was limited by input coupling losses, and improving the input coupling by 10 dB
leads to small-signal gains exceeding 150 dB/cm, putting the on-chip OPA in direct competi-
tion with the largest single mode SOA gains reported.
4. OPG SIMULATION
In the main text we argue that operating the amplifier in the OPG regime (with a strong pump
and no input signal) provides a practical way of extracting the gain of the amplifier avoiding
saturation effects and also revealing its full gain-bandwidth. In this section we support these
claims with semi-classical simulations of parametric generation.
3
Seeding the OPA with an input noise having an energy of half-a-photon per frequency mode,
and a uniformly distributed random phase, has been known to provide the same average signal
output power as the quantum mechanical solution [
3
]. This is equivalent to neglecting thermal
excitations and modeling the remaining vacuum fluctuations as complex Gaussian random
variables with zero mean and a half-a-photon variance [
4
]. Fig. S2a shows the simulated average
output power (integrated from 1,600 nm to 3,000 nm) for the 6-mm-long dispersion-engineered
waveguide, along with the expected curve from the CW theory and our measured data. The
simulation deviates from the theory at pump energies larger than 4 pJ due to efficient parametric
generation (OPG) producing pump depletion (also shown in Fig. S2a). The simulations results
shown in Fig. S2a are the ensemble average of 100 simulations, each simulation producing a
different result due to the stochastic nature of the input signal. This is illustrated in Fig. S2b.,
where the output spectra OPG for a pump energy of 3 pJ is shown for the first 20 simulations
along with the average of 100 simulations.
0
1
2
3
4
5
6
Pump Energy (pJ)
10
2
10
0
10
2
10
4
OPG Energy (fJ)
CW Theory
OPG Simulation (Energy)
Measured data
1500
1750
2000
2250
2500
2750
3000
Wavelength (nm)
50
40
30
20
10
PSD (dBm/THz)
Average of 100 simulations
0
10
20
30
40
50
Pump Depletion (%)
OPG Simulation (Depletion)
a
b
Fig. S2.
Numerical simulation of optical parametric generation. a
, Simulated OPG energy
as a function of pump energy along with the CW theory and measured data. The pump is a
100-fs-long hyperbolic secant pulse center at 1045 nm. The results are the average of 100 sim-
ulations. In each simulation, the input signal is a realization of a complex Gaussian random
variable with zero mean and half-a-photon variance. The simulation results deviate from the
CW theory for pump energy levels above ~4 pJ beyond which efficient parametric generation
occurs and the no-pump-depletion approximation is no longer valid. The simulated pump de-
pletion level (
P
OPG
/
P
pump
) is shown on the right hand axis.
b
, Output power spectral density
for 20 OPG simulations illustrating its stochastic nature, as well as the average for 100 simula-
tions.
5. DEVICE FABRICATION
We used a commercial wafer (NANOLN), with a 700-nm-thick X-cut MgO-doped LN thin-film
on 2-
μ
m-thick SiO
2
. The fabrication process begins with periodically poling the chip. The poling
electrodes (15 nm Cr/55 nm Au) were patterned using e-beam lithography, e-beam evaporation
and metal lift-off. Then ~300 V pulses were applied across the electrodes to produce periodic
domain inversion over a 6-mm length with a period of ~5
μ
m. We visually inspected the poling
quality using second harmonic microscopy (an example image is shown in an inset of Fig. 2a.
The metal electrodes were removed by chemical etching. The waveguides were patterned by
e-beam lithography using hydrogen silsesquioxane (HSQ) as the e-beam resist. The pattern was
transferred to the LN layer by dry etching with Ar
+
plasma. Finally, the waveguide facets were
polished to reduce the coupling losses.
6. DISPERSION ENGINEERING MAPS
After selecting a thin-film thickness of 700 nm, the width and etch depth of the waveguide can be
varied to create maps of GVD at 2
μ
m and GVM between the 1
μ
m pump and 2
μ
m signal. We
can also factor-in the waveguide length and pulse width by the following procedure. The GVM
between the signal and pump defines a walk-off length given by
L
gvm
=
τ
/GVM
, where
τ
is
the pulse width. If the waveguide width is
L
wg
, then we want to minimize the ratio
L
wg
/
L
gvm
.
A map of this ratio, for
L
wg
=
6
mm
and
τ
=
35
fs
, is shown in Fig. S3a, along with the
corresponding contour levels at 0.5 (where the waveguide is half the walk-off length).
4
Similarly, a dispersion length can be defined as
L
gvd
=
τ
2
/GVD
, and the metric would be the
ratio
L
wg
/
L
gvd
. The corresponding map is shown in Fig. S3b, along with the contour level at 0.25
(where the waveguide is only a quarter of the dispersion length). For both contours in Fig. S3, the
black dot corresponds to our waveguide geometry.
1.5
1.6
1.7
1.8
1.9
2.0
Width (um)
300
320
340
360
380
Etch (nm)
0.5
0.5
1.5
1.6
1.7
1.8
1.9
2.0
Width (um)
300
320
340
360
380
Etch (nm)
0.25
0.0
0.3
0.6
0.9
1.2
1.5
1.8
2.1
2.4
2.7
L
wg
/
L
gvm
0.1
0.2
0.3
0.4
L
wg
/
L
gvd
a
b
Fig. S3.
Dispersion engineering mappings. a
, Waveguide length to walk-off length ratio.
b
,
Waveguide length to dispersion length ratio. Both maps are for a 700-nm thick film, 6-mm long
waveguide, and 35-fs pulse width. The black dot indicates our waveguide design.
7. WAVEGUIDE CHARACTERIZATION
For the OPA measurements we use the setup shown in Fig. 2a with both, 1
μ
m and 2
μ
m pulses,
coupled into the waveguide. The 1
μ
m source (the pump) was a 1 W Yb mode-locked laser
that produces nearly transform-limited 75-fs-long pulses at a 250 MHz repetition rate (Menlo
Systems Orange). Part of the pump was sent directly to the chip, while the rest was fed into a
near-synchronously pumped degenerate OPO [
5
] to produce 2
μ
m pulses. The 1
μ
m and 2
μ
m
pulses were combined at a dichroic mirror, and coupled into the waveguides using a reflective
objective (Newport 50102-02). The two pulses were temporally overlapped by adjusting the
optical delay line, and their relative phase was scanned by the piezoelectric transducer on the
delay line. The chip was placed on a thermoelectric cooling stage (TEC), and the temperature was
finely tuned to adjust the quasi-phase matching condition. The output of the chip was collected
with another reflective objective and the remaining pump power was filtered. A 2
μ
m detector
followed by an oscilloscope was used to monitor the entire signal power without and without
the pump beam (Fig. 2b). We used an optical spectrum analyzer (OSA) covering 1200 - 2400 nm
(Yokogawa AQ6375B) with a 2 nm resolution bandwidth (Fig. 2c) to characterize the spectral gain
distribution.
For the OPG measurements of Fig. 3 of the main text, only the 1
μ
m path was used. The output
of a 10-W Yb mode-locked laser (Menlo Systems Orange High Power 10) with 100-fs pulse length
was passed through an optical chopper to reduce thermally induced damage. The average input
power was swept using a variable ND filter and the output was recorded using the same OSA.
The input/output coupling losses were estimated based on a combination of linear and non-
linear measurements as follows. Comparing the optical power before and after the chip gives the
total loss
L
t
=
L
i
+
L
wg
+
L
o
, where
L
i
is the input coupling loss,
L
wg
is the waveguide loss, and
L
o
is the output coupling loss. We estimated the waveguide loss,
L
wg
, to be much less than 1 dB
based on Q-factor measurements in other chips using the same fabrication process. Since the total
loss
L
t
is ~29 dB (at 2
μ
m) we neglect the waveguide loss in what follows.
As explained in Section 3B of the main article, the OPG data can be used to estimate the
gain
G
s
=
exp
(
2
gL
)
without any knowledge of the input/output coefficients. This is because
this gain depends only on the rate of growth of the OPG power, and not on its absolute value.
We also know that the expected number of photons generated during OPG is given by
〈
n
〉
=
sinh
2
(
gL
)
≈
0.25
exp
(
2
gL
)
=
0.25
G
s
. Thus, by estimating
G
s
, we are also estimating the average
number of photons generated, from which the OPG power in the waveguide follows immediately:
P
OPG
=
̄
h
ω
〈
n
〉
f
rep
. Comparing this expected power with the measured power gives us a total
output collection efficiency of 5.85 dB at 2
μ
m. This corresponds to 26 % output coupling
efficiency, which compares well with output coupling losses in similar waveguides estimated by
other methods [6].
Subtracting this output coupling loss from the total throughput loss gives us an input coupling
loss at 2
μ
m of ~23 dB, which is considerably larger than the output coupling loss. This is expected
5
Table S1.
Comparison of on-chip amplifiers gain and bandwidth
Type
Length
Gain
Signal
3-dB Bandwidth (Signal + Idler)
Reference
Absolute
Normalized
Wavelength
Absolute
Relative
χ
(
2
)
6 mm
62 dB
104 dB/cm
2090 nm
380 nm (26 THz)
18.2 %
This Work
χ
(
2
)
12 mm
100 dB
83 dB/cm
2700 nm
10 nm (1.64 THz)
0.4 %
[7]
SOA
1.2 mm
25.5 dB
213 dB/cm
1575 nm
34 nm (4.11 THz)
2.2 %
[8]
SOA
2 mm
13 dB
65 dB/cm
2010 nm
75 nm (5.6 THz)
3.7 %
[9]
SOA
2 mm
25 dB
125 dB/cm
1550 nm
69 nm (8.1 THz)
4.5 %
[10]
FWM
χ
(
3
)
4 mm
25.4 dB
63.5 dB/cm
2170 nm
50 nm (3.18 THz)
2.3 %
[11]
FWM
χ
(
3
)
2 cm
45 dB
22.5 dB/cm
2170 nm
150 nm (9.5 THz)
6.9 %
[12]
FWM
χ
(
3
)
1.7 cm
13.9 dB
8.2 dB/cm
1550 nm
40 nm (5 THz)
2.7 %
[13]
SBS
2.9 cm
5.2 dB
1.8 dB/cm
1550 nm
< 50 MHz
0.1 %
[14]
SRS
4.6 cm
2.3dB dB
0.5 dB/cm
1545 nm
80 GHz
< 0.1 %
[15]
Er
+
doped
3.6 cm
18 dB
5 dB/cm
1530 nm
20 nm (2.56 THz)
1.3 %
[16]
C
χ
(
2
)
2 cm
38.3 dB
19.2 dB/cm
1550 nm
+14 THz
7.2 %
[17]
0.0
2.5
5.0
7.5
10.0
12.5
15.0
17.5
20.0
3-dB Relative Bandwidth (%)
0
50
100
150
200
250
Gain (dB/cm)
% dB/cm
This Work:
Dispersion-Engineered
Quadratic OPA
SOA
FWM
RPE
Er
+
C
(2)
SBS
SRS
250
500
1000
2000
Fig. S4.
Comparison of the gain and bandwidth of quadratic OPA in dispersion-engineered
LN waveguides with other gain mechanisms in integrated photonics.
This is an alternative
representation of the data in Fig. 4 of the main text using relative bandwidth as the horizontal
axis.
since only the power coupled to the fundamental TE mode is considered at the input, while
most of the radiated modes are expected to be collected by the objective at the output. At 1
μ
m,
we assume that the output coupling loss is also 5.85 dB, since we use a low-dispersion metallic
collective objective. The measured throughput loss at 1
μ
m is ~31 dB, so the input coupling loss
was estimated to be ~25 dB. Note that the input coupling loss at 1
μ
m is expected to be larger
than that at 2
μ
m due to the corresponding mode sizes.
8. ON-CHIP AMPLIFIER STATE OF THE ART COMPARISON
Detailed bandwidth and gain numbers used to generate Fig. 4 from the main text and Fig. S3
are available in Table S1 along with the corresponding references. Previous works include only
on-chip traveling-wave amplifiers. We have striven to include the best and most recent results,
but not all publications report enough data to extract 3 dB bandwidth values and couldn’t be
added to the comparison. To estimate the bandwidth of the FWM cases we have added together
the signal and idler bandwidths.
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