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Research Article

Vol. 28, No. 7 / 30 March 2020 / Optics Express

9521

Efficient second harmonic generation in
nanophotonic GaAs-on-insulator waveguides
E RIC J. S TANTON , 1,* J EFF C HILES , 1 N IMA N ADER , 1 G ALAN
M OODY, 1,4 N ICOLAS VOLET, 2 L IN C HANG , 3 J OHN E. B OWERS , 3
S AE W OO N AM , 1 AND R ICHARD P. M IRIN 1
1 Applied Physics Division, National Institute of Standards and Technology, Boulder, CO 80305, USA
2 Department of Engineering, Aarhus University, 8200 Aarhus N, Denmark
3 Department of Electrical and Computer Engineering, University of California, Santa Barbara, CA 93106,

USA
4 Current affiliation: Department of Electrical and Computer Engineering, University of California, Santa
Barbara, CA 93106, USA
* eric.stanton@nist.gov

Abstract: Nonlinear frequency conversion plays a crucial role in advancing the functionality
of next-generation optical systems. Portable metrology references and quantum networks will
demand highly efficient second-order nonlinear devices, and the intense nonlinear interactions of
nanophotonic waveguides can be leveraged to meet these requirements. Here we demonstrate
second harmonic generation (SHG) in GaAs-on-insulator waveguides with unprecedented
efficiency of 40 W−1 for a single-pass device. This result is achieved by minimizing the
propagation loss and optimizing phase-matching. We investigate surface-state absorption and
design the waveguide geometry for modal phase-matching with tolerance to fabrication variation.
A 2.0 µm pump is converted to a 1.0 µm signal in a length of 2.9 mm with a wide signal bandwidth
of 148 GHz. Tunable and efficient operation is demonstrated over a temperature range of 45 °C
with a slope of 0.24 nm/°C. Wafer-bonding between GaAs and SiO2 is optimized to minimize
waveguide loss, and the devices are fabricated on 76 mm wafers with high uniformity. We expect
this device to enable fully integrated self-referenced frequency combs and high-rate entangled
photon pair generation.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1.

Introduction

Optical frequency conversion uses nonlinear interactions of light with matter to generate new
wavelengths of light. Recent efforts have improved conversion efficiencies and broadened the
application of this technology by using nanophotonic waveguides. Second harmonic generation
(SHG) is one nonlinear process of significant interest because it is critical for stabilizing optical
frequency combs [1]. In SHG, input pump light is converted to a signal with twice the frequency.
It is used to self-reference a frequency comb by comparing and controlling the difference between
two sections of the comb spectra that are separated by an octave. This is referred to as f -2f
self-referencing and it produces low-noise optical frequency combs used for optical clocks [2–4],
optical frequency synthesizers [5–7], low-phase-noise microwave generation [8], and molecular
sensing [9,10].
Increasing the application space for stabilized frequency combs requires reducing the cost,
size, weight, and power consumption while maintaining performance metrics [7]. Chip-scale
integration is an attractive way to achieve these goals, particularly using heterogeneous integration
to combine a suite of different materials into a compact package. A promising approach to develop
a chip-scale stabilized frequency comb is to pump a SiN microresonator with a continuous-wave
(CW) near-IR laser [6,7,11]. This design is scalable to large-volume and high-yield manufacturing
to reduce cost. However, exceptionally high nonlinear conversion efficiencies are required for
#389423
Journal © 2020

https://doi.org/10.1364/OE.389423
Received 30 Jan 2020; revised 4 Mar 2020; accepted 5 Mar 2020; published 18 Mar 2020

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this fully integrated system. In particular, avoiding the use of amplifiers for the microresonator
pump laser and the SHG pump light will be a significant advancement [12]. Given the limitations
presented by the SiN microresonator quality factor that restricts the pump power and by the
detector noise, extremely high SHG efficiency is necessary to achieve a sufficient signal-to-noise
ratio.
A comparison of the highest SHG conversion efficiencies with the corresponding signal
bandwidth from different waveguide platforms is shown in Fig. 1 for the non-depleted pump
regime. The pump wavelength is also indicated since the conversion efficiency is generally higher
for shorter wavelengths with a given platform [26,27]. Neglecting the effects of non-uniformity
and loss, conversion efficiency is expected to increase inversely proportional to the square of
the bandwidth. Larger SHG bandwidths are especially important for applications where the
pump is either broad or consists of multiple narrow-linewidth modes that reside within the
SHG bandwidth [28]. In certain cases, only one narrow-linewidth pump mode is relevant, so
larger SHG bandwidths are not helpful as long as the pump mode can be aligned to the SHG
conversion band. Another aspect to consider in this comparison is that SHG is most applicable as
a component of a system. Therefore, relevant SHG technologies should either support versatile
and low-loss coupling techniques or be compatible with heterogeneous integration [15,29]. For
example, to add SHG functionality to photonic integrated circuit (PIC) based on mature and
scalable Si photonics, it is advantageous to either deposit, grow, or bond the nonlinear material to
the Si substrate [29,30].

Fig. 1. Highest SHG conversion efficiencies demonstrated from different waveguide material
platforms for resonant (circles) and single-pass (triangles) devices. The signal bandwidth
is defined by the FWHM. References: AlGaAs-OI chip [13], AlN rings [14], GaAs-OI
chip [15], GaAs-OI ring [16], GaP PhC [17], LNOI [18], LNOI ring [19], LN RPE [20],
OP-AlGaAs [21], ox-AlGaAs [22], PP-GaN [23], PPKTP [24], SiN/LNOI [25].

Integrated f -2f self-referencing without an optical amplifier [31] or an assist-laser requires an
SHG conversion efficiency on the order of 40 W−1 . At this level, 5 µW of CW pump light (the
typical power from a single comb line [11,32,33]) can be converted to 1 nW of signal light. To
our knowledge, no single pass device has previously demonstrated this. While some resonant
devices have exceeded this efficiency [14,19], the narrow bandwidth and sensitive phase-matching
condition makes them impractical for a fully integrated f -2f self-referencing system. These
resonant devices are better suited for parametric down-conversion applications with narrow
bandwidths and squeezed light generation [14,19,34,35]. Furthermore, producing such systems
on a large scale requires a high-yield and reproducible fabrication. This work describes an SHG
device with the high efficiency and bandwidth suitable for an integrated f -2f self-referenced
comb. The high second-order nonlinearity of GaAs paired with the high index contrast and

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uniformity of integrating GaAs-based waveguides on Si has enabled this advancement [36]. We
use birefringent modal phase-matching [13,15] in a GaAs waveguide to produce SHG from a 2.0
pump with an efficiency of 40 W−1 and a signal bandwidth of 0.5 nm. The device is fabricated at
the 76 mm (3-inch) wafer scale and is compatible with integration on the SiN platform used for
the f -2f self-referencing [29].
In this manuscript, we present an optimized design for efficient SHG conversion with
modal phase-matching and we discuss processing techniques for wafer-scale fabrication. The
conversion efficiency is characterized to determine the peak efficiency, bandwidth, and temperature
dependence. Finally, we conclude with a summary of the results and present an outlook for
improving the SHG conversion efficiency.
2.

Design

To design the SHG waveguide we select a phase-matched geometry that maximizes the conversion
efficiency [26,27], η, defined as:
η=

P2ω (L)
2ω2 ξ κL2
=
,
P2ω (0)
n2ω n2ω 0 c3

(1)

where P2ω is the signal power, Pω is the pump power, L is the waveguide length, ω is the angular
frequency of the pump, 0 is the permittivity of free space, and c is the speed of light in vacuum.
The effective refractive indices of the pump and signal optical modes are nω and n2ω , respectively.
The ξ term in Eq. (1) accounts for phase-matching and the propagation loss: Note that in [27] a
typographical error exists: in Eq. (3), the α appearing in the cosine term should be replaced with
∆.
A2 + B2
e−(α2ω /2+αω )L ,
(2a)
ξ=
(∆α2 + ∆β2 )(L/2)2
where

A = sinh (∆αL/2) cos (∆βL/2),

(2b)

B = cosh (∆αL/2) sin (∆βL/2) .

(2c)

Parameters αω and α2ω are the propagation loss coefficients for the pump and signal. The phasemismatch, ∆β, and the loss-mismatch, ∆α, are defined as ∆β = β2ω − 2βω = (2ω/c)(n2ω − nω )
and ∆α = α2ω /2 − αω . A nonlinear coupling parameter, κ, in Eq. (1) accounts for two effects:
the mode overlap between the pump and signal light and the interaction between the optical
modes and the nonlinear optical susceptibility of the waveguide. It is defined as:
2

2 E
dx Ex,ω
y,2ω dx dy
,
κ = ∬
2 ∬
2 dx dy
2 dx dy
|E
|
|E
|
x,ω
y,2ω
2
2
R
R

∬

R2

(3)

where dx is the spatially varying effective second-order nonlinear coefficient corresponding to
components of the d tensor that interact with x-polarized pump light. The x-polarized component
of the pump mode is Ex,ω , and the y-polarized component of the signal mode is Ey,2ω . Wave
propagation is in the z-direction and the waveguide top-surface in the y-direction. Each integral
in Eq. (3) is taken over all space in the x-y plane.
The magnitude of the d14 and d36 nonlinear coefficients of GaAs are equivalent for a (100)
oriented top surface with wave propagation in the plane of the (01̄1̄) primary flat or the (01̄1)
secondary flat. However, there is a wide range of reported values for this coefficient, and these
values have large uncertainties because of the measurement techniques and differences in material
preparations [37–41]. For the calculation in this work, we use dx = 180 pm/V for GaAs to

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9524

approximately match our measurements, and we neglect its value in SiO2 and air. Guided optical
modes are calculated using a finite-difference method eigenmode solver from which κ is evaluated
by Eq. (3). The GaAs is partially etched so a 15 nm thick slab remains, and it is cladded with
SiO2 on bottom and air on top. The effective refractive indices, nω and n2ω , and the nonlinear
coupling, κ, are calculated for different heights and widths of the waveguide core. A Sellmeier
model is used for the index of each material [42,43]. The phase-matched geometries are found
from the intersection of the discretely calculated values for nω of the transverse-electric (TE)
polarized pump and n2ω of the transverse-magnetic (TM) polarized signal. Similarly, κ is found
by interpolation. Figure 2(a) shows the effective indices of the pump and signal light for a range
of heights and widths, and Fig. 2(b) shows the conversion efficiency η for a length of 2.9 mm (the
length used in this experiment) with the corresponding phase-matched geometries. The peak
simulated conversion efficiency is 50.3 W−1 , which occurs at a GaAs thickness of 149 nm and a
width of 1.3 . The pump and signal propagation losses used in this simulation are 1.5 dB/cm and
16.8 dB/cm, respectively.

Fig. 2. (a) Effective indices of the signal (solid lines) and pump (dashed lines) as a function
of waveguide width. Each curve represents a different waveguide height, from 140 nm with
the lowest indices to 156 nm with the highest indices, in steps of 2 nm between curves. (b)
SHG conversion efficiency (η) for a length of 2.9 mm plotted in circles on the left axis as a
function of waveguide height for the perfectly phase-matched widths plotted in triangles on
the right axis.

Phase error between the pump and signal light due to width and height variations of the
waveguide degrade the signal and typically limit the conversion efficiency of waveguided SHG

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devices. This effect is analyzed using the effective indices in Fig. 2(a). The FWHM bandwidths
of the SHG signal are calculated for variations with waveguide width (∆w) and height (∆h),
which are plotted in Fig. 3 for a length of 2.9 mm. For larger heights, ∆w increases, indicating
greater tolerance to width variations. However, the trend for ∆h is relatively constant for this
range of waveguide heights. Although the peak conversion efficiency may be lower, an SHG
waveguide with a height greater than 149 nm is less sensitive to geometric variations. For longer
waveguides, these phase errors accumulate and the conversion efficiency becomes more sensitive
to geometric variations. Our design targets a thickness of 154 nm for the SHG experiment, which
has a phase-matched width of 1.9 and a maximum possible conversion efficiency 40.5 W−1 , as
shown in Fig. 2(b).

Fig. 3. FWHM of the conversion efficiency (η) for variations in the width (∆w) shown in
solid blue on the left axis and the height (∆h) shown in dashed orange on the right axis.

The input facet waveguide width is 490 nm to minimize the coupling loss of the pump mode
from a lensed fiber, which is simulated to be 3.6 dB. This waveguide tapers to the SHG waveguide
width over a length of 100 . At the output facet the waveguide width is the same as the SHG
section. The wide waveguide at the output reduces the facet coupling uncertainty compared to a
tapered waveguide and the input facet taper is sufficient to suppress unwanted internal reflections.
The simulated efficiency for the output coupling of the signal mode is 7.7 dB, however, even
higher loss is expected because this calculation does not account for higher-order mode coupling
in the output fiber.
While GaAs and AlGaAs are ideal materials for SHG because of their large second-order
nonlinearity, two factors limit their performance and application space. First, the bandgap of
GaAs limits the SHG signal wavelength to >900 nm. For AlGaAs, shorter wavelengths down to
∼600 nm can be achieved depending on the Al mole fraction. The second limiting factor is the
surface defect-state absorption [13,44]. Specifically, As-As bonds formed at the surface create a
strong absorption feature with a peak near 950 nm [45,46]. For GaAs, this absorption feature
limits the minimum signal wavelength to ∼950 nm. Because of the larger refractive index, GaAs
supports higher confinement and smaller modes compared to AlGaAs compounds. Therefore,
we use GaAs since its bandgap is compatible with our application to convert ∼2.0 light to ∼1.0 .
3.

Fabrication

Figure 4(a) shows a schematic of the designed waveguide cross-section. The fabrication process
begins with molecular beam epitaxial growth on the (100) surface of a GaAs wafer. A 150 nm
thick Al0.8 Ga0.2 As etch-stop layer is grown first, followed by the 158 nm thick GaAs waveguide
layer. The waveguide layer is 4 nm thicker than is intended for the fabricated waveguide to
account for material loss in subsequent HCl and NH4 OH steps. This GaAs wafer is then bonded
to a Si wafer with a 3.0 thick thermal oxide layer. Before bonding, both wafers are cleaned with
solvents and activated with an atmospheric plasma containing metastable He and free-radicals of
H and N. The bond is initiated in air at room temperature using a custom-built wafer bonding
apparatus [47], and an anneal at 150 ◦ C on a hotplate for 1 hour completes the bond. Separate

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bonding experiments are used to measure the bond energy of (0.91 ± 0.16) J/m2 with the double
cantilever beam method [48]. While this is lower than other bonding recipes using SiN or Al2 O3
interlayers, this direct bond minimizes the waveguide loss and is strong enough to complete the
waveguide fabrication.

Fig. 4. (a) Schematic diagram of the waveguide cross-section. (b) Picture of the 76 mm
wafer shows nearly perfect yield of the transferred GaAs film. (c) SEM showing the output
waveguide facet.

The GaAs substrate is removed in two steps. First, the bulk of the substrate is etched with a
mixture of 350 mL H2 O2 (9.8 mol/L) and 25 mL NH4 OH (14.5 mol/L) using an N2 bubbler to
stabilize the reaction. After the 150 nm thick Al0.80 Ga0.20 As etch-stop layer becomes visible
(typically after 80 minutes), the wafer is transferred to a second mixture (also using an N2 bubbler)
with the same H2 O2 , but only 10 mL NH4 OH. Residue is removed with HCl (2.0 mol/L) for 20 s,
and the etch-stop is selectively etched with HF (1.4 mol/L) for 30 s. Residual fluoride compounds
produced from the HF etch are removed with N-methyl-2-pyrrolidone at 80 ◦ C for 5 min and
then NH4 OH (1.3 mol/L) for 1 min. An image of the Si wafer at this point in the process with the
transferred GaAs film is shown in Fig. 4(b). The RMS surface roughness, measured with atomic
force microscopy, on top of the GaAs is ∼0.2 nm over 9 µm2 , which is the same as the bottom
surface before bonding. The thickness uniformity is measured with optical interferometry to be
±0.2 % or ±0.3 nm within a 20 mm radius from the wafer center.
The GaAs waveguides are defined with electron-beam lithography and inductively-coupled
plasma (ICP) reactive-ion etching (RIE). The etch uses BCl3 with 200 W ICP power, 100 W bias
power, 5 mTorr chamber pressure, and a temperature of 20 ◦ C. By dividing the etch in two steps
and measuring the GaAs thickness in between, we consistently leave a 15 nm thick GaAs slab
with an error of ±1 nm, limited by the optical interferometric measurement. This GaAs slab is
important to prevent exposure of the GaAs/SiO2 interface to HCl, which etches the interface
non-uniformly, causing phase-errors. The SiO2 facets are etched with ICP-RIE using CHF3 and
O2 gases, and the chips are released from the wafer using a deep-RIE etch of the Si substrate. A
scanning electron micrograph (SEM) of the waveguide at the chip facet is shown in Fig. 4(c). We
estimate the sidewall roughness as 4.5 nm RMS using a top-view SEM. Finally, the chips are
cleaned with HCl (1.1 mol/L) for 10 s to minimize the As-As bonds.
4.

Experiment

To measure the SHG conversion efficiency, we use a CW and tunable 2 laser to pump the
waveguides in the TE-polarized mode, as shown in Fig. 5. A ∼1 % splitter is used to monitor
the input power, and lensed fibers couple light on and off the chip. After the output fiber, a
wavelength-division multiplexer (WDM) splits the TM-polarized signal and TE-polarized pump

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9527

light so they can be monitored on separate photodetectors. Similar setups are used to measure
transmission at 980 nm, 1064 nm, and 1220 nm using lasers at these discrete wavelengths.

Fig. 5. Experimental setup for SHG. The yellow connections are single-mode fibers and
the blue connections are polarization-maintaining single-mode fibers. PC: polarization
controller; Pol.: fiber-based linear polarizer; PD: photodetector; WDM: wavelength division
multiplexer for splitting the 1 and 2 light.

Propagation losses measured near the signal and pump wavelengths are shown in Fig. 6. We
fit the loss spectrum to a linear combination of a Lorentzian function and λ−4 to exemplify two
separate spectral trends: the absorption feature at 950 nm and the scattering loss. The waveguide
loss and facet coupling for both the pump and signal modes are characterized from transmission
measurements of waveguides with different lengths on a single chip. Since bends are used to
create the varying waveguide lengths, a second test-structure is used to verify that the bend loss
are negligible. Pump light coupling loss at the input is (5.0 ± 0.5) dB and signal light coupling
loss at the output is (12.1 ± 0.5) dB. The increased coupling loss compared to the simulation is
likely caused by residue from the die-release process, observed in SEM imaging. Additionally,
the signal light suffers from increased coupling loss due to the higher order modes in the output
lensed fiber. The propagation loss for the pump mode is (1.5 ± 0.2) dB/cm and for the signal
mode is (16.8 ± 1.8) dB/cm.

Fig. 6. Measured propagation loss showing the trends of a Lorentzian (dashed orange), λ−4
(dashed blue), and a linear combination of both (solid green).

Conversion efficiency spectra are shown in Fig. 7(a) for a length of 2.9 mm and a range of
temperatures. The peak efficiency is (39.5 ± 7.9) W−1 at 15 ◦ C and decreases to (28.5 ± 5.7)
W−1 at 60 ◦ C. The uncertainty is calculated from the coupling loss variation across the chip. The
change in peak SHG wavelength with temperature is 0.236 nm/◦ C, as shown in Fig. 7(b). The
wavelength uncertainty arises from mode-hopping in the laser, and it is calculated from repeated
measurements. We also measured SHG at 25 ◦ C from devices with lengths of 6.3 mm and 9.7
mm, but the efficiency decreases to (31.4 ± 6.2) W−1 and (15.2 ± 3.0) W−1 , respectively. The
complete SHG spectrum is shown in Fig. 7(c) at room temperature (∼22 ◦ C), to show greater
detail than the temperature controlled measurements in Fig. 7(a).

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Fig. 7. (a) SHG spectra for various temperatures, limited by the temperature control setup.
The standard error is indicated by the shaded areas. (b) Peak SHG wavelength for various
temperatures with a linear fit of 0.236 nm/◦ C. (c) The full spectrum of the SHG conversion
efficiency at room temperature (∼22 ◦ C), to show greater detail.

5.

Discussion

Phase errors resulting from thickness non-uniformity likely limit the nonlinear conversion for
the longer waveguides that produce lower conversion efficiency. This is evident from the SHG
spectral shape from Fig. 7(c) that does not show a clean sinc2 function. By extrapolating the
uniformity measurement from across the wafer to within the length of the SHG waveguide, we
estimate that the average GaAs thickness varies by up to ∼20 pm. Compared to ∆h from Fig. 3 of
∼7 pm, this thickness variation is very significant. We do not expect the average width of the
waveguide to vary significantly compared to ∆w of ∼1 nm, though we have not been able to
measure the width with this level of accuracy. The 2.9 mm long device may be more uniform
than the maximum variation of ∼20 pm because it was located close to the center of the wafer,
where the thickness variation is minimal.
Our key findings to reduce the waveguide loss at the signal wavelength involve minimizing
As-As bonds at the GaAs surface at two different steps in the fabrication process. First, the

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bonding surface of the GaAs waveguide layer must be allowed to form a native oxide after
the MBE growth, which is not removed before bonding. This direct bond between the native
GaAs-oxide surface and the SiO2 surface is optimal to reduce the As-As bonds. The second
step is the surface treatment of the top and sidewall of the GaAs waveguide. Similarly, a native
oxide is preferred, but only after an HCl strip of the surface reconstruction layer (the amorphous
surface layer naturally formed in air) [46,49]. This allows for a native oxide to grow with minimal
As-As bonds. We found that the propagation loss of the signal is (20.4 ± 1.4) dB/cm when the
native oxide is stripped with HCl before bonding, and the waveguides are stripped with HCl after
etching. When the bonding is performed with the native GaAs oxide, the loss of the signal is
(34.6 ± 3.7) dB/cm before treating the waveguides with HCl.
As indicated from the propagation loss spectrum, we have not completely avoided forming
As-As bonds and the loss from this defect state can still be improved. A potential process to
further reduce As-As bonds is to use atomic layer deposition (ALD) to passivate the surface
[46]. However, our attempts with ALD Al2 O3 passivation produced higher loss than our current
process. Also, the loss at the pump wavelength indicates that our sidewall roughness may have a
significant contribution. Further optimization of our lithography and etching process may reduce
the loss by decreasing scattering for both the signal and pump wavelengths.
At the elevated temperatures the decreased SHG conversion efficiency may be due to an
increase in propagation losses. Free-carrier absorption in the GaAs for the signal is expected to
increase with temperature [50], and the peak wavelength of the As-As defect state is strongly
temperature dependent [45]. The conversion efficiency does not degrade after the thermal cycling
between 15 ◦ C and 60 ◦ C, so the loss is not a permanent effect. Regardless, we find that the
conversion efficiency is consistent from 15 ◦ C to 40 ◦ C. An f -2f self-referenced comb can
reasonably be stabilized in this temperature range.
Similar SHG results were measured immediately following and several months after the
fabrication, showing no measurable degradation in performance. The peak efficiency, bandwidth,
and operation wavelength did not vary more than our measurement error within this time frame.
These results show that the SHG performance is not affected by oxidation or other chemical
reactions with the GaAs.
6.

Conclusions

We present a 76 mm wafer-scale GaAs-on-insulator platform and an SHG device compatible
with heterogeneous integration on a SiN PIC for f -2f self-referencing and optical frequency
synthesis [15,29]. Record-high SHG conversion efficiency of 40 W−1 (470 W−1 cm−2 ) in a 0.5
nm bandwidth is demonstrated from a single-pass device because of improvements made to
the propagation loss and the waveguide uniformity. Efficient SHG operation is confirmed over
a temperature range of 45 ◦ C. With these new conversion efficiency achievements, it is now
possible to realize a fully-integrated chip-scale f -2f system. In future work to further improve
SHG conversion efficiency, quasi-phase-matching would allow for optimization of the nonlinear
mode-overlap and for implementing greater phase-matching tolerance. Applying orientation
patterning to the GaAs platform is expected to produce even higher SHG conversion efficiencies,
exceeding 100 W−1 with a broader bandwidth of >103 GHz.
Funding
Defense Advanced Research Projects Agency (HR0011-15-C-0055).
Acknowledgments
We thank Kevin L. Silverman and Travis M. Autry for helpful discussions on the GaAs fabrication;
Scott B. Papp for helpful discussions about applications; and Saeed Khan, Abijith S. Kowligy,

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and Lynden K. Shalm for useful discussions and inputs on the manuscript. This is a contribution
of NIST, an agency of the U.S. government, not subject to copyright.
Disclosures
The authors declare no conflicts of interest.
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</reference>

<statements>
1. In a phase-matched waveguide, the conversion efficiency within the undepleted pump approximation follows the second-harmonic generation efficiency equation, where eta0 represents the normalized intrinsic coupling efficiency, L is the physical interaction length, and alpha_omega and alpha_2omega denote the linear loss coefficients at the fundamental and second-harmonic frequencies, respectively.
</statements>

Begin the assessment now. Output only the JSON list, without any conversational text or explanations.