GeoThermalCloud: Machine Learning for Geothermal Resource Exploration M. K.
Mudunuru, B. Ahmmed, and V. V. Vesselinov [1]Maruti K. Mudunuru
<EMAIL_ADDRESS>Earth System Measurement & Data Group, Atmospheric Sciences &
Global Change Division, Pacific Northwest National Laboratory, Richland, WA
99352, USA. Email<EMAIL_ADDRESS>
mm/dd/yyyy mm/dd/yyyy

# GeoThermalCloud: Machine Learning for Geothermal Resource Exploration

Velimir V. Vesselinov  Bulbul Ahmmed  Earth System Measurement & Data Group,
Atmospheric Sciences & Global Change Division, Pacific Northwest National
Laboratory, Richland, WA 99352, USA. Email<EMAIL_ADDRESS>EnviTrace LLC.
Santa Fe, NM 87501, USA. Email<EMAIL_ADDRESS>Earth and
Environmental Sciences Division, Los Alamos National Laboratory, Los Alamos,
NM 87545, USA. Email<EMAIL_ADDRESS>
###### keywords:

geothermal energy exploration, unsupervised machine learning, play fairway
analysis, SmartTensors AI Platform, hidden signatures

††volume: Volume x, Issue x, (pre-print submitted to JMLMC) 2022

## Abstract

Geothermal is a renewable energy source that can provide reliable and flexible
electricity generation for the world. In the past decade, the U.S. Geological
Survey’s resource assessments, Play Fairway Analyses (PFA), and GeoVision
report by the U.S. Department of Energy’s Geothermal Technologies Office
provided insights on enormous untapped potential for geothermal energy to
contribute to the U.S. domestic energy needs. Past studies identified that
geothermal resources without surface expression (e.g., blind/hidden
hydrothermal systems) have vast potential. These blind systems can
significantly increase power generation. But a primary challenge is locating
and quantifying these hidden resources, which do not have any thermal
manifestations on the surface. PFA has successfully identified some blind
systems in the western USA (e.g., specific locations in the Great Basin region
within Nevada). However, a comprehensive search for these blind systems can be
time-consuming, expensive, and resource-intensive, with a low probability of
success. Accelerated discovery of these blind resources is needed with growing
energy needs and higher chances of exploration success. Recent advances in
machine learning (ML) have shown promise in shortening the timeline for this
discovery. This paper presents a novel ML-based methodology for geothermal
exploration towards PFA applications. Our methodology is provided through our
open-source ML framework, GeoThermalCloud
https://github.com/SmartTensors/GeoThermalCloud.jl. The GeoThermalCloud uses a
series of unsupervised, supervised, and physics-informed ML methods available
in SmartTensors AI platform https://github.com/SmartTensors. Here, the
presented analyses are performed using our unsupervised ML algorithm called
NMF$k$, which is available in the SmartTensors AI platform. Our ML algorithm
facilitates the discovery of new phenomena, hidden patterns, and mechanisms
that helps us to make informed decisions. Moreover, the GeoThermalCloud
enhances the collected PFA data and discovers signatures representative of
geothermal resources. Through GeoThermalCloud, we could identify hidden
patterns in the geothermal field data needed to discover blind systems
efficiently. Crucial geothermal signatures often overlooked in traditional PFA
are extracted using the GeoThermalCloud and analyzed by the subject matter
experts to provide ML-enhanced PFA, which is informative for efficient
exploration. We applied our ML methodology to various open-source geothermal
datasets within the U.S. (some of these are collected by past PFA work). The
results provide valuable insights into resource types within those regions.
This ML-enhanced workflow makes the GeoThermalCloud attractive for the
geothermal community to improve existing datasets and extract valuable
information often unnoticed during geothermal exploration.

##  1 Introduction

Geothermal energy is a nearly inexhaustible renewable resource that harnesses
Earth’s heat Brown et al. (2012). Broad deployment of geothermal energy can
provide essential benefits to the U.S. domestic energy needs, including
electricity production, thermal energy storage, mining of critical materials
(e.g., lithium), and efficient heating and cooling solutions Wicks (2022);
Cheng (2022). As of 2021, the U.S. installed geothermal powerplant capacity is
more than 3.8 GigaWatts (GW) Hamm et al. (2021). In the last decade, the U.S.
Department of Energy (DOE) Geothermal Technologies Office (GTO) recently
conducted a study called the GeoVision to understand and assess the geothermal
potential in our country Geo (2019). Based on the state-of-the-art geothermal
technology (e.g., recent advances in drilling and data analytics), the 2019
GeoVision analysis report suggests that more than 30 GW of new and
undiscovered hydrothermal resources is recoverable. Additionally, we can
access 100+ GW of new geothermal energy by developing enhanced geothermal
systems (EGS). However, various technical and non-technical barriers exist to
successful exploration, development, and widespread geothermal energy
deployment Imolauer et al. (2010); Levine and Young (2017); Young et al.
(2017). A significant barrier is efficient discovery and accelerated
exploration of hidden or blind geothermal resources Dobson (2016b).

Blind systems are shown to have three times more resource potential within the
same region than existing and identified hydrothermal systems Dobson (2016b).
However, these systems lack surface thermal expressions at the ground surface,
making it more challenging to discover them compared to traditional, surface-
manifested hydrothermal resources. As a result, the probability of
successfully locating and developing these resources without a comprehensive
and expensive exploration and research is very low Gehringer and Loksha
(2012). To overcome these challenges of discovering blind systems, the U.S.
DOE-GTO has funded multiple projects through three phases to develop a
systematic approach to quantify geothermal prospects while reducing overall
exploration costs Hamm et al. (2021). This approach to improving geothermal
development projects’ success rate is called Play Fairway Analysis (PFA). The
PFA concept borrowed from oil& gas industries allows us to identify potential
locations of blind hydrothermal systems and quantify geothermal potential in
the regions of interest Holmes and Fournier (2022). Traditional PFA was able
to identify various regions of interest for geothermal exploration Faulds et
al. (2016). However, the main limitation of this traditional analysis is
effectively combining sparse and multi-source data streams with missing values
to explore geothermal resources better Vesselinov et al. (2020); Mudunuru et
al. (2020); Siler et al. (2021); Smith (2021); Moraga et al. (2022).

In this study, we present GeoThermalCloud, a novel ML methodology to enhance
and accelerate PFA by overcoming these limitations. Unsupervised (self-
supervised) ML algorithms based on non-negative matrix factorization and
custom $k$-means clustering (NMF$k$) provide means to overcome this challenge
Alexandrov and Vesselinov (2014); Vesselinov et al. (2019). The NMF$k$
algorithm is one of the tools available in the SmartTensors AI platform
https://github.com/SmartTensors to preprocess sparse datasets with missing
values. The GeoThermalCloud is developed based on this powerful AI platform
for extracting hidden geothermal signatures from field data. Specifically, the
GeothermalCloud facilitates curation and analysis of geothermal datasets
https://github.com/SmartTensors/GeoThermalCloud.jl. The uniqueness of the
GeoThermalCloud is that it is flexible, open-source, and provides
site/regional data, scripts, examples, figures, and developed geothermal
prospectivity maps for the geothermal community. It has in-build
preprocessing, postprocessing, and state-of-the-art visualization tools for
non-experts. Therefore, experts and non-experts can equally utilize this
capability without going through a steep learning curve. Based on subject
matter expertise (SME), the discovered signatures using the GeoThermalCloud
are assessed and correlated to geothermal resource type (e.g., low, medium,
and high-temperature resources) Siler et al. (2021). We can use the extracted
signatures upon SME assessment to build geothermal prospectivity, develop
exploration risk maps, and guide future data collection strategies Vesselinov
et al. (2022b).

The paper is organized as follows: Section 1 discusses state-of-the-art for
PFA and its limitations. Section 2 describes the proposed methodology for ML-
enhanced PFA. Section 3 describes the PFA data curated from open-source
geothermal repositories for NMF$k$ analysis. Sec. 4 presents the results and
discussion of ML-analyzed geothermal data for popular PFAs in the USA.
Finally, Sec. 5 presents our conclusions.

##  2 Proposed methodology

This section describes the proposed ML methodology used to build the
GeoThermalCloud. GeoThermalCloud capabilities and workflow include (1)
analyzing large field datasets, (2) assimilating model simulations (large
inputs and outputs), (3) processing sparse datasets, (4) performing transfer
learning (between sites with different exploratory levels), (5) extracting
hidden geothermal signatures in the field and simulation data, (6) labeling
geothermal resources and processes, (7) identifying high-value data
acquisition targets, and (8) guiding geothermal exploration and production by
selecting optimal exploration, production, and drilling strategies. In
Traditional PFA analysis, first, the SMEs review and analyze the data to
estimate the geothermal potential in a region Faulds et al. (2016); Lautze et
al. (2017); Siler et al. (2017); Ito et al. (2017); Lindsey et al. (2021).
Second, dataset attributes and associated evidence of geothermal system
occurrence are combined using weighted products or weighted sums. Finally,
geothermal PFA models are generated to estimate the site- or regional-scale
prospectivity in the form of maps or other data visualization tools. Our ML-
enhanced PFA workflow (ePFA) has four steps combining the strengths of SMEs
and ML to develop maps of geothermal potential Ahmmed and Vesselinov (2022b).
The site- or regional-scale data are curated and processed using ML to
discover hidden features/signatures in the first step. SMEs interpret these
signatures to associate them with a resource type (e.g., low/medium/high-
temperature geothermal systems). In the second step, ML analyses are performed
at other sites/regions to transfer knowledge (e.g., reusing discovered
geothermal patterns, signatures, and relationships between geothermal data
attributes) to the analyzed site/regional datasets. In the third step, physics
insights gained from process models and simulations are applied to enrich the
analyzed dataset (e.g., fill data gaps). Finally, the information from the
above three individual steps is combined using risk analysis and uncertainty
quantification methods to develop site-specific prospectivity maps and future
data acquisition strategies.

For this ePFA workflow development, we use an unsupervised ML algorithm called
NMF$k$ Alexandrov and Vesselinov (2014); Vesselinov et al. (2022a) to discover
hidden patterns in geothermal datasets. The NMF$k$ is a machine learning
method that combines non-negative matrix factorization (NMF) with customized
$k$-means clustering. NMF learns a parts-based representation Lee and Seung
(1999); Cichocki et al. (2009) of the analyzed geothermal data matrix. The
$k$-means clustering groups similar NMF learned pieces together and discovers
unique underlying patterns Wagstaff et al. (2001). Customization of the
$k$-means clustering provides insights on the similarity of learned parts
within a given cluster compared to other identified groups Iliev et al.
(2018); Vesselinov et al. (2019). This study builds a data matrix, $X$ of size
($n$, $m$), after curating the raw geothermal datasets. This geothermal data
matrix is developed by combining geophysical, geological, hydrological, and
thermal data sources. For instance, the DOE’s Geothermal Data Repository
(https://gdr.openei.org/) provides a venue to download such datasets for ML
analysis. The preprocessor normalizes the raw data such that the elements of
$X$ are non-negative. If needed, it also log transforms individual attributes
to reduce the effects of outliers and nonnormal distributions. Here, $n$ is
the number of geothermal data attributes, and $m$ is the number of spatial
locations of geothermal measurements. The NMF$k$ is part of SmartTensors AI
Platform and can be found at https://github.com/SmartTensors.

NMF of NMF$k$ algorithm decomposes $X_{n\times m}$ into two matrices,
$W_{n\times k}$ (basis/mixing matrix) and $H_{k\times m}$
(coefficient/attribute matrix) as:

$X=W\times H+\epsilon(k),$ (1)

where $\epsilon(k)$ is an error matrix representing the discrepancy between
$X$ and its estimate $W\times H$ for the specified $k$ while $k$ is the
unknown/hidden number of signatures present in the data and is always smaller
than $n$ and $m$. The $W$ matrix represents how the measurement locations are
related to the hidden signatures. The $H$ matrix depicts the relationship
between attributes and hidden signatures. In this methodology, the number of
hidden signatures ($k$) is unknown and is identified by performing a series of
NMF analyses for $k=2,3,\cdots,d$, where $d\leq m$. The NMF process minimizes
the following objective function, $\mathcal{L}$, based on Frobenius norm for a
specified $k$, such that the entries of the resulting $W$ and $H$ matrices are
non-negative:

$\mathcal{L}=||X-W\times H||_{F}\quad\mathrm{such\;that}\quad W,H\geq
0\quad\forall\quad n,m,k.$ (2)

In the minimization, the following rules are used to update the $W$ and $H$
matrices Lee and Seung (1999):

$W_{ij}\leftarrow W_{ij}\frac{(XH^{T})_{ij}}{(WHH^{T})_{ij}},$ (3)

and

$H_{ij}\leftarrow H_{ij}\frac{(W^{T}X)_{ij}}{(W^{T}WH)_{ij}}.$ (4)

For each $k$, NMF is solved for a series of random initial guesses (typically,
1,000 or more) for $W$ and $H$ matrices. The least value of $\mathcal{L}$ for
a given $k$ is assumed as the best value for the reconstruction error. After
completing the NMF process, the columns of the 1,000 estimated $H$ matrices
are clustered into $k$ clusters using a customized $k$-means clustering.
Alternatively, we cluster the rows of the 1,000 estimated $W$ matrices.
Typically, we cluster the smaller matrix. However, $k$ is also unknown in the
$k$-means clustering. The algorithm consecutively examines a specified $k$ by
obtaining 1,000 $H$ matrices for each feature. During clustering, the
similarity between two clusters is assessed using the Silhouette width/value
Rousseeuw (1987), which is essentially the cosine norm:

$\displaystyle\rho\left(p,q\right)=1-\frac{\sum_{i=1}^{n}p_{i}q_{i}}{\sum_{i=1}^{n}p_{i}^{2}\sum_{i=1}^{n}q_{i}^{2}},$
(5)

where $p_{i}$ and $q_{i}$ are components of vectors $\mathbf{p}$ and
$\mathbf{q}$. The Silhouette value quantifies how similar an object is to its
own cluster compared to other clusters and varies from $-1$ to $+1$; high
values indicate that the object is well matched to its own cluster and poorly
matched to neighboring clusters. The combination of reconstruction error
($\mathcal{L}(k)$) and the Silhouette value are used to determine the optimal
number of hidden signatures. If $k$ is low, the Silhouette value will be high,
but so maybe $\mathcal{L}(k)$ because of under-fitting. For high $k$, the
Silhouette value will be low and the solution may be over-fit. So, the best
estimate for $k$ is a number that optimizes both $\mathcal{L}(k)$ and the
Silhouette value.

##  3 Data

In this section, we describe the open-source geothermal datasets analyzed
using GeothermalCloud. The GeothermalCloud uses NMF$k$ not only for the
execution of ML analyses but also applies the postprocessing and visualization
tools provided by NMF$k$. Datasets include hydrological, geophysical,
geomechanical, geochemical, and geological attributes. The geothermal dataset
also covers regional conditions and corresponding resource types (e.g., low,
medium, and high temperature) in Nevada, Utah, California, Oregon, Idaho, New
Mexico, Texas, and Hawaii Islands in the USA. Figure 1 shows the spatial
locations of the analyzed data. All the data attributes are rescaled within
the range of 0.0 to 1.0 using a unit range transformation. If needed, some of
the attributes are log-transformed. In the following paragraphs, we briefly
describe the processed data for ML analysis. Additional information on these
datasets are described in references Ahmmed (2020b, a); Vesselinov et al.
(2021), which were curated from Geothermal Data Repository Weers and Anderson
(2016); Weers et al. (2022).

Figure 1: Geothermal field data: Field locations at which geothermal data is
available for ePFA analysis. This field data includes thermal, geophysical,
geomechanical, geochemical, and geological attributes. These datasets are
collected and curated at various locations in NV, UT, CA, OR, ID, NM, and HI .

The Nevada geothermal PFA Phases I, II, and III identified Great Basin as the
largest region with hidden geothermal resources. The study area comprises of
14,341 spatial locations at which geochemical data is sampled. A total of 18
different geochemical attributes are available for ML analysis. Within the
UtahFORGE region, 22 data attributes, including satellite (InSAR), geophysical
(e.g., gravity, seismic), geochemical, and geothermal attributes, are
collected at 102 locations. Twenty-one data attributes collected during the
Phase I and II at 120 different locations are analyzed in the Tularosa Basin.
In the Tohatchi Hot Spring area in Northern New Mexico, we investigated 19
data attributes observed at 41 wells. A total of 15 data attributes collected
at 247 locations during Phase I and II of Hawaiian geothermal PFA are analyzed
using NMF$k$. The data attributes employed in the current ML study are related
to geothermal processes. For instance, they can be used as a proxy or indirect
measure of rock-water interactions and infer reservoir temperatures,
geothermal conditions, reservoir boundaries, and heat source type (e.g.,
meteoric, magmatic, mixed) Dobson (2016a); Fridriksson and Ármannsson (2007);
Klein (2007); Fournier (1977).

##  4 Results and discussion

In this section, we provide the results of our ML methodology on the PFA
datasets described in Sec. 3.

###  4.1 Great basin

Great Basin includes multiple known and hidden geothermal resources, which
have been studied in the past decades Faulds et al. (2016). GeoThermalCloud is
used to analyze geochemical data, which can indirectly infer geothermal
conditions (e.g., reservoir temperatures, conditions, boundaries, and heat
source type). Geochemistry provides insight into water-rock interactions and
fluid circulation at various depths (e.g., shallow, deep). ML analysis is
performed on a curated data matrix consisting of 18 data attributes (e.g.,
temperature, pH, total dissolved solids, anions, and cation concentrations) at
14,321 locations. Our NMF$k$ algorithm identified three optimal signatures
based on the loss function, $\mathcal{L}$, and the Silhouette width value.

Figure 2 shows the dominant attributes in the discovered signatures, the
spatial distribution of these identified signatures, and their association
with a resource type (e.g., low, medium, high). Signatures A, B, and C define
modestly, highly, and moderately prospective hydrothermal systems. Signature A
represents modestly hydrothermal systems because of the low contribution of
groundwater temperature in this extracted signature. The dominant attributes
of this signature are TDS, Br-, B+, and $\delta O^{18}$. Signature B
represents highly prospective hydrothermal systems due to the high
contribution of temperature. The dominant attributes of this signature are pH,
Al3+, Be2+, and quartz and chalcedony geothermometers. Signature C defines
moderately prospective hydrothermal systems because of the medium contribution
of temperature. The dominant attributes of the signature are Mg2+ and Ca2+.
Signatures B and C distribution suggest that the significant portions of the
Great Basin region have prospective geothermal resources. Areas with a high
density of B and C locations are labeled with ellipses in this Figure 2. Some
of these locations align with existing geothermal resources and sites such as
Dixie Valley and Brady geothermal areas in Nevada Faulds et al. (2016).

Figure 2: ML analysis of Great Basin data: ML results for geothermal data
within the Great Basin region in Nevada. The left figure shows the optimal
hidden geothermal signatures discovered by NMF$k$, which is three. The right
figure provides the spatial distribution of these signatures mapped onto the
geothermal data collection locations based on resource type (e.g., modestly,
highly, and moderately prospective). The ellipses mark the regions with a high
density of similar resource-type signature locations.

###  4.2 UtahFORGE

UtahFORGE (Frontier Observatory for Research in Geothermal Energy, Utah) is a
field laboratory located in Milford, Utah, to study enhanced geothermal
systems (EGS) Moore et al. (2019); Allis et al. (2016). This field laboratory
provides a unique opportunity to develop and test new technologies for
characterizing and creating sustainable geothermal systems in a controlled
environment. During two phases of EGS site development in UtahFORGE, collected
geothermal data at 102 locations with 22 attributes. These include satellite
(InSAR), geophysical (gravity, seismic), geochemical, and geothermal
attributes. The NMF$k$ method was applied to this dataset and discovered four
optimal hidden geothermal signatures. Figure 3 shows the discovered
signatures, their relationship to data attributes, and the associated mapping
of these signatures to the sampled locations. Signatures A and B are related
to favorable geothermal conditions, which have different dominant attributes.
Signature A’s key attributes are gravity, seismic, slip rate, and specific
geochemical species. The high contribution from the slip rate indicates that
locations covered by Signature A are influenced by EGS activity. Signature B’s
dominant attributes are temperature, heat flow, pH, and ions such as K+, and
Fl-. These heat, pH, and temperature attributes are good indicators of
potential resources that may be hidden within this region. In a nutshell,
NMF$k$ analysis solidifies the idea that signature B locations can be a source
to discover blind systems in Utah.

Figure 3: ML analysis of UtahFORGE data: ML results for UtahFORGE geothermal
data. The left figure shows the four hidden geothermal signatures discovered
by NMF$k$. The right figure provides the spatial distribution of these
signatures based on low, medium, and high-temperature resource types.

###  4.3 Tularosa basin

The Tularosa basin in the southern Rio Grande Rift within New Mexico consists
of promising geothermal plays based on Phase-I and II PFA studies Brandt
(2015); Nash (2017); Nash et al. (2017). The presence of known geothermal
systems, high-temperature slim hole wells, quaternary faults, and relatively
high heat flow sheds light on possible blind geothermal systems that may be
present in the study area. During the traditional Tularosa basin PFA, various
datasets representing the heat of the Earth, such as water chemistry,
temperature gradients, 2 m temperature, and heat flow, are collected to assess
the geothermal potential. We investigate 21 attributes collected during this
PFA, including field geology (geological reconnaissance and mapping), gravity
surveys, shallow temperature surveys, well water sampling and geothermometry,
temperature logging, and magnetotelluric surveys. Figure 4 shows the results
of NMF$k$ applied to this Tularosa basin PFA dataset. Signature C defines the
potential for hidden hydrothermal resources due to favorable characteristics
associated with geological structures for geothermal exploration. Critical
attributes in this signature include heat flow, $\mathrm{SiO}_{2}$, silica
geothermometer, and fault density. The spatial distribution of this signature
suggests that resources are close to NASA and other military installations,
making geothermal energy cost-effective for power needs for several facilities
in this region.

Figure 4: ML analysis of Tularosa basin data: ML results for geothermal data
from the Tularosa Basin in New Mexico. The left figure shows the four hidden
geothermal signatures discovered by NMF$k$. The right figure provides the
spatial distribution of these signatures. Signature C has dominant attributes
(e.g., heat flow, $\mathrm{SiO}_{2}$, silica geothermometer, fault density)
that correspond to the location of hidden geothermal resources in this region.

###  4.4 Tohatchi Hot Springs

Tohatchi Hot Springs in New Mexico are favorable for enhanced geothermal
exploration Levitte and Gambill (1980); Ahmmed et al. (2021); Ahmmed and
Vesselinov (2022a). Characterizing the geothermal source and governing
mechanisms (e.g., identifying which data attributes contribute to the heat
source) that make the water hot in these springs can help us better explore
this region. NMF$k$ is a suitable method to understand the resource type
better. We investigated 19 data attributes observed at 41 wells and Tohatchi’s
hydrogeology to assess if groundwater circulates from a shallow depth ( 400 m
due to shallow geothermal system, secondary fracture permeability effects) or
from a greater depth (e.g., deep circulation of water, forced-convection
geothermal systems). Figure 5 shows the result of our NMF$k$ analysis on the
well data, which identified four hidden signatures. Signature C defines the
hidden potential geothermal resources, whose key attributes are pH, Li+,
$\mathrm{HCO}_{3}^{-}$, F+, Quartz-water-vapor geothermometer, and Na-K-Ca
geothermometer. Our ML analysis of the well data attributes and its signature
may suggest that Tohatchi Hot Springs’ geothermal reservoir is large for long-
term production of hot well water ($<100^{\mathrm{o}}$C) ML analysis in
connection with the SME technical reports allows us to hypothesize that
groundwater circulates from a shallow depth due to low geothermal gradients.

Figure 5: ML analysis of Tohatchi Hot Springs data: ML results based on
geothermal data for the Tohatchi Hot Springs region in New Mexico. The left
figure shows the four hidden geothermal signatures discovered by NMF$k$. The
right figure provides the spatial distribution of these signatures, where
signature C defines the hidden potential of high-temperature geothermal
resources within this region.

###  4.5 Hawaii islands

Hawaii is an ocean island hotspot environment with a magmatic geothermal heat
source. Currently, geothermal resources within this region provide up to 3% of
the state’s energy needs. The PFA analysis conducted across the State of
Hawaii by Lautze and co-workers Ito et al. (2017); Lautze et al. (2017) showed
that these islands have huge geothermal prospects. We have applied NMF$k$ to
discover hidden features in hydrological data linked to Hawaii’s geothermal
resources. The discovered signatures and their spatial distribution is shown
in Fig. 6. NMF$k$ identified four signatures in the PFA data that
characterized the islands and their resource types within the State of Hawaii.
Each signature has unique physical significance (e.g., water type,
alkalinity). Hidden signatures with high contributions from temperature are
essential for ML-enhanced geothermal exploration. Specifically, signatures B
and D define the hidden potential of geothermal resources within this region
as they are linked to groundwater temperature and chemistry. Signature A is
related to geochemistry (e.g., cations and anions), another proxy for blind
systems. This extracted signature shows that further data analysis of each
island in the State of Hawaii can help us better understand the geothermal
resource type and its source (e.g., meteoric, magmatic, or connate).

Figure 6: ML analysis of Hawaiian islands data: ML results based on geothermal
data for the Hawaiian islands. The left figure shows the four hidden
geothermal signatures discovered by NMF$k$. Their dominant attributes include
pH, $\delta^{18}$O, $\delta^{2}$H, and silicate. The right figure provides the
spatial distribution of these signatures. Signatures B and D define the hidden
potential of geothermal resources within this region as they are linked to
groundwater temperature and chemistry.

##  5 Conclusions

The paper presented an ML methodology developed in our open-source framework,
GeoThermalCloud https://github.com/SmartTensors/GeoThermalCloud.jl for
efficient geothermal exploration of hidden geothermal resources. The
GeoThermalCloud uses a series of unsupervised, supervised, and physics-
informed ML methods in our SmartTensors AI platform. Here, the presented
GeoThermalCloud analyses are performed using our unsupervised ML algorithm
called NMF$k$ from the SmartTensors AI platform. NMF$k$ is applied to analyze
and enhance existing public geothermal datasets. In this way, GeoThermalCloud
provides enhanced ML-based exploration results consistent with those obtained
from traditional SME-based PFA methods. The NMF$k$ discovers hidden signatures
embedded in the geothermal data, and then SMEs analyze the extracted hidden
signatures to understand the discovered knowledge. We applied the proposed
GeoThermalCloud framework to various regions in the U.S. to better
characterize the resource types based on the dominant data attributes
discovered by NMF$k$. These regions include the Great Basin in Nevada,
UtahFORGE, Tularosa basin, Tohatchi Hot Springs in New Mexico, and Hawaii
Islands. In Great Basin, NMF$k$ identified modestly, highly, and moderately
prospective hydrothermal systems, their dominant characterization attributes,
and their spatial distribution using geochemistry data Ahmmed and Vesselinov
(2022b). They also compared their results with a comprehensive PFA study
conducted in the same study area by Faulds and co-workers Faulds et al. (2016)
Fauld et al. used geophysical, geological, geochemical, etc. data yet the
prospectivities of two studies are similar with much less data using our ML.
We analyzed site prospectivity within UtahFORGE and potential exploration
options (e.g., drilling location for future geothermal field exploration). In
the Tularosa basin, Tohatchi Springs, and Hawaii islands, our NMF$k$ was able
to identify low-, medium-, and high-temperature hydrothermal systems and found
dominant attributes and spatial distribution for each hydrothermal system. To
conclude, these applications of our GeoThermalCloud for different types of
available PFA data instill confidence in our approach to analyze new datasets
(e.g., airborne geophysical, 3DEP LiDAR surveys) such as GeoDAWN Rhodes et al.
(2021), currently being collected over the parts of Nevada and California.
Figure7 shows a pictorial description of ML-enhanced PFA workflows using our
GeoThermalCloud, which we plan to enhance by combining GeoDAWN data with
multi-physics simulations. ML-enhanced GeoDAWN data will provide a deeper
understanding of the geothermal conditions (e.g., fault patterns, stress
regime) in the Great Basin that represent blind hydrothermal systems, thereby
accelerating the discovery of new and undiscovered geothermal resources for
domestic use.

Figure 7: Traditional vs. ML-enhanced PFA: This conceptual figure compares the
traditional and proposed workflow for geothermal PFA using the GeoThermalCloud
framework. In traditional PFA, geothermal play fairway models are generated
using weighted products or sums of geothermal attributes curated by SME’s from
raw field datasets. SME’s also perform data weighting in traditional PFA
models. In ML-enhanced PFA, the final prospectivity maps are generated by
combining ML analysis with subject matter expertise. ML is used at various
stages in the PFA analysis, such as extracting hidden signatures from curated
field data, transferring knowledge across sites, and discovering physics
information from site-specific process models.

###### Acknowledgements.

This research is based upon work supported by the U.S. Department of Energy’s
(DOE) Office of Energy Efficiency and Renewable Energy (EERE) under the
Geothermal Technology Office (GTO) Machine Learning (ML) for Geothermal Energy
funding opportunity, Award Number DE-EE-3.1.8.1. Los Alamos National
Laboratory is operated by Triad National Security, LLC, for the National
Nuclear Security Administration of the U.S. Department of Energy (Contract No.
89233218CNA000001). Pacific Northwest National Laboratory is operated for the
DOE by Battelle Memorial Institute under contract DE-AC05-76RL01830. The views
and opinions of authors expressed herein do not necessarily state or reflect
those of the United States Government or any agency thereof.

## Nomenclature

  * •

AI: Artificial Intelligence

  * •

DOE: Department of Energy

  * •

EGS: Enhanced Geothermal Systems

  * •

EERE: Energy Efficiency and Renewable Energy

  * •

EGS: Enhanced Geothermal Systems

  * •

ePFA: ML-enhanced Play Fairway Analysis

  * •

FORGE: Frontier Observatory for Research in Geothermal Energy

  * •

GeoDAWN: Geoscience Data Acquisition for Western Nevada

  * •

GTO: Geothermal Technologies Office

  * •

GW: GigaWatts

  * •

InSAR: Interferometric Synthetic-Aperture Radar

  * •

ML: Machine Learning

  * •

NMF: Non-negative Matrix Factorization

  * •

NMF$k$: NMF + custom $k$-mean clustering

  * •

PFA: Play Fairway Analysis

  * •

SME: Subject Matter Expertise

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#  Adapted poling to break the nonlinear efficiency limit in nanophotonic
lithium niobate waveguides

Pao-Kang Chen  J. C. Wyant College of Optical Sciences, University of Arizona,
1630 E. University Boulevard, Tucson, Arizona 85721, USA  Department of
Physics, University of Arizona, 1118 E. Fourth Street, Tucson, Arizona 85719,
USA  Ian Briggs  J. C. Wyant College of Optical Sciences, University of
Arizona, 1630 E. University Boulevard, Tucson, Arizona 85721, USA  Chaohan Cui
J. C. Wyant College of Optical Sciences, University of Arizona, 1630 E.
University Boulevard, Tucson, Arizona 85721, USA  Liang Zhang  J. C. Wyant
College of Optical Sciences, University of Arizona, 1630 E. University
Boulevard, Tucson, Arizona 85721, USA  Manav Shah  J. C. Wyant College of
Optical Sciences, University of Arizona, 1630 E. University Boulevard, Tucson,
Arizona 85721, USA  Linran Fan<EMAIL_ADDRESS>J. C. Wyant College of
Optical Sciences, University of Arizona, 1630 E. University Boulevard, Tucson,
Arizona 85721, USA

Nonlinear frequency mixing is of critical importance in extending the
wavelength range of optical sources Schliesser et al. (2012); Ghimire and Reis
(2019); Buryak et al. (2002). It is also indispensable for emerging
applications such as quantum information O’brien et al. (2009); Wehner et al.
(2018) and photonic signal processing He (2002); Cerullo and De Silvestri
(2003); Langrock et al. (2006). Conventional lithium niobate with periodic
poling is the most widely used device for frequency mixing due to the strong
second-order nonlinearity. The recent development of nanophotonic lithium
niobate waveguides promises improvements of nonlinear efficiencies by orders
of magnitude with sub-wavelength optical conferment. However, the intrinsic
nanoscale inhomogeneity in nanophotonic lithium niobate limits the coherent
interaction length, leading to low nonlinear efficiencies Zhang et al. (2022);
Chang et al. (2016); Boes et al. (2019); Wang et al. (2018a); Rao et al.
(2019); Zhao et al. (2020); Chen et al. (2022). Therefore, the performance of
nanophotonic lithium niobate waveguides is still far behind conventional
counterparts. Here, we overcome this limitation and demonstrate ultra-
efficient second-order nonlinearity in nanophotonic lithium niobate waveguides
significantly outperforming conventional crystals Umeki et al. (2010);
Kashiwazaki et al. (2020); Parameswaran et al. (2002a, b); Suntsov et al.
(2021); Cho et al. (2021); Berry et al. (2019); Carpenter et al. (2020). This
is realized by developing the adapted poling approach to eliminate the impact
of nanoscale inhomogeneity in nanophotonic lithium niobate waveguides. We
realize overall second-harmonic efficiency near $10^{4}$ %/W without cavity
enhancement, which saturates the theoretical limit. Phase-matching bandwidths
and temperature tunability are improved through dispersion engineering. The
ideal square dependence of the nonlinear efficiency on the waveguide length is
recovered. We also break the trade-off between the energy conversion ratio and
pump power. A conversion ratio over 80 % is achieved in the single-pass
configuration with pump power as low as 20 mW. Our work could provide highly
scalable and extremely efficient solutions for next-generation nonlinear
optical sources, amplifiers, and converters pivotal for a wide range of
classical and quantum applications such as ultra-fast optics and photonic
quantum networks.

Lithium niobate is one of the most important materials for photonics because
of the strong second-order nonlinearity across a broad transparency window
Weis and Gaylord (1985). Therefore, it is widely used in diverse applications
ranging from fiber communications to quantum information science Wooten et al.
(2000); Elshaari et al. (2020). Most photonic applications of lithium niobate
are based on the electro-optic effect and second-order nonlinear-optical
processes. Recently, significant efforts have been devoted to the development
of the nanophotonic lithium niobate platform. The sub-wavelength confinement
of optical fields can significantly increase the nonlinear strength and enable
unique capabilities for dispersion engineering and large-scale integration Zhu
et al. (2021); Boes et al. (2023).

The nanophotonic lithium niobate platform has achieved remarkable advances in
applications based on the electro-optic effect. Compared with conventional
electro-optic modulators, significant improvements have been demonstrated in
almost all key merits including modulation efficiency, bandwidth, long-term
stability, and communication data rate Wang et al. (2018b); Xu et al. (2020);
Li et al. (2020). Besides the performance improvement, nanophotonic lithium
niobate modulators have also been used to demonstrate novel photonic functions
such as optical comb generation, non-reciprocal propagation, and quantum
transduction Zhang et al. (2019); Shah et al. (2023); Xu et al. (2021).

Figure 1: Adapted poling for nanophotonic lithium niobate waveguides with
nanoscale inhomogeneity a, Device schematic showing the change of poling
periods depending on local structure variations. b, Phase diagrams of second-
harmonic generation in nanophotonic lithium niobate waveguides with periodic
and adapted poling under structure variations, showing random and consistent
phase accumulation respectively. c, Simulated phase-matching wavelength for
different waveguide thicknesses with a fixed poling period. Inserts:
fundamental and second-harmonic mode profiles at 1550 nm and 775 nm
respectively. d, Measured thickness of the lithium niobate device layer.
Inset: measured reflection spectrum for 607.11 nm lithium niobate layer
thickness. e, Calculated second-harmonic spectrum of nanophotonic lithium
niobate waveguides with different lengths, assuming periodic poling and
thickness variation in d. f, Calculated peak second-harmonic efficiency of
nanophotonic lithium niobate waveguides with different lengths using periodic
(red) and adapted (blue) poling approaches.

Similarly, it is expected that the nanophotonic lithium niobate platform can
also bring revolutionized performance for applications based on second-order
nonlinear-optical processes. Periodically poled lithium niobate waveguides
have been the standard device structure for second-order nonlinear-optical
processes Umeki et al. (2010); Kashiwazaki et al. (2020); Parameswaran et al.
(2002a, b); Suntsov et al. (2021); Cho et al. (2021); Berry et al. (2019);
Carpenter et al. (2020). It has been widely demonstrated that nanophotonic
lithium niobate waveguides with periodic poling can exhibit nonlinear strength
(unit %/W/cm2) orders of magnitude higher than conventional crystals Wang et
al. (2018a); Rao et al. (2019); Zhao et al. (2020); Chen et al. (2022); Chang
et al. (2016); Boes et al. (2019); Zhang et al. (2022). Unfortunately, the
increase of the nonlinear strength cannot be translated into the improvement
of the nonlinear efficiency (unit %/W), which directly determines the
performance of nonlinear processes, even after intense efforts Wang et al.
(2018a); Rao et al. (2019); Zhao et al. (2020); Chen et al. (2022); Chang et
al. (2016); Boes et al. (2019); Zhang et al. (2022).

The intrinsic nanoscale inhomogeneity in nanophotonic lithium niobate limits
the coherent interaction length of second-order nonlinear-optical processes to
a few millimeters Wang et al. (2018a); Rao et al. (2019); Zhao et al. (2020);
Chen et al. (2022); Chang et al. (2016); Boes et al. (2019); Zhang et al.
(2022). Currently, the maximum demonstrated nonlinear efficiency in
nanophotonic lithium niobate waveguides is around 940 %/W Zhao et al. (2020).
With much larger coherent interaction length, conventional lithium niobate
crystals can achieve nonlinear efficiencies 2400 %/W in spite of smaller
nonlinear strength Umeki et al. (2010). It remains an outstanding question
whether nanophotonic lithium niobate waveguides can outperform conventional
crystals to achieve efficient second-order nonlinear-optical processes.

It is noteworthy that high-efficiency second-order nonlinear processes can be
realized in lithium niobate nanophotonic cavities, since the thickness
variation is small within the length scale of integrated photonic cavities Lu
et al. (2019). However, waveguide-based second-order nonlinear processes can
be highly preferred when large bandwidth, small insertion loss, wide
tunability, and high power capability are required.

Here we overcome this challenge and demonstrate nanophotonic lithium niobate
waveguides that outperform conventional crystals in all critical merits
including nonlinear efficiency, power consumption, optical bandwidth, and
wavelength tunability. The sub-wavelength confinement of optical fields leads
to the increase of the second-order nonlinear strength but also causes the
significant sensitivity of the phase-matching condition to structure
variations. For example, a thickness change of 10 nm in the 600-nm lithium
niobate device layer can cause a phase-matching wavelength shift over 70 nm
for second-harmonic generation (Fig. 1c). Unfortunately, there is significant
thickness variation in the nanophotonic lithium niobate device layer caused by
intrinsic uncertainties during wafer manufacturing including random ion
implantation depth and non-uniform chemical-mechanical polishing rate (Fig.
1d) Bruel (1995). Non-ideal nanofabrication conditions and poor poling quality
further intensify the inhomogeneity. If the standard period poling technique
is used, second-order nonlinear-optical processes will accumulate random
phases long the nanophotonic waveguide depending on the local photonic
geometry (Fig. 1b). Such random phases prevent the constructive enhancement of
second-order nonlinear-optical processes, thus limiting the nonlinear
efficiency. The calculated second-harmonic spectrum is shown in Fig. 1e (See
Methods). Significant deviation from the theoretical sinc-squared function can
be observed. Instead of the quadratic increase, the peak second-harmonic
efficiency starts to saturate with device lengths over 3 mm and even decreases
at long device lengths (Fig. 1f). Therefore, increasing the device length
cannot improve the nonlinear efficiency. Currently, efficient second-order
nonlinear-optical processes have only been demonstrated in nanophotonic
lithium niobate waveguides with device lengths below 5 mm Chang et al. (2016);
Wang et al. (2018a); Rao et al. (2019); Boes et al. (2019); Zhao et al.
(2020); Chen et al. (2022); Zhang et al. (2022).

To compensate for the random perturbation to the phase-matching condition
caused by the nanoscale inhomogeneity, we propose the adapted poling approach
where the poling period is adjusted depending on the local photonic structure
(Fig. 1a). Then, ideal quasi-phase-matching conditions can be realized along
the entire waveguide despite the nanoscale inhomogeneity (Fig. 1b). No extra
optical loss will be introduced as there is no sudden refractive index change.
The constructive enhancement at different waveguide sections leads to the
quadratic increase of the second-harmonic efficiency with respect to the
device length (Fig. 1f).

Figure 2: Performance comparison between periodic and adapted poling
approaches. a, Optical image of fabricated poling electrodes with different
lengths. b, False-color scanning electron microscopy (SEM) image of one poling
electrode pair. c, Piezoresponse force microscopy showing the high-quality
domain inversion of lithium niobate. The waveguide position is labeled with
dashed lines. d, e, f, g, measured the second-harmonic generation spectrum of
nanophotonic lithium niobate waveguides with periodic poling and 3-mm device
length (d), adapted poling and 3-mm device length (e), periodic poling and
21-mm device length (f), and adapted poling and 21-mm device length (g). The
measured thickness of the lithium niobate device layer and poling period are
shown in the insets. The dashed lines are calculated second-harmonic spectrum
using the measured device layer thickness.

We use X-cut nanophotonic lithium niobate wafers with a 600-nm device layer to
fabricate nanophotonic lithium niobate waveguides (See Methods and Extended
Data Fig. 1). The waveguide direction is aligned with the Y-axis of the
lithium niobate crystal. We use the fundamental transverse-electric mode for
both fundamental and second-harmonic fields around 1550 nm and 775 nm
respectively (Insets of Fig. 1c). This allows the use of the largest nonlinear
coefficient $d_{33}=19.5$ pm/V. It is noteworthy that the nonlinear
coefficient changes with the wavelength Shoji et al. (1997). As the accurate
measurement of the nonlinear coefficient for lithium niobate at 1550 nm has
not been performed, we use the value measured at 1313 nm, which has the
smallest wavelength difference with our work Shoji et al. (1997). Before
fabrication, we measure the device layer thickness with broadband optical
reflectance along the designed waveguide location (Insets of Fig. 1d). Then we
pattern the electrode for ferroelectric domain inversion with different poling
regime lengths (Fig. 2a&b). We fabricate two types of nanophotonic lithium
niobate waveguides. The first type uses the standard periodic poling approach
where the distance between adjacent electrodes fingers is fixed. The second
type uses the adapted poling approach where the distance between adjacent
electrodes fingers is adjusted based on the measured device layer thickness.
High-quality ferroelectric domain inversion with average duty cycle $49\%\pm
4\%$ can be achieved by applying optimized high-voltage pulses (Fig. 2c). Then
each waveguide is patterned with multi-pass electron-beam lithography to avoid
the waveguide structure distortion at writing field edges. We also keep
sufficient distance between waveguides and chip edges to minimize the
influence of the argon-plasma etching non-uniformity.

With 3-mm-long waveguides, both periodic and adapted poling approaches show a
second-harmonic spectrum close to the theoretical sinc-squared function (Fig.
2d&e). Peak efficiencies are also similar with values of 135 %/W and 138 %/W
respectively. This agrees with previous observations that improvements in the
second-order nonlinear-optical performance can be achieved with short
waveguides Chang et al. (2016); Wang et al. (2018a); Rao et al. (2019); Boes
et al. (2019); Zhao et al. (2020); Chen et al. (2022); Zhang et al. (2022), as
the perturbation to the phase-matching condition caused by the nanoscale
inhomogeneity is still relatively small compared with the phase-matching
bandwidth. However, the situation changes dramatically with long waveguides.
We compare the second-harmonic performance of 21-mm-long waveguides with
periodic and adapted poling approaches. The second-harmonic spectrum with
periodic poling deviates significantly from the theoretical sinc-squared
function (Fig. 2f). The second-harmonic spectrum shows multiple peaks as
different waveguide sections have different phase-matching conditions. The
peak efficiency is measured as 1100 %/W, which is significantly lower than the
theoretical limit of 10500 %/W. Using the measured thickness data, we
numerically calculate the second-harmonic spectrum only considering the local
phase-matching condition change due to the nanoscale inhomogeneity. The strong
agreement between calculated and experimental results shows that the nanoscale
inhomogeneity is the major factor limiting the overall second-order nonlinear-
optical efficiency (Fig. 2f).

With the adapted poling approach, the second-harmonic spectrum still shows the
theoretical sinc-squared function. Compared with the 3-mm-long waveguides, the
second-harmonic spectrum shows much higher peak efficiency and narrower
bandwidth (Fig. 2g). This shows that the nanoscale inhomogeneity is
compensated by the poling period adjustment along the waveguide. We measure
peak efficiency as high as 9500$\pm$1000 %/W close to the theoretical limit
10500 %/W.

The bandwidth of the second-harmonic generation is measured as 0.97 nm in the
21-mm-long waveguides, which is twice larger than conventional lithium niobate
crystals with the same length. This is achieved with the smaller group-
velocity mismatch between fundamental and second-harmonic fields (164 fs/mm),
enabled by the dispersion engineering capability in nanophotonic lithium
niobate waveguides. We further observe that the peak wavelength can be tuned
more efficiently in nanophotonic lithium niobate waveguides, reaching 0.2
nm/oC temperature tunability which is better than the temperature tunability
of conventional lithium niobate crystals (HC Photonics Corp.) 0.15 nm/oC (See
Methods).

Next, we measure the peak efficiency dependence on the waveguide length with
both standard periodic poling and adapted poling approaches (Fig. 3a). For the
periodic poling approach, the peak efficiency only shows a marginal increase
at small waveguide lengths and saturates at large waveguide lengths. In
contrast, the peak efficiency with the adapted poling approach follows the
quadratic dependence on the waveguide length, matching the theoretical
prediction for ideal second-order nonlinear processes in lithium niobate
waveguides.

Figure 3:  Ultra-efficient nanophotonic lithium niobate waveguides. a, Peak
second-harmonic efficiency with different waveguide lengths. The dashed line
shows the square dependence on the waveguide length with a normalized
efficiency 2044 %/W/cm2 averaged across all waveguides (see Methods). b, Self-
calibrated efficiency ratio characterizing the impact of the nanoscale
inhomogeneity. The ideal value (unity) is labeled with the dashed line. c,
Absolute power conversion efficiency of the 21-mm-long waveguide with adapted
poling.

Difficulties in the direct measurement of fiber-waveguide coupling
efficiencies and the absolute calibration of power detectors cause the major
uncertainty source for the estimation of the second-order nonlinear
performance. Here we derive the self-calibrated parameter to characterize the
impact of the nanoscale inhomogeneity without requiring absolute power and
loss calibration (See Methods). We first integrate the measured second-
harmonic efficiency over wavelength to obtain the spectrum area $A$. Then we
assume an ideal sinc-squared second-harmonic efficiency spectrum with peak
efficiency $\eta{}_{0}$, and calculate the corresponding spectrum area $A_{0}$
which is proportional to the ideal peak efficiency $\eta{}_{0}$. As the
nanoscale inhomogeneity conserves the spectrum area up to the dispersion
correction, we can use the measured spectrum area $A$ to calibrate the ideal
spectrum area $A_{0}$, thus obtaining the ideal peak efficiency $\eta{}_{0}$
under the same loss condition with the experiment Helmfrid et al. (1993). Then
the ratio between the measured raw second-harmonic peak efficiency ($\eta{}$)
and the ideal peak efficiency ($\eta{}_{0}$) gives the self-calibrated
evaluation of the nanoscale inhomogeneity without uncertainties from optical
power and loss calibration (Fig. 3b). The efficiency ratio for the adapted
poling approach stays constant around unity for all waveguide lengths, while
the efficiency ratio for the periodic poling approach drops rapidly to 0.2
when the waveguide length is beyond 6 mm.

Figure 4:  Towards ultimate second-order nonlinearity performance. Comparison
of overall second-harmonic efficiencies with different device lengths (a) and
absolution conversion efficiency with different pump powers (b) for this work
(blue), previous nanophotonic lithium niobate waveguides (red), and
conventional lithium niobate devices (orange). In a, nanophotonic lithium
niobate waveguide limit with adapted poling (dashed blue) assumes the
theoretical normalized efficiency 2388 %/W/cm2, and the conventional limit
(dashed orange) assumes the maximum demonstrated value 150 %/W/cm2
Parameswaran et al. (2002a). In b, the state-of-the-art for nanophotonic
lithium niobate waveguides with inhomogeneity (dashed red) and conventional
crystals (dashed orange) assume 1000 %/W and 2400 %/W overall second-harmonic
efficiency Zhao et al. (2020); Umeki et al. (2010). References for the data
can be found in Extended Data Table 1.

The optimized second-harmonic generation allows us to realize high absolute
conversion efficiency with low pump power. Using the 21-mm-long waveguide with
adapted poling, we observe the onset of pump depletion and second-harmonic
saturation with pump power below 5 mW. We have measured second-harmonic power
of 16.5 mW with a pump power of 20 mW, corresponding to 82.5 % absolute power
conversion efficiency (Fig. 3c). By fitting the absolute conversion efficiency
with the pump-depletion model, we obtain the overall second-harmonic
efficiency around 9500$\pm$500 %/W, which is consistent with the value
measured in the non-depletion region.

We further compare our nanophotonic lithium niobate waveguides using adapted
poling with lithium niobate devices using periodic poling in previous works.
The overall second-harmonic efficiency of our device exceeds both the
conventional crystal limit and nanophotonic inhomogeneity limit significantly
(Fig. 4a). Furthermore, we summarize the demonstrated absolute conversion
efficiency with both conventional and nanophotonic lithium niobate devices
(Fig. 4b). In contrast to previous conventional and nanophotonic lithium
niobate platforms in which high conversion efficiency can only be achieved
with strong pump power, our nanophotonic lithium niobate waveguide with
adapted poling breaks the trade-off between conversion efficiency and pump
power. We expect that further improvement can be readily made by increasing
the length of nanophotonic lithium niobate waveguides. With 4-inch
nanophotonic lithium niobate wafers, it is practical to fabricate nanophotonic
lithium niobate waveguides with lengths over 90 mm. Using adapted poling, this
can lead to second-harmonic efficiency above $2\times 10^{5}$ %/W without
cavity enhancement. Pump power below 1 mW will be sufficient to achieve the
same conversion efficiency demonstrated in this work. Even a smaller footprint
can be realized by using meandering waveguides.

In summary, we present for the first time an nanophotonic lithium niobate
waveguide with overall second-order nonlinear efficiency beyond the
nanophotonic inhomogeneous limit and the conventional crystal limit. This is
achieved using the adapted poling approach to compensate for the nanoscale
inhomogeneity in the nanophotonic lithium niobate platform. The capability to
realize ultra-high second-order nonlinear efficiency with an integrated
platform is critical for future photonic systems with strong requirements for
low power consumption and large-scale manufacturing compatibility. The adapted
poling approach can be readily extended to other wavelength ranges and second-
order nonlinear interactions. It can also be used to enhance the Kerr
nonlinearity through the cascaded second-order nonlinear process Cui et al.
(2022a, b). Therefore, this work could have a profound impact on a wide range
of nonlinear photonic processes such as optical parametric oscillation and
supercontinuum generation in the classical regime, as well as entanglement
generation and quantum frequency conversion in the quantum regime. The
dramatic performance improvement of these second-order nonlinear processes
could open new opportunities for broad applications including ultra-short
optical pulse generation, the interconnection between diverse quantum systems,
and all-photonic signal processing.

Acknowledgments This work is supported in part by the U.S. Department of
Energy, Office of Advanced Scientific Computing Research (Field Work Proposal
ERKJ355); Office of Naval Research (N00014-19-1-2190); NSF-ERC Center for
Quantum Networks (EEC-1941583). The reactive ion ether utilized in this study
was acquired through an NSF MRI grant (ECCS-1725571). Device fabrication is
performed in the OSC cleanroom at the University of Arizona.

Author contributions P.C. and L.F. conceived the experiment. P.C. fabricated
the device, performed the measurement, analyzed the data, and developed the
simulation in useful discussion with C.C. and L.Z.. P.C. and I.B. optimized
the poling condition. I.B. and M.S. optimized the fabrication procedure. P.C.
and L.F. wrote the manuscript. L.F. supervised the work.

Competing interests All information in this work is covered by a pending
patent application (Application Number 63302331) filed by Pao-Kang Chen and
Linran Fan.

Data availability The data generated and/or analyzed in this work are
available from the corresponding author upon reasonable request.

Methods

Device fabrication

We first fabricate alignment markers for the thickness measurement, the
patterning of electrodes, and the patterning of photonic waveguides. After the
thickness measurement, nickel electrodes are patterned on top of the lithium
niobate device layer. The electrode period is obtained using Extended Data
Fig. 2. The gap between electrodes is 15 µm. We apply 350 V electric pulses
with 550 ms duration to nickel electrodes for domain inversion. Then nickel
electrodes are removed by hydrochloric acid. Electron-beam lithography is used
to define photonic waveguides with hydrogen silsesquioxane resist. Waveguides
patterns are aligned in the poling regime. Then argon-based plasma is used to
transfer waveguide patterns into the lithium niobate device layer (600 nm
total thicknesses) with 350 nm etching depth. The waveguide top width is 1.8
µm. SEM image of the fabricated waveguide is shown in Extended Data Fig. 1.

Device characterization and measurement uncertainty

Light is launched into and collected from the waveguide with a pair of lensed
fibers. A continuous-wave telecom tunable laser is used as the pump. The
polarization is controlled by an in-line fiber polarization controller.
Transmitted fundamental and second-harmonic fields are separated with a fiber
wavelength-division multiplexer, and detected with InGaAs and Si PIN
photodetectors respectively. Erbium-Doped Fiber Amplifier is used after the
telecom laser for high-power measurement, and a band-pass filter is used to
remove the amplified spontaneous emission noise.

The mean transmission of waveguides is $T_{\rm f,mean}=0.085$ at the
fundamental wavelength and $T_{\rm sh,mean}=0.040$ at the second-harmonic
wavelength. Fiber-waveguide coupling loss and waveguide propagation loss are
both included. The propagation loss is 0.76 dB/cm at the fundamental
wavelength and 0.69 dB/cm at the second-harmonic wavelength, inferred from on-
chip resonator quality factors. With fundamental power $P_{\rm f}$ in the
input fiber and second-harmonic power $P_{\rm sh}$ in the output fiber, the
nonlinear efficiency for each waveguide is calculated as

$\displaystyle\eta=\frac{P_{\rm sh}}{\sqrt{T_{\rm sh}}T_{\rm f}P_{\rm f}^{2}}$
(1)

The uncertainty is mainly due to the calibration of the transmissions for both
fundamental and second-harmonic fields. We estimate the relative uncertainty
of the on-chip input fundamental power is $\sim$7 %, the uncertainty of the
waveguide transmission is $\sim$9 %, and the relative uncertainty of the on-
chip generated second-harmonic power fluctuation is $\sim$5 %. Following the
error propagation rule, we calculated the relative uncertainty of the
nonlinear efficiency as 12.5 %. For the 21-mm-long adapted poling waveguide,
this leads to $\pm$1000 %/W absolute uncertainty.

Calculation of second-harmonic spectrum

We use the finite element method to simulate the effective reflective index of
both the fundamental field and the second-harmonic field with different
waveguide thicknesses, and then calculate the corresponding poling period for
the target fundamental wavelength $\lambda$ (Extended Data Fig. 1). Next, we
numerically solve the second-harmonic generation process in the non-depletion
regime. We consider $2N$ sections with alternating second-order nonlinearity
coefficient

$\displaystyle\chi_{n}=\chi_{0}(-1)^{n},n=1,2,...,2N$ (2)

The section length $L_{n}$ is half of the poling period. For the standard
periodic poling approach, $L_{n}$ is constant. For the adapted poling
approach, $L_{n}$ depends on the thickness of the $n$th section.

Each section is divided into $M$ cells with equal lengths $L_{n}/M$.
Therefore, the total number of cells is $2N\cdot M$. With input fundamental
wavelength $\lambda$, the fundamental and second-harmonic fields in the $m$th
cell of the $n$th section ($E_{n,m}^{\rm f}(\lambda)$ and $E_{n,m}^{\rm
sh}(\lambda)$) can be calculated from the fields in previous sections.

$\displaystyle E_{n,m}^{\rm f}(\lambda)=$ $\displaystyle E_{n,m-1}^{\rm
f}(\lambda)\cdot\exp{[ik_{n,m}^{\rm f}(\lambda)L_{n}/M]}$ (3) $\displaystyle
E_{n,m}^{\rm sh}(\lambda)=$ $\displaystyle E_{n,m-1}^{\rm
sh}(\lambda)\cdot\exp{[ik_{n,m}^{\rm sh}(\lambda)L_{n}/M]}$
$\displaystyle+\chi_{0}(-1)^{n}[E_{n,m}^{\rm f}(\lambda)]^{2}L_{n}/M$

$\displaystyle E_{n,1}^{\rm f}(\lambda)=$ $\displaystyle E_{n-1,M}^{\rm
f}(\lambda)\cdot\exp{[ik_{n,1}^{\rm f}(\lambda)L_{n}/M]}$ (4) $\displaystyle
E_{n,1}^{\rm sh}(\lambda)=$ $\displaystyle E_{n-1,M}^{\rm
sh}(\lambda)\cdot\exp{[ik_{n,1}^{\rm sh}(\lambda)L_{n}/M]}$
$\displaystyle+\chi_{0}(-1)^{n}[E_{n,1}^{\rm f}(\lambda)]^{2}L_{n}/M$

with $k_{n,m}^{\rm f}$ and $k_{n,m}^{\rm sh}$ the wavevectors for the
fundamental and second-harmonic fields respectively. Initial conditions are
set as

$\displaystyle E_{1,1}^{\rm f}(\lambda)=$ $\displaystyle E_{0}$ (5)
$\displaystyle E_{1,1}^{\rm sh}(\lambda)=$ $\displaystyle 0$

Then the overall second-harmonic efficiency at wavelength $\lambda$ is given
by

$\displaystyle\eta(\lambda)\propto\big{|}E_{N,M}^{\rm
sh}(\lambda)\big{|}^{2}/|E_{0}|^{4}$ (6)

In this calculation, we assume low pump power therefore the second-harmonic
generation is in a non-depletion regime. We also ignore the change of the
second-order nonlinear coefficient ($\chi_{0}$) due to the optical mode size.
Therefore, only the local phase-matching condition change due to the nanoscale
inhomogeneity is included.

Self-calibrated second-harmonic generation performance

The area of the second-harmonic efficiency spectrum is calculated as

$\displaystyle A=\int\eta(\lambda)\,d\lambda$ (7)

where $\eta(\lambda)$ is the measured second-harmonic efficiency. The measured
peak efficiency is $\eta=\max[\eta(\lambda)]$. We can also calculate the
spectrum area assuming the ideal phase-matching condition along the entire
waveguide Helmfrid et al. (1993)

$\displaystyle A_{0}$ $\displaystyle=\alpha\eta_{0}/L$ (8)

where $\eta_{0}$ is the peak efficiency for the ideal second-harmonic spectrum
and $\alpha=2\pi(d\Delta k/d\lambda)^{-1}$ is the dispersion factor with
$\Delta k$ the wavevector mismatch between fundamental and second-harmonic
fields. Under the same loss condition, the spectrum area is conserved under
inhomogeneous broadening Helmfrid et al. (1993).

$\displaystyle A$ $\displaystyle=A_{0}$ (9)

Therefore, we can get the ideal peak efficiency $\eta_{0}$ under the same loss
condition, but with the ideal phase-matching condition along the entire
waveguide.

$\displaystyle\eta_{0}$ $\displaystyle=AL/\alpha$ (10)

Then we can have the self-calibrated efficiency ratio between the measured and
ideal peak efficiencies

$\displaystyle R=\frac{\eta}{\eta_{0}}=\frac{\eta\alpha}{AL}$ (11)

Assuming uncalibrated extra losses $t_{1}$ and $t_{2}$ for the input
fundamental and output second-harmonic fields respectively, we have

$\displaystyle\frac{\eta}{A}=\frac{\eta^{\prime}t_{2}t_{1}^{-2}}{\int\eta^{\prime}(\lambda)t_{2}t_{1}^{-2}\,d\lambda}=\frac{\eta^{\prime}}{A^{\prime}}$
(12)

where $\eta^{\prime}$ and $A^{\prime}$ are the second-harmonic efficiency and
spectrum area with perfect loss calibration. Eq. (12) shows that the
efficiency ratio $R$ is independent of loss and power calibration.

Using the ideal peak efficiency $\eta_{0}$ (Eq. 10), we can also obtain the
intrinsic normalized efficiency without the impact of nanoscale inhomogeneity

$\displaystyle\eta_{0,norm}=\eta_{0}/L^{2}=A/L\alpha$ (13)

The average intrinsic normalized efficiency across all waveguides is
2044$\pm$150 %/W/cm2. This value is slightly smaller than the theoretical
normalized efficiency 2388 %/W/cm2 mainly due to the deviation of poling duty
cycles from 50%.

Temperature tunability

The phase-matching condition can be written as

$\displaystyle\frac{\pi}{\Lambda}=\frac{2\pi(n_{2}-n_{1})}{\lambda}$ (14)

where the poling period $\Lambda$ depends on temperature $T$ due to thermal
expansion, and fundamental and second-harmonic effective refractive indices
($n_{1}$ and $n_{2}$) depend both on the temperature $T$ and wavelength
$\lambda$. By differentiating Eq. (14), we obtain the temperature tunability

$\displaystyle\frac{d\lambda}{dT}=\frac{\frac{\partial\Lambda}{\partial
T}(n_{2}-n_{1})+(\frac{\partial n_{2}}{\partial T}-\frac{\partial
n_{1}}{\partial T})\Lambda}{\frac{1}{2}-(\frac{\partial
n_{2}}{\partial\lambda}-\frac{\partial n_{1}}{\partial\lambda})\Lambda}$ (15)

The simulated temperature tunability of nanophotonic lithium niobate
waveguides is 0.34 nm/oC, in contrast to 0.13 nm/oC for conventional lithium
niobate. This is mainly attributed to dispersion control. Instead of a
constant negative dispersion value in conventional lithium niobate
($(\frac{\partial n_{2}}{\partial\lambda}-\frac{\partial
n_{1}}{\partial\lambda})\Lambda={{{\color[rgb]{0,0,0}-0.608}}}$), integrate
waveguides show positive dispersion value ($(\frac{\partial
n_{2}}{\partial\lambda}-\frac{\partial
n_{1}}{\partial\lambda})\Lambda={{{\color[rgb]{0,0,0}0.363}}}$), boosting the
temperature tunability (Extended Data Fig. 3 and 4).

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