# Learning From High-Dimensional Cyber-Physical Data Streams for Diagnosing
Faults in Smart Grids

Hossein Hassani, , Ehsan Hallaji, , Roozbeh Razavi-Far, , Mehrdad Saif Hossein
Hassani, Ehsan Hallaji, and Mehrdad Saif are with the Department of Electrical
and Computer Engineering, University of Windsor, Windsor, ON N9B 3P4, Canada
(e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>msaif@uwindsor.ca).Roozbeh
Razavi-Far is with the Faculty of Computer Science and Canadian Institute for
Cybersecurity, University of New Brunswick, Fredericton, NB E3B 5A3, Canada,
and also with the Department of Electrical and Computer Engineering,
University of Windsor, Windsor, ON N9B 3P4, Canada (e-mail: roozbeh.razavi-
far@unb.ca).

###### Abstract

The performance of fault diagnosis systems are highly affected by data quality
in cyber-physical power systems. These systems generate massive amounts of
data that overburden the system with excessive computational costs. Another
issue is the presence of noise in recorded measurements, which prevents
building a precise decision model. Furthermore, the diagnostic model is often
provided with a mixture of redundant measurements that may deviate it from
learning normal and fault distributions. This paper presents the effect of
feature engineering on mitigating the aforementioned challenges in cyber-
physical systems. Feature selection and dimensionality reduction methods are
combined with decision models to simulate data-driven fault diagnosis in a
118-bus power system. A comparative study is enabled accordingly to compare
several advanced techniques in both domains. Dimensionality reduction and
feature selection methods are compared both jointly and separately. Finally,
experiments are concluded, and a setting is suggested that enhances data
quality for fault diagnosis.

###### Index Terms:

Feature selection, dimensionality reduction, classification, fault diagnosis,
cyber-physical power systems.

##  I Introduction

Fault diagnosis system (FDS) is a critical component of cyber-physical power
systems that is crucial for detecting malfunctions, identifying their cause,
and pinpointing their location in the grid [1]. Nevertheless, designing a
traditional model-based FDS model becomes more challenging as the grid grows
in scale and complexity. The integration of intelligent models with power
systems results in a cyber-physical system that bypasses the aforementioned
problem [2]. Making use of data-driven approaches for diagnostic purposes
provides the model with a better generalization. Furthermore, intelligent
models facilitate the decision-making process in control systems by reducing
human-machine interaction [3].

Data-driven FDS is primarily carried out using classification algorithms [4,
5, 6]. Having a set of sampled records corresponding to different system
states, one can build a statistical model that maps incoming signal patterns
to certain conditions such as different events of faults in the systems [7].
In a fault detection setting, the FDS model can be formulated as an anomaly
detector that reveals any signal pattern mismatches those of the normal or
healthy system state. The past decade has witnessed numerous advancements in
intelligent FDS models that resort to sensory data for monitoring power
systems [8, 9]. Nevertheless, the computational efficiency of intelligent
models and their dependency on data quality limit the feasibility of such
methods for FDS in power systems.

Data-driven methods only perform well if supplied with high-quality data.
These methods try to capture the characteristics of a data distribution based
on the existing relationships in the input space and form a mapping from
distribution samples to a set of states in the system [10]. However, if the
input distribution at hand is not a good representative of the system, the
statistical model cannot capture the system characteristics accordingly. In
power systems, this situation happens due to a number of reasons. Firstly, the
captured signal through sensors often contains noise, which makes it harder
for machine learning models to capture the intrinsic relationships within
data. Furthermore, the abundance of the set of measurements and the number of
sensors (e.g., phasor measurement units) result in high-dimensional data [11].
The higher the dimensionality size of the data, the more samples are needed
for obtaining an accurate statistical model. This phenomenon is also referred
to as the curse of dimensionality in the literature. In turn, processing high-
dimensional data significantly increases the computational burden of the FDS.
In addition, the utilized measurements in the power system most likely result
in some invariant and duplicate features that feed the FDS model with
irrelevant information that deteriorates the precision of the constructed
model.

Feature selection (FS) and dimensionality reduction (DR) techniques are two
main approaches that are commonly used in tackling the curse of dimensionality
and improving the data quality to ensure optimal performance of the diagnostic
model. FS is referred to the process of filtering or ranking different
features (i.e., dimensions of the feature space) to remove non-informative
features and select only a limited set of features that lead to the optimal
performance of decision-making models [12, 13, 14]. On the other hand, DR
methods transform the whole feature space into a smaller space [15, 16, 17].
Under a supervised setting, DR may additionally improve distribution by making
classes more distinguishable in the transformed space [18, 19].

In this paper, an experimental review is performed on state-of-the-art FS and
DR methods for diagnosing faults in cyber-physical power systems. Both
mentioned harsh conditions, namely noisy signals (with different levels and
frequencies) and high-dimensional data are taken into account throughout the
experiments. This comparative study is enabled by analyzing FS and DR methods
in three study groups: 1) FS, 2) DR, and 3) FS and DR. Finally, we suggest the
best methods that will most likely benefit FDS in real-world power systems,
where similar conditions are expected.

The remainder of this paper is organized as follows. Section II presents the
employed case-study and data. Section III explains the proposed methodology.
Section IV contains the experimental results and analysis. Finally, the paper
is concluded in Section V.

Figure 1: The single-line diagram of the IEEE 118-bus system [20].

##  II Case-Study: Cyber-Physical Power Systems

In the present study, the IEEE 118-bus system has been employed in order to
check for the effectiveness of the proposed diagnostic methodology. The
single-line diagram of this system has been illustrated in Fig. 1. This system
contains, as the name stands, 118 buses, 91 loads and 19 generation units. The
system is simulated using PowerFactory, and data measurements are collected
using a sampling rate of 10KHz for faulty and normal scenarios.

Three types of faults are simulated. These faults are named load-loss (LL),
generator outage (GO), and generator ground (GG) faults. To simulate an LL or
GO fault, a breaker has been placed between the load or generator and the
corresponding bus. The breaker is initially closed, and, then, it is triggered
for 25 ms to disconnect the load or generator from the bus. As for the GG
faults, three-phase short-circuit faults are simulated between the generation
units and the ground. In either fault case, data measurements are collected
from the moment the fault appears in the system until the moment the fault has
been cleared. This period of time has been set to be 25 ms and with a sampling
rate of 10KHz, therefore, a total number of 250 samples are collected for each
scenario.

The list of simulated faults is represented in Table I. As it can be observed
from this table, there exist 31 LL fault scenarios, 19 GO, and 19 GG, in
addition to the normal operational state of the system, leading to a total
number of 70 simulated scenarios, where each one could be thought of as a
class of data to be classified. Therefore, we are dealing with a multi-class
classification problem with 70 classes. It is worth noting that as there are
19 generation units, the GO and GG faults are simulated on all of the buses
connected to a generation unit. However, for the LL faults, only 31 out of 91
possible locations are considered in the construction of the fault scenarios.
The selected buses to simulate the LL faults are tried to be from different
zones of the grid as shown in Fig. 1, in order to consider the impact of the
fault location on the performance of the proposed methodology.

In general, six datasets are constructed following the aforementioned fault
scenarios. These datasets are shown by
$\\{\mathcal{D}_{1},\ldots,\mathcal{D}_{6}\\}$ and the characteristics of each
one is summarized in Table II. We aim to investigate the effect of the signal-
to-noise-ratio (SNR) and fault resistance (FR) values on the performance of
the proposed diagnostic, too. In this regard, three different SNR values
including $\\{\text{10 dB},\text{30 dB},\text{70 dB}\\}$ are considered to
model the deep noisy measurements as well as data measurements with a slight
noise. Following this, the FR value has also been set to be either $1\Omega$
or $10\Omega$. Therefore, with three SNR and two FR values, there could be a
total of six combinations to construct the set of data, as described in Table
II.

TABLE I: Summary of the simulated fault scenarios. Fault Type |  # Scenarios
---|---
LL |  31
GO |  19
GG |  19
Normal |  1
Total |  70
TABLE II: Summary of the characteristics of the constructed datasets. Dataset | SNR (dB) | FR ($\Omega$) | # Samples | # Features
---|---|---|---|---
$\mathcal{D}_{1}$ | 10 | 1 | 17,500 | 354
$\mathcal{D}_{2}$ | 10 | 10 | 17,500 | 354
$\mathcal{D}_{3}$ | 30 | 1 | 17,500 | 354
$\mathcal{D}_{4}$ | 30 | 10 | 17,500 | 354
$\mathcal{D}_{5}$ | 70 | 1 | 17,500 | 354
$\mathcal{D}_{6}$ | 70 | 10 | 17,500 | 354

In terms of the set of features, as given in Table II, there are a total
number of 354 features. Firstly, three types of measurements are collected
from each bus of the system following a simulated scenario. These measurements
are voltage, frequency, and phase angle. Due to the fact that the system has
118 buses, and three types of features are collected from each bus, therefore,
the constructed datasets contain 354 features. Table III summarizes the set of
features.

TABLE III: Collected features to construct datasets. Measurements |  Feature
---|---
voltage |  1 to 118
frequency |  119 to 236
phase angle |  236 to 354

##  III Methodology

In line with what was previously mentioned in Section I, the general framework
of the proposed diagnostic model, as shown in Fig. 2, is to perform FS and DR
techniques on the constructed datasets described in Table II, and, then, fed
the set of selected or extracted features to three classification models
including k Nearest Neighbours (kNN), Support Vector Machine (SVM) and Random
Forest (RF) for the sake of fault diagnosis. Within this general framework, we
discuss the implementation procedure and evaluation metrics in this section.

Figure 2: General framework of the proposed methodology.

###  III-A Implementation Procedure

The first step in the implementation procedure is to standardize the given
datasets. This is of paramount importance because the feature sets (i.e.,
voltage, frequency, and phase angle) take different values from different
domains. For instance, the values of voltage measurements are per unit and are
close to 1, while the frequency measurements fluctuate around 50 Hz.
Therefore, there is a need to standardize the given datasets for the sake of
eliminating the impact of scale variation from one feature to another. In this
regard, given dataset $\mathcal{D}$, we do normalize the dataset column-wise
through the following equation:

$\displaystyle\mathcal{D}^{j}_{n}=\frac{\mathcal{D}^{j}-\mathcal{D}^{j}_{min}}{\mathcal{D}^{j}_{max}-\mathcal{D}^{j}_{min}},$
(1)

where $\mathcal{D}^{j}_{n}$ is the $j$th column of the normalized dataset with
$j=1,\ldots,354$, $\mathcal{D}^{j}_{max}$ is the maximum value of a given
column, and $\mathcal{D}^{j}_{min}$ shows the minimum value of the
corresponding column.

The normalized dataset will then be fed into the FS and DR modules in order to
select or extract the set of most informative features. As for the FS
techniques, we resort to seven techniques including Infinite Feature Selection
(InfFS) [14], relief [21], Least Absolute Shrinkage and Selection Operator
(LASSO) [13], Unsupervised Feature Selection with Ordinal Locality (UFSOL)
[22], and Concrete Feature selection based on Mutual Information (CFMI) [13].
These techniques are selected to serve two purposes. First, the selected FS
methods are categorized into filters, wrappers, and embedded. To this end, we
aim to check which category of FS methods could improve diagnostic performance
through a comparative study between different categories. Second, we aim to
look for providing a ranking for the best FS-classifier combination for the
sake of diagnosis. In the same vein, the selected DR techniques are from two
broad categories namely linear and nonlinear, depending on the type of
transformation. The employed DR techniques are Principal Component Analysis
(PCA) [15], Linear Discriminant Analysis (LDA) [19], Multi-dimensional Scaling
(MDS) [17], Locally Linear Embedding (LLE) [16] and Constrained Adversarial
Dimensionality Reduction (CADR) [18]. The aforementioned two purposes are also
considered in the implementation of DR techniques. Features of the selected FS
and DR techniques are summarized in Table IV.

TABLE IV: Characteristics of the selected FS and DR techniques. Technique | FS/DR | Type | Class
---|---|---|---
InfFS | FS | Filter | Unsupervised
relief | FS | Filter | Supervised
UFSOL | FS | Wrapper | Unsupervised
LASSO | FS | Embedded | Supervised
CFMI | FS | Embedded | Unsupervised
PCA | DR | Linear | Unsupervised
LDA | DR | Linear | Supervised
MDS | DR | Nonlinear | Unsupervised
LLE | DR | Nonlinear | Unsupervised
CADR | DR | Nonlinear | Supervised

Each FS or DR technique is then combined with a classification model (kNN, SVM
and RF). In this regard, the selected set of features is fed into the
aforementioned classification models and the classification results are then
represented in terms of the accuracy and f-measure. However, the number of
selected or extracted features by means of the FS and DR techniques needs to
be adjusted carefully. To this end, in all experiments, we start with one
feature and increase the number of features to the value, for which the
performance of the classification model does not improve significantly.

###  III-B Evaluation Metrics

To measure the performance of the FS/DR-classifier combinations, we resort to
the classification accuracy $\mathcal{A}$ and f-measure $\mathcal{F}_{1}$
values. In this regard, the constructed combinations are validated through a
10-fold cross-validation manner and the attained results are collected in
terms of accuracy and f-measure. It is evident that for binary classification
with only positive and negative classes, the accuracy index captures the ratio
of the correct decisions made by the classification model. As for the
f-measure, it could be thought of as the harmonic mean of the precision
$\mathcal{P}$ and recall $\mathcal{R}$, where the precision is a measure of
the ratio of positive samples that are correctly classified into a positive
category, while the recall measures the performance of the classification
model in classifying the positive data samples. Following this, one case
defines f-measure as given below:

$\displaystyle\mathcal{F}_{1}=\frac{2\times\mathcal{P}\times\mathcal{R}}{\mathcal{P}+\mathcal{R}}.$
(2)

In terms of the multi-class classification, however, a confusion matrix of the
following form could be constructed:

$\displaystyle\mathcal{C}=\left[\begin{array}[]{ccc}cm_{11}&\ldots&cm_{1l}\\\
\vdots&\ddots&\vdots\\\ cm_{l1}&\vdots&cm_{ll}\end{array}\right],$ (6)

With $l$ being the total number of classes. Following this structure, the
precision and recall can then be defined as follows:

$\displaystyle\mathcal{P}$
$\displaystyle=\frac{1}{l}\sum_{k=1}^{l}\frac{cm_{kk}}{\sum_{i=1}^{l}c_{ki}},$
(7) $\displaystyle\mathcal{R}$
$\displaystyle=\frac{1}{l}\sum_{k=1}^{l}\frac{cm_{kk}}{\sum_{i=1}^{l}c_{ik}}.$
(8)

Having $\mathcal{P}$ and $\mathcal{R}$ calculated, the f-measure can then be
obtained based on the Eq. 2 for a multi-class classification problem.

##  IV Simulation Results and Comparative Study

In this section, a comprehensive comparative study is provided to compare the
performance of FS/DR-classifier combinations. In this regard, the ultimate
goal is to identify what combinations perform better for the sake of fault
diagnosis in cyber-physical power systems. In terms of the FS and DR
techniques, we initially investigate that what category of methods could deal
better with the constructed datasets. In particular, FS techniques are
selected from three categories including filters, wrappers, and embedded
methods, and the selected DR techniques are either linear or nonlinear.
Therefore, the goal is to investigate which category of FS and DR methods is
superior in comparison with other categories. Furthermore, by resorting to the
overall performance of FS and DR techniques, we aim to investigate whether FS
techniques are preferred over DR techniques or vice versa.

###  IV-A Baseline

The baseline models refer to the case, in which no data reduction technique is
employed. In this regard, the original datasets are only normalized and, then,
fed into the classification models. Following this setup, the classification
results are collected in terms of accuracy and f-measure through a 10-fold
cross-validation scheme. To this end, the attained average accuracy and
f-measure values by means of the baseline models are represented in Table V.
The collected results in Table V denote that kNN outperforms the SVM and RF in
terms of average accuracy and f-measure. Furthermore, it is evident that
moving from dataset $\mathcal{D}_{1}$ to $\mathcal{D}_{6}$ the attained values
are increased because of the fact that the level of noise is decreased from 10
dB to 70 dB.

TABLE V: The attained accuracy and f-measure values by means of the baseline models. Dataset | kNN | SVM | RF
---|---|---|---
| $\mathcal{A}$ | $\mathcal{F}_{1}$ | $\mathcal{A}$ | $\mathcal{F}_{1}$ | $\mathcal{A}$ | $\mathcal{F}_{1}$
$\mathcal{D}_{1}$ | 0.4288 | 0.4413 | 0.3559 | 0.3717 | 0.4012 | 0.4014
$\mathcal{D}_{2}$ | 0.4257 | 0.4424 | 0.3480 | 0.3651 | 0.1793 | 0.1797
$\mathcal{D}_{3}$ | 0.5716 | 0.5807 | 0.3514 | 0.3676 | 0.3417 | 0.3452
$\mathcal{D}_{4}$ | 0.5786 | 0.5858 | 0.5531 | 0.5692 | 0.3403 | 0.3429
$\mathcal{D}_{5}$ | 0.6778 | 0.6848 | 0.7748 | 0.7887 | 0.5037 | 0.5053
$\mathcal{D}_{6}$ | 0.6804 | 0.6869 | 0.7211 | 0.7370 | 0.5150 | 0.5176
Avg. | 0.5604 | 0.5703 | 0.5174 | 0.5332 | 0.3802 | 0.3820

###  IV-B FS-classifier Combinations

In this section, the performance of the FS techniques along with their
combinations with the given classification models is presented in terms of the
accuracy and f-measure. The aim is to first compare the performance of the FS
techniques to see which technique is superior, which is then followed by a
discussion on the superiority of FS-classifier combinations to see which
combination could be the best for the sake of fault diagnosis in power
systems.

TABLE VI: The attained average values of the accuracy and f-measure for FS-classifier combinations. Dataset | InfFS | relief | LASSO | UFSOL | CFMI
---|---|---|---|---|---
kNN | SVM | RF | kNN | SVM | RF | kNN | SVM | RF | kNN | SVM | RF | kNN | SVM | RF
$\mathcal{D}_{1}$ | 0.54 | 0.51 | 0.60 | 0.58 | 0.56 | 0.51 | 0.58 | 0.59 | 0.65 | 0.67 | 0.60 | 0.57 | 0.58 | 0.54 | 0.66 | 0.64 | 0.60 | 0.55 | 0.56 | 0.42 | 0.62 | 0.55 | 0.58 | 0.58 | 0.59 | 0.58 | 0.73 | 0.74 | 0.56 | 0.56
$\mathcal{D}_{2}$ | 0.57 | 0.56 | 0.64 | 0.64 | 0.60 | 0.56 | 0.61 | 0.56 | 0.68 | 0.65 | 0.63 | 0.56 | 0.61 | 0.56 | 0.67 | 0.64 | 0.62 | 0.56 | 0.57 | 0.47 | 0.64 | 0.57 | 0.60 | 0.60 | 0.59 | 0.57 | 0.73 | 0.73 | 0.56 | 0.55
$\mathcal{D}_{3}$ | 0.64 | 0.68 | 0.69 | 0.73 | 0.64 | 0.67 | 0.74 | 0.77 | 0.79 | 0.82 | 0.74 | 0.76 | 0.70 | 0.73 | 0.75 | 0.78 | 0.71 | 0.72 | 0.66 | 0.60 | 0.70 | 0.65 | 0.66 | 0.59 | 0.71 | 0.71 | 0.78 | 0.79 | 0.62 | 0.61
$\mathcal{D}_{4}$ | 0.71 | 0.74 | 0.76 | 0.79 | 0.71 | 0.73 | 0.73 | 0.75 | 0.78 | 0.81 | 0.73 | 0.75 | 0.72 | 0.74 | 0.77 | 0.79 | 0.73 | 0.74 | 0.66 | 0.61 | 0.71 | 0.66 | 0.67 | 0.61 | 0.73 | 0.73 | 0.79 | 0.79 | 0.64 | 0.63
$\mathcal{D}_{5}$ | 0.80 | 0.83 | 0.83 | 0.86 | 0.79 | 0.82 | 0.74 | 0.79 | 0.78 | 0.82 | 0.75 | 0.79 | 0.76 | 0.75 | 0.80 | 0.79 | 0.78 | 0.77 | 0.75 | 0.73 | 0.78 | 0.77 | 0.74 | 0.71 | 0.82 | 0.81 | 0.82 | 0.83 | 0.71 | 0.70
$\mathcal{D}_{6}$ | 0.75 | 0.79 | 0.79 | 0.82 | 0.75 | 0.78 | 0.77 | 0.81 | 0.81 | 0.84 | 0.78 | 0.82 | 0.70 | 0.75 | 0.73 | 0.77 | 0.72 | 0.76 | 0.77 | 0.74 | 0.80 | 0.77 | 0.76 | 0.70 | 0.82 | 0.82 | 0.83 | 0.84 | 0.76 | 0.75
Avg. | 0.67 | 0.69 | 0.72 | 0.74 | 0.68 | 0.68 | 0.70 | 0.71 | 0.75 | 0.77 | 0.71 | 0.71 | 0.68 | 0.68 | 0.73 | 0.73 | 0.70 | 0.68 | 0.66 | 0.60 | 0.71 | 0.66 | 0.67 | 0.63 | 0.71 | 0.70 | 0.78 | 0.79 | 0.64 | 0.63

We begin with a comprehensive study on the performance of the given FS-
classifier combinations, where the attained results are collected in Table VI
in terms of accuracy and f-measure. We initially study the effects of the SNR
values on the performance of combinations. From the presented results in Table
VI, it could be concluded that the average values of accuracy and f-measure
increases from 0.61, 0.71, 0.77 and 0.58, 0.72, 0.78, respectively, by moving
from SNR values of 10 dB to 30 dB and 70 dB. Obviously, the lower the level of
noise, the better the performance of the combinations. In order to check for
the performance of the given combinations in dealing with different fault
resistance values, by resorting to the collected results in Table VI it could
be concluded that the average values of the accuracy and f-measure are 0.69,
0.69 and 0.70, 0.70, when the fault resistance values are 10$\Omega$ and
1$\Omega$, respectively. Therefore, the attained results denote that the value
of the fault resistance has slightly impacted the performance of the given
combinations, where it shows that a higher value of the fault resistance could
negatively impact the performance.

TABLE VII: Comparison between various FS techniques in terms of The average accuracy and f-measure for the given datasets. FS | InfFS | relief | LASSO | UFSOL | CFMI
---|---|---|---|---|---
$\mathcal{A}$ | 0.69 | 0.71 | 0.70 | 0.68 | 0.72
$\mathcal{F}_{1}$ | 0.70 | 0.72 | 0.70 | 0.63 | 0.71

In terms of the comparison between the given FS techniques, the average values
of the accuracy and f-measure w.r.t. to datasets $\mathcal{D}_{1}$ to
$\mathcal{D}_{6}$ are summarized in Table VII. Firstly, it could be observed
that CFMI outperforms the other FS techniques in terms of accuracy, which is
then followed by relief, LASSO, InfFS, and UFSOL. However, in terms of the
f-measure, the collected results denote that the relief technique outperforms
the rest of the techniques, which is followed by CFMI, LASSO, InfFS, and
UFSOL. Secondly, by taking into account the average values of the accuracy for
the filter, wrapper, and embedded techniques, the attained results denote that
all categories have almost the same performance, however, in terms of the
f-measure, the filter category of methods outperforms the other two categories
by 1%.

TABLE VIII: The attained average values of the accuracy and f-measure by means of the classification models in combination with the given FS techniques. Model |  kNN |  SVM |  RF
---|---|---|---
$\mathcal{A}$ |  0.68 |  0.74 |  0.68
$\mathcal{F}_{1}$ |  0.67 |  0.73 |  0.67

In terms of the classification models, i.e., kNN, SVM, and RF, the average
values of the accuracy and f-measure w.r.t. all combinations are summarized in
Table VIII. The collected results denote that SVM outperforms the other
classification models, while kNN and RF have shown the same performance in
dealing with the given datasets and in accordance with the constructed
combinations. Further to this, by comparing the results of Table VIII with
those presented in Table V, it is evident that FS techniques have considerably
helped with improving the performance of the given classification models.
Turning into details, it could be observed that the performance of the kNN is
improved from 0.56 to 0.68, that is from 0.51 to 0.74 and from 0.38 to 0.68
for the SVM and RF, respectively, and in terms of the average accuracy values.
In the same vein and in terms of the average f-measure values, the presented
results show that the performance of the kNN is improved from 0.57 to 0.67,
that is from 0.53 to 0.73 and from 0.38 to 0.67 for the SVM and RF,
respectively.

TABLE IX: Ranking of the DR-classifier combinations based on the accuracy and f-measure. Rank |  Combination |  Combination
---|---|---
|  Accuracy |  f-measure
1 |  CFMI-SVM |  CFMI-SVM
2 |  relief-SVM |  relief-SVM
3 |  LASSO-SVM |  InfFS-SVM
4 |  InfFS-SVM |  LASSO-SVM
5 |  CFMI-kNN |  relief-kNN
6 |  UFSOL-SVM |  relief-RF
7 |  relief-RF |  CFMI-kNN
8 |  relief-kNN |  InfFS-kNN
9 |  LASSO-RF |  InfFS-RF
10 |  LASSO-kNN |  LASSO-kNN
11 |  InfFS-RF |  LASSO-RF
12 |  InfFS-kNN |  UFSOL-SVM
13 |  UFSOL-RF |  CFMI-RF
14 |  UFSOL-kNN |  UFSOL-RF
15 |  CFMI-RF |  UFSOL-kNN

Finally, we analyze the performance of the constructed combinations in order
to rank them based on the accuracy or f-measure. In this regard, the ranked
combinations are listed in Table IX w.r.t. accuracy and f-measure. Firstly,
the given ranking in terms of the accuracy and f-measure denotes that CFMI-SVM
is the best combination in order to deal with the constructed datasets
$\mathcal{D}_{1}$ to $\mathcal{D}_{6}$. Secondly, it could be observed that in
four out of the five first combinations, the FS techniques are combined with
the SVM, showing that SVM could be of a better performance in combination with
FS techniques for the sake of fault diagnosis.

###  IV-C DR-classifier Combinations

In this section, we present the attained results by means of the DR-classifier
combinations. In this regard, we first compare DR techniques in dealing with
the given datasets in terms of accuracy and f-measure, and, then, all the
possible DR-classifier combinations are ranked based on their performance.

Same as what was presented in Section IV-B for the FS-classifier combinations,
Table X summarizes the attained average values of the accuracy and f-measure
of all DR-classifier combinations w.r.t. datasets $\mathcal{D}_{1}$ to
$\mathcal{D}_{6}$. We start with the impact of the SNR on the performance of
the given DR techniques. In this regard, we resort to the attained accuracy
and f-measure values collected in Table X and calculate the average accuracy
and f-measure for all combinations w.r.t. different SNR values. To this end,
the attained average accuracy and f-measures are 0.55, 0.62, 0.68, 0.57, 0.64,
and 0.69, respectively, for datasets with the SNR value of 10 dB, 30 dB, and
70 dB. As it was expected, the lower the value of the SNR, the lower the value
of accuracy and f-measure values, denoting that noisy measurements could
degrade the performance of DR techniques. Other than the SNR, the value of the
fault resistance could also play an important role in the performance of the
given DR techniques. By resorting to the results presented in Table X, the
average values of the accuracy and f-measure w.r.t. fault resistance values of
1$\Omega$ and 10$\Omega$ are 0.62, 0.61, and 0.64, 0.63, respectively,
denoting that both the index measures are degraded by 1% when the fault
resistance value is increased from 1$\Omega$ to 10$\Omega$. This is due to the
fact that a higher fault resistance value could decrease the signature of the
faults on the collected data measurements from the grid.

TABLE X: The attained average values of the accuracy and f-measure for DR-classifier combinations. Dataset | PCA | MDS | LDA | LLE | CADR
---|---|---|---|---|---
kNN | SVM | RF | kNN | SVM | RF | kNN | SVM | RF | kNN | SVM | RF | kNN | SVM | RF
$\mathcal{D}_{1}$ | 0.49 | 0.50 | 0.63 | 0.64 | 0.58 | 0.58 | 0.50 | 0.50 | 0.62 | 0.63 | 0.58 | 0.57 | 0.40 | 0.50 | 0.43 | 0.51 | 0.42 | 0.50 | 0.41 | 0.42 | 0.58 | 0.65 | 0.46 | 0.46 | 0.73 | 0.74 | 0.81 | 0.81 | 0.72 | 0.72
$\mathcal{D}_{2}$ | 0.48 | 0.48 | 0.64 | 0.64 | 0.57 | 0.57 | 0.48 | 0.48 | 0.62 | 0.62 | 0.57 | 0.57 | 0.39 | 0.50 | 0.42 | 0.50 | 0.42 | 0.50 | 0.43 | 0.43 | 0.54 | 0.45 | 0.46 | 0.46 | 0.73 | 0.73 | 0.80 | 0.80 | 0.70 | 0.70
$\mathcal{D}_{3}$ | 0.60 | 0.61 | 0.71 | 0.71 | 0.67 | 0.67 | 0.60 | 0.60 | 0.71 | 0.71 | 0.67 | 0.67 | 0.39 | 0.50 | 0.43 | 0.51 | 0.42 | 0.50 | 0.51 | 0.52 | 0.70 | 0.75 | 0.56 | 0.56 | 0.76 | 0.77 | 0.85 | 0.86 | 0.74 | 0.74
$\mathcal{D}_{4}$ | 0.59 | 0.59 | 0.70 | 0.71 | 0.66 | 0.66 | 0.59 | 0.59 | 0.70 | 0.71 | 0.66 | 0.66 | 0.40 | 0.51 | 0.43 | 0.51 | 0.42 | 0.50 | 0.50 | 0.51 | 0.67 | 0.73 | 0.55 | 0.56 | 0.77 | 0.77 | 0.86 | 0.87 | 0.74 | 0.75
$\mathcal{D}_{5}$ | 0.68 | 0.68 | 0.78 | 0.78 | 0.74 | 0.74 | 0.68 | 0.68 | 0.78 | 0.79 | 0.74 | 0.74 | 0.39 | 0.50 | 0.43 | 0.50 | 0.42 | 0.50 | 0.56 | 0.56 | 0.87 | 0.67 | 0.62 | 0.62 | 0.82 | 0.82 | 0.90 | 0.91 | 0.79 | 0.79
$\mathcal{D}_{6}$ | 0.67 | 0.67 | 0.77 | 0.78 | 0.73 | 0.73 | 0.67 | 0.67 | 0.77 | 0.78 | 0.73 | 0.73 | 0.39 | 0.50 | 0.43 | 0.50 | 0.43 | 0.51 | 0.57 | 0.57 | 0.83 | 0.85 | 0.62 | 0.62 | 0.83 | 0.83 | 0.90 | 0.91 | 0.80 | 0.80
Avg. | 0.59 | 0.59 | 0.71 | 0.71 | 0.66 | 0.66 | 0.59 | 0.59 | 0.70 | 0.71 | 0.66 | 0.66 | 0.39 | 0.50 | 0.43 | 0.51 | 0.42 | 0.50 | 0.50 | 0.50 | 0.69 | 0.68 | 0.55 | 0.55 | 0.77 | 0.78 | 0.85 | 0.86 | 0.75 | 0.75

In terms of the comparison between the given DR techniques, Table XI
summarizes the performance of all the given techniques in terms of the average
accuracy and f-measure w.r.t. datasets $\mathcal{D}_{1}$ to $\mathcal{D}_{6}$.
Following the presented results in Table XI, it could be observed that
firstly, CADR outperforms other DR techniques in terms of accuracy and
f-measure. Secondly, for the linear DR techniques, i.e., PCA and LDA, the
average accuracy and f-measure values are 0.53 and 0.58, respectively, while
for the nonlinear techniques, i.e., MDS, LLE, and CADR, the average accuracy,
and f-measure values are 0.67 and 0.67, respectively. Therefore, the attained
results denote that nonlinear DR techniques could in general outperform linear
ones for the sake of fault diagnosis.

TABLE XI: Comparison between various DR techniques in terms of The average accuracy and f-measure for the given datasets. DR | PCA | MDS | LDA | LLE | CADR
---|---|---|---|---|---
$\mathcal{A}$ | 0.65 | 0.65 | 0.41 | 0.58 | 0.79
$\mathcal{F}_{1}$ | 0.65 | 0.65 | 0.50 | 0.58 | 0.80

In terms of the classification models combined with the given DR techniques,
the average accuracy and f-measure values are summarized in Table XII.
Firstly, it is evident that SVM results in the best classification performance
($\mathcal{A}=0.68$, $\mathcal{F}_{1}=0.69$), which is followed by the RF
($\mathcal{A}=0.61$, $\mathcal{F}_{1}=0.62$) and kNN ($\mathcal{A}=0.57$,
$\mathcal{F}_{1}=0.59$). Secondly, compared with the results presented in
Table V for the baseline models, it could be concluded that the constructed
DR-classifier combinations have improved the performance of the diagnostic
system, verifying the importance of feature extraction in dealing with the
curse of dimensionality.

TABLE XII: The attained average values of the accuracy and f-measure by means of the classification models in combination with the given DR techniques. Model |  kNN |  SVM |  RF
---|---|---|---
$\mathcal{A}$ |  0.57 |  0.68 |  0.61
$\mathcal{F}_{1}$ |  0.59 |  0.69 |  0.62

Finally, all the possible DR-classifier combinations are ranked based on their
performance in terms of accuracy and f-measure, where the ranked combinations
are collected in Table XIII. The ranked combinations in terms of accuracy, in
line with what was discussed previously, denote that CADR in combination with
any of the given classification models outperforms the rest of the DR
techniques. In addition, three out of the first five combinations are cases in
which SVM is combined with a DR technique, verifying the superiority of the
SVM classification model over the kNN and RF. Furthermore, seven out of the
first 10 combinations include a nonlinear DR technique that denotes these
techniques are superior to their linear counterparts. The provided rankings
based upon the f-measure index, there is no change in the first 10
combinations, and, therefore, the same concluding remarks could be made.

TABLE XIII: Ranking of the DR-classifier combinations based on the accuracy and f-measure. Rank |  Combination |  Combination
---|---|---
|  Accuracy |  f-measure
1 |  CADR-SVM |  CADR-SVM
2 |  CADR-kNN |  CADR-kNN
3 |  CADR-RF |  CADR-RF
4 |  PCA-SVM |  PCA-SVM
5 |  MDS-SVM |  MDS-SVM
6 |  LLE-SVM |  LLE-SVM
7 |  PCA-RF |  PCA-RF
8 |  MDS-RF |  MDS-RF
9 |  PCA-kNN |  PCA-kNN
10 |  MDS-kNN |  MDS-kNN
11 |  LLE-RF |  LLE-RF
12 |  LLE-kNN |  LDA-SVM
13 |  LDA-SVM |  LDA-kNN
14 |  LDA-RF |  LLE-kNN
15 |  LDA-kNN |  LDA-RF

###  IV-D Discussion

In Section IV-B and Section IV-C, we analyzed the performance of the FS/DR-
classifier combinations. In this section, we aim to discuss and compare the
performance of FS techniques in comparison with the DR techniques in order to
find out which type of data reduction could help more with improving the
performance of fault diagnosis for the given case-study.

In this regard, we resort to the average values of the accuracy and f-measure
attained by means of FS techniques in combination with all the given
classification models and compare them with those of DR techniques. The
attained results denote that the average accuracy and f-measure values for the
FS techniques are 0.70 and 0.70, respectively, and those are 0.62 and 0.64 for
the DR techniques. Therefore, the attained results denote that for this case-
study, the combination of FS techniques with classification models could deal
better with the classification task compared with the combinations of the DR
techniques with classification models.

##  V Concluding Remarks

In this paper, we studied the fault diagnosis problem of cyber-physical power
systems by resorting to a data-driven technique. This proposal suggested the
combination of data reduction techniques including feature selection and
dimensionality reduction and their combinations with classification models in
order to identify four types of faults including generator outage, generator
ground, and load loss. It was proposed to make use of different types of
feature selection and dimensionality reduction techniques to investigate the
impact of different models on diagnostic performance. The attained results
first denoted that in terms of feature selection and their combinations with
classification models, all the categories performed more or less the same to
deal with the classification task. However, for the dimensionality reduction
techniques, it was observed that nonlinear models could improve the
classification performance in comparison with the linear counterpart.
Furthermore, the results of the given experiments denoted that in general,
feature selection techniques could be of a better performance in comparison
with dimensionality reduction methods for the sake of fault diagnosis. In
addition to this, we studied the impact of noisy measurements and fault
resistance values, where the results showed that when the level of noise and
the value of the fault resistance decreases, the performance of the
constructed combinations could be improved. Due to the high volume of
available data that could be collected from large-scale power systems, this
study could be further extended to the use of deep learning algorithms for the
sake of classification due to their ability in dealing with large-size
datasets.

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# Arbitrarily Structured Laser Pulses

Jacob R. Pierce<EMAIL_ADDRESS>Department of Physics and
Astronomy, University of California, Los Angeles, California 90095, USA  John
P. Palastro<EMAIL_ADDRESS>University of Rochester, Laboratory for
Laser Energetics, Rochester, New York 14623, USA  Fei Li  Department of
Physics and Astronomy, University of California, Los Angeles, California
90095, USA  Bernardo Malaca  GoLP/Instituto de Plasmas e Fusão Nuclear,
Instituto Superior Técnico, Universidade de Lisboa, Lisbon 1049-001, Portugal
Dillon Ramsey  University of Rochester, Laboratory for Laser Energetics,
Rochester, New York 14623, USA  Jorge Vieira  GoLP/Instituto de Plasmas e
Fusão Nuclear, Instituto Superior Técnico, Universidade de Lisboa, Lisbon
1049-001, Portugal  Kathleen Weichman  University of Rochester, Laboratory for
Laser Energetics, Rochester, New York 14623, USA  Warren B. Mori  Department
of Physics and Astronomy, University of California, Los Angeles, California
90095, USA

###### Abstract

Spatiotemporal control refers to a class of optical techniques for structuring
a laser pulse with coupled space-time dependent properties, including moving
focal points, dynamic spot sizes, and evolving orbital angular momenta. Here
we introduce the concept of arbitrarily structured laser (ASTRL) pulses which
generalizes these techniques. The ASTRL formalism employs a superposition of
prescribed pulses to create a desired electromagnetic field structure. Several
examples illustrate the versatility of ASTRL pulses to address a range of
laser-based applications.

††preprint: AIP/123-QED

Light can exhibit structures in space, time, or coupled space-time. The
realization that these structures can be shaped using optical techniques has
led to a deeper understanding of the fundamental properties of light and
advances in multiple laser-based applications. Spatial shaping involves the
use of static optical elements, such as phase plates, deformable mirrors, or
spatial light modulators, to form transverse structures with orbital angular
momentum, Airy or Bessel profiles, or tailored speckle patterns [1, 2, 3, 4].
Temporal shaping typically employs Fourier methods, in which frequency
components are separated by diffraction gratings and separately modified, to
synthesize pulses with nearly arbitrary time-dependent waveforms and
polarizations [5, 6, 7].

Spatiotemporal pulse shaping invokes some combination of these techniques.
Examples go back as far as smoothing by spectral dispersion (1989) [8] and
pulse front tilt (1996) [9]. More modern examples include spatiotemporal
optical vortices (STOVs) [10, 11, 12], diffraction-free light sheets [13, 14,
15], and flying focus concepts [16, 17, 18, 19]. Flying focus pulses, in
particular, feature intensity peaks with programmable trajectories that can
travel distances far greater than a Rayleigh range while maintaining a near-
constant profile. The trajectory control afforded by these pulses has been
proposed as a means to overcome limitations of traditional pulses in a number
of applications, including laser wakefield acceleration [20, 21, 22], Raman
amplification [23], photon acceleration [24], vacuum electron acceleration
[25], and nonlinear Thomson scattering [26], as well as for the exploration of
fundamental physics, such as radiation reaction [27] and nonlinear Compton
scattering [28].

There are two approaches for creating spatiotemporally structured pulses. The
first approach requires identifying a specific solution to Maxwell’s
equations. Once known, this solution can be synthesized using programmable
phase modulation techniques. This is the approach taken to create light sheets
and STOVs. However, this approach does not provide a method to systematically
identify new solutions of Maxwell’s equations with features that are desirable
for a particular application. In the second approach, one imagines an
electromagnetic structure with desirable features and synthesizes this
structure using a superposition of solutions to Maxwell’s equations with known
properties. This is the approach presented here.

In this letter, we introduce a concept and theoretical formalism for
generating laser pulses with arbitrary structure. These arbitrarily structured
laser (ASTRL) pulses are synthesized using a superposition of traditional
laser pulses with controlled and varying properties. The ASTRL formalism
generalizes the creation of pulses with evolving focal points (flying foci),
spot size, orbital angular momentum, and polarization, and can also describe
pulses with exotic topological structure such as STOVs. An example of each of
these is presented. The flexibility of the ASTRL concept opens new and
previously unimagined possibilities for structured light with the potential to
improve a wide range of laser-based applications, while its simplicity
facilitates implementation into simulations.

The formulation of the ASTRL concept begins with the vacuum wave equation for
the electric field of a laser pulse:

$\left[\nabla^{2}-\frac{1}{c^{2}}\partial_{t}^{2}\right]\mathbf{E}(\mathbf{x},t)=0.$
(1)

ASTRL pulses are constructed by superposing solutions to Eq. 1 with varying
properties. The electric field of an ASTRL pulse can be expressed either as a
sum or an integral over these solutions:

$\mathbf{E}(\mathbf{x},t)=\int d\eta\;\mathbf{E}_{\eta}(\mathbf{x},t),$ (2)

where $\eta$ parameterizes the varying properties of the solutions. For
example the amplitude, spot size, focal point, pulse duration, or relative
delay of each $\mathbf{E}_{\eta}$ can all depend on $\eta$. Equation 2 is the
most general representation of an ASTRL pulse in vacuum.

Analytic solutions for $\mathbf{E}_{\eta}$ simplify the process of designing
an ASTRL pulse for a specific application. Paraxial solutions, in particular,
are expressed in terms of familiar quantities that can be parameterized in
terms of $\eta$. To derive such solutions, consider a laser pulse propagating
in the $\hat{z}$-direction with constant polarization $\hat{\bm{\epsilon}}$.
The transverse electric field can be expressed as an envelope
$A(\mathbf{x}_{\perp},z,t)$ modulated by a carrier

$\mathbf{E}_{\perp}(\mathbf{x}_{\perp},z,t)=\tfrac{1}{2}A(\mathbf{x}_{\perp},z,t)e^{i(k_{0}z-\omega_{0}t)}\hat{\bm{\epsilon}}+\mathrm{c.c.},$
(3)

where $k_{0}$ is the central wavenumber and $\omega_{0}=ck_{0}$. Substituting
Eq. 3 into Eq. 1 and performing the Galilean change of variables
$(\xi,s)=(z-ct,z)$ provides

$\left[\nabla_{\perp}^{2}+\partial_{s}^{2}+2ik_{0}\partial_{s}+2\partial_{\xi}\partial_{s}\right]A(\mathbf{x}_{\perp},s,\xi)=0.$
(4)

Upon applying the slowly-varying envelope approximation
($|\partial_{\xi}|,|\partial_{s}|\ll k_{0}$), Eq. 4 reduces to

$\left[\nabla_{\perp}^{2}+2ik_{0}\partial_{s}\right]A(\mathbf{x}_{\perp},s,\xi)\approx
0.$ (5)

Equation 5 is the paraxial wave equation and its operator is independent of
$\xi$. This independence admits separable solutions, i.e., solutions with no
spatiotemporal coupling, of the form

$A(\mathbf{x}_{\perp},s,\xi)\approx B(\xi)C(\mathbf{x}_{\perp},s),$ (6)

where $C(\mathbf{x}_{\perp},s)$ satisfies
$[\nabla_{\perp}^{2}+2ik_{0}\partial_{s}]C(\mathbf{x}_{\perp},s)=0$ and
$B(\xi)$ is an arbitrary function. For example, $C(\mathbf{x}_{\perp},s)$ may
be any Laguerre-, Hermite-, or Ince-Gaussian mode and the longitudinal profile
may be $B(\xi)=\exp[-(\xi/c\tau_{0})^{2}]$.

The transverse electric field of a coherent ASTRL pulse can be approximated by
summing or integrating over paraxial solutions:

$\mathbf{E}_{\perp}(\mathbf{x}_{\perp},s,\xi)\approx\tfrac{1}{2}e^{ik_{0}\xi}\int
d\eta\;B_{\eta}(\xi)C_{\eta}(\mathbf{x}_{\perp},s)\hat{\bm{\epsilon}}(\eta)+\mathrm{c.c.},$
(7)

where the properties of $B_{\eta}(\xi)$, $C_{\eta}(\mathbf{x}_{\perp},s)$, and
$\hat{\bm{\epsilon}}(\eta)$ may vary with respect to $\eta$. The integral over
$\eta$ may introduce spatiotemporal coupling, such that in general, Eq. 7
cannot be written as a separable function. Thus, $\mathbf{E}_{\perp}$ may
possess a high degree of spatiotemporal coupling despite the absence of
coupling in Eq. 6.

The $\eta$-dependence of $B_{\eta}(\xi)$ may encode a delay $\Delta$, duration
$\tau_{0}$, and weight $B_{0}(\eta)$ for each constituent pulse as

$B_{\eta}(\xi)=B_{0}(\eta)\exp\left[-\left(\frac{\xi-\Delta(\eta)}{c\tau_{0}(\eta)}\right)^{2}\right].$
(8)

The durations $\tau_{0}(\eta)$ set the fastest timescale over which the
properties of the ASTRL pulse can vary and determine the number of pulses
required for discrete approximation of Eq. 7 (see Supplemental). Unless
otherwise stated, $\tau_{0}$ and $B_{0}$ will be independent of $\eta$ for the
examples presented here.

Figure 1: (a): On-axis evolution of the envelope of an ASTRL flying focus
pulse found by evaluating Eq. 7 with $k_{0}w_{0}=40$, $\omega_{0}\tau_{0}=20$,
and $\beta_{f}=1.05$. The dashed line indicates the trajectory $z=\beta_{f}t$.
(b) The envelope from a quasi-3D FDTD simulation for the same parameters. The
two are in excellent agreement. (c) – (e): A generalized flying focus
constructed using the ASTRL formalism in which the focal point oscillates
transversely as it moves along the optical axis. The propagation was simulated
in 2D using the FDTD method. The pulse was initialized at $ct/z_{R}=-10$ using
Eq. 7 with $x_{\perp\text{osc}}=w_{0}$, $k_{\text{osc}}=1/z_{R}$,
$\beta_{f}=1.05$, $L=20z_{R}$, $k_{0}w_{0}=40$, and $\omega_{0}\tau_{0}=20$.
From left to right, the frames show the pulse at $ct/z_{R}=-4$, $0$, and $4$.

A natural example for the $\eta$-dependence of
$C_{\eta}(\mathbf{x}_{\perp},s)$ is an $\eta$-indexed Gaussian-beam:

$C_{\eta}(\mathbf{x}_{\perp},s)=\left[\frac{z_{R}(\eta)}{q_{\eta}(s)}\right]^{d_{\perp}/2}\exp\left[-\frac{ik_{0}|\mathbf{x}_{\perp}-\mathbf{x}_{\perp
0}(\eta)|^{2}}{2q_{\eta}(s)}\right],$ (9)

where $q_{\eta}(s)\equiv s-s_{0}(\eta)+iz_{R}(\eta)$ is the complex beam
parameter and $d_{\perp}$ is the number of transverse dimensions. In Eq. 9,
the longitudinal focal position $s_{0}(\eta)$, transverse focal position
$\mathbf{x}_{\perp 0}(\eta)$, and Rayleigh range $z_{R}(\eta)$ (and associated
spot size $w_{0}(\eta)=\sqrt{2z_{R}(\eta)/k_{0}}$) may all be expressed as
functions of the integration variable $\eta$, which allows for extended focal
ranges and evolving transverse structure. More generally, the mode numbers of
the transverse profile can depend on $\eta$, e.g., the radial or orbital
angular momentum (OAM) quantum numbers of a Laguerre-Gaussian mode. Finally,
$\hat{\bm{\epsilon}}(\eta)$ may encode an evolving polarization.

The utility and generality of the ASTRL concept will be demonstrated by using
Eq. 7 to construct several examples of spatiotemporally structured laser
pulses. To begin, consider the flying focus [16, 17]. Each time slice in a
flying focus pulse has a different focal point along the propagation axis. The
time and location at which each slice comes to focus can be controlled to
produce an intensity peak that travels at any velocity, including backwards or
faster than the speed of light. Within the ASTRL formalism, a flying focus
pulse may be described using a Gaussian transverse profile (Eq. 9) with
$\eta\in\left[-\frac{1}{2},\frac{1}{2}\right]$ and $s_{0}(\eta)=\eta L$. This
choice of $s_{0}(\eta)$ produces a focal range extending from $s=-L/2$ to
$s=L/2$, where $L$ can be much larger than $z_{R}$. At each $s$ in the focal
range, the focus will occur at a time determined by the condition
$\xi=\Delta(\eta)=\Delta(s/L)$. The delay $\Delta(\eta)$ required for a given
focal trajectory can be derived from geometric optics as in Ref [20]. For
example, motion of the focal point at a constant velocity
$\mathrm{v}_{f}=c\beta_{f}$ is attained with
$\Delta(\eta)=(1-1/\beta_{f})L\eta$.

Figure 1(a) displays the on-axis envelope of a superluminal ($\beta_{f}=1.05$)
flying focus pulse obtained by evaluating Eq. 7. The white dashed line shows
the spacetime trajectory $z=\beta_{f}t$ and demonstrates that the laser
envelope travels at the expected velocity over 10 Rayleigh ranges. A finite-
difference time-domain (FDTD) simulation of the full set of Maxwell’s
equations produces nearly identical results [Fig. 1(b)]. The simulation used
Eq. 7 to initialize the fields, but did not make the approximations leading to
Eq. 5. A slight discrepancy can be observed near $ct\approx 5z_{R}$ due to the
inexactness of Eq. 6. The simulation was conducted using the code OSIRIS [29]
with quasi-3D azimuthal decomposition [30] and the solver introduced in Ref
[31] to mitigate numerical dispersion.

The ASTRL concept can also be used to describe new types of generalized flying
focus pulses. For instance, a pulse whose focal point oscillates in the
transverse direction as it translates along the propagation axis could serve
as a controllable wiggler for generating radiation from relativistic electron
bunches or enhance direct laser acceleration of electrons through new
parametric resonances. Such a pulse can be constructed by parameterizing both
the transverse and longitudinal focal coordinates: $\mathbf{x}_{\perp
0}(\eta)=x_{\perp\text{osc}}\sin[k_{\text{osc}}s_{0}(\eta)]\hat{\mathbf{x}}$
and $s_{0}(\eta)=\eta L$, where, as before,
$\eta\in\left[-\frac{1}{2},\frac{1}{2}\right]$. Figures 1(c)–(e) display the
envelope from 2D OSIRIS simulations of this configuration. The intensity peak
undergoes the expected oscillatory motion. Here, $C_{\eta}$ is given by Eq. 9
and $\Delta(\eta)=(1-1/\beta_{f})L\eta$, i.e., the focal point moves with a
constant axial velocity.

Aside from moving focal points, many applications can benefit from pulses that
have a fixed focal point with prescribed spatiotemporal profiles or time-
dependent polarization (Fig. 2). Traditional Fourier techniques for temporal
pulse shaping, as in Ref [6], produce pulses described by Eq. 6 with a
customized $B(\xi)$. Similarly, previous polarization-structuring methods
amount to evaluating Eq. 7 with a prescribed $\hat{\bm{\epsilon}}(\eta)$ and
$B_{\eta}$, but with a $C_{\eta}$ that is independent of $\eta$. Figure 2(a)
shows an example. This particular ASTRL pulse is constructed from a discrete
sum of seven pulses, indexed by $\eta\in\\{0,1,\ldots 6\\}$, that share a
common focal point, $s_{0}(\eta)=0$. The delay and polarization of each pulse
is given by $\Delta_{\eta}=-2c\tau_{0}\eta$ and
$\hat{\bm{\epsilon}}(\eta)=[\cos(\pi\eta/5),\sin(\pi\eta/5),0]$ while
$C_{\eta}$ is given by Eq. 9. Evolving polarization structures as in this
example have led to advances in nonlinear spectroscopy [32], quantum control
[33, 34], nanophotonics [35], and a number of other areas [36, 37]. More
exotic pulses with continuous polarization evolution can also be described
using the ASTRL concept.

Figure 2: Examples of ASTRL pulses constructed from superpositions of laser
pulses focused to the same point ($s_{0}(\eta)=0$): (a) A pulse everywhere
linearly polarized, but with a time-dependent polarization angle. The duration
of each pulse in the superposition is $\omega_{0}\tau_{0}=20$. (b) A laser
pulse with a spot size that continuously decreases in time with
$k_{0}w_{0\text{min}}=40$, $w_{0\text{max}}=4w_{0\text{min}}$,
$\omega_{0}\tau_{0}=40$, and $\Delta_{\text{max}}=20c\tau_{0}$. (c) Time
slices of the transverse electric field of a laser whose angular momentum
continuously evolves in time. (d) A lattice of STOVs with $k_{0}w_{0}=40$ and
$\omega_{0}\tau_{0}=20$. All examples are evaluated using Eq. 7 and shown at
the focal point $s=0$.

A time-dependent spot size can improve the performance of applications
including laser wakefield acceleration [38] and inertial confinement fusion
[39, 40, 41]. In laser wakefield acceleration, a spot size that increases in
time can stabilize the propagation of an intense laser pulse in a plasma
channel [38]. While in inertial confinement fusion, a spot size that decreases
continuously in time can mitigate cross-beam energy transfer [39, 40, 41]. As
demonstrated in Fig. 2(b), such a pulse can be described by Eqs. 7, 8, and 9
with a spot size parameterized as
$w_{0}(\eta)=w_{0\text{min}}+\eta(w_{0\text{max}}-w_{0\text{min}})$ and the
delay $\Delta(\eta)=-\Delta_{\text{max}}\eta$, where $\eta\in[0,1]$.

The orbital angular momentum and radial mode numbers ($\ell$ and $p$,
respectively) of Laguerre-Gaussian modes provide additional degrees of freedom
to structure the time-dependence of the transverse profile. Figure 2(c)
displays a sequence of time slices in an ASTRL pulse formed by superposing
three pulses that partially overlap in time with $(\ell,p)=(0,2),(3,0),(1,1)$.
Each pulse is indexed by $\eta\in\\{0,1,2\\}$, has a delay
$\Delta_{\eta}=-2c\tau_{0}\eta$, and a weight $B_{0}(\eta)=1,0.8,1$. This
construction produces a quasi-continuous transformation in angular momentum.
In the first, third, and fifth frames, one pulse dominates, and the transverse
profile is nearly a pure Laguerre-Gaussian mode; in the second and fourth
frames, two pulses interfere to produce a hybrid mode. Investigations into
structured OAM have only recently begun [42, 43, 44], but its potential for
ultrafast probing of chiral systems has already drawn attention [44]. Many
other applications based on OAM [45], including optical trapping and
manipulation [46, 47, 48, 49], imaging [50, 51, 52], quantum optics [53, 54],
nonlinear optics [55], and laser-plasma interactions [56, 57, 58, 59, 60, 61,
62, 63], may also benefit from added control over its structure.

As a final example, the ASTRL concept can be used to construct STOVs. In fact,
the vacuum STOV solution introduced in Ref. [10],

$\begin{split}A(\mathbf{x}_{\perp},s,\xi)&=\left[\frac{\xi}{c\tau_{0}}\pm\frac{ix}{w(s)}\left(\frac{-q^{*}(s)}{q(s)}\right)^{1/2}\right]\\\
&\frac{z_{R}}{q(s)}\exp\left[-\frac{ik_{0}x_{\perp}^{2}}{2q(s)}\right]\exp\left[-\left(\frac{\xi}{c\tau_{0}}\right)^{2}\right],\end{split}$
(10)

is a superposition of two separable solutions as in Eq. 6, where the
$C_{\eta}$ functions are Hermite-Gaussian modes of order $(0,0)$ and $(1,0)$.
Here the $s$-dependence has been included, $w(s)\equiv
w_{0}\sqrt{1+(s/z_{R})^{2}}$, and $q^{*}(s)$ is the complex conjugate of
$q(s)$. The generalization of Eq. 10 to higher-order vortex topologies (see
Ref. [12]) can also be expressed as a superposition of separable solutions.
Figure 2(d) illustrates a novel STOV “lattice” assembled by superposing
solutions in the form of Eq. 10. The solutions are indexed by
$\eta\in\\{0,1,\ldots 7\\}$ with delays $\Delta_{\eta}=-2c\tau_{0}\eta$,
transverse displacements alternating between $x_{0}=\pm w_{0}$, and polarities
(the sign in Eq. 10) alternating between $\pm 1$.

The ASTRL description is particularly useful for simulations of structured
light and its applications. Previous simulations of flying focus pulses
required a Fresnel integral to propagate the field from the optic plane of a
specific experimental design to the simulation plane. In contrast, the ASTRL
description enables direct field initialization in the simulation plane, which
simplifies implementation. Further, by decoupling the structured light-matter
interaction physics from the details of a particular optical configuration,
the ASTRL description expedites prototyping of new concepts. For example, the
axiparabola-echelon pair used in Ref. [20] would not need to be redesigned for
every simulation. An optical configuration capable of delivering the desired
pulse can be designed or implemented in a simulation after a concept shows
promise (see Supplemental).

As demonstrated by the examples above, an ASTRL pulse may generally exhibit
spatiotemporal coupling despite being assembled from pulses with no
spatiotemporal coupling (i.e., Eq. 6). The separability of Eq. 6 was derived
by reducing the full wave equation (Eq. 4) to the paraxial wave equation (Eq.
5). The validity of the approximations leading to Eq. 5 may be determined by
evaluating derivatives of the approximate solution
$A(\mathbf{x}_{\perp},s,\xi)=z_{R}/q(s)\exp[-ik_{0}x_{\perp}^{2}/2q(s)]\exp[-(\xi/c\tau_{0})^{2}]$.
At any location, $|\partial_{s}^{2}A|\ll|2ik_{0}\partial_{s}A|$ when
$k_{0}w_{0}\gg 1$; this is the usual paraxial approximation, which fails only
for tightly focused laser pulses. On the other hand,
$|2\partial_{\xi}\partial_{s}A|\ll|2ik_{0}\partial_{s}A|$ when
$\omega_{0}\tau_{0}\gg 1$. This condition only fails for pulses with durations
less than a few-cycles. These two conditions, i.e., $k_{0}w_{0}\gg 1$ and
$\omega_{0}\tau_{0}\gg 1$, place constraints on $C_{\eta}$ and $B_{\eta}$,
respectively.

For an envelope that is initially separable at some $s=s_{i}$, spatiotemporal
coupling will develop beyond a range $|s-s_{i}|\gtrsim
L_{\text{stc}}\equiv\frac{1}{2}c\tau_{0}k_{0}^{2}w_{0}^{2}$ because of the
$\partial_{\xi}\partial_{z}$ term in Eq. 4 (see Supplemental). There are two
common scenarios that can be considered. In the first scenario, every pulse
comprising an ASTRL pulse is separable at its focus. Because each pulse only
contributes significantly within a Rayleigh range of its focus, the condition
on the accuracy of Eq. 6 can be expressed as $L_{\text{stc}}\gtrsim z_{R}$ or,
equivalently, $\omega_{0}\tau_{0}\gtrsim 1$. In the second scenario, which is
more applicable to field initialization in a simulation, every pulse is
separable at a specified distance $L_{\eta}$ from focus. The condition on the
accuracy of Eq. 6 is then given by $L_{\text{stc}}\gtrsim\max_{\eta}L_{\eta}$.
For example, in Fig. 1(a-b) $\max_{\eta}L_{\eta}=\frac{1}{2}L_{\text{stc}}$,
which contributes to the slight discrepancy at the end of the focal region.
However, even if $L_{\text{stc}}\lesssim\max_{\eta}L_{\eta}$ and
spatiotemporal coupling develops in the constituent pulses, the qualitative
features of the desired electromagnetic structure would remain intact.

The generality of the ASTRL concept extends well beyond Eq. 7 and the examples
presented here. The time-dependence of any combination of parameters, such as
the longitudinal focal point and OAM, can be simultaneously structured.
Further, the ASTRL concept can be extended to accommodate multi-color pulses
[64, 65], superpositions of pulses with vector polarization [66, 67, 68, 69],
and pulses structured in the spatiospectral domain. In fact, using a wavelet
transform one can show that arbitrary monochromatic laser fields may be
decomposed as a generalized version of Eq. 7 (see Supplemental).

The ASTRL formalism provides a framework for constructing laser pulses with
desired spatiotemporal structure. An ASTRL pulse is synthesized using
superpositions of known solutions to Maxwell’s equations, such as an ensemble
of traditional laser pulses. The flexibility of the ASTRL concept was
illustrated with several examples, including pulses with moving focal points,
dynamic polarization, evolving angular momentum, and nontrivial topological
structure. In addition to facilitating the creation of new and previously
unimagined electromagnetic structures, the ASTRL formalism simplifies the
injection of structured light in simulations. New pulses based on this concept
may enable or enhance techniques in a range of scientific disciplines,
including microscopy, non-linear optics, quantum optics, and laser-plasma
interactions.

###### Acknowledgements.

The authors graciously acknowledge C. Joshi, D. H. Froula, M. Vranic, E. P.
Alves, SJ Spencer, A. Di Piazza, M. Formanek, T. Carbin, and C. Barkan for
their lucid insights. Work supported by the National Science Foundation award
2108970, the Department of Energy contracts DE-SC00215057 and DE-SC0010064,
the Scientific Discovery through Advanced Computing (SciDAC) program through a
Fermi National Accelerator Laboratory (FNAL) subcontract No. 644405, and the
Unversity of Rochester, Laboratory for Laser Energetics. Computer simulations
were performed on NERSC’s Cori cluster (account m1157). J. Pierce was
supported by a scholarship from the Directed Energy Professional Society.

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