# FisHook - An Optimized Approach to Marine Species Classification using
MobileNetV2

Kohav Dey  Department of Computer Science and Engineering
PES University, Banashankari
Bangalore, India
<EMAIL_ADDRESS>Krishna Bajaj  K S Ramalakshmi  Department of Computer
Science and Engineering
PES University, Banashankari
Bangalore, India
<EMAIL_ADDRESS>Department of Computer Science and
Engineering
PES University, Banashankari
Bangalore, India
<EMAIL_ADDRESS>Samuel Thomas  Sriram Radhakrishna  Department of
Electronics and Communication Engineering
PES University, Banashankari
Bangalore, India
<EMAIL_ADDRESS>Department of Computer Science and Engineering
PES University, Banashankari
Bangalore, India
<EMAIL_ADDRESS>
###### Abstract

Marine ecosystems are vital for the planet’s health, but human activities such
as climate change, pollution, and overfishing pose a constant threat to marine
species. Accurate classification and monitoring of these species can aid in
understanding their distribution, population dynamics, and the impact of human
activities on them. However, classifying marine species can be challenging due
to their vast diversity and the complex underwater environment. With
advancements in computer performance and GPU-based computing, deep-learning
algorithms can now efficiently classify marine species, making it easier to
monitor and manage marine ecosystems. In this paper, we propose an
optimization to the MobileNetV2 model to achieve a 99.83% average validation
accuracy by highlighting specific guidelines for creating a dataset and
augmenting marine species images. This transfer learning algorithm can be
deployed successfully on a mobile application for on-site classification at
fisheries.

###### Index Terms:

Marine, MobileNetV2, classification, transfer-learning, dataset augmentation,
mobile application

##  I Introduction

Fish and other seafood have grown to comprise a large part of the world’s diet
and around 50 percent of the population have become regular consumers of the
same. Furthermore, according to the Food and Agricultural Organization of the
United Nations, the global consumption of seafood has risen above 20 kilograms
per person per year [1]. In case of a multivariate catch of marine yield,
there is a requirement for a highly efficient classification algorithm for the
identification of new subspecies with edibility constraints and nutritional
composition, as well as a database to add the said species for other
pisciculture enthusiasts to utilise. The effective classification of these
species can aid in the identification of their rarer counterparts and when
combined with market research, it can facilitate informed decision making
processes to ensure profitable sales and monitoring of marine species. Regular
monitoring helps us understand the distribution, population dynamics, and the
impact of human activities on marine species, enabling the development of
effective conservation strategies, with conventional methods of classification
currently in use being highly effort-intensive, error-prone and time-
consuming.

Figure 1: The model introduced in this paper was deployed on an Android mobile
application with a mature back-end.

Figure 2: Architecture of MobileNetV2.

The proposal aims to develop a marine species classifier using an image-based
approach, which will be implemented through transfer learning. To achieve
this, we will be utilizing the MobileNetV2 [2] architecture, which has been
proven to be effective in a variety of image classification tasks.
Additionally, we will incorporate a traditional sequential neural network to
further enhance the accuracy of the model. The dataset used for training and
testing the model consists of nine distinct fish classes [3], each with its
unique characteristics and features. By leveraging transfer learning, we are
utilising the pre-trained MobileNetV2 model initialised with ImageNet [4]
weights, as a feature extractor. This will enable the model to leverage the
pre-existing knowledge of the MobileNetV2 architecture to improve the
classification accuracy of the dataset. The primary objective of this marine
species classifier is to accurately identify and differentiate between the
various fish classes in the dataset. This classifier can be utilized in a
range of marine applications, such as monitoring and conservation efforts, to
better understand the distribution and behavior of different fish species.

##  II Related Work

At present, several machine learning models trained and tested on various
datasets have been implemented to achieve an acceptable classification
accuracy. Ulucan et al. [3] implemented a feature-based classification model
using SVMs providing an average training and validation accuracy of 98.01
percent and 88.69 percent, for 9 distinct classes. Rossi et al. [5] built a
mobile application with a remote server for cloud-based machine learning,
which implements an Artificial Neural Network(ANN) classifier, with an
accuracy of 100 percent, for 7 distinct fish classes. Cchabra et al. [6] built
a VGG16 transfer learning model achieving an accuracy of 93.8 percent for 8
different classes of fish. Liu et al. [7] proposed a MobileNetV2 based
transfer learning model, achieving an accuracy of 95 percent, for 7 categories
of fishes. The optimized approach proposes an implementation that contributes
to the latter, by specifying guidelines for the creation and augmentation of a
dataset as well as changes to the existing model to obtain an improvement in
the accuracy of the model.

This literature survey indicates that MobileNetV2-based deep learning models
have been successfully applied for fish species identification and
classification tasks with high accuracy. The studies reviewed in this survey
have used different techniques such as transfer learning, data augmentation,
feature extraction, and hybrid models to improve the performance of the
MobileNetV2-based models. The proposed research paper can build upon the
existing literature by proposing a novel approach for fish species
identification using MobileNetV2 and exploring the potential of different
techniques to improve the accuracy of the model.

##  III Concept Theory and Implementation

Marine life is essential to the planet’s ecosystem, providing food, oxygen and
regulating climate change. As it stands, innumerable species currently live
under constant threat from climate change, pollution, overfishing, and other
human activities. Accurate classification and monitoring of marine species can
help us understand their distribution, population dynamics, and the impact of
human activities on them, allowing us to develop effective conservation
strategies. The classification of marine species can be a challenging task due
to the vast diversity of species and the complex underwater environment. With
improvements in computer performance and advancements in GPU-based computing,
it has become possible to classify marine species accurately and efficiently
using deep-learning algorithms to process large quantities of data, making it
easier to monitor and manage marine ecosystems. In this paper, we present an
optimization to the most accurate image classification model for the purposes
of marine specie classification. The issue has been evaluated in the past
using the MobileNetV2 model and has already yielded a significant average
classification accuracy of 97.26 percent [8]. MobileNetV2 is a neural network
architecture that achieves state-of-the-art performance on a range of tasks,
including image classification, object detection, and semantic segmentation.
Moreover, MobileNetV2 is computationally efficient, achieving a speedup of up
to 3.4x over its predecessor, MobileNetV1, while maintaining similar accuracy.
However, we noticed that the model can be significantly improved upon by
introducing modifications to the type of dataset used for training. By
highlighting specific guidelines for the creation of such a dataset we’ve
achieved an improved validation accuracy average of 99.83 percent, while at
the same time, appending it with augmented versions of the same marine specie
images along with the enhanced model was studied and found to hinder the
effectiveness of the model. The transfer learning algorithm was implemented
with the use case of on-site classification at fisheries and deployed
successfully on a mobile application.

###  III-A Datasets

A dataset consisting of 9 species of marine fauna, was used for the training a
modified version of the classifier implemented by Ulucan et. al [3] Each of
the seven classes - red mullet, gilt head bream, horse mackerel, sea bass, red
sea bream, black sea sprat and striped red mullet, consisted of 50 distinct
fish images, while the other two - trout and shrimp, had 30 distinct images.
The dataset and the images captured were varied enough to capture all possible
physical details of the subjects by virtue of them being taken from multiple
angles to the surface as well as to the body-axis of the subjects themselves.
The images were sized at 590 x 445 pixels. The images were passed on to an
augmentation algorithm where each image is rotated with non-repeated random
angles and they are reflected, while maintaining a consistent image size.
Dataset collection and augmentation concluded with each seafood type
consisting of approximately 1,100 images.

Figure 3: Sample images along with labels in the training dataset after
augmentation.

Several approaches were undertaken to build the classifier. Any TensorFlow
based model which has to be run as an application in edge devices, needs to be
converted to a compressed .tflite model. Of all, MobileNetV2 delivered
promising prediction results, following which the Tensorflow [9] model was
converted into a .tflite [10] model format to be deployed in a Flutter
application. The paper gives an end-to-end implementation of the mentioned
approach, a brief overview of the various shortcomings identified and the
future scope for improvement as well.

Figure 4: The original dataset that the model was trained on [3].

###  III-B Image Preprocessing

Augmentation of the dataset was performed by the ImageDataGenerator [11] which
generates batches of tensor image data with real-time data augmentation [12].
Augmentation and pre-processing was done implicitly with a validation split of
0.2 in the training dataset. The pre-process-input API scales the input pixels
between -1 and 1. Tensors of varied dimensions are reshaped to 224X224 pixels,
as an appropriate input to the MobileNetV2 network for feature extraction
followed by, classification by CNN.

###  III-C Transfer Learning

Transfer learning [13] is a machine learning technique which focuses on
reusing pre-trained models for similar deep learning tasks, given the vast
compute resources and the time taken to train such complex neural networks on
massive amounts of data. Since most real-world problems do not have sufficient
labelled data points that can be used for training purposes, this kind of an
optimisation enables the training of deep neural networks with comparatively
less data, alongside reducing the training time and improving the performance
of the model.

A base network is trained on a base dataset and task, after which the learned
features are repurposed or transferred to a second target network, which is
trained for a different task, on a different target dataset with similar high-
level features to the original. Transfer learning produces better results when
the features learnt by the model are general to both the base and target
dataset. This type of transfer learning is called as inductive transfer.

###  III-D MobileNetV2 - Overview

MobileNetV2 [2], introduced by Google is a convolutional neural network
architecture that is designed to perform efficiently on edge devices. It is
based on an inverted residual structure where the residual connections are
between the bottleneck layers, which is an improvisation over MobileNetV1.
MobileNetV1 uses depth-wise separable convolution and has a reduced complexity
cost as well. The intermediate expansion layer uses lightweight depth-wise
convolutions to filter features as a source of non-linearity in the narrow
layers.

On a whole, the architecture of MobileNetV2 containing the initial fully
convolution layer with 32 filters, followed by 19 residual bottleneck layers
is used for feature extraction in the implementation to generate a feature map
as an input to the fully connected Dense layers of the traditional CNN.
Bottleneck layers are added to reduce the number of feature maps in the
network, which, otherwise, tend to increase in each layer. This is achieved by
using 1x1 convolutions with fewer output channels than input channels, thereby
reducing the model complexity and make it light-weight that can be implemented
in edge devices, like smartphones.

###  III-E Model Architecture

The paper proposes a transfer learning based implementation using MobileNetV2
where the fully convolutional pre-trained layers of the model are used for
feature extraction and a traditional neural network with fully connected Dense
layers are used for classification. MobileNetV2 takes an input tensor of size
224 x 224 x 3 in the input layer. The weights of the top-trainable layers are
frozen and the generated feature map is fed as an input to the fully connected
dense layers of the neural network. Non-linearity is introduced in the hidden
layers of the network by the RELU activation function, which allows the
stacking of multiple layers of neurons to create a deep neural network. Non-
linearity helps the network to learn complex datasets, which high accuracy.

$Relu(x)=max(0,x)\\\ $ (1)

Figure 5: RELU Activation Function for reference.

Figure 5 is a graphical representation of the RELU function (Equation 1) which
produces the same output as the input for non-zero values and all negative
values are reduced to zero.

Every layer in the neural network is trained with an associated loss function,
to quantify the performance of the model during the training phase. It is a
measure of the difference between the predicted output and the actual output
for a given input, which is used to optimize the model’s parameters in the
subsequent training iterations. Minimizing the loss function can significantly
improve the accuracy of the model and make it more effective at its intended
task.

Categorical cross-entropy loss [14] is a commonly used loss function in deep
learning for multi-class classification problems.

$-{(y\log(p)+(1-y)\log(1-p))}$ (2)

Loss for a single class label per observation is given by Equation 3.

$-\sum_{c=1}^{M}y_{o,c}\log(p_{o,c})$ (3)

If M is greater than 2 (i.e. multiclass classification), loss for each class
label per observation is calculated separately and added to the result
cumulatively. The resulting sum gives the categorical cross entropy loss for
the classifier (Equation 4).

The final output layer uses the softmax activation for better intepretation of
classification, scaling the prediction of each class in the numpy array to
either 0 or 1. Transfer learning with MobileNetV2 is used to classify
different species of fishes and the method produces acceptable results since
the base neural network has learnt the features necessary to classify fishes
from the ImageNet dataset.

$\sigma(z_{i})=\frac{e^{z_{i}}}{\sum_{j=1}^{K}e^{z_{j}}}\ \ \ for\
i=1,2,\dots,K$ (4)

Equation 2 is a mathematical representation of the softmax [15] function,
which normalises the prediction probabilities of various classes in a multi-
class classifier.

During the training phase, measuring accuracy is an essential part of
evaluating the performance of deep learning models on both the training
dataset and validation dataset. The following are the commonly used metrics
used to evaluate the model’s performance during training:

1\. Training accuracy: It measures the accuracy of the model on the training
dataset during the training process. Training accuracy is measured as a
percentage of correctly classified instances in the training dataset.

$Accuracy_{training}=\frac{1}{n}\sum_{i=1}^{n}(y_{i}-\hat{y_{i}})\\\ $ (5)

2\. Training loss: It measures the error or difference between the predicted
output and the actual output on the training dataset during the training
process. The training loss is used to optimize the model’s parameters to
improve its performance.

$L_{train}=\frac{1}{n}\sum_{i=1}^{n}L(y_{i},\hat{y_{i}})\\\ $ (6)

3\. Validation accuracy : It measures the accuracy of the model on the
validation dataset during the training process. The validation accuracy
provides an estimate of how well the model will perform on new data.

$Accuracy_{valdation}=\frac{1}{m}\sum_{i=1}^{m}(y_{i}-\hat{y_{i}})\\\ $ (7)

4\. Validation loss : It measures the error or difference between the
predicted output and the actual output on the validation dataset during the
training process. The validation loss is used to monitor overfitting, where
the model performs well on the training dataset but not on the validation
dataset.

$L_{val}=\frac{1}{m}\sum_{i=1}^{m}L(y_{i},\hat{y_{i}})\\\ $ (8)

During the training process, we aim to minimize the training loss and improve
the training accuracy while avoiding overfitting. Validation accuracy and
validation loss are used to monitor the model’s performance and prevent
overfitting. If the validation loss increases while the training loss
decreases, it indicates that the model is overfitting and needs to be
adjusted.

##  IV Deployment

The use of a mobile application for data collection, coupled with an AWS/GCP
based deployment for processing and classification, is a promising approach to
building a marine species classifier. In this section, we discuss the design
and implementation of a Flutter-based mobile application and its integration
with the cloud server.

The cross-platform mobile application was developed using Flutter and was
designed to be user-friendly, accessible and customizable. The application
allowed users to upload/capture pictures of marine species and then sent to a
Firebase shared storage, from where the images were requested by an EC2
instance for processing and classification[16].

The AWS pipeline was designed to efficiently process large volumes of data in
real-time. The algorithm was deployed using AWS’ EC2 service, an elastic
computation unit on the cloud that provided scalable, reliable, and cost-
effective model training and deployment with the aid of purpose-built
automation scripts. The model was updated continually with new data as and
when the user feeds for classification, ensuring that the algorithm remained
accurate and relevant over time.

Pipelining the process in this manner allows the distribution of the
computational load exerted by the training process across multiple remote
computers [17] at the point of exchange from the shared Firebase storage [18]
(being used as a bucket in this implementation). This would greatly increase
the cap on throughput and assist any extended user-base for the mobile
application by ensuring scalability.

Figure 6: An example of an image taken from the dataset itself, being
evaluated by the model accurately.

The mobile application provided a simple and accessible way for users to
collect data, while the cloud pipeline ensured the efficient processing and
classification of this data. The integration of a Flutter-based mobile
application with AWS represented a powerful and innovative approach to marine
species classification. This approach has the potential to contribute to more
effective marine conservation efforts by enabling more accurate and efficient
data collection and classification.

##  V Results

###  V-A Testing Specifications

TensorFlow, a popular open-source library developed by Google for machine
learning and artificial intelligence, was used to conduct the experiment along
with the use of an additional supporting package known as Keras [19] for
further code abstraction. The experimental setup used to train the model prior
to deployment adopts an Ubuntu 20.04 version computer, with specifications
including the 8-core, 16-thread, AMD Ryzen 9 5900S processor, 16GB DDR4 RAM,
1TB disk memory and a dedicated NVIDIA GeForce RTX 3060 GPU, with 6GB VRAM.

###  V-B Training Process

The model uses Categorical Cross Entropy loss function to find the error or
deviation in the learning process and the Adam optimizer to optimize the input
weights by comparing the prediction and loss function, used in the training
process.

Training accuracy, training loss, validation accuracy and validation loss are
the metrics used to evaluate the performance of the model. The neural network
was trained with a batch size of 32, for 50 epochs with early stopping
callback, where the training process terminates when there is an increase in
the validation loss of the model. Feature extraction is done by the pre-
trained layers of the MobileNetV2 model and further an artificial neural
network consisting of two fully connected dense layers has been appended to
perform multi-class fish classification. The model was trained for 312 seconds
406 milliseconds (5.223 minutes) over 3 epochs, with approximately 104 seconds
468 milliseconds per epoch.

###  V-C Training & Validation Results

For reference, the training accuracy, training loss, testing accuracy and
testing loss are quantitatively analyzed in Table 1.

Results
---
Model |  Training Accuracy |  Training Loss |  Validation Accuracy |  Validation Loss
InceptionV3 and custom non-augmented dataset based underwater |  99.73% |  0.0072 |  91.13% |  0.372
MobileNetV1 and custom non-augmented dataset based underwater |  99.29% |  0.0199 |  89.92% |  0.5113
MobileNetV2 and custom non-augmented dataset based underwater |  99.58% |  0.0085 |  92.89% |  0.3165
MobileNetV2 with ANN and custom non-augmented dataset based outside water |  99.99% |  0.0021 |  99.83% |  0.0051

TABLE I: A quantitative analysis of the test results.

Figure 7: Accuracy plots of the training and validation phase.

Figure 8: Loss plots of the training and validation phase.

From the graph curves in Figure 7 and 8, it can be observed that the
validation accuracy and loss increase with an increase in number of iterations
in the network training process, and it stabilises in a certain interval. With
the nature of the trajectory, it can be inferred that the model was trained
efficiently with appropriate hyper-parameters to produce an acceptable
accuracy and loss, with very minimal over-fitting.

Drawing parallels to the results obtained by Liu et al. [7] , the mean
accuracy of 99.83% achieved with the CNN model based on MobileNetV2
architecture is an improvisation to the model proposed with an SVM based
classifier, trained on the same dataset, which provides a mean accuracy of
98.01%. The trained model was then converted into a special efficient portable
Flatbuffer, the Tensorflow Lite format (.tflite) to be deployed in edge
devices. The model was imported in a cross platform Flutter framework
application, and was used to predict an unseen image (Figure 6), which
classified the fish as a Black Sea Sprat in this example, with a confidence
measure of 0.99.

Interestingly enough, it must be noted here that when the same model was
trained on the previously mentioned augmented dataset, it brought the
validation accuracy of the model far below 50%. This was because the original
dataset by Ulucan et al. [3] was already implicitly augmented as it contained
images of the same fish captured at multiple angles, allowing the dataset
itself to successfully capture unique low-level features [20] of the fish in
its entirety. Therefore, further augmentation introduced a large amount of
noise into the dataset, causing the network to constantly change its weights
throughout the training phase without significant convergence.

###  V-D Deployment Accuracy

Figure 9: An example of an image with significant noise evaluated successfully
by the model[21].

While the training and testing of the model yielding promising results on the
dataset provided by Ulucan et al. [3], it’s important to note here that the
model is able to make accurate predictions even upon encountering images that
never featured in said dataset, as well as those that additionally contained
noise. Refer to figure 9 for one such example.

It can be observed in this demo that the picture evaluated by the model
contained plenty of noise in the form of a human holding the Gilt-Head Bream
and occlusion of the tail joint as well as the region below the gills.
It would be fair to claim at this point that the model produced a good one-off
prediction. However, upon testing it with 38 more such images sourced
publicly, it was observed that the model accurately classified 35 images with
a confidence of 0.5 or more, yielding a 92.10% deployment accuracy.

The most reasonable hypothesis that can be drawn at this point is that the
strong lighting in the original dataset highlighted the stray marks on the
cutting board well enough for them to be picked up as low level features by
the neural network, hence training the model to account for significant noise
in its predictions as well

##  VI Conclusion and End Notes

In conclusion, the algorithm presented in this research paper has demonstrated
impressive accuracy in classifying multiple marine species based on data
collected through a mobile application. With a reported testing accuracy of
99.83%, this algorithm has the increased potential to play a crucial role in
marine conservation efforts with the improvements acheived upon the status
quo.

The implementation of the algorithm in a server-side framework on AWS allows
for continual updates and improvements to the model based on new data, which
is particularly important in a field as dynamic as marine biology. The
research has the potential to make significant contributions to ongoing
efforts to better understand and conserve marine ecosystems and provides a
promising step towards more accurate and efficient methods for marine species
classification. Further development and refinement of this algorithm could
have far-reaching implications for the field of marine conservation, helping
to ensure the preservation and protection of marine species for generations to
come.

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# Investigation of $Q$ degradation in low-loss Si3N4 from heterogeneous laser
integration

Joel Guo  Department of Electrical and Computer Engineering, University of
California, Santa Barbara, Santa Barbara, California 93106, USA  Corresponding
author<EMAIL_ADDRESS>Chao Xiang  Department of Electrical and Computer
Engineering, University of California, Santa Barbara, Santa Barbara,
California 93106, USA  Warren Jin  Department of Electrical and Computer
Engineering, University of California, Santa Barbara, Santa Barbara,
California 93106, USA  Jonathan Peters  Department of Electrical and Computer
Engineering, University of California, Santa Barbara, Santa Barbara,
California 93106, USA  Mingxiao Li  Department of Electrical and Computer
Engineering, University of California, Santa Barbara, Santa Barbara,
California 93106, USA  Theodore Morin  Department of Electrical and Computer
Engineering, University of California, Santa Barbara, Santa Barbara,
California 93106, USA  Yu Xia  Department of Electrical and Computer
Engineering, University of California, Santa Barbara, Santa Barbara,
California 93106, USA  John E. Bowers  Department of Electrical and Computer
Engineering, University of California, Santa Barbara, Santa Barbara,
California 93106, USA

###### Abstract

High-performance, high-volume-manufacturing Si3N4 photonics requires extremely
low waveguide losses augmented with heterogeneously integrated lasers for
applications beyond traditional markets of high-capacity interconnects. State-
of-the-art quality factors ($Q$) over 200 million at 1550 nm have been shown
previously; however, maintaining high $Q$s throughout laser fabrication has
not been shown. Here, Si3N4 resonator intrinsic $Q$s over 100 million are
demonstrated on a fully integrated heterogeneous laser platform. $Q_{i}$ is
measured throughout laser processing steps, showing degradation down to 50
million from dry etching, metal evaporation, and ion implant steps, and
controllable recovery to over 100 million from annealing at 250 ${}^{\circ}C$
\- 350 ${}^{\circ}C$.

††journal: ol

Silicon nitride (Si3N4) photonics has developed rapidly in the last decade,
leveraging CMOS-compatibility as well as favorable material properties such as
a high effective $\chi^{(3)}$ nonlinearity, a wide transparency window, and
low optical losses [1]. These properties allow for powerful functionalities
such as frequency combs [2], photogalvanic-induced second harmonic generation
[3, 4], magneto-optical trap beam delivery [5], feedback tolerant lasers [6,
7], low-frequency noise rivaling fiber lasers [8], and tunable lasers beyond
the silicon bandgap [9, 10, 11]. Notably, all of these functions depend
heavily on ultra-low losses, which can be equivalently represented by the
quality factor, $Q$.

Despite the powerful functionalities available in Si3N4, heterogeneous laser
integration is still required to scale in terms of economics or circuit
complexity. As photonic systems grow in complexity, electrically-pumped
optical gain on the same chip is necessary to compensate losses while allowing
for denser integration [12]; similarly in quantum computing systems,
individually addressing millions of atoms or ions necessitates dense
integration of lasers on chip [13, 14]. In terms of economic viability, Intel
demonstrated commercial deployment of millions of 800 Gbps optical
transcievers, leveraging heterogeneous laser integration via wafer bonding on
a silicon photonics platform [15].

A similar wafer bonding process was used in this work to integrate
electrically pumped optical gain with ultra-low loss Si3N4 resonators [7].
However, laser integration comes with tradeoffs; previous work showed that
processing such as O2 plasma and UV ozone cleaning can increase the loss of
Si3N4 waveguides operating at visible wavelengths by UV radiation exciting
carriers into defect states, which then become optically absorbing [16].
Furthermore, earlier work on Si3N4 charge-trapping memory devices reported
potential excitation through not only UV, but also ion, electron, and X-ray
radiation [17], all of which are possible in the reported fabrication process
here.

In this work, intrinsic $Q$ ($Q_{i}$) greater than 100 million is demonstrated
within a heterogeneous laser process for the first time. The $Q_{i}$ of ultra-
low loss Si3N4 resonators was characterized throughout laser backend
processing steps, showing degradation with etches, metal liftoffs, and ion
implant, and controllable recovery back up to over 100 million with annealing.
Previous works have dedicated much study to understanding scattering and
absorption-limits in high-$Q$ Si3N4 resonators [18, 19, 20]; however, no work
has shown influences of $Q$ degradation from heterogeneous laser integration.
Previous heterogeneous laser work has mitigated the effects of laser
fabrication by burying the ultra-low-loss Si3N4 layer over 4 $\mu$m below the
surface of the laser chips, resulting in $Q_{i}$ near 40 million (0.5 dB/m) at
wavelengths around 1550 nm [7]. To better understand how these $Q$s can be
improved to the state-of-the-art values of over 200 million [21], the process
on this heterogeneous laser platform is repeated with $Q$ measurements
performed between laser backend steps, highlighting $Q$ degradation or
recovery contributions from individual steps.

Countless groups across the world have demonstrated powerful functionalities
in low-loss Si3N4 with off-chip lasers; the results reported here bolster the
credibility of those previous reports promising heterogeneous laser
compatibility. With these advancements, laser-integrated low-loss Si3N4
photonics is uniquely positioned to address applications such as field-
deployable atom-based clocks and quantum sensors, as well as low-noise
microwave photonic oscillators. Due the significant reduction in cost, size,
weight, and power compared to traditionally used bulk-optics constrained to
lab environments, these laser-integrated Si3N4 photonic solutions can now
target emerging markets in GPS-free position, navigation, and timing (PNT), as
well as next-generation radar and commercial 5G.

Figure 1: a) Simplifed fabrication process for heterogeneously integrating
InP/Si lasers on multi-layer Si3N4 [7]. The ultra-low-loss, bottom-layer Si3N4
was processed, clad, annealed at a commercial foundry, while the top layer was
processed and clad at UCSB. The bottom layer Si3N4 was isolated by over 4
$\mu$m from the surface in the remaining fabrication steps. SOI was bonded and
processed to form Si waveguides and grating couplers. InP gain dies were
bonded and laser mesas are etched. The boxed section of the backend steps are
detailed in b) including excess bonded Si removal, SiO2 claddings, via etches,
metal liftoffs, hydrogen (H+) implantation, and rapid thermal anneal (RTA)
steps. $Q_{i}$ was measured through backend steps and plotted in Fig. 2.

The heterogeneous laser fabrication is composed of four stages: 1) Si3N4
processing, 2) Si processing, 3) InP processing, and 4) backend processing
(Fig. 1a). The $Q$ is measured throughout the backend processing steps,
further illustrated in Fig. 1b. In the Si3N4 processing stage, the bottom
layer 100 nm Si3N4 is deposited by LPCVD on 200 mm wafers at a commercial
foundry and annealed over 1000 ${}^{\circ}C$ to drive out residual hydrogen
and reduce absorption losses [21]. SiO2 cladding deposition and CMP are also
performed at the foundry. After performing another layer of 100 nm LPCVD Si3N4
and coring the wafers into 100 mm wafers through external vendors, the Si3N4
is etched at UCSB, clad with SiO2, and planarized with CMP. Notably, the top
layer of Si3N4 is vertically separated from the bottom layer Si3N4 by over 4
$\mu$m SiO2, which helps to protect the bottom layer from contaminants
introduced by further laser processing [7]. In the Si processing stage, SOI is
first bonded, then the Si substrate and buried oxide are subsequently removed
by mechanical polishing, Si Bosch etch, and BHF wet etch. The remaining 500 nm
Si is then etched into waveguides, grating couplers, and outgassing channels.
In the InP processing stage, InP gain dies are bonded. The InP substrate is
removed with mechanical polishing and HCl wet etch, then the InP laser mesa
(P-InP, QW, and N-InP) is etched using a combination of methane/hydrogen/argon
(MHA) dry etching and H3PO4-based wet etching [7].

The backend laser processing steps consist of 1) excess Si removal, 2) SiO2
deposition and via etch, 3) N-metal liftoff, 4) P-metal liftoff, 5) cladding
SiO2 deposition, 6) via etch, 7) rapid thermal anneal (RTA), 8) hydrogen (H+)
implant, 9) heater metal liftoff, and 10) probe metal liftoff. During the InP
steps, excess bonded Si is left on top to protect the Si3N4 resonators from
process contamination, so in the backend steps, the excess bonded Si is first
removed with an unbiased XeF2 dry etch. The optical mode in the Si waveguides
is then able to couple into the Si3N4 waveguides without leaking into the
excess bonded Si, thus the $Q$ in the lower-layer Si3N4 resonators can be
measured via adiabatic tapers from the Si to the Si3N4. Since grating couplers
are etched in the Si layer, the $Q$ can be measured by vertical fiber coupling
throughout the remaining process steps without dicing the wafer and edge-
coupling. After excess Si removal, the laser mesa is clad with SiO2 and dry
etched with a CF4/O2 chemistry to form vias. P- and N- metal contacts are then
formed by electron-beam evaporation and liftoff. Further SiO2 is deposited and
vias are etched; RTA is then performed to lower the resistances of the P- and
N-contacts. The wafer is then hydrogen implanted to insulate the sides of the
laser mesa and confine injected carriers. Lastly, heater and probe metals are
deposited for thermo-optic phase tuners and electrical pads.

The $Q$ was measured throughout the laser backend processing steps by sweeping
a fiber-MZI-calibrated laser through the resonances of Si3N4 ring resonators
with 10 $\mu$m-wide resonator waveguides and 20 GHz FSR. All resonances were
measured in the wavelength range of 1540 - 1560 nm, corresponding to the
bandwidth of the Si grating couplers. The transmission spectra were fit with a
coupled-mode-theory model accounting for backscattering [19], and the $Q$ was
extracted from the fitted FWHM of the doublet resonances. Coupling $Q$ was
also extracted to confirm that only intrinsic $Q$ varied.

Figure 2: $Q_{i}$ measured before, throughout, and after backend steps
consisting of 1) excess Si removal, 2) SiO2 deposition & via etch, 3) N-metal
liftoff, 4) P-metal liftoff, 5) SiO2 deposition, 6) via etch, 7) rapid thermal
anneal, 8) hydrogen implant, 9) heater metal liftoff, and 10) probe metal
liftoff. $Q_{i}$ measured on the same ring design on another wafer from the
same foundry LPCVD Si3N4 batch but without laser processing is plotted as step
0), showing approximate values before laser processing. Measurements after
laser testing several months following fabrication (without additional
annealing) are plotted as step 11).

Extracted $Q_{i}$ is plotted after each backend processing step in Fig 2.
After the excess bonded Si is removed (step 1), $Q_{i}$ over 100 million
around 1550 nm is measured, corresponding to about 0.25 dB/m. The $Q_{i}$
drops down to 75 million with laser mesa cladding and SiO2 via etch (step 2).
The $Q_{i}$ continues to fall to 50 million with electron-beam evaporation and
liftoff of N- and P-contact metals (steps 3 and 4). After further SiO2
cladding deposition (step 5), the $Q_{i}$ is restored above 100 million. The
SiO2 ICP-PECVD was performed at 250 ${}^{\circ}C$ for 20 minutes, effectively
acting as a high-temperature anneal. As previously mentioned, a similar anneal
was shown to restore low-loss Si3N4 waveguides at visible wavelengths [16],
and the measurement here corroborates this near 1550 nm. Once again, $Q_{i}$
drops with the same SiO2 via etch as previous (CF4 \+ O2), down to near 50
million at 1550 nm (step 6). Next, the $Q_{i}$ recovers with a 30 second, 370
${}^{\circ}C$ rapid thermal anneal (step 7) - again, similar to previous work
although for a significantly shorter time [16]. In the last steps of the
backend process, the $Q_{i}$ continues to degrade with hydrogen (H+) implant
(step 8) and heater metal liftoff (step 9) down to seemingly saturated values
near 50 million. Details of the hydrogen implant are provided in previous
works [22]. The $Q_{i}$ recovers slightly from probe metal deposition (step
10), most likely related to heating from the thick metal stack. To mitigate
risk associated with migrating the H+ dopants, further annealing was held off
until after laser testing. However, upon measuring after months of laser
testing, the $Q_{i}$ was found to have recovered above 100 million (step 11).
Thus, $Q_{i}$ over 100 million was found to be recoverable immediately by high
temperature annealing and also gradually without annealing.

A summary of the drops and recoveries in $Q_{i}$ is plotted in Fig. 3 by
averaging the data over wavelength and plotting with process step. Data from
two other dies on the same wafer processed simultaneously are also included to
show the same trends across a wafer. The measured Si3N4 resonators on dies 2
and 3 were protected with aluminum foil during metal liftoff steps, while the
measured resonators on die 1 were not protected, showing negligible
differences from foil protection. Significant $Q_{i}$ drops are correlated
with etches, implant, and metal liftoffs with varying strengths, while
recoveries are correlated with anneals. The $Q_{i}$ measured from another
wafer from the same foundry Si3N4 LPCVD batch but without further laser
processing is also plotted as step 0) on both Figures 2 and 3 approximating
the $Q_{i}$ values before laser processing.

Figure 3: Wavelength averages of $Q_{i}$ with process steps across three dies
from the same wafer. Dies 2 and 3 were protected with aluminum foil during
metal liftoff steps, while die 1 was not. $Q_{i}$ over 100 million was
measured after Si removal (backend step 1). The $Q_{i}$ generally dropped with
dry etches, metal liftoffs, and H+ implant and recovered with high temperature
anneals. Over time, the $Q_{i}$ recovered to its initially measured values,
measured after several months of laser testing. Again, measured Q data from a
wafer without laser processing is included as step 0) as a reference. SiO2
etches (steps 2 and 6), anneals (steps 5 and 7), and metal liftoffs (steps 3,
4, 9, 10) are grouped together by color since they are similar processes. Note
that the XeF2 etch in step 1 is unbiased and categorized differently than the
biased SiO2 etches.

In conclusion, the intrinsic $Q$ of Si3N4 resonators was characterized
throughout heterogeneous laser processing steps, dropping with dry etches, ion
implant, and metal liftoffs, while recovering with anneals and thermal
relaxation over a long time constant. Recovery values saturate near 110
million and degradation values saturate near 50 million, both at 1550 nm. All
$Q$ measurements were taken within a day of the associated processing step,
after stripping the residual photoresist with solvent and O2 plasma cleans
when necessary (i.e. SiO2 deposition and RTA did not require photoresist). As
a result, the time constant for recovery without annealing is greater than
days. Previous work showed that the mechanism of increasing loss was the
population of optically absorbing defect states in Si3N4 by UV excitation only
[16], while the work presented here introduces many more potential excitation
mechanisms due to the variety of processing necessary for laser fabrication.
As previously mentioned, SixNy has a high concentration of defect states,
which can be populated not only by UV, but also by electron, ion, and UV
radiation [17]. All of these are potentially introduced in the fabrication
process reported here; for example, electron-beam evaporation of metals is
known to eject X-rays, ionized metal vapor, and secondary electrons [23, 24].
Aluminum foil protection covering the resonators was used to test mitigation
of Q degradation in all metal evaporation steps with negligible differences
found, as shown in Fig 3; however, the foil also makes penetration of UV and
electron radiation into the Si3N4 unlikely. Future work dedicated to
separating the effects of UV, ion, electron, and X-ray radiation should start
by measuring $Q$ degradation from O2 plasma cleans of increasing times, since
this was common to nearly all steps in varying degrees depending on residual
photoresist. X-ray radiation can perhaps be tested with electron beam
evaporation systems at different beam currents necessary for evaporation of
the various metals used in the P- and N-contact metal stacks. Considering the
higher $Q$s of 160 million measured on another wafer from the same Si3N4
foundry run but without heterogeneous laser processing, further prevention
before the backend steps may be possible and will need to be investigated.
Lastly, the same suspected Si3N4 defect states responsible for the $Q$
fluctuations in this work have also been postulated to be involved in the
coherent photogalvanic effect, resulting in large effective $\chi^{(2)}$
nonlinearities in Si3N4 [4]; thus, further investigation to link these two is
necessary.

Despite the many potential degradation mechanisms, Si3N4 resonators with
intrinsic $Q$s greater than 100 million were demonstrated after full
heterogeneous InP/Si laser integration. Factoring in the powerful
functionalities of microresonator frequency combs, second harmonic generation,
and extreme laser noise reduction, this laser-integration Si3N4 photonics
platform paves the way for scalable manufacturing and field-deployment of
next-generation atomic clocks, quantum sensors, and low-noise microwave
oscillators.

Funding This research was supported by DARPA GRYPHON (HR0011-22-2-009)

Acknowledgments The authors thank Minh Tran, Demis John and Brian Thibeault
for helpful discussions. A portion of this work was performed in the UCSB
Nanofabrication Facility, an open access laboratory.

Disclosures The authors declare no conflicts of interest.

Data Availability Statement Data underlying the results presented in this
paper are not publicly available at this time but may be obtained from the
authors upon reasonable request.

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# Exact Solution of Hartemann-Luhmann Equation of Motion for a Charged
Particle interacting with an Intense Electromagnetic Wave/Pulse

Shivam Kumar Mishra<EMAIL_ADDRESS>Sudip Sengupta  Institute for
Plasma Research, Bhat, Gandhinagar, India, 382428  Homi Bhabha National
Institute, Training School Complex, Mumbai, 400094, India

###### Abstract

We report an exact solution of the Hartemann-Luhmann equation of motion for a
charged particle interacting with an intense electromagnetic wave/pulse. It is
found that the radiation reaction force has a significant affect on the
charged particle dynamics and the particle shows, on average, a net energy
gain over a period of time. Further, using a MATHEMATICA based single particle
code, the net energy gained by the particle is compared with that obtained
using Landau-Lifshitz and Ford-O’connell equation of motion, for different
polarizations of the electromagnetic wave. It is found that the average energy
gain is independent of both the chosen model equation and polarization of the
electromagnetic wave. Our results thus show, that the simpler and hence
analytically tractable Hartemann-Luhmann equation of motion ( as compared to
Landau-Lifshitz and Ford-O’connell equation of motion ) is adequate for
calculations of practical use (for e.g. energy calculation).

###### pacs:

52.27.Lw,52.30.Ex,52.35.We,51.20.+d

##  I INTRODUCTION

An accelerating charged particle is associated with electromagnetic fields
that radiates energy irreversibly out to infinity Griffiths (2013); Jackson
(1999). In addition to radiated energy, there exists an electromagnetic field
energy called the Schott energy which is localized at the particle and can be
exchanged with the particle’s mechanical energy Schott (1912); Ferris and
Gratus (2011). These fields ( radiation field and the Schott field ) carry
energy and momentum and affect the dynamics of the charged particle by
providing a self-influence called the self-force or radiation reaction (RR)
force Lorentz (1909); Abraham (1905); ”Dirac (1928); Dirac (1938); Bhabha
(1939); Wheeler and Feynman (1945); Rohrlich (1964); Teitelboim (1970); Barut
(1974); Rohrlich (2007). Radiation reaction force significantly affects the
charged particle dynamics when the power radiated by the particle becomes
comparable to the instantaneous rate of change of energy of the particle Shen
(1970); Hadad  _et al._ (2010); Kravets, Noble, and Jaroszynski (2013); Sagar,
Sengupta, and Kaw (2015). In this scenario, the Lorentz force equation is not
an appropriate choice for investigating the charged particle dynamics. For a
complete description of the charged particle dynamics, we need an equation
which incorporates the effect of radiation reaction. In this context, a long
ongoing debate has suggested a number of equations of motion viz. Lorentz-
Abraham-Dirac(LAD)Dirac (1938), Ford-O’connellConnell (2003); Ford and
O’Connell (1991, 1993), EliezerEliezer (1948), Landau-LifshitzLandau and
Lifshitz (1980), M.O.PapasMo and Papas , Caldirola(Caldirola, ), Hartemann-
Luhmann(Hartemann and Luhmann, 1995), YaghjianYaremko , SokolovSokolov  _et
al._ (2009) et. al. etc. It is worth noting here that except for the LAD
equation, the other equations have not been derived from first principles.
Thus, the LAD equation is the most basic equation for describing the
interaction of a spin less point charged particle with an electromagnetic
field and it can be written as

$\dot{u}^{\alpha}=\frac{e}{mc}F^{\alpha\beta}u_{\beta}+g^{\alpha}$ (1)

where $F^{{\alpha}{\beta}}$ is the electromagnetic field tensor,
$u^{\alpha}={\gamma(c,\vec{v})}$ is the four-velocity of the charge,
$\gamma=1/\sqrt{1-(v/c)^{2}}$, $\tau_{0}=(2/3)(e^{2}/mc^{3})$ and the dot
represents differentiation with respect to proper time $\tau$. We use the
metric convention $(+1,-1,-1,-1)$. For the LAD equation the radiation reaction
term $g^{\alpha}$ is given by,

$g^{\alpha}=\tau_{0}\left({\ddot{u}}^{\alpha}+\frac{\dot{u}^{\beta}\dot{u}_{\beta}}{c^{2}}{{u}}^{\alpha}\right)$
(2)

In the above equation, the first and the second term on the right hand side
respectively represent the radiation reaction force due to the Schott energy
and the radiated energySchott (1912); Rohrlich (2000). The presence of the
Schott term in the LAD equation results in well known unphysical problems like
pre-acceleration and runway solutionGriffiths (2013); Jackson (1999); Rohrlich
(2007); Hartemann (2001). Using the condition that the four acceleration is
orthogonal to the four velocity Landau and Lifshitz (1980), the radiation
reaction term in Eq.(2) can be rewritten as

$g^{\alpha}=\tau_{0}\left(\delta_{\beta}^{\alpha}-\frac{{u}^{\alpha}{u}_{\beta}}{c^{2}}\right){\ddot{u}}^{\beta}$
(3)

From Eq.(3), it can be seen that the Schott term is of order unity whereas the
radiated term is of order $\sim\gamma^{2}$. So in the ultra relativistic case,
where radiation reaction becomes important, Schott term may be neglected in
comparison with the radiated term. Thus, Eq.(1) can be written as

$\dot{u}^{\alpha}=\frac{e}{mc}F^{\alpha\beta}u_{\beta}+\tau_{0}\frac{{\dot{u}}^{\beta}{\dot{u}}_{\beta}}{c^{2}}{u}^{\alpha}$
(4)

Eq.(4) was first derived by Hartemann and Luhmann within the framework of
classical electrodynamics and does not suffer from unphysical problems like
pre-acceleration and runway solutionsHartemann and Luhmann (1995).

Following Landau-Lifshitz, in the non-relativistic regime, the radiation
reaction term may be considered small as compared to the Lorentz force term,
provided the typical wavelength $\lambda$ and the typical field amplitude $F$
of the external electromagnetic field fulfill the following two conditions;

$\lambda>>\frac{e^{2}}{mc^{2}},F<<\frac{m^{2}c^{4}}{e^{3}}$ (5)

In order to perform an analogous reduction of degree in the relativistic case,
the conditions (5) have to be fulfilled in the instantaneous rest frame of the
charged particleLandau and Lifshitz (1980); Bild, Deckert, and Ruhl (2019).
This allows for reduction of degree of the Hartemann-Luhmann equation, by
substituting the acceleration terms in the radiation reaction force with the
Lorentz force term. Performing this substitution, Eq.(4) can finally be
written as

$\dot{u}^{\alpha}=\frac{e}{mc}F^{\alpha\beta}u_{\beta}-\tau_{0}\frac{e^{2}}{m^{2}c{{}^{4}}}u^{\alpha}F^{\beta\gamma}F_{\gamma\delta}u^{\delta}u_{\beta}$
(6)

Eq.(6) is the modified form of Hartemann-Luhmannn equation of motion. The
purpose of this article is to study the charged particle dynamics in the
presence of a relativistically intense elliptically polarized electromagnetic
wave/pulse using the above new form of Hartemann-Luhmann equation, as given by
Eq.(6). This study gives a clearer understanding of particle dynamics and
energy gain during wave particle interaction, than previous studies that have
been done by several authors using the Landau-Lifshitz equation of motion
Hadad  _et al._ (2010); Kravets, Noble, and Jaroszynski (2013); Burton and
Noble (2014); Keitel  _et al._ (1998); Piazza (2008); Di Piazza,
Hatsagortsyan, and Keitel (2009); Ondarza-Rovira and Boyd (2020). In section
II of this article, the new form of Hartemannn-Luhmann equation (hereinafter
called modified Hartmann-Luhmann equation) has been solved analytically for
energy, momentum, and position of a particle which is interacting with a
relativistically intense light wave/pulse. The dynamics of a charged particle
interacting with a wave train and a Gaussian laser pulse is presented in
subsections III.1 and III.2 respectively. In section IV, the average energy
gained by the particle as obtained from the modified Hartemann-Luhmann
equation is compared with that obtained by numerically solving the Landau-
Lifshitz and Ford-O’connell equation of motion. Finally in section V, we
present a summary of our results.

##  II Solution of Hartemann-Luhmann Equation

In dimensionless form, the modified Hartemann-Luhmann equations, governing the
dynamics of a charged particle interacting with an intense electromagnetic
wave are

$\frac{d\vec{p}}{dt}=\left(\vec{E}+\vec{\beta}\times\vec{B}\right)-\tau_{0}\gamma^{2}\left\\{\left(\vec{E}+\vec{\beta}\times\vec{B}\right)^{2}-\left(\vec{E}.\vec{\beta}\right)^{2}\right\\}\vec{\beta}$
(7)
$\frac{d\gamma}{dt}=\vec{E}.\vec{\beta}-\tau_{0}\gamma^{2}\left\\{\left(\vec{E}+\vec{\beta}\times\vec{B}\right)^{2}-\left(\vec{E}.\vec{\beta}\right)^{2}\right\\}$
(8)

where the symbols have their usual meanings and the normalizations used are
$p\rightarrow p/mc$, $E\rightarrow eE/m\omega c$, $B\rightarrow eB/m\omega c$,
$\tau_{0}\rightarrow\omega\tau_{0}$ and $\beta=v/c$. The above equations are
respectively the spatial and temporal component of Eq.(6). Consider the
interaction of a charged particle with a light wave/pulse which is propagating
along the z-direction and defined by the normalized vector potential
$\vec{A}(\phi)$ ($\vec{A}\rightarrow e\vec{A}/mc^{2}$) where
$\phi=t-z+\phi_{0}$ ($t\rightarrow\omega t$ and $z\rightarrow kz$). Here
$\omega$ and $k$ are respectively the frequency and wave number of the light
wave, and $\phi_{0}$ is the initial phase of the light wave as seen by the
particle. Using $\vec{E}=-\partial\vec{A}/\partial t$ and
$\vec{B}=\vec{\nabla}\times\vec{A}$, Eq.(7) and Eq.(8) may be respectively
written as

$\frac{d\vec{p}}{dt}=\left(-\frac{d\vec{A}}{dt}+\vec{\nabla}(\vec{\beta}.\vec{A})\right)-\tau_{0}\gamma^{2}\left\\{\left(-\frac{d\vec{A}}{dt}+\vec{\nabla}(\vec{\beta}.\vec{A})\right)^{2}-\left(\vec{\beta}.\frac{\partial\vec{A}}{\partial
t}\right)^{2}\right\\}\vec{\beta}$ (9)
$\frac{d\gamma}{dt}=-\beta.\frac{\partial\vec{A}}{\partial
t}-\tau_{0}\gamma^{2}\left\\{\left(-\frac{d\vec{A}}{dt}+\vec{\nabla}(\vec{\beta}.\vec{A})\right)^{2}-\left(\vec{\beta}.\frac{\partial\vec{A}}{\partial
t}\right)^{2}\right\\}$ (10)

where $d\vec{A}/dt=\partial\vec{A}/\partial t+(\beta.\vec{\nabla})\vec{A}$,
and we have used the relation
$\vec{\nabla}(\vec{\beta}.\vec{A})=(\vec{\beta}.\vec{\nabla})\vec{A}+\vec{\beta}\times(\vec{\nabla}\times\vec{A})$
as $\beta$ is a function of time alone. Using $\vec{A}=\vec{A}(\phi)$ and
transverse nature of the light wave/pulse Eq.(9) and Eq.(10) may be
respectively written as

$\frac{\Delta}{\gamma}\vec{p^{\prime}}=-\left[\vec{A^{\prime}}\frac{\Delta}{\gamma}+(\vec{\beta}.\vec{A^{\prime}})\hat{z}\right]-\tau_{0}\Delta^{2}A^{\prime
2}\vec{\beta}$ (11)
$\frac{\Delta}{\gamma}\gamma^{\prime}=-\beta.\vec{A^{\prime}}-\tau_{0}\Delta^{2}A^{\prime
2}$ (12)

Here prime denotes derivative with respect to $\phi$, $\Delta=\gamma-p_{z}$ (
$p_{z}=\gamma\beta_{z}$ ) and $d/dt=(\Delta/\gamma)d/d\phi$. Subtracting the
z-component of equation of motion (Eq.(11)) from the energy equation (Eq.(12))
we get

$\Delta^{\prime}=-\tau_{0}\Delta^{2}A^{\prime 2}$ (13)

On integrating the above equation and taking $\phi=\phi_{0}$ at $t=0$ (and
$z=0$) we get

$\Delta=\frac{\Delta_{0}}{1+\tau_{0}\Delta_{0}I_{1}}$ (14)

where $I_{1}=\int_{\phi_{0}}^{\phi}A^{\prime 2}d{\phi}$ and
$\Delta_{0}=(\gamma_{0}-p_{z0})$ where $\gamma_{0}$ and $p_{z0}$ are
respectively the initial energy and z-component of initial momentum of the
particle. It is clear from the above equation that in the absence of radiation
reaction (i.e. $\tau_{0}=0$), $\Delta$ is a constant of motionKaw and Kulsrud
(1973); Landau and Lifshitz (1980); Gibbon (2005). Now the perpendicular
component of equation of motion (x,y components of Eq.(11)) gives

$\Delta\vec{p^{\prime}}_{\perp}=-\vec{A^{\prime}}\Delta-\tau_{0}\Delta^{2}A^{\prime
2}\vec{p}_{\perp}$ (15)

Multiplying equation Eq.(13) by $\vec{p}_{\perp}$, subtracting from the above
equation and dividing by $\Delta^{2}$ we get

$\left(\frac{\vec{p}_{\perp}}{\Delta}\right)^{\prime}=-\frac{\vec{A^{\prime}}}{\Delta}$
(16)

which on integration yields the perpendicular component of momentum as

$\vec{p}_{\perp}=\frac{\Delta}{\Delta_{0}}\left[\vec{p}_{\perp
0}-(\vec{A}-\vec{A}_{0})-\tau_{0}\Delta_{0}\vec{I}_{2}\right]$ (17)

Here $\vec{I}_{2}=\int_{\phi_{0}}^{\phi}\vec{A^{{}^{\prime}}}I_{1}d\phi$,
$\vec{p}_{\perp 0}$ is the initial perpendicular component of momentum and
$\vec{A_{0}}=\vec{A}(\phi_{0})$ is the vector potential seen by the particle
at $t=0$ (and $z=0$).

To obtain the longitudinal component of momentum, we begin with the
z-component of equation of motion (Eq. (11)) as

$\Delta\vec{p^{\prime}}_{z}=-\vec{\beta}.\vec{A^{\prime}}-\tau_{0}\Delta^{2}A^{\prime
2}$ (18)

Following a similar procedure as used for obtaining the perpendicular
component of momentum, i.e. multiplying Eq.(13) $p_{z}$, subtracting from the
above equation and dividing by $\Delta^{2}$ we get

$\left(\frac{p_{z}}{\Delta}\right)^{\prime}=-\frac{\vec{p}_{\perp}.\vec{A^{\prime}}}{\Delta^{2}}$
(19)

Substituting the expressions for $\Delta$ and $\vec{p}_{\perp}$ from Eqs. (14)
and (17) respectively and integrating, we arrive at the final expression for
longitudinal momentum as

$p_{z}=\frac{\Delta}{\Delta_{0}}\left[p_{z0}-\frac{1}{\Delta_{0}}\left\\{\vec{p}_{\perp
0}-\frac{1}{2}(\vec{A}-\vec{A_{0}})\right\\}.(\vec{A}-\vec{A_{0}})-{\tau_{0}}\left\\{{\vec{p}_{\perp
0}}-(\vec{A}-\vec{A_{0}})\right\\}.\vec{I_{2}}+\frac{1}{2}\Delta_{0}\tau_{0}^{2}I_{2}^{2}\right]$
(20)

Equations (17), (20) along with $\gamma=p_{z}+\Delta$ ( with $\Delta$ given by
equation (14) ) respectively give the momentum and energy as a function of
phase $\phi$, for a charged particle interacting with an electromagnetic
wave/pulse including radiation reaction effects. Eqs. (17) and (20) can be
integrated further to obtain the particle positions. The expressions for
positions of a charged particle interacting with an elliptically polarized
light wave train are presented in the Appendix.

##  III Dynamics of a charged Particle in an Electromagnetic Wave/Pulse

The well known solution of Lorentz force equation ( i.e. without the radiation
reaction term ) shows that the charged particle motion in an electromagnetic
wave/pulse results in no net energy gain by the particle from the wave/pulse.
In case of interaction with a finite duration pulse, the dynamics only results
in a net displacement of the particle from its initial positionLandau and
Lifshitz (1980); Gibbon (2005). It happens due to the well defined phase
relationship between the electric field of the electromagnetic wave and the
velocity of the particle. In the presence of radiation reaction force the
phase relationship between the electric field and the velocity of the particle
is disturbed in such a way that as a result the charged particle gains a net
amount of energy. This can be seen from the radiation reaction term in the
Hartemann-Luhmann equation which acts in a direction opposite to the velocity
vector of the particle. In the next two subsections, we respectively describe
the motion of a charged particle interacting with a wave train and a pulse.
Results are obtained by numerically solving the Hartemann-Luhmann equations
using a MATHEMATICA based single particle code and are also compared with the
analytical solutions obtained in section II.

###  III.1 DYNAMICS OF A CHARGED PARTICLE IN AN ELECTROMAGNETIC WAVE TRAIN

The vector potential representing a wave train is given by

$A(\phi)=a_{0}\left(\delta\cos(\phi)\hat{x}+g\sqrt{1-\delta^{2}}\sin(\phi)\hat{y}\right)$
(21)

The value of $\delta=(0,1)$ and $\delta=1/\sqrt{2}$ correspond to linear and
circular polarization respectively. $g=\pm 1$ respectively correspond to right
and left handedness of polarization. We now present results which clearly
exhibit the effect of radiation-reaction on charged particle dynamics moving
in an intense electromagnetic wave train for two different polarizations viz.
linear ($\delta=1$) and circular ($\delta=1/\sqrt{2}$). As stated before, the
Hartemann-Luhmann equations are numerically integrated with $a_{0}=500$ and
$\tau_{0}=1.8\times 10^{-8}$ ; the particle starts from origin with initial
momenta as $p_{\perp 0}=0$ and $p_{z0}=-10^{3}$ ( initial energy
$\gamma=10^{3}$ ).

Fig.1 and Fig.2 represent the evolution of longitudinal and transverse
momentum, energy and trajectory of the particle in linearly ($\delta=1$) and
circularly polarized ($\delta=1/\sqrt{2}$) electromagnetic wave respectively.
The red and blue curve respectively represent the solution of the Hartemann-
Luhmann and Lorentz force equations. The dashed green curve on the red and
blue curve represent the analytical solution. The clear mismatch between the
red and blue curve show the strong signature of the radiation reaction force.
In fig. 1(a), the blue line represents the longitudinal momentum of the
particle without radiation-reaction. This shows that the particle is drifting
opposite to the direction of propagation of the wave, whereas red line shows
that, inclusion of radiation reaction stops the particle within one cycle of
electromagnetic wave and the particle is pushed along the direction of
propagation of the wave. Fig 1(b) represents the transverse momentum with and
without radiation reaction, which shows that the net average transverse
momentum with radiation reaction is smaller than the transverse momentum
without radiation reaction. Fig. 1(c) represents the evolution of energy of
the particle with and without radiation reaction. The red curve shows that in
the presence of radiation reaction, energy of the particle shows a drastic
change, whereas in the absence of the radiation reaction, the average energy
of the particle remains constant. Finally Fig. 1(d) represents the trajectory
of the particle with ( red ) and without radiation reaction (blue). The
results are qualitatively same for circularly polarized light shown in fig.2.
The fig. 3 represents the net average energy gained by the particle for
different polarization of the electromagnetic wave. The red, blue and green
curves respectively correspond to $\delta=1$, $\delta=1/\sqrt{2}$ and
$\delta=1/4$. The results show that the average energy gain is independent of
the polarization of the wave.

###  III.2 DYNAMICS OF A CHARGED PARTICLE IN THE PRESENCE OF ELLIPTICALLY
POLARIZED GAUSSIAN LASER PULSE

In this subsection the effect of the radiation-reaction has been studied for a
elliptically polarized Gaussian laser pulse. The vector potential for the
laser pulse is given by

$\vec{A}(\phi)=a_{0}e^{{-\frac{(\phi-\xi_{0})^{2}}{2n_{0}}}}\left(\delta
cos(\phi)\hat{x}+g\sqrt{1-\delta^{2}}sin(\phi)\hat{y}\right)$ (22)

The effect of radiation-reaction on charged particle dynamics moving in
intense electromagnetic laser pulse has been studied for linearly ($\delta=1$)
and circularly($\delta=1/\sqrt{2}$) polarized light. The energy gain has been
studied by taking $n_{0}=10$, $\xi_{0}=n_{0}\pi$, $a_{0}=10^{3}$ and
$\tau_{0}=1.8\times 10^{-8}$; the particle initially starts from the origin
with $p_{\perp 0}=0$, and $p_{z0}=-10^{3}$.

Fig.4 and Fig.5 respectively represent the evolution of longitudinal and
transverse momentum, energy and trajectory of the particle in linearly
($\delta=1$) and circularly polarized ($\delta=1/\sqrt{2}$) electromagnetic
light pulse wave. As before, the the red and blue curve represent the solution
of the Hartemann-Luhmann and Lorentz force equations respectively. The clear
mismatch between the red and blue curve show a strong signature of the
radiation reaction force. In fig.4(a), the blue line represents the
longitudinal momentum of the particle without radiation-reaction. This shows
that the particle is drifting opposite to the direction of propagation of the
wave, whereas red line shows the inclusion of radiation reaction stops the
particle within laser pulse and the particle is pushed along the direction of
propagation of the wave. Fig. 4(b) represents the transverse momentum with and
without radiation reaction, which shows that there is net gain in transverse
momentum in the presence of radiation reaction. Fig. 4(c) represents the
change in energy of the particle with and without radiation reaction. The red
curve shows that in the presence of radiation reaction, the particle gains
energy, which can also be seen from the gain in transverse as well as the
longitudinal momentum of the particle, after the pulse has passed over the
particle. However, in the absence of the radiation reaction, there is no net
gain in energy of the particle from the pulse. Finally Fig. 4(d) represents
the trajectory of the particle with (red) and without radiation reaction
(blue). The results are qualitatively same for the circularly polarized light
as shown in fig.5. As the wave crosses over the particle, fig.6 represents the
net energy remaining with the particle in the case of different polarization
of the electromagnetic wave. The red, blue, and green curves correspond to
$\delta=1,\,\,1/\sqrt{2},\,\,1/4$ respectively.

##  IV Comparison of Harteman-Luhmann equation of motion with Landau-Lifshitz
and Ford-O’connel

The Landau-Lifshitz and Ford-O’connell equation of motion in normalized form
can respectively be written as

$\dot{u}^{\alpha}=F^{\alpha\beta}u_{\beta}+\tau_{0}\left(F^{\alpha\beta}_{,\gamma}u^{\gamma}u_{\beta}+F^{\alpha\beta}F_{\beta\gamma}u^{\gamma}-u^{\alpha}F^{\beta\gamma}F_{\gamma\delta}u^{\delta}u_{\beta}\right)$
(23)

and

$\dot{u}^{\alpha}=F^{\alpha\beta}u_{\beta}+\tau_{0}\Delta_{\beta}^{\alpha}\dot{f}^{\beta}$
(24)

The normalizations is the same as that used for Hartemann-Luhmann equation of
motion. Here $F^{\alpha\beta}_{,\gamma}=\partial F^{\alpha\beta}/\partial
x^{\gamma}$,
$\Delta_{\beta}^{\alpha}=(\delta^{\alpha}_{\beta}-u^{\alpha}u_{\beta})$ and
${f}^{\beta}$ is the Lorentz force. We solve the above equations numerically
using a MATHEMATICA based single particle code.

To test the validity of the Hartemann-Luhmann equation of motion, the energy
gained by the particle has been compared with the numerical solution of the
Landau-Lifshitz as well as Ford-O’connell equation of motion, for both the
cases viz. for a wave train and for a Gaussian pulse. Fig. 7(a) and Fig. 7(b)
respectively show the energy gain for a charged particle interacting with a
wave train (with $a_{0}=500$ ) and Gaussian pulse ( with $a_{0}=10^{3}$ ). The
initial conditions chosen are the same for both the cases; the particle starts
from the origin with initial momentum as $p_{\perp 0}=0$ and $p_{z0}=-10^{3}$.
Here red, blue and green curves respectively represent the energy gain
obtained from the solution of Hartemann-Luhmann, Landau-Lifshitz and
Ford-O’connell equations of motion. Thus comparison of energy gain shows that
the results are independent of the model equation; the contribution of the
terms eliminated in the Hartemann-Luhmann equation of motion have a negligible
influence on the final energy gain.

##  V Summary and conclusions

Present day lasers can deliver very high intensities, of the order of $\sim
10^{22}W/cm^{2}$. In future, intensities could be extended by more than two
orders of magnitudeshi (2013). It is well known that for intensities of the
order of $\sim 10^{22}W/cm^{2}$, for electrons having initial energy in the
$\sim GeV$ range, the radiation reaction force can become comparable or even
greater than the applied force Kravets, Noble, and Jaroszynski (2013). In this
scenario, neglecting radiation reaction force from charged particle dynamics
is a serious approximation.

In this article we have presented a simple and complete picture of particle
dynamics in a relativistically intense electromagnetic wave/pulse including
the effects of radiation reaction. In this context it has been shown that for
ultra-high intensities radiation-reaction force is basically due to the
influence of radiated energy, and the contribution of Schott energy is
negligibly small. In order to incorporate the effect of radiation reaction
into charged particle dynamics for ultra-high laser intensities Hartemann-
Luhmann equation has been analytically solved, for particle dynamics in an
elliptically polarized electromagnetic wave/pulse. A comparative study between
Hartemann-Luhmann equation and Lorentz Force equation shows that at ultra-high
laser intensities, radiation reaction force significantly affects the charged
particle dynamics. The radiation reaction force disturbs the phase
relationship between the velocity and the electric field of the wave in such a
way that the particle starts gaining energy. A comparison between the net
energy gain with different polarization of electromagnetic wave, shows that
the net energy gain is independent of polarization. Further, using a
MATHEMATICA based single particle code, the analytical results of the
Hartemann-Luhmann equation of motion has been verified and the energy gain by
the particle is compared with the Landau-Lifshitz and Ford-O’connell equation
of motion. A good match of energy gain for these three equations shows that
the energy gain is independent of chosen model.

##  Appendix A Analytical Expressions for Particle Positions

The solution of the integrations ${I}_{1}$, $\vec{I}_{2}$ and expressions for
the position of the particle wave train are given by

$I_{1}=\frac{1}{4}\left[-2\phi+(2\delta^{2}-1)\left\\{(sin(2\phi)-(sin(2\phi_{0})\right\\}+2\phi_{0}\right]$
(25)
$\begin{split}I_{2x}=\frac{1}{24}\delta((15-6\delta^{2})sin(\phi)+(2\delta^{2}-1)sin(3\phi)+6(2\delta^{2}-3)sin(\phi_{0})\\\
-6((2\delta^{2}-1)sin(2\phi_{0})+2(\phi-\phi_{0}))cos(\phi)-2(2\delta^{2}-1)sin(3\phi_{0}))\end{split}$
(26)
$\begin{split}I_{2y}=\frac{1}{24}g\sqrt{(1-\delta^{2})}(-3(2\delta^{2}+3)cos(\phi)-(2\delta^{2}-1)cos(3\phi)-8(2\delta^{2}-1)\\\
cos(\phi_{0})^{3}-6(2(\phi-\phi_{0})-sin(2\phi_{0}))sin(\phi)-12\delta^{2}(sin(2\phi_{0})sin(\phi)))\end{split}$
(27) $\begin{split}x(\phi)=\frac{1}{(kp_{z0}-\omega\gamma_{0})}({-p_{x0}\phi-
a_{0}\delta(-\phi+sin(\phi))})+\frac{\tau_{0}a^{2}_{0}}{72}\delta((9(-9+2\delta^{2})cos(\phi)+(1-2\delta^{2})\\\
cos(3\phi)-4(-20+4\delta^{2}+9\phi sin(\phi))))\end{split}$ (28)
$\begin{split}y(\phi)=\frac{1}{(kp_{z0}-\omega\gamma_{0})}({-p_{y0}\phi-
a_{0}\delta(-1+cos(\phi))})+\frac{\tau_{0}a^{2}_{0}}{72}\sqrt{1-\delta^{2}}(24(1-\delta^{2})\phi+36\phi
cos(\phi)\\\ -9(7+2\delta^{2})sin(\phi)+(1-2\delta^{2})sin(\phi))\end{split}$
(29) $\begin{split}z(\phi)=\frac{1}{13824(\gamma-
p_{z0})}(13824(p_{z0}\phi+a_{0}(p_{x0}\delta\phi+p_{y0}\sqrt{1-\delta^{2}}(-1+cos(\phi))-p_{x0}\delta
sin(\phi)\\\
-\frac{a_{0}(2(1+2\delta^{2})\phi-8\delta^{2}sin(\phi)+(-1+2\delta^{2})sin(\phi))}{8(p_{z0}-\gamma_{0})}))+24\tau_{0}a^{3}_{0}(a_{0}(-99+460\delta^{2}\\\
+52\delta^{4}+72\phi^{2}-24(-8+27\delta^{2}+2\delta^{4})cos(\phi)+96(-1+2\delta^{2})cos(2\phi)+8\delta^{2}(1-2\delta^{2})\\\
cos(3\phi)+(1-2\delta^{2})^{2}cos(4\phi)-288\delta^{2}\phi
sin(\phi)+72(-1+2\delta^{2})\phi sin(2\phi))-8(-p_{x0}\delta\\\
(9(-9+2\delta^{2})cos(\phi)+(1-2\delta^{2})cos(3\phi)-4(-20+4\delta^{2}+9\phi
sin(\phi)))-p_{y0}\sqrt{1-\delta^{2}}\\\ (24(1+\delta^{2})\phi+36\phi
cos(\phi)-9(7+2\delta^{2})sin(\phi)+(1-2\delta^{2})sin(\phi)+(1-2\delta^{2})sin(\phi))))\\\
-a^{6}_{0}\tau^{2}_{0}(-12\phi(105+188\delta^{2}-188\delta^{4}-64\delta^{6}+24\phi^{2})+2304(-1+\delta^{4})\phi
cos(\phi)+1152\\\ (1-2\delta^{2})\phi cos(2\phi)-36(1-2\delta^{2})^{2}\phi
cos(4\phi)-576(-7-2\delta^{2}+7\delta^{4}+2\delta^{6})sin(\phi)\\\
+9(-1+2\delta^{2})(85-20\delta^{2}+20\delta^{4}-48\phi^{2})sin(2\phi)-64(1-2\delta^{2}-\delta^{4}+2\delta^{6})sin(3\phi)\\\
+36(1-2\delta^{2})^{2}sin(4\phi)+(-1+2\delta^{2})^{3}sin(6\phi)))\end{split}$
(30)

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Figure 1: Figs. (a), (b), (c) and (d) represent the longitudinal momentum, transverse momentum, energy and trajectory of the particle respectively. The red and blue curves represent the solution of the Hartemann- Luhmann and Lorentz Force equations respectively and the dashed green curves represent the analytical solution. These graphs are plotted for the $a_{0}=500,\delta={1},p_{z0}=-1000$. |
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Figure 2: Figs. (a),(b),(c) and (d) represent the longitudinal momentum, transverse momentum, energy and trajectory of the particle respectively. The red and blue curves represent the solution of the Hartemann- Luhmann and Lorentz Force equations respectively and the dashed green curves represent the analytical solution. These graphs are plotted for the $a_{0}=500,\delta=\frac{1}{\sqrt{2}},p_{z0}=-1000$.  Figure 3: Average Energy gain with different polarizations and the red blue and green curves respectively correspond to $\delta=1,\,\,1/\sqrt{2},\,\,1/4$. These graphs are plotted for the $a_{0}=500,p_{z0}=-1000$.  |
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Figure 4: Figs. (a), (b), (c) and (d) represent the longitudinal momentum, transverse momentum, energy and trajectory of the particle respectively. The red and blue curves represent the solution of the Hartemann- Luhmann and Lorentz Force equations respectively. These graphs are plotted for the $a_{0}=1000,\delta={1},p_{z0}=-1000$.  |
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Figure 5: Figs. (a), (b), (c) and (d) represent the longitudinal momentum, transverse momentum, energy and trajectory of the particle respectively. The red and blue curves represent the solution of the Hartemann-Luhmann and Lorentz Force equations respectively. These graphs are plotted for the $a_{0}=1000,\delta=\frac{1}{\sqrt{2}},p_{z0}=-1000$.  Figure 6: Energy gain with different polarizations; the red blue and green curves respectively correspond to $\delta=1,\,\,1/\sqrt{2},\,\,1/4$. These graphs are plotted for the $a_{0}=1000,p_{z0}=-1000$. |
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Figure 7: Figs. (a), (b) respectively represent the energy gain computed for a
charged particle interacting with a wave train and a Gaussian pulse. The red,
blue and green curves respectively represent the solution of the Hartemann-
Luhmann, Landau-Lifshitz and Ford-O’connell equations of motion. Here
figure7(a) and 7(b) are plotted for the $(a_{0}=500,\delta={1},p_{z0}=-1000)$
and $(a_{0}=1000,\delta={1},p_{z0}=-1000)$ respectively.