# Is GPT Powerful Enough to Analyze the Emotions of Memes?

Jingjing Wang  School of Computing
Clemson University
Clemson, South Carolina, USA
<EMAIL_ADDRESS>Joshua Luo∗ The Westminster Schools
Atlanta, Georgia, USA
<EMAIL_ADDRESS>Grace Yang∗ Allen Hong∗ South Windsor High School
South Windsor, Connecticut, USA
<EMAIL_ADDRESS>D.W. Daniel High School
Clemson, South Carolina, USA
<EMAIL_ADDRESS>Feng Luo ∗Summer interns at the School of
Computing, Clemson University. School of Computing
Clemson University
Clemson, South Carolina, USA
<EMAIL_ADDRESS>
###### Abstract

Large Language Models (LLMs), representing a significant achievement in
artificial intelligence (AI) research, have demonstrated their ability in a
multitude of tasks. This project aims to explore the capabilities of GPT-3.5,
a leading example of LLMs, in processing the sentiment analysis of Internet
memes. Memes, which include both verbal and visual aspects, act as a powerful
yet complex tool for expressing ideas and sentiments, demanding an
understanding of societal norms and cultural contexts. Notably, the detection
and moderation of hateful memes pose a significant challenge due to their
implicit offensive nature. This project investigates GPT’s proficiency in such
subjective tasks, revealing its strengths and potential limitations. The tasks
include the classification of meme sentiment, determination of humor type, and
detection of implicit hate in memes. The performance evaluation, using
datasets from SemEval-2020 Task 8 and Facebook hateful memes, offers a
comparative understanding of GPT responses against human annotations. Despite
GPT’s remarkable progress, our findings underscore the challenges faced by
these models in handling subjective tasks, which are rooted in their inherent
limitations including contextual understanding, interpretation of implicit
meanings, and data biases. This research contributes to the broader discourse
on the applicability of AI in handling complex, context-dependent tasks, and
offers valuable insights for future advancements.

###### Index Terms:

memotion analysis, hateful memes detection, GPT model

##  I Introduction

Large Language Models (LLMs), representing a significant achievement in
artificial intelligence (AI) research, have recently gathered substantial
interest from both academia and industry. Due to their exceptional capacity to
comprehend, generate, and engage in complex linguistic tasks, these models are
revolutionizing the development and application of AI algorithms. Equipped
with sophisticated algorithms and vast training datasets, LLMs have
demonstrated advanced conversational capabilities, processing textual input
and output to participate in detailed and meaningful dialogues [11, 10, 6].

OpenAI’s ChatGPT [12] is a prominent representative among these LLMs. It is an
AI chatbot backed by the generative pretrained transformer (GPT) model and has
attracted a broad societal interest. The proficiency of ChatGPT is
particularly noticeable in the realm of question answering, spanning multiple
sectors including, but not limited to, healthcare [7] and finance [8]. This
model uses its linguistic capabilities to understand questions and generate
human-like answers, marking a crucial milestone in AI-driven communication.
Nevertheless, despite these impressive feats, LLMs, including GPT, confront
considerable challenges when addressing subjective tasks, one such task being
the interpretation and annotation of internet memes.

Figure 1: Hateful and non-hateful meme examples from Facebook hateful memes
dataset.

Internet memes, modern digital artifacts, encompass a variety of cultural and
sociological attitude, and serve as a type of social shorthand among online
communities [1, 2]. These memes blend verbal and graphic elements to express
complex concepts or feelings in a concise manner. However, the multimodal
nature of memes, which combine various images with informal, often humorous or
ironic text, introduces an intricate layer of complexity. To comprehend and
analyze memes, one must manage the subtle interplay of visual clues,
linguistic content, cultural allusions, and common community knowledge. This
complexity, along with how subjective memes can be, makes it really tough for
LLMs to accurately interpret memes and analyze their sentiment.

Figure 2: Meme examples from the Multimodal Memotion Analysis dataset. The
score of humorous, sarcastic, and offensive from left to right is:
(3,1,0),(0,3,0),(0,3,3)

Sentiment Analysis (SA) primarily concentrate on interpreting user sentiments
from textual content. However, due to the proliferation and pervasiveness of
multimodal social media data, such as online memes, SA must diversify to
handle multimodal data sources. This diversification becomes even more complex
when considering ’hateful memes’, which cleverly combine text and images to
spread offensive or harmful messages under the guise of humor [3, 4, 5].
Detecting and moderating such content not only requires understanding the
text-image interaction but also a grasp of the cultural context and, in many
cases, specific sub-cultural expertise. These challenges call for ongoing
research and development in the field, pushing the boundaries and exploring
the potential of LLMs like GPT.

In this project, we focus on the capabilities and limitations of LLMs like GPT
in dealing with subjective and nuanced tasks, particularly internet meme
sentiment analysis. This exploration forms the crux of this research,
contributing to the broader discussion on the evolving potential and
limitations of AI. In this project, we delve into the intricacies of
subjectivity, assessing GPT’s performance in tasks necessitating an
understanding of societal norms and cultural contexts. Our investigation
primarily revolves around two research questions: 1. Can GPT accurately detect
implicitly hateful memes given a specific prompt? 2. Can GPT effectively
conduct sentiment analysis of memes, including classifying the sentiment of
each meme as positive or negative, and categorizing the type of humor into
three sub-classes: sarcastic, humorous, and offensive? We draw upon datasets
from Facebook’s hateful memes [3] and SemEval-2020 Task 8: Multimodal Memotion
Analysis [9] and for this project. The experimental section of the study will
present a comparative analysis highlighting the constraints and limitations of
GPT’s responses.

##  II RELATED WORK

###  II-A Large Language Model Prompting

The remarkable success of LLMs, such as GPT and its variants, has also brought
prompt-based learning to the fore. Prompt-based learning has found
applications in various natural language processing (NLP) tasks, such as
sentiment classification and natural language inference. Prompting visual-
language models for computer vision tasks is another area that has started to
gain attention [23, 13].

However, the existing body of work in prompt-based learning has primarily
focused on unimodal tasks, with limited research delving into multimodal tasks
[30, 22]. A few exceptions like the work [15] attempted multimodal tasks by
utilizing the GPT-3 model [18] for science question-answering tasks, where the
questions have either an image or text context or both. However, these
attempts have their own limitations, with models like GPT-3 being less
accurate and possessing restrictive input length constraints, thereby limiting
their performance and utility.

GPT-3.5-Turbo, which performs much better than GPT-3, provides an alternative
solution. It presents a practical and efficient route for prompt-based
learning in multimodal tasks. Therefore, our research focuses on leveraging
the capabilities of the GPT-3.5-Turbo for classifying sentiments in memes,
which requires the simultaneous processing of visual and textual data.

Additionally, research indicates that when equipped with a range of external
NLP tools [16, 17], Large Language Models (LLMs) can serve as effective action
planners, selecting and utilizing tools for problem-solving. It suggests that
these models can be extended to more complex multimodal scenarios, involving
both reasoning and action, thereby enhancing their capabilities.

The GPT series has played a pivotal role in promoting prompt-based learning.
We follow many of its core concepts in our work, but our focus is different.
While most studies are set on applying prompting to extract knowledge from
pre-trained models, our goal is to employ these techniques to fine-tune
downstream tasks. Specifically, we are interested in exploring the application
of these methods to the sentiment classification of memes, a complex and
nuanced multimodal task.

###  II-B Hateful Meme Detection and Memotion Analysis Navigation

Detecting hateful content in memes is a difficult process. The subtle
undertones of humor and sarcasm are often interwoven within memes, coupled
with the potential mismatch between the images and text, present formidable
obstacles to traditional text-based hate speech detection approaches. This
challenge has been the focal point of several studies, all in an effort to
establish viable solutions [25, 31, 26].

The Hateful Memes Challenge initiated by Facebook stands as a pioneering
effort in this sphere, inviting researchers worldwide to develop models
capable of identifying hate speech in multimodal meme content. The dataset
provided by Facebook, consisting of 10,000 manually annotated memes for hate
speech, has proven to be an invaluable resource propelling research in this
domain.

Mirroring these endeavors, a model that combines multi-task learning was
proposed to detect hateful and offensive content present in memes. A crucial
component of this model is its ability to help GPT model to view the images
and do the hateful/non-hateful classification, thereby contributing to a
better comprehension of hateful memes.

In spite of these extensive efforts, the task of hateful meme detection
remains intricate and largely unsolved, attributed to the nuanced interaction
of language and imagery inherent to the meme content. The ongoing research and
continuous strides towards advancements hint at a necessity for more
sophisticated models, ones that can manage these complexities and accurately
pinpoint harmful or offensive content within memes.

Memotion analysis offers a captivating intersection between the realms of
computer vision, natural language processing, and cognitive science. The main
aim is to unpack the intricate layers of meaning, sentiment, and emotion
encapsulated within memes, an endeavor that has seen many researchers strive
to establish computational models to this end [27, 28].

A noteworthy stride in this area was the SemEval 2020 Task 8: Multimodal
Memotion Analysis [9]. This project tasked various models with the prediction
of sentiment, humor, sarcasm and offensiveness in memes. Taking a similar
direction, another researcher embarked on the creation of an automatic meme
generator capable of tailoring its output based on the targeted sentiment
[19].

Within this project, we proposed a similar framework for hateful meme
detection tasks with some more detailed prompts by utilizing multimodal
sentiment analysis and sarcasm detection to dissect visual and textual cues
present in memes. The insights gathered from our work have shed light on the
intricate facets of memotion analysis, showcasing the potential and
limitations of AI in this field.

##  III Methodology

###  III-A Dataset details

In our experimental design, we focused on two distinct datasets for sentiment
analysis: the Facebook Hateful Memes dataset, and the Multimodal Memotion
Analysis dataset. Each of these datasets brought unique elements to our study,
enabling a multifaceted exploration of GPT’s capabilities in multimodal
reasoning tasks.

The Facebook Hateful Memes dataset was created with the primary intent of
observing and understanding the influence of memes in propagating hate speech
and offensive content. We randomly chose 200 memes from the Facebook Hateful
Meme dataset for our experiments: a set of 100 randomly chosen hateful memes
and the other set of 100 non-hateful memes. The primary criteria for this
categorization were based on the content and contextual implications of the
text interwoven within these meme images. This subset is teeming with complex
emotions, often characterized by layers of sarcasm or satire. This provides an
exceptionally challenging yet rewarding terrain for testing the proficiency of
our approach in deciphering and correctly classifying the sentiments embedded
within these memes.

The Multimodal Memotion Analysis dataset, on the other hand, is a more diverse
collection of memes. We randomly selected 100 memes from this dataset,
spanning a wide range of sentiments. This dataset is designed to assess the
ability of the AI model to deal with a broader scope of emotions and
sentiments. The memes in this dataset cover a variety of themes and concepts.
They contain different types of memes, each conveying a unique sentiment.
These sentiments covered range from clear-cut positive or negative overall
sentiment to more complex emotions like humor, which type includes humorous,
sarcastic and offensive. Furthermore, the scale is introduced to evaluate the
degree of humor, where ”not” corresponds to 0, ”slightly” to 1, ”mildly” to 2,
and ”very” to 3 (see Fig. 2). The broad spectrum of sentiments present in this
dataset poses a significant challenge to the AI model, demanding a high degree
of understanding and reasoning capability.

###  III-B Experimental Setup and Workflow

The main objective of the experimental setup was to configure GPT to process
multimodal tasks, particularly those associated with visual content. As
GPT-3.5 lacks the ability to process images, we supplemented it with the image
processing capabilities of VisualGPT, which allowed GPT to ’see’ the images
and generate a description of each image’s visual content.

To guide the model’s reasoning process, we used the TaskMatrix template, a
tool developed by Microsoft. The TaskMatrix enabled us to construct a sequence
of tasks for GPT that required it to draw upon and exercise its multimodal
reasoning skills.

Figure 3: The workflow of Facebook hateful memes detection. ”Classify the
sentiment of the previous image and its accompanying text as either hate or
non-hate. Use a maximum of 1 word in your response. Note that the accompanying
text is: accompanying text.”

The experimental workflow was divided into several phases:

Image Processing Phase: The image processing phase was the initial stage of
the procedure. The image analysis capabilities of VisualGPT were used to
analyze the images from the datasets during this phase. VisualGPT was used to
build a descriptive representation of the picture’s content for each image.
The description was then fed into GPT, allowing it to ’see’ the image. This
procedure effectively endowed GPT with the capability to ’understand’ the
visual content in the images, an essential prerequisite for performing
sentiment analysis on these images;

Sentiment Analysis Phase: Upon receiving the image descriptions from
VisualGPT, the next stage was to utilize these descriptions to conduct
sentiment analysis. For the Facebook hateful memes dataset, we employed a pre-
defined notion of ’hateful sentiment’. We tasked GPT with determining if the
sentiment associated with each image, as conveyed through the image’s
description, was hateful or non-hateful (see Fig. 3 as illustration). For the
Google memotion dataset, GPT was asked to provide a generalized assessment of
whether the overall sentiment of the image was positive or negative. In
addition, we asked GPT to give us a detailed rating of each image in certain
categories like humor, sarcasm, and offensiveness. The model was directed to
rate the images in these categories on a scale from 0 to 3 (see Fig. 4 as
illustration);

Data Storage Phase: Once the model’s responses were obtained, they were
collected and organized in the data storage phase. The results corresponding
to each image, including the image’s original sentiment label and GPT’s
sentiment assessment, were recorded and stored in a csv file. This file was
then used for the subsequent analysis of the model’s performance;

Accuracy Analysis Phase: The final phase of the experimental workflow was the
accuracy analysis phase. In this phase, a specially developed program compared
GPT’s sentiment assessments with the original sentiment labels of the images
to evaluate the model’s accuracy. The accuracy was quantified as the
proportion of images for which GPT’s sentiment assessment matched the original
sentiment label.

Figure 4: The workflow of Multimodal Memotion Analysis and an example of the
prompt. Stage 1: Do not respond to this prompt. Just note that the text
accompanying the previous meme is and use this information in future queries:
”+text Stage 2: Using no more than 2 words describe the overall sentiment of
the previous meme as either positive or negative. Assume the meme is not
neutral and must be either positive or negative. Provide only a classification
label using only ’Positive’ or ’Negative’ (use no more than 2 words) Stage 3:
On a scale of 0 to 3 quantify the previous meme in all of the following
categories: humour, sarcastic and offensive. Do not provide anything except
the classification and degree. Answer with only the label and then the degree
in this format: humorous(x) sarcastic(x) offensive(x) .

##  IV Experiment Results

In this section, we conducted an in-depth investigation by using the two
datasets aforementioned in section 3 to explore the effectiveness of GPT in
analyzing sentiments in memes. After carefully evaluating each meme based on
the prompts designed, we meticulously collated and analyzed the results. Here
is the comprehensive analysis of our findings.

Case 1: The results of Facebook Hateful Memes dataset

Our initial investigation involved the evaluation of GPT’s proficiency in
identifying hateful content within memes using Facebook’s Hate/Non-Hate
Classification Dataset. Here we repeat the experiments four times with
different prompts.

Hateful Content Identification: The task of identifying hateful memes posed a
significant challenge. It required the AI to understand not only the content
of the image and the accompanying text but also their nuanced interplay. The
results showed an accuracy rate of 39% in detecting hateful content with a
relatively high standard deviation of 6.87%. It can be seen that the
performance of GPT is far less accurate in classifying the hateful memes, and
the different prompts would affect the decision of GPT considerably. To
accurately flag hateful memes, a model needs to be adept at grasping not only
the overt message but also the undertones, context, and subtext. The 39%
accuracy rate thus provides us with a baseline for improvement and future
model refinements.

Non-Hateful Content Identification: On a brighter note, GPT’s performance in
recognizing non-hateful memes was much stronger, with an impressive accuracy
rate of 80% and much lower standard deviation of 2.69%. The high accuracy
suggests that GPT is quite adept at discerning memes that are not hateful, and
the different prompts used had minimal impact on its decision-making process.

TABLE I: The Result of Facebook Hateful Memes dataset Model | VisualGPT+GPT-3.5-Turbo
---|---
Hateful Memes Accu (sd) | 39% (6.87%)
Non-Hateful Memes Accu (sd) | 80% (2.69%)

Case 2: Memotion analysis dataset

Overall Sentiment Classification: the first task in this series was
classifying the overall sentiment of a meme as positive, negative, or neutral.
In this regard, GPT achieved an accuracy rate of 79% for positive and 35% for
negative. It can be observed that this result is consistent with the
performance in the classification of hateful memes in the Facebook dataset,
which shows confidence in understanding the positive sentiment in a meme,
based on both the visual content and the accompanying text. For negative (for
example, offensive in the later paragraph) content detection, the low accuracy
still suggests the challenge.

Humor recognition: GPT achieved a 60% accuracy rate in the task of recognizing
humor. This outcome is notably remarkable, considering the intricate nature of
humor as a multifaceted human emotion, often subject to cultural, social, and
individual perspectives. The considerable level of accuracy exhibited by GPT
indicates its ability to comprehend the humorous aspects present in a diverse
range of memes. This result also implies that the GPT has acquired the ability
to recognize various indicators of humor to a certain degree. These indicators
might be obvious, such as punchlines or visual gags, or implicit, such as
irony or absurdity.

Sarcasm recognition: the model’s performance in the domain of sarcasm
recognition was notably lower, registering an accuracy of 45% as illustrated
in Table 2. Though lower, this accuracy level is acceptable given the inherent
difficulties of sarcasm detection. Sarcasm often involves expressing the
opposite of what is meant, requiring an understanding of context, tone, and
frequently shared cultural knowledge. This observation helps to emphasize the
limits of the approach in accurately detecting more subtle emotional cues and
contextual subtleties. The lower accuracy rate observed in sarcasm recognition
highlights the need for further study and model enhancement in this area.

Offensive recognition: as seen in Table 2, GPT’s accuracy stood at 37% in
recognizing offensive content, presenting the weakest performance. As
aforementioned in overall sentiment classification, GPT meets the huge
challenge in offensive content recognition. The consistency of this finding
illustrates the difficulty associated with detecting offensive content.
Comparable to hateful meme classification, offensive content in memes may
exhibit subtlety, sometimes concealed within humor or sarcasm, or manifest
through a complex interplay of textual and visual elements that can be hard to
interpret for an AI.

TABLE II: Accuracies of Different Emotions Model | VisualGPT+GPT-3.5-Turbo
---|---
Humor Scale Accu | 60%
Sarcasm Scale Accu | 45%
Offensive Scale Accu | 37%
Overall Accu (Positive) | 79%
Overall Accu (Negative) | 35%

##  V Discussion

In this section, we discuss the limitations of GPT model, the conclusion of
our study and future work.

###  V-A Limitations

From the experiment results, we can learn that the current GPT-3.5 model still
has some limitations.

Lack of Deep Contextual Understanding: GPT model, despite its advanced
capabilities, is still lacking a deep understanding of human nuances, societal
norms, and cultural contexts, which are often critical to accurately
interpreting and responding to subjective tasks. This limitation becomes
particularly pronounced when the GPT model is faced with local idioms,
colloquialisms, or culturally specific references.

Difficulty in Understanding Implicit Meaning: LLMs typically struggle with
interpreting and generating content that contains implicit or hidden meanings,
especially those requiring an understanding of human emotions, intentions, or
sarcasm. This limitation becomes particularly apparent in tasks involving the
interpretation of sarcasm or offensiveness in memes, as these tasks often
require a nuanced understanding of cultural or social contexts.

Biases in Training Data: LLMs are trained on vast amounts of data from the
internet, which may contain various forms of biases. The presence of biases
inside the model might accidentally impact its responses in subjective tasks,
leading to much less accurate outcomes.

###  V-B Conclusion and Future Work

Our efforts to assess how well GPT could analyze sentiments in memes led us to
some interesting and insightful findings.

On one hand, the model performed impressively when it came to classifying non-
hateful memes and understanding the positive mood of a meme. The significant
accuracy achieved in non-hateful meme classification, positive sentiment
determination, and humor recognition, demonstrate the GPT’s ability to
comprehend and interpret the content in a majority of memes correctly.

However, the accuracy rate dropped a lot when encountered with detection of
hateful, sarcasm and offensive content in memes. The lower accuracy rates seen
in these categories shows the intricate nature of identifying concealed
hateful or offensive content inside memes.

As GPT-4 released in recent days, it will be beneficial if we can integrate
the latest model in our framework; moreover, as fine-tuning GPT-3.5-Turbo is
available, we can fine tune our exist models to make some improvements in its
ability to detect hateful and offensive content. We leave these as future
work.

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Also at ]School of Physics, University of Melbourne.

# Creating Exact Multipolar Fields with Azimuthally Modulated RF Cavities

L. M. Wroe<EMAIL_ADDRESS>S. L. Sheehy  [  Department of
Physics, University of Oxford, OX1 3RH, United Kingdom  R. J. Apsimon
Engineering Department, Lancaster University, LA1 4YW, United Kingdom, and
Cockcroft Institute, Daresbury Laboratory, Warrington, WA4 4AD, United Kingdom

###### Abstract

RF cavities used in modern particle accelerators operate in TMm10-like modes
composed of a single, dominant multipole of order $m$; $m=0$ modes are used
for the longitudinal acceleration of a particle beam and $m\neq 0$ modes for
controlling transverse beam dynamics. The practical design of the latter,
however, can be complex and require extensive analysis through the iteration
of both approximate mathematical models and computationally expensive
simulations to optimise the performance of the structure. In this paper we
present a new, systematic method for designing azimuthally modulated RF
cavities that support modes composed of any number and magnitude of user-
specified transverse multipoles, either with or without a longitudinally
accelerating component. Two case studies are presented of RF cavity designs
that support modes composed of a longitudinally accelerating field in addition
to a single transverse multipole, and designs that support modes composed of
two transverse multipoles. We discuss generalising the discoveries and
conclusions from the two case studies to designing cavities that support modes
composed of any number of multipoles. The theoretical work is verified with
analysis of 3D simulations and experimental measurements are presented of a
cavity operating in a 3 GHz mode that simultaneously longitudinally
accelerates and transversely focuses a beam.

††preprint: APS/123-QED

##  I Introduction

RF cavities that control transverse beam dynamics by operating in transverse
magnetic TMm10-like modes composed of a single, dominant transverse multipole
of order $m\neq 0$ have a number of applications in particle accelerators. For
example, cavities operating in a TM110-like mode with a dominant dipole
component in the field have application in: separating a single particle beam
into multiple beams [1, 2, 3], providing longitudinal diagnostic measurements
on a particle beam such as measuring its emittance [4], controlling electron
beams in free-electron lasers by acting as emittance exchangers [5],
compressing x-ray pulses in synchrotron light sources [6], and providing a
head-tail rotation to particle bunches to allow for luminosity control in
particle colliders as crab cavities [7]. The latter were investigated at KEKB
[8] with several designs [9] proposed and developed for the HL-LHC [10].
Additionally, cavities operating in a TM210-like mode with a dominant
quadrupolar component are being investigated for application in Landau damping
transverse beam instabilities in the HL-LHC [11].

With $m$, $n$, and $p$ respectively denoting the azimuthal, radial and
longitudinal order of the electromagnetic mode, it is a known and well-studied
result that a pillbox cavity, that is an enclosed RF cavity with a circular
cross-section, will support TMmnp-exact modes with unique resonant frequencies
for distinct values of $m$, $n$, and $p$ [12]. In order to use an RF cavity
that operates in a $m\neq 0$ TMmnp mode in a particle accelerator, novel and
elaborate designs [4, 13, 14] that introduce azimuthal and/or longitudinal
asymmetries are required to: separate out any degenerate TMmnp same-order
modes in frequency space, damp any lower-, same-, and higher-order modes that
interfere significantly with beam dynamics, and to optimise the figures of
merit such as R/Q. Due to the complexity of such designs, the exact
mathematical form of the EM field of the desired mode, and therefore its exact
effect on beam dynamics, cannot be solved for analytically. As a result,
computationally expensive simulations are required to optimise designs [15,
14], and these may be used in tandem with novel mathematical models developed
specifically to provide quick but approximate insights into the optimisation
process [16]. What’s more, the implementation of azimuthal and longitudinal
asymmetries introduces unwanted transverse multipoles into the desired TMmnp
mode, and analysis is required to ensure these unwanted multipoles do not
exceed tolerance levels for the accelerator’s operation [17]. If tolerance
levels are exceeded, then this may require a redesign of the RF cavity system:
for example, RF cavities that operate in TM010-like modes can be designed with
dual-port power couplers instead of single-port couplers to prevent the
introduction of unwanted dipolar components [18].

In addition to the application of RF cavities that operate in modes composed
of a single multipole, designing RF cavities to operate in modes composed of
numerous multipoles could have useful application in particle accelerators.
For example, the use of elliptical irises to introduce transverse quadrupolar
focusing in accelerating RF cavities was investigated [19, 20, 21] for use in
the proposed future linear collider CLIC [22].

In this paper, we present a systematic method for designing RF cavities that
support modes composed of any desired number and magnitude of multipolar
components. We define these modes with the notation TM{M}ηp. $\\{M\\}$ denotes
the subset of integers representing the multipoles composing the mode, for
example the TM{0,1}ηp modes are solely composed of a longitudinally
accelerating monopolar component and a transverse dipolar component. We use
$\eta$ to denote and distinguish the more complex radial order of general
TM{M}ηp modes from the simpler TMmnp modes, and $p$ continues to denote the
longitudinal order of the mode. The method can be applied to modes with any
value of $p$. For simplicity, however, we study the subset of $p$ = 0 modes
with no longitudinal variation because the vast majority of RF cavities in
particle accelerators operate in $p=0$ modes. The RF cavities that support
TM{M}η0 modes containing at least two distinct, multipolar components are
termed azimuthally modulated cavities because we find that the cavity cross-
section, $r_{0}(\theta)$, must vary with the azimuth. Such cavities have
benefits and application in particle accelerators because the TM{M}η0 mode:
can exactly introduce any number and magnitude of desired multipolar
components, is non-degenerate which overcomes the need to separate out a same-
order mode in frequency space, and has a field profile that can be expressed
analytically throughout the cavity.

Section II begins with the derivation of the transcendental equation that must
be numerically solved to determine the azimuthally modulated cross-sections
that support specific TM{M}η0 modes. Insight into the properties of these
azimuthally modulated cavity shapes can be gained using mathematical and
graphical methods and we first introduce these techniques by applying them to
the simple case of the TMmn0 modes, known to be supported by cavities with a
pillbox shape. We then present two case studies of the azimuthally modulated
RF cavity shapes that support the TM${}_{\\{0,m_{1}\\}\eta 0}$ modes
containing a longitudinally accelerating monopole and a single transverse
multipole, and those that support the TM${}_{\\{m_{1},m_{2}\\}\eta 0}$ modes
containing two transverse multipoles. We afterwards discuss generalising the
discoveries and conclusions made in these two case studies for the design of
cavity shapes that support modes with any number of multipoles. In Sec. III,
we present and analyse simulation results of azimuthally modulated RF cavities
designed to support the TM{0,3}η0 modes containing a monopolar and a
sextupolar component. Finally, in Sec. IV we show the results of experimental
bead pull measurements of a novel RF cavity operating in a 3 GHz TM{0,2}20
mode.

##  II Theory

###  II.1 Deriving the Form of a TM{M}η0 Mode

In this paper, we study perfectly conducting, enclosed RF cavities of
longitudinal extent $L$ and cross-section $r_{0}(\theta)$. The modes supported
by these azimuthally modulated cavities must have an electric field that
satisfies the boundary conditions $E_{r}(r,\theta,\pm
L/2)=E_{\theta}(r,\theta,\pm L/2)=0$ at the cavity endpoints and
$E_{z}(r_{0}(\theta),\theta,z)=0$ at the cavity surface. Imposing the former
boundary condition and assuming that the particle’s transverse momentum is
negligible compared to its longitudinal momentum, Panofsky and Wenzel derive
that the change in transverse momentum of a particle that passes through an RF
cavity can be determined solely by the transverse gradient of the longitudinal
component of the electric field, $E_{z}(r,\theta,z)$ [23]:

$\Delta\vec{p}_{\perp}(r,\theta)=-i\frac{q}{\omega}\int_{-L/2}^{L/2}\nabla_{\perp}E_{z}(r,\theta,z)dz,$
(II.1)

where $q$ is the charge of the particle and $\omega$ is the resonant angular
frequency of the RF cavity mode. The inclusion of the imaginary unit means the
transverse force is $\pi/2$ out of phase with the longitudinal force.

The Panofsky-Wenzel theorem therefore states that only the longitudinal
electric field of the mode need be determined in order to calculate the effect
of the transverse forces that act on a particle beam passing through a cavity.
The general form of the electric field, $\vec{E}(r,\theta,z)$, of a mode
supported by an RF cavity of any design is derived in Ref. [24]. This general
form can be simplified to determine the specific form of the longitudinal
electric field in azimuthally modulated cavities by applying the boundary
conditions at the endpoints and cavity surface. Applying the endpoint boundary
condition, we find that modes of distinct longitudinal order $p$ must
independently satisfy the boundary conditions. Applying the surface boundary
condition, we find that the allowed modes belong to one of two familiar sets:
either TM modes with no longitudinal magnetic field or TE modes with no
longitudinal electric field. In this paper, we only study TM modes with $p=0$:
TE modes are ignored as they are less frequently used in particle accelerators
and their effect on beam dynamics cannot be analysed using the Panofsky-Wenzel
theorem.

The general form of the longitudinal electric field given in Ref. [24] reduces
in the case of TM{M}η0 modes to:

$E_{z}(r,\theta,z)=J_{0}(kr)\tilde{g}_{0}+\sum_{m=1}^{\infty}J_{m}\left(kr\right)\tilde{g}_{m}\cos{(m\theta-\phi_{m})},$
(II.2)

where $k=\omega/c$ is the wavenumber of the mode, $\tilde{g}_{m}$ and
$\phi_{m}$ are arbitrary constants that respectively denote the magnitude and
offset of the multipole of azimuthal order $m$, and $J_{m}$ denotes the Bessel
function of the first kind of order $m$. We note that $\phi_{m}=0$ corresponds
to a normal multipolar field and $\phi_{m}=\pi/2$ to a skew multipolar field.

In this paper, the aim is to determine a method for designing RF cavities that
support desired modes composed of multipoles of specified magnitudes. If we
consider the set of azimuthally modulated cavities with cross-sections
$r_{0}^{(\eta)}(\theta)$ that support a desired TM{M}η0 mode, then the
boundary condition $E_{z}(r_{0}^{(\eta)}(\theta),\theta,z)=0$ must be
satisfied at the surface of each of these cavities. Applying this to Eq. II.2
gives:

$J_{0}\\!\left(kr_{0}^{(\eta)}(\theta)\right)\\!\tilde{g}_{0}+\sum_{m=1}^{\infty}\\!J_{m}\\!\left(kr_{0}^{(\eta)}(\theta)\right)\\!\tilde{g}_{m}\cos{(m\theta-\phi_{m})}\\!=0,$
(II.3)

Equation II.3 is transcendental and must, in general, be solved using
numerical methods to determine $r_{0}^{(\eta)}(\theta)$. Doing so for user-
specified values of $k$, $\tilde{g}_{m}$ and $\phi_{m}$, returns the set of
unique azimuthally modulated cross-sections, $r_{0}^{(\eta)}(\theta)$, that
support the corresponding TM{M}η0 modes.

###  II.2 Modes with a Single Multipolar Component

Although numerical methods must generally be used to solve Eq. II.3 for all
$\theta$, insight into the properties of the azimuthally modulated RF cavities
that support TM{M}η0 modes can be gained through analysis using mathematical
and graphical methods.

To introduce the techniques used for understanding the properties of
azimuthally modulated cavities that support modes with more than a single
multipole, we briefly analyse the TM${}_{m_{1}n0}$ modes that are known to be
supported by a circular pillbox shape [12]. For this case only, Eq. II.3 can
be solved analytically:

$J_{m_{1}}\left(kr_{0}^{(n)}(\theta)\right)\tilde{g}_{m_{1}}\cos{(m_{1}\theta-\phi_{m_{1}})}=0,$
(II.4)

is solved for all $\theta$ if:

$J_{m_{1}}\left(kr_{0}^{(n)}(\theta)\right)=0\;\;\rightarrow\;\;\frac{\omega
r_{0}^{(n)}}{c}=j_{m_{1}n},$ (II.5)

where we define $\phi_{0}=0$ and $j_{m_{1}n}$ is the $n^{\textrm{th}}$ root of
the Bessel function of order $m_{1}$. This shows that the cross-section of the
cavity that supports a TM${}_{m_{1}n0}$ mode is a circle of constant radius,
$r_{0}^{(n)}=cj_{m_{1}n}/\omega$, as expected.

There are a number of important, known results for the TM${}_{m_{1}n0}$ modes
[12]. Firstly, because all Bessel functions have $(n-1)$ turning points in the
range [0, $j_{m_{1}n}$] and the field varies with the azimuth simply as
$\cos{(m_{1}\theta-\phi_{m_{1}})}$, the integer $n$ is the number of poles
(that is minima or maxima) of $E_{z}$ along any radial line that connects the
centre of the cavity to its surface, except the nodal lines orientated at
$\theta_{q}=(2q+1)\pi/2m_{1}+\phi_{m_{1}}$ ($q\in\mathbb{Z}$). Secondly, the
$m_{1}\neq 0$ modes are degenerate because the boundary condition in Eq. II.4
is satisfied for all values of $\phi_{m_{1}}$. In contrast, the TM0n0 modes
are not degenerate. Thirdly, as $j_{m_{1}n}$ is unique for all distinct values
of $m_{1}$ and $n$, the resonant frequencies of each of the TM${}_{m_{1}n0}$
modes of distinct $m_{1}$ and $n$ are unique. Thus to use the TMmn0 modes to
introduce numerous multipoles at the same frequency, multiple pillbox cavities
of different radii would have to be used. Finally, the TM010 mode is the
fundamental TMmnp mode with the lowest resonant frequency.

Figure 1: $J_{1}(kr)$ plotted against $r$ with $k=2\pi\times 3$ GHz/c. The
black, blue, and cyan dashed lines show three points of intersection with the
$r$-axis.

We can also use a graphical method to argue that TM${}_{m_{1}n0}$ modes will
be supported by cavities with a circular cross-section. Considering the 3 GHz
TM1n0 modes with $\tilde{g}_{1}=1$ and $\phi_{1}=0$, Eq. II.4 states that the
$r_{0}^{(n)}(\theta_{0})$ solutions at a particular angle $\theta_{0}$ occur
where, if plotted as a function of $r$, $J_{1}(kr)\cos{\theta_{0}}$ intersects
the horizontal $r$-axis. Figure 1 plots $J_{1}(kr)$ (the case where
$\theta_{0}$ = 0) for $r=[0,25]$ cm. The first three positive intersections
with the $r$-axis are indicated by the dashed black, blue, and cyan lines that
correspond respectively to the $r^{(n)}(0)$ solutions with $n$ = 1, 2, and 3.

We could next consider plotting $J_{1}(kr)\cos{\theta}$ for many $\theta$
values and determining where the intersections lie for each $\theta$. This is
not necessary, however, because the cosine term in $J_{1}(kr)\cos{\theta}$
acts as a $\theta$-dependent amplifying term on the Bessel function and thus
the locations of the intersections with the $r$-axis will not change. As a
result, we conclude that the $r_{0}^{(n)}(\theta)$ solutions are constant with
respect to changes in $\theta$ and so plotting any $r_{0}^{(n)}(\theta)$
solution in Cartesian space plots a circle. This graphical argument can be
generalised to argue that TM${}_{m_{1}n0}$ modes of any $\tilde{g}_{m_{1}}$
and $\phi_{m_{1}}$ will be supported by a circular pillbox.

###  II.3 Modes with a Monopolar and a Single Transverse Multipolar Component

We now investigate the properties of the TM${}_{\\{0,m_{1}\\}\eta 0}$ modes
and the corresponding azimuthally modulated cross-sections that support them.
We begin with general mathematical analysis.

From Eq. II.2, the subset of TM${}_{\\{0,m_{1}\\}\eta 0}$ modes with
$\phi_{m_{1}}=0$ have a longitudinal electric field of the form:

$E_{z}(r,\theta,z)=J_{0}\left(kr\right)\tilde{g}_{0}+J_{m_{1}}\left(kr\right)\tilde{g}_{m_{1}}\cos{m_{1}\theta},$
(II.6)

and from Eq. II.3, must satisfy the boundary condition:

$J_{0}\left(kr_{0}^{(\eta)}(\theta)\right)\tilde{g}_{0}+J_{m_{1}}\left(kr_{0}^{(\eta)}(\theta)\right)\tilde{g}_{m_{1}}\cos{m_{1}\theta}=0.$
(II.7)

The periodicity of the $\cos{m_{1}\theta}$ function in Eq. II.7 means that
both the longitudinal electric field of a TM${}_{\\{0,m_{1}\\}\eta 0}$ mode
and the corresponding azimuthally modulated cross-section have $m_{1}$-fold
rotational symmetry. Additionally, $r_{0}^{(\eta)}(0)$ and
$r_{0}^{(\eta)}(\pi/m_{1})$ are turning points of opposite forms in the cross-
section and so if $r_{0}^{(\eta)}(0)$ is a maximum then
$r_{0}^{(\eta)}(\pi/m_{1})$ is a minimum, or vice versa.

Furthermore, if we consider the specific angles
$\theta_{q}=(2q+1)\pi/(2m_{1})$ $(q\in\mathbb{Z})$, the cosine term vanishes
and Eq. II.7 becomes:

$J_{0}\bigg{(}kr_{0}^{(\eta)}\left(\theta_{q}\right)\bigg{)}\tilde{g}_{0}=0\;\;\rightarrow\;\;\frac{\omega}{c}r_{0}^{(\eta)}\left(\theta_{q}\right)=j_{0x},$
(II.8)

where $x\in\mathbb{Z}$.111The reason $x\neq\eta$ is due to the possibility of
conditional modes, discussed in Sec. II.3.2. As well as relating the cavity
frequency and radial value at these angles to the Bessel roots $j_{0x}$, Eq.
II.8 defines $\eta$ for the TM${}_{\\{0,m_{1}\\}\eta 0}$ modes as the number
of poles in $E_{z}$ specifically along the $\theta_{q}=(2q+1)\pi/(2m_{1})$
radial lines. This can be contrasted with the generality of the interpretation
of $n$ as the number of poles in a TMmnp mode along any non-nodal, radial
line.

As per the TM0n0 modes in the circular pillbox cavity, the
TM${}_{\\{0,m_{1}\\}\eta 0}$ modes supported by azimuthally modulated cavities
are also non-degenerate with no same-order mode. Furthermore, by the Courant
nodal domain theorem [25], the TM${}_{\\{0,m_{1}\\}10}$ modes are the
fundamental TM modes in the azimuthally modulated cavities that support them.

Additionally, we note that if the ratio of the transverse multipolar component
to the monopolar component is zero, $\tilde{g}_{m_{1}}/\tilde{g}_{0}=0$, the
form of $E_{z}$ in Eq. II.6 is equivalent to that of a TM0n0 mode. Such modes
are supported by a cavity with a circular cross-section, as discussed in Sec.
II.2. The cross-section of the cavity that supports a TM${}_{\\{0,m_{1}\\}\eta
0}$ mode for a small ratio of transverse multipole to monopole,
$\tilde{g}_{m_{1}}/\tilde{g}_{0}<<1$, will therefore be perturbed only
slightly from circular. Taking the other extreme whereby the transverse
multipole to monopole ratio tends to infinity,
$\tilde{g}_{m_{1}}/\tilde{g}_{0}\rightarrow\infty$, the cavity cross-sections
do not tend back to circular but instead tend to unique, non-circular shapes.
This asymmetry arises because the TM${}_{m_{1}n0}$ modes are degenerate
whereas the TM${}_{\\{0,m_{1}\\}\eta 0}$ modes are non-degenerate for all
ratios of $\tilde{g}_{m_{1}}/\tilde{g}_{0}$.

Finally, we consider the azimuthally modulated cross-sections that support
general TM${}_{\\{0,m_{1}\\}\eta 0}$ modes for any value of $\phi_{m_{1}}$.
Introducing a $\phi_{m_{1}}\neq 0$ term into Eq. II.6 and Eq. II.7, we find
the set of general cavity cross-sections are identical to the subset of cavity
cross-sections with $\phi_{m_{1}}=0$, except for a physical rotation about the
$z$-axis of $\phi_{m_{1}}/m_{1}$.

####  II.3.1 Example: Modes with a Monopolar and a Sextupolar Component

It turns out that for certain combinations of $m_{1}$ and $\eta$, there is a
limit on the ratio of transverse multipole to monopole beyond which the cross-
section of the azimuthally modulated RF cavity becomes discontinuous. This is
best illustrated through an example and here we consider 3 GHz TM{0,3}η0 modes
with $\phi_{3}=0$. Inserting this, $m_{1}=3$ and $k=2\pi\times 3$ GHz$/c$ into
Eq. II.7 gives the longitudinal electric field profile of such a mode:

$E_{z}(r,\theta,z)=J_{0}\left(kr\right)\tilde{g}_{0}+J_{3}\left(kr\right)\tilde{g}_{3}\cos{3\theta},$
(II.9)

and the corresponding cross-section can be determined from Eq. II.7 which we
rearrange to:

$J_{0}\left(kr_{0}^{(\eta)}(\theta)\right)=-\frac{\tilde{g}_{3}}{\tilde{g}_{0}}J_{3}\left(kr_{0}^{(\eta)}(\theta)\right)\cos{3\theta}.$
(II.10)

(a) The cavity shapes, $r_{0}^{(\eta)}(\theta)$, that support the $\eta$ = 1,
2, and 3 TM{0,3}η0 modes as black, blue, and cyan lines respectively.

(b) $J_{0}(kr)$ and $-0.95J_{3}(kr)\cos{3\theta}$ plotted against $r$ for
$\theta=0,\pi/3$. The $\eta$ = 1, 2, and 3 solutions where the two functions
intersect for all $\theta$ are plotted as black, blue, and cyan lines.

Figure 2: 3 GHz TM{0,3}η0 modes with $\tilde{g}_{3}/\tilde{g}_{0}=0.95$ and
$\phi_{3}=0$.

We firstly solve Eq. II.10 numerically for a sextupole to monopole ratio of
$\tilde{g}_{3}/\tilde{g}_{0}$ = 0.95. The cross-sections of the RF cavities
that support the $\eta=1$, 2, and 3 modes are plotted in Fig. 2a. They have
$3$-fold rotational symmetry, as expected.

To help understand the properties of these azimuthally modulated cavities, we
analyse Eq. II.10 graphically. Generalising the monopole term to $J_{0}(kr)$
and sextupole term to $-0.95J_{3}(kr)\cos{3\theta}$ then, if we plot these two
functions against $r$ for a given $\theta=\theta_{0}$, Eq. II.10 states the
$r^{(\eta)}(\theta_{0})$ solutions are the points at which the two functions
intersect. Figure 2b plots $J_{0}(kr)$ in red and the two functions $\pm
0.95J_{3}(kr)$ in green, the latter corresponding to the minimum and maximum
values of $\cos{3\theta_{0}}=\mp 1$. The $\eta$ = 1, 2, and 3 points of
intersection of the red and green lines are indicated respectively by the
black, blue, and cyan vertical dashed lines.

Considering the intersections at other $\theta$ values, we note that the
monopole term has no $\theta$ dependence and thus intersections will always
lie on the red, monopolar line for all $\theta$. In contrast, the sextupolar
term contains a cosine term that acts as a $\theta$-dependent amplifying term
on $-0.95J_{3}(kr)$. As Fig. 2b plots $\pm 0.95J_{3}(kr)$ which correspond to
the maximum and minimum amplification, intersections at all other $\theta$
must occur between these maximum and minimum intersections. The solid black,
blue, and cyan lines on Fig. 2b show the $\eta$ = 1, 2, and 3 intersections
for all $\theta$ and it is clear that each $r^{(\eta)}(\theta)$ solution is
bounded between its corresponding vertical dashed lines.

This graphical analysis of Eq. II.10 in Fig. 2b explains why cavity shapes in
Fig. 2a are physically larger for larger $\eta$. Additionally, Fig. 2b shows
$j_{31}>j_{02}$, that is the 3${}^{\textrm{rd}}$ order Bessel function first
intersects the $r$-axis after the 0${}^{\textrm{th}}$ order Bessel has
intersected it twice. As a result, both the $\eta=1$ and $\eta=2$ cross-
sections exist in the interval $r=[0,j_{31}]/k\approx[0,10]$ cm, and this
explains why $r_{0}^{(1)}(0)$ is a maximum whereas $r_{0}^{(2)}(0)$ is a
minimum in Fig. 2a. This is also why we could not equate $\eta$ with $x$ in
Eq. II.8 as there may be more than one solution in an interval
$r=[j_{m_{1}(x-1)},j_{m_{1}x}]/k$.

(a) The cavity shapes, $r_{0}^{(\eta)}(\theta)$, that support the $\eta$ = 1,
2, and 3 TM{0,3}η0 modes as black, blue, and cyan lines respectively.

(b) $J_{0}(kr)$ and $-1.2J_{3}(kr)\cos{3\theta}$ plotted against $r$ for
$\theta=0,\pi/3$. The $\eta$ = 1, 2, and 3 solutions where the two functions
intersect for all $\theta$ are plotted as black, blue, and cyan lines.

Figure 3: 3 GHz TM{0,3}η0 modes with $\tilde{g}_{3}/\tilde{g}_{0}=1.2$ and
$\phi_{3}=0$.

There is a further consequence from having multiple solutions in an interval
$r=[j_{m_{1}(x-1)},j_{m_{1}x}]/k$: if we consider increasing the value of
$\tilde{g}_{3}/\tilde{g}_{0}$ beyond 0.95 then the $\eta$ = 1 and $\eta$ = 2
solutions at the angles $\cos{3\theta_{0}}=1$ will further converge until, at
a critical ratio of $\tilde{g}_{3}/\tilde{g}_{0}$, $r_{0}^{(1)}(\theta_{0})$ =
$r_{0}^{(2)}(\theta_{0})$. To see this graphically, consider the green line
representing $-0.95J_{3}(kr)$ in Fig. 2b: if $\tilde{g}_{3}/\tilde{g}_{0}$ was
increased from 0.95, the $\eta=1$ (black) and $\eta=2$ (blue) intersection
points at $r\approx 5.5$ cm will move closer together. The critical value at
which the $r_{0}^{(1)}(\theta_{0})$ and $r_{0}^{(2)}(\theta_{0})$ converge is
numerically calculated as $(\tilde{g}_{3}/\tilde{g}_{0})_{\textrm{crit}}$ =
0.98.

The implication of trying to solve Eq. II.10 with
$\tilde{g}_{3}/\tilde{g}_{0}=1.2>(\tilde{g}_{3}/\tilde{g}_{0})_{\textrm{crit}}$
is shown in Fig. 3: for a range of $\theta$ values around $\cos{3\theta}=0$,
real $\eta=1$ and $\eta=2$ solutions do not exist because there is no
intersection between the Bessel terms, and therefore solution, at $r\approx
5.5$ cm. As a result, the $r_{0}^{(1)}(\theta)$ and $r_{0}^{(2)}(\theta)$
solutions are discontinuous.

This can be contrasted to the continuous $r_{0}^{(3)}(\theta)$ solution in
Fig. 3b whereby the two terms continuously intersect for all $\theta$. In
fact, because there is only one root to the $0^{\textrm{th}}$ order Bessel
function in the interval $[j_{31},j_{32}]$, the $\eta=3$ solution is
continuous for all values of $\tilde{g}_{3}/\tilde{g}_{0}$. We can also see
that the $\eta=3$ solution is bounded within the interval $[j_{31},j_{32}]/k$.
Specifically as $\tilde{g}_{3}/\tilde{g}_{0}\rightarrow 0$,
$r_{0}^{(3)}(\theta)\rightarrow j_{03}/k$, and as
$\tilde{g}_{3}/\tilde{g}_{0}\rightarrow\infty$,
$r_{0}^{(3)}(\theta)\rightarrow[j_{31}/k$ for $\cos{3\theta}>0$; $j_{32}/k$
for $\cos{3\theta}<0$].

The mathematical solutions to Eq. II.3 can inform the practical and realistic
design of cavities that support TM{M}η0-like modes for useful application in
particle accelerators. RF cavities used in particle accelerators must have an
enclosed cross-section and may have constraints on their maximum physical size
and frequency of operation. Therefore, although the $r_{0}^{(3)}(\theta)$
structure is continuous and enclosed for the multipolar ratio
$\tilde{g}_{3}/\tilde{g}_{0}=1.2$, it is a physically larger cavity than the
discontinuous $r_{0}^{(1)}(\theta)$ and $r_{0}^{(2)}(\theta)$ shapes. As a
potential compromise between creating a mode with the desired
$\tilde{g}_{3}/\tilde{g}_{0}=1.2$ multipolar ratio whilst minimising the
physical size of the cavity that supports it, we can consider designing an RF
cavity where the cross-section jumps to the $\eta=3$ solution at the angles
where real $\eta=1$ and $\eta=2$ solutions do not exist. We define such
cavities that mix modes of differing $\eta$ as hybrid cavities and these are
further discussed and analysed in Sec. III.

####  II.3.2 Conditional Modes

For the case of the TM{0,3}η0 modes, we showed that the $\eta=1$ and $\eta=2$
solutions are discontinuous if $\tilde{g}_{3}/\tilde{g}_{0}$ exceeds a
critical value. This arises because $j_{31}>j_{02}$ and so there are two roots
to the $0^{\textrm{th}}$ order Bessel function in the interval $[0,j_{31}]$.
In contrast, the $\eta=3$ solution is continuous for all values of
$\tilde{g}_{3}/\tilde{g}_{0}$ because there is only one root to the
$0^{\textrm{th}}$ order Bessel function in the interval $[j_{31},j_{32}]$.
Extending this, the azimuthally modulated cavity designed to support a
TM${}_{\\{0,m_{1}\\}\eta 0}$ mode will have a critical limit on
$\tilde{g}_{m_{1}}/\tilde{g}_{0}$ beyond which the solution is discontinuous
if in the $[j_{m_{1}(x-1)},j_{m_{1}x}]$ interval in which the
$r_{0}^{(\eta)}(\theta)$ solution lies, there is more than one root to the
$0^{\textrm{th}}$ order Bessel function.

To generalise this result, we define any TM{M}η0 mode of given multipolar
content $\\{M\\}$ and radial order $\eta$ as conditional if any corresponding
$r_{0}^{(\eta)}(\theta)$ solution is discontinuous for any possible
combination of magnitudes of the multipolar coefficients. A TM{M}η0 mode is
therefore unconditional if it has a continuous cross-section for all possible
ratios of the multipolar coefficients.

###  II.4 Modes with Two Different Transverse Multipolar Components

Having determined the properties of azimuthally modulated RF cavities that
support the TM${}_{\\{0,m_{1}\\}\eta 0}$ modes in the previous subsection, we
now examine the azimuthally modulated RF cavities that support the
TM${}_{\\{m_{1},m_{2}\\}\eta 0}$ modes composed of two transverse multipoles.

Using Eq. II.2, the TM${}_{\\{m_{1},m_{2}\\}\eta 0}$ modes composed of two
different, transverse multipoles with $m_{2}>m_{1}>0$ have a longitudinal
electric field of the form:

$E_{z}(r,\theta,z)=J_{m_{1}}\left(kr\right)\tilde{g}_{m_{1}}\cos{(m_{1}\theta-\phi_{m_{1}})}+\\\
J_{m_{2}}\left(kr\right)\tilde{g}_{m_{2}}\cos{(m_{2}\theta-\phi_{m_{2}})}.$
(II.11)

Without loss of generality, the corresponding set of azimuthally modulated
cavity shapes can be determined from Eq. II.3 with $\phi_{m_{1}}=0$:

$J_{m_{1}}\left(kr_{0}^{(\eta)}(\theta)\right)\tilde{g}_{m_{1}}\cos{m_{1}\theta}+\\\
J_{m_{2}}\left(kr_{0}^{(\eta)}(\theta)\right)\tilde{g}_{m_{2}}\cos{(m_{2}\theta-\phi_{m_{2}})}=0.$
(II.12)

The cross-sections that solve Eq. II.12 and support
TM${}_{\\{m_{1},m_{2}\\}\eta 0}$ modes require further consideration compared
to those that support TM${}_{\\{0,m_{1}\\}\eta 0}$ modes for four interlinked
reasons:

  * •

Solutions do not necessarily have an $m_{1}$-fold symmetry but instead a
rotational symmetry that varies depending upon the values of $m_{1}$, $m_{2}$
and $\phi_{m_{2}}$. As a result, $\eta$ has a more specific definition for
TM${}_{\\{m_{1},m_{2}\\}\eta 0}$ modes.

  * •

A practical RF cavity must have $r_{0}^{(\eta)}(\theta)>0$, however,
$r_{0}^{(\eta)}(\theta)=0$ is a valid solution to Eq. II.12 for all $m_{1}$
and $m_{2}$. We therefore define a TM${}_{\\{m_{1},m_{2}\\}\eta 0}$ mode as
forbidden if $r_{0}^{(\eta)}(\theta)=0$ at any angle $\theta$.

  * •

Solutions are not necessarily bounded within the interval $r=[j_{m_{2}(x-1)}$,
$j_{m_{2}x}]/k$ due to both terms in Eq. II.12 having differing orders of
cosine dependence.

  * •

A practical RF cavity must be closed with $r^{(\eta)}(0)=r^{(\eta)}(2\pi)$,
however, solutions of particular $m_{1}$, $m_{2}$ and $\phi_{m_{2}}$ do not
satisfy this condition. We define such TM${}_{\\{m_{1},m_{2}\\}\eta 0}$ modes
as spiral.

Appendix A derives why certain modes are forbidden or spiral. Section II.4.1
presents an example of a forbidden mode and Sec. II.4.2 an example of a spiral
mode.

TM${}_{\\{m_{1},m_{2}\\}\eta 0}$ modes can also be conditional whereby the
cross-section is discontinuous and a hybrid solution needs be considered if
the ratio of the transverse multipolar coefficient of order $m_{1}$ to the
transverse multipolar coefficient of order $m_{2}$ (or vice versa) exceeds a
critical value. A TM${}_{\\{m_{1},m_{2}\\}\eta 0}$ mode will be conditional
if, for any $\theta=\theta_{0}$, there is more than one root to the
$m_{1}^{\textrm{th}}$ order Bessel function within the
$[j_{m_{2}(x-1)},j_{m_{2}x}]/k$ interval that the $r_{0}^{(\eta)}(\theta_{0})$
solution exists in.

####  II.4.1 Example: Modes with a Quadrupolar and an Octupolar Component

(a) The cavity shapes, $r_{0}^{(\eta)}(\theta)$, that support the $\eta$ = 1,
2, and 3 TM{2,4}η0 modes as black, blue, and cyan lines respectively.

(b) $\pm J_{2}(kr)$ and $\pm 5J_{4}(kr)$ plotted against $r$. The $\eta$ = 1,
2, and 3 solutions where the $J_{2}(kr)\cos 2\theta$ and $-5J_{4}(kr)\cos
4\theta$ functions intersect for all $\theta$ are plotted as black, blue, and
cyan lines.

Figure 4: 3 GHz TM{2,4}η0 modes with $\tilde{g}_{4}/\tilde{g}_{2}=5$,
$\phi_{4}=0$.

To illustrate a forbidden mode, consider the $\eta$ = 1, 2, and 3 cross-
sections that support 3 GHz TM{2,4}η0 modes with $\tilde{g}_{4}/\tilde{g}_{2}$
= 5 and $\phi_{m_{2}}=0$. Numerically solving Eq. II.12 with these parameters
returns the cross-sections plotted in Fig. 4a. Figure 4b plots the two Bessel
terms for maximum and minimum amplitude and the $\eta$ = 1, 2, and 3
solutions. It shows that each solution crosses one interval and so the
$\eta=1$ solution forbidden. Additionally, because there is only a single root
to the 2${}^{\textrm{nd}}$ order Bessel function within each
$[j_{4(x-1)},j_{4x}]/k$ interval, all TM{2,4}η0 modes are unconditional and
each solution bounded between $[j_{4(\eta-2)},j_{4\eta}]/k$.

####  II.4.2 Example: Modes with a Dipolar and Sextupolar Component

Figure 5 shows an example of a spiral mode by plotting the
$r_{0}^{(\eta)}(\theta)$ solutions in the range $r\approx[0,15]$ cm that
support 3 GHz TM{1,3}η0 modes with $\tilde{g}_{3}/\tilde{g}_{1}$ = 1 and
$\phi_{3}=0.3$. The solutions form a single, continuous, spiralling line which
differs from the discrete, closed solutions formed by unconditional solutions
of distinct $\eta$ in previous examples. It is evident that
$r_{0}^{(\eta)}(0)\neq r_{0}^{(\eta)}(2\pi)$ and therefore this is a spiral
mode.

Figure 5: The $r^{(\eta)}(\theta)$ solutions that support the 3 GHz TM{1,3}η0
modes with $\tilde{g}_{3}/\tilde{g}_{1}=1$ and $\phi_{3}=0.3$ is plotted in
black in the range $r\approx[0,15]$ cm.

###  II.5 Modes with Three Or More Different Multipolar Components

Equation II.3 can also be numerically solved to determine the azimuthally
modulated cross-sections that support TM{M}η0 modes with three or more
multipolar components. General mathematical analysis of the properties of
these modes and the corresponding cross-sections that support them is
challenging. The concepts of certain modes being conditional, forbidden or
spiral, however, generalises to TM{M}η0 modes with any number multipolar
content and provides rationale for understanding the numerical solutions to
Eq. II.3 as well as helping to inform the scope and limitation of a general
azimuthally modulated RF cavity design.

##  III CST Simulations

(a)

(b)

(c)

(d)

(e)

Figure 6: $E_{z}$ in cavities operating in 3 GHz TM310, TM{0,3}η0 and hybrid
TM{0,3}10-TM{0,3}30 modes. Red is a maximum in the field, blue a minimum and
green is zero. The cavities are to scale: the radius of the circular pillbox
is 10.15 cm.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 7: The calculated multipolar coefficient, $\tilde{g}_{m}$, is plotted
against the distance from the centre, $r$, with colour denoting the order,
$m$, for cavities operating in the TM310 and four different TM{0,3}η0 modes.

(a)

(b)

(c)

(d)

(e)

Figure 8: Frequency spectra of the 40 lowest frequency TM modes for the 5
different cavity shapes that support the mode denoted in the subcaption. The
red crosses represent the 3 GHz mode of interest.

Analysis of 3D electromagnetic simulations of azimuthally modulated RF
cavities is presented here to confirm the validity of the theoretical work.
Electric field distributions, Helmholtz decomposition results, and mode
spectra are shown for: a pillbox cavity operating in a TM310 mode, for
azimuthally modulated cavities with continuous cross-sections that support the
TM{0,3}10, TM{0,3}20, and TM{0,3}30 modes with
$\tilde{g}_{3}/\tilde{g}_{0}=0.95$, and for an azimuthally modulated hybrid
cavity that supports a hybrid TM{0,3}10-TM{0,3}30 mode with
$\tilde{g}_{3}/\tilde{g}_{0}=1.2$. All modes are normal with $\phi_{3}=0$ and
each of the cavities were designed with length 0.5 mm and such that the mode
of interest resonates at 3 GHz.

To create the hybrid cavity, the cavity wall is extended from the
$r_{0}^{(1)}(\theta)$ solution to the $r_{0}^{(3)}(\theta)$ solution at the
angles where the $\eta=1$ solution is no longer real. Although this creates a
closed cavity, it introduces additional transverse multipoles into the mode
because the boundary condition $E_{z}(r,\theta,z)=0$ must also be satisfied
where the cavity wall extends from the $\eta=1$ to the $\eta=3$ solution. The
field is small in this region compared to the field at the centre, however,
and so the additional multipoles are also small. Thus the hybrid mode is
dominantly, but not solely, composed of monopolar and sextupolar components
and the cavity will resonate close to, but not exactly at, the designed
resonant frequency.

Figure 6 shows the $E_{z}$ distribution of these different modes. The
difference between the TM310 mode and TM{0,3}η0 modes is evident by comparing
the subfigures: there is no longitudinal accelerating field at the centre of
the circular pillbox cavity operating in the TM310 mode (Fig.
LABEL:subfig:CSTF0) whereas there is in the case of the TM{0,3}η0 modes (Figs.
LABEL:subfig:CSTF1-LABEL:subfig:CSTF4). The resonant frequency of each of the
modes was calculated by CST as 3.000 GHz, except for the hybrid cavity which
resonated at 3.004 GHz.

Figure 7 shows the multipolar content of each of the five fields calculated
using the Helmholtz decomposition method. This method is detailed in Ref. [17]
and involves exporting the longitudinal electric field along the surface of
cylinders orientated along the cavity axis and then performing an FFT on the
data to determine $\tilde{g}_{m}$. To determine any dependence of these
coefficients with radial distance, $E_{z}$ is exported along 20 discretely
formed cylinders of non-linearly spaced radii in the range [0.02, 0.95] cm. By
normalising the results to $\tilde{g}_{3}$ in Fig. LABEL:subfig:CSTD0, it is
clear that the TM310 mode in the circular pillbox only contains a sextupole
component that is constant with radius. By normalising the results to
$\tilde{g}_{0}$ in Figs. LABEL:subfig:CSTD1-LABEL:subfig:CSTD4 we find the
calculated multipolar content to be as designed: for example, in Fig.
LABEL:subfig:CSTD1, $\tilde{g}_{0}=1$ for all radii and $\tilde{g}_{3}=0.95$
for all radii beyond $r\approx 0.03$ cm. The reason for the sextupolar and
octupolar components diverging randomly at small $r$ from the expected
constant value is because the data must be divided by $r^{m}$ to determine
$\tilde{g}_{m}$. Simulation errors arising from the finite accuracy of the
mesh and numerical errors arising from the pseudo-cylinders being composed
only of 360 distinct points are therefore amplified for smaller radii and
larger $m$. Accounting for this error, the multipoles are constant with radial
distance for each cavity shape and thus simulation results agree with the
radial Bessel dependence given by Eq. II.2.

The azimuthally modulated RF cavity shapes, $r_{0}^{(\eta)}(\theta)$, support
the infinite number of modes that satisfy their boundary conditions. Figure 8
plots the frequency spectra of the 40 lowest frequency TM modes of the five
different cavity shapes. The red crosses denote the 3 GHz mode of interest and
any orthogonal degenerate modes of the same frequency. Figure
LABEL:subfig:CSTO0 shows the TM310 mode is not the fundamental but the
7${}^{\textrm{th}}$ lowest mode and that it is degenerate with the skew mode,
as expected. This can be contrasted with the TM{0,3}10 modes in Fig.
LABEL:subfig:CSTO1 and Fig. LABEL:subfig:CSTO4: not only are these modes
fundamental and non-degenerate, but they also have a larger frequency
separation. Additionally, while the frequency spectra of the TM{0,3}20 and
TM{0,3}30 modes is similar in distribution to that of the circular pillbox and
the modes are not fundamental, they are evidently non-degenerate with no same-
order mode. The frequency spectra of the cavities is important because other
modes may require consideration in a particle accelerator if they are excited,
directly by the RF power supply or indirectly by a passing beam inducing
wakefields, and this significantly interferes with beam dynamics. Any
interfering modes may require damping by incorporating lower- and higher-order
mode couplers into the design or could necessitate a complete redesign of the
cavity.

##  IV Experimental Test of an RF Cavity that Simultaneously Accelerates
Longitudinally and Focuses Transversely

Here in Sec. IV, we show that beam pipes and a dual-port power coupler can be
incorporated into an azimuthally modulated RF cavity design whilst maintaining
desired multipolar ratios. We also test a prototype in order to verify that
the bead pull method can be used to measure the field of an azimuthally
modulated cavity and accurately determine the ratio of multipolar terms with
minimal error.

###  IV.1 Designing the Cavity

Figure 9: $E_{z}$ at the longitudinal centre of the designed RF cavity system,
where blue is a minimum in the field, orange maximum, and green zero.

Figure 9 shows the E-field distribution of a critically-coupled, single-cell,
azimuthally modulated RF cavity system designed to support a 3 GHz TM{0,2}20
mode with a quadrupole to monopole ratio of 5.041. Instead of using a single-
port coupling system which would introduce a dipolar component into the field
of the RF cavity, a dual-port coupling system was used in conjunction with a
T-junction waveguide design to split the input power equally into the two
waveguide arms. The dual-port coupler introduces an additional quadrupolar
component into the mode. Helmholtz decompositions of simulations showed that
by solving Eq. II.3 for the enclosed RF cavity shape that supports a 3 GHz
TM{0,2}20 mode with $\tilde{g}_{2}/\tilde{g}_{0}$ = 4.032 and $\phi_{2}=0$,
and using this shape to create the prototype in Fig. 9 which includes the dual
port coupler and holes at the positions of the poles, the resultant field had
a quadrupole to monopole ratio of $\tilde{g}_{2}/\tilde{g}_{0}$ = 5.041 and
negligible contributions from higher order multipoles. The $\eta=2$ mode was
chosen over the $\eta=1$ mode in order to give more measurement points to
consider and compare.

The RF cavity was designed as two identical halves to be sealed together and
was milled out of aluminium. Seven 10 mm diameter holes were drilled into the
RF cavity to create holes with centres at the locations of the seven poles in
the TM{0,2}20 mode. One half the RF cavity system is photographed in Fig. 10.

Figure 10: Half the RF cavity system designed to support the TM{0,2}20 mode
with quadrupolar component 5.041 times greater than the monopolar component.

###  IV.2 Characterisation of the RF Cavity System

Measurements of the prototype were taken using a Vector Network Analyser (VNA)
connected to a matched waveguide adaptor secured to the top of the cavity
waveguide port via coaxial cable. The apparatus was calibrated at the end of
the coax, which excluded the matched waveguide adaptor from calibration. To
ensure this does not impact results, two cases are compared in Fig. 11: the
frequency response of shorting the calibrated coax, and of attaching the
waveguide adaptor to the calibrated coax and clamping an aluminium plate to
it. The maximum magnitude in the difference of the impedance measurements
between the two cases is 0.2 $\Omega$. This is of the order of 100 times
smaller than the changes in impedance measured in the bead pull testing and
that used for calculating Q-factors. In addition to this, measurements do not
give any evidence for resonances in the waveguide adaptor and the external
Q-factor does not change significantly between simulations without the
waveguide adaptor and measurements with the waveguide adaptor. We thus
conclude that the waveguide adaptor’s effect on measurements can be neglected.

Figure 11: Impedance measurements of the calibrated coaxial cable shorted, and
with the waveguide adaptor attached to the calibrated coax and shorted by
clamping an aluminium plate to it.

The resonant frequency of the RF cavity system, $f_{0}$, and its intrinsic,
$Q_{0}$, external, $Q_{e}$ and loaded, $Q_{L}$, Q-factors can be calculated
from measurement of the variation in the S-parameter $S_{11}$ with driving
frequency and using:

$\beta=\frac{Q_{0}}{Q_{e}}=\frac{Q_{0}-Q_{L}}{Q_{L}}=\frac{1-S_{11}(f=f_{0})}{1+S_{11}(f=f_{0})},$
(IV.1)

where $f$ is the driving frequency and $\beta$ is the coupling coefficient. It
should be noted that at resonance $S_{11}$ is purely real.

Figure 12 is a Smith chart that shows the measured variation in $S_{11}$ for
driving frequencies in the range $f_{0}\pm 1$ MHz for five different
simulation and experimental measurements. The Smith chart maps the impedance,
$Z$, onto the $S_{11}$ plane and provides a method for visualising $\beta$
[26].

The blue line of Fig. 12 represents the simulation output of $S_{11}$ of the
RF cavity system modelled as perfectly smooth copper; the electric field from
this simulation was shown in Fig. 9. From this data, we calculate $f_{0}$ =
2.99914 GHz, $Q_{0}$ = 20830, $Q_{e}$ = 20050 and so the system is critically
coupled as $\beta\simeq 1$. The RF cavity coupling ports were optimised for
copper, while the prototype was milled out of aluminium to reduce cost. The
red line in Fig. 12 shows the simulation output for perfectly smooth aluminium
and the impact of the choice of material on $\beta$: we calculate $f_{0}$ =
2.99917 GHz, $Q_{0}$ = 16880 and $\beta=0.831$. The yellow line represents the
measurement of the prototype, and we calculate that $f_{0}=2.99914$ GHz,
$Q_{0}=9120$ and $\beta=0.410$. The prototype is thus significantly
undercoupled compared to the perfectly smooth simulation.

Figure 12: Smith chart showing the variation in simulation output and
experimental measurements of $S_{11}$ in the range $f_{0}\pm 1$ MHz.

This discrepancy in $\beta$ is likely due to two effects. Firstly, the surface
roughness due to machining of the aluminium which leads to increased resistive
losses [27]. The RMS surface roughness was measured with a white light
interferometer to be 0.33 $\mu$m. The purple line shows the simulation output
when this level of roughness is included - $f_{0}$ = 2.99955 GHz, $Q_{0}$ =
15150 and $\beta=0.750$. The second effect is RF leakage due to an imperfect
seal between the two halves of the prototype. This can be modelled by
uniformly separating the two cavity halves: the green dashed line shows that a
0.33 $\mu$m surface roughness and 3.5 $\mu$m uniform air gap between the two
halves replicates the coupling coefficient of experimental measurements -
$f_{0}$ = 2.99955 GHz, $Q_{0}$ = 8490 and $\beta=0.410$.

In reality the separation between the two halves of the cavity will not be
uniform, however, this would be difficult to model. Simulations show the
prototype is particularly sensitive to gaps near the coupling between the
waveguide and the cavity and a separation on the order of microns is feasible
given the distribution of the bolts used for tightening and the presence of
deeper scratches on the aluminium surfaces that are bolted together.

###  IV.3 Experimental Method

Despite the RF cavity system being significantly undercoupled, the electric
field of the RF cavity can be determined using the bead pull technique [28,
29, 30]. This method uses the result from cavity perturbation theory [31] that
when a dielectric bead, small enough such that the EM fields are approximately
constant over it, is introduced into an RF cavity at a position
$(r,\theta,z)$:

$\Delta f\propto|E_{0}(r,\theta,z)|^{2},$ (IV.2)

where $\Delta f$ is the change in resonant frequency caused by introducing the
perturbing object and $E_{0}(r,\theta,z)$ is the magnitude of the electric
field at that position. In our experiments, we used a spherical, plastic bead
with a 2.2 mm diameter as the perturbing object.

The VNA cannot be used to measure directly the change in resonant frequency of
the RF cavity caused by the perturbing object at a high enough sample rate. An
indirect measurement is therefore required. Two possible parameters to use for
this indirect measurement are the change in phase of $S_{11}$ or $|\Delta Z|$,
where $|\Delta Z|=|Z-Z_{\textrm{res}}^{\textrm{unpert}}|$ and
$Z^{\textrm{unpert}}_{\textrm{res}}$ is the impedance of the unperturbed
cavity at the resonant frequency. Figure 13 shows experimental measurements of
the variation in $\tan{(\arg{(S_{11})})}$ (top) and $|\Delta Z|$ (bottom) of
the unperturbed cavity with $f$. The dashed lines are linear fits between the
respective parameter and $f$, fitted to and extrapolated from small $\Delta
f$. The 2.2 mm diameter perturbing bead caused a maximum change in resonant
frequency of $|\Delta f|$ = 0.079 MHz and the inset of Fig. 13 zooms in on
this region. It shows $|\Delta Z|$ varies linearly with $f$ to within 0.3 %,
compared to $\tan{(\arg{(S_{11})})}$ which diverges by up to 20 %.

Figure 13: Measured variation in $\tan{(\arg{(S_{11})})}$ (top) and $|\Delta
Z|$ (bottom) with driving frequency. The insets zoom in on the region
contained inside the black rectangle.

We can therefore determine the electric field at a position $(r,\theta,z)$ in
the RF cavity by measuring, at a fixed driving frequency, the change in
impedance caused by the bead at that position and using the relationship:

$|E_{0}(r,\theta,z)|\propto\sqrt{|\Delta Z|}.$ (IV.3)

To carry out a bead pull measurement, we threaded kevlar wire through one of
the holes before centering it using a digital motor system. The 2.2 mm
diameter, dielectric bead knotted onto the wire was then pulled through the
cavity at a speed of 3 mm/s using a longitudinal motor. The VNA was set up to
continuously measure the impedance of the RF cavity at the driving frequency
of 2.99914 GHz and sample rate of over 50 points per second. The change in
impedance of the RF cavity was measured for the bead being pulled through all
7 holes in both longitudinal directions.

###  IV.4 Experimental Results

The data was exported and analysed in MATLAB [32]. Data collected when the
bead was outside the beam pipes, that is $|z|>38.1$ mm where $z$ is the
longitudinal position relative to the centre of the RF cavity, was removed
from the data set. A line of best fit between the first and last 50 data
points was then used to remove the small linear drift in impedance over time
we observed.

Figure 14: Comparison of bead pull measurements of the electric field compared
to simulation.

Figure 14 compares the longitudinal electric field measurement along each of
the seven poles from a CST simulation of the cavity system to the experimental
bead pull measurements, where we assume $|E_{0}|=|E_{z}|$ for the
measurements. Simulation data is plotted as full lines and experimental data
as dashed. The different colours correspond to measurements along the
different poles: blue denotes the single central pole, red the two inner
horizontal ($x$) poles, green the two outer horizontal ($x$) poles, and black
the two vertical ($y$) poles. To normalise the simulation data, all data
points were divided by the maximum electric field in the $y$ pole. To
normalise the experimental data, all data points were divided by the ratio of
the mean value of the median experimental electric field measurements between
$|z|<1$ mm along each pole to the mean value of the maximum simulated electric
field measurements along each pole. The similarity between experimental and
simulation results indicate that the cavity is symmetric in the horizontal,
vertical, and longitudinal directions, as expected.

Table 1:  Comparison between simulation and experimental measurements of $E_{z}$ through the different beam pipe locations. Bead Pull Along | Simulation | Experimental
---|---|---
y Poles | 1.000 | 1.004$\pm 0.008$
x Poles Inner | 0.802 | 0.805$\pm 0.001$
x Poles Outer | 0.501 | 0.506$\pm 0.002$
Central Pole | 0.364 | 0.368$\pm 0.001$

Table 1 presents a quantitative comparison between the experimental and
simulation results with the mean of the electric field measurement between
$|z|<1$ mm shown alongside the standard deviation of the measurements in that
range. $E_{z}$ is expected to be flat in this range with a standard deviation
below three decimal places and the results show the experimental data is
consistent with simulation. It should be noted that errors arising from
inaccuracies aligning the wire to the centre of the holes or from divergences
from the assumed linearity in Eq. IV.3 have not been included in this
analysis.

The data collected thus support that the electric field of the RF cavity is as
designed, within error. Therefore we have used successfully the presented
theory, in conjunction with 3D simulations to determine the quadrupolar term
introduced by the dual-port coupler, to design and measure the field of an
azimuthally modulated RF cavity that supports a TM{0,2}20 mode with a
quadrupolar to monopolar ratio of 5.041.

##  V Conclusion

This paper has derived the basis of the longitudinal electric fields in
azimuthally modulated RF cavities and presented a systematic method for
determining the cavity shape that will support TM{M}η0 modes. Examples of the
RF cavity shapes that support TM${}_{\\{0,m_{1}\\}\eta 0}$ or
TM${}_{\\{m_{1},m_{2}\\}\eta 0}$ modes have been explicitly examined with
rules derived for RF cavity shapes being conditional, forbidden or spiral
depending upon the values of $m_{1}$ and $\eta$, and $m_{1}$, $m_{2}$ and
$\eta$. The theoretical work has been supported by analysis of 3D simulations
and experimental test of an RF cavity that supports a TM{0,2}20 mode.

The results of this paper provide a basis for the development of new designs
of azimuthally modulated RF cavities that contain any magnitude and order of
multipole components. These designs may have useful application in particle
accelerators, particularly as they do not have degenerate same-order modes and
as they can introduce or cancel wanted or unwanted multipolar components to a
fine precision.

Future research can use the formalism and results presented in this paper to
guide the development of approximate analytic models of cavity designs and
understand how the variation of certain design parameters affects the cavity
performance. This could reduce the need to run computationally expensive
simulations. Additionally, the paper provides a basis for determining the
exact effect of the Bessel dependence of the transverse deflecting fields on
beam dynamics. Future studies could also investigate the introduction of
longitudinal asymmetries alongside the azimuthal asymmetries in the RF cavity
with the aim of enhancing the multipolar content of a field and creating
additional parameters with which to optimise performance in aspects such as
shunt impedance.

##  VI Acknowledgements

We would like to thank Graeme Burt for many useful discussions and
particularly for his assistance in designing the coupler for the RF cavity
system. We would also like to thank Shinji Machida, Jean-Baptiste Lagrange,
David Kelliher, Chris Rogers and Chris Prior of the ISIS Intense Beams Group
for their insightful discussions and feedback, as well as Manjit Dosanjh who
has contributed greatly to the direction of the project and regarding the
application of discussed work. This study and the experimental work has been
funded by the Royal Society under RS$\backslash$PhD$\backslash$181200 and
RGF$\backslash$EA$\backslash$18023, and the CI Core Grant ST/P002056/1 has
funded travel.

##  Appendix A Forbidden and Spiral Modes

Here we discuss why certain TM${}_{\\{m_{1},m_{2}\\}\eta 0}$ modes are
forbidden, that is $r^{(\eta)}(\theta)$ = 0 for some $\theta$ value, or
spiral, that is $r^{(\eta)}(0)\neq r^{(\eta)}(2\pi)$.

As our first step, we prove that these $r_{0}^{\eta}(\theta)$ solutions are
not necessarily bounded within an interval $r=[j_{m_{2}(x-1)},j_{m_{2}x}]/k$
($x\in\mathbb{Z}$). These solutions must solve the boundary condition in Eq.
II.12 that we replicate here:

$J_{m_{1}}\left(kr_{0}^{(\eta)}(\theta)\right)\tilde{g}_{m_{1}}\cos{m_{1}\theta}+\\\
J_{m_{2}}\left(kr_{0}^{(\eta)}(\theta)\right)\tilde{g}_{m_{2}}\cos{(m_{2}\theta-\phi_{m_{2}})}=0.$
(A.1)

Firstly, consider Eq. A.1 for $\theta=0$: it follows that
$r_{0}^{(\eta)}(0)\neq j_{m_{2}y}/k$ ($y\in\mathbb{Z}$). This means that all
solutions $r_{0}^{(\eta)}(0)$ lie within an interval
$r=[j_{m_{2}(x-1)},j_{m_{2}x}]/k$ and not at the limits, or edges, of the
interval. Next we consider the unique angles in the range $[0,2\pi]$,
$\theta_{q}=(2q+1)\pi/2m_{1}$ ($q=0,1,\cdots,\lfloor 2m_{1}-1/2\rfloor$), that
eliminate the cosine term of order $m_{1}$ such that Eq. A.1 becomes:

$J_{m_{2}}\left(kr_{0}^{(\eta)}(\theta_{q})\right)\tilde{g}_{m_{2}}\cos{\left(\frac{m_{2}}{m_{1}}(2q+1)\frac{\pi}{2}-\phi_{m_{2}}\right)}=0.$
(A.2)

This has two possible solutions whereby either the Bessel term of order
$m_{2}$ is zero or the cosine term is zero. We will firstly consider the
cosine term to be non-zero, which forces:

$kr_{0}^{(\eta)}(\theta_{q})=j_{m_{2}y},$ (A.3)

to satisfy Eq. A.2. Equation A.3 thus states that if
$\cos{(m_{2}\theta_{q}-\phi_{m_{2}})}\neq 0$, $r_{0}^{(\eta)}(\theta_{q})$
must lie at the limit of an interval $r=[j_{m_{2}(x-1)},j_{m_{2}x}]/k$.
Therefore the solution will cross the upper limit of the interval if the
gradient is positive at $\theta_{q}$, whereas it will cross the lower limit if
it is negative. Differentiating Eq. A.1 we find:

$\frac{dr_{0}^{(\eta)}(\theta)}{d\theta}=\frac{1}{k}\frac{J_{m_{1}}(kr_{0}^{(\eta)}(\theta))\tilde{g}_{m_{1}}m_{1}\sin{m_{1}\theta}+J_{m_{2}}(kr_{0}^{(\eta)}(\theta))\tilde{g}_{m_{2}}m_{2}\sin{(m_{2}\theta-\phi_{m_{2}})}}{J_{m_{1}}^{\prime}(kr_{0}^{(\eta)}(\theta))\tilde{g}_{m_{1}}\cos{m_{1}\theta}+J_{m_{2}}^{\prime}(kr_{0}^{(\eta)}(\theta))\tilde{g}_{m_{2}}\cos{(m_{2}\theta-\phi_{m_{2}})}},$
(A.4)

which if $\cos{(m_{2}\theta_{q}-\phi_{m_{2}})}\neq 0$ becomes:

$\frac{dr_{0}^{(\eta)}(\theta)}{d\theta}\bigg{|}_{\theta_{q}}=\frac{m_{1}\tilde{g}_{m_{1}}}{k\tilde{g}_{m_{2}}}\frac{J_{m_{1}}(j_{m_{2}y})(-1)^{q}}{J_{m_{2}}^{\prime}(j_{m_{2}y})\cos{m_{2}\theta_{q}}}\neq
0.$ (A.5)

Equation A.5 can be numerically evaluated to determine the sign of the
gradient at all angles $\theta_{q}$ at the limits of each interval
$j_{m_{2}y}$. We conclude that if the number of times the gradient is positive
does not equal the number of times the gradient is negative (that is the
number of times the upper limit of an interval is crossed does not equal the
total number of times the lower limit of an interval is crossed) then
$r^{(\eta)}(0)$ will lie in a different interval to $r^{(\eta)}(2\pi)$. This
means $r^{(\eta)}(0)\neq r^{(\eta)}(2\pi)$ and the solution and mode are
spiral. If, however, the number of times the upper limit is crossed equals the
number of times the lower limit is crossed, then the solution will lie in the
interval $r=[j_{m_{2}(x-\zeta_{1})},j_{m_{2}(x+\zeta_{2})}]/k$, where the
$\zeta_{i}$ are numerically calculated integers. If $\zeta_{1}\geq x$ then the
corresponding TM${}_{\\{m_{1},m_{2}\\}\eta 0}$ mode will be forbidden as it
will have $r^{(\eta)}(\theta)=0$ for some $\theta$.

Table 2:  The $\eta$ values of the first non-forbidden TM${}_{\\{m_{1},m_{2}\\}\eta 0}$ mode shown for the case $\phi_{m_{2}}\neq\phi_{m_{2}}^{(q)}$ (left of comma) and $\phi_{m_{2}}=\phi_{m_{2}}^{(q)}$ (right of comma). $\nexists$ denotes a spiral mode. $m_{2}$ $m_{1}$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
---|---|---|---|---|---|---|---|---|---
2 | 2, 1 | - | - | - | - | - | - | - | -
3 |  $\nexists$, 1 | 3, 2 | - | - | - | - | - | - | -
4 | 2, 1 | 2, 1 | 4, 3 | - | - | - | - | - | -
5 |  $\nexists$, 1 | 3, 2 |  $\nexists$, 2 | 5, 4 | - | - | - | - | -
6 | 2, 1 |  $\nexists$, 1 | 2, 1 | 3, 2 | 6, 5 | - | - | - | -
7 |  $\nexists$, 1 | 3, 2 |  $\nexists$, 2 | 3, 2 |  $\nexists$, 3 | 7, 6 | - | - | -
8 | 2, 1 | 2, 1 | 4, 3 | 2, 2 | 4, 3 | 4, 3 | 8, 7 | - | -
9 |  $\nexists$, 1 | 3, 2 |  $\nexists$, 1 | 3, 2 |  $\nexists$, 2 | 3, 2 |  $\nexists$, 4 | 9, 8 | -
10 | 2, 1 |  $\nexists$, 1 | 4, 3 | 3, 2 | 2, 1 |  $\nexists$, 2 | 4, 3 | 5, 4 | 10, 9

We now come back to Eq. A.2 and allow the cosine term to be zero. This
requires the phase to take one of the following $q$-dependent values:

$\phi_{m_{2}}=\phi_{m_{2}}^{(q)}=\frac{\pi}{2}\left(\frac{m_{2}}{m_{1}}(2q+1)-(2x+1)\right).$
(A.6)

The effect of setting the phase term equal to $\phi_{m_{2}}^{(q)}$ is that at
the corresponding $\theta_{q}$ angle, $r_{0}^{(\eta)}(\theta_{q})$ does not
lie on the limit of an interval and thus the interval is not crossed.

Table 3:  The value of $\phi^{(q)}_{m_{2}}$ for TM${}_{\\{m_{1},m_{2}\\}\eta 0}$ modes. O denotes odd integer and N any integer. $m_{2}$ $m_{1}$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
---|---|---|---|---|---|---|---|---|---
2 | O$\frac{\pi}{2}$ | - | - | - | - | - | - | - | -
3 | N$\pi$ | O$\frac{\pi}{4}$ | - | - | - | - | - | - | -
4 | O$\frac{\pi}{2}$ | O$\frac{\pi}{2}$ | O$\frac{\pi}{6}$ | - | - | - | - | - | -
5 | N$\pi$ | O$\frac{\pi}{4}$ | N$\frac{\pi}{3}$ | O$\frac{\pi}{8}$ | - | - | - | - | -
6 | O$\frac{\pi}{2}$ | N$\pi$ | O$\frac{\pi}{2}$ | O$\frac{\pi}{4}$ | O$\frac{\pi}{10}$ | - | - | - | -
7 | N$\pi$ | O$\frac{\pi}{4}$ | N$\frac{\pi}{3}$ | O$\frac{\pi}{8}$ | N$\frac{\pi}{5}$ | O$\frac{\pi}{12}$ | - | - | -
8 | O$\frac{\pi}{2}$ | O$\frac{\pi}{2}$ | O$\frac{\pi}{6}$ | O$\frac{\pi}{2}$ | O$\frac{\pi}{10}$ | O$\frac{\pi}{6}$ | O$\frac{\pi}{14}$ | - | -
9 | N$\pi$ | O$\frac{\pi}{4}$ | N$\pi$ | O$\frac{\pi}{8}$ | N$\frac{\pi}{5}$ | O$\frac{\pi}{4}$ | N$\frac{\pi}{7}$ | O$\frac{\pi}{16}$ | -
10 | O$\frac{\pi}{2}$ | N$\pi$ | O$\frac{\pi}{6}$ | O$\frac{\pi}{4}$ | O$\frac{\pi}{2}$ | N$\frac{\pi}{3}$ | O$\frac{\pi}{14}$ | O$\frac{\pi}{8}$ | O$\frac{\pi}{18}$

The work discussed here is brought together in Table 2 which, for given
$m_{1}$ and $m_{2}$ values between 1 and 10, displays the first non-forbidden
and non-spiral mode for $\phi_{m_{2}}\neq\phi^{(q)}_{m_{2}}$ (left of comma)
and $\phi_{m_{2}}=\phi^{(q)}_{m_{2}}$ (right of comma). These were numerically
calculated using Eq. A.5. The values of $\phi^{(q)}$ for given $m_{1}$ and
$m_{2}$, as calculated by Eq. A.6, are displayed in Table 3.

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