# Word Ladders: A Mobile Application for Semantic Data Collection

###### Abstract

Word Ladders is a free mobile application for Android and iOS, developed for
collecting linguistic data, specifically lists of words related to each other
through semantic relations of categorical inclusion, within the Abstraction
project (ERC-2021-STG-101039777). We hereby provide an overview of Word
Ladders, explaining its game logic, motivation and expected results and
applications to nlp tasks as well as to the investigation of cognitive
scientific open questions.

Keywords: Gamification, Linguistic Data, Ratings Collection, Word Ladders

Word Ladders: A Mobile Application for Semantic Data Collection

Marianna Marcella Bolognesi, Claudia Collacciani, Andrea Ferrari,
---
Francesca Genovese, Tommaso Lamarra, Adele Loia,
Giulia Rambelli, Andrea Amelio Ravelli, Caterina Villani
University of Bologna, Italy
Corresponding<EMAIL_ADDRESS>

Abstract content

## 1\. Introduction

The adoption of gamified methodologies for collecting human data is considered
a promising path for developing linguistic resources, such as the construction
of thesauri and ontologies. These resources, in turn, can be used in NLP tasks
Jurgens and Navigli (2014); Madge et al. (2017); Bonetti and Tonelli (2021).

The present paper briefly illustrates the rationale, functioning, technical
structure of the mobile application Word Ladders, freely available on Apple
and Android devices 111Android download: http://shorturl.at/kqAOS Apple
download: http://shorturl.at/buIS2, developed by the Abstraction research
group as part of a 5-year project funded by the European Research Concil
(ERC-2021-STG-101039777).

Figure 1: Schematic representation of the workflow of data collection and
processing. (a) Data are collected through Word Ladders, anonymized and stored
into a AWS server. (b) Stored data are accessed using Postman API and
converted to generate a graph. (c) The resulting graph is post-processed to
detect typos and remove noisy ladders. (d) The final graph is used to i.)
understand the semantic organization of users (of different sociodemographic
backgrounds) and ii.) compute the Specificity rating for the given words.

Word Ladders has been developed with a scientific and educational objective in
mind. On the research side, it is designed to collect linguistic data,
specifically lists of words related to each other through semantic relations
of categorical inclusion or meaning extension, also referred to as
hypernymy/hyponymy in linguistics, or IS-A relation. For example, given the
word apple add increasingly more generic words such as fruit and food, and
more specific ones, such as Fuji. The pool of words used in Word Ladders is a
subset of the 40.000 words presented in the English dataset of concreteness
ratings collected by Brysbaert et al. (2014) and, for which several other
psycholinguistic metrics have been collected in rating tasks through the past
decade. In particular, the words that compose the Brysbaert database have been
filtered and also manually checked, to include only words that do not present
potential violent connotations, swearwords and other words that may be
sensitive, based on the guides offered by Ipsos MORI in the “Public attitudes
towards offensive language on TV and Radio: Quick Reference Guide”
commissioned by Ofcom. The mobile application therefore takes into account
sensitivities of players with different ages, including minors. The data
collected through this application is intended for constructing an ontology
containing hierarchical semantic relations of category inclusion as well as
sociodemographic data about the users that produced such associations (please
see below for more details about the sociodemographic data collected by the
app, in anonymized form). From this resource, it will be possible to extract
metrics of word Specificity for words in English and Italian. These measures,
in turn, will be used to predict word processing and comprehension together
with other psycholinguistic variables, and to explore how lexical precision
affects readability for different types of texts, written for different types
of audience.

The scientific objectives tackled by the academic programme that supported the
development of Word Ladders are described in detail on the Abstraction
website222https://www.abstractionproject.eu/.

On the user side, the game aids speakers in doing "linguistic stretching",
prompting them to recall highly specific and highly generic words from memory,
and connecting them to basic level vocabulary, thanks to the feedback that the
application provides to users after each played match. The process of
"linguistic stretching" helps users keep their active lexicon trained. The
application was published on mobile stores in early September 2023 and can be
freely downloaded.

## 2\. Related Works

Gamification implies the integration of game design elements into non-game
contexts to stimulate users’ participation Terril (2008). This principle has
proven effective in serious games, which are interactive computer-based tools
designed to achieve educational outcomes such as learning new information or
developing particular skills Ritterfeld et al. (2009). In Natural Language
Processing (NLP), where user engagement and data collection are paramount, the
application of gamification techniques is typically implemented through GWAPs
(games with a purpose), which are a type of game in which the player’s actions
contribute to scientific or anyway real-world purposes, such as semantic
annotations, image labeling and so forth (e.g., Von Ahn and Dabbish, 2008).
This approach holds significant potential to scale up data collection that is
otherwise boring for participants, in addition to being time and energy-
consuming for researchers McPherson and Burns (2007); DeRight and Jorgensen
(2015).

In language sciences, recent research indicates promising prospects for
gamification in enhancing engagement, proficiency, and interaction in language
learning and communication processes Smith et al. (2022); Shortt et al.
(2023), second-language acquisition, grammar proficiency, and advanced
linguistic issues Benini and Thomas (2021); Dehghanzadeh et al. (2021); Shortt
et al. (2023); Bonetti and Tonelli (2021). However, the effectiveness of
gamified techniques is not consistently proven Lumsden et al. (2016). Studies
show advantages of gamified approaches with participants with certain clinical
conditions, such as ADHD (Attention Deficit Hyperactivity Disorder; Lumsden et
al., 2016; Smiderle et al., 2020), or for particular tasks such as training
healthy behaviors and environmental awareness Qian and Clark (2016); Hammady
and Arnab (2022). Yet, in other cases, the addition of game elements may not
significantly enhance performance or engagement and could introduce
uncontrolled variables Gundry and Deterding (2019). Therefore, the decision to
gamify or not gamify depends on the specific user population, task nature, and
the potential for enhanced engagement and performance. In the particular case
of linguistic data for NLP tasks, a gamified approach designed to collect
linguistically relevant data can hardly be realized within the framework of a
highly recreational game, because the information requested from players
involves the commitment to provide a substantial cognitive effort, which
typically discourages users attracted by easy and fast entertainments and
rewards. We address this challenge arguing that the benefits gained by users
may target their cognitive training, the enhancement of their linguistic
proficiency, and similar educational outcomes. We, therefore, introduce Word
Ladders, a mobile application characterized by a dual objective: one oriented
towards fulfilling scientific objectives for researchers and another focused
on users’ linguistic training, achieved alongside the app’s entertaining
appeal. In the present contribution Word Ladders is illustrated, and the
challenges involved in relation with the two objectives are discussed.

## 3\. WORD LADDERS

### 3.1. Game Logic

The game revolves around eliciting a particular type of word association,
namely the semantic relation IS-A. Users are encouraged to create word ladders
by adding steps above and below an initial prompt. The game can be played in
English or Italian.

The game’s rules, presented to players, are as follows:

  1. 1.

The primary aim is to construct a ladder by adding words on each step, but not
all words yield points, so make thoughtful choices.

  2. 2.

Commence by adding progressively more general words above the provided prompt.
For instance, if the starting word is APPLE, you can add fruit above it, then
food above fruit, and finally object above food. The key is to think, "The
word I just added describes a type of…"

  3. 3.

Ensure that each added word is more general than the previous one. In the
example, APPLE is a type of FRUIT, which is a type of FOOD, which is a type of
OBJECT.

  4. 4.

Conversely, you can be more specific and precise on the steps below the
initial word. For instance, you can add Granny Smith, a specific type of
apple, below APPLE.

  5. 5.

Accumulate more points by creating longer ladders, so challenge yourself and
your friends to build the lengthiest ladder possible.

  6. 6.

To aid word selection, consider the category or type that the placed word
belongs to.

  7. 7.

If unsure about how to play, a graphic tutorial is available to guide players
through the gameplay and clarify the rules.

The Word Ladders platform offers three distinct game modes: individual,
challenge, and group. The objective remains the same in all three modes:
building the longest ladder of valid words, starting from a given prompt. The
time allocated to complete a match (namely: a ladder) is uniformly set at 120
seconds across all modes. This time allocated has been established based on
experimental sessions followed by debriefing sessions conducted at the hosting
institution, and reported in (Genovese et al., 2024).

At the end of the time allocated for the match, the proposed ladder is
automatically rated on a scale of 1 to 5 stars. The goodness of ladders is
assessed against an external knowledge base that features the semantic
relation IS-A, namely MultiWordNet 1.5.0 Pianta et al. (2002), a multilingual
lexical database333Licensed under a Creative Commons Attribution 3.0 Unported
License aligned with Princeton WordNet 1.6, hosted by FBK Foundation in Trento
(Italy). Of the initial pool of words (Brysbaert’s dataset of concreteness
ratings) filtered for sentitive words, only words listed in MultiWordNet are
retained, with the exact number of words in English and Italian, respectively.
To evaluate ladder quality and assign user ratings and scores, two parameters
are considered: the number of entered words and the validity of those words
(namely: whether they are in MultiWordNet 1.5.0). Additionally, a graph of the
words produced by users is constructed in the backend. Therefore, if users
frequently produce a word in response to a starting prompt (e.g., Fuji’
produced below apple), it becomes a valid entry even if it is not present in
MultiWordNet (after a production frequency of N=50).

Word Ladders functions as a gamified method for collecting linguistic data
that can be used for NLP tasks, as described below. Additionally, it qualifies
as a serious game by not only incorporating gamification elements into
scientific tasks but also aiming to enhance and broaden players’ vocabulary.
In a recent paper, (Genovese et al., 2024) provide a case study in which
linguistic associations of the type IS-A are collected in different
experimental conditions: 1. using a beta version of Word Ladders in Individual
mode; 2. using a beta version of Word Ladders in Challenge (1 vs. 1) Mode, and
3. using a classic online survey form (control group). Results show that the
quality of linguistic data acquired through Word Ladders is comparable with
that obtained through traditional survey-based methods. Moreover, the study
also shows that users’ motivation to complete the task and possibly repeat it
in the future is significantly higher when the task is performed through Word
Ladders, compared to the control group that used a standard survey. This was
particularly true for players who performed the task using the Challenge mode
of the app, which shows that the interactive aspect and the 1-to-1 competition
increased users’ engagement. This suggests that Word Ladders is a scalable
approach to collect large quantities of linguistic data.

### 3.2. Motivations

#### 3.2.1. Relevance for Language Sciences and NLP

The theoretical problem that led to the implementation of Word Ladders is
hereby explained. When working on conceptual abstraction from cognitive and
computational perspectives, it is often implied that two different variables
are correlated, namely: concreteness/abstractness (banana vs belief) and
specificity/genericity (banana vs food). The assumption goes in the direction
that abstract (intangible) concepts are also more generic than concrete ones
because these two variables, which for simplicity are called Concreteness and
Specificity, are thought to be highly correlated. However, recent studies
Bolognesi et al. (2020); Bolognesi and Caselli (2023) demonstrate this is not
the case: the two variables are theoretically distinct and function as
separate significant predictors of linguistic performance in lexical and
semantic decision tasks. While the cognitive scientific community and the NLP
community are now directing their attention to these two linguistic variables,
when it comes to Specificity, the resources available to operationalize and
measure it are very limited, often metrics based on WordNet.

WordNetMiller (1995) is a lexical resource that conveniently encodes the
semantic relation IS-A (e.g., cat IS A type of mammal), allowing for the
extraction of specificity metrics (as in Bolognesi et al., 2020). However, the
vocabulary known by WordNet is much larger than the lexical knowledge of any
speaker. For instance, under the node INSECT, WordNet allows up to six
subordinate levels before reaching a leaf node (e.g., insect, beetle,
lamellicorn beetle, scarabeid, dung beetle, scarabaeus sacer). Because of the
richness of WordNet’s encyclopedic knowledge, specificity scores extracted
from this taxonomy do not reflect speakers’ lexical knowledge and allow for
only partial investigations into the cognitive mechanisms of abstraction and
generalizations in language.

For this reason, Bolognesi and Caselli (2023) and Ravelli et al. (under
review) collected specificity scores from speakers in an experimental setting,
using a Best-Worst Scaling paradigm Louviere et al. (2015) through which
participants were asked to indicate the most and least generic word among
tuples of four. For instance, given four nouns like

  1. reluctant, surprised, shy, emotional

the participant should select reluctant as the more specific and emotional as
the most generic term. The Best-Worst Scaling provides a helpful method for
constructing resources to measure Specificity. However, this paradigm is time
and energy-consuming, and does not allow exploration of speakers’ variability
in the production of categorizations, because the words are arbitrarily
selected by the experimenters.

Conversely, Word Ladders is a generative task inviting users to construct
generalizations through a gamified approach. The application collects
sociodemographic data about users (which are not connected to personal
information) so that it is then possible to analyze the type of word ladders
produced in relation to age, profession, level of education, first language,
and reading habits. For each of these variables, it is possible to develop
hypotheses about their impact on the length and the accuracy of the ladders
produced by players. For instance, it can be argued that age, level of
education, and reading habits have positive effects on the length and accuracy
of word ladders produced by players. Crucially, the sociodemographic
information collected upon registration in the app is also used within the
game to enable players to measure their performance with their peers,
filtering the leaderboard based on similar types of players.

The word ladders collected through the application will be used to construct
the following two main resources:

  * •

a dataset of specificity scores, built by extracting specificity directly from
word ladders, measuring the position of a word within a ladder divided by the
ladder length;

  * •

a taxonomy of hierarchically organized categories, in which the primary
semantic relation encoded is the semantic relation IS-A of category inclusion
(but not limited to this, given that players tend to insert other hierarchical
relations, such as part-whole).

The resources listed above enable a range of downstream tasks. Primarily, with
specificity scores available for both Italian and English words, we can delve
into the role of Specificity in language processing. This includes the
investigation of Specificity in language-related tasks such as lexical
decisions, recall, semantic decisions, production, and memorization, in which
linguistic behavior is explained by the levels of word
Specificity.Additionally, these scores are required to evaluate various text-
related tasks, such as assessing text readability and comprehensibility based
on lexical features, which may include specificity. Furthermore, integrating
specificity scores into assisted-writing software allows for suggestions of
word replacements aimed at enhancing text clarity.

Finally, due to the open, generative nature of this linguistic game, which
involves semantic categorizations on various levels of inclusiveness, it would
be possible to observe the emergence of language stereotypes. In fact, during
a public engagement event where the present application was disseminated,
gender stereotypes emerged spontaneously from players asked to construct word
ladders starting from the words doctor (some indicating man as an IS-A
relationship) and caregiver (some indicating woman as an IS-A relationship).
Given that Word Ladders collects some players’ sociodemographic information,
it will be possible to associate the emergence (and decline) of stereotypes
with different types of players.

#### 3.2.2. Educational Applications

Word Ladders has also been implemented as an educational tool in public
schools, aimed at supporting students’ language training and vocabulary-
related activities. Beyond its widespread adoption in Italian public schools,
where it has been introduced and utilized in 84 classes spanning from 5th
grade (primary school) to high school across three regions, the app is
currently used in a longitudinal case study at Imola’s IC7 primary and middle
schools, Italy. Specifically, Word Ladders is integrated into three phases of
classroom interventions—beginning, middle, and end of the school year—with two
groups of parallel classes employing distinct teaching methodologies for
vocabulary development. The length and accuracy of the word ladders produced
by pupils will be compared between the two groups, each employing a different
methodology for lexical training: one group using a traditional method and the
other an experimental approach that may give pupils an advantage in developing
fine-grained lexical knowledge.

## 4\. App Documentation and Data Collection

### 4.1. Service Implementation

Word Ladders is based on a React Native444https://reactnative.dev front-end, a
NodeJS555https://nodejs.org back-end, and a MongoDB666https://www.mongodb.com
database. The code then underwent a refactoring phase with an external gaming
studio [Anon], which also finalized the publication of Word Ladder on the
mobile stores. The application’s back-end is hosted on AWS cloud server and is
accessed through APIs, some of them used by the application’s front-end, and
others developed to grant data access to researchers (using
Postman777https://www.postman.com).

For the researchers involved in the programme, it is possible to download the
collected data through the APIs. The access is limited to an individual
account, protected by a strong password (alphanumeric). The filtering API
allows the selection of the desired MongoDB collection, the specification of a
MongoDB query, and the output file format (CSV or JSON). The raw API allows
the download of the computational tree used by the system to track and
validate the users’ ladders.

Data collected through the application are stored through the back-end
architecture into a database, which is a cloud server hosted on Amazon AWS, on
which the Abstraction research group have a dedicated account protected by
strong password. The Computing machine within the AWS cloud server is private
and dedicated to the project, on a server hosted within the European
territory. Data collected from users and stored on AWS include the following
information provided by the registered users: age, level of education,
profession, mother tongue. These are associated with the username, which is an
invented nickname. The application does not collect personal information such
as email, phone number, IP, or geolocation. Data from crash/non-fatal error
management platforms are recorded, such as device model and operating system
on which the problem occurred. However, this data is not associated with the
user. This information is used by the development teams to correct any product
problems but not to monitor the user’s activities. The library that does this
is Google’s Firebase Crashlytics. In summary: some device data is collected
but is not associated with users, the data is collected only in case of
problems and is not for monitoring purposes.

##### Game levels

The pool of words from which Word Ladders randomly draws an item for each
match is divided into 50 levels of increasing difficulty, which are based on a
series of nested rankings that take into account word frequency (the more
frequent the easier), word concreteness (the more concrete the easier), word
familiarity (the more familiar the easier) and part of speech (nouns before
verbs, before adjectives). Players are exposed to 10 words randomly selected
for each level, before they move automatically to the next level, provided
that they reached a threshold of ladder accuracy. The levels are created based
on word concreteness, frequency, and familiarity. Therefore, users will play
on a word like banana or wolf, before they get to construct a ladder on words
like entropy or insinuation. As previously mentioned, words have also been
automatically filtered and subsequently manually checked for appropriateness.
All words referring to taboo domains, violent domains or sensitive domains
have been removed, since the application is used by minors as well as by
adults. Guidelines provided by a dedicated Ethics Consultant and by the
Ethics’ committee of the hosting institution were followed (Protocol of
Approval n. 0121435, May 20th 2022).

##### Data structure

To process the data of a ladder game and evaluate the validity and the score
of the ladder, the designed structure is a graph that always has a center
starting point which is the starting word of the ladder. The graph can be
described as the union of the different graphs that start from the node that
represents the starting word of the ladder, one for representing the hypernyms
and the other one to represent the hyponyms side of the ladder. Every node can
have multiple connections, the only connection that is not allowed is between
a node of the hypernyms graph and the hyponyms one. This structure is pre-
generated starting from the knowledge base of MultiWordNet and will be
populated with the data of every play. Every arc that connects two nodes
contains information about how any time other users generated that arc or if
was present inside MultiWordNet. This graph structure is compressed into a
list of nodes and two lists of arcs, one for hypernyms and the other for
hyponyms.

##### Ladder evaluation

To evaluate the ladder built by the user, both words attested in MultiWordNet
(i.e., validated words) and novel terms (i.e., not validated words) are
considered. The validated ladder quality is proportional to the amount of play
on the same ladder. The number of plays will be used to balance the two ladder
lengths, validated and not validated (the original ladder sent by the user).
Defining: np as the number of plays of the ladder for a word w, r is a
constant value comprised between 0.2 and 0.8 referring to the ratio that will
balance the two ladder lengths, and g as the number of plays needed to obtain
a good evaluation of the ladder, then

$npl=min(max(\frac{np}{g},0.2),0.8)$ (1)

and the score s of the user ladder is computed as

$s=100*npl*\frac{min(ulv,m)}{m}+(1-npl)*\frac{min(ul,m)}{m}$ (2)

where m is the greatest ladder length ever achieved, ul is the length of the
original user ladder, and ulv is the length of the validated user ladder.
After that, the score can be increased up to 10 points as the time spent by
the user in the game decreases, the upper bound of the score is always 100. To
calculate ulv, all the words and their position inside the user ladder are
considered with the already known possible ladders, and those among them that
respect the minimum threshold will be considered valid.

### 4.2. Preliminary Data Analyses

USERS
---
Number of users | 3196
Age mean (std) | 33.51 (19.77)
PLAYED GAMES
Single player | 28223
Challenge | 5142
Team | 531
Average games per user | 10.62
Italian | 31825
English | 2115
TOTAL | 33896
Table 1: Number of users and played games in Word Ladders; data dump of
February 13th, 2024.

Good ladders
---
IT | essere vivente, animale, vertebrato, mammifero, canide, VOLPE, volpe artica
EN | living being, animal, mammal, canine, FOX, grey fox
Bad ladders
IT | formaggio, latte, mammifero, felino, gatto, VOLPE, furbo
EN | world, solar system, planet, earth, race, species, apes, monkey, animal, FOX

Table 2: Examples of data collected with Word Ladders. On the top, examples of
good ladders produced in Italian and English with the starting word fox
(volpe, in Italian); on the bottom, two ladders with the same starting words
that do not express a specificity relation.

(a) Highest education level.

(b) Profession.

Figure 2: Education (a) and profession (b) information about Word Ladders
users.

(a) New users count per month.

(b) Played games per month.

Figure 3: Cumulative counts for number of users (a) and number of played games
(b).

In the first six months since its official launch at the end of August 2023,
the application has seen continuous growth both in terms of users and number
of games played, as we can see from the Figures 3. At the time of writing
(13/02/2024), there have been over 30K games played from 3K active
players888All data is collected in compliance with GDPR regulations, as
described in the data policy (see ANON_LINK).. Figure 2 shows some demographic
information about users (educational level and profession).

Table 1 reports the rough counts of users and played games, showing that the
most played game mode is single player, which represents about 83% of all the
interactions with the app, while the other two game modes, i.e., challenge and
team, gather together less than the 17%. All plays produce a ladder of words
from each player in every game mode.

Despite the instruction provided in the app, not all the submitted word
ladders do reflect the taxonomic order the players are asked to produce (i.e.,
from the most generic on top to the most specific on bottom). Sometimes users
tend to produce semantic relations other than hypernym/hyponym, as we can see
from the bad ladders reported in Table 2, which are examples of free
association. While totally useless data are easy to filter out (e.g. non-
words, blasphemy, mispellings, etc.) with simple rule-based approaches, it is
more difficult to discriminate among word ladders that contain the desired
hierarchical semantic relation among the words from those ladders that contain
other semantic relations. Current ongoing work is aimed at disentangling the
different semantic relations introduced by users within their played matches.
The semantic annotations of word-to-word relations within each produced ladder
will also be used to compare users’ abilities to identify and discriminate
between semantic relations.

## 5\. Conclusion

The data collection through Word Ladders is currently ongoing, as previously
illustrated. The final objective is to obtain around 100 played matches on
each of the words given by the application. The present paper provided an
overview of the application, its logic, the theoretical motivations that led
to its implementation, the type of data collected, the data collection
activities, and how the data will be utilized. Moreover, we illustrated the
experimental approach tuned towards an educational objective of linguistic
training, which we argue is a promising one for the motivation and involvement
of students and pupils as well as general public in large-scale data
collections in language sciences. As illustrated, Word Ladders has
demonstrated its efficacy in raising users’ motivation and engagement in data
collection activities, in contrast to conventional experimental modalities
such as online surveys.

In the forthcoming months, our efforts will be directed towards expanding the
dissemination of Word Ladders for the collection of English word ladders
within educational contexts. This initiative aims to contribute to the
development of parallel knowledge architectures in English and Italian
languages, enriched with hierarchical semantic relations between words,
supplemented by users’ sociodemographic data. These resources will be used, as
described, to measure word specificity and then investigate how categorical
generalizations and specifications function in human cognition. Moreover, a
parallel research aims at approximating the functioning of Word Ladders using
LLMs, to explore to what extent categorical generalizations and specifications
performed by LLMs compare to those performed in human cognition.

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# Stability of the Modulator in a Plasma-Modulated Plasma Accelerator

J. J. van de Wetering<EMAIL_ADDRESS>John Adams
Institute for Accelerator Science and Department of Physics, University of
Oxford, Denys Wilkinson Building, Keble Road, Oxford OX1 3RH, United Kingdom
S. M. Hooker  John Adams Institute for Accelerator Science and Department of
Physics, University of Oxford, Denys Wilkinson Building, Keble Road, Oxford
OX1 3RH, United Kingdom  R. Walczak  John Adams Institute for Accelerator
Science and Department of Physics, University of Oxford, Denys Wilkinson
Building, Keble Road, Oxford OX1 3RH, United Kingdom  Somerville College,
Woodstock Road, Oxford OX2 6HD, United Kingdom

###### Abstract

We explore the regime of operation of the modulator stage of a recently
proposed laser-plasma accelerator scheme [Phys. Rev. Lett. 127, 184801
(2021)], dubbed the Plasma-Modulated Plasma Accelerator (P-MoPA). The P-MoPA
scheme offers a potential route to high-repetition-rate, GeV-scale plasma
accelerators driven by picosecond-duration laser pulses from, for example,
kilohertz thin-disk lasers. The first stage of the P-MoPA scheme is a plasma
modulator in which a long, high-energy ‘drive’ pulse is spectrally modulated
by co-propagating in a plasma channel with the low-amplitude plasma wave
driven by a short, low-energy ‘seed’ pulse. The spectrally modulated drive
pulse is converted to a train of short pulses, by introducing dispersion,
which can resonantly drive a large wakefield in a subsequent accelerator stage
with the same on-axis plasma density as the modulator. In this paper we derive
the 3D analytic theory for the evolution of the drive pulse in the plasma
modulator and show that the spectral modulation is independent of transverse
coordinate, which is ideal for compression into a pulse train. We then
identify a transverse mode instability (TMI), similar to the TMI observed in
optical fiber lasers, which sets limits on the energy of the drive pulse for a
given set of laser-plasma parameters. We compare this analytic theory with
particle-in-cell (PIC) simulations and find that even higher energy drive
pulses can be modulated than those demonstrated in the original proposal.

##  I Introduction

In a laser-plasma accelerator (LPA), plasma oscillations are driven by pushing
free electrons away from an ultrashort laser pulse via the ponderomotive
force. The heavier ions remain approximately stationary relative to the
electrons, thus the electron-ion charge separation collectively forms a strong
electric field which can be used to accelerate charged particles. The plasma
wave driven in this way will have a phase velocity set by the group velocity
of the laser pulse, which is well suited for the acceleration of relativistic
charged particles. The accelerating gradients achievable by LPA are set by the
wavebreaking field $E_{0}=m_{e}\omega_{p}c/e$ and can be on the order of 100
GV/m [1], three orders of magnitude larger than those possible in radio-
frequency cavities. Here the plasma frequency
$\omega_{p}=(n_{e}e^{2}/m_{e}\epsilon_{0})^{1/2}$, where $n_{e}$ is the
electron density.

Efficient excitation of the plasma wave by a single laser pulse requires that
the duration of the pulse is less than half the plasma period
$T_{p}=2\pi/\omega_{p}$. For plasma densities of interest $T_{p}$ is in the
100 fs range, and hence single-pulse LPAs first became practical with the
development of chirped pulse amplification (CPA) [2], which allowed joule-
scale pulses to be compressed to sub-picosecond durations. Ever since, most
experimental demonstrations of LPAs have used high intensity ultrashort laser
pulses from Ti:sapphire CPA laser systems. However, these systems suffer from
low ($\sim$ 0.1-10 Hz) repetition rates [3] and poor ($<0.1\%$) electrical-to-
optical energy efficiencies [4]. Despite the advantages gained by being much
more compact, the low efficiency and repetition rate of the laser drivers used
today severely limit the number of applications for which LPAs offer an
advantage over conventional, radio-frequency particle accelerators.

It is important, therefore, to consider alternative laser systems and/or
develop novel approaches for driving LPAs. Contemporary thin-disk lasers are
efficient and can already provide joule-scale pulses at kHz repetition rates
[5, 6, 7]. However, they cannot drive a LPA directly since the small bandwidth
of their gain media limits the duration of the pulses they generate to [8, 9,
10] $\tau\gtrsim$1\text{\,}\mathrm{p}\mathrm{s}$$, which is much longer than
the plasma period. We note that pulses from thin-disk lasers have been
compressed to a duration below $100\text{\,}\mathrm{f}\mathrm{s}$, following
spectral broadening via self-phase modulation in a gas [11]. However, to date
this approach has been limited to pulse energies below
$120\text{\,}\mathrm{m}\mathrm{J}$.

With the objective of applying the desirable features of thin-disk lasers to
LPAs, some of the present authors recently proposed a scheme, illustrated in
Fig. 1, for converting picosecond-duration pulses to a train of shorter pulses
that could be used to resonantly excite a plasma wave [12]. In this scheme,
which we call the Plasma-Modulated Plasma Accelerator (P-MoPA), a high-energy,
picosecond-duration ‘drive’ pulse is modulated spectrally by co-propagating it
in a plasma channel with a low amplitude ($\sim 1\%$) plasma wave driven by a
low-energy, short ‘seed’ pulse. To first order, the spectral modulation takes
the form of a set of sidebands of angular frequencies
$\omega_{m}=\omega_{L}+m\omega_{p0}$, where $\omega_{L}$ is the angular
frequency of the input drive pulse, $m=\pm 1,\pm 2,\pm 3,\ldots$ is the
sideband order, and $\omega_{p0}$ is the plasma frequency on the axis of the
plasma channel. The spectrally-modulated drive pulse is converted into a
temporal modulation by passing it through a dispersive optical system that
removes the relative spectral phase, $\psi_{m}=-|m|\pi/2$, of each sideband.
This forms a train of short pulses, spaced temporally by $T_{p0}$, which can
resonantly drive a large amplitude plasma wave in a plasma accelerator stage
with the same axial density as the modulator.

In our earlier work a one-dimensional (1D) fluid model, and 2D particle-in-
cell (PIC) simulations were used to demonstrate the operation of the plasma
modulator and accelerator stages, and to show that GeV-scale energy gains
could be obtained from existing thin-disk laser systems. In this paper we
derive a full 3D theory of seeded spectral modulation and we use this to
establish the useful operating regime for the modulator stage in the P-MoPA.
We find that the range of operation of the modulator is determined by the
onset of the transverse mode instability (TMI), similar to the TMI observed in
high power fiber laser systems [13, 14, 15, 16, 17]. This analysis is used to
establish the regime of parameter space for which the modulator can be
operated successfully. The results of the 3D analytic theory are compared with
particle-in-cell (PIC) simulations, and are found to be in good agreement. We
find that even higher energy drive pulses can be modulated than those
considered in the original proposal.

Figure 1: [Color online]. Outline of the P-MoPA scheme from the original
proposal [12]. A short, low energy seed pulse excites a small wake in the
modulator stage which spectrally modulates a long, high energy drive pulse
into interleaving redshifted (Stokes) and blueshifted (anti-Stokes) pulse
trains whilst maintaining a smooth envelope. Chromatic dispersion is then
applied to the spectrally modulated drive pulse to compress it into a
multipulse train, which can then be used to resonantly drive a wakefield in
the accelerator stage with the same density as the modulator.

##  II The Plasma Modulator

###  II.1 Seeded Spectral Modulation in Plasma Channels

Propagation of the envelope of a laser pulse in an axisymmetric plasma channel
of electron density $n_{0}(r)=n_{00}+\delta n_{0}(r)$ with a small wake
$\delta n(r,\xi;|a|^{2})$ can be approximately described by the paraxial wave
equation [18, 19, 20, 21] (see Supplemental Material [22] for its derivation)

$\displaystyle\left[\frac{i}{\omega_{L}}\frac{\partial}{\partial\tau}+\frac{c^{2}}{2\omega_{L}^{2}}\Delta_{\perp}\right]a=$
$\displaystyle\frac{\omega_{p}^{2}}{2\omega_{L}^{2}n_{0}}\left[\delta
n_{0}(r)+\delta n(r,\xi;|a|^{2})-n_{0}(r)|a|^{2}/4\right]a$ (1)

where $a(r,\theta,\xi,\tau)$ is the envelope of the pulse’s normalized vector
potential, $\omega_{L}$ is the laser frequency and the propagation is
described in co-moving coordinates $\xi=z-v_{g}t$, $\tau=t$, with
$v_{g}/c=(1-\omega_{p0}^{2}/\omega_{L}^{2})^{1/2}$ defined as the group
velocity of electromagnetic plane waves in uniform plasma of density $n_{00}$,
corresponding to the on-axis plasma channel frequency
$\omega_{p0}=\omega_{p}(r=0)$. This group velocity may differ from the group
velocity of a tightly focused laser pulse [23, 24]. Nonlinearities come from
weakly relativistic effects as well as from interaction between the pulse and
its own excited wake.

Successful seeded spectral modulation requires relativistic and self-wake
effects to be negligible, which reduces Eq. (II.1) to a linear paraxial wave
equation

$\displaystyle\left[\frac{i}{\omega_{L}}\frac{\partial}{\partial\tau}+\frac{c^{2}}{2\omega_{L}^{2}}\Delta_{\perp}\right]a$
$\displaystyle=\frac{\omega_{p}^{2}}{2\omega_{L}^{2}n_{0}}\left[\delta
n_{0}(r)+\delta n(r,\xi;|a_{s}|^{2})\right]a$ (2)

where $a_{s}$ denotes the seed pulse envelope, whose intensity we assume to be
unchanging relative to the modulating drive pulse envelope $a$. We also demand
that the channel is matched to the spot size $w_{0}$ of the seed and drive
pulses. A matched parabolic channel and its respective unperturbed Gaussian
drive pulse envelope take the form [25, 26, 24]

$\displaystyle n_{0}(r)=n_{00}+\Delta n(r/w_{0})^{2}$ $\displaystyle
a(r,\xi,\tau)=a_{0}f(\xi)\exp\left(-\frac{r^{2}}{w_{0}^{2}}-i\omega_{L}\tau\frac{2c^{2}}{\omega_{L}^{2}w_{0}^{2}}\right)$
(3)

where $\Delta n\equiv(\pi r_{e}w_{0}^{2})^{-1}$ is the channel depth
parameter, $r_{e}$ is the classical electron radius and $0\leq f(\xi)\leq 1$
is the slowly-varying longitudinal envelope of the drive pulse. Assuming the
seed wake is small relative to the channel depth parameter $|\delta
n(r,\xi;|a_{s}|^{2})|\ll\Delta n$, we can apply time-independent perturbation
theory to Eq. (2), yielding the following modulation to the total phase of the
laser pulse

$\displaystyle\Phi(\xi,\tau)$
$\displaystyle=k_{L}\xi-\frac{2c^{2}\tau}{\omega_{L}w_{0}^{2}}\left(1+\left\langle\frac{\delta
n(r,\xi;|a_{s}|^{2})}{\Delta n}\right\rangle_{\perp}\right)$ (4)

where $k_{L}=\eta\omega_{L}/c$ is the laser wavenumber with the on-axis plasma
index of refraction $\eta=(1-\omega_{p0}^{2}/\omega_{L}^{2})^{1/2}$ and
$\langle(\ldots)\rangle_{\perp}=(4/w_{0}^{2})\int_{0}^{\infty}(\ldots)\exp(-2r^{2}/w_{0}^{2})rdr$
denotes the intensity-weighted transverse average. Using this expression we
can also retrieve the shift in instantaneous frequency

$\displaystyle\frac{\Delta\omega(\xi;L_{\text{mod}})}{\omega_{L}}$
$\displaystyle=-L_{\text{mod}}\frac{2c^{2}}{\omega_{L}^{2}w_{0}^{2}}\left\langle\frac{\partial}{\partial\xi}\frac{\delta
n(r,\xi;|a_{s}|^{2})}{\Delta n}\right\rangle_{\perp}$ (5)

where $L_{\text{mod}}=v_{g}\tau$ is the modulator length. This predicts that
the spectral modulation amounts to a radial averaging of the longitudinal
gradient of the wake weighted by the transverse intensity profile of the drive
pulse. This independence of the spectral modulation of radial position is
desirable as it allows the spectral modulation to be converted into a temporal
one by applying the same chromatic dispersion across the entire cross-section
of the modulated pulse. Figure 2 shows the results of a 2D PIC simulation
which demonstrates that the spectral modulation is indeed independent of the
transverse coordinate, despite the fact that the amplitude and phase of the
plasma wave _does_ depend on the radial coordinate, as predicted by Eq. (5).
Figure 2 also shows suppression of the spectral modulation towards the tail of
the drive pulse, which is most apparent in the plot of the retrieved
instantaneous frequency. This is also expected from Eq. (5), since the large
curvature of the wavefronts of the plasma wave towards the tail of the drive
pulse leads to a reduction in the spectral modulation when the longitudinal
gradient of the wave is averaged radially. Note that in the Supplemental
Material [22] we compare the results of PIC simulations in 2D and cylindrical
geometry. All PIC simulations presented in this paper were performed with
axial plasma density
$n_{00}=$2.5\text{\times}{10}^{17}\text{\,}\mathrm{c}\mathrm{m}^{-3}$$, seed
and drive laser wavelength $\lambda_{L}=$1030\text{\,}\mathrm{n}\mathrm{m}$$,
seed pulse FWHM duration
$\tau_{\text{seed}}=$40\text{\,}\mathrm{f}\mathrm{s}$$ and modulator length
$L_{\text{mod}}=$110\text{\,}\mathrm{m}\mathrm{m}$$. A complete list of
simulation parameters is included in the Supplemental Material [22].

Figure 2: [Color online]. 2D PIC simulation of the modulator stage in a
matched parabolic plasma channel of matched spot size
$w_{0}=$30\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$$ with
$W_{\text{seed}}=$50\text{\,}\mathrm{m}\mathrm{J}$$ and
$W_{\text{drive}}=$0.6\text{\,}\mathrm{J}$$,
$\tau_{\text{drive}}=$1\text{\,}\mathrm{p}\mathrm{s}$$. The top panel shows
the on-axis longitudinal intensity profiles $|a|^{2}$ for the seed and drive
pulses. The middle panel plots the on-axis instantaneous frequency calculated
by a Hilbert transform. The bottom panel displays the full 2D distributions of
the relative amplitude $\delta n/n_{00}$ of the plasma wave and the relative
frequency modulation $\Delta\omega/\omega_{L}$.

###  II.2 Channel Suppression of Spectral Modulation

As observed in Fig. 2, Eq. (5) implies that the spectral modulation of the
drive pulse is limited by wave-front curvature of the plasma wave. For low
amplitude wakes this curvature is dominated by the transverse profile of the
plasma channel. For square channels the wake has flat phase fronts over most
of the transverse profile of the lowest-order mode of the channel. However,
for channels with a curved transverse profile, such as parabolic channels, the
wave-fronts of the plasma wave are curved, which can strongly suppress the
spectral modulation.

Figure 3 shows the results of 2D PIC simulations that compare the performance
of a modulator with square and parabolic plasma channels. It can be seen that
for the square channel the wave-fronts of the plasma wave are flat across most
of the channel width, and as a consequence the pulse train generated by
removal of the sideband spectral phase exhibits strong temporal modulation
over the entire duration of the pulse train. In contrast, the wave-fronts of
the wake driven in a parabolic channel of
$30\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ matched spot size are
strongly curved, and this curvature increases towards the tail of the drive
pulse. As a consequence, the generated pulse train does not exhibit complete
intensity modulation near its tail, which would reduce the amplitude of the
plasma wave it could drive in the accelerator stage. As shown in Fig. 3,
increasing the matched spot size of the parabolic channel to
$50\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ reduces the wake curvature,
which leads to improved modulation of the generated pulse train. We note that
the deleterious effects of wave-front curvature would be even more pronounced
in 3D geometry (see Supplemental Material [22]).

Figure 3: [Color online]. Comparison of the performance of modulators with
plasma channels of different transverse profiles each with wall thicknesses of
$20\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ (see Supplemental Material
for their parameterizations [22]). The left panels show the transverse
electron density (blue) and normalized guided intensity (black) profiles of
the channels for: top, a square channel of diameter
$30\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$; middle, a parabolic
channel of matched spot size
$30\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$; bottom, a parabolic
channel of matched spot size
$50\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ (with $(50/30)^{2}\times$
more seed and drive pulse energy to account for the larger spot size). The
middle panels show the relative wake amplitudes $\delta n/n_{00}$ at the end
of the modulator, calculated by 2D PIC simulations. The right panels show the
on-axis pulse envelopes $|a|$ at the end of the modulator before (dashed blue)
and after (solid orange) the expected [27] sideband spectral phase $\psi_{m}$
was removed (see Supplemental Material [22]).

##  III Stability of the Plasma Modulator

It is important to understand the extent to which instabilities will arise in
the modulator, and the range of laser and plasma parameters for which any
deleterious effects arising from them can be avoided. Since we do not want to
waste any of the drive pulse energy within the modulator stage, we would like
its envelope to remain smooth as it propagates. Although the specific set of
parameters used in the original P-MoPA proposal [12] were shown to work in
simulation, for the scheme to be practical we want a large, well-defined
parameter space where it is stable.

(a)  $\tau_{\text{drive}}=$1\text{\,}\mathrm{p}\mathrm{s}$$,
$W_{\text{drive}}=$0.6\text{\,}\mathrm{J}$$ (b)
$\tau_{\text{drive}}=$1\text{\,}\mathrm{p}\mathrm{s}$$,
$W_{\text{drive}}=$1.2\text{\,}\mathrm{J}$$ (c)
$\tau_{\text{drive}}=$4\text{\,}\mathrm{p}\mathrm{s}$$,
$W_{\text{drive}}=$2.4\text{\,}\mathrm{J}$$

Figure 4: [Color online]. 2D PIC simulations of the intensity profiles,
$|a|^{2}$, of drive pulses at the exit of the modulator with a channel of
square cross-section with
$w_{0}=$30\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$$ and
$W_{\text{seed}}=$50\text{\,}\mathrm{m}\mathrm{J}$$, for various energies and
durations of the drive pulse. The top row displays the intensity profiles
decomposed into its redshifted Stokes $(\omega-\omega_{L})<-\omega_{p0}/2$,
central $|\omega-\omega_{L}|<\omega_{p0}/2$, and blueshifted anti-Stokes
$(\omega-\omega_{L})>\omega_{p0}/2$ components; the bottom panel displays the
full intensity profile of the drive pulse.

Increasing the drive pulse energy $W_{\text{drive}}$ eventually disrupts the
plasma modulator with a TMI via a self-modulation mechanism. This instability
drives time-varying wakes which excite higher order transverse channel modes.
The results of PIC simulations demonstrating this phenomenon are given in Fig.
4. It can be seen that the Stokes sidebands become more defocused towards the
trailing end of the pulse, thereby exciting higher order transverse modes,
whereas the anti-Stokes-shifted light remains relatively well focused. This
asymmetry between the Stokes and anti-Stokes light leads to the formation of
low-contrast pulse trains in the modulator, with the Stokes-shifted radiation
forming a train off-axis, and the anti-Stokes light forming a train on-axis.
This effect becomes worse with increasing drive pulse energy, and can be seen
to be especially bad for the $2.4\text{\,}\mathrm{J}$ pulse, which has
undergone severe transverse break-up and has been strongly redshifted. This
difference in behaviour between the Stokes and anti-Stokes sidebands has
previously been observed as the result of relativistic effects [28]. As shown
in the Supplemental material [22], a similar effect is predicted in the non-
relativistic regime when non-paraxial effects are accounted for.

###  III.1 Envelope Self-Modulation and Raman Forward Scattering

We now consider the stability of picosecond-scale pulses propagating in long
_unperturbed_ plasma channels, i.e. in the absence of a wake driven by a seed
pulse. As discussed by Mori [29], as long as
$1/k_{L}w_{0}\gg\omega_{p}^{2}/\omega_{L}^{2}$ is satisfied, the following
parameter determines whether a laser pulse will be dominated by Raman forward
scattering (RFS) or envelope self-modulation (SM) instabilities:

$\displaystyle\Gamma\equiv\frac{P}{1\text{ TW}}\cdot\frac{\tau_{L}}{1\text{
ps}}\cdot\left(\frac{n_{e}}{1\times 10^{19}\text{
cm}^{-3}}\right)^{5/2}\cdot\left(\frac{\lambda_{L}}{1\text{
$\mu$m}}\right)^{4}$ (6)

where $P$ is the peak laser power and $\tau_{L}$ is the laser duration. When
$\Gamma\geq 3$, RFS dominates, whereas SM dominates when $\Gamma\leq 0.4$.
Substituting laser-plasma parameters used by Jakobsson et al [12], we find
$\Gamma=6.3\times 10^{-5}$ and $1/k_{L}w_{0}=55\times
10^{-4}\gg\omega_{p}^{2}/\omega_{L}^{2}=2.4\times 10^{-4}$, meaning that we
are well in the SM-dominated regime. The maximum growth rate and $e$-folding
number of envelope SM in a uniform plasma is given by [30, 31]

$\displaystyle\gamma_{\text{SM}}=\frac{1}{8}a_{0}^{2}\frac{\omega_{p}^{2}}{\omega_{L}}(1+a_{0}^{2})^{-3/2},\,\,\,\,N_{e,\text{SM}}=\gamma_{\text{SM}}T_{\text{int}}$
(7)

where $T_{\text{int}}=L_{\text{mod}}/c$ is the interaction time.The parameters
used by Jakobsson et al [12] yield $\sim 1.3$ $e$-foldings, and hence SM could
become problematic, especially if we try to include more energy in the drive
pulse.

###  III.2 Transverse Mode Stability Condition

For a pulse propagating in a waveguide, the growth of SM is complicated by
oscillations in the spot size of the mode, which arise from excitation of more
than one waveguide mode. This becomes relevant when the SM growth rate is
slower than the spot size oscillation frequency [32]
$\gamma_{\text{SM}}<\omega_{w}=4c^{2}/\omega_{L}w_{0}^{2}$. In this section we
consider the effects of the plasma channel on self-modulation.

Consider coupling a slightly unmatched drive pulse into a parabolic plasma
channel so it undergoes small spot size oscillations, and assume that no
centroid oscillations are present, so that only radial Laguerre-Gaussian modes
$\text{LG}_{p0}(r)$ are excited (see Supplemental Material [22]). Neglecting
relativistic effects, applying time-dependent perturbation theory to Eq.
(II.1), the coefficients $\alpha_{p}(\xi,\tau)$ of each radial Laguerre-
Gaussian mode $p$ at longitudinal coordinate $\xi$ are found to evolve
according to:

$\displaystyle\frac{i}{\omega_{c}}\frac{\partial\alpha_{p}(\xi,\tau)}{\partial\tau}=$
$\displaystyle\sum_{n}\alpha_{n}(\xi,\tau)\Big{\langle}\text{LG}_{p0}\Big{|}\frac{\delta
n+\delta n_{\text{NL}}}{\Delta
n}\Big{|}\text{LG}_{n0}\Big{\rangle}e^{i(p-n)\omega_{w}\tau}\,,$
$\displaystyle
a(r,\xi,\tau)=e^{-i\omega_{c}\tau}\sum_{p}\alpha_{p}(\xi,\tau)\text{LG}_{p0}(r)e^{-ip\omega_{w}\tau}$
(8)

where $\delta n$ is the fixed seed wake, $\delta n_{\text{NL}}$ is the self-
wake of the drive pulse, and $\omega_{w}$ and $\omega_{c}=\omega_{w}/2$ are
the spot size and centroid oscillation frequencies respectively [32]. Coupling
the drive pulse into a slightly unmatched channel corresponds to following set
of initial conditions

$\displaystyle\alpha_{0}=a_{0}f(\xi),\quad\alpha_{1}=\epsilon_{w}a_{0}f(\xi),\quad\alpha_{p\neq
0,1}\approx 0$ (9)

where $\epsilon_{w}=-\delta w/w_{0}\ll 1$ is the channel spot size mismatch
parameter. In order to solve Eq. (III.2), the self-wake $\delta n_{\text{NL}}$
must be known. We estimate the self-wake as follows. We first assume that the
self-wake can be neglected, and calculate the intensity modulation of the
drive pulse caused by the seed wake only. We then use this intensity
modulation to calculate the self-wake it would excite. This estimate of the
self-wake can then be used to define the plasma modulator stability condition,
which sets bounds on the laser-plasma parameters to prevent nonlinear self-
modulation from exciting transverse mode transitions.

We will work in the shallow channel limit $\Delta n\ll n_{00}$, which allows
us to neglect the effects of the channel on wake structure [33]. This gives a
seed wake of the form [18]

$\displaystyle\delta n(r,\xi)=\delta
n_{s}\cos(k_{p0}\xi)\text{LG}_{00}^{2}(r)$ (10)

where $k_{p0}=\omega_{p0}/v_{g}$ and $\delta n_{s}$ denotes the on-axis seed
wake amplitude. As the drive pulse should remain primarily in the fundamental
mode, we can approximate Eq. (III.2) as a two-level system comprising the
$\text{LG}_{00}$ and $\text{LG}_{10}$ modes with transitions between them
driven by the seed wake

$\displaystyle i\frac{\partial\alpha_{0}}{\partial\tau}$
$\displaystyle=2\Omega_{s}\cos(k_{p0}\xi)\left(\alpha_{0}+\tfrac{1}{2}\alpha_{1}e^{-i\omega_{w}\tau}\right)\,,$
$\displaystyle i\frac{\partial\alpha_{1}}{\partial\tau}$
$\displaystyle=2\Omega_{s}\cos(k_{p0}\xi)\left(\tfrac{1}{2}\alpha_{0}e^{i\omega_{w}\tau}+\tfrac{1}{2}\alpha_{1}\right)$
(11)

where $\Omega_{s}=(\omega_{p0}^{2}/8\omega_{L})(\delta n_{s}/n_{00})$ is the
rate of spectral modulation parameter. Note that since we are using the
paraxial description, we are implicitly assuming symmetry between the Stokes
and anti-Stokes sidebands (see Supplemental Material [22] for the non-paraxial
description). We can already see from these expressions that the first radial
mode spectrally modulates half as fast as the fundamental, as it is more
sensitive to the radial drop-off of the seed wake amplitude
$\sim\text{LG}_{00}^{2}(r)$. This asymmetry, coupled to spot size
oscillations, is one of two effects contributing to plasma-resonant
modulations of the drive pulse intensity which excite a self-wake. Since the
spot size oscillation frequency is necessarily much higher than the spectral
modulation rate, i.e. $\omega_{w}\gg\Omega_{s}$, as a consequence of the seed
wake being small relative to the channel depth parameter, we can integrate Eq.
(III.2) by assuming most of the light remains in the fundamental mode to find
a first order solution
$\alpha_{0}=\alpha_{0}^{(0)}(\xi,\tau)+\alpha_{0}^{(1)}(\xi,\tau),\,\alpha_{1}=\alpha_{1}^{(1)}(\xi,\tau)$.
This yields the following intensity modulation

$\displaystyle\frac{|a(r,\xi,\tau)|^{2}}{|a_{0}|^{2}f^{2}(\xi)}=\left\\{1-2\left(\epsilon_{w}+\frac{\Omega_{s}\cos(k_{p0}\xi)}{\omega_{w}}\right)\left(\cos[\Omega_{s}\tau\cos(k_{p0}\xi)]-1\right)\right\\}$
$\displaystyle\text{LG}_{00}^{2}(r)$
$\displaystyle+\left\\{2\epsilon_{w}\cos\left[\Omega_{s}\tau\cos(k_{p0}\xi)-\omega_{w}\tau\right]+\frac{2\Omega_{s}\cos(k_{p0}\xi)}{\omega_{w}}\left(\cos\left[\Omega_{s}\tau\cos(k_{p0}\xi)-\omega_{w}\tau\right]-1\right)\right\\}$
$\displaystyle\text{LG}_{00}(r)\text{LG}_{10}(r)\,.$ (12)

We see from Eq. (III.2) that the drive pulse is modulated radially,
longitudinally, and temporally. This modulation can be physically understood
by splitting it into three effects which can be isolated by setting certain
terms to zero. The first is a longitudinally uniform spot size oscillation of
amplitude $\delta w$, which can be recovered by setting $\Omega_{s}=0$. The
second comes from coupling between the $\delta w$ spot size oscillations and
the seed wake spectral modulation due to the effect mentioned earlier where
higher order radial modes spectrally modulate slower than the fundamental mode
in the $\sim\text{LG}_{00}^{2}(r)$ seed wake. The third depends purely on the
seed wake, which can be seen when setting $\epsilon_{w}=0$. The seed wake
introduces local variations in the matched spot size as it perturbs the
channel plasma density, varying the spot size longitudinally in $\xi$ and
temporally in $\tau$.

To estimate the self-wake $\delta n_{\text{NL}}$ that would be excited by this
intensity modulation, we are only interested in keeping terms resonant with
the plasma $\sim\cos(k_{p0}\xi+\phi)$ which excite the largest amplitude self-
wake. We also only consider propagation times up to
$\Omega_{s}\tau_{\text{mod}}\sim 1/2$, as this provides sufficient spectral
modulation for compression into a pulse train which roughly coincides with the
minimum modulation required to reach the accelerator stage wake amplitude
plateau discussed by Jakobsson et al [12]. With both of these considerations
in mind, we can Taylor expand the
$\cos\left[\Omega_{s}\tau\cos(k_{p0}\xi)-\omega_{w}\tau\right]$ terms in Eq.
(III.2) to first order in $\Omega_{s}\tau$. This gives the plasma-resonant
part of the intensity modulation and the approximate self-wake it would excite
[34]

$\displaystyle|a(r,\xi,\tau)|_{\text{res}}^{2}=|a_{0}|^{2}f^{2}(\xi)$
$\displaystyle\left(2\epsilon_{w}\Omega_{s}\tau\sin(\omega_{w}\tau)+\frac{2\Omega_{s}\left[\cos(\omega_{w}\tau)-1\right]}{\omega_{w}}\right)\cos(k_{p0}\xi)\text{LG}_{00}(r)\text{LG}_{10}(r)\,,$
$\displaystyle\frac{\delta
n_{\text{NL}}(r,\xi,\tau)}{n_{00}}=\frac{e^{2}}{8\pi^{2}m_{e}^{2}\epsilon_{0}c^{5}}\frac{\omega_{p0}W_{\text{drive}}(\xi)\lambda_{L}^{2}}{\pi
w_{0}^{2}}$
$\displaystyle\left(2\epsilon_{w}\Omega_{s}\tau\sin(\omega_{w}\tau)+\frac{2\Omega_{s}\left[\cos(\omega_{w}\tau)-1\right]}{\omega_{w}}\right)\sin(k_{p0}\xi)\text{LG}_{00}(r)\text{LG}_{10}(r)$
(13)

where $W_{\text{drive}}(\xi)$ indicates the total energy of the drive pulse
contained between its head and coordinate $\xi$. To prevent self-modulation
driving transverse mode transitions (and to make this calculation self-
consistent), we require that the self-wake effect on the
$\text{LG}_{00}\rightarrow\text{LG}_{10}$ transition must be negligible
throughout the full propagation in the modulator, resulting in the constraint

$\displaystyle\int_{0}^{\tau_{\text{mod}}}\\!\\!\\!\frac{d\tau}{\tau_{\text{mod}}}\big{\langle}\text{LG}_{10}\big{|}\delta
n_{\text{NL}}e^{i\omega_{w}\tau}\big{|}\text{LG}_{00}\big{\rangle}\ll\big{\langle}\text{LG}_{10}\big{|}\delta
n\big{|}\text{LG}_{00}\big{\rangle}\,.$ (14)

Substituting $\Omega_{s}\tau_{\text{mod}}=1/2$ and Eq. (III.2) into this
constraint yields the plasma modulator transverse mode stability condition

$\displaystyle\left|\frac{\delta w/w_{0}}{\delta
n_{s}/n_{00}}+\frac{k_{p0}^{2}w_{0}^{2}}{8}\right|\frac{\omega_{p0}W_{\text{drive}}\lambda_{L}^{2}}{\pi
w_{0}^{2}}\ll P_{\text{mod}}\,,$ $\displaystyle
P_{\text{mod}}=\frac{32\pi^{2}m_{e}^{2}\epsilon_{0}c^{5}}{e^{2}}\approx$220\text{\,}\mathrm{G}\mathrm{W}$\,.$
(15)

This sets a limit on the total energy of the drive pulse $W_{\text{drive}}$
for a given plasma density, spot size, laser wavelength, seed wake and channel
spot size mismatch. For the laser-plasma parameters used by Jakobsson et al
[12], taking $\delta n_{s}/n_{00}\sim 2.5\%$ and $\delta w/w_{0}\sim 12\%$
from the PIC simulations therein, the requirement of Eq. (III.2) becomes
$$37\text{\,}\mathrm{G}\mathrm{W}$\ll$220\text{\,}\mathrm{G}\mathrm{W}$$,
which is satisfied. Hence, in this regime we do not expect the self-modulation
effects to be debilitating to the plasma modulator. In PIC simulations, we
have found that even letting the LHS of Eq. (III.2) go up to
$\sim\,$$70\text{\,}\mathrm{G}\mathrm{W}$ remains stable enough for
compression into a pulse train despite the excitation of the first radial mode
being non-negligible. Figure 5 shows the calculated intensity profiles,
$|a|^{2}$, of drive pulses at the exit of the modulator before and after
compression into a pulse train (by removing the expected [27] sideband
spectral phase $\psi_{m}$ of a pulse without TMI (see Supplemental Material
[22])) for drive pulses of energy $W_{\text{drive}}=$1.2\text{\,}\mathrm{J}$$
and various pulse durations. It can be seen that, despite the pulse duration
varying by a factor of 16, they each undergo the same amount of spot size
oscillation driven transitions to higher order transverse modes. This is in
agreement with Eq. (III.2), which is independent of the drive pulse duration.
As a consequence, for drive pulses of duration 1, 2 and
$4\text{\,}\mathrm{p}\mathrm{s}$, the transverse structure of the pulse at the
end of the modulator is identical. For drive pulses of
$250\text{\,}\mathrm{f}\mathrm{s}$ duration, the structure is also similar,
but in this case the pulse duration is only approximately two plasma periods
long, and hence the assumption of small bandwidth breaks down. We note also
that Fig. 5 shows that in each case the spectrally-modulated drive pulse can
still be compressed into a well-defined pulse train suitable for the
accelerator stage, despite the transverse structure that it has developed.

(a)  $\tau_{\text{drive}}=$250\text{\,}\mathrm{f}\mathrm{s}$$ (b)
$\tau_{\text{drive}}=$1\text{\,}\mathrm{p}\mathrm{s}$$ (c)
$\tau_{\text{drive}}=$2\text{\,}\mathrm{p}\mathrm{s}$$ (d)
$\tau_{\text{drive}}=$4\text{\,}\mathrm{p}\mathrm{s}$$

Figure 5: Calculated intensity profiles, $|a|^{2}$, of drive pulses at the
exit of the modulator before (top) and after (bottom) compression into a pulse
train for drive pulses of energy $W_{\text{drive}}=$1.2\text{\,}\mathrm{J}$$
and FWHM duration: (a) $0.25\text{\,}\mathrm{p}\mathrm{s}$, (b)
$1\text{\,}\mathrm{p}\mathrm{s}$, (c) $2\text{\,}\mathrm{p}\mathrm{s}$ and (d)
$4\text{\,}\mathrm{p}\mathrm{s}$. For these 2D PIC simulations the modulator
was taken to have a square cross-section with
$w_{0}=$30\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$$, and the seed pulse
to have an energy of $W_{\text{seed}}=$50\text{\,}\mathrm{m}\mathrm{J}$$.

For most practical applications, the spot size oscillations will be determined
by the mismatch between the transverse amplitude profile of the lowest-order
channel mode, and that of the drive pulse at the channel entrance. However,
even in the limit of a perfectly matched channel, small spot size oscillations
can arise from other sources, such as ponderomotive and relativistic self-
focusing [35, 36]. In addition, the $k_{p0}^{2}w_{0}^{2}/8$ term in Eq.
(III.2), which comes from the seed wake forming a plasma-resonant variation in
matched spot size, ensures that there will always be an upper limit on the
drive pulse energy.

There are two ways that excitation of higher order transverse modes can be
mitigated, other than ensuring that Eq. (III.2) is satisfied. However, each of
these comes at a cost. First, the treatment above assumed propagation in a
shallow channel. For deeper channels, i.e. $\Delta n\sim n_{00}$, the self-
wake will be partially suppressed by the off-resonant plasma wave, and the
difference in spectral modulation rate of the higher-order mode and the
fundamental will be decreased due to the radial component of the seed wake.
This allows for more energy to propagate in the modulator without transitions
to higher order modes. However, channels of this form also suppress the
spectral modulation towards the tail of the pulse, as shown in Fig. 3, which
is detrimental to pulse compression. Another option would be to use a leaky
channel that leaks higher order modes faster than the fundamental mode [37,
38]. However, this approach would have reduced efficiency, since drive pulse
energy transferred to higher order modes would be lost.

##  IV Conclusion

We have derived a full 3D analytic theory of seeded spectral modulation, and
have used this to establish the useful operating regime for the modulator
stage in the P-MoPA. This model is found to be in very good agreement with
those obtained from 2D PIC simulations of the modulator.

The analytic theory leads to several important conclusions. First, the
spectral modulation of the drive pulse is independent of radial distance from
the axis of the modulator channel. This ensures that, after leaving the
modulator, the entire spectrally-modulated pulse can be compressed by a simple
optical system that removes the spectral phase accumulated in the modulator.
Second, curvature of the seed-pulse-driven plasma waves is shown to reduce the
degree of spectral modulation, and hence the modulation of the pulse train
that is generated after compression. This finding establishes limits on the
shape of the channel used in a P-MoPA modulator.

We also explored limits to the operating parameters of a seeded modulator set
by the self-modulation of the drive pulse and excitation of higher-order
transverse channel modes. We found that the operation of the modulator is
limited by the onset of the transverse mode instability (TMI), similar to the
TMI observed in high power fiber laser systems. An analysis of the excitation
of higher-order modes allowed the identification of a condition on the energy
of the drive pulse, the relative amplitude of oscillations in its spot size,
and the relative amplitude of the seed-pulse-driven wake, that must be
satisfied for stable operation.

Finally we emphasize that the results presented here show that the modulator
in a P-MoPA can exhibit stable operation over a much broader range of
operating parameters than considered in the original proposal [12]. This
includes operation at higher drive pulse energies, which bodes well for the
development of high-repetition-rate, GeV-scale P-MoPAs.

###### Acknowledgements.

This work was supported by the UK Engineering and Physical Sciences Research
Council (EPSRC) (Grant No. EP/V006797/1), the UK Science and Technologies
Facilities Council (Grant No. ST/V001655/1]), InnovateUK (Grant No. 10059294),
UKRI funding (ARCHER2 Pioneer Projects) and the Ken and Veronica Tregidgo
Scholarship in Atomic and Laser Physics. This publication arises from research
funded by the John Fell Oxford University Press Research Fund. This work used
the ARCHER2 UK National Supercomputing Service (ARCHER2 PR17125)
https://www.archer2.ac.uk. This research used the open-source particle-in-cell
code WarpX [39] https://github.com/ECP-WarpX/WarpX, primarily funded by the US
DOE Exascale Computing Project. Primary WarpX contributors are with LBNL,
LLNL, CEA-LIDYL, SLAC, DESY, CERN, and Modern Electron. We acknowledge all
WarpX contributors. This research was funded in whole, or in part, by EPSRC
and STFC, which are Plan S funders. For the purpose of Open Access, the author
has applied a CC BY public copyright licence to any Author Accepted Manuscript
version arising from this submission. The input decks used for the PIC
simulations presented in this paper are available at
http://doi.org/10.5281/zenodo.7734261

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