11institutetext: Shenzhen International Graduate School, Tsinghua University,
Shenzhen, China 11email: {shuib22<EMAIL_ADDRESS>
11email: {yang.yujiu<EMAIL_ADDRESS>
# ContextVis: Envision Contextual Learning and Interaction with Generative
Models

Bo Shui  0009-0001-0981-9608 Chufan Shi  0009-0005-7889-5187
Yujiu Yang  0000-0002-6427-1024 Xiaomei Nie Corresponding Author.
0000-0002-8901-8981

###### Abstract

ContextVis introduces a workflow by integrating generative models to create
contextual learning materials. It aims to boost knowledge acquisition through
the creation of resources with contextual cues. A case study on vocabulary
learning demonstrates the effectiveness of generative models in developing
educational resources that enrich language understanding and aid memory
retention. The system combines an easy-to-use Dashboard for educators with an
interactive Playground for learners, establishing a unified platform for
content creation and interaction. Future work may expand to include a wider
range of generative models, media formats, and customization features for
educators.

###### Keywords:

Contextual Learning Interactive Learning Tools User Interaction Generative
Models.

##  1 Introduction

Recently, generative models have made significant strides in the field of
artificial intelligence, demonstrating remarkable capabilities in creating
realistic and coherent content across various media formats. These models,
such as language models and image generation models, have been widely adopted
in the creation of creative resources, offering unique opportunities to enrich
materials with contextual information. The deployment of generative models in
educational resource development holds considerable promise for augmenting the
learning process. It provides learners with an immersive framework that aids
in knowledge comprehension and retention.

In this regard, we present the ContextVis system, a workflow that harnesses
the capabilities of generative models to craft tailored learning materials and
experiences. Through a case study on vocabulary learning, we demonstrate the
utility of generative models in generating educational content that supports
language development and enhances learning outcomes. The system represents a
promising approach to elevate contextual and exploratory learning using
generative models and interactive visualizations.

##  2 Related Work

###  2.1 Leveled and Themed Language Learning

Leveled reading, a pedagogical approach characterized by evaluating children’s
reading skills using specialized instruments and subsequently supplying
reading materials commensurate with their ability, has garnered international
acceptance and has continued to evolve in recent years [22]. Optimal selection
of reading content that presents an appropriate challenge is crucial to
fostering advancement in reading proficiency [31].

Empirical evidence suggests that the use of leveled reading materials in
English for primary school students learning English as a second language can
substantially improve their linguistic capabilities, comprehension skills, and
cultural awareness [33]. Renowned publications such as Reading A-Z [14],
Geronimo Stilton [28], and Oxford Reading Tree [24] adeptly integrate
repetitive sentence structures and captivating illustrations that resonate
with the developmental stages of a child’s cognition [15, 20].

Theme and context play a vital role in language learning. The idea of
contextualized learning emphasizes that knowledge is best acquired within the
context in which it is used and applied [4]. Learners apply their knowledge in
various themes and contexts, deconstructing complex concepts, identifying
relationships between elements, and adeptly engaging in problem analysis and
resolution [12, 27]. For effective learning, the language input should be
comprehensible, engaging, relevant, and abundant, focusing more on meaning
rather than on grammatical rules [13].

The recent English Curriculum Standards for Compulsory Education issued by the
Ministry of Education of China in 2022 advocate for English language teaching
through leveled and contextual exploration, offering level guidelines and
theme examples [18]. Our work utilizes the standards as a foundation for
leveled vocabulary and theme references to develop a system that supports both
teachers and students in the language instruction.

###  2.2 Generative Models for Education

Recent advancements in generative artificial intelligence models have been
propelled by the growth of available data and model scaling. These models now
exhibit capability to generate high-quality content across multi-modalities
and demonstrate improved understanding of user instructions, enabling content
creation that aligns with human intentions [5]. Notably, empirical evidence
has suggested that large generative models are now able to synthesize emergent
content that is not only contextually pertinent but also coherent, contingent
upon specific prompts [32].

The integration of generative models in the creation of educational resources
offers a unique opportunity to enrich learning materials with contextual cues,
thereby enhancing the learning experience. They have so far been used in
various educational applications, such as storytelling co-creation with
specific knowledge [34], generating reading comprehension quizzes [8],
authoring data-driven articles [30], generating coherent comics [11], and
creating stories [1, 25]. Moreover, the collaboration with human teachers in
the creation of educational resources has been shown to be effective in
enhancing the quality and alleviating teachers’ workload [10].

These studies demonstrate the potential of generative models in creating
educational resources that are tailored to specific learning objectives and
contexts, as well as integrating generation with human teachers to enhance the
quality of the contents. Our work focuses on maintaining the consistency of
the learning context in generated story scripts and images for the learning
vocabulary.

###  2.3 Interaction Design for Human-AI Collaboration

Human-AI collaboration is the study of how humans and AI agents work together
to accomplish a shared goal [29]. In order to understand and improve this
emergent modality of interaction, HCI researchers have proposed a variety of
principles, frameworks, and guidelines in recent years [2, 6, 7, 16]. Notably,
Amershi et al. have proposed 18 guidelines for Human-AI Interaction,
categorized within the sequential phases of initial stage, active interaction,
error response, and post-interaction reflection [2]. Cimolino et al. proposed
a framework as an analysis tool with four dimensions of the role, supervision,
influence and mediation of AI [7].

The domain of AI-assisted design, particularly in graphical and interaction
design, has witnessed a significant increase in interest and application since
2016 [26].

Shi et al. proposed that AI can assist designers in Discovering, Visualizing,
Creating and Testing, and designers can reciprocally augmenting AI by Training
and Regulating [26]. In the teaching process, educators harness AI assistance
in lesson planning, thereby liberating themselves from repetitive pedagogical
tasks, which in turn fosters innovation in educational content and
methodologies [35]. The guidelines for Human-AI interaction laid out in the
literature are highly pertinent to our work, as we focus on leveraging AI in
the creation, visualization, and validation phases of educational content
generation.

##  3 The ContextVis System

We present the ContextVis system, which leverages generative AI to develop
contextual learning materials and experiences. As illustrated in Fig. 1, the
workflow comprises three essential components: the back-end platform for
processing user input and generating contextual data using Large Language
Models (LLMs), as well as generating multi-modal assets through generative
models; the database, where the metadata and generated resources are stored;
and the front-end platform, featuring a Dashboard for educators and a
Playground for learners. It is worth noting that our vision for this system
encompasses the ability to receive inputs and generate outputs in multiple
media formats, as the models employed can be substituted to accommodate
diverse tasks.

Figure 1: The general workflow of ContextVis system.

##  4 Case Study: Contextual Vocabulary Learning

Acquiring English vocabulary can be challenging due to its inherent complexity
characterized by contextual nuances, semantic diversity, and an extensive
lexicon. Understanding and memorizing grammar rules alongside vocabulary
demands a coherent and persistent learning strategy, often requiring prolonged
exposure to a variety of linguistic stimuli. This extended engagement with
both language inputs and outputs is crucial for reinforcing linguistic
knowledge.

The process of vocabulary acquisition can be effectively integrated into the
proposed workflow, where the integration of contextual factors plays a crucial
role in fostering language development. To this end, we conducted a case study
for vocabulary learning which exemplifies the significance of the ContextVis
system, shown in Fig. 2. In the workflow, with selected vocabulary in a unit
and an optional theme input in the Dashboard, the generative models in the
back-end automatically generates a contextually coherent a story script and
stickers for each word as output. The generated data and assets are stored in
the database on the server for learners to access and explore in the
Playground.

Figure 2: A case workflow of ContextVis system.

###  4.1 Dashboard for Educators

To facilitate seamless integration of vocabulary with generative contents, we
developed a Dashboard for educators, shown in Fig. 3, with multiple panels
allowing educators to input and select the vocabulary, then generate, preview
and refine the generated educational resources. By assigning a theme as
contextual cue, the generated assets of scripts and stickers are stored in the
server and can be exported as teaching materials for classroom use. The
Dashboard serves as a pivotal interface within the ContextVis system, offering
user-friendly tools and features that empower educators to effectively create
tailored educational content.

Figure 3: An overview of the Dashboard.

####  4.1.1 Story Script

Stories serve as powerful media for contextualizing vocabulary, integrating
words organically within cohesive narratives. We extracted vocabulary from
English textbooks used in primary schools in China to generate story scripts
using the state-of-the-art large language model, GPT-4 [21]. Educators select
vocabulary and input in the Theme panel to send request for generation on the
server. A model status indicator is shown in the header next to the unit
selector to inform the process of generation on the server. When the script is
generated, it is displayed in the Script panel with the words highlighted to
show their contextual usage. The prompt template and an example script is
shown below, in which the words are from a Grade 2 textbook for primary
schools:
Prompt: Generate sentences consisting of the following words with the theme of
<school trip>, make sure the sentences are easy and s
hort for primary school students learning English as a second lang
uage, one sentence for one word and all sentences are contextually
related to form an article. And generate a prompt for each word ba
sed on the sentence for image generation in stable diffusion. Here
are the words: <spring, ..., cool>.

Output:
Word: "spring"
Sentence: "In spring, the flowers bloom and we go on a school trip
to the park."
Sticker Prompt: "Children on a school trip in a park full of bloom
ing flowers representing spring."
...
Word: "cool"
Sentence: "It’s cool in the forest, and we love exploring it."
Sticker Prompt: "A group of students exploring a cool, shaded fore
st."

####  4.1.2 Word Sticker

To further enhance the expressiveness of the generated story script, we
incorporated word stickers into the narrative. The Stable Diffusion XL (SDXL)
model [23] with two Low-Rank Adaptation (LoRA) modules [9] is employed to
accomplish the task, where the LoRA modules fine-tune the SDXL base model to
create emoji-like images based on the sticker prompts [19] and accelerate the
generation [17]. Using the sticker prompts generated with the script, example
of which shown in the Output section above, the system generates stickers of
coherent theme for each word. These stickers are subsequently transmitted to
the Dashboard, populating the Stickers panel.

####  4.1.3 Data Refine and Theme Variants

To encourage divergent thinking and foster customization, educators can
further generate variant scripts and utilize alternative prompts, expanding
the possibilities for tailoring the output content to specific learning
objectives. Upon selecting an item in the Stickers panel, the current prompt
associated with the chosen sticker is displayed in the Refine panel. Educators
can edit the prompt and regenerate stickers to better fit the intended
context, while simultaneously mitigating the risk of generating potentially
inappropriate content.

Moreover, by leveraging varied themes and contexts, educators can create more
engaging learning contents that not only reinforce vocabulary acquisition but
also promote linguistic adaptability, as shown in Fig. 4. By offering a
diverse array of learning materials, educators can stimulate more dynamic
forms of vocabulary engagement and deepen learners’ connections with the
language.

Figure 4: Stickers generated for the same vocabulary in different contexts.

###  4.2 Playground for Learners

In addition to the generated learning contents, the Playground serves as a
platform for learners to engage in interactive exploration. This interactive
space offers learners the opportunity to actively review with the concepts and
vocabulary introduced in the generated content and to explore more by
themselves. It consists of four panels: Vocabulary, Script, Selected and the
interactive visualization in the center, as shown in Fig. 5. The Playground is
seamlessly synchronized with the Dashboard as the two platforms are
interconnected by the data and assets they share in the database on the
server. This integration ensures that the learning materials sophisticatedly
prepared by educators are available for learners to review and explore extra
contents aligning with the theme of the original materials.

Figure 5: Explorer view of the Playground.

####  4.2.1 Interactive Visualization

Learners select the unit in the header, and the Playground send queries to the
database to retrieve the corresponding script and stickers. The Script and
Visualization panels are updated with incoming data, so the learner can
interact with the stickers to explore the connections between them. The
visualization utilizes the D3.js [3] library to create an interactive network,
where the stickers attract each other to mimic the context, aiding learners in
comprehending the contextual relationship between words and their
corresponding usage within the script.

####  4.2.2 Exploratory Learning

The visualization additionally facilitates a deeper investigation of the
interconnections between vocabularies. When the stickers are selected, they
are highlighted and appear in the Selected panel, where a maximum of two
stickers can be selected at the same time, prompting the user to explore
further into the relationship between them. Upon clicking the Explore button,
additional vocabulary and stickers will be generated under the same context,
to establish a connection between the selected items in a pop-up Exploration
panel, as shown in Fig. 6. These newly generated elements either act as
supplementary materials for the original resources or diverge from the
original learning materials, thereby fostering divergent thinking and
exploratory learning.

Figure 6: Example of a relation explored in the Playground.

The prompt template and an example output is shown below:

Prompt: Find the relationship between the following words based on the
semantic relevance with the theme of <Switzerland>. Add rel
ated words to make the semantic relevance more consistent. Add a p
rompt for each word based on the theme for image generation in sta
ble diffusion. Ensure the first and last two words are the input w
ords, for example: "Monitor, Mouse" - "Monitor-Computer-Computer A
ccessories-Mouse". Here are the two input words: <lake, hill>.

Output:
Word: "geneva"
Sticker Prompt: "Cityscape of Geneva, Switzerland, with the iconic
Jet d’eau fountain and lake Geneva in the foreground."
Word: "chocolate"
Sticker Prompt: "Swiss chocolate bars with the Swiss alps mountain
in the background."
Word: "alps"
Sticker Prompt: "The majestic Swiss alps on a sunny day, with pict
uresque ski resorts and chalets"

##  5 Discussion

In this article, we presented the ContextVis system, a workflow that leverages
generative models for creating contextual learning materials and experiences.
Through a case study on contextual vocabulary learning, we have demonstrated
the utility of generative models in resource creation and the effectiveness of
the system in fostering language development.

The user-friendly Dashboard for educators and an interactive Playground for
learners together provide a comprehensive platform for both content creation
and engagement. The integration of generative models, contextual cues, and
interactive visualizations offers unique opportunities for learners to
understand, explore, and retain knowledge. During the development stage, we
found that providing competent information such as theme and context to
generative models is essential to generate high-quality learning materials.

Future directions for the ContextVis system include expanding the range of
generative models and media formats, as well as more in-depth customization
options for educators. Overall, the ContextVis system represents a promising
approach to enhance contextual and exploratory learning using generative
models and interactive visualizations.

####  5.0.1 Acknowledgements

This work was supported by a research grant from Shenzhen Key Laboratory of
Next Generation Interactive Media Innovative Technology (Funding No:
ZDSYS20210623092001004) and the Center for Social Governance and Innovation at
Tsinghua University, a major research center for Shenzhen Humanities & Social
Sciences Key Research Bases.

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# Interplay of magnetic field and geometry in magneto-transport of mesoscopic
loops with Rashba and Dresselhaus spin-orbit interactions

Shreekantha Sil  Department of Physics, Visva-Bharati, Santiniketan, West
Bengal-731 235, India  Santanu K. Maiti<EMAIL_ADDRESS>School of
Chemistry, Tel Aviv University, Ramat-Aviv, Tel Aviv-69978, Israel  Arunava
Chakrabarti  Department of Physics, University of Kalyani, Kalyani, West
Bengal-741 235, India

###### Abstract

Electronic transport in closed loop structures is addressed within a tight-
binding formalism and in the presence of both the Rashba and Dresselhaus spin-
orbit interactions. It has been shown that any one of the spin-orbit (SO)
fields can be estimated precisely if the other one is known, by observing
either the transmission resonance or anti-resonance of unpolarized electrons.
The result is obtained through an exact analytic calculation for a simple
square loop, and through a numerically exact formulation for a circular ring.
The sensitivity of the transport properties on the geometry of the
interferometer is discussed in details.

###### pacs:

73.23.Ad, 71.70.Ej

##  I Introduction

Spintornics is a recent field of utmost research interest that include
magnetic memory circuits, quantum computers zutic ; datta ; ando ; aharony1
magnetic nano-structures and quasi one-dimensional semiconductor rings which
have been acknowledged as ideal candidates for investigating the effects of
quantum coherence in low-dimensions, and have been examined as the prospective
quantum devices moldo ; engels ; vlaminck ; ando1 . A central mechanism
governing the physics in the meso- and nano-length scales are the Rashba and
Dresselhaus spin-orbit interactions which result from a structural inversion
asymmetry rashba , and bulk inversion asymmetry dresselhaus ; meier
respectively. The effects are pronounced in quantum rings formed at the
interface of two narrow gap semiconductors, as already discussed in the
literature koga ; premper .

Needless to say, an accurate estimation of the spin-orbit interaction (SOI)
strengths is crucial in the field of spintronics. The Rashba spin-orbit
interaction (RSOI) can be controlled by a gate voltage placed in the vicinity
of the sample premper ; cmhu ; grundler  and hence, can be ‘measured’.
Comparatively speaking, reports on the techniques of measurement of the
Dresselhaus spin-orbit interaction (DSOI) are relatively few premper ; cmhu ;
grundler . Very recently we have put forward an idea of estimating the DSOI
strength provided the RSOI is known by measuring a minimum in the Drude weight
santanu . A minimum in the Drude weight appears only when the strengths of the
RSOI and DSOI are exactly identical.

From a closer look at the Rashba and Dresselhaus SO interactions it turns out
that both the interactions are equivalent to the $SU(2)$ gauge field which
introduces a phase in the wave function. In this communication we describe the
role of this phase and considering its effect on quantum interference we
develop a simple idea about how one of the two SOI’s can be estimated while
the other is known. This quantum interference effect in presence of SO
interactions has not been described in our previous work santanu  and the
present analysis may provide a basic understanding of designing switching
devices for spintronic applications in near future. A simple version of a
quantum ring, in the form of a loop with a rhombic geometry is considered for
an analytical attack on the problem, while numerically exact results are
provided for circular rings with and without disorder and with a magnetic flux
threading these polygonal structures. We adopt a tight-binding formalism in
contrast to a recently proposed scheme where a continuous version of the model
is presented ramaglia  to consider the combined effect of the RSOI and the
DSOI. When strengths of

Figure 1: (Color online). Schematic view of a mesoscopic square loop subjected
to RSOI and DSOI and connected to the leads (source and drain) at its two
extremities.

the RSOI and DSOI are equal, the end to end transmission across the rhombic
loop is shown to be equal to unity and vanishes when the loop is threaded by a
magnetic flux $\phi$ equal to the half flux-quantum ($\phi_{0}/2$). Thus, one
can estimate the DSOI by observing the peak when $\phi=0$ or dip when
$\phi=\phi_{0}/2$ in the transmission (conductance) spectrum when the RSOI is
known, and vice versa.

The idea is extended to the case of rings with circular geometry, where we
have evaluated the transmission coefficient numerically. The essential
differences in the cases of a rhombic loop and a circular ring are discussed
to highlight the sensitivity of the results on the loop geometry.

In what follows we describe the procedure and the results. In section II, we
present the model and the method. In section III we discuss the sensitivity of
the results on the geometry of the closed loop structures, and in section IV
we draw our conclusions.

##  II The model and the method

The Hamiltonian: Let us consider the rhombic loop depicted in Fig. 1 which is
threaded by a magnetic flux $\Phi$. Each side of the loop is of length $L$,
and the loop contains $N$ number of equispaced atomic sites with ‘lattice
constant’ $a$. Within a tight-binding framework the Hamiltonian for this
network in the presence of the RSOI and DSOI reads,

$\mbox{\boldmath$H$}=\mbox{\boldmath$H_{0}$}-i\alpha\mbox{\boldmath$H_{R}$}+i\beta\mbox{\boldmath$H_{D}$}$
(1)

where,

$\mbox{\boldmath$H_{0}$}=-\sum_{i}\left(\mbox{\boldmath$c_{i}^{\dagger}t$}\mbox{\boldmath$c_{i+1}$}e^{i\phi}+\mbox{\boldmath$c_{i+1}^{\dagger}t$}\mbox{\boldmath$c_{i}$}e^{-i\phi}\right)$
(2)

The Rashba and Dresselhaus spin-orbit parts of the Hamiltonian, viz, $H_{R}$
and $H_{D}$, are given by,

$H_{R}$ $\displaystyle=$
$\displaystyle\sum_{i}\left(\sin\theta\mbox{\boldmath$c_{i}^{\dagger}$}\mbox{\boldmath$\sigma_{x}$}\mbox{\boldmath$c_{i+1}$}\right.-\left.\cos\theta\mbox{\boldmath$c_{i}^{\dagger}$}\mbox{\boldmath$\sigma_{y}$}\mbox{\boldmath$c_{i+1}$}\right)e^{i\phi}+h.c.$
$H_{D}$ $\displaystyle=$
$\displaystyle\sum_{i}\left(-\cos\theta\mbox{\boldmath$c_{i}^{\dagger}$}\mbox{\boldmath$\sigma_{x}$}\mbox{\boldmath$c_{i+1}$}\right.+\left.\sin\theta\mbox{\boldmath$c_{i}^{\dagger}$}\mbox{\boldmath$\sigma_{y}$}\mbox{\boldmath$c_{i+1}$}\right)e^{i\phi}+h.c.$
(3)

where, $i$ refers to the atomic sites in the arms of the loop.
$\phi=2\pi\Phi/N\phi_{0}$, and $\phi_{0}=hc/e$, the fundamental flux-quantum.
The other operators in Eq. 3 are as follows.
$c_{i}$=$\left(\begin{array}[]{c}c_{i,\uparrow}\\\
c_{i,\downarrow}\end{array}\right)\hskip 5.69046pt\mbox{and}\hskip 5.69046pt$
$t$=$t\left(\begin{array}[]{cc}1&0\\\ 0&1\end{array}\right)$.

Here the site energy of an electron at any $i$-th site is assumed to be zero
throughout the geometry. $t$ is the nearest-neighbor hopping integral.
$\alpha$ and $\beta$ are the isotropic nearest-neighbor transfer integrals
which measure the strengths of Rashba and Dresselhaus SOI, respectively.
$\sigma_{x}$, $\sigma_{y}$ and $\sigma_{z}$ are the Pauli spin matrices.
$c_{i,\sigma}^{\dagger}$ ($c_{i,\sigma}$) is the creation (annihilation)
operator of an electron at the site $i$ with spin $\sigma$
($\uparrow,\downarrow$).

Eigenvalues and eigenfunctions of the Hamiltonian: These are obtained by
adopting a $\bf{k}$-space description of the Hamiltonian given in Eq. 3, viz,
${\bf H}=\sum_{k}{\bf c_{k}^{\dagger}H_{k}c_{k}}$. Using discrete Fourier
transform $c_{k}=\frac{1}{\sqrt{N}}\sum_{n}c_{n}\exp(-i{\bf k}.n{\bf a})$, the
Hamiltonian matrix reads,

$\mbox{\boldmath$H_{k}$}=\left(\begin{array}[]{cc}\epsilon_{k}&\gamma_{k}+i\delta_{k}\\\
\gamma_{k}-i\delta_{k}&\epsilon_{k}\end{array}\right)\\\ $ (4)

where,

$\displaystyle\epsilon_{k}$ $\displaystyle=$ $\displaystyle-2t\cos(ka+\phi)$
$\displaystyle\gamma_{k}$ $\displaystyle=$ $\displaystyle
2\left(\alpha\sin\theta+\beta\cos\theta\right)\sin(ka+\phi)$
$\displaystyle\delta_{k}$ $\displaystyle=$ $\displaystyle
2\left(\alpha\cos\theta+\beta\sin\theta\right)\sin(ka+\phi)$

While writing the above expressions, we have set the lattice constant $a=1$.
The energy eigenvalues are obtained from Eq. 4, and are given by,
$E_{k_{\pm}}=\epsilon_{k}\pm\sqrt{\gamma_{k}^{2}+\delta_{k}^{2}}$.

Let us denote the left vertex of the loop in Fig. 1 as the ‘origin’ $(0,0)$. A
simple but lengthy algebra now allows one to write the wave function for an
energy $E$ at a distance $L$ along any arm of the rhombic loop as,

$|\Phi_{E}(L,a)\rangle=\mathcal{R}_{E}(L,\alpha,\beta,\theta)|\Phi_{E}(0)\rangle$
(5)

where, the elements of the matrix $\mathcal{R}_{E}(L,\alpha,\beta,\theta)$
are,

$\displaystyle\mathcal{R}_{E}(L,\alpha,\beta,\theta)_{11}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(e^{ik_{+}La}+e^{ik_{-}La}\right)$
$\displaystyle\mathcal{R}_{E}(L,\alpha,\beta,\theta)_{12}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(e^{i(k_{+}La+\nu_{k+})}-e^{i(k_{-}La+\nu_{k-})}\right)$
$\displaystyle\mathcal{R}_{E}(L,\alpha,\beta,\theta)_{21}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(e^{i(k_{+}La-\nu_{k+})}-e^{i(k_{-}La-\nu_{k-})}\right)$
$\displaystyle\mathcal{R}_{E}(L,\alpha,\beta,\theta)_{22}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(e^{ik_{+}La}+e^{ik_{-}La}\right)$ (6)

Here, $\nu_{k\pm}=\tan^{-1}(\delta_{k\pm}/\gamma_{k\pm})$. $k_{\pm}$ are the
wave vectors corresponding to the energy values
$\epsilon_{k}\pm\sqrt{\gamma_{k}^{2}+\delta_{k}^{2}}$, as already mentioned.
Transmission of unpolarized electrons: Let us now assume that the electrons
enter the loop at the point $A$ through the source, and are drained out at
$B$. In addition, without any loss of generality, and to get a relatively
simple set of equations, we take our loop to be a square one with
$\theta=\pi/4$. For an electron traveling in the loop in the clockwise sense
with a specified energy $E$, the wave vector $k_{\pm}$ are the solutions of
the equation $E=E_{k_{\pm}}$, and can be explicitly written as,

$\displaystyle k_{\pm}a$ $\displaystyle=$
$\displaystyle\cos^{-1}\xi_{\pm}(E)-\phi$ (7)

where,

$\displaystyle\xi_{\pm}(E)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{1+\frac{(\alpha+\beta)^{2}}{t^{2}}}}\left[-\frac{E}{2t\sqrt{1+\frac{(\alpha+\beta)^{2}}{t^{2}}}}\right.$
(8) $\displaystyle\pm$
$\displaystyle\left.\frac{\alpha+\beta}{t}\sqrt{1-\frac{E^{2}}{4t^{2}(1+\frac{(\alpha+\beta)^{2}}{t^{2}}}}\right]$

In a similar manner, we need to work out the wave vectors $k^{\prime}_{\pm}$
for the electrons with the same energy $E$, and traveling in the counter-
clockwise sense. The result is,

$\displaystyle k^{\prime}_{\pm}a$ $\displaystyle=$
$\displaystyle\cos^{-1}\xi^{\prime}_{\pm}(E)-\phi$ (9)

where,

$\displaystyle\xi^{\prime}_{\pm}(E)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{1+\frac{(\alpha-\beta)^{2}}{t^{2}}}}\left[-\frac{E}{2t\sqrt{1+\frac{(\alpha-\beta)^{2}}{t^{2}}}}\right.$
(10) $\displaystyle\pm$
$\displaystyle\left.\frac{|\alpha-\beta|}{t}\sqrt{1-\frac{E^{2}}{4t^{2}(1+\frac{(\alpha-\beta)^{2}}{t^{2}}}}\right]$

The probability that the electrons travel in the clockwise or the counter-
clockwise sense are assumed to be equal. The transmission amplitude is given
by the matrix,

$\tau=\left(\begin{array}[]{cc}\tau_{\uparrow\uparrow}&\tau_{\uparrow\downarrow}\\\
\tau_{\downarrow\uparrow}&\tau_{\downarrow\downarrow}\end{array}\right)$ (11)

which, for $\theta=\pi/4$, simplifies to,

$\displaystyle\tau$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[\mathcal{R}_{E}(L,\alpha,\beta,-\pi/4)\mathcal{R}_{E}(L,\alpha,\beta,\pi/4)\right.$
(12)
$\displaystyle+\left.\mathcal{R}_{E}(L,\alpha,\beta,\pi/4)\mathcal{R}_{E}(L,\alpha,\beta,-\pi/4)\right]$

$\mathcal{R}_{E}$ matrices can be cast in to the forms,

$\displaystyle\mathcal{R}_{E}(L,\alpha,\beta,\pi/4)$ $\displaystyle=$
$\displaystyle\frac{e^{i\phi
L}}{2}\left(\mathcal{A}_{1}I+\frac{\mathcal{B}_{1}\sigma_{x}}{\sqrt{2}}-\frac{\mathcal{B}_{1}\sigma_{y}}{\sqrt{2}}\right)$
$\displaystyle\mathcal{R}_{E}(L,\alpha,\beta,-\pi/4)$ $\displaystyle=$
$\displaystyle\frac{e^{i\phi
L}}{2}\left(\mathcal{A}_{2}I+\frac{\mathcal{B}_{2}\sigma_{x}}{\sqrt{2}}+\frac{\mathcal{B}_{2}\sigma_{y}}{\sqrt{2}}\right)$

with,

$\displaystyle\mathcal{A}_{1}$ $\displaystyle=$
$\displaystyle\exp(ik_{+}La)+\exp(ik_{-}La)$ $\displaystyle\mathcal{A}_{2}$
$\displaystyle=$
$\displaystyle\exp(ik_{+}^{\prime}La)+\exp(ik_{-}^{\prime}La)$
$\displaystyle\mathcal{B}_{1}$ $\displaystyle=$
$\displaystyle\exp(ik_{+}La)-\exp(ik_{-}La)$ $\displaystyle\mathcal{B}_{2}$
$\displaystyle=$
$\displaystyle\exp(ik_{+}^{\prime}La)-\exp(ik_{-}^{\prime}La)$ (14)

The transmission amplitude given by Eq. 12 can now be explicitly written as,

$\displaystyle\tau$ $\displaystyle=$
$\displaystyle\frac{1}{8}\left[(2\mathcal{A}_{1}\mathcal{A}_{2}+\sqrt{2}(\mathcal{A}_{2}\mathcal{B}_{1}+\mathcal{A}_{1}\mathcal{B}_{2})\sigma_{x}\right.$
(15) $\displaystyle+$
$\displaystyle\left.\sqrt{2}(\mathcal{A}_{1}\mathcal{B}_{2}-\mathcal{A}_{2}\mathcal{B}_{1})\sigma_{y})\right.\cos
2\phi L$ $\displaystyle+$
$\displaystyle\frac{i}{4}(\sigma_{x}\sigma_{y}-\sigma_{y}\sigma_{x})\mathcal{B}_{1}\mathcal{B}_{2}\sin
2\phi L$

The coefficient of transmission for an incoming up-spin electron is
$T_{\uparrow}=|\tau_{\uparrow\uparrow}+\tau_{\downarrow\uparrow}|^{2}$, and
the transmission coefficient for an incoming down-spin electron is
$T_{\downarrow}=|\tau_{\downarrow\downarrow}+\tau_{\downarrow\uparrow}|^{2}$,
so that the final transmission coefficient for spin unpolarized electrons
turns out to be,

$\displaystyle T$ $\displaystyle=$
$\displaystyle\frac{1}{2}(T_{\uparrow}+T_{\downarrow})$ (16) $\displaystyle=$
$\displaystyle\frac{1}{2}\left[\frac{|\mathcal{A}_{1}\mathcal{A}_{2}|^{2}}{8}+\frac{|\mathcal{A}_{2}\mathcal{B}_{1}+\mathcal{A}_{1}\mathcal{B}_{2}|^{2}}{16}+\frac{|\mathcal{A}_{1}\mathcal{B}_{2}-\mathcal{A}_{2}\mathcal{B}_{1}|^{2}}{16}\right]$
$\displaystyle=$
$\displaystyle\cos^{2}\left(\frac{(k^{\prime}_{+}-k^{\prime}_{-})}{2}La\right)+\sin^{2}\left(\frac{(k_{+}-k_{-})}{2}La\right)$
$\displaystyle\times\cos^{2}\left(\frac{(k_{+}-k_{-})}{2}La\right),\hskip
14.22636pt\mbox{for}\hskip 4.26773pt\phi=0$ $\displaystyle=$ $\displaystyle
4\sin^{2}\left(\frac{(k_{+}-k_{-})}{2}La\right)\sin^{2}\left(\frac{(k^{\prime}_{+}-k^{\prime}_{-})}{2}La\right),$
$\displaystyle\mbox{for}\hskip 4.26773pt\phi=\pi/4L$

When $\alpha=\beta$, from Eq. 10 we observe that
$\xi^{\prime}_{+}=\xi^{\prime}_{-}$ or $k^{\prime}_{+}=k^{\prime}_{-}$ and it
gives $\mathcal{B}_{2}=0$. Therefore, for $\phi=0$, the transmission
coefficient $T=1$, and, $T=0$ for $\phi=\pi/4L$. Thus we get perfect
transmission for $\phi=0$ while a vanishing of transmission coefficient for
$\phi=\pi/4L$.

##  III Effect of loop geometry

Electronic transport turns out to be sensitive to the geometry of the
mesoscopic loop. To this end, we have numerically calculated the

Figure 2: (Color online). Schematic view of a mesoscopic ring subjected to
RSOI and DSOI and threaded by a magnetic flux $\Phi$. The ring is
symmetrically connected to the leads (source and drain).

two-terminal spin transport in a ring geometry threaded by a magnetic flux
$\Phi$ (Fig. 2). The role of $\Phi$ will be discussed later. Here, for the
time being, we ignore the flux.

Figure 3: (Color online). Two-terminal transmission coefficient of a $40$-site
ordered ring for different values of the RSOI ($\alpha$) and the DSOI
($\beta$). $\Phi$ is set at $0$.

Figure 4: (Color online). Two-terminal transmission coefficient of a $40$-site
disordered ring for different values of the RSOI ($\alpha$) and the DSOI
($\beta$). The results have been averaged over $60$ disorder configurations.
$\Phi$ is fixed at $0$.

For a ring like structure, the azimuthal angle keeps on changing as one
traverses the perimeter of the ring. This generates an effective site
dependent hopping integral in the Hamiltonian santanu . As a result,
scattering takes place as the electron travels across the sites on the ring.
The scattering becomes a maximum when $\alpha=\beta$ santanu , and naturally,
the two-terminal transport is expected to show up a minimum at $\alpha=\beta$.
We use exact numerical methods. In Fig. 3 we show the variation of the two-
terminal transmission

Figure 5: (Color online). Two-terminal transmission coefficient of a $40$-site
ordered ring for different values of RSOI ($\alpha$) and the DSOI ($\beta$).
$\Phi$ is set equal to $0.5$.

Figure 6: (Color online). Two-terminal transmission coefficient of a $40$-site
disordered ring for different values of the RSOI ($\alpha$) and the DSOI
($\beta$). The results have been averaged over $60$ disorder configurations.
$\Phi$ is fixed at $0.5$.

coefficient for an ordered ($W=0$, $W$ measures the strength of disorder) ring
of $40$ sites. Similar observations are presented in Fig. 4 for a $40$-site
ring with random diagonal disorder. The results in the latter case have been
averaged over $60$ disorder configurations. In both the figures the
transmission minimum as $\alpha$ equals $\beta$ are obvious. It is interesting
to note that the random disorder does not destroy the minima, which speaks for
the robustness of the results from the standpoint of experiments.

Before we end this section, it should be appreciated that, in an experiment
the transmission minimum is not easy to locate, and hence an inaccuracy in the
value of the SOI might be introduced. This difficulty may be circumvented if
the transmission minimum becomes precisely equal to zero. This is easily
achieved if the ring is threaded by a magnetic flux which is set equal to half
the flux-quantum $\phi_{0}=hc/e$. In Figs. 5 and 6 the results are presented
for a $40$-site ordered ring and a randomly disordered ring of the same size.
The flux threading the rings is set at $\Phi=\phi_{0}/2$. The two-terminal
transmission coefficient exhibits clear zeros in both the cases as soon as the
DSOI equals the strength of the RSOI. Once again, the transmission zero in
this special situation is independent of the disorder configuration.

##  IV Conclusion

In conclusion, we have presented exact analytical results to show that the
spin-orbit interactions present in a mesoscopic sample can be measured by
observing two-terminal transmission resonance in a simple square network. The
transmission coefficient is shown to be exactly equal to unity when the Rashba
and the Dresselhaus interactions become equal in strength. Thus, knowing the
Rashba interaction, for example, a determination of the Dresselhaus term is
possible. For a multi-site ring, the transmission maximum observed for the
single square loop gets converted in to transmission minimum. This happens
again when the Rashba and the Dresselhaus interactions are equal. With a
magnetic flux equal to half the flux-quantum, the transmission minimum becomes
an exact transmission zero, and hence facilitates a possible experimental
measurement.

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# SYNCA: A Synthetic Cyclotron Antenna for the Project 8 Collaboration

A. Ashtari Esfahani  S. Böser  N. Buzinsky  M. C. Carmona-Benitez  C.
Claessens  L. de Viveiros  M. Fertl  J. A. Formaggio  L. Gladstone  M. Grando
J. Hartse  K. M. Heeger  X. Huyan  A. M. Jones  K. Kazkaz  M. Li  A. Lindman
C. Matthé  R. Mohiuddin  B. Monreal  R. Mueller  J. A. Nikkel  E. Novitski  N.
S. Oblath  J. I. Peña  W. Pettus  R. Reimann  R. G. H. Robertson  L. Saldaña
P. L. Slocum  J. Stachurska  Y.-H. Sun  P. T. Surukuchi  A. B. Telles  F.
Thomas  M. Thomas  L. A. Thorne  T. Thümmler  L. Tvrznikova  W. Van De
Pontseele  B. A. VanDevender  T. E. Weiss  T. Wendler  E. Zayas  A. Ziegler,
11footnotetext: Corresponding author.

###### Abstract

Cyclotron Radiation Emission Spectroscopy (CRES) is a technique for measuring
the kinetic energy of charged particles through a precision measurement of the
frequency of the cyclotron radiation generated by the particle’s motion in a
magnetic field. The Project 8 collaboration is developing a next-generation
neutrino mass measurement experiment based on CRES. One approach is to use a
phased antenna array, which surrounds a volume of tritium gas, to detect and
measure the cyclotron radiation of the resulting $\beta$-decay electrons. To
validate the feasibility of this method, Project 8 has designed a test stand
to benchmark the performance of an antenna array at reconstructing signals
that mimic those of genuine CRES events. To generate synthetic CRES events, a
novel probe antenna has been developed, which emits radiation with
characteristics similar to the cyclotron radiation produced by charged
particles in magnetic fields. This paper outlines the design, construction,
and characterization of this Synthetic Cyclotron Antenna (SYNCA). Furthermore,
we perform a series of measurements that use the SYNCA to test the position
reconstruction capabilities of the digital beamforming reconstruction
technique. We find that the SYNCA produces radiation with characteristics
closely matching those expected for cyclotron radiation and reproduces
experimentally the phenomenology of digital beamforming simulations of true
CRES signals.

##  1 Introduction

Neutrinos are the most abundant standard model fermions in our universe, but
due to weak interaction cross-sections with other particles, neutrinos are
particularly difficult to study. Consequently, many fundamental properties of
neutrinos are still unknown including the absolute scale of the neutrino mass
[1]. Direct, kinematic measurements of the neutrino mass are particularly
valuable due to their model independent nature [2]. To date the most sensitive
direct neutrino mass measurements have been performed by the KATRIN
collaboration [3], which measures the molecular tritium $\beta$-decay spectrum
to infer the neutrino mass. Current data from neutrino oscillation
measurements [1] allow for neutrino masses significantly smaller than the
design sensitivity of the KATRIN experiment; therefore, there is a need for
new technologies for performing direct neutrino mass measurements to probe
lower neutrino masses.

Figure 1:  A sketch of an antenna array large-volume CRES experiment.
Electrons from $\beta$-decays are confined in a magnetic field using a set of
trap coils. The cyclotron radiation produced by the motion of the trapped
electrons can be detected by a surrounding antenna array to determine the
electron energies. Measuring the energies of many electrons produces a
$\beta$-decay spectrum.

The Project 8 collaboration is developing new methods for neutrino mass
measurement based on Cyclotron Radiation Emission Spectroscopy (CRES) [4, 5,
6, 7], with the goal of measuring the absolute scale of the neutrino mass with
a 40 $\textrm{meV}/\textrm{c}^{2}$ sensitivity [8, 2]. This sensitivity goal
will require the development of two separate technical capabilities. First is
the development of an atomic tritium source, which avoids significant spectral
broadening due to molecular final states [9]. Second is the technology for
performing CRES in a multi-cubic-meter experimental volume with high combined
detection and reconstruction efficiency, which is required in order to obtain
sufficient event statistics near the tritium spectrum endpoint.

One approach for a large-volume CRES experiment is to use an array of
antennas, which surrounds a volume of tritium gas, to detect the cyclotron
radiation produced by the $\beta$-decay electrons when they are trapped in a
background magnetic field using a set of magnetic trapping coils (see Figure
1). Project 8 has developed a conceptual experiment design to study the
feasibility of this approach. The design consists of a single circular array
of antennas with a radius of 10 cm and 60 independent channels positioned
around the center of the magnetic trap. The motivation behind this antenna
array design is to first develop an understanding of the antenna array
approach to CRES with a small scale experiment before attempting to scale the
technique to large volumes by using multiple antenna rings to construct the
full cylindrical array. The development of the antenna array approach to CRES
has largely proceeded through simulations using the Locust software package
[10, 11], which is used to model the fields emitted by CRES events and predict
the signals received by the surrounding antenna array. To validate these
simulations, a dedicated test stand is being constructed to perform
characterization measurements of the prototype antenna array developed by
Project 8 (see Figure 2) and benchmark signal reconstruction methods using a
specially designed transmitting calibration probe antenna.

Figure 2:  A schematic of the antenna array test stand. The circular antenna
array has a radius of 10 cm with 60 independent channels (limited number shown
for clarity). The test stand includes an arbitrary waveform generator (AWG),
local oscillator (LO), and data acquisition (DAQ) hardware. Finally, a
specialized Synthetic Cyclotron Antenna (SYNCA) is used to inject signals to
test the antenna array.

We call this probe antenna the Synthetic Cyclotron Antenna or SYNCA. The SYNCA
is a novel antenna design that mimics the cyclotron radiation generated by
individual charged particles trapped in a magnetic field, which will be used
in the antenna test stand to perform characterization measurements, simulation
validation, and reconstruction benchmarking. This paper provides an overview
of the design, construction, and characterization measurements of the SYNCA
performed in preparation for its usage as a transmitting calibration probe.

In Section 2 we provide a description of the cyclotron radiation field
characteristics that we recreate with the SYNCA. In Section 3 we give an
overview of the simulations performed to develop an antenna design that mimics
the characteristics of cyclotron radiation. In Section 4 we outline
characterization measurements to validate that the fields generated by the
SYNCA match simulation, and finally in Section 5 we demonstrate an application
of the SYNCA to test phased array reconstruction techniques on the bench-top.

##  2 Cyclotron Radiation Phenomenology

To understand the cyclotron radiation phenomenology that the SYNCA should
mimic, we consider a charged particle moving at relativistic speed in the
presence of an external magnetic field (see Figure 3). In the special case we
shall examine, the entirety of the electron’s momentum is directed
perpendicular to the magnetic field; therefore, the trajectory of the electron
is confined to the cyclotron orbit plane. Because the momentum vector is
oriented perpendicular to the magnetic field, electrons with these
trajectories are said to have pitch angles of $90^{\circ}$.

Figure 3:  An electron (red dot) performing cyclotron motion in the x-y plane.
The resulting cyclotron radiation is observed by an antenna located at the
field point of interest.

The cyclotron radiation fields generated by this circular trajectory are those
which we aim to reproduce with the SYNCA. We can describe the electromagnetic
(EM) fields using the Liénard-Wiechert equations [12, 11], which in non-
covariant form express the electric field as

$\vec{E}=e\left[\frac{\hat{n}-\vec{\beta}}{\gamma^{2}(1-\vec{\beta}\cdot\hat{n})^{3}|\vec{R}|^{2}}\right]_{t_{\textrm{r}}}+\frac{e}{c}\left[\frac{\hat{n}\times[(\hat{n}-\vec{\beta})\times\dot{\vec{\beta}}]}{(1-\vec{\beta}\cdot\hat{n})^{3}|\vec{R}|}\right]_{t_{\textrm{r}}},$
(2.1)

where $e$ is the particle’s charge,
$\hat{n}=(\vec{r}-\vec{r_{s}})/|\vec{r}-\vec{r_{s}}|$ is the unit vector
pointing from the electron to the field measurement point,
$\vec{\beta}=\dot{\vec{r_{s}}}/c$ is the velocity of the particle divided by
the speed of light, and $\gamma$ is the relativistic Lorentz factor. The
equation is meant to be evaluated at the retarded time as indicated by
$t_{\textrm{r}}=t-|\vec{R}|/c$, which accounts for the time delay due to the
finite speed of light between the point where the field was emitted and the
point where the field is detected.

In Appendix A we show that the electric fields received by an antenna
positioned in the plane of the cyclotron orbit and located at a distance that
is much greater than the radius of the cyclotron orbit is a sum of a radially
and azimuthally polarized fields. For an electron with a kinetic energy near
the 18.6 keV Q-value of the tritium $\beta$-decay [9] moving in a 1 T magnetic
field, we show that the time-averaged amplitude of the radially polarized
electric field is a factor of 8 smaller than the azimuthally polarized
component. Additionally, we make the assumption that the receiving antenna is
designed to have a gain that is significantly larger for the azimuthal
electric fields, which is true for the antenna array in the test stand
mentioned in Section 1. Therefore, we can approximate the electric fields from
Equation 2.1 as purely azimuthally or $\phi$-polarized. The simplified
expression for the electric field received by an antenna becomes

$\vec{E}=E_{\phi}\hat{\phi}=\frac{e\frac{r_{c}\omega_{c}}{c}}{4r_{a}r_{c}}\left[\frac{\frac{r_{c}\omega_{c}}{c}-\cos{\omega_{c}t}-\frac{4r_{c}}{r_{a}}\sin{\omega_{c}t}}{(1-\frac{r_{c}\omega_{c}}{c}\cos{\omega_{c}t})^{3}}\right]_{t_{r}}\hat{\phi},$
(2.2)

where the radius of the cyclotron orbit is called $r_{c}$, the cyclotron
frequency is called $\omega_{c}$, and the radial position of the receiving
antenna is called $r_{a}$. Equation 2.2 has been evaluated in the non-
relativistic limit where $\gamma\simeq 1$, which is justified by the fact that
$|\vec{\beta}|\simeq\frac{c}{4}$ for an electron with an 18.6 keV kinetic
energy in a 1 T magnetic field.

The rather complicated expression for the electric field can be simplified
using Fourier analysis. In Appendix B we show that Equation 2.2 can be
represented by a sum of pure sinusoids with frequencies that are harmonics of
the main cyclotron carrier frequency,

$\vec{E}=\frac{e\frac{r_{c}\omega_{c}}{c}}{4r_{a}r_{c}}\left[\sum_{k=0}^{7}{A_{k}e^{i\omega_{k}t}}\right]_{t_{r}}\hat{\phi}.$
(2.3)

The frequencies ($\omega_{k}=k\omega_{c}$) are uniformly spaced by integer
values of the electron’s cyclotron frequency of $25.898$ GHz and include a
constant term from the electron’s coulomb field at zero frequency. The
relative amplitudes ($A_{k}$) are dimensionless complex numbers which encode
the relative amplitudes of the harmonics and the overall starting phase of the
cyclotron motion. Because magnitude of the relative amplitudes exponentially
decreases for higher harmonics, it is usually sufficient to consider only the
terms up to $k=4$ where the relative amplitude of the hamonics has decresed
from the main carrier by a factor of approximately 100. However, for
completeness we include harmonics up to 7th order in Equation 2.3. The range
of frequencies to which the receiving antenna array in the antenna test stand
is sensitive is defined by the antenna’s transfer function. The receptive
bandwidth for the antennas used in the test stand is a range of frequencies
with a bandwidth on the order of a few GHz centered around the main cyclotron
carrier frequency of 25.898 GHz. Therefore, the higher order harmonics as well
as the zero frequency term can be ignored when considering only the signals
that will be received by the antenna array.

Considering only the 1st order harmonic term from Equation 2.3, which
represents the portion of the electric field that will be detected by the
array, and evaluating this at the retarded time we obtain the following for
the $\phi$-polarized electric fields

$E_{\phi}\propto\cos{\left(\omega_{c}\left(t-|\vec{R}|/c\right)-\Delta\right)},$
(2.4)

where the arbitrary phase $\Delta$ is defined by $A_{k}=|A_{k}|e^{i\Delta}$.
We are interested in the characteristics of the amplitude of the electric
field as a function of the radial distance component ($|\vec{R}|$) of the
retarded time. In particular, the maximum of $E_{\phi}$ occurs when the
argument of the cosine function is equal $n\pi$ where $n\in\\{0,\pm 2,\pm
4,...\\}$; however, the solutions where $n$ is negative can be discarded since
they represent unphysical negative overall phases. Applying this condition to
Equation 2.4 gives a condition on the radial position of the maximum of
$E_{\phi}$

$\displaystyle\omega_{c}(t-|\vec{R}|/c)-\Delta$ $\displaystyle=n\pi,$ (2.5a)
$\displaystyle|\vec{R}|$
$\displaystyle=\frac{c}{\omega_{c}}\left((\omega_{c}t-\Delta)-n\pi\right),$
(2.5b)

which is a function of time in the frame of the moving electron ($t$).
Equation 2.5 can be further simplified by noticing that the azimuthal position
of the electron ($\phi_{e}(t)$) as a function of time is defined by
$\phi_{e}(t)=\omega_{c}t-\Delta$ which reduces Equation 2.5 to

$|\vec{R}|=\frac{c}{\omega_{c}}(\phi_{e}(t)-n\pi).$ (2.6)

Equation 2.6 represents an archimedian spiral which is formed when plotting
the amplitude of $E_{\phi}$ in the x-y plane. The solution where $n=0$
represents the leading edge of the radiation spiral which propagates outward
from the electron at the speed of light. The additional solutions for $n>0$
represent the persistent spiral at radii inside the leading edge of the
radiated fields that have not yet been detected by the receiver at the current
time. In Figure 4(a) we show the expected spiral pattern for the maxima of the
cyclotron radiation.

In particular, we note that for the circular array geometry of the test stand,
depicted as the series of circles in Figure 4(a), each antenna receives a
linearly polarized wave with a phase offset that corresponds to the azimuthal
angle for that antenna element. Therefore, as we show in Figure 4(b), when the
relative phase of the received signal is plotted as a function of the
recieving antenna’s azimuthal position the result is also an Archimedean
spiral.

(a)

(b)

Figure 4:  The amplitude maxima of the cyclotron radiation form an Archimedean
spiral as the radiation propagates outward from the cyclotron orbit center
(a). A circular antenna array located at a fixed radius from the orbit center
will receive electric fields with equal magnitude in each of its channels, but
the phase of the electric field incident on each array channel will be
linearly out of phase from its neighbor antennas by an amount equal to the
angular separation of the two channels (b).

Based on these analytical calculations we can characterize the magnitude,
polarization, and phase of the signals received by the antenna array using
three criteria. These criteria are the basis of comparison for the radiation
produced by the SYNCA and cyclotron radiation emitted by electrons and will be
used to evaluate the performance of antenna designs. The criteria are:

  1. 1.

Electric fields that are $\phi$-polarized near $\theta=90^{\circ}$

  2. 2.

Uniform time-averaged electric field magnitudes around the circumference of a
circle centered on the antenna

  3. 3.

Electric fields whose phase is equal to the azimuthal angle at the point of
measurement plus a constant

The Locust simulation package [10] can be used to directly simulate the EM
fields generated by electrons performing cyclotron motion to validate the
analytical calculations. Locust simulates the EM fields by first calculating
the trajectory of the electrons in the magnetic trap using the Kassiopeia
software package [13]. The trajectory can then be used to solve for the EM
fields using the Liénard-Wiechert equations directly with no approximations.
The resulting electric field solutions drive a receiving antenna by convolving
the time-domain fields with the finite-impulse response filter of the antenna
or they can be examined directly to study the field characteristics that the
SYNCA must reproduce. In the next section we compare the radiation field
patterns for electrons simulated with Locust to patterns from a SYNCA antenna
design.

##  3 Design of the SYNCA

One potential SYNCA design is the crossed-dipole antenna [14]. A crossed-
dipole antenna consists of two dipole antennas, one of which is rotated
$90^{\circ}$ with respect to the other, which are fed with signals that are
out of phase from the opposite dipole by $90^{\circ}$ (see Figure 5).

Figure 5:  An idealized crossed-dipole antenna consists of two electric dipole
antennas oriented perpendicular to each other and is fed with four signals
with a quadrature phase relationship. An example antenna feed circuit is shown
which is composed of a chained combination of a quadrature hybrid-coupler
(Quad) and two baluns.

This arrangement causes the signals fed to each arm of the dipole to be out of
phase from each of the neighboring arms by $90^{\circ}$, which mirrors the
spatial phase relationship of cyclotron radiation fields.

A potential drawback of this design is that standard crossed-dipole antennas
do not radiate uniform electric fields near the $\theta=\pi/2$ plane. Typical
crossed-dipole antennas use dipole arm lengths equal to $\lambda/4$ or larger
[14], where $\lambda$ is the wavelength at the desired operating frequency.
Such large arm lengths cause the electric field magnitude to vary
significantly around the circumference of the antenna. However, making the
antenna electrically small by shrinking the arm length can improve the antenna
pattern uniformity.

In general, the criterion for an electrically small antenna is that the
largest dimension of the antenna ($D$) obey $D\lesssim\lambda/10$ [15]. In our
application, we are attempting to mimic the cyclotron radiation emitted by
electrons produced from tritium $\beta$-decay with energies near the spectrum
endpoint. For a background magnetic field of 1 T, the corresponding cyclotron
frequency of tritium endpoint electrons is approximately 26 GHz. Therefore,
the electrically small condition would require that the largest dimension of
the crossed-dipole antenna be smaller than 1.2 mm.

A crossed-dipole antenna with an overall size of 1.2 mm is challenging to
fabricate due to the small dimensions of the dipole arms that, in practice,
are fragile and unsuitable for use as a calibration probe. To mitigate some of
the challenges with the fabrication of such a small antenna, a variant
crossed-dipole antenna design using printed circuit board (PCB) technology
(see Figure 6) was developed in partnership with an antenna prototyping
company, Field Theory Consulting 222https://fieldtheoryinc.com/.

Figure 6:  A model of the PCB crossed-dipole antenna with dimensions. The
design has an inside diameter of 2.16 mm for the central circular trace, which
is 0.13 mm wide. The dipole arms each have a width of 1.27 mm and protrude
beyond the circular trace by 1.40 mm, which gives an overall width of 4.96 mm
for the length of the antenna PCB trace from end-to-end. The overall size of
the antenna is 20.0 mm the majority of which is the PCB dielectric material.
This design was observed in simulation to maintain the field characteristics
of the idealized crossed-dipole while being simpler to fabricate due to the
increased size of the antenna.

The PCB crossed-dipole design uses four rectangular pads to represent the
dipole arms, which are connected by a thin circular trace. The circular trace
both adds mechanical stability to the antenna and improves the azimuthal
uniformity of the electric fields compared to a more standard crossed-dipole
geometry. Futhermore, the circular trace allows for a greater separation
between dipole arms than standard crossed-dipoles, which is required to
accommodate the coaxial connections to each pad. The pads each contain a
through-hole solder joint to connect coaxial transmission lines using hand
soldering. The antenna PCB has no ground plane on the bottom layer as this was
observed in simulation to significantly distort the radiation pattern in the
plane of the PCB. The only ground planes present in the model are the outer
conductors of the four coaxial transmission lines which feed the antenna.
These are left unterminated on the bottom of the PCB dielectric material.

The antenna design development utilized a combination of Locust electron
simulations and antenna simulations using ANSYS HFSS [16], a commercial
finite-element electromagnetic simulation software. Two antenna designs were
simulated: an idealized electrically small crossed-dipole antenna with an arm
length of 0.40 mm and an arm separation of 0.05 mm, as well as a PCB crossed-
dipole antenna with the dimensions shown in Figure 6. Plotting the magnitude
of the electric fields generated by the antennas across a 10 cm square located
in the same plane as the respective antennas reveals the expected cyclotron
spiral pattern (see Figure 7) which closely matches the prediction for
simulated electrons. The spiral pattern demonstrates that the electric fields
have the appropriate phases to mimic cyclotron radiation, which fulfills SYNCA
criterion 3 identified in Section 2.

Figure 7: A comparison of the electric field magnitudes, normalized by the
maximum value of the electric field in each simulation, plotted on a 10 cm
square to visualize the Archimedean spirals formed by the electron (a), the
crossed-dipole antenna (b), and a PCB crossed-dipole antenna (c). The matching
patterns indicate that the electric fields have similar phase characteristics.
These images were generated using Locust simulations for the electron and
ANSYS HFSS for both antennas.

As we can see from Figure 8, the crossed-dipole antenna, which uses an
idealized geometry, exhibits good agreement with simulation. The antenna has a
maximum deviation from a simulated electron of approximately 0.5 dB in the
total electric field, 1 dB for the $\phi$-polarized electric field and 1 dB
for the $\theta$-polarized electric field.

In comparison, the pattern of the PCB crossed-dipole antenna, because the
simulation incorporates the geometry of the coax transmission lines, exhibits
some distortion from the idealized cross-dipole simulations. The vertically
oriented ground planes of the coax lines introduce more $\theta$-polarized
electric fields than are observed for simulated electrons near
$\theta=90^{\circ}$. The significant $\theta$-polarized field minimum is still
present but shifted to approximately $\theta=65^{\circ}$.

Figure 8: A comparison of the normalized electric field magnitudes for the
ideal crossed-dipole, PCB crossed-dipole, and a simulated electron as a
function of the polar angle ($\theta$). (a) Shows the total electric field,
(b) shows the $\phi$-polarized electric field component, and (c) shows the
$\theta$-polarized electric field component. These images were generated using
Locust simulations for the electron and ANSYS HFSS for both antennas.

The $\theta$-polarized field deviations of the PCB crossed-dipole antenna
should not greatly impact the performance of the antenna because the receiving
antenna array is primarily $\phi$-polarized. Therefore deviations in the
$\theta$-polarized fields will be suppressed due to the polarization mismatch.
More importantly, the $\phi$-polarized electric field pattern generated by the
PCB crossed-dipole closely matches simulated electrons across the polar angle
range of $50^{\circ}<\theta<150^{\circ}$. In this region the PCB crossed-
dipole differs by less than $0.5$ dB from simulated electrons. This range
greatly exceeds the beamwidth of the receiving antenna array which is designed
to be most sensitive to fields produced near $\theta=90^{\circ}$. Therefore,
we conclude that the PCB crossed-dipole antenna generates a $\phi$-polarized
radiation pattern that fulfills SYNCA criterion 1 from Section 2.

The final SYNCA criterion is related to the uniformity of the electric fields
when measured azimuthally around the antenna. As we saw for real electrons in
Section 2 it is expected that the magnitude of the electric field be
completely uniform as a function of the azimuthal angle due to the symmetry of
the cyclotron orbit. In Figure 9 we plot the total electric field as a
function of azimuthal angle for an electron, the crossed-dipole antenna, and
the PCB crossed-dipole antenna.

Figure 9: A comparison of the normalized electric field magnitudes for the
crossed-dipole, PCB crossed-dipole, and a simulated electron as a function of
the azimuthal angle ($\phi$) evaluated at $\theta=90^{\circ}$. This image was
generated using Locust simulations for the electron and ANSYS HFSS for both
antennas.

The crossed-dipole antenna exhibits perfect uniformity around the azimuthal
angle, whereas the PCB crossed-dipole has a small periodic deviation with a
maximum difference of 0.3 dB caused by the coaxial transmission lines below
the PCB. Such a small deviation from uniformity is acceptable since it is
smaller than the expected variation in uniformity caused by imperfections in
the antenna fabrication process, which modifies the antenna shape in an
uncontrolled manner by introducing solder blobs with a typical size of a few
tenths of a millimeter on the dipole arms (see Figure 10). Additionally, the
SYNCA will be separately calibrated to account for azimuthal differences in
the electric field magnitude. Therefore we see from the simulated performance
of the PCB crossed-dipole antenna that this antenna design meets all three of
the SYNCA criteria.

##  4 Characterization of the SYNCA

Two SYNCAs were manufactured using the PCB crossed-dipole design (see Figure
10). The antenna PCB (Matrix Circuit Board Materials, MEGTRON 6) is connected
to four 2.92 mm coaxial connectors (Fairview Microwave, SC5843) using semi-
rigid coax (Fairview Microwave, FMBC002), which also physically support the
antenna PCB. The antenna PCB consists only of two layers which correspond to
the copper antenna trace and the PCB dielectric. Each coax line is connected
to the associated dipole arm using through-hole soldering and phase matched to
ensure that the electrical length of each of the transmission lines is
identical at the operating frequency. The antenna PCB is further reinforced
using custom cut polystyrene foam blocks, which have an electrical
permittivity nearly identical to air. A custom 3D printed mount is included at
the base of the antenna to support the coax connectors and to provide a sturdy
mounting base.

(a)

(b)

Figure 10:  (a) A cartoon schematic which highlights the routing of the semi-
rigid coax transmission lines. (b) A photograph of a SYNCA constructed using
the modified crossed-dipole PCB antenna design. Visible in the photograph of
the SYNCA are four blobs of solder which are an artifact of the SYNCA’s hand-
soldered construction. These solder blobs are the most significant deviation
from the SYNCA design shown in Figure 6 and are responsible for a significant
fraction of the irregularities seen in the antenna pattern.

Characterization measurements were performed using a Vector Network Analyzer
(VNA) to measure the electric field magnitude and phase radiated by the SYNCA
to verify the radiation pattern (see Figure 11). The VNA is connected to the
SYNCA at one port through a hybrid-coupler whose outputs are connected to two
baluns to generate the signals with the appropriate phases to feed the SYNCA
(see Figure 5). The other port of the VNA is connected to a single reference
horn antenna that serves as a field probe. To position the SYNCA, a
combination of translation and rotation stages are used to characterize the
antenna’s fields across the entire radiation pattern circumference. This
measurement scheme is equivalent to measuring the fields generated by the
SYNCA using a full circular array of probe antennas.

The antenna measurement space is surrounded by RF anti-reflective foam to
isolate the measurements from the lab environment (see Figure 11(b)) and
remaining reflections are removed using the VNA’s time-gating feature. The
SYNCA is affixed to the stages by a custom RF transparent mount made of
polystyrene foam. The coaxial cables deliver the antenna feed signals
generated by the VNA to the SYNCA while still allowing unrestricted rotation.
The horn antenna probe is nominally positioned in the plane formed by the
antenna PCB ($\theta=90^{\circ}$ or $z=0$ mm) at a distance of 10 cm from the
SYNCA, to match the expected position of the antenna array relative to the
SYNCA in the antenna array test stand. The horn antenna can be manually raised
or lowered to different relative vertical positions to characterize the
radiation pattern at different polar angles.

(a)

(b)

Figure 11:  A schematic of the VNA characterization measurements (a). This
setup allows for antenna gain and phase measurements across a full
$360^{\circ}$ of azimuthal angles using a motorized rotation stage and control
of the radial position of the SYNCA using a translation stage. A photo of the
setup in the lab is shown in (b).

Several $360^{\circ}$ scans were performed with probe vertical offsets of
-10.0 mm, -5.0 mm, 0.0 mm, 5.0 mm, and 10.0 mm relative to the antenna PCB
plane. These probe offsets cover a 2 cm wide vertical region centered on the
SYNCA PCB, approximately equal to $\pm 6$ degrees of polar angle. The
measurements show that the SYNCA is generating fields with nearly isotropic
magnitude across the probed region. The standard deviation of the electric
field magnitude measured around the antenna circumference is approximately 2.9
dB for a typical rotational scan. The presence of a significant pattern null
is noted near $45^{\circ}$ (see Figure 12), which we attribute to small
imperfections in the antenna PCB that could be introduced from the hand
soldered terminations connecting the coax cables to the antenna. There is no
significant difference in the radiation pattern when measured across the 2 cm
vertical range. The measured relative phases closely follow the expectation
for an electron, being linear with the measurement rotation angle and forming
the expected spiral pattern. Other than the small phase imperfections there is
a slight sinusoidal bias to the phase data, which we determined is the result
of a small ($\lesssim 1$ mm) offset of the antenna’s phase center from the
rotation axis of the automated stages.

(a)

(b)

Figure 12:  Linear interpolations of the measured electric field magnitude (a)
and phase (b). The data was acquired using a VNA at 120 points spaced by 3
degrees from 0 to 357 degrees of azimuthal angle. The different color lines
indicate the vertical offset of the horn antenna relative to the SYNCA PCB and
the dashed line shows the expected shape from electron simulations. No
significant difference in the antenna pattern is observed for the measured
vertical offsets.

The characterization measurements confirm the simulated performance of the
SYNCA. As expected the fields generated by the antenna are nearly isotropic in
magnitude, $\phi$-polarized, and are linearly out of phase around the
circumference of the antenna as predicted for cyclotron radiation in Section
2. Small imperfections in the magnitude and phase of the antenna are expected,
particularly at the antenna’s high operating frequency of 26 GHz where small
geometric changes can have significant impacts on electrical properties.
However, calibration through careful characterization measurements can be used
to remove the majority of these pattern imperfections, including the
relatively large pattern null near $45^{\circ}$, which will allow for the
usage of the SYNCA as a test source for free-space CRES experiments utilizing
antenna arrays. In the next section we use the VNA measurements obtained here
as a calibration for signal reconstruction using digital beamforming.

##  5 Signal Reconstruction with the SYNCA

(a)

(b)

Figure 13:  (a) A depiction of the relative phase differences for signals
received by a circular antenna array from an isotropic source. The phases
correspond to a unique spatial position. (b) A schematic of the setup used to
perform digital beamforming.

Digital beamforming is a standard technique for signal reconstruction using a
phased array [17]. The SYNCA, since it exhibits the same cyclotron phases as a
trapped electron, can be used to perform simulated CRES digital beamforming
reconstruction experiments on the bench-top without the need for the magnet,
cryogenics, and vacuum systems required by a full CRES experiment. The fields
received by the individual elements of the antenna array will have phases
dependent on the spatial position of the source relative to the antennas.
Therefore, a simple summation of the received signals will fail to reconstruct
the signal due to destructive interference between the individual channels in
the array. However, applying a phase shift associated with the source’s
spatial position removes phase differences and results in a constructive
summation of the channel signals (see Figure 13). We can summarize the digital
beamforming operation succinctly using the following equation

$y[t_{n}]=\sum_{m=0}^{N-1}x_{m}[t_{n}]A_{m}e^{i\phi_{m}},$ (5.1)

where $y[t_{n}]$ represents the summed array signal at time $t_{n}$,
$x_{m}[t_{n}]$ is the signal received by channel $m$ at time $t_{n}$,
$\phi_{m}$ is the phase shift applied to the signal received at channel $m$,
and $A_{m}$ is an amplitude weighting factor that accounts for the different
signal power received by individual channels. By changing the digital
beamforming phases, the point of constructive interference can be scanned
across the sensitive region of the array to search for the location of a
radiating source, which is identified as the point of maximum summed signal
power above a specified threshold. The digital beamforming phases consist of
two components,

$\phi_{m}=2\pi d_{m}/\lambda+\theta_{m},$ (5.2)

where $d_{m}$ is the distance from the $m$-th array element to the source, and
$\theta_{m}$ is the relative angle between the source position and the $m$-th
antenna. The first component is the standard digital beamforming phase that
corresponds to the spatial position of the source, and the second component is
the cyclotron phase that corresponds to the relative azimuthal phase offset.

Figure 14:  Digital beamforming maps generated using a simulated 60 channel
array and electron simulated using the Locust package. (a) and (b) show the
beamforming maps for simulated electrons without the cyclotron spiral phases
and with the cyclotron spiral phases respectively. (c) and (d) show the
beamforming maps produced from SYNCA measurements. We observe good agreement
between simulated electrons and the SYNCA measurements.

With a small modification to the hardware used to characterize the SYNCA (see
Figure 11), we can perform a digital beamforming reconstruction of a synthetic
CRES event. By replacing the VNA with an arbitrary waveform generator (AWG),
the SYNCA can be used to generate cyclotron radiation with an arbitrary signal
structure, which can then be detected by digitizing the signals received by
the horn antenna. Rotational symmetry allows us to use the rotational stage of
the positioning system to rotate the SYNCA to recreate the signals that would
have been received by a complete circular array of antennas.

Using this setup, signals from a 60 channel circular array of equally spaced
horn antennas were generated with the SYNCA positioned 10 mm off the central
array axis, reconstructed using digital beamforming, and compared to Locust
simulation (see Figure 14). When the cyclotron spiral phases are not used,
which is equivalent to setting $\theta_{m}$ in Equation 5.2 to zero, the
SYNCA’s position is reconstructed as a relatively faint ring as predicted by
simulation. However, when the appropriate cyclotron phases are used during the
beamforming procedure, both the simulated electron and the SYNCA appear as a
single peak of high relative power corresponding to the source position.
Therefore, we observe good agreement between the simulated and SYNCA
reconstructions. While it may seem that for the case with no cyclotron phase
corrections the ring reconstructs the position of the electron as effectively
as beamforming with the cyclotron phase corrections, it is important to note
that the simulations and measurements were generated without a realistic level
of thermal noise. The larger maxima region and lower signal power, which
occurs without the cyclotron phase corrections, significantly reduce the
probability of detecting an electron in a realistic noise background.

(a)

(b)

Figure 15:  A plot of the SYNCA’s reconstructed position using the synthesized
horn-antenna array and digital beamforming. (a) Shows the reconstructed
position of the SYNCA compared with the target position indicated by the
positioning system readout. (b) Shows the reconstruction error, which is the
difference between the target and reconstructed positions. The error bars in
(b) are the uncertainty in the mean position of the 2D Gaussian used to fit
the digital beamforming reconstruction peak obtained from the fit covariance
matrix. The mean fit position uncertainty of 0.02 mm is an order of magnitude
smaller than the typical reconstruction error of 0.3 mm obtained by
calculating the standard deviation of the difference between the reconstructed
and target position.

To bound the beamforming capabilities of the synthetic array of horn antennas,
we performed a series of beamforming reconstructions where the SYNCA was
progressively moved off the central axis of the array (see Figure 15). To
extract an estimate of the position of the SYNCA using the digital beamforming
image we apply a 2-dimensional (2D) Gaussian fit to the image data and extract
the estimated centroid value. We find that the synthetic horn antenna array
reconstructs the position of the SYNCA with a $1\sigma$-error of 0.3 mm with
no apparent trend across the 30 mm measurement range. This reconstruction
error is an order of magnitude larger than mean fit position uncertainty of
0.02 mm indicating that systematic effects related to the SYNCA positioning
system could be contributing additional uncertainty to the measurements. Note
that the current mean reconstruction error of 0.3 mm is a factor of 20 smaller
than the full width at half maximum of the digital beamforming peak (6 mm),
which could be interpreted as a naive estimate of the position reconstruction
performance of this technique. Because these measurements are intended as a
proof-of-principle demonstration, we do not investigate potential sources of
systematic errors further; however, we expect that a similar and more thorough
investigation will be performed using the Project 8 antenna array test stand,
where typical reconstruction errors can be used to estimate the energy
resolution limits of antenna array designs.

##  6 Conclusions

In this paper we have introduced the SYNCA, which is a novel antenna design
that emits radiation that mimics the unique properties of the cyclotron
radiation generated by charged particles moving in a magnetic field. The
characterization measurements of the SYNCA validated the simulated performance
of the PCB crossed-dipole antenna design. Additionally, the SYNCA was used to
estimate the position reconstruction capabilities of a synthesized array of
horn antennas and experimentally reproduced the simulated digital beamforming
reconstruction of electrons.

While the SYNCA performs well, there exist discrepancies in the phase and
magnitude of the radiation pattern compared to the simulated SYNCA design that
are related to the small geometric differences in the soldered connections.
Future design iterations that replace the soldered connections with a fully
surface mount design could improve the radiation pattern at the cost of some
complexity and expense. Furthermore, improving the design of the antenna PCB
and mounting system would allow the antenna to be inserted into a cryogenic
and vacuum environment where in-situ antenna measurement calibrations could be
performed.

The discrepancies in the radiation pattern and phases exhibited by the as-
built SYNCA should not greatly impact its performance as a calibration probe.
Both magnitude and phase variations can be accounted by applying the SYNCA
characterization measurements as a calibration to the data collected by the
antenna array test stand. The separate calibration of the SYNCA radiation does
not impact the primary goals for the antenna array test stand which are array
calibration and signal reconstruction algorithm performance characterization,
because it can be performed with standard reference horn antennas with well
understood characteristics.

The SYNCA antenna technology advances the CRES technique by providing a
mechanism to characterize free-space antenna arrays for CRES measurements
without the need for a magnet and cryogenics system, which would be required
for calibration using electron sources. Both the Project 8 collaboration as
well as future collaborations which are developing antenna array based CRES
experiments can make use of SYNCA antennas as an important component of their
calibration and commissioning phases.

##  Appendix A Dominant Electric Field Polarization

Consider an electron following a circular cyclotron orbit in a uniform
magnetic field whose guiding center is positioned at the origin of the
coordinate system. The equation of motion can be expressed as

$\vec{r}_{s}=(r_{c}\cos{\omega_{c}t_{r}})\hat{x}+(r_{c}\sin{\omega_{c}t_{r}})\hat{y}.$
(A.1)

For single antenna located along the y-axis at position $\vec{r}=r_{a}\hat{y}$
we are interested in the incident electric fields from the electron. The
electric field is given by Equation 2.1, which we evaluate in the regime where
$r_{a}\gg r_{c}$. This limit can justified by comparing the radius of the
cyclotron orbit for an electron with the tritium beta-spectrum endpoint energy
of 18.6 keV in a 1 T magnetic field to the typical ($r_{a}\simeq 100$ mm)
radial position of the receiving antenna. We find that the cyclotron orbit has
a radius of 0.46 mm which is approximately a factor of 200 smaller than the
typical antenna radial position. In this regime we can make the approximation
$\vec{R}\simeq r_{a}\hat{y}$ and the expression for the electric field at the
antenna’s position becomes

$\vec{E}=\frac{e}{\gamma^{2}r_{a}^{2}}\frac{\hat{x}(\frac{r_{c}\omega_{c}}{c}\sin{\omega_{c}t_{r}})+\hat{y}(1-\frac{r_{c}\omega_{c}}{c}\cos{\omega_{c}t_{r}})}{(1-\frac{r_{c}\omega_{c}}{c}\cos{\omega_{c}t_{r}})^{3}}-\frac{e}{cr_{a}}\frac{\hat{x}(\frac{r_{c}^{2}\omega_{c}^{3}}{c^{2}}-\frac{r_{c}\omega_{c}^{2}}{c}\cos{\omega_{c}t_{r}})}{(1-\frac{r_{c}\omega_{c}}{c}\cos{\omega_{c}t_{r}})^{3}}.$
(A.2)

Since the receiving antenna is part of a circular array of antennas, it is
useful to rewrite Equation A.2 in terms of the azimuthal ($\hat{\phi}$) and
radial ($\hat{r}$) polarizations. Making use of the fact that for an antenna
located at $R=r_{a}\hat{y}$ that $\hat{\phi}=-\hat{x}$ and $\hat{r}=\hat{y}$
we find

$\displaystyle\vec{E}$ $\displaystyle=\hat{\phi}E_{\phi}+\hat{r}E_{r}$ (A.3)
$\displaystyle E_{\phi}$
$\displaystyle=\frac{e}{(1-\frac{r_{c}\omega_{c}}{c}\cos{\omega_{c}t_{r}})^{3}}\left[-\frac{\frac{r_{c}\omega_{c}}{c}\sin{\omega_{c}t_{r}}}{\gamma^{2}r_{a}^{2}}+\frac{\omega_{c}\left(\frac{r_{c}^{2}\omega_{c}^{2}}{c^{2}}-\frac{r_{c}\omega_{c}}{c}\cos{\omega_{c}t_{r}}\right)}{cr_{a}}\right]$
(A.4) $\displaystyle E_{r}$
$\displaystyle=\frac{e(1-\frac{r_{c}\omega_{c}}{c}\sin{\omega_{c}t_{r}})}{\gamma^{2}r_{a}^{2}(1-\frac{r_{c}\omega_{c}}{c}\cos{\omega_{c}t_{r}})^{3}}.$
(A.5)

For the purposes of designing a synthetic cyclotron radiation antenna we are
interested in the dominant electric field polarization emitted by the
electron. The antenna is being designed to mimic the cyclotron radiation
produced by electrons with kinetic energies of approximately 18.6 keV in a 1 T
magnetic field. Since the relativistic beta factor for an electron with this
kinetic energy is $|\vec{\beta}|\simeq\frac{1}{4}$, the approximations
$\gamma\simeq 1$ and $\frac{r_{c}\omega_{c}}{c}\simeq\frac{1}{4}$ are
justified. Inserting these expressions into the equations for the electric
field components above simplifies the comparison of the magnitudes of the two
components. Additionally, we compare the time-averaged magnitudes to evaluate
the root mean squared electric field ratio. The time-averaged ratio of the
radial and azimuthally polarizied electric fields with the above
simplifications is given by

$\frac{\left<|E_{r}|\right>}{\left<|E_{\phi}|\right>}=\frac{8-\sqrt{2}}{\left|1-\frac{r_{a}}{r_{c}}\frac{1-2\sqrt{2}}{8}\right|}\simeq\frac{r_{c}}{r_{a}}\frac{8(8-\sqrt{2})}{2\sqrt{2}-1}=0.13,$
(A.6)

where we have made use of the fact that for these magnetic fields and kinetic
energies the cyclotron radius is much smaller than the radius of the antenna
array.

From Equation A.6 we see that the time-averaged azimuthal polarization is
larger than the radial polarization by about a factor of 8, which makes it the
dominant contribution to the electric fields at the position of the antenna.
We must also consider the directivity of the receiving antenna which can have
a gain that is disproportionately large for a specific polarization component.
Because the $E_{\phi}$ component is dominant, the receiving antenna array is
designed with an azimuthal polarization, which negates the voltages induced in
the antenna from the radially polarized fields. Therefore, we conclude that
for the purpose of designing the SYNCA antenna it is sufficient to specify
that the emitted electric fields have a large azimuthal polarization.

##  Appendix B Simplification of the Expression for the Azimuthally Polarized
Electric Fields

Returning to Equation A.4 we can rearrange the terms into a more compact form

$E_{\phi}=\frac{e\frac{r_{c}\omega_{c}}{c}}{4r_{a}r_{c}}\left[\frac{\frac{r_{c}\omega_{c}}{c}-\cos{\omega_{c}t_{r}}-\frac{4r_{c}}{r_{a}}\sin{\omega_{c}t_{r}}}{(1-\frac{r_{c}\omega_{c}}{c}\cos{\omega_{c}t_{r}})^{3}}\right],$
(B.1)

where we have used the relation $\frac{c}{\omega_{c}}\simeq 4\gamma r_{c}$,
which is true for electrons with these kinetic energies, to replace the extra
factor of $\omega_{c}$ and replaced the relativistic gamma factor with $1$.
Recall, that this equation is valid for an electron moving in a circular
trajectory with radius $r_{c}$ whose electric fields are being received by an
antenna located in the plane of the cyclotron orbit at a distance $r_{a}$ from
the electron’s guiding center. This rather complicated expression can be
simplified using Fourier analysis. Assuming a background magnetic field of 1 T
and a kinetic energy of 18.6 keV we calculate numerically the electric field
using Equation B.1 and apply a discrete Fourier Transform to visualize the
frequency spectrum (see Figure 16).

Figure 16: A plot of the numeric solution to Equation B.1. The time-domain
representation of the signal (a) is composed of a zero frequency term and a
series of harmonics separated by the main cyclotron frequency as shown in the
plot of the frequency spectrum (b). We can see that the relative amplitude of
the harmonics beyond $k=7$ are smaller than the main carrier by a factor of
about $10^{-5}$ and are completely negligible.

We observe that the azimuthally polarized electric field is periodic with a
base cyclotron frequency of 25.898 GHz corresponding to the highest power
frequency component in Figure 16. The frequency spectrum reveals that the
signal is composed of a constant term with zero frequency and a series of
harmonics separated by 25.898 GHz. Therefore, we can represent the azimuthal
electric fields from the electron as a linear combination of pure sinusoids
with frequencies given by $\omega_{k}=k\omega_{c}$ ($k\in 0,1,2...$) and
amplitudes extracted from the Fourier representation. The exponential decay of
the amplitudes implies that it will usually be sufficient to truncate the sum
after the 4th harmonic whose amplitude is approximately a factor of 100 less
than the 1st order frequency term. Using this representation we can transform
the equation for the azimuthally polarized electric fields in Equation B.1
into

$E_{\phi}=\frac{e\frac{r_{c}\omega_{c}}{c}}{4r_{a}r_{c}}\sum_{k=0}^{7}{A_{k}e^{i\omega_{k}t_{r}}},$
(B.2)

where we have truncated the sum over harmonics at the 7th order for
completeness. The amplitudes $A_{k}$ are dimensionless complex numbers, which
encode the relative powers of the harmonics as well as the starting overall
phase of the cyclotron radiation.

## Acknowledgments

A special thanks to Field Theory Consulting, which helped to design the PCB
crossed-dipole antenna and fabricated the SYNCA’s.

This material is based upon work supported by the following sources: the U.S.
Department of Energy Office of Science, Office of Nuclear Physics, under Award
No. DE-SC0020433 to Case Western Reserve University (CWRU), under Award No.
DE-SC0011091 to the Massachusetts Institute of Technology (MIT), under Field
Work Proposal Number 73006 at the Pacific Northwest National Laboratory
(PNNL), a multiprogram national laboratory operated by Battelle for the U.S.
Department of Energy under Contract No. DE-AC05-76RL01830, under Early Career
Award No. DE-SC0019088 to Pennsylvania State University, under Award No. DE-
FG02-97ER41020 to the University of Washington, and under Award No. DE-
SC0012654 to Yale University; the National Science Foundation under Award No.
PHY-2209530 to Indiana University, and under Award No. PHY-2110569 to MIT;
This work has been supported by the Cluster of Excellence "Precision Physics,
Fundamental Interactions, and Structure of Matter" (PRISMA+ EXC 2118/1) funded
by the German Research Foundation (DFG) within the German Excellence Strategy
(Project ID 39083149); the Karlsruhe Institute of Technology (KIT) Center
Elementary Particle and Astroparticle Physics (KCETA); Laboratory Directed
Research and Development (LDRD) 18-ERD-028 and 20-LW-056 at Lawrence Livermore
National Laboratory (LLNL), prepared by LLNL under Contract DE-AC52-07NA27344,
LLNL-JRNL-840701; the LDRD Program at PNNL; Indiana University; and Yale
University. A portion of the research was performed using the HPC cluster at
the Yale Center for Research Computing. Computations for this research were
performed on the Pennsylvania State University’s Institute for Computational
and Data Sciences’ Roar supercomputer.

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