# THz-Empowered UAVs in 6G:
Opportunities, Challenges, and Trade-Offs

M. Mahdi Azari, , Sourabh Solanki, ,
Symeon Chatzinotas, , Mehdi Bennis M. M. Azari, S. Solanki, S. Chatzinotas are
with the SnT, University of Luxembourg. M. Bennis is with the University of
Oulu.

###### Abstract

Envisioned use cases of unmanned aerial vehicles (UAVs) impose new service
requirements in terms of data rate, latency, and sensing accuracy, to name a
few. If such requirements are satisfactorily met, it can create novel
applications and enable highly reliable and harmonized integration of UAVs in
the 6G network ecosystem. Towards this, terahertz (THz) bands are perceived as
a prospective technological enabler for various improved functionalities such
as ultra-high throughput and enhanced sensing capabilities. This paper focuses
on THz-empowered UAVs with the following capabilities: communication, sensing,
localization, imaging, and control. We review the potential opportunities and
use cases of THz-empowered UAVs, corresponding novel design challenges, and
resulting trade-offs. Furthermore, we overview recent advances in UAV
deployments regulations, THz standardization, and health aspects related to
THz bands. Finally, we take UAV to UAV (U2U) communication as a case-study to
provide numerical insights into the impact of various system design parameters
and environment factors.

##  I Introduction

In the envisioned 6G networks, UAVs are going to play a prominent role in
several aspects such as network intelligence and autonomy [1] thanks to their
flexibility in 3D positioning and their technological advancement in cost-
effective manufacturing. On the other hand, to satisfy the proliferating
demands in 6G, terahertz (THz) band stands among the candidate enablers [1,
2]. This is owing to the underlying potential of THz band in characterizing
higher throughput, lower latency, accurate localization, and precise sensing
and imaging capabilities. Recent advancements in the field of semiconductor
industries enable design of more compact THz devices which was traditionally
one of the major barriers in THz deployment [3]. Therefore, towards the
progression of 6G networks, THz-empowered UAVs may offer several
opportunities, opening the door for new services and novel applications [1].
Regarding this, Section III presents the prospective use cases and
opportunities enabled by THz-empowered UAVs.

Despite the huge potential of THz band, there is still a long way to go before
we see its practical applications in wireless communications. However, the
persistent efforts across the research community over the past few years
helped this speculative technology become increasingly attainable. The roadmap
towards advancing the technology further has also been actively discussed in
various scenarios [4, 5]. THz-empowered UAVs not only face the challenges
coming from the nature of such frequencies, they are also affected by the
intricacies of efficient UAVs deployment. For example, molecular noise,
absorption loss, and uncertain medium are major issues with the THz
transmissions which become more aggravating in outdoor scenarios with UAVs. On
the other hand, UAVs flexibility of positioning them in line-of-sight (LoS)
and adjusting their distances have great potential in using THz bands
effectively. This paper, in Section IV, highlights important design challenges
of THz-empowered UAVs along with induced trade-offs to provide relevant
insights into the new system design.

Certain barriers in the deployment of THz-empowered UAVs include regulatory
and health aspects pertaining to the use of THz frequencies, which are
overlooked in most academic reports. Such challenges are elaborated in Section
V, where the advancement of regulations, standardization, and safety aspects
are reviewed. This section is followed by Section VI where several system
design parameters and environment factors for quality communication between
UAVs are recognized and their impacts are investigated.A few works have
analyzed the performance of THz communications for UAV-ground links [6, 7],
but the underlying intricacies of combining the two technologies were not
discussed. To our best knowledge, this work is the first one that provides a
comprehensive study on UAVs empowered by THz functionalities where potential
use cases, new challenges, design trade-offs, regulatory and health aspects,
as well as numerical insights are discussed.

##  II Fundamental Features of Terahertz Communications

THz propagation exhibits certain distinct attributes reviewed in the sequel.

####  II-1 Channel and Propagation Characteristics

Unlike lower frequencies, wavelengths of THz are comparable to the size of
rain, dust, or snow, which makes attenuation due to molecular absorption more
severe. Specifically, the atmospheric absorption due to oxygen
($\textrm{O}_{2}$) and water ($\textrm{H}_{2}$O) molecules is significant in
the THz channel. Since this absorption is dependent on the concentration and
composition of the molecules along the path of the waves, the losses can vary
based on the meteorological conditions thereby making THz band highly
frequency-selective. THz waves also encounter spreading loss due to expansion
of EM waves along the medium. The path loss due to absorption and spreading
account for the LoS propagation of the THz waves. The path loss becomes more
significant with increase in the transmission distance, however, the molecular
absorption loss can be minimized by selecting a suitable bandwidth (0.38-0.44
THz, for example) where the losses are well below 10 dB per km [4]. In
addition to aggravating the attenuation, molecular absorption also introduces
a non-white noise depending on the mixture of molecules.

On the other hand, when LoS propagation is not feasible due to obstacles, THz
waves undergo diffuse scattering, specular reflections, diffraction etc.,
which account for the non-LoS (NLoS) propagation. These characteristics are
dependent on the geometry and material of the impinging surface as well as the
incident angle of EM wave. Essentially, the THz channel models should
encapsulate the free space and absorption losses which are dependent on
frequency, distance, altitudes, and relative air composition. Also, the path
loss is a function of 3D LoS/NLoS probability. Nonetheless, channel modelling
for THz propagation is yet to be fully understood. Several measurement
campaigns and research efforts are ongoing for its characterization [5].

####  II-2 Large Antenna Arrays for Pencil Beamforming

To mitigate the high transmission loss at THz, the adoption of large antenna
arrays for directional beamforming is inevitable. In fact, extremely small
wavelengths can facilitate the small-sized antenna design that can be deployed
on a large scale to form huge antenna arrays for sharped pencil beams. Such
directional transmission can compensate for the attenuation up to a certain
extent. However, it brings about additional challenges such as mobility and
handover management and efficient beam tracking and alignment. Further,
wideband beamforming is also a critical problem primarily caused by spatial-
wideband effect due to the use of large antenna arrays and frequency-
selectivity due to multipath delay [8]. Wideband effects are also prevalent at
mmWave band but they are more prominent at THz and require extensive research
efforts to explore the appropriate solutions.

Figure 1: Representative use cases of THz-empowered UAVs.

##  III Selected Use Cases and Opportunities

In the following, we discuss several important applications of THz-empowered
UAVs as illustrated in Figure 1.

####  III-1 Safe and Reliable Operation of UAVs

Large antenna arrays in small sizes at THz can enable cm-level localization
accuracy and sensing solutions with finer range Doppler and angular
resolutions [9]. Thus, THz-empowered UAVs possess the capability of accurate
localization and sensing which can facilitate the instantaneous perception of
the environment. Such functionalities lead to a safe and reliable navigation
of THz-empowered UAVs. Note that the safe operation of UAVs is crucial for
complying with the regulations in air-space and also for their practical
application. Empowering UAVs with THz can help in achieving this objective.
For instance, leveraging accurate sensing and localization, THz-empowered UAVs
can flexibly position them with the intended transmitters/receivers to improve
the reliability and safety. Moreover, by detecting the location and presence
of any obstacles or other aerial nodes, they can be safely navigated while
seamlessly avoiding any conflict or accident.

####  III-2 Interactive Aerial Telepresence

It is possible to deploy UAVs and control them remotely to perform unsafe,
costly, or time-sensitive tasks for humans. This is referred to as aerial
telepresence and it becomes even more powerful when augmented reality (AR) is
used which can offer 3D visuals and real-time tele-interaction with the
environment. In fact, such haptic guidance enhances UAV capability, opens up
new applications and, perhaps most importantly, provides access to experts
from anywhere at any time. For such applications, THz deployment is essential
particularly for real-time interactions where AR data cannot be compressed [2]
and high data rates are desired. For highly precise interactive tasks and
immersive experience, UAVs with THz sensors can provide accurate instantaneous
perception of the environment thereby offering remote cognitive capabilities.

####  III-3 Advanced Swarm of UAVs

To enhance the performance of UAVs networks and tackle the limits of an
individual UAV, UAVs may perform in a swarm [10]. In these scenarios, UAVs may
need to exchange large amount of data, particularly given the significant
growth of 6G data demand and increasing number of sensors per UAV for
integrated services in 6G e.g., sensing, localization, mapping, etc.
Accordingly, UAVs need to boost their interconnection rate capacity beyond the
current technology, compatible with the upcoming generations and services.
Ultra-high-throughput U2U communication link can be provided in THz given the
favorable LoS dominant propagation and the fact that UAVs can establish
sufficiently close communication range to mitigate high path loss and
molecular absorption.

####  III-4 Time-Critical MEC-Empowered Operation

Due to limited on-board computation capacity, UAVs may rely on mobile edge
computing (MEC) servers which impose significant delay in obtaining required
information. Providing sufficiently high-speed data link improves the time-
critical remote computations capability, enhances the freshness of data, and
eventually enables instantaneous environmental cognition. MEC-empowered
operation through THz links allows to reduce the UAVs payload resulting in
lower propulsion energy consumption and longer operational lifetime. A real-
time remote processing in turn creates new means of online NLoS UAVs control
and navigation through instantaneous perception of the environment. In this
application, when the environment is unknown, three steps of sensing,
processing, and navigating can be performed in real-time thanks to ultra-high-
throughput THz links between UAVs and the computing-and-controlling servers.

####  III-5 Pollution Monitoring and Tracking

Frequency scanning spectroscopy at THz band allows air quality monitoring and
detection of some chemicals since various materials at certain frequencies
have vibrational absorption. The specific absorption characteristics of
different gaseous medium significantly boost the THz’s sensing capability.
THz-spectroscopy coupled with UAV’s flexibility can offer effective solutions
for pollution monitoring and tracking. For instance, UAVs equipped with THz
spectrometer can fly near the source of pollution (e.g., industrial chimneys)
to detect the concentration of various pollutants. Also, THz-empowered UAVs
can fly up to the mountain peaks to detect the water vapour concentrations.
Such utilities of THz-empowered UAVs can help in combating the air pollution
and assist to keep track of climate change.

####  III-6 Rescue and Surveillance

THz characteristics combine the advantages of microwaves and visible light for
a practical implementation of NLoS imaging system. They involve large
bandwidths and small wavelengths for high spatial resolutions, and also
moderate scattering features for image reconstruction algorithms that are
computationally less expensive than optical systems [5]. Although the range of
THz NLoS imaging might be yet short, UAVs flexibility in 3D portioning can
address such limit by hovering in the proximity of a target area. UAVs
equipped with THz waves can view NLoS objects during rescue and surveillance,
expanding the benefits of THz imaging systems.

##  IV Design Challenges and Trade-Offs

In the following, we list specific challenges and potential trade-offs in the
deployment of THz-empowered UAVs for different services.

###  IV-A Challenges

####  IV-A1 UAVs Wobbling and Motion

Effective use of THz frequencies requires highly directional antennas at both
transmitter and receiver sides to compensate the large propagation loss.
However, UAVs wobbling/fluctuation due to wind can result in beam misalignment
which deteriorates the communication quality. Also, due to uncontrolled tilts
and rotations from UAVs motion, maintaining a perfect beam alignment would be
intricate thus, resulting into several beam hopping and handover issues. Such
impact gets worse as antenna footprint shrinks with larger antenna gain. Even
if such phenomena does not heavily reduce the overall rate, more frequent
handovers results in significant delay and adversely impact the link
reliability. To guarantee a certain level of reliability under delay
constraint, some handovers may be skipped at the expense of rate reduction
Further, we recall that sensing and localization capability in THz can assist
UAVs to operate reliably by precisely positioning the intended
transmitters/receivers. In addition, intelligent reflective surface (IRS)
enabled THz architecture can ease effective beam alignment by offering
enhanced sensing and localization [11].

####  IV-A2 Power-Hungry Payload

High hardware power consumption in THz, essentially corresponding to the
analog-to-digital converters (ADCs), is one of the main challenges for UAVs
with limited power budget. Although ultra-wide bandwidths in THz band appears
to be the major motivation to move towards higher frequencies, the power
consumption of ADCs remarkably increases when sampling rate surpasses 100 MHz
which grows exponentially with resolution [12]. At lower frequencies, it is
widely accepted that the propulsion power consumption is dominant over the
electrical part which facilitates several energy-efficient designs. In THz,
given the high number of antennas with the corresponding RF chains and ADCs
the total payload power consumption may not be negligible as compared to that
of propulsion. Therefore, it is important to update the design guidelines for
THz-empowered UAVs. To plan the THz-empowered UAV trajectory, longer traveling
is needed for a meaningful data transfer in contrast to low frequencies.

####  IV-A3 Limited Design Space

Optimal height of UAVs in low frequencies can be primarily achieved by
transitions from NLoS to LoS for more favorable propagation conditions, which
compensate for the adverse impact of longer link lengths at altitudes [1].
However, such findings should be re-visited for THz communication since the
communication quality drastically drops with the link length. For instance,
though the THz band can support Tbps links for designing an integrated access
and backhaul for a UAV BS, the benefits are only reachable within a short
range of communication, which limits the space for efficient and optimal UAV
placement. Accordingly, a more strategic and opportunistic planning of 3D
placements is needed. The new strategic planning could take advantage of
accurate sensing, mapping, and localization capabilities of THz to plan a
smarter deployment and establish a more favorable communication environment.
For instance, high resolution sensing in THz can be exploited to provide
predictive control of the serving users mobility. Although sensing in THz is
highly accurate, the range of sensing is short. To overcome the short range of
THz wave, multi-frequency operation can be helpful. UAVs may exploit mmWave
sensing capability to detect major environmental changes in longer range and
rely on THz for high-resolution sensing capability to track subtle changes in
shorter range.

####  IV-A4 Dense Deployment of UAVs

UAVs have to be closer to receivers/transmitters in THz and hence a higher
density of UAVs may be needed for specific task accomplishment such as
monitoring or sensing. However, each UAV can offload/disseminate larger data
and hence higher throughput can be achieved by each UAV making a higher sum-
rate capacity. Furthermore, although molecular absorption and high path loss
force the dense deployment of UAVs, it may create considerable LoS
interference (get worse under frequent beams misalignment) and handovers.

####  IV-A5 Multi-Functionality Design

As discussed earlier, THz-empowered UAVs significantly benefit from the
integration of various functionalities such as communication, sensing,
localization, and computing. However, a co-design of such multi-function
system is not an easy task. In addition to the general challenges such as
waveform design for integrated sensing and communication [11], the development
of novel models and powerful algorithms with low-complexity that can be
executed on UAVs with limited power and computation capability is a major
challenge. Furthermore, the algorithms should take into account the
sensitivity of several UAVs applications to delay caused by the integrated
design.

####  IV-A6 Uncertain Medium Condition

Air composition and its dependency on the meteorological condition make the
THz channel highly uncertain and unreliable particularly in an uncontrolled
outdoor scenario. The molecular absorption is highly dependent on the
percentage of water vapor in the air and the level of humidity. Therefore,
offline optimal network deployment and design aspects can only be valid under
certain circumstances, limiting the benefits. To some extent, an online
instantaneous sensing and communication can alleviate such issue.

###  IV-B Trade-offs

Since THz transmitters require highly directional transmission, they comprise
large antenna arrays to have pencil sized beams. UAV’s motion and wobbling can
lead to frequent beam misalignment and thus desiring frequent handovers. Such
frequent handovers can cause delay and hence adversely affect the latency as
well as reliability performance. On one hand, higher antenna gain leads to
higher data rate, while on the other, it may unfavorably affect the latency
and reliability. Moreover, large antenna arrays also result in high energy
consumption due to large number of ADCs and RF chains. Further, longer flights
are desirable for a meaningful data transfer in THz-empowered UAVs at the cost
of increased energy consumption. To counterbalance the shorter communication
range, the UAVs can be densely deployed which, however, may lead to high
interference and thus affect the reliability. As a consequence, the underlying
trade-offs between Latency, Data Rate, Energy consumption, and Reliability can
be established.

Furthermore, integration of various functionalities such as communication,
sensing, and localization can improve the reliability and data rate by having
a better perception of the environment and beam tracking. However, such multi-
functionality design can also lead to an increased payload energy consumption
and processing delay. Thus, trade-off between energy consumption, and
reliability exists. In addition, this integrated design may also contribute
somewhat to latency as various operations are involved. Hence, the reliability
can be increased at the expense of latency resulting in a trade-off. All these
interlinked challenges and trade-offs are also illustrated in Figure 2.
Considering the unique features of THz-empowered UAVs, their performance can
be characterized by analyzing various performance metrics viz., rate-energy,
delay-rate, delay-reliability trade-offs, etc., in addition to the traditional
individual key performance indicators (KPIs).

Figure 2: Challenges and Trade-offs in THz-empowered UAVs.

##  V Standardization, Regulations, and Health Issues

This section reviews standardization and regulation aspects of UAVs deployment
and standardization and health issues related to THz deployment.

###  V-A UAVs Deployment

Standardization efforts for the UAV deployment started in 2017 when 3GPP in
its release-15 studied the LTE-supported UAVs. Under this release, it
published a technical report TR 36.777 which identified the problems of
uplink/downlink interference and provided UE- and network-based solutions for
the performance enhancements. In Release 17, the technical specification TS
22.125 identified the UAVs operation requirements via 3GPP system. In the
continuation of TS 22.125, TR 22.829 defined new service requirements and KPIs
for various 3GPP supported UAV applications. Further, TR 23.754 addressed the
mechanisms to support the unmanned aerial systems (UASs) connectivity,
identification, and coverage, TR 23.755 identified the application layer
support for UAS and provided relevant architectural solutions, and TR 33.854
studied the security aspects of UASs. Technical report TR 22.261 from
release-17 and 18 discusses the 5G enhancements for UAV. Specifically, it
describes the UAVs application based on the laid out KPIs in TS 22.125 and
also plan for the enhanced architecture to support UAS.

On the other hand, international telecommunication union (ITU) has also
provided various recommendations to facilitate the UAV based operations and
applications. For example, ITU-T F.749.10 specified the requirements for
communication services of civilian UAVs while ITU-T Y.4421 (ex. Y.UAV.arch)
determined the recommendation for functional architecture of UAV controllers
using IMT-2020 networks. Further, ITU-T X.677 specified the identification
mechanism for UAVs using object identifiers, ITU-T Y.4559 identified the
functional architecture and other requirements of base station inspection
services using UAVs. Another recommendation ITU-T Q.3060 describes the
application of UAVs for fast deployment telecommunication networks in case of
any natural disaster.

The IEEE standardization body has also been actively involved in the
regulation of the UAV communications and its applications. For instance, the
draft standard IEEE P1936.1 established a framework to support the drone
applications where it specifies typical application classes and scenarios and
required execution environments. IEEE P1939.1 draft defines a low altitude
structure for efficient UAV traffic management. Furthermore, IEEE P1920.1
defines the air-to-air communication for the self-organized aerial adhoc
networks where aerial platforms can form a network to communicate with each
without requiring the supporting cellular/satellite infrastructure. Another
project IEEE P1920.2 identifies the protocol to exchange the information
between the vehicles for UASs.

One of the additional challenges for UAVs deployment is the legislative
process. Until a few years ago, the operation of small UAVs was not strictly
regulated and they were legally classified as remote controlled toys. But
increasing number of UAVs and their potential applications demand for the
strict regulation in low-altitude air space to ensure the safe operation and
avoid the conflict with manned aircraft. While the large UAVs operating in
high-altitude air space can be governed by the same air traffic control (ATC)
mechanism as for the conventional aircraft systems, UAVs in low-altitude are
difficult to capture by the radar surveillance. Therefore, various countries
set specific guidelines that need to be adhered while flying the UAVs to
ensure their safe operation. For example, in the USA, Federal Aviation
Administration (FAA) and National Aeronautics and Space Administration (NASA)
has been jointly framing the concepts of operations for UAV traffic management
(UTM) under 400 feet above ground level where ATC services are not available.
In Europe, under the project Single European Sky ATM Research (SESAR), U-space
programme aims to establish the necessary requirements for integrating UAVs
into European airspace. As per the instructions from such governing bodies,
for example, the UAVs first need to apply for the air space, the flight plan
should be submitted before flying, and the pilot should be certified. However,
some exceptions can be made to the regulations and they can vary depending on
the applications e.g., commercial use, hobbyist flying, research and
development purposes etc.

(a)

(b)

(c)

Figure 3: (a) Impact of distance, molecular noise, molecular absorption
coefficient, frequency of operation, altitude, and interference in U2U THz
communication. (b) Number of antennas and UAV wobbling effects where U2U link
distance is 100 m. The wobbling degree shows the maximum variation around the
point where the beams are perfectly aligned. (c) THz performance comparison
with mmWave and sub-6 GHz for different path loss exponents (PLEs).

###  V-B THz Deployment

Although the development of THz communication systems is still in fledgling
stage, some standardization and regulation processes have already been
initiated. For instance, as an amendment to 802.15.3-2016, IEEE 802.15.3d-2017
is the first standard for THz which aims at alternate PHY layer at frequencies
between 252 and 321 GHz. Further, the outcomes of world radiocommunication
conference (WRC) 2019 provide a regulatory framework for the operation of
fixed service (FS) and land mobile service (LMS) applications in frequency
bands between 275 GHz and 450 GHz.

On the flip side, the increasing interest for the THz communications naturally
raises the concerns of associated health risks due to high frequency
radiation, especially when some non-scientific population still hold the
negative perception about the 5G technology, specifically, mmWave and
beamforming. In fact, the electromagnetic fields (EMFs) of radio frequencies
below $3\times 10^{16}$ GHz is non-ionizing in nature and thus do not possess
sufficient energy to ionize the cells and thus avoiding cancer and death risks
[13]. However, they do have sufficient energy to push the electrons and ions
to higher energy states. The resulting thermal effects may pose certain health
issues and therefore, the maximum RF exposure limits are regulated by the
regulatory bodies (ICNIRP, FCC, EC, IEEE, etc.) to prevent the harmful heating
effects. Nevertheless, it is more important for large antenna arrays in THz to
obligate the RF exposure regulations as beamforming can result into elevated
power density levels compared with omnidirectional transmissions. Albeit, UAVs
supported communications can increase the separation between the THz radiation
and the human bodies to reduce the RF exposure to a certain extent. To date,
there is no evident proof of any major health concerns which is supported by
scientific community [14]. However, this aspect requires significant research
and experimental studies across various domains to analyze and discover any
health related issues which may arise due to THz applications.

Figure 4: The UAV flying time significantly decreases with the payload related
power requirement.

##  VI Case Study: U2U THz Communication

This section provides important insights into various aspects of U2U THz
communications. Note that this is an illustrative case study and THz
communications are not just limited for U2U. The simulated scenario includes
multiple THz U2U links operating at $\mathrm{f_{c}}=350$ GHz sharing same
spectrum. UAVs are distributed according to Poisson point process of density
$\lambda_{\mathrm{U}}$. To avoid collision and in compliance with regulatory
aspects, there is a minimum distance between any two UAVs. UAVs are equipped
with antenna arrays of $\mathrm{N_{a}}$ elements and are flying at altitude
$\mathrm{h_{U}}=100$ m above a built-up urban area unless otherwise is
mentioned.

Figure 3(a) illustrates the spectral efficiency (SE) of a U2U link operating
in THz with $\mathrm{N_{a}}=64$ antenna elements for various system
parameters. From this figure, following conclusions are drawn:

  * •

THz communication performance significantly drops as the communication
distance increases due to severe path loss in THz. At the lower altitude,
i.e., $\mathrm{h_{u}}=30$ m, the LoS U2U link is likely to be blocked and
hence the performance is worse.

  * •

Molecular noise notably reduces the performance of THz links. This can be
observed by comparing the solid blue and solid orange curves corresponding to
$\mathrm{f_{c}}=350$ GHz at 200 m. As the distance increases, the thermal
noise becomes dominant and hence the molecular noise is less involved.

  * •

High non-monotonic dependency of molecular absorption to frequency can be
drastic and results in an extremely low performance. For example, a peak of
molecular absorption at around $\mathrm{f_{c}}=380$ GHz results in a
remarkably low SE.

  * •

Co-channel interference is negligible thanks to the high path loss, molecular
absorption loss, and beamforming. As can be seen there is no gap between the
circle markers and the solid curve corresponding to
$\lambda_{\mathrm{U}}=0,10$ interfering UAVs per $\mathrm{km^{2}}$
respectively. Accordingly, spectrum sharing management could be less complex
compared to lower frequencies with large interference impact.

Figure 3(b) shows the impact of number of antennas on the SE. In this figure,
the wobbling is characterized through variations in the elevation angles of
each UAV array, which follows a uniform distribution with the specified
maximum degree, i.e., $1^{o}$ or $2^{o}$. By comparing the dashed and solid
blue curves, it can be seen that after a certain point (e.g.,
$\mathrm{N_{a}}=50$), the slope is much lower in the presence of molecular
noise. This means that the increased SE per antenna element is significantly
lower which considerably reduces the final performance gain of more
concentrated beams in THz. Indeed, the narrower beam and more concentrated
energy increases the received power yet it leads to higher re-radiated power
by the medium, which acts as molecular noise and degrades the performance.
Next, in the presence of UAVs wobbling, increased number of elements and
narrower beams could be even detrimental from a certain point onward. This is
due to the fact that the wobbling imposes frequent beam misalignment. When the
beam is too narrow, the total antenna gain variation with even a little
misalignment could be significant resulting in remarkable performance
deterioration. Accordingly, there could be an optimal number of UAVs antenna
elements depending on the severity of wobbling.

Figure 3(c) compares the achievable throughput of a single U2U link adopting
different technologies, i.e., 4G LTE, 5G mmWave, and 6G THz. In the LTE
underlay, the uplink spectrum of ground cellular users are re-used by the UAV
and, in LTE overlay, the available cellular uplink spectrum is split into two
orthogonal portions assigned to ground users and the U2U link. Accordingly, in
the former case, ground cellular users generate interference on the UAV,
however, in the latter case, there is no ground to air interference at the
expense of sacrificing the spectrum. Details of the simulated scenarios for
LTE U2U link can be found in [15]. This figure reveals that the THz U2U
achievable throughput is higher than the mmWave U2U communication up to a
certain distance.

Figure 4 illustrates the impact of payload weight and power consumption (due
to communication and/or processing) on the overall UAV power consumption and
flying time. Apparently, the potential high payload weight for carrying THz-
related hardware, e.g., 2.5 Kg for THz spectrometer, significantly reduces the
UAV life-time. A non-negligible THz payload power consumption leads to even
lower flying time, however, a dedicated payload battery can make up for it.

##  VII Conclusion

Synergies between THz and UAV technologies open the doors for new
applications, provide significant performance enhancement, and enable
intelligent interactions with the environment. However, the benefits are only
reachable if several unique challenges are well addressed. For instance, high
payload power consumption and weight of a THz-empowered UAV require new design
approaches. Integrated functionalities for enhanced performance needs to be
handled onboard with limited computation power. And limited design space leads
to very dense deployments or much longer flight time with increased latency
and energy consumption. This paper provided detailed discussions of the
opportunities, challenges, and design approaches as well as important factors
which determine the benefits of the combination of THz and UAV.

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#  Helicity transfer in strong laser fields via the electron anomalous
magnetic moment

Yan-Fei Li<EMAIL_ADDRESS>Department of Nuclear Science and Technology,
Xi’an Jiaotong University, Xi’an 710049, China  Yue-Yue Chen
<EMAIL_ADDRESS>Department of Physics, Shanghai Normal University,
Shanghai 200234, China  Karen Z. Hatsagortsyan  Christoph H. Keitel  Max-
Planck-Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany

###### Abstract

Electron beam longitudinal polarization during the interaction with
counterpropagating circularly-polarized ultraintense laser pulses is
investigated, while accounting for the anomalous magnetic moment of the
electron. Although it is known that the helicity transfer from the laser
photons to the electron beam is suppressed in linear and nonlinear Compton
scattering processes, we show that the helicity transfer nevertheless can
happen via an intermediate step of the electron radiative transverse
polarization, phase-matched with the driving field, followed up by spin
rotation into the longitudinal direction as induced by the anomalous magnetic
moment of the electron. With spin-resolved QED Monte Carlo simulations, we
demonstrate the consequent helicity transfer from laser photons to the
electron beam with a degree up to 10%, along with an electron radial
polarization up to 65% after multiple photon emissions in a femtosecond
timescale. This effect is detectable with currently achievable laser
facilities, evidencing the role of the leading QED vertex correction to the
electron anomalous magnetic moment in the polarization dynamics in ultrastrong
laser fields.

The development of modern ultraintense laser facilities, with a record
intensity already reaching $10^{23}$ W/cm2 [1], brings about new possibilities
for testing predictions of strong-field quantum electrodynamics (QED) theory.
The typical field strength characterizing the strong-field QED regime is the
Schwinger critical field $E_{S}=1.3\times 10^{16}$ V/cm (corresponding to the
intensity of $I_{S}=4.6\times 10^{29}$W/cm2) [2, 3]. While directly not
attainable by lasers, the critical field can be achieved using the Lorentz
boost of ultrarelativistic electrons in a head-on collision geometry [4],
which enables the experimental study of nonlinear regimes of strong-field QED
processes. In particular, recently such experiments are proposed at DESY
(LUXE) [5], and at FACET-II (E320) in the Stanford Linear Accelerator Center
(SLAC) [6].

In strong background fields electrons can be polarized due to the spin-flip
during photon emissions, which was first discovered for synchrotron radiation
[7, 8, 9, 10, 11] and termed as radiative polarization. Recently, the
possibility has been proven of efficient radiative polarization using
ultrastrong laser fields, applied to produce polarized electrons [12, 13] and
positrons [14, 15, 16] in a femtosecond timescale, as well as for polarization
transfer from electrons to $\gamma$-rays in laser fields [17, 18]. The
polarization effects in strong laser fields have a capability of detecting the
quantum stochastic nature of electron dynamics [19], diagnosing magnetic
fields of plasma [20], and providing ultra-short, high-brilliance, low-
emittance polarized beam sources for fundamental studies in high-energy
physics [21, 22, 23] and material science [24, 25]. The completely spin- and
photon-polarization-resolved probability rates for nonlinear Compton
scattering have been derived from strong-field QED theory in the Furry picture
for a plane-wave laser field [26], and for the locally constant fields [27],
as well as via the quantum operator method [28], and employed for a deep
analysis of all polarization channels, helicity transfer in the perturbative
regime, and investigation of the polarization dependent energy and angle
distributions [29]. Furthermore, the study of polarization effects has been
extended to higher-order QED processes [30, 31, 32, 33, 34, 35], and QED
cascades [36].

In radiative polarization the electron spin-flip is preferable along the
instantaneous magnetic field in the rest-frame of the electron. Because of
that in a storage ring or in ultrastrong laser fields initially unpolarized
electrons are mostly polarized transversely after the interaction.
Nevertheless, high-precision high-energy physics at accelerators demands
longitudinal beam polarization, e.g., the Qweak experiment at Jefferson Lab
[37] and E158 at SLAC [38]. Presently, one common way of producing circularly
polarized electrons in accelerators is via photoemission induced by a circular
polarized laser field from a solid [39, 40]. There were a series of attempts
to employ Compton scattering for this purpose. While here the helicity of the
laser photons is efficiently transferred to emitted $\gamma$-rays [41, 42],
the longitudinal polarization of the scattered electrons is not efficient in
the linear regime and suppressed in the nonlinear regime. Thus, the electron
longitudinal polarization can reach only $P_{\parallel}\sim 10^{-3}$ during a
single photon emission in a circularly polarized laser field [43] at the laser
strong field parameter $a_{0}=100$. Moreover, while scattered electrons in the
Compton process are weakly polarized, the total longitudinal polarization of
the electron beam is vanishing. This is due to the polarization of unscattered
electrons, which exactly cancels that of scattered ones [44, 45]. The latter
is explained as interference of the incoming electron wave function with that
of the forward scattered one.

Recently, it has been demonstrated that QED radiative corrections, i.e., the
interaction of the electron with its own radiation field, can also affect the
electron spin dynamics in intense background fields [31, 32, 33]. In
particular, due to QED loop corrections, the electrons exact spin-dependent
wave function becomes unstable inside a strong background field, leading to 1%
polarization for unscattered electrons [31, 32]. The latter provides the QED
description of the polarization of unscattered electrons discussed in [44].
The spin effects resulted from the electron mass loop are also described as a
spin rotation, which appeared to be more significant within the tail of a
tightly focused laser beam [33]. Furthermore, there are experimental plans to
reach the fully nonperturbative regime of QED employing beam-beam collisions
in TeV-class lepton colliders [46, 47, 48, 49, 50, 51], when the effective
field in the rest frame of electrons could be supercritical, and radiative
corrections to QED processes nonperturbative and substantial.

In this Letter, we investigate the role of the electrons anomalous magnetic
moment on the helicity transfer from a circularly polarized (CP) laser pulse
to an ultrarelativistic electron beam for the nonlinear Compton scattering
process in the radiation reaction dominated regime. The electron three-
dimensional polarization properties are analyzed using numerical Monte Carlo
simulations based on the spin-resolved radiation probabilities in the local
constant field approximation (LCFA). While previous studies neglecting the QED
radiative corrections came to a conclusion that the helicity transfer from
laser photons to electrons is forbidden in the nonlinear Compton scattering
process,we obtain a sizable longitudinal polarization of electrons when the
one-loop QED vertex correction [52, 8] to the anomalous contribution to the
magnetic moment is accounted for. A longitudinal polarization degree close to
$3\%$ is shown, which could be further improved up to 10% with post-selection
techniques. We prove the scenario of the helicity transfer. Initially, spin-
flips during photon emissions induce electron transverse polarization which is
phase-matched with the laser field. Due to the latter property and the
anomalous correction to the magnetic moment [53, 54, 55], the oscillating
transverse polarization is transformed into accumulated longitudinal
polarization during the interaction. The latter demonstrates a signature of
QED radiative corrections for electron polarization dynamics in ultrastrong
laser fields. Additionally, the electron transverse radiative polarization
with a degree over 60% is shown after the interaction, of interest for high-
energy applications.

We model the laser-electron interaction process with the Monte Carlo method
[56, 57, 58, 59], which treats the photon emissions quantum mechanically with
the spin-resolved photon emission probabilities in the LCFA [12, 17]. The LCFA
[60, 11, 61, 62, 63, 64] is valid at $a_{0}\equiv|e|E_{0}/(m\omega_{0})\gg 1$,
when the coherence length of the photon emission, $l\sim\lambda_{L}/a_{0}$, is
much smaller than the typical length of the trajectory (here the laser
wavelength $\lambda_{L}$). Furthermore, $E_{0}$ is the laser field amplitude,
$\omega_{0}$ the laser frequency, and $e(<0)$, $m$ are the electron charge and
mass, respectively. Relativistic units $\hbar=c=1$ are used throughout. The
photon emission probability is determined by the local value of the quantum
strong-field parameter
$\chi_{e}\equiv|e|\sqrt{-(F_{\mu\nu}p^{\nu})^{2}}/m^{3}$, where $F_{\mu\nu}$
is the field tensor and $p^{\nu}$ the four-vector of electron momentum. The
simulation method is the following [65]: The common statistical event
generator is conducted at each simulation step to determine whether or not a
photon-emission occurs. If a photon-emission occurs, the emitted photon energy
is determined via the stochastic procedure and spectral probability, while the
electron and photon polarizations via the averaged algorithm involving the
density matrix for the mixed state of an electron ensemble [59, 66] to reduce
the statistical fluctuation. If the photon-emission event is rejected, the
electron spin changes according to the non-radiation probability [59, 14].
This no-emission spin variation originates from the radiative correction of
the one-loop propagator correction where the electron propagator is modified
by the process that a virtual photon is emitted and reabsorbed by the electron
[31]. At first order in the fine-structure constant $\alpha$, it cancels the
longitudinal polarization of the electrons induced by emitting soft photons in
a circularly polarized field [44, 67, 34]. In the regime $a_{0}\sim 1$, the
cancellation is broken after including radiation reaction, leading to a
nonzero longitudinal polarization of the final electrons [35].

The spin precession between photon emissions is governed by the Thomas-
Bargmann-Michel-Telegdi equation [68]: ${\rm d}{\bf S}/{\rm
d}t=\bm{S}\times\bm{F}$, with

$\displaystyle{\bm{F}}$ $\displaystyle=$
$\displaystyle\frac{e}{m}\left[-\left(\frac{g}{2}-1\right)\frac{\gamma}{\gamma+1}\left({\bm{\beta}}\cdot{\bf
B}\right){\bm{\beta}}\right.$ (1)
$\displaystyle\left.+\left(\frac{g}{2}-1+\frac{1}{\gamma}\right){\bf
B}-\left(\frac{g}{2}-\frac{\gamma}{\gamma+1}\right){\bm{\beta}}\times{\bf
E}\right],$

where ${\bf E}$ and ${\bf B}$ are the laser electric and magnetic fields,
respectively, $\beta\approx-1$ the electron velocity, and $g$ the electron
gyromagnetic factor. Taking into account the radiative correction to the first
order of $\alpha$ in the interaction with the radiation field and being exact
with respect to the external field [9],

$g\left(\chi_{e}\right)=2+2\mu\left(\chi_{e}\right),\,\,\,\,\,\mu\left(\chi_{e}\right)=\frac{\alpha}{\pi\chi_{e}}\int_{0}^{\infty}\frac{y}{\left(1+y\right)^{3}}{\bf
L}_{\frac{1}{3}}\left(\frac{2y}{3\chi_{e}}\right){\rm d}y,$ (2)

with ${\bf L}_{\frac{1}{3}}\left(z\right)=\int_{0}^{\infty}{\rm
sin}\left[\frac{3z}{2}\left(x+\frac{x^{3}}{3}\right)\right]{\rm d}x$. At
$\chi_{e}\ll 1$, one obtains the Schwinger result
$g=2+\frac{\alpha}{\pi}\approx 2.00232$. The electron dynamics is described by
the Newton equation with the Lorentz force. The modification of the equation
of motion due to the anomalous magnetic moment [69, 70] does not change the
electron dynamics [65].

(a)log10($\mathcal{N}_{e}$)(b)$P_{\parallel}$(c)$|P_{\perp}|$(d)

log${}_{10}($d$\widetilde{N_{e}}/[$d$\theta$sin$(\theta)$])

$P_{\parallel}|P_{\perp}$

$\theta$

Figure 1: (a) Distribution of electron number density
log10($\mathcal{N}_{e}$), with $\mathcal{N}_{e}=$
d$N_{e}/[$d$\varphi$d$\sin\theta$ d$\theta$ d$\phi$]; (b) Average longitudinal
polarization $P_{\parallel}={\bm{\beta}}\cdot{\bm{S}_{f}}$; (c) Average
transverse polarization
$|P_{\perp}|=|{\bm{S}}_{f}-{\bm{\beta}}\cdot{\bm{S}_{f}}|$, vs polar angle
$\theta$ in [$155^{\circ},180^{\circ}$] and the azimuthal angle of $\varphi$
in [$0^{\circ},360^{\circ}$], respectively; (d) $P_{\parallel}$ (red-solid),
$P_{\perp}$ (black-dashed) and
log${}_{10}($d${N_{e}}/[$d$\theta$sin$(\theta)$]) (yellow-dotted) at
$\varphi=180^{\circ}$ vs $\theta$, respectively.

The polarization effect of electrons is illustrated in Fig. 1. A right-hand CP
tightly-focused Gaussian laser pulse is used, with peak intensity
$I_{0}\approx 2\times 10^{22}$ W/cm2 $(a_{0}=100\sqrt{2})$, pulse duration
(FWHM) $\tau=5T_{0}$, with the laser period $T_{0}$, $\lambda=1\mu$m, and
focal radius $w_{0}=5\lambda$. The counterpropagating cylindrical electron
bunch has a length of $L_{e}=5\lambda$ and radius of $w_{e}=1\lambda$.
$N_{e}=10^{6}$ unpolarized electrons are distributed longitudinally uniformly.
The transverse distribution is Gaussian with the variance of
$\sigma_{x,y}=0.3\lambda$. The initial electron kinetic energy is
$\varepsilon_{0}=1$ GeV, the energy spread
$\Delta\varepsilon_{0}/\varepsilon_{0}=10\%$, and the angular divergence
(FWHM) $\Delta\theta=0.1$ mrad. The feasibility of our scheme for larger beam
spreadings is shown in [65]. As the quantum parameter for pair creation
$\overline{\chi}_{\gamma}\equiv|e|\sqrt{-(F_{\mu\nu}k^{\nu})^{2}}/m^{3}\approx
0.005\ll 1$, with $k^{\mu}=(\omega,{\bm{k}})$ being the four-momentum of laser
photons, the pair creation effect can be neglected [65].

After the interaction the scattered electrons concentrate in the center of the
angle distribution, and have longitudinal, as well as transverse polarization,
see Figs. 1(a)-(c). The average longitudinal polarization degree is not large
$\overline{P}_{\parallel}=\frac{1}{N_{e}}\sum\limits_{i=1}^{N_{e}}{\bm{S}_{i}}\cdot\frac{\bm{\beta}_{i}}{|\bm{\beta}_{i}|}\approx$
2.65%, nevertheless exceeding by an order of magnitude the QED tree-level
result $\overline{P}_{\parallel}\approx 0.1\%$ [43]. The electrons are highly
polarized in the transverse plane with polarization vector pointing to the
center of the beam. The electron number-density decreases exponentially from
the center to the peripheries [Fig.1(d)], while the polarization
$P_{\parallel}$ and $P_{\perp}$ increase with the deflection angle. Therefore,
the polarization purity can be increased by selecting large angle electrons
with post-momentum-angle selection techniques. Meanwhile, the longitudinal
polarization degree can be increased by post-energy-selection due to the
$P_{\parallel}$ dependency on the electrons energy [Fig. 2]. Higher
$P_{\parallel}$ can be obtained by collecting low energy electrons. For
instance, by collecting the electrons with energies less than 45 MeV, we can
get a polarization degree of $P_{\parallel}=10\%$ with the percentage of
number of 1%.

$P_{\parallel}$

log10(d$N_{e}$/d$\varepsilon_{e}$)

$\varepsilon_{e}$(MeV)

Figure 2: Longitudinal polarization $P_{\parallel}$, and number density
log10(d$N_{e}$/d$\varepsilon_{e}$) (MeV-1), versus final electron energy
$\varepsilon_{e}$, respectively.

(a)

a.u.

$t/T_{0}$

$S_{x}$

$E_{x}$$p_{y}$$S_{x}$(b)

a.u.

$t/T_{0}$

$S_{y}$

$E_{y}$$p_{x}$$S_{y}$(c)

$S_{z}$

$t/T_{0}$$g(\chi_{e})$$g=2$w/o
RP(d)$\omega_{\gamma}/\varepsilon_{0}$$\mathcal{T}_{1}$$\mathcal{T}_{2}$$\mathcal{T}_{3}$

Figure 3: The evolution of the field and electron parameters: (a) $E_{x}$,
$p_{y}$ and $S_{x}$; (b) $E_{y}$, $p_{x}$ and $S_{y}$; (c) $S_{z}$ for
$g=g(\chi_{e})$ (red solid), $g=2$ (blue dotted), and $g=g(\chi_{e})$ but
without radiative polarization (magenta dashed), for an initially unpolarized
electron beam. The electron spin dynamics is calculated numerically with
Eq.(3). At $t=0$ the peak of the laser pulse reaches the focal spot. The field
and momentum components are normalized to their maximum. Radiation reaction
for the electron momenta is neglected for simplicity [71]. (d) The values of
the three terms in Eq.(5) are defined as
$\mathcal{T}_{1}=\frac{1}{a}\\{u^{2}\textrm{K}_{2/3}-u\textrm{K}_{1/3}[{\bm{S}}_{i}\cdot({\bm{\beta}}\times\bm{s})]\\}$,
$\mathcal{T}_{2}=\frac{1}{a}\\{u^{2}\textrm{IntK}_{1/3}-u\textrm{K}_{1/3}[{\bm{S}}_{i}\cdot({\bm{\beta}}\times\bm{s})]\\}$
and
$\mathcal{T}_{3}=\frac{1}{a}u(1+u)\textrm{K}_{1/3}({\bm{\beta}}\times\bm{s})$,
with
$a=(1+u)\mathrm{IntK_{1/3}}-(2+2u+u^{2})\mathrm{K_{2/3}}+\bm{S}_{i}\cdot({\bm{\beta}}\times\bm{s})u\mathrm{K_{1/3}}$
and $S_{i}=0.01$, $\bm{S}_{i}$ is parallel to $({\bm{\beta}}\times\bm{s})$.

The reason for the electron beam polarization is analyzed in Fig. 3. Without
radiation reaction, the electrons typically move along a spiral trajectory in
the CP laser pulse, with the transverse momentum perpendicular to the electric
field [Figs. 3(a,b)]. In each time step, the spin vector of an electron flips
to the direction parallel or anti-parallel to the instantaneous spin
quantization axis according to the quantum probabilities [14]. The evolution
of polarization of an electron ensemble including both effects of spin-flip
and spin precession is described by the following equation [9]:

$\displaystyle\frac{d\bm{S}}{dt}$ $\displaystyle=$
$\displaystyle{\bm{S}}\times{\bm{F}}-\frac{\alpha
m}{\sqrt{3}\pi\gamma}\int_{0}^{\infty}\frac{u^{2}du}{\left(1+u\right)^{3}}\left(\mathrm{K_{2/3}}\bm{S}\right.$
(3)
$\displaystyle\left.+\left(\mathrm{IntK_{1/3}}-\mathrm{K_{2/3}}\right)\left(\bm{S}\cdot\bm{\beta}\right)\bm{\beta}+\left(\bm{\beta}\times\bm{a}\right)\mathrm{K_{1/3}}\right),$

where ${\rm IntK}_{\frac{1}{3}}\equiv\int_{u^{\prime}}^{\infty}{\rm d}z{\rm
K}_{\frac{1}{3}}(z)$, ${\rm K}_{2/3}={\rm K}_{2/3}(u^{\prime})$, ${\rm
K}_{1/3}={\rm K}_{1/3}(u^{\prime})$, ${\rm K}_{n}$ is the $n^{\rm th}$-order
modified Bessel function of the second kind, $u^{\prime}=2u/3\chi_{e}$,
$u=\omega_{\gamma}/(\varepsilon_{0}-\omega_{\gamma})$, $\varepsilon_{0}$ the
electron energy, $\omega_{\gamma}$ the photon energy, and
$\bm{a}=\dot{\bm{\beta}}/|\dot{\bm{\beta}}|$. As shown in Figs. 3 (a)-(c), the
transverse polarization $\bm{S_{\perp}}$ oscillates synchronously with
$\bm{E}$, and $|S_{z}|$ builds up exponentially. The phase matching between
$S_{x,y}$ and $E_{x,y}$ follows from the domination of the last term in Eq.
(3) for $\bm{S_{\perp}}$. In fact,
$\mathrm{K_{2/3}}|\bm{S_{\perp}}|\ll\mathrm{K_{1/3}}$, and the spin precession
role [first term in Eq. (3)] is known to be minor for the transverse radiative
polarization. Consequently, $d\bm{S_{\perp}}/dt$ is parallel to
$-\bm{\beta}\times\bm{a}$, and thus, $\bm{S_{\perp}}$ is parallel to $\bm{E}$.
We analyze the accumulation of the longitudinal polarization:

$\displaystyle\frac{dS_{\parallel}}{dt}$ $\displaystyle=$
$\displaystyle-\frac{e}{m}{\bm{S}_{\perp}}\cdot\left[\left(\frac{g}{2}-1\right){{\bm{\beta}}}\times{\bf
B}+\left(\frac{g\beta}{2}-\frac{1}{\beta}\right){\bf E}\right]$ (4)
$\displaystyle-\frac{\alpha
m}{\sqrt{3}\pi\gamma}S_{\parallel}\int_{0}^{\infty}\frac{u^{2}du}{\left(1+u\right)^{3}}\mathrm{IntK_{1/3}},$

with the use of Eq. (11.171) in [72]. For an initially unpolarized electrons
beam, the longitudinal polarization arises due to the $\bm{S_{\perp}}$ term,
connected to the spin procession, which can be approximated as
$dS_{\parallel}/dt\approx-\frac{2e}{m}(\frac{g}{2}-1){\bm{S}_{\perp}}\cdot{\bf
E}$, taking account of $S^{i}_{\parallel}=0$, $\beta\approx 1$ and
${\bm{\beta}}\parallel-{\bm{k}}$. As well known, the Dirac theory predicts
precisely g=2, which leads to vanishing longitudinal polarization [Fig.3(c)].
However, the QED loop corrections induce an anomalous contribution to the
electron’s magnetic moment, $g\neq 2$ [Eq. (2)], which results in rotation of
the transverse polarization to longitudinal direction in the case when
$\bm{S}_{\perp}$ and $\bm{E}$ are phase matched. Note that, the phase matching
of $\bm{S}_{\perp}$ and $\bm{E}$ is a unique feature of electrons in a counter
propagating circularly polarized laser field, and consequently such a
configuration is essential for producing $P_{\parallel}$ [65]. Therefore, the
electron anomalous magnetic moment is the origin of helicity transfer. Our
estimation of ${\bm{S}_{\parallel}}$ via Eq. (3) yields $P_{\parallel}=3.26\%$
after the interaction, which is in accordance with the simulation result.
Thus, even though the spin precession is trivial for the well-studied
transverse polarization in strong-field QED [12, 9, 13], it plays an essential
role in generating longitudinal polarization. The correlation between
anomalous magnetic moment and longitudinal polarization may consequently
provide new potential of accurately measuring $g/2-1$. For instance, the
maximum and the changing rate of the averaged polarization degree exclusively
depend on the anomalous magnetic moment, which would be sensitive measures of
the anomaly $g/2-1$ [73].

Moreover, the Monte-Carlo simulation reveals a radial polarization feature of
the electron beam with absolute value $P_{\perp}$ up to 65% [Fig. 1(c)]. Even
though the averaged transverse polarization is negligible due to the
cancellation from opposite angles, which coincides with the prediction of
modified BMT equations [Fig. 3(a,b)], it is possible to collect electrons in a
certain angle to obtain a high transverse polarization, as well as a high
longitudinal polarization by applying a spin-rotating system [12]. We can give
a simple estimation of the radial polarization using the spin-flip transition
probabilities, which determine the spin change after a photon emission [12,
14]:

$\Delta{\bm{S}^{R}}=\frac{\left[u^{2}\textrm{K}_{2/3}-u\textrm{K}_{1/3}\left[\bm{S}_{i}\cdot\left(\bm{\beta}\times\bm{a}\right)\right]\right]\bm{S}_{\perp}+\left[u^{2}\textrm{IntK}_{1/3}-u\textrm{K}_{1/3}\left[\bm{S}_{i}\cdot\left(\bm{\beta}\times\bm{a}\right)\right]\right]\bm{S}_{\parallel}+u\left(1+u\right)\textrm{K}_{1/3}\left(\bm{\beta}\times\bm{a}\right)}{(1+u)\mathrm{IntK_{1/3}}-(2+2u+u^{2})\mathrm{K_{2/3}}+\bm{S}_{i}\cdot({\bm{\beta}}\times\bm{a})u\mathrm{K_{1/3}}}.$
(5)

The initial spin $\bm{S}_{i}$ is taken as the average polarization of the
ensemble shown in Fig.3 (a)-(c), i.e. $|{\bm{P}_{\perp}}|\sim 10^{-2}$ and
$|{\bm{P}_{\parallel}}|\sim 10^{-2}$. In this case, the fist two terms of Eq.
(5) are negligible compared with the last term since $|\bm{S}|\ll 1$, and the
remaining terms with modified Bessel functions are comparable, see Fig.3 (d).
Consequently, the change of transverse spin is estimated as
$\Delta{\bm{S}^{R}_{\perp}}\propto-{\bm{\beta}}\times\bm{a}$, anti-parallel to
the momentum of the scattered electron after one photon-emission [65]. This
feature is preserved during multiple photon emissions, yielding the radial
transverse electron polarization [Fig. 1(c)].

Since the transverse polarization $|\Delta\bm{S}^{R}_{\perp}|$ increases with
the emitted photon energy [65], the electrons that experience more energy loss
obtain higher transverse polarization, which subsequently contributes to
higher longitudinal polarization according to Eq. (4). Meanwhile, as the
deflection angle of electron $\theta_{D}\sim 1/\gamma$, the transverse
polarization degree increases with the decrease of $\theta$ [Fig.1(d)].
Therefore, $P_{\parallel}$ is inversely proportional to $\varepsilon_{e}$ and
$\theta$ [Figs. 1(d), 2]. The above analysis is not relevant for scattered
electrons with energy higher than 135 MeV, corresponding to those within angle
of $\theta>167^{\circ}$. These electrons are polarized opposite to the laser
helicity and the polarization degree increases with $\varepsilon_{e}$ and
$\theta$ [Fig. 2]. This counterintuitive polarization feature highlights two
different contributions of QED loop effects to the electron polarization in
background fields. Firstly, the loop effects induce anomalous magnetic moment
$g>2$, which yields the rotation of the radiative transverse polarization into
the longitudinal direction. Secondly, the loop effects result in varying the
electron spin even without photon emissions [31]. The latter effect is
included in our Monte Carlo simulations by means of an additional no-photon-
emission probability for the spin-flip [59]. We estimate this effect via
turning off artificially the photon-emission in the modified BMT equation
[65]:

$\displaystyle\frac{d\bm{S}^{NR}}{dt}$ $\displaystyle=$
$\displaystyle\frac{e}{m}\left[\bm{S}\times\bm{F}\right]-\frac{\alpha
m}{\sqrt{3}\pi\gamma}\int_{0}^{\infty}\frac{du}{\left(1+u\right)^{3}}\left\\{\left[\bm{S}\cdot({\bm{\beta}}\times\bm{a})\right.\right.$
(6)
$\displaystyle\left.\left.u\mathrm{K_{1/3}}\right]\bm{S}-({\bm{\beta}}\times\bm{a})u\mathrm{K_{1/3}}\right\\}.$

Compared with Eq. (3), the dominant term $({\bm{\beta}}\times\bm{a})$ has
opposite sign. Thus, if the transverse spin dynamics is governed by no-
emission polarization, the phase of ${\bm{S}}_{\perp}$ is opposite to
${\bm{E}}$. Consequently, the energetic electrons obtain a negative
longitudinal polarization [Fig.3(c)] due to polarization rotation associated
with anomalous magnetic moment, i.e. $P_{\parallel}=-P_{z}\propto{\bf
S_{\perp}}\cdot{\bf E}<0$, which is opposite to the low-energy electrons in
the remaining contribution to the spectrum, as shown in Fig. 2.

(a)

$\tau/T_{0}$

$\varepsilon_{0}$$P_{\parallel}$(b)

$\tau/T_{0}$

$\varepsilon_{0}$$P_{\parallel}$

Figure 4: The dependence of the longitudinal polarization $P_{\parallel}$ on
the laser duration $\tau$ and the initial electron energy $\varepsilon_{0}$ at
the laser intensity of $a_{0}=100\sqrt{2}$ and $50\sqrt{2}$, respectively.

The impacts of the laser and electron beam parameters on the longitudinal
polarization are analyzed in Fig. 4. With the increase of $\varepsilon_{0}$
and $a_{0}$, the transverse polarization $P_{\perp}$ increases since radiative
polarization is enhanced for a larger radiation loss scaled by
$\chi_{e}\approx 5\times 10^{-6}a_{0}\gamma$ , and consequently, the
longitudinal polarization $P_{\parallel}$ grows. While for a certain
$\varepsilon_{0}$ and $a_{0}$, $P_{\parallel}$ rises at first and then
declines with respect to $\tau$. Without radiation reaction, one would expect
a monotonously increase of $P_{\parallel}$ with the increase of interaction
time from Eq.(3). Unfortunately, radiation reaction breaks the phase
correlation between ${\bm{S}}_{\perp}$ and ${\bm{E}}$, and disrupts the
longitudinal polarization built at the preliminary stage of the interaction,
resulting in the decrease of $P_{\parallel}$ for a long laser pulse [65].

Concluding, we have analyzed the role of the anomalous magnetic moment and no-
photon-emission spin dynamics, i.e. effects which both are consequences of QED
radiative corrections, for electron polarization in the radiation dominated
regime with multiple photon emissions. We showed that exclusively due to these
effects helicity transfer is possible from CP laser photons to electrons in
ultrastrong field regime $a_{0}\gg 1$,

which challenged the belief that circularly polarized laser beams cannot
induce high longitudinal polarization. This signature is robust with respect
to the laser and electron parameters and measurable with currently available
experimental technology. It could serve for testing QED predictions on
radiative corrections.

Acknowledgement: The authors thank Y.-T. Li, W.-M. Wang and Q.-Z. Lv for
helpful discussions. This work is supported by the National Natural Science
Foundation of China (Grants No. 12075187, and No. 11804269), the Program for
Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of
Higher Learning, Shanghai Rising-Star Program.

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