# X-ray pulse generation with ultra-fast flipping of its orbital angular
momentum

J. Morgan<EMAIL_ADDRESS>SLAC National Accelerator Laboratory,
Menlo Park, California, 94025, USA  B. W. J. McNeil  University of Strathclyde
(SUPA), Glasgow G4 0NG, United Kingdom  Cockcroft Institute, Warrington, WA4
4AD, UK  ASTeC, STFC Daresbury Laboratory, Warrington, WA4 4AD, UK

###### Abstract

A method to temporally tailor the properties of X-ray radiation carrying
Orbital Angular Momentum (OAM) is presented. In simulations, an electron beam
is prepared with a temporally modulated micro-bunching structure which, when
radiating at the second harmonic in a helical undulator, generates OAM light
with a corresponding temporally modulated intensity. This method is shown to
generate attosecond pulse trains of OAM light without the need for any
additional external optics, making the wavelength range tunable. In addition
to the OAM pulse train, the method can be adapted to generate radiation where
the handedness of the OAM mode may also be temporally modulated (flipped).

††preprint: APS/123-QED

##  I Introduction

Light-matter interactions play an important role in our understanding of
material interactions across most of science, from how viruses function to the
properties of matter at the centre of gas giant planets. The material
properties that can be observed depend critically upon the form of the light
used: wavelength, intensity, temporal duration etc. Much research is therefore
devoted to improving upon, or developing new forms of light, that can access
previously unobserved material properties.

Increasing attention is being given to light with helical wave-fronts, as its
transverse phase rotation means the light carries orbital angular momentum
(OAM). This OAM is distinct from the spin momentum of the light and can be
significantly greater, allowing another dimension of light-matter interactions
to be explored and utilised. Allen et al. [1] was first to recognise OAM is a
property belonging to light with an azimuthal phase dependence given by
$\exp{i\ell\phi}$, where $\phi$ is the azimuthal angle and $\ell$ the mode
index. Since publication of this seminal paper in 1992, there has been
significant research interest in the scientific applications of this light
which has benefited numerous fields [2] such as imaging [3, 4] and
communication [5]. As OAM can be exchanged with atoms and molecules, there has
also been great interest in the study of OAM light with matter [6, 7, 8, 9,
10]. This is being further developed by investigating new methods to generate
OAM light pulses that can interact with matter at its fundamental spatial
(X-ray wavelengths) and temporal (attosecond) scales [11, 12, 13, 14].

There has been specific interest in the generation of pulses of OAM light at
XUV and X-ray wavelengths due to predicted experimental applications [15, 16].
To meet this need, short individual pulses and trains of pulses have been
generated at XUV using high harmonic generation [17, 18, 19, 20]. Although
these methods are predicted to work into the soft X-ray, this is yet to be
demonstrated. OAM light can be generated by manipulating the phase profile of
Gaussian beams with $l=0$ using optical elements which convert the light
phase. These require precise fabrication which can be particularly challenging
at shorter wavelengths, although elements which work at XUV and X-ray
wavelengths are becoming increasingly available [21]. However, these optical
elements are not tuneable, so different elements are required at different
wavelengths. Such conventional optical elements can also pose a barrier to the
rate at which the OAM mode of the light can be controlled. Routine OAM
switching optics, such as spatial light modulators, only achieve a mode
switching rate in the kHz range, although this could possibly be improved on
with more complex techniques.

In recent developments at short wavelengths, a temporal variation in the OAM
of a light beam (a ‘self-torqued’ beam) has been achieved using harmonic
generation [22] and opens up another new branch of potential applications.

Free Electron Lasers (FELs) provide a method for generating intense OAM light
without the need for additional optical elements. There has been significant
progress in the generation of light with OAM in a FEL including experimental
demonstration [23, 24] and the proposed combination of OAM modes to generate
Poincaré X-ray beams [25]. However, current FEL studies have not yet
considered the detailed control of the temporal properties of the OAM light.

In this article, a new method is proposed to use a FEL to generate high power,
X-ray OAM light with attosecond intensity modulation. The approach uses a FEL
‘afterburner’ configuration [21]. This consists of a main FEL lasing section
that first develops a strongly modulated electron beam micro-bunching which is
then injected into the helical undulator afterburner where the attosecond
timescale, temporally modulated OAM light is generated. A small modification
to this approach, the addition of a second afterburner undulator, allows the
generation of light pulses which alternate between $l=\pm 1$ OAM modes, again
at attosecond timescales. The alternation of OAM light at such fast timescales
and short wavelengths would be the first time such output would be available
for users.

##  II Orbital Angular Momentum From Relativistic Electron Beams

OAM was shown to be carried by the harmonic radiation emitted from electrons
propagating in a spiral path through a helical undulator [26, 27]. The light
is characterised by the OAM index:

$\ell=\pm(h-1).$ (1)

Here, $h$ is the harmonic number and the mode index $\ell$, is sign dependent
upon the handedness of the undulator. This theoretical work was subsequently
experimentally corroborated with OAM demonstrated to be carried by both
incoherent and coherent undulator radiation [23, 28]. Many methods have been
proposed to generate OAM light with relativistic electron beams. Some methods
do not require emission at the harmonics [24, 29] while another method is to
simply filter out the harmonic OAM emission from a helical undulator.

Here, a FEL afterburner arrangement is used [24, 30] in which a FEL amplifier
first bunches the electron beam before it is propagated into a downstream
helical undulator, the afterburner, where the OAM light emission takes place.
The pre-prepared electron bunching is at the second harmonic of the helical
undulator afterburner resonance and generates the coherent harmonic emission
of radiation which carries the desired OAM.

##  III Generation of light pulses with alternating OAM

A method is now proposed to generate a train of ultra-short light pulses with
alternating OAM of $\ell=+1,-1,+1,...$. The method uses a similar setup as the
2nd harmonic helical undulator OAM afterburner, but with an electron beam
prepared so that its micro-bunching is modulated into a periodic comb
structure. This can be obtained by using an energy modulated electron beam in
the first FEL amplifier that induces the electron micro-bunching. When
emitting in a short helical afterburner undulator, the modulated micro-
bunching of the electron beam causes emission of a train of coherent OAM light
pulses spaced at the modulation frequency, in a similar way to that described
in [30]. If a second OAM afterburner undulator, with the same tuning but
opposite handedness as the first, is added, a second interweaving light pulse
train can be superimposed so that the final light generated alternates between
OAM mode pulses of approximately the same power.

###  III.1 Modulating the electron beam micro-bunching

When a relativistic electron beam is prepared with a sinusoidal energy
modulation, $\gamma(t)=\gamma_{0}+\gamma_{m}\cos({\omega_{m}t})$, the energy
gradients about the mean energy $\gamma_{0}$ ‘spoil’ the FEL interaction
there. No seeding of the FEL is required and the process starts up from noise.
If the modulation amplitude, $\gamma_{m}$, is sufficiently large, the only
regions that develop any significant electron FEL micro-bunching are about the
minima of the energy modulated beam [30]. After a preparatory process in a FEL
amplifier, the energy modulated electron beam has regions of relatively high
micro-bunching separated by the modulation period
$\lambda_{m}=2\pi/\omega_{m}$.

When the electron beam with modulated electron micro-bunching structure is
propagated through a short OAM afterburner undulator module, the power of the
coherent OAM light it radiates has the same modulation period.

The modulation in the temporal power profile of the radiation corresponds to a
frequency spectrum which is broader than the typical FEL output and is
discretized into frequency modes. To examine the effect of this on OAM mode
purity we first present analysis of the spectral output of a single undulator
module. Following this, simulations of the method to generate pulse trains of
OAM light, as well as pulse trains with alternating OAM, are presented.

###  III.2  Spontaneous emission spectrum of a single undulator module

To examine the OAM content of a helical undulator at the second harmonic, the
far field radiation emitted by a single electron propagating in a helical
undulator [31] is multiplied by the term
$\exp(-\sigma_{x}^{2}\omega^{2}/c^{2}\sin^{2}\theta)$, which captures the
restriction imposed by the beam’s transverse dimensions on the angle of
emission $\theta$. A Gaussian transverse profile is assumed, where
$\sigma_{x}$ is the rms beam radius. The radiation spectrum is then considered
in terms of OAM content. Fig. 1 plots the radiation spectrum decomposed into
different OAM modes (as in [24]) for different number of undulator periods,
$N_{u}$. An undulator parameter of $a_{u}>1$ is used, increasing the
fractional power emitted into the $\ell=1$ mode [31].

Figure 1: Spectrum of helical undulator radiation for different number of
undulator periods $N_{u}=9,12,18,22$ with $a_{u}=2.634$, $\lambda_{u}=3.9$cm,
$\gamma=7869$ and $\sigma_{x}=26\mathrm{\mu m}$.

It can be seen that undulators with a lower number of periods $N_{u}$, have
significant emission into the fundamental Gaussian mode with $\ell=0$. For a
long electron beam bunched uniformly at the second harmonic, the bunching
picks out a narrow region of the FEL bandwidth about the peak of the $\ell=1$
emission. There is therefore little emission into the Gaussian mode when the
micro-bunching structure is not modulated as assumed in the simple OAM
afterburner.

When the electron beam micro-bunching is modulated to have a comb structure,
as described in Section III.1, then radiation pulses are generated at the
modulation period. For a small number of undulator periods $N_{u}$, it can be
seen from Fig. 1 that a significant fraction of the power can be emitted off-
resonance into the $\ell=0$ mode. Increasing $N_{u}$ both increases the
relative fraction of power in the $\ell=1$ mode, and decreases the bandwidth
with FWHM spectral width of harmonic, $h$ given by:

$\Delta\omega_{h}=\frac{0.88}{hN_{u}}\omega_{h}(\theta),$ (2)

where $\omega_{h}(\theta)$ is the peak of the spectrum.

##  IV Numerical Simulations

Using the FEL simulation code Puffin [32, 33], the generation of OAM pulse
trains in the soft X-ray is demonstrated. Fig 2 shows a schematic of the
layout used to generate two different types of pulse trains: a) each pulse has
an OAM with index of $\ell=1$; and b) the pulses have an OAM index that
alternates between $\ell=\pm 1$.

Figure 2: The electron’s energy is modulated before injection into an
amplifier undulator where the electron beam develops a similarly modulated
micro-bunching structure. A pulse train of radiation carrying OAM radiation is
then emitted in the short helical afterburner tuned to the second harmonic of
the electron micro-bunching. In $a)$ a single helical undulator afterburner
will produce a single handed OAM pulse train, while in $b)$ a second undulator
with opposite handedness is added. The pulse trains emitted within each
afterburner can then be made to alternate spatially, resulting in the
generation of OAM pulses that alternate between $\ell=\pm 1$.

The method is modeled using parameters inspired by the LCLS-II project at SLAC
[34]. A $40$ fs long electron beam is considered with a flat top beam current
profile and current $I=1$ kA. The energy is $4\ \mathrm{GeV}$ and the radius
and emittance are $\sigma_{x}=22\ \mathrm{\mu m}$ and $\epsilon_{x,y}=3.5\
\mathrm{mm-mrad}$ respectively.

###  IV.1 OAM pulse train generation

The electron beam is first prepared in a seeded undulator modulator to give it
an energy modulation of amplitude ${\gamma_{m}/\gamma_{0}}=0.0014\approx\rho$
(where $\rho$ is the FEL parameter [35]) and period $\lambda_{m}=100\
\mathrm{nm}$. This relatively long wavelength energy modulation is imposed as
initial conditions on the electron beam without the need for simulations (it
is relatively easy to achieve such modulation as detailed in [36, 37].)

The energy modulated electron beam is then micro-bunched in a planar undulator
FEL amplifier with rms undulator parameter $\bar{a}_{u}=1.72$, period
$\lambda_{u}=3.9$ cm and at the resonant wavelength $\lambda_{r}=1.25\
\mathrm{nm}$. The micro-bunching at the resonant wavelength occurs at the
minima of the energy modulations where the gain is greater. Fig. 3 shows the
bunching structure which has been established at the end of the FEL amplifier.
The regions of the electron beam with high electron bunching will emit
coherently in the afterburner so that a modulation in the electron bunching
parameter leads to a corresponding modulation in the radiation intensity.

Figure 3: Electron bunching parameter ($0<|b|<1$) as a function of relative
time for a section of electron beam captured near the end of the FEL
amplifier.

The first amplifier stage has a small reverse taper in the undulator parameter
$\bar{a}_{u}$ [38]. This reduces the intensity of radiation emitted by the
electrons without any significant reduction to the electron bunching at
$\lambda_{r}$. In addition to using a reverse taper to suppress the radiation
output from the modulator section, a small steering of the electron beam away
from it before entry into the afterburner sections can also be used. In the
simulations presented here, the steering of the electron beam is not modeled
explicitly - the radiation generated in the modulator is simply removed before
injection into the afterburner sections.

Following the preparation of the $4$ GeV electron beam, with a micro-bunching
at $1.25$ nm modulated at $\lambda_{m}=100$ nm, it is then injected into the
afterburner section where the OAM output is generated. The first undulator
module is a right-handed helical undulator with $\bar{a}_{u}=2.634$,
$\lambda_{u}=3.9$ cm and $N_{u}=20$. The fundamental resonant wavelength is
then $\lambda_{1}=2.5$ nm, so that the electrons are micro-bunched at the 2nd
harmonic. The relative propagation distance, or ‘slippage’, of a wavefront
through the electrons is then $N_{u}\lambda_{1}=\lambda_{m}/2$, one half of
the electron beam modulation period.

In Fig. 4 it is seen that the modulated micro-bunching of the electron beam at
the 2nd harmonic of resonant wavelength on the helical afterburner emits a
pulse train of OAM light of mode $\ell=1$ and wavelength $\lambda_{OAM}=1.25$
nm. Each pulse has a FWHM pulse duration of $\tau\approx 167$ as,
approximately 40 optical cycles corresponding to twice the number of helical
undulator periods.

It is seen that in addition to the OAM emission there is also some coherent
emission into the $\ell=0$ mode. This emission is expected due to the
relatively short length of the afterburner undulators needed to produce short
pulses as discussed in section III.2. The length of the undulator is fixed to
generate a specific pulse length. However, reducing the crossectional radius
of the electron beam is one method that will increase the emission into the
OAM mode.

Figure 4: Temporal plot of the radiation power output from the $\ell=1$
helical afterburner decomposed into OAM modes. The radiation wavelength
$\lambda_{OAM}\approx 1.25$ nm and the pulse FWHM durations are $\tau\approx
40\;\lambda_{OAM}/c=167$ as.

###  IV.2 Time Varying OAM

A left-handed helical undulator is now added to the right-handed afterburner
(see Fig 2 b)) with the same length and tuning. Previous versions of this
crossed undulator configuration have been used to enable polarization control
at the fundamental [39, 40], spatially varying polarization [41] and the
generation of Poincaré beams [25]. Recently, crossed undulators have been
proposed as a method to modulate the polarization much in the same way we
alternate the OAM here [42, 43].

The additional undulator emits a second pulse train with OAM mode $\ell=-1$.
The slippage in the first undulator means that this $\ell=-1$ pulse train is
shifted relative to the first by half a modulation period. The combined pulse
train from the two afterburners then consists of a sequence of pulses that
alternate between $\ell=1$ left-hand circularly polarized light and $\ell=-1$
and right-hand circularly polarized light. Fig. 5 shows the power radiation
pulse train after the second afterburner decomposed into the different OAM
modes. The individual pulses have the same pulse length, $\tau_{p}=167$, as
before. The time between pulse peaks is also $167$ as, corresponding to a
switching rate of $6$ PHz.

The intensity profile and phase of the radiation of an $\ell=1$ and $\ell=-1$
is shown in Fig 6. The intensity (averaged over a single spike) shows the
typical OAM ‘doughnut’ profile in the transverse plane. The phase of the
radiation is captured near the intensity peaks of two neighboring radiation
spikes. There is a clear azimuthal phase distribution which changes handedness
in the $167$ as interval between snapshots.

Figure 5: (top) Power Vs relative time at the end of the afterburner for FEL
radiation decomposed into OAM different modes.(bottom) Stokes parameter
$s_{3}$ vs time.

Figure 6: (Left) Pulse intensity averaged over a single pulse and (right)
instantaneous phase for the pulses found as relative times (top) $t=2516$ as
and (bottom) $t=2683$ as in Fig 5.

Also included in the Fig. 5 is the normalized Stokes parameter for circular
polarization

$s_{3}=\frac{|E_{R}|^{2}-|E_{L}|^{2}}{|E_{R}|^{2}+|E_{L}|^{2}},$ (3)

The polarization of the light alternates along with the OAM mode. This then
provides an interesting case as the polarization rotates in the same direction
as the OAM. The total angular momentum contribution from the OAM plus the spin
momentum add and therefore the pulse train alternates between total angular
momentum $\pm 2\hbar$. We also note that tuning one of the afterburner
undulators so that emission is at the fundamental frequency will produce
pulses with no OAM. This $\ell=0$ mode will be emitted from helical and planar
undulators and thus a variety of OAM and spin combinations are available.

If OAM pulses with only one polarization are required, a linear polarizer
which will filter out one linear polarization component may be a solution.
Filtering out linear polarization corresponding to the polarization of the
amplifying FEL section would also help ensure light emitted in this section
does not interfere with the final pulse train delivered to experiment.

##  V Conclusion

Ultra-fast manipulation of the temporal properties of FEL generated OAM light
have been proposed and simulated for the first time. The method can be used to
generate a pulse train of light carrying OAM and also alternate the handedness
of the OAM at ultra-fast timescales. It was shown that the temporal profile of
OAM radiation emitted from electrons can be controlled by tailoring the
structure of micro-bunching in the electron beam. In addition, since the
handedness of the OAM is determined by the magnetic undulator and not external
optics, FELs are uniquely positioned to offer ultra-fast flipping of OAM
handedness at X-ray wavelengths. As the method uses a relatively simple
afterburner configuration, it is relatively cost effective to implement at
existing FEL facilities.

##  VI Acknowledgments

The authors would like to thank E. Hemsing for providing useful comments in
the preparation of this paper. This work was supported by U.S. Department of
Energy Contract No. DE-AC02-76SF00515 and Award No. 2021-SLAC-100732. We are
grateful to funding from the Science and Technology Facilities Council
(Agreement Number 4163192 Release#3); ARCHIEWeSt HPC, EPSRC grant
EP/K000586/1; EPSRC Grant EP/M011607/1; John von Neumann Institute for
Computing (NIC) on JUROPA at Jülich Supercomputing Centre (JSC), project
HHH20.

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