![color online). 3D plots of simulations at 0 setting of the phase shifter and positive helicity of the upstream undulator [Fig. 4(c)] calculated for the actual (left) and zero (right) emittance, respectively.](https://www.researchgate.net/profile/Karsten-Holldack/publication/255174362/figure/fig4/AS:667689018208265@1536200875618/color-online-3D-plots-of-simulations-at-0-setting-of-the-phase-shifter-and-positive_Q320.jpg)
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First Observation of Photons Carrying Orbital Angular Momentum in Undulator Radiation
J. Bahrdt, K. Holldack, P. Kuske, R. Mu
¨
ller, M. Scheer, and P. Schmid
Helmholtz-Zentrum Berlin, Albert-Einstein-Straße 15, 12489 Berlin, Germany
(Received 26 February 2013; published 15 July 2013)
Photon beams of 99 eV energy carrying orbital angular momentum (OAM) have been observed in
the 2nd harmonic off-axis radiation of a helical undulator at the 3rd generation synchrotron radiation
light source BESSY II. For detection, the OAM carrying photon beam was superimposed with a reference
beam without OAM. The interference pattern, a spiral intensity distribution, was recorded in a plane
perpendicular to the propagation direction. The orientation of the observed spiral structure is related to
the helicity of the undulator radiation. Excellent agreement between measurements and simulations has
been found.
DOI: 10.1103/PhysRevLett.111.034801 PACS numbers: 41.60.Ap, 41.60.Cr
Introduction.—For an axially symmetric geometry the
solution of the Helmholtz equation in paraxial approxima-
tion can be expanded in Laguerre-Gaussian (LG) polyno-
mials. In 1992, Allen et al. demonstrated analytically that
beams consisting of an ith LG mode with a cork-screw-like
phase distribution and an annular intensity carry an orbital
angular momentum of l@ [1] per photon. Since then, vari-
ous experiments using OAM-photons were performed in
the visible and in the infrared regime [2,3]. OAM photons
were utilized for the micromanipulation of small particles
in optical tweezers [4] or for channel multiplexing in
telecommunication [5]. Also, OAMs lead to other selection
rules for electronic transitions, e.g., quadrupole transitions,
forbidden for Hermite Gaussian (HG) beams are allowed
if spin and orbital angular momentum add up to two [6].
So-called OAM dichroism spectroscopy is feasible if the
beam size is of the same magnitude as the sample region
[6]. Already today, focus sizes well below 20 nm are
achievable with high quality zone plates as described in
[7,8]. However, a widespread use of OAM beams is ham-
pered by the difficulties to generate such beams.
The first transformation of the more common HG beams
into LG beams has been performed with cylindrical lenses
[9]. Today, sophisticated computer-controlled spatial light
modulators are used to generate forked holographic pat-
terns which transform HG modes into LG modes in the
1st diffraction order of the forked hologram [10].
Singular photon beams—another name for OAM beams
emphasizing the phase singularities—have been obtained
in the x-ray regime as well. So far, they were produced
either with a circular phase plate [11] or with a simple
rectangular aperture [12] from transversal coherent spheri-
cal waves. Both experiments were performed at 3rd gen-
eration storage rings (APS and SPRING-8, respectively).
The light generated in these experiments was a mixture of
many modes and the energy tunability was limited.
Various schemes of singular photon beam production
in free electron lasers (FELs) based on helical undulators
have been proposed. One method uses a CO
2
seed laser,
resonant to the 2nd harmonic of a helical undulator, which
is superimposed to the electron bunch while passing the
undulator [13]. The FEL mechanism in the presence of
the CO
2
laser field gradient forces the electrons on a cork-
screw-like charge distribution with a period of half the
resonance wavelength. Only recently, a spiral or helical
bunching in the 2nd harmonic has been observed [14].
Helical bunching on the 1st FEL harmonic can be achieved
in a harmonic generation scheme where a seed laser is
superimposed to the electron beam in a helical modulator.
A chicane converts this energy modulation into a helical
bunching which is transferred to the FEL radiator [15]. At
short wavelengths, the required strong seed lasers are not
available and alternative schemes are proposed. The echo
enabled harmonic generation (EEHG) seeding scheme is
predicted to generate intense x-ray OAM beams at the FEL
fundamental [16].
A direct and efficient method to generate Laguerre-
Gaussian beams of high mode purity in accelerator based
synchrotron radiation light facilities was theoretically pre-
dicted in 2007 [17,18]. A relativistic particle passing
through a helical undulator is supposed to produce
Laguerre-Gaussian modes in the off-axis radiation of the
2nd undulator harmonic. Here, we report on the first
experimental proof of existence of these OAM modes.
Theory.—Higher harmonic photon beams of helical
undulators carry an orbital angular momentum of
ðn 1Þ@ as described in [17], where n is the harmonic
number. The OAM sign depends on the helicity of the 1st
undulator harmonic. For these cases, the on-axis intensity
is zero (annular beam) and the phase on a circle in a plane
perpendicular to the photon beam varies with ðn 1Þ’,
where ’ is the azimuthal angle. Figure 1 shows the phase
characteristics of OAM carrying beams and beams without
OAM photons. OAM photons exist in a region of the
parameter space where helical undulators usually are not
operated because the brightness of helical undulators
is reduced at higher harmonics, though the flux is still
rather high.
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The intensity distribution of a singular photon beam
does not exhibit the peculiar feature of a cork-screw-like
phase distribution. Without a wave front sensor, solely
relying on pure intensity measurements, the observation
of photons carrying orbital angular momentum is difficult.
An azimuthal intensity variation may be a hint [18]but
the modulation is smeared out by electron beam emittance
and photon beam line effects. A superior method is an
interference experiment which utilizes the corkscrew-like
phase distribution. The photon beam carrying OAMs is
superimposed on a reference beam without OAM photons
(or with OAM of another quantum number) monitoring the
interference fringes in the intensity distribution perpen-
dicular to the propagation direction.
Experimental setup.—The experiment was carried out at
the 3rd generation electron storage ring BESSY II. The
layout of the experimental setup is depicted in Fig. 2. The
double undulator UE-56 [19] was used for the generation
of the two light beams, the singular photon beam and the
reference beam. The UE-56 double undulator consists of
two APPLE II type modules [20 ]. Each module can be
tuned individually for photon energy and polarization. The
1st undulator (upstream) was tuned to pure helical mode
and a photon energy of 49.5 eV in the 1st harmonic. The
reference undulator (downstream) was tuned to 99 eV and
horizontal linear polarization. Other polarization states can
be used for reference, as well, including circular polariza-
tion with zero or finite topological charge, as long as the
topological charges of the two beams are different.
A permanent magnet phase shifter consisting of eight
rotatable magnets between the two undulators [19] permits
the realization of an additional phase shift between the two
light beams. A monochromator behind the double undu-
lator [21] selects a specific wavelength and narrows the
bandwidth while elongating the two wave packages. Thus,
the overlap of the two originally separated light beams
happens only behind the monochromator (the arrangement
is similar to the crossed undulator as proposed indepen-
dently by Nikitin [22] and Kim [23]). The monochromator
resolution was 2500, providing a longitudinal overlap of
>98%. Small fractions of the interfering light beams are
cut out with a 100 m pinhole upstream of the first
optical element and measured with a 4mmGaAsP diode
downstream of the monochromator behind the exit slit and
refocusing chamber. The transverse intensity distribution
of the interfering beams was measured by so-called ‘‘pin-
hole maps’’ where the pinhole is moved in the transverse
plane while monitoring the diode signal.
The basic pattern of the expected intensity distribution
can be evaluated from a simple model where the OAM
carrying beam and the linearly polarized beam are both
described by point sources (far field approximation). Only
the horizontal electric field components interfere whereas
the vertical component of the helical undulator contributes
to an independent intensity background. The horizontal
field amplitudes of both beams, A and B, are described
by the real expressions
Aðr; ’Þ¼
aðrÞ
L þ d
cos
d
2
þ
ðL þ dÞ
r
2
ðn 1Þ’
þ
2L
!t
; (1)
Bðr; ’Þ¼
bðrÞ
L
cos
L
r
2
þ
2L
!t
(2)
with L ¼ distance between the linear undulator and
plane of the pinhole (12 237 mm), d ¼ distance between
the helical and linear undulator (2126 mm), r ¼
transverse distance to beam axis, ¼ photon wavelength,
! ¼ detection frequency, t ¼ time, aðrÞ and bðrÞ radial part
of the field amplitudes and ¼ Lorentz factor of the rela-
tivistic electrons.
The intensity distribution of the horizontal field compo-
nents is evaluated from Eqs. (1) and (2) via
Iðr; ’Þ¼
!
2
Z
2
!
0
ðA þ BÞ
2
dt (3)
leading to
Iðr; ’Þ¼
a
2
2ðL þ d Þ
2
þ
b
2
2L
2
þ
ab
LðL þ dÞ
cos
d
2
d
L
2
r
2
ðn 1Þ’
: (4)
FIG. 2 (color online). Setup of the experiment with two un-
dulators and a phase shifter in between (not shown), a movable
pinhole in front of the monochromator and the detector behind
the refocusing optics.
FIG. 1 (color online). Surfaces of equal phase for beams with
various orbital angular momenta.
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We call the 3rd term the spiral term. The spiral sense of
rotation depends on the helicity of the undulator and the
orientation is determined by the relative phase between the
two light beams (constant phase term of the cos argument).
The observation of the spiral term is an unambiguous proof
for the presence of photons carrying OAM. The spiral term
has a maximum if the argument of the cos function is zero
which gives the correlation
’ ¼
d
2
þ
d
L
2
r
2
ðn 1Þ: (5)
The term (n 1) represents the topological charge of
the beam. A comparison of Eq. (5) with measurements
permits the derivation of the sign and the absolute value of
the topological charge from the sense of rotation and the
shape of the spiral.
It must be emphasized that the electron beam emittance
(i.e., transverse phase space volume) may have a signifi-
cant impact on the OAM beam. Assuming the electron
beam emittance being much larger than the OAM carrying
photon beam of an individual electron, each electron still
produces an OAM beam; however, the vortices of these
FIG. 3 (color online). Measured (symbols) and simulated
(solid lines) intensity cuts of the individual undulators: vertical
cuts of the linear undulator (dots and solid line) and diagonal
cuts of the helical undulator (diamonds and solid line) are
plotted. The unused undulator is set to a magnetic gap of
100 mm. Inset: Measured color-coded intensity profiles from
the linear (upper) and the helical (lower) undulator, respectively.
FIG. 4 (color online). Measurement [(a) and (d)] and simulation [(b) and (c)] for the case of 10
[(a) and (b)] and 0
[(d) and (c)]
setting of the phase shifter. In the numeric simulations the spiral orientation is not a fitted parameter and the agreement with
measurements demonstrates a realistic modeling of the undulators and the phase shifter. The dashed spirals in the upper plots are
evaluated from the analytic equation Eq. (5) where the constant phase term is set to zero to match the observed orientation of the spiral.
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individual beams are widely spread and the electron bunch
does not produce a photon beam with a well-defined single
vortex and a phase distribution as presented in [17].
The described interference experiment can only be per-
formed with a sufficiently small electron beam emittance:
Usually, BESSY II is operated at an electron energy of E ¼
1:72 GeV and a horizontal beam emittance of "
x
¼ 6:20
nm rad (superconducting wavelength shifters switched
off). The vertical emittance is a factor of 100 smaller (1%
coupling ratio). These values have to be compared to the
wavelength-dependent diffraction limited source size of the
undulator radiation. Simulations with the synchrotron radia-
tion code
WAVE [24] demonstrate that the spiral intensity
distribution is completely smeared out by the large horizon-
tal emittance at E ¼ 1:72 GeV. The horizontal emittance of
a storage ring scales with the square of the electron energy
[25]. To achieve the required degree of transverse coher-
ence, the storage ring was ramped down with stored electron
beam from E ¼ 1:72 GeV to E ¼ 917 MeV. The electron
energy of this optics, which was established 10 years ago for
the Physikalisch Technische Bundesanstalt, was determined
earlier by means of Compton backscattering [26]. More
recent measurements by one of the authors (P.S.) obtained
a horizontal emittance of "
x
¼ 1:66 nm rad and a natu-
ral fractional energy spread of
E
¼ 3:67 10
4
for this
setting. The typical beam current and lifetime during the
measurements was 1 mA and 8 h, respectively.
Results.—The measured intensity distributions of the
individual, noninterfering beams as emitted by the linear
and helical undulator, respectively, are plotted in Fig. 3.
As expected, the individual profiles do not show any spiral
features. When overlapping the two beams coherently, a
pronounced spiral intensity distribution becomes visible
(Fig. 4). The spiral orientation depends on the angular
settings of the phase shifter magnets and a rotation is
visible when tuning the magnets from 0
to 10
. From
magnetic measurements a path lengthening of 4.4 nm is
expected for the 10
rotation. This corresponds to a relative
phase change of 127
between the two undulator beams
which is consistent with observation.
The measurements are reproduced by
WAV E simulations
utilizing realistic three-dimensional undulator and phase
shifter fields. The excellent agreement between measure-
ments and calculations validates the model underlying the
simulations. The line of maximum intensity of the spiral
structure is well reproduced by the analytical model of
Eq. (5) [Fig. 4(a) and 4(b)].
Comparing Fig. 5 (negative helicity) and Fig. 4(a) (posi-
tive helicity) an opposite sense of rotation of the spiral
intensity distributions for the two cases is observed. This
is another proof of the existence of OAM photons. The spiral
shape is slightly distorted for negative helicity due to elec-
tron beam steering caused by residual undulator field errors.
A comparison of simulations at zero and at actual
emittance demonstrates, that even at E ¼ 917 MeV the
experimental results are still influenced by emittance
effects (Fig. 6) due to lack of complete transversal
coherence.
Under usual operation conditions of 3rd generation light
sources with a typical emittance of "
x
¼ 3–6 nm rad
OAM carrying photon beams do not have a single, well-
defined vortex with a distinct phase distribution of the
wave front. Hence, they can hardly be observed utilizing
interference methods. FEL arrangements for the produc-
tion of x-ray photons carrying OAMs are complicated to be
operated as pointed out earlier. This will be different for the
next generation light sources currently under discussion:
energy recovery linacs (ERLs) and ultimate storage rings
(USRs), i.e., diffraction limited light sources. ERLs [27]
are single (or few) turn storage rings which are filled with
low emittance beams from a linac. The beam is extracted
before being damped into an equilibrium state of larger
emittance. USRs [28–31] on the other hand provide a low
emittance even in the equilibrium.
Conclusion.—Our measurements clearly support predic-
tions that the new accelerator-based diffraction limited
light sources will deliver intense, clean, and energy-
tunable OAM beams over a wide energy range with a
single standard helical undulator under normal user con-
ditions. The high energy limit of the extended photon beam
parameter space is defined by the electron energy and the
FIG. 6 (color online). 3D plots of simulations at 0
setting of
the phase shifter and positive helicity of the upstream undulator
[Fig. 4(c)] calculated for the actual (left) and zero (right)
emittance, respectively.
FIG. 5 (color online). Measured intensity distribution at 0
setting of the phase shifter and negative helicity. The spiral
sense of rotation is reversed as compared to the results for
positive helicity (Fig. 4).
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emittance. It is expected that new exciting experiments
with OAM carrying photon beams will become reality in
many fields of research as soon as these radiation sources
become operational.
The authors thank A. Gaupp, E. Gluskin, I. McNulty,
and S. Sasaki for many helpful discussions.
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