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OPEN

Received: 12 January 2017
Accepted: 13 June 2017
Published: xx xx xxxx

Helical Phase Structure ofRadiation from an Electron in
Circular Motion
M. Katoh1,2, M. Fujimoto1,2, N. S. Mirian1, T. Konomi1,2, Y. Taira3, T. Kaneyasu4, M. Hosaka5,
N. Yamamoto5, A. Mochihashi5, Y. Takashima5, K. Kuroda6, A. Miyamoto7, K. Miyamoto7 &S. Sasaki7
We theoretically show that a single free electron in circular motion radiates an electromagnetic wave
possessing helical phase structure, which is closely related to orbital angular momentum carried by it.
We experimentally demonstrate it by interference and double-slit diffraction experiments on radiationfrom relativistic electrons in spiral motion. Our results indicate that photons carrying orbital angularmomentum should be created naturally by cyclotron/synchrotron radiations or Compton scatterings invarious situations in cosmic space. We propose promising laboratory vortex photon sources in various
wavelengths ranging from radio wave to gamma-rays.
Twisted photons possessing helical wave front carry orbital angular momentum other than well-known spin
angular momentum1. Such photons have been investigated theoretically and experimentally, in particular,
towards applications in the information technologies, the nanotechnologies and the imaging technologies2, 3.
Their interactions with nuclei, atoms, molecules, materials or plasma have been also being explored theoretically rather than experimentally2–4, that might be due to the lack of the laboratory photon sources other than in
the laser wave length range where twisted photon beams can be readily obtained by using conventional opticaldevices. Twisted photons in nature, particularly in astrophysics, have been reviewed5, 6. However, the authors did
not discuss the sources of twisted photons explicitly because they could not find promising radiation processes.
Instead, they discussed detection schemes of twisted photons in astronomical observations. Other authors proposed twisted radiation from rotating black holes or inhomogeneous interstellar media7, 8, however, they discussed modification of radiation rather than radiation processes itself.
In this paper, we show that a single free electron in circular motion emits twisted photons. This is one of themost fundamental radiation processes by a free electron and is the basis of synchrotron/cyclotron radiations orCompton scattering, which play important roles in various situations in nature or in laboratories, such as aroundmagnetized neutron stars, in supernova explosions with magnetic fields, in nuclear fusion plasma, electron accelerators and so on. The radiations from such electrons cover the entire range of wave lengths, from radio wave
to gamma-rays, depending on the physical parameters such as the electron energy or radius of motion. They
possibly play important roles with their angular momentum in nature and in laboratories, which have not beenrecognized by researchers. Moreover, this radiation process can be the basis of laboratory vortex photon sourcesin the entire wavelength range, which will open new research opportunities.
Radiation generated by electrons in circular motion was first studied by O. Heaviside in 19049 and has since
been addressed in many scientific reports10, 11 and textbooks12, 13. In the following, we derive an analytic expression that contains a term representing the spatial phase structure explicitly. We show that the radiation field possesses a spiral phase structure. We explain intuitively the mechanism which produces the structure.
We assume that the electron trajectory draws a circle in a plane perpendicular to the z-axis as shown in Fig. 1.
The radiation field is given by the following formulae, which are directly derived from the Liénard–Wiechert
potentials12:1

Institute for Molecular Science, National Institutes of Natural Sciences, Okazaki, 444-8585, Japan. 2Sokendai (The
Graduate University for Advanced Studies), Okazaki, 444-8585, Japan. 3National Institute of Advanced Industrial
Science and Technology (AIST), Tsukuba, 305-8565, Japan. 4Saga Light Source, Tosu, 841-0005, Japan. 5Nagoya
University, Nagoya, 464-0814, Japan. 6University of Tokyo, Kashiwa, 277-0882, Japan. 7Hiroshima University,
Higashi-Hiroshima, 739-0046, Japan. Correspondence and requests for materials should be addressed to M.K.
(email: mkatoh@ims.ac.jp)
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www.nature.com/scientificreports/e-Figure 1. Coordinate system. The electron is rotating in the x-y plane around the origin with an initial positionon the x-axis. The azimuthal angle of the position of the observer is measured from the y-axis. The observer
frame is defined as a spherical coordinate.

→e
E (t ) =
cR

{

}

→
→

→
n × (→n −β)×β→n ⋅ β )3
(1 − →t′→→
H (t ) = →n × E t′t′ = t −

(1)
(2)

R
c

(3)

→

Here, E (t ) and H (t ) are the electric and magnetic fields, respectively, c is the speed of light, e is the elemental→→
charge, n is the unit vector, R is the distance from the origin to the observer (see Fig. 1), β is the electron velocity
normalised by the velocity of light, and ω is the electron angular velocity. The right sides of Eqs (1) and (2) represent the emitter time t ′, which is related to the observer time t by Eq. (3). An expression for the electric field in theobserver frame is obtained by inserting adequate expressions for the vectors in Eq. (1):→eβωcos θ sin(ωt ′ − ϕ)→eθ − {cos(ωt ′ − ϕ) − β sin θ}→
E (R , θ , ϕ , t ) =eϕcR {1 − β sin θ cos(ωt ′ − ϕ)}3 (4)Because the magnetic field can be obtained using Eq. (2), we do not show its explicit form.
The waveforms of the radiation fields calculated using Eq. (4) are shown in Fig. 2. As the electron velocity
increases, the sinusoidal waveforms become distorted and develop kinks around the phase, corresponding to
the time when the instantaneous electron motion is directed to the observer (Fig. 1). The deformed wave can be
decomposed into harmonic components with frequencies that are integer multiples of ω9.
The spatial distribution of the propagating electric field is also shown in Fig. 2. The relativistic kink, where thefield is strengthened, is distributed as a spiral. This suggests that the phase of the higher harmonic components
also depends on the azimuthal angle. Such a behavior of the radiation field can be seen in the textbooks describing synchrotron radiation13, 14. However, the authors did not discuss it in the context with the phase structure or
the angular momentum. We decompose Eq. (4) into a Fourier series to demonstrate the spiral phase structure
analytically, following the previous literature12, and we express the result by unit vectors in Cartesian coordinates
to make the phase structure explicit:
∞

R0
→
→
→
e
lω{ε+l (θ )e i (l−1)ϕ e+ + ε−l (θ )e i (l+1)ϕ e− + iεzl(θ ) eze ilϕ}e−il(ωt− c )l = 1 cR→E (R 0 , θ , ϕ , ωt ) = Re∑
ε±l (θ ) ≡εxl(θ ) ± ε yl (θ )2= βJl′(lβ sin θ ) ±εzl ≡ cos θJl(lβ sin θ )ScIeNTIfIc REPortS | 7:6130 | DOI:10.1038/s41598-017-06442-2

cos2 θ
Jl(lβ sin θ )
sin θ

(5)

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Figure 2. Left: Waveform of the electric field propagating towards the polar angle (θ) of 30° and the azimuthal
angle (φ) of 0° for a range of electron velocities, β. Blue lines represent the θ components, and red lines represent
the φ components (see Fig. 1). Black dotted lines show the field intensities, given as the square summation of
the electric field components. To emphasise the change in the waveform, the electric fields are divided by β inthe calculation (see Eq. (4)). Right: Radiation field intensity propagating towards 30° from the z-axis and its
projection on the x-y plane. The electron velocity β is 0.5. The brightness is the magnitude.

Figure 3. Electric field distribution in the upper hemisphere viewed from the z-direction (see Fig. 1), fromleft to right, for the fundamental (l = 1), second (l = 2) and third (l = 3) harmonics calculated from Eq. (5).
The colour represents the field intensity. The fundamental frequency has an intensity maximum in the centre,
whereas the harmonics show zero intensity at the centre. Arrows represent the direction of the electric field at a
specific time.

where Jl and Jl ′ are a Bessel function of the first kind and its derivative, respectively. Here, we have introduced
rotation vectors related to the unit vectors in the x and y directions as →e± = →
ex ± i→ey / 2 . The first term in theparentheses represents circular polarised components with the same helicity as that of the electron motion, andthe second term represents components with the reverse helicity. The third term arises from the spherical natureof the field. The summation of these three terms represents the elliptical polarisation. The electric fields of thefundamental, second and third harmonics calculated from Eq. (5) are shown in Fig. 3. The vortex nature is clearlyobserved in the electric field distribution of the second and third harmonics but not in the electric field distribution of the fundamental component.
In Eq. (5), when θ is small, the first term becomes dominant and the field is accurately represented by paraxial
approximation. Its polarisation is circular and has a phase term exp{i(l − 1)φ} that is a common feature of a vortex beam. According to Allen et al.1, such a field possesses an orbital angular momentum of l−1. In the generalcase of θ, the field represented by Eq. (5) is non-paraxial. It has been argued that the radiation field of such a casepossesses angular momentum, even though the spin and orbital angular momentum are difficult to separate15.
Recently we have successfully derived an expression for the ratio of the angular momentum density to the energy
density for the radiation emitted by an electron in circular motion in a separate paper, taking a different mathematical approach16. The expression shows that a photon of the l-th harmonic carries the total angular momentum
whose z-component is equal to l. It may be interpreted as that the first component in Eq. (5) has spin angularmomentum of +1 and orbital angular momentum of l−1, whereas the spin and orbital angular momenta of thesecond component are −1 and l + 1, respectively. The total angular momentum in the z direction is always l.
When an electron in circular motion drifts along the z-axis with high relativistic velocity, the radiation field
perpendicular to the z-axis is strengthened by the Lorentz factor γ, which is given by the electron energy divided
by the electron rest mass energy, and is collimated into a narrow cone around the z-axis12. Consequently, the field(

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215

Intensity

13.3 mm

216

0

215

Intensity

13.3 mm

216

0

215

Intensity

13.3 mm216

013.3 mm13.3 mm13.3 mm13.3 mmFigure 4. Interference between two undulator radiations. From the left column to the right: interference
patterns between the fundamental and third harmonics and between the fundamental and second harmonics
for left-handed polarisation, and between the fundamental and second harmonics and between thefundamental and third harmonics for right-handed polarisation. From the top row to the bottom, the raw CCD
images, those with the analytic calculation results (red dotted lines) following Bahrdt et al.19 and the numericalsimulation results by SRW25. The handedness is defined along the electron beam direction. The centres of theanalytic results are fitted to the measurements.

is well represented by the paraxial approximation. Such radiation fields can be produced in the laboratory using
a helical undulator. These devices are widely used as synchrotron light sources17, wherein a high-energy electron
beam executes spiral motion in a specially designed magnetic field, radiating circularly polarised light. This has
been identified as a vortex photon source both mathematically18 and experimentally19, although the origin of thetwisting has not been addressed. We acknowledge that these pioneering works inspired us to carry out this work.
The helical undulator radiation corresponds to the case where an electron in circular motion travels towards
the z-axis at relativistic velocity. Because the phase is Lorentz invariant, the harmonic components of the helicalundulator radiation should preserve the helical phase structure, which is consistent with the conservation ofangular momentum along the direction of motion of the frame in the Lorentz transformation12.
Here, we present experimental results that provide clear evidence of the twisted nature of the harmonics andthe non-vortex nature of the fundamental as expressed by Eq. (5). To observe the spatial phase structure, a significant fraction of the radiation field should be observed simultaneously. The collimated nature of the undulator
radiation provides an ideal experimental condition. We have carried out a series of experiments on the opticalvortex beam from an undulator at the UVSOR-III electron storage ring20, 21. Some relevant results are shown
below.
First, we discuss the interference between the fundamental radiation and harmonics. The interference measurement between a reference beam and a vortex beam is an established method to show its phase structure22.
Bahrdt et al. proposed a novel technique to apply this method for undulator radiation and successfully demonstrated it for the fundamental and second harmonics19. Following their approach, we carried out interference
experiment including the higher harmonics. Figure 4 shows the interference patterns between the fundamental
radiation from one undulator and the second or third harmonics from another undulator. They are comparedwith analytic calculations based on the formulae in the references19, 23 and a simulation code SRW24. Single- anddouble-spiral structures are clearly observed and they are well reproduced by the analytic calculations and thenumerical simulations. This result clearly shows that the orbital angular momentum increases as the harmonic
number increases, which is one of the important prediction of Eq. (5). Moreover, the directions of the spiral structures are reversed with the reversal of the electron circulation direction. These results indicate that the relative
phase difference between the light beams is in agreement with the theoretical prediction of Eq. (5). However, at
this stage, the absolute phase structure of radiation generated by a single undulator is unclear.
Next, we present the results from a double-slit diffraction experiment. This is the first experimental result todemonstrate the vortex nature of radiation from a single undulator using longitudinally incoherent but spatially
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Figure 5. Double-slit diffraction patterns of undulator radiation. From the left column to the right; thefundamental, the second harmonic (left-handed) and the second harmonic (right-handed), from the top row tothe bottom; schematic drawings of the double-slits (red rectangles) and the phase distributions, the raw CCD
camera images, those with the analytic calculation following Sztul and Alfano24 (red dotted lines) and the SRW
simulation results25. The handedness is defined as same as in Fig. 4. The centres of the analytic calculation results
are fitted to the measurements.

coherent radiation. The fundamental and second harmonic components in the ultraviolet range were extracted
and irradiated onto a double-slit apparatus. The diffraction patterns were observed using a charge-coupled device(CCD) camera, as shown in Fig. 5. An ordinary stripe pattern was observed for the fundamental radiation, as
expected for plane waves. However, for the second harmonic, we observed a singularity in the middle of the pattern, as has been demonstrated for a Laguerre–Gaussian beam created by a laser25. This singularity arises fromthe change in the phase difference between the two slits, as illustrated in Fig. 5. The singularity appeared only
when the centre of the beam was located between the slits. The observed patterns were accurately reproduced by
the analytic expression25 and SRW code24. We noted that diffraction experiments were carried out for coherent
vortex radiation from a single undulator26, 27. However, these experiments are inadequate to show the propertyof radiation from a single electron, because, in case of coherent radiation, the radiation field is strongly affected

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Figure 6. Experimental setup of the interference experiment (upper) and the double-slit diffraction experiment(lower). Electrons travel from left to right while executing spiral motion.

by the distribution of electrons26, 28. We also noted that a vortex X-ray beam was successfully produced by using
spiral phase plate and undulator beam, which may be an alternative approach for practical applications29.
We have shown that the harmonic components of an electromagnetic field radiated by electrons in circularmotion naturally have a helical phase structure, which suggests the presence of orbital angular momentum. We
demonstrated this experimentally by observing helical undulator radiation. The generation of a twisted photon
beam from a helically micro-bunched electron beam was reported27. However, a vortex radiation field was formedby the constructive interference of the non-twisted radiation from the electrons that were helically aligned inspace. This is analogous to the generation of monochromatic radiation from a micro-bunched beam travelling ina uniform magnetic field, even though each electron produces broadband synchrotron radiation28.
This work allows us to predict the conditions under which twisted radiation will be produced. We proposethat cyclotron/synchrotron radiation, particularly from an electron cyclotron maser30, should be re-examined
as twisted radiation. Another candidate is the non-linear inverse Compton scattering of circular-polarised light.
In this case, the intense incoming light field causes relativistic circular motion of electrons, thereby producing
twisted harmonic radiation, as described in a separated paper31. These radiation processes may play important,unexplored roles in the solar magnetosphere10 or planetary magnetosphere32, around magnetised neutron stars33,
around active galactic nuclei34, in tokamaks used for nuclear fusion35, or in particle accelerators17. Observations
using phase information may provide new approaches to the analysis of such systems. Several methods have been
proposed for detecting vortex photons in nature, but there has been no explicit discussion of the sources of suchradiation5, 6, 36. Our work suggest some possible vortex photon sources in nature.
This work also suggests possible technologies for producing twisted photon beams in laboratories at wavelengths ranging from radio waves to gamma rays. In the ultraviolet and X-ray ranges, helical undulators canprovide high-brightness twisted radiation. In the microwave and terahertz ranges, gyrotrons, where high-energy
electrons execute circular motion and produce cyclotron radiation with harmonics, are potentially powerful sources of twisted radiation37. Another candidate is the non-linear inverse Compton scattering of intense
circular-polarised laser radiation by a relativistic electron beam provided by an accelerator, which could potentially provide a twisted X-ray or gamma-ray source31. The development of these light sources will expand theapplication of twisted radiation to the entire electromagnetic wavelength range where alternative methods38–40
are not applicable.

Methods

This experiment was conducted at the BL1U beamline of the UVSOR-III electron storage ring, which is equippedwith two polarisation-variable undulators in tandem. The undulators had 10 magnetic periods with an 88-mm
period length. Two undulators are separated by a space of approximately 0.5 m. The undulators were operated incircular-polarised mode during the experiments. In the undulators, the electron beam executes spiral motion. We
define the handedness of electron circulation and light polarisation along the beam direction, such that clockwise circulation around the beam axis is right-handed. The electrons produce quasi-monochromatic synchrotron
radiation and harmonics in the ultraviolet wavelength range. This major advantage of our experiment allowed
us to perform all of the experiments in the air using ordinary optical components and devices. The UVSOR-III
electron storage ring was operated at 500 MeV for the experiment including up to 2nd harmonic and 400 MeV
for those including up to 3rd harmonic, which are lower than the nominal electron energy of 750 MeV. The electron beam emittance at this energy was estimated to be 8 nm-rad and 5nm-rad, respectively. The electron beam
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was diffraction-limited in the UV wavelength range; therefore, the undulator radiation was spatially coherent,
which was essential for the experiments described below. The typical electron beam current used in the experiment was 1 mA, much smaller than the normal operating current but sufficient for these measurements. The
undulator radiation was extracted from the accelerator through the SiO2 window, without the use of mirrors or
monochromators.
In the interference experiment, two undulators in tandem were tuned to produce either the fundamental
radiation or harmonics at the same wavelength, i.e., 355 nm. The two interfered light beams were observed via a
CCD camera (BITRAN BK50-UV) with a bandpass filter at 355 nm and a width of 1.3 nm, which was positioned
approximately 7.5 m downstream of the centre of the second undulator and was directly viewing the undulators.
The CCD is 13.3 mm × 13.3 mm in size with 1024 × 1024 pixels. In the diffraction experiments, a double slit witha width of 200 microns and separation of 2 mm was positioned approximately 7.5 m from the centre of the helicalundulator and was aligned so that the optical beam centre was in the middle of the slits. The diffraction image wasobserved via the same CCD camera as in the interference experiment with a 355-nm bandpass filter placed 3 m
downstream from the slit. A diffraction pattern with a singularity, as shown in Fig. 6, appeared when the beam
centre was located between the slits but not when the slit position was shifted vertically by 3 mm in either direction. To obtain a clear diffraction pattern for the images presented in this paper, a linear polarised filter was placedin the vertical direction to eliminate contamination from the horizontally polarised radiation caused by the bending magnets. Neutral density (ND) filters were added as necessary to prevent saturation of the CCD camera.

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Acknowledgements

Part of this work was supported by JSPS KAKENHI Grant Numbers 26286081 and 26390112. The constructionof BL1U at UVSOR was supported by the Quantum Beam Technology Program of MEXT/JST. We appreciate the
IMS Equipment Development Center for their efforts to design and fabricate the double slit. We thank to Profs.
S. Kubo, Y. Yoshimura, K. Ohmi, and T. Saito and Drs. K. Tsuchiya and T. Hayakawa for their helpful discussions.

Author Contributions

M.K. contributed to all the theoretical work. T.K., M.H., Y. Tai, Y. Tak, A. Mo, and M.F. verified the analytic
calculations and contributed to the execution of the numerical calculations. M.K. and M.H. contributed to all
experiments. S.S., T.K., N.Y., A. mi, K.M. and K.K. contributed to the interference experiment. M.F. and N.S.M.
contributed to the diffraction experiments. The manuscript was prepared by M.K. and was checked by all otherauthors. The current affiliation of N.Y. and T.K. is the High Energy Accelerator Research Organization (KEK).

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