PRL 118, 094801 (2017)

PHYSICAL REVIEW LETTERS

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Angular Momentum of Twisted Radiation from an Electron in Spiral Motion
M. Katoh,1,2,* M. Fujimoto,1,2 H. Kawaguchi,3 K. Tsuchiya,4 K. Ohmi,4 T. Kaneyasu,5 Y. Taira,6 M. Hosaka,7A. Mochihashi,7 and Y. Takashima7
1

Institute for Molecular Science, National Institutes of Natural Sciences, Okazaki 444-8585, Japan
2
Sokendai (the Graduated University for Advanced Studies), Okazaki 444-8585, Japan3
Muroran Institute of Technology, Muroran 050-0071, Japan
4
High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801, Japan
5
Saga Light Source, Tosu 841-0005 Japan
6
National Institute of Advanced Industrial Science and Technology (AIST), Tsukuba 305-8568, Japan
7
Nagoya University, Nagoya 464-0814, Japan(Received 10 October 2016; published 27 February 2017)
We theoretically demonstrate for the first time that a single free electron in circular or spiral motion emits
twisted photons carrying well-defined orbital angular momentum along the axis of the electron circulation,in adding to spin angular momentum. We show that, when the electron velocity is relativistic, the radiation
field contains harmonic components and the photons of lth harmonic carry lℏ total angular momentum foreach. This work indicates that twisted photons are naturally emitted by free electrons and are more
ubiquitous in laboratories and in nature than ever thought.
DOI: 10.1103/PhysRevLett.118.094801

Twisted photons that carry an orbital angular momentum(OAM) in addition to a spin angular momentum (SAM) havebeen intensively studied for applications in information,
nanotechnologies, and imaging technologies [1]. Their interactions with atoms [2], molecules [3], materials [4,5], andplasmas [6] are also being explored. A photon with OAM was
originally discussed with respect to a specific mode ofelectromagnetic wave called the Laguerre-Gaussian mode[7], and the discussion was later extended to more generalcases [8]. It was shown that, when the radiation field has aphase term represented by einφ, each photon carries an OAM
of nℏ, where φ is the azimuthal angle around the propagation
axis, n is an integer, and ℏ is the Planck constant.
These previous works addressed mathematical models forthe twisted electromagnetic waves without considering theradiation processes themselves. Under the requirement that
their models satisfy the Helmholtz equations, the authors
discussed the existence of AM. In the laboratories, researchers have developed various technologies to produce electromagnetic waves of the Laguerre-Gaussian mode. Currently,
twisted photons in the visible and infrared wavelength
ranges can be readily produced using conventional laser
sources and holographic filters [1]. On the other hand, there
is much less effort to seek physical processes that produce
twisted photons in Nature. Some authors have reviewed thepossible roles of twisted photons in astrophysics [9,10], butthey mainly discussed how to detect the twisted photons
instead of how to create them. Other authors have addressed
the modifications of radiation fields by a gravitational fieldaround a rotating black hole [11] or inhomogeneousinterstellar media [12,13] away from the radiation processes.
Here, we show theoretically for the first time that anelectromagnetic wave radiated by a single free electron in
0031-9007=17=118(9)=094801(5)circular motion has a helical phase structure and carries
OAM. This work will bring the vortex photon science to a
new stage where the vortex photons are discussed in thecontext of real physical processes, how they are created,
how they propagate, and what role they play in laboratory
systems and in natural systems.
The radiation from an electron in circular motion was
first addressed by Heaviside in 1904 [14]. It is the basis of
the radiation processes by electrons in a magnetic field oran intense circular polarized light field, which are known as
cyclotron (synchrotron) radiation or Thomson (Compton)scattering. Depending on the physical parameters, theirradiation wavelengths are ranging from radio wave togamma rays. Because they play important roles in astrophysics, plasma physics, and accelerator physics, theradiation from an electron in circular or spiral motion
has been addressed in many textbooks [15,16] and literature [17,18]. However, surprisingly, no study has discussed its phase structure or angular momentum for morethan 100 years. In this paper, we discuss this radiation
process paying a special attention to the phase structure of
the radiation field. We derive its vector potential in a
spherical coordinate. Then, we demonstrate its helicalphase structure and its AM.
Generally, the real part of the time-averaged AM density~j of an electromagnetic field can be represented as a cross~ and the position
product of the linear momentum density pvector ~r as follows [19]:
1
~ ×H
~ Þ
~r × ðE
h~ji ¼ ~r × h~
pi ¼
4πc
r ~
~ Þ − H~  ð~n · EÞÞ:~ðEð~n · H
ð1Þ
¼
4πc

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PHYSICAL REVIEW LETTERS

spherical coordinate in contrast with previous works [7,8],
where the cylindrical coordinate was selected. Generally,
radiation propagates in a free space as a spherical wave;
therefore, it seems best described in the spherical coordinate.
The electromagnetic field radiated by a moving electron
can be represented by a retarded potential. When theelectron motion is periodic, it can be decomposed into
Fourier series as follows [15]:

∞ ⌒
X

β~
 ¼
~
~ l ð~rÞe−ilωt : ð4ÞAð~r;tÞ ¼ e
Aj~r − ~re j − ð~r − ~re Þ · β~ tl¼1eFIG. 1. Coordinate System. The azimuthal angle φ is takenfrom the y axis.

~ and H~ are the electric andHere, c is the light velocity, E
magnetic fields, the asterisk on the shoulder represents theircomplex conjugate, and n~ is the unit vector that directs theobserver, which is defined as n~ ¼ ~r=r. The brackets hi
indicate time average. In the radiation zone, the electric andmagnetic fields are approximately perpendicular to n~ [15].
~  and n~ · E,
~ andIn this case, the inner products n~ · H
therefore AM, are approximately zero. It should be notedthat an electromagnetic field with AM should have anonzero component along the direction of propagation.
By integrating Eq. (1) on a spherical surface thatsurrounds the source, the AM that the radiation field carries
away can be expressed as [19]
Z
Z
dJ~
1 ~
2
~
~ Þ − H~  ð~n · EÞÞ:~
¼ chjir dΩ ¼ r3 dΩ ðEð~
n·H
dt
4π
ð2Þ
Here, Ω is the solid angle, and the integration is taken over aspherical surface at distance r from the origin. The energythat the radiation carries away can be expressed as [16]ZdU
c ~ ~
¼ r2 dΩ ðE× H Þ:
ð3Þdt
4π
We can see in Eq. (3) that for the energy flow, it issufficient to consider terms in the integrant up to the orderof 1=r2 because the higher-order terms disappear at a
sufficiently large distance (r → ∞). However, for AM, wemust consider the terms up to the order of 1=r3 .
We treat the electron motion and radiation field in acoordinate system shown in Fig. 1. An electron travels on acircular orbit around the origin. Here, we do not assume thecause of this circular motion. The electron motion can be
specified by only two parameters: its velocity β and its
angular frequency ω. The radius re of the motion can be
expressed with these parameters as re ¼ cβ=ω. Although
the direction of electron motion is another parameter, forsimplicity, here, we select it as counterclockwise around thez axis as shown in Fig. 1. We treat this problem in theHere, ~r is the position vector of the observing point, ~re andβ~ are the electron position and velocity, respectively, as
previously described, which should be expressed with theelectron time te ¼ t − j~r − ~re j=c. It should be noted thatwhen the electron velocity is much smaller than the light
velocity, only the fundamental component exists. When theelectron velocity is comparable to the light velocity, e.g., it
is relativistic, the harmonic components appear.
The Fourier components of the vector potential can be
expressed as follows [15]:⌒ikr~ l ð~rÞ ¼ e 1 eA
c 2π=ω rIn~ ·~re ðte Þc Þilωðte −e 
1d~re ðte Þ þ o 2 : ð5ÞrHere, we separate the term with the first order of 1=r fromthe other terms for convenience in later discussion. The
position vector of the electron can be represented in thespherical coordinate as follows:~re ¼ re fsin θ sinðωte − φÞ~er þ cos θ sinðωte − φÞ~eθ− cosðωte − φÞ~eφ g:

ð6Þ

By inserting Eq. (6) into Eq. (5), we can obtain thefollowing expression:
0 1
Alr
B
~ l ð~r;tÞ¼ @ Alθ CAAAlφH iðlϕ−lβsinθsinϕÞ1
01
sinθ 2πcosϕdϕeH iðlϕ−lβsinθsinϕÞe eiðlω=cÞr ilφ B
C1e re ω@cosθ 2π¼
cosϕdϕAec r
H iðlϕ−lβsinθsinϕÞ1
sinϕdϕ2π e⌒ ð2Þ

~ l þþA¼eeiðkr−lωtþlφÞr0

Jl ðlβsinθÞ

1

 
1
C
B
@ cotθJl ðlβsinθÞ A þo 2r
iβJ0l ðlβsinθÞð7ÞHere, we introduced a new parameter ϕ ¼ ωte − φ andexpress the integration in Eq. (5) by Bessel functions of thefirst order. We also used the relation k ≡ lω=c.

094801-2

The electric and magnetic fields can be expressed by thevector potential as follows:
0
1
1 0
0
Hlr
B H C B − 1 ∂ ðrAð1Þ Þ C
lφ A@ lθ A ¼ @ r ∂r
ð1Þ1∂
r ∂r ðrAlθ Þ

Hlφ

1

0

ð1Þ
1 ∂
1 ∂ ð1Þðsin θAlφ Þ − r sinθ ∂φ Alθ C
B r sin θ ∂θBþB@

þoðr12 Þþoðr12 Þ

 
1
C
þ
o
;
C
A
r3

we can find nonzero electric and magnetic field components in the propagation direction, which would contributeto AM in Eq. (2). In addition, all necessary field components in the following discussion can be expressed by thederivatives of the lowest-order terms of the vector potential
in Eq. (7).
Before calculating AM, we will show the helical phasestructure of the radiation field. The electric and magnetic
fields in the lowest order of 1=r can be derived from
Eqs. (8) and (9) as follows:
0

ð8Þ

0

ð1Þ 1

Elr

Elr

1

0

0

1

ð9Þ

~ ¼ Eð1Þ e~x þ Eð1Þ e~y þ Eð1Þ e~z ¼
E
lx
ly
lz

ð1Þ

1

0

ð1Þ

ikAlφ
1

0

0
C eiðkr−lωtþlφÞe BB¼ lω@ icotθJl ðlβ sin θÞ C
:
A
c
r
0
−βJl ðlβ sin θÞ

0

ð1Þ 1

Elx

0

− sin θ sin φ

B ð1Þ C BB E C ¼ B sin θ cos φ@ ly A @
ð1Þ
cos θ
E
0

0

− cos θ sin φcos θ cos φ− sin θ− cos φ1

C
− sin φ CA01

B ð1Þ C× @ Elθ A:

ð11Þ

ð1Þ
Elφ

The final result can be expressed as a sum of circularpolarized components of positive and negative helicities,
which are proportional to the rotation vectors defined as

ð1Þ

ð1Þ

ð1Þ

ðElx − iEly Þ e~x þ i~ey ðElx þ iEly Þ e~x − i~ey
pﬃﬃﬃ þ
pﬃﬃﬃ þ Eð1Þ
pﬃﬃﬃ
pﬃﬃﬃ
~z
lz e
2
2
2
2
0 n
o
cos2 θ
0sin θ J l ðlβ sin θÞ þ βJ l ðlβ sin θÞ

1

iðl−1Þφ ~

e
eþ C
B
C
o
e eiðkr−lωtÞ B
C
B n
ð1Þ
ð1Þ
ð1Þ
≡ Elþ e~þ þ El− e~− þ Elz e~z ¼ lωB þ − cos2 θ Jl ðlβ sin θÞ þ βJ0 ðlβ sin θÞ eiðlþ1Þφ e~− C:
l
C
B
c
r
sin θ
@

As shown in Eq. (12), the electric field can be
decomposed into two circular polarized components
with respect to the z axis in the positive andnegative helicities with the phase terms eiðl−1Þφ andð10Þ

To clearly show the phase structure, we express theseterms in the Cartesian coordinate as follows:lz

Here, we present the results by separating the terms of
the order of 1=r. In Eqs. (8) and (9), the first bracket on theright-hand side is the first order of 1=r, and the second
bracket is the second order. Later, we represent those terms
as EðiÞ and HðiÞ , and the index on the shoulder represents
the order of 1=r. Some components are not explicitly
shown because they are not necessary in the following
discussion. It should be noted that in the order of 1=r2 ,

0

ð1Þ
1 ∂
r ∂r ðrAlφ ÞElφB E C B 1 ∂ ðrAð1Þ Þ Clθ A@ lθ A ¼ @ r ∂r
ð1Þ
1
∂
Elφ
r ∂r ðrAlφ Þ
1
0
ð1Þ
1 ∂
1 ∂ ð1ÞðsinθAÞ−A− rsinθ lθ∂θrsinθ ∂φ lφ C
B
1
C
B
1
þB
Cþo 3 :
oðr2 Þ
A@r
oðr12 Þ

1

0

C
B
ð1Þ C
1 ∂
B ð1Þ C B
ðrAlθ Þ C ¼ B ikAð1ÞClθ@ Elθ A ¼ B
r
∂r
@
A
A @
ð1Þ

0

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PRL 118, 094801 (2017)−i cos θJl ðlβ sin θÞeilφ e~z

ð12Þ

A

eiðlþ1Þφ , respectively. We will discuss this phenomenon later.
By inserting Eqs. (8) and (9) into Eqs. (2) and (3), theAM density can be expressed as

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PRL 118, 094801 (2017)r
ð1Þ ð2Þð1Þ ð2Þ ð1Þ ð2Þð1Þ ð2Þ
ð0;Elθ Hlr −H lθ Elr ;Elφ H lr −H lφ Elr Þr;θ;φh~jl i ¼
4πc
 
1
ð13Þ
þo 3 :r
From the symmetry of the system, the projected AM to
the z axis is expected to be well defined [16]:hjlz i ¼ −hjlθ i sin θ

 
r
1
ð2Þ ð1Þð1Þ ð2Þ
sin θ Elr Hlθ − Elθ H lr þ o 4
ð14Þ
¼
4πc
r
The total angular momentum carried away by theradiation can be obtained by integrating Eq. (14) on thesphere surrounding the source as follows:

 Z
dJ lz
ð15Þ
¼ cr2 hjlz idΩ:dt
By inserting Eqs. (8) and (9) into Eq. (14) and using
Eq. (15), we obtain

 Z

dJ lz
cr2
ð1Þ ð1Þð1Þ ð1Þ¼ dΩklðAlθ Alθ þ Alφ Alφ Þ
dt
4πc
ð1Þ ∂

− ik Alθ
¼

r2
kl
4π

Z

∂θ

ð1Þ

ð1Þ ∂

ðsin θAlφ Þ þ Alφ
ð1Þ ð1Þ∂θð1Þðsin θAlθ Þ

ð1Þ ð1ÞdΩðAlθ Alθ þ Alφ Alφ Þ:ð16ÞHere, the terms in the brace in Eq. (16) vanish in the
integration regarding to θ.
On the other hand, the energy that the electromagnetic
wave carries away can be obtained from the Poynting
vector, whose radial component is expressed as follows:c ~c ð1Þ ð1Þ
ð1Þ ð1Þ~ lhSlr i ¼
El × H¼ ðElθ Hlφ − Elφ Hlθ Þ
4π
4πrc 2 ð1Þ ð1Þ
ð1Þ ð1Þð17Þ¼ k ðAlθ Alθ þ Alφ Alφ Þ
4πThen, the carried energy is


Z
dU l
ck2
ð1Þ ð1Þð1Þ ð1Þ¼
r2 dΩðAlθ Alθ þ Alφ Alφ Þ:
dt
4π

ð18Þ

Using Eqs. (16) and (18), the ratio of AM to the energythat the radiation field carries is obtained as
R 2
ð1Þ ð1Þð1Þ ð1Þkl
4π r dΩðAlθ Alθ þ Alφ Alφ Þ¼ 2R
ð1Þ ð1Þð1Þ ð1Þlck
hdU
r2 dΩðAlθ Alθ þ Alφ Alφ Þdt i
4πhdJdtlz i¼

N p ℏlll¼¼:
ck lω N p ℏlωð19Þ

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Here, we have introduced Plank constant ℏ and thenumber of photons N p as in the previous works [7,8].
Equation (19) clearly show that the radiation field carries
nonzero AM. Moreover, it is consistent with a quantum
mechanical picture that each photon carries lℏ AM. In thequantum electrodynamics, vector potentials of electromagnetic fields are considered as photon wave functions [20]. By
applying the z component of the AM operator, −iℏ∂=∂φ[16], to the vector potential shown in Eq. (7), it isstraightforward to show that its eigenvalue is equal to lℏ,which is consistent with Eq. (19). Here, the SAM and OAM
are not clearly separated, in contrast with a previous work
adopting the paraxial approximation [7], but in common
with other works adopting the nonparaxial approximation[8] and also with the discussions on AM in the multipole
expansion of electromagnetic fields [16,19,21].
We have shown that the ratio of the AM density in the z
direction to the energy density is expressed in a simple formin Eq. (19). Previous works assumed mathematical models
for the radiation field, where the polarization was treated asa free parameter. However, in our case, we began from theelectron motion, which is counterclockwise as previously
described. Therefore, the polarization is defined by theelectron motion and not a free parameter. We see thepolarization of the radiation field from Eq. (12), which ismostly circular polarized when the polar angle θ is small.
The direction of the polarization is identical to the electron
motion (we will call this direction positive for convenience). When θ increases, the polarization becomes elliptical. In other words, the field becomes a mixture of
circularly polarized components with positive and negative
helicities. When θ is π=2, it is linearly polarized, which
implies that the intensities of both helicity components are
identical. When θ is larger than π=2, the negative helicity
components becomes dominant. It should be noted that thepositive helicity component has the phase term represented
by eiðl−1Þφ and the negative eiðlþ1Þφ . We speculate that, if we
consider the z component of AM, in the former case, SAM
is þ1 and OAM is l − 1, and in the latter case, SAM is -1,
and OAM is l þ 1. In any case, the total AM is always l.
It may be instructive to consider the angular momentum
in the nonlinear inverse Compton process of circularly
polarized light, which is also an example of the radiationfrom an electron in circular motion. In classical mechanics,
intense circularly polarized light induces a circular motionof an electron. When the light is sufficiently intense, theelectron motion becomes relativistic, and higher harmonics
are radiated. In quantum mechanics, this process can be
described as a multiphoton process. Let us consider that nphotons with the energy of ℏω are coming to the electron.
Because they are circularly polarized, they bring ℏ AM for
each; therefore, they bring nℏAM to the system in total.
One photon with energy nℏω is outgoing, which implies
that the energy is conserved. However, the outgoing photon
can bring at most ℏ AM as SAM. For the conservation of

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angular momentum, it should carry the remaining AM asanother form of AM, which is OAM. If the outgoing photon
carries ℏ as SAM, then it should carry other ðn − 1Þℏ as
OAM. When it carries −ℏ as SAM, then it should carryðn þ 1Þℏ as OAM. This speculation seems consistent withthe above discussion on Eq. (12).
In the Lorentz transformation, a spherical wave shouldbe transformed to a spherical wave. Indeed, by using
Eq. (7), it is straightforward to obtain the vector potentials
for the radiation from an electron in spiral motion that drifts
along the z axis at a relativistic speed:eiðkr−lωtþlφÞ
1
K
sin θ
Alr ¼ eJ l
1 − βz cos θ l γ 1 − βz cos θr 1
þo 2 ;reiðkr−lωtþlφÞ cos θ − βz 1
K
sin θ
Jl l
Alθ ¼ eγ 1 − βz cos θr
1 − βz cos θ sin θ 1
þo 2 ;riðkr−lωtþlφÞ
e
iK=γ
K
sin θ
Alφ ¼ eJ l
1 − βz cos θ l γ 1 − βz cos θ
r
 
1
ð20Þ
þo 2 :
r
Here, βz is the electron velocity along the z axis, γ is theLorentz factor, and K=γ is the pitch angle of the electron
motion to the z axis. The angular frequency ω is related tothat in the electron frame ω as follows:
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
1 þ K2ω¼
ð21Þω :γð1 − βz cos θÞ
By applying the operator, −iℏ∂=∂φ, to Eq. (20) as in
the discussion about Eq. (19), it is straightforward toshow that its eigenvalue is equal to lℏ; therefore, theAM in the z direction is assured to be invariant to theLorentz transformation [15]. Using Eq. (21), one can derive
the electromagnetic field for a helical undulator [22,23],
inverse Compton scattering of circular polarized light [24]or cyclotron or synchrotron radiation [17,18] with the phaseterms which represent their vortex nature.
In this paper, we have theoretically shown that thephotons radiated by an electron in a spiral motion are
twisted and carry OAM. As far as we know, this is the first
theoretical demonstration that radiation field by a single
charged particle carries well-defined OAM. This process is
one of the most fundamental radiation processes in laboratories and in Nature. Therefore, we conclude that photons
with OAM are much more ubiquitous than previously
thought. They may play unexplored, important roles withrespect to their OAM. In addition, this process will be thebasis of laboratory vortex photon sources that cover the

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entire wavelength range from radio waves to gamma rays,
and they will open completely new research opportunities.
M. K. contributed to the entire derivation of the analytic
formulas with helps by H. K., K. T., and K. O. on themathematical approaches. The analytic formulas werechecked by M. F., T. K., M. H., A. M., Y. Tak, and Y.
Tai. The manuscript was prepared by M. K. based on thediscussions with all others and was checked by all others.
This work was partially supported by JSPS KAKENHI
Grant No. 26286081.

*

Corresponding author.
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