# Transverse spin and orbital angular momenta in rotating-wave structured
light

Zhen-Lai Wang<EMAIL_ADDRESS>Center for Fundamental Physics and
School of Mathematics and Physics, Hubei Polytechnic University, Huangshi
435003, China

###### Abstract

Within monochromatic optical fields, we demonstrate the rotating-wave
structured light with wave vortex carrying an intrinsic transverse orbital
angular momentum orthogonal to the propagation direction of light. Remarkably,
we find that such a rotating-wave structured light reveals highly nontrivial
features of transverse spin and orbital angular momentum densities. The
normalized total angular momentum density is conserved universally, suggesting
a mutual conversion of the intrinsic transverse spin and orbital angular
momentum in free space. Despite such mutual conversion at local level, the
integral intrinsic orbital angular momentum can be well defined with the
topological charge of the vortex per photon. Moreover, the orientation of
transverse spin density is governed by the direction of the Poynting momentum
density, manifesting a spin-momentum-locking effect in free space.

##  I Introduction

Optical angular momentum manifests itself in two distinct forms [1]. One is
spin angular momentum (SAM), which originates from the intrinsic polarization
degrees of freedom [2]; the other is orbital angular momentum (OAM), which
arises from the spatial phase degrees of freedom [3]. The former is purely
intrinsic, or, more specifically, independent of the coordinate origin. The
latter is allowed to make a clear distinction between the intrinsic and
extrinsic ingredients [4]. SAM and OAM can be separately observable by the
distinct effects upon test particles [5]. For the conventional light beams,
both SAM and OAM are essentially longitudinal since they are aligned with the
direction of propagation.

Nevertheless, there has been a growing interest in the transverse angular
momentum of light [6]. Unlike the conventional and longitudinal angular
momentum of light, the optical transverse angular momentum is perpendicular to
the propagation direction of light. The transverse SAM (t-SAM) has been
generated by evanescent waves [7, 8, 9], interference fields [10, 11] and
focused beams [12, 13, 14]. The appearance of t-SAM requires the transverse
inhomogeneity of the field intensity [6] and can be closely linked to the curl
of the energy flow density [15, 16]. The t-SAM plays crucial roles in the
intrinsic spin-momentum-locking effect [17, 18] and the intrinsic quantum spin
Hall effect of light [19].

The extrinsic transverse OAM (t-OAM) is the most straightforward and trivial
form of transverse angular momentum, which is analogous to the classical
angular momentum [6]. Despite its trivial derivation, the extrinsic t-OAM can
make remarkable physical effects including the spin-Hall and orbital-Hall
effects of light [20, 21, 22, 23, 24, 25, 26, 27, 28, 29]. Recently, the
intrinsic t-OAM is studied intensively in theory [30, 31, 32, 33] and in
experiment [34, 35, 36, 37] with a new type of localized optical pulses
characterized by spatiotemporal vortex structure. However, unlike the typical
light beams with angular momentum, such pulses are fundamentally polychromatic
as opposed to monochromatic.

Thus the intrinsic t-OAM stands for the last missing piece of transverse
angular momentum within monochromatic optical fields. Recently, the intrinsic
t-OAM of monochromatic optical fields is reported to be generated through the
interference of two parallel and counter-propagating linearly-polarized
focused beams, and to be observed directly by using an optomechanical sensor
[38].

The motivation for this work is to find the intrinsic t-OAM in the
monochromatic optical fields. In this work, we propose that the rotating-wave
monochromatic structured light can carry the intrinsic t-OAM. In Sec. II, the
rotating wave is introduced concisely, which is a monochromatic wave with wave
vortex, propagating along the azimuthal direction and consisting of infinite
plane waves. In Sec. III, we construct two modes of the rotating-wave
structured light in the vector formulation of electromagnetic field. In Sec.
IV, we show that the rotating-wave monochromatic structured light can possess
an integral intrinsic t-OAM of well-defined value per photon and exhibits
locally unusual t-SAM and t-OAM properties. Finally, the conclusion are
presented.

##  II Rotating wave

(a) (b)

Figure 1: (a) The wave-vectors of the plane waves point radially outward from
the origin and rotates on the $k_{y}$-axis to form a circle in the plane
$(k_{z},k_{x})$ plane of $\bm{k}$ space. (b) Surface plot of the rotating
wave, calculated via the real part of the complex wavefunction in Eq. (1) with
$l=3$. The arrow denotes the wave’s direction of propagation.

Throughout this paper, let us restrict ourselves to the monochromatic wave
field propagating in free space. The typical and simple wave field with OAM is
Bessel beam. For simplicity, we can consider specifically a scalar Bessel beam
propagating in the $(z,x)$ plane. As illustrated in Fig. 1(a), such beam can
be generated by superposing plane waves with the same frequency $\omega$, wave
vectors $\bm{k}$ pointing radially outward from the origin and forming a
circle in the $(k_{z},k_{x})$ plane, and with the corresponding azimuthal
phase difference $l\phi$. Here $l=0,\pm 1,\pm 2,...$ is the topological charge
of the vortex and $\phi$ is the azimuthal angle in $\bm{k}$ space. Thus, the
resulting scalar wave function can be written as the integral over $\phi$ from
$0$ to $2\pi$:

$\displaystyle\psi(\bm{r},t)$ $\displaystyle=\int_{0}^{2\pi}e^{ik\cos\phi
z+ik\sin\phi x+il\phi-i\omega t}{\rm d}\phi$
$\displaystyle=2\pi{i}^{l}J_{l}(k\rho)e^{il\varphi-i\omega t}.$ (1)

Here $J_{l}(k\rho)$ denotes the $l$th-order Bessel function of the first kind,
$k=\omega/c$ is the wave number and $c$ is the phase velocity of wave, and
$(\rho,\varphi)$ indicate the polar coordinates in the $(z,x)$ plane
associated with $z=\rho\cos\varphi$ and $x=\rho\sin\varphi$.

Note that Eq. (1) stands for a pure rotating wave [39]. The term
$e^{il\varphi-i\omega t}$ bears resemblance to the typical term
$e^{ikx-i\omega t}$ in a traveling wave, enabling the wave to preserve its
waveform while propagating along the azimuthal direction in the $(z,x)$ plane.
As an illustration, Fig. 1(b) shows the rotating wave for the mode $l=3$.
Unlike traveling waves, rotating waves are confined to specific modes due to
their circular propagation. For mechanical waves, such as surface waves in
water [39] and acoustic waves [40, 41, 42], rotating waves offer a clear
understanding of angular momentum, a concept commonly encountered in quantum
waves or electromagnetic waves [43, 44]. More interestingly, rotating waves
can provide us with new insights into the intrinsic t-SAM and t-OAM of light,
as we will discuss in the following sections.

##  III Rotating-wave structured light

We now consider the rotating-wave structured light. Light is a form of
electromagnetic waves which manifest as vector fields, so there is a need for
constructing vector beams of rotating wave. For simplicity and more
concentration on the intrinsic t-OAM, we only consider the rotating-wave beam
consisting entirely of linearly polarized plane waves. Due to the
transversality of electromagnetic waves, the electric and magnetic fields
$\bm{\mathcal{E}}$, $\bm{\mathcal{B}}$ and the wave vector $\bm{k}$ of each
plane wave in the beam spectrum constitute a right-handed orthogonal triad. As
a consequence, two basic modes of polarization should be considered in the
problem. One mode is that $\bm{\mathcal{B}}$ is set along the positive
$y$-direction and so $\bm{\mathcal{E}}$ is put in the $(z,x)$ plane; see Fig.
2(a). Conversely, the other one is that $\bm{\mathcal{E}}$ is fixed along the
negative $y$-direction and so $\bm{\mathcal{B}}$ is placed in the $(z,x)$
plane; see Fig. 2(b).

(a) (b)

Figure 2: Schematic diagrams of the two basic modes of polarization. (a)
$\bm{\mathcal{B}}$ is polarized in the positive $y$-direction and then
$\bm{\mathcal{E}}$ is located in the $(z,x)$ plane. (b) $\bm{\mathcal{E}}$
points in the negative $y$ direction and then $\bm{\mathcal{B}}$ lies in the
$(z,x)$ plane. Dots and crosses are used to represent a field coming out of or
going into the plane of the paper, respectively.

With the help of Maxwell’s equations, the resulting electric and magnetic
fields of the first mode can be given by

$\bm{B}=\psi(\bm{r},t)\bm{e}_{y},~{}~{}~{}\bm{E}=\frac{i}{k}\bm{\nabla}\times\bm{B}.$
(2)

Here $\bm{e}_{y}$ is the unit vector in the $y$-direction. For the convenience
of future use and with a slight computation cost, the explicit components of
the resulting electric field can be obtained

$\displaystyle E_{x}$
$\displaystyle=-i^{l+1}\pi\big{[}J_{l-1}e^{i(l-1)\varphi}-J_{l+1}e^{i(l+1)\varphi}\big{]}e^{-i\omega
t},$ (3) $\displaystyle E_{z}$
$\displaystyle=-i^{l}\pi\big{[}J_{l-1}e^{i(l-1)\varphi}+J_{l+1}e^{i(l+1)\varphi}\big{]}e^{-i\omega
t}.$ (4)

For notational simplicity, from now on the argument $k\rho$ will be leave out
from the Bessel functions.

In the similar way, the second mode can be represented by

$\bm{E}=-\psi(\bm{r},t)\bm{e}_{y},~{}~{}~{}\bm{B}=-\frac{i}{k}\bm{\nabla}\times\bm{E}.$
(5)

Clearly, the second mode is able to be fulfilled properly from the first one
by the exchange $\bm{E}\to\bm{B}$ and $\bm{B}\to-\bm{E}$. In fact, such
exchange is a typical transformation of electromagnetic duality in the free
space [45, 46]. As a direct result of the transformation of electromagnetic
duality, the case of the second mode can be derived from that of the first
one. For this reason, we will concentrate on the first mode in the following
section.

##  IV Spin and orbital angular momenta

In this core section, we turn now to analyze the t-SAM and t-OAM of the
rotating-wave structured light. We start with the time-averaged densities of
energy $W$, SAM $\bm{S}$ and OAM $\bm{L}$ in a monochromatic optical field
with separate electric- and magnetic-field contributions

$\displaystyle W$ $\displaystyle=$ $\displaystyle
W^{e}+W^{m}=\frac{g\omega}{2}\big{(}|\bm{E}|^{2}+|\bm{B}|^{2}\big{)},$ (6)
$\displaystyle\bm{S}$ $\displaystyle=$
$\displaystyle{\bm{S}}^{e}+{\bm{S}}^{m}=\frac{g}{2}{\rm
Im\,}\big{(}\bm{E}^{*}\times\bm{E}+\bm{B}^{*}\times\bm{B}\big{)},$ (7)
$\displaystyle\bm{L}$ $\displaystyle=$
$\displaystyle{\bm{L}}^{e}+{\bm{L}}^{m}=\frac{g}{2}{\rm
Re\,}\big{(}E^{*}_{j}\hat{\bm{\mathcal{L}}}E_{j}+B^{*}_{j}\hat{\bm{\mathcal{L}}}B_{j}\big{)}.$
(8)

Here $g=(8\pi\omega)^{-1}$ in Gaussian units, the OAM operator
$\hat{\bm{\mathcal{L}}}=\bm{r}\times\frac{1}{i}\bm{\nabla}$ and Einstein
summation convention is used for the three-component vector.

Substituting Eqs. (2)-(4) into Eqs. (6)-(8), we can calculate the time-
averaged densities of energy, SAM and OAM. The transverse angular momentum is
the focus of attention in this study. Consequently, only the $y$-components of
the SAM and OAM densities should be taken into consideration for the light
propagating in the $(z,x)$ plane. In fact, the SAM density is totally
transverse because it is non-zero solely along the $y$-direction. Omitting the
common factor $g\pi^{2}$ and writing in explicit contributions from electric
and magnetic fields, we obtain finally

$\displaystyle W^{e}$
$\displaystyle=\omega(J_{l-1}^{2}+J_{l+1}^{2}),~{}\qquad\qquad~{}W^{m}=2\omega
J_{l}^{2},$ (9) $\displaystyle S^{e}_{y}$
$\displaystyle=J_{l-1}^{2}-J_{l+1}^{2},~{}~{}\qquad~{}~{}~{}~{}\qquad~{}S^{m}_{y}=0,$
(10) $\displaystyle L^{e}_{y}$
$\displaystyle=(l-1)J_{l-1}^{2}+(l+1)J_{l+1}^{2},~{}~{}L^{m}_{y}=2lJ_{l}^{2}.$
(11)

From Eq. (10), we see that the t-SAM density is completely provided by the
roatation of the electric field. The normalized electric SAM density $\omega
S_{y}/W^{e}$ can change sign from $-$ to $+$ and its minimum and maximum
values are $-1$ and $1$, respectively; see Fig. 3(a). The rotation direction
of the electric field then converts from the negative to the positive
$\varphi$-direction relative to the $y$-axis. Namely, the electric field can
shift from the left-handed to the right-handed circular polarization, and vice
versa. For the magnetic sector, we can identify the relation
$L_{y}^{m}=lW^{m}/\omega$, and so we can confirm that the t-OAM coincides well
with $l\hbar$ per photon as the quantization of photon’s energy $\hbar\omega$.
However, for the t-OAM density of the electric contribution, there does not
exist a similarly direct correlation between $L^{e}_{y}$ and $W^{e}$.
Interestingly, from Eqs. (9)-(11) we can find the total transverse angular
momentum density of the electric sector satisfies

$\mathcal{J}^{e}_{y}=L^{e}_{y}+S^{e}_{y}=\frac{l}{\omega}W^{e}.$ (12)

Again, for the electric sector the total transverse angular momentum still
corresponds to $l\hbar$ per photon. By Eq. (12), the normalized electric t-OAM
density is then given by $\omega L_{y}/W^{e}=l-\omega S_{y}/W^{e}$ and so it
can be regarded as varying with $\omega S_{y}/W^{e}$ if $l$ is given, and vice
versa. It achieves its minimum and maximum values of $l-1$ and $l+1$; see Fig.
3(c).

In addition, the densities of energy, t-SAM and t-OAM display strong electric-
magnetic asymmetry [6], especially the t-SAM density. Not coincidentally, the
electric-magnetic asymmetry can be found in the second mode (5) as well.
However, in such case the electric field will make no contribution to the
t-SAM density and so only the magnetic field will generate the t-SAM density.
In particular, if $l=0$, the optical structure is so trivial that the vortex
does not occur and both the densities of SAM and OAM vanish.

(a) (b)  (c) (d)

Figure 3: Spatial distributions of the normalized t-SAM and t-OAM. (a) $\omega
S_{y}^{e}/W^{e}$, calculated via Eqs. (9) and (10). (b) $\omega S_{y}/W$, Eqs.
(13) and (14). (c) $\omega L_{y}^{e}/W^{e}$, Eqs. (9) and (11). (d) $\omega
L_{y}/W$, Eqs. (13) and (15). The parameter is $l=3$.

Evidently, the densities of energy, t-SAM and t-OAM contributed by the
electric and magnetic fields together can be written as

$\displaystyle W$ $\displaystyle=\omega(J_{l-1}^{2}+2J_{l}^{2}+J_{l+1}^{2}),$
(13) $\displaystyle S_{y}$ $\displaystyle=J_{l-1}^{2}-J_{l+1}^{2},$ (14)
$\displaystyle L_{y}$
$\displaystyle=(l-1)J_{l-1}^{2}+2lJ_{l}^{2}+(l+1)J_{l+1}^{2}.$ (15)

The normalized t-SAM and t-OAM density are depicted in Fig. 3(b) and (d). From
Eqs. (13)-(15), we obtain easily the total transverse angular momentum density

$\mathcal{J}_{y}=L_{y}+S_{y}=\frac{l}{\omega}W.$ (16)

Once again, the total transverse angular momentum corresponds to $l\hbar$ per
photon. It is natural to interpret Eqs. (12) and (16) as representing
conservation of the transverse angular momentum. Such conservation manifests a
new type of mutual conversion of the t-SAM and t-OAM without light-matter
interaction [18]. Integrating Eq. (16) over the $(z,x)$ plane, we obtain the
normalized integral values of total transverse angular momentum

$\frac{\langle\mathcal{J}_{y}\rangle}{\langle W\rangle}=\frac{l}{\omega}.$
(17)

Here $\langle\cdots\rangle$ means the integration over the $(z,x)$ plane,
$\langle\cdots\rangle=\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}\cdots{\rm
d}z{\rm d}x=\int^{\infty}_{0}\int^{2\pi}_{0}\cdots\rho{\rm d}\rho{\rm
d}\varphi$. As seen from Eq. (17), the integral total transverse angular
momentum can be defined well with $l\hbar$ per photon. More specifically, such
integral transverse angular momentum merely stands for the intrinsic t-OAM,
which will be elucidated in the remainder of this section.

(a) (b)  (c)

Figure 4: Spin-momentum locking for the rotating-wave structured light.
(a)-(b) The spatial distributions of the Poynting momentum density with $l=\pm
3$, respectively, as given by Eq (20). Arrows are used to indicate the
direction of the Poynting momentum density. (c) The spatial distributions of
the t-SAM $S_{y}$ along the $z$-axis with $l=\pm 3$, respectively, Eq (14).

It is worth further looking into the t-SAM. Following Refs. [15] and [16], the
connection between the t-SAM density and the Poynting momentum density for an
arbitrary surface evanescent wave can be established as follows:

$\bm{S}=\frac{1}{2k^{2}}\bm{\nabla}\times\bm{P},$ (18)

where the Poynting momentum density takes the form

$\bm{P}=gk{\rm Re\,}\big{(}{\bm{E}}^{*}\times{\bm{B}}\big{)}.$ (19)

Plugging Eqs. (2)-(4) into Eq. (19), the Poynting momentum density is given by

$\bm{P}=\frac{4l}{\rho}J_{l}^{2}\cos\varphi\bm{e}_{x}-\frac{4l}{\rho}J_{l}^{2}\sin\varphi\bm{e}_{z}=\frac{4l}{\rho}J_{l}^{2}\bm{e}_{\varphi}.$
(20)

Here the common factor $g\pi^{2}$ has been omitted. From Eq. (20), we find
that the Poynting momentum density is directed along purely the azimuthal
direction in the $(z,x)$ plane, as illustrated in Fig. 4(a)-(b). Therefore,
the $y$-components of SAM and OAM are indeed transverse because of their
orientations perpendicular to the Poynting momentum density as well as the
propagation direction.

With elementary algebra, we can verify that the connection (18) still holds
for the t-SAM density (14) and the Poynting momentum (19) despite non-
evanescent wave. Such connection offers a guideline for understanding the
t-SAM from the perspective of the transverse gradient of Poynting momentum.
The t-SAM here is the interference result of plane waves with different wave-
vector directions in the same plane, even taking place in the very simple
interference systems of two [10] or three [11] plane waves. In our instance
here, the t-SAM density reversals the local orientations universally once the
Poynting momentum changes from the positive to the negative
$\varphi$-direction, as illustrated in Fig. 4(c). Thus, the direction of the
t-SAM density is determined by the direction of Poynting momentum density,
which can be derived clearly from the connection (18). From the connection
between the t-SAM density and the Poynting momentum, we show that the
rotating-wave structured light reveals an intrinsic spin-momentum-locking
effect [17, 18] without light-matter interaction.

Let us coming back to the integral transverse angular momentum. In general,
the t-SAM of helicity independence exists locally, and yet its integral value
vanishes [6]. The integral t-SAM in Eq. (14) can be verified to go to exactly
zero:

$\displaystyle\langle S_{y}\rangle$ $\displaystyle=\int^{2\pi}_{0}{\rm
d}\varphi\int^{\infty}_{0}(J_{l-1}^{2}-J_{l+1}^{2})\rho{\rm d}\rho$
$\displaystyle=\frac{8\pi l}{k^{2}}\int^{\infty}_{0}J_{l}(k\rho)\frac{{\rm
d}J_{l}(k\rho)}{{\rm d}(k\rho)}{\rm d}(k\rho)=0.$ (21)

Then the integral total transverse angular momentum can be written as

$\langle\mathcal{J}_{y}\rangle=\langle L_{y}\rangle+\langle
S_{y}\rangle=\langle L_{y}\rangle=\frac{l}{\omega}\langle W\rangle.$ (22)

Hence the integral transverse angular momentum belongs entirely to the t-OAM.

An alternate approach to derive the above result is giving by performing
directly the integrations of energy $\langle W^{e}\rangle$ and $\langle
W^{m}\rangle$ and the t-OAM $\langle L^{e}_{y}\rangle$ and $\langle
L^{m}_{y}\rangle$. Using the closure equation of Bessel Functions of the first
kind [47], we have

$\langle W^{e}\rangle=\langle W^{m}\rangle,~{}~{}\langle
L^{e}_{y}\rangle=\langle L^{m}_{y}\rangle.$ (23)

Immediately, we obtain [48]

$\frac{\langle L_{y}\rangle}{\langle W\rangle}=\frac{\langle
L^{e}_{y}\rangle}{\langle W^{e}\rangle}=\frac{\langle
L^{m}_{y}\rangle}{\langle W^{m}\rangle}=\frac{l}{\omega}.$ (24)

Thus, the relation (22) between the integral t-OAM and energy is recovered,
and the integral t-SAM vanishes.

We are now in the position to interpret why the integral t-OAM is intrinsic.
For a translation of the coordinate origin $\bm{r}\to\bm{r}+\bm{r}_{0}$, the
integral t-OAM is changed by $\langle L_{y}\rangle\to\langle
L_{y}\rangle+\langle L_{y}^{\rm ext}\rangle=\langle
L_{y}\rangle+\big{(}\bm{r}_{0}\times\langle\bm{P}\rangle\big{)}_{y}$. From Eq.
(20) we find $\langle\bm{P}\rangle=0$ and so the extrinsic t-OAM $\langle
L_{y}^{\rm ext}\rangle$ is equal to zero. The integral t-OAM is unchanged
under the translation of the coordinate origin and therefore it is intrinsic.

##  V Discussion and Summary

In summary, we have shown that the rotating-wave structured light carries
purely integral intrinsic t-OAM within monochromatic optical fields.
Furthermore, the rotating-wave structured light with wave vortex, consisting
of linearly-polarized plane waves, exhibits locally unique t-SAM and t-OAM
features. The sum of the locally-normalized t-SAM and t-OAM is conserved
universally, which manifests a strong coupling of t-SAM and t-OAM. The t-SAM
and t-OAM variations are heavily dependent on each other. Such strong coupling
uncovers a new type of mutual conversion of the intrinsic t-SAM and t-OAM
without light-matter interaction. In spite of such locally mutual conversion,
the integral t-SAM and t-OAM are rather robust, and the integral intrinsic
transverse angular momentum belong purely to the intrinsic t-OAM.

Impressively, the t-SAM density of the rotating-wave structured light also can
be directly linked to the curl of the Poynting momentum density even though it
is not associated with evanescent waves. As a consequence, the orientation of
the local t-SAM is made directly subject to the local direction of the
Poynting momentum density, which can be recognized as an intrinsic spin-
momentum-locking effect of the rotating-wave structured light in free space.

We look forward to the future experimental verification of such nontrivial
properties of the spin and orbital angular momentum in the rotating-wave
structured light. The rotating-wave structured light might offer high
potential for applications in the optical manipulation of matter.

## Acknowledgments

The author is grateful to Z. Xiong and Y. Liu for fruitful discussions. This
work was funded by the Scientific Research Project of Hubei Polytechnic
University (No. 20xjz02R).

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