# The Azimuthally-Radially Polarized Beam: Helicity and Momentum Densities,
Generation and Optimal Chiral Light

Albert Herrero-Parareda and Filippo Capolino ⋆ A. Herrero-Parareda and F.
Capolino are affiliated with the Department of Electrical Engineering and
Computer Science, University of California, Irvine, CA, 92697 USA e-mail:
<EMAIL_ADDRESS>received MONTH DD, YYYY; revised MONTH DD, YYYY.
https://orcid.org/0000-0002-8501-5775 https://orcid.org/0000-0003-0758-6182

###### Abstract

We investigate the optical properties of the azimuthally-radially polarized
beam (ARPB), a superposition of an azimuthally polarized beam and a radially
polarized beam, which can be tuned to exhibit maximum chirality at a given
energy density. This condition is called "optimal chiral light" (OCL) since it
represents the maximum local chirality at a given energy density. The
transverse fields of an ARPB dominate in the transverse plane but vanish on
the beam axis, where the magnetic and electric fields are purely longitudinal.
This spatial separation between transverse and longitudinal components leads
to vanishing linear and angular momentum densities on the axis, where only the
energy and helicity densities associated with the longitudinal fields persist.
The ARPB does not have a phase variation around the beam axis and nonetheless
exhibits a power flow around the beam axis that causes a longitudinal orbital
momentum density. We introduce a concise notation for the ARPB and provide
field quantities, especially for the optimally chiral configuration. The ARPB
shows promise for precise one-dimensional chirality probing and
enantioseparation of chiral particles along the beam axis. We propose a setup
to generate ARPBs with controlled chirality and orbital angular momentum.

##  I Introduction

Chiral molecules, also known as enantiomers, are not superimposable on their
mirror images [1, 2]. They have the same constitution but follow different
chemical processes [3]. When interacting with chiral light, which also has
non-superimposable mirror images, enantiomers also have different optical
properties [4, 5]. In 2016 it was reported that $50\%$ of FDA-approved drugs
have chiral active ingredients [6], where one enantiomer initiates the desired
chemical reaction and the other one acts as an antagonist [7]. Therefore, it
is of supreme importance to develop efficient techniques to probe the
chirality of enantiomers [8, 9, 10] and to separate them [11, 12]. Ref. [13]
reports that a system combining chirality and optical trapping could be used
to separate enantiomers. The chiroptical response of natural chiral systems,
however, is very weak [14, 15]. These responses depend on the chirality of
light, as observed theoretically in Refs. [16, 17, 18], and experimentally in
Ref. [19]. The optical chirality is quantified by the helicity density [20],
and the concepts of helicity maximization and optimally chiral light (OCL)
were introduced in Ref. [21]. A monochromatic field propagating in free space
is optimally chiral (OC) when its helicity density $h$ reaches its upper
bound, $|h|=u/\omega$, where $u$ is the energy density of light and $\omega$
its angular frequency. When light is not optimal chiral, one has
$|h|<u/\omega$. As shown in Ref. [21], circularly polarized light (CPL) is an
example of OCL, and that is why CPL has been used so successfully to detect
the chirality of matter. However, the results in Ref. [21] show that there are
countless other examples of structured light that are OCL, and the special
beam studied in this paper is a useful and simple example.

Emerging strategies attempt to maximize chiral light-matter interactions by
enhancing the optical chirality using dielectric [22, 23, 12, 24] or plasmonic
nanostructures [25, 26, 27]. It is shown in [28] that the average chirality of
the near field of plasmonic nanostructures cannot be significantly higher than
that in freely-propagating plane waves. These studies so far have only
considered nanostructures illuminated with CPL, which only displays optical
chirality (i.e., helicity) in the transverse plane perpendicular to the beam
propagation direction, or with linearly polarized light [29] for enhanced
optical rotation measurements. However, Gúzman Rosales et al. [30] highlighted
the importance of longitudinal (i.e., along the beam propagation direction)
field components in chiral interactions, and Kamandi et al. [31] showed that
beams with only longitudinal chirality are useful for unambiguous chirality
probing. In this context, we define transverse and longitudinal optical
chirality as the contributions associated solely with the transverse and
longitudinal field components, respectively, as in Refs. [31, 32, 33]. A beam
with transverse chirality can only reveal the transverse chirality of the
sample, which could be mistaken with its anisotropy as the sample is probed in
two directions ($x$ and $y$) [31, 34]. In contrast, a beam with just
longitudinal chirality only probes the sample along one direction ($z$).

Figure 1: Conceptual representation of the APB, the RPB, and the ARPB as their
combination with a phase delay of $\psi=\pm\pi/2$. The electric field is
represented in blue and the magnetic field in red. The radial components of
the fields are not shown. The bell curve denotes the Gaussian-like parameter
$f$. In the ARPB with $\psi=\pi/2$, the electric field $\mathbf{E}$ (blue bell
curve) leads in time to the magnetic field $\mathbf{H}$ (red bell curve), and
the opposite (lagging) occurs when $\psi=-\pi/2$.

A beam with purely longitudinal chirality on the optical axis is the
azimuthally-radially polarized beam (ARPB) [31], a superposition of an
azimuthally polarized beam (APB) [35, 36, 37, 38, 39] and a radially polarized
beam (RPB) [40, 41, 42]. In Ref. [43], the authors tightly focus an achiral
beam resulting from the in-phase superposition of an APB and an RPB to excite
a chiral dipole moment in an achiral nanostructure. Here, instead, we examine
out-of-phase superpositions of these two beams, which results in a chiral
ARPB. The phase-shift $\psi$ and the ratio of amplitudes $\hat{V}$ between the
APB and the RPB can be tuned to produce an ARPB with optimal chirality in all
directions, which we refer to as an optimally chiral ARPB (OC-ARPB). This
occurs when $\psi=\pm\pi/2$ and $\hat{V}=1$ [21]. Figure 1 displays schematic
views of an APB, an RPB, and the two configurations of an OC-ARPB. For
$0<\psi<\pi$, the electric field leads the magnetic field, while the opposite
occurs for $-\pi<\psi<0$. The transverse fields of the ARPB vanish on the beam
axis, where the longitudinal fields persist [31, 21, 44, 45]. This spatial
separation is shown in Figures 2(a) and (b), which respectively depict the
propagation of the magnitude of the transverse
$\mathbf{E}_{\perp}=E_{\rho}\hat{\bm{\rho}}+E_{\varphi}\hat{\bm{\varphi}}$ and
longitudinal $E_{z}$ electric fields of the ARPB. The ARPB only displays
longitudinal chirality on the beam axis and can be used to confidently probe
the chirality of an anisotropic sample (as opposed to CPL). Moreover,
preliminary theoretical results show that illuminating dielectric nanospheres
with an OC-ARPB results in almost twice the helicity density enhancement than
using a CP Gaussian beam [21]. Other investigations into beams with
longitudinal chirality, including Gúzman Rosales et al [30], study the optical
chirality of beams with orbital angular momentum (OAM) [30, 46, 47, 48, 49,
50, 51].

(a)

(b)

Figure 2: Schematic representation of propagation and localization of the (a)
transverse
$\mathbf{E}_{\perp}=E_{\rho}\hat{\bm{\rho}}+E_{\varphi}\hat{\bm{\varphi}}$,
and (b) longitudinal $E_{z}$ fields of an ARPB with $w_{0}=\lambda=400$ nm.
The $E_{z}$ field is localized on the axis. The same happens for the magnetic
field, with $H_{z}$ localized on the axis.

In this paper, we further advance the concept of a chiral ARPB, defined in
terms of an arbitrary relative amplitude $\hat{V}$ and phase shift $\psi$
between the APB and the RPB, as in Refs. [21, 44, 52, 45]. We introduce a
concise notation for the APB and the RPB in Section II. The ARPB is described
in Section III, and the analytical expressions for its field quantities are
provided in Section IV. In Section V, we focus on the OC-ARPB and its
exceptional electric-magnetic symmetry. To support future experimental
studies, we discuss setups that generate an ARPB in Section VI. The
conclusions are found in Section VII.

##  II Cylindrical polarization basis

The non-zero field components of the APB in the cylindrical basis ($\rho$,
$\varphi$, $z$) are

$\begin{array}[]{c}E_{\varphi}^{\mathrm{APB}}=\frac{\rho}{w^{2}}V_{A}f,\\\
H_{\rho}^{\mathrm{APB}}=-\frac{\rho}{w^{2}\eta_{0}}V_{A}f\left(A_{\rho}+iB_{\rho}\right),\\\
H_{z}^{\mathrm{APB}}=-\frac{2i}{kw^{2}\eta_{0}}V_{A}f\left(A_{z}+iB_{z}\right),\end{array}$
(1)

and those of the RPB

$\begin{array}[]{c}H_{\varphi}^{\mathrm{RPB}}=\frac{\rho}{w^{2}\eta_{0}}V_{R}f,\\\
E_{\rho}^{\mathrm{RPB}}=\frac{\rho}{w^{2}}V_{R}f\left(A_{\rho}+iB_{\rho}\right),\\\
E_{z}^{\mathrm{RPB}}=\frac{2i}{kw^{2}}V_{R}f\left(A_{z}+iB_{z}\right),\end{array}$
(2)

where $V_{A}$, $V_{R}$ are their complex amplitudes (with unit of Volts),
respectively. The dimensionless shorthand parameters are

$\begin{array}[]{c}f=\frac{2}{\sqrt{\pi}}e^{-(\rho/w)^{2}\zeta}e^{-2i\tan^{-1}(z/z_{\text{R}})}e^{ikz},\\\
A_{\rho}=1+\frac{1}{kz_{R}}\frac{\rho^{2}-2w_{0}^{2}}{w^{2}}+\left(\frac{2z\rho}{w^{2}kz_{R}}\right)^{2},\\\
B_{\rho}=-\frac{4}{w^{2}}\frac{1}{k^{2}}\frac{z}{z_{R}}\left(1-\frac{\rho^{2}}{w^{2}}\right),\\\
A_{z}=1-\frac{\rho^{2}}{w^{2}},\\\
B_{z}=\frac{z}{z_{R}}\frac{\rho^{2}}{w^{2}},\end{array}$ (3)

where $w=w_{0}\sqrt{1+(z/z_{R})^{2}}$ represents the beam radius, and $w_{0}$
is defined as half the beam waist parameter at $z=0$. The Gouy phase is given
by $\zeta=1-iz/z_{R}$, and the Rayleigh range is expressed as $z_{R}=\pi
w_{0}^{2}/\lambda$, where $\lambda$ is the free-space wavelength. The
wavenumber is $k=2\pi/\lambda$. The characteristic impedance of free space is
given by $\eta_{0}=\sqrt{\mu_{0}/\varepsilon_{0}}$, where $\mu_{0}$ and
$\varepsilon_{0}$ are the absolute permeability and permittivity in vacuum,
respectively. In this paper, we consider monochromatic fields with the
implicit time dependence $e^{-i\omega t}$, where $\omega$ is the angular
frequency of the fields and $c$ is the speed of light in vacuum.

The dimensionless parameters $A_{\rho}$, $B_{\rho}$, $A_{z}$, and $B_{z}$
described in Equation (3) are real, but the parameter $f$ is complex and
related to the normalized Laguerre-Gaussian (LG) modes [53, 35]

$u_{\pm
1,0}=\frac{2\rho}{\sqrt{\pi}w^{2}}e^{-\left(\rho/w\right)^{2}\zeta}e^{-2i\tan^{-1}\left(z/z_{R}\right)}e^{\pm
i\varphi},$ (4)

as $f=\frac{w^{2}}{\rho}u_{\pm 1,0}e^{\mp i\varphi}e^{ikz}$. Further
elaboration on this subject can be found in Section A of the Supporting
Information, where we show the APB electric field in different polarization
bases. Comparing Equations (1) and (2) shows that the APB and the RPB are
electromagnetically dual. When they have the same beam parameters, their
fields are related as [45]

$\begin{array}[]{c}\mathbf{E}^{\text{APB}}=\eta_{0}\mathbf{H}^{\text{RPB}}\left(\frac{V_{\text{A}}}{V_{\text{R}}}\right),\\\
\mathbf{E}^{\text{RPB}}=-\eta_{0}\mathbf{H}^{\text{APB}}\left(\frac{V_{\text{R}}}{V_{\text{A}}}\right).\end{array}$
(5)

The APB/RPB only has non-zero longitudinal magnetic/electric field on the axis
($\rho=0$).

##  III The azimuthally-radially polarized beam

The ARPB is formed by adding an APB and an RPB, with a complex amplitude ratio
given by $V_{A}/V_{R}=\hat{V}e^{i\psi}$. The parameter $\hat{V}$ monitors the
relative magnitudes of the two beams, while their relative phase shift $\psi$
is referred to as the phase parameter as in Ref. [31]. The combined fields of
the ARPB are

$\begin{array}[]{c}\mathbf{E}^{\text{ARPB}}=\mathbf{E}^{\text{APB}}+\mathbf{E}^{\text{RPB}}=\frac{fV_{R}}{kw^{2}}\left[k\rho\left(A_{\rho}+iB_{\rho}\right)\,\hat{\bm{\rho}}+k\rho\hat{V}e^{i\psi}\,\hat{\bm{\varphi}}+2i\left(A_{z}+iB_{z}\right)\,\hat{\bm{z}}\right],\\\
\mathbf{H}^{\text{ARPB}}=\mathbf{H}^{\text{APB}}+\mathbf{H}^{\text{RPB}}=-\frac{fV_{R}}{kw^{2}\eta_{0}}\left[k\rho\hat{V}e^{i\psi}\left(A_{\rho}+iB_{\rho}\right)\,\hat{\bm{\rho}}-k\rho\,\hat{\bm{\varphi}}+2i\hat{V}e^{i\psi}\left(A_{z}+iB_{z}\right)\,\hat{\bm{z}}\right],\end{array}$
(6)

which are purely longitudinal on the beam axis. Note that the field components
in cylindrical coordinates do not show an azimuthal dependence. In Ref. [35],
the authors demonstrate that as $w_{0}$ of an APB decreases, its longitudinal
magnetic field component $H_{z}$ increases at a faster rate compared to its
transverse fields $\mathbf{E}_{\perp}$ and $\mathbf{H}_{\perp}$. As a result
of the APB/RPB duality shown in Eq. (5), this phenomenon is also observed in
the longitudinal electric field component $E_{z}^{RPB}$ of an RPB.
Consequently, the longitudinal fields of the ARPB increase more rapidly than
the transverse fields as $w_{0}$ decreases.

##  IV ARPB field quantities

We provide the cycle-averaged densities of specific field quantities of the
ARPB that play a critical role in local chiral light-matter interactions [4,
17]. The cycle-averaged energy density of a monochromatic EM field is
$u=\varepsilon_{0}|\mathbf{E}|^{2}/4+\mu_{0}|\mathbf{H}|^{2}/4$ [54], and the
complex Poynting vector is
$\mathbf{S}=\left(\mathbf{E}\times\mathbf{H}^{*}\right)/2$ [55]. Its real part
is associated with the linear momentum density
$\mathbf{p}=\Re(\mathbf{S})/c^{2}$ as in [55], while the imaginary part is
associated with the reactive energy flux density of the field. The spin
angular momentum (SAM) density is
$\bm{\sigma}=-\frac{\varepsilon_{0}}{4i\omega}\left(\mathbf{E}\times\mathbf{E}^{*}\right)-\frac{\mu_{0}}{4i\omega}\left(\mathbf{H}\times\mathbf{H}^{*}\right)$,
and its helicity density is $h=\frac{1}{2\omega
c}\Im\left(\mathbf{E}\cdot\mathbf{H}^{*}\right)$ [21]. While the energy and
SAM densities naturally split into electric and magnetic field contributions,
i.e., $u=u_{e}+u_{m}$ and $\bm{\sigma}=\bm{\sigma}_{e}+\bm{\sigma}_{m}$, the
Poynting vector $\mathbf{S}$ and the helicity density $h$ depend on the
interaction between the electric and magnetic fields of the beam. In this
section, our focus is exclusively on the final expressions for the field
quantities of the ARPB. Additional information about the field quantities of
the APB, the RPB, and the ARPB can be found in Section B of the Supporting
Information. To make the field quantities easier to comprehend, we express
them as a combination of a normalization constant (denoted by the subscript
$0$) and a dimensionless term. The energy density of an ARPB is

$\begin{array}[]{c}u^{\mathrm{ARPB}}=\frac{1}{2}u_{0}\left(1+\hat{V}^{2}\right)\left[(k\rho)^{2}\left(1+A_{\rho}^{2}+B_{\rho}^{2}\right)+4\left(A_{z}^{2}+B_{z}^{2}\right)\right],\end{array}$
(7)

where $u_{0}=\frac{\varepsilon_{0}|f|^{2}}{2k^{2}w^{4}}|V_{R}|^{2}$. The
energy density of an ARPB describes an annular ring in the transverse plane.
The components of the Poynting vector $\mathbf{S}^{\mathrm{ARPB}}$ of an ARPB
are

$\begin{array}[]{c}S_{\rho}^{\mathrm{ARPB}}=iS_{0}\left[\right.\hat{V}^{2}(A_{z}-iB_{z})-(A_{z}+iB_{z})\left.\right],\\\
S_{\varphi}^{\mathrm{ARPB}}=-2S_{0}\hat{V}\left(\sin\psi+i\cos\psi\right)\left(A_{\rho}A_{z}+B_{\rho}B_{z}\right),\\\
S_{z}^{\mathrm{ARPB}}=\frac{1}{2}S_{0}k\rho\left[\right.\hat{V}^{2}(A_{\rho}-iB_{\rho})+(A_{\rho}+iB_{\rho})\left.\right].\end{array}$
(8)

where $S_{0}=\frac{\rho|f|^{2}}{k\eta_{0}w^{4}}|V_{R}|^{2}=2\rho\omega u_{0}$.
Note that $S_{0}$ vanishes on the beam axis, while $u_{0}$ does not. When
$\hat{V}=1$, the longitudinal and radial energy flux density,
$S_{z}^{\mathrm{ARPB}}$ and $S_{\rho}^{\mathrm{ARPB}}$, are purely real.
Moreover, near its focal plane ($z\approx 0$), the radial component of the
Poynting vector of an ARPB with $\hat{V}=1$ is negligible:
$S_{\rho}^{\mathrm{ARPB}}\propto B_{z}\approx 0$ $\text{W/m}^{2}$ (see Eq.
(3)). Therefore, most of the energy density flows in the $z$ direction and the
ratio of active versus reactive energy flux density in the azimuthal direction
depends on the phase parameter $\psi$. For $\psi\neq 0,\pi$, which is the case
for chiral ARPBs, there is a nonzero azimuthal linear momentum density
$p_{\varphi}=\Re(S_{\varphi})/c^{2}$. While the ARPB is not a conventional
vortex beam since it does not have a phase variation around the axis (usually
characterized by $e^{il\varphi}$) [56], it exhibits a power flow around the
axis that can be controlled by $\psi$. Therefore, it necessarily implies the
existence of an orbital angular momentum (OAM) density in the longitudinal
direction [57, 58, 59]. The longitudinal OAM density $l_{z}^{\mathrm{ARPB}}$
of the ARPB arises from the propagation of energy in the azimuthal direction
$p_{\varphi}^{\mathrm{ARPB}}$, which is, in essence, a rotation of the beam
around the axis. This rotation appears as a result of the out-of-phase
combination of an APB and an RPB, which do not individually exhibit an
azimuthal power flow (see Eq. (S5) of the Supplementary Information, Section
B, for further details). The longitudinal OAM density of a field in
cylindrical coordinates is calculated as $l_{z}=p_{\varphi}\rho-\sigma_{z}$
(see Eq. (S8) and (S9) of the Supplementary Information, Section B, for more
information), which for an ARPB is

$l_{z}^{\mathrm{ARPB}}=l_{0}\hat{V}\sin(\psi)\left[2(A_{\rho}A_{z}+B_{\rho}B_{z})+A_{\rho}\right],$
(9)

where $l_{0}=-\frac{\varepsilon_{0}\rho^{2}|f|^{2}}{\omega w^{4}}|V_{R}|^{2}$.
The longitudinal OAM density vanishes on the beam axis, where the fields are
purely longitudinal. Remarkably, the OAM density-carrying ARPB is generated
from a phase-shifted superposition of two beams with zero longitudinal OAM
densities, the APB and the RPB. A chiral ARPB carries an $l_{z}$ that reaches
a maximum for $\psi=\pm\pi/2$, which is the case for OCL as discussed in
Section V. Additionally, the OAM density of an ARPB can be continuously tuned
by varying the phase parameter $\psi$, in contrast with the discrete azimuthal
phase dependence $e^{il\varphi}$ of LG vortex beams, where $l$ is the
topological charge. As shown later in Section VI, the value of $\psi$ is
readily accessible by means of standard optical elements (beam splitters,
mirrors, and wave plates), while varying $l$ generally requires elaborate
optical elements, including Q-plates [60], spatial light modulators
(SLMs)[61], spiral phase plates [62], or cylindrical lenses [63, 64].

The components of the SAM density $\bm{\sigma}^{\mathrm{ARPB}}$ of the ARPB
are

$\begin{array}[]{c}\sigma_{\rho}^{\mathrm{ARPB}}=2\sigma_{0}\hat{V}\sin(\psi)B_{z},\\\
\sigma_{\varphi}^{\mathrm{ARPB}}=-\sigma_{0}\left(1+\hat{V}^{2}\right)\left[A_{\rho}A_{z}+B_{\rho}B_{z}\right],\\\
\sigma_{z}^{\mathrm{ARPB}}=\sigma_{0}k\rho\hat{V}\sin(\psi)A_{\rho},\end{array}$
(10)

where $\sigma_{0}=\frac{\varepsilon_{0}\rho|f|^{2}}{\omega
kw^{4}}|V_{R}|^{2}=S_{0}/(\omega c)$ vanishes on axis. While the azimuthal SAM
density $\sigma_{\varphi}^{\mathrm{ARPB}}$ is the sum of the SAM densities of
the individual APB and RPB (see Supporting Information, Section B), the radial
and longitudinal components of the SAM density arise from the interaction
between the two beams and therefore depend on the phase parameter $\psi$.

The cycle-averaged helicity density $h$ is naturally split between the
contributions from each field component, i.e., $h=h_{\rho}+h_{\varphi}+h_{z}$,
and also between the transverse $h_{\perp}=h_{\rho}+h_{\varphi}$ and
longitudinal $h_{z}$ helicity densities. The components of the helicity
density $h^{\mathrm{ARPB}}$ of an APRB are

$\begin{array}[]{c}h_{\rho}=h_{0}\hat{V}\sin(\psi)(k\rho)^{2}\left(A_{\rho}^{2}+B_{\rho}^{2}\right),\\\
h_{\varphi}=h_{0}\hat{V}\sin(\psi)(k\rho)^{2},\\\
h_{z}=4h_{0}\hat{V}\sin(\psi)\left(A_{z}^{2}+B_{z}^{2}\right),\end{array}$
(11)

where $h_{0}=\frac{\varepsilon_{0}|f|^{2}}{2\omega k^{2}w^{4}}|V_{R}|^{2}$.
The normalization parameters of the energy $u_{0}$ and helicity $h_{0}$
densities are related as $h_{0}=u_{0}/\omega$. Note that $h_{0}$ and $u_{0}$
do not vanish on the axis, while $S_{0}$, $\sigma_{0}$, and $l_{0}$ do vanish
there, and that is why there is a special relation between $h_{0}$ and
$u_{0}$. The total helicity density of an ARPB is also related to its energy
density (shown in Eq. (7)) as

$h^{\mathrm{ARPB}}=\frac{u^{\mathrm{ARPB}}}{\omega}\frac{2\hat{V}}{1+\hat{V}^{2}}\sin(\psi).$
(12)

For $\hat{V}=1$, the helicity density is in agreement with the expressions in
Eq. (13.28) and (13.29) of Ref. [45]. For $\psi=0,\pi$, the interaction
between the fields of the APB and the RPB is minimized; the energy $u$, linear
momentum $\mathbf{p}$, SAM $\bm{\sigma}$, longitudinal OAM $l_{z}$, and
helicity $h$ densities of the resulting ARPB are the sum of those of the APB
and the RPB (see Section B of the Supporting Information for details).
Conversely, for $\psi=\pm\pi/2$, which corresponds to the OC-ARPB as shown in
Section V, the interaction between the APB and the RPB is maximized. The OC-
APRB displays maximum linear momentum, SAM, OAM, and helicity densities. In
essence, the phase parameter $\psi$ mediates the interaction between the
fields of the APB and the RPB. While the longitudinal OAM $l_{z}$ and helicity
densities are proportional, the latter is not a by-product of the former. In
fact, both field quantities originate from the out-of-phase superposition of
the APB and the RPB, which is determined by the phase parameter $\psi$, in
contrast to what occurs for focused LG beams $u_{l,p}$ [50], where the
helicity density is proportional to the topological charge $l$.

Moreover, the ARPB displays an exceptional spatial separation between the
longitudinal and transverse fields, which vanish on the beam axis. As a
consequence, the energy $u$, linear momentum $\mathbf{p}$, SAM $\bm{\sigma}$,
longitudinal OAM $l_{z}$, and helicity $h$ densities describe an annular ring
in the transverse plane. On the $z$ axis, only the contribution of the
longitudinal fields to the energy and helicity densities, shown respectively
in Eqs. (7) and (12), persist.

(a)

(b)

(c)

Figure 3: Normalized helicity density $\hat{h}$ of an ARPB in terms of
$\hat{V}$ and $\psi$, calculated as $\hat{h}=hu/\omega$. (a) Colormap of
$\hat{h}$ for $\hat{V}\in[0,2]$ and $\psi\in[-\pi,\pi]$. The maximum magnitude
for the normalized helicity density occurs for OC-ARPBs that is $V_{A}=\pm
iV_{R}$. (b) The normalized helicity density for $\hat{V}=1$ and
$\psi\in[-\pi,\pi]$. The maximum values are found for $\psi=\pm\pi/2$. (c) The
normalized helicity density for $\psi=\pi/2$ and $\hat{V}\in[0,2]$. The
magnitude of the normalized helicity density reaches unity ($\hat{h}=1$) for
$\hat{V}=1$ and $\psi=\pm\pi/2$, i.e., $V_{A}=\pm iV_{R}$.

##  V Optimally chiral ARPB

It was demonstrated in Ref. [21] that the helicity density of a chiral field
is locally bounded by its energy density as

$-\frac{u}{\omega}\leq h\leq\frac{u}{\omega}.$ (13)

By definition, OCL is a local descriptor that structured light reaches the
upper or lower bound of the helicity density. It occurs for fields whose
phasors locally satisfy $\mathbf{E}=\pm i\eta_{0}\mathbf{H}$, which is
referred to as the optimal chirality condition [21]. This is equivalent to the
condition that the field phasors of OCL must satisfy

$\begin{array}[]{c}\nabla\times\mathbf{E}=\pm k\mathbf{E},\\\
\nabla\times\mathbf{H}=\pm k\mathbf{H}.\end{array}$ (14)

Thus, the optical chirality density
$C=(1/4)(\varepsilon_{0}\mathbf{E}^{*}\cdot\nabla\times\mathbf{E}+\mu_{0}\mathbf{H}^{*}\cdot\nabla\times\mathbf{H})$
[65], which evaluates the collinearity between the fields and their curls, is
also maximum for OCL. For time-harmonic fields in free space, the optical
chirality and the helicity densities are proportional, with
$C=\omega^{2}c^{-1}h$ [21]. As a result, the magnitude of optical chirality
has an upper bound given by the energy density, $|C|\leq ku$, which is reached
for OCL.

In this paper, we find convenient to use the normalized (and dimensionless)
helicity density $\hat{h}=h\omega/u$, whose magnitude is bounded by unity,
i.e., $-1\leq\hat{h}\leq 1$. It is naturally split into its transverse
$\hat{h}_{\perp}=h_{\perp}\omega/u$ and longitudinal
$\hat{h}_{z}=h_{z}\omega/u$ components (with
$|\hat{h}_{\perp}+\hat{h}_{z}|\leq 1$). The normalized helicity density
$\hat{h}^{\mathrm{ARPB}}$ of the ARPB is reduced to

$\hat{h}^{\mathrm{ARPB}}=\frac{2\hat{V}}{1+\hat{V}^{2}}\text{sin}(\psi),$ (15)

which does not depend on the parameters of the beam or on the position where
the field is evaluated. This result is confirmed in Figure 3(a), which shows
the normalized helicity density (computed as $h\omega/u$ and evaluated at an
arbitrary position with $\rho\leq 20w_{0}$ and $z\leq 10z_{R}$) versus
$\hat{V}$ and $\psi$. The respective maximum and minimum normalized helicity
densities are found for $\hat{V}=1$, $\psi=\pm\pi/2$ rad. Figure 3(b) denotes
the vertical cross-section of Figure 3(a) for $\hat{V}=1$, which is identified
as the sinusoidal term $\hat{h}=\text{sin}(\psi)$ in Eq. (15). Figure 3(c)
shows the horizontal cross-section of Figure 3(a) for $\psi=\pi/2$, described
by $\hat{h}=2V/(1+V^{2})$. The ARPB is optimally chiral ($\hat{h}=\pm 1$) when
the APB leads or lags the RPB by a quarter of a period and the beams have
amplitudes of the same magnitude, i.e., $V_{A}=\pm iV_{R}$, in agreement with
what was observed in Ref. [21]. This condition is equivalent to the optimal
chirality condition $\mathbf{E}=\pm i\eta_{0}\mathbf{H}$. Indeed, the fields
of an ARPB with $V_{A}=\pm iV_{R}$ are

$\begin{array}[]{c}\mathbf{E}_{\perp}=\frac{\rho}{w^{2}}fV_{R}\left[\left(A_{\rho}+iB_{\rho}\right)\,\hat{\bm{\rho}}\pm
i\,\hat{\bm{\varphi}}\right],\\\ \mathbf{H}_{\perp}=\mp
i\mathbf{E}_{\perp}/\eta_{0},\\\
E_{z}=\frac{2i}{kw^{2}}fV_{R}\left(A_{z}+iB_{z}\right),\\\ H_{z}=\mp
iE_{z}/\eta_{0}.\end{array}$ (16)

Even though the OC-ARPB attains its maximum helicity density on an annular
ring around the beam axis (with the same topology as the energy density $u$),
the relation $\mathbf{E}=\pm i\eta_{0}\mathbf{H}$, and therefore
$\hat{h}=\hat{h}_{\perp}+\hat{h}_{z}=1$, hold locally everywhere in space.

(a)

(b)

(c)

Figure 4: Comparison of the energy $u$ and helicity $h$ densities of an OC-
ARPB (of $1$ mW power) with $w_{0}=\lambda$, $w_{0}=2\lambda$, and
$w_{0}=5\lambda$. (a) Depicts the energy density at focus in the transverse
plane and (b) shows the energy density on the beam axis. In both figures, the
energy density increases as the ratio $w_{0}/\lambda$ decreases. (c) Depicts
the transverse $\hat{h}_{\perp}$ and longitudinal $\hat{z}$ normalized
helicity density of the three OC-ARPB. Even though the focused ARPB has a
higher helicity density, the region of space where its longitudinal component
dominates over the transverse one remains approximately constant at
$\rho\approx 0.225\lambda$.

As discussed in Section III, when $w_{0}$ decreases, the relative increase in
the magnitude of the longitudinal fields compared to the transverse fields
becomes more pronounced. Therefore, focusing an ARPB through a lens increases
its energy density, and consequently, the upper bound of its chirality. Figure
4 illustrates this behavior by comparing the energy densities and the
normalized helicity densities of two OC-ARPBs of $1$ mW power with
$w_{0}=\lambda$ and $w_{0}=2\lambda$, where $\lambda=400$ nm. Figures 4(a) and
(b) show the energy density $u$ of both cases, at the focus and on the $z$
axis, respectively. Focusing the ARPB by decreasing the ratio $w_{0}/\lambda$
increases the energy density and, consequently, the helicity density. Figure
4(c) displays the normalized helicity densities of the transverse
$\hat{h}_{\perp}$ and longitudinal $\hat{h}_{z}$ components for both beams.
Although the focused OC-ARPB has a larger helicity density, the region where
the longitudinal component dominates over the transverse component remains
roughly the same, at approximately $\rho\approx 0.225\lambda$.

As a result of the collinearity between the electric and magnetic field
phasors, OCL displays an exceptional electric-magnetic symmetry in its energy
and spin densities, i.e., $u_{e}=u_{m}$ and $\bm{\sigma}_{e}=\bm{\sigma}_{m}$
[21]. The Poynting vector $\mathbf{S}^{\text{OC-ARPB}}$ of an OC-ARPB is

$\begin{array}[]{c}\mathbf{S}^{\text{OC-
ARPB}}=S_{0}\left[\right.2B_{z}\hat{\bm{\rho}}\mp
2\left(A_{\rho}A_{z}+B_{\rho}B_{z}\right)\hat{\bm{\varphi}}+k\rho
A_{\rho}\hat{\bm{z}}\left.\right],\end{array}$ (17)

which is real-valued globally and, consequently, all the energy flux density
contributes to the linear momentum density; an OC-ARPB has no reactive energy
flux density. Moreover, the Poynting vector of an OC beam is collinear with
its SAM density, as demonstrated in the Supplementary Information of Ref.
[21]. For an OC-ARPB, the spin density $\bm{\sigma}^{\text{OC-ARPB}}$ is

$\begin{array}[]{c}\bm{\sigma}^{\mathrm{ARPB}}=\sigma_{0}\left[\right.\pm
2B_{z}\hat{\bm{\rho}}-2\left(A_{\rho}A_{z}+B_{\rho}B_{z}\right)\hat{\bm{\varphi}}\pm
k\rho A_{\rho}\hat{\bm{z}}\left.\right].\end{array}$ (18)

As demonstrated here for the OC-ARPB, although it is easily verified that it
applies for all OCL, $\mathbf{S}=\pm\omega c\bm{\sigma}$. Since the spin
density is a real quantity, OCL has a purely real energy flux density
$\mathbf{S}\in\Re$, and therefore a maximized linear momentum density
$\mathbf{p}=\pm k\bm{\sigma}$, which changes sign for $\psi=+\pi/2$ or
$\psi=-\pi/2$. The normalization constants of the Poynting vector and the spin
density of the ARPB also follow the same relation $S_{0}=\omega c\sigma_{0}$.
Figure 5 displays the magnitudes of the (a) azimuthal and (b) longitudinal
components of the Poynting vector of an OC-ARPB $\mathbf{S}^{\text{OC-ARPB}}$
on the transverse plane at the beam focus ($z=0$). Its radial component
$S_{\rho}$, which is approximately zero near the beam’s focus (see Eq. (17),
where $B_{z}\approx 0$). The azimuthal component $S_{\varphi}$ has a peak
described by a small ring around the beam axis. This occurs because it is
partly associated with the longitudinal fields of the OC-ARPB, which peak on
the beam axis. The longitudinal component of the Poynting vector $S_{z}$ is
associated with the transverse fields and therefore describes a bigger ring
around the beam axis.

(a)

(b)

Figure 5: Magnitudes of the (a) azimuthal and (b) longitudinal components of
the Poynting vector $\mathbf{S}$ of an OC-ARPB (of $1$ mW power at
$\lambda=400$ nm) in the transverse plane at the focus of the beam, i.e.,
$z=0$. The radial component $S_{\rho}$ has not been plotted since it is
practically zero at the focus of the beam. The azimuthal component
$S_{\varphi}$ peaks in a small ring close to the beam axis, since it is
associated with the radial and longitudinal fields. The longitudinal component
$S_{z}$ is the largest of them since it is only associated with the transverse
fields, which are significantly larger than the longitudinal components. Due
to azimuthal symmetry and vanishing transverse fields on the beam axis, the
components of the Poynting vector describe an annular ring around it.

In summary, the OC-ARPB displays the enhanced chirality-probing capabilities
characteristic of OCL, together with an exceptional spatial separation of
longitudinal and transverse properties. Unlike CPL, which only measures
transverse chirality, the ARPB measures exclusively longitudinal chirality
along the beam axis, making it preferable because transverse chirality
measurements can be mistaken for sample anisotropy [31]. Using the ARPB beam
as well as some other beams with longitudinal axial chirality, such as those
discussed in Refs. [46, 47, 48, 49, 50, 51] whose chirality is proportional to
the topological charge $l$, eliminate the chiral-anisotropy ambiguity issue by
only measuring the chirality of a particle located on the beam axis. The
advantage of the OC-ARPB over these beams is that its chirality can be easily
tuned by adjusting the phase parameter $\psi$, as shown in Section VI, and
therefore it is a flexible and versatile option for achieving controlled
probing or enantioseparation of chiral nanoparticles.

(a)

(b)

(c)

(d)

Figure 6: Schematic of a linearly polarized beam traveling through an SWP. (a)
An $x$-polarized beam is transformed into an RPB. (b) An $y$-polarized beam is
transformed into an APB. (c) A beam polarized at $\theta=\pi/4$, with $\theta$
the angle between the $x$ and $y$ axis, is transformed into the sum of an RPB
and an APB with opposite phases. (d) A beam polarized at $\theta=-\pi/4$ is
transformed into a sum of an RPB and an APB with the same phase.

##  VI ARPB generation

We show an optical setup that generates an ARPB with arbitrary $\hat{V}$ and
$\psi$ from an arbitrarily polarized laser source. An option to transform
familiar wave polarization types (such as linear, circular, and elliptical)
into vector modes (such as radial or azimuthal) is the use of an S-waveplate
(SWP). The SWP is a combination of half-wave plate (HWP) segments whose slow
axis is changed to maintain an azimuthal angle with respect to a reference
axis [66], and can produce an APB, an RPB, or a combination of both by
selective input polarization [42]. Its Jones matrix is [67]

$\underline{\mathbf{H}}_{S}=\left(\begin{array}[]{cc}\cos\varphi&\sin\varphi\\\
\sin\varphi&-\cos\varphi\end{array}\right).$ (19)

$\mathbf{J}_{in}$ | $\to$ | $\mathbf{J}_{out}=\underline{\mathbf{H}}_{S}\mathbf{J}_{in}$
---|---|---
$\left(1,\ 0\right)^{T}$ | $\to$ | $\mathbf{J}_{RPB}=\left(\cos(\varphi),\ \sin(\varphi)\right)^{T}$
$\left(0,\ -1\right)^{T}$ | $\to$ | $\mathbf{J}_{APB}=\left(-\sin(\varphi),\ \cos(\varphi)\right)^{T}$
$\frac{1}{\sqrt{2}}\left(1,\ -e^{i\psi}\right)^{T}$ | $\to$ |  $\mathbf{J}_{ARPB}=\mathbf{J}_{RPB}+e^{i\psi}\mathbf{J}_{APB}$,
Table I: SWP relevant input and output pairs. It provides the inputs required
to obtain an RPB, an APB, and an ARPB with $\hat{V}=1$ and arbitrary $\psi$,
respectively. The superscript $T$ denotes the transpose operator and $\varphi$
is the azimuthal coordinate.

The Jones vectors for relevant input/output pairs are summarized in Table I
and depicted in Figure 6. In particular, Figure 6(a) shows that an
$x$-polarized beam becomes an RPB, and Figure 6(b) that a $y$-polarized beam
becomes an APB. Figure 6(c) and (d) illustrate the conversion of linearly
polarized beams at angles $\theta=\mp\pi/4$ with respect to the $x$ axis into
achiral ARPBs with $\hat{V}=1$ and either $\psi=0$ or $\psi=\pi$,
respectively. An ARPB with arbitrary $\hat{V}$ and $\psi$ can be obtained by
setting a linear polarizer at an angle $\theta$ from the $x$ axis and adding a
phase delay equal to $\psi$ on the $y$-polarized component through a delay arm
of length $L$, as shown in Figure 7. In this arrangement, the amplitude ratio
$\hat{V}$ between the APB and the RPB is controlled by the angle $\theta$ of
the linear polarizer relative to the $x$ axis, which is given by

$\hat{V}=|\tan(\theta)|.$ (20)

To obtain $\hat{V}=1$, the linear polarizer must be set to $\theta=\pm\pi/4$,
where choosing the minus sign removes a $180\degree$ phase from the resulting
APB upon conversion by the SWP. The length $L$ of the delay arm sets the phase
parameter $\psi$, and the two are related by the equation

$kL-\tan^{-1}\left(\frac{L}{z_{R}}\right)=\psi,$ (21)

which depends on $\lambda$ and $w_{0}$.

An alternative method for introducing the phase delay $\psi$ between the $y$\-
and $x$-polarized components of a linearly polarized beam relies on using a
HWP and a quarter-wave plate (QWP). The Jones matrix of a HWP at angle
$\theta_{h}$ with respect to the $x$ axis is [68]

$\underline{\mathbf{HWP}}(\theta_{h})=\left(\begin{array}[]{cc}-\cos(2\theta_{h})&-\sin(2\theta_{h})\\\
-\sin(2\theta_{h})&\cos(2\theta_{h})\end{array}\right),$ (22)

and that of a QWP at angle $\theta_{Q}$ is

$\underline{\mathbf{QWP}}(\theta_{q})=\left(\begin{array}[]{cc}i\cos^{2}\theta_{q}+\sin^{2}\theta_{q}&(i-1)\cos\theta_{q}\sin\theta_{q}\\\
(i-1)\cos\theta_{q}\sin\theta_{q}&i\sin^{2}\theta_{q}+\cos^{2}\theta_{q}\end{array}\right).$
(23)

The resulting transformation reads

$\mathbf{J}_{out}=\underline{\mathbf{HWP}}(\theta_{h})\underline{\mathbf{QWP}}(\theta_{q})\mathbf{J}_{in}.$
(24)

The angle combinations that provide the $\psi$ phase delay depend on the
initial and final position of the polarization on the Poincaré sphere (see
Ref. [68] for details). Replacing the delay arm in Figure 7 with the HWP and
the QHP, as depicted in Figure 8, indicates that the incident polarization is
$\mathbf{J}_{in}=(1,\ 1)^{T}$ and the final is $\mathbf{J}_{out}=(1,\
e^{i\psi})^{T}$. Examples of angle combinations for the most relevant $\psi$
values for this transformation are shown in Table II.

$\psi$ (rad) |  $\theta_{H}$ (rad) |  $\theta_{Q}$ (rad)
---|---|---
$0$ | $7\pi/4$ | $5\pi/4$
$\pi/2$ | $0$ | $0$
$\pi$ | $3\pi/2$ | $5\pi/4$
$-\pi/2$ | $0$ | $-\pi/2$
Table II: Examples of angle combinations that provide the most relevant phase
delay $\psi$ values for the polarization transformation of a beam with
$\mathbf{J}_{in}=(1,\ 1)^{T}$ into a beam with $\mathbf{J}_{out}=(1,\
e^{i\psi})^{T}$. Figure 7: Optical setup to obtain an ARPB with $\hat{V}=1$
and arbitrary $\psi$. The beam is polarized at an angle $\theta=-\pi/4$ with
respect to the $x$-axis, and the $y$-component is added a phase-shift $\psi$
using a delay arm of length $L$. The resulting beam travels through an SWP,
resulting in the desired ARPB. Figure 8: Optical setup to obtain an ARPB with
$\hat{V}=1$ and arbitrary $\psi$ using a QWP and a HWP at angles $\theta_{q}$
and $\theta_{h}$, respectively, instead of the delay arm from Figure 7.
Possible angle combinations that provide the most relevant $\psi$ values are
shown in Table II.

Using two cascaded HWPs at angles $\theta_{1}=\delta/2$ and $\theta_{2}=0$
rotates the polarization by an angle $\delta$ with respect to the input vector
[69]. The Jones matrix of the resulting polarization rotator, i.e.,
$\underline{\mathbf{PR}}(\delta)=\underline{\mathbf{HWP}}(\delta/2)\underline{\mathbf{HWP}}(0)$,
is

$\underline{\mathbf{PR}}(\delta)=\left(\begin{array}[]{cc}\cos\delta&-\sin\delta\\\
\sin\delta&\cos\delta\end{array}\right).$ (25)

This device can be adapted for a variety of purposes. For example, a radial-
to-azimuthal polarization rotator is obtained by setting $\delta=\pi/2$, as
depicted in Fig. 9. The order of the HWPs is inverted in Ref. [70], which adds
a $\pi$ phase to the resulting APB. Whilst this phase difference is not
relevant in measurements, it may affect light manipulation. Using the
polarization rotator with $\delta=\pi$ instead changes the handedness of an
incident OC-ARPB.

Figure 9: Schematic for a radial-to-azimuthal polarization rotator using two
HWPs with an angle difference of $\theta_{1}-\theta_{2}=\pi/4$.

Although the main focus in this section is on generating both configurations
of the OC-ARPB for chirality probing and enantioseparation purposes, the
setups depicted in Figure 7 and 8 can also be used to smoothly adjust the OAM
density of the ARPB, either by varying the length $L$ of the delay arm or the
angles of the wave plates, which has practical applications in conducting OAM-
related experiments.

##  VII Conclusion

We have introduced a concise notation to describe the APB, the RPB, and the
ARPB. We have investigated the properties of the ARPB and derived the
analytical expressions for its field quantities. Our findings demonstrate that
an in-phase superposition of an APB and an RPB results in an achiral ARPB. An
out-of-phase superposition, however, results in a chiral ARPB with larger
linear momentum and spin densities (both in terms of magnitude and number of
components) than those of achiral beams. Moreover, a chiral ARPB carries a
longitudinal OAM density despite having no azimuthal phase dependence, which
can be continuously adjusted by varying the phase delay $\psi$ between the APB
and the RPB. Indeed, the ARPB can be optimally chiral and exhibits maximum
linear momentum, angular momentum, and helicity densities. The transverse
fields of the ARPB vanish on the beam axis, and therefore the same occurs to
its linear and angular momentum densities, where only the energy and helicity
densities associated with the longitudinal fields persist. We have shown that
focusing an ARPB through a lens boosts its longitudinal fields more than the
transverse ones, which results in an enhanced longitudinal chirality.
Therefore, the ARPB provides a controlled environment that may be used to trap
and probe chiral nanoparticles. Accordingly, the next step would be to
calculate the photoinduced forces on a chiral dipolar particle illuminated by
an ARPB, with an emphasis on the optimally chiral configuration. To aid future
research experiments, we have also provided the schematics for an optical
setup that generates an ARPB and provided a polarization rotator that changes
the handedness of an OC-ARPB.

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