# Focused linearly-polarized light scattering from a silver nanowire:
Experimental characterization of optical spin-Hall effect

Diptabrata Paul  Department of Physics, Indian Institute of Science Education
and Research (IISER), Pune 411008, India  Deepak K. Sharma  Department of
Physics, Indian Institute of Science Education and Research (IISER), Pune
411008, India  Present address: Laboratoire Interdisciplinaire Carnot de
Bourgogne, UMR 6303 CNRS, Université de Bourgogne Franche-Comté, 9 Avenue
Alain Savary, 21000 Dijon, France  G V Pavan Kumar<EMAIL_ADDRESS>Department of Physics, Indian Institute of Science Education and Research
(IISER), Pune 411008, India  Center for Energy Science, Indian Institute of
Science Education and Research (IISER), Pune 411008, India

###### Abstract

Spin-orbit interactions (SOI) are a set of sub-wavelength optical phenomenon
in which spin and spatial degrees of freedom of light are intrinsically
coupled. One of the unique example of SOI, spin-Hall effect of light (SHEL)
has been an area of extensive research with potential applications in spin
controlled photonic devices as well as emerging fields of spinoptics and
spintronics. Here, we report our experimental study on SHEL due to forward
scattering of focused linearly polarized Gaussian and Hermite-Gaussian
($\textrm{HG}_{10}$) beams from a silver nanowire (AgNW). Spin dependent anti-
symmetric intensity patterns are obtained when the polarization of the
scattered light is analysed. The corresponding spin-Hall signal is obtained by
computing the far-field longitudinal spin density ($s_{3}$). Furthermore, by
comparing the $s_{3}$ distributions, significant enhancement of the spin-Hall
signal is found for $\textrm{HG}_{10}$ beam compared to Gaussian beam. The
investigation of the optical fields at the focal plane of the objective lens
reveals the generation of longitudinally spinning fields as the primary reason
for the effects. The experimental results are corroborated by 3-dimensional
numerical simulations. The results lead to better understanding of SOI and can
have direct implications on chip-scale spin assisted photonic devices.

##  I Introduction

Along with energy and linear momentum, angular momentum (AM) represents the
most important dynamical parameters of light Allen  _et al._ (1992); Franke-
Arnold  _et al._ (2008); Bekshaev  _et al._ (2011). The AM of light can be
decomposed into two parts, spin angular momentum (SAM), which is related to
the circular polarization of a light beam and orbital angular momentum (OAM),
related to the helical phase front of a beam Allen  _et al._ (1992); Yao and
Padgett (2011); Bekshaev  _et al._ (2011); Molina-Terriza  _et al._ (2007). In
recent times, these concepts have been exploited in context of spin-orbit
interaction (SOI), an optical phenomenon in which polarization and position of
light are intrinsically coupled Liberman and Zel’dovich (1992); Bliokh  _et
al._ (2015); Aiello  _et al._ (2015).

One of the unique examples of SOI is spin-Hall effect of light (SHEL) Onoda
_et al._ (2004), where equivalent to its electronic counterpart Hirsch (1999),
a transverse shift in the scattered beam location is perceived due to the
transverse spin flow of the impinging light beam on a dielectric interface
Hosten and Kwiat (2008); Bliokh and Aiello (2013); Bekshaev (2012); Goswami
_et al._ (2014) or a scatterer Haefner  _et al._ (2009); Sharma  _et al._
(2018). Further, the transverse shift in the scattered beam is opposite for
opposite spins of the impinging beam, allowing us to distinguish between them
Sharma  _et al._ (2018). But, since SOI effects are weak for paraxial beams,
observation of transverse shift in scattering of linearly polarized beams
becomes difficult due to absence of net transverse spin flow. The situation
changes drastically in case of non-paraxial light beams, where depending on
the numerical aperture (NA), higher AM can be generated Bliokh  _et al._
(2015, 2010), thus leading to enhanced SOI Bomzon  _et al._ (2006); Nieminen
_et al._ (2008). To this end, strong focusing of light has been an area of
extensive research in recent years with implications in studies such as
optical manipulation Zhao  _et al._ (2009); Yan and Scherer (2013), sub-
wavelength position sensing Rodríguez-Herrera  _et al._ (2010); Neugebauer
_et al._ (2019a), AM inter-conversion Zhao  _et al._ (2007); Nechayev  _et
al._ (2019). In this context, the interaction of the optical fields at the
focus with a nanoscopic object and the resultant SOI at sub-wavelength scale
can have potential utilization in various photonic applications such as
generation of structured fields, controlling optical wave propagation, optical
nano-probing and refractive index sensor Mueller and Capasso (2013); O’Connor
_et al._ (2014); Cardano and Marrucci (2015); Rodríguez-Herrera  _et al._
(2010); Neugebauer  _et al._ (2019a); Zhou  _et al._ (2018).

Figure 1:  (color online) Schematic of the measurement scheme of SHEL. The
focal longitudinal spin density ($s_{z}$) is scattered by an AgNW and can be
observed in the Fourier plane image as far-field longitudinal spin
distribution ($s_{3}$).

Motivated by this, we study the longitudinal spin density through optical
interaction between a quasi one dimensional scatterer, namely a single
crystalline silver nanowire (AgNW) on a glass substrate and focused linearly
polarized Gaussian and $\textrm{HG}_{10}$ beams as shown in Fig. 1.
Polarization analysis of the forward scattered light in the Fourier plane (FP)
reveals anti-symmetric distribution of intensity for opposite circular
polarization states with respect to long axis of the AgNW. The pattern inverts
for left circular polarization (LCP) analyzed case with respect to that of
right circular polarization (RCP) analyzed one, revealing intrinsic SHEL
Sharma  _et al._ (2018). The difference between the two FP intensity
distributions quantify the spin-Hall signal as well as far-field longitudinal
spin density ($s_{3}$). Furthermore, the comparison of $s_{3}$ for Gaussian
beam with respect to that of $\textrm{HG}_{10}$ beam reveals higher spin-Hall
signal for $\textrm{HG}_{10}$ beam. The enhancement factor of longitudinal
spin distribution for $\textrm{HG}_{10}$ beam with respect to that of Gaussian
beam is further quantified from experimentally measured and numerically
simulated data.

In the following sections, we describe the nature of the optical fields at the
focus and show the existence of longitudinally spinning fields at focal plane.
Furthermore, by analyzing the scattering pattern of the focal optical fields
from a AgNW, emerging SHEL is observed and resulting far-field longitudinal
spin density is measured.

##  II Theoretical Description

###  II.1 Description of focal optical fields

Focusing of paraxial optical beams through an objective lens leads to complex
distribution of optical fields at the focal plane. The nature of polarized
optical fields at the focus of an objective lens can be theoretically
calculated using Debye-Wolf integral, originally proposed by Richards and Wolf
Wolf and Gabor (1959); Richards  _et al._ (1959). When an paraxial light beam
with electric field $\mathbf{E_{in}}$ and wave vector
$\mathbf{k}=k\mathbf{\hat{z}}$ passing through medium of refractive index
$n_{1}$ is incident on an objective lens of numerical aperture,
$\textrm{NA}=n_{1}\sin(\theta_{\mathrm{max}})$ and having focal length $f$,
then the complex field distribution at the focal plane can be obtained by,

$\displaystyle\mathbf{E}(\rho,\varphi,z)=$
$\displaystyle-\frac{\mathrm{i}kf\mathrm{e}^{-\mathrm{i}kf}}{2\pi}\int\displaylimits_{0}^{\theta_{\mathrm{max}}}\int\displaylimits_{0}^{2\pi}\mathbf{E_{ref}(\theta,\phi)}\mathrm{e}^{\mathrm{i}kz\cos\theta}$
(1a)
$\displaystyle\mathrm{e}^{\mathrm{i}k\rho\sin\theta\cos(\phi-\varphi)}\sin\theta\,\mathrm{d}\phi\,\mathrm{d}\theta$

$\displaystyle\mathbf{E_{ref}}=$
$\displaystyle[t^{\mathrm{s}}[\mathbf{E_{in}}\cdot\mathbf{n_{\phi}}]\mathbf{n_{\phi}}+t^{\mathrm{p}}[\mathbf{E_{in}}\cdot\mathbf{n_{\rho}}]\mathbf{n_{\theta}}]\sqrt{\cos\theta}$
(1b)

Figure 2:  (color online) Schematic of the focusing configuration. The
incident field is denoted by $\mathbf{E_{in}}$ and arrow indicate the incident
polarization. Spherical lens surface is shown by the reference sphere arc with
focal length $f$. The electric field after refraction at the lens surface is
given by $\mathbf{E_{ref}}$. $\mathbf{E}=(E_{x},E_{y},E_{z})$ represent the
focal electric field components.

The geometric representation of the system is given in Fig. 2 and closely
follows ref. Novotny and Hecht (2006). The formulation takes into account both
the refraction of the paraxial beam at the spherical lens surface (reference
sphere) given by $\mathbf{E_{ref}}(\theta,\phi)$ and the non-paraxial effects
there after. $t^{\mathrm{s}}$ and $t^{\mathrm{p}}$ are the transmission
coefficients corresponding to $\mathbf{s}$ and $\mathbf{p}$ polarized electric
field components respectively. $\mathbf{n_{\rho}}$ and $\mathbf{n_{\phi}}$
represents the unit vectors of a cylindrical coordinate system whereas the
unit vectors of a spherical polar coordinate are given by
$\mathbf{n_{\theta}}$ and $\mathbf{n_{\phi}}$, origin of the coordinate system
being the focal point $(x,y,z)=(0,0,0)$, as shown in Fig. 2. $n_{1}$ and
$n_{2}$ represent the medium refractive index.

Using Eq. 1(a), the electric field profiles, $\textbf{E}=(E_{x},E_{y},E_{z})$,
at the focal plane of a 0.5 NA lens is calculated for linearly polarized ($x$
polarized) paraxial Gaussian(G) and Hermite-Gaussian beam of order $m=1$,
$n=0$ ($\mathrm{HG_{10}}$) at wavelength $\lambda=633$ nm, propagating along
$z$ axis. Here $m$ and $n$ represent number of nodal lines along $x$ and $y$
axis respectively. The intensity distribution corresponding to the $x$
polarized ($I_{x}=\varepsilon_{0}{|E_{x}|}^{2}$) and $y$ polarized
($I_{y}=\varepsilon_{0}{|E_{y}|}^{2}$) component of the total focal optical
field for Gaussian(G) and $\mathrm{HG_{10}}$ (HG) beams are given in Figs.
3(a) and 3(b) respectively. The intensity distribution values are normalized
with respect to maximum value of total intensity,
$I=\varepsilon_{0}{|E|}^{2}$. The paraxial polarization profile of the beams
are given in 3(a) insets. Close inspection of the focal intensity profile in
Fig. 3(b) reveal that compared to $I_{y}$ of Gaussian beam, $I_{y}$ of
$\mathrm{HG_{10}}$ beam is significantly higher, although their distributions
differ. The ratio of maximum value of $I_{y}$ of $\mathrm{HG_{10}}$ beam to
that of Gaussian beam turns out to be $\approx 2.97$.

Figure 3:  (color online) Theoretically calculated focal intensity profiles
corresponding to (a) $x$ polarized field component, (b) $y$ polarized field
component of the total optical field for Gaussian and $\mathrm{HG_{10}}$
beams. Insets in (a) shows the $x$ polarized paraxial beam profiles. (c)
represents longitudinal spin density ($s_{z}$) at the focus for Gaussian and
$\mathrm{HG_{10}}$ beams. (d) Comparative plot of longitudinal spin density
distribution ($S_{z}$) for Gaussian (dashed blue line) and $\mathrm{HG_{10}}$
beam (solid red line).

###  II.2 Spin density of optical fields

One of the unique features that emerge due to focusing of linearly polarized
paraxial beam is presence of spinning electric fields. This can be quantified
using spin density Bliokh and Nori (2015), given by,

$\mathbf{s}\propto\textrm{Im}\left[\varepsilon_{0}(\mathbf{E^{*}}\times\mathbf{E})+\mu_{0}(\mathbf{H^{*}}\times\mathbf{H})\right].$
(2)

The symmetric nature of Eq. (2) allows us to consider the contribution of
either the electric field (E) part or the magnetic field (H) part. Since a
plasmonic scatterer responds more towards the electric field part of the
incident field, we consider the electric field contributions only. The
spinning fields represented by Eq. (2) can be decomposed into three components
which can be either longitudinal or transverse with respect to the propagation
axis ($z$ axis) Aiello and Banzer (2016). The longitudinally spinning optical
fields, given by:

$s_{z}\propto 2\,\textrm{Im}[E_{x}E_{y}^{*}]\propto s_{3}.$ (3)

can be quantified in the far-field through measuring the Stokes parameter
$s_{3}$, which determines the degree of circular polarization of a planar
optical field Setälä  _et al._ (2002).

Theoretically calculated distribution of $s_{z}$ at the focus of $x$ polarized
Gaussian and $\mathrm{HG_{10}}$ beams are given in Fig. 3(c). For simplicity,
we have ignored the proportionality constants in the equations. The values
have been normalized with respect to maximum of $I$. A quantitative measure of
the $s_{z}$ at the focus can be obtained by computing longitudinal spin
density distribution ($S_{z}$) of one half of the focal plane, i.e. by
integrating over the one of the spatial coordinates $y=0$ to $3\lambda$:

$S_{z}(x)=\int\displaylimits_{y=0}^{3\lambda}s_{z}(x,y)dy.$ (4)

Comparing the $s_{z}$ in Fig. 3(c) for Gaussian and $\mathrm{HG_{10}}$ beams,
it is evident that the $x$ polarized $\mathrm{HG_{10}}$ beam possesses
significantly higher $s_{z}$ with respect to that of $x$ polarized Gaussian
beam. The same is reflected in Fig. 3(d), which show the comparative magnitude
of $S_{z}$ for $x$ polarized Gaussian (dashed blue line) and
$\mathrm{HG_{10}}$ beams (solid red line) as a function of $x$ coordinates at
the focal plane. The enhancement of $s_{z}$ for $\mathrm{HG_{10}}$ beam with
respect to that of Gaussian beam can be obtained by calculating the ratio of
extremum value of $S_{z}$, which in this case is $\eta_{theory}=1.29$. The
enhancement factor, $\eta_{theory}$ is dependent on the NA of the objective
lens.

##  III Experimental Implementation

The experimental implementation in observing the SHEL as well as measuring the
longitudinal spin density relies on elastic scattering of the focal optical
fields. Various probe geometries can be used for studying such light matter
interactions such as, spherical particles Neugebauer  _et al._ (2015, 2018);
Yang  _et al._ (2018); Singh  _et al._ (2018), gold chiral geometry Woźniak
_et al._ (2019). We use a pentagonal-cross sectional silver nanowire (AgNW)
for our study. Chemically synthesized single crystalline AgNWs of diameter
$\sim 350$ nm were drop casted onto a glass substrate Sun  _et al._ (2003).
Fig. 4(a) shows an optical image of the AgNW used for our experiments. Fig.
4(b) shows scanning electron micrograph of a nanowire section. The AgNW is
illuminated by Gaussian beam and $\mathrm{HG_{10}}$ beam, prepared by
projecting Gaussian beam onto a spatial light modulator with blazed hologram,
at wavelength $\lambda=632.8$ nm. The paraxial (Gaussian/$\mathrm{HG_{10}}$)
beam, with polarization parallel to the long axis of the AgNW ($x$ axis), is
focused at the center of the AgNW placed at ($x=0$, $y=0$), with a $0.5$ NA
objective lens.

Figure 4:  (color online) (a) Optical bright-field image of the AgNW used for
our experiments. (b) Magnified FESEM image of top view of an AgNW section. The
schematic of experimental setup for the elastic forward scattering is given in
(c).

The forward scattered light is collected using an oil immersion objective lens
having 1.49 NA and then analyzed using a combination of linear polarizer and
quarter wave plate. The schematic of forward scattering setup is depicted in
Fig. 4(c). The scattered light in the Fourier plane (FP) is captured by
relaying the back-focal plane of the collection objective lens onto a CCD
Kurvits  _et al._ (2015); Vasista  _et al._ (2019). Detailed experimental
setup used for the study is similar to our previous reports and is given in
Appendix C Sharma  _et al._ (2018, 2019). The use of low NA (0.5 NA)
excitation and high NA (1.49 NA) collection objective lens facilitates the
collection of scattered light in higher angles in the FP. The un-scattered
light, which dominates the incident NA part of the FP intensity distribution,
is omitted for analysis.

##  IV Results and Discussion

###  IV.1 Nanowire as a scatterer

An AgNW placed at the focal plane of an objective lens mimics the effect of a
nanoscopic strip diffraction Bekshaev (2017); Bekshaev  _et al._ (2020).
Scalar diffraction theory and Babinet’s principle can be applied to obtain the
diffraction pattern in case of strip diffraction of paraxial beams. But due to
the complex and vectorial nature of the non-paraxial optical fields near to
the focal plane, full scattering theory Born  _et al._ (1999) has to be
considered for nanowire diffraction. To this end, the scattered optical field
intensity distribution in FP from AgNW has been previously used for probing
the SOI with the incident optical fields Sharma  _et al._ (2018, 2019).

The scattered optical field from the AgNW can be collected in two distinct
regions in the far-field, depending on its origin. These are sub-critical or
allowed region ($\textrm{NA}<1$) and super-critical or forbidden region
($\textrm{NA}>1$) in FP. The longitudinally polarized field ($E_{x}$)
component of the total optical field gets maximally scattered in the sub-
critical region. The transversely polarized optical field ($E_{y}$) component
leads to near-field (NF) accumulation at the AgNW edges Song  _et al._ (2011).
The evanescent NF gets partially converted into propagating waves at the air-
glass interface and can be observed in the super-critical region of the far-
field Novotny and Hecht (2006). Hence, elastic scattering from an AgNW allows
us to route a portion of the scattered light either in the sub-critical region
or in the super-critical region by engineering the incident linear
polarization state as longitudinal or transverse to the wire. To this end, in
our previous report, the resultant SOI effects due to forward scattering of
incident circularly polarized light beams in similar experimental
configuration have been reported to be prominent within the sub-critical
region of FP Sharma  _et al._ (2018). Hence, by analyzing of the FP intensity
distribution in the sub-critical region due to scattering of longitudinally
polarized paraxial beams, we can probe the resultant SOI effects as well as
extract the characteristics of the focal optical field.

Figure 5:  (color online) Experimentally measured far-field intensity patterns
for (a) Gaussian beam, (b) $\textrm{HG}_{10}$ beam. $I$ represents total
intensity, $I_{+}$ and $I_{-}$ represents LCP and RCP analyzed FP intensity
distribution respectively. Inner and outer circles in (a) represent the
critical angle at air-glass interface and the collection limit of the
objective lens NA respectively. White arrows in $I_{+}$ and $I_{-}$ indicate
the bias of the intensity lobe(s). Black disk at the center of the FP indicate
the blocked incident NA region. Far-field longitudinal spin density ($s_{3}$)
patterns are shown in (c) Gaussian beam and (d) $\textrm{HG}_{10}$ beam.
Numerically simulated far-field intensities $I$, $I_{+}$, $I_{-}$ for Gaussian
and $\textrm{HG}_{10}$ beams are given in (e) and (f) respectively. (g) and
(h) represent numerically calculated $s_{3}$ for Gaussian and
$\textrm{HG}_{10}$ beams respectively. The inner dashed white circle in (e)
and dashed black circle in (g) represent the incident NA region equivalent to
experimental counterpart.

###  IV.2 Experimentally measured scattering intensity

First, we examine the elastic scattering of optical fields due to focusing of
longitudinally polarized paraxial Gaussian and $\textrm{HG}_{10}$ beams from
an AgNW. Experimentally measured FP intensities of forward scattered Gaussian
and $\textrm{HG}_{10}$ beams are shown in Figs. 5(a) and 5(b) respectively.
The inner and outer white circles in Fig. 5(a) indicate the critical angle at
air-glass interface and collection limit of the objective lens respectively.
$I$ represents the un-analyzed scattered FP intensity distribution. $I_{+}$
and $I_{-}$ denotes the left circular polarization (LCP) and right circular
polarization (RCP) analyzed FP intensity distribution respectively. Absence of
light in the super-critical region ($\textrm{NA}=k_{x}/k_{0}=k_{y}/k_{0}>1$)
in the measured FP intensity distribution indicates the minimal accumulation
of NF at the NW edges. The incident NA is blocked and is shown by a black disk
at the center.

The FP intensity distribution corresponding to $I$ of Gaussian beam in Fig.
5(a) has only one lobe where as the scattering intensity pattern due to
$\textrm{HG}_{10}$ beam in Fig. 5(b) has two intensity lobes due to the
presence of nodal line in its incident beam intensity profile (see Fig. 3(a)).
Both the intensity patterns corresponding to $I$ in Figs. 5(a) and 5(b) show
symmetric pattern with respect to $k_{x}/k_{0}$ axis (parallel to AgNW long
axis). In contrast, the FP intensity distribution corresponding to $I_{+}$ and
$I_{-}$ of Gaussian beam exhibit shift along $k_{x}/k_{0}<0$ and
$k_{x}/k_{0}>0$ respectively for $k_{y}/k_{0}>0$ region (vice versa for
$k_{y}/k_{0}<0$ region), revealing anti-symmetric scattering pattern, as
depicted by the white arrows in Fig. 5(a) (see Appendix B). For
$\textrm{HG}_{10}$ beam, the analysed $I_{+}$ and $I_{-}$ component exhibit
shift along $k_{y}/k_{0}<0$ and $k_{y}/k_{0}>0$ direction for $k_{x}/k_{0}>0$
region respectively (and vice versa for $k_{x}/k_{0}<0$ region), as depicted
by the white arrows in Fig. 5(b). The opposite shifts of the intensity
distribution of $I_{+}$ with respect to that of $I_{-}$ indicates the presence
of partial circular polarization in the optical fields Neugebauer  _et al._
(2019b). This spin dependent opposite angular shift of the lobe(s) in the FP
corresponding to $I_{+}$ and $I_{-}$ is analogous to SHEL Onoda  _et al._
(2004); Neugebauer  _et al._ (2019b); Sharma  _et al._ (2018). Thus, the
magnitude of this spin-Hall signal and the consequent presence of the spin in
the FP can be obtained by calculating far-field longitudinal spin density
($s_{3}$) as, $s_{3}\propto I_{+}-I_{-}$. Experimentally measured $s_{3}$ for
forward scattered Gaussian and $\textrm{HG}_{10}$ beams are shown in Figs.
5(c) and 5(d) respectively. The experimentally measured intensity values are
normalized with respect to the maximum value of $I$.

###  IV.3 Numerically simulated scattering intensity

The experimental results are corroborated by full wave 3-dimensional finite
element method (FEM) simulation. A geometry having pentagonal cross-section
with diameter $350$ nm and length $5$ $\mathrm{\mu m}$ is used for modelling
of the NW. Material of the NW is mimicked by matching the refractive index
with that of Ag at $633$ nm wavelength Johnson and Christy (1972). Meshing of
the system is done by free tetrahedral geometry of size $35$ nm. The NW is
illuminated with excitation beams (Gaussian, $\textrm{HG}_{10}$) at
$\lambda=633$ nm. The beam waist at the air-glass interface is fixed to $633$
nm to mimic similar focusing conditions as in our experiments. Reciprocity
arguments are used for near field to far-field transformation of the scattered
optical field Yang  _et al._ (2016).

Numerically simulated FP intensity distribution of forward scattered light for
Gaussian and $\textrm{HG}_{10}$ beams are shown in Figs. 5(e) and 5(f)
respectively. $I$ represents the total scattered intensity. $I_{+}$ and
$I_{-}$ represent the LCP and RCP analyzed FP intensities respectively.
Similar to Fig. 5(a), the numerically calculated intensity distribution for
$I_{+}$ and $I_{-}$ of Gaussian beam in Fig. 5(e) have opposite shifts of the
intensity lobe(s) along $k_{x}/k_{0}$ axis, hence exhibiting anti-symmetric
scattering pattern, as shown by the white arrows (see Appendix B). The
numerically simulated FP intensity distribution corresponding to $I_{+}$ and
$I_{-}$ of $\textrm{HG}_{10}$ beam in Fig. 5(f), similar to its experimental
counterpart in Fig. 5(b), exhibit shift along $k_{y}/k_{0}<0$ and
$k_{y}/k_{0}>0$ direction for $k_{x}/k_{0}>0$ region respectively (and vice
versa for $k_{x}/k_{0}<0$ region), indicated by the white arrows. The opposite
shifts for $I_{+}$ and $I_{-}$ components are analogous to angular spin-Hall
shift. Corresponding to this angular shift, the $I_{+}$ and $I_{-}$ components
of the the scattered field at the $x$-$y$ plane ($z=0$ plane, air-glass
interface plane) also exhibit opposite positional shift for Gaussian and
$\textrm{HG}_{10}$ beams (see Appendix A). Finally, the spin-Hall signal can
be is obtained by computing the far-field longitudinal spin density ($s_{3}$)
as, $s_{3}\propto I_{+}-I_{-}$. Numerically simulated $s_{3}$ for Gaussian and
$\textrm{HG}_{10}$ beams are given in Figs. 5(g) and 5(h) respectively. The
intensity values are normalized with respect to the maximum value of $I$. The
simulated FP intensity patterns are in good agreement with their experimental
counterparts.

###  IV.4 Enhancement of SHEL and longitudinal spin density

As discussed in the previous sections, the spin-Hall signal for $I_{-}$ and
$I_{+}$ can be obtained through $s_{3}$. The enhancement of the spin-Hall
signal can be further quantified by comparing the $s_{3}$ distribution for
Gaussian and $\textrm{HG}_{10}$ beams. Since the wire is centered at ($x=0$,
$y=0$) with its long axis is aligned along $x$ axis and the scattering is
prominent in $k_{y}/k_{0}$ direction, we consider the
$s_{3}(\kappa_{x},\kappa_{y})$ distribution along $\kappa_{x}$, where
$\kappa_{x}=k_{x}/k_{0}$ and $\kappa_{y}=k_{y}/k_{0}$. Due to the symmetrical
pattern, we consider only one half of the FP i.e., $\kappa_{y}\geq 0$ half. In
addition, since the scattering is dominant within the critical angle, we
consider the sub-critical region. Thus, similar to Eq. (4), quantitative
measure of far-field longitudinal spin density distribution ($S_{3}$) can be
obtained by integrating $s_{3}$ over $\kappa_{y}$ from $0$ to $1$:

$S_{3}(\kappa_{x})=\int\displaylimits_{\kappa_{y}=0}^{1}s_{3}(\kappa_{x},\kappa_{y})d\kappa_{y}.$
(5)

Figure 6:  (color online) Comparative plot of the far-field longitudinal spin
density distribution ($S_{3}(\kappa_{x})$) of Gaussian (dashed blue line) and
$\textrm{HG}_{10}$ beam (solid red line) extracted from (a) experimentally
measured and (b) numerically simulated FP intensity distributions.

The far-field longitudinal spin density distribution, $S_{3}(\kappa_{x})$,
extracted from the experimentally measured $s_{3}$ in Figs. 5(c) and 5(d) is
plotted in Fig. 6(a). Similar analysis is also performed for the numerically
simulated $s_{3}$ patterns given in Figs. 5(g) and 5(h). Fig. 6(b) shows the
corresponding plot of $S_{3}(\kappa_{x})$. Both the $S_{3}(\kappa_{x})$
extracted from the experimentally measured and numerically calculated $s_{3}$
exhibit higher longitudinal spin density values and hence higher spin-Hall
signal for $\textrm{HG}_{10}$ beam compared to Gaussian beam. The enhancement
factor in the spin-Hall signal as well as the longitudinal spin density can be
obtained by calculating the ratio of extremum (maximum or minimum) values of
$S_{3}(\kappa_{x})$ of $\textrm{HG}_{10}$ to that of Gaussian beam. From Fig.
6(a) the average enhancement factor value turns out to be, $\eta_{exp}=1.31$.
While from the numerically simulated data in Fig. 6(b), the calculated value
of enhancement factor is, $\eta_{sim}=2.39$. Since focal $s_{z}\propto s_{3}$,
the enhancement can be attributed to higher $s_{z}$ of $\textrm{HG}_{10}$ beam
with respect to that of Gaussian beam (From Figs. 3(c) and 3(d):
$\eta_{theory}=1.29$). The difference between the measured $s_{3}$ enhancement
values $\eta_{exp}$ and $\eta_{sim}$ can be attributed to omission of incident
NA region in the experimental data as well as subtle difference of focusing in
the numerical simulations with the experimental focusing conditions.

Hence, our experiments demonstrate SHEL in scattering of linearly polarized
focused beam as well as its enhancement for $\textrm{HG}_{10}$ beam compared
to Gaussian beam. The enhancement factor is quantified by obtaining the ratio
extremum values of spin density distribution $S_{3}(\kappa_{x})$. The complex
nature of the optical field at the focus of an objective lens allows
generation of longitudinally spinning electric fields which results in such
effect.

##  V Conclusion

In summary, we demonstrate experimental observation of SHEL by analyzing the
forward elastic scattered light of focused linearly polarized Gaussian and
$\textrm{HG}_{10}$ beams from a single crystalline nanowire. The measure of
the spin-Hall signal is obtained by computing far-field longitudinal spin
density ($s_{3}$). Furthermore, by comparing the spin-density distribution
($S_{3}$) we infer enhancement of spin-Hall signal for $\textrm{HG}_{10}$ beam
with respect to Gaussian beam, quantified by the factor $\eta_{exp}=1.31$ from
experimental measurements and $\eta_{sim}=2.39$ from numerically simulated
results. By studying the focal optical fields, we attribute the observed
effect to the generation of longitudinally spinning optical fields due to
focusing, hinting the geometrical origin of the effect. Our experiments reveal
the very intricate nature of the SOI between a focused optical beam and a
nanoscopic object Bliokh  _et al._ (2015). In recent times spin-Hall effect of
light has gained a lot of traction in research due to great potential for spin
assisted photonic devices Ling  _et al._ (2017); Lodahl  _et al._ (2017) as
well as fundamental study of optical energy flow Bekshaev  _et al._ (2011);
Aiello  _et al._ (2009). Additionally, the emerging SOI with a single
crystalline AgNW can also pave way for on-chip photonic device applications.

###### Acknowledgements.

This work was partially funded by Air Force Research Laboratory grant
(FA2386-18-1-4118 R&D 18IOA118), DST Energy Science grant (SR/NM/TP-13/2016)
and Swarnajayanti fellowship grant (DST/SJF/PSA-02/2017-18). DP and DKS
acknowledge Vandana Sharma, Chetna Taneja, Sunny Tiwari, Shailendra Kumar
Chaubey, Surya Narayan Banerjee and Utkarsh Khandelwal for fruitful
discussion.

##  Appendix A Scattered electric fields at the focal plane

Figure 7:  (color online) Numerically simulated scattered normalized electric
field intensity at the focal plane ($z=0$) corresponding to (a) Gaussian beam
and (b) $\textrm{HG}_{10}$ beam. $I$ represents total scattered electric
field, $I_{+}$ and $I_{-}$ indicates the LCP and RCP polarized scattered
electric fields respectively. The intensity distribution plot (green line) of
lobes in $y>0$ and $y<0$ half as a function of $x$ coordinates is super
imposed on the figures.

Numerically simulated scattered electric field from an AgNW at focal plane
($z=0$ plane) for Gaussian and $\textrm{HG}_{10}$ beams are shown in Fig. 7.
The NW positioned at ($x=0$, $y=0$) and along $x$ axis is indicated by the
yellow dashed lines. $I$ represents total scattered intensity, $I_{+}$ and
$I_{-}$ represents the LCP and RCP analyzed components respectively. The
intensity distribution plot of the lobe(s) as a function of $x$ coordinates
(green lines) in both $y>0$ and $y<0$ halves are superimposed in the Fig. 7.

The scattered intensity profile $I$ reveal that the intensity lobe(s) gets
split into two equal halves due to the scattering from the NW. Although there
is very less NF accumulation, but with respect to Gaussian, the NF
accumulation for $\textrm{HG}_{10}$ beam is more due to higher $E_{y}$
component (see Fig. 3(b)). Close inspection of intensity profiles
corresponding to $I_{+}$ and $I_{-}$ exhibit that with respect to symmetric
intensity profile of $I$, the intensity lobe(s) in $y>0$ and $y<0$ regions of
$I_{+}$ and $I_{-}$ show opposite shifts, as indicated by the white arrows.
The spatial shift ($\delta$) along $x$ axis can be quantified by calculating
the shift of $I_{+}$ (or $I_{-}$) distribution maxima of each of the lobe(s)
in $y>0$ half with respect to the corresponding lobe(s) in $y<0$ half. For
scattered Gaussian beam the average shift is $\delta_{G}=64.0$ nm and that for
$\textrm{HG}_{10}$ beam is $\delta_{HG}=92.2$ nm. The enhancement of the
positional shift can be obtained as the ratio $\delta_{HG}/\delta_{G}=1.44$.
The average shift calculated here are discussed in more general terms as shift
vectors in different systems Shi and Song (2019); Liu  _et al._ (2020). The
average shifts are consequence of breaking of symmetry in the $x$-$y$ plane
due to the presence of an extended geometry such as a NW. This spin dependent
shift is analogous to SHEL Onoda  _et al._ (2004); Hosten and Kwiat (2008).

##  Appendix B Wave vector shift in the Fourier plane

Figure 8:  (color online) Intensity distribution plot of $I$ (solid black
line) and corresponding $I_{+}$ (solid red line) and $I_{-}$ (dashed blue
line) components in FP for integrated $\kappa_{y}=[0,1]$ along $\kappa_{x}$
for Gaussian beam from (a)experimentally measured data and (b) from
numerically simulated FP images. The corresponding distribution plot for
$\textrm{HG}_{10}$ beam from experimentally measured data and numerically
simulated data is given in (c) and (d) respectively.

The FP image of total intensity ($I$) for Gaussian and $\textrm{HG}_{10}$ beam
exhibit symmetrical characteristic about $k_{x}/k_{0}$ axis, in contrast to
anti-symmetrical distribution of $I_{+}$ and $I_{-}$. To probe the shift of
the intensity lobe(s) of $I_{+}(I_{-})$, we plot the distribution
$I_{+(-)}^{tot}(\kappa_{x})=\int\displaylimits_{\kappa_{y}=0}^{1}I_{+(-)}(\kappa_{x},\kappa_{y})d\kappa_{y}$
and compare it with the distribution
$I^{tot}(\kappa_{x})=\int\displaylimits_{\kappa_{y}=0}^{1}I(\kappa_{x},\kappa_{y})d\kappa_{y}$
for Gaussian and $\textrm{HG}_{10}$ beam in Fig. 8.

For Gaussian beam, we can see from Fig. 8(a) and Fig. 8(b) that the
$I^{tot}_{+}(\kappa_{x})$ (solid red line) and $I^{tot}_{-}(\kappa_{x})$
(dashed blue line) analysed distributions exhibit angular shift along
$\kappa_{x}<0$ and $\kappa_{x}>0$ direction respectively. The measured shift
from the experimental data in Fig. 8(a) is $\delta\kappa_{x,expt}^{G}=0.34$,
where as from the numerically simulated FP image the shift measured in Fig.
8(b) is $\delta\kappa_{x,sim}^{G}=0.05$.

For $\textrm{HG}_{10}$ beam, the shift along $\kappa_{x}$ is small compared to
Gaussian beam as can be seen in Figs. 8(c) and 8(d). The shift along
$\kappa_{x}$, obtained from experimental data in Fig. 8(c) is
$\delta\kappa_{x,expt}^{HG}=0.01$ and that obtained from numerically simulated
data in Fig. 8(d) is $\delta\kappa_{x,sim}^{HG}=0.01$. For $\textrm{HG}_{10}$
beam, the shift is more along $\kappa_{y}$ as can be determined from examining
the FP intensity distribution corresponding to $I_{+}$ and $I_{-}$ in Fig.
5(f) of manuscript. The presence of nodal line of $\textrm{HG}_{10}$ beam
restricts angular shift along $\kappa_{x}$ axis Sharma  _et al._ (2018).

The shift obtained from the experimentally data and numerically simulated data
depict a qualitative comparison of the angular shift between Gaussian and
$\textrm{HG}_{10}$ beams. The values differ due to difference in exact
focusing conditions as well as due to omission of excitation NA region for
experimentally measured FP intensity distribution.

##  Appendix C Detailed schematic of the experimental setup

Figure 9:  (color online) Detailed schematic of the experimental setup.

Detailed schematic of our forward scattering experimental setup is given in
Fig. 9. M1-M6 are mirrors, L1-L4 are lenses, HWP: Half-wave plate, QWP:
Quarter wave plate, P: Polarizer, A: Analyser, BS1: Beam-splitter.
Illumination objective lens ($50\times$, $0.5$ NA) focuses incident
Gaussian/$\textrm{HG}_{10}$ beam produced through spatial light modulator
(SLM). The scattered light by the sample placed on a piezo-stage is collected
using a $100\times$, $1.49$ NA objective lens and back focal plane is imaged
at the CCD by projecting it through lenses L2, L3. The collected light is
analysed using quarter wave plate and an analyser. A pin hole is placed at the
image plane to select the excitation part of the wire and hence to avoid
scattering from the other part of the sample. A Flip lens L4 is used to get
the real plane image of the sample.

## References

  * Allen  _et al._ (1992) L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
  * Franke-Arnold  _et al._ (2008) S. Franke-Arnold, L. Allen, and M. Padgett, Laser & Photonics Reviews 2, 299 (2008).
  * Bekshaev  _et al._ (2011) A. Bekshaev, K. Y. Bliokh, and M. Soskin, Journal of Optics 13, 053001 (2011).
  * Yao and Padgett (2011) A. M. Yao and M. J. Padgett, Adv. Opt. Photon. 3, 161 (2011).
  * Molina-Terriza  _et al._ (2007) G. Molina-Terriza, J. P. Torres, and L. Torner, Nature Physics 3, 305 (2007).
  * Liberman and Zel’dovich (1992) V. S. Liberman and B. Y. Zel’dovich, Phys. Rev. A 46, 5199 (1992).
  * Bliokh  _et al._ (2015) K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, Nature Photonics 9, 796 (2015).
  * Aiello  _et al._ (2015) A. Aiello, P. Banzer, M. Neugebauer, and G. Leuchs, Nature Photonics 9, 789 (2015).
  * Onoda  _et al._ (2004) M. Onoda, S. Murakami, and N. Nagaosa, Phys. Rev. Lett. 93, 083901 (2004).
  * Hirsch (1999) J. E. Hirsch, Phys. Rev. Lett. 83, 1834 (1999).
  * Hosten and Kwiat (2008) O. Hosten and P. Kwiat, Science 319, 787 (2008).
  * Bliokh and Aiello (2013) K. Y. Bliokh and A. Aiello, Journal of Optics 15, 014001 (2013).
  * Bekshaev (2012) A. Y. Bekshaev, Phys. Rev. A 85, 023842 (2012).
  * Goswami  _et al._ (2014) S. Goswami, M. Pal, A. Nandi, P. K. Panigrahi, and N. Ghosh, Opt. Lett. 39, 6229 (2014).
  * Haefner  _et al._ (2009) D. Haefner, S. Sukhov, and A. Dogariu, Phys. Rev. Lett. 102, 123903 (2009).
  * Sharma  _et al._ (2018) D. K. Sharma, V. Kumar, A. B. Vasista, S. K. Chaubey, and G. V. P. Kumar, Opt. Lett. 43, 2474 (2018).
  * Bliokh  _et al._ (2010) K. Y. Bliokh, M. A. Alonso, E. A. Ostrovskaya, and A. Aiello, Phys. Rev. A 82, 063825 (2010).
  * Bomzon  _et al._ (2006) Z. Bomzon, M. Gu, and J. Shamir, Applied Physics Letters 89, 241104 (2006).
  * Nieminen  _et al._ (2008) T. A. Nieminen, A. B. Stilgoe, N. R. Heckenberg, and H. Rubinsztein-Dunlop, Journal of Optics A: Pure and Applied Optics 10, 115005 (2008).
  * Zhao  _et al._ (2009) Y. Zhao, D. Shapiro, D. Mcgloin, D. T. Chiu, and S. Marchesini, Opt. Express 17, 23316 (2009).
  * Yan and Scherer (2013) Z. Yan and N. F. Scherer, The Journal of Physical Chemistry Letters 4, 2937 (2013).
  * Rodríguez-Herrera  _et al._ (2010) O. G. Rodríguez-Herrera, D. Lara, K. Y. Bliokh, E. A. Ostrovskaya, and C. Dainty, Phys. Rev. Lett. 104, 253601 (2010).
  * Neugebauer  _et al._ (2019a) M. Neugebauer, S. Nechayev, M. Vorndran, G. Leuchs, and P. Banzer, Nano Letters 19, 422 (2019a).
  * Zhao  _et al._ (2007) Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, Phys. Rev. Lett. 99, 073901 (2007).
  * Nechayev  _et al._ (2019) S. Nechayev, J. S. Eismann, G. Leuchs, and P. Banzer, Phys. Rev. B 99, 075155 (2019).
  * Mueller and Capasso (2013) J. P. B. Mueller and F. Capasso, Phys. Rev. B 88, 121410 (2013).
  * O’Connor  _et al._ (2014) D. O’Connor, P. Ginzburg, F. J. Rodríguez-Fortuño, G. A. Wurtz, and A. V. Zayats, Nature Communications 5, 5327 (2014).
  * Cardano and Marrucci (2015) F. Cardano and L. Marrucci, Nature Photonics 9, 776 (2015).
  * Zhou  _et al._ (2018) X. Zhou, L. Sheng, and X. Ling, Scientific Reports 8, 1221 (2018).
  * Wolf and Gabor (1959) E. Wolf and D. Gabor, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 253, 349 (1959).
  * Richards  _et al._ (1959) B. Richards, E. Wolf, and D. Gabor, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 253, 358 (1959).
  * Novotny and Hecht (2006) L. Novotny and B. Hecht,  _Principles of Nano-Optics_, 1st ed. (Cambridge University Press, 2006).
  * Bliokh and Nori (2015) K. Y. Bliokh and F. Nori, Physics Reports 592, 1 (2015), transverse and longitudinal angular momenta of light.
  * Aiello and Banzer (2016) A. Aiello and P. Banzer, Journal of Optics 18, 085605 (2016).
  * Setälä  _et al._ (2002) T. Setälä, A. Shevchenko, M. Kaivola, and A. T. Friberg, Phys. Rev. E 66, 016615 (2002).
  * Neugebauer  _et al._ (2015) M. Neugebauer, T. Bauer, A. Aiello, and P. Banzer, Phys. Rev. Lett. 114, 063901 (2015).
  * Neugebauer  _et al._ (2018) M. Neugebauer, J. S. Eismann, T. Bauer, and P. Banzer, Phys. Rev. X 8, 021042 (2018).
  * Yang  _et al._ (2018) A. P. Yang, L. P. Du, F. F. Meng, and X. C. Yuan, Nanoscale 10, 9286 (2018).
  * Singh  _et al._ (2018) A. K. Singh, S. Saha, S. D. Gupta, and N. Ghosh, Phys. Rev. A 97, 043823 (2018).
  * Woźniak  _et al._ (2019) P. Woźniak, I. D. Leon, K. Höflich, G. Leuchs, and P. Banzer, Optica 6, 961 (2019).
  * Sun  _et al._ (2003) Y. Sun, B. Mayers, T. Herricks, and Y. Xia, Nano Letters 3, 955 (2003).
  * Kurvits  _et al._ (2015) J. A. Kurvits, M. Jiang, and R. Zia, J. Opt. Soc. Am. A 32, 2082 (2015).
  * Vasista  _et al._ (2019) A. B. Vasista, D. K. Sharma, and G. P. Kumar, “Fourier plane optical microscopy and spectroscopy,” in  _digital Encyclopedia of Applied Physics_ (American Cancer Society, 2019) pp. 1–14.
  * Sharma  _et al._ (2019) D. K. Sharma, V. Kumar, A. B. Vasista, D. Paul, S. K. Chaubey, and G. V. P. Kumar, ACS Photonics 6, 148 (2019).
  * Bekshaev (2017) A. Y. Bekshaev, Journal of Optics 19, 085602 (2017).
  * Bekshaev  _et al._ (2020) A. Bekshaev, L. Mikhaylovskaya, S. Patil, V. Kumar, and R. P. Singh, J. Opt. Soc. Am. A 37, 780 (2020).
  * Born  _et al._ (1999) M. Born, E. Wolf, A. B. Bhatia, P. C. Clemmow, D. Gabor, A. R. Stokes, A. M. Taylor, P. A. Wayman, and W. L. Wilcock,  _Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light_, 7th ed. (Cambridge University Press, 1999).
  * Song  _et al._ (2011) M. Song, A. Bouhelier, P. Bramant, J. Sharma, E. Dujardin, D. Zhang, and G. Colas-des Francs, ACS Nano 5, 5874 (2011), pMID: 21682318.
  * Neugebauer  _et al._ (2019b) M. Neugebauer, P. Banzer, and S. Nechayev, Science Advances 5 (2019b), 10.1126/sciadv.aav7588.
  * Johnson and Christy (1972) P. B. Johnson and R. W. Christy, Phys. Rev. B 6, 4370 (1972).
  * Yang  _et al._ (2016) J. Yang, J.-P. Hugonin, and P. Lalanne, ACS Photonics 3, 395 (2016).
  * Ling  _et al._ (2017) X. Ling, X. Zhou, K. Huang, Y. Liu, C.-W. Qiu, H. Luo, and S. Wen, Reports on Progress in Physics 80, 066401 (2017).
  * Lodahl  _et al._ (2017) P. Lodahl, S. Mahmoodian, S. Stobbe, A. Rauschenbeutel, P. Schneeweiss, J. Volz, H. Pichler, and P. Zoller, Nature 541, 473 (2017).
  * Aiello  _et al._ (2009) A. Aiello, N. Lindlein, C. Marquardt, and G. Leuchs, Phys. Rev. Lett. 103, 100401 (2009).
  * Shi and Song (2019) L.-k. Shi and J. C. W. Song, Phys. Rev. B 100, 201405 (2019).
  * Liu  _et al._ (2020) Y. Liu, Z.-M. Yu, C. Xiao, and S. A. Yang, Phys. Rev. Lett. 125, 076801 (2020).