# Orientation Averaging of Optical Chirality Near Nanoparticles and Aggregates

Atefeh Fazel-Najafabadi<EMAIL_ADDRESS>School of Chemical
and Physical Sciences, Victoria University of Wellington, PO Box 600,
Wellington 6140, New Zealand  The MacDiarmid Institute for Advanced Materials
and Nanotechnology  Sebastian Schuster<EMAIL_ADDRESS>School of Chemical and Physical Sciences, Victoria University of Wellington,
PO Box 600, Wellington 6140, New Zealand  The MacDiarmid Institute for
Advanced Materials and Nanotechnology  School of Mathematics and Statistics,
Victoria University of Wellington, PO Box 600, Wellington 6140, New Zealand
Baptiste Auguié<EMAIL_ADDRESS>School of Chemical and Physical
Sciences, Victoria University of Wellington, PO Box 600, Wellington 6140, New
Zealand  The MacDiarmid Institute for Advanced Materials and Nanotechnology

(27 August 2024)

###### Abstract

Artificial nanostructures enable fine control of electromagnetic fields at the
nanoscale, a possibility that has recently been extended to the interaction
between polarised light and chiral matter. The theoretical description of such
interactions, and its application to the design of optimised structures for
chiroptical spectroscopies, brings new challenges to the common set of tools
used in nano-optics. In particular, chiroptical effects often depend crucially
on the relative orientation of the scatterer and the incident light, but many
experiments are performed with randomly-oriented scatterers, dispersed in a
solution. We derive new expressions for the orientation-averaged local degree
of optical chirality of the electromagnetic field in the presence of a
nanoparticle aggregate. This is achieved using the superposition _T_ -matrix
framework, ideally suited for the derivation of efficient orientation-
averaging formulas in light scattering problems. Our results are applied to a
few model examples, and illustrate several non-intuitive aspects in the
distribution of orientation-averaged degree of chirality around
nanostructures. The results will be of significant interest for the study of
nanoparticle assemblies designed to enhance chiroptical spectroscopies, and
where the numerically-efficient computation of the averaged degree of optical
chirality enables a more comprehensive exploration of the many possible
nanostructures.

##  I Introduction

Chiral materials naturally respond with a slight asymmetry to left or right
circularly-polarised light [1, 2], and the minute difference in spectroscopic
signals – in absorption, fluorescence, Raman scattering etc. – can be
exploited to trace back precious 3D structural information at the nanoscale,
such as the relative orientation of molecular groups (handedness). This is
particularly important in the realm of biochemistry [3, 4], where mirror-image
molecules (enantiomers) can differ dramatically in their interaction with
other chiral molecules. The ability to characterise the handedness of chiral
molecules with optical spectroscopy, in the form of circular dichroism or
related techniques, remains a challenging pursuit with important practical
applications [5]. The main limitation is that the chiroptical activity of most
molecules is very weak; most naturally-occurring chiroptical effects have thus
been restricted in their study and application to bulk samples such as highly-
concentrated chemical solutions.

Artificial chiral nanostructures have opened remarkable new perspectives in
this area [6], with demonstrations of enhanced chiroptical signals [7, 8]. The
rising interest in nano-optics in the past few decades has highlighted the
potential of nano-materials and nano-structures engineered to bridge the gap
between the wavelength of free propagating light and localised optical near-
fields exciting molecules. Acting as nano-antennas, metallic or dielectric
nanoparticles supporting resonant electromagnetic modes can funnel light into
subwavelength regions associated with greatly amplified electromagnetic fields
[9]. This effect has been a cornerstone in the development of surface-enhanced
Raman (SERS) or fluorescence spectroscopies [10]. Such nanostructures can
similarly enhance the chiroptical response of a chiral molecule [11, 12, 13,
14, 15, 16, 17]

Cohen and coworkers [18, 19] recognised that engineered nanostructures may
enhance the interaction between light and chiral matter not just through a
locally-enhanced magnitude of electric ($\mathbf{E}$) and magnetic
($\mathbf{B}$) fields, but also conjointly by shaping the local degree of
optical chirality $\mathscr{C}\propto\Im(\mathbf{E}^{*}\cdot\mathbf{B})$,
where $\Im$ stands for ‘imaginary part’ — a quantity requiring full
consideration of the vectorial nature of both fields and their complex phase
[20].

Specifically, let us consider the time-averaged rate of absorption $A$ of a
chiral molecule excited by local electric $\mathbf{E}$ and magnetic fields
$\mathbf{B}$ at a frequency $\omega$, and characterised by its electric dipole
moment $\mathbf{p}=\alpha\mathbf{E}-i\gamma\mathbf{B}$, and magnetic dipole
moment $\mathbf{m}=\beta\mathbf{B}+i\gamma\mathbf{E}$ [21],

$A\mathrel{\mathop{:}}=\frac{\omega}{2}\Im\left({\mathbf{E}}^{*}\cdot{\mathbf{p}}+{\mathbf{B}}^{*}\cdot{\mathbf{m}}\right),$
(1)

where $\alpha$ is the electric dipole polarisability, $\beta$ the magnetic
dipole polarisability, and $\gamma$ characterises the mixed electric-magnetic
dipole polarisability and is responsible for optical activity. Under the
common simplifying assumption of monochromatic fields and a homogeneous
embedding medium, this expression simplifies to [21]:

$A=\frac{\omega}{2}\left[\Im\left(\alpha\right)|{\mathbf{E}}|^{2}+\Im\left(\beta\right)|{\mathbf{B}}|^{2}+\Im\left(\gamma\right)\mathscr{C}\right],$
(2)

where

$\mathscr{C}=\frac{-\omega\epsilon_{0}}{2}\Im(\mathbf{E}^{*}\cdot\mathbf{B})$
(3)

is referred to as the local degree of optical chirality (LDOC), first
introduced by Lipkin [22].

While $\alpha,\beta,\gamma$ are intrinsic properties of the scattering
particle or molecule, $\mathscr{C}$ encapsulates the chirality of the
electromagnetic field at a given position in space [22]. The differential
absorbance of a molecule in response to a plane wave with left and right
circular polarisations will be proportional to
$\Im\left(\gamma\right)\mathscr{C}$, where
$\mathscr{C}=\pm\frac{k\varepsilon_{0}E_{0}}{2}$, with $k$ the wavenumber,
$\varepsilon_{0}$ the permittivity of free space, and $E_{0}$ the amplitude of
the incident electric field. By placing the molecule in the vicinity of a
nanostructure, the local electromagnetic field, and therefore $\mathscr{C}$,
can be amplified beyond this value [19, 20]. This offers the possibility to
engineer nanostructures that optimise the chiroptical response of analytes.
Such designs can be modelled by solving Maxwell’s equations for a given
nanostructure, calculating $\mathbf{E}$ and $\mathbf{B}$ which readily provide
$\mathscr{C}$; this procedure has been used to estimate the enhancement of
chiroptical signals for molecules in a variety of configurations [23, 21].

Although the design of useful structures benefits from extensive research and
advances in surface-enhanced spectroscopies and nano-optics in the past
decades, the consideration of chirality at the nano-scale, and its interaction
with polarised light, brings a host of new challenges to our intuition, and
our modelling capabilities. Where surface-enhanced spectroscopies such as SERS
rely primarily on the amplification of the electric field intensity in the
immediate vicinity of nanostructures, in pursuing enhanced optical activity
one must also consider the polarisation state of near-field regions, which
remains generally less intuitive, and for which we have fewer helpful rule-of-
thumbs to guide us in the design of efficient chiroptical platforms [24, 25].

An additional challenge presented by chiral nanostructures is the particular
importance of the relative orientation between incident light and the
scatterer. For example, a 2D structure fabricated by lithography may present a
differential response to circularly polarised light known as ‘extrinsic’
chirality, though the same structure would display no circular dichroism if
illuminated from all directions of incidence. This distinction is particularly
relevant as many 3D chiral nanostructures are characterised in solution, where
they are randomly-oriented with respect to incident light. To describe the
optical response of such samples it is essential to be able to reliably
compute the orientation-averaged response of the structure to incident light
of a given polarisation (Fig. 1). This can present a time-consuming hurdle for
theoretical calculations. In the context of optical activity, a comparison
with experiments may involve computing far-field orientation-averaged circular
dichroism [26], and for experiments testing the effect of superchiral fields
on molecular species, the orientation-averaged local degree of optical
chirality $\langle\mathscr{C}\rangle$; the two quantities are not necessarily
directly related to each other [27].

Figure 1: Schematic illustration of the light scattering problem under
consideration. A rigid cluster is formed by $N$ particles arranged in a fixed
spatial configuration; each particle may be non-spherical and oriented
arbitrarily with respect to the common ‘cluster’ reference frame $(O,x,y,z)$.
The orientation and position $\mathbf{r_{i}}$ of each particle ($i=1\dots N$)
is arbitrary but fixed in this reference frame; in practice such particles
would be held in place by a template [28, 29, 30], our calculations assume
that the template has no impact on the optical properties of the structure.
Incident light takes the form of a plane wave with wavevector $\mathbf{k}$
from an arbitrary direction $(\varphi,\theta)$ in the cluster reference frame,
and we consider both possible states of circular polarisation (left, $L$, or
right $R$). For each direction of incidence and either polarisation we can
calculate the local degree of chirality $\mathscr{C}$ in the vicinity of the
particles, and map its spatial distribution (schematically represented by
purple and green gradients). The quantity $\langle\mathscr{C}\rangle$ derived
in this work represents the average value for the chosen polarisation when the
cluster as a whole is randomly oriented with respect to incident light, or
equivalently, when $\mathbf{k}(\varphi,\theta)$ spans the full solid angle in
the cluster reference frame.

Many common tools used in nano-optics, such as the Finite-Element Method (FEM)
[31], Finite-Difference Time Domain (FDTD) [32], Discrete-Dipole Approximation
(DDA) [33], Volume or Surface Integral Equation methods (VIE/SIE) [34, 35,
36], solve the Maxwell equations for a given configuration of particles and a
given incident field. These computations can be demanding even for just a few
particles, and improving the methods is still an active area of research. This
computational burden limits the ability to explore the vast landscape of
possible structures and seek those that approach optimum bounds [37, 38]; it
also often leads to shortcuts where accuracy is traded for practical
computational time. For example, orientation-averaged properties may be
approximated by considering three orthogonal directions of incidence ($x$,
$y$, $z$ axes), though the validity of this approximation is not clear and has
received little attention to date. A rigorous result for the orientation-
averaging of observable quantities such as far-field cross-sections, near-
field intensity, or in our case here the LDOC $\langle\mathscr{C}\rangle$
considers the integral over Euler angles $\varphi,\theta$ describing the
direction of the wavevector incident on the scatterer (or collection of
scatterers) (Fig. 1),

$\langle\mathscr{C}(x,y,z)\rangle=\frac{1}{4\pi}\int_{0}^{\pi}\int_{0}^{2\pi}\mathscr{C}(x,y,z;\varphi,\theta)\sin\theta\mathop{}\\!\mathrm{d}\varphi\mathop{}\\!\mathrm{d}\theta.$
(4)

This averaging over the full solid angle of possible illumination directions
is often complemented by a further averaging over the direction of
polarisation for linearly-polarised light [39, 40, 41, 42]. In the case of
circular polarisation, however, the rotations of the wavevector do not affect
the helicity of the light and it is therefore useful to consider the angular
averaging over $(\varphi,\theta)$ for left ($L$) or right ($R$) polarisations
separately [43]. The fully averaged optical response to unpolarised light will
be the average of both.

In contrast to the methods previously mentioned, the _T_ -matrix framework
affords a relatively efficient solution to the Maxwell equations in a basis of
vector spherical waves [44]; intrinsic to the method is therefore the
description of a scatterer or collection of scatterers, to waves incident from
any direction. This is a particular strength of the method which has led to
analytical formulas for orientation-averaged quantities such as far-field
cross-sections [41, 45]. Such results were recently extended by Rockstuhl and
coworkers for optical activity [26], and by Stout and co-workers for near-
field quantities in clusters of spherical particles [46, 47]. We extend here
these previous results to the calculation of orientation-averaged near-field
optical chirality. The rest of this manuscript is divided as follows. First,
we summarise the essential formulas and our notations for the superposition
_T_ -matrix method, followed by our newly-derived formulas. In a second part,
we illustrate these results by applying the formulas to model clusters of
nanoparticles, and discuss the distribution and enhancement of near-field
optical chirality around various nanostructures. These results illustrate the
power of analytical results enabled by the _T_ -matrix method, which can
provide rapid, accurate, and physically-insightful simulations of a vast range
of nanostructures. We hope to encourage more researchers to adopt this method
in their comparisons to experiments, but also to further improve the method
beyond its current limitations [48, 49].

##  II _T_ -matrix formalism and main formulas

We follow standard notations [50, 10] for vector spherical waves and describe
the geometry in the usual spherical coordinates $(r,\varphi,\theta)$ (Fig. 1);
for the extension to $N$ scatterers we follow closely the scatterer-centred
superposition _T_ -matrix approach presented by Auger and Stout [51, 47]. This
section provides a brief summary of our choice of definitions for
completeness.

###  II.1 A single scatterer

Waterman’s _T_ -matrix method [52, 53] offers a rigorous solution of
Helmholtz’ wave equation by describing the light scattering process in terms
of an incident field and a scattered field, both expanded into bases of
regular and irregular Vector Spherical Wave Functions (VSWFs), respectively
[46],

$\displaystyle\mathbf{E}_{\text{inc}}(k\mathbf{r})$
$\displaystyle=\sum_{n=1}^{\infty}\sum_{m=-n}^{n}[$
$\displaystyle\widetilde{a}_{1,nm}\widetilde{\mathbf{M}}_{nm}\\!(k\mathbf{r})$
$\displaystyle+\widetilde{a}_{2,nm}\widetilde{\mathbf{N}}_{nm}\\!(k\mathbf{r})]$
(5) $\displaystyle\mathbf{E}_{\text{sca}}(k\mathbf{r})$
$\displaystyle=\sum_{n=1}^{\infty}\sum_{m=-n}^{n}[$
$\displaystyle{f}_{1,nm}{\mathbf{M}}_{nm}(k\mathbf{r})$
$\displaystyle+f_{2,nm}{\mathbf{N}}_{nm}(k\mathbf{r})]$ (6)

where $\widetilde{\mathbf{M}}_{nm}$ and $\widetilde{\mathbf{N}}_{nm}$ are
normalised vector spherical wave functions composed of spherical harmonics
($Y_{nm}(\varphi,\theta)$) and spherical Bessel ($j_{n}(k\mathbf{r})$
functions for the ingoing wave, while ${\mathbf{M}}_{nm}$ and
${\mathbf{N}}_{nm}$ are composed of spherical harmonics and spherical Hankel
functions ($h_{n}(k\mathbf{r})$) of the first kind for the scattered wave (See
Appendix (A) for their explicit definition, following Stout and coworkers [46,
51, 47]). $\widetilde{a}_{1,nm}$, $\widetilde{a}_{2,nm}$, $f_{1,nm}$, and
$f_{2,nm}$ are the associated expansion coefficients.

In a more compact form, we write the incident and scattered electric fields
as:

$\displaystyle\mathbf{E}_{\text{inc}}(k\mathbf{r})=\widetilde{\boldsymbol{\Psi}}(k\mathbf{r})\widetilde{\mathbf{a}}$
(7)
$\displaystyle\mathbf{E}_{\text{sca}}(k\mathbf{r})=\boldsymbol{\Psi}(k\mathbf{r}){\mathbf{f}}$
(8)

where ${\widetilde{\mathbf{a}},\mathbf{f}}$ are (infinite-dimensional) column
vectors of incident and scattered field coefficients.
$\widetilde{\boldsymbol{\Psi}}=[\widetilde{\mathbf{M}}\quad\widetilde{\mathbf{N}}],\boldsymbol{\Psi}=[\mathbf{M}\quad\mathbf{N}]$
are the corresponding, infinite-dimensional row vectors for 3D-vector-valued
basis functions. We note here that all the fields expressed in the spherical
basis can be easily converted to another coordinate system, such as the
cartesian reference frame $(O,x,y,z)$ in Fig. 1, simply with a suitable
conversion matrix (the conversion to cartesian coordinates is given in
Appendix (A) for reference); this transformation will be left implicit in some
formulas below to avoid cluttering already-lengthy formulas.

Given the linearity of the Helmholtz equation governing the scattering
problem, the expansion coefficients of the incident field,
$\widetilde{\mathbf{a}}$, and of the scattered field, ${\mathbf{f}}$, obey a
linear relationship:

$\mathbf{f}=\mathbf{T}\widetilde{\mathbf{a}},$ (9)

where $\mathbf{T}$ is the so-called ‘transition’ or ‘transfer’ matrix (_T_
-matrix for short). This quantity is an inherent property of the scatterer at
a given frequency, and is independent of the incident field. It is this
characteristic which leads to analytical formulas for orientation-averaged
properties [47].

The elements of the _T_ -matrix can be determined through a variety of methods
[54, 55, 56]; for simple shapes the Extended Boundary Condition Method,
obtained by enforcing the boundary conditions on the scatterer’s surface, is
arguably the most accurate and efficient [57]. In particular, for a
homogeneous, spherical scatterer-centred at the origin, $\mathbf{T}$ is a
diagonal matrix whose elements coincide with the Mie theory. For a system
composed of several scatterers, the collective _T_ -matrix can be built up
from the individual ‘one-body _T_ -matrices’ of a single scatterer [46, 50].
Following the notations of Ref. [46] and Ref. [50] we briefly recapitulate
this ‘multi-scatterer _T_ -matrix’ method.

###  II.2 The _T_ -matrix method for _N_ scatterers

We now consider the situation depicted in Fig. 1, where $N$ particles are
present. With $\mathbf{r}$ we denote the position vector in the global
‘cluster’ reference frame. The position of the centre of the $j$th particle,
$\mathbf{r}_{0}^{(j)}$, induces the position vector in this particle’s frame,
$\mathbf{r}_{j}=\mathbf{r}-\mathbf{r}_{0}^{(j)}$.

The one-body _T_ -matrix of the $j$th particle is denoted $T_{1}^{(j)}$. To
keep the formulas relatively concise we assume in the following that each
scatterer’s _T_ -matrix is provided in the orientation of the common reference
frame; in practice, this means that the _T_ -matrix of nonspherical particles,
typically calculated in a high-symmetry orientation ($z$ axis coinciding with
the symmetry axis for axisymmetric particles such as spheroids), needs to be
rotated via the standard Wigner D-matrices [50].

In a system composed of $N$ scatterers, the net field exciting each particle
is composed of the original incident field, plus the field scattered by other
particles. If we denote by $e_{N}^{(j)}$ the coefficients of the electric
field exciting particle $j$, and $f_{N}^{(j)}$ the scattered field
coefficients from particle $j$,

$\displaystyle f_{N}^{(j)}=T_{1}^{(j)}e_{N}^{(j)},$ (10) $\displaystyle
e_{N}^{(j)}=J^{(j,0)}\widetilde{a}+\sum_{\mathclap{l=1,l\neq
j}}^{N}H^{(j,l)}f_{N}^{(l)}.$ (11)

where the regular translation matrix $J^{(j,0)}$ transforms the vector of
incident field coefficients from the global origin ($\mathbf{r}_{0}$) to the
position $\mathbf{r}_{j}$. The irregular translation matrices $H^{(j,l)}$
convert from an irregular VSWF expansion at $\mathbf{r}_{l}$ to a regular VSWF
expansion at $\mathbf{r}_{j}$.
We can also formally consider the expansion coefficients $f_{N}^{(j)}$ of the
field scattered by the $j$th particle in response to the incident field
$\widetilde{a}$, now in the presence of all other particles. Specifically, we
define the $N$-body _T_ -matrix $T_{N}^{(j)}$ of the $j$th particle to relate
$f_{N}^{(j)}$ with $\widetilde{a}$ through

$\displaystyle f_{N}^{(j)}$ $\displaystyle=\qquad\quad
T_{N}^{(j)}J^{(j,0)}\widetilde{a},$ (12)
$\displaystyle\stackrel{{\scriptstyle\mathclap{\text{inc. plane
waves}}}}{{=}}\qquad\quad
T_{N}^{(j)}\left(e^{i\mathbf{k}\cdot\mathbf{r}_{0}^{(j)}}\widetilde{a}\right).$
(13)

These $N$-body _T_ -matrices can be obtained by replacing Eqs. (12) and (11)
in Eq. (10) and solving the following coupled system of equations

$T_{N}^{(j)}=T_{1}^{(j)}\left(I+\sum_{\mathclap{l=1,l\neq
j}}^{N}H^{(j,l)}T_{N}^{(l)}J^{(l,j)}\right).$ (14)

where $I$ is the identity matrix. Equation (14) can be used to implicitly
define scatterer centred $N$-body _T_ -matrices $T_{N}^{(j,l)}$ [45, 46] by
setting

$T_{N}^{(j)}=\sum_{l=1}^{N}T_{N}^{(j,l)}J^{(l,j)}.$ (15)

In this way, $T_{N}^{(j,l)}$ described the full $N$-body contribution of
particle $l$ to the _T_ -matrix of particle $j$.

After calculation of the one-body _T_ -matrices, by replacing Eqs. (15) and
(12) in Eq. (8) for a system of $N$ particles, the scattered electric field
can be expressed as

$\mathbf{E}_{\text{sca}}(k\mathbf{r})=\sum_{j=1}^{N}\sum_{l=1}^{N}\boldsymbol{\Psi}(k\mathbf{r})T_{N}^{(j,l)}J^{(l,0)}\widetilde{a}.$
(16)

A great advantage of this particle-centred formalism is that the fields may be
evaluated even inside the particle cluster, which is not possible when the
whole cluster is described by a single global _T_ -matrix [47].

We can now proceed to evaluate
$\mathscr{C}\propto\Im(\mathbf{E}^{*}\cdot\mathbf{B})$ from the knowledge of
the local fields. First, according to the Maxwell equation
$\mathbf{B}=-i\omega^{-1}\nabla\times\mathbf{E}$, and using Eq. (5) and the
relation between $\widetilde{\mathbf{M}}$ and $\widetilde{\mathbf{N}}$ [58],
we have
$\mathbf{B}_{\text{inc}}=-ik\omega^{-1}\sum[\widetilde{a}_{1,nm}\widetilde{\mathbf{N}}_{nm}+\widetilde{a}_{2,nm}\widetilde{\mathbf{M}}_{nm}]$.
We re-write $\mathbf{B}$ in the compact form
($\mathbf{B}_{\text{inc}}(k\mathbf{r})=\widetilde{\boldsymbol{\Phi}}(k\mathbf{r})\widetilde{\mathbf{a}}$)
analogous to Eq. (7) for the electric field by defining
$\widetilde{\boldsymbol{\Phi}}=[\widetilde{\mathbf{N}}\quad\widetilde{\mathbf{M}}]$
for the incident field and similarly
${\mathbf{\Phi}}=[{\mathbf{N}}\quad{\mathbf{M}}]$ for the scattered magnetic
field. Expanding Eq. (3) into incident and scattered fields as
$\mathscr{C}=\frac{-\omega\varepsilon_{0}}{2}\Im\left(\textbf{E}_{\text{inc}}^{*}\textbf{B}_{\text{inc}}+\textbf{E}_{\text{inc}}^{*}\textbf{B}_{\text{sca}}+\textbf{E}_{\text{sca}}^{*}\textbf{B}_{\text{inc}}+\textbf{E}_{\text{sca}}^{*}\textbf{B}_{\text{sca}}\right)$
and inserting the corresponding expressions yields

$\begin{split}\mathscr{C}=&\frac{k\varepsilon_{0}E_{0}^{2}}{2}\Re\Bigg{[}\widetilde{a}^{\dagger}\widetilde{\boldsymbol{\Psi}}^{\dagger}(k\mathbf{r}){\widetilde{\boldsymbol{\Phi}}}(k\mathbf{r})\widetilde{a}+\\\
{}&\sum_{j=1}^{N}\sum_{l=1}^{N}\widetilde{a}^{\dagger}{\widetilde{\boldsymbol{\Psi}}}^{\dagger}(k\mathbf{r})\boldsymbol{\Phi}(k\mathbf{r}_{j}){T}_{N}^{(j,l)}{J}^{(l,0)}\widetilde{a}+\\\
{}&\sum_{j=1}^{N}\sum_{l=1}^{N}\widetilde{a}^{\dagger}J^{(0,l)}{T^{\dagger}}_{N}^{(j,l)}\boldsymbol{\Psi}^{\dagger}(k\mathbf{r}_{j})\widetilde{\boldsymbol{\Phi}}(k\mathbf{r})\widetilde{a}+\\\
{}&\sum_{j=1}^{N}\sum_{l=1}^{N}\sum_{i=1}^{N}\sum_{k=1}^{N}\widetilde{a}^{\dagger}J^{(k,l)}{T^{\dagger}}_{N}^{(j,l)}\boldsymbol{\Psi}^{\dagger}(k\mathbf{r}_{j})\boldsymbol{\Phi}(k\mathbf{r}_{i}){T}_{N}^{(i,k)}\widetilde{a}\Bigg{]}\end{split}$
(17)

where $E_{0}$ is the incident electric field’s amplitude and $\Re$ stands for
‘real part’. Note that each of the four terms is of the form
$\widetilde{a}^{\dagger}\mathbf{M}\widetilde{a}$; this will be important in
the derivation of the orientation-averaged $\langle\mathscr{C}\rangle$ below.

We can simplify Eq. (17) by translating the incident field to the centre of
each particle via $J^{(l,0)}$ in the second and third terms of Eq. (17), and
noting that for plane wave illumination
$J^{(l,0)}=e^{i\mathbf{k}\cdot\mathbf{r}_{0}^{(l)}}I$, such that

$\widetilde{\boldsymbol{\Psi}}(k\mathbf{r})\equiv\widetilde{\boldsymbol{\Psi}}(k\mathbf{r}_{l})J^{(l,0)},\quad\widetilde{\boldsymbol{\Phi}}(k\mathbf{r})\equiv\widetilde{\boldsymbol{\Phi}}(k\mathbf{r}_{l})J^{(l,0)}.$
(18)

Introducing these relations into Eq. (17) leads to

$\begin{split}\mathscr{C}=&\frac{k\varepsilon_{0}E_{0}^{2}}{2}\Re\Bigg{[}\widetilde{a}^{\dagger}\widetilde{\boldsymbol{\Psi}}^{\dagger}(k\mathbf{r})\widetilde{\boldsymbol{\Phi}}(k\mathbf{r})\widetilde{a}+\\\
{}&\sum_{j=1}^{N}\sum_{l=1}^{N}\widetilde{a}^{\dagger}\widetilde{\boldsymbol{\Psi}}^{\dagger}(k\mathbf{r}_{l})\boldsymbol{\Phi}(k\mathbf{r}_{j}){T}_{N}^{(j,l)}\widetilde{a}+\\\
{}&\sum_{j=1}^{N}\sum_{l=1}^{N}\widetilde{a}^{\dagger}{T^{\dagger}}_{N}^{(j,l)}\boldsymbol{\Psi}^{\dagger}(k\mathbf{r}_{j})\widetilde{\boldsymbol{\Phi}}(k\mathbf{r}_{l})\widetilde{a}+\\\
{}&\sum_{j=1}^{N}\sum_{l=1}^{N}\sum_{i=1}^{N}\sum_{k=1}^{N}\widetilde{a}^{\dagger}J^{(k,l)}{T^{\dagger}}_{N}^{(j,l)}\boldsymbol{\Psi}^{\dagger}(k\mathbf{r}_{j})\boldsymbol{\Phi}(k\mathbf{r}_{i}){T}_{N}^{(i,k)}\widetilde{a}\Bigg{]}\end{split}$
(19)

where the addition rule of the translation matrices $I=J^{(l,0)}J^{(0,l)}$ is
used.

This expression for $\mathscr{C}$ makes no assumption on the polarisation
state of the incident field. We now turn to the description of optical
chirality under illumination with circularly-polarised plane waves.

###  II.3 Helicity versus parity bases for VSWFs

In the standard basis of transverse-electric and transverse-magnetic VSWFs,
the _T_ -matrix of Eq. (9) is usually written in the $2\times 2$ block-matrix
form

$\displaystyle\begin{bmatrix}T_{11}&T_{12}\\\
T_{21}&T_{22}\end{bmatrix}\begin{bmatrix}\widetilde{a}_{1}\\\
\widetilde{a}_{2}\end{bmatrix}=\begin{bmatrix}f_{1}\\\ f_{2}\end{bmatrix}$
(20)

where subscripts 1 and 2 refer to electric and magnetic multipolar
contributions, respectively. The sub-matrices $T_{11}$ and $T_{22}$ describe
the coupling of electric-electric and magnetic-magnetic multipolar components,
while $T_{12}$ and $T_{21}$ describe cross-coupling of electric and magnetic
multipolar components.

For circularly-polarised light, it is more useful to consider a different
basis of VSWFs [26],

$\displaystyle\begin{bmatrix}T_{\scriptscriptstyle LL}&T_{\scriptscriptstyle
LR}\\\ T_{\scriptscriptstyle RL}&T_{\scriptscriptstyle
RR}\end{bmatrix}\begin{bmatrix}\widetilde{a}_{\scriptscriptstyle L}\\\
\widetilde{a}_{\scriptscriptstyle
R}\end{bmatrix}=\begin{bmatrix}f_{\scriptscriptstyle L}\\\
f_{\scriptscriptstyle R}\end{bmatrix}$ (21)

where the subscripts ($R$) and ($L$) refer to right and left circularly
polarised light. This matrix describes the scattering of circularly-polarised
incident fields in the helicity basis. The transformation from the parity
basis can be obtained via the helicity operator
($\Lambda=\frac{\nabla\times}{k}$) and leads to,

$\displaystyle\mathbf{Z}_{R,nm}=\frac{1}{\sqrt{2}}(\mathbf{M}_{nm}-\mathbf{N}_{nm}),\quad\Lambda\mathbf{Z}_{R,nm}=-\mathbf{Z}_{R,nm}$
(22a)
$\displaystyle\mathbf{Z}_{L,nm}=\frac{1}{\sqrt{2}}(\mathbf{M}_{nm}+\mathbf{N}_{nm}),\quad\Lambda\mathbf{Z}_{L,nm}=\mathbf{Z}_{L,nm}$
(22b) .

These definitions lead to the following relation between the _T_ -matrix
components in parity and helicity bases

$\displaystyle\begin{bmatrix}T_{\scriptscriptstyle LL}&T_{\scriptscriptstyle
LR}\\\ T_{\scriptscriptstyle RL}&T_{\scriptscriptstyle
RR}\end{bmatrix}=\frac{1}{2}\begin{bmatrix}I&I\\\
I&-I\end{bmatrix}\begin{bmatrix}T_{11}&T_{12}\\\
T_{21}&T_{22}\end{bmatrix}\begin{bmatrix}I&I\\\ I&-I\end{bmatrix},$ (23)

where $I$ is the identity matrix with the same size as the 4 matrix blocks
($T_{11}$, etc.). Consequently, electric and magnetic fields in the helicity
form become

$\displaystyle\mathbf{E}_{\text{tot}}=E_{0}\left(\widetilde{\boldsymbol{\Gamma}}(k\mathbf{r})\widetilde{a}+\sum_{j=1}^{N}\boldsymbol{\Gamma}(k\mathbf{r}_{j})T_{N}^{(j)}J^{(j,0)}\widetilde{a}\right),$
(24)
$\displaystyle\mathbf{B}_{\text{tot}}=E_{0}\frac{-ik}{\omega}\left(\widetilde{\boldsymbol{\Delta}}(k\mathbf{r})\widetilde{a}+\sum_{j=1}^{N}\boldsymbol{\Delta}(k\mathbf{r}_{j})T_{N}^{(j)}J^{(j,0)}\widetilde{a}\right),$
(25)

where we introduce
$\widetilde{\boldsymbol{\Gamma}}=[\widetilde{\mathbf{Z}}_{L,nm}\quad\widetilde{\mathbf{Z}}_{R,nm}]$,
$\widetilde{\boldsymbol{\Delta}}=[\widetilde{\mathbf{Z}}_{L,mn}\quad-\widetilde{\mathbf{Z}}_{R,mn}]$,
and the same convention for $\boldsymbol{\Gamma}$ and $\boldsymbol{\Delta}$
(composed of irregular VSWFs). Note that the $T_{N}^{(j)}$ as well as
$J^{(j,0)}$ in Eq. (24) need to be also in the helicity basis, following the
same transformation as Eq. (23).

With the general expression for $\mathscr{C}$ and the transformations between
parity and helicity bases, we can now proceed to calculate the orientation-
averaged $\mathscr{C}$ for circularly polarised incident light.

###  II.4 Orientation-averaging of $\mathscr{C}$

A distinctive advantage of the _T_ -matrix framework is that any variation of
the incident field is captured in the $\widetilde{a}$ expansion coefficients,
while the _T_ -matrix itself is unchanged. From this property and the
orthonormality of VSWFs over the 2-sphere, the orientation average of any
optical property that can be written as a bilinear product of the form
$\widetilde{a}^{\dagger}\mathbf{M}\widetilde{a}$ reduces to a simple trace
formula [47, 43],

$\langle\widetilde{a}^{\dagger}\mathbf{M}\widetilde{a}\rangle=4\pi\operatorname{Tr}(\mathbf{M})$
(26)

where $\operatorname{Tr}$ stands for ‘trace of’ and $\mathbf{M}$ is any
bilinear operator whose matrix coefficients are independent of the incident
field. This formula can be directly applied to evaluate the orientation-
average of Eq. (19). We consider an incident field described in the helicity
basis, and restrict the coefficients to a single helicity (corresponding to
either left or right polarisation). This corresponds to $\widetilde{a}=[0\quad
a_{\scriptscriptstyle R}]^{t}$ for $R$ polarisation, and
$\widetilde{a}=[a_{\scriptscriptstyle L}\quad 0]^{t}$ for $L$ polarisation.
The result in both cases contains four different terms (See Appendix (B) for
more details),

$\langle\mathscr{C}\rangle=2\pi
k\varepsilon_{0}E_{0}^{2}\,\Re\left(A_{0}+B_{0}+C_{0}+D_{0}\right)$ (27)

with, For $R$ polarisation:

$\begin{split}A^{\text{(R)}}_{0}=&{-1}/{4\pi}\\\
B^{\text{(R)}}_{0}=&\operatorname{Tr}\left(\sum_{j=1}^{N}\sum_{l=1}^{N}\widetilde{\mathbf{Z}}_{\scriptscriptstyle
R}^{\dagger}(k\mathbf{r}_{l})\left[\mathbf{Z}_{\scriptscriptstyle
L}(k\mathbf{r}_{j})T_{\scriptscriptstyle
LR}^{(j,l)}-\mathbf{Z}_{\scriptscriptstyle
R}(k\mathbf{r}_{j})T_{\scriptscriptstyle RR}^{(j,l)}\right]\right)\\\
C^{\text{(R)}}_{0}=&\operatorname{Tr}\left(\sum_{j=1}^{N}\sum_{l=1}^{N}\left[-T_{\scriptscriptstyle
LR}^{\dagger(j,l)}\mathbf{Z}_{\scriptscriptstyle
L}^{\dagger}(k\mathbf{r}_{j})-T_{\scriptscriptstyle
RR}^{\dagger(j,l)}\mathbf{Z}_{\scriptscriptstyle
R}^{\dagger}(k\mathbf{r}_{j})\right]\widetilde{\mathbf{Z}}_{\scriptscriptstyle
R}(k\mathbf{r}_{l})\right)\\\
D^{\text{(R)}}_{0}=&\operatorname{Tr}\left(\sum_{j=1}^{N}\sum_{l=1}^{N}\sum_{i=1}^{N}\sum_{k=1}^{N}J_{\scriptscriptstyle
RR}^{(k,l)}\left(T^{\dagger(j,l)}_{\scriptscriptstyle
LR}\mathbf{Z}_{\scriptscriptstyle
L}^{\dagger}(k\mathbf{r}_{j})+T^{\dagger(j,l)}_{\scriptscriptstyle
RR}\mathbf{Z}_{\scriptscriptstyle
R}^{\dagger}(k\mathbf{r}_{j})\right)\left(\mathbf{Z}_{\scriptscriptstyle
L}(k\mathbf{r}_{i})T_{\scriptscriptstyle
LR}^{(i,k)}-\mathbf{Z}_{\scriptscriptstyle
R}(k\mathbf{r}_{i})T_{\scriptscriptstyle
RR}^{(i,k)}\right)\right).\end{split}$ (28)

Notice that the sum of $B^{\text{(R)}}_{0}$ and $C^{\text{(R)}}_{0}$
simplifies to,

$\Re(B^{\text{(R)}}_{0}+C^{\text{(R)}}_{0})=-2\operatorname{Tr}\left(\sum_{j=1}^{N}\sum_{l=1}^{N}\widetilde{\mathbf{Z}}_{\scriptscriptstyle
R}^{\dagger}(k\mathbf{r}_{l})\mathbf{Z}_{\scriptscriptstyle
R}(k\mathbf{r}_{j})T_{\scriptscriptstyle RR}^{(j,l)}\right).$ (29)

The corresponding formulas for $L$ polarisation read,

$\begin{split}A^{\text{(L)}}_{0}=&{+1}/{4\pi}\\\
B^{\text{(L)}}_{0}=&\operatorname{Tr}\left(\sum_{j=1}^{N}\sum_{l=1}^{N}\widetilde{\mathbf{Z}}_{\scriptscriptstyle
L}^{\dagger}(k\mathbf{r}_{l})\left[\mathbf{Z}_{\scriptscriptstyle
L}(k\mathbf{r}_{j})T_{\scriptscriptstyle
LL}^{(j,l)}-\mathbf{Z}_{\scriptscriptstyle
R}(k\mathbf{r}_{j})T_{\scriptscriptstyle RL}^{(j,l)}\right]\right)\\\
C^{\text{(L)}}_{0}=&\operatorname{Tr}\left(\sum_{j=1}^{N}\sum_{l=1}^{N}\left[T_{\scriptscriptstyle
LL}^{\dagger(j,l)}\mathbf{Z}_{\scriptscriptstyle
L}^{\dagger}(k\mathbf{r}_{j})+T_{\scriptscriptstyle
RL}^{\dagger(j,l)}\mathbf{Z}_{\scriptscriptstyle
R}^{\dagger}(k\mathbf{r}_{j})\right]\widetilde{\mathbf{Z}}_{\scriptscriptstyle
L}(k\mathbf{r}_{l})\right)\\\
D^{\text{(L)}}_{0}=&\operatorname{Tr}\left(\sum_{j=1}^{N}\sum_{l=1}^{N}\sum_{i=1}^{N}\sum_{k=1}^{N}J_{\scriptscriptstyle
LL}^{(k,l)}\left(T^{\dagger(j,l)}_{\scriptscriptstyle
LL}\mathbf{Z}_{\scriptscriptstyle
L}^{\dagger}(k\mathbf{r}_{j})+T^{\dagger(j,l)}_{\scriptscriptstyle
RL}\mathbf{Z}_{\scriptscriptstyle
R}^{\dagger}(k\mathbf{r}_{j})\right)\left(\mathbf{Z}_{\scriptscriptstyle
L}(k\mathbf{r}_{i})T_{\scriptscriptstyle
LL}^{(i,k)}-\mathbf{Z}_{\scriptscriptstyle
R}(k\mathbf{r}_{i})T_{\scriptscriptstyle
RL}^{(i,k)}\right)\right),\end{split}$ (30)

where the sum of $B^{\text{(L)}}_{0}$ and $C^{\text{(L)}}_{0}$ simplifies to,

$\Re(B^{\text{(L)}}_{0}+C^{\text{(L)}}_{0})=2\operatorname{Tr}\left(\sum_{j=1}^{N}\sum_{l=1}^{N}\widetilde{\mathbf{Z}}_{\scriptscriptstyle
L}^{\dagger}(k\mathbf{r}_{l})\mathbf{Z}_{\scriptscriptstyle
L}(k\mathbf{r}_{j})T_{\scriptscriptstyle LL}^{(j,l)}\right).$ (31)

In some circumstances we may also be interested in averaging over both
possible states of circular polarisation, with
$\tfrac{\langle\mathscr{C}\rangle^{\text{(R)}}+\langle\mathscr{C}\rangle^{\text{(L)}}}{2}$,
which should vanish everywhere for achiral cluster geometries.

The terms $A_{0}^{(R)}$ and $A_{0}^{(L)}$ of Eqs. (28) and (31), respectively,
are equal and opposite and correspond to the results for circularly-polarised
plane waves. This highlights the fact that in the absence of a scatterer, and
for linearly polarised light, $\langle\mathscr{C}\rangle$ vanishes
identically: $a_{\text{R}}$ and $a_{\text{L}}$ are of the same magnitude, only
$A_{0}^{(R)}$ and $A_{0}^{(L)}$ contribute to $\langle\mathscr{C}\rangle$, and
cancel each other. This is also clear from Eq. (19) where the first term
contains
$\widetilde{\boldsymbol{\Psi}}^{\dagger}(k\mathbf{r})\widetilde{\boldsymbol{\Phi}}(k\mathbf{r})$.
An integration over the 2-sphere for linearly-polarised incoming plane waves
will necessarily make this term vanish by orthogonality of different VSWFs.

The value of $\mathscr{C}$ is often normalised with respect to the value of
circularly-polarised plane waves with unit incident field, $\mathscr{C}=\pm
k\varepsilon_{0}/2$, and we introduce

$\langle\overline{\mathscr{C}}\rangle=\frac{2}{k\varepsilon_{0}E_{0}^{2}}\langle\mathscr{C}\rangle$
(32)

for easier comparisons in the examples below.

##  III Application to model nanostructures

We have applied the newly-derived formula Eqs. (28)–(31) to a range of cluster
geometries, with both spherical and nonspherical scatterers, metallic and
dielectric. Our custom-built computer program for the scatterer-centred
superposition _T_ -matrix method is based on the algorithms described in Refs.
[58, 46] and has been tested against FEM simulations [48]. The validity of
Eqs. (28) and (31) was confirmed by comparing the results to orientation-
averaged values of $\langle\mathscr{C}\rangle$ computed by numerical
integration of Eq. (4) over the full solid angle. The _T_ -matrix of spheres
is obtained from Mie theory, while for spheroids we use the smarties program
[57]. Numerical implementations of the _T_ -matrix method require us to
truncate the series of VSWFs to a maximum multipolar order $N_{\text{max}}$;
in our calculations a value of $N_{\text{max}}=8$ was used, which was
sufficient to ensure convergence of the results.

The following examples were chosen to illustrate the results in simple
geometries but the method is equally applicable to more complex structures,
for which an in-depth study is beyond the scope of this manuscript.

###  III.1 Monomer and dimer of silver and silicon nanospheres

Figure 2: (a, b) 2D colour maps of the normalised orientation-averaged local
degree of optical chirality $\langle\overline{\mathscr{C}}\rangle$ around a
single Si sphere with radius $R=$50\text{\,}\mathrm{nm}$$ excited at
$\lambda=$480\text{\,}\mathrm{nm}$$ (a), and a Ag sphere with
$R=$30\text{\,}\mathrm{nm}$$ at $\lambda=$468\text{\,}\mathrm{nm}$$ (b). Note
that the patterns are spherically symmetric. (c, d, e) Corresponding plot of
the radial dependence of $\langle\overline{\mathscr{C}}\rangle$,
$\langle|\mathbf{E}|^{2}\rangle$, $\langle|\mathbf{B}|^{2}\rangle$ with origin
at the surface of the sphere. Note that $\langle|\mathbf{E}|^{2}\rangle$ is
displayed on a log scale to better compare the relative decay profiles, and
that we scale the magnetic field by the speed of light
$c=$299\,792\,458\text{\,}\mathrm{m}$/s$ to retain the same order of magnitude
as the electric field. (f) Radial dependence of the phase angle between
$\mathbf{E}$ and $\mathbf{B}$ at the location $P$ along the $x$ direction as
shown in the inset, for three particular incidence directions, $k_{x}$ (solid
line) and $k_{y}$ or $k_{z}$ (dashed line, both directions are equivalent).

As the simplest case study, we start with a single nanosphere immersed in
water (refractive index $n=1.33$), and consider both a plasmonic (Ag) and a
high-index dielectric (Si) spheres, with dielectric functions from Refs. [10]
and [59], respectively. Silver nanoparticles have been used extensively in
surface-enhanced spectroscopies such as SERS, while silicon nanospheres have
been shown to support strong electric and magnetic resonances in the visible
spectrum [60, 61, 62]. The radii are chosen as $30$ nm for Ag and $50$ nm for
Si, to display a relatively high value of optical chirality.

As expected the local degree of optical chirality reaches its highest value at
the particle surface where electric and magnetic fields are strongest, and
decays to the value of $\pm 1$ far from the scatterer, corresponding to the
degree of optical chirality of a circularly polarised plane wave (Fig.
2(a–b)). The distributions are identical for either $L$ or $R$ polarisation as
the structure is achiral, with opposite sign; for simplicity we only present
the results for $R$ polarisation. $\langle\overline{\mathscr{C}}\rangle$ for
Si is nearly 4 times higher at the surface than
$\langle\overline{\mathscr{C}}\rangle$ for the Ag sphere.

Figure 3: (a) Map of the normalised local degree of optical chirality
$\langle\overline{\mathscr{C}}\rangle$ for a dimer of silver spheres with
radius $R=$30\text{\,}\mathrm{nm}$$ and gap $5\text{\,}\mathrm{nm}$,
calculated on a 2D plane passing through the sphere centres. The wavelength of
excitation is $\lambda=$413\text{\,}\mathrm{nm}$$, corresponding to the peak
extinction. (b) Same configuration as in (a) for a dimer of
$50\text{\,}\mathrm{nm}$-radius Si spheres with $5\text{\,}\mathrm{nm}$ gap,
illuminated at $\lambda=$490\text{\,}\mathrm{nm}$$ (peak extinction). (c, d)
Calculated $\langle\overline{\mathscr{C}}\rangle$ spectra at four different
locations around each dimer, as depicted in the inset. The point M is in the
middle of the gap and points A, B, C are $0.1\text{\,}\mathrm{nm}$ away of the
nanoparticle’s surface. (e, f) Surface-averaged of
$\langle\overline{\mathscr{C}}\rangle$ across the particles’ surface for each
dimer. The inset in (f) traces the maximum surface-averaged
$\langle\overline{\mathscr{C}}\rangle$ for a Si dimer with varying sphere
radius ($$20\text{\,}\mathrm{nm}$-$200\text{\,}\mathrm{nm}$$) and constant gap
($5\text{\,}\mathrm{nm}$).

Less intuitive is the decay of $\langle\overline{\mathscr{C}}\rangle$ with
distance from the particle surface, which is very different for both cases
(Fig. 2(c)), and does not directly correlate with the decay profile of the
electric field intensity $|\mathbf{E}|^{2}$, more familiar in this context.
The degree of optical chirality depends on both electric and magnetic fields,
and therefore on the relative phase of each of their vector components. A
general case could be difficult to interpret, but in this highly-symmetric
configuration we can gain some insight by looking at the relative phase of
$\mathbf{E}$ and $\mathbf{B}$, calculated as
$\angle\left(\mathbf{E},\mathbf{B}\right)=\cos^{-1}\tfrac{\Re(\mathbf{E}^{*}.\mathbf{B})}{|\mathbf{E}||\mathbf{B}|}$,
at a fixed location and for a given direction of incidence. The relative
phases versus distance from the particle surface are plotted in Fig. 2(f) for
three perpendicular incident directions, with the observation point $P$ along
the $x$ axis. We observe that although the electric field is more intense near
the Ag sphere (with nearly equal magnetic fields (Fig. 2(e))), the phase
difference between $\mathbf{E}^{*}$ and $\mathbf{B}$ for a silver sphere is
more variable with the incident direction. Moreover, the phase difference for
a silicon sphere is nearly 90 degrees, which in turn leads to the
comparatively higher value of $\langle\overline{\mathscr{C}}\rangle$. These
considerations confirm the potential benefit of using high-index dielectric
resonators instead of plasmonic particles [62, 61, 63], as they can offer
better control over magnetic fields [24] and preserve the helicity of the
incident light [64, 25]

The next obvious structure to consider is a dimer of nanospheres, the simplest
multi-particle configuration which nevertheless has served as the workhorse of
many SERS studies, as the gap between nanospheres can support highly-localised
electromagnetic ‘hotspots’. In Fig. 3 we consider the orientation-averaged
$\langle\overline{\mathscr{C}}\rangle$ for both Si and Ag dimers, with the
dimer axis along $x$ and a $5\text{\,}\mathrm{nm}$ gap. At the hotspot for the
Si dimer the absolute value of $\langle\overline{\mathscr{C}}\rangle$ reaches
$21.8$, and $18.5$ for the Ag dimer, and for the latter the helicity has
changed sign with respect to the incident right-handed circularly-polarised
light.

In this configuration the value of $\langle\overline{\mathscr{C}}\rangle$
varies with position on either sphere. Since analytes cannot generally be
placed at will in the optimum region [65], we present the spatial variation at
three particular locations on the sphere (by symmetry both spheres in a dimer
present the same pattern)(Fig. 3(c,d)). We also present in Fig. 3(e,f) the
surface-averaged value of $\langle\overline{\mathscr{C}}\rangle$, evaluated
numerically with a 38-point Lebedev spherical quadrature rule. We observe that
although the maximum absolute values of $\langle\overline{\mathscr{C}}\rangle$
are comparable in both cases, the surface-average is considerably weaker for
the Ag dimer than for the Si dimer ($-1.7$ and $-6$, respectively), due to the
sign change of $\langle\overline{\mathscr{C}}\rangle$ across the Ag dimer.

Since the formulas (28)–(31) are analytical and can be evaluated with
negligible computational overhead once the particle-centred _T-_ matrices have
been obtained, we can explore the effect of various parameters on the value
and distribution of $\langle\mathscr{C}\rangle$, as a first step toward
identifying optimised structures for practical applications. Such an example
is shown in inset of Fig. 3(f), for the maximum value of the surface-averaged
$\langle\overline{\mathscr{C}}\rangle$ as a function of particle radius, for a
fixed $5\text{\,}\mathrm{nm}$ gap. By increasing the radius of the Si
nanospheres the magnitude of both electric and magnetic fields increases, and
up to $50\text{\,}\mathrm{nm}$ the phase difference between them remains
nearly $\pi/2$; beyond $50\text{\,}\mathrm{nm}$ however the phase difference
varies more and results in a decrease of the surface-averaged
$\langle\overline{\mathscr{C}}\rangle$.

###  III.2 Analytical vs numerical orientation-averaging

Figure 4: Predicted spectrum of the local degree of optical chirality at the
point $P$ half-way between the last two particles, as depicted in the inset.
The structure consists of a helix of five prolate gold spheroids, with helix
axis $z$, radius $100\text{\,}\mathrm{nm}$, pitch $700\text{\,}\mathrm{nm}$,
angular-step $\delta=\pi/4$. The spheroids are oriented along the helix, and
have semi-axes $a=b=$30\text{\,}\mathrm{nm}$$ and
$c=$50\text{\,}\mathrm{nm}$$. The structure is illuminated with a right
circularly-polarised plane wave along the $x$ direction (blue dashed line),
$y$ direction (green dashed line), or $z$ direction (purple dashed line). The
average over these three incident directions (red dashed line) is compared to
the rigorous orientation-averaged value $\langle\overline{\mathscr{C}}\rangle$
(solid black line).

Our last example is a helix of gold nanorods, depicted in Fig. 4. Elongated
nanoparticles have been shown to amplify the far-field circular dichroism in
chain-like [66] or helical structures [67], and changing the particles’ aspect
ratio provides a convenient means to tune the resonance position. This system
also provides a useful test for our numerical implementation of the formalism,
requiring _T-_ matrices of spheroidal particles [57], and also the rotation of
each individual particle.

The helix is composed of five identical Au spheroids with semi-axes
$a=$30\text{\,}\mathrm{nm}$,c=$50\text{\,}\mathrm{nm}$$; the radius of the
helix is $100\text{\,}\mathrm{nm}$ and pitch $700\text{\,}\mathrm{nm}$, and
the angular step between each particle is $\delta=\pi/4$. The orientation-
averaged value of $\langle\overline{\mathscr{C}}\rangle$ at a particular
‘hotspot’ position $P$ between the last two particles is compared in Fig. 4 to
the value obtained for three particular directions of incidence
($k_{x},k_{y},k_{z}$, respectively). The value of $\mathscr{C}$ is very
different for the three incident directions, and their ‘naive’ averaging
provides an unreliable estimate of the true orientation-averaged value
$\langle\overline{\mathscr{C}}\rangle$. This points to the importance of
proper orientation-averaging when comparing simulations to experiments where
samples are randomly-oriented with respect to the incident light. The size and
symmetry of the cluster as well as the strength of the interparticle coupling
will contribute to the difficulty in estimating numerically the orientation-
average with discrete simulations at specific directions of incidence. The
analytical results presented here can serve as a useful benchmark to evaluate
the number of quadrature points needed for a numerical evaluation of Eq. (4).
A comparison of spherical quadrature rules for both far-field and near-field
quantities is beyond the scope of this study but will be presented elsewhere.

##  IV Conclusion

We have derived analytical formulas for the orientation-averaged degree of
optical chirality $\langle\mathscr{C}\rangle$ around clusters of nanoparticles
illuminated with circularly-polarised light, using the rigorous superposition-
_T_ -matrix framework. The formulas, although lengthy, are readily implemented
in software, and provide an efficient method to calculate rigorously-averaged
results, in contrast to numerical estimates obtained from several directions
of incidence, as is commonly done with other numerical methods. The local
degree of optical chirality informs us on the subtle interplay between complex
electric and magnetic fields, and how their combination may enhance the
chiroptical response of molecules in the vicinity of a cluster of particles.
We illustrated the behaviour of $\langle\mathscr{C}\rangle$ by mapping its
distribution around single spheres and dimers of Ag and Si, with less
intuitive results compared to the better-known electric field intensity. The
importance of the analytical formula for $\langle\mathscr{C}\rangle$ was
illustrated for a helix of elongated particles, where we show that a simple
averaging of the results from three orthogonal directions provides a poor
estimate of the true value.

These few examples illustrate the broad range of studies that become tractable
with an efficient and accurate formulas for $\langle\mathscr{C}\rangle$,
notably the search for optimal structures that maximise the chiroptical
response of adsorbed species, either at a specific location in space, or as
averages over the surface of the particles.

The _T_ -matrix framework used to arrive at these results is ideally-suited
for many studies requiring orientation-averaging, and we hope this paper
serves to demonstrate its particular usefulness in the context of near-field
chiroptical properties where it has not received much attention to date, with
most of the literature relying on purely numerical solutions such as FDTD, or
coupled-dipole approximations with limited range of validity. Although the
superposition _T_ -matrix method provides a rigorous solution to the light
scattering problem, it also suffers some limitations, in particular where the
circumscribed spheres of neighbouring particles overlap [48]. We hope this
study will also trigger further work in improving the method and overcoming
its current limitations [49].

## Acknowledgements

The authors would like to thank Dmitri Schebarchov and Eric Le Ru for helpful
discussions, the Royal Society Te Apārangi for support through a Rutherford
Discovery Fellowship (B.A.), and the MacDiarmid Institute for additional
funding (A.F.N., S.S.).

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##  Appendix A Vector Spherical Harmonics

The vector spherical harmonics
($\widetilde{\mathbf{M}},\widetilde{\mathbf{N}},{\mathbf{M}},{\mathbf{N}}$)
are defined as follows

$\begin{matrix}\widetilde{\mathbf{M}}_{nm}(k\mathbf{r},\theta,\varphi)\\\
{\mathbf{M}}_{nm}(k\mathbf{r},\theta,\varphi)\end{matrix}=\gamma_{nm}\nabla\times\begin{pmatrix}\mathbf{r}\widetilde{\xi}_{nm}(k\mathbf{r},\theta,\varphi)\\\
\mathbf{r}{\xi}_{nm}(k\mathbf{r},\theta,\varphi)\end{pmatrix},\\\ $ (33)

and

$\begin{matrix}\widetilde{\mathbf{N}}_{nm}(k\mathbf{r},\theta,\varphi)\\\
{\mathbf{N}}_{nm}(k\mathbf{r},\theta,\varphi)\end{matrix}=\frac{1}{k}\nabla\times\begin{pmatrix}\widetilde{\mathbf{M}}_{nm}(k\mathbf{r},\theta,\varphi)\\\
{\mathbf{M}}_{nm}(k\mathbf{r},\theta,\varphi)\end{pmatrix},$ (34)

where

$\displaystyle\begin{pmatrix}\widetilde{\xi}_{nm}(k\mathbf{r},\theta,\varphi)\\\
{\xi}_{nm}(k\mathbf{r},\theta,\varphi)\end{pmatrix}=\quad\begin{pmatrix}j_{n}(k\mathbf{r})\\\
h_{n}^{(1)}(k\mathbf{r})\end{pmatrix}Y_{nm}(\theta,\varphi)$ (35)
$\displaystyle\gamma_{nm}=\left[\frac{(2n+1)(n-m)!}{4\pi
n(n+1)(n+m)!}\right].$ (36)

In Eq. (35), $j_{n}(k\mathbf{r})$ and $h_{n}^{(1)}(k\mathbf{r})$ are regular
and irregular spherical Bessel functions, respectively and the spherical
harmonics are defined as

$Y_{nm}(\theta,\varphi)=P_{n}^{m}(\cos(\theta))e^{im\varphi},$ (37)

$P_{n}^{m}(\cos(\theta))$ are the associated Legendre functions, defined using
the Condon–Shortley phase

$P_{n}^{m}(x)=\frac{(-1)^{m}}{2^{n}n!}(1-x^{2})^{m/2}\frac{d^{n+m}}{dx^{n+m}}(x^{2}-1)^{n},m\geq
0$ (38)

and

$P_{n}^{(-m)}=(-1)^{m}\frac{(n-m)!}{(n+m)!}P_{n}^{m}.$ (39)

### Transformation to cartesian coordinates

Since we seek vector fields expressed in the cartesian basis of the cluster
reference frame, for the calculation of different terms of $\mathscr{C}$ (Eqs.
(28)-(31)) we must convert electric and magnetic field components from a
spherical to a cartesian basis. The matrix $C_{j}$ to perform this
transformation for particle $j$ reads:

$C_{j}=\begin{pmatrix}\sin\theta_{j}\cos\varphi_{j}&\cos\theta_{j}\cos\varphi_{j}&-\sin\varphi_{j}\\\
\sin\theta_{j}\sin\varphi_{j}&\cos\theta_{j}\sin\varphi_{j}&\cos\varphi_{j}\\\
\cos\theta_{j}&-\sin\theta_{j}&0\end{pmatrix}.$ (40)

$C_{0}$ is the corresponding matrix for the global ‘cluster’ coordinate
system. Note that the $C_{j}$ matrices are needed even if all particle frames
are parallel, to transform field components from spherical unit vectors to
cartesian ones.

##  Appendix B Derivation of Eq. (28)

We provide here a step-by-step derivation to arrive at equations (28) and
(31). The key ingredient is to carefully keep track of the block matrix
structure of the involved quantities in the helicity basis for VSWFs. For the
sake of brevity, we will explicitly perform this derivation for right
circularly-polarised light, Eq. (28). The modifications needed to derive Eq.
(31) are minimal.

As discussed in the main text, right circularly-polarised light is described
with an expansion coefficient $\widetilde{a}^{\text{R}}$ that has the block
structure $\widetilde{a}^{\text{R}}_{\scriptscriptstyle LR}=[0\quad
a_{\scriptscriptstyle R}]^{t}$ in the helicity basis (more generally, we use
the $LR$ subscript to refer to helicity basis in the following). Then,
starting from Eq. (19), we convert each term to its helical basis
representation and insert the expansion coefficient for $\widetilde{a}$. The
resulting equation is the sum of four terms
$\mathscr{C}=\frac{k\varepsilon_{0}E_{0}^{2}}{2}\,\Re\left(A+B+C+D\right)$,
where we named the terms in a way indicative of their contributions to the
final results, and we derive them sequentially below.

First, notice that the basis transformation changing $\widetilde{a}$ to
$\widetilde{a}_{\scriptscriptstyle LR}$ is simply given by

$\widetilde{a}_{\scriptscriptstyle LR}=\frac{1}{\sqrt{2}}\begin{bmatrix}I&I\\\
I&-I\end{bmatrix}\widetilde{a},$ (41)

which holds independent of whether we look at regular or irregular VSWFs. This
transformation is self-adjoint, real, and involutive.

As a result, it is straightforward to get the precursor of
$A^{\text{(R)}}_{0}$ from Eq. (19):

$\displaystyle A^{\text{(R)}}$
$\displaystyle=(\widetilde{a}^{\text{R}})^{\dagger}\widetilde{\boldsymbol{\Psi}}^{\dagger}(k\mathbf{r})\widetilde{\boldsymbol{\Phi}}(k\mathbf{r})\widetilde{a}^{\text{R}},$
(42) $\displaystyle=(\widetilde{a}^{\text{R}}_{\scriptscriptstyle
LR})^{\dagger}\widetilde{\boldsymbol{\Gamma}}^{\dagger}(k\mathbf{r})\widetilde{\boldsymbol{\Delta}}(k\mathbf{r})\widetilde{a}^{\text{R}}_{\scriptscriptstyle
LR}.$ (43)

At this point, we insert the definition of $\widetilde{\boldsymbol{\Gamma}}$
and $\widetilde{\boldsymbol{\Delta}}$. When doing this, it is useful to
remember that to transpose a block matrix, one first treats the blocks as
entries of a matrix, transposes this matrix as usual, and then additionally
transposes each of the component blocks. This gives:

$A^{\text{(R)}}=\widetilde{a}_{\scriptscriptstyle
R}^{\dagger}\widetilde{\mathbf{Z}}_{\scriptscriptstyle
R}^{\dagger}\left(-\widetilde{\mathbf{Z}}_{\scriptscriptstyle
L}\right)\widetilde{a}_{\scriptscriptstyle R}.$ (44)

Orientation-averaging of this expression relies on the orthogonality relations
between VSWFs on the sphere and must lead to a constant. The main difficulty
is to keep track of the normalisation factors, but here we can also realise
that the average value of $\mathscr{C}$ for the incident field must equal the
value of a single circularly-polarised plane wave, which is independent of its
direction of propagation. This gives $A^{\text{(R)}}_{0}=-\tfrac{1}{4\pi}$.

This combination of matrix algebra before performing the orientation average
repeats in the other terms of Eq. (28). In the following we therefore focus on
the matrix structure. For $B^{\text{(R)}}$, we write

$B^{\text{(R)}}=\\\
{}\quad\sum_{j,l=1}^{N}[0,\widetilde{a}_{\scriptscriptstyle
R}^{\dagger}]\begin{bmatrix}\widetilde{\mathbf{Z}}_{\scriptscriptstyle
L}^{l,\dagger}\\\ \widetilde{\mathbf{Z}}_{\scriptscriptstyle
R}^{l,\dagger}\end{bmatrix}[\widetilde{\mathbf{Z}}_{\scriptscriptstyle
L}^{j},-\widetilde{\mathbf{Z}}_{\scriptscriptstyle
R}^{j}]\begin{bmatrix}(T^{(l,j)}_{N})_{\scriptscriptstyle
LL}&(T^{(l,j)}_{N})_{\scriptscriptstyle LR}\\\
(T^{(l,j)}_{N})_{\scriptscriptstyle RL}&(T^{(l,j)}_{N})_{\scriptscriptstyle
RR}\end{bmatrix}\begin{bmatrix}0\\\ \widetilde{a}_{\scriptscriptstyle
R}\end{bmatrix}\\\ {}\quad=\sum_{j,l=1}^{N}\widetilde{a}_{\scriptscriptstyle
R}^{\dagger}\widetilde{\mathbf{Z}}_{\scriptscriptstyle
R}^{l,\dagger}\left(\widetilde{\mathbf{Z}}_{\scriptscriptstyle
L}^{j}(T^{(l,j)}_{N})_{\scriptscriptstyle
LR}-\widetilde{\mathbf{Z}}_{\scriptscriptstyle
R}^{j}(T^{(l,j)}_{N})_{\scriptscriptstyle
RR}\right)\widetilde{a}_{\scriptscriptstyle R},$ (45)

which, upon integration, leads to $B^{\text{(R)}}_{0}$.

In the expressions for $C^{\text{(R)}}$ we need to account for the
transposition of block matrices:

$C^{\text{(R)}}=\\\
{}\quad\sum_{j,l=1}^{N}[0,\widetilde{a}_{\scriptscriptstyle
R}^{\dagger}]\begin{bmatrix}(T^{(l,j)}_{N})_{\scriptscriptstyle
LL}^{\dagger}&(T^{(l,j)}_{N})_{\scriptscriptstyle RL}^{\dagger}\\\
(T^{(l,j)}_{N})_{\scriptscriptstyle
LR}^{\dagger}&(T^{(l,j)}_{N})_{\scriptscriptstyle
RR}^{\dagger}\end{bmatrix}\begin{bmatrix}\widetilde{\mathbf{Z}}_{\scriptscriptstyle
L}^{l,\dagger}\\\ \widetilde{\mathbf{Z}}_{\scriptscriptstyle
R}^{l,\dagger}\end{bmatrix}[\widetilde{\mathbf{Z}}_{\scriptscriptstyle
L}^{j},-\widetilde{\mathbf{Z}}_{\scriptscriptstyle R}^{j}]\begin{bmatrix}0\\\
\widetilde{a}_{\scriptscriptstyle R}\end{bmatrix}\\\
{}\quad=-\sum_{j,l=1}^{N}\widetilde{a}_{\scriptscriptstyle
R}^{\dagger}\left((T^{(l,j)}_{N})_{\scriptscriptstyle
LR}^{\dagger}\widetilde{\mathbf{Z}}_{\scriptscriptstyle
L}^{l,\dagger}+(T^{(l,j)}_{N})_{\scriptscriptstyle
RR}^{\dagger}\widetilde{\mathbf{Z}}_{\scriptscriptstyle
R}^{l,\dagger}\right)\widetilde{\mathbf{Z}}_{\scriptscriptstyle
R}^{j}\widetilde{a}_{\scriptscriptstyle R},$ (46)

which, after orientation-averaging, yields $C^{\text{(R)}}_{0}$. Finally, the
term $D^{\text{(R)}}$ is derived as follows:

$\displaystyle D^{\text{(R)}}$
$\displaystyle=\sum_{\begin{subarray}{c}i,j,k,l=1\\\ \mathclap{\text{pairwise
distinct}}\end{subarray}}^{N}[0,\widetilde{a}_{\scriptscriptstyle
R}^{\dagger}]\begin{bmatrix}J^{(k,l)}_{\scriptscriptstyle LL}&{0}\\\
{0}&J_{\scriptscriptstyle
RR}^{(k,l)}\end{bmatrix}\begin{bmatrix}(T^{(j,l)}_{N})_{\scriptscriptstyle
LL}^{\dagger}&(T^{(j,l)}_{N})_{\scriptscriptstyle RL}^{\dagger}\\\
(T^{(j,l)}_{N})_{\scriptscriptstyle
LR}^{\dagger}&(T^{(j,l)}_{N})_{\scriptscriptstyle
RR}^{\dagger}\end{bmatrix}\begin{bmatrix}\widetilde{\mathbf{Z}}_{\scriptscriptstyle
L}^{j,\dagger}\\\ \widetilde{\mathbf{Z}}_{\scriptscriptstyle
R}^{j,\dagger}\end{bmatrix}[\widetilde{\mathbf{Z}}_{\scriptscriptstyle
L}^{i},-\widetilde{\mathbf{Z}}_{\scriptscriptstyle
R}^{i}]\begin{bmatrix}(T^{(i,k)}_{N})_{\scriptscriptstyle
LL}&(T^{(i,k)}_{N})_{\scriptscriptstyle LR}\\\
(T^{(i,k)}_{N})_{\scriptscriptstyle RL}&(T^{(i,k)}_{N})_{\scriptscriptstyle
RR}\end{bmatrix}\begin{bmatrix}0\\\ \widetilde{a}_{\scriptscriptstyle
R}\end{bmatrix},$ (47) $\displaystyle=\sum_{\begin{subarray}{c}i,j,k,l=1\\\
\mathclap{\text{pairwise
distinct}}\end{subarray}}^{N}\widetilde{a}_{\scriptscriptstyle
R}^{\dagger}J_{\scriptscriptstyle
RR}^{(k,l)}\left((T^{(j,l)}_{N})_{\scriptscriptstyle
LR}^{\dagger}\widetilde{\mathbf{Z}}_{\scriptscriptstyle
L}^{j,\dagger}+(T^{(j,l)}_{N})_{\scriptscriptstyle
RR}^{\dagger}\widetilde{\mathbf{Z}}_{\scriptscriptstyle
R}^{j,\dagger}\right)\left(\widetilde{\mathbf{Z}}_{\scriptscriptstyle
L}^{i}(T^{(i,k)}_{N})_{\scriptscriptstyle
LR}-\widetilde{\mathbf{Z}}_{\scriptscriptstyle
R}^{i}(T^{(i,k)}_{N})_{\scriptscriptstyle
RR}\right)\widetilde{a}_{\scriptscriptstyle R}.$ (48)

As above, the orientation average of this expression will result in
$D^{\text{(R)}}_{0}$. The proof for left-handed incident fields works the same
way with appropriate index changes.


# Sharp endpoint estimates for the $X$-ray transform and the Radon transform
in finite fields

Doowon Koh  Department of Mathematics
Chungbuk National University
Cheongju city, Chungbuk-Do 361-763 Korea<EMAIL_ADDRESS>
###### Abstract.

This note establishes sharp $L^{p}-L^{r}$ estimates for $X$-ray transforms and
Radon transforms in finite fields.

###### 2010 Mathematics Subject Classification:

43A32, 43A15

Key words and phrases:$k$-plane transform, $X$-ray transform, Radon transform,
finite fields.
This research was supported by Basic Science Research Program through the
National Research Foundation of Korea(NRF) funded by the Ministry of
Education, Science and Technology(2012010487)

##  1\. Introduction

Let $\mathbb{F}_{q}^{d},d\geq 2,$ be a $d$-dimensional vector space over the
finite field $\mathbb{F}_{q}$ with $q$ elements. We endow $\mathbb{F}_{q}^{d}$
with a normalized counting measure $dx.$ For each $q$, we denote by $M_{q}$ a
collection of certain subsets of $\mathbb{F}_{q}^{d}.$ Recall that a
normalized surface measure $d\sigma$ on $M_{q}$ can be defined by the relation

$\int\Omega(w)d\sigma(w)=\frac{1}{|M_{q}|}\sum_{w\in M_{q}}\Omega(w)$

where $|M_{q}|$ denotes the cardinality of $M_{q}$ and $\Omega$ is a complex-
valued function on $M_{q}.$ For any complex-valued function $f$ on
$(\mathbb{F}_{q}^{d},dx)$ and $w\in M_{q}$, we consider an operator
$T_{M_{q}}$ defined by

$T_{M_{q}}f(w)=\frac{1}{|w|}\sum_{x\in w}f(x).$

We are interested in determining exponents $1\leq p,r\leq\infty$ such that the
following inequality holds:

$\|T_{M_{q}}f\|_{L^{r}(M_{q},d\sigma)}\lesssim\|f\|_{L^{p}(\mathbb{F}_{q}^{d},dx)}\quad\mbox{for
all}~{}~{}f~{}~{}\mbox{on}~{}~{}\mathbb{F}_{q}^{d},$

where the operator norm of $T_{M_{q}}$ is independent of $q$, the size of the
underlying finite field $\mathbb{F}_{q}.$ If $M_{q}$ is $\Pi_{k},$ a
collection of all $k$-planes in $\mathbb{F}_{q}^{d}$ with $1\leq k\leq d-1$,
then the operator $T_{\Pi_{k}}$ is called as the $k$-plane transform. In
particular, $T_{\Pi_{1}}$ and $T_{\Pi_{(d-1)}}$ are known as the $X$-ray
transform and the Radon transform respectively. In Euclidean space, the
complete mapping properties of $k$-plane transforms were proved by M. Christ
in [1]. Readers may refer to [9], [8], [3] for the description of $k$-plane
transforms in the Euclidean setting. In 2008, Carbery, Stones, and Wright [2]
initially studied the mapping properties of $k$-plane transforms in finite
fields. Using combinatorial arguments, they proved the following theorem.

######  Theorem 1.1.

Let $d\geq 2$ and $1\leq k\leq d-1$ be an integer. If

(1.1) $\|T_{\Pi_{k}}f\|_{L^{r}(\Pi_{k},d\sigma)}\leq
C\|f\|_{L^{p}(\mathbb{F}_{q}^{d},dx)}$

holds with $C$ independent of $|\mathbb{F}_{q}|$, then $(1/p,1/r)$ lies in the
convex hull $H$ of

$((k+1)/(d+1),1/(d+1)),(0,0),(1,1)~{}~{}\mbox{and}~{}~{}(0,1).$

Conversely, if $(1/p,1/r)$ lies in
$H\setminus\left\\{((k+1)/(d+1),1/(d+1))\right\\},$ then (1.1) holds with $C$
independent of $|\mathbb{F}_{q}|.$ Finally, if
$(1/p,1/r)=((k+1)/(d+1),1/(d+1))$, then the restricted type inequality

(1.2) $\|T_{\Pi_{k}}f\|_{L^{d+1}(\Pi_{k},d\sigma)}\leq
C\|f\|_{L^{\frac{d+1}{k+1},1}(\mathbb{F}_{q}^{d},dx)}$

holds with $C$ independent of $|\mathbb{F}_{q}|.$

Notice from Theorem 1.1 that if one could show that the restricted type
inequality (1.2) can be replaced by the strong type inequality, then the
mapping properties of the $k$-plane transforms in finite fields would be
completely established. Namely, our task would prove the following conjecture.

######  Conjecture 1.2.

Let $d\geq 2$ and $1\leq k\leq(d-1)$ be integers. Then, we have

$\|T_{\Pi_{k}}f\|_{L^{d+1}(\Pi_{k},d\sigma)}\leq
C\|f\|_{L^{\frac{d+1}{k+1}}(\mathbb{F}_{q}^{d},dx)}\quad\mbox{for
all}~{}~{}f~{}~{}\mbox{on}~{}~{}\mathbb{F}_{q}^{d},$

where $C$ is independent of $|\mathbb{F}_{q}|.$

###  1.1. Statement of main results

In this paper we prove that Conjecture 1.2 is true for the $X$-ray transform
and the Radon transform. More precisely, we obtain the following theorem.

######  Theorem 1.3.

Let $d\geq 2$ be any integer. If $k=1$ or $k=d-1,$ then

$\|T_{\Pi_{k}}f\|_{L^{d+1}(\Pi_{k},d\sigma)}\leq
C\|f\|_{L^{\frac{d+1}{k+1}}(\mathbb{F}_{q}^{d},dx)}\quad\mbox{for
all}~{}~{}f~{}~{}\mbox{on}~{}~{}\mathbb{F}_{q}^{d},$

where $C$ is independent of $|\mathbb{F}_{q}|.$

In order to prove Theorem 1.3 for the $X$-ray transform ($k=1$), we shall
adapt both the combinatorial arguments in [2] and the skills in [6] for
endpoint estimates. On the other hand, a Fourier analytic argument will be
required to prove Theorem 1.3 for the Radon transform ($k=d-1$) .

######  Remark 1.4.

After writing this paper, the author realized that our result for the $X$-ray
transform is a corollary of Theorem 1.1 in the paper [4]. This was pointed out
by R. Oberlin.

##  2\. Proof of the mapping properties of the $X$-ray transform

In this section, we restate and prove Theorem 1.3 in the case of the $X$-ray
transform. Namely, we prove the following statement which implies the sharp
boundedness of the $X$-ray transform.

######  Theorem 2.1.

Let $d\geq 2$ be any integer.

$\|T_{\Pi_{1}}f\|_{L^{d+1}(\Pi_{1},d\sigma)}\leq
C\|f\|_{L^{\frac{d+1}{2}}(\mathbb{F}_{q}^{d},dx)}\quad\mbox{for
all}~{}~{}f~{}~{}\mbox{on}~{}~{}\mathbb{F}_{q}^{d},$

where $C$ is independent of $|\mathbb{F}_{q}|.$

###### Proof.

We begin by following the argument in [6]. Without loss of generality, we may
assume that $f$ is a non-negative real-valued function and

(2.1) $\sum_{x\in\mathbb{F}_{q}^{d}}f(x)^{\frac{d+1}{2}}=1.$

Thus, it is natural to assume that $\|f\|_{\infty}\leq 1.$ Furthermore, we may
assume that $f$ is written by a step function

(2.2) $f(x)=\sum_{i=0}^{\infty}2^{-i}E_{i}(x),$

where the sets $E_{i}$ are disjoint subsets of $\mathbb{F}_{q}^{d}$, and here,
and throughout the paper, we write $E(x)$ for the characteristic function on a
set $E\subset\mathbb{F}_{q}^{d}.$ From (2.1) and (2.2), we also assume that

(2.3) $\sum_{j=0}^{\infty}2^{-\frac{(d+1)j}{2}}|E_{j}|=1\quad\mbox{and
}~{}~{}|E_{j}|\leq 2^{\frac{(d+1)j}{2}}~{}~{}\mbox{for
all}~{}~{}j=0,1,\cdots.$

Since $dx$ is the normalized counting measure on $\mathbb{F}_{q}^{d},$ the
assumption (2.1) shows that we only need to prove

(2.4) $\|T_{\Pi_{1}}f\|^{d+1}_{L^{d+1}(\Pi_{1},d\sigma)}\lesssim q^{-2d},$

where $f$ satisfies (2.2) and (2.3). Since we have assumed that $f\geq 0$, it
is clear that $T_{\Pi_{1}}f$ is also a non-negative real-valued function on
$\Pi_{1}.$ By expanding the left-hand side of (2.4) and using the facts that
$|w|=q$ for $w\in\Pi_{1}$ and $|\Pi_{1}|\sim q^{2(d-1)}$, we see that

$\|T_{\Pi_{1}}f\|^{d+1}_{L^{d+1}(\Pi_{1},d\sigma)}=\frac{1}{|\Pi_{1}|}\sum_{w\in\Pi_{1}}\left(T_{\Pi_{1}}f(w)\right)^{d+1}$
$\sim\frac{1}{q^{d+1}}\frac{1}{q^{2(d-1)}}\sum_{i_{0}=0}^{\infty}\dots\sum_{i_{d}=0}^{\infty}2^{-(i_{0}+\dots+i_{d})}\sum_{(x_{0},\dots,x_{d})\in
E_{i_{0}}\times\dots\times E_{i_{d}}}\sum_{w\in\Pi_{1}}w(x_{0})\dots w(x_{d})$
$\sim\frac{1}{q^{d+1}}\frac{1}{q^{2(d-1)}}\sum_{0=i_{0}\leq i_{1}\leq\dots\leq
i_{d}<\infty}2^{-(i_{0}+\dots+i_{d})}\sum_{(x_{0},\dots,x_{d})\in
E_{i_{0}}\times\dots\times E_{i_{d}}}\sum_{w\in\Pi_{1}}w(x_{0})\dots
w(x_{d}),$

where the last line follows from the symmetry of $i_{0},\cdots,i_{d}.$ We now
follows the argument in [2]. Notice that we can write

$\sum_{(x_{0},\dots,x_{d})\in E_{i_{0}}\times\dots\times
E_{i_{d}}}=\sum_{s=0}^{\infty}\sum_{(x_{0},\dots,x_{d})\in\Delta(s,i_{0},\dots,i_{d})},$

where $\Delta(s,i_{0},\dots,i_{d})=\\{(x_{0},\dots,x_{d})\in
E_{i_{0}}\times\dots\times E_{i_{d}}:[x_{0},\dots,x_{d}]~{}\mbox{is
a}~{}s\mbox{-plane}\\}$ and $[x_{0},\dots,x_{d}]$ denotes the smallest affine
subspace containing the elements $x_{0},\dots,x_{d}.$ In addition, observe
that if $s>1$ and $(x_{0},\dots,x_{d})\in\Delta(s,i_{0},\dots,i_{d}),$ then
the sum over $w\in\Pi_{1}$ vanishes. On the other hand, if $s=0,1,$ then the
sum over $w\in\Pi_{1}$ is same as the number of lines containing the unique
$s$-plane, that is $\sim q^{(d-1)(1-s)}.$ From these observations and (2.4),
it is enough to prove that for all $E_{i},i=0,1,\dots,$ satisfying (2.3),

(2.5) $\sum_{i_{0}=0}^{\infty}\sum_{i_{1}\geq
i_{0}}^{\infty}\cdots\sum_{i_{d}\geq
i_{d-1}}^{\infty}2^{-(i_{0}+i_{1}+\cdots+i_{d})}\sum_{s=0}^{1}|\Delta(s,i_{0},\dots,i_{d})|q^{-s(d-1)}\lesssim
1.$

Namely, it suffices to prove that for every $d\geq 2$ and $s=0,1$,

$\mbox{A}=\sum_{i_{0}=0}^{\infty}\sum_{i_{1}\geq
i_{0}}^{\infty}\cdots\sum_{i_{d}\geq
i_{d-1}}^{\infty}2^{-(i_{0}+i_{1}+\cdots+i_{d})}|\Delta(s,i_{0},\dots,i_{d})|q^{-s(d-1)}\lesssim
1,$

where $\Delta(s,i_{0},\dots,i_{d})$ is defined as before, and the sets
$E_{i}$, i=0,1,…, satisfy (2.3). Suppose that $s=0.$ Since
$|\Delta(0,i_{0},\dots,i_{d})|\leq|E_{i_{0}}|$, it follows

$\mbox{A}\leq\sum_{i_{0}=0}^{\infty}\sum_{i_{1}\geq
i_{0}}^{\infty}\cdots\sum_{i_{d}\geq
i_{d-1}}^{\infty}2^{-(i_{0}+i_{1}+\cdots+i_{d})}|E_{i_{0}}|.$

Since the sum of a convergent geometric series is similar to the value of the
first term, we have the desirable conclusion for $s=0$:

$\mbox{A}\lesssim\sum_{i_{0}=0}^{\infty}|E_{i_{0}}|2^{-(d+1)i_{0}}\leq\sum_{i_{0}=0}^{\infty}|E_{i_{0}}|2^{-\frac{(d+1)i_{0}}{2}}=1,$

where the last equality is obtained from (2.3).

Next, we assume that $s=1.$ We must show that for all $E_{i},i=0,1,\dots$,
satisfying (2.3), we have

(2.6) $\sum_{i_{0}=0}^{\infty}\sum_{i_{1}\geq
i_{0}}^{\infty}\cdots\sum_{i_{d}\geq
i_{d-1}}^{\infty}2^{-(i_{0}+i_{1}+\cdots+i_{d})}|\Delta(1,i_{0},\dots,i_{d})|q^{-(d-1)}\lesssim
1.$

We estimate the upper bound of $|\Delta(1,i_{0},\dots,i_{d})|.$ Fix
$x_{i_{0}}\in E_{i_{0}}$ which has $|E_{i_{0}}|$ choices. Notice that if
$(x_{i_{0}},\dots,x_{i_{d}})\in\Delta(1,i_{0},\dots,i_{d}),$ then all points
$x_{i_{0}},\dots,x_{i_{d}}$ must lie on a line, which is determined by at
least two different points of them . Therefore, for each $l=1,2,\dots,d,$, we
can define

$L(l)=\\{(x_{i_{0}},\dots,x_{i_{d}})\in\Delta(1,i_{0},\dots,i_{d}):[x_{i_{0}},\dots,x_{i_{l}}]~{}\mbox{is
a line, and}~{}[x_{i_{0}},\dots,x_{i_{l-1}}]~{}\mbox{is a point}\\},$

where we recall that $[\alpha_{1},\dots,\alpha_{s}]$ means the smallest affine
subspace containing all points $\alpha_{1},\dots,\alpha_{s}$ in
$\mathbb{F}_{q}^{d}.$ It is clear that
$\Delta(1,i_{0},\dots,i_{d})=\cup_{l=1}^{d}L(l),$ which implies that

(2.7) $|\Delta(1,i_{0},\dots,i_{d})|\leq\sum_{l=1}^{d}|L(l)|.$

By the definition of $L(l),l=1,\dots,d,$ it follows that for every
$l=1,\dots,d,$

(2.8) $|L(l)|\leq|E_{i_{0}}||E_{i_{l}}|q^{d-l}.$

To see this, first fix $x_{i_{0}}\in E_{i_{0}}$ which has $|E_{i_{0}}|$
choices. For each fixed $x_{i_{0}}\in E_{i_{0}},$ if
$(x_{i_{0}},\dots,x_{d})\in L(l)$, then all points
$x_{i_{1}},\dots,x_{i_{l-1}}$ are automatically chosen as $x_{i_{0}},$ and
there are at most $|E_{i_{l}}|$ choices for $x_{i_{l}}\in E_{i_{l}}.$ Since
$x_{0}$ and $x_{i_{l}}$ determine a fixed line, all points
$x_{i_{l+1}},\dots,x_{i_{d}}$ must lie on the line. Thus, there are at most
$q$ choices for each $x_{i_{l+1}},\dots,x_{i_{d}}$, because a line contains
exactly $q$ points in $\mathbb{F}_{q}^{d}.$

From (2.6), (2.7), and (2.8), it suffices to prove that for every
$l=1,\dots,d,$

$\mbox{B}=\sum_{i_{0}=0}^{\infty}\sum_{i_{1}\geq
i_{0}}^{\infty}\cdots\sum_{i_{d}\geq
i_{d-1}}^{\infty}2^{-(i_{0}+i_{1}+\cdots+i_{d})}|E_{i_{0}}||E_{i_{l}}|q^{1-l}\lesssim
1.$

Since $|E_{i_{l}}|\leq q^{d}$ and $l\geq 1,$ it is easy to see that
$|E_{i_{l}}|^{(l-1)/d}q^{1-l}\lesssim 1.$ Therefore, it follows that

$\mbox{B}\lesssim\sum_{i_{0}=0}^{\infty}\sum_{i_{1}\geq
i_{0}}^{\infty}\cdots\sum_{i_{d}\geq
i_{d-1}}^{\infty}2^{-(i_{0}+i_{1}+\cdots+i_{d})}|E_{i_{0}}||E_{i_{l}}|^{\frac{d+1-l}{d}}.$

Since $\frac{d+1-l}{d}>0$, applying (2.3) gives

$\mbox{B}\lesssim\sum_{i_{0}=0}^{\infty}\sum_{i_{1}\geq
i_{0}}^{\infty}\cdots\sum_{i_{d}\geq
i_{d-1}}^{\infty}2^{-(i_{0}+i_{1}+\cdots+i_{d})}|E_{i_{0}}|2^{\frac{(d+1-l)(d+1)i_{l}}{2d}}.$

Compute the inner summations by checking that each of them is a convergent
geometric series. It follows that

$\mbox{B}\lesssim\sum_{i_{0}=0}^{\infty}|E_{i_{0}}|2^{\frac{(-d^{2}-dl-l+1)i_{0}}{2d}}\leq\sum_{i_{0}=0}^{\infty}|E_{i_{0}}|2^{-\frac{(d+1)i_{0}}{2}}=1,$

where the last equality follows from (2.3). Thus, we complete the proof of
Theorem 2.1. ∎

######  Remark 2.2.

It seems that the similar arguments as above work for settling Conjecture 1.2,
but it may not be simple to estimate $|\Delta(s,i_{0},\dots,i_{d})|.$

##  3\. Proof of mapping properties of the Radon transform

In this section, we prove Theorem 1.3 in the case of the Radon transform.
Namely, we shall prove the following.

######  Theorem 3.1.

Let $d\geq 2$ be any integer. Then,

$\|T_{\Pi_{d-1}}f\|_{L^{d+1}(\Pi_{d-1},d\sigma)}\leq
C\|f\|_{L^{\frac{d+1}{d}}(\mathbb{F}_{q}^{d},dx)}\quad\mbox{for
all}~{}~{}f~{}~{}\mbox{on}~{}~{}\mathbb{F}_{q}^{d},$

where $C$ is independent of $|\mathbb{F}_{q}|.$

###### Proof.

First, notice that if the dimension $d$ is two, then the statement of Theorem
3.1 follows immediately from Theorem 2.1. We therefore assume that $d\geq 3.$
As before, we may assume that $f$ is a non-negative real function and

(3.1) $\sum_{x\in\mathbb{F}_{q}^{d}}[f(x)]^{(d+1)/d}=1.$

Moreover, we may assume that the function $f$ is a step function:

(3.2) $f(x)=\sum_{i=0}^{\infty}2^{-i}E_{i}(x),$

where the sets $E_{i}$ are disjoint subsets of $\mathbb{F}_{q}^{d}.$ Notice
that (3.1) and (3.2) imply that

(3.3) $\sum_{j=0}^{\infty}2^{-\frac{(d+1)j}{d}}|E_{j}|=1\quad\mbox{and
}~{}~{}|E_{j}|\leq 2^{\frac{(d+1)j}{d}}~{}~{}\mbox{for all}~{}~{}j=0,1,\dots.$

We write $\Pi_{d-1}=H\cup\Theta$ where $H$ and $\Theta$ are defined by

$H:=\\{w\in\Pi_{d-1}:(0,\dots,0)\notin w\\}$

and

$\Theta:=\\{w\in\Pi_{d-1}:(0,\dots,0)\in w\\}.$

It is clear that $H$ and $\Theta$ are disjoint. Notice that we can identify
$H$ with $\mathbb{F}_{q}^{d}\setminus\\{(0,\dots,0)\\}$ in the sense that if
$w\in H$, then there exists a unique
$w^{\prime}\in\mathbb{F}_{q}^{d}\setminus\\{(0,\dots,0)\\}$ such that

$w=\\{x\in\mathbb{F}_{q}^{d}:w^{\prime}\cdot x=1\\}.$

Thus, if $w\in H$, then we may assume that

$T_{\Pi_{d-1}}f(w)=\frac{1}{|w|}\sum_{x\in\mathbb{F}_{q}^{d}:w^{\prime}\cdot
x=1}f(x)=\frac{1}{q^{d-1}}\sum_{x\in\mathbb{F}_{q}^{d}:w^{\prime}\cdot
x=1}f(x).$

On the other hand, for a fixed $w\in\Theta,$ there is a unique line passing
through the origin, say $L_{w}$, such that

$w=\\{x\in\mathbb{F}_{q}^{d}:w^{\prime}\cdot x=0~{}\mbox{for
all}~{}w^{\prime}\in L_{w}\setminus\\{(0,\dots,0\\}\\}.$

By selecting one specific $w^{\prime}\in L_{w}\setminus\\{(0,\dots,0)\\}$ we
can identify $w\in\Theta$ with the specific point $w^{\prime}\in
L_{w}\setminus\\{(0,\dots,0)\\}.$ Throughout the paper, we denote by $S$ the
collection of the specific points each of which is chosen from
$L_{w}\setminus\\{(0,\dots,0)\\}$ for every $w\in\Theta.$ 111In the Euclidean
setting, one can consider the set $S$ as a half part of the unit sphere.
However, it is not true in general in the finite field setting. For example,
if the dimension $d$ is four and $-1\in\mathbb{F}_{q}$ is a square number,
then the line $l=\\{t(i,1,i,1):t\in\mathbb{F}_{q}\\}$ does not intersect the
set $\\{x\in\mathbb{F}_{q}^{4}:x_{1}^{2}+\cdots+x_{4}^{2}=1\\}.$ Thus, we also
assume that if $w\in\Theta,$ then

$T_{\Pi_{d-1}}f(w)=\frac{1}{q^{d-1}}\sum_{x\in\mathbb{F}_{q}^{d}:w^{\prime}\cdot
x=0}f(x),$

where $w^{\prime}\in S.$ Since $\Pi_{d-1}=H\cup\Theta$ and
$H\cap\Theta=\emptyset,$ the Radon transform $T_{\Pi_{d-1}}$ can be viewed as

$T_{\Pi_{d-1}}f(w)=T_{0}f(w)+T_{1}f(w)\quad\mbox{for}~{}~{}w\in\Pi_{d-1},$

where the operators $T_{0}$ and $T_{1}$ are defined as

$T_{0}f(w)=\frac{\Theta(w)}{q^{d-1}}\sum_{x\in\mathbb{F}_{q}^{d}:w^{\prime}\cdot
x=0}f(x)$

and

$T_{1}f(w)=\frac{H(w)}{q^{d-1}}\sum_{x\in\mathbb{F}_{q}^{d}:w^{\prime}\cdot
x=1}f(x).$

In order to prove Theorem 3.1, it therefore suffices to show that the
following two inequalities hold:

(3.4)
$\|T_{0}f\|_{L^{d+1}(\Pi_{d-1},d\sigma)}\lesssim\|f\|_{L^{\frac{d+1}{d}}(\mathbb{F}_{q}^{d},dx)},$

and

(3.5)
$\|T_{1}f\|_{L^{d+1}(\Pi_{d-1},d\sigma)}\lesssim\|f\|_{L^{\frac{d+1}{d}}(\mathbb{F}_{q}^{d},dx)},$

where the functions $f$ satisfy (3.1), (3.2), and (3.3).

###  3.1. Proof of the inequality (3.4)

Let us denote by $\chi$ the canonical additive character of $\mathbb{F}_{q}$
(see [7] or [5]). Recall that the orthogonality relation of $\chi$ holds:

$\sum_{s\in\mathbb{F}_{q}}\chi{(as)}=\left\\{\begin{array}[]{ll}0&\mbox{if}~{}~{}a\in\mathbb{F}_{q}^{*}=\mathbb{F}_{q}\setminus\\{0\\}\\\
q&\mbox{if}~{}~{}a=0.\end{array}\right.$

Using the orthogonality relation of $\chi$, we have

$T_{0}f(w)=\frac{\Theta(w)}{q^{d-1}}\sum_{x\in\mathbb{F}_{q}^{d}}\left[\frac{1}{q}\sum_{s\in\mathbb{F}_{q}}\chi(s(w^{\prime}\cdot
x))\right]f(x)$
$=\frac{\Theta(w)}{q^{d}}\sum_{x\in\mathbb{F}_{q}^{d}}\sum_{s=0}\chi(s(w^{\prime}\cdot
x))f(x)+\frac{\Theta(w)}{q^{d}}\sum_{x\in\mathbb{F}_{q}^{d}}\sum_{s\in\mathbb{F}_{q}^{*}}\chi(s(w^{\prime}\cdot
x))f(x)$
$=\frac{\Theta(w)}{q^{d}}\sum_{x\in\mathbb{F}_{q}^{d}}f(x)+\frac{\Theta(w)}{q^{d}}\sum_{x\in\mathbb{F}_{q}^{d}}\sum_{s\in\mathbb{F}_{q}^{*}}\chi(s(w^{\prime}\cdot
x))f(x)$ $:=T_{0}^{\star}f(w)+T_{0}^{\star\star}f(w).$

Since $|T_{0}^{\star}f(w)|\leq\|f\|_{L^{1}(\mathbb{F}_{q}^{d},dx)}$ for all
$w\in\Pi_{d-1},$ we see that

$\|T_{0}^{\star}f\|_{L^{d+1}(\Pi_{d-1},d\sigma)}\leq\|f\|_{L^{1}(\mathbb{F}_{q}^{d},dx)}\leq\|f\|_{L^{\frac{d+1}{d}}(\mathbb{F}_{q}^{d},dx)},$

where we used the facts that $dx$ and $d\sigma$ are the normalized counting
measure on $\mathbb{F}_{q}^{d}$ and the normalized surface measure on
$\Pi_{d-1}$ respectively. To prove the inequality (3.4), it remains to prove
that for all functions $f$ satisfying (3.1), (3.2), and (3.3),

(3.6)
$\|T_{0}^{\star\star}f\|_{L^{d+1}(\Pi_{d-1},d\sigma)}\lesssim\|f\|_{L^{\frac{d+1}{d}}(\mathbb{F}_{q}^{d},dx)}=q^{-\frac{d^{2}}{d+1}},$

where the last equality follows from (3.1). We need the following lemma.

######  Lemma 3.2.

Let $d\geq 3$. Then, for every subset $E\subset\mathbb{F}_{q}^{d},$ we have

$\|T_{0}^{\star\star}E\|_{L^{\frac{d+1}{2}}(\Pi_{d-1},d\sigma)}\lesssim
q^{-\frac{(d^{2}+1)}{d+1}}|E|^{\frac{d-1}{d+1}}.$

###### Proof.

Since $d\geq 3,$ we see that the statement of Lemma 3.2 follows immediately by
interpolating the following two estimates: for all indicator functions $E(x)$
on $\mathbb{F}_{q}^{d},$

(3.7) $\|T_{0}^{\star\star}E\|_{L^{\infty}(\Pi_{d-1},d\sigma)}\lesssim
q^{-d+1}|E|$

and

(3.8) $\|T_{0}^{\star\star}E\|_{L^{2}(\Pi_{d-1},d\sigma)}\lesssim
q^{-d+\frac{1}{2}}|E|^{\frac{1}{2}}.$

To obtain (3.7), notice that

$\|T_{0}^{\star\star}E\|_{L^{\infty}(\Pi_{d-1},d\sigma)}=\|T_{0}E-T_{0}^{\star}E\|_{L^{\infty}(\Pi_{d-1},d\sigma)}$
$\leq\|T_{0}E\|_{L^{\infty}(\Pi_{d-1},d\sigma)}+\|T_{0}^{\star}E\|_{L^{\infty}(\Pi_{d-1},d\sigma)}\leq
q^{-d+1}|E|+q^{-d}|E|\sim q^{-d+1}|E|.$

It remains to prove that (3.8) holds. It follows that for every set
$E\subset\mathbb{F}_{q}^{d},$

$\|T_{0}^{\star\star}E\|^{2}_{L^{2}(\Pi_{d-1},d\sigma)}=\frac{1}{|\Pi_{d-1}|}\sum_{w\in\Pi_{d-1}}\left|\frac{\Theta(w)}{q^{d}}\sum_{x\in\mathbb{F}_{q}^{d}}\sum_{s\in\mathbb{F}_{q}^{*}}\chi(s(w^{\prime}\cdot
x))E(x)\right|^{2}$ $=\frac{1}{|\Pi_{d-1}|}\sum_{w^{\prime}\in
S}\left|\frac{1}{q^{d}}\sum_{x\in
E}\sum_{s\in\mathbb{F}_{q}^{*}}\chi(s(w^{\prime}\cdot
x))\right|^{2}:=\frac{1}{|\Pi_{d-1}|}\sum_{w^{\prime}\in
S}\Gamma(w^{\prime}),$

where we used the fact that $S$ can be identified with $\Theta.$ Using a
change of variables, we see that for each $w^{\prime}\in
S,~{}\Gamma(w^{\prime})=\Gamma(tw^{\prime})$ for all $t\in\mathbb{F}_{q}^{*}.$
By the definition of $S,$ it therefore follows that

$\|T_{0}^{\star\star}E\|^{2}_{L^{2}(\Pi_{d-1},d\sigma)}\leq\frac{1}{|\Pi_{d-1}|}\frac{1}{q-1}\sum_{w^{\prime}\in\mathbb{F}_{q}^{d}}\left|\frac{1}{q^{d}}\sum_{x\in
E}\sum_{s\in\mathbb{F}_{q}^{*}}\chi(s(w^{\prime}\cdot x))\right|^{2}.$

Since $|\Pi_{d-1}|\sim q^{d}$, if we expand the square term and apply the
orthogonality relation of $\chi$ to the sum over
$w^{\prime}\in\mathbb{F}_{q}^{d}$, then we see that

$\|T_{0}^{\star\star}E\|^{2}_{L^{2}(\Pi_{d-1},d\sigma)}\lesssim\frac{1}{q^{3d+1}}\sum_{w^{\prime}\in\mathbb{F}_{q}^{d}}\sum_{x,x^{\prime}\in
E}\sum_{s,s^{\prime}\in\mathbb{F}_{q}^{*}}\chi(w^{\prime}\cdot(sx-s^{\prime}x^{\prime}))$
$=\frac{1}{q^{2d+1}}\sum_{x,x^{\prime}\in
E,s,s^{\prime}\in\mathbb{F}_{q}^{*}:sx=s^{\prime}x^{\prime}}1\leq\frac{|E|}{q^{2d-1}}.$

Thus, the proof of Lemma 3.2 is complete. ∎

We now prove (3.6). Since we have assumed that $f$ is considered as a step
function (3.2), it follows that

$\|T_{0}^{\star\star}f\|^{2}_{L^{d+1}(\Pi_{d-1},d\sigma)}=\|(T_{0}^{\star\star}f)(T_{0}^{\star\star}f)\|_{L^{\frac{d+1}{2}}(\Pi_{d-1},d\sigma)}$
$\leq\sum_{i=0}^{\infty}\sum_{j=0}^{\infty}2^{-i-j}\|(T_{0}^{\star\star}E_{i})(T_{0}^{\star\star}E_{j})\|_{L^{\frac{d+1}{2}}(\Pi_{d-1},d\sigma)}$
$\sim\sum_{i=0}^{\infty}\sum_{j\geq
i}^{\infty}2^{-i-j}\|(T_{0}^{\star\star}E_{i})(T_{0}^{\star\star}E_{j})\|_{L^{\frac{d+1}{2}}(\Pi_{d-1},d\sigma)},$

where the last line follows from the symmetry of $i,j.$ By Hölder’s
inequality, the inequality (3.7), and Lemma 3.2, (3.6) will follow if we prove
that

$\sum_{i=0}^{\infty}\sum_{j\geq
i}^{\infty}2^{-i-j}|E_{i}||E_{j}|^{\frac{d-1}{d+1}}\lesssim 1.$

This can be justified by making use of (3.3) and computing the summation over
$j$ variable:

$\sum_{i=0}^{\infty}\sum_{j\geq
i}^{\infty}2^{-i-j}|E_{i}||E_{j}|^{\frac{d-1}{d+1}}\leq\sum_{i=0}^{\infty}\sum_{j\geq
i}^{\infty}2^{-i-j}|E_{i}|\left(2^{\frac{(d+1)j}{d}}\right)^{\frac{d-1}{d+1}}\sim\sum_{i=0}^{\infty}|E_{i}|2^{-\frac{(d+1)i}{d}}=1,$

which completes the proof of (3.4).

###  3.2. Proof of the inequality (3.5)

By showing that the inequality (3.5) holds, we shall complete the proof of
Theorem 3.1. We shall take the same steps as in the previous subsection. From
the orthogonality relation of $\chi,$ it follows that for all $w\in\Pi_{d-1},$

$T_{1}f(w)=\frac{H(w)}{q^{d}}\sum_{x\in\mathbb{F}_{q}^{d}}f(x)+\frac{H(w)}{q^{d}}\sum_{x\in\mathbb{F}_{q}^{d}}\sum_{s\in\mathbb{F}_{q}^{*}}\chi(s(w^{\prime}\cdot
x-1))f(x):=T_{1}^{\star}(w)+T_{1}^{\star\star}(w).$

As before, it is easy to see that

$\|T_{1}^{\star}f\|_{L^{d+1}(\Pi_{d-1},d\sigma)}\leq\|f\|_{L^{\frac{d+1}{d}}(\mathbb{F}_{q}^{d},dx)}.$

Thus, it is enough to prove that for all functions $f$ satisfying (3.1),
(3.2), and (3.3),

$\|T_{1}^{\star\star}f\|_{L^{d+1}(\Pi_{d-1},d\sigma)}\lesssim\|f\|_{L^{\frac{d+1}{d}}(\mathbb{F}_{q}^{d},dx)}=q^{-\frac{d^{2}}{d+1}},$

where the last equality follows from (3.1). From the same arguments as in the
proof of (3.4), our task is only to obtain Lemma 3.2 for the operator
$T_{1}^{\star\star}.$ As in the proof of Lemma 3.2, it suffices to prove the
following two equalities: for every subset $E$ of $\mathbb{F}_{q}^{d},$

(3.9) $\|T_{1}^{\star\star}E\|_{L^{\infty}(\Pi_{d-1},d\sigma)}\lesssim
q^{-d+1}|E|$

and

(3.10) $\|T_{1}^{\star\star}E\|_{L^{2}(\Pi_{d-1},d\sigma)}\lesssim
q^{-d+\frac{1}{2}}|E|^{\frac{1}{2}}.$

The inequality (3.9) follows immediately from the same argument as before. To
prove (3.10), we observe that

$\|T_{1}^{\star\star}E\|^{2}_{L^{2}(\Pi_{d-1},d\sigma)}=\frac{1}{|\Pi_{d-1}|}\sum_{w\in\Pi_{d-1}}\left|\frac{H(w)}{q^{d}}\sum_{x\in\mathbb{F}_{q}^{d}}\sum_{s\in\mathbb{F}_{q}^{*}}\chi(s(w^{\prime}\cdot
x-1))E(x)\right|^{2}$ $=\frac{1}{|\Pi_{d-1}|}\sum_{w\in
H}\left|\frac{1}{q^{d}}\sum_{x\in
E}\sum_{s\in\mathbb{F}_{q}^{*}}\chi(s(w^{\prime}\cdot x-1))\right|^{2}.$

Recall that $H\subset\Pi_{d-1}$ can be identified with
$\mathbb{F}_{q}^{d}\setminus\\{(0,\dots,0)\\}.$ Thus, if we dominate the sum
over $w\in H$ by the sum over $w^{\prime}\in\mathbb{F}_{q}^{d}$, expand the
square term, and use the orthogonality relation of $\chi$ over the variable
$w^{\prime}\in\mathbb{F}_{q}^{d},$ then it follows that

$0\leq\|T_{1}^{\star\star}E\|^{2}_{L^{2}(\Pi_{d-1},d\sigma)}\leq\frac{1}{|\Pi_{d-1}|q^{d}}\sum_{x,x^{\prime}\in
E,s,s^{\prime}\in\mathbb{F}_{q}^{*}:sx=s^{\prime}x^{\prime}}\chi(-s+s^{\prime}).$
$=\frac{(q-1)|E|}{|\Pi_{d-1}|q^{d}}+\frac{1}{|\Pi_{d-1}|q^{d}}\sum_{x,x^{\prime}\in
E,s,s^{\prime}\in\mathbb{F}_{q}^{*}:sx=s^{\prime}x^{\prime},s\neq
s^{\prime}}\chi(-s+s^{\prime})=\mbox{I}+\mbox{II}.$

Since $|\Pi_{d-1}|\sim q^{d}$, it is clear that
$\mbox{I}\sim\frac{|E|}{q^{2d-1}}.$ We claim that $\mbox{II}\leq 0.$ Indeed,
if we use a change of the variables by putting $s=t,\frac{s^{\prime}}{s}=u,$
then we see that

$\mbox{II}=\frac{1}{|\Pi_{d-1}|q^{d}}\sum_{x,x^{\prime}\in
E,t,u\in\mathbb{F}_{q}^{*}:x=ux^{\prime},u\neq 0,1}\chi(-t(1-u)).$

Since $u\neq 1,$ the summation over $t\in\mathbb{F}_{q}^{*}$ is exactly $-1.$
Thus, our claim follows from the observation:

$\mbox{II}=\frac{-1}{|\Pi_{d-1}|q^{d}}\sum_{x,x^{\prime}\in
E,u\in\mathbb{F}_{q}^{*}:x=ux^{\prime},u\neq 0,1}1\leq 0.$

Therefore, we conclude that

$\|T_{1}^{\star\star}E\|^{2}_{L^{2}(\Pi_{d-1},d\sigma)}\leq\mbox{I}+\mbox{II}\leq\mbox{I}\sim\frac{|E|}{q^{2d-1}},$

which implies that the inequality (3.10) holds . We have finished proving
Theorem 3.1. ∎

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