# Capturing near-field circular dichroism enhancements from far-field
measurements

Jorge Olmos-Trigo<EMAIL_ADDRESS>Donostia International Physics
Center, Paseo Manuel de Lardizabal 4, 20018 Donostia-San Sebastian, Spain.
Centro de Física de Materiales, Paseo Manuel de Lardizabal 5, 20018 Donostia-
San Sebastian, Spain.  Jon Lasa-Alonso  Donostia International Physics Center,
Paseo Manuel de Lardizabal 4, 20018 Donostia-San Sebastian, Spain.  Centro de
Física de Materiales, Paseo Manuel de Lardizabal 5, 20018 Donostia-San
Sebastian, Spain.  Iker Gómez-Viloria  Centro de Física de Materiales, Paseo
Manuel de Lardizabal 5, 20018 Donostia-San Sebastian, Spain.  Gabriel Molina-
Terriza  Donostia International Physics Center, Paseo Manuel de Lardizabal 4,
20018 Donostia-San Sebastian, Spain.  Centro de Física de Materiales, Paseo
Manuel de Lardizabal 5, 20018 Donostia-San Sebastian, Spain.  IKERBASQUE,
Basque Foundation for Science, María Díaz de Haro 3, 48013 Bilbao, Spain.
Aitzol García-Etxarri  Donostia International Physics Center, Paseo Manuel de
Lardizabal 4, 20018 Donostia-San Sebastian, Spain.  IKERBASQUE, Basque
Foundation for Science, María Díaz de Haro 3, 48013 Bilbao, Spain.

###### Abstract

Molecular Circular dichroism (CD) spectroscopy faces significant limitations
due to the inherent weakness of chiroptical light-matter interactions. In this
view, resonant optical antennas constitute a promising solution to this
problem since they can be tuned to increase the CD enhancement factor,
$\mathit{f}_{\rm{CD}}$, a magnitude describing the electromagnetic near-field
enhancement of scatterers associated with a given helicity. Here, we derive an
exact multipolar expansion of $\mathit{f}_{\rm{CD}}$, which is valid to deduce
the integrated near-field CD enhancements of chiral molecules in the presence
of scatterers of any size and shape under general illumination conditions.
Based on our analytical findings, we show that the near-field
$\mathit{f}_{\rm{CD}}$ factor can be related to magnitudes that can be
computed in the far-field, i.e., the scattering cross-section and the helicity
expectation value. Moreover, we show that in the case of lossless
cylindrically symmetric samples, the near-field $\mathit{f}_{\rm{CD}}$ factor
can be inferred experimentally only from two far-field measurements at
specific scattering angles. Our contribution paves the way for the
experimental characterization of devices capable of enhancing molecular CD
spectroscopy.

Chirality is a geometrical property of objects which are not superimposable
with its mirror image. Chiral objects are ubiquitous in nature. Many organic
molecules, such as glucose and most biological amino acids, are chiral. Not to
mention the DNA double helix, which, in its standard form, always twists like
a right-handed screw Watson (2012).

In the pharmaceutical industry, chiral speciﬁcity is critical because opposite
enantiomers, i.e., mirror pairs of chiral molecules, can have beneficial or
detrimental biological effects on our organism depending on their handedness.
Perhaps the most representative case of this is the so-called “Thalidomide
tragedy” which took place during the late 50s McBride (1961). Thalidomide was
extensively prescribed to pregnant women due to its benefits in alleviating
nausea. Instead, this drug prevented the proper growth of the fetus Lenz and
Knapp (1962). The outcome was that thousands of infants were born with severe
congenital malformations. That fatality occurred because Thalidomide is a
chiral molecule that was marketed as a 50/50 mixture of R and S enantiomers.
While the S enantiomer is effective in alleviating nausea, the R enantiomer is
toxic. Thus, drugs consisting of chiral molecules may indeed have different
therapeutic or toxicological effects.

Even if they share the same atomic composition, enantiomer pairs are
indistinguishable when measuring their scalar molecular properties. Their
chiral nature is revealed only when interacting with other chiral entities. In
electromagnetism, the most common chiral observable is helicity Calkin (1965).
Chiral molecules show a preferential absorption for fields of opposite
helicities (left or right-handed polarized waves with helicity eigenvalues of
$\sigma=\pm 1$, respectively). In a conventional Circular dichroism (CD)
spectroscopy setup, a chiral molecular solution is sequentially illuminated by
fields of opposite helicities, and the total transmitted power is recorded for
each case. The CD signal is then computed by taking the difference between
these two power measurements in transmission. However, the inherent weakness
of chiroptical responses strongly limits the sensitivity of CD spectroscopy.

Optical antennas, designed to control the properties of light Yu and Capasso
(2014), are promising candidates to enhance the spectroscopic signals of
chiral samples. The underlying phenomenon is that optical antennas can be
engineered to enhance electromagnetic fields while preserving the
electromagnetic helicity Wen  _et al._ (2016); Abendroth  _et al._ (2020);
Zhang  _et al._ (2020). Many works have explored optical antennas Zambrana-
Puyalto and Bonod (2016); García-Etxarri and Dionne (2013); Czajkowski and
Antosiewicz (2022) and metasurfaces, namely, flat planar arrays of optical
antennas, made of metallic Fan and Govorov (2010); García-Guirado  _et al._
(2018); Poulikakos  _et al._ (2018) or/and high refractive index materials
García-Etxarri  _et al._ (2011); Olmos-Trigo  _et al._ (2020a) for enhanced
chiral sensing Tseng  _et al._ (2020); Warning  _et al._ (2021). Examples of
such works can be found, for instance, in the ultraviolet Hu  _et al._ (2019),
visible Hendry  _et al._ (2010); Kelly  _et al._ (2018); Zhang  _et al._
(2017); Yao and Liu (2018); Hanifeh and Capolino (2020); A. Paiva-Marques  _et
al._ (2020); Mohammadi  _et al._ (2021) or near -and far-infrared spectral
range Hendry  _et al._ (2012); Solomon  _et al._ (2018); Graf  _et al._
(2019); Garcia-Guirado  _et al._ (2019); Lasa-Alonso  _et al._ (2020);
Droulias and Bougas (2020); Rui  _et al._ (2022). Moreover and quite recently,
optical cavities have been also proposed as efficient platforms for enhanced
chiral sensing Feis  _et al._ (2020); Khanbekyan and Scheel (2022); Olmos-
Trigo and Zambrana-Puyalto (2022).

However, researchers rely exclusively on numerical methods to design enhanced
chiral sensing devices, as capturing the vector character of the near-field
contribution can be challenging. Here, we derive an exact multipolar expansion
of $\mathit{f}_{\rm{CD}}$, which is valid to deduce the integrated near-field
CD enhancements of chiral molecules in the presence of scatterers of any size
and shape under general illumination conditions. From our analytical results,
we show that $\mathit{f}_{\rm{CD}}$ is proportional to the scattering cross-
section and the helicity expectation value, which are experimentally
measurable magnitudes in the far-field. In addition to this, we also show that
for lossless and cylindrically symmetric scatterers, it is possible to infer
the $\mathit{f}_{\rm{CD}}$ factor only with two far-field measurements: the
extinction cross-section and the helicity density at specific scattering
angles.

To describe the excitation of chiral molecules, we adopt the formalism
introduced by Tang and Cohen Tang and Cohen (2010). The CD signal of a chiral
molecule under the illumination of a well-defined helicity field ($\sigma=\pm
1$) can be computed in vacuum as

$\rm{CD}_{\rm{inc}}({\bf
r})=-\frac{4}{\epsilon}\text{Im}\\{G\\}C^{\sigma}_{\rm{inc}}({\bf r}),$ (1)

where $\rm{G}$ is the chiral polarizability of the molecule and
$\rm{C}^{\sigma}_{\rm{inc}}({\bf r})$ is the incident local density of optical
chirality Poulikakos  _et al._ (2019),

$\displaystyle\rm{C}^{\sigma}_{\rm{inc}}({\bf
r})=\frac{k\epsilon}{2}\text{Im}\\{{\bf E}^{\sigma}_{\rm{inc}}({\bf r})\cdot
Z{{\bf H}_{\rm{inc}}^{\sigma}}^{*}({\bf r})\\}.$ (2)

Here, $k$ is the radiation wavevector, $\epsilon$ is the electric permittivity
of the medium and $Z=\sqrt{\mu/\epsilon}$ its electromagnetic impedance.
Moreover, ${\bf E}^{\sigma}_{\rm{inc}}({\bf r})$ and ${{\bf
H}_{\rm{inc}}^{\sigma}}({\bf r})$ refer to incident electromagnetic fields
with well-defined helicity 111Notice that under the widely used illumination
of circularly-polarized plane-waves, we get $|C^{\sigma}_{\rm{inc}}({\bf
r})|=k\epsilon|E_{0}|^{2}/2\textasciidieresis$, $E_{0}$ denoting the amplitude
of the electric field Tang and Cohen (2010). That is, we obtain a scalar that
does not depend on the spatial coordinates. (see Appendix A.1 for more
details).

In the presence of achiral antennas (see Fig. 1) and in the helicity basis
Olmos-Trigo  _et al._ (2019), the total electromagnetic fields can be
generally written as ${\bf E}^{\sigma\sigma^{\prime}}_{\rm{tot}}({\bf r})={\bf
E}^{\sigma\sigma^{\prime}}_{\rm{sca}}({\bf r})+{\bf
E}^{\sigma}_{\rm{inc}}({\bf r})\delta_{\sigma\sigma^{\prime}}$. Here ${\bf
E}^{\sigma\sigma^{\prime}}_{\rm{sca}}({\bf r})$ is the scattered
electromagnetic field written in terms of the electromagnetic modes with well
defined helicity $\sigma^{\prime}=\pm 1$ (see Appendix A.2 for more details).
In analogy with Eq. (1), we can express the total CD signal of a chiral
molecule in the presence of an achiral optical antenna as Fernandez-Corbaton
_et al._ (2016)

$\rm{CD}_{\rm{tot}}({\bf
r})=-\frac{4}{\epsilon}\text{Im}\\{G\\}C^{\sigma}_{\rm{tot}}({\bf
r})=\frac{k}{2}\text{Im}\\{G\\}\sum_{\sigma^{\prime}=\pm
1}\sigma^{\prime}|{\bf E}^{\sigma\sigma^{\prime}}_{\rm{tot}}({\bf r})|^{2}.$
(3)

We seek to maximize $\rm{CD}_{\rm{tot}}({\bf r})$ with Eq. (3). However, such
enhancements cannot be achieved through $\rm{G}$ since it is a ﬁxed chiral
molecular parameter that cannot be engineered upon illumination. In contrast,
the total density of optical chirality, $\rm{C}^{\sigma}_{\rm{tot}}({\bf r})$,
can be tuned to enhance light-matter interactions and thus, increase the
sensitivity of CD spectroscopy 222For the sake of simplicity and hereinafter,
we assume the factor $(k\epsilon)/2$ in the definition of the density of
optical chirality.. To get a deeper insight into the total density of optical
chirality, let us split $\rm{C}^{\sigma}_{\rm{tot}}({\bf r})$ into three
contributions; namely, $\rm{C}^{\sigma}_{\rm{tot}}({\bf
r})=\rm{C}^{\sigma}_{\rm{inc}}({\bf r})+\rm{C}^{\sigma}_{\rm{sca}}({\bf
r})+\rm{C}^{\sigma}_{\rm{int}}({\bf r})$, with

$\displaystyle\rm{C}^{\sigma}_{\rm{inc}}({\bf r})$ $\displaystyle=$
$\displaystyle\sigma|{\bf E}^{\sigma}_{\rm{inc}}({\bf r})|^{2},$ (4)
$\displaystyle\rm{C}^{\sigma}_{\rm{sca}}({\bf r})$ $\displaystyle=$
$\displaystyle|{\bf E}^{\sigma+}_{\rm{sca}}({\bf r})|^{2}-|{\bf
E}^{\sigma-}_{\rm{sca}}({\bf r})|^{2},$ (5)
$\displaystyle\rm{C}^{\sigma}_{\rm{int}}({\bf r})$ $\displaystyle=$
$\displaystyle 2\sigma\text{Re}\\{{{\bf E}^{\sigma}_{\rm{inc}}}^{*}({\bf
r})\cdot{{\bf E}^{\sigma\sigma}_{\rm{sca}}}({\bf r})\\}.$ (6)

It is experimentally challenging to place enantiomers at will, namely, at a
desired spatial coordinate ${\bf r}$. Accordingly, it is more convenient to
introduce an averaged expression of the local CD enhancement factor that gives
insight into how efficient the nanoantenna might be at enhancing CD
spectroscopy Lasa-Alonso  _et al._ (2020). To that end, let us first integrate
both Eq. (1) and Eq. (3) over an imaginary sphere of radius $r$ surrounding
the achiral antenna to calculate then the ratio between these integrals,
namely,

$\mathit{f}_{\rm{CD}}=\frac{\int{\rm{CD}}_{\rm{tot}}({\bf
r})dS}{\int{\rm{CD}}_{\rm{inc}}({\bf
r})dS}=1+\frac{\int{\rm{C}}^{\sigma}_{\rm{sca}}({\bf
r})dS+\int{\rm{C}}^{\sigma}_{\rm{int}}({\bf
r})dS}{\int{\rm{C}}^{\sigma}_{\rm{inc}}({\bf r})dS},$ (7)

where $dS=r^{2}\sin\theta d\varphi d\theta$. To compute Eq. (7), we need the
orthogonality relations that the incident and scattered electromagnetic fields
satisfy when written in terms of the multipoles with well-defined helicity.
Fortunately, these can be derived from Jackson’s book in its third edition
Jackson (1999) (see Appendix B.1 for the explicit derivation). Now, by
considering these relations, and after some algebra (see Appendix B.2 for more
details), we arrive at

$\mathit{f}_{\rm{CD}}=1+\frac{\tilde{\rm{C}}^{\sigma}_{\rm{sca}}+\tilde{\rm{C}}^{\sigma}_{\rm{int}}}{\tilde{\rm{C}}^{\sigma}_{\rm{inc}}},$
(8)

where

$\displaystyle\tilde{\rm{C}}^{\sigma}_{\rm{inc}}$ $\displaystyle=$
$\displaystyle\sigma\sum_{lm}|C^{\sigma}_{lm}|^{2}G_{j_{l}j_{l}},$ (9)
$\displaystyle\tilde{\rm{C}}^{\sigma}_{\rm{sca}}$ $\displaystyle=$
$\displaystyle\sigma\sum_{lm}|C^{\sigma}_{lm}|^{2}\text{Re}\\{a_{lm}{b_{lm}}^{*}\\}G_{h_{l}h_{l}},$
(10) $\displaystyle\tilde{\rm{C}}^{\sigma}_{\rm{int}}$ $\displaystyle=$
$\displaystyle\sigma\sum_{lm}|C^{\sigma}_{lm}|^{2}\text{Re}\\{\left(a_{lm}+b_{lm}\right)G_{j_{l}h_{l}}\\}.$
(11)

Here $\tilde{C}$ denotes the average of $C$ over a spherical surface, $a_{lm}$
and $b_{lm}$ are the so-called electric and magnetic scattering coefficients
of an arbitrary sample while $C^{\sigma}_{lm}$ denotes the incident
coefficients characterizing the nature of the wave. Moreover,

$G_{f_{l}g_{l}}=\frac{1}{2}\left[2f^{*}_{l}(u)g_{l}(u)+\frac{1}{u^{2}}\frac{\partial}{\partial
u}\left(uf^{*}_{l}(u)\frac{\partial}{\partial
u}\left(ug_{l}(u)\right)\right)\right].$ (12)

Here $G_{f_{l}g_{l}}$ is a scalar function that depends on spherical Bessel
and Hankel functions. In particular, we may have
$\\{f_{l},g_{l}\\}=\\{j_{l},j_{l}\\},\\{j_{l},h_{l}\\},\\{h_{l},h_{l}\\}$,
$j_{l}$ and $h_{l}$ being the spherical Bessel and Hankel functions,
respetively Jackson (1999). Moreover, notice that $u=kr$ is the radius of the
imaginary sphere, normalized by $\lambda/2\pi$, surrounding the object where
the averaging integral is performed. For more details, please check Appendix
B.1.

Figure 1: Scattering process in which an incident field with well-defined
helicity (red beam with $\sigma=+1$) impinges on an achiral antenna,
represented by a glossy cube. Both $\rm{R}$-and-$\rm{S}$ Thalomide enantiomers
are also depicted close to the achiral antenna. Figure 2: Circular dichroism
enhancements for a multipolar sphere under the illumination of a circularly
polarized plane-wave. a) $\mathit{f}_{\rm{CD}}$ (exact) and b)
$\mathit{f}_{\rm{CD}}-{\tilde{\rm{C}}^{\sigma}_{\rm{int}}}/{\tilde{\rm{C}}^{\sigma}_{\rm{inc}}}$
(scattering). Here $u=x+\delta_{x}$, with $\delta_{x}\ll 1$, being $x=ka$ the
optical size and $m$ the index contrast.

Equation (8), together with Eqs. (9)-(12), is the first main result of this
paper. These equations describe the integrated CD enhancement in the presence
of scatterers of any size and shape under the excitation of fields with well-
defined helicity. Our results overcome previous approximations, such as the
widely used system of a circularly-polarized plane-wave illuminating dipolar
objects Zhang  _et al._ (2017); Yao and Liu (2018); Hanifeh and Capolino
(2020); A. Paiva-Marques  _et al._ (2020); Hendry  _et al._ (2012); Solomon
_et al._ (2018); Graf  _et al._ (2019); Garcia-Guirado  _et al._ (2019); Lasa-
Alonso  _et al._ (2020); Droulias and Bougas (2020); Mohammadi  _et al._
(2021); Rui  _et al._ (2022). Thus, our findings may find applications in
chiral sensing and chiral spectroscopy techniques beyond the current state-of-
the-art.

At this point and for didactic purposes, we provide the steps to find
$\mathit{f}_{\rm{CD}}$ around any optical antenna, which can be organized as
follows:

  1. 1.

First, we need the solution of the electromagnetic fields under the
illumination of a well-defined helicity beam. These can be obtained by any
Maxwell’s solver 333We notice that we can directly compute the
$\mathit{f}_{\rm{CD}}$ factor for the exact solution of the electromagnetic
fields under the illumination of a well-defined helicity beam. However, we
would lose track of the role of the multipoles contributing to the
$\mathit{f}_{\rm{CD}}$ factor, Moreover, imaginary integrating spheres
surrounding the object would be needed at each calculation of the fields in
order to perform the averaging integral surrounding the object..

  2. 2.

Then, we project in the far field the exact solution of the electromagnetic
fields scattered by the object to obtain the scattering coefficients (see Eq.
(9.123) in Jackson’s book in its third edition Jackson (1999)). For
convenience, we provide in Appendix C the conversion between our scattering
coefficients $a_{lm}$ and $b_{lm}$ to the ones employed in Jackson’s book.
Also notice that for spherical objects, we might use Mie’s theory to directly
jump to step 3.

  3. 3.

Finally, we can compute $\mathit{f}_{\rm{CD}}$ via Eq. (8), together with Eqs.
(9)-(12).

As an illustrative example of the abovementioned recipe, we depict the exact
expression of $\mathit{f}_{\rm{CD}}$ for a sphere sustaining several
multipoles under plane-wave illumination (see Fig. 2a)). Moreover, we also
depict in Fig. 2b) the scattering contribution to $\mathit{f}_{\rm{CD}}$. That
is,
$\mathit{f}_{\rm{CD}}-{\tilde{\rm{C}}^{\sigma}_{\rm{int}}}/{\tilde{\rm{C}}^{\sigma}_{\rm{inc}}}$.
In fact, we infer, by comparing Fig. 2a) with Fig. 2b), that the exact
solution of $\mathit{f}_{\rm{CD}}$ can be fairly approximated to just
scattering in near-field, namely, $\mathit{f}_{\rm{CD}}\sim
1+{\tilde{\rm{C}}^{\sigma}_{\rm{sca}}}/{\tilde{\rm{C}}^{\sigma}_{\rm{inc}}}$.
Now, we understand the validity of this approximation based on the following
facts:

  * •

On mathematical grounds, and according to Eq. (34), it can be checked that
$G_{h_{l}h_{l}}\gg|G_{j_{l}h_{l}}|$ for $u<l$. That is, fundamental properties
of spherical Bessel functions dictate that the scattering contribution
dominates over the interference term for $u<l$.

  * •

On physical grounds, we notice that there are no strong resonances of the
$l$-th multipole for $u>l$. For example, dipolar and quadrupolar resonances
typically arise for $u<1$ and $1<u<2$, respectively Coe  _et al._ (2022).

Table 1: Analytic expressions for the CD enhancement factor, $\mathit{f}_{\rm{CD}}$, depending on the interaction between an incident plane-wave and an achiral scatterer. Here $a_{lm}$ and $b_{lm}$ denote the electric and magnetic scattering coefficients and $\\{G_{h_{l}h_{l}},G_{j_{l}h_{l}}\\}$ can be computed from Eq. (34); $\langle\Lambda\rangle$ denotes the electromagnetic helicity expectation value and $\Lambda_{\theta}$ the helicity density; $\sigma_{\rm{sca}}$ and $\sigma_{\rm{ext}}$ are the scattering and extinction cross-sections; and $\lambda$ the radiation wavelength. Approximation in the calculation of $\mathit{f}_{\rm{CD}}$ | Plane wave illumination |
---|---|---
Exact multipolar expansion | $\mathit{f}_{\rm{CD}}=1+\sum_{lm}\left(2l+1\right)\left(\text{Re}\\{a_{lm}{b_{lm}}^{*}\\}G_{h_{l}h_{l}}+\text{Re}\\{\left(a_{lm}+b_{lm}\right)G_{j_{l}h_{l}}\\}\right)$ |
Scattering approximation | $\mathit{f}_{\rm{CD}}\sim 1+\sum_{lm}\left(2l+1\right)\text{Re}\\{a_{lm}{b_{lm}}^{*}\\}G_{h_{l}h_{l}}$ |
Scattering approximation for arbitrary samples well-described by a single multipolar order $l$ | $\mathit{f}_{\rm{CD}}\sim 1+\dfrac{\pi G_{h_{l}h_{l}}}{\lambda^{2}}\langle\Lambda\rangle{\sigma}_{\rm{sca}}\underbrace{\longrightarrow}_{\text{Lossless}}\mathit{f}_{\rm{CD}}\sim 1+\dfrac{\pi G_{h_{l}h_{l}}}{\lambda^{2}}\langle\Lambda\rangle{\sigma}_{\rm{ext}}$ |
Scattering approximation for cylindrical samples well-described by a single multipolar order $l$ | $\mathit{f}_{\rm{CD}}\sim 1+\dfrac{\pi G_{h_{l}h_{l}}}{\lambda^{2}}\Lambda_{\theta}{\sigma}_{\rm{sca}}\underbrace{\longrightarrow}_{\text{Lossless}}\mathit{f}_{\rm{CD}}\sim 1+\dfrac{\pi G_{h_{l}h_{l}}}{\lambda^{2}}\Lambda_{\theta}{\sigma}_{\rm{ext}}$ |

At this point, we will delve into Eq. (10), which approximates the efficiency
of an optical antenna to enhance the $\mathit{f}_{\rm{CD}}$ factor, as it has
just been previously explained. In particular, let us examine its physical
meaning when just one multipole order (electric and magnetic) contributes to
the optical response of the antenna. In this regard, it is essential to note
that the one multipole approximation includes the most studied scenario by the
nanophotonic community devoted to enhanced chiral sensing: a circularly
polarized plane wave incident on dipolar objects Zhang  _et al._ (2017); Yao
and Liu (2018); Hanifeh and Capolino (2020); A. Paiva-Marques  _et al._
(2020); Hendry  _et al._ (2012); Solomon  _et al._ (2018); Graf  _et al._
(2019); Garcia-Guirado  _et al._ (2019); Lasa-Alonso  _et al._ (2020);
Droulias and Bougas (2020); Mohammadi  _et al._ (2021); Rui  _et al._ (2022).
Now, when the scattering can be described by just one multipole order, $l$,
Eq. (10) reads

$\displaystyle\tilde{\rm{C}}^{\sigma}_{\rm{sca}}$ $\displaystyle=$
$\displaystyle\sigma
G_{h_{l}h_{l}}\sum_{m=-l}^{m=l}|C^{\sigma}_{lm}|^{2}\text{Re}\\{a_{lm}{b_{lm}}^{*}\\},$
(13)

By inspecting Eq. (13), we notice that $\tilde{\rm{C}}^{\sigma}_{\rm{sca}}$ is
proportional to the interference between the electric and magnetic scattering
coefficients $\text{Re}\\{a_{lm}{b_{lm}}^{*}\\}$. Now, let us introduce the
normalized (and unit-less) electromagnetic helicity expectation value, which
reads Olmos-Trigo  _et al._ (2020b, c); Lasa-Alonso  _et al._ (2022)

$\langle\Lambda\rangle=\frac{\int_{\Omega}\left({|{\bf
E}^{\sigma+}_{\rm{sca}}}|^{2}-{|{\bf
E}^{\sigma-}_{\rm{sca}}}|^{2}\right)\;d\Omega}{\int_{\Omega}\left({|{\bf
E}^{\sigma+}_{\rm{sca}}}|^{2}+{|{\bf
E}^{\sigma-}_{\rm{sca}}}|^{2}\right)\;d\Omega}.$ (14)

Computing $\langle\Lambda\rangle$ in the case in which the optical response
can be described by a single multipolar order $l$ Olmos-Trigo  _et al._
(2020b, c); Lasa-Alonso  _et al._ (2022), we obtain

${\langle\Lambda\rangle}=2\sigma\frac{\sum_{m}|C^{\sigma}_{lm}|^{2}\text{Re}\\{a_{lm}{b_{lm}}^{*}\\}}{\sum_{m}|C^{\sigma}_{lm}|^{2}\left(|a_{lm}|^{2}+|b_{lm}|^{2}\right)}=\frac{\sigma\sum_{m}|C^{\sigma}_{lm}|\text{Re}\\{a_{lm}{b_{lm}}^{*}\\}}{k^{2}\sigma_{\rm{sca}}},$
(15)

where $\sigma_{\rm{sca}}$ is the scattering cross section Bohren and Huffman
(2008). At this point, we notice that the expression for
$\tilde{\rm{C}}^{\sigma}_{\rm{sca}}$ resembles $\langle\Lambda\rangle$. In
fact and without loss of generality, we can write
$\tilde{\rm{C}}^{\sigma}_{\rm{sca}}={G_{h_{l}h_{l}}}\langle\Lambda\rangle
k^{2}\sigma_{\rm{sca}}$. As a result, $\mathit{f}_{\rm{CD}}$ yields

$\displaystyle\mathit{f}_{\rm{CD}}\sim
1+\frac{{\tilde{\rm{C}}^{\sigma}_{\rm{sca}}}}{\tilde{\rm{C}}^{\sigma}_{\rm{inc}}}=1+\frac{{{G_{h_{l}h_{l}}}\langle\Lambda\rangle{k^{2}\sigma_{\rm{sca}}}}}{\tilde{\rm{C}}^{\sigma}_{\rm{inc}}}.$
(16)

This is another significant result of the present work. We have linked the
averaged optical chirality associated with scattered fields,
$\tilde{\rm{C}}^{\sigma}_{\rm{sca}}$, which is usually computed in the near-
field, with quantities that can be evaluated or measured in far-field, i.e.
the helicity expectation value, $\langle\Lambda\rangle$ and the scattering
cross-section, ${{\sigma}_{\rm{sca}}}$. In addition, and for helicity-
preserving objects, $\langle\Lambda\rangle\sim 1$, the averaged optical
chirality associated with scattered fields is simply given by the scattering
cross-section. That is, $\tilde{\rm{C}}^{\sigma}_{\rm{sca}}\sim
G_{h_{l}h_{l}}k^{2}{\sigma}_{\rm{sca}}$. It is also essential to notice that,
for both lossless and helicity-preserving objects,
$\tilde{\rm{C}}^{\sigma}_{\rm{sca}}\sim
G_{h_{l}h_{l}}k^{2}{\sigma}_{\rm{ext}}$, with
${\sigma}_{\rm{ext}}={\sigma}_{\rm{sca}}$, ${\sigma}_{\rm{ext}}$ being the
extinction cross-section Bohren and Huffman (2008). That is, the CD
enhancement captured by the $\mathit{f}_{\rm{CD}}$ factor is related to a
single measurement of the extincted power in the forward direction whenever
$\langle\Lambda\rangle\sim 1$. The relation between $\mathit{f}_{\rm{CD}}$ and
${\sigma}_{\rm{ext}}$ for helicity-preserving achiral objects greatly reduces
eventual experimental and numerical calculations devoted to enhanced chiral
sensing.

So far, we have shown an alternative way to infer local CD enhancements in the
near-field limit by computing far-field magnitudes such as the helicity
expectation value, the scattering cross-section, or the extinction cross-
section. In particular, a scenario of major interest for the purpose of
enhanced chiral detection occurs when the antenna preserves the helicity of
the incident illumination, i.e. whenever $\langle\Lambda\rangle\sim 1$. These
objects satisfy $|{\bf E}^{\sigma\sigma^{\prime}}_{\rm{sca}}({\bf r})|\sim 0$
for $\sigma\neq\sigma^{\prime}$. This phenomenon is desirable since the local
sign of optical chirality is preserved. Experimentally, identifying helicity-
preserving scatterers requires measuring the polarization of all the scattered
field components, something which is not feasible in practice. Thus, our
question is: can we infer the helicity expectation value,
$\langle\Lambda\rangle$, from a single measurement in the far-field? In the
last part of this work, we will discuss scenarios in which the helicity
density at a given scattering angle can be identical to its expected value. In
particular, we will focus on cylindrically symmetric scatterers which preserve
the total angular momentum in the incident direction ($m=m^{\prime}$) and
whose optical response can be well-described by a single multipolar order
($l=l^{\prime}$), e.g., nanodisks at normal incidence or spherical particles
under the illumination of tightly-focused Laguerre-Gaussian beams Sanz-
Fernández  _et al._ (2021).

Mathematically, we can express the aforementioned condition as
${\langle\Lambda\rangle}=\Lambda_{\theta}$, where $\Lambda_{\theta}$ denotes
the helicity density at an angle $\theta$, where $\theta$ is the scattering
angle. After some algebra (see Appendix D), we obtain:

$P^{m}_{l}(\cos\theta)\frac{\partial
P^{m}_{l}(\cos\theta)}{\partial\cos\theta}=0~{}~{}~{}\Longrightarrow~{}~{}~{}{\langle\Lambda\rangle}=\Lambda_{\theta},$
(17)

where $P^{m}_{l}(\cos\theta)$ are the associated Legendre Polynomials Jackson
(1999). This is another key result of the present work. The helicity
expectation value can indeed be computed from a single measurement of the
helicity density at a specific angle $\theta$. Our result implies that for a
cylindrically symmetric scatterer whose response can be well-described by a
single multipolar order, $l$, if we excite it with an illumination with a
fixed total angular momentum, $m$, Eq. (17) specifies the angle at which the
helicity density is equal to its expected value. For instance, if we consider
the typical case of a cylindrically symmetric dipolar target ($l=1$) under a
circularly polarized plane-wave illumination ($m=\pm 1$), Eq. (17) yields that
the angle we should look at is $\theta=\pi/2$. That is, the expectation value
of the electromagnetic helicity can be inferred from a single measurement at
the right angle. Thus, and according to Eq. (16), for lossless and
cylindrically symmetric targets, we can infer the $\mathit{f}_{\rm{CD}}$
factor by just considering two far-field measurements: extinction cross-
section, in the forward direction, and helicity density, at an angle $\theta$
specified by Eq. (17) 444In terms of the usual Stokes parameters, extinction
is proportional to the $s_{0}$ parameter and helicity density is the ratio
$s_{3}/s_{0}$ Lasa-Alonso  _et al._ (2022)..

Table 1 resumes the main results of this work, particularized for plane-wave
illumination, as it is the most common external excitation for enhanced chiral
sensing to date. In short, we have derived an exact multipole expansion of the
integrated CD enhancement factor, $\mathit{f}_{\rm{CD}}$, for scatterers of
any form and shape under general illumination conditions. In addition, we have
established a roadmap to infer local CD enhancements from far-field
measurements. That is, $\mathit{f}_{\rm{CD}}$ can be extracted by calculating
the helicity expectation value and the scattering cross section, two Stokes
parameters that can be evaluated in the far-field limit. Finally, we have
shown an even more practical route to deduce $\mathit{f}_{\rm{CD}}$ for
cylindrically symmetric objects by means of measuring the extinction cross-
section and the local density of electromagnetic helicity at specific angles.
Our results pave the way for experimental verification and characterization of
building blocks for CD enhancement from far-field measurements, and thus, may
give rise to novel developments in the field of chiral light-matter
interactions.

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##  Appendix A Multipolar electromagnetic fields in a well-defined helicity
basis

###  A.1 Incident electromagnetic fields

To start with the calculation of the surface-enhanced circular dichroism (CD),
we need first to write down the electromagnetic fields. The most generic
expression of the incident electromagnetic fields are given by

$\displaystyle{\bf E}_{\rm{inc}}({\bf r})$ $\displaystyle=$
$\displaystyle\sum_{l=0}^{\infty}\sum_{m=-l}^{+l}g^{e}_{lm}\bm{N}^{j}_{lm}({\bf
r})+g^{m}_{lm}\bm{M}^{j}_{lm}({\bf r}),$ (18) $\displaystyle{\rm i}Z{\bf
H}_{\rm{inc}}({\bf r})$ $\displaystyle=$
$\displaystyle\sum_{l=0}^{\infty}\sum_{m=-l}^{+l}g^{e}_{lm}\bm{M}^{j}_{lm}({\bf
r})+g^{m}_{lm}\bm{N}^{j}_{lm}({\bf r}),$ (19)

where $g^{e}_{lm}$ and $g^{m}_{lm}$ stands for the incident electric and
magnetic beam’s shape coefficients, respectively, and

$\displaystyle{\bm{M}}^{j}_{lm}({\bf r})=j_{l}(kr)\bm{X}_{lm}({\bf
r}),\qquad{\bm{N}}^{j}_{lm}({\bf
r})=\frac{\bm{\nabla}\times{\bm{M}}^{j}_{lm}({\bf
r})}{k},\qquad\bm{X}_{lm}({\bf
r})=\frac{{\bf{L}}Y_{lm}(\theta,\varphi)}{\sqrt{l(l+1)}}.$ (20)

Here $\bm{M}^{j}_{lm}({\bf r})$ and $\bm{N}^{j}_{lm}({\bf r})$ are (incident)
Hansen’s multipoles Jackson (1999), $j_{l}(kr)$ are the spherical Bessel
functions, $k$ being the radiation wavelength, and ${\bf r}$ the observation
point. Moreover, $Y_{lm}(\theta,\varphi)$ are the spherical harmonics,
$\theta$ and $\varphi$ being the polar and azimuthal angles, and
${\bf{L}}=\left\\{-{\rm i}{\bf r}\times\bm{\nabla}\right\\}$ is the total
angular momentum operator. At this point, let us consider an arbitrary
incident electromagnetic field with well-defined helicity, $\sigma=\pm 1$.
Mathematically, we can write this well-defined helicity field as

$\displaystyle{\bf E}^{\sigma}_{\rm{inc}}({\bf r})$ $\displaystyle=$
$\displaystyle\frac{{\bf E}_{\rm{inc}}({\bf r})+\sigma{\rm i}Z{\bf
H}_{\rm{inc}}({\bf
r})}{2}=\sum_{l=0}^{\infty}\sum_{m=-l}^{+l}\left(\frac{{g^{e}_{lm}+\sigma
g^{m}_{lm}}}{\sqrt{2}}\right)\left(\frac{{\bm{N}}^{j}_{lm}({\bf
r})+\sigma{\bm{M}}^{j}_{lm}({\bf
r})}{\sqrt{2}}\right)=\sum_{l=0}^{\infty}\sum_{m=-l}^{+l}C_{lm}^{\sigma}\bm{\Psi}_{lm}^{\sigma}({\bf
r}),$ (21)

where

$\displaystyle C_{lm}^{\sigma}=\frac{{g^{e}_{lm}+\sigma
g^{m}_{lm}}}{\sqrt{2}},$ and $\displaystyle\bm{\Psi}_{lm}^{\sigma}({\bf
r})=\frac{{\bm{N}}^{j}_{lm}({\bf r})+\sigma{\bm{M}}^{j}_{lm}({\bf
r})}{\sqrt{2}}.$ (22)

Let us recall that the multipoles $\bm{\Psi}_{lm}^{\sigma}({\bf r})$ are
eigenvectors of the squared angular momentum $L^{2}$, the projection of the
angular momentum on one direction, $L_{z}$, and helicity
$\bm{\Lambda}=(1/k)\nabla\times$ Fernandez-Corbaton  _et al._ (2013) with
eigenvalues $l(l+1)$, $m$, $\sigma$, respectively.

###  A.2 Scattered and total electromagnetic fields

At this point, let us consider the scattered electromagnetic fields. The most
generic expression of these fields is given by

$\displaystyle{\bf E}^{\sigma}_{\rm{sca}}({\bf r})$ $\displaystyle=$
$\displaystyle\sum_{l=0}^{\infty}\sum_{m=-l}^{+l}a^{\sigma}_{lm}\bm{N}^{h}_{lm}({\bf
r})+b^{\sigma}_{lm}\bm{M}^{h}_{lm}({\bf r}),$ (23) $\displaystyle{\rm i}Z{\bf
H}^{\sigma}_{\rm{sca}}({\bf r})$ $\displaystyle=$
$\displaystyle\sum_{l=0}^{\infty}\sum_{m=-l}^{+l}a^{\sigma}_{lm}\bm{M}^{h}_{lm}({\bf
r})+b^{\sigma}_{lm}\bm{N}^{h}_{lm}({\bf r}),$ (24)

where $a^{\sigma}_{lm}$ and $b^{\sigma}_{lm}$ stand for the electric and
magnetic scattering coefficients, respectively. Notice that $a^{\sigma}_{lm}$
and $b^{\sigma}_{lm}$ depend on the incident illumination. As a result, we
have explicitly indicated the $\sigma$-dependency. Moreover,

$\displaystyle{\bm{M}}^{h}_{lm}({\bf r})=h_{l}(kr)\bm{X}_{lm}({\bf
r}),\qquad{\bm{N}}^{h}_{lm}({\bf
r})=\frac{\bm{\nabla}\times{\bm{M}}^{h}_{lm}({\bf r})}{k},$ (25)

where $\bm{M}^{h}_{lm}({\bf r})$ and $\bm{N}^{h}_{lm}({\bf r})$ are
(scattered) Hansen’s multipoles Jackson (1999) and $h_{l}(kr)$ are the
spherical Hankel functions. Following the steps done in Eq. (21), the electric
field reads as

$\displaystyle{\bf E}^{\sigma\sigma^{\prime}}_{\rm{sca}}({\bf r})=\frac{{\bf
E}^{\sigma}_{\rm{sca}}({\bf r})+\sigma^{\prime}{\rm i}Z{\bf
H}^{\sigma}_{\rm{sca}}({\bf r})}{2}$ $\displaystyle=$
$\displaystyle\sum_{l=0}^{\infty}\sum_{m=-l}^{+l}\left(\frac{{a^{\sigma}_{lm}+\sigma^{\prime}b^{\sigma}_{lm}}}{\sqrt{2}}\right)\left(\frac{{\bm{N}}^{h}_{lm}({\bf
r})+\sigma^{\prime}{\bm{M}}^{h}_{lm}({\bf
r})}{\sqrt{2}}\right)=\sum_{l=0}^{\infty}\sum_{m=-l}^{+l}D_{lm}^{\sigma\sigma^{\prime}}\bm{\Phi}_{lm}^{\sigma^{\prime}}({\bf
r}),$ (26)

where

$\displaystyle
D_{lm}^{\sigma\sigma^{\prime}}=\frac{{a^{\sigma}_{lm}+\sigma^{\prime}b^{\sigma}_{lm}}}{\sqrt{2}},$
and $\displaystyle\bm{\Phi}_{lm}^{\sigma^{\prime}}({\bf
r})=\frac{{\bm{N}}^{h}_{lm}({\bf r})+\sigma^{\prime}{\bm{M}}^{h}_{lm}({\bf
r})}{\sqrt{2}}.$ (27)

At this stage, let us write down the relation between the incident and
scattering amplitudes. These are given by Cramer’s rule of the tangential
Maxwell boundary conditions Jackson (1999), $a^{\sigma}_{lm}=a_{lm}g^{e}_{lm}$
and $b^{\sigma}_{lm}=b_{lm}g^{m}_{lm}$ where $a_{lm}$ and $b_{lm}$ are the so-
called electric and magnetic scattering coefficients, respectively. Notice
$a_{lm}$ and $b_{lm}$ do not depend on the incident illumination but on the
optical properties and geometry of the target, e.g., the electric and magnetic
Mie coefficients. Now, and since we are dealing with a well-defined helicity
field, we can notice from Eq. (21) that $g^{e}_{lm}=\sigma g^{m}_{lm}$.
Accordingly, we can write

$\displaystyle
a^{\sigma}_{lm}=a_{lm}\left(\frac{C^{\sigma}_{lm}}{\sqrt{2}}\right),$
$\displaystyle b^{\sigma}_{lm}=\sigma
b_{lm}\left(\frac{C^{\sigma}_{lm}}{\sqrt{2}}\right).$ (28)

Now, by inserting Eq. (28) into the left side of Eq. (27), we arrive to

$\displaystyle
D^{\sigma\sigma^{\prime}}_{lm}=C^{\sigma}_{lm}\frac{\left({a_{lm}+\sigma\sigma^{\prime}b_{lm}}\right)}{2}.$
(29)

From now on, this will be our choice for the representation of the scattered
coefficients. To conclude Appendix A, let us write the total electromagnetic
fields. These are given by the additive sum of the incident and scattered
electromagnetic fields. Hence, by taking into account both Eq. (21) and Eq.
(26), we can straightforwardly write,

$\displaystyle{\bf E}^{\sigma}_{\rm{tot}}({\bf r})$ $\displaystyle=$
$\displaystyle\sum_{\sigma^{\prime}=\pm 1}{\bf
E}^{\sigma\sigma^{\prime}}_{\rm{tot}}({\bf r}),\qquad{\rm i}Z{\bf
H}^{\sigma}_{\rm{tot}}({\bf r})=\sum_{\sigma^{\prime}=\pm
1}\sigma^{\prime}{\bf E}^{\sigma\sigma^{\prime}}_{\rm{tot}}({\bf r}),$ (30)

with

$\displaystyle{\bf E}^{\sigma\sigma^{\prime}}_{\rm{tot}}({\bf r})$
$\displaystyle=$ $\displaystyle{\bf E}^{\sigma\sigma^{\prime}}_{\rm{sca}}({\bf
r})+{\bf E}^{\sigma}_{\rm{inc}}({\bf r})\delta_{\sigma\sigma^{\prime}},$ (31)

where $\delta_{\sigma\sigma^{\prime}}$ is a Kronecker delta. Next, we will use
the orthogonality expressions that satisfy both the incident and scattered
electromagnetic fields to compute the exact multipolar expansion of the CD
enhancement factor.

##  Appendix B An exact multipolar expansion of $f_{\rm{CD}}$ beyond the
plane-wave picture

###  B.1 An exact multipolar expansion of $f_{\rm{CD}}$: Orthogonality
relations of well-defined helicity multipoles

To derive an exact multipolar expansion of the CD enhancement factor,
$\mathit{f}_{\rm{CD}}$, beyond the plane-wave picture, we need to know the
orthogonality relations that satisfy both incident and scattered
electromagnetic fields over an integrating sphere surrounding the object under
illumination. To that end, we need first to calculate the set of orthogonality
relations that fulfill the multipoles with well-defined helicity. That is, we
need the orthogonality relations between incident
$\\{\bm{\Psi}_{lm}^{\sigma},\bm{\Psi}_{l^{\prime}m^{\prime}}^{\sigma}\\}$,
interference
$\\{\bm{\Psi}_{lm}^{\sigma},\bm{\Phi}_{l^{\prime}m^{\prime}}^{\sigma}\\}$, and
scattering
$\\{\bm{\Phi}_{lm}^{\sigma},\bm{\Phi}_{l^{\prime}m^{\prime}}^{\sigma}\\}$
terms, according to Eq. (7). Fortunately, all these relations can be derived
from Jackson’s third edition book. Let us start this section by transcribing
Eq. $10.48$ that can be found on page $472$ of Ref. Jackson (1999):

$\displaystyle\int_{\Omega}{N^{f}_{lm}}^{*}\cdot
N^{g}_{l^{\prime}m^{\prime}}d\Omega=\delta_{ll^{\prime}}\delta_{mm^{\prime}}\left(f^{*}_{l}(u)g_{l}(u)+\frac{1}{u^{2}}\frac{\partial}{\partial
u}\left(uf^{*}_{l}(u)\frac{\partial}{\partial
u}\left(ug_{l}(u)\right)\right)\right),$
$\displaystyle\int_{\Omega}{M^{f}_{lm}}^{*}\cdot
M^{g}_{l^{\prime}m^{\prime}}d\Omega=f^{*}_{l}(u)g_{l}(u)\delta_{ll^{\prime}}\delta_{mm^{\prime}}.$
(32)

Here $u=kr$ denotes the optical radius of the integrating sphere and
$\\{\delta_{ll^{\prime}},\delta_{mm^{\prime}}\\}$ are Kronecker deltas. Notice
that $\\{f_{l}(u),g_{l}(u)\\}$ denote either Bessel or Hankel spherical
functions, namely, $j_{l}(u)$ and $h_{l}(u)$ Jackson (1999), depending on the
nature of the field: incident or scattered, respectively.

At this point, we have already learned that multipoles with well-defined
helicity $\\{\bm{\Psi}_{lm}^{\sigma},\bm{\Phi}_{lm}^{\sigma}\\}$ are
constructed by a linear combination of the Hansel multipoles (see the right
side of Eq. (22) and Eq. (27) . As a result, we can write from Eq. (32)

$\displaystyle\int_{\Omega}({\bm{\Psi}^{\sigma}_{l^{\prime}m^{\prime}}})^{*}\cdot\bm{\Psi}_{lm}^{\sigma}\;d\Omega=G_{j_{l}j_{l}}\delta_{ll^{\prime}}\delta_{mm^{\prime}},$
$\displaystyle\int_{\Omega}({\bm{\Phi}^{\sigma}_{l^{\prime}m^{\prime}}})^{*}\cdot{\bm{\Phi}^{\sigma}_{lm}}\;d\Omega=G_{h_{l}h_{l}}\delta_{ll^{\prime}}\delta_{mm^{\prime}},$
$\displaystyle\int_{\Omega}({\bm{\Psi}^{\sigma}_{l^{\prime}m^{\prime}}})^{*}\cdot\bm{\Phi}_{lm}^{\sigma}\;d\Omega=G_{j_{l}h_{l}}\delta_{ll^{\prime}}\delta_{mm^{\prime}},$
(33)

with

$G_{f_{l}g_{l}}=\frac{1}{2}\left[2f^{*}_{l}(u)g_{l}(u)+\frac{1}{u^{2}}\frac{\partial}{\partial
u}\left(uf^{*}_{l}(u)\frac{\partial}{\partial
u}\left(ug_{l}(u)\right)\right)\right].$ (34)

Now, we can re-write Eq. (34) to get rid of second derivatives by making use
of fundamental properties of the Ricatti-Bessel functions Watson (1995). In
particular, we can write,

$\displaystyle
G_{j_{l}j_{l}}=\frac{1}{2}\left[j^{2}_{l}(u)\left(1+\frac{l(l+1)}{u^{2}}\right)+\frac{1}{u^{2}}\left(\frac{\partial}{\partial
u}\left(uj_{l}(u)\right),\right)^{2}\right]$ $\displaystyle
G_{h_{l}h_{l}}=\frac{1}{2}\left[|h_{l}(u)|^{2}\left(1+\frac{l(l+1)}{u^{2}}\right)+\frac{1}{u^{2}}\left|\frac{\partial}{\partial
u}\left(uh_{l}(u)\right)\right|^{2}\right],$ (35)

$\displaystyle
G_{j_{l}h_{l}}=\frac{1}{2}\left[j_{l}(u)h_{l}(u)\left(1+\frac{l(l+1)}{u^{2}}\right)+\frac{1}{u^{2}}\left(\frac{\partial}{\partial
u}\left(uj_{l}(u)\right)\frac{\partial}{\partial
u}\left(uh_{l}(u)\right)\right)\right],$ (36)

where $\frac{\partial}{\partial
u}\left(uf_{l}(u)\right))=uf_{l-1}(u)-lf_{l}(u)$ is satisfied for
$f(u)=\\{j_{l}(u),h_{l}(u)\\}$.

###  B.2 An exact multipolar expansion of $f_{\rm{CD}}$: From the helicity
basis to the standard electric and magnetic multipolar expansion

At this stage, we have all ingredients to calculate the exact multipolar
expansion of $f_{\rm{CD}}$. The starting point of this section will be Eq.
(7). According to Eq. (7), we need the orthogonality relations that satisfy
the Rieman-Silberstein representation of the incident and scattered
electromagnetic fields:

$\displaystyle\tilde{\rm{C}}^{\sigma}_{\rm{inc}}$ $\displaystyle=$
$\displaystyle\int_{\Omega}{\rm{C}}^{\sigma}_{\rm{inc}}({\bf
r})d\Omega=\int_{\Omega}\sigma|{\bf E}^{\sigma}_{\rm{inc}}({\bf
r})|^{2}d\Omega,$ (37) $\displaystyle\tilde{\rm{C}}^{\sigma}_{\rm{sca}}$
$\displaystyle=$ $\displaystyle\int_{\Omega}{\rm{C}}^{\sigma}_{\rm{sca}}({\bf
r})d\Omega=\int_{\Omega}|{\bf E}^{\sigma+}_{\rm{sca}}({\bf r})|^{2}-|{\bf
E}^{\sigma-}_{\rm{sca}}({\bf r})|^{2}d\Omega,$ (38)
$\displaystyle{\rm{C}}^{\sigma}_{\rm{int}}$ $\displaystyle=$
$\displaystyle\int_{\Omega}\tilde{{\rm{C}}}^{\sigma}_{\rm{int}}({\bf
r})d\Omega=2\sigma\int_{\Omega}\text{Re}\\{{{\bf
E}^{\sigma}_{\rm{inc}}}^{*}({\bf r})\cdot{{\bf
E}^{\sigma\sigma}_{\rm{sca}}}({\bf r})\\}d\Omega.$ (39)

These orthogonality relations can be computed by combining Eq. (21) and Eq.
(26) with Eq. (33). In fact and after some algebraic manipulations, it can be
shown that

$\displaystyle\tilde{\rm{C}}^{\sigma}_{\rm{inc}}=\sigma\sum_{lm}|C^{\sigma}_{lm}|^{2}G_{j_{l}j_{l}},$
$\displaystyle\tilde{\rm{C}}^{\sigma}_{\rm{sca}}=\sum_{lm}\left(|D^{\sigma+}_{lm}|^{2}-|D^{\sigma-}_{lm}|^{2}\right)G_{h_{l}h_{l}},$
$\displaystyle\tilde{{\rm{C}}}^{\sigma}_{\rm{int}}=2\sigma\sum_{lm}\text{Re}\\{{C^{\sigma}_{lm}}^{*}D^{\sigma\sigma}_{lm}G_{j_{l}h_{l}}\\}.$
(40)

Now, let us insert Eq. (29) into Eq. (40) to obtain a closed-relation of the
optical chirality enhancements in terms of the electric and magnetic
scattering coefficients. After some algebra, we arrive to

$\mathit{f}_{\rm{CD}}=1+\frac{\tilde{\rm{C}}^{\sigma}_{\rm{sca}}+\tilde{\rm{C}}^{\sigma}_{\rm{int}}}{\tilde{\rm{C}}^{\sigma}_{\rm{inc}}},$
(41)

with

$\displaystyle\tilde{\rm{C}}^{\sigma}_{\rm{inc}}=\sigma\sum_{lm}|C^{\sigma}_{lm}|^{2}G_{j_{l}j_{l}},$
$\displaystyle\tilde{\rm{C}}^{\sigma}_{\rm{sca}}=\sigma\sum_{lm}|C^{\sigma}_{lm}|^{2}\text{Re}\\{a_{lm}{b_{lm}}^{*}\\}G_{h_{l}h_{l}},$
$\displaystyle\tilde{\rm{C}}^{\sigma}_{\rm{int}}=\sigma\sum_{lm}|C^{\sigma}_{lm}|^{2}\text{Re}\\{\left(a_{lm}+b_{lm}\right)G_{j_{l}h_{l}}\\}.$
(42)

##  Appendix C Conversion from our conventions to those in Jackson’s book

The multipolar expansion of the scattered electromagnetic fields provided in
Jackson’s book reads as Jackson (1999)

$\displaystyle{\bf E}^{\rm{Jack}}_{\rm{sca}}$ $\displaystyle=$ $\displaystyle
Z\sum_{l=0}^{\infty}\sum_{m=-l}^{+l}{\rm i}a_{E}(l,m)\bm{N}^{h}_{lm}({\bf
r})+a_{M}(l,m)\bm{M}^{h}_{lm}({\bf r}),$ (43) $\displaystyle{\rm i}Z{\bf
H}^{\rm{Jack}}_{\rm{sca}}$ $\displaystyle=$ $\displaystyle
Z\sum_{l=0}^{\infty}\sum_{m=-l}^{+l}{\rm i}a_{E}(l,m)\bm{M}^{h}_{lm}({\bf
r})+a_{M}(l,m)\bm{N}^{h}_{lm}({\bf r}).$ (44)

Now, by inspecting Eqs. (43)-(44), we notice that the conversion from our
conventions to those in Jackson’s book are given by

$\displaystyle a_{lm}=\sqrt{2}{\rm
i}Z\left[\frac{a_{E}(l,m)}{C^{\sigma}_{lm}}\right],$ $\displaystyle
b_{lm}=\sqrt{2}\sigma Z\left[\frac{a_{M}(l,m)}{C^{\sigma}_{lm}}\right].$ (45)

where $Z=\mu/\epsilon$ is the medium impedance. Notice that the electric and
magnetic coefficients, provided in Jackson’s book, can be computed by
conventional far-field projections of the scattered electromagnetic fields
(see Eq. (9.123) in Ref. Jackson (1999)). For completeness, we transcribe
these expressions,

$\displaystyle Za_{E}(l,m)h_{l}(kr)=-\frac{k}{\sqrt{l(l+1)}}\int
Y^{*}_{lm}(\theta,\varphi){\bf r}\cdot{\bf E}^{\rm{Jack}}_{\rm{sca}},$
$\displaystyle a_{M}(l,m)h_{l}(kr)=\frac{k}{\sqrt{l(l+1)}}\int
Y^{*}_{lm}(\theta,\varphi){\bf r}\cdot{\bf H}^{\rm{Jack}}_{\rm{sca}}.$ (46)

##  Appendix D Extracting the helicity expectation value from a single
measurement of its local density

In this Appendix, we derive the condition given in Eq. (17), that relates the
local density of helicity at a certain angle, $\Lambda_{\theta,\phi}$, with
the helicity expectation value, $\langle\Lambda\rangle$. For that aim, we
should first define the local density of helicity, which we consider to be
given in far-field by:

$\Lambda_{\theta,\varphi}\equiv\lim_{kr\rightarrow\infty}\frac{|\mathbf{E}^{\sigma+}_{\rm{sca}}(r,\theta,\varphi)|^{2}-|\mathbf{E}^{\sigma-}_{\rm{sca}}(r,\theta,\varphi)|^{2}}{|\mathbf{E}^{\sigma+}_{\rm{sca}}(r,\theta,\varphi)|^{2}+|\mathbf{E}^{\sigma-}_{\rm{sca}}(r,\theta,\varphi)|^{2}},$
(47)

where $\mathbf{E}^{\sigma\sigma^{\prime}}_{\rm{sca}}(r,\theta,\varphi)$ is the
scattered field written in terms of electromagnetic modes with well-defined
helicity. From Eq. (47), we notice that we require the asymptotic behavior of
$\mathbf{E}^{\sigma\sigma^{\prime}}_{\rm{sca}}(r,\theta,\varphi)$ in the far-
field limit. For that aim, we need first to calculate how Hansel multipoles
behave in the far-field limit. After some algebra, we arrive to

$\displaystyle\lim_{kr\rightarrow\infty}\bm{N}^{h}_{lm}({\bf
r})=-\frac{e^{ikr}}{kr}\frac{(-i)^{l+1}}{\sqrt{l(l+1}}\bm{\xi}_{lm}(\theta,\varphi),$
$\displaystyle\lim_{kr\rightarrow\infty}\bm{M}^{h}_{lm}({\bf
r})=-\frac{e^{ikr}}{kr}\frac{(-i)^{l+1}}{\sqrt{l(l+1}}i\bm{\eta}_{lm}(\theta,\varphi),$
(48)

where $\bm{\xi}_{lm}(\theta,\varphi)=r\nabla Y_{lm}(\theta,\varphi)$ and
$\bm{\eta}_{lm}(\theta,\varphi)=\hat{r}\times\bm{\xi}_{lm}(\theta,\varphi)$.
Now, the far-field expression of the electromagnetic field scattered by an
arbitrary sample can be computed Carrascal  _et al._ (1991):

$\mathbf{E}^{\sigma\sigma^{\prime}}_{\rm{sca}}(r,\theta,\varphi)=-\frac{e^{ikr}}{kr}\sum_{lm}(-i)^{l+1}\frac{C_{lm}^{\sigma}}{\sqrt{2l(l+1)}}\left[\frac{a_{lm}+\sigma\sigma^{\prime}b_{lm}}{\sqrt{2}}\right]\left[\frac{\bm{\xi}_{lm}+i\sigma^{\prime}\bm{\eta}_{lm}}{\sqrt{2}}\right].$
(49)

Substituting the scattered field in Eq. (49) into the expression of the
helicity density given by Eq. (47), we obtain for fixed $l$ and $m$ values:

$\Lambda_{\theta}=2\sigma\frac{\text{Re}\left(a_{lm}{b_{lm}}^{*}\right)\left[|\bm{\xi}_{lm}|^{2}+|\bm{\eta}_{lm}|^{2}\right]-\left[|a_{lm}|^{2}+|b_{lm}|^{2}\right]\text{Im}\left(\bm{\xi}_{lm}^{*}\cdot\bm{\eta}_{lm}\right)}{\left[|a_{lm}|^{2}+|b_{lm}|^{2}\right]\left[|\bm{\xi}_{lm}|^{2}+|\bm{\eta}_{lm}|^{2}\right]-4\text{Re}\left(a_{lm}{b_{lm}}^{*}\right)\text{Im}\left(\bm{\xi}_{lm}^{*}\cdot\bm{\eta}_{lm}\right)}.$
(50)

The type of scatterers which may be described by fixed $l$ and $m$ values are
cylindrically symmetric particles, illuminated by a beam with a well-defined
angular momentum, $m$, and with a non multipolar response. Due to the
cylindrical symmetry of the scatterers, helicity density cannot depend on
$\varphi$ variable. This is the reason why we have chosen to write helicity
density as $\Lambda_{\theta}$ in Eq. (50). Crucially, it can be checked that
whenever $\text{Im}\left(\bm{\xi}_{lm}^{*}\cdot\bm{\eta}_{lm}\right)=0$, one
recovers the expression of the helicity expectation value, i.e.

$\text{Im}\left(\bm{\xi}_{lm}^{*}\cdot\bm{\eta}_{lm}\right)=0~{}~{}~{}\Longrightarrow~{}~{}~{}\Lambda_{\theta}=\langle\Lambda\rangle=2\sigma\frac{\text{Re}\left(a_{lm}{b_{lm}}^{*}\right)}{|a_{lm}|^{2}+|b_{lm}|^{2}}.$
(51)

Importantly, the condition
$\text{Im}\left(\bm{\xi}_{lm}^{*}\cdot\bm{\eta}_{lm}\right)=0$ is purely
geometrical, i.e. does not depend on the particular response of the scatterer.
This fact makes the expression completely general and applicable to any type
of cylindrical sample whose response is well-described by a fixed $l$. Thus,
for this type of scatterers, there are locations in the far-field at which the
helicity density is equal to the helicity expectation value.

The specific sites at which $\langle\Lambda\rangle=\Lambda_{\theta}$ are
obtained by finding the solutions to the equation
$\text{Im}\left(\bm{\xi}_{lm}^{*}\cdot\bm{\eta}_{lm}\right)=0$. More
explicitly, we have that the vector and scalar spherical harmonics are written
as:

$\displaystyle\bm{\xi}_{lm}(\theta,\varphi)$ $\displaystyle=\frac{\partial
Y_{lm}(\theta,\varphi)}{\partial\theta}\hat{u}_{\theta}+\frac{1}{\sin\theta}\frac{\partial
Y_{lm}(\theta,\varphi)}{\partial\varphi}\hat{u}_{\varphi}$ (52)
$\displaystyle\bm{\eta}_{lm}(\theta,\varphi)$ $\displaystyle=\frac{\partial
Y_{lm}(\theta,\varphi)}{\partial\theta}\hat{u}_{\varphi}-\frac{1}{\sin\theta}\frac{\partial
Y_{lm}(\theta,\varphi)}{\partial\varphi}\hat{u}_{\theta}$ (53) $\displaystyle
Y_{lm}(\theta,\varphi)$
$\displaystyle=\sqrt{\frac{2l+1}{4\pi}\frac{(l-m)!}{(l+m)!}}P_{l}^{m}(\cos\theta)e^{im\varphi},$
(54)

where $P_{l}^{m}(\cos\theta)$ are the associated Legendre polynomials. With
the definitions above it is straightforward to check that

$P_{l}^{m}(\cos\theta)\frac{\partial
P_{l}^{m}(\cos\theta)}{\partial\cos\theta}=0~{}~{}~{}\Longrightarrow~{}~{}~{}\text{Im}\left(\bm{\xi}_{lm}^{*}\cdot\bm{\eta}_{lm}\right)=0.$
(55)

In conclusion, for fixed values of $l$ and $m$, there is an angle $\theta$,
given by the transcendental equation above, at which the helicity density,
$\Lambda_{\theta}$, is equal to the helicity expectation value,
$\langle\Lambda\rangle$.