figuret

# Extremize Optical Chiralities through Polarization Singularities

Weijin Chen  Authors contribute equally to this work. School of Optical and
Electronic Information, Huazhong University of Science and Technology, Wuhan,
Hubei 430074, P. R. China  Qingdong Yang  Authors contribute equally to this
work. School of Optical and Electronic Information, Huazhong University of
Science and Technology, Wuhan, Hubei 430074, P. R. China  Yuntian Chen  Email:
<EMAIL_ADDRESS>School of Optical and Electronic Information, Huazhong
University of Science and Technology, Wuhan, Hubei 430074, P. R. China  Wuhan
National Laboratory for Optoelectronics, Huazhong University of Science and
Technology, Wuhan, Hubei 430074, P. R. China  Wei Liu  Email:
<EMAIL_ADDRESS>College for Advanced Interdisciplinary Studies, National
University of Defense Technology, Changsha, Hunan 410073, P. R. China

###### Abstract

Chiral optical effects are generally quantified along some specific incident
directions of exciting waves (especially for extrinsic chiralities of achiral
structures) or defined as direction-independent properties by averaging the
responses among all structure orientations. Though of great significance for
various applications, chirality extremization (maximized or minimized) with
respect to incident directions or structure orientations have not been
explored, especially in a systematic manner. In this study we examine the
chiral responses of open photonic structures from perspectives of quasi-normal
modes and polarization singularities of their far-field radiations. The
nontrivial topology of the momentum sphere secures the existence of
singularity directions along which mode radiations are either circularly or
linearly polarized. When plane waves are incident along those directions, the
reciprocity ensures ideal maximization and minimization of optical
chiralities, for corresponding mode radiations of circular and linear
polarizations respectively. For directions of general elliptical
polarizations, we have unveiled the subtle equality of a Stokes parameter and
the circular dichroism, showing that an intrinsically chiral structure can
unexpectedly exhibit no chirality at all or even chiralities of opposite
handedness for different incident directions. The framework we establish can
be applied to not only finite scattering bodies but also infinite periodic
structures, encompassing both intrinsic and extrinsic optical chiralities. We
have effectively merged two vibrant disciplines of chiral and singular optics,
which can potentially trigger more optical chirality-singularity related
interdisciplinary studies.

Optical responses of photonic structures are generally anisotropic, depending
on both polarizations and incident directions of exciting waves. Chiralities
emerge when optical responses are distinct for circularly polarized waves of
opposite handedness while the same incident direction BARRON_2009__Molecular ;
BORISKINA__Singular . Chiral effects can be observed for both intrinsically
chiral and achiral structures HENTSCHEL_2017_Sci.Adv._Chiral ;
COLLINS_AdvancedOpticalMaterials_chirality_2017 ; LUO_Adv.Opt.Mater._Plasmonic
; QIU_Adv.Funct.Mater._3D ; MUN_LightSci.Appl._Electromagnetic . As for chiral
structures, (intrinsic) chiralities are generally present and can be
quantified as direction-independent properties through orientation-averaging
chiral effects BARRON_2009__Molecular ; BORISKINA__Singular ; FERNANDEZ-
CORBATON_2015_ACSPhotonics_Dual ; VINEGRAD_ACSPhotonics_Circular ;
SURYADHARMA_Phys.Rev.B_Quantifying ; FAZEL-
NAJAFABADI_ArXiv201210010Phys._Orientation ; while for achiral structures,
(extrinsic) chiralities are manifest for some symmetry-breaking incident
directions PAPAKOSTAS_Phys.Rev.Lett._optical_2003 ;
PLUM_Appl.Phys.Lett._Optical ; PLUM_Phys.Rev.Lett._metamaterials_2009 ;
SERSIC_Phys.Rev.Lett._Ubiquity ; RIZZA_Phys.Rev.Lett._OneDimensional , which
nevertheless would be extinguished for symmetry-preserving directions or when
orientation-averaged. For both scenarios, chiral responses are generally
dependent on structure orientations, and thus it is interesting and
significant to clarify systematically how chiralities can be extremized with
respect to incident directions.

Besides chiral optics, another photonic branch of singular optics has also
gained enormous momentum from two-decade explosive developments of
nanophotonics, fertilizing many related disciplines GBUR_2016__Singular ;
BERRY_2017__HalfCentury ; DENNIS_2009_ProgressinOptics_Chapter ;
LIU_ArXiv201204919Phys._Topological . Vectorial optical singularities
correspond to states of circular and linear polarizations, for which semi-
major axes and orientation planes of polarization ellipses are undefined,
respectively NYE_1983_ProcRSocA_Polarization ; NYE_1983_Proc.R.Soc.A_Lines ;
BERRY_2001_SecondInt.Conf.Singul.Opt.Opt.VorticesFundam.Appl._Geometry .
Polarization singularities are skeletons of general electromagnetic waves,
which are robust against perturbations and generically manifest in both
natural and artificial fields GBUR_2016__Singular ; BERRY_2017__HalfCentury ;
DENNIS_2009_ProgressinOptics_Chapter ; LIU_ArXiv201204919Phys._Topological .
Despite the ubiquity of polarization singularities, it is rather unfortunate
and surprising that for decades both disciplines of chiral and singular optics
develop almost independently with rare crucial interactions, especially when
fundamental entities of circular polarizations are shared by both fields.

Here we investigate chirality extremization, for both achiral and chiral
structures, with plane waves of variant incident directions. Chiral effects
are examined from perspectives of quasi-normal modes (QNMs) and their
radiation polarization singularities. The nontrivial topology of momentum
sphere has secured existences of singularity directions, as protected by the
Poincaré-Hopf theorem CHEN_2019__Singularities ; CHEN_2019_ArXiv190409910Math-
PhPhysicsphysics_Linea ; CHEN_ACSOmega_Global . For a structure supporting a
dominant QNM at a spectral regime, with waves incident along singularity
directions of circular and linear mode radiations, chiral effects are ideally
maximized and minimized respectively. For directions of general elliptical
polarizations, the equality of a Stokes parameter and circular dichroism is
discovered, demonstrating that an intrinsically chiral structure can
unexpectedly manifest no chirality or even chiralities of opposite handedness
for different orientations. The chirality extremization is generically
protected by reciprocity and thus broadly applicable to both extrinsic and
intrinsic chiral responses.

Without loss of generality, we study reciprocal nonmagnetic structures of
relative permittivity $\mathbf{\epsilon}(\mathbf{r},\omega)$ in vacuum
background of refractive index $n=1$. An open structure supports a set of QNMs
characterized by eigenfield $\tilde{\mathbf{E}}_{j}(\mathbf{r})$ and complex
eigenfrequency $\tilde{\omega}_{j}$ LALANNE__LaserPhotonicsRev._Light . The
incident plane wave is ${\mathbf{E}}_{\rm{inc}}(\mathbf{r})$, with wavevector
$\mathbf{k}_{\rm{inc}}$, real frequency $\omega$ and wavelength $\lambda$.
Excited fields can be expanded into QNMs as
$\mathbf{E}=\sum\alpha_{j}(\omega)\tilde{\mathbf{E}}_{j}(\mathbf{r})$, with
excitation coefficients LALANNE__LaserPhotonicsRev._Light :

$\alpha_{j}(\omega)\propto\iiint_{\mathbf{V}}-i\omega[\varepsilon(\mathbf{r},\omega)-1]\mathbf{E}_{\rm{inc}}(\mathbf{r})\cdot\tilde{\mathbf{E}}_{j}(\mathbf{r})d^{3}\mathbf{r},$
(1)

where $\mathbf{V}$ denotes regions the structure occupies. When only one non-
degenerate QNM is dominantly excited, as is the case through this study, the
mode subscript $j$ can be dropped. It is well-known that far-field radiations
of QNMs and thus the corresponding scattered fields are divergent. Several
approaches exist that can overcome this difficulty
LALANNE__LaserPhotonicsRev._Light , and a direct technique is to treat
$\tilde{\mathbf{J}}(\mathbf{r})=-i\omega[\varepsilon(\mathbf{r},\omega)-1]\tilde{\mathbf{E}}(\mathbf{r})$
as source currents. They can then be expanded into electromagnetic multipoles
(spherical harmonics) at real $\omega$, based on which convergent radiated (or
scattered) far-fields $\tilde{\mathbf{E}}_{\rm{rad}}(\mathbf{r})$ can be
directly calculated POWELL_Phys.Rev.Applied_interference_2017 ;
GRAHN_NewJ.Phys._electromagnetic_2012 .

This current-radiation perspective can simplify Eq. (1) as:

$\alpha(\omega)\propto\iiint_{\mathbf{V}}\mathbf{E}_{\rm{inc}}(\mathbf{r})\cdot\tilde{\mathbf{J}}(\mathbf{r})d^{3}\mathbf{r}.$
(2)

Let’s assume that ${\mathbf{E}}_{\rm{inc}}$ comes from a point-dipole moment
$\mathscr{P}$ ($\mathscr{P}$ locates on the transverse plane
$\mathscr{P}\cdot\mathbf{k}_{\rm{inc}}=0$ and is in-phase with
${\mathbf{E}}_{\rm{inc}}$: $\mathscr{P}\propto{\mathbf{E}}_{\rm{inc}}$) in the
far zone. According to Lorentz reciprocity LANDAU_1984__Electrodynamicsb ;
POTTON_Rep.Prog.Phys._reciprocity_2004 :

$\iiint_{\mathbf{V}}\mathbf{E}_{\mathrm{inc}}(\mathbf{r})\cdot\tilde{\mathbf{J}}(\mathbf{r})d^{3}\mathbf{r}=\tilde{\mathbf{E}}_{\mathrm{rad}}\cdot\mathscr{\dot{P}}=-i\omega\tilde{\mathbf{E}}_{\mathrm{rad}}\cdot\mathscr{{P}},$
(3)

where time-derivative $\mathscr{\dot{P}}$ denotes the dipolar current and
$\mathbf{E}_{\mathrm{rad}}$ is the radiated field along
$\mathbf{k}_{\rm{rad}}=-\mathbf{k}_{\rm{inc}}$. Since
$\mathbf{E}_{\mathrm{inc}}\propto\mathscr{{P}}$, Eq. (3) can be converted to
$\iiint_{\mathbf{V}}\mathbf{E}_{\mathrm{inc}}(\mathbf{r})\cdot\tilde{\mathbf{J}}(\mathbf{r})d^{3}\mathbf{r}=-i\omega\tilde{\mathbf{E}}_{\mathrm{rad}}\cdot\mathscr{{P}}\propto\tilde{\mathbf{E}}_{\mathrm{rad}}\cdot\mathbf{E}_{\mathrm{inc}}$,
which simplifies Eq. (2) to:

$\alpha(\omega)\propto\tilde{\mathbf{E}}_{\mathrm{rad}}\cdot\mathbf{E}_{\mathrm{inc}},~{}~{}\mathbf{k}_{\rm{inc}}=-\mathbf{k}_{\rm{rad}}.$
(4)

It means that QNM excitation efficiency for incident plane waves can be
calculated directly, through the dot product of radiated and incident fields.
Alternative derivations for Eq. (S8), without referring to source currents of
either radiated or incident fields, are presented in Ref.
Supplemental_Material .

The validity of Eq. (S8) can be checked against the simplest example of a
metal bar supporting solely an electric dipole. Electric fields radiated are
shown in Fig. 1(a): the dipole is vertically oriented and the fields are
parallel to the lines of longitude CHEN_ACSOmega_Global ;
JACKSON_1998__Classical . Along the dipole, there are no radiations
$\tilde{\mathbf{E}}_{\mathrm{rad}}=0$ and thus $\alpha=0$ according to Eq.
(S8). It means plane waves incident parallel to the bar will not excite it,
which agrees with our conventional method of projecting the incident electric
field onto the bar axis. Similar analysis can be conducted along other
directions, where one linear polarization (parallel to the radiated field)
excites the dipole while the other orthogonal polarization does not. The
superiority of Eq. (S8) to Eq. (1) is not obvious for this example but would
become apparent for sophisticated bodies.

Radiations can be always expanded into right- and left-handed circularly
polarized (RCP and LCP, denoted respectively by $\circlearrowright$ and
$\circlearrowleft$) components along $\mathbf{k}_{\rm{rad}}$:
$\tilde{\mathbf{E}}_{\mathrm{rad}}=\mu\tilde{\mathbf{E}}_{\mathrm{rad}}^{\circlearrowright}+\nu\tilde{\mathbf{E}}_{\mathrm{rad}}^{\circlearrowleft}$.
Here $\mu$ and $\nu$ are generally complex numbers ($|\mu^{2}|+|\nu^{2}|=1$),
the ratio of which decides properties of the polarization ellipse of
$\tilde{\mathbf{E}}_{\mathrm{rad}}$ YARIV_2006__Photonics . For incident RCP
and LCP waves along $\mathbf{k}_{\rm{inc}}=-\mathbf{k}_{\rm{rad}}$, according
to Eq. (S8) excitation coefficients are respectively:

$\alpha^{\circlearrowright}\propto\mu,~{}~{}\alpha^{\circlearrowleft}\propto\nu.$
(5)

When only one QNM is dominantly excited, all cross sections of
extinction,scattering and absorption are proportional to excitation efficiency
$|\alpha|^{2}$:
$\rm{C}_{\rm{ext,sca,abs}}^{\circlearrowright,\circlearrowleft}\propto|\alpha^{\circlearrowright,\circlearrowleft}|^{2}$.
The circular dichroism (CD) is:

$\rm{CD}=(\rm{C}_{\rm{abs}}^{\circlearrowright}-\rm{C}_{\rm{abs}}^{\circlearrowleft})/(\rm{C}_{\rm{abs}}^{\circlearrowright}+\rm{C}_{\rm{abs}}^{\circlearrowleft}),$
(6)

which can then be simplified through Eq. (5) as:

$\rm{CD}=|\mu^{2}|-|\nu^{2}|=S_{3},~{}~{}\mathbf{k}_{\rm{rad}}=-\mathbf{k}_{\rm{inc}}.$
(7)

Here $S_{3}$ is nothing but exactly one of the Stokes parameters widely
employed for polarization characterizations YARIV_2006__Photonics : $S_{3}=\pm
1$ corresponds respectively to RCP and LCP waves; $S_{3}=0$ corresponds to
linear polarizations and other values of $S_{3}$ to elliptical polarizations.
Equation (7) constitutes the main result of this study, which reveals the
subtle connection between $S_{3}$ of radiations and CD when only one QNM is
dominantly excited: $|\rm{CD}|$ can be maximized and minimized antiparallel to
the C-direction ($S_{3}=\pm 1$) and L-direction ($S_{3}=0$) to ideal values of
$|\rm{CD}|=1$ and $\rm{CD}=0$, respectively. We emphasize that those chirality
extremization directions are exactly where generic polarization singularities
are present NYE_1983_ProcRSocA_Polarization ; NYE_1983_Proc.R.Soc.A_Lines ;
BERRY_2001_SecondInt.Conf.Singul.Opt.Opt.VorticesFundam.Appl._Geometry . The
nontrivial topology of the momentum sphere secures the existence of those
singularity directions CHEN_2019__Singularities ;
CHEN_2019_ArXiv190409910Math-PhPhysicsphysics_Linea ; CHEN_ACSOmega_Global ,
which guarantees broad applicability of our model.

We can generalize definitions of CD to
$\rm{CD_{ext,sca}}=(\rm{C}_{\rm{ext,sca}}^{\circlearrowright}-\rm{C}_{\rm{ext,sca}}^{\circlearrowleft})/(\rm{C}_{\rm{ext,sca}}^{\circlearrowright}+\rm{C}_{\rm{ext,sca}}^{\circlearrowleft})$,
and identical to Eq. (7) we have $\rm{CD_{ext,sca}}=S_{3}$
Supplemental_Material . Based on this relation we can deduce polarization
properties of mode radiations from general scattering properties of the
scatterer. For example, it has been proved that for oppositely incident waves
on a reciprocal scatterer SOUNAS_Opt.Lett.OL_extinction_2014 ;
CHEN_2020_Phys.Rev.Research_Scatteringa :
$\rm{CD_{ext}}(\mathbf{k}_{\rm{inc}})=\rm{CD_{ext}}(-\mathbf{k}_{\rm{inc}})$,
which immediately implies:
$S_{3}(\mathbf{k}_{\rm{rad}})=S_{3}(-\mathbf{k}_{\rm{rad}})$ and thus Eq. (7)
is actually valid for both directions of
$\mathbf{k}_{\rm{inc}}=\pm\mathbf{k}_{\rm{rad}}$. It is further proved
CHEN_2020_Phys.Rev.Research_Scatteringa  that for inversion-symmetric
scatterers, $\rm{CD_{ext}}=0$ for arbitrary incident direction, which requires
that QNM is everywhere linearly-polarized ($S_{3}=0$) throughout the momentum
sphere Supplemental_Material .

It is clear from Eq. (7) that CD (including its generalized versions of
$\rm{CD_{ext,sca}}$) is solely decided by polarizations while has nothing to
do with intensities of radiations. As a result, an alternative approach for
expansions of the source currents $\tilde{\mathbf{J}}(\mathbf{r})$ into
multipoles of complex frequencies does not affect our results, since such an
expansion changes only the radiation intensity but not the polarization
distributions (ratios between different multipoles are the same for real and
complex frequency multipolar expansions) WU_Phys.Rev.A_Intrinsic .

Figure 1: (a) A metal bar supports an electric dipole mode and its radiated
electric fields on the momentum sphere. (b) A gold SRR with the spherical
coordinate system $(r,\theta,{\varphi})$, and (c) $S_{3}$ distribution for the
QNM supported. Two singularity directions are marked:
$\mathbf{k}_{\rm{rad}}^{c}(\theta=67.3^{\circ},{\varphi}=175.1^{\circ})$
($\star$) and $\mathbf{k}_{\rm{rad}}^{l}(\theta=0)$ ($\times$). (d) and (e) CD
and absorption spectra for RCP and LCP incident waves, for
$\mathbf{k}_{\rm{inc}}=-\mathbf{k}_{\rm{rad}}^{c},-\mathbf{k}_{\rm{rad}}^{l}$
respectively [directions also marked in (f)]. (f) Angular CD spectra for waves
incident antiparallel to directions on a great momentum circle marked in (c)
by a dashed arc.

To confirm our theory, we begin with the widely employed split ring resonator
(SRR) that exhibits two symmetry mirrors (x-y and y-z planes) and thus achiral
[Fig. 1(b)]. The SRR is made of gold with permittivity listed in Ref.
Johnson1972_PRB . Neighbouring $\omega_{1}=7.4875\times 10^{14}$ rad/s
($\lambda_{1}=2.516~{}\mu$m), an individual QNM is excited with eigenfrequency
$\tilde{\omega}_{1}=(7.4875\times 10^{14}-2.0414\times 10^{13}\rm{i})$ rad/s
(numerical results are obtained using COMSOL Multiphysics in this study). The
$S_{3}$ distribution for this mode is presented in Fig. 1(c), where
polarizations singularizes with $S_{3}=\pm 1$ or $0$ are generically manifest
along various directions GBUR_2016__Singular ; BERRY_2017__HalfCentury ;
DENNIS_2009_ProgressinOptics_Chapter ; LIU_ArXiv201204919Phys._Topological .
Two singularity directions are marked (one C-direction of
$\mathbf{k}_{\rm{rad}}^{c}$ and one L-direction of
$\mathbf{k}_{\rm{rad}}^{l}\|\mathbf{z}^{+}$) and circularly-polarized plane
waves are incident antiparallel to those directions. The corresponding spectra
of absorption and CD are demonstrated respectively in Figs. 1(d) and (e),
showing clearly that $|\rm{CD}|$ is maximized and minimized respectively. The
deviation of CD from its ideal absolute values of $1$ and $0$ at some spectral
positions is induced by some marginal contributions of other QNMs excited, as
is also the case in following studies of other structures. We emphasize that
zero CD in Fig. 1(e) is observed along $\mathbf{k}_{\rm{rad}}^{l}$ that
preserves the mirror symmetry of the whole scattering configuration (parity
conservation requires zero CD BARRON_2009__Molecular ;
CHEN_2020_Phys.Rev.Research_Scatteringa ). This is consistent with the
conception that extrinsic chiralities are present along symmetry-breaking
directions PAPAKOSTAS_Phys.Rev.Lett._optical_2003 ;
PLUM_Appl.Phys.Lett._Optical ; PLUM_Phys.Rev.Lett._metamaterials_2009 ;
SERSIC_Phys.Rev.Lett._Ubiquity ; RIZZA_Phys.Rev.Lett._OneDimensional .

To verify the validity of Eq. (7) for general non-singularity directions
$0<|S_{3}|<1$, a great circle on the momentum sphere (parameterized by
$0\leq\beta\leq 2\pi$) that contains the marked C-direction and L-direction is
selected [also marked in Fig. 1(c)]. The angular CD spectra
($\omega=\omega_{1}$) along directions antiparallel to points on this circle
are shown in Fig. 1(f), where two sets of results are presented: one
calculated from $S_{3}$ [Eq. (7)] and the other through $\rm{C}_{\rm{abs}}$
obtained in direct scattering simulations [Eq. (6)]. Both sets of results
agree well, the symmetry of which
[$\rm{CD}(\beta)\approx\rm{CD}(\beta\pm\pi)$] reconfirms our previous claims
of $\rm{CD}(\mathbf{k}_{\rm{inc}})=\rm{CD}(-\mathbf{k}_{\rm{inc}})$ and
$S_{3}(\mathbf{k}_{\rm{rad}})=S_{3}(-\mathbf{k}_{\rm{rad}})$ for single mode
excitations.

Now that achiral structures do exhibit chiral responses along some incident
directions, why should they be classified as achiral? The conventional
standard is purely geometric: they exhibit mirror or inversion symmetries and
thus can be superimposed onto their mirror images BARRON_2009__Molecular .
Alternatively, CD can be employed to categorize structures as chiral or not,
but only when it is orientation-averaged among all incident directions:

$\mathbb{CD}=\frac{1}{4\pi|\mathbf{k}_{\rm{inc}}|^{2}}\iint\rm{CD}~{}d^{2}\mathbf{k}_{\rm{inc}}.$
(8)

The law of parity conservation BARRON_2009__Molecular ;
CHEN_2020_Phys.Rev.Research_Scatteringa  ensures that the mirror-symmetry or
inversion-symmetry operation maps
$\rm{C}_{\rm{abs}}^{\circlearrowright,\circlearrowleft}$ to
$\rm{C}_{\rm{abs}}^{\circlearrowleft,\circlearrowright}$, flipping the sign of
CD according to Eq. (6). As a result, $\mathbb{CD}=0$ for mirror-symmetric or
inversion-symmetric structures, which agrees with the geometric standard of
image superimposition.

Figure 2: (a) An intrinsically chiral SRR dimer and (b) $S_{3}$ distribution
for the QNM supported. Two singularity directions are marked:
$\mathbf{k}_{\rm{rad}}^{c}(\theta=63.4^{\circ},{\varphi}=7.3^{\circ})$
($\star$) and
$\mathbf{k}_{\rm{rad}}^{l}(\theta=88.1^{\circ},{\varphi}=7.3^{\circ})$
($\times$). (c) and (d) CD and absorption spectra for circularly-polarized
incident waves), for
$\mathbf{k}_{\rm{inc}}=-\mathbf{k}_{\rm{rad}}^{c},-\mathbf{k}_{\rm{rad}}^{l}$
respectively [directions also marked in (e)]. (e) Angular CD spectra for waves
incident antiparallel to directions on a great momentum circle marked in (b)
by a dashed arc.

Now we turn to a SRR dimer that is intrinsically chiral [Fig. 2(a)], which
supports a QNM at eigenfrequency $\tilde{\omega}_{2}=(7.0765\times
10^{14}-1.6125\times 10^{13}\rm{i})$ rad/s. The $S_{3}$ distribution of this
QNM is presented in Fig. 2(b), where one C-direction and one L-direction are
marked. The corresponding spectra of absorption and CD are demonstrated
respectively in Figs. 2(c) and (d) for plane waves incident antiparallel to
those directions. We have also selected a great momentum circle [as marked in
Fig. 2(b), containing C-direction, L-direction and the z-axis] and the
corresponding angular $\rm{CD}$ spectra [$\omega=\rm{Re}(\tilde{\omega}_{2})$
and $\lambda=~{}2.662~{}\mu$m, where Re denotes the real part] along
directions antiparallel to points on this circle are shown in Fig. 2(e). It is
worth noting that along the dimer twisting direction (z-axis with $\beta=0$
and $\pi$), CD is actually quite small ($\rm{CD}=S_{3}=0.12$) and far from
being maximal.

For chiral structures consisting of twisted elements that are similar to that
shown in Fig. 2(a), though not stated explicitly, it is widely believed that
the incident direction of maximal $|\rm{CD}|$ should be parallel to the
twisting axis BORISKINA__Singular ; HENTSCHEL_2017_Sci.Adv._Chiral ;
COLLINS_AdvancedOpticalMaterials_chirality_2017 ; QIU_Adv.Funct.Mater._3D ;
MUN_LightSci.Appl._Electromagnetic ; GANSEL_Science_Gold ;
SCHAFERLING_2012_Phys.Rev.X_Tailoring ; YIN_NanoLett._Interpreting ;
CUI_NanoLett._Giant ; WANG_ACSPhotonics_Circular ;
NAJAFABADI_2017_ACSPhotonics_Analytical ; GORKUNOV_Phys.Rev.Lett._Metasurfaces
. In fact, this is not the case and CD along other directions can be much more
significant as shown in Fig. 2(e). A structure being achiral or chiral is
decided by whether or not $\mathbb{CD}=0$, and the sign of $\mathbb{CD}$
rather than CD determines overall structure handedness. For achiral
structures, $\mathbb{CD}=0$ does not require $\rm{CD}=0$ everywhere, and
extrinsic chiralities emerge along directions of $\rm{CD}\neq 0$
PAPAKOSTAS_Phys.Rev.Lett._optical_2003 ; PLUM_Appl.Phys.Lett._Optical ;
PLUM_Phys.Rev.Lett._metamaterials_2009 ; SERSIC_Phys.Rev.Lett._Ubiquity ;
RIZZA_Phys.Rev.Lett._OneDimensional . In a similar fashion, for chiral
structures, $\mathbb{CD}\neq 0$ requires neither $\rm{CD}\neq 0$ nor
$\mathbb{CD}$ and $\rm{CD}$ being of the same sign everywhere. This means that
an intrinsically chiral structure can manifest no chirality at all
($\rm{CD}=0$) or even chiralities of opposite handedness (CD of different
signs) among different incident directions [Fig. 2(e)].

Up to now, we have discussed how to extremize CD to its ideal absolute maximum
or minimum values with respect to incident directions. Since $|\mathbb{CD}|$
is automatically minimized ($\mathbb{CD}=0$) for achiral structures, it is
both interesting and significant to ask how to maximize $|\mathbb{CD}|$
ideally to $\mathbb{CD}=\pm 1$? According to Eq. (8), the ideal maximization
of $|\mathbb{CD}|$ requires $\rm{CD}=\pm 1$ along all incident directions [or
more accurately some isolated directions can be exempted, as they do not
affect the overall integration in Eq. (8)]. This requires mode radiations are
circularly-polarized with the same handedness throughout the momentum sphere
except for some isolated directions. This is possible, for example, for a pair
of parallel electric and magnetic dipoles (or higher-order multipoles of the
same order) of the same magnitude (in terms of total scattered power) and
$\pm\pi/2$ phase contrast, as secured by the electromagnetic duality symmetry
(along the isolated directions parallel to the dipole orientation direction
there are no radiations and thus the polarizations are not defined)
JACKSON_1998__Classical . Our conclusions for $|\mathbb{CD}|$ maximization are
consistent with the results presented in Ref. FERNANDEZ-
CORBATON_2016_Phys.Rev.X_Objects , where the problem has been approached from
a very different perspective. However up to now, most if not all realistic
reciprocal structures are anisotropic in terms of CD responses and thus ideal
$\mathbb{CD}=\pm 1$ has not been obtained so far, though not theoretically
impossible.

Figure 3: (a) The dispersion curve of one TE-like QNM band for square-latticed
PCS (show schematically as inset). The $S_{3}$ distribution of the QNM band is
also shown and a singularity direction $\mathbf{k}_{\rm{rad}}^{c}$ is marked
($\star$). (b) Angular reflection spectra for RCP and LCP incident waves
without material losses, and (c-d) corresponding near-field distributions when
$\mathbf{k}_{\rm{inc}}=\mathbf{k}_{\rm{rad}}^{c}$. (e) Angular absorption and
reflection spectra with material losses.

As a final step, we proceed to apply our framework to infinitely-extended
photonic crystal slabs (PCSs), which support Bloch QNMs
JOANNOPOULOS_2008__Photonic ; HSU_Nat.Rev.Mater._bound_2016 ;
LIU_ArXiv201204919Phys._Topological . It is well known that through symmetry
breaking, some bound states in the continuum HSU_Nat.Rev.Mater._bound_2016
can be broken into circularly-polarized radiating states
LIU_ArXiv201204919Phys._Topological ; LIU_2019_Phys.Rev.Lett._Circularly ;
GUO_2020_Phys.Rev.Lett._Meron ; YIN_2020_Nature_Observationa ;
YE_2020_Phys.Rev.Lett._Singular ; CHEN_2019_ArXiv190409910Math-
PhPhysicsphysics_Linea ; YODA_Phys.Rev.Lett._Generationa . One such square-
latticed (periodicity $p=380$ nm) PCS of index $n=2.02$ is shown as inset of
Fig. 3(a), which is achiral (symmetric with respect to the x-y and y-z planes)
while the symmetry respect to the x-z plane is broken. The dispersion curve
[Re($\tilde{\omega}$) versus $\mathbf{k}_{\rm{rad}}^{x}$, with
$\mathbf{k}_{\rm{rad}}^{y}=0$] that is colored according to the $S_{3}$
distribution of QNM radiations for one non-degenerate TE-like (electric fields
distributed on the x-y plane) band are shown in Fig. 3(a). One C-direction is
marked ($\star$) with $\tilde{\omega}_{3}=(3.8409\times 10^{15}-1.8686\times
10^{13}\rm{i})$ rad/s and
$\mathbf{k}_{\rm{rad}}^{c}=(\mathbf{k}_{\rm{rad}}^{c,x},\mathbf{k}_{\rm{rad}}^{c,y},\mathbf{k}_{\rm{rad}}^{c,z})$,
with $\mathbf{k}_{\rm{rad}}^{c,x}p/2\pi=0.1332$,
$\mathbf{k}_{\rm{rad}}^{c,y}=0$,
$\mathbf{k}_{\rm{rad}}^{c,z}=\sqrt{[\rm{Re}(\tilde{\omega}_{3})/c]^{2}-(\mathbf{k}_{\rm{rad}}^{c,x})^{2}-(\mathbf{k}_{\rm{rad}}^{c,y})^{2}}$
and $c$ is the speed of light. Now we shine circularly-polarized plane waves
onto the PCS with fixed frequency $\omega$=Re$(\tilde{\omega}_{3})$ along
different directions (variant $\mathbf{k}_{\rm{inc}}^{x}$) on the x-z plane
($\mathbf{k}_{\rm{inc}}^{y}=0$), with
$\mathbf{k}_{\rm{inc}}=(\mathbf{k}_{\rm{inc}}^{x},\mathbf{k}_{\rm{inc}}^{y},\mathbf{k}_{\rm{inc}}^{z})$.
Angular reflection (R) spectra is presented in Fig. 3(b), where the critical
direction $\mathbf{k}_{\rm{inc}}=\mathbf{k}_{\rm{rad}}^{c}$ is marked.

Figure 3(b) shows clearly that along C-direction RCP and LCP waves are almost
fully reflected and transmitted (no higher-order diffractions), respectively.
Near-field distributions ($\mathbf{E}_{y}$ on the x-z plane; dashed rectangles
denote the cross sections of the PCS) are presented in Figs. 3(c) and (d),
where the formation of standing wave upon reflection in Fig. 3(d) obscures the
information of oblique incidence. Those properties agree with our result in
Eq. (S8), which requires that circularly-polarized Bloch QNM is maximally and
not excited by LCP and RCP waves, leading to perfect reflection and
transmission respectively. When material losses are incorporated
($n=2.02+0.001i$), both spectra of reflection and absorption (A) efficiency
are summarized in Fig. 3(e). Along the marked critical direction, as expected,
there is considerable and almost no absorption for incident LCP and RCP waves
respectively, which is also consistent with our previous analyses.

To conclude, we revisit optical chiralities from perspectives of QNMs and
their polarization singularities. When plane waves are incident parallel to
singularity C\- and L-directions, the chiral responses in terms of CD would be
ideally maximized and minimized, respectively. For general incident
directions, we discover the subtle equivalence of CD and $S_{3}$ of the QNM
radiation, showing that an intrinsical chiral structures can surprisingly
manifest no chirality at all or even chiralities of opposite handedness among
different incident directions. The validity of our conclusions resides on the
approximation of single QNM excitation, which is indeed applicable to many
structures at various spectral regimes, especially to the chiral ones that do
not exhibit any geometric symmetries (e.g. rotational symmetries that protect
at least one pair of degenerate counter-rotating QNMs
GORKUNOV_Phys.Rev.Lett._Metasurfaces ). When several QNMs (including
degenerate ones) are co-excited, observable optical properties (such as
$\rm{C}_{\rm{ext,sca,abs}}$) are neither proportional to the excitation
efficiency of any QNM nor their direct sum, since QNMs are generally not
orthogonal and inter-mode interference terms have to be carefully analysed
LALANNE__LaserPhotonicsRev._Light ; POWELL_Phys.Rev.Applied_interference_2017
; LEUNG_1994_Phys.Rev.A_Completeness . Besides QNM-based approaches
LALANNE__LaserPhotonicsRev._Light ; POWELL_Phys.Rev.Applied_interference_2017
; ALPEGGIANI_Phys.Rev.X_quasinormalmode_2017-1 , techniques such as the
scattering matrix method MISHCHENKO_2002__Scattering  can be also employed to
tackle more general scenarios, for which to extract simple and general
principles like the ones unveiled here constitutes a promising future research
direction. Our results are directly applicable to elliptical dichroism
Supplemental_Material , and singularity based approaches we introduce can be
potentially extended to structured non-plane beams, especially those carrying
both spin and angular momentum ZAMBRANA-PUYALTO_Nat.Commun._Angular ;
NI_PNAS_Gigantic .

_Acknowledgment_ : We acknowledge the financial support from National Natural
Science Foundation of China (Grant No. 11874026 and 11874426), and several
other Researcher Schemes of National University of Defense Technology. W. L.
is indebted to Sir Michael Berry and Prof. Tristan Needham for invaluable
correspondences.

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Supplemental Material for:

Extremize Optical Chiralities through Polarization Singularities

Weijin Chen1,∗, Qingdong Yang1,∗, Yuntian Chen1,2,†, and Wei Liu3,‡

1 _School of Optical and Electronic Information, Huazhong University of
Science and Technology, Wuhan, Hubei 430074, P. R. China_
2 _Wuhan National Laboratory for Optoelectronics, Huazhong University of
Science and Technology, Wuhan, Hubei 430074, P. R. China_
3 _College for Advanced Interdisciplinary Studies, National University of
Defense Technology, Changsha, Hunan 410073, P. R. China_

The Supplemental Material includes the following three sections: (I).
Alternative derivations for Eq. (4) without referring to source currents of
radiations or incident fields; (II). Verifications of
$\rm{CD_{ext,sca}}=S_{3}$ for both achiral and chiral scatterers; (III).
$S_{3}$ distributions of an inversion-symmetric scatterer and its CD
responses; (IV). Applications for elliptical dichroism maximization.

##  S1 (I). Alternative derivations for Eq. (4) without referring to source
currents of radiations or incident fields

According to de Hoop’s formulations DEHOOP_Appl.sci.Res._reciprocitya , the
direct dot product of $\tilde{\mathbf{E}}_{\mathrm{rad}}$ and
$\mathbf{E}_{\mathrm{inc}}$ in the far zone can be expressed as an integral
form (Eq. (4.10) of Ref. DEHOOP_Appl.sci.Res._reciprocitya ) :

$\tilde{\mathbf{E}}_{\mathrm{rad}}\cdot\mathbf{E}_{\mathrm{inc}}\propto\int_{\mathbf{S}}\left({\mathbf{E}}^{\mathrm{s}}\times\mathrm{\mathbf{H}}_{\mathrm{inc}}^{s}-\mathrm{\mathbf{E}}_{\mathrm{inc}}^{s}\times{\mathbf{H}}^{\mathrm{s}}\right)\cdot\bm{n}\mathrm{d}\mathbf{S},~{}~{}\mathbf{k}_{\rm{inc}}=-\mathbf{k}_{\rm{rad}}.$
(S1)

Here S denote the boundary of the scatterer and $\bm{n}$ denotes its normal
vector pointing outwards; (${\mathbf{E}}^{\mathrm{s}}$,
${\mathbf{H}}^{\mathrm{s}}$) are radiated electric and magnetic fields on S of
the QNM excited; ($\mathbf{E}_{\mathrm{inc}}^{s}$,
$\mathbf{H}_{\mathrm{inc}}^{s}$) are incident electric and magnetic fields on
the boundary. Apparently (${\mathbf{E}}^{\mathrm{s}}$,
${\mathbf{H}}^{\mathrm{s}}$) originate from the excited QNM and thus for
single-mode excitation we have:

$({\mathbf{E}}^{\mathrm{s}},{\mathbf{H}}^{\mathrm{s}})\propto\alpha(\tilde{{\mathbf{E}}},{\tilde{\mathbf{H}}}),$
(S2)

where $(\tilde{{\mathbf{E}}},~{}{\tilde{\mathbf{H}}})$ are the eigenfields of
the QNM excited. Through Eqs. (S1) and (S2) we obtain:

$\tilde{\mathbf{E}}_{\mathrm{rad}}\cdot\mathbf{E}_{\mathrm{inc}}\propto\alpha\int_{\mathbf{S}}\left(\tilde{{\mathbf{E}}}\times\mathrm{\mathbf{H}}_{\mathrm{inc}}^{s}-\mathrm{\mathbf{E}}_{\mathrm{inc}}^{s}\times{\tilde{\mathbf{H}}}\right)\cdot\bm{n}\mathrm{d}\mathbf{S},$
(S3)

which can be simplified as:

$\alpha\propto\tilde{\mathbf{E}}_{\mathrm{rad}}\cdot\mathbf{E}_{\mathrm{inc}},~{}~{}\mathbf{k}_{\rm{inc}}=-\mathbf{k}_{\rm{rad}}.$
(S4)

It is exactly Eq. (4) in the main text.

We further note that the field-field surface integral form on the right hand
side of Eq. (S1) can be converted to the field-current volume integral form
through the Lorentz reciprocity LANDAU_1984__Electrodynamicsb :

$\int_{\mathbf{S}}\left({\mathbf{E}}^{\mathrm{s}}\times\mathrm{\mathbf{H}}_{\mathrm{inc}}^{s}-\mathrm{\mathbf{E}}_{\mathrm{inc}}^{s}\times{\mathbf{H}}^{\mathrm{s}}\right)\cdot\bm{n}\mathrm{d}\mathbf{S}=\iiint_{\mathbf{V}}\mathbf{E}_{\rm{inc}}(\mathbf{r})\cdot\tilde{\mathbf{J}}(\mathbf{r})d^{3}\mathbf{r},~{}~{}\mathbf{k}_{\rm{inc}}=-\mathbf{k}_{\rm{rad}},$
(S5)

which is consist with Eqs. (2-4) in our main text. As a result, the
derivations formulated through Eqs. (S1)-(S4) are effectively equivalent to
those in the main text, despite that here the source currents of neither the
radiated nor the incident fields are involved.

##  S2 (II). Verifications of $\rm{CD_{ext,sca}}=S_{3}$ for both achiral and
chiral scatterers

The generalized versions of CD in terms of extinction and scattering cross
sections $\rm{C}_{ext,sca}$ are:

$\rm{CD_{ext,sca}}=(\rm{C}_{\rm{ext,sca}}^{\circlearrowright}-\rm{C}_{\rm{ext,sca}}^{\circlearrowleft})/(\rm{C}_{\rm{ext,sca}}^{\circlearrowright}+\rm{C}_{\rm{ext,sca}}^{\circlearrowleft}),$
(S6)

and it has been clarified in the main text that similar to CD, we have

$\rm{CD_{ext,sca}}=S_{3}.$ (S7)

Here we aim to verify Eq. (S7) numerically and show the results in Fig. S1.
The structures we study [Figs. S1(a) and (d)] are the same as those achiral
and chiral scatterers already investigated in Figs. 1 and 2. The angular
$\rm{CD_{ext,sca}}$ spectra are shown in Figs. S1(b-c) for the achiral
scatterer and in Figs. S1(e-f) for the chiral scatterer. Other parameters
(such as the orientations of the great momentum circles and their
parameterization angle $\beta$) for Figs. S1(b-c) and Figs. S1(e-f) are the
same as those for Fig. 1(f) and Fig. 2(e), respectively. The two sets of
results [obtained according to Eq. (S6) and Eq. (S7), respectively] shown for
both scenarios agree well, confirming the equivalence between $S_{3}$ and
generalized $\rm{CD_{ext,sca}}$.

Figure S1: (a) The achiral SRR and (d) the chiral SRR dimer, which are the
same as those studied in Figs. 1 and 2, respectively. The angular
$\rm{CD_{ext,sca}}$ spectra are shown in (b-c) and (e-f), for the structures
in (a) and (d), respectively.

##  S3 (III). $S_{3}$ distributions of an inversion-symmetric scatterer and
its CD responses

Our analyses in the main text show that if an inversion-symmetric structure
supports an individual QNM, then the mode is everywhere linearly-polarized
($S_{3}=0$) on the momentum sphere and thus $\rm{CD}=0$ for all incident
directions. This is confirmed numerically in Fig. S2: the SRR dimer exhibits
inversion symmetry [Fig. S2(a)] and supports dominantly one QNM at
eigenfrequency $\tilde{\omega}_{4}=(8.68\times 10^{14}-2.599\times
10^{13}\rm{i})$ rad/s (the corresponding wavelength for the real part of
$\tilde{\omega}_{4}$ is $\lambda=2.17~{}\mu$m); the $S_{3}$ distribution of
the QNM is presented in Fig. S2(b), where a random direction is indicated
($\times$). It is clear that $S_{3}\approx 0$ throughout the momentum sphere
and thus radiations are linearly polarized along arbitrary directions.
Antiparallel to this direction the corresponding CD and absorption spectra are
summarized in Fig. S2(c).

Figure S2: (a) An inversion-symmetric SRR dimer and (b) the $S_{3}$ parameter
distribution for radiations of the QNM supported at
$\tilde{\omega}_{4}=(8.68\times 10^{14}-2.599\times 10^{13}\rm{i})$ rad/s. A
random singularity direction
$\mathbf{k}_{\rm{rad}}^{l}(\theta=58^{\circ},{\varphi}=110^{\circ})$ is marked
($\times$). (c) CD and absorption spectra for RCP and LCP incident waves are
shown with $\mathbf{k}_{\rm{inc}}=-\mathbf{k}_{\rm{rad}}^{l}$.

##  S4 (IV). Applications for elliptical dichroism maximization

In our main text we have discussed only CD for incident circularly-polarized
waves of opposite handedness. Similar investigations can be extended to
elliptical dichroism (ED) for incident waves of orthogonally elliptical
polarizations. Equation (4) in our main text is reproduced here as follows:

$\alpha(\omega)\propto\tilde{\mathbf{E}}_{\mathrm{rad}}\cdot\mathbf{E}_{\mathrm{inc}},~{}~{}\mathbf{k}_{\rm{inc}}=-\mathbf{k}_{\rm{rad}},$
(S8)

which indicates that for generally elliptically-polarized
$\tilde{\mathbf{E}}_{\mathrm{rad}}$, there must be two elliptical
polarizations which can maximally or not excite the QNM when incident
antiparallel to the radiation direction
$\mathbf{k}_{\rm{inc}}=-\mathbf{k}_{\rm{rad}}$. Let us assume that the
radiation $\tilde{\mathbf{E}}_{\mathrm{rad}}$ is along the direction
$\mathbf{k}_{\rm{rad}}$ and characterized by normalized Jones vector (in
linear basis of $\mathbf{e}^{\theta}$-$\mathbf{e}^{\varphi}$, where
$\mathbf{e}^{\theta}$ and $\mathbf{e}^{\varphi}$ are unit vectors in the
spherical coordinate system) YARIV_2006__Photonics :
$\mathbf{J}_{\rm{rad}}=(\varsigma,\tau)$, with $|\varsigma|^{2}+|\tau|^{2}=1$.
For light incident antiparallel to the radiation direction
$\mathbf{k}_{\rm{inc}}=-\mathbf{k}_{\rm{rad}}$, according to Eq. S8: the
incident wave that can maximally excite the QNM is characterized by the Jones
vector $\mathbf{J}_{\rm{inc}}^{\uparrow}=(\varsigma^{*},\tau^{*})$ with
normalized excitation coefficient $\alpha^{\uparrow}=1$, where ∗ denotes
complex conjugate; while the incident wave that cannot excite the QNM
corresponds to a Jones vector
$\mathbf{J}_{\rm{inc}}^{\downarrow}=(\tau,-\varsigma)$ with null excitation
coefficient $\alpha^{\downarrow}=0$. We emphasize that all the Jones vectors
for both radiations and incident waves are defined in the same linear basis of
$\mathbf{e}^{\theta}$-$\mathbf{e}^{\varphi}$ defined for the radiation
direction. The aforementioned two incident waves are obviously orthogonal
since
$(\mathbf{J}_{\rm{inc}}^{\uparrow})^{*}\cdot\mathbf{J}_{\rm{inc}}^{\downarrow}=0$.
The absorption cross sections associated with two such incident polarizations
are:

$\rm{C}_{\rm{abs}}^{\uparrow,\downarrow}\propto|\alpha^{\uparrow,\downarrow}|^{2}=1,0$
(S9)

Figure S3: Structures and mode radiation $S_{3}$ parameters for achiral and
chiral scatterers in (a) and (b), which are respectively reproduced from Figs.
1 and 2. For each scenario a radiation direction is selected along which the
radiation is elliptically polarized:
$\mathbf{k}_{\rm{rad}}^{\rm{ach}}(\theta=33^{\circ},{\varphi}=0^{\circ})$ and
$\mathbf{k}_{\rm{rad}}^{\rm{ch}}(\theta=21^{\circ},{\varphi}=196^{\circ})$. ED
and absorption spectra for the achiral and chiral cases are shown respectively
in (c) and (d).

The ED is defined as:

$\rm{ED}=(\rm{C}_{\rm{abs}}^{\uparrow}-\rm{C}_{\rm{abs}}^{\downarrow})/(\rm{C}_{\rm{abs}}^{\uparrow}+\rm{C}_{\rm{abs}}^{\downarrow}),$
(S10)

which obviously reaches its ideal maximal value $\rm{ED}=1$ for the two
incident orthogonal polarizations characterized by Jones vectors of
$\mathbf{J}_{\rm{inc}}^{\uparrow,\downarrow}$. This means that our theory can
be employed for ED maximization.

To confirm our analyses we revisit the achiral and chiral scatterers already
investigated in Figs. 1 and 2, which together with the $S_{3}$ parameter
distributions for the QNMs supported are reproduced respectively in Figs.
S3(a) and (b). For the achiral scenario in Fig. S3(a), a radiation direction
$\mathbf{k}_{\rm{rad}}^{\rm{ach}}$ is selected and the radiation Jones vector
is $\mathbf{J}_{\rm{rad}}^{\rm{ach}}=(0.963,0.269e{{}^{-1.37i}})$. The
corresponding spectra of absorption and ED are demonstrated in Figs. S3(c),
for incident waves of Jones vectors
$\mathbf{J}_{\rm{inc}}^{\uparrow,\downarrow}$. While for the chiral scenario
in Fig. S3(b), a radiation direction $\mathbf{k}_{\rm{rad}}^{\rm{ch}}$ is
chosen and the radiation Jones vector is
$\mathbf{J}_{\rm{rad}}^{\rm{ch}}=(0.969,0.247e{{}^{-1.51i}})$. The
corresponding spectra of absorption and ED are presented in Fig. S3(d). It is
clear that for both scenarios, ED are ideally maximized close to the resonant
frequency.

## References

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