# Multi-Mode Optical Chirality Extremizations on Incident Momentum Sphere

Pengxiang Wang  1 Yuntian Chen  1,${\dagger}$ and Wei Liu2,3  1School of
Optical and Electronic Information, Huazhong University of Science and
Technology, Wuhan, Hubei 430074, People’s Republic of China
2College for Advanced Interdisciplinary Studies, National University of
Defense Technology, Changsha, Hunan 410073, P. R. China.
3Nanhu Laser Laboratory and Hunan Provincial Key Laboratory of Novel Nano-
Optoelectronic Information Materials and Devices, National University of
Defense Technology, Changsha, Hunan 410073, P. R. China.
<EMAIL_ADDRESS><EMAIL_ADDRESS>
††journal: opticajournal††articletype: Research Article

We study the momentum-space evolutions for chiral optical responses of multi-
mode resonators scattering plane waves of varying incident directions. It was
revealed, in our previous study [Phys. Rev. Lett. 126, 253901 (2021)], that
for single-mode resonators the scattering optical chiralities characterized by
circular dichroism ($\mathbf{CD}$) are solely decided by the third Stokes
parameter distributions of the quasi-normal mode (QNM) radiations:
$\mathbf{CD}=\mathbf{S}_{3}$. Here we extend the investigations to multi-mode
resonators, and explore numerically the dependence of optical chiralities on
incident directions from the perspectives of QNM radiations and their circular
polarization singularities. In contrast to the single-mode regime, for multi-
mode resonators it is discovered that $\mathbf{CD}$s defined in terms of
extinction, scattering and absorption generally are different and cannot reach
the ideal values of $\pm 1$ throughout the momentum sphere. Though the exact
correspondence between $\mathbf{CD}$ and $\mathbf{S}_{3}$ does not hold
anymore in the multi-mode regime, we demonstrate that the positions of the
polarization singularities still serve as an efficient guide for identifying
those incident directions where the optical chiralities can be extremized.

##  1 Introduction

Optical and geometric chiralities are profoundly different concepts: the
former generally corresponds to distinct optical responses of photonic
structures that are excited by sources of opposite handedness; while the
latter is a pure mathematical (geometric) one, which indicates that an object
is not invariant under the mirror operation [1, 2]. Nevertheless, those
concepts have almost always been discussed together, signifying the widely-
spread dogma that they are inextricably linked. Chiral optics dates back
partly to Louis Pasteur’s investigations into chiral molecules, which are
randomly orientated in the solution and thus the optical chiralities observed
were effectively orientation-averaged and thus independent on the incident
direction of the source [1]. For such systems containing many randomly-
orientated molecules, the absence of geometric chirality of each consisting
molecule inevitably results in the absence the optical chirality, which is
required by the fundamental law of parity conservation of Maxwell equations
[1, 3]. While for other systems, either individual structures or ensembles of
ordered or partially-ordered consisting molecules, the connection between
optical and geometric chiralities are generally broken: (extrinsic) optical
chiralities can be observed for achiral structures [2, 4, 5, 6] and a chiral
structure of fixed geometric chirality can manifest no optical chirality or
even optical chiralities of opposite handedness for different incident
directions [6].

Optical chiralities are generally dependent on the incident directions of
exciting sources, and all those directions constitute a closed momentum
sphere. According to the extreme value theorem [7], irrespective of the
specific physical configurations, there must be directions along which the
optical chirality reaches its extremized (maximum or minimum) values. In the
single-QNM regime, that is when there is only one QNM effectively excited, the
optical chirality is extremized to its ideal values along the directions where
the polarization singularities of the QNM radiations locate (independent on
the geometric chirality of the photonic structures investigated)
$\mathbf{CD}=\mathbf{S}_{3}$ [6]: for QNM radiation directions of circular
polarizations ($\mathbf{S}_{3}=\pm 1$), the CD defined
($\mathbf{CD}\in[-1,1]$) reaches its ideal values of $\mathbf{CD}=\pm 1$;
while for directions of linear polarizations ($\mathbf{S}_{3}=0$), optical
chiralities are absent ($\mathbf{CD}=0$). It is worth mentioning that though
the principles revealed in Ref. [6] and the exact correspondence
$\mathbf{CD}=\mathbf{S}_{3}$ are secured by the fundamental laws of
electromagnetic reciprocity and energy conservation (optical theorem), they
are in principle only applicable in the single-QNM regime.

In this paper we study numerically momentum-space chirality extremizations for
scattering photonic structures that support simultaneously multi-QNMs,
including both non-degenerate QNMs that are spectrally close and degenerated
QNMs that are induced by geometric symmetry. The optical chiralities are
characterized by CDs defined in terms of distinctions between extinction,
scattering and absorption cross sections with incident circularly-polarized
plane waves of opposite handedness (Section 2). Similar to the single-QNM
regime, on the closed momentum space, there must be directions along which the
CD defined can reach their extremized values $\mathbf{CD}^{\rm{max,min}}$:
$\mathbf{CD}^{\rm{min}}\leq\mathbf{CD}\leq\mathbf{CD}^{\rm{max}}$.
Nevertheless, in contrast to the single-QNM scenario, in the multi-QNM regime
(Section 3): $\mathbf{CD}$ generally cannot reach its ideal values of
$\mathbf{CD}=\pm 1$; CDs defined in terms of extinction, scattering and
absorption are generally not identical; the correspondence between
$\mathbf{CD}$ and $\mathbf{S}_{3}$ ($\mathbf{CD}=\mathbf{S}_{3}$) does not
hold anymore, implying that $\mathbf{CD}$ is extremized along the directions
where the polarizations of the QNM radiations are not necessarily singular.
Despite those differences between single- and multi-QNM excitations, we have
demonstrated numerically that in the multi-QNM regime, the positions of the
radiation polarization singularities in the momentum sphere still provide an
efficient guide for identifying the directions incident along which the
$\mathbf{CD}$s can be extremized. Our study can stimulate further explorations
to establish a comprehensive QNM-based theoretical model that can
systematically deal with not only reciprocal but also non-reciprocal multi-
mode chirality extremizations, with incident waves structured both spatially
and temporally beyond the plane-wave excitation regime.

##  2 Definitions and Single-Mode Optical Chirality Extremizations

Figure 1: (a) A gold structure [half of the four-fold rotionally symmetric
structure shown in Fig. 4(a)] with all geometric paramters and its orientation
within the spherical coordinate system specified. The azimuthal and polar
angles are $\varphi$ and $\theta$, respectively. (b) $\mathbf{S}_{3}$
distributions for the QNM supported. (c) The dependence of three
$\mathbf{CD}$s on $\varphi$ along the latituide circle of
$\theta=100^{\circ}$. (d)-(f) The dependence of three $\mathbf{CD}$s on
incident directions. In (b) and (d)-(f) the positions of polarization
singularities are also marked (stars: RCP; crosses: LCP).

Throughout this study, we focus on scattering structures, the optical
responses of which are characterized by cross sections of extinction,
scattering and absorption are $\rm{C}_{\rm{ext,sca,abs}}$ [8]. For right- and
left-handed circularly polarized incident plane waves (RCP and LCP, denoted
respectively by $\circlearrowright$ and $\circlearrowleft$) with wavevector
$\mathbf{k}_{\rm{inc}}$, those cross sections are denoted more specifically as
$\rm{C}_{\rm{ext,sca,abs}}^{\circlearrowright,\circlearrowleft}$. Then the
three CDs defined in terms of extinction, scattering and absorption are [6]:

$\rm{\mathbf{CD}}_{\rm{ext,sca,abs}}=\frac{\rm{C}_{\rm{ext,sca,abs}}^{\circlearrowright}-\rm{C}_{\rm{ext,sca,abs}}^{\circlearrowleft}}{\rm{C}_{\rm{ext,sca,abs}}^{\circlearrowright}+\rm{C}_{\rm{ext,sca,abs}}^{\circlearrowleft}},$
(1)

which is bounded as $\mathbf{CD}_{\rm{ext,sca,abs}}\in[-1,1]$.

In the single-QNM excitation regime, it is revealed the three CDs are
identical $\mathbf{CD}_{\rm{ext,sca,abs}}=\mathbf{CD}$ and inextricably linked
to the Stokes parameter $\mathbf{S}_{3}$ of the QNM radiation (radiation
wavevector $\mathbf{k}_{\rm{rad}}$) along both directions parallel or
antiparallel to $\mathbf{k}_{\rm{inc}}$:

$\rm{\mathbf{CD}}=\mathbf{S}_{3},~{}~{}\mathbf{k}_{\rm{rad}}=\pm\mathbf{k}_{\rm{inc}}.$
(2)

That is, in the single-QNM regime, the $\mathbf{S}_{3}$ for radiations along
both directions of $\pm\mathbf{k}_{\rm{rad}}$ are the same. As a result, the
$\mathbf{CD}$ reaches its ideal extremized values of $\pm 1$ along directions
where the QNM radiations are circularly-polarized (polarization singularities)
[6]. It is worth mentioning that beyond the single-QNM regime, such defined
three CDs do not have to be identical and could be quite different. For
example, for an arbitrary inversion-symmetric structure that is simultaneously
reciprocal, along any incident direction $\mathbf{CD}_{\rm{ext}}=0$ while
generally $\mathbf{CD}_{\rm{sca,abs}}\neq 0$ [9].

To further exemplify the conclusions above, we study a gold structure
(effective bulk permittivity is fitted from data in Ref. [10]; all numerical
results presented in this study are obtained using COMSOL Multiphysics)
schematically shown in Fig. 1(a), with all the geometric parameters specified.
This structure is essentially half of the structure shown in Fig. 4(a) which
exhibits rotation symmetry and thus supports degenerate modes [11]. With the
rotation symmetry broken, for this structure a single QNM [characterized by a
complex eigenfrequency of $\tilde{\omega}_{A}=(8.196\times 10^{14}-3.5\times
10^{13}\rm{i})$ rad/s] can be excited with negligible excitations of other
QNMs. $\mathbf{S}_{3}$ distributions for radiations of this QNM are shown in
Fig. 1(b), where the positions of circular polarization singularities are also
marked (stars: RCP; crosses: LCP). To show clearly the orientation of the
structure, in Fig. 1(a) a spherical coordinate system with the azimuthal angle
$\varphi$ and polar angle $\theta$ is also included.

For all the following discussions, the incident direction
$\mathbf{k}_{\rm{inc}}$ is described by the angle doublet ($\varphi$,
$\theta$) and the corresponding radiation direction is anti-parallel to it
$\mathbf{k}_{\rm{rad}}=-\mathbf{k}_{\rm{inc}}$. Such a setting is required to
directly apply the electromagnetic reciprocity principle [12, 13], built on
which the subtle connection between CD and $\mathbf{S_{3}}$ of QNM radiations
[Eq. (2)] can be established. The dependence of CDs on ($\varphi$, $\theta$)
are shown in Figs. 1(d)-(f), with incident angular frequency
$\omega=\mathbf{Re}(\tilde{\omega}_{A})$ and the corresponding wavelength
$\lambda_{A}=2.3~{}\mu$m. As we can see, all three CD are identical and they
reach their ideal values of $\mathbf{CD}=\pm 1$ at the positions where the QNM
radiations are singularly polarized [marked by stars (RCP;
$\mathbf{CD=S_{3}}=1$) and crosses (LCP; $\mathbf{CD=S_{3}}=-1$)]. As has been
mentioned already, the angular positions for the radiation singularities are
($\pi+\varphi$, $\pi-\theta$) rather than ($\varphi$, $\theta$) directly. The
CD distributions on a latitude circle ($\theta=100^{\circ}$) are also shown in
Fig. 1(c), further confirming Eq. (2) and the equality of all CDs.

Figure 2: (a) A gold SRR with all geometric parameters specified and (b) its
scattering spectra where two QNMs are pinpointed. (c) $\mathbf{S}_{3}$
distributions for the QNM-B. (d)-(f) The angular dependence of three
$\mathbf{CD}$s. In (c)-(f) the positions of polarization singularities are
also marked (stars: RCP; crosses: LCP).

##  3 Multi-Mode Optical Chirality Extremizations

As a next step, we extend our investigations to photonic structures that
support spectrally close non-degenerate modes and also degenerate modes. We
start with a gold split-ring resonator (SRR) schematically shown in Fig. 2(a),
with all geometric parameters specified. In Fig. 2(b) we show its scattering
spectra (in terms of scattering cross sections) for an RCP wave incident along
+z axis ($\theta=0$). For the spectral regime shown, two QNMs are marked the
corresponding complex eigenfrequencies are: $\tilde{\omega}_{B}=(1.4359\times
10^{15}-7.7\times 10^{13}\rm{i})$ rad/s and $\tilde{\omega}_{C}=(1.1204\times
10^{15}-2.538\times 10^{14}\rm{i})$ rad/s. The $\mathbf{S}_{3}$ distributions
for the QNM-B are shown in Fig. 2(c). We now fix the angular frequency of the
incident wave as $\omega=\mathbf{Re}(\tilde{\omega}_{B})$
($\lambda_{B}=1.313~{}\mu$m) and show in Figs. 2(d)-(f) the angular dependence
of CDs, where the polarization singularities of QNM-B also marked. In contrast
to the single-QNM excitations [Figs. 1(d)-(f)]: (i) The three CDs are not
identical anymore (see the extreme values of the color bar); (ii) None of the
CDs defined can reach their ideal values of $\pm 1$; (iii) The positions of
the extremized CDs ($\mathbf{CD}^{\rm{min}}$ and $\mathbf{CD}^{\rm{max}}$) do
not overlap exactly at those of the polarization singularities. Despite those
discrepancies, it is clear from Figs. 2(d)-(f) that the positions of the QNM
singularities still provide an excellent guide for identifying the directions
along which the extremized values of all CDs can be obtained.

Figure 3: (a) A gold SRR-pair with a twisting angle $30^{\circ}$ [each SRR is
idential to that shown in Fig. 2(a)] and (b) its scattering spectra where two
QNMs are pinpointed. (c) $\mathbf{S}_{3}$ distributions for the QNM-D. (d)-(f)
The angular dependence of three $\mathbf{CD}$s. In (c)-(f) the positions of
polarization singularities are also marked (stars: RCP; crosses: LCP).

We now proceed to the more sophisticated configuration of a coupled SRR pair (
twisted by $30^{\circ}$ with respect to each other) shown in Fig. 3(a), where
each SRR is identical to the one shown in Fig. 3(a). Its scattering cross
section spectra (RCP wave incident along +z axis) are shown in Fig. 3(b),
where two spectrally close QNMs are marked. Their corresponding complex
eigenfrequencies are: $\tilde{\omega}_{D}=(1.357\times 10^{15}-2\times
10^{14}\rm{i})$ rad/s and $\tilde{\omega}_{E}=(1.0279\times 10^{15}-3.23\times
10^{14}\rm{i})$ rad/s. The $\mathbf{S}_{3}$ distributions for the QNM-D are
shown in Fig. 3(c). Figures 3(d)-(f) demonstrate the angular dependence of CDs
with $\omega=\mathbf{Re}(\tilde{\omega}_{D})$ ($\lambda_{D}=1.39~{}\mu$m),
where the positions of the polarization singularities for QNM-D are also
marked. Similar to the single SRR scenario, though the positions of the
polarization singularities do not overlap exactly with those of extremized
CDs, they still served as an efficient guide to identify the optimal incident
directions for chirality extremizations.

Figure 4: (a) A gold structure exhibiting four-fold rotation symmetry, with
all geometric parameters specified. (b)-(c) $\mathbf{S}_{3}$ distributions for
the two degenerate QNMs. (d)-(f) The angular dependence of three
$\mathbf{CD}$s. In (b)-(f) the positions of LCP singularities are also marked
by crosses. In (d)-(f) the actual positions of the minimum values of
$\mathbf{CD}$ ($\mathbf{CD}^{\rm{min}}$) are also marked by circles.

At the end, we turn to the gold structure [shown in Fig. 4(a)] that exhibits
four-fold rotation symmetry. Such a symmetry secures a pair of degenerate
modes [11]. The $\mathbf{S}_{3}$ distributions of both are shown in Figs. 4(b)
and (c) and their degenerate complex eigenfrequency is
$\tilde{\omega}_{F}=(7.9645\times 10^{14}-4.17\times 10^{13}\rm{i})$ rad/s.
With the angular frequency of the incident wave fixed as
$\omega=\mathbf{Re}(\tilde{\omega}_{F})$ ($\lambda_{F}=2.367~{}\mu$m), the
angular dependence of CDs is summarized in Figs. 4(d)-(f), with the LCP
singularities of both QNMs marked. Since all CDs are negative, half of the
polarization singularities (RCP; $\mathbf{S_{3}}=1$) have totally lost their
connections with $\mathbf{CD}^{\rm{max}}$. This is probably due to the co-
excitation of both QNMs and that none of them plays a overwhelming role, while
a precise clarification for it, from the perspectives of QNMs with explicit
formalism, is yet to be explored. Despite this, the positions of LCP
singularities still provide reasonable guide for identifying the optimal
directions to obtain $\mathbf{CD}^{\rm{min}}$ [positions marked by circles in
Figs. 4(d)-(f)], especially efficient for $\mathbf{CD}_{\rm{sca}}$ shown in
Fig. 4(f).

##  4 Conclusions

In this work, we extend our pervious investigations of single-mode chirality
extremizations to the multi-mode regime, studying scattering structures when
more than one QNMs are simultaneously excited. The optical chiralities are
characterized by CDs defined in terms of extinction, scattering and
absorption, which are not identical anymore beyond the single-mode regime. We
have further revealed that though the exact correspondence between CDs and
$\mathbf{S}_{3}$ distributions of the QNM radiations is broken, the positions
of the polarization singularities of the QNMs excited still provide an
efficient guide to identify those directions incident along which the CDs can
be extremized. Our numerical studies presented in this work can stimulate
further in-depth explorations for a theoretical model that can systematically
deal with the problem of momentum-space optical chirality extremizations, not
only beyond the single-QNM regime, but also covering spatially and spectrally
structured incident waves (e.g. those carrying both spin and angular momentum
[14]), for scattering photonic structures that are not necessarily reciprocal.

## Funding

National Natural Science Foundation of China (Grant No. 11404403 and
61405067); National Key Research and Development Program of China (Grant No.
2021YFB2800303);Outstanding Young Researcher Scheme of National University of
Defense Technology of China; Innovation Project of Optics Valley Laboratory.

## Acknowledgments

We thank Weijin Chen and Qingdong Yang for their help with numerical
simulations.

## References

  * [1] L. D. Barron, _Molecular Light Scattering and Optical Activity_ (Cambridge University Press, 2009).
  * [2] S. Boriskina and N. I. Zheludev, eds., _Singular and Chiral Nanoplasmonics_ (Jenny Stanford Publishing, Singapore, 2014), 1st ed.
  * [3] T.-D. Lee and C.-N. Yang, “Question of parity conservation in weak interactions,” Phys. Rev. 104, 254 (1956).
  * [4] A. Papakostas, A. Potts, D. M. Bagnall, S. L. Prosvirnin, H. J. Coles, and N. I. Zheludev, “Optical manifestations of planar chirality,” Phys. Rev. Lett. 90, 107404 (2003).
  * [5] E. Plum, X.-X. Liu, V. A. Fedotov, Y. Chen, D. P. Tsai, and N. I. Zheludev, “Metamaterials: Optical activity without chirality,” Phys. Rev. Lett. 102, 113902 (2009).
  * [6] W. Chen, Q. Yang, Y. Chen, and W. Liu, “Extremize Optical Chiralities through Polarization Singularities,” Phys. Rev. Lett. 126, 253901 (2021).
  * [7] R. W. Rudin, _Principles of Mathematical Analysis_ (McGraw-Hill Publishing Company, Auckland, 1976), 3rd ed.
  * [8] C. F. Bohren and D. R. Huffman, _Absorption and Scattering of Light by Small Particles_ (Wiley, 1983).
  * [9] W. Chen, Q. Yang, Y. Chen, and W. Liu, “Scattering activities bounded by reciprocity and parity conservation,” Phys. Rev. Research 2, 013277 (2020).
  * [10] P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370 (1972).
  * [11] B. Hopkins, A. N. Poddubny, A. E. Miroshnichenko, and Y. S. Kivshar, “Circular dichroism induced by Fano resonances in planar chiral oligomers,” Laser Photon. Rev. 10, 137–146 (2016).
  * [12] L. D. Landau, L. P. Pitaevskii, and E. M. Lifshitz, _Electrodynamics of Continuous Media: Volume 8_ (Butterworth-Heinemann, Amsterdam u.a, 1984), 2nd ed.
  * [13] R. J. Potton, “Reciprocity in optics,” Rep. Prog. Phys. 67, 717 (2004).
  * [14] Y. Chen, W. Du, Q. Zhang, O. Ávalos-Ovando, J. Wu, Q.-H. Xu, N. Liu, H. Okamoto, A. O. Govorov, Q. Xiong, and C.-W. Qiu, “Multidimensional nanoscopic chiroptics,” Nat. Rev. Phys. 4, 113–124 (2022).