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A general model for designing the chirality of exciton-polaritons

Nanophotonics

February 2025
14(3):407-416

DOI:10.1515/nanoph-2024-0662

License
CC BY 4.0

Authors:

Siying Peng

Westlake University

Chirality of exciton-polaritons can be tuned by the chirality of photons, excitons, and their coupling strength. In this work, we propose a general analytical model based on coupled harmonic oscillators to describe the chirality of exciton-polaritons. Our model predicts the degree of circular polarization (DCP) of exciton-polaritons, which is determined by the DCPs and weight fractions of the constituent excitons and photons. At the anticrossing point, the DCP of exciton-polaritons is equally contributed from both constituents. Away from the anticrossing point, the DCP of exciton-polaritons relaxes toward the DCP of the dominant constituent, with the relaxation rate decreasing as the coupling strength increases. We validate our model through simulations of strongly coupled topological edge states and excitons, showing good agreement with model predictions. Our model provides a valuable tool for designing the chirality of strong coupling systems and offers a framework for the inverse design of exciton-polaritons with tailored chirality.

(color online). Analytical model predictions of the energies of the lower and upper polariton bands in dispersion, with their DCP values ρ c color coded into the bands. The following assumptions are made: (a) ρ c e = 0 , ρ c p = 0 ${\rho }_{c}^{e}=0,\enspace {\rho }_{c}^{p}=0$ , (b) ρ c e = 1 , ρ c p = 1 ${\rho }_{c}^{e}=1,\enspace {\rho }_{c}^{p}=1$ , (c) ρ c e = 0 , ρ c p = 1 ${\rho }_{c}^{e}=0,\enspace {\rho }_{c}^{p}=1$ , (d) ρ c e = 1 , ρ c p = 0 ${\rho }_{c}^{e}=1,\enspace {\rho }_{c}^{p}=0$ , (e) ρ c e = − 1 , ρ c p = 1 ${\rho }_{c}^{e}=-1,\enspace {\rho }_{c}^{p}=1$ , (f) ρ c e = − 1 , ρ c p = 0.5 ${\rho }_{c}^{e}=-1,\enspace {\rho }_{c}^{p}=0.5$ . ρ c e ${\rho }_{c}^{e}$ and ρ c p ${\rho }_{c}^{p}$ are color coded into the exciton energy (E e = 2.4 eV) lines and the parabolic photon curves. The coupling strength is set g = 0.2 eV.

…

(color online). New criterion of chiral strong light–matter interaction. (a) The DCP values of LP ρ c LP ${\rho }_{c}^{\text{LP}}$ and UP ρ c UP ${\rho }_{c}^{\text{UP}}$ as a function of coupling strength g at the anticrossing point for an exciton with ρ c e = − 1 ${\rho }_{c}^{e}=-1$ coupled with a photon with ρ c p = 1 ${\rho }_{c}^{p}=1$ . (b)–(e) Predicted energies of the LP and UP bands in dispersion, with their DCP ρ c color coded for g indicated in (a) as black, orange, green, and purple dots, respectively. Exciton linewidth γ e = 0.1 eV and photon linewidth γ p = 0.1 eV.

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(color online). Contour plots of the LP DCP ρ c L P $\left({\rho }_{c}^{LP}\right)$ with the decay rate difference between the exciton and photon (|γ e − γ p | (eV)). The plots represent the following cases: (a) ρ c e = 0 , ρ c p = 1 ${\rho }_{c}^{e}=0,\enspace {\rho }_{c}^{p}=1$ and g = 0.2 eV; (b) ρ c e = 1 , ρ c p = 0 ${\rho }_{c}^{e}=1,\enspace {\rho }_{c}^{p}=0$ and g = 0.2 eV; (c) ρ c e = 1 , ρ c p = − 1 ${\rho }_{c}^{e}=1,\enspace {\rho }_{c}^{p}=-1$ and g = 0.2 eV; (d) ρ c e = 1 , ρ c p = − 1 ${\rho }_{c}^{e}=1,\enspace {\rho }_{c}^{p}=-1$ and g = 0.3 eV. (e) The LP DCP ρ c LP $\left({\rho }_{c}^{\text{LP}}\right)$ as a function of coupling strength g between a fully RCP exciton ρ c e = − 1 ${\rho }_{c}^{e}=-1$ and a fully LCP photon ρ c p = 1 ${\rho }_{c}^{p}=1$ . The LP DCP ρ c LP $\left({\rho }_{c}^{\text{LP}}\right)$ as a function of g between a fully LCP photon ρ c p = 1 $\left({\rho }_{c}^{p}=1\right)$ and an exciton with ρ c e ${\rho }_{c}^{e}$ in color coded for (f) k ‖/k 0 = 0.25, (g) anticrossing point k ‖/k 0 = 0.2828, and (h) k ‖/k 0 = 0.32. (i) The LP DCP ρ c LP $\left({\rho }_{c}^{\text{LP}}\right)$ dispersion as a function of the uncoupled exciton DCP ρ c e ${\rho }_{c}^{e}$ under strong coupling with a fully LCP photon at a coupling strength of 0.2 eV. The DCP inheritance of the LP mode from the photon DCP is illustrated for (j) k ‖/k 0 = 0.25, (k) anticrossing point k ‖/k 0 = 0.2828, and (l) k ‖/k 0 = 0.32. (m) The LP DCP dispersion ρ c LP $\left({\rho }_{c}^{\text{LP}}\right)$ as a function of the uncoupled photon DCP ρ c p ${\rho }_{c}^{p}$ under strong coupling with a fully LCP exciton at a coupling strength of 0.2 eV. The DCP inheritance of the LP mode from the exciton DCP is illustrated for (n) k ‖/k 0 = 0.25, (o) anticrossing point k ‖/k 0 = 0.2828, and (p) k ‖/k 0 = 0.32. The dashed lines in (e), (i), and (m) indicate the anticrossing point.

…

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Simulation of DCP of exciton-polaritons in topological photonic crystals (color online). (a) Schematic of the strong coupling system: a 2D halide perovskite exciton strongly coupled to topological edge states formed at the boundary between shrunk (R = a/3.5) and expanded (R = a/2.8) hexagonal Si3N4 photonic lattices. (b) Simulated energy band structure of the topological edge states in photonic crystals, with a lattice constant a = 0.4 µm, Si3N4 pillar height h = 1 µm, and diameter d = 0.08 µm. The white dashed line marks the exciton energy of 2D halide perovskite, and black dashed curves represent edge state energies. (c) Simulated DCP values for the topological edge states. (d) Calculated exciton-polaritons energies and their DCP values by using a three-coupled harmonic oscillator model with coupling strength g = 24 meV and ρ c e = 0 ${\rho }_{c}^{e}=0$ . (e) Calculated DCP inheritance rate from topological edge states to exciton-polariton bands. (f) Energy band structure of the strong coupling system. The fits to the LP and UP bands are obtained with a three-coupled harmonic oscillator model that describes the interaction between the topological edge states (black dashed curves) and the exciton of 2D halide perovskite (white dashed line), are given as white-solid curves. (g) Simulated DCP ρ c for the strong coupling system. (h) Energies of the exciton-polaritons extracted from (g), with the DCP values color coded into the bands. (i) Simulated DCP inheritance ratio from topological edge states to exciton-polariton bands.

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Nanophotonics 2025; 14(3): 407– 416

Research Article

Ping Bai and Siying Peng*

A general model for designing the chirality

of exciton-polaritons

https://doi.org/10.1515/nanoph-2024-0662

Received November 21, 2024; accepted January 14, 2025;

published online February 3, 2025

Abstract:Chirality of exciton-polaritons can be tuned by the

chirality of photons, excitons, and their coupling strength.

In this work, we propose a general analytical model based

on coupled harmonic oscillators to describe the chirality

of exciton-polaritons. Our model predicts the degree of

circular polarization (DCP) of exciton-polaritons, which is

determined by the DCPs and weight fractions of the con-

stituent excitons and photons. At the anticrossing point, the

DCP of exciton-polaritons is equally contributed from both

constituents. Away from the anticrossing point, the DCP of

exciton-polaritons relaxes toward the DCP of the dominant

constituent, with the relaxation rate decreasing as the cou-

pling strength increases. We validate our model through

simulations of strongly coupled topological edge states and

excitons, showing good agreement with model predictions.

Our model provides a valuable tool for designing the chi-

rality of strong coupling systems and oers a framework

for the inverse design of exciton-polaritons with tailored

chirality.

Keywords: chirality; exciton-polaritons; degree of circular

polarization

1 Introduction

Exciton-polaritons are quasi-particles that are half-light

half-matter, arising from the strong coupling of photons in

optical microcavities and excitons in semiconductor mate-

rials. Due to their photonic component, exciton-polaritons

exhibit propagation characteristics that can be described

*Corresponding author: Siying Peng, Research Center for Indus-

tries of the Future and School of Engineering, Westlake University,

Hangzhou, Zhejiang 310030, China, E-mail: pengsiying@westlake.edu.cn.

https://orcid.org/0000-0002-1541-0278

Ping Bai, School of Engineering, Westlake University, Hangzhou, Zhejiang

310030, China, E-mail: baiping@westlake.edu.cn.

https://orcid.org/0000-0002- 7004-6501

by a well-deﬁned dispersion relation [1],[2]; their excitonic

component, on the other hand, imparts stronger nonlinear

behavior compared to pure photons [3]–[5].Thegenera-

tion of exciton-polaritons facilitates the modulation of the

spontaneous emission rate [6],[7], spectral linewidth [8],

and polarization characteristics of emitting materials [9],

[10]. Investigating the chiral emission of exciton-polaritons

is essential for gaining deeper insights into quantum optical

phenomena in strong light–matter coupling, serving as a

foundation for the development of spin-based active optical

devices and tunable optoelectronic systems. This research

uncovers new physical phenomena, such as quantum entan-

glement and chiral quantum light ﬁelds [11], while also oer-

ing practical implications for the advancement of cutting-

edge optoelectronic devices and quantum information tech-

nologies. Chiral exciton-polaritons, for instance, hold poten-

tial for realizing all-optical switches [12]–[14], optical logic

gates [15], novel quantum light sources [16],[17], and optical

isolators [18], furthering advancements in quantum com-

puting and communication.

The chirality of exciton-polaritons can originate from

asymmetric chiral semiconductor materials [19]–[21] or

from chiral optical modes in the microcavities [22]–[28].The

chirality of semiconductor materials refers to their struc-

tural property of being dierent from their mirror image

[29]–[31]. This asymmetry results in dierent responses

to circularly polarized light, such as left- or right-handed

polarization. Chiral materials are important in many ﬁelds,

including optics, pharmaceuticals, biomolecules, and nan-

otechnology, as their physical, chemical, and biological

properties depend on their chiral form. Chiral microcavities

are specially designed optical cavities that can dierentiate

and enhance the circular polarization states of light (left-

or right-handed). These cavities control and manipulate

the chirality of light through their chiral structural prop-

erties [24],[32]–[35], resulting in optical modes with spe-

ciﬁc handedness. Chiral microcavities have applications in

chiral molecule detection [36]–[38], quantum information

processing [39], and optical signal isolation [40],[41].

In this work, we propose a general analytical model

to investigate the mechanisms underlying chirality of

exciton-polaritons in strong light–matter coupling systems,

408 —P. Bai and S. Peng: The chirality of exciton-polaritons

initiated from the coupled harmonic oscillator model. The

proposed model determines the DCP of exciton-polaritons

and introduces a new criterion for chiral strong coupling

upon their formation. Speciﬁcally, the model addresses

three key behaviors: (a) At the anticrossing point, the DCP

of the exciton-polaritons is equally derived from the DCPs

of both exciton and photon. (b) Away from the anticross-

ing point, the DCP of exciton-polaritons gradually shifts

toward the dominant constituent (either exciton or photon),

with the rate of change decreasing as the coupling strength

increases. (c) The same exciton-polariton band is expected

to detect a full range of DCP values from left-handed to

right-handed luminescence simultaneously.

To validate the model, we simulate a photonic crystal

superlattice with topological edge states, which are strongly

coupled with excitons in 2D halide perovskites. Our results

show that the chiral response of the topological edge states

is transferred to the exciton-polariton bands. The simulated

DCP for exciton-polaritons agrees closely with predictions

from our analytical model. Our study provides the under-

standing of chiral inheritance in exciton-polaritons, and

our analytical model enables the inverse design of exciton-

polaritons with speciﬁc chirality.

2 Theoretical model for the DCP

of exciton-polaritons

We commence our theoretical analysis by investigating the

strong light–matter interaction between a single exciton in

the material and a photon state conﬁned within a micro-

cavity. This interaction can be modeled using a two coupled

harmonic oscillator Hamiltonian, expressed as:

H=

Ee−i𝛾e

p−i𝛾p

2

,(1)

where Eeand Epcorrespond to the energies of the exci-

ton and cavity photon, respectively, and 𝛾eand 𝛾pdenote

their decay rates, given by the full-width at half-maximum

(FWHM). The coupling strength g, which represents the

coherent interaction between the excitons and photons, is

determined by the exciton’s dipole moment 𝜇, the electric

ﬁeld direction of the cavity mode ue, the number of excitons

N, and the cavity mode volume V. It is expressed as: g=

𝝁⋅ueNEp

𝜖0V. The energies of the lower and upper polariton

bands for zero detuning are given by:

EUP/LP =E0−i(𝛾e+𝛾p)

4±1

2(2g)2−(𝛾e−𝛾p)2

4.(2)

By tuning g, we can distinguish the energies and DCP of

chiral exciton-polaritons in strong coupling chiral cavities

[42],[43].

The eigenvectors and eigenvalues of the system can be

determined by diagonalizing the Hamiltonian matrix. This

process is represented as:

H=MEM−1,(3)

where Eis a diagonal matrix containing the eigenvalues,

which correspond to the exciton-polariton energies,

E=ELP 0

0EUP.(4)

Here, ELP and EUP represent the energies of the lower

and upper polaritions, respectively, arising from the split-

ting at zero detuning (Ee=Ep), with the energy dierence

given by the Rabi splitting (ℏΩ). The matrix Mconsists of

the eigenvectors of H, which are the Hopﬁeld coecients,

i.e.,

M=𝛼1𝛼2

𝛽1𝛽2.(5)

These coecients describe the weight fractions of exci-

ton and photon in the hybrid exciton-polaritons. In the

lower polariton (LP) branch, the exciton and photon contri-

butions are 𝛼12and 𝛽12, respectively, with the normal-

ization condition 𝛼12+𝛽12=1. Similarly, in the upper

polariton (UP) branch, the exciton and photon contributions

are 𝛼22and 𝛽22, satisfying 𝛼22+𝛽22=1.

For both chiral excitons and chiral photons, the dier-

ent response to LCP and RCP light can be quantiﬁed by the

DCP, which is deﬁned as:

𝜌c=(ILCP −IRCP )∕(ILCP +IRCP ),(6)

where ILCP and IRCP represent the intensities of LCP and RCP

light, respectively. For excitons, these intensities include the

absorption or emission of circularly polarized light [44],

[45], as well as the transmission or reﬂection of LCP and RCP

waves in a chiral medium with a polarization-dependent

refractive index [27],[46]. For cavity photons, the intensities

relate to the propagation of LCP and RCP light. The DCP is

deﬁned to distinguish between LCP and RCP components.

In a strong coupling system with chiral exciton-

polaritons, both chiral exciton and chiral photon are

involved. We deﬁne the DCP of the uncoupled exciton and

photon as 𝜌e

cand 𝜌p

c, respectively, which will be inherited

into the exciton-polaritons with their respective weight frac-

tions. Therefore, the DCPs of the LP and UP modes can be

expressed as:

𝜌LP

c=𝜌e

c×𝛼12+𝜌p

c×𝛽12,(7)

P. Bai and S. Peng: The chirality of exciton-polaritons —409

𝜌UP

c=𝜌e

c×𝛼22+𝜌p

c×𝛽22,(8)

where 𝛼12and 𝛽12represent the weight fractions of the

exciton and photon in the LP mode, while 𝛼22and 𝛽22

represent the respective weight fractions in the UP mode.

These values described in matrix Min Eq. (5) are calculated

by diagonalizing the Hamiltonian.

3 Analytical model predictions

of DCP in exciton-polaritons

To demonstrate the DCP of exciton-polaritons in a strong

coupling system, we assume a parabolic dispersion for the

chiral photon mode in a cavity, with photon energy deﬁned

as Ep=(k∕k0)2∕0.2+2, where k∕k0ranges from 0 to 0.5.

The exciton energy is Ee=2.4 eV. For 𝛾e=0, 𝛾p=0, and

g=0.2 eV, we show the polariton energies and their DCP

in Figure 1. When both the exciton and photon are achiral

(𝜌e

c=0and𝜌p

c=0), the hybrid LP and UP modes are also

achiral, as shown in Figure 1(a). Similarly, when both are

fully LCP (𝜌e

c=1and𝜌p

c=1), the LP and UP modes are

fully LCP, as shown in Figure 1(b). These cases conﬁrm the

(a) (b)

(e) (f)

Figure 1: (color online). Analytical model predictions of the energies of

the lower and upper polariton bands in dispersion, with their DCP values

𝜌ccolor coded into the bands. The following assumptions are made:

(a) 𝜌e

c=0,𝜌

c=0, (b) 𝜌e

c=1,𝜌

c=1, (c) 𝜌e

c=0,𝜌

c=1,

(d) 𝜌e

c=1,𝜌

c=0, (e) 𝜌e

c=−1,𝜌

c=1, (f) 𝜌e

c=−1,𝜌

c=0.5.

𝜌e

cand 𝜌p

care color coded into the exciton energy (Ee=2.4 eV) lines and

the parabolic photon curves. The coupling strength is set g=0.2 eV.

validity of the model.

In Figure 1(c) and (d), we show the DCPs of exciton-

polaritons for strong coupling between an achiral exciton

and a fully LCP photon, and between a fully LCP exciton

and an achiral photon. The DCP of the polaritons varies

from 0 to 1, reaching 0.5 at zero detuning, indicating an LCP

and RCP intensity ratio of 3:1 for both LP and UP modes. As

the system deviates from the anticrossing point, the DCP of

the polaritons gradually returns to the DCP of the exciton

or photon, and their energies move closer to those of the

exciton or photon.

In Figure 1(e) and (f),weconsidercaseswherethechi-

rality of the exciton and photon is opposite. For example, in

Figure 1(e), a fully LCP exciton couples with a fully RCP pho-

ton, resulting in a full range of DCP values, i.e., 𝜌cfrom −1to

1, at the same exciton-polariton band and achiral polaritons

at the anticrossing point. The LCP and RCP intensity ratio

of polaritons is the sum of that for both the exciton and

the photon. In this case, the ratio is 1:0 for the exciton and

0:1 for the photon, leading to an equal intensity of LCP and

RCP, thus an achiral response. In Figure 1(f ), the photon has

aDCPof𝜌p

c=0.5, with a 3:1 intensity ratio of LCP to RCP.

When this photon strongly couples with a fully RCP exciton,

the DCP of the LP and UP modes at the anticrossing point

is the average of the DCPs of the uncoupled exciton and

photon, i.e., 𝜌e

c+𝜌p

c∕2.

Strong coupling and the exciton-polariton formation

occur when 2g>𝛾e−𝛾p

2. Our theoretical model shows the

DCP of LP and UP with visible changes in the absence of

(a)

(b)

Figure 2: (color online). New criterion of chiral strong light– matter

interaction. (a) The DCP values of LP 𝜌LP

cand UP 𝜌UP

cas a function of

coupling strength gat the anticrossing point for an exciton with 𝜌e

c=−1

coupled with a photon with 𝜌p

c=1. (b)– (e) Predicted energies of the LP

and UP bands in dispersion, with their DCP 𝜌ccolor coded for gindicated

in (a) as black, orange, green, and purple dots, respectively. Exciton

linewidth 𝛾e=0.1 eV and photon linewidth 𝛾p=0.1 eV.

410 —P. Bai and S. Peng: The chirality of exciton-polaritons

visible energy splitting. For instance, Figure 2(a) demon-

strates an exciton with 𝜌e

c=−1coupledwithaphotonwith

𝜌p

c=1 results in zero DCP at the anticrossing point when

2g>𝛾e−𝛾p

2.InFigure 2(b), the predicted energies show

small splitting; however, notable DCP changes are observed.

Energy splitting is only visible with larger g(orange, green,

and purple dots in Figure 2(a)), as shown in Figure 2(c)–(e).

Therefore, our model has introduced a new indicator for

chiral strong coupling.

The criterion for strong coupling is 2g>𝛾e−𝛾p∕2,

indicating that the dierence in decay rates between the

exciton and the photon plays a critical role in their inter-

action. In Figure 3, we show how the decay rate dierence

aects the DCP of the LP mode across the in-plane wave

(a) (b) (c) (d)

(e)

(i) (j) (k) (l)

(m) (n) (o) (p)

(f) (g) (h)

Figure 3: (color online). Contour plots of the LP DCP 𝜌LP

cwith the decay rate difference between the exciton and photon (𝛾e−𝛾p(eV)).

The plots represent the following cases: (a) 𝜌e

c=0,𝜌

c=1andg=0.2 eV; (b) 𝜌e

c=1,𝜌

c=0andg=0.2 eV; (c) 𝜌e

c=1,𝜌

c=−1and g=0.2 eV;

(d) 𝜌e

c=1,𝜌

c=−1and g=0.3 eV. (e) The LP DCP 𝜌LP

cas a function of coupling strength gbetween a fully RCP exciton 𝜌e

c=−1 and a fully LCP

photon 𝜌p

c=1. The LP DCP 𝜌LP

cas a function of gbetween a fully LCP photon 𝜌p

c=1and an exciton with 𝜌e

cin color coded for (f) k∕k0=0.25,

(g) anticrossing point k∕k0=0.2828, and (h) k∕k0=0.32.(i)TheLPDCP𝜌LP

cdispersion as a function of the uncoupled exciton DCP 𝜌e

cunder

strong coupling with a fully LCP photon at a coupling strength of 0.2 eV. The DCP inheritance of the LP mode from the photon DCP is illustrated for

(j) k∕k0=0.25, (k) anticrossing point k∕k0=0.2828, and (l) k∕k0=0.32. (m) The LP DCP dispersion 𝜌LP

cas a function of the uncoupled photon

DCP 𝜌p

cunder strong coupling with a fully LCP exciton at a coupling strength of 0.2 eV. The DCP inheritance of the LP mode from the exciton DCP is

illustrated for (n) k∕k0=0.25, (o) anticrossing point k∕k0=0.2828, and (p) k∕k0=0.32. The dashed lines in (e), (i), and (m) indicate

the anticrossing point.

P. Bai and S. Peng: The chirality of exciton-polaritons —411

vector dispersion. Since the UP mode behaves similarly, we

refer to Figure S1(a)–(d) in Supplementary Material S1 for

those results. Figure 3(a) shows the LP DCP inherited from

a fully LCP photon. The DCP remains largely unaected by

the decay rate dierence until the photon couples with the

exciton at k∕k0=0.2828. Beyond the anticrossing point,

the LP DCP rapidly decreases as 𝛾e−𝛾pincreases due to

the achiral exciton fraction in the LP mode. In Figure 3(b),

when a fully LCP exciton couples with an achiral photon, the

LP DCP inherited from the exciton stays largely unchanged

for larger wave vectors, even with signiﬁcant decay rate

dierences. For smaller wave vectors, the DCP decreases

as the decay rate dierence increases. Figure 3(c) shows

thecasewheretheLPmodeinheritsDCPfromboththe

exciton and photon. In this case, the DCP returns to the

values of the photon or the exciton at wave vectors smaller

or larger than the anticrossing point, with the transition

accelerating as 𝛾e−𝛾pincreases. However, as shown by

comparing Figure 3(c) and (d), this recovery is slowed down

by an increase in the coupling strength g.

Next, we examine the eects of coupling strength g

on the LP DCP formed by coupling a fully LCP photon

with a fully RCP exciton, as shown in Figure 3(e).Whenno

strong coupling occurs (g=0), the DCP follows the photon’s

DCP for wave vectors smaller than the anticrossing point

(dashed line) and the exciton’s DCP for larger wave vec-

tors. As gincreases, the photon and exciton interact more

strongly, forming a hybrid LCP and RCP LP mode. At the anti-

crossing point (k∕k0=0.2828), the LP mode inherits equal

chirality from the LCP photon and the RCP exciton, resulting

in an achiral LP mode with equal LCP and RCP intensities.

Figure 3(g) shows the LP DCP at the anticrossing point as a

function of gfor a fully LCP photon and an exciton with 𝜌e

varying from −1 to 1. The LP DCP is independent of gand

equals 1+𝜌e

c∕2. In Figure 3(f),atk∕k0=0.25, the LP DCP

starts with the photon’s DCP 𝜌p

c=1and decreases as the

exciton becomes more RCP and coupling strength increases.

In Figure 3(h),atk∕k0=0.32, the LP DCP starts with the

exciton’s DCP 𝜌e

cand increases with stronger coupling to

the LCP photon (similar results for the UP DCP results are

shown in Figure S1(e)–(h) in Supplementary Material S1).

In Figure 3(i), we show the LP DCP formed by coupling a

fully LCP photon with an exciton 𝜌e

c. The LP DCP depends

more on 𝜌p

cfor k∕k0<0.2828,wheretheLPmodeiscloser

to the photon, and more on 𝜌e

cfor k∕k0>0.2828, where

the LP mode is closer to the exciton. At the anticrossing

point, the contributions from both the photon and exciton

are equal. This is also shown in Figure 3(m),whichdisplays

the LP DCP formed by coupling a fully LCP exciton with a

photon 𝜌p

c.

Figure 3(j)–(l) show the LP DCP normalized to the

uncoupled photon DCP 𝜌LP

c∕𝜌p

cat k∕k0=0.25,0.2828

and 0.32, respectively. At k∕k0=0.25, 𝜌LP

c∕𝜌p

cis more com-

pact, indicating a greater dependence on the photon DCP. In

the inset of Figure 3(j),theratio𝜌LP

c∕𝜌p

cis approximately 0.6

for the coupling systems, where 𝜌p

cis represented by color

coding, and the exciton has 𝜌e

c=0. This indicates that the

photon fraction in the LP mode exceeds that of the exciton.

In the inset of Figure 3(l), the ratio is about 0.36 for 𝜌e

c=0,

indicating that the photon fraction is less than the exciton’s.

At the anticrossing point, shown in the inset of Figure 3(k),

the ratio is 0.5.

Similarly, Figure 3(n)–(p) show the LP DCP normal-

ized to the uncoupled exciton DCP 𝜌LP

c∕𝜌e

cat k∕k0=

0.25,0.2828 and 0.32, respectively. 𝜌LP

c∕𝜌e

cis more compact

at k∕k0=0.32, indicating a greater dependence on the

exciton DCP. 𝜌LP

c∕𝜌e

cdiverges for 𝜌e

c=0. This ratio is 0.5 at

the anticrossing point for all 𝜌e

cand 𝜌p

c=0, as shown in the

inset of Figure 3(o).𝜌LP

c∕𝜌e

cshows a smaller value of 0.4 at

k∕k0=0.25, and a larger value of 0.63 at k∕k0=0.32 for

all 𝜌e

cand 𝜌p

c=0, as shown in the insets of Figure 3(n) and

(p), respectively. Similar results for the UP DCP are shown in

Figure S1(i)–(p) in Supplementary Material).

The proposed general analytical model provides

insights into the mechanisms driving chirality in

exciton-polaritons within strongly coupled systems.

Importantly, this model enables the inverse engineering

of exciton-polariton bands with tailored chirality in their

dispersion for coupled harmonic oscillator models. This

indicates that the DCP values of the exciton-polariton states

in the in-plane wave vector dispersion can be precisely

tuned by speciﬁc DCP values of the uncoupled exciton and

photon, as well as their coupling strength. As shown in

Figure 4(a), in a two-coupled harmonic oscillator model,

the LP and UP DCP values vary systematically with the

DCPs of their uncoupled components at the anticrossing

point; for example, achieving an achiral polariton DCP

𝜌LP/UP

c=0 requires the exciton and photon DCPs to be

opposites 𝜌p

c=−𝜌e

c, as shown by the red line in Figure 4(b).

Conversely, a DCP 𝜌LP/UP

c=0.5 is achieved by coupling a

fully RCP exciton 𝜌e

c=1 with an achiral photon 𝜌p

c=0, as

illustrated by the black line in Figure 4(b).

Furthermore, our model is extendable to systems with

three or more coupled harmonic oscillator models, oering

ﬂexibility in designing exciton-polaritons with complex chi-

ral characteristics. Figure 4(c)–(e) illustrate the dependence

of LP, middle polariton (MP), and UP DCP values on the

uncoupled exciton and photons DCPs in a three-coupled

harmonic oscillator model with two photons and an exciton.

The Hamiltonian for this system is given by:

412 —P. Bai and S. Peng: The chirality of exciton-polaritons

H=

Ee−i𝛾e

2g1g2

g1Ep1−i𝛾p1

g20Ep2−i𝛾p2



. (9)

In this conﬁguration, exciton and photon energies

were set to Ee=Ep1=2.4 eV, Ep2=2.5 eV, respectively.

The linewidths are 𝛾e=0.12 eV, 𝛾p1=𝛾p2=0.16 eV and

thecouplingstrengthsareg1=g2=0.2 eV. The DCPs

of the exciton-polariton states vary in response to the

DCPs of exciton and photon, with the UP DCP showing

a stronger dependence on the DCP of the second pho-

ton due to its energy being farther from the anticrossing

point.

In Figure 4(f), we present the relationship between 𝜌e

and 𝜌p

cbased on data in Figure 4(c)–(e). This relationship

facilitates the design of LP, MP, and UP with DCP values

(a) (b)

(e) (f)

Figure 4: Inverse design the exciton-polaritons with precise chirality

(color online). (a) The LP and UP DCP for a two coupled harmonic

oscillator model at anticrossing point can be inverse designed by strong

couplingofanexcitonwithDCP𝜌e

cand a photon with DCP 𝜌p

c.

(b) The exciton DCP 𝜌e

cversus photon DCP 𝜌p

cfrom (a) used to design

exciton-polaritons with DCP values of −0.5 (blue), 0 (red), and 0.5 (black)

at the anticrossing point. For a three coupled harmonic oscillator model

with two photons and an exciton, the LP DCP 𝜌LP

c(c), middle-polariton

(MP) DCP 𝜌MP

c(d), and UP DCP 𝜌UP

c(e) at anticrossing point can be

inverse designed by strong coupling of an exciton with DCP 𝜌e

c, photon

1withDCP𝜌p1

cand photon 2 with DCP 𝜌p2

c. The coupling strength

for both coupled harmonic oscillator model is 0.2 eV. (f) 𝜌e

cversus 𝜌p2

from (c)– (e) used to design of LP (solid), MP (dashed), and UP (dotted)

with DCP of −0.5 (blue), 0 (red), and 0.5 (black) at the anticrossing point.

of −0.5 (blue lines), 0 (red lines), and 0.5 (black lines) at

the anticrossing point. For example, an achiral LP can be

obtained by strongly coupling photon 1 with 𝜌p1

c=−0.5,

photon 2 with 𝜌p2

c=0, and an exciton with 𝜌e

c=0.2899

(red line in Figure 4(f)). A UP with 𝜌UP

c=−0.5 cannot be

achieved by setting 𝜌p1

c=0 for photon 1 (no dotted blue line

in Figure 4(f)). Achiral UP with 𝜌UP

c=0 are achievable by

setting 𝜌p1

c=−0.5 for photon 1, 𝜌p2

cbetween 0.63 and 0.5 for

photon 2, and 𝜌e

cbetween −1 and 1 for the exciton (red line

in Figure 4(f)).

Our model can be extended to include dynamic eects

such as spin or polarization relaxation, resulting in time-

dependent variations in the exciton-polaritons’ DCP. By

introducing a time-dependent coupling strength g(t), we can

predict the evolution of DCP during spin relaxation, aligning

our theoretical predictions with experimental studies of

cavity polariton spin dynamics in quantum microcavities

[47],[48].

4 Simulation of DCP of

exciton-polaritons in topological

photonic crystals

To validate our analytical model, we performed 3D FDTD

simulations to design a superlattice of Si3N4dielectric

cylinders arranged in an expanded and shrunken honey-

comb pattern. Additional simulation details of the photonic

crystals and the strong coupling system are provided in

Supplementary Material S2 and S3. The simulated energy

band structure of the superlattice is shown in Figure 5(b),

which displays a pair of topological edge states. Placing

the 2D halide perovskite layer on the photonic crystals

(Figure 5(a)) induced strong coupling, resulting in two split

modes (Figure 5(f)): one at a lower energy and one at a

higher energy than the exciton (Ee=2.427 eV, marked by

the white-dashed line in Figure 5(b)). By ﬁtting the peak

energies of these bands using a three-coupled harmonic

oscillator model, we determined a strong coupling strength

of 24 meV.

To determine the DCP of the topological edge states

in the photonic crystals, we used the Lorentz reciprocity

principle [49], which allows us to calculate emitted light by

evaluating ﬁeld enhancements under circularly polarized

(CP) incident light. This approach lets us switch between the

source and detector of electromagnetic ﬁelds, enabling the

calculation of far-ﬁeld emitted power and polarization from

randomly placed dipoles by assessing ﬁeld enhancements at

their locations (see Supplementary Material S4).

P. Bai and S. Peng: The chirality of exciton-polaritons —413

(a)

(b) (c) (d)

(f) (g) (h)

(e)

(i)

Figure 5: Simulation of DCP of exciton-polaritons in topological photonic crystals (color online). (a) Schematic of the strong coupling system: a 2D

halide perovskite exciton strongly coupled to topological edge states formed at the boundary between shrunk (R=a∕3.5) and expanded (R=a∕2.8)

hexagonal Si3N4photonic lattices. (b) Simulated energy band structure of the topological edge states in photonic crystals, with a lattice constant

a=0.4 μm, Si3N4pillar height h=1μm, and diameter d=0.08 μm. The white dashed line marks the exciton energy of 2D halide perovskite, and black

dashed curves represent edge state energies. (c) Simulated DCP values for the topological edge states. (d) Calculated exciton-polaritons energies and

their DCP values by using a three-coupled harmonic oscillator model with coupling strength g=24 meV and 𝜌e

c=0. (e) Calculated DCP inheritance

rate from topological edge states to exciton-polariton bands. (f) Energy band structure of the strong coupling system. The fits to the LP and UP bands

are obtained with a three-coupled harmonic oscillator model that describes the interaction between the topological edge states (black dashed curves)

and the exciton of 2D halide perovskite (white dashed line), are given as white-solid curves. (g) Simulated DCP 𝜌cfor the strong coupling system.

(h) Energies of the exciton-polaritons extracted from (g), with the DCP values color coded into the bands. (i) Simulated DCP inheritance ratio from

topological edge states to exciton-polariton bands.

In the simulations, CP plane waves are directed onto the

photonic crystal from angles 𝜃=−30◦to 30◦and polariza-

tions 𝜙=−180◦to 180◦. The ﬁeld enhancement factor for

each angle and polarization is calculated by integrating the

total electric ﬁeld intensity within the 2D halide perovskite

region and normalizing it to the intensity in free space, i.e.,

I(𝜆)=

∭

VE(x,y,z,𝜆)2dxdydz

∭

VEref (x,y,z,𝜆)2dxdydz,(10)

where E(x,y,z,𝜆) is the electric ﬁeld with the photonic crys-

tals present, Eref (x,y,z,𝜆) is the ﬁeld without the photonic

crystals, and the integrals are evaluated over the unit cell

area V.

The simulated ﬁeld enhancements for the topological

edge states and the exciton-polaritons under LCP and RCP

light, i.e., ILCP and IRCP,areshowninFigure S5 in Supple-

mentary Material S4.Figure 5(c) shows the resulting DCP

𝜌cacross the energy spectra, with reciprocity-based calcu-

lations highlighting a chiral response from the topological

edge states, achieving a high DCP (𝜌c=∓0.847) at ky=

±0.1145k0.

To examine DCP transfer from topological edge pho-

tons to hybrid exciton-polaritons under strong coupling, we

used a three-coupled harmonic oscillator model to calcu-

late the exciton-polaritons energies and their DCP values as

shown in Figure 5(d).Figure 5(e) shows that the inheritance

𝜌EP

c∕𝜌p

cis about 0.5 for bands xand yat the strong anti-

crossing point (ky∕k0=±0.2114), indicating equal photon

contribution. For the highest energy band (z), the ratio

𝜌EP

c∕𝜌p

capproaches 1, indicating that this band is mostly

photon-based, as it lies farther from the exciton energy. To

compare the model-calculated DCP of exciton-polaritons, we

simulated the DCP of the strong coupling system, shown

in Figure 5(f), where chiral exciton-polaritons appear as

blue and red regions. By extracting the energies and DCP

peak values in Figure 5(g), we derived the energy bands

and DCPs of the exciton-polaritons in Figure 5(h).Figure 5(i)

shows the DCP inheritance for bands x,y,andz, consistent

overall with the theoretical model results in Figure 5(e).

414 —P. Bai and S. Peng: The chirality of exciton-polaritons

However, the highest energy band, inﬂuenced more by

the achiral exciton, inherits only about 0.7 of the photon’s

chirality. At the anticrossing point (ky∕k0=±0.2114), the

lower two energy bands equally inherit the DCP of topolog-

ical edge states, resulting in an inheritance 𝜌EP

c∕𝜌p

cof 0.4.

Additionally, our theoretical model accurately predicted the

transmission dissymmetry gtrans for lower and upper polari-

tons in chiral (R-MBA)PbI3and (R-MBA)2PbI4ﬁlms within

an FP cavity, closely matching the experimental data (see

Supplementary Material S5).

5 Conclusions

In conclusion, we have developed a generally applicable

analytical model that provides deep insight into the mech-

anisms responsible for chirality in exciton-polaritons. Built

upon the framework of coupled harmonic oscillators, this

model oers reliable predictions of the DCP of exciton-

polaritons and introduces a new criterion for chiral strong

coupling upon their formation, making it a valuable tool to

aid the simulation and experimental approaches in the ﬁeld.

The model captures three key behaviors that are critical

for understanding and designing chiral exciton-polaritons:

(a) at the anticrossing point, the DCP of exciton-polaritons

results from an equal contribution of the DCPs from both

the exciton and photon components. This ﬁnding highlights

the balanced interaction between these two constituents at

the strong anticrossing point. (b) As the system moves away

from the anticrossing point, the DCP of exciton-polaritons

transitions toward the dominant component, whether it be

the exciton or the photon. Notably, these transitions occur

more gradually as the coupling strength increases, pro-

viding insight into how varying coupling conditions inﬂu-

ence the chirality of the system. And (c) the model pre-

dicts that a single exciton-polariton band can exhibit a full

range of DCP values, from −1to+1, which enables the

design of exciton-polaritons with tailored chirality. More-

over, our model may be applied for designing the chirality

of complex systems, for instance, systems involving multi-

ple photons and excitons, exciton-phonon-polaritons, and

exciton-plasmon-polaritons.

To validate our model, we conducted simulations of

a photonic crystal superlattice featuring topological edge

states, which are strongly coupled to a 2D halide perovskite

exciton. Using the reciprocity principle, we observed a chi-

ral response in the topological edge state bands, which was

eectively transferred to the exciton-polariton bands. The

simulated DCP values demonstrated good agreement with

the model’s predictions by a three-coupled harmonic oscil-

lator model. Our study enhances the understanding of chiral

inheritance in exciton-polaritons and reveals how chirality

is transferred and maintained in these hybrid systems. Addi-

tionally, our analytical model oers a practical tool for the

inverse design of exciton-polaritons with precise chirality.

Using this model, researchers can predict and tailor the chi-

ral properties of exciton-polaritons for speciﬁc applications,

advancing the development of chiral optoelectronic devices.

Research funding: This work was supported by National

Natural Science Foundation of China (12204384), Research

Center for Industries of the Future, and Westlake Education

Foundation.

Author contributions: The manuscript was written with

the contributions of all authors. All authors have accepted

responsibility for the entire content of this manuscript and

approved its submission.

Conflict of interest: Authors state no conﬂict of interest.

Ethical approval: The conducted research is not related to

either human or animals use.

Data availability: The datasets generated and/or analyzed

during the current study are available from the correspond-

ing author upon reasonable request.

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