Planar chirality and optical spin-orbit coupling for chiral Fabry-Perot cavities

Jerome Gautier,1 Minghao Li,1 Thomas W. Ebbesen,1 and Cyriaque Genet1, ∗

1Universite de Strasbourg, CNRS, Institut de Science et d’Ingenierie Supramoleculaires, UMR 7006, F-67000 Strasbourg, France

We design, in a most simple way, Fabry-Perot cavities with longitudinal chiral modes by sand-wiching between two smooth metallic silver mirrors a layer of polystyrene made planar chiral bytorsional shear stress. We demonstrate that the helicity-preserving features of our cavities stemfrom a spin-orbit coupling mechanism seeded inside the cavities by the specific chiroptical featuresof planar chirality. Planar chirality gives rise to an extrinsic source of three-dimensional chiralityunder oblique illumination that endows the cavities with enantiomorphic signatures measured ex-perimentally and simulated with excellent agreement. The simplicity of our scheme is particularlypromising in the context of chiral cavity QED and polaritonic asymmetric chemistry.

The design of chiral cavities with modes preserving op-tical helicity has recently become a major goal in the fieldof light-matter interactions. Coupling matter to chiraloptical modes indeed enriches the field with symmetry-breaking effects that have a great potential, which is

FIG. 1. Panel (a) schematizes the spin-orbit coupling mech-anism at play through a planar chiral polymer system -hereconceptually represented by a 2D spiral. Planar (2D) chirality(in its most general form, that is without any rotational invari-ance –see [1]) is characterized by circular polarization conver-sions that depend on the direction of the probe beam trans-mitted through the spiral. Panel (b) illustrates the breakingof left- vs. right-handed polarization in a Fabry-Perot cavitycomposed of two usual metallic mirrors but enclosing a 2Dchiral medium. Panel (c) describes how a planar chiral sys-tem viewed under oblique illumination yields signatures of 3Dchirality (i.e. circular dichroism). Two opposite ±θ obliqueillumination angles are connected by a simple mirror symme-try in the (x, y) plane and this corresponds to the sequence oftransformations detailed as the succession of a flip (a C2 rota-tion along the y−axis) and of a mirror reflection with respectto the (x, z) plane. The result of this sequence is to show,as detailed in the main text, that the optical activity asso-ciated with this extrinsic 3D chirality induced on the planarchiral system at oblique illumination is reversed for oppositeincidence angles ±θ.

well recognized in the context of high-resolution chirop-tical sensing [2–4], polaritonic physics [5, 6], chiral quan-tum optics [7] and quantum materials [8]. In the spe-cific context of light-matter strong coupling studies, suchchiral cavities are expected to give rise to chiral polari-tonic states that open uncharted paths for driving new,asymmetric, chemical syntheses and material properties[9, 10].

Such cavities however are challenging to design becauseoptical helicity changes sign at each mirror reflection,so that helicity densities are eventually brought to zerothrough the multiple paths that determine the modalstructure of the cavity. Fundamentally, the difficultystems from the pseudoscalar nature of the optical helic-ity, itself rooted in the pseudovector nature of the opticalspin [11]. For this reason, a cavity cannot be chiral with-out spin-orbit coupling at play, explaining immediatelywhy filling a Fabry-Perot cavity with an optically activematerial cannot meet the challenge. At the cost of com-plexity therefore, various schemes have been proposed forrealizing helicity-preserving optical cavities, involving forinstance intracavity polarization optics [12, 13] or elabo-rate designs ranging from Bragg resonant twisted sculp-tured thin films [14] to optical metamaterial metasurfaces[15, 16] difficult to scale down to the visible range.

In this Letter, we take another route, simple yet gen-eral, by exploiting a deep connection between chiralityand optical spin-orbit interactions. We show indeed thatoptical spin orientations can be locked to intracavitypropagation directions when a seed of planar (2D) chiral-ity is present inside the cavity. This seed is given by in-serting between the two metallic mirrors of a Fabry-Perotcavity a layer of polystyrene made 2D chiral under tor-sional shear stress. By taking advantage of the extrinsicproperties associated with planar chirality under obliqueillumination, we demonstrate how the Fabry-Perot cav-ity can be endowed with an helicity-preserving modal re-sponse. This is a clear asset of our chiral cavities that,combined with the simplicity of our approach, immedi-ately makes our systems particularly relevant for appli-cations involving polaritons built on chiral light-matterstates hybridized throughout the cavity mode volume.

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The dispersive chiroptical features of our cavities areanalyzed in the framework of the Jones-Mueller formal-ism. We start by noting that, unlike three dimensional(3D) chirality associated with optical activity, planar(2D) chirality in its most general expression (withoutpoint symmetry like rotational invariance [1]) is char-acterized by polarization transfers from left- to right-handed circular polarization, i.e. from positive to nega-tive helicities, that are flipped when exchanging the enan-tiomeric form of the 2D chiral structure through whichlight is transmitted [1, 17, 18].

These peculiar features are most clearly describedwithin the Jones formalism, starting in the circular basisof polarization |R〉, |L〉 with the Jones matrix of a bire-fringent (linear LB and circular CB) and dichroic (linearLD and circular CD) optical system: [19]

J =

(Jll JlrJrl Jrr

)=

(cos T2 + iC

T sin T2 − (iL+L′)

T sin T2

− (iL−L′)T sin T

2 cos T2 −iCT sin T

2

)(1)

where T =√L2 + L′2 + C2 for C = CB − iCD, L =

LB− iLD measured along the linear |x〉, |y〉 polarizationaxes, and L′ = LB′−iLD′ along π/4-tilted (|x〉±|y〉)/

√2

linear polarization axes [20, 21]. The difference χ be-tween diagonal elements of the Jones matrix is a measureof the optical activity of the system, with a real part pro-portional to circular dichroism (CD) and an imaginarypart associated with circular birefringence (CB) accord-ing to:

χ = (Jll − Jrr)/2 = (CD + iCB)× sin (T/2)

T. (2)

For 2D chirality, reciprocity imposes Jll = Jrr, thatis χ = 0, from the interconversion of the planar enan-tiomeric forms of the 2D chiral system when the prop-agation of the probing light beam is reversed [22–24].But in the absence of any point symmetry, the squarenorm difference ρ = |Jrl|2−|Jlr|2 of off-diagonal elementsis non-zero and characterizes 2D chirality through whatis known as the circular conversion dichroism (CCD)[1, 25, 26]:

ρ = (LB × LD′ − LB′ × LD)×(

2 sin (T/2)

T

)2

(3)

that stems from the misalignment between LB and LD.One key point for this work is that CCD associated

with 2D chirality couples optical spin with the propa-gation direction of the light beam, in the sense that re-versing the direction of propagation and the helicity as ithappens after reflection on one cavity’s end-mirror makesthe light beam transmitted through the enantiomorphicJones matrix for which Jrl and Jlr are exchanged. Instriking contrast with 3D chirality, this implies that af-ter one round-trip inside the cavity, this spin-orbit cou-pling leads to a different left- vs. right-handed circular

polarization balance that depends on the initial choice ofhelicity, as illustrated in Fig. 1 (a) and (b).

There is second key aspect associated with a planarchiral system that, viewed at an oblique angle of inci-dence, yields optical signatures that engage both 2D and3D chirality. These chiroptical features can be under-stood by a point group symmetry analysis. A planarchiral system is indeed of C1h symmetry with only oneplane of symmetry perpendicular to the optical axis, andthus contrasts with a 3D chiral system having C2 symme-try with an axis of rotation perpendicular to the opticalaxis. But when observed under oblique incidence +θ,a planar chiral object, described with the Jones matrixJ+θ, looses C1h symmetry with no additional C2 symme-try, as clearly seen on the left hand side of Fig. 1 (c). Asa consequence of the tilt therefore, a flip of the systemdescribed by the operation ΠxJ

T+θΠ

−1x performed on the

Jones matrix written in the linear polarization basis, doesneither transform it into its initial configuration by anyrotation along the optical z−axis (3D chirality) nor intoits (x, y)−plane mirror symmetrical 2D enantiomer (2Dchirality). This means that the system under obliqueincidence must be described by a combination of bothchiralities with different signatures viewed from both ±θincidence angles.

The connection between opposite incidence angles±θ can be made most straightforwardly through a(x, y)−plane mirror symmetry noted σh on Fig. 1 (c).This simple operation however cannot be directly ex-pressed within the Jones formalism. To do so, we decom-pose σh into two successive transformations: one C2 rota-tion along the x− axis (flipping operation) followed by a(x, z)−plane mirror reflection, yielding J+θ = JT−θ whereT is the matrix transpose. Within the circular basis

of polarization, this relation becomes J+θ = σ1JT

−θσ−11

with σ1 the first Pauli matrix.The most important consequence of this analysis is

that the optical signatures associated with 3D chiralitywill be reversed under opposite oblique incidence anglewhile those associated with 2D chirality will be preserved

χθ = −χ−θ ρθ = ρ−θ, (4)

following the definitions of Eqs. (2) and (3). As seen,the manifestation of 3D chirality is angle-dependent andas such, is totally different from intrinsic 3D chiralitygenerally rotationnaly invariant. This illustrates how ex-trinsic are these 3D chiral features that emerge from 2Dchirality at oblique incidence [18, 27]. Below, we exploitthese relations (4) as a way to characterize the 2D chi-rality of a system, particularly relevant when the sourceof planar chirality remains weak.

Our approach to induce 2D chirality inside a Fabry-Perot cavity is to use atactic polymers such aspolystyrene [28]. When a torsional shear stress is appliedto such an atactic polymer, chiroptical features arise in

3

the polymer matrix that are induced by a macroscopicchiral conformation of the chains. We generated thestress inside the polymer matrix by spin-coating clock-wise or anticlockwise a thin layer (ca. 150 nm) of dis-solved polystyrene solution (molecular weight of 195 K,diluted 4% in weight in toluene) on a 30 to 60 nm thicksilver mirror -details are provided in Appendix A. Fric-tion forces the polymer chains to take a macroscopic chi-ral arrangement close to the surface of the mirror viain-plane spinning of the chains, adopting a macroscopicC1h symmetry of 2D chirality. Far from the surface, theconformation of the polymer chain is not hindered byfriction and the chains are simply randomly distributedwithin the volume [29]

Within it, these structural changes correspond to thespecific chiroptical features that we analyzed above. Ex-perimentally, the CD signal is measured as the (0, 3) co-efficient of the cumulated differential Mueller matrix, asexplained in Appendix C. The Mueller matrix (MM) it-self is acquired on a home-build optical setup that yieldscalibrated, angle-resolved (Fourier space) MM describedin details in Appendix B. Remarkably, the MM gives thepossibility to separate linear birefringences and dichro-isms from circular ones and thus to measure true planarchiroptical features and artifact free CD [21, 30].

The first feature observed for a polymer layer spin-coated on a glass substrate is the absence of CD, a traitexpected from the cryptochiral nature of the polymerlayer [31, 32], and consistent with 2D chirality that doesnot yield any CD as shown by the green curve in Fig. 2(c). We then form a Fabry-Perot cavity by sandwichingthe polymer layer between two Ag mirrors of the samethickness (as explained in Appendix A) and measure itsCD in transmission. This time, as seen in Fig. 2 (a), clearsignatures of CD are observed under oblique illumination.These signatures are remarkable in that they correspondto optically active transverse electric (TE) and trans-verse magnetic (TM) modes of the Fabry-Perot cavity.At normal incidence, the opposite helicity between TEand TM modes is a direct consequence of 2D chirality,as explained in Appendix D. At fixed illumination an-gles where the degeneracy between TE and TM modesis lifted, this yields the bi-signated signatures observedexperimentally through the cavity at fixed illuminationangles and displayed in Fig. 2 (c). Remarkably, as seenin particular in Fig. 2 (c) and (d), the contrast of the±k‖ angularly averaged bi-signated profiles are reversedbetween opposite enantiomorphic cavities.

It is also clear from the data that reveal CD signs ex-changed from both sides of the normal incidence that thespin-coated polymer thin film yields a zero CD at normalincidence inside the cavity. As discussed further below,we interpret the tilt of the whole chiral landscape as aneffect of intertwined 2D chirality and extrinsic 3D chiral-ity. This results in the more intense CD signals observedin Fig. 3 (c) in one angular sector in relation with the

enantiomorphism of the cavity.

FIG. 2. (a) Measured CD dispersions for both enantiomor-phic cavities: clockwise shear stress -left panel- and anticlock-wise shear stress -right panel. The TE (continuous line) andthe TM (dashed line) modes are superimposed to the mea-sured CD. (b) Simulated CD dispersions for both enantiomor-phic cavities for the same clockwise and anticlockwise shearstresses. (c) Associated experimental traces averaged over thetwo ±k‖ angular sub-spaces for both forms where the red andblue traces correspond to averaging performed over the posi-tive and negative sub-spaces, respectively. Once converted inmdeg, the CD values reported correspond to ca. 600 mdeg.(d) With the same color coding, simulated traces averagedfrom the simulations shown in (b) over both ±k‖ angularsub-spaces.

The properties of our cavities can be simulated usingthe transfer matrix approach presented in the AppendixE. In this approach, a first approximation describes ourpolymer film under shear stress as a Pasteur medium, i.e.as a chiral isotropic medium [33, 34], with the constitu-tive relations

D(r) = εE(r) + iκ

cH(r) (5)

B(r) = −iκcE(r) + µH(r) (6)

where the permittivity (ε = ε0εr), the permeability (µ =µ0µr) are the usual isotropic parameters (c2 = 1/ε0µ0)of the polymer medium and κ the (complex) parameterassociated with its chiral response. This model captureswell the experimental features observed in Fig. 2 (a) and(c) when describing the chiral response of our materialwith a (θ, λ)−dispersive chiral parameter

κeff(θ, λ) = κ(λ)[a× cos(θ) + b× sin(θ)]. (7)

In this effective model, the wavelength dependent com-plex parameter κ(λ) is taken to be only weakly dispersivein the visible range, in agreement with the cryptochiral-ity of the polymer itself –see Appendix E. Then the tilt of

4

the 2D chiral material is described by involving the twosignatures given in Eq. (4). The parity-even response inθ associated with 2D chirality is gauged by the a parame-ter while the 3D chiral parity-odd response in θ is gaugedby the b parameter, driven by the extrinsic 3D chiralityemerging from 2D chirality under oblique illumination.By choosing a b/a ' 10 ratio, the model reproduces wellthe (θ, λ) dispersion of the MM measured experimentallyin strict relation with the enantiomorphism of the cavity,as shown in Fig. 2 (b). There is a very good agreementbetween theory and experiment in the angular evolutionof the chiroptical properties of the TE and TM modes,with the bi-signation and the asymmetry in the CD sig-nal measured between the two positive and negative ±θangular sectors observed in Fig. 2 (d). These features,both measured and simulated, illustrate the role of planarchirality when present inside a Fabry-Perot cavity.

FIG. 3. (a) Global helicity αi(λ) normalized to the maximumintensity of the intracavity electric field for the i = TE cavitymode calculated for enantiomorphic cavities with clockwise-left panel- and anticlockwise -right panel- shear stresses. Forboth forms, the in-plane wavevector k‖ values for zero he-licity of the mode are marked with a filled and empty cir-cle, respectively. (b) Intracavity δGi(z) calculated at a cho-sen k‖ = +4 µm−1 by placing inside the cavity a 2D chiralmedium described by Eq. (7) (top row) or (bottom row) withthe cavity uniformly filled with 3D chiral medium describedby a corresponding, constant, κ3D (see main text) for bothi = TE (left side) and i = TM (right side) modes. Note thatthe contrast of the bottom row is adjusted for clarity but withvalues one order of magnitude smaller than those of the toprow.

As we now show, the unique chiroptical properties thatplanar chirality yields under oblique illumination leadto the possibility to store a preferred helicity within agiven mode in one cavity round-trip. To demonstratethis, we quantify the chirality of a cavity mode usingthe metric (used for instance in [2, 35, 36]) δG(r) =(|G+(r)|2−|G−(r)|2)/

√2, where G±(r) = E(r)±iηH(r)

are the Riemann-Silberstein vectors and η is the usualimpedance of the field. As explained in Appendix E,this impedance within a chiral medium can be simpli-fied to the local difference between left and right electricfield intensities [37]. We chose this metric because it isdirectly linked to the optical chiral density and thus di-rectly measures the predominance of one spin-polarized

field over the other [38]. Integrating δG(r) along thez−propagation direction inside the chiral film gives theglobal helicity of the cavity mode i = TE, TM within ah = z2 − z1 thick layer

αi(λ) =1

h

∫ z2

z1

δGi(r)dz. (8)

Those quantity are displayed in panels (a) and (b) in Fig.3 where αi(λ) has been normalized by the field maximumintensity inside the cavity. They demonstrate that thecavity modes defined in our designer Fabry-Perot cavityare characterized by finite helicity densities, whose hand-edness is opposite in each ±θ angular sector. Here too,the combination of 2D and 3D chiralities contributes tothe tilt of the chiral landscape and the change of helicitythat we expect for the extrinsic 3D chirality is shifted tonon-zero incidence angles. The fact that αi(λ) is non-zero along the i = TE, TM modes at normal incidenceis a central result of the Letter, with the sign of the he-licity of the resonator at normal incidence that dependson the clock/anticlockwise spin-coating direction. Thisgives our cavities a real potential for exploring resonantstrong coupling signatures in chiral polaritonic chemistryand material science.

The angular evolution of αi(λ) is related to the profileof δGi inside the cavity as shown in Fig. 3 (b). The δGiprofiles, shown in Fig. 3 (b), reveal that when the cavityis modelled with a 2D chiral layer of the polymer film,chosen here to correspond to a 20% volume fraction ofthe cavity, the local helicity of the cavity modes can beenhanced by ca. one order of magnitude in comparisonwith an intrinsically 3D chiral cavity. [39]

In conclusion, we demonstrated that a polymer film onwhich a chiral stress is imposed can seed planar chiralitywithin a Fabry-Perot cavity. This seed enables a spin-orbit coupling mechanism that shapes, for each round-trip inside the cavity, transverse electric and transversemagnetic modes with a preferred helicity density. An-alyzed using the Jones-Mueller formalism, the proposedmechanism for shaping such chiral modes results from thecombination between 2D and 3D chiralities under obliqueillumination and as such, is a universal mechanism thatcan be involved in a great variety of system, in particularsoft, polymeric media, and over large optical bandwidths.This universality, combined with the simplicity in the im-plementation, paves the way to exploit such chiral modesin the context of chiral cavity QED [4, 8] and polaritonicchemistry [9]. There, the chiral nature of the polaritonicstates that can be created within our cavities yields thecore ingredient needed for inducing a new type of selec-tivity for asymmetric syntheses performed in the regimeof strong coupling. This will yield original strategies thatwe are currently exploring in the endeavor to draw a newlandscape for asymmetric chemistry driven by chiral po-laritonic states.

5

ACKNOWLEDGMENTS

This work was supported by the French NationalResearch Agency (ANR) through the Programmed’Investissement d’Avenir under contract ANR-17-EURE-0024, the ANR Equipex Union (ANR-10-EQPX-52-01), the Labex NIE (ANR-11-LABX-0058 NIE) andLabex CSC (ANR-10-LABX-0026 CSC) projects, theUniversity of Strasbourg Institute for Advanced Study(USIAS) (ANR-10-IDEX-0002-02) and the European Re-search Council (ERC project no 788482 MOLUSC).

APPENDIX

A –SAMPLE FABRICATION

The fabrication process for our cavities is simple. Wefirst thoroughly clean a 2.5× 2.5 cm2 glass substrate bysonication in a 0.5%wt Hellmanex solution in ultra-purewater for 8 minutes. We then sonicate the substrate ina bath of ultra-pure water only, for 30 minutes followedby a sonicated bath in pure ethanol for 30 other minutes.We finally rinse the substrate using a series of 20 dips ina solution of ultra-pure water.

In order to form the cavity itself, we first sputter a layerof silver using an Emitech K575X tabletop sputterer at60 mA (for 45 s for 30 nm and 95 s for 60 nm Ag layers).On top of the Ag mirror thus sputtered, we spin-coata solution of dissolved Polystyrene,4%wt in Toluene at1400 RPM for 2 min to obtain a thin film of 150 nm.The RPM speed was calibrated using profilometry. Wefinally “close” the cavity by sputtering another Ag layeron top, using the same sputter parameters for the samethickness.

B – EXPERIMENTAL SETUP

Our experimental setup used for the full determina-tion of the Mueller matrix element is schematized in 4.The first part is made of a polarization state generator(PSG) composed of a Glan-Taylor (GT) linear polarizerand a motorized quarter-wave plate. The light beam isinjected through the sample using a Nikon ELWD 40×(NA=0.6) objective and collected using a Nikon ELWD100× (NA=0.9) objective. The collected light passesthrough a polarization state analyzer (PSA) composed ofa quarter-wave plate and a linear polarizer (LP). The twolinear polarizers were chosen different because of the neg-ative bias induced on the Mueller matrix elements whenusing a GT within the PSA. The last part is made of alens set at the focal distance from the back focal plane(BFP) of the second objective, associated with a secondlens at the entry of a spectrometer (Teledyne PrincetonInstrument, SpectraPro HRS-300) for imaging this BFP

–referred below as “Fourier space imaging”. Removingthis lens gives us the ability to image the focal plane ofthe objective on the CCD –referred as “real space imag-ing”. Spatially resolved, the spectra are recorded usinga PIXIS 1024 CCD camera.

FIG. 4. Experimental setup used for the Mueller matrix deter-mination, composed of a polarization state generator (PSG)and analyzer (PSA). The two lenses give us the ability to im-age the BFP of the objective, i.e. the Fourier space of oursample.

In order to resolve the Mueller matrix for a given wave-length, we build a system of equation linking the mea-sured intensity for a given state of polarization (SOP) toits Mueller matrix (MM) element. The SOP are gener-ated using a carefully chosen combination of angles forthe quarter waveplates in both the PSA and PSG.

Because the Mueller matrix is a 4×4 matrix, there areat least 16 linearly independent equations to solve for oursystem. Experimentally, we overestimate this minimalset by doing 64 measurements, solving the system by theleast-square method. This approach was already detailedby us in [40] and is simply summarized here.

Let MS, MPSA, MPSG be the Mueller matrix of thesample, the PSA, and respectively the PSG. We can formthe following system of equations represented in the fol-lowing matrix form:

Sout = MPSAMSMPSGSin (9)

Sout = MPSAMSG (10)

The intensity recorded, for one experiment, by the CCD,corresponds to the first element of Sout, Iout, which canbe expressed as:

Iout =

4∑j=1

4∑i=1

mPSA1,i × gi ×mS

j−1,i (11)

where mPSA1,i is the known first line of the PSA Mueller

matrix element, gi the known element of the vector re-sulting from MPSG.Sin, and mS

j,i the unknown Muellermatrix element of the sample. We write our set of 64equations linking the intensity and the Mueller matrixelement in the following and most convenient matrix for-mulation:

b64×1 = A64×16.X16×1 (12)

6

where b is a containing each intensity of the 64 measure-ment, and X is a vector containing all the Mueller matrixelements. Because we overestimate our system, there isno unique solution but there is a unique solution thatminimize the residue ν2:

ν2 = (b−AX)T .(b−AX) (13)

The vector that minimize the residue can be expressedas:

X = (ATA)−1ATb. (14)

From Eq. (14), one can uniquely define the MM asso-ciated with the change of the incident light SOP throughthe medium. Alone however, the Muller matrix is hardlyuseful in a chiroptical context. In order to access genuinechiroptical observables, some data filtering is necessary inorder to remove any optical artifacts that would preventus from recovering the relevant chiroptical features of thesample, in particular its circular dichroism.

C – DATA FILTERING

From an experimental Mueller matrix, one can developan algebra, which allows to isolate the CD signal of thesample and remove artifact signals in the system. Thiswell-known algebra is detailed for instance in [41]. Forour experiments, two different data filtering steps wereused when imaging real space vs. Fourier space.

In the real space, we first identify the polarization re-sponses of the objectives by measuring the MM of (i)a setup with two M40× and no sample, and (ii) of thesetup described in 4. We find indeed that experimen-tally, M40× 'M100×. One can then remove the responseof the objective by noting that when measuring an emptysetup, i.e. with no sample, one effectively measures:

Mempty = M40×M40× (15)

(16)

We can then computeM40× = (Mempty)12 and remove the

responses of our objectives in the real space. In Fourierspace the same procedure is applied.

The second filtering step consists in removing the con-tribution of the glass substrate of the sample from ourexperimental Mueller matrix. To do so, we first mea-sure the response in both real and Fourier spaces of thecleaned substrate, Mglass. Then, we can determine theMueller matrix of the sample alone without the glass sub-strate contributions by using the following serial Muellerdecomposition:

MS = MglassMFP (17)

where MFP is the Mueller matrix of the Fabry-Perotcavity (without substrate) which can be rewritten asMFP = M−1

glassMS.

The third filtering step is to decompose the previouslyobtained Mueller matrix using the Cloude decomposition[42]. The goal of the method is to give an estimation ofthe equivalent non-depolarizing Mueller matrix, knownas Mueller-Jones matrix, necessary for the last filteringstep below. Following [41, 43], Cloude decompositionconsists in computing the 4× 4 hermitian coherency ma-trix T , which will have the following matrix elements:

t11 =1

4(m00 +m11 +m22 +m33)

t12 =1

4(m01 +m10 − i(m23 −m32))

t13 =1

4(m02 +m20 − i(m31 −m13))

t14 =1

4(m03 +m30 − i(m12 −m21))

t22 =1

4(m00 +m11 −m22 −m33)

t23 =1

4(m12 +m21 − i(m30 −m03))

t24 =1

4(m13 +m13 − i(m02 −m20)

t33 =1

4(m00 −m11 +m22 −m33)

t34 =1

4(m23 +m32 − i(m10 −m01))

t44 =1

4(m00 −m11 −m22 +m33)

The coherency matrix can be computed from any givenexperimental matrix. By considering that any depolariz-ing Mueller matrix M can be considered as a convex sumof non-depolarizing Mueller-Jones matrix, denoted MJi,one can link the eigenvalues λi of the coherency matrixto MS by the following:

M =

3∑i=0

= λiMJi (18)

MJi = A.(Ji ⊗ Ji∗)A−1 (19)

where Ji is the Mueller-Jones matrix associated to MJi

and A is the passage matrix that can be written as

A =

1 0 0 11 0 0 −10 1 1 00 −i i 0

(20)

We then can rank the MJi in terms of their respectiveweight. Generally, this decomposition is dominated bythe first term, ie λ0 >> λ1, λ2, λ3, and one can considerλ0MJ0 as a good estimate of the non-depolarizing Mullermatrix associated to MS . From this estimate, one candirectly extract the CD following [20]. We first compute

7

the cumulated differential Mueller matrix Lm as

Lm =ln(MJ0) (21)

=1

2(L−GLTG) (22)

where G is the Minkowski tensor G=diag(1,-1,-1,-1) andln the matrix logarithm. In this manner, the CD simplycorresponds to the Lm(0, 3) matrix element.

D –BI-SIGNATED CD SIGNALS FOR TE ANDTM MODES

In order to explain the bi-signated CD signal for theTE and TM modes measured through the cavity underoblique illumination, we look at the helicity of the trans-mitted beam under TE and TM polarizations at normalincidence, where the two associated modes are degen-erated. Within the Stokes-Mueller formalism, TE andTM modes are expressed by Stokes vectors as STM =(1,−1, 0, 0)T and STE = (1, 1, 0, 0)T . Since our systemat normal incidence is a 2D chiral system, it is simplydescribed by a Jones-Mueller matrix given by

M2D =

1 −a1 −a2 0−a1 d1 0 b2−a2 0 d2 −b1

0 −b2 b1 d3

, (23)

and therefore yielding the TE and TM transmittedStokes vectors:

SoutTM = M2D · STM =

1 + a1

−a1 − α−a2

b2

(24)

SoutTE = M2D · STE =

1− a1

−a1 + α−a2

−b2

. (25)

The S3 elements ±b2 for each transmitted Stokes vec-tor are opposite. This implies that the helicities associ-ated with the TE and TM modes, degenerated at nor-mal incidence, are opposite. When the optical activityemerges at an oblique angle accompanied by a lifting ofdegeneracy, the CD for the TE and TM branches willhave opposite sign accordingly.

E – CHIRAL TRANSFER MATRIX

In the liquid crystal community, the usual approachfollowed for simulating transmission spectra is the Berre-man matrix formalism [44]. But this method suffers from

the rise of singularities in certain specific cases [45]. Toovercome this problem, we model, in a first approxima-tion, our polymer film under shear stress as a typicalPasteur medium, i.e. as an isotropic chiral medium. Wehowever introduce a spatially dispersive response of thechirality parameter of the medium in order to mimic itsreal response associated with extrinsic/intrinsic chirality.

In a Pasteur medium, one derives the constitutiveequations as [34]:

D = εE + iκ

cH B = −iκ

cE + µH, (26)

where the permittivity (ε), the permeability (µ) and thechiral parameter (κ) are usual isotropic parameters. Thesource-free wave equation for such a Pasteur medium isgiven by [34]:

∇2E− iωκc∇∧E + iω(

κ

cηµ− εµ)E = 0. (27)

In this case, the eigenstates are circularly polarized planewaves and the electromagnetic field inside the chiralmedium can be written as a superposition of right andleft polarized waves going in the forward and backwarddirections:

E =E+Re−i(k+.r+ωt) + E+Le

i(k+.r−ωt)+

E−Re−i(k−.r+ωt) + E−Le

i(k−.r−ωt), (28)

where R and L denote the right-going and left-goingplane waves. These eigenstates feel the chirality of themedium as a standard medium without electromagneticcoupling, i.e.:

D± = ε±E± B± = µ±H±, (29)

where one can derive ε± and µ± from the constitutive pa-rameters and the wavenumber associated with each po-larization state [34]:

µ± = µ± κ

c

õ

εε± = ε± κ

c

√ε

µk± = ω(

√εµ± κ

c).

(30)

Our approach is similar to [46] and to other transfermatrix computations. Once the field is characterized inthe chiral medium, the field continuity equation can bewritten in a convenient matrix form En−1 = An−1,nEn

where A is a bloc symmetric 4 × 4 matrix linking theright and left polarized electric field going forward andbackward from the (n) layer to the (n − 1) layer whichare gathered in the following quadrivector En:

E+R

E−RE+L

E−L

n−1

=

(aT aRaR aT

)E+R

E−RE+L

E−L

n

, (31)

8

where:

aT =

(ηr+1

4 (1 +cos(θ+,n)cos(θ+,n−1) ) ηr−1

4 (1− cos(θ−,n)cos(θ+,n−1) )

ηr−14 (1− cos(θ+,n)

cos(θ−,n−1) ) ηr+14 (1 +

cos(θ−,n−1)cos(θ−,n−1) )

)(32)

aR =

(ηr+1

4 (1− cos(θ+,n)cos(θ+,n−1) ) ηr−1

4 (1 +cos(θ−,n)cos(θ+,n−1) )

ηr−14 (1 +

cos(θ+,n)cos(θ−,n−1) ) ηr+1

4 (1− cos(θ−,n)cos(θ−,n−1) )

)(33)

with ηr = ηn−1

ηnthe ratio between the usual wave

impedance in their respective layer and θ±,n the of thewave associated with the left or right polarized field thatone can recover by imposing continuity of the phase atthe interface, according to:

θ±,n = asin(k±,n−1sin(θ±)

k±,n). (34)

To take into account the phase gained by the electricfield inside one layer, we introduce the Pn matrix:

Pn =

e−ib+ 0 0 0

0 e−ib− 0 00 0 eib+ 00 0 0 eib−

(35)

where b± = k±,ndncos(θ±,n). With this, the final to-tal transfer matrix, Ttot, can be written as the followingproduct of matrix:

Ttot = A0,1P1A1,2...PNAN−1,N . (36)

Finally, in order to measure the total field intensitytransmitted by our sample, we set E+L,N = E−L,N = 0and compute:(

E+R

E−R

)out

=

(ttot11 ttot12

ttot21 ttot22

)−1(E+R

E−R

)in

. (37)

From this equation, one can easily calculate the totaltransmission of our sample. Moreover, probing the sam-ple’s medium with four linearly independent Stokes vec-tors leads to compute the Mueller matrix of the sampleusing the chiral parameter indicated in Fig.5. In orderto model this, we assume a resonance in the UV that isoptically active. This resonance yields a non-zero imagi-nary part for the chiral parameter, which fixes the disper-sive nature of our polymer, even far from the resonancethrough the (broad-band) Kramers-Kronig relation.

Using the previously calculated matrix An,n+1 and Pnand their relative z−positions inside the layer, one cancompute the electric field intensity at any point withinour multilayer system. With this, we compute the theRiemann-Silberstein vectors inside the chiral medium asdefined in the main text:

G±(r) = E(r)± iηH(r) (38)

FIG. 5. Real -panel (a)- and imaginary -panel (b)- parts ofκeff .

where η is the usual impedance of the medium. By as-suming that H = i

ηE, we write:

δG(r) = |G+(r)|2 − |G−(r)|2. (39)

Our transfer matrix simulation allows monitoring δG(r)with respect to its position along the z−axis for both TEand TM modes. The raw results are presented in Fig. 6.

FIG. 6. TE (a) and TM (b) intracavity mode electric fieldintensities evaluated at an angle of -35 deg (k‖ = −5 µm−1).Corresponding δGTE(r) (c) and δGTM (r) (d). The thin filmboundaries are indicated with red dashed lines.

This variable leads us to monitor the local helicity ofthe field inside the cavity. In addition, we compute thepredominant helicity of the mode by integrating alongthe z axis, denoting it by αλ = 1/h

∫ z2z1δG(r)dz. The

two simulations presented on Fig. 3 (a) in the mainpaper are obtained by changing the enantiomeric formassociated with the extrinsic 3D chirality coming fromthe planar chiral structure. The value for the TE andTM modes are indicated in 7

A key result is the change of the preferential helic-ity of the field at normal incidence as we change the

9

FIG. 7. (a) Global helicity αi(λ) for the i = TE (a)-(b) andi = TM (c)-(d) cavity mode calculated for enantiomorphiccavities with clockwise - (a)and (c)- and anticlockwise -(b)and(d)- shear stresses.

enantiomeric form of the planar structure. The pointwhere the helicity flips sign is determined by the rela-tive strength between 2D chirality (parameter a in themodel) and 3D chirality (parameter b in the model). Inour simulations, we choosed b/a = 10, as discussed in themain text.

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