Observation of second order meron polarisation textures in optical microcavities

Multicomponent Bose-Einstein condensates, quantum Hall systems, and chiral magnetic materials display twists and knots in the continuous symmetries of their order parameter, known as Skyrmions. Originally discovered as solutions to the nonlinear sigma model in quantum field theory, these vectorial excitations are quantified by a topological winding number dictating their interactions and global properties of the host system. Here, we report the first experimental observation of a stable individual second order meron, and antimeron, appearing in an electromagnetic field. These complex textures are realised by confining light into a liquid-crystal filled cavity which, through its anisotropic refractive index, provides an adjustable artificial photonic gauge field which couples the cavity photons motion to its polarisation resulting in formation of these fundamental vectorial vortex states of light. Our observations take a step towards bringing topologically robust room-temperature optical vector textures into the field of photonic information processing and storage.

Twists in the SO(3) order parameter of magnetic systems lead to topologically protected excitations known as skyrmions which are characterised by nontrivial spin textures [1][2][3][4]. Just like quantised singular vortices in superfluid Helium or Bose-Einstein condensates these skyrmionic excitations are topologically robust against external perturbation since they cannot smoothly relax into the defect free ground state of the system, thus becoming highly important to understand phase transitions and critical behaviour in ordered many-body systems down to the quantum level [5]. This robustness has also led to innovative proposals in the field of spintronics of stable information storage and processing with skyrmions at unprecedented spatial scales. They have been observed in chiral magnets [6], non-centrosymmetric magnets [7], surface plasmons [8,9], exciton-polaritons [10] to name a few, and have reached room temperature conditions in magnetic thin films [11,12] and liquid crystals [13,14].
Skyrmion textures appear as natural excitations in multicomponent quantum systems since a surjective homomorphism links the SU(2) unitary symmetry group to the SO(3) rotational symmetry group. In a photonic system, the two orthogonal polarisation components of the electromagnetic field can be described by a threedimensional Stokes (pseudospin) vector located on the surface of the Poincaré sphere. Therefore, such topological knots and twists in an electromagnetic field can in-principle exist in the same sense as skyrmions in thinfilm magnetic materials. Of special interest are spin textures known as magnetic vortices or "merons" which orig- * Jacek.Szczytko@fuw.edu.pl inate from Yang-Mills theory [15]. Due to their similarity to skyrmions they are sometimes referred to as half skyrmions or baby skyrmions since they can possess half of the skyrmion topological integer charge Q defined in two-dimensional system as where S is the order parameter. Alternatively, the charge of the meron can be determined through Q = vp/2 from its vorticity (v) and polarity (p) which describe the inplane and out-of-plane order parameter orientation respectively [16]. The simplest configurations are those composed of v = ±1 and p = ±1 referred to as "merons" (Q = 1/2) and "antimerons" (Q = −1/2) (Fig. 1a). In fact, twists in the Hamiltonian parameter space can be regarded as merons whose textures relate to the Berry curvature and charge determines the topological character of condensed matter systems [17][18][19][20]. Interestingly, merons cannot exist as isolated objects unless spatially constrained [21,22]. They either form in lattices [23][24][25][26][27][28][29] or as paired objects only observed before in magnetic thin films [30]. There also exist higher order merons, with vorticity v = ±2 which are referred as "second order merons" (Q = 1) and "second order anti-merons" (Q = −1) (Fig. 1a).
To the best of our knowledge, these second order twists in order parameter have not been observed in any system to date.
In this study, we present experimental and numerical evidence of second order merons and antimerons in the photonic field of an optical microcavity filled with a liquid crystal (LC) at room temperature. The second order merons appear as the natural eigenmodes of the The arrows represent the order parameter S from equation (1). b, Schematic of a microcavity filled with liquid crystal media. The liquid crystal microcavity can be tuned to contain perpendicularly polarised, degenerate modes c with the same mode numbers or d with different mode numbers. The electric field distribution of X (Y ) polarized mode is plotted in green (pink) colour.
system due to its tunable optical anisotropic structure. We demonstrate that a pattern of merons (antimerons) can smoothly merge to form a second order meron (antimeron). Effective Hamiltonians describing the two distinct meron textures are derived linking our observations to alternative low-dimensional condensed matter systems and paving the way towards synthesising fundamental order parameter twists on nonlinear optical fluids in the strong light-matter coupling regime.
We investigated microcavities with a birefringent LC layer enclosed between two parallel distributed Bragg reflectors (DBRs), as schematically shown in Fig. 1b. The birefringent medium is characterised with two refractive indices: extraordinary n e , parallel to the director of the LC molecules defining the long optical axis, and ordinary n o , perpendicular to the director. This molecular director can be altered by application of external bias to transparent indium tin oxide (ITO) electrodes on the sample. We investigated a configuration in which the director rotates in the x-z plane with applied field. Different effective refractive indices n for linearly polarised light along x and y axes lead to splitting of the optical modes fulfilling standing wave condition for an optical path length nd = N λ/2 along the width of the cavity d, for incident wavelength λ and mode number N . In a sufficiently wide cavity multiple optical modes with different mode numbers can be confined. The unique property of LC-filled microcavity is the control over the en-ergies of linearly x-polarised optical modes (X) with respect to y-polarised modes (Y ) which allows to tune them in-and-out of resonance with respect to each other. In this work we concentrate on two different regimes where both X and Y modes have the same parity corresponding to (N X , N Y ) = (N, N ) and (N X , N Y ) = (N + 2, N ) (see Figs. 1c,d) which possess uniquely different photonic spin-orbit coupling mechanisms leading to meron and antimeron textures.
The optical eigenmodes, in the XY polarisation basis, can be described by the following Hamiltonian with similar structure to the one describing TE-TM splitting in optically isotropic semiconductor microcavities [31]: x /m x +k 2 y /m y )/2 describes cavity photons with masses m x,y along the x, y direction respectively, δ x , δ y , δ xy are parameters proportional to the birefringence ∆n = n e − n o [32],σ x,y,z are the Pauli matrices, and ∆E = E Y,N Y − E X,N X is the XY mode splitting at normal incidence (k = 0). Notably, this splitting is equivalent to the presence of an effective magnetic field (Zeeman splitting) which plays the role of an artificial photonic gauge field applied to the structure. In this sense, the polarisation of the cavity photons plays the same role as a two-level spinor for massive particles. The derivation of equation (2)  dielectric medium is presented in the Supplementary Information.
In the (N, N ) regime the molecular director is oriented along the z axis, so θ = 90 • , and the refractive indices of the cavity medium are the same for normal-incident light polarised along x or y axis (i.e, m x = m y ). Here, we have δ x = δ y = δ xy > 0 and ∆E = 0 which gives rise to the standard optical spin Hall effect (see Figs. 2a,b) observed for microcavity exciton-polaritons and bare cavity photons [33][34][35]. This unique interplay between the photons motion and polarisation results in a spatial polarisation texture composed of a meron-antimeron lattice, as previously observed in a microcavity exciton-polariton condensate [27].
On the other hand, the (N + 2, N ) regime is obtained by changing the molecular director θ < 90 • which tunes the refractive index of cavity for light polarised along x axis (see Fig. 1). In this regime one has m x = m y , δ x , δ xy > 0, δ y < 0. By detuning the modes slightly, ∆E < 0, leads to severely different artificial spin-orbit coupling of the cavity photons (see Figs. 2c,d).
To illustrate the difference between the two regimes we show in Fig. 3 the real-space polarisation textures of light transmitted through a LC microcavity, calculated using the Berreman method [36] (see Methods). Figures 3a-d show the adiabatic evolution of the polarisation texture in the (N, N ) regime for an excitation polarisation going from linear to circular. Figure 3a shows the previously reported half-skyrmion lattice [27]. Here, four merons of charge Q = ±1/2 -two with positive and two with negative polarity -can be observed in the quadrants of the system. When polarisation of the excitation beam changes to elliptical, two merons start merging and create a single second-order meron (also referred to as bimeron) of Q = +1 as the laser excitation becomes fully circularly polarised.
In the (N + 2, N ) regime, corresponding to Fig. 3e-h, a very different behaviour is observed. Starting with linearly polarised incident light (Fig. 3e) we observe again four antimerons with the same polarity but inverse vorticity compared to the (N, N ) regime. When the excitation polarisation is gradually changed to circular, two of these antimerons merge creating one second order antimeron Q = −1 presented in Fig. 3h.
This dramatic change in the topological integer charge Q of these spin textures is precisely captured by equation (2). The charge Q has a different sign between the (N, N ) and (N + 2, N ) regimes because of the polarisation structure of parabolic eigenmodes in momentum space (see Fig. 2). The splitting between XY modes in both regimes is inverted. The fixed energy of the excitation laser selects an approximate circle in momentum space. Traversing the k-space circle of excited modes results in spin rotation which is in opposite direction between the two regimes.
The real-space polarisation textures of the eigenmodes of equation (2) can be investigated at room temperature using polarisation-resolved imaging of light transmitted through the LC microcavity. The exact polarisation state of light can be determined by a measurements of Stokes parameters S 1 , S 2 , S 3 defined as the degree of linear [S 1 for X (horizontal) and Y (vertical) linear polarisations, S 2 for diagonal and antidiagonal linear polarisations] and circular polarisation (S 3 ). The Stokes parameters corresponding to a second order meron, given by its analytical form [see equation (3) in Methods], are presented in Fig. 4a-c. The overlaid black arrows in the S 3 maps correspond to S = (S 1 , S 2 ). The same symmetry can be observed experimentally in the spatially resolved polarisation pattern of circularly polarised light transmitted through the LC microcavity in the (N, N ) regime as shown in Fig. 4d-f.
Similarly, analytical pseudospin texture of a second order antimeron is depicted in Fig. 4g-i. As expected from numerical modelling, such polarisation texture can be observed in (N + 2, N ) regime. Experimental results, presented in Fig. 4j-l, reveals second order antimeron texture. In the case of circularly polarised σ + incident light the LC microcavity, in (N + 2, N ) regime, acts as a fullwaveplate and the σ + light is directly transmitted, which gives a maximum for S 3 in the centre of the incidence spot at x = y = 0. Off-centre polarisation becomes linear far from the centre of the topological texture. The rotation of arrows around the centre in Fig. 4d,j indicates the ro-  showing the analytical spin texture of a second order meron given by equation (3). Black arrows correspond to S = (S1, S2). d-f, Experimental spatial polarisation texture of σ + polarised light transmitted through a LC microcavity in (N, N ) regime. g-i, S3, S1, and S2 Stokes parameters showing the analytical spin texture of a second order antimeron given by equation (3). j-l, Experimental spatial polarisation texture of σ + polarised light transmitted through a LC microcavity in (N + 2, N ) regime.
tation of the axis of linear polarisation. The difference between the second order meron (Fig. 4d) and antimeron ( Fig. 4j) is associated with the direction of rotation of linear polarisation axis. Along a clockwise directed path around the centre of the light spot the polarisation axis rotates clockwise for a meron and anticlockwise for an antimeron. The difference is clearly visible in the real-space patterns of S 2 depicted in Fig. 4 and reveals exactly the same rotation of polarisation in the reciprocal space in Fig. 2. It is straightforward to derive from equation (1) that the two opposite vorticities correspond to opposite topological integer charge Q. The precise size and orientation of the merons depends on the birefringence of the LC filling the cavity and the energy of the optical mode relative to the centre of the stopband (see Suppl. Inform. Fig. S7 and Fig. S8).
In our study we have provided the first experimental observation of a second order meron and antimeron in an electromagnetic field. The meron and antimeron polarisation textures result from the anisotropic refractive index of our optical liquid-crystal filled cavity. The artificial photonic gauge field which couples the cavity photon motion with its polarisation enables the emergence of vortical polarisation patterns. The flexibility in designing topological spin textures of light can be further combined in optical lattices mimicking magnetic order [37] or integrated with photonics devices. Furthermore, our findings are of fundamental interest to other systems described by models hosting analogous textures such as the Yang-Mills gauge theory or non-linear sigma models. These cavity merons can be described as a novel high order optical vector vortex state, providing a new element of structured light for study in the field of optical physics with potential application in communication, and high resolution imaging [38]. Our work opens new perspectives on using merons as topologically robust optical quaternary memory elements determined by combination of two orthogonal flows of spin (polarisation) vorticity and two opposite orientations of spin polarity.
The polarisation of light coming from the cavity is described through the standard definition of the Stokes parameters, Here, I X,Y , I d,a , I σ + ,σ − correspond to the intensities of horizontal, vertical, diagonal, antidiagonal, right-hand circular and left-hand circular polarised light. Simulations Berreman method [36,39] was used to calculate electric field transmitted at different incidence angles corresponding to varying in-plane wave vectors. Electric field in real space was obtained as a Fourier transform of the results in reciprocal space multiplied with a Gaussian envelope with dispersion σ x = 0.9 µm in real space.
Simulations in Fig Experiment Experimental results were obtained in a polarisation-resolved tomography measurement. Light from a broadband halogen lamp was circularly polarised and focused on a given sample with a 100× microscope objective. Transmitted light was collected by a 50× microscope objective, polarisation resolved and focused with a 400 mm lens on a slit of a monochromator equipped with a CCD camera. Full image was obtained by movement of the lens parallel to the slit. Experimental spatial polarisation textures presents constant energy cross sections around 10 meV above the resonances of the cavities at normal incidence, as shown in Fig. S3 and Fig. S4. (N, N ) sample Experimental results presented in Fig. 4d-f were obtained on a cavity made of DBRs with 6 pairs of SiO 2 /TiO 2 layers designed for maximum reflectance at ≈ 700 nm. ≈ 2 µm thick cavity is filled with birefringent liquid crystal with n o = 1.504 and n e = 1.801 with director oriented along z direction (HT alignment). Cavity mode resonance occurs at 768.5 nm. Transmission wavelength was equal to 763.3 nm.
(N + 2, N ) sample Experimental results presented in Fig. 4j-l were obtained on a cavity made of DBRs with 5 pairs of SiO 2 /TiO 2 layers designed for maximum reflectance at ≈ 580 nm. ≈ 2 µm thick cavity is filled with birefringent liquid crystal with n o = 1.539 and n e = 1.949 with director oriented along x axis (HG alignment). Experiments were performed with square waveform with frequency 1 kHz and peak-to-peak amplitude of 1.425 V applied to ITO electrodes which rotates LC molecules towards z axis resulting in close to degenerate cavity modes in horizontal and vertical polarisations at 583.9 nm and 584.3 nm. Transmission wavelength was equal to 581.5 nm.

Role of symmetry
The eigenvalue problem for the modes in the birefringent cavity can be analysed from the point of view of the symmetry. Since we are dealing with the coupling of two modes we wish to express the relevant Hamiltonians as second order polynomials in k x and k y with coefficients given by combinations of Pauli matrices. In our considerations we have to take into account the fact that the transformation law for the Pauli matrices in each case reflects the symmetry of the basis functions under discussion.
1) In the case of the (N, N ) resonance ( xz = 0) the symmetry of the system is given by the group D ∞h with rotation symmetry about the z axis and reflection plane perpendicular to the z axis.
It easy to verify, that under the reflection in the mirror xy plane all the Pauli matrices remain invariant while under the rotation by the angle φ about the z− axis only theσ y matrix remains invariant while (σ z ± iσ x ) → e ∓2iφ (σ z ±iσ x ). Taking into account that under this rotation k x ±ik y → e ∓iφ (k x ±ik y ) and that the only invariant of second order is equal to k 2 x + k 2 y we can postulate the following form of the Hamiltonian: with all coefficients α i -real, due to the hermiticity requirement. Under the rotation by π around the x axis we have E x → E x , E y → −E y , soσ z remains invariant andσ x changes sign. Under the same transformation also the term k x k y changes sign so the term proportional to α 5 is not invariant and we have to set α 5 = 0. Finally, the time reversal symmetry which in this representation is equivalent to the complex conjugation requires that α 0 = α 2 = 0. If we also set α 1 = 0 we obtain the most general form of the Hamiltonian admitted by the symmetry: . (6) with two parameters related to xx and zz .
2) In the case of the (N + 2, N ) resonance xz = 0 and the relevant symmetry group is C 2h with the twofold rotation symmetry about the y-axis. In this caseσ z is invariant under all symmetry operations whileσ x and σ y change sign under rotation and reflection in the xzplane. The possible invariants are thereforeσ 0 k 2 x ,σ 0 k 2 y , σ z k 2 x ,σ z k 2 y ,σ x k x k y andσ y k x k y . However, the last term is excluded due to the time reversal symmetry so the most general form of the Hamiltonian admitted by the C 2h symmetry for a pair of modes of the same parity has six parameters which can be expressed in terms of n o , n e , θ and mode order N :  Figure S1 presents simulated angle-resolved spectra corresponding to the data shown in Fig. 3 in the main text. Fig. S1a shows intensity of unpolarised light transmitted through the cavity in (N, N ) regime (θ = 90 • , Fig. 2a-d). We remind that θ is the angle of the liquid crystal (LC) molecular director.  Fig. S2p-r). Exact parameters of the simulated structures were optimised to match with experimental angle-resolved spectra for a given sample, shown in Fig. S3 and Fig. S4. Figure S3a,b presents experimental transmission intensity and S 1 parameter from cavity in (N, N ) regime corresponding to data shown in Fig. S2d-f. Fig. S3c,d shows simulated spectra for a cavity that consists of two DBRs made of 5 pairs of layers with refractive indices n low = 1.45 and n hi = 2.2 centred at λ 0 = 700 nm. Simulated cavity is 1855 nm thick and filled with birefringent liquid crystal with n o = 1.504 and n e = 1.801 with director oriented along z direction. Figure S4a,b presents experimental transmission intensity and S 1 parameter from cavity in (N + 2, N ) regime * Jacek.Szczytko@fuw.edu.pl corresponding to data shown in Fig. S2m-o. Fig. S4c,d shows simulated spectra for a cavity that consists of two DBRs made of 4 pairs of layers with refractive indices n low and n hi centred at λ 0 = 580 nm. 1902 nm thick cavity is filled with birefringent liquid crystal with n o = 1.539 and n e = 1.949 with molecules rotation angle θ = 26.27 deg.

III. COUPLING OF CAVITY MODES IN
(N + 2, N ) REGIME Figure S5 presents experimental angle-resolved transmittance spectra for a cavity tuned around (N + 2, N ) crossing (varying external voltage). Fig. S5a-e presents dispersion relation for wave vectors along x direction, Fig. S5f-j along y direction and Fig. S5k-o along diagonal direction. For wave vectors along the x and along y axes the X-polarised mode gradually crosses the Y -polarised mode. However for the antidiagonal wave vector direction (k x = −k y ) an anticrossing behaviour between the modes can be observed, which is an evidence on coupling between them. This anticrossing can be better illustrated in Fig. S6, showing transmission intensity at different voltages at a fixed 4.5 µm −1 wave vector value oriented in different directions: Fig. S6a for k x , Fig. S6b for k y , Fig. S6c for k d and Fig. S6d for k a . With wave vector along x and y directions are polarised accordingly to the main axes of LC molecules as shown in Fig. S6e,f presenting intensity difference between X-polarised transmission intensity (I X ) and Y -polarised intensity (I Y ). At those directions modes crosses each other. Detection along the diagonal and antidiagonal directions (Fig. S6c,d) reveals coupling between the modes observable as anticrossing behaviour. For these wave vector orientations there is significant difference between intensity detected in diagonal (I d ) and antidiagonal (I a ) linear polarisations as presented in (Fig. S6g,h). Experimentally observed results are in a good agreement with Berreman matrix simulations shown in Fig. S6i-l.

IV. MERON ORIENTATION AND SIZE
Size and orientation of the meron polarisation texture depends on the exact properties of a given LC microcavity. Fig. S7 Fig. S7d-f. With varying birefringence both spatial size and orientation of the second order meron polarisation texture changes, as summarised in Fig. S7g. With increasing birefringence meron texture rotates clockwise with the steepest change when ∆n is close to zero. Low optical anisotropy of the LC layer results also in increasing size of the meron texture. Due to low light intensity far away from the excitation spot simulation range is limited to ≈ ±100 µm.
Size and orientation of the meron textures depends also on the energy position of the mode within the photonic stopband region of the DBRs, which is summarised in Fig. S8. Calculations were performed for analogous cavity as in Fig. S7, with ∆n = 0. Energy of the mode is changed in simulations by adjusting thickness of the LC layer filling the cavity by −300 nm to 350 nm from initial value 2437 nm resulting in a cavity resonance at central wavelength λ 0 . Such thickness range allows to tune cavity mode energy by ≈ 0.3 eV, as shown in the angleresolved reflectance spectra in Fig (Fig. S8g) follows qualitatively the same dependence as when varying the birefringence shown previously in Fig. S7g.

V. EFFECTIVE HAMILTONIANS FOR COUPLED X AND Y POLARISED MODES
The eigenmodes inside the cavity are represented by plane waves propagating in the plane of the cavity per-pendicular to the z axis: The vector E k can by found from the following effective wave equation in the birefringent medium characterised by a dielectric tensor ij : where k = k = (k x , k y ) and k 0 = ω/c. Assuming that xy = yx = zy = yz = 0, we have up to the second order in k x and k y : Here, yy = n 2 o and for the given angle θ between the director of the LC molecules and the x axis we havẽ and xz = zx = (n 2 e − n 2 o ) sin θ cos θ. We wish to find the approximate dispersion relations of modes almost perfectly confined between the mirrors. Therefore the electric field is expanded as follows: where the basis states: with n = 1, 2, 3 . . ., correspond to the electric field polarised parallel to the x and y axis, respectively. In this representation the matrix elements: At k x = k y = 0 we have simple modal solutions with the electric field E x,n = |X, n polarised along the x axis with frequency ω Xn = ck Xn = cπn/(Ln ef f ) and E y,n = |Y, n modes polarised along y direction with ω Y n = ck Y n = cπn/(Ln o ). The degeneracy of modes occurs when ω Xn ≈ ω Y n ≈ ω 0 = (ω 2 Xn + ω 2 Y n )/2. In order to find the approximate dispersion relation for frequencies in the vicinity of ω 0 we solve the system of linear equations for expansion coefficients f sn : (Ŵ ) sn,s n f s n = 0 (12) where (Ŵ ) sn,s n = π 2 L 2 n 2 δ ss + (B 1 ) ss − k 2 0 (B 0 ) ss δ nn + sn|Â∂ z |s n . (13) In the matrix form we have: Note that the last term in Eq. (13) is linear in k so the coupling of modes of different parity can be treated perturbatively. In particular, when the degenerate modes are of the same parity, for example n = n = N or n = N and n = N + 2, this last term will lead to the correction of the second order and higher in k. In order to see this we can introduce the projection operatorP on the modes of the same parity as N (P -parity), andQ -the projection operator on the modes of opposite parity (Q-parity).
Then of course f =P · f +Q · f where the first term constitutes the dominant part of f and the other represents the admixture from the states of opposite parity. Since we are interested mainly in the dispersion relation, we are looking for the solution for the dominant partP · f : The matrixQŴQ is limited to the subspace of states withQ-parity and so is its inverse (QŴQ) −1 . To the lowest (zeroth) order in k: The matrixQŴP which couples modes of different parity has the form: (QŴP ) sn,s n = (Â) ss 4nn The electric field in the vicinity of the degeneracy point can be approximated by: 2) In the case of degeneracy of two modes of different order but the same parity the mixing term is different from zero so the effective equation where the prime over summation sign means that only m with parity different from the parity of n and n which is the same as the parity of N are included. In this way the denominator is always different form zero. Approximating we obtain the following equation for f in the case of the resonance of modes of the order N + 2 and N : Note that the effective equations in the vicinity of the resonance of modes of the same order (N, N ) [eq. (19)] and for the case of different orders, (N, N + 2) [eq. (22)] have similar structure. However the origin of the term proportional to k x k y , which is responsible for coupling between the modes is different in each situation. In the (N, N ) case we have a direct coupling between the TE and TM modes whereas the coupling between modes of different order is of indirect character and is mediated by modes with opposite parity. By standard manipulations, both equations can be transformed into an eigenvalue problem with a Hamiltonian presented in the main text.

VI. SPIN STRUCTURE AND MERON ORIENTATION FROM MOMENTUM-SPACE HAMILTONIAN
The meron and antimeron spin structure results from transmission of light through cavity modes, which can be approximately described with Hamiltonian (2) in the main text. The emergence of such structures and the topological charge Q can be predicted from the eigenmodes of the Hamiltonian taking into account that the system is excited with resonant laser light with a Gaussian envelope in space. In Fig. S9 we show the spin polarisation of one of the the Hamiltonian eigenmodes in the meron (N, N ) and antimeron (N + 2, N ) case. The shaded ring in momentum space corresponds to the approximate area excited with resonant light, which results from the parabolic dispersion relation of the cavity (see Fig. 2 in the main text). The second order meron spin structure of can be observed on this ring, and is retained after performing Fourier transform into real space, assuming that the excitation laser beam is Gaussian-shaped.
This simple explanation, however, is incomplete as it neglects the second, orthogonal eigenmode and does not explain the meron rotation angle discussed in the previous section. To take into account the second mode, we estimate the amplitude and polarisation of light transmitted through microcavity. The amplitude of input light can be written as where A(k) is a Gaussian shaped amplitude, A(ω) is ap-proximately δ-shaped laser frequency spectrum, u in is the polarisation of input light, e.g. in linear polarization basis u in = (1, 0) T for a horizontally polarised light. In the considered cases (N, N ) and (N + 2, N ) the cavity acts as a full-wave plate, so the polarisation of cavity mode at the output is not rotated by the cavity. The output amplitude is where we approximate the cavity transmission coefficient t as a sum of two eigenmodes i = 1, 2, each corresponding to a peak in transmission t i (k) with a similar amplitude and a Gaussian shape. The operator P (u in , u i ) = u in · u i is the projection of input light polarisation on the eigenmode of the Hamiltonian (2) polarisation. The shape of t i (k) in momentum space is ring-like for each mode, with slightly differing radii. This results from the parabolic dispersion relation of the in-plane photonic cavity modes as shown in Fig. 3 in the main text. Calculations of the above simplified Hamiltonian model are compared with Berreman method simulations in the case of (N + 2, N ) antimeron with σ + excitation in Fig. S10. The approximate 45 degrees orientation of the antimeron results from the overlap of the two rings in momentum space, with the phase of the transmission coefficients t i differing by π/2. Such phase difference is explained by the dependence of the phase of the transmission coefficient on transverse momentum. This additional phase shift leads to rotation of input circular polarisation into horizontal or vertical polarisation in the diagonal directions (k x = ±k y ), which results in the whirling polarisation structure in momentum space and the corresponding rotation of the meron orientation.