## Introduction

The creation and recombination of excitons, electron–hole pairs bound by Coulomb forces1, mediates light–matter interaction in semiconductors. The exciton transition dipole moment can be highly anisotropic, with the dipole’s strength and orientation dictated by the particular electronic properties of the host semiconductor and its dipole selection rules2,3. This fundamental relationship implies that different low-dimensional semiconductor structures with varying electronic properties can be found to yield in-plane (IP), out-of-plane (OP) or mixed dipole orientations. Further, beyond purely electronic effects, an anisotropic crystal shape also induces additional dielectric effects, which affect the intrinsic transition dipole moment4.

The orientation of radiative excitons determines their potential in optoelectronic and integrated photonic applications. IP dipoles are more easily accessed in experiments, but emerging applications in integrated photonic chips can be enabled with OP orientations to efficiently mediate light–matter interactions with planar and cylindrical waveguides5,6,7,8,9,10. Recently, two-dimensional (2D) transition metal dichalcogenide (TMD) semiconductors, which sustain robust excitons, have emerged11,12, presenting new opportunities to engineer light–matter interaction at the nanoscale. Typically, these 2D semiconductors present bright luminescent intra-layer excitons with IP transitions, which favour directional outcoupling of radiation, as has been demonstrated in MoS2 at room temperature13. In TMDs, OP transitions can only be found in weakly luminescent (the so-called grey) excitons14,15,16 or engineered structures such as chemically transformed Janus monolayers (MLs)17,18 or interlayer excitons in type II heterostructures, which have small oscillator strength19,20,21,22.

In contrast to TMDs, layered III–VI chalcogenides have a band-gap defined by intra-layer electronic states with a major pz atomic orbital composition. This endows InSe23,24,25,26, GaSe27,28 and their heterostructures28 with large quantum-confinement effects: as the layer thickness varies from the bulk to ML, the band-gaps tune from the infrared to violet. The large band-gap tunability can be exploited for optoelectronic applications29,30,31, while the broken inversion symmetry of InSe crystals leads to strong optical nonlinearities even at the atomic thickness limit32. Narrow quantum dot-like emissions that have been observed in both GaSe33,34 and InSe35 offer promise for future quantum photonic devices. Electrons in the conduction band of InSe have small effective mass36,37 and weak electron–phonon scattering38, which combine to yield high electron mobility, in particular in devices encapsulated by hexagonal boron nitride39. These features make InSe uniquely attractive among the wide array of layered van der Waals materials. Additionally, the pz atomic orbital nature, combined with the symmetry of the valence and conduction band states at the band-gap of layered InSe, may naturally provide for excitons with large oscillator strengths and large OP dipole orientation23,24,25,26,39,40.

Recent experimental works have attempted to demonstrate the OP dipole orientation of InSe26,41. Unfortunately, those experiments do not allow unambiguous observation of the OP dipole orientation. The strong effects of strain42 and dielectric-induced luminescence enhancement43,44 arise, which can be competing explanations. More importantly, the concept behind these works does not allow a quantitative determination of the exact dipole orientation. Here we report the far-field photoluminescence (PL) intensity distribution from ML WSe2 and MoSe2, representative of the TMD semiconductor family, and InSe samples of varying thicknesses down to the quantum-confinement regime. We find that the room-temperature PL from WSe2 and MoSe2 originates exclusively from IP dipoles, which indicates that grey excitons in these semiconductors (with OP dipoles) do not contribute significantly to the room-temperature emission. This result contrasts with the reported low-temperature PL of W-based TMDs such as WSe2 and WS2, in which OP dipoles contribute significantly to the emitted PL signal14. In contrast to 2D TMDs, our experimental results reveal that layered InSe flakes sustain luminescent excitons with an intrinsic OP orientation. We perform ab initio calculations of the electronic band structure of bulk and 2D InSe to reveal that the dipole orientation is due to the behaviour under the inversion symmetry operation of the electronic states, unique to the III–VI semiconductor family. Our results represent an unambiguous quantification of the orientation of the emission dipole of InSe flakes with different thicknesses. The experimental technique used in this work allows us to disentangle the dipole orientation from several additional effects that have prevented a reliable and quantitative determination of the dipole orientation of InSe to date.

## Results

### Orientation of luminescent excitons in bulk InSe

Figure 1a summarises the experimental concept used to determine the three-dimensional orientation of luminescent excitons in the van der Waals materials we investigate. The far-field PL emission pattern at room temperature is recorded by collecting the emitted photons with a microscope objective and imaging the intensity distribution in the objective’s back-focal plane on a charge-coupled device (CCD) camera. The inclusion of a linear polariser between the objective and the CCD allows separation of s- and p-polarised wave components to the back-focal plane (k-space). The numerical aperture (NA) of the objective was 0.95. A more detailed scheme of the experimental set-up can be found in Supplementary Fig. 1. This experimental technique has been extensively used to determine the radiation patterns of different fluorescent systems such as single molecules45 and single photon emitters46,47,48. More recently, back-focal-plane imaging has also been employed to resolve the orientation of luminescent excitons in 2D semiconductors such as MoS213 and CdSe nanoplatelets4,49,50.

Figure 1b shows the normalised k-space emission pattern of a mechanically exfoliated 90-nm-thick InSe bulk flake deposited on top of a SiO2/Si substrate with a SiO2 thickness of 105.3 ± 0.1 nm, as measured by nulling ellipsometry (see Methods). The vertical (ky/k0) and horizontal (kx/k0) axes represent the orthogonal components of the in-plane photon wavevector (k0 sin (θ), with θ being the emission angle) normalised to the photon wavevector in air (k0). Radiation with an in-plane wavevector larger than the NA of the objective is not collected by the objective (black region). The white arrow in the top right corner indicates the orientation of the transmission axis of the linear polariser used during the experiment. The vertical and horizontal cross-sections of the experimental k-space emission pattern are plotted next to the CCD image, which correspond to directions parallel () and perpendicular () to the polariser, respectively. Consequently, and cuts in Fig. 1b can be referred to as p- and s-polarised emissions, respectively. Dashed lines indicate the selected regions of the corresponding and cuts.

In order to determine the intrinsic emission dipole orientation of excitons in InSe (i.e. the IP-to-OP ratio) and to disentangle the effects that the dielectric multilayer environment has on the k-space emission intensity, we employ the analytical model proposed by Benisty et al.51 to model the k-dependent emission profile of bulk InSe. This model has been recently applied to simulate the back-focal plane emission for both the PL and the Raman emission of different 2D materials44 (see Methods). Figure 1c shows normalised k-space emission patterns calculated for pure IP (left panel) and pure OP (right panel) distributions of dipoles, respectively, emitting at an energy of ~1.244 eV (see Fig. 2a) and distributed all along the thickness of a 90-nm-thick InSe flake deposited on top of a SiO2/Si substrate with a SiO2 thickness of 105.3 nm. In the calculations, the effect of a detection polariser oriented along the ky direction has been included in order to reproduce the experimental conditions in Fig. 1b. As observed in these figures, the calculated k-space patterns for pure IP and pure OP distribution of dipoles are very different from each other. First, an overall higher emission intensity is observed for the k-space pattern corresponding to a pure IP distribution of dipoles, which can be understood by the larger emission outcoupling of IP dipoles in comparison to OP ones. Second, the k-space emission pattern for a distribution of pure OP dipoles shows a dark axis along the direction perpendicular to the polariser, which can be understood by the fact that a pure OP dipole does not emit s-polarised light. Consequently, only p-polarised emission contains information about both IP and OP dipoles. The presence of such a dark axis in the k-space pattern after a detection polariser represents a characteristic signature of a pure OP emission dipole.

Blue and red dashed lines in the left and bottom panels of Fig. 1b represent the results of a simultaneous fit of the cross-sections of the experimental k-space pattern to the analytical model described in Methods. The fits reveal a 97% OP and 3% IP intrinsic emission dipole distribution in bulk InSe, which demonstrates the nearly complete OP emission characteristics of luminescent excitons in bulk InSe.

### Orientation of luminescent excitons in 2D InSe

In order to investigate whether 2D InSe retains the OP emission characteristic of the bulk material, we carried out measurements of the k-space emission pattern for an 8-nm-thick InSe flake. Figure 2a shows a comparison of the intensity-normalised room-temperature PL of bulk and 8-nm-thick InSe. A blueshift of ~5 meV in the central emission energy is observed for the 2D InSe flake, a consequence of the quantum-confinement-induced increase of the electronic band-gap of InSe with reducing thickness24,25,26, which confirms the 2D nature of the studied flake. Figure 2b shows the measured normalised k-space emission pattern of the 8-nm-thick InSe flake deposited on top of a SiO2/Si substrate with a SiO2 thickness of 104.0 ± 0.1 nm. The labelling in this figure is the same as the one employed in Fig. 1b. We observe that the s- and p-polarised k-space cross-sections for this sample (bottom and left panels of Fig. 2b, respectively) are much more similar to each other than for the bulk sample (see Fig. 1b). The origin of this behaviour lies in the angular redistribution of the emitted light induced by the different thicknesses and emission wavelengths of the bulk and 8-nm InSe flakes52. A simultaneous fit of the measured k-space intensity profiles to the analytical model described in Methods reveals a 95  ± 1% OP intrinsic emission dipole distribution in the 8-nm-thick 2D InSe flake. The shadowed regions around the calculated values represent a confidence interval of 1% for the dipole orientation. These results demonstrate that 2D InSe retains the nearly complete OP dipole emission characteristics previously demonstrated for the bulk counterpart. Finally, the polar plot in Fig. 2c shows the measured and calculated far-field emission patterns for the studied InSe flake as a function of the emission angle θ. The black dashed lines indicate the maximum collection angle (θmax) = 71.8° provided by the the objective with NA = 0.95 used in our experiments.

The orbital nature of the electronic states at the band-gap edges makes 2D InSe a valuable building block in the design of van der Waals heterostructures with tailored optoelectronic properties. On the one hand, the strong pz orbital nature of the top valence band endows InSe with one of the largest band-gap tunability ranges found in a 2D semiconductor when its thickness varies between the bulk and the ML. On the other hand, electronic states of pz orbital nature are even under mirror symmetry operation; combining these states with the odd symmetry of the bottom of the conduction band determines the OP orientation of the luminescent excitons in this 2D semiconductor.

### Role of grey excitons in the emission of 2D MoSe2 and WSe2

Such an OP emission characteristic contrasts with the strongly IP-localised excitons of other 2D semiconductors such as 2D MoS213, 2D black phosphorus53 and 2D CdSe nanoplatelets4,49,50. For 2D MoS2, it has been demonstrated that the room-temperature PL originates solely from in-plane excitons13. Consequently, the PL emission of ML MoS2 can be described by an isotropic distribution of incoherently radiating dipoles lying in planes parallel to the layer interfaces. The analogy of the nature of the orbitals involved in the optical band-gaps of other Mo-based 2D TMDs such as MoSe2 suggests analogously strong IP localisation of luminescent excitons in this 2D van der Waals semiconductor at room temperature. Supplementary Fig. 2 in the Supplementary Note 1 shows the experimental and calculated k-space emission pattern of ML MoSe2 deposited on top of a gold substrate. Fitting of the experimental data to the analytical model reveals a pure IP dipole orientation also for 2D MoSe2.

Similar to 2D MoSe2, it is usually assumed that the room-temperature PL emission of ML WSe2 also originates from purely IP dipoles, due to the analogy of the electronic band structures of group-VI TMDs. However, Mo- and W-based TMDs present spin–orbit split conductions bands with different sign at the ±K points of the Brillouin zone, which leads to an excitonic ground state with an IP transition dipole moment that is bright for Mo-based TMDs, and two energetically split dark exciton states for W-based TMDs with OP transition dipole moments54. The upper dark energy state (grey exciton) is electric dipole allowed for electric fields polarised along the z-direction, while the low energy state is strictly dipole forbidden (perfectly dark)16,55. Low-temperature PL measurements of ML WSe2 and WS2 with high NA objectives have shown that grey excitons contribute significantly to the emitted PL signal14, making the low-temperature PL of these semiconductors originating not exclusively from IP dipoles but from a combination of IP and OP dipoles. However, a quantitative and unambiguous determination of the contribution of the grey excitons to the room-temperature PL of W-based TMDs is still missing.

Therefore, we next experimentally quantify the contribution of OP dipoles to the room-temperature PL of ML WSe2. Figure 3a, b show the experimental k-space emission profiles of ML WSe2 flakes deposited on top of SiO2/Si substrates with SiO2 thicknesses of 104.0 ± 0.1 and 304.0 ± 0.1 nm, respectively, for directions (blue solid line) and (red solid line) to the collection polariser. Due to interference effects, the k-space emission profiles are strongly affected by the SiO2 thickness44. Dashed lines in Fig. 3a, b represent the result of a simultaneous fit of the experimental k-space profiles to the analytical model described in Methods. Figure 3c, d show the measured and calculated far-field emission patterns shown in Fig. 3a, b, respectively, as a function of the emission angle θ. As can be seen in these figures, a very good agreement is observed between the experimental and the calculated values. The fits reveal a 100% IP intrinsic dipole distribution for ML WSe2 deposited on both substrates. These results demonstrate the validity of the analytical model employed in this work to unravel the intrinsic emission dipole orientation of InSe. The good agreement between the experimental and calculated values observed in Fig. 3 proves the capability of the employed analytical model to disentangle the dipole orientation from the angular redistribution of emitted light induced by the underlaying multilayer structure. More importantly, the results shown in Fig. 3 indicate that the contribution of the grey exciton states (with OP dipoles) to the room-temperature PL of ML WSe2 is negligible, since the far-field emission patterns can be reproduced with a distribution of pure IP dipoles. These results are supported by very recent theoretical calculations56.

### Electronic nature of the vertical dipole in InSe

In order to explain quantitatively the orientation of luminescent excitons in few-layer InSe, we have performed ab initio calculations using density functional theory (DFT) within the local-density approximation as implemented in Quantum Espresso57. We have included the spin–orbit interaction with fully relativistic norm-conserving pseudopotentials58. The details of the converged parameters are: energy cutoff 90 Ry; k-point sampling 12 × 12 × 1 in ML and 15 × 15 × 10 in bulk. In our calculations, we have assumed that the thickness of a ML InSe is, nominally, 0.833 nm59.

Figure 4a, b show the band structure of ML and bulk InSe, respectively, obtained without spin–orbit coupling. Regarding the ML InSe band structure, we can recognise the singular camel back around the Γ point25. We also indicate the symmetries of the bottom of the conduction band state (A2) and the two upper valence band states ($$A_1^\prime$$ and E″), which are key to identify allowed optical transitions. The selection rules for the dipole operator establish that, without spin–orbit splitting, transitions from the top of the valence band to the bottom of the conduction band are only allowed for z-axis (OP) polarised light. In the case of (x, y)-axis (IP) polarised light, only the transition from the E″ to A2 is allowed26,40,60. Table 1 shows the group representation of the valence and conduction band states at Γ. The even (1) or odd (−1) character is an intuitive way to understand the selection rules. Thus, the product with the dipole operator is 1 (−1) for bright (dark) transitions. In the case of bulk InSe, the stacking order changes the symmetry of the crystal but keeps the same selection rules61.

Figure 4c, d show a detailed view of the valence band structure calculated with (solid lines) and without (dashed lines) spin–orbit interaction for ML and bulk InSe, respectively. The wave functions at Γ of the conduction band state A2(c1) and the valence band states $$A_1^\prime (v_1)$$ and E″($$v_2$$) are plotted in Fig. 4e–g for the ML case, which reflect their main In s, Se pz and Se pxpy orbital character, respectively. It is well known that spin–orbit interaction makes IP optical transitions weakly allowed in bulk InSe by mixing with deeper lying valence bands26. In fact, some mixing can be observed to occur (Fig. 4d) between the highest valence band (with pz character) and deeper valence bands (with pxpy character) when spin–orbit interaction is considered, although these bands are separated by ~1 eV from each other. In ML InSe, band-structure calculations predict that these two set of valence bands are ~0.3 eV apart from each other (Fig. 4c) and a stronger mixing of states between them is expected to occur. Consequently, our results reveal that the inclusion of spin–orbit interaction favours a mixing of valence band states with pz and pxpy orbital character that depends on the thickness of InSe, partly relaxing the restricted panorama of IP/OP-allowed optical transitions in InSe described above (Table 1).

In order to quantify the orientation of the luminescent excitons, we have also computed the optical absorption including excitonic effects, within the framework of the Bethe–Salpeter Equation (BSE), as implemented in the Yambo code62,63, with the aim to assure an accurate determination of the IP/OP character of excitons (see Supplementary Figs. 4 and 5 and related discussion in Supplementary Note 3). At first approximation, we can consider the emission proportional to the absorption. Figure 5a shows the absorption of ML, bilayer, eight-layer and bulk InSe for IP (σx) and OP (σz) polarised light, with (solid lines) and without (dashed lines) spin–orbit interaction. Consistent with the previous discussion, optical absorption processes are either IP-allowed or OP-allowed in InSe without spin–orbit interaction. Spin–orbit interaction makes optical absorption processes involving states at the conduction and valence band edges slightly IP-allowed. Our calculations reveal that, when spin–orbit interactions are considered, a 2% IP absorption can be expected at the band-gap energies (Fig. 5a), close to the value of 3% obtained experimentally from our results in Fig. 1b. A similar situation is predicted to occur for InSe samples with important quantum-confinement effects. In particular, for 2D InSe of 8-layers thick, the IP contribution goes up to 4.7%, which is also in very good agreement with the experimental increase of the IP contribution observed for thin InSe layers (5% for 8-nm InSe; Fig. 2b). In general terms, the IP absorption at band-gap energies becomes larger when the number of layers decreases. The band-gap is underestimated within the local-density approximation, and we have applied a scissor operator to match the experimental values. However, the GW correction just produces a rigid shift without implications in the spin–orbit interaction and the selection rules for IP and OP light64. Nevertheless, it is worth noting that for the absorption at the limit of a ML or bilayer, other effects like interaction with the substrate can also have a strong effect.

We also show the excitonic wave function of the first excitonic state of ML InSe in Fig. 5b, c. We can appreciate the strong localisation in plane and the electronic density surrounding the selenium atoms. The consequence is a strong dependence on the number of layers due to the coupling of the wave functions of different layers25. In the case of bilayer InSe, shown in Fig. 5d, the wave function spans over the different layers, supporting our hypothesis of the strong interlayer interaction.

It is worth mentioning that alternative calculation methods based on DFT cannot determine the IP/OP exciton absorption ratio since DFT calculates optical spectra only within the independent-particle approximation excluding excitonic effects, although DFT can provide for an accurate estimate of the exciton binding energy. In contrast to this, the IP/OP absorption ratio, as estimated by BSE, is not changed due to excitonic effects. However, one of the drawbacks of the BSE is the convergence of the exciton binding energies with the number of k-points. While this convergence is not critical in 2D materials like ML or few-layer InSe, systems such as bulk InSe require a very large number of k-points. The convergence studies shown in Supplementary Figs. 4 and 5 show the impact that a particular choice of the number of k-points has on the IP/OP absorption ratio and the binding energy for excitons in bulk InSe, respectively. These studies show that, although the exciton binding energy of bulk InSe converges very slowly with the number of k-points, the IP/OP exciton absorption ratio is already converged with the k-grid employed in our calculations (15 × 15 × 10), validating our results within the BSE framework.

## Discussion

The strong IP localisation of room-temperature luminescent excitons in some 2D semiconductors such as 2D MoS213, MoSe2, WSe2, black phosphorus53 or 2D CdSe nanoplatelets4,49,50 is well suited for vertically emitting devices due to the large outcoupling efficiency of the radiation emitted by IP dipoles. In contrast, the IP orientation of the transition dipoles of these 2D semiconductors is not convenient for some planar photonics applications and for the study of light–matter interactions where dipole orientation is critical (such as dipole-dipole correlations). For some photonics applications it is instead desirable to have semiconductors with an OP orientation of the transition dipoles that benefit at the same time from the integration capabilities offered by the van der Waals platform. Examples of photonic applications in which OP dipoles outperform IP ones can be found for radiating dipoles embedded inside or placed at the surface of dielectric5,65, semiconductor6 and metal-cladding optical waveguides7 with planar symmetry. In such planar photonics applications, OP dipoles not only present a higher coupling efficiency to the waveguide modes but also show a higher enhancement of the spontaneous emission rate (Purcell factor) than IP dipoles. Finally, the advantages of OP dipoles over IP ones for photonics applications are not restricted to planar waveguides. OP dipoles present higher Purcell enhancements and couple more efficiently than IP ones to multimode cylindrical waveguides and optical fibres when placed in the vicinity of the waveguides8,9,10.

Our results show that the van der Waals semiconductor InSe fulfills the aforementioned properties. By performing k-space imaging of the far-field PL signal of InSe flakes of different thicknesses, we demonstrate that not only bulk InSe but also 2D InSe flakes present dipole-allowed transition dipoles with a large intrinsic OP orientation. These findings are supported by ab initio calculations of the electronic band structure and the absorption coefficient of both bulk and 2D InSe. These results, together with the large band-gap tunability and the high electron mobility offered by 2D InSe, position the 2D forms of this semiconductor as valuable building blocks in the design and fabrication of van der Waals heterostructures with tailored optoelectronic properties that can be harnessed in the next generation of photonics applications.

## Methods

### Sample preparation

The InSe and WSe2 flakes studied in this work were obtained by employing the mechanical exfoliation technique and then transferred on top of SiO2/Si substrates. The thickness of the SiO2 layers and the bulk InSe flake used in this work was determined by means of nulling ellipsometry measurements using a Nanofilm EP4 ellipsometer from Accurion. The ML thickness of the WSe2 flake used in our experiments was confirmed by means of room-temperature PL measurements (see Supplementary Fig. 3 in Supplementary Note 2). In a similar way, the thickness of the 2D InSe flake studied in this work was confirmed to be ~8 nm by means of room-temperature PL, since it has been previously reported that the room-temperature PL emission energy of InSe can be used to efficiently determine the thickness of InSe flakes ranging between 1 and ~14 nm24,26.

### k-space spectroscopy

k-Space spectroscopy (or back-focal-plane imaging) was used to determine three-dimensional orientation of luminescent excitons in InSe. This technique allows to measure the far-field intensity distribution of PL, that is, the PL signal as a function of the emission angle. For this, we used the back-focal plane imaging set-up described in Supplementary Fig. 1. A 532-nm continuous-wave excitation laser is collimated and reflected towards the sample with a 10% reflective plate. A microscope objective with a NA = 0.95 then focuses the excitation laser on the sample in a diffraction limited spot. The PL emission from the material is then collected by the same microscope objective and collimated in the main path towards the collection optics and instruments. The two lenses in the main path focus the image of the back-focal plane of the microscope objective on a CCD camera, while keeping the beam collimated. A flip mirror in front of the CCD can be lifted to send the PL beam to a fibre and a spectrometer.

To obtain a reliable result, the far-field measurement corresponding to the 2D material is performed behind a polariser, which is rotated over 180°, with a k-space pattern acquired every 10°. The sample is then moved to image an area with bare substrate and the measurement is repeated without the 2D material to obtain a background measurement. After subtraction of the background, the 19 images obtained are rotated and their average weighted by their integrated intensity is calculated. This process is used to mean out any possibly existing anisotropy in the polarisation response of our set-up, and also to mean out the imperfections and defaults present on the optics, to which the measurement is very sensitive.

### Model analysis of dipole emission

We calculate the k-dependent emission profile of the different 2D materials by employing the analytical model proposed by Benisty et al.51, which has been recently applied to simulate the k-vector-dependent emission intensity for both the PL and the Raman emission of different 2D materials with very good results44. This model is based in the combination of a transfer matrix method and a dipole emission source term, and allows calculation of the emission of a thin source plane in an arbitrary planar structure in which the source plane is modelled by incoherent electrical oscillating dipoles. In this formalism, the k-space intensity profiles measured for directions perpendicular () and parallel () to the collection polariser can be calculated as:

$$\tilde I^ \bot \left( {k_\parallel } \right) = C\alpha \tilde I_{{\mathrm{IP}}}^s\left( {k_\parallel } \right)$$
(1)

and

$$\tilde I^\parallel \left( {k_\parallel } \right) = C[\alpha \tilde I_{{\mathrm{IP}}}^p\left( {k_\parallel } \right) + \left( {1 - \alpha } \right)\tilde I_{{\mathrm{OP}}}^p\left( {k_\parallel } \right)],$$
(2)

respectively. In these equations, C is a proportionality constant that depends on different experimental parameters (such as integration time and excitation intensity), α represents the ratio between IP and OP emission dipoles, k is the in-plane photon momentum (k = k0 sin (θ)) and the s and p superscripts denote TE and TM polarisations, respectively. Terms $$\tilde I_{{\mathrm{IP}}}^{s,p}\left( {k_\parallel } \right)$$ and $$\tilde I_{{\mathrm{OP}}}^p\left( {k_\parallel } \right)$$ in Eqs. (1) and (2) represent the k-space intensities calculated for distributions of purely IP and purely OP dipoles, respectively, in the corresponding experimental configuration. In our calculations, the effect of the excitation electric field has also been considered by following the formalism described in refs. 44,66. This formalism assumes the conservation of energy, and consequently, that the emission intensity of the dipole is proportional to the power dissipated by the corresponding dipole at the excitation energy. In the calculation of $$\tilde I_{{\mathrm{IP}}}^{s,p}\left( {k_\parallel } \right)$$ and $$\tilde I_{{\mathrm{OP}}}^p\left( {k_\parallel } \right)$$ we have also taken into account the influence of the thickness of the active material (important when dealing with thick flakes such as the bulk InSe flake used in our experiments). This was done by dividing the thickness of the active material into thin layers (sheets) of dipoles with thickness Δz (with Δz being the thickness of a ML for the corresponding material) and summing the emitted power from discrete subsources with the corresponding weighting coefficients.

At this point, it is worth noting that a close examination of Eqs. (1) and (2) reveals that only the k-space intensity profiles measured for directions parallel to the detection polariser contain information about the OP component of the intrinsic dipole orientation.

Finally, in order to consider the broad PL emission of the materials studied in this work (in contrast to a purely monochromatic or narrow-band emission case), monochromatic k-space intensity calculations were preformed for emission wavelengths covering all the spectral range observed in the measured PL spectra and were then combined as follows:

$$\tilde I_{{\mathrm{IP}},{\mathrm{OP}}}^{s,p}\left( {k_\parallel } \right) = \mathop {\int}\limits_{\lambda _{\mathrm{min}}}^{\lambda _{\mathrm{max}}} {C_{{\mathrm{PL}}}} \left( \lambda \right)I_{{\mathrm{IP}},{\mathrm{OP}}}^{s,p}\left( {k_\parallel ,\lambda } \right){\mathrm{d}}\lambda ,$$
(3)

where $$I_{{\mathrm{IP}},{\mathrm{OP}}}^{s,p}\left( {k_\parallel ,\lambda } \right)$$ is the k-space intensity calculated for a monochromatic emission wavelength λ and CPL(λ) is a normalisation constant calculated from the experimental PL spectrum (IPL(λ)) as

$$C_{{\mathrm{PL}}}\left( \lambda \right) = \frac{{I_{{\mathrm{PL}}}\left( \lambda \right)}}{{{\int}_{\lambda _{\mathrm{min}}}^{\lambda _{\mathrm{max}}} {I_{PL}} \left( \lambda \right){\mathrm{d}}\lambda }}.$$
(4)