Relaxing symmetry rules for nonlinear optical interactions via strong light-matter coupling


 Transition Metal Dichalcogenides (TMDCs) have been in the limelight for the past decade as a candidate for several
optoelectronic devices, and as a versatile test bed for various fundamental light-matter interaction phenomena thanks to their exceptional linear optical properties arising from their large binding energy, strong spin-orbit coupling and valley physics in the monolayer (ML) limit. They also boast strong non-linear properties fortied by excitonic responses in these systems. However, the strong second order non-linear responses are mostly restricted to the ML limit owing to crystal symmetry requirements, posing several limitations in terms of smaller interaction length and lower damage threshold. Here we demonstrate a self-hybridized exciton-polariton system in bulk WSe2 that allows us to relax the crystal symmetry rules that govern second order
non-linearities. The demonstrated polariton system shows intense Second Harmonic Generation (SHG) when the fundamental
wavelength is resonant with the lower polariton, with an efficiency comparable to the one from a ML WS2 when excited at
the same fundamental wavelength and intensity. We model this phenomenon by considering a system with alternating second-
order susceptibilities under an asymmetric electric field profile determined by the polariton mode. Helicity resolved polarization experiments show very similar non-linear response as the one from a ML where the helicity of the SHG 
flips with respect to the fundamental harmonic. This polaritonic system offers a platform to leverage robust second order non-linear response from centrosymmetric systems, while at the same time allowing access to third-order non-linearity inherent in strongly coupled
systems.


Introduction
Transition Metal Dichalcogenides (TMDCs) are Van der Waals(vdW) materials that exhibit a vista of much sought after optical and electronic properties. They have been studied extensively in the monolayer (ML) limit due to their direct bandgap, large exciton binding energy, valley properties, and oscillator strength [1][2][3][4]. Due to these properties, strong coupling between cavity photons in microcavities and excitons in ML TMDCs can been demonstrated at room temperature [5]. In addtion in such a strongly coupled system, the polaritons retain the valley excitonic properties of the host material [6][7][8]. More recently, there has been interest in polaritons formed in bulk TMDCs due to their high refractive index, which enables Fabry-Perot modes to be sustained in TMDC slabs which strongly couples with the exciton modes of the bulk TMDC [9][10][11]. This coupling results in a self-hybridized system where the bulk TMDC itself provides both the photonic and excitonic components required for strong coupling, without the need for an external cavity.
In addition to their exceptional linear optical properties, there have also been numerous reports on their nonlinear optical response. Second harmonic generation (SHG) can be observed in TMDCs owing to the lack of inversion symmetry in the ML and odd layer limit [12][13][14][15].More recently, SHG in TMDCs has emerged as a powerful spectroscopic tool to characterize the layer number as well as the crystal orientation, the latter being an important parameter for achieving precise twist angle in vdW heterostructures [16][17][18][19]. In addition to intense SHG, other second order susceptibility χ (2) mediated processes like sum frequency generation at continuous wave pump,optical parametric amplification and signatures of spontaneous parametric down-conversion have also been realized in these systems [20][21][22]. These nonlinear optical responses can be further enhanced by engineering lightmatter interactions with nanophotonic tools. Since 2D TMDCs show high values of nonlinear susceptibility, (e.g.,two-photon absorption in MoS 2 is 4 orders of magnitude larger than III-V group semiconductors such as GaAs) they represent an ideal platform for nonlinear optical applications [23]. Despite this advantage, 2D TMDCs suffer from the limitation that second-order nonlinear response is limited to the ML/odd few layer limit, which proves to be a challenge for device integration. Several attempts have been put forward to increase the interaction time via auxiliary systems where a ML or few-layer TMDCs couple to plasmonic or photonic modes in passive [24][25][26] or active [27] nanostructures.
The restoration of inversion symmetry in bulk TMDC crystals with naturally occurring 2H stacking, prohibits second order non-linearity under the electric dipole approximation, consequently creating a limitation in terms of the available interaction length. In this work, we achieve strong nonlinear χ (2) response, despite the presence of inversion symmetry, in self-hybridized exciton polaritons formed via strong coupling of Fabry-Perot modes sustained by a bulk ( 140 nm) WSe 2 and the exciton resonances of each layer. The existence of these modes creates a non-zero phase difference between the SHG signal generated in each layer, owing to the asymmetry of the electric field of the fundamental laser at the polariton mode, which when added coherently, does not cancel out in the far field. We compare the polarization response of this polariton system and find they exhibit similar behavior as their ML counterparts. A comparison between a ML TMDC system and these self hybridized polariton systems show that the latter is more efficient at lower powers than the former. The strong coupling assisted approach presented here allows to simultaneously use material systems where crystal symmetry dependent selection rules can be relaxed to recover latent non-linearity whilst leveraging the additional photonic control knobs and the inherent non-linearity of polariton fluids. [28]. Figure 1a shows the schematic of the structure used in this work, which consists of a bulk WSe 2 flake dry transferred on top of . We detect no discernible signal from the region outside the square box, and estimate a lower bound for the enhancement factor based on the detector noise levels. Inside the box the SHG signal is enhanced at least by three orders of magnitude. Figure 2b compares the SHG intensity as a function of detuning of the fundamental wavelength from the one at k = 0 . The laser has a linewidth of approximately 5nm and the polariton linewidth is 20nm. We find that the SHG signal is reduced by 80% for a laser-cavity detuning of 15 nm.

Results
To further characterize the nature of the SHG in the polariton system we resort to polarization dependent studies. We measure the linear polarization response of the SHG by rotating the plane of input linear polarization of the fundamental while keeping the bulk TMDC and an output analyzer stationary. Under this measurement configuration the D 3 h symmetry structure should give a four-fold symmetry [29]. See Supplementary Note 3 for details. The inset of Figure 3a. shows the fourfold symmetry as a function of the input polarization of the fundamental excitation. This gives distinctive evidence that the SHG originates from a crystal with a D 3 h symmetry group and it can be attributed to the residual phase that the SHG and the fundamental pick up due to multiple reflections within the structure, thereby producing a non-zero SHG in the far field. After confirming the crystal symmetry, we excite the sample with a circularly polarized fundamental. As seen in ML TMDCs, the polariton system also produces a circularly polarized SHG signal. Figure 3a shows that the measured SHG signal when excited with σ + fundamental. The helicity of the SHG signal is flipped from the one of the fundamental thereby confirming the D 3 h symmetry.
We define degree of circular polarization as ρ = | I σ + −I σ − I σ + +I σ − |, where I σ + is the intensity of right-handed circularly polarized light and I σ − is the intensity of left-handed circularly polarized light when analyzed through a quarter wave plate and linear polarizer at the output. After normalizing with the degree of circular polarization of the laser,we obtain a value of ρ = 0.87 for the case of bulk WSe 2 polaritons, which is comparable with the reported value ρ = 0.94 obtained for SHG from a ML WS 2 coupled to a passive photonic structure [25].
Next, we compare the efficiency of SHG from polaritons with a ML WS 2 that was transferred on 300nm thermal oxide on Silicon. WS 2 was chosen as an archetypal ML as it has one of the largest value of second order permitivity.The power dependent measurements in Figure 3b. show the ratio between the SHG signal from the WSe 2 self-hybridized polariton sample and a reference WS 2 ML which is pumped at the same wavelength. The plot can be divided into three distinct regimes of power. At low pump powers (region I), it can be seen that the efficiency of SHG from the polariton system is higher than the one from a  [30][31][32][33][34] A detailed study of this third order non-linearity is beyond the scope of this work.
We now examine the physical processes that give rise to such intense SHG from a center-symmetric material. One could posit that the SHG is generated from an odd layered system with the last layer generating an intense SHG owing to loss of inversion symmetry. However this would result in a broadband response of the SHG and not be limited to the polariton resonance, as shown in Fig. 2b. Moreover this strong fundamental wavelength dependence has been noted in several TMDC flakes of various thickness when they are pumped on and off the polariton resonance. Another possibility is the enhancement of the residual surface SHG rate due to an enhanced density of photonic states owing to a cavity mode that is resonant with either the SH or the fundamental wavelength. To investigate this possibility we perform FDTD simulations for a bulk material with a fixed χ (2) using a broadband fundamental source, where we replicate the configuration used in our experiment and obtain an enhancement of ≈ 16 times. See Supplementary Note 5 and Supplementary Figure S4b for more details. A similar enhancement has been reported in systems where a TMDC flake is coupled to passive resonant structures [24][25][26]. This passive enhancement factor however fails to explain the drastic enhancement of at least three orders of magnitude when the system is excited at the polariton resonance. To fully explain the experimental results we use a model with layers of alternating χ (2) [15]. Any crystal with inversion symmetry should possess zero second order non-linearity because the macroscopic polarization, p (2) (E) = −p (2) (−E) leads to vanishing of all even nonlinear susceptibility under the electric dipole approximation. This is indeed true for 2H stacking of even number of layers in 2D TMDCs in which the inversion symmetry is restored. This cancellation of SHG for an even number of layers can be envisioned macroscopically as shown in Fig. 4a, where two successive MLs have alternative signs of the induced second order polarization, i.e., χ (2) in the first layer is equivalent to −χ (2) in the next layer, leading to a cancelling of the SHG for even number of layers in the far field. However, recent reports have shown that despite the existence of geometrical inversion symmetry in some systems, e.g., bilayer WSe2, SHG can arise by breaking the inversion symmetry of the induced charges. If the field is asymmetric across different layers of the 2D TMDC, the nonlinear polarizations add coherently, generating SHG. This is depicted in Fig. 4b where the nonlinear polarizations represented by arrows, are in opposite directions, but not of the same magnitude [35] . Recent work has additionally shown that artificially stacking 2D layers with a controllable twist angle between the layers can lead to enhancing or suppressing SHG, described by a superposition relation shown in Fig. 4b but with the arrows now pointing along different directions [36]. For the given polariton system, if we consider a pump wavelength at the fundamental frequency ω impinging from the top, the electric field distribution is not symmetric. In this case the SHG from different layers does not cancel each other in the far field as shown in Fig. 4c thereby resulting in the SHG signal that is observed in the experiment. This effect is further enhanced by the higher photonic density of states at the polariton branches.
While this model captures the qualitative behavior of the SHG, it overestimates the value of the SHG response seen previously in a related work [15]. We classically model the enhancement in the nonlinear process by solving coupled linear equations at the fundamental frequency ω, and at the SHG frequency 2ω. To describe the SHG response of the system we model the WSe 2 as a mutlilayer system with layer thickness of 0.5 nm. The only non-vanishing susceptibility terms are, where y is the armchair direction of the crystal lattice. We assume in the simulation that the incident polarization is aligned with the armchair direction, giving rise only to SHG polarized in the armchair direction. We model the nonlinear polarization in each layer as, where l is the layer number, and χ (2) l is positive for odd layers, and negative for even layer counted from a fixed reference, and l,y is the field distribution in layer l at the fundamental frequency. We then use this polarization current as the source for the SH signal. We use this model to describe the SHG from a ML and the polariton system. Interestingly, using this simple model we can observe an enhancement of around four orders of magnitude between SHG from ML, and the SHG from bilayer, which perfectly matches our experimental results. The polariton system shows enhanced SHG at lower polariton resonance with SHG efficiency close to that of a ML. We attribute this large SHG in the cavity coupled system near resonance due to the combination of the field enhancement and its spatial profile, which leads to an asymmetric distribution of nonlinear polarization. At the same time in bulk WSe 2 outside the lower polariton resonance we found an almost negligible SHG owing to near perfect cancellation of the SH field of individual layers in the far field as shown in Fig 4d. The model used here does not invoke non-local effects and remain well within the dipole approximation. The analysis shows that in layered systems like TMDCs, SHG should be treated as a collective effect of its constituent non-centrosymmetric ML.
Finally we comment on the advantages of having a dielectric spacer layer. In the structure described in the text above, the use of air as a spacer layer allowed us to access narrower polariton modes in addition to the advantages mentioned earlier. However similar SHG responses can also be found in structures with PMMA spacers and a metal mirror. The resonances obtained in those systems are broader and have a lower enhancement due to their lower quality factor. We could do away with the bottom mirror and form self-hybridized polaritons in bulk TMDCs on glass or silicon substrates as reported in several works [9,27].
However in these systems we are limited to bulks of thickness ranging only about from about 60 nm to 100 nm. This is shown via various transfer matrix calculation for different device geometries in Supplementary Note 6 and Supplementary Figure 5.
In summary, here we have demonstrated a platform to relax crystal symmetry requirements for second order non-linear response in bulk TMDC crystals via formation of exciton polaritons. The fact that we achieve strong nonlinear responses in bulk TMDCs reduces challenges with fabrication and integration into passive photonic platforms. Although we focus solely on the SHG process in the current work other χ (2) processes like sum/difference frequency generation and optical parametric amplification can be explored in these systems. In addition, these polariton systems inherently possess a χ (3) response that can lead to saturation of absorption, tunable index of refraction and single photon non-linearity via polariton blockade, thereby forming a versatile platform for a plethora of applications in the field of non-linear photonics and quantum optics. Furthermore, the power dependent saturation of the SHG can be used to estimate polariton-polariton interaction strengths in systems where traditional techniques of measuring contrasts in resonant laser reflection as a function of power are not feasible. For the SHG measurements, a Titanium-Sapphire laser (Coherent Mira) was used to pump at the fundamental wavelength of 832 nm, while the SHG signal was detected through the Princeton Instruments Monochromator. A 500 nm short pass filter was used in the collection side to filter out the excitation beam. Linear polarization measurements were recorded by using a linear polarizer (LP) in the excitation path and a collinear LP in the collection path. For helicity resolved SHG measurements, the excitation circular polarization was determined by using an LP followed by a quarter-wave plate (QWP) to determine the left and right circularly polarized states of the excitation beam. In the collection path, an analyzer of a QWP followed by an LP was used to resolve the chiral response of the system.

Coupled Mode Theory
In this work, the nonlinear response of bulk WSe 2 is underpinned by coupling it to a photonic cavity in form of a Fabry-Perot resonator with its two reflective surfaces are a bulk WSe 2 , and the DBR as shown in Fig. 1a. The system represents two eigenmodes, represented by their respective frequencies ω a and ω b . Here, ω a represents the exciton resonance of the WSe 2 , and ω b represents the resonance frequency of the photonic cavity which can be actively tuned with changing its height h. For an external excitation, the photonic cavity mode is indirectly excited, and the bulk WSe 2 mode is excited directly from an impinging wave as shown in Fig. 1a. These dynamics can be captured by employing the coupled mode theory (CMT) to describe the coupled oscillators system. The two oscillators are the material resonance ω a and the photonic cavity mode ω b . The material resonance mode is coupled to the radiation incident from above with radiation rate γ r , while the cavity mode is coupled indirectly to the incident radiation. The two modes are coupled with coupling rate κ thus forming the half matter half-light particles when κ > γ r . The CMT governing equations are: where γ a and γ b are the absorption rates of the 2D TMDC, and the photonic cavity respectively while γ r is the radiative coupling rate. For oblique incidence, the material resonance ω a does not change, the cavity resonance changes as with c is the light speed, and k ⊥ is the normal wavenumber inside the cavity. We can solve the differential equations under slowly varying approximation and give the reflection as, Here, ω b0 = 2.23 × 10 15 rad/s, ω a = 2.55 × 10 15 rad/s, γ a = 0.015ω b , γ b = 0.0011ω a , γ r = 0.05ω a , κ = 0.004ω b , calculated to match the experimental results.