Introduction

Symmetry serves as a cornerstone in many areas of physics. Nonetheless, the disruption of this symmetry is a process known as symmetry breaking and introduces novel physics and potentially catalyses transformative breakthroughs in many device designs and performance1. Time-reversal symmetry is a fundamental symmetry characteristic and stipulates that a system’s response remains invariant to that of the original system after a time-reversal transformation2. Generally, time modulation3,4, nonlinearity5,6,7,8,9 and magnetic bias10 have been extensively utilized on different optical platforms to break the time-reversal symmetry, which leads to nonreciprocity. In addition to time-reversal symmetry breaking, another consequence of symmetry disruption is spontaneous symmetry breaking of two modes; this is a transition from a symmetric state to an asymmetric state11. Nonlinearity in optics establishes a distinctive framework for the examination of symmetry breaking12. Kerr nonlinear materials arise from third-order susceptibility \({\chi }^{(3)}\) and produce an intensity-dependent refractive index. This specific property facilitates the manipulation of the optical attributes, such as the resonance frequency of an optical structure embedded with the Kerr materials, by simply modulating the incident power. In the solitary mode, the refractive index and consequent phase shift are governed by the incident light intensity; this effect is known as the self-phase modulation (SPM). However, when two modes co-propagate in the same medium, one of the modes induces a refractive index change that affects not only its phase but also the phase of the other mode. This effect is known as cross-phase modulation (XPM)13,14,15,16. The SPM is the self-influence of an individual mode, whereas the XPM is the interaction modulation between different modes. Furthermore, for two co-propagating competing optical modes, the SPM and XPM factors of the system are different14. Spontaneous symmetry breaking primarily originates from the imbalance of phase modulation between these two modes, which experience different refractive index shifts within the confinement of the Kerr materials. Therefore, Kerr effect light-matter interaction dynamics can produce nonreciprocity as well as the spontaneous mode of symmetry breaking17.

To date, diverse spontaneous symmetry breaking events have been investigated in ring cavities, including clockwise and counterclockwise propagation18,19,20,21,22 and cavity solitons23,24,25,26,27,28. Recent experimental work has demonstrated the occurrence of circularly polarised splitting with linearly polarised input light in a single Kerr cavity above a threshold power, with one of the circularly polarised components being transmitted. Concurrently, in the splitting region, a new linear polarisation orthogonal to the initial input is generated29,30. Furthermore, Kerr-induced nonreciprocal transmission and bistability in coupled nanocavities have been theoretically investigated8. However, research on handedness polarisation splitting within Kerr-coupled cavities remains limited. In this study, we theoretically and experimentally examine the handedness polarisation splitting of a coupled heterostructure due to the Kerr effect. The spontaneous symmetry breaking for the incident light is theoretically predicted, results in the transmitted fields randomly diverging into either a right- or left-handed circular polarisation (RCP or LCP) dominant field, and produces an additional linearly polarisation output that is orthogonal to the incident light polarisation. This theoretical result is experimentally confirmed. This finding enables the optical manipulation of the output handedness by varying the incident power. The random splitting characteristics could provide novel applications in quantum optics. Additionally, the magnet-free structure potentially facilitates the progression of all-optical polarisation controllers and sensors.

Results

Theory of spontaneous symmetry breaking

The coupled Fabry–Pérot (cFP) nanocavities are based on a multilayer thin-film heterostructure. As depicted in Fig. 1, the structure consists of five layers, denoted as \(l1\) to \(l5\), with corresponding thicknesses \(L1\) to \(L5\), respectively. Within this structure, \(l1\), \(l3\) and \(l5\) are thin silver films. The non-Hermitian system is an open system that exchanges energy with its surrounding environment. For the cFP nanocavities studied here, the refractive index of silver plays a crucial role in the coupled modes within the nanocavities as well as in scattering/dissipation. Therefore, the proposed cFP nanocavities are non-Hermitian31,32,33. The second layer \(l2\) is a Kerr nonlinear layer, whose permittivity is defined by \({\varepsilon }_{2}={\varepsilon }_{l}+{\chi }^{(3)}{\left|E\right|}^{2}\), where \({\varepsilon }_{l}\) is the linear term, \({\chi }^{(3)}\) is the third-order nonlinear susceptibility and \(E\) denotes the local electric field. When the incident power is too weak to induce a substantial permittivity change (\(|\frac{{\chi }^{(3)}{|E|}^{2}}{{\varepsilon }_{2}}|\le {10}^{-4}\)), the nonlinearity is negligible; therefore, the system is in the linear regime. However, when the incident power is sufficiently high to induce a substantial permittivity change, the system transitions into a nonlinear regime. The fourth layer, \(l4\), is a passive dielectric layer. When a linearly x-polarised light beam with frequency \(\omega\) is incident into the system, from \(l1\), the heterostructure can be considered a coupled nanocavity (cavity \(a\) and cavity \(b\)) with resonant frequencies \({\omega }_{a}\) and \({\omega }_{b}\) due to the confinement of light in cavities \(a\) and \(b\). In fact, the coupling strength \(\kappa\) between the two cavities is primarily dictated by L334,35,36, and the decay rates, \({Y}_{a(b)}\), for the two nanocavities are mostly influenced by \(L1\) and L537. Conventionally, linearly polarised light can be viewed as the superposition of equal pumping of RCP and LCP fields; thus, the roles of SPM and XPM become critical in dictating the propagation states of light. Note that, the effect of XPM is twice as strong as that of SPM in this system22,29,38,39.

Fig. 1: Schematic of a multilayer thin film heterostructure.
figure 1

Each layer is denoted as \(l1\), \(l2\), \(l3\), \(l4\) and \(l5\) with thicknesses of \(L1\), \(L2\), \(L3\), \(L4\) and \(L5\), respectively. where \(l1\), \(l3\) and \(l5\) are silver layers, \(l2\) is the Kerr nonlinear layer and \(l4\) is the passive dielectric layer. A linearly x-polarised light beam, that can be viewed as the superposition of equal pumping of right- or left-handed circular polarisation (RCP and LCP) fields, is incident from \(l1\). An additional polarisation component, orthogonal to the incidence, appears in the output field due to the spontaneous symmetry breaking.

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The derivation of the electric field is described by nonlinear coupled mode theory. Kerr nonlinearity introduces a power-dependent resonant frequency shift through the characteristic power Pa40. By incorporating the joint effect of the SPM and XPM within the Kerr material on the polarisation and neglecting fast-time effects, the slow-time (\(\tau \)) equation of motion, with an input injected into cavity a, can be written as follows16,29,41,42,43:

$$\frac{{{\mbox{d}}}}{{{\mbox{d}}}\tau }\left(\begin{array}{c}\widetilde{{a}_{\pm }}\\ \widetilde{{b}_{\pm }}\end{array}\right)= -{i}\left(\begin{array}{cc}{\omega }_{a}-\frac{{\gamma }_{a}\left({\left|{\widetilde{{a}}_{\pm }}\right|}^{2}+2{\left|{\widetilde{{a}}_{\mp }}\right|}^{2}\right)}{{P}_{a}}-{{{{\rm{i}}}}\gamma }_{a} & \kappa \\ \kappa & {\omega }_{b}-{{{\rm{i}}}}{\gamma }_{b}\end{array}\right)\left(\begin{array}{c}\widetilde{{a}_{\pm }}\\ \widetilde{{b}_{\pm }}\end{array}\right)\\ -\left(\begin{array}{c}\sqrt{{\kappa }_{{ea}}}\,{f}_{\pm }\\ 0\end{array}\right)$$
(1)

where \({\gamma }_{a(b)}={Y}_{a(b)}/2\) and \({\kappa }_{{ea}}\) is the loss rate for the coupling between cavity \(a\) and the environment. \(\widetilde{a}(\widetilde{b})\) represents the field amplitude inside the cavity \(a(b)\), and \(f\) is the incident field, in which the subscript \(+(-)\) denotes the right- (left-) handed circular polarisation. The term \(({|\widetilde{{a}_{\pm }}|}^{2}+2{|\widetilde{{a}_{\mp }}|}^{2})\) represents the effects of SPM and XPM, and \({{{{\rm{i}}}}}^{2}=-1\). In the steady state, Eq. (1) can be written into the frequency domain39,44,45:

$$ \left(\begin{array}{cc}\omega -[{\omega }_{a}-\frac{{\gamma }_{a}\left({\left|{\widetilde{{a}}_{\pm }}\right|}^{2}+2{\left|{\widetilde{{a}}_{\mp }}\right|}^{2}\right)}{{P}_{a}}]+{{{\rm{i}}}}{\gamma }_{a} & -\kappa \\ -\kappa & \omega -{\omega }_{b}+{{{\rm{i}}}}{\gamma }_{b}\end{array}\right)\left({\widetilde{{a}_{\pm }}}\atop{\widetilde{{b}_{\pm }}}\right) \\ \;=\left({-{{{\rm{i}}}}\sqrt{{\kappa }_{{ea}}}\,{f}_{\pm }}\atop{0}\right),$$
(2)

In the linear regime, by neglecting the nonlinear term, the transmission coefficient of the system t is proportional to \({\chi }_{t}\), where is \({\chi }_{t}\) defined as follows:

$${\chi }_{t}=\frac{-\kappa }{(\omega -{\omega }_{a}+{{{\mbox{i}}}{\gamma }}_{a})(\omega -{\omega }_{b}+{{{\mbox{i}}}{\gamma }}_{b})-{\kappa }^{2}}.$$
(3)

Therefore, the total transmittance \(T={P}_{{out}}/{P}_{{in}}={\left|t\right|}^{2}\propto {|{\chi }_{t}|}^{2}\), where \({P}_{{out}}\) is the output power and \({P}_{{in}}\) is the incident power8. Similarly, in the nonlinear regime including SPM and XPM, the transmittance of RCP and LCP (\({T}_{+}\) and \({T}_{-}\)) are defined as the ratio of the respective output power of RCP and LCP (\({P}_{{out}+}\) and \({P}_{{out}-}\)) to the incident power (\({P}_{{in}}={|{f}_{+}|}^{2}+{|{f}_{-}|}^{2}\)) and are scaled with \({|{\chi }_{t\pm }^{{nl}}|}^{2}\): \({T}_{\pm }={P}_{{out}\pm }/{P}_{{in}}\propto {|{\chi }_{t\pm }^{{nl}}|}^{2}\), where \({\chi }_{t\pm }^{{nl}}\) is defined as follows:

$${\chi }_{t\pm }^{{nl}}=\frac{-\kappa }{(\omega -[{\omega }_{a}-\frac{{\gamma }_{a}\left({\left|{\widetilde{{a}}_{\pm }}\right|}^{2}+2{\left|{\widetilde{{a}}_{\mp }}\right|}^{2}\right)}{{P}_{a}}]+{{{\mbox{i}}}{\gamma }}_{a})\,(\omega -{\omega }_{b}+{{{\mbox{i}}}{\gamma }}_{b})-{\kappa }^{2}},$$
(4)

By solving Eq. (4), we can analytically deduce the characteristic of the transmittance under different \({P}_{{in}}\) from the spectrum of \({|{\chi }_{t\pm }^{{nl}}|}^{2}\), as shown in Fig. 2. In addition, analytic calculations of Eq. (4) at \({P}_{{in}}=0.82{{{\rm{GW}}}}{{{{\rm{cm}}}}}^{-2}\) with different \(\kappa \) values are shown in Supplementary Note 1. Here, \({f}_{+}={f}_{-}\) due to the equal pumping of the two handedness polarisation lights. As depicted in Fig. 2a, b the symmetric solution curves of \({|{\chi }_{t+}^{{nl}}|}^{2}={|{\chi }_{t-}^{{nl}}|}^{2}\) (black) exhibit two resonant peaks as a function of wavelength \(\lambda\). These peaks exhibit discernible bending due to the influence of the Kerr effect. The peaks situated at longer wavelengths have a greater degree of bending. In fact, the emergence of asymmetric solution curves (red) indicates splitting for \({|{\chi }_{t+}^{{nl}}|}^{2}\) and \({|{\chi }_{t-}^{{nl}}|}^{2}\) for the peaks at longer wavelengths. In these instances, \({|{\chi }_{t+}^{{nl}}|}^{2}\) or \({|{\chi }_{t-}^{{nl}}|}^{2}\) becomes dominant, while the other becomes subordinate. This dynamic results in random handedness polarisation splitting, imparting handedness to the system. Moreover, the polarisation splitting gap is more substantial at \({P}_{{in}}=1.1\,{{{\rm{GW}}}}{{{{\rm{cm}}}}}^{-2}\) (Fig. 2b) than at \({P}_{{in}}=0.82\,{{{\rm{GW}}}}{{{{\rm{cm}}}}}^{-2}\) (Fig. 2a).

Fig. 2: Analytic calculations of Eq. (4).
figure 2

The spectrum of \({\left|{\chi }_{t\pm }^{{nl}}\right|}^{2}\) as a function of wavelength at a \({P}_{{in}}=0.82{{{\rm{GW}}}}{{{{\rm{cm}}}}}^{-2}\) and b \({P}_{{in}}=1.1{{{\rm{GW}}}}{{{{\rm{cm}}}}}^{-2}\) and c as a function of \({P}_{{in}}\) at λ=539 \({{\mbox{nm}}}\). The black curve indicates the symmetric solution, while the red curve indicates the asymmetric solution. The skewed curve in the blue-shaded region of (c) denotes the optical bistable loop. The green and brown dots represent the switch-on and switch-off thresholds, respectively. The green and brown dashed arrows represent the abrupt increases and decreases from the lower and upper branches, respectively. The parameters used for the analytic calculations are \({\omega }_{a}={\omega }_{b}=2.5133\times {10}^{15}{{\mbox{rad}}}{{{{\rm{s}}}}}^{-1}\), group velocity \({v}_{g}=2\times {10}^{8}{{\mbox{m}}}{{{{\rm{s}}}}}^{-1}\), \({\gamma }_{a}={\gamma }_{b}={10}^{13}-9\times {10}^{12}{{{\rm{i}}}}\) \({{\mbox{rad}}}\) \({{{{\rm{s}}}}}^{-1}\), \(\kappa =1.5\times {10}^{14}\) \({{\mbox{rad}}}\) \({{{{\rm{s}}}}}^{-1}\) and \({P}_{a}=1.25\times {10}^{-14}{{{\rm{GW}}}}\,{{{{\rm{cm}}}}}^{-2}\).

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\({|{\chi }_{t\pm }^{{nl}}|}^{2}\) is further investigated against \({P}_{{in}}\) at \(\lambda =539\,{{{\rm{nm}}}}\); at this wavelength, strong optical bistability is observed when the following equation is satisfied: \(\left|({\lambda }_{{res}}-\lambda )/(w/2)\right|\, > \,\sqrt{3}\), where \({\lambda }_{{res}}\) is the resonance wavelength and \(w\) is the full width at half-maximum (FWHM) of the resonance46. As depicted in Fig. 2c, the symmetric solution is indicated by the black curve, while the red curve is the asymmetric solution. The skewed-shape curve in the blue-shaded region is the optical bistable loop8 and has three branches: stable upper and lower branches and an inherently unstable middle branch. Due to the instability of the middle branch, an abrupt increase/drop at the endpoint of the lower/upper branch is observed and indicated by the green/brown dashed arrow. The endpoints are the switch-on (green dots) and switch-off (brown dots) thresholds. In addition, the disparity between the switch-on and switch-off thresholds is defined as the bistable width. \({|{\chi }_{t\pm }^{{nl}}|}^{2}\) fails to preserve its symmetry and, instead, diverges spontaneously at \({P}_{{in}}=0.89\,{{{\rm{GW}}}}{{{{\rm{cm}}}}}^{-2}\). This value is the spontaneous symmetry breaking threshold; beyond this value, the one polarisation invariably and randomly takes precedence over the other. However, the handedness polarisation splitting does not persist. Beyond \({P}_{{in}}=22.92\,{{{\rm{GW}}}}{{{{\rm{cm}}}}}^{-2}\), \({|{\chi }_{t+}^{{nl}}|}^{2}\) and \({|{\chi }_{t-}^{{nl}}|}^{2}\) reconverge to equality, indicating the end of the polarisation splitting phase. Thus, in different incident power regions, different symmetry effects, such as optical bistability and handedness polarisation splitting, are present.

Numerical simulations

In the following section, analytical calculations are cross-verified with numerical simulations using COMSOL Multiphysics. In the simulation, \({\varepsilon }_{l}\) and \({\chi }^{(3)}\) for \(l2\) are 2.2 and \(4.4\times {10}^{-18}\,{{{{\rm{m}}}}}^{2}\,{{{{\rm{V}}}}}^{-2}\), respectively46. The refractive indices for \(l4\) and the substrate are 1.47 and 1.52, respectively. Silver is described by the Drude model, in which the high-frequency dielectric constant equals 3.7, the plasma frequency is assumed to be \(2.2\times {10}^{15}{{{\rm{Hz}}}}\) and the damping coefficient is \(4.3524\times {10}^{12}\) Hz47. The thickness of each layer is as follows: \(L1=L5=40\,{{{\rm{nm}}}}\), \(L2=L4=400\,{{{\rm{nm}}}}\) and \(L3=15\,{{{\rm{nm}}}}\). The continuous-wave linearly x-polarised incident light is set at a normal incidence. Figure 3a, b show the transmittance spectra for RCP (\({T}_{+}\)) in black solid curves and LCP (\({T}_{-}\)) in red dashed curves at different \({P}_{{in}}\). At Pin = \(0.115\,{{{\rm{GW}}}}{{{{\rm{cm}}}}}^{-2}\) (Fig. 3a), a subtle splitting is observed at the resonance peak situated at longer wavelengths between \({T}_{+}\) and \({T}_{-}\), and \({T}_{-}\) surpasses \({T}_{+}\). When the power is increased to \(0.132\,{{{\rm{GW}}}}{{{{\rm{cm}}}}}^{-2}\) (Fig. 3b), the splitting becomes more prominent. In addition, \({T}_{-}\) become dominant with a more pronounced peak, while \({T}_{+}\) significantly decreases, forming a valley-like shape. Figure 3c illustrates the dependence of \({T}_{+}\) and \({T}_{-}\) on \({P}_{{in}}\) at \(\lambda =475.4\,{{{\rm{nm}}}}\), as derived from the curves representing a gradually increasing (superscript ‘in’) or decreasing (superscript ‘de’) incident power from a low or high value. \({T}_{+}^{{in}}\) and \({T}_{+}^{{de}}\) are represented by black solid and black dashed curves, respectively, and \({T}_{-}^{{in}}\) and \({T}_{-}^{{de}}\) are represented by red dashed and red dotted curves, respectively. Within the blue-shaded region, an optical bistable loop is formed by the bistability from increasing and decreasing power curves. As the power gradually increases, a sudden leap occurs from a lower to a higher transmittance at the switch-on threshold \({P}_{{in}}=0.099\,{{{\rm{GW}}}}{{{{\rm{cm}}}}}^{-2}\). However, during the gradual power decreasing step, the transmittance abruptly decreases from a higher to a lower value at the switch-off threshold \({P}_{{in}}=0.097\,{{{\rm{GW}}}}{{{{\rm{cm}}}}}^{-2}\) instead of at \({P}_{{in}}=0.099\,{{{\rm{GW}}}}{{{{\rm{cm}}}}}^{-2}\). Note that \({T}_{+}\) and \({T}_{-}\) remain constant until the power reaches the spontaneous symmetry breaking threshold \({P}_{{in}}=0.11\,{{{\rm{GW}}}}{{{{\rm{cm}}}}}^{-2}\). Beyond this threshold, a distinct splitting between RCP and LCP is obtained. Furthermore, the near coincidence of RCP and LCP for the increasing and decreasing curves in the splitting state indicates that dominance could be assumed by either polarisation, hence, the randomness of the splitting trend. The simulation results are in good agreement with the analytic calculations of the handedness polarisation splitting.

Fig. 3: Numerical simulations of T+ and T_.
figure 3

\({T}_{+}\) (black solid curves) and \({T}_{-}\) (red dashed curves) as a function of wavelength at a \({P}_{{in}}=0.115{{{\rm{GW}}}}{{{{\rm{cm}}}}}^{-2}\) and b \({P}_{{in}}=0.132{{{\rm{GW}}}}{{{{\rm{cm}}}}}^{-2}\). c \({T}_{+}\) and \({T}_{-}\) against \({P}_{{in}}\) at \(\lambda =475.4\,{{{\rm{nm}}}}\), where the superscript \({in}\)/\({de}\) denotes the gradually increasing/decreasing incident power from a low/high value. \({T}_{+}^{{in}}\)/\({T}_{+}^{{de}}\) are represented by black solid/black dashed curves, and \({T}_{-}^{{in}}\)/\({T}_{-}^{{de}}\) are represented by red dashed/red dotted curves, respectively. The optical bistable loop is in the blue-shaded region.

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The spontaneous symmetry breaking between RCP and LCP results in the deviation from linear behaviour; here, the output polarisation does not retain the optical linearity. As a result of the spontaneous symmetry breaking, an orthogonal polarisation component of the output light is generated, and this partial conversion of the incident polarisation occurs above the threshold power. When the balance between the RCP and LCP is disrupted, the constructive interference along the original axis decreases, and at the same time, an enhancement develops along the orthogonal axis. This is a consequence of the uneven contributions from the RCP and LCP waves. This imbalance causes a reorientation in the polarisation direction, culminating in the creation of an orthogonal polarisation component. Thus, spontaneous symmetry breaking occurs. As shown in Fig. 4a, a y-polarised state is discernible at the resonance peak, which is located at longer wavelengths. As the incident power increases from \(0.115\,{{{\rm{GW}}}}{{{{\rm{cm}}}}}^{-2}\) (red solid curve) to \(0.132\,{{{\rm{GW}}}}{{{{\rm{cm}}}}}^{-2}\) (black solid curve), the transmittance peak of this y-polarised state becomes more pronounced and is slightly redshifted. Furthermore, the y-polarised output powers are derived from \({P}_{{out}}=T\times {P}_{{in}}\) at \(\lambda =475.4\,{{{\rm{nm}}}}\) for gradually increasing/decreasing incident power (red solid curve/black dashed curve), as shown in Fig. 4b. From Fig. 3c, the polarisation splitting is not in the bistable loop region, and the red solid curve and black dashed curve are almost coincident in Fig. 4b. Above the threshold power at \(0.11\,{{{\rm{GW}}}}{{{{\rm{cm}}}}}^{-2}\), the y-polarised state appears and has a distinct increase. This new generated state cannot be observed when the power is below the threshold value. Above \(0.15\,{{{\rm{GW}}}}{{{{\rm{cm}}}}}^{-2}\), the output power spectrum remains relatively stable, which is attributed to the redshift of the transmittance peak. Thus, for a specific wavelength, the newly generated polarised light will saturate with increasing incident power. The transmittance spectra for both the x-polarised and y-polarised outputs in the wavelength range include two resonance peaks at \(0.115\,{{{\rm{GW}}}}{{{{\rm{cm}}}}}^{-2}\) and \(0.132\,{{{\rm{GW}}}}{{{{\rm{cm}}}}}^{-2}\) (see Supplementary Note 2 for more details).

Fig. 4: Nonlinear simulation of y-polarised light, with x-polarised incidence.
figure 4

a Transmittance when \({P}_{{in}}=0.115\,{{{\rm{GW}}}}{{{{\rm{cm}}}}}^{-2}\) (red solid curve) and \({P}_{{in}}=0.132\,{{{\rm{GW}}}}{{{{\rm{cm}}}}}^{-2}\) (black solid curve). b \({P}_{{out}}\) against \({P}_{{in}}\) at \({{{\rm{\lambda }}}}=475.4{{{\rm{nm}}}}\) for gradually increasing/decreasing incident power, indicated by the red solid curve and black dashed curve, respectively.

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Experimental results

To experimentally verify the spontaneous symmetry breaking, a multilayer thin film heterostructure was fabricated (c.f.: Method). The linear optical measurements and simulation results of the fabricated sample are shown in Supplementary Note 3. While the splitting patterns of the RCP and LCP components are random, the appearance of the orthogonal polarised output is a clear indicator of spontaneous symmetry breaking. Thus, the spontaneous symmetry breaking characteristics were experimentally measured. The nonlinear measurement result is shown in Fig. 5, and the behaviour of the transmitted output for vertically polarised light in relation to horizontally polarised incident light at \(\lambda =653{{{\rm{nm}}}}\) is analysed. However, no generation of vertical component output at \(\lambda =640\,{{{\rm{nm}}}}\) was observed, as shown in Supplementary Note 4. The measurement was carried out in the forward direction across a range of incident power. In Fig. 5a, at low incident power levels, e.g., \(3\,{{{\rm{mW}}}}\), the output of the vertically polarised component is virtually negligible, and this minimal response is maintained as the incident power increases to \(8\,{{{\rm{mW}}}}\). When the incident power is increased to \(9\,{{{\rm{mW}}}}\), the vertically polarised component is discernible, and the output intensity of the horizontally to vertically polarised light is \({\sim }{10}^{2}{:}1\). These results show that above the threshold power of \(8\,{{{\rm{mW}}}}\), the spontaneous symmetry breaking of RCP and LCP triggers the appearance of vertically polarised output. From the vertical output of the incident power values, the power dependence of this generated polarisation is shown in Fig. 5b. After the parabolic increase above the threshold power, the output’s growth rate begins to saturate. This observed behaviour agrees well with our analytic calculations and numerical simulations. These results provided valuable insight into the effect of incident power on the spontaneous symmetry breaking of RCP and LCP, as well as the subsequent generation of vertically polarised output.

Fig. 5: Measurement of the newly generated polarised output at λ = 653 nm.
figure 5

a Output characteristics of the vertical component at \(3\,{{{\rm{mW}}}}\) (red solid curve), \(8\,{{{\rm{mW}}}}\) (blue solid curve), \(9\,{{{\rm{mW}}}}\) (black solid curve), \(13\,{{{\rm{mW}}}}\) (purple solid curve) and \(19\,{{{\rm{mW}}}}\) (green solid curve) as the incident light is horizontally polarised. b Output of the vertical component as a function of the incident power. The uncertainty in the measurements at different incident powers is represented by the error bars (s.d.), and the solid blue curve is the theoretical fit.

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Conclusion

In conclusion, the optical properties of cFP nanocavities under different incident power conditions were studied. The intrinsic effects of SPM and XPM within Kerr materials could have a profound impact on the output polarisation. A linearly polarised incident light could be considered a superposition of equal pumping of RCP and LCP, with the interaction of SPM and XPM potentially leading to a spontaneous symmetry breaking of polarisation. From the derivation of the nonlinear coupled mode theory equations, clear bifurcations of RCP and LCP in the \({\left|{\chi }_{t}^{{nl}}\right|}^{2}\) spectra were discernible at a specific wavelength range and as a function of the incident power. Numerical simulations revealed a handedness polarisation splitting in transmittance when the incident power was increased to a high value. Within this random splitting pattern, either the RCP or LCP became dominant, while the other component weakened. This phase imbalance led to the generation of an output whose polarisation was orthogonal to the incident light. The appearance of orthogonal polarisation was confirmed by the experimental results, indicating the occurrence of spontaneous symmetry breaking in the coupled nanocavities. Experimentally, the spontaneous symmetry breaking threshold power was \(8\,{{{\rm{mW}}}}\). When horizontally polarised input was incident into the cFP nanocavities, both horizontally and vertically polarised output light was observed, showing spontaneous symmetry breaking. This vertically polarised transmitted light became more significant with increasing incident light power. Both the theoretical and experimental results clearly showed the spontaneous symmetry breaking characteristics of the cFP nanocavities: polarisation splitting. This non-Hermitian cFP nanocavity device has potential applications in all-optical polarisation controllers, polarisation sensors, and quantum computer development.

Methods

Optical measurements

In the linear regime, the forward reflectance \({R}_{f}={|{r}_{f}|}^{2}\), backward reflectance \({R}_{b}={|{r}_{b}|}^{2}\) and transmittance \(T={\left|t\right|}^{2}\) of the samples were measured with a spectrometer (Ocean Optics USB2000) from 400 nm to 800 nm, and the normal incident halogen light (Ocean Optics HL-2000) was the light source directed onto the samples where \({r}_{f}\) and \({r}_{b}\) are the forward and backward reflection coefficients, respectively (see Supplementary Note 3 for more details). For nonlinear characterization, a mode-locked Ti-sapphire femtosecond laser (Mira HP-D, Coherent) as a seed laser source, with a wavelength of 800 nm, a repetition rate of 76 MHz and a pulse width of 120 fs, was utilized to generate visible excitation light. The excitation light in the tuneable visible region was produced from an optical parametric oscillator (Mira OPO-X, Coherent). A single pulse was sufficient to cover the duration of one or even several round-trip processes. The \(653\,{{{\rm{nm}}}}\)-wavelength excitation light was focused onto the sample via an objective lens. Then, the output signal light transmitted from the sample was collected using a homo-parametric objective lens. The signal light was then coupled into a spectrometer (Andor Shamrock 500i, Oxford Instruments) via an optical fibre. To observe spontaneous symmetry breaking induced by Kerr nonlinearity, the incident light was horizontally polarised. In addition, a polariser (GL10-A, THORLABS) was used to discern the generated vertically polarised output light (see Supplementary Note 5 for more details).

Fabrication

Non-Hermitian metal/nonlinear/metal/insulator/metal thin film structures were fabricated by vacuum thermal evaporation. 45 mm × 45 mm glass substrates with \(30\,{{\mbox{mm}}}\times 30\,{{\mbox{mm}}}\) masks were used. The thin film thickness and deposition rate were monitored by a quartz crystal microbalance. Silver (Ag) with 99.99% purity and NPB (N,N’-di(1-naphthyl)-N,N’-diphenyl-4,4’-biphenyldiamine), a passive dielectric, were used in the deposition. PFO (poly(9,9-dioctylfluorenyl-2,7-diyl)), an optically active material, was deposited using spin coating. The metal/nonlinear/metal/insulator/metal layer thicknesses were \(10\,{{{\rm{nm}}}}\)/\(85\,{{{\rm{nm}}}}\)/\(9\,{{{\rm{nm}}}}\)/\(85\,{{{\rm{nm}}}}\)/\(45\,{{{\rm{nm}}}}\).