Symmetry Breaking of Counter-Propagating Light in a Nonlinear Resonator

Spontaneous symmetry breaking is a concept of fundamental importance in many areas of physics, underpinning such diverse phenomena as ferromagnetism, superconductivity, superfluidity and the Higgs mechanism. Here we demonstrate nonreciprocity and spontaneous symmetry breaking between counter-propagating light in dielectric microresonators. The symmetry breaking corresponds to a resonance frequency splitting that allows only one of two counter-propagating (but otherwise identical) states of light to circulate in the resonator. Equivalently, this effect can be seen as the collapse of standing waves and transition to travelling waves within the resonator. We present theoretical calculations to show that the symmetry breaking is induced by Kerr-nonlinearity-mediated interaction between the counter-propagating light. Our findings pave the way for a variety of applications including optically controllable circulators and isolators, all-optical switching, nonlinear-enhanced rotation sensing, optical flip-flops for photonic memories as well as exceptionally sensitive power and refractive index sensors.


Fabrication of microresonators
Our microresonator is produced by milling a 2.7 mm diameter fused silica rod with a commercial CO2 laser (25 W) to create a rotationally-symmetric protrusion around the rod 1 .
The rod is fixed on a spindle and the laser is focused onto its surface, steered using two galvo mirrors to obtain the desired microresonator profile. The machining is divided into three steps: first we use high power to evaporate the glass and make a cylindrical surface coaxial with the spindle, then we reduce the power and start alternating the position of the laser between two different focal spots 120 µm apart until two grooves appear in the rod and the central prominence is well defined (see Fig. S1). Finally, we reduce the laser power to melt the glass surface instead of evaporating it, in order to obtain the highest possible smoothness and thus a high Q factor. This fabrication process produces an almost perfect rotational symmetry that nearly cancels any kind of backscattering due to imperfections.
This can be seen from the absence of mode splitting when sweeping the laser frequency across any resonance at low power. In addition, we don't measure a significant amount of backscattering (less than -10 dB) when we send light in the resonator just in one direction.
Fig. S1 | Microrod resonator used in the experiment. The resonator is machined from a 2.7 mm diameter fused silica rod by evaporating and melting its surface with a CO2 laser while spinning it on a motorised spindle.

Fabrication of tapered fibres
The tapered fibre in our experiment is obtained from a commercial 125 µm diameter optical fibre. A 3 cm section of the fibre, stripped of the plastic coating, is heated up to the softening point of glass and stretched by 6 cm using two linear motors. By optimising the temperature and pulling speed, we obtain tapered fibres that allow over-coupling to the resonators. The diameter of the tapered fibre is thus estimated to be of the order of, or less than, 1 µm. After tensioning, the tapered fibre is glued to an aluminium bracket that will be fixed to the experiment setup, and FC-APC connectors are spliced onto each end. The resonator is fixed to a 3-axis micrometer translation stage that allows us to move it with respect to the fixed tapered fibre to tune the coupling strength.

Experimental setup
We send light into the resonator in opposite directions by splitting a fibre-coupled and amplified mode-hop-free diode laser in the 1.55 µm wavelength range into two paths. To be able to detect the transmission we use circulators at both ends of the tapered fibre to separate out the transmitted light in each direction. We also employ 1% couplers to measure the incident power in each direction. A variable attenuator and a fibre-coupled electro-optic intensity modulator control power in the two arms, allowing us to tune and modulate the pump power imbalance. A fibre-loop interferometer fed with light from the laser provides the calibration for the frequency offset (see "Calibrating the laser frequency axis" below).
Measuring the Q factor, finesse and effective mode cross-sectional area It is possible to determine the intrinsic and loaded Q factors and finesses of a resonance via its linewidth and coupling efficiency (in the limit of high Q factors). This is done by modulating sidebands onto the laser using an electro-optic modulator (EOM), sweeping the frequency across the resonance and fitting the sum of three Lorentzians to the transmission profile (see the red curve in Fig. S2). By measuring the distance between the sidebands and scaling this to twice the modulation frequency, we are able to calibrate the linewidth of the carrier. This measurement is performed at very low power in order to avoid Kerr and thermal broadening. To ensure that this is the case, we take measurements while scanning the laser frequency in both directions across the resonance and verify the absence of thermal or Kerr-induced resonance broadening when scanning from high to low laser frequencies. Knowing the laser frequency and the mode family's free spectral range (FSR) we find the loaded factor and finesse , and using the coupling efficiency obtain their intrinsic counterparts as where the + sign is for an undercoupled resonator and thesign is for an overcoupled one.
The Kerr broadening is linked to the Kerr effect that shifts the resonance to lower frequencies with increasing circulating power by increasing the refractive index. If we sweep the laser frequency from above to below the resonance at a sufficiently high speed to avoid thermal broadening, but slower than the cavity build-up time, the power transmitted by the tapered fibre will have the shape of a tilted Lorentzian satisfying: where Δ is the full linewidth in terms of frequency , is the height of the peak and is the degree of tilt. By fitting this curve, it is possible to derive the effective mode crosssectional area eff from as explained below.  is calibrated using a fibre loop resonator formed by connecting a 1% fibre coupler to itself.
Calibrating the laser frequency axis The data for Fig. 2b, 3a-b and 4b are taken by modulating the current of the externalcavity diode laser (ECDL) to sweep its frequency. During this modulation, the optical power is kept constant by subsequently sending the light through an erbium doped fibre amplifier, which is operated in saturation. The ECDL laser current modulation is produced by feeding a triangular wave voltage from a signal generator to the laser controller. By recording this voltage at the same time as the transmission of the tapered fibre, we are able to use it to associate a frequency offset with each data point.
However, the laser frequency does not vary linearly with the ECDL current. We therefore perform a further step in order to extract the frequency offset. To get an equally spaced frequency reference we send a small amount of the pump light through a fibre loop cavity consisting of 1% coupler connected to itself as shown in Fig. S3. The mode spacing of the fibre loop interferometer was itself calibrated against the sidebands generated at known frequencies using the EOM. Using the fibre loop cavity as a reference we are able to produce a calibration curve between the modulation voltage and the laser frequency offset, which we apply to the microresonator data to generate the laser frequency axis.

Symmetry breaking
Under the condition of equal pump powers in, CW = in, CCW ≡ in , the solutions to the coupled equations (1, 2) are as shown in Fig. S4. For in / 0 greater than the threshold of 1.54, there is a range of over which the solution where CW = CCW is unstable; instead the symmetry breaks and the coupled powers split, accompanied by a splitting between the resonance frequencies in the two directions. Producing the hysteresis graph In Fig. 2c we display the power coupled into the resonator vs. the pump power ratio, taken while holding the laser frequency constant and modulating one of the pump powers up and down using a Mach-Zehnder EOM. To obtain the coupled power, we measured the power transmitted by the tapered fibre vs. the voltage applied to the EOM both with and without the resonator, and then subtracted the two traces. The measurements of the output power of the EOM taken via a 1% coupler were used to represent the horizontal axis as a pump power ratio.
Mach-Zehnder interferometer for Fig. 4b When using coherent counter-propagating light derived from the same source in fibre optics it is important to consider interference effects due to unwanted reflections off connectors and optical components. Since the light is travelling in both directions, any reflected light interferes with the light travelling in the opposite direction. The interference changes from constructive to destructive with the laser frequency since the relative phase of the two waves changes as: where 1 and 2 are the two path lengths from the point where the light splits (in the 3dB splitter right after the amplifier) to where it recombines, i.e. the reflection point, and is the refractive index of the fibre. Even a tiny reflection will cause a relatively large power fluctuation, e.g. a -40 dB reflection will cause a ±2% fluctuation as the interference phase changes.
It is possible to reduce this effect by choosing high-return-loss components and ensuring that any fibre-to-fibre connection is clean. The residual power oscillation can be dealt with by setting the interference phase to be almost constant by balancing the two path lengths to the reflection point, although this does not work if there are multiple reflection points. Optical isolators can help to remove reflections from parts of the setup which are far from the tapered fibre, though they may introduce reflections of their own.
In our measurements we took advantage of this effect to create the ±3% pump power imbalance oscillations for Fig. 4b. Furthermore, tuning the path length difference allowed us to choose the power oscillation period with respect to the laser frequency change.

Comb generation and other non-linear effects
The threshold power for symmetry breaking ( in, CW 0 ⁄ > 8 (3√3) ≈ 1.54 ⁄ ) is identical to the minimum possible threshold for sideband generation by four-wave mixing 2 .
However, the actual threshold for sideband generation is only this low if the Kerr shift of the sidebands due to cross-phase modulation from the pump exactly cancels out the resonator dispersion, which is rarely the case. As a result, the nonlinearity-induced symmetry breaking between counter-propagating light usually appears before four-wave mixing sidebands are generated. To avoid this unwanted effect we simply chose a mode family that does not exhibit any other nonlinearities, which we confirm with an OSA. In addition, other nonlinearities are very obvious in the transmitted power through the tapered fibre since they change the amount of light in the pump mode. We could also observe competing nonlinear optical effects such as Raman scattering and Brillouin scattering, which can lead to more complicated dynamics of the optical system. However, Brillouin scattering has a quite narrow gain bandwidth with ~10 GHz shift from the pump laser, which is not resonant in our microresonator. Raman scattering can be quenched using the right coupling regime 3 .

Derivation of equations 2 and 3
In high-Q dielectric microresonators, it is possible to reach sufficient circulating intensities to see the effects of the Kerr, or 3 , nonlinearity. The refractive index of a mode in a cicular resonator increases linearly with the circulating power circ in that mode: at which Kerr nonlinear effects occur, wherein = res /(2 ) is the loaded quality factor.