### Non-linear spin-wave excitations

In the experiments, we utilize super-Nyquist sampling Kerr microscopy (SNS-MOKE)^{40} to investigate coherent non-linear spin-wave excitations in 20 nm thick Ni_{80}Fe_{20} elements placed on top of a coplanar wave guide (CPW), as depicted in Fig. 1a. At low bias fields, the rf-magnetic field generated by the CPW causes magnetization precession with large ellipticity. For low rf-driving field amplitudes the imaginary part of the dynamic susceptibility results in a Lorentzian resonance line shape as a function of the external bias field, as demonstrated in Fig. 1b for the case of a 30 μm × 15 μm elliptical Ni_{80}Fe_{20} element. As the excitation amplitude is increased, the in-plane excursion angle of the magnetization becomes very large (of the order of 20°)^{41}. The elliptical precession trajectory caused by the in-plane shape anisotropy results in an inherent frequency modulation due to the anisotropic nature of the spin-wave dispersion. A good example where these conditions are satisfied is a 20 nm thick Ni_{80}Fe_{20} layer, as predicted in ref. 38. For large driving amplitudes the phase of the response changes as soon as a threshold rf-field of ≈0.30 mT is exceeded (inset in Fig. 1b). At the same time the resonance condition shifts towards lower magnetic bias fields^{41}.

In order to capture the physics of non-linear spin waves excited at low bias fields and large driving amplitudes, it is necessary to treat the magnetization dynamics explicitly within the limit of large modulation. For this, the Landau–Lifshitz–Gilbert equation can be considered in k-space. Using algebraic transformations, it assumes the form of well-known Mathieu or Hill equations. The advantage of this procedure is that the critical spin-wave modes oscillating at half-integer multiples of the pumping frequency can be quickly and reliably identified numerically^{38}. The obtained solutions for the critical spin waves are characterized by strong amplitude-phase-oscillations. These amplitude-phase-oscillations are connected with frequency components at several odd half-integer multiples of the driving frequency. Our aim is now to directly image these spin waves and to resolve their frequency components. For this, a phase-stable non-linear response is required. In comparison to more common magneto-optical imaging approaches which are typically sensitive to the magnitude of the magnetic excitation, the SNS-MOKE technique provides a unique combination of phase-resolved lock-in detection while offering an arbitrary high-frequency resolution^{40}. This feature allows obtaining the real and imaginary parts of the dynamic susceptibility at arbitrary frequency components simultaneously, such as 3/2 of the driving frequency as shown in Fig. 1c (see methods for further details). However, when this method is applied to a continuous magnetic layer, spin waves oscillating at 3/2 of the driving frequency are expected, but no corresponding signal is observed by means of SNS-MOKE (see Supplementary Fig. S1). The most likely explanation is that in this case the parametric spin waves are not phase stable during the measurement. To promote phase stability, an elliptical magnetic element is prepared. Here the fixed boundary conditions facilitate the formation of a standing spin-wave mode pattern that is fixed in space. Indeed, above a threshold rf-amplitude, a coherent signal starts to appear (cf. Fig. 1c). Simultaneously, we detect additional coherent non-linear signals at other integer and half-integer multiples as a direct consequence of the strongly anharmonic precession of the magnetization, as demonstrated in Fig. 2. This behavior is a hallmark of the amplitude-phase-oscillations expected for non-linear processes at low bias fields^{38}. While the integer multiples^{13,16} cover a field range as broad as the ferromagnetic resonance line shape, the odd half-integer harmonics are restricted to a very narrow field window in which they emerge coherently. Using additional micromagnetic simulations for a comparable sample geometry we find, that the additional half-integer multiples with integer index *n* indeed arise from the precessional motion, spanning a frequency comb *n*⋅f_{rf}/2 with amplitudes matching the experimental results, as shown Fig. 2i.

For the non-linear spin-wave process shown in Fig. 2c, one clearly observes a threshold behavior which is in good agreement with the prediction from our analytical k-space model^{38} (see Supplementary Fig. S2). However, as already discussed, the detection of coherent NLSWs at half-integer harmonics strongly depends on the sample geometry. Thus, the values obtained by SNS-MOKE might be larger compared to the simulations due to the lack of phase stability near the threshold. While one can observe phase-stable NLSW generation in larger elliptical elements, even smaller structures, such as a 5 μm × 4 μm rectangle, shows no phase-stable NLSW response in our measurements. We attribute this effect to rapid thermal fluctuations of the phase of the non-linear response.

### Phase stability

The fact that a direct observation of NLSWs is only possible within a narrow field window and in certain element geometries requires further attention. Especially since indirect evidence for the onset of these non-linearities is observed for all investigated structures and in agreement with micromagnetic simulations. For conventional parametric spin-wave processes it is known that the resulting non-linear magnon pair populates one of two energetically equivalent phase states either in-phase or out-of-phase with respect to the rf-excitation^{42}. In the case of three-magnon scattering in parallel pumping geometry randomly occurring phase transitions between the 0-state and the *π*-state appear stochastically with equal probability. In our experiments, we identify such degenerate phase states for the transverse pumping geometry, as depicted in Fig. 3a and reveal their stochastic switching behavior in phase-resolved measurement of the 3/2 f_{rf} non-linearity as randomly emerging sign reversals of the signal. Note that for SNS-MOKE experiments phase stability of the excited NLSWs with respect to the excitation on time scales larger than the measurement bandwidth is essential to obtain a coherent signal. For this purpose, we investigate how experimental parameters such as the excitation amplitude and the bias field influence phase stability.

Interestingly, near the threshold condition, the NLSW signals appear to be less stable in phase over time, resulting in randomly occurring phase flips between the two distinct states most likely caused by thermal fluctuations. A further increase of the rf-driving power stabilizes the NLSWs over a broader field range allowing for phase-stable spatially-resolved imaging, as demonstrated in Fig. 3e. However, even in this stable regime, the initial phase of the NLSWs with respect to the excitation is set randomly to one of the degenerate states. This is demonstrated in Fig. 3b, c where the NLSW is prepared in the stable regime while a brief interruption of the rf-power results either in a 180° phase flip or leaves the phase unchanged with equal probability. This observation illustrates the existence of two degenerate phase states from which one is randomly chosen as soon the non-linearity at 3/2 f_{rf} sets in. In fact, whenever half-integer harmonics of the driving signal are generated a twofold degeneracy of the possible phase state occurs (cf. Fig. 3a). Thus, in contrast to SNS-MOKE, conventional TR-MOKE experiments with rf-amplitude or phase modulation would be unable to detect such NLSW signals due to a periodic resetting to a random phase state. In the next step, we aim for a better understanding of the stability criteria of the NLSWs. Therefore, we slowly sweep the external magnetic field and detect the phase stability for each step on the time scale of 50 s (cf. Fig. 3f–h). Here, we reveal a regime of enhanced stability where phase flips are less likely (gray area). For slightly larger (or smaller) bias fields the stability is reduced as indicated by the occurrence of switching events on the time-scale of a few seconds, as recently reported for three-magnon scattering processes^{42}. In Fig. 3f–h we demonstrate how the increasing number of switching events in the vicinity of the stable regime results in a decreased average amplitude and an enhanced standard deviation of the measured signal. In comparison to this small field range of enhanced stability micromagnetic simulations indicate that the field range in which 3/2 f_{rf} non-linear spin waves are generated is as broad as the line width of the ferromagnetic resonance (see Supplementary Fig. S3). Moreover, the formation of this stable regime, where phase flips are mostly prevented, strongly depends on the chosen element geometry. In SNS-MOKE experiments, therefore only a small fraction of the entire NLSW spectrum is observable as random phase flips due to thermal fluctuations occur on a time scales faster than the measurement bandwidth, resulting in a loss of signal.

### Spin-wave imaging

The experimental observation of phase-stable NLSWs at 3/2 f_{rf} implies that the predicted amplitude-phase-oscillations^{38} occur coherently with respect to the driving frequency in a certain bias field window. In the phase-stable regime (cf. Fig. 3), SNS-MOKE allows for spatially-resolved imaging of the NLSWs and directly reveals the in-plane wave vector components of the parametrically generated magnons at 3/2 f_{rf}. For this type of measurement, all experimental parameters are fixed while real and imaginary parts of the polar magnetic response are recorded in a point-wise fashion. Here, a mostly uniform signal is obtained at the driving frequency f_{rf} = 2.3 GHz, as shown in Fig. 4a, while above the threshold condition a phase-stable pattern of NLSWs is detected, oscillating at a frequency of 3/2 f_{rf} = 3.45 GHz (see Fig. 4b). At the same time, additional non-linear signal components due to the amplitude-phase-oscillations are detected at other half-integer multiples, as demonstrated in Supplementary Fig. S4. Corresponding micromagnetic simulations of the non-linear magnetization dynamics are carried out and compared to the experimental results, as shown in Fig. 4d, e, respectively. The direct comparison of the excited wave vectors in the standing spin-wave patterns is achieved by performing a two-dimensional fast Fourier transform (2D-FFT) for experimental data and the simulation. For both cases, the 2D-FFT pattern shows the predicted NLSW pairs at 3/2 f_{rf}^{38}. Furthermore, we find a good agreement between our results and the analytical model as we compare the obtained non-linear wave vector components from spatially-resolved imaging with our calculations as a function of the rf-frequency (see Supplementary Fig. S5). The non-linear effects we observe convert the uniform mode into short-wavelength dipole-exchange spin waves within the ferromagnetic material without the need for patterning while the wavelength can be tuned by external parameters. This is an advantage compared to recent concepts of generating short-wavelength spin waves^{19,20} using periodic metallic coupling structures.

We would like to point out that the observed spin-wave patterns emerge in a frequency-locked regime^{38}, additional signal components are visible and follow the iso-frequency line of the non-linear dispersion at 3/2 f_{rf} obtained from the analytical model (red line in Fig. 4). We explain this population of different modes by elastic magnon–magnon scattering processes facilitated by slight spatial variations of magnetic properties as it can be expected for a polycrystalline Ni_{80}Fe_{20} sample at low bias fields. To support this assumption, the impact of spatially varying magnetic properties is demonstrated using micromagnetic simulations, as shown in Supplementary Fig. S6. In the following, we aim to achieve phase control by stabilizing the NLSWs in one of the degenerate phase states.

### Phase-locking of non-linear spin waves

The integration of NLSWs into magnon-based devices requires direct control of propagation direction, amplitude or phase. Furthermore, the synchronization of excited non-linear magnons to an external reference frequency has enormous potential for magnon-based computing^{23,25,26,27}. Phase control in such a non-linear system would allow to precisely set the NLSW’s initial phase and prevent the phase state from random switching events by increasing its robustness against thermal fluctuations. For this, we modify the excitation scheme by integrating a second rf-source^{43} operating at 3/2 f_{rf}, as shown in Fig. 5a. This allows to seed the magnetic system with an additional frequency component at the desired phase. Physically, phase-locking is achieved due to the coupling between the homogeneous rf-field and the NLSWs for the following reasons: (i) As NLSWs are generated within a confined element their standing spin-wave pattern is fixed in space even if the phase state randomly switches. As a result of these boundary conditions only two degenerate phase states exist. (ii) The interplay of a uniform rf-field with the boundaries of the elements results in non-uniform dynamic demagnetization fields^{44}. Here, the non-uniformity of the internal fields provides a spectrum of wave vectors in the range of the NLSWs with an relative amplitude of a few percent with respect to the uniform contribution. The presence of this rf-field breaks the symmetry of the initially degenerate states and allows to select only one of them. As soon as coherent magnons with the corresponding wave vectors are provided by the seed rf-field, the NLSWs will parametrically amplify this specific mode with the phase of the seed source.

The applied power level of the seed frequency component is typically two orders of magnitude smaller compared to the pump excitation. This modified excitation scheme is now applied to one of the elements which do not show a coherent NLSW signal without a seed excitation, namely a 5 μm × 4 μm rectangular element (cf. Supplementary Fig. S1). In the following, we analyze the signal amplitude at 3/2 f_{rf} while the power level of rf-pump and rf-seed were varied in a step-wise fashion. Figure 5b shows the obtained 2D map. In agreement with our analysis above, a regime exists where the coherent excitation of NLSWs only occurs when an additional seed amplitude (at f_{seed} = 3/2 f_{rf}) is applied. We would like to point out, that due to the amplitude-phase-oscillations, the introduced stabilization process can also be performed at other odd half-integer seed frequencies, as demonstrated in Supplementary Fig. S10. Here, the separation of seed and detection frequency reduces the influence of direct excitations at the detection frequency, and thus provides a pure NLSW signal. Clearly, the seed frequency sets and stabilizes the phase of the obtained NLSW signal.

Next, we study the phase control of the excited modes for the 5 μm × 4 μm rectangular magnetic element. In a discrete structure, one can expect that only two phase states with respect to the excitation are possible for a standing spin wave. For this purpose spatially-resolved imaging is performed with both power levels set to the stable regime above threshold, while the phase of the seed (at 3/2 f_{rf}) was varied with respect to the phase of 3/2 of the pump frequency for the images shown in Fig. 5c–e. Here, the amplitude of the NLSW signal is strongly suppressed for a relative phase Δ*θ* = 180° (cf. Fig. 5d), while the sign of the signal changes from c to e corresponding to the two distinct phase states. Measuring the magnitude of the signal in Fig. 5f as a function of Δ*θ* in the center of this device results in a 180° periodicity. At the same time the phase of the signal *ϕ* shifts by 180°, as shown in g. We link the rapid phase jump in g to the transition from one of the two possible phase states to the other while the magnitude returns to the same level as the phase of the seed frequency is varied. For the transition region the two phase states are equally probable and the signal vanishes. The observation of these two possible phase states and their accessibility via phase shifts of the seed frequency is further corroborated by micromagnetic simulations (cf. Fig. S11). Note that the observed phase-locking effect strongly depends on the applied pump (f_{rf}) and seed (3/2 f_{rf}) power levels. For example, as the pump power is slightly decreased, the transition between the states becomes less sharp and at the same time, the magnitude is reduced. However, the applied scheme allows to initialize the NLSW in a defined and stable phase state, and provides a direct control for setting the phase state.