The spin-wave waveguide (magnon conductor) was a ferrimagnetic yttrium iron garnet (YIG) film with a thickness of *t* = 10 µm. The YIG film was grown using INNOVENT e.V. Technologieentwicklung Jena (Germany). The length and width of the waveguide were 20 mm and 2 mm, respectively, and the edges of the waveguide were cut at a \({45}^{^\circ }\)angle to prevent edge scattering. As shown in Fig. 1, spin waves (magnon current) were generated by an excitation antenna comprising a 450-µm-wide microwave strip line connected to a signal generator (Agilent technology 83732 B). Another end of the antenna was connected to a low-noise ground line (GND). The spin-wave signal was detected by an antenna comprising a 75-µm-wide microwave strip line connected to a spectrum analyser (Agilent technology E4440A). The distance between the antennas, that is, the propagation length of the spin waves, was optimised to be 4 mm. The YIG film allows the long-distance propagation of spin waves over 10 mm and the antenna distance changes the effect of direct electromagnetic interference (EMI) between antennas. These characters are crucially important for the accurate evaluation of the spin-wave amplitude, which is evaluated as an induced voltage on the antenna. If the antennas are close to each other, the EMI-induced voltage dominates the signal. If the antennas are too separated, the spin-wave amplitude is too attenuated by the magnetic damping. An external magnetic field *H*x = 66.0 kA/m (830 Oe) was applied parallel to the waveguide (*x*-direction) to generate BVMSWs25, whereas *H*y = 68.4 kA/m (860 Oe) was applied normally to the waveguide (*y*-direction) to generate MSSWs. To eliminate unfavourable EMI from the power supply of the electromagnet, a pair of permanent magnets was used in our setup.

To obtain the magnonic noise signal, the excitation frequencies of the BVMSWs and MSSWs were fixed at *f*p = 4.0900 GHz and *f*p = 4.2110 GHz, respectively. The local oscillator (LO) of the spectrum analyser remained fixed at these frequencies, and zero-span measurements were performed. The zero-span signal *V*(t) comprised spin-wave amplitude *V*sw and magnonic noise, as represented by \(V\left(t\right)={V}_{SW}+\varDelta {V}_{noise}\left(t\right)\). The spin-wave amplitude *V*sw (magnitude of the magnon current) was determined using the mean value \(\stackrel{-}{V\left(t\right)}\).

As shown in Fig. 2, magnonic noise appeared in both the BVMSWs and MSSWs. As shown in Fig. 2(a), magnonic noise was observed as a fluctuation of the spin-wave amplitude in the zero-span measurement. At a relatively weak excitation power *P*in= 0 dBm, the fluctuation of *V*(*t*) of the MSSW, indicated by red squares, was stronger than that of the BVMSW, indicated by the blue squares. The mean value of \(\stackrel{-}{V\left(t\right)}\) (= *V*sw) of the MSSW was 7.3 mV and the standard deviation was \(\sigma =4.2\times {10}^{-6}\). The mean value of \(\stackrel{-}{V\left(t\right)}\) (= *V*sw) of the BVMSW was 1.3 mV, and the standard deviation was \(\sigma =1.3\times {10}^{-6}\). Standard deviation is a good parameter for comparing the magnitude of \(\varDelta {V}_{noise}\left(t\right)\). To understand the contribution of background noise, the reference signal was measured without microwave (RF) injection, as indicated by the green squares, which proved that the electronic noise in our system was negligible. The power spectrum of the magnonic noise was deduced by calculating the fast Fourier transform (FFT), as shown in Fig. 2(b). The background system noise was − 135 dBm and remained unchanged over the entire frequency region. At a higher frequency of *f* > 2 Hz, the noise spectra of the BVMSW and MSSW became constant (white noise) at -130 dBm, as calculated by averaging the values from 10 Hz to 100 Hz. At a lower frequency of *f* < 2 Hz, the noise spectra showed \(1/{f}^{\alpha }\) dependences, where \(\alpha\) is an index number. Using the fitting analysis, the index numbers for the BVMSW and MSSW were deduced to be \(\alpha =1.05\) and \(\alpha =1.88\), respectively.

The magnonic noise \(\varDelta {V}_{noise}\left(t\right)\) became stronger when the excitation power was increased to *P*in= 9 dBm. As shown in Fig. 2(c), the mean value of \(\stackrel{-}{V\left(t\right)}\) (= *V*sw) of the MSSW was 20 mV, and the standard deviation was \(\sigma =1.2\times {10}^{-5}\). The mean value of \(\stackrel{-}{V\left(t\right)}\) (= *V*sw) of the BVMSW was 3.6 mV, and the standard deviation was \(\sigma =1.6\times {10}^{-6}\). As shown in Fig. 2(d), the noise spectra of the BVMSW at *f* < 2 Hz were enhanced to the index \(\alpha =1.65\), whereas those of the MSSW were almost unchanged (\(\alpha =2.00\)). At a stronger excitation power of *P*in= 15 dBm, as shown in Fig. 2(e), the mean value of \(\stackrel{-}{V\left(t\right)}\) (= *V*MSSW) was 34 mV and the standard deviation was larger (\(\sigma =1.9\times {10}^{-5}\)), whereas the mean value of \(\stackrel{-}{V\left(t\right)}\) (= *V*BVMSW) was 7.1 mV and the standard deviation was \(\sigma =5.8\times {10}^{-6}\). The index number of the BVMSW and MSSW was \(\alpha =1.99\) and \(1.98,\) respectively, as depicted in Fig. 2(f). As shown, the standard deviation increased as the excitation power increased for both spin-wave modes. Standard deviation corresponds to the magnitude of fluctuation and means that the spin wave becomes unstable. In other words, the population of magnons fluctuates. Figure 2(g) shows the power dependence of the index \(\alpha\)of the low-frequency magnonic noise \(1/{f}^{\alpha }\). The index \(\alpha\) of the BVMSW increased significantly from 1.0 to 2.0, whereas that of the MSSW remained unchanged at approximately 2.0.

The index \(\alpha\) could reflect the magnon dynamics at the surface of the YIG waveguide. It is commonly known that in CMOS devices, the \(1/{f}^{\alpha }\) noise originates from the surface carrier (electron) states26–28. Similarly, as reported in a previous work29, a harshened YIG surface increases the low-frequency spectrum, revealing the effect of the surface state. Therefore, the enhancement of \(\alpha\) in the BVMSW can reflect the increased contribution of the magnonic surface state (magnon scattering), or, in the word of fluid dynamics, spin-wave turbulence30 at the surface. As shown in Fig. 3, the BVMSW could change its spatial distribution owing to high-power excitation. The excited BVMSW at a lower power was mainly distributed in the central region of the YIG waveguide, as shown in Fig. 3(a). The profile was reproduced by simulations (see supplemental materials). However, as the excitation power increased, the BVMSW encountered internal turbulence and broadened the spatial distribution toward the surface [Fig. 3(b)]. The BVMSW would then be subjected to strong surface turbulence, similar to the case of a normal fluid, such as water. On the contrary, the spatial distribution of the MSSW remained almost unchanged by the excitation power owing to the well-known nonreciprocal magnetic potential, and the contribution of surface turbulence was always higher than that of internal turbulence [Figs. 3(c) and 3(d)]. In this context, although it is possible to explain these phenomena qualitatively, the detailed formulation of surface magnon dynamics (surface spin-wave turbulence) is beyond the scope of this study and remains to be elucidated.

The magnonic noise at *f* > 10 Hz is also sensitive to internal spin-wave turbulence. As shown in Fig. 4(a), the baseline of white magnonic noise in MSSW increased as the excitation power increased; however, the baseline of *P*in =14.6 dBm decreased from the baseline of *P*in=14.2 dBm. Strikingly, the white magnonic noise spectrum at *P*in=14.2 dBm [Fig. 4(a)] contained multiple peaks, which were observed up to 500 kHz. To understand this phenomenon, the noise power was calculated by averaging the spectrum over 10 Hz < *f* < 500 Hz, as shown in Fig. 4(b). The spin-wave amplitude *V*MSSW increased continuously, as indicated by the blue closed circles. As indicated by the red closed circles, the magnonic noise initially increased continuously; however, it exhibited discontinuous changes at *P*in =14.1 dBm and 14.6 dBm. In the case of large-amplitude spin waves, magnon-magnon scattering became stronger and caused complicated phase interference coupled with excited magnons. The number of magnons (*V*MSSW) increased while the spin-wave instability increased (internal turbulence). Under these conditions, the magnonic system has been understood to show collective oscillations and multiple spectrum peaks of spin-wave AO31–33. The noise experiment directly yielded the threshold of energy flux in spin waves as \({P}_{AO}^{MSSW}\)= -16.2 dBm. The abrupt change in the magnonic noise was the fingerprint of internal spin-wave turbulence. Above *P*in=14.6 dBm, the system could turn into a metastable state and the noise spectrum could lose the AO peaks, or the AO frequency could shift toward a higher frequency31 (*f* > 500 Hz) and disappear from the detection frequency range (10 < *f* < 500 Hz).

The noise spectra at *f* > 10 Hz for the BVMSW are shown in Fig. 4(c). As indicated by the red circles, the BVMSW contains the fingerprint of AO. The power dependence of magnonic noise is shown in Fig. 4(d). The magnonic noise also exhibited abrupt changes at *P*in = 11.7 dBm and 12.1 dBm. Notably, the threshold energy flux \({P}_{AO}^{BVMSW}\)= -26.2 dBm in the BVMSW was smaller than that in the MSSW. This can be explained by four-magnon scattering34–38. This point is discussed later.

To check the validity of our experiment, four-magnon scattering was measured by the spin-wave spectrum using the normal operation of the spectrum analyser with a resolution bandwidth (RBW) of 10 kHz. As shown in Fig. 5(a), the MSSW spectrum exhibited specific satellite peaks near the main resonant frequency *f*p = 4.2110 GHz. The spectral peak frequencies are listed in Table 1.

Table 1

Satellite-peak frequencies in MSSW (*f**p* =4.2110 GHz and *P*in = 15 dBm).

| 1st satellite *n* = 1 | 2nd satellite *n* = 2 |

*f*n+ (GHz) *f*n− (GHz) *Δf*n (MHz) =| *f*n±*- f**p* *|* | 4.2274 4.1946 16.4 | 4.2192 4.2028 8.2 |

Note that the peak frequency satisfies the relation:

$$\begin{array}{c}2{f}_{p}={f}_{n}^{+}+{f}_{n}^{-} , \left(2\right)\end{array}$$

where *n* is the labelling number of the satellite peak. This is the energy-conservation law and represents the nonlinear four-magnon scattering that leads to spin-wave instability. The resonance power at *f**p* (number of excited magnons) and the satellite peak power at *f*1+ (number of scattered magnons) were deduced, as shown in Fig. 5(b). The number of excited magnons increased continuously, while the number of scattered magnons abruptly increased at *P*in = 13.8 dBm. This resulted in the threshold energy flux in the MSSW being \({P}_{4mag}^{MSSW}\)= -16.2 dBm, which induced four-magnon scattering.

Similarly, the BVMSW spectrum is shown in Fig. 5(c) and the peak frequencies of the BVMSW spectrum indicated by solid triangles are summarised in Table 2.

Table 2

Satellite-peak frequencies in BVMSW (*f**p* =4.0900 GHz and *P*in=12.1 dBm).

| 1st satellite *n* = 1 | 2nd satellite *n* = 2 | 3rd satellite *n* = 3 | 4th satellite *n* = 4 | 5th satellite *n* = 5 |

*f*n+ (GHz) *f*n− (GHz) *Δf*n (MHz) = | *f*n±*- f**p* *|* | 4.0909 4.0891 0.9 | 4.0907 4.0893 0.7 | 4.0905 4.0895 0.5 | 4.0904 4.0896 0.4 | 4.0902 4.0898 0.2 |

The number of satellite peaks increased and the frequency shift *Δf*n = | *f*n±*- f**p* *|* became smaller than the MSSW; however, the energy-conservation law (four-magnon scattering) was valid for any pair of magnons. Figure 5(d) shows the excitation-power dependence of the number of excited and scattered magnons, revealing the threshold energy flux in the BVMSW to be \({P}_{4mag}^{BVMSW}\)= -26.4 dBm, which induced four-magnon scattering.

Finally, we discuss the relationships between \({P}_{4mag}^{BVMSW}\), \({P}_{4mag}^{MSSW}\), \({P}_{AO}^{BVMSW}\), and \({P}_{AO}^{MSSW}\). Our experimental results revealed \(P_{{4mag}}^{{BVMSW}}= - 26.4\,{\text{dBm}}\,\,<\,\,P_{{4mag}}^{{MSSW}}= - 16.2\,{\text{dBm}}\) and agreed with the theoretical frameworks of four wave processes (modulation instability30 or 2nd order Suhl instability34,39). In general, the threshold of the oscillation amplitude \({\left( {{{\Delta {M_z}} \mathord{\left/ {\vphantom {{\Delta {M_z}} M}} \right. \kern-0pt} M}} \right)_{{\text{th}}}}\)or critical driving field *h**c* depends on the relative angle \(\theta\) between magnetisation M and wavevector k. Both thresholds become the smallest when \(\theta =0\). The theory also predicts that instability in an MSSW (\({\mathbf{k}} \bot {\mathbf{M}}\)) could be induced with a larger *Δf*n than that in the case of a BVMSW (\({\mathbf{k}}\parallel {\mathbf{M}}\)), which is consistent with our experimental results (Tables 1 and 2). Because four-magnon scattering (instability) occurs before AO, the relationships\(P_{{4mag}}^{{BVMSW}}= - 26.4\,{\text{dBm}}\,\, \leqslant \,\,P_{{AO}}^{{BVMSW}}= - 26.2\,{\text{dBm}}\), \(P_{{4mag}}^{{MSSW}}= - 16.2\,{\text{dBm}}\,\, \leqslant \,\,P_{{AO}}^{{MSSW}}= - 16.2\,{\text{dBm}}\), and \(P_{{AO}}^{{BVMSW}}= - 26.2\,{\text{dBm}}\,\,<\,\,P_{{AO}}^{{MSSW}}= - 16.2\,{\text{dBm}}\)hold.