We demonstrate the nonreciprocal transmission of magnonic frequency combs (MFCs) in a dissipative cavity-magnonic system. We utilize the recently emerged pump-induced magnon (PIM) mode in YIG spheres to generate an MFC, as the PIM mode exhibits excellent nonlinearity under coherent pumping. Meanwhile, the dissipative cavity magnonic device is prepared to critical bound states in the continuum (BIC), providing clear nonreciprocity. Based on the different absorption efficiencies of the device in two opposite directions, we have demonstrated a clear difference in the number of frequency comb teeth for forward and reverse transmission, showcasing the ability to generate unidirectional combs. The nonreciprocal MFCs can be systematically tuned by modulating the detuning of the pump and BIC, the magnon and cavity modes, as well as the pump and perturbation tone. This research promotes the combination of MFCs and functional non-Hermitian cavity-magnon electronic devices, realizing new applications for nonreciprocal magnonic devices.

Due to the ultrahigh frequency and time resolution, the optical frequency combs (OFCs)1–5 have become a highly researched and trending topic for years, which has produced extensive applications in atomic clocks,6 satellite navigation,7 and molecular fingerprinting.8 These advancements have inspired the exploration of frequency combs based on other forms of matters. The recently emerged magnonic frequency combs (MFCs),9–15 which are built on magnonic dynamics that possesses the advantages of large spin density16 and avoiding the Joule heating, become attractive for the advantage of high frequency resolution,17 tunability,10 and integrability.18 Most current studies focus on searching for the initiation methods and further improving the conversion efficiency of MFCs,19 while the transmission of MFCs, especially the nonreciprocal emission, remains rarely explored. A recent study20 has theoretically demonstrated the non-reciprocal MFCs and has inspired us to observe non-reciprocal MFCs experimentally. However, due to the difficulty of realizing nonlinearity and nonreciprocity in one device, the nonreciprocal transmission of MFCs has not yet been experimentally studied. To solve this problem, the Walker modes (WMs)21–23 in the yttrium iron garnet (YIG) sphere stand out as a key element for realizing both nonreciprocity and nonlinearity. The WM as a magnon mode provides a symmetry-breaking system24 that allows for unidirectional energy transmission and suppression of backscattering.25–28 By coupling the WM in a YIG sphere with a cavity mode and continuous waves, one can construct an on-chip BIC system with notable nonreciprocity.25,29,30 BIC allows for highly directional energy transmission and enhanced frequency selectivity in frequency comb transmission, which is of great significance for designing nonreciprocal magnon devices with advanced performance.31 Meanwhile, the WM can nonlinearly couple to other magnon modes10 and then generates MFCs in YIG spheres. Benefitting from the high tunability and nonlinearity of the pump-induced magnon (PIM) mode,10,32,33 which can couple with the WM, we can easily generate MFCs in a YIG sphere by a pump and a perturbation tone.19 The YIG sphere, as well as the WM, connects the nonlinear MFC and nonreciprocal BIC34,35 together with an on-chip device. The flexible tunability of magnon and PIM provides additional dimensions for modulating the nonreciprocal transmission of MFCs,31 which is prospective for establishing a solid tuning method of MFC transmission and opening up promising applications in metrology.

In our work, we experimentally demonstrate a nonreciprocal MFC system in an on-chip device by combining the BIC and PIM through the WM. The WM is coupled to a planar microwave device through dipole interaction. Meanwhile, a pump is fed from one port of transmission line to excite the WM with coherent microwaves, thereby inducing PIM. We utilize the nonlinearity of PIM in a YIG sphere to generate an MFC through a pump and a perturbation tone19 and utilize the resonant cavity and waveguide structure on the device to couple with the WM and generate a BIC with nonreciprocal transmission. By simultaneously implementing the BIC and PIM with the same medium, the WM, we realize the nonreciprocal MFCs based on the different absorption efficiencies of the device in two opposite directions. Due to the flexible tunability of PIM and the magnetic field dependence of WM, we can systematically tune the nonreciprocity of MFC system in three dimensions, which is the detuning between the pump and BIC, the magnon and cavity mode, as well as the pump and perturbation tone. Finally, we achieve the nonreciprocal transmission of MFCs in opposite directions with a difference in the number of comb teeth at 14. Our study can be applied in the MFC system with stronger nonlinearity and denser comb teeth, which can also be generalized to frequency combs in other physical systems so as to create opportunities for the nonlinear device.

The schematic of our nonreciprocal setup is shown in Fig. 1(a), which consists of a T-shaped microwave cavity with an antenna and a 1-mm YIG sphere. The experimental device is fabricated on a 0.78 mm-thick RT/duroid 5880 substrate. It consists of a 50-Ω transmission line, a perpendicular stripline, and a microwave antenna on the one side of the transmission line, as shown in Fig. 1(a). Standing waves are formed in the perpendicular stripline due to the reflections from both its open end and its intersection with the transmission line. The fundamental mode frequency of the cavity is 2.923 GHz. The signal from the microwave source is fed into the antenna from port 3 to excite the PIM in the YIG sphere. Coupling the YIG sphere to the planar cavity establishes an open-cavity magnetic subsystem driven by the pump. Pumps are, respectively, fed into the cavity from ports 1 and 2 to measure the transmission spectra S21 and S12. A uniform magnetic field is applied in the normal direction of the cavity surface. The high-frequency alternating magnetic field from the microstrip line drives the WM. We precisely tune the WM’s frequency to find the BIC where the spectral linewidth infinitely approaches zero. In our device, we define the transmission direction from port 2 to port 1 as the forward direction, while the transmission direction from port 1 to port 2 as the reverse direction. The forward signal is almost impervious due to BIC, while the reverse signal mostly transmits to port 2, forming the nonreciprocal transmission. Our experiments are conducted based on this nonreciprocal device, and the overall experimental setup is schematically shown in Fig. 1(b).

FIG. 1.

(a) Schematic diagram of the setup of the nonreciprocal MFCs used in the experiment. Port 3 is connected to the antenna, ports 1–2 are transmission lines, and the cavity connected perpendicular to them is a microwave resonant cavity. Signals can be transmitted nonreciprocally in both directions. The purple sphere located in the middle of the plate represents the YIG sphere, which is in a constant magnetic field perpendicular to the plate face up. (b) Schematic diagram of the overall wiring of the experimental system. VNA (Vector Network Analyzer) is used to measure S21 as well as S12 of the system, and SA (Spectrum Analyzer) measures the radiation spectra in two opposite directions. The perturbation tone is fed into the system through the antenna. (c) Schematic of nonreciprocal MFCs from port 1 to 2 and port 2 to 1.

FIG. 1.

(a) Schematic diagram of the setup of the nonreciprocal MFCs used in the experiment. Port 3 is connected to the antenna, ports 1–2 are transmission lines, and the cavity connected perpendicular to them is a microwave resonant cavity. Signals can be transmitted nonreciprocally in both directions. The purple sphere located in the middle of the plate represents the YIG sphere, which is in a constant magnetic field perpendicular to the plate face up. (b) Schematic diagram of the overall wiring of the experimental system. VNA (Vector Network Analyzer) is used to measure S21 as well as S12 of the system, and SA (Spectrum Analyzer) measures the radiation spectra in two opposite directions. The perturbation tone is fed into the system through the antenna. (c) Schematic of nonreciprocal MFCs from port 1 to 2 and port 2 to 1.

Close modal

The transmission spectra and the radiation spectra of the device in both directions are measured separately. The VNA is used to measure the S-parameters of the device that includes S21, S12, S11, and S22. During the measurement, the probe power is set at −25 dBm. To characterize the nonreciprocal emission of the MFCs, we first connect port 1 to the microwave source to feed a pump and measure the radiation spectrum from port 2. A perturbation tone is fed to the YIG through the antenna to create a perturbation of the pump to generate MFCs.19 Then, we switch the cable connection of ports 1 and 2, i.e., sending the pump from port 2 and measuring the radiation spectrum from port 1.

Comparing these two radiation spectra, we can characterize the nonreciprocal MFCs in two opposite directions [Fig. 1(c)], of which the properties are totally contrary to the nonreciprocal transmission of the system. This is because BIC improves the absorption rate of the YIG sphere, which enhances the nonlinear effect of the WM-PIM coupling, leading to an increase in the tooth number of MFC. We tune the WM frequency (ωm) to produce the BIC and measure the S21 and S12 parameters [black dashed line in Fig. 2(a)]. As shown in this figure, there is a clear nonreciprocity in the transmission spectra of the system near the BIC. We additionally measure S21 when a −10.7 dBm pump fed from port 1 and S12 when the pump fed from port 2 to take the influence of pump in consideration. After feeding the pump, S12 undergoes a tiny split. At this point, there is still a significant nonreciprocity with an isolation of about 25 dB. This indicates that the system maintains a robust nonreciprocal behavior after the input of the pump, which establishes a fundamental prerequisite for achieving nonreciprocity in the MFC.

FIG. 2.

(a) Measured S21 as well as S12 for the system under the BIC condition (black dashed line) and S21 as well as S12 for the system with pump turned on (blue solid line vs red solid line). In this experiment, the blue line is the measurement data of the incoming signal from port 1, and the red line is the measurement data of the incoming signal from port 2. (b) Absorption rate of the system in both directions. (c) and (d) MFCs measured in both directions at the BIC frequency. (e) and (f) MFCs measured in both directions away from the BIC frequency.

FIG. 2.

(a) Measured S21 as well as S12 for the system under the BIC condition (black dashed line) and S21 as well as S12 for the system with pump turned on (blue solid line vs red solid line). In this experiment, the blue line is the measurement data of the incoming signal from port 1, and the red line is the measurement data of the incoming signal from port 2. (b) Absorption rate of the system in both directions. (c) and (d) MFCs measured in both directions at the BIC frequency. (e) and (f) MFCs measured in both directions away from the BIC frequency.

Close modal

Then, we obtain the absorption rate of the system in two opposite directions [Fig. 2(b)] from S12 and S21 in Fig. 2(a) and their corresponding reflection parameters S11 and S22, with the relations 1 − |S11|2 − |S21|2 and 1 − |S22|2 − |S12|2. We can also see an obvious nonreciprocity in the absorption rate. The difference in absorption rates increases significantly as the pump frequency gets closer to the BIC condition and reaches a maximum of over 50% under the BIC condition. However, the nonreciprocal absorption rate is opposite to the transmission nonreciprocity. The forward direction with a low transmission rate possesses a high absorption rate, while the reverse direction with a high transmission rate possesses a low absorption rate. Since the spin number of the PIM increases with the pump power,10 the pump microwave fed in the forward direction with a high absorption rate excites a stronger PIM than the case with the pump fed in the reverse direction. It improves the nonlinearity of the system, leading to the enhancement of MFCs. We have developed a theoretical model to describe the asymmetric absorption rate and its nonreciprocal MFC. Please see the supplementary material for details.

To verify this expectation in experiment, we feed a −12.6 dBm perturbation tone through port 3 to perturb the PIM and measure the radiation spectra in two opposite directions separately with pump frequency at BIC [Figs. 2(c) and 2(d)] and away from BIC ((ωuωBIC)/2π = 6 MHz) [Figs. 2(e) and 2(f)]. The detuning between the pump and the perturbation tone is fixed at (ωuωr)/2π = 0.04 MHz throughout the measurements to make sure that the perturbation generates MFCs in the system. The power of pump keeps at −10.7 dBm as previous. As shown in Figs. 2(c) and 2(d), there is an obvious nonreciprocal transmission of MFCs under BIC conditions, where the difference of absorption rate in two opposite directions is large. The tooth number, which can also be described as the radiation linewidth, exhibits a great difference. In addition, there is also about one order of magnitude difference in radiation intensity. However, when (ωuωBIC)/2π = 6 MHz, the absorption difference in two directions becomes smaller. The nonreciprocity of the MFC is weak, and there is no tooth number difference. It can be seen that when the system is in BIC, the nonreciprocity of the MFCs increases when the pump frequency used to generate MFCs is at the position where the absorption rate of the coupled system is higher. In fact, not only the pump and perturbation tone can generate MFCs, but also the high order combs contribute to MFCs. Hence, the difference in the absorption rate at the pump frequency serves only as a qualitative reference for the nonreciprocity of MFC, while a quantitative relation is beyond the scope of our work.

Furthermore, we investigate the effect of BIC conditions on the nonreciprocal MFCs by means of modulating the frequency of the hybrid mode ω in the coupled system. We have calculated the dependency of the imaginary part of the hybrid mode Im(ω) on the WM frequency ωm [Fig. 3(a)]. When Im(ω) = 0, BIC occurs in the system.26 As the WM frequency gradually approaches the frequency corresponding to the BIC condition, the nonreciprocity of the system gradually increases.25 To further verify whether BIC is the strongest condition for MFC nonreciprocity experimentally, we vary the WM frequency from the BIC condition (ωm = ωm3) to lower frequencies and measure the transmission characteristics of MFC in different directions under BIC conditions (ωm3) and under ωm1, ωm2 [Figs. 3(b) and 3(c)]. We kept the detuning of the pump and perturbation tone to Δ/2π = 0.04 MHz and fixed the pump frequency at ω. The power of the pump and perturbation tone remained at −10.7 and −12.6 dBm, respectively. The difference in the tooth number of MFCs in opposite directions is also extracted and displayed in Fig. 3(d). When ωm = ωm1, the eigenfrequency of the hybrid modes is far from the BIC condition, and the nonreciprocity of the MFCs is insignificant, with the difference in tooth numbers of only 1. As ωm approaches the BIC condition (ωm = ωm2), the nonreciprocal performance of the MFC in the system gradually increases, where the number of teeth differences in the MFC increases to 6. When ωm is under the BIC condition (ωm3), the nonreciprocal performance of the MFC reaches a maximum value under this modulating dimension, and the MFC teeth difference increases to 10. As a result, the system under the BIC condition produces the maximum MFC nonreciprocal performance.

FIG. 3.

(a) Variation of the imaginary part of the hybridization mode ω with the WM frequency in a cavity photon–magnon coupled system obtained by theoretical calculations. When the imaginary part reaches 0, the system reaches BIC. ωm1, ωm2, and ωm3 are the frequency conditions of the WM for the measurements. (b) Input pump from port 1 and measure the MFC at port 2. ωm1, ωm2, and ωm3 correspond to the three experimental conditions in (a). (c) Input pump from port 2 and measure the MFC at port 1. ωm1, ωm2, and ωm3 correspond to the three experimental conditions in (a). (d) The difference in the tooth number of MFCs measured at each port with different frequencies of the WM.

FIG. 3.

(a) Variation of the imaginary part of the hybridization mode ω with the WM frequency in a cavity photon–magnon coupled system obtained by theoretical calculations. When the imaginary part reaches 0, the system reaches BIC. ωm1, ωm2, and ωm3 are the frequency conditions of the WM for the measurements. (b) Input pump from port 1 and measure the MFC at port 2. ωm1, ωm2, and ωm3 correspond to the three experimental conditions in (a). (c) Input pump from port 2 and measure the MFC at port 1. ωm1, ωm2, and ωm3 correspond to the three experimental conditions in (a). (d) The difference in the tooth number of MFCs measured at each port with different frequencies of the WM.

Close modal

We can also modulate the nonlinear effects of the coupled system by modifying the detuning between two PIMs Δ = ωuωp, which can also tune the nonreciprocity of MFCs. Based on this, we adjust the WM frequency to the BIC to construct the nonreciprocity. The frequency of the perturbation tone is fixed at ωp = ω, and the power is set to −12.6 dBm. We then scan the pump frequency from Δ/2π = −1.5 to Δ/2π = 1.5 MHz with the power set to −10.7 dBm to observe the nonreciprocity of MFCs with different nonlinearities of the system. The radiation spectra of the system in two opposite directions are measured by SA [Figs. 4(a) and 4(b)], and the tooth number of MFC at different Δ is shown in Fig. 4(c). When pump is fed from the reverse direction, the number of MFC teeth in the system stays within 5 regardless of Δ. This is because the excitation of PIM is weak. The change in Δ has tiny effect on enhancing the MFC. However, when the pump is fed from the forward direction, the MFC is sensitive to Δ. In experiment, as the absolute value of Δ gradually decreases, the nonlinear effects of the system increase dramatically. The number of teeth of MFC increases to a maximum of 18, which demonstrates a clear tooth difference in both directions. In addition, the radiation linewidth in opposite directions varies from 60 to 340 kHz, exhibiting about one order of magnitude difference. PIM is induced by microwave photons, with each MFC tooth capable of exciting PIM that absorbs or emits microwaves of specific wavelengths, thus being observable by our measurements. Since PIM weakly interacts with each other to form a spin ensemble, disturbs the uniform precession of WM, and acts as defects in such dynamics, it resembles a special type of color centers. Due to the clear difference in forward and backward scattering behaviors of our devices, the overall linewidth and intensity variation of its radiation can be controlled by an order of magnitude.

FIG. 4.

(a) and (b) Modulate the cavity photon–magnon coupling system to BIC, and fix the frequency of the perturbation tone under the BIC condition. Then, we sweep the detuning between the pump and perturbation tone from −1.5 to 1.5 MHz and use the MFCs measured by SA in both directions to plot the mapping. (c) The difference in the tooth number of MFCs in each direction at different detuning.

FIG. 4.

(a) and (b) Modulate the cavity photon–magnon coupling system to BIC, and fix the frequency of the perturbation tone under the BIC condition. Then, we sweep the detuning between the pump and perturbation tone from −1.5 to 1.5 MHz and use the MFCs measured by SA in both directions to plot the mapping. (c) The difference in the tooth number of MFCs in each direction at different detuning.

Close modal

We have experimentally achieved the nonreciprocal transmission of MFCs in a nonreciprocal BIC system with the advantage of WM as a carrier of nonreciprocity and nonlinearity. The MFCs are generated by the nonlinearity of PIMs, which couple to the WM, and the nonreciprocity of MFCs is realized through the BIC system with different absorption rates in two opposite directions. Based on our device, we demonstrate three controlling dimensions to enhance the nonreciprocal performance of MFCs. Through appropriate tuning, the nonreciprocity of MFCs becomes strongest when the system is in BIC and the pump frequency is equal to the BIC frequency, where the system possesses the maximum absorption rate difference. In addition, the nonreciprocal characteristic of MFC transmission can be further enhanced by tuning the detuning of the two PIMs to enhance the nonlinearity of the system itself. Ultimately, we achieve a nonreciprocal behavior of 4 comb teeth vs 18 comb teeth in both directions. This nonreciprocal behavior is very obvious. The difference in tooth numbers of MFCs in two directions is several times the tooth number of the MFC with fewer teeth, which clearly demonstrates the feasibility of nonreciprocal transmission and tuning of MFCs. Our method is promotable and is expected to make the nonreciprocity increase further with a more optimized design of power, detuning, and nonlinearity of the device.

Expanding MFCs into the nonreciprocal regime can effectively shield external disturbances and suppress backscattering, which facilitates the stable unidirectional transmission. This could stimulate the ideas of spin-wave accumulation and signal amplification. By incorporating techniques such as increasing the tooth number,19 they can together contribute to higher precision in frequency-domain measurements. In addition, nonreciprocal MFCs could further enable the design of multi-channel magnon-based diodes, paving the way for constructing complex logic operations.36 Inspired by the theoretical progress,20,36 our experimental realization could demonstrate the advantages of enhancement of PIM nonlinearity at low power levels, offering the feasibility for external controlling approaches due to the open electromagnetic environment and the prospect of being extended to various systems where interference is controllable. This study of nonreciprocal MFCs is expected to provide strong support for high-precision measurements and transmission and processing of coherent information.

See the supplementary material for the theoretical model to describe the asymmetric absorption rate and its nonreciprocal MFC.

This work was supported by the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDB0580000), the National Natural Science Foundation of China (Grant Nos. 12122413, 12227901, 12204306, U23A6002, and 12474120), STCSM (Grant Nos. 23JC1404100 and 22JC1403300), the National Key R&D Program of China (Grant Nos. 2022YFA1404603), and the Shandong Provincial Natural Science Foundation, China (Grant No. ZR2024YQ001).

The authors have no conflicts to disclose.

Kaixin Zhao: Conceptualization (lead); Data curation (lead); Investigation (lead); Methodology (lead); Writing – original draft (lead); Writing – review & editing (lead). Fan Yang: Data curation (equal); Investigation (equal); Writing – review & editing (equal). Chenxiao Wang: Data curation (equal); Investigation (equal); Writing – review & editing (equal). Zhijian Chen: Data curation (equal); Investigation (equal); Methodology (equal); Writing – review & editing (equal). Jiantao Song: Data curation (equal); Investigation (equal). Shuhuan Ma: Data curation (equal); Investigation (equal). Zixuan Yue: Data curation (equal); Investigation (equal). Weihao Liu: Data curation (equal); Investigation (equal). Liaoxin Sun: Data curation (equal); Investigation (equal). Jinwei Rao: Data curation (equal); Investigation (equal); Writing – review & editing (supporting). Bimu Yao: Data curation (supporting); Project administration (equal); Validation (lead); Writing – review & editing (supporting). Wei Lu: Funding acquisition (lead); Resources (equal); Supervision (equal); Validation (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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