The growing divide between the demand for computing resources and the performance of digital hardware necessitates the development of post-complementary metal–oxide–semiconductor (CMOS) architectures that can achieve ultra-high computational throughput at ultra-low energies. An extreme example of this comes from the field of deep learning, where the computation required to train state-of-the-art deep neural networks grew by over 300,000× between 2015 and 2020, doubling every 3.4 months^1,2, whereas the efficiency of graphics processing units (GPUs) grew^3,4,5 by only 300-fold. Many distinct photonic architectures have been proposed and demonstrated to attempt to address this major bottleneck facing the field of AI and computing⁶, namely, matrix-vector multiplication (MVM). The vast majority of these approaches can be classified as weight-stationary photonic processors⁷, where a matrix of programmable optical weights—typically encoded in a 2D array of non-volatile memory elements or optical modulators^{8,9,10,11,12,13,14,15}—is used to perform a linear transformation on a vector of optical inputs. This design approach has the benefit of performing computation in the memory array itself, not unlike analogue computing in the electrical domain using crossbar arrays of resistive random access memory (RRAM), which can substantially reduce data movement and latency while improving energy efficiency¹⁶.

However, a crucial (yet seldom mentioned) limitation of these weight-stationary approaches is the time and energy required to update the fixed weights of the matrix to implement useful computing algorithms. Unlike electronic crossbar arrays, which aim to fully map the entire neural network into analogue weight banks to minimize reprogramming of the array, optical weight banks have much lower storage density (for example, ~0.01 bits μm^–2; ref. ¹⁷) and can only store a tiny fraction of the necessary parameters on-chip (for example, a 16 × 16 phase-change memory array¹⁸ or 64 × 64 Mach–Zehnder interferometer array⁸). Thus, to accommodate the large number of parameters required by real-world applications, the photonic weight bank must be reprogrammed many times for each matrix operation that exceeds the dimensions of the weight bank. This requires photonic memory cells that can be deterministically programmed quickly, efficiently and with high endurance, or they will dramatically reduce throughput and energy efficiency of the entire system^7,19,20.

To address these challenges, we propose a resonance-based photonic architecture (inspired by the broadcast and weight design⁹) that leverages the non-reciprocal phase shift in magneto-optical materials to implement photonic in-memory computing. In this architecture (Fig. 1a) we excite both the clockwise (CW) and counterclockwise (CCW) modes of a micro-ring resonator (MRR) with a magneto-optic cladding layer (cerium-substituted yttrium iron garnet or Ce:YIG). The interaction between the optical mode and Ce:YIG causes a non-reciprocal phase shift for the two counter-propagating modes, which is visible as a split resonance shift with opposite signs, dependent on the direction and strength of an applied magnetic field. This approach has several benefits. First, we can efficiently achieve programming speeds of ~1 GHz, non-volatility, and multi-level encoding, as demonstrated in this work (see Fig. 1b and Table 1 for a comparision with the state of the art). Second, the cycling endurance of magnetic-based memory is known to be orders of magnitude greater than competing technologies²¹, an outstanding challenge for many non-volatile optical memories²². Finally, unlike add–drop MRR weights that are based on reciprocal optical effects⁹ (for example, thermo-optic or plasma-dispersion effects), the differential signal is measured from the through-port transmission for both the CW and CCW modes, improving the symmetry and extinction ratio for both positive and negative weighting.

a, Our vision for a non-reciprocal photonic computing platform that leverages high-speed magneto-optic memory arrays to enable fast weight updates. In this platform, radio frequency (RF), analog or digital electronic signals are converted to the optical domain through electro-optic (E/O) modulators. A single row from our proposed architecture, which leverages non-reciprocal memory cells to compute a dot-product optically, is illustrated below. Balanced photodetectors (BPDs) convert the differential optical signal back to the electrical domain which then can be further processed using CMOS logic coupled to high-speed, static random-access memory (SRAM). b, Speed versus energy comparison of state-of-the-art non-volatile photonic memory technologies that have been demonstrated on-chip. c, Cross-section of non-reciprocal magneto-optic memory. Either wafer bonding and chemical mechanical polishing (left) or growth and patterning of amorphous silicon on Ce:YIG substrates (right) can be used to heterogeneously integrate Ce:YIG with silicon waveguides. In both cases, high-quality Ce:YIG is guaranteed by growing the garnet on a native substrate of substituted gadolinium gallium garnet (SGGG). SOI, silicon-on-insulator. d, Illustration of two implementations used to demonstrate magneto-optic memory cells where both counter-propagating modes are excited in the MRR. Optical circulators are used to prevent back reflection to the optical source during measurements. e, Positive and negative weights are encoded by switching the magnetization direction and amplitude, which results in opposite resonance shifts for the CW and CCW counter-propagating modes.

Integration of the Ce:YIG layer with the photonic circuit can be achieved either through wafer bonding and polishing²³, or by deposition and patterning of an amorphous silicon layer²⁴ (Fig. 1c). Programming the state of the memory cell requires a radial in-plane magnetic field that is supplied by an integrated gold electromagnet. To maintain a non-volatile state without dissipating power, a ferromagnetic thin-film (CoFeB) patterned into an array of bar magnets can be integrated with the electromagnet on-chip^24,25. A magneto-optic memory cell can be implemented using either the add–drop MRR or all-pass MRR configurations (Fig. 1d). Figure 1e shows example spectra from a non-reciprocal memory cell under a negative (left) and positive (right) radial-applied magnetic field. In the case of positive magnetization (({M}_{x} > 0)), the forward propagating CCW mode (that is, same propagation direction as current flow in the electromagnet) red-shifts while the CW mode blue shifts. If the optical probe is red detuned from resonance when ({M}_{x}=0), the resulting differential transmission encodes a negative weight. The opposite is true for negative magnetization (({M}_{x} < 0)). For a critically coupled MRR, this approach can achieve high transmission contrast, limited by the extinction ratio of the CW and CCW modes.

The functionality of our non-reciprocal memory cell can be extended beyond the single dot-product shown in Fig. 1a to MVM operations. In Fig. 2 we compare two broadcast and weight architecture designs featuring non-reciprocal (Fig. 2a) and reciprocal (Fig. 2b) MRR-based weights. Here, the matrix operation ({W}bf{x}=bf{b}) is achieved through fan-out of the optical input vector to each row of (W). This input vector is then multiplied by the wavelength-dependent transmission of each resonator in the row before being summed through incoherently combining the transmitted power at each row’s pair of output waveguides as originally proposed by Tait and co-workers⁹. An important distinction between the two approaches can be visualized in the transmission spectra of the bus waveguides. Although the differential photocurrent compares the CW and CCW through ports in Fig. 2a, the differential transmission of the through and drop ports is used to compute (bf{b}) in Fig. 2b. This is an important distinction because the drop port of the reciprocal MRR weight reaches its maximum extinction ratio at a phase shift of π when the optical probe is centred at resonance. Thus, to achieve high optical contrast between the through and drop ports (that is, to improve the bit precision of the weight), a much larger phase shift is required in the case of a reciprocal MRR compared with a non-reciprocal MRR. Although this may not be a limiting factor for a single memory cell, reducing the phase shift required to achieve the full range of positive and negative weight values in the array reduces the optical cross-talk between neighbouring resonances²⁶ and alleviates the challenge of achieving strong optical modulation on-chip.

a, Non-reciprocal photonic computing platform leveraging an integrated magneto-optic memory array. Matrix-vector multiplication is achieved by taking the differential transmission between the CW and CCW propagating modes. Scale bar, 30 μm. b, Reciprocal ‘Broadcast and weight’ architecture, which uses the difference between the through and drop ports of an add–drop MRR (scale bar, 15 μm) to encode the values of matrix W. In both approaches, a wavelength multiplexer (MUX) is used to combine the input optical signals into a single waveguide. c, Simulated map of differential through-port transmission for a non-reciprocal memory cell with Q = 10,000. d, Encoded weight value of non-reciprocal memory cell for an optical probe spaced 0.5, 1.0 and 1.5 FWHM away from resonance (dashed lines in d). Notably, the encoded value is an anti-symmetric function centred at zero phase shift. e, Simulated map of the difference between the through and drop ports of a reciprocal memory cell with Q = 10,000. f, Encoded weight value of reciprocal memory cell when the optical probe is held at resonance (white dashed line in e). A larger phase shift is needed to achieve symmetric weighting compared to the non-reciprocal case.

This distinction is highlighted in Fig. 2c–f where we simulate the differential transmission of both reciprocal and non-reciprocal optical memory with the same quality factor (Q = 10,000). In Fig. 2c we see that the differential transmission for both positive and negative values is an anti-symmetric function centred at (phi =0). This function is shown in Fig. 2d for three different optical probe wavelengths: 0.5×, 1.0× and 1.5× the full-width half maximum (FWHM) linewidths red detuned from resonance when (phi =0). As the detuning of the probe increases, the maximum and minimum weight values increase in magnitude, while the linearity of the weighting function near (phi =0) decreases. The differential transmission for the case of a reciprocal memory cell is shown in Fig. 2e. The reciprocal weighting function is symmetric and centred at (phi =0), requiring a resonance shift of ~0.5 × FWHM to reach negative values when the probe is centred at resonance (Fig. 2f). Thus, to achieve an equal range of positive and negative weights (that is, minimal power penalty²⁶), a phase shift of (Delta phi =0.17uppi) is required for an ideal reciprocal MRR with the same quality factor of Q = 10,000.

The current in the integrated electromagnet used to control the resonance of the MRR gives rise to a magnetic field and a Joule heating effect. A comprehensive model of the magneto-optic and thermo-optic effect is provided in Supplementary Section 1. The modelling results of Fig. 2 were experimentally verified for a MRR with a Ce:YIG layer (Q ≈ 10,000) and an integrated electromagnet. For this demonstration, the ring radius is 35 µm and the waveguide cross-section is a 600 nm × 220 nm silicon ridge with a 400-nm-thick top-cladding of Ce:YIG. A 10-nm-thin silicon oxide layer separates the silicon from the Ce:YIG layer (see Supplementary Section 1.4). Figure 3a shows the optical transmission spectra for the fundamental transverse magnetic CW and CCW propagating modes as a function of applied current. The resonance position of the spectra shows both a linear and quadratic dependence on the applied current corresponding to the magneto- and thermo-optic effects, respectively. In Fig. 3b,c we separate the resonance shifts of the CW and CCW modes into their non-reciprocal (magneto-optic) and reciprocal (thermo-optic) components. Although the thermo-optic effect red-shifts both the CW and CCW spectra, the magneto-optic effect induces shifts in opposite directions for the two modes. For a set current, the thermo-optic shift is estimated as the average shift of the two spectra compared with the no-current scenario. The magnitude of the magneto-optic shift is half of the measured resonance split between the CW and CCW modes. In Fig. 3b,c we overlay the mathematical model with the measurement results, showing an excellent agreement between theory and experiments. Figure 3d plots the FWHM linewidth of the resonator for the CW and CCW modes, showing a similar quality factor for both modes. By varying the applied current in the electromagnet, we observe slight changes in the extinction ratio in Fig. 3a and linewidth in Fig. 3d. These variations are caused by the non-reciprocal loss in the Ce:YIG, where the optical loss changes depending on the direction of light propagation and the transverse magnetic field^27,28,29. Please refer to Supplementary Section 1 for more details.

a, Spectral response of a non-reciprocal magneto-optic memory cell for different fixed currents (Q ≈ 10,000). Both linear and quadratic resonance shifts are observed due to the magneto-optic and thermo-optic effects, respectively. b, Magneto-optic spectral shift for the CW and CCW modes. Dashed lines correspond to our analytical model of the expected magneto-optic phase shift excluding thermal effects. c, Thermo-optic phase shift (extracted from a) resulting from heating the electromagnet while a constant current is applied. d, Extracted resonance linewidth as a function of applied current for the CW and CCW modes. Slight changes in the non-reciprocal loss for different propagation directions change the quality factor of the MRR. e, Encoded weight for a probe wavelength (λ_p) red-detuned 0.5×, 1.0× and 1.5× FWHM away from the central resonance when no magnetic field is applied. Experimental weight values are in good agreement with the modelled results in Fig. 2d.

In Fig. 3e we plot the differential optical transmission of the CW and CCW modes for three different optical probe wavelength positions: 0.5×, 1.0× and 1.5× the FWHM linewidths red detuned from resonance when (phi =0) (that is, no magnetic field). As expected, there is good overlap between the experimental results and the simulated weighting functions for the scenario of an ideal resonator with a similar quality factor (Fig. 2d). Again, we see that increasing the red detuning of the optical probe increases the dynamic range of the weight function due to reduced insertion loss while also decreasing the linearity between the maximum and minimum weight values. At larger current values, we observe a deviation from the expected weight value. We attribute this to the fact that, for large detuning, the two modes have different FWHM, as shown in Fig. 3d, resulting in a distinct weighting for the two directions.

Having characterized the steady-state response of our non-reciprocal optical memory, we next demonstrate high-speed weight updates through characterization of the memory cell’s dynamic response. For high-speed characterization beyond ~1 MHz, the magneto-optic response dominates while the dynamic thermo-optic response becomes negligible. This can be seen in Fig. 4a, in which the dynamic optical transmission of a CW probe for two current pulses can be observed at different time scales (1 ms versus 10 ns pulse width). In the case of a slow 1 ms current pulse, the optical transmission includes both a blue shift (increase in transmission) from the fast magneto-optic response and a red shift (decrease in transmission) from the slow thermo-optic response. For a fast current pulse and red-detuned probe, the slow decay from the thermal response of the ring disappears (rise time ≈ 50 μs and fall time = 92 μs), and we only observe the fast magneto-optic response with a rise and fall time of less than 1 ns, and a ferromagnetic resonance of 0.55 GHz. As the estimated time response of the integrated electromagnet circuit is 6 ps, the ringing observed in the optical response is attributed to magneto-optic response of the Ce:YIG, resulting in a rise/fall time of 0.95 ns (see Supplementary Section 3 for more details).

a, Comparison of dynamic response of thermo- and magneto-optic effects, demonstrating a five-orders-of-magnitude difference in response time (red-detuned CW probe used for both measurements). From the thermo-optic response, the fall time is 92 µs, whereas the rise time is 50 µs. For high-speed modulation above ~1 MHz, the thermo-optic effect is negligible provided the average power dissipated remains constant. The magneto-optic response can be fit with a second-order response with a natural angular frequency of 3.6 Grad s^–1 and a dimensionless damping factor equal to 0.29. From these results, we estimate a rise/fall time of 0.95 ns and a ferromagnetic resonance of 0.55 GHz (see Supplementary Section 3 for more details). b,c, Eye diagrams for clockwise and counterclockwise propagating modes for pseudorandom binary sequence modulation at bit rate r_b = 500 Mbps (b) and 1 Gbps (c) speeds. d, Simultaneous measurement of CW and CCW transmission for PAM4 modulation at 500 Mbps. Optical transmission of the CW and CCW modes are shown in red and blue, respectively, while the current applied to the electromagnet is shown in green. e, Plot of differential optical power between CW and CCW signals demonstrating the ability to rapidly update non-reciprocal multibit optical weights. f, Simultaneous measurement of CW and CCW transmission for binary modulation (on–off keying (OOK)) at 1 Gbps. g, Differential optical power between CW and CCW signals demonstrating programming speeds up to 1 ns.

Due to the high-speed magneto-optic response and the soft in-plane magnetic axis of the Ce:YIG, the memory cell can be programmed with low energy. Figure 4b,c shows open eye diagrams measured for both the CW and CCW optical probe for 500 Mbps and 1 Gbps modulation speeds (see Supplementary Section 2 and Supplementary Fig. 5 for details on the measurement set-up). From Fig. 4b we can see that for a weight update rate of 500 Mbps (that is, 2 ns pulse width), we can achieve an open eye diagram with an extinction ratio as high as 8.3 dB for a programming energy as low as 2.28 pJ. Reducing the pulse width to 1 ns and the amplitude by half reduces the programming energy to a mere 298 fJ (Fig. 4c). This corresponds to a ~8× improvement in energy efficiency because ({E}_{rm{b}}={I}^{2}R times Delta t+L{I}^{2}/2), where (E_{rm{b}}) is the energy per bit, (I) is the programming current, (R) is the resistance of the electromagnet (~1.43 Ω), (L) is the inductance (0.3 nH) and (Delta t) is the duration of the programming pulse. It is also worth noting that for our non-reciprocal memory cell, we achieve substantial modulation of the optical signal with a peak voltage as low as ±21.5 mV for ±13.8 mA current pulses, representing an extremely low programming voltage that is compatible with the most advanced CMOS nodes.

In Fig. 4d,e we demonstrate multi-level optical weighting using a four-level (2-bit) pulse amplitude modulation (PAM4) programming signal with maximum current amplitude of ±13.8 mA, corresponding to a record low programming energy of 143 fJ per bit. In these experiments we capture the transmission of the CW and CCW modes simultaneously using the experimental set-up described in Supplementary Fig. 6. The differential transmitted power between the CW and CCW modes is shown in Fig. 4e, in which we clearly observe four distinct transmission levels, allowing us to achieve to two positive and two negative optical weights given our 2-bit electrical input. Using the high-speed magneto-optic effect, we can achieve optical weight updates as fast as 1 ns, as shown in Fig. 4f,g. In our case, the maximum programming speed is limited by the ferromagnetic resonance of the Ce:YIG. Although in this device the maximum time response is 1 GHz, a much faster programming speed can be reached using other magneto-optic material systems that can support modulation rates of tens of gigahertz^30,31.

In a final demonstration, we show the non-volatile response of our non-reciprocal memory cell when integrated with a switchable ferromagnetic layer. In this experiment, patterned CoFeB magnetic stripes are integrated in the cladding above the MRR to provide a programmable, non-volatile magnetic field. The shape anisotropy and orientation of the micrometre-sized CoFeB bar magnets provides the static radial magnetic field needed to induce a non-reciprocal optical phase shift in the memory cell²⁴. In this demonstration, the silicon MRR has a cross-section of a 1,000 nm × 220 nm with a radius of 50 µm. The measurement set-up and cross-section of our device are shown in Fig. 5a. To program the state of the memory cell, current is applied to the integrated electromagnet that encodes the magnetic field strength in the CoFeB magnetic domains. After the current is removed, the laser is swept to obtain the non-volatile MRR spectral shift of the CW mode (Fig. 5c–e). For an optical probe positioned at the resonance dip of the MRR (dashed lines in Fig. 5c,d), we observe the hysteretic behaviour in the optical transmission shown in Fig. 5b. When the programming current is increased from negative to positive values, we observe an increase in optical transmission for currents greater than 0 mA (red points in Fig. 5b). The transmission eventually saturates above 200 mA and remains constant due to saturation of the CoFeB magnetic layer (see the illustration in Fig. 5a). As we change the direction of the applied current from positive to negative values, the transmission remains constant until negative current values, highlighting the non-volatile response of the memory cell. The results in Fig. 5b show at least 11 distinct optical transmission levels corresponding to a non-volatile memory cell capable of storing ~3.5 bits; however, this value is limited by our experimental set-up rather than the device itself. In Supplementary Section 4 we provide a theoretical analysis of the magnetic, thermal and optical noise and expect the maximum bit precision of our device to exceed 13 bits.

a, Experimental set-up for non-volatile weight encoding. The magnetic field from the gold electromagnet aligns the magnetic domains of the CoFeB ferromagnetic layer according to the amplitude and polarity of the applied current. b, Hysteresis of optical transmission for a non-reciprocal memory cell with CoFeB bar magnets integrated on-chip. For these transmission measurements, the probe wavelength was fixed (black dashed lines in c and d). c, Negative to positive programming current sweep showing a blue shift for positive currents. d, Positive to negative current sweep showing a return back to the original resonance position. e, Waterfall plot of transmission spectra for sequential programming currents sweeping first in the positive and then in the negative directions. f, Demonstrated cycling endurance for over 2.4 billion write and erase cycles. Mean values of the non-volatile write and erase states are shown as black data points, whereas grey bands represent the standard deviation. Time domain measurements in g show the operation of the memory cell without observing any degradation. Variation in the observed extinction ratio is attributed to thermal drift of the unpackaged device over the three-day measurement.

The intermediate spectra obtained for a sweep from negative to positive current values is shown in Fig. 5c, whereas the reverse sweep from positive to negative is shown in Fig. 5d. The dashed lines in Fig. 5c,d represent the position of the optical probe used to plot the hysteresis of the through-port transmission (Fig. 5b). At this probe wavelength, we observe a maximum extinction ratio of 16.2 dB between the minimum and maximum non-volatile states. To better visualize the spectral dependence on programming current, we sequentially plot the optical transmission spectra for the full current sweep as a waterfall graph in Fig. 5e. As the current increases from negative to positive values, the MRR resonance blue shifts until saturation. Reversing the current from positive to negative values causes a red shift back to the original resonance position.

To demonstrate the ultra-high cycling endurance of our non-reciprocal memory cell, we programmed an arbitrary function generator to cycle between write and erase pulses at a rate of 10 kHz, with an amplitude of ±5 V and a pulse width of 500 ns. The optical transmission was captured manually during the three-day experiment and is shown in Fig. 5f (see Supplementary Section 2.2 for more experimental details). After 2.4 billion write and erase cycles, the device continued to function without any sign of degradation (Fig. 5g). This is a greater than three orders of magnitude improvement over past photonic memory technologies, highlighting the benefit of using optically coupled magnetic media for non-volatile data storage. Although we do see a variation in the extinction ratio of the device in Fig. 5f,g, we attribute this to slight thermal and mechanical drift of the unpackaged device during the three-day experiment. To confirm the long-term non-volatile stability of our device, we also compared the non-reciprocal spectral shift of the CW and CCW modes after programming and see data retention over a four-day measurement (see Supplementary Section 2.2). Opto-electronic packaging with active thermal control is expected to counter drift of the MRR resonance peak, which would address this observed variation in the extinction ratio.

We have demonstrated the first instance of a non-volatile magneto-optical memory cell that features non-reciprocity for in-memory computing in the optical domain. This unique combination of fast, fatigue-free programmability with non-volatile weights addresses current limitations of existing integrated approaches to optical memory that have yet to combine: (1) non-volatility, (2) multibit storage, (3) high switching speed, (4) low switching energy and (5) high endurance, in a single platform. In Table 1 we compare state-of-the-art demonstrations of various non-volatile photonic memory technologies capable of being integrated on-chip. It is worth noting that although some recent demonstrations of waveguide-integrated electronic memristors (termed memresonators) have shown great promise for fast and efficient non-volatile switching between two or three states^32,33, the maximum cycling endurance of such devices has not been demonstrated beyond 1,000 write–erase cycles. These devices are also not well suited for photonic computing applications where multi-level, non-volatile storage is necessary. Although phase-change memory has shown great promise for optical computing platforms due to its compact footprint, multi-level storage and ease of integration^15,34,35,36, the limited cycling endurance, high switching energies and limited switching speeds remain outstanding challenges. Other non-volatile photonic memory cells leveraging charge trapping, MEMS, or ferroelectric materials have sub-megahertz programming speeds which are not yet practical for computing applications due to considerable write latencies.

Compared with these competing photonic memory technologies, our non-reciprocal magneto-optic memory cell offers an efficient non-volatile storage solution that could provide unlimited read/write endurance at sub-nanosecond programming speeds. Our initial results used integrated electromagnets to switch a non-volatile ferromagnetic layer, however, we expect future implementations employing spin–orbit–torque or spin–torque-transfer effects could further improve the switching efficiency of our magneto-optic memory cells and provide a direct optical interface to emerging magnetic and spintronic memory technologies. Although bonding Ce:YIG on silicon-on-insulator wafer and depositing amorphous silicon on Ce:YIG are currently the best fabrication techniques for integrating high-quality Ce:YIG in silicon photonics, recent strides in the monolithic integration of magneto-optic garnet on silicon and silicon nitride substrates offer a pathway to further enhancements in the near future³⁷. By precisely depositing magneto-optic materials on specific areas, we can further reduce the insertion loss and achieve higher integration density for non-reciprocal photonic computing.

The transmission spectra of magneto-optic memory cell in Fig. 3 is modelled using the transfer matrix method, in which the effective index of the CW and CCW modes is computed using a finite-element method. The resonance of the MRR is controlled by the current in the integrated electromagnet, giving rise to a magnetic field and a Joule heating effect. The magneto- and thermo-optic effects alter the effective index of the modes, and their impact is described using a perturbative approach. A comprehensive description of the model employed is provided in Supplementary Section 1.

To fabricate the magneto-optic material used for all devices, a 500-nm-thick single-crystalline Ce:YIG (Ce:Y₂Fe₅O₁₂) was epitaxially grown on a wafer of (111)-oriented (Ca, Mg, Zr) SGGG using an radio frequency sputtering method at 750 °C. This magneto-optic garnet has a large Faraday rotation of 4,800 degrees per centimetre at 1,550 nm and was used for all of the devices presented in this work.

The devices characterized in Figs. 3 and 4 were fabricated by bonding a Ce:YIG/SGGG on a 220-nm-thick SOI wafer with 2 μm buried oxide. The SOI wafer is patterned using a 248 nm ASML 5500 deep-ultraviolet stepper, and dry etched using a Bosch process (Plasma-Therm 770) to form the waveguides and resonators. Patterned SOI and Ce:YIG/SGGG samples are rigorously cleaned, and activated with O₂ plasma (EVG 810). Ce:YIG is directly bonded onto the SOI wafer using a flip-chip bonder (Finetech) and then annealed at 200 °C for 6 h under 3 MPa to strengthen the bond. The required alignment accuracy is fairly tolerant (~200 μm). After bonding, a 1 μm layer of SiO₂ is sputtered everywhere on the chip as an upper cladding. The SGGG substrate is then thinned by mounting the sample against a flat chuck and polishing (Allied Technologies) using a series of increasingly fine lapping films. The thickness of SGGG is monitored using a micrometre and confirmed to be ~5 μm with a separate Dektak profilometry measurement. The variation of thickness across the sample is roughly ±1.5 μm due to imperfect levelling of the chuck. The patterns for gold coils and contacts are defined on the backside of the SGGG with a 365 nm GCA i-line wafer stepper. Then, using electron-beam evaporation, 22 nm of titanium is deposited as an underlayer, followed by 1.5 μm of gold, and the metal coils and contacts are released by a lift-off procedure. Finally, the sample is diced and the facets are polished.

Non-volatile magneto-optic memory cell characterized in Fig. 5 is fabricated growing amorphous silicon (a-Si) on a Ce:YIG/SGGG wafer. A 10-nm-thick SiO₂ buffer layer is deposited onto a Ce:YIG/SGGG wafer via plasma-enhanced chemical vapour deposition (PE-CVD). Next, a 220-nm-thick a-Si:H guiding layer is deposited via PE-CVD with a gas mixture of SiH₄ at 300 °C. Subsequently, a 200-nm-thick SiO₂ layer is deposited as a hard mask to protect the a-Si:H layer, and a 300-nm-thick positive resist (ZEP-520A), as well as a charge-dissipating agent (ESPACER), are coated onto the substrate. Waveguide patterns are exposed to the resist using an electron-beam lithography system. The waveguide patterns are transferred to a SiO₂ hard mask via reactive ion etching using CF₄ and a-Si:H waveguides are formed using SF₆. A 750-nm-thick SiO₂ layer is deposited on the top of the waveguide core to isolate the guided mode from the integrated thin-film magnet to avoid optical absorption. Electron-beam lithography is performed to transfer the magnet patterns of an array of 20 µm × 5 µm stripes. A 10-nm-thick ruthenium buffer layer followed by a 300-nm-thick CoFeB thin-film magnet are deposited using an radio frequency facing target sputtering method at room temperature with argon. Next, the stripe array of thin-film magnets is formed using the lift-off process. The longer side of each stripe, which is the easy magnetization axis of CoFeB, is aligned perpendicular to the waveguide. Finally, after the deposition of an 80-nm-thick SiO₂ layer using PE-CVD, a 25-µm-wide and 700-nm-thick Cr/Au electromagnet for magnetizing integrated magnets was formed via electron-beam vapour deposition.

The complete simulation code and all simulation files required to reproduce the results presented in Fig. 2 is available at https://nonreciprocalringresonators.github.io.