Angular-momentum-biased magnetless nonreciprocity at terahertz and optical frequencies: modulation trade-offs, actuator-minimal integrated designs, and modeling beyond the omega_m much smaller than omega_1 coupled-mode approximation

Sounas, Caloz and Alù (Nature Communications 4, 2407, 2013) proposed 'angular momentum biasing' - an azimuthal, traveling-wave spatiotemporal modulation of the permittivity of a resonant ring - as a route to giant, linear, magnet-free nonreciprocity in subwavelength components, and demonstrated the concept through coupled-mode theory and full-wave simulations from radio to optical frequencies. The paper itself left three issues open: (1) the scheme with azimuthal modulation order $L_m=1$ requires a modulation frequency $\omega_m$ close to the signal resonance frequency $\omega_1$, which 'may be challenging to achieve especially at terahertz and optical frequencies'; (2) the proposed optical isolator design (azimuthal order $L_m=22$, conceptually 88 pin diodes (fast semiconductor switching elements used as electrically driven permittivity modulators) at 65 nm pitch along a silicon ring) was explicitly not optimized, with a 'significantly lower number' of actuators conjectured to suffice; (3) the coupled-mode analysis neglects coupling to higher-order azimuthal modes under $\omega_m\ll\omega_1$ and assumes approximately uniform fields and permittivity in the modulated regions, leaving the behavior when these approximations are violated unexplored for practical layouts. Later work has realized angular-momentum-biased circulators in CMOS microwave electronics and demonstrated the first optical isolators based on angular momentum biasing (17 dB in silicon nitride at about 4 W of RF drive) and on closely related dynamic-modulation schemes (15-30 dB in thin-film lithium niobate and travelling-wave electro-optic architectures). What remains open is the completion of the original program: magnet-free angular-momentum-biased isolators and circulators at terahertz frequencies, where no time-modulation-based demonstration exists; power-efficient, broadband, low-insertion-loss integrated implementations with minimized actuator counts; and validated, unified design theory for the regimes in which the 2013 approximations fail.

Background

Reciprocity is the symmetry of a linear, time-invariant transmission channel between two ports: interchanging source and observer leaves the response unchanged (the Lorentz reciprocity theorem of electromagnetism). Devices that break this symmetry - isolators (one-way transmitters) and circulators (ports connected only in a cyclic order) - are essential for protecting sources from back-reflections and for full-duplex operation. Today they are almost exclusively realized with magneto-optical materials biased by static magnetic fields via the Faraday effect (non-reciprocal polarization rotation), which is hard to integrate on semiconductor chips and incompatible with, e.g., superconducting circuits. The Onsager-Casimir principle states that any quantity that is odd under time reversal (a magnetic field, but also linear or angular momentum) can bias a medium into a non-reciprocal state.

Sounas, Caloz and Alù (2013) exploited this by 'angular momentum biasing': a ring-shaped resonator supporting degenerate clockwise/counter-clockwise azimuthal modes $|\pm k\rangle$ (eigenmodes proportional to $e^{\pm ik\phi}$, where $k$ is an integer azimuthal order and $\phi$ the azimuthal angle) is modulated by a permittivity perturbation $\Delta\varepsilon(\mathbf{r},t)=\Delta\varepsilon_m\cos(\omega_m t-L_m\phi)$, a traveling modulation wave that circulates around the ring with effective angular momentum $L_m$ and modulation frequency $\omega_m$. Because the modulation carries angular momentum, it couples $|+k\rangle$ and $|-k\rangle$ asymmetrically, splitting the degenerate pair by $\pm\kappa_m$ (with coupling rate $\kappa_m\approx\Delta\varepsilon_m/2\varepsilon$), which combined with a high quality factor $Q$ (ratio of stored energy to dissipated energy per cycle, $Q=\omega_1\tau$ for energy relaxation time $\tau$) yields strong nonreciprocal transmission - giant Faraday-like rotation and isolation - at subwavelength footprints without magnetic bias. The analysis used temporal coupled-mode theory (CMT, the standard first-order description of amplitudes of nearly isolated resonant modes), valid when $\omega_m\ll\omega_1$ (so that coupling to higher-order azimuthal modes, which would be triggered by the intermodulation products at $\omega_k\pm\omega_m$, is negligible) and when the modulation occupies regions where the field and permittivity are nearly uniform. Because amplitude modulation generates intermodulation sidebands at $\omega_k\pm\omega_m$, one might instead choose $L_m=1$ with $\omega_m=\omega_2-\omega_1\approx\omega_1$ to resonantly couple the neighboring azimuthal orders $|+1\rangle$ and $|+2\rangle$ asymmetrically; the authors note that such a modulation frequency, close to the signal frequency, is hard to reach, especially at terahertz (THz) and optical frequencies. For $L_m=2$ the two counter-propagating states couple resonantly already at $\omega_m=0$, so the modulation frequency can be arbitrarily small - but the modulation strength $\Delta\varepsilon_m$ still must be large enough, relative to the mode linewidth $1/\tau$, to separate the split states. In their optical proof of concept, a channel-drop filter (a resonant ring that transfers a narrow, selected wavelength band from one straight waveguide to another) with a silicon ring of radius comparable to the wavelength was azimuthally modulated with order $L_m=22$, conceptually realized by 88 pin diodes (fast semiconductor switches used as electrically driven permittivity modulators) spaced 65 nm apart along the ring; the authors state this design was not optimized and that far fewer diodes may suffice.

Since 2013, spatiotemporally modulated (space-time) media have become a large research area, surveyed in the review literature (e.g. Sounas and Alù, Nature Photonics 2017, on time-modulation routes to non-reciprocal photonics; Caloz and co-workers, Physical Review Applied 2018, a tutorial on electromagnetic nonreciprocity). Angular-momentum-biased circulators have been realized and integrated in CMOS at microwave frequencies; magnet-free optical isolation based on time modulation has been demonstrated experimentally, alongside adjacent magnetless routes (optomechanics, i.e. coupled optical and mechanical resonances; stimulated Brillouin scattering, i.e. scattering of light from coherent traveling acoustic waves; staggered commutation, i.e. time-interleaved capacitor switching that emulates a rotating non-reciprocal element; and Josephson parametric circuits, i.e. superconducting resonators with parametric frequency conversion). At terahertz frequencies, by contrast, demonstrated isolators still rely on magneto-optics (e.g. magnetized indium antimonide (InSb)), and the time-modulation route remains without a demonstration. The open issues identified in the 2013 paper therefore map today onto a narrower but still precise frontier: extending angular-momentum-biased nonreciprocity to THz frequencies, and making the optical implementations practical (low RF drive power, broadband, low insertion loss, few actuators), supported by modeling that remains valid when $\omega_m$ is not small compared with $\omega_1$ and when the modulation is strongly non-uniform or spatially discrete.

Problem Statement

The source-faithful problem, as left open by Sounas, Caloz and Alù (Nature Communications 4, 2407, 2013), is to complete the practical development of angular-momentum-biased (azimuthally spatiotemporally modulated) magnetless nonreciprocity, whose three linked open issues share one objective - making the giant subwavelength non-reciprocal response usable in integrated devices, especially where the modulation is hardest, at terahertz and optical frequencies:

1. Modulation-frequency and modulation-strength trade-offs: for azimuthal modulation order $L_m=1$, resonant asymmetric coupling of neighboring azimuthal modes requires the modulation frequency $\omega_m$ to be close to the ring resonance frequency $\omega_1$ (more precisely $\omega_m=\omega_2-\omega_1$, where $\omega_2$ is the resonance frequency of the neighboring azimuthal mode of order two), which the authors flag as challenging especially at terahertz and optical frequencies; for $L_m=2$ the frequency can be small but the modulation strength $\Delta\varepsilon_m$ must still split the degenerate pair beyond the linewidth. Determine the realizable routes and quantitative trade-offs (isolation, bandwidth, footprint, drive power) for angular-momentum-biased non-reciprocity when the modulation frequency is not small compared with the signal frequency or when the available modulation amplitude is weak - in particular, design and quantitatively validate a linear, magnet-free, integrable non-reciprocal isolator or circulator at terahertz frequencies, a regime in which no time-modulation-based isolator or circulator has yet been experimentally realized.

2. Optimized, fabricable integrated designs: the 2013 optical isolator concept (a silicon channel-drop filter ring with azimuthal order $L_m=22$, conceptually 88 pin diodes spaced 65 nm along the ring) was explicitly presented as unoptimized, with a 'significantly lower number' of actuators conjectured to suffice. Develop and demonstrate optimized device architectures that minimize the number, pitch and drive power of the discrete actuators while preserving isolation at the level of the demonstrated magnet-free isolators (about 17 dB, with related dynamic-modulation schemes reaching 15-30 dB), and establish their fabrication feasibility at nanoscale pitch and at terahertz/optical frequencies, together with the system-level verification (insertion loss, noise, spurious sidebands, drive power) that would make them practical alternatives to magneto-optical components; where an architecture trades isolation away for fewer actuators, lower drive power or larger bandwidth, report the achieved isolation, bandwidth and insertion loss explicitly against these benchmarks rather than against an unquantified 'large isolation'.

3. Theory beyond the 2013 approximations: the original coupled-mode analysis neglects coupling to higher-order azimuthal modes under the assumption $\omega_m\ll\omega_1$ and treats the modulation as concentrated where the field and permittivity are approximately uniform. Establish and validate a quantitative description - and the associated design constraints - of angular-momentum-biased devices in the regimes where these approximations fail: $\omega_m$ comparable to $\omega_1$, strongly non-uniform or spatially discrete modulation, and parasitic intermodulation-induced coupling to higher-order modes, for realistic integrated layouts.

Items 1-3 are facets of one scientific objective: a practically usable, magnet-free, linear, subwavelength non-reciprocal technology based on angular momentum biasing across the electromagnetic spectrum. Microwave-frequency integration is largely done; the problem's open core sits at terahertz frequencies, in the efficiency and actuator-minimization of optical implementations, and in the validated modeling of the regimes excluded by the original approximations. The three items are independently standing sub-goals of this one objective: the modulation-frequency and modulation-strength trade-off study together with a terahertz isolator or circulator (item 1), the optimized, fabricable, actuator-minimal architecture with its system-level performance metrics (item 2), and the validated theory beyond the 2013 approximations (item 3). The terahertz isolator-or-circulator deliverable belongs to item 1; item 2 concerns the actuator-minimal architecture itself. Throughout, 'realize', 'demonstrate', 'verify' and 'device' denote a quantitatively validated design - with isolation, bandwidth, insertion loss, noise, spurious sidebands and drive power computed from a self-consistent coupled-mode, Floquet or full-wave treatment, as in the 2013 paper's coupled-mode-theory and full-wave-simulation demonstration - rather than a fabricated and measured component, so a validated theoretical/numerical result is the intended deliverable.

References

1. Dimitrios L. Sounas, Christophe Caloz, Andrea Alù. "Giant non-reciprocity at the subwavelength scale using angular momentum-biased metamaterials." Nature Communications 4, 2407 (2013). DOI: 10.1038/ncomms3407. https://doi.org/10.1038/ncomms3407
2. Nicholas A. Estep, Dimitrios L. Sounas, Jason Soric, Andrea Alù. "Magnetic-free non-reciprocity and isolation based on parametrically modulated coupled-resonator loops." Nature Physics 10, 923–927 (2014). DOI: 10.1038/nphys3134. https://doi.org/10.1038/nphys3134
3. Nicholas Aaron Estep, Dimitrios L. Sounas, Andrea Alu. "Magnetless Microwave Circulators Based on Spatiotemporally Modulated Rings of Coupled Resonators." IEEE Transactions on Microwave Theory and Techniques 64, 502–518 (2016). DOI: 10.1109/TMTT.2015.2511737. https://doi.org/10.1109/TMTT.2015.2511737
4. Ahmed Kord, Dimitrios L. Sounas, Zhicheng Xiao, Andrea Alu. "Broadband Cyclic-Symmetric Magnetless Circulators and Theoretical Bounds on Their Bandwidth." IEEE Transactions on Microwave Theory and Techniques 66, 5472–5481 (2018). DOI: 10.1109/TMTT.2018.2860023. https://doi.org/10.1109/TMTT.2018.2860023
5. Ahmed Kord, Mykhailo Tymchenko, Dimitrios L. Sounas, Harish Krishnaswamy, Andrea Alu. "CMOS Integrated Magnetless Circulators Based on Spatiotemporal Modulation Angular-Momentum Biasing." IEEE Transactions on Microwave Theory and Techniques 67, 2649–2662 (2019). DOI: 10.1109/TMTT.2019.2915074. https://doi.org/10.1109/TMTT.2019.2915074
6. Negar Reiskarimian, Harish Krishnaswamy. "Magnetic-free non-reciprocity based on staggered commutation." Nature Communications 7, 11217 (2016). DOI: 10.1038/ncomms11217. https://doi.org/10.1038/ncomms11217
7. Benjamin J. Chapman, Eric I. Rosenthal, Joseph Kerckhoff, Bradley A. Moores, Leila R. Vale, J. A. Mates, Gene C. Hilton, Kevin Lalumière, Alexandre Blais, K. W. Lehnert. "Widely Tunable On-Chip Microwave Circulator for Superconducting Quantum Circuits." Physical Review X 7, 041043 (2017). DOI: 10.1103/PhysRevX.7.041043. https://doi.org/10.1103/PhysRevX.7.041043
8. Romain Fleury, Dimitrios L. Sounas, Caleb F. Sieck, Michael R. Haberman, Andrea Alù. "Sound Isolation and Giant Linear Nonreciprocity in a Compact Acoustic Circulator." Science 343, 516 (2014). DOI: 10.1126/science.1246957. https://doi.org/10.1126/science.1246957
9. Dimitrios L. Sounas, Andrea Alù. "Angular-Momentum-Biased Nanorings To Realize Magnetic-Free Integrated Optical Isolation." ACS Photonics 1, 198 (2014). DOI: 10.1021/ph400058y. https://doi.org/10.1021/ph400058y
10. Dimitrios L. Sounas, Andrea Alù. "Non-reciprocal photonics based on time modulation." Nature Photonics 11, 774 (2017). DOI: 10.1038/s41566-017-0051-x. https://doi.org/10.1038/s41566-017-0051-x
11. Christophe Caloz, Andrea Alù, Sergei Tretyakov, Dimitrios Sounas, Karim Achouri, Zoé-Lise Deck-Léger. "Electromagnetic Nonreciprocity." Physical Review Applied 10, 047001 (2018). DOI: 10.1103/PhysRevApplied.10.047001. https://doi.org/10.1103/PhysRevApplied.10.047001
12. Nima Chamanara, Sajjad Taravati, Zoé-Lise Deck-Léger, Christophe Caloz. "Optical isolation based on space-time engineered asymmetric photonic band gaps." Physical Review B 96, 155409 (2017). DOI: 10.1103/PhysRevB.96.155409. https://doi.org/10.1103/PhysRevB.96.155409
13. Ian A. D. Williamson, S. Hossein Mousavi, Zheng Wang. "Dual-Carrier Floquet Circulator with Time-Modulated Optical Resonators." ACS Photonics 5, 3649–3657 (2018). DOI: 10.1021/acsphotonics.8b00576. https://doi.org/10.1021/acsphotonics.8b00576
14. Arezoo Zarif, Khashayar Mehrany, Mohammad Memarian, Hesam Heydarian. "Optical isolation enabled by two time-modulated point perturbations in a ring resonator." Optics Express 28, 16805–16821 (2020). DOI: 10.1364/OE.392914. https://doi.org/10.1364/OE.392914
15. Arezoo Zarif, Khashayar Mehrany, Mohammad Memarian, Kambiz Jamshidi. "Design Guidelines for a Tunable SOI Based Optical Isolator in a Partially Time-Modulated Ring Resonator." IEEE Photonics Journal 14, 1–11 (2022). DOI: 10.1109/JPHOT.2022.3207876. https://doi.org/10.1109/JPHOT.2022.3207876
16. Hao Tian, Junqiu Liu, Bin Dong, J. Connor Skehan, Michael Zervas, Tobias J. Kippenberg, Sunil A. Bhave. "Hybrid integrated photonics using bulk acoustic resonators." Nature Communications 11, 3073 (2020). DOI: 10.1038/s41467-020-16812-6. https://doi.org/10.1038/s41467-020-16812-6
17. Manav Shah, Ian Briggs, Pao-Kang Chen, Songyan Hou, Linran Fan. "Visible-telecom tunable dual-band optical isolator based on dynamic modulation in thin-film lithium niobate." Optics Letters 48, 1978–1981 (2023). DOI: 10.1364/OL.482635. https://doi.org/10.1364/OL.482635
18. Oğulcan E. Örsel, Gaurav Bahl. "Electro-optic non-reciprocal polarization rotation in lithium niobate." APL Photonics 8, 096107 (2023). DOI: 10.1063/5.0146238. https://doi.org/10.1063/5.0146238
19. Kyunghun Han, Yiliang Bao, Junyeob Song, David A. Long, Sean Bresler, Daron Westly, Jason Gorman, Thomas LeBrun, Kartik Srinivasan, Vladimir Aksyuk. "Integrated optical isolators for broadband multi-laser operation." Nature Photonics 20, 1107–1115 (2026). DOI: 10.1038/s41566-026-01978-0. https://doi.org/10.1038/s41566-026-01978-0
20. Freek Ruesink, Mohammad-Ali Miri, Andrea Alù, Ewold Verhagen. "Nonreciprocity and magnetic-free isolation based on optomechanical interactions." Nature Communications 7, 13662 (2016). DOI: 10.1038/ncomms13662. https://doi.org/10.1038/ncomms13662
21. JunHwan Kim, Seunghwi Kim, Gaurav Bahl. "Complete linear optical isolation at the microscale with ultralow loss." Scientific Reports 7, 1647 (2017). DOI: 10.1038/s41598-017-01494-w. https://doi.org/10.1038/s41598-017-01494-w
22. Huanan Li, Tsampikos Kottos, Boris Shapiro. "Floquet-Network Theory of Nonreciprocal Transport." Physical Review Applied 9, 044031 (2018). DOI: 10.1103/PhysRevApplied.9.044031. https://doi.org/10.1103/PhysRevApplied.9.044031
23. Michael Kreiczer, Yakir Hadad. "Green's function theory of spatiotemporally modulated loaded wire surface." Physical Review Research 5, 043256 (2023). DOI: 10.1103/PhysRevResearch.5.043256. https://doi.org/10.1103/PhysRevResearch.5.043256
24. Raana Sabri, Hossein Mosallaei. "High-quality-factor space-time metasurface for free-space power isolation at near-infrared regime." Advanced Photonics Nexus 2, 066008 (2023). DOI: 10.1117/1.APN.2.6.066008. https://doi.org/10.1117/1.APN.2.6.066008
25. Adam Mock, Dimitrios Sounas, Andrea Alù. "Tunable Orbital Angular Momentum Radiation from Angular-Momentum-Biased Microcavities." Physical Review Letters 121, 103901 (2018). DOI: 10.1103/PhysRevLett.121.103901. https://doi.org/10.1103/PhysRevLett.121.103901
26. Hadiseh Nasari, Hady Moussa, Yoshiaki Kasahara, Arno Thielens, Andrea Alù. "Observation of Floquet rotational super-radiance." Nature 655, 608–616 (2026). DOI: 10.1038/s41586-026-10725-y. https://doi.org/10.1038/s41586-026-10725-y
27. Amirhossein Babaee, Zachary Fritts, Steve M. Young, Anthony Grbic. "Analysis of Multi-Tone, Multi-Conductor, Spatially Discrete Traveling-Wave Modulated Loop Networks." IEEE Transactions on Antennas and Propagation (2026). DOI: 10.1109/TAP.2026.3707304. https://doi.org/10.1109/TAP.2026.3707304