Introduction

Non-reciprocity in ferromagnetic materials originates from the unidirectional precession of electron spins1 or the Zeeman splitting of atomic orbitals2 in the presence of a static magnetic bias. There is a large interest in mimicking this effect in artificial materials without requiring applied magnetic fields, but instead relying on bias with other odd vectors under time reversal in compliance with the general principle of microscopic reversibility3. Such possibility would relax the current requirements of large magnetic biasing devices in non-reciprocal components, with the potential of revolutionizing the microwave and photonic component industry. Kodera et al.4, for instance, present a non-reciprocal metamaterial composed of transistor-loaded ring resonators, where suitably designed active loads, biased with direct electric current, can suppress one of the two azimuthally propagating eigenstates, producing a unidirectionally rotating magnetic moment functionally equivalent to ferromagnetic effects. Although this approach can be used to realize some non-reciprocal components5,6, it is limited to microwave and millimetre-wave frequencies, where transistors are available, and it involves significant power consumption in the transistors’ biasing network. Another transistor-based, current-biased non-reciprocal metamaterial was proposed in ref. 7, consisting of cross-polarized dipoles interconnected to each other via transistors to achieve symmetry breaking and microwave Faraday rotation, with similar limitations.

8,9,10,11,12,13Refs 8–13 present a different class of non-reciprocal components. In this case, biasing is provided by the linear momentum vector imparted by longitudinal spatiotemporal modulation suitably tailored to break time-reversal symmetry. This concept can provide optical isolation by means of direct or indirect interband photonic transitions in spatiotemporally modulated waveguides. Direct transitions10 can be described as the photonic equivalent of the Aharonov–Bohm effect, whereas indirect transitions8 occur in the presence of a traveling wave modulation breaking space-inversion and time-reversal symmetries. The proposed devices can be broadband but their size is determined by the coherence length of the transitions, which, for reasonable modulation amplitudes, is several wavelengths, making these devices rather bulky. Nonlinearities have also been proposed to induce non-reciprocity14,15,16,17,18 but the operation of the resulting devices is limited to specific ranges of input power. All these solutions rely on inherently weak effects, requiring large volumes, and are therefore not necessarily preferable to conventional solutions based on magneto-optical effects. The field of optical metamaterials has also offered opportunities for boosting the non-reciprocal response of magneto-optical materials by means of large field enhancement and localization. Two recent examples are plasmonic-enhanced Faraday rotation19 and isolation enabled by parity-time symmetry20.

In the following, we suitably combine the strong wave–matter interaction of resonant metamaterial inclusions with the non-reciprocal properties of a new form of spatiotemporal modulation aimed at imparting a bias based on the angular momentum vector. We show that this combination can induce giant non-reciprocal response at the subwavelength scale, leading to the introduction of a new type of linear, non-reciprocal phenomenon based on angular momentum-biased metamaterials, which can overcome the limitations of the aforementioned approaches with potentially groundbreaking applications in microwave and nanophotonic integrated systems.

Results

Principle of angular momentum-biased metamaterials

The simplest constituent inclusion suited to form the proposed metamaterial is a ring resonator with an azimuthal spatiotemporal permittivity modulation Δε(φ,t)=Δεm cos (ωmt−Lmφ), as depicted in Fig. 1a, where φ is the azimuthal coordinate in a cylindrical reference system co-centred with the inclusion and Lm is the modulation orbital angular momentum. In absence of modulation, Δεm=0, the ring supports degenerate counter-propagating resonant states with azimuthal dependence eiφ, resonating when the ring circumference is times the guided wavelength, which implies that, for the fundamental |±1 states used herein, the ring dimensions are smaller than the wavelength. The resonant size can be further reduced by adding capacitances along the loop, as in split-ring resonator designs21. As it will be shown shortly, introducing the aforementioned spatiotemporal azimuthal modulation lifts the degeneracy between the |±1 states and produces non-reciprocity, an effect that can be interpreted as the metamaterial analogue of a static magnetic bias removing the degeneracy between atomic states of opposite orbital angular momenta in magnetic materials.

Figure 1: Principle of non-reciprocal angular momentum-biased metamaterials.
figure 1

(a) Azimuthally symmetric ring resonator with a spatiotemporal modulation of permittivity. The modulation follows the form of an azimuthally propagating wave in the +φ direction. In the absence of modulation, the resonator supports two degenerate counter-propagating states |±1. The applied modulation lifts the degeneracy and produces non-reciprocity. (b) Transformation of the |±1 states in the frequency and angular momentum plane for Lm=1 (left) and Lm=2 (right). Red crosses: states of the unmodulated ring. Blue crosses: intermodulation products. (c) Frequency diagram of the ring eigenstates without (left and right columns) and with (middle column) modulation for Lm=2. The states of the modulated ring are hybridizations of the |±1 states. Each hybrid state consists of a dominant (thick line) and a secondary (thin line) sub-state. (d) Sub-state eigenfrequencies versus the modulation frequency for Lm=2. (e) Sub-state energies versus the modulation frequency for Lm=2.

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The proposed permittivity modulation is a type of amplitude modulation and, as such, it results in the generation of two intermodulation products and for each state |k. If any of them overlaps in frequency with another state , resonant coupling between the |k and state occurs, significantly affecting both resonances. As our goal is to lift the degeneracy between |±1 states, Lm=1 and ωm=ω2−ω1 might appear the most reasonable choice, so that the |+1 state gets resonantly coupled to the |+2 state, whereas no coupling occurs for the |−1 state, as illustrated in Fig. 1b, left panel. However, ω2 is usually close to 2ω1 and, as a result, ωm should be close to ω1, which may be challenging to achieve especially at terahertz and optical frequencies. For Lm=2, in contrast, the states |±1 resonantly couple to each other for ωm=0, as illustrated in Fig. 1b, right panel. For ωm identically zero, the structure is obviously reciprocal but any small departure from zero can break reciprocity and, no matter how large ω1 is, strong non-reciprocal response may be obtained by properly choosing Δεm and the resonator Q-factor, as will be discussed in more detail in the following.

This concept may be analysed using coupled-mode theory as in ref. 22: the amplitudes a1 of the |±1 states satisfy the equations (see Methods for details)

where

is the coupling coefficient between the |±1 states, with St being the resonator cross-section and Et1 the corresponding normalized electric field distribution. For homogeneous resonators (Δεm and ε uniform, where ε is the background permittivity), κm=Δεm/2ε. The solution of equation (1) provides the eigenstates of the modulated ring (see Supplementary Note 1 for details):

where ωα=ω1−Δω/2, ωβ=ω1+Δω/2 and Δω=−ωm. This solution may be extended to take into account the presence of loss and coupling with the excitation signals, as discussed in the Methods section.

The states |α and |β are hybridizations of the non-modulated ring states |+1 and |−1, which are generally characterized by different frequencies and amplitudes, as illustrated in Fig. 1c. In the absence of modulation (ωm=0), the sub-states |±1 of each hybrid state share the same frequency and energy, as shown in Fig. 1d,e, respectively, and the system is reciprocal as expected. However, when modulation is introduced (ωm≠0), the sub-states split (Fig. 1d) and non-reciprocity arises. This splitting follows from the simultaneous spatial and temporal nature of the proposed modulation, generating the states and from |ke−iωt (Fig. 1b). Therefore, if the sub-state |+1 exists at frequency ω, the sub-state |−1 can only exist at frequency ω−ωm. For ωm≠0, the energy is unevenly distributed between sub-states (Fig. 1e) with the unbalance increasing with ωm. The dominant sub-states for |α and |β are |+1 and |−1, respectively, as indicated in Fig. 1c–e with thick lines.

The amount of non-reciprocity is determined by the minimum distance between sub-states of opposite handedness Δωmin=min{ωm,ωβ−ωα=Δω} and by the resonance width ω1/Q, where Q is the resonance quality factor, corresponding to the inverse of the fractional bandwidth. In practical devices such as polarization rotators and circulators, which are based on interference between states, Δωmin and ω1/Q must be of the same order23. In the Supplementary Note 1, we prove that Δωmin≤Δω1κm/, with the maximum value holding for ωm=Δω=ω1κm/. Therefore, Qκm=, consistently with the expectation that a lower Q resonator requires a higher κm and subsequently a higher Δεm.

Microwave Fano-resonant metasurface

Consider now a spatiotemporally modulated metasurface consisting of periodically arranged pairs of broadside-parallel metallic rings patterned on both sides of a thin dielectric layer, as in Fig. 2a. Permittivity modulation is effectively obtained by loading the rings with time-variable capacitors ΔCn=ΔCm cos (ωmt−2φn) at the azimuthal positions φn=nπ/4, where n=0…7, which is equivalent to applying a continuous capacitance modulation ΔC=(4ΔCm/π)cos(ωmt−2φ), as shown in the Methods section. It is noteworthy that the modulation amplitude 4ΔCm/π is the average of the localized capacitance ΔCm over the discretization period π/4, as it may be intuitively expected. A possible practical implementation of this capacitance modulation is illustrated in Fig. 2b: it consists of a varactor, as its core element, a direct current-biasing source, an alternating current modulation source with frequency ωm and appropriate filters that minimize the interference between the ring and the biasing network (see Supplementary Note 2 for details). Such a circuit may be easily integrated into the ring substrate within conventional printed circuit technology. Further, as varactors and filters are low-loss components, the overall power consumption is expected to be very low (in principle zero for ideally lossless circuit elements).

Figure 2: Angular momentum-biased metasurface.
figure 2

(a) Constituent inclusion of the metasurface: a ring-pair resonator periodically loaded with time-variable capacitors. (b) Practical implementation of the time-variable capacitors: a band-stop filter (BSF) and a band-pass filter (BPF) are used to minimize the leakage of ring and modulation signals to the biasing network and to the ring, respectively. The two filters are designed to have zero susceptance and zero reactance at the ring resonance ω1, respectively. (c) Transmission through the unmodulated metasurface. A Fano-type resonance is supported at 8.9 GHz, resulting from the interference of a high-Q magnetic (anti-parallel currents) resonance and a low-Q electric (parallel currents) one. (d) Transmission across the metasurface in the +z direction with ΔCm=0.02 pF and fm=0.1 GHz. Blue line: RHCP incidence. Red line: LHCP incidence. (e) Same as in d but for fm=0.5 GHz. If the propagation direction is reversed (from +z to −z), the RHCP and LHCP curves flip. At 8.91 GHz (dashed line), the metamaterial operates as an electromagnetic diode for CP waves; RHCP and LHCP waves can only propagate along the −z and +z directions, respectively. The geometrical parameters of the structure are R=2 mm, w=1 mm, t=0.4 mm, εs=9 and p=3.4R.

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In the absence of modulation, the metasurface exhibits two resonances, shown in the inset of Fig. 2c, a low-Q ‘bright’ mode at 23 GHz with parallel currents induced in the two rings and a coupled high-Q ‘dark’ mode at 9 GHz with anti-parallel currents. Suitable coupling between these two modes results in a peculiar Fano-resonant response24,25 at 8.9 GHz, with a sharp transition from full to no transmission (Fig. 2c). This response is ideal for our purpose, as its sharp frequency response relaxes the requirements on the modulation capacitance and, at the same time, leads to strong non-reciprocal effects because of the associated anti-parallel currents in the rings, maximizing the excitation of the modulation capacitors.

Figure 2d shows the transmission of circularly polarized (CP) waves through the metasurface (along +z) for ΔCm=0.02 pF and fm=100 MHz. The chosen value of ΔCm corresponds to an effective capacitance modulation of 0.026 pF rad−1, which, considering the ring-pair capacitance 0.48 pF rad−1, leads to κm=0.027. This yields ω1κm0.24 GHz, which is enough for a clear separation between |α and |β states, consistent with the discussion in the previous subsection. Small variations of ΔCm that may occur in practice can be easily compensated by adjusting ωm as long as the |α and |β states are distinguishable.

As expected, the response is different for right-handed CP (RHCP) and left-handed CP (LHCP) excitations, and in each case two resonant dips are observed, with the stronger one resulting from coupling with the state whose dominant sub-state is of the same handedness as the incident wave. RHCP incident waves strongly couple to |α at frequency fα, whereas LHCP waves strongly couple to |β at fβ. The weaker resonant dips correspond to the secondary sub-states at fβ+fm (RHCP excitation, substate |+1 of |β) and fα−fm (LHCP waves, |−1 of |α). If the structure is excited from −z, the transmission curves of Fig. 2d switch handedness as the incident wave feels opposite modulation spin, a clear demonstration of non-reciprocity. This polarization transmission asymmetry can be exploited to realize, for instance, a CP isolator by placing the transmission null of one polarization at the same frequency as the transmission peak of the other polarization. This condition is fulfilled for fm=0.5 GHz, as shown in Fig. 2e; at 8.91 GHz RHCP and LHCP waves can penetrate the metasurface only from −z and +z, respectively. This operation may be the basis of different types of polarization-dependent microwave isolators.

Another important non-reciprocal effect, common in ferromagnetic materials, is Faraday rotation, that is, the non-reciprocal rotation of the polarization plane of a wave, as it propagates through the material. The rotation is opposite for opposite propagation directions, as it is determined by the (fixed) bias direction. We can achieve the same effect in the proposed spatiotemporally modulated metasurface, as shown in Fig. 3a, which plots the polarization rotation angle θ for different fm and ΔCm=0.02 pF. As the modulation frequency increases, θ increases and the bandwidth decreases. The bandwidth reduction is clearly because of the decrease of Δω as fm increases, but the monotonic increase of θ may seem contradictory with the fact that the separation between states, which determines the amount of non-reciprocity, is actually reduced as fm increases. As shown in Fig. 3b, this peculiar monotonic increase of θ results from the fact that at resonance, for fm>0.5 GHz, the transmission coefficient for x-polarized waves Txx decreases faster than the transmission coefficient of x- to y-polarized waves Tyx, so that Tyx/Txx, which is proportional to θ, actually increases. For fm=0.5 GHz, Tyx is maximum and θ=60°, corresponding to a giant rotation of 6,000° per free-space wavelength, without the need of any magnetic bias. In Fig. 3c, we show the ellipticity angle χ at the output for linearly polarized inputs. It is easy to see that this angle is zero at the frequency of maximum θ, implying that the transmitted field is linearly polarized, making the proposed metasurface particularly exciting for applications requiring non-reciprocal linear polarization rotation.

Figure 3: Faraday rotation for the angular momentum-biased metasurface.
figure 3

(a) Rotation angle. (b) Transmission coefficient from x- to x-polarized waves, Txx, and from x- to y-polarized waves, Tyx. (c) Ellipticity angle: at the frequency of maximum rotation the ellipticity angle is zero, corresponding to ideal Faraday rotation. All the results were derived for the same ring parameters as in Fig. 2.

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Optical isolator

As pointed out above, the proposed scheme of spatiotemporal modulation with Lm=2 poses no restriction on the modulation frequency, opening the possibility to apply the proposed concept to optical frequencies. As a proof of concept, an optical isolator based on a spatiotemporally modulated channel-drop filter is presented in Fig. 4, with the design details provided in the caption. Without modulation, the power entering the structure through the channel waveguide (right-hand-side waveguide) from either port 1 or 2 couples to the right- or left-handed ring resonance, respectively, creating a transmission dip at resonance (Fig. 4a). Splitting the ring resonances with proper azimuthal spatiotemporal modulation moves the transmission dips to different frequencies for opposite propagation directions, thus creating non-reciprocity. The isolator can be realized on silicon (Si), which exhibits the strongest electro-optic effect observed to date, with typical values around Δεm=5 × 10−4εs, where εs is its permittivity, leading to κm2.5 × 10−4 (refs 26, 27). Such permittivity modulation can be obtained using pin diodes, as analytically described in refs 9, 26, 27. According to our bandwidth criterion QCm=, a Q-factor ~7,000 would be sufficient for adequate separation of the |α and |β states. Such level of Q-factor is common in Si-photonics integrated systems, and in the design of Fig. 4a it is achieved using the 11th azimuthal resonance of the ring. Although the theory above was derived for the |±1 states, it can be easily extended to any pair of states by substituting Lm=2 with Lm=2. Therefore, Lm=22 was used in the design of Fig. 4, which may be achieved by uniformly integrating 88 pin diodes along the ring perimeter, leading to a separation of 65 nm between consecutive diodes. It should be noted that this design has not been optimized, and our theory indicates that a significantly lower number of pin diodes may still be sufficient to achieve a similar effect in optimized geometries.

Figure 4: Angular momentum-biased optical isolator based on a channel-drop filter.
figure 4

(a) Reciprocal transmission versus frequency of a conventional channel-drop filter. Waves entering from ports 1 and 2 couple to the right- and left-handed resonances of the ring, as illustrated in the left and right insets, respectively, creating a transmission dip at the ring resonance. (b) Non-reciprocal transmission versus frequency when spatiotemporal modulation is applied to the ring. The insets show the electric field amplitude distribution at the frequency indicated by the dashed line for excitation from port 1 (left) or 2 (right). The geometrical parameters of the structure are R=0.88a, w=0.2a and g=0.3a, where a is an arbitrary reference length. The modulation frequency for the case of panel b is fm=2 × 10−4 (c/a). For operation at 1.55 μm, the corresponding absolute values are a=1.04 μm, R=0.92 μm, w=0.21 μm and fm=60 GHz.

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The simulated scattering parameters of the structure without and with modulation are presented in Fig. 4a,b, respectively. In the absence of modulation, S21=S12 (Sij being the transmission coefficient from port j to port i), indicating that the system is reciprocal. When the modulation is applied, the right- and left-handed resonances of the ring split and non-reciprocity occurs, as shown in Fig. 4b. For instance, at the right-handed resonance, indicated in Fig. 4b with the dashed line, transmission from port 1 to 2 is significantly lower than transmission from port 2 to 1, an effect that can also be seen in the corresponding field plots in the inset. This operation is obtained within a ring structure that is comparable in size to the operation wavelength λ=1.55 μm, and without the need of magnetic bias.

Discussion

A new approach to achieve magnetic-free non-reciprocity via angular momentum biasing was presented here on the basis of resonant rings with specifically tailored spatiotemporal azimuthal modulation. The proposed form of modulation removes the degeneracy between opposite resonant states, which, combined with suitably induced high-Q response, realizes giant non-reciprocity in subwavelength components with moderate modulation frequencies and amplitudes. A few applications based on this approach have been presented, including an ultrathin radio frequency isolator, giant Faraday rotation and an optical isolator, all realized without requiring bulky magnetic biasing elements. The proposed approach opens pathways towards many non-reciprocal integrated microwave and nanophotonic components, without requiring magnetic bias, for a variety of applications.

Methods

Coupled-mode equations neglecting loss and excitation

The starting point of our analysis is the coupled-mode theory from Winn et al.22 for a resonator with a small permittivity perturbation Δε(r,t). In particular, if ak, ωk and Ek are the amplitude, the eigenfrequency and the electric field of the kth mode,

where

is the coupling coefficient between the kth and th modes. The modes are normalized as

where Hk is the magnetic field of the kth mode. Noticing that the integral in equation (6) is essentially the energy of the kth mode and that the electric and magnetic energies of a resonator are equal,

The eigenspectrum of the ring of Fig. 1a consists of pairs of counter-propagating modes |±k with electric fields

where Etk is the electric field of the kth pair in the transverse ρ−z plane. Substituting equation (8) into equation (7), we arrive at the following normalization condition for Etk

where St is the ring cross section. For a permittivity modulation Δε(r,t)=Δεm cos (ωmt−2φ) with ωm<<ω1, the coupling between the |±1 modes and the higher-order modes can be neglected and the coupled-mode equations (4) and (5) reduce to equations (1) and (2). Dividing equations (2) and (9) yields

If the modulation is applied to a region, which concentrates most of the field and where ε varies very little, κm=Δεm/(2ε).

Coupled-mode equations including loss and excitation

In the presence of loss, described via a relaxation time τ, the −iω1 term multiplying a1 in equation (1) should be substituted by −i ω1−τ−1 (ref. 28). In addition, when excitation is taken into account the terms −iμ+1s+1 and −iμ−1s−1 should be added to the right-hand side of equation (1), where s+1 and s−1 are the components of the incident wave that couple to the |+1 and |−1 states, respectively, and μ+1 and μ−1 are the corresponding coupling coefficients, which are generally functions of the geometry. Then, the coupled-mode equations read

These equations support a family of solutions similar to those of equation (3).

Discrete capacitance modulation

The ring of Fig. 2a is best analysed in terms of voltage V and current I instead of the electric and magnetic fields E and H. Then, the coupled-mode equations (4) and (5) are expected to be valid under the substitutions E→V, H→I, ε→C and μ0→L, so that the coupling coefficient is given by

where Vk=Vk0eikφ is the voltage of the kth mode, with Vk0 an appropriate normalization coefficient. For the discretely modulated ring,

Then, the coupling coefficients between the −1 and +1 states read

If the modulation was continuous, ΔC=ΔCm cos (ωmt−2φ) and

Comparing equation (14) with equation (15), we conclude that the discrete modulation is equivalent to a continuous one with amplitude 4ΔCm/π.

Full-wave simulations

All the numerical simulations of the varactor-loaded ring were performed in the frequency domain by combining a full-wave finite-element electromagnetic simulation (CST Microwave Studio) with a harmonic balance circuit simulation (Agilent Advanced Design System). The numerical domain was terminated with unit cell boundary conditions along the x and y directions and with wave ports along the z direction. The wave ports were used to excite the incident wave and absorb the reflected and transmitted waves. Further, lumped ports were inserted where the variable capacitors were supposed to be connected. The S-parameters obtained through the electromagnetic simulations were imported into the circuit simulator. Ideal variable capacitors, implemented in Agilent Advanced Design System via equation-based non-linear components as shown in the Supplementary Fig. S1, were connected to the lumped ports of the electromagnetic model and the structure was solved for the S-parameters at the wave ports.

The optical ring of Fig. 4 was simulated with a home-made finite-difference time-domain code, where the time-domain permittivity modulation was realized by continuously changing the permittivity of the structure at each time step. The computational domain was terminated with perfectly matched absorbing layers.

Additional information

How to cite this article: Sounas, D.L. et al. Giant non-reciprocity at the subwavelength scale using angular momentum-biased metamaterials. Nat. Commun. 4:2407 doi: 10.1038/ncomms3407 (2013).