Nonreciprocal metasurface with space–time phase modulation

Creating materials with time-variant properties is critical for breaking reciprocity that imposes fundamental limitations on wave propagation. However, it is challenging to realize efficient and ultrafast temporal modulation in a photonic system. Here, leveraging both spatial and temporal phase manipulation offered by an ultrathin nonlinear metasurface, we experimentally demonstrated nonreciprocal light reflection at wavelengths around 860 nm. The metasurface, with travelling-wave modulation upon nonlinear Kerr building blocks, creates spatial phase gradient and multi-terahertz temporal phase wobbling, which leads to unidirectional photonic transitions in both the momentum and energy spaces. We observed completely asymmetric reflections in forward and backward light propagations over a large bandwidth around 5.77 THz within a sub-wavelength interaction length of 150 nm. Our approach highlights a potential means for creating miniaturized and integratable nonreciprocal optical components.


S1.1 Design of the spatiotemporal phase modulated metasurface
Numerical simulations were carried out using a commercially available finite element method (FEM) solver package -COMSOL Multiphysics -with periodic boundary conditions for a single building block. Third-order finite elements and at least 10 mesh steps per wavelength were used to ensure the accuracy of the calculated results. The experimentally obtained optical constants of silver and amorphous silicon were used to model the back reflector and the nanoantennas. The refractive index of the spacer layer (SiO2) was chosen to be 1.45. The scattered field formulation was used to calculate the reflected light at λ = 860 nm. By sweeping the size of nanoantennas (lx and ly), we mapped out the static phase shifts which cover over 2π range, and selected three nanoantennas to construct a unit supercell (Fig. 2b). Fig. S1a illustrates the simulated electric field distribution of a supercell at normal incidence (λ = 860 nm), showing a smoothly slanted wavefront. We also simulated the diffraction efficiency under different incident angles which are transformed into wavevectors along the x axis. As shown in Fig. S1b, the metasurface keeps a high diffraction efficiency above 84% near normal incidence (within +/− 10 degrees).
The nonlinear simulation takes two major steps in a customized model using COMSOL Multiphysics: (1) We simulated the spatially dependent permittivity change using an iterative scheme -We first calculated the field distribution in the computational domain with the pump beam incidence, then updated the permittivity in the Kerr medium (n2 = 5 × 10 -13 cm 2 W -1 1 ) with the calculated inhomogeneous field, and after that we calculated the field again with modified permittivity. We iterated over the steps above until the change of the field distribution was within a predefined tolerance. As shown in Fig. S1c, pumped at a peak intensity of 15 GWcm -2 by 800 nm light, the nanoantenna shows a significant permittivity change thanks to the resonanceenhanced local field and large nonlinear Kerr index of amorphous silicon. (2) We then simulated the structure with the pump-induced permittivity changes with a probe light incidence. With the nonlinear simulation, we obtained the reflected field distribution of an 860 nm probe beam upon incidence on an arrayed nanoantennas whose permittivity has been modified by the pump beam.
An observable wavefront shift, d = (λ/2π)Δφ, related to the phase shift change is shown in Fig.   S1d. With increasing pumping intensity, the phase shifts induced by the three elemental nanoantennas show a uniform change (Fig. S1e), which is important for keeping a well-aligned dynamic phase modulation across the whole antennas array (Fig. S1f).
In order to map the position of the pickup mirror relative to the tangential momentum kx of the reflected beam, we used a raytracing software package (Zemax) to simulate the ray trajectory of light focused by the aspheric lens used in our experiments. By varying the distance between the incident beam and the axis of the lens, different focusing angle θ after the lens are obtained and transformed into kx by . As shown in Fig. S1g, this method has a better precision than the calculation made by treating the aspheric lens as a perfect thin lens, especially for off-axis rays.

S1.2. Finite-difference time-domain (FDTD) simulations of the dynamic phase modulation induced nonreciprocal photonic conversions
To verify the theoretical predictions of equation (2) In the backward propagation scenario, we sent back the two converted signals from the forward propagation case. Here, as we had two input fields, the field distribution of the outgoing waves would be a superposition of multiple plane waves. In order to clearly visualize the outgoing waves, we separated the converted waves arisen from the two backward inputs and presented their electric field distributions at modulated wavelengths in Fig S2c). None of the backreflections return to the state of the original forward incidence, which proves that the spatiotemporal phase modulated metasurface is nonreciprocal. In the second set of simulations, we studied the case where only one direction of transition is allowed. Fig. S3 shows the simulation results of dipoles array with ks = 0.78kprobe, kM = 2/3 ks = 0.52kprobe , Δω = 2.8 THz, and Δφ = 0.2 radians. In the forward propagation, the normal incidence (λ = 860 nm) is converted to two outgoing waves: the static diffraction and the first-order downconversion. The up-conversion corresponds to a non-propagating mode that cannot radiate into free space. As shown in the field plot, this evanescent mode cannot carry energy away from the metasurface and no propagating field can be observed in free space. In the backward case, the incident wave is the backward of the down-conversion (Fig. S3a). Only the down-conversion at λ = 874 nm can be observed at the normal direction. Similarly, the up-conversion results in a nonpropagating mode that cannot carry energy away.

S4. Scattering matrix of the space-time phase modulated metasurface
In our design, nonreciprocity is achieved through asymmetric photonic transitions induced by the spatiotemporal phase modulation. In the main text, we demonstrated in theory and experiment that unidirectional photonic transitions can break the reciprocity. Although the up and down conversions can coexist under certain conditions, e.g. small kM and small ks, the nonreciprocal effect arisen from the time-dependent phase modulation of metasurface will not be compromised.
We will demonstrate the nonreciprocity by deriving the scattering matrix of the system. As shown in Fig. S7, we considered our metasurface as a four-port system, where s1+ is the amplitude of the incident wave (ωi, kix), s2-is the amplitude of the static diffraction (ωi, kix + ks), s3-is the amplitude of the down-conversion (ωi -Δω, kix + ks -kM) and s4-is the amplitude of the up-conversion (ωi + Δω, kix + ks + kM). Table 1 summarizes the frequency and wavevector of each port.

Port number Input s+
Output s- As mentioned in the main text, the reflected light from the metasurface will acquire a dynamic phase and be decomposed into components of different frequencies and wavevectors, as expressed below: where ζ η = and η is the static diffraction efficiency of the metasurface, i E  is the electric field of the incident wave, Δφ is the amplitude (or modulation depth), ks is the static phase gradient of the metasurface, Δω is the dynamic modulation frequency and kM is the modulation wavevector.
High-order modes are not considered here since they are very weak with small phase modulation depth. We also omitted the specular reflection because the metasurface diffraction efficiency is close to unity around the designed wavelength. According to equation (S1), we can relate the si-to si+ by using the following equations: Therefore, the scattering matrix of the four-port system can be written as: The common factor s ik x e is omitted since it does not influence the form of the scattering matrix.
It is evident that the scattering matrix is asymmetric (S13 ≠ S31 and S14 ≠ S41) due to the asymmetric photonic transitions arisen from spatiotemporal phase modulation. Thus, the system breaks reciprocity. S12 (S21) represents the insertion loss to the nonreciprocal system. Under ideal condition, it could be reduced to zero when J0(Δφ) = 0 (e.g. Δφ = 2.405 radians).
In order to validate the scattering matrix obtained directly from field calculation, we also studied the dynamic system based on the coupled mode theory 3,4 . Considering the same four-port system shown in Fig. S7b, the incident wave through port 1 will directly excite the resonant mode a0: Therefore, the scattering matrix of the system can be expressed as: Because Δω (~ 2.8 THz × 2π) is much smaller than ω0 and the bandwidth of the metasurface resonance is much larger than Δω, we can assume 0 γ γ γ γ − + = = = . In addition, we can also assume By comparing equation (S9) with equation (S3), we found that the scattering matrices derived from two difference methods are of the same form, in which S12 = S21, |S31| = |S41|, S13 = S14 = 0, as well as S13 ≠ S31 and S14 ≠ S41. Therefore, both derivations reach the same conclusion -the dynamically phased modulated metasurface possesses an asymmetric scattering matrix and is nonreciprocal.
In order to prove the validity of the scattering matrix we derived above, we experimentally demonstrated the nonreciprocal light reflection when both up and down-conversions are presented.
As shown in Fig. S8a, the normal incident light (green) has Stokes (red) and anti-Stokes (blue) shifts, whose frequencies are shifted down and up by Δω, respectively. The reverse of the case above will consist of two paths: (a) the backward propagation of the up-converted signal (blue) and (b) the backward propagation of the down-converted signal (red). First, we consider path (a).

The up-converted signal (blue) is sent back towards the metasurface and again has Stokes (green)
and anti-Stokes (purple) shifts, where the Stokes (green) reflects at an angle ( 0 sin 2 M k k θ = − ) different from the normal direction, and the anti-stokes (purple) exits at the same angle of the original forward incidence but with frequency shifted up by 2Δω. We can see that not all light returns to the angle same as the original incidence even if we do not consider the shifts in frequency. It is sufficient to conclude that our system is nonreciprocal because the reciprocity of the system requires both the Stokes and anti-Stokes shifts go back to the original incident state. Path (b) will have similar processes, and we will not elaborate here for the sake of conciseness. therefore cannot be detected in our experiment. In the backward propagation case, we sent back the anti-Stokes signal (λ = 865 nm and kx = 0.17 kprobe) onto the metasurface. It was partially upconverted to a mode of λ = 857 nm and kx = 0, and partially down-converted to a mode λ = 873 nm and kx = -0.62kprobe. Although the frequency of the down-converted signal coincides with that of original forward incidence, its transverse wavevector is different. Therefore, none of the backconverted signals go back to the state of the original forward incidence (λ = 873 nm and kx = 0), and hence the system is nonreciprocal.

S5. Tunable dynamic modulation
As briefly mentioned in the main text, the optical dynamic modulation is very flexible, where Δω, kM and ks can be easily modified. Different Δω can be obtained by varying the size of block attached to the center of the V-shaped split mirror. For instance, as shown in Fig. S9a, by using different size blocks, we get wavelength differences of 6 nm and 5 nm. Due to the experimental limitation, we have a limited range of Δω. However, we can realize a large and tunable Δω with two different optical parametric amplifiers (OPA) 6 , which can be used to build a tunable frequency-shifting optical isolator.
Meanwhile, kM can be changed by adjusting the angle between the two pumps. Fig. S9b displays the Fourier transform analysis of interference patterns (insets) with varying kM. It should be noted that the interference pattern is produced by the common frequency components in each pump, otherwise it would not be possible to observe the interference fringes because the pattern moves at high speed.
To demonstrate the importance of modifying kM, we measured the dispersion diagram with kM = 0.23kprobe and ks = −0.48kprobe shown in Fig. S9c. It is clear that the reflected light experiences both upward and downward transitions because they both satisfy the phase matching condition and their tangential momenta are allowed in free space. By calculation, kM must be at least 0.5kprobe for a normal incident input to achieve unidirectional photonic transitions in both forward and backward propagations. This is readily attainable by our experiment setup. In addition, we demonstrated arbitrary photonic transitions by using different kM and ks. As shown in Fig. S10 a-d, at fixed ks = 0.72 kprobe, the tangential wavevector of the down converted signal decreases with increased kM, as determined by kx = ks -kM . By the same token, when kM is fixed at 0.31kprobe, kx can be changed by varying ks (Fig. S10, e-g). This provides us with great flexibility in designing angle-shifting optical isolators.
We note that in Fig. S10 the conversion efficiency decreases from10 -4 to 10 -5 as kM increases. Since the two pumps are separated further apart to create increased kM, the focusing spot of the pumps is broadened as a result of the larger intersection angle between two pumps and distorted due to worse lens aberrations. Therefore, the peak pump intensity gradually decreases at the same input pump power, resulting in a reduced efficiency. Meanwhile, at larger kM, the period of dynamic modulation decreases and covers fewer nanoantennas in a single period, which further decreases the efficiency. In addition, with the same kM, the conversion efficiency increases as ks increases. A larger ks corresponds to a smaller supercell period; therefore more nanoantennas can be accommodated in a unit dynamic modulation period, which leads to a stronger modulation strength and higher efficiency. The converted signals are magnified by 10 4 or 10 5 for better illustration.

S6. Analysis of the operational bandwidth of space-time metasurfaces
The bandwidth of time dependent nonreciprocal systems is usually determined by the phase matching conditions and modulation frequencies (more accurately, dynamic modulation induced frequency shift). For example, in ref. 7,8 , bandwidths of a few hundred GHz were achieved by dispersion engineering at the expense of large device footprint, so that the phase matching condition was fulfilled over a larger bandwidth comparing with their counterparts 9 . It should be noted that this bandwidth is valid under monochromatic excitation. For broadband pulses the bandwidth is limited by the modulation frequency since we need to avoid overlap between the main pulse and its temporal-modulation-induced sidebands, which will cause signal distortion [10][11][12] . Our space-time metasurface works with optical modes in the continuum (not discrete guided modes) and naturally fulfills phase matching conditions in a broad frequency range. Besides, the meta-atoms in our system has a low quality factor (< 50) which allows the device to operate at a relatively broad bandwidth. In addition, our ultrafast modulation method features a modulation frequency Δf ~ 2.8 THz, which ensures at most 5.6 THz (2× Δf) bandwidth with broadband excitation (Fig. S13). To sum up, with either a narrowband or broadband excitation, our metasurfaces exhibit several THz bandwidth, which is at least one order of magnitude greater than the largest ones(a few hundred GHz) 7,8 reported on time dependent nonreciprocal systems.
We measured the dispersion diagrams for both forward and backward reflections with a monochromatic probe ranging from 854 to 914 nm (Fig.S11). By fitting the converted signal efficiencies at different wavelengths, we obtained a − 3 dB bandwidth (full-width at halfmaximum, FWHM) of ~5.77 THz (875 -890 nm), which is determined by the broad optical resonance linewidth (Fig. S12). It should be noted that here we used a different sample from the one, which has degraded over time, used in the main text. The resonance of this sample shifted from 860 nm (Fig. S5) to 880 nm (Fig.S12). The backward reflection suffers lower conversion efficiency, because of the poor spatial overlap between probe and pump beams and the decreased diffraction efficiency of metasurface at oblique incidence. But these dispersion diagrams unequivocally show nonreciprocal reflections across a large wavelength range.   (frequency shift between the forward incidence and the back-reflection) is larger than ΔfFWHM of the pulse, the back-reflection (solid red line) can be distinguished from the forward incident light (dashed blue line). Therefore, robust nonreciprocity is preserved with a bandwidth of at most 2Δf.

S7. Analysis of the conversion efficiency and optimization methods
Roughly speaking, the first order conversion efficiency is proportional to J1(∆φ) 2 according to our theory (equation (2)). We compared the efficiency at different peak pump intensities with theory as shown in Fig. S14. The efficiency was on the order of 10 -4 with the pump intensity below the damage threshold of the samples. It started to deviate from the theoretical predication when the peak power intensity increases beyond 1 GWcm -2 using our high repetition rate (80 MHz) laser due to the possible thermal damages. Nevertheless, the conversion efficiency (dashed line) increases super-linearly with increasing peak power intensity, which leads to a tremendous boost in efficiency if pumped with a low-repetition-rate and high energy laser. In addition, the pulse width of the probe is stretched to 2 ps due to a series of nonlinear effects in the PCF. The focal spot of the probe is about 5 times larger than that of the pumps. Therefore, the converted signal only comes from a small fraction of the incident probe. By improving the temporal and spatial quality of probe, the conversion efficiency can be further increased. Despite being limited by the experimental conditions, the conversion efficiency we achieved is still two orders of magnitude greater than the efficiency of the third-order nonlinear generation reported to date in amorphous silicon nanostructures at comparable pump intensities. 13 We also carried out a control experiment on a bare amorphous Si film but were not able to detect any converted signals, which confirmed that the amorphous Si resonant nanoantennas greatly enhanced the nonlinear induced dynamic phase change. In addition, as shown in our FDTD simulation results (Fig. S4), the conversion efficiency reaches up to 100% when J0(∆φ = 2.405) 2 equals to zero, which is attainable with optimized nonlinear materials and metasurface designs. Here we propose to increase modulation depth and conversion efficiency with a low pump power requirement using the following methods.
First, we can use materials with large nonlinearity to construct the meta-atoms. It has been reported 14 that ITO in the ENZ region has extremely large nonlinear Kerr index that is around two orders of magnitude greater than that of amorphous silicon. A recent study 15 combined ITO thin film with gold nanoantennas and demonstrated even higher effective Kerr index (~ 3.73 cm 2 GW -1 ). To achieve a similar nonlinear phase shift (~ 0.03 radians) as demonstrated in our experiment, the required pump power intensity is below 100 MWcm -2 with the ITO-antenna system. In addition, this ITO-nanoantenna system achieves a maximum nonlinear phase shift of 0.68 radians at pump intensity of 3.27 GWcm -2 (energy density ~ 6.1 pJµm -2 ). According to equation (2) in our paper, the conversion efficiency can be: This conversion efficiency is about three orders of magnitude greater than that achieved in our experiment. Furthermore, by using high-quality-factor resonant nanoantennas and ENZ materials with low damping factors, the nonlinear phase shift can be further increased even at a moderate pump intensities 16,17 . It is worth noting that this conversion efficiency is realized with a subwavelength interaction length ~ 50 nm (total thickness of gold nanoantenna and ITO film). In comparison, a recent demonstration 8 on travelling modulation induced nonreciprocal system shows mode conversion efficiency of 1% at the cost of on-chip optical driving power of 90 mW and an interaction length of 2.39 cm.
Second, with further optimized meta-atom designs using amorphous silicon, we can get much larger phase modulation depth at lower pump intensity. Therefore, we are able to achieve same nonreciprocal effect with much lower pump power. We designed a new amorphous silicon metaatom of which the working wavelength is around 1.55 µm. Our full-wave simulations show that we can achieve a very large phase shift change (Δφ) with a relatively low pump intensity (Fig.   S15). Notably, at intensity of 0.4 GWcm -2 (resulting Δφ = 2.4 radians) with this new design, we achieve 100% mode conversion efficiency (no residual 0 th order reflection remains). In addition, doubly resonant nanoantennas can be designed to have enhanced local field intensity at both pump and probe wavelengths, leading to a relaxed pump power requirement. Last but not least, interaction time/length can be increased by stacking two space-time modulated metasurfaces to form a cavity. Photons are trapped inside the cavity to enable much longer interaction time with the travelling-wave modulation, leading to a boosted nonreciprocal conversion efficiency. In addition, a metasurface integrated resonator system can be used to increase the effective interaction length, For example, we can integrate metasurfaces on top of a micro-ring resonator, and apply a travelling wave modulation across the metasurface. Photons circulating inside the resonator will interact with the travelling-wave modulation for a time determined by the quality factor (Q-factor ~ 10 3 ~ 10 6 ) of the ring resonator, which is much larger than that of our current resonant antennas (Q-factor ~ 40).