Abstract
Creating materials with timevariant 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 travellingwave modulation upon nonlinear Kerr building blocks, creates spatial phase gradient and multiterahertz 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 subwavelength interaction length of 150 nm. Our approach highlights a potential means for creating miniaturized and integratable nonreciprocal optical components.
Introduction
Reciprocity is a fundamental principle rooted in linear physical systems with timereversal symmetry, requiring that the received–transmitted field ratios are the same when the source and detector are interchanged^{1}. However, it is preferable to break reciprocity in many practical applications, such as lasers and fullduplex communication systems, so that backscattering from defects or boundaries can be avoided^{2}. Traditionally, nonreciprocity has been realized through magnetooptic materials that are too bulky and lossy to be integrated into modern photonic systems^{3}. In addition, nonlinear materials^{4,5,6} are employed to achieve nonreciprocity at the cost of a high intensity requirement, but they suffer from poor isolation and are reciprocal to noises^{7}. To circumvent these limitations, an increasing number of studies have focused on developing materials with timevariant properties in which timereversal symmetry is explicitly broken to achieve nonreciprocity^{1,8}. To date, based on the strong electrooptic^{9,10}, acoustooptic^{11,12,13,14}, or optomechanical effects^{15,16} of different materials, proofofconcept temporal modulation has been demonstrated at frequencies ranging from kilohertz to gigahertz, which are much lower than the optical frequency as a result of the slow carrier injection of electrooptic modulation and lowfrequency acoustic or mechanical modes in acoustooptic or optomechanical modulation. In addition, these dynamic systems suffer from limited bandwidth either due to the group velocity mismatch among photonic modes or the intrinsic narrow linewidths of acoustic and mechanical modes. Moreover, they require long interaction lengths to observe the desired effect. Nonreciprocity with a subwavelength interaction length and an ultrafast modulation frequency over the THz bandwidth is technically challenging and has not been realized to date.
Here, we experimentally demonstrate a new approach that achieves nonreciprocity on an ultrathin metasurface with space–time phase modulation. A metasurface^{17} is an optically thin nanostructured twodimensional material that can manipulate light through its subwavelengthsized building blocks. Designed specifically to achieve a controlled optical response, metasurfaces have been used to create novel optical devices, including invisibility cloaks^{18}, flat lenses^{19,20,21}, and ultrathin holograms^{22,23}, to enhance nonlinear generation^{24,25}, and to explore interesting physical effects^{26,27}. While these concepts have opened up a new paradigm for manipulating light with an ultrathin layer, there are fundamental limitations that a spatially modulated metasurface alone cannot overcome. In particular, a timevarying response is required to violate reciprocity in a nonpower and nonmagneticfielddependent fashion. Metasurfaces with spatiotemporal modulation^{28} in the index^{29,30} or directly in their phase profile^{31} have been theoretically proposed recently and experimentally demonstrated at radio frequencies^{32,33,34,35,36}, offering an opportunity to overcome this limitation. We experimentally demonstrate this new dynamic metasurface with an additional fast temporal phase modulation. Our space–time modulated metasurface (Fig. 1a) consists of a set of specifically designed nonlinear nanoantennas that not only provide abrupt static phase shifts for the incident light but also are capable of changing the phase shifts dynamically in the presence of an external travellingwave modulation. To introduce the dynamic phase change to the metasurface, we incorporate a heterodyne interference of two coherent waves to generate multiterahertz (~2.8 THz) timevarying modulation^{37}. This breaks the timereversal symmetry of the metasurface. Our experiments demonstrate nonreciprocal light propagation in free space at around λ = 860 nm across an ultrathin (150 nm) layer. Furthermore, we achieve completely asymmetric photonic transitions over a wavelength range of ~15 nm (corresponding to a bandwidth of 5.77 THz). We believe that the space–time modulated metasurface demonstrates a viable way to obtain ultrafast timevariant material responses and potentially motivates a new direction in constructing miniaturized and integratable magneticfree nonreciprocal optical components.
Results
A conventional spatially modulated metasurface with a phase gradient (e.g., φ = k_{s}x) on the surface is capable of imparting additional linear or orbital angular momentum to incident light, which breaks the inversion symmetry and enables full control over the photonic transitions in the momentum space. However, this process is linear and timereversal symmetric and is inherently reciprocal (Fig. 1b, c). In contrast, our space–time modulated metasurface has a phase modulation of the form
where Δφ is the temporal modulation depth, Δω the modulation frequency, k_{M} is the modulation spatial frequency, and k_{s} is the static phase gradient introduced by the spatial distribution of the nanoantennas. This modulation acting upon the incident light field gives an additional space–time varying phase factor, expressed as exp[iφ(x,t)]. Applying the Jacobi–Anger expansion, this phase term can be rewritten as a series of Bessel functions of the first kind, which enables the reflected field to be expressed as
where ω_{i} and \(\vec k_{\mathrm {i}}\) are the frequency and freespace wavevector of the incident wave, \(\zeta = \sqrt \eta\), and η is the static diffraction efficiency of the metasurface. Note that only the zeroth and firstorder Bessel functions are retained since the phase modulation depth Δφ is small, which leads to negligible contributions from higherorder functions. It is evident from the second term on the righthand side of Eq. (2) that a sinusoidal phase component will be decomposed into two photonic transitions (i.e., sidebands) in the energy space with resulting frequencies ω_{r} = ω_{i} ± Δω, where ‘ + ’ and ‘–’ denote an upward and downward transition, respectively. The dynamic phase modulation breaks reciprocity and leads to timereversalasymmetric photonic transitions^{38}. Different from the recently reported phononassisted nonreciprocal waveguides^{13,14} based on indirect interband photonic transitions^{38}, our system is naturally phase matched for all free space modes and does not require complex design of the acoustic and photonic modes to fulfil stringent momentum/energy matching conditions. Furthermore, our space–time metasurfaces exhibit agile control over both momentum and energy conversions. To achieve unidirectional frequency conversion, the metasurface can be designed to either fulfil k_{ix} + k_{s} + k_{M} > k_{i} (k_{ix} is the projection of incident wavevector k_{i} along the x direction), 2k_{M} > k_{i}, and –k_{i} < k_{ix} + k_{s} – k_{M} < k_{i} to ensure unidirectional downconversions (Fig. 1d, e) or fulfil k_{ix} – k_{s} – k_{M} < –k_{i}, 2k_{M} > k_{i} and –k_{i} < k_{ix} – k_{s} + k_{M} < k_{i} to ensure unidirectional upconversions. As an example, we depict the case of unidirectional downconversions in Fig. 1d, e. The optical paths of forward and backward propagation are shown schematically in the top panel, and the photonic states represented by different colour dots are shown in the bottom dispersion diagrams. The photonic transition from the blue dot state to green occurs in the forward propagation, whereas green to red occurs in the backward propagation. With a space–time modulated metasurface, the frequency transitions arise from the parametric processes caused by the temporal modulation, while the momentum transitions arise from both temporal and spatial modulation. As a result, the allowed transitions (i.e., downward transitions) can be selected by pushing a given output state (i.e., upward transitions) to the forbidden (i.e., nonpropagative) region with a unidirectional momentum transfer, k_{s}, provided by the spatial phase modulation of the metasurface. Note that the reflection angle of the backward propagating light is not necessarily the same as that of the forward propagating light even though they have the same x component of the wavevector (k_{x}), because the frequency shift also changes the length of the wavevector. The paths of the incident and returning beam overlap only in the special case of normal incidence (i.e., k_{ix} = 0), which is particularly useful for freespace optical isolators. For completeness, we also presented bidirectional photonic transitions on the space–time metasurfaces, which also exhibit nonreciprocity (Fig. S2). In all cases, the spatiotemporal phase modulation leads to completely asymmetric reflections in the forward and backward directions. The back reflected wave does not return to the same state as the forward incident wave, and therefore, the process is nonreciprocal.
We used a set of nanobar antennas (Fig. 2a) made of amorphous silicon (αSi), which has a large Kerr index and low optical loss, as the building blocks of the metasurface. With the adoption of a 50nmthick SiO_{2} spacer layer and a silver backreflector plate to create a gap resonance, the nanoantenna can induce a large phase shift (over 2π) upon arrival of the incident light. The permittivity of the αSi nanoantennas can be modulated by an intense optical field due to the nonlinear Kerr effect. This is an ultrafast process (Fig. S6) and is the key to obtaining the THz temporal phase modulation. Subsequently, the lightinduced permittivity change of the nanoantennas will detune their resonances and therefore lead to a change in the phase shift upon arrival of the incident light at the operational wavelength. The static phase shifts (φ) and the changes in the phase shifts (Δφ) induced by the pump light (λ_{pump} = 800 nm, I_{pump} = 15 GW/cm^{2}) are simulated separately and mapped out for the selection of designs in a twodimensional parameter space spanned by nanoantenna dimensions l_{x} and l_{y}. Three different nanoantennas with static phase shifts covering a 2π range while preserving a uniform phase shift change under pump light illumination were chosen to build a supercell of the metasurface. The parameters for the nanoantenna designs used in our experiment are indicated as red diamonds in Fig. 2b. The spatial distribution of these nanoantennas creates a static phase gradient k_{s} = 2π/p_{x}, where p_{x} is the supercell period in the x direction. Note that the nanoantennas transform the small permittivity modulation from the Kerr effect into relatively large dynamic phase modulation, leading to an efficient nonreciprocal photonic transition at moderate pump power. Moreover, the local field is enhanced by those resonant elements. As a result, the designed nanoantennas significantly boost the temporal phase modulation depth.
We fabricated the metasurface as shown in Fig. 2c and characterized its linear performance with a kspace imaging microscope (Fig. S5a). Our metasurface attains an overall diffraction efficiency of 84% experimentally around the designed operational wavelength λ = 860 nm (Fig. S5c). The measured reflection angles agree with the theory at different wavelengths and different supercell periods (Fig. 2d).
Next, we developed a fast temporal modulation technique that uses a heterodyne interference between two laser lines that are closely spaced in frequency (Fig. 3)^{39}. In contrast to a homodyne interference setup, the heterodyne interference pattern results in a travellingwave intensity distribution given by I(x,t) = I_{0}[1 + cos(Δωt−k_{M}x)], where k_{M} = 2π/Λ_{M} (Λ_{M} is the period of the interference fringes) and Δω = ω_{p2} – ω_{p1} is the frequency difference between the two pump laser lines. Projecting this interference pattern on the designed nonlinear metasurface imprints a travellingwave phase profile, Δφcos(Δωt−k_{M}x), onto the reflected wave. Together with the spatial phase modulation φ = k_{s}x created by the distribution of nanoantennas on the metasurface, we obtained a space–time modulation of the form described by Eq. (1). Therefore, the incident wave acquires the energy and momentum shifts when reflected by the metasurface. This process shares some similarities with conventional fourwave mixing in a homogeneous nonlinear material^{40}. However, there are subtle but essential differences between them. In our case, we used the metaatoms to tailor locally both the linear and nonlinear responses—collectively, the metasurface exhibits large temporal phase wobbling under a driving field in addition to a static phase shift, which cannot be achieved using a natural material.
The experimental setup for creating the temporal modulation is illustrated in Fig. 3b and described in the Methods section. In our experiment, the centre wavelength difference was ~6 nm, and the frequency difference was ~2.8 THz (Fig. 3c). In addition, k_{M} was adjusted by changing the angle between the two pump beams impinging onto the metasurfaces. Fourier transform analysis of the interference pattern shows a k_{M} equal to 0.54k_{probe} (Fig. 3d), where k_{probe} is the length of the freespace wavevector of the probe beam.
To demonstrate the nonreciprocal light propagation, the metasurface with k_{s} = 0.72k_{probe} was imprinted by the interference pattern at a peak pump intensity of ~1 GW/cm^{2}, which generated a dynamic phase modulation with ∆f = 2.8 THz and k_{M} = 0.54k_{probe}. For the forward propagation experiment, the 860 nm probe light hit the metasurface at normal incidence and resulted in a reflected wave with a shift in the energymomentum space. This shift was captured by collecting the reflected spectra with a fibre aperture scanning spatially on the Fourier plane (along the k_{x} direction) of the focusing lens before the metasurface. Figure 4a displays a static diffraction at 348.6 THz and 0.72k_{probe} determined by k_{s} and a downshifted signal at 345.8 THz and 0.18k_{probe} produced by the spatiotemporalmodulationinduced ∆f, k_{M}, and k_{s}. The zerothorder reflection can also be detected, as the diffraction efficiency dropped due to edge effects and slight polarization misalignment. The aspheric lens used to focus the pump and probe beams and to collect converted signals has an effective NA of 0.76; k_{x}/k_{probe} is therefore bounded by the limited collection angle. On the other hand, the accumulated k_{x}/k_{probe} ~1.26 of the upward transition is greater than unity. Therefore, it is evanescent and cannot carry energy away from the metasurface. For the backward propagation experiment, we sent a probe beam (f = 345.8 THz and k_{x} = −0.18k_{probe}) with the same frequency as that of the previous downconverted signal but in the opposite direction onto the metasurface. We observed another downward transition at 343.0 THz exit along the normal direction (Fig. 4b). Similarly, the upward transition is nonexistent since its accumulative k_{x} is greater than k_{probe}. Therefore, the backward propagating light cannot return to the initial state. We also performed a fine scan over the ω − k_{x} regions where highorder conversions may exist. However, no converted signals were observed since these processes have much lower efficiency. In addition, we performed a control experiment on amorphous silicon film (thickness ~150 nm) using a similar pump intensity, but no converted signal was detected. We experimentally realized nonreciprocal light reflections on the space–time phase modulated metasurface. The experimental results are also confirmed by our finitedifference time domain (FDTD) simulations (Fig. S3). In addition, we investigated the scattering matrix of our space–time phase modulated metasurface. It is asymmetric and hence is direct evidence that our system is nonreciprocal (see S4 of the Supplementary information). To experimentally show the nonreciprocal operation wavelength range of our metasurface, we conducted additional measurements on the nonreciprocal processes by mapping out the dispersion diagrams (Fig. S11) with a narrowband probe, of which the centre wavelength ranges from 854 to 914 nm. By extracting the conversion efficiencies at the tested wavelengths (Fig. S12), a 3dB bandwidth (fullwidth at halfmaximum, FWHM) of ~5.77 THz was demonstrated (see S6 of the Supplementary information). It is worth noting that in contrast to the waveguidebased systems, the bandwidth of our space–time metasurface is not constrained by the phasematching conditions. Our experimentally obtained bandwidth is at least one order of magnitude greater than the largest ones reported (a few hundred gigahertz)^{9,14} on timedependent nonreciprocal systems.
Using our space–time metasurface, we are able to independently control the static phase gradient k_{s}, the dynamic spatial frequency k_{M}, and the temporal modulation frequency Δω, which provides unpreceded flexibility in manipulating the photonic transitions. As shown in Fig. 5, by changing k_{s}, we can selectively enable downward or upward transitions. Additionally, we demonstrated nonreciprocal reflections with arbitrary transverse momenta by tuning both k_{s} and k_{M} along the metasurfaces (Fig. S10). Moreover, we showed that the modulation frequency can be changed by adjusting the frequency splitting between the two pumps (Fig. S9a). It is worth noting that this level of controllability has not been achieved in previous timevariant nonreciprocal systems^{9,10,11,12,13,14,15,16,41}.
Discussion
We experimentally demonstrated nonreciprocal light reflection based on the space–time modulated nonlinear metasurface. The heterodyne interference created by frequencyshifted pump beams provides robust and controllable spatiotemporal modulation, of which ∆ω and k_{M} can be readily tuned as desired. It is worth noting that more complex twodimensional spatiotemporal modulation can be constructed from the heterodyne interference of three or more pump beams. The spatiotemporal phase modulation greatly expands the functions of conventional static phase gradient metasurfaces, providing an additional degree of freedom for manipulating the temporal properties of light and achieving nonreciprocal light propagation. Particularly, we achieved in our experiment a 2.8THz modulation frequency, a huge step towards optical frequencies, and an ~5.77THz 3dB bandwidth, which is orders of magnitude greater than that of current timevariant nonreciprocal systems to the best of our knowledge.
Although travellingwavemodulationinduced nonreciprocity has already been explored in theory and demonstrated in waveguide systems, such as optical fibre and silicon waveguides, experimental realizations in free space optical systems remain scarce due to the lack of fast, efficient dynamic modulation techniques. Previous demonstrations rely on long interaction lengths (centimetre to metre scale) to amplify the weak dynamic modulation, which makes it difficult to realize in a free space optical system. However, through the use of nonlinear optical metasurfaces to greatly enhance the dynamic modulation strength, optical nonreciprocity is achieved at a subwavelengthscale interaction length on metasurfaces, making this approach potentially compatible with integrated nanophotonic and quantum optical systems. Moreover, by using dynamic phase modulation of a metasurface working in free space, our operation bandwidth is not limited by the group velocity mismatch among guided modes^{10,14}. Instead, it is the resonant linewidth of the metasurface that determines the nonreciprocal bandwidth, which is approximately in the range of tens of terahertz.
The conversion efficiency, which is defined as P_{signal}/P_{probe} and is proportional to J_{1}(Δφ)^{2}, is on the order of 2.5 × 10^{−4} in our experiments due to limited experimental conditions. The temporal mismatch (Fig. S6 a) between the pump (~400 fs) and probe (~2 ps) beams limits the effective interaction time between the probe and the dynamic modulation. Due to the poor spatial profile of the probe beam and lens aberration, only ~20% of the probe beam overlaps with the pump beams, resulting in a reduced effective interaction area. In addition, due to the high repetition rate of our laser system (80 MHz), we need to maintain a low pump intensity to avoid significant thermal effects (Fig. S14).
The conversion efficiency of our space–time metasurface can be improved in the following aspects. We find that the conversion efficiency increases superlinearly with the pump intensity in our system (Fig. S14). Using pump light with a low repetition rate and optimized spatiotemporal overlap, we will be able to achieve a higher conversion efficiency. In addition, highly nonlinear materials, e.g., indium tin oxide (ITO), which has large thirdorder nonlinearity in its epsilonnearzero region^{42,43}, can be used to greatly increase the dynamic phase modulation depth. Furthermore, an optimized nanoantenna design, e.g., a design involving doubly resonant nanoantennas, will improve the temporal phase modulation depth and relax the pump power requirement. In addition, by stacking a pair of space–time phase modulated metasurfaces to form a cavity or integrating metasurface with resonator structures, i.e., microring resonators, the effective interaction length/time will be tremendously increased without sacrificing the device footprint, leading to enhanced nonreciprocal mode conversion efficiency. (A detailed quantitative analysis is included in S7 of the Supplementary information.)
In conclusion, we have demonstrated nonreciprocal light reflection on an ultrafast space–time phase modulated metasurface. This approach exhibits excellent flexibility in controlling optical mode conversions. Moreover, the dynamic system features broken time reversal symmetry compatible with topological insulators, which inspires a new route for isolation using topologically protected edge states^{44} that are immune to disorders. We believe that it will provide a new platform for exploring interesting physics in timedependent material properties and will open a new paradigm in the development of scalable and integratable nonreciprocal devices.
Materials and methods
Sample fabrication
A 200nm layer of silver was deposited onto a silicon substrate with a 5nm Ge adhesion layer by electronbeam (ebeam) physical vapour deposition (SEMICORE EGun Thermal Evaporator). A 50nm SiO_{2} dielectric spacer layer and 150nm amorphous silicon layer were then grown by plasma enhanced chemical vapour deposition (PECVD). The metasurface nanoantennas were created using a sequential process of ebeam lithography (EBL), liftoff of a chromium mask, and inductively coupled plasmareactive ion etching (ICPRIE). In the EBL process, we employed a 1:1 diluted ZEP 520A ebeam resist to achieve high resolution. The total pattern size was 200 × 200 μm^{2}, written using a Vistec 5200 100 kV ebeam writer. A chlorinebased plasma RIE recipe involving Cl_{2} and Ar gas was used to etch amorphous Si, creating the nanoantennas. The sample was finally immersed in a chrome etchant to remove the chrome mask.
Simulation method
We developed a twostep fullwave model in a commercial finite element method solver (COMSOL Multiphysics) to obtain the nonlinear Kerreffectinduced phase shift change (Δφ). The first step iteratively calculates the pumpbeammodified effective permittivity of the structure. The second step simulates a weak probe beam incident onto the structure with the calculated effective permittivity in the first step. By varying the geometrical parameters of the nanoantennas, we obtained the corresponding phase shift changes.
We also performed timedomain simulations using the FDTD to study the spatiotemporalphasemodulationinduced nonreciprocal photonic transitions. We updated the fields in each time step, taking into account the temporal changes in permittivity. By varying the spatiotemporal modulation parameters Δφ, k_{s}, k_{m}, and Δω, we simulated different conditions corresponding to the experimental demonstrations.
Please refer to section S1 of the Supplementary information for more detailed information on the simulation method.
Nonreciprocal reflection experiment
To perform measurements of nonreciprocal reflection, the experimental setup shown in Fig. 3b was used. To create two pump beams with a frequency separation for the travellingwave modulation, the output of a Ti:Sapphire pulsed laser (140fs pulse width, 80MHz repetition rate) at λ = 800 nm was dispersed by a transmission grating and then spatially separated by a customized split mirror with a variablesize block attached in the centre to tune the frequency separation. The spatially separated beams were then reflected back through the grating and recombined into two pump beams with shifted central frequencies. By adjusting delay line 1, the temporal delay between the two pump beams can be tuned. To create the probe beam, the other small portion of the Ti:Sapphire pulsed laser radiation was sent to a nonlinear photonic crystal fibre (PCF) to generate a supercontinuum. Probe light with a 860nm wavelength was selected using a monochromator. By adjusting delay line 2, the probe can be synchronized with the pumps. An aspheric lens with an effective NA of 0.76 focused the three beams onto the metasurface, which was mounted on a threedimensional (3D) translation stage. Due to aberration of the aspheric lens, the focal spot of the pump beams was an ellipse with major and minor axis lengths of 50 μm and 45 μm, respectively. The reflected signal was directed to a fibre coupler by a Dshaped pickup mirror, of which the position was adjusted by a linear translation stage to collect the kspace information of the output signal.
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Acknowledgements
The work is partially supported from the Gordon and Betty Moore Foundation and the Penn State MRSEC, the Center for Nanoscale Science, under award number NSF DMR1420620. The authors gratefully acknowledge Zhiwen Liu for fruitful discussions.
Author contributions
X.N. and X.G. conceived the project, designed the simulations and experiments. X.G. performed the numerical simulations, fabricated the samples, and conducted the measurements. Y. Ding helped with the kspace imaging and performed the raytracing simulations. Y. Duan calculated the metasurface diffraction efficiency. X.N. and X.G. analysed the data and wrote the manuscript. All authors contributed to the discussion of the results. X.N. supervised the project.
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Guo, X., Ding, Y., Duan, Y. et al. Nonreciprocal metasurface with space–time phase modulation. Light Sci Appl 8, 123 (2019). https://doi.org/10.1038/s413770190225z
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