Nonreciprocal flat optics with silicon metasurfaces

Letter Cite This: Nano Lett. 2018, 18, 1104−1109

pubs.acs.org/NanoLett

Nonreciprocal Flat Optics with Silicon Metasurfaces Mark Lawrence,* David R. Barton, III, and Jennifer A. Dionne* Department of Materials Science and Engineering, Stanford University, Stanford, California 94305, United States S Supporting Information *

ABSTRACT: Metasurfaces enable almost complete control of light through ultrathin, subwavelength surfaces by locally and abruptly altering the scattered phase. To date, however, all metasurfaces obey time-reversal symmetry, meaning that forward and backward traveling waves will trace identical paths when being reflected, refracted, or diffracted. Here, we use full-field calculations to design a passive metasurface for nonreciprocal transmission of both direct and anomalously refracted near-infrared light over nanoscale optical path lengths. The metasurface consists of a 100 nm-thick, periodically patterned Si slab. Owing to the high-quality-factor resonances of the metasurface and the inherent Kerr nonlinearities of Si, this structure acts as an optical diode for free-space optical signals. This structure also exhibits nonreciprocal anomalous refraction with appropriate patterning to form a phase gradient metasurface. Compared to existing schemes for breaking time-reversal symmetry, our platform enables subwavelength nonreciprocity for arbitrary free-space optical inputs and provides a straightforward path to experimental realization. The concept is also generalizable to other metasurface functions, providing a foundation for one-way lensing and holography. KEYWORDS: Nonreciprocal optics, all-dielectric metasurface, nonlinear metasurface, beam-steering

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splitting, leading to an asymmetric permittivity tensor.22 A more recent proposal involves a spatiotemporal refractive index modulation used to mimic the presence of a traveling wave, which can impart a fixed amount of linear or angular momentum to an incident photon, similar to the Doppler effect for sound.23,24 In both cases, the induced change in the material properties is generally very small at optical frequencies, so long optical path lengths are required to observe reasonable levels of nonreciprocity. Though a great many schemes for miniaturization have been proposed, involving plasmonics,25−28 photonic crystals,29,30 microring resonators,24,31 experimental demonstrations remain limited to the scale of a few hundred microns.32−34 Moreover, generating the control bias, that is, creating a magnetic field with a solenoid or a nonuniform voltage bias with electrical contacts, adds further to the complexity and footprint of such devices. Here, we propose an approach to nonreciprocally manipulate free-space near-infrared light with ultrathin metasurfaces. By exploiting high quality factor (high Q) resonances in periodic silicon nanostructures, we theoretically demonstrate one-way transmission through devices as thin as 100 nm. By engineering the distribution of these Si nanostructures, as to generate a spatial variation in the transmitted phase, we also show that nonreciprocal beam steering can be achieved. Our approach relies on the nonlinear Kerr effect: with sufficiently high power-

he traditional rules of optics have undergone a revolution in recent years, owing in part to the emergence of metasurfaces: two-dimensional arrays of carefully designed nanoantennas.1−3 Tasks that previously demanded the path lengths of many complex light rays be accounted for simultaneously, such as lensing,4,5 holography,6−8 and polarization manipulation,9−14 can now be achieved with devices just tens to hundreds of nanometers thick. Such “flat” optics is possible thanks to the ability for the amplitude, polarization, and, most importantly, phase of light scattered from each subwavelength antenna to be precisely engineered. Almost every conceivable transfer function has been realized, from diffraction-limited and chromatic aberration-corrected lensing15,16 to retroreflection17 and even invisibility cloaking,18 but one important constraint that has yet to be addressed is reciprocity. In other words, for all demonstrations thus far, the measurement gives the same result if the roles of source and detector are reversed. While seemingly innocuous, this symmetry places strict limits on a number of key optical technologies including telecommunications, photovoltaics, and even quantum information.19−21 The difficulty of achieving nonreciprocal transmission in the linear regime, that is, for low light intensities, stems from a need to break time reversal symmetry. To break time reversal symmetry, an effective motion within a device is needed, allowing it to behave differently for an identical wave propagating in forward or backward directions. Traditionally, magneto-optical effects have been employed, in which the application of a DC magnetic field causes molecular Zeeman © 2018 American Chemical Society

Received: November 1, 2017 Revised: January 22, 2018 Published: January 25, 2018 1104

DOI: 10.1021/acs.nanolett.7b04646 Nano Lett. 2018, 18, 1104−1109

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Nano Letters densities, photon−photon interactions mediated by anharmonic oscillations of the electron cloud cause the refractive index of a medium to change in proportion to the local electric field intensity. Our high Q resonators are designed to have highly asymmetric internal fields in the linear regime so that the incorporation of a nonlinear refractive index projects the local asymmetry to the far-field, generating nonreciprocal transmission. This phenomenon provides a means to break reciprocity without the need for any sort of external biasing mechanism. Moreover, this platform works for arbitrary optical mode inputs, promising the potential for nonreciprocal beam steering, one-way flat lensing, and efficient holography. The nonlinear Kerr effect can be described by including a term in the permittivity of the dielectric material, which is proportional to the square of the local electric field strength, ε(r) = ε0(εr + χ(3)|E(r)|2). Trapping light over many optical cycles causes the electric field inside a resonator to become greatly amplified compared to the incident wave, leading to far higher index changes compared to traveling wave platforms. Moreover, the scattering efficiency varies dramatically with wavelength across a resonance, and therefore, small shifts in the central wavelength, coming from a change in refractive index, can produce large differences in transmission. Just how pronounced both of these effects are for a given system depends on the associated Q factor. As χ(3) of most materials is very small, large Q is typically required to reduce the modulation power threshold to reasonable levels. Thankfully, some of the largest nonlinear indices are found in semiconductors such as silicon, which, as well as being easy to pattern, also has negligible absorption coefficients at infrared frequencies.35 By utilizing the nonlinear Kerr effect, high-Q cavities of Si (and InP) have exhibited optical diode action for waveguide modes.36,37 Nonreciprocal transmission has also been realized in similar arrangements with active materials, based on gain saturation,38,39 optomechanics,40−43 and parametric wavemixing.44 However, these systems achieved low radiative loss rates through physical isolation from other scattering structures; that is, they relied on evanescent coupling between a nearby waveguide and the cavity. Passive nonlinear nonreciprocity, which relies on the precise placement of a subwavelength scatterer within a Fabry−Perot cavity, has also been proposed.45 Although useful in the microwave regime, the much weaker nonlinear coefficients and smaller wavelengths make this approach less practical at optical frequencies. When dealing with free-space optical radiation modes, the cavity’s position is no longer a useful degree of freedom. Instead, the symmetries of a structure must be exploited, suppressing the radiative loss via destructive interference. Figure 1a illustrates our chosen geometry to controllably reduce free-space radiative coupling and realize a nonreciprocal high Q cavity. It consists of shallow periodic grooves cut into both sides of a 100 nm thick poly crystalline Si slab with a refractive index of 3.7.46 Without grooves, the plane-wave illumination of such a slab, suspended in vacuum, reveals only broad Fabry−Perot (FP) fringes in the transmittance spectrum, as seen in the solid black curve in Figure 1b. However, the system also supports guided waves, which are completely isolated from the vacuum solutions because of the need for momentum conservation (gray curve in Figure 1b). Superpositions of these guided waves can form bound states, an example of which is denoted by the pink star in Figure 1b; this state is calculated by folding the continuous slab dispersion

Figure 1. Guided mode resonance. (a) Schematic of proposed setup, periodically patterned silicon slab, t = 100 nm, w = 40 nm, and period p = 650 nm, suspended in vacuum and illuminated with plane waves. A single unit cell is highlighted blue. (b) Transmittance spectrum and guided wave dispersion for Δ = t and δ = 0. Dotted lines indicate projections of the dispersion into the first Brillouin zone, based on p = 650 nm, and pink star marks the first bound state with field plotted in inset. (c) Transmittance spectrum for t − Δ = 20 nm. Inset maps the peak electric field normalized to incident plane wave, scale bar is 100 nm. (d) Quality factor of resonance with increasing groove depth.

about the fictitious period, p = 650 nm, represented by the dashed gray curve in Figure 1b. Inserting grooves with a period of 650 nm causes the bound state to leak, appearing as a resonance in transmission (see Figure 1c). By keeping the depth of the grooves small compared to the slab thickness, coupling to free-space remains weak, which leads to a huge amplification of the internal electric field. The local field is seen to reach 130-times that of the incident plane wave for 10 nm deep grooves in Figure 1c. Importantly, silicon can be considered lossless in the infrared and so the enhancement associated with these so-called guided mode resonances (GMR) can be made arbitrarily large, as indicated by Figure 1d, which shows the Q factor diverging as the groove depth tends to zero. As such, they have been shown to make an excellent platform for sensing,47 modulation,48,49 and lasing applications.50−52 While the in-plane geometry of a patterned slab is responsible for the overall GMR lifetimes, the relative contribution from leakage above and below the slab can be controlled by altering the mirror symmetry in the out-of-plane 1105

DOI: 10.1021/acs.nanolett.7b04646 Nano Lett. 2018, 18, 1104−1109

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average of the electric field within the silicon, shows the ratio of forward to reverse field enhancement growing monotonically with δ, until δ = 14 nm. In this configuration, the direct and FP mediated GMR excitation strengths are matched, producing maximal asymmetry. While the field asymmetry peaks at δ = 14 nm, the maximum enhancement in the forward direction occurs at δ = 4 nm, after which it begins to fall. This result is consistent with the decrease in Q factor, shown in the inset of Figure 2c, which can be compensated for by revisiting the in-plane design of the grooves. The maximum transmission drops inversely with increasing field asymmetry, which has previously been shown to be a result of time reversal symmetry.54,55 Each of these features must be considered when building a nonreciprocal device to achieve the desired threshold power, operational power range and insertion loss. To realize nonreciprocity over nanoscale optical paths, we include the nonlinear behavior of the slabs. We incorporate the third-order susceptibility of poly-silicon, χ(3) = 2.79 × 10−18 m2/V2, into our frequency domain simulations and search for the steady state solutions with increasing incident power.46 Figure 3a shows the transmittance

direction. As the period has been chosen to align the first TE polarized GMR with the first quarter-wave minimum of the FP background, a completely symmetric perturbation, δ = 0, can be seen to produce ideal Fano interference. This highly asymmetric line shape is a signature of a simple superposition between resonant and nonresonant scattering,53 illustrated schematically in the left panel of Figure 2a. Starting with the

Figure 2. Guided mode resonance in asymmetric structure. (a) Schematic of modal interactions. For a symmetric structure, the guided mode and Fabry−Perot resonances simply interfere in the farfield, whereas for asymmetric structures, the two resonances couple in the nearfield. (b) Transmittance spectrum evolution with increasing δ, in 2 nm steps. Inset maps the electric field normalized to incident plane wave in forward and reverse directions, bordered with red and blue respectively, for δ = 10 nm (dashed curve). Scale bar is 100 nm. (c) Red dots plot the volume average of the electric field amplitude normalized to incident wave. Black stars plot the ratio of the volume averages of the electric field for forward versus reverse illumination. Inset plots Q factor of resonance for same δ range. Δ = 80 nm throughout. Field plots and volume averages are plotted at central resonant wavelength, as defined in the Supporting Information.

Figure 3. Nonreciprocity from nonlinear Kerr effect. (a, c) Transmittance spectra for incident plane waves with (a) 10 kW/cm2 and δ = 10 nm, (c) 5 kW/cm2 and δ = 4 nm. Black curves represent linear spectra for reference. (b) Transmittance as a function of input power at wavelength of 1.47269 μm and δ = 10 nm (dashed curves), 1.47315 μm and δ = 4 nm (solid curves). Red and blue curves represent forward and reverse illumination, respectively, as defined in Figure 1a.

spectra through an asymmetric corrugated membrane with δ = 10 nm when illuminated from each side; we consider an incident intensity of 10 kW/cm2. By comparing the dip in forward transmission with the linear case, the GMR is seen to red-shift by roughly one full-width at half-maximum at this intensity. However, the reverse transmission spectrum remains essentially unchanged. Consequently, the device becomes highly nonreciprocal, behaving as a diode for λ ≈ 1472.69 nm. Details regarding the discontinuity visible in the forward spectrum are provided in the Supporting Information. We next consider the change in transmittance as a function of input power. Analogous to a p−n junction diode, the dashed curves in Figure 3b, represent effective optical I−V curves for light with a wavelength of λ = 1472.69 nm. In the linear regime, the slab is highly reflective from both sides. As the incident

configuration investigated in Figures 1c and 2b, the evolution of the transmittance as the notched region is shifted by δ with respect to the symmetric configuration. The transmittance spectrum exhibits both a blue-shift and a reduced maximum. This change can be explained by a direct coupling pathway emerging between the FP mode and the GMR. Importantly, unlike directly exciting it with a plane wave, the FP-mediated excitation of the GMR depends on the incident direction. Consequently, destructive and constructive interference between the various pathways leads to a marked difference in the field enhancement when illuminating the slab from below (i.e., “forward” direction) and above (i.e., “reverse” direction), respectively, as depicted schematically in the right panel of Figure 2a. Indeed, Figure 2c, which tracks the peak volume 1106

DOI: 10.1021/acs.nanolett.7b04646 Nano Lett. 2018, 18, 1104−1109

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Nano Letters intensity is increased beyond 2 kW/cm2, the transmittance begins to climb in the forward direction, while remaining very low in the reverse direction. One-way transparency persists until around 20 kW/cm2 when the membrane also starts to transmit in the reverse direction. This behavior, which is functionally equivalent to the response of an electrical diode at the breakdown voltage, results from an illumination intensity, which is high enough to shift the GMR across the wavelength of interest, even in the reverse direction. This structure with δ = 10 nm exhibits highly nonreciprocal behavior over a large range of incident powers, completely blocking light from one side while allowing it to pass through from the other. However, the transmittance in the forward direction is only around 25%. While comparable to previous demonstrations of nonreciprocal transmission, it still represents a fairly significant loss for the incoming signal. To increase the peak transmittance, the field asymmetry must be reduced (see Figure 2). Nevertheless, small amounts of asymmetry can still be sufficient to achieve diode action. Figure 3b and c show the results for a slab with δ = 4 nm. In this case, above a threshold of about 3 kW/cm2, high transmittance of around 75% is observed in the forward direction. At the same time, the range of incident powers for which the transmission is suppressed in the reverse direction is reduced compared with δ = 10 nm, (∼6 kW/cm2, compared to ∼20 kW/cm2). This result makes sense, given that under reverse illumination a higher intensity is reached within the silicon for the δ = 4 nm membrane. Accordingly, modification of the structural asymmetry in this metasurface provides continuous tunability of the nonreciprocal response, from exhibiting diode action over a wide range of illuminating powers to exhibiting low insertion loss over a reduced power range. We now show how this platform can be used to realize a nonreciprocal metasurface. Instead of a continuous silicon sheet, we consider a periodic structure with period p ≈ √2 λ, where λ is the freespace wavelength. Here, two new solutions emerge alongside the zeroth-order transmission, with diffraction angles θ ≈ ∓ 45°. The scattering into one of these off normal diffraction orders becomes optimal for a transmitted field with uniform intensity and a linearly varying phase; by using nanoantenna designs that approximate a linear phase function, many groups have observed this phenomenon, which is known as anomalous refraction or beam steering.1,5,56,57 Here, we employ the structure depicted in Figure 4a, consisting of a simple Si bar with a height of 700 nm and a variable width. Figure 4c plots the intensities of the various diffraction orders when combining bars with widths of 210 nm, 240 nm, and 400 nm, taken from Figure 4b, into a single unit cell with a period of 2121 nm. As expected, around the design wavelength of 1500 nm, a normally incident plane wave is redirected to the left (+1st order) with very little energy diffracted into other orders. This behavior is also verified in the field map shown in the right panel of Figure 4d. Importantly, as for the continuous films considered in Figure 1, the phase gradient metasurface in Figure 4d also supports bound states made up of guided modes which propagate orthogonally to the plane of diffraction. Next, we introduce a high Q mode into the diffraction spectrum. Following the above prescription, we add periodic grooves, 70 nm in width and 25 nm deep, to each side of the 400 nm wide silicon bar, as illustrated in Figure 5a. Subtly perturbing the translationally invariant grating gives rise once again to a sharp GMR, which perturbs the linear phase gradient, scattering light in to other diffraction orders. Because the

Figure 4. Phase gradient metasurface. (a) Periodic array of silicon structures, used to design metasurface. (b) Transmitted intensity (gray) and phase (black) through geometry shown in panel, as a function of width ‘w’. Stars denote parameters chosen for metasurface elements, w1 = 210 nm, w2 = 240 nm, and w3 = 400 nm. (c) Diffraction transmittance spectrum. (d) Schematic of phase gradient metasurface, left panel, and H field map verifying anomalous refraction. Element colors correspond to stars in panel b.

Figure 5. Nonreciprocal anomalous refraction. (a) Phase gradient metasurface with periodic grooves of period p2 = 570 nm, 70 nm width, and 25 nm depth. (b) Metasurface diffraction intensity for incident power of 8.3 kW/cm2. Black curve represents the linear forward +1st intensity for reference, while the colors of the other curves correspond to arrows shown in panel c. (c) Schematic of diffraction setup through metasurface, with incident angle for reverse illumination matching +1st diffraction angle for forward illumination θ = sin −1λ/p. (d) Diffraction intensity at wavelength of 1.50981 μm as a function of input power, curve designation same as panel c.

overall change to the geometry is very small, away from the sharp spectral feature the diffraction spectrum remains unaffected. As a result, we are free to adjust the period of the grooves to center the GMR close to the operating wavelength of 1500 nm. From the black curve in Figure 5b, it can be seen that a period of 570 nm produces a resonant wavelength of around 1509 nm. Unlike the slab investigated in the first part of this paper, which, without grooves, is invariant under a transformation converting between forward and reverse illumination, the metasurface designed in Figure 4 is highly 1107

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asymmetric with respect to the normal and oblique incident waves being considered. Because of this inherent asymmetry, different electric field strengths build up inside the Si for forward and reverse illumination, defined in Figure 5c, even with symmetric grooves, that is, equal groove depths on each side. Details regarding the consequences of tuning the geometric asymmetry can be found in the Supporting Information. The nonlinear, nonreciprocal response of this metasurface is shown in Figure 5b and d. With an illumination intensity of 8.3 kW/cm2, forward illumination (normal incidence) generates a pronounced red-shift of the GMR. For reciprocal illumination (i.e., oblique incidence matching the angle of the +1st diffracted order), a much smaller red shift is observed. Therefore, the Kerr modulation of the GMR can be seen to produce nonreciprocal beam steering for λ ≈ 1509.8 nm. Specifically, highly efficient beam steering occurs in the forward direction, while diffraction into both the zeroth and −1st orders is found in the reverse direction. Figure 5d plots the power dependence of this system, showing highly nonreciprocal behavior at λ = 1509.81 nm for 6 kW/cm2 to 12 kW/cm2. In summary, dielectric metasurfaces represent an exciting platform for breaking the reciprocity of free-space electromagnetic signals. Here, we designed a subwavelength grating, which, thanks to the nonlinear Kerr effect, acts as an ultrathin (100 nm thick) optical diode for illumination intensities of a few kW/cm2. Then, using the same platform but starting with a three-element phase gradient metasurface, we achieved large angle (∼45°) nonreciprocal beam steering. While our calculations focused on nonreciprocal beam steering as a proof of principle, this scheme could be easily extended to more complicated transfer functions such as nonreciprocal lensing and holography. Furthermore, the proposed scheme of embedding a high Q resonance within a phase gradient metasuarface is generalizable to other nonlinear phenomena such as parametric amplification and lasing. While high Q structures require precise fabrication, extremely high Q GMRs, at and beyond those found in the current manuscript, have been experimentally observed.52,58−60 Therefore, with the simplicity of the structures presented and the maturity of Silicon processing, the relevant tolerances for our scheme are well within current technological capabilities. As a final note, we have only considered systems that rely on translational symmetry in at least one dimension, with the excitation being planar in that dimension. However, a number of recent studies have shown that trapped states can be found in structures confined in all three directions.61−63 By carefully tailoring a nonlinear resonator to a particular, nonplanar incident wave, nonreciprocal freespace devices could be miniaturized even further.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. ORCID

Mark Lawrence: 0000-0002-0822-5453 Author Contributions

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We acknowledge the generous funding from the AFOSR PECASE grant (FA9550-15-1-0006) and NSF EFRI grant (1641109) as well as funding from Northrop Grumman. D.B. acknowledges support from a Stanford Graduate Fellowship.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.nanolett.7b04646. Numerical specifications, simulations confirming robustness of scheme to variations in fabrication and characterization conditions, effects of two photon absorption, structural symmetry breaking with phase gradient metasurface (PDF) 1108

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DOI: 10.1021/acs.nanolett.7b04646 Nano Lett. 2018, 18, 1104−1109