Tunable metasurfaces via subwavelength phase shifters with uniform amplitude

Metasurfaces with tunable spatial phase functions could benefit numerous applications. Currently, most approaches to tuning rely on mechanical stretching which cannot control phase locally, or by modulating the refractive index to exploit rapid phase changes with the drawback of also modulating amplitude. Here, we propose a method to realize phase modulation at subwavelength length scales while maintaining unity amplitude. Our device is inspired by an asymmetric Fabry-Perot resonator, with pixels comprising a scattering nanopost on top of a distributed Bragg reflector, capable of providing a nearly 2π nonlinear phase shift with less than 2% refractive index modulation. Using the designed pixels, we simulate a tunable metasurface composed of an array of moderately coupled nanopost resonators, realizing axicons, vortex beam generators, and aspherical lenses with both variable focal length and in-plane scanning capability, achieving nearly diffraction-limited performance. The experimental feasibility of the proposed method is also discussed.

Extending this to an infinite number of outgoing beams, the total E-field outside the cavity can be expressed as below: Since | ′| < 1, the series converges, and utilizing ′ − ′ = 1 and = − ′ by energy conservation, the expression simplifies to: In the lossless case, we have { } = 0, and we can find the magnitude and phase of the expression as below: We can also determine how rapidly the phase changes with respect to the cavity phase by taking the derivative and evaluating at = , giving: For the lossy case, we get: By letting = 0, we neglect the loss and recover the lossless equations.

Quality Factor and Loss
The quality factor of a resonator is an important quantity for characterizing resonators and how efficiently they store and dissipate energy. For an asymmetric Fabry-Perot cavity excited by a transient electric field which decays with time, the amplitude within the cavity can be related to the resonator's Q by: where is time, 0 is the initial electric field, and 0 is the resonant angular frequency.
In one roundtrip in the cavity, we can express the field amplitude as a product of the top mirror reflectivity (the bottom mirror has = 1) and the attenuation due to material absorption (i.e. a fraction 1 − of the field amplitude is lost per roundtrip) such that: where denotes the field after one roundtrip. Evaluating (12) at = and equating to (13), we obtain: Now utilizing the relations 0 = 2 and = 2 , where is the cavity length, is the refractive index, and is the speed of light, we can express the Q as below: where, as previously, = 4 denotes the phase accumulated in one roundtrip. The value of is dependent on the structure of the cavity and for an arbitrary value of = 10, we plot the Q as a function of the loss fraction 1 − for different values of ( Fig.   S1). We observe that the Q remains relatively flat for low losses, but decays rapidly as the loss increases.  [ Solving (16) for gives: Due to the antisymmetric nature of the phase with respect to the = axis, we can express the total phase modulation range in terms of the phase evaluated at : More generally, we can express the phase modulation range for both regimes as a piecewise function as below:  In addition to characterizing the achievable phase shift, it is important to quantify the necessary cavity phase modulation range. To reduce the necessary change in refractive index, this range should be minimized. From Fig. 1 of the main text we saw that by increasing the top mirror reflectivity we could reduce the necessary cavity phase range.
We can quantify this range by calculating the range over which 80% of a 2 shift is achieved, as a full 2 shift would always require a 2 cavity phase range. Fig. S4 shows the required cavity phase range as a function of the fractional amplitude loss for = 0.7 and we see that the required range decreases as the loss increases, but the overall change is minimal and in this particular case only corresponds to a 11.5 % change. While we could minimize this required cavity modulation range by introducing lossy materials, in terms of realizing a practical device, the modest benefit provided by this reduction would likely be offset by the drop in amplitude efficiency due to material absorption.

Focusing Efficiency
To characterize the efficiency of our tunable metasurface aspherical lenses, we calculated the focusing efficiency as a function of on-axis focal length (Fig. S5). In particular, we took the ratio of the total power in the focal plane within a circle of radius three times the focal spot's FWHM to the total power incident on the metasurface 1 . We report a trend of increasing efficiency with higher focal lengths, and found focusing efficiencies as high as 41% at 280 focal length. Figure S5: Focusing efficiency as a function of focal length for the tunable aspherical metasurface lens. The blue line is an eye guide.

Design of Rectangular Scattering Elements
The asymmetric structure of our nanoposts can be generalized to other scattering element geometries for which electrical routing for index tuning would be simpler. A 1-D grating of rectangular lines with a unit cell as in Fig. S6a is possible and could be used in focal line scanning applications. Fig. S6b shows the phase characteristics for such an example device structure with lines of width 620 nm, period 800 nm, and height 516 nm, achieving nearly 2π phase modulation with a change in refractive index of 0.0949 (< 3%). The incident light is polarized orthogonal to the length of the lines, which in RCWA are modeled as infinitely long rectangles. With this structure, an 80 80 tunable cylindrical lens was designed, demonstrating focal scanning (Fig. S6c).

Index Modulation by Thermal Tuning
One route to tune the refractive index of individual scatterers is by exploiting the thermooptic effect by heating either electrically or optically. Based on the thermal coefficient of refractive index for silicon at room temperature ( = 1.86 × 10 −4 −1 ), we calculate a temperature change of ~317 K for our nanoposts:

Index Modulation by Free Carrier Injection
To calculate the change in carrier density ( ) necessary for nearly 2π phase modulation of our nanoposts ( = 0.059), we use the Drude Expansion Model 2 : where is the electron charge, λ is the operating wavelength, is the speed of light, 0 is the vacuum permittivity, is the nominal refractive index, and * and ℎ * give the electron and hole conductivity effective masses respectively. Substituting appropriate parameter values, we get a carrier density change of = 2.93 × 10 19 −3 . Under such high carrier densities, Auger recombination will dominate relative to the radiative and Shockley-Read-Hall pathways, allowing us to calculate the recombination lifetime as: where is the Auger recombination coefficient. With the determined recombination lifetime, we can find the generation rate of electron-hole pairs in the steady state using: If we assume the incident laser used for exciting free carriers has a wavelength of 500 nm, the absorption coefficient corresponds to an average penetration depth of nearly 1 , which is greater than the height of our nanoposts, allowing us to approximate a depth-uniform generation rate 3 : where 0 is the incident photon flux and is the absorption coefficient at 500 nm.
Combining (23) and (24) and then multiplying the flux by the photon energy, we calculate a required incident laser intensity of 1.26 / 2 if thermo-optic effects are neglected.
We could also achieve carrier injection electrically by forward biasing a nanopost configured as a p-n or p-i-n junction diode (Fig. S7). With a p-n junction configuration, the highly doped p and n regions would enable very high changes in carrier density near the junction, which would fall off rapidly as you move away from the center. Instead, with a p-i-n junction configuration, via Sentaurus simulation we calculate that with our 504 nm thick posts, with 50 nm thick p and n regions with dopant concentrations on the order of 3 × 10 19 −3 , we can achieve the necessary carrier modulation nearly uniformly across the entire intrinsic region with only 5 V applied bias. We emphasize that reverse biased p-n junctions (as opposed to our forward biasing here) generally used in silicon photonic modulators usually produce very small changes in carrier concentration over a small volume, which is not ideal for tunable metasurface applications.

Phase Modulation with Multi-Element Unit Cells
As discussed in the main text, in going to subwavelength resolution with a single scattering nanopost per unit cell, we must carefully handle the resultant coupling between adjacent pixels. Depending on the level of variance in phase as a function of the lattice constant, the designed scatterers may be more well-suited for implementing particular classes of phase profiles. Our heuristically determined moderately coupled nanoposts in Fig. 2d of the main text lend themselves well to implementing lenses, axicons, and vortex beam generators, profiles with relatively low phase gradients; however, for the rapidly varying profiles necessary for generating holograms, such as that in Fig. S8a, the resultant hologram in the far-field (Fig. S8b) exhibits a missing upper corner, bright spots, and gaps indicative of errors in phase modulation. If instead we reduce the spatial resolution and utilize pixels of double the area which consist of a 2x2 set of identical nanoposts (Fig. S8d), then we can ensure all phase gradients are reduced by a factor of 2.
With this reduced spatial resolution, the generated hologram in the far-field (Fig. S8c) is cleaner. While there are still bright spots, the generated hologram matches the ideal result

Simulation by the Angular Spectrum Method
As simulating large regions using the finite-difference time-domain (FDTD) method is memory and time-intensive, we only use FDTD to simulate the fields propagated to a few microns off our devices and then use a wave optics propagator to simulate the fields at much further distances. In particular, as our propagation distances are much smaller or on the order of our aperture size, the Fresnel number of our system satisfies ≫ 1, The longitudinal wavevector is a function of the wavenumber and transverse wavevectors, defined as: