SUPPLEMENTARY INFORMATION

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1 SUPPLEMENTARY INFORMATION doi:0.038/nature727 Table of Contents S. Power and Phase Management in the Nanophotonic Phased Array 3 S.2 Nanoantenna Design 6 S.3 Synthesis of Large-Scale Nanophotonic Phased Array 8 S.4 Phase Noise Analysis of Large-Scale Nanophotonic Phased Array List of Figures S Power and phase management in the nanophotonic phased array S2 Nanoantennadesign... 7 S3 Phasedarraysynthesis... 0 S4 Phase noise tolerance in the nanophotonic phased array

2 RESEARCH SUPPLEMENTARY INFORMATION S. Power and Phase Management in the Nanophotonic Phased Array In phased arrays, it is essential to have all of the pixels emit with a desired amplitude pattern, which is uniform here in our demonstration, so that all of the emissions can create an ideal interference condition for the phase interactions to take effect in the far field. Especially when it comes to large arrays with thousands of pixels, it is crucial to design the power feeding mechanism carefully and reliably to have a large number of nanoantennas emit with precise amplitudes. This is one of the most important reasons that limit previous nanophotonic phased array demonstrations to a smaller number of antennas. In our work, a new optical power feeding network, which is fundamentally different from previous nanophotonic phased array demonstrations, is used to achieve uniform emission across the large number of nanoantennas. Figure Sa shows the way in which the optical power from the input fibre is equally split into M row bus waveguides. The length Lc(m) of the directional couplers is varied to change the coupling ratio in such a way that the m th ( m M) row bus waveguide has a coupling efficiency of /(M +2 m), as shown by Fig. Sa and the blue line in Fig. Sb. In our phased array, the bus-to-row coupler length varies from 3.53µm (coupling efficiency=.54%) in the first coupler to 8.05µm (coupling efficiency=50%) in the last coupler with a constant coupling gap of 20nm, as shown by the red line in Fig. Sb. Note that the last part of the optical power in the silicon bus is discarded after the M th row through a gradual waveguide taper, in order to avoid the necessity to achieve 00% coupling efficiency in the M th row using an over-sized coupler. This causes a negligible power loss of.54%. The optical power is coupled in the same way from each row bus waveguide to the unit cells along it. The green line in Fig. Sb shows the row-to-unit coupler length varying from 2.23µm (coupling 2

3 SUPPLEMENTARY INFORMATION RESEARCH efficiency=.54%) to 8.2µm (coupling efficiency=50%). Notice that we use the coupler length instead of the coupler gap to change the coupling efficiency since the coupling efficiency is less sensitive to coupler length variation and is thus more controllable in fabrication. The measured uniform near-field emission across all the 4096 nanoantennas shown in Fig. 3b in the main text validates the effectiveness of the proposed power feeding network and the accuracy of the FDTD simulation. Phase management is another important aspect that needs to be carefully designed in the phased array to ensure all of the pixels to emit with the desired phases. This desired phase, ϕ mn in pixel (m, n) for example, is achieved by two identical optical delay lines where each provides a propagation phase of ϕ mn /2, as shown in Fig. Sc. This geometric design makes the position of the nanoantenna (which is connected to the second delay line via a waveguide bend with constant phase) independent on the phase delay ϕ mn, so that all of the nanoantennas can be placed on a periodic grid that is required for phased arrays. It is also important to notice that the varied coupler length slightly affects the phases of the transmitted light and the coupled-out light, This effect was also taken into account in the design when calculating how much phase shift ϕ mn is needed for pixel (m, n). To sum up, in order to achieve arbitrary patterns in a phased array, it is critical for a large number of nanoantennas to emit with desired phases and desired amplitudes at the same time. In our nanophotonic phased array, because the amplitude control using the coupler length and the phase control using the delay lines are decoupled from each other, it is straightforward to have any pixel emit with any desired phase from 0 to 2π and any desired amplitude from 0 to (normalized), and more important, any arbitrary combination of the two. This enables the proposed nanophotonic phased array to generate truly arbitrary far-field patterns for the first time. 3

Fibre RESEARCH SUPPLEMENTARY INFORMATION a M+ M M- 2 L () C To unit cells Row bus waveguide To unit cells Row bus waveguide2 To unit cells L (2) C L (3) Row bus waveguide 3 C L C (M) To unit cells

a, Power feeding network where the optical power in the bus waveguide is equally coupled to M row waveguides. The green boxes indicate the optical power in each segment of the network.

4 Fibre RESEARCH SUPPLEMENTARY INFORMATION a M+ M M- 2 L () C To unit cells Row bus waveguide To unit cells Row bus waveguide2 To unit cells L (2) C L (3) Row bus waveguide 3 C L C (M) To unit cells Row bus waveguide M Silicon bus waveguide (Discard) b c Incoming transmitted Directional coupler mn /2 mn /2 constant Figure S. Power and phase management. a, Power feeding network where the optical power in the bus waveguide is equally coupled to M row waveguides. The green boxes indicate the optical power in each segment of the network. b, The coupling efficiency (Blue), and coupler length for the bus-to-row couplers (Red) and the row-to-unit couplers (Green) in the nanophotonic phased array. The length of row-to-unit couplers is a little different from the bus-to-row couplers because different bend radii are used in the two cases. The coupler lengths are obtained through a 3D-FDTD simulation. c, Optical phase management in the unit cell (m, n) to achieve a desired phase shift ϕ mn. 4

5 SUPPLEMENTARY INFORMATION RESEARCH S.2 Nanoantenna Design The silicon dielectric nanoantenna is used as an emitter in each pixel for the direct integration with CMOS process. The nanoantenna measures 3.0µm 2.8µm consisting of 5 grating etches, as shown in Fig. S2a. The first grating etch is half-way through the 220nm-thick silicon layer to create an up-down asymmetry in order to have more power emit up, as shown in Fig. S2b where a total emission efficiency of 86% is achieved at a wavelength of.55µm with 5% emitting up and 35% emitting down. More efficient up-emission can be realized if a more optimized partial etch depth is used (the partial etch depth was fixed to 0nm in our process for the consideration of other devices on the same mask), or a reflective ground plane is implemented underneath the grating to reflect the downward emission. It is also noted that the 3dB bandwidth of the emission exceeds 200nm, an inherent nature of the short grating length that could find potential applications such as broadband vertical couplers. The grating period is 720nm, a little detuned from the period of a second order grating (58nm for Si-SiO 2 gratings at λ 0 =.55µm) that would emit vertically, in order to suppress the resonant back-reflections (5% as shown in Fig. S2b) that would interfere with the light propagation in the phased array. This period detuning also creates a non-vertical emission as shown in Fig. 2b in the main text. 5

RESEARCH SUPPLEMENTARY INFORMATION a Full etch (220 nm) b Partial etch (0 nm) 720 nm 2.8 m 3.0 m Figure S2. Nanoantenna design. a, A scanning electron micrograph (SEM) of the fabricated nanoantenna.

6 RESEARCH SUPPLEMENTARY INFORMATION a Full etch (220 nm) b Partial etch (0 nm) 720 nm 2.8 m 3.0 m Figure S2. Nanoantenna design. a, A scanning electron micrograph (SEM) of the fabricated nanoantenna. The bright color represents silicon with a height of 220nm while the dark color is the buried oxide (BOX), and the color in-between is the partially etched silicon with a height of 0nm. b, The simulated grating emission efficiency. 6

7 SUPPLEMENTARY INFORMATION RESEARCH S.3 Synthesis of Large-Scale Nanophotonic Phased Array The objective of nanophotonic phased array synthesis is to generate a specific far-field radiation pattern by assigning the optical phase of each pixel in the phased array. As shown in equation () in the main text, the far-field radiation pattern is the multiplication of the far field of an individual nanoantenna S(θ, φ) and that of the array factor F a (θ, φ). While the far field of an individual nanoantenna is fixed, the array factor F a (θ, φ) is related to the emitting phase of all the pixels in the array 23 M N F a (θ, φ) = w mn e j2π(xmu+ynv) = F (w mn )=F(e jϕmn ) m= n= (S) where M N is the size of the array, and (x m,y n ) describes the position of each nanoantenna. The emitting amplitude and phase of the nanoantenna is described by w mn and ϕ mn respectively, so that w mn = w mn e jϕmn. In a phased array, all the nanoantennas emit with a desired amplitude pattern, which is uniform as we used here ( w mn =), to create an ideal interference condition in the far field for the phase (ϕ mn ) interaction to take effect properly. The parameters u = sin(θ) cos(φ)/λ 0 and v = sin(θ) sin(φ)/λ 0 are related to the far-field coordinates (θ, φ), and λ 0 is the optical wavelength in free space. As shown in equation (S), the array factor F a (θ, φ) is a simple discrete Fourier transform (denoted by "F ") of the emitted phase of the array. This provides an efficient way to find the optical phase ϕ mn to generate a given radiation pattern F a (θ, φ) using the Gerchberg-Saxton algorithm 23,24 as shown by the block diagram Fig. S3a. At the k th iteration, an approximated array factor Fa k (θ, φ) consisting of the desired amplitude F a (θ, φ) and a trial phase Φ k (θ, φ) is inversely Fourier-transformed to get the corresponding wmn k of each nanoantenna. The far-field trial phase Φ k (θ, φ) is not of interest and is arbitrarily chosen since only the 7

8 RESEARCH SUPPLEMENTARY INFORMATION amplitude of the array factor F a (θ, φ) affects the final far-field radiation image. The amplitude of w k mn is then set to while its phase e jϕk mn is kept so that the amplitude of the nanoantennas is uniform across the array. Therefore the updated array factor Fa k (θ, φ) is obtained through Fourier transform whose phase Φ k (θ, φ) is passed to the (k + ) th iteration as the new trial phase Φ k+ (θ, φ). The initial trial phase of the radiation field is set to Φ (θ, φ) =0or any arbitrary values in the st iteration. After several iterations, the final array factor Fa k (θ, φ) generated by the phase e jϕk mn converges to the desired pattern Fa (θ, φ). Figure S3b shows the array factor pattern produced by a nanophotonic phased array with the MIT-logo "MIT" in the far field while Fig. S3d gives the corresponding phase ϕ mn calculated from above-described method. The pixel pitch here was chosen to be λ 0 /2 to produce an unambiguous pattern in the far field, while the pixel pitch used in the simulation and experiment of the main text is a multiple of λ 0 /2 resulting in a replication of the fundamental pattern, as shown in Fig. 2c and Fig. 3 in the main text. Figure S3c shows the simulated array factor pattern that aims to generate multiple beams with different angles in the far field to show versatile patterns can be generated with the large-scale nanophotonic phased array, while Fig. S3e shows the corresponding phase in the array. 8

9 Row index n Row index n SUPPLEMENTARY INFORMATION RESEARCH a AF k (, ) k (, ) AF k (, ) F - w k mn w k mn k mn k=k+ AF *k (, ) k (, ) AF *k (, ) F e j k mn k mn b Intensity (db) 0 c d d Phase ( rad).0 e Column index m Column index m Figure S3. Phased array synthesis. a, Block diagram showing the antenna synthesis method for large-scale nanophotonic phased arrays. b, The simulated far-field array factor pattern - the MIT-logo "MIT" by the nanophotonic phased array. c, The simulated far-field array factor pattern aiming to generate multiple beams with different angles in the far field, which may prove useful in optical free space communications. A pixel pitch of λ 0 /2 was chosen in b and c. d, The colour plot of the phase of the array corresponding to the "MIT" pattern. e, The colour plot of the phase of the array corresponding to the multiple-beam pattern. 9

10 RESEARCH SUPPLEMENTARY INFORMATION S.4 Phase Noise Analysis of Large-Scale Nanophotonic Phased Array In a nanophotonic phased array, the far-field generation relies on the precise optical phase ϕ mn of each nanoantenna. However, due to random fabrication imperfections, the actual phase at each nanoantenna usually differs from its ideal value ϕ mn. We refer to this random error as a phase noise ɛ mn whose impact on the array factor pattern is to be analyzed. Assuming the random phase noise has a Gaussian probability distribution with zero mean <ɛ mn >=0and standard deviation σ which is usually the case for noise introduced by fabrication. The actual resulting array factor pattern under the presence of phase noise is again given by equation (S), with the phase noise added to the ideal phase F ac a (θ, φ) =< F (e jɛmn e jϕmn ) >=< F (e jɛmn ) > F id a (θ, φ) (S2) where F ac a (θ, φ) stands for the actual array factor pattern with noise while F id a (θ, φ) is the ideal pattern. is the convolution operator. Note that the expectation value (denoted by the angle brackets) is used here, meaning that the average value is taken for the stochastic variables and functions. The discrete Fourier transform of phase noise is given by < F (e jɛmn ) >= m <e jɛmn > e j(xmu+ynv) (S3) n And the expectation value in equation (S3) is by definition calculated as <e jɛmn >= + e jɛ 2πσ e ɛ2 2σ 2 dɛ = e σ2 /2 (S4) Substituting equation (S4) into equation (S3) and then into equation (S2) yields F ac a (θ, φ) =e σ2 /2 F id a (θ, φ) (S5) 0

11 SUPPLEMENTARY INFORMATION RESEARCH The significance of equation (S5) lies in that it shows the shape of the far-field array factor pattern is preserved while its amplitude is reduced by a factor of e σ2 /2 due to the presence of the phase noise. To validate the above noise analysis, a Gaussian phase noise with various noise levels (i.e. standard deviation σ) is added to the nanophotonic phased array whose phase ϕ mn is set to generate the MIT-logo "MIT", then the far-field array factor pattern F a (θ, φ) is simulated. Figure S4 shows the array factor pattern with different phase noise levels, σ =0(Fig. S4a, no phase noise), σ = π/6 (Fig. S4b), σ = π/8 (Fig. S4c), and σ = π/4 (Fig. S4d). It is seen that in all cases, as the phase noise level increases, the shape of the desired pattern remains, as indicated by equation (S5); but its signal-to-noise ratio (SNR) drops, compared to the unperturbed pattern in Fig. S4b. The rising background noise comes from the lost power in the desired pattern area since the ideal interference conditions designed to generate the desired pattern are no longer completely satisfied under the presence of the phase noise. The simulation results are consistent with the theoretical analysis in equation (S5). It is noted that even with a considerably large phase noise (σ = π/4), the designed pattern is still distinguishable. Therefore, the phased array shows high tolerance to phase errors, which largely relaxes the stringent accuracy requirements on fabrication, and hence ensures such a large-scale nanophotonic phased array can be reliably produced with moderate fabrication requirements and function properly. Moreover, this high error-tolerance does not depend on the scale of array. In fact, the more nanoantennas the array has, the more genuine the above analysis is, from statistical considerations. Therefore, there exist no major obstacles to scale up the pixels of the nanophotonic phased array beyond to even millions of pixels.

12 RESEARCH SUPPLEMENTARY INFORMATION 9 9 a Intensity (db) 60 b c 9 60 Intensity (db) d Figure S4. Phase noise tolerance. The far-field array factor patterns with different phase noise levels, simulated by adding Gaussian phase noise mn with standard deviations of a, σ = 0 (no phase noise) b, σ = π/6 c, σ = π/8 and d, σ = π/4 to the ideal phase ϕmn. The designed pattern is still distinguishable even with a considerably large phase noise level of σ = π/4. 2 W W W. N A T U R E. C O M / N A T U R E

Plane wave excitation by taper array for optical leaky waveguide antenna

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