Optical phased array (OPA) technology has undergone rapid progress and been adapted to a number of platforms, prominently silicon-on-insulator (SOI) [1] and silicon nitride (SiN) [2] photonic integrated circuits (PIC). They have found applications in a range of fields in sensing and displays, such as light detection and ranging (LIDAR) [3], augmented reality [4], microscopy [5], and optogenetic probes [6]. Dynamic steering of beams generated by large-scale two-dimensional (2D) OPAs remains, however, a daunting challenge due to the large number of required phase tuners. This is particularly true for the SiN platform, due to its reduced thermo-optic coefficient resulting in higher thermal phase shifter power consumption. Despite substantial progress [7], SiN phase shifters remain larger, consume more power, and have lower switching speed than their silicon counterparts. A number of strategies alleviate the complexity of 2D beam steering. A common approach is to only steer the beam in 1D with the OPA per se and to use the wavelength dependence of the emission angle of the grating couplers (GC) forming the antenna elements for steering in the other [8]. Waveguide delay loops [9,5] or arrayed waveguide gratings [6] have also been adopted to increase OPA sensitivity. These techniques require, however, a bulky and expensive tunable laser and are restricted to applications compatible with a wide enough wavelength range, restricted, e.g., by the absorption spectrum of molecules in microscopy or optogenetics. Free-space communication or frequency-modulated continuous wave (FMCW) LIDAR are further applications for which chirping the laser frequency for beam steering is problematic. Recently, some of us have proposed to implement beam steering independently in the two (xy) directions along the surface of the chip [10,11] using a crossbar configuration. Phase increments are applied across a set of vertical (horizontal) waveguides and determine the steering in the x (y) direction. At each horizontal and vertical waveguide cross point, light is tapped off from both and sent to a common GC, which radiates light containing phase information for both the x- and y-direction steering. Two-dimensional beam forming and steering then results from the linear superposition of the diffraction patterns of the two source beams. Despite successful 2D beam-steering demonstration and significant tuning related power consumption and chip area reduction, this architecture still has shortcomings that will be addressed in this Letter.

Figure 1(a) shows the far-field emission of a square-shaped OPA with the same phase applied to each of the GCs, resulting in a beam emitted straight up. For compactness, similar assumptions are made as for the structures reported in the rest of this Letter. The wavelength is 520 nm, as supported by SiN, the GCs are spaced by 15 µm, and each emit a Gaussian beam with a 1/e² beam diameter of 8 µm. The OPA comprises 15 GCs to a side. Rescaling of the color map [Fig. 1(b)] reveals the cross-shaped beam diffraction induced by the square OPA aperture. When combining two such beams, one tilted along x and one along y, a third beam is formed where the diffraction patterns cross, that can be steered in two dimensions. When constant phase increments $\Delta {\varphi _x}$, $\Delta {\varphi _y}$ are applied across the waveguides, most of the power remains in the mainlobes of the source beams, resulting in inefficient beamforming [Fig. 1(c)]. Also, the generated beam is surrounded by the diffraction patterns of the two main beams from which it is generated. Without further refinements, it only has approximately twice the amplitude (four times the intensity) than its surroundings and is relatively weak. To overcome this, static phase offsets were applied to the GCs in [10], optimized to uniformly distribute the power between the main and sidelobes to significantly increase the power of the formed beam [Fig. 1(d)]. For this OPA, it is approximately 5.4 dB lower than the mainlobe in Fig. 1(a), with a 6–9-dB sidelobe suppression. In addition to the power distribution, the number of sidelobes can also be adjusted. The phase shifter currents required for beam-steering to desired angles are iteratively optimized and stored in a look-up table. While this yields good results, the OPA performance is prone to process and temperature variations. The beam also cannot be continuously steered, but only stepped to pre-calibrated sidelobe positions.

 

Fig. 1. Emitted intensity profile of a non-apodized square-shaped OPA with (a), (b) a single generated beam (different color scales) and (c) two beams deflected in x and y. (d) Beam formed after phase optimization following [10]. Peak intensities are normalized to 1. Axes indicate the emitted light’s k-vector.

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In the approach proposed here, nonlinear interaction between the mainlobes instead allows simplified and continuous tuning with constant phase increments, as well as a higher optical power within the formed beam. The intensity emitted by individual GCs is apodized with a Gaussian profile according to their position (300-µm 1/e² width), since the diffraction pattern is no longer helpful.

We have first searched for a suitable material or structure that can be added to an OPA to implement the required nonlinearity. Other authors have used add-on planar structures to overcome other shortcomings of OPAs, e.g. a metasurface increasing the angular steering range [12]. Here, the mainlobes of two linearly steered beams are mixed by a strongly nonlinear layer to generate a third beam steered in both directions. Partially oxidized graphene sheets dispersed in polymer film matrices have e.g., been shown to possess a giant nonlinear absorption [13], which could in principle serve to operate the required transformation. However, even this uncommonly large response proved insufficient given the ∼14 W required per grating coupler in continuous wave operation and the <1-W power levels that can be injected in an SiN PIC via a single input. While slightly more complex, we found optically addressable (OA) liquid crystal (LC) based spatial phase modulators (SPM) [14] (light valves) to be a suitable candidate that can be modulated at kHz speeds [15]. In these, a layer cladding the LC absorbs a small portion of the transmitted light, modifying the electric field applied to the LC sandwiched between it and a glass slide coated with indium-tin-oxide (ITO), reorienting the LC and applying a phase shift [14]. The glass slide can be straightforwardly replaced by a PIC, yielding a compact device with integrated nonlinear beam mixing (see the Supplementary material for a schematic and enlarged figures). Here, we assume the characteristics of a commercially available device [15], to which the PIC can be attached, which uses bismuth silicon oxide (BSO) as a photo-absorber. Depending on its thickness, it can apply phase shifts of one to several times 2π. It is sensitive to optical power levels from 10 µW/cm² to a few tens of mW/cm² depending on its electrical bias. The induced phase shift is highly linear relative to the applied optical power until it reaches approximately two-thirds of its maximum [14] and we assume staying in this linear range. As a consequence of the optically addressable spatial phase modulator (OASPM) nonlinearity, two beams generated with in-plane wave vectors $[{{k_x},0} ]$, $[{0,{k_y}} ]$ mix and create harmonics $[{m \cdot {k_x},({-m + 1} )\cdot {k_y}} ],\; m \in \mathrm{\mathbb{Z}}$, as illustrated by Fig. 2(a). It shows the real part of the E-field resulting from the two original beams, along $[{-{k_y},{k_x}} ]$, before and after transmission through the LC, assuming a maximum phase shift of π at the field maxima. Figure 2(b) shows the corresponding Fourier transforms as a power spectral density (PSD). While the strength of the first harmonics is larger for slightly higher phase shifts, at approximately 1.4π in this simplified 1D problem, higher order harmonics grow disproportionally, which is problematic (see below). The abscissas in Fig. 2(a) correspond to half the relative phase between the two beams, $[{{k_x} \cdot x\; -{k_y} \cdot y} ]/2$.

 

Fig. 2. (a) Real part of the E-field as first emitted (black) and after transformation by the OASPM (red). (b) Corresponding spectra.

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Due to the finite size of the GCs, higher order harmonics are aliased and each create a multitude of sidelobes, crowding k-space and reducing the range in which a single beam can be steered interference free. In a regular OPA, this is given by a region measuring $\Delta k = 2\pi /d$ referred to as an irreducible Brillouin zone (BZ), with $d = 15$ µm corresponding here to a ${\pm} {1^o}$ steering range. Here, this is not limited by the size of the GCs, which can be much smaller at visible wavelengths [16], but rather by the OASPM. The free carrier diffusion length inside the BSO layer is approximately 20 µm and the LC only reacts to the average optical intensity in a corresponding area, so that shrinking the GCs further no longer helps to increase the steering range. The interference-free range is reduced further due to one diffraction order being aliased into the chosen BZ for each harmonic order. It is thus important to minimize the power converted into higher orders other than the targeted one, chosen as $m =-1$, and its unavoidable counterpart at $m = 2$. This limits the maximum nonlinearity. We calculate the position of these two orders in a single BZ, see Fig. 3(a). The locus of the targeted beam ($m =-1$) is varied in a test pattern consisting in concentric circles of increasing diameter (up to $2\Delta k/3$) centered on ${\boldsymbol k} = [{0.5\Delta k,0.65\Delta k} ]$, chosen to have the maximum interference-free surrounding range. The locus of the aliased $m = 2$ beam (red dashed lines) is confined in the lower third of the BZ. The upper two-thirds can thus be used without suffering from interference. We simulate an OPA whose dimensions have been increased to 20 × 20 GCs to support the apodized emission pattern described above. Their emission angle is assumed to be 3.4° from normal, oriented such that the emitted power is maximized at ${\boldsymbol k} = [{-\pi /d,3.3\pi /d} ]$ (center of the blue circles in Fig. 3(a) after shifting the diffraction order by $[{-\Delta k,\Delta k} ]$). The assumed maximum phase shift applied by the LC where the emitted power is maximized (${\varphi _{LC,max}}$) is varied between $\pi $ and $3\pi $ to determine the optimum LC thickness.

 

Fig. 3. (a) Locus of the harmonic components $m =-1$ (solid blue) and $m = 2$ (dashed red) within a single BZ. They are nonlinearly generated from the beams with $m = 0$ and $m = 1$ (cyan, after aliasing). (b) Far-field emission pattern. White, green, red, magenta, and cyan boxes indicate the position of the source beams, the target beam ($m =-1$), its counterpart at $m = 2$, the next higher harmonic, and the aliased source beams. The dashed white square shows the single BZ in which the targeted beam is steered. Color scales are normalized relative to the strongest beam peak power.

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The LC is first assumed to be immediately on top of the PIC with the PIC part of the LC-cell. The photoconductor, located on the other side of the cell, is modeled to respond to the optical intensity after convolution with a 2D Gaussian with a 25-µm 1/e² diameter, modeling the carrier diffusion limiting the spatial resolution of the OASPM. After applying the LC-induced phase shift, the field is far-field transformed to determine the emission pattern of the OPA.

Figure 3(b) shows the result of a simulation in which the OPA is programmed to generate two beams with ${\boldsymbol k} = [{0.6\Delta k,0} ]$ and ${\boldsymbol k} = [{0,0.825\Delta k} ]$ (white boxes), which are intended to mix into a third beam at $[{-{k_x},2{k_y}} ]$ (green box). Somewhat surprisingly, this works even though one of the two generating beams contains a vanishing amount of power, as a consequence of it being too far outside of the emission cone of the GCs. To understand this, one has to bear in mind that the spacing between GCs is of the order of the spatial resolution of the OASPM and the GCs themselves are actually somewhat smaller. Consequently, the phase applied by the OASPM is approximately constant across individual GCs and only depends on the average intensity emitted by the GC without following its local intensity profile. The emission is thus adequately described by

Figure 4(a) shows an overview of the generated pattern with ${\varphi _{LC,max}} = \pi $ and Fig. 4(b) a detailed view of the main BZ, whose upper two-thirds is confirmed to remain free of other beams. Second-order harmonics also follow the contours in the lower third, as predicted. When ${\varphi _{LC,max}}$ is increased to $2.2\pi $, as shown in Fig. 4(c), the target beam intensity increases, but third-order harmonics appear and clutter the diagram, limiting the interference-free range. In Fig. 4(d), we explore what happens when the PIC is not integrated into the OASPM (adjacent to the LC layer), but attached to an off-the-shelf device. Emitted light then first propagates through 500 µm of BSO. The target beam power goes down some and the pattern is slightly less regular, but the scheme still fundamentally works. More quantitatively, in Fig. 5(a), the power contained in the target beam and a second-order harmonic are shown as a function of ${\varphi _{LC,max}}$. The target beam power is maximized for ${\varphi _{LC,max}} = 2.2\pi $ and reaches ∼72% of the strongest pump beam diffraction order. However, as in any OPA, the power efficiency is also limited by spreading of the power over several diffraction orders, so that overall, 4.7% of the emitted power is converted into the target. For this level of phase shifts, the undesirable second harmonic already reaches 31% of the target beam power. At a lower ${\varphi _{LC,max}}$ of 1.4$\pi $, the power conversion efficiency drops to 3.3%, but the power in the second harmonic remains a factor of 10 smaller. At ${\varphi _{LC,max}} = \pi $ (2% conversion), the second harmonic is negligible. Figure 5(b) shows the power in the target beam as it is scanned across the two inner contours from Fig. 3(a). The power stays quite homogeneous as the beam is scanned, with the hybrid solution consisting in assembling the PIC with an off-the-shelf OASPM slightly less efficient and a bit less homogeneous. An OPA in which the beam is conventionally steered with an $N \times N$ phase shifter array but that is otherwise identical would result in 30% of the light in the target beam, which is 9× and 6× more than here respectively for ${\varphi _{LC,max}} = 1.4\pi $ and ${\varphi _{LC,max}} = 2.2\pi $. This power reduction is the price to pay for the much-simplified steering. Conventional beam steering, however, requires a phase shifter to be integrated in each unit cell, constraining its size and the achievable steering range. While a 9 × 9-µm² unit cell has been achieved with silicon tuners [1], resulting in a steering range of ±5° at 1550 nm, this would be particularly limiting in SiN at the visible wavelengths targeted here due to the larger phase tuners. While the OASPM resolution currently constrains the GC pitch, segmenting the photo-absorber to suppress interpixel carrier diffusion might increase the steering range significantly beyond the state of the art.

 

Fig. 4. Far-field emission pattern. Thirty-six beams are sequentially generated along the target contours shown in Fig. 3 (here in white circles). The plotted intensity corresponds to the max. taken over all simulations. (a) Overview with ${\varphi _{LC,max}} = \pi $. Detailed views in the main BZ for (b) ${\varphi _{LC,max}} = \pi $ and (c) ${\varphi _{LC,max}} = 2.2\pi $. The red dashed lines show the predicted locus of the $m = 2$ mixing product.

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Fig. 5. (a) Fraction of power in the targeted beam, one of the two second harmonics, and the strongest pump beam diffraction order, as a function of ${\varphi _{LC,max}}$. (b) Fraction of power in the target beam as a function of k, scanned on the two inner circles in Fig. 3, for ${\varphi _{LC,max}} = \pi $ and $2.2\pi $. The x axis gives the angle of k relative to the center of the circles. Light propagation through the photo-absorber is taken into account when the PIC is not integrated (“assembled”).

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Applying disorder to GC positions is a well-known technique to suppress higher diffraction orders [17]. While for the scheme investigated here GCs are required to remain arranged along rows and columns, the distances between rows (columns) can be randomized. As a proof-of-concept, we investigate the previously described OPA, in which the size of the GCs has been reduced from 10-µm diameter (8-µm beam mode field diameter) to 5 µm (4 µm) to create a sparser array leaving more room for displacing rows (columns). These are randomly displaced within ${\pm} 9$ µm relative to their neighboring left column / bottom row as the array is built up. The GC emission angle is increased to 10.3° and the device oriented to target a cone centered on $[{-2.5\Delta k,4.5\Delta k} ]$, sufficiently far from the nominal source beams at $[{2.5\Delta k,0} ]$ and $[{0,2.25\Delta k} ]$ so that these are not generated. The PIC is again assumed to be integrated in the LC-cell, so that Eq. (1) holds. Here, ${\varphi _{LC,max}}$ is set to $2.2\pi $, which would result in strong second-order harmonics in ordered OPAs. Results are shown in Fig. 6 for a beam targeted at $[{-2.5\Delta k,4.5\Delta k} ]$. Any beam that would need to be aliased to fall within the GC emission cone is smeared out, which applies to all but the target one, thus the only generated one. It is also generated even though the source beams are not present in the emission profile, following Eq. (1). This method suffers, however, from a couple of well-known challenges associated with disordered OPAs. The other beams are not suppressed, their power is merely distributed, resulting in a speckled background (here, suppressed by at least 6 dB). Due to the GC filling a smaller portion of the unit cell, their emission cone is also increased relative to the BZ, resulting in a larger number of smeared out diffraction orders capturing a larger portion of the emitted power. As a consequence, the 4.7% power efficiency is reduced to 1.2%. A disordered OPA thus comes with its own set of limitations.

 

Fig. 6. Emission profile of the disordered OPA. The BZ delimited by dashed lines provides a scale for comparison with the other figures.

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We have introduced a beam-steering method converting two beams, each steered in 1D, into a third beam steered in two dimensions, by means of a nonlinear layer added over the PIC. The control complexity of an N × N OPA is reduced from N² to 2N phase tuners. This beam can be continuously steered within two-thirds of the irreducible BZ interference free. Using an optically addressable LC, this scheme can be implemented with practical power levels. Using a disordered OPA, a single beam can be generated, subjected to the usual speckled background and reduced peak power.