The automotive industry has driven the research in long-range light detection and ranging (LiDAR) engines based on solid-state beam scanning, as part of a “sensor fusion” process [1]. The goal of a LiDAR system is to construct a 3D point cloud of the surrounding environment by combining two functions: beam steering and ranging. The beam steering requires sending a narrow beam to specific directions, while the ranging determines the distance of any target in the beam path through the time elapsed between the beam emission and the arrival of the back-scattered light [2]. Solid-state 2D beam steering can be performed using optical phased arrays (OPAs), where an array of optical antennas shapes and directs the beam to a certain far-field direction based on the phase delay of the light signal fed to each antenna [3]. A possible OPA implementation is to place the antennas in a 2D periodic [4] or sparse [5–7] array and control the phase of the input signal using phase shifters in the light distribution network. Such fully active tuning is challenging in terms of the circuit density, the control complexity of the individual phase tuners, and the high power consumption (in the case of thermal phase shifters).

In order to reduce the number of phase shifters, 2D beam steering can also be implemented with a 1D array of long waveguide grating-based antennas; the beam is scanned in one direction by tuning the phase shifters of the individual antennas, while a tunable laser is used to scan the beam in the other direction (as a result of the wavelength-dependent emission angle of the waveguide gratings) [8–12]. However, the complexity of the phase control can be entirely eliminated by using a single passive control variable: the wavelength of the tunable laser. In this fully passive approach, differential delay lines are incorporated into the light distribution network instead of the active phase shifters [13–17], where the phase delay of each input light signal is now a function of the laser wavelength. Such a dispersive optical phased array (DOPA) is in essence an arrayed waveguide grating (AWG), whereas instead of imaging on an output star coupler into the waveguide modes, light radiates off-chip into the far field, using grating-based antennas. As opposed to electronically controlled OPAs, DOPAs can have faster scan rates (limited only by the scanning rate of the laser wavelength), lower control complexity (only a single control variable, i.e., the laser wavelength), and lower power consumption (the phased array is passive).

Scaling these classical DOPAs to aperture sizes large enough for projecting the light beam over a long distance with high angular resolution, is challenging due to the DOPA architecture itself, namely the delay lines feeding the antennas. Increasing the number of antennas requires adding progressively longer delay lines in the light distribution network, which in turn increases the circuit footprint and insertion loss. In addition, the inevitable accumulation of phase errors along these long waveguides [18] causes the deterioration of the beam quality. In other words, the challenge of control complexity (in the active approach) is now replaced by the challenge of the excessively large passive circuit design and high-quality fabrication (in the passive approach).

In this paper, we present the first experimental demonstration of the pixelated dispersive optical phased array concept as a solution to the bottleneck of the delay line network scaling. In addition, we conduct a comprehensive evaluation of its performance with respect to a corresponding baseline classical DOPA. Both devices use an unbalanced splitter tree architecture for the light distribution network, which is, to our knowledge, also a first experimental demonstration of this architecture in the literature. The pixelated DOPA is shown to be a promising implementation of the scanning engine that can achieve realistic long-range LiDAR requirements on a chip-scale level, with high beam quality.

DOPAs provide passive 2D beam scanning using a single variable, i.e., the laser wavelength, as shown in Fig. 1, which also indicates the parameters used in this section. In one direction, denoted here as the $y$ axis, scanning is due to the dispersion of the long grating antennas. The far-field angle follows the grating equation, where the field of view $\mathrm{\Delta }{\theta }_y$ is a function of the grating dispersion $\frac{\mathrm{d}{\theta }_g}{\mathrm{d}\lambda }$ and the available wavelength tuning range $\mathrm{\Delta }\lambda$: $$\mathrm{\Delta }{\theta }_y=\frac{\mathrm{d}{\theta }_g}{\mathrm{d}\lambda }·\mathrm{\Delta }\lambda .$$(1)

In the other direction, denoted here as the $x$ axis, the grating antennas are arranged in a 1D array, fed by waveguide delay lines. The far-field angle is a function of the constant phase difference $\mathrm{\Delta }\varphi$ between each two consecutive antennas, following the phased array principle [19]. The phase difference is in turn a function of the wavelength $\lambda$, determined by the optical path length difference: $$\mathrm{\Delta }\varphi =2\pi \frac{\mathrm{\Delta }L·n_{\mathrm{eff}}(\lambda )}{\lambda },$$(2)where $n_{\mathrm{eff}}(\lambda )$ is the effective refractive index of the delay line waveguide, and $\mathrm{\Delta }L$ is the delay line length difference between each two consecutive antennas. In the $x$ direction, the unambiguous steering field of view (FOV) $\mathrm{\Delta }{\theta }_x$ is a function of the pitch between the antennas $p_x$ [19]: $$\mathrm{\Delta }{\theta }_x\approx \arcsin \frac{\lambda }{p_x}.$$(3)

Since $\mathrm{\Delta }L$ is larger than the periodicity of the gratings, the scanning rate along the $x$ direction (fast axis) is much faster than along the $y$ direction (slow axis). For a specific wavelength sweep rate of the laser, a longer $\mathrm{\Delta }L$ results in $\mathrm{\Delta }\varphi$ sweeping faster through the $[0,2\pi ]$ interval. The wavelength range required for one complete $2\pi$ phase cycle is the free spectral range along the $x$ axis ${\mathrm{FSR}}_x$: $${\mathrm{FSR}}_x\approx \frac{{\lambda }^2}{n_g·\mathrm{\Delta }L},$$(4)where $n_g$ is the group index of the delay line waveguide. One full scan covering the FOV along the $x$ direction is a scan line, achieved by a wavelength range equal to ${\mathrm{FSR}}_x$. The number of resolvable points in the $y$ direction $N_y$ (the number of scan lines) is then a function of ${\mathrm{FSR}}_x$: $$N_y=\frac{\mathrm{\Delta }\lambda }{{\mathrm{FSR}}_x}=\frac{\mathrm{\Delta }{\theta }_y}{\delta {\theta }_y},$$(5)where $\delta {\theta }_y$ is the $y$ axis sampling resolution, i.e., the angular separation between each two consecutive resolvable points in the $y$ direction. In this DOPA variation, for each wavelength point, there is a corresponding far-field location where a narrow beam is formed. This means that a narrow beam is continuously scanned along the $x$ axis upon scanning the laser wavelength, which is why we name this classical DOPA variation a continuous DOPA, as shown in Fig. 1(a).

Continuous DOPA implementations define the $x$ axis far-field sampling resolution specification to be the same as the $x$ axis beam divergence specification (see Appendix B for further explanation on the specifications of resolution and beam divergence along the $x$ direction). The $x$ axis far-field sampling resolution $\delta {\theta }_x$ is the angular separation between each two consecutive resolvable points in the $x$ direction, and the beam divergence is the full width at half-maximum ${\mathrm{FWHM}}_x$ of the spot at the maximum range, i.e., the Rayleigh range. In other words, for a diffraction-limited DOPA, which can be achieved in a platform with low losses and low phase errors, the number of resolvable points in the $x$ direction, $N_x$, is equal to the total number of antennas. Consequently, for a circuit designed to fulfill the required beam divergence for LiDAR, requiring a large number of antennas, the number of resolvable spots is orders of magnitude higher than the typical LiDAR requirement. For example, for a maximum range (Rayleigh range) of 200 m and $x$ direction FOV of 50° [20], the required beam divergence is 0.01°. This translates to the number of antennas equaling 8192. On the other hand, the angular resolution requirement is 0.1°, translating into the number of antennas equaling 512, where the number of antennas is rounded to the next power of 2 [21]. However, the pixelated DOPA, shown in Fig. 1(b), decouples these two specifications by subdividing the light distribution network into smaller blocks. The total number of antennas $N_x$ is now defined as $N_x=M{N}_x^{\prime }$, where $M$ is the number of blocks and $N_x^{\prime }$ is the number of antennas per block. The blocks are fed by a $1\times M$ balanced splitter tree, while the aperture is kept the same as in the continuous DOPA. As a result, the phase relation between the last and the first antennas of adjacent blocks is broken, and a narrow far-field spot only exists at specific wavelengths, i.e., the operation wavelengths ${\lambda }_m$ [22]: $${\lambda }_m=\frac{N_x^{\prime }}{m}·\mathrm{\Delta }L·n_{\mathrm{eff}}(\lambda ).$$(6)

When scanning the laser wavelength, the constructive interference condition for a narrow beam is now satisfied for discrete wavelengths, corresponding to discrete far-field locations. For this reason, we name this DOPA variation the pixelated DOPA. Here, the number of resolvable points along the $x$ axis equals the number of antennas within a block $N_x^{\prime }$, and the angular resolution along the $x$ direction $\delta {\theta }_x$ is $$\delta {\theta }_x\approx \frac{M}{N_x}.\arcsin \frac{\lambda }{p_x}.$$(7)

This additional degree of freedom in the distribution network significantly reduces the total length of the delay lines, simplifying the circuit design and reducing both the optical losses and phase errors.

In this work, we aim to experimentally demonstrate the pixelated DOPA concept and compare it one-to-one with a continuous DOPA with the same aperture design. Hence, we designed and fabricated a circuit for each variation [schematics are shown in Figs. 1(a) and 1(b)], where the two circuits have the same aperture design, differential delay line length $\mathrm{\Delta }L$, and distribution network choice. The center wavelength is 1550 nm with a span of 100 nm.

When defining the circuit on the schematic level, our first design parameter is the choice between Si and SiN for the waveguiding layer; these are the two key materials for compact photonic circuits and wafer-scale manufacturing. Si allows for more compact waveguides and bends, and allows higher group velocity index than SiN, requiring a shorter delay length $\mathrm{\Delta }L$ for the same target number of scan lines, following Eqs. (4) and (5). On the other hand, SiN waveguides exhibit lower phase errors [23], which should result in a better beam quality in terms of diffraction-limited beam divergence and side lobe suppression ratio (SLSR), where the SLSR is the ratio between the main lobe and the highest side lobe.

The differential delay $\mathrm{\Delta }L$ between the antennas can be introduced through different architectures, namely the AWG, demonstrated in Refs. [13,15–17]; the snake, demonstrated in Ref. [14]; and the unbalanced splitter tree. The latter was suggested in Ref. [22] but has not yet been demonstrated experimentally. As discussed in Ref. [22], the unbalanced splitter tree strikes a balance between an arrayed waveguide grating with fully independent waveguides, and a snake bus waveguide with controlled power taps, which does not scale well because of the difficulty in designing power taps with a very small but controlled coupling fraction. In addition, it has an intermediate performance sitting between the AWG and the snake architectures in terms of insertion loss and footprint [21]. In this work, we opted for the unbalanced splitter tree architecture fabricated on an LPCVD SiN stack in IMEC’s 200 mm pilot line. This platform has low propagation loss and low phase errors [24]. However, this results in a relatively large bend radius which increases the size of the circuit. In addition, our focus here is on the circuit demonstration, while the individual building blocks can be further optimized. We therefore use simple and proven components to build the circuits where possible.

The emitting aperture consists of a 16-element antenna array, with a pitch of 6 μm to circumvent any possible evanescent coupling between the antennas. This gives a theoretical $x$ axis FOV $\mathrm{\Delta }{\theta }_x$ of 15°, and a beam divergence ${\mathrm{FWHM}}_x$ of 0.87° at 0° emission angle [19]. The optical antenna is implemented using weak sidebox gratings, as shown in Fig. 1(c), with a period of 1 μm. The grating antenna is designed to have a strength of $\approx 17\mathrm{dB}$/mm, resulting in an effective antenna length below 1 mm. The simulated scan rate of the grating is $\approx 0.07$ deg/nm, resulting in a $y$ axis FOV of $\approx 7°$ for a 100 nm wavelength range. However, the metrics associated with the grating antenna design can differ after fabrication, since the grating is not a proven component provided by the platform. For this demonstration, we aimed to populate the $y$ axis FOV with 25 scan lines for the selected 100 nm wavelength range. We calculated the waveguide group velocity index $n_g$ to be 2.0, based on the SiN waveguide dimensions and the material refractive indices, through mode simulations. The differential delay line length $\mathrm{\Delta }L$ is then 300 μm, calculated from Eqs. (4) and (5), around a center wavelength of 1550 nm.

A microscope image of the fabricated devices is shown in Fig. 2. On the circuit layout level, the unbalanced splitter tree architecture requires that only half of the arms in each splitter stage have a delay line. We used meander-shaped delay lines with bend radius of 60 μm, as shown in Fig. 2(d). Here, the delay lines are added at the south output arm of the splitter at each level of the splitter tree. However, we also added meanders to the north output arm to provide balance in the number of bends, as shown in Fig. 2(b). Splitters based on $1\times 2$ MMIs, with a footprint of $62.5\mathrm{\mu m}\times 5\mathrm{\mu m}$ including the tapers, are used. Due to the footprint of the meanders, the output pitch of the distribution network does not match the desired array pitch $p_x$. A fan-in routing is used to match the pitches while not adding extra delay between the antennas (delay-maintaining fan-in), as shown in Fig. 2(e). Note that for this circuit, we separated these design concerns in different sections of the circuit, resulting in a larger-than-necessary footprint.

For the pixelated DOPA, the same design choices are followed in order to compare the two variations, except for the distribution network which is subdivided into four blocks fed by a balanced splitter tree, as shown in Fig. 2(c). Thereby, the pixelated DOPA is expected to have approximately four pixels per scan line, matching the number of antennas $N_x^{\prime }$ in a single block. Moving from the continuous DOPA to the pixelated DOPA, the number of splitting levels of the unbalanced tree is reduced from four to two, and the longest delay line length path is reduced from 4.5 to 0.9 mm, respectively, which should improve the beam quality due to lower phase errors.

We perform a broadband measurement of the continuous DOPA far field as proof of the unbalanced splitter tree architecture and as a baseline for the pixelated DOPA. We extract the FOV, scan rate, and FWHM, where the far-field images are acquired through a Fourier imaging setup. The images are processed to locate the main lobe, grating lobes, and side lobes pixels through a peak detection algorithm.

For the $x$ axis, the FOV $\mathrm{\Delta }{\theta }_x$ is 15° around 1550 nm, while the scan rate $\frac{\mathrm{d}{\theta }_x}{\mathrm{d}\lambda }$ is $\approx 3.66$ deg/nm, as shown in Figs. 3(a) and 3(b). As a result, the wavelength range required to sweep over one scan line ${\mathrm{FSR}}_x$ equals $\mathrm{\Delta }{\theta }_x/(\frac{\mathrm{d}{\theta }_x}{\mathrm{d}\lambda })\approx 4.1\mathrm{nm}$ (see Visualization 1), corresponding to $\approx 24.5$ scan lines for a 100 nm wavelength range, following Eq. (5). The beam divergence is a function of the wavelength and the emission angle [19] and varies slightly with the introduction of phase errors [25]. Since it is difficult to disentangle these dependencies, we report the histogram of the beam divergence for the $x$ axis ${\mathrm{FWHM}}_x$ based on 225 data points, as shown in Fig. 3(c). The median ${\mathrm{FWHM}}_x$ is 0.92°, so it follows that the far-field sampling resolution $\delta {\theta }_x$ is $\approx 0.92$°. This indicates that we have 16 fully resolvable points per scan line, and the wavelength step required to move from one fully resolvable point to the next along a scan line is $\approx 240.4\mathrm{pm}$. Along the $y$ axis, the scan rate $\frac{\mathrm{d}{\theta }_y}{\mathrm{d}\lambda }$ is $\approx 0.072$ deg/nm [Fig. 3(b)], which means that the FOV $\mathrm{\Delta }{\theta }_y$ is 7.2° for the 100 nm tuning range. Based on the number of scan lines and the FOV, the far-field sampling along the $y$ axis $\delta {\theta }_y$ is $\approx 7.2°/24.5=0.29°$. The histogram of the beam divergence along the $y$ axis ${\mathrm{FWHM}}_y$, based on 910 data points, is shown in Fig. 3(c), where the median ${\mathrm{FWHM}}_y$ is 0.36°, meaning that the effective antenna length is $\approx 225$ μm, which deviates from the design. The deviation is expected because of the fabrication variations that are more pronounced with the small geometric perturbations required for weak gratings [26] and because of the phase errors along the grating antenna [27]. On the other hand, the number of pixels and FOV along the $x$ and $y$ directions match the design specifications. Beam steering results over a 100 nm wavelength range are shown in Fig. 4.

In order to compare both DOPA variations in terms of the phase errors, we investigate the background of side lobes between the main lobe and the grating lobes. This is especially important because these phase errors, resulting from the geometry variations, are not compensated after the fabrication of such passive devices. Phase and amplitude errors cause power to be redistributed between the main lobe, grating lobes, and the side lobes. The SLSR, that is the ratio between the main lobe and the highest side lobe, has been conventionally used as an indicator of the phase errors of active and passive OPAs. However, looking only at the highest side lobe can give a skewed picture of the phase errors in the OPA, as it ignores the light directed to the background, i.e., to all the other angles outside the main lobe. The background can be indicated by the crosstalk suppression, which we calculate by excluding the main lobe and grating lobe pixels and then taking the 90^th percentile of all other pixels (see Appendix C for further details on this method).

Considering FMCW ranging, the redistribution of light away from the main lobe due to phase errors reduces the useful signal level. This redistribution is well characterized by the crosstalk suppression. Additionally, prominent spikes in the side lobes can, from the receiver’s point of view, produce beat signals that are indistinguishable from the main lobe beat signal, resulting in erroneous or ambiguous ranging. In the worst-case scenario, this erroneous beat signal might dominate the main lobe beat signal if it is reflected from a target closer than the target along the path of the main lobe. Such spike events in the side lobes are well described by the SLSR. Therefore, we propose that in addition to using SLSR when evaluating OPAs, crosstalk suppression should also be used as an indicator for the level of phase errors and its impact. We extract the SLSR using a method similar to Ref. [10]. We calculate the SLSR values of the continuous DOPA at 790 wavelength points. The median of these points is 3.6 dB, and the histogram of the measurements is shown in Fig. 5(a). The crosstalk suppression of the continuous DOPA has a median of 9.5 dB, and the histogram of the measurements is shown in Fig. 5(a).

To demonstrate the beam steering of the pixelated DOPA, we first pinpoint the wavelengths at which the constructive interference occurs, i.e., the operation wavelengths. We extract the SLSR at different wavelengths, and it can be observed that the operation wavelengths are local maxima for the SLSR, as shown in Fig. 5(e). After identifying those wavelengths, we show the beam steering results of the far-field pixels of the pixelated DOPA in Fig. 6(a). The constructive interference wavelengths are separated by $\approx 1\mathrm{nm}$, denoting that the wavelength step required to move from one pixel to the next along a scan line is $\approx 1\mathrm{nm}$ (see Visualization 1). The FOV along the $x$ direction $\mathrm{\Delta }{\theta }_x$ is 15° around 1550 nm, while the scan rate $\frac{\mathrm{d}{\theta }_x}{\mathrm{d}\lambda }$ is $\approx 3.66$ deg/nm and the median ${\mathrm{FWHM}}_x$ is 0.97°, as shown in Figs. 7(a)–7(c). Along the $y$ axis, the scan rate $\frac{\mathrm{d}{\theta }_y}{\mathrm{d}\lambda }$ is $\approx 0.072$ deg/nm, which means that the FOV $\mathrm{\Delta }{\theta }_y$ is 7.2° for the 100 nm tuning range, and the median ${\mathrm{FWHM}}_y$ is 0.36°, as shown in Figs. 7(b) and 7(c). The far-field sampling along the $x$ direction $\delta {\theta }_x$ is then $\approx 3.66$°, corresponding to 4 pixels per scan line, as shown in Fig. 6(b), which matches the number of antennas per block $N_x^{\prime }$. We calculate the SLSR and crosstalk suppression of the pixelated DOPA at the constructive interference (operation) wavelengths, at 70 and 63 wavelength points, respectively. The median of the SLSR is 7.6 dB, while the median of the crosstalk level is 11 dB. The histogram of the measurements is shown in Fig. 5(d).

One of the key advantages of the pixelated DOPA can be understood in view of the trade-off between the number of resolvable points along the $y$ direction and the phase errors of the circuit. These phase errors can be traced in DOPAs, due to their passive operation, in the form of a degraded SLSR from the expected theoretical values [14,15,17] (see Appendix A for further elaboration on the literature of DOPAs). The fabricated DOPAs always exhibit worse SLSR than the theoretical value due to the effective refractive index variation along the waveguides introduced by the process variations. Such variations result in deviations in the optical path lengths within the distribution network, which subsequently introduce errors in the input phase to the antennas of the optical phased arrays, deteriorating the far-field beam quality. On the other hand, to achieve higher $y$ axis resolution $\delta {\theta }_y$ with a specific technology implementation and a given laser wavelength range, it is necessary to increase the differential delay length $\mathrm{\Delta }L$, following Eq. (4). A well-designed continuous DOPA should balance these performance metrics. On the other hand, the pixelated DOPA relaxes this trade-off by introducing a new design parameter, that is, the number of resolvable points along the $x$ direction, through the discretization of the distribution network into blocks, i.e., pixelation of the far field. In this work, the SLSR of the pixelated DOPA exceeds that of the continuous DOPA at the operation wavelengths where a collimated beam exists. Furthermore, the pixelated DOPA remains functionally superior to the continuous DOPA, in terms of the SLSR, within a considerable wavelength range around these operation wavelengths, as shown in Figs. 5(b) and 5(e). As for the crosstalk suppression, the pixelated DOPA has better performance than the continuous DOPA at the operation wavelengths and a similar performance within a considerable wavelength range around these operation wavelengths, as shown in Figs. 5(c) and 5(f).

The pixelated DOPA adjusts the design space of the scanning engine. For a specific group velocity index of the distribution network waveguides, the requirement of the number of pixels in the $y$ axis (the total number of scan lines) determines the shortest delay line length $\mathrm{\Delta }L$ for both the pixelated DOPA and the continuous DOPA. For a specific architecture choice and in the case of the continuous DOPA, the total number of antennas required to achieve the beam divergence in the $x$ axis determines the number of resolvable pixels per scan line in the $x$ axis, and hence the length of the longest delay line. However, in the case of the pixelated DOPA, the requirement of the number of pixels per scan line in the $x$ axis determines the degree of discretization, i.e., the number of antennas per block, regardless of the beam divergence requirement. Hence, the degree of discretization determines the length of the longest delay line. The pixelated DOPA can thereby satisfy the requirement of the number of pixels along the $y$ direction and achieve high beam quality by setting the number of pixels along the $x$ direction to more practical values, and hence reducing the total length of the delay lines.

Figure 6 shows that there are large scanning blind spots along the $x$ direction. This is due to the small scale of this proof-of-concept demonstration, where the pixelation of the far field is very strong, resulting in 3.66° resolution along the $x$ direction. Comparing Figs. 5(b) and 5(e), the wavelength range around the operation wavelengths for which the pixelated DOPA has higher SLSR than the continuous one amounts to approximately 0.4 nm, which translates into a bandwidth of 50 GHz. On the other hand, due to the large differential delay used, the resolution of 0.29° along the $y$ direction is very close to the 0.1° of LiDAR requirement, where the blind spots are very small. A clever design should balance the degree of discretization and the differential delay length to fulfill the number of pixels requirement while maintaining high beam quality.

In a scaled-up implementation, both the number of pixels in the $x$ and $y$ directions would rise. And the wavelength range around the operation wavelengths for which the pixelated DOPA maintains high SLSR would be around 8 to 24 pm, which translates into a bandwidth of 1 to 3 GHz around the operation wavelengths. This bandwidth is specifically important for swept source FMCW LiDAR, where the beam steering and ranging are done with the same wavelength variable [28]. To perform the FMCW ranging, the tunable laser needs to sweep over a bandwidth of a few GHz, while maintaining its position on the same pixel with a good beam quality. However, scaling implies the accumulation of phase errors even with the pixelated approach. Without further improvement in the process control to reduce geometry variations, phase error correction would be necessary either through adding phase tuners at an intermediate level of the tree or by phase trimming [29].

The input of the pixelated DOPA is compatible with star couplers, where the $1\times M$ splitter tree in our demonstration would be replaced by an $N\times M$ star coupler, allowing multiplexing. Tunable lasers with different wavelength bands can be connected to different input ports. Each tunable laser can cover a band that is a subsection of the full wavelength tuning range, enabling wavelength multiplexing for fast frame rates [28]. The choice of the wavelength bands and the position of the star coupler input ports can be designed such that for a single wavelength, two or more far-field locations are addressed simultaneously, allowing for spatial multiplexing, similar to Ref. [15]. Such multiplexing schemes are compatible with current hybrid 3D integration technologies using flip chip bonding [30] or transfer printing [31]. The star coupler also allows for a Gaussian power distribution over the blocks to improve the beam quality [32].

The FOV along the $x$ direction can be improved by reducing the DOPA pitch and designing the antenna such that the antenna’s element factor suppresses any remaining grating lobes. Furthermore, the scanning in the $y$ direction can be improved through the optimization of the antenna design. Long antennas can be achieved by stepping aside from the traditional weak gratings to new antenna concepts like the leaky fin antenna, proposed in Ref. [26]. In terms of the circuit layout, further optimization can be done for a compact circuit, for example, by designing compact MMIs, adiabatic bends [33], placing the meanders more compactly, and using the delay of the fan-in waveguides as part of the circuit differential delay.

In this paper, we have demonstrated the 2D beam steering of the continuous and pixelated variations of the dispersive optical phased arrays, in a 16-channel unbalanced splitter tree architecture. For a wavelength steering range of 1500 to 1600 nm, the devices have an FOV of $15°\times 7.2°$, where the continuous DOPA steers over $16\times 25$ pixels, while the pixelated DOPA steers over $4\times 25$ pixels. The pixelated DOPA shows better performance in terms of the side lobe suppression ratio and the crosstalk suppression at these pixels. The pixelated DOPA can therefore be a solution to the scaling bottleneck of dispersive optical phased arrays, paving the way for large-scale low-complexity long-range solid-state LiDAR.