Integrated silicon(-nitride)-on-insulator (SOI) photonic platforms have matured in the past decades, mainly driven by optical data-communication applications. The technology also shows great potential for more diverse applications in sensing, metrology, and information processing. One of the key characteristics of these platforms is that the waveguides have a high refractive index contrast, resulting in high confinement of the optical modes. This translates to the possibility of making narrow waveguides with small bend radii allowing for dense integration and scaling of optical functionality [1–3].

One of the emerging applications on these platforms is optical phased arrays (OPAs), which are a key element for chip-to-chip free space communication and light detection and ranging (LiDAR) systems [4]. LiDAR systems can be useful in applications such as self-driving cars, bio-metric security, virtual reality (VR), machine vision (e.g., geological surveying, satellite detection), and augmented reality (AR). In LiDAR systems for these applications, a focused beam of light is used to probe the surroundings, map the environment, and track movements [5–13].

This work introduces a new concept to create an integrated optical antenna on SOI platforms for long-range solid-state LiDAR applications, where apodization of the surface grating is not required but is replaced by a continuously tapered design. The concept illustrated in Fig. 1 shows the proposed geometry to generate a Gaussian beam with a large beam diameter. The motivation to explore this novel geometry is provided in the remainder of the introduction.

The optical part of a LiDAR system is usually constructed from two main functional subsystems [see Fig. 2(a)]: •A ranging engine to determine the distance of the reflector, i.e., using pulses with time-of-flight measurement or a frequency-modulation approach [14,19–21].•A beam-steering and beam-forming engine that can shape the beam profile and point it in the right direction. This can be done using either mechanical steering, focal plane arrays, or OPAs [10,22–26].

OPAs are a very promising technique to integrate on the surface of a chip and have the advantage that they can be implemented as a solid-state system. An OPA consists of a 1D (or 2D) array of N (or N2) antenna elements (optical emitters) periodically arranged along each dimension x (and y). Using a 2D array of discrete optical emitters, one can define phases between emitters in both the x and y directions on the chip surface allowing beam steering in two dimensions, θx and θy [see Fig. 3(a)]. This placement is equivalent to N 1D OPAs of which each 1D OPA contains N optical antennas, which results in N2 phase shifters. This means enabling 2D beam steering requires a vast number of waveguide connections, splitters, and phase shifters to control every antenna output. The 2D approach also drastically increases the number of electrical connections to the optical circuit, like [13,27,28].

A sufficient field of view [Δθx and Δθy, see Fig. 3(a)] for an antenna array requires a large number of antennas arranged with a sufficiently small pitch, Px and Py (Fig. 3). For example, to allow steering of the beam to an angle of ±27° from the vertical on the x axis (Δθx≈54°), for a wavelength of 1550 nm, the antenna pitch in the x direction Px should not be larger than 1.7 μm [13]. The high-index contrast (HIC) of Si or SiN waveguides is suited for this purpose, as dense integration of a large number of elements on a small surface area is possible in these systems (as illustrated in Figs. 3 and 2). This is incompatible with the spacing required to feed all optical and electronic signals to every optical emitter (grating coupler) in a 2D OPA [13,27,28].

To mitigate these scalability issues for 2D OPAs, more simple 1D OPAs with dispersive antennas [as illustrated in Figs. 2(a), 3(a), and 3(b)] can also be used to perform 2D beam steering. These emitting surfaces consist of N long dispersive optical antennas (as opposed to N2 antennas for 2D OPAs). The dispersive nature of the optical antennas (such as diffraction gratings) enables beam steering in the y direction (θy) with a variation of the wavelength λ, which can be centrally controlled for all N long antennas of the 1D OPA. Using wavelength steering reduces the number of required antenna elements (and, thus, phase shifters, routing waveguides, and electrical connections), which makes it easier to position the antennas with a small pitch Px [7,10,13,22,29].

For practical LiDAR applications, the optical beams need to be projected over a sufficiently large distance [see Fig. 2(b)] (i.e., automotive LiDAR, free space communication, geological mapping, spacecraft, machine vision) [5–7,9,10,13]. If we want to probe far-away objects (i.e., z=200m), the aperture size Ly needs to be in the order of 30 mm. This follows from the Rayleigh range of a Gaussian beam, zr=πw02λ,with w0 being the beam radius (the operating wavelength we target is λ=1.55μm). This would determine the required on-chip aperture size (i.e., the length of the grating) to be Ly>2w0. Since an aperture of this size would truncate the beam at 86% of the power of a full Gaussian profile, the performance of a LiDAR system can be improved by increasing the aperture size to suppress sidelobes and other imperfections in the far field. To capture 99% of the power in the Gaussian beam, the aperture size can be increased to Ly=3w0≈30mm [see Figs. 3(a) and 3(b)] [13,30].

While an HIC waveguide helps to minimize Px, making such a long optical antenna comes with challenges in such a platform. The propagating mode in the waveguide will be very sensitive to geometrical perturbations [3]; therefore, it is difficult to implement very weak, well-controlled periodic gratings in the guiding structure to create out-of-plane grating couplers. One would have to resort to very small perturbations with subnanometer precision and reproducibility, which are inherently hard to fabricate reliably. In other words, any grating in a waveguide with HIC will be too strong, resulting in beams with a waist w0 that is too narrow (and, thus, a large divergence θdiv and low resolution) for these practical LiDAR applications. Adding a collimating lens above the chip can help, but this will reduce the angular scanning range along θy [Δθy on Fig. 3(a)] when we sweep the wavelength.

In this publication, we side-step this problem to define a weak grating. We present an alternative concept for an optical antenna that mitigates many of the scalability issues of long gratings (see Section 2). In this new antenna, we adapt the concept of lateral leakage devices to explore a novel geometry that gives us continuous control over the power radiating away from a guided mode in the waveguide (see Fig. 1) [31–33].

First, we present a short overview of the current state-of-the-art grating-based antennas and their limitations in Section 2. In Section 3, we discuss possible mechanisms to create leaky wave antennas in mm-wave technology, where non-periodic antennas are presented as a fundamental alternative to periodic structures such as gratings and how we can implement a similar concept in dielectric optical waveguides. Next, the device concept is discussed in Section 4, where the cross section is discussed in more detail and the relevant parameters that can be used to tune both the leakage rate and propagation constant are presented. These parameters are swept in simulation to construct a design space, and the sensitivity to possible deviations is analyzed. In Section 5, the design space is utilized to construct a full antenna from a targeted power profile and propagation constant. The simulation results of the designed antenna derived in Section 5 are presented in Section 6, where both the resulting power profile and far-field are analyzed.

Directly defining gratings in the waveguide core on either Si or SiN platforms reports beam diameters well below the target of 30 mm [34–36], and the longest reported near-field beam diameter (3 mm) in literature (to our knowledge) uses a sidewall corrugated SiN waveguide [37].

The fundamental challenge to create long (weak) optical antennas using periodic perturbations in an HIC material system (i.e., long gratings) is illustrated by Bruck and Hainberger in Ref. [38], where an analysis of the optimum spot-size (beam diameter) in relation to the grating etch depth is analyzed in a low index-contrast system with increasing thickness of an HIC layer. The study shows that reducing the index contrast of the geometrical perturbations is crucial to increase the beam diameter, and this can be achieved by using a thinner layer of HIC material.

To overcome this problem, one can fabricate a weak grating with an HIC material using a SiO2 gap between the waveguide core and the perturbations to diminish the strength of the grating. This is demonstrated in Refs. [39–41]. Another mechanism to decrease grating strength with these overlays employs a lower index contrast material (SiN) on the silicon waveguide to define the grating overlay, as demonstrated in Refs. [42–44]. Both with or without a SiO2 separation layer, all these mechanisms report beam diameters well below the target of 30 mm and even below the sidewall corrugated devices presented in Ref. [37].

If we want this grating antenna to emit a Gaussian beam profile rather than the exponential decay obtained from a conventional grating with uniform strength [see Fig. 3(b)], one has to resort to apodization of the geometrical perturbations creating the grating [34,37,45]. This technique gradually increases the perturbation strength (i.e., etch depth) of the grating from 0 to a maximum value, creating an increasing index contrast along the length of the grating. This has the effect of increasing the strength of the grating, increasing the coupling from a guided mode to free space, and creating a Gaussian beam shape [Fig. 3(b)].

The long double-layer SiN antennas reported by Raval et al. [37] (with the largest near field beam diameter in literature) also implemented the apodization technique (to our knowledge, over the longest length found in literature). The two layers of nitride have a geometrical perturbation that increases with the distance along the antenna. It can be noticed that the antenna presented in Ref. [37] needs a perturbation depth up to 10 nm, starting from 0 nm, within the first 1000 μm of the device. This implies an increase in the order of 10 pm/μm, which is challenging on a large scale with high yield since a silicon atom has a van der Waals radius of 210 pm. Increasing the length of the antenna will even result in more gradual apodization curves, making it even harder to realize. This shows that apodization for longer beams would require more gradual and precise control over the apodization than what can be reliably fabricated.

Because both the Si and SiN material systems (or combinations) exhibit fundamental challenges to achieve antenna lengths (and corresponding beam widths) in the order of 30 mm and demand unfeasible control on fabrication due to the apodized periodicity (especially in the beginning of the emitter where the grating should be very weak), it can be interesting to investigate an alternative vertical coupling mechanism to decouple the apodization (grating strength) from the out-of-plane grating coupler.

In mm-wave technology, the concept of traveling wave antennas (or leaky wave antennas) has been known for decades [46–49]. This class of antennas uses a traveling wave on a guiding structure as the main radiating mechanism. This usually relies on the knowledge that a “fast wave” (meaning the phase velocity vp is higher than the speed of light in vacuum) allows phase matching to occur between the propagation constant of the waveguide and free-space propagation at a certain angle [49]. The most basic example of a mm-wave fast-wave antenna implementation is an air-filled metal waveguide with a slot opening in the top surface, acting as an antenna aperture [46–49].

A grating coupler antenna is an example of a “slow” leaky wave antenna, where a slow wave (vp<c) has a fast spatial harmonic mode due to a periodic perturbation [49]. This is the only way to achieve vertical emission out of a dielectric optical waveguide.

It is possible to arrive at an intermediate scheme without resorting to a direct periodic perturbation of the guided optical waveguide mode, where a dielectric radiating channel can be implemented using dielectric slab waveguides in the waveguide cladding leading to leaky conditions similar to that of a fast wave antenna in mm-wave. Because the k-vector of a TE-polarized slab mode in the cladding of a ridge waveguide is larger than the k-vector of the fundamental TM-polarized waveguide mode, phase matching allows the guided mode to couple to the TE-polarized slab mode propagating at a certain angle. These modes radiate away from the waveguide core in the lateral direction, hence the name lateral leakage [31–33].

The leaky behavior can be geometrically tuned using the waveguide width because the coupling to these slab modes physically originates from interference between two radiating discontinuities on the left and right side of the waveguide cross section (where the overlap of longitudinal field components yields coupling between the guided TM-polarized mode and the radiating TE-slab modes). This mechanism can alter the leakage from extremely high values (decay lengths of only a few wavelengths) to almost zero (no leakage). The waveguide width values where this high and low lateral leakage occurs are called the anti-magic and magic widths, respectively [31–33].

It has to be noted here that, in order to achieve any sort of vertical coupling from this slab to free space, a periodic perturbation still would be required because the k-vector of the slab modes still is situated below the light cone. So this mechanism only couples light from the waveguide core to the slab in the cladding, but the geometrical parameters that influence the coupling strength are all much easier to control. The grating that can then couple light to free space can be designed to be very strong and uniform (unapodized), which is much easier to achieve than a weak grating. In other words, we can avoid the use of precise periodic features that require subnanometer fabrication precision. This exact technique was actually used to observe the lateral leakage in Ref. [33].

Inspired by this lateral leakage mechanism, we propose a novel antenna concept where we rotate the waveguide core and the slab 90°. The in-plane “slab” now becomes a vertical “fin,” and the leakage is no longer lateral (in the plane of the chip) but out of the plane of the chip, at an angle from the vertical. The guided mode is now the TE-polarized mode in the waveguide core region, coupling to the TM mode of the vertical slab (from here on referred to as fin), as illustrated in Fig. 4. Note that there is only a single fin on top, in contrast with the lateral leakage waveguide, which can leak in both directions. This will have a significant impact on the mechanism that we use to control the leakage rate.

The structure resembles the cross section of a groove antenna used in mm-wave technology with metallic waveguides (either filled with dielectric or air). Here, a vertical fin (in this context sometimes referred to as “stub”) is placed on top of a rectangular waveguide. When this fin is off-centered with respect to the waveguide core, the asymmetry will cause the guided mode to become leaky [46,50–53]. For the optical implementation of this antenna and the simulations presented here, we chose SiN as the guiding layer, but the antenna could also be implemented in Si or other materials used for making dielectric optical waveguides.

Generally, a leaky mode has a complex effective index which can be written [54] as neff=βλ02π+jαλ02π,which we rewrite as neff=n+jκ.This allows us to extract the leakage rate, which in this context we call radiation strength (which can be intuitively interpreted as the rate at which the TE-polarized mode in the waveguide core couples to the TM polarized mode in the fin as is explained later), as follows: α[dBm]=20log10(e)2πκλ0.While this complex propagation constant follows directly from rigorously solving Maxwell’s equations fully in this cross section of lossless materials, an intuitive way of understanding this leakage behavior is to see it as a coupling between a lossless propagating waveguide mode and off-axis slab modes in the fin, which we call the radiating modes. When the structure is fully symmetric, the guided TE mode in the waveguide core and the radiating TM mode in the fin have different symmetries. Therefore, their modal overlap is zero, and they do not couple (see Fig. 6). Changing the position of the vertical fin relative to the waveguide core center (referred to as the offset) breaks this symmetry, which results in a coupling between the TE-polarized core mode and TM-polarized fin mode (meaning κ increases with a larger offset). This coupling is enabled by the fact that both modes have a field component along the y axis (the propagation direction of the waveguide), which is strongest near the corners where the fin meets the core, so the two modes can couple if their overlap integral becomes non-zero. Indeed, as the symmetry of this Ey component is different for the TE and TM modes, the coupling cancels out completely when the fin is centered on the waveguide core. However, once the fin is off-center, the coupling increases, and this can be very accurately controlled. Note that the Ey component, and therefore the magnitude of this effect, become stronger when the core and fin material has a high refractive index contrast with the surrounding cladding

This leakage principle makes it possible to define a cross section where the offset monotonically maps to the radiation strength of the structure, giving the possibility to vary the cross section along the propagation direction y from a fully bound mode to a strongly radiative mode, increasing κ along the length of the antenna.

Another requirement for antennas in an OPA is a constant emission angle over the entire length of the antenna to guarantee a flat phase front. This requires a constant phase velocity of the waveguide structure. Otherwise, the emitted wavefront will be curved (either converging or diverging), which would be equivalent of having a smaller emitting surface, negating the advantage of having a long antenna.

The phase velocity is expressed as the real part of the propagation constant β of the guided waveguide mode, where β=nk0. As the Kramers–Kronig relations couple the real and imaginary part of neff, a change in n is induced whenever κ (the radiation strength of the mode) changes, so we need to add a second degree of freedom in the cross section to compensate for this change in n. This can be done by changing wcore at the same time as changing the offset, creating a 2D design space spanned by the offset and the core width. In theory, other parameters such as tcore or wfin could be used for the same purpose, but we chose the two parameters that are most convenient to control in planar fabrication processes. The other parameters, including the width of the fin, are kept constant over the entire antenna length.

The design space of the cross section was simulated at a wavelength λ=1.55μm using a film-mode matching (FMM) technique with transparent boundary conditions. The FMM-solver used here was REME, specifically developed at RMIT to study lateral leakage behavior of ridge waveguides [31]. When we use transparent boundary conditions on the top and bottom of the cross section, the FMM-solver is able to determine the κ of a guided mode with neff. We simulated the cross section on a parameter grid of 20×20 discrete design coordinate pairs (offset,wcore) to determine both κ and n, and we used a bi-variate smooth spline interpolation algorithm from the scipy package to interpolate the results. The result is a 2D design space for the radiation strength, plotted as a color map in Fig. 5, with overlayed iso-n contours.

In this simulation, transparent boundary conditions are used to allow power to leak away from the simulation window. The FMM technique is one of the few techniques that allow for true transparent boundaries along one lateral axis, not requiring perfectly matched layers, which always have some residual reflections [31]. This means that for our simulations we assume that the fin extends indefinitely and light does not (yet) couple to free space. In real fabricated devices, the height of the fin needs to be finite, and the coupling from the fin to free space will require a periodic structure (as mentioned in Section 3). It has to be noted here that the need for a periodic structure is not fundamentally required since the light is already propagating vertically in the fin, as opposed to the lateral leakage in the slab. This means a tilted top surface at Brewster’s angle could also be used here. Since this approach would lack a strong angular dispersion required for wavelength steering (as is discussed later; see Section 6.B), the most obvious method of coupling the light from the fin to free space is a strong uniform (blazed) grating designed to divert all incident light to the first diffraction order. There is no need for apodization in this grating because the beam profile is determined by the leakage rate from the waveguide to the fin (i.e., by the waveguide/fin offset, see Section 5).

For a smooth Gaussian-shaped emission profile, we want a continuous design with increasing leakage rate α but exhibiting a constant n in the longitudinal direction (y direction). So we can select one of the iso-n contours in Fig. 5 to give us a mapping of the core width and offset to radiation strength of the cross section for a given propagation constant β.

Figure 5 shows that, for each given value of wcore, increasing the offset results in a higher radiation strength (α). The maximum absolute value of the offset increases for larger values of wcore, but it maintains a similar α relatively.

This behavior can be intuitively explained by examining the imaginary part of the longitudinal component (ℑ{Ey}) of the TE mode in the cross section on the top surface of the waveguide (i.e., the bottom aperture of the fin) shown in Fig. 6.

In the perfectly symmetric case [meaning offset=0, Fig. 6(a)], the anti-symmetric nature of Ey in the guiding structure causes a cancellation effect in this mode coupling because the overlap with the TM mode in the fin is zero. When the offset increases, the symmetric Ey profile of the fin will have an increasing overlap with the guided mode of the structure, and coupling to the fin is possible. This coupling increases in strength as Ey has the largest values near the unperturbed waveguide edges [Figs. 6(a) and 6(b)].

Because the design space was interpolated using a bi-variate spline algorithm, we also have access to the local derivatives. These can learn us something about the theoretical sensitivity to certain fabrication parameters such as wcore and the offset. The derivatives are plotted in Fig. 7, together with the iso-n-contour used for the antenna design (see Section 5) in this work.

This analysis reveals the fundamental advantage of using this approach rather than the apodization of a periodic structure. As pointed out in the introduction, the most crucial and challenging part of the antenna design is at the start of the propagation, where radiation has to be weak. Making such a weak grating requires very precise control on fabrication and will be sensitive to a deviation from the target design value.

If we consider Figs. 7(c) and 7(d), we see that for low values of the offset (and thus α) the sensitivity to a geometrical deviation of both the design parameters, offset and wcore, is low. This implies that this cross section is less sensitive to fabrication variation of the design parameters in the weak regime at the start of the antenna. The sensitivity increases if we follow the iso-n contour determining the antenna design parameters for a given β, meaning that higher values of α will be more susceptible to any deviation. This means the relative “error” will be always small, and the most significant deviations will be found toward the end of the antenna, where most optical power has already radiated out of the waveguide. This built-in robustness in the weak radiation regime is a clear advantage over apodization methods used on periodic structures.

Figures 7(a) and 7(b) show the deviation of n (and thus β) with a change in offset and wcore. A similar trend with the sensitivity of α to the design parameters can be observed in Fig. 7(a). The value of n will change the least at the start of the antenna in the weak radiation regime (low values of offset). This is also the section of the device with most of the power still present in the guiding structure. As light propagates along the contour, sensitivity increases. Figure 7(b) reveals that the sensitivity of n to wcore will be similar for all values on the design contour.

When we follow the iso-n-contour for the design wavelength of 1.55 μm, the value of n remains constant, as can be seen in Fig. 8. If the values of the index contour are used at another wavelength, the contour will not produce constant n values. We solved the modes for the cross sections on the n-contour with λ=1.5μm and λ=1.6μm to analyze the sensitivity of the radiation angle to the wavelength. As can be seen in Fig. 8, the values remain relatively constant when the offset <200nm, but when we increase the value of n, and thus the propagation constant β, the corresponding emission angle θy starts to deviate from its initial value. This will result in a slightly converging phase front for shorter wavelengths and a diverging phase front for longer wavelengths. But, as we will show in Section 5, the offset only goes to this region toward the very end of the antenna design, where most power has already radiated from the device and the impact on the center of the beam is limited. Still, to compensate for this effect, we could consider unlocking one or more additional design variables, such as the width of the fin wfin.

To design an antenna with a target power output profile, the first step is back-calculating the radiation strength at each y position along the antenna (which would be the apodization function when using a periodic perturbation as discussed in the introduction).

We start from a desired output power profile of the antenna, which in the context of LiDAR will be a Gaussian beam as mentioned in the introduction. We use a 100 μm discretization to define this profile with N segments (profile[1] to profile[N]), which means we can determine the required radiation ratio per segment as illustrated in Fig. 9.

The antenna should dissipate all input power, P0, which implies PN=0. This means the last segment should radiate all incoming power from the previous antenna segment, meaning PαN=PN−1 or radN=1. This determines PN−1, so we can determine radN−1 and continue this process for all preceding antenna segments. Because αn=10log(1−radn), we now know the required radiation strength α(y). The output of this process is illustrated in Fig. 10, where Fig. 10(b) shows the resulting α(y) for the Gaussian power profile with w0=8mm plotted in Fig. 10(a).

As mentioned before, the antenna requires a constant propagation constant β for a constant emission angle θy along the propagation direction y. This means we can select one of the iso-n contours from the interpolated design space for a specific antenna design with given β, shown in Fig. 11. These curves were obtained by sampling points with n≈1.662 from the interpolated design space (Fig. 5). On these sampled points, a polynomial was fitted for wcore(offset) [Fig. 11(a)] and a uni-variate spline was fitted for offset(α) [Fig. 11(b)]. The function offset(α) has a very steep start, illustrating the robustness of cross sections with low radiation strength.

Combining the curve α(y) [Fig. 10(b)] with the function obtained in Fig. 11(b), offset(α), results in the required offset at each y coordinate of the antenna [offset(y)], plotted in Fig. 12(a). With offset(y) and the function wcore(offset) [Fig. 11(a)], we can determine the accompanying width of each cross section [wcore(y)] shown in Fig. 12(b). The offset increases quasi-linearly with y for the largest part of the device.

With the antenna design functions from Fig. 12, the cross section of each y coordinate is determined. The resulting structure is conceptualized in Figs. 1 and 13. The fin is kept uniform and straight over the entire antenna length, while the waveguide core is tapered according to Fig. 12(b) in such a way that the offset changes according to Fig. 12(a). As mentioned in Section 4.C, a strong uniform grating on top of the fin is still required to couple the light from the fin to free space. No apodization is needed because the radiation strength is continuously tuned by the tapered waveguide position underneath.

It should be noted that the strong grating on top should be sufficiently far away from the guiding structure, meaning the fin would require a thickness of at least 2 μm. While fabricating such a fin is still not trivial, the fact that it is uniform over the entire antenna length makes it easier to optimize a fabrication process specifically for this structure. We believe this is an improvement on the scalability of long-distance optical antenna designs since the core layer and fin structures can be fabricated in separate steps and the design is inherently robust to deviations of the offset between the fin and core.

The geometry of the antenna design with n=1.662 determined in Section 5 is constructed within the REME eigenmode propagation engine [31]. Using the design functions from Fig. 12, the corresponding cross section for each segment is determined, and the corresponding modes are calculated using the FMM algorithm with transparent top and bottom boundary conditions. The REME mode solver calculates the overlap between all sections to determine mode couplings between each segment, and we launch the TE-polarized waveguide mode (λ=1.55μm) from the left (y=0).

We then monitor the leaky TM component of the fields (Ez), just before it hits the transparent upper boundary (i.e., just below where we would position a strong grating as illustrated in Fig. 1).

The extracted 2D power profile (|Ez|2) in Fig. 14(a) was constructed using 100 μm discretization segments, which means the number of cross sections to be analyzed is in the order of 300. The fin and core layers are visualized to help with the interpretation of the plot. Note the very distorted aspect ratio: the device is approximately 1.5 μm wide in the x direction, while 30 mm long in the y direction.

The data on the black dotted line represented in Fig. 14(a) normalized to peak power were used to create a 1D power profile plot. The larger discretization size introduces a ripple in the power profile, which is strongest at the peak of the profile (y=15mm). The target Gaussian from Fig. 10(a) normalized to peak power is also shown to see if they match. At most positions, the target profile is approached very well (the peak position matches quasi perfectly), even with this larger discretization size.

The strongest deviation from the target profile is in the beginning just before y=5mm. This is related to numerical artifacts and errors that are introduced by the two subsequent curve-fitting procedures. The first one is used in Section 4.C to interpolate the 2D design space. This generates enough samples to select an iso-n-contour with sufficient precision from the design space. On this sampled contour, functions are again fitted in Fig. 11, as described in Section 5. The effect of these numerical errors can also be seen by a bump in the curve in Fig. 12(b) just before y=5mm. This shows the offset has to increase linearly to obtain a perfect Gaussian power profile, but the numerical fitting errors cause a deviation from this linear increase. This is not a fundamental limitation, and it could be further fine-tuned by sampling the design space in more detail in the low-leakage regime.

To determine the quality of the phase profile, we sample the field Ez on the aperture of the fin and calculate the far field. As mentioned before, these simulations do not include the grating structure on top of the antenna (it is replaced by a transparent boundary), so we simulate the far field in an infinitely high fin structure in the z direction. As a comparison, we calculate the far field of the target intensity Gaussian [Fig. 10(a)] with a perfect phase ramp radiating at the analytically calculated phase matching angle θy, which determines the direction of radiation in the fin (see Fig. 1).

This angle can be determined as θy=arccosββTM,fin=arccosnNTM.Here, NTM is the effective index of the TM polarized slab mode in the fin, and β is the propagation constant of the antenna. For the design wavelength of the antenna (λ=1.55μm), we find θy≈arccos1.6621.73845≈17.055°.The results are shown in Fig. 15(a). The simulated field on the aperture matches the target field quite well and agrees with the phase-matching angle. There is a small difference in peak angle (almost negligible, this can be attributed to numerical artifacts), but the far-field of the sampled field has the same full width at half-maximum (FWHM) value as the discretized version. This shows that the phase profile of the simulated field in the aperture almost approximates a perfect phase ramp and deviations can be attributed to the discretization of the design with 300 cross sections.

Since both the numerator and denominator in the analytical expression for θy are susceptible to dispersion, θy will also be wavelength dependent. This is not a problem, as we wish to use the wavelength to perform beam steering along the θy axis anyway. We simulated the antenna designed for λ=1.55μm at other wavelengths and calculated the corresponding far-field profiles. The results are shown in Fig. 15(b). When the wavelength increases, so does the direction of radiation θy, but only with a relatively small value (Δθy≈0.8°). This is insufficient for wide-angle wavelength steering but reveals robustness in terms of performance when deviating from the design wavelength. We also see that the beam divergence angle increases 5%–15% for the longer and shorter wavelengths, which corresponds to the nonuniform n profile plotted in Fig. 8.

To enable wavelength steering, a low-reflection blazed diffraction grating without any required apodization could be implemented on top of the fin (as mentioned in Section 4.C). This grating will have a much stronger angular dispersion than the antenna, allowing the beam to change direction (θy) significantly with wavelength steering [see Fig. 3(a)]. The K-vector of the grating should be determined such that the wavelength of 1.55 μm is diffracted upward (θy=0°), requiring a grating period Λ in the order of Λ=λ0NTMcosθy=λ0n≈933nm.The robustness of the radiation angle θy to the wavelength also means that a misalignment between core and fin in fabrication would not affect the incidence angle of the emitted beam toward the strong grating on top.

Long-range LiDAR requires Gaussian beams with a wide waist, and it is not trivial to tailor such emission profiles in an HIC waveguide platform with an OPA antenna. We need the high contrast material system because a small antenna pitch Px is necessary, but conventional techniques utilizing apodized periodic perturbations require impossible precise fabrication that does not scale well to larger antennas. Therefore, it would be beneficial to avoid periodic, grating-like structures for the apodization of the radiation profile and instead use a continuous design.

Since leaky wave antennas without periodicity are only possible in “fast” guiding structures (vp>c) and dielectric optical waveguides only guide “slow” waves (vp<c), an intermediate radiation channel can be engineered in the form of a vertical slab waveguide called a “fin.” This is inspired by the same principle as lateral leakage where power can leak from a guiding structure laterally. Since no periodic grating is used to construct the emission intensity profile, we eliminate unfeasible fabrication requirements.

The proposed cross section of our leakage-based antenna is based on a single geometrical parameter (the offset of the fin relative to the waveguide core) to tune the radiation strength. The core width of the waveguide, wcore, is then changed accordingly to maintain the propagation constant and ensure a constant angle of radiation θy to end up with a collimated beam. This leads to a design that has a continuous tapered waveguide core where both wcore and the offset to the fin change simultaneously (see Figs. 1 and 13).

The proposed structure can be fabricated in two separate steps, i.e., one defining the waveguide layer and the other defining a subsequent deposition of the fins. This would require the alignment of two different masks, with relaxed alignment conditions for both the x and y directions. This tolerance to alignment originates from the fundamental robustness of the cross section to the offset in the most crucial part of the design, the beginning of the antenna. The device also exhibits robustness to changes in wavelength, meaning that the far-field radiation angle θy is not strongly dependent on the wavelength and remains constant for wavelengths other than the design wavelength λ=1.55μm.

The simulations in this work use a transparent boundary condition where a strong uniform grating should be placed to diffract the light from the fin into free space and to provide a strong wavelength dependence for steering in the y direction.

This device concept comes with its own fabrication challenges but has built-in tolerance and robustness that provide the scalability required for precise long-range LiDAR systems. The main fabrication challenge to overcome is the height of the fin, more specifically the high aspect ratio. This must be sufficient to decouple the leakage profile from the diffraction at the top grating. The fin, therefore, needs to be patterned in a thick layer of a high-index material (here, silicon nitride) with sufficiently vertical sidewalls. The fin profile also needs to be very uniform along the length of the antenna and between the many antennas in a large OPA. For this reason, we kept the fin geometry as simple as possible to facilitate the process development. Alternatively, the width of the fin could be used as an additional design parameter by tapering the fin along the length of the antenna, i.e., to perform a first-order compensation of the wavelength dispersion that gives rise to a slightly curved wavefront (and beam widening) for longer and shorter wavelengths.

At this point, we did not yet include the strong uniform grating in the simulations (a transparent boundary condition was used instead). Due to the large length of the device (in these simulations Ly=30mm), including the grating in the entire device was not computationally feasible at the time. One possibility is to use the FMM simulation output presented in this work and sample the vectorial field components on the surface of the TBC. Using this field as input, the near-field of the grating can be obtained by performing an finite-difference time-domain (FTDT) method in a much smaller simulation window (it can severely be limited in size in the z dimension). The far field from the grating can then again be obtained using a plane wave decomposition analysis.

Even though it is much easier to fabricate a strong uniform grating compared to the apodized gratings with a very weak radiating section, it is still not trivial. The efficiency of the device is reliant on the efficiency of the periodic grating on top, but the leakage-based antenna provides a mechanism to avoid apodization of the grating teeth. The grating ideally should have no reflections and only a single diffraction order, which introduces its own challenges in fabrication. A strong grating will be less sensitive to fabrication variation, but it is still challenging to have a uniform structure on this length scale. For instance, it could be implemented as a blazed grating or a volume-phase holographic grating.

We believe that this device presents a promising alternative to conventional periodic gratings to facilitate scalable 1D OPAs with dispersive antennas as an emitting surface for long-distance solid-state LiDAR integrated on a chip.

Acknowledgment. We thank Thach Nguyen and Arnan Mitchell for the use of the FMM-solver (REME) to simulate the cross section and propagation of leaky modes [31].