With the rise of autonomous vehicles^[1–3] and unmanned drones^[4], light detection and ranging (lidar) has become an indispensable device for them. So far, the mechanical lidar^[5] is still one of the most mature solutions, but its high cost and difficult assembly have plagued researchers. Besides, the short detection distance limits the wide application of flash lidars^[6]. The integrated on-chip silicon (Si) optical phased array (OPA)^[7–11], as an advanced solid-state beam steering device, can overcome the above defects and has gained significant interest for its energy saving and miniaturization. In application, the grating-emitter-based OPA is considered a viable candidate to achieve two-dimensional (2D) optical steering, i.e., phase steering in one direction and wavelength steering in the other direction. For phase steering, the OPA with a pitch close to a half-wavelength along the lateral direction is constructed to realize a wide beam steering range with low crosstalk, which is usually built upon metamaterial waveguides^[12,13], corrugated waveguides^[14], nano-structured Si waveguide arrays^[15], etc. Notably, an OPA with a phase mismatched unequal width waveguide distribution has been applied, implementing a steering range of 110° and a maximum peak power of 720 mW^[16]. Wavelength steering is often enabled by the grating dispersion when the wavelength of the input laser light is scanned. A macroscopic emitting aperture with its size of $>1\mathrm{mm}$ is typically required for ranging distances of interest for autonomous vehicles^[17] because a larger aperture would generally enable a narrower width of the main beam lobe and thus a high angular imaging resolution. The Si waveguide grating with shallow etching is one possible approach to realizing long-length emitting for its weak emission rate^[18]. However, in practice, they are challenging to fabricate. A more promising approach of integrating a Si nitride (${\mathrm{Si}}_3{\mathrm{N}}_4$) overlayer on a Si waveguide has gained significant interest because of its low nonlinearity, broad transparency range, low propagation loss, and low index contrast characteristics^[4]. On this basis, several surface gratings with novel structures, such as strip-line grating^[19] and fishbone grating^[20], have been realized in the modulation of the emission profile and the improvement of radiation beam quality. In addition, the downward radiant power of the grating always introduces destructive interference, which reduces the performance of the grating emitter. An effective approach of dual-layer grating misalignment has been proposed, consequently achieving more than 95% unidirectional radiation^[21].

In this Letter, we propose a ${\mathrm{Si}}_3{\mathrm{N}}_4/\mathrm{Si}$ grating emitter structure with a specific variation in widths and duty cycles of ${\mathrm{Si}}_3{\mathrm{N}}_4$ overlayers. This structure provides an approximate Gaussian emission profile and long effective coupling length of 2.247 mm. The design is mainly centered around two targets: (1) to increase the far-field sideband suppression ratio (SSR), thus reducing the interference of the scanning light sidelobes in the far-field to the steering detection, which is realized by constructing the Gaussian near-field emission profile; (2) to increase the steering accuracy, which is directly governed by the angular full width at half-maximum (FWHM) beamwidth related to the effective grating length. The coupled-mode theory (CMT) is employed to analyze the coupling between the guided mode and radiation mode. In addition, a genetic algorithm (GA)^[22–24] drives the coupled-mode model to search a large parameter space containing the two geometric freedoms of ${\mathrm{Si}}_3{\mathrm{N}}_4$ overlayers (with fixed height) to produce the large-scale approximate Gaussian emission profile on the grating surface. Our theoretical investigation suggests that when the working wavelength is 1550 nm, the designed grating emitter can obtain a far-field angular FWHM beamwidth of less than 0.026° and an SSR of larger than 32.622 dB. In addition, the proposed grating emitter can realize longitudinal beam steering of 3.94° by wavelength tuning within the wavelength range of 1530–1570 nm. Our design here provides a practical and feasible approach for beam steering, shedding light on the possibilities of realizing high-performance solid-state lidars with a high signal-to-noise ratio (SNR) and detection accuracy.

Purely numerical methods such as finite difference time domain (FDTD) are theoretically feasible for the grating design. However, accurate calculation results can only be obtained by the ultra-high-precision meshing of the millimeter (mm)-length rectangular structure, which undoubtedly consumes a significant amount of computing resources. By contrast, the CMT avoids the tedious meshing progress and can give quantitative predictions of the coupling between the guided mode and the radiation mode and a physical understanding of grating radiation. In the coupled-mode model employed here, by modulating the widths and duty cycles of ${\mathrm{Si}}_3{\mathrm{N}}_4$ overlayers, we can precisely control the coupling between the guided mode and radiation mode. In prior studies, radiation modes can be constructed in relatively simple forms for 2D slab waveguides^[25] and optical fibers^[26]. They cannot apply to our design of the ${\mathrm{Si}}_3{\mathrm{N}}_4/\mathrm{Si}$ grating emitter with a rectangular structure. We adopt the semi-analytical method proposed by Poulton^[27] to construct fully 3D radiation modes of the ideal rectangular Si waveguide (see Appendix A for details), where the radiation modes are deduced from the response of the waveguide to an incoming extended field with a given symmetry and polarization to calculate power density functions of the radiation modes more explicitly and conveniently.

Shown schematically in Fig. 1, the customized grating is designed on a Si-on-insulator (SOI) platform. The Si waveguide has a height of 220 nm and a width $W_{\mathrm{Si}}$ of 500 nm. A ${\mathrm{Si}}_3{\mathrm{N}}_4$ overlayer with a height $h_0$ of 50 nm is deposited above the Si waveguide with ${\mathrm{SiO}}_2$ cladding. The 2.247 mm grating contains 2500 grating periods totally with a period $\mathrm{\Lambda }$ of 899.07 nm. The coupling between the guided mode and the radiation mode is constrained by the (quasi-)phase matching condition, and the radiation field consequently emerges at an angle of 30° (referenced to the grating facet normal). As a traveling wave with a fixed radiation angle, the radiation field has a planar phase front. In practice, 2500 independently parametrized ${\mathrm{Si}}_3{\mathrm{N}}_4$ overlayers undoubtedly increase the difficulty of actual fabrication and reduce the robustness of the device. An effective optimization strategy is proposed here by (1) grouping a series of 25 neighboring grating periods as an optimization group, where the widths and duty cycles of the ${\mathrm{Si}}_3{\mathrm{N}}_4$ overlayers are set to be identical; (2) constructing two fitting functions $g_1$ and $g_2$, representing, respectively, the mapping relations from the group order S = 1–100 to the common widths and duty cycles of the ${\mathrm{Si}}_3{\mathrm{N}}_4$ overlayers in each group, to reduce the number of optimal variables. In particular, $g_1$ represents the percentage of the ${\mathrm{Si}}_3{\mathrm{N}}_4$ overlayer width to the Si core width. The fitting functions $g_1$ and $g_2$ can be expressed as $$\{\begin{array}{c}{g}_1(S)=\sum _{j=0}^{40}{a}_j{[(S-50)/a_{41}]}^j+a_{42}{e}^{-{(S-a_{43})}^2/a_{44}}\\ g_2(S)=\sum _{j=0}^{40}{b}_i{[(S-50)/b_{41}]}^j+b_{42}{e}^{-{(S-b_{43})}^2/b_{44}}\end{array},$$(1)where $a_j$ and $b_j$ ($j=0,1,\dots ,44$) are the independent fitting parameters to be optimized. So, the width of the ${\mathrm{Si}}_3{\mathrm{N}}_4$ overlayer can be given by $$W(z)=\{\begin{array}{ll}{g}_1(S)·W_{\mathrm{Si}},& \mathrm{\Lambda }(n-1)<z\le \mathrm{\Lambda }(n-1)+{\Lambda g}_2(S),\\ 0,& \text{others,}\end{array}$$(2)where $n=1+25(S-1),\dots ,25S$. Along the propagation direction, the $z$ component of the coupling coefficient can be ignored, so it can be approximately expressed as $${\kappa }_{\mu \nu }(z)=\frac{j\omega {\epsilon }_0}{4P_{\xi }}{\int }_0^{W(z)}\mathrm{d}x{\int }_0^{h_0}\mathrm{d}y(n^2-n_0^2){\mathit{E}}_{\mu t}^{*}·{\mathit{E}}_{\nu t},$$(3)where $P_{\xi }$ ($\xi$, a label indicating radiation modes) is the power spectral density. $n$ and $n_0$ are the refractive index of ${\mathrm{Si}}_3{\mathrm{N}}_4$ and cladding medium, respectively. ${\mathit{E}}_{\mu t}$ and ${\mathit{E}}_{\nu t}$ are the transverse components of the electric field of the guided mode and radiation mode, respectively. Taking the coupling coefficients ${\kappa }_{\mu \nu }$ and ${\kappa }_{\nu \mu }$ as the core, a set of coupled-mode equations display the power transfer process between the guided mode and radiation mode. Actually, the continuity and infinity of the radiation mode derived from the semi-analytical model reduce the feasibility of mathematically solving its amplitude coefficients directly. We turn to investigate the attenuation efficiency of the guided mode to simplify solving the coupled-mode equations. The phase matching condition or quasi-phase matching condition stipulates that the guided mode can only be effectively coupled with some radiation modes with a special propagation constant ${\beta }_{\nu }$. Hence, it can be approximately considered that the coupling coefficient ${\kappa }_{\mu \nu }$ no longer depends on the propagation constant ${\beta }_{\nu }$. Under the semi-analytical model, the coupled-mode equations can be written as $$\frac{{\mathrm{d}A}_m(z)}{\mathrm{d}z}\approx \sum _{\nu }\int a_{\nu }(z,{\beta }_{\nu }){\kappa }_{m\nu }(z)\exp (-j\delta z)\mathrm{d}{\beta }_{\nu },$$(4a)$$\frac{{\mathrm{d}a}_{\nu }(z,{\beta }_{\nu })}{\mathrm{d}z}=\sum _m{\kappa }_{\nu m}(z)A_m(z)\exp (j\delta z),$$(4b)where $A_m(z)$ and $a_{\nu }(z)$ are the model amplitudes of the guided mode and radiation mode, respectively. $\delta ={\beta }_m-{\beta }_{\nu }$, where ${\beta }_m$ is the propagation constant of the guided mode. The integral form of Eq. (4b) can be written as $$a_{\nu }(z,{\beta }_{\nu })={\int }_0^z\sum _m{\kappa }_{\nu m}(z^{\prime })A_m(z^{\prime })\exp (j\delta z^{\prime })\mathrm{d}{z}^{\prime }.$$(5)

In our design, it is assumed that the guided mode is the ${\mathrm{TE}}_{00}$ mode, so $m=1$. Then, substituting Eq. (5) into Eq. (4a), together with is ${\kappa }_{m\nu }=-{\kappa }_{\nu m}^{*}$, the attenuation of the guided mode can be expressed as $$\frac{{\mathrm{d}A}_m(z)}{\mathrm{d}z}=-\pi \sum _{\nu }{|{\kappa }_{m\nu }(z)|}^2{A}_m(z).$$(6)

The attenuation of the ${\mathrm{TE}}_{00}$ mode per unit length along the $z$ direction can be given by $$\frac{{\mathrm{d}P}_m(z)}{\mathrm{d}z}=-2\pi \sum _{\nu }{|{\kappa }_{m\nu }(z)|}^2{|A_m(z)|}^2,$$(7)where the right part contains all the power of the guided mode coupled to all radiation modes per unit length, that is, the emission profile. GA has excellent adaptive optimization global search capability^[28], which is suitable for our tricky optimization problem without any prior knowledge to realize an approximate Gaussian emission profile along the grating emitter. We take the quantitative similarity between the calculated emission profile and a Gaussian model as the loss function.

The optimized width and duty cycle distribution of the ${\mathrm{Si}}_3{\mathrm{N}}_4$ overlayers in each grating period are plotted in Fig. 2(a) as the solid black line and the solid red line, respectively. Assuming that the amplitude of the normalized fundamental guided mode is 1 (a.u.), the corresponding optimized emission profile at the wavelength of 1550 nm along the 2.247 mm grating length is plotted in Fig. 2(b). The solid black line represents the calculation result, agreeing well with a fitted Gaussian curve, having the expected value $\mu =0.001$ and variance ${\sigma }^2=9\times {10}^{-8}$, plotted with the solid red line. Their quantitative similarity can be expressed using the following cross-correlation coefficient formula, where $F_1$ and $F_2$ represent our optimized distribution of the emission profile and the corresponding fitted Gaussian profile, respectively: $$C=\frac{{\int }_0^L{F}_1·F_2\mathrm{d}z}{\sqrt{{\int }_0^L{|F_1|}^2\mathrm{d}z}·\sqrt{{\int }_0^L{|F_2|}^2\mathrm{d}z}}=0.9992,$$(8)where the integral variable $z$ is the propagation distance of the light in the longitudinal direction, and $L$ indicates the total propagation distance. The cross-correlation coefficient of 0.9992 strongly suggests that our designed grating has an emission profile nearly approaching an ideal Gaussian profile. The emission profile in the grating periods of 1145–1365 is investigated using a 3D FDTD method. The corresponding major electric field distribution of the guided mode is depicted in Fig. 2(c), and the calculated emission profile, shown in Fig. 2(d), approaches the results simulated by 3D FDTD, which further illustrates the effectiveness and accuracy of our design. Unfortunately, through the simulated result, the unidirectionality of our device only exceeds 46.2%, which can be improved by dual-layer grating misalignment^[21].

At the wavelength of 1550 nm, the far-field beam profile for our proposed grating with the approximate Gaussian near-field emission intensity is shown in Fig. 3(a). The simulated FWHM beamwidth is as low as 0.026°, and the SSR is as high as 32.622 dB. Figure 3(b) illustrates the far-field intensity distribution for beam steering within the 1530–1570 nm wavelength range with a 5 nm step. A steering angle of 3.94° centered at 30° (working wavelength of 1550 nm) is achieved with approximately 0.1° per nanometer angular steering dispersion. Moreover, the far-field SSR for the 1530–1570 nm wavelength range is greater than 32 dB, suggesting that high-performance beam steering can be maintained during wavelength tuning.

In practice, deviations in fabricated dimensions of the ${\mathrm{Si}}_3{\mathrm{N}}_4$ overlayers from designed dimensions can lead to the radiation pattern change in our proposed grating antenna. To simulate the impact of fabrication errors as accurately as possible, we apply the Monte Carlo method to characterize the radiation fluctuations in the far field, assuming that the fabrication errors of the width and length (that is duty cycle) of the ${\mathrm{Si}}_3{\mathrm{N}}_4$ overlayer satisfy the Gaussian random distribution with mean value $\mu =0$. Table 1 presents the averaged far-field SSR with the standard deviation $\sigma$ varying from 10 nm to 60 nm, and Fig. 4 shows the error maps at $\sigma =30\mathrm{nm}$, 40 nm, 50 nm, and 60 nm, respectively. The results show that the grating antenna can still maintain good performance with reasonable fabrication errors, e.g.,  $\sigma$ below 30 nm, which is a quite exaggerated error for any state-of-the-art CMOS photonics foundry. Hence, our design could hold considerable merit and value in developing practical large aperture solid-state phase arrays.

In conclusion, we have proposed a practical design of a ${\mathrm{Si}}_3{\mathrm{N}}_4$ overlayer assisted Si grating emitter with an approximate Gaussian emission profile over $>2\mathrm{mm}$ length, which has a narrow angular FWHM of 0.026° and a large far-field SSR of 32.622 dB. The theory clearly reveals the general way to design a grating with a special emission profile by using CMT and GA. Further analysis shows that our emitter can also function well with almost undeteriorated angular divergence and SSR performance for the wavelength range from 1530 nm to 1570 nm, while achieving a longitudinal steering angle of about 3.94°. Our theoretical analysis indicates that our design is robust to practical fabrication errors for the width and length of the ${\mathrm{Si}}_3{\mathrm{N}}_4$ overlayer. We believe that the high-performance solid-state lidar systems can benefit from our demonstrated device.