Simultaneous Enhancement of Viewing Angle and Angular Resolution of Integrated Lidar With Bidirectional Signals and Orthogonal Polarizations

Coarse wavelength division multiplexing (CWDM) technology has been widely used in short-reach optical communication systems. By using the same technology in a solid-state photonics-based Lidar, one can achieve significant cost reduction. Beam steering is realized through diffraction of light wave upon grating couplers. However, current design has a small viewing angle and a low angular resolution. We propose a configuration based on Silicon-on-Insulator (SOI) waveguide grating coupler. The viewing angle is increased by 2.5 times with light signals propagating bidirectionally. The angular resolution is doubled through interleaving beams with trans-electrical (TE) polarization and beams with trans-magnetic (TM) polarization. By optimizing the design of a single grating coupler, the output beams are evenly distributed and the transmittance is improved. With 4 CWDM lasers being the source, a 34° beam angle of view and an angular resolution of 1.9° are demonstrated.


I. INTRODUCTION
Light detection and ranging (Lidar) has garnered significant interest as a tool for sensing over a wide range, which uses photons as the sensing medium.It has been seen extensive usages in autonomous driving, robotics, three-dimensional sensing, and aerial mapping [1], [2], [3], [4].The Lidar typically consists of multiple discrete optical components and utilizes a mechanical beam steering mechanism, resulting in high cost, slow scan rates, large size, and poor reliability.
The associate editor coordinating the review of this manuscript and approving it for publication was Mohammad Zia Ur Rahman .
The integrated Lidar, which is based on solid-state photonics, integrates all the components onto a single silicon substrate via photonics integration.It is low in price, chipscale, energy-efficient, and has a good system durability [5], [6], [7], [8], [9].An integrated photonics system can provide a path for low-cost Lidar systems.
We choose a wavelength division multiplexing (WDM) laser array as the light source.Short-reach optical interconnect modules are widely used within massive web-scale data centers.Developed for low cost and driven by demands, the short-reach optical modules have the largest volume among components used for optical communication.
Silicon photonics components in the short-reach optical modules are growing rapidly and account for a large share of the total market of silicon photonics.WDM laser arrays and WDM photodiode arrays, in the 1310-nm wavelength band (O-band), are key components for the short-reach optical modules.The same components can also be utilized for solidstate Lidars [10].It is not only highly cost-effective, but also simple to gain functionality of beam steering through wavelength tuning [11], [12], [13], [14], [15], [16], [17].Similar to the current sixteen-line Lidar, it is capable of simultaneously measuring several sites and achieves a high level of measurement precision.The SOI waveguides can be etched to form a single grating coupler (GC).Through a WDM multiplexer, multiple lasers are combined into one signal.The signal is diffracted out of the SOI waveguides by the GC [18], [19].The output angles are determined from their wavelengths [20], [21].Reflected signals from surrounding environment pass through a WDM de-multiplexer.Multiple photodiodes can simultaneously detect the signals separated by different wavelengths [22], [23], [24].Lidar's point-ofcloud data can reach a high throughput by simultaneously operating multiple lasers and multiple photodiodes.
The most common WDM laser array is CWDM4 laser array operating at λ 1 = 1271 nm, λ 2 = 1291 nm, λ 3 = 1311 nm, λ 4 = 1331 nm.By using CWDM4 laser array and CWDM4 photodiode array in a solid-state Lidar, we leverage the most mature design with the highest volume.It is believed that this is the most cost-effective solution.However, one challenge is to increase the angle of view in the total 80-nm wavelength tuning range due to the limited dispersion of the SOI materials.The other challenge is that an angular resolution is approximately 4 • given that there are only four channels in the CWDM4 laser array.This is far from acceptable for most Lidar applications.It has recently been demonstrated that integrated Lidar may have a wide viewing angle or a high angular resolution [25], [26].One can also increase the wavelength range and reduce the channel spacing, but those solutions increase the cost and complexity [27].
In this paper, a structure is designed to simultaneously increase the viewing angle by 2.5 times and enhance the angular resolution by 2 times.There is a significant disparity between the effective index of the TE mode and the TM mode, which results in the output angles of the TE beams being different from those of the TM beams.Through optimal design, a TE beam can be located right in the middle of two TM beams, and a TM beam can be located right in the middle of two TE beams.This leads to a two-fold increase of the angular resolution [28].The signal can also propagate bidirectionally, forward or backward in the GC.There is a mirror symmetry between the output beams of the forward-propagating signal and the output beams of the backward-propagating signal.Two beams can be seamlessly stitching together, and the total viewing angle can be doubled more than two times.With CWDM4 laser array as the signal source, sixteen output beams are diffracted from a single GC, with four beams generated from each wavelength.In this way, it is possible to achieve both high angular resolution and wide viewing angles.

II. CONCEPT AND FIBER STRUCTURE
The schematic diagram of the proposed configuration is shown in Fig. 1, where the forward-propagating beam and the backward-propagating beam are transmitted bidirectionally towards each other in the SOI waveguides [29].We integrate an optical switch, a shallow-etched GC and two polarization elements (PE) in the same substrate [30].The outputs of four lasers are set in the TE mode, combined with a WDM multiplexer and coupled into the SOI waveguide.Next, an optical switch determines the light's propagation direction, allowing the output signal to be directed to one of the two paths as needed [31].As shown in Fig. 1, each path is equipped with a polarization element (PE), which consists of an optical switch nested within the PE, a polarization rotator, and a polarization beam combiner.While the TE polarization is needed, the nested optical switch directs the signal to the bottom path.When the TM polarization is required, the nested optical switch directs the signal to the top path.The polarization rotator changes the TE polarization to the TM polarization [32], [33], [34], [35].Both TE and TM polarization are combined again with the polarization beam combiner.In this manner, the propagation direction and the polarization mode can be both adjusted for the CWDM4 laser signal.There are four stages of operation.In the first stage, the light is directed to the left path and its polarization remains in the TE mode.Four beams are diffracted from the TE GC, denoted as − → λ TE n with n = 1,2,3,4 corresponding to four wavelengths of the CWDM4 lasers.The arrow on λ indicates the direction of the propagating beam.In the second stage, the polarization on the left path is rotated from the TE mode to the TM mode, and four beams with TM polarization − − → λ TM n are generated.In the third stage, the light is directed to the right path and its polarization remains in the TE mode.Four beams ← − − λ TM n with output angles are generated accordingly.Fourth, the polarization on the right path is rotated from the TE mode to the TM mode, and four beams ← − − λ TM n are generated.We can obtain sixteen diffracted beams (four beams for each wavelength) by sequentially performing the four stages described above [22], [36], [37].A continuous operation is carried out by repeating the above steps.Figure 2 (a-d ← − λ TE 1 ) do.So, the total viewing angle is increased by more than two times.In a later section, we will show that the increase is 7/3 ≈ 2.3 times.This is another noticeable advantage of using the CWDM laser array instead of the continuously tunable laser.However, there is one limitation of configuration (I).The GC transmittance is low when the diffracted beam is along the vertical line, more details will be provided in the latter section.To improve the transmittance, a second configuration is further introduced, which is defined as configuration (II) and shown in Fig. 2(f).As long as 16 beams are evenly distributed with a constant angle difference between two adjacent beams, there is no need to align the diffracted TE beam at λ 4 along the vertical line.Here, one can introduce a small angle between ( ← − λ TE 4 ) and the vertical line.Thus, we avoid the region with low transmittance when the output beams are along the vertical line.In a following section, it will also show that the viewing angle is increased by 2.5 times, which is another improvement against the configuration (I).
One-dimensional beam steering can be realized using this structure and wavelength tuning.Two-dimensional beam steering can be achieved with optical phase arrays (OPAs), which are implemented by duplicating GC configurations and integrating multiple GCs on a single substrate.By using a binary tree structure composed of cascading broadband Y-splitters, the input signal can be split and fed into multiple GCs [38].Adjusting the phase between adjacent GCs allows beams to be diffracted out perpendicularly to the GC.

III. LIDAR WITH WDM LASER IN O BAND A. CALCULATIONS IN THEORY
Configuration (I) is examined in detail below.The following parameters are chosen for configuration (I): In the SOI waveguide, the height and width are set to 295 nm and 386 nm, respectively.To diffract the signal out of the SOI waveguide, the GC is partially shallow etched.During one grating period, one fraction of the SOI waveguide retains its 295 nm height, whose width is L GC , while the other fraction is etched to 240 nm.The period of the GC is = 500 nm, and the crucial parameter f , the filling factor, is defined as the ratio between L GC and , which is set to 0.55.The wide waveguide width and shallow etching of the GC structure are conducive to reducing the scattering loss, which is the main source of silicon waveguide's loss [39].In the actual fabrication, reducing the side wall roughness of the waveguide can effectively reduce the transmission loss of the waveguide.For the etched waveguides, the sidewalls can be smoothed by thermal oxidation, laser beam, hydrogen annealing and other methods [40].Among them, hydrogen annealing can effectively avoid some damage caused by the cleaning process after the thermal oxidation.
The critical parameters of the GC are determined through theoretical analysis.The GC parameters are adjusted to make sure that the output angles for ← − λ TE 4 are along the vertical line and the remaining fourteen beams are evenly distributed.When the physical size of an underlying structure in the GC is much smaller than the optical wavelength under consideration, the effective index of the GC can be determined by the effective mode theory (EMT), as shown in Eq. ( 1) [41], [42].
where n H is the effective index of intact portion of SOI (296 nm), n L represents the effective index of etched portion of SOI (240 nm).We first calculate both n H and n L for the TE and TM modes within the wavelength range of 1.25 µm to 1.35 µm using a finite difference eigenmode solver.For n H calculation, the dimension of buried oxide (BOX) SOI waveguide is 386 nm × 295 nm, and the dimension of BOX SOI waveguide is 386 nm ×240 nm for n L calculation.The results are illustrated in Fig. 3.It is well known that SOI waveguides are highly birefringent.The effective index of a TE mode can be significantly different from that of a TM mode, which can Moreover, Eq. ( 2) establishes a relationship between the diffracted beam angle and the effective index n eff , as shown below, where k = 2π /λ, n c is the refractive index of SiO 2 cladding, and q represents the order of diffraction beam which is equal to 1 for the GC.The output angle ϕ is defined in a Cartesian coordinate, where the X axis is from the left to the right, and the Y axis is from the bottom to the top of the SOI waveguide.In Eq (2), we assume that the light is propagating from left to right along the plus X direction.ϕ refers to the angle between the SOI surface normal (Y axis) and the diffracted beam in the far field.When the diffracted beam falls into the first quadrant, ϕ is positive; When the diffracted beam falls into the second quadrant, ϕ is negative.When the propagating direction of light is reversed, we first use Eq. ( 2) to calculate the output angle, and then reverse the sign of the output angle given that the light is propagating in the minus X direction.By simplifying Eq. ( 2), one can obtain Eq. ( 3) as shown below: To satisfy our requirement, the output beam of the input beam 4 should diffract along a vertical line and its diffraction angle is zero: Eq. ( 3) is rewritten by combining Eq. ( 4) and Eq. ( 5): The GC design must satisfy Eq. ( 6) so that the viewing angle can be doubled by leveraging beams propagating bidirectionally.Next, we will derive the conditions so that the TE and TM beams are combined in an interleaving way which leads to the increase of angular resolution by two-fold.Because the sixteen diffracted beams are symmetrically distributed around the vertical line, we only carry out theoretical analysis on the eight beams to the right of the vertical line (the first quadrant).
In the current design, the diffraction beams − → λ TE n and the diffraction beams n fall into the first quadrant.To combine these beams in an interleaving way which doubles the angular resolution, it needs to meet the following conditions derived below.In general, the output angle can be approximated by a linear curve as a function of wavelength, as shown in Eq. ( 7) and Eq. ( 8).
Slope L,TE and Slope R,TM are defined as the slopes of the linear functions of the TE mode propagating from left to right and TM mode propagating from right to left, respectively.λ 0 is the wavelength on which TE beam and TM beam have the same non-zero output angle.To make sure that the eight beams are evenly distributed in an interleaving manner, two conditions need to be satisfied, Slope L,TE = −Slope R.TM = sl 0 (9) λ 0 = λ 3 − 0.75 * sp (10) where sl 0 is defined as the absolute value of the linear function slope, sp is the channel spacing between adjacent wavelength channels.Four wavelengths of CWDM4 laser array satisfy the following condition: Combined with the above equations, the output angles of the eight beams (four TE polarized beams and four TM polarized beams) are obtained as shown below, Taking the equations in order from the smallest to the largest output angle is as follows: As seen, a difference in output angle exists between two adjacent output beams equals to |0.5× sl 0 × sp|.Thus, it improves the angular resolution by a factor of two.Based on the considerations above, we choose the filling factor for the GC as f = 0.55.

B. 2D SIMULATION RESULTS
Numerical simulation is conducted using finite difference time domain (FDTD) method, which is the standard method for modeling nanophotonic devices.As the twodimensional (2D) simulation time is much smaller than the three-dimensional (3D) simulation time, we choose the 2D simulation to validate our assumptions and establish roughly the critical parameters of the GC.After that, a threedimensional simulation is performed in order to obtain an accurate result.
A boundary condition is needed to solve the Maxwell equation, and we choose the perfect matching layer (PML) to minimize the back-reflection at the edge of the simulation area.The 2D FDTD simulation assumes a slab waveguide, where only the height of the waveguide is taken into consideration when calculating n H and n L .The variation of refractive index along the waveguide width is ignored.All the GC parameters are scanned and analyzed in 2D simulation to gain an understanding of how each parameter affects the simulation results, and to determine the approximate range of the parameter values.

C. 3D SIMULATION RESULTS
After the 2D simulation, we perform 3D FDTD to fine tune the GC parameters, which is quite complicated and takes a considerable amount of time.This simulation model is configured similarly to the one shown in Fig. 1.Multiple lasers, all of which are set to the TE polarization, are combined through a WDM multiplexer.The output of WDM multiplexer is injected into the GC which is sandwiched between two silica layers.A monitor is placed over the top silica layer to collect the near-field electromagnetic fields.The height and width of the SOI waveguide are 300 nm and 386 nm, respectively.Figure 4 depicts two representative results for waveguide width w = 360 nm and w = 405 nm, respectively.Based on the width-scan results, the width is set to 386 nm so that the output angle is right along the 0 • vertical line with a 1.331 µm incident light source.For the GC, its thickness in intact section is 300 nm and the thickness in etched section is 250 nm, respectively.The filling factor f is 0.55 and the period is 500 nm.
Figure 5 displays the output angles of sixteen beams (eight TE beams in triangle symbol and eight TM beams in circular symbol) on the left when the GC is etched by 55 nm.The ϕ value is determined by the maximum location of the electrical field emitted from the GC in the far field.The total view angle is |3× sl 0 × sp| in the plain vanilla scenario when there is no adjustment on the polarization state and propagation direction.As discussed earlier, the propagation direction is adjusted with an optical switch and the polarization state is altered with a PE.From the simulation result, one can see that two TE beams propagate along the vertical line at λ 4 = 1.331 µm.When the wavelength is λ 4 , the left-propagating TM mode diffracts at a maximum angle away from the vertical line, which is equal to 15 • , and the output angle of the  A combination of these counter-propagating beams creates a 30 • viewing angle.For any two adjacent diffracting beams, the spacing between their diffracting angles is approximately 1.95 • .It is proved that the output beam is uniformly distributed.It is therefore possible to increase the angular resolution twice by combining the TE and TM output beams in an interleaving manner.Additionally, Fig. 5 illustrates the output angles of the eight beams (four TE beams are marked with the black triangles and four TM beams are marked with the red squares), arranged from the smallest to the largest.The linear regression was also performed on the output angles of these eight beams, and the linear regression has an R 2 value of 0.995.It has been demonstrated that the TE beams and the TM beams are linearly distributed, and the GC can improve the angular resolution by a factor of 2. Figure 6 shows the far field beam distribution for both modes at the CWDM4 wavelengths.We use the Fresnel-Kirchhoff diffraction formula to calculate the distribution of far-field beam at the distance of one meter.When describing a beam originating from a point source, the spherical coordinate system is used.As wavelength decreases, the output angle for the TE mode increases, while the output angle for the TM mode decreases.When the wavelength of the TE mode is λ 4 =1331nm, the output beam will be diffracted along the vertical line.When the wavelength is λ 1 =1271nm, the TM mode is diffracted at the largest angle away from the vertical line, which is approximately 14.85 • .By using our design with counter-propagating signals, an approximate 29 • angle of view can be achieved.Furthermore, the TE beams and the TM beams are interleaved evenly to double the angular resolution.
The GC transmittance for both polarized modes and different wavelength is displayed in Fig. 10 (a).The transmittance is defined as the ratio between the energy of the upper-propagating beam and the energy injecting into the GC [43], [44], [45].As shown in Fig. 10 (a), the transmittance decreases steadily as the diffracted position of the outgoing beam approaches the vertical line.When the TE beam diffracts along the vertical line, the transmittance is only 0.05, which is just about to one-sixth of the maximal transmittance.This is one drawback of configuration (I) as discussed earlier.
Thus, the configuration (II) is introduced to address this issue.The GC structure is optimized, and the TE beam at λ4 diffracts at a small angle.We also ensure that the diffracted beams are uniformly distributed.Given that the transmittance decreases significantly when ϕ is close to zero, one needs to avert the undesired situation where the diffracted beam propagates in the direction perpendicularly to the GC.

IV. RESULTS OF THE CONFIGURATION (II)
An illustration of the diffracted beam distribution obtained from this model can be found in the box labelled II in Fig. 1.The configuration (I) and (II) are almost identical in principle.Against the configuration (I), the configuration (II) avoids the region with low transmittance and increases the viewing angle by 2.5 times.It should be noted that when the wavelength is 1331 nm, the absolute values of the diffraction angles are the smallest.No matter whether the beam is propagating from left to right or from right to left, the diffraction angle is the closest to the vertical line.To make sixteen beams distribute uniformly, the angle difference between two beams which are the closest to the vertical line should be 0.5× sl 0 ×sp.In this configuration, the TM mode diffracts at the biggest angle away from the vertical line when the wavelength is λ 4 .
The important GC parameters are also scanned in order to determine the optimal parameter combination.The width of the SOI waveguide is 390 nm and its height is 295 nm.A portion of the SOI (length L GC ) remains intact at the grating period of 500 nm, while the other is etched by 60 nm to realize the optimal GC.The filling factor of the GC is f = 0.4.The variation of the etching depth will affect the effective refractive index of the guide mode, which in turn will change the steering angle of the far-field beam.Using different etching depths, the beam scanning ranges of the fundamental TE mode and TM mode are calculated and the results are shown in Fig. 7.We show two representative sets of results taken from the scanning process of the etch depth.On the left side, the waveguide is etched by 30 nm and the angle between the two diffracted beams nearest the vertical line is larger than |0.5× sl 0 × sp|.While on the right, the waveguide has been etched by 70 nm.In this case, the angle between two diffracted beams closest to the vertical line is smaller than |0.5× sl 0 × sp|.Based on the optimization results of the GC parameters, the etching depth was eventually set to 60 nm, where the angular spacing between the two diffracted beams closest to the vertical line is exactly |0.5× sl 0 × sp|.   Figure 8 (b) also depicts the output angles of eight beams that are all to the right of the vertical line.The R 2 value of the linear regression is 0.999, which shows that the output angles are linearly distributed.It is also demonstrated that the GC can improve the angular resolution to an ideal level.Figure 9 illustrates the far-field beam profiles for both modes at the CWDM4 wavelengths.The TE beams and TM beams are interleaved together, effectively doubling the angular resolution.The angle between two adjacent beams is approximately 1.95 • .When the wavelength is λ 4 , the TM mode diffracts at the largest angle away from the vertical line, whereas the TE mode diffracts at the smallest angle.The spatial resolution will be doubled when the TE and TM beams are combined.As shown in Fig. 10, when the configuration (II) is used, the maximum transmittance is approximately 0.37 and the minimum transmittance is around 0.18.Thus, we can avoid the undesired situation where the diffracted beams propagate in the direction vertically to the GC and reduce the transmittance variation.In addition, the total viewing angle is |7.5× sl 0 × sp|.Compared with the plain vanilla scenario, the viewing angle is improved by 2.5 times and the angular resolution is improved by 2 times.For the convenience of comparison, the diffraction angles and transmittance corresponding to each wavelength under the two configurations are shown in Table 1.As can be seen from Table 1, configuration (II) has an obvious improvement over configuration (I).
The angular resolution is not only determined by interleaving the beams and the beam width.For the one-dimensional uniform line, the beam width can be calculated by the following equation: where k is the beamwidth factor, N is the number of linear array elements, d is the array spacing, and ϕ is the steering angle.When N * d is much larger than λ, k of the beam width of the half-power point is 0.886.The calculation above demonstrates that the beam width is determined by the incident wavelength λ, antenna aperture N * d and the steering angle ϕ.As the number of array elements and the spacing between them increase, the resulting antenna aperture N * d also grows, resulting in a narrower beam.We only show the case of the single GC for its simplicity.As part of the actual implementation, multiple GCs can be further integrated on a single substrate and suitable array spacing can be selected.
As the steering angle ϕ increases, the beam width additionally broadens.The cosine value of the steering angle at plus or minus 60 degrees is 0.5, which is twice that of the cosine value at zero degrees.In our manuscript, the total steering   individual lasers within laser arrays.For example, the output power of the particular laser can be increased to compensate the low transmittance without influencing the other lasers.This is a much easier method for compensating the wavelength dependence of the transmittance.If the CWDM laser array is replaced by a TLS, one must adjust the output power of TLS while tuning the TLS wavelength.When the intensity of the semiconductor laser is adjusted, its linewidth will be larger and its wavelength will be less accurate.This is known as linewidth enhancement effect for semiconductor laser.This illustrates another benefit of using a CWDM laser array as the laser source.
Furthermore, it is well known that variations in temperature significantly impact the emitted wavelength of the laser.As the temperature rises, the threshold current of the laser will also increase, causing the output wavelength to move towards longer wavelengths.Conversely, as the temperature decreases, the center wavelength will shift towards shorter wavelengths.The CWDM systems often employ distributed feedback (DFB) lasers as the light sources, which function within a temperature range of −5 to 70 • C. Each 1 • C variation in the light source temperature causes the wavelength of the DFB laser to shift by 0.08 to 0.1 nm.When considering the impact of laser source temperature on wavelength harmonics, we can initially utilize the current CWDM4 laser array operating at λ 1 = 1271 nm, λ 2 = 1291 nm, λ 3 = 1311 nm, λ 4 = 1331 nm for a broad scanning range.After the desired target area or object is detected, the temperature of the laser source can be adjusted to perform a more precise and narrower scan.By combining the two kinds of precision scanning, it is beneficial to achieve a fast scanning of a wide range and a fine scanning of the target object.
The GC spot size can be further optimized by introducing apodization of the grating period along the beam propagation direction [46], [47].An important parameter to be considered is the upward emission efficiency, which can be improved by adding a top cladding or bottom mirror to the system [33].

V. CONCLUSION
In conclusion, a concept of using CWDM laser array as the light source is proposed for solid-state photonics-based Lidar.It addresses two fundamental issues, including the steering angle and angular resolution.The steering angle has a 2.5× increase by using the counter-propagating beams within a single GC.Simultaneously, the angular resolution is improved by 2 times by controlling the polarization state of the beam within a single GC.When a CWDM4 laser array is used as the light source, sixteen diffracted beams are generated with uniform distribution.This design is demonstrated through theoretical analysis and simulation study.Furthermore, the spatial distribution of diffracted beams is optimized to reduce the transmittance variation.

FIGURE 1 .
FIGURE 1. System configuration using counter-propagating orthogonally-polarized beams.TE/TM outputs are shown with solid/dashed lines.The arrow on top of the wavelength indicates the propagating direction of the signal.Different wavelengths are shown in different colors.

4 .
) illustrate the operations in those four stages.When the input signal is changed to a CWDM4 laser array, two innermost TE beams ( almost overlapped.The rest 14 beams are evenly distributed.As discussed earlier, TE beams and TM beams can be interleaved to increase the angular resolution.Without bidirectional signals and orthogonal polarizations, the total viewing angle is the angular difference between As seen in Fig.2(e), the two outmost TM beams ( more diverged from the vertical line than the two outmost TE beams (

FIGURE 2 .
FIGURE 2. (a-d) Diagrams of polarized beam produced by each of the four stages of operation when using the first configuration; (e) outgoing beams distribution of configuration (I) and (f) outgoing beams distribution of configuration (II).

FIGURE 3 .
FIGURE 3. Numerical results of n H , n L , and n eff for (a) TE mode and (b) TM mode.

FIGURE 4 .
FIGURE 4. 3D numerical simulation results of the output angles for counter-propagating TE beams and TM beams when (a) w = 360 nm and (b) w = 405 nm.

FIGURE 5 .
FIGURE 5. (a) Numerical simulation results of the output angles; (b) linear regression of the output angles of eight beams on the right side of the vertical line (perpendicular to the SOI waveguide plane).

FIGURE 7 .
FIGURE 7. 3D numerical simulation results of the output angles of counter-propagating TE beams and TM beams when (a) etching depth = 30 nm and (b) etching depth = 70 nm.

Figure 8 (
a) shows numerical simulation results of output angles using 3D-FDTD.When the wavelength is λ 4 , the output angles of the TM mode are the biggest and approximately 16.9 • .The angle between two adjacent beams is about 2 • .As the light enters the GC from different directions, the output angle is symmetrically distributed along the vertical axis.We can achieve a wide angle of view, about 33.8 • , by combining all the output beams.Unlike the configuration (I), this new distribution takes full advantage of the diffracted beam in space and prevents a large reduction in the outgoing energy.

FIGURE 8 .
FIGURE 8. (a) Numerical simulation results of the output angles; (b) linear regression of the output angles of eight beams on the right side of the vertical line (perpendicular to the SOI waveguide plane).

Figure 8 (
Figure 8 (a) shows numerical simulation results of output angles using 3D-FDTD.When the wavelength is λ 4 , the output angles of the TM mode are the biggest and approximately 16.9 • .The angle between two adjacent beams

FIGURE 9 .
FIGURE 9. Far-field beam profiles of the configuration (II) for the TE-polarized beam propagating from right at (a1) λ 4 = 1331 nm, (a2) λ 3 = 1311 nm, (a3) λ 2 = 1291 nm, (a4) λ 1 = 1271 nm, and for the TM-polarized beam propagating from left at (b1) λ 4 = 1331 nm, (b2) λ 3 = 1311 nm, (b3) λ 2 = 1291 nm, (b4) λ 1 = 1271 nm respectively.angleranges from −14.85 • to 14.85 • when configuration (I) is used and from −16.9 • to 16.9 • when configuration (II) is applied.The maximum and lowest values of cos ϕ are 1 and 0.95, respectively.This facilitates the formation of a narrower beam.When the antenna aperture N * d and the steering angle ϕ remain constant, increasing the wavelength leads to a wider beam width.When utilizing WDM laser arrays in the O-band rather than in the C-band, a narrower beam width can be achieved.Since CWDM laser array is composed of many individual lasers, one can precisely control the output power of

FIGURE 10 .
FIGURE 10.Transmission for the TE and TM beams in the 3D simulation for different diffraction angle distributions.

TABLE 1 .
Performance comparison of integrated lidar with two design schemes s.