- Chinese Optics Letters
- Vol. 22, Issue 7, 071301 (2024)

Abstract

1. Introduction

LiDAR is in high demand and is under rapid development owing to burgeoning autonomous driving technology. Optical phased arrays (OPAs)^{[1]} have great potential in the field of LiDAR^{[2]} due to their lack of mechanical components and ability to achieve precise and fast beam scanning. Recently, benefiting from the advancement of photonic integrated circuit (PIC) technology^{[3]}, integrated OPAs have been devised and reported. Such devices can realize a wide scanning range with no aliasing beam^{[4,5]}, high resolution^{[6,7]}, compact size^{[8]}, and low power consumption^{[9]}, which are highly desirable features in LiDAR applications^{[10]}. Furthermore, OPAs also show promising applications in optical wireless communications (OWC) and sensing.

However, most integrated OPAs typically use a single beam for detection and ranging^{[11]}, which does not fully utilize the OPA’s flexible beam control ability. In principle, an OPA can independently control multiple beams^{[12]}, which helps reduce the scanning range required for a single beam and improve the frame rate^{[13,14]}, simultaneously track multiple objects and communicate with multiple users in OWC. Some efforts have been made in the special design of system hardware, such as cascading multiple OPAs^{[15]} or introducing the Butler matrix into the OPA system^{[5]}, to achieve multi-beam scanning. In addition, specially designed gratings^{[13]} and polarization multiplexing^{[16,17]} can also help realize multi-beam forming in OPAs. Usually, multi-beam generation tends to be more complicated/difficult^{[18]}. It is not easy to make directions of multiple beams independent from each other in the experiment, and the relative difference of beam angles (or their sine values) is often fixed, which does not offer the flexibility needed for handling multiple objects moving independently in advanced application scenarios.

In this work, we demonstrate independent control of dual-beam by dividing the OPA into non-uniform sub-arrays of elements. This paper is organized as follows. In Sec. 2, theoretical considerations of this multi-beam generation approach will be presented based on the working principle of OPAs. Then, in Sec. 3, we present the simulation of optical multiple beamforming and the schematic of a uniform division of array elements for generating dual independent beams. In Sec. 4, experimental work of dual-beam generation will be reported with uniform or non-uniform division. In Sec. 5, we discuss the effect of residual phase distribution errors on beamforming through simulation.

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2. Principle of Optical Multiple Beamforming

For an OPA with $N$ uniformly distributed emitters, the electric field distribution $E$ of its far field satisfies ^{[19]}. Ideally, the pitch of the emitters should be half-wavelength^{[1]} to avoid generating grating lobes. Grating lobes are secondary beams generated by an OPA with large emitter pitches. They are steered together with the primary beam and, hence, are not independent beams. For independent multi-beam generation, an OPA with a half-wavelength emitter pitch would be preferred; otherwise, mixing the dependent and independent multi-beam would cause complications in many scenarios.

In order to achieve multiple beams, the entire array elements can be divided into several sub-arrays, each of which can be regarded as an independently controlled sub-OPA. By applying phase distributions with varying degrees of linear variation to different parts, thereby multi-beam scanning is achieved. To illustrate the principle, here we consider the case of dual beams generated by the OPA with a uniform radiation element for simplification. In such situations, the far-field intensity can be calculated as

Here, ${I}_{1}$ and ${I}_{2}$ represent the direct contributions of the two sub-arrays to the intensity at angle ${\theta}_{1}$, while ${I}_{1\&2}$ stems from the interference between the two parts as mentioned before. Due to the symmetry, the equation for the beam intensity at the other angle ${\theta}_{2}$ [equals to $-{\mathrm{sin}}^{-1}(\mathrm{\Delta}{\phi}_{2}/kd)$] is formally similar to Eq. (3). Obviously, the contribution of ${I}_{1}$ is the most significant at angle ${\theta}_{1}$ when ${N}_{1}$ is large, and the direct contribution ${I}_{2}$ from the intensity of beam 2 is small at angle ${\theta}_{1}$. The effect of the term ${I}_{1\&2}$ tends to be small because ${I}_{1\&2}/{I}_{1}\sim 1/{N}_{1}$. Additionally, it is easy to verify that when the phase distribution returns to the situation corresponding to a single beam, $I$ will evolve into ${({N}_{1}+{N}_{2})}^{2}$, which is consistent with the theory of single-beam formation^{[20]}.

3. Simulation and Analysis

To analyze the characteristics of dual-beam generation, we have simulated the far-field intensity distribution of a 32-element OPA. Representative single-beam pointing at several angles is shown in Fig. 1(a). Note that here we have assumed an OPA with a half-wavelength emitter pitch to avoid the generation of grating lobes (dependent beams). Full consideration of the beam intensity also needs to consider the influence of the envelope, ${I}_{r}(\theta )=F(\theta )\xb7I(\theta )$, where ${I}_{r}(\theta )$ denotes the real intensity and $F(\theta )$ represents the envelope function. The $F(\theta )$ in Fig. 1(a) is simulated from a waveguide with a size of $450\text{\hspace{0.17em}}\mathrm{nm}\times 220\text{\hspace{0.17em}}\mathrm{nm}$ through the 3D-FDTD simulation software. The envelope effect originates from the angular dependence of waveguide radiation intensity^{[4]}. To illustrate the dual beam generation, we first consider two single beams at $-10\xb0$ and 30° in Fig. 1(b). When we divide the OPA into two halves, one half uses the phase setting for the beam pointing at $-10\xb0$, and the other uses the phase setting for the beam pointing at 30°. As such, dual-beam pointing at $-10\xb0$ and 30° is obtained as shown in Fig. 1(c). Similarly, dual-beam pointing at $-20\xb0$ and 20° is illustrated in Fig. 1(e) based on phase settings of the two single beams shown in Fig. 1(d). The OPA division scheme is illustrated in Fig. 1(f).

Figure 1.The simulation of far-field beamforming. (a) Far-field patterns of beams pointing at different angles (each beam intensity is represented by a unique color), whose peak intensity follows an envelope function. (b) Two single beam points at −10° (red) or 30° (green), respectively; and (c) the dual-beam points at both angles (blue). (d) Two single beam points at −20° (red) or 20° (green), respectively; and (e) the dual-beam points at both angles (blue). (f) Schematic diagram of a uniform dividing scheme for generating independent dual beams.

4. Experimentation

4.1. Experimental setup

A schematic for the silicon-based integrated OPA chip used here is shown in Fig. 2(a), which consists of a single input grating coupler, power splitting components, phase shifters, and an output emitter array. Power splitting is based on cascaded $1\times 2$ multimode interference couplers (MMIs) and splits the input light evenly into 32 channels. Then the light is sent to phase shifters, thus controlling the phase of each channel. Phase shifters are wire-bonded to a printed circuit board (PCB) to establish the electrical connection as shown in Fig. 2(b). The measured resistance of each shifter is about 540 Ω, and the power needed for thermo-optically generating a $\pi $ phase shift ${P}_{\pi}$ is about 18.5 mW. Meanwhile, a trench is added between the adjacent phase shifters to reduce thermal crosstalk. The structure is designed with a 220 nm silicon-on-insulator (SOI) wafer and a 2 µm thick buried oxygen layer. Here, the OPA adopts waveguide-based broadband end-fire emitters, and a waveguide superlattice (SC5b type, consisting of waveguides with widths of 480, 420, 360, 450, and 390 nm, and uniform heights of 220 nm) with a spacing of 0.8 µm is periodically arranged on the end face of the chip to suppress the crosstalk^{[4,21]}. Therefore, the aperture linearity of the emitter array is about 25 µm. It is worth noting that this method is actually applicable to OPA devices utilizing grating emission^{[22-24]}. The emitter radiation efficiency is estimated at about 80%. As shown in Ref. [4], the change of the envelope for different waveguide widths in the waveguide superlattice is small and can be neglected. The experimental system setup is shown in Fig. 2(c). The emitted laser of 1550 nm is coupled into the grating coupler through a polarization-maintaining fiber. Finally, the far-field intensity distribution of the OPA is measured by a photodetector that scans the far-field angles through mechanical rotation.

Figure 2.(a) Schematic of the OPA structure. (GC, grating coupler; MMIs, multi-mode interference couplers; PS, phase shifter; WEA, waveguide emitter array). Not drawn to exact scale or to exact number of elements. (b) An optical microscope image of the OPA chip after wire-bonding and wires are protected by black resin. (c) Schematic of the experimental setup used to characterize the OPA. (TL, tunable laser; FPC, fiber polarization controller; SMF, single-mode fiber; DUT, device under test; PD, photodetector; MVS, multi-channel voltage source, PC, personal computer.)

Ideally, the phase difference between the adjacent elements of the OPA is usually a constant according to theory. Due to the fabrication variation, random phase errors are present in all channels of an OPA. This requires the compensation of such phase errors (via phase calibration) and makes the phase allocation in beam generation a non-trivial process, which can be even more complicated for the multi-beam generation case. Simultaneous phase calibration for multiple beams from multiple sub-arrays is difficult. Here we develop a phase data allocation method using phase data based on multiple steps of phase calibration. The phase data allocation method can be explained with the help of Fig. 1(f). Two single beams are generated first for the entire OPA to build up the phase data of each beam (which has the phase errors compensated), which can be obtained with the help of, for example, an iterative phase calibration method such as particle swarm optimization (PSO)^{[4]}. Then the uniform or symmetrical division scheme is used to assign half of the phase data of the previous two single beams to the two sub-arrays, resulting in the two independent beams simultaneously.

4.2. Experimental results of independent dual-beam generation

Based on the forementioned uniform dividing scheme, we have implemented a dual-beam that can be independently controlled using an OPA chip, as can be seen in Figs. 3(a)–3(g). Here the cases of two beams on different sides and the same side of 0° are presented, with the maximum scanning range reaching 100° (from $-50\xb0$ to 50°). The required phase data are collected and organized from single beams, which are obtained from PSO method. The comparison of peak intensity between the two beams is consistent with theory, that is, if the beam position is closer to 0°, its value will be higher. The interference will exist, no matter where the beams are located, as shown in Eq. (2). It also can be seen that some beams partially deform, which should be attributed to the residual phase errors and is discussed in Sec. 5.

Figure 3.Experimental far-field patterns of dual beams through uniform division. The dual-beam points at the same side: (a) −40° & −10°, (b) 10° & 30°, and (c) −40° & −20° simultaneously; at different sides: (d) −10° & 20°, (e) −20° & 30°, (f) −30° & −30°, and (g) −50° & −50°. Beam patterns are normalized to their peak intensities.

In order to reduce the possible difference in peak light intensity due to the envelope effect when multiple beams are generated, a non-uniform division scheme can be introduced. Specifically, in the case of dual beams, the peak intensity [$I({\theta}_{1})$ and $I({\theta}_{2})$] at two pointing angles ${\theta}_{1}$ and ${\theta}_{2}$ will roughly follow the relationship as $I({\theta}_{1})/I({\theta}_{2})={N}_{1}^{2}\text{\hspace{0.17em}}{\mathrm{cos}}^{2}\text{\hspace{0.17em}}{\theta}_{1}/{N}_{2}^{2}\text{\hspace{0.17em}}{\mathrm{cos}}^{2}\text{\hspace{0.17em}}{\theta}_{2}$. Here, ${N}_{1}$ and ${N}_{2}$ represent the number of elements in each submatrix, and we have ${N}_{1}+{N}_{2}=N$, where $N$ is the total element number. Therefore, as a preliminary validation of this concept, the peak intensity of different beams can be adjusted to a close level, by assigning appropriate numbers of emitter elements to each beam. For example, if we want to use the forementioned 32-channel OPA to achieve dual beams pointing at $-30\xb0$ and 10° simultaneously, with almost equal intensity, the corresponding number of array elements should be around 17 and 15. Generally, as the total element number $N$ of an OPA increases, the result of intensity balance should be better, as it can achieve more precise adjustment. Furthermore, this scheme can also be applied to achieve other beam intensity ratios between the two beams.

To validate this scheme, we conducted simulations and experiments, and the results are shown in Figs. 4(a)–4(d). Figures 4(a) and 4(c) belong to uniform division, while Figs. 4(b) and 4(d) are the corresponding ones with non-uniform division. Through the non-uniform division scheme, the intensity of the two beams becomes more balanced, and all measured multi-beam far-field patterns are in reasonably good agreement with the simulation results in terms of beam divergence. Specifically, the difference between the two beam’s intensities in Fig. 4(c) is reduced from about 0.5 to almost very small in Fig. 4(d), thus verifying the feasibility of the scheme. Furthermore, the ratio between the peak intensities of the two beams can be flexibly set, as shown in Figs. 4(e)–4(g). In such cases, the $-50\xb0$ beam has a relative intensity of 1.0, while the relatively intensity of the other beam is 1.0, 0.5, and 0.3, respectively. Furthermore, it is worth noting that this scheme is suitable for multi-beam generation instead of just two (of course, to generate more beams, an OPA with more elements is usually preferred, which is usually true for most multi-beam generation approaches).

Figure 4.Simulated and experimental intensity distributions of the dual-beam pointing at (a) −30° & 10° through uniform division, (b) −30° & 10° through non-uniform division, (c) −50° & −10° through uniform division, and (d) −50° & −10° through the non-uniform division scheme. Experimental intensity distributions of the dual-beam pointing at −50° & 50° through the non-uniform division scheme; the intensity ratios of the two beams are (e) 1.0, (f) 0.5, and (g) 0.3. All beam patterns are normalized to their peak intensities.

5. Discussion

According to the theory of wave optics, when the emission angles of two beams are relatively close, the influence of this interference term may cause severe beam deformation. By changing the overall phase of these beams, a destructive interference can be formed here, thereby reducing beam degradation. Through this method, we obtain a dual-beam pointing at $-3\xb0$ and 2° simultaneously, as shown in Figs. 5(a) and 5(b). Besides, we can even apply different phase modulations to different light beams to label them.

Figure 5.(a) Experimental far-field pattern of a dual beam pointing at −3° & 2° through uniform division. (b) The corresponding infrared image (a) when the camera is facing the chip. Simulated average similarity between the far fields with and without residual phase error, for the case of (c) single beam, (d) symmetric dual beams, and (e) asymmetric dual beams. The phase error added to the ideal value randomly varies within the range of [−Δϕ, Δϕ], and the maximum phase error Δϕ is represented as the horizontal axis. (Uni, uniform division; Non, non-uniform division.)

It is worth noting that there is a small discrepancy between the experimental and simulation results, which we think is due to the residual phase errors. Due to manufacturing errors, there may be accumulated random phase errors in the waveguide^{[25,26]}. For example, before calibration, the average phase error in this OPA to generate a 0° beam is about $0.8\pi $, and the standard deviation is about $0.6\pi $ (roughly inferred by the needed phase shift of each channel). Although most parts of the phase errors have been compensated during the phase calibration process in the phase data allocation, small residual phase errors could remain. The use of larger or more advanced optimization processes should further reduce residual errors, but complete elimination should still be difficult. Here we introduce residual phase errors of different amplitudes $\mathrm{\Delta}\varphi $ into the standard phase distribution and calculate the cosine similarity ρ between the newly generated far-field and the ideal one^{[27]}, which can be calculated as $\int {I}_{1}(\theta ){I}_{2}(\theta )\mathrm{d}\theta /\left(\sqrt{\int {I}_{1}^{2}(\theta )\mathrm{d}\theta}\sqrt{\int {I}_{2}^{2}(\theta )\mathrm{d}\theta}\right)$, where ${I}_{1}(\theta )$ and ${I}_{2}(\theta )$ are the angular intensity distributions with and without residual phase errors, respectively. A similarity value deviating more from unity indicates a larger difference between two beams, which also means a larger residual phase error. We have simulated the angular intensity distributions of a large ensemble of samples of OPAs with random residual phase errors. Figures 5(c)–5(e) show the average similarity between the actual and ideal distributions of a single/dual beam under the influence of various levels of phase errors. From this, we can see that, compared to dual beams, single beams have stronger robustness, which can maintain a similarity of approximately 0.9 under maximum phase error, but only about 0.8 remains for the dual-beam. Moreover, as the beam pointing angle increases, the sensitivity to phase error also increases.

Two ideal far-field distributions and their variation range under the influence of specific phase error are shown in Figs. 6(a) and 6(b). Obviously, there is a larger uncertain range of beam variation in the case of dual-beam generation even though its maximum phase error is smaller, which corroborates the previous conclusion that a single beam is more robust. This may be because, in the dual-beam situation, the number of elements involved in generating one beam is reduced by nearly half compared to a single beam, and such situations will become more obvious when generating more beams. Two samples of simulated dual-beams with phase errors are shown in Fig. 6(c), which illustrates the possibilities of how the beams may be affected. By comparing the similarity, we can roughly estimate that the standard deviation of the residual error in the far field of Fig. 3(f) is about $0.14\pi $. Furthermore, we have simulated the performance of OPAs with different numbers of elements $N$ for the case of single or dual-beam generation under random phase variation. The average similarity of the beam is plotted against $N$ for various levels of maximum phase errors in Figs. 6(d) and 6(e). The results show that the stability of their generation is closely related to the number of OPA channels. For a given degree of phase error, the far-field distribution will gradually approach the ideal value as the number of elements increases. Such a trend is consistent with the single beam generation case^{[2]}. With more elements embedded, the independent multi-beamforming will be more robust against phase error, and the widened beam will be compressed. Meanwhile, the improved full-field scanning speed and multi-target tracking ability are preserved. Therefore, the element division method can greatly increase the applicability range of an OPA, which can switch back to single beam operation mode with high resolution whenever needed, and it also has scalability. Besides, this method also has good scalability and is suitable for OPAs with various number of elements.

Figure 6.The variation range of OPAs’ far-field distribution with phase error and the corresponding ideal far-field distribution, for the case of (a) single beam pointing at 0° and (b) dual beams pointing at −30° and 30°. (c) Far-field samples with phase error influence (maximum phase error is 0.15π) are shown in the case of dual beams pointing at −30° and 30°. The legends show the maximum range of phase error variation, and the intensity value is normalized to the peak value of the ideal beam. The stability of the far-field distribution of the OPA under residual phase error with different total numbers of elements N, for the case of (d) single beam pointing at 0° and (e) dual beams pointing at −30° and 30°.

Here, we compared several conceptual works on generating multiple beams based on OPAs, which are shown in Table 1. The method proposed in this paper does not rely on special chip designs and, therefore, can be used more widely and readily. Moreover, independent beam control allows for handling multiple objects moving independently. Moreover, the dependency between the beams is very low, allowing for flexible modulation. Besides, compared to the integration of multiple individual OPAs (assuming the multiple OPAs have the number of elements summed up equal to the number of elements in this approach), this approach provides the flexibility of adjusting the number of elements in each subarray.

Ref. | Method | Special Chip Design | Beam Dependence |
---|---|---|---|

[ | Butler Matrix | Yes | Yes |

[ | Special Grating | Yes | Yes |

[ | Multiple OPAs | Yes | No |

Our work | Subarray Division | No | No |

Table 1. Comparison of Several Multi-Beam Schemes

6. Conclusion

We have generated dual beams that can be independently controlled with the OPA system, and the maximum scanning range reaches 100°. Phase data for the dual-beam generation can be built up from the single-beam generations through multiple steps of phase calibration. The difference in light intensity between the two beams caused by the envelope effects can be compensated by introducing a non-uniform division scheme. Through the same approach, the flexible adjustment of the intensity ratio between the two beams has also been achieved. In addition, we also investigate the impact of phase error during the beam generation process. Simulations of large ensembles of OPAs with various levels of random residual phase errors have been conducted to help understand the results. Under the same level of residual phase error control, increasing the number of elements should effectively improve the quality of beam generation, although in this work the available number of elements is limited due to fabrication/testing costs. Such a multi-beam approach is also applicable to OPAs using grating emission, not limited to waveguide end-fire emission as shown in this paper. It also should be noted that the method proposed here is not limited to the generation of dual beams, but it is generally applicable in multi-beam scenarios. This work may pave the way for handling multiple independently moving objects in solid-state LiDAR systems and optical wireless communications.

References

[14] M. Nakai, I. Takai, S. Chung *et al*. “Beam forming in integrated optical phased array enabled with deep neural network**. Conference on Lasers and Electro-Optics (CLEO), 2(2022)**.

[20] R. Mailloux**. Phased Array Antenna Handbook(2017)**.

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