Effect of Beam-Steering Angle, Operation Wavelength and Mean Inter-Element Distance on the Side-Lobe Levels of Integrated Optical Phased Arrays under Beam-Steering Operation

: Integrated optical phased arrays (OPAs) play an important role in a broad range of applications. Fabrication constraints, however, pose a limit to the minimum inter-element separation that further results in high-intensity side lobes. The intensity of these secondary lobes can be reduced by arranging the antenna elements with non-uniform separation distance, which has been addressed by different methods. In this paper we employ one of the already proven optimization algorithms, i.e., differential evolution, to optimize the element positions of linear arrays with different conﬁgurations operating under beam-steering operation and considering a minimum inter-element distance. These optimizations allowed us to derive some design guidelines that can assist in reducing the side-lobe level (SLL) of integrated linear OPAs. In particular, we found that it is necessary to optimize the positions for the broadest beam-steering angle and the shortest operation wavelength. Additionally, optimizations of different conﬁgurations reveal that, when imposing a minimum inter-element distance, there is an optimum mean distance that minimizes the SLL of the array.


Introduction
Photonic antennas have attracted increasing interest in recent years due to their critical role in areas as diverse as spectroscopy [1], optical communications [2] and light detection and ranging (LIDAR) [3].The first developed antennas operating in the range of optical frequencies exploited the plasmonic resonance of metallic surfaces [4].Such antennas can present small dimensions but they typically show low efficiency and pose serious feeding challenges.In order to overcome these issues, all-dielectric photonic antennas were proposed [5].In addition to high efficiency, all-dielectric antennas can be integrated not only with the feeding waveguide but also with other integrated devices such as phaseshifters and modulators, as well as with the associated control and signal processing electronics [6].However, these advantages can only be achieved at the expense of a significant size increase.This issue is especially critical in the design and implementation of optical phased arrays (OPAs), where multiple coherently-fed elements are used to allow the tailoring of the radiation pattern and its dynamical control.OPAs are particularly useful in applications where some kind of scanning or alignment is needed, for instance in re-configurable free-space optical cross-connects [7] and in electrically steerable LIDARs [8].
OPAs can be roughly classified into linear and planar arrays, each of them having its applications and design particularities.In particular, linear arrays typically present lower cost and smaller occupation area and, therefore, are preferable in applications where the beam steering should be performed in mainly one direction.Additionally, linear phase arrays can be designed to radiate either in the direction of the substrate plane or in a direction not contained in this plane as can be seen in Figure 1: (a) in-plane linear OPA and (b) out-of-plane linear OPA.The large antenna size of all-silicon antennas then limits the minimum separation in OPAs.In addition, the minimum inter-element separation is further increased when the OPA is implemented on integrated photonics, e.g., silicon on insulator, where the antenna elements are typically built in the same layer as the feeding network and phase and amplitude control elements [9].In this case, if antenna elements are uniformly distributed, that is, if the separation between elements is the same, the radiated fields do not interfere constructively only in the desired direction but also in other directions, leading to highintensity lobes that are denominated grating lobes.The intensity and number of these grating lobes depend on the inter-element separation and the beam-steering angle (the angle in which the main lobe is directed) and they should be minimized, as they can cause crosstalk, security faults and degrade the signal-to-noise ratio [10,11].In order to minimize the intensity of these lobes, in [12], the authors employed an array of waveguides with tailored widths specially designed to minimize the coupling; thus, allowing the reduction of inter-element separation.The most adopted solution, however, is to arrange the elements in a non-uniform configuration; that is, with different inter-element separations.
Arranging the elements non-uniformly turns the antenna design into a combinatory mathematical optimization problem that has been extensively studied in RF applications, where sparse arrays are attractive because of their good trade-off between cost and gain and the reduced near-field coupling between elements.Thus, a plethora of optimization algorithms has been proposed, both deterministic and heuristic.Among the employed deterministic algorithms, we can mention the discretization of the radiation integral [13,14], tapering using geometrical progression [10] and the optimization via cyclic difference sets [15].Nevertheless, heuristic methods have shown better performance and are more widely adopted.For instance, in [16], simulated annealing is proposed, whereas in [17] and in [18], particle swarm is employed.Evolutionary computing has also been extensively studied.In [11,[19][20][21][22][23][24], the optimization is performed using genetic algorithms and in [25][26][27], different configurations of differential evolution are employed.However, most of these works are focused on optimizing the position of the elements without analyzing the effect of structural (number of elements and mean and minimum inter-element distances) and operational parameters (operating wavelength and beam-steering direction).For instance, only [10,11,19,20,24] consider emission in out-of-broadside angles and only [11,14,16,19,20,25,27] include a minimum inter-element separation as an optimization constrain.This means that just [11] considers simultaneously beam-steering operation and minimum inter-element spacing.However, the interdependence among the different structural and operational parameters is still not clear.
In this paper, we do not intend to propose or compare optimization algorithms to find the optimum antenna position, but to analyze the effect of the beam-steering angle and the minimum separation distance on the side-lobe level (SLL) in terms of the number of elements and the average inter-element separation distance, which allows us to derive some design guidelines.In particular, we show that it is enough to optimize the positions of elements for the maximum steering angle to ensure that the SLL is lower for the complete beam-steering range.A similar behavior is observed in relation to the operating wavelength.Since the SLL reduces as the operation wavelength increases, it is necessary to optimize the positions for the shortest wavelength of the operation band.In addition, the optimizations for different array configurations reveal that when setting a minimum inter-element distance, there is an optimum average separation that minimizes the SLL of the OPA.Finally, we discuss the possibility of adopting sectorized antennas to reduce the SLL of the complete OPA, as this reduces the maximum steering angle of each sector.

Optimization of Element Positions
Assuming that the mean inter-element distance is fixed and that the separation cannot be smaller than d min , the search for optimum positions of N elements has N − 1 degrees of freedom.The problem can be mathematically formulated as: where SLL(θ BS , X) is the SLL of the array with elements located at positions x k and the main lobe is directed towards θ BS .If the coupling between the antenna elements is weak, the field radiated by an array of N identical elements located at positions z n along the z axis can be approximated by: where E el (θ, φ) is the electric field radiated by the radiating element centered at the origin, k is the wave number, θ is the azimuthal angle and φ n is the phase of the feeding of the n-th element.The total radiated field can be alternatively expressed as the product of the field of the radiating elements and a term that accounts for the element positions and their phases, denominated the array factor (AF): For a linear array, the AF only depends on θ, the set of element positions X and the beam-steering angle that sets the phases φ i .Thus, considering an array of low-directivity radiating elements, the maximum value of the norm of the AF equals the number of elements; because, in this direction, all of the elements interfere constructively.Regarding the SLL, it can be obtained from the AF as: AF(θ BS , X, θ BS ) and AF(θ sec , X, θ BS ) are the values of the array factor at the desired radiation direction and at the direction of the strongest side lobe [28].The minimization has a twofold constraint: on the one hand, the separation between the adjacent elements d k = x k − x k−1 has to be larger than d min .On the other hand, the total length of the array is set to L, which can be alternatively written in terms of the mean distance d as L = (N − 1) • d.The adopted angle criteria and the nomenclature are shown in Figure 2, alongside a block diagram of a non-uniformly distributed array antenna and a graphical representation of phases of the feeding.
x  Even for a relatively low number of elements, the SLL minimization cannot be addressed by brute force.In order to find sub-optimal but acceptable solutions, different heuristic methods have been proposed.Among the different algorithms used to optimize the element positions in non-uniform sparse linear arrays [26], differential evolution (DE) has demonstrated a good trade-off between complexity and performance, which has led many authors to use this heuristic algorithm [29][30][31].DE emulates the natural evolution of species by improving the quality of successive generations [32].Within each iteration, DE selects the solutions (individuals) out of a set of tentative solutions that present the best performance in terms of the metric to be optimized, denominated as cost function.These individuals are combined (crossed-over) and randomly modified (mutated) to generate a new improved generation of solutions.Once the best tentative solution attains an acceptable value or no further improvement is achieved, the algorithm execution stops.
The results presented in Section 3 were obtained using a DE algorithm with a crossover probability of 0.1 and a scaling factor ranging from 0.2 to 0.8 [33].The constraints in (1) were introduced by means of a normalization of the newly generated solutions.To dimension the size of the population and the number of iterations, we conveyed initial optimizations showing that an acceptable trade-off between complexity and performance is achieved by employing a population size of 40 individuals, 100 random initial populations and a number of iterations that depends on the number of antenna elements: 500 for 4 elements, 1000 for 8 elements and 3000 for 16 elements.The convergence of the algorithm can be observed in Figure 3, where each trace corresponds to the metric (in our case, the SLL) of the best individual of the evolution of an independent initial population.Figure 3a is for 4-element arrays whereas Figure 3b and Figure 3c are for 8 and 16 elements respectively.Looking at the traces, it can be concluded that some initial populations lead to poor solutions, so by choosing independent random populations, the sensitivity to initial conditions is mitigated.Comparing the traces for different numbers of antenna elements, the populations of arrays with a lower number of elements converge more rapidly.For a larger number of elements, DE requires more iterations and the final configurations present a higher variance.Both observations can be justified by the higher dimension of the solution space of higher element solutions.

Results
The SLL is influenced by a great number of parameters, of which the most important are the beam-steering angle, the operation wavelength, the inter-element distance and the number of elements.In order to analyze their effect and interplay, we divide this section into three subsections.

Effect of Beam-Steering Angle and Operation Wavelength on the Side-Lobe Level
First, we analyzed the effect of beam steering and the target steering range on the SLL value.In Figure 4a we show the positions of 8-element arrays optimized for three different steering ranges: 0 • , 22.5 • and 45 • , setting the minimum distance between elements to 2λ and a mean separation of 6λ.It can be noted that the element positions depend on the target steering angle.The SLL of the arrays designed for the three steering ranges are shown in Figure 4b.As can be observed, at 0 • , the configuration optimized for a range of 0 • (no beam steering) outperforms arrays optimized for steering ranges of 22.5 • and 45 • .However, for beam-steering angles between 0 • and 22.5 • , the array optimized for a range of 22.5 • presents the best performance.Similar behavior can be observed for the beam-steering angles up to 45 • when beam-steering angles between 22.5 • and 45 • are considered.It is important to note that, as envisaged, the SLL worsens as the beam-steering angle is increased.Thus, optimizing for a whole range is equivalent to optimizing for the maximum beam steering; that is, for the extreme value of the target steering range.In fact, the SLL presents a monotonically increasing stair-case shape, which can be understood by observing the evolution of the radiation diagram as the beam-steering angle increases, as shown in the contour plots in Figure 4b,c.It can be seen that, as the beam-steering angle increases, new side lobes enter the visible region and, consequently, the SLL increases.Furthermore, the transition to higher SLL depends on the beam width of the entering high-intensity lobe.For antennas with a length of several wavelengths, the secondary lobes present a narrow beam width and the transition between low and high SLLs is relatively abrupt.Therefore, optimizing for large beam-steering ranges prevents the appearance of high-power lobes when the beam is steered within this range.However, this is achieved at the expense of the presence of minor lobes that results in a performance penalty for narrower beam-steering angles.We also analyzed the effect of the operation wavelength on the SLL.In Figure 5, we show the SLL in terms of the wavelength for arrays of (a) 4, (b) 8 and (c) 16 elements optimized for beam-steering angles of 0, 22.5 • and 45 • .In all cases, the array element positions were optimized for an arbitrary wavelength and then, keeping the positions of the elements constant, the SLL was computed for a wavelength ranging from 0.5 to 1.5 times the design wavelength.As can be seen, when modifying the operation wavelength, the SLL also presents a step-like behavior.In this case, the longer the wavelength is, the lower the SLL.This characteristic shape can be explained using an argument similar to that in Figure 4, but when sweeping the wavelength, instead of a shift of the visible region, the visible region is expanded or compressed.Thus, for a shorter wavelength, the visible region is expanded and some high-intensity lobes that were outside the visible region now enter the visible region.We can then conclude that in terms of the wavelength, the worst-case scenario corresponds to the shortest wavelength.

Effect of the Mean Inter-Element Distance and Beam-Steering Angle on the Side-Lobe Level
One of the most important structural array parameters that affect the SLL of an array is its size, which for the 1D case, is the total length of the array, L. Recalling that the total length can be expressed in terms of the mean inter-element distance, d and the number of elements, N, according to L = N • d, in order to explicitly observe the effect of the mean inter-element distance, d was swept.To observe the dependence of the SLL on the mean inter-element distance, in Figure 6, we show the modulus of the AF for three different inter-element values.In addition, to see the effect of the beam-steering angle, the AF for a beam-steering angle of 0 • , 22.5 • and 45.0 • are shown in Figure 6a, Figure 6b and Figure 6c, respectively.As can be seen, independently of the beam-steering angle, both short and long inter-element distances, i.e., 1.1 λ and 5 λ, result in higher values of SLL, whereas an intermediate value of 1.5 λ leads to a better performance in terms of the SLL.An additonal point to note is that, as expected from the results in Section 3.1, the SLL increases as the beam is directed away from the broadside direction.

Effect of the Number of Elements and the Average and Minimum Separation on the Side-Lobe Level
Afterward, we proceeded to quantify the SLL in terms of the number of elements as well as the minimum and mean inter-element distance.We considered arrays with 4, 8 and 16 elements and minimum separations of 1 λ, 2 λ and 3 λ.Due to the large number of interacting parameters, in Figure 7 we show different contour plots of the SLL in terms of beam-steering range and mean inter-element distance.Each row corresponds to a number of elements: (a-c) are for 4 radiating elements, (d-f) are for 8 and (g-i) are for 16, whereas each column represents a minimum inter-element distance: (a,d,g) 1 λ, (b,e,h) 2 λ and (c,f,i) 3 λ.Comparing the SLL for different numbers of elements and keeping fixed the minimum inter-element separation, we can see that the larger the element count is, the lower the SLL.On the other hand, if we fixed the number of elements and increase the minimum inter-element distance, a SLL degradation is observed.These results can be interpreted in terms of the trade-off between design freedom and the inherent appearance of side lobes.The degree of freedom can be increased by increasing either the number of radiating elements or the ratio between the mean distance to the minimum distance.A similar argumentation can be applied to explain why for each configuration, there is a value of mean distance that minimizes the SLL.Thus, if the mean distance is very close to the minimum distance, the design freedom is small and poor SLL is obtained.If the mean separation is too large, the larger freedom in the design cannot compensate for the effect of large element separation.The presence of an optimum mean distance indicates that there is an optimum array length in order to minimize the SLL.This behavior only occurs when physical constraints are introduced and, to our best knowledge, has not been previously reported.The optimum length of a sparse array is an important design parameter, so we analyzed it in more detail.Its dependency on the number of elements, minimum inter-element distance and scanning range is shown in Figure 8a.This optimum mean distance increases with the minimum inter-element separation, whereas it is almost independent of the number of elements and the beam-steering range.Figure 8b shows the SLL values at optimum inter-element spacing for different numbers of elements and minimum separations, revealing that generally speaking, at optimum distance the SLL penalty due to extending the beam-steering range is relatively small.However, for the particular case of a 1 λ separation distance, a significant deterioration of the SLL can be observed for no-null beam-steering ranges.These results have a deep consequence in the design of sectorized arrays.If the initial observation of the dependency of SLL on the beam-steering angle could lead to thinking that sectorization may help in reducing the SLL of the array, when an optimum mean separation is adopted, a marginal SLL reduction can be expected, at least until a four-fold sectorization.

Figure 1 .
Figure 1.Examples of integrated linear OPAs: (a) in-plane emission and (b) out-of-plane emission.

Figure 2 .
Figure 2. Schematic of the linear sparse antenna alongside an illustrative polar and Cartesian representation of the AF indicating the employed nomenclature.A graphical representation of the relation between the phase of the feeding and the element positions is also shown.

Figure 3 .
Figure 3. Evolution of the SLL of the best individual in terms of the iteration for (a) 4-element, (b) 8-element and (c) 16-element antenna.In each subfigure, the different traces correspond to independent random initial populations.

Figure 4 .
Figure 4. (a) Optimized element positions for steering ranges of 0 • , 22.5 • and 45 • for an 8-element array with a minimum separation of 2 λ and a mean distance of 6 λ.(b) SLL in terms of the beam-steering angle for the three beam-steering ranges.(c-e) Modulus of the AF in terms of the beam-steering angle and azimuthal angle θ for the three optimized cases.

-Figure 5 .
Figure 5. SLL in terms of the operation wavelength for (a) 4 elements, (b) 8 elements and (c) 16 elements.In each case, traces for maximum beam-steering angles of 0 • , 22.5 • and 45 • are presented.

Figure 7 .
Figure 7. SLL in terms of the mean inter-element distance and beam-steering range for different number of radiating elements and minimum inter-element distance.(a-c) 4 elements with minimum distances of 1 λ, 2 λ and 3 λ, (d-f) 8 elements with minimum distances of 1 λ, 2 λ and 3 λ and (g-i) 16 elements with 1 λ, 2 λ and 3 λ.