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Abstract
We consider the impact of intra-wafer systematic spatial variation, pattern density mismatch, and line edge roughness on splitter-tree-based integrated optical phased arrays. These variations can substantially affect the emitted beam profile in the array dimension. We study the effect on different architecture parameters, and the analysis is shown to be consistent with experimental results.
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1. Introduction
Several studies [1–8] in recent years have explored manufacturing process variations and their impact on integrated photonics, for the goal of achieving high yield and performance of photonic circuits. For example, Melati et al. [1] present a thorough theoretical and experimental study of how roughness affects the performance of photonic waveguides, e.g., loss and back reflection; Chrostowski et al. [2] study the effect of spatial variation on a Mach-Zehnder interferometer (MZI) and a dual phase shifted waveguide Bragg grating, as illustrations of the methodology proposed for photonic circuit variation analysis; El-Henawy et al. [3] study the effect of line edge roughness on a splitter; Xing et al. [5–7] use MZIs as test structures to study and evaluate the spatial variations and pattern density effect from inverse measurement. However, while a relatively wide range of process variation types have been covered, most of the studies are still limited to single photonic components (e.g., waveguides, splitters, resonators) or small-scale simple photonic circuits (e.g., MZIs, coupled ring-resonator optical waveguides [8]).
There is a benefit to exploring process variations on system-level performance. One impactful system is the integrated optical phased array (OPA), an emerging class of photonic systems that enable manipulation and dynamic control of free-space light in a compact form factor, at low cost, and in a non-mechanical way [9–16]. However, being large-scale photonic circuits spanning across millimeters and consisting of hundreds or thousands of device components, these integrated phased arrays can be highly susceptible to various types of process variations. It is desirable to study the impact of process variations on phased arrays so that we can obtain more robust designs that achieve high performance given actual manufacturing limitations.
In this work, we focus on a splitter-tree-based optical phased array [11,13,16] and study three types of process variations that can potentially degrade the performance of a phased array. More specifically, we explore the effects of intra-wafer systematic spatial variation, pattern density mismatch, and line edge roughness, as well as how OPA architecture parameters affect the impact of these three process variations. We identify the behaviors and mechanisms by which these spatial variations affect the shape of the radiation pattern emitted from the phased array. Furthermore, we validate these simulation results with experimental characterization of fabricated splitter-tree-based OPAs.
2. Splitter-tree architecture and sources of variation
As shown in Fig. 1(a), a splitter-tree-based phased array consists of a splitter-tree structure that distributes the input power equally to each antenna row, phase modulators that control the beam steering, and antenna rows with designed periodic perturbations [11,13,16]. Furthermore, with this architecture, beam steering in the antenna dimension can be achieved via wavelength tuning [11,12]. Here, we focus on the case with no active beam steering and, hence, ignore the effect from phase modulators. It is important to understand the impact of process variations before introducing modulation and beam steering, as this will serve as the basis for more comprehensive studies involving active components in the future.
Fig. 1. (a) Simplified top-view schematic of a splitter-tree-based phased array with two splitter layers. The array ($y$-axis) and antenna ($x$-axis) dimensions are also shown in the figure. (b) Workflow to simulate the impact of process variations on OPAs.
The following simulation results are based on a silicon phased array with $L=9$ sequential splitter layers, 512 antennas, an antenna pitch of $P = 2\,\mathrm{\mu}\mathrm {m}$, and a design wavelength of 1550 nm. We also set the OPA aperture to be the same width in the array and antenna dimensions, with apodized gratings, so that the emitted beam pattern is the same along the two dimensions for the nominal case. We center the beam profile at the peak of the main lobe for better visualization.
In this paper, we present three types of process variations that can potentially have a significant impact on the emitted beam profile of the splitter-tree-based phased arrays: (i) intra-wafer systematic (IWS) spatial variation [7], (ii) spatial variation due to layout pattern density mismatch [6,17], and (iii) line edge roughness (LER) [18]. Furthermore, we present how OPA architecture parameters interact with these three process variations. We mainly consider the impact of these variations on the emitted beam profile around the main lobe of the far-field radiation pattern, in both the array and antenna dimensions. Most of the results focus on the change in side-lobe level (SLL), as it is the most significant behavior observed for all three sources of variation.
Figure 1(b) shows the workflow for simulating the impact of process variations on the emitted beam profile of OPAs. For the first step, we simulate the impact of process variations on the device components within OPAs, which can be broken into four components: phase variation in the waveguides of the splitter-tree structure, amplitude imbalance of the individual splitters, phase variation at the antennas, and amplitude variation along the antenna perturbations. These simulations are performed by adding geometric deviations based on the particular variation being considered to the nominal geometry, and carrying out numerical FDTD or eigenmode photonic simulations to evaluate the device-level output impact. However, in some special cases, an abstract device-level impact model is sufficient for analysis without intricate numerical photonic simulations. The second step is to combine these device-level impacts into the full OPA system and calculate the resulting emitted beam profile.
3. Effect of intra-wafer systematic spatial variation
Intra-wafer systematic (IWS) spatial variation, specifically the systematic variations in waveguide thickness or width across the wafer that arise from deposition and etching processes, can be modeled as a low-order polynomial function [7], $\zeta (x,y)=\sum _{i,j\geq 0}^{i+j\leq K}q_{i,j}x^i y^j$, where $\zeta (x,y)$ is the thickness or width variation at location $(x,y)$, $q_{i,j}$ is the polynomial coefficient, and the order $K$ is a small integer, e.g., $K=2$. Typically, this spatial function varies at a length scale ($\sim 100\,\mathrm {mm}$) that is much longer than the length scale of the OPA [5] ($\sim 1\,\mathrm {mm}$); therefore, the variation can be approximated as a linear function $\zeta (x,y)=g_x x + g_y y + \zeta _0$, where $g_x$ and $g_y$ are variables that represent the gradient of the geometric variation (thickness or width) in each of the two dimensions. Since IWS variation is systematic, $g_x$ and $g_y$ are not random, and only dependent on the location of the OPA on the wafer.
For the device-level impact analysis, all the splitters have the same amplitude imbalance. Our analysis results show that the impact from the splitters is insignificant even if the imbalance is large. Similarly, we can show that the amplitude variation at the antenna grating perturbations is also insignificant compared to the effect of phase variation, even if the variation is large at the device level. Furthermore, variation along the $x$-axis ($g_x$) has no significant impact compared to variation along the $y$-axis ($g_y$). Figure 2(b) shows the emitted beam pattern when applying $g_y=0.5\,\mathrm {nm/mm}$ to the entire phased-array system, where we observe an increase in the side-lobe level (SLL) on one side of the main lobe in the array dimension (along the $y$-axis in Fig. 1(a)).
Fig. 2. (a) Summary of parameters used in the simulations of IWS impact. (b) The emitted radiation pattern in the array dimension (blue) and antenna dimension (orange) when applying a waveguide width variation spatial gradient of 0.5 nm/mm along the $y$-axis. (c) The emitted radiation pattern in the array dimension when applying variation to only the splitter tree (blue) versus only the antennas (orange).
To determine which part of the OPA is most susceptible to IWS variations, we separate the effect due to variations in the splitter-tree section versus variations along the antennas by applying the spatial variation only to certain parts of the phased-array system. As shown in Fig. 2(c), the variation in the splitter-tree structure dominates the impact on the beam shape along the array dimension.
Since we identified that the dominant sensitivity is due to phase change in the splitter-tree structure, we can describe the effect of the spatial variation by the phase variations $p_i$ at the beginning of each antenna row $i$ (shown in Fig. 1(a)). We find that $p_i = g_yf_i + C_1i+C_0$, where the phase variation pattern $f_i$ depends on the design of the splitter tree, the type of variation (thickness or width), and the material. The additional linear term $C_1i+C_0$ only changes the location of the lobes; thus the pattern $f_i$ determines the effect on the beam shape.
We show in Fig. 3(a) the phase variation pattern $f_i$ for thickness and width variation on the nominal phased array. Notably, the pattern has the same behavior for thickness and width variation, with a difference of approximately just a scaling factor. This is because the pattern is dominated by the phase change of the routing waveguides with identical specifications, and different types of geometric variation all affect the phase by perturbing the effective index of the waveguide mode.
Fig. 3. (a) The phase variation pattern $f_i$ of the thickness (solid blue) and width (dashed orange, scaled) variation on the nominal phased array. (b) Change in SLL as a function of thickness gradient $g_y$.
Given this behavior, it is sufficient to only focus on the analysis of one type of variation in the following discussion of spatial variations, since the other type will only differ by a scaling factor; hence we focus on thickness variation. Figure 3(b) shows the relationship between $|g_y|$ of thickness variation and increase in SLL; this can be approximated as a linear model for small values of $|g_y|$: $\Delta _\mathrm {SLL} = \alpha |g_y|$, where $\alpha$ is a coefficient determined by the pattern $f_i$.
In summary, IWS spatial variation will increase one side of the SLL along the array dimension of the emitted radiation pattern of OPAs, primarily by affecting the phase in the splitter-tree structure. We also introduce the phase variation pattern $f_i$ for the study of this type of variation.
4. Effect of layout pattern density mismatch
Another form of spatial variation that can potentially affect the far-field radiation pattern is variation in layout pattern density. This can affect the etch rate during fabrication and is correlated with changes in the width and partial etch depths of waveguides [6]. Within the splitter-tree-based OPA, the splitter-tree structure is relatively sparse, hence the pattern density can be maintained via a grid of dummy fill shapes inserted in between the routing waveguides. On the other hand, the emitting aperture of the OPA is made up of densely packed antennas, and hence dummy fill shapes cannot be efficiently used without affecting the scattering behavior. This results in a layout pattern density mismatch between the OPA emitting aperture and the fill-shape grid, which may result in undesired variation in waveguide width.
Assuming the layout pattern density varies in a small range, we can use a linear model for the change in waveguide width: $\Delta w = k_w(\rho _\mathrm {eff} - \rho _0)$, where $k_w=\left.\frac {\partial w}{\partial \rho }\right |_{\rho _0}$ is the linear coefficient, $\rho _0$ is the pattern density of the fill-shape grid, and $\rho _\mathrm {eff}$ is the average pattern density when applying a Gaussian filter $G(x, y) = \frac {1}{2\pi \sigma }\exp \left (-\frac {x^2+y^2}{2\sigma ^2}\right )$ [6]. We can also assume that $\sigma$ is much greater than the antenna pitch $P$, but small compared to the length scale of the OPA, so that we can equivalently set the antenna area to have a constant pattern density $\rho _0 + \Delta \rho$, while ignoring the difference along each antenna row and only focusing on $\Delta w$ in the $y$ direction. Then we have
Similar to the IWS spatial-variation case, on the device level, phase variation is also the dominant factor in the layout pattern density effect. In Fig. 4(b) we show the beam shape when applying a pattern-density mismatch of $w_\mathrm {max}=0.5\,\mathrm {nm}$ (Here we pick this value as an example. In the actual case as shown in Section 6, our example of $P=2\,\mathrm{\mu}\mathrm {m}$ silicon OPAs have negligible pattern density mismatch) and ratio $r = D/\sigma = 5$, where we observe a symmetric increase in SLL on both sides of the main lobe in the array dimension.
Fig. 4. (a) Summary of parameters used in the simulations of pattern density impact. (b) The emitted radiation pattern in the array dimension (blue) and antenna dimension (orange) when applying pattern density effect with $w_\mathrm {max}=0.5\,\mathrm {nm},\,r=5$. The functions (c) $f_w(w_\mathrm {max})$ and (d) $f_r(r)$ in the separable model of pattern density effect.
The form of $\Delta w(y)$ indicates that a change in SLL is controlled by two parameters, $w_\mathrm {max}$ and ratio $r=D/\sigma$, and it must be a symmetric function of $w_\mathrm {max}$. For small $w_\mathrm {max}$, we find that the response is approximately separable: $\Delta _\mathrm {SLL}(w_\mathrm {max}, r)=f_r(r)f_w(w_\mathrm {max})$. This separable model has a maximal error of 6% at $w_\mathrm {max}=1\,\mathrm {nm}$ compared to the complete form. For larger $w_\mathrm {max}$, the error increases exponentially, but the separable model still maintains a similar trend as in the complete form.
We show the estimated $f_r$ and $f_w$ in Fig. 4(c) and 4(d), where we normalize $f_r$ to be a scalar function with a maximal value of 1. Here $f_w$ is a monotonous increasing function for $w_\mathrm {max}>0$, which means that SLL increases as $|w_\mathrm {max}|$ increases. The function $f_r$ reaches the largest impact at around $r\approx 4.5$.
In summary, pattern density mismatch will cause a symmetric increase in SLL along the array dimension of the emitted radiation pattern of OPAs, primarily by affecting the phase in the antennas. We also build a separable model for the impact of the two parameters, $w_\mathrm {max}$ and $r$.
5. Effect of line edge roughness
A third type of process variation that can affect the performance of OPAs is line edge roughness (LER). LER is a limitation of photolithography processes related to non-ideal light interactions with photoresists. It introduces random perturbations on the sidewall of the waveguides or other structures, which can introduce phase variation, back reflection, and scattering loss in waveguides [1] and imbalanced transmission power of splitters [3]. The geometric perturbations can be modelled as zero-mean spatially correlated random Gaussian noise with covariance
where $\xi (x,y)$ is the perturbation in the normal direction at the layout boundary location $(x,y)$, and amplitude $A$ and correlation length $L_c$ are the parameters that control the magnitude and length scale of the Gaussian noise.For the device-level analysis, the length scales of components ($\sim 10\,\mathrm{\mu}\mathrm {m}$) are generally much larger than the correlation length of LER ($\sim 10\,\mathrm {nm}$), so we model the impact to be independent for each device within the OPA. Based on this independent model, simulations show that only the transmission imbalance of the splitters cause a significant impact on the emitted beam profile, where we model the transmission imbalance $\delta$ of each splitter as independent and identically distributed (i.i.d.) random normal variables with variance $s^2$: $\delta \sim \mathcal {N}(0, s^2)$. Since the variation at the splitter-tree structure only affects the array dimension, we ignore the antenna dimension in the visualization.
We show the emitted radiation pattern of two example instantiations of a phased array with transmission imbalance of $s=0.1$ in Fig. 5(b). The random variation can cause the SLL to increase or decrease, but it is always symmetric, as for all of the transmission variations. We also separate the effect of each layer by applying only the transmission imbalance on the splitters at a certain layer. As shown in Fig. 5(c), the variations at the splitters of the second and third layers contribute over 97% of the variance in the SLL, while variations after the fifth layer have almost no impact on the emitted beam shape.
Fig. 5. (a) Summary of parameters used in the simulations of LER impact. (b) Emitted beam shape along the array dimension when applying LER transmission imbalance of $s=0.1$. (c) Percentage of variance of SLL when applying LER to only splitters at a specific layer. (d) Mean (solid blue) and standard deviation (dashed orange) of the SLL as functions of transmission imbalance standard deviation $s$.
In a previous study [4] we showed that by using a linear sensitivity model from the adjoint method, we have the variance of imbalance $s^2=\beta A^2L_c$ for small $L_c$, where $\beta$ is determined by the splitter design and can be estimated from the adjoint analysis. We also validated this relationship by comparing the adjoint prediction to the actual simulated random instantiations of a splitter with LER. Figure 5(d) shows the mean and standard deviation of $\Delta _{SLL}$ for different $s$ calculated from 20000 random samples. For small $s$, we can empirically fit the model as $\Delta _{SLL}\sim \mathcal {N}(q_0s^2+q_1s^4, q_2^2s^2)$, where $q_0 = 0.31\,\mathrm {dB},\, q_1=1.42\,\mathrm {dB},\, q_2=4.45\,\mathrm {dB}$.
In summary, LER can cause a random symmetric change in SLL along the array dimension of the emitted radiation pattern of OPAs, primarily by introducing random amplitude imbalance to the splitters of the second and third layers. We also show how the distribution of the SLL changes with respect to the LER parameters $A$ and $L_c$.
6. Effect of architecture parameters
In this section, we explore the effect of the OPA architecture parameters, e.g., number of layers $L$, antenna pitch $P$, antenna row length $l$, etc. We also briefly discuss the impact of waveguiding material, specifically two commonly used materials, silicon and silicon nitride.
For the case of intra-wafer systematic spatial variations, the phase variation pattern $f_i$ shown in Fig. 3(a) is dominated by the sharp jump at the center, which is the phase difference between the middle two antenna rows. This phase jump is also the dominant factor for the increase in SLL. Intuitively, splitter trees with more layers (and hence longer paths) or a larger antenna pitch (and hence greater difference in variation) will increase this phase jump, and thus result in a larger impact on SLL. This is validated by the simulation results of various phased array designs with different numbers of splitter layers ($L$) and antenna pitches ($P$), as shown in Fig. 6(a). Since SLL is mostly determined by the variation at the splitter-tree structure, the design of the antenna rows (length and shape) does not have a significant effect. Silicon-nitride waveguides are less sensitive to variation compared to silicon waveguides; therefore, a silicon-nitride OPA exhibits less impact on SLL.
Fig. 6. (a) The predicted $\alpha$ for thickness variation of different splitter-tree architectures, estimated by applying a gradient of $g_y=1\,\mathrm {nm/mm}$. (b) Experimentally measured data of fabricated OPAs with various architecture properties showing the emitted beam intensity along the array dimension. The first side lobes on each side of the main lobes are noted with arrows of corresponding colors.
For the case of layout pattern density mismatch, the impact is dominated by the value of $w_\mathrm {max}$, which is heavily dependent on the antenna pitch $P$ and the antenna waveguide width $w_\mathrm {ant}$ (both of which depend on the waveguiding material), since $\Delta \rho =w_\mathrm {ant}/P - \rho _0$. In our case, silicon OPAs with $P=2\,\mathrm{\mu}\mathrm {m}$ have almost no density mismatch, and therefore are not susceptible to the pattern density effect, whereas silicon-nitride OPAs with $P=2\,\mathrm{\mu}\mathrm {m}$ have $\Delta \rho = 0.45$. Silicon and silicon-nitride OPAs with $P=4\,\mathrm{\mu}\mathrm {m}$ have $\Delta \rho = -0.15$ and $0.075$, respectively. If we use the coefficient $k_w=33.3$ in the model of pattern density and waveguide width relationship for both materials, as estimated in [6], this will give $w_\mathrm {max}=-15\,\mathrm {nm}$, $-5\,\mathrm {nm}$, and $2.5\,\mathrm {nm}$, for the latter three cases respectively. However, this could be an overestimation, as the model from [6] is from data in a much narrower layout pattern density range.
On the other hand, for OPAs with the same pattern density mismatch, we need to consider the effect of the ratio $r=D/\sigma$. If we use the estimates $\sigma =69\,\mathrm{\mu}\mathrm {m}$ from [6], the $L=7,\,P=2\,\mathrm{\mu}\mathrm {m}$ (or $L=6,\,P=4\,\mathrm{\mu}\mathrm {m}$) phased array will have $r=2^LP/\sigma =3.7$ which is close to the maximum impact shown in Fig. 4(d). A phased array with a larger $L$ or $P$ will have a larger $r$, and therefore will be less vulnerable to the layout pattern density effect.
It should also be noted that the separable layout pattern density effect model in Section 4 does not take into account the changes in antenna row design. Instead of the maximum possible width difference $w_\mathrm {max}$, the complete model will depend on the maximum average phase difference $p_\mathrm {max}$, which is proportional to $w_\mathrm {max}$ and $l$. Besides these two parameters, $p_\mathrm {max}$ is also relatively smaller for silicon-nitride antennas or antennas with uniform perturbations, as the former is less sensitive to width variation and the latter has a smaller effective antenna length since more light is scattered from the beginning of the row.
For the case of line edge roughness, since the model assumes that the variations at each splitter are not spatially correlated, the number of layers $L$ is the only architecture parameter that can possibly affect the emitted beam profile. However, simulations show that there is no significant difference in the results for varying $L\geq 6$. This is because the impact distribution in Fig. 5(c) is true even when varying $L$, so adding more layers beyond $L=6$ has negligible impact on the SLL. Similar to the cases of other variations, silicon-nitride splitters are generally more robust to LER, and thus have a smaller $\beta$.
7. Validation through experimental results
Next, we validate our theoretical findings by experimentally measuring the output far-field beam profiles for multiple splitter-tree-based integrated optical phased arrays. The phased arrays were fabricated in a CMOS-compatible 300-mm wafer-scale silicon-photonics process at the State University of New York Polytechnic Institute’s (SUNY Poly) Albany NanoTech Complex [13,16].
Since the spatial variations depend on the location on the wafer, and LER is random, we do not have a sufficiently large sample size to do a thorough analysis. However, we are still able to examine the trend suggested by our theory in the experimental characterization. We show a few emitted far-field beam profiles for OPAs with different numbers of layers and antenna pitches in Fig. 6(b). As expected, the emitted patterns have generally asymmetric side lobes and an increase in SLL with respect to the nominal simulated case.
To separate the effect of IWS spatial variation from other possible sources of impact, we can look at the difference in SLL at the two sides of the main lobe, as it is the only variation of the three that causes an asymmetric profile. Notably, this difference in SLL with respect to layer number and antenna pitch roughly follows the same trend as shown in our variation simulations. The only large deviation from our theoretical prediction is the $L=8$, $P=4\,\mathrm{\mu}\mathrm {m}$ result (green line), where the side lobes on the two sides of the main lobe are almost at the same level. However, this is most likely only because the phased array for this sample happens to be located where $g_y$ is very small.
Additionally, we can examine the other two variations, namely layout pattern density effect and LER, by looking at the lower SLL (LSLL) between the two sides of the main lobe. Figure 6(b) shows that OPAs with $P=4\,\mathrm{\mu}\mathrm {m}$ have larger LSLL compared to OPAs with $P=2\,\mathrm{\mu}\mathrm {m}$, which is consistent with our analysis for the pattern density effect. Notably, between the two $P=4\,\mathrm{\mu}\mathrm {m}$ arrays, the $L=8$ variant shows a greater LSLL because it has a longer antenna length.
We do not have a direct observation for the LER in our existing measurements, but a design with $L=7$, $P=2\,\mathrm{\mu}\mathrm {m}$ should be able to eliminate the effects of IWS spatial variation and pattern density effect, and serve as the validation sample for LER impact. Among the measurements shown in Fig. 6(b), the increase in LSLL of the $L=8$, $P=2\,\mathrm{\mu}\mathrm {m}$ sample (orange line) may demonstrate the contribution from LER, since spatial variation will cause a lower LSLL, while the OPAs with $P=2\,\mathrm{\mu}\mathrm {m}$ have negligible layout pattern density mismatch.
In summary, we observe the effect of IWS spatial variation and pattern density mismatch in the experimental results by observing the difference in SLL at the two sides of the main lobe and the LSLL, respectively, which is consistent with our simulated analysis. Due to the sample size of our fabricated OPAs, we are unable to directly observe the effect of LER or to separate each type of variation thoroughly, which could be a direction for future work.
8. Conclusions
We evaluate the impact of three process variations – intra-wafer systematic spatial waveguide geometric variations, layout pattern density mismatch, and line edge roughness – on splitter-tree-based integrated optical phased arrays, and show that all of these variations cause increased side-lobe levels along the array dimension. The process variations contribute to increased side-lobe levels through different mechanisms: IWS spatial variations primarily affect the phase distributions in the splitter tree, layout pattern density primarily affects the phase distributions in the antenna rows, and LER primarily affects the amplitude distributions of the second and third layers of splitters. We show how varying the number of splitter layers and antenna pitch affects the side-lobe suppression and validate the theoretical findings of the spatial variations and pattern density effect by observing a similar trend via experimental measurements. These results can be used as a guideline for phased array designs in the future, and elements of the analysis and model (e.g., the phase variation pattern $f_i$ for the spatial variations, the conclusions that over 80% of LER impact comes from the second layer splitters, etc.) can be useful for fast calibration of active phased arrays.
Future directions could consider an inverse model that is able to separate the different process variation effects from the measured emitted beam profile, for additional validation and application of the theoretical findings in this work. Future work could also include modeling and analysis of the impact of variations on phase modulators, thus expanding to the case with active beam steering.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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