A phasor is a scalar, complex value sufficient to describe the steady state of a mono-frequency sinusoidal waveform. In the context of an optical signal, this represents the electric field component of the monochromatic laser. A phasor term takes the form of: E i = E e j ϕ i (5) In Equation 5, E i is the phasor term for signal i , E is its real amplitude, and ϕ i represents the incoming phase seen by the subsequent optical component on that signal’s path. The transformation matrices of size n , representing the effect of any combination of optical components, apply directly to the vector composed of n incoming signals’ phasor. As an example, the incoming signal I n 1 and I n 2 to an MZI’s input ports will become O u t 1 and O u t 2 at the output ports, related by Equation 6: O u t 1 O u t 2 = D MZI I n 1 I n 2 (6)

Due to difficulties of controlling the absolute phase of optical signal (Ip et al., 2008), the incoming data (i.e., feature vectors of each data sample) will be solely represented by the intensity ( P ) of the signal, given by P i ∝ | E i | 2 , this implies that in an array of incoming signal represented as: E ⃗ = E 1 E 2 … E N = E in 1 e j ϕ 1 E in 2 e j ϕ 2 … E in N e j ϕ N (7) In Equation 7, only the E in i terms vary across different data, and each ϕ i takes the same value in the range [ 0,2 π ) for all i ∈ 1,2 , … , N − 1 . In theory, the zero phase difference ensures each MZI can split power to an arbitrary ratio between its two outputs ( P out1 P out2 ∈ [ 0 , ∞ ] ) . To achieve so, a layer of phase shifters is assumed to be present before the MZI mesh, albeit this is omitted in the diagram.

For the correct functioning of any trained network, not only the zero phase difference across input channels is required, but the absolute phase of each input signal should also remain constant throughout the network’s training and operation. As the MZI mesh works by the principle of interference, an incoherent or varying initial phase difference between signals will affect the intended splitting ratio learned from network training, making the resulting output signal array drastically different from the expectation.

During ONN inference, when light couples through the waveguide, the processor suffers from inherent propagation loss. The propagation loss eventually leads to a challenging optical power budget, limiting the signal-to-noise ratio at the photodetectors and reducing the classification accuracy of ONN. The linear loss values ( L linear ) , transformed from loss in dB-scale ( L dB ) by L linear = 1 0 − L dB / 10 , are applied to the ONNs and remain constant at a per MZI basis. The lossy transformation matrix can be expressed as D MZI L = L linear ⋅ D MZI (8)

The programmed phase shift can deviate from its intended value due to thermal crosstalk (Shafiee et al., 2024). When programming a targeted waveguide, the heat from resistive heaters can propagate to other waveguides, creating unintended phase changes. To capture these imperfections, we model the programmed phases with a Gaussian distribution ( θ , ϕ ) ∼ N ( ( θ ̂ , ϕ ̂ ) , ( σ θ 2 , σ ϕ 2 ) ) where θ ̂ , ϕ ̂ [rad] are phases obtained after training and quantization and ( σ θ 2 , σ ϕ 2 ) are the phase variations due to thermal crosstalk.

We first evaluate networks with input sizes ( N ) of 8, 16, 32, and 64 to understand the required network complexity given the two datasets under consideration (to be introduced in Section 3.2). We perform each simulation (training followed by evaluation with the test set) five times with five different random seeds. Each model with a random seed is trained for 50 epochs, and the phase values obtained from the epoch that gives the highest validation accuracy are kept for final tests. Except for special pruned cases, the first two output ports ( O 0 and O 1 in Figure 1A) are used for final decision calculations. We use the test set to calculate each model’s test accuracy and F1 score. While test accuracy provides an overall measure of the model’s ability to classify test samples, the F1 score offers a balanced assessment of the model’s performance in correctly predicting both positive and negative classes in a binary classification task.

The limited size of the ONN we are evaluating and the resolution of the images in the selected datasets mean that data must first be compressed in some way prior to inference. Therefore, we use Principal Components Analysis (PCA) to perform dimensionality reduction. Mathematically, PCA maps n -dimensional data to a k -dimensional subspace ( k ≪ n ) by finding the eigenvectors that best represent the feature distributions in the data. These eigenvectors are decided by sorting their corresponding singular values obtained from SVD. The higher the singular value, the more variance in data points in the direction of the eigenvector, making the eigenvector more representative. The top- k eigenvectors are combined and multiplied with the original data matrix to complete the transformation, resulting in k features that are used as the input signal to the k × k ONN.

We make two assumptions about the input range of ONN based on laser power consumption. The first assumption assumes a fixed per-channel laser input range ( P i ∈ [ 0,1 ] mW per channel). This ensures the same input range to all channels regardless of the input feature size ( N ) and the resultant overall laser power increases with N . On the other hand, the second assumption fixes the total laser power to 10 mW. For each input channel, P i ∈ [ 0 , 10 N ] mW. As a result, the per-channel input range will decrease as N increases.

With these training setups, training time ranged from two to three minutes for 8 × 8 meshes to nearly half an hour for 64 × 64 meshes on an Apple M2 Pro Processor (10 cores, 16 GB memory and 200 GB/s memory bandwidth).

The MNIST dataset (Deng, 2012) consists of 70,000 28 × 28 grayscale images of handwritten digits 0-9. Given that our focus is binary classification, we modify the 10-class MNIST dataset by aggregating samples with labels 0 − 4 and labels 5 − 9 into samples with labels [ 1,0 ] and [ 0,1 ] , respectively. Compared to using any two out of the 10 classes, this approach allows us to maximally utilize the available dataset and test the networks’ generalizability across diverse samples while avoiding biased task complexity caused by choosing two specific classes out of ten. The dataset is split 50,000 : 10,000 : 10,000 to form the training, validation, and test sets.

The CIFAR-10 dataset (Krizhevsky and Hinton, 2009) contains 32 × 32 images in 10 classes (airplane, automobile, bird, cat, deer, dog, frog, horse, ship, truck), each with three colour channels (red, green, and blue). Similar to how we process the MNIST dataset, we rearrange the CIFAR-10 labels to make the classification binary by aggregating the original label of “airplanes”, “cars”, “ships”, and “trucks” into a new group called “vehicles”; “birds”, “cats”, “deer”, and “dogs” into a new group called “animals”. The images originally labelled as “frog” and “horse” are removed from the dataset to ensure the balance between data samples in the two classes. This reduces the total image number to 48,000 with 24,000 images in each category. The dataset is split 32,000 : 8,000 : 8,000 to form the training, validation, and test sets.

In both datasets, the original images have sufficient pixels to clearly depict the represented objects. This ensures that the complexity of any formulated task comes from the intrinsic difficulty of distinguishing objects across different classes rather than from low image resolution. Depending on the specific task complexity and the degree of over-parametrization in the network model, varying levels of pruning can be carried out. As a result, our aggregated classification tasks of both datasets provide meaningful task complexity and serve as effective benchmarks for evaluating the ONN capability and the efficacy of the pruning process.

The binary optical trigger structure is anticipated to be used in event-triggered structures, where the ONN activates the rest of a system when a pre-defined event takes place (e.g., a vehicle is detected by ONN after training on the CIFAR-10 dataset). A key challenge in the implementation of such a system is the optical trigger false negatives: the pre-defined event happens, but the ONN does not send a trigger signal, and the rest of the system fails by default. Therefore, one goal of our work is to minimize the number of False Negatives (FN) while maintaining the classification accuracy of ONN.

The FN reduction method considered in this work changes the weight assigned to each label class in the loss function. We penalize FN more severely, and the binary cross-entropy loss becomes L BCE = − β y ⋅ log y ̂ − 1 − y log 1 − y ̂ (9) where β is a constant greater than 1, and y ̂ is the output from classifier ( y ) after Sigmoid activation. Consequently, the gradient of the loss function with respect to the network weight ( w ) becomes ∂ L BCE ∂ w = − β y + β − 1 ⋅ y y ̂ + y ̂ * ⋅ z (10) where z is the input to the ONN layer and the complex conjugate is taken for complex-valued neural networks.

During implementations in this work, in order to strike a trade-off between FN reduction and the classification performance, we train ONN models with different weights ( β ∈ [ 1,2 ] ) assigned to the positive class and record their effects on the FN numbers and the test accuracy.

Programming the MZI-based building block of ONNs involves configuring their phase shifters to form a desired transfer matrix. In this work, we consider MZIs with phase shifters controlled by thermally changing the phase using resistive heaters tuned by a voltage supply (Masood et al., 2013). The relationship between the heater control voltage ( V bias ) and the intended phase shift ( θ , ϕ ) can be formulated as θ , ϕ = γ V bias 2 , (11) where γ = π / V π 2 (Gu et al., 2020b) and V π refers to the required bias voltage for programming { θ , ϕ } = π (Shokraneh et al., 2020).

Practical voltage sources have limited resolution, meaning they can be adjusted only to a finite number of discrete voltage levels equally spaced between the maximum and minimum values. A b -bit voltage supply has 2 b achievable voltage levels spaced apart by V res = V max / ( 2 b − 1 ) with the i -th voltage level being V i = i ⋅ V res volts. V max denotes the maximum supply voltage. As we require a phase setting range over 2 π ( { θ , ϕ } ∈ [ 0,2 π ) ), all the voltage levels beyond V 2 π are not used and the effective bit-resolution of the voltage supply further drops by ⌊ log 2 V max V 2 π ⌋ . The resultant quantized phases are obtained by mapping the sampled voltage levels ( V i ) back to V bias in Equation 11.

Models with selected hyperparameters from the previous steps are quantized by rounding the trained phase shifts to their nearest quantized phase values. We choose the least voltage resolution (in [ 4,16 ] bits) that enables the closest ONN test accuracy to those obtained from full resolution (32-bit) training settings.

To gain insight into the trade-off between reducing component usage and disruption in network expressivity, we look for a suitable metric to evaluate the pruned mesh’s expressivity. In previous works, the concept of fidelity was employed (Feng et al., 2022; Zhang et al., 2021) for evaluating the similarity between two complex density matrices. Similar metrics include the Frobenius norm, cosine similarity, and correlation coefficients. However, we note the unsuitability of a simple similarity metric in our particular case, as the goal of pruning is not to produce an optical mesh capable of approximating the original unitary matrix. Rather, given the relative simplicity of the binary trigger task and monitoring of only two entries in the output vector, fully unitary ONNs are likely over-parameterized, and completely different sets of optimal weight may exist in the sub-unitary space that have little to no relation to the unitary weight matrices producing similar classification accuracy.

For this reason, we employ a sampling-based benchmark to evaluate the signal routing ability of 8 × 8 sub-unitary meshes: 10,000 random sub-unitary weight matrices implemented by MiniBokun mesh are generated, each is provided with three sets of random input vectors (10 per set), plus an input vector whose power is equally distributed to all ports. Although random, the vectors in each of the three sets have distinct optical power concentrations at specific input ports (with the highest power at ports 0, 3, and 5, respectively). The input vectors are assumed to be coherent with an absolute phase of 0 rad, and the total input power is fixed at 10 mW. For each topology under test, the power distribution at each output port when subjecting the ONN to the aforementioned artificial input vectors is recorded. The expressivity of each topology can then be inferred based on the attained distribution.

Following the method discussed in Section 3.4, the monitored average binary MNIST accuracy per 10 × 10 network is shown in Figure 2. As indicated by the plateauing part of the curve, up to four MZIs columns can be pruned with less than 0.5 % accuracy drop, pruning six layers leads to the minimal triangle topology discussed in Section 3.4, in which significant accuracy degradation is observed. We thus restore two pruned columns, and remove the top and bottom MZIs in the left-most restored column (MZI 23,27 in Figure 1C, or equivalently, MZI 19,23 in Figure 2) to create diagonal access concerning paths I 0 → O 4 and I 9 → O 5 in practical chip-calibration (Mojaver et al., 2023). With the above steps, we obtain the MiniBokun Topology.

Similar to the full-size Bokun Mesh proposed in (Mojaver et al., 2023), MiniBokun provides diagonal access for all MZIs in the mesh for practical calibration consideration, yet with no MZI wasted due to being used solely for calibration purposes. Two simple observations can be made for a sufficient formal definition of MiniBokun topology, regardless of network size N :

• There are always two MZI columns before the widest column, each containing N 2 − 1 and N 2 − 2 MZIs.

The placement of each MZI is thus well-defined, and the number of MZIs in a N -input MiniBokun mesh is 1 8 ( N 2 − 10 N + 32 ) , as shown in Table 1.

Using the benchmark method presented in Section 3.4.1. We perform statistical analysis on the collected power distribution at each output port for 8 × 8 1) Clements (Unitary), 2) Triangle (Over-Pruned) and 3) MiniBokun topologies, Reck topology is omitted as it covers the same unitary space as Clements. The box plot of the result is shown in Figure 3.

As we are not assuming any loss, the average total output power, as expected, sums to 10 mW, which matches the total input power assumption. All tested topology-input combinations managed to achieve near 0 mW output in all output ports. The maximum registered maximum power difference among tested samples were 1.330, 6.205 and 4.080 mW for Clements (port 0), triangle (port 3) and MiniBokun (port 5), respectively. Unitary structures such as the Clements mesh provide full signal routing between any input-output waveguide pairs, thus giving a relatively uniform power distribution profile across each port, even when facing input with power concentrated on particular ports. On the other hand, sub-unitary topologies provide limited signal routing paths, in triangle topology, given the imbalanced number of MZI across different paths and the unbiased random phase setting, the biased weight space manifests as mismatching of maximum detected output power across different output, as well as the varying interquartile ranges. In particular, the low maximum power on edge ports (0, 7) indicates an impaired ability to discard unwanted power as part of the inference process. MiniBokun topology also shows such bias in its power distribution, but to a lighter extent, attributing to the two additional columns providing extra paths to disregard power from inputs 1 through 6, leading to a larger expressible space. This can be validated by the classification accuracy difference between a triangle mesh and a MiniBokun mesh in Figure 2.

We observe that topology sizes of N = 8 and N = 16 work best for the binary optical trigger, especially under the assumption of fixed input laser power.

Under the assumption of constant per-channel input power, all three metrics, the accuracy of each model on both validation and test set and the F1 score, increase as N increases. As shown in Figure 4A, on average, the accuracy of the model prediction on the MNIST dataset improves by an absolute 6.9% as N reaches 64. The models’ accuracy on the CIFAR-10 dataset, which is more complex than MNIST, only increases by an absolute 2.2%, as shown in Figure 4B.

When the assumption changes to fixed laser power, the actual per-channel input power range decreases as the optical mesh scales up. As N grows from 8 to 64, the maximum input optical power P i to a channel drops from 1.25 mW to 0.16 mW. This input range reduction significantly undermines the ability of larger ONNs to learn. As seen in Figure 4, although the MNIST test accuracy increases with N , the magnitude of the growth in all models drops to an average maximum of 4.5%. On the CIFAR-10 dataset, model accuracy starts to drop after N = 16 and eventually falls below the accuracy of N = 8 .

The increase in ONN classification accuracy with N is subject to the perfect operating conditions assumed in the simulations. In reality, the optical loss of an ONN increases linearly with its size and becomes especially significant when N ≥ 32 (Shafiee et al., 2024). Moreover, the power consumption of configuring the phase values increases quadratically with the optical mesh sizes (Al-Qadasi et al., 2022). Finally, under the fixed channel input power assumption, the laser input power obviously increases linearly with the number of input channels. Therefore, given our goal of finding a robust model that balances the overall accuracy and power efficiency, only ONNs of N = 8 and N = 16 trained with the fixed laser power assumption are considered.

The weighted class method effectively reduces the number of FNs made by ONNs after training. As shown in Figures 5A, C, the FN count decreases from more than 1,000 to fewer than 10 as β in Equations 9, 10 increases from 1 to 2. Increasing β forces the model to make more positive predictions. Subsequently, there are more false positives, and the overall test accuracy drops, as shown in Figures 5B, D. In this case, sharp declines in overall test accuracy are observed for MNIST ( β > 1.4 ) and CIFAR-10 ( β > 1.2 ) . To ensure a balance between the decrease in FN and the drop in accuracy, we finally selected β ∈ [ 1,1.4 ] for the rest of our discussions. Models trained with β in this range achieve at most a 75% reduction in FN but less than a 5% drop in accuracy.

We also find that an 8-bit voltage supply resolution is sufficient for models with the selected hyperparameters to achieve similar accuracy to those trained with full precision (32-bit), using a voltage supply setting of V max = 4 V , V π = 1.92 V (Shokraneh et al., 2020). According to Figure 6, ONN accuracy increases significantly as the voltage supply resolution increases from 4 to 8 bits and gradually converges to the full-resolution test accuracy. At the 8-bit resolution point, most 8 × 8 models show < 0.5 % deviation from the full-resolution accuracy while most 16 × 16 models show < 0.8 % deviation. Therefore, we assume an 8-bit voltage supply resolution for the rest of our discussions.

Based on the selected hyperparameters and optimization parameters, we summarized the accuracy and F1 score of all the models with different topologies in Table 2. The numbers labeled in bold are the best-performing topology in each category.

In the N = 8 case, despite having fewer programmable phase shifters due to the small input size and the pre-training pruning performed, the MiniBokun mesh only experiences a 1.57% drop in accuracy and a 0.7% drop in F1 score on average compared with other meshes. Even with the slightly undermined learning ability, the performance gap between the best-performing model and MiniBokun is not large. MiniBokun mesh preserves a good balance in classifying both the positive and the negative classes of a dataset.

The performance gap between MiniBokun and the other meshes further closes when N increases to 16. The average performance degradation drops to 1.42% in accuracy and 0.63% in F1 score. In certain cases, the MiniBokun mesh outperforms the other two in terms of both accuracy and F1 score.

The pruned MiniBokun mesh has demonstrated a strong capability in reducing overall power consumption. Compared to the conventional Clements and Reck topology, the MiniBokun mesh saves 4.6% power in the aggressive case and 24.2% in the conservative case when N = 8 . These numbers further grow to 18.0% and 47.6% when N increases to 16. On the other hand, the benefits of pruning in saving latency are insignificant, only up to 0.1 μs when N = 16 , as the latency of the slowest components (phase programming) is large per se and invariant to the topology.

If we take a closer look at the component-wise power and latency consumption, the phase value programming dominates both calculations. Assuming uniform phase distribution, the programming power is directly proportional to the number of MZIs in a mesh (Al-Qadasi et al., 2022). Without insulation (the “conservative” approach), the programming power can take up to 39.8% of the total power consumption in Reck and Clements topology when N = 8 (as shown in Figure 8A), and this proportion continues to grow as the size of the optical mesh increases. Using the pruning strategy introduced in this work, we can effectively reduce the number of MZIs in the optical mesh by more than half. This subsequently relaxes the power requirement for programming the phase values and reduces the proportion it takes in the total power consumption, as shown in Figure 8B. The power savings by pruning becomes more evident when the optical meshes scale up, as indicated by the growing gap between the two lines in Figure 8C. Using insulated heaters with smaller P π (the “aggressive” approach) can effectively reduce the overall power consumption. However, this comes at a cost of 5 × more time spent on the programming stage.

In the latency calculation, the pruned MiniBokun mesh effectively shortens the optical path length that light propagates through and lowers the number of memory read by reducing the number of phase values to be programmed. However, these savings ( ≤ 0.06 μs in total) are comparatively negligible to the programming time itself ( [ 30,150 ] μs). Disregarding the programming latency, the speed of the optical mesh computation is bottlenecked by the electrical ADCs and DAC. As the pruning technique does not alter the parameters of these devices, the speed of the MiniBokun mesh is still limited by them.

The pruning strategy significantly reduces the area of the optical mesh by placing fewer MZIs horizontally along the optical path. As shown in Table 3, the MiniBokun mesh employs 50% fewer MZIs when N = 8 , and 60% fewer MZIs when N = 16 . Subsequently, the optical path length reduces from 2 N − 3 in Reck mesh and N in Clements mesh to N 2 + 1 in MiniBokun, bringing the total layout area down by at least 40%. As shown in Figure 8D, when the optical mesh size increases, the area of the MiniBokun mesh grows less rapidly than the other two topologies. When reaching the N = 64 limit, it saves 73.6% and 48.4% layout area when compared to the Clements and Reck topologies.

For practical deployment, a typical smart lock uses a microcontroller comparable in size to an Arduino chip (Arduino, 2014), implying area constraints on the order of several square centimeters (Motwani et al., 2021). In contrast, modern smartphone processors, facing stricter area limitations, typically have a footprint of over 100 mm² (Yang et al., 2024). By comparison, our estimated mesh area is 2.4 mm² for the 16 × 16 MiniBokun, which meets the area constraint of both of these target applications.