1. INTRODUCTION

Artificial neural networks (ANNs) have emerged as a transformative force in the realm of artificial intelligence (AI), exhibiting remarkable capabilities in various applications such as image and voice recognition, medical treatment, and autonomous systems [1–9]. However, the escalating demand for computational power and energy efficiency has driven research into advanced hardware solutions to accelerate computations within ANNs [10–12]. Optical neural networks (ONNs) have garnered significant attention as promising alternatives to conventional electronic processors, drawing on several key advantages over their electronic counterparts [11–24]. The utilization of photons as information carriers enables ONNs to engage in high-speed parallel information processing. Additionally, photonic circuits, characterized by their exceptional attributes like ultra-wide bandwidth, high frequency, and low energy consumption, offer an ideal foundation for the execution of complex AI computations.

Matrix-vector multiplications (MVMs) are fundamental mathematical operations in ONNs essential for data transformation, weight adjustments, and learning processes [5,25–27]. The weighted interconnections between adjacent neurons in the network are mathematically represented by a matrix whose entries are the weight values. Due to the abundant orthogonal channels in the wavelength domain, a wavelength division multiplexing (WDM) protocol known as “broadcast-and-weight” has been proposed for neural network models [20,28–32]. The input vector is simultaneously fed into a single waveguide and evenly distributed to a reconfigurable filter bank in order to load weights. WDM-MVMs offer a practical approach to expand the number of neurons with spectrum reuse strategies, demonstrating significant potential for high computational density. Currently, WDM-MVMs predominantly utilize microring resonator (MRR) arrays [12,16,24,33–38]. On the one hand, however, the footprint of MRR is constrained by waveguide bending losses, impeding the further reduction of chip size and power consumption. On the other hand, the free spectral range (FSR) limits the number of channels, posing a barrier to a large-scale cascade [39].

 

Fig. 1. (a) Schematic representation of the PCNC matrix core. (b) Advantages of the PCNC scheme compared with MRR. (c) Principle of PCNC weight bank.

Download Full Size | PDF

To address the challenges in WDM-MVMs, here we propose a novel and cost-effective photonic crystal nanobeam cavity (PCNC) matrix core. As a one-dimensional structure based on band engineering, PCNCs exhibit notable advantages over MRRs to execute WDM-MVMs. Our PCNC unit has a compact size of ${20}\;{\unicode{x00B5}{\rm m}} \times {0.5}\;{\unicode{x00B5}{\rm m}}$, which is comparable to that of a single mode waveguide and is conducive to mitigation of footprint expansion. Consequently, PCNCs naturally hold higher thermal-tuning efficiency than MRRs, leading to reduced power consumption when loading the same weights in ONNs. Furthermore, owing to the properties of finely tailored Bloch modes, PCNCs are not constrained by the limitation of free spectral range (FSR) within the required bandwidth. This practical attribute enables feasible scaling of channels for enhanced computational capacity of the matrix core. We demonstrate a ${3} \times {3}$ PCNC matrix core, showcasing its matrix computation capabilities through image convolution and cat-dog face recognition. Moreover, to highlight its scalability, we fabricate a chip with six wavelength channels, realizing a modified National Institute of Standards and Technology (MNIST) recognition task as a general competency demonstration. This compact, efficient, and scalable device paves the way for scaling up MVMs. Its compatibility with the standard complementary metal-oxide semiconductor (CMOS) processes and flexibility for implementation on other material platforms imbue our solution with potential for application in high-speed, large-capacity networks on-chip (NoCs) and commercial optical computing modules.

2. RESULTS

Figure 1(a) illustrates the schematic diagram of the proposed PCNC matrix core. The $N$-dimensional vector (with its components represented as ${x_i}$) obtained from preliminary modulation is loaded as intensity signals of different wavelengths into the chip and distributed to the $N \times N$ PCNC weight bank through a directional coupler (DC). Similar to microrings, the PCNC array selectively weights the input vector [30]. Each PCNC processes a specific wavelength channel, with the transmission corresponding to that wavelength serving as the synaptic weight (${w_{\textit{ij}}}$). All units in the array achieve reconfigurable transmission through metal heaters located above each PCNC (not indicated in Fig. 1), and the mapping between weights and driving voltages is accomplished through predefined calibration. At the end of each row, the total power obtained from the parallel bus waveguides through a multiplication and summation process is detected by a photodetector (PD), corresponding to a component (${y_i}$) of the output vector. This process is depicted in Fig. 1(c), which matches the principle of WDM-MVM [28].

As a distinctive design based on one-dimensional photonic crystals, a nanobeam core exhibits three typical features superior to the MRR core for matrix computations in ONNs, shown in Fig. 1(b). Nanobeams achieve resonance enhancement not through the ring waveguide required for the whispering gallery mode (WGM) of MRRs, but rather by etching elliptic holes on a compact one-dimensional waveguide to create a bandgap reflection [40]. The smaller footprint also allows nanobeams to generate the same resonant frequency shift as MRRs with less thermal power (${P_t}$) under equivalent conditions, leading to lower energy consumption in optical MVMs [41]. Furthermore, through optimization of the intracavity Bloch modes [42–44], an individual nanobeam unit exhibits no multi-wavelength resonant peaks within the required bandwidth, which is crucial for the scalability of the matrix core.

The proposed PCNC matrix core is composed of an array of $N \times N$ tunable nanobeam filters as shown in Fig. 2(a). It is fabricated on a silicon-on-insulator (SOI) chip with a top silicon layer thickness of 220 nm. The detailed fabrication process can be found in Section 1 of Supplement 1. The photonic crystal structure of the nanobeam cavity is centrally symmetric, and for convenience, we have labeled the elliptical holes from ${-}{20}$ to 20 and will only discuss the region with positive serial numbers. Each hole is characterized by three parameters that define its position and shape: ${r_x}$ and ${r_y}$ correspond to the semi-axes of the ellipse, while $a$ represents the distance between adjacent ellipses, as shown in Fig. 2(b). For instance, $r_x^{(4)}$ represents the half of the horizontal axis length of the hole with serial number 4. The photonic band of the hole closest to the center allows for the presence of resonant wavelength light, while holes 9 to 20 are designed identical to form a strong photonic crystal bandgap, confining the resonant light to the central region. Holes 1 to 8 are arranged in a structure with a quadratic gradient, to reduce scattering loss from reflections.

 

Fig. 2. Design and characteristics of PCNC units. (a) Schematic diagram of a single PCNC unit. (b) Key structural parameters and elliptical hole distribution characteristics of the PCNC unit. The absolute values of serial numbers represent the distance of holes from the center, while the positive and negative signs indicate different directions. (c) Scanning electron microscope (SEM) image of one PCNC unit. (d) Manufactured ${3} \times {3}$ PCNC matrix core, with an additional row for backup. (e) Photonic crystal band diagram of a PCNC, retaining only bandgap information. (f) Optical field distribution of resonant wavelengths (top) and non-resonant wavelengths (bottom) obtained through three-dimensional time-domain finite-difference (3D-FDTD) simulations. (g) Measured transmission spectra of one row (top) and thermal tuning characteristics of the transmission spectrum of a single unit (bottom).

Download Full Size | PDF

As a conceptual demonstration, we have designed a ${3} \times {3}$ array of nanobeam resonators and fabricated the matrix core chip, and an image of one unit captured by a scanning electron microscope (SEM) is depicted in Fig. 2(c). The chip is packaged with optical input/output (I/O) and the thermoelectric cooler (TEC), as is shown in Fig. 2(d). A zoom-in view of the device and the content regarding the structure of multi-channel configurations can be found in Section 2 of Supplement 1. Each unit is individually parameterized through the aforementioned procedure, and each column of PCNCs corresponds to a distinct wavelength channel. Metal electrodes deposited on the silica cladding are employed to manipulate the effective refractive index distribution and band structure of the PCNC, thereby thermally modulating the signal light intensity. The mechanism of thermo-optic tuning is presented in Section 3 of Supplement 1.

With this configuration, Fig. 2(e) shows that the photonic bandgap formed by each hole varies with the increasing serial number. More details are given in Supplement 1, Section 2. Wavelengths allowed (i.e., on the band) at the center hole are positioned within the bandgap of the edge holes and are thus reflected. Essentially, this introduces a defect mode within the bandgap of the edge holes, allowing coupled light to stably resonate at the center hole. By employing this band engineering approach, light that meets the resonance wavelength condition can be coupled into the PCNC through a bus waveguide and stimulate Bloch mode [45], while light of other wavelengths directly propagates to the next level of units, as depicted in Fig. 2(f).

The transmission spectra of one row of tested PCNCs are depicted in Fig. 2(g). The three resonant wavelengths are ${\lambda _1} = {1544.40}\;{\rm nm}$, ${\lambda _2} = {1554.20}\;{\rm nm}$, and ${\lambda _3} = {1561.08}\;{\rm nm}$. In fact, it can be observed that there are still numerous wavelength channels available due to the FSR-free feature. In view of the flexible structural design, channels can be set at wavelengths that align with the requirements of the light source band, rather than being restricted solely to the conventional band (C-band).

In order to characterize the advantages of the PCNC device, we have fabricated and tested an MRR unit under the same conditions, and the results are presented in Table 1. The relation between resonance wavelength and tuning power of an individual channel can be found in Supplement 1, Section 4. Compared to microrings, PCNC reduces one dimension in size and exhibits a tuning coefficient over three times larger. Furthermore, there is no FSR within the detected wavelength band, indicating excellent scalability.

Table 1. Performance Comparison of the Fabricated PCNC and MRR Unit

View Table

For practical purposes, we experimentally demonstrated the applications of the PCNC matrix core chip in a convolutional neural network (CNN). First, we select standard test images to validate the chip’s capability in extracting image features. A standard ${256} \times {256}$ grayscale “Cameraman” image with 8-bit color depths is utilized. The principle of the convolution process involves the weighted summation of element-wise products between a small matrix (referred to as the convolutional kernel) and corresponding image patches in the input image. Specifically, the weights of the kernel are assigned to the pixel values of the input image. During each image convolution operation, the convolutional kernel is slid over the input image with a predefined step length, generating weighted summations that are reshaped into feature maps. Several commonly used convolutional kernels are employed to extract various features from the image, including blur, bottom sobel, emboss, and motion blur. A tunable laser source (TLS) is employed to provide multi-channel laser inputs that match the resonant wavelengths of PCNCs. The grayscale values of the input image’s pixels are normalized and arranged as a serial data flow, which is then fed into an electro-optic modulator and weighted by the PCNC matrix core during light propagation, as illustrated in Fig. 3. The optically weighted results are detected by photodetectors (PDs), and subsequently reshaped into the final feature maps. The experimental results closely resemble the theoretical feature maps generated by a 64-bit digital computer. Section 5 of Supplement 1 provides further insights into the photonic matrix multiplication, encompassing the treatment of negative numbers.

 

Fig. 3. Image convolution implemented by PCNC core. (a) Experimental setup diagram. (b) Matrix convolution results using different kernels, including several classical image processing techniques. EOM: electro-optical Mach–Zehnder modulator. MUX, wavelength multiplexer. PD, photodetector. FPGA, field programmable gate array.

Download Full Size | PDF

Next, we conduct an experiment to validate the practicality of our chip in CNNs using an animal faces dataset (AFHQ) [46], in which we demonstrate the recognition of cat and dog faces. Figure 4(a) illustrates the CNN model employed, where the first layer of convolutional computations is accomplished using the PCNC matrix core. Due to the intricate details present in animal faces and the fact that the actual images are compressed versions of the original dataset images, the complexity of this task is challenging. Leveraging a pre-encoded dataset, the matrix core is trained through a gradient descent optimizer, and the learned parameters are mapped onto the driving voltages of the PCNC core. A total of 6000 images with dimensions of ${28} \times {28}$ pixels are utilized for training, and a test set consisting of 120 images (60 cats and 60 dogs) is employed. Input features are intensity-modulated onto lasers of different wavelengths and processed through convolutional, non-linear, and fully connected layers to generate classification information. The PCNC synapses enable parallel convolutional computations on optical data streams of image patches, yielding feature maps as depicted in Figs. 4(b) and 4(c). The accuracy of the convolutional layer computations is depicted in Fig. 4(d), where the experimentally obtained values closely cluster near the diagonal, indicating precise computations with a high degree of reliability. It is worth noting that a calibration method is employed to correct errors caused by thermal crosstalk [47], and a detailed explanation of this calibration approach can be found in Section 6 of Supplement 1. The normalized standard deviation obtained from the Gaussian fit for the residuals is 0.0128. This result indicates that the PCNC core holds competitive computational accuracy. For the cat-dog face test dataset, we achieved a recognition accuracy of 85.9%, which is close to the result obtained using a 64-bit computer (88.3%). The relatively low precision in both optics and electronic computing attribute to substantial compression of the input image to accommodate the current scale of optical computing.

 

Fig. 4. Cat and dog image recognition experiment using a ${3} \times {3}$ PCNC core. (a) Architecture of the convolutional neural network employed. (b) Subset of the test dataset images and (c) corresponding output feature maps. (d) Scatter plot for measuring convolution accuracy. The inset presents a histogram of the residual distribution, illustrating the differences between measured and computed values. (e) Confusion matrices for cat and dog image classification on the AFHQ test dataset. Conv., convolution layer. ReLU, rectified linear unit. FC, fully connected layer.

Download Full Size | PDF

Finally, to demonstrate the excellent scalability of PCNC, we have fabricated a matrix core chip with six wavelength channels. Microscope images of the chip and the transmission spectrum of one row are depicted in Figs. 5(a) and 5(b), respectively. The six resonant wavelengths, namely, ${\lambda _4}$ to ${\lambda _9}$, are 1539.48 nm, 1544.08 nm, 1548.70 nm, 1553.38 nm, 1558.08 nm, and 1563.40 nm. Similarly, the diverse resonant wavelengths are designed based on the aforementioned band-engineering mapping. Moreover, this approach can still be further extended for additional wavelength channels, primarily attributed to PCNC’s freedom from FSR limitations across a considerable frequency range. We have utilized the extended chip to recognize the more demanding MNIST handwritten digit images. The multi-layer network model architecture used in this experiment is identical to the one employed in the previous AFHQ dataset experiment, and the parameters were trained in the electrical domain using the gradient descent algorithm and then loaded into the optical domain; 10,000 images are employed for training the gradient descent algorithm, while 500 images form the test set. The outcomes, depicted in Fig. 5(c), reveal that the classification accuracy of the PCNC core is 87.0%, closely aligned with the 64-bit computer’s performance of 87.8%.

 

Fig. 5. Handwritten digit recognition experiment using a six-channel PCNC core. (a) Manufactured six-channel PCNC matrix core chip. (b) Transmission spectra of one of the rows. (c) Confusion matrices for handwritten digit classification on the MNIST test dataset.

Download Full Size | PDF

3. DISCUSSION

As a foundational device distinct from MRRs, the PCNC approach, in addition to the demonstrated attributes of compact size, high efficiency, and scalability, also holds untapped potential for further enhancement when employed as a matrix computation core. In this paper, the theoretical computational domain for the nanobeam utilized is within the domain of positive real numbers. Alternatively, employing an add-drop type of waveguide coupling scheme and performing differential processing at the output end can extend both the transmission matrix and the output vector to encompass the negative number domain [43]. Notably, the Fano-enhanced nanobeam proposed in Refs. [48,49] can be adopted to enhance port transmission efficiency and the extinction ratio, ensuring a higher computational dynamic range. Designing thermal electrodes tailored to the small mode volume of PCNCs or employing phase-change material (PCM) [20] to switch output states holds the potential to further enhance energy efficiency. Etching thermal isolation trenches between PCNC units can also be employed to increase the thermal tuning coefficient and aid in eliminating thermal crosstalk, thereby further reducing the overall chip size [50].

Although we have demonstrated the multi-channel capabilities of the PCNC matrix core, it is evident from the transmission spectrum that there are still numerous wavelength channels available for selection. Previous analyses [30,39] have examined the expansion limits of the MRR scheme, with its maximum channel count (${N_c}$) constrained by FSR as

The flexibility of the PCNC scheme extends beyond these aspects. In scenarios where stringent precision is not required, the lateral dimensions of the nanobeams can be reduced by decreasing the number of holes in the bandgap regions on both sides [52], which enables achieving a higher computational density. Similar to MRRs, PCNCs can be employed for on-chip front modulation, substituting the need for external modulators to load vectors [22]. As a structurally adaptable design, the PCNC can even be transposed onto other material platforms, such as lithium niobite [53,54], to achieve faster and more efficient modulation. In future endeavors, the integration of non-linear activation functions could be pursued to achieve all-optical neural network chips. When compared to systems like Mach–Zehnder interferometer (MZI) mesh and spatial diffraction, the remarkably compact dimensions and compatibility with CMOS processes of our proposed approach offer advantages for cost-effective deployment within miniaturized commercial computing modules.

In conclusion, we have introduced the PCNC matrix core as a novel solution that addresses key challenges in WDM-MVMs. The compact dimensions facilitate high-density computations, while its superior thermal tuning efficiency, surpassing MRRs by more than threefold, offers energy-efficient operation. Notably, the PCNC matrix core’s design enables utilization of a multitude of independent wavelength channels in WDM, overcoming the limitations of FSR. Our experiments in animal face recognition and handwritten digit classification highlight its practicality and scalability. With its low-cost attributes, inherent flexibility, and compatibility with CMOS processes, the PCNC scheme emerges as an ideal choice for optical accelerator chips and paves the way for a new generation of hardware to advance AI applications.