Matrix inversion is a fundamental and widely utilized linear algebraic operation but computationally expensive in digital-clock-based platforms. Optical computing is a new computing paradigm with high speed and energy efficiency, and the computation can be realized through light propagation. However, there is a scarcity of experimentally implemented matrix inverters that exhibit both high integration density and the capability to perform complex-valued operations in existing optical systems. For the first time, we experimentally demonstrated an iterative all-optical chip-scale processor to perform the computation of complex-valued matrix inversion using the Richardson method. Our chip-scale processor achieves an iteration speed of 10 GHz, which can facilitate ultra-fast matrix inversion with the assistance of high-speed Mach–Zehnder interferometer modulators. The convergence can be attained within 20 iterations, yielding an accuracy of 90%. The proposed chip-scale all-optical complex-valued matrix inverter represents a distinctive innovation in the field of all-optical recursive systems, offering significant potential for solving computationally intensive mathematical problems.

Matrix inversion, recognized as a fundamental yet computationally intensive mathematical problem, often constitutes the most time-intensive portion of numerous computational tasks.1–3 In practical engineering, matrix inversion modules serve as fundamental components for an extensive array of problems, such as machine learning4–6 and signal processing.7,8 The time complexity of matrix inversion on traditional computing systems typically scales as ∼O(N2.37) even with the most advanced algorithms.9,10 As traditional computing systems encounter increasing difficulties in enhancing their capacities through further miniaturization,11–13 the exploration of new platforms, such as integrated photonics, emerges as a potential solution to meet the demand for high-speed computation.

Owing to the inherent characteristics of optical systems, such as low latency, minimal energy loss, and compact footprint, coupled with the intrinsic properties of photons, including natural parallelism, optical systems offer substantial advantages in executing mathematical computations.14–17 Several groups have reported experimental implementations of on-chip optical computing,18–21 which can be adapted and modified to perform on-chip matrix inversion operations. Various computational domains have witnessed the substantial potential of optical techniques, such as reservoir computing,22–26 convolution operations,27–30 and neural networks.31–34 It is feasible to achieve integrated high-speed parallel processors in complex optical networks. Recently, there has been an exploration into the feasibility of employing iterative solvers on mainstream integrated photonic platforms.35 Despite the advancements, experimental designs in this field remain underexplored and limited.

Early research on matrix inversion through optical methods was reported employing a free space optical design36 or optical fiber network.10 This proposed network has been experimentally successful in executing the inversion of 3 × 3 matrices, thereby reducing the overall solving time to O(N2).10 In addition, an iterative scheme combining photonic integration technologies and fiber loops has been proposed and verified to solve real-valued matrix inversion.37,38 Limited by the large phase fluctuations, fiber-optic systems and free-space optical systems are only appropriate for real-valued calculations and exhibit a large system footprint. The utilization of integrated platforms has provided a novel methodology for inverting complex-valued matrices. With the advent of metamaterial applications in analog computing,39 a significant milestone was achieved in the microwave domain that the inversion of a 5 × 5 complex-valued matrix with a size of several tens of centimeters was successfully proved.40,41 The kernel matrix in this work, which is configured by the distribution of dielectric constants through inverse design, is not reconfigurable. Consequently, the implementation of a reconfigurable matrix inversion system using the Miller or Direct-Complex-Matrix (DCM) system has been theoretically proposed via system-level simulation.35 While chip-scale integrated complex-valued matrix inversion offers advantages, including scalability, phase stability, and high speed, there has been no research conducted to experimentally implement this approach.

Here, we design and experimentally confirm the feasibility of implementing the inversion of a 4 × 4 complex-valued matrix using on-chip iterative photonics computing chips. We demonstrate an all-optical complex-valued matrix inverter (OCMI) on a silicon-on-insulator (SOI) platform based on the Richardson method for the first time. The computational circuit establishes a resonant cavity with an optical path length of ∼10 mm, enabling the OCMI chip to achieve convergence at a rate of 10 GHz within 20 iterations. The utilization of on-chip implementations for rapid complex matrix inversion presents a prospective avenue for augmenting the processing velocity of wireless multiple-input multiple-output (MIMO) systems, reducing the energy dissipation of computational problems and the computational resource requirements in optimization problems. It is possible to achieve the inversion of larger matrix dimensions by employing the concept of block matrix inversion. Moreover, the unprecedented level of integration achieved through advanced design methodologies has concomitantly led to a reduction in the dimensions of functional modules.

The basic model for solving matrix inversion has a similar form to the Richardson iteration equation.42 The process of solving the inverse of a matrix using iteration can be equivalently understood as solving a specific system of linear equations,
AX=I.
(1)
A is an N × N matrix to be inverted, while I is an identity matrix with the same dimension as A. The solution to the linear equation is the inverse of matrix A. By introducing a variable parameter ω and decomposing it, such linear equations can be transformed into a form known as the Richardson iteration equation. This equation is suitable for implementation on hardware systems,
X=IωAX+ωI.
(2)
The convergence rate of the inversion algorithm can be adjusted by the parameter ω. X is the output matrix of the iterative system. The iteration matrix G can be defined as
G=IωA.
(3)
The convergence of iterative methods is determined by the spectral radius of the iteration matrix G. The necessary and sufficient condition for the convergence of an iterative method is that the spectral radius of its iteration matrix is <1.43 Furthermore, the relationship between the convergence rate of the iterative method and the spectral radius of the iteration matrix is as follows:44,
RG=lnρ(G).
(4)
In the realm of iterative methods, the Richardson method stands out due to its simplicity and minimal constraints compared to other iterative inversion method.45 It facilitates the computation of the inverse matrix by selecting an initial guess and iteratively updating it until convergence is achieved. In addition to its simple structure, the Richardson iteration method possesses the potential for implementation in physical systems. The primary computational operations involved in the iteration equation are matrix–vector multiplication and vector addition. By utilizing suitable coding technology, it is feasible to achieve spontaneous iteration in physical systems. This capability allows for rapid calculations of matrix inversion, thereby facilitating efficient computations.
The architecture of the proposed OCMI is illustrated in Fig. 1(a). The OCMI consists of the input region, coupling region, kernel region, and probe region. The optical path integrated on the chip can be categorized into two coherent segments: the reference light for detection and the signal light for computation. To ensure stability and precision, the reference light and the signal light originate from a common laser and are split on-chip. The input region modulated the input on the chip, resulting in the generation of a series of unit vectors as input. In the all-optical approach, the output optical signal is directly fed back to the input of the mathematical kernel with a tunable coupling factor, eliminating the need for time-consuming optical-to-electronic and electronic-to-optical conversions. The coupling region is formed of four parallel Mach–Zehnder interferometers (MZIs) with an adjustable transmission matrix. These MZIs exert control over the factor denoted as ω. The relationship between the phase shift and the coupling factor can be mathematically expressed as follows:
ω=ieiθ/2cosθ/2,θ0.
(5)
FIG. 1.

Principles of the OCMI chip. (a) Conceptual drawing of the proposed on-chip system, including input region, kernel region, coupling region, and probe region. The purple blocks represent the heaters, while the yellow lines represent the heater connections. The black arrows indicate the input light, which is encoded as the sequential unit vectors I1∼4 in the input region. The red arrows represent the output, corresponding to the input, and are the four temporal sequence vectors O1∼4. (b) Box diagram of the proposed iterative process. (c) Simulation results of the iterative equations corresponding to different coupling factors.

FIG. 1.

Principles of the OCMI chip. (a) Conceptual drawing of the proposed on-chip system, including input region, kernel region, coupling region, and probe region. The purple blocks represent the heaters, while the yellow lines represent the heater connections. The black arrows indicate the input light, which is encoded as the sequential unit vectors I1∼4 in the input region. The red arrows represent the output, corresponding to the input, and are the four temporal sequence vectors O1∼4. (b) Box diagram of the proposed iterative process. (c) Simulation results of the iterative equations corresponding to different coupling factors.

Close modal
The introduction of a tunable coupling factor enhances the method’s stability as well as capability, enabling the inversion of a more extensive range of matrices. Combining Euler’s formula with the transmission characteristics of MZI, the transmission matrix of MZI can be represented by ω as
TMZI=iω1ωωiω+1.
(6)

The kernel matrix K dominates the largest area on the chip, utilizing the network introduced by Clements et al.46 This implicit MZI matrix network requires configuration through an iterative process. The process can be achieved after several voltage iterations. The solution of such an equation is generated as the output electromagnetic field. The amplitude and phase of the optical output correspond to the magnitude and angle of the complex elements. The probe region, which is composed of four multimode interferences (MMIs), represents the final segment of the on-chip computing structure and is responsible for capturing and analyzing the amplitude and phase of the output.

The block diagram of the proposed OCMI chip is clearly shown in Fig. 1(b) and can be derived from Eq. (6) and characterized as
Xk+1=iω+1KXk+ωIO=ωKX(iω+1)I
(7)
where the superscript k represents the number of iterations, K is the matrix encoded on chip, ω is the tunable coupling factor ranging from −1 to 1, and the variable O represents the output of the system, which can be obtained through coherent detection. The iteratively convergent outcomes X of the system can correspond to the inverse of the matrix A. The equation’s form undergoes variations based on different values of the coupling factor. According to Eqs. (2) and (7), a linear correlation exists between K and A, which can be succinctly formulated as
K=1iω+1Iωiω+1A.
(8)
In the experiment, the kernel matrix K can be obtained through linear calculations [Eq. (8)] and loaded onto the OCMI chip. When the iterative process convergences, X(k) and X(k+1) can be regarded approximately equal. Therefore, by eliminating terms containing KX of Eq. (7), a linear relationship between the output of the OCMI chip and X after convergence can be obtained as follows:
iω+1O=2iω1I+ωX.
(9)

The symbol X denotes the iterative variable matrix after convergence. Based on this linear relationship, the inverse of matrix A can be obtained through linear calculations by detecting and recording four sets of output vectors (O).

The simulation results of the system are depicted in Fig. 1(c), where different curves represent the iterative processes of a specific matrix for various coupling factors. Notably, the system is capable of converging to the inverse matrix within 20–30 iterations.

Silicon photonics networks can be easily scaled to the size of the targeted matrix owing to their high integration and polarization stability. Here, a silicon photonic chip fabricated with the standard 220-nm-thick silicon-on-insulator foundry process is used to perform matrix inversion. The on-chip implementation of the principle is designed using a combination of MMI and MZI. Figure 2(b) shows the model of MZI with a phase shifter in the inner arm, which is the basic unit of the processor. The input vector modulation section comprises two-stage cascaded MZIs for efficient input routing, along with four MZIs serving as optical switches. The kernel matrix can be calculated and loaded into the mathematical kernel through an electronic control system, which controls the transmission of the MZIs. The phase shift θ of the MZI plays a crucial role in determining the coupling factor ω, which directly impacts the convergence of the system.

FIG. 2.

Fabricated photonic chip. (a) Optical micrograph of a fabricated photonic chip. The black arrow on the right represents the input port for the optical signal, which passes through the cascaded MZI to encode a four-channel time-sequenced input vector. Each input produces four outputs. The black arrow on the left represents the output port, which is connected to an external multi-channel high-speed photodetector for detection. (b) Structure diagram of the MZI kernel matrix. (c) Schematic diagram of a single MZI. (d) Optical and electronic package of the on-chip iterative system. FA: fiber array.

FIG. 2.

Fabricated photonic chip. (a) Optical micrograph of a fabricated photonic chip. The black arrow on the right represents the input port for the optical signal, which passes through the cascaded MZI to encode a four-channel time-sequenced input vector. Each input produces four outputs. The black arrow on the left represents the output port, which is connected to an external multi-channel high-speed photodetector for detection. (b) Structure diagram of the MZI kernel matrix. (c) Schematic diagram of a single MZI. (d) Optical and electronic package of the on-chip iterative system. FA: fiber array.

Close modal

The core architecture of the system constitutes an all-optical iterative matrix processor, as illustrated in Fig. 1. By leveraging iterative algorithms, we can effectively solve matrix inversion optically. The experimental setup of the system is depicted in Fig. 3. Input pulses are generated using continuous wave (CW) lasers and subsequently introduced into the iterative system to initiate the computation process. Upon traversing the on-chip iterative system, the corresponding iterative equation is successfully executed. The on-chip system consistently generates output light through MMI, and the output exhibits a linear relationship with the iterative vector X. Simultaneously, the output undergoes coherent operations with the pre-split reference light on the chip to facilitate coherent detection. The external power supply plays a crucial role in voltage regulation, achieved by connecting to the on-chip electrodes through the printed circuit board (PCB). Following the coherent operations, the light is directed through an optical fiber to an external photodetector.

FIG. 3.

(a) Experimental setup of the matrix inversion system based on OCMI chip. (b) and (c) Two parts of the experimental process, detecting and coding the initial matrix and the matrix inversion, respectively. PC: polarization controller.

FIG. 3.

(a) Experimental setup of the matrix inversion system based on OCMI chip. (b) and (c) Two parts of the experimental process, detecting and coding the initial matrix and the matrix inversion, respectively. PC: polarization controller.

Close modal

Before computation, it is necessary to encode the kernel matrix K onto the chip’s MZI array. In the experiments, the response speed of the phase shifters is measured to be 47 kHz, and the operation speed of the electrical power supply is about 1 kHz. It needs about 300 operations to load the kernel matrix for the 4 × 4 MZI network we utilized, and then, the iteration time required for encoding the kernel is ∼0.3 s.

The experimental procedure of the matrix inversion is comprised of two fundamental components: the measurement of the original matrix and the computation of the inverse matrix. The sole distinguishing factor between the two stages lies in the state of the coupling region. Different coupling factors are associated with varying proportions of input light coupled into the system.

In the first stage of the computational experiment, depicted in Fig. 3(b), the phase of the coupling MZI should be set to 0. Under such circumstances, the coupling type of the loop can be classified as “crossing” and the output of the system can be probed and iterated to get the kernel matrix K. Theoretically, the input light is entirely coupled into the system for matrix–vector multiplication. After a single computation, it is fully coupled into the output waveguide. The matrix K can be probed and calculated in this manner. According to Eq. (7), the processor essentially simplifies to matrix–vector multiplication when ω = i,
O=K.
(10)
Then, the operating state was set as “partially coupled.” Incorporating Eq. (9) into Eq. (10), the output O at this point exhibits a linear relationship with the inverse of A:
O=2iω1iω+1I+ωA1iω+1.
(11)

Based on the linear relationship between K and A described in the principle, as well as the loading method for matrix K, the inverse of matrix A, which is determined by the mathematical problem, can be solved by the OCMI chip. Hence, while the on-chip implementation of the kernel matrix K corresponds to a unitary matrix, the matrix A is not confined to being unitary. The two experimental stages probe both matrix A and its inverse, thereby enabling the successful execution of the on-chip matrix inversion process.

The experimentally extracted solutions of the matrix inversion are shown in Fig. 4, compared with the theoretical solutions. Each paired top and bottom graph signifies one set of experimental results. The depicted blue line represents the inverse results of the known complex-valued matrix computed via a computer, while the red dots symbolize the results measured within the system. The 16 points on the x-axis correspond to the 16 elements of the matrix.

FIG. 4.

Experimental results for matrix inversion. Panels (a) and (b) represent the results of solving the inversion of two matrices. The blue line is the result of matrix inversion calculated in a computer for the predetermined matrix, whereas the dark red dots represent the experimental result from the iterative system.

FIG. 4.

Experimental results for matrix inversion. Panels (a) and (b) represent the results of solving the inversion of two matrices. The blue line is the result of matrix inversion calculated in a computer for the predetermined matrix, whereas the dark red dots represent the experimental result from the iterative system.

Close modal
The inversion of various matrices has been achieved by altering the configuration of the kernel matrix, which demonstrates the replicability of the scheme and the reconfigurability of the system. The accuracy of each experiment result can be quantified using the loss function,
Loss=1XexpXsimXsim,
(12)
where Xexp and Xsim represent the values of the inverse matrix obtained through experimental calculations on the OCMI chip and through computer, respectively. In the experiment, the complex amplitude outcomes are obtained via coherent detection. Adhering to the prescribed error evaluation methodology, a meticulous analysis was undertaken across a dataset consisting of 100 instances. The derived outcome revealed an average loss function value of 0.90, thereby signifying a discernible degree of robustness inherent within the implemented OCMI chip.

We present a comparison of the reported matrix inverters with our work in Table I. The major advantage of the OCMI chip lies in the chip-scale full integration of the computational modules and its elevated level of reconfiguration, which is departure from previous methodologies such as inverse designed dielectric kernel structure.40 The function of the dielectric kernel structure is fixed once the fabrication is completed, while the MZI mesh in our work accomplishes the implementation of a reconfigurable kernel matrix, facilitating the creation of a versatile 4 × 4 unitary matrix. This capability adequately fulfills the matrix-scale necessities within optical communication systems (such as MIMO descramblers). By employing the concept of block matrix inversion, it is also possible to achieve the inversion of larger matrix dimensions. In addition, reconfigurability is manifested within the feedback loop, which utilizes four MZIs with adjustable coupling factor ranging from −1 to 1. This innovation extends the range of solvable matrices beyond unitary matrices in practical computations, enhancing the versatility of the system. Moreover, compared with solutions employing slot couplers or fiber, this feature also contributes to the compactness of the system. Moreover, compared with solutions employing slot couplers35 or fiber,32 this coupling MZI array offers a higher level of integration, which contributes to the compactness of the system.

TABLE I.

Comparison of our OCMI scheme with three other schemes.

Signal typeIntegrationReconfigurabilityData typeSize
Metastructure40,41 Microwave Full No Complex 5 × 5 
Simulation35  All-optical Simulation Yes Complex 3 × 5 
Fiber network36  All-optical None No Real 3 × 3 
Microring37  Optoelectronic Partial Yes Real 3 × 3 
Chip-based38  All-optical Partial Yes Real 2 × 2 
This work All-optical Full Yes Complex 4 × 4 
Signal typeIntegrationReconfigurabilityData typeSize
Metastructure40,41 Microwave Full No Complex 5 × 5 
Simulation35  All-optical Simulation Yes Complex 3 × 5 
Fiber network36  All-optical None No Real 3 × 3 
Microring37  Optoelectronic Partial Yes Real 3 × 3 
Chip-based38  All-optical Partial Yes Real 2 × 2 
This work All-optical Full Yes Complex 4 × 4 

Another strength of the work is the implementation of coherent computing on a SOI platform. The SOI platform offers mature fabrication processes and compatibility with CMOS technology, both of which are beneficial for achieving large-scale optical computing. In addition, phase decoherence can be avoided when light propagates in silicon-based waveguides. The on-chip split reference light ensures coherence and a stable phase difference with the iteratively output signal light, ensuring the reliability of complex-valued calculations and detections. Currently, the hybrid integrated system comprising the SOI platform, two-dimensional materials,47,48 and phase-change material (PCM)49 exhibits reduced crosstalk. The proposed OCMI system enables higher accuracy and lower energy consumption through the introduction of this hybrid system. Furthermore, the integration of on-chip laser and photodetector can be considered to realize higher integration in the future.

We have validated the feasibility of on-chip matrix inversion through our model; however, both the speed and the range of matrix A are limited by the unitary matrix K. By optimizing the matrix architecture and adopting a programmable crossbar structure,15,27,50,51 matrix elements can be directly assigned one by one, circumventing the time-consuming iterative process. This enhancement could enable the speed of OCMI. On the other hand, the crossbar structure can perform any transmission matrix to extend the range of A matrix.

The OCMI chips are composed of four ring resonators with equal loop lengths that are mutually coupled. Consequently, the experimental computation results are related to the computing wavelength. Perturbations in the laser's wavelength as well as shifts in the response spectrum caused by surrounding environmental conditions may affect the precision of the experimental calculations. To intuitively explain this issue and analyze the impact of these disturbances on the results, we measured the four-channel output spectrum of the system when a broadband light source was input from port 1, which is shown in Fig. 5(a). The Free Spectral Range (FSR) of the circuit is about 10 GHz. For example, the yellow curve in Fig. 5(a) shows an output response with an extinction ratio of about 3 dB, where a single element can produce a relative error of up to 50% with the perturbations of computing wavelength or the shift of spectral response.

FIG. 5.

Spectral response of the OCMI chip. (a) Four-channel output spectrum of the system when a broadband light source was input from port 1. (b) Spectrum fluctuations of 6 h interval.

FIG. 5.

Spectral response of the OCMI chip. (a) Four-channel output spectrum of the system when a broadband light source was input from port 1. (b) Spectrum fluctuations of 6 h interval.

Close modal

In addition, we conducted tests on the stability of the system’s output by recording the system’s output again after 6 h intervals. It can be observed that the system’s output spectrum only exhibited minimal deviations under stable environment. This is attributed to our temperature control of the OCMI chip. This comparison is shown in Fig. 5(b), which demonstrates the stability of the system. Each pair of almost overlapping curves represents the measurement results before and after a 6 h interval.

Furthermore, the precision of this work is affected by factors such as the accuracy of the coupling factors and the kernel matrix and the amplitude and phase detection accuracies of the receivers. Future improvements could involve designing thermal insulation slots on the chip to reduce crosstalk errors. In addition, the coupler coefficients can be calibrated by scanning the voltage and measuring the output, while the receivers can be optimized for linearity or calibrated post-reception in the electrical domain.

In summary, we propose and experimentally demonstrate a chip-scale iterative chip computing the inversion of complex-valued matrix. Our design boasts a high degree of integration, representing the first instance of chip-scale computational units. Employing the iteration method, the system can converge within a mere 20 iteration cycles. When considering the iteration speed, we exclude the time required for loading the kernel matrix and for coherent detection, focusing solely on the natural iterations occurring during the optical transmission process. Given the impressively short optical path and the spontaneity of the iterative system, an iteration speed of 10 GHz is achieved. The proposed OCMI computing unit is anticipated to be employed in future dense computing systems to accelerate matrix inversion operations. This holds significant implications in the fields of optical communication and intelligent photonic systems.

We are grateful for the financial support from the Natural Science Foundation of China (Grant Nos. U21A20511, 62075075, and 62275088), the Innovation Project of Optics Valley Laboratory (Grant No. OVL2021BG001), and the Knowledge Innovation Program of Wuhan Basic Research (Grant No. 2023010201010049).

The authors have no conflicts to disclose.

Jianji Dong, Xinyu Liu and Hailong Zhou proposed the original idea. Xinyu Liu designed the chip and performed the experimental verification. Xinyu Liu wrote the original manuscript, revised by Hailong Zhou and Junwei Cheng. Jianji Dong, Hailong Zhou and Xinliang Zhang provided resource support and supervised the project.

Xinyu Liu: Data curation (equal); Investigation (equal); Validation (equal); Writing – original draft (equal). Junwei Cheng: Validation (equal); Writing – review & editing (equal). Hailong Zhou: Methodology (equal); Validation (equal); Writing – review & editing (equal). Jianji Dong: Funding acquisition (lead); Resources (lead); Supervision (lead). Xinliang Zhang: Funding acquisition (equal); Project administration (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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