Integrated photonics modular arithmetic processor

To recover the phase difference between two paths, $\varphi\triangleq\Delta\varphi+\varphi_{0}$, from multi-channel optical power signals $\vec{I}$, it is necessary to have knowledge of the corresponding multi-channel optical power states at different values of $\varphi$ beforehand. That requires utilizing voltage as an intermediate quantity and obtaining two mappings:

1. 1.

   Mapping between the applied voltage to PM and resulting $\vec{I}$

2. 2.

   Mapping between the voltage and its produced $\varphi$

An experimental procedure was conducted to access the relationship between $\vec{I}$ and $\varphi$. The voltage applied to PM0 was varied from 0V to 10V with a voltage increment of 0.01V. Then the first mapping is obtained by recording the multi-channel optical power values at different voltage levels. The optical power for the $k$-th channel and produced phase variance at voltage index $i$ are denoted as $I_{k,i}$ and $\Delta\varphi_{i}$. Moreover, the establishment of the second mapping is based on the prior knowledge of the photonics device. Theoretically, the variation of output optical power in $K$-channels optical quantizer follows a cosine function with respect to the phase [31] :

, where $P_{\rm chip}$ is the power of the input laser coupled to the chip. The multiple-channel signals are analyzed individually. Firstly, The signal from the $k$-th channel, denoted as $I_{k}$, is scaled to [-1,1] to conform to the input range of the arccosine function:

As the arccosine function is not monotonically changing with respect to the phase $\Delta\varphi$ , the relationship of $\varphi_{i}^{k}$ is estimated based on the absolute value of the variation of the arccos function:

As that accumulation-based approach introduces errors that accumulate over time, a further process is employed to enhance the precision. This process utilizes the ratio of normalized intensity signals with adjacent voltages, denoted as $R_{k,i}=\widehat{I}_{k,i+1}/\widehat{I}_{k,i}$ :

where $\delta_{i}^{k}=\varphi_{i+1}^{k}-\varphi_{i}^{k}$ is the finite difference of phase with respect to voltage, which along with the integer value $N$ is determined by the previous estimation obtained through Eq.7. This process provides finer estimates within the $\pi$ interval based on the coarse-grained results provided in the previous step. Given the availability of multiple channels, the final step involves an averaging of the phase relationships obtained from each channel :

With the obtained mapping between $\vec{I}$ and $\varphi$, the phase at a given moment can be determined from the measured optical power. Expressly, the current $\varphi$ is specified as the phase value corresponding to the previously recorded optical power with the minimum Euclidean norm compared to the currently measured optical power. It is worth noting that in Fig.2f, Fig.3d, and Fig.4, there are waveform spikes observed in the obtained phase signals at slot boundaries. These spikes result in significant deviations in 1 to 2 sampling points. These deviations are due to synchronization issues related to the multi-channel voltage sources and multi-channel optical detectors in our experimental setup.

Given the multi-channel optical power signals $\vec{I}(\varphi)=\left[I_{1}(\varphi),I_{2}(\varphi),\dots,I_{K}(\varphi)\right]$, which were obtained from the experimental procedure in AppendixA, and the thresholds of comparators for the $K$ channels $\vec{t}=\left[t_{1},\dots t_{K}\right]$. The intersection of $t_{k}$ with its corresponding $I_{k}(\varphi)$ determines two boundary points in the 360-degree phase range. The sorted boundaries of all channels is denoted as $\vec{p}_{{\rm expt}}=\vec{p}_{{\rm expt}}(\vec{I},\vec{t})$, which represents the actual phase boundary points. And in the ideal scenario, the phase boundaries $\vec{p}_{{\rm ideal}}$ should shifted by half interval $\frac{2\pi}{2m}$ to the discrete phase points $\varphi_{0}^{{}^{\prime}}+\Delta\varphi_{\mathbb{Z}}$, where the total static phase $\varphi_{0}^{{}^{\prime}}$ is adjusted by applied an extra bias phase $\varphi_{b}$. Thus the ideal phase boundaries can be described as $p_{ideal}^{i}=\varphi_{0}+\varphi_{b}+i\cdot\frac{2\pi}{m}+\frac{\pi}{m}$.

To accurately extract the phase information and reduce the probability of incorrect quantization operation, an optimization procedure was performed to find the optimal values of $\varphi_{b}$ and $\vec{t}$ that minimize the discrepancy between $\vec{p}_{{\rm ideal}}(\varphi_{b})$ and $\vec{p}_{{\rm expt}}(\vec{I},\vec{t})$. This optimization aims to enhance the robustness of OPDCs by placing the most error-prone phase point in a less vulnerable position. The optimization problem can be formulated as follows:

The optimization objective is to minimize the loss $\triangleq\lVert\vec{p}_{{\rm ideal}}(\varphi_{b})-\vec{p}_{{\rm expt}}(\vec{I},\vec{t})\rVert_{\infty}$. Solving this optimization problem using gradient-based methods can be challenging. Therefore, a heuristic optimization method, the differential evolution (DE) algorithm [44] is employed. The DE solver provides multiple trial values for $\vec{t}$ at each iteration. A search algorithm was performed for each trial threshold to determine the minimized loss under a specific search range of $\varphi_{b}$ with a given precision. Then those minimum values were passed back to the DE solver for further exploration in the next iteration.

In the practical experiments, the DE solver had a population size of 20 and was run for 40 iterations. Fig.6 presented the iteration process of threshold optimization for $m=7$ and $m=10$ in the case of 5-channels OPDC. When modulus $m=2K$, the intervals tend to be evenly divided. In our previous works [31], we adopted a uniform threshold approach, where the thresholds for all channels were set to half of the normalized power. This approach equally divides the phase range $2\pi$ into $2K$ intervals in the ideal case. However, as shown in Fig.6a, in real scenarios where manufacturing variations exist, the optimized threshold approach performs better than the uniform threshold approach. Moreover, the optimized threshold approach can adopt different moduli with the same optical device. When modulus $m<2K$ (Fig.6b), there is a need to merge the remaining phase intervals to unify them into uniformly divided intervals of $m$. This merging process involves mapping multiple different codewords to the same integer.

Fig.7a,b displays the optimized results for $m=9$ and $m=7$. It is worth noting that for $m=7$, it is possible to utilize information from 5 optical power signals to quantify the phase information. However, signals from only channels 1, 3, 4, and 5 were employed to reduce the length of the codewords in our experimental setup. The experimental results demonstrate that this approach still yields satisfactory outcomes. Moreover, in Fig.7c, we attach the multi-channel optical power signal of the experiment in Section2.3, which uses the optimized parameters corresponding to figure7b. The time-domain waveforms and the digital results of threshold decision confirm the effectiveness of the optimization method.

The incorporation of high linearity phase modulators, e.g., lithium niobate modulators [45], enhances the capabilities of the ODPC by allowing it to be combined with low-bit-width DACs, thus achieving a larger overall bit width in the limitation of given the number of PMs. To accomplish this, the $L$-bit digital operand is partitioned into G groups corresponding to G DACs, where the $i$-th DAC has a bit width of $W_{i}$ (with $\Sigma_{i=1}^{G}W_{i}=L$). The phase response of the $i$-th PM to the least significant bit (LSB) voltage of the $i$-th DAC is set to $\Delta\varphi_{i}^{{}^{\prime}}=2\pi|2^{w_{i}}|{m}/m$, where $w_{i}=\Sigma_{j=1}^{i-1}W_{j}$ represents the bit position of the LSB of the i-th DAC in the original L-bit operand. By mapping the digital operand $X$ using the DACs and the corresponding PMs, the resulting $\Delta\varphi_{X}$ is corresponding in Eq.3:

The basic design illustrated in Fig.4 is the specific case in which $W_{i}$ is limited to 1. The PPMAP unit, which incorporates the generalized ODPCs, is depicted in Fig.8a.

For a given electro-optic coefficient of the electro-optic material in the modulator and a given modulus, $m$, the objective of the grouping strategy is to minimize the magnitude of the modulation phase, which contributes to reduceing the size and power consumption of the modulator [45].

According to Eq.11, the LSB phase response of the $i^{{\rm th}}$ modulator for the DAC should be $2\pi|2^{w_{i}}|_{m}/m$. To achieve this, we need to choose a grouping method for the L-bit digital signal such that $|2^{w_{i}}|_{m}$ is as close to zero as possible.

Being close to zero implies being in proximity to either 0 or $m$ because a digital value $|2^{i}|_{m}$ close to m can be achieved by modulating the phase of the digital value $m-|2^{i}|_{m}$ on the opposite waveguide. That is achieved by subtracting a small number from a large number in the modulo operation. Fig.8b illustrates examples of bit grouping for moduli of 15, 19, 21, 30, 31, and 33, where the maximum modulation phase of any PM, $(2^{W_{i}}-1)\times\min\left(|2^{i}|_{m},m-|2^{i}|_{m}\right)$, is designed to around $2\pi$. We have observed that the bit weights $|2^{i}|_{m}$ can be close to zero by employing an appropriate grouping strategy. And for some moduli, $|2^{i}|_{m}$ exhibits periodicity. The specific analysis for the ODAC design depends on the bit width and performance requirements of the DAC, the characteristics of modulators, and the specific modulus.

In Fig5a, the relative error (RE) is defined as the average absolute value to error over the dynamic range. The RE of a calculation with $n$-bit precision is defined as the $n$-bit relative quantization error of the calculation result:

And the theoretical RE of the intensity-based method under Gaussian noise with uniform intensity is given by:

where $f(x)$ represents the probability density function of the normal distribution and $A_{\max}$ is the maximum signal amplitude. The ratio of $\sigma$ and $A_{\max}$ is determined by its relationship with SNR and peak-to-average power ratio (PAPR):

Consequently, the relationships of RE between SNR as well as the calculation precision expressed in the form of the bit length, $n$, is inferred:

The logarithm of the RE decreases linearly with SNR and $n$. Assuming the calculation results follow the uniform distribution, it has a PAPR of 4.77 dB (i.e., $A_{\max}=\sqrt{3}A_{{\rm rms}}$), Eq.15 can be simplified to:

This equation shows that as $n$ increases, the required SNR increases linearly with a slope of 6.02, corresponding to the numerical simulation shown in Fig5a. It is worth noting that the constant term in the equation may differ depending on the specific metrics of calculation error used. However, the coefficient of the linear term remains similar. For example, in the reference [22], they directly employ ${\rm SNR}=20\log_{10}2^{n}=6.02\cdot n$, which aligns with the derived relationship in Eq.16. That linear term indicates that as the calculation precision increases, the required SNR will reach an unattainable level and imply an exponential increase in energy consumption [21].

For the PPMAP architecture, the probability of the occurrence of errors, denoted as ${\rm P}$, is equal to 1 minus that of the PPMAP system successfully operating, which means that there should be no errors in any of the $N$ PPMAP units. Moreover, the probability of the $i$-th PPMAP unit not encountering errors, denoted as ${\rm P}^{{\rm unit}}_{i}$, is determined by accurately achieving threshold decisions for the output optical power on all the $K_{i}$ channels. The success rate of the $j$-th channel is represented as ${\rm P}^{\rm chan}_{i,j}$.

In the simulation, each chosen modulus, i.e., 7, 11, 19, 23, is prime, denoted as $p$. In this case, an index-sum modular multiplication operation is implemented via a modular addition operation with moduli $p-1$ [46], equal to twice the designed $K_{i}$ in each PPMAP unit. That makes for all discrete phase points in $\Delta\varphi_{\mathbb{Z}}$, the relative distance, $\left\lvert d_{j}\right\rvert/A_{\max}$, between their corresponding ideal output optical power and decision threshold in all $K_{i}$ channels are given by [32].

Considering the relative variance noise $A_{\max}/\sigma=\sqrt{3}A_{{\rm rms}}/\sigma=\sqrt{3}\cdot 10^{\frac{{\rm SNR}}{20}}$, the success rate for the $j^{{}^{\prime}}$-th channel in the $i$-th unit:

where ${\Phi}(x)=[1+$erf$(x/\sqrt{2})]/2$, i.e., the cumulative distribution function of standard Gaussian distribution $\mathcal{N}(0,1)$. Then considering the commutative law of the continued multiplication in Eq.17, $\Pi_{j=1}^{K_{i}}{\rm P}_{i,j}^{{\rm chan}}=\Pi_{j^{{}^{\prime}}=1}^{K_{i}}{\rm P}_{i,j^{{}^{\prime}}}^{{\rm chan}}$, the error model described in Eq.4 can be derived.

The deviation between the simulated relative frequency and the probability model around 31 dB in Fig.5 is since only a few samples out of the 40,000 experienced errors near an SNR of 31 dB, which does not satisfy the prerequisites of the law of large numbers. Apart from this specific range, the relative frequency obtained in our simulations aligns well with our probability model, confirming our model’s accuracy.

Although all the above experiment is carried out on the same PPMAP chip with five-channel OPDC, the number of the output channel in OPDC is adjustable. To demonstrate the expansibility of OPDC, we design and fabricate an OPDC module with nine output channels. Fig.9 shows its structural rending, micrograph, and test results. Taking the modulus equal to 18 as an example, the deviation between the experimental and ideal boundaries is relatively small, indicating that the expanded OPDCs can reliably extract the calculation result from phase information.

Besides horizontal expansion by adding the number of output channels, OPDC can be vertically expanded with a cascaded structure [47], which possesses relatively low wavelength sensitivity. Furthermore, the OPDC fabricated on a thin-film lithium niobate platform exhibits the same performance on phase identifying [48], indicating the potential for monolithic integration of the overall PPMAP system on lithium niobate process platform. It should be emphasized that the Gray code generated by OPDC requires an additional codeword conversion unit to transform into the natural binary code for subsequent calculations. However, specific-designed photonic structure [41, 42, 43, 20] has the potential to implement the fixed codeword conversion task.

We employed a Tunable Semiconductor Laser (Santec TSL 570) to provide the incident CW laser at the wavelength of 1535 nm, and a Multi-port Optical Power Meter (Santec MPM 210) to measure the multi-channel optical power signals. After the electrical packaging of the integrated photonic chip, metal wires connect the electrodes on the chip to the FPC (Flexible Printed Circuit) adapter board. We employed a multi-channel programmable voltage source (nicslab XDAC-40MUB-R4G8) to control the applied voltages on the electrodes. The experimental equipment and the host computer were connected via GPIB interface and Ethernet cables, which facilitated the transmission of control signals and the acquisition of experimental data. In-house software was employed to perform real-time monitoring of the optical power signals from each port, which allowed evaluate system stability and monitor the polarization state of the incident light for controlling the coupling efficiency.