Diffraction Neural Network for Multi‐Source Information of Arrival Sensing

All‐optical diffractive neural networks (DNNs) based on passive structures have been shown to perform complicated functions by optical acceleration, thus reforming in situ applications requiring digital computing processing. Its stable, compact, and passive features make it suitable for many conventional applications in complex electromagnetic environments. Here, a DNN based on a multilayer passive metasurface array is presented to estimate the information of arrival (IOA) over a wide range of frequency bands and incident angles. Unlike traditional methods, multiple correlated or coherent electromagnetic signals interfere with each other, resulting in difficult separation which is a fundamental obstacle in multi‐source IOA detection. Here, the proposed diffraction system creates separate virtual channels to isolate and process different incoming waves. It recognizes and classifies the beams, focuses them on the output plane divided by the arrival angle and working frequency, and displays the IOA and number of sources in real time at the speed of light. Furthermore, the massive‐parallelism and data‐post‐processing‐free strategy allows for broader applications in harsh environments, such as seismic detection and underwater circumstances.


Introduction
Neural networks that can learn, associate, remember, and analyze vast amounts of information quickly and efficiently without human intervention have developed into powerful tools for handling various challenging and informative tasks.Deep learning, which is a key example, has found diverse application scenarios, DOI: 10.1002/lpor.202300202 from computer science (e.g., computer vision, [1][2][3] target tracking, [4][5][6] speech recognition [7][8][9] ) to scientific research (e.g., cloaks, [10][11][12] holography, [13][14][15] wave sensing, [16][17][18] and beyond [19][20][21][22][23] ).The computational speed of computerdependent neural networks is constrained by the performance and memory of the hardware and results in high energy consumption.To eliminate such defects, the usage of electromagnetic (EM) waves without any processing as the input and directly managing the incident wave in the physical layer leads to the formation of a diffraction neural network (DNN).Light-speed-operation and data-post-processing-free features save the DNN from demanding memory and time-consuming computation.The implementation of DNN extends from recognition and classification [24,25] to pulse shaping [26] and spatially controlled wavelength demultiplexing [27] in the terahertz band; however, previous works rarely considered broadband and large incidence angles or cases of multiple beam incidence.
Information of arrival (IOA) detection determines the frequency and orientation of a target by analyzing the received echo.It is a key technology for adaptive arrays, which occupies a pivotal position in the fields of mobile communications, radar, and medical imaging.[30][31] The system needs to be precisely designed to reduce noise errors and interference caused by uneven hardware conditions, and the algorithm is restricted by various fundamental factors such as the data length, received signal purity, and source direction. [32]To alleviate these deficiencies, IOA detection approaches that rely on neural networks and metasurfaces [16,17] have emerged with high efficiency and integration levels.Nevertheless, the addition of active devices to the metasurface barely allows broadband work, let alone multisource detection. [16]Active devices are prone to be interfered and unstable in complex and changeable EM environments.A generalized regression neural network traverses a pre-built original database to indirectly determine the IOA, which is unsuitable for multisource determination. [17]The real obstacles of multisource incoming wave detection are the interference of partially correlated or even coherent signals with each other and the difficulty in separating them.[35][36][37] The emergence of these new approaches cannot fundamentally avoid the issues of slow speed, time-consuming computation, difficult separation of multiple incident sources, and unintuitive output.
In this work, we propose an efficient IOA estimation method for single and multiple EM wave incident sources using a diffractive optical network with fixed physical parameters optimized in advance by a deep learning-based strategy.This hardware design utilizes several metasurfaces and EM wave detectors to quickly identify and classify EM wave information at different frequencies and incident directions.Instead of relying heavily on software-based computation, our diffraction network automatically classifies EM wave information and intuitively displays it on the detection plane through a light-speed operation when it interacts with the metasurfaces.In the presence of multiple incident sources, waves with different incidence angles and frequencies are focused separately through the IOA estimation system without interference.Additionally, the observer can simultaneously determine the number of sources from the output plane.Besides its simplicity and explicitness, the sensing system can perform massive parallel processing without data post-processing, thereby overcoming the necessity of customizing the data algorithm and physical structures according to the incident sources.Our perceptual system can also extend the working range and refine the detection resolution to meet higher estimation requirements.It should be noted that this approach is a general design strategy not only for microwave band, but can also be extended to other frequencies such as visible and infrared bands.We believe that this study expands the flexibility and freedom of EM field regulation and facilitates the application of information carried by broadband, multidirectional incident waves.

Results
A schematic of the wave-sensing DNN is shown in Figure 1.Our diffractive optical network comprises two physical layers, each composed of numerous phase modulation units.Incident waves with multiple directions and different frequencies irradiate the wave-sensing diffractive network and are automatically recognized and classified when the EM waves interact with the diffractive physical layers.Finally, all the transmitted waves converge into the output plane, instantly revealing the IOA.Considering the incident angle and frequency as horizontal and vertical coordinates, the output plane is redivided into numerous sub-regions; thus, the position and number of focusing points highlight the IOA and number of incident sources.
To form discrepant phase accumulation for manipulation, the diffraction layers in previous works [24][25][26][27] required varying the thickness of the dielectric pillars to change the propagation paths of EM waves.The frequency and incident angle affected the range of height variation required for the dielectric pillar to obtain 2 phase coverage.The wide band and large incident angle aggravated the thickness variation, which enhanced the back-and-forth reflection between pillars.As shown in Figure 2a,b, we chose the polarization conversion unit [38] with a stable and high conversion rate to obtain better effects.The top and bottom layers each have two parallel metal gratings, in which two pairs are orthogonal to each other.The middle layer contains a circular ring with an oblique metal bar structure.Figure 2c,d indicates that it can operate at large incident angles and broad bands, covering  phase (another  phase is achieved at  = −45 • ) and maintaining conversion rates higher than 0.8 (see Section S1, Supporting Information for more details).
In general, our perceptual diffraction network design assumes an input bandwidth between f min and f max , with incident angles ranging from − to +.For better and clearer location of each incident angle and frequency during training, the continuous incident angle range is discretized into M points, and the bandwidth is divided into N frequency points.With  and f as the x-axis and y-axis, respectively, the output plane is divided into several subregions.Each discrete IOA point occupies a specific focusing region; thus, there are altogether M × N focusing targets to be optimized.Next, we consider the wave propagation model in a multilayer metasurface system, and select a broadband optical forward model based on the Rayleigh-Sommerfeld diffraction formula. [39]The errors between the targets and network predictions are measured using a customized loss function.In each iteration, plane waves at all frequencies and incident angles propagate through the multilayer metasurface; thus, the   Without loss of generality, the energy distribution in the focusing region conforms to the physical law, [40,41] and the size of the focal spot is positively correlated with the wavelength  (see Section S2, Supporting Information for details).The region (, f ) has the strongest energy concentration, at least 3 dB higher than the sub-bright region.Adjacent spots are spaced 42 and 36 mm apart in  and f directions, respectively, leaving sufficient space for untrained frequency points and incident angles.As shown in Figure 3a, the one-dimensional ( = 0 • ) plane wave under different , f is directly considered as the input, which is a complex form of amplitude and phase.An input of the same size as a metasurface has a uniform intensity.In the design of our diffraction neural network, each physical layer comprises multiple units, and the connection of units between two adjacent layers mimics the relationship between neurons in deep learning.The physical parameter optimized for the diffraction layer is the arm length , which enables phase control of each diffraction layer (see Section S3, Supporting Information for details of DNN training).The transmission coefficients, comprising phases and amplitudes, produce a unique complex modulation applicable to all diffraction layers.Because our goal is to achieve zero-error detection, all (M × N) incident cases form the training set.For different incoming waves, if the peak values of the spots on the imaging plane falls in the corresponding subregions, it indicates that the detection is accurate.Figure 3b shows that during training, the accuracy rate increases steadily and reaches 100%.The transmitted waves are focused precisely onto the pre-designed posi-tion, with only one bright spot on the output plane, as shown in Figure 3c.

Realization of Single Source Detection
To demonstrate the advantages of the sensory optical diffractive neural network in terms of compactness and real-time performance, we first tested the IOA sensing ability of the diffraction system under all trained combinations of frequencies and incident angles for a single source.The frequency band ranged from 8-12 GHz and was split into nine detection points (at intervals of 0.5 GHz).The incident wave scanned from −30 • to 30 • in steps of 10 • (seven discrete angle values).Therefore, our diffraction system could perform zero-error sensing for at least 63 groups (, f ).We compared the optimization results of different layers, and finally selected two layers of metasurface to construct the estimation system.Taking the wavelength at the center frequency (10 GHz) as the standard, we set the size of the metasurface to be about 12 10G , and the distance between layers to be about 7 10G .Our perceptive DNN was implemented using two metasurfaces, with each layer comprising 59 × 65 units.The target was defined more accurately by redividing the output plane into 177 × 195 pixel grids with a unit of 2 mm and calculating the changing energy distribution in the focusing subdomain (34 × 34 mm 2 ) by the unit of pixel.Therefore, we applied the nearest-neighbor upsampling algorithm to extend the output of the last metasurface to ensure consistency with the number of pixels in the output plane.In the computer training, the learning rate of the DNN was set to 10 −3 to ensure that it did not fall into  the local optimal solution.We applied the Adam optimizer to update the weight smoothly, and the modified MSE loss function to evaluate the electric field strength error between the output plane and targets.
Due to simple structure, the metasurfaces are fabricated by standard printed circuit board technology.Figure 4a shows a schematic of the experimental setup (see Section S4, Supporting Information for the detailed experimental equipment).In the experiment (Figure 4b), the two metasurfaces were separated by 200 mm.A horn antenna, which covered the entire working frequency band, was placed at the center of the metasurface at 600 mm (600 mm = 16 8G , where  8G is the maximum wavelength, 37.5 mm).The spherical waves emitted by the transmitting antenna can be approximated as plane waves when they reach the metasurface.A probe fixed 200 mm away from the scanning platform was connected to a vector network analyzer (VNA) to determine the amplitude of the electric field.The probe scanned the output plane with a fine-grained step size of 4 mm to ensure an accurate field intensity distribution of the whole imaging plane.The scanning area was 404 × 384 mm 2 , which was slightly larger than the output plane during training.The entire experiment was conducted in an anechoic microwave chamber.Meanwhile, the influence of stray and diffracted waves on the detection effect was avoided by placing absorbing materials tightly around the metasurfaces.The position of the horn antenna in each group of experiments, namely, the direction of the incident wave, is shown in Figure 4c (top view).The perceived capabilities of the sensing system at −30 • to 30 • and 8-12 GHz were experimentally verified (Figure 4d).The experimental results were normalized from the maximum field intensity on the plane, while the areas with energy less than 0.3 were filtered to 0. The energy profiles in the y and z directions at the focal points are shown in Figure 4e,f.The brightest circular spots are clearly visible on the plane, which is in strong contrast with the rest of the region.The experimental results show that our sensing system can accurately recognize incoming wave information within a preset range, which is consistent with the numerical results (see Section S5, Supporting Information for more example).We note that all the electronics here were used only to illustrate the results obtained at microwave frequencies, whereas at visible frequencies, the results can be observed with the naked eye.

IOA Perception of Multiple Incident Sources
Next, we focus on a sensing system for detecting the frequency and incident angle from multiple sources.In real-world scenarios, wave-sensing systems cannot always work with ideal EM measurement devices such as VNA.The experimental setup is described in Section S6 (Supporting Information).To mimic an actual multi-source detection working environment, we used horn antennas connected with signal generators to emit multiple single-frequency EM waves.The two antennas were positioned as horizontally as possible at the middle of the system so that the plane waves covered the whole metasurface, and the intensity of the focal points was uniform (Figure 5a).The perceived frequencies and directions of the incident waves are shown in the top view of Figure 5b.We used a probe, frequency mixer, and ZYNQ (an ultra-high-performance embedded system produced by Xilinx) integrated with an analog-digital converter to analyze  the field intensity data.The received voltage amplitude indicated the variation and strength of the incident wave signal in the time domain.We scanned 63 positions at 36 mm (f direction) and 42 mm ( direction) intervals using the field scanning platform to determine the EM strength and incident wave information.Voltage signals of 1024 ps were sampled at each position, and the varying curves are shown in Figure 5d.The detected signals in the target region display regular variations in periodic stability over time whereas those collected in other regions (non-incoming information points) are random and irregular noise.To quantify the results visually, we define an indicator based on the root mean square (RMS) of the measured voltage: the absolute variance (normalized by the maximum RMS in the plane) between the current RMS and background noise RMS, namely,|RMS − RMS bg noise |∕RMS max .Figure 5c shows the distribution of the RMSs on the output plane, which exhibits oneto-one correspondence with the incident case in Figure 5b.The curves in the target regions show the coherence and correlation of the incident signals.According to the definition of Pearson's correlation coefficient, the correlation coefficients of the two sources in the target areas were 0.5598, 0.4028, and 0.7485, respectively.The first set of experiments in Figure 5 adopts coherent beam incidence, and sinusoidal signals with the same period (frequency) are measured at the incoming information point.This achieves the desired separation of the coherent signals.Additionally, the non-incoming frequency is filtered in the target area, and only the pure incident frequency is observed (Figure 5e).The experimental results illustrate the high accuracy and purity of the incident-frequency detection, and reveal remarkable consistency with the optimization objective.Different incoming waves, even coherent and strongly correlated beams, arriving at our sensing system were equivalent to automatically entering a separate channel distinguished by the frequency and incidence angle, without connection and interference.Here, the voltage information sampled in the experiment was in exact agreement with the actual incident wave information (see Section S7, Supporting Information for more example).It is worth noting that our detection system has no pressure to deal with multiple (more than two) sources and can accurately identify multiple sources at the same frequency or incident angle without any interference.

Discussion
We aimed at designing an independent virtual channel for each group of (, f ) incident waves which can isolate and process (in parallel) incoming waves under different incident angles and frequencies, even for correlated or coherent signals.According to statistical standards, similar sinusoids reflect the incident signals with varying degrees of correlation.It was confirmed that strongly or weakly correlated incoming waves could be accurately estimated using our detection system.Moreover, for coherent EM waves incident from different directions, (0, 10) and (10,10), the energy converged to the corresponding target regions, which also clearly indicated the number of sources.It is of great theoretical significance and engineering value to study the method of source number estimation, which is of paramount importance for actual direction-finding systems.The proposed system avoids the challenge of conventional detection which requires a prior condition of the number of sources, but dissociates each incident sig-nal and focuses on the output plane.To clearly demonstrate the physical model and facilitate its practical implementation, we primarily considered source detection from 8-12 GHz, −30 • to 30 • .However, the system designed here can perform more advanced tasks, operate in a wider range of frequency bands and incident angles, and also detect an unlimited number of incident sources in parallel and real time.Further, the results of incoming wave detection in the optical band are instantly visible to the naked eye in the output plane.The experiments on multi-source incoming wave information detection and estimation of the source number show the excellent recognition and classification ability of the sensing system for coherent and correlated beams, which eliminates the trouble of the traditional detection algorithm in processing the signals collected into a channel with significant time and energy consumption.As desired, our sensing system can effectively address the shortcomings in incoming wave detection, such as the separation of coherent and correlated beams, IOA acquisition without additional data processing, and absence of algorithms necessitating prior data (such as the number of incident sources).
In conclusion, we present a metasurface-based incoming information sensing system designed by a computer-based DNN and experimentally implemented using passive meta-neurons.In contrast to the approach of converging all signals into one channel, our strategy fundamentally divides the channel by frequency and incident angle and virtually isolates different beams to identify the inseparable coherent and correlated beams.It achieves real-time, multi-source, and instantly available IOA detection without participation of high-cost computers, thus eliminating the heavy hardware burden caused by conventional deep learning.Besides our proposed efficient real-time frequency and incidence angle detection, polarization and distance detection can also be included as desired targets. [42,43]It is noteworthy that its sensor-scanning-free and power-supply-free features can be adapted to perform more sophisticated tasks in harsher environments in seismic detection, radar, and sonar.

Experimental Section
Implementation of Loss Function: The output plane is divided into focused and unfocused regions.The focusing region refers to the fixed region specified for each incoming wave angle and frequency, namely, the region bounded by the white dotted circle on the left side of Figure 3c.The non-focusing region is not occupied by a fixed frequency or incident angle specified.The size of the meta-neurons was 6 mm.To maintain the physical size of the output plane, the output plane was re-partitioned with a step size of 2 mm, so that it was composed of 3M × 3N pixels.I m (m = 0, 1, 2, … < 3M × 3N) was used to represent the power of each pixel point.The energy of all pixels involved in a focusing position (white circle) was added separately to obtain the power I ′ n (n = 0, 1, 2, … < 63) of the focusing position under each group of incident waves.The power loss term includes the loss of the focusing region and the entire output plane, which is defined as: where W 1 = 0.6, W 2 = 0.4.

Figure 2 .
Figure 2. Metasurface element and processing of the influence of the frequency and incoming wave direction on the phase and amplitude.a) Schematic of two metasurfaces with the polarization transformation function.b) Metasurface element in the DNN.c) Phase affected by the incoming wave direction, frequency, and .The polarization conversion unit can obtain phase coverage from − to .Phase coverage from 0 to  at  = 45 • is shown.d) The amplitude at any incident angle,  and frequency exceeds 0.8.
computed loss values propagate backward to update the parameters until the prediction results converge to the desired spectral distribution.

Figure 3 .
Figure 3. Design of wide-band, multi-directional incident wave sensing diffraction networks and output results of single source incidence.a) Schematic of the wave-sensing system architecture.b) Loss and accuracy of the training set versus epoch.c) Optimization results of the DNN at various frequencies and incident angles.

Figure 4 .
Figure 4. Experimental design and results of single source incidence.a) X-polarized plane waves incident through the first metasurface are converted to y-polarized waves, and then restored to x-polarized waves after passing through the second metasurface.b) A single source of each frequency and direction is incident on the estimation system, and the probe scans the electric field in the output plane.c) Schematic of the frequency and direction of a single incident source during experiment (top view).d) Experimental detection results of a single EM wave at different frequencies and incident angles.e,f) Profile electric field distribution of the focal points in (d) along the y and z directions.

Figure 5 .
Figure 5. Experimental results for the detection of multiple simultaneously incident sources.a) Experimental setup of two incident source information estimation.b) Schematic of frequency and direction for multiple source incidence in the experiment, top view.c) Experimentally detected signal intensity distribution of the output plane with multiple incident sources.d) Time-varying signal in the target area (red) in (c).e) Experimental detection of frequency.The incident frequencies can be clearly observed in the target areas, while no other frequencies exist.