1.

Introduction

Machine learning, and in particular deep learning, has drastically impacted the area of information and data processing in recent years.^1–5 Research on optical machine learning has a very rich history,^6–12 due to its advantages in terms of power efficiency, scalability, computational capacity, and speed. With today’s substantial computational power, advances in manufacturing technologies [e.g., three-dimensional (3-D) printing and lithography], and increasing availability of machine learning-related programming tools (e.g., TensorFlow), there has been remarkable progress on the use of machine learning in optics and photonics, focusing on, e.g., the development of new integrated photonics devices^13–24 or the design of application-specific free-space optical neural networks.^25–27

The task of object recognition and classification is an important application area of machine learning. It is conventionally realized in two main steps. First, a lens-based imaging system followed by a CMOS/CCD array captures a scene at hand. The digitized and stored image of the scene is then fed into an all-electronic artificial neural network (ANN) pretrained for the task. The sampling density, and thus the number of detectors on the optoelectronic sensor plane, are dictated by the desired spatial and/or temporal resolution of the designed system.²⁸ In a classification system, high spatial resolution is generally desired due to the vital importance of spatial features for the performance of ANNs, forcing the pixel count and density of the sensor arrays to be relatively high, which, consequently, increases the requirements on the size of the memory as well as the computational power, inevitably hampering the effective frame-rate. The compressive sensing/sampling field has broadly aimed to overcome some of these resource inefficiencies in conventional optical systems. However, computationally demanding recovery algorithms associated with compressive sensing frameworks partially hinder its application for a wide range of areas in need of real-time operation.

In earlier work, we introduced diffractive deep neural networks,^25,27 which are composed of successive diffractive optical layers (transmissive and/or reflective), trained and designed using deep learning methods in a computer, and then physically fabricated to all-optically perform statistical inference based on the trained task at hand. In this framework, complex wave field of a given scene or object, illuminated by a coherent light source, propagates through the diffractive layers, which collectively modulate the propagating light such that the intensity at the output plane of the diffractive network is distributed in a desired way; i.e., based on the specific classification or imaging task of interest, these diffractive layers jointly determine the output plane intensity in response to an input. The applications of this concept for the design of optical imaging systems, as well as all-optical object classification, were experimentally realized.²⁵

Unlike traditional, imaging-based machine vision systems, a diffractive optical neural network trained for a classification task needs only a few optoelectronic detectors, as many as the number of individual classes in a given dataset. Following their design and fabrication, diffractive optical neural networks execute classification with passive optical components, without the need for any power except the illumination beam and a simple max operation circuitry at the backend. Unless optical nonlinearities are utilized, diffractive optical neural networks are linear in nature, except the final optoelectronic detector plane; despite its linearity, additional diffractive layers have been shown to improve the generalization and inference performance of the network, indicating the depth advantage that comes with the increasing number of diffractive neural layers in the optical network.^25,27 With a single photodetector assigned to each individual class of objects, Ref. 27 demonstrated a blind testing accuracy of 97.18% for all-optical classification of handwritten digits (MNIST database, where each digit was encoded in the amplitude channel of the input) and achieved 89.13% for all-optical classification of fashion-products (Fashion-MNIST database, where each object was encoded in the phase channel of the input).

In spite of the promising performance of the earlier work on diffractive optical networks, these architectures suffer from a well-known limitation in optics: the optoelectronic detectors are only sensitive to the incident optical power rather than the complex optical field, which limits the range of realizable values to nonnegative real numbers. In this work, this nonnegativity of the detected signal at the output plane of diffractive neural networks is mitigated through a differential detection scheme, which employs two optoelectronic detectors per data class at the output plane [see Fig. 1(b)]. In this differential detection scheme, the output signal for each class is represented by the normalized difference of the signals of the corresponding detector pair. Therefore, half of the optoelectronic detectors are accounted for in the positive part of the output signal, and the other half represents the negative part. The final inference of the diffractive optical network is simply made based on the maximum differential signal detected by these positive and negative detector pairs, each representing a separate class. Using this differential measurement scheme together with five diffractive layers having a total of 0.2 million neurons, we numerically achieved blind testing accuracies of 98.54%, 90.54%, and 48.51%, for MNIST, Fashion-MNIST, and grayscale CIFAR-10 datasets, respectively (see Fig. 2 and Table 1). For comparison, without using the differential detection scheme, similar diffractive optical neural networks achieve blind testing accuracies of 97.51%, 89.85%, and 45.20% for the same datasets, respectively.

Fig. 1

Illustration of different diffractive neural network design strategies. (a) Standard design refers to $D([M,0],[1,L,P])$, where $M$ is the number of classes in the target dataset, which in this specific design is also equal to the number of detectors per diffractive neural network, $L$ is the number of diffractive layers per optical network, and $P$ refers to the number of neurons per diffractive layer. In the examples shown in this figure, $L=5$, $P=40\mathrm{k}$, meaning 0.2 million neurons in total. (b) Differential design shown on the left refers to $D([M,M],[1,L,P])$, whereas the one on the right refers to $D([M][M],[2,L,P])$ as it uses two different jointly optimized diffractive networks, separating the positive and the negative detectors by placing them at different output planes without optical coupling between the two. (c) Class-specific design shown here refers to $D([M/N,0],[N,L,P])$, where $N>1$ is the number of class subsets (in this example, $N=2$ case is shown). (d) Class-specific differential design shown here refers to $D([M/N,M/N],[N,L,P])$ where $N=2$ is illustrated. In general, there can be another version of a class-specific differential design where each diffractive neural network has only positive or negative detectors at the corresponding output plane; this special case is denoted with $D([M/N][M/N],[2N,L,P])$, where $2N>2$ refers to the number of jointly designed diffractive neural networks. $N=1$ case, i.e., $D([M][M],[2,L,P])$ is included as part of (b) right panel, and we do not consider it under the class-specific neural network design since there is no class separation at the output/detector planes.

Fig. 2

Operation principles of a differential diffractive optical neural network. (a) Setup of the differential design, $D([M,M],[1,L,P])$. In the example shown in this figure, $M=10$, $L=5$, $P=40\mathrm{k}$. (b) A correctly classified test object from the MNIST dataset is shown. Subparts of (b) illustrate the following: (i) target object placed at the input plane and illuminated by a uniform plane wave, (ii) normalized intensity distribution observed at the output plane of the diffractive optical neural network, (iii) normalized optical signal detected by the positive (red) and the negative (blue) detectors, (iv) differential class scores computed according to Eq. (1) using the values in (iii). (c) and (d) are the same as in (b), except for Fashion-MNIST and CIFAR-10 datasets, respectively. Note that while the input object in (b) is modeled as an amplitude-encoded object, the gray levels shown in (c) and (d) represent phase-encoded perfectly transparent input objects. Since diffractive optical neural networks operate using coherent illumination, phase and/or amplitude channels of the input plane can be used to represent information.

Table 1

Blind testing classification accuracies of nondifferential (top row) and differential diffractive optical networks, without any class specificity or division. M=10 classes exist for each dataset: MNIST, Fashion MNIST, and gray-scaled CIFAR-10. For each data point, the training of the corresponding diffractive optical neural network model was independently repeated six times with random initial phase modulation variables and random batch sequences; therefore, each data point reflects the mean blind testing accuracy of these six trained networks, also showing the corresponding standard deviation.

In addition to the introduction of differential detection per class, in this work, we also made use of parallel computation capability of passive diffractive layers, and jointly optimized separate diffractive optical neural networks for positive and negative detectors (see e.g., Fig. 3), which are designed to work in parallel for differential inference of data classes. In some other implementations, we jointly optimized a group of diffractive optical networks, where each diffractive network was specialized to infer a subset of classes (see e.g., Fig. 4). Our results demonstrate the significant advantages of these design strategies that use a combination of differential detection and class-specific optimization of individual diffractive neural networks that work all in parallel. For example, with class-specific differential diffractive optical neural networks that are jointly optimized, we numerically achieved blind testing accuracies of 98.52%, 91.48%, and 50.82% for MNIST, Fashion-MNIST, and grayscale CIFAR-10 datasets, respectively, coming close to the inference performance of, e.g., LeNet, an all-electronic deep neural network. Apart from these jointly optimized diffractive neural networks, we also designed independently optimized multiple diffractive neural networks, inspired by ensemble methods used in deep learning. For example, using three independently optimized differential diffractive neural networks that separately project their diffracted light onto a common output/detector plane, we numerically achieved blind testing accuracies of 98.59%, 91.06%, and 51.44% for MNIST, Fashion-MNIST, and grayscale CIFAR-10 datasets, respectively.

Fig. 3

Operation principles of a diffractive optical neural network using differential detection scheme, where the positive and the negative detectors are split into two jointly optimized networks based on their sign. (a) Setup of the differential design, $D([M][M],[2,L,P])$. In the example shown in this figure, $M=10$, $L=5$, $P=40\mathrm{k}$. (b) A correctly classified test object from the MNIST dataset is shown. Subparts of (b) illustrate the following: (i) target object placed at the input plane and illuminated by a uniform plane wave, (ii) normalized intensity distribution observed at the output plane of the diffractive optical neural network, (iii) normalized optical signal detected by the positive (red) and the negative (blue) detectors, (iv) differential class scores computed according Eq. (1) using the values in (iii). (c) and (d) are the same as in (b), except for Fashion-MNIST and CIFAR-10 datasets, respectively. Note that while the input object in (b) is modeled as an amplitude-encoded object, the gray levels shown in (c) and (d) represent phase-encoded perfectly transparent input objects.

Fig. 4

Operation principles of a diffractive optical neural network using class-specific detection scheme, where the individual class detectors are split into separate networks based on their classes. Unlike Figs. 2 and 3, there are no negative detectors in this design. (a) Setup of the class-specific design, $D([M/\mathrm{2,0}],[2,L,P])$. In the example shown in this figure, $M=10$, $L=5$, $P=40\mathrm{k}$. (b) A correctly classified test object from the MNIST dataset is shown. Subparts of (b) illustrate the following: (i) target object placed at the input plane and illuminated by a uniform plane wave, (ii) normalized intensity distribution observed at the two output planes of the diffractive optical neural networks, (iii) normalized optical signal detected by the detectors. (c) and (d) are the same as in (b), except for Fashion-MNIST and CIFAR-10 datasets, respectively. Note that while the input object in (b) is modeled as an amplitude-encoded object, the gray levels shown in (c) and (d) represent phase-encoded perfectly transparent input objects.

Because of the passive nature of diffractive neural networks, at the cost of optical set-up alignment complexity as well as illumination power increase, one can create scalable, low-power, and competitive solutions to perform optical computation and machine learning through these jointly optimized diffractive neural network systems.

2.

Results and Discussion

Figure 1 shows the examples of the input–output plane configurations for various diffractive neural network designs investigated and compared in this manuscript. For the sake of clarity, we devised a notation to represent these different optical network designs based on (1) the number of positive and negative detectors at the output plane, (2) the number of jointly trained but independent diffractive networks (i.e., there is no electromagnetic coupling among individual diffractive networks), (3) the number of layers constituting each one of these individual diffractive neural networks, and (4) the number of neurons at each diffractive layer of an individual optical network. According to this notation, if a diffractive optical classifier has $Q^{+}$ positive detectors, $Q^{-}$ negative detectors at each output plane of $N$ jointly optimized diffractive networks, each of which contains $L$ layers with $P$ neurons per layer, this optical classifier design is denoted by $D([Q^{+},Q^{-}],[N,L,P])$. If the $Q^{+}$ positive detectors and $Q^{-}$ negative detectors are not at the same diffractive network output plane, i.e., decoupled from each other by distributing them to different optical neural networks, then the total number of jointly optimized diffractive networks needs to be doubled to $2\times N$, where each network will either have $Q^{+}$ or $Q^{-}$ detectors at the corresponding output plane. To reflect such a configuration, our notation is defined as $D([Q^{+}][Q^{-}],[2N,L,P])$ where the brackets separating the number of positive and negative detectors indicate that they are not placed on the same output plane, but rather they follow different diffractive optical networks that are all jointly optimized, but operate individually without optical coupling from others. According to this notation, the standard diffractive optical neural network architectures [Fig. 1(a)] used in, e.g., Refs. 25 and 27 can be denoted as $D([M,0],[1,L,P])$ for a dataset with $M$ classes (e.g., $M=10$ for MNIST). As another example, for a dataset with $M$ classes, a combination of class division and differential detection can lead to $2\times M$ jointly trained diffractive neural networks, with a single detector at the output plane of each diffractive network (corresponding to either positive or negative portion of a class signal), and this diffractive neural system is denoted, based on our notation, as $D([1][1],[2M,L,P])$.

After the introduction of our notation to symbolize different diffractive neural systems ($D$), we now focus on quantifying the impact of some of these different designs on the inference and generalization performance of a diffractive classifier. First, we start our analysis by comparing the performance of standard diffractive optical networks used earlier,^25,27 where there is a single optical network with $M=10$ detectors at the output plane [e.g., $D([10,0],[1,5,40\mathrm{k}])$], against the performance of a differential detection network with the same diffractive configuration, except the output plane, i.e., $D([10,10],[1,5,40\mathrm{k}])$. In Table 1, first and second rows reveal that maximizing the normalized differential signal for the target class improves the blind inference accuracy of a diffractive optical network (composed of five diffractive layers with a total of 0.2 million neurons) to 98.54%, 90.54%, and 48.51% for MNIST, Fashion-MNIST, and grayscale CIFAR-10 datasets, respectively, compared to the corresponding accuracies achieved by $D([10,0],[1,5,40\mathrm{k}])$, i.e., 97.51%, 89.85%, and 45.20%, respectively. For a dataset with $M$ classes, this performance gain comes at the expense of a twofold increase in the number of optoelectronic detectors ($2M$ instead of $M$), together with the use of additional but simple electronic read-out circuitry, composed of, e.g., $M$ differential amplifiers and normalization logic; this extra computation at the output plane is rather straightforward, with a computational complexity of $O(M)$.

When the optical path is divided into two as shown in Fig. 3(a), we further increase the number of degrees of freedom of the diffractive neural system and decouple the optical signals detected by the positive and negative detector pairs; as a result of this, the blind testing accuracy of $D([10][10],[2,5,40\mathrm{k}])$ shown in Fig. 3(a) increases further to 90.94% and 49.10% for Fashion-MNIST and CIFAR-10 datasets, respectively, as shown in the last row of Table 1. For MNIST dataset, on the other hand, the average performance of $D([10][10],[2,5,40\mathrm{k}])$ remains approximately at the same level as $D([10,10],[1,5,40\mathrm{k}])$ architecture, i.e., 98.49% versus 98.54%, respectively.

Table 2 summarizes our results on an alternative diffractive classifier design strategy: we jointly trained a group of diffractive neural networks, where each one of them specialized on a subgroup of classes, and the optoelectronic detectors were placed at the output plane of the corresponding network. For example, as part of this design strategy, $D([2,0],[5,5,40\mathrm{k}])$ of Table 2 refers to a diffractive neural system that is composed of five jointly trained diffractive neural networks, each having five diffractive layers (40k neurons per layer) and two detectors are placed at the corresponding output plane, where each detector represents one class of the dataset. Each one of these five diffractive neural networks is jointly optimized together with the others but does not have any optical coupling from the other networks. Based on our comparative analysis reported in Table 2 (nondifferential row), the best performance among the nondifferential diffractive designs is achieved when each diffractive optical network of a neural system specializes on only one class: $D([1,0],[10,5,40\mathrm{k}])$ achieved blind testing accuracies of 97.61%, 90.34%, and 48.02%, for MNIST, Fashion-MNIST, and grayscale CIFAR-10 datasets, respectively. The same conclusion regarding the success of class-specific diffractive neural networks also holds for differential detection strategy; Table 2 (differential rows) reports that $D([1,1],[10,5,40\mathrm{k}])$ and $D([1][1],[20,5,40\mathrm{k}])$ achieved the winner performance in this comparison for each differential row, with blind testing accuracies of 98.59% (98.52%), 91.37% (91.48%), and 50.09% (50.82%), for MNIST, Fashion-MNIST, and grayscale CIFAR-10 datasets, respectively, where the values in parentheses refer to the performance of $D([1][1],[20,5,40\mathrm{k}])$.

Table 2

Blind testing classification accuracies of different class division architectures combined with nondifferential and differential diffractive neural network designs. For each data point, the training of the corresponding diffractive optical neural network model was independently repeated six times with random initial phase modulation variables and random batch sequences; therefore, each data point reflects the mean blind testing accuracy of these six trained networks, also showing the corresponding standard deviation.

A direct comparison between the “differential” and “nondifferential rows of Table 2 further emphasizes the importance of the differential detection scheme. Not only the differential diffractive network designs show significantly better performance compared to their nondifferential counterparts when the number of neurons and the number of diffractive layers are the same but also even $D([1,0],[10,5,40\mathrm{k}])$ architecture with 2 million neurons in total cannot outperform $D([10,10],[1,5,40\mathrm{k}])$ that has 0.2 million neurons in total, despite having 10 times more number of neurons in the diffractive classifier design.

Figure 5 shows the general conclusions that are revealed by our analysis: (1) differential diffractive neural systems outperform their nondifferential counterparts; (2) diffractive neural networks that subspecialize on a single class as part of a neural system outperform their counterparts that specialize on multiple classes per diffractive network; and (3) the combination of both design strategies, i.e., class-specific differential detection outperforms other counterpart diffractive neural system designs. These mean that for a dataset with $M$ classes, $D([1][1],[2M,L,P])$ would be a winner design, with $L$ diffractive layers per class-specific optical network, and $P$ neurons per diffractive layer (see Tables 2 and 3). For example, Table 2 reports that, for MNIST, Fashion-MNIST, and grayscale CIFAR-10 datasets, $D([1][1],[2M=20,5,40k])$ design achieves blind testing accuracies of 98.52%, 91.48%, and 50.82%, respectively; to the best of our knowledge these values report the highest accuracy levels achieved so far for an all-optical neural network design.

Fig. 5

Performance comparison of different diffractive neural network systems as a function of $N$, the number of class subsets. $M=10$ classes exist for each dataset: MNIST, Fashion MNIST, and grayscale CIFAR-10. Based on our notation, $N=M=10$ refers to a jointly optimized diffractive neural network system that specializes to each one of the classes separately. These results confirm that class-specific differential diffractive neural networks ($D([M/N][M/N],[2N,L,P]$) for $N>1$ outperform other counterpart diffractive neural network designs. For each data point, the training of the corresponding diffractive optical neural network model was repeated six times with random initial phase modulation variables and random batch sequences; therefore, each data point reflects the mean blind testing accuracy of these six trained networks, also showing the corresponding standard deviation.

Table 3

Comparison of blind testing accuracies of different types of neural networks, including optical, hybrid, and electronic.

So far, in our differential diffractive neural network designs, we considered balanced differential detection between the optical signals of [$Q^{+}$] and [$Q^{-}$], i.e., $[Q^{+}]-[Q^{-}]$. To further explore if this balanced differential detection is indeed an ideal choice, we considered a more general case, where the two detectors of a pair assigned to a class can be merged with arbitrary scaling factors, $p_m$ and $n_m$, respectively ($m$ represents the class number). We can generally denote this broader diffractive network design as $D(p_m[M/N]n_m[M/N],[2N,L,P])$, where $p_m$ and $n_m$ can be any real number that can vary from class to class. For example, $p_m=n_m=1$ refers to the standard balanced differential detection case considered so far, whereas $p_m=1$, $n_m=-1$ refers to a simple summation of the signals of the two detectors assigned to class $m$. By treating $p_m$ and $n_m$ as additional independent learnable parameters of a diffractive neural network, $D(p_m[M/N]n_m[M/N],[2N,L,P])$, we trained different designs that were initialized with random $(p_m,n_m)$ values, which quickly converged to a solution with $p_m\approx n_m$ for each class of the corresponding dataset, proving empirically that a balanced differential detection is indeed preferred. We also noticed that the general design $D(p_m[M/N]n_m[M/N],[2N,L,P])$ with learnable detector coefficients did not improve our blind inference performance compared to the case of $p_m=n_m=1$.

Another method to benefit from the parallel computing capability of passive diffractive neural networks is to create independently optimized diffractive neural networks that optically project their diffracted light onto the same output/detector plane. Unlike the jointly optimized diffractive neural systems described earlier, here in this alternative design strategy we select a diffractive network design, $D$, and independently optimize replicas of this design, where each network separately projects its diffracted pattern onto the same (i.e., common) detector plane. Not to interfere with the inference results of each diffractive neural network, here we considered intensity-only summation of the optical signals of each diffractive network at the common output plane, as opposed to coherent summation of the diffracted fields, which could perturb the predictions of each independent network due to constructive and destructive interference at the output plane. This can easily be achieved in a diffractive neural system by adjusting the relative optical path length differences between the individual diffractive networks to be larger than the temporal coherence length imposed by the bandwidth of the illumination source, ensuring that each detector at the common output plane sums up the optical intensities of all the individual diffractive neural networks. For each diffractive network of the ensemble, coherent operation is still maintained since the layer-to-layer separation in a given diffractive network is very small (e.g., $40\lambda$ for the designs considered in this work).

This design strategy of using independently optimized diffractive networks is in fact similar to ensemble methods^32,33 that are frequently used in machine learning literature. Figure 6 shows the blind testing accuracies achieved by this strategy using either nondifferential or differential diffractive neural networks, i.e., $D([10,0],[1,5,40\mathrm{k}])$ or $D([10,10],[1,5,40\mathrm{k}])$. For example, using three independently optimized differential diffractive neural networks that optically project their light onto a common output plane with 10 detector pairs (one for each class), we numerically achieved blind testing accuracies of 98.59%, 91.06%, and 51.44% for MNIST, Fashion-MNIST, and grayscale CIFAR-10 datasets, respectively (see Fig. 6). Further increasing the number of independently trained differential diffractive neural networks combined in an ensemble system brings diminishing return to the inference performance of the ensemble. For example, for CIFAR-10 dataset, optical classifier models that are composed of 2, 3, and 5 independently optimized differential diffractive neural networks, $D([10,10],[1,5,40\mathrm{k}])$, achieve blind testing accuracies of 50.68%, 51.44%, and 51.82%, respectively. This diminishing return behavior stems from the increasing correlation between the output intensity distributions generated by the ensemble model and an additional independently optimized diffractive neural network (to be added to the ensemble).

Fig. 6

The comparison between the classification accuracies of ensemble models formed by 1, 2, and 3 independently optimized diffractive neural networks that optically project their diffracted light onto the same output/detector plane. Blue and orange curves represent $D([\mathrm{10,}0],[\mathrm{1,}5,40\mathrm{k}])$ and $D([10,10],[1,5,40\mathrm{k}])$ designs, respectively. (a) MNIST, (b) Fashion-MNIST, and (c) grayscale CIFAR-10. Not to perturb the inference results of each diffractive network due to constructive/destructive interference of light, incoherent summation of the optical signals of each diffractive network at the common output plane is considered here, which can be achieved by adjusting the relative optical path length differences between the individual diffractive networks to be larger than the temporal coherence length of the illumination source.

After reporting the results of various different design strategies for diffractive neural networks, in Table 3, we present a quantitative comparison of diffractive neural systems against some of the earlier hybrid (i.e., optical and electronic) neural networks as well as some of the widely known all-electronic machine learning models used in the literature. This comparison once again highlights the importance of class-specific differential detection for improving the blind inference performance and the generalization of diffractive neural network systems. For example, $D([1][1],[20,5,40\mathrm{k}])$ matches the blind inference performance of convolutional deep neural networks such as LeNet and AlexNet for MNIST and Fashion-MNIST datasets and falls short of the performance of LeNet for CIFAR-10 dataset only by 4.39%. A similar conclusion can be drawn from Table 3 for our comparison against the hybrid systems reported in Refs. 26 and 27.

While the presented systematic advances in diffractive neural network designs have helped us achieve a competitive inference performance, with classification accuracies that are among the highest levels achieved so far for optical neural networks, there is still a considerable performance gap with respect to the state-of-the-art all-electronic deep learning models such as ResNet (see e.g., Table 3, CIFAR-10 performance comparisons). Despite its inferior performance compared to such all-electronic deep learning models that set the state-of-the-art in machine learning, class-specific differential diffractive neural networks still present several important advantages in terms of scalability, memory usage, computation speed, and power efficiency since the main computation occurs all-optically and at the speed of light through diffraction within passive optical layers without the need for external power, except for the illumination light and a few detectors and related circuitry at the network output. Having underlined these important advantages, we should also note that significantly higher classification accuracies of state-of-the-art electronic deep neural networks such as ResNet once again emphasize the vital role of multichannel convolutional layers and nonlinearity inherent in these networks; as discussed in earlier work,^25,27 the use of nonlinear optical materials or optical resonances in diffractive neural networks can potentially improve the inference capabilities of diffractive neural systems beyond the currently presented results. Our results also reinforce an earlier conclusion regarding diffractive optical neural networks: their inference and generalization capabilities improve with additional diffractive layers jointly designed and optimized by gradient-based learning, which illustrates the depth feature of diffractive neural systems, even if there is no nonlinear optical material being employed per layer. Stated differently, the general family of functions represented in this work through $D([M/N,M/N],[N,L,P])$ or $D([M/N][M/N],[2N,L,P])$ cannot be covered by a single diffractive optical layer, no matter how many neurons are employed.

Finally, we would like to also emphasize that these reported advances in the inference and generalization performance of class-specific differential diffractive neural networks come at the cost of a requirement to increase the input illumination power. For example, to keep the signal-to-noise ratio (SNR) of each photodetector that is positioned at an output plane of a class-specific differential diffractive neural network system, (e.g., $D([M/N,M/N],[N,L,P])$) at the same level as the SNR of the photodetectors of a standard diffractive neural network (i.e., $D([M,0],[1,L,P])$), the optical power of the input illumination beam must be increased by approximately $N$ fold; the exact comparison is dataset and task dependent and is actually governed by the photon efficiencies of different diffractive networks that make up of $D([M/N,M/N],[N,L,P])$. However, if $N$ is increased to $M$ (e.g., $M=10$ for the datasets considered in this work), this means each diffractive network unit that is part of $D([M/N,M/N],[N,L,P])$ has only two photodetectors at the corresponding output plane, whereas the standard diffractive neural network, $D([M,0],[1,L,P])$, has $M=10$ photodetectors. Therefore, if we include in the training phase of the diffractive neural system a photon efficiency loss term for the photodetectors of $D([M/N,M/N],[N,L,P])$, penalizing poor diffraction efficiency per detector, one can potentially reduce this $N$-fold illumination power penalty by making class-specific networks more photon efficient as they deal with much smaller number of photodetectors at their output. To list another disadvantage of class-specific differential diffractive neural networks, because of their increased parallelism the complexity of the fabrication and alignment of the optical neural network set-up would be more complicated, and the overall size of the diffractive neural system would be increased compared to a single standard diffractive neural network. However, these are challenges that can be mitigated with 3-D integrated photonic systems fabricated through, e.g., lithography, and the need for increased optical illumination power is in general not a major concern due to various high-power lasers commonly available in different formats, including portable systems.

3.

Methods