1. Introduction

Optical fibers have become one of the most popular transmission media for information delivery in optical communications and endoscopy, due to the capability of transmitting optical signals with high speed, great flexibility, and large capacity. Compared with single-mode fibers, multimode fibers (MMFs) support the simultaneous propagation of many optical modes, enabling information to be delivered in parallel. This property makes the MMF a promising candidate for delivering two-dimensional images. However, in MMFs, strong mode coupling always accompanies the propagation of light. In this scenario, the output plane always contains fully scrambled information, i.e., speckle patterns. Due to this reason, directly delivering images through the MMF remains a big challenge.

Recent developments in wavefront shaping show that information scrambling, originating from either optical scattering in biological tissue or mode coupling in MMFs, can be corrected. For example, within a certain time scale, the process of mode coupling inherent in the MMF can be treated as deterministic. Thus, the propagation of light through the MMF can be modeled as a linear transmission matrix (TM). With complete knowledge of the TM, arbitrary images can be delivered through the MMF. The initial attempt was employing feedback-based schemes to retrieve the TM row by row, enabling successful projection of a single spot [1,2] or patterned spot array [3–5] through the MMF. Later on, digital optical phase conjugation was also demonstrated to synthesize a sharp focus through the MMF with high efficiency [6–9]. One could also directly measure the TM, allowing arbitrary patterns to be projected at the distal end [10–12]. Moreover, imaging through MMFs, which is also referred to as speckle imaging, could be achieved in a similar manner [13–15]. Despite these accomplishments in projecting images through MMFs, it is notable that most projected images exhibit strong speckle features. This observation is likely due to the uncareful treatment of the sampling issue in the optical system, which is closely related to the pixelation of the SLM, the camera, and their matching relation. With this relationship in mind, projecting images with smooth structures could be realized, but a sophisticated alignment between the SLM and the camera is required [10]. Moreover, wavefront shaping techniques generally require a larger number of independent controls than the number of independent speckles in the projected images.

In recent years, neural networks are gradually becoming a powerful tool to tackle complicated optical systems. In 2018, deep neural networks were used to reconstruct and classify handwritten digits through a 1-km-long MMF [16]. Even under conditions with strong instability, speckle imaging for either amplitude or phase could be achieved [16]. Inspired by this work, a variety of neural networks to reconstruct input images from the output speckle patterns were proposed and demonstrated [17–23]. Neural networks have also shown strong robustness when the MMFs were bent or dynamically perturbed [24–26]. These demonstrations make neural networks a promising tool for addressing information scrambling in MMFs. Besides enabling speckle imaging, neural networks also enable the projection of images through the MMF [27]. A schematic of the image projector is shown in Fig. 1, which contains a phase-only SLM, an MMF, and a camera. Both the SLM and the camera are controlled by a computer. In the training process, paired phase maps displayed by the SLM and intensity measurements captured by the camera are then sent to the computer. Then, after being assigned a greyscale image as the target, a well-trained network can immediately instruct the SLM to display a phase map that leads to the desired greyscale image at the camera sensor. Unfortunately, however, developing such a projector network seems to be more challenging than one would expect. The foremost problem is that there is no appropriate training dataset to start with [27]. In the scenario of speckle imaging, the training dataset can be harvested by displaying structured amplitude or phase images using the SLM and capturing the resultant speckle patterns using the camera. The structured images can be, for example, digits, alphabets, and recognizable drawings, which can be freely chosen for specific reconstruction tasks. However, the condition is completely different in projecting images, as one does not have any prior knowledge regarding what kind of phase maps can produce structured images through the MMF. Attempts have been made to employ arbitrary inputs and the resultant speckle images as the training dataset [28,29]. Due to the extremely ill-posed condition, only a single focal spot could be projected [28]. With an appropriate training dataset generated by the measured TM prior, i.e., structured output images and their corresponding input phase maps, Ref. [17] successfully demonstrated projecting images through the MMF. However, if one knew the TM to generate the training dataset, the problem would have been solved [27]. As a result, projecting images through MMFs falls within a chicken-and-egg dilemma. To solve this dilemma, a projector network consisting of two subnetworks, namely Model-net (M-net) and Actor-net (A-net), was proposed [27]. The M-net learns the forward propagation, while the A-net learns the backward process. The goal of the training process is to make the A-net become the inverse of the M-net. This projector network has been demonstrated with great success to project binary intensity images through the MMF with good generalization ability. However, the generalization ability degrades for greyscale images with rich structures (many variations and details) at a large scale (many nonzero elements), while an iterative training process is required to have the to-be-projected images included in the training process [27]. In this work, we attempt to tackle this problem by developing a modified projector network. After a one-time training, it enables the projection of grayscale images through the MMF. It is worth emphasizing that the training dataset for this modified project network can be generated by pairing random phase maps and resultant speckle patterns. The modified projector network shows strong generalization ability as none of the to-be-projected images has been seen by the network before.

 

Fig. 1. Schematic of the image projector, containing a spatial light modulator, a multimode fiber, and a camera.

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2. Architecture of the projector network

We start by describing the architectural design of the projector network, which is based on the physical process of light propagation through the MMF. The projector network comprises three subnetworks named Forward-net (F-net), Backward-net (B-net), and Holography-net (H-net). Just as their names imply, the F- and B-net adopt the fully-connected layers to account for the forward and backward propagation of light in MMFs. These two subnets function similarly to the A- and M-net employed in Ref. [27], with slight differences in detailed implementation. In particular, for both the F- and B-net, four fully-connected layers are adopted with appropriate connections to explicitly mimic complex operations with real numbers. Moreover, the outputs of the F-net, i.e. the real and imaginary parts of the complex speckle field, are connected to the inputs of the B-net, which is in the reversed order of the AM-net [27]. Both the input and output of the combined FB-net are phase maps, rather than the to-be-project images. Detailed mathematical descriptions of light propagation through the MMF and architecture of the F- and B-net are provided in Eqs. (1)–(3) and Fig. 6 of the Appendix, respectively.

For intensity measurements, although the FB-net can acquire the absolute square of both their real and imaginary parts, a direct arithmetic square root leads to four possible solutions of complex fields. There exists up to ${2^{3601}}$ possibilities in terms of optical fields for a projected intensity image with a dimension of $60 \times 60$. This ambiguity may not be a big issue in projecting approximately-binary images with a limited number of nonzero elements [27], but causing challenges for the projector network when targeting greyscale images. To address this issue, the H-net consisting of two fully-connected layers and a genetic algorithm (GA) (Fig. 2) is developed. The H-net first separates the target intensity pattern into the absolute value of the real and imaginary parts. These two parts are then transformed into a real-imaginary pair, severing as the input for the GA. In particular, the role of the GA is to find the correct complex field by generating a pair of masks with values of 1 and -1. The real-imaginary pair output from the FB-net then multiplies the paired masks to form the population group. The execution of the GA can be divided into five steps [30,31]. (1) Parental selection: once the population group has been generated, two of them are selected as the parents (denoted as $pa$ and $ma$). The selection rule is that an individual with a higher cost value has a higher possibility to be selected. (2) Breeding: with two parents being selected, a new offspring is created as $pa\cdot T + ma\cdot ({1 - T} )$, leading to the formation of new real-imaginary pairs, where T is a random binary template pair generated with a parental mask ratio of 0.5. (3) Mutation: the sign of certain elements is reversed with a small probability. An adaptive mutation ratio ranging from 0.1 to 0 with an exponential behavior is chosen. After mutation, the cost values of the offspring are estimated and recorded. (4) Replacement: the offspring with larger cost values can replace the real-imaginary pairs with low cost values in the original population. The above procedures are repeated until half of the population is filled with new offspring. (5) Ordering: the cost values of the newly generated population are estimated and recorded and the next generation is prepared according to the cost values. For the GA, the cost function is the intensity-based Pearson’s correlation coefficient between the targets and the resultant outputs after successively passing through the B-net and the F-net.

 

Fig. 2. Schematic illustration of the Holography-net, which consists of two fully-connected layers and a genetic algorithm (GA). The fully-connected layers separate the intensity image into a real-imaginary pair, while the GA finds the correct complex fields. The cost function computes the intensity-based Pearson’s correlation coefficient between the targets and the resultant outputs from the network.

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Having described the architecture of the projector network, we then describe the training process, which trains three subnetworks synergetically. The training data consists of a series of randomly generated input phase maps (uniformly distributed within a range of 0 to 2π) and their corresponding output speckle patterns (normalized to values ranging from 0 to 255). The input and output dimensions are $20 \times 20$ and $60 \times 60$. Notably, these three subnetworks can be trained with either the same or different data. In this work, we adopted the same training data for all three subnetworks for simplicity. Firstly, we train the F-net with 10,000 data to optimize the choice of parameters. After the training process, the F-net is tested with 1,000 data to examine the performance of the trained F-net. Then, the trained F-net will be combined with the untrained B-net to form a semi-trained FB-net. The parameters of the trained F-net are fixed while only the parameters of the B-net are optimized during the training process. A total number of 10,000 training images are fed to FB-net. Similarly, the FB-net is tested with 1,000 data to examine the performance after training. Once trained, the F- and B-net can be used separately with parameters being fixed. Then, we proceed to train the H-net with 10,000 data which are generated from a trained F-net. Notably, although the output speckle patterns have a dimension of $60 \times 60$ to guarantee sufficient constraints, we only project greyscale images within a small area with a dimension of $20 \times 20$. Therefore, when estimating the cost function for the GA, intensity correlation is computed only for the small area. Detailed training processes are illustrated in Figs. 7–9 of the Appendix. It is worth noting that projecting greyscale images on a larger scale is possible but requires the network to be reconfigured to higher dimensions. Generally speaking, both the input and out dimensions of the training data need to be increased accordingly, at the cost of computational resources.

3. Results

3.1 Simulation results of the projector network

The performance of the combined projector network, which is referred to as the FBH-net afterward, is first examined through numerical simulations. To mimic physically projecting greyscale images through the MMF during simulations, a TM with uncorrelated transmission coefficients is pre-defined and is blind to the network. To generate the training data, the output of the MMF is calculated by multiplying the TM with the input optical field. To examine the generalization ability of the FBH-net, MNIST, EMNIST, and Fashion-MNIST are tested as targets, which were never seen by the network before. The first row of Fig. 3 illustrates typical examples of the targets, while the second row shows the projected images achieved using the FBH-net. For 1,000 testing data, the averaged intensity Pearson’s correlation coefficients between the projected images and the targets are 0.91, 0.88, and 0.76 for MNIST, EMNIST, and Fashion-MNIST, respectively. These results indicate that projecting large-scale greyscale images (Fashion-MNIST) is generally more difficult than projecting small-scale approximately-binary images. As a fair comparison, the third row shows the projected images achieved using the AM-net when being trained with the same data. Correspondingly, the averaged intensity Pearson’s correlation coefficients for MNIST, EMNIST, and Fashion-MNIST are 0.65, 0.58, and 0.38, respectively. The relatively low coefficients indicate that the AM-net cannot be trained well with a randomly generated dataset.

 

Fig. 3. Simulation results to examine the performance of the projector network. The first row illustrates typical examples of the targets, which belong to MNIST, EMNIST, and Fashion-MNIST. The second row shows the projected images using the FBH-net. The third row shows the projected images using the AM-net with the same training data. Pearson’s correlation coefficients between the projected images and the targets are shown in the upper right corner. Additional simulation results on projected images can be found in the Dataset 1 [32].

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Notably, it is worth mentioning that the AM-net performs slightly better when trained with MNIST (Same as the process in Ref. [27]) by achieving averaged correlation coefficients of 0.86, 0.78, and 0.57 for MNIST, EMNIST, and Fashion-MNIST, respectively. These results are not discussed further here due to the choice of different training datasets. Nonetheless, the projected greyscale images do not exhibit satisfactory correlation coefficients. Notably, among all realizations, the FBH-net trained with random patterns maintains the best performance with a high generalization ability to greyscale images.

3.2 Experimental results of the projector network

We then proceed to build an experimental setup to validate the proposed scheme, which is illustrated in Fig. 4. Light was generated by a continuous-wave laser (MDL-C-642-nm-30 mW, CNI). A half-wave plate and a polarizing beam splitter adjusted the power being dumped into the system. To ensure plane wave illumination, light was expanded by a pair of lenses. A square aperture was employed to control the illuminating area on the SLM (PLUTO-2-NIR-011, Holoeye, 1920 × 1080 pixels, 8 µm/pixel). After modulation, light was then coupled into an MMF (FC/PC-50/125-900 µm 1 m, Shenzhen Optics-Forest Inc) by an objective lens (OBJ1, OPLN10X). The output light at the distal end of the MMF was collimated by another objective lens (OBJ2, OPLN10X) and captured by a camera (GS3-U3-32S4C, 8 bits, FLIR). A lens and a polarizer were used to control the field of view and the polarization state of the captured speckles. The temperature was fixed at around 26 °C during experiments.

 

Fig. 4. Schematics of the experimental setup. HWP, half-wave plate; PBS, polarizing beam splitter; BB1, BB2, beam block; L1, L2, L3, biconvex lens; BS, beam splitter; SLM, spatial light modulator; M, mirror; PP, polarizing plate; OBJ1, OBJ2, objective lens; CCD, charge-coupled device.

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During experiments, the training data were generated by sending randomly generated input fields to the MMFs and measuring their corresponding output speckle patterns. With the projector network being trained, MNIST, EMNIST, and Fashion-MNIST were employed to test the generalization ability, as shown in Fig. 5. Again, the first row illustrates typical examples of the targets, while the second row shows the projected images achieved using the FBH-net. The averaged intensity-based Pearson’s correlation coefficients for MNIST, EMNIST, and Fashion-MNIST are 0.91, 0.92, and 0.87 respectively. As a comparison, the third row illustrates the projected images achieved using the AM-net with the same training data, exhibiting averaged intensity-based Pearson’s correlation coefficients of 0.77, 0.83, and 0.71 for MNIST, EMNIST, and Fashion-MNIST, respectively. Interestingly, networks generally perform better in experiments than in simulations. This observation is probably because the mode coupling within the MMF is not so severe as to fully randomize the TM. Nonetheless, the experimental observations still confirm the high generalization ability of the FBH-net.

 

Fig. 5. Experimental results to examine the performance of the projector network. The first row illustrates typical examples of the targets, which belong to MNIST, EMNIST, and Fashion-MNIST. The second row shows the projected images using the FBH-net. The third row shows the projected images using the AM-net with the same training data. Pearson’s correlation coefficients between the projected images and the targets are shown in the upper right corner. Additional experimental results on projected images can be found in the Dataset 1 [32].

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4. Conclusion

In this work, the developed projector network is designed primarily based on a linear model (even with nonlinear activation functions Tanh). This assumption is valid when the output intensity of the laser source is relatively low and the length of the MMF is short. In this condition, nonlinearities such as the Raman scattering and self-phase modulation can be omitted. Interestingly, some recent works have already demonstrated neural networks have the potential to handle these nonlinear phenomena with additional nonlinear ingredients [33,34]. Therefore, for scenarios where nonlinearities in MMFs are prominent, future works can further extend the capability of the network to handle different levels of nonlinearities. Moreover, all intensity correlations observed experimentally are still around 0.9, leaving plenty of room for us to further optimize the architecture and parameters.

In conclusion, we demonstrate an MMF-enabled projector that can project greyscale images with high generalization ability. Three subnetworks that account for forward transmission, backward transmission, and non-holographic phase retrieval for complex fields were designed and trained synergetically with paired random input phase maps and resultant speckle patterns. With the trained projector network, binary images (MNIST and EMNIST) and greyscale images (Fashion-MNIST), which have never been seen by the network before, can be well projected through the MMF. This work provides a promising substitute for conventional TM-based approaches in projecting greyscale images through MMFs, showing great prospects in a variety of applications.

Appendix

A1. Mathematical description of light propagating through the multimode fiber

Although various nonlinear effects do exist when light propagates through the MMF, the basic framework of the projector network is based on a linear model that describes light propagation through the multimode fiber (MMF). Using the representation of complex notation, ${\boldsymbol{E}_{\mathbf{in}}}$ and ${\boldsymbol{E}_{\mathbf{out}}}$ are the complex fields of the incident light and the output light, respectively. These two fields are mathematically connected through a transmission matrix $\mathbf{T}$ of the multimode fiber, as follows

(1)$$ {{\boldsymbol{E}_{\mathbf{out}}} = \mathbf{T}{\boldsymbol{E}_{\mathbf{in}}}} $$

Assuming the spatial light modulator (SLM) has N segments and the camera has M pixels, $\mathbf{T}$ has a dimension of $M \times N$. We rewrite Eqs. (1) in an element-wise computation format. For example, for the m-th pixel of the camera,

(2)$$\begin{aligned} {\boldsymbol{E}_{\textrm{ou}{\textrm{t}_m}}} &= \mathop \sum \nolimits_n {A_n}{t_{mn}}\textrm{exp}({i({{\varphi_n} + {\theta_{mn}}} )} )\\ &{ = \mathop \sum \nolimits_n ({{A_n}{t_{mn}}} )[{\textrm{cos}({{\varphi_n} + {\theta_{mn}}} )+ i\textrm{sin}({{\varphi_n} + {\theta_{mn}}} )} ]}\end{aligned}$$

Here, ${A_n}$ and ${\varphi _n}$ are amplitude and phase of the optical field that corresponds to the n-th segment of the SLM. For phase-only SLM, ${A_n} = 1$ is set for all n. ${t_{mn}}$ and ${\theta _{mn}}$ refer to the amplitude and phase of the matrix element in the m-th row and n-th column. The goal of the projector network is to determine the unknown variables ${t_{mn}}$ and ${\theta _{mn}}$. By rearranging Eqs. (2), we establish a linear relationship between the input and the output

(3)$$\begin{aligned}{\boldsymbol{E}_{\textrm{ou}{\textrm{t}_m}}} &= \mathop \sum \nolimits_n {t_{mn}}[{\textrm{cos}({{\varphi_n}} )\textrm{cos}({{\theta_{mn}}} )- \textrm{sin}({{\varphi_n}} )\textrm{sin}({{\theta_{mn}}} )} ]\\ &{ + \mathop \sum \nolimits_n i{t_{mn}}[{\textrm{sin}({{\varphi_n}} )\textrm{cos}({{\theta_{mn}}} )+ \textrm{cos}({{\varphi_n}} )\textrm{sin}({{\theta_{mn}}} )} ]}\end{aligned}$$

This mathematical modeling of the propagation of light through the MMF helps us design the architecture of the projector network. Before proceeding, we note that the neural network only supports operations with real numbers. Thus, the real part and imaginary part of Eqs. (3) need to be handled and separated with special care.

A2. Architecture of the Forward-net and Backward-net

Forward-net and Backward-net are approximately symmetric but differ in functionalities. The Forward-net (F-net) is used to describe the forward transmission process through the MMF, while the Backward-net (B-net) is used to consider the inverse process. A schematic of the F-net is provided in Fig. 6(a). According to Eqs. (3), an input phase map ${\varphi _{\textrm{in}}}$ is firstly split into two sub-maps of $\textrm{cos}({{\varphi_{\textrm{in}}}} )$ and $\textrm{sin}({{\varphi_{\textrm{in}}}} )$. Then, both $\textrm{cos}({{\varphi_{\textrm{in}}}} )$ and $\textrm{sin}({{\varphi_{\textrm{in}}}} )$ are sent into two fully-connected layers. Thus, the four terms summed in Eqs. (3) are represented by four independent fully-connected layers. Among them, layers enclosed in blue boxes, i.e., F1 associated with $\textrm{cos}({{\varphi_{\textrm{in}}}} )$ and F3 associated with $\textrm{sin}({{\varphi_{\textrm{in}}}} )$, are used to account for the real part. Similarly, layers enclosed in red boxes, i.e., F2 associated with $\textrm{cos}({{\varphi_{\textrm{in}}}} )$ and F4 associated with $\textrm{sin}({{\varphi_{\textrm{in}}}} )$, are used to account for the imaginary part. After passing through a Tanh layer, the blue group becomes the real part of ${\boldsymbol{E}_{\textrm{out}}}$, while the red one becomes the corresponding imaginary part. The network then calculates the square of the modulus of ${\boldsymbol{E}_{\textrm{out}}}$ as the intensity output.

 

Fig. 6. Architecture of the Forward-net (F-net) and the Backward-net (B-net). (a) A schematic of the F-net. (b) A schematic of the combined FB-net.

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Right after the F-net has been trained, it is directly combined with an untrained B-net to form a semi-trained FB-net, as shown in Fig. 6(b). The architect of the B-net is to mimic the inverse process of the F-net. In particular, B1 and B3, which are enclosed in blue boxes, are connected and pass through a Tanh layer to denote $\textrm{cos}({{\varphi_{\textrm{in}}}} )$. Similarly, B2 and B4, which are enclosed in red boxes, are connected and pass through another Tanh layer to denote $\textrm{sin}({{\varphi_{\textrm{in}}}} )$. An operation of argument is then performed to compute the phase map ${\varphi _{\textrm{in}}}$.

A3. Illustration of the training process for the projector network

Detailed training processes for the F-net, the semi-trained FB-net, and Holography-net (H-net) are illustrated in Fig. 7, Fig. 8, and Fig. 9, respectively. All three F-, B-, and H-net adopt the mean square error as the loss function. An adaptive moment estimation optimization algorithm with a learning rate of ${10^{ - 4}}$ and a minibatch size of 32 is set for training. The number of epochs used for training the F-net, the semi-trained FB-net, and the H-net are 50, 100, and 50, respectively. All neural networks are generated through the Deep-learning Tool-Box of MATLAB 2020a on a personal computer equipped with a graphics processing unit (NVIDIA RTX 2060s), a computing processing unit (i7-10600 k 3.6 GHz), and a 32-GB random access memory.

 

Fig. 7. Training process of the F-net. MSE: mean square error.

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Fig. 8. Training process of the semi-trained FB-net. MSE: mean square error.

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Fig. 9. Training process of the H-net. MSE: mean square error.

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Funding

National Key Research and Development Program of China (2019YFA0706301); National Natural Science Foundation of China (12004446, 92150102); Fundamental and Applied Basic Research Project of Guangzhou (202102020603).

Disclosures

The authors have no conflicts to disclose.

Data availability

Data underlying the results presented in this paper are available in Dataset 1 [32].

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