Fiber optical networks provide high capacity and long-haul data transmission for global communication systems [1]. Using fibers with multiple transverse mode support, such as FMFs, multiplexing over spatial communication channels can be enabled [2]. Space division multiplexing (SDM), among others, is one technology for the satisfying the exponentially increasing demand for globally exchanged data [3]. However, the amount of sensitive data is also increasing. Along with the technological progress of connected devices, information security becomes more and more important.

Cryptographic algorithms are vulnerable against attacks with sufficient computing power, especially quantum attacks [4]. However, the physical behavior of FMFs offers security techniques at the physical layer of transmission. Eavesdroppers can be significantly hampered in their probability of deciphering transmitted messages by employing sophisticated wiretap codes; this is called physical layer security [5]. Physical layer security has its origin in wireless communication and its feasibility in optical fiber-communication systems has already been experimentally demonstrated [6,7]. In addition to actively preventing the unauthorized decoding of transmitted messages by exploiting physical properties, passive monitoring for the detection of possible attackers can also be applied to optical FMF communication systems.

Eavesdropping of FMFs can be accomplished by, e.g., splitting, bending, and v-groove cutting [8,9]. Whereas splitting disrupts the overall transmission, which leads to a higher bit error rate [10], bending and v-groove cutting affect FMF physical transmission properties, such as power loss and mode-dependent power loss (MDL). In both cases, mode-dependent transmission will be disturbed. In particular, modes with high power close to the core–cladding boundary are assumed to be tapped more easily [11]. This directly affects the FMF transmission properties, which can be monitored to enable surveillance.

As pointed out in Ref. [12], eavesdropping detection can be realized by monitoring the physical properties and transmission performance of the fiber. Whereas tracking of physical properties is strongly environmentally sensitive, the transmission performance can be evaluated by measuring the power of the received optical signal [13,14]. When eavesdropping appears through tapping the fiber, light is coupled unevenly with respect to the modes, i.e., MDL. The optical field measured at the receiver can be used to recognize this unauthorized tapping, enabling surveillance of the whole communication channel. In our work, the recognition is enabled by artificial intelligence (AI).

AI has been widely used with FMFs. With deep learning, FMF intensity patterns could be decomposed into their corresponding mode domain [15,16]. In addition, it is possible to reconstruct the optical fields of the fiber input based on the fiber output, even for images with only intensity variations [17,18]. In Ref. [19], different NN designs were compared for image reconstruction. Due to faster training and comparable performance, the single hidden layer NN is used in our work (see Fig. 1).

 

Fig. 1. NN architecture for FMF phase reconstruction. FMF output is used as NN input. Intensity image is reshaped into a vector, followed by two fully connected layers with rectified linear unit (ReLU) activation. NN output is reshaped back into an image. The NN has 2 million parameters, optimized by Adam-algorithm (learning rate: 0.001; batch size: 32). Mean squared error (MSE) is used as loss function and correlation coefficient [Eq. (6)] and MSE are used as metrics.

Download Full Size | PDF

For AI-driven systems, training data must be collected. A suitable method for data collection is to use a simulation model. Commonly, the FMF is treated as a linear and unitary system. Thus, modes propagate through the waveguide with reversible crosstalk and power conservation. Following these arrangements, a concise mathematical model can be created that can be used to calculate input and output light fields. Suitable training data consist of simple grayscale images of these field distributions. Any distribution can be represented as a mode-weight vector $\vec {x}$. The complex FMF spatial modes $U_\text {k}$ are calculated analytically [20]. Each mode $k \in [1;N]$ is associated with a complex weight $x_\text {k}$:

Using the aforementioned model, simulation data can be generated. First, we create both random mode weights regarding the FMF input (Eq. (1)) and a synthetic transmission matrix with which weights at the FMF output can be calculated. By applying singular value decomposition to a random matrix, a unitary linear operator $T$ is modeled. The image showing the intensity of the FMF output is used as the NN input, whereas the image of the phase at the FMF input is used as the ground truth. By creating a number of random input mode vectors, a dataset with variable size is generated and the NN shown in Fig. 1 can be trained.

The achievable performance of a NN depends on the representability of the data it is trained with. Ambiguities, for example, can lead to crucial performance drops. This is the case for complex-conjugated optical fields, as they have equivalent intensity distributions. Thus, two identical camera images could be assigned to different phase distributions. Avoiding ambiguities, phase values of the mode weights at the fiber input are determined within the interval $\psi \in [0;\pi ]$, exclusively. In addition, the relative phase distribution with respect to the first mode is considered [15,16].

Ensuring a fast training, low-resolution images ($32 \times 32$ pixels) are used for a FMF supporting $N=10$ modes. The training was performed with $10k$ image pairs within a time of 12 min. The course of the correlation coefficient $\Gamma$ during the training is shown in Fig. 2. It is given by

 

Fig. 2. Training performance of NN with $10k$ FMF intensity–phase image pairs. At the NN input, we used 10-mode images each. The case in which 10-mode images are used at the NN output has the lowest performance. However, the NN performance can be improved significantly, if the SDOF at the NN output are reduced to, for example, 5-mode images in both simulation and experiment. Simulation performs slightly better than experiment. Solid lines indicate curves for training data, whereas dashed lines indicate test data.

Download Full Size | PDF

Within our investigations, we have observed a significant dependence between NN performance and spatial degrees of freedom (SDOF) of the phase images used. In Ref. [18], for example, a reconstruction to one of ten different digit elements at the input of a multi-mode fiber (MMF) is shown by using a NN, based on its output speckle patterns. Each digit sample is represented by far fewer modes than the MMF supports. However, due to modal crosstalk, the MMF output speckle pattern consists of a superposition of the whole MMF mode domain. To ensure high-quality image reconstruction, it is crucial that the number of modes, i.e., SDOF, at the NN output is significantly lower than the SDOF at the NN input. In our work, using a $10$-mode FMF, the NN input consists of $10$-mode intensity images and the output of $5$-mode phase images. This improves the performance significantly, compared with the case where both input and output of the NN consist of $10$-mode images. The difference is $ {15}\,\%$, as shown in Fig. 2.

For experimental data generation, the phase (Eq. (5)) of the FMF input is modulated using a spatial light modulator (SLM). A piston-like phase-only SLM with $240\times 200$ pixels and a refresh rate of ${200}\, \textrm{Hz}$ is used [24]. The phase-modulated light is then imaged with a 4f-Kepler telescope and a $40\times$ microscope objective onto the FMF input facet. The intensity (Eq. (4)) of the emerging light field is measured using a monochromatic camera (light path P1 in Fig. 3).

 

Fig. 3. Optical setup for characterization of FMF with path $P1$. $P2$ is used for alignment purposes; $P3$ allows holographic phase measurement of the SLM for calibration. BS: beam splitter; M: mirror; L: lens; MO: microscope objective; CAM: camera; FMF: few-mode fiber ($l= {20}\,\textrm{cm}$).

Download Full Size | PDF

We have investigated the temporal behavior of the optical system. Within 1 h, light fields could be transmitted on each individual mode channel with a variance of less than $ {5}\,\%$. This provides reliable datasets up to 70k image pairs within 1 h of time.

The trained NN reconstructs the physical relation between the input and the output of the FMF it is trained on. Due to the influence of eavesdropping on the FMF under test, a drop in image reconstruction performance is expected. For our investigations, the FMF transmission model is extended by MDLs, representing an eavesdropper. This can be realized, for example, by weighting each mode with a linear transmission coefficient $l_\text {i}$. The coefficients can be represented as the main diagonal elements of a loss matrix $L_{{N}\times {N}}$ ([25]), such as

The aforementioned MDL influences the optical field at the legitimate receiver, resulting in a change in the measured intensity image. This affects the reconstruction performance of the NN, which can be evaluated by a statistical $t$ test to get a binary result about the eavesdropper’s presence. An illustration of this procedure is shown in Fig. 4. Within every single $t$ test, 1k input mode vectors and their associated images are calculated. By determining the average of the reconstructed correlations, the $t$ test observes the hypothesis No eavesdropper by testing the equality of the reconstruction by means of correlation. The significance level is set to $ {5}\,\%$.

 

Fig. 4. Scheme for eavesdropper detection considering MDL. Without eavesdropping, phase reconstruction with $\Gamma = {0,92}$ is possible. The performance drops when eavesdropping occurs to, e.g., $\Gamma = {0,89}$ and can be recognized by a statistical $t$ test.

Download Full Size | PDF

When an eavesdropper gets access to the FMF channel, the influence on the physical properties depends on the type of tapping. As pointed out earlier, eavesdroppers are able to access the outer area of the fiber first, e.g., by v-groove cutting [8]. It can be concluded that modes with high intensity close to the core–cladding boundary are more likely to be tapped. To these modes, a higher leakage, i.e., lower loss factor, is assigned, using the loss matrix $L$ (Eq. (7)).

Our investigations assume that, for low leakage, only the highest-order mode is tapped. By increasing the leakage, both the highest- and second-highest-order modes are tapped (see Table 1). With this approach, we investigate how much a tap on individual modes influences the overall result and whether the $t$ test can recognize the changed transmission properties. In addition, we examine the difference of the tap position (see Eqs. 8 and 9). Results are shown in Fig. 5.

 

Fig. 5. Detection performance. The correlation coefficient $\Gamma$ is shown as a function of the respective MDL. The $t$ test allows detection at $l=0.95$ (right-hand area), which corresponds to a MDL leakage of −0.45 dB on the highest-order modes for MDL close to the output of the fiber. For MDL close to the input, a detection is possible at $l=0.80$ (center area), which corresponds to a MDL leakage of −1.94 dB on the highest-order modes.

Download Full Size | PDF

Table 1. Scheme for MDL Simulation^a

View Table

Taps of $ {5}\,\%$ ($\widehat ={-0,45}\,\textrm{dB}$) of the power on the highest-order modes change the transmission characteristics, so that an image reconstruction of the NN fails a $t$ test. In this example, we have assumed that only the two highest-order modes are tapped at a maximum of $ {5}\,\%$ randomly (see Table 1). Thus, we have filled the loss matrix $L$ with the corresponding values $l_\text {9,10}=0.95$. This means that these mode channels are attenuated through multiplication of the transmission matrix by the loss matrix (Eq. (7)), since it is assumed that the tapped power ends up at the eavesdropper. In this case, the tap occurred close to the receiver, i.e., after the modal crosstalk is fully developed (see Eq. (9)). If the tapping takes place close to the transmitter (Eq. (8)), i.e., before the crosstalk is fully developed, the $t$ test fails at a tapping of $ {20}\,\%$ ($\widehat ={-1,94}\,\textrm{dB }$) on the highest-order modes. In addition to the two highest-order modes, the second-highest-order modes are also tapped with $ {10}\,\%$ each ($l_\text {7,8}=0.9$, $l_\text {9,10}=0.8$). The remaining modes were not affected in this case. These examples show that, in both cases, it is possible to detect an eavesdropper with a single hidden layer NN. With our approach, it must be ensured that the influence on the transmission channel via a tap can be distinguished from other influences, such as mechanical stress, vibration, or temperature fluctuations. These also lead to changes in the transmission properties.

In a practical communication implementation, certain mode channels will be dedicated to surveillance. Thus, data would be transmitted through the remaining channels. Sequentially, test images would be transmitted and image reconstruction could be performed to detect a possible eavesdropper. Simulations show that MDL induced on surveillance channels affects modal crosstalk. Thus, with increasing MDL, the reconstruction quality drops. If this is below a certain threshold, the eavesdropper causing MDL will be detected. The MDL assumed needs to be verified experimentally in future. This might be challenging, because even after the eavesdropping position, where MDL occurs, additional modal crosstalk appears. Due to the FMF temporal instability, the reconstruction quality will drop as well. Even for small temperature gradients, mechanical strokes, or bending, the modal crosstalk will be affected. The NN then has to be trained again, including new data collection.

In conclusion, the detection of eavesdropping is possible by analyzing the intermodal dependencies of FMFs. For the first time, to our knowledge, an FMF data link was measured over mode channels using a NN. With the representative data of an FMF, the NN was trained, which meets real-time requirements for predictions, used for eavesdropping detection. A potential eavesdropper reduces the quality of the reconstruction. Based on a statistical test, eavesdropping detection was demonstrated. It has to be highlighted that, among SDM, FMF has additional advantages in terms of information security compared with single-mode fiber transmission systems.