1. Introduction

Modal power analysis is usually performed after separating modes by spatial mode sorting technologies [1–8]. Integrated multi-mode waveguides and fibers are two typical optical interconnection platforms that can implement spatial-division multiplexing and demultiplexing. Modal power analysis is necessary in this new multiplexing technology, when optical data carried by different modes need to be detected. On the SOI platform, spatial mode sorting utilizes mode coupling in optical devices such as directional couplers [9,10]. The inter-waveguide mode coupling process can be designed to emulate an eigenmode projection operation. This is done by selective phase matching between a specific optical mode in a deliberately designed multi-mode waveguide and the optical mode in a single-mode waveguide, where the optical power carried by the mode in the multi-mode waveguide can be very efficiently coupled or projected into the single-mode waveguide. By repeating this process for the rest of the modes in the multi-mode waveguide, spatial mode sorting can be carried out [11–14]. On the fiber platform, tilted Bragg gratings imprinted to the fiber can be used to realize spatial mode sorting [15,16]. Some approaches capable of extracting both amplitudes and phases have also been demonstrated [17–21]. By mapping the fiber mode to different frequencies, both amplitude and phase of the modes can be extracted without any reference light [17]. Nevertheless, all these methods rely on spatial mode sorting technique that are implemented on the device level. Here we would like to report a novel modal power analyzing method that does not rely on spatial mode sorting, which eliminates the complexity on device level and shifts the burden to the data processing level. Simply speaking, our technique does not involve the actual spatial separation of the modes. Instead, the coherent far-field diffraction image contributed from all the modes is processed as a whole, i.e., as a gray-scale image. The key concept behind our method relies on the fact that there exists certain correlation between the modal power distribution and the corresponding far-field intensity pattern that can be in practice conveniently captured with a charge-coupled device (CCD) or receiver arrays. To uncover this correlation, CNNs are trained with theoretically synthetic sample sets consisting of the far-field intensity patterns with the corresponding modal power distribution as label.

Convolutional neural networks have been shown to be very efficient for processing data and discovering patterns with grid data structures, e.g., images [22–25]. It has features called sparse interaction and parameter sharing, which can significantly reduce the storage burden of parameters [26]. A typical convolutional layer consists of three stages. The convolution stage performs convolutions to produce linear activations, the detector stage performs a nonlinear activation, and the pooling stage modifies the output of the layer further and gives naturally invariant to translation when using max-pooling [27]. In our case, a CNN architecture consisting of two convolutional layers and three fully connected layers is first designed to deal with thin optical wavaguides, and later it is extended to include five convolutional layers and three fully connected layers to tackle a more general waveguide. The convolution operation takes the pixelated image pattern and the kernel function or filter as two input parameters [28]. Our trained CNNs are very successful when taking either processed or unprocessed far-field intensity patterns as input. It is capable of predicting its modal power distribution with very high accuracy. Notice that our method focuses on analyzing the power of different modes rather than spatially separating them, which has potential applications in cases, e.g., at the optical output of an optical route or an optical-electrical-optical repeater, where the optical data need to be detected before further actions are taken. We believe our proposed approach is novel and clearly belongs to a different paradigm that has recently attracted a lot of interests in other field and will impact the future optical system.

2. Theories and methods

2.1 Theoretical model

Our training set consists of samples of far-field intensity patterns with known modal power distribution as label. They can be collected either in experiments or numerical computations. Here they are obtained numerically by projecting a variety of random superpositions of a set of orthogonal waveguide modes into far-field and recording their corresponding intensity patterns. The performance of our trained neural networks is benchmarked with criteria that are the percentage of test samples that are predicted precisely within a predetermined square error or absolute error tolerance.

Herein we consider an air-cladded silicon-on-insulator (SOI) waveguide shown in Fig. 1(a) as an example to demonstrate our concept of machine learning assisted modal power analysis. The SOI waveguide is assumed to be a multimode bus waveguide commonly found in the emerging spatial-division multiplexing scheme. The width and height of this SOI waveguide are chosen, for instance, as $w=1.6\mu m$ and $h=0.22\mu m$ respectively. This thin waveguide supports only three quasi-TE optical modes at a wavelength$\lambda \text{=1}\text{.55}\mu m$, which are $T{E}_{00}$, $T{E}_{10}$, $T{E}_{20}$ mode, respectively, as shown in Fig. 1(b). The arbitrary guided optical field near the exit of the SOI waveguide, $\stackrel{\rightharpoonup }{E}(x,y)$, could thus be written as a linear superposition, $\stackrel{\rightharpoonup }{E}(x,y)={\displaystyle \sum _i{a}_i{\stackrel{\rightharpoonup }{E}}_i}(x,y)$, where $a_i$ and ${\stackrel{\rightharpoonup }{E}}_i(x,y)$ is the complex amplitude and normalized electric field of mode i. Owing to orthogonality, i.e., ${\displaystyle \int {\stackrel{\rightharpoonup }{E}}_i}(x,y)\cdot {\stackrel{\rightharpoonup }{E}}_j(x,y)dxdy={\delta }_{ij}$, the complex amplitude of the three modes can then be calculated by projecting total electric field $\stackrel{\rightharpoonup }{E}(x,y)$ to each waveguide mode. However, in practice, the far-field intensity pattern $I(x\text{'},y\text{'})$, described in Eq. (1), rather than the field inside the waveguide is more conveniently captured by using an imaging device (e.g., CCD).

 

Fig. 1 SOI waveguide under consideration. (a) A schematic with a superimposed mode at the exit of the SOI waveguide and the corresponding far-field pattern that can be easily recorded by CCD and analyzed by CNN. (b) Pseudo-color images showing the calculated mode profile for the three quasi-TE modes of this SOI waveguide with w = 1.6µm, h = 0.22µm, and λ = 1.55µm.

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In this scheme, the phase information of the field is unrecoverable and thus the full knowledge of the field is unknown, rendering the projection method discussed above inapplicable. In order to retrieve the modal power distribution, a mapping function or operation that maps the intensity pattern I to the modal power distribution (i.e., $[{\overline{A}}_1,{\overline{A}}_2,{\overline{A}}_3]$ in our three-mode system, where ${\overline{A}}_i=|a_i{|}^2/{\displaystyle \sum _j|a_j{|}^2}$) needs to be investigated. The essence of this investigation involves finding feature maps, where machine learning based CNNs have clear advantages over other approaches.

2.2 Steps to generate and preprocess synthetic data set

2.3 Network architecture

In this work, max(n) = 0 is satisfied and the preparation of the input data is shown in Fig. 2(a). The far-field intensity pattern of our waveguide is imaged and pixelated into a [16], [64], [1] array, where 16, 64 and 1 represent the number of pixels along vertical direction and horizontal direction and the number of color channels, respectively. Notice that only one color channel is required to describe our intensity pattern, which is basically a greyscale map. Because all three modes considered here are single-moded along vertical axis, the far-field pattern is manifested with a collection of stripes along vertical axis as shown in Fig. 2(a). It is thus convenient to compress this [16], [64], [1] array of intensity data by summing up the values along the direction of the stripe, i.e., vertical direction to form a simplified [1], [64], [1] array. This 64 intensity values denoted by I in this array is further normalized, logarithmized and multiplied by −1 to become the final input data to our CNN. Preprocessing the input data in this fashion provides two key advantages, i.e., 1) great simplification of the CNN which takes effectively a 1D array as input, and 2) boost in training speed due to a reduced dynamic range of the data value.

 

Fig. 2 Input data and the CNN architecture. (a) Far-field intensity distribution data is compressed by summing up all the intensity values along the vertical axis leaving behind a horizontal-position (or $x\text{'}$-axis) dependent grayscale map, I, and -log(I) is used as final input data to our CNN. (b) CNN architecture with two convolutional layers and three fully connected layers.

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As shown in Fig. 2(b), our specialized CNN consists of 2 convolutional layers, denoted as Conv1 and Conv2, and 3 fully connected layers, denoted as FC3, FC4 and FC5. Both the convolutional layers use filters with shape of [1,3], stride of [1,1] and same-padding, use ReLU [30] as activation function and use max-pooling with shape of [1,2] and stride of [1,2]. Number of filters of Conv1 and Conv2 are 16 and 32 separately. The number of features shrinks while the number of channels is multiplied owning to the [1,2] stride of max-pooling and multiplication of filter number. The feature map after two convolutional layers is flattened to one dimension and then fed to fully connected layers. The activation functions of FC3 and FC4 are ReLU while that of FC5 is softmax [31]. Number of neurons used for FC3, FC4 and FC5 are 64, 32 and 3, separately. The final output of FC5 corresponds the predicted modal power distribution of a given far-field intensity pattern.

In training phase, the output will be compared with label to compute the mean square error (MSE) of m samples as shown in Eq. (2),

3. Results and discussion

A data set containing 9000 samples is classified as a training set and another data set containing 1000 samples is classified as a cross-validation set. In an epoch, mini-batches with size of 16 out of 9000 samples in the training set are fed to the network in turn until the entire set is traversed. The training set is then shuffled and used in a new epoch. After repeating this process 10000 times, the MSE of training set ends up with 2.72e-5 while that of cross-validation set is 9.04e-5. The difference between the training set and the cross-validation set is a result of the natural regularization of CNN and is completely acceptable, suggesting that our trained CNN performs a superb job in predicting modal power distribution. Next, in order to prevent overfitting to the cross-validation set, the trained neural network is tested for a second time with 10000 randomly generated samples and the MSE ends up with 9.07e-5, which is very close to the MSE of the cross-validation set.

To further examine the efficacy of our trained CNN, square error (SE) and absolute error (AE) of the modal power distribution of the i^th sample are investigated. Their definitions are shown in Eqs. (3) and (4),

 

Fig. 3 Histograms of square error distribution of 10000 testing samples. Insert: samples with square error larger than 1e-3.

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For samples whose maximum squared errors are less than $to{l}_{se}$, their maximum possible absolute error can be calculated by $A{E}_{\max }=\sqrt{2\times to{l}_{se}/3}$. When $to{l}_{se}$ is set to 1e-3, the maximum possible absolute error is approximately equal to 0.0258. A typical histogram for actual distribution of absolute error is shown in Fig. 4. A zoom-view plot for AE larger than 0.02 is illustrated in the insert for clarity. It is obvious that the absolute errors for modal power distribution of the three modes are less than 1% for more than 85% of all the samples and the absolute error is hardly greater than 2%. If maximum absolute error is set to $to{l}_{ae}=0.02$, the prediction accuracy can be as high as 98.95%.

 

Fig. 4 Histogram of absolute error distribution of 10000 testing samples. Insert: samples with absolute error larger than 0.02.

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It is worth noting that the modes concerned in this study are those from a thin waveguide that is single mode along its vertical direction. For simplicity, a technique has been then applied to compress the intensity data as explained above and in Fig. 2(a). However, it is neither mandatory nor a limiting factor for our concept if stricter analysis is required and more general cases are under consideration. To illustrate this, a full-scale CNN, taking a matching input data of the size [16], [64], [1] (e.g., the left bottom image of the pixelated far-field pattern shown in Fig. 2(a)) instead of [1], [64], [1] in the specialized case, is trained and investigated. The prediction accuracy for this full-scale CNN is found to be 99.02% with a maximum square error tolerance $to{l}_{se}=1e-3$ and 97.94% with maximum absolute error tolerance $to{l}_{ae}=0.02$. Compared with the results from the specialized CNN above, the performance remains nearly unchanged but the training time doubles when using the same computation resources. Hence, the preprocessing of data so as to facilitate the construction of a specialized CNN may be preferred in scenarios where the modal power distribution in thin waveguide is of interest.

To verify the general feasibility of our concept, a more general condition, e.g., concerning waveguides with multiple mode orders in both vertical and horizontal directions, is also investigated. For the illustration purpose, a SOI waveguide with 8 quasi-TE eigenmodes corresponding to max(m) = 1 and max(n) = 3 is used. Considering the fact that the far-field intensity patterns are determined simultaneously by as many as 16 variables (8 for amplitude and 8 for phase), the difficulty of accurately predicting the modal power distribution increases dramatically, compared with the case shown above. Nevertheless, after a few rounds of optimization, an appropriate CNN structure for this difficult task is shown in Fig. 5(a). It can be broken into 8 hidden layers. Each of the layers denoted by “Conv1-5” consists of a sequence of convolutional stage, batch-normalization stage [33], activation stage and max-pooling stage. Each layer uses convolutional filters with the size of [3,3], stride of [1,1] and same-padding, uses ReLU as activation function, and uses max-pooling with size of [2,2] and stride of [2,2]. The activation functions of layers denoted by “FC1” and “FC2” are ReLU while that of “FC3” is softmax. The size of 2D images and number of channels after each layer are indicated in Fig. 5(a). After training with 396000 samples from the training set (4000 from the cross-validation set) for 1000 epochs, the MSE of the training set ends up with 8.90e-6 while that of the cross-validation set is 2.35e-5. Performance of this CNN is benchmarked statistically with 10000 testing samples and the corresponding square error and absolute error distributions are shown in Fig. 5(b) and (c), respectively. The prediction accuracy for this CNN turns out to be excellent, about 98.70% with a maximum square error tolerance $to{l}_{se}=1e-3$ and 97.42% with maximum absolute error tolerance $to{l}_{ae}=0.02$. Therefore, compared with the previous case shown above, although the number of modes has been more than doubled (8 vs. 3), the prediction performance by this newly trained CNN remains nearly unchanged. This finding confirms the generality of using CNNs to confidently predict modal power distribution. Our concept thus may have a very wide application scope in the related field.

 

Fig. 5 (a) Structure of proposed CNN for predicting mode power distribution for waveguides with multiple mode orders in both vertical and horizontal directions. (b) Performance of this CNN with 10000 testing samples benchmarked with square error distribution. Insert: samples with square error larger than 1e-3. (c) Performance benchmarked with absolute error distribution. Insert: samples with absolute error larger than 0.02.

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In reality, the performance of CNNs can be impaired by many factors. Among them, noise data could have the most direct impact. So in order to verify the robustness of our general CNN shown in Fig. 5(a), noisy testing samples are prepared and fed into the trained CNN. In detail, the intensity value of each pixel of testing samples is multiplied by its own factor, f, which equals to $1+N(0,1)\cdot \sigma$. Here $N(0,1)$ is the standard normal distribution and $\sigma$ can be regarded as the simulated noise intensity. The prediction accuracy against the noise intensity with absolute error < 0.02 is shown in Fig. 6. Overall, the prediction accuracy slowly degrades with the increasing noise intensity. It, however, still remains larger than 95% even when the simulated noise intensity reaches 0.08. Notice that this kind of noise level can be hardly reached in any practical scenarios. This study thus confirms that our CNN model can be very tolerant to noises.

 

Fig. 6 Prediction accuracy against different noise intensities with absolute error smaller than 0.02 as criteria.

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4. Conclusion

In summary, a modal power distribution analyzing scheme using CNNs has been proposed and its performance has been evaluated using synthetic data. A specialized CNN and two full-scale CNN are trained separately to discriminate modal power based on the preprocessed far-field intensity pattern. These trained CNNs have performed excellently on predicting modal power distribution with a very high accuracy > 98% even at the maximum SE tolerance $to{l}_{se}=1e-3$ and even for waveguides that are heavily multi-moded. In addition, these CNNs can be very efficient and robust in predicting mode power distribution even when exaggerated noises are incorporated in the model. We believe that this new approach to analyze modal power for modes in multimode waveguides holds great promises for potential applications in spatial-division multiplexing and structured light, and can be confidently extended to few mode fibers.