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Abstract
We propose a polarimeter that combines a plasmonic spiral structure and the machine learning algorithm with an ultra-compact footprint of 10*10µm2. Being different from previous similar schemes working only as circular polarization analyzers, arbitrary states of polarization (SOPs) can be retrieved via the spiral structure for the first time to our best knowledge, by analyzing the near-field intensity distribution through machine learning. A 3-layer neural network (NN) is successfully trained to correlate intensity patterns to the SOPs of incident light. Based on simulation, a low estimation error benchmarked by mean-squared error (MSE) of only 1.23e-3 is achieved. In this way, without the conventional bulky optical system or complex nano-structures, SOP detection is achieved via such a simple and ultra-compact spiral. The proposed scheme not only pushes the application limits of the device based on plasmonic spiral structures but also provides a new insight for SOP detection.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The state of polarization (SOP) is one of the basic properties of monochromatic electromagnetic wave along with phase and frequency, and it has been studied and implemented in various fields ranging from astronomy [1], remote sensing [2] to material analysis [3]. In order to quantify the SOP, relative Stokes vector [4] with three Stokes parameters has been introduced. Conventionally, a series of polarizers and wave plates have to be arranged properly and several round measurements are needed to obtain Stokes parameters, increasing inconvenience of the system. Furthermore, those systems are usually inevitably complex and bulky.
In order to detect SOP with only one chip, different types of metastructures were proposed [5–8]. Typically, incident light with different SOPs can be guided into different directions. Other designs use polarization sensitive nano-structures that generate unidirectional surface plasmon polariton (SPP) wave [9,10]. However, devices mentioned above are usually spatially inefficient or require complex nano-structures, making them difficult to design and fabricate. Without utilizing complex structures, the plasmonic spiral lens consisting of only an Archimedes’ spiral has attracted great interests for years, mainly for the ability to distinguish circular polarization [11–15]. Using a left-hand spiral plasmonic lens, a bright focus spot appears at the center of the transmission pattern for right-handed circular polarization (RCP) illumination, while a dark center can be observed for left-handed circular polarization (LCP) illumination. However, this property cannot be used to directly detect SOPs of elliptically polarized light.
To tackle this issue, the machine learning method is introduced here. Recently, machine learning is widely implemented in the field of nanophotonics such as using geometric deep learning to facilitate the design process [16]. In this paper, machine learning algorithm is used to uncover the correlation between near-field intensity patterns and input SOPs. Specifically, a 4-turn Archimedes’ spiral slot is designed on a 600 nm thick Au film with a footprint of 10 × 10µ${m^2}$. By using the near-field intensity images with pre-known incident SOPs to train a three-layer neural network (NN), this spiral structure can be implemented to detect any SOP in one shot, which is different from previous similar schemes worked only for circular polarization. Although the device is designed for the wavelength of 980 nm, it is still applicable for a broadband ranging from 900 to 1100 nm, showing a high tolerance to wavelength shift. Simulated results show that the training process takes only 250 epochs to successfully converge and the detection accuracy is 1.23e-3, benchmarked by mean-squared error (MSE) between the predicted SOPs (represented by Stokes parameters) and real values.
2. Theories and methods
In this part, the theoretical SPP field distribution of a designed spiral structure is derived, in order to verify its potential to use machine learning for SOP detection. Also, the architecture of the NN and data generation process will be discussed.
2.1 Theoretical models
A left hand 4-turn Archimedes spiral slot is penetrated in Fig. 1(a), based on a thin Au film with a thickness of 600 nm on the glass substrate. The whole footprint is 10µ$\textrm{m}$ in length and 10µ$\textrm{m}$ in width, illuminated normally by a plane wave at 980 nm. The slit width of each turn of the spiral is chosen to be 250 nm. As shown in Fig. 1(b), it is described in the cylindrical coordinate as :
Fig. 1. (a) The Au spiral structure on a glass substrate illuminated normally by different polarization states. (b) A left hand 4-turn spiral slot penetrated through an Au film. (c) Single turn spiral.
In order to prove that such a spiral structure is sensitive to different SOPs theoretically, in other words, its transmission intensity pattern will change along with the incident SOP, the theoretical field distribution is derived. For brevity, we only calculate the condition of a single-turn spiral, as shown in Fig. 1(c), considering the major difference between multi-turn and single-turn spiral is only the enhancement of plasmonic field [13]. According to Ref. [11], the near-field distribution of this spiral structure illuminated by LCP is given as:
2.2 Data set generation and network architecture
As shown in Figs. 2(a) to 2(d), the data set contains pixelated near-field intensity patterns with pre-known SOPs as labels. Specifically, 700 near-field intensity patterns of this spiral device under arbitrary polarized illumination are collected, using FDTD method (commercial simulation software FDTD). 300 samples are randomly selected as training-set, 100 as validation-set and 300 as testing-set. In simulation, a plane wave at 980nm is chosen to be the light source and the calculation region is 15µ$\textrm{m} \times \; $15µ$\textrm{m} \times {\; }$5µ$\textrm{m}$, surrounded by a perfect matched layer (PML) which serves as the absorbing boundary. The monitor with exactly the same footprint of the spiral is located 100 nm above the Au film. Then, the center region (2µ$\textrm{m} \times {\; }$2µ$\textrm{m}$) of each nearfield intensity pattern consisting of 2500 pixels is retrieved. Finally, every pixel value of each sample is normalized into a range of [0-2], and flattened to be a one-dimensional vector. Here, the relative Stokes vector is utilized to label each sample. To reduce the complexity of design and computation, a feed-forward network with single hidden layer as shown in Fig. 2(e) is developed via the Neural Network Toolbox in matlab, which consists of three layers, namely input layer, hidden layer and output layer. The activation function of the input and hidden layer is sigmoid and that of the output layer is linear. There are 2500 nodes in the input layer which will be fed by the data set mentioned before. As the near-field intensity pattern of this spiral device is highly sensitive to incident SOP, a feed forward NN with high detection accuracy can already be obtained using a training-set consisting of only 300 samples, and thus the required computation for training is acceptable even when there are large number of input nodes. However, it should be noted that appropriate dimensionality reduction (DR) methods of the input such as autoencoder architecture [16–18] are necessary when huge amount of training data is required and the number of input features is also large to reduce computation complexity. The last layer has 3 nodes to output the predicted relative Stokes parameters. As for the hidden layer, the detection accuracy of NNs with different number of nodes in the hidden layer should be tested via the testing-set, which will be discussed later.
Fig. 2. a∼d are simulated near-field intensity distributions for horizontal, 45°, left hand and right hand circularly polarized wave. Intensity patterns illuminated by arbitrary SOPs will be utilized as the training-set. (e) NN architecture with three layers including one hidden layer. Input layer has 2500 nodes that correlate to 2500 pixels of each intensity image and the output layer’s 3 nodes correlate to the 3 Stokes parameters.
3. Results and discussion
Here, the NN is optimized via certain methods. Furthermore, the detection accuracy of Stokes parameters as well as operation bandwidth will be discussed to demonstrate the reliability of our method. Firstly, the network will be trained by the training-set through the scaled conjugate gradient algorithm [19] typically used for fast training. Using an early stop technique, the validation-set is utilized to measure network generalization error of each training iteration in order to halt training when generalization error stops decreasing. A testing-set consisting of randomly selected 300 samples (intensity patterns of this spiral device under different polarized illumination) is utilized to test the robustness and reliability of NN with different node number in the hidden layer. Specifically, in order to quantify the estimation error and reflect the performance of the trained NN, the MSE of N samples (N = 300) in the testing set is computed as shown in Eq. (7),
Fig. 3. (a) The MSE of testing-set for NNs with different number of nodes in the hidden layer. (b) Training process represented by MSE of the training-set and validation-set for every training iteration.
The final results of Stokes parameter’s detection are given in Figs. 4(a) and 4(b), where the predicted ${[{{S_1},{S_2},{S_3}} ]^T}$ of 300 samples in the testing-set are compared with the real values. For clarity, Fig. 4(b) is plotted in the form of Poincare sphere. In all cases, the testing results have good agreement with their real Stokes parameters. Although the detection error still exists, it can be reduced by further optimizing the machine learning algorithm or collecting more training samples.
Fig. 4. (a) Measurement of the state of polarization ${[{{S_1},{S_2},{S_3}} ]^T}$ of 300 selected polarizations using the spiral polarimeter (orange) with their real Stokes parameters (blue). (b) Comparison between measured Stokes parameters (orange) using our Stokes analyzer and their real values (blue) in a Poincare sphere.
To be noted, even though the plasmonic spiral structure is designed for the wavelength of 980nm, it is still applicable for other wavelengths. Concretely, we calculate the correlation coefficients (R) [20] between the intensity distribution matrix of λ=980nm and other wavelength conditions, as shown in Fig. 5. When R is close to 1, the intensity patterns of other wavelengths are similar to the condition of 980 nm. Oppositely, as the wavelength moves away from the designed 980 nm, R is close to 0 and the device can no longer be used to detect SOPs because it gets insensitive to SOPs under such wavelength. It turns out that the same algorithm for polarization analysis can be shared, within a broad wavelength ranging from 900 to 1100nm.
Fig. 5. Correlation coefficients (R) between the intensity distribution matrix at wavelength of 980nm and other wavelength conditions.
4. Conclusion
In summary, we propose a plasmonic spiral device to retrieve Stokes parameters of arbitrary SOP incidence, assisting by the machine learning algorithm. The whole chip has a footprint of 10*10µ${m^2}$, which is extremely space efficient comparing with traditional polarimeters. After mathematically proving the strong polarization sensitivity of the proposed structure, machine learning algorithm is used to find the hidden correlation between near-field intensity patterns and Stokes parameters. We successfully use such a simple spiral structure to achieve any SOP detection, and a bandwidth larger than 200 nm is found for the proposed scheme.
Funding
National Natural Science Foundation of China (61775073, 61911530161, 61922034); Program for HUST Academic Frontier Youth Team (2018QYTD08).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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