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摘要
Objective Ultrashort laser pulses have been widely used as essential tools in many scientific research fields, such as ultrahigh intense laser physics, ultrafast spectroscopy, and nonlinear optical microscopy. The key aspect of measuring the temporal profile of an ultrafast laser pulse is the accurate characterization of its spectral phase. Self-referenced spectral interferometry (SRSI) is a relatively new characterization technique for measuring the intensity and phase of ultrashort laser pulses with attractive capacity introduced in 2010. SRSI is an analytical, sensitive, accurate, and fast method. The development of SRSI in recent years is to simplify the setup, optimize the reference pulse, or adapt to different situations. However, SRSI still uses the initially proposed algorithm, Fourier transform spectral interferometry ( FTSI), based on spectral interferometry and few iterations. Thus, the approximate calculation is used in this algorithm to simplify the calculation process, leading to the loss of some details, and the calculation accuracy is not sufficiently high. Therefore, research on new algorithms that can improve the measurement performance and accuracy of SRSI is of great significance to promote the development of SRSI technology and ultrafast laser technology. With the rapid improvement of computer computing power, deep learning has recently achieved great success. This study proposes a deep learning method using a neural network called Dense-ID-U-Net used for one-dimensional signal processing to measure spectral phases of femtosecond pulses with the SRSI method. Furthermore, on our simulated datasets, the measurement of spectral phase accuracy using Dense-ID- U-Net is at least about one order of magnitude improved than that of the traditional SRSI algorithm. Additionally, measured data are used to verify that Dense-ID-U-Net, trained by simulated data, can calculate experimental data. Methods A one-dimensional U-Net neural network structure combined with self-designed dense blocks, called Dense-ID-U-Net (Fig. 2), was designed for one-dimensional data, and its weights were initialized according to the features of SRSI. The classical encoder-decoder network structure with added dense blocks and skip connections was used in the measurement to improve the network's performance. Based on the principle of SRSI (Fig. 3), three different datasets with many analog data close to real spectral phases of ultrafast pulses were simulated (Fig. 6). The first dataset used a randomly generated phase curve of the highest order of third and a fixed perfect Gaussian curve to simulate the spectrum and a fixed delay time to generate the spectrum of interference fringes. In the second dataset, we simulated the spectra of interference fringes using curves of the highest order of third and spectra simulated by random Gaussian curves with a delay time of random values from 500 to 1500 fs. Finally, we simulated the spectra of interference fringes in the third dataset. Each set of data contains a phase curve, a spectral amplitude, and a delay time. The phase curve consists of two lines added together, a main curve simulated by three random Taylor coefficients and a carrier simulated by five Taylor coefficients multiplied by a random coefficient between 0.3 and 0.5. The spectral amplitude was simulated by a random Gaussian curve. The delay time were randomly valued between 500 and 1500 fs (Fig. 6). Dense-ID-U-Net with specially initialized weight was trained on those datasets. Then, we compared the measurement results using the traditional SRSI algorithm FTSI with trained Dense-ID-U-Net on those datasets. We also compared the result and train process of Dense-ID-U-Net with other neural networks without dense blocks or skip connections. Consequently, we used the experimental data to verify whether Dense-ID-U-Net trained on simulated data can calculate experimental data. Results and Discussions The measurement results using the traditional SRSI algorithm FTSI and trained Dense-ID- U-Net of the three experiments are shown in Figs. 7,8, and 9. The accuracy of the spectral phase measurement using Dense-ID-U-Net is at least about one order of magnitude improved more than that of the traditional SRSI algorithm. Compared with other neural networks without dense blocks or skip connections, the measurement results with dense structures or skip connections show obvious superiority. Finally, the measured experimental data confirm that Dense-ID-U-Net trained on simulated data can calculate experimental data (Fig. 10). Conclusions This study proposed a one-dimensional convolutional encoder-decoder neural network called Dense-ID- U-Net based on the encoder-decoder structure with our design of dense blocks and added skip connections. Dense-lD-U-Net can adapt to various studies by modifying neural network parameters and changing weights initialization methods. Here, it is used in the SRSI method based on deep learning. End-to-end learning of the relationships between spectral interference fringes and real spectral phases utilizes input information without intermediate calculation, which is the advantage of deep learning. The fitting ability of the neural network is significantly improved using our design of dense blocks. The added skip connections can make good use of the primary information. The accuracy of spectral phase measurement using Dense-ID-U-Net is at least about one order of magnitude improved more than that of the traditional SRSI algorithm. It is verified that Dense-ID-U-Net, trained by simulated data, can calculate measured data (Fig. 10). However, laser pulses are more diverse in practice. In future studies, we will consider various conditions of laser pulses to enhance the dataset to adapt to the specific situation. The advantage of Dense-1D-U-Net is that it is robust and can adapt to different studies by training it on different datasets and initializing its weights in different ways. This neural network can be extended to ultrafast spectroscopy and related studies based on one-dimensional information.
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