# Improvement in Signal Phase Detection Using Deep Learning with Parallel Fully Connected Layers
## 2 Methods
### Acquisition of Experimental Training Data  
Figure 3 shows the experimental optical setup to validate our proposed method. A single-mode semiconductor laser (LM405-PLR40, ONDAX, Inc., California, United States) that operates at a wavelength of 405 nm is first passed through a spatial filter to form a clean spherical wave. Subsequently, it is phase-modulated using a spatial light modulator (SLM) (X10468-05, Hamamatsu Photonics K.K., Shizuoka, Japan) to add a phase signal to the laser beam. The phase-modulated signal light is focused by a lens with a focal length of 200 mm and then low-pass-filtered using a variable square aperture (SLX-1, SIGMAKOKI Co., Ltd., Tokyo, Japan) inserted into the Fourier plane. Finally, the filtered signal light is imaged by a CCD camera (PL-B953U, Pixelink, Ontario, Canada) that detects the intensity distribution with an 8-bit resolution. Note that we directly detect the intensity pattern passing through the aperture, omitting the holographic recording material. This means that we neglect the influence of holographic reconstruction on the spatial frequency components of the signal image. However, it is a reasonable assumption if the aperture size determines the in-plane hologram size. In this case, even though holographic reconstruction will change the spatial frequency components owing to the off-Bragg diffraction, its influence is identical to that caused by the aperture. The pixel pitch of the SLM and CCD camera is 20 and 4.65 mm, respectively. Each data pixel of the input page data is composed of 4 \(\times\) 4 pixels in SLM.  
Figure 2: Preparation of the training data from the intensity image detected by the imager. The image with \(3\times 3\) pixels that surround the target signal pixel is extracted and added with the label of the corresponding signal phase.  
Thus, the pixel pitch of the data pixel is 80 \(\upmu\)m. Nyquist size \(w\)[29; 30] is expressed by Equation (1).  
\[w=\frac{f\lambda}{a}, \tag{1}\]  
where \(f\) is the focal length of the lens, \(\lambda\) is the wavelength of the light source, and \(a\) is the pixel pitch. The Nyquist size of our experimental setup is approximately 1 mm. The aperture size normalized by this Nyquist size is defined as the Nyquist ratio. The experiment is conducted at various Nyquist ratios using apertures that can be adjusted in 10 \(\upmu\)m increments. The optical lens system magnifies the signal image by 1.5. One data pixel corresponds to 26 \(\times\) 26 pixels in the CCD camera. After the data acquisition, one data pixel is resampled from 26 \(\times\) 26 pixels to 4 \(\times\) 4 pixels. Each page data sample consists of 20 \(\times\) 20 data pixels in which 162 data pixels are assigned as signal pixels. A total of 900 page data pixels are acquired at each Nyquist ratio: 300 for training, 300 for validation, and 300 for test data. The acquired output images are cropped to prepare the input image for deep learning, whose pixel size is changed according to the number of signal pixels simultaneously output in a single classification process. For example, when one signal pixel is output in a single classification process, the acquired output image is divided into 3 \(\times\) 3 data pixels and input into a neural network to predict the original signal phase. Note that the relative phase between the signal and known phase reference pixels represent crucial information in this study, and each image is rotated or flipped so that all input images have the same arrangement of the known phase reference pixel.