Simulated dataset for training, validation, and testing

Here, simulated data were employed to train the CAN network. It will be shown later that the trained network can be employed to analyze experimental data and provide the expected phase profile. The dataset simulation is based on Fresnel’s equation³¹ and the transfer matrix approach^31,45, calculating the reflection and transmission coefficients (Fig. ^S1). Materials’ parameters, including the gold refractive index (n_m) and the thickness (d_m), the sample refractive index (n_s), were varied to generalize the SPR phenomenon for network training accommodating for errors and discrepancies in experimental measurements. The parameters consist of random gold film thickness in a range of 20 to 60 nm (Fig. ^S2), random incident wavelength in a range of 550 to 650 nm, the refractive index of the gold⁴⁶ n_m of 0.18 + 3.44i with ± 10% error in real part and imaginary part, and random refractive index of the sample n_s in a range of 1.0 to 1.4 as labeled in Fig. 2. The simulated BFP was cropped to only one quadrant (256 × 256 pixels), as shown in Fig. 4. All four quadrants carry redundant information due to symmetry in the BFP of the uniform sample. There were 1000 input and output image pairs in each dataset for training and validation. The dataset was further separated to 90% and 10% ratio for training and validation, respectively.

Simulated data for n₀ of 1.52, n_m of 0.18 + 3.44i, d_m of 45 nm, n_s of 1.00, and l₀ of 633 nm for (a) full BFP amplitude image (512 × 512 pixels) before cropping, (b) BFP amplitude input (256 pixels × 256 pixels), and (b) BFP phase output (256 pixels × 256 pixels).

The dataset consists of 2 types of simulated images, including the BFP image input data and the corresponding phase output of the BFP. The phase profiles of the BFP were employed as the label for supervised learning.

The dataset preparation process is shown in Fig. 5. Firstly, the amplitude of BFP and phase of BFP were computed using the Fresnel equations and the transfer matrix approach. These images were then read during CAN network training and validation.

Dataset preparation flowchart.

For testing the networks, the gold thicknesses of 30 nm, 40 nm, 45 nm, and 50 nm, the incident wavelength of 633 nm, and the sample refractive index range from 1.00 (air) to 1.372 (liquid BSA-protein¹¹) were excluded from the dataset for training and validation to test the performance of the trained network.