Fig. 1. Schematic diagram of the frameworks of (a) deep and (b) SL-DNN; (c) the entire diffraction and multi-layer phase modulation process can be regarded as a matrix multiplication by diffraction matrix M. (d) The diffraction matrix of SL-DNN with different values of Fresnel number can be represented by MA (∼101), MB (∼10−3), and MC (∼10−5).

Fig. 2. Schematic experimental setup of SL-DNN. A laser beam at 515 nm was used. The linearly polarized beam was incident on the DMD and images of digits in the MNIST data set were illuminated by DMD. After that, light was normally reflected and propagated to the SLM. SLM modulates the phase of light field and it was reflected by a beam splitter (BS). The output layer is shown by the incoming light received by a CMOS camera. The image dimensions of digits are resized to N and Nr to show different F when the diffraction distance d is fixed. Colors of two light paths are only to distinguish between two SL-DNNs with different F.

Fig. 3. (a) Images of MNIST handwritten input digits are binarized. Ten light intensity detector regions I0,I1,⋯,I9 are set on the output plane, respectively. The detector with maximum sum of intensity shows the predicted number. (b) The confusion matrix and energy distribution percentage of F show numerical test results of blindly testing 10,000 images, and it achieves the max accuracy rate of 94.94%. (c) The confusion matrix and energy distribution percentage for the experimental results. We use 1000 different handwritten digits in the test set as input and achieve an accuracy rate of 92.70%.

Fig. 4. Accuracy of SL-DNN as an MNIST handwritten digit classifier with changing Fresnel number F. For different working wavelengths, SL-DNN has a same range of F approximately from 10−4 to 10−2, which shows SL-DNN’s good performance.

Fig. 5. Optical intensity of single-pixel illumination at different F.

Fig. 6. Classification accuracy, MSE, and SCE loss of SL-DNN trained with MSE and SCE loss function for MNIST handwritten recognition.

Fig. 7. Accuracy of SL-DNN in MNIST handwritten recognition within a certain range of phase error.

Fig. 8. Accuracy of SL-DNN in MNIST handwritten recognition within a certain range of diffraction distance error.

Fig. 9. Confusion matrix and energy distribution of SL-DNN at MNIST recognition task using SCE loss function only.

Fig. 10. Confusion matrix and energy distribution of SL-DNN at fashion MNIST recognition task.

Fig. 11. Confusion matrix and energy distribution of SL-DNN with modReLU nonlinear activation function at MNIST and fashion MNIST recognition task.

Fig. 12. Experimental setup of SL-DNN.

Fig. 13. Phase values modulated by SLM without calibration (red line) and the desired shifted phase (green line).

Fig. 14. Experiment results of resizing the images of input digits to 50, 500, and 800, respectively, and equivalent Fresnel number F is approximately 1×10−2, 8×10−4, and 5×10−5.