Fig. 1. Schematic and working principle of multiplane QPI using a wavelength-multiplexed diffractive processor. Illustration of a wavelength-multiplexed diffractive multiplane QPI processor. The diffractive QPI processor is composed of $K$ diffractive layers, which are jointly optimized using deep learning to simultaneously perform phase-to-intensity transformations for $M$ phase-only objects that are successively positioned along the axial direction ($z$), while also routing QPI signals of these objects to the designated wavelength channels at the same output FOV.

Fig. 2. The lateral separation settings of the input objects and the blind testing results of the diffractive multiplane QPI processors. (a) Input volume visualization for six different diffractive designs under different input lateral separation distances ($r$) spanning $\{0,0.2R,0.4R,0.6R,0.8R,R\}$. The color map represents the input object distribution in the axial range. (b) Examples of the blind testing results for multiplane QPI using six different diffractive processors under different input lateral separation distances ($r$).

Fig. 3. Impact of the input lateral separation and the input object thickness on multiplane QPI performance. (a) PCC values of the resulting multiplane QPI measurements with $H_{\text{test}}=0.6$ under different input lateral separation distances ($r$) spanning {0, 0.4R, R}. (b) The same as (a), except for $H_{\text{test}}=1$. (c) Average PCC values of the resulting multiplane QPI measurements as a function of $H_{\text{test}}$. These six curves refer to the blind testing performances of the six diffractive processors trained under different input lateral separation distances ($r$).

Fig. 4. Impact of the input axial separation on the output multiplane QPI performance. (a) Average PCC values of the diffractive multiplane QPI processor outputs with different input lateral separation distances ($r$) covering $\{0,0.2R,0.4R,0.8R\}$ and different input axial separation distances ($Z$) covering $\{128{\lambda }_m,64{\lambda }_m,32{\lambda }_m,16{\lambda }_m\}$. (b) The corresponding output examples of the diffractive multiplane QPI results.

Fig. 5. Cross-talk analysis of multiplane QPI under different input lateral separation distances. Output image matrix demonstrating the cross talk from one input plane to the output wavelength channels, represented by the off-diagonal images. Each row corresponds to a set of input (ground-truth) phase objects alongside the resulting diffractive output images. The diagonal images represent the diffractive output images at the target wavelengths.

Fig. 6. Lateral resolution and phase sensitivity analysis for the diffractive multiplane QPI processor designs. (a) Images of the binary phase grating patterns encoded within the phase channels of the input object, along with the $r=R$ diffractive processor’s resulting output QPI signals (${\mathit{\Phi }}_w$) at the target input plane. The grating has a linewidth of $5.2{\lambda }_m$, and the thickness range parameter $(H_{\text{test}})$ of the input phase object is selected from {0.2, 0.6, 1}. (b), (c) The same as (a), except for $r=0.4R$ in panel (b) and $r=0$ in panel (c).

Fig. 7. Results for testing the external generalization performance of the $r=R$ diffractive multiplane QPI processor design using blind testing images from a new data set composed of Pap smear images. (a) PCC values of the diffractive multiplane QPI processor outputs as a function of the input thickness range. (b) Examples of the ground-truth phase images at different input planes, which are compared to their corresponding diffractive QPI output images.

Fig. 8. Analysis of the trade-off between the imaging performance and the output diffraction efficiency of diffractive multiplane QPI processors. (a) The PCC values of the diffractive multiplane QPI outputs with various levels of diffraction efficiency penalty, plotted as a function of the output diffraction efficiency values. Two sets of diffractive QPI designs using $r=R$ and $r=0.4R$ were trained and blindly tested. Specifically, purple markers (①, ②, ③ and ④) depict different $r=R$ designs, where ${\beta }_{\mathrm{Eff}}=0$ was used for ① and ${\beta }_{\mathrm{Eff}}=100$, ${\eta }_{\text{thresh}}=1\%$, 5%, 10% were used for ②, ③, and ④, respectively, in the training loss function [see Eqs. (15 and 16)]. Gold markers (①, ②, ③, and ④) represent their counterparts using $r=0.4R$. (b) Visualization of the diffractive output fields produced by diffractive QPI processor designs with different input lateral separation distances and various levels of diffraction efficiency-related penalty term.

Fig. 9. Experimental setup and validation of the diffractive multiplane QPI processor for phase-to-intensity transformations. (a) Illustration of a diffractive multiplane QPI processor composed of three diffractive layers $(L_1,L_2,L_3)$ to perform QPI operation on multiplane phase objects. (b) Thickness profiles of the optimized diffractive layers (upper row) and the photographs of their fabricated versions using 3D printing (lower row). (c) Photographs of the experimental setup, including the fabricated diffractive QPI processor. (d) Numerically simulated and experimentally measured intensity patterns at the output plane, compared with the ground-truth input objects, successfully demonstrating experimental phase-to-intensity transformations.