GoKawiilExperimental set-up
The set-up consists of the spatial DOP modulator interfaced separately with the high-dimensional PNN and the encryption–decryption system. These two blocks make a different use of scattering media and digital neural networks and are described hereafter in dedicated subsections.
The DOP modulator is composed of a phase-only liquid-crystal-on-silicon SLM (Hamamatsu X13138, 1,280 × 1,024 pixels, 12.5 μm pixel pitch, 60 Hz frame rate) sandwiched between an input half-waveplate (HWP) and an output pair composed of a quarter-waveplate (QWP) and a HWP. An expanded beam from a continuous-wave laser (λ = 532 nm, 250 mW), with diagonal (D) polarization set by the input HWP, illuminates the SLM. The output QWP and HWP are mounted on high-speed motorized rotation stages and oriented at angles α and β, which are programmed together with the SLM. The WPs convert the phase delay ϕ imparted by a SLM pixel into a SOP set by (ϕ, α, β), as detailed in Supplementary Note 1. The DOP and SOP of a macromode are controlled through the four parameters (δϕ, \(\bar{\phi }\), α, β). In the two-SLM implementation (Supplementary Fig. 4), two identical SLMs (Hamamatsu X15213-16L, 1,280 × 1,024 pixels, 12.5 μm pixel pitch, 60 Hz frame rate) are cascaded pixel-to-pixel by means of a 4f lens system with an inserted HWP at a fixed angle γ = 22.5°. The modulator is calibrated using polarimetry measurements performed by a rotating-WP polarimeter (Thorlabs PAX1000VIS, 0.25° accuracy) that measures S 1 , S 2 , S 3 and ρ.
Spatial modulation of the DOP and SOP is realized in two different configurations. In the first, the modulated beam is observed in a far-field plane located at a distance z from the SLM, whereas in the second, it is observed in the Fourier plane. We detail here the first configuration, as the experimental set-up is more versatile and does not require further optical components, and the Fourier implementation by means of a microlens array is detailed in Supplementary Note 6. The working distance z is set according to the micromode size l, which determines the diffraction length after which micromodes mix by propagation. At full resolution (N = 32 × 32), z is set to approximately 5 cm. For this z, the size of the macromode formed in the far field is comparable with its size L on the SLM.
The modulator is validated by a non-full-Stokes polarization camera (method 1) and a full-Stokes imaging system (method 2). The polarization camera (Thorlabs Kiralux, 2,448 × 2,048 pixels) acquires images (Fig. 3) of the linear polarization degree \(
u =\sqrt{{S}_{1}^{2}+{S}_{2}^{2}}/{S}_{0}\), azimuth θ = arctan(S 2 /S 1 )/2 and intensity S 0 (x, y). Full-Stokes and DOP imaging is performed by carrying out Stokes measurements with the camera in intensity mode, that is, sequentially acquiring intensity projections of the beam profile through a QWP and a polarizer at different orientations54. The accuracy of the spatial DOP and SOP modulation is evaluated by the \({\rm{RMSE}}=\frac{1}{N}{\sum }_{i}^{N}\sqrt{{\sum }_{k}|{S}_{k}^{{\rm{m}}}-{S}_{k}^{{\rm{p}}}{|}^{2}/3}\), in which superscripts ‘m’ and ‘p’ denote measured and programmed values, respectively.
Programming the spatial DOP modulator
The SLM active area is divided into N square macromodes (blocks of pixels). A macromode is further divided into M square micromodes, each consisting of l × l pixels, with l properly set to fill the SLM active area for a target resolution N. For instance, we use l = 12 pixels for M = 256 and N = 25 (Fig. 3d), that is, the micromode size is 150 μm in this case. The minimum macromode size required for accurate spatial modulation of the DOP and SOP is L = 25 pixels, achieved by using M = 25 micromodes of length l = 5 pixels (Fig. 3h). For high-resolution modulation (Fig. 3i), a few blank pixels of constant polarization are used to separate the macromodes and avoid their overlap owing to diffraction. The phase mask is constructed by assigning to all the pixels of the jth micromode a constant phase ϕ j in the interval [0, 2π]. The value ϕ j is randomly extracted from a Gaussian PDF that characterizes the ith macromode, \({{\mathcal{N}}}^{(i)}(\phi )=(1/\sqrt{2{\rm{\pi }}\delta {\phi }_{i}^{2}})\exp [-{(\phi -{\bar{\phi }}_{i})}^{2}/2\delta {\phi }_{i}^{2}],\) with standard deviation δϕ i in [0, π/2] and mean \({\bar{\phi }}_{i}\) in [0, 2π]. By varying \(\bar{\phi }\), the SOP spans a trajectory on the Poincaré sphere that is tunable by the WP angles.
We calibrate the modulator by performing the analysis in Fig. 2 at different values of (δϕ, \(\bar{\phi }\), α, β). In Fig. 2, each data point corresponds to a single-mask experiment. Note that, as ρ tends to zero, the polarized component becomes less definite and, consistently, the measurement error on the Stokes parameters is larger. Averaging over several statistically equivalent masks allows us to reduce the noise observed in single-mask experiments (Supplementary Fig. 2). The DOP is calibrated using the average modulation and the fitting function ρ = aexp(−bδϕ2) + c. The measured SOP (Fig. 2b) is in close agreement with the polarization matrix model (Supplementary Note 2). We then construct a mapping between (S 1 , S 2 , S 3 , ρ) and the four parameters (δϕ, \(\bar{\phi }\), α, β) = X. A target beam, spatially modulated in DOP and SOP, is generated by setting the vectors X(i) accordingly. We study the dependence on the number of micromodes M in Supplementary Fig. 3. In the two-SLM implementation, the WP angles α and β are replaced by a second tunable phase \({\phi }_{2}^{(i)}\), which is set independently for each macromode and remains constant within it. In this case, the modulator is programmed by the vectors \({X}^{(i)}=(\delta \phi ,\bar{\phi },{\phi }_{2})\). The calibration of the two-SLM modulator is reported in Supplementary Fig. 5. The modulator is programmed using custom MATLAB codes.
Avoiding macromode crosstalk
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