ENCODING and decoding images
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is consequently modulated. Specifically, the associated two-photon wavefunction, expressed in the momentum basis, takes exactly the form of the object:
Ψ kski = t( k s + k i),
where k s and k i are the momenta of the signal and idler photons, respectively, and t is the object. These correlations are measured by Fourier imaging the down-converted field onto a single-photon-sensitive camera and detecting coincidences between all pixel pairs [ 9 ]. Note that in our experiment, the crystal is cut to produce frequency-degenerate( 810 nm) photon pairs with the same polarization( Type I SPDC).
To intuitively understand the shaping of correlations, we first consider the case without the object and lens before the crystal. The collimated pump at the crystal implies a near-zero transverse momentum( k p ≈ 0), leading to strong anti-correlations between the signal and idler photons momenta: k s + k i ≈ 0. Each time a photon is detected at a certain position on the camera( placed in the Fourier plane of the crystal), its twin appears in the symmetrical position. These strong anti-correlations form a sharp central peak in the measured correlation image( Fig. 2b), constructed by summing the signal and idler photon positions for each detected pair( k s + k i).
With the lens, the pump is focused on the crystal, broadening its transverse momentum( k p = Δk, with | Δk | >> 0). Momentum conservation now gives k s + k i = Δk, reducing the strength of anti-correlations and widening the correlation peak( Fig. 2c). Note that only the spatial correlations are affected- the direct intensity remains uniform.
Placing an object( e. g. a‘ sleeping cat’) in the back focal plane of the lens projects its Fourier transform onto the pump field at the crystal. Conversely, this means the pump field transverse spatial momentum in the crystal plane has exactly the cat shape. Momentum conservation effectively‘ encodes’ the cat shape into the photon pairs spatial correlations. Then, measuring coincidences in the Fourier plane‘ decodes’ the cat in the correlation image, while the intensity image remains uniform. Our approach enables shaping photon-pair correlations into arbitrary objects, without any information appearing in the intensity image.
CORRELATION-BASED IMAGING A key aspect of this approach is the ability to reconstruct correlation images by measuring photon coincidences across all camera pixel pairs. In the work presented above and reported in Ref. [ 10 ], we used an EMCCD camera, requiring 2 to 10 hours to reconstruct a high-quality correlation image, depending on the object complexity. While suitable for proof-of-principle demonstrations, this duration is impractical for real-world applications.
In recent years, new technologies have emerged, significantly accelerating this process. Among them, single photon avalanche diode( SPAD) cameras or intensified
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