Fig. 1. Schematic diagram of triple correlation. (a) Simplified beam-emission model of traditional thermal light (left) versus EPR-type thermal light (right). (b) Sketch of the triple correlation among the injected pump beam and a pair of converted beams in SPDC or the RH source. (c) An example of the diffracted twin beam produced via a commercial SLM imprinted with random holograms (RHs). (d) General theoretical model based on Fourier optics, mainly analyzing the second-order correlation function between the twin beams driven by a structured pump in this paper. (e), (f) Two specific cases of geometrical optics illustrate how to harness such triple correlations to enable rich “nonlocal” optical information interaction in the RH source.

Fig. 2. Experimental results of human face recognition via triple correlations in the RH source. (a) Experimental setup [according to Fig. 1(e)] used for pattern recognition and correlation manipulation. L: lens; I: iris; BS: 50/50 non-polarizing beamsplitter; D: detector. (b), (c) Seven human-face patterns form a custom phase pattern library or dataset {Φ}; we utilize seven different colors and capital letters in (b) to represent the seven human-face patterns, and the corresponding lowercase letters as well as their complementary colors represent their negative patterns {−Φ}; the notations A and a, for instance, in (c), denote {Φ/2} and {−Φ/2}, respectively. (d), (e) Experimental results of phase pattern recognition when two facial patterns are placed in different configurations according to the inset at the top. The graphical horizontal and vertical coordinates in (d), (e) refer to the marking rules in (b), (c). (f) Two confusion matrices corresponding to (d), (e). The two correlation arrays in (d), (e) contain 14×14 and 7×14 measured correlation functions (151 × 151 pixels), respectively. 4000 realizations contribute to each correlation function g(2)(r2;ϕp,ϕ1,ϕ2).

Fig. 3. Experimental results of correlation manipulation. By engineering specific spatial phase structure in the pump beam, optical correlations can be induced between two arbitrary independent phase patterns, for example, matched correlation generation between facial patterns and helical patterns (a) or random-grid patterns (b); (c) the corresponding 3D confusion matrices (7×7×7). The correlation array with 7×7 cells (151×151pixels) in (a), (b) only exhibits the correlation functions g(2)(r2;ϕp,ϕ1,ϕ2) in the diagonal section including the body diagonal of (c).

Fig. 4. Experimental results of nonlocal spiral phase contrast imaging of different targets with the RH source pumped by a vortex beam (ℓ=−0.5). (a) Experimental setup. (b) Amplitude target and its edge-enhanced imaging under (c) ℓ=0 and (d) ℓ=−0.5. (e) Phase target and (f) its vortex mapping imaging under ℓ=−0.5. 500,000 realizations contribute to these correlated patterns (801×801pixels).

Fig. 5. Nonlocal intelligent image classification with the RH source. (a) Experimental setup used for nonlocal classification of two types of handwritten digits (0 and 1). (b) “Learned” single-layer phase-only filter (decoder) placed in the pump beam. See Section 5.C and Fig. 9 for more details. (c) Desired output mode TM. The two focal points (from left to right) correspond to digit 1 and digit 0, respectively. (d), (e) Experimental results. The first column: targets placed in SLM1; the second column: retrieved correlation function g(2)(r) (801×801pixels) over 20,000 realizations; the third column: g(2)(r)·TM for an intuitive comparison. The inserted cross-section can directly infer the category of the input target.

Fig. 6. Comparison of optical metrology of relative phase shift under three light sources: HBT-type thermal light, coherent beam, and EPR-type thermal light. (a) Experimental apparatus used to observe far-field stripe patterns of a double slit with Δφ (to be measured) between two single slits. (b) When Δφ=0,π/5,2π/5, the yielded stripe patterns and their cross-sections of the three cases. (c) Curves of experimentally measured transverse offset Δx as a function of relative phase Δφ in the three light sources. Each panel in (b) is composed of three observed patterns (140×461pixels), and 8000 realizations contribute to the correlation patterns.

Fig. 7. Based on the Klyshko’s advanced-wave picture, the three cases of two-detector joint correlation detection in Section 3 can be well interpreted by the single-detector observations.

Fig. 8. Experimental observation of correlation and conservation of radial momentum in EPR-type thermal light (±1-order diffracted waves in the RH source). (a) Experimental apparatus; (b) radial phase profile of pr=−3 (upper) and the corresponding hologram (bottom) carrying the radial momentum of pr; (c) experimental measurement of five correlation functions of radial momentum g(2)(ppr,p1r,p2r). 4000 realizations contribute to each correlation coefficient.

Fig. 9. Geometrical framework and numerical results of the single-layer Fourier deep diffractive neural network. (a) The unfolded diagram of the binary classification of two classes of handwritten digits (amplitude targets). (b) The well-trained phase mask is then used in Fig. 5 for correlated image filtering. (c), (d) Numerical results of optical classification of training and test sets, and the recognition accuracy reaches 99%. The four patterns, from left to right, in each panel of (c), (d) are the amplitude target, the complex amplitude in the Fourier-transform plane, the resultant image Iout filtered by the phase mask, and Iout·TM, respectively.