Fig. 1. (a) Schematic of a PCL array at the source with N=3. Relative CV spectrum of the array at (b) z=10zr and (c) z=20zr. The beam parameters are λ=632.8  nm, w=0.157  mm, σ=0.1w, r0=1.5w, and L0=1. Here, zr=πw2/λ=0.122  m is the Rayleigh distance.

Fig. 2. Numerical simulation of (a) spatial distribution of the averaged intensity for N=3 and z=0; (b) spatial distribution of the averaged intensity for N=3 and z=20zr; (c) correlation phase of a radial array for N=3 and z=0; and (d) correlation phase of a radial array for N=3 and z=20zr. The other beam parameters are the same as those in Fig. 1.

Fig. 3. (a) Experimental setup to generate a CV with a radial array of N fundamental PCL beams. Examples of the CGH displayed on the SLM for the production of the complex field (N=3) under the following conditions: (b) total coherence with c0=0, δp=0; and (c) poor coherence with c0=3w, δp∈(0,2π). (d) Instantaneous intensity I(k)(x,y) of the PCL array produced with c0=3w, δp∈(0,2π). (e) Phase distribution of the complex light field E(k)(x,y) produced with the CGH of (c). The array is in the random state of independent perturbation. The beam parameters are w=0.157  mm, r0=1.5w, and L0=1. BE, beam expander (Newport T81-3X); BS, beam splitter; SLM, spatial light modulator; NF, neutral density filter (Absorptive, optical density 1.0); RM, reflecting mirror; P1, pinhole; MO, microscope objective; CMOS, complementary metal oxide semiconductor camera; PC1, PC2, personal computers.

Fig. 4. Experimental results for the averaged intensity of the independently perturbed PCL array at different propagation distances: (a) z=0.2  m, (b) z=0.57  m, and (c) z=1.14  m. (d) The averaged intensity of the PCL array focused by the MO with a propagation distance z=1.14  m. The beam parameters are the same as those in Fig. 3(c). In the 2D graphs, the x and y axes are in the unit of pixels, and 1 pixel is 5.2 μm. In the 1D graphs, the vertical axes representing the averaged intensity are normalized. The diameters of the dark centers are marked in the 1D graphs. The camera is a CinCam CMOS-1201 (1280×1024 at 19.8 fps, 5.2 μm).

Fig. 5. Experimental results of the PCL array under different perturbation conditions. (a) and (b) are for c0=0, δp=0 and L0=1 with (a) z=0.57  m, (b) z=1.14  m; (c) and (d) are for the case of independent perturbation with c0=3w and L0=1 at (c) z=0.57  m, (d) z=1.14  m; (e) and (f) are for the case of uniform perturbation with c0=3w and L0=1 at (e) z=0.57  m, (f) z=1.14  m; and (g) and (h) are for the independent perturbation with c0=3w and L0=0 at (g) z=0.57  m, (h) z=1.14  m. Other parameters are w=0.157  mm and r0=1.5w. The horizontal and vertical axes representing the x and y axes are scaled in pixels with 1  pixel=5.2  μm. The camera is a CinCam CMOS-1201 (1280×1024 at 19.8 fps).

Fig. 6. Computational realizations of (a) the second-order correlation G(2)(x,y,−x,−y) of the random array; (b) the magnitude of the CCF, |W(x,y,−x,−y)|; and (c) the Fourier transform of the averaged intensity of the array. (d) Experimental realization of the second-order correlation obtained with a Mach–Zehnder interferometer with two Dove prisms. The parameters of the array are the same as those in Fig. 5(d). In (a) and (b), the shooting camera is a TUCSEN ISH500 (600×800 at 49.6 fps), and 1 pixel is 2.2 μm; while in (d), the shooting camera is a CinCam CMOS-1201 (1280×1024 at 19.8 fps), and 1 pixel is 5.2 μm. (c) is in spatial frequency domain where 1 pixel is 1/(5.2×1280)  μm−1.