Vortex Speckles with Customized Symmetry and Spatial Correlations

A family of optical speckles with propagation-invariant annular morphology are generated via a phase-only mask and free-space evolution, termed vortex speckles, for which phase design is inspired by the vortex beam’s helical wave front carrying orbital angular momentum. Vortex beams’ graceful symmetry (central and/or axial) can be naturally inherited into vortex speckles by adjusting the parity of the topological value $\ell$ and the consistency of the initial phase angle ${\varphi }_0$. By sequentially imprinting such a series of phase profiles onto a spatial light modulator, the resultant stochastically fluctuating vortex speckles naturally constitute an annular pseudothermal light field, and the customized symmetry in intensity profiles leads to spatial correlations, which are exploited for Hanbury Brown and Twiss interferometry and beam-splitter-free lensless ghost imaging and diffraction experiments. Symmetric annular pseudothermal light also has diverse phase correlations, so in a proposed vortex self-interferometer, we can manipulate photon correlations (parity-induced anticorrelation) and observe intriguing correlation states (spatially varying photon correlation oscillations). This work not only presents a flexible light-source option for applications requiring ring-shaped speckle illumination, but it also paves the way for the customization of nontrivial classical correlation modes.

Figure 1

(a1),(a2) Typical helical wave fronts of $\ell =+4$ and $-5$ with initial phase ${\varphi }_0=\pi /4$; (b1),(b2) phase of two multiplexed OAM modes; (c) generation of a phase-only mask, composed of $n$ circular statically irrelevant two OAM-multiplexed phase profiles, e.g., the shaded regions in (b1),(b2).

Figure 2

Experimental results of vortex speckles after propagating 0.9 m in free space compared to a typical optical speckle pattern (a1). By adjusting the ring spacing $s$ and the fluctuation range of the random variable ${\ell }^i$, the propagation scale of the vortex speckle can be customized: when ${\ell }^i\in [\pm 20,\pm 60]$, $s=20$(a2), 10(a3), and 40(a4) pixels, respectively; when $s$ is fixed at 20 pixels, ${\ell }^i\in [\pm 10,\pm 25]$(a5) and ${\ell }^i\in [\pm 45,\pm 60]$(a6). The first row displays the single-shot speckles, and the second row (b1)–(b6) their ensemble averages over 10 000 times. (c1)–(c6) Symmetrical vortex speckles and their intensity correlation results (d1)–(d6); centrosymmetry: (c1) ${\ell }^i\in odd$, (c2) ${\ell }^i\in even$; axial symmetry (${\ell }_1^i+{\ell }_2^i=0$): (c3) ${\varphi }_0$=0, (c4) ${\varphi }_0=\pi /4$; supersymmetry: (c5),(c6). The curves inserted in the second and fourth rows are the 1D profiles and the HBT curves pointed by arrows, and the inserted values in the fourth row is the correlation coefficient $g^{(2)}$.

Figure 3

Experimental results of quantum mimic ghost imaging and diffraction using the symmetrical annular pseudothermal light. The targets are placed on the upper left side of the annular light field in (a) [(c)], and (b1)–(b3) [(d1)–(d3)] correspond to the optical correlations in Figs. 2(d1), 2(d3), 2(d5), respectively.

Figure 4

Experimental results of the ring-shaped pseudothermal light with tunable photon correlations in a vortex self-interferometer. (a) Phase profile design. (b) Symmetrical angular coordinates. (c1),(c2) Vortex beams with ${\ell }_0=31,32$ and corresponding vortex speckles (c3),(c4) in this interferometer. Centrosymmetric: (d1)–(d4) parity-induced anticorrelations, (d5) plots the correlation coefficients $g^{(2)}$ in (d3) as a function of radial position; axisymmetric: (e1)–(e4) photon correlation oscillations; supersymmetric: (f1)–(f4) diverse nontrivial photon correlations; (f5),(f6) plot $g^{(2)}$ in (f3),(f4) as a function of angular position. A total of eight correlation states are displayed, and a simplified four-quadrant hexagrams are inserted with red for photon bunching and blue for anticorrelation; the values of $g^{(2)}$ and the angular position are also inserted, and the angular intervals, $\mathrm{\Delta }{\phi }_1$ and $\mathrm{\Delta }{\phi }_2$, are $2\pi /620$ and $2\pi /640$, respectively.

Figure 5

Extensions to speckle customization with other convex morphological structures. By manipulating the spatial curvature of the phase mask (a1),(b1),(c1), hollow speckles of elliptical rings (a2), square rings (b2), and pentagonal rings (c2) can be generated likewise.

Figure 6

Experimental details of vortex speckles’ generation. (a) Experimental setup. (b) Generation of the phase mask with a grating structure. (c) By adjusting the angle of rotation of the NPBS to allow only the +1-order diffraction pattern to pass through the diaphragm. All phase profiles in this panel are scaled down to $201\times 201$ pixels for display convenience.

Figure 7

Experimental details of (a1),(a2) ghost imaging and (b1),(b2) ghost diffraction experiments using symmetrical annular pseudothermal light. (c1),(c2) Preparation of three-point object used in the ghost diffraction experiments: amplitude modulation is achieved by the transmissive SLM and a polarizer. Unfolded equivalent geometric optics [58, 59] of (d1) classical ghost imaging with centrosymmetric annular pseudothermal light and (d2) quantum ghost imaging with entangled photon pairs in the I-type SPDC. A lens must be required in (d2) to construct the conjugate relationship between the object plane and the detection plane, while in (d1), no lens is required, and only the equidistant relationship needs to be satisfied.

Figure 8

Statistical properties of vortex speckles. In the annular region of interest in the vortex speckle (a),(b1),(c1) are the PDFs of (a) in intensity and phase, respectively; (b2),(c2) are the ensemble-averaged intensity and phase PDFs over 2500 vortex speckle patterns; (d) is the ensemble-averaged complex-amplitude joint PDF for the speckle field that displays circular Gaussian statistics. Simulation parameters: ${\ell }^i\in [\pm 10,\pm 70]$ and 1-m free-space propagation.

Figure 9

Photon correlations and their manipulation of centrosymmetric annular pseudothermal light in the vortex self-interferometer (case 1). When $k>0.5$, the photon correlations can be manipulated to jump between bunching and antibunching by adjusting the parity of ${\ell }_0$. Simulation parameters: ${\ell }^i\in [\pm 10,\pm 70]\mathrm{\&}even$, $s=20$ pixels, and 1-m free-space propagation; ${\ell }_0=40$ for the top curve (same parity) and ${\ell }_0=41$ for the bottom curve (opposite parity).

Figure 10

Photon correlation oscillations of axisymmetric annular pseudothermal light in the vortex self-interferometer (case 2, $x$ axis). (a) The 2D theoretical distribution of oscillating photon correlations as angular position ${\phi }_1$ and average intensity ratio $k$ change; (b) 1D theoretical profile and numerical results of oscillating photon correlations ranging from 2/3 to 14/9 when $k=2$ corresponding to the green dashed line in (a). Simulation parameters: ${\ell }^i\in [\pm 10,\pm 70]\mathrm{\&}{\ell }_1^i+{\ell }_2^i=0$, 1-m free-space propagation, $s=20$ pixels, and ${\ell }_0=40$.