Complementary Speckle Patterns: Deterministic Interchange of Intrinsic Vortices and Maxima through Scattering Media

Intensity maxima and zeros of speckle patterns obtained behind a diffuser are experimentally interchanged by applying a spiral phase delay of charge $\pm 1$ to the impinging coherent beam. This transform arises from the expectation that tightly focused beams, which have a planar wave front around the focus, are so changed into vortex beams and vice versa. The statistics of extrema locations and the intensity distribution of the so-generated “complementary” patterns are characterized by numerical simulations. It is demonstrated experimentally that the incoherent superposition of the three “complementary speckle patterns” yield a synthetic speckle grain size enlarged by a factor of $\sqrt{3}$. A cyclic permutation of optical vortices and intensity maxima is unexpectedly observed and discussed.

Figure 1

Experimental generation of complementary speckle patterns. (a) Spiral phase delays of charge $-1$ and $+1$ (${\mathrm{SP}}^{\mp 1}$) are introduced by a spatial light modulator on the beam impinging onto the diffuser. Speckle patterns corresponding to each configuration are recorded on a camera. The intensity (b) and phase (c) patterns of $A_0$ were measured experimentally by phase-stepping digital holography. $+1$ and $-1$ vortices of $A_0$ are identified with $+$ and $\circ$ symbols, respectively. Intensity patterns $I_1$ (cyan) and $I_{-1}$ (red) resulting from the addition of ${\mathrm{SP}}^{\pm 1}$ are also compared to vortices of $A_0$ (d). The maxima of $I_1$ and $I_{-1}$ exhibit a high correlation with vortices of $A_0$.

Figure 2

RPDFs of separation distances between one set of critical points and the closest points of another set, obtained from numerical simulations. The abscissa unit is $\lambda /(2\mathrm{NA})$. The distance between $V_1^{+}$ and the closest $M_0$ is notated $d(V_1^{+},M_0)$. Vertical thick dashed gray lines correspond to the correlation length $l_c\approx 0.78$.

Figure 3

Speckle patterns $I_0$, $I_1$, and $I_{-1}$ measured through a ground glass diffuser and overlapped, coded in saturated red green blue colors (a). The locations of the maxima are marked with squares, illustrating a dense mapping of the plane. Cross sections of the (normalized) autocorrelation functions of $I_0$, the sum $I_0+I_1$, and the sum $I_0+I_1+I_{-1}$ demonstrate the complementarity of speckle patterns (b).

Figure 4

From experimental data shown in Fig. 1, the sign of $I_{-1}-I_1$, represented in black (negative) and white (positive), identifies the sign of vortices $V_0^{\pm }$ (a). PDFs of intensities $I_1$ at optical vortices of $A_0$ (b) deduced from numerical simulations (solid lines) and from fits of first-order approximation functions. The PDFs are compared with the PDF of speckle patterns $I_1$.

Figure 5

Joint PDF of distances $d(M_0,V_1^{+})$ and $d(M_0,V_1^{-})$ in the logarithmic color scale (a) illustrating that some maxima are not changed into vortices. The fraction of maxima populations is indicated in each quadrant. The RPDF of the distance between vortices $d(V_0^{+},V_1^{-})$ showing a correlation of the locations of these two populations of vortices (b).