## Abstract

Deterministic control of coherent random light is highly important for information transmission through complex media. However, only a few simple speckle transformations can be achieved through diffusers without prior characterization. As recently shown, spiral wavefront modulation of the impinging beam allows permuting intensity maxima and intrinsic $\pm 1$-charged optical vortices. Here, we study this cyclic-group algebra when combining spiral phase transforms of charge $n$, with ${D}_{3}$- and ${D}_{4}$-point-group symmetry starlike amplitude modulations. This combination allows for statistical strengthening of permutations and controlling of the period to be 3 and 4, respectively. Phase saddle points are shown to complete the cycle. These results offer new tools to manipulate critical points in speckles.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. INTRODUCTION

The propagation of coherent light through scattering media yields random wave fields with typical intensity structures called optical speckles. The control of light distribution inside and through complex media via wavefront modulation of the impinging beam is of critical importance for applications ranging from bio-imaging [1] to telecommunications [2], for instance, by multiplexing information with orbital angular momentum [3]. Information transmission through diffusers is typically characterized in terms of field and intensity correlations [4]. For diffusers exhibiting so-called memory effect correlations, important invariants were identified under specific spatial (tilt and shift) transformations [5–8]. Additionally, regardless of the wavefront of the impinging beam, critical points in random wave fields exhibit many topological correlations [9], which thus demand the development of specific tools to be analyzed. Optical vortices are especially important critical points, since they are centered on singular phase points coinciding with nodal points of the intensity. They spontaneously appear in random wave fields [10], and thereby allow efficient super-resolution microscopy [11,12]. The present work explores the possibility of manipulating topological correlations between critical points in random wave fields under symmetry control and spiral wavefront modulation, in a Fourier plane of the impinging beam.

Critical points are characterized by their topological charge and their Poincaré number [13]. They may typically be controlled by applying phase or amplitude masks in a Fourier plane. Any smooth and regular transform of the wave field (either in phase or amplitude) induces changes preserving both the topological charge and the Poincaré number [9,14]. Of note, these conservation rules account for the topological stability of isolated vortices of charge 1 in speckles, since the creation or annihilation of vortices can only occur in pairs [11,14]. As opposed to smooth phase transforms, the addition of a spiral phase mask in a Fourier plane is a *singular transform* and results in a change in the total orbital angular momentum [15,16]. Recently, while considering correlations in the spatial distribution of critical points in a speckle under such spiral phase transforms [17], we observed a strong interplay between intensity maxima and optical vortices. More precisely, the results obtained suggested that the topological charges of these critical points were all incremented by applying a $+1$ spiral phase mask in the Fourier plane. The impossibility of spontaneously getting $+2$-charged vortices (which are unstable and thus unlikely in random light structures [18]) resulted in the observation of a partial cyclic permutation of the three populations of critical points (namely, maxima and $\pm 1$-charged vortices). Furthermore, as a third kind of possible transform, it was observed that the orbital angular momentum may be not conserved when using amplitude masks with a high degree of symmetry [19,20]. As a result, optical vortices can be created using simple amplitude masks [21,22]. This property proved to be of interest for using imaging applications to reveal symmetry in an imaged object [23–25] and for allowing topological charge measurements [26], especially in astronomy [27].

Here, by combining spiral phase transforms of order $n$ with starlike amplitude masks having discrete point group symmetries ${D}_{3}$ and ${D}_{4}$, we study experimentally the topological correlations between intensity maxima and optical vortices in speckles. A new co-localization criterion is proposed, inspired by statistical mechanics. Although random wave fields do not possess any symmetry, such a combination allows us to strengthen periodicity and even to control the period of the cyclic permutation. It is worth noting that for an amplitude mask of symmetry ${D}_{4}$, a phase saddle point appears as a complementary critical point to complete a cycle of period 4. A transposition between vortices of charge $-1$ and vortices of charge $+1$ is also revealed when adding a 2-charged spiral phase mask.

## 2. EXPERIMENTAL PROCEDURE

The experimental procedure consisted in modulating a random phase pattern in a Fourier plane with an amplitude mask and a spiral phase mask. Here, spiral phase masks of order $n$, ${\mathrm{SP}}_{n}(\theta )={e}^{i.n.\theta}$ (in polar coordinates), were applied for $n\in \u27e6-6;6\u27e7$. As amplitude masks, three binary amplitude (BA) masks were used: a disk and two periodic angular slits with point group symmetries ${D}_{3}$ and ${D}_{4}$. They are defined by the following angular transmission function (in polar coordinates):

The experimental configuration is detailed in Fig. 1 (See Section 1 of Supplement 1 for further details on the experimental methods). A spatial light modulator (SLM) (LCOS, X10468, Hamamatsu) was illuminated with a collimated laser beam at 635 nm and Fourier conjugated to a camera ($768\times 1024$ pixels, pixel size $4.65\times 4.65\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\mathrm{\mu m}}^{2}$) with a converging lens. The phase ${\mathrm{\Phi}}_{n}$ and amplitude ${A}_{n}$ of the modulated (${\mathrm{SP}}_{n}$ mask) random wave were measured at the camera plane with phase stepping interferometry [29]. To do so, the SLM ($792\times 600$ SLM pixels, pixel size $20\times 20\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\mathrm{\mu m}}^{2}$) was split into two parts to generate both the modulated random wave (or signal wave) on one side and a reference wave on the other side [see Fig. 1(a)]. The signal wave was generated by simultaneously adding the scattering random phase pattern, the spiral phase modulation ${\mathrm{SP}}_{n}$, and the amplitude mask ${\mathrm{BA}}^{N}$. Adding a blazed grating achieved spatial separation of the imprinted signal wavefront from undiffracted light (the latter being sent to a beam block). The signal speckle intensity ${I}_{n}$ could be measured directly by removing the contribution of the reference beam.

For phase-stepping interferometry, an additional Fresnel lens was added to the reference beam in order to cover the camera surface. The latter spherical contribution, as well as the relative phase tilt between the signal and the reference beams, was removed in a numerical postprocessing step. A stack of eight images was sequentially recorded by phase shifting the reference beam by $2\pi /8$ phase steps. All BA masks had the same radius of $r=170$ pixels at the SLM, thus yielding the same speckle grains size on the camera plane: $\lambda /(2\mathrm{NA})=70\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$ full width at half-maximum (FWHM), where $\lambda $ is the wavelength and $\mathrm{NA}\simeq r/f\simeq 4.53\times {10}^{-3}$ the numerical aperture of illumination [with $f=750\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{mm}$ being the focal length of the lens L in Fig. 1(a)]. The speckle grain size thus covered 15 camera pixels and ensured a fine sampling of the speckle patterns. Hereafter, all distances and spatial densities are expressed with $\lambda /(2\mathrm{NA})$ as the length unit.

In Figs. 1(b)–1(d), an illustration of the speckle intensity and phase maps obtained for the three different geometries of BA masks is shown. In the following study, intensity maps ${I}_{n}$ and phase maps ${\mathrm{\Phi}}_{n}$ were measured for all three ${\mathrm{BA}}^{N}$ masks ($N\in \{3,4,\infty \}$) and for each ${\mathrm{SP}}_{n}$ mask ($n\in \u27e6-6;6\u27e7$). For comparison, the intensity map ${I}_{\mathrm{rand}}$ and the phase map ${\mathrm{\Phi}}_{\mathrm{rand}}$ obtained for a noncorrelated scattering random pattern were acquired for all the ${\mathrm{BA}}^{N}$ masks independently.

## 3. STATISTICAL ANALYSIS OF THE TOPOLOGICAL CORRELATIONS BETWEEN CRITICAL POINTS

#### A. Critical Points Studied

With the field at the camera being linearly polarized, optical fields are studied here as scalar fields. The location of the main critical points of the experimental intensity and phase maps were measured (See Section 1.C of Supplement 1 for details on the detection of critical points), and their statistical correlation distances were analyzed. Importantly, for a given BA mask, adding ${\mathrm{SP}}_{n}$ masks preserves all statistical properties of the speckle patterns, such as the number density of critical points. Phase saddle points of ${\mathrm{\Phi}}_{n}$ are denoted as ${S}_{n}^{p}$ and vortices of charge as $\pm 1\text{:}{V}_{n}^{\pm}$. Vortices of charge higher than 1 do not appear in Gaussian random wave fields [18]. Maxima and saddle points of ${I}_{n}$ are denoted as ${M}_{n}$ and ${S}_{n}^{I}$, respectively. Nonzero minima and phase extrema have not been considered here, since they have significantly lower densities [30]. All the notations are summarized in Table 1.

The measured average number densities of the critical points are presented in Table 2. The density of the critical points of type $X$ ($X={V}^{\pm}$, $M$, ${S}^{p}$ or ${S}^{I}$) is notated $\rho (X)$. As expected, $\rho ({V}^{-})$ and $\rho ({V}^{+})$ are equal [9], and $\rho (X)$ depends on both the type of critical point and the BA mask.

#### B. Statistical Tools for the Analysis of Topological Correlations

To study statistical transformations of critical points quantitatively, specific tools are presented. What we discuss as the transformation of critical points by the addition of spiral phase masks refers to the mean nearest neighbor distances between populations of critical points and calls for a discrimination parameter. Two specific statistical tools are then described below: the radial distribution function (RDF) and the weighted median normalized distance (WMND).

In our previous study [17], correlations between critical points could be characterized by computing the radial probability density function (RPDF) of the nearest-neighbor distance. Figure 2(a) presents RPDFs of the distance $d({V}_{0}^{-},X)$ in the case of ${\mathrm{BA}}^{\infty}$. We define $d(Y,X)$ as the distance between a $Y$ point and the closest $X$ point. The $\mathrm{RPDF}(r)$ corresponds to the probability of finding the closest $X$ point at the distance $r$ from a $Y$ point, per unit area (${\int}_{0}^{\infty}\mathrm{RPDF}(r)2\pi r\mathrm{d}r=1$). One drawback associated with the use of the RPDF is that it may suggest paradoxes if improperly interpreted. Considering $d({V}_{0}^{-},{V}_{2}^{+})$ and $d({V}_{0}^{-},{S}_{\mathrm{rand}}^{I})$, it seems that ${V}_{0}^{-}$ correlates with both ${V}_{2}^{+}$ and ${S}_{\mathrm{rand}}^{I}$, since both RPDFs reach high values at zero distances. While a correlation is expected in the former case (due to topological charge incrementation), no correlation is expected from the latter, which involves two independent sets of random points. The reason that the amplitude of the RPDF of $d({V}_{0}^{-},{S}_{\mathrm{rand}}^{I})$ is higher than that of $d({V}_{0}^{-},{V}_{2}^{+})$ at zero distances is just due to the $\sim 3$-times higher spatial density of the intensity saddle points ${S}_{\mathrm{rand}}^{I}$ as compared to the vortices ${V}_{2}^{+}$ (see Table 2): the probability of finding a saddle point at a close distance is thus larger. To quantitatively characterize topological correlations, we thus need to normalize the RPDFs by the number densities $\rho (X)$.

Our first statistical tool, the RDF—well known in statistical mechanics [31]—was extended here for nearest neighbor by normalizing the RPDF of $d({V}_{0}^{-},X)$ by $\rho (X)$, and the distances $d({V}_{0}^{-},X)$ by the mean $X$-interpoint half-distance ${(2\sqrt{\rho (X)})}^{-1}$ [32]. Figure 2(c) shows the RDF of the same data as in Fig. 2(a). As a result, all the RDFs of $d({V}_{0}^{-},{X}_{\mathrm{rand}})\xb72\xb7\sqrt{\rho (X)}$are superimposed for every ${X}_{\mathrm{rand}}$, and the spatial correlation between ${V}_{0}^{-}$ and ${V}_{2}^{+}$ clearly appears.

To obtain a single binary parameter discriminating the spatial correlation between ${V}_{0}^{-}$ and $X$, we further define the WMND as a second statistical tool: the WMND $({V}_{0}^{-},X)$ is the 50% weighted percentile of $d({V}_{0}^{-},X)\xb72\xb7\sqrt{\rho (X)}$, with weights corresponding to the RDF values. A $\mathrm{WMND}({V}_{0}^{-},X)$ value around 0.5 means that no spatial correlation exists between ${V}_{0}^{-}$ and $X$, while $\mathrm{WMND}<0.5$ and $\mathrm{WMND}>0.5$ represent an attraction and a repulsion, respectively. A zero WMND value means perfect correlation, while $\mathrm{WMND}=1$ means perfect anticorrelation.

Figure 2(d) presents the WMND $({V}_{0}^{-},X)$ values for all the critical points considered in this study and for $n\in \u27e60;6\u27e7$. For comparison, the weighted median distance (WMD) associated with the RPDFs—defined as the 50% weighted percentile of $d({V}_{0}^{-},X)$, with weights corresponding to the RDPF values—is also computed and displayed in Fig. 2(b). To validate this tool, taking ${\mathrm{BA}}^{\infty}$ as an illustrative example, we notice that WMND $({V}_{0}^{-},{X}_{\mathrm{rand}})$ is around 0.5 for all values of ${X}_{\mathrm{rand}}$, as expected. By comparison, WMD $({V}_{0}^{-},{S}_{\mathrm{rand}}^{I})=0.35$, which irrelevantly suggests correlations as discussed above. Moreover, for $n>3$, the RDFs of $d({V}_{0}^{-},{X}_{n})$ are observed to match the RDFs of $d({V}_{0}^{-},{X}_{\mathrm{rand}})$; no noticeable spatial correlation is obtained for $n>3$. Again as expected, WMND $({V}_{0}^{-},{X}_{n})$ values are around 0.5 for $n>3$. Conversely, we get WMD $({V}_{0}^{-},{S}_{n}^{I})<0.38$, which would falsely suggest correlations. All these observations validate the WMND as a parameter for assessing the spatial correlation between pairs of critical points in a speckle pattern.

## 4. TOPOLOGICAL CORRELATIONS BETWEEN CRITICAL POINTS FOR THE DIFFERENT AMPLITUDE MASKS

Figure 3 presents the WMND(${\mathrm{Y}}_{0}$, ${X}_{n}$) values for all the critical points (${\mathrm{Y}}_{0}$ and ${X}_{n}$) screened in this study, for ${\mathrm{SP}}_{n}$ masks with $n\in \u27e6-6;6\u27e7$ and for the three considered BA apertures. The WMND was verified to be around 0.5 for all the amplitude masks and all the pairs (${\mathrm{Y}}_{0}$, ${X}_{\mathrm{rand}}$), for which there is obviously no spatial correlation. For the sake of readability, in the following, we only discuss the interplay between critical points when adding positively charged SP masks, but symmetrical behaviors are observed for negatively charged SP masks (Fig. 3).

For the aperture ${\mathrm{BA}}^{\infty}$, the WMND reveals several noticeable features (reported in Table 3). First, as expected from our previous study [17], we notice some spatial correlations for the triplets $({V}_{m-1}^{-},{M}_{m},{V}_{m+1}^{+})$. Because $\rho ({V}_{0}^{-})<\rho ({M}_{1})$, a one-to-one transformation is impossible between vortices and maxima. Although ${V}_{0}^{-}$ and ${V}_{2}^{+}$ have the same number density, we also notice that WMND $({V}_{0}^{-},{V}_{2}^{+})$ is significantly different from 0, indicating that the rate of the macroscopic transformation from ${V}_{0}^{-}$ to ${V}_{2}^{+}$ is below 1.

In agreement with the three-point cyclic permutation algebra observed in [17], a weak attraction is found for the pair $({V}_{0}^{+},{V}_{+1}^{-})$, corresponding to the topological charge equation $1+1=-1$. However, as an alternative transformation for ${V}_{0}^{+}$, a similar attraction is now also observed for $({V}_{0}^{+},{S}_{+1}^{p})$. ${V}_{0}^{+}$ is thus subject to a bifurcation between ${V}_{+1}^{-}$ and ${S}_{+1}^{p}$, which implies two different mechanisms.

As a first possibility, some ${V}_{0}^{+}$ transform into ${S}_{+1}^{p}$. This transformation inspires the following interpretation: when adding an ${\mathrm{SP}}_{+1}$ phase mask to an isolated Laguerre–Gaussian beam with the topological charge $+1$, a $+2$-charged vortex is obtained. Under weak perturbation, this $+2$ vortex splits into two $+1$ vortices, accompanied by the creation of both an intensity saddle point and a phase saddle point in between [14]. The creation of this pair of saddle points is governed by the Poincaré number conservation. In the frame of this model, a ${V}_{0}^{+}$ vortex is expected to colocalize with both ${S}_{+1}^{I}$ and ${S}_{+1}^{p}$ and to anticorrelate with the two ${V}_{1}^{+}$ that split away. Although no noticeable spatial attraction was found for $({V}_{0}^{+},{S}_{+1}^{I})$, a colocalization is observed for $({V}_{0}^{+},{S}_{+1}^{p})$ and a weak repulsion is observed for the pair $({V}_{0}^{+},{V}_{+1}^{+})$, consistent with this interpretation. In speckles, where $+2$-charged vortices cannot be encountered since they are unstable [18], the weak perturbation approximation cannot be fully valid, potentially accounting for the remaining discrepancy between experimental observations and the proposed model.

As a second possible transformation, the more surprising attraction of the pairs $({V}_{0}^{+},{V}_{+1}^{-})$ is observed, which calls for another mechanism. Since no such transformation can be imagined from isotropic ${V}_{0}^{+}$, it may only be interpreted by a mechanism dominated by strong perturbations. The statistically uniform mesh created by vortices and maxima in speckles [33], together with strong correlations observed for pairs $({M}_{0},{V}_{+1}^{+})$ and $({V}_{0}^{-},{M}_{1})$, seems to constrain ${V}_{0}^{+}$ to colocalize with ${V}_{+1}^{-}$. This transformation would deserve further analytical investigation, but we anticipate that the creation mechanism of ${V}_{+1}^{-}$ from ${V}_{0}^{+}$ can only be a many-body problem involving the field structure (maxima, phase saddles, and vortices) surrounding the initial ${V}_{0}^{+}$ of interest.

When adding an ${\mathrm{SP}}_{+2}$ mask for ${\mathrm{BA}}^{\infty}$, ${V}_{0}^{+}$ is not observed to significantly colocalize with any remarkable critical point (see Fig. 3 and Table 3), whereas two possible transformations might have been expected for ${V}_{0}^{+}$. On the one hand, from the three-point cyclic permutation, we could expect that ${V}_{0}^{+}$ would transform into ${M}_{+2}$. On the other hand, since in Table 3, maxima and phase saddle points are noted to be simply exchanged [see pairs $({M}_{0},{S}_{+2}^{p})$ and $({S}_{0}^{p},{M}_{+2})$], a similar symmetrical exchange between $-1$ and $+1$ vortices could be expected, yielding a transformation of ${V}_{0}^{+}$ into ${V}_{2}^{-}$ (as ${V}_{0}^{-}$ is transformed into ${V}_{2}^{+}$). However, no such correlation is observed either for the pair (${V}_{0}^{+}$, ${M}_{2}$) or for (${V}_{0}^{+}$, ${V}_{2}^{-}$). Conversely, these correlations appear when applying amplitude masks ${\mathrm{BA}}^{3}$ and ${\mathrm{BA}}^{4}$, respectively, as detailed in the following.

For $|n|>3$, no significant spatial correlation with the addition of ${\mathrm{SP}}_{n}$ is observed for ${\mathrm{BA}}^{\infty}$. This aperture has a circular symmetry. Therefore, its spiral spectrum contains only the fundamental spiral mode $n=0$, and is not invariant with the addition of any SP masks. All the described topological correlations associated with ${\mathrm{BA}}^{\infty}$ are summarized in Table 3.

As a possible solution to strengthen the three-point cyclic permutation, we used the ${\mathrm{BA}}^{3}$ amplitude mask, making the Fourier plane almost invariant with respect to the addition of ${\mathrm{SP}}_{\pm 3k}$ (so long as $3k$, with $k$ being an integer, remains small enough). Here, four main observations can be noted. First, as expected, the pairs $({\mathrm{Y}}_{0},{\mathrm{Y}}_{\pm 3})$ and $({\mathrm{Y}}_{0},{\mathrm{Y}}_{\pm 6})$ are observed to have a WMND very close to zero, indicating that the macroscopic transformation rate is close to 1 for all these pairs. Second, we observe that the cycle of period 3 reinforces the spatial correlations of the triplet $({V}_{m-1}^{-},{M}_{m},{V}_{m+1}^{+})$ and even extends it to the third and sixth spiral harmonics. Third, no noticeable correlation is observed between ${V}_{0}^{+}$ and phase saddle points ${S}_{+1}^{p}$ (although an anticorrelation is obtained between ${V}_{0}^{+}$ and ${V}_{+1}^{+}$), contrary to the case of ${\mathrm{BA}}^{\infty}$. The periodicity of 3 induces a strong correlation for the pairs $({V}_{0}^{+},{V}_{+1}^{-})$ and establishes a cyclic permutation of three populations of critical points ${V}^{-}$, $M$, and ${V}^{+}$. Fourth, as a consequence, the pairs $({V}_{0}^{+},{M}_{2})$ also exhibit strong spatial correlations, contrary to the case of ${\mathrm{BA}}^{\infty}$. In these two latter permutations, it must be restated that not all intensity maxima $M$ may transform into vortices, because of the difference in spatial densities (Table 2) of these two populations of critical points [17].

Next, we constrained the period to be equal to 4 by using the ${\mathrm{BA}}^{4}$ mask. In this case, the WMND $({\mathrm{Y}}_{0},{\mathrm{Y}}_{\pm 4})$ values are close to zero (transformation rate close to 1). As expected, this periodicity enhances the spatial correlation for the quadruplet $({V}_{m-1}^{-},{M}_{m},{V}_{m+1}^{+},{S}_{m+2}^{p})$ and extends it to their fourth spiral harmonics. Furthermore, in Fig. 3, strong correlations are observed for the pairs $({V}_{0}^{-},{M}_{1})$, $({M}_{0},{V}_{+1}^{+})$, and $({S}_{0}^{p},{V}_{+1}^{-})$. However, ${V}_{0}^{+}$ is still observed to bifurcate between ${V}_{+1}^{-}$ and ${S}_{+1}^{p}$ with the same likelihood, similar to the ${\mathrm{BA}}^{\infty}$ case. Therefore, the cyclic permutation of the four populations of critical points ${S}^{p}$, ${V}^{-}$, $M$, and ${V}^{+}$ is not clearly established for ${\mathrm{BA}}^{4}$ (unlike the permutation obtained for ${\mathrm{BA}}^{3}$).

When considering the addition of $\pm 2$-charged spiral masks for ${\mathrm{BA}}^{4}$, a transposition (two-cycle permutation) between ${V}^{+}$ and ${V}^{-}$ is clearly obtained (with a transposition rate below 1). A strong spatial correlation, reinforced as compared to ${\mathrm{BA}}^{\infty}$, is also clearly observed between $M$ and ${S}^{P}$ with the addition of ${\mathrm{SP}}_{\pm 2k}$.

In summary, the results displayed in Fig. 3 reveal the fundamental topological transformations of critical points in a speckle with the addition of an SP mask for a single spiral mode aperture (${\mathrm{BA}}^{\infty}$), and demonstrate the possible modification of topological transformation via the addition of BA masks with dihedral symmetry. (See Section 2.B of Supplement 1 for numerical confirmation of the experimental results, and Section 2.C of Supplement 1 for details on transformations induced by BA masks with dihedral symmetries of orders higher than 4.) For the sake of simplicity, we chose here starlike amplitude masks with a dihedral symmetry, which comprise spiral harmonics with equal amplitudes (at a small enough spiral mode number). (See Section 2.D of Supplement 1 for details on the influence of the width of the star branches.)

## 5. WAVEFIELD CONTROL IN THE VICINITY OF CRITIAL POINTS

On a local scale, the transformations of critical points shown in Fig. 3 arise from the convolution in the imaging plane of the scattered field with the point spread function (PSF) associated with the combined amplitude and spiral phase masks. Controlling the transformation of a critical point ${\mathrm{Y}}_{0}$ of the complex wave field ${A}_{0}$, by adding an ${\mathrm{SP}}_{n}$ mask $(n\in {\mathbb{Z}}^{*})$ in a Fourier plane, requires that the PSF associated with the combination of the ${\mathrm{SP}}_{n}$ and BA masks has a significant amplitude in the coherence area surrounding the critical point, that is, the area where the randomness of the speckle pattern has a limited influence as compared to the control by the incident wave field. As a definition for the coherence area, we use the one proposed by Freund [34]: ${C}_{\mathrm{area}}={(\rho ({V}^{+})+\rho ({V}^{-}))}^{-1}/2$. This definition avoids issues related to the shape of the aperture [34] encountered when considering the area of the intensity autocorrelation peak [35]. In our case, for all three BA masks, the coherence length was measured to be ${C}_{\text{length}}=\sqrt{{C}_{\mathrm{area}}}\simeq \lambda /(2\xb7\mathrm{NA})$.

Experimentally, the PSFs can be obtained [Fig. 4(a)] by computing the intensity cross correlations of the measured speckle pattern ${I}_{0}$ and the measured speckle patterns ${I}_{n}$ associated with ${\mathrm{SP}}_{n}$ masks ($n\in \u27e60;6\u27e7$) and for the three BA masks. The mean values of ${I}_{n}$ were subtracted before computing the cross correlations. The intensity cross correlations $\mathrm{xcorr}({I}_{0},{I}_{n})$ are identical to the PSF of the combined BA and ${\mathrm{SP}}_{n}$ masks. The centered spot of the autocorrelation $\mathrm{xcorr}({I}_{0},{I}_{0})$ illustrates the spatial extent of the coherence area, and has the same dimension for all three BA masks since they have the same radial aperture.

For ${\mathrm{BA}}^{\infty}$, we observe that $\mathrm{xcorr}({I}_{0},{I}_{n})$ has a circular symmetry with the highest values distributed on a ring whose radius (marked with a green line) increases with $n$ [Fig. 4(b)], as observed for simple Laguerre–Gaussian beams [36]. Interestingly, for $n>3$, not only do we observe that the ring radius is larger than twice ${C}_{\text{length}}$, but also that its amplitude is reduced to below 1/10 of the autocorrelation peak value [Fig. 4(c)]. As a consequence, achieving the transformation of critical points by applying ${\mathrm{SP}}_{n}$ masks is inefficient (or “unlikely”) and dominated by the surrounding random field. For this reason, no spatial correlation between pairs of critical points $({\mathrm{Y}}_{0},{X}_{n})$ could be found for $n>3$. For ${\mathrm{BA}}^{3}$ and ${\mathrm{BA}}^{4}$, the cross-correlation patterns $\mathrm{xcorr}({I}_{0},{I}_{n})$ have dihedral symmetries ${D}_{3}$ and ${D}_{4}$ and periodicities of $N=3$ and 4, respectively. In both cases, the radial distance of the strongest peak remains below $1.4\times {C}_{\text{length}}$, and its amplitude always remains above 1/3 of the auto-correlation maximum value [Fig. 4(c)]. The addition of the ${\mathrm{SP}}_{n}$ mask thus allows us to control the field inside the coherence area surrounding critical points ${\mathrm{Y}}_{0}$, even for $n>3$.

## 6. CONCLUSION

The critical points that naturally appear in a random wave field can be transformed by the addition of a spiral phase mask in a Fourier plane. Here we studied these transformations experimentally by imprinting spiral phase masks with a charge $n\in \u27e6-6;6\u27e7$ to a laser beam impinging on a randomly scattering surface. In addition, these phase masks were combined with starlike amplitude masks with dihedral symmetries ${D}_{3}$ and ${D}_{4}$ in order to better control critical point transformations.

For a simple disk-shaped aperture carrying a single spiral mode $n=0$, we experimentally demonstrated the topological correlation existing between the critical points of the initial wave field ${A}_{0}$ and the corresponding spiral transformed field ${A}_{n}$. A partial transformation of vortices ${V}_{0}^{-}$ into maxima ${M}_{+1}$ was observed, as well as a transformation of maxima ${M}_{0}$ into vortices ${V}_{+1}^{+}$. Vortices ${V}_{0}^{+}$ were observed to either correlate with phase saddle points ${S}_{+1}^{p}$ or with vortices of the opposite sign ${V}_{+1}^{-}$. For this statistical bifurcation, two transformation interpretations were suggested, calling for further analytical studies. No simple topological correlation was found between the critical points of the wave fields ${A}_{0}$ and ${A}_{n}$ for $|n|>3$. This result could be explained by the weak influence of spiral phase masks with a charge higher than 2 in the coherence area surrounding the critical points.

Furthermore, by adding centered BA masks with dihedral symmetry ${D}_{3}$ or ${D}_{4}$ and *Dirac-comb*-like spiral spectra (of period 3 and 4), we demonstrated that it is possible to deeply modify the topological correlation between critical points. The observed changes arise from the introduction of a periodicity in the transformation between the critical points. We could thereby extend the correlation to spiral phase masks with charges higher than 3, and reinforce some spatial correlation intrinsically present with a circular aperture symmetry. For the amplitude mask with a ${D}_{3}$ symmetry, a cyclic permutation between negatively charged vortices ${V}^{-}$, maxima $M$, and positively charged vortices ${V}^{+}$ is observed. For the amplitude mask with a ${D}_{4}$ symmetry, phase saddle points participate as complementary points to complete the 4-period cycle. Considering the addition of 2-charged spiral masks, transpositions between ${V}^{-}$ and ${V}^{+}$ and between $M$ and ${S}^{p}$ were also revealed for ${D}_{4}$ symmetry. The enhancement of the spatial correlation between the critical points of the wave fields ${A}_{0}$ and ${A}_{n}$ (compared to ${\mathrm{BA}}^{\infty}$) could be explained by the strong influence of the spiral phase mask in the coherence area surrounding each critical point when the BA masks are added.

Here, cyclic permutations were controlled using BA masks enforcing periodicity. Interestingly, our study may extend to other amplitude masks with a dihedral symmetry, such as polygonal [20] and triangular apertures [26], whose interactions with vortex beams were studied in free space. For $N$-gons, the $N$th spiral harmonics have a much lower amplitude than the fundamental spiral mode ($n=0$). As a result, the spatial correlation between critical points for $|n|>3$ is weaker than for ${\mathrm{BA}}^{N}$, and vanishes with the increasing charge of the SP mask (See Section 2.E of Supplement 1).

In a nutshell, we showed here that it is possible to manipulate the topological correlation between critical points and to control the transformation of critical points in random wave fields by combining amplitude masks and spiral phase transforms. Topological manipulation of critical points in random wave fields is of high importance in understanding and controlling light propagation through scattering and complex media. The statistical study of correlations between permuted critical points provides a new tool for analyzing seemingly informationless and random intensity patterns and thus to transmit information through complex media.

## Funding

Agence Nationale de la Recherche (ANR) (ANR-18-CE42-0008-01, ANR-CE09-0015-01).

See Supplement 1 for supporting content.

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