Generation of coherence vortex by modulating the correlation structure of random lights

Min-Jie Liu; Jun Chen; Yang Zhang; Yan Shi; Chun-Liu Zhao; Shang-Zhong Jin

doi:10.1364/PRJ.7.001485

1 INTRODUCTION

A coherence vortex (CV) [1–3] is a common phenomenon in various laser speckles [4–8]. The study of CVs began in 1998, when Gori et al. proposed that a partially coherent light source would possess helicoidal modes in its spectral degree of coherence [9]. For speckle fields that are partially coherent lights, the random fluctuations of the light fields tend to hide the singular points of the averaged intensity [10]. However, in the picture of two-point correlation, the singularities survive even when the fields are of low coherence [11]. A CV field, which is a special optical vortex, possesses a spiral phase structure in a two-point correlation function formed as cross-spectral density (CSD) [12]. The accumulated phase change around the singularity of the helical correlation structure is $2 m π$ , where $m$ is the topological charge of the CV field and is linked to the orbital angular momentum (OAM) of the photon according to $m ℏ$ (where $ℏ$ is the Planck constant). The two-point correlation is responsible for the coherence of light, hence the term CV. A CV is a point singularity in the coherence of the optical field where the intensity is unnecessarily zero. The benefit from the low coherence is that a CV is less influenced than an optical vortex when it propagates in turbulence [13] and through obstacles [14]; thus, it has been applied in areas such as object identification and imaging [7,8,15], reduction in turbulence-induced degradation during propagation [16], optical manipulation [17,18], and self-reconstruction in obstacle imaging [14,19]. In quantum optics, the correlated photons can be used to transmit and process information with respect to the physical dimension of correlation [20]. Similar to its quantum counterpart, a partially coherent laser beam also possesses a spatial correlation of freedom, which is called the spectral degree of coherence and is embedded in CSD functions [21]. This property enables the use of the correlation of freedom of the light field to perform imaging tasks [22,23]. Information on correlation structures can also be transferred by CV beams when encoded with topological charges [24,25]. In communication, the generation and multiplexing strategy of multiple information channels, i.e., propagating several transmission modes within a common medium and without interacting with one another, is important to increase the transmission capacity of light [26,27]. Accordingly, correlation structures provide a routine to increase the transmission capacity with consideration of coherence as an inherent property of light beams.

Recently, the combination of mixed CV fields in the low coherence limit has been distinguished by designing a ghost imaging system [28]. Given that CV is a special mode in the coherence dimension, CV communication is possible as long as the generation of various coherence structures has been conducted. As a result, methods of CV generation and correlation modulation warrant investigation.

Previous works have revealed the generation of various structured lights that utilize the combination of special optical beams [29–34], and sometimes ultrafast pulses [35]. Applications have been demonstrated in optical manipulation of microparticles [36,37]. Among them [29,30], it is pointed out that an optical vortex with a net OAM can be produced by completely coherent light arrays during free-space propagation. Whether a CV can be produced in the same way is unclear. In fact, these vortices are produced differently. An optical vortex exists in the phase of the wave function, whereas a CV is embedded in spiral structures in the phase of the correlation function [38]. In the former case, the decrease in coherence or the addition of turbulence results in OAM spectrum dispersion [5,39]. Even in free-space propagation, some special slits that cause wavefront distortion by diffraction also result in the distortion of OAM spectra [40,41]. For the latter case in which spiral structures exist in the dimension of coherence, the variation in the CV spectrum during propagation has not yet been studied. To the best of our knowledge, few works have focused on CV generation by directly configuring the phase of the correlation function.

2 THEORETICAL ANALYSIS

We consider a partially coherent light (PCL) array with $N$ fundamental beams. At the source plane ( $z = 0$ ), the mixed field is $E (x, y; 0) = e^{i δ} \sum_{j = 1}^{N} E_{j} (x, y; 0),$ (1)where $δ$ is a random phase and $E_{j}$ is the light field of the $j$ -indexed beamlet, $E_{j} (x, y; 0) = \exp [- \frac{{(x - a_{j})}^{2} + {(y - b_{j})}^{2}}{w^{2}}] \exp (i φ_{0 j}),$ (2)and $w$ is the waist width of the beamlet. $φ_{0 j} = L_{0} α_{j}$ is an initial phase, $α_{j} = π (2 j - 1) / N$ is the azimuthal angle of the j-indexed beamlet, and $L_{0}$ is a constant. We restrict that $0 \leq L_{0} < N$ due to periodicity. Parameters $a_{j}$ and $b_{j}$ illustrate the displacement of the center of the j-indexed beamlet and are related to the radius $r_{0}$ of the radial array according to $a_{j} = r_{0} \cos α_{j}$ and $b_{j} = r_{0} \sin α_{j}$ . The positions of the fundamental beamlets are shown in Fig. 1(a), where the number of the fundamentals is chosen as $N = 3$ for concreteness.

Figure 1.(a) Schematic of a PCL array at the source with $N = 3$ . Relative CV spectrum of the array at (b) $z = 10 z_{r}$ and (c) $z = 20 z_{r}$ . The beam parameters are $λ = 632.8 nm$ , $w = 0.157 mm$ , $σ = 0.1 w$ , $r_{0} = 1.5 w$ , and $L_{0} = 1$ . Here, $z_{r} = π w^{2} / λ = 0.122 m$ is the Rayleigh distance.

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The statistical distribution of the random phase $δ$ corresponds to a correlation within the combined field, $⟨ E^{*} (ρ_{1}) E (ρ_{2}) ⟩ = E^{*} (ρ_{1}) E (ρ_{2}) g (ρ_{1}, ρ_{2})$ . Specifically, we use a Schell model [42] to mathematically represent the correlation function, i.e., $g (ρ_{1}, ρ_{2}) = \exp (- {| ρ_{1} - ρ_{2} |}^{2} / 2 σ^{2})$ , where $σ$ is the transverse coherence width of the beam. Hence, the CSD of the PCL array reads $W (ρ_{1}, ρ_{2}; 0) = ⟨ E^{*} (ρ_{1}; 0) E (ρ_{2}; 0) ⟩ = \sum_{s = 1}^{N} \sum_{q = 1}^{N} W_{s q} (ρ_{1}, ρ_{2}; 0),$ (3)with $W_{s q} (ρ_{1}, ρ_{2}; 0) = \exp [- \frac{{(x_{1} - a_{s})}^{2} + {(y_{1} - b_{s})}^{2}}{w^{2}}] \exp [- \frac{{(x_{2} - a_{q})}^{2} + {(y_{2} - b_{q})}^{2}}{w^{2}}] \times \exp (i φ_{s q}) g (ρ_{1}, ρ_{2}; 0),$ (4)where $ρ_{1} \equiv (x_{1}, y_{1})$ and $ρ_{2} \equiv (x_{2}, y_{2})$ are two arbitrary spatial points expressed in Cartesian coordinates at the source plane; * indicates the complex conjugate; and $φ_{s q =} L_{0} (α_{q} - α_{s})$ is the phase difference between the numbering index $s$ and $q$ .

According to coherence theory and the paraxial propagation law, the propagated CSD at $z > 0$ can be obtained by [43] $W_{s q} (r_{1}, r_{2}; z) \propto \iint \iint W_{s q} (ρ_{1}, ρ_{2}; 0) h_{1}^{*} (r_{1}, ρ_{1}) h_{2} (r_{2}, ρ_{2} {) d}^{2} ρ_{1} d^{2} ρ_{2},$ (5)where $r_{1} \equiv (u_{1}, v_{1})$ and $r_{2} \equiv (u_{2}, v_{2})$ are two arbitrary spatial points at the output plane, and $h_{p}$ with $p = 1, 2$ is the response function: $h_{p} (r_{p}, ρ_{p}) = \exp [\frac{i k}{2 B} (A x_{p}^{2} - 2 x_{p} u_{p} + D u_{p}^{2}) + \frac{i k}{2 B} (A y_{p}^{2} - 2 y_{p} v_{p} + D v_{p}^{2})],$ where $k = 2 π / λ$ is the wavenumber and $λ$ is the wavelength, and $A$ , $B$ , and $D$ are transfer-matrix elements of the optical system. For the case of free-space propagation, $A = D = 1$ and $B = z$ . Substituting Eqs. (3) and (4) into Eq. (5) yields the CSD function of the PCL array [Fig. 1(a)] at the output plane as follows: $W (r_{1}, r_{2}; z) = \sum_{s = 1}^{N} \sum_{q = 1}^{N} W_{s q} (r_{1}, r_{2}; z),$ (6)where $W_{s q} (r_{1}, r_{2}; z)$ reads $W_{s q} (r_{1}, r_{2}; z) = \frac{π^{2}}{M_{1} M_{2}} {(\frac{1}{λ | B |})}^{2} \exp (i φ_{s q}) \exp (- \frac{2 r_{0}^{2}}{w^{2}}) \exp (\frac{N_{2}^{2} + N_{1}^{2}}{4 M_{2}}) \times \exp [\frac{1}{4 M_{1}} {(\frac{2 b_{s}}{w^{2}} + \frac{i k}{B^{*}} v_{1})}^{2} + \frac{1}{4 M_{1}} {(\frac{2 a_{s}}{w^{2}} + \frac{i k}{B^{*}} u_{1})}^{2}] \times \exp [- \frac{i k D^{*}}{2 B^{*}} (u_{1}^{2} + v_{1}^{2}) + \frac{i k D}{2 B} (u_{2}^{2} + v_{2}^{2})],$ (7)with $M_{1} = \frac{1}{w^{2}} + \frac{1}{2 σ^{2}} + \frac{i k A^{*}}{2 B^{*}}, M_{2} = \frac{1}{w^{2}} + \frac{1}{2 σ^{2}} - \frac{1}{4 σ^{4} M_{1}} - \frac{i k A}{2 B}, N_{1} = \frac{1}{w^{2}} (2 a_{q} + \frac{a_{s}}{σ^{2} M_{1}}) + i k (\frac{u_{1}}{2 σ^{2} M_{1} B^{*}} - \frac{u_{2}}{B}), N_{2} = \frac{1}{w^{2}} (2 b_{q} + \frac{b_{s}}{σ^{2} M_{1}}) + i k (\frac{v_{1}}{2 σ^{2} M_{1} B^{*}} - \frac{v_{2}}{B}) .$ (8)

To examine the existence of CVs in the propagated radial array, the output CSD can be projected to OAM modes [44,45]: $P (L) \propto | \int_{0}^{\infty} \int_{0}^{\infty} \int_{0}^{2 π} \int_{0}^{2 π} W (r_{1}, r_{2}; z) \exp (i L \cdot Δ θ) r_{1} r_{2} d r_{1} d r_{2} d θ_{1} d θ_{2} |,$ (9)where $r_{1} \equiv (r_{1}, θ_{1})$ and $r_{2} \equiv (r_{2}, θ_{2})$ are re-expressed in polar coordinates, and $Δ θ = θ_{1} - θ_{2}$ . $P (L)$ is a weight coefficient that represents the probability for an OAM mode with the topological charge as $L$ . The center correlation function of the field is the CSD taken at points of (0, 0) and $(r_{2}, θ_{2})$ , i.e., $W (0, 0, r_{2}, θ_{2})$ . On this basis and by substituting Eqs. (6)–(8) into Eq. (9), under the assumption of the paraxial propagation $w^{2} ≪ λ z$ and low coherence source $σ ≪ w$ , we obtain the mode coefficient as $P (L) \propto (\frac{1}{2 π}) | \sum_{s = 1}^{N} \sum_{q = 1}^{N} P_{s q} (L) |,$ (10)where $P_{s q} (L) = Q_{1} \times \exp [i (φ_{s q} - L \cdot α_{s q}^{(+)})] \exp (\frac{4 r_{0}^{2} c o s^{2} α_{s q}^{(-)}}{w^{4} M_{2}}) \times \sum_{n = 0}^{\infty} {[\frac{- 4 k^{2} r_{0}^{2} \cos^{2} α_{s q}^{(-)}}{w^{4} M_{2} k (k - 4 i M_{2} z)}]}^{n + | L | / 2} \frac{Γ (n + | L | / 2 + 1)}{Q_{2} \times n! Γ (n + | L | + 1)},$ (11)with the parameters $Q_{1}$ and $Q_{2}$ as $Q_{1} = \frac{2 π^{3}}{M_{1} M_{2} λ^{2} z^{2}} \exp (- \frac{2 r_{0}^{2}}{w_{0}^{2}} + \frac{r_{0}^{2}}{M_{1} w_{0}^{4}}), Q_{2} = \frac{π^{2}}{M_{2} λ^{2} z^{2}} - \frac{i π}{λ z},$ and $α_{s q}^{(+)} = (α_{s} + α_{q}) / 2$ , $α_{s q}^{(-)} = (α_{q} - α_{s}) / 2$ , $Γ (\cdot)$ as the Euler gamma function. During the derivation of Eq. (11), the following formulas are used [45,46]: $\int_{0}^{2 π} \exp [- i n ϕ_{1} + ξ \cos (ϕ_{1} - ϕ_{2})] d ϕ_{1} = 2 π \exp (- i n ϕ_{2}) I_{n} (ξ), I_{n} (ξ) = \sum_{k = 0}^{\infty} \frac{1}{k! Γ (k + n + 1)} {(\frac{ξ}{2})}^{2 k + n}, with Re (n) > - 1, \int_{0}^{\infty} x^{2 m + 1} \exp [- q x^{2}] d x = \frac{1}{2 q^{m + 1}} Γ (m + 1), with Re (q) > 0 and m > - 1 .$

In Eq. (11), the magnitude of the series summation converges over $n$ when $σ ≪ w$ and $w^{2} ≪ λ z$ are satisfied. Therefore, a finite summation can be used to replace the infinite one during the calculation of $P_{s q} (L)$ under low-coherence conditions. According to the formula of $P (L)$ , one can simulate the OAM spectrum of the PCL array at different propagation distances. Numerical simulations reveal that $P (L)$ comprises items at $L = m N$ with $m$ integers. As shown in Figs. 1(b) and 1(c), the largest contribution to the CV spectrum originates from the item with $L = 0$ . Side modes are distributed symmetrically at $L = m N$ with $m$ integers. This symmetry does not vary with the change in propagation distance. Hence, the total OAM of the combined field is always zero.

Figure 2 simulates the averaged intensity distribution $⟨ I (r, z) ⟩$ and the correlation phase distribution $ψ (x, y)$ of a radial array of three fundamentals ( $N = 3$ ). $ψ (x, y)$ is the phase of the center correlation function $W (0, 0, x, y)$ . The simulation process is as follows. To obtain the averaged intensity in Figs. 2(a) and 2(b), one makes use of Eq. (6) by setting $r_{1} = r_{2} = r$ , i.e., $W$ $(r, r; z) = ⟨ I (r, z) ⟩$ . To construct the correlation phase in Figs. 2(c) and 2(d), one extracts the phase of $W (0, 0, x, y)$ by setting $r_{1} = (0, 0)$ and $r_{2} = r = (x, y)$ in Eq. (6). The numerical results of Figs. 2(a) and 2(b) show that the combined field cannot maintain its initial irradiance distribution after far-field propagation. Figures 2(c) and 2(d) show that phase modulation among different beamlets at the source does not lead to a spiral phase structure at the far field.

Figure 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 $z = 20 z_{r}$ ; (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 $z = 20 z_{r}$ . The other beam parameters are the same as those in Fig. 1.

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The numerical simulations correlating the PCL array in a Schell model of Gaussian function cannot produce a CV. The reason is that in free-space propagation, the irradiance distributions of PCL beams at the far field are determined by their correlation functions [47]. The correlation among different spatial points ruins the initial phase configuration among fundamental beamlets. Hence after far-field propagation, a vortex phase structure cannot be formed at the output plane. Although many coherence models have been proposed in the literature, experimental demonstrations are limited. Methods for the direct modulation of the complex field in the physical dimension of correlation are required.

3 EXPERIMENTAL METHOD

We present an experiment to demonstrate a strategy of generating a CV utilizing the previous radial arrays by modulating the correlation structure itself. Our approach is based on the work of Ref. [48] where a spatial light modulator (SLM) is utilized to produce a partially coherent light. Different from the work of Ref. [48], in this paper a PCL array is produced which provides a way to modulate the correlation functions azimuthally.

The experimental setup is shown in Fig. 3(a) [microscope objective (MO) is excluded], which digitally generates PCL fields by utilizing a spatial light modulator [48]. The PCL field of interest is a radial array with N fundamental beamlets [Eqs. (1) and (2)]. Here, we set $N = 3$ . A linearly polarized light from a He–Ne laser source ( $λ = 632.8 nm$ , 2 mW; Newport R-30989) is expanded with a beam expander and then goes through a beam splitter before shining on the SLM (phase modulation type, $1280 \times 720$ , 6.3 μm, 60 Hz; UPOLabs, HDSLMM63R). The polarization of the incident light matches the SLM requirement. The first order of the reflected light of the SLM is selected by a pinhole (P1) and then propagates through a neutral density filter. The light is reflected by a reflecting mirror to increase the propagation distance on the optical table and is then detected by a camera. The free-space propagation distance from the SLM to the camera is z. The SLM displays a movie at a certain rate. The movie consists of a series of computer-generated holographs (CGHs). The $k$ th frame of the movie is taken as an example. It is a CGH that is designed to produce a superposed field $E^{(k)}$ formulated as [48] $E^{(k)} (x, y) = \sum_{p = 1}^{M_{1}} [E_{1}^{(p)} (x, y) + E_{2}^{(p)} (x, y) + E_{3}^{(p)} (x, y)] \exp (i δ_{p}),$ (12)where $δ_{p}$ is a random phase that ranges from 0 to $2 π$ , and $E_{j}^{(p)}$ is the $j$ th beamlet ( $j = 1, 2, 3$ ) of the $p$ th array ( $p = 1, 2, \dots, M_{1}$ ) in the superposed field $E^{(k)}$ [Eq. (12)]. We have $E_{j}^{(p)} = u_{j}^{(p)} \exp (i φ_{0 j})$ with the amplitude defined as $u_{j}^{(p)} (x, y) = \exp {- \frac{{[x - a_{j} - x_{j}^{(p)}]}^{2} + {[y - b_{j} - y_{j}^{(p)}]}^{2}}{w^{2}}} .$ (13)Parameters $a_{j}$ , $b_{j}$ , $φ_{0 j}$ , and $w$ are the same as those in Eq. (2). The shifted coordinates $(x_{j}^{(p)}, y_{j}^{(p)})$ representing the positions of the perturbation centers are random numbers that are uniformly distributed within a circle of radius $c_{0}$ . Notably, the fundamental beamlets are perturbed separately. In other words, the perturbation centers $(x_{j}^{(p)}, y_{j}^{(p)})$ have no connection with each other. We denote this kind of randomness modulation as independent perturbation. Figures 3(b) and 3(c) show examples of the CGHs displayed on the SLM to produce the PCL array under different coherence conditions. Notably, the CGH modulation for the three beamlets merges together with increased $c_{0}$ . The parameter $c_{0}$ is inversely proportional to the coherence width of the fundamental PCL beamlet [48]. If the array is of total coherence, then $c_{0} = 0$ and $δ_{p} = 0$ ; if the array is of poor coherence, then $c_{0} > w$ and $δ_{p}$ is a random phase. As shown in Fig. 3(d), the instantaneous intensity of the array with $c_{0} = 3 w$ , $δ_{p} \in (0, 2 π)$ appears as speckles. As shown in Fig. 3(e), which is calculated according to Eq. (12), the phase of the field $E^{(k)} (x, y)$ has a random distribution. The speckles and phases change randomly with time, which guarantees that the light field is not totally coherent in the time domain.

Figure 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 $c_{0} = 0$ , $δ_{p} = 0$ ; and (c) poor coherence with $c_{0} = 3 w$ , $δ_{p} \in (0, 2 π)$ . (d) Instantaneous intensity $I^{(k)} (x, y)$ of the PCL array produced with $c_{0} = 3 w$ , $δ_{p} \in (0, 2 π)$ . (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 $w = 0.157 mm$ , $r_{0} = 1.5 w$ , and $L_{0} = 1$ . 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.

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By incoherent superposition of a large number of realizations of the field $E^{(k)}$ , where $k = 1, 2, \dots, M_{2}$ , the averaged CSD at the detection plane can be obtained according to $W (x_{1}, y_{1}, x_{2}, y_{2}) = ⟨ E^{(k) *} (x_{1}, y_{1}) E^{(k)} (x_{2}, y_{2}) ⟩ = \frac{1}{M_{2}} \sum_{k = 1}^{M_{2}} E^{(k) *} (x_{1}, y_{1}) E^{(k)} (x_{2}, y_{2}) .$ (14)The magnitude of the CSD can be obtained according to the Hanbury Brown and Twiss (HBT) effect [25]. A camera records the intensity pattern from each CGH frame, i.e., $I^{(k)} (x, y)$ . With the data of the instantaneous intensities, the second-order correlation can be obtained by computation, $G^{(2)} (x_{1}, y_{1}, x_{2}, y_{2}) = ⟨ I^{(k)} (x_{1}, y_{1}) I^{(k)} (x_{2}, y_{2}) ⟩$ . Accordingly, the magnitude of the CSD can also be computed, ${| W (x_{1}, y_{1}, x_{2}, y_{2}) |}^{2} = G^{(2)} (x_{1}, y_{1}, x_{2}, y_{2}) - ⟨ I^{(k)} (x_{1}, y_{1}) ⟩ \cdot ⟨ I^{(k)} (x_{2}, y_{2}) ⟩$ . To reconstruct the cross-correlation function (CCF), i.e., $W (x, y, - x, - y)$ , the transformation $(x_{1}, y_{1}) \to (x, y)$ , $(x_{2}, y_{2}) \to (- x, - y)$ is necessary in Eq. (14).

In the experiment, the number of coherently superposed fields is $M_{1} = 500$ . The frame rate of the SLM is 60 fps. The shooting frequency of the camera should be below 60 Hz; for example, we have utilized 19.8 and 49.6 Hz for the setting of the cameras. The number of CGH frames playing in the SLM movie is 1000. The number for ensemble average in Eq. (14) is $M_{2} = 200$ . In the experiment of the production of the PCL array, the power ratio of the first-order diffracted beam from the SLM is around 10%.

Figure 4 presents the experimental results of the averaged intensity of the independently perturbed random array at different propagation distances. The parameter $z$ presents the distance from the SLM to the camera. As shown in Fig. 4(a), the random array has a single-beam shape at a near field ( $z = 0.2 m ≃ 1.6 z_{r}$ ). At a longer propagation distance $z = 0.57 m ≃ 4.7 z_{r}$ , a dark center appears inside the light field, as shown in Fig. 4(b). As the generated beam propagates to $z = 1.14 m ≃ 9.3 z_{r}$ [Fig. 4(c)], the dark hollow structure maintains its shape and the center of the beam darkens. The structure appearing at a far field reveals the shape of the correlation function at the source. The far-field property has been influenced by the randomness at the source [Eqs. (12) and (13)]. In other words, the perturbation method and the initial phase configuration among different beamlets modulate the correlation structure of the field. In view of applications in optical trapping or manipulation, one may use a microscope objective to compress the vortex structure. As shown in Fig. 3(a) (MO is included), after a far-field propagation ( $z = 1.14 m$ ) the CV is focused with a microscope objective ( $10 \times$ , $NA = 0.25$ ) with a focal length of 17.1 mm and a working distance of 7.3 mm. Figure 4(d) presents the averaged intensity of the CV at the focal plane, where the diameter of the vortex ring is compressed to approximately 250 μm and the maximum intensity at the ring is $5.17 mW / {mm}^{2}$ . Comparatively, the sizes of the central hollow section in Figs. 4(b) and 4(c) are 650 μm and 1.46 mm, respectively. The intensities at the ring of Figs. 4(b) and 4(c) are $1.65 mW / {mm}^{2}$ and $0.34 mW / {mm}^{2}$ , respectively. The data of the peak intensities in Figs. 4(b)–4(d) is obtained when the neutral density filter is not used, and the total power of the incident beam on the detection plane is 2 mW.

Figure 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 ( $1280 \times 1024$ at 19.8 fps, 5.2 μm).

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Figure 5 presents the experimental results for the propagation of the array with different perturbation models. Figures 5(a) and 5(b) show the case of a phase modulated radial array with total coherence, where the array holds the shape of fundamental beamlets from $z = 0.57 m$ to 1.14 m. Figures 5(c) and 5(d) are the case of the independent perturbation presented in Fig. 4. A comparison of Figs. 5(a) and 5(b) with Figs. 5(c) and 5(d) demonstrates that when the number of the fundamental beamlets is small, i.e., $N = 3$ , the totally coherent array does not form a dark hollow structure according to propagation, but the specially designed random array does. In Figs. 5(e) and 5(f), we use an uniform perturbation to replace the independent one, which means that the perturbation centers in Eq. (13) transform as $(x_{j}^{(p)}, y_{j}^{(p)}) \to (x^{(p)}, y^{(p)})$ . This finding means that the fundamental beamlets of the array are perturbed uniformly. In the uniform-perturbation model, a dark hollow beam is not produced. A comparison of Figs. 5(e) and 5(f) with Figs. 5(c) and 5(d) shows that the independent perturbation model leads to the hollow structure in the far field. To verify the effect of the initial phase on the propagation of the independently perturbed array, Figs. 5(g) and 5(h) present a case without phase modulation $L_{0} = 0$ , which reveals that the propagated random array is in a single-beam shape and no vortex structure is formed. A comparison of Figs. 5(g) and 5(h) with Figs. 5(c) and 5(d) shows that the dark hollow structure at the far field is a direct result of the phase modulation at the source.

Figure 5.Experimental results of the PCL array under different perturbation conditions. (a) and (b) are for $c_{0} = 0$ , $δ_{p} = 0$ and $L_{0} = 1$ with (a) $z = 0.57 m$ , (b) $z = 1.14 m$ ; (c) and (d) are for the case of independent perturbation with $c_{0} = 3 w$ and $L_{0} = 1$ at (c) $z = 0.57 m$ , (d) $z = 1.14 m$ ; (e) and (f) are for the case of uniform perturbation with $c_{0} = 3 w$ and $L_{0} = 1$ at (e) $z = 0.57 m$ , (f) $z = 1.14 m$ ; and (g) and (h) are for the independent perturbation with $c_{0} = 3 w$ and $L_{0} = 0$ at (g) $z = 0.57 m$ , (h) $z = 1.14 m$ . Other parameters are $w = 0.157 mm$ and $r_{0} = 1.5 w$ . 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 ( $1280 \times 1024$ at 19.8 fps).

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One may notice that the averaged intensities in Figs. 5(c) and 5(d) are not symmetric. The reason is that at the far field the irradiance distribution of a partially coherent light beam is a result of its correlation function at the source plane. The total correlation function of the radial array is a superposition of the correlations of the beamlets. High-ordered interferences among the beamlets of the array lead to the irradiance distribution of the combined field at the far field. When the number of the beamlets is as small as three, the interfering patterns are not symmetrically distributed and as a result Figs. 5(c) and 5(d) present asymmetric distributions of the averaged intensity. Increasing the number of beamlets at the source, the symmetry of the generated dark hollow structure would be improved.

The findings based on Fig. 5 demonstrate that the PCL array designed according to the strategy of Eqs. (12) and (13) can produce a dark hollow beam in the far field despite the poor coherence and the small number of beamlets at the source plane. We then further confirm that the dark hollow structure generated by this random array possesses a CV.

The dark rings inside the contour graph of the CCF are known to indicate the existence of an OAM in a partially coherent vortex beam [25]. For a beam with good coherence, the existence of the vortex can also be revealed by the ring dislocations in the Fourier transform of the intensity [49]. Figures 6(a)–6(c) present the computational realizations of the second-order correlation $G^{(2)} (x, y, - x, - y)$ , the magnitude of the CCF $| W (x, y, - x, - y) |$ , and the Fourier transform of the averaged intensity of the random array in Fig. 5(d). A dark ring is observed in the distributions of the second-order correlation, the CCF, and the Fourier transformed intensity. Thus, the results of Figs. 6(a)–6(c) indicate a CV embedded in the dark hollow beam produced by the special random array designed by Eqs. (12) and (13). The topological charge is 1. Notably, the computational quantities of $G^{(2)} (x, y, - x, - y)$ and $| W (x, y, - x, - y) |$ are obtained based on the HBT relation and the instantaneous intensities obtained experimentally [Eq. (14)]. To demonstrate the correctness of the computational realizations, we perform wavefront folding interferometry [setup not shown in Fig. 3(a)] in which two Dove prisms are used to produce the correlation between the fields that are diametrically opposed [48]. A Mach–Zehnder interferometer is inserted into the path between P1 and the detector. The experimental result of the interferometry is shown in Fig. 6(d). A ring dislocation is embedded in the interference fringes. This result agrees with that of the computational realizations in Fig. 6(a).

Figure 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 ( $600 \times 800$ at 49.6 fps), and 1 pixel is 2.2 μm; while in (d), the shooting camera is a CinCam CMOS-1201 ( $1280 \times 1024$ at 19.8 fps), and 1 pixel is 5.2 μm. (c) is in spatial frequency domain where 1 pixel is $1 / (5.2 \times 1280) {μm}^{- 1}$ .

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The finding in Fig. 6 confirms the existence of a CV in the light field generated by the random array. Two approaches are adopted for the justification. One approach relies on measuring the instantaneous intensity of the structured light and then reconstructing the second-order correlation function. The magnitude of the CCF can be computed according to the HBT relation and Eq. (14). The result for the second-order correlation is shown in Fig. 6(a). The second approach is to measure the interference pattern in a Mach–Zehnder interferometer with two Dove prisms inverting the beams with respect to the horizontal and vertical axes, respectively. The output intensity of the interferometer is the second-order correlation function between ( $x$ , $y$ ) and $(- x, - y)$ , as shown in Fig. 6(d). From the results of both approaches, i.e., Figs. 6(a) and 6(d), one can observe a dark ring embedded in the interference fringes. It means that a CV exists inside the structured light field.

It is worth noting that there are other situations that spatially incoherent sources are used to create vortices [50,51] where people utilized rotating ground-glass disk (RGGD) to destroy the coherence of the light source and produce special vortex speckles. The difference between the RGGD method and the one proposed here is that in this paper the coherence of the complex light field is designed in a digital way. Correlation modulations can be carried out in the azimuthal direction. Moreover, fewer optical elements are needed in the experiment setup.

Considering the limitations of this method, the following issues are listed. (1) The quality of the structured light produced with the method here is limited by the spatial resolution of the digital modulator, i.e., SLM; and (2) the temporal evolution of the light source is controlled by the CGHs, and hence the duration of the light source relies on the amount of the calculation carried out in advance.

4 CONCLUSION

We proposed an experimental strategy to modulate the correlation structure of light fields in a digital and controllable way. By utilizing PCL arrays, we provided a routine to control the correlation structure according to phase configuration. A vortex-beam structure that cannot be produced using conventional laser arrays nor uniformly perturbed PCL arrays was evidenced through experiments by digitally modulating the randomness of the array in a special way. The CCF and the Fourier transformed contours of the produced beam revealed the existence of a CV embedded therein. Although we demonstrated this method with the use of an SLM in the experiment, it is not limited to this device. Considering the emergence of digital coding metamaterials [52] that also manipulate light fields in a programmable way, more optional routines for experimental realization may be available.

The method in this study is a general one and can be useful in other studies on optical manipulation and communication where the correlation structure plays a key role.

Category: Physical Optics

Received: Jul. 23, 2019

Accepted: Oct. 18, 2019

Published Online: Nov. 21, 2019

The Author Email: Jun Chen (chenjun.sun@gmail.com)

DOI:10.1364/PRJ.7.001485