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Spatial correlated vortex arrays

Zhangrong Mei, Yonghua Mao, Jixian Wang, and Xiaohui Shi

Open Access Open Access

Abstract

Spatial correlated vortex arrays may form in the same beam when a random source contains multiple helical phase structures. We introduced two types of partially coherent sources with Cartesian and polar symmetric helical phase structure and reveal the characteristics of their radiated fields, respectively. It is demonstrated that far fields generated by these families of sources carry interesting features through the joint regulation of coherence and topological charge, being lattice-like vortex patterns with adjustable dimension and shape.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Optical beams carrying a helical phase structure around the axis with the phase singularity on the axis, known as optical vortices [1,2], have recently become the focus of many theoretical investigations and found applications in optical tweezers, micromanipulation [3], optical communications [4] and quantum information processes [5]. In addition to a single optical vortex, vortex arrays are of particular interest in manipulating the systems of particles [6], driving micro-optomechanical pumps [7], processing quantum information and micro-lithography [8]. Vortex arrays can be generated by plane or spherical wave interference method [9], helical phase-space filtering method [10], hollow multimode waveguide method [11], fractional Talbot effect method [12], etc. However, many of these studies are restricted to the domain of fully coherent vortex beams, leaving aside possible fluctuations of light.

Compared with the fully coherent fields, the partially coherent vortex beams can effectively overcome the light intensity flicker and light spot diffusion effects caused by propagation [13]. A partially coherent source or field is typically characterized by its second-order, two-point (generally complex-valued) correlation function [14], and hence, the phase singularity may arise as a vortex-like structure in its argument, rather than in the argument of the filed itself [15]. The extension of the concept of optical vortex from coherent field to correlation singularity was shown to be far from trivial [16]. Subsequent theoretical studies of correlation vortices have resulted in a number of mathematical models [17–19]. It is shown that partially coherent vortex beams with special correlation functions, have more freedom for beam shaping, have unique characteristics such as self-splitting, self-shaping and self-focusing [20–22], so they have more advantages in practical applications. However, except for a recent paper on a vortex array embedded in a partially coherent beam which leads to a very special array structure [23], no other models are available for generating partially coherent vortex arrays with different geometries. This present work was motivated by the desire to generate vortex array embedded in a partially coherent beam. The single optical vortex is characterized by the helical phase structure with the phase singularity. To achieve this aim of generating vortex arrays in the same beam, we present a class of planar sources with cross-spectral density (CSD) possessing phase structures with multiple phase singularities. As we shown, such sources can form vortex arrays with different periodic structures. Furthermore, we demonstrate how both the topological charge and the coherence length of source field affect the far-field spectral densities.

2. Spatial correlated rectangular vortex arrays

For fully coherent beams, the field amplitude is generally used to characterize them. The electric field of a coherent scalar beam containing a single optical vortex in the initial transverse plane may be expressed as [18]

$$E(\mathbf{\rho }) = A(\rho )\exp (i\beta )\exp (il\phi ),$$
where $\mathbf{\rho }$ is the position vector of field point with polar and Cartesian coordinates of $\mathbf{\rho }$ being $(\rho ,\phi )$ and $(x,y)$, $A(\rho )$ is the electric field amplitude of the beam, $\beta $ is an arbitrary phase and l is the topological charge.

If multiple vortices exist in the same beam and present the $S \times T$ rectangular array, one may select the following form of the initial field

$$E(\mathbf{\rho }) = A(\rho )\prod\limits_{s = 1}^S {\prod\limits_{t = 1}^T {\exp (i{l_{st}}{{\phi ^{\prime}}_{st}})} } ,$$
where ${l_{st}}$ is the topological charge of the $s \times t$th vortex, and ${\phi ^{\prime}_{st}} = \arctan [{(y - {y_{st}})/(x - {x_{st}})} ]$, $({x_{st}},{y_{st}})$ is the location of the $s \times t$th phase singularity in the source plane. Here, the topological charge ${l_{st}}$ of each vortex is set to the same value l. In order to arrange vortex core uniformly and axis-symmetrically, we can set
$${x_{st}} ={-} \frac{{(2S - 1){d_x}}}{2} + (s - 1){d_x},\quad s = 1,2, \cdots S,$$
$${y_{st}} ={-} \frac{{(2T - 1){d_y}}}{2} + (t - 1){d_y},\quad t = 1,2, \cdots T,$$
where ${d_x}$ and ${d_y}$ are the distances between phase singularities in the x and y directions, respectively.

For partially coherent beams, they are usually characterized by cross-spectral density (CSD) function in space-frequency domain, i.e,

$$W({{\mathbf \rho }_1},{{\mathbf \rho }_2}) = \langle {E^\ast }({{\mathbf \rho }_1})E({{\mathbf \rho }_2})\rangle ,$$
where ${{\mathbf \rho }_1}$ and ${{\mathbf \rho }_2}$ are transverse position vectors of a pair of points in the source plane, $\langle \textrm{ }\rangle $ denote an ensemble average and star stand for complex conjugate. The CSD for a spatial correlation rectangular vortex array may therefore be expressed by substituting from Eq. (2) into Eq. (5):
$$W({{\mathbf \rho }_1},{{\mathbf \rho }_2}) = A({\rho _1})A({\rho _2})g({{\mathbf \rho }_1},{{\mathbf \rho }_2})\prod\limits_{s = 1}^S {\prod\limits_{t = 1}^T {\exp [i{l_{st}}({{\phi ^{\prime}}_{1st}} - {{\phi ^{\prime}}_{2st}})]} } ,$$
where $g({{\mathbf \rho }_1},{{\mathbf \rho }_2})$ is the statistical distribution of the phase $\beta $ which represents the spatial correlation function between the two source points ${{\mathbf \rho }_1}$ and ${{\mathbf \rho }_2}$. For a conventional Carter-Wolf type light source corresponding to a Gaussian Schell-model correlator, the function $g({{\mathbf \rho }_1},{{\mathbf \rho }_2})$ may be expressed as
$$g({{\mathbf \rho }_1},{{\mathbf \rho }_2}) = \exp [{ - {{({{\mathbf \rho }_1} - {{\mathbf \rho }_2})}^2}/{\delta^2}} ],$$
where $\delta $ is the coherence length in the initial plane. Setting the Gaussian profile for function $A(\rho )$:
$$A(\rho ) = \exp ({ - {\rho^2}/{\sigma^2}} ),$$
and substituting Eqs. (7) and (8) into Eq. (6), the CSD may be written
$$W({{\mathbf \rho }_1},{{\mathbf \rho }_2}) = \exp \left( { - \frac{{\rho_1^2 + \rho_2^2}}{{\sigma_0^2}}} \right)\exp \left[ { - \frac{{{{({{\mathbf \rho }_1} - {{\mathbf \rho }_2})}^2}}}{{{\delta^2}}}} \right]\prod\limits_{s = 1}^S {\prod\limits_{t = 1}^T {\exp [{il({{\phi^{\prime}}_{1st}} - {{\phi^{\prime}}_{2st}})} ]} } .$$
To construct the correlation phase of the light fields, one extracts the phase of the field by setting the observation point ${{\mathbf \rho }_1} = 0$ and another point ${{\mathbf \rho }_2} = {\mathbf \rho } = (x,y)$ in Eq. (9). Figures 1 and 2 shows the initial phase profiles of rectangular vortex array for different topological charge l and different dimensions $S \times T$, respectively. Multiple phase singularities are uniformly arrayed along the x-axis and y-axis, with adjustable separation and number of vortex cores. Each phase singularity also has l phase ramps from $- \pi $ to $\pi $.

 figure: Fig. 1.

Fig. 1. Phase profiles of rectangular vortex arrays with $S \times T = 2 \times 2$ and ${d_x} = {d_y} = 0.4\textrm{mm}$ for different topological charge. (a) $l = 1$; (b) $l = 2$; (c) $l = 3$; (d) $l = 4$.

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 figure: Fig. 2.

Fig. 2. Phase profiles of rectangular vortex arrays with $l = 2$ and ${d_x} = {d_y} = 0.4\textrm{mm}$ for different vortex cores. (a) $S \times T = 2 \times 2$; (b) $S \times T = 4 \times 2$; (c) $S \times T = 4 \times 4$.

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The propagation law of the CSD function at a pair of points $({\mathbf{r}_1},z)$ and $({\mathbf{r}_2},z)$ across a transverse plane of the half-space $z > 0$ are related to those in the source plane as [14]

$$W({\mathbf{r}_1},{\mathbf{r}_2},z) = {\left( {\frac{k}{{2\pi z}}} \right)^2}\int\!\!\!\int {W({{\mathbf \rho }_1},{{\mathbf \rho }_2})} \exp \left\{ { - \frac{{\textrm{i}k}}{{2z}}[{{{({\mathbf{r}_1} - {{\mathbf \rho }_1})}^2} - {{({\mathbf{r}_2} - {{\mathbf \rho }_2})}^2}} ]} \right\}{\textrm{d}^2}{{\mathbf \rho }_1}{\textrm{d}^2}{{\mathbf \rho }_2},$$
where k denotes the wave number related to the wavelength $\lambda $ by $k = 2\pi /\lambda $. Numerical integration of Eq. (10) allows us to visualize the propagation characteristics of the light field generated by the source ( Eq. (9)). Although the four dimensional numerical integrate generally takes more time, it is possible to expand the cross-term of variables that allow Eq. (10) to be computed from the product of two two-dimensional integrals:
$$W({\mathbf{r}_1},{\mathbf{r}_2},z) = \prod\limits_{n = 1}^2 {\frac{k}{{2\pi z}}} {\sum\limits_{{S_n} = 0}^\infty {\frac{1}{{{S_n}!}}\left( { - \frac{2}{{{\delta^2}}}} \right)} ^{{S_n}}}\int_{ - \infty }^\infty {{K_n}} \textrm{d}{x_n}\textrm{d}{y_n},$$
where
$${K_n} = \exp [{ - g\rho_n^2 + {{( - 1)}^n}ik{{({\mathbf{r}_n} - {\mathbf{\rho }_n})}^2}} ]x_n^{{S_1}}y_n^{{S_2}}\prod\limits_{s = 1}^S {\prod\limits_{t = 1}^T {\exp [{{{( - 1)}^{n - 1}}il{{\phi^{\prime}}_{nst}}} ]} } ,$$
$$g = {\sigma ^{ - 2}} + {\delta ^{ - 2}}.$$
Performing numerical integration of Eq. (11) with Eq. (12), the far-field spectral densities radiated by a source in Eq. (9) with relative coherence length ${L_c} = \delta /\sigma = 2$ for different topological charge and different dimensions $S \times T$ are shown in Figs. 3 and 4, respectively. In this paper, the far-field plane is set at $z = 10{z_R}$, ${z_R} = k{\sigma ^2}/2$ is the Rayleigh length. It is clear from these plots that assigning different topological charges and phase singularities, it is possible to achieve rectangular vortex arrays with different intensity profiles. For smaller dimensions and topological charges, as shown Fig. 3(a), the profile of rectangular intensity vortex is more intact. As the topological charge increases, each vortex splits into multiple vortices whose number is equal to topological charge. It is precisely because of the increase in the number and dimension of vortices that they overlap each other, and the Gaussian intensity distribution of the initial plane, that it looks as if the profiles of rectangular vortex array in not very complete.

 figure: Fig. 3.

Fig. 3. Far-field spectral densities generated by a random source (9) corresponding to Fig. 1.

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 figure: Fig. 4.

Fig. 4. Far-field spectral densities generated by a random source (9) corresponding to Fig. 2.

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The case of $l = 1$ and $S \times T = 4 \times 4$ is shown in Fig. 5 for different states of coherence. For low coherence cases, each vortex core fills will diffuse light. With the increase of the relative coherence length, the diffuse light in the core gradually weakens, and finally forms a rectangular array of dark vortex cores with minimum intensity that is close to zero.

 figure: Fig. 5.

Fig. 5. Far-field spectral densities for various relative coherence length. (a) ${L_c} = 0.5$; (b) ${L_c} = 1$; (c) ${L_c} = 2$; (d) ${L_c} = 5$.

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3. Spatial correlated radial vortex arrays

If multiple vortices exist in the same beam and are uniformly distributed in a ring with radius d, the initial field can be represented as

$$E(\mathbf{\rho }) = A(\rho )\exp (i\beta )\prod\limits_{j = 1}^M {\exp (i{l_j}{{\phi ^{\prime}}_j})} ,$$
where ${l_j}$ is the topological charge of the jth vortex, and ${\phi ^{\prime}_j} = \arctan [(y - {y_j})/(x - {x_j})]$, $({x_j},{y_j})$ is the location of the jth phase singularity, ${x_j} = d\cos {\theta _j},$ $\textrm{ }{y_j} = d\sin {\theta _j},$ ${\theta _j} = (2j - 1)\pi /M.$ Here, the topological charge ${l_j}$ of each vortex is set to the same value l. The CSD for a spatial correlation ring vortex array may be expressed by substituting from Eq. (14) into Eq. (4):
$$W({{\mathbf \rho }_1},{{\mathbf \rho }_2}) = A({\rho _1})A({\rho _2})g({{\mathbf \rho }_1},{{\mathbf \rho }_2})\prod\limits_{j = 1}^M {\exp [{il({{\phi^{\prime}}_{1j}} - {{\phi^{\prime}}_{2j}})} ]} .$$
Also setting Schell-model correlator (7) for the spatial correlation function $g({{\mathbf \rho }_1},{{\mathbf \rho }_2})$ and Gaussian profile (8) for the light field $A(\rho )$, then the CSD takes on the form
$$W({{\mathbf \rho }_1},{{\mathbf \rho }_2}) = \exp \left( { - \frac{{\rho_1^2 + \rho_2^2}}{{{\sigma^2}}}} \right)\exp \left[ { - \frac{{{{({{\mathbf \rho }_1} - {{\mathbf \rho }_2})}^2}}}{{{\delta^2}}}} \right]\prod\limits_{j = 1}^M {\exp [{il({{\phi^{\prime}}_{1j}} - {{\phi^{\prime}}_{2j}})} ]} .$$
As examples of the initial phase profiles of such fields are shown in Figs. 6 and 7 for different topological charge l and different order M, respectively. Multiple phase singularities are uniformly distributed on a circle, with adjustable number of vortex cores and ring radius. There are l phase ramps from $- \pi $ to $\pi $ around each phase singularity, which only exist on the outside of the circle.

 figure: Fig. 6.

Fig. 6. Phase profiles of radial vortex arrays with $M = 6$ and radius $d = 0.3\textrm{mm}$ for different topological charge. (a) $l = 1$; (b) $l = 2$; (c) $l = 3$; (d) $l = 4$.

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 figure: Fig. 7.

Fig. 7. Phase profiles of radial vortex arrays with $l = 2$ and radius $d = 0.4\textrm{mm}$ for different vortex cores. (a) $M = 4$; (b) $M = 5$; (c) $M = 6$; (d) $M = 7$.

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Substituting Eq. (16) into Eq. (10), converting the four dimensional integrate into the product of two two-dimensional integrals, we can arrive at the same formula as Eq. (11), but the integral function is given by the expression

$${K_n} = \exp [{ - g\rho_n^2 + {{( - 1)}^n}ik{{({\mathbf{r}_n} - {\mathbf{\rho }_n})}^2}} ]x_n^{{S_1}}y_n^{{S_2}}\prod\limits_{j = 1}^M {\exp [{{{( - 1)}^{n - 1}}il{{\phi^{\prime}}_{nj}}} ]} .$$
Figures 8 and 9 show the far-field spectral densities radiated by a source in Eq. (16) with relative coherence length ${L_c} = \delta /\sigma = 5$ for different topological charge and different order M, respectively. One can see from those figures that the intensity profiles exhibit a uniformly distributed circular vortex array as expected. The number of dark vortex cores is equal to the product of order M and topological charge l.

 figure: Fig. 8.

Fig. 8. Far-field spectral densities generated by a random source corresponding to Fig. 6.

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 figure: Fig. 9.

Fig. 9. Far-field spectral densities generated by a random source corresponding to Fig. 7.

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The far-field spectral density generated by a random source with $M = 8$ and $l = 2$ for different states of coherence is shown in Fig. 10. For the low coherence case, each vortex core fills with diffuse light. The high coherence case produces a circular vortex array in which the central intensity of each dark core is close to zero.

 figure: Fig. 10.

Fig. 10. Far-field spectral densities for various relative coherence length. (a) ${L_c} = 0.5$; (b) ${L_c} = 1$; (c) ${L_c} = 2$; (d) ${L_c} = 5$.

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4. Conclusions

In this article, we have considered a new family of partially coherent sources, which is characterized by multiple helical phase structures. As practical examples, we modeled two types of such sources with either Cartesian or polar symmetric helical phase structure and investigated numerically the significant features of the radiated far-field intensities. We have found that such sources with any initially intensity distribution, as Gaussian in our examples, can generate lattice-like vortex radiation patterns. We also demonstrated how their features of radiated fields depend on their topological charge, array dimensions and relative coherence length. The results show the joint regulation of phase and coherence length to optical field can be used as a mechanism of generating spatial correlated vortex arrays.

Funding

National Natural Science Foundation of China (11974107); Natural Science Foundation of Zhejiang Province (LY23A040006).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (10)
Fig. 1.
Fig. 1. Phase profiles of rectangular vortex arrays with $S \times T = 2 \times 2$ and ${d_x} = {d_y} = 0.4\textrm{mm}$ for different topological charge. (a) $l = 1$; (b) $l = 2$; (c) $l = 3$; (d) $l = 4$.

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Fig. 2.
Fig. 2. Phase profiles of rectangular vortex arrays with $l = 2$ and ${d_x} = {d_y} = 0.4\textrm{mm}$ for different vortex cores. (a) $S \times T = 2 \times 2$; (b) $S \times T = 4 \times 2$; (c) $S \times T = 4 \times 4$.

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Fig. 3.
Fig. 3. Far-field spectral densities generated by a random source (9) corresponding to Fig. 1.

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Fig. 4.
Fig. 4. Far-field spectral densities generated by a random source (9) corresponding to Fig. 2.

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Fig. 5.
Fig. 5. Far-field spectral densities for various relative coherence length. (a) ${L_c} = 0.5$; (b) ${L_c} = 1$; (c) ${L_c} = 2$; (d) ${L_c} = 5$.

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Fig. 6.
Fig. 6. Phase profiles of radial vortex arrays with $M = 6$ and radius $d = 0.3\textrm{mm}$ for different topological charge. (a) $l = 1$; (b) $l = 2$; (c) $l = 3$; (d) $l = 4$.

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Fig. 7.
Fig. 7. Phase profiles of radial vortex arrays with $l = 2$ and radius $d = 0.4\textrm{mm}$ for different vortex cores. (a) $M = 4$; (b) $M = 5$; (c) $M = 6$; (d) $M = 7$.

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Fig. 8.
Fig. 8. Far-field spectral densities generated by a random source corresponding to Fig. 6.

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Fig. 9.
Fig. 9. Far-field spectral densities generated by a random source corresponding to Fig. 7.

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Fig. 10.
Fig. 10. Far-field spectral densities for various relative coherence length. (a) ${L_c} = 0.5$; (b) ${L_c} = 1$; (c) ${L_c} = 2$; (d) ${L_c} = 5$.

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Equations (17)

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E ( ρ ) = A ( ρ ) exp ⁡ ( i β ) exp ⁡ ( i l ϕ ) ,
E ( ρ ) = A ( ρ ) ∏ s = 1 S ∏ t = 1 T exp ⁡ ( i l s t ϕ ′ s t ) ,
x s t = − ( 2 S − 1 ) d x 2 + ( s − 1 ) d x , s = 1 , 2 , ⋯ S ,
y s t = − ( 2 T − 1 ) d y 2 + ( t − 1 ) d y , t = 1 , 2 , ⋯ T ,
W ( ρ 1 , ρ 2 ) = ⟨ E ∗ ( ρ 1 ) E ( ρ 2 ) ⟩ ,
W ( ρ 1 , ρ 2 ) = A ( ρ 1 ) A ( ρ 2 ) g ( ρ 1 , ρ 2 ) ∏ s = 1 S ∏ t = 1 T exp ⁡ [ i l s t ( ϕ ′ 1 s t − ϕ ′ 2 s t ) ] ,
g ( ρ 1 , ρ 2 ) = exp ⁡ [ − ( ρ 1 − ρ 2 ) 2 / δ 2 ] ,
A ( ρ ) = exp ⁡ ( − ρ 2 / σ 2 ) ,
W ( ρ 1 , ρ 2 ) = exp ⁡ ( − ρ 1 2 + ρ 2 2 σ 0 2 ) exp ⁡ [ − ( ρ 1 − ρ 2 ) 2 δ 2 ] ∏ s = 1 S ∏ t = 1 T exp ⁡ [ i l ( ϕ ′ 1 s t − ϕ ′ 2 s t ) ] .
W ( r 1 , r 2 , z ) = ( k 2 π z ) 2 ∫ ∫ W ( ρ 1 , ρ 2 ) exp ⁡ { − i k 2 z [ ( r 1 − ρ 1 ) 2 − ( r 2 − ρ 2 ) 2 ] } d 2 ρ 1 d 2 ρ 2 ,
W ( r 1 , r 2 , z ) = ∏ n = 1 2 k 2 π z ∑ S n = 0 ∞ 1 S n ! ( − 2 δ 2 ) S n ∫ − ∞ ∞ K n d x n d y n ,
K n = exp ⁡ [ − g ρ n 2 + ( − 1 ) n i k ( r n − ρ n ) 2 ] x n S 1 y n S 2 ∏ s = 1 S ∏ t = 1 T exp ⁡ [ ( − 1 ) n − 1 i l ϕ ′ n s t ] ,
g = σ − 2 + δ − 2 .
E ( ρ ) = A ( ρ ) exp ⁡ ( i β ) ∏ j = 1 M exp ⁡ ( i l j ϕ ′ j ) ,
W ( ρ 1 , ρ 2 ) = A ( ρ 1 ) A ( ρ 2 ) g ( ρ 1 , ρ 2 ) ∏ j = 1 M exp ⁡ [ i l ( ϕ ′ 1 j − ϕ ′ 2 j ) ] .
W ( ρ 1 , ρ 2 ) = exp ⁡ ( − ρ 1 2 + ρ 2 2 σ 2 ) exp ⁡ [ − ( ρ 1 − ρ 2 ) 2 δ 2 ] ∏ j = 1 M exp ⁡ [ i l ( ϕ ′ 1 j − ϕ ′ 2 j ) ] .
K n = exp ⁡ [ − g ρ n 2 + ( − 1 ) n i k ( r n − ρ n ) 2 ] x n S 1 y n S 2 ∏ j = 1 M exp ⁡ [ ( − 1 ) n − 1 i l ϕ ′ n j ] .
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