Coherence singularity and evolution of partially coherent Bessel–Gaussian vortex beams

Junan Zhu; Hao Zhang; Zhuoyi Wang; Xuechun Zhao; Xingyuan Lu; Xingyuan Lu; Yangjian Cai; Yangjian Cai; Chengliang Zhao; Chengliang Zhao

doi:10.1364/OE.483647

1. Introduction

Vortices are ubiquitous in the natural world and also exist in optics, where they are referred to as vortex beams. Vortex beams usually exhibit an annular intensity ring, and the phase takes the form of exp(ilφ), where l represents the topological charge (TC) and φ represents the azimuthal coordinate [1]. Characteristically, the central point is called a phase singularity, as the phase is undefined. Since Allen et al. [2] proposed that each photon of the vortex beam carries orbital angular momentum with a discrete value of lħ (where ħ is the reduced Planck’s constant), vortex beams have attracted increasing attention [3–8]. In particular, as a precise solution of the Helmholtz equation, the Bessel vortex beam has non-diffractive and self-healing properties [9,10]. Because of the power limitation, it is impossible to generate the ideal Bessel beam; however, an approximate Bessel beam with a gradually decreasing ring intensity can be generated, which is called a Bessel–Gaussian (BG) beam [11,12]. Thus, the distribution of a BG beam can be described as a Bessel function limited by a Gaussian function [11]. Benefiting from the existence of the Bessel term, BG beams are nearly non-diffractive and have a self-healing capability [13,14]. These properties endow BG beams with various applications, such as optical micromanipulation [15], high-capacity free-space optical communication [16], and rotation-speed detection [17].

However, when a coherent beam propagates through turbulent media, such as fog and heat flow, the beam will deform, scintillate or drift. Compared with fully coherent beam, partially coherent beam has advantages in anti-turbulence [18,19]. The Gaussian Schell-model partially coherent vortex beam was introduced by Gori et al. in 1998 [20]. Its intensity degenerates from a hollow-ring shape into a Gaussian distribution during propagation. At the same time, the phase singularity associated with the central zero intensity vanishes, which has been verified for other vortex beams, except the vortex preserving random beam [21–24]. The partially coherent Bessel–Gaussian (PCBG) vortex beam is a typical partially coherent vortex beam [25]. Numerous studies have been performed on the propagation characteristics and applications of PCBG vortex beams [26,27]. Compared with fully coherent BG beams, PCBG vortex beams have significant advantages, such as scintillation and drift in atmospheric turbulent transmission [26] and less crosstalk in free-space communication [28].

The disappearance of phase singularities brings challenges to the measurement of TC for partially coherent vortex beams. The methods proposed for fully coherent vortex beams based on diffraction patterns [29], interference patterns [30,31], or other diffraction-related phenomena [32,33] are applicable to the measurement of TC for BG vortex beams [34,35]. However, they become invalid when the degree of coherence decreases. Recently, the coherence singularities have been proved to appear in the cross-spectral density (CSD) structure [36]. Thus, several methods for determining the TC of partially coherent vortex beams have been proposed, e.g., measuring the cross-correlation function [37] and reconstructing the amplitude [38] and phase [39,40] of the complex degree of coherence or CSD. All these methods are designed for measurement on the focal plane. To the best of our knowledge, no method has been proposed for simultaneously measuring the TC magnitude and sign of a PCBG vortex beam during free-space propagation. Considering the particularity of the intensity distribution of PCBG vortex beams, the effectiveness of the aforementioned schemes for PCBG vortex beams is worth investigating. The findings of the present study support this. There are non-negligible differences between Laguerre–Gaussian (LG) and PCBG vortex beams when their TC is measured according to the quantitative relationship between the CSD and TC modulus.

In this study, we theoretically and experimentally investigated the TC measurement method for a PCBG vortex beam on several planes during free-space propagation. First, we simulated the intensity, CSD amplitude, and phase with on- and off-axis reference points for the PCBG vortex beam under different propagation distances and coherence widths. With the on-axis reference point, there was no quantitative relationship between the dark rings of the CSD amplitude and the TC modulus. However, for the off-axis reference point, the simulation and experimental results both indicated that the number of coherence singularities was equal to the magnitude of the TC. In addition, the phase winding direction around each coherence singularity reversed when the TC sign reversed. Furthermore, the results indicated that the location and distance of coherence singularities were related to the coherence, propagation distance, and reference.

2. Theory

The electric field of a fully coherent BG vortex beam in the source plane can be expressed as follows [11]:

(1)$$E({u,\varphi } )= {J_l}({{k_r}u} )\exp \left( { - \frac{{{u^2}}}{{w_0^2}}} \right)\exp ({il\varphi } ), $$

where u and φ represent the radial and azimuthal coordinates, respectively, in the source plane; l represents the TC; w₀ represents the initial beam waist; J_l(·) represents the l^th-order Bessel function of the first kind; and k_r is a radial wave vector. With a reduction in the coherence, the second-order statistical properties of the beam can be characterized by the CSD function, i.e., the correlation of two points described as $W({u_1},{u_2},{\varphi _1},{\varphi _2}) = \left\langle {{E^*}({u_1},{\varphi _1})E({u_2},{\varphi _2})} \right\rangle $. In the source plane, the CSD function of a PCBG vortex beam can be expressed as

(2)$$\begin{aligned} W({{u_1},{u_2},{\varphi_1},{\varphi_2},0} )&= {J_l}({{k_r}{u_1}} ){J_l}({{k_r}{u_2}} )\exp \left( { - \frac{{u_1^2 + u_2^2}}{{w_0^2}}} \right)\\ &\quad \times \exp ({il({{\varphi_1} - {\varphi_2}} )} )\exp \left( { - \frac{{u_1^2 + u_2^2 - 2{u_1}{u_2}\cos ({{\varphi_1} - {\varphi_2}} )}}{{2\sigma_0^2}}} \right) \end{aligned}, $$

where σ₀ represents the coherence width. According to Collins’ formula [41,42] and paraxial approximation, the CSD function in the observation plane of free-space propagation evolves into

(3)$$\begin{aligned} W({{r_1},{\theta_1},{r_2},{\theta_2},z} )&= {\left( {\frac{1}{{\lambda |B |}}} \right)^2}\int\!\!\!\int {{u_1}{u_2}d{u_1}d{u_2}d{\varphi _1}d{\varphi _2}W({{u_1},{u_2},{\varphi_1},{\varphi_2},0} )} \\ &\quad \times \exp \left\{ { - \frac{{ik}}{{2{B^\ast }}}[{{A^\ast }r_1^2 - 2{r_1}{u_1}\cos ({{\varphi_1} - {\theta_1}} )+ {D^\ast }u_1^2} ]} \right\}\\ &\quad \times \exp \left\{ {\frac{{ik}}{{2B}}[{Ar_2^2 - 2{r_2}{u_2}\cos ({{\varphi_2} - {\theta_2}} )+ Du_2^2} ]} \right\} \end{aligned}, $$

where r₁, r₂ and θ₁, θ₂ represent the radial and azimuthal coordinates, respectively, at the observation plane; λ represents the wavelength; and k is the wave vector. Here, we ignore the frequency term. If only the spatial dimension is considered, the CSD function is four-dimensional. To show the CSD distribution, a reference point (r₂ = r₀, θ₂ = θ₀) was selected for two-dimensional plotting, that is, W(r₁,θ₁,r₀,θ₀,z). For a free-space propagation system, we can set A = 1, B = z, C = 0, and D = 1, where z represents the propagation distance [41]. The average intensity distribution I(r,θ) can be calculated using the CSD function by setting r₁ equal to r₂ and θ₁ equal to θ₂.

Figure 1 presents the simulated intensity patterns of the PCBG vortex beam with varying TC at different propagation distances in free space. The simulation was conducted with pseudo-mode superposition [43]. Here, the wavelength was λ = 532 nm, the initial waist was w₀ = 2 mm, the radial wave vector was k_r = 10, and the coherence width was σ₀ = 0.4 mm, i.e., σ₀ = 0.2w₀. Additionally, three different propagation distances (z = 500, 750, and 1000 mm) and three TC values (l = + 2, + 3, –3) were considered. As shown in Fig. 1, the intensity patterns of the PCBG vortex beam were still close to the BG profile at a short propagation distance (z = 500 mm). As the beam propagation distance increased, the dark core at the center and dark rings gradually disappeared, and the beam shape gradually degenerated into a Gaussian distribution—particularly in the case of a small TC (l = + 2), as shown in Fig. 1(g). For the same coherence width, the intensity with a larger TC (l = + 3) did not completely degenerate into the Gaussian distribution. These results confirm that the vortex had the effect of anti-decoherence [44].

Fig. 1. Simulation results for the intensity distribution of a PCBG vortex beam at different propagation distances with varying TC.

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Although the TC affects the intensity distribution of the PCBG vortex beam, this is only a qualitative relationship, and the specific TC value cannot be directly determined from the intensity distribution of the PCBG vortex beam. For a partially coherent beam, the CSD function has been verified to be more robust and includes richer information. Previous studies on partially coherent LG vortex beams [45] indicated that the TC magnitude can be determined according to the number of dark rings in the CSD amplitude with an on-axis reference point (r₀ = 0). The amplitude distributions of the CSD function of the PCBG vortex beam at different propagation distances in this study are shown in Fig. 2. The simulation results indicated that the number of dark rings of the CSD amplitude with an on-axis reference point was not always equal to the TC magnitude. For example, the amplitude distribution had three dark rings for l = + 2 (Fig. 2(d)) and two dark rings for l = + 3 (Fig. 2(e)) at z = 750 mm. At z = 1000 mm, there were three dark rings for both |l| = + 2 and |l| = + 3. Thus, there was no explicit and reliable relationship between the TC value and the number of dark rings. The parameters used were identical to those for Fig. 1. The difference in results between the partially coherent LG vortex beam and the PCBG vortex beam is reasonable, because the light intensity of a PCBG vortex beam has multiple rings, and the distribution of the CSD is affected by the light intensity. Therefore, the TC magnitude of a PCBG vortex beam cannot be determined by observing the amplitude of the CSD function with an on-axis reference point.

Fig. 2. Simulation results for the amplitude distribution of the CSD function with an on-axis reference point (r₀ = 0) of a PCBG vortex beam at different propagation distances with varying TC.

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As the CSD function is complex-valued, we further investigated the CSD phase distribution. For a fully coherent vortex beam, the phase gradient around phase singularities can help quantify the TC, whereas for a partially coherent beam, the phase singularities vanish because of overlapping modes. Recent studies have indicated that as the coherence width decreases, these phase singularities transform into coherence singularities, where the CSD amplitude is zero and the phase is uncertain [46]. This inspires us to measure the TC of the PCBG vortex beam from its CSD phase.

To obtain the locations of coherence singularities from the equation W(r₁,r₂) = 0, we fix the reference point as r₀. Then, the four-dimensional CSD function degenerates into a two-dimensional function, i.e., W(r,r₀), and the locations of coherence singularities can be calculated from W(r,r₀) = 0. These coherence singularity locations are related to the selection of reference point r₀. Here, we define r₀ ≠ 0, i.e., the off-axis reference point, as the phase with an on-axis reference point cannot help determine the TC value (details are omitted).

As the CSD function contains both the intensity and degree of the coherence term [47], the reference point should be chosen in the area illuminated by the beam on the observation plane. Here, we set the reference point at r₀ = 1 mm, θ₀ = 0, and the other parameters were identical to those for Fig. 2. The simulation results for the CSD phase distribution with an off-axis reference point at different propagation distances are shown in Fig. 3. Clearly, for TC values of +2, + 3, and –3, the number of coherence singularities in the phase pattern was equal to the TC magnitude. By comparing the phase distributions of two inverse TCs, i.e., l = + 3 and –3, we found that the sign of the TC was related to the winding direction of the phase from 0 to 2π around these coherence singularities. When the phase winding direction was clockwise and counterclockwise, the TC sign was negative and positive, respectively. As the propagation distance increased, this feature held, and the distance between coherence singularities increased as the beam broadened. Therefore, the magnitude and sign of the TC can be simultaneously determined by observing the CSD phase distribution of a PCBG vortex beam with an off-axis reference point.

Fig. 3. Simulation results for the phase distribution of the CSD function with an off-axis reference point (r₀ ≠ 0) for a PCBG vortex beam at different propagation distances with varying TC.

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Next, we investigated the properties of a PCBG vortex beam with different spatial coherence widths. Figure 4 shows the phase distribution with an off-axis reference point (r₀ ≠ 0) for a PCBG vortex beam with different coherence widths (σ₀ = 2, 0.6, and 0.2 mm). We set the reference point at r₀ = 1 mm and θ₀ = 0, and the propagation distance was 1000 mm. The TC was l = + 2, + 3, and –3, and the initial waist and wave vector were identical to those for Fig. 1. When the coherence width was large, i.e., Figs. 4(a)–(c), the PCBG vortex beam at the observation plane maintained an approximate spiral phase, from which we determined the TC magnitude and sign for the partially coherent beam. With a reduction in the coherence width, the spiral phase gradually split. Fortunately, the coherence singularities remained, and the magnitude and sign of the TC could be determined with low coherence width. Clearly, the distance between these coherence singularities increased with a reduction in the coherence width, indicating that the degree of coherence affected the coherence singularity distribution of the PCBG vortex beam.

Fig. 4. Simulation results for the phase distribution of the CSD function with an off-axis reference point (r₀ ≠ 0) for a PCBG vortex beam with variations in the TC and coherence width.

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3. CSD measurement method

As indicated by the above simulation results, we determined the TC of the PCBG vortex beam during free-space propagation by obtaining the phase distribution of the CSD function with an off-axis reference point. Next, it was important to measure the TC of a PCBG vortex beam experimentally. Generally, it is difficult to directly obtain the CSD function via a charge-coupled device (CCD), because the CSD function includes real and imaginary parts and the CCD can only capture the magnitude of the light.

Recently, Shao et al. [48] reported a robust and efficient method for detecting the phase distribution of the CSD function via self-referencing holography. Therefore, we introduced two different values of phase perturbation at the same position of the PCBG vortex beam, and the Fourier intensities of an unperturbed beam and two perturbed beams were recorded via a CCD for reconstructing the complex CSD function.

First, we define a transmission function as T(r), where the value is 1 and 0 inside and outside the area of T(r), respectively. After this function is applied to the PCVB vortex beam to be measured (observation plane), the intensity distribution on the Fourier plane, i.e., the CCD plane, can be expressed as

(4)$${I_0}(\boldsymbol{\mathrm{\rho}} )= {\int\!\!\!\int {W({{{\mathbf r}_{\mathbf 1}},{{\mathbf r}_{\mathbf 2}}} )T({{{\mathbf r}_{\mathbf 1}}} )[{T({{{\mathbf r}_{\mathbf 2}}} )} ]} ^\ast }\exp [{ - i2\pi \boldsymbol{\mathrm{\rho}}({{{\mathbf r}_{\mathbf 1}} - {{\mathbf r}_{\mathbf 2}}} )} ]d{{\mathbf r}_{\mathbf 1}}d{{\mathbf r}_{\mathbf 2}}, $$

where ρ is the position vector in the Fourier plane.

Second, we introduce a perturbation point at r₀ in the observation plane (identical to the reference point in the previous section). The perturbation point is far smaller than the area of T(r); thus, it can be approximated as a Dirac function C₁·δ(r–r₀), where C₁ is an arbitrary complex-valued constant. Then, the transmission function T(r) of Eq. (4) evolves into

(5)$$T^{\prime}(r )= T({\mathbf r} )+ {C_1}\delta ({{\mathbf r} - {{\mathbf r}_{\mathbf 0}}} ). $$

Accordingly, the intensity of the perturbed beam on the Fourier plane can be expressed as

(6)$$\begin{aligned} {I_1}(\boldsymbol{\mathrm{\rho}} )&= \int\!\!\!\int {d{{\mathbf r}_{\mathbf 1}}d{{\mathbf r}_{\mathbf 2}}W({{{\mathbf r}_{\mathbf 1}},{{\mathbf r}_{\mathbf 2}}} )} \exp [{ - i2\pi \boldsymbol{\mathrm{\rho}}({{{\mathbf r}_{\mathbf 1}} - {{\mathbf r}_{\mathbf 2}}} )} ]\\ &\quad \times [{T({{{\mathbf r}_1}} )+ {C_1}\delta ({{{\mathbf r}_1} - {{\mathbf r}_{\mathbf 0}}} )} ]{[{T({{{\mathbf r}_2}} )+ {C_1}\delta ({{{\mathbf r}_2} - {{\mathbf r}_{\mathbf 0}}} )} ]^\ast } \end{aligned}. $$

Taking the inverse Fourier transformation (FT) of I₁(ρ)–I₀(ρ) yields

(7)$${F^{ - 1}}[{{I_1}(\boldsymbol{\mathrm{\rho}} )- {I_0}(\boldsymbol{\mathrm{\rho}} )} ]= C_1^\ast W({{{\mathbf r}_{\mathbf 0}} + {\mathbf r},{{\mathbf r}_{\mathbf 0}}} )T({{{\mathbf r}_{\mathbf 0}} + {\mathbf r}} )+ {C_1}{[{W({{{\mathbf r}_{\mathbf 0}} - {\mathbf r},{{\mathbf r}_{\mathbf 0}}} )T({{{\mathbf r}_{\mathbf 0}} - {\mathbf r}} )} ]^\ast }. $$

To extract “twin images” of the CSD, i.e., W(r₀+ r, r₀) and W(r₀–r,r₀), from Eq. (7), we must obtain another intensity distribution at the Fourier plane with a different value of C₁. Assume constants of C₁ for the first perturbed intensity and C₂ for another perturbed intensity. Then, the following two equations can be obtained:

(8)$$\left\{ \begin{array}{l} {F^{ - 1}}[{{I_1}(\boldsymbol{\mathrm{\rho}} )- {I_0}(\boldsymbol{\mathrm{\rho}} )} ]= C_1^\ast W({{{\mathbf r}_{\mathbf 0}} + {\mathbf r},{{\mathbf r}_{\mathbf 0}}} )T({{{\mathbf r}_{\mathbf 0}} + {\mathbf r}} )+ {C_1}{[{W({{{\mathbf r}_{\mathbf 0}} - {\mathbf r},{{\mathbf r}_{\mathbf 0}}} )T({{{\mathbf r}_{\mathbf 0}} - {\mathbf r}} )} ]^\ast }\\ {F^{ - 1}}[{{I_2}(\boldsymbol{\mathrm{\rho}} )- {I_0}(\boldsymbol{\mathrm{\rho}} )} ]= C_2^\ast W({{{\mathbf r}_{\mathbf 0}} + {\mathbf r},{{\mathbf r}_{\mathbf 0}}} )T({{{\mathbf r}_{\mathbf 0}} + {\mathbf r}} )+ {C_2}{[{W({{{\mathbf r}_{\mathbf 0}} - {\mathbf r},{{\mathbf r}_{\mathbf 0}}} )T({{{\mathbf r}_{\mathbf 0}} - {\mathbf r}} )} ]^\ast } \end{array} \right., $$

where I₁ and I₂ represent the FT intensities with perturbation values C₁ and C₂, respectively. By solving the above equations, the CSD term W(r,r₀) can be extracted after coordinate shifting on W(r₀+ r,r₀):

(9)$$W({{{\mathbf r}_{\mathbf 0}} + {\mathbf r},{{\mathbf r}_{\mathbf 0}}} )= \frac{{{C_2}{F^{ - 1}}[{{I_1}(\boldsymbol{\mathrm{\rho}} )- {I_0}(\boldsymbol{\mathrm{\rho}} )} ]- {C_1}{F^{ - 1}}[{{I_2}(\boldsymbol{\mathrm{\rho}} )- {I_0}(\boldsymbol{\mathrm{\rho}} )} ]}}{{C_1^\ast {C_2} - {C_1}C_2^\ast }}. $$

4. Experimental setup and results

With the method introduced in Section 3, we experimentally measured the CSD of a PCVB vortex beam. Figure 5(a) shows the experimental setup for generating the PCBG vortex beam and measuring its CSD function at any transmission plane. A coherent laser beam was emitted by a pump solid-state laser (λ = 532 nm) and expanded by a beam expander (BE). Then, the expanded beam was focused on a rotating ground glass disk (RGGD) by the lens L1 (f = 100 mm), and the transmitted beam became incoherent. According to the van Cittert–Zernike theorem, the beam collimated by L2 (f = 100 mm) produced a partially coherent beam, and its coherence width is related to the beam size on the RGGD. A smaller beam corresponds to a higher degree of coherence [49]. After the beam was transmitted through the Gaussian amplitude filter (GAF), the output beam became a Gaussian Schell-model (GSM) beam, whose intensity and CSD function both conformed to the Gaussian distribution [50]. The GSM beam passed through the first transmissive spatial light modulator (SLM1), in which a designed hologram was loaded for generating the BG beam. Then, a 4f system, including lenses L3 and L4 and a circular aperture (CA), was used to filter out useless diffractive orders. Then, a PCBG vortex beam with a controllable TC, coherence width, and radial wave vector was obtained after lens L4. To study the beam propagation properties, the posterior focal plane of L4 was taken as the source plane.

Fig. 5. Schematic of the experimental CSD measurement of a PCVB vortex beam. (a) Experimental setup for generating a PCBG vortex beam and measuring its CSD function in the free-space propagation plane (BE, beam expander; L1–L5, thin lenses; RGGD, rotating ground glass disk; GAF, Gaussian amplitude filter; CA, circular aperture; MI, mirror; BS, beam splitter; SLM1, transmissive spatial light modulator; SLM2, reflective spatial light modulator; CCD1 and CCD2, charge-coupled devices). The experimental results for the (b) intensity and (c) CSD phase distribution with an off-axis perturbation point at different propagation distances. (d) Evolution of coherence singularities.

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A beam splitter (BS) was used to split the PCBG vortex beam into two parts; one was reflected to CCD1 for intensity recording, and the other was directed to the reflective SLM2 for CSD measurement. The distances from the source plane to SLM2 and CCD1 were equal. SLM2 was used to load the phase perturbation. Because the perturbation point was approximated as the Dirac function, the size of the perturbation point was infinitely small in theory. However, as infinite smallness does not exist in real situations, we used a perturbation point approximately one order of magnitude smaller than the area of the transmission function T(r) in our experiments. According to Eq. (9), the reconstruction of the CSD function requires three different FT intensities. Thus, we used a transmission function without a perturbation point and two transmission functions with two different perturbation points (i.e., C₁ = exp(–2πi/3) and C₂ = exp(2πi/3)). Then, we captured the FT intensities (I₀, I₁, and I₂) via CCD2, which was located at the Fourier plane of SLM2, and L5 (f = 350 mm) was used to realize this FT. The position of the perturbation point could be freely changed with SLM2; that is, we could recover the CSD function with any on-axis or off-axis reference point.

Figure 5(b) shows the experimental intensity distributions of the PCBG vortex beam at different propagation distances (500, 750, 1000, and 1250 mm). The initial beam waist was 2 mm, the TC was +3, the coherence width was 0.6 mm, and the other parameters were identical to those for Fig. 1. The recovered CSD phase distribution is presented in Fig. 5(c). As shown, three coherence singularities appeared in the CSD phase patterns. Clearly, the number of coherence singularities was equal to the TC magnitude. The counterclockwise winding direction around the coherence singularities indicated that the TC was positive. These results are consistent with the aforementioned simulation results, indicating that the self-reference holography method can be used to measure the TCs of PCBG vortex beams.

In addition, the evolution of the coherence singularities with the beam propagation was examined, as shown in Fig. 5(d). Here, the red, yellow, and blue curves denote the positions of the three different coherence singularities, and the black dashed curve denotes the reference points. As the reference points spiral forward, the positions of the three coherence singularities rotate. On a certain observation plane, the positions of the coherence singularities were determined by the reference point.

Finally, we experimentally measured the phase of the PCBG vortex beam with different TCs and coherence widths. The coherence widths were 1.2, 0.6, and 0.4 mm, and the other parameters were identical to those for Fig. 3. The experimental intensity patterns are presented in Figs. 6(a)–(i). The CSD phase distributions with off-axis perturbation points are shown in Figs. 6(j)–(r). The experimental results agreed well with the simulation results. As indicated by the CSD phase patterns, the magnitude and sign of the beam’s TC were determined accurately using the number of coherence singularities and the winding direction of the phase, respectively. Even if the PCBG vortex beam degenerates into the Gaussian distribution, i.e., with very low coherence (Fig. 6(g)), we can determine the magnitude and sign of the TC. Additionally, the experimental results indicated that the distribution of coherence singularities was related to the coherence, which was consistent with the simulation results.

Fig. 6. Experimental results for the intensity patterns (a)–(i) and phase distribution of the CSD function with an off-axis perturbation point (j)–(r) for a PCBG vortex beam with different coherence widths.

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5. Conclusion

We developed a method for determining the TC magnitude and sign for a PCBG vortex beam by measuring its CSD phase with an off-axis reference point. The effects of the propagation distance and coherence on the coherence singularity distribution of PCBG vortex beam were investigated theoretically and experimentally. Our results indicated that there is no quantitative relationship between the TC and CSD amplitude with an on-axis reference point. However, the TC of the PCBG vortex beam can be recovered from the CSD phase distribution with an off-axis reference point. Its magnitude is equal to the number of coherence singularities, and its sign is negative and positive when the phase winding direction is clockwise and counterclockwise, respectively. Additionally, the distribution of coherence singularities is related to the selection of reference points, propagation distance, and degree of coherence. A longer propagation distance and a lower degree of coherence correspond to larger spacing between coherence singularities. This research provides guidance for the TC measurement of partially coherent vortex beams. Our finding may be useful in applications where the TC must be measured in free-space propagation, such as optical manipulation and optical communication.

Funding

Postgraduate Research & Practice Innovation Program of Jiangsu Province (KYCX22_3183); Key Lab of Modern Optical Technologies of Jiangsu Province (KJS2138); Local Science and Technology Development Project of the Central Government (YDZX20203700001766); Tang Scholar; Priority Academic Program Development of Jiangsu Higher Education Institutions; China Postdoctoral Science Foundation (2022M722325); National Natural Science Foundation of China (11974218, 12174280, 12192254, 12204340, 92250304); National Key Research and Development Program of China (2019YFA0705000, 2022YFA1404800).

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.

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Coherence singularity and evolution of partially coherent Bessel–Gaussian vortex beams

Abstract

1. Introduction

2. Theory

3. CSD measurement method

4. Experimental setup and results

5. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (6)

Equations (9)

Optics Express