Partially coherent fractional vortex beam

Jun Zeng; Xianlong Liu; Fei Wang; Chengliang Zhao; Yangjian Cai

doi:10.1364/OE.26.026830

1. Introduction

As pointed out by Allen in 1992, a vortex beam with helical phase front, described by a phase structure of $\exp (i l θ)$ , carries an orbital angular momentum (OAM) of $l ℏ$ per photon, where l is the topological charge (TC), $θ$ is the azimuthal angle, and $ℏ$ is Planck’s constant divided by 2π. At the center of a vortex beam, the intensity is zero and the phase is undermined. The TC of a vortex beam is a robust quantity upon propagation through atmospheric turbulence, which could be used as an information carrier in optical communications [1,2]. Due to above features, increasing interests have been paid to optical vortices embedded in Gaussian-like beams [3,4], nondiffracting beams [5,6] and perfect vortex beam [7] due to their important applications in quantum information processing [8], free-space optical communications [9], optical manipulation [10], super-resolution imaging [11] and optical measurements [12]. Yang et al. introduced a novel vortex beam named anomalous vortex beam [13], which has an extra beam order parameter, and the electric field of such beam evolves into that of an elegant Laguerre-Gaussian beam in the far field in free space. The evolution properties of an anomalous vortex beam passing through paraxial optical system and nonlinear media have been explored in [14] and [15], and anomalous vortex beam was used for optical trapping [16].

Above mentioned literatures are limited to vortex beam with integral TC, and in fact, the value of TC can be non-integral (i.e., fractional TC). The vortex beam with fractional TC (i.e., fractional vortex beam), which possesses a radial opening in the annular intensity ring, was first observed by Basisty [17]. Later, many studies on fractional vortex beams have been reported, including fractional plane-wave vortex beam, fractional Gaussian-like vortex beam and fractional Bessel beam [18–24]. To date, besides some conventional methods including non-integer forked diffraction gratings [22], spiral phase plate [25], and some other special methods, such as weighted superposition [26], optically uniaxial crystals [27] and so on, have been used for generating fractional vortex beams. Recently, by using Fermat’s spiral slit, an easy way was proposed to modulate the vortex beam’s topological charge to be an arbitrary value, both integral and fractional, within a continuous range [23]. In contrast with the integral vortex beam, the fractional vortex beam has unique advantages in practical applications. For example, using fractional OAM can increase the degree of entanglement of photons in quantum information processing [28]. According to its specific radial opening gap in the intensity distribution, the fractional vortex beam could be used for optical sorting [29] as well as guiding and transporting particles [30]. The fractional vortex beam can be applied to achieve anisotropic edge enhancement based on its asymmetric intensity profile [31]. The observed fractional charges in the experiment were found to follow the same OAM algebra that was earlier established for Laguerre–Gaussian beam with integral charge, which makes fractional vortex beam more advantageous in optical excitations [32]. Topological charge transfer in frequency doubling of fractional OAM state was explored in [33].

It is well known that one of the most important properties of a laser beam is its spatial high coherence, and we treat it as a fully coherent beam. However, after propagating through random phases, turbulence media and rotating scatter, the spatial coherence of a light beam decreases, and we call light beam with low spatial coherence as partially coherent beam [34,35]. Gaussian Schell-model (GSM) beam is a typical partially coherent beam, whose intensity and degree of coherence both satisfy Gaussian distribution. GSM beam has unique advantages over coherent beam in many applications. For example, GSM beam can effectively overcome speckle in laser nuclear fusion [36]. GSM beam can be used to reduce the bit error rate as well as to improve signal to noise ratio in free-space optical communication [37]. GSM beam can be applied to realize classic ghost interference [38]. Since Gori and collaborators introduced helical phase structure to partially coherent beam [39], considerable interests have been paid to partially coherent vortex (PCV) beam [40–50]. It was shown in [43] that one can determine the topological charge of a PCV beam from its cross-spectral density in the far field (or in the focal plane). PCV beam has shown advantages over coherent vortex beam in some applications, such as self-reconstruction, free-space optical communications and optical trapping [47–50]. However, it is worth noting that the TC of partially coherent vortex beam in all literatures is an integer. To our knowledge, no work has been reported on the PCV beam with fractional TC, namely partially coherent fractional vortex (PCFV) beam.

In this paper, we propose a vortex beam with both fractional TC and low coherence, namely partially coherent fractional vortex (PCFV) beam, as a natural extension of fractional vortex beam. The propagation properties of a PCFV beam focused by a thin lens have been studied and compared with those of the conventional partially coherent vortex beam with integral TC. The most interesting property of PCFV beam is that the opening gap of the intensity pattern and the rotation of the beam spot disappear gradually and the CSD distribution becomes more symmetric and more recognizable with decreasing coherence. We also report experimental generation of PCFV beam.

2. Partially coherent fractional vortex beam and its paraxial propagation

In this section, we will introduce the theoretical model for a PCFV beam, and derive its paraxial propagation formula. It is known that a partially coherent beam can be characterized by either the mutual intensity in the space-time domain or the cross-spectral density (CSD) function in the space-frequency domain [34]. The CSD function of the field in the source plane is defined as a two-point correlation function:

W_{0} (r_{1}, r_{2}) = 〈 E^{*} (r_{1}) E (r_{2}) 〉,

where E denotes the field fluctuating in a direction perpendicular to the z-axis,

r_{i} (i = 1, 2)

are the transverse position vectors in the source plane, the asterisk denotes the complex conjugate and the angular brackets denote an ensemble average.

To be a physically genuine CSD function, the CSD function should satisfy the condition of nonnegative definiteness and can be expressed in the following form [51,52]:

W_{0} (r_{1}, r_{2}) = \int p (v) H^{*} (r_{1}, v) H (r_{2}, v) d^{2} v,

where

p (v)

is a nonnegative function and

H

is an arbitrary kernel. By choosing suitable expressions of H and

p (v)

, various partially coherent beams with different spatial correlation functions have been introduced [44,53–58]. In this paper, we set:

p (v) = e x p (- 2 v^{2} / σ_{g}^{2}),

and

H (r, v) = {(\frac{\sqrt{2} r}{w_{0}})}^{l} \exp (- \frac{r^{2}}{w_{0}^{2}}) \exp (- i l φ) e x p (- i k r \cdot v),

then applying Eqs. (2)-(4), the CSD function is obtained as follows

W_{0} (r_{1}, r_{2}) = {(\frac{2 r_{1} r_{2}}{w_{0}^{2}})}^{l} \exp (- \frac{r_{1}^{2} + r_{2}^{2}}{w_{0}^{2}}) \exp [- i l (φ_{1} - φ_{2})] e x p (- \frac{{(r_{2} - r_{1})}^{2}}{2 σ_{g}^{2}}) .

Equation (5) denotes the CSD function of a typical partially coherent vortex beam-partially coherent Laguerre-Gaussian beam, l denotes the topological charge, which can be an arbitrary value, both integral and fractional,

k = 2 π / λ

is the wavenumber with

λ

being the wavelength,

σ_{g}

is the spatial coherence width,

w_{0}

is the beam width,

φ

is the azimuthal (angle) coordinates. By choosing a fractional value of l, Eq. (5) represents the CSD function of a PCFV beam.

Due to fractional value of l, it’s hard to derive analytical propagation formula for a PCFV beam, we have to resort to numerical integration. Paraxial propagation of a PCFV beam through a stigmatic ABCD optical system can be treated by the following Collins formula [42,59]:

\begin{array}{l} W (ρ_{1}, ρ_{2}) = \frac{k^{2}}{4 π^{2} B^{2}} \int \int W_{0} (r_{1}, r_{2}) \exp (- \frac{i k A}{2 B} r_{1}^{2} + \frac{i k}{B} r_{1} \cdot ρ_{1} - \frac{i k D}{2 B} ρ_{1}^{2}) \\ \times \exp (\frac{i k A}{2 B} r_{2}^{2} - \frac{i k}{B} r_{2} \cdot ρ_{2} + \frac{i k D}{2 B} ρ_{2}^{2}) d^{2} r_{1} d^{2} r_{2}, \end{array}

where

W_{0} (r_{1}, r_{2})

and

W (ρ_{1}, ρ_{2})

are the CSD functions of the PCFV beam in the source plane and output plane, respectively. A, B and D are the transfer matrix elements of the ABCD optical system.

In order to evaluate Eq. (6), we introduce the following “sum” and “difference” coordinates, i.e.:

r_{s} = (r_{2} + r_{1}) / 2, r_{d} = r_{2} - r_{1},

ρ_{s} = (ρ_{2} + ρ_{1}) / 2, ρ_{d} = ρ_{2} - ρ_{1} .

On substituting from Eq. (5) and Eqs. (7)-(8) into Eq. (6), after some rearrangement, we obtain:

\begin{array}{l} W (ρ_{1}, ρ_{2}) = \frac{k^{2}}{4 π^{2} B^{2}} \exp (\frac{i k D ρ_{s} \cdot ρ_{d}}{B}) \int \int \int \int A^{*} (r_{s} - r_{d} / 2) A (r_{s} + r_{d} / 2) p (v) \\ \times \exp (- \frac{i k}{B} (r_{s} \cdot ρ_{d} + r_{d} \cdot ρ_{s})) \exp (- i k r_{d} \cdot v) d^{2} r_{s} d^{2} r_{d} d^{2} v, \end{array}

where

A (r)

is defined as:

A (r) = {(\frac{\sqrt{2} r}{w_{0}})}^{l} \exp (- \frac{r^{2}}{w_{0}^{2}}) \exp (- i l φ) \exp (i k A r^{2} / 2 B) .

Then, $A^{*} (r_{s} - r_{d} / 2)$ and $A (r_{s} + r_{d} / 2)$ can be expressed in terms of their Fourier transforms, i.e.,

A^{*} (r_{s} - r_{d} / 2) = {(\frac{k}{2 π})}^{2} \int {\tilde{A}}^{*} (u) \exp (- i k u \cdot (r_{s} - r_{d} / 2)) d^{2} u,

A (r_{s} + r_{d} / 2) = {(\frac{k}{2 π})}^{2} \int \tilde{A} (u) \exp (i k u \cdot (r_{s} + r_{d} / 2)) d^{2} u .

On substituting from Eqs. (11)-(12) into Eq. (9), and after some mathematical operations and integrations over $r_{s}$ , $r_{d}$ and $u$ , Eq. (9) becomes

\begin{array}{l} W (ρ_{1}, ρ_{2}) = \frac{k^{2}}{4 π^{2} B^{2}} \exp (\frac{i k D ρ_{s} \cdot ρ_{d}}{B}) \int \int {\tilde{A}}^{*} (v +(2 ρ_{s} - ρ_{d}) / 2 B) \\ \times \tilde{A} (v +(2 ρ_{s} + ρ_{d}) / 2 B) p (v) d^{2} v . \end{array}

2.1 Average intensity of partially coherent fractional vortex beam

First, we focus on the average intensity of a PCFV beam passing through a stigmatic ABCD optical system. By setting $ρ_{1} = ρ_{2} = ρ$ in Eq. (13), the average intensity of a PCFV beam on propagation is obtained as

I (ρ) = W (ρ, ρ) = \frac{k^{2}}{4 π^{2} B^{2}} \int \int {| \tilde{A} (v + ρ / B) |}^{2} p (v) d^{2} v .

In the next step, we rewrite Eq. (14) in terms of convolution [60]

I = f (- ρ / B) \otimes p (- ρ / B),

where “

\otimes

” denotes the convolution,

f (u) = k^{2} {| \tilde{A} (- u) |}^{2} / 4 π^{2} B^{2}

with

u = - ρ / B

,

\tilde{A}

denotes the Fourier-transform operation of A and can be written in the following form

\tilde{A} (v) = \int A (r) \exp (- i k v \cdot r) d^{2} r .

Note that MATLAB can perform the following fast Fourier transform

\tilde{F} (ε) = \int F (x) \exp (- i 2 π ε x) d x .

Comparing Eqs. (16) and (17), the spatial frequency “ $ε$ ” equals to “ $v / λ$ ” where $v$ is a dimensionless parameter. Using this new parameter, Eq. (14) becomes

I (ρ) = λ^{2} \frac{k^{2}}{4 π^{2} B^{2}} \int \int {| \tilde{A} (\frac{v}{λ} + \frac{ρ}{λ B}) |}^{2} p (\frac{v}{λ}) d^{2} \frac{v}{λ} .

2.2 Cross-spectral density of a partially coherent fractional vortex beam

We will consider the CSD of a PCFV beam passing through the ABCD optical system at two specified points $ρ_{1}$ and $ρ_{2}$ . If $ρ_{1} = ρ$ and $ρ_{2} =0$ , Eq. (13) turns out to be

W (ρ, 0) = \frac{k^{2}}{4 π^{2} B^{2}} \exp (\frac{i k D ρ^{2}}{2 B}) \int \int {\tilde{A}}^{*} (v) p (v) \tilde{A} (v + ρ / B) d^{2} v .

Following above discussions, we could introduce a function, i.e.:

F_{A p} (v) = {\tilde{A}}^{*} (v) p (v) .

Then the CSD of the PCFV beam can be written in the following alternative form

W (ρ, 0) = \frac{1}{B^{2}} \exp [i π λ B D {(\frac{ρ}{λ B})}^{2}] \int \int F_{A P} (\frac{v}{λ}) \tilde{A} (\frac{v}{λ} + \frac{ρ}{λ B}) d^{2} \frac{v}{λ} .

When

σ_{g} \to \infty

, Eqs. (18) and (21) reduce to the average intensity and CSD of a fully coherent fractional vortex (FCFV) beam, respectively. When l equals to an integral value, Eqs. (18) and (21) reduce to the average intensity and CSD of the conventional partially coherent vortex beam with integral TC (i.e., PCIV beam), respectively.

3. Propagation properties of a partially coherent fractional vortex beam focused by a thin lens

In this section, applying Eqs. (18) and (21), we explore the propagation properties (e.g., average intensity and CSD) of a PCFV beam focused by a thin lens numerically.

We assume that the PCFV beam is focused by a thin lens with focal length f located in the source plane (z = 0). The distance between the source plane and the receiver plane is z. Then the transfer matrix between the source plane and the receiver plane reads as

(\begin{matrix} A & B \\ C & D \end{matrix}) = (\begin{matrix} 1 & z \\ 0 & 1 \end{matrix}) (\begin{matrix} 1 & 0 \\ - 1 / f & 1 \end{matrix}) = (\begin{matrix} 1 - z / f & z \\ - 1 / f & 1 \end{matrix}) .

Applying Eqs. (18) and (22), we calculate in Figs. 1 and 2 the normalized intensity distributions $I (ρ) / I_{\max} (ρ)$ of a PCIV beam ( $l = 1$ ) and a PCFV beam ( $l = 1.5$ ) focused by a thin lens at several propagation distances z for different values of the spatial coherence width $σ_{g}$ , respectively. The parameters for calculations are set as $λ = 532 n m$ , $w_{0} = 1.2 m m$ , $f = 400 m m$ . We infer from Figs. 1-2 that the intensity distribution of the PCIV beam ( $l = 1$ ) displays a full-doughnut-shaped pattern near the thin lens, and the intensity profile maintains circular symmetry with the increase of z when $σ_{g}$ is large (i.e., $σ_{g} = 3 m m$ ). With the decrease of $σ_{g}$ , the hollow profile of the focused beam spot disappears gradually, and finally becomes a Gaussian profile ( $z = f$ , $σ_{g} = 0.1 m m$ ), this phenomenon is consistent with those reported in [43]. Figure 2 shows that the intensity pattern of a PCFV beam is much different from that of a PCIV beam. The intensity pattern of a PCFV beam possesses a radial opening in the annular ring encompassing the dark core near the source plane ( $z = 0.1 f$ ), and the opening gap rotates clockwise as z increases (or rotates anticlockwise for negative l, which is not shown here), and up to 90 degree at the focal plane ( $z = f$ ). The opening gap profile at the focal plane disappears gradually with the decrease of $σ_{g}$ , and finally also becomes a Gaussian profile ( $σ_{g} = 0.1 m m$ ). Thus, it is hard to distinguish the intensity distributions of a focused PCIV beam ( $l = 1$ ) and a focused PCFV beam ( $l = 1.5$ ) at the focal plane when $σ_{g}$ is small.

Fig. 1 Normalized intensity distribution $I (ρ) / I_{\max} (ρ)$ of a PCIV beam ( $l = 1$ ) focused by a thin lens at several propagation distances z for different values of $σ_{g}$ .

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Fig. 2 Normalized intensity distribution $I (ρ) / I_{\max} (ρ)$ of a PCFV beam ( $l = 1.5$ ) focused by a thin lens at several propagation distances z for different values of $σ_{g}$ .

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To learn more about the influences of spatial coherence width $σ_{g}$ and topological charge on the focused intensity distribution, we calculate in Fig. 3 the normalized intensity distribution $I (ρ) / I_{\max} (ρ)$ of a PCV beam with different l focused by a thin lens at focal plane ( $z = f$ ) for different values of $σ_{g}$ , and the other calculation parameters are the same as in Figs. 1-2 . One finds that when $σ_{g}$ is large ( $σ_{g} = 3 m m$ ), the topological charge l plays a decisive role in determining the focused intensity, namely, as l varies, the focused intensity profile of the PCV beam varies and displays unique distribution. In particular, the intensity profile of the PCV beam with opposite topological charge will be reversed (see $l = 1 .2$ and $l = - 1 .2$ ). With the decrease of $σ_{g}$ , the difference between the focused intensity distributions of the PCV beams with different topological charge l disappears gradually, and finally all intensity profiles becomes Gaussian profiles when $σ_{g}$ is small ( $σ_{g} = 0.1 m m$ ). Thus, the method for discriminating the value of topological charge l of a PCV beam from its intensity profile will be invalid when its spatial coherence width is small [61].

Fig. 3 Normalized intensity distribution $I (ρ) / I_{\max} (ρ)$ of a PCV beam with different l (both integral and fractional) focused by a thin lens at focal plane ( $z = f$ ) for different values of $σ_{g}$ .

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Here, we propose a method to discriminate the PCFV beams with different topological charges based on the CSD. With the help of Eq. (21), we calculate in Figs. 4-5 the density plot of the modulus of the CSD $| W (ρ, 0) |$ distribution of a PCIV beam ( $l = 1$ ) and a PCFV beam ( $l = 1.5$ ) focused by the thin lens at several propagation distances z for different values of the spatial coherence width $σ_{g}$ and topological charge l, and the other parameters are the same as in Figs. 1-2. One see from Fig. 4 that the distribution of $| W (ρ, 0) |$ of a focused PCIV beam with high coherence ( $σ_{g} = 3 m m$ ) transforms gradually from a full-doughnut-shaped profile ( $z = 0.1 f$ ) to a solid profile with concentric bright rings around at the focal plane (z = f). Furthermore, decreasing the value of $σ_{g}$ will accelerate the speed of this transformation, and the dark rings (i.e., ring dislocations) more obvious. The absolute value of l of a PCIV beam is equal to the number of the dark rings at the focal plane [43]. Similar to the evolution process of intensity distribution in Fig. (2), we find from Fig. 5 that the distribution of $| W (ρ, 0) |$ of a focused PCFV beam with high coherence ( $σ_{g} = 3 m m$ ) also possess an intrinsic opening gap near the source plane ( $z = 0.1 f$ ), which will rotate up to 90 degrees at the focal plane ( $z = f$ ), and the rotation direction is consistent with Fig. 2. In addition, the distribution of $| W (ρ, 0) |$ at the focal plane displays transverse bilateral symmetry. With the decrease of $σ_{g}$ , as well as the increase of $z$ , the distribution of $| W (ρ, 0) |$ gradually becomes more symmetric and characteristic, in particular, the symmetry at the focal plane evolves from transverse bilateral symmetry for the case of $σ_{g} = 3 m m$ to rectangular symmetry for the case of $σ_{g} = 0.1 m m$ .

Fig. 4 Density plot of the modulus of the CSD $| W (ρ, 0) |$ of a PCIV beam ( $l = 1$ ) focused by a thin lens at several propagation distances z for different values of $σ_{g}$ .

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Fig. 5 Density plot of the modulus of the CSD $| W (ρ, 0) |$ of a PCFV beam ( $l = 1.5$ ) focused by a thin lens at several propagation distances z for different values of $σ_{g}$ .

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To learn more about the dependence of the CSD distribution on the topological charge l, we calculate in Fig. 6 the density plot of the modulus of the CSD $| W (ρ, 0) |$ of a PCV beam with different values of l focused by the lens at focal plane ( $z = f$ ) for different values of the spatial coherence width $σ_{g}$ . The other parameters are the same as in Figs. 1-2. It is interesting to find from Fig. (6) that whether high spatial coherence width or low spatial coherence width, the PCV beams with different l have their own unique $| W (ρ, 0) |$ distributions (i.e., the value of l of the PCV beam can be characterized by the distribution of $| W (ρ, 0) |$ ). Moreover, with the decrease of $σ_{g}$ , the distribution of $| W (ρ, 0) |$ becomes more symmetric and more exquisite. In particular, the distribution of $| W (ρ, 0) |$ of the focused PCFV beam with half-integral topological charge (e.g. $l = 0.5$ and $l = 1.5$ ) evolves from bilateral symmetry to rectangular symmetry. Through comparison between Figs. 1-6, we find that just by measuring the focused intensity distribution, it is difficult to discriminate the value of l (e.g., $l = 1$ and $l = 1.5$ ) when spatial coherence width is low (e.g., $σ_{g} = 0.1 m m$ ), but we can solve this problem by measuring the distribution of the modulus of the CSD $| W (ρ, 0) |$ . However, the analysis of this paper is based on qualitative analysis, and the next step is to quantify the corresponding relation between the value of l of PCFV beam and the distribution of $| W (ρ, 0) |$ . The determination of fractional topological charge are exploited to guide and transport microscopic particles [30].

Fig. 6 Density plot of the modulus of the CSD $| W (ρ, 0) |$ of a PCV beam with different l (both integral and fractional) focused bya lens at focal plane ( $z = f$ ) for different values of $σ_{g}$ .

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4. Experimental generation of a partially coherent fractional vortex beam

In this section, to verify the theoretical prediction, now we carry out our experimental generation of a PCFV beam and measure its focusing properties.

Figure 7 shows our experimental setup for generating PCFV beam and measuring its focused intensity distribution and the modulus of the CSD. After passing through a beam expander, the laser beam (_{$λ = 532 n m$}) focused by the thin lens L₁ (f₁ = 15cm) illuminates a rotating ground-glass disk (RGGD), producing a partially coherent beam with Gaussian statistics. Here the speed of the RGGD is controlled by a motion controller. After passing through the collimation thin lens L₂ (f₂ = 25cm) and the Gaussian amplitude filter (GAF), the transmitted beam becomes a Gaussian Schell-model beam [34,59], whose intensity and degree of coherence satisfy Gaussian distributions, and then is reflected by a beam splitter (BS). The reflected beam goes toward a spatial light modulator (SLM), which acts as a phase grating designed by the method of computer-generated holograms (CGH), and the generated beam becomes a PCFV beam. The value of topological charge l of the generated PCFV beam is determined by the phase grating. The spatial coherence width of the generated PCFV beam is controlled by the focused beam spot on the RGGD and the roughness of the RGGD together. In our experiment, the roughness of the RGGD is fixed, so we mainly modulate the spatial coherence width by varying the focused beam spot size on the RGGD. The spatial coherence width is approximated as $λ f_{2} / (π ω_{0})$ (here $ω_{0}$ depends on focused beam spot size). Large focused beam spot corresponds to low coherence. We have chosen three different beam spots on the RGGD to generate the PCFV beam with three different spatial coherence widths ( $σ_{g} = 3 mm, 0 .7mm, 0 .1mm$ ). The detailed measurement method of spatial coherence width can be found in [62]. The generated PCFV beam passes through a thin lens L₃ (just behind SLM) with the focal length $f = 400 m m$ , and then arrives at a charge coupled device (CCD). Because the lens L₃ is placed close to the SLM, here we can approximately treat the lens L₃ plane as the source plane (z = 0), the distance between the thin lens L₃ and the CCD is z. The CCD is be used to measure not only the intensity distribution but also the modulus of the CSD distribution. The principle and detailed procedure for measuring $| W (ρ, 0) |$ distribution can be found in [53,63,64].

Fig. 7 Experimental setup for generating PCFV beam, measuring the focused intensity distribution and the modulus of the CSD distribution. Laser, Nd: YAG laser; BE, beam expander; L₁, L₂ and L₃, thin lenses; RGGD, rotating ground-glass disk; GAF, Gaussian amplitude filter; BS, beam splitter; SLM, spatial light modulator; CGH, computer-generated holograms; CCD, charge-coupled device.

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In our experiment, the beam width of the generated PCV beam is fixed with $w_{0} = 1.2 mm$ , and the spatial coherence width are $σ_{g} = 3 mm$ , $σ_{g} = 0.7 mm$ and $σ_{g} =0 .1mm$ , respectively. We can obtain $σ_{g}$ precisely according to the method proposed in [62]. Figure 8 shows our experimental results of the normalized intensity distribution of the generated PCFV beam focused by a thin lens L₃ with focal length $f = 400 m m$ at several propagation distances z for different values of $σ_{g}$ . Our experimental results clearly demonstrate the clockwise rotation of the opening gap of PCFV beam with $l = 1.5$ upon propagation when the $σ_{g}$ is large ( $σ_{g} = 3 mm$ ), and the rotation angle of the opening gap becomes 90 degrees at the focal plane ( $z = f$ ). When the $σ_{g}$ is small ( $σ_{g} = 0.1 m m$ ), the opening gap disappears gradually and the intensity distribution becomes a Gaussian profile, which agrees well with the numerical results in Fig. 2. Figures 9 and 10 show our experimental results of the normalized intensity distribution and the modulus of the CSD $| W (ρ, 0) |$ distribution of the generated PCV beam with different l (both integral and fractional) focused by a thin lens with focal length $f = 400 m m$ at the focal plane ( $z = f$ ) for different values of $σ_{g}$ . One infers from the Fig. 9-10 that the PCV beams with different l have different focused intensity distributions as well as different distributions of the modulus of the CSD $| W (ρ, 0) |$ , namely, it indicates that we can distinguish the value of l by measuring not only the focused intensity distribution but also the distribution of $| W (ρ, 0) |$ when $σ_{g}$ is large ( $σ_{g} = 3 mm$ ). However, with the decrease of $σ_{g}$ , the focused intensity distribution gradually becomes the same Gaussian profile for all values of l, in other word, the method for determining the value of l of the PCFV beam from the focused intensity distribution will be invalid when $σ_{g}$ is small. It is exciting to find that even with small $σ_{g}$ , the distributions of $| W (ρ, 0) |$ of the PCFV beams with different l have their own unique distribution characteristics, and can still be used to characterize the value of l. Furthermore, decreasing $σ_{g}$ makes the modulus of the CSD $| W (ρ, 0) |$ distribution more symmetric and more exquisite, and the $| W (ρ, 0) |$ distribution of PCFV beam with half-integral topological charge ( $l = 0.5$ and $l = 1.5$ ) evolves from bilateral symmetry to central symmetry with the decrease of $σ_{g}$ , thus, one can easily determinate the half-integral value of l by varying (i.e., decreasing) $σ_{g}$ . Our experimental results are consistent with our theoretical predictions.

Fig. 8 Experimental results of the normalized intensity distribution of the generated PCFV beam ( $l = 1.5$ ) focused by a thin lens with focal length f = 400mm at several propagation distances z for different values of $σ_{g}$ .

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Fig. 9 Experimental results of the normalized intensity distribution of the generated PCV beam with different l focused by a thin lens with focal length f = 400mm at focal plane (z = f) for different values of $σ_{g}$ .

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Fig. 10 Experimental results of the modulus of the CSD $| W (ρ, 0) |$ distribution of the generated PCV beam with different l (both integral and fractional) focused by a thin lens with focal length f = 400mm at focal plane (z = f) for different values of $σ_{g}$ .

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5. Summary

We have introduced PCFV beam as a natural extension of coherent fractional vortex beam, and we have carried out theoretical and experimental studies of the effect of both spatial coherence width and the topological charge on the propagation properties (e.g., average intensity and CSD) of a focused PCFV beam. In contrast with the integral vortex beam, which displays a full-doughnut-shaped intensity pattern, the PCFV beam exhibits intensity distribution that possesses a radial opening in the annular intensity ring encompassing the dark core, and the opening gap rotates gradually up to 90 degrees as propagation distance increases. In contrast with full coherent fractional vortex beam, the PCFV beam at the focal plane exhibits more symmetric and exquisite CSD distribution depending on the magnitude of spatial coherence width. In addition, we have found that the method for determining the value of topological charge l based on focused intensity distribution becomes invalid when its spatial coherence width is small, but one can discriminate the value of l based on the exquisite CSD distribution with low spatial coherence width. Our experimental results verify our theoretical predictions. Consequently, manipulating spatial coherence properties and topological charge of a PCFV beam paves a way for manipulating its propagation properties and for beam shaping. Our results will be useful for guiding and transporting microscopic particles and for information transfer.

Funding

National Natural Science Foundation of China (91750201, 11525418, 11774251, 11474213, 11804198, 11874046); Project of the Priority Academic Program Development (PAPD) of Jiangsu Higher Education Institutions; Qing Lan Project of Jiangsu Province.

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Partially coherent fractional vortex beam

Abstract

1. Introduction

2. Partially coherent fractional vortex beam and its paraxial propagation

2.1 Average intensity of partially coherent fractional vortex beam

2.2 Cross-spectral density of a partially coherent fractional vortex beam

3. Propagation properties of a partially coherent fractional vortex beam focused by a thin lens

4. Experimental generation of a partially coherent fractional vortex beam

5. Summary

Funding

References

Cited By

Figures (10)

Equations (22)

Optics Express