1 Introduction

The beams with autofocusing properties, such as circular Airy beams, Pearcey beams, circle Pearcey beams, aberration laser beams, radial sublinear and superlinear chirp beams [1,2,3,4,5,6], have gained tremendous interest. These abruptly autofocusing (AAF) fields can maintain a low-intensity profile during propagation until the focal point caused by the radial caustic collapse. The first AAF beams involve circular Airy beams proposed in 2010 [1] and experimentally demonstrated in 2011 [7, 8], have proved advantageous in medicine [9], optical manipulations [10,11,12], laser processing [7], and the generation of light bullets [13]. In a similar way, the combination of vortex beams [14] with circular Airy beams, commonly known as circular Airy vortex (CAV) beam, has also received a similar attention, for example, to drive micro-particles around the primary ring at the focal plane [15]. CAV beams are being also exploited in applications such as image edge enhancement, optical spanners and optical communications [16,17,18]. Previous studies, which have only considered the generation of single CAV beams and circular Airy Gaussian vortex (CAGV) beams, have demonstrated that the CAGV beam ring's diameter in the focal plane depends on the order of the topological charge (TC) of the vortex [19, 20]. In addition, their initial launch angle, which is directly associated with the ballistic propagation trajectory, is also an important parameter [21]. Crucially, despite many studies reported along this line, to date, the simultaneous generation of multiple CAGV beams still represents a real challenge. Nonetheless, their successful generation will pave the path in various research fields. Even though the multiplexing of vortex beams with the same or different TCs can be realized using multi-order diffraction phase elements (DPEs) [22,23,24,25,26,27,28], the simultaneous generation of multiple CAGV beams with independent properties has not been implemented with these techniques.

In this paper, we put forward a method for the spatial multiplexing of CAGV beams, each with a unique topological charge and tunable launch angle. The CAV beams are special cases of the CAGV beams. Our technique comprises the use of conjugate symmetric extension (CSE) computer-generated holography (CGH), which provides a resolution limited only by the generating device, a spatial light modulator (SLM) in our case. Different from multi-order DPEs based on the superposition of interference patterns, this CGH algorithm has a higher efficiency [29]. The simultaneous multiplexing of different CAGV beams with independent properties provides great flexibility in various areas of research, for example, it enables the simultaneous manipulation of multiple microparticles in 3-Dimensions.

2 Theoretical analysis

Using the Cartesian coordinates (x, y), the electrical field of CAGV beams in the initial plane can be written as,

$$u(x,y) = A_{0} \exp \left[ {\frac{a}{bw}\left( {r_{0} - r} \right)} \right]Ai\left( {\frac{{r_{0} - r}}{bw}} \right)\exp \left[ {in \, atan\left( {\frac{{y - y_{d} }}{{x - x_{d} }}} \right)} \right]\exp \left[ { - \frac{{\left( {r_{0} - r} \right)^{2} }}{{w^{2} }}} \right]r^{n} \exp \left( {iv\frac{{r_{0} - r}}{bw}} \right)$$
(1)

where A0 is a normalization constant, Ai (x) is the Airy function, a is an exponential truncation factor that ensures a finite energy and b is a distribution factor parameter. Further, w is the radius of the beam waist, r0 represents the radius of the CAGV beam in the initial plane, \(r = \sqrt {\left( {x - x_{d} } \right)^{2} + \left( {y - y_{d} } \right)^{2} }\), where, xd and yd are the displacement relative to the beam's axis, n is the TC of the vortex, and v is a parameter associated with the initial launch angle of the parabolic trajectory, that is, \(\beta { = }{\nu \mathord{\left/ {\vphantom {\nu {(kx_{0} )}}} \right. \kern-\nulldelimiterspace} {(kx_{0} )}}\), where \(\beta\) is the initial launch angle, k is the wavenumber of the optical wave, and x0 is an arbitrary transverse scale [30]. In this way, for v > 0 the beam propagates following an upward parabolic trajectory whereas for v < 0 it propagates following a downward parabolic trajectory.

To multiplex several CAGV beams, we make use of the superposition principle which allows the addition of many beams, each with independent properties. As such, the overall optical field wave function u0(x0, y0) can be given by,

$$\begin{gathered} u_{0} (x_{0} ,y_{0} ) = \sum\limits_{j = 1}^{k} {A_{0} \exp [\frac{a}{bw}(r_{0} - r_{j} )]Ai\left[ {\frac{{r_{0} - r_{j} }}{bw}} \right]\exp \left[ { - \frac{{\left( {r_{0} - r_{j} } \right)^{2} }}{{w^{2} }}} \right]} \\ \times \exp \left\{ {in_{j} a\tan \left[ {\left( {y - y_{dj} } \right)/\left( {x - x_{dj} } \right)} \right]} \right\}r_{j}^{{n_{j} }} \exp \left[ {iv_{j} \frac{{\left( {r_{0} - r_{j} } \right)}}{bw}} \right] \\ \end{gathered}$$
(2)

To demonstrate our technique and without the loss of generality, we will consider two specific cases of five CAGV beams generated by combinations of nj ∈ [1, 5] and v = 0, as well as combinations of vj ∈ [− 1, − 0.5, 0, 0.5, 1] and n = 2, arranged along a circular ring. Here, the center location (xdj, ydj) of each CAGV beam, in the cylindrical coordinates (R, θ) are defined as (\(R\mathrm{cos}{\theta }_{j}, R\mathrm{sin}{\theta }_{j}\)) for θj = (2j-1) × 2π/5, where j = 1, 2, 3, 4, 5. R is the distance from the optical axis to the center of each CAGV beam, in our case, R = 0.5 mm. For the sake of clarity, the intensity profiles of such two arrays in the plane z = 0 are shown in Fig. 1a, b, respectively, along with their phase profile shown as an inset. Notice that the intensity profile of all CAGV beams at this plane is almost identical, even though their topological charge or initial launch angle is different.

To experimentally generate the multiplexed set of CAGV beams described above, we employ the technique of CGH based on CSE. To this end, the optical wave function described in Eq. (2) can be extended by symmetric conjugation into the expression [29]

$$U_{0} (m_{0} ,n_{0} ) = \left\{ {\begin{array}{*{20}c} {u_{0} (m_{0} ,n_{0} )} \\ {u_{0}^{*} (M - m_{0} ,N - n_{0} )} \\ 0 \\ \end{array} } \right.\begin{array}{*{20}c} {(m_{0} = 1,2,...M/2 - 1;n_{0} = 1,2,...,N - 1)} \\ {(m_{0} = M/2 + 1,...,M - 1,n_{0} = 1,2,...N - 1)} \\ {m_{0} = 0,{\kern 1pt} {\kern 1pt} {\kern 1pt} or{\kern 1pt} {\kern 1pt} n_{0} = 0,{\kern 1pt} {\kern 1pt} {\kern 1pt} or{\kern 1pt} {\kern 1pt} m_{0} = M/2} \\ \end{array}$$
(3)

This function then Fourier transformed to generate a real-valued distribution containing both the amplitude and the phase information of the light wave. The obtained real distribution is coded to give a gray-scale hologram, which can be used to reconstruct the original light wave.

The discrete Fourier transform of U0 (m0, n0) can be given by,

$$F(\eta ,\zeta ) = \frac{1}{MN}\sum\limits_{{m_{0} = 0}}^{M - 1} {\sum\limits_{{n_{0} = 0}}^{N - 1} {U_{0} (m_{0} ,n_{0} )} } \times \exp \left[ { - i2\pi (\frac{{m_{0} \eta }}{M} + \frac{{n_{0} \zeta }}{N})} \right]$$
(4)

The complex object light can be expressed in the form U0(m0, n0) = A(m0, n0)exp[iφ(m0, n0)] , where A(m0, n0) and φ(m0, n0) represent the amplitude and the phase, respectively. We can obtain the multiplexed hologram to generate multiple CAGV beams, namely,

$$F(\eta ,\zeta ) = \frac{2}{MN}\sum\limits_{{m_{0} = 1}}^{M/2 - 1} {\sum\limits_{{n_{0} = 1}}^{N - 1} {A(m_{0} ,n_{0} )} } \cos [2\pi (\frac{{m_{0} \eta }}{M} + \frac{{n_{0} \zeta }}{N}) - \varphi (m_{0} ,n_{0} )],$$
(5)

where η = 0, 1, …, M-1 and ζ = 0, 1, …, N-1.

To produce an 8-bit gray-scale hologram, the real-valued distribution F(η, ζ) is linearly mapped to the range [0, 255]. Experimentally, the inverse Fourier transform, performed by a lens located a distance f (where f is the focal length of the lens) from the SLM, allows the generation of the desired light field. Figure 1c, d show examples of the hologram displayed on the SLM for two cases of five CAGV beams. It is worth mentioning that even though the number of generated CAGV beams is not theoretically limited, experimentally this is limited by the resolution of the SLM since an increase in the number of beams leads to a degradation of the quality of the modes [31].

Fig. 1
figure 1

Examples of two cases of five CAGV beams arranged along a circular ring of radius R. a and b correspond to intensity profiles of the five CAGV beams given by combinations of nj ∈ [1, 5] and v = 0, as well as vj ∈ [− 1, − 0.5, 0, 0.5, 1] and n = 2, whose phase distribution is shown on the insets of each beam. c and d are the corresponding digital holograms required to generate the beams shown in (a) and (b), respectively. For these beams, we used the parameters a = 0.1, b = 0.2 and w = 0.1 mm

Full size image

In order to investigate the propagation characteristics of the CAGV beam array in free space, we further employ the Fresnel diffraction integral, which allows to obtain the field U1(x1, y1, z) at arbitrary positions as function of the initial electric field u0(x0, y0), that is,

$$U_{1} (x_{1} ,y_{1} ,z) = \frac{{e^{ikz} }}{\lambda z}\int_{ - \infty }^{\infty } {\int_{ - \infty }^{\infty } {u_{0} (x_{0} ,y_{0} )} } \exp \left\{ {\frac{ik}{{2z}}\left[ {(x_{0} - x_{1} )^{2} + (y_{0} - y_{1} )^{2} } \right]} \right\}dx_{0} dy_{0} .$$
(6)

3 Numerical simulation and experimental results

Figure 2 shows the experimental setup to generate CAGV beams. Here, a linearly polarized light beam from a He–Ne laser (\(\lambda =\) 632.8 nm) illuminates an SLM (Holoeye PLUTO VIS, 1920X1080 pixels, 8 \(\mu\)m pixel size), where the required hologram is displayed. The input power is adjusted through the combination of a half-wave plate (HWP) and a polarization beam splitter (PBS). A biconvex lens of focal length f = 150 mm (25.4 mm in diameter) placed at a distance f from the SLM is employed to perform the Fourier transform required to produce the encoded beams, which are recorded at a distance f from the lens with a Charge-Couple Device (CCD) camera (Thorlab, BC106N-VIS, 1360X1024 pixels, 4.5 \(\mu\)m pixel size).

Fig. 2
figure 2

Schematic representation of our experimental setup. HWP, half-wave plate; PBS, polarization beam splitter; SLM, spatial light modulator; M, mirror; CCD, charge-coupled device camera; the focal length of the lens is 150 mm; the initial plane (z = 0 mm) is located in the back focal plane of the lens. The intensity distribution can be observed when moving CCD along z axis

Full size image

As a first example, we consider the set of CAGV beams given by combinations of nj ∈ [1, 5] and v = 0. We generate such beams using the digital hologram shown in Fig. 1c and investigate their propagation properties. Figure 3 shows the theoretical and experimental intensity profiles of such beams at distances of z = 0 mm (Fig. 3a1, b1, c1, 20 mm (Fig. 3a2, b2, c2), 30 mm (Fig. 3a3, b3, c3 and 40 mm (Fig. 3a4, b4, c4). In the simulation, Eq. (2) and Eq. (6) as well as the parameters a = 0.1, b = 0.2 and w = 0.1 mm are used. As can be seen, the intensity distributions of all CAGV beams at the generation plane is almost the same. Note that, the size of each CAGV beam can be decreased by employing a lens with a smaller focal length. Upon propagations, the CAGV beams experience an autofocusing at a distance of about 30 mm (Fig. 3a3, b3, c3), where the power of the CAGV beam rings is concentrated in a small area and the maximum intensity near the center increases sharply. The lateral acceleration of the CAGV beams are attained and energy rushes in an accelerated fashion towards the focus. However, the center intensity of the beams remains zero because of the vortex embedded in these CAGV beams. Importantly, as can be seen, the beam carrying larger values of n possesses a larger hollow radius. After the focal plane, the maximum intensity begins to reduce, and the center radius of CAGV beams increases in proportion to n. The interference fringes observed between adjacent CAGV beams are produced by interference between higher-order rings. These interference effects can be removed, for example by increasing the radius R of the ring or by adding a digital spatial filter to each beam. Notice that our experimental and numerical results agree well each other.

Fig. 3
figure 3

Transverse intensity distribution of five CAGV beams at the propagation planes z = 0, 20, 30, and 40 mm. a and b correspond to numerical simulations, 3D and 2D views, respectively, while c corresponds to experimental results

Full size image

For the sake of completeness, we also simulate the dependence of the peak intensity of CAGV beams mentioned above on the propagation distance z, as shown in Fig. 4. Here, the increase of the maximum intensity during propagation can be defined as the ratio of Imax to I0, where Imax and I0 are the maximum intensity at arbitrary propagation distance and the initial plane, respectively. We can see that these five CAGV beams autofocus, reaching a maximum Imax/I0, after a propagation distance of about z = 30 mm. This indicates that our technique enables simultaneous and independent manipulation at the same plane by multiplexing different modes of CAGV beams. We can see from Fig. 4 that the experimental results for z = 0, 20, 30, and 40 mm agree well with our simulation.

Fig. 4
figure 4

Imax/I0 as a function of propagation distance z. Solid lines correspond to simulations while data points to experiment. For these beams we used the parameters a = 0.1, b = 0.2, w = 0.1 mm, v = 0

Full size image

As a second example, we generate five CAGV beams with the same TC (n = 2) but different initial launch angles (v = − 1, − 0.5, 0, 0.5, 1) and study their propagation dynamics. Figure 5 illustrates the theoretical and experimental intensity profiles of the beams described above. The parameters a, b, and w are the same as those of the first example. Figure 5a1, 5b1 and 5c1 show the intensity distribution of the five CAGV beams in the initial plane z = 0. As shown in Fig. 5a2–5a4, 5b2–5b4 and Fig. 5c2–5c4, the autofocusing of these CAVG beams with different launch angles is reached at different propagation distances. It can be noted that the focal length decreases monotonically as v increases. The beam with positive v parameter has the strongest autofocusing property. On the contrary, when the v parameter is negative, the autofocusing property becomes weak and even disappear. This technique offers an effective method to independently modulate the focal property of each CAGV beam in the array, i. e., one could control the focal position and the focal intensity of five CAGV beams at the same time by varying the initial launch angle.

Fig. 5
figure 5

The intensity pattern of the CAGV beam array with different initial launch angle (v = − 1, − 0.5, 0, 0.5, 1) as z increases. a1–a4 simulated results of z = 0, 20, 25 and 30 mm; b1–b4 corresponding top view of row a, c1–c4 corresponding experimental results

Full size image

Again, we study abruptly autofocusing property of such CAGV beam array, i. e., the dependence of Imax/I0 on the propagation distance z for v =  + 1, + 0.5, 0, − 0.5, − 1. As shown in Fig. 6, the CAGV beams exhibit a maximum peak intensity at autofocusing positions. As expected, larger values of v can feature a faster autofocusing process, which is more intense than negative values of v. If the parameters n and v are controlled simultaneously, the higher focal intensity and shorter focal length can be obtained. Experimental results of z = 0, 20, 25 and 30 mm agree well with simulation results.

Fig. 6
figure 6

Peak intensity distribution of CAGV beams with different v parameters as a function of propagation distance z, solid lines correspond to simulations, while data points to experiment. For these beams, we used the parameters a = 0.1, b = 0.2, w = 0.1 mm, n = 2

Full size image

4 Conclusion

In summary, here we put forward a technique for the simultaneous generation of multiple CAGV beams with independent launch angles and topological charges. Our technique is based on the use of conjugate symmetric extension computer-generated holography. The intensity distributions and propagation properties of CAGV beam arrays have been theoretically and experimentally investigated. It was demonstrated that the focusing distance and peak intensity can be adjusted as desired by appropriately selecting the parameters of CAGV beams. In this way, the focal position can be controlled by adjusting the initial launch angle while the focal intensity by changing the topological charge. The simultaneous generation of multiple CAGV beams will be of great relevance in, for example, optical micromanipulation, since it provides a simple way to manipulate many particles simultaneously, each with different powers and/or in different planes. It will also be of interest in laser biomedical treatment or optical communications.