Interplay between topological phase and self-acceleration in a vortex symmetric Airy beam

Zhao-Xiang Fang; Yue Chen; Yu-Xuan Ren; Lei Gong; Rong-De Lu; An-Qi Zhang; Hong-Ze Zhao; Pei Wang

doi:10.1364/OE.26.007324

1. Introduction

Water vortex features with a region in the fluid where the flow rotates around an axial line, which may be straight or curved. Such phenomenon in fluid dynamics is associated with the velocity singularity. Besides, the vortex in natural waves with phase singularity are dominant in various physical settings. Those natural phenomena span across macroscopic to microscopic scales, including, fluid dynamics, surface plasmon polariton [1], electrons [2, 3], acoustics [4], and neutrons [5]. For example, the optical vortex is associated with spatial phase singularity. The optical vortex and singular beam has brought a new branch, i.e., singular optics, which describes the physical mechanism, controlled generation and applications of vortex beam. The optical vortex has a helical wavefront surrounding the singularity point, around which the circular integration of the spatial phase equals to a multiple of $2 π$ . Such integer is related to the topological structure of the light beam, and the orbital angular momentum(OAM) of a single photon in the beam is $l ℏ$ [6], where $ℏ$ is reduced Planck constant [7, 8]. The optical phase singularity can be found in the pattern of beam interference (e.g., Young’s triple slit interference). Controlled generation of the phase singularity can be done with the spatial light modulators [7–10]. Novel metamaterial structures and the evanescent waves can also generate and annihilate the phase singularities [11, 12], and the linear and nonlinear propagation of the vortices also brings lots of new phenomenon, such as vortex collision, four wave mixing, and modulational instability [13–15]. The optical vortex beam has broad applications in various areas, including the particle manipulation [16], multiplex free-space communication [17] and high-dimensional quantum cryptography [18].

Early studies on vortex beams focused on Laguerre-Gaussian modes [8, 10]. Recently, the vortices are demonstrated to be embedded in propagation invariant beams, such as Mathieu and Bessel beams [13], and the vector modulation of the vortex optical beams has been extensively studied both in linear and nonlinear media [19–22]. Further, the optical vortex beam can also be delivered to various positions assisted by the path of the higher order spatial modes, e.g., the Hermite-Gaussian and the Ince-Gaussian beams [23, 24]. A fundamental question raises, how to deliver the vortex beam in curved traces? Although there were some efforts to deliver the vortex beam along arbitrary curve [25], we explore the possibility to deliver the topological phase with the naturally self-accelerating symmetric Airy beam(SAB) [26, 27]. Initially predicted by the catastrophe theory, Airy beam is one kind of higher order spatial waves, following a self-bending trajectory with self-healing and non-diffracting features [28–30], and has also demonstrated the ability to illuminate highly resolved structures in fluorescence microscopy [31], to excite optical routing combined with optically induced photonic structures [32], to micromanipulate the curved structures [33] and to clear particles in optical micromanipulation [34]. To satisfy different applications, variant types of optical modes were experimentally created based on Airy functions, for instance the dual Airy beam [35], the abruptly autofocusing Airy beam [36, 37]. In the temporal domain, the femtosecond laser can also be tailored into temporal Airy pulses enabling the light bullets in linear system [38].

We first derive the beam amplitude for an SAB and the general form for the VSAB. Then the VSAB was experimentally produced by modulating the collimated beam with a synthesized binary amplitude Lee hologram projected on a digital micromirror device (DMD). The interplay of vortex and SAB was experimentally investigated by mapping the beam propagation trajectory using an axial scan scheme as well as the beam propagation simulation. The on-axis VSAB results in a rotation of the SAB pattern against propagation, while the SAB can guide the propagation of a slightly misaligned off-axis vortex beam. The experimental findings were corroborated by the propagation simulation. The rotation of the SAB pattern imbedded with a single on-axis vortex is associated with the energy flow near the bottle-like focal point, which is verified by calculating the Poynting vector in the transverse planes at discrete positions. Finally, we demonstrate the ability to carry and deliver multiple vortices with the self-accelerating symmetric Airy beam.

2. Vortex symmetric Airy beam

The naturally autofocusing SAB was produced by including the even cubic phase in the spectral domain. Specially, the autofocusing SAB displays a single on-axis lobe at the focus, and the beam splits into four off-axis identical major lobes with outward acceleration during further propagation. Although an incomplete analytical form for the SAB has been reported [39], we provide a general analytic expression for the electric field envelope of the SAB in the spatial domain. The complex field of one-dimensional (1D) finite Airy beam is written as [29],

u_{0} (s, ξ = 0) = A i (s) \exp (a s)

where

u_{0}

is the electric field envelope,

s

represents a dimensionless coordinate (

s_{x} = x / x_{0}

and

s_{y} = y / y_{0}

with

x_{0}

and

y_{0}

the transverse length scales),

ξ = z / k x_{0}^{2}

is a normalized propagation distance,

k = 2 π / λ

is vacuum wavenumber,

A i (s)

represents the Airy function, whose integral form is

A i (η) = \frac{1}{π} \int_{0}^{\infty} \cos (\frac{1}{3} t^{3} + η t) d t

, and

a

is a decaying factor. The angular spectrum for 1D Airy beam (Eq. (1)) is

U_{0} (k) = \exp (- a k^{2}) \exp (i (k^{3} - 3 a^{2} k - i a^{3}) / 3)

. Even symmetric phase in spectral space induces the angular spectrum of 1D SAB [26],

U_{1} (k) = \exp (- a k^{2}) \exp (\frac{i}{3} ({| k |}^{3} - 3 a^{2} | k | - i a^{3}))

The Inverse Fourier transform of Eq. (2) determines the complex amplitude of 1D SAB,

\begin{array}{l} u_{1} (s) = \frac{1}{2 π} \int_{- \infty}^{\infty} U_{1} (k) \exp (i k s) d k \\ ​ = \frac{\exp (- a s)}{2 π} \int_{0}^{\infty} \exp (\frac{i k^{3}}{3} - k s) d k + \frac{\exp (a s)}{2 π} \int_{0}^{\infty} \exp (\frac{i k^{3}}{3} + k s) d k \\ = \frac{\exp (- a s)}{2} [A i (- s) + i G i (- s)] + \frac{\exp (a s)}{2} [A i (s) + i G i (s)] \end{array}

where Gi represents a Scorer function [40], the integral representation of which is,

G i (η) = \frac{1}{π} \int_{0}^{\infty} \sin (\frac{1}{3} t^{3} + η t) d t

The two-dimensional (2D) SAB can be naturally expressed in the following,

u_{2} (x, y, z = 0) = u_{1} (s_{x}) u_{1} (s_{y})

The electric field of VSAB, originated from the 2D SAB superimposed by spiral phase in the Cartesian coordinates, can be expressed as:

u (x, y) = u_{2} (x, y) \prod_{j = 1}^{N} \frac{{[(x - x_{j}) + i (y - y_{j})]}^{l}}{{[{(x - x_{j})}^{2} + {(y - y_{j})}^{2}]}^{l / 2}}

where

(x_{j}, y_{j})

are the coordinates of the jth vortex phase dislocation, l represents the topological charge for each optical vortex, and N is the number of vortices.

In contradistinction to traditional symmetric Airy beam, which shows needle-like focus, the on-axis VSAB autofocuses to a bottle-like focal line. This is confirmed by the beam propagation simulation for an on-axis VSAB with l = 1 and $(x_{1}, y_{1}) = (0, 0)$ , shown in Fig. 1. Figure 1(a) is the 3D slice plot of the beam profile. Slices at various propagation positions show the transverse profiles at the respective locations. The side-view slices at the center are projected to the back and the bottom surfaces as indicated by the dashed arrows. Clearly, the bottle-like structure appears between the first two transverse slices, as observed from Figs. 1(a) and 1(b). For better visualization, we plot in 2D the bottom projection of the side-view slice in Fig. 1(b). The on-axis VSAB inherits the autofocusing property from SAB, but focuses into a doughnut other than concentrates into a needle [Fig. 1(b2)]. Due to its vortex structure that exhibits a strong axial binding force, the autofocused doughnut beam is suitable for simultaneous trapping and rotating micro-particles with either high or low refractive indices [41]. More importantly, the focus of the on-axis SAB has a much lower power density and thus causes less optically-induced thermal damage on an object in the beam center [41, 42].

Fig. 1 Numerical simulation for on-axis VSAB (l = 1) propagating in free space. (a)The sliced maps that are surrounded by two side-view profiles, the blue dashed arrows and blue lines indicate the beam propagation dynamics; The side-view profile of an on-axis VSAB (l = 1), its corresponding intensity profiles (b1)~(b4) and topological phase (b5)~(b8) at the planes marked by dashed lines in (b) with ξ = 0.05, 1.08, 2, and 2.7, respectively; Panels b1-b4 correspond to slices a1, a2, a4 and a5, respectively.

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The topological charge not only determines the OAM carried by the photons, but also regulates the size of the doughnut shape in the autofocusing plane and causes backward shift of autofocusing position [Fig. 1(a2)]. The propagation distance ξ is normalized with the autofocusing plane $(ξ = 1)$ of the conventional SAB. The autofocus for VSAB (l = 2) is found at plane away from the SAB autofocus with a new position of $ξ = 1.2$ . The autofocus position will shift more forward (along beam propagation) for vortex with large topological charge. The inclusion of the spiral phase rotates the transverse pattern of the SAB during propagation [Figs. 1(a3) and 1(a4)]. The rotational effect is also augmented as the topological charge increases.

3. Experimental results

A single-mode He-Ne laser (Thorlabs, 633nm) is expanded and collimated by a telescope composed of two lenses (focal distances for L1 and L2 are 2.5 cm and 20 cm respectively), then tailored by an amplitude digital micromirror device (DLP 7000, XGA, Texas Instrument) to produce the VSAB. The DMD pixel with two angular states can either register or discard the microbeam on the camera. In effect, each mirror unit leads to an on-off modulation of the reflected beam at that specific location. This results in a binary amplitude modulation of the incident light beams according to the micromirror state [43]. Compared with the standard spin-to-orbital angular momentum converters [44], and the rotationally symmetric metamaterial OAM mode selection nanosieves [45], the DMD provides erasable modulation and allows encoding both the vortex and SAB simultaneously [7]. The complex amplitude of the VSAB beam is encoded as binary amplitude hologram using the Lee method [27, 46].

Figure 2 shows the schematic layout of the experimental setup. A Fourier lens L3 (20cm) collects the modulated light and transforms to the modulated beam to the spectrum space at the back focal plane. A pinhole at the spectrum space filter out the first diffraction order [47]. The produced intensity patterns are further projected by lens L4 (focal length 10 cm) and captured by a CMOS camera (DS-CFM300-H, resolution 2048 × 1536, pixel size 3.2μm). In order to prevent the camera from being saturated, two polarizers with adjustable polarization angles are utilized to attenuate the beam power.

Fig. 2 Schematic of the experimental setup. Insets near the DMD and camera show the displayed binary amplitude hologram and the experimental beam profile of an off-axis VSAB (l = 1).

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The propagation dynamics of the on-axis VSABs with l = 1 and 2 are experimentally verified by mapping the 3D beam intensity [Figs. 3(a) and 3(b)]. The propagation distance is normalized to the distance $z_{0}$ from the autofocusing plane at the back focal plane (BFP) of the focusing lens. The spiral phase structure superimposed into the autofocusing region enables the SAB beam to form a cylindrically hollow optical guide channel. Although the traditional Airy beam embedded with an optical vortex experiences beam collapse in linear system and suppression due to nonlinear media [48–50], the interaction of the spiral phase and the Airy structure is still unclear. Interestingly, we found that the non-zero topological charge shifts the autofocusing plane of the VSAB presenting a non-identical focal position [second dashed lines in Figs. 3(a) and 3(b)]. In general, the autofocusing plane is shifted more forward (away from the light source) for larger topological charge. The ratios of autofocusing positions between VSAB and SAB are 1.08:1 and 1.2:1 for charges 1 and 2 respectively as predicted by simulation (see Fig. 1 for l = 1). The ratios become 1.1:1 and 1.25:1 as measured in the experiment (Fig. 3), which shows good consistency with the theoretical prediction.

Fig. 3 Experimental demonstration of on-axis VSABs. (a) a side view of an on-axis VSAB (l = 1); (a1)-(a4) transverse beam profiles at the planes $z = 0.05 z_{0}$ , $1.1 z_{0}$ , $2 z_{0}$ , and $2.7 z_{0}$ $(z_{0} = 1.8 c m)$ , marked by the dashed lines in (a), respectively; (b) the side view of an on-axis VSAB (l = 2) and the corresponding snapshots (b1-b4) at the planes $z = 0.05 z_{0}$ , $1.25 z_{0}$ , $2.1 z_{0}$ and $2.8 z_{0}$ , respectively. The scale bars for all figures are the same as shown in (b4). The rotation angles as a function of propagation distance are shown in (c) for l = 1 and (d) l = 2.

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Furthermore, the self-rotation of on-axis VSAB [predicted by theory, Fig. 1(b4)] is also corroborated and quantified in the beam measurement [Figs. 3(c) and 3(d)]. The rotational effect is attributed to the interaction between the SAB and a spiral phase front carried by a coaxial vortex. Before the focal plane, the beam experiences a strong self-focusing, thus the self-rotation is not so apparent. However, the on-axis VSABs rotate anticlockwise more significantly after the focal plane. As the topological charge increases, the rotation effect is more evident, which shows good consistency with the simulation shown in Fig. 1. The direction of rotation will be reversed for negative-charge vortices as evidenced by clockwise rotation in the SAB pattern with negative charge for the imbedded on-axis vortex. There is no significant difference on the magnitudes of the rotation angle for on-axis VSABs with opposite topological charges.

We also investigate how the off-axis vortex interacts with the SAB. As a demonstration, the parameters for the off-axis VSAB (Eq. (6)) are $(x_{1}, y_{1})$ = (−0.96mm, 0.96mm) and topological charge l = 1. A vortex beam was projected into one of the four major lobes of the SAB. The propagation dynamics for the off-axis VSAB is characterized both theoretically [Fig. 4(a)] and experimentally [Fig. 4(b)]. The transverse beam profiles are simulated in Figs. 4(a1)-4(a4) for positions marked by the dashed lines in Fig. 4(a), and the corresponding phase structures are mapped in Figs. 4(a5)-4(a8). The positions of the vortex phases are marked by the red circles on each panel. For the existence of an off-axis vortex, the null intensity originally occurs at the top-left area [Fig. 4(a1)], and due to the interplay between topological vortex structure and the self-acceleration, the singularity point gradually moves along the diagonal line. After autofocusing position, the off-axis vortex still shifts with outward acceleration, thus inducing central annihilation of the bottom-right main lobe. That means the SAB carries the vortex and forms an energy potential well in the major lobe for a long distance, providing potentials especially in the particle trapping. As a comparison, the autofocusing position of the off-axis VSAB is the same as that of the SAB, and it keeps constant in the presence of the off-axis vortex. Further, we discover that the autofocusing position basically keeps unchanged regardless of the off-axis VSAB with larger topological charge. The experimental beam profile [Fig. 4(b)] shows good consistency with the theoretical prediction [Fig. 4(a)].

Fig. 4 Propagation dynamics of an off-axis VSAB in free space. (a) Simulation and (b) experimental side-view profiles, panels (a1-a4, b1-b4) show the transverse beam profiles at the positions $ξ$ = 0.05, 1, 2.1 and 2.8 marked by the dashed lines, and panels (a5-a8) map the transverse phase structure at the corresponding positions. The fork-like phase structures are marked by a red circle on each phase panel.

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4. Discussions

Intuitively, the interplay between topological phase structure and self-accelerating beam can be understood by studying the internal transverse power flow. The local energy flow of the copropagating beam is expressed in terms of Poynting vector [51], which is defined as $\vec{S} = \frac{c}{4 π} \vec{E} \times \vec{B}$ , where c is the speed of light in vacuum, $\vec{E}$ and $\vec{B}$ are the electric and magnetic fields, respectively. In the Lorenz gauge, the time-averaged Poynting vector reads,

〈 \vec{S} 〉 = \frac{c}{4 π} 〈 \vec{E} \times \vec{B} 〉 = \frac{c}{8 π} [i ω (u \nabla_{⊥} u^{*} - u^{*} \nabla_{⊥} u) + 2 ω k {| u |}^{2} \vec{e_{z}}]

where

\nabla_{⊥} = \frac{\partial}{\partial x} \vec{e_{x}} + \frac{\partial}{\partial y} \vec{e_{x}}

,

\vec{e_{x}}

,

\vec{e_{y}}

, and

\vec{e_{z}}

are the unit vectors along the x, y, and z directions in sequence, and * denotes the complex conjugate. The

\vec{e_{z}}

component in the last term is the major contribution to the Poynting vector in the Gaussian optics. Here, in the discussion, we focus more on the non-zero

\vec{e_{x}}

and

\vec{e_{y}}

contribution to the energy flow.

The change of the Poynting vector can be used to explain the physics behind the self-rotation phenomenon and the change of the autofocusing position for the on-axis VSABs. Figures 5(a1)-(a4) and (b1)-(b4) show the energy flow map superimposed on the intensity patterns in the transverse planes with ascending propagation distances for on-axis VSABs (l = 1, 2). The direction and magnitude of the arrows (shown in white) demonstrate the counterpart of the energy flow in the transverse planes, and they change dynamically during propagation. Near the self-focus, for both on-axis VSABs (l = 1, 2), the beam energy flows from the outer region towards the central region, while the central energy flow exhibits counter-clockwise rotation [Figs. 5(a1) and 5(b1)]. Interestingly, the energy density of VSAB with l = 2 in the outer region is weaker than that of VSAB with l = 1, while the rotation energy in the central area exhibits a stronger and larger density distribution. After focusing, the energy for on-axis VSABs concentrates more on the four major lobes but less in the central region. Besides, the entire beam shape is rotated counter-clockwise, and this phenomenon gets more obvious with the increase of topological charge and propagation distance [Figs. 5(a3) and 5(a4) and Figs. 5(b3) and 5(b4)]. Due to the existence of the on-axis vortex, it not only induces rotating energy distribution, but also weakens the focusing energy density that flows towards the central area, thus causing the backward movement of the autofocusing position and counter-clockwise rotation when the beam diffracts outwardly. As the topological charge increases, such effect becomes more obvious. In summary, the autofocusing effect of the symmetric Airy beam is compromised by the presence of the on-axis vortex, leading to the forward shift of the autofocusing plane. Meanwhile, the diffraction effect from the vortex dominates the autofocusing effect of the symmetric Airy beam when the topological charge number reach a certain value.

Fig. 5 Numerical demonstrations of the energy flow (white arrows) superimposed on the transversal intensity maps for VSABs at different distances. The Poynting vectors for on-axis VSAB with topological charge (a) l = 1, and (b) l = 2 are pointing to the azimuthal direction near the self-focusing plane, and away from the beam axis near the major lobe at a position far away from the beam focus. In contrast, the Poynting vector for an off-axis VSAB points to the beam axis before and near focus with a small amount of component circulating the beam axis. At far field, the major lobe without vortex shows outward energy flow, while the lobe imbedded with vortex exhibits circulation on the Poynting vector map. The propagation coordinates for (a, b) are the same as described in Fig. 2, while positions in (c) are the same as those in Fig. 4.

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The Poynting vector map for the off-axis VSAB at the focal plane flows towards the central area with no significant energy rotation [Figs. 5(c1) and 5(c2)], and it is even stronger than that of on-axis VSAB with l = 1. Once the beam travels, the VSAB also splits into four major parts [Figs. 5(c3) and 5(c4)] similar to the SAB. At far field, the vortex-free major lobe shows outward energy flow, while the major lobe imbedded with vortex exhibits circulation on the Poynting vector map. In this particular case, the off-axis vortex travels with the bottom right lobe of the SAB. Although the occurrence of off-axis vortex influences the energy density on the top-left corner, it basically has no effect on the energy that flows towards the center, while the energy flux density at the bottom-right lobe is reduced as compared with that of the on-axis VSABs.

Further, we demonstrate that the major lobes of SAB are able to carry multiple vortices, and no significant interaction among the carried vortices is observed along a considerably long distance. The VSABs with multiple off-axis vortices are generated according to Eq. (7) with $| x_{j} | = | y_{j} | = 0.96 m m$ and l = 1. Figure 6(a) shows the transverse profiles of the multiple VSAB with dual vortices on one side, on the diagonal, triple vortices, and four vortices in sequence, and the phase structures for those patterns are correspondingly shown in Fig. 6(b). Similarly, the central intensity annihilation occurs in the corresponding main lobes. The experimental intensity patterns are respectively demonstrated in Fig. 6(c). Different from the vortex interaction in the nonlinear system [15, 52], the vortices carried by the major lobes of the SAB are independent instead of showing any significant interaction between adjacent or the diagonal vortex pairs despite sharing the same autofocus position. We also investigated the multiple vortices with opposite topological charges. The intensity distribution of dual vortices with opposite topological charges either on the side or the diagonal shows no significant difference with those bearing identical topological charges. The results for triple vortices with one positive and two negative charges show similar behavior as that for identical positive-charge triple vortices. The interplay between topological phase structure and the self-acceleration in off-axis VSAB is insensitive to the sign of the topological charge, while the magnitude of the topological charge determines the vortex size. The proposed VSAB modes with multiple off-axis vortices could facilitate the application in the particle manipulation requiring the VSAB dynamically reshaped in a deformed fashion or reducing the opto-thermal damage in specific region.

Fig. 6 The simulation (a) intensity and (b) phase, (c) experimental intensity profiles for multiple vortices imbedded in the symmetric Airy beam. Each column shows the results for VSAB with vortex pair on the side, vortex pair along the diagonal, triple vortices, and four vortices. Red circles in (b) mark the positions of phase singularities.

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5. Conclusion

In summary, we have proposed a new class of VSAB that exhibits autofocusing property and vortex structure. The on-axis vortex assists the SAB to form a hollow focal channel due to topological phase structure and autofocusing, and also rotates the pattern of SAB during propagation. The rotation direction of the SAB pattern can be reversed by using an on-axis vortex with opposing topological charge. The propagation dynamics for VSAB with vortex at different locations is investigated both experimentally and theoretically for the off-axis vortex imbedded on SAB. For single vortex, the off-axis VSAB shows the projection of the vortex into one of the major lobes of the SAB, and one hollow channel forms in the corresponding major lobe during propagation. Off-axis VSAB with vortices imbedded on multiple major lobes reveals that the interplay between topological phase and the self-acceleration does form a doughnut shape in the corresponding major lobe. Once the vortices are imbedded in the major lobes of the SAB, they are protected from each other and show no significant interaction between the vortices in different major lobes. Further, the interplay of the topological structure and the self-acceleration can be understood by the energy flow expressed in the form of Poynting vector, which offers a more intuitive explanation of the beam dynamics and describes the effect from the on or off-axis vortex embedded into the SAB. The proposed VSAB can be extensively applied in particle trapping, biological micromanipulation and optical communications.

Funding

Anhui Natural Science Foundation (1708085MF143); National Natural Science Foundation of China (NSFC) (31670866, 60974038); Popularization of Science Foundation of Chinese Academy of Sciences (KP2015C10).

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Interplay between topological phase and self-acceleration in a vortex symmetric Airy beam

Abstract

1. Introduction

2. Vortex symmetric Airy beam

3. Experimental results

4. Discussions

5. Conclusion

Funding

References and links

Cited By

Figures (6)

Equations (7)

Optics Express