Open Access
Add to BibSonomy
Abstract
In recent years, the manipulation of structured optical beam has become an attractive and promising area. The Gaussian beam is the most common beam as the output beam of the laser, and the Airy beam is recently proposed with fascinating properties and applications. In this paper, for the first time to our knowledge, the polarization is used as a tool to design a new kind of Airy-Gaussian vector beam by connecting the Gaussian and Airy functions, which opens a new avenue in designing new beams based on the existed beams. We realize the Airy-Gaussian vector beam with space-variant polarization distribution in theory and experiment, and find that the vector beam can autofocus twice during propagation. The optical chains with flexible intensity peaks are achieved with the Airy-Gaussian vector beam, which can be applied in trapping and delivering particles including biological cells and Rydberg atoms. Such optical chains can significantly improve the trapping efficiency, reduce the heat accumulation, and sweep away the impurity particles.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
In recent years, the manipulation of structured optical beams has become an attractive and promising area. Among various beams, there are two kinds of beams in two opposite extreme conditions: Gaussian beam and Airy beam. The Gaussian beam is the most traditional and common beam, which is playing an essential role in most laser applications [1–3]. Meanwhile, the Airy beam proposed in 2007 [4] is one of the most attractive beams, which is also one of the beams as the solutions of Maxwell’s equation [5]. Although it is proposed only 16 years ago, the Airy beam attracts great attention since its proposal [6], and exhibits various novel properties including non-diffraction [4], self-healing [7], and self-acceleration [8]. Numerous studies have demonstrated that the Airy beam has great potential for application in many areas, such as particle manipulation [9], filamentation [10,11], Raman scattering control [12], sixth-generation (6G) communication technology [13], and high-resolution light-sheet microscopy [14,15].
As introduced above, the Gaussian beam is the most common beam as the output beam of the laser [1–3], and the Airy beam is recently proposed with fascinating properties and applications [4,7–15]. Gaussian and Airy functions are closely related in many research fields. For example, Airy transformation of Gaussian beam plays an essential role in optical research, which is a useful technique in modulating amplitude and phase of a light beam [16–18]. Obviously, it is interesting to connect the two traditional and novel beams. How to add the novel and fascinating properties of the Airy beam into the most common Gaussian beam becomes an interesting and significant topic. Polarization, as an intrinsic property of light, plays an essential role in manipulating and applying light, and the vector beam with space-variant polarizations on the wave front shows great potential in a variety of scientific and engineering applications [19–24]. Thus, the polarization manipulation can become a new avenue to connect the Gaussian and Airy beams/functions.
Here, we propose the Airy-Gaussian vector beam with space-variant polarizations, based on the famous Gaussian and Airy functions. The beam is experimentally realized by the setup with the spatial light modulator (SLM) and 4f system. The proposition of the Airy-Gaussian vector beam is the first time to build a polarization relation between the Gaussian and Airy beams/functions, which enriches the family of vector beams and adds new thoughts in designing new beams based on the existing beams. For the propagation property, the Airy-Gaussian vector beam modulated by an annulus amplitude mask can autofocus twice during propagation, and the optical chains with good quality are achieved. The flexibly modulated optical chains can be applied in trapping and delivering particles such as cells and Rydberg atoms, and the controllable peak spots in the optical chains can improve the trapping efficiency, reduce the heat accumulation, and sweep away the impurity particles.
2. Theoretical design and experimental generation of the Airy-Gaussian vector beam
In the Cartesian coordinate system, the two-dimensional Airy beam is described by [4]:
Recently, the array of Airy beams has gained increasing interest because of their unique arrangement [25–28]. The array of four Airy beams becomes a research hotspot because of its properties such as self-healing [27], self-focusing [26], and scintillation reduction [25]. Therefore, we choose the array of four Airy beams arrangement in this paper. Based on the two-dimensional Airy beam, four Airy beams can be put in an array to make the beam with four-fold symmetry, as shown in Fig. 1(a). The expression of the Airy beams array is
where $s = (x, y)^{\operatorname {T}}$ with the superscript T being the symbol of the transposed matrix, $d$ denotes the transverse displacement parameter. $M_j$ is a rotation matrix that can rotate the Cartesian coordinate system. On this basis, we can change the rotation angle of the Airy beam and get the Airy beams array by $M_j$. The expression of the rotation matrix $M_j$ is
Fig. 1. The schematic of designing Airy-Gaussian vector beam. (a) Airy beams array; (b) orthogonal complementary mode; (c) Airy-Gaussian vector beam and the corresponding Stokes parameters.
To generate the Airy-Gaussian vector beam with a Gaussian intensity profile, we superimpose the Airy beams array to an orthogonal complementary mode. The expression of the basic Gaussian beam is
where $w_{\mathrm {G}}$ is the waist radius of the Gaussian beam. The complementary mode in Fig. 1(b) can be expressed asThe resulting Airy-Gaussian vector beam is the superposition of the above two beams of $U_{\mathrm {A}}$ and $U_{\mathrm {GA}}$ with orthogonal polarization states. For simplicity, we choose the Airy beams array and the complementary mode to be with $x$- and $y$-polarizations, and the Airy-Gaussian vector beam can be written as
where $\hat {\textbf {e}}_x$ and $\hat {\textbf {e}}_y$ are the unit vectors in $x$- and $y$-directions. Equation (6) is the mathematical expression of the Airy-Gaussian vector beam. In this equation, it is clear that the two orthogonally-polarized components correspond to the Airy beams array component and the complementary component, which is consistent with the schematic shown in Fig. 1. The intensity patterns and Stokes parameters of the Airy-Gaussian vector beam are shown in Fig. 1(c). It should be pointed out that besides the Airy beams array, there are also other kinds of special Airy beams such as circular Airy beam and Airyprime beam. We will use these special Airy beams to design new kinds of vector beams and study their properties in the future work.The experimental setup for generating the Airy-Gaussian vector beam is shown in Fig. 2, which is a common-path interferometric configuration with the aid of a 4f system [29–33]. The linearly polarized Gaussian beam output from the 532 nm laser is homogenized and expanded by a beam expander (BE). The parallel light formed by the lens L1 ($f$ = 120 mm) incidents on a phase-only spatial light modulator (SLM) (Pluto-Vis, Holoeye System Inc.) with 1920 $\times$ 1080 pixels (each pixel has a dimension of 8 $\times$ 8 $\mathrm{\mu}$m$^2$). The SLM is located in the input plane of the first 4f system which is composed of a pair of lenses L2 and L3 whose focal lengths are 300 mm and 150 mm, respectively. The designed two-dimensional binary phase holographic grating is loaded on the SLM, and the amplitude and phase modulation implemented on the diffraction beams are controlled by the duty ratio and grating structure of the holographic grating. After the SLM, a linear polarized light is diffracted into various diffraction orders, and the influence of the redundant orders should be eliminated. Only $\pm$1st orders (in $x$- and $y$-directions) selected by a spatial filter (SF) are converted into a pair of orthogonal circularly polarized base vectors by the quarter-wave plates at the Fourier plane of the 4f system. The $\pm$1st orders are recombined to form the Airy-Gaussian vector beam by Ronchi grating (RG) placed in the output plane of the 4f system. The generated Airy-Gaussian vector beam can be detected by the charge-coupled device (CCD), and the radius of the beam is 1.8 mm.
Fig. 2. Schematic of the experimental setup for generating the Airy-Gaussian vector beam. BE, beam expander; L1 ($f$ = 120 mm), L2 ($f$ = 300 mm), and L3 ($f$ = 150 mm), lenses; PBS, polarization beam splitter; SLM, spatial light modulator; RM, reflecting mirror; SF, spatial filter; WP, wave plates; G, Ronchi phase grating.
Figure 3 shows the simulated and experimentally generated intensity patterns of the total field, $x$-component and $y$-component of the Airy-Gaussian vector beam when $w_{\mathrm {A}}=0.25 \mathrm {~mm}$, $\alpha =0.05$, $d=0.3 \mathrm {~mm}$, $C=0.91$. The polarization states of the Airy beams array and the complementary mode are $x$- and $y$-polarizations, respectively. The total intensity distribution of the Airy-Gaussian vector beam is consistent with the Gaussian mode. It can be seen from Fig. 3 that the experimental results agree with the simulation results.
Fig. 3. The total intensity, $x$-component, and $y$-component of the Airy-Gaussian vector beam when $w_{\mathrm {A}}$ = 0.25 $\mathrm {~mm}$, $\alpha$ = 0.05, $d$ = 0.3 $\mathrm {~mm}$, $C$ = 0.91. The two rows show the simulated and experimental results, respectively. The three columns also correspond to the intensity patterns of the Airy-Gaussian vector beam, Airy beams array, and complementary mode, respectively.
3. Propagation properties of the Airy-Gaussian vector beam
As is known, the Airy beam exhibits fascinating properties in propagation, including self-acceleration (or self-bend), self-healing, and diffraction free. We certainly hope to find good properties in the propagation of the Airy-Gaussian vector beam after adding the Airy function to the Gaussian beam. We now study the propagation properties of the generated Airy-Gaussian vector beam with the beam propagation method (BPM) [34], which is a general beam propagation algorithm based on the wave equation under Fresnel approximation. Among different BPMs, we take the split-step spectral-domain method [35], which is a common numerical technique for simulating beam propagation [36]. Then, the amplitude of the Airy-Gaussian vector beam at distance $\Delta z$ is given by
Here we do not consider the longitudinal electric field component of the beam during propagation, since it is known that this component can be ignored under the paraxial condition [37]. The beam coherence-polarization (BCP) matrix provides the information of polarization and spatial correlation, and the BCP matrix of a vector beam passing through different positions is defined as follows [38]:
Figure 4 shows the propagation of the Airy-Gaussian vector beam when $w_{\mathrm {A}}=0.18 \mathrm {~mm}$, $\alpha =0.05$, $d=0.3 \mathrm {~mm}$, $C=0.93$, and $(r_{1}, r_{2})=(0 \mathrm {~mm}, 3.6 \mathrm {~mm})$. Figure 4(a) displays the cross section of the beam intensity in the $x$–$z$ plane. Figures 4(b)–4(d) indicate the transverse intensity distributions at different planes which are marked by the dashed lines in Fig. 4(a). The corresponding simulated results are shown in the insets of Figs. 4(b)–4(d). An interesting feature of the Airy-Gaussian vector beam in free-space propagation is the two-fold self-focusing effect, and the two foci locate in planes 2 and 3 marked in Fig. 4(a). The intensity of the first focus is weaker than that of the second, and the distance between the two focal points is 65 mm. The field at the first focus is mainly with vertical polarization, which means it mainly comes from the $y$-polarized complementary mode. The field at the second focus is mainly with horizontal polarization, which mainly comes from the $x$-polarized Airy beams array. The maximum intensity of the second focus is 4.35 times larger than that of the first focus, meaning that the self-focusing ability of the Airy beams array is stronger than that of the complementary mode. It is worth mentioning that the reason of this two-fold self-focusing effect is the lateral acceleration property of the Airy beams. In this case, different components of the Airy-Gaussian vector beam attain transverse velocities and energy sprints in an accelerated fashion towards the focus, and the energy with transverse velocities in different components of the Airy-Gaussian vector beam is unequable, leading to the different energy distributions and different locations of the two foci.
Fig. 4. Propagation of Airy-Gaussian vector beam when $w_{\mathrm {A}}$ = 0.18 $\mathrm {~mm}$, $\alpha$ = 0.05, $d$ = 0.3 $\mathrm {~mm}$, $C$ = 0.93, and $(r_{1}, r_{2})$ = $(0 \mathrm {~mm}, 3.6 \mathrm {~mm})$. (a) Cross section of the beam intensity in $x$–$z$ plane. (b)-(d) The transverse intensity patterns of the generated Airy-Gaussian vector beam at 0 mm, 55 mm, and 120 mm, respectively. The corresponding simulated results are shown in the insets.
We further adjust the parameters of the Airy-Gaussian vector beam when $w_{\mathrm {A}}=0.18 \mathrm {~mm}$, $\alpha =0.05$, $d=0.6 \mathrm {~mm}$, $C=0.26$, and $(r_{1}, r_{2})=(0.8 \mathrm {~mm}, 2.5 \mathrm {~mm})$, as shown in Fig. 5. It can be seen in Fig. 5(a) that an optical chain is achieved, and eight bright spots and seven tubes appear from 48 mm to 168 mm along the propagation direction. The distance between adjacent bright spots increases during propagation. Figures 5(b1)-(f1) show the transverse intensity patterns of the optical chain at different planes along the dashed lines in Fig. 5(a). From 96 mm to 104 mm, the peak intensity of the beam decreases, and the shape of the beam changes from a solid spot to a ring. From 104 mm to 118 mm, the peak intensity of the beam increases, and the shape of the vector beam changes from a ring to a solid spot. When the beam propagates from 118 mm to 152 mm, the evolution of the beam is similar to the case from 96 mm to 118 mm. It should be pointed out that the sizes of the bright spots and the tubes increase along with the propagation, and the distance between adjacent bright spots increases during propagation. Figures 5(b2)-(f2) show the corresponding experimental results, which agree with the simulated results.
Fig. 5. Propagation of Airy-Gaussian vector beam when $w_{\mathrm {A}}$ = 0.18 $\mathrm {~mm}$, $\alpha$ = 0.05, $d$ = 0.6 $\mathrm {~mm}$, $C$ = 0.26, and $(r_{1}, r_{2})$ = $(0.8 \mathrm {~mm}, 2.5 \mathrm {~mm})$. (a) Cross section of the beam intensity in $x$–$z$ plane. (b1)-(f1) Snapshots view of the transverse intensity patterns of the optical chain when $z$ = 96 mm, 104 mm, 118 mm, 134 mm, and 152 mm. (b2)-(f2) The corresponding experimental results.
After achieving the optical chain, we hope to manipulate it in a more flexible way. Here, we generate the optical chain with controllable peak bright spot at arbitrary positions by modulating the polarization state of the Airy-Gaussian vector beam. This means that the position of the peak bright spot with the strongest intensity can be flexibly controlled in the optical chain. Figure 6 shows that when $w_{\mathrm {A}}$ = 0.103 mm, 0.11 mm, 0.12 mm, 0.128 mm, 0.14 mm, and 0.157 mm, the peak bright spots in the optical chains are at the positions of 64.6 mm, 71.4 mm, 80.1 mm, 90.3 mm, 104.1 mm, and 123.1 mm, respectively. It can be observed from Fig. 6 that with the increasing $w_{\mathrm {A}}$, the position that the peak intensity appears in the optical chain gradually moves from the first spot to the last spot. The acceleration property of the Airy component promotes the convergence of Airy-Gaussian vector beam towards the beam center during the propagation, and the beam finally interferes at the beam center. This leads to the enhancement of the peak intensity of a specific bright spot in the optical chain.
Fig. 6. The normalized intensity (NI) patterns of the optical chains with controllable peak bright spots when $d$ = 0.6 $\mathrm {~mm}$, $C$ = 0.26, and $(r_{1}, r_{2})$ = $(0.8 \mathrm {~mm}, 2.8 \mathrm {~mm})$. The six patterns show the cases when $w_{\mathrm {A}}$ = 0.103 mm, 0.11 mm, 0.12 mm, 0.128 mm, 0.14 mm, and 0.157 mm, respectively.
Figures 7(a) and 7(d) show two cases of the experimental results of the optical chains with controllable peak bright spots when $w_{\mathrm {A}}$ = 0.148 mm and 0.185 mm. By comparing Figs. 7(a) and 7(d), we can see clearly that the peak bright spots in the chains with the strongest intensity are at different positions. In Fig. 7(a), the intensity of the peak bright spot at 111 mm is significantly stronger than that at other positions. In Fig. 7(d), the intensity of the peak bright spot at 156 mm is significantly stronger than that at other positions. Figures 7(b1)-(b5) and 7(c1)-(c5) show the theoretically simulated and experimentally generated transverse intensity patterns of the optical chain at different planes along the dashed lines in Fig. 7(a). Figures 7(e1)-(e5) and 7(f1)-(f5) show the theoretically simulated and experimentally generated transverse intensity patterns of the optical chain at different planes along the dashed lines in Fig. 7(d). Figures 7(g) and 7(h) show the contour plots of the normalized intensity on the axis corresponding to Figs. 7(a) and 7(d), which also proves the feasibility of manipulating controllable peak intensity of the bright spots in the optical chain.
Fig. 7. The intensity of the optical chains with different peak bright spots when $d$ = 0.6 $\mathrm {~mm}$, $C$ = 0.61, and $(r_{1}, r_{2})$ = $(0.8 \mathrm {~mm}, 2.2 \mathrm {~mm})$. (a) Side view of optical chain when $w_{\mathrm {A}}$ = 0.148 mm. (b1)-(b5) Transverse intensity patterns along the dashed lines in (a) when $z$ = 88 mm, 98 mm, 111 mm, 131 mm, and 156 mm. (c1)-(c5) The experimental results corresponding to (b1)-(b5). (d) Side view of optical chain when $w_{\mathrm {A}}$ = 0.185 mm. (e1)-(e5) Transverse intensity patterns of the optical chain along the dashed lines in (d) when $z$ = 88 mm, 98 mm, 111 mm, 131 mm, and 156 mm. (f1)-(f5) The experimental results corresponding to (e1)-(e5). (g) and (h) show the dependence of the NI on the propagation distance along the axial direction of the region in (a) and (d).
In order to get insights into the influence of the $w_{\mathrm {A}}$ on the positions of the peak bright spots in different optical chains, we further discuss the positions of the peak bright spots in the optical chains with different $w_{\mathrm {A}}$. Figure 8(a) shows the positions of the peak bright spots in the optical chains when $w_{\mathrm {A}}$ = 0.103 mm, 0.11 mm, 0.12 mm, 0.128 mm, 0.14 mm, and 0.157 mm. Obviously, the positions of the peak bright spots in the optical chain can be controlled by changing the polarization of Airy-Gaussian vector beam. Furthermore, Fig. 8(b) shows the dependence of the normalized intensity of the peak spots on $C$ when $C$ = 0.26, 0.38, 0.50, 0.61, 0.71, 0.79, and 0.87. It is found that the intensity of the peak spots can also be adjusted by the amplitude coefficient of Airy beam. The simulated and experimental results in Figs. 8(a) and 8(b) agree with each other. It can be seen from Fig. 8 that there is a slight error between the experimental and simulated results. This error is mainly caused by the limited resolution of the SLM in experimental setup, as this causes slight errors in the polarization distribution of the input beam, which leads to errors of the optical chain in Fig. 8. In a word, the positions and the intensity of the peak spots can both be flexibly modulated, which adds a new degree of freedom for controlling the optical chains. These controllable optical chains can be applied in various areas such as optical manipulation.
Fig. 8. The positions and NI of the optical chains with controllable peak bright spots when $d$ = 0.6 $\mathrm {~mm}$, and $(r_{1}, r_{2})$ = $(0.8 \mathrm {~mm}, 2.8 \mathrm {~mm})$. (a) Dependence of the positions of the peak bright spots in the optical chains on $w_{\mathrm {A}}$ when $d$ = 0.6 $\mathrm {~mm}$, $(r_{1}, r_{2})$ = $(0.8 \mathrm {~mm}, 2.8 \mathrm {~mm})$, $C$ = 0.5, and $w_{\mathrm {A}}$ = 0.103 mm, 0.11 mm, 0.12 mm, 0.128 mm, 0.14 mm, and 0.157 mm. (b) Dependence of the NI of the peak spots on $C$ when $d$ = 0.6 $\mathrm {~mm}$, $(r_{1}, r_{2})$ = $(0.8 \mathrm {~mm}, 2.8 \mathrm {~mm})$, $w_{\mathrm {A}}$ = 0.14 mm, and $C$ = 0.26, 0.38, 0.50, 0.61, 0.71, 0.79, and 0.87.
Besides the manipulation of the intensity of the optical chain, the polarization evolution of the optical chain is also important. Figure 9 shows the intensity and Stokes parameters of different peak spots in the optical chain when $d$ = 0.6 $\mathrm {~mm}$, $C$ = 0.5, $w_{\mathrm {A}}$ = 0.148 $\mathrm {~mm}$, and $(r_{1}, r_{2})$ = $(0.8 \mathrm {~mm}, 2.2 \mathrm {~mm})$. By comparing the second to fourth columns in Fig. 9, we can see that the Stokes parameters of the peak bright spot are different from the adjacent spot. The Stokes parameter $S_1$ is much stronger than $S_2$ and $S_3$ for the spots at 88 mm and 160 mm, which means the polarization is mainly $x$-polarization. Meanwhile, the values of the Stokes parameters $S_1$, $S_2$, and $S_3$ are close to each other for the spot at 110 mm, which means the polarization contains part of the elliptical or circular polarizations. As is known, the circular polarization is relative to the spin angular momentum (SAM) of photon [39,40]. SAM is related to the circular polarizations with two possible quantized values of ${\pm }\hbar$, which can make the particle spin around its own axis [39–42]. Recently, much attention has been paid to the manipulation of SAM, which can be used in areas including nanophotonics [43], plasmonics [44], quantum communications [45], and so on. Apparently, the Airy-Gaussian vector beam with space-variant linear polarizations carries no SAM, but SAM appears in the peak bright spot of the optical chain at 110mm, which can be derived as the Stokes parameter $S_3$ is proportional to the longitudinal SAM density $S_z$ by the formular $S_3=j\left (E_x E_y^*-E_y E_x^*\right ) \propto S_z$. This means that we can manipulate not only the intensity but also the polarization state of the optical chain, and the optical chain with SAM provides a new tool for applying SAM. The appearance of SAM in the optical chain can be used in realms such as trapping and spinning particles.
Fig. 9. The intensity and Stokes parameters of the optical chain when $d$ = 0.6 $\mathrm {~mm}$, $C$ = 0.5, and $(r_{1}, r_{2})$ = $(0.8 \mathrm {~mm}, 2.2 \mathrm {~mm})$. (a) Side view of the optical chain when $w_{\mathrm {A}}$ = 0.148 $\mathrm {~mm}$. (b)-(d) Transverse intensity patterns along the dashed lines in (a) when $z$ = 88 mm, 110 mm, and 160 mm, respectively. The Stokes parameters $S_1$, $S_2$, and $S_3$ are shown in the second to fourth columns. The corresponding simulated results are shown in the insets.
The optical chain can be applied in various areas, and one representative application is the optical trapping [46,47] and manipulation [48,49]. Compared with ordinary single-beam optical tweezers, the optical chain plays an essential role in optical trapping owing to its advantage of high efficiency in multi-particle trapping, and each particle can be accurately confined three-dimensionally. Since the optical chain can be regarded as a one-dimensional array of either bright spots or dark traps, multiple high-index or low-index micro-particles can be trapped simultaneously at the nodal regions. The flexibly controllable optical chains we propose can be applied in trapping multiple particles such as cells and Rydberg atoms, and the controllable peak spots in the optical chain can improve the trapping efficiency, reduce the heat accumulation, and sweep away the impurity particles.
In biological experiments, the common optical tweezers are widely used in the manipulation of biological particles such as cells [50,51]. There is a limitation of applying the optical trapping in the areas of biology and medicine, as the photodamage can be inflicted on trapped cells in the focusing process [52], and the damage is related to the power of optical tweezers and the accumulated dose [53,54]. Obviously, reducing the accumulated dose is an effective way to reduce the cell damage caused by optical tweezers. On one hand, we hope the energy to be high enough to guarantee the strength of the trap. On the other hand, we also want to avoid the photodamage caused by the accumulated dose of energy. Compared with the traditional optical chains using uniform bright spots to trap cells, the optical chains we propose can guarantee the capture strength and reduce the photodamage. To realize this, we need to switch the holographic grating on the SLM, and the optical chains with different $w_{\mathrm {A}}$ can be generated. In this way, the positions of the peak spots change over time, and the heat accumulation is weakened. When the peak spots move away, the cells or particles may do random Brownian motions, but they will be captured again if the peak intensity moves back in a short time. Based on this, bright spots in the optical chain can simultaneously capture particles under the rapid movement of peak bright spots. Take the experimental setup in this paper as an example, the SLM (Pluto-Vis, Holoeye System Inc.) has a frame rate of 60 Hz, which means we can switch 60 gratings on the SLM in one second. If we choose 6 peak spots as shown in Fig. 6, each peak spot will appear 10 times in one second. This can guarantee the capture strength, and the heat accumulation is reduced to 1/6 compared with the case of using the six peak spots with high energy, which reduces the damage to biological particles caused by optical chains.
In addition, the optical chain we propose can also be used in trapping particles with the sweeping function. Impurity particles will make uncontrollable influences on target particles. In order to accurately capture the target particles, we can sweep the impurity particles by the peak bright spot in the chain. For instance, the array of dark traps in the optical chain we propose can be used in trapping the Rydberg atom, which is a highly excited atom containing at least one electron far away from its nucleus [55]. The strong, controllable interactions between Rydberg atoms make them become an intriguing booth for quantum information [56], quantum computing [57], and microwave electric field detection [58]. D. Barredo et al. have demonstrated the three-dimensional trapping of individual Rydberg atoms in holographic optical bottle beam traps [47]. Since the Rydberg polarizability is that of a free electron and is negative, a stable trap must be a dark region surrounded by light [47,59]. It has been found that Rydberg atoms will collide with impurity atoms in the environment [60–63]. This will increase the loss rate of the Rydberg atoms [62,63]. When the optical chain with controllable peak spot is applied, the peak spot can capture the impurity particles in a more exact way, and the heat accumulation can also be reduced. In this way, this novel optical chain can sweep away deexcited Rydberg atoms and other impurity particles theoretically when trapping Rydberg atoms in the dark trap, which can reduce Rydberg atoms’ loss rate by reducing the collisions between atoms. In addition, we can increase the accuracy of Rydberg atoms’ lifetime detection by sweeping away impurity particles with this novel optical chain. Besides the optical trapping, the nonlinear effects of the optical beam are also interesting, so we will study the nonlinear effects of the Airy-Gaussian vector beam in the future after we apply a laser with high power and study it in the nonlinear medium such as a Kerr medium.
4. Conclusion
In conclusion, we propose the Airy-Gaussian vector beam with space-variant polarizations, applying the polarization as a bridge to connect the famous Gaussian and Airy beams/functions. We modulate the initial polarization of Gaussian beam by Airy function to design the Airy-Gaussian vector beam, and experimentally realize the beam with the setup based on the SLM and 4f system. The propagation property is further studied, and it is found that the Airy-Gaussian vector beam modulated by an annulus amplitude mask can autofocus twice during propagation. The optical chains with good quality are achieved, and the intensity distribution of the optical chain can be flexibly controlled by manipulating the initial polarization of Airy-Gaussian vector beam. The proposition of the Airy-Gaussian vector beam is the first time to build a polarization relation between the Gaussian beam and the Airy beam, which adds new thoughts and provides new methods for designing and manipulating the optical beams. Meanwhile, the flexibly modulated optical chains can be applied in trapping and delivering particles such as cells and Rydberg atoms, and the controllable peak spots in the optical chain can improve the trapping efficiency, reduce the heat accumulation, and sweep away the impurity particles.
Funding
National Natural Science Foundation of China (11904199, 12174217); Higher Educational Youth Innovation Science and Technology Program Shandong Province (2022KJ175).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. A. E. Siegman, “Physical properties of Gaussian beams,” in Lasers, (University science books, 1986), pp. 663–695.
2. A. Forbes, “Structured light from lasers,” Laser Photonics Rev. 13(11), 1900140 (2019). [CrossRef]
3. U. Levy, Y. Silberberg, and N. Davidson, “Mathematics of vectorial Gaussian beams,” Adv. Opt. Photonics 11(4), 828–891 (2019). [CrossRef]
4. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007). [CrossRef]
5. M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979). [CrossRef]
6. N. K. Efremidis, Z. Chen, M. Segev, and D. N. Christodoulides, “Airy beams and accelerating waves: an overview of recent advances,” Optica 6(5), 686–701 (2019). [CrossRef]
7. C. Ament, P. Polynkin, and J. V. Moloney, “Supercontinuum generation with femtosecond self-healing Airy pulses,” Phys. Rev. Lett. 107(24), 243901 (2011). [CrossRef]
8. E. Greenfield, M. Segev, W. Walasik, and O. Raz, “Accelerating light beams along arbitrary convex trajectories,” Phys. Rev. Lett. 106(21), 213902 (2011). [CrossRef]
9. J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008). [CrossRef]
10. P. Polynkin, M. Kolesik, and J. Moloney, “Filamentation of femtosecond laser Airy beams in water,” Phys. Rev. Lett. 103(12), 123902 (2009). [CrossRef]
11. P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324(5924), 229–232 (2009). [CrossRef]
12. Y. Hu, A. Tehranchi, S. Wabnitz, R. Kashyap, Z. Chen, and R. Morandotti, “Improved intrapulse Raman scattering control via asymmetric Airy pulses,” Phys. Rev. Lett. 114(7), 073901 (2015). [CrossRef]
13. J. C. Zhang, G. B. Wu, M. K. Chen, X. Liu, K. F. Chan, D. P. Tsai, and C. H. Chan, “A 6g meta-device for 3d varifocal,” Sci. Adv. 9(4), eadf8478 (2023). [CrossRef]
14. J. Wang, X. Hua, C. Guo, W. Liu, and S. Jia, “Airy-beam tomographic microscopy,” Optica 7(7), 790–793 (2020). [CrossRef]
15. Y. Zhou, P. Zammit, V. Zickus, J. M. Taylor, and A. R. Harvey, “Twin-Airy point-spread function for extended-volume particle localization,” Phys. Rev. Lett. 124(19), 198104 (2020). [CrossRef]
16. V. Y. Fedorov, D. Papazoglou, and S. Tzortzakis, “Transformation of ring-Airy beams during efficient harmonic generation,” Opt. Lett. 44(12), 2974–2977 (2019). [CrossRef]
17. G. Zhou, F. Wang, and S. Feng, “Airy transform of Laguerre-Gaussian beams,” Opt. Express 28(13), 19683–19699 (2020). [CrossRef]
18. X. Zang, W. Dan, F. Wang, Y. Zhou, Y. Xu, and G. Zhou, “Generation of finite energy Airyprime beams by Airy transformation,” Opt. Express 30(14), 24948–24967 (2022). [CrossRef]
19. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1–57 (2009). [CrossRef]
20. T. Bauer, P. Banzer, E. Karimi, S. Orlov, A. Rubano, L. Marrucci, E. Santamato, R. W. Boyd, and G. Leuchs, “Observation of optical polarization möbius strips,” Science 347(6225), 964–966 (2015). [CrossRef]
21. R. C. Devlin, A. Ambrosio, N. A. Rubin, J. B. Mueller, and F. Capasso, “Arbitrary spin-to-orbital angular momentum conversion of light,” Science 358(6365), 896–901 (2017). [CrossRef]
22. H. Larocque, D. Sugic, D. Mortimer, A. J. Taylor, R. Fickler, R. W. Boyd, M. R. Dennis, and E. Karimi, “Reconstructing the topology of optical polarization knots,” Nat. Phys. 14(11), 1079–1082 (2018). [CrossRef]
23. A. Forbes, M. de Oliveira, and M. R. Dennis, “Structured light,” Nat. Photonics 15(4), 253–262 (2021). [CrossRef]
24. Z. Zhang, H. Zhao, S. Wu, T. Wu, X. Qiao, Z. Gao, R. Agarwal, S. Longhi, N. M. Litchinitser, L. Ge, and L. Feng, “Spin–orbit microlaser emitting in a four-dimensional hilbert space,” Nature 612(7939), 246–251 (2022). [CrossRef]
25. Y. Gu and G. Gbur, “Scintillation of Airy beam arrays in atmospheric turbulence,” Opt. Lett. 35(20), 3456–3458 (2010). [CrossRef]
26. L. Wang, X. Ji, X. Li, T. Wang, H. Yu, and Q. Li, “Focusing and self-healing characteristics of Airy array beams propagating in self-focusing media,” Appl. Phys. B 125(9), 165 (2019). [CrossRef]
27. C. Y. Hwang, D. Choi, K. Y. Kim, and B. Lee, “Dual Airy beam,” Opt. Express 18(22), 23504–23516 (2010). [CrossRef]
28. X. Zang, W. Dan, F. Wang, Y. Zhou, Y. Cai, and G. Zhou, “Dependence of autofocusing ability of a ring Airyprime beams array on the number of beamlets,” Opt. Lett. 47(21), 5654–5657 (2022). [CrossRef]
29. X. L. Wang, J. Ding, W. J. Ni, C. S. Guo, and H. T. Wang, “Generation of arbitrary vector beams with a spatial light modulator and a common path interferometric arrangement,” Opt. Lett. 32(24), 3549–3551 (2007). [CrossRef]
30. Y. Pan, S. M. Li, L. Mao, L. J. Kong, Y. Li, C. Tu, P. Wang, and H. T. Wang, “Vector optical fields with polarization distributions similar to electric and magnetic field lines,” Opt. Express 21(13), 16200–16209 (2013). [CrossRef]
31. X. Z. Gao, J. H. Zhao, M. S. Wang, J. J. Liu, G. B. Zhang, and Y. Pan, “Bipolar-variant spin angular momentum and its evolution in a tight focusing process,” Phys. Rev. A 102(6), 063514 (2020). [CrossRef]
32. J. H. Zhao, Y. Pan, X. Z. Gao, R. Ma, Z. X. Man, Z. C. Ren, C. Tu, Y. Li, and H. T. Wang, “Five-dimensional poincaré sphere system for representing azimuthally varying vector optical fields,” Phys. Rev. A 106(2), 023506 (2022). [CrossRef]
33. X. F. Sun, X. Z. Gao, G. B. Zhang, F. Yang, J. J. Liu, R. Ma, Z. X. Man, and Y. Pan, “The appearance and annihilation of the spin angular momentum for the multi-polar vector optical field in the focal plane,” APL Photonics 8(5), 056101 (2023). [CrossRef]
34. M. Feit and J. Fleck, “Light propagation in graded-index optical fibers,” Appl. Opt. 17(24), 3990–3998 (1978). [CrossRef]
35. T. R. Taha and M. I. Ablowitz, “Analytical and numerical aspects of certain nonlinear evolution equations. ii. numerical, nonlinear schrödinger equation,” J. Comput. Phys. 55(2), 203–230 (1984). [CrossRef]
36. A. K. Glaser, Y. Chen, and J. T. Liu, “Fractal propagation method enables realistic optical microscopy simulations in biological tissues,” Optica 3(8), 861–869 (2016). [CrossRef]
37. Y. Cai, X. Lu, and Q. Lin, “Hollow Gaussian beams and their propagation properties,” Opt. Lett. 28(13), 1084–1086 (2003). [CrossRef]
38. J. Tervo, “Azimuthal polarization and partial coherence,” J. Opt. Soc. Am. A 20(10), 1974–1980 (2003). [CrossRef]
39. L. Allen, M. Padgett, and M. Babiker, “Iv the orbital angular momentum of light,” in Progress in optics, vol. 39 (Elsevier, 1999), pp. 291–372.
40. Y. Zhao, J. S. Edgar, G. D. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99(7), 073901 (2007). [CrossRef]
41. M. E. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394(6691), 348–350 (1998). [CrossRef]
42. A. Lehmuskero, R. Ogier, T. Gschneidtner, P. Johansson, and M. Kall, “Ultrafast spinning of gold nanoparticles in water using circularly polarized light,” Nano Lett. 13(7), 3129–3134 (2013). [CrossRef]
43. F. Cardano and L. Marrucci, “Spin–orbit photonics,” Nat. Photonics 9(12), 776–778 (2015). [CrossRef]
44. F. Feng, G. Si, C. Min, X. Yuan, and M. Somekh, “On-chip plasmonic spin-hall nanograting for simultaneously detecting phase and polarization singularities,” Light: Sci. Appl. 9(1), 95 (2020). [CrossRef]
45. Y. Shen, I. Nape, X. Yang, X. Fu, M. Gong, D. Naidoo, and A. Forbes, “Creation and control of high-dimensional multi-partite classically entangled light,” Light: Sci. Appl. 10(1), 50 (2021). [CrossRef]
46. B. Melo, I. Brand ao, B. P. da Silva, R. B. Rodrigues, A. Z. Khoury, and T. Guerreiro, “Optical trapping in a dark focus,” Phys. Rev. Appl. 14(3), 034069 (2020). [CrossRef]
47. D. Barredo, V. Lienhard, P. Scholl, S. de Léséleuc, T. Boulier, A. Browaeys, and T. Lahaye, “Three-dimensional trapping of individual Rydberg atoms in ponderomotive bottle beam traps,” Phys. Rev. Lett. 124(2), 023201 (2020). [CrossRef]
48. V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419(6903), 145–147 (2002). [CrossRef]
49. Y. Zhao, Q. Zhan, Y. Zhang, and Y. P. Li, “Creation of a three-dimensional optical chain for controllable particle delivery,” Opt. Lett. 30(8), 848–850 (2005). [CrossRef]
50. A. Ashkin and J. M. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science 235(4795), 1517–1520 (1987). [CrossRef]
51. A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared laser beams,” Nature 330(6150), 769–771 (1987). [CrossRef]
52. M. A. de Oliveira, D. S. Moura, A. Fontes, and R. E. de Araujo, “Damage induced in red blood cells by infrared optical trapping: an evaluation based on elasticity measurements,” J. Biomed. Opt. 21(7), 075012 (2016). [CrossRef]
53. S. Ayano, Y. Wakamoto, S. Yamashita, and K. Yasuda, “Quantitative measurement of damage caused by 1064-nm wavelength optical trapping of escherichia coli cells using on-chip single cell cultivation system,” Biochem. Biophys. Res. Commun. 350(3), 678–684 (2006). [CrossRef]
54. T. Aabo, I. R. Perch-Nielsen, J. S. Dam, D. Z. Palima, H. Siegumfeldt, J. Glückstad, and N. Arneborg, “Effect of long-and short-term exposure to laser light at 1070 nm on growth of saccharomyces cerevisiae,” J. Biomed. Opt. 15(4), 041505 (2010). [CrossRef]
55. N. Šibalić and C. S. Adams, “Rydberg lifetimes,” in Rydberg physics, (IOP Publishing, 2018), p. 5.
56. D. Jaksch, J. I. Cirac, P. Zoller, S. L. Rolston, R. Côté, and M. D. Lukin, “Fast quantum gates for neutral atoms,” Phys. Rev. Lett. 85(10), 2208–2211 (2000). [CrossRef]
57. M. Saffman, T. G. Walker, and K. Mølmer, “Quantum information with Rydberg atoms,” Rev. Mod. Phys. 82(3), 2313–2363 (2010). [CrossRef]
58. M. Jing, Y. Hu, J. Ma, H. Zhang, L. Zhang, L. Xiao, and S. Jia, “Atomic superheterodyne receiver based on microwave-dressed Rydberg spectroscopy,” Nat. Phys. 16(9), 911–915 (2020). [CrossRef]
59. S. Zhang, F. Robicheaux, and M. Saffman, “Magic-wavelength optical traps for Rydberg atoms,” Phys. Rev. A 84(4), 043408 (2011). [CrossRef]
60. B. Olmos, W. Li, S. Hofferberth, and I. Lesanovsky, “Amplifying single impurities immersed in a gas of ultracold atoms,” Phys. Rev. A 84(4), 041607 (2011). [CrossRef]
61. G. Fields, F. B. Dunning, S. Yoshida, and J. Burgdörfer, “Destruction of high-n (n ˜ 300) Rydberg atoms in Rydberg-Rydberg collisions,” Phys. Rev. A 99(2), 022710 (2019). [CrossRef]
62. Z. Feng, H. Zhang, J. Che, L. Zhang, C. Li, J. Zhao, and S. Jia, “Collisional loss of cesium Rydberg atoms in a magneto-optical trap,” Phys. Rev. A 83(4), 042711 (2011). [CrossRef]
63. S. K. Kanungo, J. D. Whalen, Y. Lu, T. C. Killian, F. B. Dunning, S. Yoshida, and J. Burgdörfer, “Loss rates for high-n, $49 \lesssim n \lesssim 150$, 5sns (3S1) Rydberg atoms excited in an 84Sr Bose-Einstein condensate,” Phys. Rev. A 102(6), 063317 (2020). [CrossRef]
