Generation and measurement of irregular polygonal perfect vortex optical beam based on all-dielectric geometric metasurface

Yue Liu; Chengxin Zhou; Kuangling Guo; Zhongchao Wei; Hongzhan Liu

doi:10.1364/OE.488434

1. Introduction

Optical vortex (OV) with a spiral wavefront as a structured light that carries orbital angular momentum (OAM) of $l\hbar$ per photon, where l is the topological charge (TC) and can theoretically be taken as any integer, indicating the repeating rate of 2π phase shifts azimuthally along the beam cross section, and $\hbar$ is the reduced Planck constant [1,2]. Since its unique optical properties were discovered by Allen et al. in 1992, OV beams have received extensive attention in optical communications [3], optical tweezers [4], microscopic imaging [5], and nonlinear optics [6]. Because of the phase singularity at the beam center, conventional OV beams have a doughnut-shaped uniform intensity profile. To meet the demands of diverse OAM distribution, the spiral OAM distribution [7], structured OAM distribution [8], nonuniform hollow circular distribution [9], and grafted vortex beams [10] are shown successively. Essentially, OV beams with TC-dependent intensity distributions are not ideal for applications requiring the spatial superposition of different beams [11]. Therefore, Ostrovsky et al. propose perfect optical vortex (POV) beam with light intensity distribution insensitive to TC [12], and has great potential in optical communication [13], particle capture [14], and quantum optics [15]. POV beam can be generated by axicon [16], spatial light modulators (SLM) [17], interferometers [18], and digital microscopy devices [19]. In general, POV beam presents a uniform light intensity distribution, while the single shape is not conducive to form some complex structured light fields. To overcome this limitation, elliptic perfect optical vortex (EPOV) beam [20], fractional order POV beam [21] and perfect vector vortex beam [22] are studied. Recently, a high-order cross-phase (HOCP) based on astigmatism has been demonstrated that can shape and manipulate various OV beams [23–25]. Combined with the POV beam, the polygonal perfect vortex beams are produced using SLM [26]. These diverse POV beams described above possess promising opportunities for some advanced applications, e.g., optical microfluidic sorting [27] and micro-particles regulation and acceleration [28]. In addition, it is important and necessary to determine the OAM in advance, and measuring the OAM has also driven the development of many methods and techniques with different levels of complexity and performance [29–31]. The most widely used methods mainly include optical interferometry [32] and diffractometry [33]. However, the generation and detection of various POV beams requires a series of bulky and tightly aligned optical elements in the optical path due to the additional phase distribution involved-making the space requirements excessive and costly. Therefore, a compact, simple, and efficient method to generate and measure diverse POV beams is urgently needed.

Metasurface, consisting of customized planar nanostructures, exploits the abrupt phase discontinuities of structured surfaces to modulate the amplitude, polarization and phase of incident electromagnetic waves at the subwavelength scale [34,35]. Benefiting from the flexible controllability and extraordinary performance, optical metasurfaces are considered as a prime choice for next-generation intelligent optical devices and have applied in metalenses [36,37], optical holography [38,39], and OV beam generators [40–43]. Indeed, attributed to the unique optical properties of POV beam, from beam generation and measurement to beam shaping and phase manipulation, researchers have also conducted in-depth studies of POV beams based on metasurfaces. Initially, POV beams are successfully generated at various wavelengths using metasurface [32,44–49], but the shape of the resultant beam is relatively single, i.e. an annular ring, which limits the capture and manipulation of multiple particles. Recently, EPOV beams with asymmetric optical field are demonstrated based on all-dielectric metasurface and ellipticity as a new modulation dimension has potential applications in structured optical communication [50,51]. Inspired by plant grafting, Ahmed et al. generate multi-channel grafted POV beams and achieve asymmetric singularity distribution through a simple metasurface, further enriching the diversity of POV beams [52]. Lately, a new kind of POV beam called composite perfect vortex beam has been reported, where the TC can be directly identified by analyzing a rosette-like intensity pattern [53]. In fact, the POV beam modes studied above are too single, whereas multi-state POV beams are ideal light sources for particle manipulation, microfluidic acceleration, and structured optical communication. In addition, the detection method for POV beam is not convenient and mostly requires interference with a co-propagating Gaussian beam to further determine the TC.

In this paper, we originally combine the HOCP and EPOV beam to generate irregular polygonal perfect optical vortex (IPPOV) beam using a geometric metasurface composed of TiO₂ nanostructures. By altering the parameters of HOCP, both the shape of the irregular polygon (triangle, quadrangle, pentagon, etc.) and the intensity distribution can be manipulated flexibly. Furthermore, the free-space propagation characteristics of the IPPOV beam are studied, which first focuses and then diverges, and finally gradually evolves into IPPOV beams, as shown in Fig. 1. It is worth mentioning that for the first time the TC of POV beam can be obtained directly by observing the number and direction of bright spots at the focal plane without a reference beam. Compared to the reported studies, we introduce parameters both ellipticity γ and u, adding modulation dimensions while maintaining the beam properties, enriching the spatial mode distribution of the POV beam, improving the beam modulation capability, and playing an important role in particle capture and manipulation.

Fig. 1. Schematic diagram of the IPPOV beam generation and measurement based on all-dielectric geometric metasurface.

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2. Theory and design

It has been experimentally proved that the POV beam can be obtained by Fourier transformation of the higher-order Bessel Gaussian (BG) beam used as an approximation to the Bessel beam. Actually, the incident Gaussian beam is successively passed through a spiral phase plate and an axicon to generate a BG beam, whose transverse electric field distribution can be expressed in Cartesian coordinate system (x, y) as [17,51]:

(1)$${E_{BG}}(x,y) = {J_l}\left[ {{k_r}\sqrt {{x^2} + {{(\gamma y)}^2}} )} \right]\exp \left[ {il\arctan (\frac{{\gamma y}}{x})} \right]$$

where ${J_l}$ is the first kind of l-th order Bessel function, ${k_r}$ denotes the radial wave vector related to the axicon period, γ represents a scalar factor used to control the ellipticity of the POV beam, and l is the TC carried by the beam. Next, the resultant BG beam is Fourier transformed through a lens with focal length f. Then the POV beam can be observed at the back focal plane of the lens, and its transverse electric field can be written in the polar coordinate system as:

(2)$${E_{PV}}(r,\vartheta ) = {i^{l - 1}}\frac{{{w_0}}}{{{w_g}}}\exp (il\vartheta )\exp ( - \frac{{{{(r - {R_\gamma })}^2}}}{{w_g^2}})$$

where ${w_0}$ is the beam radius of the incident Gaussian beam, ${w_g} = {{2f} / {k{w_0}}}$ represents the beam waist of the Gaussian beam at the Fourier plane, ${R_\gamma }$ defines the ring radius of the POV beam, and k is the wave vector in free-space.

Based on theoretical analysis and experimental verification, HOCP can regulate all types of optical vortices at the far-field [24–26]. To increase the diversity of POV beams and further improve the beam modulation capability, we introduce HOCP to the EPOV beam, which allows polygonal shaping of the optical field to generate IPPOV beams. Instead of the traditional method with a series of optical devices, here a simple geometric metasurface is constructed with the combined phase distributions of helical phase plate, axicon, Fourier lens, and HOCP, as shown in Fig. 2. The total phase profile encoded on the metasurface ${\varphi _{meta}}$ is specifically described as [26]:

(3a)$${\varphi _{meta}}(x,y) = {\varphi _{spiral}}(x,y) + {\varphi _{axicon}}(x,y) + {\varphi _{lens}}(x,y) + {\varphi _{HOCP}}(x,y)$$

(3b)$${\varphi _{spiral}} = l \cdot \arctan (\frac{{\gamma y}}{x})$$

(3c)$${\varphi _{axicon}} ={-} 2\pi \frac{{\sqrt {{x^2} + {{(\gamma y)}^2})} }}{d}$$

(3d)$${\varphi _{lens}} = \frac{{ - \pi ({x^2} + {y^2})}}{{\lambda f}}$$

(3e)$${\varphi _{HOCP}} = u \cdot {x^p} \cdot {y^q}$$

where Eq. (3b) indicates the spiral phase and l is the TC carried by the beam. Equation (3c) is the phase distribution of the axicon and d is the axicon period, controlling the radius of the POV beam. Equation (3d) describes the phase of the Fourier-transform lens, λ is the incident light wavelength and f is the focal length. While Eq. (3e) is the expression for the HOCP, the parameter u controls the conversion rate and affects the degree of astigmatism of the OV beam, the indices p and q are positive integers and $p + q > 2$. Notably, the sum of the indices represents the order of the HOCP and determines the number of sides of the produced polygon beam.

Fig. 2. Phase distribution of metasurface as a superposition of spiral phase plate, axicon, Fourier transformation lens and HOCP.

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The Jones matrix of an anisotropic nano-unit is known to be [34]:

(4)$$J(x,y) = R( - \theta )\left( {\begin{array}{cc} {{t_x}}&0\\ 0&{{t_y}} \end{array}} \right)R(\theta ) = \left( {\begin{array}{cc} {{t_x}{{\cos }^2}\theta + {t_y}{{\sin }^2}\theta }&{{t_x}\sin \theta \cos \theta - {t_y}\sin \theta \cos \theta }\\ {{t_x}\sin \theta \cos \theta - {t_y}\sin \theta \cos \theta }&{{t_x}{{\sin }^2}\theta + {t_y}{{\cos }^2}\theta } \end{array}} \right)$$

where R(θ) is the rotation matrix, θ is the rotation angle of the nanopillar with relative to the reference coordinate system, ${t_x} = {T_x}{e^{i{\varphi _x}}}$ and ${t_y} = {T_y}{e^{i{\varphi _y}}}$ represent the transmission response of the meta-atom to x-polarized and y-polarized lights, and ${T_x}$, ${T_y}$, ${\varphi _x}$ and ${\varphi _y}$ denote the nanopillar transmission amplitude and phase delay, respectively. When circularly polarized light ${E_{in}} = \frac{{\sqrt 2 }}{2}\left[ {\begin{array}{c} 1\\ { \pm i} \end{array}} \right]$ is incident on the metasurface, the output electric field is governed by:

(5)$${E_{\textrm{out}}} = J(x,y) \cdot {E_{in}} = \frac{{\sqrt 2 }}{4}({t_x} + {t_y})\left[ {\begin{array}{c} 1\\ { \pm i} \end{array}} \right] + \frac{{\sqrt 2 }}{4}({t_x} - {t_y}){e^{ {\pm} i2\theta }}\left[ {\begin{array}{c} 1\\ { \mp i} \end{array}} \right]$$

where $\left[ {\begin{array}{c} 1\\ i \end{array}} \right]$ and $\left[ {\begin{array}{c} 1\\ {\textrm{ - }i} \end{array}} \right]$ indicate left-handed circularly polarized (LCP) and right-handed circularly polarized (RCP) lights, respectively. According to Eq. (5), the transmitted light contains both the co-polarized component with the original spin term and the converted spin state carrying an additional phase shift of ±2θ, that is, cross-polarized component. Additionally, to achieve high cross-polarized component, it is essential to satisfy ${T_x} = {T_y} = 1$ and $|{{\varphi_x} - {\varphi_y}} |= \pi$, i.e. the nanopillar is used as a half-wave plate. Here, the meta-atom consists of a rectangular TiO₂ nanopillar with height H = 600 nm and a fused silica square substrate with lattice constant P = 380 nm as shown in Fig. 3(a). TiO₂ is chosen because of its high refractive index and low loss properties at visible frequencies. In order to quantify the transmitted cross-polarized component, the important parameter polarization conversion efficiency (PCE), defined as the proportion of the cross-polarized component in the total transmitted wave, is introduced. The nanopillar need to be optimized using 3D-finite-difference-time-domain (FDTD) to get the maximum cross-polarized component with periodic boundary conditions are applied in the x- and y-directions and perfectly matched layers (PMLs) are implemented in the z-direction. We calculate the transmission amplitude and the corresponding phase delay of nanostructure from 50 nm to 350 nm under the x- and y-polarized incident lights at 633 nm. The simulated incident light is plane wave and propagates along the + z direction. Next, the transmittance under the circularly polarized incident light are calculated based on ${|{{t_{cross}}} |^2} = \frac{1}{4}{({T_x}{e^{i{\varphi _x}}} - {T_y}{e^{i{\varphi _y}}})^2}$ and ${|{{t_{co}}} |^2} = \frac{1}{4}{({T_x}{e^{i{\varphi _x}}} + {T_y}{e^{i{\varphi _y}}})^2}$. Thus, the PCE at the design wavelength can be obtained as $PCE = \frac{{{{|{{t_{cross}}} |}^2}}}{{{{|{{t_{co}}} |}^2} + {{|{{t_{cross}}} |}^2}}}$ in Fig. 3(b). To realize a high PCE, the length and width of the nanopillar are set to L = 260 nm and W = 110 nm, respectively (marked with white pentagram in Fig. 3(b)). We also calculate the PCE of the optimized TiO₂ nanopillar as well as the corresponding phase delay under the x- and y-polarized lights across a broad wavelength range, as displayed in Figs. 3(c) and (d). It can be seen that the meta-atom we choose has high PCE and cross-polarization component at most wavelengths of 520-720 nm. The PCE of this TiO₂ nanopillar is close to 100% (black dashed line in Fig. 3(c)), and the cross-polarization component is up to 90% at the design wavelength λ=633 nm. In addition, the phase difference between x- and y-polarized lights approaches π and the meta-atom can be regarded as a half-wave plate. Therefore, it can be used as the ideal basic unit to construct the geometric metasurface. More importantly, we can flexibly and effectively modulate the incident light by adjusting the orientation angle of the nanopillar without changing its geometric parameters.

Fig. 3. (a) Perspective and top view of a meta-atom consisting of a rectangular TiO₂ nanopillar with height H = 600 nm, length L = 260 nm width W = 110 nm arranging spatially a square silica substrate with lattice constant P = 380 nm. (b) Simulated PCE as a function of nanopillars geometrical parameters L and W at 633 nm. (c) Calculated efficiency of co-, cross-polarized transmitted components and PCE at wavelengths of 520-720 nm. The black dashed line represents the wavelength of 633 nm. (d) Phase shifts ${\varphi _x}$, ${\varphi _y}$ for x- and y- polarized lights and their difference Δ${\varphi}$ for the wavelength ranging 520 nm from 720 nm.

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3. Results and discussion

Based on the above analysis, all-dielectric geometric metasurfaces composed of TiO₂ (L = 260 nm and W = 110 nm) with dimensions of 50 µm × 50 µm are constructed to generate IPPOV beams. All numerical simulations are implemented using the FDTD method, where PML boundary conditions are employed in the x-, y-, and z-axes. First, we generate IPPOV beams with different shapes by controlling the order of the HOCP (3, 4, 5, and 6 respectively). The conversion rates u are 0.2 × 10¹⁵, 0.2 × 10²⁰, 0.2 × 10²⁵, and 0.2 × 10³⁰ and the combinations of p and q are 1-2, 2-2, 2-3, and 3-3, respectively. The specific parameters are d = 4 µm, f = 80 µm, l = 3, λ=633 nm, and γ=0.8. Figures 4(a₁)-(a₄) give the electric field intensity distributions of transmitted beams with different order in the xy plane at z = 80 µm under LCP light incident. Obviously, the resultant beams show irregular triangle (Fig. 4(a₁)), quadrangle (Fig. 4(a₂)), pentagon (Fig. 4(a₃)), and hexagon (Fig. 4(a₄)) intensity distributions respectively (marked with white dash lines), verifying that the order of the HOCP is equal to the number of sides of the beam. In addition, the order also affects the electric field intensity distribution of the beam; the higher the order, the more the light intensity will be concentrated towards the horizontal vertices. Therefore, the particle trajectory can be precisely controlled by adjusting the order of the HOCP, offering more possibilities to manipulate particles. The intensity distributions of the transmitted IPPOV beams in the xz plane are shown in Figs. 4(b₁)-(b₄). When the IPPOV beam propagates in free-space, the beam is first focused at a certain plane (the focusing position is influenced by the order of the HOCP), and then the light wave gradually diverges and evolves into the IPPOV beam with different shapes at the Fourier plane (z = 80 µm) depending on the modulation phase, and the beam size gradually increases as the beam propagates. In order to quantify the dimensions of the generated beams, we also extract normalized cross-sections intensity distributions of different IPPOV beams along the x- and y-directions at z = 80 µm. From Figs. 4(c₁)-(d₄), the horizontal/vertical radii of order 3, 4, 5, and 6 are respectively: 14.2 µm/11.3 µm, 14.2 µm/11.4 µm, 14.4 µm/11.4 µm, and 14.2 µm/11.4 µm, indicating that the size of the IPPOV beam is not affected by the order of the HOCP and still conforms to the “perfect” characteristics.

Fig. 4. When make the ellipticity γ=0.8, the simulated electric field intensity distributions of the IPPOV beam at z = 80 µm with the order is 3 (a₁), 4 (a₂), 5 (a₃), and 6 (a₄). The electric field intensity profiles of resultant beam in the xz plane when the order of HOCP is 3 (b₁), 4 (b₂), 5 (b₃), and 6 (b₄). Normalized cross-sections of the intensity profiles of the IPPOV beams along the horizontal/vertical with the order is 3 (c₁)/(d₁), 4 (c₂)/(d₂), 5 (c₃)/(d₃), and 6 (c₄)/(d₄).

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Of course, there are different combinations of exponents p and q for the same order, which have different modulation effects on the generated IPPOV beam. For further observation and comparison, Fig. 5(a) shows the IPPOV beam without HOCP (order = 0), and the electric field intensity profile presents elliptical. Then based on Fig. 4, we show the electric field intensity distributions of different combinations for the same order in the xy plane at z = 80 µm. When the absolute value of the difference between p and q is equal, the symmetry of the electric field intensity distribution will change, but the light intensity is still concentrated at the horizontal vertexes as shown in Figs. 5(b₁)-(b₂), (c₁)-(c₃), (d₁)-(d₃), and (e₁)-(e₃). When the exponent difference is zero, i.e., p = q, the electric field intensity profile is relatively symmetrical (in Figs. 5(c₂) and (e₂)). When the difference is greater, from Figs. 5(d₁), (e₁) and (e₃) can be seen that the asymmetry and irregularity of the electric field intensity distribution will be more obvious, mainly because the ellipticity is introduced by stretching the coordinate axis and the difference between p and q of the HOCP leads to asymmetric phase distributions. Therefore, the optical field can be modulated by controlling the order and exponent of the HOCP in practical applications, further extending the beam modulation range.

Fig. 5. The electric field intensity distribution of IPPOV beam at z = 80 µm for the order of the HOCP is 0 (without HOCP) (a). The electric field intensity distribution of IPPOV beam at z = 80 µm for the order of the HOCP is 3 with p = 1, q = 2 (b₁) and p = 2, q = 1 (b₂). The intensity profile of irregular quadrilateral beam in the xy plane when p = 1, q = 3 (c₁), and p = 2, q = 2 (c₂) and p = 3, q = 1 (c₃). The electric field intensity distribution of IPPOV beam with the order is 5 when p = 1, q = 4 (d₁), p = 2, q = 3 (d₂), and p = 3, q = 2 (d₃). The electric field intensity profiles of resultant beam at z = 80 µm with the order is 6 when p = 2, q = 4 (e₁), p = 3, q = 3 (e₂), and p = 4, q = 2 (e₃).

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Essentially, the IPPOV beam is a kind of POV beam with TC-independent light intensity distribution at a specific plane. To verify this property, IPPOV beams with different TCs are studied and the parameters are set to d = 4 µm, f = 80 µm, λ=633 nm, γ=0.8, u = 0.2 × 10²⁰, p = q = 2, and l = -3, 4, and 5. Figures 6(a₁)-(a₃) show the electric field intensity of cross-polarized components carrying different TCs at z = 80 µm for LCP illumination. It can be found that the electric field intensity distributions exhibit quadrangle, and the maximum intensity is located at the horizontal vertexes while being influenced by the sign of the TC. In order to observe the evolution of the beam, we also show the electric field distributions of the beams with different TCs in the xz plane. As can be seen from Figs. 6(b₁)-(b₃), the beams are first focused at z = 43 µm, and gradually diverge as the propagation distance increases. Figures 6(c₁)-(d₃) demonstrate the cross-section of normalized intensity curves along y = 0/x = 0, and the horizontal/vertical radii of the beams carrying TCs l = -3, 4, and 5 are obtained as 14.2 µm/11.4 µm, 14.2 µm/11.5 µm, 14.5 µm/11.7 µm, respectively. The characteristics that the size of the generated IPPOV beam is independent of the TC is further verified by combining the dimensions of l = 3 in Figs. 4(c₂) and (d₂).

Fig. 6. The electric field intensity distribution of IPPOV beam at z = 80 µm with l = -3 (a₁), 4 (a₂), 5 (a₃). The electric field profile of the generated IPPOV beam carrying the TC of l = -3 (b₁), 4(b₂), 5 (b₃) in the xz plane. The Normalized cross-sections of the intensity profiles of the IPPOV beam along the horizontal /vertical direction with l = -3 (c₁)/(d₁), 4 (c₂)/(d₂), 5 (c₃)/(d₃).

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Furthermore, keeping other parameters constant and making the ellipticity γ=1.2, the light intensity distributions of the IPPOV beams carrying different TCs for LCP illumination are also demonstrated. Since the order of the HOCP remains 4 (u = 0.2 × 10²⁰, p = q = 2), the electric field intensity distributions of IPPOV beams with different TCs at z = 80 µm display quadrangle from Figs. 7(a₁)-(a₄), while the maximum light intensity switches from the horizontal position to the vertical vertexes due to ellipticity change. Alternatively, the manipulation of the optical field intensity vertexes based on the ellipticity can further accurately control the particle acceleration and realize more complex particle motion [54]. The electric field intensity profiles of the resultant beams in the xz plane (see Figs. 7(b₁)-(b₄)) are similar to Fig. 6, where the beam first focused at z = 41 µm, then gradually diverging and finally evolving into an irregular quadrangle POV beam. Figures 7(c₃)-(d₄) give the normalized intensity distributions of the IPPOV beams along the x- and y-directional cross-sections for l = -3, 3, 4, and 5, and the corresponding horizontal/vertical diameters are 14.4 µm/17.0 µm, 14.4 µm/17.1 µm, 14.5 µm/17.3 µm and 14.6 µm/17.5 µm, respectively. The reason for the slightly different radius is mainly the ring width change caused by the increasing TC [32,47]. Combined with Fig. 6, the beam size and light intensity distribution can be adjusted by changing the ellipticity factor, increasing the modulation dimension of the POV beam.

Fig. 7. When set the ellipticity γ=1.2, the electric field intensity distribution of IPPOV beam at z = 80 µm with l = -3 (a₁), 3 (a₂), 4 (a₃), and 5 (a₄). The electric field profile of the generated IPPOV beam carrying the TCs of l = -3 (b₁), 3 (b₂), 4 (b₃), and 5 (b₄) in the xz plane. The Normalized cross-sections of the intensity profiles of the IPPOV beam along the horizontal /vertical direction with l = -3 (c₁)/(d₁), 3 (c₂)/(d₂), 4 (c₃)/(d₃) and 5 (c₄)/(d₄).

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It is known that the traditional POV beam converts the incident light into a hollow non-diffractive Bessel-like beam with a certain TC at the center of focal plane, showing a donut-shaped intensity pattern and the ring radius is gradually broadened as TC increases at this position [45,46]. The IPPOV beam we propose is different from the traditional POV beam at the focusing position. We further extract the electric field distributions of the IPPOV beam carrying different TCs in the xy plane of Fig. 6 (γ=0.8) at z = 43 µm and Fig. 7 (γ=1.2) at z = 41 µm, as shown in Fig. 8. Since we introduce the ellipticity by stretching the coordinate axis, the optical field at the focal plane has a certain rotation. As can be seen from the Fig. 8, both the ellipticity and TC can manipulate the electric field intensity distribution of the generated beam at the focal plane. In general, the number of bright spots minus one is equal to the absolute value of TC, and the ellipticity affects the rotation direction of these spots. Specifically, for γ<1, if the TC is negative, the angle between the electric field intensity profile and the positive direction along x-axis is less than 90° in Fig. 8(a₁), and the corresponding angle is greater than 90° when the sign of the TC carried is positive as shown in Figs. 8(a₂)-(a₄). While γ>1, the angle of the electric field intensity to the positive x-axis is greater than 90° if the sign of the TC carried is negative from Fig. 8(b₁), and the angle to the positive direction along x-axis is less than 90° if the TC is positive as demonstrated in Figs. 8(b₂)-(b₄). Therefore, we provide a simple and efficient way to determine the TC of the IPPOV beam by observing the number and direction of the bright spots at the focal plane, which has not been mentioned in previous studies [32,45–51].

Fig. 8. The electric field intensity distribution of IPPOV beam at z = 43 µm with l = -3 (a₁), 3(a₂), 4 (a₃), and 5 (a₄) when set the ellipticity γ=0.8. The electric field intensity profile of IPPOV beam at z = 41 µm with l = -3 (b₁), 3 (b₂), 4 (b₃), and 5 (b₄) when the ellipticity γ=1.2.

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In addition to regulating the light intensity distribution of the IPPOV beam by controlling the ellipticity, the conversion rate u of the HOCP can also modulate the electric field intensity. Making d = 4 µm, f = 80 µm, λ=633 nm, order = 4 (p = q = 2), l = 3 and varying u from 0.1 × 10²⁰ to 0.6 × 10²⁰, the electric field intensity distributions of beams in the xy plane at z = 80 µm with different ellipticities γ=0.8 and γ=1.2 are demonstrated in Fig. 9. As u increases, the electric field intensity is concentrated towards the horizontal vertexes for γ=0.8 from Figs. 9(a₁)-(a₄) and the electric field intensity tends to the vertical vertexes when γ=1.2 as shown in Figs. 9(b₁)-(b₄). Furthermore, the shape of the generated beams become more significant with increasing u. Therefore, the parameter u can be chosen reasonably in practical applications, opening up new paths for modulating the electric field intensity distribution of the beam and further enriching the POV beam.

Fig. 9. The electric field intensity distribution of IPPOV beam at z = 80 µm with u = 0.1 × 10²⁰ (a₁), 0.2 × 10²⁰ (a₂), 0.4 × 10²⁰ (a₃), and 0.6 × 10²⁰ (a₄) when the ellipticity γ=0.8. The electric field intensity profile of resultant beam in the xy plane at z = 80 µm with u = 0.1 × 10²⁰ (b₁), 0.2 × 10²⁰ (b₂), 0.4 × 10²⁰ (b₃), and 0.6 × 10²⁰ (b₄) for the ellipticity γ=1.2.

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4. Conclusion

In summary, we originally introduce the ellipticity and HOCP to the conventional POV beam, using the all-dielectric geometric metasurface consisted of TiO₂ nanopillars to generate novel IPPOV beams at visible light. Versatile shapes of the IPPOV beam can be achieved flexibly by controlling the order of the HOCP, and the electric field intensity distribution of the generated beam at the Fourier plane can be modulated via changing the ellipticity γ and the parameter u of the HOCP. In addition, the “perfect” properties of IPPOV beams are verified by quantitatively analyzing the beam size with different TCs under various parameters. More importantly, the propagation characteristics of IPPOV beams in free-space are studied. The beam first focuses on a certain position, gradually diverges with the increase of propagation distance, and finally evolves into different IPPOV beams according to the modulation phase. We extract the electric field intensity distribution at the focal plane. It can be found that the number of bright spots minus one gives the magnitude of the TC, and the sign of TC can be determine easily by investigating the rotation direction of these spots, providing a simple and effective method to measure the TC of POV beams. Moreover, this work enriches the mode distribution of POV beams and increases the modulation dimension of beams.

Funding

National Natural Science Foundation of China (62175070, 61875057); Natural Science Foundation of Guangdong Province (2021A1515012652); Science and Technology Program of Guangzhou (2019050001).

Disclosures

The authors declare that there are no conflicts of interest related to this article.

Data availability

Date 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. P. Coullet, L. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73(5), 403–408 (1989). [CrossRef]

2. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992). [CrossRef]

3. J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, and M. Tur, “Terabit free-space data transmission employing orbital angular momentum multiplexing,” Nat. Photonics 6(7), 488–496 (2012). [CrossRef]

4. M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5(6), 343–348 (2011). [CrossRef]

5. S. Fürhapter, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “Spiral phase contrast imaging in microscopy,” Opt. Express 13(3), 689–694 (2005). [CrossRef]

6. D. Gauthier, P. R. Ribič, G. Adhikary, A. Camper, C. Chappuis, R. Cucini, L. DiMauro, G. Dovillaire, F. Frassetto, and R. Géneaux, “Tunable orbital angular momentum in high-harmonic generation,” Nat. Commun. 8(1), 14971 (2017). [CrossRef]

7. P. Li, S. Liu, T. Peng, G. Xie, X. Gan, and J. Zhao, “Spiral autofocusing Airy beams carrying power-exponent-phase vortices,” Opt. Express 22(7), 7598–7606 (2014). [CrossRef]

8. J. A. Rodrigo and T. Alieva, “Freestyle 3D laser traps: tools for studying light-driven particle dynamics and beyond,” Optica 2(9), 812–815 (2015). [CrossRef]

9. Y. Zhang, Y. Xue, Z. Zhu, G. Rui, Y. Cui, and B. Gu, “Theoretical investigation on asymmetrical spinning and orbiting motions of particles in a tightly focused power-exponent azimuthal-variant vector field,” Opt. Express 26(4), 4318–4329 (2018). [CrossRef]

10. H. Zhang, X. Li, H. Ma, M. Tang, H. Li, J. Tang, and Y. Cai, “Grafted optical vortex with controllable orbital angular momentum distribution,” Opt. Express 27(16), 22930–22938 (2019). [CrossRef]

11. H. Yan, E. Zhang, B. Zhao, and K. Duan, “Free-space propagation of guided optical vortices excited in an annular core fiber,” Opt. Express 20(16), 17904–17915 (2012). [CrossRef]

12. A. S. Ostrovsky, C. Rickenstorff-Parrao, and V. Arrizón, “Generation of the “perfect” optical vortex using a liquid-crystal spatial light modulator,” Opt. Lett. 38(4), 534–536 (2013). [CrossRef]

13. W. Shao, S. Huang, X. Liu, and M. Chen, “Free-space optical communication with perfect optical vortex beams multiplexing,” Opt. Commun. 427(15), 545–550 (2018). [CrossRef]

14. M. Chen, M. Mazilu, Y. Arita, E. M. Wright, and K. Dholakia, “Dynamics of microparticles trapped in a perfect vortex beam,” Opt. Lett. 38(22), 4919–4922 (2013). [CrossRef]

15. J. Pinnell, I. Nape, M. de Oliveira, N. TabeBordbar, and A. Forbes, “Experimental demonstration of 11-dimensional 10-party quantum secret sharing,” Laser Photonics Rev. 14(9), 2000012 (2020). [CrossRef]

16. M. Jabir, N. Apurv Chaitanya, A. Aadhi, and G. K. Samanta, “Generation of “perfect” vortex of variable size and its effect in angular spectrum of the down-converted photons,” Sci. Rep. 6(1), 21877 (2016). [CrossRef]

17. P. Vaity and L. Rusch, “Perfect vortex beam: Fourier transformation of a Bessel beam,” Opt. Lett. 40(4), 597–600 (2015). [CrossRef]

18. T. Wang, S. Fu, F. He, and C. Gao, “Generation of perfect polarization vortices using combined gratings in a single spatial light modulator,” Appl. Opt. 56(27), 7567–7571 (2017). [CrossRef]

19. C. Zhang, C. Min, and X.-C. Yuan, “Shaping perfect optical vortex with amplitude modulated using a digital micro-mirror device,” Opt. Commun. 381(15), 292–295 (2016). [CrossRef]

20. A. Kovalev, V. Kotlyar, and A. Porfirev, “A highly efficient element for generating elliptic perfect optical vortices,” Appl. Phys. Lett. 110(26), 261102 (2017). [CrossRef]

21. G. Tkachenko, M. Chen, K. Dholakia, and M. Mazilu, “Is it possible to create a perfect fractional vortex beam?” Optica 4(3), 330–333 (2017). [CrossRef]

22. P. Li, Y. Zhang, S. Liu, C. Ma, L. Han, H. Cheng, and J. Zhao, “Generation of perfect vectorial vortex beams,” Opt. Lett. 41(10), 2205–2208 (2016). [CrossRef]

23. C. Wang, Y. Ren, T. Liu, C. Luo, S. Qiu, Z. Li, and H. Wu, “Generation and measurement of high-order optical vortices by using the cross phase,” Appl. Opt. 59(13), 4040–4047 (2020). [CrossRef]

24. Y. Ren, C. Wang, T. Liu, Z. Wang, C. Yin, S. Qiu, Z. Li, and H. Wu, “Polygonal shaping and multi-singularity manipulation of optical vortices via high-order cross-phase,” Opt. Express 28(18), 26257–26266 (2020). [CrossRef]

25. C. Wang, Y. Ren, T. Liu, Z. Liu, S. Qiu, Z. Li, Y. Ding, and H. Wu, “Measurement and shaping of circular Airy vortex via cross-phase,” Opt. Commun. 497, 127185 (2021). [CrossRef]

26. C. Wang, Y. Ren, T. Liu, Z. Liu, S. Qiu, Z. Li, Y. Ding, and H. Wu, “Generating a new type of polygonal perfect optical vortex,” Opt. Express 29(9), 14126–14134 (2021). [CrossRef]

27. M. P. MacDonald, G. C. Spalding, and K. Dholakia, “Microfluidic sorting in an optical lattice,” Nature 426(6965), 421–424 (2003). [CrossRef]

28. A. A. Kovalev, V. V. Kotlyar, and A. P. Porfirev, “Optical trapping and moving of microparticles by using asymmetrical Laguerre–Gaussian beams,” Opt. Lett. 41(11), 2426–2429 (2016). [CrossRef]

29. M. A. Cox, L. Cheng, C. Rosales-Guzmán, and A. Forbes, “Modal diversity for robust free-space optical communications,” Phys. Rev. A 10(2), 024020 (2018). [CrossRef]

30. B. Salgado, E. Peters, and G. Funes, “Detection of Hermite-Gaussian modes in vortex beams affected by convective turbulence,” Waves Random Complex Media 32(4), 1879–1889 (2022). [CrossRef]

31. S. Ast, S. Di Pace, J. Millo, M. Pichot, M. Turconi, N. Christensen, and W. Chaibi, “Higher-order Hermite-Gauss modes for gravitational waves detection,” Phys. Rev. D 103(4), 042008 (2021). [CrossRef]

32. Y. Zhang, W. Liu, J. Gao, and X. Yang, “Generating focused 3D perfect vortex beams by plasmonic metasurfaces,” Adv. Opt. Mater. 6(4), 1701228 (2018). [CrossRef]

33. S. Fu, C. Gao, T. Wang, Z. Zhang, and Y. Zhai, “Detection of topological charges for coaxial multiplexed perfect vortices,” in IEEE Opto-Electronics and Communications Conference (OECC) and Photonics Global Conference (PGC) (IEEE, 2017), pp.1–2.

34. H. T. Chen, A. J. Taylor, and N. Yu, “A review of metasurfaces: physics and applications,” Rep. Prog. Phys. 79(7), 076401 (2016). [CrossRef]

35. N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011). [CrossRef]

36. M. Khorasaninejad, W. T. Chen, R. C. Devlin, J. Oh, A. Y. Zhu, and F. Capasso, “Metalenses at visible wavelengths: Diffraction-limited focusing and subwavelength resolution imaging,” Science 352(6290), 1190–1194 (2016). [CrossRef]

37. M. Wang, J. S. Lee, S. Aggarwal, N. Farmakidis, Y. He, T. Cheng, and H. Bhaskaran, “Varifocal Metalens Using Tunable and Ultralow-loss Dielectrics,” Adv. Sci. 10(6), 2204899 (2023). [CrossRef]

38. P. Georgi, Q. Wei, B. Sain, C. Schlickriede, Y. Wang, L. Huang, and T. Zentgraf, “Optical secret sharing with cascaded metasurface holography,” Sci. Adv. 7(16), eabf9718 (2021). [CrossRef]

39. L. Huang, X. Chen, H. Mühlenbernd, H. Zhang, S. Chen, B. Bai, Q. Tan, G. Jin, K.-W. Cheah, and C.-W. Qiu, “Three-dimensional optical holography using a plasmonic metasurface,” Nat. Commun. 4(1), 2808 (2013). [CrossRef]

40. Y. Liu, L. Chen, C. Zhou, K. Guo, X. Wang, Y. Hong, X. Yang, Z. Wei, and H. Liu, “Theoretical study on generation of multidimensional focused and vector vortex beams via all-dielectric spin-multiplexed metasurface,” Nanomaterials 12(4), 580 (2022). [CrossRef]

41. F. Yue, D. Wen, C. Zhang, B. D. Gerardot, W. Wang, S. Zhang, and X. Chen, “Multichannel polarization-controllable superpositions of orbital angular momentum states,” Adv. Mater. 29(15), 1603838 (2017). [CrossRef]

42. A. Ma, Y. Intaravanne, J. Han, R. Wang, and X. Chen, “Polarization detection using light’s orbital angular momentum,” Adv. Opt. Mater. 8(18), 2000484 (2020). [CrossRef]

43. Y. Ming, Y. Intaravanne, H. Ahmed, M. Kenney, Y. Q. Lu, and X. Chen, “Creating composite vortex beams with a single geometric metasurface,” Adv. Mater. 34(18), 2109714 (2022). [CrossRef]

44. W. Liu, Q. Yang, Q. Xu, X. Jiang, T. Wu, J. Gu, J. Han, and W. Zhang, “Multichannel terahertz quasi-perfect vortex beams generation enabled by multifunctional metasurfaces,” Nanophotonics 11(16), 3631–3640 (2022). [CrossRef]

45. J. He, M. Wan, X. Zhang, S. Yuan, L. Zhang, and J. Wang, “Generating ultraviolet perfect vortex beams using a high-efficiency broadband dielectric metasurface,” Opt. Express 30(4), 4806–4816 (2022). [CrossRef]

46. J. Xie, H. Guo, S. Zhuang, and J. Hu, “Polarization-controllable perfect vortex beam by a dielectric metasurface,” Opt. Express 29(3), 3081–3089 (2021). [CrossRef]

47. Y. Liu, C. Zhou, K. Guo, Z. Wei, and H. Liu, “Generation of multi-channel perfect vortex beams with the controllable ring radius and the topological charge based on an all-dielectric transmission metasurface,” Opt. Express 30(17), 30881–30893 (2022). [CrossRef]

48. Z. Shen, L. Teng, Z. Xiang, L. Li, Y. Rui, and Y. Shen, “Generation of polarization-insensitive perfect vortices by dielectric metasurfaces with diverging or converging axicon phases,” Opt. Laser Technol. 155, 108409 (2022). [CrossRef]

49. J. Han, Y. Intaravanne, A. Ma, R. Wang, S. Li, Z. Li, S. Chen, J. Li, and X. Chen, “Optical metasurfaces for generation and superposition of optical ring vortex beams,” Laser Photonics Rev. 14(9), 2000146 (2020). [CrossRef]

50. K. Cheng, Z. Liu, Z.-D. Hu, G. Cao, J. Wu, and J. Wang, “Generation of integer and fractional perfect vortex beams using all-dielectric geometrical phase metasurfaces,” Appl. Phys. Lett. 120(20), 201701 (2022). [CrossRef]

51. Q. Zhou, M. Liu, W. Zhu, L. Chen, Y. Ren, H. J. Lezec, Y. Lu, A. Agrawal, and T. Xu, “Generation of perfect vortex beams by dielectric geometric metasurface for visible light,” Laser Photonics Rev. 15(12), 2100390 (2021). [CrossRef]

52. H. Ahmed, Y. Intaravanne, Y. Ming, M. A. Ansari, G. S. Buller, T. Zentgraf, and X. Chen, “Multichannel superposition of grafted perfect vortex beams,” Adv. Mater. 34(30), 2203044 (2022). [CrossRef]

53. B. Zhang, J. Wang, J. Wu, and S. Tian, “Creating perfect composite vortex beams with a single all-dielectric geometric metasurface,” Opt. Express 30(22), 40231–40242 (2022). [CrossRef]

54. H. Wang, L. Tang, J. Ma, X. Zheng, D. Song, Y. Hu, Y. Li, and Z. Chen, “Synthetic optical vortex beams from the analogous trajectory change of an artificial satellite,” Photonics Res. 7(9), 1101–1105 (2019). [CrossRef]

Generation and measurement of irregular polygonal perfect vortex optical beam based on all-dielectric geometric metasurface

Abstract

1. Introduction

2. Theory and design

3. Results and discussion

4. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (9)

Equations (9)

Optics Express