Vector vortex beam generation with dolphin-shaped cell meta-surface

Zhuo Yang; Deng-Feng Kuang; Fang Cheng

doi:10.1364/OE.25.022780

1. Introduction

Vortex beams carrying orbital angular momentum are attracting more and more attention [1–3]. The characteristics of vortex beams are utilized to expand their applications such as optical communications [3–6], optical tweezers [7–9] and quantum computing [10,11]. Vortex beams are generated by spiral phase plates [12–14], spatial light modulators [15,16] and computer generated holograms [17–19] in the early decade. Nowadays, sub-wavelength scale artificial optical elements are utilized tentatively to generate vortex beams with the microminiaturization and integration of optical elements [20–26].

Nonetheless, despite this success, vector beam generation by sub-wavelength scale artificial optical elements faces two challenges. Firstly, elements converting linearly polarized beams into vortex beams are extremely difficult to fabricate. The integration can be achieved with the utilization of elements merely to convert linearly polarized laser beams into vortex beams directly without polarization converters. Metallic helical nanocone, which is a combination of helical structure and conic structure, is presented in vector vortex beams generation when the incident waves are linearly polarized fields [20,21]. However, the apex of helical nanocone, of which the scale is only a few nanometers, is difficult to fabricate. Helical nanocone with low precision apex is difficult to generate vortex beam. Secondly, in most cases, sub-wavelength scale optical elements that are easily fabricated can only convert circularly polarized light into vortex beam. Nowadays, meta-surfaces, which are two-dimensional artificial optical elements with sub-wavelength scale [22], are utilized to tune vortex beams [23–25]. Nevertheless, only circularly polarized light can be transformed into vortex beam by using meta-surfaces in most cases on the basis of Pancharatnam-Berry phase principle [24,26]. Therefore, it’s still rarely mentioned that vortex beam generation with the illumination of linearly polarized beams.

Dolphin-shaped cell meta-surface (DSCMS), which is a combination of dolphin-shaped metallic cells and dielectric substrate, is presented to generate localized vector vortex beam with the illumination of linearly polarized beam. Surface plasmon polaritons are excited at the boundary of the metallic cells, then guided by the metallic structures and finally squeezed to the tips to form highly localized strong electromagnetic fields which generate the intensity of vector vortex beams at z component. The aim of DSCMS is to realize the transmission, characterization and manipulation of the nano-scale localized vector vortex beams by using new composite artificial electromagnetic elements, promoting the development of communication, bioscience and materiality. DSCMS has important application values in broadband optical communication [27,28] and nano-particles controlling [29] on the basis of the principle of localized vector vortex beam.

2. Structure of DSCMS and simulation method

DSCMS is a circular array of metallic dolphin-shaped micron-nano-sized structures as shown in Fig. 1. The dolphin-shaped cell geometrically consists of two semi-crescent-shaped structures constituted by two semi-cylinders as shown in Fig. 1(a). Dolphin-shaped cell structures are angularly arranged to form a circular array as shown in Fig. 1(b). The connection between the geometric center of the dolphin-shaped cell and the geometric center of the circular array divides the circle into $N$ equal parts. The central axis of the dolphin-shaped cell structure is always pointing to the center of the circular array of which the radius is $R$ . $α$ is the angle between the geometric centers of the two adjacent dolphin-shaped cell structures. $R_{1}$ and $R_{2}$ are the radii of the two semi-cylindrical structures that intercept the structures of the dolphin-shaped cells ( $R_{1} < R_{2}$ ). The numerical relationship between $R_{1}$ and $R_{2}$ is $R_{2} = 2 R_{1}$ which the center distance is $d = R_{1}$ . The distance between two tips of the dolphin-shaped cell structures is $d' = 6 R_{1}$ . The thickness of dolphin-shaped structures is $d_{1}$ , while the thickness of the substrate is $d_{2}$ . The material of dolphin-shaped structures is silver and the substrate is comprised of silica.

Fig. 1 The sketch of (a) dolphin-shaped cell and (b) dolphin-shaped cell meta-surface.

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We simulate the electric fields of DSCMSs with three-dimensional finite-difference time domain (3D-FDTD) method [30] and perfectly matched layers are used as simulation boundaries when the DSCMS is illuminated by linearly polarized light.

3. Frequency selection of DSCMS

The incident linearly polarized wave propagates in the z direction. The structural parameters of DSCMS are initialized as follows: $R_{1} = 150 n m$ , $R_{2} = 2 R_{1} = 300 n m$ , $R = 1000 n m$ , $d_{1} = 100 n m$ , $d_{2} = 300 n m$ . The length and width of the substrate made by glass are much larger than that of the metallic circular array. The vortex phase generation is closely related to the metallic circular array. Therefore, the thickness of the substrate does not affect the vortex phase generation at sub-wavelength scale. It is not necessary to set the height for optimization rigorously. We only need to ensure that the thickness of the meta-surface can be thin enough in order to the miniaturization of the optical elements. Drude models are used for both the permittivity $ε (ω)$ and permeability $μ (ω)$ of materials with identical dispersion forms as follows [31,32]

ε (ω) = ε_{0} (1 - \frac{ω_{p e}^{2}}{ω^{2} - j ω γ_{e}})

μ (ω) = μ_{0} (1 - \frac{ω_{p m}^{2}}{ω^{2} - j ω γ_{m}})

where

ω_{p e}

and

ω_{p m}

are the electric and magnetic plasma frequencies,

γ_{e}

and

γ_{m}

are the corresponding collision frequencies, respectively.

The spectral transmittance of DSCMS is illustrated in Fig. 2(a). The maximum transmittance is 0.8782 with the incident wavelength of 321 nm, while the minimum transmittance is 0.4598 with the incident wavelength of 448 nm. The distributions of electromagnetic fields around the cells with different incident wavelength are shown in Figs. 2(b)-2(d). The relative permittivity of silver is shown in Table 1, which $ε'$ is the real part and $ε''$ is the imaginary part of the relative permittivity. The unit of the intensity of the electric fields is taken as 1 a.u., which is normalized to the intensity of the incident electric field. The electronic fields remain loosely bound to the interface between silver and air with the maximum transmittance, while the electronic fields remain strongly bound to the interface between silver and air with the minimum transmittance. Then the distribution of electromagnetic fields with the incident wavelength of 660 nm is discussed because the transmittance with the wavelength of 660 nm is about 0.65 which is middle of 0.5 (minimum transmittance) and 0.8 (maximum transmittance). The electronic fields remain moderately bound to the interface between silver and air with the incident wavelength of 660 nm. Surface plasmon polaritons are excited at the boundary of the metallic cells, then guided by the metallic structures and finally squeezed to the tips to form highly localized strong electromagnetic fields. The localized vector electric field is wavelength selective with the maximum intensity enhancement up to 1000 a.u. with the incident wavelength of 660 nm. Transmitted fields behind DSCMS are tuned prospectively when the electronic fields are radiated and bounded simultaneously by the silver dolphin-shaped structures.

Fig. 2 (a) Transmission spectrum of DSCMS and distributions of total electronic fields intensity around the cells with the incident wavelength of (b) 321nm, (c) 448nm and (d) 660nm.

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Table 1. Relative Permittivity of Silver According to Drude Mode

View Table

4. Vector vortex beam generated by the DSCMS

We simulate the electromagnetic fields of DSCMS with 3D-FDTD method and perfectly matched layers are used as simulation boundaries when the DSCMS is illuminated by linearly polarized light. The incident wave propagates in the z direction. The intensity and phase distributions of the transmission fields at different distances behind DSCMS are presented in Fig. 3 with the incident wavelength of 660 nm. The distance between the meta-surface and calculated planes is D, which can be thought as the distance between the calculated planes and the interface of metallic structures and dielectric substrate for the reason that the effects of silica in the range of visible light are few. Figure 3(a), (c) and (e) show the electric field intensity distributions of the transmission fields in xy plane at D = 1000 nm, D = 2000 nm and D = 3000 nm. As the distance increases, the hollow ring of the vortex beam intensity distribution becomes apparent. Figure 3(b), (d) and (f) show the z components electric field phase distributions of the transmitted fields in xy plane at D = 1000 nm, D = 2000 nm and D = 3000 nm. The central phase distributions exhibit vortex characteristic. The helical phase light field gradually diverges and the central vortex is becoming even more obvious with the increase of the propagation distance. The light fields radiated by the tips of the inner circle form vector vortex beams. Simultaneously, the light fields radiated by the tips of the outer circle are incorporated into the vortex fields formed by the tips of the inner circle. As the distance continues to increase, the vortex characteristic is no longer obvious though the area of central vortex phase becomes larger. This means that the transmission fields are localized vortex beams.

Fig. 3 The total electric field intensity distributions of the total transmission fields at different distances (a) D = 1000 nm, (c) D = 2000 nm, (e) D = 3000 nm and the phase distributions of the transmission fields in z components at different distances (b) D = 1000 nm, (d) D = 2000 nm, (f) D = 3000 nm behind DSCMS.

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The ratios of the intensity of the light fields at z component to the total intensity of the light fields at different distances at the rear of the DSCMS and the phase distributions of the transmission fields at z component are set out in Fig. 4. Figure 4(a), (c) and (e) show the z components electric field phase distributions of the transmission fields in xy plane at D = 400 nm, D = 800 nm and D = 2000 nm. The area of central vortex phase becomes larger when the distances are increase as we discussed before. Figure 4(b), (d) and (f) the ratios of the intensity of the light fields at z component to the total intensity of the light fields at D = 400 nm, D = 800 nm and D = 2000 nm. When the distance is 400 nm, the maximum value of the ratio of transmission fields intensity at z component to the total field intensity is about 80%. The maximum value of the ratio is approximately 45% when the distance is 800 nm, likewise 20% when the distance is 2000 nm. It is inferred that the ratios of transmission fields intensity at z component to the total field intensity decrease as the distances increase. In other words, localized transmitted fields of DSCMS, which are also called localized vector vortex beams, are utilized to achieve the effects of vortex beams. However, the effect of phase vortex at the center of the transmission field is obviously enhanced by the increase of distance. Therefore, the relations between the vortex characteristics of transmitted fields in z components and the proportion of z components transmitted electric fields are dialectical. Here, we define orbital angular momentum (OAM) mode purity as the ratio between the power of the OAM mode with topological charge to that of the entire beam [33]. The OAM mode purity is approximately 9.61%, 9.47% and 3.67% when D is 400 nm, 800 nm and 2000 nm according to the simulation in Fig. 4. It is inferred that the OAM mode purity decreases as the distance increases. Simultaneously, the velocity of the purity diminution becomes larger when the distance increases.

Fig. 4 The phase distributions of the z components transmission fields in xy plane at different distances (a) D = 400 nm, (c) D = 800 nm, (e) D = 2000 nm and the ratios of the intensity of the light fields at z component to the total intensity of the light fields at different distances (b) D = 400 nm, (d) D = 800 nm, (f) D = 2000 nm behind DSCMS.

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The spot size is the most practical problem we need to consider. The light emitted by ordinary lasers could not meet our needs. We want to get a micro-size focused beam. One idea is to use a microscopic objective system to focus incident light. This method was widely utilized in optical tweezers and nano-particles controlling. Increasing the structural dimensions of the dolphin-shaped cells would affect the results of frequency selection reported in Section 2.

One of the problems that could appear is the localized vector vortex beam generation when the transmitted field is performed. This technical problem could become an advantage in principle when the coupling into a single-mode fiber is needed. According to our previous study, the combined configuration of DSCMS and metallic granary-shaped nano-tip (MGSN) [34] can be proposed to achieve and manipulate three-dimensional vector vortex optical field at the scale of several tens nanometers for the applications of extremely localized vector vortex beam generation.

We simulate the electromagnetic fields of DSCMS with three-dimensional finite-difference time domain (3D-FDTD) method when the DSCMS are illuminated by linearly polarized light. The physical model of the optical vortex generation is provided on the basis of the time-dependent Maxwell’s equations in electrodynamics, which are rigorous physical models in electromagnetic solutions. It could not be the focus of this paper that providing the analytical solutions of electromagnetic fields. Simultaneously, we provide the explanation according to the existing similar theoretical models in order to make physical models easier to be understood.

The abrupt phase change of the meta-surfaces is utilized to explain the vortex phase generated by DSCMS. Currently, arbitrary surfaces with sub-wavelength thickness can be regarded as meta-surfaces which have different properties. The DSCMS can be thought as planar elements whose properties are characterized by its effective permittivity $ε$ and permeability $μ$ while the complex cells and substrate structures of the meta-surfaces can be ignored. Such meta-surfaces can be described in terms of a permittivity tensor as follow [35, 36]

\bar{\bar{ε}} = \frac{ε_{1} d_{1} + ε_{2} d_{2}}{d_{1} + d_{2}} (\bar{x x} + \bar{y y}) + {(\frac{ε_{1}^{- 1} d_{1} + ε_{2}^{- 1} d_{2}}{d_{1} + d_{2}})}^{- 1} \bar{z z}

The transmission phase can be directly obtained as follow [34]

ϕ_{t} = \arg (\frac{2 Y_{0} - Y_{e}}{2 Y_{0} + Y_{e}} + 1)

The horizontal admittances for each medium is defined as

Y_{i} = H_{x i} / E_{y i}

for TE polarization and

Y_{i} = H_{y i} / E_{x i}

for TM polarization (here

i

is the index of the layers). Synchronously, surface plasmon polaritons propagate forward along the surface of the structures with the incident wavelength of 660nm, constantly being compressed to the tips to form hot spots. Figure 5 shows that the phase distributions of the transmission fields at D = 2000 nm generated by DSCMS and its mirrored structure are adverse because of the opposite rotating direction of tips. It demonstrates that the intensity of central vortex beams can be generated by strong fields on the tips. As the propagation distance increases, the central vortex beams diverge outward.

Fig. 5 (a) The phase distributions of the transmission fields at D = 2000 nm generated by DSCMS and (b) its mirrored structure.

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Structural factor $m$ , which is proposed to analyze the optical response of the system for different values of the ratio between R₂ and R₁, is defined as follow

m = R_{1} / R_{2}

The phase distributions of the z component transmitted fields generated by DSCMS with different structural factors at D = 2000 nm are shown in Fig. 6. As the structural factor decreases, it has better results of the central optical vortex.

Fig. 6 The phase distributions of the z component transmission fields at D = 2000 nm generated by DSCMS with the structural factor (a) m = 1.5, (b) m = 2 and (c) m = 3.

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5. Conclusion

In summary, we put forward the DSCMS, which is the combined configuration of metallic dolphin-shaped cell structures and dielectric substrate, to produce localized vector vortex beam. The intensity and phase of the electric fields of the optical field generated by the DSCMS with the illumination of linearly polarized beam are calculated by 3D-FDTD method. We can obtain the localized vector vortex beams of DSCMS when the bottom dielectric edge of DSCMS is illuminated by linearly polarized beam. The electronic fields remain moderately bound to the interface between silver and air with the incident wavelength of 660 nm. As the distance increases, the hollow ring of the vortex beam intensity distribution becomes apparent, while the vortex characteristic is no longer obvious though the area of central vortex phase becomes larger. We also obtain the dialectical relations between the vortex characteristics of transmitted fields in z components and the proportion of z components transmitted electric fields. The localized vector vortex beam generated by DSCMS can be utilized for communication, bioscience and materiality.

Funding

National Natural Science Foundation of China (NSFC) (11274186)

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Vector vortex beam generation with dolphin-shaped cell meta-surface

Abstract

1. Introduction

2. Structure of DSCMS and simulation method

3. Frequency selection of DSCMS

4. Vector vortex beam generated by the DSCMS

5. Conclusion

Funding

References and links

Cited By

Figures (6)

Tables (1)

Equations (5)

Optics Express

Material	321 nm			448 nm			660 nm
Material	$ε'$	$ε''$		$ε'$	$ε''$		$ε'$	$ε''$
Silver (Ag)	0.15403	0.91960		−5.3159	0.63067		−16.065	1.1853