High-efficiency, broadband all-dielectric transmission metasurface for optical vortex generation

Heng Zhou; Jiaqi Yang; Chunqing Gao; Shiyao Fu

doi:10.1364/OME.9.002699

1. Introduction

Beams carrying orbital angular momentum (OAM), have received extensive attention since Allen demonstrated it in 1992 [1]. Because of excellent properties, OAM beams have many applications in optical communications [2–5], rotation detection [6–8], optical tweezers [9–11] and so on. Many methods have been studied to generate OAM beams. Compared with the traditional ways such as the spiral phase plates (SPPs) [12], cylindrical lenses [13], q-plate [14] and computer-generated holograms [15], metamaterial devices attach significant attention in recent years. Metamaterials are subwavelength periodic nanoscale structures [16]. Different from conventional bulky optical components, metamaterial devices are smaller in size and more compact that can be easily integrated into electronic and mechanical systems. But three-dimensional metamaterials with multilayer structures have also drawbacks such as high loss, dispersion and difficult to manufacture, which limit their applications.

With the development of nanotechnology, metasurface has been proposed to overcome the limits mentioned above. Metasurface, consists of subwavelength antenna arrays resulting in ultrathin thickness, is regarded as two-dimensional surface. It changes the amplitude and phase of an incident beam by exciting strong resonance of each unit structure [17]. Harvard university used metasurfaces to generate OAM beam in 2011 [18]. They used electron beam lithography to fabricate Gold V-antenna arrays on intrinsic silicon wafers and implement phase control from 0 to 2π of the antenna that is very significant near the resonant wavelength. This plasmonic metasurface generated OAM beam with m = 1, where m is the topological charge and can be taken as an arbitrary integer. Based on the same theory, different plasmonic metasurfaces are proposed to generate OAM beam [19]. Due to severe ohmic losses and strong absorption, these plasmonic based metasurfaces showed low energy efficiency.

Now, all-dielectric metasurfaces with low losses and high refractive index have proved of great potential for generating OAM beams, because each resonator consisting of dielectric metasurfaces can support strong electric and magnetic dipole resonances due to the Mie scattering theory [20,21]. These metasurfaces have wide resonant wavelengths and are robust in tailoring the phase and amplitude of wavefront, and suitable to generate OAM beams. Quite a few literatures have been published in this topic by using dielectric metasurface with different resonator shapes such as Z-shape [21], rectangle [22] and elliptical [23]. In addition, to explore new mechanisms for generating OAM beams with improved quality [24–28], expanding the operation bandwidth of OAM beams is meaningful to reduce dispersion-induced strong distortions and exploit new applications such as broadband object detection [29,30] and broadband OAM imaging [31]. Some groups have investigated broadband OAM generators in microwave region (6.95-18 GHz in [32], 14-16 GHz in [33]), millimeter wave region (59-70 GHz [34]) and optical regime (670 to 1100 nm [35], 1500nm-1600 nm [23]). However, broadband transmissive all-dielectric OAM generator in near infrared region (NIR) has rarely been reported. Exploration of the NIR region has profound implications for a multitude of applications, such as enhancing spectrum efficiency for OAM communications and increasing available spectrum.

In this letter, we demonstrate a high-efficiency all-dielectric metasurface working in a broadband wave range from 1300 nm to 1700nm (nearly covers all optical communication waveband) for optical vortex generation by utilizing structured silicon resonators resting on fused silica substrate. Based on Mie scattering theory, this transmissive metasurface has extremely high efficiency up to 97.38% across a 400 nm bandwidth and present good mode purity up to 99.99%. To render the resultant phase-shift exhibit proper features required for broadband OAM beams, we use Jones matrix to make a detailed derivation of the propagation process. Simulation results of three representative wavelengths chosen from the operation waveband are presented to show the OAM generator performance, including phase and electric field distribution which are consistent well with expectations. Furthermore, this flexible phase control can be utilized to implement other broadband optical elements with high transmittance such as lenses, quarter wave plates and holograms.

2. Theoretical analysis

To simulate broadband all-dielectric metasurface for optical vortex generating, Jones matrix is employed firstly to describe the mathematics behind the metasurface. For arbitrary E_in and E_out, each dielectric resonator can be seen as a pixel which has the general relation between the input and output electric field that can be described as E_out = TE_in, with

(1)$$T = \left[ {\begin{array}{{cc}} {{T_{xx}}}&{{T_{xy}}}\\ {{T_{yx}}}&{{T_{yy}}} \end{array}} \right]$$

where T_xx(_yy) and T_xy(_yx) are the transmission of co-polarized light and cross-polarized light. Any arbitrary E_in and corresponding desired E_out can be mapped using a symmetric and unitary Jones matrix. For a birefringent metasurface, if the phase shifts along x- and y- directions and the rotation angle can be chosen freely, it can implement the corresponding Jones matrix [36]. The particularly relationships between E_in, E_out and T are

(2)$$\left[ {\begin{array}{{cc}} {E_x^{out\ast }}&{E_y^{out\ast }}\\ {E_x^{in}}&{E_y^{in}} \end{array}} \right]\left[ {\begin{array}{{c}} {{T_{xx}}}\\ {{T_{yx}}} \end{array}} \right] = \left[ {\begin{array}{{c}} {E_x^{in\ast }}\\ {E_x^{out}} \end{array}} \right]$$

and

(3)$$\left\{ {\begin{array}{{c}} {{T_{xy}} = {T_{yx}}}\\ {{T_{yy}} = - \exp (2i\angle {T_{yx}}){T_{xx}}^\ast } \end{array}} \right.$$

We choose a pair of polarization orthogonality basis, left circular polarization and right circular polarization, as $|{\lambda ^ + \rangle }$ and $|{\lambda ^ - \rangle }$:

(4)$$\begin{array}{{lc}} {|{{\lambda^ + }} \rangle = \frac{{\sqrt {2} }}{{2}}\left[ {\begin{array}{{c}} {1}\\ i \end{array}} \right]}&{|{{\lambda^ - }} \rangle = \frac{{\sqrt 2 }}{2}\left[ {\begin{array}{{c}} 1\\ { - i} \end{array}} \right]} \end{array}$$

The dielectric metasurface transformations in our design are

(5)$$|{{\lambda^{ + }}} \rangle \to {e^{im\varphi }}|{{\lambda^ - }} \rangle \quad \ |{{\lambda^ - }} \rangle \to {e^{i( - m)\varphi }}|{{\lambda^ + }} \rangle $$

with m the topological charge of the OAM beam and φ the azimuthal angle. Based on Eq. (5), we can get two sets of E_in and E_out and put them into Eq. (2) and Eq. (3), the transmission matrix gives:

(6)$$T = \frac{1}{2}\left[ {\begin{array}{{cc}} {{e^{im\varphi }} + {e^{ - im\varphi }}}&{ - i{e^{im\varphi }} + i{e^{ - im\varphi }}}\\ { - i{e^{im\varphi }} + i{e^{ - im\varphi }}}&{ - {e^{im\varphi }} - {e^{ - im\varphi }}} \end{array}} \right]$$

T is a symmetric and unitary matrix that can be eigen decomposition into the following eigenvalues as:

(7)$${\lambda_{1}} = 1 = {e^0} \quad {\lambda _2} = - 1 = {e^{ - \pi }}$$

and eigenvector as:

(8)$$|{{v_1}} \rangle = \left[ {\begin{array}{{c}} {\cos (\frac{1}{2}m\varphi )}\\ {\sin (\frac{1}{2}m\varphi )} \end{array}} \right] \quad |{{v_2}} \rangle = \left[ {\begin{array}{{c}} { - \sin (\frac{1}{2}m\varphi )}\\ {\cos (\frac{1}{2}m\varphi )} \end{array}} \right]$$

So, T can be translated into

(9)$$\begin{aligned} T &= \left[ {\begin{array}{{cc}} {\cos (\frac{1}{2}m\varphi )}&{\sin (\frac{1}{2}m\varphi )}\\ {\sin (\frac{1}{2}m\varphi )}&{ - \cos (\frac{1}{2}m\varphi )} \end{array}} \right]\left[ {\begin{array}{{cc}} {{e^{i\ast 0}}}&0\\ 0&{{e^{i\ast ( - \pi )}}} \end{array}} \right]\left[ {\begin{array}{{cc}} {\cos (\frac{1}{2}m\varphi )}&{\sin (\frac{1}{2}m\varphi )}\\ {\sin (\frac{1}{2}m\varphi )}&{ - \cos (\frac{1}{2}m\varphi )} \end{array}} \right]\\ & = R(\theta )\left[ {\begin{array}{{cc}} {{e^{i{\phi_x}}}}&0\\ 0&{{e^{i{\phi_y}}}} \end{array}} \right]R( - \theta ) \end{aligned}$$

ϕ_x and ϕ_y are the phase shifts along x- and y- directions and θ is the rotation angle of each metasurface pixel’s principal axis [37]. We can easily find

(10)$${\phi _x} = 0 \quad {\phi _y} = - \pi \quad \theta = \frac{1}{2}m\varphi$$

According to Eq. (10), in order to design an all-dielectric metasurface generating OAM beam with topological charge m, we need to optimize the geometric parameters of each resonator.

3. Element design

The schematic of the high-efficient all-dielectric metasurface for OAM beam generation is shown in Fig. 1(a). The structure is composed of elliptical silicon resonators laying on 500 nm thickness fused silica substrate. Its thickness is expected, due to the different resonator filling fractions of antennas [38]. Geometric parameter definition of a basic unit is shown in Fig. 1(b), the lattice constant is chosen as 650 nm. In our work, we employ the finite difference time domain (FDTD) method in Lumerical software package for numerical simulations. In simulations, the background refractive index is set to be a constant value as n = 1, fused silica substrate here has a refractive index n₁= 1.4, and silicon resonator has a changeable refractive index depending on wavelength (adopted from Palik: 1.2µm/3.519-1.8µm/3.458). The boundaries are set as perfect matched layer (PML).

Fig. 1. (a) The schematic of the high-efficient all-dielectric metasurface. (b) Geometric parameter definition for a basic unit of the metasurface.

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For our proposed dielectric metasurface, silicon shows high quality electromagnetic resonant properties and avoids ohmic losses, due to its high refractive index. In the NIR region, both silicon resonators and silica substrate have low absorption loss. These properties result in a high-efficient transmissive metasurface. To realize polarization conversion and impart a π phase shifts for incident wave, the electric and magnetic dipole resonances need to be optimized to generate orthogonal polarization with respect to the incident wave. The optimized geometric parameters of the resonators are r₁= 250nm, r₂= 120 nm, h = 1020 nm with θ = 45°. As a result, the elliptical resonators with high index can couple x- or y-pol incident wave and induce displacement currents forming orthogonal electromagnetic dipole resonances that effectively producing orthogonal polarization with respect to the incident wave. To give a better description, the performance of the optimized resonator is plotted in Fig. 2, with an example of x-pol incident. In a broad waveband from 1300 nm to 1700nm, T_xx of the output wave is very low comparing to T_xy as shown in Fig. 2(a). To further evaluate the polarization conversion efficiency, we utilize the overall polarization conversion rate (PCR) to characterize the conversion efficiency, as follows:

(11)$${PCR} = {{I}_y}/({I_x} + {I_y}),$$

where I_x and I_y are the intensity of the x- and y-polarization components of the transmissive wave. The PCR as total transmittance are plotted in Fig. 2(b), in which both PCR and transmittance greater than 0.84 have a 400 nm bandwidth (1300 nm – 1700nm), as shown in Fig. 2(c). The phases of orthogonal polarizations (ϕ_x_, ϕ_y) and phase shift (_Δϕ) in a broad waveband are shown in Fig. 2(d), where the phase shift is nearly constant as -π responding to Eq. (10).

Fig. 2. The (a) transmittance coefficients, (b) polarization conversion rate, (c) total transmission and (d) phase difference of the optimized resonator in the operation waveband from 1300 nm to 1700nm.

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Based on the model discussed in section 2, by tiling rotation θ of the resonator, the resonator can alter the geometric phase of the output wave from 0 to 2π. Furthermore, the proposed broadband dielectric metasurface for generating optical vortex can be achieved by rotating the Si antenna corresponding the azimuthal angle, due to the same theory.

4. Results and discussions

To give a clear demonstration of the proposed broadband OAM beam generator, we have a circularly polarized Gaussian beam incidence on the structure for simulation, which has a broad wave band from 1300 nm to 1700nm. The design of the all-dielectric metasurface that generate OAM for m = 1 is consisting of elliptical silicon resonators with varying rotation angles (θ =ϕ/2). The simulation radius of the circular dielectric metasurface is 25µm. For the geography phase, it gradient change from 0 to 2π corresponding to the topological charge m = 1 with high efficiency, shown in Fig. 3.

Fig. 3. In a broadband wavelength from 1300 nm to 1700nm, simulated transmission beam (a) geography phase delay and (b) transmittance for Si resonator with varying rotation angle θ.

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We adopt three representative wavelengths in the wave range: 1310 nm, 1550 nm and 1645 nm, which have the applications in optical communications and high-power lasers. Amplitude and phase information of the transmissive field with orthogonal polarization are recorded by monitor 3000 nm away from the dielectric metasurface at three wavelengths. It is noteworthy that the monitor is not too far away from the substrate due to the limitation of available computational resources. As shown in Fig. 4, the optical field distribution of OAM beam exhibits a “donut” shape and at the center there is a phase singularity. For the phase, it exhibits apparent spiral nature and fits well with the topological charge m = 1. And this presented phase is the superposition of geography phase and propagation phase [39]. When the position of monitor is immobilized, different wavelengths correspond to different propagation phases so Fig. 4 shows discrepancy in phase at the same azimuthal angle. In addition, we analyze the OAM spectrum to study the purity of generated OAM beams quantitatively, crosstalk between different modes and stability of this OAM generator [40,41], shown in Fig. 4. The mode purity (MP) is defined as the ratio of the dominant mode intensity over the overall intensity distribution in all the modes, as expressed as

(12)$${\textrm{MP}} = \frac{{{{|{{{\textrm{E}}_{\textrm{m}}}} |}^2}}}{{{{\sum {|{{{\textrm{E}}_{\textrm{i}}}} |} }^2}}}$$

with ${E_m}$ the dominant mode amplitude with topological charge m and ${E_i}$ the amplitude of the i-th mode. It can be clearly seen that the m = 1 mode is predominant for all the three wavelengths. The recorded mode purity for the three wavelengths are 99.98%, 99.99% and 99.99%, respectively.

Fig. 4. Electric field distribution, phase distribution and mode purity of the OAM beam with topological charge m = 1 at three representative wavelengths, (a)-(c) 1310 nm, (d)-(f) 1550 nm, (g)-(i) 1645 nm.

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For an all-dielectric transmissive broadband OAM generator, transmittance is also an important property to evaluate its energy efficiency. The transmittances of the three representative wavelengths are shown in Table 1. It is worth mentioning that wavelength and transmittance are not liner relationship, there are several possible reasons for this fluctuation in transmission. Firstly, the resonator itself is different for the wavelength due to the characteristic of metasurface. The extinction index of silicon also depends on the wavelength causing different transmittance. These lead to a small fluctuation of the transmittance. Besides, transmittance can be improved by changing the number and the boundary dimensions of the resonators. Figure 4 and Table 1 show a good resemblance at three significant wavelengths, explicitly demonstrating the broadband nature of 1300 nm to 1700nm m = 1 OAM beam generator.

Table 1. The transmittances of the three representative wavelengths with the topological charge m = 1.

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In addition, we change the number and arrangement of the resonators, that consisting OAM beam generator, trying to generate high order OAM beam which shows great potential in optical communication [3], rotation Doppler effect [6] and so on. Low order OAM with topological charge 1 to 9 are already in mass production. However, high order (greater than 10) are usually generated by holograms, which faces some boundedness like low efficiency and bandwidth limitation, for example a transmissive spatial light modulator (Holoeye, LC 2012) with a bandwidth of 420 nm to 850 nm is only 28% efficient. Those limitations can be solved by using broadband dielectric metasurfaces. Here, we choose high order topological charge as m = 20 to design our metasurface for simulation. The simulation condition is set to same with the simulation work of m = 1. To evaluate the performance of the proposed high order OAM beam generator, the electric field distribution and phase distribution are recorded at three representative wavelengths, as shown in Fig. 5, high order OAM beam is also a “donut” shape and with increasing of topological charge, the ring will become more and more big. For the phase distribution, it has 20 times change from 0 to 2π corresponding to the topological charge m = 20 and in the middle, there is a phase singularity. The recorded mode purity for the three wavelengths are 99.28%, 99.49% and 99.59%, respectively, which can be clearly seen from Fig. 5 that the m = 20 mode is predominant for all the three wavelengths. The transmittance of three significant wavelengths are shown in Table 2, which present high energy efficiency comparing with transmissive metallic metasurfaces.

Fig. 5. Electric field distribution, phase distribution and mode purity of the OAM beam with topological charge m = 20 at three representative wavelengths, (a)-(c) 1310 nm, (d)-(f) 1550 nm, (g)-(i) 1645 nm.

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Table 2. The transmittances of the three representative wavelengths with the topological charge m = 20.

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5. Conclusion

In summary, we propose an approach to design high-efficient OAM beam generator based on all-dielectric metasurface. This proposed dielectric metasurface shows high energy efficiency up to 97.38% with a broad operation bandwidth from 1300 nm to 1700nm. The electric field distribution and phase distribution fit well with the expectation in three significant wavelengths. The recorded mode purities are all above 99.28%. Meanwhile, the proposed metasurface can generate OAM beams with any order with rearranging the resonators, based on the design method we propose. This work may provide a paradigm for exploiting new broadband optical devices and present great potential to the generation of more complex structured light.

Funding

National Natural Science Foundation of China (NSFC) (11834001); National Postdoctoral Program for Innovative Talents of China (BX20190036); China Postdoctoral Science Foundation (2019M650015).

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Wavelength (nm)	1310	1550	1645
Transmittance	85.66%	76.40%	74.19%

Wavelength (nm)	1310	1550	1645
Transmittance	85.66%	76.40%	74.19%

High-efficiency, broadband all-dielectric transmission metasurface for optical vortex generation

Abstract

Corrections

1. Introduction

2. Theoretical analysis

3. Element design

4. Results and discussions

5. Conclusion

Funding

References

Cited By

Figures (5)

Tables (2)

Equations (12)

Optical Materials Express