Visible-wavelength metalenses for diffraction-limited focusing of double polarization and vortex beams

Hongguang Dong; Faqiang Wang; Ruisheng Liang; Zhongchao Wei; Hongyun Meng; Li-Hua Jiang; Haifeng Cen; Ling Wang; Shijie Qin; Chenglong Wang

doi:10.1364/OME.7.004029

1. Introduction

Recently, Pancharatnam–Berry (PB) phase metasurfaces, as one well-known technique, achieve a full phase control by adjusting the orientation angle of antennas with identical geometry [1], which has attracted many researchers’ extensive attentions and was being used to fabricate various optical components with different function [2, 3]. Such as conventional metalens which provide the identical phase at the same radius distance to the center of the device and give a gradual phase along the radial direction [4–12]. And, for the vortex beam generators, on the contrary, there is a spiral-phase achieved along the azimuth angle and the same phase change at the radial direction, which enable to manipulate the orbital angular momentum of the incident light [13–24]. However, no matter for metalens or vortex beam generators, the efficient operation wavelength range of the light scattered by transmission-mode ultrathin PB metasurfaces has been limited to the microwave and near infrared [4, 7, 8, 11–24] because of the high intrinsic losses of the metamaterials based on plasmonic materials in visible range. Other than manipulating the phase by control the resonance of the transverse electric (TE) or transverse magnetic (TM) wave with plasmonic metasurfaces [4, 14–17], an intrinsic phase difference between ordinary and extraordinary light was generated by transmitting incident light through the high-refractive-index biaxial crystal materials, which is also considered to reduce the significant ohmic loss by supporting both electric and magnetic resonances at visible and NIR frequencies [2, 3, 5, 6]. But, the metalenses based on the PB phase metasurfaces are polarization-dependent [5–8] and we need to design the new device for different polarized light. What’s more, most devices only have the single function, for example, traditional optical vortex generators always just generate the vortex light and requires additional lens to shape the beam in the practical experiment, which is a cumbersome process [15–24]. So, it is a contribution for the development of the integrated devices with multiple functions based on metasurfaces.

Thus, in this paper, we chose the biaxial crystal materials of zinc sulfide (ZnS) [25, 26], which is the material with the high refractive index and transmittance in the visible range but cheaper than TiO_2, and then designed several metalenses. From the simulation of the electric field distribution by finite difference time domain (FDTD) method, we discovered that these metalenses have the ability of diffraction-limited focusing at visible wavelengths λ = 405,532 and 633 nm with the transfer efficiency as high as 72%, 65%, 67%, respectively. And then, we still developed a double-polarized light transmission metalens enable focusing light at the same point with a full-hybrid structure at the wavelength λ = 532 nm. In addition, based on the design principle of the helical phase plate and metalens, we introduce the phase gradient concept when we design metalens. Interestingly, we realized the conversion of the circularly polarized plane light into a vortex beam and focusing the vortex beam simultaneously by one device.

2. Diffraction-limited focusing metalenses at visible wavelengths

The metalenses are consist of ZnS nanofins on a glass substrate and the reason we chose ZnS is for it’s high refractive index ( $~ 2.4$ ), low power losses at visible spectrum. Polarized plane wave is upright incident from one side of the lens and focusing into a spot on the other side (Fig. 1(a)). In order to achieve this goal, every nanofin must contribute a $φ (x, y)$ phase at its position $(x, y)$ . What important is that the $φ (x, y)$ must follow the principle [4]:

φ (x, y) = \frac{2 π}{λ_{d}} (f - \sqrt{x^{2} + y^{2} + f^{2}})

Where

λ_{d}

is the design wavelength, f is the focal length. We accomplish this

φ (x, y)

phase by rotating the nanofins an angle

θ (x, y)

(Fig. 1(d)). On the basis of PB geometric phase concept,

θ (x, y)

should satisfy the relationship:

φ (x, y) = 2 θ (x, y)

to guarantee the right handed circularly polarized incident light change to left handed circularly polarized light after transmitting via the structure [1]. Thus, the angle

θ (x, y)

can be expressed as:

Fig. 1 (a) The scenograph and schematic of the metalens. (b) Top view of the metalens showing radius R. (c and d) Side and top view of the unit cell consists of ZnS on a glass substrate showing height H, width W and length L of the nanofin, with unit cell dimensions P $\times$ P and height h of the substrate. According to the geometric Pancharatnam-Berry phase concept, we rotate the nanofin by an angle $θ$ to obtain the required phase.

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θ (x, y) = \frac{π}{λ_{d}} (f - \sqrt{x^{2} + y^{2} + f^{2}})

In addition, we design the nanofins working as half-waveplates by regulating the nanofins’ width W, length L, height H and the unit size P on the glass substrate (the refractive index n $\sim$ 1.45) which height is h = 200 nm (Fig. 1(c)). We have investigated the phase shift and given the charts corresponding to each unit. As it was shown in Fig. 2, we separately collected the phase shift with the size changes of each nanofin and calculated their phase difference between the electric field in the x and y direction(phase (Ex)- phase(Ey)). Then in order to make sure that the unit work as half-waveplates, the phase difference must be $π$ . Finally, three different modes are selected for wavelengths $λ_{d}$ = 405, 532 and 633 nm and corresponding size L = 40, 70, 110 nm, W = 170, 290, 350 nm, P = 200, 320, 400 nm and height H = 600, 550, 600 nm (Fig. 1(c) and 1(d)). Still, the minimum size of the device should satisfy full phase changes. The minimum length of the radius R of the ordinary metalens should follow the principle $R \geq \sqrt{{(f + λ_{d})}^{2} - f^{2}}$ . In our work, for the sake of saving simulated time, the size of the metalenses we designed is slightly bigger than the minimum length. Perfectly matched layers at the x, y, z boundaries are used for all simulations.

Fig. 2 Phase shift corresponding to each unit with different lengths of L and W. (a) and (b) Computed phase shift (in x and y direction) as a function of the nanofin width and length at wavelength of 405 nm. (c) is the phase difference between (a) and (b). Similarly, (d-f) are the phase shift corresponding to the unit designed at wavelength of 532 nm. (g-i) are the phase shift corresponding to the unit designed at wavelength of 633 nm.

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And then, we numerically simulated three planar lenses with the designed wavelengths $λ_{d} =$ 405, 532 and 633 nm, and these components have different radius R (1.5 $μm$ , 1.76 $μm$ , 1.8 $μm$ ), but have the same focal length f = 2 $μm$ . Here, we define the focal length as the distance from the focal spot to the exit plane of metalenses. A beam of right-handed circularly polarized plane wave is normal incident to lenses from negative direction of z axis and then focusing at the location z = 2.3 $μm$ (for matelens designed at $λ_{d} =$ 532 nm, the focal spot at the location z = 2.275 $μm$ ). Simulated intensity profiles in the focal region(x-z plane) are shown in Fig. 3(a-c), a slightly deviation of the focus length ( ± 50 nm) is emerged because of the different density of the nanofins and the short focal length and thus the corresponding numerical aperture (NA) is calculated as 0.6, 0.66, 0.67.

Fig. 3 (a-c) The intensity profiles (x-z plane) of the metalenses designed at wavelengths of (a) 405, (b) 532, (c) 633 nm, and (d-f) showing the focal spot(x-y plane), respectively. (g-i) Corresponding horizontal cuts of focal spots in (d-f) with full width at half-maximum of 330, 422 and 468 nm. All metalenses were illuminated by right-handed circularly polarized light.

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The focal spots (x-y plane) and corresponding horizontal cuts are shown in Fig. 3(d-i), and diffraction-limited ~ $λ / (2 N A)$ full width at half-maximum (FWHM) are labeled in the Fig. 3(g-i), respectively. For metalens with the designed wavelength $λ_{d} =$ 405, 532 and 633 nm, the FWHM≈0.5 $λ / N A$ and the focusing efficiencies as high as 72%, 65%, 67% are obtained, respectively. We define the efficiency as the ratio of the optical power in the focal spot area (circle of radius FWHM spanning the center of the focal spot) to the incident light power [5]. Though these metalenses’ numerical aperture are smaller and lead to their focal spots larger than other papers’ work [4, 5], their spots are much closer to the ideal situation.

It is important to note that the metalenses not only work at the specific wavelengths, they also perform well at other wavelength. For example, we use the light with wavelength of λ = 405 and 633 nm to illuminate the metalens designed at $λ_{d}$ = 532. The simulated electric field intensity at the position x = 0, y = 0 along the z axis was shown in Fig. 4. Because the metalens is wavelength-dependent, chromatic aberration result in a variational focal length for different wavelength and it is a problem need to solve.

Fig. 4 Utilized different RCP light with wavelength of 405 and 633 nm to illuminate the metalens designed at $λ_{d} =$ 532 and measured electric field intensity at the position x = 0, y = 0 along the z axis.

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3. Double polarization effective metalens by hybrid structure

Insufficiently, all above metalenses are only living on RCP incident light and useless for the opposite polarized light which is severely restricting their application. Thus, in this section, we utilize a full-hybrid structure (Fig. 5) enable work effectively for RCP and LCP light. Here, we rotate the nanofins by an angle $θ_{R} (x_{R}, y_{R}) = φ (x_{R}, y_{R}) / 2$ for RCP incident light and the same to LCP incident light but the angle $θ_{L} (x_{L}, y_{L}) = - φ (x_{L}, y_{L}) / 2$ (Fig. 5(c)), here $(x_{R}, y_{R})$ and $(x_{L}, y_{L})$ is the center position of the cell for corresponding circularly polarized light and then two cells with different working mechanism are alternatively distributed. We should note that the different cells (green and blue) are arranged one by one which is the most different between with other structure row by row or column by column [6]. Thus, this full-hybrid concept provides a more flexible way for designing other optical device as holograms. Here, the radius of metalens is R = 1.6 $μm$ and the green nanofins are equal in number to the red nanofins.

Fig. 5 (a and b) Perspective view and top view of the full-hybrid metalens show that a square metalens instead of the discal structure, with radius R. Every green (blue) nanofin is surrounded by four blue (green) nanofins, except that at the corner and edge area. (c and d) Top view and perspective view of the hybrid unit cell showing it consists of two parts. The green and blue nanofins are working for focusing RCP light and LCP light, respectively and rotating nanofins to satisfy phase realization.

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Then, we use different circularly polarized light with wavelength $λ_{d} = 532 nm$ illuminate metalens. And the simulated intensity profiles are shown in the focal region(x-z plane) (Fig. 6(a) and 6(b)). It is obviously that they have the similar focal length (Fig. 6(c)) and almost same horizontal cuts at the position z = f (Fig. 6(d)) with the same FWHM = $432 nm$ . For this unique characteristic, we can see that it has realized the function of analogous polarization-insensitive focusing. At the same time, the transfer efficiency is about ~24% for both RCP and LCP light. Though it is much lower than our foregoing work, it is still higher than that of the metalenses based on the material of silicon and plasmonic metasurfaces [23]. Furthermore, we simulated the focal length as a function of wavelength in the range of 100 nm (Fig. 6(e)). Here, we define the focal length as the distance from the focal spot to the origin of coordinates(x = 0, y = 0, z = 0).

Fig. 6 (a and b) Simulated the intensity profiles(x-z plane) in the focal region of the hybrid metalens with designed wavelength $λ_{d}$ = 532 nm and the metalens was illuminated by right-handed circularly polarized light and left-handed circularly polarized light, respectively.(c and d) Corresponding vertical cuts in (a and b) at the location x = 0 nm and horizontal cuts at the location z = f . (e) Measured the focal length as a function of wavelength.

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4. Optical vortex generation and focus

In this section, we introduce a spatially varying azimuthal angle $α$ of the nanofins around the singular point when design the device. And the integral of the phase difference along the circle path C around the vortex axis is $2 l π$ ( $l$ is integer), which can be expressed as:

\oint_{c} \nabla β d α = 2 l π

Here

β = l α

is the phase gradient. So the device will introduce a spiral-like phase shift with respect to the planar wavefront of the incident light to create a vortex beam with topological l [11, 16]. Next, we add this azimuthal angle to the Eq. (1), and Eq. (1) will be changed as:

φ' (x, y) = \frac{2 π}{λ_{d}} (f - \sqrt{x^{2} + y^{2} + f^{2}}) + l α

Thus, the rotation angle

θ

of the nanofines will denoted by:

θ (x, y) = \frac{π}{λ_{d}} (f - \sqrt{x^{2} + y^{2} + f^{2}}) + \frac{1}{2} l α

Mathematically, the azimuthal angle

α

is given by

\tan (α - π / 2) = y / x

for

y < 0

and

\tan α = y / x

for

y > 0

. Then, the device will create a vortex beam and focusing the beam at the same time.

Because the vortex beam is characterized by an $\exp (i l β)$ azimuthal phase dependence, the orbital angular momentum in the propagation direction has the discrete value $l ℏ$ per photon. Here, as an example, we design the devices with a specific topological $l = 2$ , which can be identified from the phase distributions on an x-y plane 200 nm above the metasurfaces illuminated by RCP light with the wavelength of 405 nm (Fig. 7(b)). And the distribution of electromagnetic fields is shown by Fig. 7(c), a hollow dark spot is attainable in the focal plane and the corresponding FWHM of doughnut focus is ~330 nm, which is equal to the FWHM of the spot focused by our prefer designed metalens, and the peak-to-peak distance is ~670 nm (Fig. 7(d)). In order to analysis the conversion efficiency $η$ , we define it as the ratio of the optical power E_ring in the ring area (double FWHM as the width of the ring) to the incident light power E_in: $η = E_{r i n g} / E_{i n}$ . At the last, we get the conversion efficiency as 73% by numerical calculation. We also have studied the viability of higher orbital angular momentum (OAM) modes and design another two devices with $l$ = 3 and $l$ = 4 at wavelength of 405 nm. And then we analyzed their phase changes of the devices and collected the light field distribution in the focal plane and the results were shown in Fig. 8(a) and 8(b). In order to analyze the purity of the OAM mode, we encode and measure optical OAM via phase holograms [27, 28]. As an example, here, we chose the beam created by our device which is assumed with a topological charge $l'$ = 2. First, we get a standard phase holographic board by the interference of the plane wave with a standard vortex beam which with the topological charge $l$ = 2, and then, let the beam created by our device go through the phase holograms. If $l'$ = $l$ , the −1 order of the far-field diffraction will be a bright dot, and the 0 order will be a bright ring. Thus, we can judge the topological charge of the vortex beam created by our device is consistent with the topological charge we guessed, and the results were shown in Fig. 8(c).

Fig. 7 (a) Top view of the metalens designed at the wavelength of 405 nm and corresponding radius and focal length of the metalens are R = 3 $μm$ and f = 4 $μm$ , respectively. (b) Measured the x direction phase distributions on an x-y plane 200 nm above the metasurfaces. (c) Simulated the intensity profiles(x-y plane) in the focal region. (d) Corresponding horizontal cuts of doughnut focal in (c) with full width at half-maximum of 330 nm and the peak-to-peak distance is ~670 nm.

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Fig. 8 (a and b) Simulated the intensity profiles(x-y plane) in the focal region and measured the x direction phase distributions on an x-y plane 200 nm above the metasurfaces corresponding to the devices with topological charge $l$ = 3 and $l$ = 4; (c) is the far field diffraction of the detected light through phase holograms.

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5. Conclusion

In summary, we demonstrated the metalenses based on the zinc sulfide with high efficiency (as high as 72%) in the visible spectrum and all metalenses own the capability to focus light into diffraction-limited spots which makes them have a prospect of imaging. In addition to the full-hybrid structure, it’s also able to break through the limit to the single polarization and realized an analogous polarization-insensitive metalens. At the third section, we combine the metalens with the vortex beam generators and introduce the azimuthal angle in our metalens which could generate a vortex beam and focus at the same time. Thus, for these unique properties, these metalenses have the promising applications in miniaturized holographic optical components, near-field microscopy, integrated optics, wearable optics devices.

Funding

National Natural Science Foundation of China (NSFC) (61275059, 11674109, 61774062).

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Visible-wavelength metalenses for diffraction-limited focusing of double polarization and vortex beams

Abstract

1. Introduction

2. Diffraction-limited focusing metalenses at visible wavelengths

3. Double polarization effective metalens by hybrid structure

4. Optical vortex generation and focus

5. Conclusion

Funding

References and links

Cited By

Figures (8)

Equations (5)

Optical Materials Express