1. Introduction

Metalenses, composed of inhomogeneous optical antenna arrays in thin plane, have emerged as versatile platforms for the development of miniaturized optical lenses because of its unprecedented control over phase of transmitted and/or reflected electromagnetic waves. Based on discontinue phase principle, metasurface could manipulate light locally and impart space-variant phase shift in a desirable manner [2–5], which breaks the dependence on gradual phase accumulation along the optical path. Metalens with two foci in longitudinal or transverse direction, called bifocal metalens, has significant applications in optical tomography technique [6–8], optical data storage [9,10] due to its specific characteristics of double foci with one incident beam. To make bifocal metalenses feasible in various applications, metalenses with the following functions are urgently needed: first, relative focal intensity between two foci should be modulated conveniently, say, through altering the polarization of incident beam; second, the focusing efficiency should be as high as possible; last but not the least, the separate distance between those two foci should be designed at will.

Up to now, tremendous high-performance plasmonic and dielectric metalenses with one focus have been realized theoretically [11,12] and experimentally [13–21], while less work on bifocal or multifocal metalens is involved. Multifocal metalens focuses one incident beam into different positions longitudinally or transversely, thus, playing important role in optical communications [22,23], multi-imaging systems [24,25] and micro-manipulating optics [1]. Though several works on bifocal or multifocal metalenses have been realized [1,25], they are all based on such technique that divides metalens into multi-ring bands with each band corresponding to one focus, thus resulting in low focusing efficiency, poor signal-to-noise ratio and inconvenience in controlling relative intensity between different foci. For instance, Chen et al. [25], designed a multifoci metalens consisting of three concentric ring bands with each band applied to each kind of polarization, such as linear polarization, right- and left-handed circular polarization. That is, for given polarized incident beam, only part of it can be concentrated by counterpart ring band into the corresponding position, while the other of incident beam illuminating on the ring band is diverged as background scattering, leading to low efficiency and low signal-to-noise ratio. In addition, for each designed metalens by Guo et al. [1], the relative focal intensity of each focus is fixed, not changeable. If you wanted to get other kind of intensity metalens you have to repatterned metalens, which is time-consuming and inefficient in practical applications.

Due to above problems for reported bifocal metalenses [1,25], here, we proposed the phase equation of longitudinal bifocal metalens by combining geometric phase and propagation phase of each unit cell of metalens [26], because incident beams with different helicities experience opposite-sign geometric phase that can be used to control relative intensity and separate distance between two foci. To demonstrate the proposed phase equation, some numerical simulations are performed using commercial FDTD Solutions. The results show that the carefully designed metalens can focus one incident beam into one or two spots in tandem along longitudinal direction, depending on the polarization of incident beam, one focus for circular polarization and two foci for ellipse polarization. Unlike fixed relative intensity between the two foci of previously reported literatures [1], the relative intensity between two foci of the present metalens can be flexibly modulated by changing the ellipticity of incident beam. Under extreme case, circularly polarized beam will be focused to one spot, but in different position for different circular polarization due to different sign of geometric phase. In our method, the metalens is made of dielectric materials, transparent in visible range, and the whole cells in the plane of metalens contribute to the focusing of incident beam, so the focusing efficiency is as high as 72%, much higher than those of metal-based plasmonic metalenses. Furthermore, the separate distance between two foci can be designed at will. In Results and discussions section, three kinds of separate distances ($2\mu m$, $3\mu m$, $4\mu m$) are manifested. We believe the presented longitudinal bifocal metalens may provide potential applications in multi-imaging systems, optical tomography technique, and optical data storage.

2. Theoretical analyses

Normally, two kinds of phase are involved in the art of metalens, propagation phase and geometric phase. The propagation phase originates from optical path of incident beam transmitting through metalens. For incident beam, each nanofin in the array of metalens can be optically taken as a waveguide [15], different size nanofin exhibits different propagation constant, leading to different phase even for same-height nanofin. So, by controlling the cross size of nanofin, propagation phase covering the range from 0 to $2\pi$ can be obtained, which is essential for converging incident plane wave into a spot. Differently, geometric phase, also called Pancharatnam-Berry (PB) phase, stems from the rotation of nanofin around the propagation direction relative to, say, $x$ axis when elliptically polarized beam incident on nanofin; and geometric phase has the form of ${\varphi }_{PB}=\pm 2\sigma \theta$ [27-28], where $\theta$ is rotation angle of nanofin relative to $x$ axis and $\sigma$ is helicity of incident beam. Thus, geometric phase can also cover the range from 0 to $2\pi$ for circular polarization ($\sigma =\pm 1$) because $\theta$ can take any value in the range of (0,$\pi$).

Theoretically, to focus incident plane wave the phase profile of metalens has the form of [29]:

For bifocal metalens in longitudinal direction, though several works [1,25] have been reported, only one kind of phase is involved in their designs, In [1] and [25], Wang et al. use only propagation phase and Chen et al. use only geometric phase, respectively. In addition, they are all based on such technique that divides metalens into multi-ring bands with each band corresponding to one focus, so each focus receives only part of incident beam, leading to low focal efficiency and high background noise. In a word, to date, no longitudinal bifocal metalens using both propagation and geometric phases is achieved. To obtain bifocal metalens, a convenient and effective method is to make metalens exhibiting different focal effect (i.e. exhibiting different focal length) for beams with different helicity, because propagation phase is independent on polarization. Thus, the essential difficulty is how to organically design a metalens that is valid to both opposite-helicity polarized beams.

Here, we overcome this difficulty by introducing helicity of incident beam into phase equation, more precisely, into the focal length of spot. Here, we assume that the time dependence is $\exp (-j\omega t)$ and the incident light propagates along $+z$ axis. Hence, the phase profile of bifocal metalens appears like this:

It is clear that the focus of metalens now becomes polarization-dependent, i.e. $F=z-\sigma d$. Here, $2d$ is separate factor presenting separate distance between those two foci, which can be set at will. On the other hand, metalens using both propagation and geometric phases presents total phase$\psi =\varphi \pm 2\sigma \theta$. According to principles of discontinue phase involved in metasurfaces [30–32], a beam will normally be divided into two parts in the transmission field, one with same polarization (i.e. co-polarized) and the other with orthogonal polarization (i.e. cross-polarized) as incident beam [33]. If all nanofins are designed working as local transparent half-wave plates, then the co-polarized part disappears, only cross-polarized part exists. Under this thought, incidence of circularly polarized (CP) light will be converted into opposite-helicity CP light. So, the propagation phase and rotation angle of geometric phase in Eq. (2) can be written as [23]:

3. Results and discussions

As is shown in Fig. 1(a), a metalens focuses collimated incident light into two different focal spots, $F_1$ and $F_2$ under transmission mode. The nanostructures of the bifocal metalens are high-aspect-ratio $Ti{O}_2$ nanofins on a glass substrate as is shown in Figs. 1(b)-1(d). To achieve the phase profile of bifocal metalens in Eq. (2), each nanofin at position $(x,y)$ must impart the required propagation phase $\varphi$ and geometric phase ${\varphi }_{PB}$ given by Eqs. (3) and (4), respectively. The propagation phase$\varphi$, covering the range from 0 to$2\pi$, is achieved by same height nanofin with different cross size (width $W$ and length $L$). As is shown in Fig. 1(d), the geometric phase${\varphi }_{PB}$, covering the range from 0 to $2\pi$ similarly, is implemented via rotating nanofins with the rotation angle $\theta =\pm {\varphi }_{PB}/2$ relative to $x$ axis. To maximize the polarization conversion efficiency, i.e., maximum proportion of cross-polarized in transmission field, each nanofin is designed to work as a half-wave plate. Meanwhile, to ensure high efficiency, the nanofin height $H$ and unit cell size $S$ are optimized at the design wavelength of 532 nm. Here, we carefully designed eight nanofin with an incremental propagation phase of $\pi /4$ between adjacent nanofins. We set the nanofin height $H=600nm$ so that it can provide $2\pi$ phase coverage through a range of cross size (width $W$ and length $L$), and unit cell size $S=350nm$ to meet the Nyquist sampling criterion ($S<\frac{\lambda }{NA}$). A more intuitive top view of the designed metalens is shown in Fig. 1(e). The simulated amplitudes and phases of the cross-polarized radiation using commercial FDTD Solutions is shown in Fig. 2, where eight nanofines with almost unity amplitude and required phases are achieved.

 

Fig. 1 Schematic illustration of the designed bifocal metalens and its unit cell, the TiO₂ nanofin. (a). The metalens focuses collimated incident light into two different focal points under transmission mode. (b-c) Side and top views of the unit cell showing height H, width W, and length L of the nanofin, with unit cell dimensions $S\times S$. (d) The geometric phase is imparted by rotation of the nanofin by angle $\theta$, according to the Pancharatnam-Berry phase. (e) Top view of the designed metalens consisting of nanofins with same height but different cross size and orientation.

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Fig. 2 The simulated propagation phases and amplitudes of the y-polarized transmittance of eight unit cells for x-polarized incident light at the wavelength of 532 nm. The nanofins sizes ($L$ and $W$) of unit cells from 1 to 4 are L = 278, 280, 82, 104, 118, 136, 274, and 268 nm, and W = 118, 136, 274, 268, 278, 280, 82, and 104 nm.

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To investigate the focusing characteristics and the ability of intensity adjustment of the bifocal metalens in detail, we firstly, designed a metalens whose focal lengths are $10\mu m$ and $13\mu m$ with an aperture of $22.4\times 22.4\mu m^2$. The metalens consists of 64 nanofins along both $x$ and $y$ direction. The simulated results are discussed in section 3.1 and 3.2. Then, in section 3.3, to demonstrate the separate factor of the metalens can be set at will according to phase profile in Eq. (2), we designed another two metalenses with different separate distance between two foci and calculated their focusing performance in detail.

3.1 Single-focusing metalens with two alternative focal lengths

According to theoretical analyses section [33], when the metasurface is illuminated by pure circularly polarized light ($\sigma =\pm 1$), for example, left-handed circularly polarized (LCP, $\sigma =-1$), there is only cross-polarized part as incident light, i.e. right-handed circularly polarized (RCP) in the transmission field, because the nanofin, working as local half-wave plates, convert the incident light into opposite-heicity CP light. So, according to Eq. (2), the metalens becomes a single-focusing metalens with focal length$F=z+d$. Similarly, when the polarization of incident light switches to RCP ($\sigma =1$), the focal length of the metalens changes to$F=z-d$. Here, we set $z=11.5\mu m$ and$d=1.5\mu m$, so the theoretical focal lengths of the metalens are $10\mu m$ and $13\mu m$ for RCP and LCP incident light, respectively, and the two foci of the metalens can be switched by controlling the helicity of CP incident light. The intensity distributions of the metalens in x-z plane are shown in Figs. 3(a) and 3(b). From the pictures, we can see that our designed metalens has good focusing characteristics, and the simulated focal lengths of the metalens are $10.4\mu m$ and $13.3\mu m$ for RCP and LCP incident light, respectively, which agree with the theoretical analysis.

 

Fig. 3 The simulated results of the metalens under the CP incident light. (a-b) The intensity profiles at the x-z plane under the RCP and LCP incident, respectively. (c and e) Simulated focal spots intensity profiles of the metalens in two different focal planes under the RCP and LCP incidence, respectively. (d and f) Corresponding horizontal cuts of focal spots shown in c and e with full width at half-maxima of 320 and 360 nm, respectively.

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The full width at half maximum (FWHM) is a parameter commonly used to describe the spot size of a focal point. Figures 3(c) and 3(e) show the simulated focal spot of the metalens in the case of RCP and LCP incident light and their corresponding FWHMs are shown in Figs. 3(d) and 3(f), respectively. There is little difference in the maximum intensity of the two foci because of their different numerical aperture (NA), which is determined by equation of $NA=\sin [{\tan }^{-1}(D/2f)]$. Here, the NAs of the present metalens are 0.74 and 0.65 due to the limitations of unit cell size. In fact, the relative intensity between two foci can be tuned easily via modulating the ellipticity of incident light according to practical applications. And this will be discussed in detail in section 3.2. FWHMs of the horizontal cuts of the two focal spots are 360 nm and 320 nm for metalens when illuminated by RCP and LCP light beam. Smaller spot size can be achieved by designing higher numerical aperture. The present longitudinal bifocal metalens is very significant in practical applications since it has high focusing efficiency, which is defined as the ratio of the optical power of the focus (circle of radius 3 × FWHM spanning the center of the focal spot) to that of the incident beam. Using dielectric materials with a transparency window in the visible spectrum, the focusing efficiencies of the present metalens are 64% and 72% under RCP and LCP incident light, respectively, much higher than those of metal-based plasmonic metalens. Higher efficiency can be achieved by improving polarization conversion efficiency, which largely determined by the cross size of the nanofins. Another important point worth mentioning is that whole cells in the plane of the present metalens, rather than specific region, contribute to the focusing of incident beam [1, 25], which can avoid introducing background noise arising from other focal points and hence improve focusing quality.

3.2 Bifocal metalens with different relative intensity ratio of two focal spots

Since an EP light beam can be decomposed into two orthogonal states of polarization, such as LCP and RCP, there are two corresponding cross-polarized parts, RCP and LCP in the transmission field when the metalens is normally illuminated by the EP light. According to Eq. (2), two different focal spots, $F_1$ and $F_2$ appear simultaneously along longitudinal direction in image space, where $F_1$ corresponds to RCP portion of incidence, and $F_2$ corresponds to LCP portion of incidence. It is not difficult to understand that adjusting the proportion of RCP and LCP, i.e. the ellipticity of incident light, will alter the proportion of their cross-polarized, LCP and RCP in transmission field, leading to the focal spots, $F_1$ and $F_2$ with different relative intensity. In other words, the relative intensity between two foci of the bifocal metalens can be achieved by flexibly changing the ellipticity of the incident light. We calculated the relationship between the relative intensity of the two foci and ellipticity of the incident light. The results are shown in Fig. 4. For simplicity, intensity distributions in x-z plane of the bifocal metalens with different ellipticity are not shown here. In Fig. 4, the vertical axis represents the relative intensities of the two foci, and the horizontal axis represents the ellipticity of incident light, which is defined as $\gamma =(E_R-E_L)/(E_R+E_L)$ with $\gamma =\pm 1$ representing RCP and LCP, $\gamma =0$ representing linear polarization. From the picture, by changing the ellipticity of incident light, the polarization state can be tuned continuously from LCP/RCP, through elliptical polarization and linear polarizations, to RCP/LCP. Correspondingly, the relative intensity of focal spot$F_2/F_1$, relative to LCP/RCP incidence, gradually changed from 1/0, through about 0.5 to 0/1. The simulated results confirmed that the relative intensity between two foci of our present metalens can be flexibly modulated by changing the ellipticity of incident beam. In practical applications, two different focal spots with arbitrary relative intensity can be obtained by rotating the quarter-wave plate put before the metalens. Compared with previously reported the method of redesigning another pattern [1], such ellipticity tunable strategy is very convenient, efficient, and universal in the fields of optical tomography technique, optical data storage, and so on.

 

Fig. 4 The simulated results of relative intensities of $F_1$ and $F_2$ of the present metalens under incident light with different ellipticity $\gamma$, where $\gamma =0$ represents linear polarization, $\gamma =\pm 1$ represent left- and right-handed circular polarization. The symbols, round and diamond, show the relative intensity of $F_1$ and $F_2$ VS. ellipticity ranging from [-1, 1].

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3.3 Bifocal metalens with different focal lengths

For bifocal metalens, the separate distance between two foci can be designed at will which is vital in various applications. In our proposed phase equation in Eq. (2), changing the value of separate factor $2d$or center position $z$ between two foci can realize bifocal metalens with arbitrary focal lengths. Here, we designed other two metalenses with focal lengths of $10\mu m$ and $12\mu m$, $10\mu m$and $14\mu m$, respectively, and calculated their intensity distributions in the x-z plane (the third one is the metalens with focal lengths of $10\mu m$ and $13\mu m$, which is mentioned above) when illuminated by LP incident light. Since a LP light can be decomposed into LCP and RCP components with the same amplitudes, there are two focal spots with nearly equal intensity in image space. The simulated results shown in Fig. 5 confirms that all metalenses have good focusing performance.

 

Fig. 5 The intensity distributions in the x-z plane of three bifocal metalenses with (a) focal lengths are $10\mu m$ and $12\mu m$, (b) focal lengths are $10\mu m$ and $13\mu m$, (c) focal lengths are $10\mu m$ and $14\mu m$ under the illumination of XLP incident light beam.

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In addition, we calculated the simulated focal lengths of the designed three bifocal metalenses, and the results are shown in Table 1. From the table, we can see that there are little difference between theoretical focal lengths and simulated focal lengths of the designed metalens, and the difference decreases with the increase of separate factor. In our proposed phase profile, we only considered the linear factor of the polarization-dependent focal length. If non-linear factors were taken into account, such as higher-order derivative terms, the simulated focal lengths would be more accurate. But on the whole, the simulated focal lengths of our designed metalenses are in good agreement with theoretical focal lengths.

Table 1. The theoretical focal lengths and simulated focal lengths of the designed three bifocal metalenses.

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4. Conclusions

In conclusion, we have designed an intensity-adjustable dielectric longitudinal bifocal metalens based on two modes of phase manipulation, geometric phase and propagation phase. The metalens can focus one incident light into one or two spots in tandem along longitudinal direction, depending on the polarization of incident beam. When the metasurfaces is illuminated by CP light beam, it can work as a single-focusing metalens with two alternative focal lengths. Meanwhile, the focal lengths can be switched by changing the helicity of the CP incident light. When the metasurfaces is illuminated by LP light beam and EP light beam, it can act as a bifocal metalens which can focus a single light beam to two different points in the longitudinal direction. More importantly, the relative intensities ratio of the two focal points can be easily tuned by adjusting the ellipticity of the EP incident light without redesigning another phase profiles, which is very convenient, high-efficient, and time-saving in practical applications. Our designed metalens only used one phase profile, so each nanofin of the metalens can contribute to all focal points equally in the case of CP incident light, LP incident light, and EP incident light, leading to low background noise and high focusing quality. We envision that this intensity-adjustable dielectric longitudinal bifocal metalens may find potential applications in multi-functional metasurfaces, multi-imaging systems, and shared-aperture devices.