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Abstract
A polarization-dependent metasurface that consists of nanobrick arrays with spatial varying dimensions in two orthogonal directions has shown independent phase control ability, which paves a new way to design a reconfigurable step-zoom lens with two different focal lengths depending on the polarization states of an incident beam. In this paper, we report a highly integrated step-zoom metalens with dual focal lengths based on double-sided metasurfaces sitting on a transparent substrate. By assigning the focal power and balancing the aberrations between the front and rear metasurfaces, a large field-of-view ( ± 20°) step-zoom metalens corrected for monochromatic aberrations was designed, and its high performance (nearly diffraction-limited image quality for both on-axis and off-axis imaging) was verified by full-wave numerical simulations. More interestingly, the image plane of the designed metalens keeps unchanged after the zoom switching, which will bring great convenience for practical applications. With the advantages such as ultra-compactness, flexibility, and simplicity, the proposed metalens indicates the potential in the fields that require highly integrated zoom imaging and beam focusing without optical and mechanical compensations.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
In the past several years, metasurface-based planar lens or so called metalens has been intensively studied for various applications [1–22]. There are many significant breakthroughs in this field such as development of high-aspect-ratio dielectric metalens [23], corrected chromatic aberration metalens [24–31] and wide field-of-view metalens [32,33]. Unfortunately, as one of the most important functionalities in an optical imaging system, research on zoom metalens is still extremely rare. Since design of a conventional zoom lens is always challengeable because it requires not only sophisticated optical design strategy, but also complex and precise mechanical structures for lens adjustment [34–36], it is hard to apply conventional design ideals to a planar metalens. As a result, zoom metalenses realized by many other methods such as stretching a flexible substrate [37,38], changing the focal length by lateral actuation [39] or using MEMS [40] have been proposed. These ideals have promoted the innovation of zoom metalens research; however, their demands for precise mechanical structures and complicated fabrication process may limit practical applications.
In 2017, we proposed a dual field-of-view step-zoom metalens [41] based on geometric metasurfaces (GEMSs) [42,43]. In this design, two GEMSs are combined to focus an incident circularly polarized beam with two different focal lengths depending on the beam’s handedness. Since GEMS can only provide either negative or positive focal length with the same absolute value, GEMS-based zoom metalens has to suffer from the paradox of performance balance between short and long focal length conditions. Therefore, it only works well for a very small field-of-view and numerical aperture.
In this paper, we further develop the concept of dual step-zoom metalens with polarization-dependent metasurfaces [44–50] instead of GEMSs. We find both by analysis and numerical simulations that metasurfaces consisting of nanobrick arrays with spatial varying dimensions in the long and short axes have unique and independent phase control to the incident beam in orthogonal polarization directions. Therefore, we can design a metalens with two different focal lengths in orthogonal directions and correct their aberrations independently. More interestingly, the back-focal length (defined as the distance between the last lens surface and image plane) can be fixed after the zoom switching, which can be used to design a zoom metalens without complex optical and mechanical compensations that conventional optical zoom system always suffers, as the combination of two different metasurfaces can provide enough design freedoms. This unique approach features both the combination of optical step-zoom lens system and correcting monochromatic aberrations through a single metalens, which can offer a new method for the development of ultracompact and reliable optical step-zoom systems.
2. Metasurfaces design
The schematic diagram of a unit-cell structure is shown in Fig. 1(a), whereby a nanobrick with length Lx, width Ly, height H and cell size C sitting on a fused silica substrate. Figure 1(b) schematically shows the highly integrated step-zoom metalens composed of two metasurfaces with the silicon nanobricks aligned along the x and y axes on both sides of a fused silica substrate.
Fig. 1 (a) Schematic view of the polarization-dependent dielectric unit-cell structure used to implement the proposed metalens. (b) Schematic view of the metalens composed of two metasurfaces sitting on two sides of a fused silica substrate. The silicon nanobricks have the same height H = 310 nm and cell size C = 250 nm, but different dimensions in two orthogonal directions.
We use x-polarized and y-polarized light to refer to a linearly polarized plane wave whose polarized direction is along the x and y axes, respectively. Since each nanobrick can be considered as a truncated waveguide, nanostructures with different dimensions can generate different effective refractive indices by changing the size of the nanostructures in the x and y directions, and thus providing different phase distributions in the x and y directions (as illustrated in Figs. 2(a) and 2(b)). Therefore, by carefully designing the phase distributions of the nanostructures, they can act as effective phase modulators. More importantly, the functionality of phase modulation is independent in the two orthogonal directions and therefore the x-polarized and y-polarized incident light can be independently controlled. This forms the basic principle of realizing step-zoom metalens with two different focal lengths in the orthogonal directions.
Fig. 2 (a, b) Simulated phase delay and (c, d) transmittance as a function of nanobrick size Lx and Ly in two orthogonal directions. The operation wavelength is 658 nm.
We designed and simulated the nanostructures by using CST microwave studio software. The incident light is assumed to be x-polarized or y-polarized light and operated at a wavelength of 658 nm, propagating along the z-axis with electric and magnetic field vectors lying in the x-y plane. The transmitted light is collected by field ports to retrieve the phase delay and transmission efficiency. To optimize the performance of nanostructures, we swept Lx and Ly from 40 nm to 200 nm in steps of 5 nm while the cell size C and height H of nanostructures are fixed at 250 nm and 310 nm, respectively. The corresponding phase delays and transmittances for the electric field aligned along the x and y axes of the nanostructures are shown in Fig. 2. As shown in Figs. 2(a) and 2(b), the nanostructures can generate phase delays varying from 0 to 2π in two orthogonal directions. At the same time, most of the transmittances are larger than 55%, as shown in Figs. 2(c) and 2(d). To obtain a 4-step metasurface for example, we carefully chose 16 different nanostructures which can provide phase delays of 0, π/2, π and 3π/2 in two orthogonal directions. More details of the parameters of nanostructures are listed in Table 1.
Table 1. Nanobrick dimensions vs phase delays and transmittances
3. Optical design for a step-zoom metalens
The step-zoom metalens composes of two metasurfaces and the phase profiles of the first and second metasurfaces are defined as even order polynomials of the radial coordinate ρ as [33]
where R is the radius of the metalens, Ai is the coefficient which determines the shape of the phase profile and n is the number of polynomial coefficients. Since the fabrication and testing of a metasurface is quite different from conventional aspherical lens, n can be large enough depending on the aberration correction requirements.As an example, we designed a dual step-zoom metalens having a long focal length of 80 µm and a short focal length of 40 µm with a corresponding zoom ratio being 2. The thickness of the silica substrate is assumed to be 50 µm and R is 12.625 µm. Since two metasurfaces with phase profiles described in Eq. (1) can provide enough design freedoms for lens performance optimization, we can fix the back-focal length with a value of 60 µm in the design example. In the design, the second metasurface is set as the aperture stop of the metalens with a diameter of 25.25 µm, thus corresponding to a numerical aperture NA = 0.21 in the imaging space.
We used the commercial optical design software Zemax Optic Studio to optimize the phase profiles. The phase coefficients Ai are optimized for minimizing the focal spot size (root mean square spot size) with the incident angle increasing from zero to twenty degrees. In the optical design, the first metasurface operates as a negative lens and the second one as a positive lens to form a metalens for the short focal length mode, shown in Figs. 3(a) and 3(b); and two positive lenses to form a metalens for the long focal length mode, shown in Figs. 3(c) and 3(d).
Fig. 3 (a, c) Optical layout of the two different imaging modes. (b) Schematic diagram of the short focal length mode characterized by an x-polarized incident wave and (d) the long focal length mode characterized by a y-polarized incident wave. Both modes have the same back-focal length (60 µm).
The designed phase coefficients are listed in Table 2, and the corresponding phase profiles are shown in Fig. 4.
Table 2. The phase profile parameters of two metasurfaces
Fig. 4 Phase profiles of the two metasurfaces composing the step-zoom metalens. (a, b) Phase profiles of metasurface 1 and 2 for short focal length (40 µm) and (c, d) for long focal length (80 µm).
4. Numerical simulations
To verify the validity of the design approach proposed, electromagnetic propagation through the dielectric metalens was analyzed using the finite-difference time-domain (FDTD) method. The incident wave is assumed to be x and y-polarized in visible (the operation wavelength is 658 nm), propagating along the z-axis with electric and magnetic field vectors lying in the x-y plane. Due to limitations of our computer memories and saving the calculation time, the transmitted light is obtained by field ports close to the back surface of the metalens to retrieve the electric field distribution and then the follow-up propagation of light is calculated based on Rayleigh-Sommerfeld diffraction [51]. The simulation results of two different working modes for different incident angles are shown in Figs. 5 and 6, respectively. As results, the locations of focal spots at the image plane for different incident angles are shown in Table 3.
Fig. 5 Simulated results of the short-focal-length situation with the metalens illuminated by x-polarized light (on-axis and off-axis) at 658 nm incident wavelength. (a, c, e) Electric field distribution in the x-z plane and (b, d, f) the focal plane intensity in the x-y plane for different incident angles. The white dotted circles represent the diffraction-limited focal spot size (Airy disk) of the metalens.
Fig. 6 Simulated results of the long-focal-length situation with the metalens illuminated by y-polarized light (on-axis and off-axis) at 658 nm incident wavelength. (a, c, e) Electric field distribution in the x-z plane and (b, d, f) the focal plane intensity in the x-y plane for different incident angles. The white dotted circles represent the diffraction-limited focal spot size (Airy disk) of the metalens.
Table 3. Transverse location of the focal spots vs incident angle
The simulation results presented in Figs. 5 and 6 and Table 3 show that the transmitted beams are strongly focused at the image plane, and the spots quality and positions agree well with our design. The good agreement between the simulated results and the design values indicates that the proposed step-zoom metalens can realize two focal length modes and correct monochromatic aberrations efficiently, thus has satisfactory zoom and imaging performance over a large field-of-view ( ± 20°). The slight differences between the design and simulation values are due to the quantizing error of a 4-step phase-only metasurface.
One can further observe that the dimensions of the focus spots are almost equal for the two working modes (as shown in Figs. 5(b) and 6(b)). The resolution of a diffraction-limited lens is given by
where NA is the numerical aperture of the metalens. According to Eq. (2), for the same lens aperture (25.25 µm) and back-focal length (60 µm) we used in the design, the two imaging modes of the metalens correspond to the same NA = 0.21 and Res = 1.95 µm, which is consistent with Figs. 5 and 6.The MTF is derived from the PSF of focus spot at image plane by Fourier transform [52]. The calculated results of the MTF from the focusing profiles (presented in Figs. 5 and 6) are shown in Fig. 7. The MTF curves show that the step-zoom metalens has excellent image quality for the short focal length mode. For the off-axis situation of the long focal length mode, with the increasing of the incident angle, the aperture angle decreases sharply in the image space which leads to the reduction of resolution and MTF.
Fig. 7 MTFs of the metalens with (a) short focal length mode and (b) long focal length mode for different incident angles. The solid and dashed lines show the MTFs in the tangential plane and sagittal plane, respectively.
According to the simulated results, we calculated the focusing efficiencies of the metalens for two focal length modes on-axis and off-axis and list them in Table 4. The focusing efficiency is defined as the ratio of focus spot energy at image plane and incident energy. All of the focusing efficiencies are larger than 23% and the highest efficiency reaches 35.1%. In the simulations, the silicon nanobrick used has a complex refractive index of 4.12 with an imaginary part of 0.05 at the operating wavelength 658 nm, leading to a relatively large optical loss. As shown in Table 1, one can observe that the average transmission of nanostructures is about 70%, which means half of the incident energy is lost when the incident light passes through two surfaces of the metalens, and the energy loses more under oblique incidence. On the other hand, the quantizing error of a 4-step phase-only metasurface also influences the focusing efficiency. The focusing efficiency can be improved by using some materials with low optical loss or increasing steps of phase.
Table 4. The focusing efficiencies for two modes
The reconfigurable step-zoom metalens as we show here, provides an ingenious approach for realization of micro zoom lens corrected for monochromatic aberrations over a large field-of-view. It overcomes the shortcomings of the current bulky optical zoom lens system and exhibits unique advantages, such as simplicity, ultra-compactness, and flexibility, and it is promising to be integrated into a miniature optical system that demands high-performance and low-cost zoom lens. Although the zoom metalens we demonstrated here contains only two step-zooming, it can be easily extended to multi-zooming situation by assigning more metasurfaces into the metalens. That is, we can add several half-wave plates into the imaging system to transform an x-polarized beam into a y-polarized one or vice versa.
5. Summary
In summary, we have proposed an efficient approach to achieve both zoom imaging and monochromatic aberration correction. Based on polarization-dependent metasurfaces, we designed a highly integrated and reconfigurable step-zoom metalens whose focal length can be switched by flipping the polarization of an incident wave. More importantly, the combination of two metasurfaces can provide enough design freedoms to keep the back-focal length constant after the zoom switching without complex optical and mechanical compensations, which will bring a great convenience to develop an ultra-compact and reliable optical zoom system. The reconfigurable step-zoom metalens with the advantages such as ultra-compactness, flexibility and simplicity, has great promising perspectives for a variety of applications in information technology, biomedical sciences, integrated optics, optical communications, imaging, flat displays, or wearable consumer electronics.
Funding
National Natural Science Foundation of China (11774273, 11574240, 61640409, 61805184); Outstanding Youth Funds of Hubei Province (2016CFA034); Postdoctoral Innovation Talent Support Program of China (BX20180221); Guangxi Natural Science Foundation of China (2017GXNSFAA198048); Open Foundation of State Key Laboratory of Optical Communication Technologies and Networks, Wuhan Research Institute of Posts and Telecommunications (OCTN-201605).
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