## Abstract

We report the experimental demonstration of a two-dimensional (2D) metasurface with ordered rotation of elements and show that it can be used to control the symmetry of light-matter interaction. A 2D lens is demonstrated in the visible region by transforming the rotational symmetry associated with the off-axis incident light to the translational symmetry, allowing an extraordinarily large field of view (FOV) as well as optical Fourier transformation. Furthermore, such a planar lens has a long focal depth, with polarization selectivity and subwavelength resolution. The scheme presented here may provide many new perspectives on the design of novel 2D optical devices.

© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## Corrections

5 December 2017: A typographical correction was made to Fig. 3.

## 1. Introduction

The wide-angle view, as an elaborately evolved technique for arthropods and many vertebrates, is one of the most fascinating optical properties of animal visual systems. Owing to the spherical arrangement of ommatidia or spherically-shaped camera-type eyes [1], the FOV of these systems can be close or even larger than 180°. By analogy to these natural prototypes, a plethora of optical devices, such as the artificial lenses enabled by the elastomeric materials [2] have been developed. Unfortunately, the operating principle of these 3D lenses is based on the refractive optics, which cause them to be bulky and not easy to be implemented in 2D flat optical systems.

Recently, 2D metasurfaces have emerged as a promising alternative to traditional bulky components [3–7]. However, virtually all previous demonstrations of flat lenses are limited by the off-axis aberration [8,9]. Although multilayered metasurfaces have been demonstrated to be able to increase the FOV [10,11], the fabrication process becomes more complex. In a general theory, such angle-induced aberration could be considered as a direct result of the breaking of rotational symmetry of the light-lens configuration. As illustrated in Figs. 1(a) and 1(b), the pronounced Luneburg lens and compound eyes possess spherical symmetry in refractive index distribution, and therefore light rays coming from different orientation angles could be perfectly directed to predefined spherical focal surfaces. Nevertheless, the rotational symmetry of these near-optimal lenses make them to be not compatible with flat optics as well as current planar fabrication technologies [12,13]. Here, we present a conceptually new strategy that is able to control the symmetry of light-matter interaction in 2D structured media. We prove that an ultrathin flat lens with rapid phase gradient could enable almost perfect conversion from rotational symmetry to translational symmetry, resulting in superior wide-angle lensing performance.

## 2. Principle of design

The phase of the super-symmetric lens presented here follows a quadratic form. Therefore, the lens can be termed “quadratic lens”:

where*k*

_{0}is the wavenumber in free space,

*f*the corresponding focal length, and $r\in [0,2f]$ the radial distance to the lens center. Interestingly, Eq. (1) is just the phase for a normal thin lens in the paraxial regime. Assuming that the incident collimated light beam lies in the

*xz*plane with an arbitrary angle of

*θ*to the normal axis of the lens, the phase carried by the outgoing light should be:

*k*

_{0}

*x*sin

*θ*is the gradient phase induced by the oblique incidence. Since the last term in the right hand is independent of

*r*and can be neglected, there is only a transversal shift of

*f*sin

*θ*in the

*x-*direction with respect to normal incidence. In this regard, the rotational effect of the oblique incidence light is perfectly converted to the translational symmetry of the focusing beam (see Fig. 1(c)). As a result, this lens can be used in either wide-angle imaging or Fourier analysis. It is noteworthy that the symmetry transformation is also related to the transformation optics [14], but the quadratic lens is much thinner than that obtained with a direct coordinate transformation.

Although the theory of quadratic lens seems straightforward, such lens could not be easily realized with either the refractive or diffractive optics. The main challenge lies in the fact that the radial wavenumber induced by the phase gradient (*k _{r} = $\partial \Phi /\partial r$*) is rather larger than that of conventional case. For instance, there are

*k*

_{r}= k_{0}and 2

*k*

_{0}at

*r*=

*f*and 2

*f*, respectively. At normal incidence, the light fields passing through region where

*r*>

*f*will become evanescent and therefore do not contribute to the focal spot. At oblique incidence with angle of

*θ*, the evanescent zone would shift horizontally. This dynamical excitation of the evanescent wave demonstrates an interest phenomenon different from previous results of angle-selective transmission [15].

Besides the wide FOV resulted from the symmetry transformation and angle-dependent excitation of evanescent wave, the quadratic lens possesses an extraordinarily large depth of focus (DOF) compared to an ordinary lens. This fact can be clarified using the geometric optics view as shown in Figs. 2(a) and 2(b), where the propagation direction of the light ray is calculated using the metasurface-assisted law of refraction [16,17]. In contrast to the ordinary lens that focus all light rays into a single point, distinct parts of the quadratic lens contribute to slightly vertically shifted focal spots. Consequently, the formed light spots resemble a needle in 3D space, which are of particular importance in applications such as optical storage and nanolithography [18,19].

As noted in above discussion, the quadratic lens needs a rapid phase variation at its periphery (*k _{r} =* 2

*k*

_{0}), which in turn requires that the period of the phase-changing unit cell to be

*p*=

*λ*/2N (

*N*is the number of discrete levels within [0, 2π]). If we choose

*N*= 2 and

*λ*= 600 nm, the period should be as small as 150 nm, with the characteristic dimensions of the nanostructures to be much smaller. Besides the fabrication challenge, the deep-subwavelength patterns are less effective in energy efficiency because the electromagnetic resonances corresponding to the phase variation are tightly related with the dimension of these nanostructures [20].

To achieve a better phase modulation in the deep sub-wavelength scale, 2D quasi-crystal composed of elliptical nano-aperture array in an ultrathin metallic film are adopted (see Fig. 2(c)). By rotating theses apertures in a hexagonal lattice, the predefined phase profile can be realized accompanying with a strong spin-orbit interaction [21–23]. Different from the sparsely arranged nano-antennas [3,24], the nanoholes are compactly arrayed so that some of them are connected together, leading to a much wider bandwidth [25] (Note that the efficiency of single-layered metasurface has an upper limit of 25%, while the experimental efficiency is typically below 10%, which can be increased by using dielectric or reflective structures [26–28]).

## 3. Experimental validation

As a proof-of-concept, a sample was fabricated by defining quasi-continuous apertures in a 100 nm thick gold film using focus ion beam (FIB) milling following the approach shown in [25] (Fig. 3(a)). The radius of the sample is 20 μm and the predefined focal length *f* is chosen as 10 μm at λ = 532 nm (8.407 μm at 632.8 nm). To demonstrate the wide-angle performance, four plane waves with different orientation angles with respect to the *x*-axis are combined and used as the incident light. Finite difference time domain method and vectorial angular spectrum theory are adopted in the numerical simulations. In all the calculations, we assume that light is propagating along the + *z* direction. As can be seen in the simulated results at λ = 632.8 nm (Figs. 3(b) and 3(c)), the light fields for different plane waves are just the same except that there are some translational shifts corresponding to Δ = *f*sin*θ*. Owing to the long DOF, the maximal intensity shift to *z* = 7.5 μm. The values of Δ are −4.4, 5.9, and −8.3 μm for *θ* = −32°, 45°, and −80°, agreeing with the theoretical expectations. Using two He-Ne lasers and a home-made microscope, we measured the focusing behavior of the sample. While one laser beam illuminates at normal incidence, the incidence angle of the other laser beam was tuned at −32° and −80° successively. As shown in Figs. 3(d)-3(f), the measured results are in good agreement with the numerical ones.

The “optical needles” formed by the quadratic lens do not maintain their width at different positions along the *z-*direction. Instead, both the intensity and the field distribution vary at each horizontal plane before and after approaching the focal plane, as a result of the interference of the complex electromagnetic fields. Such interference may provide a mean to break the diffraction limit in classic optics, although the large side lobes are still need to be suppressed [29,30]. Another important characteristic of the formula shown in Eq. (1) is that the phase can be simply scaled for different operating wavelengths. That is to say, when the wavelength of light is altered, the focal length changes in a linear way. Recalling the fact that the quadratic lens has a rather long DOF, we expect that such lens could operate at a constant image distance within a wide spectrum. Such property may help to overcome the chromatic dispersion in traditional flat lens [18]. To demonstrate this, three lasers operating at red (632.8 nm), green (532 nm) and blue (490 nm) wavelengths are used as the light sources, as shown in Fig. 4(a). The incidence angle for the three lasers are set as: (*θ*_{1}, *φ*_{1}) = (0, 0), (*θ*_{2}, *φ*_{2}) = (32°, 0), and (*θ*_{3}, *φ*_{3}) = (80°, 270°), where *φ =* atan(*y*, *x*) denotes azimuthal angle in the *xy* plane. Figures 4(b) and 4(c) show the cross-sections of the intensity distribution. It is clear that the centroids of the focused beam would shift to larger *z* when the wavelength is decreased from 632.8 to 532 nm. Nevertheless, the three focal spots can be clearly observed in a common cross plane at *z* = 7.5 μm (see Fig. 4(c)).

## 4. Flat lens for wide-angle beam steering

Besides the novel properties mentioned above, the translational symmetry of the quadratic lens also makes the flat lens suitable for lens antennas. As shown in Fig. 5, a proof-of-concept simulation shows that the radiation direction and side lobe of the output beams can be readily tuned by adjusting the horizontal position and the distance between the source and the flat lens. Again, the side lobe of the radiation is dependent on the distance between the lens and point source. Figure 5(a) shows the radiation patterns with side lobes below −20dB. By decreasing the distance from 50 μm to 46 μm, the side lobes increase to about −12dB, while the 3dB beam widths decreased from 1.1° to 0.6° for normal incidence. In principle, further reduction of the beam width is possible by increasing of the aperture, although it will be a bit larger than an ideal lens antenna since the quadratic lens has a long depth of focus. Compared with previous beam steering techniques such as Luneburg lens and rotational prisms, the current design provide great advantages such as low profile, easy implementation and dramatically reduced side lobe.

## 5. Conclusion

In summary, we have presented the design principle and experimental validation of flat lenses with wide range of operation angles, which may find applications in many scenarios ranging from surveillance and security in the optical range to beam steering in the microwave regime, owing to its planar and ultrathin profile and outstanding wide-angle scanning ability. Moreover, the concept of symmetry transformation may provide a powerful tool for dealing with similar physical problems, leading to either improvement of the performance or completely new innovations in related areas associated with all kinds of waves.

## Funding

973 Program of China (2013CBA01700); National Natural Science Foundation of China (61622508, 61575201).

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