# Design of a metasurface lens

The design of any metasurface requires the optimization of unit elements, which can be done through sweeps and specialized algorithms designed for a specific problem, which are sometimes present in commercial software (most of the times algorithms are written by the user). The optimization of unit elements needs to focus on the transmission (reflection if the lens is designed to be reflection based) and phase distribution of these unit elements. The material selection is also very important in the design since the extinction factor or imaginary part of refractive index dictates the maximum transmission a material can achieve. After the material selection and optimization part, a lensing metasurface is designed and tested first on simulations before it can be fabricated.

## Selection of material

The material selection should be fundamentally based on the complex refractive index of the material. By employing subwavelength unit elements, we can effectively control the permittivity (both real and imaginary components) of a material, which directly correlates to complex refractive index of the material through Eq. 1 (for non-magnetic materials).

$N=n+i \kappa=\sqrt{\epsilon_{o} \epsilon_{r}}\hspace{5cm}(1)$

For lensing and other imaging application, we need maximum transmission/reflection efficiency from our unit elements, which in turn provides maximum efficiency from our designed metasurface incorporating a multitude of such unit elements. To get maximum transmission, the material from which unit elements are designed must be lossless around operating/ working wavelengths. Moreover, the real part of the refractive index of selected material should also be high to achieve greater phase shift, as discussed in previous discussion. For example, if we consider Fig. 1 (providing Complex refractive indices of amorphous silicon (a-Si), gallium nitride (GaN), and carbon (diamond-C)), we can see that a-Si remains lossy till 0.9 μm or 900 nm, but after that it almost becomes lossless. Moreover, it provides highest real part of refractive index than any of the other two materials, that is why it should be chosen for application above 800-900 nm. Whereas, if we look at n and κ curves of GaN and C, we can see that they provide almost similar real part of refractive index, but GaN becomes lossless after 400 nm and C remains lossless throughout 200-1000 nm. Therefore, for applications falling in range of 200-400 nm, C is preferable, but for application in the regime of 400-800 nm, both GaN and C can be used. And lastly, for applications above 800-900 nm, a-Si is preferable due to its higher real part of refractive index. Figure 1: Complex refractive indices of amorphous silicon (a-Si), gallium nitride (GaN), and carbon (diamond-C). The values of complex refractive indices are taken from the this source.

## Design and optimization of unit elements

After selection of the material, the design of unit elements is required. The designed unit element must be highly transmissive or reflective (based on type of lens, transmission type of reflection type), and changing in unit dimensions must yield change in phase that must range from 0 to 2π. There is a restriction on the change of height of these unit elements, such that the height should be constant in order to achieve the desired phase so that the metasurface (combination of these unit elements) can be fabricated in a single run. There are vast numbers of designs that can be used for unit elements. Depending on the target application, the unit elements can be designed as polarization sensitive or insensitive.

The polarization insensitive designs include nano-disks/cylinders, square, square pyramid, and conical. These designs usually operate on linear polarization and the desired phase is achieved by changing the lengths and widths of the resonating unit elements. For the case of disks, the diameter or radius of the disks can be changed to attain the phase distribution from 0-2π as depicted in Fig. 2. However, the design of polarization sensitive unit elements leads to somewhat different design schemes. In polarization sensitive designs, such as rectangular bars or ellipse, the desired phase change is achieved by rotating the unit elements rather than changing the dimensions of the unit elements. These designs operate for circular polarizations, where rotation of an element can give a phase change of 2xrotation angle for a cross-polarization component. This phase change is known as PancharatnamBerry (PB) phase. A depiction of polarization sensitive designs is also shown on Fig. 2. Figure 2: A visual depiction of phase coverage in polarization sensitive and insensitive unit element design for metasurface lens.

The polarization sensitive designs have an edge of higher efficiency on the polarization-insensitive designs. This higher efficiency is due to the fact that changing the dimensions (except height) of the unit elements varies their transmission, reflection and absorption capabilities, thus decreasing (since the design is optimized for highest performance under one dimensions) the overall transmission from a metasurface.

## Metasurface lens

Designing the metasurface lens after optimizing the unit element is simple. We just need to calculate the phase required by each unit element to make a lens in the metasurface. The phase required by each unit element can be calculated by the Eq. 2 below.

$\phi=\frac{2\pi}{\lambda}(f-\sqrt{x^2+y^2+f^2})\hspace{5cm}(2)$

In the above equation, λ is the target wavelength, f is the focal point, and x and y mark the central position of the unit element. The center point of this metasurface lens is at coordinates of (x,y)=(0,0). It is important to keep in mind the units of each factor in Eq. 2, and care must be taken such that each length unit is written in same dimensions. It is better to keep everything in meter and write all factors in x10^-9 (if we are working in dimensions of nm).

In order to check test the lens in simulations, usually a lens of numerical aperture (N.A) of 1 is taken and from here, focal length f can be calculated using the formula given in Eq. 3.

$f\approx\frac{D}{2N.A}\hspace{5cm}(3)$

Here, D is the diameter of the metasurface. The Eq. 3 specifies that the focal length will be half of the diameter of total metasurface. This means that if, for example, the size of the metasurface is 30 μm (square dimension), and the period of the unit element is 400 nm, then the focal length f can be assumed as 15 μm for N.A of 1. This metasurface will be able to hold 75 unit elements.

After the calculation of phase, we need to design the metasurface in such a way that the phase needed to make lens at a particular point should should be filled by unit element that yields a similar phase (or close to the required phase). For example, if we consider the Fig. 2, if the phase required by lens at a particular point is 3π/2, we can place disk with a diameter of 250 nm (for polarization insensitive design) or place a rectangle with a rotation of 135˚ (for polarization insensitive design). A visual depiction of metasurface lens is provided in Fig. 3.