In Fig. 1, the structure of our metamerials nanoslit lens is clearly displayed. The lens is composed of metal grating with very fine slits in between. The grey parts are metals, and the white parts represent the slits. All slits are exactly the same in width, and the distance between every two slits are also exactly the same. In our simulations, for the metal parts, we use gold, silver, as well as aluminum. Here, we need to define several parameters: the slit width is a, and the periodicity is d. Both a and d is much smaller than the incoming source wavelength, so the lens as a whole has a width that is in the same scale of the input wavelength. Meanwhile, L represents the thickness of the lens, which will be a variable in our simulations. It also has been shown that the refractive index of this kind of metamaterials lenses is entirely determined by the d/a ratio /cite{three}. Therefore, we can get an effective lens that has a thickness equal to L/n but with a artificially high refractive index n = d/a, which is not actually available in real world materials. By defining a material like this in COMSOL, we can actually compare the advantage of our metamaterials lens with the lens with a artificially high refractive index.

COMSOL is a finite element analysis, solver and simulation software. In COMSOL, our setup is shown as Fig. 2 above. We use perfect matched layer (PML) all around the workspace to eliminate any reflectance that could interfere with the propagating field. The source wave is a Gaussian beam excited by a surface current input. Because here we use a source beam that is larger than the lens, we actually do the simulation in an opposite way to the real-world setup. The right part of Fig. 2 is the screenshot we get after enlarging the lens part in our setup domain. Obviously, we use a spherical lens instead of the rectangular one we shown in Fig. 2. However, the periodic structure is exactly the same.

As what I have stated in the previous section, there are three main criteria that can judge wether our design of the metal nanoslit lens is successful or not: 1. focal length, 2. full width at half maximum, and 3. transmittance. In order to verify the universality of the metal nanoslit lenses, we run the simulations with different input wavelengths at 355[nm], 637[nm], 1.064[μm] and 10.6[μm]. Thus, at least we need to collect these three groups of data at four different wavelengths. In COMSOL, the focal length of a lens is pretty easy for us to get. Firstly, we can find out the focal point by searching the maximum electromagnetic field amplitude at the central line of the computational domain using the built-in function Max/Min Line. Because the base line of the lens is manually set by ourselves, we can simply calculate the distance between the center of the base line to the maximum point. This is the focal length we want. Getting FWHM in COMSOL is a little bit more difficult. We need to add a plot of the electromagnetic field amplitude at the cross section of the focal point. The dependent variable (x-axis variable) is the distance from the central line of the lens. Then we can label two extra data points by typing in the half value of the maximum field amplitude. By enlarging the plot and recording the exact corresponding x-axis values, we can measure the FWHM. Lastly, for the transmittance, we actually use two sets of values to define it. On the one hand, we measure the intensity at the cross section of the focal point by using the built-in line integration function in COMSOL. Then we will eliminate the lens and measure the intensity of the blank computational space at the same position. The ratio of these two intensities as we believe gives the transmittance of the metal nanoslit lenses. One the other hand, we also care about the maximum intensity at the focal point, because it implies the quality of the images produced by our lenses. Therefore, we will record this value as well.