1. Introduction

In recent years, efforts to miniaturize and improve the performance of microscale sized optical lenses have allowed the development of different imaging systems for the characterization of complex media. These systems, now accessible on almost all commercial spectrometers, have opened new fields of investigation, making it easier to characterize inhomogeneous samples. One can probe into two or even three-dimensional objects of macroscopic size (at the scale of mm or cm), while keeping a spatial resolution of the order of a micrometer, limited mainly by diffraction. Cylindrical lenses [1], random phase plates [2], segmented wedge arrays [3], and scanning mirrors [4] are often used as line focus generators for applications such as Raman spectroscopy [5] or X-rays imaging [6]. Metasurfaces have been investigated as potential alternatives for integrated optical free space components [7-11]. Metasurfaces are subwavelength nanostructured devices that enable the control of optical wave fronts, polarization, and phase to build a large variety of flat optical components [12-29]. To date, a large collection of metasurfaces have been proposed including metasurfaces that focus an incident circularly polarized light to line [29–34]. Efforts to maximize the efficiency of metasurfaces need not only to perfectly implement the theoretical geometries but also to account for the sensitivity of the results to fabrication imperfections. Indeed most experimentally reported metasurfaces have exhibited performances smaller than the theoretically predicted ones. We propose and experimentally demonstrated a polarization independent planar cylindrical lens that focuses a normal incident wave to a line and systematically analyze the sensitivity of the devices to fabrication. Using numerical simulations, we investigate the influence of relevant geometrical parameters on the performance of the device and classify the parameters with respect to their influence on the intercept factor of the lens. The proposed lens generates line focusing with intercept factor larger than 0.85. The line-focusing planar lens paves the way to applications such hyperspectral Raman spectroscopy, line-focusing planar solar collectors, or wave probes.

2. Numerical simulation

To achieve the desired line-focusing, the phase profile of the wave front as a function of position 𝑥 along the metasurface lens must satisfy the parabolic equation [23]:$\Phi =k0(f-\sqrt{x^2+f^2})$ where k₀ is the free space wave-vector, x is the distance from the considered element to the center of the metasurface lens and f is the focal length. The metasurface is made of TiO₂ cylinders and operates at 800 nm wavelength.

The inset of Fig. 1 (a) shows the dielectric metasurfaces unit cell used to implement line focusing. It consists of a TiO₂ cylindrical resonator on a SiO₂ (h_s = 440 nm) – silver (h_g = 100 nm) multilayer. Silicon dioxide (SiO₂) is the spacer and silver (Ag) [35] serves as ground plane for the lens to work in reflection. The thickness (h) of the cylindrical resonators is fixed to 250 nm and the radius (R) changes to achieve the required phase. The metasurface is periodic along the y direction with a sub-wavelength unit cell (p_x = p_y = 510 nm). Due to the low loss of glass and TiO₂ at the desired frequency (λ = 800 nm), the metasurface is considered lossless. Using ellipsometry, the refractive indices of TiO₂ and the SiO₂ were measured and found to be ~2.325 and ~1.49 respectively (average of 5 measurements). To obtain the required 2π phase-shift (see Fig. 1), the scattering parameters of the resonators are calculated using the commercial software CST with periodic boundary conditions. Figure 1(a) shows that the 2π phase-shift is achieved by changing the radius from 90 nm to 200 nm. We designed a cylindrical metasurface lens with a focal length of 400 µm and a size of 200 µm by 200 µm. By interpolating Fig. 1 and using the phase equation, the radius of each element fitting the required phase is obtained (see Fig. 1 (b)). To have a line focus along the y direction, elements in the y-direction are identical with a periodicity p_y, and, the radius of resonators is varied only along the x direction.

 

Fig. 1 (a) Phase shift for different radii and schematic of the unit cylindrical elements. P (period), h (thickness of the resonator), R (radius), h_s (thickness of the spacer), h_g (thickness of the ground plane). Materials: TiO₂ (brown), SiO₂ (blue), Ag (grey). (b) Phase distribution of a planar cylindrical lens under normal incidence with a 40 µm focal length and a 200 µm by 200 µm aperture size (blue). Required radius (green).

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3. Fabrication

Figure 2 illustrates the fabrication flowchart of the structures using top-down deposition and etching processes. The dielectric planar cylindrical lens is fabricated on a BK7 glass substrate [36]. It should be noted that the BK7 substrate has a crystallographic mesh parameter incompatible with that of gold or silver, resulting in a quasi-zero adhesion to the substrate. Therefore, we deposit, using electron beam metal deposition, a thin layer (5 nm) of Germanium (Ge) on the substrate since Ge improves adhesion and surface smoothness. A silver metal layer of 100 nm (larger than the skin depth of light in the visible) is then deposited without opening the chamber by changing the crucible [Fig. 2(a)]. Then, a 440 nm thick SiO₂ layer was deposited on the Ag layer using PECVD (Plasma-enhanced chemical vapor deposition) as shown in Fig. 2(b). An electron beam resist (PMMA), is spun on the sample followed by baking at 170 °C in an oven. The metasurfaces pattern is written in the resist using electron beam lithography (EBL) [Fig. 2(c)] on an area of 200 µm by 200 µm and subsequently developed in solution to remove the exposed e-beam resist (EBR). This pattern is the inverse of our final metasurfaces. The exposed sample is transferred to an atomic layer deposition (ALD) chamber producing amorphous TiO₂ while avoiding the contamination of the ALD chamber by the EBR. The ALD process (TiCl₄ precursor) deposits 400 nm of TiO₂ so that all features are filled with TiO₂ [Fig. 2(d)]. The residual TiO₂ film that coats the top surface of the resist is removed by RIE (reactive-ion-etching) process [Fig. 2(e)]. After removing PMMA, periodic TiO₂ metasurfaces were obtained [Fig. 2(f)].

 

Fig. 2 Fabrication process: (a) metal deposition, (b) SiO₂ deposition using PECVD, (c) E-beam lithography (d) Thin layer TiO₂ deposition using ALD, (e) TiO₂ RIE process to expose the underlying PMMA pattern, (f) removal of the PMMA. (g) Top view SEM images of metasurfaces and zoom-in containing 6 x 5 cylindrical structures imaged in the xy plane and clearly evidencing the gradient in structures size. (h) Atomic Force Microscope (AFM) measurements of the shape at the top a cylinder.

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Scanning electron microscope (SEM) images, presented in Fig. 2(g), shows successful fabrication of the structures with good quality. The fabrication process of Fig. 2 results in a height of TiO₂ slightly higher on top of PMMA, resulting in curved TiO₂ particles when PMMA is removed. The flatness of the top of the fabricated cylindrical elements is characterized using AFM measurements (Fig. 2(h)). These measurements confirm a curvature of the top of the cylinders resulting from the ALD deposition process with δ = 25 nm.

Non-ideal particle shapes or fabrication imperfections are unavoidable, and, they may limit the performance of optical devices. In order to quantify the impact of fabrications imperfections on line-focusing for the fabricated metasurface, numerical simulations were performed. The investigated geometrical parameters are the curvature (δ) on the top of the cylinders and the radius of the cylinders (R). Figure 3 (a) shows the simulated phase distribution for periodic structures with different δ and R. We set R = 200 nm and δ = 0 as phase reference. Figure 3 (b) shows phase from R = 90 nm to R = 200 nm with δ = 0 nm and δ = 25 nm. One can observe that the curvature affects less resonators of small radius. The phase variation as function curvature (δ) for different R is presented in Fig. 3 (c). One can see that the maximum phase variation is about 30° at about R = 158 nm. Figure 3 (d) presents the profile of Fig. 3 (c) at R = 150 nm and clearly shows that the phase variation increases with δ. To further investigate the impact of such fabrication imperfections on the metasurface efficiency, the intercept factor is analyzed. The intercept factor is defined as the ratio of the integrated power within the focal spot with fabrication imperfections to the one with no fabrication imperfections [23]. Fabrication imperfections are modelled as a random fabrication noise (ε |ΔR|) that adds to the ideal dimensions. Hence, the total phase of each element is given by, R_real = R + ε |ΔR|, where R_real is the real dimension with fabrication imperfections, R is the ideal dimension. ε is a random number between −0.5 and 0.5, chosen from a uniform distribution. |ΔR| is the magnitude of fabrication imperfection. Dimension mismatches will cause phase mismatches that reduce the intercept factor. Modeling each resonator as a point source with the given parabolic phase with phase noise, we calculated 100 metasurfaces with random fabrication imperfections and obtained an average intercept factor. Figure 3 (e) presents the average intercept factor for different curvature (δ) and different magnitude of fabrication imperfection (|ΔR|) and Fig. 3 (f) is the profile with δ = 0 nm and δ = 25 nm. The intercept factor is more sensitive to |ΔR| and it drops to about 0.85 when magnitude of fabrication imperfection is about 10 nm.

 

Fig. 3 (a) Phase distribution for a periodic structure for as a function of the radius (R) and the curvature (δ). (b) The phase profile of (a) at δ = 0 (nm) and δ = 25(nm). (c) The phase variation due to the curvature (δ) with different radius (R). (d) The phase variation with different δ at R = 150 nm. (e) Average intercept factor as function of fabrication imperfections (|ΔR|) and curvature (δ). (f) Average intercept factor as function of fabrication imperfection (|ΔR|) at δ = 0 nm and δ = 25 nm.

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The calculated intercept factor in Fig. 3 (f) shows that the metasurfaces are robust towards fabrication imperfections (radius and curvature of holes). This robustness can be understood by analyzing the confinement of optical modes in the resonators. Figure 4 presents the field distribution in resonators of various size at λ = 800 nm. One can observe that while the field is barely confined in the curved region for small radius (radius = 90 nm), the field is strongly confined in the center of the curved region when the radius is 160 nm. This explains the small sensitivity of smaller particles to the curvature of the resonators.

 

Fig. 4 Total electric field (x-polarization) for different radii without fabrication imperfections (δ = 0 nm): (a) Radius R = 90 nm, (b) R = 160 nm, (c) R = 200 nm. Total electric field distribution for different radii with fabrication imperfections (δ = 25 nm): (d) Radius R = 90 nm, (e) R = 160 nm, (f) R = 200 nm.

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4. Experiment

To optically characterize the fabricated lenses, a custom setup was used. The experimental setup is composed of two main systems dedicated to illumination and imaging. The illumination system comprises a supercontinuum laser (NKT photonics) and an acousto-optic tunable filter (Super K) to select the operating wavelength (800 nm). A beam at λ = 800 nm was transmitted through a broadband polarizing beam (50:50) splitter to illuminate the lens. The focused reflected beam is collected by the imaging system via the beam splitter. The imaging system consists of an extra-long working distance objective (x50), a corresponding lens, and a camera. Due to the limited space between the objective and the lens, an extra-long working distance (WD = 10 mm) was used. The lens was also mounted on a stage to adjust the distance between the lens and objective (50 × objective).

A clear and uniform line focus is observed for both the x and y polarization demonstrating polarization independence. The measured intensity profile illustrates the line focus at the focal distance (0.4 mm). The small discrepancies between numerical simulations and experimental results are mainly attributed to the fabrication imperfections inducing slight non-uniformity in experimental results. To better quantify the uniformity of the focus, we calculated full width at half maximum (FWHM) of light intensity along x for different values of y. The average and standard deviation of the FWHM are then calculated and the uniformity is defined as 1 – standard deviation / average. Using plots in Fig. 5, the average FWHM is 1.85 and the standard deviation is 0.12 resulting in field uniformity of 93%. In our design at the wavelength of 800 nm, the numerical aperture (NA) is 0.247, and the diffraction limit is thus d = 1.65 µm.

 

Fig. 5 (a), (b), and (c) present numerical simulations of the normalized electric field (x-y plane) for x, y polarizations and normalized electric field in x-z plane respectively. (d) and (e) present the experimental normalized electric field intensity at the focus in the xy plane for x and y polarizations. (f) compares the normalized intensity profile along x in numerical simulation and in experiment.

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5. Summary

We have experimentally demonstrated a polarization independent planar dielectric cylindrical lens made of TiO₂ for line focusing. The fabrication method results in curved cylindrical resonators with a depth of about 25 nm that was experimentally characterized using an atomic force microscope (AFM). We have quantified the impact of fabrication imperfections (including both the curved shape and non-ideal radius) on the performance of the device. The intercept factor of the lens, i.e., the ratio of the amount of energy at the focus with and without fabrication imperfections revealed that the radius of resonators plays a more critical role than the curvature at the top of the resonators. The reported lens not only provides line focusing but also uniform intensity distribution along a line with a width mainly limited by diffraction. This work will pave the way to detailed analysis of nanoscale devices accounting for various geometrical imperfections originating from nano-structuring and to a selection of architectures based on sensitivities for various applications.