Random-phase metasurfaces at optical wavelengths

Abstract

Random-phase metasurfaces, in which the constituents scatter light with random phases, have the property that an incident plane wave will diffusely scatter, hereby leading to a complex far-field response that is most suitably described by statistical means. In this work, we present and exemplify the statistical description of the far-field response, particularly highlighting how the response for polarised and unpolarised light might be alike or different depending on the correlation of scattering phases for two orthogonal polarisations. By utilizing gap plasmon-based metasurfaces, consisting of an optically thick gold film overlaid by a subwavelength thin glass spacer and an array of gold nanobricks, we design and realize random-phase metasurfaces at a wavelength of 800 nm. Optical characterisation of the fabricated samples convincingly demonstrates the diffuse scattering of reflected light, with statistics obeying the theoretical predictions. We foresee the use of random-phase metasurfaces for camouflage applications and as high-quality reference structures in dark-field microscopy, while the control of the statistics for polarised and unpolarised light might find usage in security applications. Finally, by incorporating a certain correlation between scattering by neighbouring metasurface constituents new types of functionalities can be realised, such as a Lambertian reflector.

Figure 1. Sketch of metasurface configuration.

The metasurface is positioned at z = 0 and interacts with a plane incident wave propagating along the −z-axis. The reflected electric field at the metasurface E_r is related to the scattered far-field E_ff via a two-dimensional spatial Fourier integral.

Figure 2. Numerical modelling of scattering from ideal random-phase metasurfaces.

Probability density function of the far-field intensity from an array of 200 × 200 unit cells for excitation by (a) x-polarised incident light when unit cells feature random phases described by equation (3); (b) x-polarised incident light when the random phases of the unit cells are represented by a discrete random variable that takes on values (0, π/2, π, 3π/2), each with the probability p_ϕ = 1/4; (c) unpolarised incident light when the phases for orthogonal reflection are completely correlated [i.e., ] and described by equation (3); (d) unpolarised incident light when both ϕ_x and ϕ_y are described by equation (3) but are statistically independent. The histogram plots show the PDF of the calculated far-field intensity in the upper half-space within an NA of 0.9, while the red lines correspond to the theoretical predictions by equations (5) and (6). The insets show the associated Fourier images of the far-field intensities within the NA = 0.9.

Figure 3. Reflection from GSP-based metasurfaces.

(a) Drawing of the unit cell in a GSP-based metasurface, consisting of a dielectric spacer sandwiched between an optically thick metal film and an array of metallic nanobricks with subwavelength periodicity. (b) Calculated reflection coefficient as a function of nanobrick widths for a gold-SiO₂-gold configuration with geometrical parameters t = t_s = 40 nm and Λ = 250 nm at a wavelength of 800 nm. Colour map shows the reflection coefficient amplitude for x-polarised normal incident light, while lines are contours of the reflection phase for both x- and y-polarisation.

Figure 4. Numerical modelling of scattering from random-phase GSP-based metasurfaces.

(a) Calculated (histogram) and theoretical (red line) PDF of the far-field intensity (when neglecting specular reflection) from an array of 200 × 200 unit cells for excitation by x-polarised incident light when the four types of nanobricks, defined by the intersection of contour lines in Fig. 3b for L_x = L_y, appear with equal probability in the array. The inset shows the associated Fourier image of the far-field intensity within the NA = 0.9. The image is oversaturated in order to visualise the diffusion of light. (b) Similar calculation as in a, but the frequency of occurrence of the four nanobricks is scaled by the inverse of their reflection amplitudes. (c,d) Effective reflectivity (i.e., limited by the NA) as a function of wavelength and NA for the metasurfaces described in (a,b) respectively.

Figure 5. Reflection from random-phase GSP-based metasurfaces.

(a,b) Representative scanning electron microscopy images of the fabricated GSP-based metasurfaces featuring 4 and 16 different nanobrick elements, respectively, thus approximating random-phase metasurfaces with fully correlated and statistical independent reflection phases for orthogonal polarisations. (c,d) Measured reflectivity as a function of wavelength and NA for the metasurfaces in (a,b) respectively. The incident light is x-polarised. The insets are bright-field images of the metasurfaces and surrounding gold film.

Figure 6. Statistics of scattered light.

Measured (histogram plot) and theoretical (red line) PDF of the reflected far-field intensity (when neglecting specular reflection) from GSP-based metasurfaces with (a–c) fully correlated and (d–f) statistical independent reflection phases for orthogonal polarisations. The wavelength of the incident light is 800 nm and it is either (a,d) x-, (b,e) y- or (c,f) un-polarised. The insets show the associated Fourier images of the far-field intensity within the NA = 0.55, with the area bounded by the dashed circles indicating the scattered light used for the statistics. It should be noted that the Fourier images in (c,f) are the average of (a,b,d,e), respectively, weighted so that the average intensity in the x- and y-polarised images is the same.