Generalized Brewster effect in dielectric metasurfaces

Polarization is a key property defining the state of light. It was discovered by Brewster, while studying light reflected from materials at different angles. This led to the first polarizers, based on Brewster's effect. Now, one of the trends in photonics is the study of miniaturized devices exhibiting similar, or improved, functionalities compared with bulk optical elements. In this work, it is theoretically predicted that a properly designed all-dielectric metasurface exhibits a generalized Brewster's effect potentially for any angle, wavelength and polarization of choice. The effect is experimentally demonstrated for an array of silicon nanodisks at visible wavelengths. The underlying physics is related to the suppressed scattering at certain angles due to the interference between the electric and magnetic dipole resonances excited in the nanoparticles. These findings open doors for Brewster phenomenon to new applications in photonics, which are not bonded to a specific polarization or angle of incidence.

The present section aims to present the not-so-well-known rich phenomenology associated with reflection of plane waves at an interface between an ordinary medium and one having simultaneous electric and magnetic responses. Instead of analyzing a single interface, let us focus on the case of a slab, located either in air or standing over a semi-infinite glass, since these cases arguably model more accurately the system studied in the main manuscript.
Consider the general case represented in Supplementary where the amplitude reflection coefficient (between media 1 and media 2) is given by: (2) and the amplitude reflection coefficient (between media 2 and media 3) is given by: In a totally analogous way, the reflection coefficient is given by = / , and the reflectivity is ℜ = | | . In this case: where: Note that amplitude coefficients of reflection and in the limiting case = 0 differ in sign, as E represents a polar, and H an axial vector 2 .
From this analysis it is readily seen that with appropriate variations of and it is possible to obtain arbitrary values for the Brewster angle, corresponding to the vanishing value of reflectivity, both for p-polarized light and for s-polarized light.
We illustrate now the phenomenology associated with the generalized Brewster's effect by considering a slab with and standing in air. Results, shown in Supplementary  For the sake of completeness we illustrate, in Supplementary Fig.3 the case of a magneto-electric slab on top of a glass semi-infinite medium (incidence from the side of air). As readily seen, this configuration retains all major characteristics, and serves to illustrate the phenomenology found in the case of silicon nanoparticle array over glass substrate presented in section 2 of the main text.
Supplementary Note 2: Electric far-field radiated by a pair of electric and magnetic dipoles.
Consider a pair of electric and magnetic dipoles. The electric far-field radiated in the direction given by the unit vector can be written as: with the unitary polar vector. It is clear that, in this situation, the electric field is suppressed if: with being the phase difference between the two dipoles. Whenever the phase difference between dipoles is a multiple of the field exactly vanishes. It is clear that, when an infinite array of spheres is considered in the xy-plane, this situation represents the p-polarization incidence case, and the plane of incidence coincides with the xz-plane.
Analogously, when the case depicted in Fig.2b holds, the induced electric and magnetic dipoles can be described by = (− • , , • ) and = ( , − , ). In this situation, the radiated (scattered) field in the plane containing the magnetic dipole (highlighted in Fig.2b) is purely azimuthal and reads: with the unitary azimuthal vector. It vanishes if: and will represent the s-polarization case for infinite arrays.

Supplementary Note 3: Phased arrays of point scatters.
It is known from the phased array antennas theory that the total intensity from an array of identical emitters can be expressed as: where ( ) is the so called form factor of the array, which describes the phase retardation from different elements in the lattice and ( , ) is the far-field of each identical constituent. An analogous formula holds to describe the scattering properties of an array of identical point-like scatters. As in the case of phased array antennas, ( ) carries information about the geometry of the array and does not depend on the particular scatters considered. It reads: in which = + , with = / being the wavenumber, the lattice period and = the phase difference due to oblique incidence at an angle (we consider the plane of incidence as = ). Here N is the number of particles in the array. In the limit → ∞one has: Fixing the scattering plane to = 0, ( ) is non-zero only when: This implies that, if no higher diffracted order are present, ( ) is non-zero when = − ≡ or = + ≡ . In Supplementary Fig.4a, | ( )| is plotted for = 6 ⁄ , = 730 nm and = 300 nm for several increasing number of particles . As seen, it quickly converges to the limit above, vanishing everywhere except in the reflection and transmission directions.
Let us now assume that each single element in the array is a pair of electric ( ) and magnetic ( ) dipoles. The radiated far-field ( , ) will be given by equation (8), see Supplementary Note 2 above.
Consider the two main situations presented there. In the first the electric dipole is contained in the plane of incidence ( = 0) with = (− cos , 0, sin ), and = (0, • , 0). In this case, the radiated field in this plane is given by (9). Clearly, this situation will represent the case of p-polarized incidence. In the second case, the magnetic dipole is contained in the incidence plane and reads = (− • cos , 0, • sin ) while = (0, − , 0). In this situation the radiated field in the plane of incidence is given by (11) and will represent the case of s-polarized incidence.
From (10) and (12) one can compute the relative values of and for which the field at = = − will be zero. In this case, no intensity at all will be radiated in the reflection direction (as follows from equation (13) nm and = 500). It is clear from the calculation that, due to the modulation of the form factor | ( )| , radiation in any other direction rather than those of transmission and reflection is totally inhibited due to interference from different lattice sites, even if the single particles radiate in those directions. It is also immediately seen that the suppression of radiation in the reflection direction from each single element implies the suppression of radiation from the whole array. Finally, to stress the origin of the effect in the inhibition of radiation from single elements, we plot in Supplementary Fig.4c the case = /3. This ratio does not lead to zero radiation in the reflection direction and, thus, no Brewster is obtained.

Supplementary Note 4: Absorption and higher order multipoles in arrays of spheres at p-polarized
incidence.
It is our intention here to complete the picture given in Section 1 of the main manuscript, regarding the analysis of the resonances excited in the array of silicon (Si) spheres with diameter = 180 nm and pitch = 300 nm for different wavelengths and angles of incidence for p-polarized light. As mentioned in the main text, the electric and magnetic dipole contributions are the dominant ones in the range of wavelengths and angles of incidence studied. Those are shown in the whole simulated range in Supplementary Fig.5a and b, respectively. Also, the electric quadrupole partial scattering cross section, as computed through the multipole decomposition, is shown in Supplementary Fig.5b, while the corresponding plot for the magnetic quadrupole is shown in Supplementary Fig.5c. As readily seen, both resonances appear for wavelengths much shorter than those for which the generalized Brewster effect is observed. Supplementary Figure 5d  Let us start by analyzing the p-polarized case. Thus, to achieve the zero reflection effect we must restrict ourselves to the red shaded regions where electric dipole dominates. At normal incidence the first Kerker's condition, indicated by ① and ③ in Supplementary Fig.5, leads to zero reflection. Let us first analyze the spectral region around ①. For increasing angles of incidence, equation (4) implies that / should correspondingly decrease. This is achieved at longer wavelengths with respect to ①, which manifests as the slight redshift in the zero of reflection in Fig.3b (region indicated by the white dashed curve 1) for angles below 45 degrees. Above 45 degrees, the dipoles have to be in opposite phases to cancel radiation in the reflection direction, thus crossing ❶ in Supplementary Fig.6 and entering in the green region, as it is observed in the zero of reflection in Fig.3b (region indicated by the white dashed curve 3). In order to satisfy equation (4) now the rate / should increase instead, which is again possible at longer wavelengths. In this region, however, the range of wavelengths is wider going up to ② (above ② MD contribution starts to dominate), leading to a more pronounced redshift in Fig.3b. Thus, the sequence ① → ❶ → ② always implies a redshift to fulfill equation (4), as observed in Fig.3b. Interestingly, if now ③ is chosen as the starting point, fulfilling equation Having analyzed the p-polarized case, the corresponding analysis of s-polarization is straightforward.
We are now restricted to move within the blue shaded region. Starting again in Kerker's first condition at normal incidence ③, fulfilling equation (5) now implies a blue-shift. Since at 45 degrees the dipoles must change from in-phase to anti-phase, the complete sequence is now ③ → ❷ → ②, which implies a constant blue-shift in the whole range, as observed in Fig.4b (region indicated by the white dashed curve 2). Since the blue area is narrower, this directly translates in a narrow spectral band for zero reflection, which in the real system gets reinforced by a narrowing of the magnetic resonance due to the lattice interactions.
Let us conclude showing that the main features observed in the reflectivity of the arrays can be obtained in a simple way from equations (8)-(12), which describe the radiation of a pair of electric and magnetic dipoles.
For that, let us assume that the electric ( ) and magnetic ( ) polarizabilities of the dipoles are those of a Si sphere according to Mie theory (i.e., = 6 / and = 6 / , with and the electric and magnetic dipolar scattering coefficients, respectively 3 ). One important assumption is made to correctly reproduce the results. The dipoles are assumed to change their phase abruptly around the resonance peak. For single spheres this only holds approximately but it correctly models the effect of interactions in the array. Of course one could fully take into account the effect of the lattice by computing the self-consistent field at each dipole position and computing the effective polarizabilities. However, it is enough to have good results to consider that the effect of the lattice manifests just as a steeper phase change in the polarizabilities of the particles. Thus, we take the amplitudes as given by Mie theory but assume a step function for the phases, as depicted in Supplementary Fig.7a. Then, by simply applying equations (9)  In the present section we demonstrate that, as mentioned in the main text, the differences observed between experiment and simulations in the angular reflection of a sample of silicon nanodisks on top of silica substrate are almost entirely due to the lower absorption of the fabricated sample compared to values tabulated 4 for amorphous silicon (a-Si).
As seen in Fig.5 in the main text, the differences are more pronounced above 600nm. Below that limit the agreement is fairly good (see the case at 590nm). Above, however, experiment and theory quickly depart, and reflection is higher in the fabricated sample, indicating a quick drop of absorption as compared to the tabulated data used for simulations.
In Supplementary Fig.8  show good agreement for low angles of incidence (thus fitting the spectrum at normal incidence). It is readily seen that the agreement between experiment and simulations obtained in this way is excellent.
In Supplementary Table 1  In order to complete the analysis of the generalized Brewster's effect for the Si nanodisks metasurface with pitch = 300 nm, diameter = 170 nm and height = 160 nm, given in section 2 of the main manuscript, we present here some additional results. In particular, the electric and magnetic dipolar contributions to the scattering from a single element in the array as a function of wavelength and angle of incidence under irradiation with p-polarized light are shown in Supplementary Fig.9a and b. Also, we plot in Supplementary show the radiation patterns of the dipoles at wavelengths of 590 nm and 735 nm leading, respectively, to a minimum reflection at 25° and 60° of incidence, respectively. The radiation patterns were computed with Stratton-Chu equations 5 taking into account the presence of the substrate. For these calculations, a sphere enclosing the dipoles and the substrate was considered. Although the solution is not exact, convergence against variations in the radius of the sphere was checked, yielding almost the same results. Both radiation patterns show minima in the direction of the reflected wave, thus confirming the interference origin of the observed vanishing reflection effect also in the case of silicon disks on substrate.
Supplementary Note 9: Explicit expressions used in the Multipole Decomposition.
Multipole decomposition technique was employed to analyze the different modes excited inside the particles. For particles in an array embedded in air, multipoles can be computed through the polarization currents induced inside them: where ε is the permittivity of the particle and = ( ) the electric field inside it.
This approach fully takes into account mutual interactions in the lattice 6 where only the magnetic and toroidal components are considered, since the electric one does not contribute to radiation 7 . For the quadrupolar moments we have the following expressions: