Highly efficient and ultra-broadband graphene oxide ultrathin lenses with three-dimensional subwavelength focusing

Nanometric flat lenses with three-dimensional subwavelength focusing are indispensable in miniaturized optical systems. However, they are fundamentally challenging to achieve because of the difficulties in accurately controlling the optical wavefront by a film with nanometric thickness. Based on the unique and giant refractive index and absorption modulations of the sprayable graphene oxide thin film during its laser reduction process, we demonstrate a graphene oxide ultrathin (∼200 nm) flat lens that shows far-field three-dimensional subwavelength focusing (λ3/5) with an absolute focusing efficiency of >32% for a broad wavelength range from 400 to 1,500 nm. Our flexible graphene oxide lenses are mechanically robust and maintain excellent focusing properties under high stress. The simple and scalable fabrication approach enables wide potential applications in on-chip nanophotonics. The wavefront shaping concept opens up new avenues for easily accessible, highly precise and efficient optical beam manipulations with a flexible and integratable planar graphene oxide ultrathin film.


Supplementary Note 1 Analytical Method
The intensity distribution in the focal region of the ultrathin lens can be calculated using the Rayleigh-Sommerfeld diffraction theory 2 based on the Fresnel approximation with a circular symmetry as:  (1) where the subscripts '1' and '2' indicate the parameters in the lens plane and in the focal region, respectively. As a result, the field distributions in the focal region (E 2 ) at different lateral (r 2 ) and axial (z) positions can be calculated by integrating the field distribution (E 1 (r 1 )) over the entire lens plane. One example showing here is a GO lens design with three rGO zones, as shown in Supplementary Figure 5.
When a uniform plane wave (E 1 (r 1 )=1) impinges the GO lens, part of the beam is absorbed and refracted by the rGO zones, experiencing substantial amplitude as well as phase modulations. The other part of the beam propagates through the GO zones only experiencing ignorable amplitude modulation. The modulated E-field becomes: film air where T(r 1 ) is the transmission distribution, which can be calculated using the Beer-Lambert equation 3 : where α(r 1 ) is the absorption coefficient which can be calculated from extinction coefficient K(r 1 ) through α(r 1 )=4π×K(r 1 )/λ. Ф film (=n(r 1 )t(r 1 )) and Ф air (=n air [t GO -t(r 1 )]) are the phase modulations provided by the film and the air, respectively. Meanwhile, the modulated refractive index n(r 1 ), thickness t(r 1 ) and extinction coefficient K(r 1 ) due to the laser photoreduction can be formulated as: Here n GO =2.2 is the refractive index of the GO film at 700 nm. t GO = 200 nm is the thickness of the GO film. K GO = 0.07 is the extinction coefficient of GO at 700 nm. All these parameters are measured experimentally. The modulation function M is expressed as: where C is a constant depending on the femtosecond laser power. a m is the position of m th rGO zone, and N is the total number of rGO zones (N=3 in this design). Note that the modulation function M shows a Gaussian shape governed by the intensity distribution of the laser focus, with w controlling the full width at a half maximum (FWHM). To this end, we have successfully applied the Gaussian profiles of material properties as well as the geometries of the GO lens into the analytical model, which is ready to calculate the performance of the GO lens made of materials with different physical properties at various geometries and is used all through this paper.

Effects of amplitude and phase modulations.
To understand the role of amplitude and phase modulations, the interference of two coherent beams with certain phase and amplitude modulations is studied firstly. Assuming that the electric fields of beam 1 and beam 2 are:  Figure 6). Moreover, the unique surface contour shows the dependence of the lens performance (e. g. lens peak focusing intensity normalized to the incident beam intensity in Supplementary Figure 7) on the phase and amplitude modulations. As a result, given the attainable material properties, both the amplitude and phase modulations, and the geometry of the lens can be optimized to achieve the best performance.
Compared to the conventional amplitude type Fresnel zone plate, more light is transmitted for our GO lens. The predicated maximum efficiency reaches >50%, which breaks the theoretical limitation of focusing efficiency of amplitude-type Fresnel zone plate (~10%) 3 .
Although 2π phase shift is not achieved between the adjacent GO and rGO zones, the Gaussian profile of each rGO boosts the overall GO lens performance over phase type Fresnel lens in subwavelength 3D resolution, ultrathin lens thickness and smaller lens sizes with a focusing efficiency (>50%) exceeding the theoretical limitations of phase type Fresnel lens (40%).

Derivation of optimized ring position a m
For our GO lens, the focusing is the interference of light passing through the GO and rGO zones, which provide phase and amplitude modulations. The phase modulation between the adjacent GO and rGO zones (only zone 1 and zone 2 are shown here) can be expressed as: where Δφ is the phase modulation between GO and rGO zones. Given the fact that the phase modulation within one rGO zone is comparatively weak, it is physically sound to approximate the phase modulation Δφ as a constant across each rGO zone, which is taken as the average of the Gaussian function. To guarantee a constructive interference between all the GO zones and rGO zones,   should be fixed at 2π, corresponding to constructive interference. As a result, we can get: where f is the focal length of GO lens, as indicated in Supplementary Figure 5. Therefore, by solving the Supplementary Equation 7 using the relation 2 , we can easily obtain that: During the laser-induced reduction process of GO, three physical properties (film thickness, refractive index and extinction coefficient) are correlated and all dependent on the reduction extent, which is eventually controlled by the laser power. Therefore, for a given phase modulation, the corresponding amplitude modulation is determined. As a result, we are able to consider both the amplitude and phase modulations at the same time by only including the phase modulation factor in the equation (Supplementary Equation 8) for the lens design.
The amplitude and phase modulations were optimized according to the following two criteria. 1): Maximizing the constructive interference between all the GO zones and all the rGO zones. 2): Minimizing the destructive interference between the adjacent GO and rGO zones. In this way, it is possible to achieve the best focusing/interference condition for a given phase/amplitude modulation to eventually optimize both the focal spot size and the focusing efficiency.

Derivation of wavelength independence focusing
The effective wave vector along the lateral direction which can be formulated as: which is independent of the incident wavelength.