## Abstract

We theoretically and experimentally demonstrate that the diffraction of microstructures based on silver nanowires leads to very efficient microfocusing effects. Pairs of parallel nanowires act as ultrasmall cylindrical microlenses with diffraction-limited resolution in the Fresnel region. This is a new diffraction scheme to make micron-sized optical lenses with higher transmittance than plasmonic microlens based on nano-aperture arrays. Calculations based on the scalar Rayleigh-Sommerfeld integral highlights the pure scalar diffractive contribution. Thus, the plasmon contribution is negligible in such micron-sized metallic geometry. We demonstrate that two-dimensional grids of nanowires can be used to fabricate dense arrays of microlenses, i.e. 10000x10000 DPI (dots per inch).

© 2012 OSA

## 1. Introduction

Progress in nanotechnology leads to nanostructured solid-state detectors and light sources that need microlens functions fabricated at the same scale. The size of conventional refractive microlenses cannot be reduced in the micron scale to avoid strong diffractive effects. Recently, arrays of nanoscale slits in metallic films have been designed to obtain micron-sized plasmonic lenses [1–6]. Each nano-aperture induces a phase shift on the transmitted electromagnetic field. The aperture phase shift depends on its geometrical parameters, and on material properties. The nano-apertures are arranged to obtain the curved phase front that is needed to focus the transmitted beam [7–9]. The transmittance of each sub-wavelength aperture is mediated by the surface plasmon resonance occurring inside the aperture. The overall transmittance of such opaque metallic films with nanoholes is intrinsically small, i.e. in the few-percent range.

In this present work, we demonstrate that diffractive micro structures are an alternative to these plasmonic nanoslits and lead to higher transmittance efficiency. We show that the light diffracted by parallel silver nanowires can lead to constructive interferences that induce efficient focusing effects in the Fresnel region. This is a new micro-optics design to fabricate ultrasmall microlenses with overall size of a few micrometers. They have focal lengths of several wavelengths, and diffraction limited resolutions. In addition, they are intrinsically more transparent than equivalent plasmonic lenses that are based on nano-aperture arrays. Pair of parallel nanowires acts as cylindrical microlenses. In fact, they focus the light with an intensity enhancement at the focal point up to 2.2 times the incoming intensity.

The geometry parameters are the wire widths that set the diffraction efficiency, and the distance between the wires that determine the focal lengths, and the resolutions. Experimental results with separating distances of 1, 2 and 4 microns are in excellent agreement with calculations based on the scalar Rayleigh-Sommerfeld integral** ^{10}**. In such micron-sized geometry, the plasmon contribution is confined close to the nanowires, and is negligible in the Fresnel zone. The metallic nanowires are only used to block efficiently the incident light.

We have also demonstrated that two-dimensional single microlenses and dense arrays of such ultrasmall microlenses are easily obtained with grid geometry of nanowires. The field distribution patterns analysis of a unique cell performed with FDTD simulations are in excellent agreement with experimental results. We have shown that the focal spots are located at the same plane for the 1D and 2D cylindrical lenses. Furthermore, the normalized intensity at the focal point of the 2D configuration is more than doubled compared to the 1D case.

## 2. Experimental considerations

The silver nanowires are manufactured on microscope cover slips using a laser writing technique based on photoreduction chemistry [11,12]. The fabricating set-up is based on a tri-dimensional microfabrication module (TeemPhotonics [13]**)** coupled to a Zeiss Axiovert 200 inverted microscope, and including a Q-switched Nd:YAG microlaser (NP-15010 TeemPhotonics) operating at λ = 1064 nm. The focalization of the laser beam is achieved using an 1.3 numerical aperture oil immersion objective (A-plan x100). The sample is moved with a 3D piezo-nanopositioning system (PI nanocube) of a scanning range of 100 x 100 x 100 μm. Metallic cations of a silver nitrate solution (AgNO_{3}) are dispersed in a PSS (4-styrenesulfonic acid) matrix that includes a two-photon sensitive photoreductor tris(2,2’-bipyridil)dichloro-ruthenium (II) hexahydrate, Ru(Bipy)_{3} [14]. The **0.**7 ns laser pulses are used to trigger the photoreduction process by two-photon absorption. Upon reduction, silver cations precipitate as neutral silver nanoparticles that merge as nanowires by scanning the laser focal point along arbitrary geometries. The fabricated wires are 20µm long and around 300 nm width. The samples are then observed by a collimated non polarized halogen light, and the 3D diffraction patterns are obtained with a second wide field inverted motorized microscope (Zeiss Axiovert 200M) using an x100 oil immersion objective. The light coherence is improved by using a small irradiation aperture (Fig. 1(a)
).

## 3. Rayleigh Sommerfeld model

The theoretical diffracted fields are calculated using the scalar Rayleigh-Sommerfeld integral. The electric field amplitude A(M) diffracted in the direction **u** at the point M on the observation screen by an object placed in the plane of points P is given by [10]:

_{i}and

**k**are respectively the incoming field magnitude and the wave vector.

The advantage of this model is to suitably estimate the complex field amplitude even at close distances of the diffracting object (d ≈λ) [15]. For these calculations, nanowires are located in the xy plane at z = 0, with infinite lengths along the y-axis and 300 nm width. They are illuminated by a light beam centered at λ = 546 nm, i.e. the green wavelength of the CIE-RGB system, with a propagation direction parallel to the z- axis. In such model, the refractive index of the wires is not considered. The wires act as black objects which serve only to block the incident light. Behind the structure, the light is focused in the cover slip with a refractive index of 1,53.

## 4. Results and discussions

Figure 1(b) shows the experimental light patterns transmitted by pairs of parallel silver nanowires separated respectively by 1, 2 and 4 µm for the green wavelength. However, the focusing process is exactly the same for the red, the green and the blue radiations with a slightly dispersion effect. Identical patterns are obtained by the scalar electric field model (Fig. 1(c)). These experimental and theoretical figures demonstrate clearly the efficient focusing effect of these nanowires cylindrical microlenses. Indeed, the light distribution diffracted by the nanowires lead to constructive interferences which induce a focusing process in the near field region.

Figure 2
shows the diffracted light intensity distributions normalized to the incident intensity I_{o} (I_{z}/I_{o}) along the z-propagation axis at x = 0, and along the transversal direction x-axis (I_{x}/I_{o}) when z is located on the focal point. Experimental curves are in good agreement with the scalar simulations, which confirms the pure diffractive contribution of this focusing effect.

The optical characteristics of these cylindrical microlenses are summarized in the Table 1
. Z_{f}, D_{f}, ΔX, and I_{max}/I_{o} are respectively the focal length, the depth of focus (FWHM_{z}), the lateral resolution (FWHM_{x}), and the normalized intensity at the focal point. The resolution ΔX_{d} is the calculated far field diffraction-limited resolution, i.e. $\Delta {X}_{d}=\frac{\lambda}{2NA}$. The numerical apertures $NA$ of such microlenses is defined by their nanowire separating distance D and focal lengths Z_{f}:

Table 1 shows that such microlenses are highly efficient with high numerical apertures, and with diffraction-limited resolutions. These resolutions are even slightly narrower because the focusing process occurs in intermediate region between near and far fields. The microlenses focal lengths are comparable to their nanowire separating distances. In addition, they have high transmittance efficiency with normalized maximum intensity up to 2.2 at the focal point. This is an important advantage in comparison to plasmonics lenses that are limited by their nanoslit transmittances** ^{1}**.

Figure 3
shows the calculated variations of the normalized focal length Z_{f}/λ, depth of focus D_{f}/λ, lateral resolution ΔX/λ and I_{max}/I_{o} as function of the normalized nanowire interdistance D/λ. The focal length Z_{f} and the depth of focus D_{f} increase quadratically while the resolution ΔX increases quasi linearly with D/λ. The maximum intensity at the focal point is reached when D is about three times the value of the wavelength λ.

Such diffraction scheme can also be used with a grid geometry of nanowires to design single and dense arrays of two-dimensional microlenses. Figure 4
shows the experimental and Finite-Difference Time-Domain (Lumerical FDTD solution [16]) focusing patterns of a single 2-µm square. This square is formed by two perpendicular pairs of nanowires of 2 µm length each pair is separated by 2µm.The well know Johnson and Christy silver table was used to simulate the metallic refractive index of the structure in the FDTD simulation [17]. Both patterns are similar with focal spots at 2 µm, and transversal resolution of 0.32 µm (Fig. 6
). This square microlens has the same optical characteristics that the 2µm cylindrical lens except for the focal-point light intensity that is 4.8 Io, i.e. 2.2 times greater. Dense arrays of diffractive microlenses are obtained using the same strategy with two-dimensional metallic microgrid. Figure 5(a)
shows the xy-grid plane of the microlens array. Each microlens is a square formed by silver nanowires separated by D = 2µm. This corresponds to a microlens density of about 12700x12700 DPI (dots per inch). Figure 5(b) displays the xy light pattern at the focal position Z_{f} = 2 microns, that is identical to the focal lengths of the square and cylindrical microlenses. The lateral resolution is slightly degraded to 0.43 µm, but the light intensity is further increased to 5,6 I_{o} (Fig. 6). We can then see that the optical characteristics of a unique square and one gird cell are similar. The microlens optical properties are almost not affected by this array geometry.

## 4. Conclusions

In conclusion, we have demonstrated experimentally and theoretically that parallel metallic nanowires with micrometer separating distance act as ultrasmall microlenses with high transmittance. The focusing of such microlenses occurs in the Fresnel region with a resolution that is diffraction limited. This is a more efficient alternative, and simpler to fabricate, than plasmonic microlenses based on nanoslit arrays. The focal lengths and resolutions can be tuned by varying the nanowire inter-distances. The experimental results are well simulated by the scalar electric field model. Thus, the focusing process results only from the diffraction of the masking nanowires. The plasmon contribution is negligible in this micron-sized geometry. Such approach can be extended to a grid geometry to design single and arrays of two-dimensional microlenses. These square microlenses keep the same optical characteristics than the parallel nanowire microlenses except for the focal-point light intensity that about 2 times greater. This micro-optics concept to design microlenses with typical dimensions of several wavelengths open the way to the fabrication of microlens arrays with densities greater than 10000 x10000 DPI.

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