1. Introduction

The possibility of nanoscale confinement of light is important for laser nanoprocessing of materials and ultrahigh-resolution optical microscopy. The laser processing [1] supports various fields of modern nanotechnology, such as nanoelectronics, photonics, plasmonics, information storage, biomedical applications, including cell manipulation and surgery.

Due to the light diffraction limit, the minimal size of the domain, in which the freely-propagating light can be localized, is about one-half of the wavelength. This limit is overcome in near-field optics (or nanooptics) [2], where the dominant role is played by the evanescent waves. Such waves are typically excited near the interface of different media and decay when going away from the interface. The high near-field enhancement is gained near the sub-wavelength scaled objects like nanoparticles, sharp tips and rough surfaces.

A promising technique employs transparent dielectric micro- and nanospheres [3]. Being irradiated by a wide laser beam, the sphere can form the field enhancement in a small region beyond itself, thus acting as a near-field microlens. The micro- and nanospheres of silica, titania, polystyrene and other dielectric materials are commercially available as colloidal water solutions. Drying of such solutions on a surface may result in a self-assembly of the spheres in a regular closely packed two-dimensional array (colloidal particle-lens array or CPLA) [4]. With such arrays, the parallel processing of millions of spots at the surface is possible. Alternatively, for precise positioning of individual spheres, optical tweezers can be employed [5]. Here, Bessel beam [6] provides precise positioning of the sphere on a surface without active feedback.

Within CPLA, the spheres do no act as independent lenses. The field pattern near each of the spheres within the array is influenced by its neighbors. The energy exchange between the neighboring spheres can affect both the shape and the magnitude of the produced near-field distributions. So far, several cooperative phenomena have been reported. Hexagonal structures on a gold-coated colloidal microsphere array have been observed after the irradiation by a femtosecond laser beam [7]. This is explained by the hexagonal shape of the field distribution formed at the coating.

Paper [8] demonstrates the energy redistribution within a planar cluster of microspheres. It is shown that when the surface with deposited CPLA is ablated by a laser beam, deeper nanodents are formed under the spheres located at the boundary of the array. Corresponding calculation shows that due to the energy flow from the inner spheres to the ones at the boundary, the latter produce higher field enhancement. The decrease in the field enhancement due to the neighboring spheres is also studied in [9].

In this paper, we focus on two practically important phenomena, caused by both the electrodynamical interactions between neighboring spheres and the interactions between the spheres and the substrate. The simulations of the field distributions near the small clusters of dielectric spheres irradiated by a wide laser beam are performed. The cooperative effect of the neighboring spheres and the substrate is interpreted in terms of an effective immersion medium.

2. Calculations and discussion

The problem of light diffraction by a single sphere [10] (i.e. Mie problem) is solved by means of expansion of the incident field, the scattered field and the field inside the sphere into the spherical modes – the vector spherical wave functions (VSWFs). The relations between the corresponding expansion coefficients are obtained from the boundary conditions at the spherical surface.

When several closely placed microspheres are laser-irradiated, an overlap of the modes that correspond to different spheres occurs, leading to the interrelation of the expansion coefficients. The small overlap means that the spheres focus and scatter light almost independently. In this paper we consider the situations when the overall scattered field cannot be found as the superposition of independent contributions from the constituent spheres.

With the help of specially engineered clusters of microspheres, the near-field distributions which are not accessible with single spherical particles can be obtained. Various configurations of microparticle clusters can be produced with holographic optical tweezers or by means of colloidal self-assembly.

In this paper, we address several effects of the mode coupling that are practically important for the laser material micro- and nanomodification technology. We analyze small planar close-packed clusters of microspheres with configurations that resemble parts of the colloidal arrays. A typical modeling setup is presented in Fig. 1 . It consists of a plane hexagonal cluster of seven dielectric non-absorbing spheres either situated in free space or placed on a semi-infinite substrate. The incident plane monochromatic wave propagates normally to the cluster.

 

Fig. 1 Modeling setup: a closely packed cluster of seven transparent dielectric spheres on a dielectric transparent substrate. The spheres are irradiated by a plane monochromatic wave normal to the substrate boundary plane.

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We calculate distributions of $|\overrightarrow{E}{|}^2$ (where $\overrightarrow{E}$ is the amplitude of electric field) near the cluster by means of different calculation techniques. When the substrate is present in the modeling setup, the FDTD (Finite Difference Time Domain) method is employed. The calculations are made in a rectangular cell using the MEEP code [11]. When the setup contains only the spheres, the multiparticle generalization of Mie solution is obtained [12]. Namely, with GMM01F [13] code, the electromagnetic fields are found as an expansion by VSWFs, taking into account the mode coupling.

One of the phenomena being discussed in this work concerns the focal volume length, which is important for applications in laser material micro- and nanoprocessing. By tuning the length of the focal volume one can change the aspect ratio of the laser irradiated zone. The depth to which the intense laser radiation penetrates the transparent material determines the depth to which the material modification occurs. This can determine the shape of the laser drilled hole and change the acoustical response of the medium. The latter can significantly influence the role of laser-induced tensile-stress-driven processes (spallation [14], and cavitation bubbling [15]) with respect to other mechanisms of laser ablation [16] and swelling [17, 18].

Our calculations show that the focal length of the sphere can be significantly increased due to the influence of the surrounding objects. The calculated distribution of $|\overrightarrow{E}{|}^2$ (normalized by the value of $|{\overrightarrow{E}}_0{|}^2$ in the incident wave) near seven polystyrene spheres placed on a glass substrate is given in Fig. 2 . For comparison, we present similar distributions for a single sphere in free space, for a single sphere on the substrate, and for the seven spheres without the substrate.

 

Fig. 2 Calculated distributions of $|\overrightarrow{E}{|}^2$ at the central y-z plane of the sphere (a, c) and the cluster of seven spheres (b, d) irradiated by a plane wave. The spheres are either located in free space (a, b) or placed on substrate (c, d). Both the central sphere and the substrate are indicated with white lines. Axial (e) field distribution at the center line (x = 0, y = 0) and lateral (f) field distribution near the substrate surface (at x = 0, z/d_sp = 0.525) are plotted in the bottom graphs. The field distributions are normalized by the value of $|{\overrightarrow{E}}_0{|}^2$ in the incident wave. In graph (f), the field distributions are normalized by their values in the maximum. Polarization of the incident wave is linear along x. Refractive indices of the spheres and the substrate are 1.59 and 1.46, respectively. The diameter of the spheres is 1000 nm, and the wavelength is 800 nm.

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The distributions of $|\overrightarrow{E}{|}^2$ at the central z-axis of the inner sphere are plotted in Fig. 2(e). Here, for comparison, we also present the result of the field simulation at the central axis of a larger hexagonal cluster that consists of 19 microspheres placed on the substrate. Figure 2(f) shows the lateral confinement of the field beneath the central sphere.

The pictures show that the influence of both the substrate and the surrounding spheres causes a significant elongation of the focal length of the central sphere along with a decrease in the field enhancement and the lateral confinement. From the phenomenological point, this can be understood in a way that the surrounding of the inner sphere acts as an immersion medium that weakens the focusing capabilities of the spherical lens.

The refractive index of such an immersion medium is determined by the average dielectical properties of the surrounding objects. It is shown in Ref. 8 that within the closely packed planar cluster, the spheres located at the boundary of the cluster produce a higher field enhancement than those located inside. The more neighbors the sphere has, the higher the average refractive index of the immersion medium is, and the lower field enhancement is.

To analyze this phenomenon in more detail, let us consider separately the part of the total field that is produced by the polarization currents within the dielectric spheres, i.e. the scattered field. The role of the mode coupling in the focal length elongation can be revealed by the comparison of the scattered field produced by the cluster of seven spheres with the superposition of seven scattered field distributions produced by the individual spheres which constitute the cluster. The latter case corresponds to the field scattered by the same cluster but with the mode coupling “switched off”. The comparison is shown in Fig. 3 .

 

Fig. 3 Scattered fields at the central z-axis calculated for the cluster of 7 spheres on the substrate (black line) and for a single sphere on the substrate (blue line). Coherent superposition of scattered field solutions for 7 individual spheres that constitute the cluster on the substrate (magenta line). The parameters of the setup are the same as in Fig. 2.

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Comparison of the scattered field near the single sphere and the scattered field near the inner sphere of the cluster shows that the “long focus” phenomenon occurs in the latter case due to a simultaneous decrease of the main field maximum at the sphere boundary ($z/d_{sp}=0.5$) and due to the appearance of another field maximum at $z/d_{sp}\approx 1$. This additional maximum appears as a shoulder in the field distribution calculated with the mode coupling “switched off”. Hence this maximum is the result of the interference of independent contributions from the individual spheres. Conversely, the decrease of the main maximum is the result of the mode coupling.

In Fig. 4 , another effect of the spherical mode coupling, which is similar to the effect of the immersion medium, is shown. Here, we compare $|\overrightarrow{E}{|}^2$ distributions at the central z-axis of a single sphere (acrylic glass, PMMA) in vacuum and the same sphere surrounded by 6 similar neighbors on substrate (glass). Both the neighboring dielectric objects and the immersion medium cause the global field maximum to weaken and shift along the z direction away from the spherical surface. This substantially changes the field distribution inside the sphere causing $|\overrightarrow{E}{|}^2$ to reach the maximum not at the surface but rather at the inner point.

 

Fig. 4 Calculated field distributions at the central z-axis of the cluster of seven PMMA spheres on glass substrate (black), the cluster of seven polystyrene spheres on glass substrate (black dashed), the single PMMA sphere in vacuum (red), and the single PMMA sphere in immersion media with refractive indices of 1.1 (green), and 1.2 (blue). The diameter of the spheres is d_sp = 2700 nm, the wavelength is 800 nm, refractive indices of PMMA, polystyrene, and glass are 1.49, 1.59 and 1.46, respectively.

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The strong field inside the sphere can cause dielectric breakdown and thus local heating and melting. Possibly, the heat can lead to detachment of the molten part of the sphere and deposition of the melt onto the surface. Such a process can be potentially employed for nanopatterning of materials.

The ratios of the main maxima inside and outside the sphere can be different in the cases of the sphere in the immersion medium and the sphere surrounded by neighbors. For instance, in the case of the polystyrene spheres on a glass substrate, the maximum inside the sphere becomes even stronger than the one outside.

3. Conclusions

The closely packed CPLAs are employed in laser micro- and nanostructuring. Two main effects of the mode coupling in clusters of dielectric microspheres are considered. We show that the electrodynamic interaction can significantly elongate the focus volume thus changing the aspect ratio of the laser irradiated zone and can shift the maximum of the laser field towards the inner part of the sphere. Both of these effects can have a significant impact on material response and thereby on the nanostructuring process.