1. Introduction

Elastic light scattering is a classical process by which the incident oscillating electric field of light interacts with an inhomogeneity in refractive index, such as a small particle, by exciting charge dipoles, which then re-radiate at the same frequency as the incident light. The re-radiated (scattered) field interferes with the incident field to create near-field and far-field intensity distributions which are extremely sensitive to the shape, size, and refractive index of the particle and surrounding medium [1].

Recent work has renewed interest specifically in the Fresnel zone (FZ) surrounding individual dielectric microspheres and cylinders with a diameter between 1 and 20 times the wavelength of the incident light [2]. This size regime is too large for the approximation of Rayleigh scattering, and too small for geometrical optics [3]. The FZ is defined as the part of the electromagnetic field containing both propagating and non-propagating components at distances shorter than the Fraunhofer distance $d_f=2D^2/\lambda$ from the source of the diffraction (up to 15 μm for a D = 2 μm diameter sphere and λ = 532nm). It has been shown both theoretically [4–7] and experimentally [8–11] that elongated beams of light referred to as Photonic nanojets (PNJ) can be created in the FZ, which break the classical far-field diffraction relation between spatial light confinement and divergence angle. Analytical treatments for microspheres using Mie theory, angular spectrum decomposition [4] and Debye series [5] show that the low divergence features are primarily the result of interference of high spatial frequency propagating light in the FZ. Evanescent components have only a minor role increasing the overall intensity at the scatterer surface. This phenomenon in microspheres has proven extremely versatile with applications in direct write lithography, producing features with minimum sizes 100 nm with light of wavelength 355 nm [12,13], optical data storage with resolution higher than BluRay^TM [14, 15], single molecule detection and fluorescence enhancement [16, 17], and nanoparticle detection [18, 19]. The practical result is that for many techniques requiring focused illumination or detection, simply placing a scattering particle in the focus can decrease the point spread function (PSF), and increase resolution. Due to the scale invariance of Maxwell’s equations these effects have also important in the terahertz regime, where controlling the particle shape allows added control of the intensity distribution in the Fresnel zone [20]. Recent work has explored elliptical particles [21], graded refractive index [22], coated particles [23], and micro-axicons [24].

In this work, we experimentally map the morphology and polarization dependence of the scattered intensity distribution in the both the Fresnel and far-field regions. Using a model system of dielectric particles defined lithographically in silicon nitride (Si₃N₄), we demonstrate intensity features in the FZ with a unique combination of high intensity spots and low divergence angles. The experimental results are in excellent agreement with 2D and 3D finite element method (FEM) simulations. We demonstrate that this scattering effect is not unique to particles with spherical or cylindrical symmetry, but is a more general feature of scattering particles in this intermediate size regime. Furthermore the intensity distribution in the FZ can be controlled by the direction (angle in-plane and out-of-plane) and polarization of the incident light, as well as by controlling the shape and index contrast of the particle. Together these parameters provide a previously unattained level of control over the near-field and far-field intensity distribution.

2. Experimental methods

To fully investigate this phenomenon scattering particles with square, triangular and circular cross section were defined lithographically in a 400 nm thick layer of Si₃N₄ (n = 2.1) on a 2 μm thick layer of SiO₂ (n = 1.45). The finite height of the structures was chosen to be comparable to the incident wavelength resulting in strong scattering both in the plane and out of the plane of the substrate. The material layer structure, an optical micrograph of typical structures, and an SEM image of a scatter are shown in Fig. 1(a). SEM images show high quality structures with 90° sidewall angle.

 

Fig. 1 (a) Optical micrographs of square, circular and triangular cross section particles. Scale bar 5 μm. Schematic of material layer structure, 400 nm Si₃N₄ (n = 2.1), 2 μm SiO₂ on Si base wafer. Angled SEM of 6 μm diameter microdisk. Scale bar 3 μm. Inset shows sidewall angle close to 90°. Scale bar 2 μm. (b) Two dimensional slice of intensity distribution through a 2 μm diameter disk calculated using 3D FEM. Black lines indicate geometrical boundaries and co-ordinate system used throughout is inset. S-polarized light incident along the x-axis. (c) Experimental setup: Linearly polarized light is introduced at an angle of θ_i = 80° to normal. The scattered light is collected with a NA = 0.7 long working distance objective and imaged onto CCD1 at 100x magnification using a 200 mm focal length tube lens. A non-polarizing beam splitter allows simultaneous imaging of the back focal plane on CCD2. (d) Schematic of light scattering processes. The majority of light incident with wavevector k_i will be reflected specularly along the direction k_r. Only light which is scattered by the particle will enter the collection angle of the objective lens. This light either scatters directly from the particle into the collection cone of the objective (process 1), or is scattered by the particle in the plane and scatters a second time from the rough SiO₂ surface to create a much weaker signal (process 2).

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Arrays of well-spaced scatterers (50 μm separation), with characteristic sizes ranging from 1 to 10 μm were fabricated. Figure 1(b) shows the expected near-field intensity distribution normalized to the incident intensity calculated from 3D FEM for a 2 μm diameter particle with circular cross section. The co-ordinate system used throughout the work is inset. S-polarized (electric field along y-axis) light incident from the left with wavelength of 532 nm is focused to a single spot at the surface of the disk. The spot is roughly Gaussian in the y direction with a 1/e² radius of 488 nm (0.917 λ) and Lorentzian in shape along the z-axis with 1/e² radius of 203 nm (0.38 λ). The main spot falls off exponentially along the x-axis with a 1/e² distance of 348 nm (0.654 λ). The experimental setup used is outlined in Fig. 1(c). Linearly polarized light from a diode pumped solid state laser (DPSS) with a wavelength of 532 nm is incident on the scatterer at an angle θ_i = 80° from the surface normal. The incident beam is spatially filtered to give a Gaussian profile with 1.5 mm 1/e² diameter, effectively creating a plane wave excitation on the scale of a single scatterer. The incident light is controlled to be s or p-polarized with respect to the substrate. Scattered light was collected from above using a long working distance microscope objective (MO), with a moderate numerical aperture (NA) of 0.7, resulting in a 45° half angle of collection. The 6 mm working distance of the objective and NA of 0.7 ensure that the incident and reflected beams pass freely without obstruction, and only light scattered by the structures into the cone of the objective is collected. The collected scattered light is then split using a non-polarizing beam splitter. One arm contains a 200 mm tube lens which images the particle onto CCD1 with a magnification of 100x providing information on the Fresnel zone. The second arm images the back focal plane (BFP) of the objective onto CCD2 giving the far field angular scattering distribution. A pinhole is inserted in a conjugate image plane to isolate light from a single scattering structure. This technique was used to measure phase functions of single microspheres in vacuum [25], and has recently been used to map the orientation of single molecules [26] and examine the far-field characteristics of nano-antennae [27]. This proposed indirect far-field technique presents multiple advantages over scanning near-field optical microscopy (SNOM) in this particular case. Introducing a large metallic scatterer into the region makes it extremely difficult to measure the propagating components of the near-field region. Two scattering processes are identified and depicted schematically in Fig. 1(d). Light is either directly scattered from structure into the cone of the objective (process 1), or scattered from the structure and then re-scattered from the SiO₂ substrate (process 2). This second process results in a significantly weaker signal.

3. Results and discussion

The resulting image and BFP image for a scattering particle with circular cross section of diameter 4 μm are displayed on linear and log scales in Fig. 2(a). Care must be taken in interpretation of this image. On the linear scale the observed pattern consists of bright spots at the scatterer interface from scattering process 1. These effects were previously observed in optically trapped microspheres, referred to as glare points [28], and explained as the Fourier transform of the Mie scattering amplitude over the range of angles collected by the microscope objective [29, 30]. Light originating from outside the particle, however, is due to secondary scattering from the SiO₂ substrate. This gives a direct map of the Fresnel zone convolved with the point spread function (PSF) of the collection objective. This secondary scattering allows us to make quantitative measurements of the angular dependence of the near-field intensity distribution despite the fact that the image is convolved with the PSF of the NA = 0.7 collection optics. The secondary process, however, results in a signal two orders of magnitude lower than process 1. Light scattering routinely produces signals with intensity varying over 6 orders of magnitude. For this reason a high dynamic range measurement technique is needed. Four images are taken at different shutter and gain settings, scaled and summed to give a final HDR image. A log scale is employed to display scattering from both processes in the same image.

 

Fig. 2 (a)Fresnel zone and (b) back focal plane images of a 4 μm diameter particle with circular cross section on linear and log scales. On linear scale only light from process 1 is observed, log scale allows us to see both processes simultaneously. The high intensity features on the edge and inside of the scatterers are due to light scattered directly into the collection angle of the objective. The intensity profile outside of the scatterer is due to secondary scattering from the substrate and gives a map of the Fresnel zone intensity distribution. (c) Intensity map of disk of diameter 6 μm. The insets in bottom left show scatterer orientation relative to the incident light (d) Square of side 6 μm illuminated on face. (e) Same square rotated 45° illuminated on vertex. (f) Equilateral triangle of side 6 μm illuminated on face. (g) Same triangle rotated 30° illuminated on vertex. Scale bar 3 μm. Insets in (c) to (g) are back focal plane images representing the far field angular scattering with from −45° to 45°.

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The image plane and back focal plane results for a set of scattering particles are displayed in Fig. 2(c)-2(g). All images are on a log scale over 3.5 decades. The multi-lobed Fresnel zone scattering patterns can be easily observed in the image plane. Particles with circular cross section produce a single focal spot and multiple lobes similar to the intensity distribution expected from Mie theory for an infinite cylinder [31]. Particles with square and triangular cross sections however produce more complicated distributions which show a drastic change in scattering under rotation. Particular cases of interest shown are when the incident light is aligned with one of the symmetry axis of the particle resulting in a symmetric intensity pattern. The BFP images for each case are inset and show the angular dependence of the scattering in the far-field. The angular pattern of the Fresnel zone distribution is mirrored in the far-field out of plane scattering. The well-known preference for forward scattering for particles in this size regime is observed with the BFP images brighter on the shadow side of the particle. These BFP images also change dramatically under rotation of the particle. For example a square particle illuminated on a flat face (Fig. 2(d)) tends to scatter light into lobes propagating out of plane from the surface, whereas when rotated by 45° and illuminated on a vertex (Fig. 2 (e)) the scattering is predominantly in the plane.

These images can now be used to extract quantitative information on the angular dependence of the scattering. Figure 3(a) shows the angular intensity profile taken 1 μm from the surface of a circular particle with diameter of 6 μm. Sixteen distinct scattering lobes can be resolved in the 0−180° range. The forward scattering is 2 orders of magnitude larger than the high angle scattering. This procedure was repeated for a range of characteristic sizes from 1 μm to 10 μm. A separation of variables (SoV) approach can be used to find an exact solution of the near field scattering for spheres and infinite cylinders (Mie theory). This theory suggests that the scattering depends only on two parameters, the size parameter x_s = 2π/λ and the refractive index contrast m = n_s/n₀, with n_s and n₀ refractive indices of the scatterer and the background respectively. The SoV approach is not applicable for shapes of finite height or complex cross section and a full numerical technique must be used. It is not immediately evident how this dependence will hold for complex shapes. By varying the characteristic size at fixed wavelength we are effectively varying the size parameter of the scattering. Figure 3(b) shows the number of scattering lobes as a function of characteristic size with input wavelength of 532 nm. The solid markers represent the scatterer shape and orientation relative to an incident beam from the left. The solid line shows the calculated dependence from Mie theory for an infinite cylinder. The finite height disk follows closely this trend. Square and triangular cross section particles show also a linear trend, but with slope which depends not only on particle cross section but also orientation. A similar dependence is observed in the BFP images.

 

Fig. 3 (a) Experimentally measured angular dependence of Fresenl zone intensity 1 μm from the surface of a 6 μm diameter scatterer with circular cross section. (b) Number of lobes as a function of characteristic scatterer size. The solid line is the expected dependence of an infinite cylinder calculated from Mie theory. Dark blue series is for a disk, the orange series is for square cross section orientated as in Fig. 2(d), green for square orientated as in Fig. 2(e), pink for triangle orientated as in Fig. 2(f) and light blue for triangle orientated as in Fig. 2(g).

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The Fresnel zone intensity of these scatterers shows high intensity low divergence features, termed Photonic Nanojets in previous work. These features have much lower divergence for their waist than previously thought possible from the diffraction limit in free space. The effect is a result of a peak in the angular spectrum at high k (high angles); scattering of light into high angled lobes allows the main lobe to remain with low divergence. Figure 4(a) shows an experimental image of light scattering from a 4 μm side square cross section particle, compared with the intensity distribution calculated using 2D FEM. The angular features of the intensity outside the scatterer match well, although the 2D FEM does not take into account out of plane divergence which results in the intensity fall off from the particle observed experimentally. On further inspection of this image, cross sections of the intensity profile can fitted. The main lobe has a Gaussian profile with a 1/e² waist of 691 nm and a Lorentzian falloff with a 1/e² length of 7.59 μm. This results in a total divergence of 9.9° represented as a dashed line in Fig. 4(e). For comparison the 1/e² envelope of a Gaussian focused beam with the same waist is superimposed (solid line), showing a total divergence angle of 28.1°.

 

Fig. 4 (a) Experimental image showing in plane near-field intensity for a 6μm side square. (b) 2D FEM shows good agreement outside scatter, but does not account for intensity fall-off due to out of plane divergence. Scale bar 5μm. (c) Transverse cross section at the waist of the main lobe showing 1/e² waist of 691nm (d) Axial falloff of the main lobe from the point of highest intensity. (e) Close up of main lobe in (a). The dashed line shows the 1/e² envelope of the intensity with a divergence full angle of θ = 9.9°. The solid line represents the divergence of a focused free space Gaussian beam with the same waist which has a divergence half angle of θ_g = 28.1°.

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Finally, we explore the polarization dependence of the Fresnel zone and far-field scattering. Image plane results are shown for a disk, square and triangle of 4 μm in side with s-polarized input light in Fig. 5. When a crossed polarizer is used to analyse the scattered light a dark line of zero intensity is seen along the mirror symmetry plane of the particle Fig. 5 (second row). This is because there can be no coupling between an instantaneous electric field in this plane and a field normal to it, since both normal directions are equivalent. Furthermore scattering from parts of the particle on either side of this plane must cancel out exactly, due to the mirror symmetry. These maps compare well with the 2D FEM calculated intensities of the vectorial components along the polariser axis ($E_x^2$and$E_y^2$). Therefore, this techniques provides a method of mapping the vectorial components of the Fresnel zone close to the scatterer.

 

Fig. 5 The experimental images show light scattering from a disk, square and triangle of diameter / side 4 μm. Images are displayed on a log scale. The incident light is s polarized (along the y axis), shown in black. In the first row all the scattered light is collected and imaged onto CCD1. In the second row the scattered light is passed through a polarizer with axis along the x-axis, shown in red. The asymmetric pattern is a result of the mirror symmetry in the scattering shape causing interference at the image plane. The third row then shows the image plane with the analyzer axis along the y-axis. Scale bar 2 μm.

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4. Conclusion

We have successfully demonstrated a simple imaging method to map the angular scattering intensity in the Fresnel zone of particles on a substrate. We have applied the method to finite height dielectric particles with circular, square and triangular cross section (although the method is applicable to arbitrary shape). We choose a size regime intermediate to Rayleigh scattering and geometrical optics which is difficult to describe from an analytical and numerical approach. We show that for particles with non-circular cross section the size parameter scaling depends on both the orientation and the shape of the particle. Scattering particles in this size regime are experimentally shown to produce low divergence photonic nanojets, with 1/3 of the angular divergence of a focused Gaussian beam. Finally we show that an analyser can be used to map the vectorial components of the Fresnel zone and compare with FEM. For fundamental physics, this technique can be used to experimentally verify the applicability of analytical and numerical techniques to describe light scattering close to an interface in this intermediate size regime. The phase functions of individual finite height particles can be extracted experimentally and used as source functions to describe scattering from large collections of arbitrary cross sectioned particles [32]. Furthermore, the ability to control the near-field intensity on the nanoscale with dielectric scatters can be employed to multiplex and increase the resolution of optical imaging techniques which rely on minimizing focal volume, such as fluorescence correlation spectroscopy (FCS) [33]. In particular the observed strong angle and polarization response of these particles have exciting prospects for stimulated emission depletion (STED), or modulated optical trapping techniques [34]. Numerous applications have been identified for microspheres, including high resolution direct write lithography, FCS, enhanced coupling and backscatter detection of individual nanoparticles, and high precision long depth of field medical laser scalpels. The use of lithographically defined particles with arbitrary shape allows much greater control for these applications and the possibility for cheap large scale, integrated manufacture of planar dielectric focusing elements [35].