Overview of the electromagnetic response of nanoparticles

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Mapping optical elds with DHH

Among the numerous applications of o-axis DHH microscopy, in 2011 our group rst addressed the characterization of plasmonic structures using this far eld technique [82, 83]. Plasmonics is a branch of nanophotonics that is primarily concerned about the coupling of light to electronic charges in metals. Under specic illumination conditions, charge oscillations inside nanostructurated metals enable strongly localized eld enhancements of subwavelength dimensions. This ability to squeeze light into subdi raction volumes has stimulated many attractive applications such as extremely high sensitivity spectroscopy and sensing of chemical agents, novel drug-delivery designs, original strategies for high-resolution microscopy, compact metal-based waveguides for miniaturized photonic devices and more ecient solar cells, among others [80]. A characteristic ngerprint of plasmonic nanostructures is their ability to work as nanoantennas, being able to convert freely propagating optical radiation into strongly localized energy, and vice versa [66]. Therefore, in order to understand and exploit the plasmonic properties of nanostructures, it is fundamental to obtain the full knowledge of the three-dimensional electromagnetic eld around a nanoobject interacting with a light source. Although being diraction-limited, DHH microscopy has been shown to be a useful tool to image nanoantennas’ three-dimensional scattering patterns from a single snapshot. In addition, just as with back focal plane microscopy [54], the angular radiation pattern can also be obtained straightforwardly by computing a simple Fourier transform of the hologram [82]. Heterodyning is indeed another driving force of DHH as it oers the possibility to perform phase measurements and to investigate frequency modulated phenomena, although it has not been the main purpose of my work, presented hereafter.

Comparison between DHH and NSOM images

Let us rst look at the resonant case, i.e. under a polarization along the chain axis, shown in Fig. 1.12. On the one hand, we acquired in-plane xy images with both techniques. While with DHH we only needed to reconstruct the xy-plane containing the nanohole antenna by propagating the demodulated hologram to the corresponding z distance, with the NSOM device the tip had to be laterally scanned along the whole surface, with the apex of the hollow pyramid permanently in contact with the surface. This way, near-eld in-plane images as well as topographic images were collected simultaneously. The images obtained using each of these techniques were very similar (see Fig. 1.12(b)), despite the fact that the holographic image was obviously less resolved, the NSOM lateral resolution being determined by the tip aperture, around 100 nm. Both showed an homogeneous intensity distribution, suggesting coupling between neighbouring nanoholes.

Overview of the electromagnetic response of nanoparticles

When light waves encounter an object of any size, their energy propagation changes. Matter being composed by discrete electrical charges, these charges oscillate upon illumination by the incident electromagnetic eld and radiate in turn electromagnetic energy, what is called scattered light. Besides this elastic re-emission of light, matter canalso absorb part of the incident energy and convert it into other forms such as heat or vibrations.
Before we go into any mathematical formalism, it is useful to discuss which parameters determine the amplitude and phase relations between the multiple individual dipoles induced in an object by an incident light wave. One can easily imagine that the number of possible phase relations between elementary dipoles increases for bigger particles and that the shape of the object also plays a major role (see Fig. 2.1). Strictly speaking, it is the size and shape of the particle versus the wavelength and the polarization of the incident light that determines the particle’s response. But phase relations will also be governed by the object composition, which determines the medium permittivity, i.e., the resistance of the medium to be polarized. To summarize, the factors that determine the scattering and absorption eciencies of the object under study are:
the object characteristic size, a, and shape.
« 1(!) and « 2(!) the dielectric functions of the object and the surrounding medium, respectively.
the wavelength of the incident light, , the polarization and the wavevector.

Oscillating electric dipole elds

In the dynamic case, e.g. under plane-wave illumination with ~E(~r; t) = E0e􀀀i!t, the incident eld induces an oscillating dipole moment ~p = « 0 »m(!)~E0e􀀀i!t . Here, the frequency dependence of the permittivity (« (!)) has to be taken into account. The radiation of this dipole leads to the scattering of the incident plane wave by the sphere. Many reference books, which describe the extinction eciency (energy removal from the incident beam due to the presence of the particle) by small spherical particles, consider the dipolar approximation in the electrostatic limit. However, the electrostatic polarizability 0 given by Eq. 2.4 violates the principle of energy conservation.
Let us brie y review the basics of the electromagnetic elds associated with an oscillating electric dipole. The total elds ~H (t) = ~H e􀀀i!t and ~E (t) = ~Ee􀀀i!t in any point in space are (see Ref.[41] for demonstration): ~E = 1 4″0″m n k2(~n ~p) ~n eikr r + [3~n(~n ~p) 􀀀 ~p] 1 r3 􀀀 ik r2 eikr o.

Table of contents :

Acknowledgements
Abstract
Resume
Resume Substantiel
Introduction
1 Holographic microscopy for far-eld optical mapping 
1.1 Principles of digital holography
1.1.1 Experimental suppression of parasite diraction orders
1.1.2 Experimental setup
1.1.3 Digital reconstruction process
1.2 Mapping optical elds with DHH
2 Metallic nanoparticles 
2.1 Overview of the electromagnetic response of nanoparticles
2.1.1 Quasi-static approximation
2.1.2 Oscillating electric dipole elds
2.1.3 Scattering and absorption cross-sections
2.1.4 Localized surface plasmon resonances
2.2 Electrochemistry studies coupled to holographic imaging
3 Holography for particle localization, tracking and superresolution imaging 
3.1 Wide-eld microscopy: beyond the diraction limit
3.1.1 3D localization microscopy
3.1.2 Holographic microscope: localization accuracy on immobilized NPs
3.1.2.1 Signal-to-noise ratio
3.1.2.2 Axial range
3.2 Superresolution imaging by point-by-point data accumulation: from 2D to 3D
3.2.1 Densely labeled samples
3.2.2 Our approach: moving labels
3.2.2.1 Brownian motion
3.2.2.2 Covering time for 2D stochastic image formation
3.3 Speeding up data processing: parallel programming
4 Gold NPs for superresolution stochastic optical mapping 
4.1 Framework
4.1.1 Propagating versus evanescent waves
4.1.2 Spatial resolution versus spatial frequency bandwidth
4.1.3 Near-eld optical microscopy
4.2 Optical mapping by holographic localization of Brownian scatterers
4.2.1 Imaging an evanescent wave
4.2.2 Imaging a laser beam distribution
4.3 Present challenges: optical mapping around nanostructures
4.3.1 Background suppression
4.3.1.1 Fourier space spatial ltering
4.3.1.2 Heterodyne ltering of static objects
Conclusions and prospects
A Centiles of the coupon-collector problem 
A.1 Time for r observations of each n pixels, one hit at a time (k = 1)
A.2 Time for r observations of each n pixels, with random number of observations at a time (k 1)
B Scientic Activities 
Bibliography 

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