Distance dependence of the radiative and non-radiative LDOS distributions 

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Experimental setup and results

Here, we describe the experimental setup that was realized by Etienne Castani´e, Da Cao, Valentina Krachmalnicoff and Yannick De Wilde at Institut Langevin. Our aim is not to enter the details, but to understand the important phenomena to take into account in the numerical model presented in section 2.2. Details of the setup are given in Etienne Castani´e’s PhD thesis [66], or in Refs. [3, 57, 16]. First, we explain the principle of the LDOS measurement using a fluorescent bead. Then, we present the experimental setup. Finally, we comment on the LDOS and fluorescence intensity maps measured on the metallic nanoantenna.

Fluorescent beads probe the LDOS

In the experiment, beads containing a few thousand of identical fluorescent molecules (dyes) are used as probes of the LDOS. These beads are composed of a polystyrene matrix inside which the emitters are embedded. Importantly, each fluorescent molecule is randomly oriented. Here, we explain why such sources are good candidates to perform a direct measurement of the LDOS.

Spontaneous decay rate of an emitter

A fluorescent emitter can be modeled by a three-level system (see Fig. 2.2). |gi is the ground state and |e1i and |e2i are two vibrational levels of an excited electronic state. We denote by ωexc the frequency of the transition |gi → |e1i, and ωfluo the frequency of the transition |e2i → |gi. ωexc corresponds to the frequency of the incident laser used to excite the emitters. ωfluo is the frequency of the fluorescence emission. We denote by K and 􀀀 respectively the rates of the transitions |e1i → |e2i and |e2i → |gi. We make the assumption that the transition rate K is very large compared to 􀀀. In these conditions, if the emitter is excited – i.e. put in the state |e1i – at time t = 0, it immediately decays to the lower vibrational state |e2i. Then, the system behaves like a two-level system [10]. Let us introduce the following notations.

Model for the LDOS

An ideal measurement of the LDOS requires a point-like emitter averaged over orientations. However, to understand the resolution of the experimental maps, one needs to take into account the influence of the finite size of the bead.

Calculation using a point-like source dipole

The numerical method to compute the LDOS is actually very intuitive if one understand the concept of the experiment. As in the experiment, a point source dipole is located at rs to probe the LDOS. Numerically, this is done by using an illuminating field in the Lippmann-Schwinger equation that corresponds to the radiation of a source dipole p located at rs E(r, ω) = μ0ω2G0(r, rs, ω)p + k2 Z V [ǫ(ω) − 1]G0(r, r′, ω)E(r′, ω)dr′. (2.11).
Solving this equation for three orthogonal orientations2 of the source dipole p gives access to the complete dyadic Green function G of the system. From the Green function, one can retrieve the decay rate of the emitter for one dipole orientation from Eq. (2.8), or the LDOS by averaging the decay rate over three dipole orientations (see section 2.1.1). In our calculations, we compute the LDOS for the emission frequency ωfluo of the experimental molecules.

Numerical maps of the LDOS and fluorescence intensity

We show in Fig. 2.10 the numerical maps of the fluorescence enhancement factor and the normalized LDOS computed according to the model presented in section 2.2. The experimental fluorescence intensity map (Fig. 2.6) is well recovered by the computed directional fluorescence enhancement factor F(r, ωfluo, ωexc). The fluorescence intensity is reduced by a factor of the order of 3 on top of the disks. The agreement is almost quantitative. In the experimental map, the LDOS increases by about 30% in three regions presenting an extension of about 60 nm each and separated by 100 nm. The two regions located between the gold disks are predicted by the numerical simulations. As in the case of the fluorescence intensity map, numerical and experimental data are in almost quantitative agreement regarding the expected change of the decay rate in the region between the disks with respect to a region far away from the nanoantenna. The presence of the third lateral region of enhanced LDOS in the experimental map is more speculative. The numerical simulations predict the presence of two such regions of enhanced LDOS, on the external sides of the nanoantenna. A possible explanation for this is an asymmetry of the gold structure, caused for example by a defect of the lift-off process, that would translate in an asymmetry of the structured the electromagnetic file on the surface of the nanoantenna. Numerical calculations with asymmetric shaped nanoantennas have been done and produce similar asymmetries in the LDOS images. However, since the exact shape of the nanoantenna is not accessible at the required level of resolution, having an exact matching between theory and experiment is a very speculative task and the discussion is therefore limited here to a comparison between the experimental results with numerical simulations made on an ideal antenna formed by three regularly spaced circular disks.

Resolution of the LDOS maps

One interesting feature of both the experimental and numerical LDOS maps is that both seem to exhibit variations on scales well below 100 nm, the size of the fluorescent bead. To explain this phenomenon, already observed in [64], we compare the contribution to the decay rate of the emitters located in the lower and upper half of the bead. Figure 2.11 shows the LDOS maps averaged respectively over 100 emitters located at random positions inside a 100 nm diameter bead, over the emitters located in the lower half of the bead and over the emitters located in its upper half. Each map is normalized by the value of the LDOS in vacuum to allow for the comparison between the maps. Every map is computed for a distance d = 20 nm between the bottom of the bead and the top of the trimer. A detailed observation allows us to assert that the resolution of the LDOS map is not limited by the size of the bead. Indeed, the similarity between the top and bottom maps clearly shows that the measured LDOS is driven by the emitters situated on the lower half of the bead. The two hot spots which are visible on the right and on the left side of the nanoantenna are smeared out when considering only the contribution of the emitters populating the upper part of the sphere. More insight can be given by plotting the section of the LDOS maps along the lines drawn in every map. The obtained profiles are shown in Fig. 2.12. Each curve is normalized by the maximum value of the corresponding map ρmax, in order to quantify the contrast of each hotspot. The lateral hot-spot is clearly resolved when the LDOS signal is averaged on the emitters located on the bottom of the sphere or over all the sphere, while it is washed out when the signal is averaged over the top of the sphere.
Therefore the resolution of this detail is clearly due to the bottom emitters. Consequently, the effective resolution is not limited by the size of the bead but is smaller and in the case presented in this thesis is of the order of 50 nm.

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Table of contents :

I Introduction and basic concepts 
General introduction
1 Light-matter interaction: a classical formalism
1.1 Electromagnetic radiation: the dyadic Green function
1.1.1 Green formalism
1.1.2 Eigenmode expansion of the dyadic Green function
1.2 Small particle in vacuum: the dynamic polarizability
1.2.1 Polarizability of a small spherical particle
1.2.2 Resonant scatterer polarizability
1.3 Light-matter interaction: weak and strong coupling regimes
1.3.1 Dressed polarizability in the presence of an environment
1.3.2 Coupling to one eigenmode: Weak and strong coupling regimes
1.3.3 General formulas in the weak-coupling regime
1.4 Conclusion
II Light localization in complex metallic nanostructures 
2 Characterization of a nanoantenna
2.1 Experimental setup and results
2.1.1 Fluorescent beads probe the LDOS
2.1.2 Experimental setup
2.1.3 Experimental results
2.2 Numerical model of the experiment
2.2.1 The Volume Integral Method
2.2.2 Model for the LDOS
2.2.3 Model for the fluorescence intensity
2.3 Numerical results
2.3.1 Numerical maps of the LDOS and fluorescence intensity
2.3.2 Resolution of the LDOS maps
2.4 Conclusion
3 Spatial distribution of the LDOS on disordered films
3.1 Simulation of the growth of the films
3.1.1 Numerical generation of disordered metallic films
3.1.2 Percolation threshold
3.1.3 Apparition of fractal clusters near the percolation threshold
3.2 Spatial distribution of the LDOS on disordered films
3.2.1 Statistical distribution of the LDOS
3.2.2 Distance dependence of the LDOS statistical distribution
3.2.3 LDOS maps and film topography
3.3 Radiative and non-radiative LDOS
3.3.1 Definition
3.3.2 Statistical distributions of the radiative and non-radiative LDOS
3.3.3 Distance dependence of the radiative and non-radiative LDOS distributions
3.4 Conclusion
4 The Cross Density Of States
4.1 The Cross Density Of States (CDOS)
4.1.1 Definition
4.1.2 CDOS and spatial coherence in systems at thermal equilibrium
4.1.3 Interpretation based on a mode expansion
4.2 Squeezing of optical modes on disordered metallic films
4.2.1 Numerical maps of the CDOS on disordered metallic films
4.2.2 Intrinsic coherence length
4.2.3 Finite-size effects
4.3 Conclusion
III Speckle, weak and strong coupling in scattering media 
5 R-T intensity correlation in speckle patterns
5.1 Intensity correlations in the mesoscopic regime
5.1.1 The mesoscopic regime
5.1.2 Dyson equation for the average field
5.1.3 Bethe-Salpether equation for the average intensity
5.1.4 Long range nature of the reflection-transmission intensity correlation .
5.2 Reflection-Transmission intensity correlations
5.2.1 Geometry of the system and assumptions
5.2.2 Ladder propagator for a slab in the diffusion approximation
5.2.3 Diffuse intensity inside the slab
5.2.4 Intensity correlation between reflection and transmission
5.2.5 Discussion
5.3 Conclusion
6 Nonuniversality of the C0 correlation
6.1 C0 equals the normalized fluctuations of the LDOS
6.1.1 The C0 correlation equals the fluctuations of the normalized LDOS
6.1.2 Physical origin of the C0 correlation
6.2 Long-tail behavior of the LDOS distribution
6.2.1 The “one-scatterer” model
6.2.2 Asymmetric shape of the LDOS distribution: Numerical results
6.3 C0 is sensitive to disorder correlations
6.3.1 The effective volume fraction: a “correlation parameter”
6.3.2 LDOS distribution and correlation parameter
6.3.3 C0 and correlation parameter
6.4 Conclusion and perspectives
7 Strong coupling to 2D Anderson localized modes
7.1 An optical cavity made of disorder: Anderson localization
7.1.1 LDOS spectrum of a weakly lossy cavity mode
7.1.2 Numerical characterization of a 2D Anderson localized mode
7.2 Strong coupling to a 2D Anderson localized mode
7.2.1 Strong coupling condition for a TE mode in 2D
7.2.2 Numerical observation of the strong coupling regime
7.3 Alternative formulation of the strong coupling criterion
7.4 Conclusion
General conclusion and perspectives
Appendices
A Lippmann-Schwinger equation 
B Regularized Green function and eigenmode expansion 
B.1 Regularized Green function
B.1.1 General case of an arbitrary volume δV
B.1.2 Case of a spherical volume δV
B.2 Eigenmode expansion of the regularized Green function
B.2.1 Case of a closed non-absorbing medium
B.2.2 Phenomenological approach of lossy environments
C Coupled Dipoles method 
D Simulation of the growth of disordered films 
D.1 Description of the algorithm
D.1.1 Vocabulary and notations
D.1.2 Interaction potential
D.1.3 Energy barrier for particle diffusion
D.1.4 Choice of a process
E Volume Integral method 
E.1 Weyl expansion of the Green function
E.1.1 Spatial Fourier transform
E.1.2 Weyl expansion
E.2 The Volume Integral method
E.2.1 The Lippmann-Schwinger equation
E.2.2 Analytical integration of the Green function over the unit cells
E.3 Energy balance
E.3.1 Power transferred to the environment
E.3.2 Absorption by the medium (non-radiative channels)
E.3.3 Radiation to the far field (radiative channels)
F T-T speckle intensity correlations in the diffusive regime 
F.1 Leading term for the long-range correlation
F.2 Useful integrals
Bibliography 

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