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## Evidences of the occurrence of energy transfer

Using this experimental setup, we can characterise the energy transfer process by measuring either the decay histogram of the donor bead or the fluorescence spectrum of the acceptors. These two approaches give different insights on the energy transfer process: while the modification of the decay rate is mainly due to dipole-dipole interaction and characterises the occurrence of energy transfer between donors and close-by acceptors, we can demonstrate the occurrence of long-range plasmon-mediated energy transfer using spectral measurements.

### Decay rate of the donor

In Sect. 1.3, we highlighted that the energy transfer rate is directly given by the increase in the LDOS at the position of the donor due to the presence of the acceptor. The energy transfer rate is thus given by D!A = DA − D , (2.3) where DA and D are respectively the decay rate of the donor in the presence and in the absence of the acceptor. Furthermore, the energy transfer efficiency, defined as the probability for the donor of exciting the acceptor among all the possible decay processes, is directly given by et = D!A DA .

#### Distance dependence of the energy transfer rate

From the fluorescence spectra of the acceptors measured at several donor-to-acceptor distances, we can estimate the distance dependence of the energy transfer process. Since the energy transfer process relies on the propagation of surface plasmons excited by the donor bead, we can find a relation between the propagation length of the surface plasmons excited by the donor and the characteristic distance of the energy transfer process, which we call the energy transfer range.

**Surface-plasmon propagation length**

In order to estimate the spectrally averaged propagation length of the surface plasmons excited by the donor, we study the donor-only sample and we measure plasmonic radiative losses due to surface plasmons excited by a donor bead. Figure 2.13 shows the total intensity of plasmonic radiative losses as a function of the distance d. For each distance d, we also calculate the standard error on the intensity estimates by taking into account the statistical error on the number of photons determined from spectral measurements. To do so, we can assume that the number of photons Nd measured in the spectral range of the donor at each distance d is Poisson distributed. As we perform background subtraction on the data, it is also relevant to consider the number of measured background photons noted N d . Thus, we can estimate the standard error d on each intensity estimate using the addition property of the variance for uncorrelated variables. We obtain d = q Nd + N d . (2.5) In this analysis, we neglect the error on the distance d. Indeed, we can accurately control the distance between the excitation and detection point in the experiment via the piezoelectric nanopositionners. Moreover, we monitor in real time the drift of the experiment using the EM-CCD camera and the SPAD collecting the fluorescence photon from the donor bead. During the 200 s acquisition time of one spectrum, this drift is negligible.

**Efficiency of the energy transfer process**

As already pointed out, the low efficiency of the energy transfer process for large distances d prevents any experimental determination of the energy transfer rate in this regime. Nevertheless, we can theoretically estimate the plasmon-mediated energy transfer efficiency using the Green formalism.

**Modelling of the experiment**

In order to theoretically estimate the energy transfer efficiency, we must define a model for the experiment. As discussed in Sect. 2.3.2, we can model the geometry of the sample by two semi-infinite media that are respectively silver and PVA. This model is represented in Fig. 2.15. Since energy transfer occurs for the frequencies of maximum overlap between the emission spectrum of the donor and the absorption spectrum of the acceptor, the transfer frequency ! can be approximated by using the free-space wavelength 0 = 663 nm which corresponds to the maximum of the acceptor absorption spectrum. Moreover, a Drude- Lorentz model can be used to describe the properties of silver. We can thus calculate the dielectric constant of silver from Eq. (2.1) which gives us m = −16.2 + 1.2i. The relative permittivity of PVA is set to d = 2.25 in the model.

**Table of contents :**

**1 An introduction to fluorescence **

1.1 Spontaneous emission in the dipole approximation

1.1.1 Probability distribution of the excited-state lifetime

1.1.2 The dipole approximation

1.2 Emission rate of an electric dipole

1.2.1 Power emitted by an oscillating dipole

1.2.2 Decay rate of a two-level system

1.2.3 Intrinsic quantum yield

1.3 Energy transfer between two emitters

1.3.1 Energy transfer rate

1.3.2 Expression of the polarisability

1.3.3 Förster resonance energy transfer

1.4 Fluorescence microscopy

1.4.1 Angular spectrum representation of electromagnetic waves

1.4.2 Far-field microscopy

1.4.3 Near-field microscopy

1.5 Conclusion

**I Micrometre-range plasmon-mediated energy transfer **

**2 Plasmon-mediated energy transfer above a silver film **

2.1 Introduction

2.2 Properties of surface plasmons

2.2.1 Dispersion relations

2.2.2 Propagation length

2.3 Sample preparation and experimental setup

2.3.1 Selection of a donor-acceptor pair

2.3.2 Sample preparation

2.3.3 Optical setup

2.4 Evidences of the occurrence of energy transfer

2.4.1 Decay rate of the donor

2.4.2 Spectral measurements

2.5 Distance dependence of the energy transfer rate

2.5.1 Surface-plasmon propagation length

2.5.2 Energy transfer range

2.6 Efficiency of the energy transfer process

2.6.1 Modelling of the experiment

2.6.2 Distance between the mirror and the emitters

2.6.3 Energy transfer rate

2.6.4 Energy transfer efficiency and enhancement factor

2.7 Conclusion

**3 Energy transfer mediated by single plasmons **

3.1 Introduction

3.2 Sample preparation and experimental setup

3.2.1 Donor-acceptor pair

3.2.2 Optical setup

3.2.3 Determination of the donor-to-acceptor distance

3.3 Generation of single surface plasmons

3.3.1 Demonstration of photon antibunching from single quantum dots

3.3.2 Statistical properties of the second-order correlation function

3.3.3 Quantitative characterisation of single-photon emission

3.3.4 Observation of single plasmons on silver nanowires

3.4 Study of decay histograms

3.4.1 Decay histogram of the quantum dot

3.4.2 Decay histogram of the acceptor bead under laser excitation

3.4.3 Evidence of the occurrence of energy transfer

3.5 Intensity fluctuations due to blinking

3.5.1 Blinking of the quantum dot

3.5.2 Characterisation of blinking by second-order correlations

3.5.3 Correlated blinking of the donor and the acceptor

3.6 Towards a demonstration of photon antibunching

3.6.1 Condition required to demonstrate photon antibunching

3.6.2 Comparison with the current experimental conditions

3.7 Conclusion

**II Super-resolution imaging of the local density of states **

**4 Spontaneous emission in the near field of silicon nanoantennas **

4.1 Introduction

4.2 Far-field analysis of resonant modes in silicon antennas

4.2.1 Dielectric antennas

4.2.2 Description of the sample and dark-field measurements

4.3 Experimental setup for near-field measurements

4.3.1 Description of the near-field fluorescence microscope

4.3.2 Fluorescent source

4.4 Near-field measurements

4.4.1 Methods

4.4.2 Spatial variations of the fluorescence decay rate

4.4.3 Observation of directional emission

4.5 Conclusion

**5 Single-molecule super-resolution microscopy for lifetime imaging **

5.1 Introduction

5.2 Sample preparation and experimental setup

5.2.1 Sample preparation

5.2.2 Optical setup

5.2.3 Data acquisition

5.3 Drift correction

5.3.1 Correction in the sample plane

5.3.2 Defocus correction

5.4 Position and decay rate association

5.4.1 Position and decay rate estimations

5.4.2 Temporal and spatial correlations

5.4.3 Association conditions

5.5 Experimental results

5.5.1 Reconstruction of the decay rate map

5.5.2 Density of detected molecules

5.5.3 Decay rate enhancement

5.6 Conclusion

**6 Fundamental limit on the precision of position and lifetime estimations**

6.1 Introduction

6.2 Estimation theory

6.2.1 Estimators and sampling distributions

6.2.2 Cramér-Rao lower bound

6.2.3 Data modelling

6.3 Precision of position estimations

6.3.1 Point spread function

6.3.2 EM-CCD data model

6.3.3 Calculation of the information matrix

6.3.4 Experimental conditions

6.3.5 Numerical results

6.4 Precision of decay rate estimations

6.4.1 SPAD data model

6.4.2 Calculation of the information matrix

6.4.3 Experimental conditions

6.4.4 Numerical results

6.5 Towards an optimisation of the experimental setup

6.5.1 Beamsplitter transmission

6.5.2 Optimisation of the TCSPC setup

6.5.3 TCSPC models with several unknown parameters

6.6 Conclusion

General conclusion and perspectives

Appendices

**A Dyadic Green function at an interface **

A.1 Definition of the problem

A.2 Notations

A.3 Fresnel coefficients

A.4 Angular spectrum representation

A.5 Simplified expression

**B Numerical evaluation of the LDOS **

B.1 Power dissipated by a dipole

B.2 Poynting theorem

B.3 Case of a continuous source

B.4 Case of a Gaussian pulse

**C Fisher information matrix for decay rate estimations **

C.1 Definition of the problem

C.2 Discrete formulation

C.3 Limiting cases for the discrete formulation

C.4 Integral formulation

C.5 Limiting cases for the integral formulation