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Spontaneous emission in the dipole approximation
In this section, we present a model to describe the electromagnetic interaction between a quantum emitter and its environment, in the electric dipole approximation and in the weak-coupling regime.
Probability distribution of the excited-state lifetime
Let us conceptually isolate a system of interest from the surrounding environment. Spon-taneous emission is one of the possible interaction processes between this system and its environment: from an excited state, the system can emit a quantum of energy ΔE in the form of a photon and decay to a lower energy state. Quantum mechanics associate an angular frequency ω to such transitions so that ΔE = ~ω where ~ is the reduced Planck constant.
This thesis primarily focuses on the emission rate noted and defined as the average number of emitted photons per unit time. It is straightforwardly related to the average excited-state lifetime τ = 1/ . In general, a probability density function (PDF) can be estimated by repeating many times the same measurement on the same system. Based on this principle, time-correlated single-photon counting (TCSPC) can be used to estimate the PDF of the excited-state lifetime of an emitter. Figure 1.1a illustrates the principle of the measurement: a pulsed laser excites the emitter at a high repetition rate and a single-photon avalanche diode (SPAD) detects photons coming from the emitter. For each detection event, an electronic board determines the time delay Δt between the excitation pulse and the emission of a photon. These delays are then histogrammed and used to estimate the decay rate of the emitter.
As an illustration, we perform a measurement on a semiconductor quantum dot (QD) placed on a glass substrate. Its emission is characterised by an wavelength in free space λ0 of the order of 600 nm, which corresponds to an angular frequency ω of approximately 3 × 1015 rad/s. The resulting decay histogram (Fig. 1.1b, light red curve) follows an exponential distribution that is a characteristic of photon emission by a two-level system in the weak-coupling regime. In this regime, a photon emitted in the electromagnetic field has a low probability of being absorbed back by the emitter. In contrast, in the strong-coupling regime, the emitter strongly interacts with one mode of the electromagnetic field, resulting in mode splitting and leading to the apparition of the so-called Rabi oscillations. Such effects are outside the scope of this thesis, which specifically focuses on photon emission in the weak-coupling regime.
Figure 1.1 – (a) Principle of time-correlated single-photon counting. (b) Normalised decay histogram measured on a QD on a glass substrate (red curve) and in the near field of a silver nanowire (dark red curve).
For comparison purposes, we also measure the decay histogram of a second QD located a few nanometres away from a silver nanowire (Fig. 1.1b, dark red curve). This measurement clearly demonstrates that the fluorescence decay rate of an emitter strongly depends on its environment, since the decay rate of the second QD is much larger than the decay rate of the first one. In order to give a quantitative description of this effect, and more generally of the influence of the environment on the decay rate of an emitter in the weak-coupling regime, we can assume the emitted radiation to be monochromatic. Note that, in practice, the linewidth associated with a given transition is broadened by several effects. Indeed, any emitter is subject to lifetime broadening since frequency and time are Fourier transform duals. Furthermore, in solids at room temperature, broadening occurs due to the interaction between the emitter and its environment through various dissipation processes, such as phonon emission – the emission of a quantum of vibrational motion. As a first approximation, we can consider that the emitter radiates at a single frequency if it does not interact with a strongly dispersive environment. In addition, we assume that the emission frequency does not depend on the environment. In general, the energy levels can be modified under external influence, such as a strong electric field (Stark effect) or magnetic field (Zeeman effect). However, frequency shifts due to the emitter’s own field are extremely small in optics. Thus, in the weak-coupling regime and in the absence of strong external fields, the emission frequency can be considered as an intrinsic property of the emitter. Finally, we assume that the electromagnetic field generated by the emitter is weak, so that the response of the environment is linear.
The dipole approximation
The most complete description of spontaneous emission, and more generally of light-matter interaction, is provided by the theory of quantum electrodynamics (QED), in which both radiation and matter are quantised. In contrast, in the classical theory, light is described as a wave and matter is treated with an effective theory. In most experiments, a classical formalism can predict light-matter interaction with a good accuracy using Maxwell’s equations, that are the same in both the classical and the quantum pictures.
In both the classical and the quantum pictures, light-matter interaction is usually described by decomposing the electric scalar potential and the magnetic vector potential into a Taylor series with origin at the centre of the charge distribution [6]. Let us consider the first terms involved in this expansion, called the multipolar expansion:
• The electric monopole interaction term describes the motion of a charged system submitted to an electric field. For such systems, a common approach is to find the equilibrium position and then to solve the problem for higher-order interaction terms. There is no magnetic monopole interaction term in the multipolar expansion.
• The electric and magnetic dipole interaction terms depend on the fields at the centre of the charge distribution. These terms also depend on the intrinsic properties of the emitting system. At optical frequencies, emitters preferentially interact with the electric field because of selection rules that constrain the possible transitions of the system. For this reason, the magnetic dipole interaction term is often not considered in optics.
• The electric and magnetic quadrupole interaction terms depend on the gradient of the fields at the centre of the distribution. We consider that the fields are sufficiently homogeneous over the dimensions of the emitting system so that these terms do not contribute. This argument also holds for higher order interaction terms.
Hence, the electric dipole interaction usually prevails for small emitters at optical frequen-cies. This does not hold for a few specific emitters such as lanthanide ions that have a forbidden transition under the electric dipole approximation. In such cases, electric and magnetic dipole interaction terms have the same order of magnitude, and it is possible to detect the interaction between these emitters and the magnetic field [7–9]. However, the electric dipole approximation is generally sufficient to characterise light-matter interaction at optical frequencies. Then, the intrinsic properties of an emitter can be described by its dipole moment noted µ.
Emission rate of an electric dipole
We now calculate the average value of the power dissipated in the electromagnetic field by a classical oscillator and relate it to the decay rate of a two-level quantum system.
Power emitted by an oscillating dipole
For a classical oscillator in the linear regime, we can describe the emitting system by a current density noted j(r, ω) and the environment by a relative permittivity noted (r, ω). This situation is represented in Fig. 1.2. Using this model, we can study light-matter interaction between an emitter and an arbitrary environment by solving Maxwell’s equa-tions for the electric field E(r, ω). This amounts to solve the Helmholtz equation in the frequency domain expressed by ω2 r × r × E(r, ω) − c2 (r, ω) E(r, ω) = iωµ0 j(r, ω) . (1.1)
Importantly, the relative permittivity (r, ω) is a complex number, whose real part is related to the energy scattered by the medium and whose imaginary part is related to the energy gain or loss within the medium.
Assuming that the emitter can be modelled by a pointlike electric dipole, the associated current density is expressed in terms of its dipole moment µ and the Dirac delta function δ as follows: j(r, ω) = −iωµ δ(r − r0) . (1.2)
By definition, the electric dyadic Green function G(r, r0, ω) is the impulse response of Eq. (1.1) that verifies the outgoing-wave boundary condition as well as the interface con-ditions for the electric field. The dyadic Green function is therefore related to the electric field radiated by the dipole through E(r, ω) = µ0ω2G(r, r0, ω)µ . (1.3)
The power transferred from the dipole to the field can directly be found by considering the Lorentz force acting on charges due to the electric field. The average value of the power transferred from the dipole to the electromagnetic field at the frequency ω is then P(ω) = ω Im {µ∗ • E(r0, ω)} . (1.4) In the weak-coupling regime, and assuming that the field generated by the dipole does not modify the dipole moment µ, it follows that the power transferred from the dipole to the field is related to the imaginary part of the electric dyadic Green function through the relation
µ0ω3 Im {u • G(r0, r0, ω) u} ,
P(ω) = 2 |µ|2 (1.5)
where u is the unit vector in the direction of the dipole moment. This expression shows that the power transferred from the dipole to the field at a given frequency depends on both the intrinsic properties of the emitter and its environment. While the dipole moment µ describes the intrinsic properties of the emitter, the influence of the environment is accounted for using the electric dyadic Green function G(r0, r0, ω). The dependence of dipole emission upon the environment can be interpreted as follows: the electric field radiated by the dipole can polarise the surrounding medium, that in turn radiates and creates a field that can interfere with the original field. Due to this effect, the electric field at the dipole position can be either enhanced or reduced, thus modifying the power transferred from the dipole to the field. If the dipole is embedded in a homogeneous medium of refractive index n = √ , the power transferred from the dipole to the field simplifies to P0 (ω) = nµ0ω4|µ|2 . (1.6)
Decay rate of a two-level system
While the oscillating dipole is the representation of an emitter in the classical picture, the derivation of the lifetime PDF requires a QED treatment. Indeed, energy states are not included in the classical picture. In the quantum picture, we may represent the emitter by a two-level system. Let us study the overall system defined by the two-level system and the electromagnetic field. In the weak-coupling regime, this system can be described by an excited state |ii of energy Ei as well as a set of final states |fi of identical energy Ef . Assuming that the transition probability from |ii to |fi is low, the lifetime PDF is a decreasing exponential function and Fermi’s Golden rule can be used to calculate the decay rate . The decay rate reads in this limit [10] 2µ0ω2 Im {u • G(r0, r0, ω) u} .
This important formula shows that a specific environment can either enhance or reduce the decay rate of a quantum emitter. In a homogeneous medium of refractive index n, it simplifies to 0 (ω) = nµ0ω3|µ|2 . (1.8) 3π~c
We can establish a correspondence between the power P (ω) emitted by the classical oscil-lator and the decay rate (ω) of the two-level system. Indeed, the power emitted by the oscillator at the frequency ω corresponds to the product of the decay rate of the two-level system times the quantum of energy ~ω. To make the correspondence, the classical dipole moment must also be replaced by twice the quantum dipole moment. This factor of two can be taken as a correspondence rule for the classical calculation to give the same result as the quantum one.
As a consequence, the enhancement of the decay rate of the quantum emitter due to its environment is exactly the same as the enhancement of the power dissipated by a classical oscillator. In many cases, the reference situation is a homogeneous medium, and the modification of the decay rate due to the environment is given by (ω) = P(ω) = 6πc Im {u • G(r0, r0, ω) u} . (1.9)
0(ω) P0(ω) nω
We can go one step further and express the influence of the environment on the emission process in terms of the local density of states (LDOS). The LDOS is defined as the density of electromagnetic modes at the frequency ω and at the position r0. Because the physical meaning of the LDOS is ambiguous for an open absorbing medium, we follow the work of Carminati et al. [11] and use the subsequent expression as a definition of the LDOS:
ρu(r0, ω) = 2ω Im {u • G(r0, r0, ω) u} . (1.10)
πc2
This definition coincides with the usual definition of the LDOS if a discrete set of eigen-modes can be defined. As we can see from Eqs. (1.9) and (1.10), the modification of the decay rate of a quantum emitter due to its environment is proportional to the modification of the LDOS at the position of the emitter.
Intrinsic quantum yield
In general, the electromagnetic interaction is not the only possible decay process for an emitter. As an example, we already pointed out that an emitter can decay from an ex-cited state to a lower energy state due to vibrational transitions by emitting a phonon. Such processes increase the decay rate of the emitter without increasing the number of emitted photons. Hence we consider these processes as losses and we characterise them by a decay rate losses that can depend on the environment, through the temperature of the surrounding medium for instance. However, it is generally assumed that the environment has a weaker influence on the decay rate losses than on the electromagnetic decay rate given by Eq. (1.7). As a first approximation, it is usually considered that losses is an in-trinsic property of the emitter that can be experimentally determined using measurements performed in a homogeneous, non-absorbing reference environment. For this reason, the intrinsic losses are commonly characterised by the intrinsic quantum yield of the emitter defined by 0 ηi = , (1.11) 0 + losses where 0 is the decay rate of the emitter due to spontaneous emission in the homogeneous environment of reference. Note that 0 can be calculated by using Eq. (1.8).
We now consider that this emitter is located in a non-homogeneous, dispersive environ-ment. The total decay rate tot of the emitter is then given by tot = + losses where is the decay rate due to the electromagnetic interaction and losses is the decay rate due to other interaction processes1. This can equivalently be expressed in terms of the intrinsic quantum yield, which reads tot = + 1 − ηi 0 . (1.12)
This reads = R + NR, where R and NR are respectively the radiative and non-radiative decay rates.
Using these notations, the total decay rate reads tot = R + NR + losses.
As a consequence of lossy transitions, the influence of the environment on the decay rate of an emitter is more difficult to observe if the emitter has a low intrinsic quantum yield. Indeed, the modification of the total decay rate due to a given environment is tot = 1 + ηi − 1 , (1.13) 0tot 0
where tot0 = 0 + losses. If the intrinsic quantum yield is of unity, the modification of the total decay rate due to the environment is directly given by Eq. (1.9). In contrast, the modification of the total decay rate is less pronounced if the quantum yield is low because the total transition rate is also driven by the competing decay processes.
Energy transfer between two emitters
We previously studied the spontaneous emission of an emitter in an arbitrary environment. Now, we consider the specific situation in which a second emitter is likely to be excited by the first emitter. We refer to the first emitter as the donor and to the second emitter as the acceptor.
Energy transfer rate
In order to calculate the rate of energy transfer from a donor to an acceptor, we describe both emitters by their dipole moments noted µD for the donor and µA for the acceptor, as represented in Fig. 1.3. Assuming that the acceptor dipole moment µA is entirely induced by the donor field, it can be expressed by µA = αA ED(rA, ω) , (1.14) where αA = αA uA ⊗ uA is the polarisability tensor of the acceptor and uA is the unit vector defining the orientation of the acceptor dipole moment. In the linear regime, the polarisability of the acceptor αA does not depend on the excitation field. Note that this does not hold in the saturation regime: for high excitation fields, the rate of absorption by a two-level system is indeed limited by the rate of spontaneous emission.
Table of contents :
General introduction
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
