Hirsh Majid

Hassane Abouaïssa

Daniel Jolly

Gildas Morvan

State Reconstructor for Real-Time Freeway Ramp Metering

Keywords: Reconstructor, sliding mode, ramp metering

The work presented in this paper aims to solve the problem of real-time ramp metering in the case of unavailable (not measured) traffic flow at an on-ramp or measurements provided by a faulty detector which lead to creation of unpredictable traffic states. A state reconstructor is then proposed and a systematic approach is developed for real time freeway ramp metering. The state reconstructor is developed using a second order sliding mode observer, while the control algorithm is based on the concept of differential flatness. Numerical simulations demonstrate the relevance of the provided algorithm for both state reconstructor and control of freeway networks.

I. INTRODUCTION

The main objective of real-time traffic control is to deal with a daily problem of congestion. Such problem is well identified as a source of continuous growth of the direct and indirect costs (drivers health and stress problems, pollution, increase in energy consumption, . . . etc.) [START_REF] Kostialos | Motorway network traffic control systems[END_REF], [START_REF] Smaragdis | A flow maximizing adaptive local ramp metering strategy[END_REF]. The principle of traffic control is based on the use of a variety of adequate actions such as: dynamic speed limits using variable message signs, driving information and alerts, tidal control, dynamic route guidance and ramp metering.

Ramp metering or admissible control, represents the most efficient and direct way to solve congestion problems and to upgrade freeway traffic [START_REF] Papageorgiou | Review of road traffic control strategies[END_REF], [START_REF] Papageorgiou | Traffic Control[END_REF]. This control action consists in regulating the ramp flow at the entrance of the freeway [START_REF] Kostialos | Motorway network traffic control systems[END_REF]. As stated in [START_REF] Papageorgiou | Freeway ramp metering: An overview[END_REF], ramp metering strategies are a valuable tool for an efficient traffic management that can be classified into: reactive strategies that aim to maintain the traffic conditions in freeway close to pre-specified set values using real-time measurements, and proactive strategies, aiming at specifying optimal traffic conditions for a whole freeway network based on the demand and the model prediction over a time horizon [START_REF] Smaragdis | A flow maximizing adaptive local ramp metering strategy[END_REF].

Ramp metering strategies can be implemented locally (isolated ramp metering) in the vicinity of each ramp to calculate the corresponding ramp metering values. It can be implemented simultaneously (coordinated ramp metering) when the objective is to use the available traffic measurements from larger freeway sections.

However, the performance of any control strategy is closely related to the ability to have all needed information about the hirshmajid@yahoo.com, (hassane.abouaissa,daniel.jolly, gildas.morvan)@univ-artois.fr traffic state 1 . Indeed, traffic control problems usually require the knowledge of both the present and the future traffic behaviours. Such information is provided via a set of sensors already installed along the freeway infrastructure. The future traffic behaviour can be computed using a suitable traffic model.

The main objective of this paper is to deal with isolated ramp metering problems when no information about the traffic state variable at the merge segment is available. In other words, we assume that the measurements just downstream of the controlled ramp are provided by a faulty detector. Having the state reconstructor, a systematic approach for real time isolated ramp metering is provided in order to maintain the traffic conditions at a desired optimal operating level.

The paper is organized as follow:

• Section II, recalls the used macroscopic model and provides the basis of the state reconstructor scheme. • Section III, presents the freeway ramp metering algorithm based on the concept of differential flatness . • Section IV, provides some numerical simulations and demonstrates the relevance of the proposed approach. • The conclusion, section V, summarizes the main obtained results and lists some promising perspectives and further researches.

II. TRAFFIC FLOW MODELING AND STATE

RECONSTRUCTOR

Several studies have demonstrated the relevance of macroscopic models for freeway traffic modeling, supervision and control [START_REF] Mammar | Systmes de transport intelligents: Modlisation, information et contrle[END_REF], [START_REF] May | Traffic Flow Fundamentals[END_REF] and [START_REF] Papageorgiou | Some remarks on macroscopic traffic flow modelling[END_REF].

A. Macroscopic models

The first macroscopic model was developed by Lighthill and Whitham [START_REF] Lighthill | On kinematic waves ii: A theory of traffic flow in long crowded roads[END_REF] and Richard [START_REF] Richards | Shock waves on the highway[END_REF] which is based on the conservation law:

∂ ∂t ρ(x,t) + ∂ ∂ x q(x,t) = g(x,t) (1)
where, ρ(x,t), q(x,t) and g(x,t) are, the traffic density in vehicles/km/lanes, the traffic volume in vehicles/h, the ramp generation term in vehicles/h/km, respectively. Equation ( 1) is supplied by q(x,t) = ρ(x,t)v(x,t)β . v(x,t) is the mean speed and β , the number of lanes. The mean speed v(x,t) and the traffic density ρ(x,t) are related by

V e (ρ(x,t)) = v f exp - 1 a ρ ρ c a (2)
where, a is a parameter, v f is the free-flow speed and, ρ c represents the critical density. Such fundamental diagram allows to identify the free flow zone and congested one (See Fig. 1).

As the first order macroscopic model is based on a static relationship between the main traffic variables, it presents several drawback such as the inability to describe the dynamic of the traffic behavior. In this context, the second order models seem to be more adapted. One of these models is the Payne's model [START_REF] Payne | Models of freeway traffic and control[END_REF]. In this work, we use the model proposed by [START_REF] Cremer | An approach for traffic state estimation using in-vehicle information[END_REF], (See also [START_REF] Papageorgiou | Application of automatic control concepts to traffic flow modeling and control[END_REF]) which is discretized in time and space. Consider then the following freeway segment divided into n links (Fig. 2).

The studied freeway segment is then described by the following ordinary differential equations for each link i

[15] ρi = 1 L i (q i-1 -αρ i v i+1 -(1 -α)ρ i+1 v i+2 + r i (t) -s i (t)) (3) vi = 1 τ [V e (ρ i ) -v i (t)] + η L i v i (t)[v i-1 -v i+1 (t)] - ν L i (ρ i+1 -ρ i ) - δ L i (r i (t) -s i (t)) v i (4)
where, r i (t) and s i (t) are the on-ramp and off-ramp flows, respectively. τ, η, ν and δ are model parameters. L i is the length of the link i.

B. State reconstructor

All current freeway traffic control and monitoring algorithms require an accurate knowledge of traffic states along the studied segments [START_REF] Papageorgiou | Relating time-occupancy measurements to space-occupancy and link vehiclecount[END_REF]. From the macroscopic point of view, the main variables of interest are the traffic density in vehicules/km/lane and the average speed in km/h [START_REF] Mammar | Coordinated ramp metering via second order sliding mode control[END_REF]. These important traffic data are usually obtained via a set of sensors, for example, such as inductive loop detector, camera video.

Nevertheless, The high cost of installation and maintenance of a large number of sensors or the occurrence of sensor's failures, led many researchers to develop traffic control methods

∑ obs y = x2
x1 -

e 1 = x 1 -x1
x 1 + Fig. 3. Structure of the super twisting algorithm and algorithms based on state estimation techniques in order to reduce the number of sensors or to reconstruct the missing state information. Several methods were proposed, in [START_REF] Papageorgiou | Relating time-occupancy measurements to space-occupancy and link vehiclecount[END_REF], a state estimator was developed using Extended Kalman filters [START_REF] Sun | Highway traffic state estimation using improved mixture kalman filters for effective ramp metering control[END_REF], other works use the well known Luenberger observer [START_REF] David | An introduction to observers[END_REF].

In [START_REF] Rashid | Robust State Estimation and Control of Highway Traffic Systems[END_REF], a first order sliding mode observer has been proposed in order to design a state observer. Such robust variable structure observer (see [START_REF] Utkin | Sliding modes in control and optimization[END_REF] and [START_REF] Isidori | A sufficient condition for full linearization via dynamic state feedback[END_REF], for more explanation about Sliding mode control and observation) suffers from the high frequency chattering phenomenon generally present near the sliding surface. In this paper, a second order sliding mode observer is proposed in order to overcome this problem.

1) Short overview on the observers: The sliding mode technique has been used for the observer's design in many applications. Consider the following nonlinear system

             ẋ1 = x 2 ẋ2 = x 3 . . . ẋn = f (x , . . . , x n-1 ) y = x 1 (5)
The sliding mode differentiator of order 2 (super twisting algorithm) is given in its general form by [START_REF] Fridman | Sliding modes of higher order as a natural phenomenon in control theory[END_REF] (figure 3). Where, λ 1 and π 1 are positive tuning parameters of the differentiator whose the output is y. Applying the exact differentiator to system (5) when n=2, one obtains:

ẋ1 = x2 + λ 1 | x 1 -x1 | 1/2 sign(x 1 -x1 ) (6) ẋ2 = f q (x 1 , x2 ) + π 1 sign(x 1 -x1 ) (7)
f q (x 1 , x2 ) = equation [START_REF] Papageorgiou | Relating time-occupancy measurements to space-occupancy and link vehiclecount[END_REF]. Then, the estimated value of x2 can be found from equation [START_REF] Mammar | Systmes de transport intelligents: Modlisation, information et contrle[END_REF]. For more details see ( [START_REF] Rivera | Super-twisting sliding mode in motion control systems, sliding mode control[END_REF], [START_REF] Davila | Secondorder sliding-mode observer for mechanical systems[END_REF], [START_REF] Rabhi L. Fridman | Second order sliding mode observer for estimation of velocities, wheel sleep, radius and stiffness[END_REF], [START_REF] Salgado | Generalized super-twisting observer for nonlinear systems[END_REF], [START_REF] Rabhi | Estimation of road profile using second order sliding mode observer[END_REF], [START_REF] Saadaoui | Super twisting algorithm observer for a class of switched chaotic systems[END_REF], [START_REF] Floquet | Super twisting algorithm based step-by-step sliding mode observers for nonlinear systems with unknown inputs[END_REF] and [START_REF] M'sirdi | Second order sliding-mode observer for estimation of vehicle dynamic parameters[END_REF]).

2) Traffic state estimation: Consider the following freeway segments (Fig. 4): Using the relationship between the three aggregated variables (flow q, density ρ2 and mean speed v), The output vector is:

q i (x,t) = ρ i (x,t)v i (x,t) (8)
y = [y 1 y 2 ] T = [ρ 2 v 2 ] T
As long as the outgoing traffic speed satisfies the positivity condition: v 1 , v 2 = v s V min > 0, the studied freeway section (see figure 4) can be described by the following equations:

ρ1 = 1 L 1 (q e + r -αρ 1 v 2 -(1 -α)ρ 2 v s ) (9) ρ2 = 1 L 2 (αρ 1 v 2 -(1 -α)ρ 2 v s -q s ) = ẏρ (10) v1 = 1 τ (V e (ρ 1 ) -v 1 ) + η L 1 v 1 (v e -v 2 ) - ν L 1 (ρ 2 -ρ 1 ) -δ L 1 rv 1 (11) v2 = 1 τ (V e (ρ 2 ) -v 2 ) + η L 2 v 2 (v 1 -v s ) = ẏv (12)
In expressions ( 9)-( 12), ρ 2 and v 2 are assumed to be known. V e (ρ i ) are given thanks to the May fundamental diagram (2). It should be mentioned that q s and v s form the output measurements variables which are equal to q 2 and v 2 respectively, measured on sensor St.3. q e and v e (measured on sensor St.1) are external measured inputs (see figure 4). Assuming that sensor St.2 is broken down. We try to estimate the state variables ρ 1 and v 1 which are essential for the control system from the measured states on sensor St.3 : Consider ρ and v are the estimated value of ρ and v respectively. Then, the observer errors are defined as: e ρ = ρ -ρ and

e v = v -v.
The proposed observer (called super-twisting) has the form

ρ1 = 1 L 1 (q e + r -α ρ1 v 2 -(1 -α) ρ2 v s ) + z ρ1 (13)
Where z ρ1 and z ρ2 are the output injection of the form:

ρ2 = ρ1 + z ρ2 ( 14
)
z ρ1 = πsign(ρ 2 -ρ2 ) ( 15
)
z ρ2 = λ | ρ 2 -ρ2 | 1 2 sign(ρ 2 -ρ2 ) (16)
For the speed estimation v1 , the observer reads:

v1 = 1 τ (V ( ρ1 ) -v1 ) + η L 1 v1 (v e -v 2 ) - ν L 1 (ρ 2 -ρ1 ) -δ L 1 r v1 + z v1 (17)
The output injection are of the form:

v2 = v1 + z v2 (18)
z v1 = πsign(v 2 -v2 ) ( 19
)
z v2 = λ | v 2 -v2 | 1 2 sign(v 2 -v2 ) (20)

III. REAL-TIME RAMP METERING

The ramp metering algorithm is based on the differentially flat systems concept, (see [START_REF] Fliess | Flatness and defect of non-linear systems: Introductory theory and examples[END_REF], [START_REF] Manish | Higher order sliding mode control of differentially flat systems[END_REF], [START_REF] Rudolph | Flatness based control of distributed parameter systems[END_REF] and [START_REF] Sira-Ramirez | Differentially flat systems[END_REF] for more explanations about flat systems).

Notice that the studied system ( 9)-( 12) is flat with F = ρ 2 its flat output (see [START_REF] Iordanova | Flatness-based control of traffic flow[END_REF], for the first application of flatness concept in the traffic flow control). It means that all the system variables can be parametrized in term of this flat output and a number of its time-derivative. The state variable ρ 1 reads then

ρ 1 = L 2 Ḟ αv 2 + 1 -α α F + 1 αv 2 q s ( 21
)
From the expressions above, the control variable r(t) can be parametrized in term of the flat output and its time derivatives up to two

r(t) = L 1 L 2 αv 2 F + L 2 - L 1 L 2 v2 αv 2 2 + L 1 1 -α α Ḟ
+ L 1 αv 2 qs + 1 - L 1 v2 αv 2 2 q s + 2(1 -α)v 2 F -q e (22)
Given a desired reference trajectory for the traffic density, represented by F , a linearized control law achieving an exponential asymptotic tracking of the trajectory is given by the following expression:

r(t) = L 1 L 2 2 Γ + L 2 - L 1 L 2 v2 αv 2 2 + L 1 1 -α α Ḟ + L 1 αv 2 qs + 1 - L 1 v2 αv 2 2 q s + 2(1 -α)v 2 F -q e (23)
Γ = F -λ 2 Ḟ -Ḟ -λ 1 (F -F ) (24)
F = Γ is an auxiliary control input. The set of parameters {K 1 , K 2 }, is chosen so that the roots of the closed loop characteristic polynomial, in the complex variable s: p(s) = s 2 + K 2 s + K 1 be a Hurwitz polynomial. Clearly, the tracking error, e = F -F ,

(F = ρ 2 ) satisfies ë + K 2 ė + K 1 e = 0 (25)
A suitable choice of the design parameters is obtained by matching the characteristic polynomial, p(s), to a desired characteristic polynomial p d (s), with pre-specified root locations.

IV. NUMERICAL SIMULATIONS

For numerical simulations, consider the same example of the stretch depicted in Fig. 4. We suppose that the sensor on St.2 is broken down. Thus, in order to the control system do not disrupt, we try to estimate the state variables (ρ 1 and v 1 ), which are essential for the control algorithm, from the measured state variables (ρ 2 and v 2 ) on St.3. The traffic demand (veh/h/lane) is represented in Fig. 5. For simulation, the model parameters are given in table (I).

In the no-control case, Fig. 6 and Fig. 7, show the densities and speed evolutions. Notice that in this case, the congestion appears for about 40 min. Without additive noise, we obtain a good estimation for both the traffic density and the mean speed of the first link.

Fig. 8 and Fig. 9 show the effectiveness of the proposed controller. Indeed, the proposed algorithm ensures convergence to the desired value. Notice also the good estimation results for the main state variables.

Obviously, the implementation of the ramp metering leads to a queue formation as depicted in Fig. 10.

In the presence of additive noise, Fig. 12 and Fig. 13 show a good robustness of the identification method as well as the Fig. 10. Queue length evolution control algorithm to keep the trajectory around the desired value. In table (II), the standard error occur in the estimated values of ρ and v, in the case of without additive noise and with the presence of additive noise, are given.

V. CONCLUSION

The work presented in this paper exploits the relevance of high order sliding mode observer for state estimation. The proposed super twisting algorithm seems to be very efficient for the case where any information about the state of the controlled ramp is available. On the other hand, the application of the concept of differential flatness for real time ramp metering is based on the system inversion concept, which do not requires integration of any differential equations. Notice that, the flat output is defined by the state of the downstream segment. This means that in addition of on-ramp metering of the merge section, we are able to control all the downstream ones. This fact leads to propose in future works a coordinated ramp metering algorithm which takes measurement of all studied freeway network.

Fig. 1 .

 1 Fig. 1. Fundamental diagram

  1 ) = y + λ 1 | e 1 | 1/2 sign(e 1 ) ẏ = π 1 sign(e 1 ), λ 1 , π 1 > 0

Fig. 2 Fig. 4 .

 24 Fig. 2. Freeway example

Fig. 6 .Fig. 7 .Fig. 8 .Fig. 9 .

 6789 Fig. 5. Traffic demand

Fig. 12 .Fig. 13 .

 1213 Fig. 11. Control variable

Traffic density (or occupancy) constitutes the basis state variable of the most available traffic flow models and plays an important role for both traffic flow modeling and traffic control[START_REF] Papageorgiou | Relating time-occupancy measurements to space-occupancy and link vehiclecount[END_REF].

loop-detector (sensor) measures directly the occupancy, which refers to the percentage of time a detector is occupied by vehicles. It can be transformed to density by T O = 100(L b + L v )ρ [30]. where: L b = length of the loop-detector, L v = length of the vehicle, ρ = density and T O = occupation rate.


Jiawen Liu

Agnès Maître

Laurent Coolen

Tailoring experimental configurations to probe transition dipoles of fluorescent nano-emitters by polarimetry or Fourier imaging with enhanced sensitivity

Tailoring experimental configurations to probe transition dipoles of fluorescent nano-emitters by polarimetry or

Fourier imaging with enhanced sensitivity

INTRODUCTION

Since the first solid-state detections of single-molecule luminescence [START_REF] Moerner | Optical detection and spectroscopy of single molecules in a solid[END_REF][START_REF] Orrit | Single pentacene molecules detected by fluorescence excitation in a p-terphenyl crystal[END_REF] , numerous efforts have been made to measure the properties of the optical transition dipole moment responsible for the luminescence. In particular, in order to understand the underlying physics of a nano-emitter (molecule, colored center, semi-conductor nanoparticle, two-dimensional -2D -semiconductor system etc.), its dipolar orientation is of paramount interest. A correlated crucial question is whether the emitter behaves as a single linear radiating dipole, or whether it must be described as an incoherent sum of two or three dipoles, because several degenerated transitions contribute to the emission. For instance, a "2D dipole" model (sum of two orthogonal dipoles of same oscillator strength) has been used to describe spherical colloidal quantum dots [START_REF] Empedocles | Three-Dimensional Orientation Measurements of Symmetric Single Chromophores Using Polarization Microscopy[END_REF][START_REF] Chung | Room temperature measurements of the 3D orientation of single CdSe quantum dots using polarization microscopy[END_REF][START_REF] Brokmann | Measurement of the radiative and nonradiative decay rates of single CdSe nanocrystals through a controlled modification of their spontaneous emission[END_REF] , dot-in-plates [START_REF] Cassette | Colloidal CdSe/CdS dot-in-plate nanocrystals with 2D-polarized emission[END_REF] or square nanoplatelets [START_REF] Scott | Directed emission of CdSe nanoplatelets originating from strongly anisotropic 2D electronic structure[END_REF][START_REF] Gao | CdSe nanoplatelet films with controlled orientation of their transition dipole moment[END_REF][START_REF] Feng | Consequence of shape elongation on emission asymmetry for colloidal CdSe/CdS nanoplatelets[END_REF] and could also apply to some molecules or colored centers [START_REF] Epstein | Anisotropic interactions of a single spin and dark-spin spectroscopy in diamond[END_REF] .

Rectangular nanoplatelets were shown to behave as a sum of two orthogonal dipoles of different oscillator strengths [START_REF] Feng | Consequence of shape elongation on emission asymmetry for colloidal CdSe/CdS nanoplatelets[END_REF] while three orthogonal dipoles were necessary to model a chain of selfassembled nanoplatelets [START_REF] Liu | Fourier imaging of single self-assembled CdSe nanoplatelet chains and clusters reveals out-of-plane dipole contribution[END_REF] . For 2D sheets of semiconductors, in-plane dipoles were evidenced with MoS2 [START_REF] Schuller | Orientation of luminescent excitons in layered nanomaterials[END_REF] , MoSe2 and WSe2 [START_REF] Brotons-Gisbert | Out-of-plane orientation of luminescent excitons in two-dimensional indium selenide[END_REF] while layered InSe flakes displayed a single out-of-plane dipole [START_REF] Brotons-Gisbert | Out-of-plane orientation of luminescent excitons in two-dimensional indium selenide[END_REF] . For luminescent Eu-doped films, the contribution of both electric and magnetic transition dipoles has been demonstrated [START_REF] Taminiau | Quantifying the magnetic nature of light emission[END_REF] .

Beyond its fundamental importance, the measurement of the number and orientations of the radiating dipoles is essential to understand light-matter interactions at the single-emitter scale as well as non-radiative near-field coupling mechanisms [START_REF] Novotny | Principles of Nano-optics[END_REF] . For example, in plasmonic structures, since local electric fields orient vertically to the metal substrate, a stronger plasmonic coupling between emitters and metals can be achieved when the emitters present vertical dipoles. Förster resonant energy transfer (FRET) is also more efficient between emitters of parallel dipole moments [START_REF] Gao | CdSe nanoplatelet films with controlled orientation of their transition dipole moment[END_REF][START_REF] Liu | Long range energy transfer in selfassembled stacks of semiconducting nanoplatelets[END_REF] . Furthermore, dipole anisotropy and orientation can be used to tailor the efficiency of opto-electronic devices such as light-emitting diodes [START_REF] Whi | Pushing the efficiency envelope for semiconductor nanocrystal-based electroluminescence devices using anisotropic nanocrystals[END_REF] . The orientation of a single luminescent dye can also be used to track distortions in amorphous media [START_REF] Trabesinger | Single-molecule imaging revealing the deformation-induced formation of a polecular polymer blend[END_REF] or movements of single proteins in living cells [START_REF] Brokmann | Orientational imaging and tracking of single CdSe nanocrystals by defocused microscopy[END_REF][START_REF] Ohmachi | Fluorescence Microscopy for Simultaneous Observation of 3D Orientation and Movement and its Application to Quantum Rod-Tagged Myosin V[END_REF] .

Various strategies have been proposed to probe the number and orientation of radiating dipoles in nano-emitters. A first range of methods used polarimetric analysis of the luminescence, either by decomposition into several polarization components [START_REF] Ohmachi | Fluorescence Microscopy for Simultaneous Observation of 3D Orientation and Movement and its Application to Quantum Rod-Tagged Myosin V[END_REF][START_REF] Fourkas | Rapid Determination of the Three-Dimensional Orientation of Single Molecules[END_REF] or by using a rotating polarizer to measure the degree of polarization [START_REF] Empedocles | Three-Dimensional Orientation Measurements of Symmetric Single Chromophores Using Polarization Microscopy[END_REF][START_REF] Chung | Room temperature measurements of the 3D orientation of single CdSe quantum dots using polarization microscopy[END_REF][START_REF] Lethiec | Measurement of three-dimensional dipole orientation of a single fluorescent nanoemitter by emission polarization analysis[END_REF] . A second class of methods uses the radiation pattern (angular distribution of the emission) measured by Fourier (back focal plane) imaging [START_REF] Feng | Consequence of shape elongation on emission asymmetry for colloidal CdSe/CdS nanoplatelets[END_REF][START_REF] Liu | Fourier imaging of single self-assembled CdSe nanoplatelet chains and clusters reveals out-of-plane dipole contribution[END_REF][START_REF] Lieb | Single-molecule orientations determined by direct emission pattern imaging[END_REF][START_REF] Feng | Probing the fluorescence dipoles of single cubic CdSe/CdS nanoplatelets with vertical or horizontal orientations[END_REF] or indirectly by defocused imaging [START_REF] Brokmann | Orientational imaging and tracking of single CdSe nanocrystals by defocused microscopy[END_REF][START_REF] Jasny | Single molecules observed by immersion mirror objective. A novel method of finding the orientation of a radiating dipole[END_REF][START_REF] Bartko | Imaging three-dimensional single molecule orientations[END_REF][START_REF] Böhmer | Orientation imaging of single molecules by wide-field epifluorescence microscopy[END_REF] . A widespread variant used polarized Fourier imaging to distinguish in-plane and out-of-plane dipole contributions [START_REF] Scott | Directed emission of CdSe nanoplatelets originating from strongly anisotropic 2D electronic structure[END_REF][START_REF] Gao | CdSe nanoplatelet films with controlled orientation of their transition dipole moment[END_REF][START_REF] Schuller | Orientation of luminescent excitons in layered nanomaterials[END_REF][START_REF] Brotons-Gisbert | Out-of-plane orientation of luminescent excitons in two-dimensional indium selenide[END_REF][START_REF] Taminiau | Quantifying the magnetic nature of light emission[END_REF][START_REF] Lieberherr | Optical-environment-dependent effects on the fluorescence of submonomolecular dye layers on interfaces[END_REF] . Various other protocols have been introduced, such as separating different emission directions with an annular plate [START_REF] Hohlbein | Three-Dimensional Orientation Determination of the Emission Dipoles of Single Molecules: The Shot-Noise Limit[END_REF] , comparing degrees of polarization with different collection apertures [START_REF] Nhung | Single gold bipyramid orientation measured by plasmon-resonant scattering polarimetry[END_REF] , using imaging aberrations [START_REF] Bartko | Imaging three-dimensional single molecule orientations[END_REF][START_REF] Dickson | Simultaneous imaging of individual molecules aligned both parallel and perpendicular to the optic axis[END_REF] , probing the effect of a nearby interface on decay dynamics [START_REF] Brokmann | Measurement of the radiative and nonradiative decay rates of single CdSe nanocrystals through a controlled modification of their spontaneous emission[END_REF][START_REF] Macklin | Imaging and Time-Resolved Spectroscopy of Single Molecules at an Interface[END_REF] , separating spectrally the electric and magnetic contributions [START_REF] Kim | Measuring 3D orientation of nanocrystals via polarized luminescence of rare-earth dopants[END_REF] etc.

All of these methods rely on comparing the measured observable (degree of polarization, Fourier image etc) with the result of a model involving the dipole(s) and its(their) orientation, which is the unknown quantity of interest. It is well known that this model must take into account radiation modifications introduced by the proximity of an optical interface (substrate surface, protective layer, underlying dielectric spacer etc), as it may partially reflect the radiation [START_REF] Novotny | Principles of Nano-optics[END_REF][START_REF] Lethiec | Measurement of three-dimensional dipole orientation of a single fluorescent nanoemitter by emission polarization analysis[END_REF][START_REF] Lieb | Single-molecule orientations determined by direct emission pattern imaging[END_REF] .

The objective numerical aperture is also a key parameter. Some of us have mentioned in several papers [START_REF] Feng | Consequence of shape elongation on emission asymmetry for colloidal CdSe/CdS nanoplatelets[END_REF][START_REF] Liu | Fourier imaging of single self-assembled CdSe nanoplatelet chains and clusters reveals out-of-plane dipole contribution[END_REF][START_REF] Lethiec | Measurement of three-dimensional dipole orientation of a single fluorescent nanoemitter by emission polarization analysis[END_REF][START_REF] Feng | Probing the fluorescence dipoles of single cubic CdSe/CdS nanoplatelets with vertical or horizontal orientations[END_REF] that the choice of these experimental conditions (interface within the optical environment, objective numerical aperture) can either favor or prevent the determination of the unknown quantities. However, this point was never examined in detail.

In this paper, we model the effect of the experimental conditions on three typical protocols : polarization analysis, polarized Fourier imaging and unpolarized Fourier imaging, used to determine either the emitter's number of contributing dipoles or dipole orientation. We define figures of merit to assess the sensitivity of each protocol and use them to compare several common experimental conditions as examples. We show that the experimental conditions can, in some cases, create redhibitory difficulties to extract the unknown quantities, but can also be tailored to provide optimized sensitivity. This optimization will depend on the quantity to be measured, so that it is critical to design the configuration carefully. The purpose of this paper is thus to give a guidance on how to choose the best experimental configuration according to the question to study.

The paper is organized as follows : in the first section, we outline the general experimental methods and theoretical backgrounds. In the next three sections, we successively describe the three protocols of polarization analysis, polarized Fourier imaging and unpolarized Fourier imaging. In each section, we define a relevant figure of merit to quantify the protocol's sensitivity, and then describe which experimental conditions optimize this figure of merit.

GENERAL PROBLEM

Figure 1 (a) illustrates a typical inverted fluorescence microscope. The emitters are deposited on a substrate, sometimes covered by a protective polymer layer, and excited by a focused laser beam. The fluorescence emission (always calculated for this paper at wavelength 600 nm) is collected by the same objective and sent to various detection devices, such as a photodiode preceded by a rotating polarizer for polarization analysis, or a camera conjugated to the emitter's Fourier plane for radiation pattern measurement.

Depending on how the nano-emitters are modelled, different numbers of unknown quantities will characterize their dipolar emission and be investigated experimentally. Three typical cases are presented on figs. 1(c,d,e). If the emitter is known to behave as a single linear radiating dipole (fig. 1(c)), the only quantity to be probed is its orientation (Θ, Φ). It can be obtained from the polarization analysis curve (fig. 1(b)) : the polarizer orientation for which the detection is maximal yields the in-plane (azimuthal) orientation Φ. The degree of polarization 𝛿𝛿 = 𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚 -𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚 𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚 +𝐼𝐼 𝑚𝑚𝑚𝑚𝑚𝑚 can be related to the out-of-plane (zenithal) orientation Θ. Deducing Θ from the measured 𝛿𝛿 requires the theoretical knowledge of the relation 𝛿𝛿(Θ). This calculation, which will be found in ref. 22, may involve the effect of light transmission or reflection by the optical interface between the substrate and air. It also requires to sum the radiation over all collected angles, so that the result depends on the objective numerical aperture. Throughout the paper, we will call "experimental conditions" the type of substrate used (composition, thickness, presence of a cover layer etc) and the objective numerical aperture. Details about the range of experimental conditions considered are given in Supporting Information (S.I.) section B.

A well-known case of emitter differing from the single linear dipole is the "twodimensional (2D) dipole" (sum of two orthogonal incoherent dipoles of equal oscillator strengths), which has been demonstrated for instance for semiconductor nanoparticles [START_REF] Empedocles | Three-Dimensional Orientation Measurements of Symmetric Single Chromophores Using Polarization Microscopy[END_REF][START_REF] Chung | Room temperature measurements of the 3D orientation of single CdSe quantum dots using polarization microscopy[END_REF][START_REF] Brokmann | Measurement of the radiative and nonradiative decay rates of single CdSe nanocrystals through a controlled modification of their spontaneous emission[END_REF][START_REF] Cassette | Colloidal CdSe/CdS dot-in-plate nanocrystals with 2D-polarized emission[END_REF][START_REF] Scott | Directed emission of CdSe nanoplatelets originating from strongly anisotropic 2D electronic structure[END_REF][START_REF] Gao | CdSe nanoplatelet films with controlled orientation of their transition dipole moment[END_REF][START_REF] Feng | Consequence of shape elongation on emission asymmetry for colloidal CdSe/CdS nanoplatelets[END_REF] . In this case, the "2D" dipole is characterized by its orientation (Θ, Φ), defined as the orientation of the axis normal to the 2 dipoles (fig. 1(d)). The theoretical relation between 𝛿𝛿 and Θ can also be calculated [START_REF] Lethiec | Measurement of three-dimensional dipole orientation of a single fluorescent nanoemitter by emission polarization analysis[END_REF] (S.I. section A) so that the orientation of the 2D dipole can also be obtained by polarimetry.

For some other cases, such as 2D planar materials [START_REF] Schuller | Orientation of luminescent excitons in layered nanomaterials[END_REF][START_REF] Brotons-Gisbert | Out-of-plane orientation of luminescent excitons in two-dimensional indium selenide[END_REF] or particle geometries inducing a preferred deposition orientation (nanoplatelets [START_REF] Feng | Probing the fluorescence dipoles of single cubic CdSe/CdS nanoplatelets with vertical or horizontal orientations[END_REF] , dot-in-plates [START_REF] Cassette | Colloidal CdSe/CdS dot-in-plate nanocrystals with 2D-polarized emission[END_REF] ), the orientation of the emitter is known, but one wishes to investigate the number of incoherent dipoles contributing to the emission or their ratio (fig. 1(e)). In this paper, we will define as 𝑥𝑥, 𝑦𝑦 and 𝑧𝑧 the eigenaxes of the emitter ( 𝑧𝑧 being the optical axis, normal to the sample plane) and 𝜂𝜂 𝑥𝑥 , 𝜂𝜂 𝑦𝑦 and 𝜂𝜂 𝑧𝑧 the corresponding oscillator strengths, normalized so that 𝜂𝜂 𝑥𝑥 + 𝜂𝜂 𝑦𝑦 + 𝜂𝜂 𝑧𝑧 = 1. If only two of these three dipoles are involved in the luminescence, their ratio is the only unknown quantity and it can be deduced from the degree of polarization 9 , as will be shown below. If the three orthogonal dipoles are to be probed, then the measured degree of polarization alone cannot provide enough information to answer the question. More information is then obtained by Fourier plane imaging (fig. 1(a)). In microscopy, the Fourier plane is the back focal plane of the objective, which can be conjugated by a lens onto a CCD camera (details in ref. 24). The emission intensity distribution in the Fourier plane (fig. 4(a)) is a measure of the radiation pattern (emission angular distribution in the far field) defined as 𝐼𝐼�𝑘𝑘 �⃗ � = 𝑑𝑑𝑑𝑑/𝑑𝑑Ω (where 𝑑𝑑𝑑𝑑 is the power radiated within a solid angle 𝑑𝑑Ω centered on 𝑘𝑘 �⃗ ), as each point in the Fourier plane corresponds to a specific emission angle (the radiation pattern is obtained experimentally by dividing the Fourier image by the apodization factor cos 𝜃𝜃, where 𝜃𝜃 is the angle between 𝑘𝑘 �⃗ and the 𝑧𝑧 axis, describing the ratio between an area 𝑑𝑑𝑑𝑑 in the Fourier plane and a solid angle 𝑑𝑑Ω in the radiation space). For example, in a recent paper, we used Fourier imaging combined with polarimetry to probe quantitatively the 3 dipole components 𝜂𝜂 𝑥𝑥 , 𝜂𝜂 𝑦𝑦 and 𝜂𝜂 𝑧𝑧 in stacked chains of nanoplatelets [START_REF] Liu | Fourier imaging of single self-assembled CdSe nanoplatelet chains and clusters reveals out-of-plane dipole contribution[END_REF] . The dipole components are then obtained by fitting the Fourier image with the theoretical radiation patterns, which again take into account the experimental conditions (substrate, objective etc) [START_REF] Novotny | Principles of Nano-optics[END_REF] .

Combining Fourier imaging and polarization analysis would be quite challenging for a single nano-emitter, but it has been performed for various ensembles of dipoles, for instance on a layer of nanoplatelets to analyze the ratio between in-plane and out-of-plane dipoles (which determines its efficiency as a light-emitting device) [START_REF] Whi | Pushing the efficiency envelope for semiconductor nanocrystal-based electroluminescence devices using anisotropic nanocrystals[END_REF] , or on rare-earth ions to compare the contributions of electric and magnetic dipoles [START_REF] Taminiau | Quantifying the magnetic nature of light emission[END_REF][START_REF] Kim | Measuring 3D orientation of nanocrystals via polarized luminescence of rare-earth dopants[END_REF] .

Other microscopy methods exist for analysing a radiating dipole. For instance, a slightly defocused image can be used to distinguish single linear dipoles from 2D dipoles and measure their orientation [START_REF] Brokmann | Orientational imaging and tracking of single CdSe nanocrystals by defocused microscopy[END_REF][START_REF] Jasny | Single molecules observed by immersion mirror objective. A novel method of finding the orientation of a radiating dipole[END_REF][START_REF] Bartko | Imaging three-dimensional single molecule orientations[END_REF][START_REF] Böhmer | Orientation imaging of single molecules by wide-field epifluorescence microscopy[END_REF] . This method is experimentally simple, but involves slightly more complex theoretical calculations and increased influence of aberrations [START_REF] Lieb | Single-molecule orientations determined by direct emission pattern imaging[END_REF] , so that we will not discuss it here, although the same general conclusions about the importance of the experimental conditions should also apply for defocused imaging.

To summarize this section, we have defined several measured observables (degree of polarization, Fourier image) and several unknown quantities which can be of interest (Θ, 𝜂𝜂 𝑦𝑦 , 𝜂𝜂 𝑧𝑧 etc.). A measurement protocol will thus consist in deducing the unknown quantity from its theoretical relation with the measured observable. In order for a protocol to be efficient, the measured observable must have a strong dependence on the unknown quantity. We will in the next sections, for several typical protocols, define a figure of merit quantifying this dependence, and identify under which experimental conditions this figure of merit is optimal.

POLARIZATION ANALYSIS

The measured polarization is a result of the field radiated into all directions within the objective numerical aperture. Figure 2(a) shows the two extreme cases in which the zenith angle is 0° ("out-of-plane (OP) dipole" or "z dipole") and 90° ("in-plane (IP) dipole" or "x or y dipole") respectively. As appears in the projection plane, the overall emission is not polarized for the OP dipole whereas it is maximally polarized for the IP dipole. Intermediate degrees of polarization are obtained either when the dipole orientation is between 0 and 90° or when the emitter behaves as a sum of two or three dipoles.

When the emitter is known to behave as a single linear dipole or as a "2D dipole", the degree of polarization can be expressed as a function of the dipole orientation :

𝛿𝛿(Θ) 1𝐷𝐷 = 𝐶𝐶•𝑠𝑠𝑠𝑠𝑠𝑠 2 Θ (2𝐴𝐴-2𝐵𝐵+𝐶𝐶)•𝑠𝑠𝑠𝑠𝑠𝑠 2 Θ+2𝐵𝐵 𝛿𝛿(Θ) 2𝐷𝐷 = 𝐶𝐶•𝑠𝑠𝑠𝑠𝑠𝑠 2 Θ -(2𝐴𝐴-2𝐵𝐵+𝐶𝐶)•𝑠𝑠𝑠𝑠𝑠𝑠 2 Θ+4𝐴𝐴+2𝐶𝐶 (1)
where 𝐴𝐴, 𝐵𝐵and 𝐶𝐶 are coefficients defined in ref. 22 as a function of the optical environment (substrate index etc) and of the objective numerical aperture. In most cases 𝐴𝐴 is much smaller than 𝐶𝐶 (fig. S2(a) and (b)), so that, at least for a qualitative analysis of the problem, we can take 2𝐴𝐴 𝐶𝐶 ⁄ ~0 (if this approximation is not valid, the general degree of polarization decreases but the qualitative observations below remain true : see fig. S2(c)). The effect of the experimental conditions on the function 𝛿𝛿(Θ) is then solely described by the factor 𝑓𝑓 1 = 2𝐵𝐵 𝐶𝐶 :

𝛿𝛿 1𝐷𝐷 (Θ) = 𝑠𝑠𝑠𝑠𝑠𝑠 2 𝛩𝛩 (1-𝑓𝑓 1 )𝑠𝑠𝑠𝑠𝑠𝑠 2 𝛩𝛩+𝑓𝑓 1 ; 𝛿𝛿 2𝐷𝐷 (Θ) = 𝑠𝑠𝑠𝑠𝑠𝑠 2 𝛩𝛩 (𝑓𝑓 1 -1)𝑠𝑠𝑠𝑠𝑠𝑠 2 𝛩𝛩+2
(2) Figure 2 plots the function 𝛿𝛿(𝛩𝛩) in the case of a single dipole (b) and a 2D-dipole (c) as given by equation ( 2) for different values of 𝑓𝑓 1 . The orientation Θ can be deduced with the best resolution from the measured 𝛿𝛿 if the slope of this curve is strong. For a single dipole, a configuration with 𝑓𝑓 1 ~ 1 provides a good resolution for a large range of Θ, so that 𝑓𝑓 1 ~ 1 is the condition that should be most generally looked for. However, if smaller Θ values can be anticipated (dipole almost OP), it is best to have a strong 𝛿𝛿(𝛩𝛩) slope at low Θ, which is obtained for 𝑓𝑓 1 ≪ 1. On the other hand, the protocol can be made more sensitive for larger Θ (nearly IP dipoles) by choosing 𝑓𝑓 1 ≫ 1.

In other words, in the polarimetry measurement, coefficients 𝐵𝐵 and 𝐶𝐶 (still considering that 𝐴𝐴 is negligible) describe respectively to which extent the detected intensity will depend on cos Θ (OP dipole component) and on sin Θ (IP dipole component) (see S.I. -A).

Therefore, for a large value of 𝑓𝑓 1 = 2𝐵𝐵 𝐶𝐶 ⁄ , the effect of the OP dipole on polarization will be stronger, so that a weak OP component (large Θ) will be better detected, and vice versa for a small 𝑓𝑓 1 .

A different result is found for a "2D" dipole (fig. 2(c)) : large values of 𝑓𝑓 1 lead to a very low degree of polarization, so that 𝑓𝑓 1 should be as small as possible for a better resolution (whatever the expected range of Θ values).

𝜂𝜂 𝑧𝑧 = 0).
We now turn to the case of three dipoles of known orientations 𝑥𝑥, 𝑦𝑦 or 𝑧𝑧 with unknown oscillator strengths normalized by 𝜂𝜂 𝑥𝑥 + 𝜂𝜂 𝑦𝑦 + 𝜂𝜂 𝑧𝑧 = 1. If only the degree of polarization is known, the problem cannot be solved in the general case of three dipoles, but it can be if the presence of only two dipoles can be assumed :

-for one IP dipole 𝜂𝜂 𝑥𝑥 and one OP dipole 𝜂𝜂 𝑧𝑧 (thus 𝜂𝜂 𝑦𝑦 = 0), as demonstrated experimentally for a cubic nanoplatelet standing on its edge for instance [START_REF] Feng | Probing the fluorescence dipoles of single cubic CdSe/CdS nanoplatelets with vertical or horizontal orientations[END_REF] , we find (see S.I. -A):

𝛿𝛿 = 1-𝜂𝜂 𝑧𝑧 1+(𝑓𝑓 1 -1)⋅𝜂𝜂 𝑧𝑧 (3)
A value 𝑓𝑓 1 ~1 provides a clear dependence of 𝛿𝛿 on 𝜂𝜂 𝑧𝑧 for all 𝜂𝜂 𝑧𝑧 values (fig. 2(d)), so that this case is in general the best for this measurement. However a large 𝑓𝑓 1 might be better to precisely study a very weak OP dipole, while a small 𝑓𝑓 1 would be more appropriate to probe a weak IP dipole. Again, the reason is that coefficients 𝐵𝐵 and 𝐶𝐶 respectively describe whether OP and IP dipoles will have a strong effect on the measured polarization.

-for only two IP dipoles (𝜂𝜂 𝑧𝑧 = 0), as evidenced for instance on rectangular nanoplatelets deposited horizontally [START_REF] Feng | Consequence of shape elongation on emission asymmetry for colloidal CdSe/CdS nanoplatelets[END_REF] , we have δ = �𝜂𝜂 𝑥𝑥 -𝜂𝜂 𝑦𝑦 � for any experimental condition. The degree of polarization is then solely dependent on the emitter, regardless of the experimental configurations : any value of 𝑓𝑓 1 will work (fig. 2(e)). The factor 𝑓𝑓 1 has no importance here because only IP dipoles are probed.

Eventually, the optimal choice of 𝑓𝑓 1 for each unknown information to be extracted precisely from the degree of polarization is summarized in Table 1. Different experimental configurations will lead to very different values of 𝑓𝑓 1 . For example, in Figure 3(a), we consider an emitter within a medium of index 1.5, observed with an oil objective (N.A.=1.4), at a distance 𝑧𝑧 0 from an interface with a second, upper medium (more configurations are analysed in fig. S3). This upper medium can for instance be air (emitter on a glass substrate and covered by a 𝑧𝑧 0 protective polymer layer of index 1.5) or another material such as gold or silver (corresponding to a metallic substrate covered by a 𝑧𝑧 0 silica spacer) (see S.I. section B).

Unknown quantity to be measured

We plot the 𝑓𝑓 1 values for 𝑧𝑧 0 = 100 nm as a function of the real and imaginary indices (𝑛𝑛 and 𝑘𝑘)

of the upper medium. Small values of 𝑓𝑓 1 are then obtained for a metallic substrate, especially silver, while 𝑓𝑓 1 is larger than unity when the upper medium is air, and can be tuned to unity or lower values by using a dielectric upper medium of higher index. Eventually, we compare seven common experimental cases in figure 3(b) and calculate their theoretical values of 𝑓𝑓 1 as a function of the parameter 𝑧𝑧 0 . If 𝑓𝑓 1 ~ 1 is desired, a glass substrate and an oil objective may be chosen, and a polymer cover layer of thickness 𝑧𝑧 0 = 50 nm can be added in order to achieve precisely 𝑓𝑓 1 = 1. A glass substrate with an air objective also leads to 𝑓𝑓 1 ~ 1. However, among these cases, using the oil objective with a glass substrate leads to nonnegligible values of 2𝐴𝐴 𝐶𝐶 ⁄ (fig. S2(b)), thus to an overall decrease of the degree of polarization (fig. S2(c)), so that the air objective case may be preferred (however at the cost of lower detected luminescence intensity).

Alternatively, metallic substrates offer a much broader choice of 𝑓𝑓 1 values, which can be of interest in certain cases as listed in Table 1. Interferences between direct and reflected radiations are responsible for these strong variations, as evidenced by the dependence on 𝑧𝑧 0 . If 𝑓𝑓 1 ≫ 1 is needed, a spacer thickness 𝑧𝑧 0 ~ 0 should be used (provided that not too many losses are induced by the proximity to the metal) and up to 𝑓𝑓 1 ~ 10 can be obtained with a silver substrate. If 𝑓𝑓 1 ≪

1 is requested, the spacer thickness should be adjusted between 80 and 200 nm depending on the metal and 𝑓𝑓 1 can be decreased as low as 0.006.

POLARIZED FOURIER IMAGING

We now turn to Fourier plane imaging protocols, starting with the polarization-resolved analysis (fig. 4

The numerator includes the partial derivative of 𝐼𝐼(𝑘𝑘) as a function of 𝜂𝜂 𝑧𝑧 (while keeping 𝜂𝜂 𝑥𝑥 + 𝜂𝜂 𝑦𝑦 + 𝜂𝜂 𝑧𝑧 = 1 and 𝜂𝜂 𝑥𝑥 = 𝜂𝜂 𝑦𝑦 ), integrated over the whole objective numerical aperture. The denominator is introduced for normalization. We will use this factor 𝑓𝑓 2 (which depends on 𝜂𝜂 𝑧𝑧 ) as a figure of merit of the sensitivity of the polarized Fourier imaging in a given experimental condition. Typical high-N.A. objectives lead to a maximal collection angle of about 𝜃𝜃 𝑚𝑚𝑚𝑚𝑥𝑥 = 70°.

However, collection at high angles can induce significant losses [START_REF] Kurvits | Comparative analysis of imaging configurations and objectives for Fourier microscopy[END_REF] and aberrations leading to discrepancies with theory [START_REF] Liu | Fourier imaging of single self-assembled CdSe nanoplatelet chains and clusters reveals out-of-plane dipole contribution[END_REF] . Thus we chose to calculate 𝑓𝑓 2 by integrating only up to 𝜃𝜃 𝑚𝑚𝑚𝑚𝑥𝑥 = 55°, meaning that the fit of the experimental curve will be performed most relevantly in that range and not take into account higher angles.

As a first example, Figure 5(a) considers a configuration using a gold substrate, a silica spacer and an oil objective and investigates the effect of the spacer thickness 𝑧𝑧 0 on 𝑓𝑓 2 . We identify 𝑧𝑧 0 = 190 nm as the configuration providing the highest 𝑓𝑓 2 for all values of 𝜂𝜂 𝑧𝑧 (black dashed line)

while the worst configuration corresponds to 𝑧𝑧 0 = 80 𝑛𝑛𝑛𝑛 (red dashed line). Indeed, the Fourier profile for 𝑧𝑧 0 = 80 nm is almost the same for all values of 𝜂𝜂 𝑧𝑧 , so the protocol would fail completely under such experimental conditions (fig. 5(b)). On the other hand, the optimized case 𝑧𝑧 0 = 190 nm leads to dramatic changes of the radiation pattern as a function of the IP/OP ratio (fig. 5(c)). This validates 𝑓𝑓 2 as a good figure of merit to identify optimal experimental conditions. (averaged over all values of 𝜂𝜂 𝑧𝑧 between 0 and 1). The last configuration shows no dependence on 𝑧𝑧 0 so that all values are equally good.

We performed the same analysis for various configurations and, for each, worst and best values of 𝑧𝑧 0 were identified. They are summarized, along with the corresponding factor 𝑓𝑓 2

(averaged over all values of 𝜂𝜂 𝑧𝑧 ), in fig. 5(d). For configurations using a glass substrate, the differences between the best and worst cases are not very strong (as confirmed when plotting the full radiation patterns : see fig. S4), with no effect of 𝑧𝑧 0 at all for the last configuration. This latter configuration (glass substrate and oil objective with no protective layer, as used in ref. 7) is in fact simple to implement and provides among the best factors 𝑓𝑓 2 . When gold or silver substrates are used, the value of 𝑧𝑧 0 is more critical and the worst cases must absolutely be avoided, whereas the best cases offer very good sensitivity.

Finally, we note that polarized Fourier imaging has also been used to quantify the ratio between electric and magnetic dipole contributions from an ensemble of Europium dopants [START_REF] Taminiau | Quantifying the magnetic nature of light emission[END_REF] . A similar treatment can be applied to this case and some experimental conditions are shown to be completely inefficient while others are optimal : a full treatment will be found in S.I. section H.

NON-POLARIZED FOURIER IMAGING

In this last section, we consider the use of Fourier imaging (without polarization selection) to investigate the three dipole coefficients 𝜂𝜂 𝑥𝑥 , 𝜂𝜂 𝑦𝑦 and 𝜂𝜂 𝑧𝑧 of a single nano-emitter, as performed in refs. 9, 11, 24. Because here two unknown quantities can be measured (ratios 𝜂𝜂 𝑦𝑦 𝜂𝜂 𝑥𝑥 ⁄ and 𝜂𝜂 𝑧𝑧 𝜂𝜂 𝑥𝑥 ⁄ ), this method provides more information than the two previous ones but it is also more complex. It consists in fitting with theory the experimental radiation pattern profiles taken along the two symmetry axes of the Fourier plane : resp. 𝐼𝐼(𝑘𝑘 𝑥𝑥 ) (with 𝑘𝑘 𝑦𝑦 = 0 ; green lines on fig. 6) and 𝐼𝐼�𝑘𝑘 𝑦𝑦 � (with 𝑘𝑘 𝑥𝑥 = 0 ; blue lines on fig. 6). In Figure 6(c), we show the example of a single in-plane dipole (𝜂𝜂 𝑥𝑥 = 1 ; dotted line). If we add a small portion of 𝑦𝑦-dipole (𝜂𝜂 𝑥𝑥 = 0.9, 𝜂𝜂 𝑦𝑦 = 0.1 ; left plot, full line), the 𝑥𝑥 -axis profile becomes more intense (its high-angle lobes increase with respect to the emission at normal angle) while the 𝑦𝑦-axis profile decreases. On the other hand, if we add a weak 𝑧𝑧-dipole instead (𝜂𝜂 𝑥𝑥 = 0.9, 𝜂𝜂 𝑧𝑧 = 0.1 ; right plot, full line), both 𝑥𝑥-and 𝑦𝑦-axis profiles increase (their lobes increase with respect to emission at normal angle). In the S.I. (fig. S5), we evaluate more cases and draw the following conclusion: (i) when the OP-dipole contribution (𝜂𝜂 𝑧𝑧 ) is increased, the emission lobes are increased along both profiles 𝐼𝐼(𝑘𝑘 𝑥𝑥 ) and (𝑘𝑘 𝑦𝑦 ) ; (ii) when the ratio between the two IP dipoles (ratio 𝜂𝜂 𝑥𝑥 /𝜂𝜂 𝑦𝑦 ) is changed, the ratio between the lobes of 𝐼𝐼(𝑘𝑘 𝑥𝑥 ) and 𝐼𝐼(𝑘𝑘 𝑦𝑦 ) is changed. Therefore, the typical fitting approach will be to (i) adjust 𝜂𝜂 𝑧𝑧 so that the height of the lobes matches theory, and (ii) adjust 𝜂𝜂 𝑦𝑦 so that the ratio between the lobes matches theory.

Note that, in Figure 6(c), all the curves are normalized by the value at normal angle 𝐼𝐼(𝑘𝑘 𝑥𝑥 = 𝑘𝑘 𝑦𝑦 = 0). Therefore the integral area below each curve (here labelled 𝐴𝐴 𝑥𝑥 and 𝐴𝐴 𝑦𝑦 ) will reflect the ratio of its high angle lobes with respect to the emission at normal incidence. It can be shown (see S.I.-F) that, in agreement with our qualitative observations above, (i) the sum 𝑑𝑑 = 𝐴𝐴 𝑥𝑥 + 𝐴𝐴 𝑦𝑦 will be related only to the OP component 𝜂𝜂 𝑧𝑧 , (ii) and the difference 𝐷𝐷 = |𝐴𝐴 𝑥𝑥 -𝐴𝐴 𝑦𝑦 | will be proportional to the difference |𝜂𝜂 𝑥𝑥 -𝜂𝜂 𝑦𝑦 | between the two IP dipoles, with no direct dependence on the OP dipole. A given measurement configuration will be optimal for our protocol if 𝑑𝑑 depends strongly on 𝜂𝜂 𝑧𝑧 and if 𝐷𝐷 depends strongly on |𝜂𝜂 𝑥𝑥 -𝜂𝜂 𝑦𝑦 |.

In order to identify optimal configurations, we therefore define two normalized figures of merit, respectively quantifying (i) the dependence of S on 𝜂𝜂 𝑧𝑧 and (ii) the dependence of D on 𝜂𝜂 𝑦𝑦 :

𝑓𝑓 3 = |𝜕𝜕𝜕𝜕 𝜕𝜕𝜂𝜂 𝑧𝑧 ⁄ | 𝜕𝜕 , 𝑓𝑓′ 3 = �𝜕𝜕𝐷𝐷 𝜕𝜕𝜂𝜂 𝑦𝑦 ⁄ � 𝜕𝜕 (6)

CONCLUSION

In this paper, we gave a theoretical description of three different protocols to analyse the transition dipoles in nano emitters: polarization analysis, polarized Fourier imaging and nonpolarized Fourier imaging. We demonstrated that the sensitivity with which these techniques yield transition dipole properties is significantly dependent on the experimental configuration, some conditions being completely redhibitory. We introduced figures of merit to estimate this sensitivity and compared theoretically the performance of different experimental configurations for these protocols, as summarized in Table 1 and Figs 5(d) and 7(d). These guidelines should help to avoid detrimental experimental configurations and choose the best ones to enhance measurement sensitivity.

A remaining question is whether a more general protocol can be established to obtain both the dipoles orientation and oscillator strengths of a single emitter, without any a priori knowledge of either its number of contributing dipoles or their orientation. The approach detailed here could be used to design and optimize such a protocol. Based on our results, a possibility could be to start with polarized Fourier imaging for a first estimate of the IP/OP dipoles ratio, and then switch to non-polarized Fourier imaging for a full measurement of the three dipoles.

Using an oil objective and a gold substrate covered by a 180-nm silica spacer would then be a good configuration for both polarized and non-polarized Fourier imaging. The main challenge would then be to demonstrate experimentally whether such a protocol can achieve a reasonable accuracy, given the limited photon count of single-emitter experiments.

This principle of measurement under tailored experimental conditions can be extended to any other protocol used to characterize transition dipoles, for instance defocused imaging. We Corresponding Author *laurent.coolen@insp.jussieu.fr.

Figure 1 .

 1 Figure 1. (a) Schematic of a typical microphotoluminescence setup. (b) Typical experimental

Figure 2 .

 2 Figure 2. (a) Schematic of the radiated polarization for an out-of-plane or an in-plane dipole.

Figure 3 .

 3 Figure 3. Value of the factor 𝑓𝑓 1 for an emitter at a distance 𝑧𝑧 0 from an optical interface. (a)Emitter in a medium of index 1.5, second medium of complex index 𝑛𝑛 + 𝑖𝑖𝑘𝑘, contour plot of the

  (a)) of an ensemble of emitters with random isotropic in-plane orientation (sum of three dipoles with 𝜂𝜂 𝑥𝑥 = 𝜂𝜂 𝑦𝑦 ), as performed in refs. 7,8. Figure4(b) simulates the Fourier plane images of 𝑥𝑥, 𝑦𝑦 and 𝑧𝑧 dipoles (corrected by the apodization factor cos 𝜃𝜃, where 𝜃𝜃 is the angle between 𝑘𝑘 �⃗ and the 𝑧𝑧 axis) filtered by a polarizer oriented along the 𝑥𝑥 axis. The 𝑦𝑦 -dipole contribution is completely filtered by the polarizer along the 𝑘𝑘 𝑥𝑥 axis, so that the measured image profile along the 𝑘𝑘 𝑥𝑥 axis will be a sum of the 𝑥𝑥 (IP) and 𝑧𝑧 (OP) dipoles contributions. The experimental protocol will thus consist in plotting the radiation pattern profile 𝐼𝐼(𝑘𝑘 𝑥𝑥 , 𝑘𝑘 𝑦𝑦 = 0) as a function of 𝑘𝑘 𝑥𝑥 and fit the theoretical model by adjusting the ratio between the IP and OP dipole components[START_REF] Scott | Directed emission of CdSe nanoplatelets originating from strongly anisotropic 2D electronic structure[END_REF] .

Figure 4 .

 4 Figure 4. (a) Polarized Fourier imaging schematic (electric-field shown as yellow arrows). (b)

Figure 4 (

 4 Figure 4 (c) shows the 𝑥𝑥-axis profile of the radiation pattern 𝐼𝐼(𝑘𝑘 𝑥𝑥 ) (with 𝑘𝑘 𝑦𝑦 = 0). In

Figure 5 .

 5 Figure 5. With a gold substrate and an oil objective, (a) figure of merit 𝑓𝑓 2 as a function of 𝜂𝜂 𝑧𝑧 and 𝑧𝑧 0 , showing the best (black dotted line) and worst (red dotted line) values of 𝑧𝑧 0 ; (b) and (c) radiation patterns 𝐼𝐼(𝑘𝑘 𝑥𝑥 ) (for 𝑘𝑘 𝑦𝑦 = 0) for different values of the out-of-plane contribution 𝜂𝜂 𝑧𝑧 , in

Figure 6 :

 6 Figure 6 : (a) Fourier imaging schematic (without polarization selection ; electric-field shown

Figure 7

 7 Figure 7 compares the potential of Fourier imaging for different experimental configurations

Figure 7 (

 7 a) displays values of 𝑓𝑓 3 (for 𝜂𝜂 𝑧𝑧 = 0.5) and 𝑓𝑓′ 3

Figure 7 .

 7 Figure 7. (a) Figures of merit 𝑓𝑓 3 (for 𝜂𝜂 𝑧𝑧 = 0.5) and 𝑓𝑓′ 3 (valid for any 𝜂𝜂's) as a function of the thickness 𝑧𝑧 0 for 7 different experimental configurations. (b) For an emitter on a glass slide at distance 𝑧𝑧 0 = 5 nm, (c) for various worst configurations, and (d) for various best configurations : radiation patterns 𝐼𝐼(𝑘𝑘 𝑥𝑥 ) and 𝐼𝐼(𝑘𝑘 𝑦𝑦 ) for changing values of 𝜂𝜂 𝑦𝑦 (left) and 𝜂𝜂 𝑧𝑧 (right).

  have restricted our study to nano-emitters in the vicinity of a single planar optical interface, as these situations are easier both to model and to implement, but increased flexibility would be provided by a stack of several layers of different materials. Eventually, a vast range of sophisticated experiments becomes available when non-planar optical environments are introduced, such as near-field tips or optical nano-antennas. Supporting Information. A : polarization measurement protocol and theoretical calculations. B : description of the modelled experimental conditions. C : ratio between A, B and C coefficients. D : dependence of figure of merit 𝑓𝑓 1 on the experimental configuration. E : optimized configurations for polarized Fourier imaging analysis. F : nonpolarized Fourier imaging : meaning of figures of merit S and D, dependence on factors 𝜂𝜂 𝑥𝑥,𝑦𝑦,𝑧𝑧 . G : optimized configurations for non-polarized Fourier imaging.

Table 1 :

 1

	Optimal 𝒇𝒇 𝟏𝟏

optimal condition on factor 𝑓𝑓 1 for each polarization measurement

  . Because the two figures of merit must be optimized at the same time (unless one focuses more on estimating either 𝜂𝜂 𝑧𝑧 or 𝜂𝜂 𝑦𝑦 ), a trade-off must be sought. We then plot the changes induced to the Fourier profiles 𝐼𝐼(𝑘𝑘 𝑥𝑥 ) and 𝐼𝐼(𝑘𝑘 𝑦𝑦 ) , either when adding a second 𝜂𝜂 𝑦𝑦 component to a single x dipole (red and blue curves) or when adding a third 𝜂𝜂 𝑧𝑧 component to a horizontal "2D" dipole (purple curve). From the 𝑓𝑓 3 and 𝑓𝑓′ 3 factors, we can identify potential favorable (fig.7(d)) and unfavorable (7(c)) values of 𝑧𝑧 0 for glass, gold or silver substrates. Indeed, for the favorable cases, the Fourier profiles show a very clear dependence on both the OP component and the second IP component. Under these experimental configurations, Fourier imaging can thus be used to find accurate values of the three dipole components. Moreover, the gold and silver cases show a significant improvement as compared to the glass substrate (which is why we used a gold substrate in ref.11). For the unfavorable experimental configurations (fig.7(c)), on the other hand, the Fourier profiles show negligible changes when adding a second IP dipole, and only weak changes when adding a third OP component. For an emitter on a glass substrate observed by an oil objective, all distances 𝑧𝑧 0 yield the same 𝑓𝑓 3 and 𝑓𝑓′ 3 . This

configuration is useable (fig.

7(b)

), but less efficient than the favorable gold or silver configurations (lower 𝑓𝑓′ 3 factor).

Eventually, our results indicate that examples of good configurations, in order to optimize both the 𝜂𝜂 𝑦𝑦 and 𝜂𝜂 𝑧𝑧 measurements, are to use an oil objective with gold or silver substrates with a silica spacer of around 200 nm. A glass substrate where the emitters are covered by a 100-nm polymer layer and observed with an oil objective may also be used.

Acknowledgements

This work is supported by the Agence Nationale de la Recherche -project 211668 FOENICS.