Patrick Ferrand
email: patrick.ferrand@fresnel.fr

Jérôme Wenger

Hervé Rigneault

Fluorescence Correlation Spectroscopy

Keywords: Fluorescence Correlation Spectroscopy Fluorescence correlation spectroscopy, FCS, autocorrelation, fluctuations, single molecule spectroscopy, confocal microscopy, diffusion. 1

published or not. The documents may come

1 Introduction "Intensity is not enough", that could be the motto for Fluorescence Correlation Spectroscopy (FCS) a quite mature technique nowadays that was born in the 70's (1) from the work done at that time on dynamic light scattering (DLS) [START_REF] Berne | Dynamic light scattering: with applications to chemistry, biology, and physics[END_REF]. FCS (as DLS) relies on performing a time correlation on a fluctuating light signal with the assumption that this fluctuating signal is coming from a limited and fluctuating number of particles, the size of the particles being small compared to the characteristic length of the observation volume. In principle time correlation can be performed on any kind of signal, we shall further be interested in fluorescence as the contrast mechanism used in FCS, whereas DLS relies on light scattering. To make things clear, let's consider a volume of observation, that defines a restricted area in a much larger open volume, where particles are freely moving, as illustrated in Fig. 1-A. For simplicity, we consider here a 2D diffusion process, this can be for instance what happens when a fluorescent molecule is freely moving in a biomembrane. Now looking at the fluorescence intensity recorded as a function of time in the observation volume, it is clear that molecule entering and living the volume will make this intensity fluctuate, as depicted in Fig. 1-B. Now in FCS, we are not only interested in the amplitude of these fluctuations, but also in their average time duration. In principle, one could imagine retrieving this information by measuring how much time was elapsed between an increase and a decrease of fluorescence intensity. However, this view is very simplistic and in practice, such information is very difficult to extract, because there is no way to ensure that the obtained increase and decrease are due to the same molecule. On another hand, although the signal looks chaotic, it is obvious that any molecule entering the volume will leave after a certain time, on average. Indeed, the key of the the mathematical treatment of FCS relies on this on average notion, more generally on the concept of correlation. Starting from intensity I(t), the so-called autocorrelation function (ACF) G(τ ) is defined as

G(τ ) = I(t) I(t + τ ) I(t) 2 (1)
where the brackets indicate a time average. The ACF has to be understood as a way to quantify how much a signal is still similar to itself after a given delay τ . In our example of 2D diffusion, it is clear that the signal doesn't have the time to change too much within a delay τ if τ is much smaller than the typical residence time of the molecules in the observation volume.

Mathematically, this would give I(t) I(t + τ ) I 2 (t) > I(t) I(t) and therefore G(τ ) > 1. On the contrary, if τ is much larger than the typical residence time, then signals I(t) and I(t+τ ) have no similarity and I(t) I(t+ τ ) can be written as I(t) I(t + τ ) , thus G(τ ) = 1. It is intuitive at this stage that the intermediate values of the ACF will carry the information on the duration of the fluctuation, here the residence time of the molecule in the observation volume. A typical ACF is plotted in Fig. 1-C. Although sensitive to single molecule fluctuation, FCS is intrinsically a statistic method, thanks to the huge number of individual events that are analyzed.

Although the above explanation is hand waving and limited to the simple case of 2D diffusion, for tutorial purposes, FCS is supported by a rigorous theory as we will see below. It permits to quantify a wide range of phenomena such as photophysical, photochemical, interaction, diffusion, and transport properties of molecules (3).

The scope of this chapter is very practical, "what shall I do to build and run my own FCS setup" has been our guideline when writing this chapter.

Theoretical assumptions

The most important assumption behind FCS measurements deals with the concept of stationarity. The system under study is assumed to be stationary, i. e., from a mathematical point of view, both quantities I(t) and I(t) I(t + τ ) of Eq. 1 need to be independent on time t. In other words, the total measurement duration T tot has to be much larger than the fluctuation duration [START_REF] Tcherniak | Fluorescence correlation spectroscopy: criteria for analysis in complex systems[END_REF]. A practical figure of merit of the stationarity is the value of G(τ ), that should be unity for the largest values of τ . As we will see later, a signal that is not stationary is not be suitable for a FCS analysis.

The second assumption deals with the shape of the observation volume, that is described by the point spread function (PSF) of the confocal microscope. In the case of a high numerical aperture microscope objective, the PSF is in principle a complicated distribution that depends both on the excitation and collection optics. However, for FCS studies, it is generally assumed that it can be reasonably described by a 3D gaussian distribution ( 5)

PSF(x, y, z) = I 0 exp -2 x 2 + y 2 ω 2 xy + z 2 ω 2 z , (2)
where w xy and w z denote the transverse and axial waist, respectively. With this assumption, an effective observation volume is usually defined as

V eff = π 3/2 w 2 xy w z .
Although this Gaussian assumption does not constitute a sine qua non condition for performing FCS, it allows to derive relatively simple analytical expressions for the autocorrelation function G(τ ), as we will see in section 1.5.

Amplitude of the ACF

By writing the fluctuating intensity as I(t) = I + δI(t), Eq. 1 becomes

G(τ ) = 1 + δI(t)δI(t + τ ) I 2 .
In case of diffusing molecules, the fluctuations of intensity recorded by an ideal setup are only due to the fluctuation number of molecule in the observation volume. We shall name δN this fluctuation number. For a given count rate per molecule (CRM), I = N • CRM and δI(t) can be simply written as δI(t) = δN (t) • CRM, and then, for short delays, the amplitude of the ACF is

G(τ → 0) = 1 + δN 2 N 2 = 1 + 1 N , ( 3
)
where we have used the property that the quantity N is governed by a Poisson statistics, and as such, δN 2 = N . The important result of Eq. 3 emhasizes that FCS is intrinsically designed to analyze small numbers of molecules.

Relevant time scales

Although FCS can address a large range of time scales, a priori knowledge of the fluctuation time, that we denote τ f , is necessary to set the optimum conditions for measurement, that we summarize here as

∆τ τ f T tot , ( 4
)
where ∆τ is the correlator sampling time with which the intensity I(t) is recorded. Indeed, in order to be sensitive to the fluctuations, ∆τ needs to be much smaller than τ f . Note that ∆τ will be the first channel of the computed ACF, usually named the correlogram. In addition, as discussed in section 1.1, the condition of stationarity requires that the total measurement duration T tot is much larger than τ f . Note that T tot will be the last channel of the correlogram.

Signal to noise ratio in FCS

Noise and statistical accuracy in FCS deserve specific consideration. First, understanding the origin of the signal to noise ratio (SNR) enables optimization of the experimental apparatus. Second, proper weighting procedures based on statical estimates for the noise enable more accurate analysis. Beyond technical noise, two physical phenomena bring fundamental contributions to the noise in FCS : the quantum nature of light that induces shot noise, and the stochastic nature of the fluorescence fluctuation process itself.

In the most common experimental conditions for FCS: Gaussian molecular detection efficiency, 3D Brownian diffusion, negligible background, and small sample time (∆τ → 0), but without any assumption on the average number of molecules N , it can be demonstrated that the SNR of the autocorrelation function is simply given by ( 6)

SNR τ →0 CRM √ T tot ∆τ (1 + 1/ N ) 1/2 , ( 5
)
where CRM denotes as previously the count rate per molecule. This shows that the SNR in FCS does not depend on the total detected fluorescence, but on the fluorescence count rate per molecule (CRM), on the square root of the experiment duration T tot and on the correlator channel minimum width ∆τ . This further emphasizes the single molecule nature of FCS, and provides guidelines to improve the statistical accuracy in FCS. For a fixed experimental apparatus, one can either increase the excitation power to raise the CRM, or wait for longer integration times T tot . However, both of these strategies have significant practical limitations. First, saturation and photobleaching limit the CRM increase to a certain extent. Second, due to the square root dependence of the SNR on the acquisition time, increasing T tot to a few hundred seconds has only a minor influence on the SNR.

Analytical expression of the ACF

With the assumptions of section 1.1, analytical expressions of the ACF can be derived for a large variety of physical processes. For the sake of generality, we shall write the ACF as

G(τ ) = 1 + 1 N X(τ ) G D (τ ) B(τ ), (6)
where X(τ ), G D (τ ) and B(τ ) denote the intra-or intermolecular reactions, diffusion, and background contributions, respectively, to the ACF. Note that this separation of the different dynamical contribution is only possible when molecular reactions takes place at much shorter time scale than diffusion, which is usually a reasonable assumption.

Inter-and intramolecular dynamics

The first factor in Eq. 6 accounts for changes in fluorescence yield as molecules undergo inter-or intramolecular dynamics, such as triplet blinking. The general expression for X(τ ) is

X(τ ) = 1 + T 1 -T exp - τ τ T , ( 7
)
where T is the fraction of molecule in a dark state, and τ T is the corresponding duration of the triplet fluorescence fluctuation [START_REF] Widengren | Fluorescence correlation spectroscopy of triplet-states in solution -a Theoretical and experimental-study[END_REF].

Translational diffusion

The factor G D (τ ) is derived from the diffusion equation that governs the evolution of the concentration C(r, t) of the fluorescent molecule, given by

∂C(r, t) ∂t = D∆C(r, t),
where D is the diffusion coefficient of the molecule. For a 3D translational diffusion,

G 3D D (τ ) = 1 1 + τ τ D 1 1 + s 2 τ τ D (8) with τ D = w 2 xy 4D
is the average residence time in the observation volume and s = w xy wz is a purely geometrical factor. In case of 2D diffusion the limit s → 0 gives

G 2D D (τ ) = 1 1 + τ τ D (9)

Background signal

Last but not least, the factor B(τ ) accounts for the contribution of background signal in the ACF. It has been demonstrated that ( 6)

B(τ ) = 1 - B I 2 , ( 10
)
where B is practically the intensity measured by replacing the fluorescent sample by a non fluorescent one.

Materials

FCS system

Several companies offer turn-key FCS systems. Most of them are based on confocal laser scanning microscope (CLSM) systems (see Note 1), while some others rely on a more simple optical systems, require no optical alignment but suit therefore to more diluted systems. These systems include excitation laser, appropriate filter sets, photon counting systems, and a dedicated data analysis software. Alternatively, most commercial CLSMs can be customized in order to perform FCS measurements, according to the optical scheme presented in Fig. 2. It requires however a solid background in photonics instrumentation, electronics, data acquisition and programming. The following points need to be considered 1. It is important to ensure that the scanning system can be turned off in one way or an other, in order to perform static measurements. Sometimes, this can be done by setting the scanning range to 0.

2.

A system with various available pinhole sizes will be useful is case of signal concerns (the larger the pinhole, the stronger the signal).

3.

A high numerical aperture objective (typically NA=1.2) is preferable, because it ensures the largest collection efficiency and therefore the largest CRM, in addition to the highest spatial resolution.

4. Laser excitation power will need to be measured in order to provide controlled and producible operating conditions. Photodiode power meters are usually the best choice for low power measurement, i. e., below a few mW.)

5. Because CLSM detection usually relies on photomultiplier tubes in analog mode, the detector must be replaced by one (or two, see Note 2) photon counting module(s) such as avalanche photodiodes (APD) (for instance, SPCM-AQR series by PerkinElmer Optoelectronics, PDM series by Micro-Photon-Devices, or id100 series by id Quantique). The key parameter to be considered here is quantum efficiency, because this directly affects the CRM.

6. The correlation function can be built up either by using a digital hardware correlator (by correlator.com or ALV-GmbH), or by software correlation. Although it means an additional cost, the first approach is the most easy to implement and also the most secure concerning numerical error and/or artifacts. In this case, the signal form the detectors is correlated on-line, and the ACF can be monitored in real time. Such correlator can work either in linear mode or in multi-tau mode, the later offering a wider range of accessible fluctuation times, thanks to channels of increasing width in a quasi logarithmic progression. Alternatively, correlation can be performed afterwards on the intensity series, usually in multi-tau or equivalent computation scheme. The major drawback of this approach is the huge volume of data that needs to be stored and the often prohibitive calculation time. For this reason, it is only relevant for slow processes (fluctuation time larger than 1 ms). Even if the correlation is built up on-line, we recommend to always record (with a very low time sampling, typically 100 ms), the time series, for further analysis of stationarity.

7. Last but no least, FCS data analysis strongly relies on advanced fitting procedures. The work flow will be much more effective by using data analysis software offering powerful fitting algorithms, such as Levenberg-Marquardt, as well as a complete macro programming language, such as Origin, Igor Pro, Matlab, etc.

Labeling

When choosing the fluorescent label to be used on the biological system you want to investigate, remember that the golden rule is "to maximize the CRM". Therefore, 1. Select robust dyes, labels, fluorescent proteins with the highest quantum efficiency, 2. Check carefully the filter set in order to match the spectral features of the label, 3. Always prefer labels that have been extensively documented in the literature. Even if they are not the subject of your investigations, photophysical properties such as conformal changes, triplet blinking, may be visible on the ACF, and an a priori knowledge will by a real plus when fitting the data. Diffusion coefficient of the molecule in solution is required if the label is used to calibrate the measurement volume of your FCS system.

Calibration solution

The calibration solution will be used in order to determine the measurement volume of the system.

1. Choose a solution of a robust dye those fluorescence can be measured with the same laser/filter configuration as the one that labels your biological system.

2. Choose a dye those diffusion coefficient D cal is documented (8).

3. Feel free to use the same label as for your biological system, if you think it is stable and bright enough.

4. First prepare a 1-mL aliquot of high concentration in ultrapure water, typically 1 mM. This will constitute the stock solution for future dilutions. Keep it stored at -18 • C.

5. Prepare from the stock solution four successive dilutions in 1-mL aliquots with typical concentrations 10 µM, 1 µM, 100 nM, 10 nM. Store at +4 • C in the dark.

3 Methods

Coarse optical alignment

Note that although CLSM can produce relative clear images even in case of a moderate misalignment, FCS is optically much more demanding and requires

• a perfect centering of the excitation laser on the back aperture of the microscope objective,

• a perfect conjugation of the pinhole with the excitation spot.

In case of a commercial FCS system or of a FCS-upgraded CLSM, some of the alignment settings may not be accessible to the user, and therefore require maintenance by the company. Note that this procedure needs to be followed only if the system has not been used for several weeks or if an important modification has been made on the optical system (laser replacement, filter change, etc.).

1. Switch the system on, with the correct excitation sources, microscope objective and filter set.

2. Select the largest pinhole diameter.

3. Stop the beam with an appropriate beam stopper. Set laser power to a moderate level. This should be a few hundreds µW for a onephoton fluorescence process and a few tens of mW for a two-photon fluorescence process.

4. Put a droplet of the appropriate immersion liquid on the objective.

5. Place a standard coverslip on the sample stage.

6. Put about 50 µL of the most concentrated calibration solution (10 µM).

7. Release the laser beam stopper.

8. Move the focus until the laser spot is located inside the solution. You should see the fluorescence spot when looking at the sample, your eyes been protected from laser radiation by goggles. Alternatively, remove the sample holder, and place the coverslip directly on the objective, the capillary effect of the immersion should maintain it horizontally. This ensures that the observation volume is into the solution).

9. Switch on the detector. Run a static intensity measurement. Reduce laser excitation power if the detector is saturating.

10. Optimize the signal by adjusting the pinhole alignment. Reduce laser excitation power if the detector is saturating.

11. Repeat optimization with the next dilution (1 µM).

12. Repeat optimization with a smaller pinhole diameter.

Calibration of the measurement volume

This procedure has to be followed before any series of measurements, in order to prevent any slight misalignment. In addition, it allows to determine the values of w xy and w z for the system. Steps 12-13 allow to set the best measurement conditions for a given solution on the system. It does not need to be repeated daily.

1. Switch system on, with the appropriate excitation sources, microscope objective and filter set.

2. Select the largest pinhole diameter.

3. Set laser power to a moderate level. Stop the beam with a an appropriate beam stopper.

4. Put a droplet of the appropriate immersion liquid on the objective.

5. Place a standard coverslip on the sample stage.

6. Put about 50 µL of the 100-nM calibration solution.

7. Release the laser beam stopper.

8. Move the focus until the laser spot is located inside the solution.

9. Record a ACF for T tot = 30 s, and measure simultaneously the average value of the fluorescence intensity I . If ACF is too noisy, use a larger measurement duration.

10. Fit the obtained ACF with analytical expression of Eq. 6, in order to obtain values N , τ D , as well as T and τ T , if necessary. Usually, s can be fixed to s = 0.2 for the first fitting iterations.

11. From I and N , calculate the count rate per molecule CRM = I / N .

12. Repeat steps 9 to 11 for various sizes of pinhole (see Notes 3 and 4) and various excitation powers P exc .

13. Plot variations of CRM and τ D versus P exc . The range of excitation power for which CRM has a linear dependence upon P exc and τ D does not depends on P exc is the one which suits to this calibration solution on your system. In practive, use a power 20% below the maximum suitable excitation power.

14. The lateral width of the measurement volume is given by

w xy = √ 4D ref τ D ,
where τ D is measured in the correct range of excitation power.

An example of FCS performed in a 30 nM solution of Rhodamine 6G in water [START_REF] Ferrand | A versatile dual spot laser scanning confocal microscopy system for advanced fluorescence correlation spectroscopy analysis in living cell[END_REF], under a typical excitation power of 200 µW is plotted in Fig. 3, as well as the corresponding fit, according to Eqs. 6, 7 and 8. Note that background signal was negligible and therefore B(τ ) was assumed to be unity. Given a value D ref = 280 µm 2 /s, measurement volume here is charactarized by w xy = 179 nm and w xy = 895 nm.

FCS measurements

1. Place your sample on the microscope.

2. Set the excitation power to a moderate level.

3. In the case of a cell or other specific sample, use the LSCM capabilities of the system in order to set the correct focus and set the FCS measurement area (by moving the sample, or by moving the measurement spot, depending on the system in use).

1.6 5. Record a set of short correlogram, as well as the corresponding intensity series.

6. Repeat the measurement on equivalent locations, on another part of the sample.

FCS data analysis

1. Browse all correlograms together with the corresponding intensity series that have been recorded at one location.

2. Discard all correlograms whose intensity series are not stationary (see Note 6). This often happens in complex samples such as cells because of membrane fluctuations, dye aggregation, etc.

1. FCS is not limited to one-photon absorption fluorescence, but has been successfully extended to non-linear optical contrasts, such as twophoton absorption fluorescence, etc. From an instrumental point of view, because non-linear microscopy systems are free of pinhole, detection relies usually on large area photomultiplier tubes. It is therefore subject to parasitic light, in addition to a lower quantum efficiency.

In practice two-photon FCS is subject to strong photobleaching and requires significantly higher incident power (tens of mW) than onephoton FCS (hundred of µW). Nevertheless it has several advantages like being able to excite several fluorophores with the same wavelength; it doesn't require pinhole, it has a potential deeper penetration depth in scattering samples.

2. Detectors can be subject to measurement artifacts (dead time, after pulse, etc.), that compromise the ACF for short delays (typically for τ < 1 µs). In case values at short delays are necessary for the process under study (photophysics for instance), this random noise can be efficiently rejecting by splitting the signal on two detectors, and by cross-correlating the signals obtained by the two detectors. The system illustrated in Fig. 2 is set up in this configuration.

3. Pinhole diameter is directly related to the axial extent of the PSF. Although this feature is crucial for CLSM imaging, because it determines the axial sectioning capability of the system, it can often be reasonably sacrificed while doing FCS measurement, a larger pinhole diameter providing a significantly larger CRM, and therefore a larger SNR. Indeed, enlarging the pinhole diameter produces only limited modifications of the diffusion time τ D . This is because w xy (transverse width of observation volume) is defined by the excitation beam rather than the collection optics (that defined w z ). 4. When repeating measurements with new samples, you may experience a sudden degradation of CRM and/or diffusion time, these two figures being of course much more clearly visible when measuring solutions. This is often due to a defect of the coverslip, that is sometimes outside the nominal range indicated by the manufacturer. In this case, it is usually more effective to discard the coverslip and to start a new measurement than to try to adjust the correction collar of the microscope objective. Another common problem is the drying of water between the coverslip and a water-objective lens.

Problem

5. Although FCS can address fluctuation processes at all timescales (within the limitations mentioned in section 1.3, of course), slow diffusion cannot be measured in practice, because slow species are usually bleached before leaving the measurement volume. This produces a typical drop of the recorded intensity, as illustrated in Fig. 4-D. In this case, scanning FCS approaches such as image correlation spectroscopy and its variants may be more relevant (see Note 7).

6. Fig. 4 illustrates some examples of odd-looking correlograms that constitute the usual first taste of FCS for new users and that may be quite discouraging. Possible explanations and suggestions are given for each case. Measurements like cases C and D must be definitely discarded, because they do not satisfy to the condition of stationarity required by the FCS analysis. Although these correlograms may in some cases be successfully fitted, they would produce erroneous values. More generally, these examples emphasize that FCS analysis must be operated by experienced users only, especially for measurements carried out in complex systems such as cells, where the diffusion is mostly non Brownian and the signal subjected to other spurious fluctuations.

7. Latest development of FCS focus both on hardware and software. From the hardware side the current trends are (i) dual spot measurement to measure FCS simultaneously at various locations [START_REF] Ferrand | A versatile dual spot laser scanning confocal microscopy system for advanced fluorescence correlation spectroscopy analysis in living cell[END_REF] and/or to access to absolute diffusion coefficient ( 8), (ii) multispot FCS using EMCCD cameras as detector [START_REF] Burkhardt | Electron multiplying CCD based detection for spatially resolved fluorescence correlation spectroscopy[END_REF]. From the data analysis side, recent development focus on the improvement of spatio-temporal information by means of scanning FCS [START_REF] Petrášek | Characterization of protein dynamics in asymmetric cell division by scanning fluorescence correlation spectroscopy[END_REF] and Image correlation spectroscopy techniques (12) that give access to both temporal and spatial correlations. Another powerful variant of FCS is Fluorescence Cross Correlation Spectroscopy (FCCS), which extends the capabilities of standard FCS by introducing two different fluorescent probes with distinct excitation and/or emission properties, which can be detected in the same confocal volume. It allows to probe interactions between different molecular species [START_REF] Bacia | Fluorescence crosscorrelation spectroscopy in living cells[END_REF].

Figure 1 :

 1 Figure 1: Schematic view of a typical FCS measurement. A: Diffusion of fluorescent molecules through the observation volume of a confocal microscope (represented by the dotted circle). B: The resulting fluctuating fluorescence intensity. C: The autocorrelation function calculated from the intensity, according to Eq. 1.

Figure 2 :

 2 Figure 2: A basic FCS setup, as well as a typical correlogram as computed by the correlator. The inset shows a schematic view of the 3D PSF of the system. APD stands for avalanche photodiode.

Figure 3 : 4 .

 34 Figure 3: Example of ACF recorded in a calibration solution of Rhodamine 6G. Measurement duration was 10 s. Circle are experimental points, while solid line is the fit according to Eqs. 6, 7 and 8. Inset is the intensity series, top graph is the residuals. Fitting values: N = 1.93, T = 0.133, τ T = 4.1 µs, τ D = 29 µs, and s = 0.2.

--Figure 4 :

 4 Figure 4: A non exhaustive panel of unsuitable correlograms, with the corresponding intensity series plotted in the inset. For each case the problem is detailed, diagnosis and possible solutions are given.


Julien Duboisset
email: julien.duboisset@fresnel.fr

Pascal Berto

Paulina Gasecka

Fatma-Zohra Bioud

Patrick Ferrand

Hervé Rigneault

Sophie Brasselet

Anti-Stokes Raman Scattering (CARS) and Stimulated Raman Scattering (SRS) Microscopy: A Spectral Comparative Study

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

Being able to report high level of detail in the structural molecular organization using optical microscopy is one of the prevailing goals in the field of functional imaging in biology, material sciences and molecular physics. Polarized optical microscopy is becoming a powerful tool to address molecular orientational distributions into the focal volume of a microscope. [START_REF] Brasselet | Polarization Resolved Nonlinear Microscopy: Application to Structural Molecular and Biological Imaging[END_REF] In coherent nonlinear optics, polarized second harmonic generation, [START_REF] Stoller | Polarization-Dependent Optical Second-Harmonic Imaging of Rat-Tail Tendon[END_REF][START_REF] Gusachenko | Polarization-Resolved Second-Harmonic Microscopy in Tendon Upon Mechanical Stretching[END_REF][START_REF] Duboisset | Generic Model of the Molecular Orientational Distribution Probed by Polarization Resolved Second Harmonic Generation[END_REF] polarized third harmonic generation [START_REF] Zimmerley | Probing Ordered Lipid Assemblies with Polarized Third-Harmonic-Generation Microscopy[END_REF] and polarized four wave mixing [START_REF] Munhoz | Polarization-Resolved Four-Wave Mixing for Structural Imaging in Thick Tissues[END_REF][START_REF] Bioud | Structure of Molecular Packing Probed by Polarization-Resolved Nonlinear Four-Wave Mixing and Coherent Anti-Stokes Raman-Scattering Microscopy[END_REF] have already been used to recover orientational information on endogeneous proteins and lipids in biological tissues. In addition to the orientational information, Coherent Raman Scattering (CRS) processes are sensitive to molecular bond vibrations, allowing chemical specificity without the need of fluorescence labelling. [START_REF] Cheng | Ordering of Water Molecules Between Phospholipid Bilayers Visualized by Coherent Anti-Stokes Raman Scattering Microscopy[END_REF][START_REF] Zimmerley | Molecular Orientation in Dry and Hydrated Cellulose Fibers: A Coherent Anti-Stokes Raman Scattering Microscopy Study[END_REF][START_REF] Fu | Ex Vivo and in Vivo Imaging of Myelin Fibers in Mouse Brain by Coherent Anti-Stokes Raman Scattering Microscopy[END_REF] Coherent anti-Stokes Raman scattering (CARS) microscopy is based on the interaction of two fields called pump (ω p ) and Stokes (ω s ). [START_REF] Zumbusch | Three-Dimensional Vibrational Imaging by Coherent Anti-Stokes Raman Scattering[END_REF] When their frequency difference matches a vibrational resonance (Ω R ), a blue-shifted field called anti-Stokes (ω as = 2ω p -ω s ) is generated. In this case, the intensity recovered at ω as reveals the presence of specific bonds but is quadratically dependent on the nonlinear susceptibility and can interfere with a spurious nonresonant background. [START_REF] Wang | Coherent Anti-Stokes Raman Scattering Imaging of Axonal Myelin in Live Spinal Tissues[END_REF] In recent years, the implementation of another CRS process called Stimulated Raman Scattering (SRS), has become an increasingly used tool for labelfree microscopy. [START_REF] Freudiger | Label-Free Biomedical Imaging with High Sensitivity by Stimulated Raman Scattering Microscopy[END_REF][START_REF] Saar | Video-Rate Molecular Imaging in Vivo with Stimulated Raman Scattering[END_REF] Simultaneously to the generation of an anti-Stokes photon, a gain on the Stokes beam (stimulated Raman gain, SRG) and a loss on the pump beam (stimulated Raman loss, SRL) appears when the frequency difference ω p -ω s matches Ω R . To detect the SRG signal (resp. SRL signal), the pump beam (resp. Stokes beam) is modulated at high frequency, typically in the MHz range, and the gain on the Stokes beam (resp. the loss on the pump beam) is measured by means of a lock-in amplifier. SRS is a background free process which allows to recover directly pure molecular Raman spectra. Even though CARS and SRS techniques are sensitive to the same Raman susceptibility tensor, the information recovered by both techniques differ by their underlying physical mechanism.

to retrieve molecular orientational information in organized lipid membranes, based on symmetry orders decomposition of the lipid orientational distribution function. [START_REF] Bioud | Structure of Molecular Packing Probed by Polarization-Resolved Nonlinear Four-Wave Mixing and Coherent Anti-Stokes Raman-Scattering Microscopy[END_REF] The information retrieved however neglected the non resonant background intrinsic to the CARS signal because of the high contrast obtained on the considered resonance. This nonresonant contribution is known however to perturb the interpretation of polarized signals in more general cases. [START_REF] Munhoz | High Order Symmetry Structural Properties of Vibrational Resonances Using Multiple-Field Polarization Coherent Anti-Stokes Raman Spectroscopy Microscopy[END_REF] On the other hand, SRS polarized microscopy studies have been introduced so far only in isotropic media. [START_REF] Munhoz | Polarization-Resolved Four-Wave Mixing for Structural Imaging in Thick Tissues[END_REF] A unified approach for both contrasts, encompassing the whole characteristics of both of them, has not yet been developed.

Here we propose to investigate the nature of polarization resolved CARS and SRS signals, and in particular their respective capacity to disentangle molecular orientational information in the focal volume of a microscope. We introduce an exhaustive model to analyze polarization resolved SRS signals, similarly to what has been already acheived for CARS. [START_REF] Bioud | Structure of Molecular Packing Probed by Polarization-Resolved Nonlinear Four-Wave Mixing and Coherent Anti-Stokes Raman-Scattering Microscopy[END_REF] We illustrate the method on multilamellar vesicles to extract lipid orientational organization.

Finally, we compare the CARS and SRS polarization resolved methods by extending the analysis to the frequency domain, addressing more specifically details in the vibrational lipid bond spectral region.

Macroscopic nonlinear susceptibility

The resonant third order nonlinear susceptibility χ of vibrational mode is built-up based on the sum of individual nonlinear dipoles, describing the molecular bonds contributing to this vibrational mode. [START_REF] Zimmerley | Probing Ordered Lipid Assemblies with Polarized Third-Harmonic-Generation Microscopy[END_REF][START_REF] Davis | Selection Rules and Symmetry Relation for Four-Wave Mixing Measurements of Uniaxial Assemblies[END_REF] Each of these nonlinear dipoles, defined by their direction (θ, ϕ) in the macroscopic laboratory frame, are characterized by a nonlinear susceptibility γ in the microscopic frame. In the case of CH 2 vibrational bonds studied below, only the γ zzzz is assumed non-zero . [START_REF] Zimmerley | Probing Ordered Lipid Assemblies with Polarized Third-Harmonic-Generation Microscopy[END_REF] For more complex vibrational bonds, all the components of the susceptibility tensor have to be taken into account. [START_REF] Davis | Selection Rules and Symmetry Relation for Four-Wave Mixing Measurements of Uniaxial Assemblies[END_REF] In the laboratory frame, the probability to find individual nonlinear dipoles with an orientation θ and ϕ is defined by a function f (θ, ϕ). The macroscopic nonlinear susceptibility tensor in the laboratory frame is defined as the summation over individual dipoles probed in the focal volume

a) b) c) d) Y X j e) f) X Y Z Z Y X + + j q
χ IJKL = N π 0 2π 0 f (θ, ϕ)(I.z)(J.z)(K.z)(L.z)γ sin θdθdϕ, (1)
where N is the molecular bonds density, (I.z) is the projection of the direction I along z, γ is the microscopic tensor of one elementary molecular bond, see Fig. 1.

Due to the electromagnetic fields which are lying in the XY sample plane, only the X and Y components of the macroscopic tensor are probed. The out-of plane contributions of the collection of dipoles in the focal volume will consequently contribute far less than the in-plane ones in the θ orientational average. From Eq. 1, the in-plane components of the macroscopic tensor are

χ XXXX = N γ 2π 0 p(ϕ) cos 4 (ϕ)dϕ χ Y Y Y Y = N γ 2π 0 p(ϕ) sin 4 (ϕ)dϕ χ XXY Y = N γ 2π 0 p(ϕ) cos 2 (ϕ) sin 2 (ϕ)dϕ χ XY Y Y = N γ 2π 0 p(ϕ) cos(ϕ) sin 3 (ϕ)dϕ χ Y XXX = N γ 2π 0 p(ϕ) cos 3 (ϕ) sin(ϕ)dϕ, (2) with p(ϕ) = π 0 sin 4 θf (ϕ, θ) sin θdθ, (3)
where the term sin 4 (θ) weights the contribution of molecular bonds depending on their tilt angle θ. Without loss of generality, the molecular bonds distribution in the XY plane can be described by a function of ϕ. Moreover, Eq. 2 clearly show that the tensor components are sensitive to the even part of the p(ϕ) function. Furthermore, decomposing p(ϕ) onto its orders of symmetry leads to only three non vanishing contributions of this distribution function to the third order susceptibility: its isotropic contribution as well as its second and fourth orders. Thereby, the 2D distribution function of Eq. 3 can be expanded as a truncated Fourier series limited to even order terms, see Fig. 1e-f, and modeled by 7 p(ϕ) = p 0 + p 2 cos(2ϕ) + q 2 sin(2ϕ) +p 4 cos(4ϕ) + q 4 sin(4ϕ).

Then, from Eq 2 and 4, the macroscopic susceptibility components in the OXY plane can be written with respect to the coefficients of the distribution function as

χ XXXX = N γπ 8 (6p 0 + 4p 2 + p 4 ) χ Y Y Y Y = N γπ 8 (6p 0 -4p 2 + p 4 ) χ XXY Y = N γπ 8 (2p 0 -p 4 ) χ XY Y Y = N γπ 8 (2q 2 -q 4 ) χ Y XXX = N γπ 8 (2q 2 + q 4 ). (5)
The in-plane tensor components are clearly dependent on the matter organization described by the five distribution function parameters p 0 , p 2 , q 2 , p 4 and q 4 . Henceforth, the CARS and SRS intensities have to be measured using different sets of polarization to recover these parameters, as described in the next section.

Polarization-resolved SRS intensity

The stimulated Raman gain (SRG) process is based on the interference between the electric field induced by the nonlinear dipole at ω s and the incoming field at ω s . [START_REF] Bloembergen | The Stimulated Raman Effect[END_REF][START_REF] Freudiger | Label-Free Biomedical Imaging with High Sensitivity by Stimulated Raman Scattering Microscopy[END_REF] Due to this heterodyne behavior, the gain on the Stokes beam at resonance is linear with the susceptibility, and reads 17

I SRG ∝ IJKL χ IJKL E * p,I E s,J E p,K E * s,L , (6)
where E p denotes the pump electric field, E s the Stokes electric field, and χ the resonant susceptibility tensor as previously discussed. Both incident fields E p and E s are linearly polarized. Denoting α the angle of E s and E p with respect to the X axis (see inset Fig 2), the electric fields can be expressed in the macroscopic frame as

E X = E cos α and E Y = E sin α.
Then Eq 6 becomes

I SRG (α) ∝ χ XXXX cos 4 α + 4χ XXXY cos 3 α sin α +6χ XXY Y cos 2 α sin 2 α +4χ XY Y Y cos α sin 3 α + χ Y Y Y Y sin 4 α. (7)
This equation can be rearranged as

I SRG (α) ∝ a 0 +a 2 cos(2α) + b 2 sin(2α) +a 4 cos(4α) + b 4 sin(4α), (8)
where

a 0 = 3π 4 p 0 a 2 = π 2 p 2 a 4 = π 8 p 4 b 2 = π 2 q 2 b 4 = π 8 q 4 . (9)
The polarization-resolved SRS intensity is thus described as a truncated Fourier series where the Fourier coefficients are linearly related to the distribution function parameters. This way, the polarization-resolved SRS signal allows one to retrieve directly how the unitary dipoles, representing the molecular bonds, are oriented and distributed into the focal volume.

Polarization-resolved CARS Intensity Exact calculation

The CARS dipole along the direction I (X or Y ), expressed as a function of the nonlinear susceptibility and the electric fields, is

P I (ω as ) = JKL χ IJKL E * s,J E p,K E p,L . (10)
The CARS intensity behind a polarizer oriented along X is thus proportional to the square of the CARS dipole which leads to

I CARS X ∝ JKL M N O χ XJKL χ * XM N O E * J E K E L E M E * N E * O . (11)
The total intensity is thus the summation of the intensity along X plus the intensity along

Y , I CARS = I CARS X + I CARS Y . (12)
Both incident E p and E s are linearly polarized and rotated in the OXY plane in the polarization resolved SRS operating mode. I CARS is rewritten as a truncated Fourier series where the angular frequencies are going up to the sixth order due to the presence of six electric fields in the intensity relationship. [START_REF] Bioud | Structure of Molecular Packing Probed by Polarization-Resolved Nonlinear Four-Wave Mixing and Coherent Anti-Stokes Raman-Scattering Microscopy[END_REF][START_REF] Vito | Rotating-Polarization CARS microscopy: Combining Chemical and Molecular Orientation Sensitivity[END_REF] In this case,

I CARS (α) ∝ a 0 +a 2 cos(2α) + b 2 sin(2α) +a 4 cos(4α) + b 4 sin(4α) +a 6 cos(6α) + b 6 sin(6α), (13)
where

a 0 = 1 64 (36p 2 0 + 10p 2 2 + 10q 2 2 + p 2 4 + q 2 4 ) a 2 = 1 32 (24p 0 p 2 + 3p 2 p 4 + 3q 2 q 4 ) a 4 = 3 32 (p 2 2 -q 2 2 + 2p 0 p 4 ) a 6 = 1 32 (p 2 p 4 -q 2 q 4 ) b 2 = 1 32 (p 0 q 2 + 3p 2 q 4 -3p 4 q 2 ) b 4 = 3 16 (2p 0 q 4 + p 2 q 2 ) b 6 = 1 32 (p 2 q 4 + p 4 q 2 ). (14)
In the polarization-resolved CARS signal, the retrieval of the distribution function from the Fourier coefficients is therefore not straightforward. The nonlinear dependence of the Fourier coefficients with respect to the distribution function parameters is due to the nonlinear dependence of the CARS signal with respect to the susceptibility tensor, which is not the case in SRS. To estimate the distribution function parameters, an iterative algorithm can be employed to find the right set of parameters. 7

Approximate calculation

The nonlinear dependence between the polarization-resolved CARS intensity and the distribution function leads in practice to a complex and time consuming data processing. Nevertheless, an approximation can be done in order to simplify and decrease the time of analysis.

As the CARS intensity is quadratically dependent on the susceptibility, it can be written as the square of a truncated Fourier series

ĨCARS (α) ∝ [a 0 + a 2 cos(2α) + b 2 sin(2α) +a 4 cos(4α) + b 4 sin(4α)] 2 . ( 15
)
When Eq. 15 is developed, the parameters a 0 , a 2 , a 4 , b 2 and b 4 can be expressed as functions of a 0 , a 2 , a 4 , b 2 and b 4 ,

a 0 = a 2 0 + a 2 2 /2 + b 2 2 /2 + a 2 4 + b 2 4 a 2 = 2a 0 a 2 + a 2 a 4 + b 2 b 4 a 4 = a 2 2 /2 -b 2 2 /2 + 2a 0 a 4 a 6 = a 2 a 4 -b 2 b 4 b 2 = 2a 0 b 2 + a 2 b 4 -a 4 b 2 b 4 = 2a 0 b 4 + a 2 b 2 b 6 = a 2 b 4 + a 4 b 2 . (16)
Eq. 16 shows a clear similarity to Eq. 14. From this, an approximate relationship can be found between the parameters a n , b n and the parameters p n , q n ,

a 0 ≈ 3π 4 p 0 a 2 ≈ π 2 p 2 a 4 ≈ π 8 p 4 b 2 ≈ π 2 q 2 b 4 ≈ π 8 q 4 . (17)
In a first approximation, the square root of the polarization-resolved CARS signal is very similar the polarization-resolved SRS signal. A numerical comparison was done between the exact and the approximate expressions of the CARS signal, and the two outcomes were seen to be identical within 1% whatever the set of a n , b n parameters considered. The negligible difference comes from the eighth frequency present in Ĩ, whereas I contains up to the 6th order frequency in its α dependence. In what follows, the data analysis uses the approximate calculation.

Data processing

CARS and SRS polarimetric stacks are obtained by measuring a sequence of CARS and SRS images for different polarization angles α between 0 and 175 o with a 5 o step. Consequently, I CARS (α) and I SRS (α) can be easily extracted on each pixel. According to Eq. 9 related to SRS and Eq. 17 related to CARS, the distribution function parameters p 0 , p 2 , q 2 , p 4 , q 4 are determined on each pixel. From the distribution function parameters, four parameters can be introduced to quantify the symmetry order magnitude and the main orientation of the distribution function. First, we introduce S 2 and S 4 to quantify the order of symmetry regardless of the orientation of the molecular bonds 7

S 2 = 1 p 0 p 2 2 + q 2 2 S 4 = 1 p 0 p 2 4 + q 2 4 . (18)
10 Moreover, we introduce the angles ϕ 2 and ϕ 4 to describe respectively the order 2 and the order 4 orientations of the distribution function in the sample plane

ϕ 2 =        1 2 arctan(q 2 /p 2 ), if p 2 > 0 1 2 arctan(q 2 /p 2 ) + π 2 , else (19)
ϕ 4 =        1 4 arctan(q 4 /p 4 ), if p 4 > 0 1 4 arctan(q 4 /p 4 ) + π 4 , else . ( 20
)
These four independent parameters play distinct roles in the description of the function distribution. The angle ϕ 2 reports on the main orientation of the distribution function. This can be emphasized by writing all angular dependences relative to ϕ 2 , 19

S (s) 4 = S 4 cos(4(ϕ 4 -ϕ 2 )) (21) S (a) 4
= S 4 sin(4(ϕ 4 -ϕ 2 )).

S (s) 4

describes the magnitude of the "symmetric" part of the order 4 contribution to the distribution function p(ϕ), meaning that the symmetric order four is either aligned or at π/4 with ϕ 2 . The S (a)

4 parameter is the "antisymmetric" order four, reporting a skewness in the distribution function. [START_REF] Ferrand | Ultimate Use of Two-Photon Fluorescence Microscopy to Map Orientational Behavior of Fluorophores[END_REF] The SRS and CARS polarization-resolved intensities are finally expressed as

I SRS (α) I CARS (α)            ∝ 1+ 2 3 S 2 cos(2(α -ϕ 2 )) + 1 6 S (s) 4 cos(4(α -ϕ 2 )) + 1 6 S (a) 4 sin(4(α -ϕ 2 )), ( 23
)
where it is visible that a simple Fourier analysis of the polar-resolved intensities allows one 11 to retrieve the distribution function parameters S 2 , S

4 , S (as) 4

and ϕ 2 .

Experiments

Multilamellar vesicles (MLVs) were made from DPPC lipids (1,2-dipalmitoyl-sn-glycero-3phosphocholine). Thin films, made of pure DPPC were hydrated in phosphate-buffered saline (PBS, pH 7.4) for 1 hour above the main phase transition temperature (45 o C) leading to MLVs of 1-30 µm size according to the procedure described in the literature. [START_REF] Bangham | Preparation and Use of Liposomes as Models of Biological Membranes[END_REF] The experimental setup is sketched in Fig. 2. A modelocked Nd:YVO4 MOPA laser (Plecter, Coherent, 80 MHz repetition rate) delivers synchronous 7 ps pulses at λ S = 1064 nm (used as Stokes beam) and λ = 532 nm. This frequency doubled beam is used to pump an OPO (APE Levante OPO) which generates the tunable pump/probe pulse with a maximal tuning range of λ P =730-960 nm. The two beams are overlapped in time and space and sent into a custom made scanning microscope. 21,[START_REF] Ferrand | Scanning Imaging: a General Purpose Virtual Instrument for the Control of High-Speed Data Acquisition Devices[END_REF] In the CARS operation mode the anti-Stokes signal is filtered and detected, in the forward direction, with an photomultiplier tube (R9110, Hamamatsu) coupled with a preamplifier (C7950, Hamamatsu). In the SRS operation mode, the pump beam is modulated in amplitude at the frequency of 20 MHz thanks to an acoustooptic modulator (AOM -AAOptoelectronic MT200 A0,2-800). The stimulated Raman gain (SRG) induced on the Stokes beam is detected in the forward direction by means of a high speed photodiode and a fast lock-in amplifier manufactured by APE. [START_REF] Saar | Video-Rate Molecular Imaging in Vivo with Stimulated Raman Scattering[END_REF] The collection objective lens (NA=0.6, LUCPlanFLN 40x, Olympus) has a larger numerical aperture than the excitation one (NA=0.45, LCLPLN 20x IR, Olympus) to minimize artifacts induced by cross phase modulation and thermal lensing. [START_REF] Berto | Background-Free Stimulated Raman Spectroscopy and Microscopy[END_REF] For CARS and SRS measurements, the power of the Stokes and pump beams in the sample plane was set respectively to 28 mW and 39.5 mW. Each spectral image results from an average of 3 images, using a pixel dwell time of 100 µs. To perform polarization resolved measurements, both exciting beams are linearly polarized before the excitation objective, and an achromatic half-waveplate mounted in a step rotation motor allows rotation of the incident pump and Stokes polarizations at the same time. In this configuration, polarization distortions introduced by the experimental setup, such as ellipticity, were measured as negligible.

180 90 0 α(°) α(°) ν=2836 cm -1 i) DPPC H O O O P O O O N + H H H H H H H H H H H H H H H H O O H H H H H H H H H H H H H H H H a) c) f) g) h) -1 0 1 1 0 -1 0 1 0 1 S 4 (s)

Results

Polarization-resolved CARS and SRS intensities on multilamellar vesicles

The experimental results are depicted in Fig. 3. The MLVs form spherical objects made by several concentric layers where the DPPC lipids, see Fig. 3i, are radially oriented whereas the CH 2 bonds are tangentially oriented. [START_REF] Bioud | Structure of Molecular Packing Probed by Polarization-Resolved Nonlinear Four-Wave Mixing and Coherent Anti-Stokes Raman-Scattering Microscopy[END_REF][START_REF] Wurpel | Imaging Orientational Order and Lipid Density in Multilamellar Vesicles with Multiplex CARS Microscopy[END_REF] CARS and SRS polarization resolved image stacks were recorded on the same MLV, at the CH 2 stretching mode resonance. Figure 3a,d is the summation over the whole stack and shows a strong signal from the equatorial plane of the MLV. The background in the CARS image is non zero and comes from non resonant contribution of the surrounding water whereas the SRS image is background free.

The polarization-resolved intensities in the MLV are strongly modulated as function of the incident polarization angle α as being the signature of oriented nonlinear induced dipoles, whereas the surrounding background shows no polarization dependence, as expected from an isotropic medium.

From the analysis of the polarization-resolved intensities previously discussed, the distribution function parameters are measured on each pixel (Eq 23). Superimposed to the black and white intensity image, a stick is added on each pixel whose orientation is given by ϕ 2 and whose color follows the S 2 value scale (Fig. 4b,e). The ϕ 2 orientation within the MLV shows that the CH 2 bonds are found to be oriented tangentially with respect to the MLV circular contour, as expected from the lipids geometry in a concentric arrangement and the arrangement of the CH 2 symmetric stretching bonds along the lipid chains. The We observe that the S (s) 4 values do not surpass 0.25, showing the absence of high order in the distribution function as reported in Ref. [START_REF] Bioud | Structure of Molecular Packing Probed by Polarization-Resolved Nonlinear Four-Wave Mixing and Coherent Anti-Stokes Raman-Scattering Microscopy[END_REF]

Discussion

Even though the CARS and SRS processes are different, both techniques are expected to lead to the same molecular orientational distribution. However, as seen in Fig. 3 and4, a clear difference is visible on the S 2 values measured in MLV, which are seen to be systematically higher in the SRS operating mode. Indeed, the assumption made in the theoretical part related to the CARS operating mode does not take into account the other contributions from the non resonant part. Indeed, the CARS intensity spectrum shows a non resonant contribution that can disturb the interpretation of the values obtained by the polarizationresolved technique.

In order to understand the difference observed with SRS and CARS, a model was de-

χ = χ N R + χ R (Ω) (24)
The total susceptibility read by the CARS process is then the susceptibility read by the SRS process plus an unknown non resonant contribution. If an isotropic non resonant background (S N R 2 = 0) is present, the polarization-resolved CARS intensity modulation is expected to decrease as a function of the magnitude of the non resonant part. Furthermore, if the non resonant background is itself organized (S N R 2 = 0), the interpretation can be even more difficult. [START_REF] Munhoz | High Order Symmetry Structural Properties of Vibrational Resonances Using Multiple-Field Polarization Coherent Anti-Stokes Raman Spectroscopy Microscopy[END_REF] In our simple model, a resonant susceptibility χ R is assumed to originate from an anisotropic distribution of dipoles for which S R 2 = 0.7 (as obtained from SRS measurements on MLVs in the CH2 symetric stretching bond region), and S R 4 = 0. The corresponding χ R susceptibility is calculated using Eq. 2. A non resonant susceptibility χ N R is then added to this resonant part in order to build the whole susceptibility χ read-out by the CARS contrast.

χ N R is assumed to originate from various natures of symmetry orders, from totally isotropic = 0) to highly organized (S N R 2 = 0.7), with S N R 4 = 0. These non resonant second orders are moreover assumed to lie perpendicularly to the resonant contributions (ϕ R 2 ⊥ϕ N R 2 ), consistent with the fact that non resonant signal comes mainly from electronic susceptibilities along the lipid tail, while the resonant CH 2 stretching bond orients perpendicularly to this tail. [START_REF] Bioud | Structure of Molecular Packing Probed by Polarization-Resolved Nonlinear Four-Wave Mixing and Coherent Anti-Stokes Raman-Scattering Microscopy[END_REF] From the resulting simulated polarization-dependent CARS intensity (Eq. 11), the "effective" S ef f 2 values are deduced. These values are depicted in Fig. 5 for various strengths of non resonant with respect to resonant susceptibilities, and various symmetries of the non resonant background. Figure 5 shows that S ef f 2 decreases with an increasing strength of the non resonant contribution, emphasizing the possible sources of bias induced by a non negligible non resonant background. In such case, the resulting read-out distribution, encompassing both resonant and non resonant contributions, can indeed become close to an apparent isotropic distribution (S ef f 2 = 0) in cases where S R 2 = S N R 2 .

Our experimental results described above have then been confronted to this theoretical model. The CARS non resonant background intensity is seen to be about 30% of the maximum resonant intensity, which corresponds roughly to χ N R /χ R ≈ √ 0.3 ≈ 0.55. The S N R 2 values obtained in the non resonant region are within the range [0.3-0.5] (Fig. 4), which is in agreement with previously reported values in similar MLVs . [START_REF] Bioud | Structure of Molecular Packing Probed by Polarization-Resolved Nonlinear Four-Wave Mixing and Coherent Anti-Stokes Raman-Scattering Microscopy[END_REF] Similarly to what is assumed in the model described above, it has been furthermore experimentally reported that the non resonant part in lipids is dominated by the nonlinear susceptibility of carbon-carbon bonds. [START_REF] Zimmerley | Probing Ordered Lipid Assemblies with Polarized Third-Harmonic-Generation Microscopy[END_REF] The mean direction of the C -C bonds are oriented at a right angle of the CH 2 bonds in MLV, supporting the assumption that ϕ R 2 and ϕ N R suring a reliable orientational information can be nevertheless obtained providing that the resonant strength is large enough with respect to the non resonant background. In the case of SRS, this issued does not occur providing that sources of artifact have been removed. [START_REF] Berto | Background-Free Stimulated Raman Spectroscopy and Microscopy[END_REF]

Conclusion

A polarization-resolved method was performed in order to read out the parameters of molecular orientational organization using CRS microscopy. We found that SRS and CARS are not sensitive to the same degree of organization due to their intrinsic properties. SRS provides reliable information about the molecular bonds organization, whereas CARS can be biased by a non resonant contribution. In the case of lipids organization in MLV, the non resonant contribution is known and can be incorporated to the CARS analysis in order to explain the differences observed. More generally, the method described here to analyze polarization-resolved SRS and CARS intensity will allow new, fast and accurate observation of the molecular organization in biological samples. † These authors contributed equally to this work.

Figure 1 :

 1 Figure 1: a) Assemblies of lipids into the focal volume. b) Orientation of nonlinear induced dipoles, assumed to lie along the C -H bonds probed, defined in the laboratory frame XY Z c) 3D nonlinear dipoles distribution defined into the focal volume f (θ, ϕ). d) 3D nonlinear dipoles distribution which contributes to the signal. e) p(ϕ) distribution function. f) Decomposition of the distribution function p(ϕ) on the spherical function basis. (Color online)

Figure 2 :

 2 Figure 2: Description of the polarization-resolved CRS microscope setup. The linear polarization orientation of both exciting beams is controlled by a motorized achromatic half wave plate placed before the excitation objective. AOM: Acousto-optic modulator, PD: Photodiode, PMT: Photomultiplier tube, D: Dichroic mirror, M: Metallic mirror. (Color online)

Figure 3 :

 3 Figure 3: (a) Total CARS intensity at 2836 cm -1 and polarization resolved signal in different regions of interest in the MLV. The scale bar represents 10µm. (b) The stick orientation represents ϕ 2 and the color indicates the value of S 2 . (c) Distribution function p(ϕ) for a given pixel. (d-f) Total SRS intensity at 2850 cm -1 and identical analysis as for CARS. (g-h) Plot of S 2 and S 4 values (in red for CARS, in black for SRS) for each pixel of the image within a given mask (inset Fig 3g) applied to remove all the pixels from the background. The center of the yellow cross represents the mean, and the error bar represents the standard deviation of the measured values. (i) Schematic representation of a DPPC molecule. (Color online)

S 2 Figure 4 :

 24 Figure 4: a) spectra of the MLV. b) S 2 as a function of the frequency for CARS (line) and SRS (dashed line). c) S (s) 4 as a function of the frequency for CARS (line) and SRS (dashed line). (Color online)

Fig. 4b shows the S 2

 2 Fig.4bshows the S 2 spectral dependence, obtained from a polarization analysis at each addressed wavenumber, for both CARS and SRS. The reported values are obtained from the average and standard deviation over a collection of pixels within the mask of interest depicted in Fig.3g. The values of S 2 for the CARS starts from a low value out of resonance to reach two maxima around the symmetric and asymmetric CH 2 stretching (S 2 ≈ 0.4). The S 2 values for the symmetric and asymmetric stretching are the same because the CH 2 angular distribution is identical for the two frequencies. Indeed while the microscopic nature of the

Figure 4c represents the S (s) 4

 4 Figure 4c represents the S (s) 4 values for SRS and CARS as functions of the wavenumber.

Figure 5 :

 5 Figure 5: S R 2 as function of the ratio between χ N R and χ R for different S N R 2 . The χ R is made with S R 2 = 0.7, ϕ R 2 = 0 • and S (s) 4 = 0. The χ N R is made with different values of S N R 2

2

  are perpendicular. Following Fig.5, this situation corresponds to a S ef f 2 which would lie around [0.3-0.4], which corresponds indeed to the experimental values obtained in Figs.3 and 4. This analysis shows that molecular orientational order values measured in the context of polarized CARS imaging can be easily under-estimated due to the existence of a non negligible non resonant background. Mea-

Acknowledgement

The authors acknowledge finacial support from the Centre National de la Recherche Scientifique (CNRS), Aix Marseille University A * Midex (No. ANR-11-IDEX-0001-02). This work has been supported by Contracts ReceptORIENT (No. ANR-2010-BLAN-150902), France-BioImaging (No. ANR-10-INBS-04-01), France Life Imaging (No. ANR-11-INSB-0006) and by the Conseil Regional Provence Alpes Cote dAzur Region. F.-Z.B. was supported by the ErasmusMundus Doctorate Program Europhotonics (Grant No. 159224-1-2009-1-FR-ERA MUNDUS-EMJD).

well above the background (see inset Fig. 3g). For each remaining pixel of the image, S coefficients obtained from CARS and SRS analyses. Using the obtained symmetry parameters and Eq. 4, it is possible to represent the reduced distribution function p(ϕ) read-out by polarization resolved CARS and SRS. This distribution, represented in Fig. 3c-f, shows an obvious anisotropic shape along the CH 2 vibration bond axes, with minor fourth order contributions, especially in the SRS analysis. In order to explain this difference, a finer analysis of the nature of these processes is required, in particular a spectral investigation was performed around the lipid resonance, as described in the next section.

Spectral behavior of CARS and SRS

A polarization resolved CARS and SRS analysis has been performed in the whole 2750 cm -1 to 3000 cm -1 spectral region of the lipid vibrational resonance, in the same MLV equatorial plane as described in the previous section. The total intensity depicted in Fig. 4a shows a strong signal around the lipid resonance. Three strong CARS and SRS features are distinguishable: the two bands around 2848 cm -1 and 2880 cm -1 correspond to the symmetric and asymmetric CH 2 stretching vibrations, whereas the band at 2935 cm -1 corresponds to the symmetric CH 3 stretching . 24 Fig 4a shows a typical CARS spectrum representative of a coherent superposition of a spectrally-independent non resonant signal, and a complex spectral resonance. In contrast, the SRS spectrum reproduces directly the shape of a lipid Raman spectrum. Moreover, the SRS spectrum is clearly background-free out of resonance, whereas the CARS spectrum shows a non resonant background of around 30% of the maxi-