## Abstract

In this tutorial I analyze the polarization-dependent properties of different optical contrasts widely used today in imaging, applied to biology and biomedical diagnostics. I derive the essential properties of the polarization dependence of optical processes such as two-photon fluorescence, nonlinear coherent effects in the nonresonant as well as vibrational-resonant regimes, and analyze how they can be exploited to provide information on the molecular orientational organization in a biological sample. Two examples will be detailed: the first one the measurement of lipid order in artificial and cell membranes by using fluorescent labeling, and the second one structural imaging of collagen in tissues by using second-harmonic generation.

© 2011 OSA

## 1. Introduction

Microscopy imaging has considerably evolved this past decade in order to answer more and more complex questions addressed by biologists. It is today possible to locate specific biomolecules in a cell or a tissue, *in vitro* or *in vivo*, to follow their
interaction with neighbor molecules and their environment on spatial scales down to nanometers, and to relate this information to their biological function. Microscopy imaging for biology has in particular benefited from an important development of molecular and inorganic fluorescent
nanoprobes, which provides the possibility to chemically label a protein or a lipid in order to follow its behavior by optical imaging [1]. These probes bring great flexibility in terms of label targeting, variable excitation and emission
wavelengths, and sensitivity to various properties of their local environment such as pH or cell membrane potential. Imaging techniques are also under perpetual development and benefit from ingenious inventions that are intended to improve both spatial and time resolution [2]. These techniques are able today to bring local information with a spatial resolution of submicrometric size and temporal scales down to nanoseconds by using time-resolved measurements.

While one-photon fluorescence microscopy is a standard tool for bio-imaging, nonlinear contrasts have progressively emerged as interesting alternatives for many reasons. Nonlinear excitations involve in particular near-infrared excitation wavelengths, which are less affected by scattering in tissues and therefore allow a deeper penetration of imaging in thick samples [3,4]. Owing to the nonlinear nature of the excitation in these regimes, intrinsic spatial resolution (typically 300 nm lateral) can be achieved with reduced out-of-plane photobleaching and phototoxic effects [5]. While two-photon excitation fluorescence (TPEF) microscopes were developed in the early nineties in biological samples [5], the first integration of the coherent nonlinear process of second-harmonic generation (SHG) into an optical microscope was introduced in the seventies to visualize crystalline structures [6,7]. Later demonstrations in biological tissues showed that three-dimensional SHG imaging could be achieved at more than 500 µm depth [3]. Nonlinear coherent processes of higher order such as third-Harmonic generation (THG), coherent anti-Stokes Raman scattering (CARS), and its nonresonant counterpart four-wave mixing (FWM), are also now currently introduced into biological imaging. These processes, described in Fig. 1, are detailed in Section 2. Because of their own specificities, they target different biomolecular structures:

- ■ Two-photon excitation fluorescence (TPEF): incoherent optical contrast applied to endogeneous proteins cells and tissues, synthesized fluorescent labels attached to proteins, antibodies, or embedded as lipid probes in cell membranes [1];
- ■ SHG: noncentrosymmetry specific structural contrast applied to the measurement of membrane potential using SHG active molecular lipid labels [8,9], endogeneous structural proteins such as collagen type I [10], acto-myosin, and tubulin, which are present in cells, tendons, muscle fibers or other types of tissue [11]
- ■ FWM: non-chemically-specific contrast visible in nonresonant CARS microscopy [16]

While bio-imaging is able today to bring information on complex biomolecular assemblies, the use of light polarization can bring a complementary degree of freedom in these investigations. Light–matter interaction is indeed of a vectorial nature: since the optical signals are based on the oscillation of electrons in atoms and molecules induced by an incident optical field, the radiated fields are more efficient if the incident optical polarization is parallel to the molecular transition dipole moments associated with the excited levels involved in the contrast. As a consequence, monitoring incident light polarization, as well as the direction of emitted dipoles, gives access to a way to probe the molecular orientation in a medium. In general the problem is made more complex because the focal volume of a microscope contains not one but a large quantity of active molecules. I show how it is possible to probe an orientational molecular distribution (denoted $f\left(\mathrm{\Omega}\right)$ in this tutorial), which represents the probability distribution of molecules to be present at a given angular orientation Ω relative to the macroscopic frame of the investigated sample. Obtaining information on $f\left(\mathrm{\Omega}\right)$ is the result of a steady-state, “static” measurement, time averaged over an integration time that often surpasses the millisecond time scale. In this tutorial I will not address the dynamics of the orientational diffusion of molecules, which is the scope of time-resolved measurements.

The knowledge of molecular orientational information is essential to understand the interactions that drive the structure and morphology of biomolecular assemblies, from membrane proteins aggregates to biopolymers, and its consequences in related biological functions. This organization, governed by molecular interactions such as protein–protein, protein–lipids, or lipids–lipids, plays an important role in biology. Proteins and lipids in cell membranes are, for instance, known to form structured assemblies with collective molecular order participating in cell motility [18], vesicular trafficking [19], and signaling [20–22]. Protein interactions in supramolecular complexes such as biofilaments are also highly driven by orientational order [23–25]. In particular, in the extracellular matrix, collagen and other protein filaments undergo strong interaction forces with cells that can be affected by the development of tumors. An orientational organization is quantified by the so-called molecular order, which varies between complete disorder (in an isotropic medium) to complete order (such as in a crystalline medium). The intermediate situations require models of the orientational molecular distribution $f\left(\mathrm{\Omega}\right)$ such as that governed by Boltzmann statistics. Imaging such information quantitatively by using optical microscopy is, however, a challenge.

In this tutorial, I show how light polarization can be manipulated and exploited to provide a knowledge of the molecular angular distribution in molecular and biological samples and how this information can be related to biological information. I will also emphasize that a given optical contrast only gives a partial picture of the $f\left(\mathrm{\Omega}\right)$ function, but can nevertheless be combined with other contrasts to give complementary pictures of this function. Section 2 is dedicated to the theoretical developments of different optical contrasts, accounting for their polarization-dependence specificities. Section 3 describes the essential features of a polarization-resolved microscopy setup, pointing out the different possible schemes that have been developed to retrieve polarization and orientation information in molecular media. Section 4 illustrates the application of polarization-resolved two-photon fluorescence to lipid membrane order imaging. Section 5 gives examples of application of coherent nonlinear optical processes for tissue structural imaging, as well as first developments in polarization-resolved CARS.

## 2. Polarized Light–Matter Interaction: from one Molecule to an Assembly of Molecules

Measuring an orientational molecular property in a sample requires retrieving, from a macroscopic optical measurement, microscopic information that is polarization sensitive. In this tutorial I detail how this relation operates for different contrasts exhibiting specific coherence properties. We will in particular separately consider incoherent optical processes such as one- and two-photon fluorescence from coherent optical processes such as SHG and THG and CARS (Fig. 1).

#### 2.1. Definitions and Notation

**Frame notation.** Relating the different scales from macroscopic to molecular requires defining different frames, which are depicted in Fig. 2. The *molecular frame*, attached to the molecular
structure, is denoted $(x,y,z)$ (with *z* generally along the axis of higher symmetry), for which the index notation $(u,v,w)$ will be used. The molecular distribution *microscopic frame*, defined by an ensemble of molecules, is denoted $(1,2,3)$ [with index notation $(i,j,k)$], in the same way as is generally done in crystalline media. The *macroscopic frame*, or laboratory frame, will be denoted $(X,Y,Z)$ [with index notation $(I,J,K)$], with *Z* the direction of propagation of the incident field (at the ω frequency) ${\mathbf{E}}^{\omega}$; ${\mathbf{E}}^{\omega}$, which is therefore polarized in the $(X,Y)$ plane for a planar wave, also defines the sample plane [Fig. 2(c)]. When a polarizer is used to analyze a given signal, this polarizer will therefore also be oriented along either the *X* or the *Y*
axis. The orientation of the molecular frame in the molecular distribution frame is defined by the Euler set of angles $\mathrm{\Omega}=(\theta ,\varphi ,\psi )$. These angles, used in the transformation from one frame to another, are represented schematically in Fig. 2(a). While the $(\theta ,\varphi )$ spherical coordinate angles are used to orient the axis of higher symmetry of the molecule, the *ψ* angle is used to express its rotation relative to this high-symmetry axis. Similarly, the Euler angles ${\mathrm{\Omega}}_{0}=({\theta}_{0},{\varphi}_{0},{\psi}_{0})$ will be used to orient a molecular angular distribution in the macroscopic frame [Fig. 2(b)].

Last, in all equations, bold letters correspond to vectors.

**Rotation of a tensor.** Light–matter interaction is governed by the motion of charges induced by the electromagnetic field excitation associated with light. In all the optical processes mentioned in what follows, we will employ the notion of a “susceptibility tensor”
associated with a polarizability (of first order for linear optical interactions and of higher order for nonlinear interactions). These susceptibilities need to be expressed in two frames: either the microscopic $(1,2,3)$ frame, or the macroscopic $(X,Y,Z)$ frame, when a macroscopic optical response is analyzed. A convenient tool to connect the two frames is to express the same tensor in both frames by using the rotation operator of a tensor. Considering an *n*-rank tensor ${\tau}_{{u}_{1},{u}_{2},\dots ,{u}_{n}}$ expressed in the molecular frame, its expression in the microscopic frame will be, along the same line as for a vector rotation,

#### 2.2. One- and Two-Photon Excited Fluorescence

**Fluorescence response from a single molecule.** Fluorescence is the result of two successive processes, the absorption of an incident photon and the emission of a fluorescence photon of lower frequency. These processes are separated in time owing to the delay necessary for
the molecule to relax from its high-energy excited state to its lowest (fluorescent) excited state [1]. This time delay has important consequences for the optical properties of fluorescence. First, this process is incoherent (different molecules
emit fluorescence radiations with no phase correlation between them). Second, the decorrelation between these two steps makes the fluorescence efficiency proportional to the product of two probabilities: the absorption between the ground and the excited state, and the emission from
the fluorescent state to the ground state. Since the absorption event can be performed either in a linear regime (one-photon absorption) or a nonlinear regime (*n*-photon absorption), we can write the *n*-photon fluorescence intensity from a single
molecule as the product

The *absorption probability* in a one-photon absorption process, denoted ${P}_{\mathrm{abs}}^{1\text{-ph}}$, is proportional to the imaginary part of the first order molecular susceptibility $\alpha (\omega ;\omega )$ expressed at the incident optical frequency *ω* [26]:

*•*is the tensorial scalar product similar to the scalar product for vectors.

To get more insight into the anisotropy origin of the $\alpha (\omega ;\omega )$ tensor, ${\alpha}_{IJ}(\omega ;\omega )$ can be expressed by using a quantum mechanics perturbation approach, which is generally formulated for a one-electron atom but can be extended to a complex molecular system [26]:

*I*. This quantity is a determining parameter related to the molecular orientation, since the directions of the transition dipole moments generally follow the structure of the molecule. ${\omega}_{n0}={\omega}_{n}-{\omega}_{0}$ is the frequency difference between the excited and ground levels, and ${\mathrm{\Gamma}}_{n0}$ is the spectral frequency linewidth of the excited state $|n\u3009$.

In a two-level system, where only one excited level $|n\u3009=|1\u3009$ is the dominant contribution in Eq. (4), the absorption probability can be simplified in

Equation (5) shows finally that **there are two ways to express the absorption probability in a fluorescence process: a tensorial way using the molecular polarizability tensor** $\alpha (\omega ;\omega )$ **properties, and a vectorial way using the transition moment dipoles** *μ* **properties**.

${\mu}_{01}$ will be denoted ${\mu}^{\mathrm{abs}}$ in what follows, to emphasize the contribution of this transition dipole moment in the absorption process.

The *emission probability* along the analysis axis *I*, denoted ${P}_{\mathrm{em},I}$, contains both the fluorescence quantum yield of the molecule and the characteristics of the radiated intensity ${I}_{\mathrm{em},I}$:

**I**is the unit vector along the direction

*I*.

Considering a molecule of orientation Ω in the macroscopic frame, the far field ${\mathbf{E}}^{\mathrm{em}}$ is the field radiated by the molecular emission dipole ${\mu}^{\mathrm{em}}\left(\mathrm{\Omega}\right)$, the transition dipole moment between the fluorescent and the ground state. Since it comes from the radiation from a dipole source, this radiated field in the propagation direction **k** can be expressed as

**k**.

Equation (7) can be further simplified in the case of a *planar wave illumination* propagating along the *Z* direction, and for a detection along the same *Z* direction. Then ${\mu}_{\perp}^{\mathrm{em}}$ lies in the $(X,Y)$ plane and

We will follow this approximation in this section, the use of a high-numerical-aperture (NA) objective being further detailed in Section 3.

In general ${\mu}^{\mathrm{em}}$ is different from the absorption transition dipole, because different states are involved in the absorption–emission processes, therefore involving different molecular conformations. For the sake of simplicity, we will assume that these dipoles are pointing along the same direction of orientation Ω [${\mu}^{\mathrm{abs}}={\mu}^{\mathrm{em}}=\mu \left(\mathrm{\Omega}\right)$], which will also define the orientation of the molecular structure. The fact that the two dipoles can have different angles can be addressed by using a more complete expression accounting for an additional angle.

Finally the one-photon fluorescence intensity, along the analyzing direction *I*, for a molecule oriented with an angle Ω relative to the macroscopic frame, is written as

In *multiphoton fluorescence*, the transition of the molecule to its excited state is performed by the quasi-simultaneous absorption of two or more photons through virtual nonresonant intermediate levels transitions lasting a very short period $(1{0}^{-15}\u20131{0}^{-18}\phantom{\rule{3.00003pt}{0ex}}\mathrm{s})$ (Fig. 1). Whereas the selection rules for one-photon and multiphoton absorption are different because of the different numbers of energy levels involved (virtual or not), the emission occurs from the same excited level as for the
one-photon fluorescence process. In the case of TPEF, the two-photon absorption probability ${P}_{\mathrm{abs}}^{2\text{-ph}}$, which involves a nonlinear excitation process, can be calculated as for the one-photon absorption case, using a higher order of perturbation. It is governed by the third-order nonlinear susceptibility tensor $\gamma (\omega ;\omega ,\omega ,-\omega )$ (expressed at the frequency $\omega =\omega +\omega -\omega $) [26]:

A quantum perturbative approach, calculated at a higher order of perturbation, leads to the following expression of the *γ* tensorial components, considering here only the nearly resonant terms [26]:

In the following calculation, we will assume that only one of the $|n\u3009$ levels is of the dominant transition dipole and that it is furthermore nonresonant for the *ω* frequency (${\omega}_{n0}-\omega \approx \omega $). We will principally investigate one-dimensional molecules in which we can assimilate ${\mu}_{0n}$ and ${\mu}_{n1}$ to a single vector direction ${\mu}^{\mathrm{abs}}$ along the molecular axis. The two-photon absorption can thus be defined by

Equation (13) shows finally that the two-photon absorption cross section is nonlinear with respect to the incident intensity, being proportional to its square.

Because the emission occurs from the same level as in one-photon fluorescence, the two-photon fluorescence intensity along the analyzing direction *I* (expressed by the unit vector in that direction, **I**) can then be written, in the same planar wave
approximation as for Eq. (9),

In what follows, we will focus on the polarization dependence of the fluorescence processes and therefore will consider, in a first approximation, a planar wave illumination–detection process. We will also replace the “∝” sign with an “=” sign, since only the intensity dependence with respect to the incident polarization will be investigated.

In both one- and two-photon excitation cases, one can see then that when a linearly polarized light excites a single molecule, the highest probability of absorption occurs when its transition dipole moment ${\mu}^{\mathrm{abs}}$ is oriented parallel to the incident polarization. This property, called angular photoselection, is at the origin of the polarization dependence of the fluorescence process. Letting Θ denote the angle between the
absorption–emission molecular transition dipole and the exciting polarization, the one-photon angular photoselection is proportional to ${P}_{\mathrm{abs}}^{1\text{-ph}}\propto {\left({\mu}_{01}\right)}^{2}\cdot {I}^{\omega}\cdot {cos}^{2}\mathrm{\Theta}$, whereas in the TPEF process this photoselection is proportional to ${P}_{\mathrm{abs}}^{2\text{-ph}}\propto {\left({\mu}_{01}\right)}^{4}\cdot {\left({I}^{\omega}\right)}^{2}\cdot {cos}^{4}\mathrm{\Theta}$. This makes the two-photon photoselection narrower than for a one-photon excitation, and therefore offers the possibility to measure molecular orientations with a finer precision. The incident polarization dependencies of the one- and two-photon fluorescence signals
from a fixed single molecule are plotted in Fig. 3(a) in a polar plot representation. In these graphs, the signals are computed for a varying incident polarization ${\mathbf{E}}^{\omega}$ of orientation *α* relative to *X* and analyzed along two polarization directions $(I=X)$ and $(I=Y)$, where $(X,Y)$ defines the sample plane axes and *Z* the propagation direction of the incident field ${\mathbf{E}}^{\omega}$. The molecule is assumed to lie in the sample plane. While the global orientation of the polarization responses points along the $\mu ={\mu}^{\mathrm{abs}}={\mu}^{\mathrm{em}}$ direction, the width of the polarimetric pattern is decreased when the order of the nonlinear excitation is increased, which is the consequence of the nonlinear photoselection. In addition the ${I}_{X}^{2\text{-ph}}$ and ${I}_{Y}^{2\text{-ph}}$ analyzed intensity components carry some information on the molecular orientation, their relative magnitude being a signature of the projection coefficient of the dipole along these two axes.

*Complete tensorial form of the n-photon excited fluorescence process.* The previous developments show that there are two possible ways to write an *n*-photon fluorescence process: the first one is based on the multiple product of transition dipoles,
involved either in the *n*-photon absorption probability or the one-photon emission probability; the second one based on the tensorial expression of the susceptibilities involved in these processes. Under a planar wave approximation,

*n*-rank $\mu \otimes \cdots \otimes \mu $ tensor and a one-photon emission

*α*tensor.

In particular, the one- and two-photon excitation fluorescence processes of single molecules of orientation Ω in the macroscopic frame can be written in their full tensorial form:

**From one molecule to** *n* **molecules.** So far I have detailed the principle of fluorescence for a single molecule. In a microscopy measurement, however, a great number of fluorophores is often present within the focal volume. Here I will extend
this approach to the calculation of the fluorescence signal from an assembly of *n* molecules, present in the focal volume *V* of a medium of molecular density *N*. Since the fluorescence process is based on absorption and emission events,
which are uncorrelated in time within the fluorescence lifetime scale, the radiation emitted by each molecule of the focal volume will be randomly phase shifted in time from the radiation from its neighbors. The fluorescence emitted fields should therefore be added in intensity to
account for the incoherence of this optical process. The various orientations that are experienced by the molecules are considered to lie within a molecular orientational distribution function $f\left(\mathrm{\Omega}\right)$, normalized such that $\int f\left(\mathrm{\Omega}\right)d\mathrm{\Omega}=1$. The probability density of the molecules to lie between the orientations Ω and $\mathrm{\Omega}+d\mathrm{\Omega}$ in the $(1,2,3)$ frame (Fig. 2) is therefore $N\left(\mathrm{\Omega}\right)d\mathrm{\Omega}=Nf\left(\mathrm{\Omega}\right)d\mathrm{\Omega}$ (the time dependence fluctuations are discussed below). Considering an ensemble of molecules at locations **r** within the focal volume *V* [Fig. 4(a)], the two-photon excited fluorescence intensity can
then be written as

*N*the molecular density. A similar expression could be derived for the one-photon fluorescence process. In what follows, I focus, however, on the two-photon process because of its higher photoselection power.

In Eq. (17), the integration is performed on different variables:

- ■ ${\int}_{\mathrm{NA}}$ means that the
**k**dependence of the emitted field should be integrated over the collection aperture [Fig. 4(a)], which might originate from a high-numerical-aperture (NA) objective. - ■ ${\int}_{V}$ means that the
**r**dependence of the excitation and emission fields should be integrated over the whole the focal volume [Fig. 4(a)]. In particular, the excitation polarization**E**might be distorted and depend on the location in this focal volume in the case of high-NA focusing (Fig. 4(b)) (see Subsection 3.6). - ■ ${\int}_{\mathrm{\Omega}}$ means that the dependence of the transition dipole moments involved should be integrated over the whole support of the $f\left(\mathrm{\Omega}\right)$ molecular angular distribution function.
- ■ The $\u3008\dots \u3009$ sign stands for the time average of the measured intensity over the signal fluctuations, which we will ignore in what follows, since this calculation is dedicated to a static measurement over an integration time much larger than the orientational fluctuations and the fluorescence lifetime of the molecules.

We may further simplify the calculations under several assumptions:

- ■ Planar wave illumination. We will therefore ignore as a first approximation the
**r**dependence of the excitation field**E**. The case of high-NA focusing is discussed in Subsection 3.6. - ■ Homogeneous medium (at the spatial scale of the optical resolution). We will therefore ignore the
**r**dependence of the emission fields ${\mathbf{E}}^{\mathrm{em}}$ and thus the ${\int}_{V}$ integration. - ■ Planar wave collection. We will therefore ignore the integration over ${\int}_{\mathrm{NA}}$, and consider a single
**k**detection direction along*Z*. This entails ${|{\mathbf{E}}^{\mathrm{em}}\cdot \mathbf{I}|}^{2}={|{\mu}^{\mathrm{em}}\left(\mathrm{\Omega}\right)\cdot \mathbf{I}|}^{2}$. The case of high-NA collection integration is discussed in Subsection 3.6. - ■ Absorption and emission dipoles along the same direction ${\mu}^{\mathrm{em}}={\mu}^{\mathrm{abs}}=\mu $. An angle between these dipoles, if known, can be introduced into Eq. (17).

Under these assumptions, Eq. (17) becomes

So far, we have used Ω in a general way to name the orientation of the molecules in the macroscopic frame. However, in most cases the molecular angular distribution function $f\left(\mathrm{\Omega}\right)$ is known and defined in a microscopic frame $(1,2,3)$, where the high-symmetry axis 3 can be either the principal axis of a crystalline unit-cell frame, the one-dimensional direction of a biofilament, the direction normal to a cell membrane, or so on. Therefore two unknown factors are present in Eq. (18): the orientation of the $f\left(\mathrm{\Omega}\right)$ function [denoted ${\mathrm{\Omega}}_{0}=({\theta}_{0},{\varphi}_{0},{\psi}_{0})$ as in Fig. 2], and its shape. In order to decouple these parameters, two strategies are possible depending on the dipoles or tensorial forms used to express the fluorescence process [Eq. (15)]:

(1) The *dipole moments* can be first expressed in the microscopic frame, and then expressed, by a rotation, in the macroscopic frame. Therefore new components of the excitation–emission dipole moment *μ* are used in Eq. (18):

(2) Similarly, a *tensorial* approach could be used, using the molecular two-photon fluorescence tensor $\tau =\gamma \otimes \alpha $ introduced in Eq. (15). First, this molecular tensor (called *τ*) should be expressed in the microscopic frame, then averaged over all molecular orientations, and finally rotated in the macroscopic frame
using Eq. (1). The final macroscopic tensor is denoted *T*. This operation is summarized as follows for any general tensor:

*τ*in the microscopic frame by a rotation operation, the second one calculates the averaged tensor over all molecules, and the third expresses this new macroscopic tensor

*τ*in the macroscopic frame by a new rotation operation.

Finally this leads to a macroscopic notation of the two-photon fluorescence intensity:

where ${T}_{JKLMII}$ is the macroscopic tensor associated with the two-photon fluorescence process.The two approaches detailed here lead to identical results, the second tensorial one being appropriate mostly for crystalline structures. Indeed when the crystal point group symmetry is known, the microscopic tensor can be directly deduced from the associated symmetry. To visualize
the consequence of a molecular orientational averaging on the polarization response of a fluorescence signal, we implemented the calculation of ${I}_{X}^{2\text{-ph}}\left(\alpha \right)$ and ${I}_{Y}^{2\text{-ph}}\left(\alpha \right)$ for a varying incident linear polarization in the sample plane $(X,Y)$ with an angle α relative to *X* (${E}_{X}=cos\alpha ,{E}_{Y}=sin\alpha $). Figure 3(b) shows both responses (in a polar representation as functions of *α*) from a large number of molecules oriented within an angular aperture $\mathrm{\Psi}=3{0}^{\circ}$ around the direction ${\mathrm{\Omega}}_{0}=({\theta}_{0}=9{0}^{\circ},{\varphi}_{0}=3{0}^{\circ},{\psi}_{0}={0}^{\circ})$, simulating the case of a molecular angular distribution whose shape is a cone of aperture Ψ. Contrary to the single-molecule case where the two responses ${I}_{X}^{2\text{-ph}}\left(\alpha \right)$ and ${I}_{Y}^{2\text{-ph}}\left(\alpha \right)$ point in the same direction, the mixture of molecular orientations induces a shift in their pointing direction, which is a characteristic of the presence of more than just a single molecular direction in the focal volume. In the extreme case of a random mixture of
orientations leading to an isotropic distribution [Fig. 3(c)], the polarization response point towards perpendicular directions in such a way that the sum of these two responses ${I}_{X}^{2\text{-ph}}\left(\alpha \right)+{I}_{Y}^{2\text{-ph}}\left(\alpha \right)$, is independent of the incident polarization, as expected from a medium in which no particular direction is privileged. This first example shows how a polarization read-out of the optical information can lead to a first insight into the molecular angular
distribution.

Finally, the polarization dependence of the two-photon excitation fluorescence in a molecular assembly is written in its tensorial form:

This result highlights an important property of this optical contrast: because of the incoherence of the process, the orientational averaging is performed on the molecules’ intensities and not on their radiating dipole amplitudes. Therefore the final intensity is proportional to
*N*, the molecular density.

#### 2.3. Nonlinear Coherent Optical Contrasts: Second-Harmonic Generation

**Single-molecule SHG response.** SHG at the single-molecule level originates from the scattering of twice the frequency of excitation ω, owing to a nonlinear interaction between the molecule and the optical field [26]. For symmetry reasons stemming from invariance properties upon point groups symmetry transformation [26], this process requires the molecule to be noncentrosymmetric (which means without any center of
symmetry), and therefore occurs in specific structures. For a single molecule pointing in the direction Ω in the macroscopic frame $(X,Y,Z)$, the SHG signal can be written from the radiation of the molecular nonlinear induced dipole ${\mathbf{p}}^{\mathrm{S}\mathrm{H}\mathrm{G},2\omega}=\beta (2\omega ;\omega ,\omega ):{\mathbf{E}}^{\omega}{\mathbf{E}}^{\omega}$, with *β* the molecular hyperpolarizability tensor:

*β*, following the formalism of tensorial rotation introduced in Eq. (1). In the case of statistical molecular angular distribution defined by both their shape $f\left(\mathrm{\Omega}\right)$ and orientation ${\mathrm{\Omega}}_{0}$ (Fig. 2), ${\beta}_{IJK}\left(\mathrm{\Omega}\right)$ can be replaced by ${\beta}_{IJK}(\mathrm{\Omega},{\mathrm{\Omega}}_{0})$ and rewritten similarly as for the fluorescence tensors [Eq. (1)]:

Note that the molecular $\beta (2\omega ;\omega ,\omega )$ tensor, which is an intrinsic property of the molecule, can be expressed by following a perturbative quantum approach, in the same way as for the $\alpha (\omega ;\omega )$ and $\gamma (\omega ;-\omega ,\omega ,\omega )$ tensors previously introduced for one- and two-photon absorption. Along the same lines, the second order nonlinear susceptibility also depends on the transition dipole moments of the molecule, which can be written as [26]

Note that close to resonances, the permutation of dipole moment components is not allowed anymore, which is the signature that frequencies are directly related to indices. This deviation from the Kleinman conditions implies that the tensor indices cannot be permuted, which can have important consequences for the polarization dependencies of the SHG response from complex multipolar molecules.

Finally, in a similar way as for fluorescence, the SHG process can be written either in a vectorial form, the *β* tensor being composed of a product of transition dipole moments, or in a tensorial form, keeping the tensorial information on the third-order tensor. Mostly
this latter form will be used, since one can manipulate tensorial transformation in a convenient way when changing from the microscopic to macroscopic frame.

**From one molecule to the SHG response from** *n* **molecules.** Unlike fluorescence, the SHG-induced dipole from a given molecule exhibits a determined phase relation with both the incident excitation field and the other molecules. In particular,
since the SHG radiation originates rather from a scattering process, there is no need to excite the molecule to a real excited level, and the optical process can therefore be nonresonant [Fig. 1(b)]. In the case of SHG, a macroscopic signal thus
results from the coherent addition of nonlinear molecular induced dipoles in the focal volume, induced in the focal spot of the objective. This leads to the following SHG intensity measured along the $I=(X,Y)$ polarization analysis direction:

*N*the molecular density and ${\mathbf{E}}^{\mathrm{SHG}}$ the radiated SHG field originating from the molecular dipole sources ${\mathbf{p}}^{\mathrm{SHG}}$ [Eq. (23)].

Similarly to the development of fluorescence, we will develop the calculations of polarization dependence SHG under some approximations:

- ■ Planar wave illumination: the amplitude of ${\mathbf{E}}^{\omega}$ is independent of
**r**in Eq. (23). - ■ Planar wave collection: ${E}_{I}^{\mathrm{SHG}}={p}_{I}^{\mathrm{SHG}}$.
- ■ Small focal volume relative to the nonlinear coherent lengths: this means that there will be no (or negligible) interference effect between the different dipoles positioned at different locations
**r**, and therefore the**r**dependence can be also ignored in the phase of the incident and the emitted fields in the expression of ${\mathbf{p}}^{\mathrm{SHG}}$. It is known, however, that for higher-order coherent nonlinear contrasts, phase dependencies can play a crucial role in microscopy [13].

Under these assumptions, which will be further discussed in Subsection 3.6, Eq. (26) can be simplified into the modulus square of a macroscopic nonlinear induced dipole ${P}_{I}^{\mathrm{SHG}}=N{\int}_{\mathrm{\Omega}}{p}_{I}^{\mathrm{SHG}}\left(\mathrm{\Omega}\right)f\left(\mathrm{\Omega}\right)d\mathrm{\Omega}$:

Similarly to the molecular scales, SHG requires noncentrosymmetry at the macroscopic scale, which stems from the symmetry properties of the second-order tensor ${\chi}_{IJK}^{\left(2\right)}$. This means that to provide a coherent SHG nonlinear response, a medium has to exhibit no center of symmetry. This requirement is in particular met in biological structures of helical supramolecular shapes.

Figure 5 depicts polarization dependencies of ${I}_{X}^{\mathrm{SHG}}\left(\alpha \right)$ and ${I}_{Y}^{\mathrm{SHG}}\left(\alpha \right)$ to a varying incident linear polarization in the sample plane $(X,Y)$ with an angle *α* relative to *X*, for different shapes of molecular distribution functions filling a cone aperture (as in Fig. 3) oriented along the *X* direction: illustrated are a
narrow cone aperture [Fig. 5(a)], a large [Fig. 5(b)], and isotropic [Fig. 5(c)]. In the first case, the SHG polarization dependence is seen to be quite different from the
fluorescence case, with more complex lobes or shapes appearing for ${I}_{Y}^{\mathrm{SHG}}\left(\alpha \right)$. This is characteristic of a nonlinear coherent response that is able to provide a coherent coupling between different states of polarization. The ${I}_{X}^{\mathrm{SHG}}\left(\alpha \right)$ response is seen to be nevertheless predominant because of the large number of molecules pointing in the *X* direction. When the cone aperture is enlarged, the *X* response lobes and the *Y* response magnitude are
enlarged. Finally the SHG response is canceled in the case of an isotropic distribution, owing to dipole radiations cancellations, as expected from the symmetry rule governing the SHG process. Interestingly, on replacing the cone distribution with a cone surface on which molecules
lie, the SHG polarization responses change drastically (insets in Fig. 5).

Finally, the polarization dependence of the SHG response is written in its tensorial form:

This result highlights an important property of SHG: because of the coherence of the process, the orientational averaging is performed on the radiating dipole amplitude and therefore leads to an ${N}^{2}$ molecular density dependence of the signal.

#### 2.4. Nonlinear Coherent Optical Contrasts: Third-Order Effects

### 2.4a. Third-Harmonic Generation and Four-Wave Mixing

Equation (23) can be written for higher-order effects, and in particular the THG and FWM processes that involve an additional order of interaction (Fig. 1). THG and FWM are interesting for their capacity to provide a signal in media of whatever symmetries. However, because of their high order of interaction, the spatial coherence becomes a crucial factor, and not all media will provide contrasted images by use of third-order nonlinear optical processes. In the case of pure THG, the signal is even more restricted, since it vanishes in homogeneous media and appears only at interfaces [13]. The THG and FWM can be expressed from their respective nonlinear induced dipoles at different radiating frequencies:

### 2.4b. Coherent Anti-Stokes Raman Scattering

CARS is a particular case of FWM that involves an intermediate resonance corresponding to a vibrational level (Fig. 1). In CARS the molecule is excited by two optical frequencies, the pump ${\omega}_{1}={\omega}_{2}={\omega}_{p}$, and the Stokes ${\omega}_{3}={\omega}_{s}$. The resulting microscopic induced dipole at the anti-Stokes detected frequency ${\omega}_{4}={\omega}_{AS}=2{\omega}_{p}-{\omega}_{s}$ is

The specificity of the CARS process is to exhibit, for specific frequencies ${\omega}_{s}$, a resonant contribution when $\delta \omega ={\omega}_{p}-{\omega}_{s}={\mathrm{\Omega}}_{R}$ corresponds to a vibrational energy of the molecule [Fig. 1(b)]. The advantage of this technique is to be able to address optically, with near-IR lasers, vibrational levels of low frequencies, and to exhibit superior sensitivity for imaging comparing with the Raman process [14].

The resonance specificities of the involved third-order nonlinear tensor when approaching a vibrational resonance by tuning ${\omega}_{s}$ can be written as

*a*is the oscillator strength of the vibration, and Γ the line width of the Raman resonance corresponding to the ${\mathrm{\Omega}}_{R}$ vibrational energy.

A CARS tensor out of resonance possesses 21 nonvanishing components, through index permutations: either intrinsic due to the CARS interaction (in particular the degeneracy in the ${\omega}_{p}$ frequency), or due to Kleinman symmetry, implying that out of resonance the tensor components are invariant under cyclic permutation of the indices. In nonisotropic samples, the nonresonant CARS third-order nonlinear susceptibility tensor exhibits a complex structure characteristics of the symmetry of the studied medium.

The structure of the CARS tensor close to resonances is, however, more complex and requires specific investigations in both the resonant and nonresonant regimes [27,28]. Indeed, each resonance has its own symmetry specificity, which is characterized by a tensor whose structure can be different from the nonresonant one. The complexity of the susceptibility tensor is further increased at resonance, where Kleinman symmetry conditions do not apply [26,27]. Therefore at resonance, the polarization response of the CARS signal can change drastically compared with the nonresonant case.

Finally, the polarization dependence of the THG, FWM, or CARS responses are written, following the notation previously introduced, as

*γ*might contain resonant contributions (in the case of CARS) and ${E}_{J,K,L}$ should be written for the frequency used for the measurement. Here, as above, Ω denotes the orientation of the molecules in the macroscopic frame, assuming that $f\left(\mathrm{\Omega}\right)$ is also defined in this frame (the case where $f\left(\mathrm{\Omega}\right)$ is defined in its own microscopic frame should account for an additional tensorial rotation).

#### 2.5. General Conclusion on the Different Nonlinear Optical Contrasts

In all the previous expressions, Eqs. (22), (29), and (34), one can see that it is possible to retrieve information on a sample if the
incident field polarization state, present in the ${E}_{J,K,\dots}$ coefficients, is controlled and varied. Figures 3 and 5 show in particular that a variation of the field incident polarization in the sample plane induces polarization dependencies that are
closely related to the molecular angular distribution function $f\left(\mathrm{\Omega}\right)$. In the next subsection, we will derive how microscopic information on the shape and orientation of $f\left(\mathrm{\Omega}\right)$ can be retrieved from such measurement. A question arises, however, on the degree of complexity that can be retrieved from a crystal symmetry or a molecular distribution shape. The answer to this question depends on the contrast addressed. Varying the ${E}_{J,K,\dots}$ polarization components in an independent manner in the $(X,Y)$ sample plane should be able to provide a set of independent measurements applied to each nonlinear contrast. Considering the situation where both ${I}_{X}$ and ${I}_{Y}$ are measured (without an analyzer, ${I}_{X}+{I}_{Y}$ is measured and these numbers should be divided by 2), in the case of two-photon fluorescence, the $\gamma \otimes \alpha $ tensor in the $(X,Y)$ plane contains 16 independent coefficients. In the case of SHG (or THG), the *β* (or *γ*) tensor in the $(X,Y)$ plane contains six (or eight) independent coefficients. For FWM and CARS, the number of independent parameters is increased, since three incident frequencies are different and therefore the field polarization permutations are distinguishable. This leads to 16
independent coefficients for FWM and 12 for CARS. These numbers are signatures of the maximal number of retrievable macroscopic parameters (all normalized by one of them), which is an upper limit of the unknown quantities that we might introduce in the polarization-resolved
experiments. Note that if the tensors are considered close to resonances, they are no longer one real quantity but are two unknown numbers, since both real and imaginary parts should be considered. This can considerably increase the number of unknown parameters to measure [27].

From these macroscopic coefficients, microscopic information should be retrievable. Different parameters are, however, unknown:

- ■ The molecular symmetry that governs the tensorial expression of the ${\alpha}_{uv}$, ${\beta}_{uvw}$, ${\gamma}_{uvw\u03f5}$ coefficients in the molecular $(x,y,z)$ frame and therefore their expression ${\alpha}_{ij}\left(\mathrm{\Omega}\right)$, ${\beta}_{ijk}\left(\mathrm{\Omega}\right)$, ${\gamma}_{ijkl}\left(\mathrm{\Omega}\right)$ in the microscopic frame $(1,2,3)$ (Fig. 2).
- ■ The shape of the angular distribution function $f\left(\mathrm{\Omega}\right)$ in the microscopic frame $(1,2,3)$.
- ■ The orientation ${\mathrm{\Omega}}_{0}$ of the angular distribution function $f\left(\mathrm{\Omega}\right)$ in the macroscopic frame $(X,Y,Z)$ (Fig. 2).

Solving the problem of the measurement of structural information will therefore rely on some hypotheses, either on the molecular scale (structure of the molecular tensors), on the microscopic scale (shape of the distribution function), or on the macroscopic scale (orientation of the distribution function).

Finally, retrieving information on the microscopic molecular structure in a sample will be possible by varying the incident field polarization **E**, in a few situations:

- ■ If the molecular symmetry is known, the tensorial expressions of
*α*,*β*,*γ*are known; therefore the angular distribution function $f\left(\mathrm{\Omega}\right)$ can be studied, investigating both its shape and orientation ${\mathrm{\Omega}}_{0}$. - ■ If the angular distribution function (including its orientation) is known (the most simple case being the isotropic medium), then a study of the molecular structure is possible.

These situations show that polarization-resolved studies may deal with a large number of unknown parameters, at both the microscopic and the macroscopic scales. The next subsection will detail which parameters are indeed accessible, depending on the optical contrast used.

#### 2.6. Retrieving Information on the Molecular Angular Distribution from a Polarized Measurement

A normalized molecular angular distribution function $f\left(\mathrm{\Omega}\right)$ defines the orientations explored by the molecules after time and space averaging over the integration time of the optical measurement and the focal volume of the objective.

In molecular media where one-dimensional active molecules are constrained by a known potential $U\left(\mathrm{\Omega}\right)$ at thermal equilibrium *T*, $f\left(\mathrm{\Omega}\right)$ follows the Boltzmann statistics:

Different shapes of the $U\left(\mathrm{\Omega}\right)$ potential have been introduced as models, depending on the biological medium investigated:

- ■ In lipid membranes, the orientational distribution is most often defined as lying within a cone aperture with an abrupt change of the probe potential at a defined aperture angle $\theta =\mathrm{\Psi}$ [Fig. 6(a)], [29,30]. In such a model, the fluorophores lie statistically inside the cone aperture Ψ, which contains all their possible orientations:$$f(\theta ,\varphi )=\left\{\begin{array}{cc}1\hfill & \mathrm{if}\phantom{\rule{3.00003pt}{0ex}}\left|\theta \right|\le \mathrm{\Psi}\hfill \\ 0\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$Note that the Euler angle
*ψ*is not required at this stage, since one-dimensional dipolar molecules are considered in this model.

In all of these cases the interaction potential is of cylindrical symmetry; therefore $f\left(\mathrm{\Omega}\right)$ applies as well, and only the angles $(\theta ,\varphi )$ are necessary to define it. Three unknown parameters define the shape of this distribution function: its Ψ aperture (which can be called “molecular order”), and its two orientation angles $({\theta}_{0},{\varphi}_{0})$ in the macroscopic frame.

In the most general case of a distribution function that is not necessarily of cylindrical symmetry, the number of parameters defining this function can increase considerably, and its orientation in the macroscopic frame requires an additional angle ${\psi}_{0}$, which defines the rotation angle around the higher axis of symmetry of the distribution.

In media where the molecular symmetry and angular distribution is unknown, it can be decomposed on basis functions that are invariant upon rotation: the spherical harmonics ${Y}_{m}^{J}(\theta ,\varphi )$ (in the case of $(\theta ,\varphi )$-dependent functions), or the Wigner functions ${D}_{m{m}^{\prime}}^{J}(\theta ,\varphi ,\psi )$ (in the case of $(\theta ,\varphi ,\psi )$-dependent functions) [34]. The additional *ψ* dependence is necessary only if the active molecules that define this angular distribution are not of one-dimensional cylindrical symmetry. In this new decomposition,
the unknown parameters of the distribution function are no longer its aperture but are the coefficients of the spherical decomposition that carry information on the symmetry of this distribution.

Assuming a function defined by molecules of cylindrical symmetry (such as one-dimensional elongated molecules for instance), $f\left(\mathrm{\Omega}\right)$ can be written

where the*J*coefficients are called “orders of symmetry.” The ${f}_{m}^{J}$ coefficients (also containing normalization factors) are the weights that finally determine the shape of the distribution function. ${f}_{0}^{0}$ is in particular the signature of the isotropic contribution to the distribution function, and ${f}_{m}^{J}$ with $J\ge 0$ quantifies the existence of its higher order of symmetry. In addition, all $m=0$ contributions are of cylindrical symmetry around the microscopic 3 axis defined in Fig. 2.

The ${f}_{m}^{J}$ coefficients are the parameters that define the $f\left(\mathrm{\Omega}\right)$ function. The question is now to determine which ${f}_{m}^{J}$ coefficients can be measured from a polarization-resolved experiment, and therefore how molecular symmetry can be read out by a given optical contrast. To answer this question, we will use Eqs. (22), (29), and (34), where the measured polarized optical signals are written as averaged values of quantities *A* [35], defined by

In addition, the quantities *A* are also directly related to the various *τ* tensors read-out in fluorescence and nonlinear coherent processes. One can express these tensors in a useful spherical decomposition, where in the case of symmetric tensors (for
which indices can be permuted) the calculation detailed in Appendix B leads to

*J*is the order of symmetry in this tensorial decomposition, which can be directly related to the symmetry of the crystal point group to which the molecule belongs. $(m,J)$, furthermore satisfies $-J\le m\le J$ and $0\le J\le n$, where

*J*and

*n*are of same parity (Appendix B). The ${\tau}_{m}^{J}$ coefficients are the weights of a spherical decomposition of the

*τ*tensor and depend on the optical processes evoked above. In the case of tensors for which indices cannot be permuted (under resonant conditions, for instance), then this relation takes a more complex form, and the equations developed here are no longer applicable, since the order of apparition of the ${I}_{1}...{I}_{N}$ indices is no longer undetermined. This leads to the use of a different algebra (of nonsymmetric polynomials) that leads to more complex conclusions: whereas the upper limit for a read-out

*J*order is conserved, the parity relation does not hold anymore, and

*J*can be of any parity [36].

Finally, the measured macroscopic tensor in a nonlinear process, either coherent or incoherent, can be expressed from Eqs. (22), (29), and (34), accounting for the spherical decomposition of the distribution function $f\left(\mathrm{\Omega}\right)={\sum}_{K,l}{f}_{l}^{K}{Y}_{l}^{K}\left(\mathrm{\Omega}\right)$ (with $-K\le l\le K$):

*K*orders of symmetry of the distribution function should therefore satisfy the conditions that

*K*and

*n*are of same parity and $0\le K\le n$. Equation (42) also shows that the order of symmetry of the molecular tensor should fit the order of symmetry of the distribution function in order to allow a polarization read-out of this symmetry process.

Finally, a determining conclusion can be drawn from Eq. (42) on the different optical processes: the symmetry orders of the distribution function that can be determined by polarization-resolved microscopy are those of the tensor that defines the optical process.

Therefore, under the condition that index permutation is valid

The properties demonstrated here show that polarization-resolved measurement acts as a “filtering” effect on the symmetry orders of a molecular distribution function. We can further visualize this effect by picturing the effective distribution function, the orders of which are
truncated in the measurement of such processes. Assuming, for the sake of simplicity, a distribution function of cylindrical symmetry (this situation also being widely used in biology), then $f\left(\mathrm{\Omega}\right)$ is dependent only on *θ* and $m=0$ in Eq. (38). This equation shows that $f\left(\theta \right)$ can be directly reconstructed from the ${f}_{0}^{J}$ parameters. Figure 7 shows the illustration of a cone distribution function reconstructed from a truncation of its symmetry orders, where only $J=0\u20136$ are used for the reconstruction. As expected, the cone shape is not clearly visible, especially for the low-order reconstruction, since the abrupt angle changes in this function require high-order symmetry to be reproduced. Although for a low-order truncation the
reconstructed function does not look like a cone, the aperture information is globally kept [Fig. 7(a)]. As a comparison, a Gaussian function is easier to reconstruct, since it is a smoother function containing less high-order information
[Fig. 7(b)].

Finally, polarization-resolved measurements are able to provide only a limited amount of information, and therefore the angular distribution of molecular assemblies have to be pictured by a simple function for which orders of symmetry will be filtered out by the polarization read-out process.

## 3. Polarization-Resolved Microscopy: Techniques

In this section, I review the different schemes that have been developed to retrieve microscopic information in molecular samples from a polarization-resolved experiment.

#### 3.1. Anisotropy Imaging

Polarization-resolved imaging has been widely developed based on the use of one or two incident polarization states. Although this limits the amount of information that can be retrieved (Section 2), this has lead to significant demonstrations in biology.

The idea to investigate a molecular structure by a polarization-resolved fluorescence method was introduced by Perrin in 1926 [37] and applied by Weber in 1953 to the structural study of the binding of small molecules to proteins [38]. This method is based on the fact that even though in a solution the macroscopic medium is isotropic, the fluorescence emission is polarization dependent, since the microscopic structures investigated are anisotropic. To probe this effect,
fluorescence anisotropy is the simplest scheme one can implement. In this setup, the fluorescence intensity from a solution is illuminated along the *Z* axis, using two polarization directions *X* and *Y*, and measured along the
perpendicular propagation direction *Y*. The fluorescence is then analyzed along the *X* and *Z* polarization directions by using a polarizer. The measured quantity is called ${I}_{\perp}$ or ${I}_{\parallel}$ for parallel or perpendicular input detected polarization directions. A ratiometric analysis of these two situations provides the so-called fluorescence anisotropy, independent of the intensity fluctuations and dependent solely on orientational properties:

In microscopy imaging, the geometry is different, since both the excitation and the detection directions lie along the *Z* direction [Fig. 8(a)]. A new anisotropy factor is therefore defined, which accounts for the symmetry of the
setup:

*A*measured under an incident circularly polarized light (○), in order to avoid any photosection effect. The $(X,Y)$ directions also define the sample plane depicted in Fig. 2(c). The anisotropy factor is therefore essentially sensitive to the emission probability polarization dependence. One can deduce from Eq. (44) that

*A*varies between $-1$, for transition dipoles strictly oriented along the

*Y*axis, and 1 for transition dipoles strictly oriented along the

*X*axis. In all intermediate situations, such as different orientation and different distribution functions,

*A*will take intermediate values. Figure 8(b) depicts in particular the evolution of

*A*for two-photon fluorescent molecules lying within a cone aperture Ψ, and of different orientations ${\varphi}_{0}$ in the $X,Y$ plane (with ${\theta}_{0}=\pi /2$). In this graph I assumed parallel absorption and emission dipole moments in the molecules and followed the planar wave approximation detailed in Section 2. One can immediately notice that in order to retrieve a molecular order information (a Ψ value) in a sample, knowledge of the cone orientation in the sample plane with high precision is required. Indeed determining a single experimental value

*A*does not allow retrieving more than one parameter on the sample. In addition, this measurement is applicable only when the cone lies along the

*X*or

*Y*polarization projection axes, the case ${\varphi}_{0}=\pi /4$ leading to an indetermination ($A=0$) of the molecular order, since both

*X*and

*Y*projections are equivalent. This method is therefore strongly limited in heterogeneous samples where the full range of ${\varphi}_{0}$ values needs to be explored, such as in lipid membranes containing heterogeneous molecular order domains [Fig. 8(c)].

Finally, fluorescence anisotropy is successful only in cases restricted to simple distribution functions (for instance of cylindrical symmetry) and necessitates an *a priori* knowledge of either the mean orientation of the molecular distribution or its shape [39,40]. For this reason, anisotropy imaging can been applied in specific situations:

- ■ Single-molecule studies where only one molecule lies in the focal volume of the microscope objective [41],
- ■ Isotropic viscous solutions in steady-state measurements, to obtain molecular structural information such as the ${\mu}^{\mathrm{abs}}/{\mu}^{\mathrm{em}}$ relative angle.

This last frame of studies has led many investigations since the seminal work of Axelrod in 1979 [29], in which the orientation of long chain carbocyanine dyes in spherical lipid membranes in red blood cells was determined by using steady-state fluorescence anisotropy imaging. All further studies in lipid membranes have been limited so far to simple spherical membranes geometries such as in artificial giant unilamellar vesicles (GUVs) [30], red blood cells [45], swelling cells [46], spherical cells [31,32] and spherical nuclear envelopes [47]. In these membrane studies, the measurement was performed on the perimeter of the spherical membrane (${\theta}_{0}=\pi /2$), and the in-plane average orientation ${\varphi}_{0}$ of the molecular distribution function of the fluorescent probes in the lipid membrane was assumed to follow the direction normal to the membrane. Apart from membrane studies, fluorescence anisotropy imaging has been applied to the determination of the width of the angular distributions in biopolymers of cylindrical symmetry such as actin filaments [23], muscle fibers [39], and septin filaments [25]. In these studies, the orientation ${\varphi}_{0}$ of the fibers in the sample plane was visualized in the image and used as a known parameter to reduce the complexity of the studies.

In general, the molecular and biological media exhibit much more complex angular distributions that can differ strongly from pure cylindrical symmetries [48,49], or exhibit an averaged orientation that cannot be measured in an image. Therefore more refined polarization-resolved techniques are required.

#### 3.2. Stokes Vector Polarimetry and Mueller Matrices Imaging

One standard way to refine polarization-resolved studies is to decompose the incident field polarization into several states of known polarization, in a Stokes vector formalism. The parameters of a Stokes vector (defined by George Gabriel Stokes in 1852) describe a polarization state in a convenient way, especially for incoherent or partially polarized radiation, in terms of its total intensity, partial degree of polarization, and the shape parameters of an elliptic polarization.

*α*between

**E**and the

*X*direction. $I\left(L\right)$ and $I\left(R\right)$ are, respectively, the incident/measured intensities obtained by using a left-circular and a right-circular analyzer.

This formalism has been essentially applied to study the linear transmission by a sample, modeled as a transformation of the incident Stokes vector ${\mathbf{S}}_{\mathrm{in}}$ in a new output Stokes vector ${\mathbf{S}}_{\mathrm{out}}$ (measured from the output radiated field). This determines the so-called Mueller matrix [51] characteristic of the sample: ${\mathbf{S}}_{\mathrm{out}}=\mathbf{M}\cdot {\mathbf{S}}_{\mathrm{in}}$. The Mueller matrix therefore fully characterizes the optical polarization properties of the sample (providing that the optical path characteristics are corrected for in the polarization analysis). The 16 elements of a Mueller matrix can be obtained experimentally by measurements of the optical transmission of a sample under different combinations ($\alpha ={0}^{\circ},9{0}^{\circ},4{5}^{\circ},-4{5}^{\circ},L,R$) of polarizers and quarter-wave plates in the excitation path and analyzers in the detection path (Fig. 9).

The measurement of Stokes vector transmission can lead to interesting diagnostics such as scattering in a sample under linear optics interactions (in particular, the degree of linear polarization of a sample is directly quantified by $\mathrm{DOP}=\sqrt{{S}_{1}^{2}+{S}_{2}^{2}+{S}_{3}^{2}}/{S}_{0}$). Improvements in the contrast of the measured images can also be obtained by a posttreatement in which Mueller matrices are manipulated (Fig. 9) [50]. This approach has been applied in
linear optical microscopy [50,52,53] as well as for fluorescence [54]. Although very powerful for linear optics detection of
depolarization in biological samples, this formalism has not been applied yet to nonlinear contrasts in microscopy. It has, however, been developed in the measurement of the molecular *β* tensorial properties in a solution, in a nonlinear hyper-Rayleigh-scattering
scheme [55].

#### 3.3. Linear Polarization Polarimetry

Section 1 has shown that the use of a varying linear polarization for the incident field is capable of providing intensity dependencies that contain information on the molecular angular distribution in a sample. The tensorial expansions of the different nonlinear contrasts investigated show that the greater the number of field polarizations involved in the nonlinear interaction, the more tensorial coefficients can be determined in the macroscopic framework. The interpretation of those macroscopic components in terms of microscopic information has been performed by assuming a given molecular angular distribution function, of which several parameters can be determined depending on the contrast used.

In a polarimetric setup, the linear incident polarization of the incident electric field **E** (one or different fields, if required by the nonlinear contrast) is rotated in the sample plane $(X,Y)$ of an angle *α* relative to the *X* axis. The emitted intensities can be analyzed along two directions *X* and *Y*, leading to two measurements ${I}_{X}\left(\alpha \right)$ and ${I}_{Y}\left(\alpha \right)$. Assuming an incident planar wave with no perturbations of the polarization,

As we will see in the next subsection, this polarization may be distorted by polarization-dependent optics in the setup of the sample itself.

The sample acts as a transformation of this incident field in a new radiation resulting from the interaction between molecules and light:

Polarimetric responses are represented as polar diagrams in which each measurement point is a vector pointing from the origin with an amplitude equal to the detected intensity and a tilt angle relative to *X* equal to *α* (Figs. 3 and 5). This allows a direct visualization of the signal’s polarization response relative to a rotation of the excitation field polarization.

**Two-photon polarimetry (SHG and TPEF): principle and experimental setup**. A two-photon excitation nonlinear polarimetric microscope [40,56,57] uses a pulsed
laser excitation light source such as a tunable Ti:sapphire laser, which delivers 150 fs pulses at a repetition rate of 80 MHz. The incident wavelength of such a laser can be tuned between 690 and 1080 nm with typical averaged powers of a few hundreds of milliwatts. The laser beam is
reflected by a dichroic mirror and focused on the sample by a microscope objective, for which the NA can vary between 0.6 and 1.2. The use of high apertures presents the advantage of a higher optical resolution (typically 300 nm) and a more efficient signal collection in the
epi-geometry; however it exhibits more polarization distortion and mixing for the collected signal, which should be accounted for in the signal analysis (Subsection 3.6). In an epi-geometry setup, the backward emitted signal is collected by the
same objective and directed to a polarization beam splitter that separates the beam towards two detectors (avalanche photodiodes or photomultipliers). Images are obtained by scanning either the sample on a piezoelectric stage or the beam position by galvanometric mirror piezoelectric
scanning. In the first case, the sample is imaged, and then a precise location is chosen for the measurement of the polarimetric response. The second case is more advantageous in terms of a shorter acquisition time (typically a rate of 1 image per second is reached) and consists in
recording one image per incident polarization state. In order to vary the incident laser beam linear polarization, an achromatic half-wave plate, mounted on a step rotation motor, is placed at the entrance of the microscope. An increase of the polarization switching speed can be
obtained by the use of fast linear polarization controller, such as a Pockels cell placed before a quarter-wave plate. For each value of the polarization angle *α* (relative to *X*) from 0° to
180° or 360° (the second half of the measurement being redundant), the emitted signal is recorded on the two perpendicular directions *X* and *Y* [Fig. 10(a)]. This setup, which has been applied to the investigation of molecular order in molecular and biological samples recording SHG and TPEF signals [33,40,58,59], can be extended to forward detection schemes, the forward emitted signals being generally detected through a low-NA objective ($\mathrm{NA}\approx 0.5$).

**Three-photon polarimetry (THG, FMW, and CARS): principle and experimental setup.** The specificity of higher-order sum frequency nonlinear effects is that they rely on smaller coherent lengths and therefore are quasi-inefficient in the epi-direction detection, for which the
coherent length is even smaller than in the forward direction [13]. The previous considerations can then be applied to other degenerated contrasts such as THG, which has been recently investigated under a polarization-dependence technique [60].

In the case of nondegenerate nonlinear interactions in which several different wavelengths are involved, the experimental setup provides the possibility for multiple-wavelength polarization control [Fig. 10(a)] [28,61]. For CARS in either the resonant or nonresonant regimes (FWM), both pump and Stokes beams, generated from either picosecond synchronized pulsed lasers or optical parametric oscillators (OPOs) [17], are linearly polarized, and three different schemes of polarization tuning can be used: either the Stokes (pump) polarization is fixed along the *X* axis and the pump (Stokes) polarization rotates from 0°
to 360°, or both pump and Stokes polarization rotate simultaneously (with independent angles ${\alpha}_{p}$ and ${\alpha}_{s}$ relative to *X*). The incident beams are focused in the sample through a microscope objective, the emitted anti-Stokes signal then being detected in the forward direction and split by a polarizing beam splitter, separating both the *X*
and the *Y* directions towards similar detectors as for other nonlinear contrasts.

**Data analysis**. In a nonlinear polarimetric measurement, the measured data are two polarization-dependent responses ${I}_{X}\left(\alpha \right)$ and ${I}_{Y}\left(\alpha \right)$, which can be related either to the macroscopic nonlinear tensorial coefficient or to microscopic information such as the width and orientation of a molecular angular distribution. In both cases, the general approach is to fit these *α*-dependent
functions simultaneously in an iterative approach, which can lead to very long data analysis times when this fit is performed on an image of more than 10${}^{4}$ pixels. New approaches have been introduced, based on the Fourier analysis of these *α*-dependent functions, which allow considerable gain in time analysis [62]. In order to reduce the data acquisition time, a
decrease of the number of measured *α* angles would also be required. This number is essentially dependent on the signal-to-noise ratio of the measurements and on the nonlinear contrast used (which fixes the number of unknown parameters, as mentioned in Section 2).

#### 3.4. Complete Ellipsometry

**Principle and experimental setup**. In a complete polarimetric measurement, not only the rotation angle of a linear incident polarization can be varied, but also its ellipticity. This additional parameter, which controls the phase shift between the *X* and
the *Y* components of an incident field **E**, provides more degrees of freedom for investigating in particular nonlinear tensorial responses close to resonances. In this ellipsometric approach, the output polarization is completely characterized (not just
projected onto two *X* and *Y* directions), leading to a complete diagnostics that encompasses both amplitude and phase information.

In linear ellipsometry, the measurement consists in measuring the so-called Jones Matrix **J** characterizing a sample without depolarization (if there is any depolarization, the Mueller formalism described above should be used). The Jones vector used in a $(s,p)$ basis (which correspond to the $(X,Y)$ axes in microscopy) is [64]

*X*and

*Y*components) and diattenuation (amplitude difference between

*X*and

*Y*components).

This technique, transposed to nonlinear optics in a new “nonlinear optical ellipsometry” (NOE) method, has been applied to SHG signals from organized molecular layers on surfaces [Fig. 10(b)] [65]. It follows
a procedure similar to that in linear optics, with the additional difficulty that the detected signal is not a pure phase and amplitude perturbation of the incident field, but a coherently built up radiation originating from nonlinear interaction as represented in Eq. (29). It therefore has to deal with more complex Jones matrices [65,66]. In order to create a controlled incident elliptic polarization, a variable linear
polarization is followed by a quarter-wave plate. The characterization of the nonlinear radiated polarization state, at the exit of the setup, follows the same procedure, by interposing a quarter-wave plate, followed by a half-wave plate and a polarizing beam splitter that separates
both *X* and *Y* components of the final emitted polarization. An improvement of the speed of the measurement has been obtained by rapidly modulating the incident polarization state by using a photoelastic modulator and a detection scheme with four
detectors, in a way similar to that for a Stokes vector polarimeter [65,67,68].

**Data analysis**. The real and imaginary components of nonlinear tensors can be obtained, in the same ways as linear polarization polarimetry, through curve fitting of intensities acquired as a function of rotation angles of the optical elements (either polarizers, half-wave
plates, or quarter-wave plates) [65,66]. The tensor elements are then determined by mathematically combining different values for several incident polarization states [67,68]. A determination of Fourier components analysis has also shown the possibility to provide a faster analysis, with the possibility to combine such approach with principal component analysis (PCA) in samples of unknown
composition, which allows one to retrieve information on a sample without the need to fully express its macroscopic tensorial components [65].

#### 3.5. Complex Polarization Manipulation

Last, other polarization-dependent schemes can be developed that rely on unconventional excitation polarization states.

**Spatial polarization shaping.** First, it appears from the analyzes above that one important component of the information is missing in microscopy imaging: the *Z* coupling components of the tensorial responses. These components, however, contain all
information relative to possible orientations of the molecular structures out of the sample plane, and relying on an excitation field in the $(X,Y)$ sample plane forces the study to the analysis of projected tensors. While for a multipolar tensor originating from complex multipolar symmetries the analysis of the projected tensor can provide information on this out-of-plane component [57], this is impossible in the case of a simpler structures such as one of cylindrical symmetry. In such situations, when the knowledge of possible out-of plane orientation is limited, different strategies are possible.

- ■ One strategy consists in changing the detection optical pathway focusing conditions, in order to probe the radiation pattern of the induced nonlinear dipole, which will be affected by its off-plane angle. This method, called “defocused imaging,” has been successful for determining the three-dimensional orientation of SHG active nanocrystals [69].
- ■ Another way is to force the incident polarization component along
*Z*to be predominant by manipulating the incident polarization state at the entrance of the microscope. Controlling the polarization state of the incident light over its planar wavefront is a convenient way to produce focused beams transformed in their vectorial properties. For instance, the use of radially polarized beams or phase plates with*π*phase shift quadrants produces large longitudinal components that can be exploited for out-of plane coupling [70,71]. - ■ Engineered phase plates with
*π*phase shift quadrants can be also used as a detection device for the*Z*component of radiating induced dipoles when such phase plates are placed in the back focal plane of the collection objective lens [71].

The use of spatial light modulators has led to flexible solutions for polarization manipulation and today provides a very wide range of possibilities for the manipulation of optical fields at the focus of an objective. Manipulating the excitation polarization state in single-molecule fluorescence has allowed precise determination of molecules’ three-dimensional orientation [72]. When applied to nonlinear microscopy, such flexible polarization manipulation can lead to fascinating effects, from spatial resolution enhancement [73,74] to enriched light–matter SHG interactions leading to three-dimensional information [75–78].

**Time and frequency polarization shaping.** A more recent advance has been made in the field of time and frequency pulse shaping, where an ultrashort pulse ($\approx 5\u201330\phantom{\rule{3.00003pt}{0ex}}\mathrm{fs}$) containing a large range of incident frequencies (corresponding to up to 100 nm in wavelength range) can be manipulated in its spectral domain. A pulse shaper, that is, a grating that disperses the field spectrum, before transmission through a spatial light
modulator placed in the Fourier plane of a $4f$ optical setup, allows one to control this phase profile in phase, amplitude, and polarization [79]. This last parameter results in particular in the possibility of encoding the incident field in polarization, which can be
distinct for the different incident frequencies. This possibility has been exploited to read out the whole two-dimensional projection of a SHG macroscopic tensor, in a single-pulse excitation, by the polarized measurement of the radiated SHG spectrum from molecules in a crystal
[Fig. 10(c)] [63].

#### 3.6. Controlling Polarization States in Microscopy

Although polarimetry microscopy imaging is a powerful technique, relevant information can be retrieved only if the polarization state is perfectly known at the focal spot of the objective. A linear polarization state set at the entrance of a microscope is, however, often distorted at the focus. The first origin of polarization distortions is the microscopy optical setup itself.

**First, high-NA focusing** produces strong deformations of the incident polarization state that are due to the extra components of polarization originating from highly tilted wave vector components of the focusing beam. In particular, a longitudinal component of the incident
electric field **E** along the propagation direction *Z* appears to be nonnegligible for high NAs at the border of the focal spot [Fig. 4(b)] [80]. This component will couple with
molecular transition dipoles oriented along *Z* and possibly add some new contributions in the emission radiative dipoles. Formally, accounting for this effect requires developing a rigorous vectorial expression of $\mathbf{E}\left(\mathbf{r}\right)$ in the different contrasts described in Section 2 [Eqs. (17) and (26)]. This effect, which will occur in all contrasts, originating
from either incoherent fluorescence or coherent nonlinear effects, has been particularly studied in the case of SHG [81,82]. Theoretical developments show that for about $\mathrm{NA}>0.8$, such coupling can deform the polarimetric response from a sample with off-plane orientation, which is detrimental for further data analysis if this effect is not accounted for [63].

**Second, high-NA collection** can lead to polarization mixtures because of the high tilt angle of emitted wave vectors. A theoretical development of this effect has been performed on fluorescence emission dipoles [29]. In the case of a
collection over a high aperture, then an integration over the entire aperture has to be performed in Eq. (17), accounting for the **k** dependence of the radiated field [Eq. (7)].
The emission probability (Section 2) along an analyzing direction $I=(X,Y)$, integrated over all propagation directions **k** within the NA of the collection, is then written [29] as

*X*,

*Y*, and

*Z*polarization components of the radiated dipole, increase NA is increased.

This formalism can be extended to any nonlinear optical contrast, accounting for its coherent properties: whereas in fluorescence the intensities add up, here the amplitudes have to be added to form the emission signal [56].

Figure 11 depicts calculated TPEF and SHG polarization dependencies in typical samples, accounting for both excitation and collection effects. It shows how a molecular angular distribution along a cone can lead to strong modifications of the TPEF and SHG polarimetric responses when oriented out of the sample plane. When the cone lies in the sample plane, however, the deformations are quite insignificant whatever the NA used, owing to the strong predominant coupling interaction from the in-plane components of the nonlinear tensors involved.

**Third, a reflection on mirrors and dichroic beam splitter made of dielectric multilayers** produces a phase shift between the *X* and *Y* polarization components of the electric field **E**, which induces an ellipticity in the
polarization. In addition the reflection or transmission coefficients of mirrors or the dichroic beam splitter might be different for both *X* and *Y* components, causing diattenuation (term used for the transmission property) or dichroism (term used by
extension for a reflection property). In epi-geometry microscopy imaging setups, the optics that causes the strongest deformation is known to be a dichroic beam splitter that separates the incident from the emitted signal. Considering an incident electric field polarization written as
in Eq. (46), its reflection on the dichroic beam splitter will cause a *δ* phase shift and a *γ* dichroism factor between its *X* and *Y* components. The field entering the
microscope objective can then be written [56] as

Equation (50) implies that for a rotating incident polarization with a variable angle *α* between *X* and the incident linear polarization **E**, its linear state will be preserved for the
polarization directions along *X* (*α* = 0°) and *Y* (*α* = 90°). For any intermediate polarization, the dichroism and
ellipticity parameters will cause the polarization to be elliptic, with a maximum ellipticity when the incident polarization reaches the intermediate direction *α* = 45°. This situation is illustrated in Fig. 12(a). Under strong perturbations due to large ellipticity factors, the polarization response from any sample can be strongly modified, as shown in Fig. 12(b) in the case of a TPEF response from a 1D molecular
crystal.

Characterizing polarization distortions from reflections in the setup therefore requires knowing both the *γ* and the *δ* factors, which is necessary before any polarimetric analysis. Various polarization diagnostics are possible, the most universal being
ellipsometry in the same way as for linear characterization of polarization states [83]. Ellipsometry can be implemented, in particular, in a microscope equipped with a forward detection port, working for the incident wavelength [59]. A technique using the epi-detection port of the microscope has been developed based on the fluorescence signal from an isotropic material. Indeed, assuming that the molecular distribution is isotropic, then $f\left(\mathrm{\Omega}\right)=1/\left(4{\pi}^{2}\right)$ in Eq. (17), and the resulting polarimetric data can be fitted by using *γ* and *δ* as unknown parameters. This procedure has been used in molecules dispersed in a polymer film [56]; however, this configuration presents some inconvenience. First, the knowledge of the angles between the absorption and emission dipoles of the molecule is required. Second, at high concentration the distance between molecules can be so small
that nonradiative energy transfer (homo-FRET) occurs between them, leading to a depolarization. Very dilute polymer thin films are therefore required. An alternative has been found to avoid these issues: using molecules directly diluted in a solution [84]. In this situation the molecules are freely diffusing and rotating with a rotational time scale much faster than their fluorescence lifetime. Therefore the absorption and emission steps can be considered uncorrelated in angle and position, allowing a new expression for
the fluorescence signal where the incident field characteristics are present in a much simpler expression:

*C*contains time and spatial averaging factors, which do not affect the polarization response shape because the time fluctuations of the positions

**r**and angles Ω are occurring at a much faster time scale than the fluorescence lifetime of the molecules. In Eq. (51), only $(\delta ,\gamma )$ can be considered unknown parameters. Typical responses from two-photon fluorescent solutions are shown in Fig. 12(c), where the effect of both parameters on the deformation of the isotropic polarization response are visible.

## 4. Polarization-Resolved Two-Fluorescence Microscopy: Probing Molecular Order in Cell Membranes

In this section I describe an example of application of polarization-resolved TPEF in the structural investigation of a biological system of complex architecture: the cell membrane. This membrane, constituted of different lipid types, molecules (such as cholesterol), and membrane protein, exhibits strong spatial heterogeneities, which are furthermore highly dynamic because its biomolecules are constantly interacting together and with the cell cytoplasm. Historically, the organization of cell membranes has been essentially studied in the frame of molecular spatial localization and its time dynamics. Fluorescence recovery after photobleaching (FRAP) [85], fluorescence correlation spectroscopy (FCS) [86] and single-molecule tracking (SMT) [87] have allowed a better understanding of the mobility of proteins and lipids and their perturbation by specific conditions of the cells, bringing a considerable amount of information on the nanoscale organization of the cell membrane and its associated signaling dynamics. Less attention has been directed towards protein and lipid orientations in cell membranes.

In order to answer questions such as those related to the structural organization of cell lipid membranes, polarization-resolved fluorescence imaging requires a fluorescent label that can report the orientational order behavior of the studied lipid environment, which means being rigidly attached to this system. Probing molecular order in lipid membranes has been quite successful because of the possibility to “rigidly” embed fluorescent probes within the membrane leaflets, using strongly conjugated fluorescent molecules attached to aliphatic tails. Its application to the measurement of membrane proteins’ orientational behavior is more complex. Major Histocompatibility Complex Class I (MHC I), an important cell membrane receptor protein complex, has for instance been successfully labeled in a rigid way by using a construct in which a green fluorescent protein (GFP) is included within an intermediate part of the membrane protein [88]. The orientational behavior of this system has been studied on cell blebs where the morphology of the membrane can be more easily analyzed [46].

Recently, the application of TPEF polarimetry has allowed the investigation of two issues that cannot be addressed by a pure ratiometric method such as in fluorescence anisotropy (Subsection 3.1): the investigation of the orientational organization in coexisting liquid phase with short-range order (Lo) and disordered liquid (Ld) fluid domains of micrometric sizes in artificial membranes (GUVs), and in cell membranes of nonspherical shapes.

GUVs made of lipid mixtures are in particular considered as model systems for the investigation of lipid interactions [89–92]. They can exhibit coexisting domains with different fluidity, elasticity, and polarity properties. This phase segregation into gel, liquid ordered and disordered environments is closely related to fundamental cell processes, in particular in cell signalling, where the existence of lipid-specific functional “raft” platforms is still a debate [93–99]. So far, imaging lipid domains has relied on the use of dedicated fluorescent probes, specifically partitioning in regions of known lipid composition or local polarity [100–102]. Such probes are also expected to undergo specific orientational orders detectable by fluorescence anisotropy [100,101,103].

Figure 13 shows typical TPEF polarization responses of fluorescent molecules di-8-ANEPPQ in GUVs formed from a ternary mixture of the lipids sphingomyelin, DOPC, and cholesterol [89]. In such mixtures, Lo domains are known to be essentially constituted by enriched sphingomyelin and cholesterol regions, whereas DOPC is mainly present in disordered liquid regions (Ld).

The analysis of the TPEF polarimetric measurements is based on a fit of Eq. (17), assuming a negligible effect of the high-NA focusing polarization distortion, and accounting for the high-NA collection in the epi-detection geometry. Additional effects are also accounted for, namely, the instrumental polarization distortion (Subsection 3.6) and the effects of distinct absorption and emission angles of the molecular transition dipoles. Since the region of the membrane used for polarimetric analysis lies on the equator of the membrane, the molecular angular distribution $f\left(\mathrm{\Omega}\right)$ in Eq. (17) is modeled as a filled cone of molecules oriented, after time average, within a cone aperture Ψ, which lies in the sample plane and with an azimuthal orientation ${\varphi}_{0}$. Fitting the polarimetric data on both ${I}_{X}\left(\alpha \right)$ and ${I}_{Y}\left(\alpha \right)$ analysis channels (Fig. 13(b)) allows us to give a quantitative analysis of the local molecular order Ψ in different phases of the GUVs, independently of the position investigated on the vesicle or cell contour.

Although the Lo or Ld environments cannot be identified in a pure fluorescence image, the TPEF polarimetric analysis permits us to directly create an image of the spatial distribution of molecular order [Fig. 13(a)]. Typical cone aperture values range between $30\xb0<\mathrm{\Psi}<80\xb0$ (Lo phases) and $90\xb0<\mathrm{\Psi}<170\xb0$ (Ld phases) depending on the probe molecule. In the Ld phase, where the lipid acyl chains are highly disordered, the aperture angle is therefore significantly increased. TPEF polarimetry shows overall that molecular order information is dependent on the fluorescent probe structure, primarily because it is driven by lipid–fluorophore interactions, which are influenced by the molecular head position. In particular Ψ takes larger values when the molecule inclusion localization takes place in the periphery part of the membrane [32,101,104]. Probes located near the more ordered headgroup region exhibit quasi-one-dimensional order in gel or Lo phases with cone aperture angles below 40° [30]. A similar behavior is observed for di-8-ANEPPQ, although with a slightly higher flexibility ($\mathrm{\Psi}\approx 35\xb0$ in Lo phases and $\mathrm{\Psi}\approx 90\xb0$ in Ld phases). The values obtained for di-8-ANEPPQ in Ld phases are close to the ones found previously for the widely used BODIPY-PC fluorescent lipid probe [31].

TPEF polarimetry measurements on COS-7 cells (Fig. 13c) show that the molecules’ mean orientation lies roughly along the membrane normal direction. However, at many points the membrane is of complex shape, and its global orientation difficult to visualize. The simultaneous fitting on both Ψ and ${\varphi}_{0}$ parameters makes it possible to avoid speculating on the local membrane contour as was previously done [22,31].

The measured cone aperture angle on COS-7 cell membranes shows that the di-8-ANEPPQ probes behave in a slightly more disordered way than in DOPC GUV membranes. A possible hypothesis of this relatively high degree of disorder is the membrane’s local morphology, which furthermore includes dynamics due to cell trafficking. The plasma membrane folding has been shown to occur at a spatial scale much below the diffraction limit in cells (from 20 to 100 nm) [18,19,105,106]. Polarimetry therefore also probes the properties of the membrane’s subresolution scale structure: below the 300 nm optical limit, any disorder of the membrane at nanometer scales (ruffling, vesiculation) will lead to an increase of the measured cone aperture of the probe molecules. In a more general context TPEF polarimetry can be applied to the imaging of heterogeneous membranes organization occurring in endocytosis, exocytosis [107], and cell surface ruffling [22].

Last, in the studies described here, artifacts leading to a overestimation of the cone aperture of the molecular distribution have been considered. One of them is a possible population of freely diffusing molecules in the cytoplasm below the plasma membrane. This population should be excluded by analyzing polarimetric TPEF only on the membrane contour. Second, fluorescence resonance energy transfer (homo-FRET) may occur between molecules in the membrane.

Homo-FRET is the property of an excited molecule to transfer its energy in a nonradiative way, through a dipole–dipole interaction, to a neighbor molecule that will emit fluorescence light [108]. Since the orientation of the two molecules are uncorrelated, the absorption and emission steps are decoupled in the fluorescence process, which creates depolarization and therefore a loss of orientational information. This process, which occurs at intermolecular distances of a few nanometers [109], has been used to identify possible protein clusters in cells [20,110–112]. Homo-FRET affects polarimetric TPEF data in such a way that at high transfer efficiencies (when molecules are highly concentrated), the polarimetric response from a cone aperture distribution loses its characteristics and tends to equalize the analyzed ${I}_{X}\left(\alpha \right)$ and ${I}_{Y}\left(\alpha \right)$ responses, which are occurring for a complete depolarization [40]. Therefore a preliminary knowledge of possible homo-FRET efficiency in a sample is important in order to correctly interpret fluorescence polarization-dependence data.

## 5. Polarization-resolved Coherent Nonlinear Contrasts in Tissues

In this section I describe recent work on nonlinear coherent optical contrasts performed in SHG, THG, and CARS in biological tissues. The specificities of these samples and in particular their consequence in the polarimetric analysis will be described.

#### 5.1. SHG in Fibril Structures

Since its first developments [10,113,114] and its introduction in bio-imaging [115–117], SHG microscopy is now widely used to image ordered biomolecular assemblies in complex samples at depths reaching a few hundreds of micrometers. Coherent SHG occurring naturally in noncentrosymmetric structures such as collagen type I [10], skeletal muscles [3], and microtubules [117], is today exploited as a functional contrast [118–120], possibly in conjunction with TPEF [121–124], with the ultimate goal of developing diagnostics of pathological effects related to tissues and cell architecture.

SHG polarimetry imaging was introduced decades ago [125] in ordered molecular samples. Recent work has demonstrated that rich information is contained in polarization responses recorded from a tunable incident linear polarization in the sample plane [56], for instance to distinguish specifically the local nature (symmetry, disorder) of molecular assemblies in molecular monolayers [58] and in crystals [126] down to the nanometric scale [34,57]. Its extension to biology, mostly concentrated on the study of the structure of collagen, acto-myosin, and tubulin assemblies in muscle fibers and other types of tissue, including cornea, led to a large number of current studies [127–135].

Quantifying structural factors in such fibril structures has been performed essentially by using the symmetry of the crystalline collagen type I, which belongs to the ${C}_{6}$ crystalline point group [10], stemming from its triple-helix structure. This point group is characterized by a tensor ${\chi}^{\left(2\right)}$ that exhibits seven independent coefficients in the $(1,2,3)$ crystalline frame [26]:

In the SHG process the excitation is degenerate; so the two last indices can be permuted. In addition, far from resonance, the SHG tensor follows the Kleinman symmetry rule, and therefore all indices can be permuted. This results in a two-component tensor: ${\chi}_{333}^{\left(2\right)}$, ${\chi}_{311\left(\mathrm{permut.}\right)}^{\left(2\right)}={\chi}_{322\left(\mathrm{permut.}\right)}^{\left(2\right)}$ (where “permut.” indicates that permutations on the corresponding indices have to be included).

Collagen has also been modeled by an ensemble of one-dimensional SHG-active individual one-dimensional molecules lying along a helix structure, within an angular distribution $f\left(\mathrm{\Omega}\right)$ of cylindrical symmetry [132,133] (Fig. 2). The study of the molecular origin of the large SHG signals from collagen have indeed been shown to originate from the tightly packed assembly of moderately SHG-active molecules (presumably peptide bonds), resulting in an efficient coherent buildup of the SHG signal [136,137]. The resulting distribution is therefore most generally modeled by a cone surface shape of given aperture Ψ (Fig. 2), which has also been applied to other fibril structures made of chiral moieties [134]. The resulting microscopic tensor can be calculated by applying Eq. (24) to one-dimensional molecules for which ${\beta}_{zzz}$ is the only nonvanishing component. This leads to two nonvanishing coefficients in the microscopic frame $(1,2,3)$:

The SHG polarimetric response can then be calculated by rotating the obtained microscopic tensor in the macroscopic frame $(X,Y,Z)$, using Eq. (1). Supposing a collagen fiber (possibly made of an ensemble of fibrils of the same symmetry) aligned in the sample plane $(X,Y)$, excited by a polarization in this sample plane, and with ${\varphi}_{0}$ denoting the orientation angle of the fiber relative to the *X* axis (Fig. 14), the read-out of the macroscopic tensor involves six components in the sample
plane:

For ${\varphi}_{0}=0$ in particular, the application of Eq. (56) to Eq. (27) with $({E}_{X}\left(\alpha \right),{E}_{Y}\left(\alpha \right))=(cos\alpha ,sin\alpha )$ the excitation field in the sample plane, leads to

These SHG polarimetric responses, normalized by ${\chi}_{131}^{\left(2\right)}={\chi}_{311}^{\left(2\right)}$ (in case of a nonresonant excitation allowing microscopic index permutations), are therefore only functions of the ratio ${\chi}_{333}^{\left(2\right)}/{\chi}_{311}^{\left(2\right)}$ (or of the cone aperture Ψ). Calculated responses using Eq. (57) are depicted in Figs. 5(a), 5(b) for different Ψ values. In the most general case ${I}_{X}\left(\alpha \right)$ and ${I}_{Y}\left(\alpha \right)$ are also dependent on ${\varphi}_{0}$; then the full expression of the macroscopic tensor in Eq. (56) should be used. The two parameters of the molecular distribution in the fibril, Ψ and ${\varphi}_{0}$, can be determined from the fit of a SHG polarimetric measurement as illustrated in Fig. 14. Typical ${\chi}_{333}^{\left(2\right)}/{\chi}_{311}^{\left(2\right)}$ values measured in historical works on adult collagen tendons [Fig. 14(a)] range between 1.2 [10] and 2.6 [130], depending on the spatial resolution used. More recent work has been performed in different types of fibril structures [133,134], where this ratio is found to be between 0.4 and 2.8 (corresponding to a molecular order angle Ψ between 40° and 65°) [Fig. 14(b), (c)]. Progress in image analysis also allows us today to depict cartographies of this molecular order information [Fig. 14(c)], which is particularly interesting for the investigation of heterogeneous samples.

#### 5.2. Polarization Distortions Correction in Scattering Media

Although many studies have been performed in order to investigate the microscopic architecture of collagen in tissues from such polarimetric SHG data, important issues have to be accounted for when studying polarization-dependent optical signals in such complex heterogeneous samples. Apart from the instrumental effects mentioned above, other polarization distortions can indeed originate from the sample itself, especially for propagation through micrometric scale depths where anisotropy and scattering are present. These effects are detailed below.

### 5.2a. Birefringence

One-dimensional structures, which are strongly anisotropic, can exhibit different refraction indices and absorption coefficients along their main symmetry axis (generally called 3) compared with their values along the perpendicular optical axes 1 and 2. This property can
considerably modify the optical polarization propagating through the material. In particular, in microscopy this difference in refractive indices (birefringence) introduces a phase shift in the input field polarization state between its component along the material main axis and
the perpendicular component in the sample plane. Birefringence can be significant in crystalline [138] and biological samples, in particular from dense fibril structures such as collagen, even at depths of a few micrometers [139,140]. Collagen optical anisotropy has been studied particularly in linear optics imaging, using polarized optical coherent tomography approaches [141–144]. Its influence on nonlinear optical contrasts requires investigating the polarization distortion of both excitation and radiated fields. Considering a one-dimensional structure with its main axis 3 and perpendicular axis 1 oriented in the
sample plane (the remaining axis 2 lying along the propagation direction *Z*), then the optical field propagating through this structure will have, at depth *Z*, a polarization state

In anisotropic samples in which the incident field undergoes a birefringence retardation, $E\left(\alpha \right)$ [Eq. (46)] therefore has to be rewritten to account for the consequent polarization distortions. Here we will consider a one-dimensional object projected in the sample plane as a uniaxial structure, which is
consistent with a cylindrical-symmetry distribution lying in the $(X,Y)$ plane, relevant in most of the systems imaged in nonlinear microscopy. In the other cases only a picture of the sample projection is possible, since the optical coupling is limited to the *X* and *Y* polarization directions.
Considering now that the main axis 3 of the structure is oriented at an angle ${\mathrm{\Theta}}_{b}$ relative to *X*, then the incident field polarization $\mathbf{E}(\alpha ,Z)$ at depth *Z* can be finally written in the macroscopic $(X,Y)$ frame, in a planar wave approximation:

In addition to its effect on the incident excitation field, the birefringence also affects the detected signal, which propagates back in the sample (or through the sample) in the case of a backward detection (or forward detection). An approach similar to that can be implemented to
account for this effect, assuming that the same $\Delta n$ value applies to both incident and emitted wavelengths (this is the case if no resonance is involved). The relation between the macroscopic emission dipole components (either occurring from a TPEF process or an induced nonlinear SHG process) at the focal depth
*Z* and at the exit of the sample $(Z=0)$ follows the same equation as in Eq. (59), introducing the detection wavelength in the expression of ${\mathrm{\Phi}}_{b}$.

Equation (59) shows that structures exhibiting optical axes of *a priori* unknown orientations in the sample plane can lead to erroneous deductions of the measured properties such as SHG nonlinear tensorial components.
This effect, evoked in early work on SHG microscopy in collagen structures [128], has been quantified recently [59,145]. SHG polarimetric responses from collagen
thick fibers from a rat tail tendon are seen to be strongly affected by birefringence even though the penetration depth is only a few micrometers [Fig. 15(b)]. In particular, birefringence introduces extra lobes in the polar plot shape compared
with those seen in the absence of birefringence.

It is possible to retrieve birefringence parameters ${\mathrm{\Phi}}_{b}\left(Z\right)$ and ${\mathrm{\Theta}}_{b}$ from a fit of SHG polarimetric data, providing that the nonlinear coefficients of the structures are determined from a preliminary study at depth $Z\approx 0$ where the birefringence is negligible. This approach has shown overall that (i) the expected main axis orientation ${\mathrm{\Theta}}_{b}$ lies roughly along the observed collagen fiber orientation and (ii) the measured birefringence phase shift ${\mathrm{\Phi}}_{b}\left(Z\right)$ can reach up to $\pi /2$ at a penetration depth of about $Z\approx 50\phantom{\rule{0ex}{0ex}}\mu \mathrm{m}$. This is consistent with the birefringence values ${\mathrm{\Phi}}_{b}\approx 1.35\xb0/\mu \mathrm{m}$ ($\Delta n\approx 0.003$ reported in the literature [139,140]).

Finally, an estimation of the total birefringence of the sample accumulated over its whole thickness can be also done separately by using a standard ellipsometry technique [59]. For this, the forward detection port of the microscope can be
used for a polarimetric analysis of the incident fundamental beam (at the input *ω* frequency), propagating through the sample thickness [Fig. 10(a)]. Upon rotation of the incident linear polarization, the ellipticity occurring
from the sample birefringence is expected to modify the polarimetric dependence of the measured signal through the whole sample thickness *Z* [59].

### 5.2b. Diattenuation

In addition to the refractive index difference, the propagation through a one-dimensional anisotropic structure lying in the sample plane can also lead to a different attenuation factor between the two components of the incident polarization, along the structure axis 3 and its perpendicular direction 1 lying in the sample plane:

The diattenuation $1/\Delta L=1/{L}_{3}-1/{L}_{1}$ can reach 10^{−2} µm^{−1}
in collagen from tendons (Fig. 16) [145], meaning that this effect can lead to deformation of the polarimetric responses when penetrating close to
50–100 µm in a sample.

### 5.2c. Scattering

Multiple scattering, which is due mainly to the micrometer-scale index heterogeneities in the medium, has significant consequences in tissues that can be viewed as turbid media. Scattering can lead to a strong depolarization of an incident field propagating through a medium. Many
studies have been dedicated to characterizing depolarization properties of biological tissues, in particular using the Mueller matrix formalism [53,54]. For the optical contrasts described in this work,
the effect of scattering can be written as a time and space fluctuating phase added on both the ${E}_{X}$ and ${E}_{Y}$ components of either the incident or the emitted fields. Consequently the measured intensity will be an incoherent addition of both contributions with a scaling factor *η* that quantifies the polarization cross talk occurring from scattering. This
results in a polarization cross talk for the detected intensities:

The scattering cross talk has been measured to be about ${\eta}_{XY}\approx 0.1\u20130.2$ in collagen from tendons based in experimental results at large penetration depths (Fig. 16) [145]. In such samples, it has been shown that a reliable fit of the polarization resolved SHG data could be obtained only by accounting for the conjunction of the three effects: birefringence, diattenuation and scattering. Note that the suppression or decrease of scattering can be reached by using optical clearing, performed by a treatment of the tissue with glycerol, for instance [146].

#### 5.3. Polarization-Sensitive Third-Harmonic Generation Signals in Tissues

Apart from the very extensive work that has been performed on SHG in fibril structures from biological tissues, some attempts have recently demonstrated the interest of other higher-order nonlinear coherent contrasts. THG is in particular interesting, since it is not noncentrosymmetry sensitive and can therefore give information in anisotropic, but centro-symmetric structures. The specificity of THG with respect to propagation is, however, that THG signals in microscopy occur uniquely from interfaces between species of different refractive indices, since the coherence length of this process is very small in homogeneous media (both in the forward geometry and in the epi-geometry), and therefore only a change in local environment (local refractive index) can allow THG to be nonvanishing [13]. A recent study [60] in cornea tissues made of stromal lamellae structures has shown that forward-radiated THG and SHG signals are generally anticorrelated, indicating that the THG signal originates from the lamellar interfaces whereas SHG originates from regions within the lamellae, principally from the collagen substructures. Polarization-resolved THG imaging exhibits an isotropic component (revealed upon linear excitation) representative of cellular and anchoring structures, whereas its anisotropic component (revealed upon circular excitation) is representative of an alternate anisotropy direction between the lamellae, allowing the direct imaging of their stacking and heterogeneity with a micrometer three-dimensional resolution (Fig. 17).

#### 5.4. Polarization-resolved Coherent Anti-Stokes Raman Scattering in Crystalline and Biological Samples

CARS microscopy, which provides a unique chemically selective imaging method, is becoming highly attractive in biological imaging. Its applications include studies of living cell metabolism, organelle transport *in vivo*, and viral disease [147].

Polarization-resolved CARS was used in early work to characterize symmetry properties of molecular vibrational modes in isotropic media, quantified by their Raman depolarization ratio [148,149]. Using polarization properties of CARS has also been introduced to remove the nonresonant background in CARS spectra [150] and more recently in CARS images [151]. While interpreting polarization-resolved CARS data is relatively straightforward in isotropic media, the study of order in molecular samples close to vibrational resonances is much more complex, as we will see below.

*In isotropic media,* the CARS macroscopic tensor ${\chi}^{\left(3\right)}={{\chi}^{\left(3\right)}}^{\mathrm{NR}}+{{\chi}^{\left(3\right)}}^{R}$, due to symmetry reduction properties, exhibits only two nonvanishing components ${\chi}_{XXYY}^{\left(3\right)}={\chi}_{XYXY}^{\left(3\right)}$ and ${\chi}_{XXXX}^{\left(3\right)}$, with

Since the macroscopic ${\chi}_{IJKL}^{\left(3\right)}$ components are orientational averages of molecular ${\gamma}_{uvw\u03f5}$ coefficients over an isotropic distribution $f\left(\mathrm{\Omega}\right)=1/\left(4{\pi}^{2}\right)$ [Eq. (34)], ${\rho}_{R}$ therefore contains microscopic structural information on the vibrational modes of the molecules. Note that it is possible to define a nonresonant depolarization ratio, equal to 1/3 since ${\chi}_{XXYY}^{\left(3\right)}={\chi}_{XYXY}^{\left(3\right)}$ out of resonance.

Because of the low number of macroscopic components in an isotropic medium, tuning incident polarizations for either the pump or the Stokes beams in CARS can lead to a reliable determination of the depolarization ratio ${\rho}_{R}$. This has been done recently by using a polarimetric approach, tuning either the pump, the Stokes, or the pump and Stokes linear polarization directions in a sample plane of a isotropic solution [Fig. 17(a)] [61].

Polarization-sensitive CARS has also been recently applied to unravel molecular orientation information in *anisotropic samples*, such as water molecules in phospholipid bilayers [152], and ordered biomolecular assemblies in
tissues [153] or in liquid crystals [154]. In these studies, the incident polarizations are kept parallel to each other, and only qualitative information on sample orientation is obtained. An example of
a complete polarimetric CARS response, tuning both pump and Stokes polarizations, is shown in Fig. 17(b) for collagen type I fibers extracted from a rat tail tendon. Obviously a strong difference occurs compared with the response of an isotropic
solution [Fig. 17(c)], due to the anisotropy of the structure [155].

Retrieving a refined analysis of the CARS macroscopic tensor ${\chi}^{\left(3\right)}={{\chi}^{\left(3\right)}}^{\mathrm{NR}}+{{\chi}^{\left(3\right)}}^{R}$, however, is not straightforward since it involves dealing with as many as 81 components for each resonant and nonresonant contribution, as well as the knowledge of the sample orientation ${\mathrm{\Omega}}_{0}$. While the tensor structure of the nonresonant contribution ${{\chi}^{\left(3\right)}}^{\mathrm{NR}}$ can be *a priori* retrieved knowing the point group to which the object belongs, or from a molecular angular distribution model (such as above in the case of an isotropic solution), two parameters finally characterize the ${{\chi}^{\left(3\right)}}^{R}$ contribution: (i) a specific tensor structure, related to the symmetry of the vibrational modes, and (ii) the departure from Kleinman symmetry conditions, which does no longer permits the permutation of tensorial indices. To deal with (i), the tensor structure of a
vibrational mode can be found from group theory considerations, using the character table of the point group of the studied object [156]

The extraction of symmetry information in vibrational lines of an anisotropic sample has been demonstrated by using polarization-resolved CARS, on a model crystal of octahydrosilasesquioxane (H_{8}Si_{8}O_{12}), which belongs to the ${O}_{h}$ crystallographic point group [Fig. 18(c), (d)] [28]. In this system, equivalent to a pure fourth-order symmetry, two vibrational lines were studied: its totally symmetric ${\mathrm{A}}_{1g}$ vibrational mode (Si–H stretching at 2302 cm${}^{-1}$), and its degenerate ${\mathrm{E}}_{g}$ mode (O–Si–H bending at 932 cm${}^{-1}$). The totally symmetric ${\mathrm{A}}_{1g}$ mode carries the same resonant tensorial information as the nonresonant tensor. It was in particular shown that its polarization characteristics were consistent with this symmetry as well as a derivation from the Kleinman symmetry conditions, confirming the capacity
of polarization-resolved CARS to read out a fourth-order symmetry structure ($J=4$ order of symmetry, following the notations of Subsection 2.6). In contrast, a polarization-resolved Raman process, characterized by an *α* second-rank tensor, is not able to resolve the symmetry of this
vibrational mode, and can only provide an isotropic polarization-independent response. The study of the ${\mathrm{E}}_{g}$ mode showed very different polarization characteristics of its specific symmetry, with additional signatures of Kleinman symmetry deviations close to the vibrational resonance [Fig. 18(c)].

It is therefore possible to investigate the complex multipolar nature of vibrational resonances of an object of *a priori* known symmetry by implementing a fully polarization-resolved CARS microscopy technique in which both pump and Stokes input polarizations are
controlled and tuned independently. The methodology presented here can potentially reveal not only orientational but also structural information of vibrational resonances in a general-case object, with submicrometric resolution.

The implementation of polarization-resolved CARS in biological tissues is only in its infancy [157]. Implementing polarimetric CARS in samples of unknown composition and structure would, however, significantly extend the assets of CARS microscopy as a label-free molecular analysis technique, since in this situation neither the resonances nor the structure of the molecular assemblies are known. In order to resolve this issue, approaches using spherical decompositions of molecular distributions (such as those detailed in Subsection 2.6) can allow a more general analysis of polarimetric data [155].

## 6. Conclusion

Nonlinear optical contrasts applied to imaging benefit today from the significant progress made in terms of polarization control. Since these processes involve high-order matter–field interaction, the possibility to tune polarization states is now providing determining information on the structural behavior of molecular samples. We have seen here how a methodology based on polarimetry is able to approach the full complexity of multipolar symmetries in biological samples, including vibrational mode information up to the fourth-order symmetry. These multipolar approaches allow us to distinguish features that are not accessible via linear optics polarization-resolved microscopy, which involves lower-order interaction. These approaches open new prospects towards molecular order and symmetry properties read-out imaging, such as differentiating structural behavior related to specific biological functions. The field of nanomaterials engineering could also benefit from such methods since, for instance, submicrometric crystalline structures could be differentiated based on both their structural nanometric scale properties and their intrinsic vibrational responses.

## Appendix A

When transforming a tensor component from a molecular basis $({u}_{1},\dots ,{u}_{n})=(x,y,z)$ to a new basis of vectors components $({i}_{1},\dots ,{i}_{n})=(1,2,3)$, a rotation operator ${\Re}_{\mathrm{\Omega}}$ is applied based on projector components $(\mathbf{u}\cdot \mathbf{i})$ between the two transformation frames. Following the notation of the operation written in Eq. (1),

**u**in the frame of vectors

**i**. The components of this operator are therefore written as

Any other rotation can be expressed the same way.

## Appendix B

Let us limit the analysis to molecules of cylindrical symmetry around a high-symmetry axis, an orientation that can be defined by the $\mathrm{\Omega}=(\theta ,\varphi )$ orientation (only two Euler angles are necessary). The rotation of a susceptibility component ${\tau}_{{i}_{1}...{i}_{n}}$ (in the molecular frame) by an angle $\mathrm{\Omega}=(\theta ,\varphi )$ in a new frame, ${\tau}_{{I}_{1}...{I}_{n}}$, can be written as

Therefore

It is known that the multiplication of two spherical harmonics can be expressed by a linear combination of new spherical harmonics using the Wigner 3-*j* symbols (also related to the Clebsch–Gordan coefficients) [158]:

The Wigner 3-*j* symbols $\left(\begin{array}{ccc}\hfill {J}_{1}\hfill & \hfill {J}_{2}\hfill & \hfill J\hfill \\ \hfill {m}_{1}\hfill & \hfill {m}_{2}\hfill & \hfill m\hfill \end{array}\right)$ are integers or half-integers that verify the following properties [158]:

- ■ ${m}_{1}+{m}_{1}=m$
- ■ $|{J}_{1}-{J}_{2}|\le J\le |{J}_{1}+{J}_{2}|$
- ■ $\left(\begin{array}{ccc}\hfill {J}_{1}\hfill & \hfill {J}_{2}\hfill & \hfill J\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right)=0$ if ${J}_{1}+{J}_{2}+J$ is odd.

This last property implies that as ${J}_{1}={J}_{2}=1$, *J* must be even to retrieve a nonvanishing product.

The product of two spherical harmonics can therefore be simplified in

*j*Wigner symbol terms. The product of

*n*spherical harmonics of first order can therefore be written as

*J*and

*n*of same parity, $-J\le m\le J$ and $0\le J\le n$, since ${J}_{1}+\cdots +{J}_{n}=n$.

Finally, the tensor expression in the microscopic frame can be written in a new expression:

The coefficients can be simplified and written as

*J*and

*n*are of same parity, $-J\le m\le J$ and $0\le J\le n$.

## Acknowledgments

This work has been supported in part by the French Ministry of Research, the Agence Nationale de la Recherche (ANR), the Centre National pour la Recherche Scientifique (CNRS), the Région Provence Alpes Côte d’Azur (PACA), and the European Union (Grant CARSExplorer FP7 Health). I thank the students and colleagues from Institut Fresnel and Ecole Normale Supérieure de Cachan (France) who strongly contributed to the advances in polarized imaging microscopy: at Laboratoire de Photonique Quantique et Moléculaire (ENS Cachan, France), V. Le Floc’h, C. Anceau, S. Bidault, I. Ledoux, J. Zyss, D. Chauvat; and at Institut Fresnel (Marseille, France), A. Gasecka, T.-Y. Han, P. Schön, F. Munhoz, D. Aït-Belkacem, A. Kress, H. Ranchon, X. Wang, S. Brustlein, P. Ferrand, M. Roche, P. Réfrégier, and H. Rigneault.

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**Sophie Brasselet** obtained her Ph.D. in 1997 at University of Paris-Sud, France, with the Centre National des Etudes en Télécommunications, in the study of new types of multipolar molecules for nonlinear optics in polymers. She then spent a 2-year period as a postdoc at the
University of California, San Diego, and Stanford University, USA, in the detection of isolated fluorescent molecules on cell membranes by confocal fluorescence microscopy. After 6 years at Ecole Normale Supérieure de Cachan, France, as an assistant professor working on nonlinear
microscopy and optical manipulation of single molecules, she is now working as a CNRS researcher at Institute Fresnel, Marseille, France. She is currently developing new instrumentation concepts and tools based on nonlinear optics coupled with polarization resolution, dedicated to
biological imaging in cells and tissues, and to nanoplasmonics on metallic nanostructures.