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]
  • ■ THG: interface-sensitive contrast [12], applied to dense-structure imaging such as in lipid bodies [13] and other cellular structures
  • ■ CARS: chemically specific contrast applied to vibrational modes imaging in biomolecular structures such as aliphatic C–H stretching vibration of lipids [14,15]
  • ■ FWM: non-chemically-specific contrast visible in nonresonant CARS microscopy [16]

 

Figure 1 (a) Schematic drawing of a molecule excited by different frequencies ω. The radiation from the molecule originates either from an emission transition dipole moment (μem, fluorescence), or from a nonlinear induced dipole moment (p2ω, a coherent second-order nonlinear optical process). (b) Energy diagram scheme of the different contrasts addressed in this tutorial: one- and two-photon excited fluorescence (TPEF), second-harmonic generation (SHG), FWM, THG, and CARS. (c) Images and spectra obtained from such contrasts in a molecular crystal that is active for second-order nonlinear effects (more generally sum frequency generation), TPEF and CARS. Incident wavelengths: 816.8, 888.7, 1064 nm [17].

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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(Ω) 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(Ω) 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 [2022]. Protein interactions in supramolecular complexes such as biofilaments are also highly driven by orientational order [2325]. 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(Ω) 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(Ω) 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) Eω; Eω, 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 Ω=(θ,ϕ,ψ). These angles, used in the transformation from one frame to another, are represented schematically in Fig. 2(a). While the (θ,ϕ) 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 Ω0=(θ0,ϕ0,ψ0) 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.

 

Figure 2 Definition of the different frames, axes, and notation used in this tutorial: (a) (x,y,z), molecular frame; (1,2,3), microscopic frame, which is either the crystal unit-cell or molecular distribution frame; (b) (X,Y,Z), macroscopic frame; Ω=(θ,ϕ,ψ), Euler set of angles defining the orientation of the molecule in the microscopic frame; Ω0=(θ0,ϕ0,ψ0), Euler set of angles defining the orientation of the microscopic frame in the macroscopic frame. (c) Geometry of microscopy in the EPI epi- and forward-detection schemes.

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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 τu1,u2,,un expressed in the molecular frame, its expression in the microscopic frame will be, along the same line as for a vector rotation,

Ω(τ)i1,,in=τi1,,in(Ω)=u1,,unτu1,,un(u1i1)(unin)(Ω),
where Ω is the rotation operator of angle Ω relative to the microscopic axes, also defining the orientation of the molecular frame relative to the microscopic one. (u1i1)(Ω) is the projector (rotation matrix component) between the initial frame (u1,,un) and the frame (i1,,in). The molecular (u1,,un)=(x,y,z) axes are oriented by the Euler set of angles Ω=(θ,ϕ,ψ) in the (i1,,in)=(1,2,3) microscopic frame (Fig. 2); therefore (z3)(Ω)=cosθ, for instance. All (uiii)(Ω) components are given in Appendix A. Note that Eq. (1) can also be applied to the rotation between the microscopic (1,2,3) frame and the microscopic (X,Y,Z) frame (Fig. 2).

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

In-phPabsn-phPem.
The proportionality coefficient contains collection efficiencies and normalization factors. Since polarization analysis does not depend on these coefficients, we will often omit them in the future and replace “∝” with the sign “=.”

The absorption probability in a one-photon absorption process, denoted Pabs1-ph, is proportional to the imaginary part of the first order molecular susceptibility α(ω;ω) expressed at the incident optical frequency ω [26]:

Pabs1-ph=Im(α(ω;ω))(EωEω)=IJIm(αIJ(ω;ω))EIωEJω,
where Eω is the excitation polarization of the incident electromagnetic. ⊗ is the tensorial product, defined as (AB)IJ=AIBJ, and is the tensorial scalar product similar to the scalar product for vectors.

To get more insight into the anisotropy origin of the α(ω;ω) tensor, αIJ(ω;ω) 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]:

αIJ(ω;ω)=1ε0ħnμ0nIμn0J(ωn0ω)iΓn0,
where |n denotes the different excited levels of the molecule involved in the interaction process. μ0n=ψn(r)(er)ψ0(r), with ψn(r) being the wave function of the level |n of the molecule, is the transition dipole moment between its ground |0 and excited |n levels. μ0nI is thus the component of this transition dipole moment along 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. ωn0=ωnω0 is the frequency difference between the excited and ground levels, and Γn0 is the spectral frequency linewidth of the excited state |n.

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

Pabs1-phIm(αIJ)EIωEJωIJμ01Iμ10JEIωEJω=|μ01Eω|2.
This expression of the absorption probability can also be seen as the product of the incident intensity Iω (proportional to |Eω|2) and the absorption cross section of a single molecule σ011-ph (proportional to |μ01|2), a property that can be obtained from the Fermi golden rule [26].

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 α(ω;ω) properties, and a vectorial way using the transition moment dipoles μ properties.

μ01 will be denoted μabs 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 Pem,I, contains both the fluorescence quantum yield of the molecule and the characteristics of the radiated intensity Iem,I:

Pem,IIem,I|EemI|2,
where I is the unit vector along the direction I.

Considering a molecule of orientation Ω in the macroscopic frame, the far field Eem is the field radiated by the molecular emission dipole μem(Ω), 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

Eem(Ω,k)k×(k×μem(Ω))=μem(Ω),
where μem is the projection of the emission dipole on the direction perpendicular to the emission direction 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 μem lies in the (X,Y) plane and

|EemI|2|μIem(Ω)|2=|μem(Ω)I|2.

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 μem 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 [μabs=μem=μ(Ω)], 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

II1-ph(Ω)|μabs(Ω)Eω|2|μem(Ω)I|2.

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 (10151018s) (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 Pabs2-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 γ(ω;ω,ω,ω) (expressed at the frequency ω=ω+ωω) [26]:

Pabs2-phIm(γ(ω;ω,ω,ω))(EωEωEωEω)Im(γIJKL(ω;ω,ω,ω))EIωEJωEKωELω.

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]:

γIJKL(ω;ω,ω,ω)=16ε0ħ3n,m,νPIJ,KL[μ0nIμnmLμmνKμν0J][(ωn0ω)iΓn0][(ωm02ω)iΓm0][(ων0ω)iΓν0]PIJ,KL[μ0nIμnmJμmνLμν0K][(ωn0ω)+iΓn0][(ωm0ω)iΓm0][(ων0ω)iΓν0],
where the permutation operator PIJ,KL means that the sum should account for permutations on the indices (IJ) and (KL). In the two-level model (|1 and |0) approximation,
Pabs2-phnIJKLμ0nIμn1Jμ1nKμn0LEIωEJωEKωELω,
where the quantities μ0n=μn0 ( denoting the complex conjugate) and μ1n=μn1 involve additional |n levels in the system.

In the following calculation, we will assume that only one of the |n levels is of the dominant transition dipole and that it is furthermore nonresonant for the ω frequency (ωn0ωω). We will principally investigate one-dimensional molecules in which we can assimilate μ0n and μn1 to a single vector direction μabs along the molecular axis. The two-photon absorption can thus be defined by

Pabs2-ph|μabsEω|4.

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),

II2-ph|μabs(Ω)Eω|4|μem(Ω)I|2.

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 μabs 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 Pabs1-ph(μ01)2Iωcos2Θ, whereas in the TPEF process this photoselection is proportional to Pabs2-ph(μ01)4(Iω)2cos4Θ. 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 Eω 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 Eω. The molecule is assumed to lie in the sample plane. While the global orientation of the polarization responses points along the μ=μabs=μem 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 IX2-ph and IY2-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.

 

Figure 3 (a) Polarization dependence of the one-photon and two-photon fluorescence processes for one single molecule oriented in the sample plane (X,Y). The polar plots are representations of the fluorescent intensity as functions of the angle of rotation α of the incident polarization Eω. (b) Case of a wide molecular distribution along a cone, oriented along 30 and of aperture Ψ=30. (c) Case of an isotropic distribution.

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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,

IIn-ph(Ω)I1I2nμI1...μI2n(Ω)EI1...EI2n(μIμI(Ω))I1...I2nτI1...I2nII(Ω)EI1...EI2n,
where τ=μμα is a molecular fluorescence tensor containing both a multiphoton absorption n-rank μμ 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:

II1-ph(Ω)JKαJK(Ω)αII(Ω)EJEK(αα)JKII(Ω)EJEKII2-ph(Ω)JKLMγJKLM(Ω)αII(Ω)EJEKELEM(γα)JKLMII(Ω)EJEKELEM,
where αα and γα are the tensors associated with one- and two-photon fluorescence processes, defined at the molecular scale but expressed here in the macroscopic scale [the relation between tensorial expressions in different frames can be found by using Eq. (1)].

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(Ω), normalized such that f(Ω)dΩ=1. The probability density of the molecules to lie between the orientations Ω and Ω+dΩ in the (1,2,3) frame (Fig. 2) is therefore N(Ω)dΩ=Nf(Ω)dΩ (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

II2-ph=NNAVΩ|μabs(Ω,r)E(r)|4|Eem(Ω,r,k)I|2f(Ω)dΩdrdk
with ΩdΩ=θ=0πϕ=02πψ=02πsinθdθdϕdψ and 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:

  • 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.
  • 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).
  • Ω means that the dependence of the transition dipole moments involved should be integrated over the whole support of the f(Ω) molecular angular distribution function.
  • ■ The 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.

 

Figure 4 (a) Radiation from a dipole excited in the excitation volume. (b) Map of the Z and X amplitude components of the excitation field in the focal plane when the incident light is polarized along X, using a NA = 1.2 objective.

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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 Eem and thus the V integration.
  • ■ Planar wave collection. We will therefore ignore the integration over NA, and consider a single k detection direction along Z. This entails |EemI|2=|μem(Ω)I|2. The case of high-NA collection integration is discussed in Subsection 3.6.
  • ■ Absorption and emission dipoles along the same direction μem=μabs=μ. An angle between these dipoles, if known, can be introduced into Eq. (17).

Under these assumptions, Eq. (17) becomes

II2-ph=NΩ|μ(Ω)E|4|μem(Ω)I|2f(Ω)dΩ=NΩ(γα)JKLMII(Ω)EJEKELEMf(Ω)dΩ,
where both multivectorial and tensorial forms are expressed.

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(Ω) 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(Ω) function [denoted Ω0=(θ0,ϕ0,ψ0) 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):

μ(Ω,Ω0)=Ω0(μ(Ω))=cosϕ0cosθ0sinϕ0cosϕ0sinθ0sinϕ0cosθ0cosϕ0sinϕ0sinθ0sinθ00cosθ0sinθcosϕsinθsinϕcosθ.

(2) Similarly, a tensorial approach could be used, using the molecular two-photon fluorescence tensor τ=γα 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:

τu1....unτi1...in(Ω)=Ω(τ)=u1....unτu1....un(u1i1)(unin)(Ω),τi1...in(Ω)Ti1...in=NΩτi1...in(Ω)f(Ω)dΩ,Ti1...inTI1...In(Ω0)=Ω0(T)I1...In=i1...inTi1...in(i1I1)(inIn)(Ω0),
where all projector coefficients (u1i1), and (i1I1), are given in Appendix A. (u1....un)=(u,v,w) are indices of the molecular frame with n=6 (τ=γα being a six rank tensor), (i1...in)=(1,2,3) are indices of the microscopic (molecular distribution) frame, and (I1...In)=(X,Y,Z) are indices of the macroscopic frame. The first operation expresses 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:

II2-ph=JKLMTJKLMIIEJEKELEM,
where TJKLMII 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 IX2-ph(α) and IY2-ph(α) for a varying incident linear polarization in the sample plane (X,Y) with an angle α relative to X (EX=cosα,EY=sinα). 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 Ψ=30 around the direction Ω0=(θ0=90,ϕ0=30,ψ0=0), 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 IX2-ph(α) and IY2-ph(α) 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 IX2-ph(α)+IY2-ph(α), 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:

II2-ph(α)JKLMTJKLMIIEJEKELEM(α),TJKLMII=NΩγJKLM(Ω)αII(Ω)f(Ω)dΩ,
where Ω denotes here, for the sake of simplicity, the orientation of the molecules in the macroscopic frame, supposing that f(Ω) is also defined in this frame [the case where f(Ω) is defined in its own microscopic frame should account for an additional tensorial rotation, as detailed in Eq. (20)].

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 pSHG,2ω=β(2ω;ω,ω):EωEω, with β the molecular hyperpolarizability tensor:

pISHG(Ω)=JKβIJK(Ω)EJωEKω
(below we will omit the 2ω frequency specificity of this optical response for the sake of simplicity). βIJK(Ω) is the expression, in the macroscopic frame, of the molecular susceptibility 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(Ω) and orientation Ω0 (Fig. 2), βIJK(Ω) can be replaced by βIJK(Ω,Ω0) and rewritten similarly as for the fluorescence tensors [Eq. (1)]:
βuvwβijk(Ω)=Ω(β)βIJK(Ω,Ω0)=Ω0(β(Ω)).

Note that the molecular β(2ω;ω,ω) 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 α(ω;ω) and γ(ω;ω,ω,ω) 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]

βIJK(2ω;ω,ω)=12ε0ħ2mnPJK[μ0nIμnmJμm0K][(ωm0ω)iΓm0][(ωn02ω)iΓn0]+PJK[μ0nJμnmIμm0K][(ωn0+ω)+iΓn0][(ωmn2ω)iΓmn]+PJK[μ0nJμnmIμm0K][(ωn0ω)iΓn0][(ωmn+2ω)+iΓmn]+PJK[μ0nJμnmKμm0I][(ωn0+ω)+iΓn0][(ωm0+2ω)+iΓm0].

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:

IISHG=|NNAVΩEISHG(Ω,r,k)f(Ω)dΩdrdk|2
with N the molecular density and ESHG the radiated SHG field originating from the molecular dipole sources pSHG [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 Eω is independent of r in Eq. (23).
  • ■ Homogeneous medium: βIJK(Ω) is independent of r in Eq. (23).
  • ■ Planar wave collection: EISHG=pISHG.
  • ■ 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 pSHG. 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 PISHG=NΩpISHG(Ω)f(Ω)dΩ:

IISHG=|PISHG|2=|J,KχIJK(2)EJωEKω|2
with the macroscopic susceptibility defined in a general form by
χIJK(2)=NΩβIJK(Ω)f(Ω)dΩ.
As for fluorescence, if the molecular angular distribution is defined in its microscopic frame, then the indices (ijk) should be used in Eq. (28) and a tensorial rotation should be applied following the approach of Eq. (20) to express this tensor in the macroscopic frame with (IJK) indices.

Similarly to the molecular scales, SHG requires noncentrosymmetry at the macroscopic scale, which stems from the symmetry properties of the second-order tensor χIJK(2). 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 IXSHG(α) and IYSHG(α) 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 IYSHG(α). This is characteristic of a nonlinear coherent response that is able to provide a coherent coupling between different states of polarization. The IXSHG(α) 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:

IISHG(α)|JKχIJK(2)EJEK(α)|2,χIJK(2)=NΩβIJK(Ω)f(Ω)dΩ,
where here Ω denotes the orientation of the molecules in the macroscopic frame, supposing that f(Ω) is also defined in this frame (the case where f(Ω) is defined in its own microscopic frame should account for an additional tensorial rotation).

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 N2 molecular density dependence of the signal.

 

Figure 5 SHG and THG polarimetric responses IX(α), IY(α) of a molecular angular distribution within a cone of aperture Ψ (“filled cone” distribution), oriented in the sample plane with its main axis along X (θ0=π/2, ϕ0=0, ψ0=0). (a) Ψ=40, (b) Ψ=70, (c) Ψ=90 (isotropic distribution). The SHG polarimetric responses are also represented for a molecular distribution lying along a cones of the same apertures (“cone surface” distribution).

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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:

pITHG,3ω(Ω)=JKLγIJKL(3ω;ω,ω,ω)(Ω)EJωEKωELω,pIFWM,ω4(Ω)=JKLγIJKL(ω4;ω1,ω2,ω3)(Ω)EJω1EKω2ELω3,
with ω4=ω1+ω2ω3. Like SHG, the polarization dependence of expected THG responses IXTHG(α) and IYTHG(α) are represented in Fig. 5 in the planar wave approximation, for a molecular distribution filling an increasing cone aperture. The responses are seen to be more peaked than for SHG, evolving towards a polarization-independent response in the case of an isotropic molecular distribution.

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 ω1=ω2=ωp, and the Stokes ω3=ωs. The resulting microscopic induced dipole at the anti-Stokes detected frequency ω4=ωAS=2ωpωs is

pICARS,ωAS=JKLγIJKL(ωAS;ωp,ωp,ωs)EJωpEKωpELωs.

The specificity of the CARS process is to exhibit, for specific frequencies ωs, a resonant contribution when δω=ωpωs=ΩR 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 ωs can be written as

γ(δω)=γNR+γR(δω),
where γNR is a nonresonance FWM type tensor and γR(δω) an additional resonant contribution with
γR(δω)=a(δωΩR)+iΓ,
where a is the oscillator strength of the vibration, and Γ the line width of the Raman resonance corresponding to the ΩR 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 ωp 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

IITHG,FWM,CARS(α)|JKLχIJKL(3)EJEKEL(α)|2,χIJKL(3)=NΩγIJKL(Ω)f(Ω)dΩ,
where γ might contain resonant contributions (in the case of CARS) and EJ,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(Ω) is also defined in this frame (the case where f(Ω) 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 EJ,K, 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(Ω). In the next subsection, we will derive how microscopic information on the shape and orientation of f(Ω) 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 EJ,K, 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 IX and IY are measured (without an analyzer, IX+IY is measured and these numbers should be divided by 2), in the case of two-photon fluorescence, the γα 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 αuv, βuvw, γuvwϵ coefficients in the molecular (x,y,z) frame and therefore their expression αij(Ω), βijk(Ω), γijkl(Ω) in the microscopic frame (1,2,3) (Fig. 2).
  • ■ The shape of the angular distribution function f(Ω) in the microscopic frame (1,2,3).
  • ■ The orientation Ω0 of the angular distribution function f(Ω) 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(Ω) can be studied, investigating both its shape and orientation Ω0.
  • ■ 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(Ω) 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(Ω) at thermal equilibrium T, f(Ω) follows the Boltzmann statistics:

f(Ω)exp(U(Ω)/kBT).

Different shapes of the U(Ω) 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 θ=Ψ [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(θ,ϕ)=1if|θ|Ψ0otherwise.
    Note that the Euler angle ψ is not required at this stage, since one-dimensional dipolar molecules are considered in this model.
  • ■ In lipid membranes, other smoother functions are used such as a Gaussian distribution [Fig. 6(b)] [31,32]:
    f(θ,ϕ)=ln2/Ψexp(ln(2)θ2/Ψ2).
  • ■ In biofilaments and fibers, molecules are considered that decorate the fiber and therefore lie along a cone surface, which therefore defines an open cone distribution [Fig. 6(c)] [25]. This distribution has been used for membranes as well [32].
  • ■ In such a case (cone surface) if molecular-scale orientational disorder is present, the cone aperture can be completed by a statistical width [Fig. 6(d)] [33].

In all of these cases the interaction potential is of cylindrical symmetry; therefore f(Ω) applies as well, and only the angles (θ,ϕ) 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 (θ0,ϕ0) in the macroscopic frame.

 

Figure 6 Different distribution functions. (a) Filled cone and (b) Gaussian function in a lipid membrane. (c) Open cone in a biofilament. (d) Cone width in a disordered medium.

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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 ψ0, 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 YmJ(θ,ϕ) (in the case of (θ,ϕ)-dependent functions), or the Wigner functions DmmJ(θ,ϕ,ψ) (in the case of (θ,ϕ,ψ)-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(Ω) can be written

f(θ,ϕ)=m,JfmJYmJ(θ,ϕ),
where the J coefficients are called “orders of symmetry.” The fmJ coefficients (also containing normalization factors) are the weights that finally determine the shape of the distribution function. f00 is in particular the signature of the isotropic contribution to the distribution function, and fmJ with J0 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 fmJ coefficients are the parameters that define the f(Ω) function. The question is now to determine which fmJ 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

A=02π0πA(θ,ϕ)f(θ,ϕ)sinθdθdϕ.
Using this expression, and the orthogonality properties of spherical harmonics,
ΩYmJ(Ω)YmJ(Ω)dΩ=δmmδJJ,
one can deduce an important relation: YmJ(Ω)=fmJ, which now defines the fmJ coefficients as “order parameters.”

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

τI1...In(Ω)=i1...inτi1...in(i1I1)...(inIn)(Ω)=i1...inτi1...inm,JCm,Ji1...in,I1...InYmJ(Ω)=m,JτmJYmJ(Ω)
with the Cm,Ji1...in,I1...In projection coefficients defined in Appendix B. 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 JmJ and 0Jn, where J and n are of same parity (Appendix B). The τmJ 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 I1...IN 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(Ω)=K,lflKYlK(Ω) (with KlK):

TI1...In=ΩτI1...In(Ω)f(Ω)dΩ=τI1...In(Ω)=m,Jl,KτmJflKΩYmJ(Ω)YlK(Ω)dΩ=m,JτmJfmJ
owing to the orthogonality property of the spherical harmonics [Eq. (40)]; the K orders of symmetry of the distribution function should therefore satisfy the conditions that K and n are of same parity and 0Kn. 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

  • ■ In the case of multiphoton fluorescence [Eq. (22)]: τ=γα is a six-rank tensor (n=6 in Eq. (42); then 0J6 with J even; so only fmJ=0,2,4,6 can be measured (with JmJ).
  • ■ In the case of SHG [Eq. (29)]: τ=β is a third-rank tensor (n=3 in Eq. (42); then 1J3 with J odd; so only fmJ=1,3 can be measured (which is consistent with the noncentrosymmetry specificity of SHG).
  • ■ In the case of THG and FWM [Eq. (34)]: τ=γ is a fourth-rank tensor [n=4 in Eq. (42); then 0J4 with J odd, so only fmJ=0,2,4 can be measured.

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(Ω) is dependent only on θ and m=0 in Eq. (38). This equation shows that f(θ) can be directly reconstructed from the f0J parameters. Figure 7 shows the illustration of a cone distribution function reconstructed from a truncation of its symmetry orders, where only J=06 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)].

 

Figure 7 Spherical decomposition of cone and Gaussian aperture distribution functions. (a) Representation of the functions and their filtered decomposition. (b) Representation of the norm of the even-order spherical harmonics used for the decomposition. Only m=0 is present in the decomposition, since the functions are of cylindrical symmetry around their principal axis of symmetry.

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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 or I 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:

A=III+2I.

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=IX()IY()IX()+IY()
with 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 ϕ0 in the X,Y plane (with θ0=π/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 ϕ0=π/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 ϕ0 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 samples [f(Ω)=1/4π2] in time-resolved measurements, in order to retrieve orientational diffusion rates in intracellular media [42,43] or quantify fluorescence resonance energy transfer (FRET) rates by depolarization effects in viscous solutions [44].
  • ■ Isotropic viscous solutions in steady-state measurements, to obtain molecular structural information such as the μabs/μem relative angle.
  • ■ Simplified orientational distribution function where (i) the symmetry is of cylindrical distribution and (ii) the mean orientation of the distribution can be known a priori [29,30].

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 (θ0=π/2), and the in-plane average orientation ϕ0 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 ϕ0 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.

 

Figure 8 Fluorescence anisotropy imaging. (a) Experimental scheme: the IX and IY analyzed intensities are recorded for an incident circular polarization. The sample is schematized as a spherical lipid membrane in which molecules are assembled within a cone aperture, normal to the membrane. (b) Theoretical dependence of the anisotropy factor A(Ψ) as a function of the cone aperture Ψ of a filled cone distribution, for several tilt angles ϕ0 of the cone in the sample plane. (c) Anisotropy factor A measured for (left) a GUV made of 1,2-dioleoyl-sn-glycero-3-phosphocholine (DOPC) and (right) a mixture of DOPC, sphingomyelin and cholesterol (1:1:1) (see Section 4 for a more detailed description). The points of measurement M and N at the position ϕ0=90 are represented on the theoretical A(Ψ) graph, leading to an estimation of the molecular order value at this position of the sample [40].

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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.

S=S0S1S2S3=I(α=0)+I(α=90)I(α=0)I(α=90)I(α=45)I(α=45)I(L)I(R),
where I(α) is the incident/measured intensity for an incident/analyzed polarization angle α between E and the X direction. I(L) and I(R) are, respectively, the incident/measured intensities obtained by using a left-circular and a right-circular analyzer.

 

Figure 9 (a) Principle of Mueller matrix imaging (P1, P2, linear horizontal polarizers; QWP1, QWP2, rotating quarter-wave plates). (b) Four of the Mueller component images measured from a retina in a living human eye [50]. (c) Best image obtained from a reconstruction procedure using the different Mueller components in (b) [50].

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This formalism has been essentially applied to study the linear transmission by a sample, modeled as a transformation of the incident Stokes vector Sin in a new output Stokes vector Sout (measured from the output radiated field). This determines the so-called Mueller matrix [51] characteristic of the sample: Sout=MSin. 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 (α=0,90,45,45,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 DOP=S12+S22+S32/S0). 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 IX(α) and IY(α). Assuming an incident planar wave with no perturbations of the polarization,

E(α)=cosαsinα0.

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:

Eω1(α1)...Eωn(αn)...IX(α1...αn),IY(α1...αn).

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 (NA0.5).

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 αp and αs 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 IX(α) and IY(α), 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 104 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).

 

Figure 10 (a) Principle of a polarimetric nonlinear microscopy setup. APD, avalanche photodiodes. (b) Nonlinear ellipsometry setup. Pol, polarizer; λ/2, λ/4: half- and quarter-wave plates; PMT, photomultiplier [65]. (c) Nonlinear pulse shaping for second-harmonic generation. The wide spectral band of the incident pulse is separated into two crossed polarized components, giving access to six nonlinear macroscopic tensorial coefficients in the sample plane [63].

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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]

E(α,δX,δY)=cosαeiδXsinαeiδY0,
where δ=δYδX characterizes the ellipticity of the polarization. In the linear regime, the sample is studied by means a new output transmitted field Eout=JEin, which allows us to study the perturbations imposed by the sample in terms of retardance (phase shift between 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 [7578].

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 (530fs) 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 E(r) 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 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

|EemX|2κ1|μXem|2+κ2|μYem|2+κ3|μZem|2,|EemY|2κ2|μXem|2+κ1|μYem|2+κ3|μZem|2,
with κ1>κ2>κ3 which are directly related to the NA of the objective [29]. The κ2 and κ3 coefficients, the signature of a mixture of the 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.

 

Figure 11 Effect of polarization state distortion at the focal spot [Fig. 4(b)]. Left: polarimetric SHG calculated from a cone distribution (schematic representation of a collagen fiber) of aperture angle Ψ=50° oriented along X in the sample plane, for two different objective NAs. Right: polarimetric TPEF calculated for a one-dimensional crystal (schematic representation of a molecular crystal) for two different NAs, and a different off-plane orientation θ0 of the crystal.

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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

E(α)=11+(1γ)2cosα(1γ)sinαeiδ0.

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(Ω)=1/(4π2) 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:

II2-ph(α)=VΩ|μabs(Ω,r)E(r,α,δ,γ)|4dΩdrNAVΩ|Eem(Ω,r,k)I|2dΩdrdk=CΩ|μabs(Ω)E(α,δ,γ)|4dΩ,
where the constant 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 (δ,γ) 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.

 

Figure 12 Effect of the polarization distortions from the reflective dichroic mirror on polarimetry. (a) Ellipticity distortion on an α=45 incident linear polarization for different δ phase shifts imposed by the dichroic mirror. (b) Effect of the δ parameter on TPEF polarimetry in a model tilted one-dimensional fluorescent molecular crystal (only IX is represented in the calculated responses). Both experimental polarimetric data and the fluorescent image are represented for a situation in which the dichroic mirror is not imposing large distortions. (c) Effect of both δ and γ parameters on TPEF polarimetry in an isotropic depolarized solution. The polarimetric IX TPEF response measured in a Rhodamine solution is also represented for a highly distorting dichroic mirror.

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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 [8992]. 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 [9399]. 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 [100102]. 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(Ω) 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 ϕ0. Fitting the polarimetric data on both IX(α) and IY(α) 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°<Ψ<80° (Lo phases) and 90°<Ψ<170° (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 (Ψ35° in Lo phases and Ψ90° 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].

 

Figure 13 Polarimetric TPEF in lipid membranes labeled with di-8-ANEPPQ (imaging conditions: incident wavelength 780 nm, detection wavelength 500 nm, NA = 1.2). (a) GUVs made of Ld and Lo phases from a DOPC:sphingomyelin:cholesterol (1:1:1) mixture. (b) The fit (continuous line) of the TPEF polarimetric data (markers) for marked points on the GUV contour is performed by using a filled cone model of orientation ϕ0 in the sample plane and aperture Ψ. Two populations could be found in the GUV fluorescent image (left image), characteristics of ordered and disordered phases (right image). (c) The same methodology applied to doped cell membranes shows high aperture angles, characteristics of membrane folding at the subwavelength scale [40]. The schematic membrane surface image is taken from [105].

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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 ϕ0 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 [20,110112]. 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 IX(α) and IY(α) 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 [115117], 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 [118120], possibly in conjunction with TPEF [121124], 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 [127135].

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 C6 crystalline point group [10], stemming from its triple-helix structure. This point group is characterized by a tensor χ(2) that exhibits seven independent coefficients in the (1,2,3) crystalline frame [26]:

χ333(2)χ311(2)=χ322(2);χ113(2)=χ223(2);χ131(2)=χ232(2)χ123(2)=χ213(2);χ132(2)=χ231(2);χ312(2)=χ321(2).

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: χ333(2), χ311(permut.)(2)=χ322(permut.)(2) (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(Ω) 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 βzzz is the only nonvanishing component. This leads to two nonvanishing coefficients in the microscopic frame (1,2,3):

χ333(2)=Ncos3θβzzz,χ311(2)=χ131(2)=χ113(2)=Ncosθsin2θcos2ϕβzzz,
with the orientational average defined in Eq. (39). This model, which implicitly allows index permutations, leads therefore to a tensor similar to that for the C6 crystalline point group mentioned above. The microscopic tensor components of Eq. (53) can be furthermore expressed as a function of the cone aperture Ψ:
cos3θ=02π0πcos3θδθΨ2sinθdθdϕ=2πcos3Ψ2sinΨ2,cosθsin2θcos2ϕ=02π0πcosθsin2θcos2ϕδθΨ2sinθdθdϕ=πsin3Ψ2cosΨ2,
where the Dirac function δ(θΨ/2) indicates that the distribution function f(Ω) of the molecules is equal to 1 for θ=Ψ/2 and 0 otherwise. This leads, after simplifications, to a relation between the cone aperture Ψ and the ratio χ333(2)/χ311(2):
tan2Ψ2=χ333(2)χ311(2).

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 ϕ0 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:

χXXX(2)χXXY(2)χXYY(2)χYXX(2)χYXY(2)χYYY(2)=cos3ϕ0cosϕ0sin2ϕ02sin2ϕ0cosϕ0cos2ϕ0sinϕ0cos2ϕ0sinϕ0sinϕ0(sin2ϕ0cos2ϕ0)cosϕ0sin2ϕ0cos3ϕ02sin2ϕ0cosϕ0cos2ϕ0sinϕ0sin3ϕ02cos2ϕ0sinϕ0cosϕ0sin2ϕ0sin2ϕ0cosϕ0cosϕ0(cos2ϕ0sin2ϕ0)sin3ϕ0sinϕ0cos2ϕ02cos2ϕ0sinϕ0χ333(2)χ311(2)χ131(2).

For ϕ0=0 in particular, the application of Eq. (56) to Eq. (27) with (EX(α),EY(α))=(cosα,sinα) the excitation field in the sample plane, leads to

IX(α)=(χ333(2)cos2α+χ311(2)sin2α)2IY(α)=(2χ131(2)cosαsinα)2.

These SHG polarimetric responses, normalized by χ131(2)=χ311(2) (in case of a nonresonant excitation allowing microscopic index permutations), are therefore only functions of the ratio χ333(2)/χ311(2) (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 IX(α) and IY(α) are also dependent on ϕ0; 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 ϕ0, can be determined from the fit of a SHG polarimetric measurement as illustrated in Fig. 14. Typical χ333(2)/χ311(2) 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.

 

Figure 14 Polarimetric SHG in collagen type I contained in tissues. (a), (b) Collagen type I extracted from rat tail tendon. (a) The SHG radiation is measured in the forward direction through the tissue. The incident polarization is kept fixed (colored arrow) and the analyzer is rotated in front of the detector [130]. Fits (continuous curves) are performed following a model similar to Eq. (57). (b) The SHG radiation is measured in the epi-direction at the surface of the tissue. The incident polarization is rotated (SHG polarimetry), and the analyzer is set along the horizontal direction (red markers) or vertical (green markers). The black curves represent fits using Eqs. (56) and (27), with ϕ0 and Ψ as unknown parameters. (c) SHG polarimetry image analysis in the body wall muscles of Caenorhabditis elegans ventral quadrants [134].

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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 [141144]. 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

E3(Z)E1(Z)=E3(Z=0)eiΦb(Z)E1(Z=0)
with E(Z=0) the field polarization at the entrance of the sample and Φb(Z)=2π/λΔnZ the birefringence phase shift, with Δn=n3n1 the material birefringence.

In anisotropic samples in which the incident field undergoes a birefringence retardation, E(α) [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 Θb relative to X, then the incident field polarization E(α,Z) at depth Z can be finally written in the macroscopic (X,Y) frame, in a planar wave approximation:

EX(α,Z)EY(α,Z)=cosΘbsinΘbsinΘbcosΘb.cosΘbsinΘbsinΘbeiΦb(Z).cosΘbeiΦb(Z)EX(α,Z=0)EY(α,Z=0)
with EX,Y(α,Z=0) the optical field polarization components in the macroscopic (X,Y) frame at the sample surface (Z=0).

 

Figure 15 SHG polarimetry in collagen type I fibers extracted from rat tail tendons, attached on a coverslip and immersed in 0.15M NaCl [59]. (a) SHG image (incident wavelength 800 nm, detected wavelength 400 nm, NA 0.6). (b) SHG polarimetric data recorded for a position on the collagen fiber, for two different depths in the tissue. The fit (continuous line) accounts for birefringence parameters Δn0.003. (c) Theoretical SHG polarimetry in a birefringent sample modeled by a cone of aperture Ψ=50°, for an increasing birefringent phase shift (Φb) and for two different orientations of the cone ϕ0=Θb in the sample plane.

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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 Δn 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 Φb.

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 Φb(Z) and Θb from a fit of SHG polarimetric data, providing that the nonlinear coefficients of the structures are determined from a preliminary study at depth Z0 where the birefringence is negligible. This approach has shown overall that (i) the expected main axis orientation Θb lies roughly along the observed collagen fiber orientation and (ii) the measured birefringence phase shift Φb(Z) can reach up to π/2 at a penetration depth of about Z50µm. This is consistent with the birefringence values Φb1.35°/µm (Δn0.003 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:

E3(Z)E1(Z)=E3(Z=0)eZ/2L3E1(Z=0)eZ/2L1.

The diattenuation 1/ΔL=1/L31/L1 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 EX and EY 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:

IX(Z)IY(Z)=IX(Z=0)+ηXYIY(Z=0)IY(Z=0)+ηYXIX(Z=0).

The scattering cross talk has been measured to be about ηXY0.10.2 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].

 

Figure 16 SHG polarimetry on collagen type I from a rat tail tendon, at large depth. Experimental polarimetry data represented as a magnitude map for (a) IXSHG(α) and (b) IYSHG(α). (c) and (d) show, respectively, the fit (continuous line) of IX and IY at the tendon surface including scattering cross talk, diattenuation, and birefringence (blue line) and without accounting for these parameters (red line). These results are taken from [145].

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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).

 

Figure 17 Polarization-sensitive THG imaging. THG imaging of a cornea with (a) linear and (b) circular incident polarization revealing anisotropic structures. Scale bar 100 µm. (c) Principle of the experiment with polarization-resolved detection [60].

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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 χ(3)=χ(3)NR+χ(3)R, due to symmetry reduction properties, exhibits only two nonvanishing components χXXYY(3)=χXYXY(3) and χXXXX(3), with

χIJKL(3)NR=χXXYY(3)NR(δIJδKL+δIKδJL+δILδJK)χIJKL(3)R=χXXYY(3)RδIJδKL+δIKδJL+2ρR1ρRδILδJK
where ΩR is the vibrational line frequency and ρR is the CARS depolarization ratio, ranging from 0 for totally symmetric vibrational modes to 3/4 for depolarized modes. ρR can also be written as
ρR=χXYYX(3)RχXXXX(3)R.

Since the macroscopic χIJKL(3) components are orientational averages of molecular γuvwϵ coefficients over an isotropic distribution f(Ω)=1/(4π2) [Eq. (34)], ρR 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 χXXYY(3)=χXYXY(3) 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 ρR. 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].

 

Figure 18 Polarimetric CARS. (a) Response to varying αp and αs angles simultaneously in a solution of toluene at the 787 cm1 resonance (ρR=0) [61]. (b) Polarimetry in collagen fibers from a rat tail tendon [155]. (c) Response to different polarimetric schemes in a H8Si8O12 crystal of known orientation (Eg O–Si–H bending mode at 932 cm1). The rotating polarization scheme is indicated below the corresponding polarimetric responses. (d) IX polarimetric response of the αp=αs scheme at various spectral positions around the Eg mode resonance. A strong variation of the response is observed that is due to the deviation from Kleinman conditions [28].

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Retrieving a refined analysis of the CARS macroscopic tensor χ(3)=χ(3)NR+χ(3)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 Ω0. While the tensor structure of the nonresonant contribution χ(3)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 χ(3)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 (H8Si8O12), which belongs to the Oh 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 A1g vibrational mode (Si–H stretching at 2302 cm1), and its degenerate Eg mode (O–Si–H bending at 932 cm1). The totally symmetric A1g 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 Eg 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 (u1,,un)=(x,y,z) to a new basis of vectors components (i1,,in)=(1,2,3), a rotation operator Ω is applied based on projector components (ui) between the two transformation frames. Following the notation of the operation written in Eq. (1),

Ω(τ)i1,,in=τi1,,in(Ω)=u1,,unτu1,,un(u1i1)(unin)(Ω),
where (Ω=θ,ϕ,ψ) is the set of Euler angles (represented in Fig. 2) defining the orientation of the frame of vectors u in the frame of vectors i. The components of this operator are therefore written as
Ω=x1(Ω)y1(Ω)z1(Ω)x2(Ω)y2(Ω)z2(Ω)x3(Ω)y3(Ω)z3(Ω)=cosθcosϕcosψsinϕsinψcosθcosϕsinψsinϕcosψcosϕsinθcosθsinϕcosψ+cosϕsinψcosθsinϕsinψ+cosϕcosψsinϕsinθsinθcosψsinθsinψcosθ.

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 Ω=(θ,ϕ) orientation (only two Euler angles are necessary). The rotation of a susceptibility component τi1...in (in the molecular frame) by an angle Ω=(θ,ϕ) in a new frame, τI1...In, can be written as

τI1...In(Ω)=i1...inτi1...in(i1I1)..(inIn)(Ω),
where (i1I1)(Ω) is the component of the unit vector i1 along the I1 axis. This vector component can also be expressed in terms of first-order spherical harmonics:
(i1I1)(Ω)=1m1Am,J=1i1,I1YmJ=1(Ω)
with Am,Ji1,I1 the projection coefficients of this new basis decomposition.

Therefore

τI1...In(Ω)=i1...inτi1...inm1,,mnAm1,J1i1,I1...Amn,Jnin,InYm1J1(Ω)...YmnJn(Ω)
with J1...n=1 and 1m1...n1.

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]:

Ym1J1(Ω)Ym2J2(Ω)=J,m(2J1+1)(2J2+1)(2J+1)4πJ1J2Jm1m2mJ1J2J000(1)mYmJ(Ω)
with JmJ.

The Wigner 3-j symbols J1J2Jm1m2m are integers or half-integers that verify the following properties [158]:

  • m1+m1=m
  • |J1J2|J|J1+J2|
  • J1J2J000=0 if J1+J2+J is odd.

This last property implies that as J1=J2=1, J must be even to retrieve a nonvanishing product.

The product of two spherical harmonics can therefore be simplified in

Ym1J1=1Ym2J2=1=J=0,2;mBm1,m2m,J(1)mYmJ
with JmJ and Bm1,m2m,J the coefficients containing the previous 3-j Wigner symbol terms. The product of n spherical harmonics of first order can therefore be written as
Ym1J1=1...YmnJn=1=J,mBm1...mnm,JYmJ
with J and n of same parity, JmJ and 0Jn, since J1++Jn=n.

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

τI1...In(Ω)=i1...inτi1...inm1,,mnAm1,J1=1i1,I1...Amn,Jn=1in,InJ,mBm1,...,mnm,JYmJ(Ω).

The coefficients can be simplified and written as

τI1...In(Ω)=i1...inJ,mτi1...inCm,Ji1...in,I1...InYmJ(Ω)
with Cm,Ji1...in,I1...In encompassing all coefficient summations over Am1,J1=1i1,I1, and Bm1...mnm,J appearing in the previous equation. In this expression, J and n are of same parity, JmJ and 0Jn.

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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aop-3-3-205-i001 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.

References

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2011 (2)

S. Brustlein, P. Ferrand, N. Walther, S. Brasselet, C. Billaudeau, D. Marguet, and H. Rigneault, “Optical parametric oscillators based light source for Coherent Raman Scattering microscopy: practical overview,” J. Biomed. Opt. 16, 021106 (2011).
[CrossRef] [PubMed]

J. T. Madden, V. J. Hall, and G. J. Simpson, “Mining the polarization-dependence of nonlinear optical measurements,” Analyst (London) 136, 652‒ 662 (2011).
[CrossRef]

2010 (13)

D. Aït-Belkacem, A. Gasecka, F. Munhoz, S. Brustlein, and S. Brasselet, “Influence of birefringence on polarization resolved nonlinear microscopy and collagen SHG structural imaging,” Opt. Express 18, 14859‒ 14870 (2010).
[CrossRef] [PubMed]

N. Olivier, F. Aptel, K. Plamann, M.-C. Schanne-Klein, and E. Beaurepaire, “Harmonic microscopy of isotropic and anisotropic microstructure of the human cornea,” Opt. Express 18, 5028‒ 5040 (2010).
[CrossRef] [PubMed]

I. Amat-Roldan, S. Psilodimitrakopoulos, P. Loza-Alvarez, and D. Artigas, “Fast image analysis in polarization SHG microscopy,” Opt. Express 18, 17209‒ 17219 (2010).
[CrossRef] [PubMed]

P. Schön, M. Behrndt, D. Ait-Belkacem, H. Rigneault, and S. Brasselet, “Polarization and phase pulse shaping applied to structural contrast in nonlinear microscopy imaging,” Phys. Rev. A 81, 013809 (2010).
[CrossRef]

A. L. Mattheyses, M. Kampmann, C. E. Atkinson, and S. M. Simon, “Fluorescence anisotropy reveals order and disorder of protein domains in the nuclear pore complex,” Biophys. J. 99, 1706‒ 1717 (2010).
[CrossRef] [PubMed]

A. Anantharam, B. Onoa, R. H. Edwards, R. W. Holz, and D. Axelrod, “Localized topological changes of the plasma membrane upon exocytosis visualized by polarized TIRFM,” J. Cell Biol. 188, 415‒ 428 (2010).
[CrossRef] [PubMed]

F. Munhoz, H. Rigneault, and S. Brasselet, “High order symmetry structural properties of vibrational resonances using multiple-field polarization coherent anti-Stokes Raman spectroscopy microscopy,” Phys. Rev. Lett. 105, 123903 (2010).
[CrossRef] [PubMed]

A. Gasecka, L.-Q. Dieu, D. Brühwiler, and S. Brasselet, “Probing molecular order in zeolite L inclusion compounds using two-photon fluorescence polarimetric microscopy,” J. Phys. Chem. B 114, 4192‒ 4198 (2010).
[CrossRef] [PubMed]

J. Adler, A. I. Shevchuk, P. Novak, Y. E. Korchev, and I. Parmryd, “Plasma membrane topography and interpretation of single-particle tracks,” Nat. Methods 7, 170‒ 171 (2010).
[CrossRef] [PubMed]

R. Cicchi, A. Crisci, A. Cosci, G. Nesi, D. Kapsokalyvas, S. Giancane, M. Carini, and F. S. Pavone, “Time- and spectral-resolved two-photon imaging of healthy bladder mucosa and carcinoma in situ,” Opt. Express 18, 3840‒ 3849 (2010).
[CrossRef] [PubMed]

V. Nucciotti, C. Stringari, L. Sacconi, F. Vanzi, L. Fusi, M. Linari, G. Piazzesi, V. Lombardi, and F. S. Pavone, “Probing myosin structural conformation in vivo by second-harmonic generation microscopy,” Proc. Natl Acad. Sci. U.S.A. 107, 7763‒ 7768 (2010).
[CrossRef] [PubMed]

I. Gusachenko, G. Latour, and M.-C. Schanne-Klein, “Polarization-resolved second harmonic microscopy in anisotropic thick tissues,” Opt. Express 18, 19339‒ 19352 (2010).
[CrossRef] [PubMed]

M. Zimmerley, R. Younger, T. Valenton, D. C. Oertel, J. L. Ward, and E. O. Potma, “Molecular orientation in dry and hydrated cellulose fibers: a coherent anti-Stokes Raman scattering microscopy study,” J. Phys. Chem. B 114, 10200‒ 10208 (2010).
[CrossRef] [PubMed]

2009 (15)

J. Park, N. J. Kemp, H. G. Rylander, and T. E. Milner, “Complex polarization ratio to determine polarization properties of anisotropic tissue using polarization-sensitive optical coherence tomography,” Opt. Express 17, 13402‒ 13417 (2009).
[CrossRef] [PubMed]

O. Nadiarnykh and P. J. Campagnola, “Retention of polarization signatures in SHG microscopy of scattering tissues through optical clearing,” Opt. Express 17, 5794‒ 5806 (2009).
[CrossRef] [PubMed]

S. Psilodimitrakopoulos, S. I. C. O. Santos, I. Amat-Roldan, A. K. N. Thayil, D. Artigas, and P. Loza-Alvarez, “In vivo, pixel-resolution mapping of thick filaments’ orientation in nonfibrilar muscle using polarization-sensitive second harmonic generation microscopy,” J. Biomed. Opt. 14, 014001 (2009).
[CrossRef] [PubMed]

A. Deniset-Besseau, J. Duboisset, E. Benichou, F. Hache, P.-F. Brevet, and M. C. Schanne-Klein, “Measurement of the second order hyperpolarizability of the collagen triple helix and determination of its physical origin,” J. Phys. Chem. B 113, 13445 (2009).
[CrossRef]

G. Recher, D. Rouède, P. Richard, A. Simon, J.-J. Bellanger, and F. Tiaho, “Three distinct sarcomeric patterns of skeletal muscle revealed by SHG and TPEF microscopy,” Opt. Express 17, 19763‒ 19777 (2009).
[CrossRef] [PubMed]

O. Nadiarnykh, R. LaComb, M. Brewer, and P. J. Campagnola, “Second harmonic generation imaging microscopy of ovarian cancer,” Biophys. J. 96(3), 296a (2009).
[CrossRef]

O. Masihzadeh, P. Schlup, and R. A. Bartels, “Enhanced spatial resolution in third-harmonic microscopy through polarization switching,” Opt. Lett. 34, 1240‒ 1242 (2009).
[CrossRef] [PubMed]

O. Masihzadeh, P. Schlup, and R. A. Bartels, “Control and measurement of spatially inhomogeneous polarization distributions in third-harmonic generation microscopy,” Opt. Lett. 34, 1090‒ 1092 (2009).
[CrossRef] [PubMed]

R. K. P. Benninger, B. Vanherberghen, S. Young, S. B. Taner, F. J. Culley, T. Schnyder, M. A. A. Neil, D. Wüstner, P. M. W. French, D. M. Davis, and B. Önfelt, “Live cell linear dichroism imaging reveals extensive membrane ruffling within the docking structure of natural killer cell immune synapses,” Biophys. J. 96, L13‒ L15 (2009).
[CrossRef] [PubMed]

D. Akimov, S. Chatzipapadopoulos, T. Meyer, N. Tarcea, B. Dietzek, M. Schmitt, and J. Popp, “Different contrast information obtained from CARS and nonresonant FWM images,” J. Raman Spectrosc. 40, 941‒ 947 (2009).
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B. Huang, M. Bates, and X. Zhuang, “Super-resolution fluorescence microscopy,” Annu. Rev. Biochem. 78, 993‒ 1016 (2009).
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A. Gasecka, T.-J. Han, C. Favard, B. R. Cho, and S. Brasselet, “Quantitative imaging of molecular order in lipid membranes using two-photon fluorescence polarimetry,” Biophys. J. 97, 2854‒ 2862 (2009).
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F. Munhoz, S. Brustlein, D. Gachet, F. Billard, S. Brasselet, and H. Rigneault, “Raman depolarization ratio of liquids probed by linear polarization coherent anti-Stokes Raman spectroscopy,” J. Raman Spectrosc. 40, 775‒ 780 (2009).
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N. J. Begue, A. J. Moad, and G. J. Simpson, “Nonlinear optical Stokes ellipsometry 1: Theoretical framework,” J. Phys. Chem. C 113(23), 10158‒ 10165 (2009).
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N. J. Begue, R. M. Everly, V. J. Hall, L. Haupert, and G. J. Simpson, “Nonlinear optical Stokes ellipsometry 2: Experimental demonstration,” J. Phys. Chem. C 113(23), 10166‒ 10175 (2009).
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2008 (11)

G. M. Lerman and U. Levy, “Effect of radial polarization and apodization on spot size under tight focusing conditions,” Opt. Express 16, 4567‒ 4581 (2008).
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M. R. Foreman, C. M. Romero, and P. Török, “Determination of the three-dimensional orientation of single molecules,” Opt. Lett. 33, 1020‒ 1022 (2008).
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C. K. Haluska, A. P. Schröder, P. Didier, D. Heissler, G. Duportail, Y. Mély, and C. M. Marques, “Combining fluorescence lifetime and polarization microscopy to discriminate phase separated domains in giant unilamellar vesicles,” Biophys. J. 95, 5737‒ 5747 (2008).
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E. J. Gualtieri, L. M. Haupert, and G. J. Simpson, “Interpreting nonlinear optics of biopolymers: finding a hook,” Chem. Phys. Lett. 465(4–6), 167‒ 174 (2008).
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R. P. Davis, A. J. Moad, G. S. Goeken, R. D. Wampler, and G. J. Simpson, “Selection rules and symmetry relations for four-wave mixing measurements of uniaxial assemblies,” J. Phys. Chem. B 112(18), 5834‒ 5848 (2008).
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P. Schön, F. Munhoz, A. Gasecka, S. Brustlein, and S. Brasselet, “Polarization distortion effects in polarimetric two-photon microscopy,” Opt. Express 16, 20891‒ 20901 (2008).
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R. LaComb, O. Nadiarnykh, and P. J. Campagnola, “Quantitative second harmonic generation imaging of the diseased state osteogenesis imperfecta: experiment and simulation,” Biophys. J. 94(11), 4504‒ 4515 (2008).
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M. Strupler, M. Hernest, C. Fligny, J. L. Martin, P.-L. Tharaux, and M. C. Schanne-Klein, “Second harmonic microscopy to quantify renal interstitial fibrosis and arterial remodeling,” J. Biomed. Optics 13, 054041 (2008).
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E. Ralston, B. Swaim, M. Czapiga, W. L. Hwu, Y. H. Chien, M. G. Pittis, B. Bembi, O. Schwartz, P. Plotz, and N. Raben, “Detection and imaging of noncontractile inclusions and sarcomeric anomalies in skeletal muscle by second harmonic generation combined with two-photon excited fluorescence,” J. Struct. Biol. 162, 500‒ 508 (2008).
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C. Odin, Y. Le Grand, A. Renault, L. Gailhouste, and G. Baffet, “Orientation fields of nonlinear biological fibrils by second harmonic generation microscopy,” J. Microsc. 229, 32‒ 38 (2008).
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A. V. Kachynski, A. N. Kuzmin, P. N. Prasad, and I. I. Smalyukh, “Realignment-enhanced coherent anti-Stokes Raman scattering and three-dimensional imaging in anisotropic fluids,” Opt. Express 16, 10617‒ 10632 (2008).
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2007 (9)

F. Tiaho, G. Recher, and D. Rouède, “Estimation of helical angles of myosin and collagen by second harmonic generation imaging microscopy,” Opt. Express 15, 12286‒ 12295 (2007).
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K. Yoshiki, K. Ryosuke, M. Hashimoto, N. Hashimoto, and T. Araki, “Second-harmonic-generation microscope using eight-segment polarization-mode converter to observe three-dimensional molecular orientation,” Opt. Lett. 32, 1680‒ 1682 (2007).
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E. Y. S. Yew and C. J. R. Sheppard, “Second harmonic generation polarization microscopy with tightly focused linearly and radially polarized beams,” Opt. Commun. 275, 453‒ 457 (2007).
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H. M. Kim, H.-J. Choo, S.-Y. Jung, Y.-G. Ko, W.-H. Park, S.-J. Jeon, C. H. Kim, T. Joo, and B. R. Cho, “A two-photon fluorescent probe for lipid raft imaging: C-laurdan,” ChemBioChem 8, 553‒ 559 (2007).
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T. Baumgart, G. Hunt, E. R. Farkas, W. W. Webb, and G. W. Feigenson, “Fluorescence probe partitioning between Lo/Ld phases in lipid membranes,” Biochim. Biophys. Acta 1768, 2182‒ 2194 (2007).
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S. Brasselet and J. Zyss, “Nonlinear polarimetry of molecular crystals down to the nanoscale,” C. R. Phys. 8, 165‒ 179 (2007).
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T. E. Schaus, E. W. Taylor, and G. G. Borisy, “Self-organization of actin filament orientation in the dendritic-nucleation/array-treadmilling model,” Proc. Nat. Acad. Sci. U.S.A. 104, 7086‒ 7091 (2007).
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N. Sandeau, L. Le Xuan, D. Chauvat, C. Zhou, J. F. Roch, and S. Brasselet, “Defocused imaging of second harmonic generation from a single nanocrystal,” Opt. Express 15, 16051‒ 16060 (2007).
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P. Lemaillet, F. Pellen, S. Rivet, B. Le Jeune, and J. Cariou, “Optimization of a dual-rotating-retarder polarimeter designed for hyper-Rayleigh scattering,” J. Opt. Soc. Am. B 24, 609‒ 614 (2007).
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2006 (13)

M. R. Beversluis, L. Novotny, and S. J. Stranick, “Programmable vector point-spread function engineering,” Opt. Express 14, 2650‒ 2656 (2006).
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D. Lara and C. Dainty, “Axially resolved complete Mueller matrix confocal microscopy,” Appl. Opt. 45, 1917‒ 1930 (2006).
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C. E. Bigelow and T. H. Foster, “Confocal fluorescence polarization microscopy in turbid media: effects of scattering-induced depolarisation,” J. Opt. Soc. Am. A 23, 2932‒ 2943 (2006).
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B. Corry, D. Jayatilaka, B. Martinac, and P. Rigby, “Determination of the orientational distribution and orientation factor for transfer between membrane-bound fluorophores using a confocal microscope,” Biophys. J. 91, 1032‒ 1045 (2006).
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A. M. Vrabioiu and T. J. Mitchison, “Structural insights into yeast septin organization from polarized fluorescence microscopy,” Nature 443, 466‒ 468 (2006).
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D. R. Fooksman, G. K. Grönvall, Q. Tang, and M. Edidin, “Clustering class I MHC modulates sensitivity of T cell recognition,” J. Immunol. 176, 6673‒ 6680 (2006).
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D. Débarre, W. Supatto, A.-M. Pena, A. Fabre, T. Tordjmann, L. Combettes, M.-C. Schanne-Klein, and E. Beaurepaire, “Imaging lipid bodies in cells and tissues using third-harmonic generation microscopy,” Nat. Methods 3, 47‒ 53 (2006).
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L. Polachek, D. Oron, and Y. Silberberg, “Full control of the spectral polarization of ultrashort pulses,” Opt. Lett. 31, 631‒ 633 (2006).
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E. Y. S. Yew and C. R. J. Sheppard, “Effects of axial field components on second harmonic generation microscopy,” Opt. Express 14, 1167‒ 1174 (2006).
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Y. Sun, W.-L. Chen, S.-J. Lin, S. H. Jee, Y.-F. Chen, L.-C. Lin, P. T. C. So, and C.-Y. Dong, “Investigating mechanisms of collagen thermal denaturation by high resolution second-harmonic generation imaging,” Biophys. J. 91, 2620‒ 2625 (2006).
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A. M. Pena, T. Boulesteix, T. Dartigalongue, M. Strupler, E. Beaurepaire, and M. C. Schanne-Klein, “Chiroptical effects in the second harmonic generation from collagens I and IV: applications in nonlinear microscopy,” Nonlinear Opt. Quantum Opt. 35, 1‒ 3 (2006).

X. S. Xie, J. Yu, and W. Y. Yang, “Living cells as test tubes,” Science 312, 228‒ 230 (2006).
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S.-W. Teng, H.-Y. Tan, J.-L. Peng, H.-H. Lin, K. H. Kim, W. Lo, Y. Sun, W.-C. Lin, S.-J. Lin, S.-H. Jee, P. T. C. So, and C.-Y. Dong, “Multiphoton autofluorescence and second-harmonic generation (SHG) imaging of ex-vivo porcine eye,” Invest. Ophthalm. Vis. Sci. 47, 1216‒ 1224 (2006).
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2005 (11)

K. Komorowska, S. Brasselet, G. Dutier, J. Zyss, I. Pourlsen, Ledoux Jazdzyk, L. Egelhaaf, M. Gierschner, and H. J. Hanack, “Nanometric scale investigation of the nonlinear efficiency of perhydrotriphynylene inclusion compounds,” Chem. Phys. 318, 12‒ 20 (2005).
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T. Yasui, K. Sasaki, Y. Tohno, and T. Araki, “Tomographic imaging of collagen fiber orientation in human tissue using depth-resolved polarimetry of second-harmonic-generation,” Opt. Quantum Electron. 37, 1397‒ 1408 (2005).
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R. M. Williams, W. R. Zipfel, and W. W. Webb, “Interpreting second-harmonic generation images of collagen I fibrils,” Biophys. J. 88, 1377‒ 1386 (2005).
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N. J. Kemp, H. N. Zaatari, J. Park, H. G. Rylander, and T. E. Milner, “Form-biattenuance in fibrous tissues measured with polarization-sensitive optical coherence tomography (PS-OCT),” Opt. Express 13, 4611‒ 4628 (2005).
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K. Yoshiki, M. Hashimoto, and T. Araki, “Controlled polarization pattern to determine three-dimensional molecular orientation,” Jpn. J. Appl. Phys. 44, L1066‒ L1068 (2005).
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F. Helmchen and W. Denk, “Deep tissue two-photon microscopy,” Nat. Methods 2, 932‒ 940 (2005).
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R. K. Benninger, B. Onfelt, M. A. Neil, D. M. Davis, and P. M. French, “Fluorescence imaging of two-photon linear dichroism: cholesterol depletion disrupts molecular orientation in cell membranes,” Biophys. J. 88, 609‒ 622 (2005).
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T. H. Foster, B. D. Pearson, S. Mitra, and C. E. Bigelow, “Fluorescence anisotropy imaging reveals localization of meso-tetrahydroxyphenyl chlorin in the nuclear envelope,” Photochem. Photobiol. 81, 1544‒ 1547 (2005).
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R. M. Plocinik, R. M. Everly, A. J. Moad, and G. J. Simpson, “Modular ellipsometric approach for mining structural information from nonlinear optical polarization analysis,” Phys. Rev. B 72, 125409 (2005).
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C. Anceau, S. Brasselet, and J. Zyss, “Local orientational distribution of molecular monolayers probed by nonlinear microscopy,” Chem. Phys. Lett. 411(1–3), 98‒ 102 (2005).
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H. Wang, Y. Fu, P. Zickmund, R. Shi, and J.-X. Cheng, “Coherent anti-Stokes Raman scattering imaging of axonal myelin in live spinal tissues,” Biophys. J. 89, 581‒ 591 (2005).
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2004 (4)

S. Brasselet, V. Le Floc’h, F. Treussart, J.-F. Roch, J. Zyss, E. Botzung-Appert, and A. Ibanez, “In situ diagnostics of the crystalline nature of single organic nanocrystals by nonlinear microscopy,” Phys. Rev. Lett. 92, 207401 (2004).
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A. S. Brack, B. D. Brandmeier, R. E. Ferguson, S. Criddle, R. E. Dale, and M. Irving, “Bifunctional rhodamine probes of myosin regulatory light chain orientation in relaxed skeletal muscle fibers,” Biophys. J. 86, 2329‒ 2341 (2004).
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D. Oron, D. Yelin, E. Tal, S. Raz, R. Fachima, and Y. Silberberg, “Depth-resolved structural imaging by third-harmonic generation microscopy,” J. Struct. Biol. 147, 3‒ 11 (2004).
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A. A. Asatryan, C. J. R. Sheppard, and C. M. de Sterke, “Vector treatment of second-harmonic generation produced by tightly focused vignetted Gaussian beams,” J. Opt. Soc. Am. B 21, 2206‒ 2212 (2004).
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2003 (12)

D. Scherfeld, N. Kahya, and P. Schwille, “Lipid dynamics and domain formation in model membranes composed of ternary mixtures of unsaturated and saturated phosphatidylcholines and cholesterol,” Biophys. J. 85, 3758‒ 3768 (2003).
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T. Baumgart, S. T. Hess, and W. W. Webb, “Imaging coexisting fluid domains in biomembrane models coupling curvature and line tension,” Nature 425, 821‒ 824 (2003).
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N. Kahya, D. Scherfeld, K. Bacia, B. Poolman, and P. Schwille, “Probing lipid mobility of raft-exhibiting model membranes by fluorescence correlation spectroscopy,” J. Biol. Chem. 278, 28109‒ 28115 (2003).
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H. M. McConnell and M. Vrljic, “Liquid–liquid immiscibility in membranes,” J. Biol. Chem. 32, 469‒ 492 (2003).

M. Edidin, “The state of lipid rafts: from model membranes to cells,” Annu. Rev. Biophys. Biomol. Struct. 32, 257‒ 283 (2003).
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S. Munro, “Lipid rafts: elusive or illusive?,” Cell. 115, 377‒ 388 (2003).
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D. A. Dombeck, K. A. Kasischke, H. D. Vishwasrao, M. Ingelsson, B. T. Hyman, and W. W. Webb, “Uniform polarity microtubule assemblies imaged in native brain tissue by second-harmonic generation microscopy,” Proc. Natl. Acad. Sci. U.S.A. 100, 7081‒ 7086 (2003).
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L. Moreaux, T. Pons, V. Dambrin, M. Blanchard-Desce, and J. Mertz, “Electro-optic response of second harmonic generation membrane potential sensors,” Opt. Lett. 28, 625‒ 627 (2003).
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V. Le Floc’h, S. Brasselet, J.-F. Roch, and J. Zyss, “Monitoring of orientation in molecular ensembles by polarization sensitive nonlinear microscopy,” J. Phys. Chem. B 107, 12403‒ 12410 (2003).
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D. Oron, E. Tal, and Y. Silberberg, “Depth-resolved multiphoton polarization microscopy by third-harmonic generation,” Opt. Lett. 28, 2315‒ 2317 (2003).
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J. V. Rocheleau, M. Edidin, and D. W. Piston, “Intrasequence GFP in class I MHC molecules, a rigid probe for fluorescence anisotropy measurements of the membrane environment,” Biophys. J. 84, 4078‒ 4086 (2003).
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J.-X. Cheng, S. Pautot, D. A. Weitz, and X. S. Xie, “Ordering of water molecules between phospholipid bilayers visualized by coherent anti-Stokes Raman scattering microscopy,” Proc. Natl. Acad. Sci. U.S.A. 100, 9826‒ 9830 (2003).
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2002 (8)

S. Jiao and L. V. Wang, “Two-dimensional depth-resolved Mueller matrix of biological tissue measured with double-beam polarization-sensitive optical coherence tomography,” Opt. Lett. 27, 101‒ 104 (2002).
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J. M. Bueno and M. C. W. Campbell, “Confocal scanning laser ophthalmoscopy improvement by use of Mueller-matrix polarimetry,” Opt. Lett. 27, 830‒ 832 (2002).
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P. J. Campagnola, A. C. Millard, M. Terasaki, P. E. Hoppe, C. J. Malone, and W. A. Mohler, “Three-dimesional high-resolution second-harmonic generation imaging of endogenous structural proteins in biological tissues,” Biophys. J. 81, 493‒ 508 (2002).
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J.-X. Cheng, Y. K. Ji, G. Zheng, and X. S. Xie, “Laser-scanning coherent anti-Stokes Raman scattering microscopy and applications to cell biology,” Biophys. J. 83, 502‒ 509 (2002).
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P. Stoller, B. M. Kim, A. M. Rubenchik, K. M. Reiser, and L. B. Da Silva, “Polarization-dependent optical second harmonic imaging of a rat-tail tendon,” J. Biomed. Opt. 7, 205‒ 214 (2002).
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P. Stoller, K. M. Reiser, P. M. Celliers, and A. M. Rubenchik, “Polarization-modulated second harmonic generation in collagen,” Biophys. J. 82, 3330‒ 3342 (2002).
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S. L. Veatch and S. L. Keller, “Organization in lipid membranes containing cholesterol,” Phys. Rev. Lett. 89, 268101 (2002).
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J. Van Rheenen and K. Jalink, “Agonist-induced PIP2 hydrolysis inhibits cortical actin dynamics: regulation at a global but not at a micrometer scale,” Mol. Biol. Cell 13, 3257‒ 3267 (2002).
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2001 (7)

E. Ikonen, “Roles of lipid rafts in membrane transport,” Curr. Opin. Cell Biol. 13, 470‒ 477 (2001).
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A. Gidwani, D. Holowka, and B. Baird, “Fluorescence anisotropy measurements of lipid order in plasma membranes and lipid rafts from RBL-2H3 mast cells,” Biochemistry 40, 12422‒ 12429 (2001).
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C. Dietrich, L. A. Bagatolli, Z. N. Volovyk, N. L. Thompson, M. Levi, K. Jacobson, and E. Gratton, “Lipid rafts reconstituted in model membranes,” Biophys. J. 80, 1417‒ 1428 (2001).
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A. V. Samsonov, I. Mihalyov, and F. S. Cohen, “Characterization of cholesterol-sphingomyelin domains and their dynamics in bilayer membranes,” Biophys. J. 81, 1486‒ 1500 (2001).
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I. Gautier, M. Tramier, C. Durieux, J. Coppey, R. B. Pansu, J. C. Nicolas, K. Kemnitz, and M. Coppey-Moisan, “Homo-FRET microscopy in living cells to measure monomer-dimer transition of GFT-tagged proteins,” Biophys. J. 80, 3000‒ 3008 (2001).
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T. Pentcheva and M. Edidin, “Clustering of peptide-loaded MHC class I molecules for endoplasmic reticulum export imaged by fluorescence resonance energy transfer,” J. Immunol. 166, 6625‒ 6632 (2001).
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J.-X. Cheng, L. D. Book, and X. S. Xie, “Polarization coherent anti-Stokes Raman scattering microscopy,” Opt. Lett. 26, 1341‒ 1343 (2001).
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2000 (3)

L. Moreaux, O. Sandrea, and J. Mertz, “Membrane imaging by second harmonic generation microscopy,” J. Opt. Soc. Am. B 17, 1685‒ 1689 (2000).
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G. J. Schütz, G. Kada, V. Ph. Pastushenko, and H. Schindler, “Properties of lipid microdomains in a muscle cell membrane visualized by single molecule microscopy,” EMBO J. 162, 892‒ 901 (2000).

G. H. Patterson, D. W. Piston, and B. G. Barisas, “Förster distances between green fluorescent protein pairs,” Anal. Biochem. 284, 438‒ 440 (2000).
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1999 (8)

S. E. Sund, J. A. Swanson, and D. Axelrod, “Cell membrane orientation visualized by polarized total internal reflection fluorescence,” Biophys. J. 77, 2266‒ 2283 (1999).
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P. Schwille, U. Haupts, S. Maiti, and W. W. Webb, “Molecular dynamics in living cells observed by fluorescence correlation spectroscopy with one- and two-photon excitation,” Biophys. J. 77, 2251‒ 2265 (1999).
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L. A. Bagatolli and E. Gratton, “Two-photon fluorescence microscopy observation of shape changes at the phase transition in phospholipid giant unilamellar vesicles,” Biophys. J. 77, 2090‒ 2101 (1999).
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D. Marguet, E. T. Spiliotis, T. Pentcheva, M. Lebowitz, J. Schneck, and M. Edidin, “Lateral diffusion of GFP-tagged H2Ld molecules and of GFP-TAP1 reports on the assembly and retention of these molecules in the endoplasmic reticulum,” Immunity 11, 231‒ 240 (1999).
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P. J. Campagnola, M. Wei, A. Lewis, and L. M. Loew, “High-resolution nonlinear imaging of live cells by second harmonic generation,” Biophys. J. 77, 3341‒ 3349 (1999).
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D. A. Cheresh, J. Leng, and R. L. Klemke, “Regulation of cell contraction and membrane ruffling by distinct signals in migratory cells,” J. Cell Biol. 146, 1107‒ 1116 (1999).
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A. Zumbusch, G. R. Holtom, and X. S. Xie, “Three-dimensional vibrational imaging by coherent anti-Stokes Raman scattering,” Phys. Rev. Lett. 82, 4142‒ 4145 (1999).
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R. E. Dale, S. C. Hopkins, U. A. van der Heide, T. Marszalek, M. Irving, and Y. E. Goldman, “Model-independent analysis of the orientation of fluorescent probes with restricted mobility in muscle fibers,” Biophys. J. 76, 1606‒ 1618 (1999).
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1998 (2)

R. Varma and S. Mayor, “GPI-anchored proteins are organized in submicron domains at the cell surface,” Nature 394, 798‒ 801 (1998).
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S. M. Blackman, D. W. Piston, and A. H. Beth, “Oligomeric state of human erythrocyte band 3 measured by fluorescence resonance energy homotransfer,” Biophys. J. 75, 1117‒ 1130 (1998).
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1997 (5)

K. Simons and E. Ikonen, “Functional rafts in cell membranes,” Nature 387, 569‒ 572 (1997).
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M. Flörsheimer, M. Bösch, C. Brillert, M. Wierschem, and H. Fuchs, “Second-harmonic microscopy—a quantitative probe for molecular surface order,” Adv. Mater. 9, 1061‒ 1065 (1997).
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J. F. de Boer, T. E. Milner, M. J. C. van Gemert, and J. S. Nelson, “Two-dimensional birefringence imaging in biological tissue by polarization-sensitive optical coherence tomography,” Opt. Lett. 22, 934‒ 936 (1997).
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D. J. Maitland and J. T. Walsh, “Quantitative measurements of linear birefringence during heating of native collagen,” Lasers Surg Med. 20, 310‒ 318 (1997).
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Y. Guo, P. P. Ho, H. Savage, D. Harris, P. Sacks, S. Schantz, F. Liu, N. Zhadin, and R. R. Alfano, “Second-harmonic tomography of tissues,” Optics Lett. 22, 1323‒ 1325 (1997).
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1996 (4)

I. Ben-Oren, G. Peleg, A. Lewis, B. Minke, and L. Loew, “Infrared nonlinear optical measurements of membrane potential in photoreceptor cells,” Biophys. J. 71, 1616‒ 1620 (1996).
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S. M. Blackman, C. E. Cobb, A. H. Beth, and D. W. Piston, “The orientation of eosin-5-maleimide on human erythrocyte band 3 measured by fluorescence polarization microscopy,” Biophys. J. 71, 194‒ 208 (1996).
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T. Ha, Th. Enderle, D. S. Chemla, P. R. Selvin, and S. Weiss, “Single molecule dynamics studied by polarization modulation,” Phys. Rev. Lett. 77, 3979‒ 3982 (1996).
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M. Flörsheimer, C. Radüge, H. Salmen, M. Bösch, R. Terbrack, and H. Fuchs, “In-situ imaging of Langmuir monolayers by second-harmonic microscopy,” Thin Solid Films 284, 659‒ 662 (1996).
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1995 (1)

M. Gurp, “The use of rotation matrices in the mathematical description of molecular orientations in polymers,” Colloid Polym. Sci. 273, 607‒ 625 (1995).
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1993 (2)

J. Borejdo and S. Burlacu, “Measuring orientation of actin filaments within a cell: orientation of actin in intestinal microvilli,” Biophys. J. 65, 300‒ 309 (1993).
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1990 (4)

I. Ledoux, C. Lepers, A. Périgaud, J. Badan, and J. Zyss, “Linear and nonlinear optical properties of N-4-nitrophenyl L-prolinol single crystals,” Opt. Commun. 80, 149‒ 154 (1990).
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K. Florine-Casteel, “Phospholipid order in gel- and fluid-phase cell-size liposomes measured by digitized video fluorescence polarization microscopy,” Biophys. J. 57, 1199‒ 1215 (1990).
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W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248, 73‒ 76 (1990).
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J. A. Dix and A. S. Verkman, “Mapping of fluorescence anisotropy in living cells by ratio imaging. Application to cytoplasmic viscosity,” Biophys. J. 57, 231‒ 240 (1990).
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1989 (1)

1986 (1)

I. Freund, M. Deutsch, and A. Sprecher, “Connective tissue polarity. Optical photo microscopy, crossed-beam summation, and small-angle scattering in rat-tail tendon,” Biophys. J. 50, 693‒ 712 (1986).
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1979 (2)

D. Axelrod, “Carbocyanine dye orientation in red cell membrane studied by microscopic fluorescence polarization,” Biophys. J. 26, 557‒ 573 (1979).
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J.-L. Oudar, R. W. Smith, and Y. R. Shen, “Polarization-sensitive coherent anti-Stokes Raman spectroscopy,” Appl. Phys. Lett. 34, 758‒ 760 (1979).
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1978 (2)

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