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Journal of the Optical Society of America B

Journal of the Optical Society of America B

| OPTICAL PHYSICS

  • Editor: Henry van Driel
  • Vol. 29, Iss. 6 — Jun. 1, 2012
  • pp: 1541–1550
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Polarization-resolved four-wave mixing microscopy for structural imaging in thick tissues

Fabiana Munhoz, Hervé Rigneault, and Sophie Brasselet  »View Author Affiliations


JOSA B, Vol. 29, Issue 6, pp. 1541-1550 (2012)
http://dx.doi.org/10.1364/JOSAB.29.001541


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Abstract

We present a polarimetric analysis of the four-wave mixing (FWM) signal emitted by thick tissues, in order to extract structural information on molecular order and orientation. A careful analysis of the polarization distortions introduced by the birefringence of the sample is conducted for the proper interpretation of the results. FWM, compared to other well-known nonlinear optical methods such as second-harmonic generation, gives access to additional information on the symmetry of the molecular distribution. Furthermore, it brings the advantage of being generated in any kind of sample, even when centrosymmetry is present. The model developed here is applied to thick rat-tail tendon samples, composed essentially of collagen fibers. We show that, once the birefringence of the sample is completely characterized, it is possible to retrieve the even-order components of the molecular orientational distribution up to the fourth order of symmetry.

© 2012 Optical Society of America

1. INTRODUCTION

Contrary to SHG, which requires a noncentrosymmetric medium in order to take place, four-wave mixing (FWM) microscopy can be generated in any sample structure, with arbitrary organization. Because of favorable phase-matching conditions [22

22. W. Min, S. Lu, M. Rueckel, G. R. Hotom, and X. S. Xie, “Near-degenerate four-wave-mixing microscopy,” Nano Lett. 9, 2423–2426 (2009). [CrossRef]

25

25. P. Mahou, N. Olivier, G. Labroille, L. Duloquin, J.-M. Sintes, N. Peyriéras, R. Legouis, D. Débarre, and E. Beaurepaire, “Combined third-harmonic generation and four-wave mixing microscopy of tissues and embryos,” Biomed. Opt. Express 2, 2837–2849 (2011). [CrossRef]

], FWM is efficient in biological tissues, as illustrated in recent works on epithelial lipids imaging in C. elegans worms [24

24. R. Selm, G. Krauss, A. Leitenstorfer, and A. Zumbusch, “Simultaneous second-harmonic generation, third-harmonic generation, and four-wave mixing microscopy with single sub-8 fs laser pulses,” Appl. Phys. Lett. 99, 181124 (2011). [CrossRef]

,25

25. P. Mahou, N. Olivier, G. Labroille, L. Duloquin, J.-M. Sintes, N. Peyriéras, R. Legouis, D. Débarre, and E. Beaurepaire, “Combined third-harmonic generation and four-wave mixing microscopy of tissues and embryos,” Biomed. Opt. Express 2, 2837–2849 (2011). [CrossRef]

], or for cytoplasmic cells imaging in zebrafish embryos during early division [25

25. P. Mahou, N. Olivier, G. Labroille, L. Duloquin, J.-M. Sintes, N. Peyriéras, R. Legouis, D. Débarre, and E. Beaurepaire, “Combined third-harmonic generation and four-wave mixing microscopy of tissues and embryos,” Biomed. Opt. Express 2, 2837–2849 (2011). [CrossRef]

]. Contrary to third-harmonic generation (THG), another FWM process, which is interface specific [25

25. P. Mahou, N. Olivier, G. Labroille, L. Duloquin, J.-M. Sintes, N. Peyriéras, R. Legouis, D. Débarre, and E. Beaurepaire, “Combined third-harmonic generation and four-wave mixing microscopy of tissues and embryos,” Biomed. Opt. Express 2, 2837–2849 (2011). [CrossRef]

28

28. 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. Meth. 3, 47–53 (2006). [CrossRef]

], FWM is bulk sensitive and can potentially reveal structural information from macroscopic systems using a polarization-resolved analysis. In particular, polarization-resolved FWM allows one, in principle, to probe the even-order symmetries of the molecular orientational distribution, up to the fourth order [19

19. S. Brasselet, “Polarization resolved nonlinear microscopy: application to structural molecular and biological imaging,” Adv. Opt. Photon. 3, 205–271 (2011). [CrossRef]

], giving complementary information, as compared to SHG, on the microscopic organization of complex samples.

In this work, we develop the principles of polarization-resolved FWM microscopy to retrieve quantitative information on molecular order and orientation in a model system made of collagen fibers. First, we develop a general theoretical model to determine the fourth-order nonresonant susceptibility tensor of collagen, which can be applied more generally to any kind of molecular system where a degree of molecular order is present. Then, we describe the optical setup to perform polarization-resolved FWM microscopy and the method to correct from polarization distortions that can be introduced either by the optical setup or the sample itself. Indeed, biological tissues can exhibit strong scattering and birefringence, which can lead to misinterpretation of the polarimetric results, especially in thick samples [29

29. 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]

,30

30. 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]

]. Finally, we propose a fitting procedure to quantify orientation and symmetry orders of the molecular orientational distribution in the sample. All the studies developed here are performed in the nonresonant regime; however, they can be, in principle, extended to a vibrational resonant process, such as coherent anti-Stokes Raman scattering (CARS) to probe the orientation and symmetry of specific molecular vibrational modes [31

31. 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]

].

2. THEORY

A. Microscopic Fourth-Order Susceptibility Tensor of a Molecular Assembly

Biomolecular organizations, such as collagen fibers, can be described as an assembly of elementary nonlinear-active molecules with a given statistical orientational distribution. This molecular angular distribution is defined by a normalized probability distribution function f(Ω), with Ω=(θ,ϕ) being the spherical angles denoting the orientation of the molecular frame (u,v,w) in the microscopic frame (x,y,z) [Fig. 1(a)]. The use of two orientation Euler angles (θ,ϕ) supposes that the molecules within the angular distribution are of uniaxial symmetry, which has been largely used in the case of collagen fibers [5

5. 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). [CrossRef]

,13

13. S. V. Plotnikov, A. C. Millard, P. J. Campagnola, and W. A. Mohler, “Characterization of the myosin-based source for second-harmonic generation from muscle sarcomeres,” Biophys. J. 90, 693–703 (2006). [CrossRef]

,16

16. F. Tiaho, G. Recher, and D. Rouède, “Estimation of helical angles of myosin and collagen by second harmonic generation imaging microscopy,” Opt. Express 15, 4054–4065 (2007). [CrossRef]

,18

18. 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]

,32

32. I. Rocha-Mendoza, D. R. Yankelevich, M. Wang, K. M. Reiser, C. W. Frank, and A. Knoesen, “Sum frequency vibrational spectroscopy: the molecular origins of the optical second-order nonlinearity of collagen,” Biophys. J. 93, 4433–4444 (2007). [CrossRef]

] and can be considered as a first approximation in a unknown sample. We assume furthermore in what follows that the molecular third-order susceptibility γ has only one nonvanishing component along w in the molecular frame (u,v,w), γwwww. More complex molecular susceptibility tensors can however be easily introduced in the following model. Under this molecular 1D assumption, the microscopic third-order nonlinear susceptibility tensor χijkl(3) can be written, in the microscopy frame as
χijkl(3)=γwwww[(w·i)(w·j)(w·k)(w·l)](Ω)f(Ω)dΩ,
(1)
with (w·i) being the rotation matrix components between the microscopic and molecular frames and dΩ=02π0πsinθdθdϕ.

Fig. 1. Definition of the molecular, microscopic, and macroscopic frames. The 3D plot represents a molecular distribution function. (a) Definition of the angles (θ,ϕ) that characterize the orientation of the molecular w axis in the microscopic (x,y,z) frame. We assume that the molecules (represented as arrows) are one-dimensional and oriented along the w axis. The z axis is set along the axis of higher symmetry of the molecular orientational distribution function. (b) Orientation of the microscopic frame (e.g., collagen fiber) in the macroscopic frame (X,Y,Z), defined by angles (ϕ0,θ0).

In this work we address the most general case, where the shape of the orientational distribution function is unknown, by decomposing it on a series of Ω-dependent orthonormal functions [19

19. S. Brasselet, “Polarization resolved nonlinear microscopy: application to structural molecular and biological imaging,” Adv. Opt. Photon. 3, 205–271 (2011). [CrossRef]

]. In the context of fibrillar systems, one can furthermore assume that the molecular orientational distribution is of cylindrical symmetry; then f(θ,ϕ) does not depend on ϕ. The distribution function can thus be expanded in a series of the Legendre polynomials, along the same lines of previously developed models in molecular doped poled polymers [33

33. M. G. Kuzyk, K. D. Singer, H. E. Zahn, and L. A. King, “Second-order nonlinear-optical tensor properties of poled films under stress,” J. Opt. Soc. Am. B 6, 742–752 (1989). [CrossRef]

]:
f(θ)=JfJPJ(cosθ),
(2)
where PJ(cosθ) is the J-order Legendre polynomial, with J representing the orders of symmetry of the orientational distribution function. The coefficients fJ, called order parameters [34

34. M. Gurp, “The use of rotation matrices in the mathematical description of molecular orientations in polymers,” Colloid Polym. Sci. 273, 607–625 (1995). [CrossRef]

], correspond to the weights of the function PJ in this decomposition. Increasing orders of the polynomials correspond to narrower distributions in relation to θ, together with an increasing complexity of their multipolar nature.

In the case of nonresonant FWM, a symmetric fourth-order susceptibility tensor is involved. It is thus possible to probe even components of the orientational distribution function until the fourth-order of symmetry corresponding to J=4 [19

19. S. Brasselet, “Polarization resolved nonlinear microscopy: application to structural molecular and biological imaging,” Adv. Opt. Photon. 3, 205–271 (2011). [CrossRef]

]. FWM is therefore a complementary probe relative to SHG, with a sensitivity to higher details of the molecular distribution. In the context of FWM, Eq. (1) therefore becomes
χijkl(3)=[(w·i)(w·j)(w·k)(w·l)](θ,ϕ)×[1+f23cos2θ12+f435cos4θ30cos2θ+38]sinθdθdϕ,
(3)
where f2 and f4 are the second- and fourth-order parameters, respectively. These coefficients are normalized by the weight f0 of the zeroth-order symmetry, which corresponds to the isotropic contribution of the distribution. f0 might contain all depolarized contribution to the FWM signal, including scattering from the sample. It means that the relevant parameter in the molecular order quantification, free from unwanted isotropic contributions, is the ratio f4/f2. The expression of the microscopic susceptibility tensor in Eq. (3) is further normalized by γwwww.

The decomposition of the molecular orientational distribution function in a series of the Legendre polynomialscorresponds to the multipolar expansion [20

20. J. Zyss, “Octupolar organic systems in quadratic nonlinear optics: molecules and materials,” Nonlin. Opt. 1, 3–18 (1991).

] of a one-dimensional structure with cylindrical symmetry. The zeroth-order term J=0 is the monopole and corresponds to an isotropic distribution. When the coefficient f2 is zero, then the distribution is purely hexadecapolar, with fourth-order symmetry. In the same way, when f4 is zero, then the distribution is purely quadrupolar, with second-order symmetry (J=2). Figure 2 shows the effect of different sets of f2, f4 values on the theoretical molecular angular distribution function f(θ) projected into the plane xz as a function of θ. Note that this figure does not depict the whole distribution function, but only a truncation of its even terms up to the fourth order. When both coefficients f2 and f4 vanish, the distribution is isotropic and no direction is privileged, as expected from the zeroth-order term of the multipolar expansion. When f4=0, the distribution function exhibits a two-lobe shape, that can be either along the z axis or in the plane perpendicular to it (xy plane), depending on the sign of f2. When f2=0, the angular distribution function exhibits a four-lobe shape, which characterizes the higher multipolar order, with an orientation that also depends on the sign of f4. Intermediate cases exhibit generally four-lobe patterns with more pronounced four lobes when f4 is higher than f2.

Fig. 2. Truncated molecular angular distribution function f(θ) representing its even-order terms up to J=4, projected into the xz plane, for different f2 f4 values.

B. From Microscopic to Macroscopic Susceptibility

We denote Ω0=(θ0,ϕ0) as the orientation angles of the higher symmetry axis of the microscopic frame in the macroscopic frame (X,Y,Z), i.e., the laboratory frame where the components of the incident electrical fields are defined [Fig. 1(b)]. This high symmetry axis corresponds typically to the direction of a collagen fiber in the case of collagen-based samples. The macroscopic susceptibility tensor can be obtained by projecting the microscopic susceptibility χijkl(3) into the macroscopic frame, according to
χIJKL=(X,Y,Z)(3)(θ0,ϕ0)=ijkl=(x,y,z)χijkl(3)[(i·I)(j·J)(k·K)(l·L)](θ0,ϕ0),
(4)
where (i·I) are the components of the rotation matrix between the macroscopic and microscopic frames. In what follows, we suppose that the symmetry axis of the microscopic structure (collagen fiber direction) lies in the XY plane; therefore, the Euler angle θ0 is fixed to 90°.

Finally, the FWM signal is generated from the induced third-order nonlinear polarization PFWM, that couples the incident electric fields with the macroscopic susceptibility tensor of the sample. In this work, we use a degenerate scheme, where two input fields have the same angular frequency ω1=ω2 and the third one is set to the angular frequency ω3. In the planar wave approximation, which is found to be valid in this study [19

19. S. Brasselet, “Polarization resolved nonlinear microscopy: application to structural molecular and biological imaging,” Adv. Opt. Photon. 3, 205–271 (2011). [CrossRef]

,35

35. 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]

], the measured intensity is proportional to the modulus square of the induced nonlinear polarization PFWM, at angular frequency ω0=2ω1ω3, and can be written as
II=(X,Y)|PIFWM(ω0)|2=|IJKLχIJKL(3)(ω0;ω1,ω1,ω3)EJ(ω1)EK(ω1)EL*(ω3)|2,
(5)
with I=X or Y, the analysis direction along which the signal is detected, and EJ,K(ω1) and EL(ω3) the components of the incident fields along the macroscopic directions J, K, and L.

3. EXPERIMENT

A. Optical Setup

The incident pulse trains are delivered by two picosecond tunable mode-locked lasers (Coherent Mira 900, 76 MHz, 3 ps), pumped by a Nd:vanadate laser (Coherent Verdi). The lasers are electronically synchronized (Coherent SynchroLock System) and are externally pulse picked (APE pulse picker) to reduce their rate to 3.8 MHz. Achromatic half-wave plates mounted on step rotation motors, allow rotation of the incident linear polarizations separately for both beams. The beams are spatially recombined through a dichroic filter, injected into a commercial inverted microscope (Zeiss Axiovert 200 M), and focused in the sample through a low-numerical-aperture microscope objective (Olympus LUCPLFLN 40×, NA=0.6), in order to avoid any contribution from the Z-polarized component of the excitation field in the experiment, while keeping a micrometric scale optical resolution [35

35. 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]

]. The sample is placed on a piezoelectric stage (Physik Instrument) ensuring XYZ nanometric scale precision positioning. The generated signal is forwardly collected by another microscope objective (Olympus LMPLFLN 50×, NA=0.5) and is filtered by two filters (a low-pass filter to reject the incident lasers and a bandpass filter spectrally centered at the wavelength of the emitted beam). The signal is split by a broadband polarizing cube beamsplitter (Newport), and the two resulting perpendicularly polarized beams are finally detected by two avalanche photodiodes (PerkinElmer SPCM-AQR-14) used in the photon counting mode. The incident wavelengths are set to λ1=724nm and λ3=857nm, in order to avoid any vibrational resonance from the collagen. In this work, two different schemes of polarization tuning are used: either the linear polarization of the field E(ω3) is fixed along the X axis and the linear polarization of the degenerate beam E(ω1) rotates with an angle α1 relative to X varying from 0° to 360°, or both input polarizations rotate simultaneously with an angle α1,3=α1=α3 relative to X from 0° to 360°. The use of different incident polarization configurations, together with a polarized detection, is necessary in the case of birefringent samples, in order to avoid ambiguous results (see Section 4).

B. Sample Preparation

In this work we study collagen Type I fibers, of about 100–140 μm thick, extracted from rat tail tendons. In the sample preparation (University of Exeter, UK), adult Sprague Dawley rats were euthanized for purposes unconnected with the present research. Tails were removed and immediately snap frozen in liquid-nitrogen-cooled isopentane. At the time of use, the tissue was thawed and the tendon exposed. Individual fibers were teased out by microdissection and either examined immediately or stored frozen until required. Control Raman spectra were identical in either case, contained none of the peaks characteristic of proteoglycans, and were indistinguishable from those obtained from fibers purified by enzymatic extraction.

The sample is sandwiched between two glass coverslips glued together by a thickened double-sided tape at the edge. This procedure prevents motion of the fibers during the measurements. The volume between the two coverslips is filled in with a 0.15 M NaCl solution. Recording the FWM signal emitted by the solution allows performing of all the fine optical settings and alignments of the setup.

C. Experimental Protocol

The experimental protocol consists in the following steps. First, both incident polarizations are set parallel to each other (either α1=α3=0° or α1=α3=90°) and two FWM images of the collagen fiber are recorded simultaneously, one polarized along the X axis and the other along the Y axis. The image size is typically 40μm×40μm (100×100 pixels), and the pixel dwell time is 20 ms. The average power of each of the incident beams at the focal spot position is 2 mW. The focal plane Z is fixed at the bottom surface of the sample [Z=0μm, according to Fig. 3(a)]. In this case, only the emitted signal is affected by the collagen fiber’s birefringence. The second step consists of choosing different (X,Y) positions of the acquired image in order to perform FWM polarization-resolved measurements, as described earlier in Subsection 3.A. Two experiments are performed: one for a varying α1 angle with α3=0 fixed, and the other for varying α1=α3=α1,3 angles. For each chosen point, the polarization response is also recorded for the input laser at frequency ω1, as described in Subsection 3.D, in order to characterize the local birefringence averaged over the whole thickness of the sample [29

29. 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]

]. We repeat the same procedure for another Z position of the sample [see Fig. 3(b)], by focusing the incident beams deeper into the collagen fiber, at a distance Z=d from the bottom surface. In this case, both the incident and emitted beams are affected by birefringence. Polarization-resolved FWM measurements are then performed at the same in-plane (X,Y) position for different depths.

Fig. 3. FWM detection scheme. The incident beams are linearly polarized and propagate along the Z axis. (a) Pump and Stokes fields are focused at Z=0μm (bottom surface of the sample). In this case, the polarization of the FWM signal emitted in the forward direction becomes elliptical after it traverses the whole thickness L of the sample. (b) The incident fields are focused at a distance Z=d from the surface of the sample. In this case, they are elliptical at the excitation point. The FWM signal is emitted in the forward direction, and it travels a distance Z=Ld in the sample. (c) Definition of the birefringence axis orientation Θb, which might differ from the local collagen fiber orientation ϕ0.

D. Experimental Polarization Distortions

The optical apparatus, in particular the dichroic filter recombining the excitation beams, may introduce some distortions in the incident polarizations. We use two photon fluorescence of a solution of Rhodamine 6G to characterize the ellipticity and dichroism of the incident polarizations, according to [36

36. 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). [CrossRef]

]. The parameters obtained (and used in all fitting procedures below) are 144° ellipticity and 1.09 dichroism factor between the X and Y field directions at λ1=724nm, and 55° ellipticity and 1.04 dichroism at λ3=857nm.

Moreover, collagen fibers are highly birefringent, which can lead to an erroneous interpretation of the collected signal and, by consequence, a wrong determination of the sample properties. This anisotropy is completely characterized by two parameters: (i) the orientation of the fast optical axis of the fiber Θb, and (ii) the phase shift between its fast and slow optical axes Φb(d), after the field penetrates a distance d into the sample. These birefringence parameters, defined in Fig. 3, can be estimated from a polarimetric method described in [29

29. 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]

]. Briefly, we rotate the linear polarization angle α1 of the incident field E(ω1) from 0° to 360° and record the transmitted intensity at the laser wavelength λ1 along the X and Y directions, after the beam has traversed the whole thickness of the fiber d=L. The birefringence parameters, averaged over the whole sample thickness, can be determined from a fit of the obtained data, as described in Subsection 4.A. The birefringence of the sample at wavelength λ3 is considered identical, because the sample does not exhibit resonances in this wavelength regime.

4. RESULTS

FWM images of a collagen fiber sample, acquired at Z=0μm, are shown in Fig. 4. The three images correspond to different regions of the fiber, distant by a few millimeters, which exhibit visibly different local fiber orientation. The three depicted points (labeled, respectively, positions 1, 2, and 3) represent the (X,Y) spot positions chosen to perform polarization-resolved measurements.

Fig. 4. FWM images of the studied collagen fibers, at different in-plane positions in the sample. The images show the total FWM intensity (I=IX+IY) at Z=0μm, normalized to 1. In the left image, incident polarizations are set parallel to the Y axis, while in the other two images, input polarizations are along the X axis. Points 1, 2, and 3 correspond to the spots where the polarization measurements are performed. Black lines, orientation of the principal symmetry axis of the orientational distribution function in the XY plane (ϕ0), given by the best fit of the polarization-resolved FWM intensities; black curves, in-plane projection of the orientational distribution function, truncated to its even orders (0, 2, and 4) and oriented in the macroscopic sample plane XY; dashed lines, orientation of the fast optical axis of the fiber (Θb), obtained by the fit of the sample birefringence. Scale bar: 10 μm.

A. Determination of the Sample Birefringence

Figure 5(a) shows the intensities of the degenerate field at frequency ω1 transmitted throughout the fiber and detected along the X and Y axes, as a function of α1, for the position 3 of the sample. Considering that the collagen fiber acts as a single birefringent plate of thickness L, Θb and Φb(L) can be determined by a fitting of these polarization dependencies, using the method developed in [29

29. 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]

]. This fit accounts for the instrumental distortions mentioned in Subsection 3.D. Figure 5(b) depicts the fitting cartography of the mean square error (MSE) as a function of (Θb,Φb) for the data of Fig. 5(a). As already pointed out in [29

29. 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]

], the solution [Θb,Φb(L)] is not unique when polarization distortions are already present in the experimental setup: there are four distinct solutions (labeled 1, 2, 3, 4) in the range 0Θbπ and 0Φb(L)2π, of which only two (1, 2) are independent. The other two solutions, 3 and 4, are related, respectively, to 1 and 2 by the π/2 periodicity on Θb [(Θb)3(4)=(Θb)1(2)+π/2; thus, (Φb)3(4)(L)=2π(Φb)1(2)(L)].

Fig. 5. Fitting of the birefringence parameters of the collagen fiber sample at position 3. (a) Experimental polarization-dependent intensity of the incident beam at frequency ω1 (dots) along X (black) and Y (gray), as a function of the incident polarization α1, given in polar representation. Solid lines correspond to the respective best fits, accounting for instrumental polarization distortions. (b)  (Θb,Φb) cartography of the MSE. The four labeled minima correspond to the four possible solutions.

The four possible values for the set [Θb,Φb(L)] obtained in Fig. 5 are (58°,179°)1, (40°,228°)2, (148°,181°)3, and (130°,132°)4, the subscripts being used only to label the different possible results. The obtained birefringence phase shifts for both solutions 1 and 2 can be used to deduce the approximate sample thickness L at the measured sample location. Assuming that the refractive index difference Δn between the long axis of the collagen fibers and its perpendicular direction is approximately 0.003 [37

37. F. P. Bolin, L. E. Preuss, R. C. Taylor, and R. J. Ference, “Refractive index of some mammalian tissues using a fiber optic cladding method,” Appl. Opt. 28, 2297–2303 (1989). [CrossRef]

], a phase shift of Φb(L)180° leads to L120μm, which is reasonable in the studied sample. Solutions 1 and 2 can, in principle, be used as known parameters in the future polarization-resolved FWM analysis. These solutions are, however, independent and are therefore expected to lead to different polarization-resolved FWM dependencies, which might thus result in different values for the retrieved order parameters. In practice, the polarization-resolved FWM intensities are fitted using these two independent solutions and we chose to retain the solution which leads to the best fit (Subsection 4.B).

B. Orientation and Symmetry Oorder Properties of Collagen Fibers at Z=0μm

Fitting the polarization responses of the FWM signal (Section 2) allows characterization of the symmetry order of the molecular assemblies in the collagen fiber, as well as the mean orientation of the molecular distribution in the fiber, with respect to the macroscopic coordinate system. The fitting parameters are (i) the order parameters f2 and f4 (normalized by the isotropic contribution f0) and (ii) the angle ϕ0 introduced in Eq. (4), which corresponds to the orientation of the molecular distribution in the XY plane.

The fitting procedure consists of finding simultaneously, for the two polarization configurations described in Subsection 3.A (α1 varying with α3=0° and α1,3 varying), the set (f2,f4,ϕ0) that minimizes the MSE function for the FWM intensities in the X and Y directions, normalized by the maximum of the total intensity, IX+IY. With such definitions, the MSE function is given as follows:
MSE(f2,f4,ϕ0)=1Nαi{[IXth(α1i,f2,f4,ϕ0)IXexp(α1i)]2+[IYth(α1i,f2,f4,ϕ0)IYexp(α1i)]2+[IXth(α1,3i,f2,f4,ϕ0)IXexp(α1,3i)]2+[IYth(α1,3i,f2,f4,ϕ0)IYexp(α1,3i)]2},
(6)
where the sum runs over the different incident polarization angles αi. The first two terms in the sum correspond to the first polarization configuration (α1 rotates and α3=0°), while the last two terms stand for the second configuration (α1=α3=α1,3 rotate simultaneously). The superscript “th” stands for the theoretical intensity calculated from Eq. (5), with the FWM susceptibility obtained from Eqs. (3) and (4). The superscript “exp” corresponds to the FWM intensity acquired experimentally. Finally, the error function is normalized by the number of the incident polarization angles over which the sum is done, here Nα=73 (this number is set to ensure a sufficient signal-to-noise ratio in the experimental data). This expression shows that the estimation of the measured parameters (f2,f4,ϕ0) relies on constraining conditions that leave no ambiguity on their determination: indeed, it relies on the simultaneous fitting of two set of experiments, both measured along the X and Y directions.

Note that before fitting, the excitation and emitted field polarizations must be corrected for the polarization distortions introduced by the optical setup and by the birefringence of the sample, as described in Subsection 3.D. In the present situation, because the incident beams are focused at the bottom surface of the sample (Z=0μm), only the emitted field is affected by the sample birefringence. In this way, the theoretical FWM field used in Eq. (6) must take into account the orientation of the optical axis Θb and the birefringence phase shift Φb between the fast and slow axes of the collagen fiber. As we already pointed out in Subsection 4.A, when the incident fields are affected by dichroism and ellipticity, the solution (Θb,Φb) is not unique. The two independent solutions do not result in the same theoretical FWM intensities, for the same parameters (f2,f4,ϕ0). Here, we fit independently the parameters (f2,f4,ϕ0) for both birefringence solutions that are independent and we choose the couple (Θb,Φb) that gives the smallest MSE. Figure 6 shows the experimental results and best fits for the three positions depicted in Fig. 4. The retrieved parameters found using this procedure are summarized in Table 1, together with the birefringence angles (Θb,Φb) used for these fits. For the three positions explored, the best solution for the birefringence parameters is solution 1 [(Θb,Φb)1 defined in Subsection 4.A], which gives identical results as its equivalent solution 3 [(Θb,Φb)3]. Note that not accounting for birefringence and instrumental polarization distortions induces a strong decrease in the fit quality, together with a bias on the obtained fitting parameters.

Fig. 6. FWM intensities along X (black) or Y (gray) as a function of the incident polarization at Z=0μm. (a) α1 rotates and α3=0°. (b) Both polarizations α1 and α3 rotate simultaneously. Solid lines, theoretical intensities given by the best fit; dots, experimental data. From left to right, the polar plots correspond, respectively, to positions 1, 2, and 3 (Fig. 4).

Table 1. Set of Parameters (f2,f4,ϕ0) Obtained by the Fitting Procedure, for Different Positions of the Collagen Fiber in the Same XY Plane for Z=0μma

table-icon
View This Table

Once the parameters f2 and f4 are determined, it is possible to build the corresponding even-term truncation of the orientational distribution functions. Their 3D plots in the microscopic frame (x,y,z) are shown in Fig. 7. In the three cases f40, which means that the angular distribution has a high-order symmetry (hexadecapolar) contribution represented by a four-lobe shape. Position 2, which shows the less defined fibrillar structure on the FWM image (Fig. 4), exhibits the most isotropic molecular angular distribution with the lowest values for f2 and f4. Note that a complete view of the orientational distribution function f(θ), until the fourth order of the series expansion would require the knowledge of the order parameters f1 and f3, which could be probed by PSHG. The diversity of shapes found in this sample could be most likely due to the different arrangements of fibers depending on the sample location: indeed, we do not probe here single isolated crystalline collagen fibrils, but rather a macroscopic organization of them in fiber bundles that constitute the tendon tissue. Nevertheless, the shape found here for the even-order truncated distribution is compatible with the picture found for collagen in tissues, supposed to be composed of nonlinear-active molecules lying along a cone surface pointing in the fiber direction [16

16. F. Tiaho, G. Recher, and D. Rouède, “Estimation of helical angles of myosin and collagen by second harmonic generation imaging microscopy,” Opt. Express 15, 4054–4065 (2007). [CrossRef]

,18

18. 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]

].

Fig. 7. 3D plots, in the microscopic frame, of the even-order terms of the multipolar expansion of the molecular angular distribution functions, built from the fitted parameters f2 and f4 for positions 1, 2, and 3 (from left to right).

The obtained results are represented schematically in Fig. 4, where the black lines superimposed to the FWM images of collagen represent the projection in the XY plane of the deduced molecular distribution oriented at ϕ0. The orientation of the local collagen fiber, observed from the acquired images, agrees qualitatively with the molecular averaged orientation ϕ0, for the three studied positions. The dashed lines in Fig. 4 correspond to the orientation of the sample fast optical axis Θb. This direction, which is deduced from an average information over the whole thickness of the collagen fiber, does not necessarily coincide with its local direction (black line). This is a signature of the heterogeneity of the sample along the Z direction. The measured birefringence phase shift measured in the three positions Φb(L)180200° (Table 1) agrees well with the expected thickness of the sample (L100140μm).

C. In-Depth Studies of Collagen Fibers: Z>0μm

Fig. 8. Experimental FWM intensities, measured at Z60μm, along X (black) or Y (gray), as a function of the incident polarization. (a) α1 rotates and α3=0°. (b) Both polarizations α1 and α3 rotate simultaneously. From left to right, the polar plots correspond, respectively, to positions 1, 2, and 3.

Fig. 9. Theoretical FWM intensities built from the fit set of parameters (f2,f4,ϕ0) of position 3, when focusing at different Z=dμm into the fiber (d/L varies from 0 to 1 with a step of 1/4). (a), (b) α1 and α3 rotate simultaneously. (c), (d) α1 rotates while α3=0°. (a), (c) accounting for the birefringence parameters (Θb,Φb)1=(58°,179°); (b), (d) (Θb,Φb)3=(148°,181°).

A direct qualitative comparison of these theoretical curves can be done with the experimental data of Fig. 8. In the case of the polarization scheme where only α1 rotates [Figs. 9(c), 9(d)], a four-lobe shape response appears for IX [similar to what is observed experimentally in Fig. 8(a)] when d/L0.5 and Θb=(Θb)3=148°. In the case where both polarizations α1 and α3 rotate simultaneously, a rough qualitative agreement with the experimental data [Fig. 8(b)] is found for d/L0.75 and Θb=(Θb)3. We deduce therefore that solution 3 of the birefringence parameters is the most appropriate one and that the explored position is close to a depth between 0.5L and 0.75L, which is expected from the experimental setting Z60μm. Finally, the fact that the theoretical and experimental curves agree qualitatively well means that the molecular order parameters (f2,f4,ϕ0) at this explored depth are likely to be close to the ones found previously for Z=0. We find similar results for the other positions 1 and 2 of the sample.

In addition to birefringence properties, a tissue can exhibit scattering and diattenuation, which should come into play in the physical interpretation of the nonlinear interaction with the collagen fibers. Following a procedure identical to [30

30. 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]

], we noticed that the addition of such parameters does not affect or improve significantly the fits of the measured polarization-resolved FWM responses, which means that birefringence is the dominant process responsible for the distortion of these responses at increasing depths in the sample. In general, these parameters should be in principle added, at best from a preliminary characterization based on ellipsometry performed at the incident and FWM wavelengths.

In summary, we notice that in highly birefringent samples (possibly affected in addition by scattering or diattenuation), the choice of more than one configuration polarization together with a polarized detection is determined for the interpretation of the polarimetric data. For instance, in the presence of strong birefringence, the changes in the polar plots acquired at different depths in the sample, observed in Figs. 6 and 8, are more prominent for the intensities acquired along the X direction and for the polarization scheme in which only α1 rotates. The presence of birefringence adds new unknowns to the problem of determining the molecular order of a medium and therefore requires a greater number of data. Nevertheless, this multiple configuration allows us, in principle, to explore cases where the local sample orientation contains out-of-plane components (θ0), because the number of parameters is still, in this case, compatible with the number of measurements performed. Finally, in nonbirefringent samples, the analysis can be considerably simplified. Indeed, in this situation only the detection of the total FWM intensity (IX+IY) in the case where both polarizations rotate simultaneously would be sufficient to determine the parameters (f2,f4,ϕ0) without ambiguity.

5. CONCLUSION

We have shown that polarization-resolved FWM is a powerful technique to retrieve the even orders of symmetry up to the fourth order in a molecular statistical ensemble, such as for thick collagenous tissues. In particular, we propose a fitting procedure that allows estimating the microscopic molecular orientational distribution function and its orientation in the macroscopic frame. Careful analysis concerning the birefringence of the sample must be performed before fitting the experimental FWM signals, in order to avoid erroneous or biased results. This microscopy technique, enabling imaging of local symmetry orders and molecular organization, brings new possibilities for high-contrast structural spatial investigation with submicrometric resolution. Polarization-resolved multimodal nonlinear microscopy can also be performed, by combining FWM with second-order nonlinear optical processes, such as SHG, in order to obtain complimentary information on the symmetry orders of the molecular distribution (in a nonresonant regime, one would expect the same microscopic structures to be responsible for both SHG and FWM signals). Finally, this method can be enlarged to resonant polarization-resolved nonlinear techniques, such as CARS, by probing specific vibrational resonances of the molecular ensembles and allowing an imaging tool combining both structural and chemical selectiveness.

ACKNOWLEDGMENTS

We thank Prof. C. Peter Winlove (Biomedical Physics Group, University of Exeter) for providing the rat-tail tendon collagen Type I sample. We also thank Dr. Franck Billard (Institut Carnot de Bourgogne, Dijon) and Dr. Sophie Brustlein (Institut Fresnel, Marseille) for help in the FWM polarization-dependent microscopy instrumentation developments. This work has been made possible by the support of the Ecole Polytechnique, CNRS and the region Provence Alpes Côte d’Azur.

REFERENCES

1.

M. Florsheimer, M. Bosch, C. Brillert, M. Wierschem, and H. Fuchs, “Second-harmonic imaging of surface order and symmetry,” Thin Solid Films 327–329, 241–246 (1998). [CrossRef]

2.

C. Anceau, S. Brasselet, and J. Zyss, “Local orientational distribution of molecular monolayers probed by nonlinear microscopy,” Chem. Phys. Lett. 411, 98–102 (2005). [CrossRef]

3.

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). [CrossRef]

4.

I. Freund, M. Deutsch, and A. Sprecher, “Connective tissue polarity. Optical second-harmonic microscopy, crossed-beam summation, and small-angle scattering in rat-tail tendon,” Biophys. J. 50, 693–712 (1986). [CrossRef]

5.

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). [CrossRef]

6.

P. Stoller, B.-M. Kim, A. M. Rubenchik, K. M. Reiser, and L. B. Da Silva, “Polarization-dependent optical second-harmonic imaging of rat-tail tendon,” J. Biomed. Opt. 7, 205–214 (2002). [CrossRef]

7.

G. Cox, E. Kable, A. Jones, I. Fraser, F. Manconi, and M. D. Gorrell, “3-Dimensional imaging of collagen using second harmonic generation,” J. Struct. Biol. 141, 53–62 (2003). [CrossRef]

8.

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). [CrossRef]

9.

P. J. Campagnola, A. C. Millard, M. Terasaki, P. E. Hoppe, C. J. Malone, and W. A. Mohler, “3-Dimesional high-resolution second harmonic generation imaging of endogenous structural proteins in biological tissues,” Biophys. J. 82, 493–508 (2002). [CrossRef]

10.

M. Both, M. Vogel, O. Friedrich, F. von Wegner, T. Künsting, R. Fink, H. A. Rainer, and D. Uttenweiler, “Second harmonic imaging of intrinsic signals in muscle fibers in situ,” J. Biomed. Opt. 9, 882–892 (2004). [CrossRef]

11.

T. Boulesteix, E. Beaurepaire, M.-P. Sauviat, and M.-C. Schanne-Klein, “Second-harmonic microscopy of unstained living cardiacmyocytes: measurements of sarcomere length with 20 nm accuracy,” Opt. Lett. 29, 2031–2033 (2004). [CrossRef]

12.

A. Zoumi, X. Lu, G. S. Kassab, and B. J. Tromberg, “Imaging coronary artery microstructure using second-harmonic and two-photon fluorescence microscopy,” Biophys. J. 87, 2778–2786 (2004). [CrossRef]

13.

S. V. Plotnikov, A. C. Millard, P. J. Campagnola, and W. A. Mohler, “Characterization of the myosin-based source for second-harmonic generation from muscle sarcomeres,” Biophys. J. 90, 693–703 (2006). [CrossRef]

14.

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). [CrossRef]

15.

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). [CrossRef]

16.

F. Tiaho, G. Recher, and D. Rouède, “Estimation of helical angles of myosin and collagen by second harmonic generation imaging microscopy,” Opt. Express 15, 4054–4065 (2007). [CrossRef]

17.

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). [CrossRef]

18.

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]

19.

S. Brasselet, “Polarization resolved nonlinear microscopy: application to structural molecular and biological imaging,” Adv. Opt. Photon. 3, 205–271 (2011). [CrossRef]

20.

J. Zyss, “Octupolar organic systems in quadratic nonlinear optics: molecules and materials,” Nonlin. Opt. 1, 3–18 (1991).

21.

J. Zyss, “Molecular engineering implications of rotational invariance in quadratic nonlinear optics: from dipolar to octupolar molecules and materials,” J. Chem. Phys. 98, 6583–6599 (1993). [CrossRef]

22.

W. Min, S. Lu, M. Rueckel, G. R. Hotom, and X. S. Xie, “Near-degenerate four-wave-mixing microscopy,” Nano Lett. 9, 2423–2426 (2009). [CrossRef]

23.

D. Akimov, S. Chatzipapadopoulos, T. Meyer, N. Tarce, B. Dietzek, M. Schmitt, and J. Popp, “Different contrast information obtained from CARS and nonresonant FWM images,” J. Raman Spectrosc. 40, 941–947 (2009). [CrossRef]

24.

R. Selm, G. Krauss, A. Leitenstorfer, and A. Zumbusch, “Simultaneous second-harmonic generation, third-harmonic generation, and four-wave mixing microscopy with single sub-8 fs laser pulses,” Appl. Phys. Lett. 99, 181124 (2011). [CrossRef]

25.

P. Mahou, N. Olivier, G. Labroille, L. Duloquin, J.-M. Sintes, N. Peyriéras, R. Legouis, D. Débarre, and E. Beaurepaire, “Combined third-harmonic generation and four-wave mixing microscopy of tissues and embryos,” Biomed. Opt. Express 2, 2837–2849 (2011). [CrossRef]

26.

Y. Barad, H. Eisenberg, M. Horowitz, and Y. Silberberg, “Nonlinear scanning laser microscopy by third harmonic generation,” Appl. Phys. Lett. 70, 922–924 (1997). [CrossRef]

27.

J. A. Squier, M. Müller, G. J. Brakenhoff, and K. R. Wilson, “Third harmonic generation microscopy,” Opt. Express 3, 315–324 (1998). [CrossRef]

28.

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. Meth. 3, 47–53 (2006). [CrossRef]

29.

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]

30.

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]

31.

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]

32.

I. Rocha-Mendoza, D. R. Yankelevich, M. Wang, K. M. Reiser, C. W. Frank, and A. Knoesen, “Sum frequency vibrational spectroscopy: the molecular origins of the optical second-order nonlinearity of collagen,” Biophys. J. 93, 4433–4444 (2007). [CrossRef]

33.

M. G. Kuzyk, K. D. Singer, H. E. Zahn, and L. A. King, “Second-order nonlinear-optical tensor properties of poled films under stress,” J. Opt. Soc. Am. B 6, 742–752 (1989). [CrossRef]

34.

M. Gurp, “The use of rotation matrices in the mathematical description of molecular orientations in polymers,” Colloid Polym. Sci. 273, 607–625 (1995). [CrossRef]

35.

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]

36.

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). [CrossRef]

37.

F. P. Bolin, L. E. Preuss, R. C. Taylor, and R. J. Ference, “Refractive index of some mammalian tissues using a fiber optic cladding method,” Appl. Opt. 28, 2297–2303 (1989). [CrossRef]

OCIS Codes
(190.1900) Nonlinear optics : Diagnostic applications of nonlinear optics
(190.4380) Nonlinear optics : Nonlinear optics, four-wave mixing
(180.4315) Microscopy : Nonlinear microscopy

ToC Category:
Nonlinear Optics

History
Original Manuscript: March 6, 2012
Manuscript Accepted: April 12, 2012
Published: June 1, 2012

Virtual Issues
Vol. 7, Iss. 8 Virtual Journal for Biomedical Optics
June 27, 2012 Spotlight on Optics

Citation
Fabiana Munhoz, Hervé Rigneault, and Sophie Brasselet, "Polarization-resolved four-wave mixing microscopy for structural imaging in thick tissues," J. Opt. Soc. Am. B 29, 1541-1550 (2012)
http://www.opticsinfobase.org/josab/abstract.cfm?URI=josab-29-6-1541


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References

  1. M. Florsheimer, M. Bosch, C. Brillert, M. Wierschem, and H. Fuchs, “Second-harmonic imaging of surface order and symmetry,” Thin Solid Films 327–329, 241–246 (1998). [CrossRef]
  2. C. Anceau, S. Brasselet, and J. Zyss, “Local orientational distribution of molecular monolayers probed by nonlinear microscopy,” Chem. Phys. Lett. 411, 98–102 (2005). [CrossRef]
  3. 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). [CrossRef]
  4. I. Freund, M. Deutsch, and A. Sprecher, “Connective tissue polarity. Optical second-harmonic microscopy, crossed-beam summation, and small-angle scattering in rat-tail tendon,” Biophys. J. 50, 693–712 (1986). [CrossRef]
  5. 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). [CrossRef]
  6. P. Stoller, B.-M. Kim, A. M. Rubenchik, K. M. Reiser, and L. B. Da Silva, “Polarization-dependent optical second-harmonic imaging of rat-tail tendon,” J. Biomed. Opt. 7, 205–214 (2002). [CrossRef]
  7. G. Cox, E. Kable, A. Jones, I. Fraser, F. Manconi, and M. D. Gorrell, “3-Dimensional imaging of collagen using second harmonic generation,” J. Struct. Biol. 141, 53–62 (2003). [CrossRef]
  8. 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). [CrossRef]
  9. P. J. Campagnola, A. C. Millard, M. Terasaki, P. E. Hoppe, C. J. Malone, and W. A. Mohler, “3-Dimesional high-resolution second harmonic generation imaging of endogenous structural proteins in biological tissues,” Biophys. J. 82, 493–508 (2002). [CrossRef]
  10. M. Both, M. Vogel, O. Friedrich, F. von Wegner, T. Künsting, R. Fink, H. A. Rainer, and D. Uttenweiler, “Second harmonic imaging of intrinsic signals in muscle fibers in situ,” J. Biomed. Opt. 9, 882–892 (2004). [CrossRef]
  11. T. Boulesteix, E. Beaurepaire, M.-P. Sauviat, and M.-C. Schanne-Klein, “Second-harmonic microscopy of unstained living cardiacmyocytes: measurements of sarcomere length with 20 nm accuracy,” Opt. Lett. 29, 2031–2033 (2004). [CrossRef]
  12. A. Zoumi, X. Lu, G. S. Kassab, and B. J. Tromberg, “Imaging coronary artery microstructure using second-harmonic and two-photon fluorescence microscopy,” Biophys. J. 87, 2778–2786 (2004). [CrossRef]
  13. S. V. Plotnikov, A. C. Millard, P. J. Campagnola, and W. A. Mohler, “Characterization of the myosin-based source for second-harmonic generation from muscle sarcomeres,” Biophys. J. 90, 693–703 (2006). [CrossRef]
  14. 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). [CrossRef]
  15. 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). [CrossRef]
  16. F. Tiaho, G. Recher, and D. Rouède, “Estimation of helical angles of myosin and collagen by second harmonic generation imaging microscopy,” Opt. Express 15, 4054–4065 (2007). [CrossRef]
  17. 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). [CrossRef]
  18. 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]
  19. S. Brasselet, “Polarization resolved nonlinear microscopy: application to structural molecular and biological imaging,” Adv. Opt. Photon. 3, 205–271 (2011). [CrossRef]
  20. J. Zyss, “Octupolar organic systems in quadratic nonlinear optics: molecules and materials,” Nonlin. Opt. 1, 3–18 (1991).
  21. J. Zyss, “Molecular engineering implications of rotational invariance in quadratic nonlinear optics: from dipolar to octupolar molecules and materials,” J. Chem. Phys. 98, 6583–6599 (1993). [CrossRef]
  22. W. Min, S. Lu, M. Rueckel, G. R. Hotom, and X. S. Xie, “Near-degenerate four-wave-mixing microscopy,” Nano Lett. 9, 2423–2426 (2009). [CrossRef]
  23. D. Akimov, S. Chatzipapadopoulos, T. Meyer, N. Tarce, B. Dietzek, M. Schmitt, and J. Popp, “Different contrast information obtained from CARS and nonresonant FWM images,” J. Raman Spectrosc. 40, 941–947 (2009). [CrossRef]
  24. R. Selm, G. Krauss, A. Leitenstorfer, and A. Zumbusch, “Simultaneous second-harmonic generation, third-harmonic generation, and four-wave mixing microscopy with single sub-8 fs laser pulses,” Appl. Phys. Lett. 99, 181124 (2011). [CrossRef]
  25. P. Mahou, N. Olivier, G. Labroille, L. Duloquin, J.-M. Sintes, N. Peyriéras, R. Legouis, D. Débarre, and E. Beaurepaire, “Combined third-harmonic generation and four-wave mixing microscopy of tissues and embryos,” Biomed. Opt. Express 2, 2837–2849 (2011). [CrossRef]
  26. Y. Barad, H. Eisenberg, M. Horowitz, and Y. Silberberg, “Nonlinear scanning laser microscopy by third harmonic generation,” Appl. Phys. Lett. 70, 922–924 (1997). [CrossRef]
  27. J. A. Squier, M. Müller, G. J. Brakenhoff, and K. R. Wilson, “Third harmonic generation microscopy,” Opt. Express 3, 315–324 (1998). [CrossRef]
  28. 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. Meth. 3, 47–53 (2006). [CrossRef]
  29. 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]
  30. 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]
  31. 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]
  32. I. Rocha-Mendoza, D. R. Yankelevich, M. Wang, K. M. Reiser, C. W. Frank, and A. Knoesen, “Sum frequency vibrational spectroscopy: the molecular origins of the optical second-order nonlinearity of collagen,” Biophys. J. 93, 4433–4444 (2007). [CrossRef]
  33. M. G. Kuzyk, K. D. Singer, H. E. Zahn, and L. A. King, “Second-order nonlinear-optical tensor properties of poled films under stress,” J. Opt. Soc. Am. B 6, 742–752 (1989). [CrossRef]
  34. M. Gurp, “The use of rotation matrices in the mathematical description of molecular orientations in polymers,” Colloid Polym. Sci. 273, 607–625 (1995). [CrossRef]
  35. 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]
  36. 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). [CrossRef]
  37. F. P. Bolin, L. E. Preuss, R. C. Taylor, and R. J. Ference, “Refractive index of some mammalian tissues using a fiber optic cladding method,” Appl. Opt. 28, 2297–2303 (1989). [CrossRef]

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