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Virtual Journal for Biomedical Optics

Virtual Journal for Biomedical Optics

| EXPLORING THE INTERFACE OF LIGHT AND BIOMEDICINE

  • Editor: Gregory W. Faris
  • Vol. 5, Iss. 11 — Aug. 25, 2010
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Influence of birefringence on polarization resolved nonlinear microscopy and collagen SHG structural imaging

D. Aït-Belkacem, A. Gasecka, F. Munhoz, S. Brustlein, and S. Brasselet  »View Author Affiliations


Optics Express, Vol. 18, Issue 14, pp. 14859-14870 (2010)
http://dx.doi.org/10.1364/OE.18.014859


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Abstract

We analyze the influence of the anisotropy of molecular and biological samples on polarization resolved nonlinear microscopy imaging. We show in particular the detrimental influence of birefringence on Second Harmonic Generation (SHG) and Two-Photon Excited Fluorescence (TPEF) polarization resolved microscopy imaging, which, if not accounted for, can lead to an erroneous determination of the sample properties and thus to a misinterpretation of the read-out information. We propose a method to measure this birefringence and account for this effect in nonlinear polarization resolved experiments.

© 2010 Optical Society of America

1. Introduction

In addition to their unique imaging capabilities, these contrasts are dependent on the incident light polarization state, providing an interesting way to probe molecular orientation and disorder. Structural information, from membrane architecture and proteins aggregates to biopolymers and tissue assemblies, is a key parameter in a large variety of biological phenomena. Polarization probe imaging has been exploited a long time ago in fluorescence [11

11. D. Axelrod, “Carbocyanine dye orientation in red cell membrane studied by microscopic fluorescence polarization,” Biophys. J. 26, 557–573 (1979). [CrossRef] [PubMed]

, 12

12. 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). [CrossRef] [PubMed]

] and SHG [13

13. 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). [CrossRef]

, 14

14. 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]

] in ordered molecular samples. Its extension to SHG in tissues leads to a large amount of current studies [15–22

15. 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 (2000). [CrossRef]

].

Recent works in polarization resolved nonlinear microscopy (or nonlinear polarimetry) have pointed out the need to rely on a variable state of incident polarization to retrieve relevant information on the sample structure. In fluorescence, measuring the signal from a continuous variation of the incident linear polarization state circumvents the limitations of the traditional fluorescence anisotropy imaging, which explores a limited number of polarization states and therefore requires a priori knowledge information on the studied system [11

11. D. Axelrod, “Carbocyanine dye orientation in red cell membrane studied by microscopic fluorescence polarization,” Biophys. J. 26, 557–573 (1979). [CrossRef] [PubMed]

, 12

12. 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). [CrossRef] [PubMed]

]. TPEF polarimetry has shown in particular its ability to reveal non-ambiguous information on molecular angular distributions in cell membranes and crystals [23

23. A. Gasecka, T.-J. Han, C. Favard, B. R. Cho, and S. Brasselet, “Quantitative imaging of molecular order in lipid membranes using two-photon fuorescence polarimetry,” Biophys. J. 97, 2854–2862 (2009). [CrossRef] [PubMed]

, 24

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

], with a typical 300nm lateral resolution.

In SHG, polarimetry is particularly relevant since the complex nature of the probed χ (2) tensor requires multiple polarization coupling. SHG polarimetry imaging has allowed retrieving molecular orientation and order information in molecular materials [14

14. 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]

,25

25. 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). [CrossRef]

] and crystals down to the nanometric size [26–28

26. 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, 207401–204405 (2004). [CrossRef] [PubMed]

], as well as collagen and muscle structural quantitative information in tissues [15–22

15. 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 (2000). [CrossRef]

].

Although polarimetry microscopy imaging is a powerful technique, relevant information can be retrieved only if particular care is brought on the possible polarization distortions occurring in the instrument. The degree of ellipticity appearing from reflections on mirrors and dichroic mirrors can lead to a deformation of the retrieved information, in particularly when using the epi-geometry [29

29. 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] [PubMed]

]. In addition, the excitation and detection polarization are scrambled due to the use of high numerical aperture objectives [11

11. D. Axelrod, “Carbocyanine dye orientation in red cell membrane studied by microscopic fluorescence polarization,” Biophys. J. 26, 557–573 (1979). [CrossRef] [PubMed]

]. Finally an extra coupling along the propagation direction appears to be non-negligible when using high aperture focusing [30

30. B. Richards and E. Wolf, “Electromagnetic Diffraction in Optical Systems. II. Structure of the Image Field in an Aplanatic System,” Proc. R. Soc. London Ser. A. 253, 358–379 (1959). [CrossRef]

]. These different polarization distortion factors have been thoroughly addressed and can be readily accounted for in microscopy polarization analysis [25

25. 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). [CrossRef]

, 31–33

31. 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). [CrossRef] [PubMed]

].

2. Effect of birefringence on polarimetric microscopy imaging

For both the TPEF and SHG contrasts, varying the incident polarization angle α allows retrieving information on the unknown molecular angular distribution denoted f (θ, ϕ, ψ), with (θ, ϕ, ψ) the Euler set of angles defining the molecular frame orientation in the sample. In what follows, we discard the ψ angle since cone-shape distributions are considered where only the (θ, ϕ) angle components are relevant by cylindrical symmetry, as often applied to biological media such as in cell membranes [11

11. D. Axelrod, “Carbocyanine dye orientation in red cell membrane studied by microscopic fluorescence polarization,” Biophys. J. 26, 557–573 (1979). [CrossRef] [PubMed]

] and fibrillar structures [2

2. 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] [PubMed]

]. Nevertheless this model can be extended to more general distribution functions.

The TPEF polarimetric signal measured from a molecular ensemble within an orientational distribution f (θ, ϕ) (with (θ, ϕ) the molecular dipole orientation angle), analyzed along a given polarization state I =(X,Y), can be expressed as the incoherent sum over individual fluorescent intensities [23

23. A. Gasecka, T.-J. Han, C. Favard, B. R. Cho, and S. Brasselet, “Quantitative imaging of molecular order in lipid membranes using two-photon fuorescence polarimetry,” Biophys. J. 97, 2854–2862 (2009). [CrossRef] [PubMed]

, 25

25. 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). [CrossRef]

]:

II=(X,Y)TPEF(α)=02π0πμabs(θ,ϕ)·E(α)4μem(θ,ϕ)·I2f(θ,ϕ)sinθdθdϕ
(1)

where μ⃗abs is the molecular absorption dipole, and μ⃗em is its emission dipole, both fixed relative to the molecular microscopic frame. (α) the incident linear polarization state vector oriented with an angle α relative to X. This expression, written in the planar wave approximation, assumes a small collection angle of the fluorescence radiation. In the case of a high aperture collection angle, the |μ⃗em(θ, ϕ) · I⃗|2 emission contribution in Eq. (1) has to be replaced by the JI(θ, ϕ) corrected function, previously introduced to account for the broad range of collection angles within a high numerical aperture (NA) objective [11

11. D. Axelrod, “Carbocyanine dye orientation in red cell membrane studied by microscopic fluorescence polarization,” Biophys. J. 26, 557–573 (1979). [CrossRef] [PubMed]

]. In the present work, the use of a low-NA objective (NA=0.6) allows considering the planar wave approximation to be relevant [29

29. 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] [PubMed]

, 33

33. 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]

].

In the case of SHG, the signal results from the coherent radiation of nonlinear molecular induced dipoles, induced in the focal spot of the objective. This leads to the following SHG intensity measured along the I = (X, Y) polarization analysis direction:

II=(X,Y)SHG(α)=ΣJ,KχIJK(2)EJω(α)EKω(α)2
(2)

Note that due to the small size of the focal volume relative to the nonlinear coherent lengths, the phase matching relations occurring usually at a large propagation distance are not accounted for here.

In both expressions [Eqs. (1) and (2)], the optical field E⃗(α) is expressed at the dipoles positions (either fluorescent or nonlinear induced) in the focal spot of the microscope objective, at a given depth d from the sample surface [Fig. 1(c)]. In anisotropic samples where this incident field undergoes a birefringence retardation, its expression has to be re-written in order to account for the consequent polarization distortions. In the present work, we consider that the object projection in the sample plane is uni-axial, which is consistent with a cylindricalsymmetry distribution lying in the (X,Y) plane as mentioned above. This is also 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. Denoting Θb the angle between the X macroscopic axis and the fast optical axis Xb of this object [Fig. 1(b)], and Φb the phase shift between its fast and slow optical axes, then the new expression of the optical field polarization state at the focal depth distance d is, in the planar wave approximation:

[EX(Z=d)EY(Z=d)]=[exp(iΦb(d))·cosΘbsinΘbexp(iΦb(d))·sinΘbcosΘb]·[cosΘbsinΘbsinΘbcosΘb]·[EX0(α)EY0(α)]
(3)

with E 0 X,Y the optical field polarization components in the macroscopic (X,Y) frame at the sample surface (Z = 0) [Fig. 1(c)], and Φb(d)=2πλ.Δn.d , with λ the incident wavelength and Δn the refractive index difference between the fast and slow axis of the object in the sample plane.

In addition to its effect on the incident excitation field, the birefringence also affects the detected signal, which propagates back in the sample in the case of a backward detection. A similar approach as above 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 d and at the exit of the sample (Z = 0) follows the same equation as in Eq. (3), introducing the detection wavelength in the expression of Φb.

Fig. 1. (a) Experimental set-up. D: dichroic mirror, APD: avalanche photodiodes, P: polarizer, M: mirror, S : sample, FLP : long pass filter, FBP : band pass filter. (b) Definition of the input polarization angle α and the birefringence slow axis direction θb relative to X. (c) Experimental configuration: an incident linear polarized field at the ω frequency is focused at a distance d from the sample surface in the optical axis Z direction. The transmitted fields at ω at both the distance d and the sample output distance L are elliptical if the sample is birefringent. The emitted field (fluorescence or 2ω for SHG) is measured in the epi direction.

3. In situ characterization of the sample local birefringence

Equation (3) shows that for a varying angle α of the incident polarization E⃗(α), the sample birefringence parameters can be deduced from a polarimetric measurement performed in the forward direction for the ω frequency of the excitation field [Fig. 1(a)]. In such geometry, the field propagates through the whole thickness of the sample L and therefore Φb(L) is measured. Theoretical polarimetric responses of the incident intensity IωX (α) are depicted in Fig. 2(a) in a polar representation, for a sample tilt angle of the optical axis of Θb = 30°, and for different birefringence phase shifts. If no birefringence occurs in the sample, the polarization response is a cos2 α two-lobes pattern with a maximum intensity along X (α = 0°) and a vanishing intensity along Y (α = 90°). In the presence of birefringence, this pattern tends to open (no extinction occurs anymore) and rotates progressively when Φb increases. Similar responses are expected for IωY, rotated by 90° with respect to IωX.

As seen in Fig. 2(b), experimental IωX,Y polarimetric measurements performed at a given position on a collagen type I fiber of roughly 70–100µm thickness exhibit such behavior (the sample preparation details can be found in the SHG section below). Figure 2(c) shows a map of the mean square error to the theoretical model [Eq. (3)] as a function of the deduced parameters. The minima of the fitting error can be found for Θb with a π/2 periodicity, and Φb(L) with a π periodicity, the periodicity being consistent with the fact that this technique does not discriminate the fast or slow axes of the system (this indetermination does not affect the results for the detected signals, as seen below). Note that accounting for the ellipticity and dichroism introduced by the dichroic mirror as mentioned in [29

29. 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] [PubMed]

] leads to a slight modification of the fit solutions [Fig. 2(d)] which has to be ultimately introduced in the models for TPEF or SHG contrasts as detailed below. Figure 2(d) shows in particular that introducing the instrumental distortions in the data analysis is crucial before any sample analysis. The birefringence parameters Φb(L) and Θb determined from the fit of [Fig. 2(b)] shows that (i)Θb lies roughly along the observed collagen fiber orientation (here 45°) and (ii) the measured birefringence phase shift Φb(L) = 97° can be used to deduce the approximate sample thickness L at the measured sample location. Assuming Φb≈1.35°/µm (Δn≈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] [PubMed]

, 38

38. D. T. Poh, “Examination of refractive index of human epidermis in-vitro and in-vivo.,” Proc. Inter. Conf. Lasers. 96, 118–125 (1996).

]) between the long axis of collagen fibers and their orthogonal direction, this measurement leads to L≈72µm, which is reasonable in the studied sample.

Fig. 2. (a) Theoretical linear polarimetric responses IωX through a birefringent sample at different values of Φb, for an optical axis tilt of Θb = 30°. (b) Experimental linear polarimetric responses IωX,Y (dots) through a collagen type I fiber from a rat-tail tendon, oriented at 45° relative to X. The fit (black lines) uses the model detailed in the text. (c,d) Maps of the mean square error between fitted and experimental intensities: (c) with no dichroic mirror polarization distortion, (d) with polarization distortions (dichroism factor γ = 0.043, ellipticity factor δ = 55°, with the notations of [29]). Fit solutions: Θb = 44° and Φb = 97°.

In the following sections, both TPEF and SHG contrasts are studied with known instrumental factors, Θb and Φb being introduced as fitting parameters in anisotropic molecular assemblies. Such systems can be viewed as an ensemble of excitation/emission dipoles lying within a cylindrical symmetry angular distribution f(θ, ϕ). This distribution is most generally modeled by a cone shape of given aperture and mean direction angles. This model has been applied to a large number of situations in polarization resolved fluorescence such as in labeled lipid membranes [11

11. D. Axelrod, “Carbocyanine dye orientation in red cell membrane studied by microscopic fluorescence polarization,” Biophys. J. 26, 557–573 (1979). [CrossRef] [PubMed]

], labeled Actin fibers [12

12. 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). [CrossRef] [PubMed]

] or disordered molecular crystals [28

28. S. Brasselet and J. Zyss, “Nonlinear polarimetry of molecular crystals down to the nanoscale,” C. R. Physique 8, 165–179 (2007). [CrossRef]

, 33

33. 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]

] as well as for SHG in collagen where the traditionally used C 6 crystal point group model leads to a nonlinear tensor equivalent to that of a cone [2

2. 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] [PubMed]

, 20–22

20. 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]

]. In such uni-axial samples the molecular order imposes an anisotropy along the main axis that can lead to non-negligible birefringence.

4. TPEF polarization resolved microscopy in crystalline samples

In order to visualize the effect of birefringence on TPEF, a model is applied to a cone lying in the (X,Y) plane, which main orientation φ 0 relative to X also corresponds to the optical axis of the uni-axial object. Figure 3 shows the effect of birefringence on the IX and IY TPEF polarimetric response in a system of large cone aperture. An increasing birefringence leads to a deformation of the polarimetric polar plots, in particular when the optical axis is away from the experimental projection axes X or Y. This would clearly lead to a misinterpretation of the polarimetric data, even for slight birefringence phase shifts. In particular for large values of Φb, the polarimetric data resemble those of a much smaller cone aperture than what is is actually.

Fig. 3. Theoretical TPEF polarimetric responses ITPEFX (red) and ITPEFY (green) for a molecular distribution within a large cone aperture (half angle 50°), for different orientations φ 0 of the cone in the (X,Y) frame (corresponding here also to the optical axis orientation Θb). (a) φ 0 = Θb = 0° and (b) φ 0 = Θb = 45°, with different values of the birefringence phase shift Φb.

Fig. 4. Experimental birefringence and TPEF polarimetric measurements in a PHTP-DANS 1D crystal. (a) Laser polarimetric response IωY of the non-birefringent glass substrate (no crystal). (b,c) Laser polarimetric response through the crystal oriented along the (b) X axis and (c) tilted at about 135° relative to X (as sketched). (d) Fit of the laser polarimetric response (solutions: Θb = 138°, Φb = 116°). (e) Fit of the TPEF ITPEFX polarimetric response including birefringence, at a 5 µm penetration depth (solutions: φ 0 = 164°, Θb = 138°, Φb = 100°). (f) Fit of the TPEF polarimetric response with no birefringence included. All the fits include the dichroism and ellipticity parameters of the dichroic mirror (dichroism γ = 0.009, ellipticity factor δ = 13° with the mirror used for this experiment [29]).

5. SHG polarization resolved microscopy in collagen

In this part the effect of birefringence is explored on the SHG polarimetric data on collagen type I fibers from rat-tail tendons, attached at both ends on a coverslip and immersed in 0.15M NaCl. To prepare the fibers, Adult Sprague Dawley rats were euthanased 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.

SHG imaging has been widely studied on similar systems. Analysis of SHG polarization resolved experiments is traditionally done assuming the collagen fibers to be of C6 symmetry, equivalent to considering a collection of nonlinear individual dipoles within a cone distribution of typical half apertures angles ranging from 50° to 65° [15

15. 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 (2000). [CrossRef]

, 16

16. 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] [PubMed]

, 18

18. 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]

, 21

21. 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] [PubMed]

, 22

22. 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]

]. Calculated SHG polarimetric responses are depicted in Fig. 5(a) for different tilt angle φ 0 of theoretical collagen fibers. The effect of a theoretical birefringence is seen to be detrimental to the polarimetric responses, introducing extra lobes in the polar plot shape comparing to the absence of birefringence. Again in this case, analyzing SHG polarimetry not accounting for birefringence leads to erroneous tensorial analysis of the SHG response.

Experimental analyzes performed in collagen type I fibers of thickness ranging from 70µm to 100µm confirm the strong deformation induced on the polarimetric responses, even for a few µm penetration depths in the sample [Fig. 6(a)–6(c)]. In a fiber close to horizontal along the X direction, two lobes appear clearly in the ISHGX polarimetric response, that cannot be explained in the absence of birefringence. The fit of the IωX response [Fig. 6(d)] confirms that the birefringence originates from a uni-axial system oriented close to the X axis (Θb = 18°), with a phase shift Φb = 119°. This birefringence is consistent with Δn ≈ 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] [PubMed]

, 38

38. D. T. Poh, “Examination of refractive index of human epidermis in-vitro and in-vivo.,” Proc. Inter. Conf. Lasers. 96, 118–125 (1996).

] in a ≈ 88µm thick fiber. The fitting of the SHG polarimetric responses at different penetration depths are performed using Θb and Φb as fitting parameters for both IX and IY. Excellent fitting quality is obtained for a cone distribution of 52° half aperture angle, oriented at φ 0 = 10°, which corresponds to the fiber direction observed in the SHG image (Fig. 6). The SHG fitting results show that the orientation of the optical axis Θb = 2–9°, measured locally, lies close to φ 0. The difference with the averaged Θb = 18° measured over the whole sample thickness is most probably due to the sample heterogeneity in the Z propagation direction. At last, the measured birefringence phase shifts increase when further penetrating in the sample, as expected. Supposing a birefringence Δn ≈ 0.003, the deduced theoretical penetration depth (see values in brackets given in the Fig. 6 legend) are consistent with the expected depths of focus d, including error margins due to the focal volume averaging over several micrometers in the Z direction.

Fig. 5. Theoretical SHG polarimetric responses (ISHGX (red) and ISHGY (green)) for a collagen fiber which nonlinear tensor corresponds to a cone half angle aperture distribution of the nonlinear molecular dipoles of 50°. The orientation of the fiber φ 0 also corresponds to the optical axis orientation Θb. (a) φ 0 = Θb = 0° and (b) φ 0 = Θb = 45°, with different values of the birefringence phase shift Φb.

In the data shown in Fig. 6, the direction of the optical axis Θb, measured locally, is still relatively close to its averaged value over the whole sample thickness. This is not always the case as shown in another area of the same sample (Fig. 7).
Fig. 7. Experimental SHG polarimetric responses for collagen type I fibers oriented at about 45° relative to the macroscopic X axis. The experimental parameters are the same as in Fig. 6. The corresponding birefringence parameters obtained from the fits are 52° and φ 0 = −15° for respectively the cone distribution orientation and aperture, and in addition: (a) Θb = −23°, Φb = 4°[4µm]; (b) Θb = −16°, Φb = 13°[11µm]; (c) Θb = −12°, Φb = 75°[64µm]. (d) Birefringence measurement and fit on the IωX component of the incident field, leading to birefringence parameters of Θb = 36°, Φb = 115°[85µm], when the sample is propagated through. The found birefringence axes are represented in the inset scheme.
In this situation the SHG polarimetric fit gives Θb = (−23)°–(−12)° in the explored depths of focus [with consistent birefringence phase shifts, see Fig. 7(a)–7(c)], whereas Θb = 36°≡(−54)° on average. This is a clear signature of a progressive rotation of the collagen fiber / optical axes orientation when penetrating in the sample depth. Note that still accounting for the variations of φ 0 seen on the SHG images does not notably change these results.

These results show finally, that particular care has to be brought on the local properties of the sample, which can be very distinct from averaged birefringence since the optical axis of the system can typically rotate progressively through the whole sample thickness. At last, a complete fitting procedure on both the cone shape and birefringence parameters at each measurement point is therefore necessary to lead to reliable data on the nonlinear tensor of the structure. These experimental SHG polarimetric data, performed at distances still relatively close to the sample surface, indeed clearly evidence the presence of birefringence and demonstrate the importance of accounting for such effect to retrieve sample structural information.

Fig. 6. Experimental SHG polarimetric responses for collagen type I fibers oriented close to the macroscopic X axis. The SHG images (image scale is in counts/s) represent a 30µm×30µm area of the sample scanned at different depth d: (a) 2µm, (b) 20µm, (c) 60µm. Both SHG experimental intensities IX (red dots) and IY (green dots) are depicted (after normalization for more visibility), together with the corresponding fit according to the model detailed in the text. The parameters obtained from the fits are 52° and φ 0 = 10° for respectively the cone distribution orientation and aperture, and: (a) Θb = 7°, Φb = 19°[14µm];(b) Θb = 9°, Φb = 29°[22µm];(c) Θb = 2°, Φb = 88°[65µm]. (d) Birefringence measurement and fit on the IωX component of the incident field, leading to birefringence parameters of Θb = 18°, Φb = 119°[88µm], when the sample is propagated through. The found birefringence axes are represented in the inset scheme.

6. Conclusion

We have demonstrated that birefringence from molecular and biological samples is a detrimental factor in polarization resolved information in microscopy imaging applied to anisotropic samples. We have shown that this birefringence can be however fully characterized and accounted for. In a context where SHG microscopy constantly evolves towards the exploration of complex tissues where birefringence is present at increasing depths, this analysis is more and more crucial to distinguish the local nature (symmetry, orientational disorder) of molecular and biomolecular assemblies. This study can be extended to other contrasts such as Third Harmonic Generation (THG) or Coherent Anti-Stokes Raman Scattering (CARS) for which polarization-resolved microscopy has also been implemented.

Acknowledgements

We thank Prof. C. Peter Winlove (Biomedical Physics Group, University of Exeter) for providing the collagen Type I sample. We thank Patrick Ferrand and Hervé Rigneault for help in the experimental setup development and useful discussions. The authors acknowledge the European Commission through the Human Potential Program (Marie-Curie Research Training Network NANOMATCH, contract MRTN-CT-2006-035884). This work is also supported by the French Agence Nationale de la Recherche under the program ANR JC 2007, project NLO-Shaping JC07-195504.

References and links

1.

J. Gannaway and C. J. R. Sheppard, “Second harmonic imaging in the scanning optical microscope,” Opt. Quantum Electron. 10, 435–439 (1978). [CrossRef]

2.

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] [PubMed]

3.

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). [CrossRef]

4.

L. Moreaux, O. Sandrea, and J. Mertz, “Membrane Imaging by Second Harmonic Generation Microscopy,” J. Opt. Soc. Am. B 17, 1685–1689 (2000). [CrossRef]

5.

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. 81, 493–508 (2002). [CrossRef]

6.

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 (2003). [CrossRef] [PubMed]

7.

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. Opt. 13, 054041 (2008). [CrossRef] [PubMed]

8.

O. Nadiarnykh, R. LaComb, M. Brewer, and P. J. Campagnola, “Second Harmonic Generation Imaging Microscopy of Ovarian Cancer,” Biophys. J. 94, 4504–4514 (2008). [PubMed]

9.

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. Ophthalmol. Vis. Sci. 47, 1216–1224 (2006). [CrossRef] [PubMed]

10.

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 non-contractile inclusions and sarcomeric anomalies in skeletal muscle by second harmonic generation combined with two-photon excited fluorescence,” J. Struct. Biol. 162, 500–508 (2008). [CrossRef] [PubMed]

11.

D. Axelrod, “Carbocyanine dye orientation in red cell membrane studied by microscopic fluorescence polarization,” Biophys. J. 26, 557–573 (1979). [CrossRef] [PubMed]

12.

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). [CrossRef] [PubMed]

13.

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). [CrossRef]

14.

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]

15.

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 (2000). [CrossRef]

16.

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] [PubMed]

17.

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]

18.

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]

19.

S. Yen, W. L. Chen, H. Y. Wu, F. C. Li, W. Lo, S. J. Lin, S. H. Jee, Y. F. Chen, P. T. C. So, and C. Y. Dong, “Collagen Thermal denaturation study with high resolution second harmonic generation imaging and polarization optical microscopy,” Biophys. J. 91, 2620–2625 (2006). [CrossRef]

20.

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]

21.

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] [PubMed]

22.

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]

23.

A. Gasecka, T.-J. Han, C. Favard, B. R. Cho, and S. Brasselet, “Quantitative imaging of molecular order in lipid membranes using two-photon fuorescence polarimetry,” Biophys. J. 97, 2854–2862 (2009). [CrossRef] [PubMed]

24.

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

25.

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). [CrossRef]

26.

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, 207401–204405 (2004). [CrossRef] [PubMed]

27.

K. Komorowska, S. Brasselet, J. Zyss, L. Pourlsen, M. Jazdzyk, H. J. Egelhaaf, J. Gierschner, and M. Hanack, “Nanometric scale investigation of the nonlinear efficiency of perhydrotriphynylene inclusion compounds,” Chem. Phys. 318, 12–20 (2005). [CrossRef]

28.

S. Brasselet and J. Zyss, “Nonlinear polarimetry of molecular crystals down to the nanoscale,” C. R. Physique 8, 165–179 (2007). [CrossRef]

29.

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] [PubMed]

30.

B. Richards and E. Wolf, “Electromagnetic Diffraction in Optical Systems. II. Structure of the Image Field in an Aplanatic System,” Proc. R. Soc. London Ser. A. 253, 358–379 (1959). [CrossRef]

31.

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). [CrossRef] [PubMed]

32.

N. Sandeau, L. L. 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). [CrossRef] [PubMed]

33.

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]

34.

D. Lara and C. Dainty, “Axially resolved complete Mueller matrix confocal microscopy,” Appl. Opt. 45, 1917–1930 (2006). [CrossRef] [PubMed]

35.

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 (2006). [CrossRef]

36.

I. Ledoux, C. Lepers, A. Prigaud, J. Badan, and J. Zyss, “Linear and nonlinear optical properties of N-4-nitrophenyl L-prolinol single crystals,” Opt. Commun. 80, 149–154 (1990). [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] [PubMed]

38.

D. T. Poh, “Examination of refractive index of human epidermis in-vitro and in-vivo.,” Proc. Inter. Conf. Lasers. 96, 118–125 (1996).

OCIS Codes
(190.1900) Nonlinear optics : Diagnostic applications of nonlinear optics
(180.4315) Microscopy : Nonlinear microscopy

ToC Category:
Microscopy

History
Original Manuscript: May 3, 2010
Manuscript Accepted: May 26, 2010
Published: June 28, 2010

Virtual Issues
Vol. 5, Iss. 11 Virtual Journal for Biomedical Optics

Citation
Sophie Brasselet, Dora Aït-Belkacem, Alicja Gasecka, Fabiana Munhoz, Sophie Brustlein, and S. Brasselet, "Influence of birefringence on polarization resolved nonlinear microscopy and collagen SHG structural imaging," Opt. Express 18, 14859-14870 (2010)
http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-18-14-14859


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References

  1. J. Gannaway and C. J. R. Sheppard, “Second harmonic imaging in the scanning optical microscope,” Opt. Quantum Electron. 10, 435–439 (1978). [CrossRef]
  2. 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] [PubMed]
  3. M. Fl¨orsheimer, 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). [CrossRef]
  4. L. Moreaux, O. Sandrea, and J. Mertz, “Membrane Imaging by Second Harmonic Generation Microscopy,” J. Opt. Soc. Am. B 17, 1685–1689 (2000). [CrossRef]
  5. 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. 81, 493–508 (2002). [CrossRef]
  6. 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 (2003). [CrossRef] [PubMed]
  7. 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. Opt. 13, 054041 (2008). [CrossRef] [PubMed]
  8. O. Nadiarnykh, R. LaComb, M. Brewer, and P. J. Campagnola, “Second Harmonic Generation Imaging Microscopy of Ovarian Cancer,” Biophys. J. 94, 4504–4514 (2008). [PubMed]
  9. 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. Ophthalmol. Vis. Sci. 47, 1216–1224 (2006). [CrossRef] [PubMed]
  10. 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 non-contractile inclusions and sarcomeric anomalies in skeletal muscle by second harmonic generation combined with two-photon excited fluorescence,” J. Struct. Biol. 162, 500–508 (2008). [CrossRef] [PubMed]
  11. D. Axelrod, “Carbocyanine dye orientation in red cell membrane studied by microscopic fluorescence polarization,” Biophys. J. 26, 557–573 (1979). [CrossRef] [PubMed]
  12. 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). [CrossRef] [PubMed]
  13. 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). [CrossRef]
  14. 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]
  15. 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 (2000). [CrossRef]
  16. 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] [PubMed]
  17. 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]
  18. 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]
  19. S. Yen, W. L. Chen, H. Y. Wu, F. C. Li, W. Lo, S. J. Lin, S. H. Jee, Y. F. Chen, P. T. C. So, and C. Y. Dong, “Collagen Thermal denaturation study with high resolution second harmonic generation imaging and polarization optical microscopy,” Biophys. J. 91, 2620–2625 (2006). [CrossRef]
  20. F. Tiaho, G. Recher, and D. Rouéde, “Estimation of helical angles of myosin and collagen by second harmonic generation imaging microscopy,” Opt. Express 15, 4054–4065 (2007). [CrossRef]
  21. 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] [PubMed]
  22. 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]
  23. A. Gasecka, T.-J. Han, C. Favard, B. R. Cho, and S. Brasselet, “Quantitative imaging of molecular order in lipid membranes using two-photon fuorescence polarimetry,” Biophys. J. 97, 2854–2862 (2009). [CrossRef] [PubMed]
  24. A. Gasecka, L.-Q. Dieu, D. Bruehviler, and S. Brasselet, “Probing molecular order in zeolite L inclusion compounds using two-photon fluorescence polarimetric microscoy,” J. Phys. Chem. B 114, 4192–4198 (2010). [CrossRef] [PubMed]
  25. 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). [CrossRef]
  26. 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, 207401–204405 (2004). [CrossRef] [PubMed]
  27. K. Komorowska, S. Brasselet, J. Zyss, L. Pourlsen, M. Jazdzyk, H. J. Egelhaaf, J. Gierschner, and M. Hanack, “Nanometric scale investigation of the nonlinear efficiency of perhydrotriphynylene inclusion compounds,” Chem. Phys. 318, 12–20 (2005). [CrossRef]
  28. S. Brasselet, and J. Zyss, “Nonlinear polarimetry of molecular crystals down to the nanoscale,” C. R. Physique 8, 165–179 (2007). [CrossRef]
  29. P. Sch¨on, F. Munhoz, A. Gasecka, S. Brustlein, and S. Brasselet, “Polarization distortion effects in polarimetric two-photon microscopy,” Opt. Express 16, 20891–20901 (2008). [CrossRef] [PubMed]
  30. B. Richards, and E. Wolf, “Electromagnetic Diffraction in Optical Systems. II. Structure of the Image Field in an Aplanatic System, ” Proc. R. Soc. London Ser. A. 253, 358–379 (1959). [CrossRef]
  31. 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). [CrossRef] [PubMed]
  32. N. Sandeau, L. L. 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). [CrossRef] [PubMed]
  33. 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]
  34. D. Lara and C. Dainty,“Axially resolved complete Mueller matrix confocal microscopy,” Appl. Opt. 45, 1917–1930 (2006). [CrossRef] [PubMed]
  35. 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 (2006). [CrossRef]
  36. I. Ledoux, C. Lepers, A. Prigaud, J. Badan, and J. Zyss, “Linear and nonlinear optical properties of N-4-nitrophenyl L-prolinol single crystals,” Opt. Commun. 80, 149–154 (1990). [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] [PubMed]
  38. D. T. Poh, “Examination of refractive index of human epidermis in-vitro and in-vivo.,” Proc. Inter. Conf. Lasers. 96, 118–125 (1996).

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