Generic model of the molecular orientational distribution probed by polarization-resolved second-harmonic generation

In this work we investigate a generic method able to extract information on molecular organization in biological samples from polarized second harmonic generation (SHG) microscopy, without the need to infer an a priori model for the molecular orientational distribution. The mean orientation of this distribution, as well as its first and third orders of symmetry, are estimated by monitoring SHG intensity signals under a varying incident polarization. We introduce, in particular, a reduction of the problem to a two-dimensional approach appropriate to the microscopy geometry. This method allows us to retrieve determining information which is not available in the traditional model-oriented methods, as illustrated in molecular-order imaging in collagen fibrils. The precision of the parameters estimation is evaluated by a Monte Carlo analysis, based on the Poisson noise statistics of the measured signal.

Figure 1

(a) Geometry of the SHG polarized microscopy. $(X,Y)$ is the sample plane and $Z$ is the direction of propagation of the fundamental beam. $\alpha$ is the angle of the incident polarization $\mathbf{E}$ relative to $X$. In one-dimensional symmetry molecules, $(\theta ,\varphi )$ defines the orientation of the molecular axis $\mathbf{m}$ in the $(X,Y,Z)$ frame. (b) Orientation of the 2D distribution (${\varphi }_0$) represented as a gray shape. $\varphi$ is the orientation of the molecular axis $\mathbf{m}$ relative to $X$ in the sample plane. (c) Example of a three-dimensional molecular distribution $f(\theta ,\varphi )$ (left) and its modified shape when read by a SHG polarized contrast $f(\theta ,\varphi ){\sin }^4\theta$ (right).

Figure 2

(a) SHG Intensity image of isolated collagen type I fibrils coated on the sample plane surface. The intensity scale is the sum of SHG signals over the 32 incident polarizations (threshold: 60 photons/pixel). (b)–(d) Images of the parameters obtained from a Fourier decomposition of the intensity polarization dependence: (b) ${\varphi }_0$ (degrees), (c) $p$, (d) $q$, (e) histogram of $p$ on the whole image, (f) histogram of $q$ on the whole image. For the histograms, the threshold of the analyzed pixels is 150 photons/pixel. (g) Polar representation of the truncated distribution function $\tilde{P}\left(\varphi \right)$ deduced from the averaged $p$ and $q$ values obtained over the whole image.

Figure 3

(a) SHG image of a selected region of Fig. 2. (b) Corresponding image of $p$. (c) Polar representation of the averaged distribution function $\tilde{P}\left(\varphi \right)$ obtained from the retrieved (${\varphi }_0$, $p$, $q$) parameters on three regions of interest represented as white rectangles in the SHG image. These functions are obtained using the average of the retrieved parameter within the region of interest (only active pixels are used): 1 ($p=-0.4$, $q=0.04$, ${\varphi }_0=8^{\circ }$), 2 ($p=-0.98$, $q=0.04$, ${\varphi }_0={42}^{\circ }$), and 3 ($p=-0.38$, $q=0.01$, ${\varphi }_0={80}^{\circ }$). The error of estimation on the parameters produces an error bar on the angular distribution which is smaller than the thickness of the line used to draw the distribution function; therefore, it is not represented here.

Figure 4

Variances of the $p$, $q$, and ${\varphi }_0$ parameters as functions of the total number of photon (summed over the 32 incident polarizations). Circles $p$, squares $q$, triangles ${\varphi }_0$. Dashed lines show Monte Carlo simulations using the initial parameters $p=-0.4$, $q=0$, ${\varphi }_0={45}^{\circ }$. Continuous lines show experimental variances measured over seven experiments performed for a same region of interest on a fibril of orientation ${\varphi }_0\sim {45}^{\circ }$ relative to $X$. Each point represents the average of the variances obtained in a range of photon number: [0–10] photons, [10–20] photons, etc.

Figure 5

(a) Dependency of the $p$ parameters as a function of the cone half aperture ${\psi }_0$ for two models of the molecular 2D distribution function $\tilde{P}\left(\varphi \right)$: filled cone (continuous line), cone surface (dashed line). (b) Different distributions $\tilde{P}\left(\varphi \right)$ represented as polar plots in the sample plane, for $p=0.5$, 0, $-1/3$, $-1$, $-3$.