Second Harmonic Generation of Unpolarized Light

A Mueller tensor mathematical framework was applied for predicting and interpreting the second harmonic generation (SHG) produced with an unpolarized fundamental beam. In deep tissue imaging through SHG and multiphoton fluorescence, partial or complete depolarization of the incident light complicates polarization analysis. The proposed framework has the distinct advantage of seamlessly merging the purely polarized theory based on the Jones or Cartesian susceptibility tensors with a more general Mueller tensor framework capable of handling partial depolarized fundamental and/or SHG produced. The predictions of the model are in excellent agreement with experimental measurements of $z$-cut quartz and mouse tail tendon obtained with polarized and depolarized incident light. The polarization-dependent SHG produced with unpolarized fundamental allowed determination of collagen fiber orientation in agreement with orthogonal methods based on image analysis. This method has the distinct advantage of being immune to birefringence or depolarization of the fundamental beam for structural analysis of tissues.

Figure 1

SHG transmittance microscope capable of delivering both a purely polarized and a depolarized fundamental beam.

Figure 2

(a) Polarization dependent measurement of SHG signal from $z$-cut quartz at an arbitrary angle under (1) vertically polarized and (2) depolarized incident light overlay with theoretical (3) fitting and (4) prediction. (b) Measurements of horizontally polarization SHG signals from $z$-cut quartz at different azimuthal angle under (1) vertically polarized and (2) depolarized incident light, with theoretical (3) fitting and (4) prediction.

Figure 3

Orientation images of the azimuthal angle for a single FOV of mouse tail section from (a) the pixel-by-pixel nonlinear fit analysis and (b) OrientationJ. Scale bar: $100\mu \mathrm{m}$. (c) The histogram of the orientation distribution achieved from pixel-by-pixel fit and OrientationJ. (d) Nonlinear fitting results of depolarized light excitation SHG for signal random pixel.

Figure 4

(a) The measured laboratory-frame ratio $\rho$ images for a single FOV of mouse tail section from the per-pixel fit analysis. (b) The polar tilt angle $\theta$ image recovered from the measured laboratory-frame ratio $\rho$ for the same FOV with ${\rho }_l=1.7$. (c) The distribution of the ratio $\rho$, with the maximum peak marked at 1.69. Scale bar: $100\mu \mathrm{m}$.