Backward emission angle of microscopic second-harmonic generation from crystallized type I collagen fiber

A theoretical model that deals with SHG from crystallized type I collagen fiber formed by a bundle of fibrils is established. By introducing a density distribution function of dipoles within fibrils assembly into the dipole theory and combining with structural order (m,l) parameters revealed by quasi-phase-matching (QPM) theory, our established theoretical model comprehensively characterizes both biophysical features of collagen dipoles and the crystalline characteristics of collagen fiber. This new model quantitatively reveals the 3-D distribution of second-harmonic generation (SHG) emission angle (θ,ϕ) in accordance with the emission power. Results show that fibrils diameter d(1) and structural order m, which describes the structural characteristics of collagen fiber along the incident light propagation direction has significant influence on backward∕forward SHG emission. The decrease of fibrils diameter d(1) induces an increase of the peak SHG emission angle θ(max). As d(1) decreases to a threshold value, in our case it is around d(1) = 150 nm when (m,l) = (1,0), θ(max) > 90 deg, indicating that backward SHG emission appears. The SHG may have two symmetrical emission distribution lobes or may have only one or two unsymmetrical emission lobes with unequal emission power, depending on the functional area of (m,l) on d(1).