Microscopic second-harmonic generation emission direction in fibrillous collagen type I by quasi-phase-matching theory

Abstract

Quasiphase matching (QPM) is a widely used theory in crystal to analyze the character of second-harmonic generation (SHG) emitted from it. Based on the structural features of collagen type I, where the constituted fibrils in collagen function as a crystal which has the structure of two-dimensional (2D) quasicrystalline, in this paper, we use the QPM theory on collagen for SHG emission direction study under the excitation of laser light through a microscope. The effects of numerical aperture NA, as well as the structural parameters, such as QPM order (m,l) and collagen period a=d1+d2 associated with the fibrils diameter (d1), packing density and interfibrils structure (d2), etc., on SHG emission angle φ have been investigated. Our theoretical results show that collagen period a has threshold effect on φ to present forward or backward SHG emission and NA has minor influence on this threshold value a. Collagen period of a has more significant influence on SHG emission angle φ when a is smaller than the threshold value. In reality, we realize that diameter of collagen fibrils d1 plays a major role on forward or backward emission of SHG. Here, for example, (we assume d1=a/2), when d1≤95 nm [(m,l)=(1,0)], the backward SHG shows up at any magnitude of NA, while when d1≥150 nm [(m,l)=(1,0)], SHG presents forward emission feature under all circumstances. Between them, SHG emits from forward direction to backward direction as the increase in NA. The QPM order (m,l) has nonlinear impact on SHG emission angle φ and has different degrees of influence on different collagen period a. Our theoretical results are highly consistent with the experiments results demonstrated by other researchers and provide a proper explanation of the phenomenon of appreciable backward SHG signals appearing in collagen type I. Our established theoretical model of applying QPM theory in 2D quasicrystalline fibrils is therefore confirmed to be a suitable model for dealing with SHG in type I collagen.