Connection of Jones and Mueller Tensors in Second Harmonic Generation and Multi-Photon Fluorescence Measurements

Abstract

Despite the rapidly growing use of second harmonic generation (SHG) and two-photon excited fluorescence (TPEF) microscopy, opportunities for relating polarization-dependent measurements back to local structure and molecular orientation are often confounded by losses in polarization purity. In this work, connections linking Mueller tensor and Jones tensor descriptions of polarization-dependent SHG and TPEF are shown to substantially simplify partially depolarized microscopy measurements. These connections were facilitated by the derivation of several new tensor identity relations, based on generalization of established transformations of matrices and vectors. Methods are described for integrating local-frame symmetry and azimuthal rotation angle for simplifying the Mueller tensor. Through simple expressions bridging the Mueller and Jones formalisms, mathematical models for partial depolarization can greatly simplify interpretation of SHG and TPEF measurements to reconstruct the more general Mueller tensors using the much more concise Jones descriptions for the purely polarized components. Integrating the Mueller architecture allows polarization-dependent SHG and TPEF measurements to be connected back to a relatively small set of free parameters related to local structure and orientation.

Second Harmonic Generation of Unpolarized Light

“…However, a large number of observables with many different incident polarization states are required in order to populate all 36 unique elements of the super-Mueller matrix. In a recent complementary framework [17], the Mueller tensors in partially depolarizing assemblies were greatly simplified by directly bridging Jones and Mueller tensors. The role of partial depolarization can in principle be incorporated by a single additional adjustable parameter relative to analogous measurements in nondepolarizing assemblies.…”

“…In a model system, predictions from a mathematical framework connecting Jones and Mueller tensors [17] were compared with observations for z -cut quartz. With this framework in place, polarization-dependent SHG from partial or complete depolarization of the incident beam can be quantitatively described using the intuitive Jones tensor.…”

“…In the theoretical framework developed previously [17], these connections between Mueller and Jones matrices in linear optics were shown to be directly extendable to nonlinear optics, in which the elements of the 4 × 4 × 4 Mueller tensor describing SHG M (2) driven by partially or fully depolarized incident light are connected back to combinations of simpler 2 × 2 × 2 Jones tensor elements ${\chi }_J^{false(2false)}$ using the matrix A [Eq. (1) and Supplemental Material, Eqs.…”

“…(1) and Supplemental Material, Eqs. (S6)–(S9) [18]] [17,19]. $$M^{false(2false)}=A·false({\chi }_J^{false(2false)\ast }\otimes {\chi }_J^{false(2false)}false):A^{-1}{A}^{-1}.$$…”

Generalized Stokes vector for three photon process

“…Later, Simpson J G developed a formalism for nonlinear optical ellipsometry using Jones matrices. 34, 35 4 Â 1 Stokes vector and 4 Â 4 Mueller matrix are used [36][37][38][39] in second harmonic generation (SHG) microscopy to understand molecular level structure of biological materials. SHG is studied recently using Double Stokes-Double Mueller Matrix formalism by di®erent groups.…”