  The selection rule for angle-resolved polarized Raman (ARPR) intensity of
phonons from standard group-theoretical method in isotropic materials would
break down in anisotropic layered materials (ALMs) due to birefringence and
linear dichroism effects. The two effects result in depth-dependent
polarization and intensity of incident laser and scattered signal inside ALMs
and thus make a challenge to predict ARPR intensity at any laser incidence
direction. Herein, taking in-plane anisotropic black phosphorus as a prototype,
we developed a so-called birefringence-linear-dichroism (BLD) model to
quantitatively understand its ARPR intensity at both normal and oblique laser
incidences by the same set of real Raman tensors for certain laser excitation.
No fitting parameter is needed, once the birefringence and linear dichroism
effects are considered with the complex refractive indexes. An approach was
proposed to experimentally determine real Raman tensor and complex refractive
indexes, respectively, from the relative Raman intensity along its principle
axes and incident-angle resolved reflectivity by Fresnel$'$s law. The results
suggest that the previously reported ARPR intensity of ultrathin ALM flakes
deposited on a multilayered substrate at normal laser incidence can be also
understood based on the BLD model by considering the depth-dependent
polarization and intensity of incident laser and scattered Raman signal induced
by both birefringence and linear dichroism effects within ALM flakes and the
interference effects in the multilayered structures, which are dependent on the
excitation wavelength, thickness of ALM flakes and dielectric layers of the
substrate. This work can be generally applicable to any opaque anisotropic
crystals, offering a promising route to predict and manipulate the polarized
behaviors of related phonons.
