  The physical properties of two-dimensional (2D) materials may drastically
vary with their thickness profiles. Current thickness profiling methods for 2D
material (e.g., atomic force microscopy and ellipsometry) are limited in
measurement throughput and accuracy. Here we present a novel high-speed and
high-precision thickness profiling method, termed Transmission-Matrix
Quantitative Phase Profilometry (TM-QPP). In TM-QPP, picometer-level optical
pathlength sensitivity is enabled by extending the photon shot-noise limit of a
high sensitivity common-path interferometric microscopy technique, while
accurate thickness determination is realized by developing a
transmission-matrix model that accounts for multiple refractions and
reflections of light at sample interfaces. Using TM-QPP, the exact thickness
profiles of monolayer and few-layered 2D materials (e.g., MoS2, MoSe2 and WSe2)
are mapped over a wide field of view within seconds in a contact-free manner.
Notably, TM-QPP is also capable of spatially resolving the number of layers of
few-layered 2D materials.
