  Second-order nonlinear optical interactions, including second harmonic
generation (SHG) and sum-frequency generation (SFG), can reveal a wealth of
information about chemical, electronic, and vibrational dynamics at the
nanoscale. Here, we demonstrate a powerful and flexible new approach, called
phase-modulated degenerate parametric amplification (DPA). The technique, which
allows for facile retrieval of both the amplitude and phase of the second-order
nonlinear optical response, has many advantages over conventional or
heterodyne-detected SHG, including the flexibility to detect the signal at
either the second harmonic or fundamental field wavelength. We demonstrate the
capabilities of this approach by imaging multi-grain flakes of single-layer
MoS2. We identify the absolute crystal orientation of each MoS2 domain and
resolve grain boundaries with high signal contrast and sub-diffraction-limited
spatial resolution. This robust all-optical method can be used to characterize
structure and dynamics in organic and inorganic systems, including biological
tissue, soft materials, and metal and semiconductor nanostructures, and is
particularly well-suited for imaging in media that are absorptive or highly
scattering to visible and ultraviolet light.
