  Topological photonics has attracted widespread research attention in the past
decade due to its fundamental interest and unique manner in controlling light
propagation for advanced applications. Paradigmatic approaches have been
proposed to achieve topological phases including topological insulators in a
variety of photonic systems. In particular, photonic lattices composed of
evanescently coupled waveguide arrays have been employed conveniently to
explore and investigate topological physics. In this article, we review our
recent work on demonstration of topological phenomena in reconfigurable
photonic lattices established by site-to-site cw-laser-writing or multiple-beam
optical induction in photorefractive nonlinear crystals. We focus on the study
of topological states realized in the celebrated one-dimensional
Su-Schrieffer-Heeger lattices, including nonlinear topological edge states and
gap solitons, nonlinearity-induced coupling to topological edge states, and
nonlinear control of non-Hermitian topological states. In the two-dimensional
case, we discuss two typical examples: universal mapping of momentum-space
topological singularities through Dirac-like photonic lattices and realization
of real-space nontrivial loop states in flatband photonic lattices. Our work
illustrates how photorefractive materials can be employed conveniently to build
up various synthetic photonic microstructures for topological studies, which
may prove relevant and inspiring for exploration of fundamental phenomena in
topological systems beyond photonics.
