  We propose a new design on integrated optical devices on-chip with an extra
width degree of freedom by using a photonic crystal waveguide with Dirac points
between two photonic crystals with opposite valley Chern numbers. With such an
extra waveguide, we demonstrate numerically that the topologically protected
photonic waveguide keeps properties of valley-locking and immunity to defects.
Due to the design flexibility of the width-tunable topologically protected
photonic waveguide, many unique on-chip integrated devices have been proposed,
such as energy concentrators with a concentration efficiency improvement by
more than one order of magnitude, topological photonic power splitter with
arbitrary power splitting ratio. The topologically protected photonic waveguide
with the width degree of freedom could be beneficial for scaling up photonic
devices, which provides a new flexible platform to implement integrated
photonic networks on chip.
