  3-dB couplers, which are commonly used in photonic integrated circuits for
on-chip information processing, precision measurement, and quantum computing,
face challenges in achieving robust performance due to their limited 3-dB
bandwidths and sensitivity to fabrication errors. To address this, we introduce
topological physics to nanophotonics, developing a framework for topological
3-dB couplers. These couplers exhibit broad working wavelength range and
robustness against fabrication dimensional errors. By leveraging valley-Hall
topology and mirror symmetry, the photonic-crystal-slab couplers achieve ideal
3-dB splitting characterized by a wavelength-insensitive scattering matrix.
Tolerance analysis confirms the superiority on broad bandwidth of 48 nm and
robust splitting against dimensional errors of 20 nm. We further propose a
topological interferometer for on-chip distance measurement, which also
exhibits robustness against dimensional errors. This extension of topological
principles to the fields of interferometers, may open up new possibilities for
constructing robust wavelength division multiplexing,
temperature-drift-insensitive sensing, and optical coherence tomography
applications.
