  We present an integrated design to precisely measure optical frequency using
weak value amplification with a multi-mode interferometer. The technique
involves introducing a weak perturbation to the system and then post-selecting
the data in such a way that the signal is amplified without amplifying the
technical noise, as has previously been demonstrated in a free-space setup. We
demonstrate the advantages of a Bragg grating with two band gaps for obtaining
simultaneous, stable high transmission and high dispersion. We numerically
model the interferometer in order to demonstrate the amplification effect. The
device is shown to have advantages over both the free-space implementation and
other methods of measuring optical frequency on a chip, such as an integrated
Mach-Zehnder interferometer.
