  Optical spectroscopic sensors are a powerful tool to reveal light-matter
interactions in many fields, such as physics, biology, chemistry, and
astronomy. Miniaturizing the currently bulky spectrometers has become
imperative for the wide range of applications that demand in situ or even in
vitro characterization systems, a field that is growing rapidly. Benchtop
spectrometers are capable of offering superior resolution and spectral range,
but at the expense of requiring a large size. In this paper, we propose a novel
method that redesigns spectroscopic sensors via the use of programmable
photonic circuits. Drawing from compressive sensing theory, we start by
investigating the most ideal sampling matrix for a reconstructive spectrometer
and reveal that a sufficiently large number of sampling channels is a
prerequisite for both fine resolution and low reconstruction error. This number
is, however, still considerably smaller than that of the reconstructed spectral
pixels, benefitting from the nature of reconstruction algorithms. We then show
that the cascading of a few engineered MZI elements can be readily programmed
to create an exponentially scalable number of such sampling spectral responses
over an ultra-broad bandwidth, allowing for ultra-high resolution down to
single-digit picometers without incurring additional hardware costs.
Experimentally, we implement an on-chip spectrometer with a fully-programmable
6-stage cascaded MZI structure and demonstrate a < 10 pm resolution with a >
200 nm bandwidth using only 729 sampling channels. This achieves a
bandwidth-to-resolution ratio of over 20,000, which is, to our best knowledge,
about one order of magnitude greater than any reported miniaturized
spectrometers to date. We further illustrate that by employing
dispersion-engineered waveguide components, the device bandwidth can be
extended to over 400 nm.
