  When light passes through a multimode fiber, two-dimensional random intensity
patterns are formed due to the complex interference within the fiber. The
extreme sensitivity of speckle patterns to the frequency of light paved the way
for high-resolution multimode fiber spectrometers. However, this approach
requires expensive IR cameras and impedes the integration of spectrometers
on-chip. In this study, we propose a single-pixel multimode fiber spectrometer
by exploiting wavefront shaping. The input light is structured with the help of
a spatial light modulator, and optimal phase masks, focusing light at the
distal end of the fiber, are stored for each wavelength. Variation of the
intensity in the focused region is recorded by scanning all wavelengths under
fixed optimal masks. Based on the intensity measurements, we show that an
arbitrary input spectrum having two wavelengths 20 pm apart from each other can
be reconstructed successfully (with a reconstruction error of ~3%) in the
near-infrared regime, corresponding to a resolving power of $ R \approx 10^5 $.
We also demonstrate the reconstruction of broadband continuous spectra for
various bandwidths. With the installation of a single-pixel detector, our
method provides low-budget and compact detection at an increased
single-to-noise ratio.
