  Light scattering in disordered media has been studied extensively due to its
prevalence in natural and artificial systems [1]. In the field of photonics
most of the research has focused on understanding and mitigating the effects of
scattering, which are often detrimental. For certain applications, however,
intentionally introducing disorder can actually improve the device performance,
e.g., in photovoltaics optical scattering improves the efficiency of light
harvesting [2-5]. Here, we utilize multiple scattering in a random photonic
structure to build a compact on-chip spectrometer. The probe signal diffuses
through a scattering medium generating wavelength-dependent speckle patterns
which can be used to recover the input spectrum after calibration. Multiple
scattering increases the optical pathlength by folding the paths in a confined
geometry, enhancing the spectral decorrelation of speckle patterns and thus
increasing the spectral resolution. By designing and fabricating the
spectrometer on a silicon wafer, we are able to efficiently channel the
scattered light to the detectors, minimizing the reflection loss. We
demonstrate spectral resolution of 0.75 nm at a wavelength of 1500 nm in a 25
{\mu}m by 50 {\mu}m random structure. Furthermore, the phenomenal control
afforded by semiconductor nanofabrication technology enabled us to engineer the
disorder to reduce the out-of-plane scattering loss. Such a compact,
high-resolution spectrometer that is integrated on a silicon chip and robust
against fabrication imperfections is well suited for lab-on-a-chip spectroscopy
applications.
