  We present compact integrated speckle spectrometers based on monofractal and
multifractal scattering media in a silicon-on-insulator platform. Through both
numerical and experimental studies we demonstrate enhanced optical throughput,
and hence signal-to-noise ratio, for a number of random structures with
tailored multifractal geometries without affecting the spectral decay of the
speckle correlation functions. Moreover, we show that the developed
multifractal media outperform traditional scattering spectrometers based on
uniform random distributions of scattering centers. Our findings establish the
potential of low-density random media with multifractal correlations for
integrated on-chip applications beyond what is possible with uncorrelated
random disorder.
