A formalism for giant Goos-H\"anchen shift in metasurface sensors with phase singularity

Abstract

The Goos-H\"anchen (GH) shift becomes giant in resonant photonic structures, making it promising for refractive index sensors with ultimate sensitivities. We provide here a complete formalism to analytically describe the GH shift and its associated sensitivity around the critical coupling regime in photonic structures. This analytical framework quantitatively connects physical parameters such as the quality factor, the angular dispersion, the beam size and the phase singularity to the GH shift. We numerically confirm this theory in two practical designs: a surface plasmon resonance sensor and a Bloch surface wave (BSW) metasurface sensor. Coupling our theory with numerical simulations, we design a BSW metasurface whose GH sensitivity ($10^{13} \mu m/RIU$) is more than 5 orders of magnitude higher than the current state-of-the art.We also reveal that the main practical limitation to reach ultimate GH sensitivities is the beam size. However, taking into account realistic beam sizes and introducing engineering dispersion for the metasurface, we calculate limits of detection for GH sensors as low as $10^{-13} RIU$ that still surpass current sensors. These results open the way for new sensing application needing high sensitivity and low limit of detection.