A formalism for giant Goos-Hänchen shift in metasurface sensors with phase singularity

TL;DR

This work develops a complete temporal coupled mode theory framework to describe the giant Goos-Hänchen shift in resonant photonic structures near the critical coupling regime. It derives analytic expressions showing GH shift scales with $Q/r_{min}$ and GH sensitivity scales with $(Q/r_{min})^2$, highlighting the roles of angular dispersion $a$ and phase singularity. Numerical validation on SPR and Bloch surface wave metasurfaces demonstrates giant shifts up to tens of millimeters and GH sensitivities up to $S_{GH}\sim 10^{13}~\mu m/RIU$, with BSW metasurfaces offering superior performance due to larger $Q/r_{min}$. A key practical insight is that finite beam size critically limits observable shifts, motivating a figure of merit $FOM_{GH}$ and design guidelines favoring quasi-BIC flat-band metasurfaces to achieve LODs as low as $10^{-13}$ RIU, enabling ultra-sensitive sensing for gas, biomolecule detection, and metrology.

Abstract

The Goos-Hänchen (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 ($10^{13} μm/RIU$) 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.

Figures (8)

• Figure 1: Goos-Hänchen effect ; lateral beam displacement of the reflected beam by a metasurface
• Figure 2: GH shift and sensitivity versus the incident angle for several values of $Q/r_{min}$ drawn with the temporal coupled mode theory with the following parameters, $Q=20000$, $\lambda_{res}=1539.53 nm$, $\theta_{res}=7.20^{\circ}$ and $r_{min}=0.01~or~0.05~or~0.0025$
• Figure 3: Goos-Hänchen effect for surface plasmon resonance (in a) and for Bloch Surface Wave metasurface (in b)). In a) between the 49.5 nm gold layer and the substrate (refractive index 1.62) a 5 nm thin film of GST is added. In b) the grating is a dielectric material of refractive index 1.8 : with width w=414 nm and period p=920 nm and height h=356 nm
• Figure 4: SPR shift due to the refractive index variation of the medium of analyte by $\Delta n = 0.001$; a) Intensity and phase of the reflectivity in the two media of analyte. The dashed black curves is the fit using the CMT. b) GH shift, $D_{GH}$ and variation of the GH shift $\Delta D_{GH}$
• Figure 5: a) Giant Goos-Hänchen shift for SPR by crystallization of the GST layer. The blue curve is the analytical relation (\ref{['eq:DH_parTMC']} ) $D_{GH}=cst/r_{min}$ with $cst=\frac{3a\lambda^{2}Q}{8\pi^{}c}=5e-6$ which is determined by the physical parameters, $\lambda_{res}=750~nm$, $Q=6.5$, $a=9.62\cdot10^{15}~s^{-1}$, in black circle circle the simulation done by the transfer matrix method. In b) The GH sensitivity S$_{GH}$ with in blue line the analytical relation (\ref{['eq:Sensibilite_parTMC']}), with the same physical parameters $\lambda_{res}$, $Q$, $a=$, and $S_{\lambda}=4362~nm/RIU$, and in black circle the numerical simulation. GH shift and GH sensitivity are drawn in absolute value for logarithmic representation which removes the asymmetry of the phase singularity effect.