Exploring phase sensitivity and limit of detection near the critical coupling of metasurfaces and its phase singularity

TL;DR

This paper analyzes phase sensitivity and limit of detection for optical Tamm metasensors near the critical coupling point. Using a Temporal Coupled Mode Theory model and a common-path interferometric setup, it derives explicit relations between phase sensitivity $S_{ obreak{\phi}}$, resonance quality factor $Q$, wavelength sensitivity $S_{\lambda}$, and minimum reflectivity $|r_{min}|$, and validates them experimentally by tuning nano-patterned gold gratings. The key finding is that while phase sensitivity increases dramatically near the phase singularity, the limit of detection (LOD) does not improve with proximity to critical coupling; the LOD is governed by $LOD_{\phi} = \frac{3}{2}\cdot \frac{\lambda_{res}}{Q\,S_{\lambda}}\,\sigma_R$. The work suggests that leveraging high topological charge phase singularities could further improve LOD, offering a direction for designing more sensitive phase-based photonic sensors.

Abstract

It is commonly accepted that phase singularities in refractive index sensors can provide highly sensitive detection. To address this issue, we studied the phase sensitivity and the limit of detection of Tamm photonic crystals used as temperature sensors, taken here as a model system by exploring critical coupling and its associated phase singularity. To finely tune the optical Tamm mode to critical coupling, the top metal layer is periodically nanostructured and controlled, enabling the investigation of optical Tamm resonances around phase singularities. We use a highly stable common-path interferometry setup based on digital holography, which allows us to measure extremely high phase sensitivity even at very low light intensities, while also providing information about reflectivity in the complex domain. We experimentally validate an analytical model based on Temporal Coupled Mode Theory, which fully explains the phase sensitivity response and the limit of detection of such resonant photonic sensors. We demonstrate that, although approaching a phase singularity drastically enhances phase sensitivity, no improvement of the limit of detection is expected. Furthermore, phase singularity has no effect on the limit of detection, and therefore on the efficiency of such sensors for practical applications. This work provides a comprehensive description of phase sensors operating at critical coupling, and challenges the commonly accepted assumption regarding the role of phase singularities in sensing.

Figures (9)

• Figure 1: Simulated and fabricated Tamm structures consisting of a Bragg mirror (4 pairs $Si/SiO_2$) and a 1D gold grating. (a) Schematic of the optical Tamm structures with period $a$ and thickness $t_{Au}$. $w$ is the width of the gold rods. (b) Electric field magnitude $E_y$ with $a$ = 660nm, $t_{Au}$ = 50nm, and $w$ = 560nm: green borders are the edges of the gold pattern and dashed lines locate the $Si/SiO_2$ interfaces. (c) Scanning Electron Microscope image of a fabricated 1D gold pattern.
• Figure 2: FDTD simulations of optical Tamm structures. All numerical simulations are performed with $a$=660nm and $t_{Au}$=50nm for s-polarized (TE) incident light. (a) Reflectivity amplitude. (b) Relative phase shift. (c) Values of $\tau_{rad}$ and $\tau_i$ calculated by fitting panels (a) and (b) using the complex reflection coefficient obtained from TCMT.
• Figure 3: Experimental reflectivity spectra as a function of the measured FF for $a$ = 660nm. (a) FF around the first critical point and (b) around the second critical point.
• Figure 4: Optical setup for the phase sensitivity measurements. (a) Schematic of the setup. The light polarization is controlled using a polarizer ($P$) and a half-wave plate ($HWP$). Hologram generation and beam shaping are achieved with three lenses ($L_1$, $L_2$, $L_3$), a spatial filter (SF), and a spatial light modulator (SLM). The signal is detected after passing through a slit ($S$) of 100µm width in front of a photodetector (PD). (b) Camera image of the two Gaussian beams at the sample plane. (c) Young's interference pattern before the slit $S$.
• Figure 5: Experimental complex reflectivity spectra in intensity (a) and phase (b) for Tamm plasmon photonic crystals with different FF, with $a\approx$ 600nm. (c) Time constants (absorption and radiative losses) from the TMCT model are extracted by fitting the real and imaginary parts of the complex reflectivity versus the filling factor FF. (d) Representation of each spectrum in the complex plane. The resonance position (minimum of reflectivity) is marked with a green dot, whose distance from the origin O(0,0) is minimal. Each fit is represented by a dark thin circle.