# Plasmonic Sensors beyond the Phase Matching Condition: A Simplified Approach

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Lossless HPWG Sensor

#### 2.2. Lossy HPWG Sensor

## 3. Results

## 4. Discussion

#### 4.1. Operate at the Phase Matching Wavelength

#### 4.2. Calculate the Nominal Sensitivity

#### 4.3. Operate above the Exceptional Point

#### 4.4. Identify the Nominal Device Length

#### 4.5. Calculate the FOM

#### 4.5.1. “Conventional” Mode Approach

#### 4.5.2. Coupled Mode Theory Approach

#### 4.6. Towards Optical Fibre Plasmonic Sensors

- The present dielectric waveguide is formed by a high-index, sub-wavelength silicon core and a silica cladding: its higher propagation constant provides access to the short-range SPP, which is supported at all wavelengths shown and does not cut off. In contrast, fibre plasmonic sensors typically use a wavelength-scale lower-index silica (SiO
_{2}) core, wherein the effective index of the dielectric mode is close to the refractive index of silica (${n}_{\mathrm{eff}}\approx {n}_{\mathrm{SiO}2}=1.45$). This mode typically phase-matches to the weakly confined long-range surface plasmon (LR-SPP) [56] for an analyte refractive index close to ${n}_{a}=1.45$, and typically cuts off close to regions where the supermodes anti-cross [48]. High-order plasmonic modes in metallic nanowires also cut off across the visible and infrared spectrum [57,58]. The present formalism can only only be applied in regions of the parameter space where the uncoupled bound states are supported, i.e., below modal cutoff. - The present plasmonic sensor is a two-mode system, because each uncoupled waveguide is single mode. Fibre plasmonic sensors, on the other hand, typically have core sizes of several wavelengths in diameter, and can be highly multi-mode. In multi-mode dielectric fibres, the dimensions of the matrix in Equation (6) must therefore be increased to account for the additional modes and coupling coefficients [59].
- Finally, we wish to point out that, in order to achieve sharp resonances and high FOMs in multi-mode sensors, a single-mode waveguide/fibre at input- and output- is required, which filters out higher-order modes, because these have the effect of washing out sharp resonant dips and lowering the FOM [25,48].

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Concept schematic of the challenge of calculating resonances in plasmonic sensors. (

**a**) The simple Otto configuration relies on monitoring the reflectivity R of plane waves propagating in semi-infinite media as a function of angle $\theta $. At the angle ${\theta}_{\mathrm{SPP}}$ a SPP is excited. (

**b**) $\theta $-dependent reflectance spectrum for ${n}_{a}=1$, $\lambda =800\phantom{\rule{0.166667em}{0ex}}\mathrm{nm}$, and w as labelled. Also shown is the full colourmap of the reflectance as a function of $\theta $ and w for (

**c**) ${n}_{a}=1$ and (

**d**) ${n}_{a}=1.33$. Note that the spectral maps are subtly dependent on both ${n}_{a}$ and w.

**Figure 2.**Schematic of the HPWG sensor and the coupled mode theory picture. The modes in the dielectric and plasmonic regions, ${\psi}_{1}$ and ${\psi}_{2}$ respectively, couple linearly as described by Equation (6). The power in the dielectric at output is given by $T=|{\psi}_{1}{|}^{2}$. The periodic exchange of power between waveguides can lead to a resonant spectrum that in general depends on both the length of the device L and the analyte index ${n}_{a}$ [25].

**Figure 3.**Effective index ${n}_{\mathrm{eff}}=\beta /{k}_{0}$ as a function of wavelength for the geometry shown in Figure 2 when (

**a**) ${n}_{a}=1.3$, (

**b**) ${n}_{a}=1.4$, (

**c**) ${n}_{a}=1.5$ in the lossless case. The dashed line shows the isolated plasmonic- and dielectric- modes, respectively. The solid lines show the hybrid eigenmodes. (

**d**–

**f**) show the associated calculated coupling coefficients, following the simple expression in Equation (8) (black line). Top row shows a schematic of the magnetic field for the plotted isolated- or hybrid-/super-modes.

**Figure 4.**Real part of the effective index $\Re e\left({n}_{\mathrm{eff}}\right)=\Re e(\beta /{k}_{0})$ as a function of wavelength for the geometry shown in Figure 2 when (

**a**) ${n}_{a}=1.3$, (

**b**) ${n}_{a}=1.4$, (

**c**) ${n}_{a}=1.5$, using the lossy Drude model for the gold permittivity. The dashed line shows the isolated plasmonic- and dielectric- modes, respectively. The solid lines show the hybrid eigenmodes according to the “exact” solution (dark) and obtained from CMT via the eigenvalues of Equation (9) (light). (

**d**–

**f**) show the associated $\Im m\left({n}_{\mathrm{eff}}\right)$.

**Figure 5.**Transmitted power by the plasmonic sensor as a function of $\lambda $ and ${n}_{a}$ for $L=10\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}\mathrm{m}$ using (

**a**) CMT, and (

**b**) FEM. (

**c**,

**d**): same as (

**a**,

**b**) for $L=15\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}\mathrm{m}$. (

**e**,

**f**): same as (

**a**,

**b**) for $L=20\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}\mathrm{m}$. (

**g**,

**h**): same as (

**a**,

**b**) for $L=50\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}\mathrm{m}$. EP: exceptional point.

**Figure 6.**Calculated colour maps of (

**a**) $|{\beta}_{1}-{\beta}_{2}^{R}|/{k}_{0}+|\kappa -{\beta}_{2}^{I}/2|/{k}_{0}$ using CMT and (

**b**) $|\tilde{{\beta}_{1}}-\tilde{{\beta}_{2}}|/{k}_{0}$ using the exact supermodes. The global minima in the phase space show the location of the exceptional point using our CMT model and the exact solution, as per Equations (10) and (11).

**Figure 7.**(

**a**) Green (right axis): phase matching wavelength ${\lambda}_{\mathrm{PM}}$ where ${\beta}_{1}={\beta}_{2}^{R}$, and associated half beat length ${L}_{b}$ according to the supermodes obtained with CMT (orange) and “exact” calculations (blue). (

**b**) Associated absorption length ${L}_{a}$. Solid lines indicate the average ${L}_{a}=({L}_{a}^{1}+{L}_{a}^{2})/2$; shaded regions encompass the ${L}_{a}^{1}$ and ${L}_{a}^{2}$ boundaries.

**Figure 8.**(

**a**) Transmission spectrum using the “conventional” approach of Equation (14), as a function of wavelength, for the three analyte indices as labelled, using $L=10\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}\mathrm{m}$. Also shown are the resonant wavelength ${\lambda}_{R}$, corresponding to the spectral minimum and the $\delta \lambda $, corresponding to the FWHM. (

**b**) Associated ${\lambda}_{R}$ vs. ${n}_{a}$ (green circles, left axis), second order polynomial fit (green line), and resulting sensitivity S (orange line, right axis.) Also shown in (

**c**) are the $\delta \lambda $ vs. ${n}_{a}$ (orange curve, left axis) and the total FOM = $S/\delta \lambda $. (

**d**–

**f**): same as (

**a**–

**c**), obtained from the CMT approach, using a subset of the data shown in Figure 5a as labelled. (

**g**–

**i**): same as (

**d**–

**f**), obtained from FEM calculations, using a subset of the data shown in Figure 5b as labelled.

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**MDPI and ACS Style**

Tuniz, A.; Song, A.Y.; Della Valle, G.; de Sterke, C.M.
Plasmonic Sensors beyond the Phase Matching Condition: A Simplified Approach. *Sensors* **2022**, *22*, 9994.
https://doi.org/10.3390/s22249994

**AMA Style**

Tuniz A, Song AY, Della Valle G, de Sterke CM.
Plasmonic Sensors beyond the Phase Matching Condition: A Simplified Approach. *Sensors*. 2022; 22(24):9994.
https://doi.org/10.3390/s22249994

**Chicago/Turabian Style**

Tuniz, Alessandro, Alex Y. Song, Giuseppe Della Valle, and C. Martijn de Sterke.
2022. "Plasmonic Sensors beyond the Phase Matching Condition: A Simplified Approach" *Sensors* 22, no. 24: 9994.
https://doi.org/10.3390/s22249994