Metal-insulator-metal waveguide-based optical pressure sensor embedded with arrays of silver nanorods

Infiter Tathfif; Ahmad Azuad Yaseer; Kazi Sharmeen Rashid; Rakibul Hasan Sagor

doi:10.1364/OE.439974

1. Introduction

The ability of plasmonic structures to confine light into an enhanced subwavelength scale has provided myriads of fundamental insights into nanophotonic research endeavors. Researchers are now exploring new regimes of nano-metallic structures to analyze the light-matter interaction, and the theory of Surface Plasmon Polaritons (SPPs) helps elucidate this synergy. SPPs are electromagnetic (EM) waves propagating along the metal-dielectric juncture due to collective oscillations of electrons and photons at the resonant frequency [1–5]. The waves bind firmly at the metal surface, acquiring a unique capability to overcome the diffraction limit of light [6–9]. These properties of SPPs have pervaded a wide area of study, including sensors [10–13], absorbers [14–16], filters [17], splitters [18], photonic integrated circuits [19], and switches [20].

An SPP-based pressure sensor is an optical device that intercepts applied pressure and translates it into optical signals. Optical pressure sensors have the advantage of being impervious to electromagnetic interference, have high sensitivity, signal transmission flexibility, and inexpensive manufacturing costs [21,22]. In recent times, optical pressure sensing devices have been designed in a variety of ways to facilitate different lab-on-a-chip activities. Osorio et al. [23] illustrated a photonic-crystal fiber (PCF) pressure sensor for dual environment monitoring. A hybrid dual-core PCF was employed as a hydrostatic pressure sensor with a sensitivity of 0.0116 nm/MPa [24]. A nano-pressure sensor utilizing a high-quality photonic crystal cavity resonator exhibited maximum sensitivity of 0.0117 nm/MPa [25]. Another highly sensitive optical fiber pressure sensor based on a thin-walled oval cylinder stated a sensitivity of 1.198 nm/MPa [26]. Zhao et al. [27] suggested a nano-optomechanical pressure sensor based on a ring resonator with a sensitivity of 1.47 nm/MPa. The sensitivity of a polarization-maintaining PCF was 3.42 nm/RIU [28]. Yao et al. [29] developed a fiber-tip pressure sensor with a maximum sensitivity of 4.29 nm/MPa.

The MIM waveguide is one of the most prevalent SPP waveguides due to its long propagation length, strong field confinement at the subwavelength scale, and fabrication ease [30–32]. Therefore, distinctive designs of optical pressure sensors utilizing MIM waveguide are also investigated. For example, Wu et al. [33] reported a MIM H-shaped resonator with a sensitivity of 2.12 nm/MPa. The maximum sensitivity of a MIM pressure sensor made up of concentric square disk and ring resonators was 5 nm/MPa [4]. A $\pi$-shaped resonator based on MIM geometry demonstrated a sensitivity of 8.5 nm/MPa [34]. Palizvan et al. [35] proposed an optical MIM pressure sensor based on a double square ring resonator with a sensitivity of 16.5 nm/MPa. However, the pressure sensors mentioned above lack the high sensitivity needed for diverse mechanical, electrical, and biomedical applications [36].

An ultra-compact optical pressure sensor based on MIM geometry is presented in this article. The proposed structure encompasses a straight waveguide and a ladder-shaped resonator, loaded with silver NRs. The transmittance spectrum of the proposed sensor is analyzed numerically and the relation between deformation and wavelength shift is determined when pressure is applied. The suggested layout exhibits maximum pressure sensitivity of 25.4 nm/MPa. To the best of the authors’ knowledge, this is the first-ever study of a pressure sensor utilizing NRs, and the sensitivity demonstrated is the highest to date in the literature.

2. Setup and MIM theory

The two-dimensional (2D) schematic representation of the proposed structure is shown in Fig. 1, where the white and grey regions represent air and silver, respectively. The 2D domain is chosen for structure modeling for efficient and fast computation [37]. The suggested sensor consists of a straight waveguide coupled with a ladder-shaped resonator. The resonator contains two horizontal slots and two vertical slots. Moreover, the waveguide and the lower horizontal slot are loaded with silver NRs. The initial structure parameters labeled in Fig. 1 are listed in Table 1.

Fig. 1. Two-dimensional setup of the modeled pressure sensor.

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Table 1. Initial structural parameters.

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To cover the range of the resonant wavelength, a polychromatic light source, such as a halogen lamp, can be used in the practical setup. The light reaches the input side of the pressure sensor through a single-mode fiber (SMF) [38]. The elastic membranes of a diaphragm are attached to the silver layer where the pressure will be exerted (Fig. 2(a)). Injecting fluid or gas into the diaphragm displaces its arms, generates pressure, and deforms the resonator by d nm (Fig. 2(b)) [39]. The output side of the pressure sensor is connected to an optical spectrum analyzer (OSA) through another SMF to track the changes in resonant wavelength due to the deformation. During the simulation, the width of upper horizontal slot is decreased from 40 nm to 30 nm to emulate deformation of 0 nm to 10 nm. Furthermore, W is set to 50 nm to ensure the propagation of fundamental transverse-magnetic (TM₀) mode [40]. The dispersion relation of a MIM waveguide can be characterized through the following equation [41],

(1)$$\tanh \left( {{\raise0.7ex\hbox{${{k_1}W}$} \!\mathord{\left/ {\vphantom {{{k_1}W} 2}}\right.} \!\lower0.7ex\hbox{$2$}}} \right) = - {\raise0.7ex\hbox{${{\varepsilon _{air}}{k_2}}$} \!\mathord{\left/ {\vphantom {{{\varepsilon _{air}}{k_2}} {{\varepsilon _{silver}}{k_1}}}}\right.} \!\lower0.7ex\hbox{${{\varepsilon _{silver}}{k_1}}$}},$$

where, W denotes the width of the waveguide, and the wave vectors k₁ and k₂ are defined by momentum conservations as [33],

(2)$$\begin{aligned} {k_1} &= \sqrt {{\beta ^2} - {\mu _{air}}{\varepsilon _{air}}k_0^2} ,\\ {k_2} &= \sqrt {{\beta ^2} - {\mu _{silver}}{\varepsilon _{silver}}k_0^2} , \end{aligned}$$

where, $\beta$ = n_eff $\times$ k₀ denotes the propagation constant of the MIM waveguide (n_eff = effective refractive index). k₀ refer to the free-space wave vector which can be defined as,

(3)$${k_0} = \frac{{2\pi }}{\lambda } = \frac{\omega }{c},$$

where, $\lambda$ is the wavelength of the light, $\omega$ is the angular frequency, and c is the light speed in free space. Using Eq. (1) and Eq. (2), the resulting dispersion plot for the modeled sensor at W = 50 nm is plotted in Fig. 3.

Fig. 2. a) Schematic of the resonator when no pressure is applied. b) Deformation of the resonator by d nm upon applied pressure, P Pa.

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Fig. 3. Dispersion plot for the illustrated sensor at W = 50 nm.

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Three models are generally used to describe the frequency dependence of silver: Drude, Drude-Debye (DD), and Lorentz-Drude (LD) [42–44]. Among these, the Drude model is the most classical method for obtaining the dielectric constant of the medium as it fits well with the experimental data at short wavelengths [45]. However, the Drude model generates significant errors in the long wavelengths. The DD model employs the dielectric constant as a frequency-dependent parameter. The calculation of the LD model is complex. Nonetheless, LD is the only model that considers the interband transitions and the obtained numerical results match closely with the experimental data [46,47]. Therefore, this article adopts the LD model to express the frequency dependence of silver through the following equation [48],

(4)$$\varepsilon_{silver} \left( \omega \right) = 1 - \frac{{\omega _p^2}}{{{\omega ^2} - i\omega {\Gamma _0}}} + \sum_{n = 1}^6 {\frac{{{f_n}\omega _n^2}}{{\omega _n^2 - {\omega ^2} + i\omega {\Gamma _n}}}},$$

where, $\varepsilon _{silver} \left ( \omega \right )$ represents the complex permittivity of silver. $\omega _p$ and $\omega _n$ are the plasma and resonant frequencies, respectively. In addition, $f_n$, $\Gamma _n$, and $\Gamma _0$ mean the oscillator strength, damping frequency, and collision frequency, respectively. Table 2 lists the numerical values of these parameters.

Table 2. LD parameters for Ag [48].

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The numerical analysis of the modeled sensor is performed in the commercial EM simulator COMSOL Multiphysics 5.3, employing the Finite Element Method (FEM). The designed setup incorporates scattering boundary condition to absorb the outgoing EM waves and avoid reflections. Moreover, extra-fine triangular meshing is deployed around the waveguide and the resonator for the discretization of the structure (Fig. 4). Two ports labeled P_i and P_o are placed at the input and output sides, respectively to measure power flow and calculate transmittance as, T = P_o/P_i [6].

Fig. 4. Extra-fine triangular meshing of the proposed model.

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The electron-beam lithography (EBL) can be used to fabricate the proposed nanosensor. EBL is equipped with sub-10 nm resolution and can write custom designs. On the silicon substrate, a thin coating of silver is deposited, followed by the desired pattern. Wet chemical etching, such as the use of water and dilute nitric acid, can be used to remove the surplus silver [49,50].

3. Numerical analysis and discussion

Figure 5 displays the transmittance spectrum when d = 0 nm (no pressure is applied). A resonant dip exists at the wavelength, $\lambda _{res}$ = 1482.47 nm. To comprehend the resonance within the proposed structure, both magnetic field intensity, $\lvert$H$\rvert$ = (H_z²)^1/2 and electric field intensity, $\lvert$E$\rvert$ = (E^x²+E^y²)^1/2 are plotted in Fig. 6(a) and Fig. 6(b), respectively. It is observed that the dominant magnetic field exists within the upper horizontal slot where pressure is applied, whereas the dominant electric field exists around the NRs. Since most of the input energy is trapped within the resonator, the transmittance value reduces (Fig. 5).

Fig. 5. Transmittance spectrum for d = 0 nm.

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Fig. 6. a) $\lvert$H$\rvert$ = (H_z²)^1/2 and b) $\lvert$E$\rvert$ = (E^x²+E^y²)^1/2 profiles at $\lambda _{res}$ = 1482.47 nm.

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Figure 2(b) illustrates the bending down of the top silver layer upon the application of pressure, P Pa. It is observed that the bent silver portion reduces the volume of the resonator. The deformed resonator causes a corresponding wavelength shift due to the coupling of the input light to the resonator on its way to the output port. Slater’s law relates the deformation of the resonator and the wavelength shift through the following equation [51]:

(5)$${\raise0.7ex\hbox{${\delta f}$} \!\mathord{\left/ {\vphantom {{\delta f} f}}\right.} \!\lower0.7ex\hbox{$f$}} = - {\raise0.7ex\hbox{${\left( {{\varepsilon _0}{E^2} - {\mu _0}{H^2}} \right)\delta V}$} \!\mathord{\left/ {\vphantom {{\left( {{\varepsilon _0}{E^2} - {\mu _0}{H^2}} \right)\delta V} {\int\limits_V {\left( {{\varepsilon _0}{E^2} + {\mu _0}{H^2}} \right)\delta V} }}}\right.} \!\lower0.7ex\hbox{${\int\limits_V {\left( {{\varepsilon _0}{E^2} + {\mu _0}{H^2}} \right)\delta V} }$}},$$

where, f, E, H, and V refer to the resonant frequency, electric field intensity, magnetic field intensity, and volume of the resonating cavity, respectively and the values are listed in Table 3. Since the designed schematic is 2D, the height of the device is infinity. However, for practical fabrication, any height greater than 800 nm will result in close match with the simulations [37]. Hence, the height is taken as 900 nm in Table 3 to calculate the volume.

Table 3. Slater’s law parameters at the resonant condition.

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Equation (5) states that if the resonating cavity is dominated by the magnetic field and undergoes deformation, the resonant wavelength experiences a redshift. Otherwise, when the resonating cavity is dominated by the electric field and undergoes deformation, the resonant wavelength experiences a blueshift.

4. Pressure sensing

The deformation amount d is varied from 0 nm to 10 nm, with a step size of 2 nm to establish the relationship between the resonant wavelength and d. With increasing d, the resonant wavelength experiences a redshift (Fig. 7) due to the enhancement of cavity plasmon resonance (CPR). The absorption and strong coupling of light within the resonator cavity generate this CPR effect [52,53]. The redshift can also be explained using Eq. (5). The upper horizontal slot is dominated by the magnetic field (Fig. 6(a)). Hence, when d increases, the resonant wavelength moves toward longer wavelength region. Moreover, the resonant wavelength and d exhibit a linear relationship, as demonstrated in Fig. 8.

Fig. 7. Transmittance spectrum for d = 0 nm to d = 10 nm.

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Assuming the upper silver layer as a plane plate with a thickness of h and a length of l, applied pressure P and deformation d can be related by the following equation [33],

(6)$$P = {\raise0.7ex\hbox{${2{Y_{Ag}}{h^3}d}$} \!\mathord{\left/ {\vphantom {{2{Y_{Ag}}{h^3}d} {{l^4}}}}\right.} \!\lower0.7ex\hbox{${{l^4}}$}},$$

where, Y_ag is Young’s modulus of silver with a value of 7.5$\times$10¹⁰ Pa. Furthermore, from Fig. 8, the resonant wavelength shift $\Delta \lambda _{res}$ and d is related with a constant of proportionality $\beta$ and the resulting equation is as follows [33],

(7)$$P = {\raise0.7ex\hbox{${2{Y_{Ag}}{h^3}\Delta {\lambda _{res}}}$} \!\mathord{\left/ {\vphantom {{2{Y_{Ag}}{h^3}\Delta {\lambda _{res}}} {\beta {l^4}}}}\right.} \!\lower0.7ex\hbox{${\beta {l^4}}$}}.$$

The performance of a pressure sensor is evaluated by the sensitivity value. Pressure sensitivity (S) can be expressed as [35],

(8)$$S = {\raise0.7ex\hbox{${\Delta {\lambda _{res}}}$} \!\mathord{\left/ {\vphantom {{\Delta {\lambda _{res}}} {\Delta P}}}\right.} \!\lower0.7ex\hbox{${\Delta P}$}} = {\raise0.7ex\hbox{${{l^4}\Delta {\lambda _{res}}}$} \!\mathord{\left/ {\vphantom {{{l^4}\Delta {\lambda _{res}}} {2{Y_{Ag}}{h^3}\Delta d}}}\right.} \!\lower0.7ex\hbox{${2{Y_{Ag}}{h^3}\Delta d}$}},$$

where, $\Delta d$ is the corresponding difference between two deformation values. From Fig. 8, for $\Delta d$ = 10 nm, the resonant wavelength shift is 92.93 nm. Hence, using Eq. (8), maximum sensitivity of 25.4 nm/MPa is obtained. Table 4 compares the suggested nanosensor with existing literature. It is observed that the proposed schematic displays the highest sensitivity reported to date ($\approx$ 53.94% higher than [35]).

Fig. 8. Resonant wavelength vs. deformation graph.

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Table 4. Comparison of pressure sensitivity with recent literature.

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5. Influence of NRs on the proposed structure

Previously reported pressure sensors did not include any NRs. Thus, to further elucidate the effect of NRs on pressure sensitivity, numerous structures are simulated (Fig. 9(a) to Fig. 9(g)), and Table 5 summarizes the corresponding results. Figure 9(a) shows the transmittance profile for d = 0 nm and d = 10 nm when NRs are absent within the proposed schematic. Observing Table 5, the absence of NRs results in the lowest pressure sensitivity.

Fig. 9. Transmittance spectra for d = 0 nm and d = 10 nm when a) there are no NRs, b) R = 2.5 nm, c) R = 5 nm, d) n₁ = 3, e) n₁ = 10, f) n₂ = 3, and g) n₂ = 10.

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Table 5. Influence of NRs on pressure sensitivity.

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Afterward, transmittance profiles for d = 0 nm and d = 10 nm are plotted for different NR parameters, for example, R (Fig. 9(b) and Fig. 9(c)), n₁ (Fig. 9(d) and Fig. 9(e)), and n₂ (Fig. 9(f) and Fig. 9(g)) while the remaining parameters stay constant. Table 5 indicates that increasing R, n₁, and n₂ generate longer resonant wavelength shifts, enhancing pressure sensitivity. The strong coupling of light within the gaps around NRs intensifies the energy of the resonating SPPs and hence causes redshift. This phenomenon is known as gap plasmon resonance (GPR) [54–58].

Considering all these aspects and after extensive simulations, the NR parameters of the illustrated sensor have been fixed as R = 10 nm, n₁ = 17, and n₂ = 17 to generate maximum pressure sensitivity of 25.4 nm/MPa.

6. Conclusion

In this article, a MIM optical pressure sensor decorated with silver NRs is proposed and numerically analyzed through the FEM. Transmittance spectra display a redshift for increasing pressure. Moreover, the resonant wavelength and deformation exhibit a linear association. The correlation between pressure and equivalent deformation is also explored, as well as a maximum pressure sensitivity of 25.4 nm/MPa is calculated. The article also discusses the influence of NRs on pressure sensitivity and summarizes the findings. Due to high sensitivity and compactness, the proposed sensor is a potential contender for different on-chip sensing tasks.

Acknowledgment

The authors would like to thank the Department of Electrical and Electronic Engineering of the Islamic University of Technology for their utmost support.

Disclosures

The authors declare no conflicts of interest.

Data availability

No data were generated or analyzed in the presented research.

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Parameter	Symbol	Value
Waveguide width	W	50 nm
Thickness of top silver layer	h	100 nm
Horizontal slot length	l	800 nm
Horizontal slot width	w₁	40 nm
Vertical slot width	w₂	40 nm
Gap between two vertical slots	g₁	100 nm
Gap between two horizontal slots	g₂	100 nm
Gap between waveguide and lower horizontal slot	g₃	10 nm
Radius of silver NRs	R	10 nm
Number of silver NRs in waveguide	n₁	17
Number of silver NRs in lower horizontal slot	n₂	17
Period of silver NRs	-	4.5R = 45 nm

Parameters	Values	Unit
Plasma frequency ( $ℏ ω_{p}$ )	9.01	eV
Collision frequency ( $Γ_{0}$ )	0.048	eV
Oscillator strength ( $f_{n}$ )	[0.845; 0.065; 0.124; 0.011; 0.840; 5.646]	-
Damping frequency ( $Γ_{n}$ )	[0.048; 3.886; 0.452; 0.065; 0.916; 2.419]	eV
Resonant frequency ( $ω_{n}$ )	[0; 0.816; 4.481; 8.185; 9.083; 20.29]	eV

Case	Resonant wavelength shift (nm)	Pressure sensitivity (nm/MPa)
No NRs	57.71	15.76
R = 2.5 nm	59.43	16.23
R = 5 nm	67.52	18.44
n₁ = 3	74.28	20.29
n₁ = 10	80.3	21.93
n₂ = 3	77.77	21.24
n₂ = 10	80.98	22.12

Parameter	Symbol	Value
Waveguide width	W	50 nm
Thickness of top silver layer	h	100 nm
Horizontal slot length	l	800 nm
Horizontal slot width	w₁	40 nm
Vertical slot width	w₂	40 nm
Gap between two vertical slots	g₁	100 nm
Gap between two horizontal slots	g₂	100 nm
Gap between waveguide and lower horizontal slot	g₃	10 nm
Radius of silver NRs	R	10 nm
Number of silver NRs in waveguide	n₁	17
Number of silver NRs in lower horizontal slot	n₂	17
Period of silver NRs	-	4.5R = 45 nm

Parameters	Values	Unit
Plasma frequency ( $ℏ ω_{p}$ )	9.01	eV
Collision frequency ( $Γ_{0}$ )	0.048	eV
Oscillator strength ( $f_{n}$ )	[0.845; 0.065; 0.124; 0.011; 0.840; 5.646]	-
Damping frequency ( $Γ_{n}$ )	[0.048; 3.886; 0.452; 0.065; 0.916; 2.419]	eV
Resonant frequency ( $ω_{n}$ )	[0; 0.816; 4.481; 8.185; 9.083; 20.29]	eV

Metal-insulator-metal waveguide-based optical pressure sensor embedded with arrays of silver nanorods

Abstract

1. Introduction

2. Setup and MIM theory

3. Numerical analysis and discussion

4. Pressure sensing

5. Influence of NRs on the proposed structure

6. Conclusion

Acknowledgment

Disclosures

Data availability

References

Data availability

Cited By

Figures (9)

Tables (5)

Equations (8)

Optics Express

Reference	Sensitivity (nm/MPa)	Year
This work	25.4	2021
[35]	16.5	2018
[34]	8.5	2016
[4]	5	2020
[29]	4.29	2018
[28]	3.42	2008
[33]	2.12	2016
[27]	1.47	2012
[26]	1.198	2020
[25]	0.0117	2012
[24]	0.0116	2020

Infiter Tathfif	https://orcid.org/0000-0002-3123-8383
Ahmad Azuad Yaseer	https://orcid.org/0000-0003-2836-6920
Kazi Sharmeen Rashid	https://orcid.org/0000-0002-3546-1427