1 Introduction

Sensing the stress has many application including health monitoring as many mechanical stimuli in the human body, such as blood pressure, muscles movement, pulse contain important health parameter signals, which can be easily captured by flexible wearable stress sensors (Cheng et al. 2020).

The manipulation of surface Electromagnetic (EM) waves has garnered increasing attention across a diverse range of applications, including optical filtering (Wen et al. 2020; Shaukat et al. 2020), biomedical application (Lin et al. 2021; Antenna Al-Gburi et al. 2021) and terahertz applications (Qi et al. 2020).

Two primary categories of EM surfaces exist: Surface Electromagnetic Waves (SEMW) and Nanostructure Surfaces (NS). Table 1 provides a comparison between the two.

SEMW primarily deals with electromagnetic waves that are guided by interfaces between two or more distinct media. In contrast, Nanostructure Surfaces are usually assemblies of identical elements, arranged periodically in one- or two-dimensional arrays.

Recent studies have explored these surfaces under various nomenclatures, such as Surface Plasmons (Yesudasu et al. 2021), frequency selective surface (FSS) (Anand and Mittal 2020), Spoof surface plasmon polariton (SPP) (Ren et al. 2021), as shown in Table 1 and Fig. 1.

Table 1 Categorizing EM surfaces
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Fig. 1
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Categorizing EM surfaces according to medium material

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FSS, in particular, are engineered surfaces consisting of periodically patterned thin metallic layers. Their periodic structure enables them to function as frequency filters for propagating waves with comparable wavelengths. FSS finds common usage in EM wave filtering applications like band-pass, band-stop, and perfect absorbers (Anwar et al. 2018). Notably, FSS designs can achieve transmittance that is both angle and polarization-independent, as they operate not on surface modes but on cavity angle-independent modes (Van Labeke et al. 2006).

Using origami fold/unfold tuning transforms a single-band static FSS into a dual-band dynamic FSS (Biswas et al. 2020). This technique enhances the degrees of freedom by exciting both symmetric and anti-symmetric modes due to strong coupling and the appropriate orientation of the elements.

Frequency range considerations play a role in FSS modeling. At low frequencies, the plasmonic metal can be approximated as a perfect electric conductor (PEC) with negligible resistance. In contrast, at high frequencies, the metal’s frequency-dependent parameters become significant.

Recently, multi-layers FSS has attracted increasing attention as it allows more degrees of freedom and flexibility for modeling, and various applications including antennas, wireless and radar applications (Yao et al. 2022; Al-Gburi et al. 2021; Sengupta et al. 2023; Pandhare et al. 2023).

Various techniques underpin color filtering, such as plasmonic color filtering (PCF), dielectric color filtering (DCF), nanowires (NW), and quantum dots (QD) (Shaukat et al. 2020; Gildas and Dan 2019). Thanks to Surface Plasmon Polaritons (SPP), hole-array PCFs feature sub-wavelength holes and extraordinary transmission, exceeding the diffraction limit of light. PCFs offer high-resolution and integration compatibility, and their frequency response can be tuned by modifying the size, pattern, or material (Wen et al. 2020).

In contrast to PCFs, dielectric-based color filters (DCF) minimize the absorption of visible light by metallic layers. DCFs operate based on Mie resonance and have lower losses (Lee et al. 2018). However, they face practical limitations due to material performance and fabrication challenges.

Re-configurable intelligent surfaces (RIS) serve as programmable interfaces for controlling EM wave propagation (Tapio et al. 2021). These surfaces can replace conventional antenna arrays, offering advantages like energy efficiency and simplified transceiver architecture, particularly in next-generation communications (beyond 5 G) (Liaskos et al. 2019; Zhao 2019).

Reconfiguration can be achieved through various techniques, as shown in Fig. 2 . Electrical control mechanisms include PIN diodes, varactors, and piezoelectric elements (Fan et al. 2020; Sirci et al. 2011; Lee et al. 2021; Motovilova and Huang 2020). Mechanical-based RISs do not rely on electrical signals and can be adjusted using actuators or manual reconfiguration (Mirmozafari et al. 2021; McMichael 2018; Pavia et al. 2020). Although slower, mechanical reconfiguration may be preferable in environments with high electrical noise or where electrical connectivity is undesirable.

Fig. 2
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Categorization of reconfiguration techniques

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Distinct from existing stress sensor designs utilizing reconfigurable filters (Pavia et al. 2020), our proposed model focuses on stress applied perpendicularly to the FSS surface, rather than laterally, as illustrated in Fig. 3. Applying stress in the normal direction preserves the filter’s lateral shape and dimensions, while only altering the properties of the inter-space medium. This approach facilitates more straightforward fabrication processes and enables more controlled stress sensing through the elasticity properties of the material in use. Furthermore, stress applied in this manner does not disrupt the design of the unit cell, thereby mitigating a significant source of element loss commonly found in electronically reconfigurable RIS.

Fig. 3
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(a) Lateral force, (b) Normal force

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Plasmonic nanostructures have been modelled by different tools, either numerically by finite difference time domain (FDTD) or finite element method (FEM) (Mansouri et al. 2021; Malkiel et al. 2018; Haidari 2022), or analytically by atomistic modeling (Lafiosca et al. 2021), Green function integral method (Søndergaard 2007), density function theory (DFT) (Manzhos et al. 2021) or equivalent circuit method (EC) (Das and Varshney 2022; Sadafi et al. 2021; Zhu et al. 2020) . In our study, FEM method is used by COMSOL simulation for its accurate and reliable simulation deals with different structures, besides, equivalent circuit model is chosen for its simplicity yet sufficiency in estimating the main parameters of the transmission spectrum.

Aerogels constitute a unique class of materials, characterized by a continuous 3D porous network. They have garnered considerable interest for their thermal, electrical, and mechanical properties. Traditional aerogels are generally fabricated from inorganic polymers such as silica, clay, and graphene (Abdullah et al. 2022; Zhao et al. 2018). While they offer benefits like low density, large porosity, high specific surface area, and low thermal conductivity, they are limited by their lack of flexibility, extensibility, and fragile network structures (Tafreshi et al. 2022).

In recent developments, next-generation aerogels have emerged with enhanced features. For example, aerogel fibers show improved stretchability, flexibility, tensile strength, and extensibility (Du et al. 2020; Li and Zhang 2022). Nanocellulose-based aerogels, fabricated from abundant and sustainable cellulose, combine the high porosity and large specific surface area of traditional aerogels with the impressive mechanical properties of cellulose (Chen et al. 2021). The main objectives of this research are to design a double-layer tunable infrared filter that leverages the properties of aerogels and to validate this design through numerical simulations.

Aerogel was used as elastic-electrochemical dielectric layer for sensitive pressure sensors, in 2021 (Long et al. 2021). The study didn’t depend on the transmission change by FSS structure, but on the capacitive-pressure dependence and achieved sensitivity about \(0.9 KPa^{-1}\) for a stress range \(<3 KPa\). Also, strain sensor was designed based on the FSS reflection characteristics (Wang et al. 2023). In the study, the strain degree was estimated by measuring the shift in the original frequency and achieved, experimental, a sensor sensitivity of 1.28GHz/mm when the FSS is stretched by \(0-3\,mm\) and analytically a sensitivity of 200MHz for \(1.64\%\) radial expansion.

SPP sensors based on aerogels exhibit high sensitivity such as, a metal grating with period of 740 nm imprinted on an aerogel substrate resulted in a sensitivity of 740.2 nm/RIU (refractive index unit) (Kim et al. 2019).

In the current study, we introduce a design for a double-layer tunable infrared filter. This design controls transmission by adjusting the distance between the two layers. The model relies on numerical simulations, incorporating wavelength-domain electromagnetic analyses to determine the maximum transmission peak and corresponding wavelengths applicable in filtering or stress-sensing designs. In addition to the numerical work, we offer a simplified equivalent circuit model that expedites the estimation of various parameters. The filter configuration comprises a double layer of plasmonic frequency-selective surface (FSS) made of Indium tin oxide (ITO), separated by an aerogel material in the inter-space region and achieves sensitivity of 20nm/MPa with average applied force of \(14 MN/m^2\). The significance of this work lies in its potential to revolutionize stress-sensing applications by offering a flexible, efficient, and accurate filtering mechanism. Moreover, the use of aerogels could pave the way for more sustainable and versatile sensor technologies.

2 Structure and model

The structure is shown in Fig. 4a, where two FSS surfaces of square hole array in indium tin oxide (ITO) layer with an inter-space region filled with Silica Aerogel. The main design parameters are the square hole side length d, gap g, thickness t, and inter-space distance \(t_d\).

In the model, the force is assumed to be exerted in the direction normal to the filter surface, as shown in Fig. 4b.

Fig. 4
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(a)Schematic of double layer FSS, (b) Applied force to control the inter-space distance

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The three-dimensional model is built using numerical COMSOL multi-physics simulator (BV and OY 1998), where the electromagnetic Maxwell’s equations, are solved by finite element method (FEM) for transmission pattern estimation with periodic boundary conditions assuming normal incidence of light. FEM is one of the most efficient method to deal with different structures and is convenient to simulate the proposed design under discussion.

Besides, a simple analytical model is built based of equivalent circuit (EC) model, as previously presented (Ali and Ali 2023). The transmission line EC is widely used in practical engineering application, it uses a circuit component to characterize the operations of the model, the circuit components are the discrete elements (resistors, inductors, and capacitors) that aim to estimate the transmission by calculating the S-parameter of the transmission line. The model calculates the impedance elements for each FSS layer (\(Z_{FSS}\)), taking into consideration the frequency-dependant parameter in infrared spectrum, where the metal region is dominated by the inductive impedance \(Z_{L_1}\) and \(Z_{L_2}\), whereas the air region is dominantly a capacitive impedance \(Z_c\). Besides, the metal in the optical frequencies introduces an additional resistance \(R_1\) and \(R_2\) , as shown in Fig. 5.

Fig. 5
figure 5

(a)Top view of the square hole unit cell, (b) equivalent circuit of the unit cell, reprinted from (Ali and Ali 2023)

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The impedance of the air region is approximated as a parallel plate capacitor impedance as follows:

$$\begin{aligned} Z_c=\dfrac{j}{\omega \epsilon _0 t} \end{aligned}$$
(1)

where \(\omega\) is the angular frequency and \(\epsilon _0\) is the free space permittivity. On the other hand, the impedance in the metal regions is estimated as Zhang et al. (2015):

$$\begin{aligned} Z_{L_1}=j \dfrac{d}{\omega \epsilon _0\epsilon _m g t} \end{aligned}$$
(2)
$$\begin{aligned} Z_{L_2}=j \dfrac{g}{\omega \epsilon _0 \epsilon _m (g+d) t} \end{aligned}$$
(3)

where \(\epsilon _m\) is the frequency-dependant relative permittivity in the ITO metal layer which is estimated using Drude model as Blemker et al. (2020), Johns et al. (2020)

$$\begin{aligned} \epsilon _m=\epsilon _{\infty }-\dfrac{\omega _p^2}{\omega ^2+i\omega \Gamma } \end{aligned}$$
(4)

where \(\omega _p\) is the plasma frequency, \(\epsilon _{\infty }\) is the high-frequency dielectric constant, and \(\Gamma\) is the damping coefficient. The used values are \(\omega _p=2.65\times 10^{15} rad/s\), \(\epsilon _{\infty }=3.9\) and \(\Gamma =2.05\times 10^{14} rad/s\) (Solieman and Aegerter 2006). Then, S-parameters are calculated to estimate both the double layer total reflection and transmission coefficients from Xu and He (2019), Pozar (2011):

$$\begin{aligned} S_{11}=\dfrac{(A+B/Z_0)-(Z_0C+D)}{(A+B/Z_0)+(Z_0C+D)} \end{aligned}$$
(5)
$$\begin{aligned} S_{21}=\dfrac{2}{(A+B/Z_0)+(Z_0C+D)} \end{aligned}$$
(6)

where the transmission parameters A,B,C, and D are calculated from the transfer matrix method TMM as Pozar (2011), Xu and He (2019):.

$$\begin{aligned} \begin{bmatrix} A&{}B\\ C&{}D \end{bmatrix} = \begin{bmatrix} 1&{}0\\ 1/Z_{FSS}&{}1 \end{bmatrix}\begin{bmatrix} cosh(jk_d t_d)&{}Z_csinh(jk_d t_d)\\ sinh(jk_d t_d)/Z_c&{}cosh(jk_d t_d) \end{bmatrix}\begin{bmatrix} 1&{}0\\ 1/Z_{FSS}&{}1 \end{bmatrix} \end{aligned}$$
(7)

where \(Z_0=\sqrt{\mu _0/\epsilon _0}\) is the free space impedance, \(Z_c=Z_0/\sqrt{\epsilon _r}\) is the wave impedance in the dielectric (Aerogel) layer of dielectric constant \(\epsilon _r\) and thickness \(t_d\), and \(k_d=\omega \sqrt{\epsilon _0 \epsilon _r\mu _0}\) is the wave number in it.

Transmission matrix is a sufficient method to estimate the optical transmission, absorption and reflection by studying the propagation of optical waves in multi-layers structure, as shown in Fig. 6 considering \(t_i\) and \(r_i\) are the transmission and reflection coefficients at each surface, respectively.

Fig. 6
figure 6

Schematic of wave propagation in multilayer structure

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Dielectric constant of the materials changes with pressure as the material’s response to the electric field differs. When pressure rises, the ability to polarize in response to the field decreases. As Silica aerogels are highly porous materials where most of its volume consists of air, so that the dielectric constant change with pressure can be ignored. Besides, the new dielectric constant of aerogel can be estimated by \(\epsilon _r'=\epsilon _{air} V_{air}'+\epsilon _{aerogel} V_{aerogel}'\) (Long et al. 2021) where \(V_{air}'\) and \(V_{aerogel}'\) are the volume proportion of both air and aerogel

3 Results

For validation purpose, the model was compared with previously reported single and triple layers FSS filter (Zhang et al. 2015), shown in Fig. 7. The reported analysis is based on the equivalent circuit (EC) model besides a numerical finite difference time domain (FDTD) simulation. For the triple layers model, we chose the separation distance (\(t_d\)) between the layers to be one quarter of the central wavelength of the incident light. This to let the quarter-wavelength region between the layers to act as an admittance inverter converting alternate shunt resonators to series resonators. The square filter dimensions are as follows: the square side length \(d=120 ~nm\), the gap \(g=60 ~nm\), and the height \(t=130 ~nm\)

Fig. 7
figure 7

(a) Single and (b) triple layers FSS transmission estimated by EC model, numerical simulation, reported theoretical model (Zhang et al. 2015), and reported numerical simulation(Zhang et al. 2015)

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The results of both theoretical and numerical models have shown good agreement with the reported ones. Higher deviation is noticed in triple layers FSS EC model at very high wavelength where the conductivity calculated by Drude model is almost constant, this effect is notable in triple layers more than single layer where the coupling is higher between layers.

3.1 Effect of distance \(t_d\) between layers on the transmission

After the validation of the model, different distance between the layers is applied. The model is applied to the structure with constant filter dimension as \(g=100~nm, d=100~ nm, t=100~ nm\), and limit the wavelength at \(\lambda > 800~nm\) in the available range of Drude model with \(\omega _p=2.65\times 10^{15} rad/s\), \(\epsilon _{\infty }=3.9\) and \(\Gamma =2.05\times 10^{14} rad/s\) . The transmission pattern depends on the distance between the two layers as shown in Fig. 8.

The figure shows that both peak wavelength and peak transmission vary with the distance \(t_d\) through which the incident wave travels till transmitted. As will be discussed in Sect. 3.3, the second peak in the transmission pattern is due to surface plasmon (SPP) in each layer, that’s why it is insensitive to the distance between the layers but varies with the hole dimensions. On the other hand, the first peak varies with the distance \(t_d\) as it depends on the diffraction by the hole edges (Fig. 9).

Fig. 8
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Transmission at different distance \(t_d\)

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Fig. 9
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Transmission at different distance \(t_d\) by both FEM and ECM

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As previously reported (Feyz and Neshati 2022; Zhao et al. 2022; Cong et al. 2023), despite the simplicity and speed of EC model, its main disadvantage is the wavelength deviation. This disadvantage is overcomed by the aid of the numerical simulation. The simulator calculations at few points (different values of \(t_d\)) are used to estimate an additional high-order impedance as fitting parameters then EC model becomes sufficient to generate other points in no time.

3.2 Estimating the stress from the transmission pattern

For stress sensor applications, the force applied to the FSS surface is estimated from the transmitted pattern. The sensor design is based on the elasticity properties of the inter-space material (Aerogel) where the transmission depends on the distance between the two surfaces. As the strain (\(\varepsilon\)) results from the subjected stress (Force per unit area) depends on the Young’s Modulus (Elasticity modulus, E) of the inter-space material as follow (Serway and Jewett 2018)

$$\begin{aligned} \varepsilon =Stress/E \end{aligned}$$
(8)
$$\begin{aligned} Stress=F/A \end{aligned}$$
(9)

where A is the normal area to the applied force F.

According to the experimental reported values of aerogel elasticity modulus of 0.042 GPa and yield stress of 0.023 GPa for density \(\rho =0.09 ~g/cm^3\) (Ma et al. 2018), the maximum relative change in the thickness is \(\Delta t/t=0.55\). So, for initial aerogel thickness \(t_d=900~ nm\), the distance between the two filter surfaces may vary between 500 and 1300 nm. Correspondingly, as shown in Fig. 10, exerting an average stress of \(14 ~MN/m^2\) can lead to thickness elongation about 300 nm and relative change in both peak wavelength \(\lambda _{peak}\) and transmission \(T_{peak}\) of \(33\%\) and \(17.3\%\), respectively. Positive force in Fig. 10 represents a tensile force where negative force represents compression

As shown in Fig. 10, both peak wavelength and maximum transmission can be controlled by changing the distance between the two surfaces. The oscillating pattern is explained by modes constructive and destructive interfaces shown by Eq. 7. Different modes are seen in Fig. 11 which shows electric field distribution at different distance \(t_d\) at the same wavelength \(\lambda =1000~nm\) for an example where it is noticed that at \(t_d=1000~nm\), transmitted electric field is higher than other values of \(t_d\).

Considering the stress sensing application, sensitivity of the sensor can be estimated as the ratio between the change in output (transmission peak value and wavelength) to the change in input (stress). The average sensitivity of the transmission peak is about 0.02/MPa and of the wavelength is about 20 nm/MPa.

Fig. 10
figure 10

(a) Peak wavelength and (b) maximum transmission at different distances \(t_d\)

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Fig. 11
figure 11

Electric field distribution at \(\lambda =1000~nm\) at different distance \(t_d\)

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3.3 Sensitivity to the FSS layer dimensions

To investigate the effect of the hole size, periodicity and surface thickness to the transmission pattern, ECM is used while keeping the inter-space distance constant \(t_d = 900 ~nm\) and change each parameter individually .

As shown in Fig. 12a and b, the first peak wavelength \(\lambda _{peak}\) is almost insensitive to the hole length and gap. This matches with the previous reported studies (Wang et al. 2014; Medina et al. 2017; Ali and Ali 2023) where transmission is related to both hole radius and light wavelength by factor \((r/\lambda )^4\) when light scatters through subwavelength apertures in very thin perfect metal film and diffracts at the edges (Bethe 1944). On the other hand, second peak (mode) highly affected by the hole side length and gap because of the electric field concentrations at the hole edges due to SPP, where SPP is stronger when the hole length to periodicity ratio increased (increased hole side length d or decreased gap g).

As shown in Fig. 12c, increasing the metal surface thickness leads to less coupling between the two surfaces SPP and correspondingly less transmission \(T_{peak}\). This is explained according to the Surface plasmon in one FSS layer which is considered insulator-metal-insulator problem (IMI), the electric field is maximum at the interface between metal and dielectric and decays in the direction perpendicular to the interface, the field decays by factor of 1/e at the penetration depth inside the metal (Homola 2006). So, when increasing the thickness of the metal films (thicker than penetration depth), the SPP consists of two weakly coupled propagating plasmons on the two surfaces of the metal film.

Fig. 12
figure 12

Transmission vs wavelength for (a) different hole side length d while \(g=100~nm, t=100~nm\), (b) different gap g while \(d=100~nm,t=100~nm\) and c) different thickness t while \(d=100~nm, g=100~nm\)

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4 Conclusion

In this study, we introduced a novel reconfigurable infrared filter design that serves as a stress sensor. The design employs a double-layer plasmonic frequency-selective surface (FSS) using indium tin oxide (ITO) with an aerogel material filling the inter-space region. Our approach stands out for its dual-method analysis: we utilized both equivalent circuit theory and numerical simulations to estimate the transmission pattern and achieved a sensitivity of 20nm/MPa for average stress of 14MPa. According to the materials mechanical and electrical properties, our model has some limitation on maximum allowed strain (\(<0.55\)) and spectrum range (\(\lambda >800 nm\)). However, the investigation extended to exploring the impacts of several key parameters, including the inter-space distance and FSS dimensions, as well as the elasticity of the inter-space material, on the transmission and peak wavelength characteristics.

Notably, our design addresses certain limitations present in existing models, offering improved flexibility in fabrication and reduced element loss. It also provides a comprehensive framework for optimizing the filter’s response based on the material properties of aerogel, adding a new dimension to stress sensing applications.

As future work, it would be beneficial to explore the applicability of this design in various environmental conditions and to validate it through experimental studies. Overall, our research contributes to the growing body of knowledge on reconfigurable filters and paves the way for innovations in stress sensing technologies.