To describe the optical response of our proposed system, we developed a theoretical framework based on a nine-layer-stack model (Figure 2), which combines our effective medium theory with multi-layer Fresnel reflection scheme [32], [41]. The relative permittivities of layers 1 and 9, representing the media through which light enters and exits the FPI device, are denoted by *ε*_{1} and *ε*_{9}, respectively. These two layers may be identical or different, depending on the application. Layers 2 and 8 represent the thin metallic film electrodes of thickness *h*_{f}, which are denoted by frequency-dependent permittivities *ε*_{2}(*ω*) and *ε*_{8}(*ω*), respectively. Layers 3 and 7, representing the spacer layers of thickness *h*_{s} between the NPs and their substrates (thin metallic films), are assigned permittivities *ε*_{3} and *ε*_{7}, respectively. Layers 4 and 6 emulate the monolayer of NPs, which are modeled as effective films, each of thickness *d*, with frequency-dependent permittivities *ε*_{4}(*ω*) and *ε*_{6}(*ω*), respectively. Layer 5 denotes the medium filling in the FPI cavity, with relative permittivity *ε*_{5}.

Figure 2: (A) Schematic of the proposed Fabry–Perot interferometer cavity and (B) its equivalent theoretical nine-layer-stack model. Parameters: *R*, radius of the NPs; *g*, inter-NP separation; *h*_{s}, thickness of the spacer layer; *h*_{f}, thickness of the metallic film electrodes; *L*, cavity length; *d*, thickness of the effective film emulating an NP monolayer; and *k*, the wave vector of light incident at angle *θ*.

For simplicity, here we consider *ε*_{1}=*ε*_{9}, *ε*_{2}(*ω*)=*ε*_{8}(*ω*), *ε*_{3}=*ε*_{7}, and *ε*_{4}(*ω*)=*ε*_{6}(*ω*). In specific calculations we will assume *ε*_{3}=*ε*_{7}=*ε*_{5}, as the separation *h*_{s} is mostly determined by NP-capping ligands that are embedded in the electrolytic solution. The critical step then is to determine *ε*_{4}(ω) and *ε*_{6}(*ω*) for estimating the optical response of each monolayer of NPs coupled to a thin metallic electrode. Within quasi-static dipolar approximation, the monolayer can be modeled as an effective film with anisotropic dielectric permittivity, which is determined by the optical polarizability of NPs, considering their mutual interactions within the array in the presence of the metallic substrate.

While forming the monolayer, the functionalized NPs, when substantially charged, would prefer to self-assemble in a hexagonal lattice. With *a* (=2*R*+*g*) as the lattice constant (where *R* is the NP’s radius and *g* is the inter-NP gap), we obtain the parallel and perpendicular components of the dielectric tensor for the effective film of thickness $d=\hspace{0.17em}\frac{4\pi {R}^{3}}{3{a}^{2}}$ as

$${\epsilon}_{6}^{\parallel}\mathrm{(}\omega \mathrm{)}=\hspace{0.17em}{\epsilon}_{5}+\hspace{0.17em}\frac{8\pi}{\surd 3{a}^{2}d}{\beta}^{\parallel}\mathrm{(}\omega \mathrm{)},$$(1a)

$$\frac{1}{{\epsilon}_{6}^{\perp}\mathrm{(}\omega \mathrm{)}}=\frac{1}{{\epsilon}_{5}}-\frac{1}{{\epsilon}_{5}^{2}}\frac{8\pi}{\surd 3{a}^{2}d}{\beta}^{\perp}\mathrm{(}\omega \mathrm{)},$$(1b)

respectively [32]. Here, *β*^{||,⊥} (*ω*) denotes the effective quasi-static dipolar polarizability of each NP in a monolayer while interacting with all other NPs of the 2D array along with their dipolar images on the metallic substrate, and is given by

$${\beta}^{\parallel}\mathrm{(}\omega \mathrm{)}\text{\hspace{0.17em}}=\frac{\alpha \mathrm{(}\omega \mathrm{)}}{1+\alpha \mathrm{(}\omega \mathrm{)}\frac{1}{{\epsilon}_{5}}\hspace{0.17em}\left[\frac{-1}{2}\frac{{U}_{A}}{{a}^{3}}\hspace{0.17em}-\xi \hspace{0.17em}\mathrm{(}\frac{f\mathrm{(}h,\text{\hspace{0.17em}}a\mathrm{)}}{{a}^{3}}\hspace{0.17em}-\frac{3}{2}\frac{{g}_{1}\mathrm{(}h,\text{\hspace{0.17em}}a\mathrm{)}}{{a}^{3}}\hspace{0.17em}+\frac{1}{8{h}^{3}}\mathrm{)}\right]},$$(2a)

$${\beta}^{\perp}\mathrm{(}\omega \mathrm{)}=\frac{\alpha \mathrm{(}\omega \mathrm{)}}{1+\alpha \mathrm{(}\omega \mathrm{)}\frac{1}{{\epsilon}_{5}}\hspace{0.17em}\left[\frac{{U}_{A}}{{a}^{3}}\hspace{0.17em}-\xi \hspace{0.17em}\mathrm{(}\frac{f\mathrm{(}h,\text{\hspace{0.17em}}a\mathrm{)}}{{a}^{3}}\hspace{0.17em}-12\frac{{h}^{2}{g}_{2}\mathrm{(}h,\text{\hspace{0.17em}}a\mathrm{)}}{{a}^{5}}\hspace{0.17em}-\frac{1}{4{h}^{3}}\mathrm{)}\right]},$$(2b)

where $\alpha \mathrm{(}\omega \mathrm{)}={\epsilon}_{5}{R}^{3}\frac{{\epsilon}_{\text{NP}}\mathrm{(}\omega \mathrm{)}-{\epsilon}_{5}}{{\epsilon}_{\text{NP}}\mathrm{(}\omega \mathrm{)}+2{\epsilon}_{5}}$ is the dipolar polarizability of an individual spherical NP of material with permittivity *ε*_{NP} (*ω*) surrounded by a medium with permittivity *ε*_{5}; $\xi =\frac{{\epsilon}_{5}-{\epsilon}_{\text{8}}\mathrm{(}\omega \mathrm{)}}{{\epsilon}_{5}+{\epsilon}_{\text{8}}\mathrm{(}\omega \mathrm{)}}\hspace{0.17em}$ is the image charge screening factor; and *U*_{A}, *f*(*h*, *a*), *g*_{1}(*h*, *a*), and *g*_{2}(*h*, *a*) are the “lattice sums” (over a hexagonal lattice), given by

$${U}_{A}={\displaystyle \sum}_{i\ne 0}{\displaystyle \sum}_{j\ne 0}\frac{1}{{\mathrm{(}{i}^{2}+{j}^{2}-ij\mathrm{)}}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}=11.031,$$(3a)

$$f\mathrm{(}h,\text{\hspace{0.17em}}a\mathrm{)}={\displaystyle \sum}_{i\ne 0}{\displaystyle \sum}_{j\ne 0}\frac{1}{{\mathrm{(}{i}^{2}+{j}^{2}-ij+{\mathrm{(}\frac{2h}{a}\mathrm{)}}^{2}\mathrm{)}}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}},$$(3b)

$${g}_{1}\mathrm{(}h,\text{\hspace{0.17em}}a\mathrm{)}={\displaystyle \sum}_{i\ne 0}{\displaystyle \sum}_{j\ne 0}\frac{\mathrm{(}{i}^{2}+{j}^{2}\mathrm{)}}{{\mathrm{(}{i}^{2}+{j}^{2}-ij+{\mathrm{(}\frac{2h}{a}\mathrm{)}}^{2}\mathrm{)}}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}},$$(3c)

$${g}_{2}\mathrm{(}h,\text{\hspace{0.17em}}a\mathrm{)}={\displaystyle \sum}_{i\ne 0}{\displaystyle \sum}_{j\ne 0}\frac{1}{{\mathrm{(}{i}^{2}+{j}^{2}-ij+{\mathrm{(}\frac{2h}{a}\mathrm{)}}^{2}\mathrm{)}}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}},$$(3d)

where *h*=*h*_{s}+*R*. Note that, in Eqs. (2a) and (2b), the term *U*_{A} in the denominator essentially incorporates the plasmonic interactions of each individual NP with all other NPs in the 2D hexagonal array. Considering a “reference” NP positioned at the origin, these calculations of lattice sums are carried out. The terms in the denominator, shown within parentheses, are multiplied by *ξ*, which accounts for the image-charge interactions effect. The functions *f*(*h*, *a*), *g*_{1}(*h*, *a*), and *g*_{2}(*h*, *a*) sum up the coupling effects originating from the dipolar images of the all other NPs in the array on the “reference” NP. The last term in the parentheses with 1/*h*^{3} dependence includes the effects coming from the own dipolar image charges of the NP.

With the knowledge of permittivity for all the layers in our nine-layer-stack model, we then obtain the transfer matrix $\tilde{\text{M}}\text{,}$ which allows the calculation of optical transmittance through the FPI. By defining the wave vectors at each layer as

$${k}_{1}\mathrm{(}\omega \mathrm{)}=\frac{\omega}{c}\sqrt{{\epsilon}_{1}}\mathrm{cos}\theta ,$$(4a)

$${k}_{2}\mathrm{(}\omega \mathrm{)}=\frac{\omega}{c}\sqrt{{\epsilon}_{2}\mathrm{(}\omega \mathrm{)}-{\epsilon}_{1}{\text{sin}}^{2}\theta},$$(4b)

$${k}_{3}\mathrm{(}\omega \mathrm{)}=\frac{\omega}{c}\sqrt{{\epsilon}_{3}\mathrm{(}\omega \mathrm{)}-{\epsilon}_{1}{\text{sin}}^{2}\theta},$$(4c)

$${k}_{4}^{\parallel}\mathrm{(}\omega \mathrm{)}=\frac{\omega}{c}\sqrt{{\epsilon}_{4}^{\parallel}\mathrm{(}\omega \mathrm{)}-{\epsilon}_{1}{\text{sin}}^{2}\theta},$$(4d)

$${k}_{4}^{\perp}\mathrm{(}\omega \mathrm{)}=\frac{\omega}{c}{\mathrm{(}\frac{{\epsilon}_{4}^{\parallel}\mathrm{(}\omega \mathrm{)}}{{\epsilon}_{4}^{\perp}\mathrm{(}\omega \mathrm{)}}\mathrm{)}}^{1/2}\sqrt{{\epsilon}_{4}^{\perp}\mathrm{(}\omega \mathrm{)}-{\epsilon}_{1}{\text{sin}}^{2}\theta},$$(4e)

$${k}_{5}\mathrm{(}\omega \mathrm{)}=\frac{\omega}{c}\sqrt{{\epsilon}_{5}\mathrm{(}\omega \mathrm{)}-{\epsilon}_{1}{\text{sin}}^{2}\theta},$$(4f)

$${k}_{6}^{\parallel}\mathrm{(}\omega \mathrm{)}=\frac{\omega}{c}\sqrt{{\epsilon}_{6}^{\parallel}\mathrm{(}\omega \mathrm{)}-{\epsilon}_{1}{\text{sin}}^{2}\theta},$$(4g)

$${k}_{6}^{\perp}\mathrm{(}\omega \mathrm{)}=\frac{\omega}{c}{\mathrm{(}\frac{{\epsilon}_{6}^{\parallel}\mathrm{(}\omega \mathrm{)}}{{\epsilon}_{6}^{\perp}\mathrm{(}\omega \mathrm{)}}\mathrm{)}}^{1/2}\sqrt{{\epsilon}_{6}^{\perp}\mathrm{(}\omega \mathrm{)}-{\epsilon}_{1}{\text{sin}}^{2}\theta},$$(4h)

$${k}_{7}\mathrm{(}\omega \mathrm{)}=\frac{\omega}{c}\sqrt{{\epsilon}_{7}\mathrm{(}\omega \mathrm{)}-{\epsilon}_{1}{\text{sin}}^{2}\theta},$$(4i)

$${k}_{8}\mathrm{(}\omega \mathrm{)}=\frac{\omega}{c}\sqrt{{\epsilon}_{8}\mathrm{(}\omega \mathrm{)}-{\epsilon}_{1}{\text{sin}}^{2}\theta},$$(4j)

$${k}_{9}\mathrm{(}\omega \mathrm{)}=\frac{\omega}{c}\sqrt{{\epsilon}_{9}\mathrm{(}\omega \mathrm{)}-{\epsilon}_{1}{\text{sin}}^{2}\theta},$$(4k)

and the phase shifts as *δ*_{2}=*k*_{3}*h*_{f}, *δ*_{3}=*k*_{3}*h*_{s}, ${\delta}_{4}^{\mathrm{(}\parallel ,\perp \mathrm{)}}=\hspace{0.17em}{k}_{4}^{\mathrm{(}\parallel ,\perp \mathrm{)}}d,$ ${\delta}_{5}=\hspace{0.17em}{k}_{5}\mathrm{(}L-2{h}_{\text{s}}-2d\mathrm{)},$ ${\delta}_{6}^{\mathrm{(}\parallel ,\perp \mathrm{)}}=\hspace{0.17em}{k}_{6}^{\mathrm{(}\parallel ,\perp \mathrm{)}}d,$ *δ*_{7}=*k*_{7}*h*_{s}, *δ*_{8}=*k*_{8}*h*_{f}.

A transfer matrix for the entire system can be evaluated as

$$\begin{array}{l}\tilde{\text{M}}=\frac{1}{{t}_{1,2}}\left(\begin{array}{cc}{e}^{-i{\delta}_{2}}& {r}_{1,2}{e}^{i{\delta}_{2}}\\ {r}_{1,2}{e}^{-i{\delta}_{2}}& {e}^{i{\delta}_{2}}\end{array}\right)\cdot \frac{1}{{t}_{2,3}}\left(\begin{array}{cc}{e}^{-i{\delta}_{3}}& {r}_{2,3}{e}^{i{\delta}_{3}}\\ {r}_{2,3}{e}^{-i{\delta}_{3}}& {e}^{i{\delta}_{3}}\end{array}\right)\cdot \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{1}{{t}_{3,4}}\left(\begin{array}{cc}{e}^{-i{\delta}_{4}}& {r}_{3,4}{e}^{i{\delta}_{4}}\\ {r}_{3,4}{e}^{-i{\delta}_{4}}& {e}^{i{\delta}_{4}}\end{array}\right)\cdot \frac{1}{{t}_{4,5}}\left(\begin{array}{cc}{e}^{-i{\delta}_{5}}& {r}_{4,5}{e}^{i{\delta}_{5}}\\ {r}_{4,5}{e}^{-i{\delta}_{5}}& {e}^{i{\delta}_{5}}\end{array}\right)\cdot \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{1}{{t}_{5,6}}\left(\begin{array}{cc}{e}^{-i{\delta}_{6}}& {r}_{5,6}{e}^{i{\delta}_{6}}\\ {r}_{5,6}{e}^{-i{\delta}_{5}}& {e}^{i{\delta}_{6}}\end{array}\right)\cdot \frac{1}{{t}_{6,7}}\left(\begin{array}{cc}{e}^{-i{\delta}_{7}}& {r}_{6,7}{e}^{i{\delta}_{7}}\\ {r}_{6,7}{e}^{-i{\delta}_{7}}& {e}^{i{\delta}_{7}}\end{array}\right)\cdot \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{1}{{t}_{7,8}}\left(\begin{array}{cc}{e}^{-i{\delta}_{8}}& {r}_{7,8}{e}^{i{\delta}_{8}}\\ {r}_{7,8}{e}^{-i{\delta}_{5}}& {e}^{i{\delta}_{8}}\end{array}\right)\cdot \frac{1}{{t}_{8,9}}\left(\begin{array}{cc}1& {r}_{8,9}\\ {r}_{8,9}& 1\end{array}\right),\end{array}$$(5)

where for s- and p-polarized light, the reflection and transmission coefficients, ${r}_{i,j}^{\mathrm{(}\text{s},\text{p}\mathrm{)}}$ and ${t}_{i,j}^{\mathrm{(}\text{s},\text{p}\mathrm{)}},$ at any interface between layers *i* and *j*, read

$${r}_{ij}^{\text{s}}=\frac{{k}_{i}^{\parallel}\mathrm{(}\omega \mathrm{)}-{k}_{j}^{\parallel}\mathrm{(}\omega \mathrm{)}}{{k}_{i}^{\parallel}\mathrm{(}\omega \mathrm{)}+{k}_{j}^{\parallel}\mathrm{(}\omega \mathrm{)}},$$(6a)

$${r}_{ij}^{\text{p}}=\frac{{\epsilon}_{i}^{\parallel}{k}_{j}^{\perp}\mathrm{(}\omega \mathrm{)}-{\epsilon}_{j}^{\parallel}{k}_{i}^{\perp}\mathrm{(}\omega \mathrm{)}}{{\epsilon}_{i}^{\parallel}{k}_{j}^{\perp}\mathrm{(}\omega \mathrm{)}+{\epsilon}_{j}^{\parallel}{k}_{i}^{\perp}\mathrm{(}\omega \mathrm{)}},$$(6b)

$${t}_{ij}^{\text{s}}=\frac{2{k}_{i}^{\parallel}\mathrm{(}\omega \mathrm{)}}{{k}_{i}^{\parallel}\mathrm{(}\omega \mathrm{)}+{k}_{j}^{\parallel}\mathrm{(}\omega \mathrm{)}},$$(7a)

$${t}_{ij}^{\text{p}}=\frac{2\sqrt{{\epsilon}_{i}^{\parallel}}\sqrt{{\epsilon}_{j}^{\parallel}}{k}_{i}^{\perp}\mathrm{(}\omega \mathrm{)}}{{\epsilon}_{i}^{\parallel}{k}_{j}^{\perp}\mathrm{(}\omega \mathrm{)}+{\epsilon}_{j}^{\parallel}{k}_{i}^{\perp}\mathrm{(}\omega \mathrm{)}}.$$(7b)

The transmission coefficient *t*^{(s,p)} from the nine-layer-stack model can be found as ${t}^{\mathrm{(}\text{s},\text{p}\mathrm{)}}=\hspace{0.17em}\frac{1}{{\tilde{\text{M}}}_{11}}.$ Finally, the percentage transmittance *T*^{(s,p)} is calculated using

$${T}^{\mathrm{(}\text{s},\text{p}\mathrm{)}}=100\times \text{|}{t}^{\mathrm{(}\text{s},\text{p}\mathrm{)}}{\text{|}}^{2}\times \hspace{0.17em}\frac{{n}_{9}}{{n}_{1}}\frac{\mathrm{cos}{\theta}_{t}}{\mathrm{cos}{\theta}_{i}}\hspace{0.17em},$$(8)

where ${\theta}_{t}=\hspace{0.17em}{\mathrm{sin}}^{-1}\mathrm{(}\frac{{n}_{1\hspace{0.17em}}\mathrm{sin}{\theta}_{i}}{{n}_{4}}\hspace{0.17em}\mathrm{)}$ is the angle of transmission, *θ*_{i} is the angle of incidence, ${n}_{1}\hspace{0.17em}\mathrm{(}=\hspace{0.17em}\sqrt{{\epsilon}_{1}}\mathrm{)},$ and ${n}_{9}\hspace{0.17em}\mathrm{(}=\hspace{0.17em}\sqrt{{\epsilon}_{9}}\mathrm{)}\hspace{0.17em}.$

Throughout this work, we deployed spherical silver NPs and thin silver electrodes, with silver (Ag) permittivity values taken from the literature [42], while considering the cavity to be filled with an aqueous solution. Unless otherwise specified explicitly, light is assumed to enter and exit the FPI through the aqueous medium. The spacer layer is considered to have a thickness *h*_{s}=2 nm. With this configuration, the results obtained using this framework are presented in the subsequent sections.

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