Any electromagnetic surface wave guided by the planar interface of a metal and a dielectric material is called a surface-plasmon-polariton (SPP) wave^[1,2]. SPP waves are important due to their applications in optical sensing^[3,4], imaging^[5,6], and communication^[7,8]. The electromagnetic fields of an SPP wave are strong on and in the proximity of the interface but decay away from the interface. This localization property makes them useful for optical sensors because these surface waves are sensitive to small changes in the electromagnetic properties of the partnering dielectric material near the interface. Surface-plasmonic (i.e., SPP-wave-based) sensors can thus be used to sense molecules in analytes, pollutants, and small concentrations of proteins or assays in a solution^[3,9,10].

The surface-plasmonic sensors operating in the angular interrogation mode^[1,3] measure the change in the direction of propagation of an incident plane wave that excites the SPP wave. However, the SPP wave cannot be excited merely by illuminating a metallic film on top of the partnering dielectric material. The excitation of an SPP wave is due to a resonance phenomenon^[11] that is engendered by a match of the SPP wavenumber to the magnitude of the component of the wavevector of the incident plane wave parallel to the interface plane. This match has to be achieved using prisms^[12–14], waveguides^[15], or surface-relief gratings^[1,16,17]. The grating-coupled configuration is particularly attractive since direct illumination of the partnering dielectric material can be used to excite SPP waves. The grating-coupled configuration can even be used for multiple excitations of an SPP wave^[18,19], thereby providing the opportunity to enhance the reliability and sensitivity of the sensor. The higher reliability is due to the fact that two or more manifestations of surface-plasmonic resonance can be used to sense the same analyte. Therefore, SPP-wave-based sensing using the grating-coupled configuration has been studied extensively^[20–24].

The optical characteristics of both the metallic and dielectric partnering materials affect the characteristics of the SPP waves that can be guided by the interface. In the sensing application, the dielectric material plays a critical role not just because the fluid-to-be-sensed usually infiltrates it, but also because of the variety of choices available for it. The partnering dielectric material can be either isotropic^[1,2] or anisotropic^[1,25] and either homogeneous^[1,2] or nonhomogeneous^[1,26]. The usual choice is an isotropic dielectric material^[3,9]. However, anisotropic partnering dielectric materials^[27,28] offer flexibility in designing optical sensors because the permittivity dyadic has more than one scalar parameter to tune the sensitivity.

Therefore, we chose a biaxial dielectric material for this paper. A biaxial dielectric material that is also porous is a columnar thin film (CTF), which is an ensemble of parallel nanocolumns grown by physical vapor deposition^[29–31]. The inter-columnar void regions of a CTF have to be infiltrated with the fluid to be sensed^[27,28].

The plan of this paper is as follows: the boundary-value problem for the grating-coupled configuration is briefly discussed in Section 2, with detailed treatment being available elsewhere^[19]. Numerical results are presented and discussed in Section 3, and concluding remarks are provided in Section 4. An $\exp (-i\omega t)$ dependency on time $t$ is used, with $\omega$ as the angular frequency and $i=\sqrt{-1}$. The free-space wavenumber is denoted by $k_0=\omega \sqrt{{\mu }_0{\epsilon }_0}$ and the free-space wavelength by ${\lambda }_0=2\pi /k_0$, where ${\epsilon }_0$ is the permittivity and ${\mu }_0$ is the permeability of free space. All vectors are in boldface and dyadics are underlined twice. The unit vectors in the Cartesian coordinate system are identified by ${\widehat{\mathbf{u}}}_{\mathrm{x}}$, ${\widehat{\mathbf{u}}}_{\mathrm{y}}$, and ${\widehat{\mathbf{u}}}_{\mathrm{z}}$.

A schematic of the boundary-value problem is shown in Fig. 1. The region $0<z<L_{\mathrm{c}}$ is occupied by a CTF (whether infiltrated with a fluid or not), the region $L_{\mathrm{c}}+L_{\mathrm{g}}<z<L_{\mathrm{c}}+L_{\mathrm{g}}+L_{\mathrm{m}}$ by a metal of relative permittivity ${\epsilon }_{\mathrm{m}}$, and the half-spaces $z<0$ and $z>L_{\mathrm{c}}+L_{\mathrm{g}}+L_{\mathrm{m}}$ are vacuous. The intermediate region $L_{\mathrm{c}}<z<L_{\mathrm{c}}+L_{\mathrm{g}}$ is occupied by a one-dimensional metallic grating with the CTF inside the troughs of the grating. The $xz$ plane is the grating plane because the grating profile is wholly describable in this plane, and $L$ is the period along the $x$ axis.

The as-deposited CTF is made of a material of refractive index $n_{\mathrm{s}}$, and a fluid of refractive index $n_{\mathrm{L}}$ is present in the void regions of the CTF. The relative permittivity dyadic of the CTF can be written as^[18]$${\underline{\underline{\epsilon }}}_{\mathrm{CTF}}={\underline{\underline{S}}}_{\mathrm{z}}·{\underline{\underline{S}}}_{\mathrm{y}}·({\epsilon }_{\mathrm{a}}{\widehat{\mathbf{u}}}_{\mathrm{z}}{\widehat{\mathbf{u}}}_{\mathrm{z}}+{\epsilon }_{\mathrm{b}}{\widehat{\mathbf{u}}}_{\mathrm{x}}{\widehat{\mathbf{u}}}_{\mathrm{x}}+{\epsilon }_{\mathrm{c}}{\widehat{\mathbf{u}}}_{\mathrm{y}}{\widehat{\mathbf{u}}}_{\mathrm{y}})·{\underline{\underline{S}}}_y^{-1}·{\underline{\underline{S}}}_z^{-1},$$(1)where the principal relative permittivity scalars ${\epsilon }_{\mathrm{a},\mathrm{b},\mathrm{c}}$ depend on $n_{\mathrm{s}}$, $n_{\mathrm{L}}$, and the porosity of the CTF^[32]. The dyadic $${\underline{\underline{S}}}_{\mathrm{y}}=({\widehat{\mathbf{u}}}_{\mathrm{x}}{\widehat{\mathbf{u}}}_{\mathrm{x}}+{\widehat{\mathbf{u}}}_{\mathrm{z}}{\widehat{\mathbf{u}}}_{\mathrm{z}})\cos \chi +({\widehat{\mathbf{u}}}_{\mathrm{z}}{\widehat{\mathbf{u}}}_{\mathrm{x}}-{\widehat{\mathbf{u}}}_{\mathrm{x}}{\widehat{\mathbf{u}}}_{\mathrm{z}})\sin \chi +{\widehat{\mathbf{u}}}_{\mathrm{y}}{\widehat{\mathbf{u}}}_{\mathrm{y}}$$(2)involves $\chi \in (0,\pi /2]$ as the inclination angle of the nanocolumns of the CTF with respect to the $xy$ plane, and the dyadic $${\underline{\underline{S}}}_{\mathrm{z}}=({\widehat{\mathbf{u}}}_{\mathrm{x}}{\widehat{\mathbf{u}}}_{\mathrm{x}}+{\widehat{\mathbf{u}}}_{\mathrm{y}}{\widehat{\mathbf{u}}}_{\mathrm{y}})\cos \gamma +({\widehat{\mathbf{u}}}_{\mathrm{y}}{\widehat{\mathbf{u}}}_{\mathrm{x}}-{\widehat{\mathbf{u}}}_{\mathrm{x}}{\widehat{\mathbf{u}}}_{\mathrm{y}})\sin \gamma +{\widehat{\mathbf{u}}}_{\mathrm{z}}{\widehat{\mathbf{u}}}_{\mathrm{z}}$$(3)indicates that the morphologically significant plane^[31] of the CTF is rotated by an angle $\gamma \in [0,\pi ]$ about the $z$ axis with respect to the $xz$ plane.

The relative permittivity dyadic ${\underline{\underline{\epsilon }}}_{\mathrm{g}}(x,z)={\underline{\underline{\epsilon }}}_{\mathrm{g}}(x\pm L,z)$ in the intermediate region $L_{\mathrm{c}}<z<L_{\mathrm{c}}+L_{\mathrm{g}}$ is specified as $$\begin{gathered} {\underline{\underline{\epsilon }}}_{\mathrm{g}}(x,z)=\{\begin{array}{ll}{\epsilon }_{\mathrm{m}}\underline{\underline{I}}-({\epsilon }_{\mathrm{m}}\underline{\underline{I}}-{\underline{\underline{\epsilon }}}_{\mathrm{CTF}})U[L_{\mathrm{g}}+L_{\mathrm{c}}-z-g(x)],\hfill & x\in [0,L_1),\hfill \\ {\underline{\underline{\epsilon }}}_{\mathrm{CTF}},\hfill & x\in (L_1,L],\hfill \end{array} \\ z\in (L_{\mathrm{c}},L_{\mathrm{c}}+L_{\mathrm{g}}), \end{gathered}$$(4)where $L_1\in (0,L]$, the identity dyadic $\underline{\underline{I}}={\widehat{\mathbf{u}}}_{\mathrm{x}}{\widehat{\mathbf{u}}}_{\mathrm{x}}+{\widehat{\mathbf{u}}}_{\mathrm{y}}{\widehat{\mathbf{u}}}_{\mathrm{y}}+{\widehat{\mathbf{u}}}_{\mathrm{z}}{\widehat{\mathbf{u}}}_{\mathrm{z}}$, and the unit step function $$U(\xi )=\{\begin{array}{cc}1,& \xi \ge 0,\\ 0,& \xi <0.\end{array}$$(5)Although there is no restriction in the theory on the shape of the corrugations, we chose the grating shape function $$g(x)=L_{\mathrm{g}}\sin (\pi x/L_1),$$(6)for all numerical results presented here.

Without loss of generality, the interface $z=0$ is illuminated by a plane wave propagating at the polar angle $\theta$ with respect to the $z$ axis and propagating in the $xz$ plane. Although the incident plane wave can be arbitrarily polarized, we fixed it to be $p$ polarized (i.e., ${\mathbf{E}}_{\mathrm{inc}}\mathbf{·}{\widehat{\mathbf{u}}}_{\mathrm{y}}=0$) because that polarization state is commonly used in SPP-wave-based sensors. Since the plane of incidence (i.e., the $xz$ plane) coincides with the grating plane, the electromagnetic field reflected in the half-space $z<0$ and the electromagnetic field transmitted in the half-space $z>L_{\mathrm{c}}+L_{\mathrm{g}}+L_{\mathrm{m}}$ are independent of $y$.

We used the rigorous coupled-wave approach (RCWA)^[1,33,34] to calculate the absorptance $A_{\mathrm{p}}$ of the metal-CTF structure as a function of the incidence angle $\theta$ of a $p$-polarized plane wave^[19]. In the RCWA, the electric and magnetic field phasors everywhere are expanded as infinite series of Floquet harmonics of both $p$- and $s$-polarization states. For the chosen problem, the Floquet harmonics of orders $n\in \{0,\pm 1,\pm 2,\pm 3\dots \}$ express the $x$ dependence of the field phasors using $\exp [i{k}_x^{(n)}x]$, where $$k_x^{(n)}=k_0\sin \theta +n2\pi /L.$$(7)The relative permittivity dyadic is expanded as a Fourier series with respect to $x$ for every $z\in (0,L_{\mathrm{c}}+L_{\mathrm{g}}+L_{\mathrm{m}})$ and substituted in the frequency-domain Maxwell curl postulates. The result is an infinite number of coupled ordinary differential equations. These are truncated so that Floquet harmonics of orders $|n|>N_{\mathrm{t}}+1$, $N_{\mathrm{t}}\ge 1$, are ignored, and a finite number of resulting ordinary differential equations are then solved by applying the piecewise-uniform approximation^[1] in the region $0<z<L_{\mathrm{c}}+L_{\mathrm{g}}+L_{\mathrm{m}}$. Specular and non-specular reflectances and transmittances of orders $n\in [-N_{\mathrm{t}},N_{\mathrm{t}}]$ are determined using a stable algorithm, from which the absorptance is obtained by applying the principle of conservation of energy^[18,19]. When the thickness $L_{\mathrm{m}}$ significantly exceeds the skin depth^[35] in the chosen metal, the transmitted electric and magnetic fields are negligibly small in magnitude. Care must be taken to ensure that convergent results are obtained as $N_{\mathrm{t}}$ is increased from unity.

The sensor considered in this paper essentially estimates the change in the refractive index $n_{\mathrm{L}}$ of the fluid infiltrating the CTF because of changes in the relative permittivity scalars ${\epsilon }_{\mathrm{a},\mathrm{b},\mathrm{c}}$. These three scalars were numerically estimated using a homogenization formalism. There are several homogenization formalisms, including the Maxwell Garnett formalism^[36], the Bragg–Pippard formalism^[29], and the Bruggeman formalism^[31]. The Bruggeman formalism is more reliable and widely used in optics^[37,38] because it treats all constituent materials equally, unlike the other two formalisms. We also used the Bruggeman formalism in this work to estimate ${\epsilon }_{\mathrm{a},\mathrm{b},\mathrm{c}}$ as functions of $n_{\mathrm{L}}$^[32].

Made of a material of refractive index $n_{\mathrm{s}}$, each nanocolumn of the CTF was represented as a string of electrically small ellipsoids with semi-axes in the ratio $1:{\gamma }_{\mathrm{b}}:{\gamma }_{\tau }$ so that their shape is characterized by the dyadic^[31]$$\underline{\underline{U}}={\underline{\underline{S}}}_{\mathrm{z}}·{\underline{\underline{S}}}_{\mathrm{y}}({\widehat{\mathbf{u}}}_{\mathrm{z}}{\widehat{\mathbf{u}}}_{\mathrm{z}}+{\gamma }_{\tau }{\widehat{\mathbf{u}}}_{\mathrm{x}}{\widehat{\mathbf{u}}}_{\mathrm{x}}+{\gamma }_{\mathrm{b}}{\widehat{\mathbf{u}}}_{\mathrm{y}}{\widehat{\mathbf{u}}}_{\mathrm{y}})·{\underline{\underline{S}}}_y^{-1}·{\underline{\underline{S}}}_z^{-1},$$(8)with ${\gamma }_{\mathrm{b}}$ in the vicinity of unity and ${\gamma }_{\tau }\gg 1$, as shown in Fig. 1. During the deposition of a CTF, collimated vapor of the evaporated material is incident on a planar substrate at an angle ${\chi }_{\mathrm{v}}\in (0,\pi /2]$ with respect to the substrate plane. This vapor condenses on the substrate in the form of nanocolumns inclined at an angle $\chi \ge {\chi }_{\mathrm{v}}$. We selected the uninfiltrated (i.e., $n_{\mathrm{L}}=1$) CTF to have been made by evaporating tantalum oxide with^[39]$$\{\begin{array}{c}{\epsilon }_{\mathrm{a}}=[1.1961+1.5439\nu -0.7719{\nu }^2{]}^2\\ {\epsilon }_{\mathrm{b}}=[1.4600+1.0400\nu -0.5200{\nu }^2{]}^2\\ {\epsilon }_{\mathrm{c}}=[1.3532+1.2296\nu -0.6148{\nu }^2{]}^2\\ \chi =\arctan (3.1056\tan {\chi }_{\mathrm{v}})\end{array},$$(9)where $\nu =2{\chi }_{\mathrm{v}}/\pi$. Adopting the inverse Bruggeman formalism devised for CTFs^[32], we set ${\gamma }_{\tau }=15$ and computed the parameters $n_{\mathrm{s}}$, $f$, and ${\gamma }_{\mathrm{b}}$ as functions of ${\chi }_{\mathrm{v}}$ using Eq. (9).

In order to numerically explore the grating-coupled excitation of SPP waves for sensing, we fixed ${\chi }_{\mathrm{v}}=15\mathrm{deg}$; hence, $\chi =39.77\mathrm{deg}$. Furthermore, the inverse Bruggeman formalism yielded $n_{\mathrm{s}}=2.2999$, $f=0.4439$, and ${\gamma }_{\mathrm{b}}=2.4322$ for ${\chi }_{\mathrm{v}}=15\mathrm{deg}$. These data were then employed in the forward Bruggeman formalism^[32] to find ${\epsilon }_{a,b,c}$ as functions of $n_{\mathrm{L}}>1$.

As mentioned previously, the basic principle of a surface-plasmonic sensor is sensing the change in the incidence angle $\theta$, where an SPP wave is excited when the refractive index $n_{\mathrm{L}}$ of the infiltrating fluid changes. The excitation of the SPP wave can be best inferred by identifying those peaks in the angular spectrum of $A_{\mathrm{p}}$ that do not change location on the $\theta$ axis when the thickness of the partnering dielectric material is changed above a threshold value^[40], since SPP waves are localized to their interface. The angular locations of the thickness-independent absorptance peaks must be matched against the SPP waves that are solutions of the underlying canonical boundary-value problem^[26,41]. In this canonical problem, only a single interface between the two partnering materials occupying half-spaces is present to rule out the excitation of waveguide modes^[16,42]. Therefore, we present the solution of the underlying canonical problem before the data calculated for the grating-coupled surface-plasmonic sensor.

In this canonical problem, one half-space is occupied by the fluid-infiltrated CTF, whereas a metal occupies the other half-space^[43]. Let the SPP wave propagate in the interface plane parallel to the unit vector ${\widehat{\mathbf{u}}}_{\mathrm{x}}$. The electric and magnetic field phasors vary spatially as $\exp [i(qx+\alpha z)]$, with $\mathrm{Im}(\alpha )<0$ in the fluid-infiltrated CTF and $\mathrm{Im}(\alpha )>0$ in the metal so that the field phasors decay as $|z|\to \infty$. The complex-valued wavenumber $q$ yields the phase speed and attenuation rate in the direction of propagation. A combination of search and Newton–Raphson methods^[44,45] was employed to solve the dispersion equation for SPP waves in order to determine the corresponding values of $q$. We assumed the metal as silver with relative permittivity ${\epsilon }_{\mathrm{m}}=-15.4+0.4i$^[46] and that the CTF is made by evaporating tantalum oxide^[39]; furthermore, $\gamma =30\mathrm{deg}$ and ${\lambda }_0=633\mathrm{nm}$.

Only one solution of the dispersion equation was found for any value of $n_{\mathrm{L}}$. Thus, only one SPP wave propagating along the $x$ axis can be excited, although it can have multiple excitations in the grating-coupled configuration^[18,19].

The real and imaginary parts of the relative wavenumber $q/k_0$ of the SPP wave propagating along the $x$ axis are presented in Figs. 2(a) and 2(b), respectively, as functions of the refractive index $n_{\mathrm{L}}$. These plots show an approximately linear relationship between $q$ and $n_{\mathrm{L}}$, which is desirable for a good sensor.

To delineate the excitation of the SPP wave in the grating-coupled surface-plasmonic sensor as a function of the fluid refractive index $n_{\mathrm{L}}$, we computed the absorptance $A_{\mathrm{p}}$ as a function of the incidence angle $\theta$ using the RCWA. We fixed $N_{\mathrm{t}}=15$ after checking that $A_{\mathrm{p}}$ converged within a tolerance limit of $\pm 0.1\%$. As in Sections 3.1 and 3.2, we fixed ${\lambda }_0=633\mathrm{nm}$, ${\chi }_{\mathrm{v}}=15\mathrm{deg}$, $\gamma =30\mathrm{deg}$, ${\epsilon }_{\mathrm{m}}=-15.4+0.4i$, and ${\gamma }_{\tau }=15$. Furthermore, we fixed $L_1=0.5L$, $L_{\mathrm{m}}=30\mathrm{nm}$, and $L_{\mathrm{g}}=20\mathrm{nm}$, but $L_{\mathrm{c}}$ was kept variable between 1000 and 4000 nm.

The plots in Fig. 3 present $A_{\mathrm{p}}$ as a function of $\theta$ for $L_{\mathrm{c}}\in \{1000,2000,3000,4000\}\mathrm{nm}$ and $n_{\mathrm{L}}\in \{\mathrm{1,1.27,1.37,1.43,1.70}\}$ when $L=500\mathrm{nm}$. Either one, two, or three absorptance peaks are present in each angular spectrum. The absorptance peaks with thickness-independent locations on the $\theta$ axis were correlated with the data available in Fig. 2. For this correlation, we decided that $|1-k_x^{(n)}/\mathrm{Re}(q)|\le 0.05$ for some $n\in [-N_{\mathrm{t}},N_{\mathrm{t}}]$ at an absorptance peak attributed to the excitation of an SPP wave as a Floquet harmonic of order $n$^[19].

When $n_{\mathrm{L}}=1$ (i.e., air infiltrates the CTF), Fig. 3(a) shows that the SPP wave with $q=(1.6561+0.0037i)k_0$ is excited at (i)${\theta }_{\mathrm{p}}\simeq 24.4\mathrm{deg}$ because of the in-plane wavenumber $k_x^{(1)}=1.7131k_0$ of the Floquet harmonic of order $n=+1$ and(ii)${\theta }_{\mathrm{p}}\simeq 62.6\mathrm{deg}$ because of the in-plane wavenumber $k_x^{(-2)}=1.7122k_0$ of the Floquet harmonic of order $n=-2$

which match $\mathrm{Re}(q)$ well according to the 5%-criterion adopted by us. This double excitation of the SPP wave is advantageous for reliable sensing, with a schema relying on artificial neural networks^[47].

The absorptance spectra in Fig. 3(b) illustrate the excitation of an SPP wave for $n_{\mathrm{L}}=1.27$ at ${\theta }_{\mathrm{p}}\simeq 40.5\mathrm{deg}$, when the in-plane wavenumber $k_x^{(1)}=1.9494k_0$ of the Floquet harmonic of order $n=+1$ matches the solution of the canonical problem with $q=(1.8753+0.0056i)k_0$. There is no evidence for the second excitation of the SPP wave for $n_{\mathrm{L}}=1.27$.

However, Fig. 3(c) again shows that the excitation of the SPP wave is possible for two values of the incidence angle as two different Floquet harmonics for $n_{\mathrm{L}}=1.37$. The first excitation occurs at ${\theta }_{\mathrm{p}}\simeq 34.6\mathrm{deg}$, when the in-plane wavenumber $k_x^{(-2)}=2.0322k_0$ of the Floquet harmonic of order $n=-2$ matches the canonical solution with $q=(1.9618+0.0064i)k_0$. The second excitation occurs at ${\theta }_{\mathrm{p}}\simeq 46.7\mathrm{deg}$, when $k_x^{(1)}=2.0278k_0$ of the Floquet harmonic of order $n=+1$ is a good match. When $n_{\mathrm{L}}=1.43$, the absorptance spectra in Fig. 3(d) demonstrate that the SPP wave with $q=(2.0142+0.0069i)k_0$ is excited at (i)${\theta }_{\mathrm{p}}\simeq 30.1\mathrm{deg}$ because of the in-plane wavenumber $k_x^{(-2)}=2.0985k_0$ of the Floquet harmonic of order $n=-2$ and(ii)${\theta }_{\mathrm{p}}\simeq 52.6\mathrm{deg}$ because of the in-plane wavenumber $k_x^{(1)}=2.0944k_0$ of the Floquet harmonic of order $n=+1$

which match $\mathrm{Re}(q)$ reasonably well.

Finally, when $n_{\mathrm{L}}=1.70$, Figs. 3(e) and 3(f) demonstrate that the SPP wave with $q=(2.2555+0.0097i)k_0$ is excited at (i)${\theta }_{\mathrm{p}}\simeq 16.5\mathrm{deg}$ because of the in-plane wavenumber $k_x^{(-2)}=2.3160k_0$ of the Floquet harmonic of order $n=-2$,(ii)${\theta }_{\mathrm{p}}\simeq 22.9\mathrm{deg}$ because of the in-plane wavenumber $k_x^{(-2)}=2.2109k_0$ of the Floquet harmonic of order $n=-2$, and(iii)${\theta }_{\mathrm{p}}\simeq 65.7\mathrm{deg}$ because of the in-plane wavenumber $k_x^{(1)}=2.2114k_0$ of the Floquet harmonic of order $n=+1$

which match $\mathrm{Re}(q)$ reasonably well. Contained in this triple excitation of the SPP wave is a doublet: the same SPP wave is excited at two different values of $\theta$ but as the same Floquet harmonic ($n=-2$ when $n_{\mathrm{L}}=1.70$). We have observed that the excitation at one angle of incidence is less efficient than at the other in a doublet. In Fig. 3(e), the doublet appears at ${\theta }_{\mathrm{p}}\simeq 16.5\mathrm{deg}$ and ${\theta }_{\mathrm{p}}\simeq 22.9\mathrm{deg}$ with higher $A_{\mathrm{p}}$ and, therefore, stronger excitation at ${\theta }_{\mathrm{p}}\simeq 16.5\mathrm{deg}$ than at ${\theta }_{\mathrm{p}}\simeq 22.9\mathrm{deg}$. Evidence of the doublet in the grating-coupled configuration has already been reported^[18]. The triple excitation of the SPP wave is going to be even more advantageous for reliable sensing than double excitation, in a schema relying on artificial neural networks^[47].

The results of Fig. 3 allow us to conclude that, as $\theta$ is varied, the SPP wave can be multiply excited, depending upon the value of the refractive index $n_{\mathrm{L}}$ of the infiltrating fluid. In order to examine the effect of $n_{\mathrm{L}}$ in detail, Fig. 4 shows the angular spectra of $A_{\mathrm{p}}$ when $L_{\mathrm{c}}=3000\mathrm{nm}$ and $L=500\mathrm{nm}$ for diverse values of $n_{\mathrm{L}}$; all other parameters are the same as that mentioned at the beginning of Section 3.3.

Figure 4(a) contains two absorptance peaks indicating SPP-wave excitation when $n_{\mathrm{L}}\in [\mathrm{1.00,1.20}]$. For each $n_{\mathrm{L}}$, one peak is for $n=+1$ when $k_x^{(1)}\simeq \mathrm{Re}(q)$, and the other peak is for $n=-2$ when $k_x^{(-2)}\simeq \mathrm{Re}(q)$, where $q$ is the wavenumber of the possible SPP wave gleaned from Fig. 2. Figure 4(b) has a solitary absorptance peak signifying the excitation of the SPP wave as a Floquet harmonic of order $n=+1$ when $n_{\mathrm{L}}\in [\mathrm{1.21,1.29}]$. A similar absorptance peak for $n=-2$ is absent, and we found that double excitation of the SPP wave is not possible for $n_{\mathrm{L}}\in [\mathrm{1.21,1.29}]$. When $n_{\mathrm{L}}=1.30$, the absorptance peak for $n=+1$ is not present in Fig. 4(c). Two absorptance peaks for each value of $n_{\mathrm{L}}$ appear again in Figs. 4(c) and 4(d) when $n_{\mathrm{L}}\in [\mathrm{1.33,1.50}]$: one peak for $n=+1$ when $k_x^{(1)}\simeq \mathrm{Re}(q)$ and the second peak for $n=-2$ when $k_x^{(-2)}\simeq \mathrm{Re}(q)$. The shifts in the angular locations of the two absorptance peaks indicate that these peaks begin far apart from each other from small values of $n_{\mathrm{L}}$ and come closer as $n_{\mathrm{L}}$ increases. At intermediate values of $n_{\mathrm{L}}$, the peaks merge, and only one peak is observed. When $n_{\mathrm{L}}$ increases further, the single peak divides into two peaks that get farther apart when $n_{\mathrm{L}}$ is increased further.

To analyze the usefulness of the peaks for optical sensing, we computed the sensitivity as $$S=\frac{\mathrm{\Delta }{\theta }_{\mathrm{p}}}{\mathrm{\Delta }{n}_{\mathrm{L}}},$$(10)where ${\theta }_{\mathrm{p}}$ is the $n_{\mathrm{L}}$-dependent angular location of an absorptance peak, and $\mathrm{\Delta }{\theta }_{\mathrm{p}}$ is the change in ${\theta }_{\mathrm{p}}$ when the refractive index of the infiltrating fluid changes by $\mathrm{\Delta }{n}_{\mathrm{L}}$. The sensitivity was computed from the absorptance plots for the excitation of the SPP wave as a Floquet harmonic of order $n\in \{-2,1\}$ and is presented in Fig. 5 as a function of $n_{\mathrm{L}}$ for $L_{\mathrm{c}}=3000\mathrm{nm}$ and $L=500\mathrm{nm}$. Additionally, $S$ was computed from the canonical problem by solving $\mathrm{Re}(q)=k_0\sin {\theta }_{\mathrm{p}}+2n\pi /L$ for ${\theta }_{\mathrm{p}}$ as a function of $n_{\mathrm{L}}$ and then using Eq. (10).

The predicted sensitivity and the sensitivity computed from the absorptance spectra are in good agreement. From Fig. 5, we observe that the sensitivities of the absorptance peaks corresponding to $n=+1$ are higher than those of the absorptance peaks corresponding to $n=-2$.

So far, we have presented the results in an analytical sense that tell us the angular location ${\theta }_{\mathrm{p}}$ of an absorptance peak (that indicates the excitation of an SPP wave) when we know $n_{\mathrm{L}}$. However, in practice, we have to accomplish the reverse task, i.e., find the value of $n_{\mathrm{L}}$ from the knowledge of the angular location of the peak absorptance. To make this easier, Fig. 6 shows ${\theta }_{\mathrm{p}}$ as a function of $n_{\mathrm{L}}$ for both types of absorptance peaks in Fig. 4. Once the angular spectrum of absorptance has been measured for an unknown fluid, we can find the angular locations of the absorptance peaks and use those locations to find $n_{\mathrm{L}}$ from Fig. 6. The requirement of matching two values of ${\theta }_{\mathrm{p}}$ (for many values of $n_{\mathrm{L}}$) with one value of $n_{\mathrm{L}}$ makes the measurement of the refractive index more reliable than the case when only one absorptance peak is present.

There is only one absorptance peak indicating SPP-wave excitation for $n_{\mathrm{L}}\in [\mathrm{1.21,1.31}]\cup [\mathrm{1.92,2.21}]$, two such absorptance peaks for $n_{\mathrm{L}}\in [\mathrm{0.3,1.20}]\cup [\mathrm{1.32,1.60}]\cup [\mathrm{1.68,1.69}]\cup [\mathrm{1.75,1.91}]$, and three absorptance peaks for $n_{\mathrm{L}}\in [\mathrm{1.61,1.67}]\cup [\mathrm{1.70,1.74}]$. When three absorptance peaks are possible, two of those peaks form a doublet because both of those peaks satisfy the 5% criterion for the same $n$^[18]. The doublet exists for $n=+1$ when $n_{\mathrm{L}}\in [\mathrm{1.61,1.67}]$ and for $n=-2$ when $n_{\mathrm{L}}\in [\mathrm{1.70,1.74}]$.

The $n_{\mathrm{L}}$-ranges for single, double, and triple excitation of the SPP wave depend upon the value of the grating period $L$. Thus, for $L=600\mathrm{nm}$, we determined that single excitation occurs for $n_{\mathrm{L}}\in [\mathrm{1.30,1.31}]\cup [\mathrm{1.92,2.45}]$, double excitation for $n_{\mathrm{L}}\in [\mathrm{0.3,1.29}]\cup [\mathrm{1.46,1.91}]$, and triple excitation for $n_{\mathrm{L}}\in [\mathrm{1.32,1.45}]$. Likewise, for $L=700\mathrm{nm}$, single excitation occurs for $n_{\mathrm{L}}=1.33$, double excitation for $n_{\mathrm{L}}\in [\mathrm{1.30,1.32}]\cup [\mathrm{1.95,2.50}]$, and triple excitation for $n_{\mathrm{L}}\in [\mathrm{0.3,1.29}]\cup [\mathrm{1.90,1.94}]$. Therefore, $L$ should be chosen to obtain double or triple excitation for the suspected range of $n_{\mathrm{L}}$ for a certain fluid.

Figure 6 indicates that multiple excitations can result in ambiguity when determining $n_{\mathrm{L}}$. For instance, if ${\theta }_{\mathrm{p}}\simeq 27\mathrm{deg}$ and ${\theta }_{\mathrm{p}}\simeq 58\mathrm{deg}$ are found for a sample, then both $n_{\mathrm{L}}=1.05$ and $n_{\mathrm{L}}=1.50$ are possible according to Fig. 6. The ambiguity can be eliminated by repeating the experiment after diluting the sample. Another way to eliminate the ambiguity is by incorporating $A_{\mathrm{p}}$-versus-$\theta$ data for a wide enough $\theta$ range in a schema comprising an artificial neural network^[47]. Yet another way may be to use two or more sensors with different values of the grating period $L$.

Before concluding this section, we must address two issues. First, the air/CTF/metal structure can function as an open-face waveguide^[42,48,49], whose modes can also be used for sensing an infiltrant fluid. However, as the propagation characteristics of a waveguide mode will depend strongly on the CTF thickness $L_{\mathrm{c}}$, the angular location of an absorptance peak due to the excitation of that waveguide mode will be highly susceptible to a change in $L_{\mathrm{c}}$. In contrast, the angular location of an absorptance peak due to the excitation of an SPP wave guided by the metal/CTF interface is weakly dependent on $L_{\mathrm{c}}$ (beyond a threshold value)^[28], which confers the advantage of reliability against manufacturing variabilities. Second, although the air/CTF interface could guide surface waves^[1], we were unable to find any pertinent solutions of the relevant canonical boundary-value problem^[50,51] for the chosen CTF, whether infiltrated or not.

An optical sensor was theoretically analyzed for the plane-wave illumination of a CTF on top of a one-dimensional metallic surface-relief grating. The incident plane wave was taken to be $p$ polarized, and the plane of incidence coincided with the grating plane but not necessarily with the morphologically significant plane of the CTF. The absorptance was computed as a function of the angle of incidence for different thicknesses of the CTF, using the RCWA. The thickness-independent absorptance peaks were identified, and the in-plane wavenumbers of the possible Floquet harmonics were compared with the wavenumber of the SPP wave predicted by the associated canonical boundary-value problem. The change in the angular location of the absorptance peak representing SPP-wave excitation as a function of the refractive index of the fluid infiltrating the CTF was determine to find the sensitivity.

Double and triple excitations of the same SPP wave were found to be possible, depending on the refractive index of the fluid, which can help increase the reliability of results by sensing the same fluid with more than one excitation of the SPP wave, possibly with a schema that incorporates artificial neural networks. In multiple excitations, the same SPP wave is excited as Floquet harmonics of various orders. It is even possible that the excitation occurs at different angles of incidence but as the Floquet harmonic of the same order; however, all excitations are not going to be equally efficient. The theoretical sensitivity reported here can be as high as 230 deg/RIU, which shows that higher sensitivity can be achieved using the grating-coupled configuration than a prism-coupled configuration^[27,28,52].