1. Introduction

Evanescent-wave absorption sensing using integrated photonics is a very promising approach for the miniaturization of sensor elements and their integration in small electronic devices. Whenever the sensing concept is based on the absorption of guided light from surrounding molecules, the performance of the sensor is strictly related to the portion of the electromagnetic energy that is in contact with the analyte and can be absorbed and to the possible enhancement of the radiation-matter interaction, which results from a reduced group velocity.

Within the last years, an extensive amount of scientific effort focused on the design and optimization of scalable microstructures suitable for selective gas sensing by light absorption in the mid-IR [1,2]. Different platforms have been demonstrated, spanning from germanium on different substrates [3], including silicon [4], silicon-on-insulator [5,6] and silicon nitride [7], to germanium alloys [8], low-stress silicon nitride [9], chalcogenide glass [10,11], silicon on sapphire [12,13], aluminum nitride on insulator [14] and silicon-based [15–18], which can be heterogeneously integrated with quantum-cascade-lasers and detectors [19]. In all cases the success of such sensing platforms is subordinated, among others, to the capability of designing and producing suitable waveguides that can confine and guide a significant amount of light for lengths up to the centimetre scale, and can simultaneously concentrate a large fraction of the electric field in the medium to be sensed.

Different quantities have been used to describe and optimize such sensing structures, including the filling factor f, the confinement factor $\mathrm{\Gamma }$ and the evanescent-field ratio EFR (sometimes called also evanescent power factor or power confinement factor) [13,20–23].

In this paper, we discuss the relevant parameters for the optimization of photonic structures for evanescent wave absorption sensing, focusing on the difference between the concepts of confinement factor and evanescent-field ratio. In particular we specify the approximations under which these quantities can be used to describe the sensitivity of the waveguide by evanescent-field absorption, and analyze the conditions when these parameters cannot be considered equivalent. Finally, we quantify the errors in the evaluation of the waveguide performance when using the different parameters for a set of realistic structures, and we compare different approaches to model the waveguide sensitivity using finite-element method (FEM) simulations for a large class of waveguide structures, showing that the relevant parameters can be retrieved very accurately from a series of two 2D simulations, without the need of evaluating the Poynting vector nor the group velocity of the mode.

2. Waveguide sensitivity

Given a dielectric waveguide immersed in an absorbing cladding medium, light absorption at a wavelength $\lambda \; $ can be described by the “modified” Beer-Lambert law

Given a photonic absorption sensor, including a light source, a guiding photonic structure (waveguide) and a detector with known noise-equivalent power (NEP), the interesting parameter for the sensor is its resolution, which is defined as the smallest concentration change $dC$ of the analyte that can be detected:

Accordingly, the best achievable resolution $d{C_{opt}}$ is obtained as

The sensitivity is obtained by evaluating Eq. (5) at a concentration $C = 0$.

Equations (4) and (5) allow to consider two scenarios. First, in absence of other damping mechanisms ($D = 0$) the optimal length of the photonic structure depends only on the concentration of the analyte (and on the detector). This case is completely analogous to standard non-dispersive infrared (NDIR) absorption sensors, where the optimal resolution is obtained when the initial intensity is reduced by a factor 1/e. In this simple case, the most relevant parameter for the photonic structure is $\eta $, since large $\eta $ allows for higher miniaturization of the sensor without loss of resolution [Eq. (4)]

Second, introducing a damping term $D \ne 0$ causes the best achievable resolution to increase. The optimal structure would have D as small as possible and $\eta $ as close as possible to 1. However, D and $\eta $ are physically related quantities, as the higher the delocalization of the electric field outside the dielectric waveguide medium (high $\eta $), the higher is typically the damping, both from absorption from non-completely transparent substrates and from scattering by surface roughness. An optimal waveguide structure, therefore, must provide a good compromise between these two parameters. If the working point for the sensing (i.e., the target substance and concentration) is known, optimization of the resolution and of the sensitivity of the structure reduces to a minimization of the “figure of merit” FOM:

The task is particularly critical in the mid-IR, where most conventional materials present high losses. Very recently, a suspended slab waveguide showing a reduced damping in the mid-IR and $\eta $ as high as 44% has been demonstrated [25]. Another promising approach is using slot waveguides. Slot waveguides are formed by two adjacent strip waveguides separated by a subwavelength gap. The gap can be empty or filled by a material different from the cladding. As the electric field of the first quasi-TE mode is mostly concentrated in the gap, slot waveguide having the slot filled with the analyte are extremely promising to achieve a large sensitivity to changes in the concentration of surrounding fluids. Theoretical design of single slot waveguides with $\eta $ exceeding 40% have been reported [21,26,27].

A final consideration regards the term ${I_0}$ in Eq. (5). It is very intuitive that the higher the incident energy at the detector compared to the NEP, the easier is to observe a small change in signal. This is of course limited by the saturation level of the detector. If with a laser the amount of light intensity in the waveguide is rarely the limiting factor, the situation differs when moving to simpler thermal sources. There is a limited amount of intensity that can be conveyed into a small waveguide starting from an extended thermal source, and in the design of a final sensor, this needs to be considered.

3. Mathematical expressions to describe ${\boldsymbol \eta }$

The effective absorption parameter $\eta $ defined in Eq. (1) is the physical quantity that allows for comparison and optimization of different photonic structures. However, for practical purposes, simulation of $\eta $ from Eq. (1) can be computationally expensive. Therefore, in literature other quantities have been used to describe the capability of the waveguides to sense a surrounding medium by light absorption: (i) the filling factor f, describing the fraction of the electric energy in the cladding to the total electric energy, (ii) the confinement factor $\mathrm{\Gamma }$, corresponding to the ratio of the integrated electric field energy density in the absorbing cladding and the power transported by the mode, and (iii) the evanescent-field ratio or power confinement factor EFR, corresponding to the fraction of the total electromagnetic power located in the medium to be sensed. The purpose of this section is to analyze the approximations under which such quantities can be used at the place of $\eta $ in the FOM.

As we will show, provided the addition of the sensing medium can be considered as a small perturbation to the system, $\eta $ is well represented by the confinement factor $\mathrm{\Gamma }$, while the filling factor f and the EFR are not directly proportional to the waveguide sensitivity.

In a waveguide that is homogeneous along the propagation direction z of the electromagnetic field, the electric field of the mode m along z can be expressed as

The imaginary part of ${\beta _m}$ relates to the propagation-dependent field absorption, since the light intensity is proportional to the square of the electric field

In the following we will omit the index m from the notation, keeping in mind that we always refer to the electromagnetic field of one mode. If we assume that introducing the sensing medium in proximity of the waveguide induces a small perturbation of the refractive index of the cladding $\Delta {n_{clad}}$ corresponding to a change of the relative electric permittivity $\Delta {\varepsilon _{clad}}$, we can calculate the corresponding change in the propagation constant $\Delta \beta $ by using the variation theorem [28]:

Neglecting material dispersion and for negligible changes in the magnetic permeability (e.g., for non-magnetic analytes) expression (9) reduces to

Substituting Eq. (12) in (11) and neglecting the higher order terms

Here we used also the relationship $\omega = {k_0} {\cdot} c$ to relate the vacuum wavevector ${k_0}$ with the angular frequency $\omega $ via the vacuum speed of light c. Since both ${n_{clad}}$ and $\Delta {n_{clad}}$ are assumed homogeneous within the cladding region, Eq. (13) gives an expression for the relative change of the effective refractive index with respect to the original perturbation:

We are looking for the change in absorption experienced by the waveguide mode due to the analyte in the cladding, so we are interested in the imaginary part of $\Delta {n_{eff}}$. Expanding ${n_{clad}}\; \Delta {n_{clad}}$ in Eq. (13) in their real and imaginary parts, four terms are obtained of which two are imaginary, proportional to $[{\textrm{Im}({\Delta {n_{clad}}} )\cdot\textrm{Re}\; ({{n_{clad}}} )} ]$ and $[{\textrm{Im}({{n_{clad}}} )\cdot\textrm{Re}\; ({\Delta {n_{clad}}} )} ],$ respectively. In realistic scenarios, $\textrm{Im}({{n_{clad}}} )\ll \; \textrm{Re}\; ({{n_{clad}}} )\; $ since a highly absorbing cladding degrades the propagation length, and $\textrm{Re}\; ({\Delta {n_{clad}}} )\ll \textrm{Re}\; ({{n_{clad}}} )$ because of the initial assumption that $\Delta {n_{clad}}$ is a small perturbation to ${n_{clad}}$. Therefore, the term $[{\textrm{Im}({{n_{clad}}} )\cdot\textrm{Re}\; ({\Delta {n_{clad}}} )} ]$ can be neglected, and we obtain:

Using the expression for the absorption coefficient due to the presence of the absorbing cladding medium [31]

Thus, $\mathrm{\Gamma }$ is an accurate representation of the effective absorption parameter $\eta $ within the validity of the variation theorem, and provided the additional approximations used in the derivation of Eqs. (11–15) hold. The integral of the Poynting vector at the denominator represents the power transported by the mode along the propagation direction. It can be demonstrated [28,32] that, in lossless and gainless media and within the validity of the variation theorem, this is equivalent to the product of the group velocity of the mode multiplied by the time-averaged electromagnetic energy density $\langle W \rangle$:

Note that high loss or gain are necessary for the approximation to break. By assuming that material dispersion can be neglected (as is usually the case for e.g. gas sensing applications), and remembering that

Defining the effective group index ${n_{g,eff}} = \frac{{{v_g}}}{c}$, the canonical expression for the confinement factor $\mathrm{\Gamma }$ can be retrieved. Summarizing, we obtain the following three expressions for $\mathrm{\Gamma }$:

Only in case ${n_{g,eff}}\; \approx \; {n_{clad}}$, i.e., when the effective group velocity approaches the phase velocity in the cladding, the value of f approaches $\mathrm{\Gamma }$ and the two definition could be used interchangeably. This would imply a linear dispersion in the medium matching that of the cladding, which is however unrealistic.

To complete the discussion, we want to address the evanescent field ratio EFR, which is widely used as a parameter to optimize waveguide-based sensing structures [11,13,22,26,33–39]. The EFR is defined as the fraction of the total electromagnetic power located in the cladding or in the medium to be sensed:

It is easy to show that expression (25) is mathematically equivalent to Eq. (23b) (and thus to $\mathrm{\Gamma }$) within the approximation of plane waves propagating in a homogeneous non-magnetic material, when

The corresponding linear relationship between the x and y component of electric and magnetic field strength allows using interchangeably the concepts of electric energy and stored electromagnetic power. It was shown that, while the approximation is reasonable for large waveguides and waveguides characterized by a low refractive-index contrast, it breaks down for high refractive-index contrast waveguides far from weak-guiding conditions [40].

4. FEM simulations

The FEM simulations described herein have been performed with COMSOL Multiphysics 5.5 using the wave optics module. The simulations focus on silicon waveguides with different geometries (see Fig. 1). For all waveguide types, the substrate was left unchanged and was composed by a 140 nm Si₃N₄ layer lying on a SiO₂ substrate. The height h of the waveguides is set to 660 nm, corresponding to structures, which were investigated recently [24,35]. For the slab waveguide a second height of 800 nm was also modelled. The refractive indexes of the materials were taken from literature (Si [41], SiO₂ [42] and Si₃N₄ [43]). For all simulations, a central wavelength of 4.24 µm was chosen since this is interesting for gaseous CO₂ sensing and allows comparison with experimental values for some of the simulated structures. The simulation domain was chosen large enough, so that no effect from the boundaries on the retrieved mode was seen, and all simulations were carried out at this configuration.

 

Fig. 1. Sketch of the simulated waveguides (grey) on the 140 nm Si₃N₄ membrane (green) and 2 µm thick SiO₂ substrate (yellow).

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The modeled structures are shown in Fig. 1. They are interesting in the context of ongoing investigations [27,35] and are well suited to highlight the aspects discussed in this paper.

The slab waveguide was assumed to be infinitely broad, therefore the related simulations have lower dimensionality than the simulations of all other structures.

For each structure, a modal analysis was performed on the cross-section of the waveguide to obtain the electromagnetic field profiles and the effective refractive indexes of the supported (guided and non-guided) modes. Then 3D simulations were performed for a given mode using the beam-envelope method. In this paper we focus mostly on the lowest order quasi-TE and quasi-TM modes, however the approach can be extended to any other mode of interest. The retrieval of the different parameters is described in detail below.

4.1 Direct simulation of $\eta $

The value of $\eta $ was simulated using a 3D FEM model (except for the slab waveguide, where the simulation was two-dimensional) by exciting the selected mode at the entrance port of the waveguide and propagating it to the exit port, where the total transmittance T was evaluated. The transmittance is used to extract the parameter $\eta $ directly from Eq. (1), remembering that the absorption coefficient $\alpha (\lambda )\; $ relates to the imaginary part of the refractive index $\textrm{Im}({\Delta {n_{clad}}} )$ as expressed in Eq. (17).

In general, at least two different values of ${n_{clad}}$ have to be simulated in order to obtain the difference $\textrm{Im}({\Delta {n_{clad}}} ).$

In our case, the simulation was performed with air as cladding in the sensing region ($\textrm{Im}({{n_{clad}}} )= \; \textrm{Im}({\Delta {n_{clad}}} )= \; 0$) and repeated by adding a diluted absorbing medium (e.g., a gas) to the sensing region. To model gas absorption, in our case CO₂, the real part of ${n_{clad}}$ was left unchanged at 1 and a small imaginary part was introduced. Although only two simulations are necessary, several configurations have been calculated to check the consistency of the procedure, with $\textrm{Im}({\Delta {n_{clad}}} )$ ranging between $1E - 5$ and $1E - 2$, corresponding to relative CO₂ concentrations of about 0.1% and 100%, respectively. No change in the estimated value of $\eta $ was observed up to the 4^th significant digit between these simulations. The values of $\eta $ obtained for the different structures are reported in Table 1, and they can be considered as the reference values, to which the simulations of the factors $\; \mathrm{\Gamma },\; f$ and EFR should be compared to.

Table 1. Effective refractive index (${{ n}_{{ eff}}})$, effective absorption parameter (${ \eta }$), confinement factor (${\Gamma }$) evaluated according to the different expressions in Eq. (22), filling factor (f) and evanescent-field ratio (${\textrm {EFR}}$) evaluated for all structures by FEM simulations according to the description in Section 4.

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4.2 Parameters retrieved via modal analysis

In the following discussion, we will first focus on parameters that can be obtained by 2D simulations.

For the evaluation of the confinement factor $\mathrm{\Gamma }$ according to Eq. (23a), a modal analysis on the waveguide cross-section should be performed for a minimum of two values of ${n_{clad}}$, the difference of which defines $\Delta {n_{clad}}$. Similar to the retrieval of $\eta $, in the present study we chose to analyze up to 9 values of $\Delta {n_{clad}}$, corresponding to an absorption coefficient spanning between 0 and 300 cm⁻¹, in order to verify the stability of the method for a large range of $\Delta {n_{clad}}.$

For the evaluation of the confinement factor from Eq. (23c), a boundary mode analysis should be performed for the desired mode at least at two wavelengths, to allow for the calculation of the effective group index ${n_{g,eff}}$

In this study we performed simulations at several wavelengths between 4.22 and 4.62 µm. For the strip waveguide, the effect of the wavelength step size on the accuracy of the retrieved $\mathrm{\Gamma }$ was investigated, showing that changes in the step size from 10 nm down to 1 nm affect the accuracy of the retrieved value up to 1%. Nevertheless, we typically observe convergence for step sizes of 1 nm. For all modes (since no mode is purely TE or TM), the integral of the electric energy should be calculated taking into account both the tangential and normal component of the electric field.

The filling factor f (Eq. (24)) can be retrieved from a single modal analysis, being just the second term of $\mathrm{\Gamma }$ in Eq. (23c).

The intrinsic damping of the desired mode is obtained automatically from the modal analysis, being the imaginary part of the effective refractive index. In the simulations presented here, the damping arises exclusively from the material parameters, neglecting other loss mechanisms that are typically important for the performance of real devices, in primis scattering due to surface imperfections [21]. Accounting for such effects may be required to fully simulate real waveguide properties, which need to be evaluated on a case-to-case basis depending on the actual fabrication processes, but is not relevant within our paper, where we focus on the basic FOM for evanescent-field absorption sensing and present general examples from defect free structures.

4.3 Parameters retrieved via 3D simulations

For the evaluation of the confinement factor from Eq. (23b), a three-dimensional FEM simulation was performed for the desired mode for the retrieval of the Poynting vector component in the propagation direction. The same is true for the EFR.

5. Results and discussion

We know from Section 3 that the effective absorption parameter $\eta \; $ is related to $\mathrm{\Gamma },\; f$ or EFR within different approximations. The purpose of this section is to quantify the error that we should expect when using a parameter different from $\eta $ as a measure for the expected sensitivity of realistic sensor structures. As outlined above, we focus on silicon waveguides for evanescent-wave absorption sensing, which have been previously discussed and experimentally produced in the context of CO₂ sensing [24,27,35].

Table 1 summarizes the results of the evaluation of the parameters $\eta ,\; \mathrm{\Gamma },\; f$ and EFR for a selection of waveguide structures using FEM simulations as described in Section 4.

As expected, the parameter $\mathrm{\Gamma }$ matches well the parameter $\eta $ for all simulated structures, since introduction of CO₂ at the place of air is indeed a small perturbation to the refractive index of the cladding ($\textrm{Re}({\Delta {n_{clad}}} )\le 1E - 4$ and $\textrm{Im}({\Delta {n_{clad}}} )\le 1E - 2)$. We note here that the 4.26 µm absorption band of CO₂ has a particularly high absorption coefficient compared to most other gases, and that the use of $\mathrm{\Gamma }$ at the place of $\eta $ is justified in general for gas sensing.

For all investigated structures and modes, retrieval of $\mathrm{\Gamma }$ from Eq. (23a) shows an excellent agreement with the simulated $\eta $ up to the fifth significant digit for changes in $\textrm{Im}({\Delta {n_{clad}}} )\; $ up to the maximum investigated value of $1E - 2$. This strategy is therefore very robust for the estimation of $\eta $. Retrieval of $\mathrm{\Gamma }$ from Eq. (23b) provides a similarly accurate description of $\eta $. Retrieval of $\mathrm{\Gamma }$ from Eq. (23c), instead, shows a systematic overestimation of $\eta $. This discrepancy originates from neglecting material dispersion in deriving Eq. (22), which was verified by additional simulations, where all material refractive indexes were assumed frequency-independent. We note here that, a priori, a similar discrepancy should be observed between Eqs. (23a) and (23b), since dispersion was ignored also in the derivation of Eq. (23b). However, in the specific case considered here, it is not observed because the cladding medium is air and, therefore, dispersionless. Such a discrepancy is however expected for other sensing schemes, e.g., in the case of liquid sensing.

We conclude that, in contrast to $\mathrm{\Gamma }$, for all examined structures, neither the filling factor nor the EFR are representative of the factor $\eta $ describing the sensitivity of the structure to its environment.

For what concerns the filling factor, f systematically (strongly) underestimates $\eta $. The discrepancy arises from ignoring the group velocity in expression (23c). A lower group velocity of the propagating mode increases the interaction time between the electromagnetic field and the sensing medium at parity of propagation length. This is at the basis of the well-known slow-light enhancement of light-matter interaction, which manifests itself in photonic structures, among others, as enhancement of gain in active media [44], enhancement of nonlinear effects [45,46] and Raman scattering [47], and enhancement of light absorption [20]. We thus conclude that the filling factor is in general not a good figure of merit for evanescent wave absorption sensing.

Concerning the EFR, the origin of the discrepancy is found, mathematically, in the fact that Eq. (26) only applies to plane waves propagating in a homogeneous material or for pure TE and TM modes of slab waveguides [48]. As discussed extensively by Robinson and coworkers [40], this is of course a poor approximation for the electromagnetic field in sub-wavelength waveguides characterized by a high refractive-index contrast between the core and the substrate/cladding. Such a high refractive-index contrast is characteristic of many materials used to realize integrated waveguides in the infrared, like silicon and germanium. The physical origin of the discrepancy can be understood by looking at Fig. 2, showing the spatial profile of electric, magnetic field and Poynting vector along the propagation direction for the fundamental quasi-TE mode (first row) and quasi-TM mode (second row) of the silicon strip waveguide, and the fundamental quasi-TE mode of the silicon slot waveguide (third row). The electric field and magnetic field are located spatially in different portions of the structure, while the total power is distributed identically between the electric and the magnetic field [28]. At optical frequencies, within the dipole approximation, only the interaction with the electric component of the electromagnetic field can be efficiently absorbed to induce molecular transitions. It follows that light absorption only occurs when the absorbing medium is located in the portion of space where the electric field is high. Estimating the absorbable power as the EFR, i.e., as the fraction of the power flowing along the propagation direction and being in contact with the medium, is equivalent of treating the electric and magnetic field as identically capable of being absorbed by the medium. This will lead to large discrepancies as soon as the magnetic and electric fields do not have similar spatial profiles in the transverse direction.

 

Fig. 2. Electric field (left), magnetic field (middle) and z component of the Poynting vector (right) for the fundamental quasi-TE mode of the strip waveguide (first row), the fundamental quasi-TM mode of the strip waveguide (second row) and the fundamental quasi-TE mode of the slot waveguide (third row). Only a fraction (h = 4.4 µm, w = 5.1 µm) of the simulated domain is shown for the sake of clarity. Note the different spatial location of the electric and magnetic field, and consequently the different profile of the electric field and the Poynting vector. Since molecular absorption depends exclusively on the spatial overlap between the sensing medium and the electric field, evaluation of the factor η using the z component of the Poynting vector results in a significant discrepancy.

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Accordingly, the discrepancy observed between $\eta $ and EFR is the least pronounced for the fundamental quasi-TE mode of the strip waveguide, where the spatial profiles of electric and magnetic field are similar, while it is more pronounced for the quasi-TM mode, where the electric field shows an important evanescent tail outside the waveguide, while the magnetic field is more confined inside the waveguide.

In the case of the slot waveguide, the electric field is mostly located within the gap. This can be understood, as the high refractive-index material is strongly polarized by the electric component of the electromagnetic radiation. When the polarization is along the horizontal direction, it builds a charge on the opposite inner faces of the two strips, creating a strong electric field inside. The magnetic field instead is mostly located within the silicon strips, and as a result, the discrepancy between $\eta $ and EFR exceeds 200% in this case.

Finally, in Table 2 we analyze the figure of merit defined in Eq. (6) using the parameter $\eta $, remembering that it reflects the dependence of the waveguide sensitivity and resolution on the waveguide parameters. Among the considered structures, the TE mode of the slot waveguide presents the lowest FOM, therefore is the best suited for evanescent wave absorption sensing. The last two columns in Table 2 compare the FOM using $\eta $ with the FOM retrieved using f and EFR, respectively. Interestingly, although f is not representative of the effective absorption parameter, the FOM evaluated using f retains mostly the trend among the considered structures, with the only exception of the TM modes of the slot, which is incorrectly evaluated as better than the TM mode of the strip. The situation differs for the FOM evaluated using the EFR, which predicts the slab waveguides as the best structure for the sensing and the slot TE mode to be even worse than the strip TE mode. This is also in contradiction with experimental results, showing the superiority of the strip waveguide to the slab waveguide for CO₂ sensing [24,35].

Table 2. Damping D and FOM evaluated according to Equation (6) using the correct value for the effective absorption parameter ${\eta }$ and the parameters f and EFR for all simulated structures and modes.

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6. Summary and conclusions

In summary, this paper discusses a figure of merit for the optimization of dielectric waveguides for evanescent-wave absorption sensing, focusing on determining which parameters are suitable to correctly describe the optical absorption process and, therefore, the sensitivity of the waveguide to its environment. In particular, we showed that the performance of photonic structures characterized by a high refractive-index contrast between core and cladding, which are typical for mid-infrared absorption sensing, can only poorly be described by the widely used evanescent field ratio (or power-confinement factor) parameter, which is inadequate to represent $\eta $ for all investigated structure types. On the other hand, the confinement factor is always in good agreement with the effective absorption parameter, thanks to the validity of the perturbative approach for realistic variations of the refractive index.

Among the possible calculation strategies for the confinement factor, we show that preference shall be given to the approach of Eq. (23a), which requires exclusively the validity of the perturbative approach. Its simulation requires two modal analyses (2D FEM simulations) and is very robust in the choice of the size of $\Delta {n_{clad}}$. Evaluation of Eqs. (23b) and (23c) is intrinsically less accurate, as they are only valid when dispersion can be neglected. In the examples shown here, Eq. (23b) is in good agreement with expression 23a because the chosen cladding material is air, however discrepancies are expected by using, for example, a liquid cladding as for the case of liquid sensing. The evaluation of expression 23c gives a measure of how large the effects of dispersion for standard materials and waveguide structures are. In general, the issue of the dispersion in Eqs. (23b)–(23c) can be solved by defining new “effective” refractive indexes $\varepsilon ^{\prime} = \frac{{d({\omega {\varepsilon_0}\varepsilon } )}}{{d\omega }}$ at the place of the material refractive indexes and using those for the simulations, as it is necessary, for example, for surface plasmon polariton modes at metal-dielectric interfaces [30,49]. In terms of calculation efficiency, evaluation of Eq. (23c) is comparable to evaluation of Eq. (23a), requiring two modal analyses at different frequencies, however some care is necessary in the choice of the frequency-step size to ensure convergence. Although Eq. (23b) was evaluated here by 3D simulations, we note that for most well-behaved guiding structures it is possible to define the z-component of the Poynting vector also using the electric and magnetic field components retrieved via the 2D FEM simulation. In this case, thus, a single simulation is sufficient to retrieve the confinement factor.

By investigating the sensing FOM for the different waveguide structures, we also observe that, despite both filling factor and EFR being a wrong representation of $\eta $, the FOM obtained using the filling factor is mostly capable of reproducing the correct evaluation of the “goodness” of the various structures, while the EFR produces a wrong classification. This is related to the fact that the filling factor only neglects the contribution of the effective group index in the calculation of $\mathrm{\Gamma }$. The effective group index is quite similar for all the structures investigated here, and its contribution can only minimally alter the ratio of the FOMs. Nevertheless, the situation is expected to be different for more complex structures, like photonic crystals, where the group velocity of the mode (and thus the effective group index) can be significantly affected by the structure geometry.

For the considered structures, the error committed using the EFR at the place of the effective absorption parameter $\eta $ is important enough to lead to a complete misevaluation of the sensitivity of the structures. This is expected for such structures, due to the strong difference between the spatial location of the electric field responsible for molecular absorption and the transported electromagnetic power.

In conclusion, we show that the discrepancies between the effective absorption parameter $\eta $ and the commonly used EFR can be high enough to prevent a correct evaluation of the structures, and we suggest to always use the confinement factor as a measurement of $\eta $, provided the perturbative limit is respected.