Generalized figure of merit for plasmonic dip measurement-based surface plasmon resonance sensors

Abstract

We propose a theoretical framework to analyze quantitative sensing performance parameters, including sensitivity, full width at half maximum, plasmonic dip position, and figure of merits for different surface plasmon operating conditions for a Kretschmann configuration. Several definitions and expressions of the figure of merit have been reported in the literature. Moreover, the optimal operating conditions for each figure of merit are, in fact, different. In addition, there is still no direct figure of merit comparison between different expressions and definitions to identify which definition provides a more accurate performance prediction. Here shot-noise model and Monte Carlo simulation mimicking the noise behavior in SPR experiments have been applied to quantify standard deviation in the SPR plasmonic dip measurements to evaluate the performance responses of the figure of merits. Here, we propose and formulate a generalized figure of merit definition providing a good performance estimation to the detection limit. The measurement parameters employed in the figure of merit formulation are identified by principal component analysis and machine learning. We also show that the proposed figure of merit can provide a good estimation for the surface plasmon resonance performance of plasmonic materials, including gold and aluminum, with no need for a resource-demanding computation.

1. Introduction

Surface plasmon resonance (SPR) is a label-free detection for monitoring biomolecular interaction in real-time, which utilizes the resonating effect of surface plasmons (SPs) or the induced electric fields and magnetic fields that leak out from the uniform surface of noble metals, such as gold (Au) and silver (Ag) [1]. These noble metals consist of metallic bonds that electrons can move freely and oscillate with the same frequency as surface plasmon due to the external electric fields causing the electromagnetic response referring to surface plasmon resonance (SPR) or surface plasmon polaritons (SPP) [2]. The coupling condition of the surface plasmons depends on the noble metal's surrounding materials; this enables the label-free measurement and gives rise to the underlining sensitivity mechanism [3]. In addition, the noble metals are very conductive and contain a complex permittivity, resulting in the lossy light reflection [4].

Gold (Au) is a noble metal widely employed in biological and biomedical measurements in biosensing applications due to its chemical stability and biocompatibility [5]. It also provides convenient coupling angles and coupling wavelengths to excite surface plasmons. Furthermore, since the natural frequency of surface plasmons of gold matches the light in the range of red to infrared (more than 520 nm) wavelength [6], the incident light in this range can be applied to surface plasmon resonance through an optical prism [7], a high numerical aperture objective lens [8] and an optical grating [9].

The configuration of the surface plasmon polariton-based measurement (SPP-based sensor) consists of two widely adopted configurations, which are the Kretschmann configuration [10] and the Otto configuration [11]. The difference between these two configurations is the arrangement of three layers: sensing area, noble metal, and a glass prism. In Otto configuration, the sensing area or spacing gap is located between the noble metal film and glass prism; meanwhile, in the Kretschmann configuration, the noble metal film is located between the sensing region and the glass prism as shown in Fig. 1(a). The Otto configuration required a sophisticated channel fabrication, and the nanoscale thickness of the air gap results in difficulty in fabrication; therefore, the Kretschmann configuration is more widely used as a surface plasmon polariton sensor, although research works have reported that the Otto configuration can provide a higher sensitivity [8,12]. Furthermore, the sensing performance parameters for different operating conditions are similar for the Kretschmann and Otto configurations. Therefore, the results for the Otto configuration are omitted to shorten the length of the manuscript.

Fig. 1.

Shows (a) Kretschmann configuration with angular scanning detection mechanism and (b) sensing performance of angular scanning detection mechanism.

The Kretschmann configuration has been employed and utilized in several sensing applications, for example, binding kinetics [13,14], refractive index sensing [15,16], pathogen detection [17,18], voltage sensing [19,20], surface plasmonic microscopy [21,22], and surface-enhanced Raman spectroscopy [23]. Recently, the SPR has demonstrated its ultra-sensitivity in measuring the kinetic activity of SAR-COV2 (COVID19) and the angiotensin-converting enzyme 2 (ACE2) [24,25]. In addition, Tang et al. [26] reported COVID-19 antibody immunoglobulin-G (IgG) measurements on hospitalized patients using SPR.

Several detection schemes for SPR sensing include wavelength scanning [27,28] using a tunable light source, optical fiber-based SPR sensors [29], and angular scanning detection [30–32]. Here, the angular scanning detection scheme is investigated as an example to demonstrate findings in this study. Therefore, applying the proposed figure of merit (FOM) is limited to neither the plasmonic gold sensor material nor the angular scanning detection scheme. It will be shown in the result section later that although the proposed FOM has been derived and proved using the plasmonic gold material, it can be applied to accurately predict responses of higher loss plasmonic material, such as aluminum (Al). For the Kretschmann-based angular interrogation detection, the light source is a p-polarized monochromatic light source with its wavelength λ illuminated a uniform gold plasmonic sample with the thickness d with the complex refractive index of gold nm, through a glass prism with refractive index n₀ of 1.52 with a varying incident angle θ₀, as shown in Fig. 1(a). The highest surface plasmon coupling strength occurs at the minimum reflectance position, so-called plasmonic angle θ_sp, as shown in Fig. 1(b). The reflectance spectra in Fig. 1(b) were calculated using Fresnel equations and the transfer matrix calculation [33] for the λ of 633 nm, d of 40 nm, and the sample region refractive indices n_s of 1.33 (water) and 1.35 (80 mg/ml concentration of Bovine serum albumin protein solution [34]) shown in solid blue and solid red curves, respectively.

Priya et al. [35] have reported that for coupling angle measurement through an optical glass prism, the incident wavelength strongly affects the plasmonic angle shift (Δθ_sp), full width at half maximum (FWHM), depth of the plasmonic intensity dip (I_sp), and FOM that the Δθ_sp, the sensitivity (S), and the FWHM are exponentially decaying at a longer wavelength. On the other hand, the SPR dip depth and the FOM are independent of the incident wavelength. Lakayan et al. [6] have reported a similar finding that the incident wavelength is crucial in the Δθ_sp and the FWHM. They reported that λ of 890 nm had a 10-time higher S than the conventional SPR excitation wavelengths of 670 nm and 785 nm.

A FOM is one of the crucial sensing performance parameters; it characterizes sensors’ predictive ability and performance. Therefore, it is widely adapted to determine sensors’ performance in different types of sensors, for example, surface plasmon resonance [19,36], Fabry–Perot resonators [37], optical ring resonators [38], and surface phonon structures [12]. There are several definitions of FOMs based on the purpose and detection mechanisms. For example, for surface plasmon resonance, the FOM usually employed is the S over the FWHM [39–41]. In addition, some other parameters, such as I_sp, intensity contrast (ΔI), have been employed in the FOM formulation [29,42–45].

There is no report on a direct FOM comparison between different expressions and formulas. Therefore, this research aims to propose a theoretical framework to provide the comparison between different expressions of FOMs and quantify the quantitative performance parameters, including S, θ_sp, I_sp, FWHM, ΔI, and FOMs of SPR sensors to analyze the sensing capability using Fresnel equations and transfer matrix approach [33]. In addition, we also formulate a generalized FOM that reflects the SPR limit of detection (LoD) calculated using Monte Carlo and the shot-noise model for a typical digital camera and input optical power to the SPR system. Principal component analysis (PCA) and machine learning (ML) models have been utilized to identify key performance parameters that contribute to the proposed FOM formulation. To the best of the authors’ knowledge, the theoretical framework to analyze and provide a direct FOMs comparison between different definitions and a generalized FOM formulation has never been investigated and reported before. One of the significant challenges in optical biosensing is to measure a low concentration sample or a small number of analytes, such as single-molecule detection [46] and DNA fragments [24,25]. These sensitivity demanding applications, of course, require an optical sensor design and engineered optical sensor surfaces, such as structured surfaces [47] and metamaterials [48]. The proposed generalized FOM can be applied to estimate the sensor's sensing performance during the optical sensor design process prior to the sensor fabrication with no need for time-consuming and resource-demanding computation.

2. Materials and methods

2.1. Optical simulations of SPR detection schemes

Fresnel equations and the transfer matrix calculation [33] were employed to simulate the reflectance of the Kretschmann SPR sensor as depicted in Fig. 1(a) to analyze the sensing performance of angular scanning detection mechanism in which the light wavelength λ was fixed, and the incident angle θ₀ was varied as depicted in Fig. 1(a). The complex refractive index of gold nm employed in the study was extracted from gold's complex dielectric dispersion function reported in Johnson and Christy 1972 [49].

2.2. Shot-noise model and Monte Carlo simulation

The reflectance spectra were shot-noise added [50] for photon energy of 5000 pJ captured using a digital camera with 1280 pixels × 720 pixels in which each pixel corresponding to 0.001 degrees assuming that the camera had the quantum efficiency of 60% for all the wavelengths for fair performance comparison and being independent of the camera's performance, as depicted in Fig. 2(a). The shot-noise model is a dominant noise source since the SPR dip has low light intensity, especially around the plasmonic angle [51].

Fig. 2.

(a) a camera frame with shot-noise added corresponding to the total energy of 5000 pJ, (b) camera rows summation to form a line scan plasmonic dip with 4th-degree polynomial curve fitting. The case shown here was d of 25 nm and λ of 850 nm.

The plasmonic angle for each noise-added camera frame was then recovered employing the following steps. (1) Sum all the intensity of 720 camera rows to a single-row angular spectrum to reduce the random shot-noise as depicted in Fig. 2(a), and (2) apply the 4th-degree polynomial curve fitting to the plasmonic dip covering sinθ₀ range of $\pm$ 0.065 to determine the θ_sp position for each camera frame as shown in Fig. 2(b).

A Monte Carlo simulation [50] with 1,500 iterations of randomized shot-noise added camera frames was then employed to analyze and calculate the plasmonic angle positions and their LoD as one standard deviation, σ.

2.3. Sensing performance parameters

The quantitative performance parameters including S, θ_sp which minimum reflectance occurs, I_sp, ΔI, FWHM, and FOMs can be computed from the reflectance spectrum calculated using the Fresnel equations and the transfer matrix calculation between two different sample refractive indices of ns of 1.33 (water) and 1.35 (BSA protein solution) [34]. The reason for choosing the two refractive indices was that they were typically employed biological samples for SPR performance characterization [1,52]. The proposed theoretical framework is not limited to the investigated sample refractive indices. It is established that the SPR measurement has a wide refractive index response range without losing its sensitivity and plasmonic dip characteristics [8,53].

• (1)
  FWHM is defined as the averaged width in sinθ₀ space of normalized reflectance dips with less than 50 percent of maximum optical intensity for the two sample refractive indices, as expressed in Eq (1) and depicted in Fig. 1(b).

  $$FWHM=\frac{FWH{M}_{1.33}+FWH{M}_{1.35}}{2}$$ (1)

Where the terms $\text{FWH}{\text{M}}_{1.33}$ and $\text{FWH}{\text{M}}_{1.35}$ are the FWHM of normalized reflectance spectra for two sample refractive indices of 1.33 and 1.35 corresponding to water and 80 mg/ml concentration of Bovine serum albumin protein solution [34], respectively.

• (2)
  ΔI is defined as the average difference between optical intensity at the plasmonic reflectance dip I_sp and the intensity at the critical wave vector (I_max) when the sample refractive indices were 1.33 and 1.35 as expressed in Eq. (2). This parameter indicates the depth of the plasmonic dip intensity compared to its intensity baseline. The reflectance baseline intensity is usually high since the SPR detection requires attenuated total internal reflection (ATR). Therefore, suppose the ΔI is low means that the plasmonic dip is not deep, and the measurement needs to detect low contrast signal in the strong background leading to a lower signal-to-noise ratio (SNR). On the other hand, a higher ΔI indicates a higher signal-to-noise ratio.

  $$\mathrm{\Delta }I=\frac{\mathrm{\Delta }{I}_{1.33}+\mathrm{\Delta }{I}_{1.35}}{2}$$ (2)

  The terms ΔI_1.33 and ΔI_1.35 are an intensity contrast compared to the intensity baseline of the two sample refractive indices.

• (3)
  S is defined as the change in sinθ_sp over the change in sensing region refractive index Δn_s, which was 0.02 (1.35–1.33) for this study, as expressed in Eq. (3) and depicted in Fig. 1(b). Note that the unit for Eq. (4) is RIU⁻¹.

  $$S=\frac{\mathrm{\Delta }sin{\theta }_{sp}}{\mathrm{\Delta }{n}_s}$$ (3)
• (4)Detection range (DR) is defined as the highest refractive index value that can measure before the plasmonic angle is greater than 90 degrees; in other words, still within the possible range of incident angles that the prism can accommodate.
• (5)
  FOMs: there are several definitions reported for angular scanning in the literature.

  • (5.1)
    the S over the FWHM [39,54–60] for plasmonic dip position measurement as expressed in Eq. (4).

    $$FO{M}_1=\frac{S}{\text{FWHM}}=\frac{\mathrm{\Delta }sin{\theta }_{sp}}{\text{FWHM}\mathrm{\Delta }{n}_s}$$ (4)
  • (5.2)
    the S times the ΔI over the FWHM [29,42–44] for the plasmonic dip position measurement, as expressed in Eq. (5).

    $$FO{M}_2=\frac{\mathrm{\Delta }I\times S}{\text{FWHM}}=\frac{\mathrm{\Delta }I\mathrm{\Delta }sin{\theta }_{sp}}{\text{FWHM}\mathrm{\Delta }{n}_s}$$ (5)
  • (5.3)
    the S over the product of the FWHM and the I_sp [45], as expressed in Eq. (6).

    $$FO{M}_3=\frac{S}{\mathrm{F}\mathrm{W}\mathrm{H}\mathrm{M}\times I_{sp}}=\frac{\mathrm{\Delta }sin{\theta }_{sp}}{\text{FWHM}\mathrm{\Delta }{n}_s{I}_{sp}}$$ (6)
  • (5.4)
    Here, we propose a generalized FOM as expressed in Eq. (7), which will be proved later that the FOM₄ can provide a better performance estimation compared to the other FOMs and similar to the inverse of the LoD [61] calculated using the shot-noise and Monte Carlo simulation as described in section 2.2.

    $$FO{M}_4=\frac{\sqrt{\mathrm{\Delta }I}\times S}{\mathrm{F}\mathrm{W}\mathrm{H}\mathrm{M}\times \sqrt[4]{I_{sp}}}=\frac{\sqrt{\mathrm{\Delta }I}\times \mathrm{\Delta }sin{\theta }_{sp}}{\sqrt[4]{I_{sp}}\times \text{FWHM}\mathrm{\Delta }{n}_s}$$ (7)

  Some other FOM definitions reported in the literature are not directly measurable from the plasmonic reflectance spectra. Those FOMs were excluded from the scope of the study. For example, Shen et al. [62] have reported an alternative approach calculating the FOM using the transmission line model and found that the FOM is proportional to the equivalent reactance over the resistance of the plasmonic sensor. Doiron et al. [63]. have reported that the FOM depends on the optical power. These parameters cannot be readily extracted from the optical reflectance spectra compared to the four FOMs in Eq. (4) to Eq. (7).

• (6)As explained earlier, LoD is defined as one standard deviation (σ) of plasmonic angles recovered from the 1,500 camera frames with random shot-noise. The LoD is frequently employed as a FOM indicator [64]. Here, the concept of FOM is related to the LoD by defining the FOM as 1/LoD.

3. Results and discussions

3.1. Performance parameters and optical reflectance

The performance parameters described in section 2.3 were computed for a uniform plasmonic gold sensor coated on a standard BK7 glass with sample refractive indices of 1.33 and 1.35. The λ and d were varied from 600 nm to 1900nm and 10 nm to 80 nm, respectively. The optical wavelength of 1900nm was far beyond the optical communication wavelength of 1500 nm and general optical applications. The wavelength of 1900nm was the longest wavelength reported by Johnson and Christy for the gold refractive index [49]. These wavelengths and the gold thickness range were chosen to ensure that the analysis covered all the characteristics of SPR reflectance curves ranging from low to high attenuation coefficient, coupling loss, and ohmic loss [65]. It will be shown in section 3.5 that the proposed FOM can be applied to analyze a higher loss plasmonic material, such as aluminum.

Figure 3(a)-(f) shows the angular interrogation measurement's average FWHM, ΔI, I_sp, sinθ_sp when n_s was 1.33, S, and DR, respectively. For every d, the FWHM decreases exponentially when the incident wavelength λ increases, like the finding highlighted by Priya et al. [35]. Like the effect of λ, d also affects the FWHM. The width of the plasmonic dip or the FWHM also exponentially increases when the thickness d decreases. Therefore, the largest FWHM occurs at the bottom left-hand corner of Fig. 3(a); on the other hand, the narrowest FWHM of 9.61×10⁻⁴ occurs at the upper right-hand corner of the FWHM contour corresponding to the operating position the d of 80 nm and λ of 1900nm. The strong ΔI contrast occurred along the diagonal line of Fig. 3(b). The maximum ΔI of 0.99 contrast occurs at λ of 1900nm and d of 26 nm. Note that the maximum intensity baseline at the critical angle I_max can be determined from the ΔI and the I_sp. Therefore, it is omitted to save the space of the manuscript. The operating position with the minimum I_sp of 8.18×10⁻⁷ was at the d of 52.5 nm and λ of 720 nm.

Fig. 3.

Shows quantitative performance parameters for angular scanning detection scheme: (a) FWHM, (b) ΔI (c) I_sp, (d) k_sp when n_s = 1.33, (e) S calculated using Eq. (3); and (f) DR.

For the plasmonic angle position sinθ_sp, the shorter λ gave a higher sinθ_sp; when λ increased, the sinθ_sp exponentially decreased. Furthermore, the metal thickness also affected the plasmonic coupling condition when the thickness was lower than 50 nm. In contrast, when the gold thickness d thicker than 50 nm, the SP coupling conditions did not significantly change. Thus, for the angular detection, the lower sinθ_sp excitation can reduce the demand for a high incident angle illumination [49], which requires a high refractive index prism; meanwhile, for the oil immersion objective lens based SPR excitation, the lower sinθ_sp can lower the numerical aperture (NA) requirement of the objective lens [37].

The highest sensitivity of 0.8241 RIU⁻¹ occurred around the bottom right-hand corner of the sensitivity contour when the gold thickness d was 80 nm and λ of 629 nm. The operating position had lower intensity contrast around 0.2. Thus, the higher sensitivity occurred at thicker metal thickness with d more than 30 nm and shorter incident wavelength. For the gold thickness d below 30 nm, the sensitivity degraded exponentially. It is essential to point out that the contour region with high sensitivity around the 50 nm to 80 nm gold thickness has relatively low optical intensity contrast ΔI, as shown in Fig. 3(b). The relationship is a classic trade-off between sensitivity S and optical contrast ΔI [66]. For the region with the plasmonic wavelength λ below 1000 nm and the gold thickness d below 30 nm, the ΔI contrast was low, and the sensitivity was also low, indicating that the incident light to SPR coupling was inefficient. The increased attenuation loss of the incident light coupled to the SPR [65] led to a broader FWHM and shorter propagation length. The propagation length SPR interacts with the fewer sample region, resulting in a sensitivity S. For the DR, the widest DR of 1.51 RIU appeared at the upper right-hand corner of the contour corresponding to the operating position d of 80 nm and λ 1900nm. Again, there is a significant trade-off between the sensitivity S and the DR. The SPR measurement conditions with higher sensitivity have a lower detection range; on the other hand, those operating positions with lower sensitivity have a more extended refractive index sensing range DR.

The following five operating positions have been chosen for performance comparison and discussion in the later sections—the five operating positions’ A’ to ‘E’ in Fig. 3. (1) ‘A’ is the operating position with d of 80 nm and λ 1900nm, corresponding to the narrowest FWHM, the widest DR, and the highest FOM₁ discussed later. (2) ‘B’ is the operating position with d of 26 nm and λ 1900nm, corresponding to the highest ΔI. (3) the operating position labeled ‘C’ with d of 40 nm and λ 1835nm, corresponding to the highest FOM₂ discussed later. (4) the operating position labeled ‘D’ with d of 52.5 nm and λ 720 nm, corresponding to the lowest I_sp and the highest FOM₃, and (5) the operating position labeled ‘E’ with d of 80 nm and λ 629 nm for the highest S. Fig. 4(a-e) show the SPR reflectance spectra and phase of the SPR reflectance spectra responses of the operating positions’ A’ to ‘E’.

Fig. 4.

Shows the reflectance spectra for the following SPR operating positions with sample refractive indices of 1.33 shown in solid blue curves for reflectance and dashed-blue curves for phase response and 1.35 shown in solid red curves for reflectance and dashed-red curves for phase response (a) the operating position labeled ‘A’ with d of 80 nm and λ of 1900nm (the narrowest FWHM, the widest DR, and the highest FOM₁), (b) the operating position labeled ‘B’ with d of 26 nm, and λ of 1900nm (the highest ΔI), (c) the operating position labeled ‘C’ with d of 40 nm and λ of 1835nm (the highest FOM₂), (d) the operating position labeled ‘D’ with d of 52.5 nm and λ of 720 nm (the lowest I_sp and the highest FOM₃); (e) the operating position labeled ‘E’ with d of 80 nm and λ of 629 nm (the highest S).

3.2. Figure of merits

Figure 5 shows different FOMs calculated using Eq. (4) to Eq. (7). Figure 5(a) is for the FOM₁ calculated using the S in Fig. 3(e) and the FWHM in Fig. 3(a). The highest FOM₁ of 694.9274 RIU⁻¹ was at the operating position of the incident wavelength of 1900nm and the gold thickness of 80 nm (the operating position ‘A’), as shown in Fig. 5(a). However, the FOM₁ did not consider the signal-to-noise ratio for intensity measurement; the maximum FOM₁ had the intensity contrast ΔI of only 0.04 or 4% in optical contrast. Therefore, the FOM₁ cannot provide a decent SPR operating position for the intensity detection and phase detection in angular scanning measurement since there was virtually no phase contrast in the phase reflectance spectra, as shown in Fig. 4(a). Note that the operating position ‘A’ is not experimentally practical for the intensity-based measurement. Later in section 3.4, it will be shown through machine learning and the PCA that the FOM₁ cannot provide a good estimation for sensing performance prediction because it does not consider the optical intensity information. The FOM₂ calculated using Eq. (5) considers the ΔI of the SPR dips by multiplying the ΔI to the FOM₁, as shown in Fig. 5(b). The maximum FOM₂ of 315.2957 RIU⁻¹ occurred at the operating position with d of 40 nm and λ 1835nm (the operating position ‘C’). The reflectance spectra of the operating position are shown in Fig. 4(c)—the operating position balanced out the dip movement and the signal contrast. Figure 5(c) shows the FOM₃ response calculated using Eq. (6); the FOM₁ in Fig. 5(a) over the I_sp in Fig. 3(c). The highest FOM₃ of 2.05×10⁸ RIU⁻¹ was at the d of 52.5 nm and λ of 720 nm (the operating position ‘D’). The plasmonic reflectance spectra for the operating position are shown in Fig. 4(d). The optimum operating position range for the FOM₃ was narrow and around the d of 50 nm to 55 nm and the incident wavelength of 650 nm to 850 nm, as shown as an inset in Fig. 5(c). Fig. 5(d) shows the response of our proposed FOM definition as expressed in Eq. (7). The highest FOM₄ of 2811 RIU⁻¹ appeared at d of 52.5 nm and λ of 720 nm; the high FOM₄ operating positions were along the diagonal line in Fig. 5(d) and significantly wider than the FOM₃. It is essential to point out that although the FOMs have the same unit of RIU⁻¹, they cannot be directly compared since the number of performance parameters involved differ. Table 1 summarizes the performance parameters for the five chosen operating conditions of the plasmonic uniform gold sensor. In Table 1, the five operating positions were then ranked based on their FOM performance, as indicated in a bracket next to the FOM. It can be seen that the FOMs did not agree with each other, except the FOM₃ and the FOM₄. The next question is how they can be compared and justified under fair criteria.

Fig. 5.

Shows the FOMs for angular scanning detection scheme: (a) FOM₁ calculated using Eq. (4), (b) FOM₂ calculated using Eq. (5), (c) FOM₃ calculated using Eq. (6), and (d) FOM₄ calculated using Eq. (7).

Table 1. Quantitative sensing performance parameters and the FOMs for operating positions in Fig. 3 and 5.

Operating position
Parameters	A	B	C	D	E
λ (nm)	1900	1900	1835	720	629
d (nm)	80	26	40	52.5	80
FWHM	9.61×10−4	3.4×10−3	1.6×10−3	1.05×10−2	1.7×10−2
ΔI	0.04	0.99	0.78	0.94	0.22
Isp	0.90	3.63×10−4	0.20	8.18×10−7	0.67
sinθsp (ns = 1.33)	0.88	0.88	0.88	0.92	0.95
S (RIU−1)	0.67	0.67	0.67	0.75	0.82
DR (RIU)	1.51	1.51	1.51	1.43	1.38
FOM1	694.93(1st)	198.39(3rd)	406.70(2nd)	71.47(4th)	48.51(5th)
FOM2	28.09(4th)	195.77(2nd)	315.30(1st)	67.43(3rd)	10.71(5th)
FOM3	768.17(4th)	5.47×105(2nd)	2079.50(3rd)	2.05×108(1st)	72.57(5th)
FOM4	143.3(4th)	1428(2nd)	538.5(3rd)	2811(1st)	25.2(5th)

3.3. Limit of detection

The LoD response was chosen as the criterion to compare and evaluate the performance of FOMs, as it has been widely utilized as a widely accepted FOM indicator [67,68]. Higher FOM means that the SPR can measure a lower refractive index range, in other words, inverse relationship to the LoD. The LoD was computed for the operating position assuming that the total reflectance beam energy of 5000 pJ was captured using 1080 pixels by 720 pixels with the quantum efficiency of 60%. Of course, one can increase the number of pixels or employ a higher well depth camera leading to an LoD improvement. These, however, only change the value of the LoD, not the shape of the LoD contour, as shown in Fig. 6(a). Fig. 6(b) to Fig. 6(e) show the inverse FOM₁, FOM₂, FOM₃, and FOM₄, respectively. The FOM₁ and the FOM₂ cannot provide a good representation of the LOD.

Fig. 6.

(a) the LoD for all the operating conditions calculated for 5000 pJ optical energy, (b) 1/FOM₁, (c) 1/FOM₂, (d) 1/FOM₃, and (e) 1/FOM₄

On the other hand, the inverse of FOM₃ and FOM₄ can provide a similar pattern to the LoD. The reciprocal of FOM₄ can provide a more similar pattern to the LoD. Here, the structural similarity index (SSI) [69] was computed comparing the similarity between the LoD and the four reciprocal FOMs giving the SSIM of 53.42%, 77.11%, 83.12%, and 95.62%, respectively. The FOM₄ performed better than the other FOMs because the FOM₄ took sufficient optical intensity information into account compared to the other FOMs. The following section will be shown and discussed through machine learning and the PCA that the parameters that can contribute to the LoD and FOM are the S, the FWHM, the ΔI, and the I_sp.

3.4. PCA and machine learning

In this section, the sensing performance parameters S, FWHM, ΔI, I_sp, DR, and LoD were converted from their two-dimensional contour to a 1D array and formed a data table with the array dimension 10,201 rows and six columns. The tabulated data were then modeled using the following ML models: linear regression, stepwise linear regression, tree, support vector machine (SVM), ensemble, and Gaussian process regression (GPR), as shown in Table 2. The root-mean-square error (RMSE) was calculated using five-fold cross-validation of all models. For the gold SPR responses, the GPR model provided the highest accuracy model with Matern 5/2 GPR algorithm with the lowest RMSE of 0.86%.

Table 2. RMSE of ML models trained using the five predictors S, FWHM, ΔI, I_sp, DR to model the LoD response.

ML model	Algorithm	RMSE
Linear regression	Linear	3.92%
	Interactions linear	1.79%
	Robust linear	4.20%
Stepwise linear regression	Stepwise linear	1.81%
Tree	Fine tree	1.37%
	Medium tree	1.62%
	Coarse tree	2.05%
Support vector machine (SVM)	Linear SVM	4.04%
	Quadratic SVM	1.83%
	Cubic SVM	4.41%
	Fine Gaussian SVM	2.84%
	Medium Gaussian SVM	1.43%
	Coarse Gaussian SVM	2.68%
Ensemble	Boosted trees	4.19%
	Bagged trees	1.34%
Gaussian process regression (GPR)	Squared exponential GPR	0.91%
	Matern 5/2 GPR	0.86%
	Exponential GPR	0.87%
	Rational quadratic GPR	0.87%

The PCA was then employed to identify which of S, FWHM, ΔI, I_sp, and DR contributed to the model by 97.8%, 1.9%, 0.2%, 0.1%, and 0.0%, respectively. Note that the PCA is a mathematical modeling approach to exclude parameters that contribute less to the model. Here, the statistical confidence of the PCA was 95% for the analysis. Therefore, the only four factors contributing to the LoD are S, FWHM, ΔI, and I_sp. Although the latter two parameters are less significant than the first two, they will be included in the formulation process in the next section. Here, the effect of different numbers of performance parameters included in the ML model is demonstrated by training the GPR model with the Matern 5/2 GPR algorithm using different performance parameters included, as shown in Table 3. The predicted contour responses are shown in Fig. 7. The GPR models trained using five, four, and three parameters can provide the contour that agrees well with the LoD contour discussed in Fig. 6(a) in the earlier section.

Table 3. RMSE of GPR model with Matern 5/2 GPR trained using different performance parameters.

ML model	Parameters	RMSE
Gaussian Process Regression (GPR)	S, FWHM, ΔI, Isp, DR	0.86%
Algorithm: Matern 5/2 GPR	S, FWHM, ΔI, Isp	0.86%
PCA output:	S, FWHM, ΔI	0.87%
S (97.8%), FWHM (1.9%), ΔI (0.2%), Isp	S, FWHM	5.03%
(0.1%), DR (0.0%)	S	10.44%

Fig. 7.

shows LoD models trained using (a) 5 parameters, (b) 4 parameters, (c) 3 parameters, (d) 2 parameters, and (e) 1 parameter.

In contrast, the two-parameters case has an artifact region at the higher plasmonic gold thickness region, and the one-parameter case cannot represent the LoD contour. It is established in artificial intelligence that the trained model can only be employed in the same range of the training data; prediction outside the scope of the training data can lead to an unrealistic value [70]. The last section will demonstrate this in predicting the LoD response of a higher loss plasmonic material aluminum. Therefore, there is a need to work an analytical LoD or FOM from the sensing parameters.

3.5. Proposed FOM formulation

Although the ML models can represent the shape of LoD contour, extracting an explicit equation from the ML is problematic due to the complexity of the underlining mathematical model. As discussed in the ML section above, the four parameters contributing to the LoD are S, FWHM, Ic, and I_sp. The four parameters were then included in Eq. (8) to formulate the proposed FOM by varying the exponent variables j,k,m, and n and comparing the Eq. (8) with LoD contour in terms of the SSI. Figure 8 shows the SSI responses comparing the LoD contour in Fig. 7(a) and Eq. (8) with i and j variables ranging from −2 to 2, m, and n from 1 to 2. Fig. 8(a) shows the case with m of 1 and n of 1. The maximum SSI of 95.62% appeared at j of 0.25, k of 0.5, m of 1, and n of 1, respectively. The other m and n cases had a lower SSI value, as shown in Fig. 8(a-b). The SSI contour enables us to formulate the proposed FOM as expressed in Eq. (7). It is essential to point out that the expression was not derived directly from mathematical derivation, however, with the aid of the PCA approach.

$$LoD=\frac{1}{FOM}=\frac{{I_{sp}}^j\times FWH{M}^m}{\mathrm{\Delta }{I}^k\times S^n}$$ (8)

Fig. 8.

Shows SSI responses comparing the LoD contour in Fig.7a and Eq. (8) with varying I and j variables from −2 to 2 and (a) m = 1 and n = 1, (b) m = 2 and n = 1, (c) m = 1 and n = 2

Apply the generalized FOM to the other plasmonic metals

To prove that the proposed FOM can predict the responses of plasmonic materials and other sensor structures that require an angular interrogation measurement. Here, the following four responses of plasmonic aluminum at four incident wavelengths λ of 1000 nm, 1200 nm, 1400 nm, and 1600 nm were computed for their LoD when the metal thickness was varied from 10 nm to 45 nm. It is well known that aluminum is a lossier plasmonic material [71] than gold, silver, and copper. Fig. 9(a) shows the normalized LoD of the four wavelengths calculated using the shot-noise model and Monte Carlo simulation described in section 2.2. Note that the 1,500 iterations were insufficient to calculate a stable LoD response due to the high attenuation nature of the SPs excited on the Al surface. Therefore, the LoD needs to be estimated using the FOM expressions; otherwise, it is a time and computing resource-demanding task. Fig. 9(b) shows normalized 1/FOM₄ calculated using the proposed FOM in Eq. (7). It can be seen that the FOM₄ can predict the performance of the plasmonic aluminum for all the investigated wavelengths.

Fig. 9.

Shows SPR responses of Al for varying d from 10 nm to 45 nm at four incident wavelengths λ of 1000 nm, 1200 nm, 1400 nm, and 1600 nm for (a) normalized LoD calculated using shot-noise and Monte Carlo model, and (b) normalized 1/FOM₄.

The ML models can only be employed to predict the SPR responses that are similar to the training cases; the aluminum SPR cases significantly differed from the gold SPR cases. Fig. 10(a-e) shows the predicted LoD using the models trained in section 3.4. All the models, even the one developed using five performance parameters, cannot provide a good estimation of the LoD.

Fig. 10.

Shows the LoD responses predicted using the GPR models with Matern 5/2 GPR algorithm trained using (a) 5 performance parameters S, FWHM, ΔI, I_sp and DR, (b) 4 performance parameters S, FWHM, ΔI and I_sp, (c) 3 performance parameters S, FWHM and ΔI, (d) 2 performance parameters S and FWHM, and (e) 1 performance parameter S.

Apply the generalized FOM to wavelength interrogation measurement

It is essential to point out that the proposed theoretical framework can be applied to optical intensity detection of SPR for both the angular interrogation and the wavelength scanning measurements. Figure 11 demonstrates the capability of the FOM₄ in predicting the LoD response for wavelength interrogation measuring the change in coupling plasmonic wavelength λ_sp. Fig. 11(a) shows the LoD responses of Au when varying d from 35 nm to 80 nm at four fixed incident angles sinθ₀ of 0.900, 0.925, 0.950, and 0.975. Fig. 11(b) shows the 1/FOM₄ responses corresponding to each illumination angle. It can be seen that the proposed FOM₄ can provide a good estimation of the theoretical LoD calculated using the shot-noise model and Monte Carlo simulation.

Fig. 11.

Shows SPR responses of Au when varying d from 35 nm to 80 nm at four incident angles sinθ₀ 0.900, 0.925, 0.950, and 0.975 for (a) normalized LoD calculated using shot-noise and Monte Carlo model, and (b) normalized 1/FOM₄.

The limitation of the proposed FOM is that it is not applicable for phase measurement since the phase detection differs from the intensity detection in terms of the nature and the noise performance of an optical interferometer. It is established that the phase measurement is more robust to the optical intensity noise than the amplitude or intensity measurement. In optical interferometry, the SPR resonant wave vector is instead detected through the displacement of interferent fringes; therefore, the optical intensity is not a crucial parameter. Kabashin et al. [72] have experimentally validated and explained this point. Roland et al. [73] also reported a similar finding when using a thinner plasmonic gold of 25 nm, which had lower ΔI contrast. They imaged and located subwavelength nanoparticles with 10 nm to 200 nm diameter through SPR phase imaging microscopy. Therefore, the FOM for the phase measurement does not depend strongly on the optical intensity.

4. Conclusion

We have proposed a theoretical framework to quantify performance parameters for angular interrogation based on surface plasmon resonance measurement under the conventional Kretschmann configuration. The different figures of merit definitions and expressions are reported in the literature. However, there was no direct comparison to evaluate and compare the figures of merit. Here, the limit of detection calculated using the shot-noise model and Monte Carlo simulation representing optical intensity noise and fluctuations typically found in a digital camera was applied to compare the figures of merits. The figures of merit reported in the literature cannot reasonably estimate the detection limit. Therefore, we proposed a generalized figure of merit definition of sensitivity×intensity contrast^0.5/(full width at half maximum×intensity at the plasmonic dip^0.25), which can provide a good estimation of the detection limit. The proposed figure of merit expression was developed using the principal component analysis to identify the four performance parameters contributing to the figure of merit performance. The proposed figure of merit can be employed to predict the SPR of higher-loss plasmonic material, including aluminum.

Funding

The Research Institute of Rangsit University10.13039/501100007939 (RSU); the School of Engineering of King Mongkut's Institute of Technology Ladkrabang10.13039/501100007120 (KMITL).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.