Improved self-referenced biosensing with emphasis on multiple-resonance nanorod sensors

Ahmed Abumazwed; Wakana Kubo; Takuo Tanaka; Andrew G. Kirk

doi:10.1364/OE.25.024803

1. Introduction

In propagating surface plasmon resonance (SPR) sensors, a reference channel is required to compensate for artifacts due to temperature drift and bulk RI changes [1]. However, the reference and sensing channels are not identical (for example: different metal thickness) due to fabrication imperfections, the difference in analyte transport in both channels [2], and the uncorrelated effects in each channel during the binding events (presence of air bubbles and changes in speed of fluid flow in either channel) [3]. These artifacts have motivated many researchers to seek alternative methods where the reference channel could be abandoned to avoid any external or intrinsic effects in the two-channel sensing platforms. A self-referenced SPR sensor has previously employed the excitation of a dual-mode SPR (with different penetration depths) using two laser sources [4], which increases the instrumental complexity. Alternative approaches included the excitation of dual-mode SPR and use it as reference/sensing channels based on a linear response model. This was achieved by either exciting two modes at different locations on the metal surface [5], or exciting the long range and short range SPR modes on the same location of the SPR surface [6,7]. The linear response model assumes that each resonance wavelength shift (Δλ) is a linear function of both adlayer thickness (d) and bulk RI change (Δn) as follows

Δ λ_{i} = S_{B_{i}} Δ n + S_{d_{i}} d

where S_{B_i} = ∂λ_i/∂n is the bulk RI sensitivity, and S_{d_i} = ∂λ_i/∂d is the adlayer sensitivity at the i^th resonance. The same approach (i.e. a linear response model) has been applied to gold nanorod structures, U-shaped structures and propagating plasmon waveguide resonance biosensor [8–10]. Although various approaches have been introduced for self-referencing SPR platforms, less effort has been previously paid to self-referencing based on localized surface plasmon resonance (LSPR) sensors. This paper presents the MLE approach to improve the precision and accuracy of the results based on the linear response model. The paper also provides a method to overcome the repetitive simulation to determine the sensitivity to adlayer thickness for various analytes and correct it based on the measured bulk RI sensitivity.

2. Concept of self referencing based on multiple resonances

As described above, estimating the unknown quantities (adlayer thickness and bulk RI change) requires at least two resonances. The multiple resonance characteristic of nanorod structures can be employed to generate multiple systems each of which provides solutions for the estimates. Herein, a three-resonance nanorod structures is considered as the first resonance is used with the second and third resonances to generate three systems of linear equations based on a linear response model [Eq. (1)], and the adlayer and bulk RI change can be estimated accordingly as follows

\underset{LM 1 : from λ_{1}, λ_{2}}{\underset{︸}{[\begin{matrix} Δ n_{12} \\ d_{12} \end{matrix}] = {[\begin{matrix} S_{B_{1}} & S_{d_{1}} \\ S_{B_{2}} & S_{d_{2}} \end{matrix}]}^{- 1} [\begin{matrix} Δ λ_{1} \\ Δ λ_{2} \end{matrix}]}}, \underset{LM 2 : from λ_{1}, λ_{3}}{\underset{︸}{[\begin{matrix} Δ n_{13} \\ d_{13} \end{matrix}] = {[\begin{matrix} S_{B_{1}} & S_{d_{1}} \\ S_{B_{3}} & S_{d_{3}} \end{matrix}]}^{- 1} [\begin{matrix} Δ λ_{1} \\ Δ λ_{3} \end{matrix}]}}, \underset{LM 3 : from λ_{2}, λ_{3}}{\underset{︸}{[\begin{matrix} Δ n_{23} \\ d_{23} \end{matrix}] = {[\begin{matrix} S_{B_{2}} & S_{d_{2}} \\ S_{B_{3}} & S_{d_{3}} \end{matrix}]}^{- 1} [\begin{matrix} Δ λ_{2} \\ Δ λ_{3} \end{matrix}]}}

For a system based on more resonances, the linear model can be employed to obtain i number of estimates (d_i and Δn_i) that are related to the true values (d̂ and $\hat{Δ n}$ ) by

d_{i} = C_{d i} \hat{d} + ε_{d i}

Δ n_{i} = C_{n i} \hat{Δ n} \pm ε_{n_{i}}

where C_di and C_ni are weighting factors, relating these estimates to the true values (d̂ and

\hat{Δ n}

), and the errors (due to the effect of noise) in the measured LM results are represented by ε_di and ε_ni, for the adlayer thickness and bulk RI change, respectively.

The estimated adlayer thickness and bulk RI change based on the application of the LM to a multiple systems of linear equations can be be represented by the following vectors

d = [\begin{matrix} d_{1} \\ ⋮ \\ d_{i} \end{matrix}], Δ n = [\begin{matrix} Δ n_{1} \\ ⋮ \\ Δ n_{i} \end{matrix}]

Herein, we use the MLE method to estimate the true values based on the multiple LM estimates, Considering that the noise in the system follows a normal distribution, the error terms can be represented by a Gaussian distribution with zero mean as ε_d ∼ 𝒩(0, R_d), and ε_n ∼ 𝒩(0, R_n) for adlayer thickness and bulk RI change, respectively. R_d and R_n denote the variance associated with the adlayer thickness and bulk RI change, respectively, determined by the LM method. Therefore, it is evident that the calculations consider the noise associated with the measured resonance wavelength shifts in real time, and the errors include covariances that account for a system of correlated noise sources. In matrix notation, this can be represented by the following symmetric matrices

R_{d} = [\begin{matrix} R_{d_{11}} & \dots & R_{d_{1 i}} \\ ⋮ & ⋱ & ⋮ \\ R_{d_{i 1}} & \dots & R_{d_{i i}} \end{matrix}]; R_{n} = [\begin{matrix} R_{n_{11}} & \dots & R_{n_{1 i}} \\ ⋮ & ⋱ & ⋮ \\ R_{n_{i 1}} & \dots & R_{n_{i i}} \end{matrix}]

Where the main diagonals of these matrices represent the variances of the LM estimates, and the covariances among them are symmetrically distributed above and below the main diagonal, i.e. R_{d_ij} = R_{d_ji} and R_{n_ij} = R_{n_ji}. These matrices are directly based on the real time experiments.

Now, we apply the MLE method, employing the LM estimates and including the effect of noise as given in Eq. (2) and Eq. (3). For simplicity, we apply the method on the adlayer thickness results, and the same steps can be followed in estimat the bulk RI change. Since we have considered a normal distribution for the noise, the likelihood of obtaining the estimated adlayer thickness based on the linear model (d_i), given the true value (d̂) can be obtained by multiplying the normal distributions for these estimates as follows

\prod_{i} 𝒩 (d_{1}, \dots d_{i} | C_{d 1} \hat{d}, \dots C_{d i} \hat{d}, R_{d}) \equiv \frac{1}{{(2 π)}^{i / 2} {| R_{d} |}^{1 / 2}} exp (- \frac{1}{2 R_{d}} \sum_{i} {(d_{i} - C_{d i} \hat{d})}^{2})

In matrix notation, this multivariate normal distribution can be represented by

P (d | \hat{d} C_{d}, R_{d}) = \frac{1}{{(2 π)}^{i / 2} {| R_{d} |}^{1 / 2}} exp (- \frac{1}{2} {(d - \hat{d} C_{d})}^{T} R_{d}^{- 1} (d - \hat{d} C_{d}))

According to the MLE technique, the estimate that maximizes this likelihood is obtained when its derivative with respect to the true value approaches zero [11]. For simplicity, we obtain the log of the above likelihood as follows

\ln (P (d | \hat{d} C_{d}, R_{d})) = - \frac{i}{2} \ln (2 π) - \frac{1}{2} \ln | R_{d} | - \frac{1}{2} {(d - \hat{d} C_{d})}^{T} {R_{d}}^{- 1} (d - \hat{d} C_{d})

Now, the true value (d̂) can be estimated such that the derivative of the log likelihood with respect to this true value approaches zero [11].

\frac{\partial}{\partial \hat{d}} \ln (P (d | \hat{d} C_{d}, R_{d})) \equiv \frac{\partial}{\partial \hat{d}} ({(d - \hat{d} C_{d})}^{T} {R_{d}}^{- 1} (d - \hat{d} C_{d})) = 0

This can be solved to obtain the true value (d̂) as follows

\hat{d} = \frac{C_{d}^{T} R_{d}^{- 1} d}{C_{d}^{T} R_{d}^{- 1} C_{d}}, R_{d}^{- 1} = [\begin{matrix} R_{d_{11}}^{- 1} & \dots & R_{d_{1 i}}^{- 1} \\ ⋮ & ⋱ & ⋮ \\ R_{d_{i 1}}^{- 1} & \dots & R_{d_{i i}}^{- 1} \end{matrix}]

The bulk RI change can be estimated in a similar manner as the above based on Δn, C_n and R_n. Herein, the adlayer thickness and bulk RI change, determined by the LM (d and Δn) are given the same weight; therefore, $C_{d}^{T} = C_{n}^{T} = [\begin{matrix} 1 & 1 \dots & 1 \end{matrix}]$ and the following formulas can be extracted for the estimated adlayer thickness and bulk RI change

\hat{d} = \frac{\sum_{m = 1}^{i} \sum_{k = 1}^{i} d_{m} R_{d_{m k}}^{- 1}}{\sum_{m = 1}^{i} \sum_{k = 1}^{i} R_{d_{m k}}^{- 1}}, \hat{Δ n} = \frac{\sum_{m = 1}^{i} \sum_{k = 1}^{i} Δ n_{m} R_{n_{m k}}^{- 1}}{\sum_{m = 1}^{i} \sum_{k = 1}^{i} R_{n_{m k}}^{- 1}}

In the case of the three-resonance nanorod structures, proposed here, substituting for i=3 in Eq. (5), the adlayer thickness and bulk RI change can be estimated as

\hat{d} = \frac{(R_{d 11}^{- 1} + R_{d 12}^{- 1} + R_{d 13}^{- 1}) d_{1} + (R_{d 12}^{- 1} + R_{d 22}^{- 1} + R_{d 23}^{- 1}) d_{2} + (R_{d 23}^{- 1} + R_{d 23}^{- 1} + R_{d 33}^{- 1}) d_{3}}{R_{d 11}^{- 1} + R_{d 22}^{- 1} + R_{d 33}^{- 1} + 2 (R_{d 12}^{- 1} + R_{d 13}^{- 1} + R_{d 23}^{- 1})}

\hat{Δ n} = \frac{(R_{n 11}^{- 1} + R_{n 12}^{- 1} + R_{n 13}^{- 1}) Δ n_{1} + (R_{n 12}^{- 1} + R_{n 22}^{- 1} + R_{n 23}^{- 1}) Δ n_{2} + (R_{n 13}^{- 1} + R_{n 23}^{- 1} + R_{n 33}^{- 1}) Δ n_{3}}{R_{n 11}^{- 1} + R_{n 22}^{- 1} + R_{n 33}^{- 1} + 2 (R_{d 12}^{- 1} + R_{d 13}^{- 1} + R_{d 23}^{- 1})}

3. Corrected sensitivity matrices for the linear response model

The sensitivity to adlayer thickness depends on the refractive index of the analyte (n_a) and the bulk RI sensitivity of the nanorods. Therefore, it needs to be recalculated if other biological samples are considered. Established methods based on the linear response model have previously considered specific analytes, and employed sensitivity factors based on simulating that specific analyte (using reported values for size and RI) [9,10]. However, this requires tedious numerical modeling to obtain new values. Here, we present a method to avoid the repeated numerical calculations, by calculating the adlayer thickness sensitivity based on the measured data.

The maximum sensor response, Δλ_max, at each resonance is achieved when the adlayer thickness reaches the saturation, d ≪ l_d, where l_d is the electromagnetic (EM) decay length associated with these resonances. The sensor response is related to the adlayer thickness by the following equation [12,13]

Δ λ (d) = Δ λ_{\max} [1 - \exp (- 2 d / l_{d})]

The linear response model is valid for a thin adlayer thickness, d ≤ l_d/10, and the sensor response is related to the adlayer thickness based on the sensitivity to adlayer thickness S_d as follows

Δ λ (d) = S_{d} d

Substituting l_d/10 for d in Eq. (8) and Eq. (9), we obtain

Δ λ (l_{d} / 10) = S_{d} l_{d} / 10 \equiv 0.18 Δ λ_{\max}

from which the adlayer sensitivity can be evaluated as follows

S_{d} = 1.8 Δ λ_{\max} / l_{d}

This can be related to the bulk RI sensitivity and the refractive indices for the buffer and analyte as follows

S_{d} = 1.8 S_{B} (n_{a} - n_{B}) / l_{d}

where n_a and n_B are the refractive indices for the adlayer and the buffer solution.

4. Methods

We used the FEM to model the gold nanorods and obtain the sensitivity and EM decay length for each resonance. The dielectric properties for gold were obtained from Johnson and Christy experimental data [14]. The longitudinal mode was excited by a vertical incident plane wave polarized along the long axis of the gold nanorods. The rods were modeled based on both perfectly matched layer and periodic boundary conditions. An adlayer of a thickness (d) was introduced to calculate the EM decay length as the resonance wavelength shift was tracked with changing the thickness until the shift is saturated. The simulation domain was discretized using triangular mesh and the nanorods and the adlayer were discretized using hexagonal mesh of 1 nm element size.

The gold nanorods were fabricated using the nanoimprint lithography method: a glass substrate was coated by 50 nm thick cyclic olefin copolymer (COC) by spin coating, and the coated substrate was imprinted by a silicon (Si) mould under 8 MPa pressure and 150 °C for 300 seconds. The imprinted substrate was then coated with 5 nm and 30 nm thick chromium (Cr) and gold, respectively. The nanorods were formed after a lift-off process. Figure 1 shows a Scanning Electron Microscopy (SEM) image for the fabricated noanord structure.

Fig. 1 SEM images for the fabricated nanorod structures of a width 70 nm and various lengths as (a) 120 nm, (b) 150 nm, and (d) 210 nm.

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Replica moulding method was used to fabricate the fluidic channels based on polydimethylsiloxane (PDMS). A silicon wafer was spin coated with SU-8 negative resist and UV photolithography was used to pattern the channels using a transparent photomask obtained from [15]. The unexposed SU-8 resist was developed in SU-8 developer and the wafer was blown dry with nitrogen and post-baked, forming the master for the soft lithography method [16]. The master was then coated with a monolayer formed by perfluorooctyltriethoxysilane (Sigma-Aldrich, Oakville, ON, Canada) by siloxane bonding in a desiccator connected to a vacuum line in a fume hood (for 30 minutes). This prevents the PDMS from sticking to the master as the surface becomes hydrophobic. A 70 g of Sylgard 184 base was mixed with 7 g of curing agent (10:1 ratio), and the mixture was poured on the master in a petri-dish. The air bubbles were removed by placing the mater in a desiccator and connecting it to the vacuum line in a fume hood. After removing the air bubbles, the petri-dish (containing the master and PDMS) was placed in an oven at 70 °C overnight. The replicated PDMS was then peeled off the master, and placed in an oxygen plasma for 60 seconds to transform its surface from hydrophobic to hydrophilic, increasing the bond with the nanorod glass substrate.

5. Numerical validation

COMSOL Multi-physics was used to calculate the shifts in the resonance wavelengths with different bulk refractive indices to calculate the sensitivity at each resonance. Figure 2(a) shows the simulation layout based on the periodic boundary condition and ports to calculate the scattering parameters ∼ S₂₁, which are translated into transmission efficiency as 10^S₂₁/20. The simulation was validated by simulating only the glass substrate of a refractive index of 1.5, and calculating the transmission efficiency. Another verification was performed by comparing the results based on nanorods without a substrate to those obtained based on nanorods in an integrating sphere with perfectly matched layer shown in Fig. 2(b).

Fig. 2 (a) Schematic for simulating periodic array of nanorods covered by an adlayer and a bulk RI of n_B. Periodic boundary conditions were enforced such that the structure is periodic in the xy plane. The structure is excited using port 1 (lower xy plane), and the transmitted light is calculated using port 2. (b) Simulating a single nanorod, by using a perfectly matched layer (PML) over an integrating sphere to calculate the extinction efficiency. The nanorod is excited by a plane wave polarized along the z axis and propagating in the negative x direction.

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After validating the simulation set-up, the sensitivity to bulk RI change (S_B) was calculated based on the corresponding transmission dip locations for all the resonances as shown in Fig. 3. As expected, the third resonance (at 1000 nm) exhibited the highest sensitivity to bulk RI change which is attributed to the long EM decay length. To estimate the EM decay length, the adlayer thickness was varied from 6 – 25 nm, and the corresponding shift in each resonance wavelength was tracked to plot the resonance shift against adlayer thickness. The EM decay lengths are then estimated by fitting the simulated data using Eq. (8), as shown in Fig. 4. As well, the simulated adlayer sensitivity can be determined as the slope of the curves in the linear regime (0 < d < l_d/10). To investigate the effect of the noise on the results, an adlayer of 5 nm thickness and 1.4 refractive index was introduced to the nanorods in the periodic simulation layout, and the corresponding shifts in the resonance wavelengths were determined. Last, noise (with different levels) was simulated by adding uncertainties (various σ_λ) to the simulated wavelength shifts. The linear response models and the proposed MLE method were then employed to estimate the input parameters used in the FEM simulation. The signal to noise ratio (SNR) based on the estimates were calculated as the ratio between their mean and standard deviation, as shown in Figs. 5(a) and 5(b), respectively, revealing that the MLE method can improve the precision of the linear response model results. The MLE method is less affected by the fluctuations in the resonance wavelengths as the overall variance becomes lower than any of those associated with the results based on the linear response model, applied to various linear systems, LM₁ (λ₁, λ₂), LM₂ (λ₁, λ₃) and LM₃ (λ₂, λ₃).

Fig. 3 Simulated transmission curves, demonstrating resonance wavelength shift with bulk RI change at (a) λ₁=705 nm, (b) λ₂=821 nm, and (c) λ₃=1000 nm. (d) Shifts in the resonance wavelengths vs bulk RI change to extract the bulk RI sensitivity for each resonance.

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Fig. 4 Resonance wavelength shift against adlayer thickness change, based on the simulated results shown in the insets, for (a) the first resonance (λ₁=705 nm), (b) the second resonance (λ₂=821 nm), and (c) the third resonance (λ₃=1000 nm). The EM decay length (l_d) for each resonance is extracted such as Eq. (8) provides the best fit to the resonance wavelength shift vs adlayer thickness, and the sensitivity to adlayer thickness change (S_d) is calculated as the slope of each curve at the linear regime (d ∼ l_d/10).

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Fig. 5 Top panel: calculated SNR based on (a) the estimated adlayer thickness to its standard deviation, and (b) the estimated bulk RI change to its standard deviation. The linear response model and the MLE method were applied to the simulated shifts in resonance wavelengths Δλ_i with added uncertainties σ_λ_i such that SNR (Δλ_i) = Δλ_i/σ_λ_i. Bottom panel: the percentage error associated with each method in (c) the estimated adlayer thickness and (d) the bulk RI change using Eq. (12) based on the true values used in the simulation.

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The accuracy of the methods can be determined based on the percentage error in the estimates compared with the true values that were used in the simulation as follows

∊_{\hat{x}} % = \frac{\hat{x} - x}{x} \times 100

where ∊_x̂ % is the percentage error in the estimates x̂ (d̂ or

\hat{Δ n}

) with respect to the true values x, representing d or Δn. The error in the estimates was calculated based on the estimated adlayer thickness and bulk RI change based on various uncertainties in the resonance shifts. The averaged error in the estimated adlayer thickness and bulk RI change is shown in Figs. 5(c) and 5(d), respectively. These results suggest that the MLE method achieves the best accuracy among the results obtained by the linear response model, based on the simulated data.

6. Measured results

This section presents the measured results based on bulk RI changes and surface binding experiments. Figure 6 shows the experimental set-up used for the sensing experiments. The fabricated gold nanorods substrate was cleaned by DI water and ethanol solution, blown dry with nitrogen, and plasma treated to remove any biological contaminant. The substrate was then incubated in 10 mM phosphate buffer solution (pH 7.2) of 200 μM biotin-hpdp for biotin immobilization as instructed in [9,17]. A 0.2 mg/mL streptavidin solution was prepared in a 50 mM Tris-buffer solution (pH 8.0) according to [9,17]. The Tris buffer was also used as a baseline for the sensing experiment. Both streptavidin and hpdp reagents were obtained from [18]. Cary 5000 spectrometer was used to measure the extinction curves while introducing the solutions into the nanorod structures via the PDMS fluidic channel and an automatic pump (Harvard Apparatus–PicoPlus) with 200μL/min flow speed. The resonance locations (centroids) were determined based on the dynamic-baseline centroid method [19].

Fig. 6 Experimental set-up used to measure the transmission spectra associated with the nanorod structures. The inset is an exploded view for the PDMS fluidic channel integration with the gold nanorod substrate for injecting the biological samples.

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Figure 7 shows the measured results for the three-resonance nanorod structures based on ethanol solution and biotin-streptavidin binding. The shift of each resonance is tracked in real time to investigate the self-referencing with bulk and surface binding experiments.

Fig. 7 Real time response to bulk RI changes and biotin-streptavin binding events based on three-resonance nanorod structures. The numbers on the graph represent the following: [1] DI water, [2] 8% ethanol solution, [3]16% ethanol solution, [4] Buffer, and [5] Streptavidin solution.

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The sensor was calibrated for the bulk and adlayer sensitivities. The bulk RI sensitivity was calculated based on the ethanol solutions of known concentrations and refractive indices. The measured sensitivities are lower than the calculated counterparts. This is attributed to the biotin layer as the distance between the nanorods and the ethanol solutions is increased after functionalizing the nanorods. A similar behaviour was previously observed for the nanorods [20]. The sensitivity to adlayer thickness is directly related to the bulk RI sensitivity, and hence it needs to be corrected accordingly. We propose a method to correct for this discrepancy using Eq. (11). Both simulated and corrected values are shown in Figs. 8(a) – 8(c) and Figs. 8(d) – 8(f).

Fig. 8 Top panel: simulated versus measured shift in resonance wavelengths against bulk RI changes. The bulk RI sensitivities, S_B and S′_B (nm/RIU), were determined as the slope of each graph. Bottom panel: simulated and measured resonance shifts versus the adlayer thickness based on the simulated (S_d) and corrected (S′_d) adlayer sensitivities. Each corrected sensitivity (S′_d) was obtained using Eq. (11) based on the measured bulk RI sensitivity S′_B for each resonance.

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Now, we obtain sensitivity matrices based on the true values, accounting for the changes due to fabrication and experimental conditions, in the form

S = [\begin{matrix} S_{B 1}^{'} & S_{d 1}^{'} \\ S_{B 2}^{'} & S_{d 2}^{'} \end{matrix}]

Similarly, two sensitivity matrices are obtained based on the bulk and adlayer sensitivities for the following combinations: (λ₁, λ₃) and (λ₂, λ₃). These matrices must be non-singular and well conditioned to be valid for the LM calculations [11]. The determinant of each sensitivity matrices is nonzero, hence non-singular. We also calculated the condition number κ(S) by first normalizing the columns of each sensitivity matrix, obtaining a normalized matrix s, and then multiplying the norms of the normalized matrix and its inverse as follows

κ (S) = ‖ s ‖ ‖ s^{- 1} ‖

The calculated condition number of the sensitivity matrices are 25.6, 13 and 25.8 for LM₁(λ₁, λ₂), LM₂(λ₁, λ₂) and LM₃(λ₂, λ₃), respectively. These values mean that the matrices are well conditioned (κ(S) < 100). These values are about the same as those obtained based on the simulated sensitivity matrices ∼25.6, 13, 25.7, implying that correcting the sensitivity to the adlayer thickness results in stable condition numbers, and hence stable numerical accuracy. This can be useful in optimizing the nanorod structures as the condition numbers based on the measured results agree well with those based on the simulated sensitivity matrices, whilst the measured bulk RI sensitivities deviated from the simulated counterparts.

It is also important to investigate the sensor figure of merit (FoM) based on the full width at half maximum (FWHM) given by ∼ FoM = S_B/FWHM, revealing FoMs of 3.1, 2.7 and 2.6 for the first, second and third resonances, respectively. These values exceed reported values for gold nanorods fabricated by the electron beam lithography ∼ 1.9 [8], and are comparable to those associated with chemically synthesized nanorods ∼ 1.7 – 2.6 [21]. This is attributed to the increased sensitivity of the nanorods presented in this paper 289 – 382.37 nm RIU −1 due to the increased width of the nanorods as increasing the rods minor axis was previously linked to increasing the EM decay length and hence the bulk RI sensitivity [20].

Another interesting parameter to consider is the figure of merit based on adlayer-bulk differentiation [6,8] that can be determined as χ = |S_B₁/S_d₁ − S_B₂/S_d₂|; the proposed multiple resonance rod structures revealed the following values based on the measured sensitivities: 0.18 with LM(λ₁, λ₂), 0.38 with LM(λ₁, λ₃) and 0.17 with LM(λ₂, λ₃) compared to 0.18, 0.38 and 0.17 based on the simulated sensitivities, exhibiting an improved stability system as compared to established dual-resonance nanorod structures whose measured figure of merit differed from the simulated counterpart ∼ 0.25 vs 1, respectively [8].

The estimated adlayer thickness and bulk RI changes using the linear response model based on the corrected sensitivity matrices are shown in Figs. 9(a) – 9(c). The estimates based on the MLE method were obtained using Eq. (6) and Eq. (7) with the mean and variance of the estimates based on the linear response models. The estimates based on the MLE estimation are shown in Fig. 9 (d), exhibiting improved accuracy and precision based on both bulk RI and surface binding experiments.

Fig. 9 Estimated adlayer thickness (left y-axis) and bulk RI change (right y-axis) based on the measured results after applying (a) LM₁(λ₁, λ₂), (b) LM₂(λ₁, λ₃), (c) LM₃(λ₂, λ₃), and (d) the MLE method. The cycles on the graph represent the following: [1] DI water, [2] 8 % ethanol solution, [3] 16 % ethanol solution, [4] Buffer, and [5] Streptavidin solution.

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We used the data retrieved from Fig. 9 to investigate both the accuracy and precision of the proposed method compared with the linear response model. Since ethanol solutions with known refractive indices were used in the first part of the experiment, we used the reported refractive indices [22] as a reference to calculate the errors in the estimated RI change. Figure 10(a) shows both the error and standard deviation of the bulk RI change estimated by the linear response model and the MLE method. The MLE method is shown to improve the accuracy in estimating RI changes of 0, 5.1 × 10⁻³ and 1.1 × 10⁻² with improved precision (improved RI resolution); the averaged error is 6.1 × 10⁻³, 1.7 × 10⁻³, 2.2 × 10⁻³ and 9.1 × 10⁻⁴ for LM₁, LM₂, LM₃ and the MLE method, respectively. This indicates that the accuracy can be increased by one order of magnitude employing the MLE method. The averaged standard deviation of the estimated RI change by the LM₁, LM₂, LM₃ and the MLE method was 3.6 × 10⁻³, 1.9 × 10⁻³, 5.4 × 10⁻³ and 1.2 × 10⁻³, respectively. We used the measured data for biotin-streptavidin binding experiment to obtain the mean and standard deviation in the estimated adlayer thickness and bulk RI change as shown in Figs. 10(b) and 10(c). The MLE method exhibits a decreased standard deviation based on both estimated adlayer thickness and bulk RI change during the baseline phase and both association and dissociation phases. The estimated adlayer thickness was 6 nm and 4 nm during the association and dissociation phases, respectively, as shown in Fig. 10(b). This suggests that the gold nanorods were not completely functionalized, and there were empty locations that have not been occupied by biotin. The linear response models with higher condition number and lower cross sensitivity figure of merit, χ, revealed the worst results in terms of accuracy and precision. This, however, did not impact the overall results based on the MLE method. The averaged standard deviation was 1.3 nm, 0.54 nm, 1.22 nm and 0.37 nm for LM₁, LM₂, LM₃ and the MLE method, respectively. The MLE method improves the precision and accuracy by factors of 3 and 3.7, respectively.

Fig. 10 (a) Error in estimated RI change after applying the linear response model (LM₁, LM₂, LM₃), and the MLE method to the measured results. The error was calculated as the difference between the estimated RI changes and the reported counterparts based on refractometer results for ethanol solutions of various concentrations (0%, 8%, and 16 %). The data is obtained from the first five steps in Fig. 9 (steps: 1, 2, 1, 3, 1). (b) Estimated adlayer thickness and (c) bulk RI change after applying the linear response model (LM₁, LM₂, LM₃) and the MLE method to the surface binding experimental results. The error bars denote the standard deviation of the estimated values obtained from the last three steps in Fig. 9 (steps: 4, 5, 4).

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7. Conclusion

This paper presented a method to improve the accuracy of estimating the adlayer thickness and bulk RI change. The method is based on multiple resonance sensors to generate more than a single system of linear equations, and applies the MLE method to the solutions obtained by these systems to maximize the likelihood of the results with lower variance. The paper also introduced a method to generate sensitivity matrices based on the experimental conditions, reducing errors based on the mismatch between the calculated and measured sensitivities. Moreover, this can reduce the numerical calculations if different biological samples are measured, or if the number of resonances is increased. The linear response model is limited to biological adlayers of a maximum thickness of ∼ l_d/10. However, LSPR biosensors are aimed to detect such small biological samples. Although the sensing experiments yielded noisy results (based on resonance shift) when compared to some reported measurements, the MLE method improved the results based on the estimated adlayer thickness and bulk RI change. The precision and accuracy were improved by factors of 3 and 3.7 when compared to the averaged results obtained by the linear response model, proving that the MLE method can leverage improved self-referenced LSPR sensors. Increasing the number of resonances would improve both accuracy and precision of the estimates. The averaged FoM associated with the fabricated nanorods was 2.8 RIU⁻¹ and the averaged figure of merit based on the adlayer/bulk RI cross sensitivity, was 0.24; increasing the FoM would further improve the precision, and increasing the adlayer/bulk RI cross sensitivity figure of merit can achieve improved accuracy. Additional improvement in the precision and accuracy can be achieved by optimizing the nanorods based on these parameters.

Acknowledgements

We would like to thank Professor Bruce Lennox and Professor David Juncker (Chemistry and Biomedical Engineering Departments, McGill University) for providing access to the spectroscopy and fabrication facilities. COMSOL Multiphysics package was provided by CMC Microsystems.

References and links

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Improved self-referenced biosensing with emphasis on multiple-resonance nanorod sensors

Abstract

1. Introduction

2. Concept of self referencing based on multiple resonances

3. Corrected sensitivity matrices for the linear response model

4. Methods

5. Numerical validation

6. Measured results

7. Conclusion

Acknowledgements

References and links

Cited By

Figures (10)

Equations (22)

Optics Express