1. Introduction

Surface plasmon resonance (SPR) continues to be the methodology of choice for many biosensing applications, as it provides rapid, label free method of monitoring bioaffinity reactions on the sensor surface. Efforts to improve its sensitivity span a range of approaches from investigations of new, increasingly sophisticated schemes such as Mach Zehnder interferometry with phase interrogation [1,2], heterodyne interferometry [3–5], waveguide coupled sensors [6] and nanoparticle enhancement methods [7]. However much less emphasis has been placed on deriving the maximum advantage from the information provided in these and other, more traditional systems. This paper addresses this deficiency and demonstrates how to extract this information from a fairly conventional SPR system with angular readout scheme. Thus we demonstrate that there is underutilized capacity to provide much more accurate data in the presence of interfering noise in a large class of SPR systems. The method developed here is applicable to a broad class of SPR systems involving a conventional angular interrogation [8,9] but could also be easily adapted for wavelength interrogation [10] and intensity measurements.

The presented study has been carried out on a traditional as well as one of the new generation SPR systems with nanoparticle enhancement. Dielectric particles such as latex beads [11–13], polymer particles [14, 15] produce an increase in the change to the angle or wavelength of the SPR reflectance minimum through the contribution of an additional term to the wave vector matching condition. The inclusion of additional absorbing metallic nanoparticles within the sensing layer significantly enhances the sensitivity of surface plasmon resonance (SPR) sensors based on angular and wavelength interrogation. Experimental work by a number of investigators has confirmed significant changes in the SPR reflectivity curve upon inclusion of metallic nanoparticles within the sensing layer [16–24]. Generally, changes include a broadening of the curve, increase in the minimum reflectance value and large shifts in the minimum angle or wavelength of resonance. The nature of these effects was found to depend on the composition of the nanoparticles and the thin metal SPR-supporting layer [20, 21], the wavelength of excitation and the size [23, 25] and distance of the nanoparticles from the surface [16, 17, 19]. The environment around the nanoparticles has additional impact on the reflectance output through layer interaction [21]. The changes in the reflectance curve are most pronounced at excitation wavelengths close to the absorption peak of the nanoparticles [17]. These large, complex changes in the SPR reflectance curve resulting from the inclusion of nanoparticle tags or structures can be employed to increase the resolution of SPR based sensing systems.

The resolution of a sensor is defined as the minimum change in the parameter of interest that can be resolved by the sensor. On the other hand, the sensitivity of an SPR sensor is defined as the change in the value of the monitored SPR parameter (for example intensity, resonant angle) with respect to the parameter of determination (for example the sample refractive index). A comparison of the sensitivity and resolution figures for angular, wavelength and intensity based SPR interrogation schemes has been reviewed in reference [26]. With a prism based attenuated total reflectance coupling scheme (the most common sensing scheme for chemical and biological sensing used in this work), resolution values of 5× 10⁻⁷, 2 × 10⁻⁵ and 5 × 10⁻⁵ refractive index units (RIU) were reported for typical angular, wavelength and intensity interrogation schemes for a wavelength of 630 nm. It has also been shown that the sensitivity of an SPR sensor is dependent upon the wavelength of light used (in conjunction with the interrogation method) [27, 28] as well as the choice of metal used to support the resonant plasmon mode [29]. The resolution values as high as 5.5 × 10⁻⁸ RIU were reported in a phase-based SPR measurement system [30]. Due to the necessary indirect measurement of phase change using interferometry (and hence light intensity), both measurements of phase change and intensity variations were fundamentally limited by photon statistics. Thus, despite the optimism expressed in the literature for unprecedented sensitivity advancement of SPR systems through use of phase-based measurements, this approach is in fact limited as far as resolution is concerned, in fundamentally the same way as traditional measurement approaches. It is therefore reasonable to focus on deriving the most from the data obtained in the more traditional intensity, angle and wavelength based SPR measurement that is simple and widely utilized.

In SPR based sensing a number of approaches (beyond a simple visual inspection) have been employed to distinguish between two SPR reflectance curves generated for the analyte of interest and the control sample without the analyte. One approach focuses on the angle (wavelength) at which the coupling to the plasmon mode is maximized. This corresponds to the minimum of the reflectance curve. The accuracy of the minimum angle (wavelength) leads to the resolution of a typical sensor in the order of 10⁻⁶ to 10⁻⁷ RIU change depending upon excitation wavelength [26]. Another approach is to investigate the difference between the reflectance values of two curves at a set angle (or wavelength) [26]. The difference is maximized at the highest gradient on the front slope of the SPR reflectance curve and has produced a resolution of 10⁻⁵ RIU change depending upon excitation wavelength [26]. Both these approaches do not utilize the full information contained in the two reflectance curves.

Hypothesis testing is a new approach to distinguishing SPR reflectance curves [31]. It provides a sensitive test for differences between reflectance curves caused by very small variations in sample refractive index, where the difference between the two curves is significantly influenced by noise present in the measurement system. It is particularly suitable for ultrasensitive detection of analytes, where the noise level compared to the difference of the two SPR curves is such that they appear no longer distinguishable to the naked eye. Hypothesis testing makes use of all the information available in the measured data points and it provides a decision-making method for these situations. This makes it possible for greater system resolution to be achieved.

2 Approach

2.1 SPR system under consideration

The SPR system under consideration is a variation of a conventional angle based SPR system. In this setup a high refractive index prism is interfaced with a thin metal film. The monochromatic light beam falls onto the metal layer from the high refractive index side under attenuated total reflection conditions. In this (Kretchmann) configuration surface plasmon waves can be excited when the propagation constant of the evanescent wave and the surface plasmon satisfy a matching condition. This occurs for a certain angle of incident light, and at the vicinity of this angle the reflected light is strongly attenuated. The matching condition is affected by any layer with a different refractive index adsorbing at the surface of the metal film, thus making it possible to detect minute changes in the refractive index at the metal surface.

In this work we used BK7 glass covered with a 50 nm gold metal film for the proof-of principle demonstration of hypothesis testing approach in a conventional SPR system. Further, a more complicated sequence of layers was used to establish the sensitivity achievable by using hypothesis testing with a nanoparticle-inclusive sensor (see Table 1 and 2 for the parameters used in our modeling). The layers in the simulated nanoparticle-inclusive structureincluded a glass layer consisting of a LaSF9 prism and a 50 nm thick metal layer, standard in traditional SPR, either gold or silver coated with gold. A nanoparticle layer has been located on top of the metal film. A common approach to introducing nanoparticles is attach one of the binding molecules to the nanoparticle and introduce this to the sensor surface where the complimentary binding molecule is located, often formed as a self-assembled monolayer (SAM). Our simulations assume this approach using the same simple biotin-streptavidin binding reaction measured by Li et al [32] using biotinylated gold nanoparticles with diameter of 24 nm. We also assumed that the detection of analytes has been taking place in water. A collimated light source of varying wavelength was assumed to be uniform and steady, that is all changes in reflectance beyond noise are assumed to be from sample layer refractive index changes.

Table 1. Summary of layers in multi-layered nanoparticle simulation system at λ = 632.8 nm.

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Table 2. Summary of layers in multi-layered nanoparticle simulation system at λ = 543 nm.

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2.2 Modeling of the SPR reflectance

We carried out the modeling of our SPR system with nanoparticles by using Fresnel equations and the transfer matrix method [33–36] to describe the reflectivity of the multilayer system. The multi layer model SPR system is defined with n optically isotropic and homogeneous parallel plane layers. These are stacked in the z-direction with layer 1 extending to −∞ and layer n extending to +∞ and dielectric constants of ε₁,ε₂...ε_n. The thickness of each layer is defined by the z-positions of the boundaries for that layer so layer j has thickness given by

 

Fig. 1 Multilayer system under consideration in the modeling

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The reflectance for layers j to l is given by:

In order to model the effects of nanoparticles on the reflectivity of an SPR system one needs to adequately describe the dielectric function in a composite medium incorporating the nanoparticles. To this aim we used an effective medium theory earlier adopted for SPR sensors including nanoparticle layers to account for their effect on reflectivity. We selected the modified Maxwell-Garnett approach described by Garcia et al [37], which includes geometry and homogeneity factors of the composite layer to more accurately represent the plasmon absorption bands associated with the nanoparticles. The effective permittivity of the composite layer is characterized by both real and imaginary components ${\epsilon }_{eff}=\epsilon {\text{'}}_{eff}+\epsilon \text{'}{\text{'}}_{eff}$. The modified Maxwell-Garnett approach describes the real and imaginary parts of the permittivity as [32]

The parameter K accounts for the effect if the electric field of one particle on another. If the particles are widely dispersed and dipole-dipole interaction may be assumed to be negligible, the value of K tends to zero. The subscripts _np and _m refer to the permittivities for the nanoparticles and the surrounding material, while f_np is the nanoparticle volume fill fraction.

The alternative models such Maxwell-Garnett effective medium theory [38] and the Lorentz oscillator model, applied to SPR systems [39, 40] were independently verified and found less accurate than the modified Maxwell-Garnett approach.

2.3. Modeling of noise

In our simulations we assumed that the reflected light signal is collected by a Larry-USB 1024 - D7231 array detector from AMES Photonics Inc.. The number of data points simulated (384) was based upon the array detector and laser specifications (1024 pixels over a length of 7.99 mm, expanded laser diameter of 3 mm). A noise function was generated using a normal random number generator in MATLAB, based on a defined mean and standard deviation. The mean was set at zero and the standard deviation noise value was calculated from the noise parameters of the detector as follows. Firstly the noise on the detector, N, was calculated based on a half full well capacity signal of $n_{signal}=1,630,000$electrons:

Here the parameters $n_{dark}$ and $n_{readout}$ were taken from the Larry-USB 1024 - D7231 detector specifications (AMES Photonics Inc. 2004) to be 113 and 1022, respectively. From this, the value of the signal to noise was calculated to be S/N=995. Further, in order to reduce the noise, signal averaging over m= 100,000 SPR curves was carried out, with an integration time of 1 ms, and with the total integration time of 100 s. This improves the signal to noise by the factor of m, thus producing S/N= 315,000 and a noise amplitude of N/S for a reflectance signal of one. Random white noise with this amplitude was added to the simulated SPR reflectance curves.

2.4. Statistical analysis of SPR curves: hypothesis testing

The detection of trace analytes by SPR depends on the ability to differentiate between reflectance curves from samples with and without the analyte of interest. The hypothesis testing is a simple method to distinguish curves that may otherwise be indistinguishable due to noise [41]. In the hypothesis testing method the two SPR curves (with and without the analyte) are represented by two series of observations; F_i describing the SPR curve for the analyte with noise and G_i describing the SPR curve without the analyte, also with noise, both with n data points per curve

The symbols ${\epsilon }_i$ and ${\eta }_i$ denote independent measurement errors due to noise that are both assumed to have zero mean and standard deviation of σ. It is also assumed that the experimental errors at various measurement angles ${\theta }_i$ are statistically independent. A new set of data $D_i$ (the sample) is obtained by taking the difference of the two curves $D_i=F_i-G_i$. The detection of the analyte is based on the judgment whether the difference between the two curves is zero or non –zero – in the presence of the interfering noise. The null hypothesis to be tested is that there is no difference between the two curves. An alternative hypothesis is that the difference between the two curves may be non-zero (in either direction). The test statistics is then established based on a large sample test. A large sample size (n) of our sample $D_i$ allows the assumption that the standard deviation of the sample is a satisfactory approximation to the true value of the standard deviation. The test statistics is the produced by calculating the Z value as

3. Results and discussion

3.1. Application of hypothesis testing to a conventional SPR system

The simulations of the SPR reflectance curve were based on the Fresnel reflectivity equations for a multilayered structure. Such curves are measured using a typical SPR experimental setup in the Kretschmann configuration. The layers in the simulated structure include a glass layer consisting of a BK7 glass prism and a microscope slide joined with index matching fluid creating a continuous layer, a 50 nm thick gold layer and a semi-infinite dielectric sample layer. A focused light source of wavelength 632.8 nm was assumed (corresponding to HeNe laser output) and the intensity of the light source was assumed to be uniform and steady, i.e. all changes in reflectance, beyond noise are assumed to be from sample layer refractive index changes. In the simulations we used isopropanol solutions as sample layers. The isopropanol solutions spanned a range of refractive index differences (between 0 and 1×10^{− 7}) with respect to water. Our hypothesis testing method was applied to the difference curves produced by subtracting a reflectance curve for water from that of an isopropanol solution reflectance curve.

It should be noted that even if the difference curve is clearly non-zero by inspection, the average value tends to be small and near zero. This problem has been overcome by applying the statistical test to selected five regions (A-E) located within the front edge of the SPR reflectance vs incidence angle curves where the difference was maximised. The region definitions were A: 66°−70.4°, B: 66.75°−70.3°, C: 67.5°−70.2°, D: 68.25°−70.1°, E: 69.0°−70.0° (Fig. 2 ). Due to the slope of the reflectance curve on the front edge these points had the potential to produce the largest difference between the two curves whilst maximising the number of data points used. Plots of the difference curves and Z-values of associated Δn’s and regions are displayed in Fig. 3a and 3b.We note that due to the presence of noise all the presented difference curves would be regarded and impossible to be distinguished from zero, however the hypothesis testing is able to make a clear distinction. The limit of sensitivity for the simulated system is found to correspond to a refractive index difference with respect to water of Δn = 4× 10^{− 8} where the test statistics produced values of Z > 1.96 resulting in the identity hypothesis being rejected in favour of the alternate hypothesis, i.e. the curve difference is found to be non-zero and the curves are distinguishable for this refractive index change. Although a change in refractive index corresponding to Δn = 3× 10^{− 8} produced Z-values greater than 1.96 corresponding to the rejection of the null hypothesis in favour of the alternative hypothesis for two regions (that is in two cases the curves could be distinguished from one another), for three regions the null hypothesis was not rejected, thus the test showed that the difference between the curves was zero in most cases and this was taken to indicate an inability to reject the null hypothesis for this refractive index change. Thus the value of Δn = 4× 10^{− 8} represents the limit of sensitivity achievable in the examined system. This is around an order of magnitude larger than the standard reported resolution values produced using a phase detection method.

 

Fig. 2 Reflectance curves with region definitions. Insert: Simulated 5% isopropanol reflectance difference curve from water showing a positive data region.

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Fig. 3 a) Difference plots for various sample refractive indes differences as shown. b) Hypothesis test Z values for simulations of a conventional SPR system.

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3.2 Applications of hypothesis testing to nanoparticle -minclusive SPR simulations - overview

In this part we applied the hypothesis testing method to a nanoparticle – inclusive SPR setup to further improve the sensitivity of such sensing system. As indicated earlier the modified Maxwell-Garnett approach was chosen to model the nanoparticle composite layer for use in the multilayered Fresnel simulations. Simulations were carried out for effective permittivities of the nanoparticle composite layer calculated for varying volume fill fractions (f_np) The hypothesis testing approach was then applied to determine whether to accept or reject the null hypothesis of no difference between the simulated curves for no nanoparticles (SAM only) and the nanoparticle inclusive sample (at some volume fraction f_np).

The simulations were carried out at wavelengths both close to (543 nm) and far from (632.8 nm) the absorption maximum of the 24 nm gold nanoparticles (at 526 nm). The layered system summarised in Table 1 with a 47 nm gold SPR - supporting layer was used for both wavelengths. For 543 nm a comparison simulation using the layered system described in Table 2 was carried out. Hypothesis testing was applied to selected regions of the simulated difference curves and z values calculated for each volume fill fraction. The Z values were used to determine whether to accept or reject the null hypothesis (for |Z| ≤ 1.96 and |Z| ≥ 1.96 conditions respectively). It needs to be noted that in the current approach the region of angles where the distinction between the two curves is being established needs to be selected as well, and it typically coincides with the left half of the SPR curve terminating near the minimum.

3.3 Simulation and hypothesis testing results at 632 nm

Figures 2(4) and 3(5) summarise the results obtained from simulations carried out at 632.8 nm, far from the maximum of the nanoparticle absorption peak. Figure 2(4) shows that for volume fill fractions less than 5 × 10⁻⁸ the difference curves are not distinguishable by observation above the noise level. The insert to Fig. 2(4) displays the raw simulated reflectance curve data. The nanoparticle-induced changes for the volume fractions considered are too small to be seen in these figures, thus a reflectance curve for a large volume fill fraction of 3% is included to indicate the trend of the changes manifesting on the reflectance curves where nanoparticles are included. From this it can be seen that the addition of nanoparticles to the system resulted in an increase in the minimum reflectance and a shift to higher angles. The width of the reflectance curve did not change appreciably. Two regions were chosen to illustrate the applicability of the hypothesis testing method; the positive difference region between 49.9° and 51.4° (31 data points) (Region 1) and the negative difference region from 51.4° to 61.4° (201 data points) (Region 2). The Z values calculated for these two regions are plotted in Fig. 3(5). For volume fill fractions of 5×10⁻⁸ and above, both regions show calculated Z values that result in comprehensive rejection of the null hypothesis in favour of the alternate hypothesis (i.e. these curves are distinguishable from the SAM curve with no nanoparticles). Region 2 also produces a Z value greater than the cut-off of 1.96 for a volume fill fraction f_np of 1 × 10⁻⁸ indicating that the null hypothesis may be rejected in this case also. Below f_np values of 1 × 10⁻⁸, the calculated Z values show acceptance of the null hypothesis (of no difference between the curves) for all regions apart from Region 1 with f_np of 1 × 10⁻⁹ .

3.4. Simulation and hypothesis testing results at 543 nm

Simulations were also carried out for an incident light wavelength of 543 nm. As this wavelength is close to the absorption maximum of the nanoparticles (of 526 nm), their influence on the reflectance curves is much stronger than at 632.8 nm. Simulations were carried out for two different layered systems. The previously analysed system with a 47 nm gold SPR supporting layer was considered as well as an SPR system optimized for the wavelength of 543 nm, with a 34 nm silver layer topped with 8 nm gold. Figures 4 and 5 show the difference curves simulated for different volume fill fractions at 543 nm for the layered systems under consideration. The reflectance curves for the SAM only samples are quite different, with the optimised silver-gold composite layer system producing a sharper curve with lower minimum reflectance than the gold only layer system. As the changes in the reflectance curves corresponding to the volume fill fractions considered are too small to be seen on these figures, a reflectance curve for a larger volume fill fraction of 3% is again included to demonstrate the type of changes manifesting on the reflectance curves. For the gold only layered system, an increase in both value of minimum reflectance and minimum reflectance angle is seen. The curve width also increases slightly. For the silver-gold layered system, the nanoparticle reflectance curve displays an increase in minimum reflectance and angle of minimum. Compared to the SAM only curve, the width of the nanoparticle curve increased markedly. This is especially so when compared to the curves produced for the layered system optimised for 632.8 nm (insert to Fig. 2(4)) where very little change in curve width was observed.

 

Fig. 4 Reflectance difference curves calculated using modified Maxwell-Garnett theory at 632.8nm, for various volume fill fractions. Insert: Raw SPR reflectance curves.

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Fig. 5 Calculated hypothesis testing Z values for nanoparticle simulations at 632.8nm

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The simulated difference curves for each layered system at 543 nm exhibit quite different profiles. Due to the width of the raw SPR curve for the 47 nm gold system, the resulting difference curves display a large region of positive difference containing 195 data points, corresponding to the front edge of the raw SPR reflectance curve. The optimised silver-gold system shows a smaller but sharper difference region with 57 data points for the same front edge region due to a sharper SPR reflectance curve for this system. A second negative difference region containing 192 data points is also available for analysis in this case, corresponding to the trailing (higher angle) region of the SPR reflectance curve. The optimised silver-gold system demonstrates a higher peak difference value for a given volume fill fraction than the 47nm gold system. The value of average difference and number of points in the region used for calculation will both affect the Z value calculated. In both cases higher values are more likely to produce a Z value greater than the test cut off.

Hypothesis testing was applied to both layered systems for each volume fill fraction difference curve. As an illustration, for the 47 nm gold system, a single positive difference region containing 195 data points was considered. For the silver-gold system, two regions were considered - the positive difference region with 57 data points and the large negative difference region with 192 data points. Z values were calculated for each of these regions for the volume fill fractions considered. Hypothesis testing was then applied and Z values calculated to determine whether to accept or reject the null hypothesis (for |Z| ≤ 1.96 and |Z| ≥1.96 conditions respectively). The calculated Z values for each volume fill fraction are displayed in Fig. 6 . For the 47 nm gold system, volume fill fractions of 1 × 10⁻⁸ and above produced Z values above the hypothesis testing cut off of 1.96. This results in rejection of the null hypothesis and acceptance of the alternative hypothesis that the nanoparticle containing sample reflectance curves are different from the SAM only reflectance curve (a positive sensing event has occurred). For the silver-gold system, both regions produced Z values above the cut off value of 1.96 for volume fill fractions of 5 × 10⁻⁹ and above. Region 1 (the positive difference region) also produced a Z value above 1.96 for the 1 × 10⁻⁹ volume fill fraction sample.

 

Fig. 6 Reflectance difference curves calculated using modified Maxwell-Garnett theory at 543 nm for a 47 nm gold SPR layer, at various volume fill fractions. Insert: Raw SPR reflectance curves.

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3.5. Discussion

We now discuss the results of the application of the hypothesis test to the simulated difference curves for SPR systems containing nanoparticles. The Z values obtained from the 632.8nm simulations (Fig. 3), show that for volume fill fractions of 5 × 10⁻⁸ and above, hypothesis test application to both Regions 1 and 2 of the difference curves result in rejection of the null hypothesis in favour of the alternate hypothesis. This indicates that a positive sensing event has occurred, as the two curves are distinguishable. Region 2 produced a Z value above the hypothesis test cut off for the next lowest volume fraction of 1 × 10⁻⁸, however Region 1 did not. Thus the 5 × 10⁻⁸ volume fill fraction represents the lowest resolution obtained in the simulations for the 632.8 nm system. A volume fill fraction of 5 × 10⁻⁸ corresponds to a molar concentration of 0.01nM (1sf), two orders of magnitude smaller than the nanomolar concentration levels detectable by the most popular BIACORE system [41]. For the same system (47 nm gold SPR supporting layer) applied at 543 nm, the hypothesis testing results (Fig. 6) showed a further drop in the minimum volume fill fraction discernable. For the region tested, the volume fill fraction of 1 × 10⁻⁸, corresponding to a molar concentration of 0.002nM (1sf), produced a Z value above the hypothesis testing cut off of 1.96. Thus, by simply changing the measurement wavelength to be closer to the absorption peak of the nanoparticles, an additional increase in resolution level can be achieved. This is due to the increased width of the reflectance curve, resulting in a far larger region for hypothesis test application from the front edge of the SPR reflectance curve (in our example 195 data points as opposed to 31 data points). The large increase in region size at 543 nm counterbalanced the reduction in average difference level, resulting in higher Z values for lower given volume fill fractions than at 632.8 nm. The optimised silver-gold SPR supporting layered system showed a further improvement in performance at this wavelength. Figure 8 shows Z values above the cut off value of 1.96 for volume fill fractions of 5×10⁻⁹ and above for both regions considered. This corresponds to a 0.001nM (1sf) molar concentration, an order of magnitude better than the volume fill fraction discernable for the 632.8 nm simulations. The improvement over the 47 nm gold layered system at 543 nm is due to the SPR reflectance curves produced from the optimised silver-gold system. As seen from the insert to Fig. 7 , the addition of nanoparticles to the SAM-only system produces substantial changes in the minimum reflectance value and angle as well as the curve width. As the SPR curve for the SAM-only sample for the 47 nm gold system has a large width at 543 nm (insert to Fig. 6), further nanoparticle-induced changes to the reflectance curve profile are not as pronounced. This shows the importance of optimising the metallic layer thicknesses for improving the potential sensitivity of a given SPR sensing setup.

 

Fig. 8 Calculated hypothesis testing Z values for nanoparticle simulations at 543 nm.

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Fig. 7 Reflectance difference curves calculated using modified Maxwell-Garnett theory at 543 nm for a 38 nm silver −8 nm gold SPR layer, at various volume fill fractions. Insert: Raw SPR reflectance curves.

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These results have shown the potential of inclusion of nanoparticles in a sensing layer in combination with hypothesis test application to dramatically increase the sensitivity of SPR measurements. Optimisation of the setup has been shown to produce improvements in the sensitivity level attainable, in the case presented here by an order of magnitude in the minimum molar concentration of analyte that could be detected. This was achieved through selection of wavelengths close to the absorption maximum of the nanoparticles and use of SPR supporting metallic layer thicknesses that produce a reflectance curve with lower width and minimum reflectance value, capable of large relative changes in both of these values. Further advances in sensitivity could be achieved by carrying out a full optimisation for a given nanoparticle sensing system. The method presented here is also applicable to the SPR systems utilizing the whole width of the evanescent field, and not only to traditional systems where the detection takes place in a single monolayer, thus offering scope to improve the resolution in these as well.

4. Conclusions

The hypothesis testing method was applied to a conventional and nanoparticle-inclusive SPR system. A modified Maxwell-Garnett approach, including homogeneity and geometric factors was found to produce the most accurate simulations of experimental results from an absorbing SPR system reported in the literature [32]. The modified Maxwell-Garnett approach was used to calculate dielectric constants for varying volume fill fractions of 24nm gold nanoparticles in the sensing layer of two different layered SPR systems. Simulations were carried out based on the Fresnel multi-layer theory for nanoparticles included in the sensing layer of a multilayer SPR system. Application of hypothesis testing to appropriate difference curve regions produced a sensitivity limit of 0.001nM of gold nanoparticles with 95% confidence, for a volume fill fraction of 5 × 10⁻⁹. This was achieved for a system with optimised metallic layers and a wavelength close to the absorption of the nanoparticles used. An order of magnitude molar concentration increase was seen for this system over an unoptimised system at wavelengths far from the nanoparticle absorption peak.

Statistical methods offer a powerful quantitative approach for high resolution detection in a conventional SPR system. The method described in this paper takes advantage of much of the data often discarded during testing and utilizes more fully the information they contain. A detection resolution of 4× 10⁻⁸ for changes in the real bulk refractive index was obtained from simulated work with 95% confidence. This compares favourably with results quoted in commercial biosensor literature that gives maximum resolution values in the range of 3× 10⁻⁷ [41, 42] for similar measurements. The large Z values obtained from simulations of a realistic experimental setup are indicative of the potential to produce far more sensitive results with a more finely tuned/developed setup. Alternative methods based on statistical sampling at selected points require much longer measurement time for similar accuracy and may rely on prior knowledge, such as the location of the SPR minimum. Our statistical method (applied to simulated and experimental SPR difference curves) has been therefore shown to produce a simple and effective quantitative measure for distinguishing minute refractive index differences in liquids using SPR.