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In this paper, we investigate detection characteristics of localized surface plasmon resonance biosensing based on a probabilistic Poisson distribution of target molecules. The model uses random nanoislands for localization of near-fields in three detection scenarios of non-specific, non-colocalized, and colocalized detection. Optical signatures were found to increase monotonically with target concentration and size regardless of the detection scenarios. The signatures were largest in colocalized detection of target interactions to localized fields, followed by non-colocalized and non-specific detection. The confidence interval was the narrowest in the colocalized detection due to the increased spatial certainty by localization. Based on the relative confidence interval, it was found that limit of detection can be enhanced by more than four orders of magnitude through colocalization.
© 2014 Optical Society of America
1. Introduction
An optical biosensor measures changes of light properties in response to a biomolecular interaction. Most optical biosensors measure changes in quantities such as intensity, wavelength, polarization, and phase in the far-field, i.e., a detector is located at a distance much longer than light wavelength from the sample where biointeractions occur. In contrast, near-field characteristics of light in the spatial range of only a few wavelengths have been increasingly important in the implementation of optical biosensors with molecular-scale detection sensitivity in various aspects of optical sensing techniques.
While understanding of near-fields may be critical to achieving extremely high detection sensitivity and other desirable far-field characteristics, the relation between near- and far-fields is often unclear, thus it has been difficult to find a direct link except for the simplest types of optical biosensors. In this regard, efforts to understand the effect of near-fields on the far-field detection sensitivity have been made in conjunction with the overlap between near-field distribution and target molecules being detected, for example, in the analysis of biosensing based on whispering gallery modes, FRET, and photonic crystals [1–5].
In this work, we focus on plasmonic biosensors to investigate the relation of near-field characteristics to the far-field properties. Plasmonic biosensors take advantage of surface plasmon (SP) that is longitudinal quanta of electron concentration waves formed on metal-dielectric interface with p-polarized light incidence. The momentum matching between incident photon and SP gives rise to SP resonance (SPR), which is sensitive to surface states and has thus been the basis of plasmonic biosensors. A SPR biosensor detects the resonance shift accompanied by molecular interactions and can thereby quantify the target biointeraction label-free in real time. Unfortunately, the label-free nature of plasmonic biosensing is accompanied by moderate detection sensitivity. The advent of nanotechnology, however, has opened and led to many approaches that attempt to improve sensing characteristics of plasmonic detection. For instance, metal nanoparticles (NPs) were used to conjugate in a target interaction to amplify index contrast and to enhance sensitivity [6–8]. Surface enhancement using various nanopatterns that include gratings, nanoposts, or nanoholes has been moderately successful for increased detection sensitivity [9–12]. The approach was particularly effective when fields are spatially colocalized with target molecules [13–17]. Despite the success of these enhancement approaches, use of NPs turns plasmonic biosensing into a labeled technique, while it is difficult to mass-produce nanopatterns for surface enhancement in case they are fabricated by low-throughput lithography processes.
In this regard, random nanopatterns such as nanoislands that can be synthesized without lithography have drawn significant interests as a plasmonic biosensing platform. The random nature of these patterns gives rise to random plasmon localization, which produces spatially random near-field distribution associated with electron dipoles and, as a result, establishes electromagnetically amplified hot spots at random. Although we focus on plasmonic biosensing based on nanoislands, the randomness goes far beyond what may be justified by the use of nanoislands, because the target molecular distribution is largely random by nature whether it is specific or non-specific detection. Although effects of random fluctuations on the transient responses of a plasmon sensor were studied earlier, the analysis was limited to the stochastic adsorption processes [18]. For plasmonic biosensing using nanoislands, randomness in the spatial distribution of nanopatterns and target molecules makes probabilistic interpretation of near-fields inevitable for estimating far-field characteristics when nanoislands or even periodic nanopatterns are employed. In this sense, the implication of this work can be extended to arbitrary plasmonic molecular detection that is based on localization of SP.
Random nanoislands were experimentally observed to excite near-field hot spots [19,20] and found uses in various applications such as surface-enhanced Raman spectroscopy [21–24], resonance energy transfer [25,26], highly luminescent light emitting diodes [27], enhancement and quenching of fluorescence and photoluminescence [28–33], solar cells [34,35], spontaneous light emission [36], and far-field super-resolution microscopy [37,38]. Nanoislands can be synthesized in many ways, for example, by temperature annealing that follows metal evaporation. The simplicity of the fabrication processes addresses the difficulties associated with fabricating surface nanopatterns, thereby significantly reducing sensing cost [39]. It is also possible to control the geometrical distribution of nanoislands to produce desired detection characteristics. These reasons have been the motivation behind using islands for nanoplasmonic detection [40–48]. However, most of these studies still lack detailed theoretical understanding of how the enhancement is achieved in the detection characteristics and, more interestingly, how much enhancement can be achievable. These issues are critical to the ultimate improvement of nanoplasmonic biosensing that uses a wide range of nanostructures and, therefore, are investigated in great detail based on probabilistic interpretation of near-fields to estimate far-field characteristics.
2. Methods and models
2.1 Numerical method
We have employed rigorous coupled wave analysis (RCWA) with 225 ( = 15 × 15) spatial harmonic orders for numerical calculation of near-field optical characteristics produced by nanoislands. RCWA has successfully explained near- and far-field distribution created by periodic patterns [49–52]. Even for aperiodic random nanostructures, RCWA can be effective if the period of a random pattern is much larger than light wavelength. Also, a pattern with correlation length that is much smaller than the simulated size contains a large number of structural variations in the constructed surface profiles, which can make the calculation statistically more relevant to average the performance of many realizations of rough surfaces [53]. 3D models of nanoislands were imported with unit calculation area set to be 2 × 2 μm2. We used MATLAB to obtain probability density and detection sensitivity from the calculated near-field characteristics.
2.2 Calculation models
A simple schematic shown in Fig. 1(a) was used to describe the model of nanoislands in the calculation: nanoislands were assumed to form on top of a 5-nm thick silver film and a 2.3-nm thick chromium adhesion layer on an SF10 glass substrate (refractive index ns = 1.723) with buffer ambiance (nbuffer = 1.33). The nanoislands were assumed to be of 25-nm thickness, which corresponds to that of nanoislands experimentally synthesized with 20-nm thick silver film. The silver layer underlying nanoislands models a film which may form as a result of thermal annealing and was selected to be 5 nm based on the volume ratio of silver. The film allows propagating plasmons for SPR detection. Material optical constants of chrome and silver at λ = 488, 633, and 760 nm were taken from [54]. Target biomolecules were assumed to have dielectric permittivity equal to that of adenovirus particles (nav = 1.366) with varying size. The target concentration followed the notation that is typically used for the number concentration of virus, i.e., En (n: integer) represents 10n /μL = 10n-9 /μm3.
Fig. 1 Schematics of numerical models: (a) non-specific detection using nanoislands for 3D RCWA calculation. Specific detection is considered in two distinct scenarios: (b) non-colocalized and (c) colocalized detection. The red spots represent localized fields. In the colocalized detection, an antibody layer is shown to exist only at the localized fields while it it covers nanoislands surface in non-colocalized detection.
Incident light is p-polarized. To explore the effect of light wavelength on the localization and detection characteristics, investigation was performed at λ = 488, 633, and 760 nm. Both 633 and 760 nm are frequently used for SPR measurement. The angle of incidence θi was fixed at 60° and the reflectance change as a result of an interaction was calculated.
We have considered both specific and non-specific detection of target objects. In the non-specific detection model, randomness arises from the target distribution and, to a lesser degree, the nature of nanoislands. In the specific detection model, the randomness is related to the way that target molecules bind to recognition elements on the surface. The specific recognition process of target molecules is stochastic [55,56], although the effect is not considered explicitly.
In the non-specific detection shown in Fig. 1(b), the 3D distribution of the target is probabilistically described. In contrast, the specific detection model of Fig. 1(c) has the target distribution axially controlled as the target molecules are bound to the recognition layer such as antibodies that are specific to the target, and the binding positions at the surface as well as the binding likelihood follow a probabilistic distribution. Finally, in the colocalized specific detection illustrated in Fig. 1(d), recognition elements exist only at localized fields and the likelihood of target binding to the recognition elements follow a probabilistic distribution. For all three models, near-field distribution created by nanoislands is spatially random.
While statistical models of nanoislands were known to have island size following normal distribution and the separation between islands in a log-normal probability density function [37], 3D models were created from SEM images of experimentally synthesized nanoisland samples, e.g., shown in Fig. 2(a) and in the bottom plane of Fig. 2(b): the SEM image was binarized into 2D patterns [middle planes of Fig. 2(b)], which were axially extended to make a cylinder of 5-nm height with the 2D shape as a profile. This process creates edges that may not exist in real samples and thus has an effect of exaggerating the near-fields produced by nanoislands. The experimental sample was synthesized by first depositing a 2-nm thick chrome adhesion layer and then a 20-nm thick silver film, which is followed by temperature annealing on a hot place at 175° for 10 minutes. The top plane of Fig. 2(b) presents the near-field distribution at an axial distance z = 25 nm from the surface produced by the nanoisland sample of Fig. 2(a) over a unit area of 2 × 2 μm2.
Fig. 2 (a) SEM image of a synthesized nanoisland sample used for the calculation. Also shown below the image is the conversion into the binarized nanoislands pattern to build a numerical model. (b) 2D near-field distribution (top) at an axial distance z = 25 nm from the surface, which is overlayed on the binarized boundary pattern (middle planes) and the SEM nanoisland sample image (bottom). The SEM image corresponds to the square inset of (a) in an area of 2 × 2 μm2.
2.3 Overlap integral
Enhanced SPR detection on nanoplasmonic island patterns was attributed to the localization and amplification of electromagnetic fields to form hot spots and the coupling of the hot spots with target molecules [48]. The enhancement by the electromagnetic coupling has been studied based on the overlap between localized fields and effective target permittivity [57–59]. By quantifying the overlap in the near-field, e.g., using an overlap integral OI given by
Et denotes normalized tangential electromagnetic field amplitude, thus OI takes the unit of energy. The studies suggest that the overall energy localized to target molecules on the sample surface can be related to the radiative energy measured in SPR signals and that far-field characteristics may be interpreted in terms of near-field quantities, i.e., limit of detection (LOD), which is defined as the number of target molecules (N) that produces a reference change in the reflectance (R), is correlated with the change in OI as a result of a biointeraction and detection sensitivity DS = ∂R/∂N ≈ΔOI/ΔN.Historically, the energy overlap integral or simply OI has been treated as a deterministic parameter: if a target molecule is much smaller than the sampling function that may be used for the numerical calculation, target permittivity may be approximated as
εtarget and εbuffer are the permittivity of target and buffer (εbuffer = nbuffer2), respectively. rk is the locations of Nt target molecules. Equation (1) can then be simplified asSince the second term on the right hand side of Eq. (3) is a constant with respect to the target,A target may not be in the regime of being much smaller than the sampling function. Also, the target modeling in the overlap integral becomes more complicated if axial distribution of the target location and target behavior, e.g., contact inhibition or aggregation, are considered. In general, the overlap integral is difficult to evaluate analytically because Et is largely unknown.In fact, OI is a random variable as will be discussed in the next section. The nature of OI being a random variable has been largely neglected in the above treatment where only an average behavior of OI is described. In this paper, we focus on the probabilistic aspect of OI and discuss the reliability in estimating the target concentration from measured OI.
2.4 Probabilistic interpretation of OI and reliablility in target concentration estimation
2.4.1 Non-specific detection
Given an electromagnetic field distribution, Eqs. (1)-(4) suggest that an overlap integral and eventually detection characteristics should be obtainable deterministically as long as target permittivity ε(r) can be uniquely defined. However, target distribution is not deterministic because the existence of a target molecule at a particular point in space can only be described probabilistically. Since the value of OI depends on the locations of the target molecules, OI can be considered as a random variable. Now, we analyze the distribution of OI. For simplicity, we assume that Et(r) takes quaternary levels with even spacing between the field minimum and maximum (average field amplitude of each level represented by E0, E1, E2, and E3 where E3 for the highest level), while ε(r) takes binary levels depending on the location r. ε(r ) = ε1 for r that belongs to target molecule and ε0 otherwise. Then, OI can be written as
Ki is the number of target molecules in the i-th region (i = 0, 1, 2, 3), an area that corresponds to one of the quaternary levels. At is the target volume and Si the total volume of the i-th region. The approximation in Eq. (5) is based on the assumption that Si >> AtKi. If target molecules do not prefer any point with uniform likelihood to exist over the space of interest, their distribution can be modeled as a homogeneous Poisson point process with χ for the average number of target points in unit area [60]. In this case, for every compact set S, the number of the points in S denoted by N(S) follows Poisson distribution. Also, since each Si is disjoint, each N(Si) is independent of each other. Therefore, Ki can be modeled as independent Poisson random variables with parameter χSi. The probability mass functions of Ki areWe normalize OI by the optical signature of ambience without any target molecules and produce normalized optical signature , i.e,where OIno target = Σi = 0~3{Ei2ε0Si}. Note that is a Poisson distributed random variable with a parameter Σi = 0~3{ai + biχSi}, where ai = Ei2ε0Si/OIno target and bi = Ei2ε1At/OIno target. Since each Ki is independent of each other, can be treated as Gaussian random variable invoking the central limit theorem (CLT) [61]. Usually, the CLT needs a sufficient number of independent random variables. Even if the number of independent random variables is insufficient, it is reasonable to assume that is a Gaussian random variable because K0 and K1 are Poisson random variables. The expectation and variance of are given byFrom Eq. (8), we can estimate the area concentration of target χ as follows:Since is a Gaussian random variable, is also a Gaussian random variable with mean and varianceThen, we can calculate the reliability of our target concentration estimation under given confidence level [61]. With 95% confidence level, the confidence interval of isAlso, the confidence interval of estimated that corresponds to a target concentration with 95% confidence level is given byIn terms of average, the probabilistic model discussed in this section was found to produce results that are consistent with the deterministic model described in Eqs. (1)-(4) when the target size ranges up to 25 nm. Notable disparity was observed between the probabilistic and the deterministic model as the size increases above 25 nm. This is because of the assumption that the target size is negligible compared to the size of localized fields. Thus, the target size was varied from 1 to 25 nm in this study.
2.4.2 Non-colocalized detection
The non-colocalized detection is performed by the process of target molecules to be captured by probe layers on the metal surface such nanoislands. The effect of a probe layer on near-field distribution is negligibly small, thus Et(r) in non-colocalized detection is assumed to be equal to that of non-specific detection defined earlier. The permittivity of the region outside the volume where target molecules exist is assumed to be ε0, which is similar to the case of non-specific detection. In other words, outside the binding layer, OI is constant at Ei2ε0Si while OI is given by Eq. (5) within the binding layer. i.e.,
Si,inside represents the volume of the i-th region within the binding layer and Si, outside that of the outside. Si in this case denotes the sum of Si,inside and Si, outside. Equations (6)-(9) can be written as where ai = Ei2ε0Si,inside/OIno target and bi = Ei2ε1At/OIno target. Then, the confidence interval can be calculated by Eqs. (10)-(12).2.4.3 Colocalized detection
It is assumed that the near-field distribution in the colocalized detection remains the same as that of non-colocalized and non-specific detection. The probe layer in colocalized detection exists inside S2 and S3. Target molecules are distributed inside S2,inside and S3,inside. OI is then
Thus, Eqs. (15)-(18) are rewritten as The confidence interval can then be calculated by Eqs. (10)-(12).2.5 Noise characteristics of nanoplasmonic detection
Noise factors are closely linked to the technical performance associated with instrumentation due, for example, to vibration or stray light and also the fundamental noise that is physically implicit in SPR detection such as thermodynamic refractive index fluctuation. If noise sources are statistically independent, the effect of the noise sources manifests itself as the fluctuations in the target permittivity and field amplitude that affect the overlap integral. Therefore,
CLT allows the noise term on the right-hand side to be described by a Gaussian random variable with mean μ = 0 and variance σ2 = σNOISE2 [62]. Since we can consider and O(NOISE) to be independent of each other and as the sum of the two independent Gaussian random variables, is also a Gaussian random variable with μ = Exp[] and . The probability density function p is given byσε and σEt denote the statistical deviation in the target permittivity and tangential field amplitude. The variance of is larger than that of without O(NOISE). As shown in Eq. (12), the confidence interval is determined by the variance, i.e., the increased variance extends the confidence interval. The noise factors affect LOD in that higher noise factor reduces LOD. In this study, we do not consider a particular detector model, thus LOD cannot be determined absolutely. More meaningful criterion, however, is to compare relative LOD of various detection scenarios because noise factors are in principle independent of the scenarios assuming an identical sensor hardware. In other words, the evaluation of LOD is based on the relative magnitude of optical signatures produced in each of the three detection scenarios.3. Results and discussion
3.1 Non-specific detection model
Suppose a volume defined by the lateral plane and the axial distance of one penetration depth from the surface. If the penetration depth is assumed to be 100 nm in a lateral plane of 2 × 2 μm2, four target molecules are obtained in the volume at E10. Figure 3(a) shows the detection sensitivity in terms of the overlap signatures as a result of non-specific detection at a particular concentration calculated for nanoislands. The calculated overlap integral was normalized by that of ambiance without target molecules to produce optical signatures.
Fig. 3 (a) Calculated optical signature (OI) for non-specific detection of targets of varying size of target. The optical signature was normalized by the overlap without target molecules. Confidence interval of the detection appears as a color band. The inset shows a magnified image for the case of target diameter ϕ = 25 nm Normalized optical signature for targets of varying size on nanoislands: (b) non-colocalized specific detection and (c) colocalized specific detection. Much reduced confidence interval is clear, particularly in the case of colocalized detection, for which the optical signature almost appears as a line.
Optical signature was found to increase monotonically with target concentration and size. The increase of optical signatures is almost linear with target concentration and significantly changes with target size. Because OI and ΔOI/ΔN may be respectively correlated with LOD and detection sensitivity, the results in Fig. 3(a) suggest that both LOD and sensitivity may improve for detecting larger target molecules. This is because the improvement is associated with the growth of effective volume occupied by the target and exposed by the near-field.
In addition to the non-specific detection characteristics that confirm what was obtained of average using effective medium, Fig. 3(a) also suggests probabilistic behavior of detection that appears as confidence intervals. Clearly, confidence interval increases with target size, which reflects an increase of uncertainty caused by stochastic inclusion of a larger target when detecting at an identical concentration. For targets of ϕ = 25 nm at a concentration of 2.03E11 [shown as the vertical arrow in the inset of Fig. 3(a)], for example, OI ranges from 0.00075195 to 0.001248 at 95% confidence level with an average of 0.001, i.e., the relative confidence interval (RCI) defined as the ratio of confidence interval to an average amounts to be 50%. On the other hand, the target presence, e.g. the target concentration that produces OI = 0.001 falls between (1.594 ~2.631)E11 with an average 2.03E11 [horizontal arrow in the inset of Fig. 3(a)], i.e., uncertainty in the target concentration is 49% in terms of the RCI. In other words, the uncertainty can be quite significant. For quantitative comparison, the RCI at an identical target concentration (2.03E11) is largely a constant at 50% for ϕ = 10, 15, and 20 nm, i.e., RCI is mostly constant regardless of the size as the confidence interval increases in proportion to optical signatures. On the other hand, as the target concentration increases, confidence interval remains largely unchanged, thus the RCI is reduced.
3.2 Specific detection model
What is potentially more interesting is a specific detection model because most biosensors detect target biomolecules in this way. In the specific detection model, the target recognition probe molecules are assumed to uniformly distribute on the surface at a density such that the detection is always target-limited while target molecules randomly bind to probes. In other words, target concentration can be regarded as equal to the number of bindings. Computationally, the main difference of the specific detection model is that the target molecules are distributed near surface on top of the probe layer, i.e., in a 2D distribution. For simplicity, we consider the specific detection model in two different ways, i.e., target-probe interaction is performed in non-colocalized and colocalized detection, where colocalization represents spatial overlap between target interaction and localized fields.
3.2.1 Non-colocalized detection
In this mode of specific detection, target molecules do not have any spatial preference on the surface as illustrated in Fig. 1(c). The calculated optical signatures with respect to the target concentration are demonstrated in Fig. 3(b). The trend is similar to what was obtained for non-specific detection in Fig. 3(a). However, the signature for specific detection is overall much larger than the case of non-specific detection, which is due to the surficial nature of target binding with probes. The strength of near-fields is much stronger near surface because of the evanescent nature, which significantly increases the optical signature compared to the case of non-specific detection. Note that the average number of target molecules binding to the probe is also larger in specific detection, which contributes to a faster increase in effective area as the target concentration increases. Therefore, larger optical signature in the case of specific detection can be largely associated with stronger near-fields at the surface and less dominantly with increased effective area.
Compared to non-specific detection discussed in Section 3.1, in non-colocalized detection is smaller than that of non-specific detection as the spatial randomness decreases. On a 95% confidence level, targets of ϕ = 25 nm at a concentration of E11 produce an optical signature ranging from OI = 0.046857 to 0.050386 with an average of 0.048622, i.e., RCI = 7.3%, which we can compare with RCI = 71% for the case of non-specific detection. In other words, uncertainty in the detection is drastically reduced by performing specific detection.
3.2.2 Colocalized detection
We now consider probabilistic treatment of colocalized detection in which target molecular distribution is spatially ‘locked’ to localized fields as illustrated in Fig. 1(d). Experimentally, colocalization on nanoislands would be difficult. Nevertheless, the results that we present here can still be important because the probabilistic interpretation can be extended to the field localization by arbitrary nanopatterns.
The effect of colocalization on the optical signature can be calculated by varying the target concentration assigned to a sample unit that corresponds to a localized field, i.e., when the overlap integral is calculated, higher target concentration is assigned to a sample unit in which the field is localized. The way that the optical signature is calculated for colocalized detection suggests that the results should estimate the highest that can be achievable using nanoislands. Such estimation is difficult to perform with average-based traditional calculation using effective medium. The results are presented in Fig. 3(c). Obviously, near-field distribution has a stronger effect on the optical signature than non-specific or non-colocalized detection, the detection characteristics of which are largely ruled by average properties represented conceptually by an effective medium. Although the overall trend of colocalized detection with target concentration is similar to the case of non-colocalized or non-specific detection, the optical signature is significantly increased because of colocalization between target-probe binding and localized fields.
Figure 3(c) also shows confidence interval that is extremely narrow compared to non-specific and non-colocalized detection. In particular, targets of ϕ = 25 nm produce an optical signature ranging from OI = 1.5206 to 1.537 with an average of 1.5288 at a concentration of E11, which leads to RCI = 1.1%. RCI at target concentration of E11 and 2.03E11 for ϕ = 25 nm is listed in Table 1 for the three detection schemes. The RCI is the smallest in colocalized detection among the three detection scenarios because of the reduced spatial uncertainty in the lateral and axial dimension. In other words, for an identical set of estimated target concentration, the confidence interval is minimal while the optical signature is the highest in the colocalized detection, as shown in Fig. 3(d). It is also noted that RCI changes little with target size, as mentioned in non-specific detection.
Table 1. Relative confidence interval (RCI) at 95% confidence level. 2.03E11 was selected as the target concentration that produces normalized OI = 0.001 in non-specific detection.
Figure 4 shows the RCI at a 95% confidence level calculated at ϕ = 25 nm with respect to the target concentration under the three detection schemes. Obviously, the RCI decreases monotonically with target concentration, since confidence interval is largely a constant while OI increases. Interestingly, the slope of decrease in RCI on a log-scale remains almost identical for the three detection schemes, which represents molecular detection based on Poisson processes. If we define LOD as the target concentration at which RCI = 1 (shown as the horizontal dashed line in Fig. 4), the LOD is determined to be 7.0E10, 7.2E8, and 0.60E7 for non-specific, non-colocalized, and colocalized detection, respectively. In other words, colocalized detection enhances LOD by 11700 and 120 times compared to non-specific and non-colocalized detection schemes. The improvement of the LOD is associated with the amplification of optical signatures, rather than the reduction of measurement noise.
Fig. 4 The relative confidence interval (RCI) calculated for target size of ϕ = 25 nm with respect to target concentration in non-specific, non-colocalized, and colocalized detection. The horizontal dashed line represents RCI = 1.
The results reported here indicate that colocalized detection provides much improved sensing characteristics in terms of both optical signatures and detection uncertainty over the other detection modes and also strongly suggest that the probabilistic interpretation should allow one to fully appreciate the potential of colocalized target detection compared to other detection scenarios, which would be difficult with conventional numerical methods.
3.3 Wavelength dependence
The sensitivity of conventional SPR detection is known to be improved at longer wavelengths [63–65]. On the other hand, near-fields tend to localize more efficiently at a shorter wavelength. It is therefore suggested that non-specific and non-colocalized detection may benefit from SPR measurement at a longer wavelength. However, colocalized SPR detection may have an optimum wavelength because the benefit of employing a longer wavelength may be compensated by weaker localization of near-fields. To explore this nature, probabilistic calculation was performed for colocalized detection of targets (ϕ = 25 nm) using light wavelength λ = 488, 633, and 760 nm.
The results presented in Fig. 5(a) suggest that the average behavior is largely identical regardless of the wavelength, i.e., optical signature increases with target concentration, despite the near-field distributions that are noticeably different [insets of Fig. 5(a)]. Optical signature increases as the wavelength shifts from 488 to 633 nm. The optical signature, however, decreases as the wavelength further shifts to 760 nm, which reflects more dominating effect of near-field localization when we compare optical signatures at λ = 633 and 760 nm. The relative difference between the optical signatures at λ = 488, 633 and 760 nm was calculated to be such that OI(633 nm)/OI(488 nm) = 1.74 and OI(633 nm)/OI(760 nm) = 1.67 for targets of ϕ = 25 nm. Interestingly, Fig. 5(b) shows that RCI at E11 was calculated to be 0.0135, 0.0108, and 0.0133 for λ = 488, 633 and 760 nm, respectively, i.e., at λ = 633 nm, OI is the largest with the smallest RCI. Our analysis shows that the trend is closely related to near-field distributions. In other words, RCI tends to decrease with more localized fields, thus the likelihood that a target molecule takes a specific region (Si) decreases, which reduces the uncertainty. In this sense, Fig. 5 suggests the intensity and the overall volume of the localized near-fields to be correlated with OI and RCI.
Fig. 5 (a) The optical signature OI with target concentration for targets of ϕ = 25 nm under the colocalized detection at different wavelengths λ = 488, 633, and 760 nm. Insets present the near-field distributions at the respective wavelength. (b) RCI at different wavelengths λ = 488, 633, and 760 nm. The RCI at λ = 488 and 760 nm almost overlaps thus is maginified in the inset, where RCI at λ = 488 nm is slightly larger than that of λ = 760 nm. Thick solid lines in the inset are the grids.
3.4 Discussion
While we have investigated the probabilistic aspect of plasmonic detection on random nanopatterns, a goal of this study is to be able to estimate probabilistically whether colocalized detection may be used for single molecule detection using nanoislands-based SPR sensing. We can extrapolate the results obtained with nanoislands to a detectable binding capacity below fg/mm2. For a molecule of 100 kDa molecular weight and assuming a spot size of 1 mm2, the LOD corresponds to detecting a few hundred molecules under the best circumstances, which indicates that molecular detection with nanoislands is yet to be a considerable challenge even based on colocalized detection. However, the concept of overlap integrals in the colocalization implies the possibility of further improvement of LOD, e.g., super-localized fields over an entire field-of-detection, which is optimized in size to the target molecule.
Among the three detection schemes that have been considered, colocalized detection was shown to provide the largest optical signatures with minimum uncertainty. Given that it takes significant experimental efforts to implement colocalization between near-field distribution and target molecules, non-colocalized or even non-specific detection can be an alternative depending on the targeted LOD and sensing precision.
4. Concluding remarks
In summary, we have analyzed surface-enhanced SPR detection of target molecules based on a probabilistic model. The need for probabilistic estimation arises from the detection using nanoislands as substrates because of the spatially random near-field distribution that nanoislands produce. The detection was considered with three distinct scenarios: non-specific, non-colocalized, and colocalized detection. Regardless of the detection scenarios, overall optical signature showed a supra-linear increase with target concentration, which is in good agreement with theoretical results using effective medium. Detection uncertainty represented by the confidence interval increased with target size and was not much affected by target concentration. The confidence interval was reduced significantly in specific detection. The largest increase of the optical signature as well as the highest confidence was achieved with colocalized detection, for which the probabilistic model estimates enhanced LOD by more than 10000 times compared to non-specific detection.
Acknowledgments
This work was supported by the National Research Foundation (NRF) grants funded by the Korean Government (2011-0017500 and NRF-2012R1A4A1029061).
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