1. Introduction

The analysis of molecular interactions by optical surface wave biosensors is of continued interest in life sciences. Surface plasmon polariton based analysis [1–3] has evolved to be the standard approach among various label-free optical methods [4]. The resonance exploited in plasmon based sensing is governed by the optical properties of the metal used, thus leaving the angle or the wavelength of operation as the only free parameters for optimizing either spectrally or angularly resolved systems, respectively. Noise estimations revealed that best performing plasmon based systems have nearly reached their theoretical limits [5]. One route to better performing sensors based on plasmons could be to laterally pattern the structure [6] in order to exploit local field enhancement. Alternatively, modes guided at the surface of a dielectric, one-dimensional photonic crystal [7, 8] have been suggested for plasmon like sensing [9, 10]. Such approach exploits the band gap in a dielectric stack as a truncated, one-dimensional photonic crystal [11] to obtain guiding of a surface mode, thus giving rise to surface sensitive evanescent field sensors [12–14]. Because appropriate dielectric structures usually exhibit much less material absorption than metal containing systems, enormous field enhancement factors [15] as well as Goos-Hänchen shifts [16] associated with narrow resonances below 0.1° width [17] have been observed.

Several different approaches can be conducted in order to optically probe the surface wave sensor’s response. Usually a reflectivity analysis is performed by probing the surface mode resonance appearing in the polarized intensity distribution [1, 2, 9, 17] measured under total internal reflection conditions. This can be extended to interferometric approaches exploiting the phase jump at the resonances position [14, 18–20] or Goos-Hänchen shifts [10, 13] due to the long propagation length [21, 22] of surface waves in low loss structures. Additionally, the reflectivity based approach can be combined with the detection of fluorescence [23] exploiting the surface field enhancement for labelled analysis, too. Here we will restrict to the basic reflectivity detection scheme.

As the mode resonance properties can be adjusted by the dielectric stack design (refractive indices and layer thicknesses) the sensor performance is much less limited by the properties of a single material and temperature insensitive sensors [24] are within reach.

An increased resolution of BSW sensors is expected due to the improved ratio of the resonance shift, resulting from a perturbation of the refractive index at the sensor surface, over its width [14, 17]. Besides residual material losses, the ultimate reduction of the resonance width is practically not desired because of two reasons. First, the associated increase of propagation length requires one to extend the measured spot sizes far above those commonly used in microarray technologies [25]. Second, narrow resonances lead to sampling problems, because a certain free measurement range generally limits the sampling. Although resolution issues have been discussed in the frame of surface plasmon sensors [2, 26, 27] the limiting case of ultra-narrow resonance needs to be discussed with regard to the measurement range required by the sensor. Such range becomes even more important when extending the analysis towards combined label-free – fluorescence approaches [23] because of the system’s dispersion.

This paper addresses the optimization of a BSW optical sensor based on a periodic one dimensional photonic crystal. We apply a semi-analytical resonance model in sec. 2 to discuss the effect of the detection noise on the resonance shift determination for angularly resolved reflection analysis. Combining the results with the values obtained for a ‘figure of merit’ (FoM) of such sensors yields the ‘limit of detection’ (LoD) in sec. 3, which is governed by the resonance properties as well as by its sampling. Besides the choice of the detector, such sampling depends on the (angular) range to be detected. Both quantities, LoD and FoM, are simulated for SPR in sec. 4 and are used for a sample optimization of a low loss dielectric thin film stack sustaining BSW. Results suggest marvelous performance of such BSW sensor in terms of FoM or LoD that is compared to that of SPR. Interesting BSW features with respect to stack design issues are discussed.

2. Basic experimental scheme and noise estimation

A sketch of our experimental scheme to measure angularly resolved reflectivity is shown in Fig. 1. Such scheme is not bound to single mode beams and can be applied to extended (incoherent) sources that do not suffer from speckles in real sensing environments and that allow one to observe extended spot sizes. Furthermore, it can be extended to interferometric schemes [14, 19] by adding appropriate polarizers in the illumination and detection light paths. The optical functions of half ball and Fourier lenses can be combined within the chip substrate to create small systems with disposable chips [28]. We restrict to a monochromatic analysis in order to leave material dispersion effects out of the discussion. Then the angular spectrum of reflected light intensity is detected by means of an array detector with N pixels. The surface mode resonance observed in the reflected angular spectrum is characterized by a depth D and an angular width W (capital letter ‘W’) given in degrees. Sampling by the detector with limited dynamic range yields a discretized curve exhibiting a width w (lower case letter ‘w’) measured in detector pixels. The latter width is assigned the unit ‘pix’ to distinguish w and W.

 

Fig. 1 Scheme of angular resonance detection by means of a Fourier imaging approach utilizing a lens with focal length f. The angular range A is discretized with N detector pixels that transform the angular width W of the resonance into a width w in terms of detector pixels. Inset A illustrates that discretization needs to resolve small signal shifts (curves 1 and 2) but is practically limited by the free measurement range (curve 3). Inset B illustrates the general stack that yields signal shifts due to organic layer adsorption as well as changes of analyte refractive index n_A.

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The sensitivity S_bulk is usually defined as the shift of the resonance position (degrees) due to bulk refractive index changes (RIU). Such value can be easily determined experimentally when using liquids with different refractive indices. In the current frame this definition is misleading. Dielectric systems allow for positioning the resonance at any (angular) position. As will be exemplified in sec. 4.3 below, the sensitivity also depends on the resonance position which governs the penetration depth of the evanescent field into the (aqueous) analyte solution. Defining the sensitivity over the bulk refractive index of this analyte solution will yield maximum sensitivity at the angle of total internal reflection because then the evanescent field is infinitely extended into this medium. Therefore, we define sensitivity S_layer as the signal given in degrees relative to an organic layer thickness change given in nanometer, while assuming a refractive index of n_org = 1.45 for such layer.

In order to evaluate the sensor performance the effect of the resonance width on the noise of the sensor output (resonance shift) needs be quantified. Many sources of noise have been discussed in detail as has been done for SPR applications [5, 26, 27]. Here, we restrict on estimating the effects of the resonance depth and width on the accuracy of the minimum position determination. A simplified square polynomial model that mimics fitting the resonance in the region of minimum reflectivity is assumed for this analysis and sketched in Fig. 2(A). It could directly represent an experiment that saturates the detector outside the angular range around minimum reflectivity. The approach

 

Fig. 2 (A) Simplified model of resonance analysis using a quadratic polynomial to mimic the resonance sampled by a 12 bit dynamic range detector. (B) Discretizing a function by spatially extended pixels corresponds to a spatial integration of the distribution and yields discretized intensity values y_i(x_i) that differ from the ideal distribution f(x_i) by an amount Δy_i.

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is used to derive x_min because of its linearity with respect to all fit parameters. The error for the minimum position determination depends on x_min itself because the uncertainty on the two parameters a₁ and a₂ is correlated and not independent. However, Eq. (1) can be applied in the case x_min~0 in order to estimate the noise effect. According to the illustration in Fig. 2(A) the depth D and width w of the resonance completely define a quadratic polynomial for a given dynamic range of the detector. So the parameters a_i and thus x_min are known exactly, and the parameter errors Δa_i are determined numerically from the diagonal elements of the covariance matrix C according to ${\sigma }^2\left(a_i\right)=C_{ii}$ [29]. This allows one to model the measurement noise by assuming an intensity uncertainty σ_i associated to each pixel’s intensity value y_i. We consider the detection shot noise ${\sigma }_i^2~y_i$, assuming it to be 0.6% of light intensity as previously suggested [5], and the digitization error, assuming one count of the analog-to-digital converter.

In the case of extremely narrow resonances one has also to consider effects due to limited sampling (discretization). This is illustrated in Fig. 2(B) for an arbitrary continuous function f(x). The intensity detected by each pixel corresponds to a spatial integration and results in the error

This discretization error is proportional to the second derivative of the function under consideration. It is worth to point out explicitly that in the square polynomial approximation a constant intensity offset (~a₂) is added to the data. So it cannot be considered noise and might be dropped for the present analysis. But, similar effects are introduced by mechanical vibrations of the setup that will yield higher order errors when fitting real data. Therefore the simulation results will be shown with and without inclusion of the discretization noise effect.

The total contribution of shot noise, digitization error and discretization error are then given by the lowest order of approximation from Eq. (2) and yield a total intensity uncertainty in each pixel

Calculating the result of Eq. (1) when assuming the noise according to Eq. (3) yields the results shown in Fig. 3. In order to cross check such error estimation an additional numerical experiment has been performed by adding numerical noise to a Lorentzian function of width w and depth D = 0.7. The standard deviation of the minimum position is deduced from fitting 1’000 different noise representations. The points shown in Fig. 3(B) almost perfectly match the results of the square polynomial model for sufficient resonance widths.

 

Fig. 3 (A) Error of minimum position determination with respect to resonance depth for different widths of the resonance; the black line illustrates a 1/D behavior. (B) Minimum position error vs. resonance width for different depths of the resonance; the black line depicts the asymptotic behavior. The dots illustrate uncertainties obtained from fitting Lorentzian curves of width w with added numerical noise. All simulations assume a detector with 12 bit dynamic range; the solid and the dashed curves give the error without or with inclusion of the discretization error, respectively.

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The results obtained by such numerical calculations can be summarized by the following three major points:

3. Figure of merit and limit of detection

A FoM has been defined recently [17] in order to compare the performance of different sensor approaches. It relates the depth D and width W of the resonance to the sensitivity S according to

Using the sensitivity S and the sensor output noise σ_S suggests using another assessment criterion referred to the LoD [5] according to $LoD={\sigma }_S/S$. Note, that the choice of S = S_bulk or S = S_layer determines the unit of the LoD, so it applies to both kinds of sensitivities. With the results of the previous section we obtain

In the limiting case, when the detection system samples exactly the resonance width, one obtains the minimum ${LoD|}_{A=W}~\alpha /\left(FoM\cdot \sqrt{N}\right)$. Combining it with the fact that FoM has been shown to apply to intensity only experiments strictly [14], the sensor performance measured by the FoM shall be used in the case of a small angular detection range $A~W$, i.e., when the resonance width (or less) is sampled by the detector like in plasmon imaging approaches [31–33]. Whenever a larger range needs to be detected, e.g. in case of parallel angularly resolved measurement [28, 34, 35], if the free measurement range including tolerances needs to be considered, or if fluorescence collection is desired as well [23], the optimization should minimize the LoD that also considers the resonance sampling. Thus, optimum sensor performance can only be achieved with good FoM in combination with a sufficient sampling of the resonance.

In the following section the different meanings of the LoD and FoM as performance measures will be illustrated by discussing different types of plasmon like sensors.

4. Application to planar sensor configurations

4.1 Numerical evaluation

All calculations reported here were carried out by means of self-written software that is based upon a transfer matrix formalism. Using this tool the angularly resolved, monochromatic reflectivity of the thin film system is simulated in the plane wave approximation when the sensor stack is illuminated from the (glass) substrate. The interface between substrate back side and air is neglected; a refractive effect of this interface could be avoided by applying a hemi-cylinder or half ball lens.

The numerical procedure is readily illustrated in the inset of Fig. 1. First, the angularly resolved reflectivity is calculated for the stack covered with water like medium of n_A = 1.33 refractive index [curve 1 in Fig. 1(A)]. From this curve the depth D and the full with at half depth W are determined. Second, a 10 nm thick organic layer (refractive index n_org = 1.45) is added to calculate curve 2 in Fig. 1(A). The sensitivity S [°/nm] is derived from the minima positions of curves 1 & 2. Third, the maximum signal is assumed to be detected with 10 nm thick organic layer and an increased solution refractive index n_A = 1.36 thus yielding resonance 3 as shown in Fig. 1(A). The angular range included between the upper limit of the half depth of curve 3 and the lower limit of the half depth of curve 1 is used as an estimation of the angular range A to be detected. These assumptions consider the standard label-free operation range only. Other approaches such as the detection of fluorescent labels [23] can require to extend this angular range.

4.2 Surface plasmon resonance

First, surface plasmon sensing will be discussed as a reference. As stated above the thickness of the metal layer is the only free parameter in the ‘stack design’ once the metal has been chosen. So different gold layer thicknesses on a glass substrate made of nBK7 are assumed to be used for sensing. The optical constants of gold are taken from [36] while the substrate glass dispersion is adapted from [37]. Calculation results are summarized in Fig. 4 for convenience.

 

Fig. 4 Calculation results obtained for sensitivity figure of merit FoM (A) as well as limit of detection LoD (B) vs. wavelength (650 nm < λ < 1’000 nm) for the case of 45 nm (dashed red), 50 nm (solid black) and 55 nm (dashdot blue) thick gold layers. LoD has been calculated assuming the value N = 1’000 pix.

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Both, the FoM obtained according to Eq. (5) and the LoD according to Eq. (7) exhibit local optima in the 800..850 nm spectral range when utilizing 45..55 nm gold layer thicknesses. This implies to preferably use this wavelength range for sensing issues as being done currently [28, 31]. With the assumptions used here the LoD of such sensors should reach the picometer range. This result needs to be cross checked with previous findings. The present calculation yields a surface sensitivity of S~0.05°/nm that corresponds to a bulk refractive index sensitivity of S~100°/RIU (data not shown). These data enable to convert Δd_min~0.8 pm into Δn_min~3⋅10⁻⁷ RIU, which agrees very well with previous experimental findings [26, 38–40]. So the noise levels applied to the LoD simulation yield a valid order of magnitude approximation.

4.3 Bloch surface waves

As pointed out in the introduction the resonance width can be dramatically decreased when working with low loss dielectric stacks supporting Bloch surface waves. Now more parameters can be accessed to tailor the sensor’s performance and their range of variation explored. Besides the wavelength of operation the choice of two materials, their thicknesses, and the number of periods need to be considered. It is well outside the scope of this work to span over all combinations of such a large number of parameters. In order not to confuse the reader with a too general and possibly heavy description, it is sufficient to focus our attention on an exemplary case to put into evidence the potential of BSW for sensing applications.

As case of study we discuss at the wavelength λ = 670 nm the following structure for the stack: substrate | L | (HL)^Np | water, where N_p is the number of the periods that will vary between 4 and 6. The materials Ta₂O₅ (n_H = 2.106 + 3.5 × 10⁻⁵i) and SiO₂ (n_L = 1.474 + 6 × 10⁻⁶i) are being used as high (H) and low (L) refractive index layers, respectively. Comparable stacks [41] have been recently applied for other BSW sensors [14, 17, 23]. The substrate index n_S = 1.514 is chosen similar to the SPR case at the corresponding wavelength. Fixing these prerequisites leaves the number of periods N_p and the two thicknesses of the high (d_H) and the low (d_L) refractive layers open for optimization. The optimization procedure developed for the present case can be extended to stacks based on different materials, different spectral ranges and also to aperiodic stacks.

At first the optimization will focus on maximum sensitivity because this value determines the resonance shift that can be observed. The maps of the calculated layer (S_layer) and bulk (S_bulk) sensitivities, obtained by scanning d_L and d_H with a 10 nm and 5 nm step size, respectively, are shown in Fig. 5 for three different values of N_p. Regions without resonance appear for very thin stacks, i.e., in the lower left corner of the diagrams in Fig. 5 (shown in black). This is caused by the fact that the BSW at the given λ is below its cut-off.

 

Fig. 5 Sensitivity with regard to thin film binding S_layer (top row) and with regard to bulk index changes S_bulk (bottom row) for the dielectric BSW systems subs | L | (HL)^Np |water with different number of periods. S_bulk is based on an external refractive index change of 0.01 starting from n_A = 1.33.

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Two interesting results can be derived from this analysis. First, the influence of the number of periods N_p on the BSW sensitivity is negligible, that is to say, the sensitivity variation with the number of periods is below 1.5% only for all points in the graph. Second, optima of S_layer appear in a narrow, nearly line shaped region near the BSW cut-off. In such region, the optimum thicknesses of high and low index materials are surprisingly correlated by the linear law d_L = - d_H + C, with C = 452.5 nm for the stack design considered here and operating at the given λ, as shown in Fig. 6(A). Figure 6(B) illustrates the sensitivity along such a line: the mean value is 0.02 °/nm and the extrema differ by 0.6% only, thus proving that along such a line a similar maximum can be found for all values of N_p. In contrast, the bulk index sensitivity S_bulk is maximized at the TIR edge, where the evanescent fields exhibit maximum penetration into the aqueous analyte medium. Figure 5 illustrates that bulk sensitivity is no proper target for BSW biosensor optimization.

 

Fig. 6 (A) Extrapolated pairs d_L, = f(d_H) from Fig. 5 with optimum sensitivity S_layer. The linear law d_L = -d_H + 452.5 nm correlates the two thicknesses is unequivocally. Along this line the sensitivity (B) varies by 0.6% only and has a mean value of 0.020 °/nm.

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The resonance position and the angular range according to Fig. 1(A) are shown in Fig. 7 for N_p = 5 only. Other N_p values yield similar maps with a change of 0.01% for the resonance angle (θ_BSW) and 2% for the angular range (A) only. A monotonic behavior is observed when increasing the layers’ thickness; the resonance position shifts towards larger angles and the angular range towards smaller values. Interestingly, the S_layer optimized stacks exhibit almost the same resonance angular position, i.e., maximum S_layer is achieved at a fixed propagation constant of the surface wave [dashed line in Fig. 7(A)].

 

Fig. 7 Resonance position (A) and angular range (B) for the BSW stack subs | L | (H|L)⁵ |water determined according to Fig. 1(A). The white dashed line indicates S_layer optimized stacks according to Fig. 6(A).

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The angular range has been determined for a 10 nm organic layer thickness increase only, therefore it is governed by bulk index effects, as confirmed by Fig. 7(B) that qualitatively resembles the S_bulk map shown in Fig. 5.

Once discussed the sensitivity, the dependency of the remaining parameters contributing to the LoD, as defined by the Eq. (7), on the layers thickness needs to be analyzed in order to optimize the performance of the BSW sensor. Especially the resonance depth D and width W are highly influenced by the number of periods, because this modulates the radiative losses into the substrate [14]. This is confirmed by the results shown in Fig. 8_. For large N_p = 5,6 the smallest W is obtained close to the TIR edge (boundary of the dark region), whereas for N_p = 4 the resonance width seems to suggest an optimum high index layer thickness in the range d_H = 90..100 nm. Generally, the W values span over two orders of magnitude for the simulated structures and decrease when increasing N_p. The region where the maximum D is obtained shifts towards smaller thicknesses of the low index layer as the number of periods increases, following a sort of parabolic shape with the high refractive index layer thickness.

 

Fig. 8 Resonance full width half maximum W (first row) and depth D (second row) for the BSW stacks subs | L | (HL)^Np |water with N_p = 4,5,6 (columns).

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In general, the number of periods should be utilized to adapt the system onto changes of the material losses. Increasing such material losses requires decreasing the number of periods, because this increases radiation losses. Such case might be desired in order to simplify the stack or to increase resonance width and is similar to the plasmon case [42].

Combining all resonance features enables one to compile FoM and LoD maps as shown in Fig. 9. Tuning the number of periods in the stack allows one to generate a large region (N_p = 5) with minimum LoD values in the range LoD_min>0.04 pm. For the case N_p = 6 small regions with comparable low values are found. The most important contribution to the formation of such extended range of well suited stacks with N_p = 5 is the fact that sensitivity S and resonance depth D are maximized in the same region of the map. Alternatively, this effect could be achieved by adjusting the absorption losses of the stack materials. But this will increase the resonance width, which in turn results in an increased LoD.

 

Fig. 9 Simulated FoM (top row) and LoD (bottom row) assuming 1000 pix detector for different number of periods (columns).

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4.4 Discussion

The interesting resonance properties obtained for the exemplary stack discussed here require some further discussion [43]. First, the sensitivity has been found to be independent of the number of periods. This can be explained by the reflection of the surface wave at the substrate side dielectric stack: for the angles under consideration, the phase of such reflection is almost independent of the number of periods. Because of the large index difference of n_L and n_H, deviations occur for lesser numbers of periods only. Second, the surface sensitivity S_layer exhibits a clear optimum near the TIR edge with the optimum angle being well separated from this edge. Such effect is associated with the TIR at the interface to the analyte medium and will be discussed in detail elsewhere.

The optimum performance for the exemplary BSW sensor achieved here reaches FoM_max~24 nm⁻¹ that is more than two orders of magnitude above that of SPR. But, minimum LoD_min~0.04 pm is ‘only’ a factor 20 below that of SPR. This illustrates that the superior finesse of BSW sensors, that could be associated with the ratio S/W, can be partially translated into LoD reduction only. As the sensitivity governs the angular range to be detected, it simultaneously limits the sampling of the angular distribution and increases the noise of resonance position determination. In our calculations a N = 1’000 pix detector has been assumed, while W~0.001° and A~1° for optimized stacks has been obtained. This illustrates that sampling will tremendously affect reflection intensity based resonance analysis. With regard to Fig. 3(B) such superior performance requires one to improve the detector sampling along with the mechanical stability of set-up. Note, that even varying the number of periods in an optimized stack (d_L~360 nm, d_H~95 nm) from N_p = 5 to N_p = 4 or 6 will reduce the LoD and FoM but still yield a performance improvement when comparing to SPR. Furthermore, the large region of layer thicknesses for optimized stacks in the N_p = 5 indicate that limited fabrication accuracy will not constrain the sensor performance.

In order to harness such high finesse BSW systems for obtaining minimum LoD sensors, resonance widths in the 0.001° range need to be detected. This statement holds for both, imaging and angularly resolved approaches. Optical schemes exploiting either polarization [14, 19, 20] or propagation [10, 22] based interference effects might claim less dependency on such resolution, because the depth of the resonance does not need to be resolved. Therefore, the design of such sensors seems to become rather independent of material extinction losses. On the contrary, the minimum resonance width in the millidegree range is associated with an imaginary part of the guided wave’s effective index of 7.7⋅10⁻⁶ that corresponds to a propagation length above 10 mm [44]. That is the basic limit of sensor performance, independent of the kind of observation. So interference based optical systems might relax sampling requirements only and cannot remove such kind of sensitivity limit.

5. Conclusions

Angularly resolved surface wave sensors have been discussed in order to achieve a qualitative and quantitative comparison of two different surface wave sensor concepts. Modelling an angular resonance as a simple quadratic function provides a numerical estimation of the noise of resonance position determination. In the limit of wide resonance widths the results compare to previous findings. Noise of minimum position determination has been converted into a limit of detection as an assessment criterion of the sensor device which takes into account the angular range that has to be detected by the optical system. Compared to the well-established figure of merit, which relates sensitivity, resonance depth and resonance width for narrow angular range detecting systems, the additional involvement of the angular range introduces considerations of free measurement range and tolerances. The limiting case of very narrow resonances has been modelled analytically by discretization errors that can be related to mechanical stability issues of the system.

Surface plasmon resonance sensors allow one to optimize the layer thickness for a given metal only. In contrast, dielectric stacks can be optimized by varying all layer thicknesses. Discussing a periodic BSW supporting stack with state of the art, low loss dielectric layers as example revealed that optimum sensitivity is reached slightly above the TIR limiting angle. Furthermore, a variety of stacks with regard to the number of periods and the layer thicknesses exhibit high sensitivity values that yield a linear relation between high and low index layer thicknesses, respectively. Then, the number of periods and layer thickness needs to be fixed for adjusting resonance depth D and angular range A in order to obtain a minimum limit of detection LoD. Due to the fact that the resonance shift versus width is improved, BSW promise a superior FoM enhancement compared to SPR. But, as the angular range to be detected decreases less than the resonance width, the practically relevant measure LoD reduces to a lesser extent.

Such performance improvement requires the detection of very narrow resonances that can be as low as 10⁻³ degree. Such angular resolution is closely connected to the propagation length of the surface mode in the several mm range. This exposes the general limit of resolution for such kinds of sensors independent of the method of optical surface wave analysis.