1.4.1 Surface plasmon resonance
First, surface plasmon sensing will be discussed as a reference. 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 [Palik (1998)]. Calculation results are summarized in Fig. 1.7 for convenience. Both, the FoM obtained according to Eq. (5) and the LoD according to Eq. 1.26 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 [Danz et al. (2011),Remy-Martin et al. (2012)]. 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 ∼ 0.05deg/nm that corresponds to a bulk refractive index sensitivity ofS ∼ 100deg/R IU . These data enable to convert
∆dmi n∼ 0.8pm into ∆nmi n∼ 3 · 10−7RIU, which agrees very well with previous experimental findings [?,Piliarik et al. (2009)]. So the noise levels applied to the LoD simulation yield a valid order of magnitude approximation.
1.4 Numerical evaluation 17
Fig. 1.7 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.
1.4.2 Bloch surface waves
As pointed out in the introduction the resonance width can be dramatically de-creased 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. In order not make confusion with a too general and possibly heavy description, it is sufficient to focus the attention on an exemplary case to put into evidence the potential of BSW for sensing applications and develop a strategy to design an optimized 1DPC.
As case of study the following structure is discusses at the wavelengthλ = 670 nm: substrate |L|(HL)Np | water, where Np is the number of the periods that will vary between 4 and 6. The materials Ta2O5(nH= 2.106 + 3.5 ·10−5i ) and SiO2(nL
= 1.474 + 6 ·10−6i ) are being used as high (H) and low (L) refractive index layers, respectively. Comparable stacks [Munzert et al. (2003)] have been recently applied for other BSW sensors [Sinibaldi et al. (2013),Sinibaldi et al. (2012),Sinibaldi et al.
(2014)]. The substrate index nS = 1.514 is chosen similar to the SPR case at the corresponding wavelength. Fixing these prerequisites leaves the number of peri-ods Np and the two thicknesses of the high (dH) and the low (dL) 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 (Sl a yer) and bulk (Sbul k) sensitivities, obtained by scanning dLand dHwith a 10 nm and 5 nm step size, respectively, are shown in Fig. 1.8 for three different values of Np. Regions without resonance appear for very thin stacks, i.e., in the lower left corner of the diagrams in Fig. 1.8 (shown in black). This is caused by the fact that the BSW at the givenλ is below its cut-off. Two interesting results can be derived from this analysis. First, the influence of the number of periods Np 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 Sl a yer 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 dL = - dH + C, with C = 452.5 nm for the stack design considered here and operating at the givenλ, as shown in Fig. 1.9(A). Fig. 1.9(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 Np. In contrast, the bulk index sensitivity Sbul k is maximized at the TIR edge, where the evanescent fields exhibit maximum penetration into the aqueous analyte medium. Fig. 1.8 illustrates that bulk sensitivity is no proper target for BSW biosensor optimization.
The resonance position and the angular range according to Fig. 1.4(A) are shown in Fig. 1.10 for Np = 5 only. Other Np values yield similar maps with a change of 0.01% for the resonance angle (θB SW) 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 Sl a yer optimized stacks exhibit almost the same resonance angular position, i.e., maximum Sl a yer is achieved at a fixed propaga-tion constant of the surface wave (dashed line in Fig. 1.10(A)). 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. 1.10(B) that qualitatively resembles the Sbul kmap shown in Fig. 1.8.
Once discussed the sensitivity, the dependency of the remaining parameters contributing to the LoD, as defined by the Eq. 1.26, on the layers thickness needs
1.4 Numerical evaluation 19
Fig. 1.8 Sensitivity with regard to thin film binding Sl a yer(top row) and with regard to bulk index changes Sbul k(bottom row) for the dielectric BSW systems subs |L|(HL)Np|water with different number of periods. Sbul k is based on an external refractive index change of 0.01 starting from nA
= 1.33.
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 [Sinibaldi et al. (2013)]. This is confirmed by the results shown in Fig. 1.11. For large Np
= 5,6 the smallest W is obtained close to the TIR edge (boundary of the dark region), whereas forNp = 4 the resonance width seems to suggest an optimum high index layer thickness in the range dH = 90..100 nm. Generally, the W values span over two orders of magnitude for the simulated structures and decrease when increasing Np. 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.
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 [Professor Dr. Heinz Raether (1988)].
Fig. 1.9 (A) Extrapolated pairs dL= f (dH) from Fig. 1.8 with optimum sensitivity Sl a yer. The linear law dL= −dH+ 452.5 nm that 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.
Fig. 1.10 Resonance position (A) and angular range (B) for the BSW stack subs |L|(HL)5| water determined according to Fig. 1.4(A). The white dashed line indicates Sl a yer optimized stacks according to Fig. 1.9(A).
Combining all resonance features enables one to compile FoM and LoD maps as shown in Fig. 1.12. Tuning the number of periods in the stack allows one to gen-erate a large region (Np =5) with minimum LoD values in the range LoDmi n>0.04 pm. For the case Np = 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 Np= 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.