Optimization of angularly resolved Bloch surface wave biosensors Riccardo Rizzo,1,2 Norbert Danz,1,* Francesco Michelotti,2 Emmanuel Maillart,3 Aleksei Anopchenko,2 and Christoph Wächter1 1 2
Fraunhofer Institute for Applied Optics and Precision Engineering IOF, A.-Einstein-Str. 7, 07745 Jena, Germany Basic and Applied Science for Engineering Department (SBAI), SAPIENZA Università di Roma, Via A. Scarpa 16, 00161 Roma, Italy 3 HORIBA Scientific, HORIBA Jobin Yvon S.A.S, Avenue de la Vauve, 91120 Palaiseau, France * [email protected]
Abstract: Bloch surface wave (BSW) sensors to be used in biochemical analytics are discussed in angularly resolved detection mode and are compared to surface plasmon resonance (SPR) sensors. BSW supported at the surface of a dielectric thin film stack feature many degrees of design freedom that enable tuning of resonance properties. In order to obtain a figure of merit for such optimization, the measurement uncertainty depending on resonance width and depth is deduced from different numerical models. This yields a limit of detection which depends on the sensor’s free measurement range and which is compared to a figure of merit derived previously. Stack design is illustrated for a BSW supporting thin film stack and is compared to the performance of a gold thin film for SPR sensing. Maximum sensitivity is obtained for a variety of stacks with the resonance positioned slightly above the TIR critical angle. Very narrow resonance widths of BSW sensors require sufficient sampling but are also associated with long surface wave propagation lengths as the limiting parameter for the performance of this kind of sensors. ©2014 Optical Society of America OCIS codes: (280.1415) Biological sensing and sensors; (240.6690) Surface waves; (230.5298) Photonic crystals; (240.6680) Surface plasmons; (240.0310) Thin films.
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#217751 - $15.00 USD Received 30 Jul 2014; revised 8 Sep 2014; accepted 8 Sep 2014; published 16 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.023202 | OPTICS EXPRESS 23202
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#217751 - $15.00 USD Received 30 Jul 2014; revised 8 Sep 2014; accepted 8 Sep 2014; published 16 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.023202 | OPTICS EXPRESS 23203
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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, onedimensional 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
#217751 - $15.00 USD Received 30 Jul 2014; revised 8 Sep 2014; accepted 8 Sep 2014; published 16 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.023202 | OPTICS EXPRESS 23204
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 nA.
The sensitivity Sbulk 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
#217751 - $15.00 USD Received 30 Jul 2014; revised 8 Sep 2014; accepted 8 Sep 2014; published 16 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.023202 | OPTICS EXPRESS 23205
evanescent field is infinitely extended into this medium. Therefore, we define sensitivity Slayer as the signal given in degrees relative to an organic layer thickness change given in nanometer, while assuming a refractive index of norg = 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 a1 Δa1 Δa 2 Δxmin = + ⋅ xmin (1) 2 ⋅ a2 2 ⋅ a2 a2 is used to derive xmin because of its linearity with respect to all fit parameters. The error for the minimum position determination depends on xmin 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 xmin~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 ai and thus xmin are known exactly, and the parameter errors Δai are determined numerically from the diagonal elements of the covariance matrix C according to σ 2 ( ai ) = Cii [29]. This allows one to model the measurement noise by assuming an intensity uncertainty σi associated to each pixel’s intensity value yi. We consider the detection shot noise σ i2 ~ yi , 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 y ( x ) = a2 ⋅ x 2 + a1 ⋅ x + a0
yi =
1 Δx
xi + Δx 2
⋅
f ( x ) ⋅ dx
xmin = −
⎯⎯ → Δy = y − f ( x ) = − i
x i − Δx 2
i
i
( Δx ) 24
2
(
f ′′ ( xi ) − O f
( 4)
; ( Δx )
4
)
.
(2)
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
σ 2 = 1+ i
6 a ⋅ yi + 2 1' 000 12
2
.
(3)
#217751 - $15.00 USD Received 30 Jul 2014; revised 8 Sep 2014; accepted 8 Sep 2014; published 16 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.023202 | OPTICS EXPRESS 23206
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 yi(xi) that differ from the ideal distribution f(xi) by an amount Δyi.
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.
The results obtained by such numerical calculations can be summarized by the following three major points: (i) According to Fig. 3(A) the accuracy of minimum determination is well proportional to the inverse depth (~1/D) of the resonance. Deviations occur for very deep resonances (D>0.7) due to the fact that low intensities near the minimum suffer from shot noise less. This law corresponds well to previous findings [5]. (ii) The effect of the width onto the noise is well approximated by a w1/2 law in the asymptotic case of large resonance widths. Such result confirms the previous discussion [26, 27] but contradicts the prediction obtained for center of mass algorithms [5]. Enormous deviations appear for bad sampling in the w0.04 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.
Fig. 9. Simulated FoM (top row) and LoD (bottom row) assuming 1000 pix detector for different number of periods (columns).
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 nL and nH, deviations occur for lesser numbers of periods only. Second, the surface sensitivity Slayer 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 FoMmax~24 nm−1 that is more than two orders of magnitude above that of SPR. But, minimum LoDmin~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 (dL~360 nm, dH~95 nm) from Np = 5 to Np = 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 Np = 5 indicate that limited fabrication accuracy will not constrain the sensor performance.
#217751 - $15.00 USD Received 30 Jul 2014; revised 8 Sep 2014; accepted 8 Sep 2014; published 16 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.023202 | OPTICS EXPRESS 23213
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 wellestablished 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. Acknowledgments
The authors acknowledge funding by the European Commission through the project BILOBA (Grant agreement 318035) and by the State of Thuringia (Germany) through the project Proexzellenz MeMa, as well as stimulating discussions with E. Descrovi (Politecnico di Torino).
#217751 - $15.00 USD Received 30 Jul 2014; revised 8 Sep 2014; accepted 8 Sep 2014; published 16 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.023202 | OPTICS EXPRESS 23214
Fraunhofer Institute for Applied Optics and Precision Engineering IOF, A.-Einstein-Str. 7, 07745 Jena, Germany Basic and Applied Science for Engineering Department (SBAI), SAPIENZA Università di Roma, Via A. Scarpa 16, 00161 Roma, Italy 3 HORIBA Scientific, HORIBA Jobin Yvon S.A.S, Avenue de la Vauve, 91120 Palaiseau, France * [email protected]
Abstract: Bloch surface wave (BSW) sensors to be used in biochemical analytics are discussed in angularly resolved detection mode and are compared to surface plasmon resonance (SPR) sensors. BSW supported at the surface of a dielectric thin film stack feature many degrees of design freedom that enable tuning of resonance properties. In order to obtain a figure of merit for such optimization, the measurement uncertainty depending on resonance width and depth is deduced from different numerical models. This yields a limit of detection which depends on the sensor’s free measurement range and which is compared to a figure of merit derived previously. Stack design is illustrated for a BSW supporting thin film stack and is compared to the performance of a gold thin film for SPR sensing. Maximum sensitivity is obtained for a variety of stacks with the resonance positioned slightly above the TIR critical angle. Very narrow resonance widths of BSW sensors require sufficient sampling but are also associated with long surface wave propagation lengths as the limiting parameter for the performance of this kind of sensors. ©2014 Optical Society of America OCIS codes: (280.1415) Biological sensing and sensors; (240.6690) Surface waves; (230.5298) Photonic crystals; (240.6680) Surface plasmons; (240.0310) Thin films.
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#217751 - $15.00 USD Received 30 Jul 2014; revised 8 Sep 2014; accepted 8 Sep 2014; published 16 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.023202 | OPTICS EXPRESS 23202
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#217751 - $15.00 USD Received 30 Jul 2014; revised 8 Sep 2014; accepted 8 Sep 2014; published 16 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.023202 | OPTICS EXPRESS 23203
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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, onedimensional 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
#217751 - $15.00 USD Received 30 Jul 2014; revised 8 Sep 2014; accepted 8 Sep 2014; published 16 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.023202 | OPTICS EXPRESS 23204
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 nA.
The sensitivity Sbulk 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
#217751 - $15.00 USD Received 30 Jul 2014; revised 8 Sep 2014; accepted 8 Sep 2014; published 16 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.023202 | OPTICS EXPRESS 23205
evanescent field is infinitely extended into this medium. Therefore, we define sensitivity Slayer as the signal given in degrees relative to an organic layer thickness change given in nanometer, while assuming a refractive index of norg = 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 a1 Δa1 Δa 2 Δxmin = + ⋅ xmin (1) 2 ⋅ a2 2 ⋅ a2 a2 is used to derive xmin because of its linearity with respect to all fit parameters. The error for the minimum position determination depends on xmin 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 xmin~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 ai and thus xmin are known exactly, and the parameter errors Δai are determined numerically from the diagonal elements of the covariance matrix C according to σ 2 ( ai ) = Cii [29]. This allows one to model the measurement noise by assuming an intensity uncertainty σi associated to each pixel’s intensity value yi. We consider the detection shot noise σ i2 ~ yi , 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 y ( x ) = a2 ⋅ x 2 + a1 ⋅ x + a0
yi =
1 Δx
xi + Δx 2
⋅
f ( x ) ⋅ dx
xmin = −
⎯⎯ → Δy = y − f ( x ) = − i
x i − Δx 2
i
i
( Δx ) 24
2
(
f ′′ ( xi ) − O f
( 4)
; ( Δx )
4
)
.
(2)
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
σ 2 = 1+ i
6 a ⋅ yi + 2 1' 000 12
2
.
(3)
#217751 - $15.00 USD Received 30 Jul 2014; revised 8 Sep 2014; accepted 8 Sep 2014; published 16 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.023202 | OPTICS EXPRESS 23206
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 yi(xi) that differ from the ideal distribution f(xi) by an amount Δyi.
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.
The results obtained by such numerical calculations can be summarized by the following three major points: (i) According to Fig. 3(A) the accuracy of minimum determination is well proportional to the inverse depth (~1/D) of the resonance. Deviations occur for very deep resonances (D>0.7) due to the fact that low intensities near the minimum suffer from shot noise less. This law corresponds well to previous findings [5]. (ii) The effect of the width onto the noise is well approximated by a w1/2 law in the asymptotic case of large resonance widths. Such result confirms the previous discussion [26, 27] but contradicts the prediction obtained for center of mass algorithms [5]. Enormous deviations appear for bad sampling in the w0.04 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.
Fig. 9. Simulated FoM (top row) and LoD (bottom row) assuming 1000 pix detector for different number of periods (columns).
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 nL and nH, deviations occur for lesser numbers of periods only. Second, the surface sensitivity Slayer 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 FoMmax~24 nm−1 that is more than two orders of magnitude above that of SPR. But, minimum LoDmin~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 (dL~360 nm, dH~95 nm) from Np = 5 to Np = 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 Np = 5 indicate that limited fabrication accuracy will not constrain the sensor performance.
#217751 - $15.00 USD Received 30 Jul 2014; revised 8 Sep 2014; accepted 8 Sep 2014; published 16 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.023202 | OPTICS EXPRESS 23213
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 wellestablished 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. Acknowledgments
The authors acknowledge funding by the European Commission through the project BILOBA (Grant agreement 318035) and by the State of Thuringia (Germany) through the project Proexzellenz MeMa, as well as stimulating discussions with E. Descrovi (Politecnico di Torino).
#217751 - $15.00 USD Received 30 Jul 2014; revised 8 Sep 2014; accepted 8 Sep 2014; published 16 Sep 2014 (C) 2014 OSA 22 September 2014 | Vol. 22, No. 19 | DOI:10.1364/OE.22.023202 | OPTICS EXPRESS 23214
