Abstract

An optical method is presented for in the situ monitoring of biomolecular films via reflection microscopy on patterned substrates. The method is based on measuring the reflection coefficient of a composite consisting of a substrate, a patterned optical layer, the thin film to be monitored and the cover medium. The optical layer is patterned so that an array of squares is surrounded by the bare substrate. The reflectance difference between the optical layer squares and the bare substrate is the observable, whose fractional changes reveal the thickness of the film through a simple analytical expression. The periodic image is recorded by a digital microscope, and through Fourier transform techniques, the normalized differential reflectance of the patterned optical composite is calculated as the contrast factor of two dimensional bit map. The method is demonstrated by measuring a protein binding assay inside a microfluidic module placed under a microscope.

© 2013 OSA

1. Introduction

Methods for in situ and accurate monitoring of molecular film growth on solid substrates are valuable tools for the analytical science. They allow study of deposition rates, adsorption or desorption processes and association/dissociation kinetics in the case of binding bioassays. Especially in the case of protein binding or DNA hybridization assays, label free detection in real time provides valuable results without the interfering effects of fluorophore or nanoparticle labels. If existing equipment already used for general purpose testing, such as optical microscopes, can be employed with minimal additions to monitor the thin film growth the benefits are obvious. There is a variety of transducers capable of measuring material deposition or film growth. We choose to work with optical transducers to take advantage of the galvanic isolation between the excitation and the detection electronics and the versatility in choosing wavelength while manipulating light through the laws of optics [1]. A wealth of data regarding biomolecular intereactions has been derived from label free methods such as surface plasmon resonance [2,3], waveguide based interferometry [4], reflectometric interference spectroscopy [5], or ellipsometric methods such as oblique-incidence reflectivity difference [6,7]. In these optical arrangements, dedicated set-ups involving light sources (lasers, LEDs or white light sources) and detecting devices (photodetectors or spectrometers) are precisely coupled through focusing optics or optical fibers while often polarizers and other optical components intervene. Several of these functions are summarized in a modern optical microscope equipped with a digital camera and if such an instrument can be employed for label free detection the benefits are obvious. Here, we demonstrate how a reflection microscope can be used to monitor the film thickness on a patterned substrate by using the digital images as reflectance bit maps that reveal the optical path changes due to the film growth. An optical layer on top of a substrate is patterned as an array of squares while bare substrate surrounds the squares. The biomolecular film is grown on top of the patterned substrate. The optical layer thickness is chosen so as to maximize, for a specific spectral band, the sensitivity of the optical composite reflection coefficient to any film deposition while in the bare substrate areas the reflection coefficient has zero sensitivity to thin film growth. This way the differential reflectance between the optical layer square islands and the bare areas will change during film growth and its fractional change provides the adlayer thickness through a simple formula. The differential reflectance is very accurately calculated by two dimensional Fourier transform of the patterned substrate digital image. In the transform inverse space the response will sharply peak at the point inversely proportional to the real space periodicity. Such response is immune to parasitic reflections originating in the cover optical windows of the fluidic structure and translates into a direct measure of the adlayer thickness. To demonstrate the method, a digital microscope will be employed as a biomolecular film growth monitor, the only additions being the use of a microfluidic structure to supply the appropriate reagents to a patterned substrate placed on the microscope chuck. The next section outlines the theoretical analysis of the differential reflectance and its optimization in terms of choosing the appropriate optical layer and substrate. In the subsequent sections the optical and fluidic setup will be described and then the experimental results will be presented and discussed.

2. Differential reflectance at normal incidence as an analytical tool

This section will outline the theory of maximizing differential reflectance changes induced by the growth of a film on top of an optical layer already in place on a substrate. The term differential refers to the reflectance dependence on the optical layer thickness. The use of patterned substrates with optimized dual layer thicknesses and the employment of microscope images to isolate the film induced normalized reflectance changes allows calculation of the grown film thickness through a simple analytical expression.

2.1. Optimization of the optical stack in terms of the reflectance sensitivity to film growth

An optical stack consisting of two semi-infinite media (cover medium, n0, and substrate, n3), the film under study (n1, d1), and the optical layer (n2, d2) is shown in Fig. 1 .

 

Fig. 1 Schematic diagram of the thin film to be monitored and the optical layer sandwiched between the cover medium and the substrate.

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The overall reflection coefficient for light incident at a normal angle through the cover medium is [8]

R(d1,d2)=r12r22r32+r12+r22+r32+F(δ1,δ2)1+r12r22+r22r32+r12r32+F(δ1,δ2)
Here,
δi=2πnidiλ0
where δi (i = 1,2) are the normalized optical lengths in the film and optical layer, and
ri=(ni1ni)(ni1+ni)
where ri (i = 1,2,3) are the Fresnel coefficients, and

F(δ1,δ2)=2r1r2[r32+1]cos(2δ1)+2r3r2[r12+1]cos(2δ2)+2r3r1cos(2δ1+2δ2)+2r22r3r1cos(2δ12δ2)

Our objective is to maximize the magnitude of the derivative ∂R/∂d1 and use this parameter in normalized terms as an observable for the calculation of the film thickness. The selection of materials to optimize ∂R/∂d1 is usually restricted to the optical layer (n2, d2) and the substrate (n3) since the cover medium is determined by the process to be followed for the film growth. For instance, if a protein binding assay is to be monitored, the cover medium will be the biological sample or assay buffer with n0≈1.34, while the protein layer has a refractive index n1 usually between 1.38 and 1.45. Therefore the emphasis from now on will be on how to maximize the magnitude of the derivative ∂R/∂d1 in terms of (n2, d2) and n3. By differentiating Eq. (1) with respect to d1, we obtain

R/d1=(2πn1/λ0)F(δ1,δ2)δ1(1r12)(1r22)(1r32)[1+r12r22+r22r32+r12r32+F(δ1,δ2)]2
We will assume that δ1 << 1, that is, we are dealing with very thin films. Then Eq. (5) becomes
R/d1=4(2πn1/λ0)r1r3sin(2δ2)(1r12)(1r24)(1r32)[1+FR(r1,r2,r3)+CR(r1,r2,r3)cos(2δ2)]2
where
FR(r1,r2,r3)=r12r22+r22r32+r12r32+2r1r2(r32+1)
and
CR(r1,r2,r3)=2r3(r1+r2)(1+r1r2)
One first conclusion from Eq. (6) is that the sensitivity of the reflection coefficient to the monitored film goes to zero when r1 = 0 (or n0 = n1) since in this case there is no optical differentiation between the cover medium and the film under growth. More importantly, the same sensitivity vanishes when the optical layer (δ2, n2) is absent since in this case either δ2 = 0 or r3 = 0 and
R(d1,0)/d1=0
In a patterned substrate, therefore, where the optical layer has been selectively removed, the bare substrate areas exhibit zero sensitivity to the film growth. On the contrary, in the areas bearing the optical layer the sensitivity of R to the grown film thickness is maximized if appropriate thickness δ2 is chosen. To find the value of δ22m) that maximizes the absolute value of the derivative in Eq. (6) we set

(R/d1)d2=0cos(2δ2m)=1+FR(r1,r2,r3)(1+FR(r1,r2,r3))2+8CR(r1,r2,r3)22CR(r1,r2,r3)

In most of the cases |FR|«1 and |CR|«1 so that Eq. (10) simplifies to cos(2δ2m)≈0, that is the optimum thickness is such that a photon that makes a round trip through the optical layer will experience a phase difference very close to π(1/2±m), with m an integer. Considering that |FR|«1 and |CR|«1 hold, or equivalently |r1|, |r2|«1, then the extrema for the derivative in Eq. (6) become

R(d1,d2m)/d1±4(2πn1/λ0)r1r3(1r32)
In Eq. (11) the sign of the derivative is minus when 2δ2m = (1/2 + 2m)π, or d2 = (1/8 + 1/2m)λ0/n2, and plus when 2δ2 = (3/2 + 2m)π, or d2 = (3/8 + 1/2m)λ0/n2, with m an integer.

2.2. Differential reflectance as an intrinsic parameter in structured substrate characterization

Looking at the simple formula in Eq. (11) one would think that it would be enough for monitoring of d1 to just measure the reflected beam on a one dimensional optical stack with optimized optical layer thickness. However, unwanted reflections in fluidic covers or various interfaces interfere with the measured sensitivity of R to d1. In fact they will underestimate of the actual value of the d1 induced reflectivity changes if normalized with respect to the initial R value at d1 = 0. In order to suppress unwanted reflection, we propose another method based on the measurement of the reflectance difference between sites with different optical layer thicknesses. In fact, the optical structure is partitioned into two interpolated array areas: one with no optical layer (bare substrate) where the initial reflectivity is R(0,0) and another bearing an optical layer with the optimized optical thickness δ2m = 2πn2d2m/λ0 where the initial reflectivity is R(0,d2m). If now the observable is the differential reflectance:

ΔR(d1,d2m)=R(d1,0)R(d1,d2m)
the effects of parasitic and unwanted reflections are suppressed, especially when normalized to the initial value of ΔR. Differentiating ΔR with respect to d1, and since the derivative of the first term of the right hand side in Eq. (12) is zero, we obtain with the help of Eq. (6)
ΔR(d1,d2m)/d1=4(2πn1/λ0)r1r3sin(2δ2m)×(1r12)(1r24)(1r32)[1+r12r22+r22r32+r12r32+2r1r2(r32+1))+2r3(r1+r2)(1+r1r2)cos(2δ2m)]2
The initial value of ΔR at the optimum d2m thickness is from Eq. (1)
ΔR0=R(0,0)R(0,d2m)=(r1,2+r3)2(r1,2r3+1)2r1,22+r32+2r3r1,2cos(2δ2m)1+r1,22r32+2r3r1,2cos(2δ2m)
where
r1,2=n0n2n0+n2
The exact normalized differential reflectance sensitivity is obtained by diving Eq. (13) and Eq. (14). Assuming again that cos(2δ2m)≈0 and that the absolute values of r1 and r2 are close to zero, an assumption usually valid for a realistic choice of materials, then

ΔR(d1,d2m)/d1ΔR0±2(2πn1/λ0)r1r1,2=±(4πn1/λ0)n0n1n0+n1n0+n2n0n2

The sign in Eq. (16) is plus when d2 = (1/8 + 1/2m)λ0/n2, and minus when d2 = (3/8 + 1/2m)λ0/n2. The accuracy of Eq. (16) is shown in Fig. 2 with δ2 = δ2m = 3π/4 and it is compared to the exact expressions obtained by dividing the right hand sides of Eq. (13) and Eq. (14). Two different values and far apart were chosen for n3: 4 (silicon) and 1.46 (quartz). The value for the optical layer refractive index n2 was swept from 1.4 to 2.2 covering the range of transparent dielectrics, while n0 = 1.34 (bioassay buffer). As shown in Fig. 2, Eq. (16) is quite accurate and points out that the normalized differential reflectance is inversely proportional to the (n0-n2) difference and nearly independent of the substrate (n3). This is true especially for n3 in the vicinity of n2, but even for n3 = 4 it is still an excellent approximation for n2 less than 1.5. Judging by the results from two different values of the film refractive index, n1 = 1.4 and 1.45, we conclude that the sensitivity is proportional to the n1-n0 difference (0.06 and 0.11 for the upper and the lower set of curves, respectively), as predicted by Eq. (16).

 

Fig. 2 Exact differential reflectance sensitivity at d2 = d2m as a function of the optical layer refractive index and for two different substrates, silicon (dash-dotted) and quartz (dotted). The solid line is the approximate expression in Eq. (16). The two sets of curves correspond to n1 = 1.4 and 1.45, respectively. Here, n0 = 1.34, optical length δ2m = 3π/4 and λ0 = 450 nm.

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The significance of Fig. 2, or Eq. (16), lies in the fact that the film thickness d1 can be calculated dividing the fractional change of the measured differential reflectance by the sensitivity values in Fig. 2, provided of course that δ1«1 as is the case for biomolecular films of few nm. Using the simplified Eq. (16) we obtain:

ΔR(d1,d2m)ΔR(0,d2m)ΔR0=ΔR(d1,d2m)ΔR0ΔR0±22πn1λ0n0n1n0+n1n0+n2n0n2d1d1|ΔR(d1,d2m)ΔR0ΔR0λ04πn1n0+n1n0n1n0n2n0+n2|
The above equation expresses the film thickness as the absolute value of the differential reflectance fractional change times a simple expression of known parameters (λ0, n0, n1, n2). Finally, the values of cos(2δ2m) from Eq. (10) are shown in Fig. 3 which proves that the approximation cos(2δ2m)≈0 is very good for the range of values explored here.

 

Fig. 3 Plot of cos(2δm) in Eq. (10) as a function of n2 for the silicon and the quartz substrate.

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2.3 Differential reflectance measurement through Fourier transform on periodic microscope images and choice of materials

Measuring ΔR requires simultaneous measurement of the reflectance at different sites and this is best done by optical reflection microscopy. The fractional changes of the differential reflectance are the same as the fractional changes of the contrast in the microscope image of the same patterned substrate. In fact, if a periodic pattern is chosen with interpolated areas having an optical layer thickness of either 0 or d2m, then Fourier transform techniques applied on the periodic image bit map can isolate the changes in ΔR with a high degree of accuracy by isolating the changes in the image contrast factor. Now, as for the choice of materials to maximize the non-normalized reflectance sensitivity to d1, Eq. (11) points to an optical stack with a relatively high |r3| provided r32<1/3. This can be achieved by choosing substrate material of high n3, such as silicon. Such a substrate will have the added benefit of being opaque to optical frequencies, so that there are no interfering beams reflected from the back substrate side. The choice of the optical layer (n2) comes from optimizing the normalized differential sensitivity in Eq. (16). Small values of the n0-n2 difference lead to higher normalized differential reflectance. When n0 = n2, ΔR0 = 0 and normalization with respect to ΔR0 is no longer possible. Also in the case r1,2 ≈0, implying ΔR0≈0, noise related uncertainties in the measured value of ΔR0 can lead to large uncertainties in determining d1 through Eq. (16). Another issue when ΔR0 approaches zero is that it will be nearly impossible to focus on the patterned optical stack under the microscope. For the aforementioned reasons small n0-n2 values are desirable provided the patterned optical stack is still imageable. If Si is chosen as a substrate, then SiO2 is a natural choice for the optical layer not only in terms of easiness of fabrication but also in terms of its refractive index proximity to the cover medium. In practical cases the most usual cover media is either vacuum or air (n0≈1) or water buffer solutions (n0≈1.34) in the case of bioassays. Therefore the 1.46 n2 value of SiO2 seems to satisfy the above mentioned conditions.

Having outlined the basic considerations in terms of geometry and choice of materials, the demonstration of the method will be presented in the sections to follow and consists of a monitoring the progress of a protein binding assay through optical microscopy on a silicon wafer with a patterned silicon dioxide film. Therefore from now on n0 = 1.34, n2 = 1.46, n3 = 4. For the protein refractive index two values were considered: n1 = 1.4 and n1 = 1.45. In a microscope a spectral band is more practically realized through the use of band pass optical filters as opposed to monochromatic light. The use of a narrow spectral band instead of monochromatic light has negligible effect on the differential reflectance sensitivity to film growth, as is shown in Fig. 4 . Here, the normalized differential reflectance sensitivity, calculated from Eq. (13) and Eq. (14) at λ0 = 450 nm, is plotted along with the averaged differential reflectance sensitivity defined as (ΔR(d1,d2m)ΔR(0,d2m))/(d1ΔR(0,d2m)) with d1 = 1nm and with ΔR calculated by averaging R(d1,d2m), R(d1,0) and R(0,d2m) from (1) over the spectral range 425-475nm. Therefore, the conclusions derived through the above equations obtained for monochromatic light still hold for narrow spectral bands where now λ0 is the band center. Finally, for maximum sensitivity the oxide thickness is chosen as d2m = 3/8(λ0/n2) or δm = 3π/4. With λ0 = 450 nm, d2m assumes the value of 115 nm. Now, Eq. (16) holds with the minus sign which implies that protein binding decreases the differential reflectance.

 

Fig. 4 Differential reflectance sensitivity for monochromatic light at 450 nm (solid line) compared to the averaged sensitivity defined as (ΔR(d1,d2m)ΔR(0,d2m))/(d1ΔR(0,d2m)) with d1 = 1nm and ΔR calculated by averaging R(d1,d2m), R(d1,0) and R(0,d2m) from (1) over the spectral range 425-475nm. Here, n0 = 1.34 and δ2m≈3π/4.

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3. Experimental method

The instrument employed for the reflectance measurements was an optical microscope equipped with a digital camera (OLYMPUS DP71). Apart from the bioreagents, the only additions to perform the experiments, was a fluidic module as described below.

3.1 Optical and bio-fluidic set up

The silicon wafer was oxidized at 1030 C° and a 100 nm SiO2 layer was grown. The oxide layer was patterned through lithography and etching in buffered HF solution, so that a periodic pattern of oxide islands was created with a pitch of 20 microns (Fig. 5 ). The wafer was silanized with APTES and coated with biotinylated Bovine Serum Albumin (b-BSA). The total combined thickness of the optical layer (SiO2 + APTES + b-BSA) is estimated near 110 nm, very close to d2m = 115 nm.

 

Fig. 5 The squares are the silicon oxide square islands surrounded by bare silicon. The island side is 10 μm and the pitch 20 μm.

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The wafer was placed under the microscope in a fluidic structure consisting of two Al plates, a quartz plate, a PDMS gasket, and plastic tubing (Fig. 6 ). The top Al plate had a quartz window so that the patterned silicon wafer could be monitored in real time by focusing through the quartz cover and the buffer solution while biomolecular solutions were pumped through the fluidic channels by a micropump. The microscope light was filtered in the blue (425-475nm, Fig. 6(a)) and an objective 10X was used. The gasket was made by casting PDMS in a mold bearing the plastic tubing for the supply of reagents. After the molding was complete the part of the tubing left in the reaction areas (shown as the four diamond shaped openings in Fig. 6) was removed by a sharp blade. Four screws were holding the two Al plates in place and provided the required pressure so that the PDMS gasket would seal the reaction areas. The digital files (12 bit vertical resolution, 1 picture (3072x3072 bits) per 4 seconds) of the pictures taken by the digital microscope camera were processed to monitor the reaction between the immobilized b-BSA and the streptavidin in the solution.

 

Fig. 6 Schematic cross section and photograph of the opto-fluidic structure. In (a) the objective light is focused on the oxide pattern within the fluidic structure. The Al plate seals the structure by applying pressure on the quartz cover against the PDMS gasket and the underlying silicon wafer. The shaded area is the assay buffer solution on top of the SiO2 squares (shown out of scale). In (b) the gasket is illustrated sandwiched between the quartz cover (shown as a disk) and the silicon wafer below. In the gasket the four diamond shaped bioreaction fluidic chambers are connected to the reagent supply tubing. Also shown is the top Al plate with the optical window and two of the clamping screws.

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3.2 Coating and Biomolecular reagents

Biotinylated BSA was prepared according to a published method [9]: The protein was reacted with sulfo-NHS-LC-biotin in a 0.3 M sodium carbonate buffer (pH 9.2) at a molar ratio of 1:120 for 2 h at room temperature. Then, the mixture was extensively dialyzed against 0.1 M NaHCO3 solution, pH 8.5, containing 0.15 M NaCl and 15 mM sodium azide and was stored at 4 °C. Biotinylated BSA was diluted at a concentration of 25 mg/l in 50 mM phosphate buffer, pH 7.4. The solution was then applied to the patterned wafer surface and incubated for two hours at room temperature. Then, the wafer was washed with 50 mM phosphate buffer, pH 7.4 and subsequently immersed in a blocking solution of 10 g/l BSA solution in 50 mM phosphate buffer, pH 7.4 for one hour in order to cover the remaining free protein binding sites. After blocking the wafer was flashed with de-ionized water and blown dry before inserting in the fluidic structure of Fig. 6. The same blocking solution was used as the assay buffer during the reaction with streptavidin.

4. Experimental results and discussion

4.1. Contrast factor through Fourier transform

The digital pictures taken are treated as bitmaps of the reflectivity across the imaged area. From the reflectivity map we isolate the differential reflectance that is, the difference in reflectance of the bare substrate from the oxide regions. This reflectance is free from interfering effects coming from the fluidic covers and the microscope internal optics and depends only on the intrinsic optical stack reflectance. For the optical stack chosen, the oxide reflectance increases with binding while the bare silicon reflectance stays practically the same resulting in a contrast decrease of the overall image. The differential reflectance is proportional to the contrast of the two-dimensional image bitmap. The two dimensional plots shown in Fig. 7(a,b) demonstrate the contrast change of the periodic pattern before and after the reaction. The blue squares are the silicon oxide square areas where the reflectance is lower compared to the reflectance of the surrounding bare silicon. The contrast shown is larger before the reaction because of the lower reflectance oxide regions (deep blue areas, Fig. 7(a)) compared to the same after the reaction (blue areas, Fig. 7(b)). Although in Fig. 7(a,b) the two dimensional plots show pictorially the reaction effects on the contrast, in a way allow visualization of the reaction, it takes digital signal processing through Fourier transform of the two dimensional image bit map to derive the reflectance difference at a good enough resolution by isolating the image contrast factor. At the point corresponding to the periodicity of the image in the direct domain, the transformed signal will contain a sharp peak (resonant peak) way above the noise level of the non resonant frequencies (Fig. 7(c)). This peak is directly proportional to the differential reflectance sought after. Parasitic signals, like the unwanted reflections from fluidic interfaces, will not affect the signal since they do not have the same periodicity. At the same time, normalization with respect to the incident light intensity is possible if the resonant peak is divided by the Fourier transform signal at zero frequency. The later signal is a measure of the average reflected light intensity and directly proportional to the incident intensity. Changes of the resonant peak over time provide a direct measure of the film built up and its thickness.

 

Fig. 7 Digital images of the patterned wafer (a,b) and Fourier transform (c). The digital image as a 2x2 array before (a) and after the reaction (b) with 20 nM streptavidin for 4000 seconds. In (b) the contrast is visibly lower than in (a). Two dimensional Fourier transform of the microscope image (c). The position of the sharp peak corresponds to the number of square islands in the X and Y direction.

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4.2 Bioanalytical results

The normalized differential reflectance obtained as the resonant peak in the Fourier transform domain (Fig. 7(c)) is shown in Fig. 8 . The magnitude of the peak stays more or less constant in the first 120 pictures where only the reaction buffer (10 g/l BSA) flows (Fig. 8). Then, after the introduction of streptavidin in the buffer solution and the beginning of the specific reaction, it sharply declines and after assuming a maximum slope it starts leveling off. In the maximum slope region the reaction is controlled by the diffusion of the streptaviding towards the coated silicon surface and by the binding rate constant. As the reaction proceeds, the available binding sites diminish and the streptavidin film growth starts leveling off. The saturation tendency is more evident in the highest molarity plot (20 nM) which saturates at a differential reflectance drop of 18%. From Fig. 2, on normalized differential reflectance sensitivity, we conclude that such a drop corresponds to a streptavidin layer thicknesses of 8.5 nm in case n1 = 1.4 or 4.6 nm in case n1 = 1.45nm.

 

Fig. 8 Experimental results for three streptavidin concentrations: 20 nM, 10 nM and 5nM. The signal plotted is the peak in the Fourier transform domain normalized by its value at 0 time. The analyte is introduced after the first 120 shots (vertical arrow). Every shot takes 4 seconds. In the 5 nM plot, the straight line segment between shots 713 and 787 is an extrapolation of missing data due to air bubbles.

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Such numbers are in excellent agreement with streptavidin layer thicknesses after saturation reaction with biotin coated surfaces [10]. The 5 nM reaction starts the saturation turn after the first 600 pictures and keeps approaching the 20 nM levels, while the 10 nm plot saturates at even higher thicknesses compared to the 20 nM. Such saturation effect anomalies appear [11,12] possibly due to second order effects favoring thicker saturation layers at more moderate analyte concentrations. Nevertheless, the maximum slopes do correspond with the analyte (streptavidin) concentration. The three slopes in descending order are: 1.4x10−3, 5.7x10−4 and 2.8x10−4 per shot, in good agreement with the concentration order.

4.3 Multianalyte capabilities

The above described method for the monitoring of biomolecular binding can be extended to cover cases where a number of different probes have been immobilized on the patterned silicon wafer leading to an equal number of different reactions taking place simultaneously, provided the appropriate mixture is supplied. In this case the field to be illuminated and monitored is segmented into the corresponding spotting areas, while the algorithm is modified so that each area is independently processed through Fourier transform to isolate the differential reflectance for each different spotting area. In such a case of course, the space bit count per area will be reduced resulting in higher noise levels. This places an upper limit to the number of different spotted areas that can be monitored simultaneously. However, such an effect can be counterbalanced if a higher spatial resolution (pixel number) camera is employed.

5. Conclusions

A method for monitoring molecular films was presented based on measurements of the differential reflectance of a substrate with a patterned optical layer. The film growth changes the reflection coefficient of the optical composite in a way decisively determined by the optical layer thickness. The optical layer is partitioned into two periodic arrays of maximum and zero reflectance sensitivity with regard to the molecular film growth. The maximum sensitivity regions are the optical layer islands and the zero sensitivity areas is bare silicon. The difference in the reflectance of the two regions is taken as the observable since it is free from interfering effects originating in fluidic cover or parasitic reflections. The tool for the simultaneous measurement of the reflectance from a multitude of points was an optical microscope equipped with a 12 bit digital camera. The differential reflectance was obtained by isolating the contrast factor of the periodic digital images processed through Fourier transform. By calculating the resonant peak fractional changes in the Fourier transform domain, the molecular film built-up was monitored in time through the theoretical analysis developed in section 1 and the simple formula of Eq. (16). As a demonstration experiment, label free protein-protein binding is monitored. Streptavidin solutions of various concentrations reacted with biotinlylated BSA coated on silicon wafer with a patterned SiO2 film. The only instrument employed here was a digital microscope plus an optical window equipped fluidic structure for the supply of reagents. After the introduction of the streptavidin solutions the signal changed proportionally to the analyte concentration and saturated at values in agreement to literature data for the same reaction. Multiplexed assay performance is possible if the illuminated field is partitioned into areas where different biomolecular probes have been immobilized and where independent resonant peaks are calculated.

Acknowledgments

We thank Dr. P. Petrou and Dr. S. Kakabakos for the supply of bioreagents.

References and links

1. A. P. F. Turner, “Biochemistry. Biosensors--sense and sensitivity,” Science 290(5495), 1315–1317 (2000). [CrossRef]   [PubMed]  

2. X. W. Guo, “Surface plasmon resonance based biosensor technique: a review,” J Biophotonics 5(7), 483–501 (2012). [CrossRef]   [PubMed]  

3. S. Herranz, M. Bockova, M. D. Marazuela, J. Homola, and M. C. Moreno-Bondi, “Surface plasmon resonance biosensor for parallelized detection of protein biomarkers in diluted blood plasma,” Anal. Bioanal. Chem. 398(6), 2625–2634 (2010). [CrossRef]   [PubMed]  

4. M. S. Luchansky and R. C. Bailey, “Silicon photonic microring resonators for quantitative cytokine detection and T-cell secretion analysis,” Anal. Chem. 82(5), 1975–1981 (2010). [CrossRef]   [PubMed]  

5. A. Brecht, G. Gauglitz, and J. Polster, “Interferometric immunoassay in a FIA-system - a sensitive and rapid approach in label-free immunosensing,” Biosens. Bioelectron. 8(7-8), 387–392 (1993). [CrossRef]  

6. X. Zhu, J. P. Landry, Y. S. Sun, J. P. Gregg, K. S. Lam, and X. Guo, “Oblique-incidence reflectivity difference microscope for label-free high-throughput detection of biochemical reactions in a microarray format,” Appl. Opt. 46(10), 1890–1895 (2007). [CrossRef]   [PubMed]  

7. J. Y. Wang, J. Dai, L. P. He, Y. Sun, H. B. Lu, K. J. Jin, and G. Z. Yang, “Label-free and real-time detections of the interactions of swine IgG with goat anti-swine IgG by oblique-incidence reflectivity difference technique,” J. Appl. Phys. 112(6), 064702 (2012). [CrossRef]  

8. O. S. Heavens, Optical Properties of Thin Solid Films (Dover Publications, 1991) Chap.4.

9. M. Zavali, P. S. Petrou, S. E. Kakabakos, M. Kitsara, I. Raptis, K. Beltsios, and K. Misiakos, “Label-free kinetic study of biomolecular interactions by white light reflectance spectroscopy,” Micro & Nano Lett. 1(2), 94–98 (2006). [CrossRef]  

10. S. Busse, V. Scheumann, B. Menges, and S. Mittler, “Sensitivity studies for specific binding reactions using the biotin/streptavidin system by evanescent optical methods,” Biosens. Bioelectron. 17(8), 704–710 (2002). [CrossRef]   [PubMed]  

11. P. Ihalainen and J. Peltonen, “Immobilization of streptavidin onto biotin-functionalized Langmuir-Schaefer binary monolayers chemisorbed on gold,” Sens. Actuat. B 102(2), 207–218 (2004). [CrossRef]  

12. M. Zavali, “Direct Study of biomolecular interactions through white light reflection spectroscopy” Senior Thesis, University of Ioannina, Greece, (2006).

References

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  1. A. P. F. Turner, “Biochemistry. Biosensors--sense and sensitivity,” Science290(5495), 1315–1317 (2000).
    [CrossRef] [PubMed]
  2. X. W. Guo, “Surface plasmon resonance based biosensor technique: a review,” J Biophotonics5(7), 483–501 (2012).
    [CrossRef] [PubMed]
  3. S. Herranz, M. Bockova, M. D. Marazuela, J. Homola, and M. C. Moreno-Bondi, “Surface plasmon resonance biosensor for parallelized detection of protein biomarkers in diluted blood plasma,” Anal. Bioanal. Chem.398(6), 2625–2634 (2010).
    [CrossRef] [PubMed]
  4. M. S. Luchansky and R. C. Bailey, “Silicon photonic microring resonators for quantitative cytokine detection and T-cell secretion analysis,” Anal. Chem.82(5), 1975–1981 (2010).
    [CrossRef] [PubMed]
  5. A. Brecht, G. Gauglitz, and J. Polster, “Interferometric immunoassay in a FIA-system - a sensitive and rapid approach in label-free immunosensing,” Biosens. Bioelectron.8(7-8), 387–392 (1993).
    [CrossRef]
  6. X. Zhu, J. P. Landry, Y. S. Sun, J. P. Gregg, K. S. Lam, and X. Guo, “Oblique-incidence reflectivity difference microscope for label-free high-throughput detection of biochemical reactions in a microarray format,” Appl. Opt.46(10), 1890–1895 (2007).
    [CrossRef] [PubMed]
  7. J. Y. Wang, J. Dai, L. P. He, Y. Sun, H. B. Lu, K. J. Jin, and G. Z. Yang, “Label-free and real-time detections of the interactions of swine IgG with goat anti-swine IgG by oblique-incidence reflectivity difference technique,” J. Appl. Phys.112(6), 064702 (2012).
    [CrossRef]
  8. O. S. Heavens, Optical Properties of Thin Solid Films (Dover Publications, 1991) Chap.4.
  9. M. Zavali, P. S. Petrou, S. E. Kakabakos, M. Kitsara, I. Raptis, K. Beltsios, and K. Misiakos, “Label-free kinetic study of biomolecular interactions by white light reflectance spectroscopy,” Micro & Nano Lett.1(2), 94–98 (2006).
    [CrossRef]
  10. S. Busse, V. Scheumann, B. Menges, and S. Mittler, “Sensitivity studies for specific binding reactions using the biotin/streptavidin system by evanescent optical methods,” Biosens. Bioelectron.17(8), 704–710 (2002).
    [CrossRef] [PubMed]
  11. P. Ihalainen and J. Peltonen, “Immobilization of streptavidin onto biotin-functionalized Langmuir-Schaefer binary monolayers chemisorbed on gold,” Sens. Actuat. B102(2), 207–218 (2004).
    [CrossRef]
  12. M. Zavali, “Direct Study of biomolecular interactions through white light reflection spectroscopy” Senior Thesis, University of Ioannina, Greece, (2006).

2012

X. W. Guo, “Surface plasmon resonance based biosensor technique: a review,” J Biophotonics5(7), 483–501 (2012).
[CrossRef] [PubMed]

J. Y. Wang, J. Dai, L. P. He, Y. Sun, H. B. Lu, K. J. Jin, and G. Z. Yang, “Label-free and real-time detections of the interactions of swine IgG with goat anti-swine IgG by oblique-incidence reflectivity difference technique,” J. Appl. Phys.112(6), 064702 (2012).
[CrossRef]

2010

S. Herranz, M. Bockova, M. D. Marazuela, J. Homola, and M. C. Moreno-Bondi, “Surface plasmon resonance biosensor for parallelized detection of protein biomarkers in diluted blood plasma,” Anal. Bioanal. Chem.398(6), 2625–2634 (2010).
[CrossRef] [PubMed]

M. S. Luchansky and R. C. Bailey, “Silicon photonic microring resonators for quantitative cytokine detection and T-cell secretion analysis,” Anal. Chem.82(5), 1975–1981 (2010).
[CrossRef] [PubMed]

2007

2006

M. Zavali, P. S. Petrou, S. E. Kakabakos, M. Kitsara, I. Raptis, K. Beltsios, and K. Misiakos, “Label-free kinetic study of biomolecular interactions by white light reflectance spectroscopy,” Micro & Nano Lett.1(2), 94–98 (2006).
[CrossRef]

2004

P. Ihalainen and J. Peltonen, “Immobilization of streptavidin onto biotin-functionalized Langmuir-Schaefer binary monolayers chemisorbed on gold,” Sens. Actuat. B102(2), 207–218 (2004).
[CrossRef]

2002

S. Busse, V. Scheumann, B. Menges, and S. Mittler, “Sensitivity studies for specific binding reactions using the biotin/streptavidin system by evanescent optical methods,” Biosens. Bioelectron.17(8), 704–710 (2002).
[CrossRef] [PubMed]

2000

A. P. F. Turner, “Biochemistry. Biosensors--sense and sensitivity,” Science290(5495), 1315–1317 (2000).
[CrossRef] [PubMed]

1993

A. Brecht, G. Gauglitz, and J. Polster, “Interferometric immunoassay in a FIA-system - a sensitive and rapid approach in label-free immunosensing,” Biosens. Bioelectron.8(7-8), 387–392 (1993).
[CrossRef]

Bailey, R. C.

M. S. Luchansky and R. C. Bailey, “Silicon photonic microring resonators for quantitative cytokine detection and T-cell secretion analysis,” Anal. Chem.82(5), 1975–1981 (2010).
[CrossRef] [PubMed]

Beltsios, K.

M. Zavali, P. S. Petrou, S. E. Kakabakos, M. Kitsara, I. Raptis, K. Beltsios, and K. Misiakos, “Label-free kinetic study of biomolecular interactions by white light reflectance spectroscopy,” Micro & Nano Lett.1(2), 94–98 (2006).
[CrossRef]

Bockova, M.

S. Herranz, M. Bockova, M. D. Marazuela, J. Homola, and M. C. Moreno-Bondi, “Surface plasmon resonance biosensor for parallelized detection of protein biomarkers in diluted blood plasma,” Anal. Bioanal. Chem.398(6), 2625–2634 (2010).
[CrossRef] [PubMed]

Brecht, A.

A. Brecht, G. Gauglitz, and J. Polster, “Interferometric immunoassay in a FIA-system - a sensitive and rapid approach in label-free immunosensing,” Biosens. Bioelectron.8(7-8), 387–392 (1993).
[CrossRef]

Busse, S.

S. Busse, V. Scheumann, B. Menges, and S. Mittler, “Sensitivity studies for specific binding reactions using the biotin/streptavidin system by evanescent optical methods,” Biosens. Bioelectron.17(8), 704–710 (2002).
[CrossRef] [PubMed]

Dai, J.

J. Y. Wang, J. Dai, L. P. He, Y. Sun, H. B. Lu, K. J. Jin, and G. Z. Yang, “Label-free and real-time detections of the interactions of swine IgG with goat anti-swine IgG by oblique-incidence reflectivity difference technique,” J. Appl. Phys.112(6), 064702 (2012).
[CrossRef]

Gauglitz, G.

A. Brecht, G. Gauglitz, and J. Polster, “Interferometric immunoassay in a FIA-system - a sensitive and rapid approach in label-free immunosensing,” Biosens. Bioelectron.8(7-8), 387–392 (1993).
[CrossRef]

Gregg, J. P.

Guo, X.

Guo, X. W.

X. W. Guo, “Surface plasmon resonance based biosensor technique: a review,” J Biophotonics5(7), 483–501 (2012).
[CrossRef] [PubMed]

He, L. P.

J. Y. Wang, J. Dai, L. P. He, Y. Sun, H. B. Lu, K. J. Jin, and G. Z. Yang, “Label-free and real-time detections of the interactions of swine IgG with goat anti-swine IgG by oblique-incidence reflectivity difference technique,” J. Appl. Phys.112(6), 064702 (2012).
[CrossRef]

Herranz, S.

S. Herranz, M. Bockova, M. D. Marazuela, J. Homola, and M. C. Moreno-Bondi, “Surface plasmon resonance biosensor for parallelized detection of protein biomarkers in diluted blood plasma,” Anal. Bioanal. Chem.398(6), 2625–2634 (2010).
[CrossRef] [PubMed]

Homola, J.

S. Herranz, M. Bockova, M. D. Marazuela, J. Homola, and M. C. Moreno-Bondi, “Surface plasmon resonance biosensor for parallelized detection of protein biomarkers in diluted blood plasma,” Anal. Bioanal. Chem.398(6), 2625–2634 (2010).
[CrossRef] [PubMed]

Ihalainen, P.

P. Ihalainen and J. Peltonen, “Immobilization of streptavidin onto biotin-functionalized Langmuir-Schaefer binary monolayers chemisorbed on gold,” Sens. Actuat. B102(2), 207–218 (2004).
[CrossRef]

Jin, K. J.

J. Y. Wang, J. Dai, L. P. He, Y. Sun, H. B. Lu, K. J. Jin, and G. Z. Yang, “Label-free and real-time detections of the interactions of swine IgG with goat anti-swine IgG by oblique-incidence reflectivity difference technique,” J. Appl. Phys.112(6), 064702 (2012).
[CrossRef]

Kakabakos, S. E.

M. Zavali, P. S. Petrou, S. E. Kakabakos, M. Kitsara, I. Raptis, K. Beltsios, and K. Misiakos, “Label-free kinetic study of biomolecular interactions by white light reflectance spectroscopy,” Micro & Nano Lett.1(2), 94–98 (2006).
[CrossRef]

Kitsara, M.

M. Zavali, P. S. Petrou, S. E. Kakabakos, M. Kitsara, I. Raptis, K. Beltsios, and K. Misiakos, “Label-free kinetic study of biomolecular interactions by white light reflectance spectroscopy,” Micro & Nano Lett.1(2), 94–98 (2006).
[CrossRef]

Lam, K. S.

Landry, J. P.

Lu, H. B.

J. Y. Wang, J. Dai, L. P. He, Y. Sun, H. B. Lu, K. J. Jin, and G. Z. Yang, “Label-free and real-time detections of the interactions of swine IgG with goat anti-swine IgG by oblique-incidence reflectivity difference technique,” J. Appl. Phys.112(6), 064702 (2012).
[CrossRef]

Luchansky, M. S.

M. S. Luchansky and R. C. Bailey, “Silicon photonic microring resonators for quantitative cytokine detection and T-cell secretion analysis,” Anal. Chem.82(5), 1975–1981 (2010).
[CrossRef] [PubMed]

Marazuela, M. D.

S. Herranz, M. Bockova, M. D. Marazuela, J. Homola, and M. C. Moreno-Bondi, “Surface plasmon resonance biosensor for parallelized detection of protein biomarkers in diluted blood plasma,” Anal. Bioanal. Chem.398(6), 2625–2634 (2010).
[CrossRef] [PubMed]

Menges, B.

S. Busse, V. Scheumann, B. Menges, and S. Mittler, “Sensitivity studies for specific binding reactions using the biotin/streptavidin system by evanescent optical methods,” Biosens. Bioelectron.17(8), 704–710 (2002).
[CrossRef] [PubMed]

Misiakos, K.

M. Zavali, P. S. Petrou, S. E. Kakabakos, M. Kitsara, I. Raptis, K. Beltsios, and K. Misiakos, “Label-free kinetic study of biomolecular interactions by white light reflectance spectroscopy,” Micro & Nano Lett.1(2), 94–98 (2006).
[CrossRef]

Mittler, S.

S. Busse, V. Scheumann, B. Menges, and S. Mittler, “Sensitivity studies for specific binding reactions using the biotin/streptavidin system by evanescent optical methods,” Biosens. Bioelectron.17(8), 704–710 (2002).
[CrossRef] [PubMed]

Moreno-Bondi, M. C.

S. Herranz, M. Bockova, M. D. Marazuela, J. Homola, and M. C. Moreno-Bondi, “Surface plasmon resonance biosensor for parallelized detection of protein biomarkers in diluted blood plasma,” Anal. Bioanal. Chem.398(6), 2625–2634 (2010).
[CrossRef] [PubMed]

Peltonen, J.

P. Ihalainen and J. Peltonen, “Immobilization of streptavidin onto biotin-functionalized Langmuir-Schaefer binary monolayers chemisorbed on gold,” Sens. Actuat. B102(2), 207–218 (2004).
[CrossRef]

Petrou, P. S.

M. Zavali, P. S. Petrou, S. E. Kakabakos, M. Kitsara, I. Raptis, K. Beltsios, and K. Misiakos, “Label-free kinetic study of biomolecular interactions by white light reflectance spectroscopy,” Micro & Nano Lett.1(2), 94–98 (2006).
[CrossRef]

Polster, J.

A. Brecht, G. Gauglitz, and J. Polster, “Interferometric immunoassay in a FIA-system - a sensitive and rapid approach in label-free immunosensing,” Biosens. Bioelectron.8(7-8), 387–392 (1993).
[CrossRef]

Raptis, I.

M. Zavali, P. S. Petrou, S. E. Kakabakos, M. Kitsara, I. Raptis, K. Beltsios, and K. Misiakos, “Label-free kinetic study of biomolecular interactions by white light reflectance spectroscopy,” Micro & Nano Lett.1(2), 94–98 (2006).
[CrossRef]

Scheumann, V.

S. Busse, V. Scheumann, B. Menges, and S. Mittler, “Sensitivity studies for specific binding reactions using the biotin/streptavidin system by evanescent optical methods,” Biosens. Bioelectron.17(8), 704–710 (2002).
[CrossRef] [PubMed]

Sun, Y.

J. Y. Wang, J. Dai, L. P. He, Y. Sun, H. B. Lu, K. J. Jin, and G. Z. Yang, “Label-free and real-time detections of the interactions of swine IgG with goat anti-swine IgG by oblique-incidence reflectivity difference technique,” J. Appl. Phys.112(6), 064702 (2012).
[CrossRef]

Sun, Y. S.

Turner, A. P. F.

A. P. F. Turner, “Biochemistry. Biosensors--sense and sensitivity,” Science290(5495), 1315–1317 (2000).
[CrossRef] [PubMed]

Wang, J. Y.

J. Y. Wang, J. Dai, L. P. He, Y. Sun, H. B. Lu, K. J. Jin, and G. Z. Yang, “Label-free and real-time detections of the interactions of swine IgG with goat anti-swine IgG by oblique-incidence reflectivity difference technique,” J. Appl. Phys.112(6), 064702 (2012).
[CrossRef]

Yang, G. Z.

J. Y. Wang, J. Dai, L. P. He, Y. Sun, H. B. Lu, K. J. Jin, and G. Z. Yang, “Label-free and real-time detections of the interactions of swine IgG with goat anti-swine IgG by oblique-incidence reflectivity difference technique,” J. Appl. Phys.112(6), 064702 (2012).
[CrossRef]

Zavali, M.

M. Zavali, P. S. Petrou, S. E. Kakabakos, M. Kitsara, I. Raptis, K. Beltsios, and K. Misiakos, “Label-free kinetic study of biomolecular interactions by white light reflectance spectroscopy,” Micro & Nano Lett.1(2), 94–98 (2006).
[CrossRef]

Zhu, X.

Anal. Bioanal. Chem.

S. Herranz, M. Bockova, M. D. Marazuela, J. Homola, and M. C. Moreno-Bondi, “Surface plasmon resonance biosensor for parallelized detection of protein biomarkers in diluted blood plasma,” Anal. Bioanal. Chem.398(6), 2625–2634 (2010).
[CrossRef] [PubMed]

Anal. Chem.

M. S. Luchansky and R. C. Bailey, “Silicon photonic microring resonators for quantitative cytokine detection and T-cell secretion analysis,” Anal. Chem.82(5), 1975–1981 (2010).
[CrossRef] [PubMed]

Appl. Opt.

Biosens. Bioelectron.

A. Brecht, G. Gauglitz, and J. Polster, “Interferometric immunoassay in a FIA-system - a sensitive and rapid approach in label-free immunosensing,” Biosens. Bioelectron.8(7-8), 387–392 (1993).
[CrossRef]

S. Busse, V. Scheumann, B. Menges, and S. Mittler, “Sensitivity studies for specific binding reactions using the biotin/streptavidin system by evanescent optical methods,” Biosens. Bioelectron.17(8), 704–710 (2002).
[CrossRef] [PubMed]

J Biophotonics

X. W. Guo, “Surface plasmon resonance based biosensor technique: a review,” J Biophotonics5(7), 483–501 (2012).
[CrossRef] [PubMed]

J. Appl. Phys.

J. Y. Wang, J. Dai, L. P. He, Y. Sun, H. B. Lu, K. J. Jin, and G. Z. Yang, “Label-free and real-time detections of the interactions of swine IgG with goat anti-swine IgG by oblique-incidence reflectivity difference technique,” J. Appl. Phys.112(6), 064702 (2012).
[CrossRef]

Micro & Nano Lett.

M. Zavali, P. S. Petrou, S. E. Kakabakos, M. Kitsara, I. Raptis, K. Beltsios, and K. Misiakos, “Label-free kinetic study of biomolecular interactions by white light reflectance spectroscopy,” Micro & Nano Lett.1(2), 94–98 (2006).
[CrossRef]

Science

A. P. F. Turner, “Biochemistry. Biosensors--sense and sensitivity,” Science290(5495), 1315–1317 (2000).
[CrossRef] [PubMed]

Sens. Actuat. B

P. Ihalainen and J. Peltonen, “Immobilization of streptavidin onto biotin-functionalized Langmuir-Schaefer binary monolayers chemisorbed on gold,” Sens. Actuat. B102(2), 207–218 (2004).
[CrossRef]

Other

M. Zavali, “Direct Study of biomolecular interactions through white light reflection spectroscopy” Senior Thesis, University of Ioannina, Greece, (2006).

O. S. Heavens, Optical Properties of Thin Solid Films (Dover Publications, 1991) Chap.4.

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Figures (8)

Fig. 1
Fig. 1

Schematic diagram of the thin film to be monitored and the optical layer sandwiched between the cover medium and the substrate.

Fig. 2
Fig. 2

Exact differential reflectance sensitivity at d2 = d2m as a function of the optical layer refractive index and for two different substrates, silicon (dash-dotted) and quartz (dotted). The solid line is the approximate expression in Eq. (16). The two sets of curves correspond to n1 = 1.4 and 1.45, respectively. Here, n0 = 1.34, optical length δ2m = 3π/4 and λ0 = 450 nm.

Fig. 3
Fig. 3

Plot of cos(2δm) in Eq. (10) as a function of n2 for the silicon and the quartz substrate.

Fig. 4
Fig. 4

Differential reflectance sensitivity for monochromatic light at 450 nm (solid line) compared to the averaged sensitivity defined as (ΔR( d 1 , d 2 m )ΔR(0, d 2 m ))/( d 1 ΔR(0, d 2 m )) with d1 = 1nm and ΔR calculated by averaging R(d1,d2m), R(d1,0) and R(0,d2m) from (1) over the spectral range 425-475nm. Here, n0 = 1.34 and δ2m≈3π/4.

Fig. 5
Fig. 5

The squares are the silicon oxide square islands surrounded by bare silicon. The island side is 10 μm and the pitch 20 μm.

Fig. 6
Fig. 6

Schematic cross section and photograph of the opto-fluidic structure. In (a) the objective light is focused on the oxide pattern within the fluidic structure. The Al plate seals the structure by applying pressure on the quartz cover against the PDMS gasket and the underlying silicon wafer. The shaded area is the assay buffer solution on top of the SiO2 squares (shown out of scale). In (b) the gasket is illustrated sandwiched between the quartz cover (shown as a disk) and the silicon wafer below. In the gasket the four diamond shaped bioreaction fluidic chambers are connected to the reagent supply tubing. Also shown is the top Al plate with the optical window and two of the clamping screws.

Fig. 7
Fig. 7

Digital images of the patterned wafer (a,b) and Fourier transform (c). The digital image as a 2x2 array before (a) and after the reaction (b) with 20 nM streptavidin for 4000 seconds. In (b) the contrast is visibly lower than in (a). Two dimensional Fourier transform of the microscope image (c). The position of the sharp peak corresponds to the number of square islands in the X and Y direction.

Fig. 8
Fig. 8

Experimental results for three streptavidin concentrations: 20 nM, 10 nM and 5nM. The signal plotted is the peak in the Fourier transform domain normalized by its value at 0 time. The analyte is introduced after the first 120 shots (vertical arrow). Every shot takes 4 seconds. In the 5 nM plot, the straight line segment between shots 713 and 787 is an extrapolation of missing data due to air bubbles.

Equations (1)

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R( d 1 ,0)/ d 1 =0

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