Responsivity of the Differential-Intensity Surface Plasmon Resonance Instrument

Surface plasmon resonance is used for the sensitive measurement of minute concentrations of bio-analytes and probing of electrochemical processes. Typical refractive index sensitivity, for the intensity approach, is around 10 -6 refractive index units (RIUs). A better sensitivity has been suggested by developing a differential-intensity detection method. This method relies on the excitation of surface plasmons using a weakly focused beam with the average angle of incidence equal to the resonance angle, while the reflected light is detected using a bi-cell photodiode. The Bi-cell signal is processed by calculating the difference between its two units, normalized to their sum. This ratio estimates the shift in the resonance angle using a model that represents the resonance curve with a quadratic function. However, this model does not explain the effects of parameters such as the angular width of the excitation beam and the specifications of the sensing structure on the system’s response. This paper presents a detailed evaluation of the responsivity using experimental and theoretical approaches, which can predict the effect of the different parameters, paving the way towards the investigation of a better sensitivity and the optimization of the system’s design for different applications.

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Introduction
Surface plasmon resonance (SPR) sensors are powerful analytical tools that are capable of detecting small concentrations of analytes, providing a wide range of medical and scientific applications [1]. This sensing technique relies on the excitation of longitudinal surface plasmon oscillations at the metal-dielectric interface by directing p-polarized light of a matching spatial frequency on the surface of the metal. The excitation beam is coupled to the interface through a prism that is commonly configured in Kretchmann-Raether total internal reflection setup [2]. Furthermore, the setup is fine-tuned by varying either the wavelength of p-polarized light or its incident angle to achieve the resonance condition. This is characterized by (i) a drop in the intensity of the reflected light and (ii) a sharp phase transition; both are located at the resonance position. These two features produce characteristic resonance curves with a resonance position that is sensitive to the properties of the interface. They provide a sensitive method for probing the interfacial processes within ~100nm of the surface. The measured signal is obtained by (i) tracking the resonance angle [3], (ii) the resonance wavelength [4] or (iii) monitoring the amplitude [5] or the phase [6] [7] [8] of the reflected light at the incident angle of the highest gradient. These detection methods feature different sensivities to refractive index change: sensitivity is defined, in this study, as the minimum detectable refractive index change. For instance, phase systems typically demonstrate a higher sensitivity (down to 10 -8 Refractive Index Units -RIUs), but usually require complex optical system design and do not usually offer a wide dynamic range [8]. On the other hand, amplitude-based detection systems are relatively simple, but their typical sensitivity is ~10 -5 to 10 -6 RIUs. Therefore, there is a need to improve their sensitivity for measuring low molecular weight biomolecules in low concentration biological samples. As a result, the differentialintensity detection (Bi-cell SPR) has been introduced which features high sensitivity (10 -7 to 10 -8 RIUs) while using a simple optical configuration [9]. Since then, it has been a popular method in different chemical and biological sensing applications [10] [11] [12] [13] [14] [15]. Moreover, similar differential approaches are used to improve the signal-to-noise ratio in a range of SPR systems based on nanohole array technology [16][17] [18]. In addition, the response time of the bicell detector is rapid compared to array detectors. This approach is particularly well-suited since our research is directed at resolving weak signals related to dynamic time-resolved processes. In this study, however, we use a pixelated camera to study the method in detail but this will be replaced with a bi-cell detector to perform dynamic measurements.
In this method (which will be called Differential-Intensity Surface Plasmon Resonance (DI-SPR) in this paper), the surface of the metal is illuminated through the prism with a focused beam centered on the resonance angle. The reflected light, which constructs a window of the resonance curve, is detected by a bi-cell photodiode or a pixelated camera as shown in Fig.1(a). First, by setting the resonance position as the center of the bi-cell detector, the optical power in each half of the curve is detected by one of the units of the detector, as illustrated in Fig.2. Then, the difference between the two signals is normalized to the total reflected power, proportional to the sum of the two units of the bi-cell detector.
This ratio is correlated to the change in the plasmon resonance angle resulting in a three-fold increase in the responsivity compared to the tracking of the resonance position [9]. Also, it has been shown that this method is more sensitive compared to tracking the resonance wavelength [4] or monitoring the intensity of the reflected light at the angle of the highest gradient of the SPR curve [19].
In order to demonstrate how the change in the differential response Δ[(A-B)/(A+B)] is related to the shift in the resonance angle (Δ 0 ), the resonance curve is modelled by the following polynomial function where θ is the angle of incidence and θ 0 is the resonance angle. Assuming that the SPR curve is symmetric about the resonance angle ( 0 ), a quadratic function has previously been used to model the SPR curve [9] using only the second order term [ 2 ( − 0 ) 2 ] from eq.1. The values of A and B are found by integrating the SPR curve using the resonance position ( 0 ) and the angular width ± , corresponding to the width of the unit of the bi-cell detector (Fig.2), as intergration limits. The integration is repeated, with the same integration limits, after shifting the resonance position by Δ 0 [ i.e. the second order term becomes 2 ( − 0 − Δ 0 ) 2 ].
The outputs A and B before and after the resonance shift are used to calculate the change in the differential response Δ[(A-B)/ (A+B)] relative to the shift in the resonance angle (Δ 0 )-(eq.2). This relation provides a quick estimate of the responsivity of the differential-intensity system. In this study, responsivity (η) is defined as the obtained differential response per unit shift in resonance angle, assigned a unit of deg -1 .
To arrive at this model, two assumptions were made: 1) the SPR curve R(θ) is symmetric about the resonance angle θ0; and 2) the sample-induced changes in R will only affect the resonance angle o; without altering the shape of the curve.
In reality neither of the assumptions is valid. The model thus estimates the responsivity, taking into consideration the angular width of the excitation beam, however, it does not reflect the effect of the shape of the resonance curve on the responsivity. For this reason, it does not depend on changes in specifications of the sensing structure. As a result, it cannot provide an accurate calibration to calculate the measured change in refractive index in the case of variations in these specifications or evaluate sensitivity enhancement approaches (e.g. replacing gold thin film with a silver thin film). The consequences of the asymmetry of the curve have been studied by Schneider et. al [19]. Also, the model has been modified to accommodate the effect of the measured sample (i.e. the absorbing sample) on the SPR curve and the responsivity [20]. Here, we investigated how the system responds to variations in (i) the properties of the excitation beam such as its angular width and its spatial profile, (ii) the specification of the sensing structure such as the adhesion layer (for example Chromium) and the thickness of the metal film (i.e. Gold) and (iii) the measured sample. As this technique is highly dependent on obtaining high responsivity, the investigation of the effect of these factors is critical. Therefore, this paper aims to provide a detailed evaluation of the responsivity of the differential-intensity surface plasmon resonance sensors using both experimental and theoretical approaches, providing a unified analytical expression for the responsivity. This analysis will also explain the trade-off between the responsivity and the dynamic range and allows users to reconfigure the setup for different experimental settings.

System Configuration
A differential-intensity surface plasmon resonance sensing platform has been constructed ( Fig.1(a)), adopting a prism-based Kretschmann-Raether configuration. A 633 nm linearly polarized laser is coupled though a prism (BK7, n = 1.515) to the sensor-sample interface. A cylindrical lens is inserted into the beam path so that the light beam is focused to a line onto the sensor chip.
Refractive index matching oil (Refractive Index 1.5150) has been used between the prism and the sensor chip to remove undesired reflections from the interfaces.
The resonance angle is set to ~72 o to excite surface plasmons at the refractive index of distilled water. Since, the incident angle is greater than the critical angle, total internal reflection occurs at the sensor-sample interface. The reflected light is collimated using a second cylindrical lens and detected using a CMOS camera (μEye, IDS GmbH). The angle of convergence of the light beam, inside the prism, is ~ ±3 degrees.

Responsivity Modelling
The responsivity of the DI-SPR has been modelled through the following steps: The resonance curve is generated using Fresnel equations [21][2] for the multilayer presented in Fig.1(b). Also, the quadratic approximation of a narrow range (±1 deg) of the resonance curve has been used to study the effect of the asymmetry as in section 5.4. The advantage of the Fresnel model is its realistic representation of the resonance curve with no assumptions, in contrast to the quadratic approximation.
(ii) For each generated curve, the outputs of the bi-cell detector (i.e. A and B) are calculated by the numerical integration of the resonance curve using the angular width as an integration limit. These outputs are used to calculate the differential response ( − ) ( + ). ⁄

(iii)
The resonance angle (i.e. the location of the minimum of the curve θ 0 ) is calculated, by fitting a narrow range of the resonance curve (±1 deg) to quadratic function, interpolating to the increase accuracy of finding the location of the minimum.
(iv) The responsivity  is the ratio of the change in the differential response to the change in the resonance angle (Δ 0 ) (eq. 2).
(v) The previous steps are repeated with different input parameters (such as the angular width of the excitation beam, the beam profile, thickness of the chromium layer, the thickness of the gold film and the refractive index of the sample).
The effect of the illumination profile is included in the Fresnel model by simulating the beam profile for a specific experimental setup. The beam profile, in one dimension, is given by the Gaussian function (eq. 3) where x is the distance from the center of the bell-shaped curve in mm, σ is the standard deviation which is calculated from the specifications of the laser and M is the magnification of the beam. The simulations considered a beam diameter of 5.1 mm (i.e. the value of M), matching the specification of the experimental setup.

Experiments
The system, described in Fig.1, has been used to perform the following experiments using sensor chips which were prepared by sputtering approximately 50 nm of gold on a glass slide without the use of the chromium adhesionpromoting layer. For these experiments, refractive index solutions have been prepared using different concentrations of sodium chloride (sigma Bioxtra 99.5%).
These solutions have been used to measure the system response to change in refractive index of the sample. For this purpose, a flow cell (with a 57 µl channel) has been mounted on the prism. The response of the system was recorded using the camera while switching the sample in the channel (distilled water n= 1.3319 at 633nm and 20 o c) to another one with a different refractive index (1.3334). Using these recorded video frames, the resonance curve has been produced for each frame using a pixel-angle calibration. In order to calculate the responsivity as a function of the angular width (Fig.3), a virtual differential detection approach (i.e. This analysis provides the means for calculating the responsivity (η) of the DI-SPR using the system response to changes in the resonance angle or the refractive index.
To study the change in responsivity as the resonance position of the SPR curve changes, the refractive index of the sample was stepped from 1.3319 to 1.3409 in increments of 0.5 mRIUs. Similar to the first experiment, the response of the system was recorded using the pixelated camera. The resulting shift in resonance position and the differential response were obtained using the minimum tracking and the virtual differential detection as described above. This experiment explored, in addition to the responsivity, the dynamic range of the system for set of angular widths as discussed later in section 5.4.

Experimental Responsivity
As described in the previous section, the responsivity has been calculated experimentally for the sensing structure described in section 4.

The Effect of the Beam Profile
We investigated the effect of the spatial profile of the excitation beam on the responsivity. In Fig.4, the solid line represents a uniform illumination profile whereas the dotted line represents a Gaussian profile with a diameter that is magnified to 5.1 mm at the input lens. It is clear from the figure that when the beam diameter is magnified, the Gaussian distribution produces a response similar to the uniform illumination, as one would expect. However, some discrepancy is observed for wider angular ranges due to the increase of the Gaussian effect. The theoretical results presented in the rest of the paper assume a uniform distribution.

The influence of the specifications of the Sensing Structure
First, since the chromium adhesion layer is commonly-used, we address its effect on the responsivity. This is performed by calculating the responsivity for a set of thicknesses, from 0nm to 3nm, of this layer. One important effect of the Cr layer is that, for any angular width, there is a reduction in the responsivity as the thickness of the Cr layer increases. This is shown clearly in Fig.5 (a). It is also observed that the angular width of the peak responsivity shifts to the right with increasing Cr thickness. In order to explain these results, we looked at the effect of the Cr layer on the SPR curve, which is shown in Fig.5(b), and more specifically, the FWHM and the minimum reflectivity of the curve (it will be called a0 in this text), in 5(c) and 5(d), respectively. As the thickness of Cr increases, both the FWHM and a0 increases.
Second, we investigated the effect of the gold layer. The responsivity of the differential technique has also demonstrated a noticeable dependence on the thickness of the gold film. Fig.6 (a) shows results obtained by changing the thickness of Au, without any undercoated Chromium layer. Peak responsivity is obtained, for all angular widths, at gold thickness around 48nm, which decreases rapidly either side of the peak. Fig.6 (a) can be explained by considering the SPR curve in Fig.6(b), and its parameters: FWHM and a0 in Fig.6(c) and Fig.6(d), respectively. Although the FWHM decreases with increasing Au thickness, the a0 response shows a turning point at 47nm, which correlates with the responsivity in 6(a).
A close look into Fig.5 and Fig.6 reveals that the minimum reflectivity a0 of the SPR curve is strongly related to the responsivity, showing a negative correlation. For instance, a deeper SPR curve, which is noticed around the thickness of 48nm, demonstrates a higher responsivity. Similarly, as the thickness of the adhesion layer increases, the minimum reflectivity becomes higher and so the responsivity decreases. By getting back to theoretical estimation of the responsivity, the above observations could be related to the zero order of the polynomial that represents the SPR curve. This zero order is not included in the model presented by eq. 2 and so it does not include the observations in Fig.3, Fig.5(a) and Fig.6(a).

The effect of the sample and the asymmetry of the SPR curve
The effect of the measured sample on the responsivity has been investigated. SPR curves were produced using the Fresnel model (i.e. asymmetric curves) and also a second order approximation of the SPR curve, with the zero order a0 included (i.e. symmetric curves). The zero order of the function has not been ignored, as Fig5.(d) and Fig6.(d) suggest a strong link between a0 and the system response, which will be discussed in more detail in section 5.5.
As seen from both the second-order approximation and the Fresnel model (see Fig.7(a) and (b), respectively), the responsivity drops when there is a change in the refractive index of the sample (expressed in the corresponding resonance shift -Δ 0 ). However, the Fresnel model demonstrated the information regarding the asymmetry of the curve about the resonance angle. As seen in Fig.7 (b), the maximum responsivity is shifted to the left and it is not located at the origin (i.e. the initial resonance angle) in contrast to the quadratic case - Fig.7(a). As the maximum responsivity is obtained where the bi-cell detector is balanced, this shift is rectified by mechanical translation of the detector so that the condition ( − ) ( + ⁄ ) equal to zero is satisfied. However, if a linear array detector [22] or a CCD [4] is used, the offset can be calculated and transformed to the corresponding number of pixels.
This analysis also presents valuable information about the trade-off between the responsivity and the dynamic range (Fig.7). For any angular width, the dynamic range could be defined with reference to the drop of the responsivity from its maximum value. Therefore, as in Fig improved. This flexibility, which is offered by the angular width, is valuable for experimental design. For example, to measure a low concentration sample, the design of the experiment favours the sensitivity over the dynamic range and therefore a small value of the angular width is selected and vice versa. The drop in responsivity, as SPR curve shifts (Fig.7), is due to the use of a fixed reference point for the differential detection (i.e. the initial resonance angle) while the resonance position shift in response to the change in refractive index. So, increasing the dynamic range by moving the reference using a pixelated detector has been suggested elsewhere [22].
The drop of the responsivity, due to changing the refractive index of the sample, has also been measured experimentally. From this experiment, shifts of the resonance position (Δ ) are determined for the corresponding changes in refractive index and compared to the response of the DI-SPR (A-B/A+B), as depicted in Fig.9. Responsivity, as defined earlier, is found from the change of the DI-SPR response divided by the resonance shift (Δ ). The experimental responsivity of the system, as a function of the resonance shift, is presented in Fig.10. This graph is calculated from the gradient of the DI-SPR response in Fig. 9 after smoothing by spline interpolation. From Fig.10, it is noticed that responsivity increases as angular width increases with a peak at ~ 0.5 deg before it drops again, similar to the theoretical results of Fig.3, 7(b) and 8(a). The dynamic range increases with the angular width confirming the previous theoretical observations (see Fig.7(b) and Fig.8(b)).
For all the angular widths, the experimental responsivity is lower compared to the theoretical one presented in Fig.7 Fig.11 compared to the theoretical one in Fig.7(b).
This discrepancy is attributed to the reflectivity fluctuations of the experimental SPR curve. In practice, the maximum responsivity is obtained by balancing the bi-cell detector or finding the reference pixel in the case of using array detectors as stated previously, so the shift does not affect the operation of the instrument.

Modifications to the Responsivity Model
The responsivity of DI-SPR depends, not only on the angular width of the excitation beam, but also on the other parameters of the SPR curve (Fig5.(d) and Fig6.(d)). Since the performance of the DI-SPR is strongly dependent on obtaining high responsivity, there is a need to develop a theoretical framework that encounters for the effect of these factors. In order to translate their effect in an analytical expression, the responsivity is derived using a second-order polynomial representation of the SPR curve including the zero order and the second order terms from eq.1. Eq. 4 gives the responsivity (η) obtained from the second order polynomial, with the a's being the coefficients of the power series that represents the SPR curve. For 50 nm gold, a0= 0.008 and a2=0.14. Eq.4 is validated against the data generated from the Fresnel model as presented in Fig.12. As seen in Fig.12, Eq.4 reproduces the responsivity of Fresnel model, but it departs at large values of the angular width, reflecting the effect of the asymmetry of the SPR curve. More accurate representation of the responsivity is achieved by fitting eq.5 to the responsivity data that is generated from the Fresnel model in Fig.12. For 50 nm gold, b0= 0.011 b2=0.2 and 3 =0.03.  Fig.5 (a)). Moreover, including these two factors makes a unified model, which could be extended to consider the measurement of samples that changes the characteristics of the curve (i.e. has absorption properties, [20]).
Prior information on how the experiment changes the SPR curve, in addition to the angular shift, helps selecting the operating point that optimises the sensitivity and the dynamic range.

Conclusions
Previous studies of the responsivity of the DI-SPR did not consider the effect of the parameters of the SPR curve. In this paper, we have presented a modified model that considers the influence of the noble metal thickness, the adhesion layer, and the inhomogeneity of the illumination profile, as reflected in the parameters of the SPR curve. The model can be used to calibrate the DI-SPR systems directly from coefficients of SPR curves, with no need to perform refractive index stepping. We also evaluated, both theoretically and experimentally, the dependence of the system performance on the angular width of the excitation beam in addition to the bi-cell detector balance. These findings shed the light on the selection of the angular width for a tradeoff between the responsivity and the dynamic range. For example, measuring transparent samples that induce only a shift in the resonance curve, the angular width is selected for the maximum responsivity if the measured refractive index change is very small (i.e. narrow dynamic range). In contrast, a wider angular window is selected when a wide dynamic range is needed.
In future, the model could be used to recommend an optimal operating point when measuring samples with absorption properties (i.e. Changing the shape of the SPR curve), by using prior information about the sample to predict the coefficients of the model (or the SPR curve). This framework can also be used to evaluate strategies of improving the sensitivity, by using a more efficient plasmonic sensor (e.g. the use of silver or silver -gold bi-layers).

Authors Biographies
Sidahmed   Muti-layer model for the sensing structure which is used for simulating the differential -intensity response to change in refractive index and other parameters.      the responsivity: responsivity drops to 95% of the peak (solid line) or 50% of the peak (dotted line). Fig.9 The response of the DI-SPR due to the shift of the resonance position. The response is measured for a set of angular widths. The inset shows real time tracking of the resonance position while the refractive index is changed. Fig.10 Experimental responsivity ( ) drops as the resonance position ( ) shifts which is presented for a set of angular widths of the excitation beam, obtained from the gradient of the curve fitted to data in Fig.9. nm using Fresnel models. The experimental SPR curve is generated by subtracting the dark current and normalising to the total internal reflection. Fig.12 Comparison between the second-order and third-order models. Numerically generated data is fitted to the third-order function (red) and compared to the second order function (blue) which is obtained from the polynomial model of the SPR curve.