## Abstract

We report on a unified theoretical model of the resolution of SPR sensors which makes it possible to predict the ultimate performance of all major configurations of SPR sensors. The theory indicates that the performance of SPR sensors is independent of the method of excitation of surface plasmons (prism or grating coupling) or the method of modulation (amplitude, angular or wavelength) and depends dominantly on the noise properties of the light source and detector. Results of the theoretical analysis are compared with the performance reported for several SPR sensors to illustrate that the best state-of-art SPR sensors are approaching their theoretical limits. Possibilities for further advances in the performance of SPR sensor technology are discussed.

©2009 Optical Society of America

## 1. Introduction

Optical biosensors based on surface plasmon resonance (SPR) represent the most advanced and developed optical label-free biosensor technology. SPR biosensors have become a central tool for the study of biomolecular interactions and have been widely applied in the detection of chemical and biological analytes [1]. Over the last two decades, SPR sensor instrumentation has made great advances. In order to achieve high performance, numerous SPR sensor platforms and data processing methods have been developed [1].

SPR sensors employ several different methods of optical excitation of surface plasmons. These methods include attenuated total reflection (prism coupling) [2–4], and diffraction on periodic metallic gratings (grating coupling) [5–7]. The most common modulation approaches used in high-performance sensors are based on spectroscopy of surface plasmons in the wavelength [2,4] or angular [3,5] domains. Amplitude measurements have also been widely used, in particular, in SPR imaging and related techniques. Although the performance of amplitude-based SPR sensors has been usually lower than that of spectroscopy-based SPR sensors, recent advances in SPR imaging have demonstrated that the development of high-performance SPR sensors based on amplitude-modulation is also possible [8,9]. Several configurations with SPR sensors-based on phase modulation have been proposed [10,11]. Although these sensors have exhibited high sensitivity, their performance does not usually exceed the performance of spectroscopy-based SPR sensors. Currently, the best SPR sensors using conventional surface plasmons achieve a performance level which, when expressed in terms of the smallest detectable refractive index change (refractive index resolution), approaches 10^{−7} refractive index units (RIU) [12].

In this paper a theoretical analysis of the performance of SPR sensors is presented. A unified theoretical model of the resolution of SPR sensors is established and used to evaluate the performance of existing SPR sensors and to explore the ultimate limits of SPR sensor technology.

#### 1.1 SPR sensors

SPR sensors utilize the evanescent field of a special mode of electromagnetic field propagating at a metal/dielectric interface - the surface plasmon - to measure changes in the refractive index of the dielectric in the proximity of the interface. The propagation constant of a surface plasmon *β _{sp}* propagating along a planar boundary between a semi-infinite metal with a complex permittivity ${\epsilon}_{m}={\epsilon}_{m}^{\prime}+i{\epsilon}_{m}^{\u2033}$ and a semi-infinite dielectric with a refractive index

*n*can be expressed as

*ω*is the angular frequency and

*c*is the speed of light in vacuum. The real and imaginary parts of the (complex) propagation constant may be expressed by means of the effective index

*n*and the attenuation coefficient

_{ef}*γ*of the surface plasmon [13].

_{i}The main methods of optical excitation of surface plasmons include the attenuated total reflection (prism coupling), diffraction on a metallic grating (grating coupling) and evanescent wave coupling between dielectric and plasmonic waveguides (waveguide coupling). In the attenuated total reflection method, a coupling prism is interfaced with a thin metal film and the evanescent wave created by the attenuated total reflection of light incident on the prism/metal interface couples with the surface plasmon if the phase matching condition is fulfilled:

where*n*is the refractive index of the prism,

_{p}*θ*is the angle of incidence and

*n*is the effective index of the surface plasmon. Excitation of surface plasmons via diffraction coupling requires that the component of the momentum of the diffracted light wave parallel to the interface and the propagation constant of surface plasmon are matched:

_{ef}*Λ*is the period of the diffraction grating,

*λ*is the wavelength and

*m*is the order of diffraction.

Figure 1(a)
shows a schematic of the optical system of an SPR sensor. The optical platform consists of a light source, an SPR coupler (prism or grating), and a detector. The light source provides a beam of light which is introduced to the SPR coupler to excite a surface plasmon. Subsequently, the (reflected or diffracted) light is detected and analyzed by the detector. Depending on which modulation approach is used, the detector records the intensity of (reflected or diffracted) light, or its wavelength or angular spectrum. As a change in the refractive index at the metal/dielectric interface results in a change in the effective index of the surface plasmon, any change in the refractive index can be measured by measuring changes in the intensity, angular or wavelength spectrum of light coupled to the surface plasmon. Alternatively, changes in the phase or polarization can also be measured. However, even in phase or polarization-modulation-based sensors, the sensor output is typically determined by measuring changes in the light intensity or a series of light intensities [14]. A typical wavelength or angular spectrum of light coupled with a surface plasmon is shown in Fig. 1(b). The figure also illustrates a change in the spectrum of light caused by a change in the refractive index of the dielectric Δ*n*.

## 2. Main performance characteristics of SPR sensors

The main performance characteristics of SPR sensors include sensitivity, resolution, linearity, accuracy, reproducibility and dynamic range [1].

*Sensitivity* of an SPR sensor is the ratio of the change in sensor output to the change in the quantity to be measured (*e.g*. the refractive index). The sensitivity of an SPR sensor to a refractive index *S* can be written as:

*Y*denotes the sensor output. The first term describes the sensitivity of the sensor output to the effective index of a surface plasmon and depends on which method of excitation of the surface plasmons and modulation approach are used. The second term describes the sensitivity of the effective index of a surface plasmon to the refractive index and is

*independent*of the modulation method and the method of excitation [1].

*Resolution* of an SPR sensor defines the smallest change in the refractive index that produces a detectable change in the sensor output. The magnitude of sensor output change that can be detected depends on the level of uncertainty of the sensor output - the output noise. Resolution of an SPR sensor ${\sigma}_{RI}$, is typically expressed in terms of the standard deviation of noise of the sensor output ${\sigma}_{so}$translated to the refractive index of the bulk medium:

*S*is the sensitivity of the sensor to a change in the refractive index [1]. The noise in the sensor output originates from the noise of individual light intensities involved in the calculation of the sensor output. The propagation of the intensity noise into the noise of the sensor output will be analyzed in the following sections for SPR sensors based on amplitude, angular and wavelength modulation. Herein, we do not discuss the case of phase and polarization modulation-based SPR sensors. It has previously been shown that the resolution of such sensors is comparable with the resolution of the SPR sensor based on amplitude modulation [15].

#### 2.1 Intensity noise

The intensity noise in the SPR system originates in the optical system and the readout electronics and is associated with random variation in the measured light intensity. The dominant sources of noise are fluctuations of light intensity emitted by the light source, statistical properties of light (shot noise), and noise generated in the electronic circuitry of the detector. The light source noise modulates the incident light beam and therefore the corresponding intensity noise is proportional to the measured intensity and affects the whole detector area or the whole spectrum in the same fashion. The shot noise is associated with the random arrival of photons on a detector and, in accordance with Poisson statistics, the standard deviation of the shot noise is directly proportional to the square root of the detected light intensity. The detector noise originates mostly from thermally-generated photoelectrons and the detector electronic circuitry, and its standard deviation is independent of the detected light intensity. The actual parameters of the intensity noise can be determined from the characteristics of the light source and the detector modules.

For the purpose of theoretical analysis it will be assumed that multiple detectors are involved in the measurement of the light intensity (e.g. pixels of a CCD array, photodiode array, spectrometer pixels). Light source noise affects all detectors in the same way producing a noise correlated in the spatial domain while the shot noise and the detector noise contribute to each detector independently. The intensity of light${I}_{j}(t)$ measured by the *j*-th detector can be written as

*t*is time, $\overline{I}$is the mean of the intensity,

*c*(

*t*) is the correlated component of noise (independent of the detector index) and ${s}_{j}(t)$is the stochastic (uncorrelated) component of the noise. The correlation between the detectors

*i*and

*j*is characterized by the Pearson correlation coefficient

*s*(

_{j}*t*) and

*c*(

*t*), respectively, are assumed to be the same for all detectors

*i*≠

*j*and ${\sigma}_{I}^{}$denotes the standard deviation of the measured light intensity

*I*(

_{j}*t*). Small values of the correlation coefficient ($\rho \ll 1$) indicate that detectors behave independently and fluctuations in the light source intensity are negligible compared to the detector and shot noise; high values of the correlation coefficient suggest that the light source noise dominates the noise of the measured light intensity.

In SPR sensors two types of averaging can be used during the data processing to reduce the light intensity noise. In *temporal* averaging, a time sequence of light intensities is averaged for each detector involved (e.g. CCD frames are averaged over time). As in the time domain, all noise contributions behave independently, the temporal averaging of *N _{t}* intensities reduces the noise ${\sigma}_{I}^{(\Sigma t)}$ of the averaged intensity as follows

*spatial*averaging, light intensities measured with

*N*detectors (e.g. detector pixels within one sensing channel) are averaged. By averaging signals from detectors, the uncorrelated portion of noise is reduced, while the light intensity fluctuations that are correlated in all averaged intensities are not affected. The standard deviation of the spatially averaged intensity ${\sigma}_{I}^{(\Sigma p)}$ from

_{p}*N*detectors can therefore be expressed as:

_{p}*r*≥ 1 describes the effect of signal correlation on the averaging product. For ideally uncorrelated intensities

_{A}*r*= 1.

_{A}## 3. SPR sensors with amplitude modulation

#### 3.1 Output noise

In SPR sensors with amplitude modulation, the sensor output is light intensity received from the SPR surface. The light intensity is averaged both temporally and over multiple detectors and the standard deviation of the sensor output ${\sigma}_{so}$ corresponds to the noise of the measured intensity as

where*N = N*is the total number of light intensities averaged in the measurement including temporal and spatial averaging.

_{p}N_{t}#### 3.2 Sensitivity

The sensitivity of an SPR sensor with intensity modulation is derived from the dependence of the intensity of light reflected (prism coupling) or diffracted (grating coupling) from the sensor surface on the effective index of the surface plasmon and consequently the refractive index of the dielectric *n*

*n*,

*I*

_{0}denotes the intensity of the incident light wave, and

*R*denotes the reflectivity (prism coupling) or diffraction efficiency (grating coupling). While $\partial {n}_{ef}/\partial n$ is independent of the method of surface plasmon excitation, the contribution $\partial I/\partial {n}_{ef}$describes the effect of the SPR instrument.

In order to calculate $\partial R/\partial {n}_{ef}$, a Lorentzian approximation of the reflectivity derived for the excitation of surface plasmons via a coupling prism can be conveniently used [13]:

*γ*and

_{i}*γ*denote the attenuation coefficients of surface plasmons due to absorption and radiation, respectively [16] and ${\Delta}_{ef}^{}$denotes the phase match parameter defined in Eq. (2) for the prism coupler. Yeatman et al. [17] showed that the maximum slope of the reflectivity as a function of refractive index occurs when

_{r}Under these conditions, the sensitivity *S*
^{(}
^{I}^{)} is calculated by differentiating the Eq. (12) and it can be express as

*γ*). Figure 2 shows the sensitivity of amplitude-based SPR sensors calculated for grating coupling using the rigorous model of diffractive coupling and prism coupling using Eq. (14) (assuming

_{r}= γ_{i}*γ*). This comparison indicates that the theoretical relationship for the sensitivity of prism-based SPR sensors can be used to model the sensitivity of SPR sensors based on grating coupling, at least for high coupling strengths.

_{r}= γ_{i}#### 3.3 Resolution

The refractive index resolution of an SPR sensor with amplitude modulation ${\sigma}_{RI}^{(I)}$can be expressed by combining Eqs. (10) and (14) which yields

## 4. SPR sensors with angular and wavelength modulation

#### 4.1 Output noise

In sensors based on spectroscopy of surface plasmons, multiple intensities corresponding to different wavelengths or angles of incidence are measured. The resulting wavelength or angular spectra are analyzed using an appropriate data processing algorithm and the position of the SPR resonance (resonant wavelength or angle of incidence) is used as a sensor output [2]. Numerous methods for calculating the sensor output have been used in SPR sensors including the polynomial fitting method [18], the centroid method [19], and optimal linear data analysis [20]. Comparative study [21] has shown that the intensity noise in angular or wavelength spectra transforms to noise in the sensor output in a similar fashion for the most common algorithms. Therefore, in this work, the centroid method is employed as a model data processing algorithm to simulate the propagation of the intensity noise into the sensor output.

The centroid method uses a simple formula which determines the geometrical centre of the portion of the SPR dip under a certain threshold. Based on the assumption that the portion of the SPR spectrum involved in the centroid calculation exhibits the Lorentzian profile Eq. (12) and that the optimum threshold level is half of the SPR dip depth [2], it is possible to calculate the noise in the sensor output analytically. It should also be noted that the correlated intensity noise influences the whole spectrum in the same fashion and therefore does not contribute to the noise of the sensor output [2]. Therefore, if we express the uncorrelated portion of the noise from Eq. (7) and use it to calculate the standard deviation of the sensor output, we obtain:

*d*is the relative depth of the SPR dip (difference of intensities between the dip minimum and threshold normalized to the intensity

_{R}*I*

_{0}of the corresponding portion of the incident light spectrum), ${w}^{(\theta ,\lambda )}$ is the width of the dip in the angular or wavelength spectrum at the threshold, and

*N = N*is the total number of intensities involved in the centroid calculation including the temporal averaging. The parameter

_{t}N_{p}*r*(0 ≤

_{s}*r*≤ 1) describes the effect of the noise correlation; for completely uncorrelated intensities

_{s}*r*= 1. The coefficient

_{s}*K*depends on the properties of the uncorrelated noise. It can be shown that

*K*=

*K*

_{1}= 0.50 for homogenous noise (independent of intensity),

*K*=

*K*

_{2}= 0.43 for the shot noise (noise proportional to the square root of the intensity), and

*K*=

*K*

_{3}= 0.38 for the noise proportional to the intensity. If the intensity noise is a superimposition of the three types of noise with weights

*g*

_{1},

*g*

_{2}, and

*g*

_{3}(

*g*

_{1}+

*g*

_{2}+

*g*

_{3}= 1), the coefficient

*K*can be calculated as $K={({g}_{1}^{}{K}_{1}^{2}+{g}_{2}^{}{K}_{2}^{2}+{g}_{3}^{}{K}_{3}^{2})}^{1/2}$.

#### 4.2 Sensitivity

The sensitivity of the resonant angle *θ _{r}* and the resonant wavelength

*λ*to the refractive index can be calculated by straightforward manipulation of the coupling condition:

_{r}*S*and

^{θ}*S*are the angular and wavelength sensitivity, respectively. The second derivative in both the equations describes the sensitivity of the effective index of surface plasmon to refractive index and the first derivatives represent instrumental factors. By differentiation of the coupling condition in Eq. (2) for the prism coupler and Eq. (3) for the grating coupler, the instrumental factors can be expressed as follows:

^{λ}*n*is the refractive index of the dielectric in which the resonant angle of diffraction is measured.

_{d}As follows from Eq. (16), the noise of the sensor output depends on the parameters of the SPR dip. These parameters can be conveniently determined for SPR sensors based on prism coupling using the Lorentzian approximation and Eq. (12). Assuming the threshold is half of the full dip depth (as in the previous discussion), the parameter *d _{R}* can be expressed as

*w*and

^{θ}*w*can be derived:

^{λ}Although the theoretical model for the width of the SPR dip has been derived for a prism coupler-based system, it can be extended to SPR systems based on grating coupling. Rigorous analysis of an SPR sensor employing 0th order diffraction on two different grating couplers was performed using PCGrate. Harmonic diffraction gratings of two different periods (400 nm and 800 nm) were used in these simulations. The angular and wavelength spectra were calculated. The depth of modulation of each diffraction grating was optimized (at each wavelength) to yield the maximum coupling between the incident light and surface plasmons (in prism coupling, this corresponds to the situation when *γ _{r} = γ_{i}*). Theoretical angular and spectral widths were calculated from Eqs. (25) and (26) (with grating coupler instrumental factors given by Eqs. (22) and (23)), assuming

*γ*2

_{r}+ γ_{i}=*γ*(which holds true in the case of high coupling strength);

_{i}*γ*was calculated from Eq. (1). Figure 3(a), (b) shows the SPR widths calculated using approximate analytical relations and rigorous analysis of the grating coupler. This comparison suggests that there is a strong correlation between the analytical expressions for the parameters of SPR dips and the results of the rigorous simulations and that the analytical expressions can be used to estimate the width of the SPR dip with grating coupler, at least for structures exhibiting a strong coupling between the light and surface plasmons.

_{i}#### 4.3. Resolution

By combining the expressions for sensitivity (Eqs. (18) and (19)), the output noise (Eq. (16)), and the depth and width of the SPR dip (Eq. (24)-(26)), the refractive index resolution of SPR sensors based on angular and wavelength modulation can be expressed as follows:

## 5. Unified model of SPR resolution

As follows form Eqs. (15) and (27), the expressions for the resolution of amplitude and spectroscopic SPR sensors exhibit a very similar structure. Therefore, it is possible to merge these expressions into a single formula providing a unified model of resolution for SPR sensors. Then the refractive index resolution can be written as

*K*is the noise distribution factor (for amplitude SPR sensors:

*K*= 0.38, for spectroscopic SPR sensors:

*K*=

*K*

_{1}= 0.50 for homogenous noise,

*K*=

*K*

_{2}= 0.43 for the shot noise, and

*K*=

*K*

_{3}= 0.38 for the noise proportional to the intensity; for superimposition of the three types of noise with weights

*g*

_{1}+

*g*

_{2}+

*g*

_{3}= 1, the coefficient $K={({g}_{1}^{}{K}_{1}^{2}+{g}_{2}^{}{K}_{2}^{2}+{g}_{3}^{}{K}_{3}^{2})}^{1/2}$),

*r*is the noise correlation factor (for amplitude SPR sensors:

*r = r*, for spectroscopic SPR sensors:

_{A}*r = r*),

_{s}*I*

_{0}is a portion of the intensity of the incident light generating surface plasmon which corresponds to one detector (e.g. light intensity of a wavelength component corresponding to one spectrometer pixel) and

*σ*

_{I}_{(}

_{max}_{)}is the standard deviation of noise of the highest intensity involved in the data analysis (the measured intensity for amplitude sensors and the intensity at the threshold of SPR dip for spectroscopic sensors). If the derivative in Eq. (28) is calculated using Eq. (1), Eq. (28) can be manipulated to yield:

*A*) depends only on the properties of noise in the SPR sensor and its reduction by averaging, the second term (

*B*) describes the strength of coupling between the light wave and the surface plasmon, and the third term (C) is associated with the material parameters of the structure.

As follows from the analysis of the noise factor (*A*), a potential difference between the resolution of amplitude and spectroscopic SPR sensors may originate from different behavior of the noise correlation factor (*r*). If the intensity measurements are completely *uncorrelated* (*r _{A}* =

*r*= 1), both amplitude and spectroscopic systems behave in the same fashion. However, if the intensity measurements are partially correlated (

_{s}*r*< 1 <

_{s}*r*) in spectroscopic systems a lower relative contribution of the correlated noise leads to an improved resolution. In contrast, in amplitude SPR sensors, the output noise and resolution increase with the correlated noise, for example, the noise generated by the light source to which spectroscopic systems are immune. This is the main reason why spectroscopic SPR sensors typically outperform SPR sensors based on amplitude modulation (e.g. SPR imaging). The contribution of the correlated light source noise in SPR imaging can be substantially reduced by referencing as demonstrated by Piliarik et al. [22,23].

_{A}The coupling factor (*B*) is a function of the ratio *γ _{i}*/

*γ*which characterizes the coupling strength between surface plasmons and the incident light wave and determines the shape of the SPR dip. In order to obtain the best sensor resolution${\sigma}_{RI}^{(all)}$, the coupling factor (

_{r}*B*) should be minimized by varying the

*γ*/

_{i}*γ*ratio. In SPR sensors with a prism coupler, the

_{r}*γ*/

_{i}*γ*ratio can be varied by changing the thickness of the metal film. In SPR sensors with a grating coupler, the coupling strength is controlled by varying the profile of the grating. It can be shown that the coupling factor reaches the minimum when

_{r}*γ*/

_{i}*γ*= 2. In SPR sensors,

_{r}*γ*/

_{i}*γ*ratios are typically between 1 and 2 [16,17].

_{r}The material factor (*C*) depends on the material constants of the metal and dielectric. The optical constants of the metal film play a significant role. The material factor can be optimized through the selection of material or technology for fabrication of the metal film. For example, the optical constants of gold, which is most often utilized in SPR sensing, depend dramatically on the parameters of the deposition process, purity of the materials and operating wavelegth. While the real part of the permittivity of gold ${\epsilon}_{m}^{\prime}$ (typically between −10 and −30 depending on wavelength) changes only slightly depending on the technology of fabrication, the small imaginary part has a significant impact on the sensor resolution. The minimum of ${\epsilon}_{m}^{\u2033}\doteq 1.1$ can be achieved with high-purity gold at a wavelength of around 700 nm (Platypus Technologies, LLC, USA); if ${\epsilon}_{m}^{\u2033}$ increases (e.g. due to contamination of the gold), the possible resolution increases proportionally. As the noise increases with a cube of the refractive index of the dielectric, SPR sensing in gases yields a more than two times lower resolution than sensing in an aqueous environment.

## 6. SPR sensors and their ultimate performance

Using Eq. (29), the ultimate resolution of an SPR sensor constructed using currently available materials and components and appropriate data processing was determined. The following parameters were used in the analysis: shot noise of 0.6% of light intensity for the light intensity corresponding to the threshold of the SPR dip (corresponds to a scientific grade CCD detector), uncorrelated intensity noise *r _{A}* =

*r*= 1,

_{S}*N*= 100 frames averaged for each time record (corresponds to a measurement period of approximately 1 second),

_{t}*N*= 400 pixels included in the data processing, coupling strength corresponding to

_{p}*γ*0.6

_{r}=*γ*and conventional surface plasmon at a gold/aqueous medium interface.

_{i}Figure 4 presents the refractive index resolution of several different SPR sensors. The sensors cover a wide variety of optical designs, modulation methods and operating wavelengths. They specifically include results obtained in (a) Stemmler et al. [24], (d) Nenninger et al. [2], and (h) Bardin et al. [25] using wavelength modulation and a prism coupling; results obtained in (b) Thirstrup et al. [26], (e) Chinowsky et al. [27], and (f) Biacore 3000 (GE Healthcare, USA) using angular modulation and a prism coupling, results obtained in (c) Piliarik et al. [9] using amplitude modulation and a prism coupling, results obtained in (g) Wu et al. [11] using heterodyne phase detection and prism coupling, and results obtained in (i) Piliarik et al. [28] using wavelength modulation and grating coupling.

Results summarized in Fig. 4 show that the predicted ultimate resolution of SPR sensors realistically describes current limits of the state of the art in SPR technology. These limits are predominantly determined by properties of the surface plasmon phenomenon and the present state of development of optical components. Future improvement of SPR sensors limits depends mostly on the development of detectors of light intensity with a higher signal to noise ratio. The detector noise is limited by the statistical noise corresponding to the number of detected photons. Therefore an accumulation of larger photon flux would be desired. This can be achieved either by increasing the capacity (e.g. size) of the detector or by increasing the frame-rate and the number of temporally averaged intensities *N _{t}*. The performance of SPR sensors can also be improved by optimizing properties of the surface plasmons. One such approach is based on the use of hybrid surface plasmons propagating along a thin symmetric metal waveguide (particularly of a long-range surface plasmon) which yields narrower resonances and higher sensitivity. Recently, enhancement of the SPR resolution by a factor of 5 through the utilization of long range surface plasmons has been demonstrated [29,30].

## 7. Summary

We have developed a theoretical model of the resolution of SPR sensors which makes it possible to predict the ultimate performance of all major configurations of SPR sensors. It is clear from the model that the resolution of the SPR sensor depends upon the noise of the used optoelectronic components as opposed to the design of the SPR optical platform. Therefore, comparable resolution can be achieved regardless of the selection of the SPR coupling principle (coupling prism or a diffraction grating) and modulation (amplitude, angular or wavelength). The theoretical model was used to explore the ultimate limits of SPR technology and these limits were compared to the performance of current SPR sensors. Paths towards overcoming the existing limitations of SPR sensors are associated with the development of advanced detectors with a higher signal to noise ratio and faster data transmission rates as well as with the exploitation of other types of surface plasmons.

## Acknowledgments

This research was supported by the Academy of Sciences of the Czech Republic under the contract KAN200670701, by the Ministry of Health of the Czech Republic (IGA MHCR) under contract NR/9322-3, and by the Czech National Science Foundation under contract 202/09/0193.

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