## Abstract

We have experimentally and theoretically compared intensity and phase measurements of surface plasmon resonance, in order to check the widely claimed superiority of the phase method. Both experiment and theory show a slightly higher sensitivity for intensity detection. Simulations confirm that this result is generally true for a wide range of resonance conditions. The basic reason is that phase measurements must be performed by measuring light intensities in some way, and therefore both modes of operation are limited in similar ways by photon statistics. Sensitivity can only be improved by using larger light intensities.

©2006 Optical Society of America

## 1. Introduction

Surface plasmons, electromagnetic waves of optical and infra-red frequencies propagating on the surface of a metal at its interface with a dielectric, have been investigated since the 1950’s [1–3]. During the past two decades, several methods using them have been developed for the detection of small changes in refractive index, and these have found application in particular in bio-detection [4,5]. Currently, instruments using surface plasmon resonance (SPR) absorption line shifts are available commercially, and have sensitivity to refractive index changes of about 10^{-6} in the evanescent layer of the dielectric adjoining the metal. During the last ten years, a number of articles [6–23] have been published which claim that by measuring the phase shift due to the resonance, and not the absorption, an additional two to three orders of magnitude in sensitivity can be achieved, after correction for photon flux and bandwidth. Surprisingly, no practical demonstration of this extra sensitivity has been published. Only one experimental demonstration of a sensitivity significantly better than 10^{-6} has to our knowledge appeared in the literature [20]. This result was characterized by a narrow measurement bandwidth and very large photon flux, both of which do have major influence on the sensitivity, and we show below that the result is consistent with our conclusions.

The claim that phase measurement can be more sensitive than amplitude measurement was suggested by Kabashin and Nikitin [6] (and further elaborated by others [7–23]) and is based on calculations of the complex reflection coefficient as a function of angle when light with p-polarization is reflected from a metal layer (Ag or Au) deposited on the hypotenuse of a prism above the critical angle (the Kretschman configuration: Fig. 1).

As the thickness of the metal layer is increased, the SPR absorption line becomes narrower and stronger, and the layer thickness can be optimized for practical detection of the resonance angle, which depends on the complex refractive index of the metal and those of both the glass used and the dielectric medium in contact with the metal. However, when the phase shift of the reflected light is calculated, an increasingly abrupt change at the resonance angle is found as the thickness is increased, the change becoming discontinuous at a particular metal thickness. Based on this phenomenon, linked with the capabilities of interferometric systems to measure phase shifts accurately, claims for enhanced sensitivity by more than two orders of magnitude have been made.

This work is the result of an extended effort to achieve the highest sensitivity of refractive index measurement using SPR. Several methods of phase detection were considered, but none of them were found to achieve a sensitivity exceeding that from intensity measurements. Most effort was put into phase-stepping interferometry [24], which is widely used in commercial interferometry. The problem is basically that phase or phase change can not be directly observed, but have to be inferred from the only quantity which is measurable optically: intensity. All methods of phase measurement require the investigated wave to be added coherently to a reference wave of some type, whose known influence on the measured intensity is then eliminated mathematically. This is done by means of an interferometric system. The need for intensity measurement inherent in the interferometric process eventually led us to the conclusion that the sensitivities of phase (interferometric) and intensity (absorbance) modes for determination of the resonance conditions are essentially identical, both being limited in the same way by photon statistics. We indeed found a marginal advantage to the intensity mode, in that the output (at a given angle) results from a single measurement, whereas phase determination requires at least three, usually four. In addition, the abrupt phase step predicted always occurs exactly at the resonance angle where the reflected intensity is minimum, and therefore less photons are available for the determination. In order to confirm the conclusions, an SPR optical imaging setup was constructed in which both modes could be applied to the same samples under the same conditions, by a single switch in the apparatus, and thus they could be directly compared. It was also possible with this system to confirm quantitatively the predicted dependence of the sensitivity on the photon flux. To negate the possibility that results were affected by instrumental limitations, the experiments were simulated numerically using ideal components, and excellent agreement was found.

## 2. Calculated reflective properties of a thin metal film

The calculation of the complex reflection coefficient of a metal layer sandwiched between glass and water is straightforward, using Maxwell’s equations or Fresnel coefficients, given the (complex) refractive indices of the three media and the thickness of the metal [25]. For incidence angles through the critical angle, the phase and intensity of the reflected p-polarized wave for gold films are shown in Fig. 2. The corresponding values for the s-polarized waves show no structure around the SPR angle. One observes in this case that there is a phase anomaly between the thicknesses 50 and 55nm, around which thickness the intensity variation is steepest. Thus a thickness of about 53nm would be judged optimal for either phase or amplitude investigation. However, it is also shown in the figure that the intensity calculated when a coherent reference wave with constant phase is added to the reflected wave shows no sign of the phase anomaly. This is an interferogram from which the phase should be deduced experimentally, and because it shows no signs of anomaly at any gold layer thickness, it leads us to doubt whether the phase anomaly is a real observable. This is emphasized in Fig. 3, where reference waves with different phases are used.

The accuracy to which any of the curves in Fig. 2 can be plotted depends on the number of photons employed. If *N* photons are incident on the investigated region of the surface during an experimental period, the intensity has an inherent proportional error of *N*
^{½} due to Poisson statistics. When the phase is determined, for example by means of a phase-stepping algorithm, the input involves at least three intensities with this inherent error. Based on this argument, typical errors for the determination of the resonance angle can be deduced. Fig. 3 shows the interferograms for reference waves with the four phases 0, *π*/2, *π*, 3*π*/2, which are used in the phase-step algorithm near the critical thickness where the Au layer thickness should yield infinite phase slope. Other methods of phase determination, such as Fourier fringe analysis [26] and heterodyne detection [18] have been reported and were considered. In this work, in order to study the photon shot-noise limitation to the measurement we used a single beam interferometer (sec. 3) which avoids mechanical noise. The phase-stepping method is most appropriate to this technique, especially when an image and not a single point is being investigated.

## 3. Experimental method

To investigate the limitations experimentally, a single beam interferometer was constructed (Fig. 4). A collimated polarized beam of light from a LED at 692nm is reflected at a variable angle of incidence from a Kretschman prism with an Au layer in contact in different parts with distilled water and a glucose solution of known concentration (an accepted standard for calibration of water-based SPR systems). Following the prism, the light passes through a liquid crystal variable wave plate aligned to the plane of incidence and an analyzing polarizer. The surface of the gold is imaged onto a 12-bit CCD.

If the polarizers are set to an arbitrary direction, the CCD measures the interference between the s- and p- waves. The intensity at the detector for this configuration is therefore:

$$\frac{1}{2}\sqrt{{R}_{(n,\theta )}^{S}{R}_{(n,\theta )}^{P}}\mathrm{sin}\left(2\alpha \right)\mathrm{sin}\left(2\beta \right)\mathrm{cos}\left({\phi}_{(n,\theta )}^{P}-{\phi}_{(n,\theta )}^{S}+\Delta \psi \right)]$$

where *n* is the measured refractive index, *θ* is the illumination angle, *I*
_{0} is the input intensity, ${R}_{(n,\theta )}^{S}$ and ${R}_{(n,\theta )}^{P}$ are the reflected intensities for the s- and p- polarizations respectively (as a function of illumination angle), *α* and *β* are the angles of the polarizers relative to the s-polarization, ${\phi}_{(n,\theta )}^{S}$ and ${\phi}_{(n,\theta )}^{P}$ are the phase changes for each polarization (as a function of illumination angle) that are introduced by the interaction with the SPR layer, and Δ*Ψ* is the phase shift that is introduced by the phase shifter.

If the polarizers are set to 90° (pure p- polarization) the measured intensity has a simple reflectance form:

On the other hand, if the polarizers are at 45° (symmetrical mixed polarization), interference between the s- and p- polarized reflected waves is measured, the phase difference between them being determined by the setting of the wave-plate. In this case the measured intensity at the detector has the form of a classical two beam interferometer:

The phase of the reflected wave is then determined by the usual phase-step algorithm [24],

where *φ*
_{(n,θ)} is the phase difference between the two waves, and *I*
_{(n,θ,j)} (*j*=1,2,3,4) are the four measured interferograms, one for each phase step (Δ*ψ*
_{j}
=(*j*-*1*)*π*/2) at a certain angle *θ*.

Clearly, the amplitude and phase reflection coefficients can easily be compared for identical physical conditions simply by rotating the polarizers by 45°. Using this arrangement, after careful calibration of the phase plate (phase difference as a function of applied voltage), the phase of the reflected p-wave could be determined to an accuracy of about 5mrad. The amplitude and interferometric reflectivities measured through a range of 3° around the resonance angle were fitted to the values calculated using the Fresnel coefficients, with the glucose solution refractive index *n* as a single fitting parameter from which the accuracy Δ*n* was determined from the fitting procedure. The responsivity and linearity of the system was measured for both modes and showed absolute accuracy with differences less than 1% from results obtained by an Abbe refractometer (model PA200 by Misco) for the same solutions, over a dynamic range of 0.04 in refractive index.

## 4. Application of photon statistics considerations

Following error propagation theory, the accuracy in the measured refractive index can be calculated, using the local derivatives of the measured intensities with respect to the refractive index (calculated numerically from Eq. 2 and 3) and by using the measured photon flux. In the model used here only shot noise and CCD readout noise were taken into account. For the intensity mode, the accuracy in refractive index Δ*n*
_{(θ)} at a specific illumination angle *θ* is calculated to be:

where *N*
_{0} is the total number of photons that impinging on the measured area (per read cycle), *R*
_{N}
is the readout noise of the CCD camera (electrons per pixel per read) and *b* is the number of averaged pixels. When the readout noise is neglected Δ*n*
_{(θ)} reduces to:

The dependence of the measured refractive index sensitivity on the photon flux is evident from the square root term in the denominator.

For the case of interferometric measurement the expression for Δ*n*
_{(θ)} is quite similar:

where *M*
_{(n,θ)} is the output response of the interferometer (as defined in Eq. 3). Neglecting the readout noise in this case yields a similar expression as Eq. (6) with *R* replaced by *M*.

If one wishes to calculate the refractive index *n* from the phase change, then an interferometric measurement has to be carried out. For a specific case of a four step phase measurement, the phase is calculated as shown in Eq. (4), from four interferometric measurements carried out at the same angle, but with incremental phase shifts of *π*/2. The accuracy in the measured refractive index becomes more complicated here if Δ*n*
_{(θ)} is expressed as a function of the errors in the interferometric measurements:

where *A* and *ΔI*
_{(n,θ,j)} are defined by:

For the case where the slope of the phase diverges as shown in Fig. 3, the reflectance ${R}_{(n,\theta )}^{P}$ of the SPR layer drops to zero, i.e., |∂*φ*/∂*n*|→∞, ${R}_{(n,\theta )}^{P}$→0, and the accuracy in the measured refractive index (after neglecting the readout noise) reduces to:

Eq. (10) shows the expected dependence on the photon flux. The phase derivative would have caused Δ*n*
_{(θ)} to vanish when the slope of the phase diverges if ${R}_{(n,\theta )}^{P}$ were not present in the denominator. Since ${R}_{(n,\theta )}^{P}$ approaches zero as the slope of the phase increases (see Fig. 3), it will suppress the influence of the slope of the phase and therefore Δ*n*
_{(θ)} will not necessarily approach zero.

In order to calculate the accuracy of the measured refractive index when the measurement is carried out by recording the reflected intensity (or interferogram) at a series of illumination angles around the resonance angle, we need to take into account the fact that for each angle the accuracy is different. This difference is due to variations in the absorption and slopes of the curves. To account for these variations, the fitting of an entire reflectance curve to the measured data can be considered as a weighted average of a series of separate measurements, each one of them having its own weight that is evaluated by its accuracy. The weighted averaged *n*
_{wav}
is given by [27]:

where the weights are given by ${w}_{k}=\Delta {n}_{\left({\theta}_{k}\right)}^{-2}$ (*θ*
_{k}
being the *k*-th illumination angle in the series) and the overall accuracy of the measurement is given by summing all the weights in the series:

The result that the steep phase gradient gives rise to no sensitivity advantage can be demonstrated in general for any two-beam interferometric arrangement, which includes all the phase detection schemes cited. We now show that the intensity minimum at resonance exactly cancels the effect of the phase gradient. We take Eq. (3) and differentiate the interferogram *M*
_{(n)} by *n* to calculate the responsivity. *R*
^{S}
and *φ*
^{S}
can be assumed to be constants (they may be the s- polarized wave or any independent reference wave). We then get:

When this is evaluate from the calculated SPR response curves, the two terms in the round bracket are found to be finite because (*R*
^{P}
)
^{1/2}
multiplies the phase derivative etc., and then the sum of the two resulting finite terms has no anomaly.

## 5. Results and discussion

All the measurements were made as a function of the number *N*
_{0} of incident photons used. In Fig. 5 the resulting Δ*n* is plotted, for both modes of measurement, as function of ${N}_{0}^{-\xbd}$, to which Δ*n* should be proportional if the CCD readout and dark noise is neglected; correction for the measured readout noise shows excellent agreement with this relationship. The results show limiting values of Δ*n* which are a little worse in the interferometric mode than in the intensity mode. Dark current noise was negligible in this system, because of the short exposure time (18 ms). It should be pointed out that *N*
_{0} in these experiments had values up to several thousands. In principle, much larger values are possible, but only a small area of the sample could be used in order to avoid averaging over inhomogeneities, and the use of a 12-bit digital CCD camera then restricted *N*
_{0} to numbers less than the well-depth (25k electrons) multiplied by the maximum number of pixels used. In all cases, the scaling of the results with *N*
_{0} was shown to be consistent with theory.

It is always possible that the reduced sensitivity of the interferometric mode was a result of non-ideal behavior of the phase plate, despite its calibration. We should point out that the phase plate operated in the same way during all the measurements, including those in the intensity mode, where its value was irrelevant, so that any errors introduced would be common to both modes. The mode of operation of the phase plate was therefore optimized, experimental and simulated results being compared in Fig. 6, and dependence on the Au film thickness used being shown in Fig. 7. From the latter figure, the overall superiority of the intensity mode is clear, although there could be some advantage to the interferometric mode when using films thinner than the optimum.

As mentioned in the introduction, the experiments reported by Wu et al. [20] used a phase-sensitive detector and an interferometric mode to achieve sensitivity enhanced by some two orders of magnitude over other experiments. In terms of the present discussion, we note that the light source was a 10mw laser source and two single element s- and p-polarized detectors were used. Moreover, the integration time of the phase-sensitive system was greater than 1sec. Assuming an optical throughput of 1 percent, including optical absorption in the SPR film, we thus estimate *N*
_{0} to be of order 10^{13}, and from Eq. (6) for the intensity mode, assuming an optimized film thickness, this leads us to a theoretical sensitivity limit of Δ*n*=6·10^{-9}, which is better than that reported in the experiments. The results of this experiment therefore appear to be quite consistent with the arguments we presented, the high sensitivity being explicable as resulting from the large photon flux and narrow bandwidth, without the use of interferometry.

## 6. Summary and conclusions

We have show, both by experiments and simulations, that similar sensitivities of an SPR detection system to small changes in refractive index can be achieved by the use of intensity detection and phase detection. There are two causes for this: the first one is the fact that as the phase change becomes steeper the absorbance becomes higher and therefore the SNR decreases. The second one is the misconception that the phase is measurable, whereas the true measurable quantity, the interferogram, shows no exceptional sensitivity. After optimization of the instrumental parameters and the thickness of Au film, the only way to improve the sensitivity is to increase the number of photons per measurement incident on the sample and received by the detector.

## Acknowledgments

This research was supported in part by the Minerva Foundation for Complex Systems.

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