1. Introduction

It is becoming increasingly important to measure the properties of surface plasmons (SPs) and surface waves in localized regions which are smaller than their propagation length. For instance, when an analyte binds to the surface of a substrate there is a change in the wave number of the SPs, which gives a measure of the quantity of material attached to the sample. The ability to measure this change over a small spatial region allows many analytes to be examined simultaneously with minimum utilization of reagents. Similarly, evaluation of the propagation properties of SPs allows accurate characterization of the thickness and properties of thin films. One of the standard methods to characterize the properties SPs layers is the so-called prism based Kretschmann configuration [3], although very sensitive this method does not allow one to achieve good spatial resolution since the SPs are free to propagate across the surface with no restriction on the point of excitation or detection, in addition, the prism geometry is not compatible with high spatial resolution. For this reason there has been increasing interest in using microscope geometries to couple and detect the SPs [4]. The principle of excitation through a high numerical aperture microscope objective is depicted in Fig. 1(a), where we note that the range of illumination angles afforded by the objective allows for excitation and detection of the SPs. We have shown previously that the lateral resolution can be greatly improved with an interferometer configuration [5–7] and more recently that a modified confocal microscope can give comparable results [1, 2]. In these papers we used the so-called V(z) response which is proportional to the field amplitude detected as a function of defocus, z. In all previous publications, however, we have used only the amplitude of the confocal or interference signal, although this has considerable advantage compared to the output from a non-confocal system, extraction of the SP phase confers even greater benefits. This measurement can be effected using the spatial light modulator which allows us to configure the confocal instrument as a phase stepping interferometer. Argoul et al. [8] have recently demonstrated some of the advantages of using the phase of the interference signal to enhance contrast. The present confocal system allows even more direct measurement of the SP contribution in that different parts of the beam can fulfill the role of reference and signal beams in the ‘embedded’ interferometer so that the phase of the SPs may be directly accessed as opposed to the phase of the resultant signal (that is the V(z) signal), which is the measure obtained when a reference beam is applied to the whole returning field. There is therefore a major distinction between an interference system where the reference beam interferes with all the returning signal and the ‘embedded’ interferometer where the plasmonic and nonplasmonic contributions are interfered with each other. We will elaborate on this point in more detail below. We demonstrate that compared to an amplitude only measurement phase measurement gives much better performance in terms of immunity to noise and lateral resolution.

 

Fig. 1 (a) Simplified schematic showing operation of confocal microscope with SP excitation; (b) Schematic of optical system showing relationship between different planes in the system. The blue waveform indicates phase modulation in the back focal plane.

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The hardware configuration used in this paper is the same as that used in [2], however, we use the versatility afforded by the configuration to perform some new phase stepping measurements. We show that the system forms a phase stepping interferometer between the paths P1 and P2 of Fig. 1(a). Data extraction from the system allows the amplitude and phase corresponding to the SP excitation to be extracted directly. This allows both the real and imaginary parts of the SP k-vector to be extracted which can, of course, be equated to the phase velocity and attenuation of the SPs.

The idea is presented in Fig. 1(a) which shows a defocused sample where the SPs are excited at a specific incident angle. Reciprocity dictates that the excitation and reradiation efficiency of surface waves is similar, so that energy will reradiate continuously along the propagation path of the SPs; indeed it is this uncertainty in the path of the detected radiation that limits the spatial resolution of most plasmonic imaging systems. Examining Fig. 1(a) we see that the presence of the confocal pinhole ensures that only light appearing to come from the focus returns to the pinhole. The presence of the confocal pinhole thus defines the allowable propagation paths (P2 of Fig. 1(a)) of the detected SPs thus ensuring that the resolution is determined by the footprint of the optical beam rather than their propagation length [6]. When the sample is moved away from the focus towards the objective there are two major contributions to the signal detected at the confocal pinhole. The first is light close to normal incidence that will return to the pinhole and the second is the SP path discussed above. As the sample defocus, z, is incremented by a distance, Δz, assuming a well beved pupil function, the phase between the normal incident beam and the SP beam changes:

Two simulated V(z) curves are presented in Fig. 2 which show the difference in periods between a bare gold layer compared to a thin indium tin oxide (ITO) (c.10 nm) coated gold. The V(z) curve is the output signal as a function of defocus, z. This is a complex quantity involving the integral of the detected field contributions [1] although in many cases the intensity signal, |V(z)|² is measured. The |V(z)|² curve can be obtained by mechanical scanning of the sample in the axial direction [1] or by imposing the same phase shifts electronically in the back focal plane using a spatial light modulator (SLM) [2]. The average period of each |V(z)| curve can be used to determine the plasmonic angle using Eq. (3). Although this equation is not exact and relies on a well apodized pupil function [9], it, nevertheless, gives an excellent measure of the change in plasmonic angle when an analyte is deposited. Here, we will show how modifications to the |V(z)| curve using phase stepping and pupil function engineering can be used to determine the properties of SPs in a convenient and effective manner. We modulate the phase of the reference while keeping the phase of the SPs fixed. This provides a quicker, more robust and more accurate method to extract the SP properties. A detailed simulation analysis is presented to compare the two methods in section 3.

 

Fig. 2 Simulated V(z) curves of uncoated (red) and sample coated with 10nm of indium tin oxide (blue) showing different periods of oscillations

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2. Experimental setup

The experimental system has been described in [2]. Essentially the system is a mechanical scanned confocal microscope with a phase SLM (BNS 512512 phase SLM) inserted in the back focal plane to control the interference signal detected at the pinhole. The pinhole was formed by projecting the 1000 times magnified focal spot onto the CCD array. The pinhole size can thus be selected by selecting the appropriate pixels from the array. The light source was a 632.8nm He-Ne laser (10mW) and 1.45 NA oil immersion objective (Zeiss) was used to excite SPs. Further details are given in [2].

3. Beam profile modulation in the back focal plane

The SLM shown in Fig. 1(b) is used to control the profile of the beam in the back focal plane. This is necessary to ensure that the reference beam gives a good approximation to the phase variation depicted by Eq. (3), if the beam is not apodized at the edges of the objective aperture there is a considerable amount of signal induced by the edge which effects the phase of the reference beam and, in turn, the accuracy with which θ_p can be recovered. The red curve in Fig. 3 shows a typical pupil function imposed on the back focal plane, where the angles around normal incidence and θ_p are allowed to pass through the lens. In principle the angles in the mid-range of the pupil can be allowed to pass but since they only contribute background it is better to block them. Note that since we only employ a phase only SLM blocking the light that passes is accomplished by setting adjacent pixels in antiphase [2]. Experimental images of back focal plane (BFP) are shown in Fig. 3. Figure 3(b) is modulated by using the phase SLM in antiphase for the mid frequencies and in Fig. 3(c) the mid frequencies are allowed to pass. The phase cancellation also allows the sharp edge of the clear aperture to be smoothed.

 

Fig. 3 (a) Pupil function distribution (red curve) and calculated reflection coefficient for p-incident polarization on an uncoated sample (blue curve); the vertical cyan lines represent the range of angles over which the phase stepping of the reference beam was imposed. (b) is the back focal plane (BFP) image by setting the adjacent pixels in antiphase on a phase-only SLM and (c) is the same with no modulation of the mid-frequencies.

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If we now assume that confocal system forms a two beam interferometer between the reference beam and the beam involving excitation and reradiation of SPs, we can use the phase shifting ability of the SLM to form a phase stepping interferometer between sample and reference. The experimental measurements and simulations will allow us to evaluate the effectiveness of this assumption and also to see the values of defocus where the assumption is valid.

Taking four phase shifts as the vector diagrams shown in Fig. 4, we form the standard interferometric expressions thus:

 

Fig. 4 Effect of phase stepping on relative phase of reference and plasmon beams, green line is the reference signal R, cyan line is the SP phasor, red line is the resultant signal V.

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4. Phase stepping to obtain the plasmon angle, θ_p

We now combine the phase stepping approach with V(z) and obtain the relative phase between reference and signal beams as a function of defocus, z. The sample was scanned axially through each defocus position, z, and at each defocus position 4 phase steps were performed. The four |V(z)| curves on bare gold obtained by shifting the phase of the reference in increments of 90 degrees are shown in Fig. 5. From lower to upper figures we have: zero phase shift (red), 90 degree phase shift (blue), 180 degree phase shift (black) and 270 degree phase shift (cyan).

 

Fig. 5 V(z) curves variations of bare gold obtained by shifting the phase of the reference in increments of 90 degrees. From lower to upper figures we have: zero phase shift (red), 90 degree phase shift (blue), 180 degree phase shift (black) and 270 degree phase shift (cyan). Successive curves are shifted by 0.1 units for clarity.

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Then the four |V(z)| curves are used to obtain the relative phase between reference and signal at each defocus, φ(z). The relation between the φ(z) and the plasmonic angle is expressed as:

Figure 6 shows φ(z) for a bare gold layer and a layer with an additional deposited layer of indium tin oxide (ITO). We can see that for positive defocus, that is the sample and objective are separated by more than the focal length (see Fig. 1) there is little systematic difference between the response for the two layers. For negative defocus close to the focal position the change φ(z) curve shows an irregular form, which is clearly not linear. This arises principally because the reference beam cannot be regarded as being a simple linear function as assumed in Eq. (3). Aberration in the lens and possibly the finite number of pixels in the SLM means that the defocus value before the slope is linear is somewhat increased. At larger defocus the unwrapped phase shows a linear form which relates to the value of θ_p. The values of θ_p obtained for the bare layer and coated layers are 43.48 deg. and 46.39 deg. respectively, which correspond to a thickness of 13.4 nm of ITO assuming a refractive index of 1.858. We also measured the thickness value of the ITO with a commercial ellispometer (alpha-SE J. A. Woollam (Inc)) and obtained a value of 11nm ± 2.3nm. The strength of our method for measuring the value of θ_p arises from several factors (i) the region of defocus where accurate measurements can be obtained is readily observed from the linearity of φ(z) where the periods of V(z) are stable, (ii) the method uses all the data points in the measurement range thus optimizing the signal to noise ratio and (iii) while clearly the signal to noise ratio improves as the defocus range increases a reliable measurement can be obtained over a very small region of defocus corresponding to less than Δz_p of Eq. (3). Such a measurement range is not practical if one measures the amplitude only the V(z) curve without phase stepping. In the next section we compare by simulation the immunity to measurement noise of different processing methods. It demonstrates clearly that the present method is very stable and robust compared to methods used without phase stepping. We reiterate that φ(z) refers to the phase of the SPs rather than the V(z). In the next section we consider the noise performance of different processing strategies and also show how even the relatively noisy measurements in Fig. 5 lead to well defined measurements of film thickness.

 

Fig. 6 Experimetnal φ(z) obtained from four V(z) curves using the phase-stepping technique. The red curve refers to the unwrapped φ(z) of bare gold and the blue line is the fitted data, while the green curve refers to the ITO coated sample case and the black line is the linear fit.

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We introduced the phase stepping measurement above and assert that this method is more accurate and robust in calculating the plasmonic angle θ_p compared to other methods compared to direct measurement of the ripple period or Fourier transform measurement. In order to validate this we carried out a set of Monte Carlo simulations to assess the performance of these three measurement methods. The definitions of the three methods are as follows: 1) Direct measurement of the ripple period; the ripple period Δz was calculated by averaging the first few ripples as shown in Fig. 7(a) and then the plasmonic angle θ_p can be calculated using Eq. (3). The minimum positions of the ripples are determined by 3rd order polynomial curve fitted to 25 data points (over a range of 200 nm) around the minimum as shown in Fig. 7(b). 2) For the Fourier Transform measurement, the average ripple period Δz was determined from Fourier transform of the windowed pattern of ripples. Details of the phase stepping measurement have been described above.

 

Fig. 7 (a) shows the method of measuring the ripple period measurement (b) shows 3rd order polynomial fit (black) to locate a minimum position for ripple period measurement; (c) shows regions on φ(z) and corresponding ripple positions.

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In order to compare the three methods fairly we used four times as many measurements for the direct measurement and the Fourier method as the phase stepping measurement requires four different measurements.

Now let us consider the case where there is noise. We will consider a shot noise source. The noise levels were calculated on the basis that the maximum signal in the V(z) curve contained obtained without phase shifting. The SNR is presented in dB and each value corresponds to a fixed number of photons. For N incident photons the optical SNR is $\sqrt{N}$, so the electrical signal to noise is N, so that 60dB corresponds to 10⁶ measured photons. Values below the peak value are scaled appropriately, so have proportionately worse SNR values. The V(z) curves were sampled at intervals of 8 nm, The optical signal to noise ratio is defined as:

where μ is the mean value and σ² is the variance, the ratio thus gives the signal to noise ratio. Monte Carlo simulations were carried out over 10⁶ cases. Standard derivation (S.D. in degrees) between the mean plasmonic angle (noiseless case) and the plasmonic angle recovered from the three methods (noisy cases) were determined in order to compare performance of each method as shown in Figs. 8(a) and 8(b). It may, of course, be argued that in many situations the noise is not shot noise limited, nevertheless, the relative performance between the different methods is retained provided each method is subject to a similar noise models. Our general conclusions are therefore valid for other independent noise processes.

 

Fig. 8 (a) Mean standard deviation in degrees versus SNR level in dB for single ripple measurements and half ripple measurement for phase stepping. Solid black is for 1 ripple phase stepping measurement, dashed black for 1 ripple period measurement, dotted black for 1 ripple FFT measurement and solid blue for half ripple phase stepping measurement. (b) Mean standard deviation in degrees versus SNR level in dB for first three ripple measurements. Solid black curve is for 3 ripples phase stepping measurement, dashed black curve is for 3 ripples period measurement, dotted black for 3 ripples FFT measurement.

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Figure 8(a) and 8(b) shows that the performance of phase stepping measurement (solid black) performs best as it has the lowest S.D. compared to the others; the FFT method is the next best presumably because all the data is used and the poorest method involves direct measurement of the ripple. Interestingly, the phase stepping method can be used to make measurements over very small defocus ranges, even though its performance improves more rapidly than the other methods as the measurement range is increased. This is expected since the uncertainty of the gradient of a line rapidly decreases as the extent of the measurement increases.

We now apply the Monte Carlo method to evaluate the signal to noise of our experimental measurement. The experimental results of Fig. 6 shows straight line fits to ϕ(z) obtained from the experimental curves of Fig. 5. We estimated the noise in the ϕ(z) curves for different noise levels in V(z) curves. We then selected those curves that gave the same variance of the deviation from the straight line as obtained in the experimental measurements presented in Fig. 6. We then used the Monte Carlo simulation with similar noise levels and sampling intervals to estimate the expected variations in ϕ(z) and the corresponding errors in the measurement of film thickness. Probability distributions of the variation in the measured thickness values were obtained by running the Monte Carlo simulation 50,000 times. These probability distributions are shown in Fig. 9 for different ranges of measurement defocus. The first thing to notice is that the phase stepping approach recovers film thicknesses with well-defined values even when the underlying measurements are relatively noisy. As expected when we extend the range over which the measurement is made the uncertainty decreases. This is presented in Table 1 which shows the standard deviation of the measurement error for the defocus ranges presented in Fig. 9. There is a considerable reduction in measurement uncertainty with increasing defocus range; this improvement arises partly from the better signal to noise expected when more data points are included and also from the fact that a larger measurement range gives superior performance when measuring the gradient of a line. Doubling the measurement range reduces the variance by a factor of greater than 6 which is considerably better than the value of 2 expected from considerations of signal to noise alone.

 

Fig. 9 Probability density function (pdf) of variation from expected value for ½ ripple period phase stepping measurement (blue), 1 ripple period phase stepping measurement (green), 2 ripple periods phase stepping measurement (red) and 3 ripple periods phase stepping measurement (cyan). These results were simulated with the noise level corresponding to the experimental results shown in Fig. 6. The ½, 1, 2 and 3 ripples are equivalent to 375 nm, 750 nm, 1,500 nm and 2,250 nm in z defocus distance respectively.

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Table 1. Standard deviation (S.D.) of the measurement error for the defocus ranges presented in Fig. 9

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5. Measurement of SP propagation length

The propagation properties of the SPs are related to its complex wave number where the real part is the parameter discussed in the previous section and the imaginary part is related to the attenuation of the wave. There are two components of the attenuation α_leaky and α_loss, the first term represents the strength of the coupling between the SPs and the excitation medium, as mentioned earlier reciprocity dictates that if the SPs are strongly coupled to the excitation medium they will, in turn, attenuate strongly as they propagate. The term α_loss is related to the ohmic losses in the metal and the total attenuation is the sum of these two terms.

The SLM allows us to make at least two distinct measurements of the SP propagation length which we can call indirect and direct. The indirect method is simply an extension of the phase stepping approach as discussed in the previous section, where the value of $\left|S\right|$ can be obtained from the phase stepping approach (red curve in Fig. 10(a)). There are small ripples present on the curves obtained by the indirect method, due, in part, to small amounts of amplitude phase crosstalk on the SLM. The direct method involves performing a V(z) measurement where only those angles close to θ_p are allowed to pass through the system; in other words the reference beam shown in Fig. 3a is blocked. If the phase is not required, there is no need for a reference beam. The detected $\left|S\right|$ is shown in blue curve in Fig. 10(a). Both methods allow a value of $\left|S\right|$ to be obtained as a function of defocus, z. These curves are then fitted to an exponential function Aexp (-Bz) which allows the attenuation to be obtained. Then the SP propagation length l_p is obtained from the recovered value of B using calculated $l_p=2\tan {\theta }_p/B.$

 

Fig. 10 (a) The |S| curves by using the direct method (add 0.1 in plot) and indirect methods; (b) Propagation attenuation comparison between the 29nm and 46nm thickness of gold layer; (c) Propagation attenuation comparison of |S| for different thickness of gold layer, the blue curve shows the direct method results and the red curve for the indirect results. The black curve represents the calculated value of attenuation for each layer thickness. The data from each measurement was fitted to a third order polynomial curve.

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In Fig. 10(a), the green curve shows the measurements obtained by the direct method and the red curve shows the measurements obtained by the indirect method. The calculated attenuation is shown by the fitted curve. Figure 10(b) shows the comparison of the attenuation values between the 29nm and 46nm gold layer. The results demonstrate clearly that the 29nm gold layer provides higher attenuation than the 46nm case. Figure 10(c) shows values of attenuation obtained for different thicknesses of gold (which affects the attenuation strongly) by using the direct and indirect methods. The simulation results are shown on the black curve. We can see, as may be expected, that the direct method shows a smaller variance around the fitted values, nevertheless, both measurements show a similar trend. The standard deviations from the third order fits are smaller for the direct method (1.09 microns) compared to the indirect method (1.63 microns) as shown in Fig. 10(c). This suggests that better measurement precision is obtained with the direct method, provided the fitted curve can be taken as a reasonable reference point. The measurement of attenuation of SP can therefore be obtained using pupil function engineering.

It is clear that the error in the attenuation measurement is far greater compared to the measurement of the real part of the wave number. Indeed similar observations were made in the measurement of velocity and attenuation of surface acoustic waves using the V(z) method in the scanning acoustic microscope, where relative accuracy of around 1 part in 10³ was obtained for the measurement of velocity but ‘a few’ per cent for attenuation [10].

6. Conclusions

This paper has presented a technique to extract the SP properties applying a phase SLM to alter the phase difference between the reference beams and SP signal in the confocal surface plasmon microscope (Fig. 1). We show how the method can be used to measure the phase velocity which corresponds to the SP k-vector and propagation attenuation caused by radiation leakage and ohmic heating. Compared to the previous measurement of the SP k-vector, the new technique provides more robust, more accurate and potentially faster results. The fact that measurements can be made over a very small scan range means that the optimum spatial resolution is superior to processing methods requiring large defocus range. The method also provides a means to evaluate the attenuation, although the relative uncertainty in this measurement is presently far greater than the uncertainty in the measurement of the real part of the SP wave number.