In this chapter we demonstrate how to obtain the ultimate lateral resolution in surface plasmon microscopy (SPM) (diffraction limited by the objective). Surface plasmon decay lengths are determined theoretically and experimentally, for wavelengths ranging from 531 to 676 nm, and are in good agreement. Using these values we can determine for each particular situation which wavelength should be used to obtain an optimal lateral resolution, i.e., where the plasmon decay length does not limit the resolution anymore. However, there is a trade-off between thickness resolution and lateral resolution in SPM. Because of the non-optimal thickness resolution, we use several techniques to enhance the image acquisition and processing. Without these techniques the use of short wavelengths results in images where the contrast is very low. In an example given, structures in a 2.5 nm SiO2 layer on a gold layer could be imaged with a lateral resolution of 2 \(\mu m\), and local reflectance curves were measured to determine the layer thickness with a resolution better than 0.1 nm (based on the angular shift of the curve). The SPM image is compared with an atomic force microscopy image of the same object.
It has previously been shown that with SPM a thickness resolution down to a few tenths of a nanometer can be obtained.1,2 However, the lateral resolution is limited by the decay length Lx of surface plasmons. This quantity is defined as the distance along the surface for which the attenuation of the plasmon field intensity is 1/e (see Chapter 1, Eq. 1.38). Here we will be concerned with the problem of how to obtain optimum lateral resolution while retaining sufficient thickness resolution. Although it was mentioned in the literature that it should be possible to improve the lateral resolution by choosing an appropriate wavelength and metal layer,2 to our knowledge no detailed study of this problem has been conducted previously.
Fig. 4.1 Reflectance difference for gold covered with 2.5 nm SiO2 and bare gold for the resonance angle of the bare gold (circles), and surface plasmon decay lengths for gold (squares) as a function of wavelength. Dielectric data were obtained from Ref. 6.
The problem can be illustrated by regarding a situation where a 2.5 nm SiO2 layer partly covers a 45 nm gold layer, and where the angle of incidence is chosen such that surface plasmons in the bare gold are resonantly excited. When the contrast, defined as the difference in reflectance of both regions, is calculated as a function of wavelength, a relation as depicted in Fig. 4.1 is found. In the same figure the wavelength dependent Lx values are given, using Eq. 1.38 and the second-order approximation for Im(\(\kappa_x\)) by Pockrand.3 It is seen that a smaller decay length is accompanied by a smaller reflectance difference. From the figure it can be determined how high the lateral resolution will be if the accompanying reflectance difference is sufficient for SPM imaging.
The resonance halfwidth for silver is appreciably smaller than that for gold. This leads to a higher dependence of the reflectance on the thickness, and thus a higher contrast and thickness resolution. However, the wavelength-dependent decay lengths are significantly larger than those in gold, due to the relatively low surface plasmon damping. Therefore, in view of the larger losses in gold, this material seems preferable for high-resolution SPM, and in the following we will restrict ourselves to this metal.
Now, if we can realize and observe decay lengths near the diffraction limit of the objectives used, a wavelength can be chosen for a SPM experiment in which the plasmon decay length will be just below the resolution of the microscope objective. The problem of how to obtain a higher lateral resolution is then converted to how to make sure that the lower contrast is sufficient for imaging.
In the next sections, a plasmon-carrying 45 nm gold layer is characterized by SPR measurements. From the data the expected plasmon propagation is calculated for various wavelengths. To verify these values, decay lengths that are much smaller than in experiments conducted previously4 are observed microscopically. The optimal wavelength for imaging with a high lateral resolution is chosen, depending on the microscope objective and the imaged cover layer. Several techniques to improve the image acquisition and processing are applied to compensate for the loss in contrast and to correct for other image degrading effects. SPM images of a 2.5 nm SiO2 layer are compared with an atomic force microscopy (AFM) image of this layer.
The SPM setup used is essentially the Kretschmann (ATR) configuration, and is schematically drawn in Fig. 4.2. A rotation table was used for easy and accurate angular adjustment (angular increments: 0.01°). Electronic switching between p and s polarization was done with a Pockels cell (PC 100/4; Electro Optic Developments Ltd., Basildon, England). The Pockels cell was used for normalization and correction of background effects. A HeNe laser and an Ar/Kr laser, which allows illumination with a number of lines in the visible spectrum (Innova 70; Coherent Inc., Palo Alto, CA), serve as light sources. Wavelengths ranging from 531 nm to 676 nm were used. The intensity profile of the reflected light beam was recorded with a microscope consisting of an objective and a charge-coupled device (CCD) camera (VCM 3250; Philips). The output of the camera is linear in light intensity. Depending on the light intensity and magnification needed, a 40× [extra long working distance; numerical aperture (NA) 0.49] or a 7× (NA 0.19) objective was used.
Fig. 4.2 Schematic representation of the setup used for the microscopical measurements. The Pockels cell, rotation table, and camera are interfaced to a computer; manual control is also possible. P: Pockels cell; S: spatial filter; R: rotation table; M: microscope objective.
Pockels cell, rotation table, and CCD videocamera can be controlled both manually and by computer. Video images (512×512 picture elements) are processed and stored in a computer (486 PC) using a frame grabber card (VisionPlus AT OFG; Imaging Technology, Inc., Woburn, MA). The computer-controlled features make the setup particularly well suited for the experimental methods described in the next sections.
4.2.2 Improvement of the image quality
There are a number of factors that determine the eventual quality of an SPM image apart from the diffraction limit of the microscope objective:
(i) Plasmon decay length. We have seen that this influence can be avoided almost completely by using an appropriate wavelength, which, however, results in a lower contrast.
(iii) Quantization during image acquisition. To improve the image acquisition we implemented an option to add a number of images using a dynamic range of 216. The large dynamic range is important, because with a low contrast one of the first steps is to increase the contrast digitally. With the normal dynamic range of 28 of the separate images, quantization levels become visible upon high amplification. The dynamic range of the eventually presented image is 28.
(iii) Noise caused by low light levels. Integrating n images also means averaging, and the signal to noise ratio consequently improves with a factor \(\sqrt n\). Although a maximum number of 256 images can be integrated, choosing \(n=50\) proved to be more than sufficient to remove any visible noise due to low light levels in all cases mentioned in this chapter. Increasing the laser power is another possibility but this option is of limited use in view of the resulting destructive effects on the sample. After integration, the noise due to the CCD camera can be neglected.
Fig. 4.3 Reflectance curves for a bare gold layer, measured with different wavelengths: (a) 676.4 nm; (b) 647.1 nm; (c) 632.8 nm; (d) 568.2 nm; (e) 514.5 nm; (f) 488.0 nm.
(iv) Lateral inhomogeneities in the incoming laser beam intensity profile. At the moment, this is usually the limiting factor for the lateral resolution of our SPM images. We have two ways of suppressing this ‘noise’. The first is using a spatial filter to create a more homogeneous incoming laser beam. This, however, does not work for inhomogeneities created by unwanted reflections at prism and objective surfaces. Therefore, another technique is used involving digital image processing. Using the Pockels cell, two images are acquired using p and s polarized light, respectively. The image with p polarization contains the SPM image together with the contrast generated by spatial fluctuations, while the image obtained with s polarized light only contains the same unwanted fluctuations. Because the camera output is linear in light intensity, the two images can be divided (after a suitable linear transformation) to suppress the errors in the image.
An important point to be made is that when contrast is increased digitally, it should be a linear operation. This means that all resulting gray values should be within the range of the displayed image. If this condition is not satisfied, a lot of pixels will be set to black or white, and while a higher resolution is suggested, in fact information is lost.
The substrates used are small microscope cover slips that were attached to the prism (BK7 glass) using a matching oil. A 45 nm gold layer was evaporated on top of this substrate (1 nm/s at 10-6 mbar). After the evaporation, a SiO2 layer with a thickness of a few nanometers was sputtered on top of the gold (0.1 nm/s at 10-2 mbar Ar). Photolithography and wet etching techniques were used to make a high-resolution test pattern with sharp edges (transition region < 1 \(\mu m\)) in this SiO2 layer.
The plasmon decay length is very dependent on the imaginary part of the dielectric function of the gold, which depends on the manufacturing process and surface roughness as well. Therefore, we have experimentally determined the dielectric function instead of using literature values. To characterize the metal layer on our substrate, the SPR reflectance as a function of the angle of incidence in an ATR experiment was measured and fitted for a number of wavelengths in the visible range. The measured reflectance curves for a bare gold layer of 44 nm are displayed in Fig. 4.3.
When the critical angles were calculated for the different wavelengths and compared to the experimental values, a systematic shift of 0.08° in the angular measurements was found. The critical angle is dependent on the prism refractive index only and can easily be included in an angular scan. Therefore, it seems to be better to use this angle as a reference angle instead of the angle found using perpendicular reflection, which by definition is the zero external angle. Four parameters were fitted: the layer thickness d, the real and imaginary part of the dielectric constant \(\varepsilon_i\), and a normalization constant relating the measured reflectance to the absolute reflection coefficient. Fresnel theory was used for the fitting, which resulted in excellent fits. Dispersions of the different media and the reflection losses at the prism sides were included in the calculations. As an example one of these fits is given in Fig. 4.4.
Fig. 4.4 (a) Fitted plasmon reflectance curve for a 44 nm gold layer at \(\lambda=676.4\) nm. The points result from the reflectance scan, the solid line represents the fitted values using Fresnel theory. (b) The relative difference between measurement and fit.
The thickness of the gold layer that resulted from the fitting was 43.6 nm. The accuracy for the fitted thickness, as determined by the spread in independent measurements, was rather high: for the three longest wavelengths, the thickness values found reproduced within 0.1 nm. For the shortest wavelengths the spread in the determined thickness values was about 1 nm, which can be explained by the less pronounced dip in the reflectance curves. From the residue it is seen that there exists a small systematic error. This is hardly of influence on the quoted parameters. Measurements with a surface profiler confirmed a thickness in between 43 and 44 nm. The resulting values for the complex dielectric function of the gold were found to be between dielectric functions found by others,5-7 although these literature values show some spread. The results of the fitting and the calculated decay lengths are presented in Table 4.I.
Table 4.I. Fit results for a gold layer of 44 nm and derived theoretical decay lengths.
aLiterature values (Ref. 6).
For microscopical plasmon decay length observations a substrate with an SiO2 test pattern on top of the gold layer was made in the way described briefly in the sample preparation section (Sec. 4.2.3). The thickness of the pattern was measured using the automated features of the setup, which make it possible to measure complete reflectance curves of several microscopically small areas simultaneously within one laser beam.
First, using the computer, two small areas (e.g., 25×25 \(\mu\)m) are defined in the microscopical image: one in an uncovered gold area and one in an SiO2-covered area. The intensity in these areas is measured for both p and s polarization: by dividing (p/s) we can correct for polarization-independent, non-SPR intensity variations and normalize the reflectance values. After the incident angle is incremented with 0.01 deg. the whole procedure is repeated. In this way, full reflectance curves for both areas are determined by measuring the average intensity in each area as a function of the angle of incidence.
Fig. 4.5 (a) Microscopic reflectance scans at \(\lambda\)=632.8 nm for the bare gold layer and for the SiO2-covered region, measured simultaneously within the same SPM image. (b) The root-mean-square value of the difference of both reflectance curves, as a function of their shift towards one another.
This fully automated process results in reflectance curves as displayed in Fig. 4.5(a). The SiO2 layer thickness can be calculated with the determined angular shift of the measured reflectance minima. Figure 4.5(b) shows the overall reflectance difference defined as the root-mean-square (rms) value of the difference of the reflectance curves when the two curves are shifted towards one another by \(\Delta\phi\):
\[\text{rms difference}(\Delta\phi)=\sqrt{\frac{1}{n}\sum\limits_{i=1}^{n}{\left[R_{Au}(\phi_i)-R_{SiO_2}(\phi_i+\Delta\phi)\right]^2} }\tag{4.1}\]
The measured shift for the SiO2 layer is \(\Delta\phi\) = 0.45 deg. (\(\lambda\) = 632.8 nm). Using Fresnel theory and assuming a refractive index of 1.46, a thickness of 2.5 nm was found. The accuracy in the measured shift is about 0.01 deg., corresponding to a thickness resolution better than 0.1 nm.
In Fig. 4.6 a 125 \(m\)m wide part of the test pattern as described in Sec. 4.2.3 is imaged using five different wavelengths. Because of the high magnification the effect of the plasmon decay length is visible for the longer wavelengths, in the horizontal direction. The surface plasmons propagate from left to right and the dark area in the middle is the bare gold at resonance. The contrasts in the images in Fig. 4.6 were made equal by analog and digital amplification, to use the full gray scale.
Fig. 4.6 A part of the SiO2 test pattern on gold, imaged with five wavelengths: 676.4, 647.1, 632.8, 568.2, and 530.9 nm, respectively (from left to right). Correction using p and s polarized light was applied. The plasmon wave vector is pointing to the right, and the dark area is the bare gold at resonance. The contrast of the images is chosen such that the gray value range is used in an optimal way. A detailed analysis of the images is given in Fig. 4.7.
Fig. 4.7 The intensity profiles of the images in Fig. 4.6 with indicated wavelengths. The solid lines represent the intensity profiles calculated with the dielectric constants and layer thicknesses obtained from the SPR experiments and the model presented in Chapter 3.
In Fig. 4.7 the intensity profile in the horizontal direction is given, averaged over a number of image lines crossing the 18 \(\mu\)m wide upper part of the bare gold in Fig. 4.6. The solid lines in the same figure correspond with the results of the model presented in Chapter 3 using the parameters derived from the ATR fits and the dispersion relation. Generally, the correspondence between model and experiments appears to be satisfactory. The distance from the index step at which the reflectance is changed to 1/e of its eventual change is also found to be proportional to the decay lengths calculated and given in Table 4.I. This points to the fact that we can predict the lateral resolution that can be obtained using a certain wavelength.
Because of the finite numerical aperture (NA 0.19) of the objective used, decay lengths shorter than that for the 530.9 nm wavelength cannot be observed in the microscopic image. It may be concluded that for this objective the 530.9 nm wavelength is the best choice for a high lateral resolution. Shorter wavelengths would diminish the contrast even more and would not contribute to the lateral resolution of the setup. The diffraction limited lateral resolution for this objective and wavelength is 2 \(\mu\)m, based on the Rayleigh criterion.
Fig. 4.8 Example of the suppression of the effect of lateral inhomogeneities in the light beam by the division of images acquired with p and s polarization. From left to right we see the p and s polarized image, and the ratio of the two.
In Fig. 4.8 the effect of using the ratio of the p polarized to the s polarized image to correct for beam inhomogeneities, is shown for an SPM image of a part of the SiO2 test pattern. The same part is imaged with both AFM in the deflection mode8 and SPM with \(\lambda\)=530.9 nm, to check the contours of the object and to test the lateral resolution of the SPM images (Fig. 4.9). The NA of the objective used was 0.49 (diffraction limited resolution: 1 \(\mu\)m). In the AFM image the edges of the test structure are very sharp and we can recognize the rounded corners that we already saw in the SPM images (this is also true for the part that was used for determination of the decay lengths). Although the effect of the lateral intensity inhomogeneities is still visible in the SPM image, we estimate the lateral resolution of the image to be 2 \(\mu\)m. The difference in reflectance for the gold and the SiO2 can be calculated and is about 0.01.
Fig. 4.9 AFM and SPM images of the same, small part of the test pattern. Comparing these images we estimate the resolution of the SPM image to be 2 mm (NA 0.49). Unresolved cracks in the SPM image are visible in the AFM image.
We have shown that it is possible to characterize SPR substrates accurately, and that this allows for making an optimal choice for the wavelength, regarding lateral resolution in SPM. Surface plasmon decay lengths were calculated from the determined dielectric function, and observed with the surface plasmon microscope. The intensity profile could be described with a phenomenological model. The trade-off relation between decay length and reflectance difference makes it necessary to apply several improved image acquisition techniques when short wavelengths are used. In this way diffraction-limited SPM images can be obtained while retaining the very high thickness resolution. A transparent SiO2 test pattern of 2.5 nm thickness was characterized using AFM and a new method to obtain reflectance curves of microscopically defined areas automatically. Comparing the AFM and SPM images led to the conclusion that the lateral resolution of the SPM image is 2 \(\mu\)m. This resolution is limited by lateral inhomogeneities in the incoming laser beam that could not be corrected completely. Some of the inhomogeneities are created in the objective, and we expect that by choosing a specialized objective (e.g., with a better coating) the resolution can be improved a little more (to the diffraction limit).
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