Chapter 3 Surface plasmon propagation near an index step

Generally, the application of Fresnel theory to the description of reflectometric measurements is straightforward and gives accurate results when the conditions mentioned in the first chapter are satisfied. In this chapter we will study propagation effects of surface plasmons on the SPM image of an area around the edge of a cover layer. Far from this index step Fresnel theory still applies, but at distances smaller than the propagation length of SPs (typically micrometers) it cannot be used anymore. Because Fresnel theory assumes that the layer system consists of infinite layers it can only provide the reflectance far from the step as a boundary condition. We will present a phenomenological model (based on that presented in Refs. 1,2) that describes the effect of plasmon propagation on the observed reflectance near an index step, and study this effect as a function of the wavelength. Silver layers were first characterized by fitting experimental SPR angular reflectance curves. Then the model was used to predict the resulting reflectance profile when an index step on the metal layer was imaged with a surface plasmon microscope. Measurements and calculations were performed for wavelengths ranging from 560 to 660 nm for a 50 nm silver layer with 30 nm thick SiO₂ pattern on top. Theoretical and experimental results will be compared.

In the following, $\kappa$ is the x-component of the wave vector, and may be complex ($\kappa ={\kappa }^{\prime }+i{\kappa }^{″}$). When a resonant plasmon with wave vector ${\kappa }_0$ propagates along x from an uncovered area into a covered area, a decaying plasmon (containing an imaginary component) with wave vector ${\kappa }_1$ results (see Fig. 3.1).³ Here ${\kappa }_1$ is the wave vector for resonant excitation of the covered area. Furthermore, the external light source with wave vector ${\kappa }_0$ excites a non-resonant surface plasmon with wave vector ${\kappa }_0$, if ${\kappa }_0$ is within the resonance width of the SPR curve in the covered area. The change of the surface plasmon electric field amplitude is

Fig. 3.1 Schematic representation of a surface plasmon with wave vector ${\kappa }_0$ propagating along a metal-air interface into a cover layer, where an exponentially decaying ${\kappa }_1$ plasmon results. The non-resonantly excited surface plasmon with wave vector ${\kappa }_0$ is indicated separately.

governed by the imaginary part of its wave vector. The constants $E_1$ and $E_2$ are the resonant and the non-resonant plasmon amplitude, respectively. Leaving out the time dependence, the plasmon electric field at the interface is^1,2

$$\begin{array}{}\text{(3.1)}& E_{pl}(x)=(E_1-E_2)e^{i({\kappa }_1^{\prime }+i{\kappa }_1^{″})x}+E_2{e}^{i{\kappa }_0^{\prime }x},\end{array}$$

for $x\ge 0$, with x=0 at the index step. We improved this model to predict the reflectance near the indexstep. Far from the index step, the assumptions of plane waves and infinite layers are valid, and if the boundary conditions are to be fulfilled the reflectance at a large distance from the step should coincide with Fresnel values. By imposing these conditions all parameters of the model are fixed.

In Refs. 1 and 2 it is assumed that the non-resonant contribution to E_pl is constant and starting at x=0. What is missing in this model is the fact that the plasmon electric field interferes with the incoming field from the light source. Because of this omission the model can give incorrect results, with a higher reflectance at resonance than off resonance. Therefore, for the total electric field reaching the detector an extra term should be introduced to account for the low reflectance resulting from resonant plasmon excitation, due to destructive interference:⁴

$$\begin{array}{}\text{(3.2)}& E_{tot}(x)=(E_1-E_2)e^{i({\kappa }_1^{\prime }+i{\kappa }_1^{″})x}+E_2{e}^{i{\kappa }_0^{\prime }x}-E_3{e}^{i{\kappa }_0^{\prime }x},\end{array}$$

where $E_3$ is the amplitude of the incoming field. With $A=E_1-E_2$ and $B=E_2-E_3$, the resulting intensity can be written as

$$\begin{array}{}\text{(3.3)}& I_{tot}(x)={|E_{tot}(x)|}^2=B^2+A^2{e}^{-2{\kappa }_1^{″}x}+2AB{e}^{-{\kappa }_1^{″}x}\cos ({\kappa }_1^{\prime }-{\kappa }_0^{\prime })x.\end{array}$$

For $x=-\mathrm{\infty }$ (or $x\le 0$) and $x=+\mathrm{\infty }$, the intensity equals $(E_1-E_3{)}^2$ and $(E_2-E_3{)}^2$, respectively. We can solve these constants putting E₃ equal to 1 $(E_3>E_1>E_2>0)$, because then $I_{tot}(-\mathrm{\infty })$ and $I_{tot}(+\mathrm{\infty })$ equal the macroscopic reflectance for uncovered and covered areas that can be calculated with Fresnel theory. Note that there are no free parameters to fit measurements.

When a non-resonant plasmon exits a covered area and becomes resonant the total electric field and intensity may be written in a similar way:

$$\begin{array}{}\text{(3.4)}& E_{tot}(x)=E_2{e}^{i{\kappa }_0^{\prime }x}+(E_1-E_2)(1-e^{i{\kappa }_0^{″}x})e^{i{\kappa }_0^{\prime }x}-E_3{e}^{i{\kappa }_0^{\prime }x},\end{array}$$

and

$$\begin{array}{}\text{(3.5)}& I_{tot}(x)={[B+A(1-e^{-{\kappa }_0^{″}x})]}^2,\end{array}$$

where the same definitions are used for the constants. The plasmon wave vectors were calculated using the second-order theoretical approximation of the dispersion relation of surface plasmons given by Pockrand⁵ and depend on the properties of the metal layer and the cover layer.

Two cavity-dumped dye lasers (3.8 MHz, Coherent 700) synchronously pumped by a mode-locked Nd:YLF laser (Antares 76-YLF, Coherent) served as a light source covering the wavelength range from 560 to 660 nm. The polarization could be modulated between p and s polarization electronically, using a Pockels cell (PC 100/4; Electro Optic Developments, Ltd., Basildon, England). A rotation table (MicroControle UR80PP; angular increments: 1 mdeg.) was used for computer controlled reflectance scans that were measured with a photodiode.

The experimental setup for the surface-plasmon microscope will be described in detail in the next chapter. It images the attenuated total reflectance in the Kretschmann configuration, and uses p and s polarized light to correct for inhomogeneities in the expanded incoming laser beam. A 7× (NA 0.19) objective was used to image the light on a video camera (VCM 3250; Philips) which has an output that is linear in light intensity.

Fig. 3.2 Ratio of the reflectance for p and s polarized light as a function of the internal incident angle and wavelength (560-660 nm) for a silver layer. Table 3.I shows the results from fitting these measurements.

Microscope glass slides were used as substrates on which a 50 nm silver layer was evaporated (1 nm/s at 10^-6 mbar). After the evaporation, a photoresist layer was spun on the substrate and a pattern was made photolithographically. A 30 nm thick SiO₂ layer was sputtered (0.1 nm/s at 10^-2 mbar Ar) over the bare and covered areas of the silver layer. After the removal of the photoresist using an ultrasonic acetone bath, a SiO₂ pattern on the silver resulted. Substrates were stored in a nitrogen atmosphere. They were attached to the prism (BK7 glass, 45 deg.) using a matching oil.

Fig. 3.3 Ratio of the reflectance for p and s polarized light as a function of the internal incident angle for wavelength $\lambda =600$ nm, together with Fresnel fit ($\epsilon =-13.95+i0.554$ and $d=50.2nm$). The residue was calculated as the relative difference of theoretical and experimental values.

All measurements were carried out for wavelengths ranging from 560 to 660 nm in steps of 10 nm. First the SPR reflectance of the bare silver layer was measured as a function of the angle of incidence. The low frequency part of the laser noise was effectively suppressed by measuring the reflectance switching the polarization between p and s, while integrating both signals. The ratio of the p and s reflectance as a function of the angle of incidence and the wavelength is displayed in Fig. 3.2. These normalized values can be directly compared to Fresnel calculations, taking into account the different transmission of the prism entrance and exit surfaces for different polarizations. The ratio of the transmissions of the prism for p and s polarized light is given by

$$\begin{array}{}\text{(3.6)}& \frac{T_p}{T_s}={⟮\frac{n_1\cos {\theta }_1+n_2\cos {\theta }_2}{n_1\cos {\theta }_2+n_2\cos {\theta }_1}⟯}^4,\end{array}$$

where ${\theta }_1$ and ${\theta }_2$ are the external and internal angles of the propagating light beam with respect to the normal of the entrance and exit surface, and ${\text{n}}_1$ and ${\text{n}}_2$ are the refractive index of the ambient medium and the prism, respectively. Fresnel formulae were used to fit the measured reflectance curves by varying the real and imaginary part of the dielectric constant $\epsilon$ and the thickness d of the silver layer. It has been demonstrated before,^5-7 that the reflectance curves depend on these three parameters in such a way that they can be determined separately. A computer program that was written (based on differential correction and the least squares criterion) needed less than 10 iterations to converge to the values presented in Table 3.I.

Table 3.I. Dielectric constant and thickness of the silver layer as a function of wavelength.

Of course d should not depend on the wavelength and indeed, for the independent measurements at different wavelengths the same value for the layer thickness was found with a high accuracy (50.3±0.1 nm). The excellent agreement of Fresnel calculation and experimental data can also be seen in Fig. 3.3, where one of the measurements is shown together with the calculated values as an example

The thickness of the SiO₂ pattern on top of the silver layer was checked with a surface profiler (Dektak) and was indeed about 30 nm (See Fig. 3.4). The steepness of the edge was obscured by the tip convolution, but with the lift-off method (as described in Section 3.2.2) the edge is expected to be narrower than one micron, which is sufficient for this study.

Fig. 3.4 Profile of the SiO₂ index step, measured with a surface profiler.

With the silver layer parameters (${\epsilon }_r,{\epsilon }_i,d$) known, the reflectance profile near the index step can be calculated using the model that was presented in Section 3.1. The parameters in Eq. 3.3 can be derived from Fresnel calculations using $\epsilon$ and d. The constants A and B follow from the boundary condition that the reflectance far from the index step [$I_{tot}(-\mathrm{\infty })$ and $I_{tot}(+\mathrm{\infty })$] should equal that given by Fresnel theory. The real part of the wave vectors ${\kappa }_1$ and ${\kappa }_0$ (corresponding to resonant SP excitation of covered and uncovered areas respectively) can be determined from the position of the minimum in calculated angular reflectance curves. Finally, the imaginary part of the wave vectors ${\kappa }_1$ and ${\kappa }_0$ can be determined from the width of the SPR dip or from Pockrand’s second order approximation.⁵

Fig. 3.5 Composition of all SPM images of the index step, with the resonant plasmon entering (left, at 50$\mu$m) and leaving the cover layer (right, at 200$\mu$m).

Fig. 3.6 Periodicity p of the profiles of the images in Fig. 3.4 as a function of wavelength together with the calculated values for d=27.9 nm and n equal to that of BK7 glass.

Experimentally, the reflectance profiles were determined from surface plasmon microscopy images of the index step. The index step was imaged with both p and s polarized light to correct for inhomogeneities in the incoming beam by dividing the images obtained with the different polarizations. In Fig. 3.5 the images resulting from resonant plasmonsentering the SiO₂ covered area and leaving it again are shown for 11 different wavelengths, with the SPs propagating from left to right. Squares of known size (50 $\mu$>m) were imaged as well (results not shown), to determine the lateral scale of the images. The reflectance profiles in the direction perpendicular to the index step were determined from the digitized SPM images averaging along the index step.

From these profiles, the periodicity was determined as a function of the wavelength. Equation 3.3 shows that this periodicity p is given by

$$\begin{array}{}\text{(3.7)}& p=\frac{2\pi }{{\kappa }_1^{\prime }-{\kappa }_0^{\prime }}.\end{array}$$

In Fig. 3.6 these values are displayed together with theoretical values for a 27.9 nm SiO₂ layer with the refractive index assumed to be equal to that of BK7 glass. As this value for d is more accurate than the one determined with the surface profiler (and lies within the experimental error of that value) it was used for the calculation of the reflectance profiles.

These are given in Fig. 3.7, together with the experimental values. The measured and calculated profiles were normalized by putting the macroscopic values for the reflectance far from the index step to either zero (at resonance) or one (off resonance). The correspondence between experiment and theory is quite good, qualitatively as well as quantitatively. Some fringes to the left of the index step seem to be caused by a reflection that is not accounted for by the model. Their dependence on the wavelength is different from that of the fringes on the right of the index step.

We have developed a model that describes the reflectance profile near an index step. A silver layer was characterized by fitting measured SPR reflectance curves, and the thickness of the cover layer was measured with a surface profiler. Experimental and theoretical values for the reflectance profile as a function of the wavelength were compared and found to agree quite well. In the next chapter this model will be used to determine which wavelength and metal layer are most suitable for surface plasmon microscopy with a high lateral and thickness resolution.

Fig. 3.7 Calculated and measured reflectance profiles as determined from the images in Fig. 3.4. The reflectance values at resonance and off resonance were normalized to zero and one, respectively.

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