# Chapter 1 Introduction to surface specific methods

In this chapter we will very briefly go through the spectrum of techniques that are available for interface characterization. Then the Fresnel theory is given, describing reflectometrical interface characterization techniques like ellipsometry, Brewster angle reflectometry and surface plasmon resonance.

## 1.1 General introduction to surface characterization methods

There is a wealth of experimental techniques available for the investigation of surfaces or interfaces, adsorbates and thin layers.1,2 They can be classified in structural imaging and spectroscopy methods, although some techniques combine these possibilities. A common feature is their surface sensitivity, which is very important since for any bulk object the number of bulk atoms exceeds the number of surface atoms by many orders of magnitude. However, they widely differ in surface sensitivity, lateral resolution, destructiveness, experimental conditions (e.g. UHV), and the kind of interface they can be used for.

### 1.1.1 Spectroscopy techniques

Most spectroscopy techniques3,4 can be classified by the kind of excitation and the emitted particles. Excitation may take place through photons, electrons, ions, neutral particles, heat, and electric fields.5,6 We will just briefly mention a number of these techniques.

X-ray photoelectron spectroscopy (XPS)7 is based on the photoelectric effect and the dependence of the emitted photoelectrons on their binding energy. It gives information describing the chemical composition of a solid surface under high-vacuum conditions.

Similar information is obtained in Auger electron spectroscopy (AES)8,9 where an electron beam is used for the excitation. The excited electrons have an energy characteristic of a certain core-level transition and are analysed. As in XPS, the surface sensitivity of AES is determined by the escape depth of the electrons and typical probing depths are in the range of 1 to 3 nm.

In secondary ion mass spectroscopy (SIMS)10,11 a surface under vacuum is bombarded with accelerated ions. Fragments of surface molecules are emitted and analyzed by measuring the intensity as a function of the mass-to-charge ratio. The chemical composition of the surface can be determined with a probing depth of about 1 nm.

Electron energy loss spectroscopy (EELS)12 uses an electron beam incident on the sample at a certain angle. When the energy of the reflected electrons is analyzed at an angle slightly off specular reflection peaks of lower energy are measured due to inelastic scattering (the electrons can lose energy to the vibrational modes of molecules). The sensitivity is about 10-3 of a monolayer, which is very high.

Optical vibrational spectroscopy techniques include raman spectroscopy and infrared spectroscopy. No vacuum is needed for these techniques rendering them useful for biological applications, but the surface sensitivity is lower, especially for Raman scattering. Fourier transform infrared spectroscopy (FTIR)13,14 can be used to characterize the surface chemical structure by using an attenuated total reflection (ATR) crystal or by using a metallic substrate. ATR can also improve the Raman signal as can surface enhanced Raman scattering (SERS)4 which uses a roughened metal surface to enhance scattering efficiency of adsorbates. Waveguide Raman scattering is another way to increase surface sensitivity taking advantage of the relatively large part of the light that propagates through a layer on top of the waveguide.

### 1.1.2 Microscopy techniques

In the 1930’s the transmission electron microscope (TEM)15 was developed, surpassing the resolution of light microscopes by using electrons instead of photons, and suitably shaped magnetic fields as lenses. Drawbacks of this technique are that the sample needs to be thin enough to allow the transmission of sufficient electrons, and that selective heavy metal staining is needed to obtain a good contrast. In the scanning electron microscope (SEM)1,15,16 the electrons reflected from the surface rather than those transmitted are producing an image of the surface topography. This technique together with countless staining and replicating techniques allows many more types of samples to be studied.

The field of scanning probe microscopy started with the invention of the scanning tunneling microscope (STM)17 in 1982, where a conductive tip is scanned over a (semi)conductive sample while measuring the tunneling electrons between tip and sample. In this way atomic resolution imaging can be achieved while varying the bias voltage can give information about the electronic structure with the same resolution. Soon the atomic force microscope (AFM)18,19 emerged, where the deflection of a cantilever on which a tip is mounted is caused by the force interaction between tip and sample. The deflection is measured while the tip is scanned over the sample in contact. While this technique can reach a similar resolution as STM it can be applied to many more types of samples, especially in the field of biology.

Scanning near field optical microscopy (SNOM)20 is a scanning probe method that takes the lateral resolution of optical microscopy past the diffraction limit. A small (sub-wavelength sized) aperture is scanned over a transparent sample with an objective on the opposite side of the sample. Optical excitation is achieved via a light beam through the objective whereas the light is detected using the aperture; of course the roles of the objective and the aperture can be interchanged.

The surface sensitivity of normal light microscopy can be enhanced in several ways. When a monomolecular layer is studied, fluorescent labels that have a different solubility for different parts of the monolayer can be added to it. By exciting fluorescence the different parts of the layer can be imaged with this light, even at an air-water interface.21,22 Another way is to make light reflect off the interface under investigation and, using that light, create an image with the microscope. The conditions for the reflection should be chosen such that the properties of the interface are represented in the intensity or polarization state of the reflected light. Reflectometrical methods will be discussed in the next sections after an introduction to the Fresnel theory.

### 1.1.3 Diffraction techniques

For the determination of lattice constants of crystals a number of diffraction techniques are available. X-rays, electrons, atoms, molecules, ions, neutrons, etc. can be used as probes, but because of the needed surface sensitivity only particles that do not penetrate too deep into the substrate can be used. The ideal probes for the interface are molecules, atoms, ions and low energy electrons. Electron diffraction can be used on solid surfaces only, and care must be taken that the electron dose does not disrupt the crystal lattice in the case of monolayers.23,24 X-ray diffraction can be applied even on monolayers at air-water interfaces, but requires a special experimental arrangement, and high intensity synchotron X-ray sources.25,26 Neutron reflectivity has been used for determination of the molecular order in films at the air-water interface as well.27 Thermodynamic measurements such as the lateral pressure as a function of molecular area, and surface potential measurements which determine the average dipole moment of the molecules in the layer are other techniques for the characterization of monolayers at an air-water interface.28

The list of techniques in section 1.1 is far from complete, as there are innumerable surface analysis methods and variants of these methods. It is meant only to provide a background for the next sections. Optical immunosensors will be discussed in the next chapter.

## 1.2 Fresnel theory

Fresnel’s theory29 can be derived from the Maxwell equations and describes the reflection and transmission of a plane monochromatic light wave incident upon a layer system. To apply Fresnel’s formulae the layer system should consist of infinite, perfectly flat layers. In practice, this condition is fulfilled for studies at a lateral scale exceeding the wavelength by an order of magnitude. In that situation, Fresnel theory should describe reflectometrical measurements within the experimental accuracy. Therefore, this theory can be used to determine the parameters of the layer structure under investigation by analytical inversion, or by varying the parameters to fit the model to the measurements. We can also use it to investigate the sensitivity and accuracy of reflectometrical methods. For this thesis our main interest lies in reflectometrical methods that are suitable for measurements on biological monolayers.

### 1.2.1 Reflection at an interface

When a monochromatic plane wave impinges on an infinite interface between media 1 and 2, in general reflection and transmission will occur. These fields (indices respectively $$i$$, $$r$$ and $$t$$) are described by

$\vec{E}_{i}=\vec{E}_{ip}e^{i(\vec{\kappa}_{i}\cdot\vec{r}-\omega t)}\;\;\;\;\; \vec{E}_{r}=\vec{E}_{rp}e^{i(\vec{\kappa}_{r}\cdot\vec{r}-\omega t)}\;\;\;\;\; \vec{E}_{t}=\vec{E}_{tp}e^{i(\vec{\kappa}_{t}\cdot\vec{r}-\omega t)},\tag{1.1}$

where $$p$$ indicates the polarization of the wave. The light is decomposed into a $$p$$ polarized component parallel to the plane of incidence (Fig. 1.1), and an $$s$$ polarized component (‘senkrecht’) perpendicular to this plane. The reflection and transmission formulae for an $$s$$ polarized wave are identical to Eq. 1.1, with every $$p$$ substituted by an $$s$$.

Fig. 1.1 Reflection and transmission of a plane wave incident on an interface between medium 1 and 2.

The length of the wave vectors is defined by

$\kappa_{i} =\kappa_{r}=n_1\frac{\omega }{c}\;\text{ and }\; \kappa_{t}=n_2\frac{\omega }{c}\tag{1.2}$

where $$n_1$$ and $$n_2$$ are the refractive indices of the respective media. The angles the reflected and transmitted waves are making with the surface normal are given by

$\theta_{i}=\theta_{r}\tag{1.3}$

and by Snell’s law:

$n_1\sin\theta_{i}=n_2\sin\theta_{t}\text{.}\tag{1.4}$

The Fresnel reflection coefficient for $$p$$ polarized light is29

$\frac{E_{rp}}{E_{ip}}= r_{12p} =\frac{n_{2}\cos{\theta_{i}}-n_{1}\cos{\theta_{t}}}{n_{2}\cos{\theta_{i}}+n_{1}\cos{\theta_{t}}}\text{,}\tag{1.5}$

and for $$s$$ polarized light

$\frac{E_{rs}}{E_{is}}= r_{12s} =\frac{n_{1}\cos{\theta_{i}}-n_{2}\cos{\theta_{t}}}{n_{1}\cos{\theta_{i}}+n_{2}\cos{\theta_{t}}}\text{.}\tag{1.6}$

Using Snell’s law, these formulae can be written as

$r_{12p}=\frac{\tan(\theta_{i}-\theta_{t})}{\tan(\theta_{i}+\theta_{t})}\tag{1.7}$

and

$r_{12s}=\frac{\sin(\theta_{i}-\theta_{t})}{\sin(\theta_{i}+\theta_{t})}\text{.}\tag{1.8}$

The reflected intensity or reflectance for $$p$$ and $$s$$ polarized light is given by

$R_{p}=\left|r_{12p}\right|^{2}\;\text{ and }\;R_{s}=\left|r_{12s}\right|^{2}\text{.}\tag{1.9}$

### 1.2.2 Reflection at a layer

Suppose medium 2 is a layer in between medium 1 and medium 3. Regardless of the polarization, the reflection coefficient for the layer can be expressed in the reflection coefficients of the interfaces (see Fig 1.2), with the $$z$$-axis perpendicular to the interface:

$r_{123}=r_{12}+t_{12}r_{23}t_{21}e^{2i\kappa_{z_{2}}d_{2}}+ t_{12}r_{23}r_{21}r_{23}t_{21}e^{4i\kappa_{z_{2}}d_{2}}+\ldots=\\ r_{12}+t_{12}r_{23}t_{21}e^{2i\kappa_{z_{2}}d_{2}} \left[\sum\limits_{j=0}^{\infty}{\left(r_{21}r_{23}e^{2i\kappa_{z_{2}}d_{2}}\right)^{j}}\right] \tag{1.10}$

where

$\kappa_{z_{i}}=\sqrt{\kappa_{i}^{2}-\kappa_{x_{1}}^{2}}=\frac{\omega}{c}\sqrt{\varepsilon_{i}-\varepsilon_{1}\sin^{2}{\theta_{1}}} \text{,}\tag{1.11}$

with $$i$$ the number of the medium, $$d_i$$ the layer thickness, and $$\varepsilon_{i}=n_{i}^{2}$$.

Since for $$0\lt q\lt 1$$

$\sum\limits_{j=0}^{\infty }{q^{j}}=\frac{1}{1-q} \text{,}\tag{1.12}$

we find

$r_{123}=r_{12}+\frac{t_{12}t_{21}r_{23}e^{2i\kappa_{z_{2}}d_{2}}}{1-r_{21}r_{23}e^{2i\kappa_{z_{2}}d_{2}}}= \frac{r_{12}+r_{23}(t_{12}t_{21}-r_{12}r_{21})e^{2i\kappa_{z_{2}}d_{2}}}{1-r_{21}r_{23}e^{2i\kappa_{z_{2}}d_{2}}}\text{.}\tag{1.13}$

Fig. 1.2 Multiple reflection and transmission in a layer, with unit incident amplitude. The amplitudes are specified by the Fresnel coefficients of the interfaces and a phase retardation $$\beta=\kappa_{z_{2}}d_{2}$$.

In the left of Fig. 1.3 we see the normal propagation of the light for reflection and refraction. On the basis of Fermat's principle the reversed propagation direction should also be physically possible (see middle). In this reversed situation we can imagine some extra transmitted and reflected beams (see right). By comparing the situation on the right to the situation on the left we find

$t_{12}t_{21}+r_{12}r_{12}=1\tag{1.14a}$

$t_{12}r_{21}+r_{12}t_{12}=0\;\;\Rightarrow\;\;r_{21}=-r_{12}\text{,}\tag{1.14b}$

Fig. 1.3 Graphical illustration of the Stokes relations.

which are known as the Stokes relations. Substituting in Eq. (1.13) we obtain

$r_{123}=\frac{r_{12}+r_{23}e^{2i\kappa_{z_{2}}d_{2}}}{1+r_{12}r_{23}e^{2i\kappa_{z_{2}}d_{2}}}\text{,}\tag{1.15}$

where the reflection coefficient of a layer is expressed in the reflection coefficients of the interfaces.

### 1.2.3 Reflection at multiple layers

The extension of the above formulae to a multiple layer system is straightforward. In the following, $$n$$ is the number of layers. $$m=0$$ for the medium from which the light is incident, and $$m=n+1$$ for the semi-infinite medium at the other side of the layer structure.

Going through the layers one by one and writing $$R_m$$ for the total reflection coefficient of all interfaces after medium $$m$$, we start with

$R_{n}=r_{n,n+1}\text{.}\tag{1.16}$

For $$m=n$$ to $$1$$:

$R_{m-1}=\frac{r_{m-1,m}+R_{m}e^{2i\kappa_{z_{m}}d_{m}}}{1+r_{m-1,m}R_{m}e^{2i\kappa_{z_{m}}d_{m}}}\tag{1.17}$

a recurrent relation eventually yielding the total reflection coefficient $$R_0$$.

## 1.3 Reflectometric methods

In this section we will discuss ellipsometry, Brewster angle microscopy and surface plasmon resonance. These reflectometrical methods have a number of features in common. First, an angle of incidence, a polarization state and a wavelength can be chosen for the incident light beam. Second, there is a layer structure under study for which a suitable substrate and ambient medium can be chosen, depending on the method used. Third, the intensity of the reflected light may be measured as a function of incident angle, polarization or wavelength.

These measurements can be related to the optical properties of the layer structure under investigation by the model described in the preceding section. A number of simplifying assumptions will be made: (i) the incident light can be approximated by a monochromatic plane wave; (ii) the incidence medium is transparent and optically isotropic; (iii) the sample surface is a plane boundary; (iv) the optical properties of the sample are laterally uniform over distances of the order of the wavelength; (v) the light-sample interaction is elastic, no frequency change occurs. Any changes in the optical properties of the sample in the direction of the surface normal can be accounted for by addition of more layers with the appropriate properties in the stratified-medium model (SMM). The dielectric functions of mixed phases can be calculated using effective medium theories (EMTs) on the basis of component volume fractions and microstructure.

### 1.3.1 Ellipsometry

In ellipsometry30 the change in the state of the polarization of light upon reflection at interfaces and thin films is used to characterize these films. This characterization can be real-time and in-situ and is nonperturbing. The beam that is incident on the layer system is usually monochromatic, collimated and polarized. The ratio $$\rho$$ of the complex reflection coefficients for $$p$$ and $$s$$ polarization is determined from the change in the polarization state:

$\rho=\frac{r_{p}}{r_{s}}\text{.}\tag{1.18}$

When $$r_p$$ and $$r_s$$ are written as

$r_{p}=\left|r_{p}\right|e^{i\delta_{p}}\;\;\text{ and }\;\;r_{s}=\left|r_{s}\right|e^{i\delta_{s}}\tag{1.19}$

then

$\rho=\tan\psi e^{i\Delta}\text{,}\tag{1.20}$

with

$\Delta=\delta_{p}-\delta_{s}\;\;\text{ and }\;\;\tan\psi=\frac{\left|r_{p}\right|}{\left|r_{s}\right|}\text{.}\tag{1.21}$

In ellipsometric measurements $$\Delta$$ and $$\psi$$ are measured, ellipsometry therefore has the advantage of measuring only relative amplitudes and phases. For completely transparent layers, $$\rho$$ becomes a periodic function of the layer thickness $$d.$$ By performing measurements at more than one angle of incidence $$\theta$$ or wavelength $$\lambda$$ the uncertainty of an integral multiple of the film thickness period can be resolved. The layer thickness $$d$$ should of course be independent of $$\lambda$$ or $$\theta$$. In multiple-angle-of-incidence ellipsometry (MAIE) $$\rho$$ is measured as a function of $$\theta$$, while in spectroscopic ellipsometry (SE) it is measured as a function of $$\lambda$$. In variable-angle spectroscopic ellipsometry (VASE) $$\rho$$ is determined varying both $$\theta$$ and $$\lambda$$.

Other, but less generally applicable ways to obtain more ellipsometric measurements to separate layer parameters include: (i) measuring several unknown thicknesses with the same dielectric constant; (ii) measuring the same film with different surrounding media; (iii) measuring the same film on different substrates; (iv) measuring in transmission as well as in reflection.31 Clearly it is also possible to measure the thickness with another method such as AFM.32

Fig. 1.4 Polarizer-compensator-sample-analyzer (PCSA) ellipsometer arrangement.

The inverse problem of determining the optical properties from the measurements can be solved by searching those model parameters that best match theoretical and experimental values of the ellipsometric function. Except for the simple cases that can be solved analytically with just one or two interfaces, this usually requires using linear regression analysis, to minimize an error function

$f=\sum\limits_{i=1}^{N}{\left[(\psi_{im}-\psi_{ic})^2+(\Delta_{im}-\Delta_{ic})^2\right]} \tag{1.22}$

where $$\psi_{im}$$, $$\psi_{ic}$$ and $$\Delta_{im}$$, $$\Delta_{ic}$$ denote the $$i$$th measured and calculated values of the $$N$$ independent ellipsometric measurements.

Ellipsometry can be performed with many different experimental configurations. The most common polarizer-compensator-sample-analyzer (PCSA) ellipsometer arrangement is shown in Figure 1.4. A monochromatic collimated beam (unpolarized, or circularly polarized) is linearly polarized by polarizer P, after which the compensator C (a quarter-wave retarder) generally renders the polarization elliptic. By adjusting the polarizer angle $$P$$ such that the light is linearly polarized after reflection, the output of the photo detector can be reduced to zero by adjusting the analyzer angle $$A$$.

The compensator angle $$C$$ is usually chosen equal to $$\pm\pi/4$$ with respect to the plane of incidence. Two independent nulls can be reached for each compensator setting, therefore these nulls define four zones. $$\Delta$$ and $$\psi$$ can now be determined directly from $$P$$ and $$A$$; e.g. when $$C=-\pi/4$$ then:

$\Delta=2P+\pi/2\;\;\text{ and }\;\;\psi=A\text{.}\tag{1.23}$

Fig. 1.5 The ellipsometric parameters $$\Delta$$ and $$\psi$$ and the reflectances $$R_p$$ and $$R_s$$ for $$p$$ and $$s$$ polarized light as a function of the incident angle $$\phi$$. The values were calculated for an air-glass interface $$(n_{glass}=1.5)$$ at wavelength $$\lambda = 546\text{ nm}$$.

In Fig. 1.5, $$\Delta$$ and $$\psi$$ are plotted as a function of the angle of incidence. Note that not the intensities but the angles for which the intensity vanishes determine the ellipsometric parameters, which adds to the precision of the method. In an ideal ellipsometer measuring one null is sufficient to determine $$\rho$$, but by measuring multiple nulls the experimental error can be decreased by averaging.

The null ellipsometer can be automated by rotating the polarizer and analyzer with stepping motors, using feedback control. To increase the speed and avoid the use of rotating parts Faraday cells can be inserted after the polarizer and before the analyzer to produce magneto-optical rotations. In the rotating analyzer ellipsometer (RAE) the polarization state of the reflected light is analyzed by rotating the analyzer and performing a Fourier analysis of the output signal. For fast measurements without moving parts a photoelastic modulator (PEM) can be used in place of the compensator.

Various experimental schemes have been applied to allow for fast (millisecond time scale) spectroscopic ellipsometry on semiconductor materials mainly. Layer thicknesses and dielectric functions were measured real time and in situ during layer growth or etching.33-36 Ellipsometry has also been applied succesfully to surfactant mono-molecular layers at the water-air interface.37-39

### 1.3.2 Brewster angle microscopy

A form of reflectometry which was derived from ellipsometry is Brewster angle microscopy (BAM)40-46 which is mostly applied to monolayers at an air-water interface. Consider a $$p$$ polarized beam which is incident from medium 1 on a plane interface with medium 2 (i.e. the five requirements mentioned at the start of Section 1.3 are met). At Brewster’s angle, the reflectance vanishes if both $$n_1$$ and $$n_2$$ are real (as can be seen in Figure 1.5). Brewster’s angle $$\theta_B$$ is defined as:

$\theta_B+\theta_t=\frac{\pi}{2}\text{,}\tag{1.24}$

for which $$r_{12p}$$ becomes zero (see Eq. 1.7). Using Snell’s law again we find

$n_{1}\sin\theta_B=n_2\sin(\pi/2-\theta_B)=n_2\cos\theta_B\text{,}\tag{1.25}$

and thus

$\tan\theta_B=\frac{n_2}{n_1}\text{.}\tag{1.26}$

Fig. 1.6 Schematic diagram of a Brewster angle microscope (BAM).

For a real interface however, the reflectance does not vanish, but has a minimum at Brewster’s angle. In the case of a monolayer at the air-water interface the depth of this minimum is determined by: (i) The presence of the monolayer at the interface, which has a refractive index different from $$n_1$$ and $$n_2$$. (ii) The roughness of the water surface caused by thermal fluctuations (about 0.3 nm).42 (iii) The optical anisotropy of ordered monolayers stemming from the anisotropic polarizability of hydrocarbon chains of monolayer molecules. Therefore, monolayer parts with the same molecular tilt angle (with respect to the surface normal) but different molecular azimuthal angles (direction in the surface plane), give rise to different reflectivities.

Fig. 1.7 Reflectance around Brewster’s angle for an air-water interface $$(n_{water} = 1.33; \lambda = 633 \text{ nm})$$ with and without a monolayer $$(d = 2.5\text{ nm; }n = 1.5)$$. Note the reflectance scale.

By simply imaging the reflected $$p$$ polarized light with an objective on a camera (see Fig. 1.6), monolayers at the air-water interface can be observed in situ in real time. The lateral resolution of this method is in the micrometer range (diffraction limited), while the thickness sensitivity is about 0.2 nm.40 Because of the small difference in the reflection coefficient (see Fig. 1.7) for the interface with or without the monolayer, a high polarization ratio and considerable laser power are needed to obtain a sufficient intensity and contrast (typically 100 mW).

BAM can be used to directly observe the anisotropy in monolayers, if areas with different reflectance (having the same molecular tilt angle but with different orientations in the monolayer plane) exceed the resolution of the BAM. It also allows for a quantitative study of the molecular tilt,41 for example as a function of the mean molecular area. When an analyzer is introduced in front of the camera, due to the anisotropy the contrast can be adjusted because the reflected light is only $$p$$ polarized if chains are tilted parallel or anti-parallel to the plane of incidence. Therefore, the azimuthal chain orientation can be deduced from the reflectance at different analyzer positions.

### 1.3.3 Surface plasmon resonance

In the case of internal reflection in a prism ($$n_p$$) at the interface with a sample medium ($$n_s$$), with $$n_p\gt n_s$$, then $$\theta_t\gt\theta_i$$ because of Snell’s law:

$\sin\theta_{i}=\frac{n_{s}}{n_{p}}\sin\theta_{t}\text{.}\tag{1.27}$

Thus, as $$\theta_i$$ increases the transmitted ray gradually approaches grazing incidence, and an increasing amount of energy is reflected. The angle of incidence for which $$\theta_t=\pi/2$$ is known as the critical angle $$\theta_c$$, and

$\sin\theta_{c}=\frac{n_{s}}{n_{p}}\text{.}\tag{1.28}$

For angles of incidence greater than or equal to $$\theta_c$$ all of the incident energy is reflected from the interface, a process which is known as total internal reflection (TIR). The reflectance as a function of the internal angle of incidence is plotted in Fig. 1.8(a), where the internal and external angles of incidence are defined as well.

The transmitted electric field can be written as

$\vec{E}_{t}=\vec{E}_{0t}e^{i(\vec{\kappa}_{t}\cdot\vec{r}-\omega t)}\text{,}\tag{1.29}$

where

$\kappa_{tx}=\kappa_{t}\sin\theta_{t}\;\;\text{ and }\;\;\kappa_{tz}=\kappa_{t}\cos\theta_{t}\tag{1.30}$

and there is no y component of $$\vec{\kappa}$$, as the plane of incidence is in the x-z plane and the interface is in the x-y plane. Using Snell’s law again we find

$\kappa_{t}\cos\theta_{t}=\pm \kappa_{t}\sqrt{1-\frac{n_{p}^2}{n_{s}^{2}}\sin^2\theta_{i}}\tag{1.31}$

and since in the case of TIR $$\sin\theta_i\gt n_s/n_p$$,

$\kappa_{tz}=\pm i\kappa_{t}\sqrt{\frac{n_{p}^2}{n_{s}^{2}}\sin^2\theta_{i}-1}\equiv\pm i\beta\text{.}\tag{1.32}$

Therefore,

$\vec{E}_{t}=\vec{E}_{0t}e^{\mp \beta z}e^{i(\kappa_{tx}x-\omega t)}\text{.}\tag{1.33}$

Fig. 1.8 Several configurations which can be used for evanescent wave optics, with the corresponding reflectances as a function of the internal incident angle $$\theta$$ ($$\lambda$$ =633 nm). (a) Total internal reflection at a glass prism base ($$\varepsilon$$=2.3), and definitions of the internal and external angles of incidence $$\theta$$ and $$\phi$$, respectively. (b) SP excitation in the Kretschmann geometry with a 53 nm silver layer $$\varepsilon_{silver}$$ = -18.35+i 0.55), seen as a dip in the reflectance. (c) Excitation of guided modes after addition of a 1.2 $$\mu$$m polymetylmethacrylate (PMMA) layer ($$\varepsilon_{PMMA}$$ = 2.18).

Because the positive exponential is physically untenable, we have a wave with an amplitude that decreases exponentially as it penetrates the less dense medium. The penetration depth $$d_p$$ as defined by the $$1/e$$ decrease in amplitude is

$d_{p}=\frac{\lambda}{2\pi}\sqrt{\frac{n_{p}^2}{n_{s}^2}\sin^2\theta_{i}-1}\text{.}\tag{1.34}$

The field propagates along the interface in the x direction as a so-called surface or evanescent wave. Because of the presence of the evanescent field at the interface, the reflected light is sensitive to optical changes within a distance of the order of the wavelength from the interface.47

Fig. 1.9 Dispersion of: incident light in the sample medium $$\kappa_{ix}$$; the evanescent wave $$\kappa_{ev}$$ (or light internally incident); and the surface plasmon $$\kappa_{SPR}$$.

By enhancement of the evanescent field, the system becomes even more sensitive to the optical parameters near the interface. This enhancement (by more than a factor of 10) can be obtained by surface plasmon (SP)48-50 excitation. In the so-called Kretschmann configuration51 a thin metal layer is introduced on top of the prism base, with light internally reflecting (see Fig. 1.8 (b)). Free electrons in the metal can be made to oscillate harmonically and coherently when excited by the evanescent wave along the interface if the incident light is $$p$$ polarized, that is, if its E field has a component perpendicular to the interface. For a certain angle of incidence $$\theta_{SPR}$$, the component parallel to the interface of the wave vector of the incident light matches that of the electron oscillations or surface plasmons:

$n_{p}\frac{\omega}{c}\sin\theta_{SPR}=\kappa_{e\nu}=\kappa_{SPR}\text{,}\tag{1.35}$

with $$n_p$$ as the refractive index of the prism. In this case surface plasmon resonance (SPR) occurs. The wave vector $$\kappa_{SPR}$$ on the interface between the metal layer ($$\varepsilon_m$$), and the sample medium ($$\varepsilon_s$$) is approximated by52

$\kappa_{SPR}=\frac{\omega}{c}\sqrt{\frac{\varepsilon_{m}\varepsilon_{s}}{\varepsilon_{m}+\varepsilon_{s}}}\text{.}\tag{1.36}$

Fig. 1.9 illustrates that light in the sample medium could not naturally excite SPs because

$\kappa_{s}=n_{s}\frac{\omega}{c}\lt \kappa_{SPR}\tag{1.37}$

for all $$\omega$$. The propagation length $$L_x$$ of the surface plasmon along the interface, is defined by the 1/$$e$$ attenuation of the SP field in the propagation direction:

$L_{x}=\frac{1}{2\cdot \text{Im}(\kappa_{x})}\text{.}\tag{1.38}$

Fig. 1.10 The increase of the reflectance at the original SPR angle $$\theta_{SPR}$$($$d=0$$), and the shift of $$\theta_{SPR}$$ as a function of the thickness of a cover layer on top of the metal layer (calculated with same parameters as for Fig. 1.8; $$n_{cover}$$ = 1.5).

As can be seen in Fig. 1.8 (b) the reflectance as a function of the incident angle shows a pronounced dip indicating the resonant excitation of surface plasmons. Because $$\theta_{SPR}$$ is past the critical angle $$\theta_c$$, this excitation method is known as an attenuated total reflection (ATR)51,53 method. The Fresnel theory can still be used to accurately describe the system. In Figure 1.10 the increase of the reflectance at the original $$\theta_{SPR}$$, and the shift of $$\theta_{SPR}$$ as a function of the thickness of a cover layer on top of the metal layer are given to indicate the surface sensitivity of the method. Analytical or numerical inversion enables one to determine the optical thickness, that is, the product of the layer refractive index and thickness from the resonance angle shift $$\Delta\theta_{SPR}$$,

$\Delta\theta_{SPR}\propto d\cdot n\text{,}\tag{1.39}$

with coverlayer refractive index $$n$$, and thickness $$d$$. Using Fresnel theory we can calculate the $$d$$,$$n$$ combinations that yield the same $$\Delta\theta_{SPR}$$ (see Fig. 1.11). The differences in the rest of the reflectance curves that can be calculated for these $$d$$,$$n$$ combinations are extremely small. Therefore, measuring $$R(\theta )$$ is not an accurate way to separate $$d$$ and $$n$$ when this is needed. The same is true for measurements of the wavelength dependent reflectance. Measurements on the same layer in different media do offer a possibility,54 but can hardly said to be a generally applicable method, especially for biological monolayers. In most cases we will use literature values for $$n$$ to obtain quantitative $$d$$ measurements, or be satisfied with qualitative measurements.

Fig. 1.11 Calculation of the $$d$$,$$n$$ combinations that yield the same $$\Delta\theta_{SPR}$$ as $$d=3\text{ nm}$$ and $$n=1.5$$.

SPR has been used in different sensor systems (see next chapter), but can also be used for imaging in a surface plasmon microscope (SPM).47 In this case the reflected light is imaged by an objective on a camera. Areas at resonance will appear dark, while areas where the resonance condition is not fulfilled will appear bright. Pioneering work in this field was done by Yeatman et al.,55,56 and Rothenhäusler et al.57-59 in 1987 and 1988. Yeatman's work was aimed at developing an SPM based on a scanning focused beam55 (lateral resolution 25 $$\mu$$m). The straightforward approach with the imaged parallel beam however, proved to be much faster and instrumentationally much simpler, while the sensitivity and lateral resolution is comparable or better58. The lateral resolution of SPM is determined by the dissipative and radiative losses that lead to a strong damping of the surface plasmons along the wave propagation direction. An appropriate choice of the metal layer and wavelength makes a lateral resolution in the range of a few micrometers possible (see Chapter 4).

While SPM was first applied to test patterns evaporated onto the metal layer,55-57 it was soon applied to laterally phase-separated lipid monolayers,58,60-62 and monolayers patterned using photodesorption as a dry etching technique.61,63 Other applications include the imaging of potential waves in electrochemical systems64 and spatial light modulation.65 SPM will be discussed in more detail in Chapter 4 and 5.

Fig. 1.12 The resonance angles of the SPs and GWSPs as a function of the waveguiding cover layer thickness (PMMA). Mode index and polarization are indicated for the first GWSPs (calculated with same parameters as for Fig. 1.8).

For sufficiently thick cover layers more than one minimum can be observed in the reflectance curve (see Fig. 1.8 (c)). In this case so-called guided wave surface polaritons (GWSPs)47,50,66 are excited, with the cover layer acting as a waveguide. The light traveling through the waveguiding coverlayer is totally reflected at both interfaces with the surrounding media and fulfills the mode equation:47

$\kappa_{m}d+\beta_{0}+\beta_{1}=m\pi\tag{1.40}$

where $$\kappa_m$$ is the wave vector of the mode of order $$m$$, $$d$$ the thickness of the waveguide, and $$2\beta_i=\text{Im}(r_i)/\text{Re}(r_i)$$ with $$r_i$$ as the complex reflection coefficient at the interface between waveguide and metal $$r_0$$, and waveguide and ambient medium $$r_i$$, respectively. Fig. 1.12 displays the results of Fresnel calculations of the number and positions of the reflectance minima as a function of the waveguide thickness. For the thinnest layers ($$d\lt$$ 200 nm) we only have the reflectance minimum due to the resonant excitation of surface plasmons, while for higher thicknesses an increasing number of guided modes is found. These GWSPs can be excited with $$p$$ as well as $$s$$ polarized light, and can be used in sensor measurements just like the SPR modes. The first guided modes offer a sensitivity comparable to that of the SPs but the sensitivity decreases with increasing mode index.66,67 An advantage is that measurements for $$p$$ and $$s$$ polarized light can be used to separate $$d$$ and $$n$$ for a layer on top of the waveguiding layer.68

Similar considerations are valid for the microscopy variant of this technique, called optical waveguide microscopy (OWM)69,47 using a configuration similar to that of an SPM. The fact that both $$p$$ and $$s$$ polarized guided modes can be used has been said to be advantageous for the study of anisotropic films.69 It is however virtually impossible to obtain a thick waveguiding layer that is homogeneous on the nanometer scale, which complicates the study of layers on top of the waveguide, and limits it mostly to the waveguide itself.

Scanning plasmon near field microscopy (SPNM)70 was introduced in 1992 as a new type of scanning near field optical microscope. The SPNM is essentially an SPM with a sharp STM tip probing the evanescent field of the resonantly excited surface plasmons. The scanned area is imaged via a pinhole onto a detector. The increased reflectance signal resulting from the tip disturbing the attenuated total reflection is measured when the tip is scanned a few nm above the sample. This makes essentially non-perturbative measurements possible with a lateral resolution of several nm.70,71 The high lateral resolution of the SPNM images (showing optical structures different from topography with the tip at an average 20 nm distance71) is surprising in view of the propagative character of surface plasmons. A similar approach with the STM tip exchanged for an AFM tip has also been used,72 but as the measurements were done with the tip in contact with the sample it is not really possible to separate topographical from optical contrast. In a photon scanning tunneling microscope (PSTM)73 a sharpened fiber tip is scanned through an evanescent field, converting it into progressive waves propagating through the fiber towards a detector. This approach has been used to study the surface plasmon evanescent field at silver and gold layers.74-76

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