1.1.1 Light Waves and Photons

Light is a transverse electromagnetic wave with frequencies ranging from about 3 × 1011 to 3 × 1016 Hz. The corresponding wavelengths in vacuum are from about 1 mm to 0.01 μm. In the range of optical frequencies, as well as below it (microwaves and radio waves), matter behaves as a continuum. Above the optical range, starting with soft X-rays, the discrete atomic structure of matter becomes important even for the phenomenology of radiation–matter interaction, since it leads to diffraction.

The light wave in a medium is linked to the induced motion of the electric charges of electrons and atomic nuclei, constituting macroscopic electric currents. On the other hand, it is mostly safe to neglect the induced magnetization, since the motion of magnetic moments of electrons and nuclei is too slow to follow the rapid optical oscillations. Thus, the most important quantity describing the light wave is the vector of its electric–field intensity, The existence of preferential directions of the action that the electric field in the wave exerts on electric charges in matter is the reason for the importance of its polarization. Early experiments with the optical behavior of calcite crystals led to the discovery of the polarization of light by Malus in 1808, well before the classical electromagnetic theory had been established. The spatial and temporal dependence of the electric field in a uniform, isotropic medium of the complex permittivity ε, is described by the wave equation[1,5]

(1.1)

where ω is the angular frequency of the light wave, and c is the light velocity in vacuum. A useful solution of the wave equation is the monochromatic plane wave propagating along the z–axis of an orthogonal coordinate system,

(1.2)

Here Ex and Ey are the complex amplitudes of along the x– and y–axes; they can be arranged conveniently in the 2 × 1 column vector. The symbols Re{f} and Im{f} mean real and imaginary parts of a complex quantity f, respectively; for example, the x component of the field intensity is We use the complex form of most of the equations, since they are simple and transparent. However, care should be taken in adopting a number of different possible conventions[6].

In order to satisfy the wave equation, Eq. (1.1), the nonvanishing component of the propagation vector of the plane wave of Eq. (1.2) assumes the values given by the dispersion equation

(1.3)

where is the complex refractive index of the medium. By choosing the plane–wave solution, Eq. (1.2), with the time dependence of exp(−iωt), we are adopting the standard physics convention, where the imaginary parts of ε and N are positive. Another convention, in which the two imaginary parts are negative (usually preferred in optics) is easily recognizable, since all expressions containing ε and N become complex conjugate.

The intensity of a light wave is its energy crossing a unit area perpendicular to the direction of propagation per unit time. It is given by the magnitude of the time average of the Poynting vector, where is the intensity of the magnetic field. In SI units, the intensity of the plane wave of Eq. (1.2) in free space is

(1.4)

where ε0 is the permittivity of vacuum. The signals measured by detectors in ellipsometric setups are proportional to the intensity, i.e., to the squared modulus of the complex amplitude of the electric intensity. Since a wave with the electric field amplitude of 1 V/m has the intensity of about 1.3 mW/m2. At low light intensities used in ellipsometric measurements, the response of materials to the light wave is linear.

The understanding of the light–matter interaction requires, as a rule, quantum mechanical description. A monochromatic wave of the angular frequency ω carries energy in the quanta of where is the Planck constant. To convert the wavelength λ to the photon energy in the practical units of electron–volts, we use the following relation,

(1.5)

The photon energy ranges from about 1 meV in the far–infrared to about 1 meV in the vacuum ultraviolet. For example, the green light of a mercury discharge lamp has the vacuum wavelength of 0.5461 μm, corresponding to the photon energy of 2.270 eV; a collimated light beam with the intensity of 1.3 mW/m2 (mentioned above as having the electric field intensity of 1 V/m in the classical picture) consists of the flux of 3.6 × 1015 photons per second per m2. A striking consequence of the quantized nature of the light wave is the onset of sensitivity of photoconductive or photovoltaic detectors. For example, silicon–based detectors provide signals only for photon energies above the bandgap (about 1.1 eV); a light beam of lower photon energy is not absorbed, irrespective of its intensity.

The second fingerprint of the quantum nature of a light wave (i.e., the flux of individual photons) is the superposition of possible polarization states of the photons, described by the quantum–mechanical probability amplitudes. This is formally identical with coherent superposition of the complex amplitudes of plane waves in the classical picture. The advantage of the quantum mechanical description becomes obvious especially in the treatment of unpolarized or partially polarized light.

1.1.2 Polarization of Light

The most general polarization of a monochromatic light wave, Eq. (1.2), is elliptic. Since the wave is transverse, the endpoint of the electric–field intensity vector precesses along an elliptic trajectory in any plane perpendicular to the direction of propagation. One revolution is achieved in the very short time interval of 2π/ω. The time evolution can be viewed as a superposition of harmonic vibrations along two perpendicular axes. If the two vibrations are shifted in phase, the resulting trajectory is elliptic. An example of the polarization ellipse is shown in Fig. 1.1. The wave is assumed to propagate along the z axis of the right–handed cartesian coordinate system x–y–z. The amplitudes of the electric field in the x and y directions are denoted by X and Y, respectively; both of them are real, non-negative quantities. The time dependence of the vector of Eq. (1.2) in the plane z = 0 can be written down conveniently using the following complex...