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# Refractive Index (including Complex description)

The parameter used to describe the interaction of electromagnetic radiation with matter is the complex index of refraction, ${\displaystyle {\tilde {n}}}$, which is a combination of a real part and an imaginary part.

${\displaystyle {\tilde {n}}=n-ik}$

Here, n is also called 'index of refraction' which sometime leads to confusion, and k is called the 'extinction coefficient'. In a dielectric material such as glass, none of the light is absorbed and therefore k = 0.

The refractive index of a material is the factor by which electromagnetic radiation is slowed down (relative to vacuum) when it travels inside the material. For a non-magnetic material, the square of the refractive index is the material's dielectric constant ε (sometimes expressed as the relative permittivity εr multiplied by the permittivity of free space, ε0). For a general material it is given by:

${\displaystyle n={\sqrt {\varepsilon \mu }}}$
where μ is the permeability of free space.

The speed of all electromagnetic radiation in vacuum is the same, approximately 3×108 meters per second, and is denoted by c. So if ${\displaystyle v}$ is the phase velocity of radiation of a specific frequency in a specific material, the refractive index is given by

${\displaystyle n={\frac {c}{v}}}$

This number is typically bigger than one: the denser the material, the more the light is slowed down. However, at certain frequencies (e.g. near absorption resonances, and for x-rays), ${\displaystyle n}$ will actually be smaller than one. This does not contradict the theory of relativity, which holds that no information-carrying signal can ever propagate faster than ${\displaystyle c}$, because the phase velocity is not the same as the group velocity or the signal velocity.

The phase velocity is defined as the rate at which the crests of the waveform propagate; that is, the rate at which the phase of the waveform is moving. The group velocity is the rate that the envelope of the waveform is propagating; that is, the rate of variation of the amplitude of the waveform. It is the group velocity that (almost always) represents the rate that information (and energy) may be transmitted by the wave, for example the velocity at which a pulse of light travels down an optical fibre.

Sometimes, a "group velocity refractive index", usually called the group index is defined:

${\displaystyle n_{g}={\frac {c}{v_{g}}}}$,
where ${\displaystyle v_{g}}$ is the group velocity. This value should not be confused with ${\displaystyle n}$, which is always defined with respect to the phase velocity.

At the microscale an electromagnetic wave is slowed in a material because the electric field creates a disturbance in the charges of each atom (primarily the electrons) proportional to the permittivity. This oscillation of charges itself causes the radiation of an electromagnetic wave that is slightly out-of-phase with the original. The sum of the two waves creates a wave with the same frequency but shorter wavelength than the original, leading to a slowing in the wave's travel.

If the refractive indices of two materials are known for a given frequency, then one can compute the angle by which radiation of that frequency will be refracted as it moves from the first into the second material from Snell's law.

Recent research has also demonstrated the existence of negative refractive index which can occur if ${\displaystyle \epsilon }$ and ${\displaystyle \mu }$ are simultaneously negative. Not thought to occur naturally this can be achieved with so called metamaterials and offers the possibility of perfect lenses and other exotic phenomena such as a reversal of Snell's law.

## Dispersion and Absorption

The refractive index of a material varies with frequency (except in vacuum, where all frequencies travel at the same speed, ${\displaystyle c}$). This effect, known as dispersion, is what causes a prism to divide white light into its constituent spectral colors, explains rainbows, and is the cause of chromatic aberration in lenses. In regions of the spectrum where the material does not absorb, the refractive index increases with frequency. Near absorption peaks, the refractive index decreases with frequency.

The Sellmeier equation is an empirical formula that works well in describing dispersion, and Sellmeier coefficients are often quoted instead of the refractive index in tables. For some representative refractive indices at different wavelengths, see list of indices of refraction.

In general, the refractive index is defined as a complex number with both a real and imaginary part, where the latter indicates the strength of absorption loss at a particular wavelength—thus, the imaginary part is sometimes called the extinction coefficient k. Such losses become particularly significant, for example, in metals at short (e.g. visible) wavelengths, and must be included in any description of the refractive index. The real and imaginary parts of the complex refractive index are related through use of the Kramers-Kronig relations. For example, one can determine a material's full complex refractive index as a function of wavelength from an absorption spectrum of the material.

## Anisotropy

The refractive index of certain media may be different depending on the polarization and direction of propagation of the light through the medium. This is known as birefringence or anisotropy and is described by the field of crystal optics. In the most general case, the dielectric constant is a rank-2 tensor (a 3 by 3 matrix), which cannot simply be described by refractive indices except for polarizations along principal axes.

In magneto-optic (gyro-magnetic) and optically active materials, the principal axes are complex (corresponding to elliptical polarizations), and the dielectric tensor is complex-Hermitian (for lossless media); such materials break time-reversal symmetry and are used e.g. to construct Faraday isolators.

## Nonlinearity

The strong electric field of high intensity light (such as output of a laser) may cause a medium's refractive index to vary as the light passes through it, giving rise to nonlinear optics. If the index varies quadratically with the field (linearly with the intensity), it is called the optical Kerr effect and causes phenomena such as self-focusing and self phase modulation. If the index varies linearly with the field (which is only possible in materials that do not possess inversion symmetry), it is known as the Pockels effect.

## Inhomogeneity

If the refractive index of a medium is not constant, but varies gradually with position, the material is known as a gradient-index medium and is described by gradient index optics. Light travelling through such a medium can be bent or focussed, and this effect can be exploited to produce lenses, some optical fibers and other devices. Some common mirages are caused by a spatially-varying refactive index of air.

# Glass Transition Temperature

A simplistic view of a material's glass transition temperature (Tg) is the temperature below which molecules have very little mobility. On a larger scale, polymers are rigid and brittle below their glass transition temperature and can undergo plastic deformation above it. Tg is usually applicable to amorphous phases and is commonly applicable to glasses and plastics.

A fuller discussion of the Tg requires an understanding of mechanical loss mechanisms (vibrational and resonance modes) of specific (and usually common in a given material) functional groups and molecular arrangements. Things like heat treatment and molecular re-arrangement, vacancies, induced strain and other factors affecting the condition of a material may have an effect on Tg ranging from the subtle to the dramatic. Tg is dependent on the viscoelastic materials properties, and so varies with rate of applied load (silly putty is a good example of this, as is stiff cornflour/water mixtures - pull slowly and they flow, pull rapidly and they shatter).

In polymers, Tg is often expressed as the temperature at which the Gibbs free energy is such that the activation energy for the cooperative movement of 50 or so elements of the polymer is exceeded. This allows molecular chains to slide past each other when a force is applied. From this definition, we can see that the introduction of side chains and relatively stiff chemical groups (such as benzene rings) will interfere with the flowing process and hence increase Tg.

In glasses (including amorphous metals and gels), Tg is related to the energy required to break and re-form covalent bonds in a somewhat less than perfect (may be regarded as an understatement) 3D lattice of covalent bonds. The Tg is therefore influenced by the chemistry of the glass. Eg. add B, Na, K or Ca to a silica glass, which have a valency less than 4 and they help break up the 3D lattice and reduce the Tg. Add P which has a valency of 5 and it helps re-establish the 3D lattice, increasing Tg.

The Space Shuttle Challenger disaster was caused by a rubber O-ring that was below its glass transition temperature and thus could not flex adequately to form a proper seal around one of the two solid rocket boosters.

Glass transition temperature of some materials:

 Polymer Tg (oC) Polyethylene (LDPE) -125 Polypropylene (atactic) -20 Poly(vinyl acetate) (PVAc) 28 Poly(ethyleneterephthalate) (PET) 69 Poly(vinyl alcohol) (PVA) 85 Poly(vinyl chloride) (PVC) 81 Polypropylene (isotactic) 100 Polystyrene 100 Poly(methylmethacrylate) (atactic) 105

# Ellipsometry

Ellipsometry is a very versatile optical technique that has applications in many different fields, from the microelectronics and semiconductor indutries (for characterizing oxides or photoresists on silicon wafers, for example) to biology. This very sensitive measurement technique provides unequalled capabilities for thin film metrology. As an optical technique, spectroscopic ellipsometry is non-destructive and uses polarised light to probe the dielectric properties of a sample.

Through the analysis of the state of polarisation of the light that is reflected from the sample, ellipsometry can yield information about layers that are thinner than the wavelength of the light itself, down to a single atomic layer or less. Depending on what is already known about the sample, the technique can probe a range of properties including the layer thickness, morphology, or chemical composition. It is commonly used to characterize with an excellent accuracy film thickness for single layer or complex multilayer stacks ranging from a few angstroms to several micrometres.

The name "ellipsometry" stems from the fact that the most general state of polarization is elliptic. The technique has been known for almost a century, and today has many standard applications. However, ellipsometry is also becoming more interesting to researchers in other disciplines such as biology and medicine. These areas pose new challenges to the technique, such as measurements on unstable liquid surfaces and microscopic imaging.

## Basic principles

An ellipsometer functions by reflecting a beam of light of known polarization off of a sample, and measuring the polarization change upon reflection. The exact nature of the polarization change is determined by the sample's properties (thickness and refractive index). Ellipsometry is a specular optical technique (the angle of incidence equals the angle of reflection). In its modern incarnation, ellipsometry uses a laser as the illumination source, usually a HeNe laser which has a wavelength of 632.8 nm. Although optical techniques are inherently diffraction limited, ellipsometry exploits phase information and the polarization state of light, and can achieve angstrom resolution.

In its simplest form, the technique is applicable to thin films with thickness less than a nanometre to a micrometre. The sample must be composed of a small number of discrete, well-defined layers that are optically homogeneous, isotropic, and non-absorbing. Violation of these assumptions will invalidate the standard ellisometric fitting procedure, although more advanced variants of the technique have been designed (such as spectroscopic or multi-angle ellipsometry).

## Details

Ellipsometry measures two of the four Stokes parameters, which are conventionally denoted by ${\displaystyle \Psi }$ and ${\displaystyle \Delta }$ . The polarization state of the light incident upon the sample may be decomposed into an s and a p component (the s-component is oscillating parallel to the sample surface, and the p-component is oscillating parallel to the plane of incidence). The intensity of the s and p component, after reflection, are denoted by ${\displaystyle R_{s}}$ and ${\displaystyle R_{p}}$. The fundamental equation of ellipsometry is then written:

${\displaystyle \rho ={\frac {R_{p}}{R_{s}}}=\tan(\Psi )e^{i\Delta }}$

Thus, ${\displaystyle \tan \Psi }$ is the amplitude change upon reflection, and ${\displaystyle \Delta }$ is the phase shift. Since ellipsometry is measuring the ratio of two values (rather than the absolute value of either), it is very robust, accurate, and reproducible. For instance, it is insensitive to scatter and fluctuations, and requires no standard or calibration.

The measured ${\displaystyle \Psi }$ and ${\displaystyle \Delta }$ can be converted to optical constants if a layer model is assumed. Directly inverting ${\displaystyle \Psi }$ and ${\displaystyle \Delta }$ is not possible. Instead, an iterative procedure (least-squares minimization) is used: various values of the optical constants are considered, ${\displaystyle \Psi }$ and ${\displaystyle \Delta }$ are then calculated using Fresnel reflection theory. The optical constants which come closest to the experimental ${\displaystyle \Psi }$ and ${\displaystyle \Delta }$ are then considered to be the correct values for the sample.