Lecture 12 Optical Properties Md Arafat Hossain Outlines

Interaction of light with solids Electromagnetic spectrum Light as EM waves that have wavelengths shorter than very roughly 100 μm but longer than long-wavelength X-rays, roughly 10 nm.

Light Waves in a Homogeneous Medium Light is an EM wave with time-varying electric and magnetic fields Ex and By, respectively, which propagate through space in such a way that they are always perpendicular to each other and the direction of propagation z.

Refractive Index When an EM wave is traveling in a dielectric medium, the oscillating electric field polarizes the molecules of the medium at the frequency of the wave. The EM wave propagation can be considered to be the propagation of this polarization in the medium. The field and the induced molecular dipoles become coupled. The net effect is that the polarization mechanism delays the propagation of the EM wave. The stronger the interaction between the field and the dipoles, the slower is the propagation of the wave. The relative permittivity εr measures the ease with which the medium becomes polarized, and hence it indicates the extent of interaction between the field and the induced dipoles. For an EM wave traveling in a nonmagnetic dielectric medium of relative permittivity εr, the phase velocity v is given by (1)

Refractive Index Frequency (ʋ) is in the optical frequency range: εr will be due to electronic polarization as ionic polarization will be too sluggish to respond to the field. Frequency (ʋ) is in the IR or below frequency range: εr also includes a significant contribution from ionic polarization. For free space, Velocity of light in free space The ratio of the speed of light in free space to its speed in a medium is called the refractive index n of the medium (2)

Refractive Index So light propagates more slowly in a denser medium which has a higher refractive index (support s Eq. 1). We should note that the frequency (ʋ) remains the same. The refractive index of a medium is not necessarily the same in all directions. (glasses and liquids) Depending on the crystal structure, the relative permittivity εr is different along different crystal directions. This means that, in general, the refractive index n seen by a propagating EM wave in a crystal will depend on the value of εr along the direction of the oscillating electric field (that is, along the direction of polarization). Typically noncrystalline solids, such as glasses and liquids, and cubic crystals are optically isotropic; they possess only one refractive index for all directions.

Refractive Index

Why does it happen?

Dispersion: Refractive Index-wavelength Behavior Relative permittivity, εr depends on the frequency of the applied field. So the refractive index of materials in general depends on the frequency, or the wavelength. What happens to an atom in the presence of an oscillating electric field E which is due to a light wave passing through this location; it may also be due to an applied external field.

Valid under static conditions (3) When the applied field is removed: There is then only the restoring force -βx, which always acts to pull the electrons toward the nucleus O. The equation of motion of the negative charge center is then (force = mass x acceleration) After the removal of the field, the electronic charge cloud executes simple harmonic motion about the nucleus with a natural frequency ω0 (resonance frequency.) (4) Solution of (4), (5) Die out with time because there is an inevitable loss of energy from an oscillating charge cloud. angular frequency of oscillation ω0 (6)

Dispersion: Refractive Index-wavelength Behavior Consider now the presence of an oscillating electric field due to an EM wave passing through the location of this atom. For simplicity we will again neglect energy losses. Newton's second law for Ze electrons with mass Zme driven by E is given by (7) Solution of (7), instantaneous displacement x(t) of the center of mass of electrons from the nucleus (C from 0), (8) (9) From the definition of electronic polarizibility and Eq. 8 & 9 (10) Electronic polarizability ae increase as ω increases

Express (12) this in terms of the wavelength. The simplest relationship between the relative permittivity and polarizability is (11) From (2) (12) Express (12) this in terms of the wavelength. Resonance wavelength (13) This type of relationship between n and the frequency, or wavelength, is called the dispersion relation.

Dispersion: Refractive Index-wavelength Behavior We considered the electronic polarization of an isolated atom with a well-defined natural frequency. In the crystal, however, the atoms interact, and further we also have to consider the valence electrons in the bonds. The overall result is that n is a complicated function of the frequency or the wavelength. One possibility is to assume a number of resonant frequencies, that is, not just X0 but a series of resonant frequencies, λ1, λ2, λ3... , and then sum the contributions arising from each with some weighing factor A1, A2, etc.,

Dispersion: Refractive Index-wavelength Behavior

FRESNEL'S EQUATIONS How much light get transmitted and reflected at the interface of two dielectric medium of different n Derivation: Watch video lectures https://www.youtube.com/watch?v=wahmW7h-AKo&t=1819s https://www.youtube.com/watch?v=oO8HBJ3SYx4 https://www.youtube.com/watch?v=wahmW7h-AKo&t=1819s https://www.youtube.com/watch?v=oO8HBJ3SYx4