Propagation
Dr.Entesar Ganash 1.Propagation of light in dense optical medium
The propagation is characterized by two parameters: 1) the refractive index 2) the absorption coefficient.
The classical model of light propagation assumes that there are different types of oscillators inside a medium, each with their own characteristic resonant frequency
At optical frequencies, the most important contribution is from the oscillations of the bound electrons within the atoms as in insulators & semiconductors. Free electron oscillators are responsible for the principal optical properties of metals.
The concept of the dipole oscillator was shown theoretically that an oscillating electric dipole would emit electromagnetic waves. What is electric dipole moment? 1.Propagation of light in dense optical medium
An atom electron The oscillator model of the atom is assumed that the electrons are held to the heavy nucleus by springs which represent the restoring forces due to the binding between them. What is restoring force? nucleus
The negatively charged electron & the positively charged nucleus form an electric dipole with a magnitude proportional to their separation. Lorentz postulated the existence of dipoles without knowing their origin.
The natural resonant frequency ω 0 of the atomic dipoles is determined by their mass & the restoring force magnitude experienced for small displacements. The reduced mass is given by: 1 1 1 = + µ me mn
ω0 coincides with one of the natural frequencies of the atoms ω0 = ks µ 1.Propagation of light in dense optical medium
For explanation, the fact that atom has many transition frequencies (absorption & emission spectra), supposing there are a number of dipoles in every atom
stationary
The oscillations generate a time varying dipole that has magnitude p(t) = −e x(t)
The oscillating dipole radiates electromagnetic waves at frequency ω0 So, the atom is emit light at its resonant frequency when enough energy is given to excite the oscillations. 1.Propagation of light in dense optical medium
Also, The dipole model can be used to understand how the atom interacts with an external electromagnetic wave at frequency ω .
The AC electric field applies forces on the electron & the nucleus & makes oscillations of the system at frequency ω .
If ω = ω 0 a resonance phenomenon Amplitude oscillation& energy transferring That means, the atom can therefore absorb energy from the light wave. If ω does not coincide with any of the resonant frequencies, then the atoms will not absorb, and the medium will be transparent. From quantum theory, absorption occurs if
h f = E2 − E1 Atom can also luminesce by re-emitting a photon. The re-radiated photons are incoherent with each other and are emitted in all directions. 2. Vibrational oscillators
An optical medium may has other types of dipole oscillators besides those producing from the bound electrons in the atoms.
If the medium is ionic, vibrations of charged atoms from their equilibrium positions in the crystal lattice will make an oscillating dipole moment. So the optical effects result from these vibrational oscillators must be considered.
The charged atom
Molecular bond
The charged atom
A polar molecule 2. Vibrational oscillators
The optical effects of vibrational oscillators are known in molecular physics.
The charged atoms can vibrate about their equilibrium positions & induce an oscillating electric dipole which will radiate electromagnetic waves at the resonant frequency. The vibrations will occur at lower frequencies WHY?
The molecule will interact with the electric field of a light wave through the forces exerted on the charged atoms.
A polar molecule 3. Free electron oscillators
The electronic & vibrational dipoles are both examples of bound oscillators
Metals have large numbers of free electrons. these are not bound to any atoms, & therefore do not experience any restoring forces when they are displaced.
ks = 0, ω0 = 0
The application of the dipole oscillator model to free electron systems is generally called the Drude-Lorentz model. 4.The Lorentz oscillator
Now, we will use the model to calculate the frequency dependence of the refractive index & absorption coefficient
We model the displacement of the atomic dipoles as damped harmonic oscillators WHY?
The damping term has the effect of reducing the peak absorption coefficient & broadening the absorption line.
The displacement x of the electron is governed by an equation of motion of the form: assuming the motion of the nucleus is ignored & we put me = m0
Acceleration force the damping force the driving force because of the restoring force the AC electric field of the light wave. 4.The Lorentz oscillator
The refractive index of a medium relates with its relative dielectric constant ε r by
n = ε r
Frequency dependence of the real & imaginary parts of the complex dialectic constant of a dipole oscillator at frequencies close to resonance.
The Lorentz oscillator 5.The Lorentz oscillator
The Lorentz oscillator
N is the number of atoms per unit volume χ is the electric susceptibility γ = 1/τ = damping rate References
Optical properties of solid state by mark fox