Interaction of Electromagnetic Radiation with Matter

Dielectric Constant and Atomic Polarizability

• Under the action of the electric field, the positive and negative charges inside each atom are displaced from their equilibrium positions.

• The induced dipole moment isp = αE

Where α is atomic polarizability.

Dielectric Constant and Atomic Polarizability

• The dielectric constant of a medium will depend on the manner in which the atoms are assembled. Let N be the number of atoms per unit volume. The polarization can be written approximately as

P=Np=NαE=ε0ΧE Where Χ is susceptibility.

Dielectric Constant and Atomic Polarizability

• The dielectric constant ε of the medium is

• If the medium is nonmagnetic (i.e., μ=μ0), the index of refraction is given by

Classical Electron Model

• In classical physics, we assume that the electrons are fastened elastically to the atoms and obey the following equation of motion

• The electric field of the optical wave in the atom can be written as

Classical Electron Model

• The equation has the following steady-state solution

• The induced dipole moment is

• The atomic polarizability is

Classical Electron Model

• If there are N atoms per unit volume, the index of refraction of such a medium is

• If the second term in this equation is small compared to 1, the index of refraction can be written as

Dispersion and Complex Refractive Index

• From the equation

• We can get that the index of refraction is higher for the waves with larger ω.

• The phenomenon that the refractive index depends on frequency is called the phenomenon of dispersion.

Dispersion and Complex Refractive Index

• The imaginary term iϒω in the equation

accounts for the damping of electron motion and gives rise to the phenomenon of optical absorption.

Dispersion and Complex Refractive Index

• The complex refractive index can be written as

Dispersion and Complex Refractive Index

• For a wave written as

• In which

• Using the complex refractive index, the wave number K becomes

Dispersion and Complex Refractive Index

• The electric field of the wave becomes

Optical Pulse and Group Velocity

• In the pulsed mode, the pump energy can be concentrated into extremely short time durations, thereby increasing the peak power.

• The propagation of a pulse due to dispersion can be described by representing the pulse as a sum of many plane-wave solutions of Maxwell’s equations.

Optical Pulse and Group Velocity

• The scalar amplitude ψ(z,t) can be thought of as one of the components of electromagnetic field vectors. If A(k) denotes the amplitude of the plane-wave component with wave number k, the pulse ψ(z,t) can be written as

Optical Pulse and Group Velocity

• A laser pulse is usually characterized by its center frequency ω0 or wave number k0 and the frequency spread Δ ω or spread in wave number Δk. In other words, A(k) is sharply peaked around k0. We expand ω(k) around the value k0 in terms of a Taylor series:

Optical Pulse and Group Velocity

• The ψ(z,t) becomes

• The integral in above equation is a function of the composite variable [z – (dω/dk)0t] only and is called the envelope function

Optical Pulse and Group Velocity

• The amplitude of the pulse can be written as

• This shows that, apart from an overall phase, the laser pulse travels along undistorted in shape

Optical Pulse and Group Velocity

• The group velocity of the pulse is

• The group velocity represents the transport of energy.