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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 is p = 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.