Summary Kramers-Kronig Relation (KK relation) Gain Saturation in Homogeneous Laser Media Gain constant uniformly drops in homogeneous broadening case Gain Saturation in Inhomogeneous Laser Media Spectral Hole Burning Effect

Lecture 8 Chapter V Theory of Laser Oscillation and Its Control in Continuous and Pulsed Regimes Highlights 1. Fabry-Perot Laser 2. Oscillation Frequency 3. Optimum Output Coupling 4. Mode Locking 5. Q-Switch Laser

§5.1 Fabry-Perot Laser : distributed passive losses of the medium Mirror 1 Mirror 2 Input Plane Ouput Plane propogate constant far away from resonant Complex dielectric susceptibility from laser transition : distributed passive losses of the medium

§5.1 Fabry-Perot Laser Oscillation In population inversion medium: However, in a Fabry-Perot etalon Oscillation

tc is call decay time of light intensity §5.1 Fabry-Perot Laser Threshold Condition (gain becomes equal to the losses) Phase condition Gain condition Population Inversion condition Other form tc is call decay time of light intensity

§5.1 Fabry-Perot Laser Numerical Example: Population Inversion He-Ne Laser Doppler Broadening Width Calculate

§5.1 Fabry-Perot Laser Transmission Reflectivity

§5.2 Oscillation Frequency Many frequencies exist, we want to know the frequency also satisfied gain condition : mth resonance frequency of the passive Fabry-Perot etalon

§5.2 Oscillation Frequency is the center frequency of the atomic lineshape function Assume that the cavity length is adjusted so that one of its resonance frequencies is very close near . We hope that the oscillation frequency will also be close to . is a slowly varying function of when Oscillation Frequency

§5.2 Oscillation Frequency Passive resonator linewidth : Frequency Pulling Round trip phase shift

In four-level laser system In three-level laser system §5.3 Three- And Four-Level Lasers Pump transition Ground state 1 2 E2 t2 3 Energy E1 t1 E1 >> kT Four-level System Three-level System In four-level laser system In three-level laser system Lifetime t2 is much longer than the lifetime t1 Population on level 1 cannot be neglected Population on level 1 can be neglected Oscillation starts at Oscillation starts as long as

§5.3 Three- And Four-Level Lasers So that the pump rate at threshold in a three-level laser must exceed that of a four-level laser Minimum expenditure power of three-level system Minimum expenditure power of four-level system When t2=tsp, the above two equations are equal to the power emitted through fluorescence by atoms within the volume at threshold. It is referred as the critical fluorescence power. In case of four-level laser

Numerical Example: Critical Fluorescence Power of an Nd3+:Glass Laser §5.3 Three- And Four-Level Lasers Numerical Example: Critical Fluorescence Power of an Nd3+:Glass Laser Calculate:

§5.4 Power in Laser Oscillators When the pumping intensity is increased beyond threshold point, laser will break into oscillation and emit power. We will derive the relation of laser output power to the pumping intensity. Rate Equations (four level system) Steady state solution Necessary condition for population inversion

§5.4 Power in Laser Oscillators Effective pumping rate Below oscillation threshold and When R increase until Gain > Loss and steady state assumption violate In order to keep population inversion unchange, assume Wi is allowed to increase once R exceeds its threshold value

§5.4 Power in Laser Oscillators In idealized model and Critical fluorescence power Threshold pumping rate Example:

Total loss can be attributed to two different sources: §5.5 Optimum Output Coupling in Laser Oscillators Total loss can be attributed to two different sources: (a) The inevitable residual loss due to absorption and scattering in the laser media and mirrors, as well as diffraction losses in the finite diameter reflectors; as small as possible (b) The useful loss due to coupling of output power through the partially transmissvie reflector. (1) Zero coupling (no transmission), threshold will be minimum, pe will be maximum, but no output; (2) Large coupling, threshold will be larger than pumping level, oscillation will cease, output is zero again; (3) Exist an optimum coupling value, where output is maximum.

Recall oscillation condition §5.5 Optimum Output Coupling in Laser Oscillators Recall oscillation condition In case of small loss Pe is the total power given off by the atoms due to stimulated emission

§5.5 Optimum Output Coupling in Laser Oscillators Total loss per pass Output power is thus as Recall Cross section

Maximizing Po with respect to T §5.5 Optimum Output Coupling in Laser Oscillators Maximizing Po with respect to T