Chapter 12. Multimode and Transient Oscillation Pulsed oscillation : Relaxation oscillation, Q-switching, Mode locking ? 12.2 Rate Equations for Intensities and Populations (10.5.8) => (10.5.14), G1=G2=0, A => G21

12.3 Relaxation Oscillation Assume, N1<<N2 => g(n) ~ s(n)N2 12.3 Relaxation Oscillation Steady-state solution ;

Perturbation method ; (12.2.5) => since Similarly,

where Sol) where Intensity : homework

12.4 Q Switching <Qualitative explanation> Q switching : A way of obtaining short, powerful pulses Sudden switching of the cavity Q(loss) from a low(high) value to high(low) value. Principle of Q switching : Suppose we pump a laser medium inside a very lossy cavity. Laser action is precluded even if the upper level population N2 is pumped to a very high value (nearly small signal value). Suddenly we lower the loss to a value permitting laser oscillation. We now have a small-signal gain much larger than the threshold gain for oscillation. <Qualitative explanation> Define, (12.2.5a) =>

Pumping and spontaneous decay of N2 during the pulse interval is negligible, since the pulse is short enough. (12.2.5b) =>

12.5 Methods of Q Switching - Rotating mirrors w ~ 10,000 rpm - Electro-optical switching - Saturable absorber (Passive Q-switching) ; saturate the absorption (bleaching)

12.7 Phase-Locked Oscillators Mode Locking : Locking together of the phases of many cavity longitudinal modes. => Even shorter laser pulse than can be achieved by Q switching. <Phase-locked harmonic oscillator> Displacements of N harmonic oscillators with equally spaced frequencies ; where, The sum of the displacements ;

# Peaks : at # Temporal width : # Maximum total oscillation amplitudes equals to N times the amplitude of a single oscillator # This maximum amplitudes occur at intervals of time T # This temporal width of each spike get sharper as N is increased

12.8 Mode Locking <Shortest pulse length> The maximum number of longitudinal modes : The shortest pulse length : Examples) 1) He-Ne laser, 2) Ruby laser, 3) Dye laser,

<Mode-locked laser oscillation> Electric field of m-th longitudinal mode : where, Assume, the mode fields all have the same magnitude and polarization, and also => Total electric field in the cavity ; where,

Similarly, where,

has maxima occurring at cavity round trip time

12.9 AM Mode Locking AM(amplitude modulation) : modulation index AM(amplitude modulation) : is modulated periodically If W=D (wm+1-wm=pc/L), each mode becomes strongly coupled to its nearest-neighbor modes, and it turns out that there is a tendency for the modes to lock together in phase.

12.10 FM Mode Locking FM(frequency modulation) : The phase of the fieldis modulated periodically : Bessel function of the first kind of order n control !

12.11 Methods of Mode Locking 1) Acoustic loss modulation (AO modulator) diffraction acoustic wave # A standing wave in a medium induces the refractive index variation ; # Diffraction angle ; # Modulation frequency ; control ! ex) L ~ 1 m => D ~ 9x108 rad/s => ns=D/4p = 75 MHz (quartz oscillator)

2) Electro-optical phase modulation (Pockels cell) Refractive index of electro-optic medium ; where, : applied electric field where, 3) Saturable absorbers Absorption coefficient of saturable absorber ; Suppose that there are two oscillating cavity modes ;

Time averaged intensity : Intensity is modulated => Absorption coefficient can be also modulated