The Generation of Ultrashort Laser Pulses The importance of bandwidth More than just a light bulb Two, three, and four levels rate equations Gain and saturation
But first: the progress has been amazing! 0ps Nd:glass Nd:YAG Nd:YLF Dye S-P Dye Diode SHORTEST PULSE DURATION ps 00fs CW Dye CPM Dye Color Center Cr:LiS(C)AF Er:fiber Nd:fiber Cr:YAG The shortest pulse vs. year (for different media) 0fs w/compression Cr:forsterite Ti:sapphire 965 970 975 980 985 990 995 000 005 YEAR
Continuous vs. ultrashort pulses of light A constant and a delta-function are a Fourier-Transform pair. Irradiance vs. time Spectrum Continuous beam: time frequency Ultrashort pulse: time frequency 3
Long vs. short pulses of light The uncertainty principle says that the product of the temporal and spectral pulse widths is greater than ~. Irradiance vs. time Spectrum Long pulse time frequency Short pulse time frequency 4
Light bulbs, lasers, and ultrashort pulses But a light bulb is also broadband. What exactly is required to make an ultrashort pulse? Answer: A Mode-locked Laser Okay, what s a laser, what are modes, and what does it mean to lock them? 5
There are 3 conditions for steady-state laser operation. Amplitude condition threshold slope efficiency Phase condition axial modes homogeneous vs. inhomogeneous gain media Transverse modes Hermite Gaussians the donut mode E(x,y) x y 6
Rate equations for a two-level system Rate equations for the population densities of the two states: Pump N Laser Absorption Stimulated emission Spontaneous emission dn dt dn dt BI( N N ) AN BI( N N ) AN Pump intensity dn BIN AN dt dn BIN AN AN dt If the total number of molecules is N: N N N N NN N ( NN) ( NN) N N 7 N
How does the population difference depend on pump intensity? dn BI N AN A N dt Pump N Laser N In steady-state: 0 BIN AN AN Solve for N: N AN /( A BI) N N I / Isat ( A BI) N AN where: Isat N B I A A/ B I sat is called the saturation intensity. 8
Why inversion is impossible in a two-level system Population difference population difference N N 0.5 N 0 0 I sat N N I / Isat a plot of this function Pump Recall that N NN N Laser For population inversion, we require N 0 4I sat Pump intensity It s impossible to achieve a steady-state inversion in a two-level system! 6I sat N N is always positive, no matter how hard we pump on the system! Why? Because absorption and stimulated emission are equally likely. Even for an infinite pump intensity, the best we can do is N = N (i.e., N = 0) 9
Rate equations for a three-level system So, if we can t make a laser using two levels, what if we try it with three? Assume we pump to a state 3 that rapidly decays to level. 3 Pump Fast decay Laser dn dt dn dt BIN BIN AN AN The total number of molecules is N: N N N N N N Level 3 decays fast and so N 3 = 0. d N BIN AN dt N N N N N N 0
Why inversion is possible in a three-level system dn BIN BI N AN A N dt 3 Pump Fast decay Laser In steady-state: 0 BIN BIN AN AN Solve for N: A BI B A I N N N A BI B A I N N I / I I / I sat sat where, as before: I sat A/ B I sat is the saturation intensity. Now if I > I sat, N is negative!
Rate equations for a fourlevel system 3 Fast decay Now assume the lower laser level also rapidly decays to a ground level 0. As before: dn dt Because dn dt BIN AN 0 BI( N N ) AN N 0, N N dn BIN BI N A N dt Pump 0 N N N Laser Fast decay The total number of molecules is N : 0 N N N 0 At steady state: 0 BIN BIN AN
Why inversion is easy in a four-level system 3 Pump Fast decay Laser 0 BIN BIN AN 0 Fast decay Solve for N: N BIN ABI N B A I B A I N N I / I sat I / I sat where: I sat A/ B I sat is the saturation intensity. Now, N is negative for any non-zero value of I! 3
Two-, three-, and four-level systems Four-level systems are best. Two-level system Three-level system Four-level system Fast decay Fast decay Pump Laser Pump Laser Pump Laser Fast decay At best, you get equal populations. No lasing. If you hit it hard, you get lasing. Lasing is easy! 4
Population inverstion in two-, three-, and four-level systems N levels population difference N 0 3 levels population inversion -N 4 levels 0 I sat 4I sat 6I sat 8I sat 0I sat intensity of the pump 5
What is the saturation intensity? Isat A/ B A is the excited-state relaxation rate: / B is the absorption cross-section,, divided by the energy per photon, ħ: / ħ Both and depend on the molecule, and also the frequency of the light. I sat 0 5 to 0 3 W/cm ħ ~0-9 J for visible/near IR light ~0 - to 0-8 s for molecules ~0-0 to 0-6 cm for molecules (on resonance) The saturation intensity plays a key role in laser theory. It is the intensity which corresponds to one photon incident on each molecule, within its cross-section, per recovery time. For Ti:sapphire, I sat ~ 300 kw/cm 6
Threshold condition: what about losses? collection of 4-level systems Steady-state condition #: E gain length L m Amplitude is invariant after each round trip r exp r glm Gain g must be large enough to compensate for reflection losses R and R at the mirrors. after g ln Ebefore Lm r r Note: Note: this ignores other sources of loss in the laser. R j r j 7
Threshold population inversion collection of 4-level systems g ln 4Lm RR Threshold inversion density: length L m But we have seen that the g gain at resonance is: N 0 0 Nthresh ln Lm 0 RR Example: A 5 mm Ti:sapphire crystal, in a cavity with 7 N - 0 cm 3 one high reflector and one 5% output coupler th This corresponds to ~0.5% of the Ti 3+ ions in the crystal. 8
A threshold value of pump intensity levels population difference N 0 3 levels lasing! population inversion N threshold - minimum pump intensity required to turn on a 4-level laser, I threshold 4 levels 0 I sat 4I sat 6I sat 8I sat 0I sat intensity of the pump 9
Gain vs. loss in a laser cavity collection of 4-level systems More generally: E E after before length L m RRe cavity gain per round trip e m 0 m c Round-trip length L rt net cavity loss per round trip, including mirrors Cavity Q-factor: Cavity lifetime: Q L rt c T c rt c the fractional power loss per optical cycle 0
Laser output power Threshold inversion condition becomes c = m or c Nthresh L Output power is determined by gain saturation: m 0 m 0 m 4gLm Icirc gain 4g I L sat c 0 oc loss I out 4g 0 m ocicirc I satoc 0 oc L I sat (valid only above threshold, I out 0)
Slope efficiency Recall that the unsaturated gain is proportional to the pumping rate R p : g 0 N 0 Thus, output power can also be written: 0 0 R p I I out sat oc Rp R pth, Slope efficiency = Output power is linear in the pump rate (above threshold but below saturation) output power pump power
Slope efficiency Above threshold (but not too far above), the output laser power is a linear function of the input pump power. output laser power threshold pump power The slope of this line is called the slope efficiency of the laser. For example, a slope efficiency of 50% means that for every two additional pump photons we add, one additional laser photon is generated. The concept of slope efficiency only applies for I th < I pump < I sat. Essentially all lasers exhibit this behavior. 3
Slope efficiency - examples diode-pumped thulium lasers P. Černý and H. Jelínková, SPIE 006 threshold: 33 ma slope efficiency: 0.74 W/A distributed feedback diode laser A. Jechow, et al., Opt. Lett (007) = 975 nm saturation regime 4