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1、The Generation of Ultrashort Laser Pulses,The importance of bandwidthMore than just a light bulbLasers,laser modes,and mode-lockingMaking shorter and shorter pulsesPulse-pumpingQ-switching and distributed-feedback lasersPassive mode-locking and the saturable absorberKerr-lensing and Ti:SapphireActiv
2、e mode-lockingOther mode-locking techniquesLimiting factorsCommercial lasers,Prof.Rick TrebinoGeorgia Tech,But first:the progress has been amazing!,The shortest pulse vs.year(for different media),Continuous vs.ultrashort pulses of light,A constant and a delta-function are a Fourier-Transform pair.,C
3、ontinuous beam:Ultrashort pulse:,Irradiance vs.time,Spectrum,time,time,frequency,frequency,Long vs.short pulses of light,The uncertainty principle says that the product of the temporal and spectral pulse widths is greater than 1.,Long pulse,Short pulse,Irradiance vs.time,Spectrum,time,time,frequency
4、,frequency,For many years,dyes have been the broadband media that have generated ultrashort laser pulses.,Ultrafast solid-state laser media have recently replaced dyes in most labs.,Solid-state laser media have broad bandwidths and are convenient.,But a light bulb is also broadband.What exactly is r
5、equired to make an ultrashort pulse?Answer:A Mode-locked LaserOkay,whats a laser,what are modes,and what does it mean to lock them?,Light bulbs,lasers,and ultrashort pulses,Stimulated emission leads to a chain reaction and laser emission.,Excited medium,If many molecules in a medium are excited,one
6、photon can become many.,This is the essence of the laser.The factor by which an input beam is amplified by a medium is called the gain and is represented by G.,Usually,additional losses in intensity occur,such as absorption,scat-tering,and reflections.In general,the laser will lase if,in a round tri
7、p:Total Gain Total Loss This called achieving Threshold.,The Laser,A laser is a medium that stores energy,surrounded by two mirrors.The output mirror is partially reflecting and so lets some light out.,A laser will lase if the beam increases in intensity during a round trip:that is,if.,in the form o
8、f excited states,Calculating the Gain:Einstein A and B Coefficients,In 1916,Einstein considered the various transition rates between molecular states(say,1 and 2)involving light of irradiance,I:Absorption rate=B N1 I Spontaneous emission rate=A N2 Stimulated emission rate=B N2 I,where Ni is the numb
9、er density of molecules in the ith state,and I is the irradiance.,Laser Gain,Neglecting spontaneous emission:,Stimulated emission minus absorption,Proportionality constant is the absorption/gain cross-section,s,If N2 N1:,If N2 N1:,There can be exponential gain or loss in irradiance.Normally,N2 N1,th
10、eres gain,and we define the gain,G:,The solution is:,g and a are the gain and absorption coefficients.,I=I(z-ct),Inversion,In order to achieve G 1,stimulated emission must exceed absorption:,Energy,Inversion,“Negative temperature”,B N2 I B N1 I,Here,there is inversion from level 4 to levels 3 and 2.
11、,4,In order to achieve inversion,we must hit the laser medium very hard in some way and choose our medium correctly.,This condition is called inversion.It does not occur naturally(its forbidden by the Boltzmann distribution).Its inherently a non-equilibrium state.,Canceling the BI factors,N2 N1,or:,
12、Achieving Inversion:Pumping the Laser Medium,Now let Ip be the intensity of(flash lamp)light used to pump energy into the laser medium:,Ip,Will this intensity be sufficient to achieve inversion,N2 N1?Itll depend on the laser mediums energy level system.,Output mirror,Back mirror,Laser medium,Rate Eq
13、uations for a Two-Level System,Rate equations for the densities of the two states:,Absorption,Stimulated emission,Spontaneous emission,If the total number of molecules is N:,Pump,Pump intensity,Why Inversion is Impossible in a Two-Level System,In steady-state:,where:,DN is always positive,no matter
14、how high Ip is!Its impossible to achieve an inversion in a two-level system!,Isat is the saturation intensity.,Rate Equations for a Three-Level System,Assume we pump to a state 3 that rapidly decays to level 2eliminating pump stimulated emission!,Absorption,Spontaneous emission,The total number of m
15、olecules is N:,Level 3 decays fast and so is zero.,Why Inversion is Possible in a Three-Level System,In steady-state:,Now if Ip Isat,DN is negative!,Rate Equations for a Four-Level System,Now assume the lower laser level 1 also rapidly decays to a ground level 0.So also!And,As before:,Because,The to
16、tal number of molecules is N:,At steady state:,Why Inversion is Easy in a Four-Level System(contd),Now,DN is negativealways!,Two-,Three-,and Four-Level Systems,Two-level system,Laser Transition,Pump Transition,At best,you get equal populations.No lasing.,It took laser physicists a while to realize t
17、hat four-level systems are best.,Four-level system,Lasing is easy!,Fast decay,Fast decay,Three-level system,If you hit it very hard,it lases.,Fast decay,What about the saturation intensity?,A is the excited-state relaxation rate:1/t,B is the absorption cross-section,s,divided by the energy per photo
18、n,w:s/w,The saturation intensity plays a key role in lasers.For example,when the intensity approaches(or exceeds)it,the medium becomes less absorbing(N1 decreases).,Both s and t depend on the molecule,the frequency,and the various states involved.,w 10-19 J for visible/near IR light,t 10-12 to 10-8
19、s for most molecules 10-9 to 10-3 s for laser molecules,s 10-20 to 10-16 cm2 for molecules(on resonance),1 to 1013 W/cm2,A dyes energy levels,Dyes are big molecules,and they have complex energy level structure.,S0:Ground electronic state,S1:1st excited electronic state,S2:2nd excited electronic stat
20、e,Energy,Laser Transition,Lowest vibrational and rotational level of this electronic“manifold”,Excited vibrational and rotational level,Dyes can lase into any(or all!)of the vibrational/rotational levels of the S0 state,and so can lase very broadband.,Pump Transition,Lasers emit trains of identical
21、pulses.,where I(t)represents a single pulse intensity vs.time and T is the time between pulses.,Every time the laser pulse hits the output mirror,some of it emerges.,The output of a typical ultrafast laser is a train of identical very short pulses:,R=100%,R 100%,Output mirror,Back mirror,Laser mediu
22、m,The Shah Function,The Shah function,III(t),is an infinitely long train of equally spaced delta-functions.,The Fourier Transform of the Shah Function,If w=2np,where n is an integer,the sum diverges;otherwise,cancellation occurs.,III(t),So:,The Shah Function and a Pulse Train,where f(t)is the shape
23、of each pulse and T is the time between pulses.,To do the integral,set:t/T=m or t=mT,But E(t)can also can be written:,An infinite train of identical pulses can be written:,Proof:,QED,An infinite train of identical pulses can be written:,The Fourier Transform of an Infinite Train of Pulses,The spacin
24、g between frequencies(modes)is then dw=2p/T or dn=1/T.,The Convolution Theorem says that the Fourier Transform of a convolution is the product of the Fourier Transforms.So:,E(t)=III(t/T)*f(t),The Fourier Transform of a Finite Pulse Train,A finite train of identical pulses can be written:,where g(t)i
25、s a finite-width envelope over the pulse train.,Use the fact that the Fourier transform of a product is a convolution.,A lasers frequencies are often called longitudinal modes.Theyre separated by 1/T=c/2L,where L is the length of the laser.Which modes lase depends on the gain and loss profiles.,Freq
26、uency,Intensity,Here,additional narrowband filtering has yielded a single mode.,Actual Laser Modes,Mode-locked vs.non-mode-locked light,Mode-locked pulse train:,Non-mode-locked pulse train:,Random phase for each mode,A train of short pulses,A mess,Generating short pulses=Mode-locking,Locking vs.not
27、locking the phases of the laser modes(frequencies),Intensity vs.time,Intensity vs.time,Locked modes,Intensities,Numerical simulation of mode-locking,Ultrafast lasers often have thousands of modes.,A generic ultrashort-pulse laser,A generic ultrafast laser has a broadband gain medium,a pulse-shorteni
28、ng device,and two mirrors(among other stuff):,Many pulse-shortening devices have been proposed and used.,Output pulse,R=100%,R 100%,Laser medium,Pump power,Pulse-shortening device,Pulsed pumping:ms pulses,Long and potentially complex pulse,Pumping a laser medium with a short-pulse flash lamp yields
29、a fairly short pulse.Flash lamp pulses as short as 1 s exist.Unfortunately,this yields a pulse as long as the excited-state lifetime of the laser medium,which can be considerably longer than the pump pulse.Since solid-state laser media have lifetimes in the microsecond range,it yields pulses microse
30、conds to milliseconds long.,I(t),Q-switching:ns pulses,Q-switching involves:Preventing the laser from lasing(by adding massive loss)until the flash lamp is finished flashing,andAbruptly allowing the laser to lase.,The pulse length is limited by how fast we can switch and the round-trip time of the l
31、aser and yields pulses 10-100 ns long.,100%,0%,Time,Cavity Loss,Cavity Gain,Output pulse intensity,Gain saturation,Q-Switching,How do we Q-switch a laser?Q-switching involves preventing lasing until were ready.A Pockels cell switches(in a few nanoseconds)from a quarter-wave plate to nothing.,Before
32、switching,After switching,Pockels cell as wave plate w/axes at 45,0 Polarizer,Mirror,Pockels cell as an isotropic medium,0 Polarizer,Mirror,Polarization becomes circular on the first pass and then horizontal on the next and is then rejected by the polarizer.,Polarization is unaffected by the Pockels
33、 cell and hence is passed by the polarizer.,The saturable absorber,Like a sponge,an absorbing medium can only absorb so much.High-intensity spikes burn through;low-intensity light is absorbed.The absorption coefficient vs.input intensity:,0,where:,for a 2-level system,The Effect of a Saturable Absor
34、ber,First,imagine raster-scanning the pulse vs.time like this,where k indicates the round-trip number:,After many round trips,even a slightly saturable absorber can yield a very short pulse.,Intensity,Round trips(k),Notice that the weak pulses are suppressed,and the strongest one is amplified(by the
35、 laser gain).,And the strong pulse shortens!,Alas,most absorbers recover slowly,T,Passive mode-locking with a slow saturable absorber,What if the absorber responds slowly(more slowly than the pulse)?Then only the leading edge will experience pulse shortening.,This is the most common situation,unless
36、 the pulse is many ps long.,Gain saturation shortens the pulse trailing edge.,The intense spike uses up the laser gain-medium energy,reducing the gain available for the trailing edge of the pulse(and for later pulses).,The gain saturates,too.This is good.,While saturable absorption shortens the lead
37、ing edge,of the price,saturable gain shortens the trailing edge.,Amplifying the pulse uses up the stored energy and hence reduces the gain.,Initial unsaturated loss,gain loss,gain loss,gain loss,Initial unsaturated gain,time,Laser pulse intensity,Saturated gain and loss,The Passively Mode-locked Dye
38、 Laser,Passively mode-locked dye lasers yield pulses as short as a few hundred fs.Theyre limited by our ability to saturate the absorber.,Some common dyes and their corresponding saturable absorbers,Colliding pulses have a higher peak intensity.,And higher intensity in the saturable absorber generat
39、es shorter pulses.This is called Colliding-Pulse Mode-locking(CPM).,The colliding-pulse mode-locked(CPM)laser,A Sagnac interferometer is ideal for creating colliding pulses.,CPM dye lasers produce even shorter pulses:30 fs.,Passive Mode-Locking Using a Mediums Nonlinear Refractive Index,Just as abso
40、rption varies with intensity(a a0 a2I),a mediums other optical characteristic,the refractive index,n,will also change with intensity:,This is called the optical Kerr effect.The nonlinear refractive index can be made to act like a saturable absorber.,L(x),A Lens and a Lens,A lens is a lens because th
41、e phase delay seen by a beam varies quadratically with x:,Now let L be constant,but let n vary quadratically with x:,n(x),In both cases,a quadratic variation of the phase with x yields a lens.,Laser beam,f(x)=n k L(x),f(x)=n(x)k L,Here,the lens thickness varies quadratically with x.,L(x)L0(1-x2/a2),
42、n(x)n0(1-x2/a2),Kerr-Lensing,A laser beam has an approximately quadratic intensity near its center,so an intense beam can induce a lens via the nonlinear refractive index.,Solid,Low intensity:,High intensity:,I0 exp(-x2/w2)I0(1-x2/w2),f(x)=n(x)k L n0+n2 I0(1-x2/w2)k L,Nothing happens.,Beam focuses!,
43、Kerr-Lens Mode-Locking,Kerr-lensing is the mode-locking mechanism of the Titanium:Sapphire(Ti:Sapph)laser.Ti:Sapph not only lases,but it has a large n2!,Losses are zero for a high-intensity fs pulse.,Large n2,Placing an aperture at the focus favors a short pulse:,Losses are too high for a low-intens
44、ity pulse to lase.,Large n2,Low intensity:,High intensity:,Large n2,Modeling Kerr-lens mode-locking,The gain region acts as the aperture.,We align the laser for the extra focusing due to the Kerr effect at high intensity,so a high-intensity pulse will have better overlap with the gain medium than a
45、low-intensity pulse.,High-intensity pulse,Low-intensity pulse,Ti:Sapph,Kerr-lensing is thus equivalent to saturable absorption!The Ti:Sapph crystal acts as the gain medium,the Kerr lens,and the aperture!,Titanium Sapphire(Ti:Sapphire),Ti:Sapphire has high gain and a very high n2.As a result,it acts
46、both as the gain medium and the mode-locker.It is currently the workhorse laser of the ultrafast community,emitting pulses as short as a few fs and average power in excess of a Watt.,Titanium Sapphire,It can be pumped with a(continuous)Argon laser(450-515 nm)or a doubled-Nd laser(532 nm).,Upper leve
47、l lifetime:3.2 msec,Ti:Sapphire lases from 700 nm to 1000 nm.,Mechanisms that limit pulse shortening,Gain narrowing:G(w)=exp(-aw2),then after N passes,the spectrum will narrow by GN(w)=exp(-Naw2),which is narrower by N1/2.Etalon effects:This yields multiple pulses,spreading the energy over time,weak
48、ening the pulses.Group-velocity dispersion:GVD spreads the pulse in time.And everything has GVDAll fs lasers incorporate dispersion-compensating components.Well spend several lectures discussing GVD!,The universe conspires to lengthen pulses.,Phase and Group Velocities,The phase velocity,vf,is that
49、of the high-frequency oscillations.The group velocity,vg,is that of the pulse envelope.,Dispersion is critical in ultrafast optics.,Dispersion,the dependence of the refractive index on wavelength,has two effects on a pulse,one in space and the other in time.,Both effects play major roles in ultrafas
50、t optics.,Group-velocity dispersion(GVD)disperses a pulse in time(“chirp”):,Angular dispersion disperses a beam in space(angle):,vg(blue)vg(red),In the visible and near-IR,GVD is positive in all materials.,Pulses get longer and longer(and become more chirped)as they pass through more and more glass
