超快光学第04章脉冲.pptx

Ultrashort Laser Pulses IIMore second-order phaseHigher-order spectral phase distortionsRelative importance of spectrum and spectral phasePulse and spectral widthsTime-bandwidth productProf.Rick TrebinoGeorgia Techwww.frog.gatech.eduFrequency-domain phase expansionRecall the Taylor series for():As in the time domain,only the first few terms are typically required to describe well-behaved pulses.Of course,well consider badly behaved pulses,which have higher-order terms in().whereis the group delay.is called the“group-delay dispersion.”The Fourier transformof a chirped pulseWriting a linearly chirped Gaussian pulse as:or:Fourier-Transforming yields:Rationalizing the denominator and separating the real and imag parts:A Gaussian witha complex width!A chirped Gaussian pulseFourier-Transforms to itself!whereneglecting the negative-frequency term due to the c.c.But when the pulse is long(a 0):which is the inverse of the instantaneous frequency vs.time.The group delay vs.w w for a chirped pulseThe group delay of a wave is the derivative of the spectral phase:So:For a linearly chirped Gaussian pulse,the spectral phase is:And the delay vs.frequency is linear.This is not the inverse of the instantaneous frequency,which is:2nd-order phase:positive linear chirpNumerical example:Gaussian-intensity pulse w/positive linear chirp,2=14.5 rad fs2.Here the quadratic phase has stretched what would have been a 3-fs pulse(given the spectrum)to a 13.9-fs one.2nd-order phase:negative linear chirpNumerical example:Gaussian-intensity pulse w/negative linear chirp,2=14.5 rad fs2.As with positive chirp,the quadratic phase has stretched what would have been a 3-fs pulse(given the spectrum)to a 13.9-fs one.The frequency of a light wave can also vary nonlinearly with time.This is the electric field of aGaussian pulse whose fre-quency varies quadraticallywith time:This light wave has the expression:Arbitrarily complex frequency-vs.-time behavior is possible.But we usually describe phase distortions in the frequency domain.Nonlinearly chirped pulsesE-field vs.timeE(t)3rd-order spectral phase:quadratic chirpLonger and shorter wavelengths coincide in time and interfere(beat).Trailing satellite pulses in time indicate positive spectral cubic phase,and leading ones indicate negative spectral cubic phase.S()tg()()Spectrum and spectral phase400 500 600 700Because were plotting vs.wavelength(not frequency),theres a minus sign in the group delay,so the plot is correct.3rd-order spectral phase:quadratic chirpNumerical example:Gaussian spectrum and positive cubic spectral phase,with 3=750 rad fs3 Trailing satellite pulses in time indicate positive spectral cubic phase.Negative 3rd-order spectral phaseAnother numerical example:Gaussian spectrum and negative cubic spectral phase,with 3=750 rad fs3 Leading satellite pulses in time indicate negative spectral cubic phase.4th-order spectral phaseNumerical example:Gaussian spectrum and positive quartic spectral phase,4=5000 rad fs4.Leading and trailing wings in time indicate quartic phase.Higher-frequencies in the trailing wing mean positive quartic phase.Negative 4th-order spectral phaseNumerical example:Gaussian spectrum and negative quartic spectral phase,4=5000 rad fs4.Leading and trailing wings in time indicate quartic phase.Higher-frequencies in the leading wing mean negative quartic phase.5th-order spectral phaseNumerical example:Gaussian spectrum and positive quintic spectral phase,5=4.4104 rad fs5.An oscillatory trailing wing in time indicates positive quintic phase.Negative 5th-order spectral phaseNumerical example:Gaussian spectrum and negative quintic spectral phase,5=4.4104 rad fs5.An oscillatory leading wing in time indicates negative quintic phase.The relative importance of intensity and phasePhotographs of my wife Linda and me:Composite photograph made using the spectral intensity of Lindas photo and the spectral phase of mine(and inverse-Fourier-transforming)Composite photograph made using the spectral intensity of my photo and the spectral phase of Lindas(and inverse-Fourier-transforming)The spectral phase is more important for determining the intensity!Pulse propagationWhat happens to a pulse as it propagates through a medium?Always model(linear)propagation in the frequency domain.Also,you must know the entire field(i.e.,the intensity and phase)to do so.In the time domain,propagation is a convolutionmuch harder.Pulse propagation(continued)using k=/c:Separating out the spectrum and spectral phase:Rewriting this expression:The pulse lengthThere are many definitions of the width or length of a wave or pulse.The effective width is the width of a rectangle whose height and area are the same as those of the pulse.Effective width Area/height:Advantage:Its easy to understand.Disadvantages:The Abs value is inconvenient.We must integrate to .tf(0)0DtefftDt(Abs value is unnecessary for intensity.)The rms pulse widthThe root-mean-squared width or rms width:Advantages:Integrals are often easy to do analytically.Disadvantages:It weights wings even more heavily,so its difficult to use for experiments,which cant scan to .tDtThe rms width is the normalized second-order moment.The Full-Width-Half-MaximumFull-width-half-maximum is the distance between the half-maximum points.Advantages:Experimentally easy.Disadvantages:It ignores satellite pulses with heights 49.99%of the peak!Also:we can define these widths in terms of f(t)or of its intensity,|f(t)|2.Define spectral widths(D Dw w)similarly in the frequency domain(t w w).tDtFWHM10.5tDtFWHMThe Uncertainty PrincipleThe Uncertainty Principle says that the product of a functions widthsin the time domain(Dt)and the frequency domain(D)has a minimum.Combining results:or:Use effective widths assuming f(t)and F()peak at 0:1Other width definitions yield slightly different numbers.For a given wave,the product of the time-domain width(Dt)and the frequency-domain width(Dn)is the Time-Bandwidth Product(TBP)Dn Dt TBPA pulses TBP will always be greater than the theoretical minimumgiven by the Uncertainty Principle(for the appropriate width definition).The TBP is a measure of how complex a wave or pulse is.Even though every pulses time-domain and frequency-domain functions are related by the Fourier Transform,a wave whose TBP isthe theoretical minimum is called Fourier-Transform Limited.The Time-Bandwidth ProductThe coherence time(tc=1/Dn)indicates the smallest temporal structure of the pulse.In terms of the coherence time:TBP =Dn Dt =Dt /tc =about how many spikes are in the pulseA similar argument can be made in the frequency domain,where theTBP is the ratio of the spectral width to the width of the smallestspectral structure.The Time-Bandwidth Product is a measure of the pulse complexity.DttcI(t)A complicated pulsetimeTemporal and spectral shapes and TBPs of simple ultrashort pulsesFWHMcyclesDiels and Rudolph,Femtosecond PhenomenaTime-Bandwidth ProductFor the angular frequency and different definitions of the widths:TBPrms 0.5TBPeff 3.14 TBPHW1/e 1TBPFWHM 2.76 Divide by 2p for the cyclical frequency Dtrms Dnrms,etc.Numerical example:A transform-limited pulse:A Gaussian-intensity pulse with constant phase and minimal TBP.Notice that this definition yields an uncertainty product of p,not 2p;this is because weve used the intensity and spectrum here,not the fields.Time-Bandwidth ProductFor the angular frequency and different definitions of the widths:TBPrms=6.09TBPeff=4.02TBPHW1/e=0.82TBPFWHM=2.57 Divide by 2p for Dtrms Dnrms,etc.Numerical example:A variable-phase,variable-intensity pulse with a fairly small TBP.Time-Bandwidth ProductFor the angular frequency and different definitions of the widths:TBPrms=32.9TBPeff=10.7TBPHW1/e=35.2TBPFWHM=116 Divide by 2p for Dtrms Dnrms,etc.Numerical example:A variable-phase,variable-intensity pulse with a larger TBP.A linearly chirped pulse with no structure can also have a large time-bandwidth product.For the angular frequency and different definitions of the widths:TBPrms=5.65TBPeff=35.5TBPHW1/e=11.3TBPFWHM=31.3 Divide by 2p for Dtrms Dnrms,etc.Numerical example:A highly chirped,relatively long Gaussian-intensity pulse with a large TBP.The shortest pulse for a given spectrum has a constant spectral phase.We can write the pulse width in a way that illustrates the relative contributions to it by the spectrum and spectral phase.If B()=S(),then the temporal width,Dtrms,is given by:Notice that variations in the spectral phase can only increase the pulse width.Contribution due to variations in the spectrumContribution due to variations in the spectral phaseNote:this result assumes that the mean group delay has been subtracted from.That is,the pulse is cen-tered at tgr=0.The narrowest spectrum for a given intensity has a constant phase.We can also write the spectral width in a way that illustrates the relative contributions to it by the intensity and phase.If A(t)=I(t),then the spectral width,Drms,is given by:Notice that variations in the phase can only increase the spectral width.Contribution due to variations in the intensityContribution due to variations in the phase