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1、Ultrashort Laser Pulses I,Description of pulsesIntensity and phaseThe instantaneous frequency and group delayZeroth and first-order phaseThe linearly chirped Gaussian pulse,Prof. Rick TrebinoGeorgia Techwww.frog.gatech.edu,An ultrashort laser pulse has an intensity and phase vs. time.,Neglecting the
2、 spatial dependence for now, the pulse electric field is given by:,Intensity,Phase,Carrierfrequency,A sharply peaked function for the intensity yields an ultrashort pulse.The phase tells us the color evolution of the pulse in time.,The real and complex pulse amplitudes,Removing the 1/2, the c.c., an
3、d the exponential factor with the carrier frequency yields the complex amplitude, E(t), of the pulse:,This removes the rapidly varying part of the pulse electric field and yields a complex quantity, which is actually easier to calculate with.,is often called the real amplitude, A(t), of the pulse.,E
4、lectric field E (t),Time fs,The Gaussian pulse,where tHW1/e is the field half-width-half-maximum, and tFWHM is the intensity full-width-half-maximum.The intensity is:,For almost all calculations, a good first approximation for any ultrashort pulse is the Gaussian pulse (with zero phase).,Intensity v
5、s. amplitude,The intensity of a Gaussian pulse is 2 shorter than its real amplitude. This factor varies from pulse shape to pulse shape.,Its easy to go back and forth between the electric field and the intensity and phase:The intensity:,Calculating the intensity and the phase,f(t) = - ImlnE(t),The p
6、hase:,Equivalently,I(t) = |E(t)|2,Also, well stop writing “proportional to” in these expressions and take E, E, I, and S to be the field, intensity, and spectrum dimensionless shapes vs. time.,The Fourier Transform,To think about ultrashort laser pulses, the Fourier Transform is essential.,We always
7、 perform Fourier transforms on the real or complex pulse electric field, and not the intensity, unless otherwise specified.,The frequency-domain electric field,The frequency-domain equivalents of the intensity and phase are the spectrum and spectral phase.Fourier-transforming the pulse electric fiel
8、d:,yields:,The frequency-domain electric field has positive- and negative-frequency components.,Note that f and j are different!,Note that these two terms are not complex conjugates of each other because the FT integral is the same for each!,The complex frequency-domain pulse field,Since the negativ
9、e-frequency component contains the same infor-mation as the positive-frequency component, we usually neglect it. We also center the pulse on its actual frequency, not zero. So the most commonly used complex frequency-domain pulse field is:,Thus, the frequency-domain electric field also has an intens
10、ity and phase.S is the spectrum, and j is the spectral phase.,The spectrum with and without the carrier frequency,Fourier transforming E (t) and E(t) yields different functions.,The spectrum and spectral phase,The spectrum and spectral phase are obtained from the frequency-domain field the same way
11、the intensity and phase are from the time-domain electric field.,or,Intensity and phase of a Gaussian,The Gaussian is real, so its phase is zero.,Time domain:Frequency domain:,So the spectral phase is zero, too.,A Gaussian transformsto a Gaussian,Intensity and Phase,Spectrum and Spectral Phase,The s
12、pectral phase of a time-shifted pulse,Recall the Shift Theorem:,So a time-shift simply adds some linear spectral phase to the pulse!,Time-shifted Gaussian pulse (with a flat phase):,What is the spectral phase?,The spectral phase is the phase of each frequency in the wave-form.,0,All of these frequen
13、cies have zero phase. So this pulse has: j(w) = 0Note that this wave-form sees constructive interference, and hence peaks, at t = 0.And it has cancellation everywhere else.,w1w2w3w4w5w6,Now try a linear spectral phase: j(w) = aw.,By the Shift Theorem, a linear spectral phase is just a delay in time.
14、 And this is what occurs!,t,j(w1) = 0,j(w2) = 0.2 p,j(w3) = 0.4 p,j(w4) = 0.6 p,j(w5) = 0.8 p,j(w6) = p,To transform the spectrum, note that the energy is the same, whether we integrate the spectrum over frequency or wavelength:,Transforming between wavelength and frequency,The spectrum and spectral
15、 phase vs. frequency differ from the spectrum and spectral phase vs. wavelength.,Changing variables:,The spectral phase is easily transformed:,The spectrum and spectral phase vs. wavelength and frequency,Example: A Gaussian spectrum with a linear spectral phase vs. frequency,Note the different shape
16、s of the spectrum and spectral phase when plotted vs. wavelength and frequency.,Bandwidth in various units,In frequency, by the Uncertainty Principle, a 1-ps pulse has bandwidth: dn = 1/2 THz,So d(1/l) = (0.5 1012 /s) / (3 1010 cm/s) or: d(1/l) = 17 cm-1,In wavelength:,Assuming an 800-nm wavelength:
17、,using dn dt ,or: dl = 1 nm,In wave numbers (cm-1), we can write:,The temporal phase, (t), contains frequency-vs.-time information.The pulse instantaneous angular frequency, inst(t), is defined as:,The Instantaneous frequency,This is easy to see. At some time, t, consider the total phase of the wave
18、. Call this quantity 0:Exactly one period, T, later, the total phase will (by definition) increase to 0 + 2p: where (t+T) is the slowly varying phase at the time, t+T. Subtracting these two equations:,Dividing by T and recognizing that 2/T is a frequency, call it inst(t):inst(t) = 2/T = 0 (t+T) (t)
19、/ TBut T is small, so (t+T) (t) /T is the derivative, d /dt. So were done! Usually, however, well think in terms of the instantaneous frequency, inst(t), so well need to divide by 2: inst(t) = 0 (d/dt) / 2While the instantaneous frequency isnt always a rigorous quantity, its fine for ultrashort puls
20、es, which have broad bandwidths.,Instantaneous frequency (contd),Group delay,While the temporal phase contains frequency-vs.-time information, the spectral phase contains time-vs.-frequency information. So we can define the group delay vs. frequency, tgr(w), given by:tgr(w) = d / dA similar derivati
21、on to that for the instantaneous frequency can show that this definition is reasonable. Also, well typically use this result, which is a real time (the rads cancel out), and never d/d, which isnt. Always remember that tgr(w) is not the inverse of inst(t).,Phase wrapping and unwrapping,Technically, t
22、he phase ranges from p to p. But it often helps to “unwrap” it. This involves adding or subtracting 2p whenever theres a 2p phase jump.Example: a pulse with quadratic phase,Wrapped phase,The main reason for unwrapping the phase is aesthetics.,Note the scale!,Phase-blanking,When the intensity is zero
23、, the phase is meaningless. When the intensity is nearly zero, the phase is nearly meaningless.Phase-blanking involves simply not plotting the phase when the intensity is close to zero.,The only problem with phase-blanking is that you have to decide the intensity level below which the phase is meani
24、ngless.,-f(wi),Re,Im,E(wi),S(wi),Phase Taylor Series expansions,We can write a Taylor series for the phase, f(t), about the time t = 0:where,where only the first few terms are typically required to describe well-behaved pulses. Of course, well consider badly behaved pulses, which have higher-order t
25、erms in (t).Expanding the phase in time is not common because its hard to measure the intensity vs. time, so wed have to expand it, too.,is related to the instantaneous frequency.,Frequency-domain phase expansion,Its more common to write a Taylor series for ():,As in the time domain, only the first
26、few terms are typically required to describe well-behaved pulses. Of course, well consider badly behaved pulses, which have higher-order terms in ().,where,is the group delay!,is called the group-delay dispersion.,Zeroth-order phase: the absolute phase,The absolute phase is the same in both the time
27、 and frequency domains.An absolute phase of p/2 will turn a cosine carrier wave into a sine.Its usually irrelevant, unless the pulse is only a cycle or so long.,Notice that the two four-cycle pulses look alike, but the three single-cycle pulses are all quite different.,First-order phase in frequency
28、: a shift in time,By the Fourier-transform Shift Theorem,Note that j1 does not affect the instantaneous frequency, but the group delay = j1.,First-order phase in time: a frequency shift,By the Inverse-Fourier-transform Shift Theorem,Time domain,Frequency domain,Note that 1 does not affect the group
29、delay, but it does affect the instantaneous frequency = 1.,Second-order phase: the linearly chirped pulse,A pulse can have a frequency that varies in time.,This pulse increases its frequency linearly in time (from red to blue).In analogy to bird sounds, this pulse is called a chirped pulse.,The Line
30、arly Chirped Gaussian Pulse,We can write a linearly chirped Gaussian pulse mathematically as:,Chirp,Gaussianamplitude,Carrierwave,Note that for b 0, when t 0, the terms add, and the phase changes more rapidly(so the frequency is larger).,The instantaneous frequencyvs. time for a chirped pulse,A chir
31、ped pulse has:where:The instantaneous frequency is: which is: So the frequency increases linearly with time.,The Negatively Chirped Pulse,We have been considering a pulse whose frequency increaseslinearly with time: a positively chirped pulse.One can also have a negatively chirped (Gaussian) pulse, whose instantaneous frequency decreases with time. We simply allow b to be negativein the expression for the pulse:And the instantaneous frequency will decrease with time:,
