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1、Ultrashort Laser Pulses I,Description of pulses Intensity and phase The instantaneous frequency and group delay Zeroth and first-order phase The linearly chirped Gaussian pulse,Prof. Rick Trebino Georgia Tech www.frog.gatech.edu,An ultrashort laser pulse has an intensity and phase vs. time.,Neglecti
2、ng the spatial dependence for now, the pulse electric field is given by:,Intensity,Phase,Carrier frequency,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
3、c.c., and 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
4、pulse.,Electric 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).,In
5、tensity vs. 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) = - Imln
6、E(t),The phase:,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.
7、,We always 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 el
8、ectric field:,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
9、the negative-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 ha
10、s an intensity 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 t
11、he same way 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 transforms to a Gaussian,Intensity and Phase,Spectrum and Spectr
12、al Phase,The spectral 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
13、 these frequencies have zero phase. So this pulse has: j(w) = 0 Note that this wave-form sees constructive interference, and hence peaks, at t = 0. And it has cancellation everywhere else.,w1 w2 w3 w4 w5 w6,Now try a linear spectral phase: j(w) = aw.,By the Shift Theorem, a linear spectral phase is
14、just a delay in time. 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
15、spectrum and spectral 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,Not
16、e the different shapes 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:,using dn dt ,or: dl = 1 nm,In wave numbers (cm-1), we can write:,The temporal phase, (t), contains frequency-vs.
