Ultrashort Laser Pulses II,More second-order phase Higher-order spectral phase distortions Relative importance of spectrum and spectral phase Pulse and spectral widths Time-bandwidth product,Prof. Rick Trebino Georgia Tech www.frog.gatech.edu,Frequency-domain phase expansion,Recall 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 ().,where,is the group delay.,is called the “group-delay dispersion.”,The Fourier transform of a chirped pulse,Writing a linearly chirped Gaussian pulse as: or: Fourier-Transforming yields: Rationalizing the denominator and separating the real and imag parts:,A Gaussian with a complex width!,A chirped Gaussian pulse Fourier-Transforms to itself!,where,neglecting 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 for a chirped pulse,The 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 chirp,Numerical 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 chirp,Numerical 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 a Gaussian pulse whose fre- quency varies quadratically with 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 pulses,3rd-order spectral phase: quadratic chirp,Longer 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(w),tg(w),j(w),Spectrum and spectral phase,400 500 600 700,Because were plotting vs. wavelength (not frequency), theres a minus sign in the group delay, so the plot is correct.,3rd-order spectral phase: quadratic chirp,Numerical 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 phase,Another 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 phase,Numerical 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 phase,Numerical 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 phase,Numerical 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 phase,Numerical 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 phase,Photographs 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 propagation,What 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 = w/c:,Separating out the spectrum and spectral phase:,Rewriting this expression:,The pulse length,There 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. Di