From: jd.deschenes@ulaval.ca   
      
   On 04/12/2010 11:48 AM, Phil Hobbs wrote:   
   > alex wrote:   
   >> Dear Dr Hobbs,   
   >>   
   >> Thank you for your indepth response, I have commented (asked more   
   >> q's :( :) ) within your text   
   >>   
   >> On Dec 4, 1:14 pm, Phil Hobbs   
   >> wrote:   
   >>> alex wrote:   
   >>>> I wish to know how I can construct a train of pulses mathematically   
   >>>   
   >>>> Do I consider it, in the time domain, as a convolution of a single   
   >>>> pulse with an array of deltas? Like a a grating, but in the time   
   >>>> domain. However, this means I must already decide the shape of the   
   >>>> single pulse to be shifted along time axis.... Presumably I must then   
   >>>> have this experimental?..   
   >>>   
   >>>> Or do I construct it from the inverse fourer transform of a bandwidth   
   >>>> limited(by gain bandwidth) comb of longitudinal modes (lorentzian   
   >>>> function)? Will this give me some tiny residue pulses (time domain)   
   >>>> either side of the main pulses similar to the the far field intentsity   
   >>>> of a grating,...   
   >>>   
   >>>> Thanks   
   >>>> Alex   
   >>>   
   >>> Taking a given single pulse waveform, e.g. g(t)=sech(t/tau), you can   
   >>> make it into a periodic pulse train in any number of ways. Convolving   
   >>> with the sha (comb) function III(t/T) (a train of unit-strength   
   >>> delta-functions with period T seconds) is mathematically convenient,   
   >>> because it modifies the transform in a well-defined way, i.e. sampling   
   >>> it at frequency spacing 1/T Hz.   
   >>   
   >>   
   >> So that the spectrum of the pulse train has the same bandwidth of that   
   >> of the   
   >> single pulse? However, it is missing some freq's due to it actually   
   >> sampling   
   >> the single pulse spectrum?   
   >   
   > Right.   
   >>   
   >>   
   >>   
   >>> In the frequency domain, the sampled transform of the convolution is   
   >>>   
   >>> H(f) = G(f) III(fT)   
   >>>   
   >>> which is very convenient if you've chosen g(t) sensibly. (III(t) is its   
   >>> own transform.)   
   >>   
   >> If I now limit the range of the comb and inverse transform back to the   
   >> time domain   
   >> does this change the width of the single pulse or the time between   
   >> pulses?   
   >> I think the latter. How can I reduce the width of the simgle pulse?   
   >>   
   > What do you mean by the 'range of the comb'--the rep rate? Or do you   
   > mean turning the comb on and off? By the sampling theorem, if the   
   > spectrum has compact support with a bandwidth less than 1/T, you can   
   > reconstruct the spectrum exactly by convolving it with a sinc function,   
   > which in the time domain corresponds to selecting the single pulse at   
   > t=0 by multiplying the pulse train by a rectangle function.   
   >   
   > Reducing the width of a laser pulse used to be really hard, but has got   
   > gradually easier due to a lot of smart people working on it. First you   
   > have to widen the spectrum a bit--often done via self-phase modulation   
   > in a fibre, and then applying an all-pass filter to make the phases of   
   > all the frequency components line up right, which is often done with a   
   > pair of diffraction gratings, which gives an optical path length that   
   > depends on wavelength in a nice adjustable way. If Prof. Siegman is   
   > following this thread, he'd be able to give more info. (I took his   
   > lasers course back in 1985-86, and he narrowly escaped being on my Ph.D.   
   > committee.)   
   >   
   >>>   
   >>> In the time domain, it causes an infinite number of shifted copies of   
   >>> your pulse function to be added together, which may not be quite what   
   >>> you want if g(t) doesn't go to zero rapidly enough for large t.   
   >>>   
   >>> There are various other sensible methods, e.g. forming h(t) = g((t mod   
   >>> T)/tau) or h(t) = g(sin(2*pi*t/T). All of these amount basically to   
   >>> modifying g(t) to give it compact support and then convolving the   
   >>> modified g(t) by the comb function. Most of them don't have the nice   
   >>> transform-preserving property, which may be a problem.   
   >>>   
   >>> Applying a filter to a comb is exactly equivalent to the first   
   >>> method--filtering is a convolution operation. Functions that go   
   >>> exponentially to zero in the wings, such as Gaussians and hyperbolic   
   >>> secants, sample extremely well--their high derivatives to exponentially   
   >>> to zero too, so the accuracy of the sampled spectrum improves faster   
   >>> than any power of 1/T. That's also why integrating them using the   
   >>> trapezoidal rule works anomalously well.   
   >>>   
   >>> If you don't actually know the pulse shape, pick something convenient   
   >>> like a Gaussian or sech, both of which are their own Fourier transforms.   
   >>>   
   >>> If you get a copy of Bracewell and read it like a novel, a lot of this   
   >>> stuff will become much clearer very fast.   
   >>   
   >> Which of Bracewells' books do you recommend, Fourier Analysis and   
   >> Imaging(2004) or The Fourier Transform& Its Applications(1999)   
   >>   
   >   
   > The Fourier Transform and Its Applications. I took Bracewell's class   
   > too, back in 1983-84. He really taught us how to think in Fourier space,   
   > which is one of the most useful skills one can possess, I think. His   
   > book does the same thing--though it isn't as entertaining as he was!   
   >   
   >>   
   >> Thank you once again. Have a good Festive Season.   
   >> Alex   
   >   
   > Merry Christmas to you too!   
   >   
   > Cheers   
   >   
   > Phil Hobbs   
   >   
   >   
      
   Does Prof. Siegman actually read this newsgroup or was that just a joke?   
      
   JD   
      
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