bea4ca32   
   From: jd.deschenes@ulaval.ca   
      
   On 24/09/2011 2:36 PM, alex wrote:   
   > I think I understand spectral correlation and how that results in a   
   > transform limited pulse in the time domain for 100% correlation   
   > between spectral components of the field and a stationary field if   
   > there is no correlation between components of the field.  And if the   
   > spectral range of the correlation is less than the bandwidth we have a   
   > pulse, but with a longer duration.   
   >   
   > I am struggling in understand the anology to spatial correlation.   
   > If I take the FT of the field I get the angular spectrum A(k) of the   
   > field.  Now if each angular component is correlated then in the space   
   > domain I must get the field confined to the smallest region of space   
   > deltaX defined as 1/deltak.  If the bandwidth of the angular spectrum   
   > is infinte then I would have a point source, and field from this   
   > source is pefectly spatially coherent.  If there is no correlation   
   > does this mean that the field is spread out infinitely in space?   
   > seems wierd, but analagous to stationary field from above?  It's odd,   
   > because it seem that we have created a plane wave, which I always   
   > thought was 100% spatially coherent like a point source.   
   >   
   > I cannot work out where I have gone wrong... Any ideas?   
   >   
   > Thanks   
   > ALex   
      
   First, let's go back to the time domain analogy to clear up one slight   
   misunderstanding you seem to have.  The time domain is analogous to the   
   spatial domain and the frequency domain is analogous to the angular domain.   
      
   In the time domain, there are two ways to make what you call a   
   'stationnary' field.  Either you have a single frequency in the spectrum   
   (monochromatic wave), then the field is a single cosine and spans from   
   -infinity to +infinity in the time domain.  The other way is if you have   
   a broadband spectrum, but with completely random spectral phase.  In   
   this case, the time domain signal is noise which also spans from   
   -infinity to +infinity, with constant average amplitude.  This is what   
   you call a 'stationary' field.  Note that the instantenous field at   
   every instant has a random amplitude and can go to 0 momentarily.   
      
   Back to the spatial and angular domains.  When you say that the field   
   from a point source is perfectly coherent, you are wrong.  You are   
   mixing up the correlation in the angular spectrum (up to that point you   
   were right) with the correlation in the spatial domain.  If you   
   propagate the field from a point source to the far field, then yes, the   
   field is now coherent, precisely because it is now an image of the   
   angular spectrum.   
      
   If there is no correlation and you have a wideband angular spectrum,   
   that means that the field in the spatial domain is wideband noise,   
   occupying -infinity to +infinity, just like in the time domain analogy.   
     A plane wave has a completely different angular spectrum: it is a   
   single dirac delta at the corresponding propagation angle; it is not   
   wideband.   
      
   JD   
      
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