From: pcdhSpamMeSenseless@electrooptical.net   
      
   On 12/07/2015 12:24 PM, Torsten Bronger wrote:   
   > Hallöchen!   
   >   
   > Phil Hobbs writes:   
   >   
   >> On 12/07/2015 04:13 AM, Torsten Bronger wrote:   
   >>   
   >>> Phil Hobbs writes:   
   >>>   
   >>>> [...]   
   >>>>   
   >>>> If I understand correctly, it's because a perfectly-centered   
   >>>> lens system is exactly symmetrical, so there are no even-order   
   >>>> terms.   
   >>>   
   >>> It is symmetrical, but in the mathematical description, this is   
   >>> not used.  Instead of using negative values for, say, theta, you   
   >>> just rotate the image around the optical axis by pi.   
   >>   
   >> Except if you put in even orders, your wavefront is   
   >> nondifferentiable at zero, which is unphysical.   
   >   
   > You mean a function like sgn(x) * x**2?  Yes, the second derivative   
   > is not defined at zero but why is this a physical problem?   
      
   Because the optical field obeys a set of second-order differential   
   equations.   
      
   > Why should a lens with a mapping like   
   >   
   > r(theta) = f * (theta + theta**2)   
   >   
   > be impossible to build?   
   >   
   > Tschö, Torsten.   
   >   
      
   No, it wouldn't be impossible to build, but it wouldn't be axisymmetric.   
    Real lenses are never perfectly symmetrical, and so have aberrations of   
   all orders.  The problem is that you have to measure them very carefully   
   to find all the coefficients, and the results would be so specific as to   
   be nearly useless in a theoretical paper.   
      
   Cheers   
      
   Phil Hobbs   
      
   --   
   Dr Philip C D Hobbs   
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