XPost: sci.math   
   From: pcdhSpamMeSenseless@electrooptical.net   
      
   On 01/10/2017 09:46 AM, Phil Hobbs wrote:   
   > On 01/10/2017 08:36 AM, Anton Shepelev wrote:   
   >> Hello, all   
   >>   
   >> Is there a symbolic method to calculate the intensi-   
   >> ty distribution on a flat screen from a  flat  wave-   
   >> front  after it reflected from a curved mirror whose   
   >> surface is described by an equation?   
   >   
   > I don't think so, in general.  Even in the scalar field approximation,   
   > you have to deal with multiple reflections to have full generality.   
   >   
   >>   
   >> The normal of the wavefront, and the relative  posi-   
   >> tion of screen and mirror are known.   
   >>   
   >> I know of two ways to solve it:   
   >>   
   >>   1.  numerically,  by  tracing  the  rays reflected   
   >>       from tiny sections of the mirror, and   
   >>   
   >>   2.  approximately, via the first N moments of  the   
   >>       distribution,  which  can be found by integra-   
   >>       tion over the mirror surface.   
   >   
   > Well, the usual approach in physical optics is to use the scalar   
   > approximation and apply either the Huyghens propagator for paraxial   
   > fields or the Kirchhoff or Rayleigh-Sommerfeld propagators for more   
   > general situations.  All of these fail to model the boundary   
   > contribution to the diffracted light, but they're okay for normal use.   
   >   
   > A more accurate method in general is PTD (physical theory of   
   > diffraction).  See the papers and books of Pyotr Y Ufimtsev and Joseph   
   > B. Keller.  It handles multiple scattering and models the boundary waves   
   > much better, but it's far from exact.   
      
   I should add that discussions of diffraction in the physical optics   
   literature are mostly erroneous.  The propagators come from the Green's   
   function for the wave equation in a half space with a planar boundary,   
   and folks blithely apply them to curved surfaces with no attention even   
   to the Jacobian (analogous to the area of a ray bundle).   
      
   Aberration theory (where the moments idea comes from) is an asymptotic   
   theory of the propagation of optical phase in the limit of large Fresnel   
   number.  It completely ignores amplitude and polarization besides.   
      
   An amplitude-only Fresnel zone plate focuses light reasonably well, but   
   aberration theory predicts that the light propagates undeviated.   
      
   Cheers   
      
   Phil Hobbs   
      
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