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Recursive issue

  • To: mathgroup at
  • Subject: [mg65404] Recursive issue
  • From: Clifford Martin <camartin at>
  • Date: Thu, 30 Mar 2006 05:30:02 -0500 (EST)
  • Sender: owner-wri-mathgroup at

  Recently I've been working on diffraction propagation problems inside passive  laser cavities.  The standard approach was pioneered by Fox and Li (Bell  Systems Journal 1961) which is a numerical Fresnel propagator that the solution  is fed back into the kernel for each pass. Numerically (which is how Fox and Li  did it), the Mathematica solution is:
  The solution is for a one dimensional infinite strip mirror in 1 D.  The  inital parameters and the kernel are given by:
      l = 0.6328
      w = 2*25*l
      z = 100*l
      Kmat = Table[(l/5)*(Exp[I*(((2*Pi)/l)*z)]/
            Sqrt[I*l*z])*Exp[((-I)*((2*Pi)/l)*(x  - y)^2)/
             (2*z)], {x, -400*(l/5), 400*(l/5), l/5}, 
          {y, -400*(l/5), 400*(l/5), l/5}];
      Do[f[n] =. , {n, 0,  20}]  ( * This sets the recursive terms  to zero to start another run *)
      f[0] =  Table[If[Abs[x] < 25*l,  1, 0], 
          {x, -400*(l/5), 400*(l/5), l/5}];     (* This defines the initial intensity we set it to 1 *)
      f[n_] := f[n] = Kmat  . (f[n - 1]*f[0]); 
       (* This is the recursive equation where we  use f[0] to normalize the recurring propagation at each mirror *)
      (* The next bit is  various cases from f[1] to f[200] which show that the solution conveges to a  ?normal? mode solution after enough passes *)
      ListPlot[Abs[f[0]],  PlotRange -> All, 
         PlotJoined -> True]; 
      ListPlot[Abs[f[1]],  PlotRange -> All, 
         PlotJoined -> True]; 
      ListPlot[Abs[f[2]],  PlotRange -> All, 
         PlotJoined -> True]; 
      ListPlot[Abs[f[5]],  PlotRange -> All, 
         PlotJoined -> True]; 
      ListPlot[Abs[f[10]],  PlotRange -> All, 
         PlotJoined -> True]; 
      ListPlot[Abs[f[20]],  PlotRange -> All, 
         PlotJoined -> True];
      ListPlot[Abs[f[200]],  PlotRange->All, PlotJoined->True];
      This works well and  as expected. Now for the problem (Sorry for the long set up!)
      This is Mathematica,  I thought, so I can write the integral propator explicity without worrying  about numerics .  So using the same  values of lamda, w and z, I write the initial intensity and the iterative kernel  as an integral equation and recursively:
      f[0, x_] = 1
      f[n_, x_] :=  (Exp[I*(((2*Pi)/l)*z)]/Sqrt[I*l*z])*
         Integrate[Exp[((-I)*((2*Pi)/l)*(q - x)^2)/(2*z)]*
           f[n - 1, q], {q, -w/2, w/2}]
      If you look at the  first pass the solution is correct exactly matching the numerical solution but  subsequent passes give the same solution as in pass 1 multiplied by a constant.  In other words, the recursion is not working to feed the solution back into the  kernel for the next step.  When you look  at the solution explicity (algebraicly ) for the first step it?s a bunch of Erf  functions with some constants. When you look at the second pass it?s the same  set of Erf functions multiplied by a larger constant.  It looks like the recursion is working outside in somehow. Is  there something I?m doing wrong in the explicit integral equation vs. the  numerical one?  Or is there a better way  to do the explicit recursive integral equation?
      It?s not that I  can?t solve this other ways. I?ve built a Fourier propagator and of course the  numerical Fox-Li solution. What I?m trying to understand is what is going wrong  with the explicit algebraic recursive solution.
      If any more detail  is needed I can write anyone off line and provide more details.

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