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Tim Waters

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  1. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  2. %For the blog 'Brute Force Physics' by Tim Waters
  3. %filename:stationary.m
  4. %reference: Feynman Lectures 26-4
  5. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  6.  
  7. clear all; %clears all variables
  8. close all; %closes all figures
  9.  
  10. eps= .00001 %input('Enter the tick-size (epsilon) ');
  11. n1 = 1 %input('Enter the index of refraction in region containing A ');
  12. n2 = 1.33 %input('Enter the index of refraction in region containing B ');
  13. mirror_axis = 0:eps:1; %cut (0,1) into grid points
  14. c = mirror_axis'; %creates a column vector
  15. a = ones(length(c),1)*rand; %ones(rows,columns) returns a matrix of 1s
  16. b = ones(length(c),1)*rand; %rand gives a random scalar in (0,1)
  17.  
  18. %Compute the optical path length, opl
  19. d_ac = sqrt(a.^2 + c.^2);
  20. d_cb = sqrt((1-c).^2 + b.^2);
  21. opl_ac=n1*d_ac;
  22. opl_cb=n2*d_cb;
  23. opl = opl_ac + opl_cb;
  24.  
  25. opl_min = min(opl);
  26. i.html">i=1;
  27. while opl(i.html">i)~= opl_min
  28.     i.html">i = i.html">i+1;
  29. end
  30. % i now contains the location of opl_min in the vector opl
  31.  
  32. %**************************************************************************
  33. %numerical analysis
  34. figure(2)
  35. m = a(1)+b(1); %slope of straiht line path connecting a to -b
  36. i_ab=floor((a(1)/m)/eps); %vector index of straight line path
  37. x_ab = a(1)/m;
  38. plot(c,opl) %plots opl vs. x
  39. hold on
  40. %place a black * at the straight line path location:
  41. plot(x_ab,opl(i_ab),'k*')
  42. %label it
  43. text(x_ab,opl(i_ab),...
  44.      ' \leftarrow Straight line opl',...
  45.      'EdgeColor','black',...
  46.      'LineWidth',1,...
  47.      'LineStyle',':');
  48. plot(c(i.html">i),opl(i.html">i),'r*')
  49. text(c(i.html">i),opl(i.html">i),...
  50.      ' \leftarrowMinimum opl',...
  51.      'EdgeColor','red',...
  52.      'LineWidth',1,...
  53.      'LineStyle',':');
  54.  
  55.  %compute the 1st and 2nd derivatives around the two points
  56.  [dx_c,ddx_c]=variation(opl,i.html">i,eps); %for point c
  57.  [dx,ddx]=variation(opl,i_ab,eps);
  58.  title({'Numerical Analysis: 1st and 2nd Derivatives:';
  59.   sprintf('1st variation at point c = %0.5f ',dx_c);
  60.   sprintf('2nd variation at point c = %0.5g ',ddx_c);
  61.   sprintf('1st variation at straight line path = %0.5g ',dx);
  62.   sprintf('2nd variation at straight line path = %0.5g ',ddx)})
  63.  
  64. xlabel({sprintf('Straight line opl = %0.5g ',opl(i_ab));
  65.     sprintf('Minimum opl = %0.5g ',opl(i.html">i))});
  66. hold off
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