Diffraction Interference by Iterating the Optical (Maxwell) or Quantum (Schrodinger) Wave Equation

This animation shows the two dimensional (2D) propagation of a gaussian envelope wave packet as governed by the Maxwell or Schrodinger equation. In this case the grid with two slits is represented by either a split potential or a split dielectric constant distribution so that there are two apertures to let some of the original packet through. The initial wave packet has gaussian rolloff on its boundaries to avoid the diffraction due to sharp edges. Therefore the algebraic form of the initial packet is E(x,y)=exp(-(x/sx)^2-(y/sy)^2+i*kx*x) where sx and sy are the widths of the envelopes and kx is the wave vector. The potentials and dielectric constants that make up the two slit grid also has gaussian rolloff to avoid the diffraction that sharp edges would cause. The animation shows how the diffracting wave packet progresses through the near field and results in the expected interference pattern in the far field. The angles, theta, of the maxima of the interference are controlled by the well known equation
d*sin(theta)=m*lambda
where d is the slit separation, m is an integer (e.g. m={-2, -1, 0, 1, 2})and theta is the angle from the horizontal (white rays). The white plot of the intensity of the diffracted patterns lines up on the right side of the graphic line up with the white rays as expected. For the Maxwell equation the packet is propagated using the equation d2E/dt2=1/epsilonRel(x,y)*(d/dx2+d/dy2)E where epsilonRel is the relative dielectric constant of the lens which is for this case the same as the index of refraction squared. This equation assumes the vacuum speed of light to be 1. For the Schrodinger equation the packet is propagated using the equation i*dE/dt=(d/dx2+d/dy2+V(x,y))E where i is sqrt(-1) and V(x,y) is the potential. The speed of the Schrodinger equation packet is kx and its energy is kx*kx/2. When the Maxwell propagation starts, the packet splits into 2 identical packets and one propagates left and the other propagates right. One can understand this by realizing that the initial packet stems from just an array of oscillating charges that generate it. Of course, the radiation from these charges has no left or right directionality so the split is expected. The rule of thumb diffraction solution (Green trace at the right hand side of the graphic) is valid only in the far field of the slits. Many parameters are variable in this animation and they are pretty well defined by the titles of the sliders.

I have deliberately allowed the learner to adjust the variables beyond the ranges where valid propagation results are obtained. Important parameters for this are the pair (Gridding Half Height and kx) which, if made small enough, will easily run the diffracting wave packet beyond the gridded vertical half widths. Effectively, beyond the gridded width and height, the potential is infinite and results in strong reflections. Just be aware that when you see significant reflections from the upper and lower bounds to the right of the gridded region that the plot data is not as valid.

It will be noticed that the Schrodinger propagated packet diverges (diffracts or diffuses) horizontally so that the agreement between the white trace and green trace at the right hand side is not mearly as good as that of the Maxwell propagated packet. The total desktop computer time for completion of the diffraction is several minutes. Before or after the diffraction is complete a movie that runs much faster can be started.