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This animation shows the two dimensional (2D) propagation of a gaussian envelope wave packet as governed by the Maxwell wave equation or the Schrodinger wave equation. The initial wave packet has learner adjustable gaussian envelopes in both x and y directions. I have chosen to make the wave complex with wave vector kx in the x direction: Therefore the algebraic form of the initial packet is

`f(x,y)=exp(-(x/s_x)^2-(y/s_y)^2+ik_(x)x)`

where sx and sy are the widths of the envelopes and kx is the wave vector. `f(x,y)` will represent either the initial electric field packet, `E(x,y)` or the initial quantum wave packet, `psi(x,y)`. For the Maxwell equation the packet, `E(x,y)`, is propagated using the equation`(d^2E)/(dt^2)=1/(epsilon(x,y))((d^2E)/dx^2+(d^2E)/(dy^2))`

where `epsilon` 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 wave function, `psi`, the packet is propagated using the equation`i(dpsi)/dt=((d^2)/(dx^2)+(d^2)/(dy^2)+V(x,y))psi`

where i is sqrt(-1) and V(x,y) is the potential and I have set the reduced Planck constant, ℏ, and the particle mass equal to one. The speed of the Schrodinger equation packet is `k=sqrt(k_x^2+k_y^2)` and its energy is `(k^2)/2`. For the Maxwell equation the lens is a shaped epsilon distribution which has either double concave or double convex boundaries. For the Schrodinger equation the lens is a shaped potential which has either double concave or double convex boundaries. The wave packet and the lens are color coded. The color bar at the left shows the learner the value of E*E at any (x,y). The color of the lens is associated with the potential energy or epslilon within its boundaries. The Schrodinger wave packet kinetic energy is kx*kx/2 and is adjustable using the kx slider and its lens potential is adjustable using the Lens Potential Slider. For the Schrodinger equation the lens potential must be less than the kinetic energy if the lens is to transmit the wave packet. Otherwise the lens becomes a mirror. 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 (Iteration Time Step and kx) which, if made large enough,
will easily run the diffracting wave packet beyond the gridded width and height. Effectively, beyond the gridded width and height, the potential is infinite and
results in strong reflections. Just be aware that when you see reflections from the bounds of the gridded region that
the plot data is no longer valid.**