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This animation shows the two dimensional (2D) propagation of s gaussian envelope wave packets 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 initial wave vector kx in the +x direction: Therefore the algebraic form of the initial packet is

`E(x,y)=exp(-(x/s_x)^2-(y/s_y)^2+ikx)`

where `s_x` and `s_y` are the widths of the envelopes and `k` is the wave vector. For this animation I have implicity assumed that the polarization of the electric field is perpendicular to the plane of incidence (thus along the z axis here) which is often labeled "s" polarization or "TE" polarization. The other case is with the electric field is in the y direction called "p" or "TM" but, for that case, the reflectivity at what is called the Brewster angle, is zero and that complicates the calulation. TE polarization means that the electric field vibrates in and out of the screen and the wave packet could be generated by charges vibrating in and out of the screen (z axis). If you think of charges vibrating along the z axis at fixed x position as the generators of the packet it is easy to understand that any wave would propagate in both the +`x` and - `x` direction and that is what happens: The original wave packet splits in two (as in cell mitosis) going in opposite directions along the x axis. By necessity, to stay with 2D, I have chosen to show the electric field variation in the plane of the screen. The packet is propagated using the Maxwell equation`(d^2E)/dt^2=1/epsilon((d^2E)/dx^2+(d^2E)/dy^2)`

where epsilon is the dielectric constant of the plate which is, for this case, the same as the index of refraction squared. This equation assumes the speed of light to be 1 when `epsilon=1` as in a vacuum. The plate is an `epsilon` distribution of a tilted plate with parallel front and back surfaces. This problem is done in many physics texts by boundary condition matching and the results are Snell's propagation angles and Fresnel's reflection and transmission coefficients for plane plate surfaces. However note that the present method is much more powerful because it accommodates any wave packet (or light pulse) shape and any transparent shape with any internal dielectric constant distribution. The wave packet and the plate are color coded. The color bar at the left shows the learner the value of `E^2` at any (x,y). The color of the plate is associated with the dielectric constant epslilon within its boundaries. When a reflection from the plate collides with any of the outer boundaries it is the same as hitting an infinite dielectric constant and 100% reflection occurs. In the upper left corner, I have provided both the Fresnel two surface transmissivity and the ratio of the `int(E^2dxdy)` on the Sink side to the one half of initial Source side integral. (Remember the packet undergoes mitosis!) The latter ratio agrees within about 1% with the Fresnel transmissivity. 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 and the plot data is no longer valid.
Just be aware, if the results look very odd then you need to adjust for less aggressive parameters**