Transparent Plate Reflectivity and Transmissivity by Iterating the Optical Wave (Maxwell) Equation

This animation shows the two dimensional (2D) propagation of s gaussian envelope wave packets as governed by the Maxwell wave equation or theSchrodinger 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/sx)^2-(y/sy)^2+i*kx*x) where sx and sy are the widths of the envelopes and kx 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 (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 and 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 d2E/dt2=1/epsilon*(d2/dx2+d2/dy2)E 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*E 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 Integral(E*E) on the Sink side to the one half of Source side Integral(E*E). (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

The ray diagram for transmission and reflection is shown along with a circle that represents a moving photon. The sharp disconituity at the plate boundaries will result in multiple reflections inside as one might expect. The total desktop computer time for completion of the focus is usually several minutes. Just start a calculation and let the computer do the work. You have the option of Start Movie to see the results at any time. The learner has the option to display either the magnitude of the wave packet or its real part and these can be changed at any time. To verify that the reflected and refracted wavefront angles agree with Snell's laws it is best to display the real part.