Refraction between Two Dielectric Solids Please click the link below for a description of the graphics. This animation shows the motion of the electric dipoles in a dielectric solid for an electromagnetic (em) wave that is polarized so that the electric field component is perpendicular to the plane of incidence (s polarized). The dipoles orient themselves so that the positive end points in the direction of the electric field thereby partially cancelling the electric field (hence the title dielectric). In general the dipoles do not experience a phase lag so that their phase is 180 degrees from that of the incident electric field. Since, in this animation, we have s polarization, the edge view of the solid shows only the end view of the dipoles so the viewer sees only dipole transitions from positive (red) to negative (black) as the animation proceeds. The top view shows the length changes of the dipoles as they oscillate in synchronism with the incident electric field of the em wave. Since this is refraction, we have chosen the incident medium to be lower index, n1, and the final medium a higher index of refraction, n2. As a result, the angle of the wavefronts relative to the horizonatal in the second medium is more shallow than that in the first medium. The equation for the polarization of the dipoles versus time, x, and y is p(t,x,y)=cos(w*t-kx*x-ky*y) where w is the radian frequency, kx is the horizontal component of the wave vector and ky is the vertical component of the wave vector. The refracted angle follows Snell's law where n1 sin(theta1)=n2 sin(theta2) and then kx=n w/c sin(theta), ky=n w/c cos(theta). The color of the dipole that the viewer sees transitions from black to red as the following function color(t,x,y)=black+red*(1+cos(phi(t,x,y)))/2 where black is the background color and red is the foreground color for the dipole and the cosine factor gradually increases or decreases the opacity of the red foreground color. The viewer should understand that the incident electric field is not entirely canceled by the dipoles in the solid but the action of the dipoles is a good way to show the refraction process in two media, In the second (bottom) medium, I have chosen to have entirely different lattice constants for the fixed dipoles, just to demonstrate that the this does not necessarily make a difference for the refraction angles. In this animation I have chosen not to show the reflected wave or to show how the amplitude of the refracted wave depends on the polarizability of the two media. It seems paradoxical that the incident wave, which propagates at speed c, suddenly is extinguished in the media and is replaced by a wave propagating at speed c/n due to the action of the dipoles in the media. This extinction and the refracted amplitude can actually be calculated without reference to Maxwell's equations as is done by the Ewald-Oseen extinction theorem (see link below).
Click here for Ewald-Oseen extinction theorem Click here Graphics Description theta1 psiV n2