Lens Focusing by Iterating the Optical Wave 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 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 the Maxwell equation the packet is propagated using the equation d2E/dt2=1/epsilon*(d/dx2+d/dy2)E where epsilon is the dielectric constant of the lens which is for this case the same as the index of refraction squared. This equation assumes the 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. 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 acjustable 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 Just be aware that when you see reflections from the bounds of the gridded region that the plot data is no longer valid.

The color bar at the left shows the learner the value of psi*psi at any (x,y). For the Schrodinger equation, after experimenting with the animation, it becomes obvious that a concave lens gives better focusing than a convex one. For the Maxwell equation, the convex lens focuses the transmitted light as expected (see the rays colored white). The left surface or the concave lens acts as as a partial mirror and focuses the light at a distance of 1/2 of the radius of curvature of the left surface. For both equations the sharp disconituity at the lens boundary 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 see the change in curvature of the wave fronts inside the lens, it is best to display the real part.