Diffraction of Polarized Light
Updated January 2016

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Scalar diffraction of one point source

Consider an "unpolarized" laser beam shining on a black screen that has a very small circular hole cut in the center. For the moment, we will ignore the radius of the hole, but say it is a small fraction of a wavelength of the incident light. (The effect of the hole is to distort the resulting field slightly, but it does not change the basic results presented here.) Down-beam of the first screen we place a second one, this time white, oriented perpendicular to the beam. This second screen can be placed at any z distance away from the first. Here is an animation of (a numerical approximation of) the intensity and phase of the laser beam at the second screen, after being diffracted by the first screen, at different z distances away from the first (x is to the right, and y is up):

The image at left shows the relative intensity on the screen at each z distance (scaled to a maximum of 1.0 on each screen). Although the overall intensity decreases away from the first screen, the distribution of intensity becomes more uniform across the screen. The image at right shows the phase of the beam at each point on the screen. In the plot, the phase goes from -pi (black) to +pi (white), and then wraps around back to -pi at each "edge". The edges are artifacts: the phase actually continues increasing across the edges, but the math code rescales the other values back into the range (-pi, +pi]. Surfaces of constant phase are hemispheres. As z increases, the screen intersects each hemisphere in smaller and smaller circles. Here is an animation of the intensity and phase in the z-y plane, with changing x (z is to the right, and y is up, for each constant x value):

And in the x-z plane, with changing y value:

Note that this is not a time simulation. The 3d field is "frozen" in time, and we are simply taking different planar cross-sections of it. If this were a time simulation, the phase surfaces would be expanding with time, rather than contracting with distance.

Scalar diffraction of two point sources

Now consider two small holes in the first screen, at equal distances on opposite sides of the center of the first screen. These form two light sources that are said to be "in phase" with one another. The following is an animation of the intensity and phase of the diffraction pattern projected on the second screen (x-y axes) at increasing z distances:

This animation shows the characteristic intensity interference pattern from two holes or slits that you are probably familiar with from physics class. The phase pattern is also quite different from the single-hole case. In this case, the surfaces of constant phase from each hole, one at a time, form two sets of concentric hemispheres, and these sets intersect at each point in 3d space. However, when both holes are open at the same time, the resulting intensity and phase at each point is given by the vector sum of the two complex values from each light source, adjusted for differences in magnitude and phase with increasing distance. The resulting phase is rescaled to between -pi and +pi. The intensity is zero whenever the two adjusted vectors cancel (are of equal magnitude and point in opposite directions). Here is the "side view" for cross-sections in the z-y plane:

Note the extreme variation in intensity in this plane near x = 0.0. Here the vertical cross-sections are passing in and out of the intensity interference pattern shown above. Here is the "top view" for cross-sections in the x-z plane. Note the similarity of the interference pattern to that in the x-y plane:

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