April 20, 1999
May 17, 1999
prior reading:  Feynman QED pages 54 - 67

part 8  NARROW AND WIDE SLITS

return to the NO WATER case and place four or five pints close together (so that they touch one another) on the  vertical dotted line, centered between Source and Detector.  This mimics light passing thourgh a slit, as in Figure 33, page 55 of QED.  The goal of this activity is to see how much light spreads out when passing through narrow and wide slits.

Now press the ADD DETECTORS button and click on different locations for the detector along the vertical dotted line through the original detector.  At each new position of the detector, the resulting arrow in the right-hand window is copied to the position of the detector.  Also, resulting arrows are shown at earlier positions of the detector.

Click on different locations for the detector along the dotted vertical line passing through the first Detector.  See how much light goes in these different directions.  Here is an example of three positions of the detector:

After each new position of the detector, you can click on the CLEAR PATHS button to redraw the picture without the paths being shown.  Continue placing detectors all the way to the bottom of the screen along the same vertical line.

Questions:
a)  With a small ruler, measure on the screen the lengths of the resulting arrows for the central detector and for the detector near the bottom of the screen. Calculate the ratio of resulting arrow lengths at the two locations:

 Ratio of probabilities =  (Length of arrow for detector near bottom)
                                           (Length of arrow at center detector)

b)  Therefore, what is the ratio of the probabilities for photon arrival at the same two locations?

Ratio of probabilities = (Probability of detection at detector near bottom)
                                       (Probability of detection at center detector)

Now explore the case that mimics a wide slit.  First, study what we mean by a “wide slit,” as follows.

Questions: 
c)  From your work with the Cornu spiral in Section B above, what determines the illumination from a slit that is infinitely wide (which results from removing the slit entirely!)?

d).  Discussion question:  Reconsider "the absurd" objection [Question (j)] and explain why a maximum of one photon arrives at the detector in the absence of a slit, even though the photon can arrive at the detector by exploring an infinite number of paths.

One definition of  “wide slit” is a slit wide enough to yield that portion of the Cornu spiral that lies more or less along a straight line.  Start at the center (where the dotted horizontal and vertical lines intersect) and clicking on a series of evenly-spaced points below the center until the spiral in the right-hand window begins to curve over.
Construction of a possible “lower half-slit” is shown here:

Complete the lower half-slit and add a mirror-image set of points above the center to complete the slit.  You now have a “wide-slit.”

Again move the detector up and down and compare the length of the resulting arrows at different positions:

We are interested in the ratio of the squared length of the arrow at the center detector to the squared length of the arrow at the detector near the bottom of the screen.  This ratio gives us the ratio of probabilities that the photon will arrive at these two spots.

Questions:
e)  Carry out a calculation similar to that requested in question (a) above.  What is the ratio of the arrow lengths at the detector near the bottom to that of the center detector?

f)  Carry out calculations similar to those requested in question (a) and (b) above.  What is the ratio of probability of detection at the two detectors?

g)  Compare your results for narrow and wide slits.  Imagine a screen stretched across the positions of the detectors described in the above exercises, as shown in the figure below.  In one experiment, many independent photons pass through a narrow slit to fall on this screen.  Then, in a second experiment, the same total number of independent photons pass through a wide slit to fall on the screen.  For the two cases, compare the variation of illumination from center to edge of the detector screen.  In summary, how do the shapes of the illumination curve differ for narrow and for wide slits?  You may want to reconstruct narrow and wide slits using OnePhoton in order to answer this question.