University of St Andrews
School of Physics and Astronomy

10.  Two-beam interference: Young's slits,
intensity variation by algebra

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We are now looking at optics again, in areas where the wave-like nature of light is important.  We will look at how two beams interfere, then how many beams interfere, and then (briefly) how an infinite number of infinitesimally small beams interfere.  We use the principle of linear superposition, as discussed earlier (ie we add up the disturbances at one place and time due to all the present waves, and the algebraic sum is the total disturbance there.  Looking at this in optics is trickier, as we can never measure the time-varying disturbance directly, only the time averaged "intensity" of light at a point.  The intensity of a wave is proportional to the time average of the square of the disturbance.

• Two-beam interference is the easiest to describe, and we will take as our main example the arrangement known as "Young's Slits"
• Our model supposes that light coming from the two slits is coherent and in phase.
• Where two waves come together in phase (in step) we get constructive interference, and a maximum in intensity.
• Where two waves come together out of phase (ie have a phase difference of pi radians) we get destructive interference, and a minimum in intensity.
• The above are the extremes of what may happen.  In between the results is ... in between!
• The intensity of a wave is proportional to the time average of the square of its disturbance.
• For places where there is a phase difference other than aero or a multiple of p, we need to add up the disturbances of the two waves to find the time-varying total disturbance.  Taking the time average of this quantity then gives us the intensity.
• We can do this to show that the intensity in the two beam interference pattern caused by two beams each of intensity I, and with a relative phase difference between them of f, is   4 I ² cos² (f / 2)
• In the Young's slits arrangement the phase difference comes from the difference in path length that the two beams travel from the slits to the distant screen.  If we are looking at a part of the screen at an angle Q from the straight through position, the path length difference is d sin(Q), and for each wavelength path difference there will be a 2p phase shift.  Thus the phase difference between the two beams is f = 2p d sin(Q) / l

Similar material to that presented in the lectures is available at:-

• Halliday, Resnick, & Walker - Chapters on:-   Interference

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Created by, and copyright of, Bruce Sinclair, University of St Andrews; last modified 18/09/01