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# 3: Interference

The most certain indication of a wave is interference. This wave characteristic is most prominent when the wave interacts with an object that is not large compared with the wavelength. Interference is observed for water waves, sound waves, light waves, and, in fact, all types of waves.

• 3.1: Prelude to Interference
If you have ever looked at the reds, blues, and greens in a sunlit soap bubble and wondered how straw-colored soapy water could produce them, you have hit upon one of the many phenomena that can only be explained by the wave character of light. The same is true for the colors seen in an oil slick or in the light reflected from a DVD. These and other interesting phenomena cannot be explained fully by geometric optics. In these cases, light interacts with objects and exhibits wave characteristics.
• 3.2: Young's Double-Slit Interference
Young’s double-slit experiment gave definitive proof of the wave character of light. An interference pattern is obtained by the superposition of light from two slits. When light passes through narrow slits, the slits act as sources of coherent waves and light spreads out as semicircular waves. Pure constructive interference occurs where the waves are crest to crest or trough to trough. Pure destructive interference occurs where they are crest to trough.
• 3.3: Mathematics of Interference
In double-slit diffraction, constructive interference occurs when d sin θ = mλ (for m=0,±1,±2,±3…), where d is the distance between the slits, θ is the angle relative to the incident direction, and m is the order of the interference. Destructive interference occurs when $$d \space sin \space \theta = (m + \frac{1}{2}) \lambda$$, for m = 0,±1,±2,±3,…
• 3.4: Multiple-Slit Interference
Analyzing the interference of light passing through two slits lays out the theoretical framework of interference and gives us a historical insight into Thomas Young’s experiments. Much of the modern-day application of slit interference uses not just two slits but many, approaching infinity for practical purposes. We start the analysis of multiple-slit interference by taking the results from our analysis of the double slit (N = 2) and extending it to configurations with numbers of slits.
• 3.5: Interference in Thin Films
When light reflects from a medium having an index of refraction greater than that of the medium in which it is traveling, a 180° phase change (or a λ/2 shift) occurs. Thin-film interference occurs between the light reflected from the top and bottom surfaces of a film. In addition to the path length difference, there can be a phase change.
• 3.6: The Michelson Interferometer
The Michelson interferometer (invented by the American physicist Albert A. Michelson, 1852–1931) is a precision instrument that produces interference fringes by splitting a light beam into two parts and then recombining them after they have traveled different optical paths.