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Interference of Light Waves
37.1 37.2 37.3 Conditions for Interference Young’s Double-Slit Experiment Intensity Distribution of the Double-Slit Interference Pattern Phasor Addition of Waves Change of Phase Due to Reflection Interference in Thin Films The Michelson Interferometer

Q37.1 (a) Two waves interfere constructively if their path difference iszero, or an integral multiple of the wavelength, according to δ = mλ , with m = 0 , 1, 2 , 3 , …. Two waves interfere destructively if their path difference is a half wavelength, or an odd multiple of λ 1 , described by δ = m + λ , with m = 0 , 1, 2 , 3 , …. 2 2

37.4 37.5 37.6 37.7





The light from the flashlights consists of many different wavelengths (that’swhy it’s white) with random time differences between the light waves. There is no coherence between the two sources. The light from the two flashlights does not maintain a constant phase relationship over time. These three equivalent statements mean no possibility of an interference pattern.


Underwater, the wavelength of the light would decrease, λ water =

λ air . Since thepositions of light n water and dark bands are proportional to λ, (according to Equations 37.2 and 37.3), the underwater fringe separations will decrease.
Every color produces its own pattern, with a spacing between the maxima that is characteristic of the wavelength. With several colors, the patterns are superimposed and it can be difficult to pick out a single maximum. Using monochromatic light caneliminate this problem. The threads that are woven together to make the cloth have small meshes between them. These bits of space act as pinholes through which the light diffracts. Since the cloth is a grid of such pinholes, an interference pattern is formed, as when you look through a diffraction grating.




If the oil film is brightest where it is thinnest, then n air < noil < n water . With this condition, light reflecting from both the top and the bottom surface of the oil film will undergo phase reversal. Then these two beams will be in phase with each other where the film is very thin. This is the condition for constructive interference as the thickness of the oil film decreases toward zero.


382 Q37.7

Interference of Light Waves

As waterevaporates from the ‘soap’ bubble, the thickness of the bubble wall approaches zero. Since light reflecting from the front of the water surface is phase-shifted 180° and light reflecting from the back of the soap film is phase-shifted 0°, the reflected light meets the conditions for a minimum. Thus the soap film appears black, as in the illustration accompanying textbook Example 37.5, “Interference ina Wedge-Shaped Film.” If the film is more than a few wavelengths thick, the interference fringes are so close together that you cannot resolve them. If R is large, light reflecting from the lower surface of the lens can interfere with light reflecting from the upper surface of the flat. The latter undergoes phase reversal on reflection while the former does not. Where there is negligible distancebetween the surfaces, at the center of the pattern you will see a dark spot because of the destructive interference associated with the 180° phase shift. Colored rings surround the dark spot. If the lens is a perfect sphere the rings are perfect circles. Distorted rings reveal bumps or hollows on the fine scale of the wavelength of visible light. A camera lens will have more than one element, tocorrect (at least) for chromatic aberration. It will have several surfaces, each of which would reflect some fraction of the incident light. To maximize light throughput the surfaces need antireflective coatings. The coating thickness is chosen to produce destructive interference for reflected light of some wavelength. To do Young’s double-slit interference experiment with light from an ordinary...
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