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I notice if I put heat under a laser beam it will cause the beam to shimmer. The heated air is causing a disturbance in the air which affects the light. Yet, I can blast a stream of air from a compressor across the beam and there is no effect on the light. What’s going on?

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Heat ripples the light beam because the temperature of the flame and its surroundings is sufficiently nonuniform that it creates significant differences in the index of refraction of the air that vary quickly with time.

The blast of air coming out of the nozzle is close enough to ambient that it doesn't create anisotropies in the index of refraction of the air near the light beam, and so you don't see any ripples.

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    $\begingroup$ If you measure very carefully, you will find the moving air produces pressure variations that affect the index of refraction. This effect can limit the accuracy of a interferometer used to measure displacement, for example. $\endgroup$ – The Photon Dec 15 '18 at 19:54
  • $\begingroup$ agree with @The Photon. but the heat-driven effects can be readily observed without interferometry. $\endgroup$ – niels nielsen Dec 16 '18 at 20:57
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Wind, and generally atmosphere, causes considerable effect on light propagation. However usually the effects are too small for human eye to detect on short length-scales and regular environment.

A good example where this effect is big and important (that is, actively funded and researched) is astronomy, and a whole branch of optics called Adaptive optics tries to compensate the phenomena to increase the resolution of telescopes located down on earth.

As @niels nielsen mentioned, the thermal effects are more prominent, since the average velocity of the air molecules is larger in comparison to the velocity of air out of the compressor, thus the compressor is just a slight perturbation to regular state. You have to compare the velocity of the air out of compressor to the the RMS velocity of "air particles" -the speed of sound, which is about $340 [\frac{m}{s}]$. Only when the velocity out of the compressor becomes comparable to the speed of sound you will start to notice effects on light trajectories. On the other hand, even the use of a crude estimation found in Wikipedia speed of sound shows that for oven at $T\approx 200[C^\circ]$ the speed of sound changes by more than $30\%$, thus disturbing more the light trajectories.

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