A question from a reader about the Rayleigh criterion

Question from a reader:

I am working my way through “An Introduction to Ocean Remote Sensing” and I have a question on the Rayleigh Criterion, page 73, equation (3.31) This gives the surface resolution of the human eye as 0.2 mm at a range of 1 m.

I must have missed something, but my question is how can I discern a human hair, width 10 micro-metres or a dust mote floating in the sunlight?

My answer:

Good question! So first let’s look at the definition of resolution. Consider two candles a small distance apart that we view through a slit.

two_candles

Each candle generates a diffraction pattern, so that if the candles have an angular resolution that is less than that determined from the Rayleigh criterion, then we will be not be able to see that there are two candles.

small_candle

Now let’s blow out one of the candles and move it far away. Can we still see it? The answer depends on the brightness of the candle relative to its background. If the candle is sufficiently bright, even though the subtended angle is much smaller than that given by the Rayleigh criterion, it will excite a retinal sensor. This is the solution to the dust mote problem (note that I am following http://en.wikipedia.org/wiki/Visual_acuity).

The human hair presents a similar problem and solution. From Wikipedia, “The smallest detectable visual angle produced by a single fine dark line against a uniformly illuminated background is also much less than foveal cone size or regular visual acuity. In this case, under optimal conditions, the limit is about 0.5 arc seconds or only about 2% of the diameter of a foveal cone.” The eye has the ability to detect small changes in brightness, even though the angular width of the object is much less than that subtended by the retinal sensor, and less than the angular width given by Rayleigh criterion. Two hairs side-by-side however, would suffer diffractional blurring.

Finally, try it! Draw a couple of lines on a sheet of white paper, then set the paper at distance greater than that determined from the Rayleigh criterion. Can you distinguish the lines?

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About Seelye Martin

Seelye Martin received his Ph.D. in engineering mechanics from Johns Hopkins University in 1967 then spent two years as a research associate in the Department of Meteorology at the Massachusetts Institute of Technology. In 1969 he took up a position in the School of Oceanography at the University of Washington where he is now an Emeritus Professor. Beginning in 1987, he taught courses on remote sensing of the oceans. Professor Martin has been involved with passive microwave, visible/infrared and radar ice research since 1979, and has served on a number of NASA and NOAA committees and panels involving remote sensing and high latitude processes. He has made many trips to the Arctic for research on sea ice properties and oceanography. From 2006-2008, he worked at NASA Headquarters as Program Manager for the Cryosphere, where he also served as program scientist for the ICESat-1 and ICESat-2 missions. After leaving Headquarters, from 2009 -2012, he worked in a variety of roles for the NASA high-latitude IceBridge remote sensing aircraft program. For this work, in 2012 he was awarded the NASA Exceptional Public Service Medal.
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