Sir George Biddell Airy was an astronomer who contributed to the field of optics during the nineteenth century.  He did so by discovering a cure for a defective eye condition known as astigmatism and deriving the equation used to predict diffraction disks, now known as Airy Disks.

Sir George Biddell Airy (1801 – 1892) was a British astronomer who made significant contributions to the field of optics throughout the nineteenth century.  Among these contributions were the discovery of an eye defect known as astigmatism and the invention of cylindrical lenses to help correct the problem, the invention of a water telescope which laid the groundwork for Einstein’s Theory of Relativity and dispelled the theory of aether substance in air through which light was thought to travel, and the discovery of the equation used to predict diffraction disks, which determine the limits of optical resolution [1].

Airy was introduced to the sciences as a child while being educated by his uncle, Arthur Biddell [2].  In 1823, Airy graduated from Trinity College, Cambridge, where he had served as a sizar (a student who works as a servant in order to obtain reduced fees) and subsequently became employed as a mathematics tutor [3].  In 1826, he became a Professor of Mathematics at Cambridge University, and published Mathematical Tracts on Physical Astronomy.  In 1828, Airy became a Professor of Astronomy and was made Director of the Cambridge Observatory.  The Observatory had previously been used primarily to provide data to the Royal Navy; however, under Airy’s direction, the Observatory became a research-oriented institution.  The Altazimuth telescope, used to calculate the moon’s altitude [4], was added to the Observatory in 1847, and the Airy Transit Circle, which provided the Observatory with its 4^th meridian line [5] was added shortly thereafter.  Departments in magnetism and meteorology were later added to the Observatory.  Throughout his time as Director, Airy added many different aspects of astronomy to the Observatory’s activities, including the study of eclipses, measuring gravity, and the observation of sunspots and spectroscopy.  In 1835 Airy was made Astronomer Royal [6], and in July 1872, Airy was recognized by Queen Victoria and knighted his honor of his accomplishments, after having previously refused the offer three times [7].

All optical systems have a finite aperture and therefore can only collect a portion of the waves coming from an object.  Consequently there will always be an apparent deviation from rectilinear propagation when light passes through an aperture, so the waves will be diffracted.  The perfect system is therefore called diffraction limited.  As the name sugests, the only factor impeding the system's resolving power or resolution is the diffraction of light.  In such a system, a pinhole source is focused as a central point of maximum intensity surrounded by alternate circles of light and darkness caused by the reinforcement and interference of diffracted waves.  The center of this diffraction pattern is called an Airy disk.

One of Airy’s most significant contributions of the field of optics was the discovery of the equation used to predict diffraction disks, or what are now known as Airy Disks.  Airy Disks can be used to determine the resolution of the diffraction-limited system.  “A common measure for resolution is Lord Rayleigh’s criterion that asserts that two point sources are just resolvable when the maximum of one Airy pattern coincides with the first dark ring of the second Airy pattern” (see Figure 1) [8].  The equation for the Airy disk pattern is as follows:

[source 8]    (1)

I is intensity, I_o is a constant proportional to the square of the amplitude of the electric field of the source.  The first order Bessel function J₁, as seen in the Airy equation, can be viewed graphically in the plot below. 

Plot 1. Plot of Bessel functions.

[source 8]    (2)

In equation (2), M is magnification, lambda is the wavelength of the source, r_i is radial distances measured in the image plane, and NA is the numerical aperture on the system.  NA is related to the f-number (f/#) of a system by the following equation.

(3)

The center disk of the diffraction pattern is the Airy disk and its radius is given by:

[source 8]    (4)

In the following example calculation use a hypothetical diffraction limited optical system and equations (1) through (4).  This system will have the following properties given: a 100 watt monochromatic point source of wavelength 650 nm, numerical aperture of .0156 (f/64), and a magnification of 100.  First, nu and the Airy radius are calculated:

(5) and (6)

The result of nu is then inserted into equation (1) to find the resulting Airy pattern.  Fortunately, this can be completed quickly with the aid of a computer.  The following depicts the result of this example calculation as an intensity plot of a line through the center of the Airy pattern.  Also, another result is shown of a similar system with a NA of .03125 (f/16) for comparison.  See if you can calculate the exact Airy radius for the f/16 system and see if it matches the plot's value.

Another of Airy’s important contributions to the field of optics was his discovery of a cure for astigmatism.  Astigmatism is caused by an oblong shape of the eye’s cornea, resulting in light rays to focus on two points in the back of the eye, rather than on just one.  This then leads to blurred vision [10].  After discovering astigmatism, and that he suffered from it, Airy invented a method to correct the problem.

As seen in Fig. 2, an astigmatic system has two focal planes, horizonatal and vertical, and each has its own Airy diffraction pattern.  In the middle of these two planes there is the circle of least confusion at which the familiar Airy disk can be seen.

To correct for astigmatism, “it is necessary to reduce the value of the astigmatic difference, or the distance between the tangential and sagital line images.  Complete removal of astigmatism is difficult, but can occur in optical systems when the two curves, S and T, become flatter and coincide (see Figure 3c), and the image is then formed in a region near the Petzval surface (P)” [11].  Airy's method to counter astigmatism was with glasses containing cylindrical lenses. Such lenses focus light to a line as opposed to a spherical lens which focuses light to a point.

References

1.  http://micromagnet.fsu.edu/optics/timeline/people/airy/html
            (accessed October 4, 2003)
2.  http://www.nahste.ac.uk/isaar/GB_0237_NAHSTE_P0320.html
            (accessed October 26, 2003)
3.  Ibid.
4.  http://greenwichpast.com/vip/astronomers/airy.htm
            (accessed November 2, 2003)
5.  http://www.nmm.ac.uk/site/request/setTemplate:singlecontent/contentTypeA/conWebDoc/contentId/395
            (accessed October 4, 2003)
6.  see source 2.
7.  see source 1.
8.  Bass, Michael, ed.  Handbook of Optics, Vol. II.  McGraw-Hill:  New York.  1995. 1.40.
9.  http://micro.magnet.fsu.edu/primer/java/aberations/astigmatism/
            (accessed October 26, 2003)
10. http://www.allaboutvision.com/conditions/astigmatism.htm
            (accessed November 2, 2003)
11. see source 9.

Closing Quote - Optional reading.

... in those parts of astronomy which ... [require] only method and judgment, with very little science in the persons employed, we have done much; while in those which depend exclusively on individual effort we have done little ... our principal progress has been made in the lower branches of astronomy while to the higher branches of science we have not added anything.

-Sir George Biddell Airy