The first paper, "Maximum Sensitivities of Optical Direction and Twist
Measuring Instruments", Nov. 1964, found the maximum precision with which
an optical instrument can determine the direction of a light source and
the orientation of a source about an axis parallel to the line of sight.
Equations are derived that include the sizes and shapes of the one or more
telescope apertures, and whether they are coherently coupled. In this
paper, it was also demonstrated that the ability to measure the phase of a
beam of light exactly equals the ability to measure the orientation of its
plane of polarization - and which also equals the ability to measure its
intensity. This is a surprising, and beautiful, result - the same equation
governing all 3 measurement limits:
Each limit = 0.5 / [square root of the number of detected photons]
Thus, if we detect ¼ million photons, we can measure the phase of a beam of light, and its plane of polarization, to .001 radian [about 3 arc minutes] and the intensity to 1 part in 1000.
The 2nd paper, "Limits to which Double Lines, Double Stars, and Disks can be Resolved and Measured", Aug. 1967, determined the limit to which we may resolve and measure, through a circular diffraction-limited aperture, the angular separation of double-line and double-star sources, and the angular diameter of disk sources. The variables considered, in order to derive the equations for these limits, were: diffraction, total number of photons detected, and the intensity ratio of the components of the double sources.
In the case of a diffraction-limited telescope, with its circular aperture, attempting to resolve a double star, the equation for determining the resolution limit is merely the product of three terms:
1. The Rayleigh Resolution Limit, in radians, given approximately by the wavelength of light divided by the diameter of the aperture. For instance, if the wavelength is half a thousandth of a millimeter [yellow-green], and the aperture diameter is 100 millimeters, then the resolution is 1/200,000th radian, or about 1 second of arc.
2. The 2nd term quantifies the effect of the intensity ratio, R, of the two stars constituting the double star we are trying to resolve. This term is merely the square root of R, plus its reciprocal, all divided by 2. When R=1, the term equals 1. But when R=14, the term, and thus the resolution of the telescope, doubles [from one second to two seconds in the above example].
3. The 3rd and last term is simply the inverse cube root of the number of detected photons. Thus, in the above example, 8 times as many photons would be required to make up for R increasing from 1 to 14. Note that just eight photons are required to resolve two stars one arc second apart, even with one star 14 times brighter than the other. But with 8 thousand million photons detected, the above optical system could theoretically resolve these two stars just 1/1000th arc second apart.
Radiation noise [the random fluctuation of detected photons] is assumed to be the only source of noise.
Regarding the diffraction pattern of a double star source, one might naturally assume that the resolution information we are seeking would emanate from the bright rings of the pattern. However, from the equations of the 2nd paper, it can be shown that most of the desired measurement and resolution information resides in the dark rings. In fact, the smaller the angular separation of the 2 sources, the greater the percentage of information coming from the dark rings. When the angular separation is, for instance, .001 arc second, very sharp peaks of information appear at the very center of the dark rings, with little noticeable information even immediately adjacent to the peaks.
© 1970 Oscar Falconi, PO Box 3345, Saratoga CA 95070, U.S.A.