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.
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