The Size, Structure, and Variability of Late-Type Stars Measured ...

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sources are not coaligned perfectly, they will not (necessarily) add constructively. In this
case, the visibility of the net fringe pattern may be somewhat less than unity depending
on the separation between the slits. As the slit separation is increased, the visibility falls
off from unity sinusoidally. When the separation between the slits reaches a characteristic
value,
D = λ/2 sin(∆θ) (1.2)
the two interference patterns are completely out of phase, the fringes add destructively, and
the visibility is zero.
This type of experiment could be used to measure the angular separation between
the stars in a binary, for instance. An interference pattern could be formed between
two telescopes placed close together. Then, the telescopes could be separated until the
fringes disappear. The angular separation of the two stars would then be given by Equation
1.2. The telescopes, separated by a distance, D, could be said to have a resolution,
∆Θ interferometer = λ/2D radians which is similar to the resolution of a diffraction limited
single aperture telescope having size on the order of the interferometer’s separation as previously
claimed.
For the case of an arbitrary intensity distribution, the fringes measured using the
two-slit interferometer will be the superposition of all the point source contributions. Since
the net fringes vary sinusoidally with the angular separation between the point sources, the
net fringe can be expressed as an integral over the intensity distribution modulated by a
sinusoidal response. Mathematically, this is equivalent to a two-dimensional Fourier Transform.
This result is known as the Van Cittert-Zernike Theorem (Born and Wolf (1965) [14]),
and it states that the complex visibility, V (x, y), is given by,
V (x, y) = 1 F
∫ ∞ ∫ ∞
−∞
−∞
I(θ, φ)e −2πi(θx/λ+φy/λ) dθdφ (1.3)
where [x,y] is the vector baseline (separation) between the two telescopes, F is the total
flux from the source, and I(θ, φ) is the intensity distribution on the sky as a function of the
angles θ and φ (which are assumed to be much less than unity). The measured (real-valued)
visibility defined in Equation 1.1 is the magnitude of the complex visibility defined above.
To reconstruct an arbitrary intensity distribution from visibility measurements,
one would need to measure the visibility (and its phase) on every possible baseline. Then,
the image could be reconstructed by inverting Equation 1.3. To accomplish this, one might