# Minimum seperation between two Airy disks as a function of the distance between two point sources of coherent light passing through the same objective

I have two coherent point sources of light, $A$ and $B$, separated by a distance $L$, which I focus down to the diffraction limit using a high-powered objective (e.g. a $\approx 100x$ objective). If I turn on $A$ and turn off $B$, I have an Airy disk at position $c_1$, and I turn off $A$ and I turn on $B$, I have an Airy disk at position $c_2$. Given that both light sources are sent through the same objective, what is the minimum distance between $c_1$ and $c_2$? Is it simply $L$ scaled down by the objective (i.e. $\frac{L}{100}$)? Or does something odd happen because of e.g. curvature of the lens in the objective?

EDIT: A restatement of this question would be the following - Assuming all of the optics are perfect, if I shine a laser at a point (x,y) on an objective, and then shine the laser at a point (x2,y2), will the peak of the Airy disk move the same distance?

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Having spent rather a lot of years designing and building adaptive optic systems, I've seen this sort of problem in a number of guises.

Certainly you can apply some fitting function to the image of each of your spots separately and determine the centroid of each spot alone. If your detector has sufficient resolution, these two measurements will yield the different locations of the two Airy discs.

The whole deal with the "Rayleigh limit" is simply that the two-spot pattern has no local minimum between the two peaks. In the absence of knowledge of the source, you can't tell whether it's two point sources or a single extended source. If you know in advance that two sources (such as neighbor stars) are producing the image, and you have a good idea of their relative brightness, you can fit the image to a different function, i.e. the sum of two Airy discs, to determine the peaks.

However, remember that all light entering the lens system at a given angle will be focussed to the same location. So it's not moving the laser beam to different positions on the lens but rather aiming them at different entrance angles. (I'm ignoring coma and other off-axis effects of a real optic system here)

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Look up the angular resolution of an optical telescope. It is given approximately by the equation

$$\theta=1.22\frac{\lambda}{D}.$$

From this you can convert to the linear separation between the two objects.

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I understand that two points cannot be distinguished (without perfect knowledge of their PSFs) below the Rayleigh limit. This question is about what the minimum distance between the stars can be if we can make them blink by shutting the light off from one or the other and individually resolving the Airy disks. – RGrey Jan 17 '14 at 21:12
In that case I think it would depend on systematics in your measurement apparatus: bit noise in your CCD, pixel size in your CCD, imperfections in your optics, etc. – Chris Mueller Jan 17 '14 at 21:19
Assuming all of the optics are perfect, if I shine a laser at a point (x,y) on an objective, and then shine the laser at a point (x2,y2), will the peak of the Airy disk move the same distance? This is a sort of restatement of my question. Or is there nothing fundamental to prevent this? – RGrey Jan 17 '14 at 21:20
@Chris Mueller, Why does laser intensity scale as 1/ distance squared compared to 1/ distance laser beam travels? – Frank May 24 at 9:07