Undergrad The 1.22 factor in the angular resolution

[greypilgrim]

Hi.

The angular resolution is calculated through
$$\theta=1.22\frac{\lambda}{D}\enspace.$$

It's the first zero of the intensity function (in small-angle approximation) of the Airy disk:
$$I\left(\alpha\right)=I_0\left(\frac{2J_1\left(\pi\cdot\alpha\cdot\frac{D}{\lambda}\right)}{\pi\cdot\alpha\cdot\frac{D}{\lambda}}\right)^2$$

So if the angle between two light sources is $\theta$, the central maximum of one source coincides with the first minimum of the other and vice versa.

Though this makes sense, I tried a different approach and tried to find the smallest $\theta$ where the central peak of
$$I\left(\alpha\right)+I\left(\alpha-\theta\right)$$
divides into two. This happens way earlier, at about $\theta\approx0.94\frac{\lambda}{D}$:
$\theta=0.90\frac{\lambda}{D}:$

$\theta=0.94\frac{\lambda}{D}:$

$\theta=1.00\frac{\lambda}{D}:$

Wouldn't it make more sense to define $\theta\approx0.94\frac{\lambda}{D}$ as angular resolution?

 

[Ibix]

It's a rule of thumb, to an extent.

But note that your graph is sampling hundreds of points across the diameter of one Airy disc. In other words, a CCD with the capability of your graph would need tens of thousands times more pixels than the one you'd normally fit to the instrument producing the Airy discs. That'll be a penny or two in R&D costs, and the resolution gain is tiny. Easier to build a bigger telescope. With adaptive optics.

You might like to try again with CCD elements that integrate over some finite area and not knowing where the discs are centred. The Nyquist criterion tells you that you'll get a decent performance gain until the detector pitch is about half the width of the Airy disc, and then diminishing returns will well and truly set in.

 

[greypilgrim]

It's a rule of thumb, to an extent.

So why not just take
$$\theta=\frac{\lambda}{D}\enspace?$$

 

[Andy Resnick]

Hi.

The angular resolution is calculated through
$$\theta=1.22\frac{\lambda}{D}\enspace.$$

It's the first zero of the intensity function (in small-angle approximation) of the Airy disk:
$$I\left(\alpha\right)=I_0\left(\frac{2J_1\left(\pi\cdot\alpha\cdot\frac{D}{\lambda}\right)}{\pi\cdot\alpha\cdot\frac{D}{\lambda}}\right)^2$$

So if the angle between two light sources is $\theta$, the central maximum of one source coincides with the first minimum of the other and vice versa.

Though this makes sense, I tried a different approach <snip>

There are different approaches to defining 'angular resolution'- the Rayleigh criterion was simply the first. The Sparrow criterion uses a different metric (the central dip is 5%, not (IIRC) 20%). The Nyquist limit is more appropriate for sampled imaging systems. Your approach simply defines yet another criterion.

 

[greypilgrim]

The Sparrow criterion uses a different metric (the central dip is 5%, not (IIRC) 20%).

Having read up on the topic a little, I think the Sparrow criterion is exactly the idea I had (appearance of a central dip). Based on a 5 % central dip, it seems to be called Dawes' limit.