# Number of actuators in adaptive optics

It is well known that diffraction resolution of a telescope depends on the exterior radius of aperture: One can, for instance, use just a narrow outer ring of a parabolic mirror and the resolution will be the same as it were the whole mirror (obviously, intensity will be much much smaller).

Now, take such a ring telescope, say, 3cm wide ring which is 100 meters in diameter.

Suppose we are using adaptive optics. How many actuators are needed to reach the maximal possible resolution?

I assume that spacing between the actuators should be of an order of Fried radius r0 (i.e. one actuator about every 10cm along the circumference of the ring). I.e. number or actuators should be of order D/r0, where D is the aperture diameter.

Is it correct?

• The first part about a narrow annular aperture having the same resolution as a full aperture is not right. I think that would make a great separate question but as a premise to your question it causes some problems. I recommend that you edit your question and just ask for some guidelines for how many actuators are necessary for a given aperture; that would be a great question by itself! Ask separately about your idea about an annular telescope because that answer will be totally different (and probably a bit disappointing). See en.wikipedia.org/wiki/Babinet%27s_principle – uhoh Dec 9 '19 at 1:45
• I personally made computation of the telescope transfer function for annular aperture and found that support of the function in the fourier space is the same as for circular aperture. This is an elementary excercise. So the resolution is the same. Moreover I read this statement in several papers. – Truffaldino Dec 9 '19 at 14:34
• I am also adding NASA report as an example of a paper that states " A ring-shaped primary mirror loses only collecting area, not resolution" (see page 2, last sentence before figure 1.) nasa.gov/sites/default/files/atoms/files/… – Truffaldino Dec 9 '19 at 14:58
• Ah, I see, the first sentence is correct "Consequentially, it is a well-known trick of sophomore physics that an obscuration of a small part of a telescope’s aperture dims the image but does not reduce its resolution." However, the next sentence drops the "small part" and overgeneralizes, and is in your case wrong: "A ring-shaped primary mirror loses only collecting area, not resolution." It's wrong in your case because they dropped those two words, and for your case the obscuration is not small, it's almost the same size as the main aperture. Their 2nd sentence is quite misleading. – uhoh Dec 9 '19 at 15:11
• Since I've now answered the premise part of the question, it might be a good idea to ask a more general question about number of actuators as a separate question. – uhoh Dec 9 '19 at 16:05

It is well known that diffraction resolution of a telescope depends on the exterior radius of aperture: One can, for instance, use just a narrow outer ring of a parabolic mirror and the resolution will be the same as it were the whole mirror (obviously, intensity will be much much smaller)

The premise of your question is false. I tried to convince you to separate it out as a separate question, but to no avail. So I'll answer it here instead.

From comment:

I personally made computation of the telescope transfer function for annular aperture and found that support of the function in the fourier space is the same as for circular aperture. This is an elementary excercise.

To help illustrate that a narrow annular aperture has much worse resolution than the full aperture, I'll do a quick comparison. The first plot is the diffraction pattern (scale is +/- 0.5 arcseconds) for a clear aperture of 2 million wavelengths (e.g. 1 meter radius with 1 micron wavelength), and the second is for the same aperture with a 0.9 meter obstruction.

I use Airy disks and Babinet's principle to subtract the E-field of the obstruction before squaring. For this symmetry there's no need to do an explicit Fourier analysis, the Airy disk is the Fourier transform of the circular aperture step function. (see also)

This is for an annulus with outer radius 1.0 and inner 0.9. If it was 1.0 and 0.99 it would look much much worse!

Now, take such a ring telescope, say, 3cm wide ring which is 100 meters in diameter.

That would give you a giant splat pattern of roughly the same size as a simple 3 cm diameter telescope would.

Python script:

from scipy.special import j1
import matplotlib.pyplot as plt
import numpy as np

def get_E(a, lam, theta):
ka  = (twopi/lam) * a
x   = ka * np.sin(theta)
return a * (2. * j1(x) / x)

halfpi, pi, twopi = [f*np.pi for f in (0.5, 1, 2)]

arcsecs = np.arange(0, 0.5, 0.0001)[1:]

r1, r2  = 1.0, 0.9
lam     = 1E-06

arcsecs   = np.linspace(-1, 1, 1000) # arcsec (skips zero to avoid singularity)
Xas, Yas, = np.meshgrid(arcsecs, arcsecs)
Ras       = np.sqrt(Xas**2 + Yas**2)
theta     = Ras * pi / 3600. / 180.

E1, E2    = [sign*get_E(r, lam, theta) for r, sign in ((r1, +1), (r2, -1))]
asec      = np.linspace(0, 1.0, 500)[1:]
th        = asec * pi / 3600. / 180.
e1, e2    = [sign*get_E(r, lam, th) for r, sign in ((r1, +1), (r2, -1))]

Etot = E1 + E2
etot = e1 + e2

if True:
plt.figure()
plt.subplot(2, 2, 1)
plt.imshow(E1**2, cmap='gray')
plt.title('r = 1.0, lam=1E-06', fontsize=16)
plt.subplot(2, 2, 3)
plt.imshow(Etot**2, cmap='gray')
plt.title ('r1, r2 = 1.0, 0.9, lam=1E-06', fontsize=16)
plt.subplot(2, 2, 2)
plt.plot(1E+06*th, th*e1**2) # weight by th for azimuthal (phi) integration