Top: Introduction

Forward: Deformable mirrors

- 1.1. Ideal telescope
- 1.2. Point Spread Function
- 1.3. Resolution criteria
- 1.4. Optical Transfer Function
- 1.5. Imaging through the atmosphere: long exposures
- 1.6. Turbulence statistics
- 1.7. Atmospheric time constant
- 1.8. Isoplanatic angle
- 1.9. Zernike modes
- 1.10. Statistics of atmospheric Zernike coefficients

The image of a point source (star) in an ideal telescope without
atmosphere is shaped by the diffraction and is described by an *
Airy function*:

(1) |

- is the light intensity in the focal plane, as a function of angular coordinate ;
- is the wavelength of light;
- is the diameter of the telescope aperture;
- is the so-called Bessel function.

The first dark ring is at an angular distance of from the center. This is often taken as a measure of resolution in an ideal telescope.

Image
of an astronomical object
can be considered as a multitude of points, each point spread into an
Airy function. This is written as a * convolution*:

(2) |

** Question: ** How the does diffraction-limited resolution depend on
the wavelength?

** Question:** Compute the diffraction-limited resolution of human
eye.

What happens if the telescope is not ideal? An image of a point source
would not be as good as Airy function, the resolution would be
degraded even more. But the imaging equation will still be
valid! So, the * Point Spread Function* (PSF)
is
all we need to know to characterize the imaging. The width of PSF is a
measure of resolution.

** Note 1.** Implicitly, we assumed in the above equations that
is an image of a star of unit intensity, i.e. the
integral of
over is equal to 1. Thus, the
imaging equation preserves the total flux of an astronomical object,
only distributes it differently between the pixels.

** Note 2.** We assumed that PSF has the same shape over the whole
field of view. This condition is called * isoplanatism*. It is not
always true in astronomical imaging, particularly with Adaptive
Optics, because PSF slowly changes in the field. In this case imaging
equation can be applied to parts of the field.

The shape of PSF may be irregular; what numeric measures of resolution are used in this case?

* 1. FWHM = Full Width at Half Maximum* of the PSF.

* 2. Strehl ratio*
, i.e. the central intensity of
PSF compared to the central intensity of the Airy function. The higher is
Strehl ratio, the better is resolution. Diffraction-limited image is the
best, hence always.

* 3. Encircled energy.* By definition, the integral of PSF
is equal to 1. The PSF integral over the circle of radius
is called the encircled energy. This characteristic is
important for observations of faint objects, when one wants to
concentrate the photons as much as possible.

The example of the PSF with compensation of turbulence is given in the picture below.

** Question:** Supposing that PSF is two times narrower, by how much
the Strehl ratio would change?

** Question:** What would be the Strehl ratio if one half of the
(ideal) telescope objective is given a phase delay of ?

Another way to look at the imaging equation is to use the Fourier Transforms (FT; we denote it by tilde). The convolution becomes a product, so

(3) |

The
is called * Optical transfer function*
(OTF). It describes the change of the modulus and phase of the object
FT in the imaging process. The modulus of OTF is called *
Modulation transfer function* (MTF). For astronomical (incoherent)
imaging,
. Typically, MTF decreases with
increasing frequencies, hence the small (high-frequency) details in
the image are weakened and eventually lost.

It is known that for any optical system
for
, where
is called * cutoff
frequency*, and is the maximal size of aperture. It means that the
information at the spatial frequencies above is irrevocably
lost. We need larger telescopes to look at smaller objects!

The relation between PSF and OTF is Fourier transform, so if you know one you know another, they are the same thing expressed differently. From the FT properties it follows that (PSF normalization), and that the Strehl ratio is proportional to the integral of OTF over frequencies.

** Question:** What is the minimum size of a telescope needed to
resolve a fence with 10 cm bar spacing from a distance of 5 km?

Atmospheric turbulence may be regarded as random phase aberrations added to the telescope. These aberrations constantly change in time, so does the PSF. Here we consider the average PSF, corresponding to long exposure times. The theory leads to the expression

(4) |

The atmospheric OTF is related to the statistics of atmospheric phase
aberrations, the so-called * phase structure function*
(see the next section):

(5) |

** Question:**
Suppose that the shape of the atmospheric PSF is Gaussian. What is
the corresponding shape of the structure function?

** Question:** Is the shape of the atmospheric PSF sensitive to the
structure function in the regions where
and
?

The atmospherically distorted wave-front can be visualized as a smashed paper sheet. The wave arriving from a star is plane before entering the atmosphere. Then some parts of it go through the hotter-than-average air (less refractive index) and are advanced, other parts are retarded, and the plane wave-front is deformed. It is the purpose of Adaptive Optics to compensate these distortions. But first, we need to describe them in a statistical sense.

Air is slightly dispersive, but this is usually neglected, so that the
perturbations of the optical path length are considered
as achromatic. However, the * phase* of the optical wave
strongly depends on
the wavelength ! Speaking of perturbations, we assume that
their average values are zero,
(angular
brackets denote statistical averaging).

Although random processes like are typically described
by correlation functions or covariances, in the atmospheric science
the * structure functions* are preferred. Structure function is
the average difference between the two values of a random process:

(6) |

** Question:** What is the relation between the structure function
and the covariance function
?

** Question:** How does the atmospheric
depend on the
wavelength ?

The * Kolmogorov model* of the turbulence distortions prescribes
the specific form of the phase structure function, namely

(7) |

The above model, although it may seem primitive, is the basis of the whole theory of imaging through the turbulence, including Adaptive Optics. Of course, at large distances (more than few meters) and small distances (less than 1 cm) the model is not good, but this turns out to be not very important.

** Question:** What is the r.m.s. (root-mean-square) atmospheric phase
difference at the baseline of in radians and wavelengths?

** Question:** If cm at 0.5 micron wavelength, what is
at 2.2 micron wavelength?

Now we put this model into long-exposure atmospheric OTF, and get it in the form:

(8) |

(9) |

The Strehl number of the atmospheric PSF is exactly the same as in an ideal telescope of diameter (this is the reason why a strange coefficient 6.88 appears). So, for a large telescope, , the Strehl ratio is simply .

** Question:** What is the Strehl ratio of long-exposure images at
the 4 m telescope under 1 arcsecond seeing at the wavelengths of 0.5
and 2.2 microns?

The Fried radius is sometimes identified with the characteristic scale of atmospheric perturbations. This is not quite true: we see that Kolmogorov law does not have any characteristic scale. However, only the perturbations of the size of the order of are relevant for long-exposure imaging. At smaller scales, the distortions are much less than , at larger scales the becomes so large that the atmospheric OTF is zero.

Locally, the strength of turbulent fluctuations of refractive index in
the air are described by a * refractive index structure constant*
which is measured in strange units, m. The dependence
of on altitude is called the turbulence profile. Seeing
depends on the integrated effect of all atmospheric layers:

(10) |

.

This figure shows an example of the turbulence profile at Cerro Paranal, plotted in relative units (full line). The fraction of the turbulent energy up to a given altitude is overplotted in dashed line. Although in this instance a significant part of turbulence was concentrated in only two laters, still some 1/3 of the total energy is distributed continuously over all altitudes.

** Question:** From this relation, find how and seeing depend
on the zenith angle .

Turbulence can often be modeled as fixed phase screens which are
driven by the wind in front of the telescope. Knowing the spatial
properties of the phase screens (structure function) and the wind
velocity, we know also the temporal behavior of the
perturbations. The * atmospheric time constant* is defined
as

(11) |

** Question:** Taking the typical value of =20 m/s, what
would be the atmospheric time constant at 0.5 and 2.2 micron
wavelengths under 1 arcsecond seeing?

The images of astronomical objects taken with exposure time
or shorter are called * short-exposure* images. They correspond to
fixed (frozen) atmospheric aberrations. At longer exposure times, the
aberrations are averaged, and for exposures * much* longer than
the long-exposure PSF is obtained.

The long-exposure atmospheric PSF is independent of the viewing direction (isoplanatic), because the turbulence and its structure function are statistically the same everywhere in the field. But the instantaneous atmospheric phase aberrations do depend on the direction: the telescope beam as projected on the atmospheric layer at 10 km shifts by 0.5 m for an angular offset of 10 arcseconds.

The standard definition of the atmospheric * isoplanatic angle*
is

(12) |

** Question:** What is the isoplanatic angle at
wavelengths of 0.5 and 2.2 micron under 1 arcsecond seeing?

This phenomenon is * very* troublesome for Adaptive Optics, because
it limits the distance between guide star and the scientific objects.
It turns out that for most objects there is no suitable (bright and
close-by) guide star, hence artificial laser guide stars are
required. Alternatively, a 3-dimensional correction of turbulence
(* Multi-Conjugate Adaptive Optics*, MCAO) may be attempted to
increase the corrected field.

In optics, the aberrations are often represented as a sum of special
polynomials, called * Zernike polynomials*. Atmospheric random
aberrations can be considered in the same way; however, the
coefficients of these aberrations (defocus, astigmatism, etc.) are now
random functions changing in time.

A Zernike polynomial
is defined in polar coordinates
on a circle of unit radius . They are characterized
by a * radial order* and an azimuthal order (for a given
, takes the values from 0 to ). Frequently, instead of two
indices and a continuous numeration with single index is
used. For a given radial order , there are a total of
Zernike polynomials.

The first Zernike modes are named as common aberrations, and have simple meaning (see the Table of the first 15 Zernike modes).

The power of Zernike modes comes from the fact that they are *
orthonormal*, that is the scalar product is equal to 1 if
and zero otherwise. The scalar product is defined as integral
over telescope aperture:

(13) |

Now, any phase aberration inside the telescope pupil can be represented as an infinite sum of Zernike polynomials

(14) |

(15) |

Often a limited number of Zernike modes gives already a good enough representation of atmospheric aberrations. If these modes are corrected by Adaptive Optics, an almost diffraction-limited image quality is obtained.

Piston mode corresponds to a constant phase, which does not influence image. Usually the piston mode is ignored.

** Question:** A 4 m telescope, , is defocussed by 1
mm. Compute the resulting aberration for wavelengths of 0.5 and
2.2 microns.

** Question:** Suppose that atmospheric aberration contains only
random tips and tilts with equal amplitudes
. Write the corresponding phase structure
function.

The orthonormality of Zernike modes gives an easy way to compute the phase variance integrated over the pupil. For one mode, it is . The variance for all modes is equal to the sum of squared coefficients starting from 2 (piston excluded).

Given the Kolmogorov turbulence model, we can obtain the statistical properties of the coefficients corresponding to the atmospheric phase aberrations. The mathematical manipulations lead to a simple formula:

(16) |

i \ j | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

2 | 0.449 | 0 | 0 | 0 | 0 | 0 | 0.0142 | 0 | 0 |

3 | 0 | 0.449 | 0 | 0 | 0 | 0.0142 | 0 | 0 | 0 |

4 | 0 | 0 | 0.0232 | 0 | 0 | 0 | 0 | 0 | 0 |

5 | 0 | 0 | 0 | 0.0232 | 0 | 0 | 0 | 0 | 0 |

6 | 0 | 0 | 0 | 0 | 0.0232 | 0 | 0 | 0 | 0 |

7 | 0 | 0.0142 | 0 | 0 | 0 | 0.00619 | 0 | 0 | 0 |

8 | 0.0142 | 0 | 0 | 0 | 0 | 0 | 0.00619 | 0 | 0 |

9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.00619 | 0 |

10 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.00619 |

As you may see, the Noll matrix is almost diagonal (less so for higher orders, however). Why the coefficient is missing? For Kolmogorov turbulence it is infinite! However, the first (piston) mode is of no importance to imaging.

** Question:** For 4m telescope and 1 arcsecond seeing, compute the
r.m.s. amplitude of tilt in radians (for 0.5 micron
wavelength). Convert it to the r.m.s. amplitude of stellar image
motion. Does this amplitude depend on the wavelength?

What happens if we correct the first modes with Adaptive Optics? The corresponding coefficients become zero, and the total phase variance is reduced. Denoting the pupil-averaged phase variance as , we write

(17) |

The full, uncorrected atmospheric phase variance (all modes except piston) corresponds to . In other words, in a telescope with aperture diameter of the atmospheric phase variance is about 1 square radians. If tip and tilt are corrected, then . It means that tip and tilt contribute 87% of the total phase variance. Correcting the modes up to radial order 2 leaves us with , the radial order 3 leaves uncorrected variance of . As you see, further reduction of phase variance demands the correction of larger and larger number of Zernike modes.

For a large number of corrected modes (, as happens in real systems), the assymptotic formula of Noll is very useful:

(18) |

** Question:** Using the values of and given
above, compute .

How many modes do we need to correct? Opticians know that when the
residual phase is less than 1 radian, the image quality approaches the
diffraction limit. You have by now all the tools to predict the
required number of modes as a function of telescope diameter, seeing,
and wavelength! It is sufficient to write
and to work back all the formulas (** try to do it!**). The
result is

(19) |

** Question:** How many Zernike modes must be corrected at the 4 m
telescope for imaging at 0.5 and 2.2 microns under 1 arcsecond seeing?

Is it necessary to correct the turbulence using Zernike modes? Of course not, the phase aberrations can be measured and corrected with any other set of basis functions, or without any modes at all, operating directly on the wave-fronts. It turns out that Zernike modes are the second best choice (the best set of modes is called Karunen-Loeve modes). The choice depends on the number of controlled parameters (modes) needed to achieve a given degree of correction; for Zernike modes it is less than for local wave-front control.

** Summary.** In this chapter, the fundamentals of imaging in ideal
and aberrated telescopes were reminded (PSF, OTF, diffraction limit,
Strehl ratio). Then the basic atmospheric parameters relevant for
Adaptive Optics (phase structure function, seeing, , time constant and isoplanatic angle)
were introduced. The de-composition of the random phase aberrations on
Zernike modes was studied. Now we are able to predict the number of
modes that need to be corrected.

TOP: Introduction

FORWARD: Deformable mirrors