1. INTRODUCTION

The performance of any optical/infrared (O/IR) telescope is determined by its point spread function (PSF). For perfect, circular optics this characteristic is entirely determined by just one parameter: the aperture diameter, which in turn determines the exact shape of the PSF for a particular observing wavelength.

In reality there are some other factors, related to environmental conditions and constructive
specifics of the telescope, which may have a strong influence on its final PSF. For
a ground-based telescope the most influential external factor is the atmospheric
optical turbulence, which leads to image blur in the telescope’s focal plane. The
quantitative criterion for assessing the quality of optical turbulence is so called
*seeing*, which represents the full-width at half maximum (FWHM)
angular size of the atmospheric long-exposure PSF. In the absence of other adverse
factors (telescope tracking, vibration, focusing, etc.) the ultimate shape of the
telescope’s PSF is determined by the pure optic PSF and the seeing.

Usually, circular apertures are assumed, when assessing the potential abilities of different
size telescopes. But actually all modern medium to large size O/IR telescopes have
annular apertures (except for solar telescopes) due to central obscuration, dictated
by the optical and mechanical design. This affects the characteristics of the
perfect optics in a few ways: firstly it simply blocks incoming light by the amount
corresponding to respective detail’s size, thereby reducing the aperture effective
area; secondly it alters the shape of the PSF; and thirdly it contaminates the focal
plane by the light scattering and diffractive effects on the respective elements
(baffles, spiders etc) (^{Kuhn & Hawley
1999}). For pure annular apertures the parameter *ε = d/D* is
used to describe the order of central obscuration. Here *d* and
*D* are, respectively, the inner and outer diameters of the
annulus. In some cases the obscuration can be relatively big and may significantly
change the PSF characteristics. This is especially the case for survey telescopes
where *ε* can reach 0.4 0.6. Typical examples of different class
instruments are the Pan−Starrs (*D* = 1.8 m, *ε* ≈
0.57), SDSS (*D* = 2.5m, *ε* ≈ 0.52), VISTA
(*D* = 4.1m, *ε* ≈ 0.41) and LSST
(*D* = 8.4 m, *ε* ≈ 0.61). In such cases the final
(telescope + seeing) PSF may also be changed noticeably and hence the optical power
of the telescope.

In this paper I consider the influence of central obscuration on the final performance characteristics of a telescope, while working under certain seeing condition. Here pure annular apertures without any spider are assumed. A brief summary of parameters and formulas is presented in § 2. The results of the analysis are given in § 3 followed by the discussion and conclusions in § 4.

2. THEORETICAL BASIS

The main optical characteristic is the point spread function (PSF). It represents the
normalized intensity *I* of the point like source as function of
coordinates in the focal plane. For a perfect annular aperture with obscuration
*ε* it can be presented mathematically as (^{Born & Wolf 1970}):

where k = 2π/λ is the wave number, λ is the observing wavelength, a is the aperture radius and *x*, *y* being the coordinates in the focal plane.

Another important characteristic of any optical system is the optical transfer function (OTF), *H*(*u, v*). Mathematically the PSF and OTF are Fourier transforms of each other:

where *u* and *υ* are spatial frequencies. The modulus of the OTF, called modulation transfer function (MTF), represents the ability of the optical system to transfer different spatial frequencies. For ideal optics with an annular aperture the OTF H_{opt}(*f*) can be represented by:

Where

otherwise;

and the spatial frequencies

Ground-based O/IR telescopes are subjected to the adverse influence of the atmospheric optical
turbulence. In essence the terrestrial atmosphere is a randomly changing optical
system in front of the telescope which can be characterized by its own parameters.
With the use of atmospheric optical turbulence theory (^{Tatarskii 1961}) it is possible to describe this system. The
coherence radius or Fried parameter, *r*_{0}
is the main characteristic of the atmospheric optical turbulence condition, defined
by ^{Fried (1965)} as

where *z* is the zenith angle, and

The seeing, *β* represents the FWHM angular size of the atmospheric optical turbulence long-exposure PSF

The angular resolution, *θ*_{0} represents the FWHM
size of the telescope’s ultimate (optics + atmosphere) PSF.

It has been shown (^{Fried 1966}) that the long exposure OTF of the atmosphere is

where *D*_{φ} is the phase structure function and for
Kolmogorov turbulence it is represented by

where *r*_{0} is normalized to the aperture size
*D*.

For the case of a finite turbulence outer scale
*L*_{0}, it can be described as (^{Jenkins 1998})

where *K* is a Bessel function of the second kind and
*L*_{0} is also normalized to the
aperture size *D*.

The ultimate OTF of the telescope is a product of pure optics H_{opt} and atmospheric H_{turb} OTFs

3. RESULTS OF NUMERICAL ANALYSIS

**3.1. Infinite Outer Scale
**

**3.1.1. Angular Resolution
**

The FWHMs are calculated by using the above mentioned formulas. At first, the OTFs of a
telescope for a particular ε and atmosphere for a certain r0 are calculated.
Then the final PSFs are determined via the Fourier transform of the ultimate
OTFs. The corresponding FWHMs are normalized to the unobstructed (ε = 0)
case. The results are presented in Figure
1. Here for convenience the horizontal axis indicates a
*D/r*_{0} relation. It can be
seen that *D/r*_{0} ≳ 1.1 the FWHMs
of obscured apertures are relatively wider and have local maxima at
*D/r*_{0} ≈ 2.2. This
corresponds to a wavefront overall rms error of
*σ*_{φ} ≈ 1.97 rad. But for
*D/r*_{0} ≲ 1.1 the angular
resolution of an obscured aperture becomes higher. The figure also indicates
that when σ_{ϕ} ≲1 rad diffraction effects dominate over the optical
turbulence in a process of long-exposure PSF formation.

**3.1.2. Integral Contrast
**

Integral contrast (IC) is defined as the integral of the telescope’s ultimate MTF over the entire spatial frequencies:

This definition is more appropriate as a performance metric, which concerns the spatial
resolution when dealing with extended objects. It can be used to
characterize quantitatively the ability of a telescope to reproduce the fine
structure of an object. In order to make a comparative analysis relative
values
*IC*_{ε}*/IC*_{ε=0}
for different obscuration parameters are calculated (Figure 2). It is seen that the relative IC curves have
local minima at *D/r*_{0} ≈ 1.25
i.e. when *σ*_{φ} ≈ 1.22 rad the
relative performance of an obscured telescope is significantly worse than it
would be without the presence of optical turbulence.

**3.1.3. Signal-to-Noise Ratio
**

It is well known that the signal-to-noise (*S/N*) ratio of a point-like source with an intensity B notably less than sky background level b is given by:

where *q* is the detector’s quantum efficiency (noise contribution is
neglected), *τ* is the system’s overall transmittance and
*t* is the exposure time. Under equal conditions, a
certain “image size” *d*_{ε} has to
be found in order to achieve maximal *S/N* for a particular
PSF. For a comparative analysis relative S/N values for a certain ε
normalized to unobstructed case are calculated by:

where *I*_{0},
*I*_{ε}, are the final
(telescope + atmosphere) PSFs and
*d*_{0},
*d*_{ε} are the optimal
“image sizes” for circular (*ε* = 0) and annular apertures,
respectively. The results are presented in Figure 3. In these calculations the coefficient *D/r*_{0} ≈ 0.67
(σ_{φ} ≈ 0.72 rad) and then asymptotically approach the levels
corresponding to the seeing-free
(*D/r*_{0} << 1)
case.

**3.2. Finite Outer Scale
**

The turbulence outer scale adds its own contribution to the telescope’s final performance. As
it follows from expression (9) accounting for L0 leads to a smaller wavefront
phase error under the same seeing. Unlike the previous case, where performance
characteristics can be introduced as a function of only one parameter
(*D/r*_{0} or wavefront overall
error as *L*_{0}*/D* ratio. In
order to avoid complexity the case with *ε* = 0.6 is considered
to show the general trends of the curves evolution. A median outer scale size of
*L*_{0} = 25 m (^{Ziad et al. 2000}) is adopted as mostly
anticipated and
*L*_{0}*/D* = ∞,
12.5, 3.125, 1.5625 are chosen to represent the conditions for different size
(*D* = 2 m−16 m) telescopes. Also here the final performance
characteristics of an 8m telescope with ε = 0.6 (like LSST) have been examined
as a typical example of large survey telescope acting under the range of
plausible optical turbulence conditions (seeing at 500nm = 0.3, 0.5, 0.8 arcsec;
*L*_{0} = 12.5, 25, 50 m). Trends of
the major characteristics for spectral region spanning from UV to IR (λ = 300 −
5000 *nm*) for this case are obtained.

It can be seen that the introduction of L0 notably changes the relative performance curves
(Figure 4). Particularly, the local
maxima/minima for all characteristics are shifted towards bigger
*D/r*_{0} values while
*L*_{0}*/D* is
decreasing. At the same time maximum points for FWHMs become relatively higher,
whereas minimum points for contrast and *S/N* become less deep.
The final relative performance characteristics of the assumed telescope are
presented in Figure 5.

4. DISCUSSION AND CONCLUSIONS

Ground-based O/IR telescopes with annular pupils acting under certain optical turbulence
condition have performance characteristics which differ from those of circular ones.
This difference is not constant, as for the seeing-free case, but apart from the
obscuration parameter ε it also depends on the seeing *β*, the
turbulence outer scale *L*_{0} and the
observation wavelength λ.

If a Kolmogorov power spectrum is assumed then the telescope relative performance for
different obscuration can be presented as function of
*D/r*_{0}. In the case of finite
outer scale *L*_{0} the situation is
complicated and each case with certain
*L*_{0}*/D* must be
considered separately.

The general trend of the performance curves is that FWHMs and IC values for
*D/r*_{0} ≫ 10 are nearly the same as
for an unobstructed telescope. At the same time, relative S/N values are smaller
initially due to the aperture’s effective area reduction. As the
*D/r*_{0} relation decreases the
characteristics decline, reach a certain extremum point and then rise gradually up
to the levels corresponding to the seeing-free case
(*D/r*_{0} ≪ 1). The interesting
result is that at certain values of
*D/r*_{0} the relative performance of an
obscured telescope becomes worse. The extrema correspond to
*D/r*_{0} ≈ 2.2, 1.25, 0.67 for the
resolution, IC and S/N respectively. For instance, under a 0.8′′ seeing the maximum
effect on the resolution will occur for telescopes with *D* ≈ 0.31,
1.64 and 4.2 m for the *V, K* and *M*-band
respectively. So in this condition the effect in the visible is strongest for
small-size telescopes but in the NIR to mid-IR medium-size telescopes are most
severely affected. Performance degradation can be as much as ≈ 21%, 47% and 49% for
the resolution, IC and S/N (after accounting for vignetting) respectively, for
*ε* = 0.6.

The presence of a finite outer scale, *L*_{0}, shifts
the extrema towards bigger *D/r0* values. As the
*L*_{0}*/D* relation
decreases the angular resolution at the extremal point continuously deteriorates up
to ≈ 34% for *L*_{0}*/D* =
1.5625 and *ε* = 0.6. (Figure
4). Meanwhile, the IC and S/N become slightly less degraded. For instance,
under a 0.8” seeing and *L _{0}* = 25 m, maximum deterioration of resolution for
telescopes with

*D*=2,4 and 8m will be at λ≈1.84, 2.9 and 4.16

*μm*respectively.

It is seen that the most severe performance deterioration of any medium to large size (D ≳
2*m*) telescope, working in the seeing-limited mode, takes place
in the NIR to mid-IR spectral bands. But the situation will dramatically change if,
for example, adaptive optics (AO) methods are applied. It can be shown (^{Martinez et al. 2010}) that low-order AO
implementation is equivalent to a reduction of the initial outer scale to the
smaller “effective” *L*_{0} size and a
simple tip-tilt compensation will lead to an “effective”
*L*_{0}*/D* of ~ 2.
For this condition the extreme points will be at
*D/r*_{0} ~ 6.5, 4.5, 2.9 for the
FWHM, IC and S/N respectively (Figure 4). In
this observational mode, under a 0.8′′ seeing, the maximum effect on resolution will
be for apertures with *D* ≈ 0.91, 4.8 and 12.3 m for the *V,
K* and *M*-band. Furthermore, the AO compensation for
higher order aberration terms will reduce “effective”
*L*_{0} futher to significantly
smaller values. So in this observational mode the effect of a central obscuration
for large size apertures may become significant even in the optical bands.

A relative performance analysis is done for an 8 m aperture with *ε* = 0.6 in
analogy to modern large size survey telescopes. It turns out that the relative
performance of such a telescope, working in the seeing limited mode under the
adopted atmospheric conditions can significantly differ from an unobstructed
aperture of the same size. In general the angular resolution,
*θ*_{0} can be relatively better or
poorer depending on the observation band and the turbulence parameters, whereas the
IC and S/N are always degraded. If we assume
*L*_{0} = 25 m as a more probable
value then performance degradation appears to be at levels less than ≈ 6% in the
*V* -band for all suggested seeing conditions. The maximum effect
occurs in the *K* to *M*-bands. Particularly, θ0 can
be decreased by up to ≈ 34% or increased by up to 15% depending on the seeing. At
the same time, for example, the angular resolution at the *M*-band
may be superior or inferior to the unobstructed case under the same
*β* = 0.8′′ seeing depending on the outer scale
*L*_{0} size (Figure 5). Contrast also gradually decreases from ≈ 0.99 in the
U-band (300 nm) to nearly 0.57 in the NIR. S/N is the most affected characteristic,
which drops from 0.8 to 0.41 from the *U*-band to the M-band
respectively. So losses of penetrating ability may reach
0.95^{m} in some spectral bands.

Modern telescopes are indeed sophisticated optical-mechanical systems which are dedicated to
solve different scientific tasks. So the ultimate choice of the telescope’s optical
scheme is a trade-off between complex technical issues and performance requirements.
Usually the central intensity ratio (CIR) (^{Dierickx
1992}) is used for a ground-based telescope’s performance evaluation. It
represents the IC drop caused by pure optical imperfections, relative to perfect
case and CIR ≥ 0.82 is widely adopted as a requirement. Now if we generalize this
criterion on the topic which is being considered, then the maximum affordable
obscuration will be determined. So the requirement on CIR leads to corresponding
requirements on
*IC*_{ε}*/IC*_{ε
= 0} ≥ 0.82. It turns out that the obscuration parameter must
be *ε* < 0.34 in order to satisfy the above mentioned precondition
through a broad spectral region. At the same time CIR also indicates the peak
intensity drop in a telescope’s actual long-exposure PSF relative to the perfect
case and *ε* < 0.2.

Actually, many modern telescopes have obscurations which exceed these values. As it follows from calculations, losses of optical power may be significant in a certain spectral region. For medium to large size telescopes, acting in the seeing-limited mode, this region mainly falls in the NIR to midIR part of the spectrum. But if an AO technique is applied then the situation will change. In particular ground-layer AO (GLAO) or multi-object AO (MOAO) techniques allow to partially compensate for wavefront errors in a relatively large field. So the “equivalent seeing” will become notably better and will push the extreme region toward shorter wavelengths. So it makes sense to seriously consider such classes of telescopes with off-axis optical design schemes, which will allow to more efficiently realize their potential capabilities.