◄ 4.8. Chromatic aberration   ▐    5.2. Misalignment and forced surface deviations ►
 

Wavefront aberrations not resulting from inherent properties of optical surfaces are more than common: to some degree, they are affecting imaging quality of telescopes at all times. According to their origin, they can be result of: (1) air non-homogeneity, (2) alignment errors, and (3) forced surface deformations. Some induced aberrations may be identical in form to those inherent to conical surfaces - for instance spherical aberration created by despace error in multi-element systems, or due to thermal expansion or contraction - but they remain dependant of factors other than inherent properties of optical surfaces in their intended mode of use, in near-optimum environment.

3.1. Air-medium errors: Atmospheric turbulence and tube currents

Wavefronts can preserve their form only if the media they move through are optically homogeneous. In other words, when every wavefront point propagates at identical speed. Thus, inhomogeneous media induce random wavefront deviations, resulting in optical aberrations. Air is by far the largest media through which light travels before forming the image in a telescope. What causes air to be inhomogeneous medium for the light are thermal effects, creating randomly changing structures consisting of streams and eddies of the air of varying temperatures. Since air optical density (that is, the speed at which the light propagates through it) changes with the temperature, such random thermal structures cause it to become an inhomogeneous optical media. In various forms, it begins in the high layers of Earth's atmosphere and extends all the way down to both, outside and the inside of a telescope tube, including air layers forming right next to the optical surfaces.

Atmospheric turbulence

The two main mechanisms inducing seeing error in telescopes are (1) vertical air movement caused by warmer air raising and mixing with cooler air above, and (2) upper level winds quickly carrying this relatively slowly changing turbulent structure consisting of pockets varying in size, temperature and refractive indici across the window of the aperture. The former is the primary source of wavefront roughness, while the latter causes motion of the image as a whole (wavefront tilt).

Wavefront deformations caused by atmospheric turbulence are ever-present, only vary in amplitude. The effect is popularly called the seeing error, and measured either through empirical scales of seeing quality - such as Pickering's 1-10 (FIG. 34) and Antoniadi 1-5 scale - or analytically, as given by the optical theory's take on random aberrations. For given level of seeing quality, an average wavefront error induced by atmospheric turbulence increases with the aperture size. While the degree of turbulence varies constantly, an average error to the wavefront passing through it can be expressed in terms of atmospheric coherence length (or "atmospheric coherence diameter"). This parameter is defined as an areal unit projected from the turbulent air

mass, over which the time-averaged error induced to the wavefront doesn't exceed one phase radian (1/2π of the full wave phase of 2π radians). It is usually called Fried parameter, and denoted by r0. In terms of r0 and the aperture diameter D, the long-exposure atmospheric error as a variation of the standard (phase) deviation in radians, averaged over the pupil, is given by:

                                                      Φ2 = (2πω)2 = 1.03(D/r0)5/3                                      (52)

with ω being the RMS wavefront deviation in units of the wavelength. Thus,

                                      ω = [1.031/2(D/r0)5/6]/2π   =  0.162(D/r0)5/6                      (53)

with the wavelength λ corresponding to that used to calculate the value of r0 (Eq.55). This wavefront error has two main components: (1) tilt, resulting in random image motion, and (2) roughness, resulting in structural deterioration of the diffraction pattern. For large apertures, the tilt component is dominant; short-exposure wavefront error - with "short exposure" defined as sufficiently short to eliminate image motion - reduces to the wavefront roughness RMS error:

                                      ωS= 0.1341/2(D/r0)5/6]/2π  =  0.058(D/r0)5/6                      (53.1)

or about 36% of the total RMS error.

However, the tilt error effect is not detrimental at very small apertures (smaller than ~4") where, in average seeing, it causes image to move about randomly as a whole, relatively slowly, without significantly  affecting its visual quality. Thus, in this aperture range, the size of seeing error in visual observing is mainly determined by the wavefront roughness component (Eq. 53.1). As the aperture increases - or the seeing worsens - the eye can't keep up with the frequency of image motion, as the diffraction pattern gradually breaks into a speckle structure, with an increasing number of  pattern components moving randomly within the speckle envelope. The result is progressive blurring of the image. The roughness error component too increases with (D/r0)5/6 until, at a sufficiently large aperture - or in sufficiently bad seeing - diffraction pattern disintegrates into a random speckle structure (FIG. 35).

The speckles, being formed by the energy coming from the fragments of the broken wavefront over the pupil that meet in phase at the points randomly scattered around Gaussian image point, approximate in size diffraction disc. The size of speckle spread, on the other hand, is approximated by λ/r0 in radians, which means that it is r0, not the aperture, that sets the resolution limit. The speckle phenomenon is not only characteristic of large professional telescopes. Even medium-size amateur telescopes in bad seeing begin to develop speckle structure. Number of speckles is approximated by N~(D/2r0)2; hence they begin to form for (D/r0)>3.

Due to a different nature of the seeing induced error, the Strehl approximation given by Eq. 56 becomes inaccurate for larger errors, considerably more than it is for conventional aberrations (FIG. 36). The effect is mainly due to the roughness error component, whose increase reduces the average relative size of wavefront irregularity, while at the same time increasing its average RMS error. Similarly to ordinary surface roughness, narrow zones and turned edge, the increase in the P-V (and thus RMS) error after a certain point begins to drain less energy out of the Airy disc, with the main effect gradually becoming merely spreading the energy out wider. As a result, the increase in the RMS (surface or wavefront) error is followed by relatively slower Strehl degradation, as given by the relation derived by Racine for long-exposure "seeing Strehl":

                               S={[1-exp(-Φ10/3/6)]/(1+(D/r0)2} + exp(-Φ2)                         (54)

where Φ=2πω is the phase variance across the pupil, with ω=0.162(D/r0)5/6 being the time-averaged (long-exposure) RMS wavefront error induced by seeing. Note that exp(...) denotes exponent (the value in brackets) over the natural logarithm base e(...), with e~2.718, rounded off to three decimals. So, for D/r0=1, the average RMS wavefront error ω=0.162 (Eq. 53), giving Φ=1.02, and the seeing Strehl S=0.445; for identical nominal RMS error ω of non-random aberrations the Strehl is 0.357.

Due to increased effect of the tilt error component in visual observing, either with the increase in aperture (for given seeing level), or in the seeing error (for given aperture), resulting visual Strehl gradually shifts from being near identical to the short-exposure Strehl (for low seeing error, or in very small apertures in average or better seeing), to be closer to the long-exposure Strehl (for large seeing errors). While the plot on Fig. 36 is a gross approximation, it does illustrate basic relations between the three main forms of seeing-induced error.

Size of r0 determines angular resolution of large telescopes as ~λ/r0 in radians (λ being the wavelength of light), as opposed to the "standardized" diffraction resolution given by ~λ/D. While the size of r0 results from a complex function, it can be expressed much more simply in terms of the resolution limit it imposes to large telescopes. For the seeing imposed stellar resolution α in radians, and the wavelength λc centered at 0.00055mm, it is given by:

                                                                      r0= 1.27lc/α                                             (55)

Hence, an average ~2 arc second seeing at the zenith, for λ=0.00055mm, results from r0~72mm (2 arc seconds in radians is 2/206,265). Since r0 is changing in proportion to λ1.2 and (cosγ)0.6, γ being the zenith angle in degrees, a more complete expression for its size is given by:

                            r0=1.27lc(l/lc)1.2(cosγ)0.6/α  = 5.7l1.2(cosγ)0.6/α                 (55.1)

Thus, the extracted stellar resolution α=5.7λ1.2(cosγ)0.6/r0, for given γ (zenith distance) and r0, also changes in proportion to λ1.2. From Eq. 55.1 it is obvious that this stellar resolution is not the actual resolution at the zenith angle γ, rather the zenith resolution for given λ, γ and r0. Actual stellar resolution at the zenith angle γ is inversely proportional to (cosγ)0.6 which, as expected, makes it directly proportional to the atmospheric coherence length r0.

So, for instance, at 30° from zenith (γ=30°) and the wavelength λ=0.0005mm, atmospheric coherence length r0=100mm corresponds to the zenith seeing α=0.000005716 radians, or 1.18 arc seconds (after multiplying by 206,265). The seeing at the zenith angle is numerically larger by a factor of 1/(cosγ)0.6, or

                                                                        αγ=5.7λ1.2/r0  =  α/(cosγ)0.6                                  (55.2)

 with r0 calculated from Eq. 55.1.

In terms of contrast transfer, the effect of atmospheric error is given by atmospheric OTF (Optical Transfer Function), which is for long-exposure given by TL=exp[-3.44(νD/r0)5/3], and for short-exposure it is given by TS=exp{[-3.44(νD/r0)5/3](1-1.042ν1/3)}. In both, ν is the normalized spatial frequency. Combined atmospheric and telescope contrast transfer is given by the product of their respective OTF's. It is the final OTF in the image plane of the objective. Obstructed apertures will have lower combined OTF due to the contrast-lowering effect of central obstruction, but not more. In fact, the larger obstruction, the larger relative size of r0 vs. annulus, resulting in a slower increase in phase variance over the annulus area and, consequently, slower decrease in the average Strehl (as a measure of wavefront quality, independent of the effect of central obstruction).
 

Tube currents

Compared to atmospheric turbulence, thermal imbalances between surrounding air and telescope tube, or any other structure enclosing optical elements usually produce more uniform stream of warmer air. Typical tube current is a slow flow of the warmer, lighter air off the tube structure to the upper part of the tube, where it forms a layer of slightly lower optical density. The light moves through it at a higher speed, resulting in the wavefront portion moving through it being advanced in regard to the portion moving through the slightly cooler air bellow.

Degree of thermally induced wavefront deformation arising from uneven air temperature around telescope structure depends on its design, material thermal properties and size. In general, larger apertures are affected more, which is rather a common point for all error sources not inherent to optical surfaces. Also, reflecting telescopes are affected more than refracting, due to the light path in the former being placed closer to the tube wall along its entire length. Given near-steady air temperature, the effect generally diminishes as a result of passive thermal balancing, but may persist if the air temperature keeps changing, or with thermally unsound mechanical designs. Unlike atmospheric turbulence, this source of error can be nearly eliminated with proper mechanical design and use of fans.

Thermal imbalance between the air and optical elements it surrounds creates thin layer of turbulent air in front of optical surfaces. Even small thermal imbalances of this kind can induce significant wavefront deformations. This effect is also influenced by the aperture size and thermal efficiency of the mechanical design. Natural (passive) thermal balancing often works well enough to diminish this source of error. However, large apertures and/or thermally closed or in some other way inefficient systems will likely require assistance of fans. Brian Greer's investigation illustrates very well thermally induced errors in a Newtonian.

There are other forms and instances of turbulent air causing wavefront deformations. Among them are local, or micro-turbulence, caused by the thermal emission off surrounding trees, roofs, pavement, the very bodies of people standing close by, including a body of the very observer. Quality observing requires knowledge of, and ways to control and minimize the effects of all sources of thermal errors.

◄ 4.8. Chromatic aberration   ▐    5.2. Misalignment and forced surface deviations ►
 

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