Wavefront aberrations not resulting from the 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 proper use and near-optimum environment.
5.1. Air-medium errors
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 air of varying temperatures.
Since air optical density, and so the speed at which 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 layers of turbulent air forming right next to the optical
5.1.1. 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 carrying this relatively slowly changing turbulent structure consisting of air pockets varying in size, temperature and refractive indici, across the window of the aperture. Smaller-scale variations in air optical properties - roughly at the level of an average turbulent cell - are the primary source of wavefront roughness. Compounding of optical path length variations (i.e. advance and retardation of points on the wavefront) at a larger scale creates wavefront tilt deformation.
Exaggerated illustration below shows the basic mechanism of wavefront distortion by turbulent air. For simplicity, it is reduced to a single layer, darker hue representing cooler, optically denser air. Small aperture suffers low-level roughness, and mainly inconsequential tilt error tending to move the image as a whole in time intervals sufficiently long for the eye to follow (depends mainly on the wind speed). Large aperture is more affected by roughness, breaking diffraction pattern into a speckle structure, with individual speckles popping up, moving around and disappearing mainly as a consequence of the sideways movement of turbulent air, producing quickly changing patterns of turbulent cells and inducing variations of the tilt component.
So the structure, shape and extent of speckle pattern changes simultaneously due to the constant change in the pattern of turbulent air-cells and change in the tilt component, the later causing random movement of larger portions of the image. Speckle pattern remains nearly unchanged over millisecond intervals. Two speckle patterns are similar only over solid angle of a few arc seconds (isoplanatic angle), depending on the strength of turbulence and the height of the dominant turbulence layer. General relations for isoplanatic angle is ι=r0/h, with r0 being the Fried parameter, and h the turbulence height. For r0 and h in meters, it gives the isoplanatic angle in arc seconds. With the average values, r0~0.07m and h~5km (the actual value of h is determined by the turbulence intensity structure from upper atmosphere to the ground layer), ι~2.9".
Also, any given speckle pattern remains nearly unchanged only for very brief period of time, so called coherence time, determined by the wind speed and direction. In the direction of wind, it is expressed by τ=r0/v, where v is the wind speed. For r0 in mm and wind speed in meters per second, it gives the coherence time in milliseconds. With a typical wind speed of ~20m/s, in average seeing (r0~70mm@550nm) τ~3.5ms.
To the left is illustration of the mechanism of wavefront deformation by turbulence; graph to the right shows Hufnagel-Valley Boundary model of the atmospheric turbulence profile. It is a generalized, very rudimentary model of the structure of atmospheric layers generating turbulence. It indicates only two distinctive layers, one gravitating toward the dividing line between troposphere and stratosphere, and the other forming toward the planetary boundary layer, most intensely within less than 500m above ground. Strength of turbulence is indicated by the quantity Cn2, called the structure constant of the refractive index fluctuations (normalized to 1 for the peak turbulence strength on the graph). The logCn2 plot shows the form of variation in turbulence strength with altitude more clearly; the Cn2 plot indicates that the contributions to the total strength of turbulence is roughly divided between the turbulence within the boundary layer, called ground turbulence, and turbulence in the higher atmosphere. The latter is generally much weaker, but it is compensated by much longer total path length through it.
Actual structure of turbulent atmospheric layers vary widely with the locality and time; in general, main contributors to the turbulence strength are the boundary layer and one or more relatively narrow layers of stronger turbulence in the higher atmosphere. Seeing is usually at its best when the upper turbulence layers are relatively weak, leaving ground turbulence as the dominant component.
Looking at the depiction of the tilt component of the seeing-perturbed wavefront directly suggests that the tilt error will be larger in larger aperture. While wavefront tilt over an extended portion of it doesn't change significantly, the nominal error is, in first approximation, proportional to the extent of the wavefront over the aperture. While there will be relatively small difference in either nominal linear or angular image motion caused by this error, the relative movement - in terms of angular size of aberration-free FWHM, will be increasing with the aperture. Also, with given wind speed, the frequency of change in the direction/magnitude of tilt error is also larger in larger aperture. The consequence for visual observing is that the eye gradually loses the ability to follow the image motion going from relatively small toward more significant seeing errors, with the image becoming increasingly more blurred, as a result.
Note that the above depiction of atmospheric coherence length (explained in more details ahead) is a statistical fiction; actual size of orthogonally projected atmospheric segments over which the error generated by wavefront doesn't exceed any given value varies more or less significantly around the average value. The actual size of turbulent structures covers wide span between so called outer and inner scale of turbulence, with the former varying from a few meters close to the ground to a few hundred meters in the free atmosphere, and the latter from a few mm above the ground to about 1cm near troposphere. At the level of inner scale of turbulence, kinetic energy starts dissipating into heat by viscous friction, dampening transfer of kinetic energy (i.e. turbulent motions) within smaller structures.
The basic principles for the statistical evaluation of the error induced to wavefront passing through atmospheric turbulence were proposed by Kolmogorov in 1941, as a scaling law, based on successive transfer of kinetic energy from the largest to smaller turbulent structures. Together with Obukhov's law, describing structure function of the refractive index under conditions of turbulence, it supplied the basic tools for expressing the effect of turbulence on the integrity of the wavefront passing through it. The theory further evolved through the works of Tatarski, Fried, Noll, Roddier, and others, making possible major advance in the performance of large ground telescopes by use of adaptive optics systems.
When the turbulence is strong enough to break up and enlarge central diffraction maxima, telescope resolution is determined by the atmosphere, rather than aperture size. The central maxima of PSF expanded by turbulence (atmospheric PSF) is called seeing disc, and its FWHM is seeing.
While the strength 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 turbulence diameter 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. The parameter is defined so that, to a first approximation, telescope's resolving power is limited by its aperture for r0>D, and by the atmosphere for r0<D.
The above mentioned structure constant Cn2 is the main determinant of the size of r0.
In terms of r0, the FWHM of long-exposure atmospheric PSF (seeing) is given by FWHML=0.98λ/r0, in radians (FWHML=202,140λ/r0 in arc seconds), with λ being the wavelength for which r0 is calculated. Other seeing parameters are also related to r0. The larger r0, the wider isoplanatic angle. In the simplest case with a single turbulent layer at the height h, isoplanatis angle is given by θ0=r0/h in radians. For, say, r0=70 mm and h=5km, isoplanatic angle is three arc seconds. Obviously, for given r0, isoplanatic angle is the largest when ground turbulence is a dominant component.
Likewise, for the simplest case with a single turbulent layer moving in the direction of wind, coherence time is given by τ0=r0/ν, in milliseconds, with ν being the wind speed in m/sec. For the same r0=60mm and wind speed of 15 m/sec, coherence time is 4 milliseconds.
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. 76) 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.
Being random aberration, seeing error comprises many different aberration forms constantly varying in magnitude. The dominant long-exposure form is wavefront tilt. Short-exposure error is corrected for tilt, and the remaining roughness component consists of a large number of aberration components.
Different components of long-exposure wavefront error caused by atmospheric turbulence can be quantified for its time-averaged form using wide range of Zernike terms of appropriate magnitudes. Table below lists first 12 components (21 counting sine and cosine terms separately, as marked in the j column) of the Zernike-modeled time-averaged turbulence error (variance source: Optical Imaging and Aberrations 2, Mahajan). The table is based on Noll's Zernike terms expansion concept.
Piston, or uniform aberration, has zero variance over the pupil (it only becomes source of aberration in a system with multiple pupils of unequal phase); thus the first row gives values for the total aberration, in units of (D/r0)5/6 for error standard deviation, in units of (D/r0)5/3 for variance (i.e. standard deviation squared), and in units of the full phase for RMS wavefront error (i.e. the actual error is a product of the nominal value in the table and the corresponding unit).
Every next row gives values after the specified aberration
component is corrected and deducted from the total error; for instance, after correcting for the tilt component
in both cosine and sine term,
the residual phase variance (φ2) is 0.134, and the corresponding RMS
phase error (analog but not equivalent to the RMS wavefront error) is 0.0583(D/r0)5/6,
D being the aperture diameter; since the RMS error differential to the error level before correction
for tilt is 0.1032, it equals the term-specific RMS phase error (for
cosine and sine tilt terms combined). The latter is given by the
differential between the residual RMS error for the previous and
* in units of
For higher orders (j>9), Noll's approximation for the residual variance after first J of Zernike j modes are removed is φ2~0.2944J-0.866(D/r0)5/3. Expectedly, the D/r0 quotient is one of the very basic parameters in imaging through turbulence. It expresses the seeing disc to aberration-free FWHM ratio, (λ/r0)/(λ/D), hence its value directly indicates the size of long-exposure seeing disc as larger from the aberraton-free FWHM by a factor of D/r0.
Since the total aberration is proportional to to 1.0148, the relative error contribution of each separate error component is closely approximated by their φ differential; for defocus it is 0.033 (3.3%) of the total aberration, and after the first 21 components are corrected, remaining aberration, as the RMS phase error, is 14.2% of the total aberration.
In terms of r0 and the aperture diameter D, the long-exposure atmospheric error as a wavefront phase variance averaged over the pupil, is given by:
φ2 = (2πφL)2 = 1.03(D/r0)5/3 (52)
in rad2, with φL being the RMS phase deviation in units of the wavelength. Thus, the long-exposure RMS phase error
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 disintegration 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 phase RMS error for roughness to:
or about 36% of the total RMS error. Note that these RMS errors cannot be used with Strehl approximations for non-random aberrations (Eq. 56, and others), due to statistical nature of the seeing error (these approximations generally underestimate the seeing Strehl). Appropriate Strehl approximations and exact values are given in 5.1.2. Seeing error: Strehl, resolution, OTF.
Following page addresses in more
details how the effect of atmospheric turbulence relates to the