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4.9. Transmission materials' effects       5.1.2. Limitations to the standard model

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 optically inhomogeneous 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 around optical surfaces.
 

5.1.1. Atmospheric turbulence

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.

Since based on the Kolmogorov's theory, it is regarded as Kolmogorov model, or theory (of turbulence), and it's the one usually applied to describe the effects of seeing on imaging with telescopes. The formalism here is from this model of 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 index across the window of the aperture. Variations in the temperature cause refractive index variations, which in turn correspondingly advance or delay points on the wavefront relative, breaking up its form. Small-scale variations in air optical properties - at the level of smaller turbulent cells - are the primary source of wavefront roughness. Temperature/index variation at larger scales creates wavefront tilt deformation.

Deviations from uniformity in the refractive index, phase and wave are statistically defined through the corresponding structure functions. The structure function of the refractive index fluctuations is the mean square of the difference in refractive index at two separated points in space. Similarly, the structure function of phase fluctuations resulting from index fluctuations is the mean square of the phase differential at these two points. And the wave structure function is a sum of the phase structure function and log-amplitude structure function, the latter describing change in wave amplitude due to turbulence.

Schematic illustration below (FIG. 75 top left) 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 former causing constant change of the speckle structure, and the later causing random movement of larger portions of the image and image as a whole. Speckle pattern remains nearly unchanged over time intervals measured in milliseconds. 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 the typical wind speed of ~20m/s, in the average seeing (r0~70mm, λ=550nm) τ~3.5ms.


FIGURE 75
: TOP: Illustration of the structural dynamics in the atmosphere creating wavefront distortions (left) and a model of turbulence strength as a function of the height in the atmosphere (right). The phase/wavefront map shows characteristic pattern of a random phase/wavefront advance and delay due to turbulence. On it, typical size of rr0 would be less than 1mm in diameter. In effect, large error variation over the wider area is greatly diminished within the r0 window, which is merely a diameter over which statistical (time-averaged) error does not exceed π radians (phase) or half a wave P-V.
 BOTTOM: Change of
r0 with the turbulence structure constant Cn2 in the common seeing range (mean Cn2, here denoted as Cn20, per cm for 20km path); also, change of r0 with zenith angle, hourly change (based on Kemp et al. measurements), and momentary change in the average and excellent seeing (from Bradford and Roberts' paper). The former has 5.8cm, and the latter 26.5cm averaged r0, corresponding to 2.3 and 0.5 arc  seconds seeing, respectively (for 0.005mm wavelength, from 1.27*0.0005/r0 in radians, times 206,265 for arc seconds). As these measurements indicate, the range of relative variation in r0 from moment to moment widens toward better seeing - and more so nominally - while the range becomes much steadier on hourly scale. Over 24-hour period, r0 is significantly smaller during daylight than in night hours. It is often stated that r0 is the largest  - and seeing the best - after and before the sunset, but this particular measurement has the minimum between midnight and 3 AM.

Hufnagel-Valley Boundary model of the atmospheric turbulence profile 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. The 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 main component of the structure function of the refractive index fluctuations due to turbulence, Dn=Cn2r, where r is the distance between two points over which the fluctuation is accumulated. Th Cn2 itself is a function of air pressure, temperature and the rate of energy generation per unit mass.

The logCn2 plot compresses wide variation in the turbulence strength with altitude; 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.

 The actual structure of the turbulent atmospheric layers vary widely with the locality and time; in general, main contributors to the turbulence strength are the boundary layer and up to several layers of stronger turbulence in the higher atmosphere. Seeing is usually at its best when the upper turbulence layers are relatively weak, leaving the lower-level 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 the 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 the outer and inner scale of turbulence, with the former increasing from about 1m close to the ground to a few hundred meters high in the atmosphere, and the latter from 1-2 mm above the ground to a few cm in higher atmosphere. 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.

In general, inner and outer scale of seeing change in inverse manner with seeing quality, with the former becoming larger in better seeing, and the latter smaller, with less energy initiating a turbulence.

When the turbulence is strong enough to break up and enlarge the 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 the  seeing disc, and its FWHM the 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 the turbulence diameter over which the time-averaged error induced to the wavefront doesn't exceed one phase radian (1/2 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 structure constant of the refractive index Cn2 is one of the determinants of the size of r0, which is for plane wave propagating through Kolmogorov turbulence given as:

r0 = 0.185λ1.2cos0.6γ[∫Cn2(L)dL]-0.6

with λ being the wavelength, γ the zenith angle, L the wave path length through the atmosphere, and quantity in brackets the integrated (summed up) values of Cn2 along this path. For the mean Cn2 value for the entire path, denoted by Cn20, it can be written as r0 = 0.185λ1.2cos0.6(γ)(Cn20L)-0.6. Alternately, r0=1.68(Cn20Lk2)-0.6, with k=2π, also for plane wave.

     In Kolmogorov turbulence, r0 can be used to define statistical changes in wave properties (the wave structure function) with a simple relation Dw=6.88(r/r0)5/3, where r is the separation in two points in space (since projected vertically, it can be .

     Typical Cn2 values in the lower 20km, or so, of the atmosphere range from 10-15 for strong turbulence to 10-19 m-2/3 (cm) for weak turbulence, with the highest values in the free atmosphere being at the height of about 10km, while reaching its maximum well inside the boundary layer. The latter is commonly dozens of times stronger, on he average, than the free atmosphere turbulence, but so its depth (path), with its error contribution of generally comparable to that of the free atmosphere.

     In terms of mean atmospheric structure constant Cn20, it is typically between 10-17 and 10-18. Taking λ=0.000055cm, γ=0 (so cosγ=1), L=2,000,000cm (20km) and assumed mean Cn20 for the atmosphere equal to these values, gives the corresponding zenith range in r0 from about 3.8cm to 15cm, respectively (in terms of seeing, with r0~14.4 cm for this wavelength in 1 arc second seeing, from about 4 to 1 arc second, in the same order).

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.


FIGURE 76
: Pickering's seeing scale uses 10 levels to categorize seeing quality, with 1 being the worst and 10 near-perfect. Matching seeing descriptions are: 1-2 "very poor", 3 "poor to very poor", 4 "poor", 5 "fair", 6 "fair to good" 7 "good", 8 "good to excellent", 9 "excellent" and 10 "perfect". Seeing becomes diffraction-limited (~0.8 Strehl) from ~8.
 It may be practical to relate this popular scale with the basic parameter of optical theory related to seeing, D/r0. While the Pickering's scale is strictly for a 5-inch refractor, it can be easily generalized based on the pattern's relative size vs. Airy disc. Verbal description of the scale according to Pickering's notes is:
1 Star image is usually about twice the diameter of the third diffraction ring, or 13 arc seconds (13") in diameter
2 Image occasionally twice the diameter of the third ring
3 Image about the same diameter as the third ring (6.7"), and brighter at the center
4 The central maxima often visible; arcs of diffraction rings sometimes seen on brighter stars
5 Central maxima smeared but always visible; arcs frequently seen on brighter stars
6 Central maxima somewhat blurred, always visible; short arcs constantly seen
7 Central maxima sometimes sharply defined; diffraction rings seen as long arcs or complete circles
8 Central maxima always sharply defined; rings seen as long arcs or complete circles, but always in motion
9 The 1st diffraction ring is stationary; outer rings momentarily stationary.
10 The complete diffraction pattern is stationary
With the perfect 5-inch FWHM of slightly over 0.9 arc seconds, 13 arc second blur at the level 1 is about 14 times larger. Assuming that Pickering refers to the entire visible blur of a brighter star, not only its bright core (seeing FWHM), it corresponds to D/
r0~6 long-exposure pattern. Since at this error level there is little difference in size between the long-exposure FWHM and the visual blur, this first - the worst - level on the scale can be assumed to be worse than D/r0=5 seeing. The rest is simply fitting in D/r0 values to the pattern descriptions.

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 the error's standard deviation, in units of (D/r0)5/3 for the variance (i.e. standard deviation squared), and in units of the full phase for the RMS wavefront error (i.e. the actual error is a product of the nominal value in the table and the corresponding unit). It should be noted that the RMS error obtained from the standard (phase) deviation by dividing it by 2p (i.e. as the averaged wavefront deviation corresponding to the averaged phase variation) is a fictional value that nearly certainly will not represent the actual wavefront's RMS error - the reason being that the linear increase in the P-V wavefront error (which is the base for the RMS error) results in a cyclic change in the corresponding phase. But being tied to the phase it is more reliable in predicting diffraction effects of the error.

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 wavefront 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 the tilt is 0.1032, it equals the term-specific RMS wavefront error (for cosine and sine tilt terms combined). The latter is given by the differential between the residual RMS error for the previous and following aberration.
 

WAVEFRONT STRUCTURING OF THE AVERAGED (LONG EXPOSURE) SEEING ERROR

CORRECTED ZERNIKE COMPONENT

RESIDUAL PHASE ERROR*

RMS WFE ERROR φ

j

Variance
φ2

Standard
deviation
φ=
2πφ

Residual

Term-specific

#

Order

Name

nominal**

%

nominal**

%

1

-

Piston

1

1.0299

1.0148

0.1615

100

0

0

2

2nd

Tilt

cosine term

2

0.582

0.763

0.1215

75.2

0.0400

25

sine term

3

0.134

0.366

0.0583

36.1

0.0632

39

3

2nd

Defocus

4

0.111

0.333

0.0530

32.8

0.0053

3.3

4

4th

Primary astigmatism

sine term

5

0.0880

0.297

0.0473

29.3

0.0057

3.5

cosine term

6

0.0648

0.255

0.0406

25.1

0.0067

4.2

5

4th

Primary coma

sine term

7

0.0587

0.242

0.0385

23.8

0.0021

1.3

cosine term

8

0.0525

0.229

0.0364

22.5

0.0021

1.3

6

6th

Elliptical coma (arrows, trefoil)

sine term

9

0.0463

0.215

0.0342

21.2

0.0022

1.4

cosine term

10

0.0401

0.200

0.0318

19.7

0.0024

1.5

7

4th

Primary spherical aberration

11

0.0377

0.194

0.0309

19.1

0.0009

<1

8

6th

Secondary astigmatism

cosine term

12

0.0352

0.188

0.0299

18.5

0.0010

<1

sine term

13

0.0328

0.181

0.0288

17.8

0.0011

<1

9

8th

Quadrafoil

cosine term

14

0.0304

0.174

0.0277

17.2

0.0011

<1

sine term

15

0.0279

0.167

0.0266

16.5

0.0011

<1

10

6th

Secondary coma

cosine term

16

0.0267

0.163

0.0259

16.0

0.0007

<1

sine term

17

0.0255

0.160

0.0255

15.8

0.0004

<1

11

8th

Secondary trefoil

cosine term

18

0.0243

0.156

0.0248

15.4

0.0007

<1

sine term

19

0.0232

0.152

0.0242

15.0

0.0006

<1

12

10th

Pentafoil

cosine term

20

0.0220

0.148

0.0236

14.6

0.0006

<1

sine term

21

0.0208

0.144

0.0229

14.2

0.0007

<1

* in units of radians/(D/r0)5/6, radians2/(D/r0)5/3 when squared (variance)
** in units of full phase (analog to the RMS wavefront error in units of full wave)

For higher orders (j>9), Noll's approximation for the residual variance after the 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 wavefront error, is 14.2% of the total aberration.

In terms of r0 and the aperture diameter D, the long-exposure atmospheric error as 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 wavefront deviation in units of the wavelength. Thus, the corresponding long-exposure RMS wavefront 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. Tilt component is dominant; as above table shows (j3), it makes 87% of the variance, and 64% of the RMS error. Short-exposure wavefront error - with "short exposure" defined as sufficiently short to eliminate image motion, i.e. the tilt error component - reduces to the RMS error due to wavefront roughness:

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). The appropriate Strehl approximations and exact values are given in 5.1.2. Seeing error: Strehl, resolution, OTF.

    These relations imply that for any given r0 (i.e. any given seeing) the seeing error changes with the aperture diameter D as D5/6. As shown at left, this rate of increase is somewhat weaker than what in proportion to D would be, with every doubling in aperture diameter resulting in 1.78 times error increase, and every halving of the aperture diameter in error reduction by the inverse value, 0.56. This applies to both, long- and short exposure errors, which would imply that it probably applies to the visual seeing error, which is a sort of the mixture of the two, with the proportion of each varying with the error magnitude.

The standard Kolmogorov model of turbulence uses certain assumptions which are not strictly met in field conditions. This can significantly affect its accuracy. In general, the actual error is smaller than what the model predicts, as outlined in more details on the next page.


4.9. Transmission materials' effects       5.1.2. Limitations to the standard model

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