vladimir p. lukin v.e. zuev institute of atmospheric optics sb ras,

Vladimir P. Lukin

V.E. Zuev Institute of Atmospheric Optics SB RAS, Zuev Sq.,1, 634055, Tomsk, Russia, [email protected]

OUTER SCALE OF TURBULENCE IN THE ANISOTROPIC BOUNDARY

LAYER

During a period of 40 years we have performed successively the investigations of the effect of low-frequency spectral range of atmospheric turbulence on the optical characteristics. The turbulence models as well as a outer scale of turbulence have been determined and its influence on the characteristics of telescopes and systems of laser beam formations has been investigated too.

Centre for Advanced Instrumentation, Durham University, UK, Sept. 2014

CONTENT

• 1. INTRODUCTION. PHASE FLUCTUATIONS MEASUREMENTS

• 2. DEVELOPMENT OF THE MODELS OF TURBULENCE SPECTRA

• 3. OUTER SCALE OF TURBULENCE AND INSTABILITY PARAMETER

• 4. INVESTIGATION OF ANISOTROPY OF THE ATMOSPHERIC TURBULENCE SPECTRUM IN LOW-FREQUENCY REGION

• 5. INVESTIGATION OF THE ATMOSPHERIC TURBULENCE DYNAMICS OF THE BASIS OF ASTROCLIMATIC OBSERVATIONS

• 6. HIGH-ALTITUDE OF ATMOSPHERIC TURBULENCE

• 7. THE MODEL “AVERAGED” OVER THE ENTIRE COLOMN ATMOSPHERE

• 8. OUTER SCALE OF ANISOTROPIC TURBULENCE

• 9. CHARACTERISTICS OF GROUND-BASED OPTICAL TELESCOPE DUE TO ATMOSPHERIC TURBULENCE

In the early 70’s the scientists in Italy (A.Consortini, M. Bertolotti, L. Ronchi), in USA (Ochs, Buser, S.Clifford) and in USSR (V.Pokasov, V.Lukin) almost simultaneously discovered the phenomenon of deviation from the power law and the effect of saturation for the structure phase function. This meant the existence of correlation for phase fluctuations and the need in applying the turbulence spectrum with a finite outer scale Lo. My main papers for subject are:

V. Lukin, V. Pokasov, S. Khmelevtsov, “Investigation of temporal characteristics of optical waves phase propagating in ground-boundary layer”, Izv. VUZov. Radiofizika, 15, No.12, pp.1861-1866, 1972.V.Lukin, V.Pokasov, “Optical wave phase fluctuations”, Applied Optics, 1981, V.20, No.1, pp.121-135.

CONTENT










2. DEVELOPMENT OF THE MODELS OF TURBULENCE

SPECTRA

2. DEVELOPMENT OF THE MODELS OF TURBULENCE SPECTRA (cont.)

2. DEVELOPMENT OF THE MODELS OF TURBULENCE SPECTRA (cont.)

• Atmospheric turbulence models with a finite outer scale, such as Greenwood-Tarazino’s, Von Karman’s, Russian model, have found wide usage.

• In 1993 I showed that it is possible to pass from one model to the other:

• V.P. Lukin, "Intercomparison of models of the atmospheric turbulence spectrum", Atmos. Oceanic Opt. 6, pp. 628–631, 1993.

• V. Lukin, “Comparison of spectral model of atmospheric turbulence”, Proc.SPIE, 1994, V.2222, pp.527-535.

• V.V. Voitsekhovich, J. Opt. Soc. Am. A12, pp.1346–1353, 1995.

• V.V. Voitsekhovich and S. Cuevas, J. Opt. Soc. Am. 12, pp.2523–2531, 1995.

,27.0 10

1 KoG

,36.0 10

1 KoR

,

.33.1 10

1 GoR

.

CONTENT










3. OUTER SCALE OF TURBULENCE AND INSTABILITY PARAMETER

• Measurements of phase structure function in saturation region may to give information for outer scale:

•

• In experiments we used a homodyne phasometer and obtained the nice tool for measurements of outer scale in near surface region of atmospheric.

• In the experiments also were employed: measurers of the temperature and wind velocity at fixed altitudes, data were used for calculating the temperature structure parameter and instability parameter

.033.05

24

2)(

3/50

222

2

LLCk

D

n

SS

25.02 /)( vTT

T

ghB

3. OUTER SCALE OF TURBULENCE AND INSTABILITY PARAMETER (cont.)

• In 1980 I found that the outer scale of turbulence Lo depends on the thermodynamic stability of the atmosphere.

• In the surface layer of the atmosphere the outer scale of the turbulence can be comparable, on the one hand, with the hight over the underlayer surface and, on the other hand, the scale is dependent on characteristics of thermodynamic instability.

CONTENT










4. INVESTIGATION OF ANISOTROPY OF THE SPECTRUM OF ATMOSPHERIC TURBULENCE IN LOW-FREQUENCY

REGIONTogether with the dependence of theouter scale of turbulence on variationsof meteorological parameters of theatmosphere, there exists anisotropy ofatmospheric turbulent spectra.The properties of inhomogeneities withdimensions exceeding several metersdepend on direction.As a consequence, the characteristics ofsome parameters of optical waves appearto be direction dependent. Three dimensional turbulentatmospheric spectra for optical wavepropagating application is convertedinto two-dimensional one.Actually we take into account two sizesof outer scale along two perpendiculardirection:

),(: 32 ),(: 03020 )},//exp(1{

))((033.0),,(203

23

202

22

6/1123

22

232

xCx nn

For example, the random two-directional displacements of an image formed along the horizontal atmospheric path on aperture with radius , which is formed by the optical radiation passed through layers, exhibit such properties:

R

),,(1 2

LSd

kF

.)( 222zyF

4. INVESTIGATION OF ANISOTROPY OF THE SPECTRUM OF ATMOSPHERIC TURBULENCE IN LOW-FREQUENCY REGION

(cont.)• In our papers • Lukin V.P., Investigation of anisotropy of the atmospheric turbulence in the low-

frequency range, Proc. SPIE, Vol.2471, 1995.• Lukin V.P., Antoshkin L.V., Botugina N.N., Emaleev O.N., Lavrinova L.N.,

Investigation of turbulence spectrum in the ground atmospheric layer, Atmospheric and Ocean Optics, Vol.8, No.12, pp.993-996, 1995.

• was been shown that for ,),( 103

102 R

)},;2;2

1,

6

1(

22{)

6

1(264.0 12

3/102

6/5

3/12222

FR

LCF ny

)}.;2;2

3,

6

1(

22{)

6

1(264.0 12

3/102

6/5

3/12222

FR

LCF nz

:)1(202

203

Hence, we may to calculate its ratios:

.

)];2;21

,61

()2

(1[

)];2;23

,61

()2

(1[

126/1

2202

126/1

2202

2

2

1

FR

FR

Ky

z,)(

)(22

22

zy

zyK


(cont.)

• If the outer scale has equal horizontal and vertical dimensions ( ), then

• and . In this case the isotropy takes place.

• Alternatively, if , then we have an anisotropy.

• Our new spectrum have two parameters in large-scale region:

011 K 0K

1

),,(: 102

103

10

).1/,(: 202

203

102

10

The data of experiments give us to calculate two parameters too:

,2

2

1y

zK

.)(

)(22

22

zy

zyK

These two experimental parameters are not sensitive to experiment and path parameters ( ).LCF n,, 2

Optical measurement were accompanied by corresponding meteorological measurements of temperature ),(hT pressure ),(hp

,

and wind velocity ).(hv

.


(cont.)

• Meteorological data were used for calculating the following characteristics at the height of optical radiation propagation:

)(/))5.0()2(()(

2 hvhThThT

ghB - Obukhov’s parameter of

stability,

4/3

572.02

2

0 })1000/()(8.2

{p

h

TC

L Tmeteo

- outer scale of turbulence (Tatarski estimation from meteorological data).

The so-called coefficient of anisotropy obtained in the experiment, varied in

the

interval from 0.82 to 2.87 with the mean value 1.85 that indirectly indicated that the optical inhomogeneities, which produce the phase fluctuations, were anisotropic in the large-scale region. The relationship is obvious between the coefficient of anisotropy value and the instability parameter.

The measured values of anisotropy can be explained with the help of the spectrum as a model with two different projections of the outer scale of turbulence into the vertical and horizontal directions.

,2

2

1y

zK

CONTENT










5. INVESTIGATION OF THE ATMOSPHERIC TURBULENCE DYNAMICS OF THE BASIS OF ASTROCLIMATIC

OBSERVATIONS

We applied the stars motion measurements in focal plane small telescope for

near horizontal atmospheric paths.

Several experiments with point source jitter for propagation along horizontal

paths near ground ( h = 2.5 m) were made.

Under the near surface conditions one can expect a strong anisotropy of the

turbulence that leads to the anisotropy of the jittering process. As a result for

horizontal propagation we have the strong anisotropy:

,)33.0(0 mLmeteo ),6.85.5()1(2*)/( 0203

Thus, the vertical size of large-scale turbulence smaller then horizontal one in (1.4 – 3) times and this level depend from parameter of stability of atmosphere

.9.24.12

2

1 y

zK


OBSERVATIONS (cont.)

Then we apply the stars motion measurements in focal plane small telescope for vertical and slant atmospheric paths we used.

• For the stars, observed at small zenith angles the jitter is

• practically isotropic – parameter

• At the same time, observed at large zenith angles is order are

• indicate of strong anisotropy too ( ).

• Thus it implies that:• in the case of vertical propagation (along the zenith direction) the effective

spectra of turbulence practically isotropic,• the anisotropy of the spectrum is maximum near ground surface (observed

optical beams jitter along the horizontal propagation) and determined by the atmospheric instability.

03020

07560

.15.19.02

2

1 y

zK

4.33.22

2

1 y

zK



• In 1983 we have made the first approach for impact the effective outer scale for entire atmosphere. We have calculating the phase structure function with Russian model turbulence spectra for vertical wave propagation and

presented the integral value (along the vertical path) as

)}./exp(1{025.0),( 20

23/113/50

2

0

rkhdh nH

We used very known A.Gurvich’s model and the Coulman-Vernin vertical profile for outer scale:

20

)2500

8500(1

4)(

hhL VC

Here are 0r is Fried’s radius for vertical path,

0 is a fitting parameter to adjust the result of calculation with simple model.



We made a measurementsof star angular displacementsvariance for different aperturesizes. Aperture averagefunction of star image jitter dataon TT-600 (Zelenchuk, 1982)presented on Figure.Experimental dots are out oftheory curves calculated forKolmogorov model of turbulencewith infinite outer scale. These calculations ofeffective size of outer scale basedon these measurements gave

10 =1.6 m.



• Under similar approximation the phase structure function presented as:

• and for have two

regions:

)]},4/;1,6

5(1[

)6

11(2)/{(88.6)(

22011

3/50

3/50

3/53/50

F

rrDS

1m

,)/(88.6)( 3/50rDS 10

,)(5.20)( 3/500

rDS 10

.

For NTT (3.6m, La Silla) data it gave us two parameters for entire atmosphere

0r

0 =17 cm (for 0.5 mkm). = (0.6-0.8) m,



Using similar formulas we may calculate the

temporal structure function of phase difference:

)()()(2)(2

)]},(),([)],0(),0({[)( 2

vDvDDvD

tStStStSD

SSSS

S

Then for v

),(2)( vDD SS

but for v

).(2)( SS DD

.

We used the data of R.Angel measurements on MMT (Arizona, 6.5 m, Mt. Graham) and obtained the next estimation:

,210 m

,

,06.1)2.2(0 mmr

,

./5.8 cmv

.

Year Author Outer scale size (m)

Instrument Site

1983 Lukin et al. 3-15 TT-600 Zelenchuk

1984 Mariotti et al. = 8 I2T CERGA

1987 Colavita et al. > 2000 Mark III Mt. Wilson

1989 Tallon 5-8 Hartmann-Shack

Mauna Kea

1991 Rigaut et al. = 50 COME ON La Silla

1991 Nightingale = 2 DIMM La Palma

1993 Ziad et al. 5-100 Hartmann-Shack

OHP

1994 Agabi et al. 50-300 GSM1 OCA

1995 Busher et al. 10-100 Mark III Mt.Wilson

1995 Fuchs 2,4-1.5 Ballons Paranal

5. INVESTIGATION OF THE ATMOSPHERIC TURBULENCE DYNAMICS OF THE BASIS OF ASTROCLIMATIC OBSERVATIONS (cont.)

CONTENT










6. HIGH-ALTITUDE OF ATMOSPHERIC TURBULENCE

• To describe the optical radiation propagation at vertical and slant optical paths it is necessary to introduce into account vertical distribution of parameters of atmospheric turbulence. Undoubtedly, the outer scale of the turbulence undergoes significant variations both in the surface layer and at the large altitudes.

• At present, there are exist a lot of models of height profiles L0(h). Some models chosen for the study are presented below:

(A) L hh m

h h m0

0 4 1

0 4 1( )

. , ,

. ,

L h

h m

h h m

h h m

0

0 4 1

0 4 1 25

2 25

( )

. , ,

. ,

, ,

L h

h m

h h m

h h m

h m

0

0 4 1

0 4 1 25

2 25 1000

2 1000 1000

( )

. , ,

. ,

, ,

, ,

(B)

(C)

6. HIGH-ALTITUDE OF ATMOSPHERIC TURBULENCE (cont.)

(D) L hh

0 2

5

17500

2000

( )

L hh

0 2

5

18500

2500

( )

(E)

The model (A) is recommended for small heights, (B) is proposed by D.L. Fried, and (C)is a generalization of (A) and (B). The models (D) and (E) are obtained by generalizing results of measurements performedin the USA, France, and Chile.

Model E is known Coulman-Vernin profile. Models (D) and (E) obtained from data of astronomical measurements, hence, its is not good for altitude below altitude of site

summits.


A modified Coulman-Vernin profileof the outer scale of turbulenceaccording to-situ sounding madeduring the PARSCA campaign(PARanal Seeing Campaign) in March1992. According the PARSCA data(see Figure), the average outer scale ofturbulence presents a maximum at ground level twice larger than maximum of the model:

)],1500

exp(8

)([3

2)(

0

00

hh

hLhL VCParanal

here is the altitude of the mountain summit. At Paranal = 2636 m.0h 0h


IAO SB RAS experimental data of CT2 (left) and outer scale Lo (right) from acoustical sounding. These data gave us the values of outer scale for small altitudes above the ground.

CONTENT










7. THE MODEL “AVERAGED” OVER THE ENTIRE COLOMN ATMOSPHERE

• The possibility to introduce an efficient outer scale as an integral characteristic of turbulence is of great interest as it can permit one to change the height profile for the outer scale.

• One of the reasons to introduce this parameter is that the applicability of the models of height profiles of atmospheric turbulence is restricted due to their dependence on geographical location. It will also permit one to simplify mathematical calculations connected with the account for influence of the atmospheric turbulence on the phase of optical wave.

• A.Gurvich’s model

02

4

68

10

12

14161820

10–18

H , km

the best

the worst

Cn2(H), m– 2/ 3

10–17 10

–1610

–15 10–14

10–13

7. THE MODEL “AVERAGED” OVER THE ENTIRE COLOMN ATMOSPHERE (cont.)

• We propose methods for determining the effective outer scale, namely, by the discrepancy between structure functions of phase fluctuations on the saturation level.

• To determine the effective outer scale by this model, minimization of the integral square discrepancy of structure functions of phase fluctuations

( ) [ ( , ) ( , ( ))]max

L d D L D L h00

0 02

This Figure presented several variants of introducing we used. max

.


• Analysis of the value obtained by different methods for the same height profile demonstrates that its growth i.e.,

• [0...10 m] < [0...Arg(90%)] [0...]) • is caused by the necessity to

compensate for increasing influence of the portions at large argument values with the increase of

• (i.e., for intervals• [0...10 m] < [0...Arg(90%)] • < [0...]). • Studying the dependence of the value

• on the model of • one can say that lower value • of for the “best” vision conditions is

caused by essential distinctions in the behavior of .

C hn2( )

L h0( )

– best

Model Method

0...10 m 0...Arg(90%) 0...

(В) 34.7 50.6 58.4

(С) 32.5 39.9 42.9

(D) 0.60 0.66 0.71

(E) 0.68 0.75 0.84

L0*

C hn2( )

L0*

C hn2( ) The “best” profile rapidly falls off

with the growth of height, and the probability of appearance of large-scale fluctuations diminishes.

C hn2( )

L0*

L h0( )

L0* L0

*

L0*

max

max


• Figure presented the structure function for the profile (C) and the family of structure functions calculated for fixed values of

• . • The structure function for the

profile – C and for the corresponding effective outer scale -for model (C).

• V. Lukin, B. Fortes, E. Nosov, “Effective “outer scale of turbulence” for imaging through the atmosphere”, Proc. SPIE, Vol.2319, pp.98-106, 1997.

4000

0 1 2 3 4 5 6 7 8 9 10

1000

2000

3000

, m

L0(h) – C

20 m

50 m

10 m

5 m

D

100 mL =0

0L


• Dr. Borgino alike us suggested to use the effective outer scale for the atmosphere as a whole as a given in formula:

• For the outer scale of turbulence, several models of its vertical profile in the atmosphere were proposed.

• Based on them I calculated the effective outer scale for the atmosphere as a whole for Gurvich model of structure parameter.

• This scale turned out to be about 10-20 meters for the best observatories of the world.


• Effect of underlying terrain on jitter of astronomic images

• As can be seen from formula the effective outer scale of turbulence depends on the receiver’s height above the surface .

• However, this height is often taken zero. • Estimates show that the relative error

introduced by substitution of zero• ( = 0) for the real receiver’s height is • on the order of .

• This error becomes significant for large ground-based telescopes, in which the center of the entrance mirror is typically located at the heights of some tens of meters from the underlying surface.

• For example, for the 25m receiver’s height

• = 25 m the error is 20%.

•

1 2 3 4 5 0

50

100

150

200

0.7-0.8 m 2.7-1.9 m

26-41 m 29-48 m

72-195 m e

d

c

b

a

L 0 ef , m

Type of outer scale model

0H3/1

0 )/( ehH

= 3 (a), 6 (b), 12 (c), 24 (d), and 48 m (e).0H 0H

0H


• 1. One can introduce the effective outer scale of turbulence as an integral parameter describing the character of atmospheric turbulence along the whole propagation path.

• 2. Introduction of the effective outer scale can considerably simplify mathematical calculations connected with the account for the influence of atmospheric turbulence on the phase of optical wave propagated along vertical atmospheric paths.

• 3. The description accuracy studied demonstrate that the error caused by the change of the height profile of the outer scale for a constant value, i.e., the effective outer scale, considerably varies depending both on the model of a parameter profile and the method of determination.

• 4. The error in determination of the Strehl parameter does not exceed 16% in the situation when the effective outer scale is larger than the diameter of a telescope.

CONTENT










8. Outer scale of anisotropic turbulence

• V.I. Tatarski determines the vertical outer scale from the condition of equality of mean squares of random difference of temperatures and its systematic difference .

• This condition gives

•

• 1.17,

• In our paper we generalized this expression for the case anisotropic turbulence:

• where is the height, is the function of similarity (

• and are the Monin-Obukhov number and scale), is the temperature scale,

• and are the energy and temperature functions of anisotropy.

• V.V.Nosov, O.N.Emaleev, V.P. Lukin, E.N.Nosov, Results of measurements of surface turbulence of the atmospheric air in the mountains of the Baikal Astrophysical Observatory. //X International symposium “Atmospheric and Ocean Optics. Atmospheric Physics”. Papers abstracts, Tomsk, 2003.

TL03/22 )( zCT

22 )()/( zdzdT 4/3222/3

0 ]})/(/[{)/()( dzdTCCCzL TT

c 3.0. 2/31

*4/14/3

0 /49.0)(])()([)]()([ dzdTzTzL VTk

TA

z )( ,/ Lz L *T

)(V )(T

8. Outer scale of anisotropic turbulence (cont.)

• From measurements of meteorological characteristics (in the mountain of the Baikal Astrophysical Observatory) on the surface values were reconstructed of outer scale of anisotropic turbulence – in the wide range of local temperature stratifications (from stable to super-strong instable).

• A comparison of experimental and theoretical results of outer scales measurements L0T in mountain region for anisotropic turbulence:

• 1 – experiment (data from spectra on 5/3 dependence),

• 2 – experiment (data from spectra in saturation regime),

• 3 – experiment (data on Tatarski determination),

• 4 – semiempirical theory for anisotropical layer,

• 5 – semiempirical theory for isotropical layer.

Bolbasova L.A., Lukin V.P., Nosov V.V. Comparison of Kolmogorov’s and coherent turbulence // Applied Optics. 2014. Vol.53. Iss.10. p.B231-B236.

CONTENT

• 1. INTRODUCTION PHASE FLUCTUATIONS MEASUREMENTS



• 4. INVESTIGATION OF ANISOTROPY OF THE SPECTRUM OF ATMOSPHERIC TURBULENCE IN LOW-FREQUENCY REGION






9. CHARACTERISTICS OF GROUND-BASED OPTICAL TELESCOPE DUE TO ATMOSPHERIC TURBULENCE

• In 1986 I completed working at the monograph, • V.P.Lukin, Аtmosfernaja adaptivnaja optika, Nauka, Novosibirsk, 1986.• V.Lukin, Atmospheric Adaptive Optics, SPIE Press, 1996.• which developed the theory of adaptive correction of laser beams and images in the

atmosphere as a turbulent, absorbing, and refractive medium.

• The following subjects were studied in this monograph for the first time:• A) capabilities of a two-color adaptive system (1979),• V.Lukin, “Efficiency of some correction systems”, Optics Letters, 4, No.1, pp.15-17,

1979.• B) possibilities of applying an artificial reference source (1979-1983) for image

correction; later on such sources were called laser guide stars, • V.P. Lukin, “Correction of random angular displacements for optical beams”, Quantum

Electronics, 1980, т.7, №6. с.1270-1279.• V.Lukin, V.Matuchin, “An adaptive image correction”, Kvantovaja Electronika, 1983,

10, No.12, pp.2465-2473.• C) dynamic characteristics of adaptive optical systems (1986), where the ideas of

“predicting” fluctuations for adaptive system operation were used for the first time• V.Lukin, V.Zuev, “Dynamic characteristics of optical systems”, Applied

Optics, 1987, 27, No.1, pp.139-147.

8. CHARACTERISTICS OF GROUND-BASED OPTICAL TELESCOPE DUE TO ATMOSPHERIC TURBULENCE (cont.)

In the period of 1991-1995, we developed a

Four dimensional numerical dynamic model

incliding the outer scale of turbulence of

atmospheric adaptive systems:

V.Lukin, B.Fortes, “Modeling of the image

observed through a turbulent atmosphere”, Proc.

SPIE, 1992, V.1688, pp.477-488.

V.Lukin, B.Fortes, F.Kanev, P.Konyaev, “Four

dimensional computer dynamic model of

Atmospheric optical systems”, Proc.SPIE, 1994,

V.2222, pp.522-526.

E.A. Vitrichenko, V.P. Lukin, L.A. Pushnoi,

V.A.Tartakovski, The problem of optical testing,

Nauka, Novosibirsk, 1990, 350 p.

V.Lukin, B. Fortes, Adaptive beaming and Imaging

in the Turbulent Atmosphere, SPIE Press, PM109,

2002, 201 p.

In 1994 we completed a design project of an adaptive system for the 10-meter combined Russian telescope AST-10. V.Lukin, “Computer modeling of adaptive optics for telescope design”, ESO Workshop Proc. No.54, 1995, pp.373-378.V.P. Lukin, B.V. Fortes, “Partial phase correction of turbulent distortions in telescope AST-10”, Applied Optics, 1998, т.37, №21, pp.4561-4568.Characteristics of long-base stellar interferometers were calculated (1997) with allowance for the interferometer base orientation, wind velocity distribution, and the outer scale of turbulence.V.P. Lukin, B.V. Fortes, Ground-based spatial interferometers and atmospheric turbulence, Pure and Applied Optics, V.5, No.1, pp.1-11, 1996.



• Using an analytical algorithm for atmospheric correction the performance of the Euro50 telescope is analyzed.

• The atmospheric model employed here is the ORM (Observatorio del Roque de los Muchachos) 7 layer model with an outer scale distribution.

• Direct estimations of the effective outer scale of turbulence by formula Borgino based on the seven layers models of Cn2 and some assumptions for the vertical distribution L0 give value about 20…40 m, including the turbulence of the ground boundary layer.

• The values of effective outer scale found that at apertures of modern telescopes of several meters the influence of the finite L0 on the energy balance between tip-tilt and higher order modes must be properly taken into account.

• V. Lukin, A. Goncharov, M. Owner-Petersen, T. Andersen, *Institute of Atmospheric Optics RAS, Tomsk, Russia, Lund Observatory, Lund, Sweden

• The effective outer scale estimation for Euro50 site. Proc. SPIE, Vol.5026, pp.112-118, 2002.

• Lukin V.P., Nosov V.V. and Torgaev A.V. Features of optical image jitter in a random medium with a finite outer scale // Applied Optics. 2014. Vol. 53. Iss.10. p.B196-B204.

• Lukin V.P. Adaptive optics in the formation of optical beams and images // Physics -Uspekhi 57 (6) 556 - 592 (2014). DOI: 10.3367/UFNe.0184.201406b.0599

Papers, relevant with this topics• 1. Bouricius G.M.B., Clifford S.F. Experimental study of atmospheric ally induced

phase fluctuations in an optical signal // Journ. Opt. Soc. Am. 1970. 60(11), 1484-1489.

• 2. Consortini A., Ronchi L. and Moroder E. Role of the outer scale of turbulence in atmospheric degradation of optical images // J. Opt. Soc. Am. 63, 1246-1248 (1973).

• 3. Takato N., Yamaguchi I. Spatial correlation of Zernike phase-expansion coefficients for atmospheric turbulence with finite outer scale // J. Opt. Soc. Am. A 12. 5, 958-963 (1995).

• 4. Winker D. M. Effect of a finite outer scale on the Zernike decomposition of atmospheric optical Turbulence // J. Opt. Soc. Am. A 8, 10, 1568-1573 (1991).

• 5. Dewan Edmond M., Grossbard Neil The inertial range “outer scale” and optical turbulence // Environ Fluid Mech (2007) 7:383–396.

• 6. Voitsekhovich V. and Cuevas S. Adaptive optics and outer scale of turbulence // J.Opt.Soc.Am. A12. pp.2523-2531. 1995.

• 7. Tokovinin A., Ziad A., Martin F., Avila R., Borgnino J., Conan R., Sarazin M. Wave-front outer scale monitoring at La Silla // Proc.SPIE. Vol.3353. pp.1155-1162. 1998.

• 8. Reinhard G. W. and Collins S. A. Outer-Scale Effects in Turbulence-Degraded Light-Beam Spectra // JOSA, Vol. 62, Issue 12, pp. 1526-1528 (1972)9. Borgnino J. Estimation of the spatial coherence outer scale relevant to long baseline interferometry // Applied Optics 29 13 (1990) 1863-1865.

• 10. Borgnino J., Martin F., Ziad A. Effect of a finite spatial-coherence outer scale on the covariances of angle-of-arrival fluctuations // Optics Communications 91 (1992) 267-279.

• 11. Buscher D.F., Armstrong J.T. et al. Interferometric seeing measurements on Mt. Wilson: power spectra and outer scales // Applied Optics 34 6 (1995) 1081-1096.

• 12. Coulman C.E., Vernin J., Coqueugniot Y. and Caccia J.L. Outer scale of turbulence appropriate to modeling refractive-index structure profiles // App. Opt. Vol.27, no 1, January 88.

• 13. Coulman C.E., Vernin J. The significance of anisotropy and the outer scale of turbulence for optical and radio "seeing" // App. Opt. 30, no 1, 1 Jan. 1991.

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vladimir p. lukin v.e. zuev institute of atmospheric optics sb ras,

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