JOSA COMMUNICATIONSCommunications are short papers. Appropriate material for this section includes reports of incidental researchresults, comments on papers previously published, and short descriptions of theoretical and experimental tech-niques. Communications are handled much the same as regular papers. Proofs are provided.

Atmospheric turbulence and the resolution limits of largeground-based telescopes: comment

V I. Tatarskii and V U. Zavorotny

Cooperative Institute for Research in Environmental Sciences, University of Colorado/National Oceanographic andAtmospheric Administration, Wave Propagation Laboratory, Boulder, Colorado 80309-0216

Received January 23, 1993; revised manuscript received March 8, 1993; accepted April 26, 1993

The paper by McKechnie [J. Opt. Soc. Am. A 9, 1937 (1992)] is discussed, in which McKechnie tries to explainastronomers' visual descriptions of bright cores in star images by using his own model of atmospheric inhomo-geneities, which is different from the conventional turbulent model. In this Communication we show that thestar images presented by McKechnie are consistent with the traditional turbulent model. The McKechniemodel of scattering, a single-scale atmosphere with the characteristic scale Lo 35 cm, contradicts all ourexperiences with turbulence: both direct in situ measurements of turbulent characteristics and indirect ex-periments of wave propagation in a turbulent atmosphere.

1. INTRODUCTIONSometimes astronomers see a star as a relatively brightstable core, associated with an unperturbed star imageand a wider and less-bright halo surrounding the core. 13

These observations attracted McKechnie's attention andraised doubts about the correctness of the conventionalmodel of optical turbulence. As a result, McKechnie 4 pro-posed a new model, i.e., a single-scale random mediumwith a characteristic scale that is small compared with thetelescope aperture diameter.

Currently, the Kolmogorov-Obukhov model of turbu-lence (described by a power-law spectrum for the so-calledinertial interval of wave numbers) is accepted as the mostreasonable one for astronomical and other purposes. 6

According to the Kolmogorov-Obukhov turbulence theory,7fully developed turbulence represents a collection of eddieswith various scales, from the largest, outer-scale (Lo) ed-dies down to the smallest, inner-scale (lo) eddies.

In a real atmosphere the mean value of 1,, is of the orderof a few millimeters near the ground and -1 cm near thetropopause. Variations of 1 with height depend on a pos-sible layered structure of turbulence. Large-scale turbu-lence has an anisotropic character. For instance, thehorizontal dimension of the outer scale may be muchlarger than the vertical one.'-l0 L grows linearly withthe height in the surface layer of the atmosphere. Thelimitation on Lo in the vertical dimension is of the order of100 m because of the possible stratification of turbu-lent layers.

The Kolmogorov-Obukhov model has been indepen-dently verified by both direct in situ measurements andindirect experiments on sound, laser irradiance, and radio-wave propagation in the atmosphere. It was found thatthis model is an excellent first approximation, althoughfor some anomalous conditions modifications and correc-tions are necessary.

To accept McKechnie's model now means that an enor-mous number of well-grounded studies based on theKolmogorov-Obukhov model would have to be rejected.This should be a sufficient reason to decline the proposedsingle-scale model from the start. We show here that thepresented paper contains no clear evidence of failure ofthe conventional Kolmogorov-Obukhov model and thatthe images presented can be easily explained by the con-ventional model.

2. ASTRONOMICAL IMAGING THROUGHATMOSPHERIC TURBULENCEFor imaging purposes, the phase fluctuations of wavesincident upon the telescope aperture are the most impor-tant. The relevant statistical characteristic for the de-scription of the phase fluctuations of the wave froman infinitely distant source (e.g., a star) is the plane-wave phase-structure function D,,(r). According to theKolmogorov-Obukhov theory of turbulence, this functioncan be written in the following form6 :

D,(r) = 2.91k2(cos y)-lr5/3fdhCN2(h), 1,, < r < L,

(1)

where r is a transverse separation in a pupil plane, k =2 fr/A is the wave number, y is the zenith angle, and CN2(h)is the height distribution of the refractive-index structureconstant. For astronomical conditions L >> D, where Dis the telescope diameter, and Eq. (1) can be used withoutany outer-scale limitation.

A long-exposure image is characterized by the averageatmospheric transfer function, which is described withthe second-order moment B(r) of the complex amplitude

(p)ll:

B(r) = ((p) T*(p + r)) = exp[- 1/2 D(r)] .

0740-3232/93/112410-05$06.00 C 1993 Optical Society of America

(2)

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It is an important fact that the atmospheric transferfunction based on the Kolmogorov-Obukhov model of tur-bulence has only one parameter, r, which is Fried's pa-rameter or the coherence length." Indeed, it can beshown that for r >> 1o

B(r) = exp[-3.44(r/r 0)'13], (3)

where

-3/5ro = 0.42k2(cos y)-' f dhCN2(h) . (4)

Therefore, when r0 < D, the long-exposure image of a starwill be spread over angles of -A/r 0. This situation is com-mon for astronomical observations.

According to Roddier,' typical values of f dhCN'(h) forastronomical observations are between 10-'5 M1/3 and2 x 10-3 Mi/3. Using these values in Eq. (4) for y = 0, weobtain 2 cm < r < 21 cm for A = 0.5 Am and 12 cm <ro < 125 cm for A = 2.2 ,um.

The instantaneous image, or its spatial spectrum, is arandom function. We use here the term instantaneousimage for a single shot, which produces one random real-ization of an image. Often the term short-exposureimage is used in the literature for the same purpose.5

But historically the term short exposure was introducedby Fried"' to describe imaging with tilt correction. Actu-ally, this short-exposure image is a long-exposure, aver-aged image but is made with simultaneous fast-trackingadaptive optics, which removes image-centroid wandering,i.e., corrects wave-front tilts on the telescope aperture foreach instantaneous image. McKechnie4 calls this tech-nique "shift-and-add with respect to the centroid of lightenergy" (p. 1943). Such an image is no longer a randomrealization and can be described with the short-exposure(improved long-exposure) modulation transfer function"valid for r < ID:

Be(r) = exp{-3.44(r/r 0)"/'[1 - (r/ID)1/3]}. (5)

The tilt correction provides improved resolution by theremoval of image-centroid motions; however, blurring ofthe instantaneous image as a result of higher wave-frontdistortions remains. Fried" showed that blurring reachesa minimum for D/r 0 = 3.7 and that tilt correction is mosteffective for Di/r0 = 3, where the maximum increase inresolution is -2. The physical meaning of this result isthat for D/r 0 c 3 the bulk of light energy in the image iscontained in a single, wandering, but diffraction-limitedspeckle. For smaller Di/r0 the resolution is determinedmostly by diffraction on the telescope aperture. Forlarger D/r 0 turbulent blurring and centroid wander reducethe resolution, and blurring progressively dominates overwander.

Let us consider now the qualitative picture of the imagedegradation of a star (a point source) for different valuesof r 0 and D, using the theoretical model discussed above.When r0 << D, we have well-developed speckle patternsfor the monochromatic instantaneous image containedwithin the so-called seeing disk, having angular size -Air 0.The number of speckles is -(D/r0 )2 , each one having adiffraction-limited (for ideal optics) angular extent of A/GD.

In this pattern it is difficult to distinguish one specklefrom another. Each one has approximately the samebrightness, its position is continuously changing, and theimage looks like a boiling pattern. The correspondinglong-exposure image of the same star has approximatelya Gaussian shape of angular size Air0 and is a result ofaveraging caused by both the random motion of the in-stantaneous image centroid and the evolution of individualspeckles. Here we should stress that for a large telescope(r0 << ID) the scattering disk of Air0 appears to be mostly aresult of the image breaking up into many speckles ratherthan a result of image-centroid motion.

Increasing r0 (e.g., using longer wavelengths) or decreas-ing D decreases the number of speckles. When we haveonly a few speckles, the total energy is distributed in anirregular way among the speckles, and at each instantwe have one brightest speckle. Such a situation hap-pens very rarely in visual astronomy, because r0 = 10 cmand ID 2-4 m. However, shifting to the infrared bandproduces more favorable conditions. For example, r0can reach -0.5 m for A = 2 [tm, according to Eq. (4). Theresulting presence of bright, dominant speckles moti-vated the use of the shift-and-add (SAA) technique.12 '14

McKechnie references this technique in his paper4 as "theshift-and-add with respect to the core" (p. 1944).

The SAA technique finds the points of maximum inten-sity (or dominant speckle) in each speckle pattern orspecklegram. If the displacement of this point from theoptical axis is known, then each frame can be shifted toplace this maximum intensity on the axis. Adding manysuch shifted frames will yield the average intensity aroundthe brightest spot in all the specklegrams. The dominantspeckle produces the diffraction peak (core), and the othersmaller speckles produce a wide pedestal or halo (seeFig. 5A of Ref. 4 and figures in Refs. 15-17). It should benoted here that, similarly to the short-exposure technique(centroid correction), a shift of the frame provided by theSAA technique also means some kind of wave-front tiltremoval. But, unfortunately, the SAA technique does nothave a theoretical representation adequate to evaluate thetilt removal.

It is logical to assume that the "core," that was observedin Refs. 1-3, represented none other than the brightestspeckle in the speckle pattern of a star. According to theconventional model, this speckle and other less-brightspeckles should be in constant random motion in order toproduce a wide blurring spot with an angular size of A/r0during long exposure.

3. ATMOSPHERIC MODULATIONTRANSFER FUNCTION PROPOSED BYMcKECHNIE

The central point of McKechnie's argument is that thecore does not move sufficiently according to visual impres-sions of the observers.'' If the core, as an element of theinstantaneous image, does not move, then we agree withthis conclusion; the core should manifest itself as a narrow,bright spot in the long-exposure image, too. The questionis, Does the author provide quantitative evidence of hisassertion? First we consider the theoretical substantia-tion of the author's hypothesis and then the experimentalmaterial.

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Generally speaking, theoretical substantiation is absentin the paper. From the beginning, the author assumesthat the real turbulent atmosphere is described by hismodel of the single-scale medium and that the entire prob-lem can be reduced to the calculation of the long-exposuremodulation transfer function. Let us consider the au-thor's arguments.

In Section 2 of Ref. 4 the author introduces a, the rmsvariation of the wave-height fluctuations, and rms quanti-ties associated with the long-exposure image and theshort-exposure image, oar and a-, respectively. The lattermeans an image obtained with a tilt or energy-centroidcorrection, as mentioned above. The author concludes[see Eq. (5) of Ref. 4] that, for large telescopes with adiameter LD 2 2 m, i.e., much larger than the suggestedscale of atmospheric irregularities, Lo -~ 20-40 cm,

as = Ol = O, (6)

and then there is no difference between long-exposureimages and short-exposure images, and both images canbe described now by the long-exposure modulation trans-fer function. [Eq. (6) of Ref. 4]. At the same time animage motion caused by the atmosphere is negligible fortelescopes of this size. Therefore McKechnie chooses aone-scale, Gaussian correlation function p(w) ( corre-sponds to the coordinate r in our notation), which givesthe following phase-structure function:

D,( = k2 [1 - p(o)], p() = exp(-()2 /C902). (7)

The scale of this function, which plays the role of somesort of outer scale L., is chosen to be very small, -35 cm.This assumption, however, is not sufficient to model theobserved bright core. To obtain a bright core, the authoris obliged to deal with very small wave-front variations a;o- < 0.3A. McKechnie's justification for this single-scalemodel and the values of the parameters a- and ),, are dis-cussed in his previous paper. 8 He obtained the smallvalue of the quantity by using a two-wavelength correla-tion of the on-axis intensity in the focal plane and invok-ing the Gaussian-field assumption. This assumption isknown to be invalid for calculations of the two-wavelengthintensity correlation, a rigorous derivation of which ispresented in a paper by Shishov.,9 The Gaussian-fieldassumption is valid only for calculation of the on-axismonochromatic intensity variance in the limit r << ID.

Thus, with an incorrect small value of a; McKechnie isforced to choose the corresponding incorrect small valuefor wO, or Lo to comply with typical long-exposure stellarimages at visible wavelengths. To obtain a core in thelong-exposure image and to be consistent with the visualobservations ' 3 the author must choose an even smallervalue of a-, namely, < 0.3A. Before we show that thismodel is unacceptable for the traditional interpretation ofa turbulent atmosphere, we turn to what the author sug-gested is observational evidence of cores at near-infraredwavelengths.

4. DISCUSSION OF THE PRESENTED STARIMAGESIn Ref. 4, McKechnie presents images of a single star inFigs. 5A-5C and of a double star in Figs. 6A-6C, obtainedat 2.2 Aum with a 4-m telescope, and images of a single star

in his Fig. 7, obtained at 3.4 gm with the same telescope.Figures 5A, 6A, and the bottom-right images of Figs. 7A,7B, and 7C demonstrate bright, almost diffraction-limitedfeatures, which are the expected result of use of the SAAtechnique. Regardless of McKechnie's predictions, we donot see any core in the long-exposure images (Fig. 5C andFigs. 7A-7C, top-right images).4 These images present awide, one-scale intensity distribution, the so-called seeingdisk, which corresponds to predictions from conventionaltheory. The author sees this contradiction also, andhe explains it as follows: The core exists in the long-exposure image of a star, but it is smeared out because oftelescope tracking errors and wind-induced oscillations ofthe telescope. These sources of image motion shouldsmear out a stable core in the long-exposure image.

In such a case, why do we not see any core in the short-exposure images (Figs. B, 6B)? The use of this tech-nique should significantly reduce the possible effect ofmechanical oscillations of the telescope. Nevertheless, wesee in these figures almost the same intensity distribu-tions as for long-exposure images, which implies that tele-scope oscillations produce image broadening that is muchsmaller than 1-1.5 arcsec, the characteristic size of thelong-exposure intensity distributions.

Another question arises: Since telescope oscillationsare removed by the short-exposure technique or areinsignificant, why do we not see in this case a predictedstable core?

The author explains this surprising result in the follow-ing way: The short- and long-exposure images coincideonly in the asymptotical limit of infinitely large telescopeapertures where all image motion vanishes. For finiteapertures, image motion is still important. In particular,from McKechnie's model it follows that the centroid oflight energy in the halo exhibits larger motions than doesthe core. Then the tracking with respect to the centroidof light energy leads to the effect of smearing out the core.McKechnie argues that this effect is less pronounced inshort-exposure star images at 3.4 pm (Figs. 7A, 7B, 7C,bottom-left images)4 because of the brighter core and thecorrespondingly weaker halo. We see that the SAA im-ages and the short-exposure images are close to the instan-taneous star images (Figs. 7A, 7B, 7C, top-left images).However, we have no evidence that these images contain astable core, because any image motion (turbulence, me-chanical, etc.) is removed.

The conventional turbulence model, presented here inSection 2, does not predict a stable core or stable specklein a stellar image. All the image speckles are in constantmotion and blurring, and this phenomenon causes the wideand smooth intensity distribution in the long-exposurestellar image. It is easy to interpret the images with useof the conventional model. For example, from McKechnie'spaper for A = 2.2 gm the value of the seeing disk is-1.5 arcsec. It translates into f dhCN2 (h) 1.5 10-12 ml/3 , which corresponds to typical astronomical see-ing conditions5 (for A = 0.5 ,m, r 7 cm). Substitutingthis value into Eq. (4) we find that r. = 0.4 m. The ratioD/r. = 10 is larger than the value of ID)/r, 3 for whichthe short-exposure technique is most effective. Thismeans that we should not expect significant improvementin the resolution for the short-exposure images in Figs. 5Band 6B.

The features observed in McKechnie's Fig. 7 also can be

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easily explained on the basis of the conventional modeland do not require the new model suggested by Eq. (7)of the present Communication. Indeed, for A = 3.4 Azmand for the seeing disk 0.7 arcsec we obtain5 thatf dhCN2(h) 7 X 10-13 m113, which is also consistent withtypical astronomical seeing conditions5 (for A = 0.5 m,r, = 10 cm). From Eq. (4) of the present Communicationwe find that r, 1 m. The ratio 1D/r 0 = 4 is favorable forusing the short-exposure technique, and we should have asingle, wandering speckle, which is supported by all theimages presented in McKechnie's Fig. 7 (Ref. 4).

5. COMPARISON OF McKECHNIE'S MODELWITH PHASE MEASUREMENTS

The model developed by McKechnie [Eq. (7) of the presentCommunication] predicts a bright core only by using smallvalues of wave-front fluctuations, < 0.3A. A compari-son of these values with measurements with use of anoptical interferometer2 0'2 1 or a telescope22 shows total dis-agreement. Using the single-scale model of McKechniefor >> wo z- 0.25 cm and for = 0.3A, D,(co) shouldsaturate to the level

D,(c) = 82 7.1(rad2 ). (8)

Measurements2 0 of the phase-structure function D,(w)with an optical interferometer at A = 2.2 m provide noevidence of saturation up to co = 20 m with the followingvalues for Dw(w) (rad2 ):

Dq(9 m) = 50-250; D,(13 m) = 100-260.

Other phase measurements,21 made with the Mark IIIinterferometer at Mt. Wilson with baselines up to 32 m,show excellent agreement with the standard Kolmogorovtheory and predict an atmospheric outer scale larger than-1 km. Even wave-front fluctuations measured withthe 1.5-m centroid-tracking telescope2 2 show the power-law behavior of the structure function D, of residualwave-front fluctuations. For separations of 1 m,D,p -40-200.

6. MEASUREMENTS OF ATMOSPHERICTURBULENCE

Astronomy is not the only field that deals with propaga-tion of light waves through a randomly inhomogeneousatmosphere. Many experiments have been conductedwith laser sources in the boundary layer, and all of themdemonstrated that the Kolmogorov-Obukhov model of op-tical turbulence is more relevant than the model pre-sented by McKechnie (see, e.g., Refs. 23-26).

The vast amount of experimental material related toturbulent spectra versus height cannot be cited here.The data from Refs. 27-30 demonstrate the power-law be-havior of the spectra and values of outer scale, which growswith height.

McKechnie quoted results by Coulman et al.3 ' and byCoulman and Vernin32 as evidence of a small atmosphericouter scale. Indeed, the authors of Refs. 31 and 32 cameto such a conclusion with the equation6

C,2= aM2L 41 3, (9)

which is valid only for surface-layer turbulence (z c100 m). Here a is a constant (-2.8) and M is the gradientof potential refractive index. Using optical measure-ments for C 2 profiles and calculating M from radiosondedata, Refs. 31 and 32 proposed an analytic approximationfor the L0 (z) profile at altitudes 2 km < z < 17 km.

First, McKechnie incorrectly extends this dependencedown to the ground (see Fig. 14 of Ref. 4). Then thevalues for L. are 1-4 m in the range 2 km < z < 11 km,much larger than is needed to approach the characteristicscale of the McKechnie model. Even though they arelarger, in our opinion these values of L. were under-estimated as result of the very rough assumption Eq. (9)of the present Communication, which has not been veri-fied with in situ measurements above 2 km.

7. CONCLUSION

Two main conclusions can be drawn from the discussionpresented here:

1. The paper by McKechnie 4 does not show phenomenain the long-exposure or short-exposure images of a starthat cannot be easily explained by the conventionalKolmogorov-Obukhov model for optical turbulence.

2. McKechnie's model of a weakly scattering, ran-domly inhomogeneous atmosphere with a single character-istic scale Lo 0 35 cm, does not explain the full range ofastronomic observations and contradicts all our previousexperiences with turbulence.

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