686 J. Opt. Soc. Am. A/Vol. 20, No. 4 /April 2003 Andrei A. Tokovinin

Polychromatic scintillation

Andrei A. Tokovinin

Cerro Tololo Inter-American Observatory, Casilla 603 La Serena, Chile

Received September 30, 2002; revised manuscript received November 19, 2002; accepted November 25, 2002

A formula to compute the index of stellar scintillation detected with a finite spectral bandpass and with arbi-trary aperture is derived in the limit of weak perturbations. It also applies to differential scintillation (rela-tive fluctuations of light intensity in a pair of apertures), where the effect of finite bandpass turns out to besignificant. The new formula is used for measurements of free-atmosphere seeing and low-resolution turbu-lence profiles with concentric-ring apertures. © 2003 Optical Society of America

OCIS codes: 010.1330, 030.6600, 280.7060, 290.5930.

1. INTRODUCTIONWhen a light beam propagates through turbulence, its in-tensity fluctuates. This phenomenon is known in as-tronomy as scintillation or twinkling of stars. Stellarscintillation is an important factor that limits the preci-sion of ground-based photometry.1

Statistical analysis of scintillation yields some informa-tion on the optical turbulence. The isoplanatic angle inadaptive optics can be readily estimated from the scintil-lation index in a telescope with 10-cm aperture.2–4 Asuccessful technique to measure turbulence profiles fromscintillation of double stars is known as SCIDAR (SCIn-tillation Detection And Ranging).5 Attempts to measureturbulence profiles from scintillation of single stars weremade by Ochs et al.6 and Caccia et al.7 Recently a newmethod of turbulence profile estimation, multi-aperturescintillation sensor (MASS), has been successfully tested.8

It is based on single-star scintillation detected with fourconcentric-ring apertures of suitable size. From the fournormal and six differential scintillation indices measuredwith such apertures, a turbulence profile with a verticalresolution Dh/h ; 0.5 is constructed.

Theoretical analysis of scintillation has been done bymany authors, as reviewed in Refs. 9 and 10. Formulasto compute the variance of relative light fluctuations (theso-called scintillation index) are well known. The differ-ential scintillation index for a pair of apertures was intro-duced in Refs. 11 and 12. However, all these papers con-sider only monochromatic light, whereas in realinstruments the light is always polychromatic. Analysisof the effect of finite bandpass in SCIDAR was done byVernin and Azouit,13 who concluded that monochromatic-light theory is a good approximation for this instrument.Krause-Polstorf et al.3 obtained polychromatic responseas a weighted combination of monochromatic responses,which is not quite correct (see below).

The problem of polychromatic scintillation is addressedin this paper. I derive in Section 2 a general expressionfor the polychromatic scintillation index that should re-place the standard formula for the monochromatic case.The effect of finite spectral bandpass is then evaluated inSection 3; in some cases, e.g., for differential scintillation,it is significant. The new formula [Eqs. (13) and (14)] has

1084-7529/2003/040686-04$15.00 ©

already found application in the interpretation of MASSdata; more applications are foreseen. Moreover, the re-placement of the sin2(plzf 2) term found in the expressionfor monochromatic scintillation by the square of theimaginary part of the Fourier transform (FT) of the spec-tral response curve is a general result that may be ap-plied to other problems of polychromatic light propagationthrough turbulence. Whenever phase effects are consid-ered, the cos2(plzf 2) diffraction term will be similarly re-placed by the square of the real part of the FT.

2. SCINTILLATION INDEX FOR FINITEBANDPASSThe classical results of the theory of wave propagation inturbulent media under the weak-perturbation (Rytov)approximation9,10 are used below. Amplitude fluctua-tions at the entrance aperture of an optical instrumentproduced by turbulence are described through the naturallogarithm of light-wave amplitude x, a Gaussian randomvariable with zero average. The logarithm reflects onlyrelative fluctuations so that source brightness, atmo-spheric transmission, etc., are all irrelevant. In theweak-perturbation regime the condition x ! 1 is im-posed. This condition is usually satisfied for verticallight propagation through the atmosphere when turbu-lence is not very strong and zenith angle is moderate.Otherwise, scintillation saturates and the weak-perturbation theory used here overestimates its power.

Let F(l) be the spectral response describing the rela-tive weighting of different wavelengths (a combination offilter transmission, detector response, and source spectralenergy distribution). The exact definition of F(l) de-pends on the type of detector: For a photon-counting de-tector the response is measured in relative numbers ofphotons, whereas for a bolometer the response is definedby the energy distribution. The response must be nor-malized as

E F~l!dl 5 1. (1)

In any instrument the light is averaged spatially in theaperture plane (e.g., by the entrance aperture of a tele-

2003 Optical Society of America

Andrei A. Tokovinin Vol. 20, No. 4 /April 2003 /J. Opt. Soc. Am. A 687

scope or by detector pixels). The aperture function W(x)describes this averaging; it is also normalized by defini-tion as

EE W~x!d2x 5 1. (2)

Let I be the detected light flux normalized by its aver-age so that ^I& 5 1. It is computed as the square of am-plitude averaged over wavelength and aperture. Forweak scintillation (x ! 1)

I 5 EEE exp@2x~x, l!#F~l!W~x!dld2x

' 1 1 2E x~l!F~l!dl, (3)

where

x~l! 5 EE x~x, l!W~x!d2x (4)

is the spatially averaged (smoothed) amplitude.The scintillation index s is defined as a variance of the

normalized light flux in

s 5 ^I2& 2 ^I&2

5 4EE ^x~l1!x~l2!&F~l1!F~l2!dl1dl2 . (5)

When two apertures W1 and W2 are considered, the dif-ferential scintillation index sd is defined as a variance ofthe flux ratio or, for weak scintillation, as a variance ofthe difference of normalized light fluxes I1 and I2 withinthe apertures in the form

sd 5 ^~I1 2 I2!2&, (6)

with ^I1& 5 ^I2& 5 1. Differential scintillation is ana-lyzed in the same manner as normal scintillation byreplacing the aperture function W with the differenceW1 2 W2 .11,12

The covariance of amplitudes under the integral in Eq.(5)—i.e., Bx(l1 , l2) 5 ^x(l1)x(l2)&—will now be com-puted for one thin turbulent layer. As in the standardtheory of scintillation,9,10,12 the amplitude x is repre-sented as an integral of the FT of phase perturbationsf(f ) in a thin layer modified by the propagation (Fresneldiffraction) and convolved with the aperture function(multiplied by its FT W):

x~l! 5 EE f~f, l!sin~plzf 2!W~f !df. (7)

Here f is the spatial frequency, f 5 uf u, and z is the propa-gation distance, also called range (z 5 h sec g for a turbu-lent layer at altitude h and zenith angle g). For differen-tial scintillation, W is replaced by W1 2 W2 .

The path-length fluctuations l (f ) are achromatic aslong as the dispersion of air is neglected. Phase is re-lated to path length as f(f, l) 5 (2p/l) l (f ). The powerspectrum of path-length fluctuations F l is related to therefractive-index structural constant Cn

2(z) and the thick-ness dz of a turbulent layer (e.g., Eq. 7.16 in Ref. 10):

F l~f ! 5 9.69 3 1023f 211/3Cn2~z !dz. (8)

Although x is real, I write the covariance Bx as an av-erage product of complex conjugates. Each amplitude isrepresented as an integral over spatial frequencies [Eq.(7)], and the product of integrals is written as a double in-tegral over two frequencies f and f 8:

Bx~l1 , l2! 5 EEE E ^f~f, l1!f* ~f 8, l2!&

3 sin~pl1zf 2!sin~pl2zf 82!

3 W~f !W* ~f 8!dfdf 8. (9)

The average product of phase FTs is related to thepower spectrum of path-length fluctuations by

^f~f, l1!f* ~f 8, l2!& 54p2

l1l2F l~f !d ~f 2 f 8!. (10)

Because of the delta function, the integral over two spa-tial frequencies is reduced to the integral over a singlespatial frequency. Hence

Bx~l1 , l2! 5 ~9.69 3 1023!Cn2~z !dz

4p2

l1l2

3 EE f 211/3 sin~pl1zf 2!

3 sin~pl2zf 2!A~f !df, (11)

where A 5 uWu2 is the aperture factor (for differentialscintillation A 5 uW1 2 W2u2).

For circularly symmetric apertures all terms in Eq. (11)depend only on the frequency modulus f. Thus the inte-gration in angle in the frequency space can be made, add-ing another 2p coefficient and changing the power of ffrom 211/3 to 28/3. I put this expression for Bx(l1 , l2)into Eq. (5), rearrange the terms, introduce the weightingfunction (WF) Q(z) that relates the scintillation index s tothe strength of turbulent layer, and obtain

s 5 Q~z !Cn2~z !dz, (12)

where

Q~z ! 5 32p3~9.69 3 1023!E0

`

f 28/3A~ f !S~z, f !df,

(13)

S~z, f ! 5 EE ~l1l2!21 sin~pl1zf 2!sin~pl2zf 2!

3 F~l1!F~l2!dl1dl2

5 F E l21F~l!sin~plzf 2!dl G2

. (14)

The simplification of Eq. (14) is done by noting that thedouble integral is in fact a product of two identical inte-grals over l1 and l2 . Moreover, these integrals resemblethe FT F(k):

F~k ! 5 E dll21F~l!exp~2pikl!, (15)

where k is the frequency (inverse wavelength). The inte-grals that enter into the expression for S equal the imagi-

688 J. Opt. Soc. Am. A/Vol. 20, No. 4 /April 2003 Andrei A. Tokovinin

nary part of the FT of the spectral response function di-

vided by l, which we denote as Fi(k). Thus

S~z, f ! 5 @Fi~zf 2/2!#2. (16)

It is apparent that S is nonnegative. It can be computednumerically for any passband function; I compute it ana-lytically below for a specific F(l).

When the joint effect of several atmospheric layers isconsidered, the scintillation index in the weak-perturbation regime can be computed as a sum (integral)over all layers:

s 5 E Cn2~z !Q~z !dz. (17)

Taken together, Eqs. (17), (13), and (16) give the com-plete formula for the polychromatic scintillation index.In the monochromatic case F(l) 5 d (l 2 l0) andS(z, f ) 5 l0

22 sin2(pl0zf 2); hence the known expressionfor the WF3,11,12 is obtained.

3. EVALUATING THE EFFECT OF FINITEBANDPASSIn this section, I evaluate the influence of finite bandpasson the scintillation index for a specific quasi-Gaussianspectral response function:

F~l! 5l

l0

1

sA2pexpF2

~l 2 l0!2

2s 2 G . (18)

Strictly speaking, normalization of this function is correctonly asymptotically for narrow bandpass, but here I ne-glect this small multiplicative error. The FT of F(l)/l isreadily obtained from the tables

Fi~k ! 51

l0sin~2pl0k !exp~22p2s 2k2!, (19)

S~z, f ! 5 l022 sin2~pl0zf 2!exp~2p2s 2z2f 4!.

(20)

Spectral response curves are often characterized bytheir FWHM L. For a Gaussian curve L 5 sA8 ln 25 2.355s. Thus S is proportional toexp(21.780z2f 4L2). The final expression for the WF be-comes

Q~z ! 5 9.62l022E

0

`

dff 28/3A~ f !

3 exp~21.780z2f 4L2!sin2~pl0zf 2!, (21)

where 9.62 5 32p3 (9.69 3 1023). For the monochro-matic case (L 5 0) it is equivalent to the known formula[e.g., Eq. (3) in Ref. 12].

Intuitively it is clear that enlarged bandpass will influ-ence scintillation index mostly for the smallest apertures(which pass higher spatial frequencies) and for higher al-titudes when the argument in the sin2(plzf 2) term be-comes larger than 1. In Fig. 1(a) the most critical case ofbandwidth influence is plotted, as computed from Eq.(21). For larger apertures the effect of the same band-width is smaller. For apertures more than 15 cm in di-ameter, scintillation approaches the geometric-optics re-

gime and becomes almost achromatic; hence classicalresults for the monochromatic case would be applicable.

4. DISCUSSIONThe results presented above might appear strange. Afterall, scintillation is almost always detected in wideband,and the practice of SCIDAR measurements seems to con-firm that bandwidth is of little importance. Indeed, Ver-nin and Azouit13 have estimated the effect of finite band-width on the scintillation amplitude. Their method isquite similar to the one used here. They also considerthe covariance of amplitudes like our Eq. (9) but neglectaperture averaging, which permits carrying out the firstintegration over frequency analytically. They show thatthe relative bandwidth of L/l , 0.25 is still acceptable,having only small influence on the scintillation ampli-tude. In Fig. 1(b) the WF for normal scintillation and arelatively large bandpass L/l 5 0.6 is computed, whichconfirms these estimates.

The different sensitivity of normal and differentialscintillation to bandpass is explained by the different spa-tial filtering. The normal scintillation index correspondsto a low-pass spatial filter. When this filter is combinedwith the propagation term and with the turbulence spec-

Fig. 1. Weighting functions Q(z) of Eq. (21) in 1010 m21/3 at500-nm wavelength. (a) Differential scintillation WFs for a pairof 2-cm and 4-cm concentric apertures: solid curve, monochro-matic; dashed curve, bandwidth 100 nm. The dotted curve isthe sum of the differential WFs with 0.027 of the single 2-cm ap-erture WF for 100-nm bandwidth. (b) Normal scintillation WFsfor a 2-cm diameter aperture: solid curve, monochromatic;dashed curve, bandwidth 300 nm.

Andrei A. Tokovinin Vol. 20, No. 4 /April 2003 /J. Opt. Soc. Am. A 689

trum’s steep rise toward low frequencies, the net effect isthat the scintillation amplitude is dominated by perturba-tions of a size comparable with the Fresnel radius Alzand is little affected by the spectral bandwidth as long asL/l ! 1. On the other hand, the differential scintilla-tion index corresponds to spatial bandpass filtering.With increasing range the small-scale perturbations indifferent colors become progressively decorrelated, reduc-ing the amplitude of polychromatic differential scintilla-tion.

The analysis presented above shows that simple aver-aging of the monochromatic WFs multiplied by the spec-tral response leads to wrong results. This is preciselythe method used in Ref. 3 to account for finite bandwidthin the scintillometer of Ochs et al.6 The scintillometer,like the MASS, is based on differential scintillation; hencefor high layers the bandpass effect becomes really impor-tant.

The WF of differential scintillation in monochromaticlight saturates at high altitudes [Fig. 1(a)]. Hence thecorresponding scintillation index provides a useful mea-sure of the Cn

2 integral and of seeing in the freeatmosphere.11,14 This is no longer the case for polychro-matic scintillation. However, the situation may be cor-rected by adding to the differential WF a small fraction ofthe normal WF that increases with altitude. In Fig. 1(a)the dotted curve shows the sum of the differential poly-chromatic WF (for 4-cm and 2-cm apertures) with the0.027 fraction of the normal WF for the inner 2-cm aper-ture and a bandwidth of 100 nm; this sum is again prac-tically constant at high altitudes. Hence direct estima-tion of free-atmosphere seeing from differentialscintillation as suggested in Refs. 11 and 12 is also pos-sible with wide spectral band-width. Similarly the mea-surement of atmospheric time constant t0 by differentialexposure scintillation is not constrained by finite spectralbandwidth: I repeated the calculations presented in Ref.12 for the polychromatic case (L 5 100 nm) and verifiedthat the estimates of t0 remain valid for real profiles ofturbulence and wind.

The WFs in the MASS instrument8 are computed withthe new formula for polychromatic scintillation. Thespectral response is carefully determined for each star bytaking into account its spectral energy distribution.Good agreement between modeled and measured scintil-lation indices (both normal and differential) was found,their ratio being equal to 1 within 65%.

ACKNOWLEDGMENTSConstructive comments of two referees helped to improvethe clarity of the presentation and are gratefully acknowl-

edged. The experimental work on turbulence profilemeasurements with concentric apertures is being con-ducted jointly with V. Kornilov, N. Shatsky, and O. Vozia-kova from the Sternberg Astronomical Institute of Mos-cow University.

The author may be reached by e-mail at atokovinin@ctio.noao.edu.

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